text
stringlengths 0
6.23M
| __index_level_0__
int64 0
419k
|
|---|---|
TITLE: Do direct limits commute with taking subgroups?
QUESTION [0 upvotes]: If I have two directed systems of abelian groups $(A_i, \mu_{ij})$ and $(B_i, \mu_{ij}|_{B_i})$ with $B_i \unlhd A_i$ for all $i$ then can I conclude that $\varinjlim B_i$ is isomorphic to a subgroup of $\varinjlim A_i$?
Here's my attempt to reason through that. Recall that $$\varinjlim A_i = \bigoplus_i A_i \bigg/D_A$$ where $D_A$ is the subgroup of $\bigoplus_i A_i$ generated by all elements of the form $x_i - \mu_{ij}(x_i)$ where $i \leq j$ and $x_i \in A_i$.
Note that $\varinjlim B_i$ can't be a true subgroup of $\varinjlim A_i$ because of how the direct limit of a directed system of abelian groups is defined.
Now if we let $D_B$ denote the corresponding subgroup of $\bigoplus_i B_i$ in the definition of $\varinjlim B_i$, then by using the universal property of the direct limit $\varinjlim B_i$ applied to $\varinjlim A_i$ we can find the existence of a (well-defined) homomorphism $\phi : \varinjlim B_i \to \varinjlim A_i$ given by $\phi(b + D_B) = b+D_A$. Now if I can show that $\ker \phi = 0$ then I would be done, but I don't see how I could show that.
REPLY [3 votes]: If by "directed limit" you mean "filtered colimit" (which I imagine is the case here) then the answer will always be yes, the main reason being that filtered colimits commute with finite limits in $\mathbf{Ab}$.
$\require{AMScd}\def\id{\operatorname{id}}$
If $A:J\to\mathbf{Ab}$ is a filtered diagram (i.e., directed system) and $B$ is a subdiagram (this is just my way of phrasing what you describe in your question), then like you mentioned the inclusion $i:B\Rightarrow A$ induces a morphism $i:\varinjlim B\to\varinjlim A$ of colimits.
Now, this is a monomorphism (i.e., injective) iff
\begin{CD}
\varinjlim B @>\id>> \varinjlim B \\
@V\id VV @VViV \\
\varinjlim B @>>i> \varinjlim A
\end{CD}
is a pullback square, which is true because this is the colimit of the system of pullback squares
\begin{CD}
B_j @>\id>> B_j \\
@V\id VV @VV\subseteq V \\
B_j @>>\subseteq> A_j
\end{CD}
for $j\in J$.
If "directed limit" just meant an arbitrary colimit, the answer is no: for example, $\Bbb Z$ is the colimit of the diagram $\Bbb Z\xleftarrow{\id}\Bbb Z\xrightarrow{\id}\Bbb Z$, but if we look at the subdiagram $\Bbb Z\leftarrow0\rightarrow\Bbb Z$ the colimit is $\Bbb Z\oplus\Bbb Z$, and the induced map $\Bbb Z\oplus\Bbb Z\to\Bbb Z$ is given by $(n,m)\mapsto n+m$, which is certainly not injective (and $\Bbb Z\oplus\Bbb Z$ simply cannot be a subgroup of $\Bbb Z$).
To see this more concretely, the direct limit $\varinjlim A$ as the result of putting all the elements of every $A_j$ together, identifying elements via the transition maps. If the diagram is not filtered, then it is possible that "the reason $a_j\in A_j$ and $a_{j'}\in A_{j'}$ sum to zero is that they both come from some $A_k$ with $k\leq j,j'$ and in there, they sum to zero" so if your subgroup $B_k$ does not include $a_j$ and $a_{j'}$, then these elements may not sum to zero in $\varinjlim B$.
However, with a filtered diagram, any $j,j'$ have an $\ell\geq j,j'$, meaning for any $a_j\in A_j$ and $a_{j'}\in A_{j'}$ that we can find these elements in $A_\ell$ and perform the addition there.
Therefore, if $a_j+a_{j'}=0$, this fact will be witnessed in some group in the directed system containing both of those elements. This resolves the original issue because if $a_j\in B_j$ and $a_{j'}\in B_{j'}$ as well, then they must both lie in $B_\ell\subseteq A_\ell$ and therefore they will still sum to zero.
In other words, an element of $\varinjlim A$ is zero precisely when it is the image of the zero in one of the $A_{j'}\to\varinjlim A$. If $\phi:\varinjlim B\to\varinjlim A$ sends some $b$ to zero, note that $b$ comes from some $B_j$, and $\phi(b)$ is the image of some zero in $A_{j'}$. Take $\ell\geq j,j'$, then $b$ gets sent via the transition map $B_j\to B_\ell$ to the same element that $0\in A_{j'}$ is sent to via its transition map (which is again $0$). That is, $b$ vanishes in $B_\ell$, and therefore $b$ must have already been zero in $\varinjlim B$, showing that $\phi$ is injective. | 31,049 |
Sleep Mellon
2,716pages onAdd New Page
this wiki
this wiki
The Sleep Mellon was spawned by an OOC Aragorn who had lost the ability to use commas. When he said, "Sleep Mellon I shall wake you later" to Rowan Black, a very glittery Sue, he accidentally created the Sleep Mellon. The Sleep Mellon looks, as said in the original mission, "like a cantaloupe with pointy ears; neither agent decided to question it." It also has a blanket. After having an altercation with a fire-lizard, the Sleep Mellon was booted out of an RC to menace the halls of HQ, and that's where you'll find it, rolling around and losing its temper.
AppearancesEdit
- "Kill It With Fire" (The Lord of the Rings), Agents Alleb and Jesse McKines (DF)
- Original appearance. | 299,742 |
God of love, you revealed Jesus
as your beloved Son
in the Jordan River.
We praise you for the gift of Christ,
our salvation and our peace.
You anointed Jesus
for the service of the world.
Strengthen the Church’s witness
to this mission in our world today.
You brought us to new life
and made us members of Christ’s Body in baptism.
May the gift of your Holy Spirit keep us
ready to meet the demands of our baptism. Amen.
Adapted from PrayerTime: Faith-Sharing Reflections on the Sunday Gospels, Cycle B.
© RENEW International | 359,537 |
\begin{document}
\begin{frontmatter}
\title{Quantum Double Models coupled with matter: \\ an algebraic dualisation approach}
\author[usp]{M. F. Araujo de Resende\corref{cor1}}
\ead{[email protected]}
\cortext[cor1]{Corresponding author}
\author[usp]{J. P. Ibieta Jimenez}
\ead{[email protected]}
\author[usp,temuco]{J. Lorca Espiro}
\ead{[email protected]}
\address[usp]{Instituto de F\'{\i}sica, Universidade de S\~{a}o Paulo, 05508-090 S\~{a}o Paulo SP, Brasil}
\address[ime]{Instituto de Matem\'{a}tica e Estat\'{\i}stica, Universidade de S\~{a}o Paulo, 05508-090 S\~{a}o Paulo SP, Brasil}
\address[temuco]{Departamento de Ciencias F\'{\i}sicas, Facultad de Ingenier\'{\i}a, Ciencias y Administraci\'{o}n,\\ Universidad de La Frontera, Avda. Francisco Salazar 01145, Casilla 54-D Temuco, Chile}
\begin{abstract}
In this paper, we constructed a new generalization of a class of discrete bidimensional models, the so called Quantum Double Models, by introduce matter qunits to the faces of the lattice that supports these models. This new generalization can be interpreted as the algebraic dual of a first, where we introduce matter qunits to the vertices of this same lattice. By evaluating the algebraic and topological orders of these new models, we prove that, as in the first generalization, a new phenomenon of quasiparticle confinement may appear again: this happens when the co-action homomorphism between matter and gauge groups is non-trivial. Consequently, this homomorphism not only classifies the different models that belong to this new class, but also suggests that they can be interpreted as a $ 2 $-dimensional restriction of the $ 2 $-lattice gauge theories.
\end{abstract}
\end{frontmatter}
\section{Introduction}
Quantum Double Models ($ D \left( G \right) $) \cite{kitaev-1,pachos,naaij} is the name given to a class of models that, since they are defined on two-dimensional lattices, have a topological order \cite{wen} that allows to perform some fault-tolerant quantum computation \cite{kitaev-1,dennis}. This topological order is due to the fact these models are constructed by associating qunits to edges of a lattice $ \mathcal{L} _{2} $ that, in general, discretizes some $ 2 $-dimensional compact orientable manifold $ \mathcal{M} _{2} $. In the case of a $ D \left( G \right) $ where $ G $ is not a Abelian group, part of this fault-tolerant quantum computation power is justified, for instance, due to presence of non-Abelian anyons among its low energy excitations \cite{nayak}.
Since there is no qunit associated with other lattice elements, some works were published recently in order to evaluate $ D \left( G \right) $ generalizations where new qunits are associated to lattice vertices. These generalizations were denoted as \emph{Quantum Double Models plus matter} ($ D_{M} \left( G \right) $) \cite{miguel-1,miguel-2,mf-cyclic}, although this term matter does not necessarily have to be thought of in the same way as elementary particle physics. Among the main properties of these generalizations, we can highlight the presence of algebraic and topological orders, as well as the presence of non-Abelian fusion rules, even when the gauge group is cyclic Abelian \cite{mf-cyclic}.
However, as unlike the $ D \left( G \right) $ \cite{aguado}, this generalization is not self-dual, one question that arises is how to use the $ D_{M} \left( G \right) $ as the basis for defining a self-dual generalization that associates matter qunits on both faces and vertices of $ \mathcal{L} _{2} $. By the way, does the construction of a new generalization, purposely defined as the algebraic dual of the $ D_{M} \left( G \right) $, show us if it is possible? In order to answer these questions, in this work we analyse a class of models that can be interpreted as the algebraic dual of the $ D_{M} \left( G \right) $: this new generalization $ D^{K} \left( G \right) $ has the same gauge structure of the $ D \left( G \right) $, but its matter qunits are associated only to the centroids of the faces of $ \mathcal{L} _{2} $, since these centroids can be interpreted as the vertices of a dual lattice $ \mathcal{L} ^{\ast } _{2} $ \cite{wenninger}.
\section{\label{QDM-review}A brief review about the Quantum Double Models plus matter}
As we previously mentioned in the Introduction, the $ D_{M} \left( G \right) $ is a class of two-dimensional lattice models that was purposely constructed to be interpreted as a generalization of the $ D \left( G \right) $ \cite{mf-cyclic}. This construction is done:
\begin{enumerate}
\item[(i)] by taking an oriented lattice $ \mathcal{L} _{2} $ that discretizes a $ 2 $-dimensional compact orientable manifold $ \mathcal{M} _{2} $;
\item[(ii)] by assigning gauge and matter qunits
\begin{equation*}
\bigl\vert \varphi _{j} \bigr\rangle = a^{\left( \varphi \right) } _{0} \left\vert 0 \right\rangle + \ldots + a^{\left( \varphi \right) } _{N-1} \left\vert N-1 \right\rangle \ \ \textnormal{and} \ \ \left\vert \chi _{v} \right\rangle = a^{\left( \chi \right) } _{0} \left\vert 0 \right\rangle + \ldots + a^{\left( \chi \right) } _{M-1} \left\vert M-1 \right\rangle
\end{equation*}
to edges and vertices of $ \mathcal{L} _{2} $ respectively; and
\item[(iii)] by defining a Hamiltonian operator
\begin{equation}
H_{D_{M} \left( G \right) } = - \sum _{v} A^{\left( G , S \right) } _{v} - \sum _{p} \ B^{\left( G , S \right) } _{p} - \sum _{j} C^{\left( G , S \right) } _{j} \label{H-qdmv}
\end{equation}
such that $ \left. D_{M} \left( G \right) \right\vert _{M=1} = D \left( G \right) $.
\end{enumerate}
The operators that make up the Hamiltonian are
\begin{eqnarray}
A^{\left( G , S \right) } _{v} = \frac{1}{\vert G \vert } \sum _{g \in G} A^{g} _{v} \ , \ \ B^{\left( G , S \right) } _{p} = B^{0} _{p} \ \ \textnormal{and} \ \ C^{\left( G , S \right) } _{j} = C_{j} \label{qdmv-operators} \ ,
\end{eqnarray}
whose components are given by the Figure \ref{QMDv-operators-components}.
\begin{figure}[!t]
\begin{center}
\begin{tikzpicture}[
scale=0.3,
equation/.style={thin},
trans/.style={thin,shorten >=0.5pt,shorten <=0.5pt,>=stealth},
flecha/.style={thin,->,shorten >=0.5pt,shorten <=0.5pt,>=stealth},
every transition/.style={thick,draw=black!75,fill=black!20}
]
\draw[equation] (-7.5,0.0) -- (-7.5,0.0) node[midway,right] {$ A^{g} _{v} $};
\draw[trans] (-4.8,2.3) -- (-4.8,-2.3) node[above=2pt,right=-1pt] {};
\draw[trans] (1.9,2.3) -- (3.1,-0.06) node[above=2pt,right=-1pt] {};
\draw[trans] (1.9,-2.3) -- (3.1,0.06) node[above=2pt,right=-1pt] {};
\draw[flecha] (-3.1,0.0) -- (-2.2,0.0) node[above=2pt,right=-1pt] {};
\draw[flecha] (-2.4,0.0) -- (0.7,0.0) node[above=2pt,right=-1pt] {};
\draw[trans] (0.2,0.0) -- (1.1,0.0) node[above=2pt,right=-1pt] {};
\draw[flecha] (-1.0,-1.7) -- (-1.0,-1.0) node[above=2pt,right=-1pt] {};
\draw[flecha] (-1.0,-1.2) -- (-1.0,1.6) node[above=2pt,right=-1pt] {};
\draw[trans] (-1.0,1.4) -- (-1.0,1.8) node[above=2pt,right=-1pt] {};
\draw[trans,fill=white] (-1.0,0.0) circle (0.9);
\draw[equation] (-1.0,0.0) -- (-1.0,0.0) node[midway] {$ \alpha $};
\draw[equation] (-1.0,2.4) -- (-1.0,2.4) node[midway] {$ a $};
\draw[equation] (2.6,0.0) -- (2.6,0.0) node[midway,left] {$ b $};
\draw[equation] (-1.0,-2.4) -- (-1.0,-2.4) node[midway] {$ c $};
\draw[equation] (-4.7,0.0) -- (-4.7,0.0) node[midway,right] {$ d $};
\draw[equation] (4.5,-0.07) -- (4.5,-0.07) node[midway] {$ = $};
\draw[equation] (6.6,0.15) -- (6.6,0.15) node[midway] {$ \sum $};
\draw[equation] (6.6,-1.1) -- (6.6,-1.1) node[midway] {$ _{\gamma } $};
\draw[equation] (11.7,0.1) -- (11.7,0.1) node[midway] {$ \delta \left( \theta \left( g , \alpha \right) , \gamma \right) $};
\draw[trans] (16.3,2.3) -- (16.3,-2.3) node[above=2pt,right=-1pt] {};
\draw[trans] (25.6,2.3) -- (26.8,-0.06) node[above=2pt,right=-1pt] {};
\draw[trans] (25.6,-2.3) -- (26.8,0.06) node[above=2pt,right=-1pt] {};
\draw[flecha] (19.9,0.0) -- (20.8,0.0) node[above=2pt,right=-1pt] {};
\draw[flecha] (20.6,0.0) -- (23.7,0.0) node[above=2pt,right=-1pt] {};
\draw[trans] (23.2,0.0) -- (24.1,0.0) node[above=2pt,right=-1pt] {};
\draw[flecha] (22.0,-1.8) -- (22.0,-1.0) node[above=2pt,right=-1pt] {};
\draw[flecha] (22.0,-1.2) -- (22.0,1.6) node[above=2pt,right=-1pt] {};
\draw[trans] (22.0,1.4) -- (22.0,1.8) node[above=2pt,right=-1pt] {};
\draw[trans,fill=white] (21.95,0.0) circle (0.9);
\draw[equation] (21.9,0.0) -- (21.9,0.0) node[midway] {$ \gamma $};
\draw[equation] (22.1,2.3) -- (22.1,2.3) node[midway] {$ ga $};
\draw[equation] (26.2,-0.2) -- (26.2,-0.2) node[midway,left] {$ gb $};
\draw[equation] (22.1,-2.3) -- (22.1,-2.3) node[midway] {$ cg^{-1} $};
\draw[equation] (16.4,0.0) -- (16.4,0.0) node[midway,right] {$ dg^{-1} $};
\end{tikzpicture} \\
\begin{tikzpicture}[
scale=0.3,
equation/.style={thin},
trans/.style={thin,shorten >=0.5pt,shorten <=0.5pt,>=stealth},
flecha/.style={thin,->,shorten >=0.5pt,shorten <=0.5pt,>=stealth},
every transition/.style={thick,draw=black!75,fill=black!20}
]
\draw[equation] (-8.8,0.0) -- (-8.8,0.0) node[midway,right] {$ B^{h} _{p} $};
\draw[trans] (-5.6,2.3) -- (-5.6,-2.3) node[above=2pt,right=-1pt] {};
\draw[trans] (-0.1,2.3) -- (1.1,-0.06) node[above=2pt,right=-1pt] {};
\draw[trans] (-0.1,-2.3) -- (1.1,0.06) node[above=2pt,right=-1pt] {};
\draw[flecha] (-4.7,1.2) -- (-2.1,1.2) node[above=2pt,right=-1pt] {};
\draw[trans] (-2.3,1.2) -- (-0.1,1.2) node[above=2pt,right=-1pt] {};
\draw[flecha] (-4.7,-1.2) -- (-2.1,-1.2) node[above=2pt,right=-1pt] {};
\draw[trans] (-2.3,-1.2) -- (-0.1,-1.2) node[above=2pt,right=-1pt] {};
\draw[flecha] (-3.8,-2.0) -- (-3.8,0.3) node[above=2pt,right=-1pt] {};
\draw[trans] (-3.8,0.0) -- (-3.8,2.0) node[above=2pt,right=-1pt] {};
\draw[flecha] (-1.0,-2.0) -- (-1.0,0.3) node[above=2pt,right=-1pt] {};
\draw[trans] (-1.0,0.0) -- (-1.0,2.0) node[above=2pt,right=-1pt] {};
\draw[equation] (-2.3,2.0) -- (-2.3,2.0) node[midway] {$ a $};
\draw[equation] (0.7,0.0) -- (0.7,0.0) node[midway,left] {$ b $};
\draw[equation] (-2.3,-2.0) -- (-2.3,-2.0) node[midway] {$ c $};
\draw[equation] (-5.5,0.0) -- (-5.5,0.0) node[midway,right] {$ d $};
\draw[equation] (6.7,-0.07) -- (6.7,-0.07) node[midway] {$ = \delta \left( h , a^{-1} b c d^{-1} \right) $};
\draw[trans] (12.2,2.3) -- (12.2,-2.3) node[above=2pt,right=-1pt] {};
\draw[trans] (17.7,2.3) -- (18.8,-0.06) node[above=2pt,right=-1pt] {};
\draw[trans] (17.7,-2.3) -- (18.8,0.06) node[above=2pt,right=-1pt] {};
\draw[flecha] (13.0,1.2) -- (15.6,1.2) node[above=2pt,right=-1pt] {};
\draw[trans] (15.4,1.2) -- (17.6,1.2) node[above=2pt,right=-1pt] {};
\draw[flecha] (13.0,-1.2) -- (15.6,-1.2) node[above=2pt,right=-1pt] {};
\draw[trans] (15.4,-1.2) -- (17.6,-1.2) node[above=2pt,right=-1pt] {};
\draw[flecha] (13.9,-2.0) -- (13.9,0.3) node[above=2pt,right=-1pt] {};
\draw[trans] (13.9,0.0) -- (13.9,2.0) node[above=2pt,right=-1pt] {};
\draw[flecha] (16.7,-2.0) -- (16.7,0.3) node[above=2pt,right=-1pt] {};
\draw[trans] (16.7,0.0) -- (16.7,2.0) node[above=2pt,right=-1pt] {};
\draw[equation] (15.4,2.0) -- (15.4,2.0) node[midway] {$ a $};
\draw[equation] (18.4,0.0) -- (18.4,0.0) node[midway,left] {$ b $};
\draw[equation] (15.4,-2.0) -- (15.4,-2.0) node[midway] {$ c $};
\draw[equation] (12.2,0.0) -- (12.2,0.0) node[midway,right] {$ d $};
\end{tikzpicture} \\
\begin{tikzpicture}[
scale=0.3,
equation/.style={thin},
trans/.style={thin,shorten >=0.5pt,shorten <=0.5pt,>=stealth},
flecha/.style={thin,->,shorten >=0.5pt,shorten <=0.5pt,>=stealth},
every transition/.style={thick,draw=black!75,fill=black!20}
]
\draw[equation] (-8.0,0.0) -- (-8.0,0.0) node[midway,right] {$ C_{j} $};
\draw[trans] (-5.6,2.3) -- (-5.6,-2.3) node[above=2pt,right=-1pt] {};
\draw[trans] (0.1,2.3) -- (1.3,-0.06) node[above=2pt,right=-1pt] {};
\draw[trans] (0.1,-2.3) -- (1.3,0.06) node[above=2pt,right=-1pt] {};
\draw[flecha] (-4.4,0.0) -- (-2.1,0.0) node[above=2pt,right=-1pt] {};
\draw[trans] (-2.3,0.0) -- (-0.2,0.0) node[above=2pt,right=-1pt] {};
\draw[trans,fill=white] (-4.4,0.0) circle (0.9);
\draw[trans,fill=white] (-0.2,0.0) circle (0.9);
\draw[equation] (-2.3,-0.8) -- (-2.3,-0.8) node[midway] {$ a $};
\draw[equation] (-4.4,0.0) -- (-4.4,0.0) node[midway] {$ \alpha $};
\draw[equation] (-0.2,-0.02) -- (-0.2,-0.02) node[midway] {$ \beta $};
\draw[equation] (2.6,-0.07) -- (2.6,-0.07) node[midway] {$ = $};
\draw[equation] (7.5,0.0) -- (7.5,0.0) node[midway] {$ \delta \left( \theta \left( a , \alpha \right) , \beta \right) $};
\draw[trans] (12.0,2.3) -- (12.0,-2.3) node[above=2pt,right=-1pt] {};
\draw[trans] (17.7,2.3) -- (18.9,-0.06) node[above=2pt,right=-1pt] {};
\draw[trans] (17.7,-2.3) -- (18.9,0.06) node[above=2pt,right=-1pt] {};
\draw[flecha] (13.2,0.0) -- (15.5,0.0) node[above=2pt,right=-1pt] {};
\draw[trans] (15.2,0.0) -- (17.4,0.0) node[above=2pt,right=-1pt] {};
\draw[trans,fill=white] (13.2,0.0) circle (0.9);
\draw[trans,fill=white] (17.4,0.0) circle (0.9);
\draw[equation] (15.3,-0.8) -- (15.3,-0.8) node[midway] {$ a $};
\draw[equation] (13.2,0.0) -- (13.2,0.0) node[midway] {$ \alpha $};
\draw[equation] (17.4,-0.02) -- (17.4,-0.02) node[midway] {$ \beta $};
\end{tikzpicture}
\end{center}
\caption{\label{QMDv-operators-components} Definition of how the components $ A^{g} _{v} $, $ B^{h} _{p} $ and $ C_{j} $, which define the vertex, face and edge operators mentioned in (\ref{qdmv-operators}) respectively, act on the Hilbert space that is associated to $ \mathcal{L} _{2} $. Here, in the same way that the symbol $ a $ is indexing a basis element of the gauge Hilbert subspace $ \mathfrak{H} _{N} $, the symbol $ \alpha $ indexes a basis element of the matter Hilbert subspace $ \mathfrak{H} _{M} $ \cite{mf-cyclic}.}
\end{figure}
These operators (\ref{qdmv-operators}) act effectively in the subspaces that are associated with the edges subsets which, as shown in Figure \ref{QMDv-rede}, give structure to the $ v $-th vertex, the $ p $-th face and the $ j $-th edge of $ \mathcal{L} _{2} $ respectively.
This reduction $ \left. D_{M} \left( G \right) \right\vert _{M=1} = D \left( G \right) $ can be easily understood if we analyse each of the operators in (\ref{qdmv-operators}) individually by noting that, as $ \bigl\vert \varphi _{j} \bigr\rangle $ and $ \left\vert \chi _{v} \right\rangle $ need to be related to each other, these qunits belong to Hilbert subspaces $ \mathfrak{H} _{N} $ and $ \mathfrak{H} _{M} $ that are a \emph{group algebra} $ \mathds{C} \left( G \right) $ and a \emph{left} $ \mathds{C} G $\emph{-module} \cite{fulton} respectively. In the case of the vertex operator $ A^{\left( G , S \right) } _{v} $, this reduction comes from the fact that it is a modified operator (in relation to the $ D \left( G \right) $ vertex operator) that performs \emph{gauge transformations} due to the presence of matter qunits at lattice vertices \cite{mf-cyclic}. After all, since $ \mathcal{B} _{j} = \big\{ \left\vert g \right\rangle : g \in G \big\} $ and $ \mathcal{B} _{v} = \big\{ \left\vert \alpha \right\rangle : \alpha \in S \big\} $ are two bases for $ \mathfrak{H} _{N} $ and $ \mathfrak{H} _{M} $ respectively, the multiplication $ \theta : G \times S \rightarrow S $ that defines $ \mathfrak{H} _{M} $ as a left $ \mathds{C} G $-module automatically defines how the gauge group acts on these matter qunits.
In relation to the face operator $ B^{\left( G , S \right) } _{p} $ there is nothing new to be said: it is exactly the same as the $ D \left( G \right) $ face operator since it does not act on the matter qunits. It measures only \emph{flat connections}, i.e. the \emph{trivial holonomies} characterised by $ h = 0 $ along the faces. However, the novelty of the $ D_{M} \left( G \right) $ is the presence of an edge operator $ C^{\left( G , S \right) } _{j} $ in the Hamiltonian (\ref{H-qdmv}) that, together with the other operators, allows to state that its ground state $ \left\vert \Psi _{0} \right\rangle $ is such that
\begin{equation}
A^{\left( G , S \right) } _{v} \left\vert \xi _{0} \right\rangle = \left\vert \xi _{0} \right\rangle \ , \ \ B^{\left( G , S \right) } _{p} \left\vert \xi _{0} \right\rangle = \left\vert \xi _{0} \right\rangle \ \ \textnormal{and} \ \ C^{\left( G , S \right) } _{j} \left\vert \xi _{0} \right\rangle = \left\vert \xi _{0} \right\rangle \ , \label{QDMv-ground-state}
\end{equation}
is valid for all values of $ v $, $ p $ and $ j $. This edge operator works literally as a \emph{comparator}; i.e., $ C^{\left( G , S \right) } _{j} $ compares two neighbouring matter qunits by checking whether they are aligned\footnote{Since they can be interpreted as vectors that define a vector field.} by according to the $ \theta $ perspective \cite{mf-cyclic}.
\begin{figure}[!t]
\begin{center}
\begin{tikzpicture}
\draw[color=blue!20,fill=blue!20] (6,5) rectangle (8,7);
\draw[color=red!20,fill=red!20] (1,2) rectangle (3,4);
\draw[color=orange!20,fill=orange!20] (5.5,2.5) rectangle (8.5,3.5);
\draw[->, color=gray, ultra thick, >=stealth] (0,0) -- (0,2.2);
\draw[->, color=gray, ultra thick, >=stealth] (0,2) -- (0,4.2);
\draw[->, color=gray, ultra thick, >=stealth] (0,4) -- (0,6.2);
\draw[-, color=gray, ultra thick] (0,6) -- (0,8);
\draw[->, color=gray, ultra thick, >=stealth] (2,0) -- (2,2.2);
\draw[->, color=gray, ultra thick, >=stealth] (2,2) -- (2,4.2);
\draw[->, color=gray, ultra thick, >=stealth] (2,4) -- (2,6.2);
\draw[-, color=gray, ultra thick] (2,6) -- (2,8);
\draw[->, color=gray, ultra thick, >=stealth] (4,0) -- (4,2.2);
\draw[->, color=gray, ultra thick, >=stealth] (4,2) -- (4,4.2);
\draw[->, color=gray, ultra thick, >=stealth] (4,4) -- (4,6.2);
\draw[-, color=gray, ultra thick] (4,6) -- (4,8);
\draw[->, color=gray, ultra thick, >=stealth] (6,0) -- (6,2.2);
\draw[->, color=gray, ultra thick, >=stealth] (6,2) -- (6,4.2);
\draw[->, color=gray, ultra thick, >=stealth] (6,4) -- (6,6.2);
\draw[-, color=gray, ultra thick] (6,6) -- (6,8);
\draw[->, color=gray, ultra thick, >=stealth] (8,0) -- (8,2.2);
\draw[->, color=gray, ultra thick, >=stealth] (8,2) -- (8,4.2);
\draw[->, color=gray, ultra thick, >=stealth] (8,4) -- (8,6.2);
\draw[-, color=gray, ultra thick] (8,6) -- (8,8);
\draw[->, color=gray, ultra thick, >=stealth] (10,0) -- (10,2.2);
\draw[->, color=gray, ultra thick, >=stealth] (10,2) -- (10,4.2);
\draw[->, color=gray, ultra thick, >=stealth] (10,4) -- (10,6.2);
\draw[-, color=gray, ultra thick] (10,6) -- (10,8);
\draw[->, color=gray, ultra thick, >=stealth] (-1,1) -- (1.2,1);
\draw[->, color=gray, ultra thick, >=stealth] (1,1) -- (3.2,1);
\draw[->, color=gray, ultra thick, >=stealth] (3,1) -- (5.2,1);
\draw[->, color=gray, ultra thick, >=stealth] (5,1) -- (7.2,1);
\draw[->, color=gray, ultra thick, >=stealth] (7,1) -- (9.2,1);
\draw[-, color=gray, ultra thick] (9,1) -- (11,1);
\draw[->, color=gray, ultra thick, >=stealth] (-1,3) -- (1.2,3);
\draw[->, color=gray, ultra thick, >=stealth] (1,3) -- (3.2,3);
\draw[->, color=gray, ultra thick, >=stealth] (3,3) -- (5.2,3);
\draw[->, color=gray, ultra thick, >=stealth] (5,3) -- (7.2,3);
\draw[->, color=gray, ultra thick, >=stealth] (7,3) -- (9.2,3);
\draw[-, color=gray, ultra thick] (9,3) -- (11,3);
\draw[->, color=gray, ultra thick, >=stealth] (-1,5) -- (1.2,5);
\draw[->, color=gray, ultra thick, >=stealth] (1,5) -- (3.2,5);
\draw[->, color=gray, ultra thick, >=stealth] (3,5) -- (5.2,5);
\draw[->, color=gray, ultra thick, >=stealth] (5,5) -- (7.2,5);
\draw[->, color=gray, ultra thick, >=stealth] (7,5) -- (9.2,5);
\draw[-, color=gray, ultra thick] (9,5) -- (11,5);
\draw[->, color=gray, ultra thick, >=stealth] (-1,7) -- (1.2,7);
\draw[->, color=gray, ultra thick, >=stealth] (1,7) -- (3.2,7);
\draw[->, color=gray, ultra thick, >=stealth] (3,7) -- (5.2,7);
\draw[->, color=gray, ultra thick, >=stealth] (5,7) -- (7.2,7);
\draw[->, color=gray, ultra thick, >=stealth] (7,7) -- (9.2,7);
\draw[-, color=gray, ultra thick] (9,7) -- (11,7);
\draw[->, ultra thick, >=stealth] (2,1) -- (2,2.2);
\draw[->, ultra thick, >=stealth] (2,2.0) -- (2,4.2);
\draw[-, ultra thick] (2,4.0) -- (2,5.0);
\draw[->, ultra thick, >=stealth] (0.0,3) -- (1.2,3);
\draw[->, ultra thick, >=stealth] (1.0,3) -- (3.2,3);
\draw[-, ultra thick] (3.0,3) -- (4.0,3);
\draw[->, ultra thick, >=stealth] (6,5.0) -- (6,6.2);
\draw[-, ultra thick] (6,6.0) -- (6,7.0);
\draw[->, ultra thick, >=stealth] (8,5.0) -- (8,6.2);
\draw[-, ultra thick] (8,6.0) -- (8,7.0);
\draw[->, ultra thick, >=stealth] (6.0,5) -- (7.2,5);
\draw[-, ultra thick] (7.0,5) -- (8.0,5);
\draw[->, ultra thick, >=stealth] (6.0,7) -- (7.2,7);
\draw[-, ultra thick] (7.0,7) -- (8.0,7);
\draw[->, ultra thick, >=stealth] (6.0,3) -- (7.2,3);
\draw[-, ultra thick] (7.0,3) -- (8.0,3);
\draw[color=black,fill=white] (2,3) circle (0.3);
\node [] (2,3) at (2,3) {$ \gamma $};
\node [right] (2.1,4) at (2.1,4) {$ a $};
\node [below] (3,2.9) at (3,2.9) {$ b $};
\node [left] (1.9,2) at (1.9,2) {$ c $};
\node [above] (1,3.1) at (1,3.1) {$ d $};
\node [above] (7,7.1) at (7,7.1) {$ r $};
\node [right] (8.1,6) at (8.1,6) {$ s $};
\node [below] (7,4.9) at (7,4.9) {$ t $};
\node [left] (5.9,6) at (5.9,6) {$ u $};
\draw[color=black,fill=white] (6,3) circle (0.3);
\node [] (6,3) at (6,3) {$ \alpha $};
\draw[color=black,fill=white] (8,3) circle (0.3);
\node [] (8,3) at (8,3) {$ \beta $};
\node [below] (7.0,2.9) at (7.0,2.9) {$ j $};
\end{tikzpicture}
\end{center}
\caption{\label{QMDv-rede} Piece of an oriented square lattice $ \mathcal{L} _{2} $ that supports the $ D_{M} \left( G \right) $ where we see the rose and light orange coloured sectors respectively centred by the $ v $-th vertex and $ j $-th edge of this lattice, whereas the baby blue coloured sector refers to the $ p $-th face whose centroid can be interpreted as one of the vertices of a dual lattice. Here, the highlighted edges (in black) correspond to Hilbert subspaces in which, for instance, the vertex (the rose-coloured sector), face (the baby blue coloured sector) and edge (the light orange coloured sector) operators act effectively \cite{mf-cyclic}.}
\end{figure}
\subsection{\label{QDMv-properties}General $ D_{M} \left( G \right) $ properties}
By virtue of the gauge structure of the $ D_{M} \left( G \right) $ is exactly the same as that of the $ D \left( G \right) $, the $ D_{M} \left( G \right) $ supports the same $ D \left( G \right) $ quasiparticles. However, although all the $ D \left( G \right) $ fusion rules are preserved in the $ D_{M} \left( G \right) $, some $ D \left( G \right) $ quasiparticles that are detectable by $ B^{\left( G , S \right) } _{v} $ acquire confinement properties when $ \theta $ is not a trivial action: that is, transporting these quasiparticles always increases the system energy and this energy increases as a function of the number of edges involved in this transport.
One of the consequences of this quasiparticle confinement is that the $ D_{M} \left( G \right) $ ground state degeneracy no longer depends on the order of the fundamental group $ \pi _{1} $ associated with $ \mathcal{M} _{2} $. In the case of the cyclic Abelian $ D_{M} \left( \mathds{Z} _{N} \right) $, they have an algebraic order and, implicitly, a topological order too: this algebraic order is characterized by the fact that this degeneracy is at least a function of the number of cycles that the action $ \theta $ defines; this implicit topological order is consequence of the fact that the $ D_{M} \left( \mathds{Z} _{N} \right) $ ground state degeneracy depends on the second group of homology.
Another notable property of the $ D_{M} \left( \mathds{Z} _{N} \right) $ is the presence of quasiparticles with non-Abelian fusion rules. These quasiparticles are always necessary when this action of the gauge group is represented by
\begin{equation}
\Theta \left( g \right) = \begin{pmatrix}
\mathcal{A} \left( g \right) & \mathbf{0} \\
\mathbf{0} ^{\mathrm{T}} & \mathds{1}
\end{pmatrix} \label{special-action}
\end{equation}
so that the lattice system can go from one vacuum state to another and vice versa. Here, $ \mathcal{A} $ is a block diagonal representation of $ \mathds{Z} _{N} $ expressed by shift matrices, whereas $ \mathds{1} $ is an identity matrix. In this fashion, since we can always define a $ D_{M} \left( \mathds{Z} _{N} \right) $ with this action representation when $ M > N \geqslant 2 $, there will always be a particular case where these non-Abelian fusion rules are present. In particular, when $ M $ and $ N $ are coprime natural numbers, the only way to represent this action is by (\ref{special-action}).
\section{Quantum Double Models plus matter via a dualisation procedure}
One notable advantage of having already constructed the $ D_{M} \left( G \right) $ is that it can be used as the basis for new generalizations, where, for example, new qunits can be assigned to the elements of the lattice $ \mathcal{L} _{2} $ that support it. And one of these generalizations is what we will denote by $ D^{K} \negthickspace \left( G \right) $, where gauge and new matter qunits
\begin{equation}
\bigl\vert \varphi _{j} \bigr\rangle = a^{\left( \varphi \right) } _{0} \left\vert 0 \right\rangle + \ldots + a^{\left( \varphi \right) } _{N-1} \left\vert N-1 \right\rangle \ \ \textnormal{and} \ \ \left\vert \chi _{v} \right\rangle = \tilde{a} ^{\left( \chi \right) } _{0} \left\vert 0 \right\rangle + \ldots + \tilde{a} ^{\left( \chi \right) } _{K-1} \left\vert K-1 \right\rangle \label{elemento-double-p-matter}
\end{equation}
are allocated only to edges and face centroids of $ \mathcal{L} _{2} $ respectively.
In order to understand how this new allocation of qunits leads to a class of models other than $ D_{M} \left( G \right) $, it is worth remembering that the $ D \left( G \right) $ has a property that the $ D_{M} \left( G \right) $ does not have: the $ D \left( G \right) $ is \emph{self dual} \cite{aguado}. From the physical point of view, this means that for each excitation detectable by a vertex operator in the $ D \left( G \right) $ there is always another, with the same properties, that is detectable by a face operator and vice versa. The reason for this is that, when we take a lattice $ \mathcal{L} _{2} $ that discretizes some $ 2 $-dimensional compact orientable manifold, each vertex (face) operator acting on $ \mathcal{L} _{2} $ can be identified as a face (vertex) operator that acts on the dual lattice $ \mathcal{L} ^{\ast } _{2} $.
Based on this finding, it is interesting to realise a dualisation procedure on the $ D_{M} \left( G \right) $ in order to evaluate the aim features of the class $ D^{K} \negthickspace \left( G \right) $ thus obtained, which is at least based on the existence of a correspondence between the faces in $ \mathcal{L} _{2} $ ($ \mathcal{L} ^{\ast } _{2} $) and the vertices in $ \mathcal{L} ^{\ast } _{2} $ ($ \mathcal{L} _{2} $) \cite{wenninger}. This correspondence implies that the $ D^{K} \negthickspace \left( G \right) $ Hamiltonian operator must be defined as
\begin{equation}
H_{D^{K} \negthickspace \left( G \right) } = - \sum _{v} A^{\left( G , \tilde{S} \right) } _{v} - \sum _{p} B^{\left( G , \tilde{S} \right) } _{p} - \sum _{j} D^{\left( G , \tilde{S} \right) } _{j} \ , \label{toric-matter-p-hamiltonian}
\end{equation}
where, as suggested by Figure \ref{QMDp-rede},
\begin{figure}[!t]
\begin{center}
\begin{tikzpicture}
\draw[color=blue!20,fill=blue!20] (6,5) rectangle (8,7);
\draw[color=red!20,fill=red!20] (1,2) rectangle (3,4);
\draw[color=green!20,fill=green!20] (6.5,1.5) rectangle (9.5,2.5);
\draw[dotted, color=gray!60, ultra thick] (1,0) -- (1,8);
\draw[dotted, color=gray!60, ultra thick] (3,0) -- (3,8);
\draw[dotted, color=gray!60, ultra thick] (5,0) -- (5,8);
\draw[dotted, color=gray!60, ultra thick] (7,0) -- (7,8);
\draw[dotted, color=gray!60, ultra thick] (9,0) -- (9,8);
\draw[dotted, color=gray!60, ultra thick] (-1,2) -- (11,2);
\draw[dotted, color=gray!60, ultra thick] (-1,4) -- (11,4);
\draw[dotted, color=gray!60, ultra thick] (-1,6) -- (11,6);
\draw[->, color=gray, ultra thick, >=stealth] (0,0) -- (0,2.2);
\draw[->, color=gray, ultra thick, >=stealth] (0,2) -- (0,4.2);
\draw[->, color=gray, ultra thick, >=stealth] (0,4) -- (0,6.2);
\draw[-, color=gray, ultra thick] (0,6) -- (0,8);
\draw[->, color=gray, ultra thick, >=stealth] (2,0) -- (2,2.2);
\draw[->, color=gray, ultra thick, >=stealth] (2,2) -- (2,4.2);
\draw[->, color=gray, ultra thick, >=stealth] (2,4) -- (2,6.2);
\draw[-, color=gray, ultra thick] (2,6) -- (2,8);
\draw[->, color=gray, ultra thick, >=stealth] (4,0) -- (4,2.2);
\draw[->, color=gray, ultra thick, >=stealth] (4,2) -- (4,4.2);
\draw[->, color=gray, ultra thick, >=stealth] (4,4) -- (4,6.2);
\draw[-, color=gray, ultra thick] (4,6) -- (4,8);
\draw[->, color=gray, ultra thick, >=stealth] (6,0) -- (6,2.2);
\draw[->, color=gray, ultra thick, >=stealth] (6,2) -- (6,4.2);
\draw[->, color=gray, ultra thick, >=stealth] (6,4) -- (6,6.2);
\draw[-, color=gray, ultra thick] (6,6) -- (6,8);
\draw[->, color=gray, ultra thick, >=stealth] (8,0) -- (8,2.2);
\draw[->, color=gray, ultra thick, >=stealth] (8,2) -- (8,4.2);
\draw[->, color=gray, ultra thick, >=stealth] (8,4) -- (8,6.2);
\draw[-, color=gray, ultra thick] (8,6) -- (8,8);
\draw[->, color=gray, ultra thick, >=stealth] (10,0) -- (10,2.2);
\draw[->, color=gray, ultra thick, >=stealth] (10,2) -- (10,4.2);
\draw[->, color=gray, ultra thick, >=stealth] (10,4) -- (10,6.2);
\draw[-, color=gray, ultra thick] (10,6) -- (10,8);
\draw[->, color=gray, ultra thick, >=stealth] (-1,1) -- (1.2,1);
\draw[->, color=gray, ultra thick, >=stealth] (1,1) -- (3.2,1);
\draw[->, color=gray, ultra thick, >=stealth] (3,1) -- (5.2,1);
\draw[->, color=gray, ultra thick, >=stealth] (5,1) -- (7.2,1);
\draw[->, color=gray, ultra thick, >=stealth] (7,1) -- (9.2,1);
\draw[-, color=gray, ultra thick] (9,1) -- (11,1);
\draw[->, color=gray, ultra thick, >=stealth] (-1,3) -- (1.2,3);
\draw[->, color=gray, ultra thick, >=stealth] (1,3) -- (3.2,3);
\draw[->, color=gray, ultra thick, >=stealth] (3,3) -- (5.2,3);
\draw[->, color=gray, ultra thick, >=stealth] (5,3) -- (7.2,3);
\draw[->, color=gray, ultra thick, >=stealth] (7,3) -- (9.2,3);
\draw[-, color=gray, ultra thick] (9,3) -- (11,3);
\draw[->, color=gray, ultra thick, >=stealth] (-1,5) -- (1.2,5);
\draw[->, color=gray, ultra thick, >=stealth] (1,5) -- (3.2,5);
\draw[->, color=gray, ultra thick, >=stealth] (3,5) -- (5.2,5);
\draw[->, color=gray, ultra thick, >=stealth] (5,5) -- (7.2,5);
\draw[->, color=gray, ultra thick, >=stealth] (7,5) -- (9.2,5);
\draw[-, color=gray, ultra thick] (9,5) -- (11,5);
\draw[->, color=gray, ultra thick, >=stealth] (-1,7) -- (1.2,7);
\draw[->, color=gray, ultra thick, >=stealth] (1,7) -- (3.2,7);
\draw[->, color=gray, ultra thick, >=stealth] (3,7) -- (5.2,7);
\draw[->, color=gray, ultra thick, >=stealth] (5,7) -- (7.2,7);
\draw[->, color=gray, ultra thick, >=stealth] (7,7) -- (9.2,7);
\draw[-, color=gray, ultra thick] (9,7) -- (11,7);
\draw[->, ultra thick, >=stealth] (2,1) -- (2,2.2);
\draw[->, ultra thick, >=stealth] (2,2.0) -- (2,4.2);
\draw[-, ultra thick] (2,4.0) -- (2,5.0);
\draw[->, ultra thick, >=stealth] (0.0,3) -- (1.2,3);
\draw[->, ultra thick, >=stealth] (1.0,3) -- (3.2,3);
\draw[-, ultra thick] (3.0,3) -- (4.0,3);
\draw[->, ultra thick, >=stealth] (6,5.0) -- (6,6.2);
\draw[-, ultra thick] (6,6.0) -- (6,7.0);
\draw[->, ultra thick, >=stealth] (8,5.0) -- (8,6.2);
\draw[-, ultra thick] (8,6.0) -- (8,7.0);
\draw[->, ultra thick, >=stealth] (6.0,5) -- (7.2,5);
\draw[-, ultra thick] (7.0,5) -- (8.0,5);
\draw[->, ultra thick, >=stealth] (6.0,7) -- (7.2,7);
\draw[-, ultra thick] (7.0,7) -- (8.0,7);
\draw[->, ultra thick, >=stealth] (8,1.0) -- (8,2.2);
\draw[-, ultra thick] (8,2.0) -- (8,3.0);
\node [right] (2.1,4.2) at (2.1,4.2) {$ a $};
\node [below] (3.2,2.9) at (3.2,2.9) {$ b $};
\node [left] (1.9,1.8) at (1.9,1.8) {$ c $};
\node [above] (0.8,3.1) at (0.8,3.1) {$ d $};
\draw[color=black,fill=white] (7,6) circle (0.3);
\node [] (7,6) at (7,6) {$ \tilde{\gamma } $};
\node [right] (7,7.3) at (7,7.3) {$ r $};
\node [below] (8.3,6) at (8.3,6) {$ s $};
\node [left] (7,4.7) at (7,4.7) {$ t $};
\node [above] (5.7,6) at (5.7,6) {$ u $};
\draw[color=black,fill=white] (7,2) circle (0.3);
\node [] (7,2) at (7,2) {$ \tilde{\alpha } $};
\draw[color=black,fill=white] (9,2) circle (0.3);
\node [] (9,2) at (9,2) {$ \tilde{\beta } $};
\node [right] (8.0,1.7) at (8.0,1.7) {$ j $};
\end{tikzpicture}
\end{center}
\caption{\label{QMDf-rede} Piece of $ \mathcal{L} _{2} $ superimposed on its dual lattice $ \mathcal{L} ^{\ast } _{2} $ (dashed). Here, the rose and baby blue sectors represent the same ones already mentioned in Figure \ref{QMDv-rede}, whereas the new green sector corresponds to the $ j $-th edge comprised between two adjacent faces that now support matter qunits.}
\label{QMDp-rede}
\end{figure}
its vertex ($ A^{\left( G , \tilde{S} \right) } _{v} $), face ($ B^{\left( G , \tilde{S} \right) } _{p} $) and edge ($ D^{\left( G , \tilde{S} \right) } _{j} $) operators can be purposely identified as a dualisation of the $ D_{M} \left( G \right) $ face, vertex and edge operators respectively. These operators are specifically defined as
\begin{equation}
A^{\left( G , \tilde{S} \right) } _{v} = \frac{1}{\vert G \vert } \sum _{g \in G} \bar{A} ^{g} _{v} \ , \ \ B^{\left( G , \tilde{S} \right) } _{p} = \bar{B} ^{0} _{p} \ \ \textnormal{and} \ \ D^{\left( G , \tilde{S} \right) } _{j} = \frac{1}{\vert \tilde{S} \vert } \sum _{\tilde{\lambda } \in \tilde{S}} D^{\tilde{\lambda }} _{j} \ , \label{qdmp-operators}
\end{equation}
whose components are defined in Figure \ref{QMDp-operators-components}. Here, $ \tilde{S} $ must be interpreted at least as the index set for the basis $ \mathcal{B} _{p} = \big\{ \left\vert \tilde{\alpha } \right\rangle : \tilde{\alpha } \in \tilde{S} \big\} $ analogously to what happens to basis $ \mathcal{B} _{v} $.
\subsection{Solvability requirements}
However, for this dualisation procedure to be consistent, it is necessary that $ D^{K} \left( G \right) $ be a class of solvable models, i.e., that the vertex, face and edge operators of each of these models have to commute between them. And by analysing these commutation rules, we conclude that, for this to happen, it is necessary that $ G $ and $ \tilde{S} $ are \emph{at least} two groups. After all, as this dualisation procedure implies that $ \bigl\vert \chi _{\tilde{\alpha }} \bigr\rangle $ and $ \bigl\vert \phi _{j} \bigr\rangle $ are related by a co-action $ \tilde{\alpha } \mapsto \mathcal{F} \left( \tilde{\alpha } \right) = \tilde{\alpha } \otimes f \left( \tilde{\alpha } \right) $, where $ f : \tilde{S} \rightarrow G $ needs be such that
\begin{equation}
f \bigl( 1 \bigr) = 1 \ , \ \ \bigl( f \bigl( \tilde{\alpha } \bigr) \bigr) ^{\dagger } = f \bigl( \tilde{\alpha } ^{-1} \bigr) = f^{-1} \bigl( \tilde{\alpha } \bigr) \ \ \textnormal{and} \ \ f \bigl( \tilde{\alpha } _{1} \bigr) \cdot f \bigl( \tilde{\alpha } _{2} \bigr) = f \bigl( \tilde{\alpha } _{1} \ast \tilde{\alpha } _{2} \bigr) \ , \label{coation-property}
\end{equation}
the double action of an edge operator (as the one that is present in Figure \ref{double-link}) requires that
\begin{eqnarray*}
a^{\prime \prime } \negthickspace & = & \negthickspace f \bigl( \tilde{\lambda } ^{\prime } \bigr) \cdot a^{\prime } = f \bigl( \tilde{\lambda } ^{\prime } \bigr) \cdot f \bigl( \tilde{\lambda } \bigr) \cdot a = f \bigl( \tilde{\lambda } ^{\prime } \ast \tilde{\lambda } \bigr) \cdot a = f \bigl( \tilde{\lambda } \ast \tilde{\lambda } ^{\prime } \bigr) \cdot a \ , \\
\tilde{\alpha } ^{\prime \prime } \negthickspace & = & \negthickspace \tilde{\alpha } ^{\prime } \ast \tilde{\lambda } ^{\prime } = \tilde{\alpha } ^{\prime } \ast \bigl( \tilde{\lambda } \ast \tilde{\lambda } ^{\prime } \bigr) \ \ \textnormal{and} \ \ \tilde{\beta } ^{\prime \prime } = \bigl( \tilde{\lambda } ^{\prime } \bigr) ^{-1} \ast \tilde{\beta } ^{\prime } = \bigl( \tilde{\lambda } ^{\prime } \bigr) ^{-1} \ast \tilde{\lambda } ^{-1} \ast \tilde{\beta } = \bigl( \tilde{\lambda } \ast \tilde{\lambda } ^{\prime } \bigr) ^{-1} \ast \tilde{\beta } \ ,
\end{eqnarray*}
and therefore
\begin{equation}
\tilde{\alpha } \ast \tilde{\beta } = \tilde{\beta } \ast \tilde{\alpha } \ \ \Rightarrow \ \ f \bigl( \tilde{\alpha } \ast \tilde{\beta } \bigr) = f \bigl( \tilde{\beta } \ast \tilde{\alpha } \bigr) \ \ \Rightarrow \ \ f \bigl( \tilde{\alpha } \bigr) \cdot f \bigl( \tilde{\beta } \bigr) = f \bigl( \tilde{\beta } \bigr) \cdot f \bigl( \tilde{\alpha } \bigr) \ . \label{abelian-point}
\end{equation}
That is, since $ f $ is a homomorphism that satisfies (\ref{abelian-point}), $ \tilde{S} $ and $ \mathsf{Im} \left( f \right) \subset G $ must be two Abelian groups. However, as the Figures \ref{dual-comut-ad} and \ref{dual-comut-bd} show that the only way to cancel $ \bigl[ A_{v} , D_{j} \bigr] $ and $ \bigl[ B_{p} , D_{j} \bigr] $ is by taking
\begin{equation}
f \bigl( \tilde{\gamma } \bigr) \cdot g = g \cdot f \bigl( \tilde{\gamma } \bigr) \ ,
\end{equation}
we conclude that $ \mathsf{Im} \left( f \right) $ must be the centre of group $ G $ \cite{james}.
\begin{figure}[!t]
\begin{center}
\begin{tikzpicture}[
scale=0.3,
equation/.style={thin},
trans/.style={thin,shorten >=0.5pt,shorten <=0.5pt,>=stealth},
flecha/.style={thin,->,shorten >=0.5pt,shorten <=0.5pt,>=stealth},
every transition/.style={thick,draw=black!75,fill=black!20}
]
\draw[equation] (-9.5,-0.2) -- (-9.5,-0.2) node[midway,right] {$ \bar{A} ^{g} _{v} $};
\draw[trans] (-6.8,2.3) -- (-6.8,-2.3) node[above=2pt,right=-1pt] {};
\draw[trans] (-0.1,2.3) -- (1.1,-0.06) node[above=2pt,right=-1pt] {};
\draw[trans] (-0.1,-2.3) -- (1.1,0.06) node[above=2pt,right=-1pt] {};
\draw[flecha] (-5.1,0.0) -- (-3.9,0.0) node[above=2pt,right=-1pt] {};
\draw[flecha] (-4.1,0.0) -- (-1.6,0.0) node[above=2pt,right=-1pt] {};
\draw[trans] (-1.8,0.0) -- (-0.9,0.0) node[above=2pt,right=-1pt] {};
\draw[flecha] (-3.0,-1.4) -- (-3.0,-0.5) node[above=2pt,right=-1pt] {};
\draw[flecha] (-3.0,-0.7) -- (-3.0,1.0) node[above=2pt,right=-1pt] {};
\draw[trans] (-3.0,0.8) -- (-3.0,1.4) node[above=2pt,right=-1pt] {};
\draw[equation] (-3.0,2.1) -- (-3.0,2.1) node[midway] {$ a $};
\draw[equation] (0.6,0.0) -- (0.6,0.0) node[midway,left] {$ b $};
\draw[equation] (-3.0,-2.1) -- (-3.0,-2.1) node[midway] {$ c $};
\draw[equation] (-6.7,0.0) -- (-6.7,0.0) node[midway,right] {$ d $};
\draw[equation] (2.5,-0.07) -- (2.5,-0.07) node[midway] {$ = $};
\draw[trans] (4.3,2.3) -- (4.3,-2.3) node[above=2pt,right=-1pt] {};
\draw[trans] (13.6,2.3) -- (14.8,-0.06) node[above=2pt,right=-1pt] {};
\draw[trans] (13.6,-2.3) -- (14.8,0.06) node[above=2pt,right=-1pt] {};
\draw[flecha] (7.9,0.0) -- (9.1,0.0) node[above=2pt,right=-1pt] {};
\draw[flecha] (8.9,0.0) -- (11.4,0.0) node[above=2pt,right=-1pt] {};
\draw[trans] (11.2,0.0) -- (12.1,0.0) node[above=2pt,right=-1pt] {};
\draw[flecha] (10.0,-1.4) -- (10.0,-0.5) node[above=2pt,right=-1pt] {};
\draw[flecha] (10.0,-0.7) -- (10.0,1.0) node[above=2pt,right=-1pt] {};
\draw[trans] (10.0,0.8) -- (10.0,1.4) node[above=2pt,right=-1pt] {};
\draw[equation] (10.1,2.0) -- (10.1,2.0) node[midway] {$ ga $};
\draw[equation] (14.2,-0.2) -- (14.2,-0.2) node[midway,left] {$ gb $};
\draw[equation] (10.1,-2.0) -- (10.1,-2.0) node[midway] {$ cg^{-1} $};
\draw[equation] (4.4,0.0) -- (4.4,0.0) node[midway,right] {$ dg^{-1} $};
\end{tikzpicture} \\
\begin{tikzpicture}[
scale=0.3,
equation/.style={thin},
trans/.style={thin,shorten >=0.5pt,shorten <=0.5pt,>=stealth},
flecha/.style={thin,->,shorten >=0.5pt,shorten <=0.5pt,>=stealth},
every transition/.style={thick,draw=black!75,fill=black!20}
]
\draw[equation] (-9.8,-0.1) -- (-9.8,-0.1) node[midway,right] {$ \bar{B} ^{h} _{p} $};
\draw[trans] (-7.1,2.3) -- (-7.1,-2.3) node[above=2pt,right=-1pt] {};
\draw[trans] (-1.6,2.3) -- (-0.4,-0.06) node[above=2pt,right=-1pt] {};
\draw[trans] (-1.6,-2.3) -- (-0.4,0.06) node[above=2pt,right=-1pt] {};
\draw[flecha] (-6.2,1.2) -- (-3.6,1.2) node[above=2pt,right=-1pt] {};
\draw[trans] (-3.8,1.2) -- (-1.6,1.2) node[above=2pt,right=-1pt] {};
\draw[flecha] (-6.2,-1.2) -- (-3.6,-1.2) node[above=2pt,right=-1pt] {};
\draw[trans] (-3.8,-1.2) -- (-1.6,-1.2) node[above=2pt,right=-1pt] {};
\draw[flecha] (-5.3,-2.0) -- (-5.3,0.3) node[above=2pt,right=-1pt] {};
\draw[trans] (-5.3,0.0) -- (-5.3,2.0) node[above=2pt,right=-1pt] {};
\draw[flecha] (-2.5,-2.0) -- (-2.5,0.3) node[above=2pt,right=-1pt] {};
\draw[trans] (-2.5,0.0) -- (-2.5,2.0) node[above=2pt,right=-1pt] {};
\draw[trans,fill=white] (-3.85,0.0) circle (0.9);
\draw[equation] (-3.8,2.0) -- (-3.8,2.0) node[midway] {$ a $};
\draw[equation] (-0.8,0.0) -- (-0.8,0.0) node[midway,left] {$ b $};
\draw[equation] (-3.8,-2.0) -- (-3.8,-2.0) node[midway] {$ c $};
\draw[equation] (-7.0,0.0) -- (-7.0,0.0) node[midway,right] {$ d $};
\draw[equation] (-3.85,0.0) -- (-3.85,0.0) node[midway] {$ \tilde{\gamma } $};
\draw[equation] (7.2,-0.1) -- (7.2,-0.1) node[midway] {$ = \ \delta \bigl( f \left( \tilde{\gamma } \right) ab^{-1} c^{-1} d , h \bigr) $};
\draw[trans] (14.5,2.3) -- (14.5,-2.3) node[above=2pt,right=-1pt] {};
\draw[trans] (20.0,2.3) -- (21.1,-0.06) node[above=2pt,right=-1pt] {};
\draw[trans] (20.0,-2.3) -- (21.1,0.06) node[above=2pt,right=-1pt] {};
\draw[flecha] (15.3,1.2) -- (17.9,1.2) node[above=2pt,right=-1pt] {};
\draw[trans] (17.7,1.2) -- (19.9,1.2) node[above=2pt,right=-1pt] {};
\draw[flecha] (15.3,-1.2) -- (17.9,-1.2) node[above=2pt,right=-1pt] {};
\draw[trans] (17.7,-1.2) -- (19.9,-1.2) node[above=2pt,right=-1pt] {};
\draw[flecha] (16.2,-2.0) -- (16.2,0.3) node[above=2pt,right=-1pt] {};
\draw[trans] (16.2,0.0) -- (16.2,2.0) node[above=2pt,right=-1pt] {};
\draw[flecha] (19.0,-2.0) -- (19.0,0.3) node[above=2pt,right=-1pt] {};
\draw[trans] (19.0,0.0) -- (19.0,2.0) node[above=2pt,right=-1pt] {};
\draw[trans,fill=white] (17.65,0.0) circle (0.9);
\draw[equation] (17.7,2.0) -- (17.7,2.0) node[midway] {$ a $};
\draw[equation] (20.7,0.0) -- (20.7,0.0) node[midway,left] {$ b $};
\draw[equation] (17.7,-2.0) -- (17.7,-2.0) node[midway] {$ c $};
\draw[equation] (14.5,0.0) -- (14.5,0.0) node[midway,right] {$ d $};
\draw[equation] (17.65,0.0) -- (17.65,0.0) node[midway] {$ \tilde{\gamma } $};
\end{tikzpicture} \\
\begin{tikzpicture}[
scale=0.3,
equation/.style={thin},
trans/.style={thin,shorten >=0.5pt,shorten <=0.5pt,>=stealth},
flecha/.style={thin,->,shorten >=0.5pt,shorten <=0.5pt,>=stealth},
dual/.style={dotted,-,shorten >=0.5pt,shorten <=0.5pt},
every transition/.style={thick,draw=black!75,fill=black!20}
]
\draw[equation] (-8.2,0.0) -- (-8.2,0.0) node[midway,right] {$ D^{\tilde{\lambda }} _{l} $};
\draw[trans] (-5.6,2.3) -- (-5.6,-2.3) node[above=2pt,right=-1pt] {};
\draw[trans] (0.1,2.3) -- (1.3,-0.06) node[above=2pt,right=-1pt] {};
\draw[trans] (0.1,-2.3) -- (1.3,0.06) node[above=2pt,right=-1pt] {};
\draw[dual] (-4.4,0.0) -- (-0.2,0.0) node[above=2pt,right=-1pt] {};
\draw[flecha] (-2.3,-1.0) -- (-2.3,0.3) node[above=2pt,right=-1pt] {};
\draw[trans] (-2.3,0.0) -- (-2.3,1.0) node[above=2pt,right=-1pt] {};
\draw[trans,fill=white] (-4.4,0.0) circle (0.9);
\draw[trans,fill=white] (-0.2,0.0) circle (0.9);
\draw[equation] (-2.3,-0.8) -- (-2.3,-0.8) node[midway,below] {$ a $};
\draw[equation] (-4.4,0.0) -- (-4.4,0.0) node[midway] {$ \tilde{\alpha } $};
\draw[equation] (-0.2,-0.02) -- (-0.2,-0.02) node[midway] {$ \tilde{\beta } $};
\draw[equation] (2.6,-0.07) -- (2.6,-0.07) node[midway] {$ = $};
\draw[trans] (4.0,2.3) -- (4.0,-2.3) node[above=2pt,right=-1pt] {};
\draw[trans] (9.7,2.3) -- (10.9,-0.06) node[above=2pt,right=-1pt] {};
\draw[trans] (9.7,-2.3) -- (10.9,0.06) node[above=2pt,right=-1pt] {};
\draw[dual] (5.2,0.0) -- (9.4,0.0) node[above=2pt,right=-1pt] {};
\draw[flecha] (7.2,-1.0) -- (7.2,0.3) node[above=2pt,right=-1pt] {};
\draw[trans] (7.2,0.0) -- (7.2,1.0) node[above=2pt,right=-1pt] {};
\draw[trans,fill=white] (5.2,0.0) circle (0.9);
\draw[trans,fill=white] (9.4,0.0) circle (0.9);
\draw[equation] (7.3,-0.8) -- (7.3,-0.8) node[midway,below] {$ a^{\prime } $};
\draw[equation] (5.2,0.0) -- (5.2,0.0) node[midway] {$ \tilde{\alpha } ^{\prime } $};
\draw[equation] (9.4,-0.02) -- (9.4,-0.02) node[midway] {$ \tilde{\beta } ^{\prime } $};
\end{tikzpicture}
\end{center}
\caption{Definition of the components $ \bar{A} ^{g} _{v} $, $ \bar{B} ^{h} _{p} $ and $ D^{\tilde{\lambda }} _{j} $ that define the vertex, face and edge operators mentioned in (\ref{qdmp-operators}) respectively. As well as in the $ D^{K} \left( G \right) $ case, the symbol $ \tilde{\alpha } $ represents the $ \tilde{\alpha } $-th basis element of the dual matter Hilbert subspace $ \mathcal{B} _{p} $. Here, $ a^{\prime } = f \bigl( \tilde{\lambda } \bigr) \cdot a $, $ \tilde{\alpha } ^{\prime } = \tilde{\alpha } \ast \tilde{\lambda } $ and $ \tilde{\beta } ^{\prime } = \tilde{\lambda } ^{-1} \ast \tilde{\beta } $, where $ f : \tilde{S} \rightarrow G $.}
\label{QMDp-operators-components}
\end{figure}
\subsection{About the dualisation of the quasiparticles properties}
Since the conditions above guarantee that the $ D^{K} \negthickspace \left( G \right) $ is solvable, many things can already be said about this model. And one of the standard things that can be said is that its ground state $ \bigl\vert \tilde{\xi } _{0} \bigr\rangle $ can be characterized by the following relations:
\begin{equation}
A^{\left( G , \tilde{S} \right) } _{v} \bigl\vert \tilde{\xi } _{0} \bigr\rangle = \bigl\vert \tilde{\xi } _{0} \bigr\rangle \ , \ \ B^{\left( G , \tilde{S} \right) } _{p} \bigl\vert \tilde{\xi } _{0} \bigr\rangle = \bigl\vert \tilde{\xi } _{0} \bigr\rangle \ \ \textnormal{and} \ \ D^{\left( G , \tilde{S} \right) } _{j} \bigl\vert \tilde{\xi } _{0} \bigr\rangle = \bigl\vert \tilde{\xi } _{0} \bigr\rangle \ . \label{QDMp-ground-state}
\end{equation}
However, the first non-standard comment we can make about the $ D^{K} \negthickspace \left( G \right) $ concerns the comparison between the $ D_{M} \negthickspace \left( \mathds{Z} _{N} \right) $ and $ D^{K} \negthickspace \left( \mathds{Z} _{N} \right) $. After all, as the matrix representations
\begin{eqnarray}
A^{\left( \mathds{Z} _{N} , \mathds{Z} _{K} \right) } _{v} \negthickspace & = & \negthickspace \frac{1}{N} \sum _{g \in G} \bigl( X^{\dagger } _{a} \bigr) ^{g} \otimes \bigl( X^{\dagger } _{b} \bigr) ^{g} \otimes \bigl( X_{c} \bigr) ^{g} \otimes \bigl( X_{d} \bigr) ^{g} \ , \notag \\
B^{\left( \mathds{Z} _{N} , \mathds{Z} _{K} \right) } _{p} \negthickspace & = & \negthickspace \frac{1}{N} \sum _{g \in G} F_{p} \left( \tilde{\alpha } : g \right) \otimes \bigl( Z^{\dagger } _{r} \bigr) ^{g} \otimes \bigl( Z_{s} \bigr) ^{g} \otimes \bigl( Z_{t} \bigr) ^{g} \otimes \bigl( Z^{\dagger } _{u} \bigr) ^{g} \ \ \textnormal{and} \label{QDMp-vertice-plaquete-link-operators} \\
D^{\left( \mathds{Z} _{N} , \mathds{Z} _{K} \right) } _{j} \negthickspace & = & \negthickspace \frac{1}{K} \sum _{\tilde{\gamma } \in \tilde{S}} \bigl( \tilde{X} _{p_{1}} \bigr) ^{\tilde{\gamma }} \otimes F_{j} \left( \tilde{\alpha } : g \right) \otimes \bigl( \tilde{X} ^{\dagger } _{p_{2}} \bigr) ^{\tilde{\gamma }} \notag
\end{eqnarray}
are such that $ F_{p} \left( \tilde{\alpha } : g \right) $ and $ F_{j} \left( \tilde{\alpha } : g \right) $ are co-action matrices, and
\begin{equation}
X = \sum _{h \in \mathds{Z} _{N}} \left\vert \left( h + 1 \right) \textnormal{mod} \ N \right\rangle \left\langle h \right\vert \ , \ \ Z = \sum _{h \in \mathds{Z} _{N}} \omega ^{h} \left\vert h \right\rangle \left\langle h \right\vert \ \ \textnormal{and} \ \ \tilde{X} = \sum _{\tilde{\alpha } \in \mathds{Z} _{K}} \left\vert \left( \tilde{\alpha } + 1 \right) \textnormal{mod} \ K \right\rangle \left\langle \tilde{\alpha } \right\vert , \label{matrices}
\end{equation}
where $ \omega = e^{i \left( 2 \pi / N \right) } $ is the generator of the gauge group, there is a duality between the properties of the $ D^{K} \negthickspace \left( \mathds{Z} _{N} \right) $ and $ D_{M} \negthickspace \left( \mathds{Z} _{N} \right) $ quasiparticles. This duality stems from the fact that the $ D^{K} \negthickspace \left( \mathds{Z} _{N} \right) $ contains the same $ D \left( \mathds{Z} _{N} \right) $ quasiparticles, but, when $ f $ is a non-trivial homomorphism, those that are detected by the vertex operator acquire confinement properties. In other words, transporting these latter quasiparticles by using an operator like
\begin{equation}
O^{z \left( g \right) } _{\boldsymbol{\gamma }} = \prod _{j \in \boldsymbol{\gamma }} Z^{\pm g} _{j} \ , \label{z-transporter}
\end{equation}
where $ \boldsymbol{\gamma } $ is a path composed by two by two adjacent edges, always increases the system energy and this energy increases as a function of the number of edges involved in this transport. \label{confinament-comment}
\begin{figure}[!t]
\begin{center}
\begin{tikzpicture}[
scale=0.3,
equation/.style={thin},
trans/.style={thin,shorten >=0.5pt,shorten <=0.5pt,>=stealth},
flecha/.style={thin,->,shorten >=0.5pt,shorten <=0.5pt,>=stealth},
dual/.style={dotted,-,shorten >=0.5pt,shorten <=0.5pt},
every transition/.style={thick,draw=black!75,fill=black!20}
]
\draw[equation] (-8.4,-0.1) -- (-8.4,-0.1) node[midway,right] {$ D^{\tilde{\lambda }} _{j} $};
\draw[trans] (-5.6,2.3) -- (-5.6,-2.3) node[above=2pt,right=-1pt] {};
\draw[trans] (0.1,2.3) -- (1.3,-0.06) node[above=2pt,right=-1pt] {};
\draw[trans] (0.1,-2.3) -- (1.3,0.06) node[above=2pt,right=-1pt] {};
\draw[dual] (-4.4,0.0) -- (-0.2,0.0) node[above=2pt,right=-1pt] {};
\draw[flecha] (-2.3,-1.0) -- (-2.3,0.3) node[above=2pt,right=-1pt] {};
\draw[trans] (-2.3,0.0) -- (-2.3,1.0) node[above=2pt,right=-1pt] {};
\draw[trans,fill=white] (-4.4,0.0) circle (0.9);
\draw[trans,fill=white] (-0.2,0.0) circle (0.9);
\draw[equation] (-2.3,-0.8) -- (-2.3,-0.8) node[midway,below] {$ a $};
\draw[equation] (-4.4,0.0) -- (-4.4,0.0) node[midway] {$ \tilde{\alpha } $};
\draw[equation] (-0.2,-0.02) -- (-0.2,-0.02) node[midway] {$ \tilde{\beta } $};
\draw[equation] (2.6,-0.07) -- (2.6,-0.07) node[midway] {$ \ = \ $};
\draw[equation] (4.2,0.0) -- (4.2,0.0) node[midway] {$ \sum $};
\draw[equation] (4.3,-1.3) -- (4.3,-1.3) node[midway] {$ _{\tilde{\lambda }} $};
\draw[trans] (5.4,2.3) -- (5.4,-2.3) node[above=2pt,right=-1pt] {};
\draw[trans] (11.1,2.3) -- (12.3,-0.06) node[above=2pt,right=-1pt] {};
\draw[trans] (11.1,-2.3) -- (12.3,0.06) node[above=2pt,right=-1pt] {};
\draw[dual] (6.6,0.0) -- (10.8,0.0) node[above=2pt,right=-1pt] {};
\draw[flecha] (8.6,-1.0) -- (8.6,0.3) node[above=2pt,right=-1pt] {};
\draw[trans] (8.6,0.0) -- (8.6,1.0) node[above=2pt,right=-1pt] {};
\draw[trans,fill=white] (6.6,0.0) circle (0.9);
\draw[trans,fill=white] (10.8,0.0) circle (0.9);
\draw[equation] (8.7,-0.8) -- (8.7,-0.8) node[midway,below] {$ a^{\prime } $};
\draw[equation] (6.6,0.0) -- (6.6,0.0) node[midway] {$ \tilde{\alpha } ^{\prime } $};
\draw[equation] (10.8,-0.02) -- (10.8,-0.02) node[midway] {$ \tilde{\beta } ^{\prime } $};
\end{tikzpicture}
\begin{tikzpicture}[
scale=0.3,
equation/.style={thin},
trans/.style={thin,shorten >=0.5pt,shorten <=0.5pt,>=stealth},
flecha/.style={thin,->,shorten >=0.5pt,shorten <=0.5pt,>=stealth},
dual/.style={dotted,-,shorten >=0.5pt,shorten <=0.5pt},
every transition/.style={thick,draw=black!75,fill=black!20}
]
\draw[equation] (11.3,-0.2) -- (11.3,-0.2) node[midway] {$ \Rightarrow $};
\draw[equation] (13.0,-0.1) -- (13.0,-0.1) node[midway,right] {$ D^{\tilde{\lambda ^{\prime }}} _{j} \circ D^{\tilde{\lambda }} _{j} $};
\draw[trans] (19.4,2.3) -- (19.4,-2.3) node[above=2pt,right=-1pt] {};
\draw[trans] (25.1,2.3) -- (26.3,-0.06) node[above=2pt,right=-1pt] {};
\draw[trans] (25.1,-2.3) -- (26.3,0.06) node[above=2pt,right=-1pt] {};
\draw[dual] (20.6,0.0) -- (24.8,0.0) node[above=2pt,right=-1pt] {};
\draw[flecha] (22.7,-1.0) -- (22.7,0.3) node[above=2pt,right=-1pt] {};
\draw[trans] (22.7,0.0) -- (22.7,1.0) node[above=2pt,right=-1pt] {};
\draw[trans,fill=white] (20.6,0.0) circle (0.9);
\draw[trans,fill=white] (24.8,0.0) circle (0.9);
\draw[equation] (22.7,-0.8) -- (22.7,-0.8) node[midway,below] {$ a $};
\draw[equation] (20.6,0.0) -- (20.6,0.0) node[midway] {$ \tilde{\alpha } $};
\draw[equation] (24.8,-0.02) -- (24.8,-0.02) node[midway] {$ \tilde{\beta } $};
\draw[equation] (27.6,-0.07) -- (27.6,-0.07) node[midway] {$ = $};
\draw[equation] (29.2,0.0) -- (29.2,0.0) node[midway] {$ \sum $};
\draw[equation] (29.3,-1.3) -- (29.3,-1.3) node[midway] {$ _{\tilde{\lambda } , \tilde{\lambda } ^{\prime } } $};
\draw[trans] (30.4,2.3) -- (30.4,-2.3) node[above=2pt,right=-1pt] {};
\draw[trans] (36.1,2.3) -- (37.3,-0.06) node[above=2pt,right=-1pt] {};
\draw[trans] (36.1,-2.3) -- (37.3,0.06) node[above=2pt,right=-1pt] {};
\draw[dual] (31.6,0.0) -- (35.8,0.0) node[above=2pt,right=-1pt] {};
\draw[flecha] (33.6,-1.0) -- (33.6,0.3) node[above=2pt,right=-1pt] {};
\draw[trans] (33.6,0.0) -- (33.6,1.0) node[above=2pt,right=-1pt] {};
\draw[trans,fill=white] (31.6,0.0) circle (0.9);
\draw[trans,fill=white] (35.8,0.0) circle (0.9);
\draw[equation] (33.7,-0.8) -- (33.7,-0.8) node[midway,below] {$ a^{\prime \prime } $};
\draw[equation] (31.6,0.0) -- (31.6,0.0) node[midway] {$ \tilde{\alpha } ^{\prime \prime } $};
\draw[equation] (35.8,-0.02) -- (35.8,-0.02) node[midway] {$ \tilde{\beta } ^{\prime \prime } $};
\end{tikzpicture}
\end{center}
\caption{Scheme related to the double action of the edge operator $ D^{\tilde{\lambda }} _{j} $, which is used to help us conclude that $ \tilde{S} $ must be an Abelian group.}
\label{double-link}
\end{figure}
\subsection{Somes examples}
Although this confinement property is completely analogous to what happens in the $ D_{M} \left( \mathds{Z} _{N} \right) $, there are some facts that seem to \textquotedblleft break\textquotedblright \hspace*{0.01cm} this dual aspect related to these two classes. And the first fact is related to the impossibility of constructing a $ D^{K} \left( G \right) $ substantially different from a $ D \left( G \right) $ when $ K $ and $ N $ are coprime numbers. In the case of a cyclic Abelian $ D^{K} \left( \mathds{Z} _{N} \right) $ the following proposition is relevant \cite{beachy}:
\begin{proposition}
Every group homomorphism $ f : \mathds{Z} _{K} \rightarrow \mathds{Z} _{N} $ can be completely determined by
\begin{equation}
f \bigl( \bigl[ x \bigr] \bigr) \ = \ \bigl[ nx \bigr] \quad , \label{regulator-homomorphism}
\end{equation}
where $ n $ is a natural number that assumes values other than zero if, and only if, $ N $ is a natural number divisible by $ nK$. \label{proposition}
\end{proposition}
\subsubsection{Example: $ G = \mathds{Z} _{2} $ and $ \tilde{S} = \mathds{Z} _{2} $}
In order to understand how the possibility of defining these several homomorphisms influences in the definition of the $ D^{K} \left( \mathds{Z} _{N} \right) $, we will take some simple examples. And the first one is the $ D^{2} \left( \mathds{Z} _{2} \right) $ whose gauge and matter groups are $ \mathds{Z} _{2} $.
According to the Proposition \ref{proposition} above, there are two ways of constructing this $ D^{2} \left( \mathds{Z} _{2} \right) $: one where $ f $ is the trivial homomorphism and, consequently, the representations (\ref{QDMp-vertice-plaquete-link-operators}) are reduced to\footnote{In these examples, we will omit the super indexes $ \bigl( G , \tilde{S} \bigr) $ associated with these operators in favour of a lighter notation that will become very useful later on. From now on, we will also index the vertex, face and edge operators that compose the Hamiltonian (\ref{toric-matter-p-hamiltonian}) with a \textquotedblleft $ 1 $\textquotedblright \hspace*{0.01cm} for a reason that will be clear later.}
\begin{eqnarray}
A_{v,1} \negthickspace & = & \negthickspace \frac{1}{2} \left( \mathds{1} _{a} \otimes \mathds{1} _{b} \otimes \mathds{1} _{c} \otimes \mathds{1} _{d} + \sigma ^{x} _{a} \otimes \sigma ^{x} _{b} \otimes \sigma ^{x} _{c} \otimes \sigma ^{x} _{d} \right) \ , \notag \\
B_{p,1} \negthickspace & = & \negthickspace \frac{1}{2} \left( \mathds{1} _{p} \otimes \mathds{1} _{r} \otimes \mathds{1} _{s} \otimes \mathds{1} _{t} \otimes \mathds{1} _{u} + \mathds{1} _{p} \otimes \sigma ^{z} _{r} \otimes \sigma ^{z} _{s} \otimes \sigma ^{z} _{t} \otimes \sigma ^{z} _{u} \right) \ \ \textnormal{and} \label{dual-expressions-z2z2-trivial} \\
D_{j,1} \negthickspace & = & \negthickspace \frac{1}{2} \left( \mathds{1} _{p_{1}} \otimes \mathds{1} _{j} \otimes \mathds{1} _{p_{2}} + \sigma ^{x} _{p_{1}} \otimes \mathds{1} _{j} \otimes \sigma ^{x} _{p_{2}} \right) \ ; \notag
\end{eqnarray}
and the other where $ F_{p} \left( \tilde{\alpha } : g \right) = \sigma ^{z} _{p} $ and $ F_{j} \left( \tilde{\alpha } : g \right) = \sigma ^{x} _{j} $, and therefore
\begin{eqnarray}
A_{v,1} \negthickspace & = & \negthickspace \frac{1}{2} \left( \mathds{1} _{a} \otimes \mathds{1} _{b} \otimes \mathds{1} _{c} \otimes \mathds{1} _{d} + \sigma ^{x} _{a} \otimes \sigma ^{x} _{b} \otimes \sigma ^{x} _{c} \otimes \sigma ^{x} _{d} \right) \ , \notag \\
B_{p,1} \negthickspace & = & \negthickspace \frac{1}{2} \left( \mathds{1} _{p} \otimes \mathds{1} _{r} \otimes \mathds{1} _{s} \otimes \mathds{1} _{t} \otimes \mathds{1} _{u} + \sigma ^{z} _{p} \otimes \sigma ^{z} _{r} \otimes \sigma ^{z} _{s} \otimes \sigma ^{z} _{t} \otimes \sigma ^{z} _{u} \right) \ \ \textnormal{and} \label{dual-expressions-z2z2} \\
D_{j,1} \negthickspace & = & \negthickspace \frac{1}{2} \left( \mathds{1} _{p_{1}} \otimes \mathds{1} _{j} \otimes \mathds{1} _{p_{2}} + \sigma ^{x} _{p_{1}} \otimes \sigma ^{x} _{j} \otimes \sigma ^{x} _{p_{2}} \right) \ . \notag
\end{eqnarray}
\begin{figure}[!t]
\begin{center}
\begin{tikzpicture}[
scale=0.3,
equation/.style={thin},
trans/.style={thin,shorten >=0.5pt,shorten <=0.5pt,>=stealth},
flecha/.style={thin,->,shorten >=0.5pt,shorten <=0.5pt,>=stealth},
every transition/.style={thick,draw=black!75,fill=black!20}
]
\draw[equation] (-13.4,0.6) -- (-13.4,0.6) node[midway,right] {$ A_{v} \circ B_{p} $};
\draw[trans] (-7.9,3.9) -- (-7.9,-2.3) node[above=2pt,right=-1pt] {};
\draw[trans] (-0.1,3.9) -- (1.3,0.68) node[above=2pt,right=-1pt] {};
\draw[trans] (-0.1,-2.3) -- (1.3,0.80) node[above=2pt,right=-1pt] {};
\draw[flecha] (-6.4,1.2) -- (-5.1,1.2) node[above=2pt,right=-1pt] {};
\draw[flecha] (-5.3,1.2) -- (-2.0,1.2) node[above=2pt,right=-1pt] {};
\draw[trans] (-2.2,1.2) -- (-0.1,1.2) node[above=2pt,right=-1pt] {};
\draw[flecha] (-6.4,-1.2) -- (-5.1,-1.2) node[above=2pt,right=-1pt] {};
\draw[flecha] (-5.3,-1.2) -- (-2.0,-1.2) node[above=2pt,right=-1pt] {};
\draw[trans] (-2.2,-1.2) -- (-0.1,-1.2) node[above=2pt,right=-1pt] {};
\draw[flecha] (-3.8,-2.0) -- (-3.8,0.0) node[above=2pt,right=-1pt] {};
\draw[flecha] (-3.8,-0.2) -- (-3.8,2.9) node[above=2pt,right=-1pt] {};
\draw[trans] (-3.8,2.7) -- (-3.8,3.6) node[above=2pt,right=-1pt] {};
\draw[flecha] (-1.0,-2.0) -- (-1.0,0.0) node[above=2pt,right=-1pt] {};
\draw[flecha] (-1.0,-0.2) -- (-1.0,2.9) node[above=2pt,right=-1pt] {};
\draw[trans] (-1.0,2.7) -- (-1.0,3.6) node[above=2pt,right=-1pt] {};
\draw[trans,fill=white] (-2.4,0.0) circle (0.8);
\draw[equation] (-7.8,1.2) -- (-7.8,1.2) node[midway,right] {$ d $};
\draw[equation] (-3.7,-0.2) -- (-3.7,-0.2) node[midway,left] {$ c $};
\draw[equation] (-5.3,2.6) -- (-5.3,2.6) node[midway,right] {$ a $};
\draw[equation] (-2.3,-2.0) -- (-2.3,-2.0) node[midway] {$ s $};
\draw[equation] (0.6,-0.2) -- (0.6,-0.2) node[midway,left] {$ g $};
\draw[equation] (-2.3,2.0) -- (-2.3,2.0) node[midway] {$ b $};
\draw[equation] (-2.4,0.0) -- (-2.4,0.0) node[midway] {$ \tilde{\alpha } $};
\draw[equation] (9.5,0.8) -- (9.5,0.8) node[midway] {$ = \delta \left( f \left( \tilde{\alpha } \right) bg^{-1} s^{-1} c , h \right) A_{v} $};
\draw[trans] (17.4,3.9) -- (17.4,-2.3) node[above=2pt,right=-1pt] {};
\draw[trans] (25.2,3.9) -- (26.6,0.68) node[above=2pt,right=-1pt] {};
\draw[trans] (25.2,-2.3) -- (26.6,0.80) node[above=2pt,right=-1pt] {};
\draw[flecha] (18.9,1.2) -- (21.2,1.2) node[above=2pt,right=-1pt] {};
\draw[flecha] (20.0,1.2) -- (23.3,1.2) node[above=2pt,right=-1pt] {};
\draw[trans] (23.1,1.2) -- (25.2,1.2) node[above=2pt,right=-1pt] {};
\draw[flecha] (18.9,-1.2) -- (20.2,-1.2) node[above=2pt,right=-1pt] {};
\draw[flecha] (20.0,-1.2) -- (23.3,-1.2) node[above=2pt,right=-1pt] {};
\draw[trans] (23.1,-1.2) -- (25.2,-1.2) node[above=2pt,right=-1pt] {};
\draw[flecha] (21.5,-2.0) -- (21.5,0.0) node[above=2pt,right=-1pt] {};
\draw[flecha] (21.5,-0.2) -- (21.5,2.9) node[above=2pt,right=-1pt] {};
\draw[trans] (21.5,2.7) -- (21.5,3.6) node[above=2pt,right=-1pt] {};
\draw[flecha] (24.3,-2.0) -- (24.3,0.0) node[above=2pt,right=-1pt] {};
\draw[flecha] (24.3,-0.2) -- (24.3,2.9) node[above=2pt,right=-1pt] {};
\draw[trans] (24.3,2.7) -- (24.3,3.6) node[above=2pt,right=-1pt] {};
\draw[trans,fill=white] (22.9,0.0) circle (0.8);
\draw[equation] (17.4,1.2) -- (17.4,1.2) node[midway,right] {$ d $};
\draw[equation] (21.5,-0.2) -- (21.5,-0.2) node[midway,left] {$ c $};
\draw[equation] (20.0,2.6) -- (20.0,2.6) node[midway,right] {$ a $};
\draw[equation] (23.0,-2.0) -- (23.0,-2.0) node[midway] {$ s $};
\draw[equation] (25.9,-0.2) -- (25.9,-0.2) node[midway,left] {$ g $};
\draw[equation] (23.0,2.0) -- (23.0,2.0) node[midway] {$ b $};
\draw[equation] (22.9,0.0) -- (22.9,0.0) node[midway] {$ \tilde{\alpha } $};
\end{tikzpicture} \\
\begin{tikzpicture}[
scale=0.3,
equation/.style={thin},
trans/.style={thin,shorten >=0.5pt,shorten <=0.5pt,>=stealth},
flecha/.style={thin,->,shorten >=0.5pt,shorten <=0.5pt,>=stealth},
every transition/.style={thick,draw=black!75,fill=black!20}
]
\draw[equation] (1.5,0.74) -- (1.5,0.74) node[midway] {$ = \ $};
\draw[equation] (3.0,0.9) -- (3.0,0.9) node[midway] {$ \sum $};
\draw[equation] (3.0,-0.25) -- (3.0,-0.25) node[midway] {$ _{r} $};
\draw[equation] (10.0,0.8) -- (10.0,0.8) node[midway] {$ \delta \left( f \left( \tilde{\alpha } \right) bg^{-1} s^{-1} c , h \right) $};
\draw[trans] (16.5,3.9) -- (16.5,-2.3) node[above=2pt,right=-1pt] {};
\draw[trans] (25.6,3.9) -- (27.0,0.68) node[above=2pt,right=-1pt] {};
\draw[trans] (25.6,-2.3) -- (27.0,0.80) node[above=2pt,right=-1pt] {};
\draw[flecha] (19.3,1.2) -- (21.6,1.2) node[above=2pt,right=-1pt] {};
\draw[flecha] (20.4,1.2) -- (23.7,1.2) node[above=2pt,right=-1pt] {};
\draw[trans] (23.5,1.2) -- (25.6,1.2) node[above=2pt,right=-1pt] {};
\draw[flecha] (19.3,-1.2) -- (20.6,-1.2) node[above=2pt,right=-1pt] {};
\draw[flecha] (20.4,-1.2) -- (23.7,-1.2) node[above=2pt,right=-1pt] {};
\draw[trans] (23.5,-1.2) -- (25.6,-1.2) node[above=2pt,right=-1pt] {};
\draw[flecha] (21.9,-2.0) -- (21.9,0.0) node[above=2pt,right=-1pt] {};
\draw[flecha] (21.9,-0.2) -- (21.9,2.9) node[above=2pt,right=-1pt] {};
\draw[trans] (21.9,2.7) -- (21.9,3.6) node[above=2pt,right=-1pt] {};
\draw[flecha] (24.7,-2.0) -- (24.7,0.0) node[above=2pt,right=-1pt] {};
\draw[flecha] (24.7,-0.2) -- (24.7,2.9) node[above=2pt,right=-1pt] {};
\draw[trans] (24.7,2.7) -- (24.7,3.6) node[above=2pt,right=-1pt] {};
\draw[trans,fill=white] (23.3,0.0) circle (0.8);
\draw[equation] (16.6,1.4) -- (16.6,1.4) node[midway,right] {$ dr^{-1} $};
\draw[equation] (22.0,0.0) -- (22.0,0.0) node[midway,left] {$ cr^{-1} $};
\draw[equation] (19.8,2.6) -- (19.8,2.6) node[midway,right] {$ ra $};
\draw[equation] (23.4,-2.0) -- (23.4,-2.0) node[midway] {$ s $};
\draw[equation] (26.3,-0.2) -- (26.3,-0.2) node[midway,left] {$ g $};
\draw[equation] (23.4,2.0) -- (23.4,2.0) node[midway] {$ rb $};
\draw[equation] (23.3,0.0) -- (23.3,0.0) node[midway] {$ \tilde{\alpha } $};
\end{tikzpicture} \\ \vspace*{0.8cm}
\begin{tikzpicture}[
scale=0.3,
equation/.style={thin},
trans/.style={thin,shorten >=0.5pt,shorten <=0.5pt,>=stealth},
flecha/.style={thin,->,shorten >=0.5pt,shorten <=0.5pt,>=stealth},
every transition/.style={thick,draw=black!75,fill=black!20}
]
\draw[equation] (-13.4,0.47) -- (-13.4,0.47) node[midway,right] {$ B_{p} \circ A_{v} $};
\draw[trans] (-7.9,3.9) -- (-7.9,-2.3) node[above=2pt,right=-1pt] {};
\draw[trans] (-0.1,3.9) -- (1.3,0.68) node[above=2pt,right=-1pt] {};
\draw[trans] (-0.1,-2.3) -- (1.3,0.80) node[above=2pt,right=-1pt] {};
\draw[flecha] (-6.4,1.2) -- (-5.1,1.2) node[above=2pt,right=-1pt] {};
\draw[flecha] (-5.3,1.2) -- (-2.0,1.2) node[above=2pt,right=-1pt] {};
\draw[trans] (-2.2,1.2) -- (-0.1,1.2) node[above=2pt,right=-1pt] {};
\draw[flecha] (-6.4,-1.2) -- (-5.1,-1.2) node[above=2pt,right=-1pt] {};
\draw[flecha] (-5.3,-1.2) -- (-2.0,-1.2) node[above=2pt,right=-1pt] {};
\draw[trans] (-2.2,-1.2) -- (-0.1,-1.2) node[above=2pt,right=-1pt] {};
\draw[flecha] (-3.8,-2.0) -- (-3.8,0.0) node[above=2pt,right=-1pt] {};
\draw[flecha] (-3.8,-0.2) -- (-3.8,2.9) node[above=2pt,right=-1pt] {};
\draw[trans] (-3.8,2.7) -- (-3.8,3.6) node[above=2pt,right=-1pt] {};
\draw[flecha] (-1.0,-2.0) -- (-1.0,0.0) node[above=2pt,right=-1pt] {};
\draw[flecha] (-1.0,-0.2) -- (-1.0,2.9) node[above=2pt,right=-1pt] {};
\draw[trans] (-1.0,2.7) -- (-1.0,3.6) node[above=2pt,right=-1pt] {};
\draw[trans,fill=white] (-2.4,0.0) circle (0.8);
\draw[equation] (-7.8,1.2) -- (-7.8,1.2) node[midway,right] {$ d $};
\draw[equation] (-3.7,-0.2) -- (-3.7,-0.2) node[midway,left] {$ c $};
\draw[equation] (-5.3,2.6) -- (-5.3,2.6) node[midway,right] {$ a $};
\draw[equation] (-2.3,-2.0) -- (-2.3,-2.0) node[midway] {$ s $};
\draw[equation] (0.6,-0.2) -- (0.6,-0.2) node[midway,left] {$ g $};
\draw[equation] (-2.3,2.0) -- (-2.3,2.0) node[midway] {$ b $};
\draw[equation] (-2.4,0.0) -- (-2.4,0.0) node[midway] {$ \tilde{\alpha } $};
\draw[equation] (2.6,0.64) -- (2.6,0.64) node[midway] {$ = $};
\draw[equation] (4.8,0.8) -- (4.8,0.8) node[midway] {$ \sum $};
\draw[equation] (4.8,-0.35) -- (4.8,-0.35) node[midway] {$ _{r} $};
\draw[equation] (6.7,0.6) -- (6.7,0.6) node[midway] {$ B_{p} $};
\draw[trans] (8.0,3.9) -- (8.0,-2.3) node[above=2pt,right=-1pt] {};
\draw[trans] (17.1,3.9) -- (18.5,0.68) node[above=2pt,right=-1pt] {};
\draw[trans] (17.1,-2.3) -- (18.5,0.80) node[above=2pt,right=-1pt] {};
\draw[flecha] (10.8,1.2) -- (13.1,1.2) node[above=2pt,right=-1pt] {};
\draw[flecha] (11.9,1.2) -- (15.2,1.2) node[above=2pt,right=-1pt] {};
\draw[trans] (15.0,1.2) -- (17.1,1.2) node[above=2pt,right=-1pt] {};
\draw[flecha] (10.8,-1.2) -- (12.1,-1.2) node[above=2pt,right=-1pt] {};
\draw[flecha] (11.9,-1.2) -- (15.2,-1.2) node[above=2pt,right=-1pt] {};
\draw[trans] (15.0,-1.2) -- (17.1,-1.2) node[above=2pt,right=-1pt] {};
\draw[flecha] (13.4,-2.0) -- (13.4,0.0) node[above=2pt,right=-1pt] {};
\draw[flecha] (13.4,-0.2) -- (13.4,2.9) node[above=2pt,right=-1pt] {};
\draw[trans] (13.4,2.7) -- (13.4,3.6) node[above=2pt,right=-1pt] {};
\draw[flecha] (16.2,-2.0) -- (16.2,0.0) node[above=2pt,right=-1pt] {};
\draw[flecha] (16.2,-0.2) -- (16.2,2.9) node[above=2pt,right=-1pt] {};
\draw[trans] (16.2,2.7) -- (16.2,3.6) node[above=2pt,right=-1pt] {};
\draw[trans,fill=white] (14.8,0.0) circle (0.8);
\draw[equation] (8.1,1.4) -- (8.1,1.4) node[midway,right] {$ dr^{-1} $};
\draw[equation] (13.5,0.0) -- (13.5,0.0) node[midway,left] {$ cr^{-1} $};
\draw[equation] (11.3,2.6) -- (11.3,2.6) node[midway,right] {$ ra $};
\draw[equation] (14.9,-2.0) -- (14.9,-2.0) node[midway] {$ s $};
\draw[equation] (17.8,-0.2) -- (17.8,-0.2) node[midway,left] {$ g $};
\draw[equation] (14.9,2.0) -- (14.9,2.0) node[midway] {$ rb $};
\draw[equation] (14.8,0.0) -- (14.8,0.0) node[midway] {$ \tilde{\alpha } $};
\end{tikzpicture} \\
\begin{tikzpicture}[
scale=0.3,
equation/.style={thin},
trans/.style={thin,shorten >=0.5pt,shorten <=0.5pt,>=stealth},
flecha/.style={thin,->,shorten >=0.5pt,shorten <=0.5pt,>=stealth},
every transition/.style={thick,draw=black!75,fill=black!20}
]
\draw[equation] (1.1,0.64) -- (1.1,0.64) node[midway] {$ = $};
\draw[equation] (3.0,0.9) -- (3.0,0.9) node[midway] {$ \sum $};
\draw[equation] (3.1,-0.25) -- (3.1,-0.25) node[midway] {$ _{r} $};
\draw[equation] (10.0,0.8) -- (10.0,0.8) node[midway] {$ \delta \left( f \left( \tilde{\alpha } \right) bg^{-1} s^{-1} c , h \right) $};
\draw[trans] (16.5,3.9) -- (16.5,-2.3) node[above=2pt,right=-1pt] {};
\draw[trans] (25.6,3.9) -- (27.0,0.68) node[above=2pt,right=-1pt] {};
\draw[trans] (25.6,-2.3) -- (27.0,0.80) node[above=2pt,right=-1pt] {};
\draw[flecha] (19.3,1.2) -- (21.6,1.2) node[above=2pt,right=-1pt] {};
\draw[flecha] (20.4,1.2) -- (23.7,1.2) node[above=2pt,right=-1pt] {};
\draw[trans] (23.5,1.2) -- (25.6,1.2) node[above=2pt,right=-1pt] {};
\draw[flecha] (19.3,-1.2) -- (20.6,-1.2) node[above=2pt,right=-1pt] {};
\draw[flecha] (20.4,-1.2) -- (23.7,-1.2) node[above=2pt,right=-1pt] {};
\draw[trans] (23.5,-1.2) -- (25.6,-1.2) node[above=2pt,right=-1pt] {};
\draw[flecha] (21.9,-2.0) -- (21.9,0.0) node[above=2pt,right=-1pt] {};
\draw[flecha] (21.9,-0.2) -- (21.9,2.9) node[above=2pt,right=-1pt] {};
\draw[trans] (21.9,2.7) -- (21.9,3.6) node[above=2pt,right=-1pt] {};
\draw[flecha] (24.7,-2.0) -- (24.7,0.0) node[above=2pt,right=-1pt] {};
\draw[flecha] (24.7,-0.2) -- (24.7,2.9) node[above=2pt,right=-1pt] {};
\draw[trans] (24.7,2.7) -- (24.7,3.6) node[above=2pt,right=-1pt] {};
\draw[trans,fill=white] (23.3,0.0) circle (0.8);
\draw[equation] (16.6,1.4) -- (16.6,1.4) node[midway,right] {$ dr^{-1} $};
\draw[equation] (22.0,0.0) -- (22.0,0.0) node[midway,left] {$ cr^{-1} $};
\draw[equation] (19.8,2.6) -- (19.8,2.6) node[midway,right] {$ ra $};
\draw[equation] (23.4,-2.0) -- (23.4,-2.0) node[midway] {$ s $};
\draw[equation] (26.3,-0.2) -- (26.3,-0.2) node[midway,left] {$ g $};
\draw[equation] (23.4,2.0) -- (23.4,2.0) node[midway] {$ rb $};
\draw[equation] (23.3,0.0) -- (23.3,0.0) node[midway] {$ \tilde{\alpha } $};
\end{tikzpicture}
\end{center}
\caption{Result of the action of the operators $ A_{v} \circ B_{p} $ and $ B_{p} \circ A_{v} $ in a site $ \left( v , p \right) $, from which all the commutativity between $ A_{v} $ and $ B_{p} $ is clear.}
\label{dual-comut-ab}
\end{figure}
\begin{figure}[!t]
\begin{center}
\begin{tikzpicture}[
scale=0.3,
equation/.style={thin},
trans/.style={thin,shorten >=0.5pt,shorten <=0.5pt,>=stealth},
flecha/.style={thin,->,shorten >=0.5pt,shorten <=0.5pt,>=stealth},
every transition/.style={thick,draw=black!75,fill=black!20}
]
\draw[equation] (-13.5,-0.1) -- (-13.5,-0.1) node[midway,right] {$ A_{v} \circ D_{j} $};
\draw[trans] (-7.9,2.7) -- (-7.9,-2.7) node[above=2pt,right=-1pt] {};
\draw[trans] (-0.1,2.7) -- (1.1,-0.06) node[above=2pt,right=-1pt] {};
\draw[trans] (-0.1,-2.7) -- (1.1,0.06) node[above=2pt,right=-1pt] {};
\draw[flecha] (-6.4,0.0) -- (-5.1,0.0) node[above=2pt,right=-1pt] {};
\draw[flecha] (-5.3,0.0) -- (-2.0,0.0) node[above=2pt,right=-1pt] {};
\draw[trans] (-2.2,0.0) -- (-0.9,0.0) node[above=2pt,right=-1pt] {};
\draw[flecha] (-3.8,-2.4) -- (-3.8,-1.2) node[above=2pt,right=-1pt] {};
\draw[flecha] (-3.8,-1.4) -- (-3.8,1.7) node[above=2pt,right=-1pt] {};
\draw[trans] (-3.8,1.5) -- (-3.8,2.4) node[above=2pt,right=-1pt] {};
\draw[trans,fill=white] (-2.4,-1.2) circle (0.8);
\draw[trans,fill=white] (-2.4,1.2) circle (0.8);
\draw[equation] (-7.8,0.0) -- (-7.8,0.0) node[midway,right] {$ d $};
\draw[equation] (-3.7,-1.4) -- (-3.7,-1.4) node[midway,left] {$ c $};
\draw[equation] (-5.3,1.4) -- (-5.3,1.4) node[midway,right] {$ a $};
\draw[equation] (-0.3,0.0) -- (-0.3,0.0) node[midway] {$ b $};
\draw[equation] (-2.4,-1.2) -- (-2.4,-1.2) node[midway] {$ \tilde{\beta } $};
\draw[equation] (-2.4,1.2) -- (-2.4,1.2) node[midway] {$ \tilde{\alpha } $};
\draw[equation] (2.5,0.0) -- (2.5,0.0) node[midway] {$ = $};
\draw[equation] (4.5,0.0) -- (4.5,0.0) node[midway] {$ \sum $};
\draw[equation] (4.6,-1.35) -- (4.6,-1.35) node[midway] {$ _{\tilde{\lambda }} $};
\draw[equation] (5.0,-0.1) -- (5.0,-0.1) node[midway,right] {$ A_{v} $};
\draw[trans] (7.6,2.7) -- (7.6,-2.7) node[above=2pt,right=-1pt] {};
\draw[trans] (18.0,2.7) -- (19.1,-0.06) node[above=2pt,right=-1pt] {};
\draw[trans] (18.0,-2.7) -- (19.1,0.06) node[above=2pt,right=-1pt] {};
\draw[flecha] (9.1,0.0) -- (10.9,0.0) node[above=2pt,right=-1pt] {};
\draw[flecha] (10.7,0.0) -- (14.0,0.0) node[above=2pt,right=-1pt] {};
\draw[trans] (13.8,0.0) -- (15.1,0.0) node[above=2pt,right=-1pt] {};
\draw[flecha] (12.2,-2.4) -- (12.2,-1.2) node[above=2pt,right=-1pt] {};
\draw[flecha] (12.2,-1.4) -- (12.2,1.7) node[above=2pt,right=-1pt] {};
\draw[trans] (12.2,1.5) -- (12.2,2.4) node[above=2pt,right=-1pt] {};
\draw[trans,fill=white] (13.6,-1.2) circle (0.8);
\draw[trans,fill=white] (13.6,1.2) circle (0.8);
\draw[equation] (7.7,0.0) -- (7.7,0.0) node[midway,right] {$ d $};
\draw[equation] (12.4,-1.4) -- (12.4,-1.4) node[midway,left] {$ c $};
\draw[equation] (10.8,1.4) -- (10.8,1.4) node[midway,right] {$ a $};
\draw[equation] (16.9,0.0) -- (16.9,0.0) node[midway] {$ f \bigl( \tilde{\lambda } \bigr) b $};
\draw[equation] (13.6,-1.2) -- (13.6,-1.2) node[midway] {$ \tilde{\beta } ^{\prime } $};
\draw[equation] (13.6,1.2) -- (13.6,1.2) node[midway] {$ \tilde{\alpha } ^{\prime } $};
\end{tikzpicture} \\
\begin{tikzpicture}[
scale=0.3,
equation/.style={thin},
trans/.style={thin,shorten >=0.5pt,shorten <=0.5pt,>=stealth},
flecha/.style={thin,->,shorten >=0.5pt,shorten <=0.5pt,>=stealth},
every transition/.style={thick,draw=black!75,fill=black!20}
]
\draw[equation] (2.5,0.0) -- (2.5,0.0) node[midway] {$ = $};
\draw[equation] (4.5,0.1) -- (4.5,0.1) node[midway] {$ \sum $};
\draw[equation] (4.6,-1.35) -- (4.6,-1.35) node[midway] {$ _{k , \tilde{\lambda }} $};
\draw[trans] (5.7,2.7) -- (5.7,-2.7) node[above=2pt,right=-1pt] {};
\draw[trans] (19.1,2.7) -- (20.2,-0.06) node[above=2pt,right=-1pt] {};
\draw[trans] (19.1,-2.7) -- (20.2,0.06) node[above=2pt,right=-1pt] {};
\draw[flecha] (9.1,0.0) -- (10.9,0.0) node[above=2pt,right=-1pt] {};
\draw[flecha] (10.7,0.0) -- (14.0,0.0) node[above=2pt,right=-1pt] {};
\draw[trans] (13.8,0.0) -- (15.1,0.0) node[above=2pt,right=-1pt] {};
\draw[flecha] (12.2,-2.4) -- (12.2,-1.2) node[above=2pt,right=-1pt] {};
\draw[flecha] (12.2,-1.4) -- (12.2,1.7) node[above=2pt,right=-1pt] {};
\draw[trans] (12.2,1.5) -- (12.2,2.4) node[above=2pt,right=-1pt] {};
\draw[trans,fill=white] (13.6,-1.2) circle (0.8);
\draw[trans,fill=white] (13.6,1.2) circle (0.8);
\draw[equation] (5.7,0.05) -- (5.7,0.05) node[midway,right] {$ dk^{-1} $};
\draw[equation] (12.4,-1.4) -- (12.4,-1.4) node[midway,left] {$ ck^{-1} $};
\draw[equation] (9.9,1.4) -- (9.9,1.4) node[midway,right] {$ ka $};
\draw[equation] (17.6,0.0) -- (17.6,0.0) node[midway] {$ k f \bigl( \tilde{\lambda } \bigr) b $};
\draw[equation] (13.6,-1.2) -- (13.6,-1.2) node[midway] {$ \tilde{\beta } ^{\prime } $};
\draw[equation] (13.6,1.2) -- (13.6,1.2) node[midway] {$ \tilde{\alpha } ^{\prime } $};
\end{tikzpicture} \\ \vspace*{0.8cm}
\begin{tikzpicture}[
scale=0.3,
equation/.style={thin},
trans/.style={thin,shorten >=0.5pt,shorten <=0.5pt,>=stealth},
flecha/.style={thin,->,shorten >=0.5pt,shorten <=0.5pt,>=stealth},
every transition/.style={thick,draw=black!75,fill=black!20}
]
\draw[equation] (-13.5,-0.1) -- (-13.5,-0.1) node[midway,right] {$ D_{j} \circ A_{v} $};
\draw[trans] (-7.9,2.7) -- (-7.9,-2.7) node[above=2pt,right=-1pt] {};
\draw[trans] (-0.1,2.7) -- (1.1,-0.06) node[above=2pt,right=-1pt] {};
\draw[trans] (-0.1,-2.7) -- (1.1,0.06) node[above=2pt,right=-1pt] {};
\draw[flecha] (-6.4,0.0) -- (-5.1,0.0) node[above=2pt,right=-1pt] {};
\draw[flecha] (-5.3,0.0) -- (-2.0,0.0) node[above=2pt,right=-1pt] {};
\draw[trans] (-2.2,0.0) -- (-0.9,0.0) node[above=2pt,right=-1pt] {};
\draw[flecha] (-3.8,-2.4) -- (-3.8,-1.2) node[above=2pt,right=-1pt] {};
\draw[flecha] (-3.8,-1.4) -- (-3.8,1.7) node[above=2pt,right=-1pt] {};
\draw[trans] (-3.8,1.5) -- (-3.8,2.4) node[above=2pt,right=-1pt] {};
\draw[trans,fill=white] (-2.4,-1.2) circle (0.8);
\draw[trans,fill=white] (-2.4,1.2) circle (0.8);
\draw[equation] (-7.8,0.0) -- (-7.8,0.0) node[midway,right] {$ d $};
\draw[equation] (-3.7,-1.4) -- (-3.7,-1.4) node[midway,left] {$ c $};
\draw[equation] (-5.3,1.4) -- (-5.3,1.4) node[midway,right] {$ a $};
\draw[equation] (-0.3,0.0) -- (-0.3,0.0) node[midway] {$ b $};
\draw[equation] (-2.4,-1.2) -- (-2.4,-1.2) node[midway] {$ \tilde{\beta } $};
\draw[equation] (-2.4,1.2) -- (-2.4,1.2) node[midway] {$ \tilde{\alpha } $};
\draw[equation] (2.4,-0.1) -- (2.4,-0.1) node[midway] {$ = $};
\draw[equation] (4.4,-0.1) -- (4.4,-0.1) node[midway] {$ \sum $};
\draw[equation] (4.6,-1.35) -- (4.6,-1.35) node[midway] {$ _{k} $};
\draw[equation] (5.0,-0.1) -- (5.0,-0.1) node[midway,right] {$ D_{j} $};
\draw[trans] (7.6,2.7) -- (7.6,-2.7) node[above=2pt,right=-1pt] {};
\draw[trans] (18.0,2.7) -- (19.1,-0.06) node[above=2pt,right=-1pt] {};
\draw[trans] (18.0,-2.7) -- (19.1,0.06) node[above=2pt,right=-1pt] {};
\draw[flecha] (10.8,0.0) -- (12.6,0.0) node[above=2pt,right=-1pt] {};
\draw[flecha] (12.4,0.0) -- (15.7,0.0) node[above=2pt,right=-1pt] {};
\draw[trans] (15.5,0.0) -- (16.8,0.0) node[above=2pt,right=-1pt] {};
\draw[flecha] (13.9,-2.4) -- (13.9,-1.2) node[above=2pt,right=-1pt] {};
\draw[flecha] (13.9,-1.4) -- (13.9,1.7) node[above=2pt,right=-1pt] {};
\draw[trans] (13.9,1.5) -- (13.9,2.4) node[above=2pt,right=-1pt] {};
\draw[trans,fill=white] (15.3,-1.2) circle (0.8);
\draw[trans,fill=white] (15.3,1.2) circle (0.8);
\draw[equation] (7.5,0.1) -- (7.5,0.1) node[midway,right] {$ dk^{-1} $};
\draw[equation] (14.1,-1.4) -- (14.1,-1.4) node[midway,left] {$ ck^{-1} $};
\draw[equation] (11.7,1.4) -- (11.7,1.4) node[midway,right] {$ ka $};
\draw[equation] (18.0,0.0) -- (18.0,0.0) node[midway] {$ kb $};
\draw[equation] (15.3,-1.2) -- (15.3,-1.2) node[midway] {$ \tilde{\beta } $};
\draw[equation] (15.3,1.2) -- (15.3,1.2) node[midway] {$ \tilde{\alpha } $};
\end{tikzpicture} \\
\begin{tikzpicture}[
scale=0.3,
equation/.style={thin},
trans/.style={thin,shorten >=0.5pt,shorten <=0.5pt,>=stealth},
flecha/.style={thin,->,shorten >=0.5pt,shorten <=0.5pt,>=stealth},
every transition/.style={thick,draw=black!75,fill=black!20}
]
\draw[equation] (2.5,0.0) -- (2.5,0.0) node[midway] {$ = $};
\draw[equation] (4.5,0.1) -- (4.5,0.1) node[midway] {$ \sum $};
\draw[equation] (4.6,-1.35) -- (4.6,-1.35) node[midway] {$ _{k , \tilde{\lambda }} $};
\draw[trans] (5.8,2.7) -- (5.8,-2.7) node[above=2pt,right=-1pt] {};
\draw[trans] (18.9,2.7) -- (20.0,-0.06) node[above=2pt,right=-1pt] {};
\draw[trans] (18.9,-2.7) -- (20.0,0.06) node[above=2pt,right=-1pt] {};
\draw[flecha] (9.1,0.0) -- (10.9,0.0) node[above=2pt,right=-1pt] {};
\draw[flecha] (10.7,0.0) -- (14.0,0.0) node[above=2pt,right=-1pt] {};
\draw[trans] (13.8,0.0) -- (15.1,0.0) node[above=2pt,right=-1pt] {};
\draw[flecha] (12.2,-2.4) -- (12.2,-1.2) node[above=2pt,right=-1pt] {};
\draw[flecha] (12.2,-1.4) -- (12.2,1.7) node[above=2pt,right=-1pt] {};
\draw[trans] (12.2,1.5) -- (12.2,2.4) node[above=2pt,right=-1pt] {};
\draw[trans,fill=white] (13.6,-1.2) circle (0.8);
\draw[trans,fill=white] (13.6,1.2) circle (0.8);
\draw[equation] (5.7,0.05) -- (5.7,0.05) node[midway,right] {$ dk^{-1} $};
\draw[equation] (12.4,-1.4) -- (12.4,-1.4) node[midway,left] {$ ck^{-1} $};
\draw[equation] (9.9,1.4) -- (9.9,1.4) node[midway,right] {$ ka $};
\draw[equation] (17.4,0.0) -- (17.4,0.0) node[midway] {$ f \bigl( \tilde{\lambda } \bigr) kb $};
\draw[equation] (13.6,-1.2) -- (13.6,-1.2) node[midway] {$ \tilde{\beta } ^{\prime } $};
\draw[equation] (13.6,1.2) -- (13.6,1.2) node[midway] {$ \tilde{\alpha } ^{\prime } $};
\end{tikzpicture}
\end{center}
\caption{Result of action of the operators $ A_{v} \circ D_{j} $ and $ D_{j} \circ A_{v} $ on the lattice, which make it clear that $ \bigl[ A_{v} , D_{j} \bigr] = 0 $ when $ f \bigl( \tilde{\lambda } \bigr) $ belongs to the centre of $ G $.}
\label{dual-comut-ad}
\end{figure}
Both possibilities lead to a model that houses the same quasiparticles already related to the Toric Code $ D \left( \mathds{Z} _{2} \right) $, which are produced in pairs by the action of the operators \cite{mf-pedagogical}.
\begin{equation*}
\sigma ^{x} _{j} \ , \ \ \sigma ^{z} _{j} \ \ \textnormal{and} \ \ \textnormal{\textquotedblleft } \ \sigma ^{y} _{j} \ \textnormal{\textquotedblright } = \sigma ^{x} _{j} \circ \sigma ^{z} _{j} = \sigma ^{z} _{j} \circ \sigma ^{x} _{j} \ .
\end{equation*}
However, it is worth noting that only the model with (\ref{dual-expressions-z2z2}) leads to a $ D_{2} \left( \mathds{Z} _{2} \right) $ substantially different from its correspondent $ D \left( \mathds{Z} _{2} \right) $. After all, in this modified Toric Code $ D \left( \mathds{Z} _{2} \right) $, the only quasiparticles that can be moved without increase the system energy are those detectable only by the face operators, i.e., the quasiparticles $ m $ that are produced by the action of $ \sigma ^{x} _{j} $.
Regardless of the topological features of the $ D^{2} \left( \mathds{Z} _{2} \right) $ ground states, it is also important to note that the different choices we have made for $ f $ also imply another kind of ground state degeneracy.
\begin{itemize}
\item[\textbf{I.}] In the case of the former $ D^{2} \left( \mathds{Z} _{2} \right) $ with (\ref{dual-expressions-z2z2-trivial}), its ground state is two-fold degenerate and given by
\begin{eqnarray}
\bigl\vert \tilde{\xi } ^{\left( 1 \right) } _{0} \bigr\rangle \negthickspace & = & \negthickspace \frac{1}{\sqrt{2}} \prod _{v^{\prime }} A_{v^{\prime }} \prod _{j^{\prime }} D_{j^{\prime }} \left( \bigotimes _{j} \left\vert 0 \right\rangle \right) \otimes \left( \bigotimes _{p} \left\vert 0 \right\rangle \right) \ \ \textnormal{and} \label{ground-state-qdmp-z2z2} \\
\bigl\vert \tilde{\xi } ^{\left( 2 \right) } _{0} \bigr\rangle \negthickspace & = & \negthickspace \frac{1}{\sqrt{2}} \prod _{v^{\prime }} A_{v^{\prime }} \prod _{j^{\prime }} D_{j^{\prime }} \left( \bigotimes _{j} \left\vert 0 \right\rangle \right) \otimes \left( \bigotimes _{p \neq p^{\prime }} \left\vert 0 \right\rangle \right) \otimes \left\vert 1 \right\rangle _{p^{\prime }} \ . \label{ground-state-qdmp-z2z2-second-trivial}
\end{eqnarray}
This two-fold degeneracy is justified as a result of (i) none of the operators in (\ref{dual-expressions-z2z2-trivial}) is able to detect any change $ \left\vert 0 \right\rangle _{p^{\prime }} \leftrightarrow \left\vert 1 \right\rangle _{p^{\prime }} $ and (ii) the operator $ \sigma ^{x} _{p^{\prime }} $ executing it cannot be expressed as a product involving the operators (\ref{dual-expressions-z2z2-trivial}).
\item[\textbf{II.}] In the case of the latter $ D^{2} \left( \mathds{Z} _{2} \right) $ with (\ref{dual-expressions-z2z2}), the ground state is non-degenerate and given by (\ref{ground-state-qdmp-z2z2}) because the face operator in (\ref{dual-expressions-z2z2}) can detect a change $ \left\vert 0 \right\rangle _{p^{\prime }} \leftrightarrow \left\vert 1 \right\rangle _{p^{\prime }} $. In this regard, in addition to the quasiparticles inherited from the $ D \left( \mathds{Z} _{2} \right) $, this $ D^{2} \left( \mathds{Z} _{2} \right) $ also admits other quasiparticles $ Q^{\left( J , K \right) } $ arising by effect of some $ \tilde{W} ^{\left( J , K \right) } $ operators such that
\begin{equation}
B_{p,J} \circ \tilde{W} ^{\left( J , K \right) } _{p} = \tilde{W} ^{\left( J , K \right) } _{p} \circ B_{p, 1} \ \ \textnormal{and} \ \ D_{j,K} \circ \tilde{W} ^{\left( J , K \right) } _{p} = \tilde{W} ^{\left( J , K \right) } _{p} \circ D_{j,1} \ , \label{comutation-rule-Z2Z2}
\end{equation}
where $ B_{p,J} $ and $ D_{j,K} $ are the elements that define the respective projector sets $ \mathfrak{B} _{p} $ and $ \mathfrak{D} _{j} $. Here, these two sets are given by
\begin{equation*}
\mathfrak{B} _{p} = \left\{ B_{p,1} , B_{p,2} \right\} \ \ \textnormal{and} \ \ \mathfrak{D} _{j} = \left\{ D_{j,1} , D_{j,2} \right\} \ ,
\end{equation*}
where
\begin{eqnarray*}
B_{p,2} \negthickspace & = & \negthickspace \frac{1}{2} \left( \mathds{1} _{p} \otimes \mathds{1} _{r} \otimes \mathds{1} _{s} \otimes \mathds{1} _{t} \otimes \mathds{1} _{u} - \sigma ^{z} _{p} \otimes \sigma ^{z} _{r} \otimes \sigma ^{z} _{s} \otimes \sigma ^{z} _{t} \otimes \sigma ^{z} _{u} \right) \ \ \textnormal{and} \\
D_{j,2} \negthickspace & = & \negthickspace \frac{1}{2} \left( \mathds{1} _{p_{1}} \otimes \mathds{1} _{j} \otimes \mathds{1} _{p_{2}} - \sigma ^{x} _{p_{1}} \otimes \sigma ^{x} _{j} \otimes \sigma ^{x} _{p_{2}} \right) \ .
\end{eqnarray*}
According to these expressions, the only satisfactory solution of (\ref{comutation-rule-Z2Z2}) is
\begin{equation}
\tilde{W} ^{\left( 1 , 1 \right) } _{p} = \mathds{1} _{p} \ , \ \ \tilde{W} ^{\left( 1 , 2 \right) } _{p} = \sigma ^{z} _{p} \ , \ \ \tilde{W} ^{\left( 2 , 1 \right) } _{p} = \sigma ^{x} _{p} \ \ \textnormal{and} \ \ \tilde{W} ^{\left( 2 , 2 \right) } _{p} = \textnormal{\textquotedblleft } \ \sigma ^{y} _{p} \ \textnormal{\textquotedblright } \ . \label{dual-commut-rules-vertex-1}
\end{equation}
Note that, by the point of view of the face operator $ B_{p,1} $, the quasiparticle $ \tilde{Q} ^{\left( 2 , 1 \right) } $ produced by $ \tilde{W} ^{\left( 2 , 1 \right) } _{p} $ behaves effectively as a monopole $ m $.
\end{itemize}
\begin{figure}[!t]
\begin{center}
\begin{tikzpicture}[
scale=0.3,
equation/.style={thin},
trans/.style={thin,shorten >=0.5pt,shorten <=0.5pt,>=stealth},
flecha/.style={thin,->,shorten >=0.5pt,shorten <=0.5pt,>=stealth},
every transition/.style={thick,draw=black!75,fill=black!20}
]
\draw[equation] (-11.8,-0.1) -- (-11.8,-0.1) node[midway,right] {$ B_{p} \circ D_{j} $};
\draw[trans] (-6.1,2.3) -- (-6.1,-2.3) node[above=2pt,right=-1pt] {};
\draw[trans] (4.0,2.3) -- (5.2,-0.06) node[above=2pt,right=-1pt] {};
\draw[trans] (4.0,-2.3) -- (5.2,0.06) node[above=2pt,right=-1pt] {};
\draw[flecha] (-5.2,1.2) -- (-2.6,1.2) node[above=2pt,right=-1pt] {};
\draw[flecha] (-2.8,1.2) -- (1.5,1.2) node[above=2pt,right=-1pt] {};
\draw[trans] (1.3,1.2) -- (3.6,1.2) node[above=2pt,right=-1pt] {};
\draw[flecha] (-5.2,-1.2) -- (-2.6,-1.2) node[above=2pt,right=-1pt] {};
\draw[flecha] (-2.8,-1.2) -- (1.5,-1.2) node[above=2pt,right=-1pt] {};
\draw[trans] (1.3,-1.2) -- (3.6,-1.2) node[above=2pt,right=-1pt] {};
\draw[flecha] (-4.3,-2.0) -- (-4.3,0.3) node[above=2pt,right=-1pt] {};
\draw[trans] (-4.3,0.0) -- (-4.3,2.0) node[above=2pt,right=-1pt] {};
\draw[flecha] (-1.5,-2.0) -- (-1.5,0.3) node[above=2pt,right=-1pt] {};
\draw[trans] (-1.5,0.0) -- (-1.5,2.0) node[above=2pt,right=-1pt] {};
\draw[flecha] (2.7,-2.0) -- (2.7,0.3) node[above=2pt,right=-1pt] {};
\draw[trans] (2.7,0.0) -- (2.7,2.0) node[above=2pt,right=-1pt] {};
\draw[trans,fill=white] (-2.85,0.0) circle (0.9);
\draw[trans,fill=white] (1.25,0.0) circle (0.9);
\draw[equation] (-2.8,2.0) -- (-2.8,2.0) node[midway] {$ a $};
\draw[equation] (0.2,0.0) -- (0.2,0.0) node[midway,left] {$ b $};
\draw[equation] (-2.8,-2.0) -- (-2.8,-2.0) node[midway] {$ c $};
\draw[equation] (-6.0,0.0) -- (-6.0,0.0) node[midway,right] {$ d $};
\draw[equation] (-2.85,0.0) -- (-2.85,0.0) node[midway] {$ \tilde{\alpha } $};
\draw[equation] (1.25,0.0) -- (1.25,0.0) node[midway] {$ \tilde{\beta } $};
\draw[equation] (6.5,-0.1) -- (6.5,-0.1) node[midway] {$ = $};
\draw[equation] (8.6,0.0) -- (8.6,0.0) node[midway] {$ \sum $};
\draw[equation] (8.9,-1.25) -- (8.9,-1.25) node[midway] {$ _{\tilde{\lambda }} $};
\draw[equation] (9.1,-0.1) -- (9.1,-0.1) node[midway,right] {$ B_{p} $};
\draw[trans] (11.5,2.3) -- (11.5,-2.3) node[above=2pt,right=-1pt] {};
\draw[trans] (23.6,2.3) -- (24.8,-0.06) node[above=2pt,right=-1pt] {};
\draw[trans] (23.6,-2.3) -- (24.8,0.06) node[above=2pt,right=-1pt] {};
\draw[flecha] (12.4,1.2) -- (15.0,1.2) node[above=2pt,right=-1pt] {};
\draw[flecha] (14.8,1.2) -- (21.1,1.2) node[above=2pt,right=-1pt] {};
\draw[trans] (20.9,1.2) -- (23.2,1.2) node[above=2pt,right=-1pt] {};
\draw[flecha] (12.4,-1.2) -- (15.0,-1.2) node[above=2pt,right=-1pt] {};
\draw[flecha] (14.8,-1.2) -- (21.1,-1.2) node[above=2pt,right=-1pt] {};
\draw[trans] (20.9,-1.2) -- (23.2,-1.2) node[above=2pt,right=-1pt] {};
\draw[flecha] (13.3,-2.0) -- (13.3,0.3) node[above=2pt,right=-1pt] {};
\draw[trans] (13.3,0.0) -- (13.3,2.0) node[above=2pt,right=-1pt] {};
\draw[flecha] (16.1,-2.0) -- (16.1,0.3) node[above=2pt,right=-1pt] {};
\draw[trans] (16.1,0.0) -- (16.1,2.0) node[above=2pt,right=-1pt] {};
\draw[flecha] (22.8,-2.0) -- (22.8,0.3) node[above=2pt,right=-1pt] {};
\draw[trans] (22.8,0.0) -- (22.8,2.0) node[above=2pt,right=-1pt] {};
\draw[trans,fill=white] (14.75,0.0) circle (0.9);
\draw[trans,fill=white] (21.45,0.0) circle (0.9);
\draw[equation] (14.8,2.0) -- (14.8,2.0) node[midway] {$ a $};
\draw[equation] (20.4,0.0) -- (20.4,0.0) node[midway,left] {$ f \bigl( \tilde{\lambda } \bigr) b $};
\draw[equation] (14.8,-2.0) -- (14.8,-2.0) node[midway] {$ c $};
\draw[equation] (11.6,0.0) -- (11.6,0.0) node[midway,right] {$ d $};
\draw[equation] (14.75,0.0) -- (14.75,0.0) node[midway] {$ \tilde{\alpha } $};
\draw[equation] (21.45,0.0) -- (21.45,0.0) node[midway] {$ \tilde{\beta } $};
\end{tikzpicture} \\
\begin{tikzpicture}[
scale=0.3,
equation/.style={thin},
trans/.style={thin,shorten >=0.5pt,shorten <=0.5pt,>=stealth},
flecha/.style={thin,->,shorten >=0.5pt,shorten <=0.5pt,>=stealth},
every transition/.style={thick,draw=black!75,fill=black!20}
]
\draw[equation] (7.0,-0.1) -- (7.0,-0.1) node[midway] {$ = $};
\draw[equation] (8.8,0.0) -- (8.8,0.0) node[midway] {$ \sum $};
\draw[equation] (8.8,-1.35) -- (8.8,-1.35) node[midway] {$ _{\tilde{\lambda }} $};
\draw[equation] (18.8,-0.1) -- (18.8,-0.1) node[midway] {$ \delta \bigl( f \bigl( \tilde{\alpha} \ast \tilde{\lambda } \bigr) ab^{-1} f^{-1} \bigl( \tilde{\lambda } \bigr) c^{-1} d , h \bigr) $};
\draw[trans] (28.0,2.3) -- (28.0,-2.3) node[above=2pt,right=-1pt] {};
\draw[trans] (40.2,2.3) -- (41.4,-0.06) node[above=2pt,right=-1pt] {};
\draw[trans] (40.2,-2.3) -- (41.4,0.06) node[above=2pt,right=-1pt] {};
\draw[flecha] (28.9,1.2) -- (31.5,1.2) node[above=2pt,right=-1pt] {};
\draw[flecha] (31.3,1.2) -- (37.7,1.2) node[above=2pt,right=-1pt] {};
\draw[trans] (37.5,1.2) -- (39.8,1.2) node[above=2pt,right=-1pt] {};
\draw[flecha] (28.9,-1.2) -- (31.5,-1.2) node[above=2pt,right=-1pt] {};
\draw[flecha] (31.3,-1.2) -- (37.7,-1.2) node[above=2pt,right=-1pt] {};
\draw[trans] (37.5,-1.2) -- (39.8,-1.2) node[above=2pt,right=-1pt] {};
\draw[flecha] (29.8,-2.0) -- (29.8,0.3) node[above=2pt,right=-1pt] {};
\draw[trans] (29.8,0.0) -- (29.8,2.0) node[above=2pt,right=-1pt] {};
\draw[flecha] (32.6,-2.0) -- (32.6,0.3) node[above=2pt,right=-1pt] {};
\draw[trans] (32.6,0.0) -- (32.6,2.0) node[above=2pt,right=-1pt] {};
\draw[flecha] (39.3,-2.0) -- (39.3,0.3) node[above=2pt,right=-1pt] {};
\draw[trans] (39.3,0.0) -- (39.3,2.0) node[above=2pt,right=-1pt] {};
\draw[trans,fill=white] (31.25,0.0) circle (0.9);
\draw[trans,fill=white] (37.95,0.0) circle (0.9);
\draw[equation] (31.3,2.0) -- (31.3,2.0) node[midway] {$ a $};
\draw[equation] (36.9,0.0) -- (36.9,0.0) node[midway,left] {$ f \bigl( \tilde{\lambda } \bigr) b $};
\draw[equation] (31.3,-2.0) -- (31.3,-2.0) node[midway] {$ c $};
\draw[equation] (28.1,0.0) -- (28.1,0.0) node[midway,right] {$ d $};
\draw[equation] (31.25,0.0) -- (31.25,0.0) node[midway] {$ \tilde{\alpha } ^{\prime } $};
\draw[equation] (37.95,0.0) -- (37.95,0.0) node[midway] {$ \tilde{\beta } ^{\prime } $};
\end{tikzpicture} \\ \vspace*{0.8cm}
\begin{tikzpicture}[
scale=0.3,
equation/.style={thin},
trans/.style={thin,shorten >=0.5pt,shorten <=0.5pt,>=stealth},
flecha/.style={thin,->,shorten >=0.5pt,shorten <=0.5pt,>=stealth},
every transition/.style={thick,draw=black!75,fill=black!20}
]
\draw[equation] (-11.7,-0.1) -- (-11.7,-0.1) node[midway,right] {$ D_{j} \circ B_{p} $};
\draw[trans] (-6.1,2.3) -- (-6.1,-2.3) node[above=2pt,right=-1pt] {};
\draw[trans] (4.0,2.3) -- (5.2,-0.06) node[above=2pt,right=-1pt] {};
\draw[trans] (4.0,-2.3) -- (5.2,0.06) node[above=2pt,right=-1pt] {};
\draw[flecha] (-5.2,1.2) -- (-2.6,1.2) node[above=2pt,right=-1pt] {};
\draw[flecha] (-2.8,1.2) -- (1.5,1.2) node[above=2pt,right=-1pt] {};
\draw[trans] (1.3,1.2) -- (3.6,1.2) node[above=2pt,right=-1pt] {};
\draw[flecha] (-5.2,-1.2) -- (-2.6,-1.2) node[above=2pt,right=-1pt] {};
\draw[flecha] (-2.8,-1.2) -- (1.5,-1.2) node[above=2pt,right=-1pt] {};
\draw[trans] (1.3,-1.2) -- (3.6,-1.2) node[above=2pt,right=-1pt] {};
\draw[flecha] (-4.3,-2.0) -- (-4.3,0.3) node[above=2pt,right=-1pt] {};
\draw[trans] (-4.3,0.0) -- (-4.3,2.0) node[above=2pt,right=-1pt] {};
\draw[flecha] (-1.5,-2.0) -- (-1.5,0.3) node[above=2pt,right=-1pt] {};
\draw[trans] (-1.5,0.0) -- (-1.5,2.0) node[above=2pt,right=-1pt] {};
\draw[flecha] (2.7,-2.0) -- (2.7,0.3) node[above=2pt,right=-1pt] {};
\draw[trans] (2.7,0.0) -- (2.7,2.0) node[above=2pt,right=-1pt] {};
\draw[trans,fill=white] (-2.85,0.0) circle (0.9);
\draw[trans,fill=white] (1.25,0.0) circle (0.9);
\draw[equation] (-2.8,2.0) -- (-2.8,2.0) node[midway] {$ a $};
\draw[equation] (0.2,0.0) -- (0.2,0.0) node[midway,left] {$ b $};
\draw[equation] (-2.8,-2.0) -- (-2.8,-2.0) node[midway] {$ c $};
\draw[equation] (-6.0,0.0) -- (-6.0,0.0) node[midway,right] {$ d $};
\draw[equation] (-2.85,0.0) -- (-2.85,0.0) node[midway] {$ \tilde{\alpha } $};
\draw[equation] (1.25,0.0) -- (1.25,0.0) node[midway] {$ \tilde{\beta } $};
\draw[equation] (6.5,-0.1) -- (6.5,-0.1) node[midway] {$ = $};
\draw[equation] (8.5,0.0) -- (8.5,0.0) node[midway] {$ \sum $};
\draw[equation] (8.6,-1.25) -- (8.6,-1.25) node[midway] {$ _{\tilde{\lambda }} $};
\draw[equation] (9.1,-0.1) -- (9.1,-0.1) node[midway,right] {$ \delta \left( f \bigl( \tilde{\alpha } \bigr) ab^{-1} c^{-1} d , h \right) D_{j} $};
\draw[trans] (24.2,2.3) -- (24.2,-2.3) node[above=2pt,right=-1pt] {};
\draw[trans] (34.3,2.3) -- (35.5,-0.06) node[above=2pt,right=-1pt] {};
\draw[trans] (34.3,-2.3) -- (35.5,0.06) node[above=2pt,right=-1pt] {};
\draw[flecha] (25.1,1.2) -- (27.7,1.2) node[above=2pt,right=-1pt] {};
\draw[flecha] (27.5,1.2) -- (31.8,1.2) node[above=2pt,right=-1pt] {};
\draw[trans] (31.6,1.2) -- (33.9,1.2) node[above=2pt,right=-1pt] {};
\draw[flecha] (25.1,-1.2) -- (27.7,-1.2) node[above=2pt,right=-1pt] {};
\draw[flecha] (27.5,-1.2) -- (31.8,-1.2) node[above=2pt,right=-1pt] {};
\draw[trans] (31.6,-1.2) -- (33.9,-1.2) node[above=2pt,right=-1pt] {};
\draw[flecha] (26.0,-2.0) -- (26.0,0.3) node[above=2pt,right=-1pt] {};
\draw[trans] (26.0,0.0) -- (26.0,2.0) node[above=2pt,right=-1pt] {};
\draw[flecha] (28.8,-2.0) -- (28.8,0.3) node[above=2pt,right=-1pt] {};
\draw[trans] (28.8,0.0) -- (28.8,2.0) node[above=2pt,right=-1pt] {};
\draw[flecha] (33.0,-2.0) -- (33.0,0.3) node[above=2pt,right=-1pt] {};
\draw[trans] (33.0,0.0) -- (33.0,2.0) node[above=2pt,right=-1pt] {};
\draw[trans,fill=white] (27.45,0.0) circle (0.9);
\draw[trans,fill=white] (31.55,0.0) circle (0.9);
\draw[equation] (27.5,2.0) -- (27.5,2.0) node[midway] {$ a $};
\draw[equation] (30.5,0.0) -- (30.5,0.0) node[midway,left] {$ b $};
\draw[equation] (27.5,-2.0) -- (27.5,-2.0) node[midway] {$ c $};
\draw[equation] (24.3,0.0) -- (24.3,0.0) node[midway,right] {$ d $};
\draw[equation] (27.45,0.0) -- (27.45,0.0) node[midway] {$ \tilde{\alpha } $};
\draw[equation] (31.55,0.0) -- (31.55,0.0) node[midway] {$ \tilde{\beta } $};
\end{tikzpicture} \\
\begin{tikzpicture}[
scale=0.3,
equation/.style={thin},
trans/.style={thin,shorten >=0.5pt,shorten <=0.5pt,>=stealth},
flecha/.style={thin,->,shorten >=0.5pt,shorten <=0.5pt,>=stealth},
every transition/.style={thick,draw=black!75,fill=black!20}
]
\draw[equation] (13.3,-0.1) -- (13.3,-0.1) node[midway] {$ = $};
\draw[equation] (15.2,0.1) -- (15.2,0.1) node[midway] {$ \sum $};
\draw[equation] (15.2,-1.15) -- (15.2,-1.15) node[midway] {$ _{\tilde{\lambda }} $};
\draw[equation] (22.0,0.0) -- (22.0,0.0) node[midway] {$ \delta \left( f \bigl( \tilde{\alpha} \bigr) ab^{-1} c^{-1} d , h \right) $};
\draw[trans] (28.0,2.3) -- (28.0,-2.3) node[above=2pt,right=-1pt] {};
\draw[trans] (40.2,2.3) -- (41.4,-0.06) node[above=2pt,right=-1pt] {};
\draw[trans] (40.2,-2.3) -- (41.4,0.06) node[above=2pt,right=-1pt] {};
\draw[flecha] (28.9,1.2) -- (31.5,1.2) node[above=2pt,right=-1pt] {};
\draw[flecha] (31.3,1.2) -- (37.7,1.2) node[above=2pt,right=-1pt] {};
\draw[trans] (37.5,1.2) -- (39.8,1.2) node[above=2pt,right=-1pt] {};
\draw[flecha] (28.9,-1.2) -- (31.5,-1.2) node[above=2pt,right=-1pt] {};
\draw[flecha] (31.3,-1.2) -- (37.7,-1.2) node[above=2pt,right=-1pt] {};
\draw[trans] (37.5,-1.2) -- (39.8,-1.2) node[above=2pt,right=-1pt] {};
\draw[flecha] (29.8,-2.0) -- (29.8,0.3) node[above=2pt,right=-1pt] {};
\draw[trans] (29.8,0.0) -- (29.8,2.0) node[above=2pt,right=-1pt] {};
\draw[flecha] (32.6,-2.0) -- (32.6,0.3) node[above=2pt,right=-1pt] {};
\draw[trans] (32.6,0.0) -- (32.6,2.0) node[above=2pt,right=-1pt] {};
\draw[flecha] (39.3,-2.0) -- (39.3,0.3) node[above=2pt,right=-1pt] {};
\draw[trans] (39.3,0.0) -- (39.3,2.0) node[above=2pt,right=-1pt] {};
\draw[trans,fill=white] (31.25,0.0) circle (0.9);
\draw[trans,fill=white] (37.95,0.0) circle (0.9);
\draw[equation] (31.3,2.0) -- (31.3,2.0) node[midway] {$ a $};
\draw[equation] (36.9,0.0) -- (36.9,0.0) node[midway,left] {$ f \bigl( \tilde{\lambda } \bigr) b $};
\draw[equation] (31.3,-2.0) -- (31.3,-2.0) node[midway] {$ c $};
\draw[equation] (28.1,0.0) -- (28.1,0.0) node[midway,right] {$ d $};
\draw[equation] (31.25,0.0) -- (31.25,0.0) node[midway] {$ \tilde{\alpha } ^{\prime } $};
\draw[equation] (37.95,0.0) -- (37.95,0.0) node[midway] {$ \tilde{\beta } ^{\prime } $};
\end{tikzpicture}
\end{center}
\caption{Result of action of the operators $ B_{p} \circ D_{j} $ and $ D_{j} \circ B_{p} $ on the lattice, which only reinforces the need for $ f \bigl( \tilde{\lambda } \bigr) $ to belong to the centre of $ G $ so that this model is solvable.}
\label{dual-comut-bd}
\end{figure}
\subsubsection{A comment on the triviality of the homomorphism}
Although we have not made any comment on the quasiparticles that can be produced by operators acting only on the faces centroids in the $ D^{2} \left( \mathds{Z} _{2} \right) $ with (\ref{dual-expressions-z2z2-trivial}), these quasiparticles exist: they are the same ones produced by the operators $ \tilde{W} ^{\left( J , K \right) } _{p} $ mentioned in (\ref{dual-commut-rules-vertex-1}). But as the operators $ B_{p,1} $ and
\begin{equation*}
B_{p,2} = \frac{1}{2} \left( \mathds{1} _{p} \otimes \mathds{1} _{r} \otimes \mathds{1} _{s} \otimes \mathds{1} _{t} \otimes \mathds{1} _{u} - \mathds{1} _{p} \otimes \sigma ^{z} _{r} \otimes \sigma ^{z} _{s} \otimes \sigma ^{z} _{t} \otimes \sigma ^{z} _{u} \right)
\end{equation*}
completing $ \mathfrak{B} _{p} $, in this $ D^{2} \left( \mathds{Z} _{2} \right) $ where $ f $ is trivial, are such that $ F \left( \tilde{\alpha } : g \right) = \mathds{1} $, these quasiparticles cannot be completely distinguished from each other. Scilicet, the quasiparticles $ \tilde{Q} ^{\left( 2 , 1 \right) } $ and $ \tilde{Q} ^{\left( 2 , 2 \right) } $ are interpreted effectively as equal to $ \tilde{Q} ^{\left( 1 , 1 \right) } $ and $ \tilde{Q} ^{\left( 1 , 2 \right) } $ respectively.
An entirely analogous comment applies to more general models where $ G = \mathds{Z} _{N} $ and $ \tilde{S} = \mathds{Z} _{K} $, since all these models support a case where $ f $ is trivial in accordance with the Proposition \ref{proposition}. And one general characteristic of these $ D^{K} \left( \mathds{Z} _{N} \right) $, where $ f \bigl( \tilde{\alpha } \bigr) = 0 $ for all $ \tilde{\alpha } \in \mathds{Z} _{N} $, is that the quasiparticles that are produced in pairs by the action of
\begin{equation}
X^{g} _{j} \ , \ \ Z^{h} _{j} \ \ \textnormal{and} \ \ \textnormal{\textquotedblleft } \ Y^{\left( g , h \right) } _{j} \ \textnormal{\textquotedblright } = X^{g} _{j} \circ Z^{h} _{j} = Z^{h} _{j} \circ X^{g} _{j} \label{traditional-operators}
\end{equation}
on the lattice edges are insensitive of those that are produced by operators that act exclusively on the face centroids. This quasiparticle insensitivity can be evidenced in both the $ D^{2} \left( \mathds{Z} _{2} \right) $ with (\ref{dual-expressions-z2z2-trivial}) and the $ D^{3} \left( \mathds{Z} _{2} \right) $, whose orthonormal sets of operators $ \mathfrak{A} _{v} $, $ \mathfrak{B} _{p} $ and $ \mathfrak{D} _{j} $ are uniquely defined by
\begin{eqnarray}
A_{v,1} \negthickspace & = & \negthickspace \frac{1}{2} \left( \mathds{1} _{a} \otimes \mathds{1} _{b} \otimes \mathds{1} _{c} \otimes \mathds{1} _{d} + \sigma ^{x} _{a} \otimes \sigma ^{x} _{b} \otimes \sigma ^{x} _{c} \otimes \sigma ^{x} _{d} \right) \ , \notag \\
A_{v,2} \negthickspace & = & \negthickspace \frac{1}{2} \left( \mathds{1} _{a} \otimes \mathds{1} _{b} \otimes \mathds{1} _{c} \otimes \mathds{1} _{d} - \sigma ^{x} _{a} \otimes \sigma ^{x} _{b} \otimes \sigma ^{x} _{c} \otimes \sigma ^{x} _{d} \right) \ , \notag \\
& & \notag \\
B_{p,1} \negthickspace & = & \negthickspace \frac{1}{2} \left( \mathds{1} _{p} \otimes \mathds{1} _{r} \otimes \mathds{1} _{s} \otimes \mathds{1} _{t} \otimes \mathds{1} _{u} + \mathds{1} _{p} \otimes \sigma ^{z} _{r} \otimes \sigma ^{z} _{s} \otimes \sigma ^{z} _{t} \otimes \sigma ^{z} _{u} \right) \ , \notag \\
B_{p,2} \negthickspace & = & \negthickspace \frac{1}{2} \left( \mathds{1} _{p} \otimes \mathds{1} _{r} \otimes \mathds{1} _{s} \otimes \mathds{1} _{t} \otimes \mathds{1} _{u} - \mathds{1} _{p} \otimes \sigma ^{z} _{r} \otimes \sigma ^{z} _{s} \otimes \sigma ^{z} _{t} \otimes \sigma ^{z} _{u} \right) \ , \label{dual-expressions-z2z3} \\
& & \notag \\
D_{j,1} \negthickspace & = & \negthickspace \frac{1}{2} \left( \mathds{1} _{p_{1}} \otimes \mathds{1} _{j} \otimes \mathds{1} _{p_{2}} + X_{p_{1}} \otimes \mathds{1} _{j} \otimes X^{2} _{p_{2}} + X^{2} _{p_{1}} \otimes \mathds{1} _{j} \otimes X_{p_{2}} \right) \ , \notag \\
D_{j,2} \negthickspace & = & \negthickspace \frac{1}{2} \left( \mathds{1} _{p_{1}} \otimes \mathds{1} _{j} \otimes \mathds{1} _{p_{2}} + i X_{p_{1}} \otimes \mathds{1} _{j} \otimes X^{2} _{p_{2}} - i X^{2} _{p_{1}} \otimes \mathds{1} _{j} \otimes X_{p_{2}} \right) \ \ \textnormal{and} \notag \\
D_{j,3} \negthickspace & = & \negthickspace \frac{1}{2} \left( \mathds{1} _{p_{1}} \otimes \mathds{1} _{j} \otimes \mathds{1} _{p_{2}} - i X_{p_{1}} \otimes \mathds{1} _{j} \otimes X^{2} _{p_{2}} + i X^{2} _{p_{1}} \otimes \mathds{1} _{j} \otimes X_{p_{2}} \right) \notag
\end{eqnarray}
because $ N = 2 $ and $ K = 3 $ are coprime numbers. In these cases where $ f \bigl( \tilde{\alpha } \bigr) = 0 $ for all $ \tilde{\alpha } \in \mathds{Z} _{N} $, if we leave aside the topological aspects related to the manifold $ \mathcal{M} _{2} $, the $ D^{K} \left( \mathds{Z} _{N} \right) $ ground states are $ N $-fold degenerate and given by
\begin{equation}
\bigl\vert \tilde{\xi } ^{\left( \tilde{\alpha } \right) } _{0} \bigr\rangle = \frac{1}{\sqrt{2}} \prod _{v^{\prime }} A_{v^{\prime }} \prod _{j^{\prime }} D_{j^{\prime }} \left( \bigotimes _{j} \left\vert 0 \right\rangle \right) \otimes \left( \bigotimes _{p \neq p^{\prime }} \left\vert 0 \right\rangle \right) \otimes \left\vert \tilde{\alpha } \right\rangle _{p^{\prime }} \label{general-ground-state}
\end{equation}
because $ \ker \left( f \right) = \mathds{Z} _{N} $.
\subsection{The ground state degeneracy and the classifiability of the $ D^{K} \left( \mathds{Z} _{N} \right) $}
Note that this insensitivity mentioned above can be \textquotedblleft broken\textquotedblright \hspace*{0.01cm} gradually as the $ D^{K} \left( \mathds{Z} _{N} \right) $ supports other homomorphisms beyond the trivial. This is the case of the $ D^{2} \left( \mathds{Z} _{4} \right) $ and $ D^{4} \left( \mathds{Z} _{4} \right) $ that exhibit a $ 4 $-fold degenerate ground state when $ f $ is trivial, but that, by taking
\begin{equation}
f \left( 0 \right) = f \left( 2 \right) = 0 \ \ \textnormal{and} \ \ f \left( 1 \right) = f \left( 3 \right) = 1 \ , \label{coaction-choice-dual}
\end{equation}
have a $ 2 $-fold degenerate ground state given by
\begin{eqnarray}
\bigl\vert \tilde{\xi } ^{\left( 1 \right) } _{0} \bigr\rangle \negthickspace & = & \negthickspace \frac{1}{\sqrt{2}} \prod _{v^{\prime }} A_{v^{\prime }} \prod _{j^{\prime }} D_{j^{\prime }} \left( \bigotimes _{j} \left\vert 0 \right\rangle \right) \otimes \left( \bigotimes _{p} \left\vert 0 \right\rangle \right) \label{ground-state-qdmp-z2z4-first} \ \ \textnormal{and} \\
\bigl\vert \tilde{\xi } ^{\left( 2 \right) } _{0} \bigr\rangle \negthickspace & = & \negthickspace \frac{1}{\sqrt{2}} \prod _{v^{\prime }} A_{v^{\prime }} \prod _{j^{\prime }} D_{j^{\prime }} \left( \bigotimes _{j} \left\vert 0 \right\rangle \right) \otimes \left( \bigotimes _{p \neq p^{\prime }} \left\vert 0 \right\rangle \right) \otimes \left\vert 2 \right\rangle _{p^{\prime }} \ . \label{ground-state-qdmp-z2z4-second}
\end{eqnarray}
After all, in these cases with (\ref{coaction-choice-dual}), the operators performing transitions between (\ref{ground-state-qdmp-z2z4-first}) and (\ref{ground-state-qdmp-z2z4-second}) cannot be expressed as a product involving the vertex, face and edge operators that define the $ D^{2} \left( \mathds{Z} _{4} \right) $ and $ D^{4} \left( \mathds{Z} _{4} \right) $ Hamiltonians.
However, in the case of the $ D^{4} \left( \mathds{Z} _{4} \right) $, for instance, when $ f : \mathds{Z} _{4} \rightarrow \mathds{Z} _{4} $ is taken as a faithful homomorphism, its ground state is reduced only to (\ref{ground-state-qdmp-z2z4-first}). This follows because, similarly to what happens in the case $ D^{2} \left( \mathds{Z} _{2} \right) $ with (\ref{dual-expressions-z2z2}), the face operator that define the $ D^{4} \left( \mathds{Z} _{4} \right) $ Hamiltonian can detect any changes $ \left\vert 0 \right\rangle _{p^{\prime }} \leftrightarrow \left\vert \tilde{\alpha } \right\rangle _{p^{\prime }} $, where $ \tilde{\alpha } \neq 0 $.
In general, what these examples make clear is that, in order for the quasiparticles produced by (\ref{traditional-operators}) to be sensitive to those produced by operators that act exclusively on the face centroids (and consequently the quasiparticles detectable by the vertex operator acquire confinement properties), it is necessary to consider non-trivial homomorphisms, which end up decreasing the algebraic degeneracy of the $ D^{K} \left( \mathds{Z} _{N} \right) $ ground states. This algebraic degeneracy is given by
\begin{equation*}
\mathfrak{d} _{\textnormal{alg}} = \left\vert \ \ker \left( f \right) \ \right\vert \ ,
\end{equation*}
since all elements belonging to the $ \ker \left( f \right) $ cause the \textquotedblleft fake holonomy\textquotedblright
\begin{equation}
h^{\prime } = f \bigl( \tilde{\gamma } \bigr) ab^{-1} c^{-1} d = f \bigl( \tilde{\gamma } \bigr) h \label{fake-holonomy-page}
\end{equation}
(which is measured by the face operator shown in Figure \ref{QMDp-operators-components}) to correspond to the true holonomy $ h $. Based on this, we conclude that all these $ D^{K} \left( \mathds{Z} _{N} \right) $ can be classified in terms of a ordered $ 3 $-tuple $ \left( N , K , n \right) $ as follows: \label{classification-page}
\begin{enumerate}
\item[\textbf{(a)}] $ \left( N , K , 0 \right) $ are those whose algebraic degeneracy is maximal and where, on each one of their vacuum states (\ref{general-ground-state}), there is a $ D \left( \mathds{Z} _{N} \right) $ that can support new quasiparticles $ Q^{\left( J , K \right) } $. These quasiparticles, which are produced by operators $ W^{\left( J , K \right) } _{p} $ that act exclusively on the face centroids when $ K > 0 $, are insensitive to those produced by (\ref{traditional-operators}).
\item[\textbf{(b)}] $ \left( N , N , N \right) $ are those that, due to their minimal algebraic degeneracy, are identified as a modified $ D \left( \mathds{Z} _{N} \right) $. Their quasiparticles $ e^{g} $ (produced by operators $ Z^{g} _{j} $) are confined and, just as the quasiparticles $ m^{h} $ and $ \varepsilon ^{\left( g , h \right) } $ (produced by operators $ X^{h} _{j} $ and $ \textnormal{\textquotedblleft } \ Y^{\left( g , h \right) } _{j} \ \textnormal{\textquotedblright } $ respectively), are sensitive to $ Q^{\left( J , K \right) } $.
\item[\textbf{(c)}] $ \left( N , K , n \right) $ have intermediate properties to those mentioned in items \textbf{(a)} and \textbf{(b)} when $ n $ is a natural number such that $ 0 < n < N $ and $ N \vert nK $. That is, on each one of their vacuum states (\ref{general-ground-state}), there is a modified $ D \left( \mathds{Z} _{N} \right) $ where only the quasiparticles $ e^{g^{\prime }} $, $ m^{h^{\prime }} $ and $ \varepsilon ^{\left( g^{\prime } , h^{\prime } \right) } $, with $ g^{\prime } = \left( g + 1 \right) \textnormal{mod} \ k $ and $ h^{\prime } = \left( h + 1 \right) \textnormal{mod} \ k $, have the properties mentioned in item \textbf{(b)}.
\end{enumerate}
Note that the classification $ \left( N , 1 , 0 \right) $ deleted from \textbf{(a)} is associated with the identification of $ D^{1} \left( \mathds{Z} _{N} \right) $ as $ D \left( \mathds{Z} _{N} \right) $, since
\begin{eqnarray*}
H_{D^{1} \left( G \right) } \negthickspace & = & \negthickspace - \sum _{v} A^{\left( G , 0 \right) } _{v} - \sum _{p} B^{\left( G , 0 \right) } _{p} - \sum _{j} D^{\left( G , 0 \right) } _{j} \\
& = & \negthickspace - \sum _{\textnormal{v}} A^{\left( G \right) } _{v} - \sum _{p} B^{\left( G \right) } _{p} - \sum _{j} \mathds{1} _{j} = H_{D \left( G \right)} + \textnormal{cte} \ .
\end{eqnarray*}
\subsubsection{A topological comment}
In addition to all these algebraic considerations, we also need to take into account something important: as the quasiparticles $ m^{h} $ can be moved (by using an operator like
\begin{equation}
O^{x \left( g \right) } _{\boldsymbol{\gamma ^{\ast }}} = \prod _{j \in \boldsymbol{\gamma ^{\ast }}} X^{\pm g} _{j} \ , \label{x-transporter}
\end{equation}
where $ \boldsymbol{\gamma ^{\ast }} $ is any path composed by two by two adjacent dual edges) without increase the system energy, the $ D^{K} \left( \mathds{Z} _{N} \right) $ ground state degeneracy also depends on the order of the fundamental group $ \pi _{1} $ associated with $ \mathcal{M} _{2} $. This additional degeneracy $ \mathfrak{d} _{\textnormal{top}} $, which is specifically of topological origin, must be taken into account to characterize, for instance, the fact that the number of vacuum states increases as the order of the gauge group $ \mathds{Z} _{N} $ increases.
In order to understand this number increasing, it should be noted that the choice of a non-trivial $ f $ causes some quasiparticles $ \tilde{Q} ^{\left( J , K \right) } $ are interpreted effectively as monopoles $ m^{h} $. That is, this effective equivalence between these quasiparticles must be discarded in the calculation of the independent vacuum states. This allows us to affirm that, if the $ D^{K} \left( \mathds{Z} _{N} \right) $ is defined in a discretization of a manifold $ \mathcal{M} _{2} $ with genus $ \mathfrak{g} $, its total ground state degeneracy is given by
\begin{equation}
\mathfrak{d} _{\left( N , K , n \right) } = \mathfrak{d} _{\textnormal{alg}} \cdot \mathfrak{d} _{\textnormal{top}} = \left\vert \ \ker \left( f \right) \ \right\vert \cdot \left\vert \ \mathds{Z} _{K} \ / \ \textnormal{Im} \ f \ \right\vert \ ^{2 \mathfrak{g}} . \label{gsd}
\end{equation}
\subsection{The behaviour of the edge operator as a comparator}
Besides the fact that we are unable to construct a $ D^{K} \left( \mathds{Z} _{N} \right) $ with a classification other than $ \left( N , K , 0 \right) $ when $ N $ and $ K $ are coprime numbers, another point deserves attention in this dualisation procedure. After all, while the $ D_{M} \left( \mathds{Z} _{N} \right) $ edge operator behaves like a comparator (i.e. as an operator that can check the alignment of two adjacent matter qunits), the $ D^{K} \left( \mathds{Z} _{N} \right) $ edge operator does something that seems different from a comparison and seems more like a kind of gauge transformation. \label{D-transformation-page}
Although it is not incorrect to think that $ D^{\left( G , \tilde{S} \right) } _{j} $ may actually be performing some kind of gauge transformation, one of the ways to understand what this operator does is to see how it acts on a diagonal basis. For this, besides taking into account that
\begin{equation}
D_{j} \bigl\vert \tilde{\alpha } , g , \tilde{\beta } \bigr\rangle = \frac{1}{\bigl\vert \tilde{S} \bigr\vert } \sum _{\tilde{\lambda } \in \tilde{S}} \bigl\vert \tilde{\alpha } \ast \tilde{\lambda } , f \bigl( \tilde{\lambda } \bigr) \cdot g , \tilde{\lambda } ^{-1} \ast \tilde{\beta } \bigr\rangle \ , \label{dual-link-operator}
\end{equation}
we must note that this basis is obtained through the unitary transformations
\begin{equation}
\left\vert g^{\prime } \right\rangle = \frac{1}{\vert G \vert } \sum _{g \in G} \omega _{g^{\prime }} \bigl( g \bigr) \left\vert g \right\rangle \ \ \textnormal{and} \ \ \left\vert \tilde{\alpha } ^{\prime } \right\rangle = \frac{1}{\bigl\vert \tilde{S} \bigr\vert } \sum _{\tilde{\alpha } \in \tilde{S}} \bar{\chi } _{\tilde{\alpha } ^{\prime }} \left( \tilde{\alpha } \right) \left\vert \tilde{\alpha } \right\rangle \ \ \textnormal{and} \label{unitary-transformations}
\end{equation}
where $ \omega _{g^{\prime }} \left( g \right) $ and $ \chi _{\tilde{\alpha } ^{\prime }} \left( \tilde{\alpha } \right) $ are the characters of $ G $ and $ \tilde{S} $ respectively. The substitution of relations (\ref{unitary-transformations}) into (\ref{dual-link-operator}) shows that
\begin{equation}
D_{j} \bigl\vert \tilde{\alpha } ^{\prime } , g^{\prime } , \tilde{\beta } ^{\prime } \bigr\rangle = \frac{1}{\bigl\vert \tilde{S} \bigr\vert } \sum _{\tilde{\lambda } \in \tilde{S}} \chi _{\tilde{\alpha } ^{\prime }} \bigl( \tilde{\lambda } \bigr) \omega _{g^{\prime }} \bigl( f \bigl( \tilde{\lambda } \bigr) \bigr) \bar{\chi } _{\tilde{\beta } ^{\prime }} \bigl( \tilde{\lambda } \bigr) \bigl\vert \tilde{\alpha } , g , \tilde{\beta } \bigr\rangle \label{exp25}
\end{equation}
Given this result, it is important to note that, since $ \tilde{S} $ and $ \mathsf{Im} \left( f \right) \subset G $ are two finite Abelian groups, the Fourier transform $ \hat{f} \in L \bigl( \tilde{S} ^{\ast } \bigr) $ is such that
\begin{equation*}
\hat{f} \left( \chi \right) = \sum _{\tilde{\lambda } \in \tilde{S}} f \bigl( \tilde{\lambda } \bigr) \chi \bigl( \tilde{\lambda } \bigr) \ \ \textnormal{and} \ \ f \bigl( \tilde{\lambda } \bigr) = \frac{1}{\bigl\vert \tilde{S} \bigr\vert } \sum _{\chi \in \tilde{S} ^{\ast }} \hat{f} \left( \chi \right) \chi \bigl( \tilde{\lambda } \bigr) \ ,
\end{equation*}
where the dual group $ \tilde{S} ^{\ast } $ is isomorphic to $ \tilde{S} $ \cite{james,rudin,barut,hall}. After all, by noting that an expression of the sort $ \chi _{\tilde{\alpha } ^{\prime }} \bigl( \tilde{\lambda } \bigr) \bar{\chi } _{\tilde{\beta } ^{\prime }} \bigl( \tilde{\lambda } \bigr) = \chi _{\left\{ \tilde{\alpha } ^{\prime } , \tilde{\beta } ^{\prime } \right\}} \bigl( \tilde{\lambda } \bigr) $ is always a character, the substitution of these relations into (\ref{exp25}) yields
\begin{eqnarray*}
D_{j} \bigl\vert \tilde{\alpha } ^{\prime } , g^{\prime } , \tilde{\beta } ^{\prime } \bigr\rangle \negthickspace & = & \negthickspace \frac{1}{\bigl\vert \tilde{S} \bigr\vert } \sum _{\chi _{\tilde{\gamma }} \in \tilde{S} ^{\ast }} \widehat{ \left[ \omega _{g^{\prime }} \circ f \right] } \left( \chi _{\tilde{\gamma }} \right) \left( \frac{1}{\bigl\vert \tilde{S} \bigr\vert } \sum _{\tilde{\lambda } \in \tilde{S}} \chi _{\left\{ \tilde{\alpha } ^{\prime } , \tilde{\beta } ^{\prime } \right\}} \bigl( \tilde{\lambda } \bigr) \bar{\chi } _{\tilde{\gamma }} \bigl( \tilde{\lambda } \bigr) \right) \bigl\vert \tilde{\alpha } , g , \tilde{\beta } \bigr\rangle \\
& = & \negthickspace \frac{1}{\bigl\vert \tilde{S} \bigr\vert } \sum _{\chi _{\tilde{\gamma }} \in \tilde{S} ^{\ast }} \widehat{ \left[ \omega _{g^{\prime }} \circ f \right] } \left( \chi _{\tilde{\gamma }} \right) \cdot \delta \left( \chi _{\left\{ \tilde{\alpha } ^{\prime } , \tilde{\beta } ^{\prime } \right\}} , \chi _{\tilde{\gamma }} \right) \bigl\vert \tilde{\alpha } , g , \tilde{\beta } \bigr\rangle \\
& = & \negthickspace \frac{1}{\bigl\vert \tilde{S} \bigr\vert } \ \widehat{ \left[ \omega _{g^{\prime }} \circ f \right] } \bigl( \chi _{\left\{ \tilde{\alpha } ^{\prime } , \tilde{\beta } ^{\prime } \right\}} \bigr) \bigl\vert \tilde{\alpha } , g , \tilde{\beta } \bigr\rangle \ .
\end{eqnarray*}
That is, although the exact form of the index $ {\bigl\{ \tilde{\alpha } ^{\prime } , \tilde{\beta } ^{\prime } \bigr\}} $ depends on the nature of $ \tilde{S} $, we conclude that the $ D^{\left( G , \tilde{S} \right) } _{j} $ can also be interpreted as an operator that compares matter qunits differently, which only becomes clear when this operator acts on a diagonal basis. This different way of comparing rests on the \emph{Pontryagin duality}, which ensures that there is a one-to-one correspondence between the characters $ \chi _{\tilde{\lambda }} $ and the elements of $ \tilde{S} $ \cite{pontryagin}.
\section{Final remarks}
According to what we saw above, it is perfectly possible to perform a dualisation procedure on the $ D_{M} \left( G \right) $ and, thus, obtain another class $ D^{K} \left( G \right) $ of solvable models that can also be interpreted as a generalization of the $ D \left( G \right) $. However, this algebraic dual class $ D^{K} \left( G \right) $, when superimposed on the $ D_{M} \left( G \right) $ to define a more general new class with Hamiltonian
\begin{equation*}
H_{\mathrm{total}} = H_{D_{M} \left( G \right) } + H_{D^{K} \left( G \right) } \ ,
\end{equation*}
does not necessarily define self-dual models. After all, unlike the $ D_{M} \left( \mathds{Z} _{N} \right) $ where $ M $ and $ N $ are coprime numbers, which supports a non-trivial case where quasiparticles with non-Abelian fusion rules are required, it is impossible to create a $ D^{K} \left( \mathds{Z} _{N} \right) $ different from the trivial when $ N $ and $ K $ are coprime numbers. That is, from the physical point of view, this means that for each excitation detectable by the face or/and edge operators in (\ref{qdmv-operators}) there will not necessarily be another, with the same properties, that is detectable by the vertex or/and edge operators in (\ref{qdmp-operators}) respectively and vice versa.
In any case, it is important to emphasize that this construction of the $ D^{K} \left( \mathds{Z} _{N} \right) $, which was based on its recognition as an algebraic dual of the $ D^{M} \left( \mathds{Z} _{N} \right) $, really allows us to recognize some dual traces between these two classes. Note that all this construction, when made by using $ G = \mathds{Z} _{N} $ and $ \tilde{S} = \mathds{Z} _{K} $ where $ N $ and $ K $ are not coprime numbers, leads to models in which the quasiparticles $ m^{h} $ are free and the $ e^{g} $ are confined: that is, while the first quasiparticles can be transported by an operator as (\ref{x-transporter}) without increasing the system energy, the latter, when transported by an operator as (\ref{z-transporter}), increase this energy.
Despite this confinement observation about $ e^{g} $ was first made on page \pageref{confinament-comment} as a result of an analysis of the $ D^{K} \left( \mathds{Z} _{N} \right) $, it is worth noting that this confinement extends to the class $ D^{K} \left( G \right) $ as a whole, provided that $ f $ is not a trivial homomorphism. That is, whenever it is possible to define a model with non-trivial $ f $, at least a part of the quasiparticles detectable by the vertex operators will be confined. Another interesting aspect of the $ D^{K} \left( \mathds{Z} _{N} \right) $, which can also be extended to the $ D^{K} \left( G \right) $ as a whole, is associated with the possibility of classifying them as presented in items \textbf{(a)}, \textbf{(b)} and \textbf{(c)} on page \pageref{classification-page}. This is a complete classification because, once the ordered $ 3 $-tuple $ \left( N , K , n \right) $ of a model is identified, it is possible to identify not only all the properties of its quasiparticles, but also to calculate its ground state degeneracy. Note that the $ D^{K} \left( G \right) $ has algebraic and topological orders: the algebraic order is due to the co-action of the gauge group on the matter qunits; the topological order is due to the fact that, as in the $ D \left( G \right) $, the $ D^{K} \left( G \right) $ ground state degeneracy depends on the order of the fundamental group $ \pi _{1} $ associated with $ \mathcal{M} _{2} $.
Although this generalization $ D^{K} \left( G \right) $ has been successful, there is no impediment, a priori, to construct others without the artifice of the dualisation. So, one question we can ask about these other generalizations is whether one of them bring the same results from a different point of view. One of the possibilities we can explore is that in which $ f $ defines a \emph{crossed module} \cite{loday}: that is, $ f $ is a homomorphism that, together with an action $ \theta : G \times \tilde{S} \rightarrow \tilde{S} $, respects two conditions
\begin{equation*}
f \left( \theta \left( g , \tilde{\alpha } \right) \right) = g f \left( \tilde{\alpha } \right) g^{-1} \ \ \textnormal{and} \ \ \theta \bigl( f \left( \tilde{\alpha } \right) , \tilde{\beta } \bigr) = \tilde{\alpha } \tilde{\beta } \tilde{\alpha } ^{-1}
\end{equation*}
where the second is known as the \emph{Peiffer condition} \cite{peiffer-paper,mantovani}. Note that the homomorphisms that define the $ D^{K} \left( G \right) $ satisfy these two conditions when this action is trivial because $ G $ and $ \tilde{S} $ are Abelian groups. And the possible advantage of taking $ f $ as the homomorphism that completes a crossed module lies in the fact that it seems possible to recover the $ D^{K} \left( G \right) $ as a particular case of the \emph{higher lattice gauge theories} \cite{baez-1}, which are based on the higher-dimensional category theory \cite{bucur,cheng,group-category-example}. A good example of this is in Ref. \cite{faria-martins}, where a $ 2 $-lattice gauge theory is defined by using a three-dimensional lattice to which we can measure $ 1 $- and $ 2 $-holonomies: after all, while $ 1 $-holonomy is identified as the same \textquotedblleft fake holonomy\textquotedblright \hspace*{0.01cm} (\ref{fake-holonomy-page}) that is preserved by the action of the operator $ A^{\left( G , \tilde{S} \right) } _{v} $ that performs gauge transformations, the $ 2 $-holonomy \cite{abba-wag} is preserved by the action of the operator
\begin{equation*}
A^{D} _{v} = \prod _{j \in S_{v}} D^{\left( G , \tilde{S} \right) } _{j} \ ,
\end{equation*}
which corroborates with the perception of $ D^{\left( G , \tilde{S} \right) } _{j} $ as the component of an operator that performs another kind of gauge transformations. Note that, if $ f $ is the homomorphism that defines the crossed module $ \mathcal{G} = \bigl( G , \tilde{S} ; f , \theta \bigr) $, the first and second homotopy groups of this crossed module can be defined as $ \pi _{1} \left( \mathcal{G} \right) = G \ / \ \textnormal{Im} \ f = \mathrm{coker} \ \left( f \right) $ and $\pi _{2} \left( \mathcal{G} \right) = \ker \left( f \right) = \pi _{2} \left( \mathcal{G} \right) $ respectively \cite{ricardo}, whose orders define the result (\ref{gsd}). We will return to this topic in a future work.
\section*{Acknowledgments}
This work has been supported by CAPES (ProEx) and CNPq (grant 162117/2015-9). We thank C. Antonio Filho, R. Figueiredo and U. A. Maciel Neto for some mathematical discussions, as well as P. Teotonio Sobrinho for some physical discussions on subjects concerning this project. In particular, M. F. also thanks L. Daros Gama and F. Diacenco Xavier for friendly support during this work.
\section*{References} | 78,297 |
Immunisations
A wide range of vaccines are available by appointment at the Medical Service for:
- Overseas travel
- Health care workers, including students/ staff who are undertaking training placements and or field work
- Animal workers and handlers
- Annual influenza campaign available to all staff and all student
Our service is able to offer specialised vaccination clinics throughout the year for groups travel, clinical placements and or field work. Please contact the service to arrange a group booking.
If available, please bring your previous vaccination history card with you to your appointment.
Available to download: | 94,949 |
Chateau de Vullierens Gardens
The Château de Vullierens gardens have plenty of hidden treasures : 30 hectares of tree and flowers-filled gardens, including the famous iris garden and its 600 varieties.
As you walk around, you’ll also be able to enjoy 75 spectacular art pieces that are displayed all around the domain.
Since 2018, the gardens created a brand new educational and find path for families : :
King Lizard’s Kingdom :. Will you find them ?
Take the map at the entrance and make this kingdom yours. Plan minimum an hour to enjoy the full parcours.
Unicorn Family Pass Privilege :
2 adults entrance tickets for the price of 1 OR
Upon presentation of your Unicorn Family Pass during your attendance to the brunch at the Café des Iris, you get 2 adults entrance tickets to the gardens for free
(Garden entrance tickets free for children up to 11 years old)
Another new attraction : the Wine Cellars of the castle are open to the public. Until 2012, the harvesting was pressed and stored by the castle’s winemaker.
Key no miss moments:
– Blooming of the irises mid may
– Blooming of the roses early june
– Photo shoot in the gardens
– Brunch at the Café des Iris, every Sunday, Mother’s Day and Bank holidays.
Jardin des Iris
King Lizard’s Kingdom
Address :Les Jardins du Château de Vullierens, 1115 Vullierens
Phone: 079/ 274 79 64
Opening hours & rates
Café des Jardins, book your table for a brunch at [email protected] or by phone at 079 776 41 15 | 79,422 |
TITLE: Are quantum fields such as electron field fundamental?
QUESTION [4 upvotes]: I am wondering do quantum fields transform into other different fields? I came across a hypothetical inflaton field which is suggested to decay into every quantum fields we see today, however most articles I read on the internet state that quantum fields are truly fundamental. So I guess maybe the condition during cosmic inflation is different than it is today so inflaton field can decay.
REPLY [10 votes]: Quantum fields can exchange energy. So for example when an electron and a positron annihilate the energy that was in the electron/positron field is transferred to the photon field. The result is that one electron and one positron disappear and two photons are created. The fields themselves are still both present - only the energy in the fields has changed.
So the fields are not decaying into each other. They just exchange energy. At the end of inflation the inflaton field transferred its energy into other quantum fields. That annihilates the particles in the inflation field and creates particles in the other fields.
I was about to say that the inflaton field doesn't disappear after it has transferred energy to the other fields, but we need to exercise some caution because we don't know what the inflaton field was so we can't say what happened to it at the end of inflation. | 217,277 |
Events in Karlskrona
Karlskrona, 22 - 23 October 2013
Art Line workshop – for future cooperation
Karlskrona, 20 November 2013
Open Lecture – The Kairos of Scholarly Multimedia: Examining the History of Webtexts through Metadata
Karlskrona, 30 September 2013
Augmented Reality and Playing with History pt 1
Karlskrona, 24 - 26 May 2013
Interview with professor Jay David Bolter on augmented reality
Karlskrona, 24 - 26 May 2013
Three installations (part of digital art festival)
Karlskrona, 24 - 26 May 2013
#MIXITUPFEST Digital Performance Festival in Karlskrona | 34,720 |
\begin{document}
\title[Coarse Reducibility and Algorithmic Randomness]{Coarse
Reducibility and \\ Algorithmic Randomness}
\date{\today}
\author[D. R. Hirschfeldt]{Denis R. Hirschfeldt}
\address{Department of Mathematics\\ University of Chicago}
\email{[email protected]}
\thanks{Hirschfeldt was partially supported by grant
DMS-1101458 from the National Science Foundation of the United
States.}
\author[C. G. Jockusch, Jr.]{Carl G. Jockusch, Jr.}
\address{Department of Mathematics\\University of Illinois at
Urbana-Cham\-paign}
\email{[email protected]}
\author[R. Kuyper]{Rutger Kuyper}
\thanks{Kuyper's research was supported by NWO/DIAMANT grant
613.009.011 and by John Templeton Foundation grant 15619: ``Mind,
Mechanism and Mathematics: Turing Centenary Research Project''.}
\address{Department of Mathematics\\Radboud University Nijmegen}
\email{[email protected]}
\author[P. E. Schupp]{Paul E. Schupp}
\address{Department of Mathematics\\University of Illinois at
Urbana-Cham\-paign}
\email{[email protected]}
\keywords{Coarse reducibility, algorithmic randomness, $K$-triviality}
\subjclass[2010]{Primary 03D30; Secondary 03D28, 03D32}
\begin{abstract}
\vspace{1.2cm}
A \emph{coarse description} of a set $A \subseteq \omega$ is a set $D
\subseteq \omega$ such that the symmetric difference of $A$ and $D$
has asymptotic density $0$. We study the extent to which noncomputable
information can be effectively recovered from all coarse descriptions
of a given set $A$, especially when $A$ is effectively random in some
sense. We show that if $A$ is $1$-random and $B$ is computable from
every coarse description $D$ of $A$, then $B$ is $K$-trivial, which
implies that if $A$ is in fact weakly $2$-random then $B$ is
computable. Our main tool is a kind of compactness theorem for
cone-avoiding descriptions, which also allows us to prove the same
result for $1$-genericity in place of weak $2$-randomness. In the
other direction, we show that if $A \leq\sub{T} \emptyset'$ is a
$1$-random set, then there is a noncomputable c.e.\ set computable
from every coarse description of $A$, but that not all $K$-trivial
sets are computable from every coarse description of some $1$-random
set. We study both uniform and nonuniform notions of coarse reducibility.
A set $Y$ is \emph{uniformly coarsely reducible} to $X$
if there is a Turing functional $\Phi$ such that if
$D$ is a coarse description of $X$, then $\Phi^D$ is a coarse
description of $Y$. A set $B$ is \emph{nonuniformly coarsely reducible} to $A$
if every coarse description of $A$ computes a coarse description of
$B$. We show that a certain natural embedding of the Turing degrees into the
coarse degrees (both uniform and nonuniform) is not surjective.
We also show that if
two sets are mutually weakly $3$-random, then their coarse
degrees form a minimal pair, in both the uniform and nonuniform cases,
but that the same is not true of every
pair of relatively $2$-random sets, at least in the nonuniform coarse
degrees.
\end{abstract}
\maketitle
\section{Introduction}
There are many natural problems with high worst-case complexity that
are nevertheless easy to solve in most instances. The notion of
``generic-case complexity'' was introduced by Kapovich, Myasnikov, Schupp,
and Shpilrain \cite{KMSS} as a notion that is more tractable than average-case
complexity but still allows a somewhat nuanced analysis of such problems.
That paper also introduced the idea of generic computability, which
captures the idea of having a partial algorithm that correctly
computes $A(n)$ for ``almost all'' $n$, while never giving an
incorrect answer. Jockusch and Schupp \cite{JS} began the general computability
theoretic investigation of generic computability and also defined the idea of
coarse computability, which captures the idea
of having a total algorithm that always answers and may make mistakes, but correctly computes $A(n)$ for
``almost all'' $n$. We are here concerned with this latter
concept. We first need a good notion of ``almost all''
natural numbers.
\begin{defn}
Let $A \subseteq \omega$. The \emph{density of $A$ below $n$}, denoted by
$\rho_n(A)$, is $\frac{|A \uhr n|}{n}$. The \emph{upper density}
$\overline{\rho}(A)$ of $A$ is $\limsup_n \rho_n(A)$. The
\emph{lower density} $\underline{\rho}(A)$ of $A$ is $\liminf_n
\rho_n(A)$. If $\overline{\rho}(A)=\underline{\rho}(A)$ then we call
this quantity the \emph{density} of $A$, and denote it by $\rho(A)$.
We say that $D$ is a \emph{coarse description} of $X$ if $\rho(D
\triangle X)=0$, where $\triangle$ denotes symmetric difference. A set
$X$ is \emph{coarsely computable} if it has a computable coarse
description.
\end{defn}
This idea leads to natural notions of reducibility.
\begin{defn}
We say that $Y$ is \emph{uniformly coarsely reducible} to $X$, and write $Y
\leq\sub{uc} X$, if there is a Turing functional $\Phi$ such that if
$D$ is a coarse description of $X$, then $\Phi^D$ is a coarse
description of $Y$. This reducibility induces an equivalence relation
$\equiv\sub{uc}$ on $2^\omega$. We call the equivalence class of $X$
under this relation the \emph{uniform coarse degree} of $X$.
\end{defn}
Uniform coarse reducibility, generic reducibility (defined in \cite{JS}), and
several related reducibilities have been termed \emph{notions of
robust information coding} by Dzhafarov and Igusa \cite{DI}. Work on
such notions has mainly focused on their uniform versions. (One
exception is a result on nonuniform ii-reducibility in Hirschfeldt
and Jockusch \cite{HJ}.) However, their nonuniform versions also seem
to be of interest. In particular, we will work with the following
nonuniform version of coarse reducibility.
\begin{defn}
We say that $Y$ is \emph{nonuniformly coarsely reducible} to $X$, and
write $Y \leq\sub{nc} X$, if every coarse description of $X$ computes
a coarse description of $Y$. This reducibility induces an equivalence
relation $\equiv\sub{nc}$ on $2^\omega$. We call the equivalence class
of $X$ under this relation the \emph{nonuniform coarse degree} of $X$.
\end{defn}
Note that the coarsely computable sets form the least degree in both
the uniform and nonuniform coarse degrees. Uniform coarse reducibility
clearly implies nonuniform coarse reducibility. We will show in the
next section that, as one might expect, the converse fails. The
development of the theory of notions of robust information coding and
related concepts have led to interactions with computability theory
(as in Jockusch and Schupp \cite{JS}; Downey, Jockusch, and Schupp
\cite{DJS}; Downey, Jockusch, McNicholl, and Schupp \cite{DJMS}; and
Hirschfeldt, Jockusch, McNicholl, and Schupp \cite{HJMS}), reverse
mathematics (as in Dzhafarov and Igusa \cite{DI} and Hirschfeldt and
Jockusch \cite{HJ}), and algorithmic randomness (as in Astor
\cite{A}).
In this paper, we investigate connections between coarse reducibility
and algorithmic randomness. In Section \ref{embed}, we describe
natural embeddings of the Turing degrees into the uniform and
nonuniform coarse degrees, and discuss some of their basic
properties. In Section \ref{Ktriv}, we show that no weakly $2$-random
set can be in the images of these embeddings by showing that if $X$ is
weakly $2$-random and $A$ is noncomputable, then there is some coarse
description of $X$ that does not compute $A$. More generally, we show
that if $X$ is $1$-random and $A$ is computable from every coarse
description of $X$, then $A$ is $K$-trivial. Our main tool is a kind
of compactness theorem for cone-avoiding descriptions. We also show
that there do exist noncomputable sets computable from every coarse
description of some $1$-random set, but that not all $K$-trivial sets
have this property. In Section \ref{further}, we give further examples
of classes of sets that cannot be in the images of our embeddings. In
Section \ref{minpairs}, we show that if two sets are relatively weakly
$3$-random then their coarse degrees form a minimal pair, in both the
uniform and nonuniform cases,
but that, at least for the nonuniform coarse degrees, the same is not
true of every pair of relatively $2$-random sets. These results are
analogous to the fact that, for the Turing degrees, two relatively
weakly $2$-random sets always form a minimal pair, but two relatively
$1$-random sets may not. In Section \ref{questions}, we conclude with
a few open questions.
We assume familiarity with basic notions of computability theory (as
in \cite{So}) and algorithmic randomness (as in \cite{DH} or
\cite{N}). For $S \subseteq 2^{<\omega}$, we write $\open{S}$ for the
open subset of $2^\omega$ generated by $S$; that is,
$\open{S} = \{X : \exists n \,(X \uhr n \in S)\}$.
We denote the uniform measure on $2^\omega$ by $\mu$.
\section{Coarsenings and embeddings of the Turing degrees}
\label{embed}
We can embed the Turing degrees into both the uniform and
nonuniform coarse degrees, and our first connection between coarse
computability and algorithmic randomness comes from considering such
embeddings. While there may be several ways to define such embeddings,
a natural way to proceed is to define a map $\mathcal{C} : 2^\omega \rightarrow
2^\omega$ such that $\mathcal{C}(A)$ contains the same information as $A$, but
coded in a ``coarsely robust'' way. That is, we would like $\mathcal{C}(A)$ to
be computable from $A$, and $A$ to be computable from any coarse
description of $\mathcal{C}(A)$.
In the case of the uniform coarse degrees, one might think that the
latter reduction should be uniform, but that condition would be too
strong: If $\Gamma^D=A$ for every coarse description $D$ of $\mathcal
C(A)$ then $\Gamma^\sigma(n)\converges \; \Rightarrow \;
\Gamma^\sigma(n)=A(n)$ (since every string can be extended to a coarse
description of $\mathcal C(A)$), which, together with the fact that for
each $n$ there is a $\sigma$ such that $\Gamma^\sigma(n)\converges$,
implies that $A$ is computable. Thus we relax the uniformity condition
slightly in the following definition.
\begin{defn}
\label{coarseningdefn}
A map $\mathcal{C} : 2^\omega \rightarrow 2^\omega$ is a \emph{coarsening} if
for each $A$ we have $\mathcal{C}(A) \leq\sub{T} A$, and for each coarse
description $D$ of $\mathcal{C}(A)$, we have $A \leq\sub{T} D$. A coarsening $\mathcal{C}$
is \emph{uniform} if there is a binary Turing functional $\Gamma$ with
the following properties for every coarse description $D$ of
$\mathcal{C}(A)$:
\begin{enumerate}[\rm 1.]
\item $\Gamma^D$ is total.
\item Let $A_s(n)=\Gamma^D(n,s)$. Then $A_s=A$ for cofinitely many $s$.
\end{enumerate}
\end{defn}
\begin{prop}\label{coarseningsprop}
Let $\mathcal{C}$ and $\mathcal{F}$ be coarsenings and $A$ and $B$ be sets. Then
\begin{enumerate}[\rm 1.]
\item $B \leq\sub{T} A$ if and only if $\mathcal{C}(B) \leq\sub{nc} \mathcal{C}(A)$.
\item If $\mathcal{C}$ is uniform then $B \leq\sub{T} A$ if and only if $\mathcal{C}(B) \leq\sub{uc}
\mathcal{C}(A)$.
\item $\mathcal{C}(A) \equiv\sub{nc} \mathcal{F}(A)$, and
\item if $\mathcal{C}$ and $\mathcal{F}$ are both uniform then $\mathcal{C}(A) \equiv\sub{uc} \mathcal{F}(A)$.
\end{enumerate}
\end{prop}
\begin{proof}
1. Suppose that $\mathcal{C}(B) \leq\sub{nc} \mathcal{C}(A)$. Then $\mathcal{C}(A)$ computes a
coarse description $D_1$ of $\mathcal{C}(B)$. Thus $B \leq\sub{T} D_1 \leq\sub{T}
\mathcal{C}(A) \leq\sub{T} A$.
Now suppose that $B \leq\sub{T} A$ and let $D_2$ be a coarse description
of $\mathcal{C}(A)$. Then $\mathcal{C}(B) \leq\sub{T} B \leq\sub{T} A \leq\sub{T} D_2$. Thus
$\mathcal{C}(B) \leq\sub{nc} \mathcal{C}(A)$.
2. Suppose that $\mathcal{C}$ is uniform and that $B \leq\sub{T} A$. Let
$D_2$ be a coarse description of $\mathcal{C}(A)$. Let $A_s$ be as in
Definition \ref{coarseningdefn}, with $D=D_2$. Then $\mathcal{C}(B) \leq\sub{T}
B \leq\sub{T} A$, so let $\Phi$ be such that $\Phi^A=\mathcal{C}(B)$. Let $X
\leq\sub{T} D_2$ be defined as follows. Given $n$, search for an $s>n$
such that $\Phi^{A_s}(n)\converges$ and let
$X(n)=\Phi^{A_s}(n)$. (Note that such an $s$ must exist.) Then
$X(n)=\Phi^A(n)=\mathcal{C}(B)(n)$ for almost all $n$, so $X$ is a coarse
description of $\mathcal{C}(B)$. Since $X$ is obtained uniformly from $D_2$, we
have $\mathcal{C}(B) \leq\sub{uc} \mathcal{C}(A)$. The converse follows immediately from
1.
3. Let $D_3$ be a coarse description of $\mathcal{F}(A)$. Then $\mathcal{C}(A) \leq\sub{T}
A \leq\sub{T} D_3$. Thus $\mathcal{C}(A) \leq\sub{nc} \mathcal{F}(A)$. By symmetry, $\mathcal{C}(A)
\equiv\sub{nc} \mathcal{F}(A)$.
4. If $\mathcal{F}$ is uniform then the same argument as in the proof of 2 shows
that we can obtain a coarse description of $\mathcal{C}(A)$ uniformly from $D_3$,
whence $\mathcal{C}(A) \leq\sub{uc} \mathcal{F}(A)$. If $\mathcal{C}$ is also uniform then
$\mathcal{C}(A) \equiv\sub{uc} \mathcal{F}(A) $ by symmetry.
\end{proof}
Thus uniform coarsenings all induce the same natural embeddings. It
remains to show that uniform coarsenings exist.
One example is given by Dzhafarov and Igusa \cite{DI}.
We give a similar example. Let
$I_n=[n!,(n+1)!)$ and let $\mathcal{I}(A)=\bigcup_{n \in A} I_n$; this map first appeared in Jockusch and Schupp \cite{JS}. Clearly $\mathcal{I}(A)
\leq\sub{T} A$, and it is easy to check that if $D$ is a coarse
description of $\mathcal{I}(A)$ then $D$ computes $A$. Thus $\mathcal{I}$ is a
coarsening.
To construct a uniform coarsening, let
$\mathcal{H}(A)=\{\langle n,i \rangle : n \in A\; \wedge\; i \in \omega\}$ and
define $\mathcal{E}(A)=\mathcal{I}(\mathcal{H}(A))$. The notation $\mathcal{E}$ denotes
this particular coarsening throughout the paper.
\begin{prop}\label{Euniform}
The map $\mathcal{E}$ is a uniform coarsening.
\end{prop}
\begin{proof}
Clearly $\mathcal{E}(A) \leq\sub{T} A$. Now let $D$ be a coarse description of
$\mathcal{E}(A)$. Let $G =\{m : |D \cap I_m|>\frac{|I_m|}{2}\}$ and let $A_s=\{n
: \langle n,s \rangle \in G\}$. Then $G =^*\mathcal{H}(A)$, so $A_s=A$ for all
but finitely many $s$, and the $A_s$ are obtained uniformly from $D$.
\end{proof}
A first natural question is whether uniform coarse reducibility and non\-uniform
coarse reducibility are indeed different. We give a positive answer by
showing that, unlike in the nonuniform case, the mappings $\mathcal{E}$ and $\mathcal{I}$ are not
equivalent up to uniform coarse reducibility. Recall that a set $X$ is
\emph{autoreducible} if there exists a Turing functional $\Phi$ such
that for every $n \in \omega$ we have $\Phi^{X \setminus \{n\}}(n) =
X(n)$. Equivalently, we could require that $\Phi$ not ask whether its
input belongs to its oracle. We now introduce a $\Delta^0_2$-version
of this notion.
\begin{defn}
A set $X$ is \emph{jump-autoreducible} if there exists a Turing
functional $\Phi$ such that for every $n \in \omega$ we have $\Phi^{(X
\setminus \{n\})'}(n) = X(n)$.
\end{defn}
\begin{prop}
\label{cjumpauto}
Let $X$ be such that $\mathcal{E}(X) \leq\sub{uc} \mathcal{I}(X)$. Then $X$ is
jump-autoreducible.
\end{prop}
\begin{proof}
We must give a procedure for computing $X(n)$ from $(X \setminus
\{n\})'$ that is uniform in $X$. Given an oracle for $X \setminus
\{n\}$, we can uniformly compute $\mathcal{I}(X \setminus \{n\})$. Now $\mathcal{I}(X
\setminus \{n\}) =^* \mathcal{I}(X)$, so $\mathcal{I}(X \setminus \{n\})$ is a coarse
description of $\mathcal{I}(X)$. Since $\mathcal{E}(X) \leq\sub{uc} \mathcal{I}(X)$ by assumption,
from $\mathcal{I}(X \setminus \{n\})$ we can uniformly compute a coarse description $D$ of
$\mathcal{E}(X)$. Since $\mathcal{E}$ is a uniform coarsening by Proposition
\ref{Euniform}, from $D$ we can uniformly obtain sets $A_0, A_1,
\dots$ with $A_s = X$ for all sufficiently large $s$. Composing these
various reductions, from $X \setminus \{n\}$ we can uniformly compute
sets $A_0, A_1, \dots$ with $A_s = X$ for all sufficiently large $s$.
Then from $(X \setminus \{n\})'$ we can uniformly compute $\lim_s
A_s(n) = X(n)$, as needed.
\end{proof}
We will now show that $2$-generic sets are not jump-autoreducible, which
will give us a first example separating uniform coarse reducibility and
nonuniform coarse reducibility. For this we first show that no
$1$-generic set is autoreducible, which is an easy exercise.
\begin{prop}
If $X$ is $1$-generic, then $X$ is not autoreducible.
\end{prop}
\begin{proof}
Suppose for the sake of a contradiction that $X$ is $1$-generic and is
autoreducible via $\Phi$. For a string $\sigma$, let $\sigma^{-1}(i)$
be the set of $n$ such that $\sigma(n)=i$. If $\tau$ is a binary
string, let $\tau \setminus \{n\}$ be the unique binary string $\mu$
of the same length such that $\mu^{-1}(1) = \tau^{-1}(1) \setminus
\{n\}$. Let $S$ be the set of strings $\tau$ such that $\Phi^{\tau
\setminus \{n\}}(n) \converges \neq \tau(n) \converges$ for some $n$.
Then $S$ is a c.e.\ set of strings and $X$ does not meet $S$. Since
$X$ is $1$-generic, there is a string $\sigma \prec X$ that has no
extension in $S$. Let $n = |\sigma|$, and let $\tau \succ \sigma$ be a
string such that $\Phi^{\tau \setminus \{n\}}(n) \converges$. Such a
string $\tau$ exists because $\sigma \prec X$ and $\Phi$ witnesses
that $X$ is autoreducible. Furthermore, we may assume that $\tau(n)
\neq \Phi^{\tau \setminus \{n\}}$, since changing the value of
$\tau(n)$ does not affect any of the conditions in the choice of
$\tau$. Hence $\tau$ is an extension of $\sigma$ and $\tau \in S$,
which is the desired contradiction.
\end{proof}
\begin{prop}
\label{genjumpauto}
If $X$ is $2$-generic, then $X$ is not jump-autoreducible.
\end{prop}
\begin{proof}
Since $X$ is $2$-generic, $X$ is $1$-generic relative to $\emptyset'$.
Hence, by relativizing the proof of the previous proposition to
$\emptyset'$, we see that $X$ is not autoreducible relative to
$\emptyset'$. However, the class of $1$-generic sets is uniformly
$\mathrm{GL}_1$, i.e., there exists a single Turing functional $\Psi$
such that for every $1$-generic $X$ we have $\Psi^{X \oplus
\emptyset'} = X'$, as can be verified by looking at the usual proof
that every $1$-generic is $\mathrm{GL}_1$ (see \cite[Lemma
2.6]{J}). Of course, if $X$ is $1$-generic, then $X \setminus \{n\}$
is also $1$-generic for every $n$. Thus from an oracle for $(X
\setminus \{n\}) \oplus \emptyset'$ we can uniformly compute $(X
\setminus \{n\})'$. Now, if $X$ is jump-autoreducible, from $(X
\setminus \{n\})'$ we can uniformly compute $X(n)$. Composing these
reductions shows that $X(n)$ is uniformly computable from $(X
\setminus \{n\}) \oplus \emptyset'$, which contradicts our previous
remark that $X$ is not autoreducible relative to $\emptyset'$.
\end{proof}
\begin{cor}
If $X$ is $2$-generic, then $\mathcal{E}(X) \leq\sub{nc} \mathcal{I}(X)$ but $\mathcal{E}(X)
\nleq\sub{uc} \mathcal{I}(X)$.
\end{cor}
\begin{proof}
We know that $\mathcal{E}(X) \leq\sub{nc} \mathcal{I}(X)$ from
Proposition \ref{coarseningsprop}. The fact that $\mathcal{E}(X) \nleq\sub{uc}
\mathcal{I}(X)$ follows from Propositions \ref{cjumpauto} and
\ref{genjumpauto}.
\end{proof}
It is natural to ask whether the same result holds for $2$-random
sets. In the proof above we used the fact that the $2$-generic sets
are uniformly $\mathrm{GL}_1$. For $2$-random sets this fact is almost
true, as expressed by the following lemma. The proof is adapted from
Monin \cite{M}, where a generalization for higher levels of randomness
is proved. Let $U_0,U_1,\ldots$ be a fixed universal Martin-L\"of test
relative to $\emptyset'$. The \emph{$2$-randomness deficiency} of a
$2$-random $X$ is the least $c$ such that $X \notin U_c$.
\begin{lem}
There is a Turing functional $\Theta$ such that, for a $2$-random $X$
and an upper bound $b$ on the $2$-randomness deficiency of $X$, we
have $\Theta^{X \oplus \emptyset',b} = X'$.
\end{lem}
\begin{proof}
Let $\mathscr{V}_e=\{Z : e \in Z'\}$. The $\mathscr{V}_e$ are uniformly $\Sigma^0_1$
classes, so we can define a function $f \leq\sub{T} \emptyset'$ such
that $\mu(\mathscr{V}_e \setminus \mathscr{V}_e[f(e,i)])<2^{-i}$ for all $e$ and $i$. Then
each sequence $\mathscr{V}_e \setminus \mathscr{V}_e[f(e,0)],\mathscr{V}_e \setminus
\mathscr{V}_e[f(e,1)],\ldots$ is an $\emptyset'$-Martin L\"of test, and from $b$
we can compute a number $m$ such that if $X$ is $2$-random and $b$
bounds the $2$-randomness deficiency of $X$, then $X \notin \mathscr{V}_e
\setminus \mathscr{V}_e[f(e,m)]$. Then $X \in \mathscr{V}_e$ if and only if $X \in \mathscr{V}_e[f(e,m)]$,
which we can verify $(X\oplus\emptyset')$-computably.
\end{proof}
\begin{prop}
If $X$ is $2$-random, then $X$ is not jump-autoreducible.
\end{prop}
\begin{proof}
Because $X$ is $2$-random, it is not autoreducible relative to
$\emptyset'$, as can be seen by relativizing the proof of Figueira,
Miller, and Nies \cite{FMN} that no $1$-random set is autoreducible.
To obtain a contradiction, assume that $X$ is jump-autoreducible
through some functional $\Phi$. It can be directly verified that there
is a computable function $f$ such that $f(n)$ bounds the randomness
deficiency of $X \setminus \{n\}$. Now let $\Psi^{Y \oplus
\emptyset'}(n) = \Phi^{\Theta^{Y \oplus \emptyset',f(n)}}(n)$. Then
$X$ is autoreducible relative to $\emptyset'$ through $\Psi$, a
contradiction.
\end{proof}
\begin{cor}
If $X$ is $2$-random, then $\mathcal{E}(X) \leq\sub{nc} \mathcal{I}(X)$ but $\mathcal{E}(X)
\nleq\sub{uc} \mathcal{I}(X)$.
\end{cor}
Although we will not discuss generic reducibility after this section,
it is worth noting that our maps $\mathcal{E}$ and $\mathcal{I}$ also allow us to
distinguish generic reducibility from its nonuniform analog. Let us
briefly review the relevant definitions from \cite{JS}. A
\emph{generic description} of a set $A$ is a partial function that
agrees with $A$ where defined, and whose domain has density $1$. A set
$A$ is \emph{generically reducible} to a set $B$, written $A
\leq\sub{g} B$, if there is an enumeration operator $W$ such that if
$\Phi$ is a generic description of $B$, then $W^{\gr(\Phi)}$ is the
graph of a generic description of $A$. We can define the notion of
\emph{nonuniform generic reducibility} in a similar way: $A
\leq\sub{ng} B$ if for every generic description $\Phi$ of $B$, there
is a generic description $\Psi$ of $A$ such that $\gr(\Psi)$ is
enumeration reducible to $\gr(\Phi)$.
It is easy to see that $\mathcal{E}(X) \leq\sub{ng} \mathcal{I}(X)$ for all $X$. On the
other hand, we have the following fact.
\begin{prop}
If $\mathcal{E}(X) \leq\sub{g} \mathcal{I}(X)$ then $X$ is autoreducible.
\end{prop}
\begin{proof}
Let $I_n$ be as in the definition of $\mathcal{I}$. Suppose that $W$ witnesses
that $\mathcal{E}(X) \leq\sub{g} \mathcal{I}(X)$. We can assume that $W^Z$ is the graph of
a partial function for every oracle $Z$. Define a Turing functional
$\Theta$ as follows. Given an oracle $Y$ and an input $n$, let
$\Phi(k)=Y(m)$ if $k \in I_m$ and $m \neq n$, and let
$\Phi(k)\diverges$ if $k \in I_n$. Let $\Psi$ be the partial function
with graph $W^{\gr(\Phi)}$. Search for an $i$ and a $k \in I_{\langle
n,i \rangle}$ such that $\Psi(k)\converges$. If such numbers are
found then let $\Theta^Y(n)=\Psi(k)$. If $Y=X \setminus \{n\}$ then
$\Phi$ is a generic description of $\mathcal{I}(X)$, so $\Psi$ is a generic
description of $\mathcal{E}(X)$, and hence $\Theta^Y(n)\converges=X(n)$. Thus
$X$ is autoreducible.
\end{proof}
We finish this section by showing that, for both the uniform
and the nonuniform coarse degrees, coarsenings of the appropriate type preserve
joins but do not always preserve existing meets.
\begin{prop}
Let $\mathcal{C}$ be a coarsening. Then $\mathcal{C}(A \oplus B)$ is the least upper bound
of $\mathcal{C}(A)$ and $\mathcal{C}(B)$ in the nonuniform coarse degrees. The same holds
for the uniform coarse degrees if $\mathcal{C}$ is a uniform coarsening.
\end{prop}
\begin{proof}
By Proposition \ref{coarseningsprop} we know that $\mathcal{C}(A \oplus B)$ is
an upper bound for $\mathcal{C}(A)$ and $\mathcal{C}(B)$ in both the uniform and
nonuniform coarse degrees. Let us show that it is the least upper
bound. If $\mathcal{C}(A),\mathcal{C}(B) \leq\sub{nc} G$ then every coarse description $D$
of $G$ computes both $A$ and $B$, so $D \geq\sub{T} A \oplus B
\geq\sub{T} \mathcal{C}(A \oplus B)$. Thus $G \geq\sub{nc} \mathcal{C}(A \oplus B)$.
Finally, assume that $\mathcal{C}$ is a uniform coarsening and let
$\mathcal{C}(A),\mathcal{C}(B) \leq\sub{uc} G$. Let $\Phi$ be a Turing
functional such that $\Phi^{A \oplus B}=\mathcal C(A \oplus B)$.
Every coarse description $H$ of $G$ uniformly
computes coarse descriptions $D_1$ of $\mathcal{C}(A)$ and $D_2$ of $\mathcal{C}(B)$.
Since $\mathcal{C}$ is uniform, there are Turing functionals $\Gamma$ and $\Delta$
such that, letting $A_s(n)=\Gamma^{D_1}(n,s)$ and
$B_s(n)=\Gamma^{D_2}(n,s)$, we have that $A \oplus B = A_s \oplus B_s$
for all sufficiently large $s$. Let $E$ be
defined as follows. Given $n$, search for an $s \geq n$ such that
$\Phi^{A_s \oplus B_s}(n)\converges$, and let $E(n)=\Phi^{A_s \oplus
B_s}(n)$. If $n$ is sufficiently large, then $E(n)=\Phi^{A \oplus
B}(n)=\mathcal C(A \oplus B)(n)$, so $E$ is a coarse description of
$\mathcal C(A \oplus B)$. Since $E$ is obtained uniformly from $H$,
we have that $\mathcal{C}(A \oplus B) \leq\sub{uc} G$.
\end{proof}
\begin{lem}
Let $\mathcal{C}$ be a uniform coarsening and let $Y \leq\sub{T} X$. Then $Y
\leq\sub{uc} \mathcal{C}(X)$.
\end{lem}
\begin{proof}
Let $\Phi$ be a Turing functional such that $\Phi^X = Y$. Let $D$ be a
coarse description of $\mathcal{C}(X)$ and let $A_s$ be as in Definition
\ref{coarseningdefn}. Now define $G(n)$ to be the value of
$\Phi^{A_s}(n)$ for the least pair $\langle s,t \rangle$ such that $s
\geq n$ and $\Phi^{A_s}(n)[t]\converges$. Then $G =^* Y$, so $G$ is a
coarse description of $Y$.
\end{proof}
\begin{prop}
Let $\mathcal{C}$ be a coarsening. Then $\mathcal{C}$ does not always preserve existing
meets in the nonuniform coarse degrees. The same holds for the uniform coarse
degrees if $\mathcal{C}$ is a uniform coarsening.
\end{prop}
\begin{proof}
Let $X,Y$ be relatively $2$-random and $\Delta^0_3$. Then $X$ and $Y$
form a minimal pair in the Turing degrees, while $X$ and $Y$ do not
form a minimal pair in the nonuniform coarse degrees by Theorem
\ref{2randthm} below. Since every coarse description of $\mathcal{C}(X)$
computes $X$ we see that $\mathcal{C}(X) \geq\sub{nc} X$ and $\mathcal{C}(Y) \geq\sub{nc}
Y$. Therefore $\mathcal{C}(X)$ and $\mathcal{C}(Y)$ also do not form a minimal pair in the
nonuniform coarse degrees.
Next, let $\mathcal{C}$ be a uniform coarsening. We have seen above that there
exists some $A \leq\sub{nc} \mathcal{C}(X),\mathcal{C}(Y)$ that is not coarsely
computable. Then $A \leq\sub{T} X,Y$, so $A \leq\sub{uc} \mathcal{C}(X),\mathcal{C}(Y)$ by
the previous lemma. Thus, $\mathcal{C}(X)$ and $\mathcal{C}(Y)$ do not form a minimal
pair in the uniform coarse degrees.
\end{proof}
\section{Randomness, $K$-triviality, and robust information coding}
\label{Ktriv}
It is reasonable to expect that the embeddings induced by $\mathcal{E}$
(or equivalently, by any uniform coarsening) are not
surjective. Indeed, if $ \mathcal{E}(A) \leq\sub{uc} X$ then the information
represented by $A$ is coded into $X$ in a fairly redundant way. If $A$
is noncomputable, it should follow that $X$ cannot be random. As we
will see, we can make this intuition precise.
\begin{defn}
Let $X^{\mathfrak c}$ be the set of all $A$ such that $A$ is
computable from every coarse description of $X$.
\end{defn}
We will show that if $X$ is weakly $2$-random then $X^{\mathfrak c} =
{\bf 0}$, and hence $ \mathcal{E}(A) \nleq\sub{nc} X$ for all noncomputable $A$
(since every coarse description of $\mathcal{E}(A)$ computes $A$). Since no
$1$-random set can be coarsely computable, it will follow that $X
\not\equiv\sub{nc} \mathcal{E}(B)$ and $X \not\equiv\sub{uc} \mathcal{E}(B)$ for all
$B$. We will first prove the following theorem. Let $\mathcal K$ be
the class of $K$-trivial sets. (See \cite{DH} or \cite{N} for more on
$K$-triviality.)
\begin{thm}
\label{1randandK}
If $X$ is $1$-random then $X^{\mathfrak c} \subseteq \mathcal K$.
\end{thm}
By Downey, Nies, Weber, and Yu \cite{DNWY}, if $X$ is weakly
$2$-random then it cannot compute any noncomputable $\Delta^0_2$
sets. Since $\mathcal K \subset \Delta^0_2$, our desired result
follows from Theorem \ref{1randandK}.
\begin{cor}
\label{weak2randcor}
If $X$ is weakly $2$-random then $X^{\mathfrak c} = {\bf 0}$, and
hence $ \mathcal{E}(A) \nleq\sub{nc} X$ for all noncomputable $A$. In
particular, in both the uniform and nonuniform coarse degrees, the
degree of $X$ is not in the image of the embedding induced by
$\mathcal{E}$.
\end{cor}
To prove Theorem \ref{1randandK}, we use the fact, established by
Hirschfeldt, Nies, and Stephan \cite{HNS}, that $A$ is $K$-trivial
if and only if $A$ is a base for $1$-randomness, that is, $A$ is computable in a set that
is $1$-random relative to $A$. The basic idea is to show that if $X$
is $1$-random and $A \in X^{\mathfrak c}$, then for each $k > 1$ there is a way to
partition $X$ into $k$ many ``slices'' $X_0,\ldots,X_{k-1}$ such that for each
$i<k$, we have $A \leq\sub{T} X_0 \oplus \cdots \oplus X_{i-1} \oplus X_{i+1}
\oplus \cdots \oplus X_{k-1}$ (where the right hand side of this
inequality denotes $X_1 \oplus \cdots \oplus X_{k-1}$ when $i=0$ and
$X_0 \oplus \cdots \oplus X_{k-2}$
when $i = k-1$). It will then follow by van Lambalgen's
Theorem (which will be discussed below) that each $X_i$ is $1$-random
relative to $X_0 \oplus \cdots \oplus X_{i-1} \oplus X_{i+1} \oplus \cdots
\oplus X_{k-1} \oplus A$, and hence, again by van Lambalgen's Theorem,
that $X$ is $1$-random relative to $A$. Since $A \in X^{\mathfrak c}$
implies that $A \leq\sub{T} X$, we will conclude that $A$ is a base
for $1$-randomness, and hence is $K$-trivial. We begin with some notation
for certain partitions of $X$.
\begin{defn}
\label{partdef}
Let $X \subseteq \omega$. For an infinite subset $Z=\{z_0<z_1<\cdots\}$ of $\omega$,
let $X \uhr Z = \{n : z_n \in X\}$. For $k>1$ and $i<k$, define $$X^k_i = X
\uhr \{n : n \equiv i \bmod k\}\quad \textrm{and}\quad X^k_{\neq i} =
X \uhr \{n : n \not\equiv i \bmod k\}.$$ Note that $X^k_{\neq i}
\equiv\sub{T} X \setminus \{n : n \equiv i \bmod k\}$ and
$\overline{\rho}(X \triangle (X \setminus \{n : n \equiv i \bmod k\}))
\leq \frac{1}{k}$.
\end{defn}
Van Lambalgen's Theorem \cite{vL} states that $Y \oplus Z$ is
$1$-random if and only if $Y$ and $Z$ are relatively $1$-random. The
proof of this theorem shows, more generally, that if $Z$ is
computable, infinite, and coinfinite, then $X$ is $1$-random if and
only if $X \uhr Z$ and $X \uhr \overline{Z}$ are relatively
$1$-random. Relativizing this fact and applying induction, we get the
following version of van Lambalgen's Theorem.
\begin{thm}[van Lambalgen \cite{vL}]
\label{vlthm}
The following are equivalent for all sets $X$ and $A$, and all $k>1$.
\begin{enumerate}[\rm 1.]
\item $X$ is $1$-random relative to $A$.
\item For each $i<k$, the set $X^k_i$ is $1$-random relative to
$X^k_{\neq i} \oplus A$.
\end{enumerate}
\end{thm}
The last ingredient we need for the proof of Theorem \ref{1randandK}
is a kind of compactness principle, which will also be used to yield
further results in the next section, and is of independent interest
given its connection with the following concept defined in \cite{HJMS}.
\begin{defn}
\label{bounddef}
Let $r \in [0,1]$. A set $X$ is \emph{coarsely computable at density
$r$} if there is a computable set $C$ such that $\overline{\rho}(X
\triangle C) \leq $1$-r$. The \emph{coarse computability
bound} of $X$ is $$\gamma(X) = \sup\{r : X \textrm{ is coarsely
computable at density } r\}.$$
\end{defn}
As noted in \cite{HJMS}, there are sets $X$ such that $\gamma(X)=1$
but $X$ is not coarsely computable. In other words, there is no principle of
``compactness of computable coarse descriptions''. (Although
Miller (see \cite[Theorem 5.8]{HJMS}) showed that one can in fact
recover such a principle by adding a further effectivity condition to
the requirement that $\gamma(X)=1$.) The following theorem shows that
if we replace ``computable'' by ``cone-avoiding'', the situation is
different.
\begin{thm}
\label{compthm}
Let $A$ and $X$ be arbitrary sets. Suppose that for each $\epsilon >
0$ there is a set $D_\epsilon$ such that $\overline{\rho}(X \triangle
D_\epsilon) \leq \epsilon$ and $A \nleq\sub{T} D_\epsilon$. Then
there is a coarse description $D$ of $X$ such that $A \nleq\sub{T} D$.
\end{thm}
\begin{proof}
The basic idea is that, given a Turing functional $\Phi$ and a string
$\sigma$ that is ``close to'' $X$, we can extend $\sigma$ to a string
$\tau$ that is ``close to" $X$ such that $\Phi^D \neq A$ for all $D$
extending $\tau$ that are ``close to'' $X$. We can take $\tau$ to be
any string ``close to" $X$ such that, for some $n$, either
$\Phi^\tau(n) \converges \neq A(n)$ or $\Phi^\gamma(n) \diverges$ for
all $\gamma$ extending $\tau$ that are ``close to" $X$. If no such
$\tau$ exists, we can obtain a contradiction by arguing that $A
\leq\sub{T} D_\epsilon$ for sufficiently small $\epsilon$, since with
an oracle for $D_\epsilon$ we have access to many strings that are
``close to" $D_\epsilon$ and hence to $X$, by the triangle inequality
for Hamming distance. In the above discussion the meaning of ``close
to" is different in different contexts, but the precise version will
be given below. Further, as the construction proceeds, the meaning of
``close to" becomes so stringent that we guarantee that $\rho(X
\triangle D) = 0$. We now specify the formal details.
We obtain $D$ as $\bigcup_e \sigma_e$, where $\sigma_e \in
2^{<\omega}$ and $\sigma_0 \subsetneq \sigma_1 \subsetneq \cdots$. In
order to ensure that $\rho(X \triangle D) = 0$, we require that for
all $e$ and all $m$ in the interval $[|\sigma_e|, |\sigma_{e+1}|]$,
either $D$ and $X$ agree on the interval $[|\sigma_e|, m)$ or $\rho_m
(X \triangle D) \leq 2^{-|\sigma_e|}$, with the latter true for $m =
|\sigma_{e+1}|$. This condition implies that $\rho_m (X \triangle D)
\leq 2^{-|\sigma_e|}$ for all $m \in [|\sigma_{e+1}|,
|\sigma_{e+2}|]$, and hence that $\rho(X \triangle D) = 0$.
Let $\sigma$ and $\tau$ be strings and let $\epsilon$ be a positive
real number. Call $\tau$ an $\epsilon$-\emph{good} extension of
$\sigma$ if $\tau$ properly extends $\sigma$ and for all $m \in
[|\sigma|, |\tau|]$, either $X$ and $\tau$ agree on $[|\sigma|,m)$ or
$\rho_m (\tau \triangle X) \leq \epsilon$, with the latter true for
$m = |\tau|$. In line with the previous paragraph, we require that
$\sigma_{e+1}$ be a $2^{-|\sigma_e|}$-good extension of $\sigma_e$
for all $e$.
At stage $0$, let $\sigma_0$ be the empty string. At stage $e+1$, we
are given $\sigma_e$ and choose $\sigma_{e+1}$ as follows so as to
force that $ A \neq \Phi_e^D$. Let $\epsilon = 2^{-|\sigma_e|}$.
\emph{Case 1}. There is a number $n$ and a string $\tau$ that is an
$\epsilon$-good extension of $\sigma_e$ such that $\Phi^{\tau}_e(n)
\converges \neq A(n)$. Let $\sigma_{e+1}$ be such a $\tau$.
\emph{Case 2}. Case 1 does not hold and there is a number $n$ and a
string $\beta$ that is an $\epsilon$-good extension of $\sigma_e$ such
that $|\beta| \geq |\sigma_e| + 2$ and $\Phi_e^\tau(n) \diverges$ for
all $\frac{\epsilon}{4}$-good extensions $\tau$ of $\beta$. Let
$\sigma_{e+1}$ be such a $\beta$.
We claim that either Case 1 or Case 2 applies. Suppose not. Let
$D_{\frac{\epsilon}{5}}$ be as in the hypothesis of the lemma, so that
$\overline{\rho}(X \triangle D_{\frac{\epsilon}{5}}) \leq
\frac{\epsilon}{5}$ and $A \nleq\sub{T} D_{\frac{\epsilon}{5}}$. Let
$c \geq |\sigma_e| + 2$ be sufficiently large so that $\rho_m (X
\triangle D_{\frac{\epsilon}{5}}) \leq \frac{\epsilon}{4}$ for all $m
\geq c$ and $\sigma_e$ has an $\frac{\epsilon}{4}$-good extension
$\beta$ of length $c$. Note that the string obtained from $\sigma_e$
by appending a sufficiently long segment of $X$ starting with
$X(|\sigma_e|)$ is an $\frac{\epsilon}{4}$-good extension of
$\sigma_e$, so such a $\beta$ exists, and we assume it is obtained
in this manner.
We now obtain a contradiction by showing that $A \leq\sub{T}
D_{\frac{\epsilon}{5}}$. To calculate $A(n)$ search for a string
$\gamma$ extending $\beta$ such that $\Phi_e^\gamma (n) \converges$,
say with use $u$, and $\rho_m (D_{\frac{\epsilon}{5}} \triangle
\gamma) \leq \frac{\epsilon}{2}$ for all $m \in [c,u)$. We first
check that such a string $\gamma$ exists. Since Case 2 does not
hold, there is a string $\tau$ that is an $\frac{\epsilon}{4}$-good
extension of $\beta$ such that $\Phi_e^\tau(n) \converges$. We
claim that $\tau$ meets the criteria to serve as $\gamma$. We need
only check that $\rho_m (D_{\frac{\epsilon}{5}} \triangle \tau) \leq
\frac{\epsilon}{2}$ for all $m \in [c,u)$. Fix $m \in [c,u)$. Then
$$\rho_m (D_{\frac{\epsilon}{5}} \triangle \tau) \leq \rho_m
(D_{\frac{\epsilon}{5}} \triangle X) + \rho_m (X \triangle \tau)
\leq \frac{\epsilon}{4} + \frac{\epsilon}{4} =
\frac{\epsilon}{2}.$$
Next we claim that $\gamma$ is an $\epsilon$-good extension of
$\sigma_e$. The string $\gamma$ extends $\sigma_e$ since it extends
$\beta$, and $\beta$ extends $\sigma_e$. Let $m \in [|\sigma_e,
|\gamma|]$ be given. If $m < c$, then $\gamma$ and $X$ agree on the
interval $[|\sigma_e|, m)$ because $\beta$ and $X$ agree on this
interval and $\gamma$ extends $\beta$. Now suppose that $m \geq c$.
Then
$$\rho_m(\gamma \triangle X) \leq \rho_m(\gamma \triangle
D_{\frac{\epsilon}{5}}) + \rho_m(D_{\frac{\epsilon}{5}} \triangle X)
\leq \frac{\epsilon}{2} + \frac{\epsilon}{4} < \epsilon.$$ Since
$\gamma$ is an $\epsilon$-good extension of $\sigma_e$ for which
$\Phi_e^\gamma(n) \converges$, and Case 1 does not hold, we conclude
that $\Phi_e^\gamma(n) = A(n)$. The search for $\gamma$ can be
carried out computably in $D_{\frac{\epsilon}{5}}$, so we conclude
that $A \leq\sub{T} D_{\frac{\epsilon}{5}}$, contradicting our choice
of $D_{\frac{\epsilon}{5}}$. (Although $\beta$ cannot be computed from
$D_{\frac{\epsilon}{5}}$, we may use it in our computation of $A(n)$
since it is a fixed string which does not depend on $n$.) This
contradiction shows that Case 1 or Case 2 must apply.
Let $D = \bigcup_n \sigma_n$. Then $\rho(D \triangle X) = 0$, and $A
\nleq\sub{T} D$ since Case 1 or Case 2 applies at every stage.
\end{proof}
\begin{proof}[Proof of Theorem \ref{1randandK}]
Let $A \in X^{\mathfrak c}$. By Theorem \ref{compthm}, there is an
$\epsilon > 0$ such that $A \leq\sub{T} D_\epsilon$ whenever
$\overline{\rho}(X \triangle D_\epsilon) \leq \epsilon$. Let $k$ be an
integer such that $k > \frac{1}{\epsilon}$. As noted in
Definition \ref{partdef}, $X^k_{\neq i}$ is Turing equivalent to such
a $D_\epsilon$ for each $i < k$, so we have $A \leq\sub{T} X^k_{\neq i}$
for all $i<k$. By the unrelativized form of Theorem \ref{vlthm}, each
$X^k_i$ is $1$-random relative to $X^k_{\neq i}$, and hence relative
to $X^k_{\neq i} \oplus A \equiv\sub{T} X^k_{\neq i}$. Again by
Theorem \ref{vlthm}, $X$ is $1$-random relative to $A$. But $A
\leq\sub{T} X$, so $A$ is a base for $1$-randomness, and hence is
$K$-trivial.
\end{proof}
Weak $2$-randomness is exactly the level of randomness necessary to
obtain Corollary \ref{weak2randcor} directly from Theorem
\ref{1randandK}, because, as shown in \cite{DNWY}, if a $1$-random set
is not weakly $2$-random, then it computes a noncomputable
c.e.\ set. The corollary itself does hold of some $1$-random sets that
are not weakly $2$-random, because if it holds of $X$ then it also
holds of any $Y$ such that $\rho(Y \triangle X)=0$. (For example, let
$X$ be $2$-random and let $Y$ be obtained from $X$ by letting
$Y(2^n)=\Omega(n)$ (where $\Omega$ is Chaitin's halting probability)
for all $n$ and letting $Y(k)=X(k)$ for all other $k$. By van
Lambalgen's Theorem, $Y$ is $1$-random, but it computes $\Omega$, and
hence is not weakly $2$-random.)
Nevertheless, Corollary \ref{weak2randcor} does not hold of all
$1$-random sets, as we now show.
\begin{defn}
Let $W_0,W_1,\ldots$ be an effective listing of the c.e.\ sets. A set
$A$ is \emph{promptly simple} if it is c.e.\ and coinfinite, and there
exist a computable function $f$ and a computable enumeration
$A[0],A[1],\ldots$ of $A$ such that for each $e$, if $W_e$ is infinite
then there are $n$ and $s$ for which $n \in W_e[s] \setminus W_e[s-1]$
and $n \in A[f(s)]$. Note that every promptly simple set is
noncomputable.
\end{defn}
We will show that if $X \leq\sub{T} \emptyset'$ is $1$-random then
$X^{\mathfrak c}$ contains a promptly simple set, and there is a
promptly simple set $A$ such that $ \mathcal{E}(A) \leq\sub{nc} X$. (We do not
know whether we can improve the last statement to $ \mathcal{E}(A) \leq\sub{uc}
X$.) In fact, we will obtain a considerably stronger result by first
proving a generalization of the fact, due to Hirschfeldt and Miller
(see \cite[Theorem 7.2.11]{DH}), that if $\mathcal T$ is a $\Sigma^0_3$
class of measure $0$, then there is a noncomputable c.e.\ set that is
computable from each $1$-random element of $\mathcal T$.
For a binary relation $P(Y,Z)$ between elements of $2^\omega$, let
$P(Y)=\{Z : P(Y,Z)\}$.
\begin{thm}
\label{sigma3thm}
Let $\mathcal S_0,\mathcal S_1,\ldots$ be uniformly $\Pi^0_2$ classes
of measure $0$, and let $P_0(Y,Z),P_1(Y,Z),\ldots$ be uniformly
$\Pi^0_1$ relations. Let $\mathcal D$ be the class of all $Y$ for
which there are numbers $k,m$ and a $1$-random set $Z$ such that $Z
\in P_k(Y) \subseteq \mathcal S_m$. Then there is a promptly simple
set $A$ such that $A \leq\sub{T} Y$ for every $Y \in \mathcal D$.
\end{thm}
\begin{proof}
Let $(\mathcal V^m_n)_{m,n \in \omega}$ be uniformly $\Sigma^0_1$
classes such that $\mathcal S_m = \bigcap_n \mathcal V^m_n$. We may
assume that $\mathcal V^m_0 \supseteq \mathcal V^m_1 \supseteq \cdots$
for all $m$. For each $m$, we have $\mu(\bigcap_n \mathcal V^n_m) =
\mu(\mathcal S_m)=0$, so $\lim_n \mu(\mathcal V^m_n)=0$ for each
$m$. Let $\Theta$ be a computable relation such that $P_k(Y,Z) \equiv
\forall l\, \Theta(k,Y \uhr l,Z \uhr l)$.
Define $A$ as follows. At each stage $s$, if there is an $e<s$ such
that no numbers have entered $A$ for the sake of $e$ yet, and an
$n>2e$ such that $n \in W_e[s] \setminus W_e[s-1]$ and $\mu(\mathcal
V^m_n[s]) \leq 2^{-e}$ for all $m<e$, then for the least such $e$, put
the least corresponding $n$ into $A$. We say that $n$ enters $A$ for
the sake of $e$.
Clearly, $A$ is c.e.\ and coinfinite, since at most $e$ many numbers
less than $2e$ ever enter $A$. Suppose that $W_e$ is infinite. Let
$t>e$ be a stage such that all numbers that will ever enter $A$ for
the sake of any $i<e$ are in $A[t]$. There must be an $s \geq t$ and
an $n>2e$ such that $n \in W_e[s] \setminus W_e[s-1]$ and
$\mu(\mathcal V^m_n[s]) \leq 2^{-e}$ for all $m<e$. Then the least
such $n$ enters $A$ for the sake of $e$ at stage $s$ unless another
number has already entered $A$ for the sake of $e$. It follows that
$A$ is promptly simple.
Now suppose that $Y \in \mathcal D$. Let the numbers $k,m$ and the
$1$-random set $Z$ be such that $Z \in P_k(Y) \subseteq \mathcal S_m$.
Let $B \leq\sub{T} Y$ be defined as follows. Given $n$, let $$\mathcal
D^n_s = \{X : (\forall l \leq s)\, \Theta(k,Y \uhr l,X \uhr l)\}
\setminus \mathcal V^m_n[s].$$ Then $\mathcal D^n_0 \supseteq \mathcal
D^n_1 \supseteq \cdots$. Furthermore, if $X \in \bigcap_s \mathcal
D^n_s$ then $P_k(Y,X)$ and $X \notin \mathcal V^m_n$. Since $P_k(Y)
\subseteq S_m \subseteq \mathcal V^m_n$, it follows that $X \notin
P_k(Y)$, which is a contradiction. Thus $\bigcap_s \mathcal D^n_s =
\emptyset$. Since the $\mathcal D^n_s$ are nested closed sets, it
follows that there is an $s$ such that $\mathcal D^n_s =
\emptyset$. Let $s_n$ be the least such $s$ (which we can find using
$Y$) and let $B(n)=A(n)[s_n]$. Note that $B \subseteq A$.
Let $T=\{\mathcal V^m_n[s] : n \textrm{ enters } A \textrm{ at stage }
s\}$. We can think of $T$ as a uniform singly-indexed sequence of
$\Sigma^0_1$ sets since $m$ is fixed and for each $n$ there is at most
one $s$ such that $\mathcal V^m_n[s] \in T$. For each $e$, there is at
most one $n$ that enters $A$ for the sake of $e$, and the sum of the
measures of the $\mathcal V^m_n[s]$ such that $n$ enters $A$ at stage
$s$ for the sake of some $e>m$ is bounded by $\sum_e 2^{-e}$, which is
finite. Thus $T$ is a Solovay test, and hence $Z$ is in only finitely
many elements of $T$. So for all but finitely many $n$, if $n$ enters
$A$ at stage $s$ then $Z \notin \mathcal V^m_n[s]$. Then $Z \in
\mathcal D^n_s$, so $s_n>s$. Hence, for all such $n$, we have that
$B(n) = A(n)[s_n] = 1$. Thus $B =^* A$, so $A \equiv\sub{T} B
\leq\sub{T} Y$.
\end{proof}
Note that the result of Hirschfeldt and Miller mentioned above follows
from this theorem by starting with a $\Sigma^0_3$ class $\mathcal
S=\bigcap_m \mathcal S_m$ of measure $0$ and letting each $P_k$ be the
identity relation.
\begin{cor}
\label{1randcor}
Let $X \leq\sub{T} \emptyset'$ be $1$-random. There is a promptly
simple set $A$ such that if $\overline{\rho}(D \triangle
X)<\frac{1}{4}$ then $A \leq\sub{T} D$. In particular, $X^{\mathfrak
c}$ contains a promptly simple set, and there is a promptly simple
set $A$ such that $ \mathcal{E}(A) \leq\sub{nc} X$.
\end{cor}
\begin{proof}
Say that sets $Y$ and $Z$ are \emph{$r$-close from $m$ on} if whenever
$m < n$, the Hamming distance between $Y \uhr n$ and $Z \uhr n$
(i.e., the number of bits on which these two strings differ) is at
most $rn$.
Let $\mathcal S_m$ be the class of all $Z$ such that $X$ and $Z$ are
$\frac{1}{2}$-close from $m$ on. Since $X$ is $\Delta^0_2$, the
$\mathcal S_m$ are uniformly $\Pi^0_2$ classes. Furthermore, if $X$
and $Z$ are $\frac{1}{2}$-close from $m$ on for some $m$, then $Z$
cannot be $1$-random relative to $X$ (by the same argument that shows
that if $C$ is $1$-random then there must be infinitely many $n$ such
that $C \uhr n$ has more $1$'s than $0$'s), so $\mu(\mathcal S_m)=0$
for all $m$. Let $P_m(Y,Z)$ hold if and only if $Y$ and $Z$ are
$\frac{1}{4}$-close from $m$ on. The $P_m$ are clearly uniformly
$\Pi^0_1$ relations.
Thus the hypotheses of Theorem \ref{sigma3thm} are satisfied. Let $A$
be as in that theorem. Suppose that $\overline{\rho}(D \triangle
X)<\frac{1}{4}$. Then there is an $m$ such that $D$ and $X$ are
$\frac{1}{4}$-close from $m$ on. If $D$ and $Z$ are
$\frac{1}{4}$-close from $m$ on, then by the triangle inequality for
Hamming distance, $X$ and $Z$ are $\frac{1}{2}$-close from $m$
on. Thus $X \in P_m(D) \subseteq \mathcal S_m$, so $A \leq\sub{T} D$.
\end{proof}
After learning about Corollary \ref{1randcor}, Nies \cite{N2} gave a
different but closely connected proof of this result, which works even
for $X$ of positive effective Hausdorff dimension, as long as we
sufficiently decrease the bound $\frac{1}{4}$. However, even for $X$
of effective Hausdorff dimension $1$ his bound is much worse, namely
$\frac{1}{20}$.
Maass, Shore, and Stob \cite[Corollary 1.6]{MSS} showed that if $A$
and $B$ are promptly simple then there is a promptly simple set $G$ such
that $G \leq\sub{T} A$ and $G \leq\sub{T} B$. Thus we have the
following extension of Ku{\v c}era's result \cite{K} that two
$\Delta^0_2$ $1$-random sets cannot form a minimal pair, which will also
be useful below.
\begin{cor}
\label{quartercor}
Let $X_0,X_1 \leq\sub{T} \emptyset'$ be $1$-random. There is a
promptly simple set $A$ such that if $\overline{\rho}(D \triangle
X_i)<\frac{1}{4}$ for some $i \in \{0,1\}$ then $A \leq\sub{T} D$.
\end{cor}
It is easy to adapt the proof of Corollary \ref{1randcor} to give a
direct proof of Corollary \ref{quartercor}, and indeed of the fact
that for any uniformly $\emptyset'$-computable family $X_0,X_1,\ldots$
of $1$-random sets, there is a promptly simple set $A$ such that if
$\overline{\rho}(D \triangle X_i)<\frac{1}{4}$ for some $i$ then $A
\leq\sub{T} D$. (We let $\mathcal S_{\langle i,m \rangle}$ be the
class of all $Z$ such that $X_i$ and $Z$ are $\frac{1}{2}$-close from
$m$ on, and the rest of the proof is essentially as before.)
Given the many (and often surprising) characterizations of
$K$-triviality, it is natural to ask whether there is a converse to
Theorem \ref{1randandK} stating that if $A$ is $K$-trivial then $A \in
X^{\mathfrak c}$ for some $1$-random $X$. We now show that is not the
case, using a recent result of Bienvenu, Greenberg, Ku{\v c}era, Nies,
and Turetsky \cite{BGKNT}. There are many notions of randomness tests
in the theory of algorithmic randomness. Some, like Martin-L\"of
tests, correspond to significant levels of algorithmic randomness,
while other, less obviously natural ones have nevertheless become
important tools in the development of this theory. Balanced tests
belong to the latter class.
\begin{defn}
Let $\mathcal W_0,\mathcal W_1,\ldots \subseteq 2^\omega$ be an
effective list of all $\Sigma^0_1$ classes. A \emph{balanced test} is
a sequence $(\mathcal U_n)_{n \in \omega}$ of $\Sigma^0_1$ classes such
that there is a computable binary function $f$ with the following
properties.
\begin{enumerate}[\rm 1.]
\item $|\{s : f(n,s+1) \neq f(n,s)\}| \leq O(2^n)$,
\item $\forall n\; \mathcal U_n = \mathcal W_{\lim_s f(n,s)}$, and
\item $\forall n\; \forall s\; \mu(\mathcal W_{f(n,s)}) \leq 2^{-n}$.
\end{enumerate}
\end{defn}
For $\sigma \in 2^{<\omega}$ and $X \in 2^\omega$, we write $\sigma X$
for the element of $2^\omega$ obtained by concatenating $\sigma$ and
$X$.
\begin{thm}[Bienvenu, Greenberg, Ku{\v c}era, Nies, and Turetsky
\cite{BGKNT}]
\label{oberthm}
There are a $K$-trivial set $A$ and a balanced test $(\mathcal U_n)_{n
\in \omega}$ such that if $A \leq\sub{T} X$ then there is a string
$\sigma$ with $\sigma X \in \bigcap_n \mathcal U_n$.
\end{thm}
We will also use the following measure-theoretic fact.
\begin{thm}[Loomis and Whitney \cite{LW}]
\label{lwthm}
Let $\mathcal S \subseteq 2^\omega$ be open, and let $k \in
\omega$. For $i<k$, let $\pi_i(\mathcal S)=\{Y^k_{\neq i} : Y \in
\mathcal S\}$. Then $\mu(\mathcal S)^{k-1} \leq \mu(\pi_0(\mathcal
S))\cdots\mu(\pi_{k-1}(\mathcal S))$.
\end{thm}
Our result will follow from the following lemma.
\begin{lem}
\label{balancedlem}
Let $X$ be $1$-random, let $k>1$, and let $(\mathcal U_n)_{n \in
\omega}$ be a balanced test. There is an $i<k$ such that $X^k_{\neq
i} \notin \bigcap_n \mathcal U_n$.
\end{lem}
\begin{proof}
Assume for a contradiction that $X^k_{\neq i} \in \bigcap_n \mathcal
U_n$ for all $i<k$. Let
$$
\mathcal S_{n,s} = \{Y : \forall i<k\;
(Y^k_{\neq i} \in \mathcal U_n[s])\}
$$
and let $\mathcal S_n=\bigcup_s \mathcal S_{n,s}$. By Theorem
\ref{lwthm}, $\mu(\mathcal S_{n,s})^{k-1} \leq \mu(\mathcal
U_n[s])^k$, so $\mu(\mathcal S_n) \leq O(2^n)2^{-\frac{nk}{k-1}} =
O(2^{-\frac{n}{k-1}})$, and hence $\sum_n \mu(\mathcal
S_n)<\infty$. Thus $\{\mathcal S_n : n \in \omega\}$ is a Solovay
test. However, $X \in \bigcap_n \mathcal S_n$, so we have a
contradiction.
\end{proof}
\begin{thm}
There is a $K$-trivial set $A$ such that $A \notin X^{\mathfrak c}$
for all $1$-random $X$.
\end{thm}
\begin{proof}
Let $A$ and $(\mathcal U_n)_{n \in \omega}$ be as in Theorem
\ref{oberthm}. Let $X$ be $1$-random. By Theorem \ref{compthm}, it is
enough to fix $k>1$ and show that there is an $i<k$ such that $A
\nleq\sub{T} X^k_{\neq i}$. Assume for a contradiction that $A
\leq\sub{T} X^k_{\neq i}$ for all $i<k$. Then there are
$\sigma_0,\ldots,\sigma_{k-1}$ such that $\sigma_iX^k_{\neq i} \in
\bigcap_n \mathcal U_n$ for all $i<k$. Let $m=\max_{i<k} |\sigma_i|$
and let $\mathcal V_n=\{Y : \exists i<k\; (\sigma_iY \in \mathcal
U_{n+k+m})\}$. It is easy to check that $(\mathcal V_n)_{n \in \omega}$
is a balanced test, and $X^k_{\neq i} \in \bigcap_n \mathcal V_n$ for
all $i<k$, which contradicts Lemma \ref{balancedlem}.
\end{proof}
\section{Further applications of cone-avoiding compactness}
\label{further}
We can use Theorem \ref{compthm} to give an analog to Corollary
\ref{weak2randcor} for effective genericity. In this case,
$1$-genericity is sufficient, as it is straightforward to show that if
$X$ is $1$-generic relative to $A$ and $A$ is noncomputable, then $A
\nleq\sub{T} X$ (i.e., unlike the case for $1$-randomness, there are
no noncomputable bases for $1$-genericity), and that no $1$-generic
set can be coarsely computable. The other ingredient we need to
replicate the argument we gave in the case of effective randomness is
a version of van Lambalgen's Theorem for $1$-genericity. This result
was established by Yu \cite[Proposition 2.2]{Y}. Relativizing his
theorem and applying induction as in the case of Theorem \ref{vlthm},
we obtain the following fact.
\begin{thm}[Yu \cite{Y}]
\label{ythm}
The following are equivalent for all sets $X$ and $A$, and all $k>1$.
\begin{enumerate}[\rm 1.]
\item $X$ is $1$-generic relative to $A$.
\item For each $i<k$, the set $X^k_i$ is $1$-generic relative to
$X^k_{\neq i} \oplus A$.
\end{enumerate}
\end{thm}
Now we can establish the following analog to Corollary
\ref{weak2randcor}.
\begin{thm}
\label{1genthm}
If $X$ is $1$-generic then $X^{\mathfrak c} = {\bf 0}$, and hence
$ \mathcal{E}(A) \nleq\sub{nc} X$ for all noncomputable $A$. In
particular, in both the uniform and nonuniform coarse degrees, the
degree of $X$ is not in the image of the embedding
induced by $\mathcal{E}$.
\end{thm}
\begin{proof}
Let $A \in X^{\mathfrak c}$. As in the proof of Theorem
\ref{1randandK}, there is a $k$ such that $A \leq\sub{T} X^k_{\neq i}$
for all $i<k$. By the unrelativized form of Theorem \ref{ythm}, each
$X^k_i$ is $1$-generic relative to $X^k_{\neq i}$, and hence relative
to $X^k_{\neq i} \oplus A \equiv\sub{T} X^k_{\neq i}$. Again by
Theorem \ref{ythm}, $X$ is $1$-generic relative to $A$. But $A
\leq\sub{T} X$, so $A$ is computable.
\end{proof}
Igusa (personal communication) has also found the following
application of Theorem \ref{compthm}. We say that $X$ is
\emph{generically computable} if there is a partial computable
function $\varphi$ such that $\varphi(n) = X(n)$ for all $n$ in the
domain of $\varphi$, and the domain of $\varphi$ has density
$1$. Jockusch and Schupp \cite[Theorem 2.26]{JS} showed that there are
generically computable sets that are not coarsely computable, but by
Lemma 1.7 in \cite{HJMS}, if $X$ is generically computable then
$\gamma(X)=1$, where $\gamma$ is the coarse computability bound from
Definition \ref{bounddef}.
\begin{thm}[Igusa, personal communication]
If $\gamma(X)=1$ then $X^{\mathfrak c} = {\bf 0}$, and hence $\mathcal{E}(A)
\nleq\sub{nc} X$ for all noncomputable $A$. Thus, if $\gamma(X) = 1$
and $X$ is not coarsely computable then in both the uniform and
nonuniform coarse degrees, the degree
of $X$ is not in the image of the embedding induced by $\mathcal{E}$. In
particular, the above holds when $X$ is generically computable but not
coarsely computable.
\end{thm}
\begin{proof}
Suppose that $\gamma(X)=1$ and $A$ is not computable. If $\epsilon>0$
then there is a computable set $C$ such that $\overline{\rho}(X
\triangle C) < \epsilon$. Since $C$ is computable, $A \nleq\sub{T}
C$. By Theorem \ref{compthm}, $A \notin X^{\mathfrak c}$.
\end{proof}
\section{Minimal pairs in the uniform and nonuniform\\ coarse degrees}
\label{minpairs}
For any degree structure that acts as a measure of information
content, it is reasonable to expect that if two sets are sufficiently
random relative to each other, then their degrees form a minimal
pair. For the Turing degrees, it is not difficult to show that if $Y$
is not computable and $X$ is weakly $2$-random relative to $Y$, then
the degrees of $X$ and $Y$ form a minimal pair. On the other hand,
Ku{\v c}era \cite{K} showed that if $X,Y \leq\sub{T} \emptyset'$ are
both $1$-random, then there is a noncomputable set $A \leq\sub{T}
X,Y$, so there are relatively $1$-random sets whose degrees do not
form a minimal pair. As we will see, the situation for the
nonuniform coarse degrees is similar, but ``one jump up''.
For an interval $I$, let $\rho_I(X)=\frac{|X \cap I|}{|I|}$.
\begin{lem}
\label{jlem}
Let $J_k=[2^{k}-1,2^{k+1}-1)$. Then $\rho(X)=0$ if and only if $\lim_k
\rho_{J_k}(X)=0$.
\end{lem}
\begin{proof}
First suppose that $\limsup_k \rho_{J_k}(X) > 0$. Since $|J_k| = 2^k$,
we have $\overline{\rho}(X) \geq \limsup_k \rho_{2^{k+1}-1}(X) \geq
\limsup_k \frac{\rho_{J_k}(X)}{2} > 0$.
Now suppose that $\limsup_k \rho_{J_k}(X) = 0$. Fix
$\epsilon>0$. If $m$ is sufficiently large, $k \geq m$, and $n \in
J_k$, then $$|X \cap [0,n)| \leq |X \cap [0,2^{k+1}-1)| \leq
\sum_{i=0}^{m-1} |J_i| + \sum_{i=m}^k \frac{\epsilon}{2}
|J_i|.$$ If $k$ is sufficiently large then this sum is less than
$\epsilon (2^k-1)$, whence $\rho_n(X) < \frac{\epsilon
(2^k-1)}{n} \leq \frac{\epsilon n}{n}=\epsilon$. Thus
$\limsup_n \rho_n(X) \leq \epsilon$. Since $\epsilon$ is
arbitrary, $\limsup_n \rho_n(X) = 0$.
\end{proof}
\begin{thm}
If $A$ is not coarsely computable and $X$ is weakly $3$-random
relative to $A$, then there is no $X$-computable coarse description of
$A$. In particular, $A \nleq\sub{nc} X$.
\end{thm}
\begin{proof}
Suppose that $\Phi^X$ is a coarse description of $A$ and let $$\mathcal
P=\{Y : \Phi^Y \textrm{ is a coarse description of } A\}.$$ Then $Y \in
\mathcal P$ if and only if
\begin{enumerate}[\rm 1.]
\item $\Phi^Y$ is total, which is a $\Pi^0_2$ property, and
\item for each $k$ there is an $m$ such that, for all $n>m$, we have
$\rho_n(\Phi^Y \triangle A)<2^{-k}$, which is a $\Pi^{0,A}_3$ property.
\end{enumerate}
Thus $\mathcal P$ is a $\Pi^{0,A}_3$ class, so it suffices to show
that if $A$ is not coarsely computable then $\mu(\mathcal P)=0$.
We prove the contrapositive. Suppose that $\mu(\mathcal P)>0$. Then,
by the Lebesgue Density Theorem, there is a $\sigma$ such that
$\mu(\mathcal P \cap \open{\sigma})>\frac{3}{4}2^{-|\sigma|}$. It is
now easy to define a Turing functional $\Psi$ such that the measure of
the class of $Y$ for which $\Psi^Y$ is a coarse description of $A$ is
greater than $\frac{3}{4}$. Define a computable set $D$ as
follows. Let $J_k=[2^{k}-1,2^{k+1}-1)$. For each $k$, wait until we
find a finite set of strings $S_k$ such that
$\mu(\open{S_k})>\frac{3}{4}$ and $\Psi^\sigma$ converges on all of
$J_k$ for each $\sigma \in S_k$ (which must happen, by our choice of
$\Psi$). Let $n_k$ be largest such that there is a set $R_k
\subseteq S_k$ with $\mu(\open{R_k})>\frac{1}{2}$ and
$\rho_{J_k}(\Psi^\sigma \triangle \Psi^\tau) \leq 2^{-n_k}$ for all
$\sigma,\tau \in R_k$. Let $\sigma \in R_k$ and define $D \uhr J_k =
\Psi^\sigma \uhr J_k$.
We claim that $D$ is a coarse description of $A$. By Lemma \ref{jlem},
it is enough to show that $\lim_k \rho_{J_k}(D \triangle A)=0$. Fix
$n$. Let $\mathcal B_k$ be the class of all $Y$ such that $\Psi^Y$
converges on all of $J_k$ and $\rho_{J_k}(\Psi^Y \triangle A) \leq
2^{-n}$. If $\Psi^Y$ is a coarse description of $A$ then, again by
Lemma \ref{jlem}, $\rho_{J_k}(\Psi^Y \triangle A) \leq 2^{-n}$ for all
sufficiently large $k$, so there is an $m$ such that $\mu(\mathcal
B_k)>\frac{3}{4}$ for each $k>m$, and hence $\mu(\mathcal B_k \cap
\open{S_k})>\frac{1}{2}$ for each $k>m$. Let $T_k = \{\sigma \in S_k :
\rho_{J_k}(\Psi^\sigma \triangle A) \leq 2^{-n}\}$. Then $\open{T_k} =
\mathcal{B}_k \cap \open{S_k}$, so $\mu(\open{T_k}) > \frac{1}{2}$ for
each $k > m$. Furthermore, by the triangle inequality for Hamming
distance, $\rho_{J_k}(\Psi^\sigma \triangle \Psi^\tau) \leq
2^{-(n-1)}$ for all $\sigma, \tau \in T_k$. It follows that, for each
$k>m$, we have $n_k \geq n-1$, and at least one element $Y$ of
$\mathcal B_k$ is in $\open{R_k}$ (where $R_k$ is as in the definition
of $D$), which implies that $$\rho_{J_k}(D \triangle A) \leq
\rho_{J_k}(D \triangle \Psi^Y) + \rho_{J_k}(\Psi^Y \triangle A) \leq
2^{-n_k}+2^{-n} < 2^{-n+2}.$$ Since $n$ is arbitrary, $\lim_k
\rho_{J_k}(D \triangle A)=0$.
\end{proof}
\begin{cor}
If $Y$ is not coarsely computable and $X$ is weakly $3$-random
relative to $Y$, then the nonuniform coarse degrees of $X$ and $Y$
form a minimal pair, and hence so do their uniform coarse degrees.
\end{cor}
\begin{proof}
Let $A \leq\sub{nc} X,Y$. Then $Y$ computes a coarse description $D$
of $A$. We have $D \leq\sub{nc} X$, and $X$ is weakly $3$-random
relative to $D$, so by the theorem, $D$ is coarsely computable, and
hence so is $A$.
\end{proof}
For the nonuniform coarse degrees at least, this corollary does not
hold of $2$-randomness in place of weak $3$-randomness. To establish
this fact, we use the following complementary results. The first was
proved by Downey, Jockusch, and Schupp \cite[Corollary 3.16]{DJS} in
unrelativized form, but it is easy to check that their proof
relativizes.
\begin{thm}[Downey, Jockusch, and Schupp \cite{DJS}]
\label{DJSthm}
If $A$ is c.e., $\rho(A)$ is defined, and $A' \leq\sub{T} D'$, then $D$
computes a coarse description of $A$.
\end{thm}
\begin{thm}[Hirschfeldt, Jockusch, McNicholl, and Schupp \cite{HJMS}]
\label{HJMSthm}
Every nonlow c.e.\ degree contains a c.e.\ set $A$ such that
$\rho(A)=\frac{1}{2}$ and $A$ is not coarsely computable.
\end{thm}
\begin{thm}
\label{2randthm}
Let $X,Y \leq\sub{T} \emptyset''$ (which is equivalent to $\mathcal{E}(X),\mathcal{E}(Y)
\leq\sub{nc} \mathcal{E}(\emptyset'')$). If $X$ and $Y$ are both $2$-random,
then there is an $A \leq\sub{nc} X,Y$ such that $A$ is not coarsely
computable. In particular, there is a pair of relatively $2$-random
sets whose nonuniform coarse degrees do not form a minimal pair.
\end{thm}
\begin{proof}
Since $X$ and $Y$ are both $1$-random relative to $\emptyset'$, by the
relativized form of Corollary \ref{quartercor} there is an
$\emptyset'$-c.e.\ set $J >\sub{T} \emptyset'$ such that for every
coarse description $D$ of either $X$ or $Y$, we have that $D \oplus
\emptyset'$ computes $J$, and hence so does $D'$. By the Sacks Jump
Inversion Theorem \cite{Sa}, there is a c.e.\ set $B$ such that $B'
\equiv\sub{T} J$. By Theorem \ref{HJMSthm}, there is a c.e.\ set $A
\equiv\sub{T} B$ such that $\rho(A)=\frac{1}{2}$ and $A$ is not
coarsely computable. Let $D$ be a coarse description of either $X$ or
$Y$. Then $D' \geq\sub{T} J \equiv\sub{T} A'$, so by Theorem
\ref{DJSthm}, $D$ computes a coarse description of $A$.
\end{proof}
We do not know whether this theorem holds for uniform coarse reducibility.
\section{Open Questions}
\label{questions}
We finish with a few questions raised by our results.
\begin{oq}
Can the bound $\frac{1}{4}$ in Corollary \ref{1randcor} be increased?
\end{oq}
\begin{oq}
Let $X \leq\sub{T} \emptyset'$ be $1$-random. Must there be a
noncomputable (c.e.)\ set $A$ such that $ \mathcal{E}(A) \leq\sub{uc} X$? (Recall
that Corollary \ref{1randcor} gives a positive answer to the
nonuniform analog to this question.) If not, then is there any
$1$-random $X$ for which such an $A$ exists?
\end{oq}
\begin{oq}
Does Theorem \ref{2randthm} hold for uniform coarse reducibility?
\end{oq} | 86,811 |
15 minutes of fame
Will you get your 15 minutes?
I have heard it said that everyone will eventually have 15 minutes of fame. It never occurred to me as I was watching one of those afternoon TV talk shows that one day I would make my presence known on the stage. I watched the people, some of the shows seemed entirely too outrageous, people fighting and yelling at each other. How could they do that? Some of the shows gave the impression that they were there to help out those who were in need. It was one of those shows that was suggested to me by a good friend, she insisted that I give them a call. She new what I was going thru with my teenage daughter, it would be just the thing we needed to set her right.
Rather than call, I did an online application giving as much detail as was possible in the little box that was provided. Within 3 days I received a phone call from the show. They asked me a few questions, I gave the best answers that I could. It was my daughter they wanted to speak with next, and she was instructed to go to a room in the house where I couldn’t interfere. I looked at it this way, if doing back flips in my back yard would help out my daughter, then watch out world, cause back flips would be on my agenda.
Whatever my daughter said, must have convinced them that she would be an appropriate candidate for a Talk Show. They got me back on the phone to make arrangements, they asked me a few questions, one of them being did I have any black teeth. Well, I embarrassingly admitted I did have one tooth, to the side, that had long since died. Due to circumstances beyond my control, I had been unable to have this tooth fixed. Not a problem they assured me, it would be taken care of by their Dentist. Well, ‘isn’t that something’ I thought, they care enough about us to look after our teeth. My bubble of comfort was quickly burst, it was explained that black teeth show up badly on television. Oh, well I guess appearances must be everything in television.
It was explained to me that we would leave the next afternoon, that didn’t have much time for planning. The next morning I realized that I had an appointment that could not be missed, on the following day. I called the number they gave me to tell them I couldn’t make the flight. They were quite upset with me and felt that I was lying. I explained that it was a court appointment and could not be changed. They wanted to know if I could provide proof, well I could but why? We didn’t have a contract, just an over the phone verbal agreement. I was repeatedly reminded that I was not going to be paid for appearing on the show, but my food expenses would be covered once I reached my destination. I wasn’t going on the show to make money, I was going for the help they said they would supply. They explained to me that taping was done on Thursdays, so they needed me there on that day. My appointment was on a Wednesday, I could leave right after the appointment.
The Big City
Arrival and excitement
I have to admit here that I was a little excited, I was going to fly on an airplane, something I had never done, so yes the thrill was there. I was also going to New York City, some place my daughter had always wanted to see. Maybe this would be a good bonding time for us, and I had the highest hopes that help would be provided for my wayward daughter.
At the designated time a Limousine arrived to pick us up, we had one stop to make, pick up my other daughter from school, make our appointment and off we would go. My son met us at the courthouse to take my other daughter home. The daughter, the one who was not going had told her friends what was happening, a few friends knew what I was doing and of course family had been informed. My youngest daughter was thrilled that we were going to be on her favorite show, that channel was tuned in every day at 4pm, when she got home from school. We left with the promise that we would get an autograph from the host, he was her favorite person in the whole world, he helped kids.
I am not an individual who seeks to be in the limelight, I’d much rather be in the background observing, but my daughter’s well being was utmost in my mind. I was doing everything humanly possible to help her over come her bad behavior.
We arrived at out hotel at 11 pm with a nice fruit basket in our room and instructions in an envelope. We were to be picked up at 7 am, so we needed to be prepared. We were to wear the clothes we brought for the show because there was no going back to the hotel once we left. We would be brought back to our hotel after the taping of the show, and be returned to the airport the next day.
The Big Day
My daughter and I were quickly separated, I was off to the dentist, she was ushered away from me, by a person who had on a head-set, with the assurance “We’ll take good care of her”. We both looked into each others eyes, mine had fear, hers had excitement. I had to trust that my daughter would be well taken care of. It went against the grain of everything I knew, to let her out of my site. If I didn’t keep a close eye on her, anything could happen after all we were in once the largest cities in the world.
The visit at the Dentist went fairly well, he explained to me that on occasion he opened his office early for this TV show. His office person had not arrived so it was just him and I. He took the x-ray and did his best to fix my tooth for the TV, he said it was only going to be a temporary fix the tooth was too far gone. He wouldn’t remove it because it would look too bad on television. I’d have to see my own dentist for that.
I was driven back to where the show was, and led through the building. I was disappointed in what I was seeing, the paint was peeling, floors were cracked, it looked quite dismal, couldn’t they afford paint or new flooring? People were standing in a line, I asked what was going on, I was told they were waiting in line to see the show. Many were what I would consider street people, they had their clothes piled on top of themselves, and holding what looked like all of their possessions. I heard someone yell out “What time is lunch served?” Another responded “They’ll send in pizza at 11”. I thought this was a show, not a lunch line, now I was suffering a little bit of the ‘nerves’
I wanted to see my daughter. It was explained to me that I would see her later as I was escorted to what I was to find out was called ‘The green room’. There were other people in this room, I soon found out they were the mothers of other girls who were to appear on the show. A lady came up to me, introduced herself as the one with whom I had spoken to on the phone. She took me into another room where we could speak privately. As soon as I started to explain what had been happening in the life of my daughter, the tears started to flow, I was a wreck. I opened up and told her everything, the drinking, the stealing of my car, the drugs, and boys, how I really needed help with this rebellious child of mine. She promised me they would do everything they could to help us. I felt some relief, we were going to get help, there was hope for her yet.
We sat for hours in this room, these parents and I, talking about our daughters and their different behaviors. Some were horrendous, it made me glad that I only had what I was dealing with. One of the moms I was to find out, she made what she called “The circuit’, meaning that this was not her first time on a talk show, she liked doing this. We discovered from talking that one of the mothers lived in a neighboring town. They had arrived the day before and had a nice time shopping and taking a tour of the town with each other. Everyone seemed quite jovial, I was just sad.
There was a hair and make-up person brought in, to get us ready. I looked a little different from what I considered ‘normal’. When everyone was ’made up’, we were led to yet another room which was ‘sound proof’. The producer looked at my clothing, I was dressed in black. I like black, black makes me comfortable, black is all I own. I can hide in the corner when I am in black and no one will see me. They made me change clothes, one of the other guests had some extra blouses that I could choose from. I wondered how many outfits did she bring. They made me wear beige pants and an orange blouse. At that point I was the most uncomfortable I had ever been in my life. I was wearing someone else’s clothes, in a strange town, about to admit to the world that I had a ‘badly behaved’ daughter, I wanted to go home, and all I could do was cry.
I mentioned to one of the other mothers that I wanted to go home, that I had changed my mind, I no longer wanted to do the show. She told me that if I left I would be charged the plane fare, and the cost of the hotel. Fantastic. I couldn’t afford to go to the Dentist, how could I possibly afford plane fare? I guess there was no hope, I had to go on. I think at that time I understood the phrase “The Show must go on.” A microphone was attached to me, it was almost my time.
I was the third one to be called in. The host shook my hand, I got a round of applause as I entered the stage. I looked around, the ‘derelicts’ that I had seen standing in line were sitting in the audience. There were a few people that were nicely dressed, they were stationed right in the center. My ‘producer’ was crouched on the floor behind the cameras with very large pieces of poster board. I was to find out these were cue cards. As the host started asking me questions he had his own cue cards, mine were held up. I suppose they didn’t want me forgetting my ‘lines’. Like I could. I’d lived it for months now.
Things are not always what they seem
When the host asked me the questions on his cue cards, I started crying yet again. I managed to cry during the whole interview, I could not stop. I must have went through ½ box of tissues while on stage. I sat there with a handful of used tissues all wadded up, there was no place to put them. Pictures of my beautiful daughter were put up on the screen of when she was a sweet, untroubled little girl. The audience ’ood and ahhd’ at her sweet face. The time came when my daughter’s name was called. Prior to her coming on the stage I was forced to watch her interview that had been taped. It was not a pretty site, she didn’t look like my daughter, she had her hair different, clothes that were not her own, she was wearing a skirt that barely covered her butt. She said a lot of things, most of which shocked me, some of which I already knew. At least she didn’t swear. Nothing that she said would have to be bleeped out. She came on stage, the people booed, she yelled and pointed her finger a lot at them, there was much confusion and I had a hard time understanding what was being said. In her unfamiliar clothing, I noticed that she kept pulling her skirt down, the shows required wardrobe left her feeling without her normal level of comfort. The last woman was called to the stage, and during the applause my daughter whispered to me, “Mom, I didn’t really do all that, I’ll explain later”. Huh?…
The show went on. More taped videos were shown, each of the girls yelling, saying they would do what they wanted, calling their Mother’s filthy names. Two of the girls had taken lie detector tests. It was revealed that they ’lied’, one mom went nuts, almost hitting her daughter. After all of us were talked to by the host, and everyone was out on the stage, a lady came out who had at one time been a Prostitute. She had been on drugs and had been shot at, and she worked with teenagers who were troubled. Now was the time we could get some help! Boot Camp maybe? A visit to jail? Something was going to get done! We were all ushered off of the stage, microphones were removed, we were to finishing taping somewhere else. I changed back into my comfortable clothes, as did my daughter.
We all went outside to wait for the van that was to take us to our new destination. I watched in horror as the Mothers handed their daughters cigarettes and they all lit up. They were talking and laughing as if they hadn’t suffered the most traumatic moment of their life. Now I am no saint, I smoke and I am aware that my daughter smokes, but I was not ready to view her doing so. My daughter being 16, was the oldest of these girls, the youngest being 13. During our wait, it was explained to the mom’s of the girls, who had taken lie detector tests, that some of the questions asked were not exactly the ones they wanted answered. What I understood to have happened was the show had been fixed.
My next disappointment was to come rapidly. We were driven to an area that could only be viewed in the daytime. It was that bad. One of the camera men ’found’ a cardboard box and wanted the girls to sit in it, individually. My daughter refused, she said that no matter how low she sunk in life, a card board box would not be in her future. So this was the help we were to get? A cardboard box?
I was dejected, embarrassed, mortified and angry. I had poured out my shame to the world with the assurance that some sort of help would be provided in exchange for doing so. The other mother’s were excited and laughing, discussing the precious day’s shopping trip. Some of them were discussing their plans of attending another talk show. One of the girls couldn’t wait to get home and tell all of her friends.
From what I could see, no help was forthcoming. Oh they gave me numbers of counselors that I could call, as if this was something I had not already done.
As my daughter and I headed back to our hotel we talked. She told me how they made her wear those ‘hooker’ clothes as though they were her own. When she was taping her video they held up cue cards with what they wanted her to say, making her repeat them over and over, unhappy that she was not calling me a ‘Bitch’. That was one of the words on her cue card that she refused to use. She then told me how one of the producers kept coming into the ‘green room’ that she was stationed in, offering her cigarettes. She had told them that she smoked, but did not smoke in front of me. She had seen enough of these shows to feel that if she did, they would ‘show’ it to me on stage. More than anything, she did not want to see me hurt more.
Regret?
We returned home, I was disappointed, she was glad that she got to see New York City and fly on an air plane, we were both embarrassed. We were asked repeatedly by friends and family when the show would air. They had told us that we would be notified when it would be on. I didn’t care, it was something that I didn’t want to see. About 2 weeks afterward we received the phone call telling us when the show was to air. The lady excitedly told me to set my VCR to record our show was going on! I told her, “This is the one thing in my life that I wish I could ‘undo’”.
It was a bad decision on my end to agree to go on the show. My youngest daughter was upset that we did not get the Autograph. I explained that I never really got to meet the host, except when I went onstage. I had asked my producer if she could get me an autograph for my daughter, she kept saying “I’ll see what I can do”. My request fell on deaf ears. When we told her how the show was ‘set up’ she decided that she no longer wanted to watch it. I guess it was back to the Disney channel for us in the afternoon.
I was to re-think this over the next few months. My daughter’s behavior changed somewhat. She quit stealing my car, I never did pick her up from a friends house drunk again. As far as the drugs, I never saw evidence of them. She was 16 almost 17 so the boys, well that wasn’t going to stop any time soon. The only thing I can think is, my daughter saw my mortification, and decided that she was going to make things better by herself.
Someone from the show called us a couple of months later and wanted to know if we wanted to come back for a ‘follow up, success story‘. I said ‘no thanks’, I’d had about as much embarrassment as I could handle in this lifetime.
I have had my 15 minutes of ‘shame‘, Most of the time I wish I could take those 15 minutes back, but yet, if I had not spent that time being shamed, would my daughter have changed? It's amazing what we as parents will go through for the sake of our children.
More by this Author
- 36
Several weeks ago my Daughter had her boyfriend over and one thing led to another and the discussion of Farts came up. At that time he proceeded to serenade us with the sounds of his armpit farts. I just...
- 32
So you think you want to be famous? You think it’s going to be great? You think with fans adoring you, all will be well? You think that more people will respect you? You think that once you...
- 388
Having a warrant for your arrest can be scary. If you find yourself in that situation, here's what to do.
I've often wondered what goes on behind the scenes. I KNEW there could not be THAT many people behaving so outrageously -- at least not normally. I suspect I know the show of which you speak. I used to watch it WAY back when... when there was more reality than theatrics. I am glad to hear that you and your daughter benefited in some way, even if it was due to factors you did not expect!
So, is this a show that is still on the air?
Wow, I have to say I have always believed "Talk Shows" to be infact real. Your story actually opened my eyes and definately gave me a different perspective on shows like that. Thanks!!!!!!!
I do have to say though I would have possibly done it too if I knew it would benefit one of my children.
5 | 241,365 |
1 to 5 of 5 results
Rental Mobile 800779 S 3300 Rd Wellston
2 days ago in Wellston, OK
COUNTRY LIVING AT ITS BEST! 80 acres of beautiful hills, trees and pasture. Deer and wildlife abunda... On Listanza via Oodle.com
Mobile Home Sites Available in Apple Village (Edmond)
2 days ago in Oklahoma City, OK
Apple Village MHC in Edmond, OK We have: Jr. Olympic sized Shared swimming pool Storm Shelters Shady... On SpreadMyAd via Oodle.com
Apollo Mobile Home Community! Lease to Own One BR One BA low down (Oklahoma City
10 days ago in Oklahoma City, OK
This is a cute 1 bedrooms one bathrooms Mobile home located in Apollo Mobile Home Community!! 1617 S... On SpreadMyAd via Oodle.com
Clearview Mobile Home Community.. Noble Schools $695 3bd 1216ft 2
19 days ago in Oklahoma City, OK
Welcome home to Cairn Communities!! This Brand new home is just right for you or the family!!! Noble... On SpreadMyAd via Oodle.com
Nice Two BR mobile home for rent (10714e 133rd perkins ok) $475 2bd
20 days ago in Stillwater, OK
This is a nice 2 bedrooms room mobile home just outside of perkins in the country. We just put a new... On SpreadMyAd via Oodle.com | 179,301 |
TITLE: Clarification of a proof in Herrlich
QUESTION [4 upvotes]: In Herrlich on page 5 he gives a proof of $\textbf{AC} \implies \textbf{WOT}$:
He does not give a definition of cardinality $|X|$ before this proof and I searched the index for a definition but couldn't find one. Hence, since we are in $\textbf{ZF}$ without $\textbf{C}$ I assume he uses the definition: $\alpha = \min \{\beta \mid \exists s \in V_\beta \text{ s.t. } s \text{ is in bijection with } X \}$ and $|X| = \{ s \in V_\alpha \mid s \text{ is in bijection with } X \} $.
My question then is the following: Why does one resort to Hartogs number for the proof? Can one prove it as follows: Let $\alpha$ as above be the rank of $X$. Then there cannot be an injection from $V_\alpha$ into $X$. Now replace $\aleph$ with $V_\alpha$ in the proof above. Et voilà, we shortened the proof by one definition. What am I missing? I am as always very grateful for your help. Thank you in advance.
REPLY [2 votes]: Using $V_\alpha$ is not good because we don't know whether or not $V_\alpha$ can be well-ordered. In fact, if $X$ cannot be well-ordered and $X\in V_\alpha$ then it is impossible that $V_\alpha$ can be well-ordered.
What Herrlich is doing here is to define a surjection from $\aleph(X)$ onto $X\cup\{\infty\}$ which has the property that the only point in the range which has more than one point in its preimage is $\infty$, so $X$ has a bijection with an ordinal.
Doing the same thing with $V_\alpha$ to begin with will not allow us to conclude that there is an ordinal whose range is exactly $X$, which is how we prove that $X$ can be well-ordered. | 169,425 |
\begin{document}
\title{Ke Li's lemma for quantum hypothesis testing in general von Neumann algebras
}
\author{Yan Pautrat}
\author{Simeng Wang}
\affil{Université Paris-Saclay, CNRS, Laboratoire de Mathématiques d’Orsay\\
91405 Orsay, France}
\maketitle
\begin{abstract}
A lemma stated by Ke Li in \cite{KeLi} has been used in e.g.\cite{DPR,DR,KW17,WTB,TT15} for various tasks in quantum hypothesis testing, data compression with quantum side information or quantum key distribution. This lemma was originally proven in finite dimension, with a direct extension to type I von Neumann algebras. Here we show that the use of modular theory allows to give more transparent meaning to the objects constructed by the lemma, and to prove it for general von Neumann algebras. This yields immediate generalizations of e.g.\cite{DPR}.
\end{abstract}
\section{Introduction}
\label{sec:introduction}
Quantum hypothesis testing is concerned with the situation where one considers a von Neumann algebra $\M$,
equipped with a state which is either $\rho$ or $\sigma$; this uncertainty in the nature of the state makes sense in particular when $\M$ is viewed as modeling the observable quantities of a quantum system, and the physical state of this system --- itself modeled by a state in the mathematical sense --- is only known to be one or the other. A natural task is then to try and determine which state is the true one by producing, in physical terms, an experimental measurement procedure such that, depending on the measurement outcome, one will conclude that the actual state is $\rho$ or $\sigma$.
In the model of orthodox quantum mechanics, a measurement procedure is determined by a self-adjoint element $X$ of the observable algebra $\M$, and this $X$ is simply called an \emph{observable}. Non-trivial measurements will have a random outcome; the set of possible outcomes is exactly the spectrum of that element, and if $\omega$ is the actual state of the system and we denote by $\xi_X$ the spectral measure of $X$, then the probability distribution for the measurement outcomes is $\rho\circ\xi_X$. In the situation described above it will suffice to consider an observable with spectrum $\{0,1\}$, that is, an orthogonal projector of $\M$. We will take advantage of this simplification and continue this preliminary discussion assuming that the observable describing the discriminating experience, is an orthogonal projector $\test$, which we call a \emph{test}.
Suppose the decision rule is that, if the measurement outcome is $0$, then the observer concludes that the true state is $\rho$, and if the measurement outcome is $1$, then they conclude that the true state is $\sigma$. There are two ways in which this conclusion can be wrong: either the actual state was $\rho$ and the measurement outcome was $1$, or the actual state was $\sigma$ and the measurement outcome was $0$. Applying the rules determining the distribution of the measurement outcomes shows that the former occurs with probability $\rho(\test)$, and the latter with probability $\sigma(\id-\test)$. Assuming that the possible states $\rho$ and $\sigma$ are fixed, we denote these two types or error
\begin{equation} \label{eq_defalphabeta}
\aT:=\rho(\test)\qquad \bT:=\sigma(\id-\test).
\end{equation}
If, to the experimenter's knowledge, both $\rho$ and $\sigma$ may be the state of the system, then one typically wishes to make both $\aT$ and $\bT$ small. There are, however, various ways in which this can be done, and we postpone the corresponding discussion to section~\ref{sec:discussion}.
The result we are concerned with gives a test $T$ such that $\aT$ and $\bT$ satisfy a pair of upper bounds. To state it, let us assume that $\M$ is a finite-dimensional matrix algebra: $\M=\cB(\cc^n)$, and denote by $\varrho$ and $\varsigma$ the density matrices associated with the states $\rho$ and $\sigma$, that is:
\begin{equation}\label{eq_matdens}
\rho = \tr(\varrho\,\cdot) \qquad \sigma= \tr(\varsigma\,\cdot).
\end{equation}
Assume for simplicity that $\rho$ and $\sigma$ are faithful states, or equivalently that $\varrho$, $\varsigma$ are invertible matrices. Consider then the vector space $\cB(\cc^n)$, on which we define the vectors
\[\orho = \varrho^{1/2}\qquad \osig=\varsigma^{1/2}\]
and the operator
\[\drs : X \mapsto \varrho X \varsigma^{-1}.\]
If $\cB(\cc^n)$ is equipped with the scalar product $\braket XY = \tr(X^*Y)$ then $\drs$ is self-adjoint. It then holds that for any $\epsilon>0$ there exists a test $\test$ such that
\begin{equation}\label{eq_keli}
\aT\leq \epsilon\qquad \bT\leq \braket{\osig}{\ind_{(\epsilon,+\infty)}(\drs)\osig}.
\end{equation}
This result was first proven in \cite{KeLi}, as a technical step towards the ``second order Stein's lemma'' which we discus in section \ref{sec:discussion}. It is generally quoted as ``Ke Li's lemma'' for quantum hypothesis testing, even though it is not the only result of Ke Li relevant to this field. The statement as written above, however, does not appear in \cite{KeLi} and in particular, there is no mention of the operator $\drs$ in that article. The result was later reformulated in \cite{DPR} (with the same goal of proving the second order Stein's lemma) to involve the operator $\drs$, which the reader may already have recognized to be the finite-dimensional instance of a \emph{relative modular operator}. From then on it was expected that a proof written solely in terms of modular theory on $\cB(\cc^n)$ should be possible, in which case, according to a standard rule of thumb discussed in \cite{JOPP} it should extend with minimal effort to general von Neumann algebras. It turns out that such an extension holds for any two faithful normal states $\rho$, $\sigma$ on a $\sigma$-finite von Neumann algebra (yielding an extension of the second order Stein's lemma as stated in \cite{DPR}, see section \ref{sec:discussion}). To state this extension we need, however, to recall how the relative modular operator $\drs$ is defined in the general case, and we therefore postpone its statement to Section~\ref{sec:a_general_proof}.
Even though our proof is written in the language of modular theory, it did not proceed from a simple ``modular translation'' of Ke Li's finite dimensional proof, as we could not find such a thing. Note that until now, no extension of inequalities \eqref{eq_keli} was found, except for the extension to separable type I von Neumann algebras in \cite{Khabbazi_Oskouei_2019}, in which case~$\rho$ and $\sigma$ were still of the form \eqref{eq_matdens} and one could simply proceed by finite-dimensional approximations of $\varrho$ and $\varsigma$.\textbf{}
\smallskip
Let us recall shortly the definition of $\test$ in the proof of \eqref{eq_keli} as in \cite{KeLi,DPR} in order to underline its connection with modular theory, and motivate our definition of $T$ below. Consider spectral decompositions of $\varrho$, $\varsigma$:
\[ \varrho=\sum_{x=1}^n\lambda_x \ketbra{a_x}{a_x}\qquad
\varsigma=\sum_{y=1}^n\mu_y \ketbra{b_y}{b_y}
\]
where $(\lambda_x)_x$ and $(\mu_y)_y$ are labeled in nondecreasing order, and both families $(a_x)_x$ and $(b_y)_y$ are orthonormal bases of $\cc^n$. Define for any $y=1,\ldots,n$:
\[Q_{y}=\ind_{[0,\epsilon \mu_y]}(\varrho)= \sum_{x=1}^n \ind_{\lambda_x \leq \epsilon\mu_y} \ketbra{a_x}{a_x},\]
so that $(Q_y)_y$ is a nondecreasing family of projectors, and
\[\xi_{y}=Q_{y}b_y = \sum_{x=1}^n \ind_{\lambda_x \leq \epsilon\mu_y} \braket{a_x}{b_y} a_x.\]
Ke Li then defines his test as the orthogonal projector $\test\kl$ onto the vector space spanned by the $\xi_y$, $y=1,\ldots,n$.
For $X$ an operator on $\cc^n$ denote now by $\L_X$ the operator on $\cB(\cc^n)$ acting by $Y\mapsto XY$, and let $\J$ be the antilinear involution $X\mapsto X^*$ on $\cB(\cc^n)$. Notice then that $(\J \L_X \J) Y = Y X^*$ and let $\M$, $\M'$ be the sets of operators of the form $\L_X$ or $\J \L_X\J$ respectively. We then have:
\[\Delta_{\rho|\sigma}=\sum_{x,y=1}^n {\lambda_x}{\mu_y}^{-1}\, \L_{\ketbra{a_x}{a_x}}\,\J \L_{\ketbra{b_y}{b_y}}\J\]
so that
\begin{equation*}\label{eq_2eexpp}
\ind_{(0,\epsilon]}(\Delta_{\rho|\sigma})=\sum_{x,y=1}^n \ind_{\lambda_x \leq \epsilon\mu_y}\, \L_{\ketbra{a_x}{a_x}}\,\J \L_{\ketbra{b_y}{b_y}}\J=\sum_{y=1}^n \L_{Q_y} \, \J \L_{\ketbra {b_y}{b_y}}\J
\end{equation*}
and
\begin{equation} \label{eq_pomegarho}
\ind_{(0,\epsilon]}(\Delta_{\rho|\sigma})\,\osig=\sum_{x,y=1}^n \ind_{\lambda_x\leq \epsilon \mu_y}\, \mu_y^{1/2} \braket{a_x}{b_y} \,{\ketbra{a_x}{b_y}}.
\end{equation}
It is then immediate to remark that $\ind_{(0,\epsilon]}(\Delta_{\rho|\sigma})\,\osig\, b_y=\mu_y \xi_y$ for any $y=1,\ldots,n$, so that
\begin{align*}
\ran\big(\ind_{(0,\epsilon]}(\Delta_{\rho|\sigma})\,\osig\big)
&=\spann\big\{\xi_y,\ y=1,\ldots,n\big\}
\end{align*}
where the last equation is due to the faithfulness of $\sigma$. Last, remark that the range $\ran\big(\ind_{(0,\epsilon]}(\Delta_{\rho|\sigma})\,\osig\big)$ is the same as $\M'\, \ind_{(0,\epsilon]}(\Delta_{\rho|\sigma})\,\osig$. Therefore, $\test\kl$ is equivalently defined by the fact that $\L_{\test\kl}$ is the orthogonal projection on $\M'\, \ind_{(0,\epsilon]}(\Delta_{\rho|\sigma})\,\osig$. Since all quantities appearing in this last sentence are well-defined in the standard representation of any von Neumann algebra (see section~\ref{sec:a_primer_in_modular_theory}), this gives a satisfactory starting point for our proof of the extension of Ke Li's lemma.
\medskip
The structure of the paper is as follows. In section~\ref{sec:a_primer_in_modular_theory} we recall the elements of the modular theory of von Neumann algebras required by the statement of our result. In section~\ref{sec:a_general_proof} we give our result and its proof. In section~\ref{sec:discussion} we discuss the merits of our result and its possible applications.
\paragraph{Acknowledgements} YP wishes to thank Nilanjana Datta and Cambyse Rouzé for initiating the pleasant collaboration that led to \cite{DPR} and the present endeavors, Ke Li and Magdalena Musat for instructive discussions and most of all Yoshiko Ogata who helped chase the modular in Ke Li's original proof. YP was supported by ANR grant ``NonStops'' ANR-17-CE40-0006. SW was partially supported by a public grant as part of the FMJH and ANR grant ``ANCG'' ANR-19-CE40-0002.
\section{Modular theory and Haagerup $\Lp p$ spaces}
\label{sec:a_primer_in_modular_theory}
We recall first a few notions on von Neumann algebras and relative modular operators. Suggested references are \cite{BR1}, and for a pedagogical introduction we recommend section 2 (and in particular sections 2.11 and 2.12) of \cite{JOPP}. We then move on to describe a few properties of Haagerup's $\Lp p$ spaces following \cite{Haagerup} and \cite{Terp}.
Let $\M$ be a von Neumann algebra, i.e.\ a $*$-algebra of bounded linear operators on some Hilbert space, that is closed in the weak operator topology and contains the identity. The set of operators commuting with all elements of $\M$ is called the commutant of $\M$, and denoted $\M'$; von Neumann's bicommutant theorem then states that a $*$-algebra of bounded linear operators containing the identity is a von Neumann algebra if and only if $\M=\M''$, the latter being the bicommutant $(\M')'$ of $\M$.
The notion of nonnegative operators induces a partial order on $\M$: by definition, $X\leq Y$ if $Y-X$ is a nonnegative operator. A map on $\M$ such that the null operator is the only nonnegative operator mapped to the zero element is called \emph{faithful}. Linear forms $\omega$ on $\M$ with the additional regularity property that $\omega(\limsup_n X_n)=\limsup_n \omega(X_n)$ if $X_n$ is an increasing sequence of self-adjoint operators are called \emph{normal}. A linear form which maps nonnegative operators to nonnegative scalars is called \emph{positive}. A positive linear form $\omega$ from $\M$ to $[0,+\infty]$ will be called a \emph{weight}. A weight $\omega$ such that the set of nonnegative elements $A$ with $\omega(A)<+\infty$ is weakly dense is called \emph{semifinite}. A weight $\omega$ with the property that $\omega(\id)=1$ is called a \emph{state} (and necessarily takes values in $[0,+\infty)$). A weight or state with the property that $\omega(XY)=\omega(YX)$ for all $X$, $Y$ in $\M$ is called \emph{tracial}, or a \emph{trace}.
A standard representation of some von Neumann algebra $\M$ is a quadruple $(\pi,\H,\H^+,\J)$ where $\H$ is a Hilbert space (with scalar product denoted $\braket\cdot\cdot$), $\pi$ is a faithful morphism of $*$-algebras from $\M$ to $\bh$, $\H^+$ is a self-dual cone of $\H$ (i.e.\ the set of $\phi$ in $\H$ such that $\braket \phi\psi\geq 0$ for all $\psi$ in $\H^+$ is $\H^+$ itself), $\J$ is an anti-unitary involution of $\H$, and these different objects satisfy the following properties:
\begin{myitemize}
\item $\J\M \J=\M'$,
\item $\J X\J=X^*$ for $X$ in $\M\cap \M'$,
\item $\J\psi=\psi$ for $\psi$ in $\H^+$,
\item $\J X\J X\, \H^+\subset \H^+$ for $X\in \H$.
\end{myitemize}
A standard representation of a von Neumann algebra $\M$ always exists (we will describe one of them below), and if $(\pi_1,\H_1,\H_1^+,\J_1)$ and $(\pi_2,\H_2,\H_2^+,\J_2)$ are two standard representations, then there exists a unitary operator $U:\H_1\to \H_2$ such that $U\pi_1(X)U^*=\pi_2(X)$ for all $X\in \M$, $U\J_1U^*=\J_2$ and $U\H_1^+=\H_2^+$.
We will now describe the modular structure associated with a von Neumann algebra $\M$: fix a standard representation $(\pi,\H,\H_+,\J)$ of $\M$. Then for any normal state $\omega$ on $\M$ there exists a unique $\oome$ in $\H_+$ such that
\begin{equation}\label{eq_defome}
\omega(X) = \braket{\oome}{\pi(X) \oome} \mbox{ for any }X\in \M.
\end{equation}
This $\oome$ is cyclic in the sense that the closure of $\pi(\M)\H$ is $\H$ itself. If in addition $\omega$ is faithful, then $\oome$ is separating in $\pi(\M)$, that is, $\pi(X)\oome=0$ if and only if $X=0$. We continue by fixing two normal faithful states $\rho$ and $\sigma$. We can then define a densely defined operator $S_{\rho|\sigma}$ by
\[S_{\rho|\sigma} X \osig = X^* \orho.\]
This operator turns out to be closable, and its closure $\overline{S}_{\rho|\sigma}$ has polar decomposition
\[\overline{S}_{\rho|\sigma} = \J {\drs}^{1/2}\]
where $\drs$ is a positive-definite operator on $\H$, which is in general unbounded.
We are now in a position to state our main result, but will first introduce additional elements required for our proof. The reader may wish to take a peek at Theorem~\ref{theo_keli} in section~\ref{sec:a_general_proof}.
\smallskip
To prove the result we consider the Haagerup $\Lp p$-spaces associated with $\M$. We will not need the detailed construction of these spaces and only recall their properties (see chapter 2 of \cite{Terp} for a complete presentation; in the rest of this section all numbered references point to those notes, with e.g.\ Theorem 2.7 meaning Theorem 7 of chapter 2 of \cite{Terp}). An important role will be played by a distinguished von Neumann algebra $\N$, equipped with a normal semifinite faithful trace~$\tau$. This algebra $\N$ then acts on the Hilbert space $\Lp 2(\rr,\H)$ which we simply denote $\K$, for its nature will not matter. For a definition of this algebra $\N$, see the beginning of chapter 2 of \cite{Terp}: between saying too much and saying too little, it seems more convenient to say too little. This $\N$ is such that there exists a faithful normal representation of $\M$ as a sub-von Neumann algebra of~$\N$; in order to spare ourselves an additional notation for this representation, we simply assume that $\M$ is realized as a subalgebra of $\N$, and therefore acts on $\K=\Lp 2(\rr,\H)$. We say that a densely defined unbounded operator on $\K$ is measurable with respect to $(\N,\tau)$ if for every $\delta>0$ there exists a projection $P$ in $\N$ such that the range of $P$ is included in the domain of~$\N$, and $\tau(\id -P)<\delta$. We say that an unbounded operator on $\K$ is affiliated with $\N$ if any Borel bounded functional of either $\tfrac12(K+K^*)$ or $\tfrac1{2\i}(K-K^*)$ is an element of $\N$. Then for all $1\leq p\leq\infty$, there exists a space $\Lp p(\M)$ which is a subspace of the set of those measurable operators with respect to $(\N,\tau)$ which are affiliated with $\N$, which satisfies the following properties. For all $1\leq p<\infty$, an operator $X$ on $\K$ is in $\Lp p(\M)$ if and only its polar decomposition $X=U|X|$ satisfies $U\in\M$ and $|X|^{p}\in \Lp 1(\M)$. Moreover there is a faithful positive linear functional $\mathrm{tr}:\Lp 1(\M)\to\cc$ such that
\[
\tr(XY)=\mathrm{tr}(YX) \mbox{ for all } X\in \Lp p(\M),Y\in \Lp q(\M)\mbox{ with }1/p+1/q=1.
\]
If we define $\|X\|_{p}=\tr(|X|^{p})^{1/p}$ then the closed product of measurable operators on these spaces satisfies the Hölder inequality, and in particular we have $\Lp p(\M)\Lp q(\M)\subset \Lp r(\M)$ for any $p,q,r\in[1,+\infty]$ with $1/r=1/p+1/q$. This induces a natural Hilbert space structure on $\Lp 2(\M)$
by taking $p=q=2$. In addition (Theorem 2.7 and Proposition 2.10), $\M=\Lp \infty(\M)$ and the normal states $\rho$, $\sigma$ on $\M$ are of the form
\begin{equation}\label{eq_matdens2}
\rho = \tr(\varrho\,\cdot) \qquad \sigma= \tr(\varsigma\,\cdot)
\end{equation}
where $\varrho$, $\varsigma$ are positive elements of $\Lp 1(\M)$ with $\tr(\sigma)=1$. This of course echoes \eqref{eq_matdens} and will allow us to mimic some of the ``spatial'' arguments in Ke Li's proof. Taking the values of $p,q,r$ appropriately, we obtain a faithful representation of $\M$ on $\Lp 2(\M)$ as
\[
\pi(X)Y=XY,\quad X\in\M,Y\in \Lp 2(\M).
\]
Denote $\J:X\mapsto X^{*}$ and $\Lp 2(\M)_{+}$ the subset of $\Lp 2(\M)$ made of nonnegative operators. Then $\big(\pi,\Lp 2(\M),\Lp 2(\M)_{+},\J\big)$ is a standard form of $\M$ (see Theorem 2.36). Last, we will use the crucial fact that $\Lp p(\M)$ is isometrically isomorphic to a subspace of the tracial weak $\Lp p$-space on $(\R,\tau)$, and more precisely we have (Lemma~2.5):
\begin{equation}
\tr(X)=\tau\big(\ind_{(1,\infty)}(X)\big) \mbox{ for any } X\in \Lp 1(\M)_{+}. \label{eq:tracial weak norm}
\end{equation}
\section{Our result and its proof}
\label{sec:a_general_proof}
We first state the general form of Ke Li's lemma. As discussed in section~\ref{sec:a_primer_in_modular_theory}, the vector $\osig$ and the operator $\drs$ are meant in any standard representation of $\M$.
\begin{theo}\label{theo_keli}
Let $\M$ be a von Neumann algebra, and $\rho$, $\sigma$ be two faithful normal states on $\M$. For any~$\epsilon>0$ there exists a test $\test\in \M$ such that
\[\aT\leq \epsilon\qquad \bT\leq \braket{\osig}{\ind_{(\epsilon,+\infty)}(\drs)\osig}.\]
\end{theo}
\begin{rema}
Our assumption that $\rho$ and $\sigma$ are faithful is unessential, simplifies the proof and the description of $\drs$ in section~\ref{sec:a_primer_in_modular_theory}, and can be dropped by considering an approximation of general $\rho$, $\sigma$ by faithful states.
\end{rema}
The rest of this section is dedicated to the proof of Theorem~\ref{theo_keli}. We consider the standard representation $\big(\pi,\Lp 2(\M),\Lp 2(\M)_{+},\J\big)$ of $\M$ that we discussed in the second part of section~\ref{sec:a_primer_in_modular_theory}. Once again we simply write $\M$ for $\pi(\M)$, so that $\M$ will act by left-multiplication on $\Lp 2(\M)$.
Let $\pp$ be the following orthogonal projection operator on $\Lp 2(\M)$:
\begin{equation*}
\pp = \ind_{(0,\epsilon]}(\drs).
\end{equation*}
We recall that if $X$ is an element of $\Lp2(\M)$ then its \emph{support} in $\M$, which we denote by $\ell(X)$ (with an $\ell$ for ``left support''), is defined as the smallest orthogonal projector $\proj$ in $\M$ such that $\proj X=X$. It is immediate to show that this projector has range $\overline{\M'X}$. Similarly we define the support of $X \in \Lp2(\M)$ in $\M'$, which we denote by $r(X)$ (with an $r$ for ``right support'') as the smallest orthogonal projector $\proj'$ in $\M'$ such that $\proj' X=X$.
First define our test $\test$ as the \emph{support} in $\M$ of $\pp(\osig)$. By definition $\pp(\osig)$ is an element of $\Lp 2(\M)$, and therefore a densely defined closed operator on $\K$ which is affiliated with $\N$; the polar decomposition of $\pp(\osig)$ as an operator on $\K$ has partial isometry $U$ which from the properties of $\Lp 2(\M)$ belongs to $\M$. Then $UU^*$ is by definition the smallest orthogonal projector $\proj$ in $\cB(\K)$ such that $\proj\,\pp(\osig)=\pp(\osig)$, and belongs to $\M$. Therefore $T$ is equivalently defined as the orthogonal projector onto $\overline{\M'\pp(\osig)}$, as $UU^*$, or as left-multiplication by the orthogonal projector onto the closed range of $\pp(\osig)$ when the latter is viewed as an operator on $\K$. Note that by definition, $\test\in \M$.
From the first expression for $\test$ we have
\[\sigma(\id - \test) = \norme{\osig-\test \osig}^2 \leq \norme{\osig-\pp(\osig)}^2=\braket{\osig}{\ind_{(\epsilon,+\infty)}(\drs) \osig}\]
where:
\begin{myitemize}
\item the first equality follows from \eqref{eq_defome} and the fact that $\id-\test$ is a self-adjoint projection,
\item the inequality follows from the inequality $\norme{\Psi - \test \Psi}\leq \norme{\Psi - \Phi}$, valid for any $\Psi$ in~$\Lp 2(\M)$ and $\Phi\in \ran \test$ by definition of an orthogonal projection, and the fact that $\pp(\osig)$ is an element of $\M'\pp(\osig)\subset\ran\test$,
\item the second equality follows from \eqref{eq_defome} and the fact that $\id-\pp$ is a self-adjoint projection.
\end{myitemize}
This proves the second bound in Theorem~\ref{theo_keli}, and we now move on to prove the first bound.
\smallskip
Let $\varrho=\int_{0}^{1}\lambda\, \d \erho(\lambda)$ and $\varsigma=\int_{0}^{1}\mu\, \d \esig(\mu)$ be spectral decompositions of $\varrho$, $\varsigma$ in $\K$. Then from the definition in the second part of section ~\ref{sec:a_primer_in_modular_theory},
\[ \drs = \varrho \, \J\varsigma^{-1}\!\J\mbox{ on }\M\osig\]
so that, denoting $\tesig :=\J \esig \J $,
\[\drs = \int_{\rr_+^2} \lambda\mu^{-1}\, \d \erho(\lambda)\, \d \tesig(\mu)\]
at least on $\M\osig$, and since the latter integral expression defines a closed operator, equality holds everywhere.
Approximating $\mathbf{1}_{(0,\epsilon]}$ by polynomials weakly, it is easy to see that
\[
\pp=\int_{\rr_+^2} \ind_{(0,\epsilon]}(\lambda\mu^{-1})\, \d \erho(\lambda) \d \tesig(\mu).
\]
Applying Fubini's theorem twice, we have
\begin{align}
\pp(\osig)
&=\int_{\rr_+^2} \ind_{(0,\epsilon]}(\lambda\mu^{-1})\mu^{1/2}\, \d \erho(\lambda) \d \tesig(\mu)\nonumber\\
&=\int_{0}^{\infty}\mu^{1/2}\ind_{(0,\epsilon\mu]}(\varrho)\, \d \tesig(\mu). \label{eq_exptest}
\end{align}
\smallskip
We now prove a regularity lemma that will allow us to approximate $T$ thanks to Riemann sums approximating the expression \eqref{eq_exptest} for $\pp (\osig)$. Assume therefore that $(X_n)_n$ is a sequence of bounded operators on $\K$ that converges to $X$ strongly, and let $\sproj_n$ (respectively $\sproj$) be the support of $X_n$ (respectively $X$) as an operator on $\K$. For any $\xi$ in $\K$ we have using $ \sproj_n \, X_n \xi = \, X_n \xi$ and $\sproj\, X\xi=X\xi$:
\begin{align*}
\| \sproj_n \sproj X\xi-\sproj X\xi\|\leq\norme{\sproj_n X\xi - \sproj_n X_n\xi}+\norme{\sproj_n X_n\xi - X\xi}\leq 2\norme{X_n\xi - X\xi}
\end{align*}
and the upper bound goes to zero as $n\to\infty$. In particular, $ \sproj_n\sproj$ converges to $\sproj$ strongly, which further implies that $(1-p) p_n p$ and $p p_n (1-p)$ converge to $0$ weakly. Now consider $\proj_n$ (respectively $\proj$) the operator of left-multiplication by $\sproj_n$ (respectively $\sproj$) on $\Lp 2 (\M)$. By the normality of $\rho$, we get
\begin{align}
\rho (P)& = \lim_n \rho (P P_n P) \nonumber\\
& = \lim_n \rho (P P_n P) + \lim_n \rho ((1-P) P_n P) + \lim_n \rho (P P_n (1-P)) \nonumber\\
&= \lim_n \big(\rho (P_n ) - \rho((1-P) P_n (1-P)) \big) \nonumber\\
&\leq \limsup_n \rho (P_n) \label{eq_regularite}.
\end{align}
\smallskip
Recall now that our test $\test$ is defined as the support $\ell(X)$ of $X:=\pp (\osig)$. Let $(X_n)_n$ be a sequence of Riemann sums of the form
\[X_n=\sum_{i}\mu_{i}^{1/2}\ind_{[0,\epsilon\mu_{i})}(\varrho)\,\big(\tesig(\mu_i)-\tesig(\mu_{i-1})\big)
\]
that converges strongly to $X$, and let $\test_n$ be the support $\ell(X_n)$ of $X_n$. From \eqref{eq_regularite}, we have $\rho(\test)\leq \limsup_n \rho(\test_n)$ and we therefore consider $\rho(\test_n)$. Obviously
\[
T_n\leq\sum_{i}\ell\big(\ind_{[0,\epsilon\mu_{i})}(\varrho)\,\big(\tesig(\mu_i)-\tesig(\mu_{i-1})\big)\big)=: \test_n',
\]
where we omitted the scalar coefficients $\lambda_{i}^{1/2}$ since $X$ and its multiple $\lambda X$ must have the same range for any $X\in \Lp 2(\M)$. Note that $\ind_{[0,\epsilon\mu_{i})}(\varrho)$ is a projection whose range contains that of $\ell\big(\ind_{[0,\epsilon\mu_{i})}(\varrho)\,\big(\tesig(\mu_i)-\tesig(\mu_{i-1})\big)\big)$, so we write
\[
\ell\big(\ind_{[0,\epsilon\mu_{i})}(\varrho)\,\big(\tesig(\mu_i)-\tesig(\mu_{i-1})\big)\big)=\ind_{[0,\epsilon\mu_{i})}(\varrho)\,\ell\big(\ind_{[0,\epsilon\mu_{i})}(\varrho)\,\big(\tesig(\mu_i)-\tesig(\mu_{i-1})\big)\big).
\]
Note that (see e.g.\ \cite{FackKosaki})
\begin{equation*} \label{eq_monotonietauind}
\tau\big(\ind_{(1,\infty)}(X)\big)\leq\tau\big(\ind_{(1,\infty)}(Y)\big) \ \mbox{ if }\ 0\leq X\leq Y.
\end{equation*}
Thus together with \eqref{eq:tracial weak norm},
\begin{align*}
\tr(\varrho \test_n') & =\sum_{i}\tr\big(\varrho \ind_{[0,\epsilon\mu_{i})}(\varrho)\,\ell\big(\ind_{[0,\epsilon\mu_{i})}(\varrho)\,\big(\tesig(\mu_i)-\tesig(\mu_{i-1})\big)\big)\big)\\
& =\sum_{i}\tau\Big(\ind_{(1,\infty)}\Big(\varrho \ind_{[0,\epsilon\mu_{i})}(\varrho)\,\ell\big(\ind_{[0,\epsilon\mu_{i})}(\varrho)\,\big(\tesig(\mu_i)-\tesig(\mu_{i-1})\big)\big)\Big) \Big)\\
& \leq\sum_{i}\tau\Big(\ind_{(1,\infty)}\Big(\epsilon\mu_{i}\,\ell\big(\ind_{[0,\epsilon\mu_{i})}(\varrho)\,\big(\tesig(\mu_i)-\tesig(\mu_{i-1})\big)\big)\Big) \Big)\\
& \leq\sum_{i} \ind_{(1,\infty)}(\epsilon\mu_{i})\, \tau\Big(\ell\big(\ind_{[0,\epsilon\mu_{i})}(\varrho)\,\big(\tesig(\mu_i)-\tesig(\mu_{i-1})\big)\big)\Big).
\end{align*}
Note that $\sproj_{i,1}:=\ind_{[0,\epsilon\mu_{i})}(\varrho)$ and $\sproj_{i,2}:=\tesig(\mu_i)-\tesig(\mu_{i-1})$ are projections. In this case it is folklore that $\ell(\sproj_{i,1}\,\sproj_{i,2})$ is equivalent to $r(\sproj_{i,1}\,\sproj_{i,2})$, i.e.\ that there exists a unitary $U$ in $\M$ such that $\ell(\sproj_{i,1}\,\sproj_{i,2})=UU^*$ and $r(\sproj_{i,1}\,\sproj_{i,2})=U^*U$, and
\[
r(\sproj_{i,1}\,\sproj_{i,2})=\sproj_{i,1}\vee \sproj_{i,2}^{\bot}-\sproj_{i,2}^{\bot}\leq1-\sproj_{i,2}^{\bot}=\sproj_{i,2}.
\]
In other words, $\ell(\sproj_{i,1}\,\sproj_{i,2})$ is equivalent to a subprojection of $\sproj_{i,2}$, whence $\tau\big(\ell(\sproj_{i,1}\,\sproj_{i,2})\big)\leq\tau(\sproj_{i,2})$. So the previous inequality reads
\begin{align*}
\tr(\varrho \test_n') & \leq \tau\Big(\sum_{i} \ind_{(1,\infty)}(\epsilon\mu_{i})\, \tesig(\mu_i)-\tesig(\mu_{i-1})\Big)
\end{align*}
Taking the limit of Riemann sums and using \eqref{eq:tracial weak norm} again, we get
\[
\tr(\varrho \test)\leq\tau\Big(\int\ind_{(1,\infty)}(\epsilon\mu)\, \d \tesig(\mu)\Big)=\tau\big(\ind_{(1,\infty)}(\epsilon\varsigma)\big)=\tr(\epsilon\varsigma)=\epsilon.
\]
The proof is complete.\qed
\section{Applications and comparison with the Neyman--Pearson test}
\label{sec:discussion}
Let us now return to the practical task of interest, which is of discriminating between the states $\rho$ and $\sigma$. As we mentioned in the introduction, one will typically try to make both error probabilities $\aT$ and $\bT$ small (recall that $\aT$ and $\bT$ are defined by \eqref{eq_defalphabeta}). However, it is expected that there is a tradeoff between one error probability and the other. One must therefore make more precise in what sense we want to ``make both $\aT$ and $\bT$'' small. One possible sense is natural if we assume that prior probabilities $p$ and $q:=1-p$ can be assigned to the states $\rho$ and $\sigma$ respectively. One may then decide to minimize the quantity $p\, \alpha(T)+q\, \beta(T)$. This is the realm of symmetric hypothesis testing, as opposed to asymetric hypothesis testing where one wishes e.g.\ to minimize $\aT$ under a specific constraint on $\bT$.
To state the relevant result for symmetric hypothesis testing we define a specific test $\test\np$ called the \emph{Neyman--Pearson} test. In the finite-dimensional case this test is defined as
\[\test\np := \ell\big(\ind_{\rr_+}(p\varrho-q\varsigma)\big).\]
In the general case the definition is formally similar:
\[\test\np := \ell\big((p\rho-q\sigma)_+\big)\]
but here one needs to rely on the (unique) Jordan decomposition
\[p\rho-q\sigma= (p\rho-q\sigma)_+ - (p\rho-q\sigma)_-\]
with $(p\rho-q\sigma)_\pm$ two positive normal forms on $\M$. We will freely denote $\test\np$ as $\test\np(p,q)$ when we need to emphasize the dependency on $p,q$.
The following result is known as the quantum Neyman--Pearson lemma and was proved in \cite{ANS} in the finite-dimensional case and in \cite{JOPS} in the general case:
\begin{equation} \label{eq_np}
p\, \alpha(\test\np)+q\, \beta(\test\np)= \inf_T\big( p\, \alpha(T)+q\, \beta(T)\big)
\end{equation}
where the infimum is indifferently over the set of orthogonal projections of $\M$, or over the set of elements $T$ of $\M$ satisfying $0\leq T \leq \id$. The associated relevant upper bound was also proven in \cite{ANS,JOPS} and is called the \emph{Chernoff bound}:
\begin{equation} \label{eq_chernoff}
p\, \alpha(\test\np)+q\, \beta(\test\np)\leq \inf_{s\in[0,1]}\, p^sq^{1-s}\, \braket{\osig}{\drs^s \osig}.
\end{equation}
Note that the Chernoff bound (with an additional lower bound, see \cite{NussbaumSzkola,JOPS}) is sufficient to prove the standard Stein's lemma and Hoeffding bounds for asymmetric hypothesis testing, see \cite{JOPS}.
Denote by $\test\kl$ (or, again, by $\test\kl(\epsilon)$ when we need to emphasize the dependence on $\epsilon$) the test constructed in section~\ref{sec:a_general_proof} (we confess a slight inconsistency in our choice of notation, as the present $\test\kl$ is the $\L_{\test\kl}$ of section~\ref{sec:introduction}). A first interesting observation is that, in the case where $\M$ is commutative, or when $\varrho$ and $\varsigma$ commute, then
\[\test\kl(\epsilon) = \test\np\big(\frac1{1+\epsilon},\frac\epsilon{1+\epsilon}\big).\]
This can be easily seen in the finite-dimensional case, i.e.\ when $\M=\{L_X, \, X\in \cB(\cc^n)\}$ as described in section~\ref{sec:introduction} (and using the same notation). Using the fact that the families $(a_x)_x$ and $(b_y)_y$ are the same (up to permutation) when $\varrho$ and $\varsigma$ commute, expression \eqref{eq_pomegarho} takes the form
\begin{equation} \label{eq_comparaison1}
\ind_{(\epsilon,+\infty)}(\Delta_{\rho|\sigma})\,\osig=\sum_{x=1}^n \ind_{\lambda_x> \epsilon \mu_x}\, \mu_x^{1/2} \ketbra{a_x}{a_x}.
\end{equation}
On the other hand, when $p =\frac1{1+\epsilon}$,
\begin{equation} \label{eq_comparaison2}
\ind_{\rr_+}(p\varrho-q\varsigma)=\sum_{x=1}^n \ind_{\lambda_x> \epsilon \mu_x}\, \ketbra{a_x}{a_x}.
\end{equation}
Therefore, the supports of the operators in \eqref{eq_comparaison1} and \eqref{eq_comparaison2} are the same, so that $\test\kl=\test\np$.
\medskip
Now let us compare the merits of Theorem~\ref{theo_keli}, in comparison to the Chernoff bound \eqref{eq_chernoff}. From now on we always consider $p =\frac1{1+\epsilon}$ so that $p$ satisfies the relation $q/p=\epsilon$. The Chernoff bound \eqref{eq_chernoff} implies
\begin{equation}\label{eq_majnp}
\rho(\test\np)\leq \epsilon \qquad
\sigma(\test\np)\leq \epsilon^{-s} \braket{\osig}{\drs^s \osig}
\end{equation}
whereas Theorem~\ref{theo_keli} gives
\begin{equation}\label{eq_majkl}
\rho(\test\kl)\leq \epsilon\qquad
\sigma(\test\kl)\leq \braket{\osig}{\ind_{(\epsilon,+\infty)}(\drs)\osig}.
\end{equation}
If the upper bound for $\sigma(\test\kl)$ in \eqref{eq_majkl} is bounded by an application of Markov's exponential inequality, then \eqref{eq_majnp} and \eqref{eq_majkl} yield the same pair of bounds. It therefore turns out that the estimates \eqref{eq_majkl} given by Theorem~\ref{theo_keli} are no better than those \eqref{eq_majnp} given by the Chernoff bound, unless one has an estimate for $\braket{\osig}{\ind_{(\epsilon,+\infty)}(\drs)\osig}$ more precise than that given by Markov's exponential inequality. It therefore seems that our result is better suited to situations where specific information on the tails of the distribution of $\drs$ with respect to the state $\osig$ is available. This was the case, in particular, in \cite{KeLi,DPR,DR}.
These three papers are concerned with the same general question, which we now describe informally. Consider two sequences $(\rho_n)_n$ and $(\sigma_n)_n$ of states on possibly different algebras $\M_n$, which one would like to discriminate ``for large $n$''. One will therefore try and construct a sequence $(\test_n)_n$ of tests $\test_n\in\M_n$ for which $\big(\alpha(\test_n)\big)_n$ and $\big(\beta(\test_n)\big)_n$ are ``small'' in some sense. The question in \cite{KeLi,DPR,DR} is essentially the following: if one only insists on having $\sup_n\alpha(\test_n)\leq \epsilon$, what is the best rate of decrease of $\beta(\test_n)$ one can hope for? The standard result is called \emph{Stein's lemma} and is well-understood (see \cite{JOPS} and references therein): it gives general assumptions under which, essentially, an exponential decrease of any rate $r$ strictly smaller than a quantity $D$ called the \emph{relative entropy} of $(\rho_n)_n$ and $(\sigma_n)_n$ is possible, but an exponential decrease of a rate strictly larger than $D$ is not possible. The papers \cite{KeLi,DPR,DR} ask (in the case where every $\M_n$ is a finite-dimensional algebra) whether a rate \emph{equal} to~$D$ is possible, and what is then the ``next order'' describing the optimal rate of decrease for $\beta(\test_n)$. In particular, these papers identify conditions under which one can construct a sequence $(T_n)$ satisfying both conditions $\sup_n\alpha(\test_n)\leq \epsilon$ and
\[-\log\beta(\test_n) = nD + \sqrt n\, t+ o(\sqrt n)\]
for any $t < \Phi^{-1}(\epsilon) V $ but not for $t>\Phi^{-1}(\epsilon) V$. Here $\Phi$ is the cumulative distribution function of a standard normal distribution, $V$ is a quantity depending on the sequences $(\rho_n)_n$ and $(\sigma_n)_n$.
The article \cite{DR} used a submultiplicativity property of the sequences $(\rho_n)_n$ and $(\sigma_n)_n$ to derive concentration inequalities for the distribution of $\Delta_{\rho_n|\sigma_n}$ with respect to the state $\Omega_{\sigma_n}$; the nature of the considered sequences $(\rho_n)_n$ and $(\sigma_n)_n$, however, imposes the assumption that every $\M_n$ is finite-dimensional. In that case Theorem~\ref{theo_keli} follows from Ke Li's original result and the present general extension is not needed.
One can say in a similar fashion that \cite{KeLi} used a multiplicativity property of the states $\rho$ and $\sigma$, but the result in \cite{KeLi} is generalized by Theorem 1 of \cite{DPR}. The proof of the latter, however, only relies on the Chernoff bound \eqref{eq_chernoff} and on Theorem~\ref{theo_keli}. Since the Chernoff bound is proved in the general case in \cite{JOPS}, the present proof of Theorem~\ref{theo_keli} immediately extends Theorem 1 of \cite{DPR} to the case of general von Neumann algebras. | 79,960 |
\begin{document}
\sloppy
\begin{center}
\textbf{New decomposition formulas associated with the Lauricella multivariable hypergeometric functions }\\[5pt]
\textbf{Ergashev T.G.\\}
\medskip
{ Institute of Mathematics, Uzbek
Academy of Sciences, Tashkent, Uzbekistan. \\
{\verb [email protected] }}\\
\end{center}
\begin{quote}
Decomposition formulas associated with the Lauricella multivariable hypergeometric functions were known, however, due to the recurrence of those formulas, additional difficulties may
arise in the applications. Further study of the properties
of the famous expansion formulas showed that it can be reduced to a
more convenient form. In addition, this paper contains applications of new expansion formulas to the solving of boundary value problems for a multidimensional elliptic equation with several singular coefficients.
\textit{\textbf{Key words:}} multiple Lauricella hypergeometric
functions; decomposition formula;summation formula; multidimensional elliptic equation with several singular
coefficients; fundamental solutions.
\end{quote}
\section{Introduction}
A great interest in the theory of multiple hypergeometric functions is motivated essentially by the fact that the solutions of many applied problems involving (for example) partial differential equations are obtainable with the help of such hypergeometric functions (see, for details, \cite[p.47 et seq. Section 1.7]{SK}; see also the works \cite{{Opps},{Padma}} and the references cited therein). For instance, the energy absorbed by some nonferromagnetic conductor sphere included in an internal mgnetic field can be calculated with the help of such functions \cite{Loh}. Hypergeometric functions of several variables are used in physical and quantum chemical applications as well \cite{{Niuk},{Opps}} . Especially, many problems in gas dynamics lead to solutions of degenerate second-order partial differential equations which are then solvable in terms of multiple hypergeometric functions. Among examples, we can cite the problem of adiabatic flat-parallel gas flow without whirlwind, the flow problem of supersonic current from vessel with flat walls, and a number of other problems connected with gas flow \cite{Frankl}.
We note that Riemann's functions, Green's functions and the fundamental solutions of the degenerate second-order partial differential equations are expressible by means of hypergeometric functions of several variables \cite{{BN1},{BN2},{BN3},{Erg},{EH},{H},{HK},{M},{U}, {UrK},{Wt}}. In investigation of the boundary-value problems for these partial differential equations, we need decompositions for hypergeometric functions of several variables in terms of simpler hypergeometric functions of (for example) the Gauss and Appell types.
The familiar operator method of Burcnall and Chaundy \cite{{BC1},{BC2},{Chaundy}} has been used by them rather extensively for finding decomposition formulas for hypergeometric functions of two variables in terms of the classical Gauss hypergeometric function of one variable.
Following the works \cite{{BC1},{BC2}}, Hasanov and Srivastava \cite{{HS6},{HS7}} introduced operators generalizing the Burcnall-Chaundy operators and found expansion formulas for many triple hypergeometric functions which were successfully applied to the solving the boundary-value problems for the second order elliptic equation with three singular coefficients \cite{{Kar},{A20},{A22}}, and they proved recurrent formulas when the dimension of hypergeometric function exceeds three. However, due to the recurrence, additional difficulties may
arise in the applications of those decomposition formulas.
In this paper for the two Lauricella hypergeometric functions in several variables we prove new decomposition formulas which are free from the recurrence and applied to the solving the boundary-value problems for the multidimensional elliptic equation with several singular coefficients.
The plan of this paper is as follows. In Section 2 we briefly give some preliminary information, which will be used later.
In Section 3, we present the well-known decomposition formulas associated with the two and more dimensional Lauricella hypergeometric functions. In Section 4, we will prove new decomposition and summation formulas and in the last section 5 we will apply the obtained formulas to the solution of boundary-value problems.
\section{Preliminaries}
Below we give some formulas for Euler gamma-function, Gauss hypergeometric function, Lauricella hypergeometric functions of three and more variables, which will be used in the next sections.
Let be ${N}$ set of the natural numbers : $N=\{1,2,3,...\}.$
It is known that the Euler gamma-function $\Gamma(a)$ has property \cite[p.17, (2)]{Erd}
$$\Gamma(a+m)=\Gamma(a) (a)_m.$$
Here $(a)_m$ is a Pochhammer symbol, for which the equality
$(a)_{m+n}=(a)_m(a+m)_n $ and its particular case $(a)_{2m}=(a)_m(a+m)_m$ are true \cite[p.67,(5)]{Erd}.
A function
$$F {\left(a,b;c;x\right)} \equiv F {\left[
{{\begin{array}{*{20}c}
{a,b} ; \hfill \\
{c}; \hfill \\
\end{array}} x} \right]}={\sum\limits_{i =
0}^{\infty} \frac{(a)_i(b)_i}{(c)_ii!}x^i }, \,c\neq 0,-1,-2,...$$
is known as the Gaussian hypergeometric function and an equality
\begin{equation}
\label{sum}
F(a,b;c;1)=\frac{\Gamma(c)\Gamma(c-a-b)}{\Gamma(c-a)\Gamma(c-b)}, c\neq 0,-1,-2,..., Re(c-a-b)>0
\end{equation}
holds \cite[p.73, (73)]{Erd}. Moreover, the following autotransformation formula \cite[p.76, (22)]{Erd}
\begin{equation}
\label{auto}
F\left(a,b;c;x\right)=\left(1-x\right)^{-b}F\left(c-a,b;c;\frac{x}{x-1}\right)
\end{equation}
is valid.
Multiple Lauricella hypergeometric functions $F_A^{(n)}$ and $F_B^{(n)}$ in $n\in N$ (real or complex) variables are defined as following (\cite{A30} and \cite[p.33]{AP})
\begin{equation*}
F_{A}^{(n)} \left( {a,b_{1} ,...,b_{n} ;c_{1} ,...,c_{n} ;x_{1} ,...,x_{n}}
\right) \equiv F_{A}^{(n)} {\left[ {{\begin{array}{*{20}c}
{a,b_{1} ,...,b_{n} ;} \hfill \\
{c_{1} ,...,c_{n} ;} \hfill \\
\end{array}} x_{1} ,...,x_{n}} \right]}
\end{equation*}
\begin{equation*}
\label{eq7}
= {\sum\limits_{m_{1} ,...m_{n} = 0}^{\infty} {{\frac{{\left( {a}
\right)_{m_{1} + ... + m_{n}} \left( {b_{1}} \right)_{m_{1}} ...\left(
{b_{n}} \right)_{m_{n}}} } {{\left( {c_{1}} \right)_{m_{1}} ...\left(
{c_{n}} \right)_{m_{n}}} } }{\frac{{x_{1}^{m_{1}}} } {{m_{1}
!}}}...{\frac{{x_{n}^{m_{n}}} } {{m_{n} !}}}}}
\end{equation*}
\[\,{\left[ {c_{k} \ne 0,-1,-2,...;\,k = \overline {1,n} ;\,{\left| {x_{1}}
\right|} + ... + {\left| {x_{n}} \right|} < 1} \right]};\]
\begin{equation*}
F_{B}^{(n)} \left( {a_1,...,a_n,\,b_{1} ,...,b_{n} ;c;x_{1} ,...,x_{n}}
\right) \equiv F_{B}^{(n)} {\left[ {{\begin{array}{*{20}c}
{a_1,...,a_n,\,b_{1} ,...,b_{n} ;} \hfill \\
{c;} \hfill \\
\end{array}} x_{1} ,...,x_{n}} \right]}
\end{equation*}
\begin{equation*}
\label{eq71}
= \sum\limits_{m_{1} ,...m_{n} = 0}^{\infty} {{\frac{{\left( {a_1}
\right)_{m_{1}}...\left( {a_1}
\right)_{m_{n}} \left( {b_{1}} \right)_{m_{1}} ...\left(
{b_{n}} \right)_{m_{n}}} } {{\left({c}\right)_{m_{1}+...+m_n} } }{\frac{{x_{1}^{m_{1}}} } {{m_{1}
!}}}...{\frac{{x_{n}^{m_{n}}} } {{m_{n} !}}}}}
\end{equation*}
\[\,\left[ {c \ne 0,-1,-2,... ;\,{max\{\left| {x_{1}}
\right|,...,{\left| {x_{n}} \right|}\} < 1} }\right].\]
\section{Decomposition formulas associated with the Lauricella functions $F_A^{(n)}$ and $F_B^{(n)}$ }
For a given multiple hypergeometric function, it is useful to fund a
decomposition formula which would express the multivariable hypergeometric
function in terms of products of several simpler hypergeometric functions
involving fewer variables.
Burchnall and Chaundy \cite{{BC1},{BC2}} systematically
presented a number of expansion and decomposition formulas for some double
hypergeometric functions in series of simpler hypergeometric functions. For
example, the Appell function
\begin{equation*}
F_{2} \left( {a,b_{1} ,b_{2} ;c_{1} ,c_{2} ;x,y} \right) = {\sum\limits_{i,j
= 0}^{\infty} {{\frac{{\left( {a} \right)_{i+j} \left( {b_{1}}
\right)_{i} \left( {b_{2}} \right)_{j}}} {{\left( {c_{1}} \right)_{i}
\left( {c_{2}} \right)_{j}}} }{\frac{{x^{i}}}{{i!}}}{\frac{{y^{j}}}{{j!}}}}
}
\end{equation*}
\begin{equation*}
{\left[ {c_{1} ,c_{2} \ne 0,-1,-2,...;\,\,{\left| {x} \right|} + {\left| {y} \right|}
< 1} \right]}
\end{equation*}
\noindent
has the expansion \cite{BC1}
\[
F_{2} \left( {a,b_{1} ,b_{2} ;c_{1} ,c_{2} ;x,y} \right)\]
\begin{equation}
\label{eq25}
= {\sum\limits_{i =
0}^{\infty} {{\frac{{\left( {a} \right)_{i} \left( {b_{1}} \right)_{i}
\left( {b_{2}} \right)_{i}}} {{i!\left( {c_{1}} \right)_{i} \left( {c_{2}
} \right)_{i}}} }x^{i}y^{i}F\left( {a + i,b_{1} + i;c_{1} + i;x}
\right)F\left( {a + i,b_{2} + i;c_{2} + i;y} \right)}}.
\end{equation}
The Burchnall-Chaundy method, which is limited to functions of two variables,
is based on the following mutually inverse symbolic operators \cite{BC1}
\begin{equation}
\label{eq8}
\nabla \left( {h} \right) = {\frac{{\Gamma \left( {h} \right)\Gamma \left(
{{\rm \delta} _{1} + {\rm \delta} _{2} + h} \right)}}{{\Gamma \left( {{\rm
\delta} _{1} + h} \right)\Gamma \left( {{\rm \delta} _{2} + h} \right)}}},
\quad
\Delta \left( {h} \right) = {\frac{{\Gamma \left( {{\rm \delta} _{1} + h}
\right)\Gamma \left( {{\rm \delta} _{2} + h} \right)}}{{\Gamma \left( {h}
\right)\Gamma \left( {{\rm \delta} _{1} + {\rm \delta} _{2} + h}
\right)}}},
\end{equation}
\noindent
where ${\rm \delta} _{1} = x{\displaystyle\frac{{\partial}} {{\partial x}}}$ and ${\rm
\delta} _{2} = y{\displaystyle\frac{{\partial}} {{\partial y}}}$.
In order to generalize the operators $\nabla \left( {h} \right)$ and $\Delta
\left( {h} \right)$, defined in (\ref{eq8}), Hasanov and Srivastava \cite{{HS6}, {HS7}}
introduced the operators
\begin{equation*}
\label{eq9}
\tilde {\nabla} _{x_{1} ;x_{2} ,...,x_{n}} \left( {h} \right) =
{\frac{{{\rm \Gamma} \left( {h} \right){\rm \Gamma} \left( {{\rm \delta
}_{1} + ... + {\rm \delta} _{n} + h} \right)}}{{{\rm \Gamma} \left( {{\rm
\delta} _{1} + h} \right){\rm \Gamma} \left( {{\rm \delta} _{2} + ... + {\rm
\delta} _{n} + h} \right)}}},
\end{equation*}
\begin{equation*}
\label{eq10}
\tilde {\Delta} _{x_{1} ;x_{2} ,...,x_{n}} \left( {h} \right) =
{\frac{{{\rm \Gamma} \left( {{\rm \delta} _{1} + h} \right){\rm \Gamma
}\left( {{\rm \delta} _{2} + ... + {\rm \delta} _{n} + h} \right)}}{{{\rm
\Gamma} \left( {h} \right){\rm \Gamma} \left( {{\rm \delta} _{1} + ... +
{\rm \delta} _{n} + h} \right)}}},
\end{equation*}
\noindent
where ${\rm \delta} _{k} = x_{k} {\displaystyle\frac{{\partial}} {{\partial x_{k}}} }\,$ $(k=\overline{1,n}),$
with the help of which they managed to find decomposition formulas for a
whole class of hypergeometric functions in several variables.
Following the works \cite{{BC1},{BC2}} Hasanov and Srivastava
\cite{HS6} found following decomposition formulas for the
Lauricella functions of three variables
\begin{equation}\label{e26}\begin{array}{l}F^{(3)}_A \left( {{
{a,b_1,b_2,b_3;}{c_1,c_2,c_3;}} x_1,x_2,x_3} \right)
={\sum\limits_{i,j,k=0}^\infty\displaystyle\frac{(a)_{i+j+k}(b_1)_{j+k}(b_2)_{i+k}(b_3)_{i+j}}{i!j!k!(c_1)_{j+k}(c_2)_{i+k}(c_3)_{i+j}}}x_1^{j+k}x_2^{i+k}x_3^{i+j}
\\
\\
\cdot
F\left( {{{a+j+k,b_1+j+k;} {c_1+j+k;} } x_1} \right)F\left( {{{a+i+j+k,b_2+i+k;}{c_2+i+k;}} x_2} \right)\\
\\
\cdot F\left( {{{a+i+j+k,b_3+i+j;}{c_3+i+j;}} x_3} \right), \end{array}
\end{equation}
\begin{equation*}
\label{eq66}
\begin{array}{l}
F_{B}^{(3)} \left( {a_{1} ,a_{2} ,a_{3} ;b_{1} ,b_{2} ,b_{3} ;c;x_{1}
,x_{2} ,x_{3}} \right) \\
\\
= {\sum\limits_{i,j,k = 0}^{\infty} {{\displaystyle\frac{{\left( { - 1} \right)^{i + j
+ k}\left( {a_{1}} \right)_{j + k} \left( {b_{1}} \right)_{j + k} \left(
{a_{2}} \right)_{i + k} \left( {b_{2}} \right)_{i + k} \left( {a_{3}}
\right)_{i + j} \left( {b_{3}} \right)_{i + j}}} {{\left( {c - 1 + j + k}
\right)_{j + k} \left( {c - 1 + 2\left( {j + k} \right) + i} \right)_{i}
\left( {c} \right)_{2\left( {i + j + k} \right)} i!j!k!}}}}}x_{1}^{j + k} x_{2}^{i + k} x_{3}^{i + j} \\
\\
\cdot F\left( {a_{1} + j + k,b_{1} + j + k;c + 2\left( {j + k} \right);x_{1}}
\right)F\left( {a_{2} + i + k,b_{2} + i + k;c + 2\left( {i + j + k} \right);x_{2}
} \right) \\
\\
\cdot F\left( {a_{3} + i + j,b_{3} + i + j;c + 2\left( {i + j + k}
\right);x_{3}} \right) \\
\end{array}
\end{equation*}
and they proved that for all $n\in N\backslash\{1\}$ are true the
recurrence formulas \cite {HS7}
\begin{equation}
\label{e27} \begin{array}{l} F_{A}^{(n)}\left(
{{{a,b_1,...,b_n;}{c_1,...,c_n;}} x_1,...,x_n} \right) \\
\\
= {\sum\limits_{m_{2} ,...,m_{n} = 0}^{\infty} {{\displaystyle\frac{{(a)_{m_{2} + \cdot
\cdot \cdot + m_{n}} (b_{1} )_{m_{2} + \cdot \cdot \cdot + m_{n}}
(b_{2} )_{m_{2}} \cdot \cdot \cdot (b_{n} )_{m_{n}}} } {{m_{2} !
\cdot \cdot \cdot m_{n} !(c_{1} )_{m_{2} + \cdot \cdot \cdot +
m_{n}} (c_{2} )_{m_{2}} \cdot \cdot \cdot (c_{n} )_{m_{n}}} }
}}} x_{1}^{m_{2} + \cdot \cdot \cdot + m_{n}} x_{2}^{m_{2}}
\cdot \cdot \cdot x_{n}^{m_{n}} \\
\\
\cdot x_{1}^{m_{2} + \cdot \cdot \cdot + m_{n}} F\left( {{{a + m_{2} + \cdot \cdot \cdot + m_{n},b_{1} + m_{2} +
\cdot \cdot \cdot
+ m_{n};}
{c_{1} + m_{2} +
\cdot \cdot \cdot
+ m_{n};}
} x_1} \right) \\
\\
\cdot F_{A}^{(n - 1)} \left( {{ {a + m_{2} + \cdot \cdot \cdot + m_{n} ,b_{2}
+ m_{2} ,...,b_{n} + m_{n} ;}
c_{2} + m_{2} ,....,c_{n} + m_{n}
; } x_{2} ,...,x_{n}} \right),
\end{array}
\end{equation}
\begin{equation}
\label{e277}
\begin{array}{l}
F_{B}^{(n)} \left( {a_{1} ,...,a_{n} ;b_{1} ,...,b_{n} ;c;x_{1} ,...,x_{n}
} \right) \\
\\
= {\sum\limits_{k_{2} ,...,k_{n} = 0}^{\infty} {{\displaystyle\frac{{\left( { - 1}
\right)^{k_{2} + ... + k_{n}} \left( {a_{1}} \right)_{k_{2} + ... + k_{n}}
\left( {b_{1}} \right)_{k_{2} + ... + k_{n}} {\prod\nolimits_{j = 2}^{n}
{{\left[ {\left( {a_{j}} \right)_{k_{j}} \left( {b_{j}} \right)_{k_{j}}
} \right]}}}} }{{\left( {c - 1 + k_{2} + ... + k_{n}} \right)_{k_{2} + ...
+ k_{n}} \left( {c} \right)_{2\left( {k_{2} + ... + k_{n}} \right)} k_{2}
!...k_{n} !}}}}} \\
\\
\cdot x_{1}^{k_{2} + ... + k_{n}} x_{2}^{k_{2}} ...x_{n}^{k_{n}
} F\left( {a_{1} + k_{2} + ... + k_{n} ,b_{1} + k_{2} + ... + k_{n} ;c +
2\left( {k_{2} + ... + k_{n}} \right);x_{1}} \right) \\
\\
\cdot F_{B}^{(n - 1)} \left( {a_{2} + k_{2} ,...,a_{n} + k_{n} ,b_{2} + k_{2}
,...,b_{n} + k_{n} ;c + 2\left( {k_{2} + ... + k_{n}} \right);x_{2}
,...,x_{n}} \right) .\\
\end{array}
\end{equation}
However, due to the recurrence of formula (\ref{e27}) and (\ref{e277}), additional difficulties may
arise in the applications of this expansion. Further study of the properties
of the Lauricella functions $F_A^{(n)}$ and $F_B^{(n)}$ showed that formulas (\ref{e27}) and (\ref{e277}) can be reduced to a
more convenient forms.
\bigskip
\section{New decomposition formulas associated with the Lauricella functions $F_A^{(n)}$ and $F_B^{(n)}$ }
Before proceeding to the presentation of the main result of this article, we introduce the notations
\begin{equation}
\label{e4111}
A(k,n)=\sum\limits_{i=2}^{k+1}\sum\limits_{j=i}^n m_{i,j},\,\,B(k,n)=\sum\limits_{i=2}^{k}m_{i,k}+\sum\limits_{i=k+1}^n m_{k+1,i},
\end{equation}
where $m_{i,j}\in \rm{N}\cap\{0\} \left(2\leq i\leq j \leq n\right).$
It should be noted here that the sum ${ {B(2,n)+B(3,n)+...+B(n,n)}} $
has the parity property, which plays an important role in the calculation of
the some values of hypergeometric functions. In fact, by virtue of equality
\begin{equation*}
{\sum\limits_{k = 2}^{n} {{\sum\limits_{i = 2}^{k} {m_{i,k}}} } } =
{\sum\limits_{k = 1}^{n - 1} {{\sum\limits_{i = k + 1}^{n} {m_{k + 1,i}}} }
}
\end{equation*}
\noindent
we obtain
\begin{equation}
\label{eq1444}
{\sum\limits_{k = 1}^{n} {B(k,n)}} = 2{\sum\limits_{k = 2}^{n}
{{\sum\limits_{i = 2}^{k} {m_{i,k}}} } } = 2{\sum\limits_{k = 1}^{n - 1}
{{\sum\limits_{i = k + 1}^{n} {m_{k + 1,i}}} } } .
\end{equation}
We present other simple properties of the functions $A\left(
{k,n} \right)$ and $B\left( {k,n} \right)$:
\begin{equation}
\label{eq15555}
A\left( {n + 1,n + 1} \right) - B\left( {n + 1,n + 1} \right) = A\left(
{n,n} \right),
\end{equation}
\begin{equation}
\label{eq16666}
A\left( {k + 1,k + 1} \right) - B\left( {k + 1,k + 1} \right) = A\left(
{k,n} \right) - B\left( {k,n} \right) + m_{2,n + 1} + ... + m_{k,n + 1} .
\end{equation}
Those properties are easily proved if we proceed from the
definitions of functions $A\left( {k,n} \right)$ and $B\left( {k,n}
\right)$.
\textbf{Lemma 1}. The following decomposition formulas hold true at $n
\in {\rm N}$
\begin{equation*}
F_{A}^{(n)} \left( {a,b_{1} ,b_{2} ,....,b_{n} ;c_{1} ,c_{2} ,....,c_{n}
;x_{1} ,...,x_{n}} \right)
\end{equation*}
\begin{equation}
\label{eq1222}
= {\sum\limits_{{\mathop {m_{i,j} = 0}\limits_{(2 \le i \le j \le n)}
}}^{\infty} {{\frac{{(a)_{A(n,n)}}} {{m_{ij} !}}}}} \prod\limits_{k =
1}^{n} [ \frac{{(b_{k} )_{B(k,n)}}}{(c_{k})_{B(k,n)}}
x_{k}^{B(k,n)}
F( a + A(k,n), b_{k} + B(k,n);c_{k} + B(k,n);x_{k})],
\end{equation}
\begin{equation*}
F_{B}^{(n)} \left( {a_{1} ,...,a_{n} ,b_{1} ,...,b_{n} ;c;x_{1} ,...,x_{n}
} \right)
\end{equation*}
\begin{equation*}
= {\sum\limits_{{\mathop {m_{i,j} = 0}\limits_{(2 \le i \le j \le n)}
}}^{\infty} {{\displaystyle\frac{{\left( { - 1} \right)^{A\left( {n,n}
\right)}}}{{\left( {c} \right)_{2A\left( {n,n} \right)} m_{ij} !}}}}
}\prod\limits_{k = 1}^{n} [{\displaystyle\frac{{\left( {a_{k}}
\right)_{B\left( {k,n} \right)} \left( {b_{k}} \right)_{B\left( {k,n}
\right)} \left( {c - 1} \right)_{A(k,n) - A(k - 1,n)}}} {{\left( {c - 1}
\right)_{2A(k,n) - 2A(k - 1,n)}}} }
\end{equation*}
\begin{equation}
\label{eq1111} \cdot x_{k}^{B\left( {k,n} \right)} F\left(
{a_{k} + B\left( {k,n} \right),b_{k} + B\left( {k,n} \right);c + 2A\left(
{k,n} \right);x_{k}} \right)].
\end{equation}
\textbf{Proof}. We carry out the proof by the method mathematical
induction. First, we prove the validity of the equality (\ref{eq1222}).
For clarity of the course of the proof, we introduce the notations
$$
N_{l} (k,n)
= {\sum\limits_{i = l}^{k + 1} {{\sum\limits_{j = i}^{n}
{m_{i,j}}} } } , \, M_{l} (k,n) = {\sum\limits_{i = l}^{k} {m_{i,k} +}}
{\sum\limits_{i = k + 1}^{n} {m_{k + 1,i}}}, l \in N.
$$
It's obvious that
$$
N_{2} (k,n) = A(k,n),\, M_{2} (k,n)=B(k,n).
$$
So we have to prove the fairness of equality
\begin{equation}
\label{e28}\begin{array}{l} F_{A}^{(n)} {\left[
{{\begin{array}{*{20}c}{a,b_{1}
,....,b_{n} ;} \hfill \\ {c_{1} ,....,c_{n} ;} \hfill \\
\end{array}} x_{1} ,...,x_{n}} \right]}
= {\sum\limits_{{\mathop {m_{i,j} = 0}\limits_{(2 \le i \le j \le n)}
}}^{\infty} {{\displaystyle\frac{{(a)_{N_{2} (n,n)}}} {{ m_{i,j}!}}}}} \hfill\\
\\
\cdot{\prod\limits_{k = 1}^{n} {{ {{\displaystyle\frac{{(b_{k} )_{M_{2} (k,n)}
}}{{(c_{k} )_{M_{2} (k,n)}}} x_{k}^{M_{2} (k,n)}
F\left[{\begin{array}{*{20}c} {a + N_{2} (k,n),b_{k} + M_{2}
(k,n);} \hfill \\ c_{k} + M_{2} (k,n);\hfill \\ \end{array} x_{k}}
\right]} } }}}.
\end{array}
\end{equation}
In the case $n=1$ the equality (\ref{e28}) is obvious.
Let $n = 2$. Since $M_2(1,2)=M_2(2,2)=N_2(1,2)=N_2(2,2)=m_{2,2}:= i,$
we obtain the formula (\ref{eq25}).
For the sake of interest, we will check the formula (\ref{e28}) in
yet another value of $n$.
Let $n=3.$ In this case
$$M_2(1,3)=m_{2,2}+m_{2,3},\,\, M_2(2,3)=m_{2,2}+m_{3,3},\,\, M_2(3,3)=m_{2,3}+m_{3,3},$$
$$N_2(1,3)=m_{2,2}+m_{2,3},\,\, N_2(2,3)= N_2(3,3)=m_{2,2}+m_{2,3}+m_{3,3}.$$
For brevity, making the substitutions
$m_{2,2}:=i,\,\,m_{2,3}:=j,\,\,m_{3,3}:=k$, we obtain the formula
(\ref{e26}).
So the formula (\ref{e28}), that is formula (\ref{eq1222}), works for $n=1,$ $n=2$ and $n=3$.
Now we assume that for $n = s$ equality (\ref{e28}) holds; that
is, that
\begin{equation}
\label{e29}
\begin{array}{l}
F_{A}^{(s)} {\left[ \begin{array}{*{20}c}{a,b_{1} ,....,b_{s} ;}\hfill\\ c_{1} ,....,c_{s} ; \hfill\\ \end{array} x_{1}
,...,x_{s} \right]}
= {\sum\limits_{{\mathop {m_{i,j} = 0}\limits_{(2 \le i \le j \le s)}
}}^{\infty} {{\displaystyle\frac{{(a)_{N_{2} (s,s)}}} {m_{ij}! }}}} \\
\\
\cdot {\prod\limits_{k = 1}^{s} {{{{\displaystyle\frac{{(b_{k} )_{M_{2} (k,s)}
}}{{(c_{k} )_{M_{2} (k,s)}}} }x_{k}^{M_{2} (k,s)} F\left[
\begin{array}{*{20}c}{a + N_{2} (k,s),b_{k} + M_{2} (k,s);}\hfill\\ c_{k} +
M_{2} (k,s);\hfill\\ \end{array} x_{k} \right]} }}} .
\\
\end{array}
\end{equation}
Let $n=s+1.$ We prove that following formula
\begin{equation}
\label{e210}
\begin{array}{l}
F_{A}^{(s + 1)} {\left[ \begin{array}{*{20}c}{a,b_{1} ,....,b_{s+1} ;}\hfill\\ c_{1} ,....,c_{s+1} ; \hfill\\ \end{array} x_{1}
,...,x_{s+1} \right]}
= {\sum\limits_{{\mathop {m_{i,j} = 0}\limits_{(2 \le i \le j \le s + 1)}
}}^{\infty} {{\displaystyle\frac{{(a)_{N_{2} (s + 1,s + 1)}}} {m_{ij}! }}}} \\
\\
\cdot {\prod\limits_{k = 1}^{s + 1} {{ {{\displaystyle\frac{{(b_{k} )_{M_{2} (k,s + 1)}
}}{{(c_{k} )_{M_{2} (k,s + 1)}}} }x_{k}^{M_{2} (k,s + 1)} F\left[
{{\begin{array}{*{20}c}
{a + N_{2} (k,s + 1),b_{k} + M_{2} (k,s + 1);} \hfill \\
{c_{k} + M_{2} (k,s + 1);} \hfill \\
\end{array}} x_{k}} \right]} }}} \\
\end{array}
\end{equation}
is valid.
We write the Hasanov-Srivastava's formula (\ref{e27}) in the form
\begin{equation}
\label{e211}
\begin{array}{l}
F_{A}^{(s + 1)} {\left[ {{\begin{array}{*{20}c}{a,b_{1} ,....,b_{s + 1} ;}\hfill \\ { c_{1} ,....,c_{s +
1};}\hfill \\
\end{array}}
x_{1} ,...,x_{s + 1}} \right]} \\
\\
= {\sum\limits_{m_{2,2} ,...,m_{2,s + 1} = 0}^{\infty} {{\displaystyle\frac{{(a)_{N_2(1,s+1)} (b_{1} )_{M_2(1,s+1)} (b_{2} )_{m_{2,2}} \cdot \cdot \cdot (b_{s + 1}
)_{m_{2,s + 1}}} } {{m_{2,2} ! \cdot \cdot \cdot m_{2,s + 1}
!(c_{1} )_{M_2(1,s+1)} (c_{2} )_{m_{2,2}} \cdot \cdot \cdot (c_{s
+ 1})_{m_{2,s + 1}}} } }}} \\
\\
\cdot x_{1}^{M_2(1,s+1)} x_{2}^{m_{2,2}} \cdot
\cdot \cdot x_{s + 1}^{m_{2,s + 1}} F\left[\begin{array}{*{20}c}{a + N_2(1,s+1) ,b_{1} + M_2(1,s+1) ;} \hfill\\ c_{1} +M_2(1,s+1);\hfill\\ \end{array} x_{1} \right] \\
\\
\cdot F_{A}^{(s)} {\left[ {{\begin{array}{*{20}c}
{a + N_2(1,s+1) ,b_{2} + m_{2,2} ,...,b_{s + 1} +
m_{2,s + 1} ;} \hfill \\
{c_{2} + m_{2,2} ,....,c_{s + 1} + m_{2,s + 1}}; \hfill \\
\end{array}} x_{2} ,...,x_{s + 1}} \right]}. \\
\end{array}
\end{equation}
By virtue of the formula (\ref{e29}) we have
\begin{equation}
\label{e212}
\begin{array}{l}
F_{A}^{(s)} \left[ \begin{array}{*{20}c}{a + N_2(1,s+1) ,b_{2} + m_{2,2} ,...,b_{s +
1} + m_{2,s + 1} ;}\hfill\\ c_{2} + m_{2,2} ,...,c_{s + 1} +
m_{2,s + 1} ;\hfill\\ \end{array} x_{2} ,...,x_{s +
1} \right] \\
\\
= {\sum\limits_{{\mathop {m_{i,j} = 0}\limits_{(3 \le i \le j \le s + 1)}
}}^{\infty} {{\displaystyle\frac{{\left( {a +N_2(1,s+1)} \right)_{N_{3} (s +
1,s + 1)}}} {m_{ij}!}}}} \prod\limits_{k = 2}^{s + 1}
{\displaystyle\frac{{(b_{k} + m_{2,k} )_{M_{3} (k,s + 1)}}} {{(c_{k} + m_{2,k}
)_{M_{3} (k,s + 1)}}}
}x_{k}^{M_{3} (k,s + 1)} \\
\\
\cdot F\left[ {{\begin{array}{*{20}c}
{a + N_2(1,s+1) + N_{3} (k,s + 1),b_{k} + m_{2,k} + M_{3} (k,s +
1);} \hfill \\
{c_{k} + m_{2,k} + M_{3} (k,s + 1);} \hfill \\
\end{array}} x_{k}} \right] . \\
\end{array}
\end{equation}
Substituting from (\ref{e212}) into (\ref{e211}) we obtain
\begin{equation*}
\begin{array}{l}
F_{A}^{(s + 1)} {\left[ {a,b_{1} ,....,b_{s + 1} ;c_{1} ,....,c_{s + 1}
;x_{1} ,...,x_{s + 1}} \right]} \\
\\
= {\sum\limits_{{\mathop {m_{i,j} = 0}\limits_{(2 \le i \le j \le s + 1)}
}}^{\infty} {{\displaystyle\frac{{\left( {a} \right)_{N_2(1,s+1) + N_{3} (s +
1,s + 1)}}} {m_{ij}! }}}} \prod\limits_{k = 1}^{s + 1}
{\displaystyle\frac{{(b_{k} )_{m_{2,k} + M_{3} (k,s + 1)}}} {{(c_{k} )_{m_{2,k}
+ M_{3} (k,s + 1)}}}
}x_{k}^{m_{2,k} + M_{3} (k,s + 1)} \\
\\
\cdot F\left[ {{\begin{array}{*{20}c} {a + N_2(1,s+1) + N_{3} (k,s
+ 1),}
{b_{k} + m_{2,k} + M_{3} (k,s + 1);}
\hfill \\
{c_{k} + m_{2,k} + M_{3} (k,s + 1)}; \hfill \\
\end{array}} x_{k}} \right]. \\
\end{array}
\end{equation*}
Further, by virtue of the following obvious equalities
$$
N_2(1,s+1) + N_{3} (k,s + 1) = N_{2} (k,s + 1),\,\,1\leq k\leq
s+1, s\in N, $$ $$ m_{2,k} + M_{3} (k,s + 1) = M_{2} (k,s + 1),
1\leq k\leq s+1, s\in N,
$$
we finally find the equality (\ref{e210}).
The equality (\ref{eq1111}) is proved similarly as proof of the equality (\ref{eq1222}). Q.E.D.
\bigskip
\textbf{Lemma 2}. Let $a,b_{1} ,$\ldots , $b_{n} $ are real numbers with $a = 0,\,
- 1,\, - 2,...$ and $a > b_{1} + ... + b_{n}.$ Then the following summation
formulas hold true at $n \in {\rm N}$
$$
{\sum\limits_{{\mathop {m_{i,j} = 0}\limits_{(2 \le i \le j \le n)}
}}^{\infty} {{\displaystyle\frac{{(a)_{A (n,n)}}} {{m_{ij}!} }}}} {\prod\limits_{k = 1}^{n}\left[ {{\frac{{\left( {b_{k}} \right)_{B(k,n)}
\left( {a - b_{k}} \right)_{A(k,n) - B(k,n)}}} {{\left( {a}
\right)_{A(k,n)}}} }}\right]}
$$
\begin{equation}
\label{eq888}
=\frac{\Gamma \left( {a - {\sum\nolimits_{k = 1}^{n} {b_{k}
}}} \right)}{\Gamma(a)}\prod\limits_{k = 1}^{n}\left[\frac{{\Gamma(a)}}{{{{\Gamma \left( {a - b_{k}} \right)}}} }\right],
\end{equation}
$$
{\sum\limits_{{\mathop {m_{i,j} = 0}\limits_{(2 \le i \le j \le n)}
}}^{\infty} {{\displaystyle\frac{{\left( { - 1} \right)^{A\left( {n,n}
\right)}}}{{\left( {a} \right)_{2A\left( {n,n} \right)} m_{ij} !}}}}
}{\prod\limits_{k = 1}^{n} {{\left[ {{\frac{{\left( {b_{k}}
\right)_{B\left( {k,n} \right)} \left( {a} \right)_{2A\left( {k,n} \right)}
\left( {a - 1} \right)_{A(k,n) - A(k - 1,n)}}} {{\left( {c - b_{k}}
\right)_{2A\left( {k,n} \right) - B\left( {k,n} \right)} \left( {a - 1}
\right)_{2A(k,n) - 2A(k - 1,n)}}} }} \right]}}} \\
$$
\begin{equation}
\label{eq8888}
= {\displaystyle\frac{{\Gamma \left( {a} \right)}}{{\Gamma \left( {a - {\sum\nolimits_{k
= 1}^{n} {b_{k}}} } \right)}}}{\prod\limits_{k = 1}^{n} {{\left[
{{\displaystyle\frac{{\Gamma \left( {a - b_{k}} \right)}}{{\Gamma \left( {a} \right)}}}}
\right]}}}.
\end{equation}
\bigskip
\textbf{Proof}. We carry out the proof by the method mathematical
induction. First, we prove the validity of the equality (\ref{eq888}).
In the case $n=1$ the equality (\ref{eq888}) is obvious.
Let $n = 2$. Since $A(1,2)=A(2,2)=B(1,2)=B(2,2)=m_{2,2}:= i,$
we obtain well-known summation formula (\ref{eq25}):
\begin{equation*}
\label{eq9}
{\sum\limits_{m_{22} = 0}^{\infty}
{{\frac{{\left( {b_{1}} \right)_{i}} \left( {b_{1}} \right)_{i}
}{{\left( {c} \right)_{i} i!}}}}}:=F\left( {b_{1} ,b_{2} ;a;1} \right)= {\frac{{\Gamma \left( {a -
b_{1} - b_{2}} \right)\Gamma \left( {a} \right)}}{{\Gamma \left( {a - b_{1}
} \right)\Gamma \left( {a - b_{2}} \right)}}}.
\end{equation*}
So the formula (\ref{eq888}) works for $n=1$ and $n=2$.
Now we denote the left side of the formula (\ref{eq888}) by
$$T_{n} \left( {a,b_{1} ,...,b_{n}} \right): =
{\sum\limits_{{\mathop {m_{i,j} = 0}\limits_{(2 \le i \le j \le n)}
}}^{\infty} {{\displaystyle\frac{{\left( {a} \right)_{A(n,n)}}} {{m_{ij}
!}}}}} {\prod\limits_{k = 1}^{n} {{\displaystyle\frac{{\left( {b_{k}} \right)_{B(k,n)}
\left( {a - b_{k}} \right)_{A(k,n) - B(k,n)}}} {{\left( {a}
\right)_{A(k,n)}}} }}} $$
\noindent
and considering fair equality
$$T_{n} \left( {a,b_{1} ,...,b_{n}} \right) = \Gamma \left( {a -
{\sum\limits_{k = 1}^{n} {b_{k}}} } \right){\frac{{\Gamma ^{n - 1}\left(
{a} \right)}}{{{\prod\limits_{k = 1}^{n} {\Gamma \left( {a - b_{k}}
\right)}}} }},$$
\noindent
we will prove that
\begin{equation}
\label{eq1001}
T_{n + 1} \left( {a,b_{1} ,...,b_{n + 1}} \right) = \Gamma \left( {a -
{\sum\limits_{k = 1}^{n + 1} {b_{k}}} } \right){\frac{{\Gamma ^{n}\left(
{a} \right)}}{{{\prod\limits_{k = 1}^{n + 1} {\Gamma \left( {a - b_{k}}
\right)}}} }}.
\end{equation}
For this aim we will put
\[
T_{n + 1} \left( {a,b_{1} ,...,b_{n + 1}} \right) =
{\sum\limits_{{\mathop {m_{i,j} = 0}\limits_{(2 \le i \le j \le n+1)}
}}^{\infty} {{\frac{{\left( {a} \right)_{A(n + 1,n + 1)}
}}{{m_{ij} !}}}}} {\prod\limits_{k = 1}^{n + 1} {{\frac{{\left( {b_{k}}
\right)_{B(k,n + 1)} \left( {a - b_{k}} \right)_{A(k,n + 1) - B(k,n + 1)}
}}{{\left( {a} \right)_{A(k,n + 1)}}} }}}
\]
and show the validity of the recurrence relation
\begin{equation}
\label{eq11}
T_{n + 1} (a,b_{1} ,...,b_{n + 1} ) = {\prod\limits_{k = 1}^{n} {{\left[
{{\frac{{\Gamma \left( {a} \right)\Gamma \left( {a - b_{k} - b_{n + 1}}
\right)}}{{\Gamma \left( {a - b_{n + 1}} \right)\Gamma \left( {a - b_{k}}
\right)}}}} \right]}}} \,T_{n} (a - b_{n + 1} ,b_{1} ,...,b_{n} ).
\end{equation}
This process consists of $n$ steps. A detailed look at the first step.
By virtue of the equalities
\[
{\sum\limits_{{\mathop {m_{i,j} = 0}\limits_{(2 \le i \le j \le n+1)}
}}^{\infty} {}}f(...) = {\sum\limits_{{\mathop {m_{i,j} = 0}\limits_{(2 \le i \le j \le n)}
}}^{\infty} {{\sum\limits_{{\mathop {m_{i,n+1} = 0}\limits_{(2 \le i \le n+1)}
}}^{\infty} {}}} }f(...) = {\sum\limits_{{\mathop {m_{i,j} = 0}\limits_{(2 \le i \le j \le n)}
}}^{\infty} {{\sum\limits_{{\mathop {m_{i,n+1} = 0}\limits_{(2 \le i \le j \le n)}
}}^{\infty} }}\sum\limits_{m_{n+1,n+1} = 0}
^{\infty}f(...) }
\]
\noindent
and the properties of functions $A\left( {k,n} \right)$ and $B\left( {k,n}
\right)$ (see formulas (\ref{eq15555}) and (\ref{eq16666})), the right side of equality
$$T_{n + 1} \left( {a,b_{1} ,...,b_{n + 1}} \right) =
{\sum\limits_{{\mathop {m_{i,j} = 0}\limits_{(2 \le i \le j \le n+1)}
}}^{\infty} {{\frac{{\left( {a} \right)_{A(n + 1,n + 1)}
}}{{m_{ij} !}}}}} {\prod\limits_{k = 1}^{n + 1} {{\frac{{\left( {b_{k}}
\right)_{B(k,n + 1)} \left( {a - b_{k}} \right)_{A(k,n + 1) - B(k,n + 1)}
}}{{\left( {a} \right)_{A(k,n + 1)}}} }}} $$
\noindent
it is easy to convert to the form
$$
\begin{array}{l}
T_{n + 1} \left( {a,b_{1} ,...,b_{n + 1}} \right) =
{\sum\limits_{{\mathop {m_{i,j} = 0}\limits_{(2 \le i \le j \le n)}
}}^{\infty} {{\displaystyle\frac{{\left( {a - b_{n + 1}} \right)_{A(n,n)}
\left( {b_{n}} \right)_{B(n,n)}}} {{m_{ij} !}}}}} \\
\\
\cdot{\sum\limits_{{\mathop {m_{i,n+1} = 0}\limits_{(2 \le i \le n)}
}}^{\infty} {{\displaystyle\frac{{\left( {b_{n + 1}} \right)_{m_{2,n + 1} +
... + m_{n,n + 1}} \left( {a - b_{n}} \right)_{A(n,n) - B(n,n) + m_{2,n +
1} + ... + m_{n,n + 1}}} } {{m_{i,n + 1} !\left( {a} \right)_{A(n,n) +
m_{2,n + 1} + ... + m_{n,n + 1}}} } }}} \\
\\
\cdot {\prod\limits_{k = 1}^{n - 1} {{\left[ {{\displaystyle\frac{{\left( {b_{k}}
\right)_{B(k,n) + m_{k + 1,n + 1}} \left( {a - b_{k}} \right)_{A(k,n) -
B(k,n) + m_{2,n + 1} + ... + m_{k,n + 1}}} } {{\left( {a} \right)_{A(k,n) +
m_{2,n + 1} + ... + m_{k + 1,n + 1}}} } }S(k,n)} \right]}}} , \\
\end{array}
$$
\noindent
where
$$
S(k,n) = {\sum\limits_{m_{n + 1,n + 1} = 0}^{\infty} {{\frac{{\left( {b_{n}
+ B(n,n)} \right)_{m_{n + 1,n + 1}} \left( {b_{n + 1} + m_{2,n + 1} + ... +
m_{n,n + 1}} \right)_{m_{n + 1,n + 1}}} } {{m_{n + 1,n + 1} !\left( {a +
A(n,n) + m_{2,n + 1} + ... + m_{n,n + 1}} \right)_{m_{n + 1,n + 1}}} } }}
}.
$$
It is easy to notice that
$$
S(k,n) = F\left[ b_{n} + B(n,n),b_{n + 1} + m_{2,n + 1} + ... + m_{n,n + 1}
;\right.$$
$$
\left. a + A(n,n) + m_{2,n + 1} + ... + m_{n,n + 1} ;1 \right].
$$
Applying now the summation formula (\ref{sum}) to the last equality after
elementary transformations we get
$$
T_{n + 1}^{(1)} \left( {a,b_{1} ,...,b_{n + 1}} \right) = {\frac{{\Gamma
\left( {a - b_{n} - b_{n + 1}} \right)\Gamma \left( {a} \right)}}{{\Gamma
\left( {a - b_{n}} \right)\Gamma \left( {a - b_{n + 1}}
\right)}}}
$$
$$
\cdot{\sum\limits_{{\mathop {m_{i,j} = 0}\limits_{(2 \le i \le j \le n+1)}
}}^{\infty} {{\frac{{\left( {b_{n}} \right)_{B(n,n)} \left( {a
- b_{n} - b_{n + 1}} \right)_{A(n,n) - B(n,n)}}} {{m_{ij} !}}}}} {\sum\limits_{{\mathop {m_{i,n+1} = 0}\limits_{(2 \le i \le j \le n)}
}}^{\infty} {{\frac{{\left( {b_{n + 1}} \right)_{m_{2,n + 1} +
... + m_{n,n + 1}}} } {{m_{i,n + 1} !}}}}}
$$
$$
\cdot {\prod\limits_{k = 1}^{n - 1}
{{\displaystyle\frac{{\left( {b_{k}} \right)_{B(k,n) + m_{k + 1,n + 1}} \left( {a -
b_{k}} \right)_{A(k,n) - B(k,n) + m_{2,n + 1} + ... + m_{k,n + 1}}
}}{{\left( {a} \right)_{A(k,n) + m_{2,n + 1} + ... + m_{k + 1,n + 1}}} } }}
}.
$$
For definiteness, we denoted the result of the first step of the process
under consideration by $T_{n + 1}^{(1)} \left( {a,b_{1} ,...,b_{n + 1}}
\right)$. We continue the process of proving the recurrence relation (\ref{eq11}).
In each next step, having consistently repeated the reasoning carried out in
the first step, we get
$$
T_{n + 1}^{(s)} \left( {a,b_{1} ,...,b_{n + 1}} \right) = {\frac{{\Gamma
^{s}\left( {a} \right)}}{{\Gamma ^{s}\left( {a - b_{n + 1}}
\right)}}}{\prod\limits_{k = n - s + 1}^{n} {{\frac{{\Gamma \left( {a -
b_{k} - b_{n + 1}} \right)}}{{\Gamma \left( {a - b_{k}} \right)}}}}}
$$
$$
\cdot {\sum\limits_{{\mathop {m_{i,j} = 0}\limits_{(2 \le i \le j \le n)}
}}^{\infty} {{\frac{{1}}{{m_{ij} !}}}{\prod\limits_{k = n - s +
1}^{n} {{\left[ {{\frac{{\left( {b_{k}} \right)_{B(k,n)} \left( {a - b_{k}
- b_{n + 1}} \right)_{A(k,n) - B(k,n)}}} {{\left( {a - b_{n + 1}}
\right)_{A(k,n)}}} }} \right]}}}} }
$$
$$
\cdot {\sum\limits_{{\mathop {m_{i,n+1} = 0}\limits_{(2 \le i \le n-s+1)}
}}^{\infty} {}} {\frac{{\left( {a - b_{n + 1}} \right)_{N(n,n)}
\left( {b_{n + 1}} \right)_{m_{2,n + 1} + ... + m_{n - s + 1,n + 1}}
}}{{m_{ij} !}}}
$$
$$
\cdot {\prod\limits_{k = 1}^{n - s} {{\left[ {{\frac{{\left( {b_{k}}
\right)_{B(k,n) + m_{k + 1,n + 1}} \left( {a - b_{k}} \right)_{A(k,n) -
B(k,n) + m_{2,n + 1} + ... + m_{k,n + 1}}} } {{\left( {a} \right)_{A(k,n) +
m_{2,n + 1} + ... + m_{k + 1,n + 1}}} } }} \right]}}}
$$
\noindent
and in the last step
$$
T_{n + 1}^{(n)} \left( {a,b_{1} ,...,b_{n + 1}} \right) = {\frac{{\Gamma
^{n}\left( {a} \right)}}{{\Gamma ^{n}\left( {a - b_{n + 1}}
\right)}}}{\prod\limits_{k = 1}^{n} {{\left[ {{\frac{{\Gamma \left( {a -
b_{n + 1} - b_{k}} \right)}}{{\Gamma \left( {a - b_{k}} \right)}}}}
\right]}}}
$$
$$
\cdot {\sum\limits_{{\mathop {m_{i,j} = 0}\limits_{(2 \le i \le j \le n)}
}}^{\infty} {{\frac{{\left( {a - b_{n + 1}} \right)_{A(n,n)}
}}{{m_{ij} !}}}}} {\prod\limits_{k = 1}^{n} {{\left[ {{\frac{{\left( {b_{k}
} \right)_{B(k,n)} \left( {a - b_{n + 1} - b_{k}} \right)_{A(k,n) - B(k,n)}
}}{{\left( {a - b_{n + 1}} \right)_{A(k,n)}}} }} \right]}}} , \\
$$
\noindent
that is
$$
T_{n + 1}^{(n)} \left( {a,b_{1} ,...,b_{n + 1}} \right) = {\frac{{\Gamma
^{n}\left( {a} \right)}}{{\Gamma ^{n}\left( {a - b_{n + 1}}
\right)}}}{\prod\limits_{k = 1}^{n} {{\left[ {{\frac{{\Gamma \left( {a -
b_{n + 1} - b_{k}} \right)}}{{\Gamma \left( {a - b_{k}} \right)}}}}
\right]}}} T_{n} \left( {a - b_{n + 1} ,b_{1} ,...,b_{n}} \right).
$$
Thus, the validity of the ratio (\ref{eq11}) is established. By the induction
hypothesis, from the (\ref{eq11}) follows the equality
$$
T_{n} \left( {a - b_{n + 1} ,b_{1} ,...,b_{n}} \right) = \Gamma \left( {a -
b_{n + 1} - {\sum\limits_{k = 1}^{n} {b_{k}}} } \right){\frac{{\Gamma ^{n -
1}\left( {a - b_{n + 1}} \right)}}{{{\prod\limits_{k = 1}^{n} {\Gamma
\left( {a - b_{n + 1} - b_{k}} \right)}}} }}.
$$
Substituting the last expression in (\ref{eq11}) we get the equality (\ref{eq1001}). Therefore, the equality (\ref{eq888}) is true.
The equality (\ref{eq8888}) is proved similarly as proof of the equality (\ref{eq888}). Q.E.D.
\bigskip
\textbf{Lemma 3}. The following equalities
$$
{\mathop{\lim} \limits_{\mathop{z_{k} \to 0,}\limits_{k =1,...,n}}}\left\{ z_{1}^{ - b_{1}} ...z_{n}^{ - b_{n}} F_{A}^{(n)} \left(
{a,b_{1} ,...,b_{n} ;c_{1} ,...,c_{n} ;1 - {\frac{{1}}{{z_{1}}} },...,1 -
{\frac{{1}}{{z_{n}}} }} \right)\right\}
$$
\begin{equation}
\label{eq12222}
= {\frac{{\Gamma \left( {a - {\sum\nolimits_{k = 1}^{n}
{b_{k}}} } \right)}}{{\Gamma (a)}}}{\prod\limits_{k = 1}^{n} \left[{{\frac{{\Gamma \left( {c_{k}}
\right)}}{{\Gamma \left( {c_{k} - b_{k}} \right)}}}}\right]}, a > \sum\limits_{k = 1}^{n}{b_{k}}, b_{k}\neq c_{k}, k=\overline{1,n};
\end{equation}
\noindent
$$
{\mathop{\lim} \limits_{\mathop{z_{k} \to 0,}\limits_{k =1,...,n}}}\left\{z_{1}^{ - b_{1}} ...z_{n}^{ - b_{n}} F_{B}^{(n)} \left( {a_{1}
,...,a_{n} ;b_{1} ,...,b_{n} ;c;1 - {\frac{{1}}{{z_{1}}} },...,1 -
{\frac{{1}}{{z_{n}}} }} \right)\right\}
$$
\begin{equation}
\label{eq12111}
= {\frac{{\Gamma \left( {c}
\right)}}{{\Gamma \left( {c - {\sum\nolimits_{k = 1}^{n} {b_{k}}} }
\right)}}}{\prod\limits_{k = 1}^{n}\left[ {{\frac{{\Gamma \left( {a_{k} - b_{k}}
\right)}}{{\Gamma \left( {a_{k}} \right)}}}}\right]}, c > \sum\limits_{k = 1}^{n}{b_{k}}, a_{k} \neq b_{k}, k=\overline{1,n}
\end{equation}
are valid.
\textbf{Proof}. By virtue of the decomposition formula (\ref{eq1222}) we obtain
$$
F_{A}^{(n)} \left( {a,b_{1} ,...,b_{n} ;c_{1} ,...,c_{n} ;1 -
{\frac{{1}}{{z_{1}}} },...,1 - {\frac{{1}}{{z_{n}}} }} \right) =
{\sum\limits_{{\mathop {m_{i,j} = 0}\limits_{(2 \le i \le j \le n)}
}}^{\infty} {{\frac{{(a)_{A(n,n)}}} {{m_{ij}!} }}}}
$$
\begin{equation}
\label{eq1333}
\cdot {\prod\limits_{k = 1}^{n}\left[ {{\frac{{(b_{k} )_{B(k,n)}}} {{(c_{k}
)_{B(k,n)}}} }\left( {1 - {\frac{{1}}{{z_{k}}} }} \right)^{B(k,n)}F\left( {a
+ A(k,n),b_{k} + B(k,n);c_{k} + B(k,n);1 - {\frac{{1}}{{z_{k}}} }} \right)}\right]
}.
\end{equation}
Applying now the familiar autotransformation formula (\ref{auto}) to each hypergeometric function included in the sum (\ref{eq1333}), we get
\[
F_{A}^{(n)} \left( {a,b_{1} ,...,b_{n} ;c_{1} ,...,c_{n} ;1 -
{\frac{{1}}{{z_{1}}} },...,1 - {\frac{{1}}{{z_{n}}} }} \right) =
z_{1}^{b_{1}} ...z_{n}^{b_{n}} {\sum\limits_{{\mathop {m_{i,j} =
0}\limits_{(2 \le i \le j \le n)}}} ^{\infty} {{\frac{{(a)_{A(n,n)}
}}{{m_{ij}!} }}}}
\]
\[
\cdot \prod\limits_{k = 1}^{n} \left[{{\frac{{(b_{k} )_{B(k,n)}}} {{(c_{k}
)_{B(k,n)}}} }\left( {z_{k} - 1} \right)^{B(k,n)}F {\left(
{{\begin{array}{*{20}c}
{c_{k} - a + B(k,n)
- A(k,n),b_{k} + B(k,n)} ; \hfill \\
{c_{k} + B(k,n)}; \hfill \\
\end{array}} 1 - z_{k}} \right)}}\right].
\]
Using the parity property of the sum ${ {B(2,n)+B(3,n)+...+B(n,n)}} $ (see formula (\ref{eq1444})), we calculate the limit
$$
{\mathop{\lim} \limits_{\mathop{z_{k} \to 0,}\limits_{k =1,...,n}}} z_{1}^{ - b_{1}} ...z_{n}^{ - b_{n}} F_{A}^{(n)} \left(
{a,b_{1} ,...,b_{n} ;c_{1} ,...,c_{n} ;1 - {\frac{{1}}{{z_{1}}} },...,1 -
{\frac{{1}}{{z_{n}}} }} \right)
$$
$$
= {\sum\limits_{{\mathop {m_{i,j} =
0}\limits_{(2 \le i \le j \le n)}}} ^{\infty} {{\frac{{(a)_{A(n,n)}
}}{{m_{ij}!} }}}} \prod\limits_{k = 1}^{n} \left[{{\frac{{(b_{k} )_{B(k,n)}}} {{(c_{k}
)_{B(k,n)}}} }F {\left(
{{\begin{array}{*{20}c}
{c_{k} - a + B(k,n)
- A(k,n),b_{k} + B(k,n)} ; \hfill \\
{c_{k} + B(k,n)}; \hfill \\
\end{array}} 1} \right)}}\right]
$$
\noindent
and applying the summation formula (\ref{sum}) to the Gauss hypergeometric functions in the last sum, we obtain the equality (\ref{eq12222}).
The equality (\ref{eq12111}) is proved similarly as proof of the equality (\ref{eq12222}). Q.E.D.
\section{Applications of new decomposition formulas to the solution of the boundary value problems}
We consider the equation
\begin{equation}
\label{eq2222}
{\sum\limits_{i = 1}^{m} {u_{x_{i} x_{i}}} } + {\sum\limits_{k = 1}^{n}
{{\frac{{2\alpha _{k}}} {{x_{k}}} }u_{x_{k}}} } = 0,
\end{equation}
where $m \ge 2,0 < n \le m;\, \alpha _{k} $ are constants with $\,0 <
2\alpha _{k} < 1 \,$ $\left(k=\overline{1,n}\right)$ in the domain $\Omega$ defined by
$$\Omega\subset {\rm{R}}_m^{n+}:=\{(x_1,...,x_m): x_1>0,...,x_n>0\}.$$ We aim at investigating a Holmgren problem for the equation (\ref{eq2222}).
Let $\Omega \subset {\rm R}_{m}^{n +} $ be a finite simple-connected domain
bounded by planes $x_{1} = 0,...,x_{n} = 0$ and by the $1/2^n$ part of the $m -
$dimensional sphere $S:$ $x_1^2+...+x_m^2=a^2$. We introduce the notation:
\[
\,\tilde {x}_{k} : = \left( {x_{1} ,...,x_{k - 1} ,x_{k + 1} ,...,x_{n}
,...,x_{m}} \right) \in S_{k} \subset {\rm R}_{m - 1}^{(n - 1) +} \subset
{\rm R}^{m - 1} \,\left(k=\overline{1,n}\right).
\]
\textbf{Holmgren problem.} To find a function $u\left( {{x}} \right) \in
C\left( {\bar {\Omega}} \right) \cap C^{2}\left( {\Omega} \right)$,
satisfying equation (\ref{eq2222}) in $\Omega $ and conditions
\begin{equation}
\label{eq17}
{\left. {\left( {x_{k}^{2\alpha _{k}} {\frac{{\partial u}}{{\partial x_{k}
}}}} \right)} \right|}_{x_{k} = 0} = \nu _{k} \left( {\tilde {x}_{k}}
\right),
\,
\,\tilde {x}_{k} \in S_{k} \,\left(k=\overline{1,n}\right),
\end{equation}
\begin{equation}
\label{eq18}
{\left. {u} \right|}_{S} = \varphi \left( {{ x}} \right),
\quad
\,{x} \in \bar {S},
\end{equation}
where $\nu _{k} \left( {\tilde {x}_{k}} \right)$ and $\varphi \left( {{ x}}
\right)$ are given functions, and, moreover, $\nu _{k} \left( {\tilde
{x}_{k}} \right)$ can reduce to an infinity of the order less than $1 -2\alpha_1-...-
2\alpha _{n} $ on the boundaries of $S_{k}\,$ $\left(k=\overline{1,n}\right)$.
We find a solution of considered problem using Green's functions method \cite{A27}.
The Green's function can be represented as
\begin{equation}
\label{eq2144}
G_0\left( {{ x};{ \xi}} \right) = q_{0} \left( {{ x};{ \xi}}
\right) + q_{0}^{ *} \left( {{x};{ \xi}} \right),
\end{equation}
\noindent
where $q_{0} \left( {{x};{\xi}} \right)$ is the fundamental
solution of equation (\ref{eq2222}), defined by \cite{A288}
\begin{equation*}
q_{0} \left( {{x};{\xi}} \right) = \gamma _{0} \,r^{ -
2\alpha _{0}} F_{A}^{\left( {n} \right)} \left( {\alpha _{0} ,\alpha _{1}
,...,\alpha _{n} ;2\alpha _{1} ,...,2\alpha _{n} ;\sigma} \right),
\end{equation*}
where
\[
{x}: = \left( {x_{1} ,...,x_{m}} \right),
{\xi} : = \left( {\xi _{1} ,...,\xi _{m}} \right),
\,
{\sigma} : = \left( {\sigma _{1} ,...,\sigma _{n}} \right);
\]
\begin{equation}
\label{eq16667}
\alpha _{0} = {\frac{{m - 2}}{{2}}} + \alpha _{1} + ... + \alpha _{n} ;\,\,\gamma _{0} = 2^{2\alpha _{0} - m}{\frac{{\Gamma \left( {\alpha _{0}}
\right)}}{{\pi ^{m / 2}}}}{\prod\limits_{k = 1}^{n} {{\frac{{\Gamma \left(
{\alpha _{k}} \right)}}{{\Gamma \left( {2\alpha _{k}} \right)}}}}} ,
\end{equation}
\[
r^{2} = {\sum\limits_{i = 1}^{m} {\left( {x_{i} - \xi _{i}} \right)^{2}}} ,
\,r_{k}^{2} = \left( {x_{k} + \xi _{k}} \right)^{2} + {\sum\limits_{i = 1,i
\ne k}^{m} {\left( {x_{i} - \xi _{i}} \right)^{2}}} ,\,
\sigma _{k} = 1 - {\frac{{r_{k}^{2}}} {{r^{2}}}} \, \left(k=\overline{1,n}\right),
\]
a function
\[
q_{0}^{ *} \left( {{x};{\xi}} \right) = - \left(
{{\frac{{a}}{{R_{0}}} }} \right)^{2{\alpha}_{0}} q_{0} \left( {{
x};{\bar {\xi}} } \right)
\]
is a regular solution of equation (\ref{eq2222}) in the domain $\Omega $. Here
\[
{ \bar {\xi}} : = \left( {\bar {\xi} _{1} ,...,\bar {\xi} _{m}} \right),
\bar {\xi} _{i} = {\frac{{a^{2}}}{{R_{0}^{2}}} }\xi _{i} \, \left(i=\overline{1,m}\right);
R_{0}^{2} = \xi _{1}^{2} + ... + \xi _{m}^{2} .
\]
Excise a small ball with its center at ${ \xi} $ and with radius $\rho >
0$ from the domain $\Omega $. Designate the sphere of the excised ball as
$C_{\rho} $ and by $\Omega _{\rho} $ denote the remaining part of $\Omega
$.
In deriving an explicit formula for solving the Holmgren problem, the calculation of the following integral plays an important role:
\[
{\int_{C_{\rho}} {{ x}^{\left( {2\alpha} \right)}{\left[ {u\left(
{{x}} \right){\frac{{\partial G_0\left( {{x};{\xi}}
\right)}}{{\partial {{\bf n}}}}} - G_0\left( {{ x};{ \xi}}
\right){\frac{{\partial u\left( {{ x}} \right)}}{{\partial {{\bf
n}}}}}} \right]}dC_{\rho}} }
\]
\begin{equation}
\label{eq22}
= - {\sum\limits_{k = 1}^{n} {{\int_{S_{k}} {G_0^{\ast}\left(\tilde{x_k}\right)\nu _{k} \left( {\tilde {x}_{k}}
\right)dS_{k}}} \,}} + {\int_{S} {{x}^{\left( {2\alpha}
\right)}{\frac{{\partial G_0\left( {{x};{\xi}}
\right)}}{{\partial { {\bf n}}}}}\varphi \left( {\vartheta}
\right)d\vartheta}}
\end{equation}
\noindent
where
\[
{x}^{\left( {2\alpha} \right)}: = x_{1}^{2\alpha _{1}}
...x_{n}^{2\alpha _{n}} ,
\,\tilde {x}_{k}^{\left( {2\alpha} \right)} : = x_{1}^{2\alpha _{1}} ...x_{k
- 1}^{2\alpha _{k - 1}} x_{k + 1}^{2\alpha _{k + 1}} ...x_{n}^{2\alpha
_{n}},
\]
\[
G_0^{\ast}\left(\tilde{x}_k\right):=\tilde {x}_{k}^{\left(
{2\alpha} \right)} G_0 \left( {x_{1} ,...,x_{k - 1} ,0,x_{k + 1} ,...,x_{m}
;{ \xi}} \right) \, \left(k=\overline{1,n}\right),
\]
${\bf n}$ is outer normal to $\partial \Omega.$
Since we want to show the application of Lemmas 1-3, therefore, without giving in to details, we discuss only the computation of the following integral
$$
I_{11} = 2\alpha _{0} \gamma _{0} \,\rho ^{ - 2\alpha _{1} - ... - 2\alpha
_{n}} {\int\limits_{0}^{2\pi} {d\varphi _{m - 1}}} {\int\limits_{0}^{\pi}
{\sin \varphi _{m - 2} d\varphi _{m - 2}}}...
$$
\begin{equation}
\label{l1111} ...{\int\limits_{0}^{\pi} {
{u\left(\xi_1+\rho\Phi_1,...,\xi_m+\rho\Phi_m\right)}\prod\limits_{i=1}^n\left[\left(\xi_i+\rho\Phi_i\right)^{2\alpha_i}\right]\,F_{A}^{\left( {n} \right)} \left[ {{\sigma}(\rho)} \right]\sin ^{m
- 2}\varphi _{1} d\varphi _{1}}} ,
\end{equation}
\noindent
where
\[
\Phi_1=\cos \varphi _{1},\,\Phi_2=\sin\varphi _{1} \cos \varphi _{2},\,\Phi_3=\sin \varphi _{1} \sin \varphi _{2} \cos
\varphi _{3},...,\,\]
\[\Phi_{m-1}=\sin \varphi _{1} \sin \varphi _{2} ...\sin
\varphi _{m - 2} \cos \varphi _{m - 1} ,\,\Phi_m=\sin \varphi _{1} \sin \varphi
_{2} ...\sin \varphi _{m - 2} \sin \varphi _{m - 1};
\]
\[
F_{A}^{\left( {n} \right)} \left( {\sigma _{1\rho} ,...,\sigma _{n\rho}} \right): =
F_{A}^{\left( {n} \right)} \left( \alpha _{0} + 1,\alpha _{1} ,...,\alpha
_{n} ;2\alpha _{1} ,...,2\alpha _{n} ;{\sigma _{1\rho} ,...,\sigma _{n\rho}} \right);
\]
\[
r_{k}^{2} = \left( {x_{k} + \xi _{k}} \right)^{2} + {\sum\limits_{i = 1,i
\ne k}^{m} {\left( {x_{i} - \xi _{i}} \right)^{2}}} ,
\,
\sigma _{k\rho} = 1 - {\frac{{r_{k\rho} ^{2}}} {{\rho ^{2}}}} \,\left(k=\overline{1,n}\right).
\]
First we evaluate $F_{A}^{\left( {n} \right)} \left( {\sigma _{1\rho} ,...,\sigma _{n\rho}}
\right)$. For this aim we use decomposition formula (\ref{eq1222}) and then
formula (\ref{auto}):
\[
F_{A}^{\left( {n} \right)} \left( {\sigma _{1\rho} ,...,\sigma _{n\rho}} \right) =
{\sum\limits_{{\mathop {m_{i,j} = 0}\limits_{(2 \le i \le j \le n)}
}}^{\infty} {{\frac{{(\alpha _{0} + 1)_{A(n,n)}}} {{m_{ij} !}}}}
}{\prod\limits_{k = 1}^{n} {{\left[ {{\frac{{(\alpha _{k} )_{B(k,n)}
}}{{(2\alpha _{k} )_{B(k,n)}}} }\left( {1 - {\frac{{r_{k\rho} ^{2}}} {{\rho
^{2}}}}} \right)^{B(k,n)}\left( {{\frac{{r_{k\rho} ^{2}}} {{\rho ^{2}}}}}
\right)^{ - \alpha _{k} - B(k,n)}} \right]}}}
\]
\[
\times {\prod\limits_{k = 1}^{n} {{\left[ {F\left( {2\alpha _{k} - \alpha
_{0} - 1 + B(k,n) - A(k,n),\alpha _{k} + B(k,n);2\alpha _{k} + B(k,n);1 -
{\frac{{r_{k\rho} ^{2}}} {{\rho ^{2}}}}} \right)} \right]}}} ,
\]
\noindent
where $A(k,n)$ and $B(k,n)$ are expressions defined in (\ref{e4111}).
After the elementary evaluations we find
\[
F_{A}^{\left( {n} \right)} \left( {\sigma _{1\rho} ,...,\sigma _{n\rho}}\right) = \rho
^{2\alpha _{1} + ... + 2\alpha _{n}} {\prod\limits_{k = 1}^{n} {{\left[
{r_{k\rho} ^{ - 2\alpha _{k}}} \right]}}} \cdot \aleph ,
\]
\noindent
where
\[
\aleph : = {\sum\limits_{{\mathop {m_{i,j} = 0}\limits_{(2 \le i \le j \le
n)}}} ^{\infty} {{\frac{{(\alpha _{0} + 1)_{A(n,n)}}} {{m_{ij} !}}}}
}{\prod\limits_{k = 1}^{n} {{\left[ {{\frac{{(\alpha _{k} )_{B(k,n)}
}}{{(2\alpha _{k} )_{B(k,n)}}} }\left( {{\frac{{\rho ^{2}}}{{r_{k\rho} ^{2}
}}} - 1} \right)^{B(k,n)}} \right]}}}
\]
\[
\times {\prod\limits_{k = 1}^{n} {{\left[ {F\left( {2\alpha _{k} - \alpha
_{0} - 1 + B(k,n) - A(k,n),\alpha _{k} + B(k,n);2\alpha _{k} + B(k,n);1 -
{\frac{{\rho ^{2}}}{{r_{k\rho} ^{2}}} }} \right)} \right]}}} .
\]
It is easy to see that when $\rho \to 0$ the function $\aleph\, $ becomes an
expression that does not depend on ${\rm x}$ and ${\rm \xi} $. Indeed,
taking into account
the parity property of the sum ${ {B(2,n)+B(3,n)+...+B(n,n)}} $ (see formula (\ref{eq1444})), we have
\[
{\mathop {\lim} \limits_{\rho \to 0}} \aleph : = {\sum\limits_{{\mathop
{m_{i,j} = 0}\limits_{(2 \le i \le j \le n)}}} ^{\infty} {{\frac{{(\alpha
_{0} + 1)_{A(n,n)}}} {{m_{ij} !}}}}} {\prod\limits_{k = 1}^{n} {{\left[
{{\frac{{(\alpha _{k} )_{B(k,n)}}} {{(2\alpha _{k} )_{B(k,n)}}} }} \right]}}
}
\]
\begin{equation}
\label{sum1}
\times {\prod\limits_{k = 1}^{n} {{\left[ {F\left( {2\alpha _{k} - \alpha
_{0} - 1 + B(k,n) - A(k,n),\alpha _{k} + B(k,n);2\alpha _{k} + B(k,n);1}
\right)} \right]}}} .
\end{equation}
Applying now the summation formula (\ref{sum}) to each hypergeometric function
$F\left( {a,b;c;1} \right)$ in the sum (\ref{sum1}), we get
\[
{\mathop {\lim} \limits_{\rho \to 0}} \aleph : = {\frac{{1}}{{\Gamma \left(
{\alpha _{0} + 1} \right)}}}{\sum\limits_{{\mathop {m_{i,j} = 0}\limits_{(2
\le i \le j \le n)}}} ^{\infty} {{\frac{{\Gamma (\alpha _{0} + 1 +
N(n,n))}}{{m_{ij} !}}}}}
\]
\[
\cdot {\prod\limits_{k = 1}^{n} {{\left[ {{\frac{{\Gamma \left( {2\alpha
_{k}} \right)\Gamma (\alpha _{k} + M(k,n))\Gamma \left( {\alpha _{0} + 1 -
\alpha _{k} + N(k,n) - M(k,n)} \right)}}{{\Gamma ^{2}\left( {\alpha _{k}}
\right)\Gamma \left( {\alpha _{0} + 1 + N(k,n)} \right)}}}} \right]}}} .
\]
Taking into account the identity (\ref{eq888}) we obtain
\begin{equation}
\label{eq3000}
{\mathop {\lim} \limits_{\rho \to 0}} \aleph = {\frac{{\Gamma \left( {m / 2}
\right)}}{{\Gamma \left( {\alpha _{0} + 1} \right)}}}{\prod\limits_{i =
1}^{n} {{\frac{{\Gamma \left( {2\alpha _{k}} \right)}}{{\Gamma \left(
{\alpha _{k}} \right)}}}}} .
\end{equation}
Now we consider an integral
\[
L_{m} = {\int\limits_{0}^{2\pi} {d\varphi _{m - 1}}} {\int\limits_{0}^{\pi}
{\sin \varphi _{m - 2} d\varphi _{m - 2}}} {\int\limits_{0}^{\pi} {\sin ^{2}\varphi
_{m - 3} d\varphi _{m - 3}}} ...{\int\limits_{0}^{\pi} {\sin ^{m - 2}\varphi
_{1} d\varphi _{1}}},
\]
with elementary transformations it is not difficult to establish that
\begin{equation}
\label{eq3111}
L_{2m} = {\frac{{2\,\pi ^{m}}}{{(m - 1)!}}},\,\,L_{2m + 1} = {\frac{{2^{m +
1}\,\pi ^{m}}}{{(2m - 1)!!}}},\,\,\,m = 1,2,3,...
\end{equation}
If we take into account (\ref{eq16667}), (\ref{l1111}), (\ref{eq3000}) and (\ref{eq3111}) , then from (\ref{eq22}) we will have
\[
{\mathop {\lim} \limits_{\rho \to 0}} I_{11} = u\left( {\xi} \right).
\]
So we can write the solution of the Holmgren problem as follows:
\begin{equation}
\label{9999}
u\left( {\xi} \right) = - {\sum\limits_{k = 1}^{n} {{\int_{S_{k}} { G_0^{\ast} \left( {\tilde
{x}_{k};{\xi}} \right)\nu _{k} \left( {\tilde
{x}_{k}} \right)dS_{k}}} \,}}
+ {\int_{S} {x^{\left( {2\alpha}
\right)}{\frac{{\partial G_0\left( {{ x};{\xi}}
\right)}}{{\partial {\rm {\bf n}}}}}\varphi \left( {{x}} \right)dS}},
\end{equation}
where
\begin{equation*}
G_0^{\ast} \left( {\tilde
{x}_{k};{\xi}} \right)=\gamma _{0}\tilde {x}_{k}^{\left( {2\alpha} \right)}{\left\{ {{\frac{{F_{A}^{\left( {n - 1} \right)} {\left[
{{\begin{array}{*{20}c}
{{\alpha} _{0} ,\alpha _{1} ,...,\alpha _{k - 1} ,\alpha
_{k + 1} ,...,\alpha _{n} ;} \hfill \\
{2\alpha _{1} ,...,2\alpha _{k - 1} ,2\alpha _{k + 1} ,...,
2\alpha _{n} ;} \hfill \\
\end{array}} \sigma _{0}} \right]}}}{{{\left[ {\xi _{k}^{2} +
{\sum\limits_{i = 1,i \ne k}^{m} {\left( {\xi _{i} - x_{i}} \right)^{2}}}}
\right]}^{\,{\alpha} _{0}}} }}} \right.}
\end{equation*}
\begin{equation*}
\left.{ - {\frac{{F_{A}^{\left( {n - 1} \right)} {\left[
{{\begin{array}{*{20}c}
{{\alpha}_{0} ,\alpha _{1} ,...,\alpha _{k - 1} ,\alpha
_{k + 1} ,...,\alpha _{n} ;} \hfill \\
{2\alpha _{1} ,...,2\alpha _{k - 1} ,2\alpha _{k + 1} ,...,
2\alpha _{n} ;} \hfill \\
\end{array}} \bar {\sigma} _{0}} \right]}}}{{{\left[\sum\limits_{i = 1,i
\ne k}^{m} \left( a - \displaystyle\frac{x_i\xi_i}{a} \right)^{2} +
\displaystyle\frac{1}{a^2}{\sum\limits_{i = 1,i \ne k}^{m} {{\sum\limits_{j=1,j
\ne i}^{m} {x_{i}^{2} \xi _{j}^{2}}}}}-(m-2)a^{2}
\right]}^{\, {\alpha} _{0}}}}}}\right\},
\end{equation*}
$G_0\left( {{ x};{\xi}}
\right)$ is the Green's function, defined by (\ref{eq2144}).
In conclusion, we note precisely because of the decomposition formula (\ref{eq1222}), the summation formula (\ref{eq888}) and the limit value (\ref{eq12222}) that we managed to write out the solution of the Holmgren problem with conditions (\ref{eq17}) and (\ref{eq18}) for the equation (\ref{eq2222}) in an explicit form (\ref{9999}).
\begin{center}
\textbf{References}
\end{center}
{\small
\begin{enumerate}
\bibitem{AP} P.Appell and J.Kampe de Feriet,Fonctions
Hypergeometriques et Hyperspheriques; Polynomes d'Hermite,
Gauthier - Villars. Paris, 1926.
\bibitem {BN1} J.J. Barros-Neto and I.M. Gelfand, Fundamental solutions for
the Tricomi operator , Duke Math.J. 98(3),1999. 465-483.
\bibitem {BN2} J.J.Barros-Neto and I.M. Gelfand, Fundamental solutions for
the Tricomi operator II, Duke Math.J. 111(3),2001.P.561-584.
\bibitem {BN3} J.J.Barros-Neto and I.M. Gelfand, Fundamental solutions for
the Tricomi operator III, Duke Math.J. 128(1)\,2005.\,119-140.
\bibitem {BC1} J.L.Burchnall and T.W.Chaundy, Expansions of Appell's
double hypergeometric functions. The Quarterly Journal of
Mathematics, Oxford, Ser.11,1940. 249-270.
\bibitem {BC2} J.L.Burchnall and T.W.Chaundy, Expansions of Appell's
double hypergeometric functions(II). The Quarterly Journal of
Mathematics, Oxford, Ser.12,1941. 112-128.
\bibitem {Chaundy} T.W.Chaundy, Expansions of hypergeometric functions, Qiart.J.Math.Oxford Ser.13(1942) 159-171.
\bibitem {Erd} A.Erdelyi, W.Magnus, F.Oberhettinger and F.G.Tricomi,
Higher Transcendental Functions, Vol.I (New York, Toronto and
London:McGraw-Hill Book Company), 1953.
\bibitem {Erg} T.G.Ergashev, On fundamental solutions for multidimensional Helmholtz
equation with three singular coefficients. Comp.and Math. with Appl. 77(2019) 69-76.
\bibitem {A288} T.G.Ergashev, {Fundamental solutions for a class of multidimensional elliptic equations with several singular coefficients}, preprint (2018),
{http://arxiv.org/abs/1805.03826}.
\bibitem {EH} T.G.Ergashev and A.Hasanov, Fundamental solutions of the
bi-axially symmetric Helmholtz equation, Uzbek Math. J., 1, 2018.
55-64.
\bibitem {Frankl} F.I.Frankl, Selected Works in Gas Dynamics, Nauka, Moscow, 1973.
\bibitem {H} A.Hasanov, Fundamental solutions bi-axially symmetric
Helmholtz equation. Complex Variables and Elliptic Equations. Vol.
52, No.8, 2007. 673-683.
\bibitem {HK} A.Hasanov and E.T.Karimov, Fundamental solutions for a
class of three-dimensional elliptic equations with singular
coefficients. Applied Mathematic Letters, 22 (2009). 1828-1832.
\bibitem {HS6} A.Hasanov and H.M.Srivastava, Some decomposition formulas
associated with the Lauricella function $F_A^{(r)}$ and other
multiple hypergeometric functions, Applied Mathematic Letters,
19(2) (2006), 113-121.
\bibitem {HS7} A.Hasanov and H.M.Srivastava, Decomposition Formulas
Associated with the Lauricella Multivariable Hypergeometric
Functions, Computers and Mathematics with Applications, 53:7
(2007), 1119-1128.
\bibitem {Kar} E.T.Karimov, A boundary-value problem for 3-D
elliptic equation with singular coefficients. Progress in analysis
and its applications. 2010. 619-625.
\bibitem {A20} E.T.Karimov, On a boundary problem with Neumann's condition for 3D singular elliptic equations, Appl. Math. Lett., 23(2010) 517-522.
\bibitem {A30} G.Lauricella, Sulle funzione ipergeometriche a pi\`{u} variabili, Rend.Circ. Mat. Palermo, 7(1893) 111-158.
\bibitem {Loh} G.Loh\"{o}fer, Theory of an electromagnetically deviated metal sphere, I:Absorbed power, SIAM J.Appl.Nath. 49 (1989) 567-581.
\bibitem {M} R.M.Mavlyaviev, Construction of Fundamental Solutions to
B-Elliptic Equations with Minor Terms. Russian Mathematics, 2017,
Vol.61, No.6, 60-65. Original Russian Text published in Izvestiya
Vysshikh Uchebnikh Zavedenii. Matematika, 2017, No.6. 70-75.
\bibitem {A22} J.J.Nieto, E.T.Karimov, On an anologue of the Holmgreen's problem for 3D singular elliptic equation, Azian-European Jour. of Math., 5(2)(2012) 1-18.
\bibitem {Niuk} A.W.Niukkanen, Generalized hypergeometric series ${}^NF\left(x_1,...,x_N\right)$ arising in physical and quantum chemical applications, J.Phys.A:Math.Gen. 16(1983) 1813-1825.
\bibitem {Opps} S.B.Opps, N.Saad and H.M.Srivastava, Some reduction and transformation formulas for the Appell hypergeometric functions $F_2$, J.Math.Anal.Appl. 302(2005) 180-195.
\bibitem {Padma} P.A.Padmanabham and H.M. Srivastava, Summation formulas associated with the Lauricella function $F_A^{(r)}.$ Appl.Math.Lett. 13(1) (2000) 65-70.
\bibitem {A27} M.Rassias, Lecture Notes on Mixed Type Partial Differential Equations, World
Scientific, (1990).
\bibitem {SK} H.M.Srivastava and P.W.Karlsson, {Multiple Gaussian
Hypergeometric Series. New York,Chichester,Brisbane and Toronto:
Halsted Press, 1985. 428 p.}
\bibitem {U} A.K.Urinov, On fundamental solutions for the some type of
the elliptic equations with singular coefficients. Scientific
Records of Ferghana State university, 1 (2006). 5-11.
\bibitem {UrK} A.K.Urinov and E.T.Karimov, On fundamental solutions for
3D singular elliptic equations with a parameter. Applied
Mathematic Letters, 24 (2011). 314-319.
\bibitem {Wt} R.J.Weinacht, Fundamental solutions for a class of
singular equations, Contrib.Differential equations, 3, 1964.
43-55.
\end{enumerate}
}
\end{document} | 163,969 |
\begin{document}
\title[]{On the possible images of the mod $\ell$ representations associated to elliptic curves over $\QQ$}
\author{David J. Zywina}
\address{Department of Mathematics, Cornell University, Ithaca, NY 14853, USA}
\email{[email protected]}
\urladdr{http://www.math.cornell.edu/~zywina}
\subjclass[2010]{Primary 11G05; Secondary 11F80}
\begin{abstract}
Consider a non-CM elliptic curve $E$ defined over $\QQ$. For each prime $\ell$, there is a representation $\rho_{E,\ell}\colon \Gal(\Qbar/\QQ) \to \GL_2(\FF_\ell)$ that describes the Galois action on the $\ell$-torsion points of $E$. A famous theorem of Serre says that $\rho_{E,\ell}$ is surjective for all large enough $\ell$. We will describe all known, and conjecturally all, pairs $(E,\ell)$ such that $\rho_{E,\ell}$ is not surjective. Together with another paper, this produces an algorithm that given an elliptic curve $E/\QQ$, outputs the set of such \emph{exceptional primes} $\ell$ and describes all the groups $\rho_{E,\ell}(\Gal(\Qbar/\QQ))$ up to conjugacy. Much of the paper is dedicated to computing various modular curves of genus $0$ with their morphisms to the $j$-line.
\end{abstract}
\maketitle
\section{Possible images} \label{S:classification}
Consider an elliptic curve $E$ defined over $\QQ$. For each prime $\ell$, let $E[\ell]$ be the $\ell$-torsion subgroup of $E(\Qbar)$, where $\Qbar$ is a fixed algebraic closure of $\QQ$. The group $E[\ell]$ is a free $\FF_\ell$-module of rank $2$ and there is a natural action of the absolute Galois group $\Gal_\QQ:= \Gal(\Qbar/\QQ)$ on $E[\ell]$ which respects the group structure. After choosing a basis for $E[\ell]$, this action can be expressed in terms of a Galois representation
\[
\rho_{E,\ell} \colon \Gal_\QQ \to \GL_2(\FF_\ell);
\]
its image $\rho_{E,\ell}(\Gal_\QQ)$ is uniquely determined up to conjugacy in $\GL_2(\FF_\ell)$. A renowned theorem of Serre \cite{MR0387283} says that $\rho_{E,\ell}$ is surjective for all but finitely many $\ell$ when $E$ is non-CM.
In this paper, we shall describe all known (and conjecturally all) proper subgroups of $\GL_2(\FF_\ell)$ that occur as the image of such a representation $\rho_{E,\ell}$. Applying our classification with earlier work, we will obtain an algorithm to determine the set $\calS$ of primes $\ell$ for which $\rho_{E,\ell}$ is not surjective and also compute $\rho_{E,\ell}(\Gal_\QQ)$ for each $\ell\in \calS$. \\
Before stating our classification in \S\S\ref{SS:applicable 2}--\ref{SS:applicable 17}, let us make some comments. We will consider each prime $\ell$ separately. For simplicity, assume that the $j$-invariant $j_E\in \QQ$ of $E/\QQ$ is neither $0$ nor $1728$. Our first step in determining $\rho_{E,\ell}(\Gal_\QQ)$ is to compute the group
\[
G:=\pm \rho_{E,\ell}(\Gal_\QQ),
\]
i.e., the group generated by $-I$ and $\rho_{E,\ell}(\Gal_\QQ)$. The benefit of studying $G$, up to conjugacy in $\GL_2(\FF_\ell)$, is that it does not change if $E$ is replaced by a quadratic twist. Moreover, if $E'/\QQ$ is a quadratic twist of $E/\QQ$, then after choosing appropriate bases, we will have $\rho_{E',\ell} = \chi \cdot \rho_{E,\ell}$ for some quadratic character $\chi\colon \Gal_\QQ \to \{\pm 1\}$. Since $j_E\notin \{0,1728\}$, all twists of $E$ are quadratic twists and hence $G$, up to conjugacy, depends only on the value $j_E$. The character $\det\circ \rho_{E,\ell}\colon \Gal_\QQ \to \FF_\ell^\times$ describes the Galois action on the $\ell$-th roots of unity, so $\det(G)=\FF_\ell^\times$.
For a subgroup $G$ of $\GL_2(\FF_\ell)$ with $\det(G)=\FF_\ell^\times$ and $-I\in G$, we can associate a modular curve $X_{G}$; it is a smooth, projective and geometrically irreducible curve defined over $\QQ$. It comes with a natural morphism
\[
\pi_G\colon X_{G} \to \Spec \QQ[j] \cup \{\infty\}=:\PP_\QQ^1
\]
such that for an elliptic curve $E/\QQ$ with $j_E\notin \{0,1728\}$, the group $\rho_{E,\ell}(\Gal_\QQ)$ is conjugate in $\GL_2(\FF_\ell)$ to subgroup of $G$ if and only if the $j_E=\pi_G(P)$ for some rational point $P\in X_{G}(\QQ)$.
Much of this paper is dedicated to describing those modular curves $X_G$ of genus $0$ with $X_G(\QQ)\neq \emptyset$. Such modular curves are isomorphic to the projective line and their function field is of form $\QQ(h)$ for some modular function $h$ of level $\ell$. Giving the morphism $\pi_G$ is then equivalent to expressing the modular $j$-invariant in the form $J(h)$ for a unique rational function $J(t)\in \QQ(t)$.
Once we have determined $G$, we know that $\rho_{E,\ell}(\Gal_\QQ)$ will either be the full group $G$ or equal to an index $2$ subgroup $H$ of $G$ for which $-I\notin H$. For each such $H$, it is then a matter of determining whether the quadratic character $\Gal_\QQ \xrightarrow{\rho_{E,\ell}} G \to G/H\cong\{\pm 1\}$ is trivial or not.
\\
We will first focus on the general case of non-CM elliptic curves over $\QQ$. In \S\ref{SS:CM}, we will give a complete description of the groups $\rho_{E,\ell}(\Gal_\QQ)$ when $E/\QQ$ has complex multiplication.
\\
\noindent{\textit{Notation.} }
We now define some specific subgroups of $\GL_2(\FF_\ell)$ for an odd prime $\ell$. Let $C_s(\ell)$ be the subgroup of diagonal matrices. Let $\epsilon=-1$ if $\ell\equiv 3\pmod{4}$ and otherwise let $\epsilon \geq 2$ be the smallest integer which is not a quadratic residue modulo $\ell$. Let $C_{ns}(\ell)$ be the subgroup consisting of matrices of the form $\left(\begin{smallmatrix}a & b\epsilon \\b & a \end{smallmatrix}\right)$ with $(a,b) \in \FF_\ell^2-\{(0,0)\}$. Let $N_s(\ell)$ and $N_{ns}(\ell)$ be the normalizers of $C_{s}(\ell)$ and $C_{ns}(\ell)$, respectively, in $\GL_2(\FF_\ell)$. We have $[N_s(\ell):C_s(\ell)]=2$ and the non-identity coset of $C_s(\ell)$ in $N_s(\ell)$ is represented by $\left(\begin{smallmatrix}0 & 1 \\1 & 0 \end{smallmatrix}\right)$. We have $[N_{ns}(\ell):C_{ns}(\ell)]=2$ and the non-identity coset of $C_{ns}(\ell)$ in $N_{ns}(\ell)$ is represented by $\left(\begin{smallmatrix}1 &0 \\0 & -1 \end{smallmatrix}\right)$. Let $B(\ell)$ be the subgroup of upper triangular matrices in $\GL_2(\FF_\ell)$.
\subsection{\underline{$\ell=2$}} \label{SS:applicable 2}
Up to conjugacy, there are three proper subgroups of $\GL_2(\FF_2)$:
\[
G_1=\{ I\}, \quad\quad G_2=\{ I, \left(\begin{smallmatrix}1 & 1 \\0 & 1 \end{smallmatrix}\right)\}, \quad\quad G_3=\{I, \left(\begin{smallmatrix}1 & 1 \\1 & 0 \end{smallmatrix}\right), \left(\begin{smallmatrix}0 & 1 \\1 & 1 \end{smallmatrix}\right)\}.
\]
For $i=1,2$ and $3$, the index $[\GL_2(\FF_2): G_i]$ is $6$, $3$ and $2$, respectively. Define the rational functions
\[
J_1(t)=256 \frac{(t^2+t+1)^3}{t^2(t+1)^2},
\quad\quad J_2(t) = 256\frac{(t+1)^3}{t},\quad\quad J_3(t)=t^2+1728.
\]
\begin{thm} \label{T:main2}
Let $E$ be a non-CM elliptic curve over $\QQ$. Then $\rho_{E,2}(\Gal_\QQ)$ is conjugate in $\GL_2(\FF_2)$ to a subgroup of $G_i$ if and only if $j_E$ is of the form $J_i(t)$ for some $t\in \QQ$.
\end{thm}
\subsection{\underline{$\ell=3$}} \label{SS:applicable 3}
Define the following subgroups of $\GL_2(\FF_3)$:
\begin{itemize}
\item
Let $G_1$ be the group $C_s(3)$.
\item
Let $G_2$ be the group $N_s(3)$.
\item
Let $G_3$ be the group $B(3)$.
\item
Let $G_4$ be the group $N_{ns}(3)$.
\item
Let $H_{1,1}$ be the subgroup consisting of the matrices of the form $\left(\begin{smallmatrix}1 & 0 \\0 & * \end{smallmatrix}\right)$.
\item
Let $H_{3,1}$ be the subgroup consisting of the matrices of the form $\left(\begin{smallmatrix}1 & * \\0 & * \end{smallmatrix}\right)$.
\item
Let $H_{3,2}$ be the subgroup consisting of the matrices of the form $\left(\begin{smallmatrix} * & * \\0 & 1 \end{smallmatrix}\right)$.
\end{itemize}
The index in $\GL_2(\FF_3)$ of the above subgroups are $12$, $6$, $4$, $3$, $24$, $8$ and $8$, respectively. Each of the groups $G_i$ contain $-I$. The groups $H_{i,j}$ do not contain $-I$ and we have $G_i = \pm H_{i,j}$.
Define the rational functions:
\begin{align*}
J_1(t) &= 27\frac{(t+1)^3(t+3)^3(t^2+3)^3}{t^3(t^2+3t+3)^3},\,\,
J_2(t) = 27\frac{(t+1)^3(t-3)^3}{t^3},\,\,
J_3(t) = 27\frac{(t+1)(t+9)^3}{t^3}, \,\,
J_4(t)=t^3.
\end{align*}
For $t\in \QQ -\{0\}$, let $\calE_{1,t}$ be the elliptic curve over $\QQ$ defined by Weierstrass equation
\begin{align*}
y^2&= x^3 -3(t+1)(t+3)(t^2+3) x -2(t^2-3)(t^4+6t^3+18t^2+18t+9).
\end{align*}
For $t\in \QQ -\{0,-1\}$, let $\calE_{3,t}$ be the elliptic curve over $\QQ$ defined by Weierstrass equation
\begin{align*}
y^2&= x^3 -3(t+1)^3(t+9)x -2(t+1)^4(t^2-18t-27).
\end{align*}
The $j$-invariant of $\calE_{i,t}$ is $J_i(t)$.
\begin{thm} \label{T:main3}
Let $E$ be a non-CM elliptic curve over $\QQ$.
\begin{romanenum}
\item \label{T:main3 a}
If $\rho_{E,3}$ is not surjective, then $\rho_{E,3}(\Gal_\QQ)$ is conjugate in $\GL_2(\FF_3)$ to one of the groups $G_i$ or $H_{i,j}$.
\item \label{T:main3 b}
The group $\rho_{E,3}(\Gal_\QQ)$ is conjugate to a subgroup of $G_i$ if and only if $j_E$ is of the form $J_i(t)$ for some $t\in \QQ$.
\item \label{T:main3 c}
Suppose that $\pm \rho_{E,3}(\Gal_\QQ)$ is conjugate to $G_1$. Fix an element $t\in \QQ$ such that $J_1(t)=j_E$. The group $\rho_{E,3}(\Gal_\QQ)$ is conjugate to $H_{1,1}$ if and only if $E$ is isomorphic to $\calE_{1,t}$ or the quadratic twist of $\calE_{1,t}$ by $-3$.
\item \label{T:main3 d}
Suppose that $\pm \rho_{E,3}(\Gal_\QQ)$ is conjugate to $G_3$. Fix an element $t\in \QQ$ such that $J_3(t)=j_E$.
\noindent The group $\rho_{E,3}(\Gal_\QQ)$ is conjugate to $H_{3,1}$ if and only if $E$ is isomorphic to $\calE_{3,t}$.
\noindent The group $\rho_{E,3}(\Gal_\QQ)$ is conjugate to $H_{3,2}$ if and only if $E$ is isomorphic to the quadratic twist of $\calE_{3,t}$ by $-3$.
\end{romanenum}
\end{thm}
\begin{remark}
\begin{romanenum}
\item
Let us briefly explain how Theorem~\ref{T:main3} can be used to compute $\rho_{E,3}(\Gal_\QQ)$; similar remarks will hold for the remaining primes (the case $\ell=2$ is particularly simple since $-I=I$). If $j_E$ is not of the form $J_i(t)$ for any $i\in \{1,2,3,4\}$ and $t\in \QQ$, then $\rho_{E,3}(\Gal_\QQ)=\GL_2(\FF_3)$. To check if $j_E$ is of the form $J(t)$, clear denominators in $J(t)-j_E$ to obtain a polynomial in $t$ which one can then determine whether it has rational roots or not.
So assume that $\rho_{E,3}$ is not surjective, and let $i$ be the smallest value in $\{1,2,3,4\}$ for which $j_E=J_i(t)$ for some $t\in \QQ$. By Theorem~\ref{T:main3}(\ref{T:main3 a}) and (\ref{T:main3 b}), we deduce that $\pm \rho_{E,3}(\Gal_\QQ)$ is conjugate to $G_i$; note that the groups $G_i$ are ordered by decreasing index in $\GL_2(\FF_3)$. After possibly conjugating $\rho_{E,3}$, we may assume that $\pm \rho_{E,3}(\Gal_\QQ)=G_i$. If $\rho_{E,3}(\Gal_\QQ)$ does not equal $G_i$, then it is equal to one of the subgroups $H_{i,j}$ and parts (\ref{T:main3 c}) and (\ref{T:main3 d}) give necessary and sufficient conditions to check this.
\item
Our rational functions $J_i(t)$ are certainly not unique. In particular, any function of the form $J_i((at+b)/(ct+d))$ will work with fixed $a,b,c,d\in \QQ$ satisfying $ad-bc\neq 0$ (though in general, one needs to also consider the value of $J_i(t)$ at $\infty$). Given $J_i(t)$, our equations for $\calE_{i,t}$ were produced by an algorithm that we will later describe; there are other possibly simpler choices.
\end{romanenum}
\end{remark}
\subsection{\underline{$\ell=5$}} \label{SS:applicable 5}
Define the following subgroups of $\GL_2(\FF_5)$:
\begin{itemize}
\item
Let $G_1$ be the subgroup consisting of the matrices of the form $\pm \left(\begin{smallmatrix}1 & 0 \\0 & * \end{smallmatrix}\right)$.
\item
Let $G_2$ be the group $C_s(5)$.
\item
Let $G_3$ be the unique subgroup of $N_{ns}(5)$ of index $3$; it is generated by $\left(\begin{smallmatrix}2 & 0 \\0 & 2 \end{smallmatrix}\right)$, $\left(\begin{smallmatrix}1 & 0 \\0 & -1 \end{smallmatrix}\right)$ and $\left(\begin{smallmatrix}0 & 6 \\3 & 0 \end{smallmatrix}\right)$.
\item
Let $G_4$ be the group $N_{s}(5)$.
\item
Let $G_5$ be the subgroup consisting of the matrices of the form $\pm \left(\begin{smallmatrix}* & * \\0 & 1 \end{smallmatrix}\right)$.
\item
Let $G_6$ be the subgroup consisting of the matrices of the form $\pm \left(\begin{smallmatrix}1 & * \\0 & * \end{smallmatrix}\right)$.
\item
Let $G_7$ be the group $N_{ns}(5)$.
\item
Let $G_8$ be the group $B(5)$.
\item
Let $G_9$ be the unique maximal subgroup of $\GL_2(\FF_5)$ which contains $N_s(5)$; it is generated by $\left(\begin{smallmatrix}2 & 0 \\0 & 1 \end{smallmatrix}\right)$, $\left(\begin{smallmatrix}1 & 0 \\0 & 2 \end{smallmatrix}\right)$, $\left(\begin{smallmatrix}0 & -1 \\1 & 0 \end{smallmatrix}\right)$ and $\left(\begin{smallmatrix}1 & 1 \\1 & -1 \end{smallmatrix}\right)$.
\item
Let $H_{1,1}$ be the subgroup consisting of the matrices of the form $\left(\begin{smallmatrix}1 & 0 \\0 & * \end{smallmatrix}\right)$.
\item
Let $H_{1,2}$ be the subgroup consisting of the matrices of the form $\left(\begin{smallmatrix}a^2 & 0 \\0 & a \end{smallmatrix}\right)$.
\item
Let $H_{5,1}$ be the subgroup consisting of the matrices of the form $\left(\begin{smallmatrix}* & * \\0 & 1 \end{smallmatrix}\right)$.
\item
Let $H_{5,2}$ be the subgroup consisting of the matrices of the form $\left(\begin{smallmatrix}a & * \\0 & a^2 \end{smallmatrix}\right)$.
\item
Let $H_{6,1}$ be the subgroup consisting of the matrices of the form $\left(\begin{smallmatrix}1 & * \\0 & * \end{smallmatrix}\right)$.
\item
Let $H_{6,2}$ be the subgroup consisting of the matrices of the form $\left(\begin{smallmatrix}a^2 & * \\0 & a \end{smallmatrix}\right)$.
\end{itemize}
The index in $\GL_2(\FF_5)$ of the above subgroups are $60$, $30$, $30$, $15$, $12$, $12$, $10$, $6$, $5$, $120$, $120$, $24$, $24$, $24$ and $24$, respectively. Each of the groups $G_i$ contain $-I$. The groups $H_{i,j}$ do not contain $-I$ and we have $G_i = \pm H_{i,j}$.
Define the rational functions:
\begin{align*}
J_1(t) &= \frac{(t^{20}+228t^{15}+494t^{10}-228t^5+1)^3}{t^5(t^{10}-11t^5-1)^5}\\
J_2(t) &= \frac{(t^2 + 5t + 5)^3(t^4 + 5t^2 + 25)^3(t^4 + 5t^3 + 20t^2 + 25t + 25)^3}{t^5(t^4 + 5t^3 + 15t^2 + 25t + 25)^5}\\
J_3(t) &= \frac{5^4 t^3 (t^2+5t+10)^3 (2t^2+5t+5)^3 (4t^4+30t^3+95t^2+150t+100)^3}{(t^2+5t+5)^5(t^4+5t^3+15t^2+25t+25)^5}\\
J_4(t) &= \frac{(t + 5)^3 (t^2 - 5)^3(t^2 + 5t + 10)^3}{(t^2 + 5t + 5)^5}\\
J_5(t) &= \frac{(t^4+228t^3+494t^2-228t+1)^3}{t(t^2-11t-1)^5}\\
J_6(t) &= \frac{(t^4-12t^3+14t^2+12t+1)^3}{t^5(t^2-11t-1)}\\
J_7(t) &= \frac{5^3 (t+1)(2t+1)^3(2t^2-3t+3)^3}{(t^2+t-1)^5}\\
J_8(t) &= \frac{5^2(t^2+10t+5)^3}{t^5} \\
J_9(t) &=t^3(t^2 + 5t + 40)
\end{align*}
\noindent For $t\in \QQ-\{0\}$, let $\calE_{1,t}$ be the elliptic curve over $\QQ$ defined by the Weierstrass equation
\begin{align*}
y^2=x^3 &-27(t^{20} + 228t^{15} + 494t^{10} - 228t^5 + 1)x \\
&+54(t^{30} - 522t^{25} - 10005t^{20} - 10005t^{10} + 522t^5 + 1).
\end{align*}
\noindent
For $t\in \QQ-\{0\}$, let $\calE_{5,t}$ be the elliptic curve over $\QQ$ defined by the Weierstrass equation
\begin{align*}
y^2=x^3-27(t^4 + 228t^3 + 494t^2 - 228t + 1)x+54(t^6 - 522t^5 - 10005t^4 - 10005t^2 + 522t + 1).
\end{align*}
For $t\in \QQ-\{0\}$, let $\calE_{6,t}$ be the elliptic curve over $\QQ$ defined by the Weierstrass equation
\begin{align*}
y^2 = x^3-27(t^4 - 12t^3 + 14t^2 + 12t + 1)x + 54(t^6-18t^5+75t^4+75t^2+18t+1)
\end{align*}
The $j$-invariant of $\calE_{i,t}$ is $J_i(t)$.
\begin{thm} \label{T:main5}
Let $E$ be a non-CM elliptic curve over $\QQ$.
\begin{romanenum}
\item \label{T:main5 a}
If $\rho_{E,5}$ is not surjective, then $\rho_{E,5}(\Gal_\QQ)$ is conjugate in $\GL_2(\FF_5)$ to one of the groups $G_i$ or $H_{i,j}$.
\item \label{T:main5 b}
The group $\rho_{E,5}(\Gal_\QQ)$ is conjugate to a subgroup of $G_i$ if and only if $j_E$ is of the form $J_i(t)$ for some $t\in \QQ$.
\item \label{T:main5 c}
Suppose that $\pm \rho_{E,5}(\Gal_\QQ)$ is conjugate to $G_i$ with $i\in \{1,5,6\}$. Fix an element $t\in \QQ$ such that $J_i(t)=j_E$.
\noindent The group $\rho_{E,5}(\Gal_\QQ)$ is conjugate to $H_{i,1}$ if and only if $E$ is isomorphic to $\calE_{i,t}$.
\noindent The group $\rho_{E,5}(\Gal_\QQ)$ is conjugate to $H_{i,2}$ if and only if $E$ is isomorphic to the quadratic twist of $\calE_{i,t}$ by $5$.
\end{romanenum}
\end{thm}
\subsection{\underline{$\ell=7$}} \label{SS:applicable 7}
Define the follow subgroups of $\GL_2(\FF_7)$:
\begin{itemize}
\item
Let $G_1$ be the subgroup of $N_s(7)$ consisting of elements of $C_s(7)$ with square determinant and elements of $N_s(7)-C_s(7)$ with non-square determinant; it is generated by $\left(\begin{smallmatrix}2 & 0 \\0 & 4 \end{smallmatrix}\right)$, $\left(\begin{smallmatrix}0 & 2 \\1 & 0 \end{smallmatrix}\right)$ and $\left(\begin{smallmatrix}-1 & 0 \\0 & -1 \end{smallmatrix}\right)$.
\item
Let $G_2$ be the group $N_s(7)$.
\item
Let $G_3$ be the subgroup consisting of matrices of the form $\pm \left(\begin{smallmatrix}1 & * \\0 & * \end{smallmatrix}\right)$.
\item
Let $G_4$ be the subgroup consisting of matrices of the form $\pm \left(\begin{smallmatrix}* & * \\0 & 1 \end{smallmatrix}\right)$.
\item
Let $G_5$ be the subgroup consisting of matrices of the form $\left(\begin{smallmatrix}a & * \\0 & \pm a \end{smallmatrix}\right)$.
\item
Let $G_6$ be the group $N_{ns}(7)$.
\item
Let $G_7$ be the group $B(7)$.
\item
Let $H_{1,1}$ be the subgroup generated by $\left(\begin{smallmatrix}2 & 0 \\0 & 4 \end{smallmatrix}\right)$ and $\left(\begin{smallmatrix}0 & 2 \\1 & 0 \end{smallmatrix}\right)$.
\item
Let $H_{3,1}$ be the subgroup consisting of the matrices of the form $\left(\begin{smallmatrix}1 & * \\0 & * \end{smallmatrix}\right)$.
\item
Let $H_{3,2}$ be the subgroup consisting of the matrices of the form $\left(\begin{smallmatrix}\pm 1 & * \\0 & a^2 \end{smallmatrix}\right)$.
\item
Let $H_{4,1}$ be the subgroup consisting of the matrices of the form $\left(\begin{smallmatrix}* & * \\0 & 1 \end{smallmatrix}\right)$.
\item
Let $H_{4,2}$ be the subgroup consisting of the matrices of the form $\left(\begin{smallmatrix} a^2 & * \\0 & \pm 1 \end{smallmatrix}\right)$.
\item
Let $H_{5,1}$ be the subgroup consisting of the matrices of the form $\left(\begin{smallmatrix}\pm a^2 & * \\0 & a^2 \end{smallmatrix}\right)$.
\item
Let $H_{5,2}$ be the subgroup consisting of the matrices of the form $\left(\begin{smallmatrix} a^2 & * \\0 & \pm a^2 \end{smallmatrix}\right)$.
\item
Let $H_{7,1}$ be the subgroup consisting of the matrices of the form $\left(\begin{smallmatrix} * & * \\0 & a^2 \end{smallmatrix}\right)$.
\item
Let $H_{7,2}$ be the subgroup consisting of the matrices of the form $\left(\begin{smallmatrix} a^2 & * \\0 & * \end{smallmatrix}\right)$.
\end{itemize}
The index in $\GL_2(\FF_7)$ of the above subgroups are $56$, $28$, $24$, $24$, $24$, $21$, $8$, $112$, $48$, $48$, $48$, $48$, $48$, $48$, $16$ and $16$, respectively. Each of the groups $G_i$ contain $-I$. The groups $H_{i,j}$ do not contain $-I$ and we have $G_i = \pm H_{i,j}$.
Define the rational functions
\begin{align*}
J_1(t) &= 3^3\cdot 5\cdot 7^5/2^7 \\
J_2(t)&= \frac{t(t + 1)^3(t^2 - 5t + 1)^3(t^2 - 5t + 8)^3(t^4 - 5t^3 + 8t^2 - 7t + 7)^3 }{(t^3 - 4t^2 + 3t + 1)^7}\\
J_3(t) & = \frac{(t^2 - t + 1)^3(t^6 - 11t^5 + 30t^4 - 15t^3 - 10t^2 + 5t + 1)^3}{(t - 1)^7 t^7(t^3 - 8t^2 + 5t + 1)}\\
J_4(t) & = \frac{(t^2 - t + 1)^3(t^6 + 229t^5 + 270t^4 - 1695t^3 + 1430t^2 - 235t + 1)^3}{(t - 1)t(t^3 - 8t^2 + 5t + 1)^7}\\
J_5(t) & = -\frac{(t^2-3t-3)^3(t^2-t+1)^3(3t^2-9t+5)^3(5t^2-t-1)^3}{(t^3-2t^2-t+1) (t^3-t^2-2t+1)^7} \\
J_6(t) &= \frac{64 t^3(t^2+7)^3(t^2-7t+14)^3(5t^2-14t-7)^3}{(t^3-7t^2+7t+7)^7}\\
J_7(t)&=\frac{(t^2+245t+2401)^3(t^2+13t+49)}{t^7}
\end{align*}
\noindent Let $\calE_1$ be the elliptic curve over $\QQ$ defined by the Weierstrass equation
\[
y^2= x^3 -5^3 7^3 x - 5^4 7^2 106
\]
\noindent For $t\in \QQ-\{0,1\}$, let $\calE_{3,t}$ be the elliptic curve over $\QQ$ defined by the Weierstrass equation
\begin{align*}
y^2=x^3 &- 27(t^2 - t + 1)(t^6 - 11t^5 + 30t^4 - 15t^3 - 10t^2 + 5t + 1) x\\
+&54(t^{12} - 18t^{11} + 117t^{10} - 354t^9 + 570t^8 - 486t^7 \\
&\quad\quad\quad\quad+ 273t^6 - 222t^5 + 174t^4 - 46t^3 - 15t^2 + 6t + 1).
\end{align*}
For $t\in \QQ-\{0,1\}$, let $\calE_{4,t}$ be the elliptic curve over $\QQ$ defined by the Weierstrass equation
\begin{align*}
y^2=x^3 &-27(t^2-t+1)(t^6+229t^5+270t^4-1695t^3+1430t^2-235t+1) x \\
+&54(t^{12}-522 t^{11}-8955 t^{10}+37950 t^9-70998 t^8+131562 t^7\\
&\quad \quad -253239 t^6+316290 t^5-218058 t^4+80090 t^3-14631 t^2+510 t+1).
\end{align*}
For $t\in \QQ$, let $\calE_{5,t}$ be the elliptic curve over $\QQ$ defined by the Weierstrass equation
\begin{align*}
y^2 = &\,\,x^3 - 27\cdot 7 (t^2 - 3t - 3)(t^2 - t + 1)(3t^2 - 9t + 5)(5t^2 - t - 1) x \\
&-54\cdot 7^2 (t^4 - 6t^3 + 17t^2 - 24t + 9)(3t^4 - 4t^3 - 5t^2 - 2t - 1)(9t^4 - 12t^3 - t^2 + 8t - 3).
\end{align*}
For $t\in \QQ-\{0\}$, let $\calE_{7,t}$ be the elliptic curve over $\QQ$ defined by the Weierstrass equation
\begin{align*}
y^2=x^3 &- 27(t^2 + 13t + 49)^3(t^2 + 245t + 2401)x\\
& +54(t^2 + 13t + 49)^4(t^4 - 490t^3 - 21609t^2 - 235298t - 823543).
\end{align*}
The $j$-invariant of $\calE_{i,t}$ is $J_i(t)$.
\begin{thm} \label{T:main7}
Let $E$ be a non-CM elliptic curve over $\QQ$.
\begin{romanenum}
\item \label{T:main7 i}
If $\rho_{E,7}$ is not surjective, then $\rho_{E,7}(\Gal_\QQ)$ is conjugate in $\GL_2(\FF_7)$ to one of the groups $G_i$ or $H_{i,j}$.
\item \label{T:main7 ii}
The group $\rho_{E,7}(\Gal_\QQ)$ is conjugate to a subgroup of $G_i$ if and only if $j_E$ is of the form $J_i(t)$ for some $t\in \QQ$.
\item \label{T:main7 iii}
The group $\rho_{E,7}(\Gal_\QQ)$ is conjugate to $H_{1,1}$ if and only if $E/\QQ$ is isomorphic to $\calE_1$ or to the quadratic twist of $\calE_1$ by $-7$.
\item \label{T:main7 iv}
Suppose that $\pm \rho_{E,7}(\Gal_\QQ)$ is conjugate to $G_i$ with $i\in \{3,4,5,7\}$. Fix an element $t\in \QQ$ such that $J_i(t)=j_E$.
\noindent The group $\rho_{E,7}(\Gal_\QQ)$ is conjugate to $H_{i,1}$ if and only if $E$ is isomorphic to $\calE_{i,t}$.
\noindent The group $\rho_{E,7}(\Gal_\QQ)$ is conjugate to $H_{i,2}$ if and only if $E$ is isomorphic to the quadratic twist of $\calE_{i,t}$ by $-7$.
\end{romanenum}
\end{thm}
\subsection{\underline{$\ell=11$}} \label{SS:applicable 11}
\begin{itemize}
\item
Let $G_1$ be the subgroup generated by $\pm \left(\begin{smallmatrix}1 & 1 \\0 & 1 \end{smallmatrix}\right)$ and $\left(\begin{smallmatrix}4 & 0 \\0 & 6 \end{smallmatrix}\right)$.
\item
Let $G_2$ be the subgroup generated by $\pm \left(\begin{smallmatrix}1 & 1 \\0 & 1 \end{smallmatrix}\right)$ and $\left(\begin{smallmatrix}5 & 0 \\0 & 7 \end{smallmatrix}\right)$.
\item
Let $G_3$ be the group $N_{ns}(11)$.
\item
Let $H_{1,1}$ be the subgroup generated by $\left(\begin{smallmatrix}1 & 1 \\0 & 1 \end{smallmatrix}\right)$ and $\left(\begin{smallmatrix}4 & 0 \\0 & 6 \end{smallmatrix}\right)$.
\item
Let $H_{1,2}$ be the subgroup generated by $\left(\begin{smallmatrix}1 & 1 \\0 & 1 \end{smallmatrix}\right)$ and $\left(\begin{smallmatrix}7 & 0 \\0 & 5 \end{smallmatrix}\right)$.
\item
Let $H_{2,1}$ be the subgroup generated by $\left(\begin{smallmatrix}1 & 1 \\0 & 1 \end{smallmatrix}\right)$ and $\left(\begin{smallmatrix}5 & 0 \\0 & 7 \end{smallmatrix}\right)$.
\item
Let $H_{2,2}$ be the subgroup generated by $\left(\begin{smallmatrix}1 & 1 \\0 & 1 \end{smallmatrix}\right)$ and $\left(\begin{smallmatrix}6 & 0 \\0 & 4 \end{smallmatrix}\right)$.
\end{itemize}
The index in $\GL_2(\FF_{11})$ of the above subgroups are $60$, $60$, $55$, $110$, $120$, $120$, $120$ and $120$, respectively. Each of the groups $G_i$ contain $-I$. The groups $H_{i,j}$ do not contain $-I$ and we have $G_i = \pm H_{i,j}$.
Let $\calE$ be the elliptic curve over $\QQ$ defined by the Weierstrass equation $y^2+y = x^3-x^2-7x+10$ and let $\OO$ be the point at infinity. The Mordell-Weil group $\calE(\QQ)$ is an infinite cyclic group generated by the point $(4,5)$. Define
\[
J(x,y):=\frac{(f_1 f_2 f_3 f_4)^3}{f_5^2 f_6^{11}},
\]
where
\begin{align*}
f_1&=x^2+3x-6,
&f_2&=11(x^2-5)y+(2x^4+23x^3-72x^2-28x+127),\\
f_3&=6y+11x-19,
&f_4&=22(x-2)y+(5x^3+17x^2-112x+120), \\
f_5&=11y+(2x^2+17x-34),
&f_6&=(x-4)y-(5x-9).
\end{align*}
We shall view $J$ as a morphism $\calE \to \AA_\QQ^1\cup\{\infty\}$.\\
\noindent
Let $\calE_{1}/\QQ$ be the elliptic curve defined by the Weierstrass equation $y^2 = x^3-27\cdot 11^4 x + 54\cdot 11^5\cdot 43$
\noindent
Let $\calE_{2}/\QQ$ be the elliptic curve defined by the Weierstrass equation $y^2=x^3-27\cdot 11^3\cdot 131 x +54\cdot 11^4\cdot 4973$.
\begin{thm} \label{T:11 main}
Let $E$ be a non-CM elliptic curve defined over $\QQ$.
\begin{romanenum}
\item \label{T:11 main a}
If $\rho_{E,11}$ is not surjective, then $\rho_{E,11}(\Gal_\QQ)$ is conjugate in $\GL_2(\FF_{11})$ to one of the groups $G_i$ or $H_{i,j}$.
\item \label{T:11 main b}
The group $\pm\rho_{E,11}(\Gal_\QQ)$ is conjugate to $G_1$ in $\GL_2(\FF_{11})$ if and only if $j_E=-11^2$.
\item \label{T:11 main c}
The group $\pm\rho_{E,11}(\Gal_\QQ)$ is conjugate to $G_2$ in $\GL_2(\FF_{11})$ if and only if $j_E=-11\cdot 131^3$.
\item \label{T:11 main d}
The group $\rho_{E,11}(\Gal_\QQ)$ is conjugate to $G_3$ in $\GL_2(\FF_{11})$ if and only if $j_E=J(P)$ for some point $P\in \calE(\QQ)-
\{\OO\}$.
\item \label{T:11 main e}
For $i\in \{1,2\}$, the group $\rho_{E,11}(\Gal_\QQ)$ is conjugate in $\GL_2(\FF_{11})$ to $H_{i,1}$ if and only if $E$ is isomorphic to $\calE_{i}$.
\item \label{T:11 main f}
For $i\in \{1,2\}$, the group $\rho_{E,11}(\Gal_\QQ)$ is conjugate in $\GL_2(\FF_{11})$ to $H_{i,2}$ if and only if $E$ is isomorphic to the quadratic twist of $\calE_{i}$ by $-11$.
\end{romanenum}
\end{thm}
\begin{remark} \label{R:ns computation}
The modular curve $X_{ns}^+(11)=X_{G_3}$ is the only one in our classification that has genus $1$ with infinitely many rational points. Halberstadt \cite{MR1677158} showed that $X_{ns}^+(11)$ is isomorphic to $\calE$ and that the morphism to the $j$-line corresponds to $J(x,y)$.
In \S\ref{S:ns section}, we give explicit polynomials $A,B,C\in \QQ[X]$ of degree $55$ such that for a non-CM elliptic curve $E/\QQ$, we have $j_E=J(P)$ for some $P\in \calE(\QQ)-\{\OO\}$ if and only if the polynomial $A(x)j_E^2+B(x)j_E +C(x) \in \QQ[x]$ has a rational root. This gives a straightforward way to check the criterion of Theorem~\ref{T:11 main}(\ref{T:11 main d}).
\end{remark}
\subsection{\underline{$\ell=13$}} \label{SS:applicable 13}
Define the following subgroups of $\GL_2(\FF_{13})$:
\begin{itemize}
\item Let $G_1$ be the subgroup consisting of matrices of the form $\left(\begin{smallmatrix}* & * \\0 & b^3 \end{smallmatrix}\right)$.
\item Let $G_2$ be the subgroup consisting of matrices of the form $\left(\begin{smallmatrix}a^3 & * \\0 & * \end{smallmatrix}\right)$.
\item Let $G_3$ be the subgroup consisting of matrices $\left(\begin{smallmatrix}a & * \\0 & b \end{smallmatrix}\right)$ for which $(a/b)^4=1$.
\item Let $G_4$ be the subgroup consisting of matrices of the form $\left(\begin{smallmatrix}* & * \\0 & b^2 \end{smallmatrix}\right)$.
\item Let $G_5$ be the subgroup consisting of matrices of the form $\left(\begin{smallmatrix}a^2 & * \\0 & * \end{smallmatrix}\right)$.
\item Let $G_6$ be the group $B(13)$.
\item Let $G_7$ be the subgroup generated by the matrices
$\left(\begin{smallmatrix}2 & 0 \\0 & 2 \end{smallmatrix}\right)$, $\left(\begin{smallmatrix}2 & 0 \\0 & 3 \end{smallmatrix}\right)$, $\left(\begin{smallmatrix}0 & -1 \\1 & 0 \end{smallmatrix}\right)$ and $\left(\begin{smallmatrix}1 & 1 \\-1 & 1 \end{smallmatrix}\right)$; it contains the scalar matrices and its image in $\PGL_2(\FF_{13})$ is isomorphic to $\mathfrak{S}_4$.
\item
Let $H_{4,1}$ be the subgroup consisting of matrices of the form $\left(\begin{smallmatrix}* & * \\0 & a^4 \end{smallmatrix}\right)$.
\item
Let $H_{4,2}$ be the subgroup generated by matrices of the form $\big(\begin{smallmatrix} b^2 & * \\0 & a^4 \end{smallmatrix}\big)$ and $\left(\begin{smallmatrix} 2 & 0 \\0 & 4 \end{smallmatrix}\right)$.
\item
Let $H_{5,1}$ be the subgroup consisting of matrices of the form $\left(\begin{smallmatrix}a^4 & * \\0 & * \end{smallmatrix}\right)$.
\item
Let $H_{5,2}$ be the subgroup generated by matrices of the form $\big(\begin{smallmatrix}a^4 & * \\0 & b^2 \end{smallmatrix}\big)$ and $\left(\begin{smallmatrix} 4 & 0 \\0 & 2 \end{smallmatrix}\right)$.
\end{itemize}
The index in $\GL_2(\FF_{13})$ of the above subgroups are $42$, $42$, $42$, $28$, $28$, $14$, $91$, $56$, $56$, $56$ and $56$, respectively. Each of the groups $G_i$ contain $-I$. The groups $H_{i,j}$ do not contain $-I$ and we have $G_i = \pm H_{i,j}$. \\
Define the polynomials
\begin{align*}
P_1(t)&= t^{12} + 231t^{11} + 269t^{10} - 3160t^9 + 6022t^8 - 9616t^7 + 21880t^6 \\ &\quad - 34102t^5 + 28297t^4 - 12455t^3 + 2876t^2 - 243t + 1\\
P_2(t)&= t^{12} - 9t^{11} + 29t^{10} - 40t^9 + 22t^8 - 16t^7 + 40t^6 - 22t^5 - 23t^4 + 25t^3 - 4t^2 - 3t + 1\\
P_3(t)&= (t^4-t^3+2t^2-9t+3)(3t^4-3t^3-7t^2+12t-4)(4t^4-4t^3-5t^2+3t-1)\\
P_4(t)&= t^8 + 235t^7 + 1207t^6 + 955t^5 + 3840t^4 - 955t^3 + 1207t^2 - 235t+ 1\\
P_5(t)&= t^8 - 5t^7 + 7t^6 - 5t^5 + 5t^3 + 7t^2 + 5t + 1\\
P_6(t)&= t^4+7t^3+20t^2+19t+1\\
Q_4(t)&=\,t^{12} - 512 t^{11} - 13079 t^{10} - 32300 t^9 - 104792 t^8 - 111870 t^7 \\
&\quad\quad\quad - 419368 t^6 + 111870 t^5 - 104792 t^4 + 32300 t^3 - 13079 t^2 + 512 t +1, \\
Q_5(t)&=\, t^{12} - 8 t^{11} + 25 t^{10} - 44 t^9 + 40 t^8 + 18 t^7 - 40 t^6 - 18 t^5 + 40 t^4 + 44 t^3 + 25 t^2 + 8 t + 1
\end{align*}
and the rational functions
\begin{align*}
J_1(t) & = \frac{(t^2 - t + 1)^3 P_1(t)^3}{(t - 1)t(t^3 - 4t^2 + t + 1)^{13}}
\quad \quad \quad \quad \quad \quad \quad
J_2(t) = \frac{(t^2 - t + 1)^3P_2(t)^3}{(t - 1)^{13} t^{13} (t^3 - 4t^2 + t + 1)}\\
J_3(t) &= -\frac{13^4(t^2-t+1)^3 P_3(t)^3}{(t^3-4t^2+t+1)^{13} (5t^3-7t^2-8t+5)}
\quad
J_4(t) = \frac{(t^4 - t^3 + 5t^2 + t + 1)P_4(t)^3}{t (t^2 - 3t - 1)^{13}} \\
J_5(t) &= \frac{(t^4 - t^3 + 5t^2 + t + 1) P_5(t)^3}{t^{13} (t^2 - 3t - 1)}
\quad \quad \quad \quad \quad \quad \,\, J_6(t) = \frac{(t^2+5t+13)P_6(t)^3}{t}.
\end{align*}
\noindent
For $t\in \QQ-\{0\}$, let $\calE_{4,t}$ be the elliptic curve over $\QQ$ defined by the Weierstrass equation
\begin{align*}
y^2= x^3-27(t^4 - t^3 + 5t^2 + t + 1)^3P_4(t) x + 54(t^2 + 1) (t^4 - t^3 + 5 t^2 + t + 1)^4 Q_4(t).
\end{align*}
For $t\in \QQ-\{0\}$, let $\calE_{5,t}$ be the elliptic curve over $\QQ$ defined by the Weierstrass equation
\begin{align*}
y^2= x^3-27(t^4 - t^3 + 5t^2 + t + 1)^3P_5(t) x + 54(t^2 + 1) (t^4 - t^3 + 5 t^2 + t + 1)^4 Q_5(t).
\end{align*}
\begin{thm} \label{T:main 13}
Let $E$ be a non-CM elliptic curve over $\QQ$.
\begin{romanenum}
\item \label{T:main 13 a}
If $\rho_{E,13}(\Gal_\QQ)$ is conjugate to a subgroup of $B(13)$, then $\rho_{E,13}(\Gal_\QQ)$ is conjugate to one of the groups $G_i$ with $1\leq i \leq 6$ or to a group $H_{i,j}$.
\item \label{T:main 13 b}
For $1\leq i \leq 6$, $\rho_{E,13}(\Gal_\QQ)$ is conjugate in $\GL_2(\FF_{13})$ to a subgroup of $G_i$ if and only if $j_E$ is of the form $J_i(t)$ for some $t\in \QQ$.
\item \label{T:main 13 c}
For an $i\in \{4,5\}$, suppose that $J_i(t)=j_E$ for some $t\in \QQ$.
\noindent The group $\rho_{E,13}(\Gal_\QQ)$ is conjugate to $H_{i,1}$ if and only if $E$ is isomorphic to $\calE_{i,t}$.
\noindent The group $\rho_{E,13}(\Gal_\QQ)$ is conjugate to $H_{i,2}$ if and only if $E$ is isomorphic to the quadratic twist of $\calE_{i,t}$ by $13$.
\item \label{T:main 13 d}
If $j_E$ is
\[
\frac{2^4\cdot 5\cdot 13^4\cdot 17^3}{3^{13}}, \quad -\frac{2^{12}\cdot 5^3\cdot 11\cdot 13^4}{3^{13}} \quad\text{ or }\quad \frac{2^{18}\cdot3^3\cdot 13^4\cdot 127^3\cdot 139^3\cdot 157^3\cdot 283^3\cdot 929}{5^{13}\cdot 61^{13}},
\]
then $\rho_{E,13}(\Gal_\QQ)$ is conjugate to $G_7$.
\end{romanenum}
\end{thm}
Up to conjugacy, there are four {maximal} subgroups $G$ of $\GL_2(\FF_{13})$ that satisfy $\det(G)=\FF_{13}^\times$; they are $G_6=B(13)$, $N_{s}(13)$, $N_{ns}(13)$ and $G_7$. The cases concerning subgroups of $B(13)$ are completely handled in Theorem~\ref{T:main 13}.\\
Baran \cite{Baran-13} has shown that the modular curves $X_{s}^+(13)$ and $X_{ns}^+(13)$ attached to $N_s(13)$ and $N_{ns}(13)$, respectively, are both isomorphic to the genus $3$ curve $C$ defined in $\PP^2_\QQ$ by the equation
\[
(-y-z)x^3 +(2y^2+zy)x^2+(-y^3+zy^2-2z^2y+z^3)x+(2z^2y^2-3z^3y)=0.
\]
In \cite{Baran-13}, the morphism from the model of the modular curves to the $j$-line is given. The seven rational points $(0 , 0, 1)$, $(0 , 1 , 0)$, $(0,3,2)$, $(1, 0, -1)$, $(1, 0, 0)$, $(1,0,1)$, $(1, 1, 0)$ of $C$ all correspond to cusps and CM points on $X_s(13)$ and $X_{ns}(13)$. Conjecturally, there are no non-CM elliptic curves $E$ over $\QQ$ with $\rho_{E,13}(\Gal_\QQ)$ conjugate to a subgroup of $N_s(13)$ or $N_{ns}(13)$; equivalently, $C$ has no other rational points.
\\
Denote by $X_{\mathfrak{S}_4}(13)$ the modular curve corresponding to $G_7$. Banwait and Cremona \cite{banwait-cremona} have shown that $X_{\mathfrak{S}_4}(13)$ is isomorphic to the genus $3$ curve $C'$ defined in $\PP^2_\QQ$ by the equation
\[
4x^3y - 3x^2y^2 + 3xy^3 - x^3z + 16x^2yz - 11xy^2z + 5y^3z + 3x^2z^2 + 9xyz^2 + y^2z^2 + xz^3 + 2yz^3 = 0
\]
and have found the morphism from the modular curve to the $j$-line. The four rational points $(0,1,0)$, $(0,0,1)$, $(1,0,0)$ and $(1,3,-2)$ of $C'$ correspond to a CM point and three non-CM points; the non-CM points give rise to the three $j$-invariants in Theorem~\ref{T:main 13}(\ref{T:main 13 d}).
Suppose $E/\QQ$ is an elliptic curve with one of the $j$-invariants from Theorem~\ref{T:main 13}(\ref{T:main 13 d}). From \cite{banwait-cremona}, we find that the image of $\rho_{E,13}(\Gal_\QQ)$ in $\PGL_2(\FF_{13})$ is isomorphic to $\mathfrak{S}_4$. Therefore, $\rho_{E,13}(\Gal_\QQ)$ is conjugate to $G_7$ since $G_7$ has no proper subgroups $H$ whose image in $\PGL_2(\FF_{13})$ is isomorphic to $\mathfrak{S}_4$ and satisfies $\det(H)=\FF_{13}^\times$. In particular, this proves Theorem~\ref{T:main 13}(\ref{T:main 13 d}).
Conjecturally, if $E$ is a non-CM elliptic curve over $\QQ$, then $\rho_{E,13}(\Gal_\QQ)$ is conjugate to a subgroup of $G_7$ if and only if $j_E$ is one of three values from Theorem~\ref{T:main 13}(\ref{T:main 13 d}); equivalently, $C'$ has no other rational points.
\begin{remark}
The case $\ell=13$ is the first for which we do not have a complete description. As explained above, it remains to determine all the rational points of the genus $3$ curves $C$ and $C'$.
\end{remark}
\subsection{\underline{$\ell\geq 17$}} \label{SS:applicable 17}
We first describe all the known cases of non-CM elliptic curves $E/\QQ$ for which $\rho_{E,\ell}$ is not surjective for some prime $\ell\geq 17$. Define the following groups:
\begin{itemize}
\item
Let $G_1$ be the subgroup of $\GL_2(\FF_{17})$ generated by $\left(\begin{smallmatrix}2 & 0 \\0 & 11 \end{smallmatrix}\right)$, $\left(\begin{smallmatrix}4 & 0 \\0 & -4 \end{smallmatrix}\right)$ and $\left(\begin{smallmatrix}1 & 1 \\0 &1 \end{smallmatrix}\right)$.
\item
Let $G_2$ be the subgroup of $\GL_2(\FF_{17})$ generated by $\left(\begin{smallmatrix}11 & 0 \\0 & 2 \end{smallmatrix}\right)$, $\left(\begin{smallmatrix}-4 & 0 \\0 & 4 \end{smallmatrix}\right)$ and $\left(\begin{smallmatrix}1 & 1 \\0 &1 \end{smallmatrix}\right)$.
\item
Let $G_3$ be the subgroup of $\GL_2(\FF_{37})$ consisting of the matrices of the form $\left(\begin{smallmatrix}a^3 & * \\0 & * \end{smallmatrix}\right)$.
\item
Let $G_4$ be the subgroup of $\GL_2(\FF_{37})$ consisting of the matrices of the form $\left(\begin{smallmatrix}* & * \\0 & a^3 \end{smallmatrix}\right)$.
\end{itemize}
\begin{thm} \label{T:17-37}
\begin{romanenum}
\item \label{T:17-37 i}
If $E/\QQ$ has $j$-invariant $-17\cdot 373^3/2^{17}$ or $-17^2 \cdot 101^3/2$, then $\rho_{E,17}(\Gal_\QQ)$ is conjugate in $\GL_2(\FF_{17})$ to $G_1$ or $G_2$, respectively.
\item \label{T:17-37 ii}
If $E/\QQ$ has $j$-invariant $-7\cdot 11^3$ or $-7\cdot 137^3\cdot 2083^3$, then $\rho_{E,37}(\Gal_\QQ)$ is conjugate in $\GL_2(\FF_{37})$ to $G_3$ or $G_4$, respectively.
\end{romanenum}
\end{thm}
\begin{thm}[Mazur, Serre, Bilu-Parent-Rebolledo] \label{T:Mazur-Serre-BPR}
Fix a prime $\ell\geq 17$ and let $E$ be a non-CM elliptic curve defined over $\QQ$. If $(\ell,j_E)$ does not belong to the set
\begin{equation} \label{E:set exceptional 17-37}
\{ (17,-17\cdot 373^3/2^{17}), \, (17, -17^2 \cdot 101^3/2), \, (37,-7\cdot 11^3),\, (37,-7\cdot 137^3\cdot 2083^3) \},
\end{equation}
then either $\rho_{E,\ell}$ is surjective or $\rho_{E,\ell}(\Gal_\QQ)$ is conjugate to a subgroup of $N_{ns}(\ell)$.
\end{thm}
\begin{proof}
The group $\GL_2(\FF_\ell)$ has either three or four maximal subgroups with determinant $\FF_\ell^\times$. They are $B(\ell)$, $N_s(\ell)$, $N_{ns}(\ell)$ and when $\ell\equiv \pm 3 \pmod{8}$, we also have a maximal subgroup $H_{\mathfrak{S}_4}(\ell)$ whose image in $\PGL_2(\FF_\ell)$ is isomorphic to the symmetric group $\mathfrak{S}_4$.
Take any non-CM elliptic curve $E$ over $\QQ$. Serre has shown that $\rho_{E,\ell}(\Gal_\QQ)$ cannot be conjugate to a subgroup of $H_{\mathfrak{S}_4}(\ell)$, cf.~\cite{MR644559}*{\S8.4}. Bilu, Parent and Rebolledo have proved that $\rho_{E,\ell}(\Gal_\QQ)$ cannot be conjugate to a subgroup of $N_s(\ell)$, cf.~\cite{1104.4641} (they make effective the bounds in earlier works of Bilu and Parent using improved isogeny bounds of Gaudron and R\'emond). The $B(\ell)$ case follows from a famous theorem of Mazur, cf.~\cite{MR482230}. The modular curves $X_0(17)$ and $X_0(37)$ each have two rational points which are not cusps or CM points and they are accounted for by the curves of Theorem~\ref{T:17-37}.
\end{proof}
We conjecture that Theorem~\ref{T:Mazur-Serre-BPR} describes all the reasons that $\rho_{E,\ell}$ can fail to be surjective for a non-CM $E/\QQ$ and a prime $\ell \geq 17$; this is a problem raised by Serre, cf.~\cite{MR644559}*{p.399}, who asked if $\rho_{E,\ell}$ is surjective whenever $\ell >37$.
\begin{conj} \label{C:main}
If $E$ is a non-CM elliptic curve over $\QQ$ and $\ell\geq 17$ is a prime such that the pair $(\ell,j_E)$ does not belong to the set (\ref{E:set exceptional 17-37}), then $\rho_{E,\ell}(\Gal_\QQ)=\GL_2(\FF_\ell)$.
\end{conj}
Even if Conjecture~\ref{C:main} is false for some $E/\QQ$ and $\ell\geq 17$, the following proposition gives at most two possibilities for the image of $\rho_{E,\ell}$ (they can be distinguished computationally by looking at the division polynomial of $E$ at $\ell$).
\begin{prop} \label{P:big inertia last}
Suppose that $\rho_{E,\ell}$ is not surjective for a non-CM elliptic curve $E/\QQ$ and a prime $\ell\geq 17$ for which $(\ell,j_E)$ does not lie in the set (\ref{E:set exceptional 17-37}).
\begin{romanenum}
\item
If $\ell \equiv 1 \pmod{3}$, then $\rho_{E,\ell}(\Gal_\QQ)$ is conjugate in $\GL_2(\FF_\ell)$ to $N_{ns}(\ell)$.
\item
If $\ell \equiv 2 \pmod{3}$, then $\rho_{E,\ell}(\Gal_\QQ)$ is conjugate in $\GL_2(\FF_\ell)$ to $N_{ns}(\ell)$ or to the group
\[
G:=\big\{a^3: a \in C_{ns}(\ell)\big\} \cup \big\{ \left(\begin{smallmatrix}1 &0 \\0 & -1 \end{smallmatrix}\right) \cdot a^3: a \in C_{ns}(\ell) \big\}.
\]
\end{romanenum}
\end{prop}
\subsection{Algorithm}
Let $E/\QQ$ be a non-CM elliptic curve (when $E/\QQ$ has complex multiplication, the groups $\rho_{E,\ell}(\Gal_\QQ)$ are all described in \S\ref{SS:CM} below). In \cite{Zywina-images}, we give an algorithm to compute the set $S'$ of primes $\ell \geq 13$ for which $\rho_{E,\ell}$ is not surjective.
Combined with the theorems from \S\S\ref{SS:applicable 2}--\ref{SS:applicable 11}, we are now able to compute the (finite) set $S$ of primes $\ell$ for which $\rho_{E,\ell}$ is not surjective. Moreover, using the results from \S\S\ref{SS:applicable 2}--\ref{SS:applicable 11}, we can give the group $\rho_{E,\ell}(\Gal_\QQ)$, up to conjugacy in $\GL_2(\FF_\ell)$, for each $\ell \in S$.\\
Sutherland has a probabilistic algorithm to determine the groups $\rho_{E,\ell}(\Gal_\QQ)$ by consider Frobenius at many primes $p$, \cite{Sutherland2015}. His algorithm can in principle be made deterministic using effective versions of the Chebotarev density theorem. Sutherland's algorithm has the advantage that it can be used for elliptic curves over a number field $K\neq \QQ$ (for our approach, we would have more modular curves to consider and those modular curves not isomorphic to $\PP^1_\QQ$ would need to be reconsidered).
The next task that needs to be completed is to consider the images of $\rho_{E,\ell^n}$ for small primes $\ell$ and $n\geq 2$. Rouse and Zureick-Brown have already done this for $\ell=2$, cf.~\cite{R-DZ}; the case $\ell=2$ is rather accessible since all the groups that occur are solvable.
\newpage
\subsection{Complex multiplication} \label{SS:CM}
Up to isomorphism over $\Qbar$, there are thirteen elliptic curves with complex multiplication that are defined over $\QQ$. In Table 1 below, we give an elliptic curve $E_{D,f}/\QQ$ with each of these thirteen $j$-invariants (this comes from \cite{MR1312368}*{Appendix A \S3} though with some different models). The curve $E_{D,f}$ has conductor $N$ and has complex multiplication by an order $R$ of conductor $f$ in the imaginary quadratic field with discriminant $-D$.
{
\renewcommand{\arraystretch}{1.1}
\begin{table}[htdp]
\begin{center}\begin{tabular}{|c|c|c|l|c|}\hline
$j$-invariant & $D$ & $f$ & Elliptic curve $E_{D,f}$ & $N$ \\ \hline\hline
$0$ &$3$ & $1$ & $y^2=x^3+16$ & $3^3$ \\
$2^4 3^3 5^3$ & & $2$ & $y^2=x^3-15x+22$ & $2^2 3^2$ \\
$-2^{15} 3 \cdot 5^3$ & & $3$ & $y^2=x^3-480x+4048$ & $3^3$ \\ \hline
$2^6 3^3=1728$ & $4$ & $1$ & $y^2=x^3+x$ & $2^6$ \\
$2^3 3^3 11^3$ & & $2$ & $y^2=x^3-11x+14$ & $2^5$ \\ \hline
$-3^3 5^3$ & $7$ & $1$ & $y^2=x^3-1715x+33614$ & $7^2$ \\
$3^3 5^3 17^3$ & & $2$ & $y^2=x^3-29155x+1915998$ & $7^2$ \\ \hline
$2^6 5^3$ & $8$ & $1$ & $y^2=x^3-4320x+96768$ & $2^8$ \\ \hline
$-2^{15}$ & $11$ & $1$ & $y^2=x^3-9504x+365904$ & $11^2$ \\ \hline
$-2^{15} 3^3$ & $19$ & $1$ & $y^2=x^3-608x+5776$ & $19^2$ \\ \hline
$-2^{18} 3^3 5^3$ & $43$ & $1$ & $y^2=x^3-13760x+621264$ & $43^2$ \\ \hline
$-2^{15} 3^3 5^3 11^3$ & $67$ & $1$ & $y^2=x^3-117920x+15585808$ & $67^2$ \\ \hline
$-2^{18} 3^3 5^3 23^3 29^3$ & $163$ & $1$ & $y^2=x^3-34790720x+78984748304$ & $163^2$ \\ \hline
\end{tabular} \caption{CM elliptic curves over $\QQ$}
\end{center}
\label{defaulttable}
\end{table}
}
We first describe the group $\rho_{E,\ell}(\Gal_\QQ)$ up to conjugacy when $E$ is a CM elliptic curve with non-zero $j$-invariant and $\ell$ odd.
\begin{prop} \label{P:CM main}
Let $E$ be a CM elliptic curve defined over $\QQ$ with $j_E\neq 0$. The ring of endomorphisms of $E_{\Qbar}$ is an order of conductor $f$ in the ring of integers of an imaginary quadratic field of discriminant $-D$. Take any odd prime $\ell$.
\begin{romanenum}
\item \label{P:CM main a}
If $\legendre{-D}{\ell}=1$, then $\rho_{E,\ell}(\Gal_\QQ)$ is conjugate in $\GL_2(\FF_\ell)$ to $N_s(\ell)$.
\item \label{P:CM main b}
If $\legendre{-D}{\ell}=-1$, then $\rho_{E,\ell}(\Gal_\QQ)$ is conjugate in $\GL_2(\FF_\ell)$ to $N_{ns}(\ell)$.
\item \label{P:CM main c}
Suppose that $\ell$ divides $D$ and hence $D=\ell$. Define the groups
\[
G=\{ \left(\begin{smallmatrix} a & b \\0 & \pm a \end{smallmatrix}\right) : a\in \FF_\ell^\times, b\in \FF_\ell\},
\]
\[
H_1 = \{ \left(\begin{smallmatrix} a & b \\0 & \pm a \end{smallmatrix}\right) : a\in (\FF_\ell^\times)^2, b\in \FF_\ell\}, \quad \text{ and }\quad H_2 = \{ \left(\begin{smallmatrix} \pm a & b \\0 & a \end{smallmatrix}\right) : a\in (\FF_\ell^\times)^2, b\in \FF_\ell\}
\]
\noindent
If $E$ is isomorphic to $E_{D,f}$, then $\rho_{E,\ell}(\Gal_\QQ)$ is conjugate in $\GL_2(\FF_\ell)$ to $H_1$.
\noindent
If $E$ is isomorphic to the quadratic twist of $E_{D,f}$ by $-\ell$, then $\rho_{E,\ell}(\Gal_\QQ)$ is conjugate in $\GL_2(\FF_\ell)$ to $H_2$.
\noindent
If $E$ is not isomorphic to $E_{D,f}$ or its quadratic twist by $-\ell$, then $\rho_{E,\ell}(\Gal_\QQ)$ is conjugate in $\GL_2(\FF_\ell)$ to $G$.
\end{romanenum}
\end{prop}
The following deals with the excluded prime $\ell=2$.
\begin{prop} \label{P: prime 2}
Let $E/\QQ$ be a CM elliptic curve. Define the subgroup $G_2=\{ I, \left(\begin{smallmatrix}1 & 1 \\0 & 1 \end{smallmatrix}\right)\}$ of $\GL_2(\FF_2)$.
\begin{romanenum}
\item \label{P: prime 2 i}
If $j_E \in \{2^4 3^3 5^3,\, 2^3 3^3 11^3,\, -3^3 5^3,\, 3^3 5^3 17^3,\, 2^6 5^3\},$
then $\rho_{E,2}(\Gal_\QQ)$ is conjugate to $G_2$.
\item
If $j_E \in \{ -2^{15} 3 \cdot 5^3,\, -2^{15},\, -2^{15} 3^3,\, -2^{18} 3^3 5^3,\, -2^{15} 3^3 5^3 11^3,\, -2^{18} 3^3 5^3 23^3 29^3\},$
then $\rho_{E,2}(\Gal_\QQ)=\GL_2(\FF_2)$.
\item
Suppose that $j_E=1728$. The curve can be given by a Weierstrass equation $y^2=x^3-dx$ for some $d\in \QQ^\times$.
\noindent
If $d$ is a square, then $\rho_{E,2}(\Gal_\QQ)=\{I\}$.
\noindent
If $d$ is not a square, then the group $\rho_{E,2}(\Gal_\QQ)$ is conjugate to $G_2$.
\item
Suppose that $j_E=0$. The curve $E$ can be given by a Weierstrass equation $y^2=x^3+d$ for some $d\in \QQ^\times$.
\noindent
If $d$ is a cube, then $\rho_{E,2}(\Gal_\QQ)$ is conjugate in $\GL_2(\FF_2)$ to the group $G_2$.
\noindent
If $d$ is not a cube, then $\rho_{E,2}(\Gal_\QQ)=\GL_2(\FF_2)$.
\end{romanenum}
\end{prop}
It remains to consider the situation where $\ell$ is an odd prime and $E/\QQ$ is an elliptic curve with $j_E=0$. That such curves have cubic twists make the classification more involved.
\begin{prop} \label{P:j=0 situation}
Let $E$ be an elliptic curve over $\QQ$ with $j_E=0$. Take any odd prime $\ell$.
\begin{romanenum}
\item \label{P:j=0 situation i}
If $\ell \equiv 1 \pmod{9}$, then $\rho_{E,\ell}(\Gal_\QQ)$ is conjugate to $N_{s}(\ell)$ in $\GL_2(\FF_\ell)$.
\item \label{P:j=0 situation ii}
If $\ell \equiv 8 \pmod{9}$, then $\rho_{E,\ell}(\Gal_\QQ)$ is conjugate to $N_{ns}(\ell)$ in $\GL_2(\FF_\ell)$.
\item \label{P:j=0 situation iii}
Suppose that $\ell$ is congruent to $4$ or $7$ modulo $9$. Let $E'/\QQ$ be the elliptic curve over $\QQ$ defined by $y^2=x^3+16 \ell^e$, where $e\in \{1,2\}$ satisfies $ \frac{\ell-1}{3} \equiv e \pmod{3}$.
\noindent
If $E$ is not isomorphic to a quadratic twist of $E'$, then $\rho_{E,\ell}(\Gal_\QQ)$ is conjugate to $N_{s}(\ell)$ in $\GL_2(\FF_\ell)$.
\noindent
If $E$ is isomorphic to a quadratic twist of $E'$, then $\rho_{E,\ell}(\Gal_\QQ)$ is conjugate in $\GL_2(\FF_\ell)$ to the subgroup $G$ of $N_s(\ell)$ consisting of the matrices of the form $\left(\begin{smallmatrix} a & 0 \\0 & b\end{smallmatrix}\right)$ or $\left(\begin{smallmatrix} 0 & a \\b & 0 \end{smallmatrix}\right)$ with $a/b \in (\FF_\ell^\times)^3$.
\item
\label{P:j=0 situation iv}
Suppose that $\ell$ is congruent to $2$ or $5$ modulo $9$. Let $E'/\QQ$ be the elliptic curve over $\QQ$ defined by $y^2=x^3+16 \ell^e$, where $e\in \{1,2\}$ satisfies $ \frac{\ell+1}{3} \equiv -e \pmod{3}$.
\noindent
If $E$ is not isomorphic to a quadratic twist of $E'$, then $\rho_{E,\ell}(\Gal_\QQ)$ is conjugate to $N_{ns}(\ell)$ in $\GL_2(\FF_\ell)$.
\noindent
If $E$ is isomorphic to a quadratic twist of $E'$, then $\rho_{E,\ell}(\Gal_\QQ)$ is conjugate in $\GL_2(\FF_\ell)$ to the subgroup $G$ of $N_{ns}(\ell)$ generated by the unique index $3$ subgroup of $C_{ns}(\ell)$ and by $\left(\begin{smallmatrix} 1 & 0 \\0 & -1\end{smallmatrix}\right)$.
\item \label{P:j=0 situation v}
Suppose that $\ell=3$. The curve $E$ can be given by a Weierstrass equation $y^2=x^3+d$ for some $d\in \QQ^\times$. Fix notation as in \S\ref{SS:applicable 3}.
\noindent
If $d$ or $-3d$ is a square and $-4d$ is a cube, then $\rho_{E,3}(\Gal_\QQ)$ is conjugate to $H_{1,1}$.
\noindent
If $d$ and $-3d$ are not squares and $-4d$ is a cube, then $\rho_{E,3}(\Gal_\QQ)$ is conjugate to $G_1$.
\noindent
If $d$ is a square and $-4d$ is not a cube, then $\rho_{E,3}(\Gal_\QQ)$ is conjugate to $H_{3,1}$.
\noindent
If $-3d$ is a square and $-4d$ is not a cube, then $\rho_{E,3}(\Gal_\QQ)$ is conjugate to $H_{3,2}$.
\noindent
If $d$ and $-3d$ are not squares and $-4d$ is not a cube, then $\rho_{E,3}(\Gal_\QQ)$ is conjugate to $G_3$.
\end{romanenum}
\end{prop}
\subsection{Overview} \label{SS:overview}
We now give a very brief overview of the paper. In \S\ref{SS:applicable}, we describe \defi{applicable subgroups} $G$ of $\GL_2(\FF_\ell)$; these groups have many of the properties that the groups $\pm \rho_{E,\ell}(\Gal_\QQ)$ do.
In \S\ref{S:modular}, we recall what we need concerning the modular curve $X_G/\QQ$; we will identify its function field with a subfield of the field of modular function for the congruence subgroup $\Gamma(\ell)$.
In \S\ref{S:main classification}, we prove the parts of our main theorems that determine $\pm \rho_{E,\ell}(\Gal_\QQ)$. We describe the rational points of $X_G$ when $\ell$ is small. When $X_G$ has genus $0$ and $X_G(\QQ)\neq \emptyset$, then the function field of $X_G$ is of the form $\QQ(h)$ for some modular function $h$. Much of this section is dedicated to describing such $h$ and determining the rational function $J(t) \in \QQ(t)$ such that $J(h)$ is the modular $j$-invariant.
Assuming that $G:=\pm \rho_{E,\ell}(\Gal_\QQ)$ is known, with $E/\QQ$ non-CM, we describe in \S\ref{S:twist 1} how to determine the (finite number of) quadratic twists of $E'$ of $E$ for which $\rho_{E',\ell}(\Gal_\QQ)$ is not conjugate to $G$. In \S\ref{S:twists 2}, we prove the parts of our main theorems that determine $\rho_{E,\ell}(\Gal_\QQ)$ given $\pm \rho_{E,\ell}(\Gal_\QQ)$.
In \S\ref{SS:CM proofs}, we prove the propositions from \S\ref{SS:CM} concerning CM elliptic curves defined over $\QQ$. The $j$-invariant $0$ case requires special attention since one has to worry about cubic twists. Finally, in \S\ref{S:big inertia last}, we prove Proposition~\ref{P:big inertia last}.
The equations in \S\ref{SS:applicable 2}--\ref{SS:applicable 17} and \texttt{Magma} code verifying some claims in \S\ref{S:main classification} and \S\ref{S:twists 2} can be found at:
\begin{center}\url{http://www.math.cornell.edu/~zywina/papers/PossibleImages/}\end{center}
\subsection*{Acknowledgments}
Thanks to Andrew Sutherland, David Zureick-Brown and Ren\'e Schoof. The computations in this paper were performed using the \texttt{Magma} computer algebra system \cite{Magma}.
\section{Applicable subgroups} \label{SS:applicable}
Fix an integer $N\geq 2$. For an elliptic curve $E/\QQ$, let $E[N]$ be the $N$-torsion subgroup of $E(\Qbar)$. After choosing a basis for $E[N]$ as a $\ZZ/N\ZZ$-module, the natural $\Gal_\QQ$-action on $E[N]$ can be expressed in terms of a Galois representation
\[
\rho_{E,N}\colon \Gal_\QQ \to \GL_2(\ZZ/N\ZZ).
\]
When $N$ is a prime, these agree with the representations of \S\ref{S:classification}. We now describe some restrictions on the possible images of $\rho_{E,N}$.
\begin{definition}
We say that a subgroup $G$ of $\GL_2(\ZZ/N\ZZ)$ is \defi{applicable} if it satisfies the following conditions:
\begin{itemize}
\item $G\neq \GL_2(\ZZ/N\ZZ)$,
\item $-I \in G$ and $\det(G)=(\ZZ/N\ZZ)^\times$,
\item $G$ contains an element with trace $0$ and determinant $-1$ that fixes a point in $(\ZZ/N\ZZ)^2$ of order $N$.
\end{itemize}
\end{definition}
This definition is justified by the following.
\begin{prop} \label{P:basic applicable}
Let $E$ be an elliptic curve over $\QQ$ for which $\rho_{E,N}$ is not surjective. Then $\pm \rho_{E,N}(\Gal_\QQ)$ is an applicable subgroup of $\GL_2(\ZZ/N\ZZ)$.
\end{prop}
\begin{proof}
The group $G:=\pm \rho_{E,N}(\Gal_\QQ)$ clearly contains $-I$. The character $\det\circ \rho_{E,N}\colon \Gal_\QQ \to (\ZZ/N\ZZ)^\times$ is the surjective homomorphism describing the Galois action on the group of $N$-th roots of unity in $\Qbar$, i.e., for a $N$-th root of unity $\zeta\in \Qbar$, we have $\sigma(\zeta)=\zeta^{\det(\rho_{E,N}(\sigma))}$ for all $\sigma\in \Gal_\QQ$. Therefore, $\det\circ \rho_{E,N}$ is surjective and hence $\det(G)=(\ZZ/N\ZZ)^\times$.
Let $c \in \Gal(\Qbar/\QQ)$ be an automorphism corresponding to complex conjugation under some embedding $\Qbar \hookrightarrow \CC$. Set $g:=\rho_{E,N}(c)$. As a topological group, the connected component of $E(\RR)$ containing the identity is isomorphic to $\RR/\ZZ$. Therefore, $E(\RR)$ contains a point $P_1$ of order $N$. We may assume that $\rho_{E,N}$ is chosen with respect to a basis whose first term is $P_1$, and hence $g$ is upper triangular whose first diagonal term is $1$. We have $\det(g)=-1$ since $c$ acts by inversion on $N$-th roots of unity. Therefore, $g$ is upper triangular with diagonal entries $1$ and $-1$, and hence $\tr(g)=0$.
Now suppose that $G=\GL_2(\ZZ/N\ZZ)$. Define $S=\rho_{E,N}(\Gal_\QQ) \cap \SL_2(\ZZ/N\ZZ)$. Since $G=\GL_2(\ZZ/N\ZZ)$, $\rho_{E,N}(\Gal_\QQ)\neq \GL_2(\ZZ/N\ZZ)$ and $\det(\rho_{E,N}(\Gal_\QQ))=(\ZZ/N\ZZ)^\times$, we deduce that $S\neq \SL_2(\ZZ/N\ZZ)$ and $\pm S=\SL_2(\ZZ/N\ZZ)$. However, this is impossible by Lemma~\ref{L:proper SL2} below, so we must have $G\neq \GL_2(\ZZ/N\ZZ)$.
\end{proof}
\begin{lemma} \label{L:proper SL2}
There is no proper subgroup $S$ of $\SL_2(\ZZ/N\ZZ)$ such that $\pm S=\SL_2(\ZZ/N\ZZ)$.
\end{lemma}
\begin{proof}
Suppose that $S$ is a subgroup of $\SL_2(\ZZ/N\ZZ)$ for which $\pm S=\SL_2(\ZZ/N\ZZ)$. By \cite{MR2721742}*{Lemma A.6}, we deduce that there is a prime power $\ell^e$ dividing $N$ such that the image $\tilde{S}$ of $S$ in $\SL_2(\ZZ/\ell^e\ZZ)$ is a proper subgroup satisfying $\pm \tilde{S} = \SL_2(\ZZ/\ell^e\ZZ)$. So without loss of generality, we may assume that $N=\ell^e$.
The group $S$ has index $2$ in $\SL_2(\ZZ/\ell^e\ZZ)$. Therefore, $S$ is normal in $\SL_2(\ZZ/\ell^e\ZZ)$ and the quotient is cyclic of order $2$. However, the abelianization of $\SL_2(\ZZ/\ell^e\ZZ)$ is a cyclic group of order $\gcd(\ell^e,12)$, cf.~\cite{MR2721742}*{Lemma A.1}. Therefore, we must have $\ell=2$. Since the abelianization of $\SL_2(\ZZ/2^e\ZZ)$ is cyclic of order $2$ or $4$, we find that $S$ is the unique subgroup of $\SL_2(\ZZ/2^e\ZZ)$ of index $2$. The group $S$ is now easy to describe; it is the group of elements in $\SL_2(\ZZ/2^e\ZZ)$ whose image in $\SL_2(\ZZ/2\ZZ)$ lies in the unique cyclic group of order $3$. However, this implies that $\pm S \neq \SL_2(\ZZ/2^e\ZZ)$ since $-I \equiv I \pmod{2}$. This contradiction ensures that no such $S$ exists.
\end{proof}
\begin{remark}
When $N$ is a prime $\ell$, which is the setting of this paper, the last condition in the definition of applicable subgroup can be simplified to say simply that $G$ contains an element with trace $0$ and determinant $-1$.
\end{remark}
\section{Modular curves} \label{S:modular}
Fix an integer $N\geq 1$; in our later application, we will take $N$ to be a prime $\ell$. In \S\ref{SS:modular functions intro}, we recall the Galois theory of the field of modular functions of level $N$. In \S\ref{SS:modular curves intro}, we define modular curves in terms of their functions fields. We take an unsophisticated approach to modular curves and develop what we need from Shimura's book \cite{MR1291394}; it will be useful for reference in future work. Alternatively, one could develop modular curves as in \cite{MR0337993}*{IV-3}.
\subsection{Modular functions of level $N$} \label{SS:modular functions intro}
The group $\SL_2(\ZZ)$ acts on the complex upper half plane $\mathfrak{h}$ via linear fractional transformations, i.e., $\gamma_*(\tau) = (a\tau+b)/(c\tau +d )$ for $\gamma= \left(\begin{smallmatrix}a & b \\ c & d \end{smallmatrix}\right) \in \SL_2(\ZZ)$ and $\tau\in \mathfrak{h}$. Let $\Gamma(N)$ be the congruence subgroup consisting of matrices in $\SL_2(\ZZ)$ that are congruent to $ I$ modulo $N$. The quotient $\Gamma(N) \backslash\mathfrak{H}$ is a Riemann surface and can be completed to a compact and smooth Riemann surface $X_{N}$. Let $\tau$ be a variable of the complex upper half plane.
Every meromorphic function $f$ on $X_{N}$ has a $q$-expansion $\sum_{n\in \ZZ} c_n q^{n/N}$; here the $c_n$ are complex numbers which are $0$ for all but finitely many negative $n$ and $q^{1/N}:= e^{2\pi i \tau/N}$. We define $\calF_N$ to be the field of meromorphic functions on $X_{N}$ whose $q$-expansion has coefficients in $\QQ(\zeta_N)$, where $\zeta_N$ is the $N$-th root of unity $e^{2\pi i/N}$. For example, $\calF_1=\QQ(j)$ where $j=j(\tau)$ is the modular $j$-invariant with the familiar expansion
\[
j=q^{-1}+ 744 + 196884q + 21493760q^2 + 864299970q^3 + \ldots.
\]
For each $d\in (\ZZ/N\ZZ)^\times$, let $\sigma_d$ be the automorphism of the field $\QQ(\zeta_N)$ for which $\sigma_d(\zeta_N)=\zeta_N^d$. We extend $\sigma_d$ to an automorphism of $\calF_N$ by taking a function with $q$-expansion $\sum_n c_n q^{n/N}$ to $\sum_n \sigma_d(c_n) q^{n/N}$. We let $\SL_2(\ZZ)$ act on $\calF_N$ by taking a modular function $f\in \calF_N$ and a matrix $\gamma\in \SL_2(\ZZ)$ to $f\circ \gamma^t$, i.e., the function $(f\circ \gamma^t)(\tau)=f(\gamma^t_*(\tau))$ where $\gamma^t$ is the transpose of $\gamma$.
\begin{prop} \label{P:modular galois}
The extension $\calF_N$ of $\QQ(j)$ is Galois. There is a unique isomorphism
\[
\theta_N \colon \GL_2(\ZZ/N\ZZ)/\{\pm I\} \xrightarrow{\sim} \Gal(\calF_N/\QQ(j))
\]
such that the following holds for all $f\in \calF_N$:
\begin{alphenum}
\item \label{P:modular galois a}
For $A\in \SL_2(\ZZ/N\ZZ)$, we have $\theta_N(A) f = f\circ \gamma^t$, where $\gamma$ is any matrix in $\SL_2(\ZZ)$ that is congruent to $A$ modulo $N$.
\item \label{P:modular galois b}
For $A=\left(\begin{smallmatrix}1 & 0 \\ 0 & d \end{smallmatrix}\right) \in \GL_2(\ZZ/N\ZZ)$, we have $\theta_N(A) f = \sigma_d(f)$.
\end{alphenum}
The field $\QQ(\zeta_N)$ is the algebraic closure of $\QQ$ in $\calF_N$ and corresponds to the subgroup $\SL_2(\ZZ/N\ZZ)/\{\pm I\}$.
\end{prop}
We will sketch Propostion~\ref{P:modular galois} in \S\ref{SS:modular proof 1}. Throughout the paper, we will let $\GL_2(\ZZ/N\ZZ)$ act on $\calF_N$ via the isomorphism $\theta_N$ (with $-I$ acting trivially).
\begin{remark}
There are other choices for an isomorphism $\GL_2(\ZZ/N\ZZ)/\{\pm I\}$; for example, one could instead replace the transpose by an inverse in (\ref{P:modular galois a}). Our choice is explained by our application to modular curves. As a warning, there are several places in the literature where incompatible choices are made with respect to modular curves.
\end{remark}
\subsection{Modular curves} \label{SS:modular curves intro}
Let $G$ be a subgroup of $\GL_2(\ZZ/N\ZZ)$ containing $-I$ that satisfies $\det(G)=(\ZZ/N\ZZ)^\times$. Denote by $\calF_N^G$ the subfield of $\calF_N$ fixed by the action of $G$ from Proposition~\ref{P:modular galois}. Using Proposition~\ref{P:modular galois} and the assumption $\det(G)=(\ZZ/N\ZZ)^\times$, we find that $\QQ$ is algebraically closed in $\calF_N^G$.
Let $X_G$ be the smooth projective curve with function field $\calF_N^G$; it is defined over $\QQ$ and is geometrically irreducible. The inclusion of fields $ \calF_N^G \supseteq \QQ(j)$ gives rise to a non-constant morphism
\[
\pi_G \colon X_G \to \Spec \QQ[j] \cup \{\infty\} =\PP^1_\QQ.
\]
The morphism $\pi_G$ is non-constant and we have
\[
\deg(\pi_G)=[\GL_2(\ZZ/N\ZZ)/\{\pm I\}: G/\{\pm I\}]=[\GL_2(\ZZ/N\ZZ): G].
\]
We will also denote the function field $\calF_N^G$ of $X_G$ by $\QQ(X_G)$. A point in $X_G$ is a \defi{cusp} or a \defi{CM point} if $\pi_G$ maps it to $\infty$ or to the $j$-invariant of a CM elliptic curve, respectively.\\
The following property of the curve $X_G$ is key to our application; we will give a proof in \S\ref{SS:modular proof 2}.
\begin{prop} \label{P:main moduli}
Let $G$ be a subgroup of $\GL_2(\ZZ/N\ZZ)$ that contains $-I$ and satisfies $\det(G)=(\ZZ/N\ZZ)^\times$. Let $E$ be an elliptic curve defined over $\QQ$ with $j_E\not\in\{0,1728\}$. Then $\rho_{E,N}(\Gal_\QQ)$ is conjugate in $\GL_2(\ZZ/N\ZZ)$ to a subgroup of $G$ if and only if $j_E$ belongs to $\pi_G(X_G(\QQ))$.
\end{prop}
The following lemma will be key to finding modular curves of genus $0$ with rational points.
\begin{lemma} \label{L:key}
Fix a modular function $h\in \calF_N - \QQ(j)$ such that $J(h)=j$ for a rational function $J(t) \in \QQ(t)$. Let $G$ be the subgroup of $\GL_2(\ZZ/N\ZZ)$ that fixes $h$ under the action on $\calF_N$ from Proposition~\ref{P:modular galois}.
\begin{romanenum}
\item \label{L:key c}
The subgroup $G$ of $\GL_2(\ZZ/N\ZZ)$ is applicable.
\item \label{L:key a}
The modular curve $X_G$ has function field $\QQ(h)$. In particular, it is isomorphic to $\PP^1_\QQ$.
\item \label{L:key b}
Let $E/\QQ$ be an elliptic curve with $j_E\notin\{0,1728\}$. The group $\rho_{E,N}(\Gal_\QQ)$ is conjugate in $\GL_2(\ZZ/N\ZZ)$ to a subgroup of $G$ if and only if $j_E=J(t)$ for some $t\in \QQ\cup \{\infty\}$.
\end{romanenum}
\end{lemma}
\begin{proof}
By the Galois correspondence coming from the isomorphism $\theta_N$ of Proposition~\ref{P:modular galois}, the field $\QQ(h)$ equals $\calF_N^{G}$ and is an extension of $\QQ(j)$ of degree
\[
[\GL_2(\ZZ/N\ZZ)/\{\pm I\}:G/\{\pm I\}]=[\GL_2(\ZZ/N\ZZ):G].
\]
The field $\QQ$ is algebraically closed in $\calF_N^G = \QQ(h)$, so $\det(G)=(\ZZ/N\ZZ)^\times$ by Proposition~\ref{P:modular galois}. Therefore, $\QQ(h)$ is the function field of $X_G$ and the field extension $\QQ(h)/\QQ(j)$ given by $j=J(h)$ corresponds to the morphism $\pi_G\colon X_G\to \PP^1_\QQ$. The modular curve $X_G$ is thus isomorphic to $\PP^1_\QQ$ and we have $\pi_G(X_G(\QQ))=J(\QQ\cup\{\infty\})$. This proves (\ref{L:key a}). Part (\ref{L:key b}) follows from Proposition~\ref{P:main moduli}.
Finally, we prove that $G$ is applicable. We have $G\neq \GL_2(\ZZ/N\ZZ)$ since the extension $\QQ(h)/\QQ(j)$ is non-trivial by our assumption on $h$. Using part (\ref{L:key b}) and Proposition~\ref{P:basic applicable}, we find that $G$ contains an applicable subgroup. Since $G\neq \GL_2(\ZZ/N\ZZ)$ and $G$ contains an applicable subgroup, we deduce that $G$ is applicable.
\end{proof}
If $X_G$ has genus $0$ and has rational points, then there are in fact curves $E/\QQ$ with $\pm\rho_{E,N}(\Gal_\QQ)$ conjugate to $G$.
\begin{lemma} \label{L:basic HIT}
Suppose that $X_G$ is isomorphic to $\PP^1_\QQ$; equivalently, the function field of $X_G$ is of the form $\QQ(h)$. We have $j=J(h)$ for a unique $J(t) \in \QQ(t)$ because of the inclusion $\QQ(h)\supseteq \QQ(j)$. Then for ``most'' $u \in \QQ$ (more precisely, outside a set of density $0$ with respect to height), the groups $\pm \rho_{E,N}(\Gal_\QQ)$ and $G$ are conjugate in $\GL_2(\ZZ/N\ZZ)$ for any elliptic curve $E/\QQ$ with $j$-invariant $J(u)$.
\end{lemma}
\begin{proof}
Let $\calG$ be the (finite) set of applicable subgroups $H$ of $\GL_2(\ZZ/N\ZZ)$ satisfying $H \subsetneq G$. For each $H \in \calG$, let $\pi_{H,G}$ be the natural morphism $X_H \to X_G$; it has degree $[G:H] >1$. To prove the lemma, it suffices to show that the set $\calS:=\cup_{H \in \calH} \pi_{H,G}(X_H(\QQ))$ has density $0$ (with respect to the height) in $X_G(\QQ) \cong \PP^1(\QQ)$. This is a consequence of Hilbert irreducibility; in the language of \cite{MR1757192}*{\S9}, the set $\calS$ is \emph{thin} and hence has density $0$.
\end{proof}
\subsection{The modular curve $X_0(N)$}
Let $X_0(N)/\QQ$ be the modular curve $X_{B(N)^t}$, where $B(N)^t$ is the transpose of $B(N)$; it consists of the lower triangular matrices and is conjugate to $B(N)$ in $\GL_2(\ZZ/N\ZZ)$. Let $\Gamma_0(N)$ be the group of matrices in $\SL_2(\ZZ)$ whose image modulo $N$ is upper triangular. A function $f\in \calF_N$ belongs to $\QQ(X_0(N))$ if and only if it has rational Fourier coefficients and $f\circ \gamma =f$ for all $\gamma\in \Gamma_0(N)$.
Define the modular curve $X_s(N) := X_{C_s(N)}$, where $C_s(N)$ is the subgroup of diagonal matrices in $\GL_2(\ZZ/N\ZZ)$.
\begin{lemma} \label{L:borel to split}
The map $\QQ(X_0(N^2)) \to \QQ(X_s(N))$, $f(\tau)\mapsto f(\tau/N)$ is an isomorphism of fields. This isomorphism induces an isomorphism between the modular curves $X_s(N)$ and $X_0(N^2)$ which gives a bijection between their cusps.
\end{lemma}
\begin{proof}
Let $\Gamma_s(N)$ be the group of matrices in $\SL_2(\ZZ)$ whose image modulo $N$ is diagonal. The function field of $X_s(N)$ then consist of the $f\in \calF_N$ with rational Fourier coefficients for which $f\circ \gamma = f$ for all $\gamma$ in $\Gamma_s(N)$.
Define $w=\left(\begin{smallmatrix} 1 & 0 \\0 & N \end{smallmatrix}\right)$; it acts on $\mathfrak{h}$ by linear fractional transformation, i.e., $w_*(\tau) = \tau/N$. Take any $f\in \calF_N$ whose Fourier coefficients are rational. We have $f \circ w$ in $\QQ(X_s(N))$ if and only if $f \circ w \circ \gamma = f\circ w$ for all $\gamma \in \Gamma_s(N)$. Since $w \Gamma_s(N) w^{-1} = \Gamma_0(N^2)$, we deduce that $f\circ w$ belongs to $\QQ(X_s(N))$ if and only if $f$ belongs to $\QQ(X_0(N^2))$. It is now straightforward to show that the map of fields is well-defined and an isomorphism. The isomorphism of function fields of course induces an isomorphism of the corresponding curves. That the cusps are in correspondence is a consequence of the map $\Gamma_0(N^2)\backslash \mathfrak{h} \to \Gamma_s(N)\backslash \mathfrak{h}$, $\tau \to w_*(\tau)=\tau/N$ being an isomorphism of Riemann surfaces.
\end{proof}
\begin{lemma} \label{L:hauptmodul}
Let $\eta(\tau)$ be the Dedekind eta function.
\begin{romanenum}
\item \label{L:hauptmodul i}
We have $\QQ(X_0(4))=\QQ(h)$, where $h(\tau) = \eta(\tau)^8/\eta(4\tau)^8$.
\item \label{L:hauptmodul ii}
We have $\QQ(X_0(9))=\QQ(h)$, where $h(\tau) = \eta({\tau})^3/\eta(9\tau)^3$.
\end{romanenum}
\end{lemma}
\begin{proof}
This is well-known; for example see \cite{modular-towers}.
\end{proof}
\subsection{Proof of Proposition \ref{P:modular galois}}
\label{SS:modular proof 1}
For $\tau\in \mathfrak{H}$, let $\Lambda_\tau$ be the lattice $\ZZ \tau + \ZZ$ in $\CC$. Set $g_2(\tau)=g_2(\Lambda_\tau)$ and $g_3(\tau)=g_3(\Lambda_\tau)$, and let $\wp(z;\tau)$ be the Weierstrass $\wp$-function relative to $\Lambda_\tau$, cf.~\cite{MR2514094}*{{\S}VI.3} for background on elliptic functions. For each pair $a=(a_1,a_2) \in N^{-1}\ZZ^2-\ZZ^2$, define the function
\[
f_{a}(\tau) := \frac{g_2(\tau)g_3(\tau)}{g_2(\tau)^3-27 g_3(\tau)^2} \cdot \wp(a_1\tau + a_2; \tau)
\]
of the upper half plane. The function $f_a$ is modular of level $N$. Moreover, Proposition~6.9(1) of \cite{MR1291394} says that
\begin{equation} \label{E:FN gen}
\calF_N = \QQ\big(j , f_a\, \big| \, a\in N^{-1}\ZZ^2-\ZZ^2 \big).
\end{equation}
For $a,b\in N^{-1}\ZZ^2-\ZZ^2$, we have $f_a=f_b$ if and only if $a$ lies in the same coset of $\QQ^2/\ZZ^2$ as $b$ or $-b$, cf.~equation (6.1.5) of \cite{MR1291394}. So for any $A \in M_2(\ZZ)$ with determinant relatively prime to $N$, the function $f_{aA}$ depends only on the image $\tilde{A}$ of $A$ in $\GL_2(\ZZ/N\ZZ)/\{\pm 1\}$. By abuse of notation, we shall denote $f_{aA}$ by $f_{a\tilde{A}}$.
By Theorem~6.6 of \cite{MR1291394}, there is a unique isomorphism
\[
\theta_N\colon \GL_2(\ZZ/N\ZZ)/\{\pm I\} \xrightarrow{\sim} \Gal(\calF_N/\QQ(j))
\]
such that $\theta_N(A) f_a = f_{aA^t}$ for all $A\in \GL_2(\ZZ/N\ZZ)/\{\pm I\}$ and $a\in N^{-1} \ZZ^2-\ZZ^2$; we have added the transpose so the map is a homomorphism (and not an antihomorphism).
Fix any $\gamma\in \SL_2(\ZZ)$ and let $A\in \SL_2(\ZZ/N\ZZ)$ be its image modulo $N$. For any $a\in N^{-1} \ZZ^2-\ZZ^2$, the function $f_a \circ \gamma^t$ agrees with $f_{a\cdot \gamma^t} = \theta_N(A) f_a$ by equation (6.1.3) of \cite{MR1291394}. Using (\ref{E:FN gen}), we deduce that $\theta_N(A) f = f \circ \gamma^t$ for all $f\in \calF_N$; this shows that (\ref{P:modular galois a}) holds.
Now take integer $d$ relatively prime to $N$ and let $A$ be the image of $\left(\begin{smallmatrix}1 & 0 \\0 & d \end{smallmatrix}\right)$ in $\GL_2(\ZZ/N\ZZ)$. Take any $a\in N^{-1}\ZZ^2-\ZZ^2$; we have $a=(r/N,s/N)$ with $r,s\in \ZZ$. Since $f_a = f_b$ when $a\equiv b \pmod{\ZZ^2}$, we may assume that $0 \leq r <N$. We have $\theta_N(A)f_{a}= f_{aA^t}=f_{(r/N,ds/N)}$.
By equation (6.2.1) of \cite{MR1291394}, we have
\begin{align*}
(2\pi)^{-2}\wp(a_1\tau +a_2 ; \tau) =& -1/12+2{\sum}_{n=1}^\infty nq^n/(1-q^n) - \zeta_N^s q^{r/N}/(1- \zeta_N^s q^{r/N})^2\\
& -{\sum}_{n=1}^\infty (\zeta_N^{ns} q^{nr/N} + \zeta_N^{-ns} q^{-nr/N}) \cdot nq^n/(1-q^n);
\end{align*}
applying $\sigma_d$ to this series gives the same thing with $s$ replaced by $ds$. The Fourier coefficients of the expansion of $g_2(\tau)/g_3(\tau)$ are all $\pi^{-2}$ times a rational number. Therefore, $\sigma_d(f_a)=\sigma_d(f_{(r/N,s/N)})$ equals $f_{(r/N,ds/N)} = f_{a A^t}$. Using (\ref{E:FN gen}), we deduce that $\theta_N(A) f = \sigma_d(f)$ for all $f\in \calF_N$; this shows that (\ref{P:modular galois b}) holds.
This explains the existence of an isomorphism $\theta_N$ as in the statement of Proposition~\ref{P:modular galois}. The uniqueness if immediate since $\GL_2(\ZZ/N\ZZ)$ is generated by $\SL_2(\ZZ/N\ZZ)$ and matrices of the form $\left(\begin{smallmatrix}1 & 0 \\0 & * \end{smallmatrix}\right)$. Theorem~6.6 of \cite{MR1291394} implies that $\QQ(\zeta_N)$ is the algebraic closure of $\QQ$ in $\calF_N$ and that $\theta_N(A) \zeta_N = \zeta_N^{\det A}$.
\subsection{Proof of Proposition~\ref{P:main moduli}}
\label{SS:modular proof 2}
We first construct the inverse of $\theta_N$ using elliptic curves; we shall freely use definitions from \S\ref{SS:modular proof 1}. Let $E$ be an elliptic curve defined over an algebraically closed field $k$ of characteristic 0. Take any non-zero $N$-torsion point $P\in E(k)$. If $P=(x_0,y_0)$ with respect to some Weierstrass model $y^2=4x^3 -c_2 x -c_3$ of $E/k$, define $h_E(P):= c_2 c_3/(c_2^3-27 c_3^2) \cdot x_0$. If $j_E \notin \{0,1728\}$, then one can show that $h_E(P)$ does not depend on the choice of model.
\\
Let $\calE$ be the elliptic curve over $\calF_1=\QQ(j)$ defined by the Weierstrass equation
\begin{equation} \label{E:generic Weierstrass}
y^2= 4x^3 - \frac{27j}{j-1728} x - \frac{27j}{j-1728};
\end{equation}
it has $j$-invariant $j$. Fix an algebraic closed field $K$ that contains $\calF_N\supseteq \QQ(j)$ and let $\calE[N]$ be the $N$-torsion subgroup of $\calE(K)$.
\begin{lemma} \label{L:basis P1P2}
There is a basis $\{P_1, P_2\}$ of the $\ZZ/N\ZZ$-module $\calE[N]$ such that $h_\calE( r P_1 +s P_2) = f_{(r/N,s/N)}$ for all $(r,s)\in \ZZ^2-N \ZZ^2$.
\end{lemma}
\begin{proof}
Let $K_0$ be the extension of $\calF_N$ generated by the functions $g_2(\tau)$, $g_3(\tau)$, $\wp({\tau}/{N}; \tau)$, $\wp'(\tau/N;\tau)$, $\wp({1}/{N}; \tau)$ and $\wp'(1/N;\tau)$. We may assume that $K \supseteq K_0$. Let $E$ be the elliptic curve over $K_0$ defined by $y^2=4x^3 -g_2(\tau) x -g_3(\tau)$; its $j$-invariant is $j=j(\tau)$. The curves $E$ and $\calE$ are isomorphic over $K$ since they both have $j$-invariant $j$. Since $j\notin\{0,1728\}$, it suffices to prove the lemma for $E$ instead of $\calE$.
Define the pairs
\[
P_1 := (\wp({\tau}/{N}; \tau), \wp'(\tau/N;\tau) ) \quad \text{ and }\quad P_2 := (\wp({1}/{N}; \tau), \wp'(1/N;\tau) ).
\]
We claim that $P_1$ and $P_2$ form a basis of the $\ZZ/N\ZZ$-module of $N$-torsion in $E(K)$. To prove the claim it suffices to prove the analogous results after specializing the coefficients of $E$ and the entries of $P_1$ and $P_2$ by an arbitrary $\tau_0 \in \mathfrak{h}$ (since the claim comes down to verifying certain polynomial equations whose variables are the coefficients of the model of $E$ and the entries of the points). So fix an arbitrary $\tau_0 \in \mathfrak{h}$. Specializing the model of $E$ at $\tau_0$ gives an elliptic curve $E_{\tau_0}$ over $\CC$ defined by $y^2=4x^3 -g_2(\tau_0) x -g_3(\tau_0)$. From Weierstrass, we know that the map
\[
\CC/\Lambda_{\tau_0} \to E_{\tau_0}(\CC),\quad z\mapsto (\wp(z;\tau_0),\wp'(z;\tau_0)),
\]
with $0$ mapping to the point at infinity, gives an isomorphism of complex Lie groups. In particular, the points $P_{1,\tau_0}= (\wp({\tau_0}/{N}; \tau_0), \wp'(\tau_0/N;\tau_0) )$ and $P_{2,\tau_0}= (\wp({1}/{N}; \tau_0), \wp'(1/N;\tau_0))$ give a basis for the $N$-torsion in $E_{\tau_0}(\CC)$. This is enough to prove our claim. Moreover, we have $r P_{1,\tau_0} + s P_{2,\tau_0} = (\wp(r/N\cdot \tau_0 + s/N;\tau_0),\wp'(r/N \cdot \tau_0 + s/N;\tau_0))$ for all $(r,s)\in \ZZ^2-N\ZZ^2$. Therefore,
\[
h_{E_{\tau_0}}(rP_1+sP_2) = g_2(\tau_0) g_3(\tau_0)/(g_2(\tau_0)^3-27g_3(\tau_0)^2) \cdot \wp(r/N \cdot \tau_0 + s/N;\tau_0) = f_{(r/N,s/N)}(\tau_0).
\]
for all $(r,s)\in \ZZ^2-N\ZZ$. Since this holds for all $\tau_0\in \mathfrak{h}$, we deduce that $h_{E}( r P_1 +s P_2 ) = f_{(r/N,s/N)}$.
\end{proof}
Let $\rho_N \colon \Gal(K/\QQ(j)) \to \GL_2(\ZZ/N\ZZ)$ be the representation describing the Galois action on $\calE[N]$ with respect to the basis $\{P_1,P_2\}$ of Lemma~\ref{L:basis P1P2}. The fixed field of the kernel of $\Gal(K/\QQ(j)) \xrightarrow{\rho_N} \GL_2(\ZZ/N\ZZ)\to \GL_2(\ZZ/N\ZZ)/\{\pm I\}$ is generated by the $x$-coordinates of the non-zero points in $\calE[N]$. By (\ref{E:FN gen}) and Lemma~\ref{L:basis P1P2}, the extension $\calF_N$ of $\QQ(j)$ is generated by the $x$-coordinates of the non-zero points in $\calE[N]$. Therefore, the representation $\rho_N$ induces an injective homomorphism
\begin{equation} \label{E:Galois FN}
\bbar\rho_N \colon \Gal(\calF_N/\QQ(j)) \hookrightarrow \GL_2(\ZZ/N\ZZ)/\{\pm I\}.
\end{equation}
In fact, (\ref{E:Galois FN}) is an isomorphism since the groups have the same cardinality by Proposition~\ref{P:modular galois}.
\begin{lemma} \label{L:inverse of thetaN}
The homomorphism $\bbar\rho_N$ is an isomorphism. Moreover, the inverse of $\bbar\rho_N$ is the homomorphism $\theta_N\colon \GL_2(\ZZ/N\ZZ)/\{\pm I\} \to \Gal(\calF_N/\QQ(j))$.
\end{lemma}
\begin{proof}
Take any $\sigma\in \Gal(K/\QQ(j))$ and set $\tilde\sigma:=\sigma|_{\calF_N}$. There are integers $a,b,c,d\in \ZZ$ such that $\sigma(P_1)=a P_1 + cP_2$ and $\sigma(P_2)=bP_1+dP_2$, so $\rho_N(\sigma)=A$, where $A\in \GL_2(\ZZ/N\ZZ)$ is the image of $\left(\begin{smallmatrix}a & b \\c & d \end{smallmatrix}\right)$ modulo $N$. Therefore, $\bbar\rho_N(\tilde\sigma)$ is the class of $A$ in $\GL_2(\ZZ/N\ZZ)/\{\pm I\}$. We need to show that $\theta_N(A)=\tilde\sigma$.
Take any pair of integers $(r,s)\in \ZZ^2-N\ZZ^2$. We have
\[
\sigma(rP_1 +sP_2)=r\sigma(P_1) +s \sigma(P_2)= (ra+sb)P_1 + (rc+sd)P_2.
\]
Comparing $x$-coordinates and using Lemma~\ref{L:basis P1P2}, we find that $\tilde\sigma(f_{(r/N,s/N)})=\sigma(f_{(r/N,s/N)})$ is equal to $f_{((ra+sb)/N,(rc+sd)/N)} = f_{(r/N,s/N) A^t}$ which is $\theta_N(A) f_{(r/N,s/N)}$ from \S\ref{SS:modular proof 1}. Since the extension $\calF_N/\QQ(j)$ is generated by the functions $f_a$ with $a\in \ZZ^2-N\ZZ^2$, we deduce that $\theta_N(A)=\tilde\sigma$.
\end{proof}
Define the $\QQ$-variety
\[
U := \AA^1_\QQ-\{0,1728\} = \Spec \QQ[j,j^{-1},(j-1728)^{-1}];
\]
note that we are now viewing $j$ as simply a transcendental variable. The equation (\ref{E:generic Weierstrass}) defines a (relative) elliptic curve $\pi\colon \scrE \to U$. The fiber of $\scrE\to U$ over the generic fiber of $U$ is the elliptic curve $\calE/\QQ(j)$.
Let $\bbar{\eta}$ be the geometric generic point of $U$ corresponding to the algebraically closed extension $K$ of $\calF_N$. Let $\scrE[N]$ be the $N$-torsion subscheme of $\scrE$. We can identify the fiber of $\scrE[N]\to U$ at $\bbar\eta$ with the group $\calE[N]$. Let $\pi_1(U,\bbar{\eta})$ be the \'etale fundamental group of $U$. We can view $\scrE[N]$ as a rank $2$ lisse sheaf of $\ZZ/N\ZZ$-modules $U$ and it hence corresponds to a representation
\[
\varrho_N \colon \pi_1(U,\bbar{\eta}) \to \Aut(\calE[N])\cong \GL_2(\ZZ/N\ZZ)
\]
where the isomorphism uses the basis $\{P_1,P_2\}$ of Lemma~\ref{L:basis P1P2}. Taking the quotient by the group generated by $-I$, we obtain a homomorphism
\[
\bbar\varrho_N\colon \pi_1(U,\bbar\eta) \to \GL_2(\ZZ/N\ZZ)/\{\pm I\}.
\]
Note that the representation $\Gal(K/\QQ(j))\to \GL_2(\ZZ/N\ZZ)/\{\pm I\}$ coming from $\bbar\varrho_N$ factors through the homomorphism $\bbar\rho_N$. So by Proposition~\ref{P:modular galois} and Lemma~\ref{L:inverse of thetaN}, the representation $\bbar\varrho_N$ is surjective and satisfies $\bbar\varrho_N(\pi_1(U_{\Qbar}))=\SL_2(\ZZ/N\ZZ)/\{\pm I\}$.
Now take any subgroup $G$ of $\GL_2(\ZZ/N\ZZ)$ that satisfies $-I\in G$ and $\det(G)=(\ZZ/N\ZZ)^\times$. Using $\bbar\varrho_N$, the group $G/\{\pm I\}$ corresponds to an \'etale morphism $\pi\colon Y_G\to U$. The smooth projective closure of $Y_G$ is thus $X_G$ and the morphism $X_G\to \PP^1_\QQ$ arising from $\pi$ is simply $\pi_G$.
Take any rational point $u\in U(\QQ)=\QQ-\{0,1728\}$. Viewed as a morphism $\Spec \QQ \to U$, the point $u$ induces a homomorphism $u_*\colon \Gal_\QQ \to \pi_1(U)$; we are suppressing base points so everything is uniquely defined only up to conjugacy. Composing $u_*$ with $\bbar\varrho_N$ we obtain a homomorphism $\beta_u\colon \Gal_\QQ \to \GL_2(\ZZ/N\ZZ)/\{\pm I\}$. Observe that the group $\beta_u(\Gal_\QQ)$ is conjugate to a subgroup of $G/\{\pm I\}$ if and only if $u$ lies in $\pi_1(Y_G(\QQ))=\pi_G(X_G(\QQ))-\{0,1728,\infty\}$.
The fiber of $\scrE\to U$ over $u$ is the elliptic curve $\scrE_u/\QQ$ obtained by setting $j$ to $u$ in (\ref{E:generic Weierstrass}). Composing $\rho_{\scrE_u,N}\colon \Gal_\QQ \to \GL_2(\ZZ/N\ZZ)$ with the quotient map $\GL_2(\ZZ/N\ZZ)\to \GL_2(\ZZ/N\ZZ)/\{\pm I\}$ gives a homomorphism that agrees with $\beta_u$ up to conjugation. Since $-I \in G$, we find that $\rho_{\scrE_u,N}(\Gal_\QQ)$ is conjugate in $\GL_2(\ZZ/N\ZZ)$ to a subgroup of $G$ if and only if $u \in \pi_G(X_G(\QQ))$.
Finally, let $E/\QQ$ be any elliptic curve with $j$-invariant $u$. The curve $\scrE_u/\QQ$ also has $j$-invariant $u$. As noted in the introduction, since $E$ and $\scrE_u$ are elliptic curves over $\QQ$ with common $j$-invariant $u\notin \{0,1728\}$, the groups $\pm \rho_{E,N}(\Gal_\QQ)$ and $\pm \rho_{\scrE_u,N}(\Gal_\QQ)$ must be conjugate. This completes the proof of Proposition~\ref{P:main moduli}.
\section{Classification up to a sign} \label{S:main classification}
In this section, we prove the parts of the theorems of \S\ref{S:classification} that involve the groups $\pm \rho_{E,\ell}(\Gal_\QQ)$ for an elliptic curve $E/\QQ$. In the notation of \S\ref{SS:applicable}, the group $\pm \rho_{E,\ell}(\Gal_\QQ)$ is either applicable or is the full group $\GL_2(\FF_\ell)$. We consider the primes $\ell$ separately and keep the notation of the relevant subsection of \S\ref{S:classification}.
One of the main tasks is to construct modular curves of genus $0$. We will do this by finding functions $h\in \calF_\ell-\QQ(j)$ such that $j=J(h)$ for some $J\in \QQ(t)$. Let $H$ be the subgroup of $\GL_2(\FF_\ell)$ consisting of elements that fix $h$ under the action from Proposition~\ref{P:modular galois}. By Lemma~\ref{L:key}, the group $H$ is an applicable subgroup of $\GL_2(\FF_\ell)$. Furthermore, $X_H$ has function field $\QQ(h)$ and the morphism $\pi_H\colon X_H\to \PP^1_\QQ$ is described by the inclusion $\QQ(h) \supseteq \QQ(j)$. So if $E/\QQ$ is a non-CM elliptic curve, then $\rho_{E,\ell}(\Gal_\QQ)$ is conjugate to a subgroup of $H$ if and only if $j_E$ belongs to $\pi_H(X_H(\QQ))=J(\QQ\cup \{\infty\})$.
We will need to recognize $H$ as a conjugate of one of our applicable subgroups $G_i$ of $\GL_2(\FF_\ell)$. The degree of $\pi_H$, which is the same as the degree of $J(t)$, is equal to the index $[\GL_2(\FF_\ell):H]$; this observation will immediately rule out most candidates. We will also make use of Proposition~\ref{P:main moduli}; observe that the set $\pi_H(X_H(\QQ))$ depends only on the conjugacy class of $H$.
Most of this section involves basic algebraic verifications (which are straightforward to check with a computer, see the link in \S\ref{SS:overview} for many such details); much of the work, which we will not touch on, is finding the various equations in the first place.
\subsection{$\ell=2$}
Fix notation as in \S\ref{SS:applicable 2}. Up to conjugacy, $G_1$, $G_2$ and $G_3$ are the proper subgroups of $\GL_2(\FF_2)$.
\begin{itemize}
\item
Define the function
\begin{align*}
h_1(\tau) &:= 16\eta(2\tau)^8/\eta(\tau/2)^8 = 16\big( q^{1/2} + 8q + 44q^{3/2} + 192q^2 + 718q^{5/2}+\cdots \big).
\end{align*}
By Lemmas~\ref{L:borel to split} and \ref{L:hauptmodul}(\ref{L:hauptmodul i}), we have $\QQ(X_s(2))=\QQ(h_1)$. We have $C_s(2)=G_1$, so $\QQ(X_{G_1})=\QQ(h_1)$. The extension $\QQ(h_1)/\QQ(j)$ has degree $6$, so there is a unique rational function $J(t) \in \QQ(t)$ such that $j=J(h_1)$. We have $J(t)= f_1(t)/f_2(t)$ for relatively prime $f_1,f_2 \in \QQ[t]$ of degree at most $6$. Expanding the $q$-expansion of $j f_2(h_1) - f(h_1) = J(h_1) f_2(h_1) - f_1(h_1)=0$ gives many linear equations in the coefficients of $f_1$ and $f_2$. Using enough terms of the $q$-expansion, we can compute the coefficients of $f_1$ and $f_2$ (they are unique up to scaling $f_1$ and $f_2$ by some constant in $\QQ^\times$). Doing this, we found that $J_1(h_1)=j$.
\item
Define $h_2:=h_1^2/(h_1 + 1)$. Since $J_2(t^2/(t + 1))=J_1(t)$, we have $J_2(h_2)=j$.
\item
Define $h_3:=F(h_1)$ where $F(t)=(-16t^3 - 24t^2 + 24t + 16)/(t^2 + t)$. Since $J_3(F(t))=J_1(t)$, we have $J_3(h_3)=j$.
\end{itemize}
For each integer $1\leq i \leq 3$, let $H_i$ be the subgroup of $\GL_2(\FF_2)$ that fixes $h_i$. By Lemma~\ref{L:key}, $H_i$ is an applicable subgroup of $\GL_2(\FF_2)$ with index equal to the degree of $J_i(t)$. By comparing the degree of $J_i(t)$ with our list of proper subgroups, we deduce that $H_i$ is conjugate to $G_i$ in $\GL_2(\FF_2)$.
Theorem~\ref{T:main2} now follows from Lemma~\ref{L:key}(\ref{L:key b}); we can ignore $t=\infty$ since $J_i(\infty)=\infty$.
\subsection{$\ell=3$}
Fix notation as in \S\ref{SS:applicable 3}. Up to conjugacy, the groups $G_i$ with $1\leq i\leq 4$ are the applicable subgroups of $\GL_2(\FF_3)$.
\begin{itemize}
\item
Define the function $h_1 := 1/3\cdot \eta({\tau/3})^3/\eta(3\tau)^3$. By Lemmas~\ref{L:borel to split} and \ref{L:hauptmodul}(\ref{L:hauptmodul ii}), we have $\QQ(X_s(3))=\QQ(h_1)$. We have $C_s(3)=G_1$, so $\QQ(X_{G_1})=\QQ(h_1)$. The extension $\QQ(h_1)/\QQ(j)$ has degree $12$, so there is a unique rational function $J(t) \in \QQ(t)$ such that $j=J(h_1)$. We have $J(t)= f_1(t)/f_2(t)$ for relatively prime $f_1,f_2 \in \QQ[t]$ of degree at most $12$. Expanding the $q$-expansion of $j f_2(h_1) - f(h_1) = J(h_1) f_2(h_1) - f_1(h_1)=0$ gives many linear equations in the coefficients of $f_1$ and $f_2$. Using enough terms of the $q$-expansion, we can compute the coefficients of $f_1$ and $f_2$ (they are unique up to scaling $f_1$ and $f_2$ by some constant in $\QQ^\times$). Doing this, we found that $J_1(h_1)=j$.
\item
Define $h_2=F_1(h_1)$ where $F_1(t)=(t^2 + 3t + 3)/t$. Since $J_2(F_1(t))=J_1(t)$, we have $J_2(h_2)=j$.
\item
Define $h_3=F_2(h_1)$ where $F_2(t)=t(t^2 + 3t + 3)$. Since $J_3(F_2(t))=J_1(t)$, we have $J_3(h_3)=j$.
\item
Define $h_4= F_3(h_2)$ where $F_3(t)= 3(t+1)(t-3)/t$. Since $J_4(F_3(t))=J_2(t)$, we have $J_4(h_4)=j$.
\end{itemize}
Fix an integer $1\leq i \leq 4$, and let $H_i$ be the subgroup of $\GL_2(\FF_3)$ that fixes $h_i$. By Lemma~\ref{L:key}, we find that $H_i$ is an applicable subgroup and the morphism $\pi_{H_i}\colon X_{H_i} \to \PP^1_\QQ$ is described by the rational function $J_i(t)$. The index $[\GL_2(\FF_3): H_i]$ agrees with the degree of $J_i(t)$. By comparing the degree of $J_i(t)$ with our list of applicable subgroups, we deduce that $H_i$ is conjugate to $G_i$ in $\GL_2(\FF_3)$.
Theorem~\ref{T:main3}(\ref{T:main3 b}) now follows from Lemma~\ref{L:key}(\ref{L:key b}); we can ignore $t=\infty$ since $J_i(\infty)=\infty$. A computation shows that if $H$ is a proper subgroup of $G_i$ satisfying $\pm H=G_i$, then $i \in \{1,3\}$ and $H$ is one of the groups $H_{i,j}$; this proves Theorem~\ref{T:main3}(\ref{T:main3 a}).
\subsection{$\ell=5$}
Fix notation as in \S\ref{SS:applicable 5}. Up to conjugacy, the applicable subgroups of $\GL_2(\FF_5)$ are the groups $G_i$ with $1\leq i \leq 9$. Recall that the \defi{Rodgers-Ramanunjan continued fraction} is
\begin{align*}
r(\tau) &:=q^{1/5}\cdot \frac{1}{1+}\, \frac{q}{1+} \, \frac{q^2}{1+}\, \frac{q^3}{1+}\, \frac{q^4}{1+} \cdots.
\end{align*}
The function
\[
h_1(\tau):=1/r(\tau) = q^{-1/5}(1 + q - q^3 + q^5 + q^6 - q^7 - 2q^8 + 2q^{10} + 2q^{11} +\cdots)
\]
is a modular function of level $5$ and satisfies $J_1(h_1)=j$; we refer to Duke \cite{MR2133308} for an excellent exposition. An expression for $h_1(\tau)$ in terms of Klein forms can be found in \cite{MR2264315}.
Set $w:=(1+\sqrt{5})/2 \in \QQ(\zeta_5)$.
\begin{itemize}
\item
Define the function $h_2=h_1 - 1- 1/h_1$. We have $J_2(t-1-1/t)=J_1(t)$, so $J_2(h_2)=j$. (As noted in equation (7.2) of \cite{MR2133308}, $h_2$ equals $\eta(\tau/5)/\eta(5\tau)$.)
\item
Define $h_3=F_1(h_2)$ where
\[
F_1(t)=\frac{(- 3+w)t - 5}{t + (3-w)}.
\]
We have $J_3(F_1(t))=J_2(t)$ and hence $J_3(h_3)=j$.
\item
Define $h_4=h_2+5/h_2$. We have $J_4(t+5/t)=J_2(t)$ and hence $J_4(h_4)=j$.
\item
Define $h_5=h_1^5$. We have $J_5(t^5)=J_1(t)$ and hence $J_5(h_5)=j$.
\item
Define $h_6= F_2(h_5)$ where
\[
F_2(t) =\frac{-(w-1)^5t+1}{t+(w-1)^5}
\]
We have $J_6(F_2(t))=J_5(t)$ and hence $J_6(h_6)=j$. (In the notation of \cite{MR2133308}*{\S8}, we have $b=h_6$.)
\item
Define $h_7=F_3(h_3)$ where
\[
F_3(t)=-\frac{t^3 + 10t^2 + 25t + 25}{2t^3 + 10t^2 + 25t + 25}.
\]
We have $J_7(F_3(t))=J_3(t)$ and hence $J_7(h_7)=j$.
\item
Define $h_8=h_5-11-h_5^{-1}$. We have $J_8(t-11-t^{-1})=J_5(t)$ and hence $J_8(h_8)=j$. (As noted in equation (7.7) of \cite{MR2133308}, $h_8$ equals $(\eta(\tau)/\eta(5\tau))^6$.)
\item
Define $h_9=F_4(h_4)$ where
\[
F_4(t)= \frac{(t+5)(t^2-5)}{t^2 + 5t + 5}.
\]
We have $J_9(F_4(t))=J_4(t)$ and hence $J_9(h_9)=j$.
\end{itemize}
Fix an integer $1\leq i \leq 9$. Let $H_i$ be the subgroup of $\GL_2(\FF_5)$ that fixes $h_i$. By Lemma~\ref{L:key}, we find that $H_i$ is an applicable subgroup and the morphism $\pi_{H_i}\colon X_{H_i} \to \PP^1_\QQ$ is described by the rational function $J_i(t)$.
\begin{lemma}
The groups $H_i$ and $G_i$ are conjugate in $\GL_2(\FF_5)$ for each $1\leq i \leq 9$.
\end{lemma}
\begin{proof}
The index $[\GL_2(\FF_5): H_i]$ agrees with the degree of $J_i(t)$. By comparing the degree of $J_i(t)$ with our list of applicable subgroups, we deduce that $H_i$ is conjugate to $G_i$ in $\GL_2(\FF_5)$ for all $i\in \{1,4,7,8,9\}$.
The groups $H_5$ and $H_6$ are not conjugate since one can check that the image of $\PP^1(\QQ)=\QQ\cup\{\infty\}$ under $J_5(t)$ and $J_6(t)$ are different. The groups $H_5$ and $H_6$ have index $12$ in $\GL_2(\FF_5)$ and are not conjugate, so they are conjugate to $G_5$ and $G_6$ (though we need to determine which is which). The elliptic curve given by the Weierstrass equation $y^2+(1-t)xy -ty = x^3-tx^2$ has $j$-invariant $J_6(t)$ and the point $(0,0)$ has order $5$. Therefore, $H_6$ is conjugate to $G_6$ and thus $H_5$ is conjugate to $G_5$.
The groups $H_2$ and $H_3$ are not conjugate since one can check that the image of $\PP^1(\QQ)=\QQ\cup\{\infty\}$ under $J_2(t)$ and $J_3(t)$ are different. The groups $H_2$ and $H_3$ have index $30$ in $\GL_2(\FF_5)$ and are not conjugate, so they are conjugate to $G_2$ and $G_3$ (though we need to determine which is which). Since $h_7=F_3(h_3)$ and $F_3(t)$ belongs to $\QQ(t)$, we find that $H_3$ is a subgroup of $H_7$. We already know that $H_7$ is conjugate to $N_{ns}(5)$ and one can check that $G_2=C_s(5)$ is not conjugate to a subgroup of $N_{ns}(5)$. Therefore, $H_3$ is conjugate to $G_3$ and thus $H_2$ is conjugate to $G_2$.
\end{proof}
Theorem~\ref{T:main5}(\ref{T:main5 b}) now follows from Lemma~\ref{L:key}(\ref{L:key b}); we have $J_i(\infty)=\infty$ for $i\notin\{3,7\}$ and we can ignore the values $J_3(\infty)=0$ and $J_7(\infty)=8000$ since they are the $j$-invariants of CM elliptic curves. A direct computation shows that if $H$ is a proper subgroup of $G_i$ satisfying $\pm H=G_i$, then $i \in \{1,5,6\}$ and $H$ is one of the groups $H_{i,j}$; this proves Theorem~\ref{T:main5}(\ref{T:main5 a}).
\subsection{$\ell=7$} \label{SS:main proof 7}
Fix notation as in \S\ref{SS:applicable 7}. Up to conjugacy, the applicable subgroups of $\GL_2(\FF_7)$ are the groups $G_i$ with $1\leq i\leq 7$ from \S\ref{SS:applicable 7} and the groups:
\begin{itemize}
\item
Let $G_8$ be the subgroup of $\GL_2(\FF_7)$ consisting of matrices of the form $\pm \left(\begin{smallmatrix}1 & 0 \\0 & * \end{smallmatrix}\right)$.
\item
Let $G_9$ be the subgroup of $\GL_2(\FF_7)$ consisting of matrices of the form $\left(\begin{smallmatrix}a & 0 \\0 & \pm a \end{smallmatrix}\right)$.
\item
Let $G_{10}$ be the subgroup of $\GL_2(\FF_7)$ generated by $\left(\begin{smallmatrix}0 & -2 \\2 & 0 \end{smallmatrix}\right)$ and $\left(\begin{smallmatrix}1 & 0 \\0 & -1 \end{smallmatrix}\right)$.
\item
Let $G_{11}$ be the subgroup $C_s(7)$ of $\GL_2(\FF_7)$.
\item
Let $G_{12}$ be the subgroup of $\GL_2(\FF_7)$ generated by $\left(\begin{smallmatrix}1 & -1 \\1 & 1 \end{smallmatrix}\right)$ and $\left(\begin{smallmatrix}1 & 0 \\0 & -1 \end{smallmatrix}\right)$.
\end{itemize}
For $i=8, 9, 10, 11$ and $12$, the index $[\GL_2(\FF_7): G_i]$ is $168$, $168$, $84$, $56$ and $42$, respectively.\\
The \defi{Klein quartic} is the curve $\calX$ in $\PP^2_\QQ$ defined by the equation $x^3y+y^3z+z^3x=0$; it is a non-singular curve of genus $3$. The relevance to us is that $\calX$ is isomorphic to the modular curve $X(7):=X_{G_8}$; we refer to Elkies \cite{MR1722413} for a lucid exposition. In \S4 of \cite{MR1722413}, Elkies defines a convenient basis $\defi{x}$, $\defi{y}$ and $\defi{z}$ for the space of cusp forms of $\Gamma(7)$ which satisfy the equation of the Klein quartic and have product expansions
\[
\defi{x},\, \defi{y},\, \defi{z} =\varepsilon q^{a/7} \prod_{n=1}^\infty (1-q^n)^3 (1-q^7) \prod_{\substack{n>0\\n\equiv \pm n_0 \,\bmod{7}}} (1-q^n)
\]
where $(\varepsilon,a,n_0)$ is $(-1,4,1)$, $(1,2,2)$ or $(1,1,4)$ for $\defi{x}$, $\defi{y}$ or $\defi{z}$, respectively. The coordinates $(\defi{x}:\defi{y}\colon \defi{z})$ then give the desired isomorphism $X(7)\to\calX$. \\
Define
\[
h_4:=-(\defi{y}^2\defi{z})/\defi{x}^3=q^{-1} + 3 + 4q + 3q^2 - 5q^4 - 7q^5 + \ldots;
\]
it is a modular function of level $7$. Define $h_7:=F_1(h_4)$ where
\[
F_1(t)=t+\frac{1}{1-t}+\frac{t-1}{t}-8.
\]
From equations (4.20) and (4.24) of \cite{MR1722413}, with a correction in the sign of (4.23) of loc.~cit., we have $J_7(h_7)=j$.
Since $J_7(F_1(t))=J_4(t)$, we have $J_4(h_4)=j$.
Define $h_3:=F_2(h_4)$ and $h_5:=F_3(h_4)$, where
\[
F_2(t)= \frac{\beta t - (\beta-1)}{t-\beta}\quad \text{ and }\quad F_3(t)=\frac{ t - \gamma}{\gamma t-(\gamma-1)}
\]
with $\beta=4+3\zeta_7+3\zeta_7^{-1}+\zeta_7^2+\zeta_7^{-2}$ and $\gamma= \zeta_7^5 + \zeta_7^4 + \zeta_7^3 + \zeta_7^2 + 1$. Since $J_3(F_2(t))=J_4(t)$ and $J_5(F_3(t))=J_4(t)$, we have $J_3(h_3)=j$ and $J_5(h_5)=j$.\\
For $i \in \{3,4,5,7\}$, let $H_i$ be the subgroup of $\GL_2(\FF_7)$ that fixes $h_i$. We have shown that $J_i(h_i)=j$. By Lemma~\ref{L:key}, we find that $H_i$ is an applicable subgroup and that the morphism $\pi_{H_i}\colon X_{H_i} \to \PP^1_\QQ$ is described by the rational function $J_i(t)$.
\begin{lemma}
The groups $H_i$ and $G_i$ are conjugate in $\GL_2(\FF_7)$ for all $i\in\{3,4,5,7\}$.
\end{lemma}
\begin{proof}
The index of $H_i$ in $\GL_2(\FF_7)$ agrees with the degree of $J_i(t)$ which is $24$ or $8$ if $i\in\{3,4,5\}$ or $i=7$, respectively. By our list of applicable subgroups, we deduce that $H_7$ is conjugate to $G_7$ in $\GL_2(\FF_7)$. The groups $H_3$, $H_4$ and $H_5$ are not conjugate in $\GL_2(\FF_7)$ (since one can show that the images of $\PP^1(\QQ)=\QQ \cup\{\infty\}$ under $J_3$, $J_4$ and $J_5$ are pairwise distinct). By our list of applicable subgroups, the groups $H_3$, $H_4$ and $H_5$ are conjugate to the three subgroups $G_3$, $G_4$ and $G_5$; we still need to identify $H_3$ with $G_3$, etc.
The modular function $h_4\in \calF_7$ is a Laurent series in $q$ and has rational coefficients. Using Proposition~\ref{P:modular galois}, this implies that $H_4$ contains the group of matrices of the form $\pm \left(\begin{smallmatrix}1 & 0 \\ * & * \end{smallmatrix}\right)$ in $\GL_2(\FF_7)$. Therefore, $H_4$ must be conjugate to $G_4$ in $\GL_2(\FF_7)$. The elliptic curve given by the Weierstrass equation $y^2+(1+t-t^2)xy+(t^2-t^3)y = x^3+(t^2-t^3)x^2$ has $j$-invariant $J_3(t)$ and the point $(0,0)$ has order $7$, so $H_3$ is conjugate to $G_3$. Therefore, $H_5$ is conjugate to $G_5$.
\end{proof}
Following Elkies (\cite{MR1722413}*{p.68}), we multiply the equation of the Klein curve to obtain $(\defi{x}^3\defi{y}+\defi{y}^3\defi{z}+\defi{z}^3\defi{x})(\defi{x}^3\defi{z}+\defi{z}^3\defi{y}+\defi{y}^3\defi{x})=0.$ Noting that the left hand side is a symmetric polynomial in $\defi{x}$, $\defi{y}$ and $\defi{z}$, one can show that $s_2^4+s_3(s_1^5-5s_1^3 s_2 +s_1 s_2^2+7 s_1^2 s_3) =0$ where $s_1=\defi{x}+\defi{y}+\defi{z}$, $s_2=\defi{x}\defi{y} + \defi{y}\defi{z}+\defi{z}\defi{x}$ and $s_3=\defi{x}\defi{y}\defi{z}$. We now deviate from Elkies' treatment. Divide by $s_1^2 s_3^2$ and rearrange to obtain
\[
\Big(\frac{s_2^2}{s_1 s_3}\Big)^2 + \Big(\frac{s_1^2}{s_2}\Big)^2 \cdot \frac{s_2^2}{s_1 s_3} - 5 \frac{s_1^2}{s_2} \cdot \frac{s_2^2}{s_1 s_3} + \frac{s_2^2}{s_1 s_3} + 7 =0.
\]
We thus have $v^2 + (h_2^2 -5 h_2 + 1)v +7 = 0$, where
\[
h_2:= s_1^2/s_2=q^{-1/7} + 2 + 2q^{1/7} + q^{2/7} + 2q^{3/7} + 3q^{4/7} + 4q^{5/7} + 5q^{6/7} + 7q + 8q^{8/7}+\cdots
\]
and $v:=s_2^2/(s_1 s_3)$ are modular functions. We claim that
\begin{equation} \label{E:h7 expression}
h_7+(h_2^3 - 4h_2^2 + 3h_2 + 1)((h_2^2 - 5h_2 + 1)v + 7) = 0.
\end{equation}
This can be verified algebraically: In the left-hand side of (\ref{E:h7 expression}), replace $h_7$ by $F_1(-y^2z/x^3)$, $h_2$ by $(x+y+z)^2/(xy+yz+zx)$, and $v$ by $(xy+yz+zx)^2/((x+y+z)xyz)$; the numerator of the resulting rational function is divisible by $xy^3+yz^3+zx^3$.
Completing the square in the equation $v^2 + (h_2^2 -5 h_2 + 1)v +7 = 0$, we have
\begin{equation} \label{E:w2 expression}
w^2= h_2^4 - 10h_2^3 + 27h_2^2 - 10h_2 - 27,
\end{equation}
where $w:=2v+(h_2^2 -5 h_2 + 1)$. From (\ref{E:h7 expression}), we find that
\begin{equation} \label{E:h7 expression 2}
h_7 = \tfrac{1}{2} (h_2^3-4h_2^2+3h_2+1)((h_2^4-10h_2^3+27h_2^2-10h_2-13)-(h_2^2-5h_2+1)w).
\end{equation}
We have $j=J_7(h_7)$, so (\ref{E:w2 expression}) and (\ref{E:h7 expression 2}) imply that $j$ can be written in the form $\alpha(h_2)+ \beta(h_2)w$ for rational functions $\alpha(t)$ and $\beta(t)$. A direct computation shows that $\alpha(t)=J_2(t)$ and $\beta(t)=0$, and hence $J_2(h_2)=j$.
Let $H_2$ be the subgroup of $\GL_2(\FF_7)$ that fixes $h_2$. We have $J_2(h_2)=j$, so Lemma~\ref{L:key} implies that $H_2$ is an applicable subgroup and that the morphism $\pi_{H_2}\colon X_{H_2} \to \PP^1_\QQ$ is described by the rational function $J_2(t)$. The index of $H_2$ in $\GL_2(\FF_7)$ is $28$ since it agrees with the degree of $J_2(t)$. By our list of applicable subgroups, we deduce that $H_2$ is conjugate to $G_2$ in $\GL_2(\FF_7)$.\\
Let $H_{11}$ be the subgroup of $\GL_2(\FF_7)$ that fixes $h_2$ and $w$. The group $H_{11}$ is an index $2$ subgroup of $H_2$ since the extension $\QQ(h_2,w)/\QQ(h_2)$ has degree $2$. The group $H_{11}$ contains $G_8$ since $\QQ(h_2,w)$ is contained in $\QQ(\defi{x}/\defi{z},\defi{y}/\defi{z})$ which is the function field of $X(7)$; in particular, $H_{11}$ is applicable. From our classification of applicable subgroups, we find that $H_{11}$ is conjugate to $G_{11}$. The modular curve $X_{G_{11}}$ thus has function field $\QQ(h_2,w)$ and is hence isomorphic to the smooth projective curve over $\QQ$ with affine model
\begin{equation} \label{E:XH11}
y^2= x^4 - 10x^3 + 27x^2 - 10x - 27.
\end{equation}
The only rational points for the smooth model of (\ref{E:XH11}) are the two points at infinity (one can show that it is isomorphic to the quadratic twist by $-7$ of the curve $E_{7,1}$ from \S\ref{SS:CM}, and that this curve has only two rational points). Using that $J_2(\infty)=\infty$, we find that the two rational points of $X_{H_{11}}$, and hence of $X_{G_{11}}$, are cusps. Therefore, there is no non-CM elliptic curve $E/\QQ$ for which $\rho_{E,7}(\Gal_\QQ)$ is conjugate to a subgroup of $G_{11}$; the same holds for the group $G_8$ since $G_8\subseteq G_{11}$.
\\
Now consider the subfield $K:=\QQ(h_2,w/\sqrt{-7})$ of $\calF_7$. Let $H_1$ be the subgroup of $\GL_2(\FF_7)$ that fixes $K$. From the inclusions $K\supseteq \QQ(h_2) \supseteq \QQ(j)$ and (\ref{E:w2 expression}), we find that $K$ is the function field of the geometrically irreducible curve
\begin{equation} \label{E:XH1}
-7y^2= x^4 - 10x^3 + 27x^2 - 10x - 27
\end{equation}
defined over $\QQ$ (with $(x,y)=(h_2, w/\sqrt{-7})$). The curve $X_{H_1}$ is defined over $\QQ$ since $\QQ$ is algebraically closed in $K$. The only rational points of the smooth projective model of (\ref{E:XH1}) are $(x,y)=(5/2,\pm 1/4)$ (one can show that it is isomorphic to the curve $E_{7,1}$ from \S\ref{SS:CM}, and that this curve has only two rational points). These two rational points on $X_{H_1}$ lie over the $j$-invariant $J_2(5/2)=3^3\cdot 5\cdot 7^5/2^7$. This shows that for an elliptic curve $E/\QQ$, $\rho_{E,7}(\Gal_\QQ)$ is conjugate to a subgroup of $H_1$ in $\GL_2(\FF_7)$ if and only if $j_E=3^3\cdot 5\cdot 7^5/2^7$. Since $X_{H_1}$ has a rational point that is not a cusp, the group $H_1$ must be applicable and not conjugate to $G_{11}$. The group $H_{1}$ is an index $2$ subgroup of $H_2$ since $[\QQ(h_2,w/\sqrt{-7}):\QQ(h_2)]=2$. From our description of applicable groups, we deduce that $H_1$ is conjugate to $G_1$.
\begin{remark}
The rational points on $X_{H_1}$ were first described by A.~Sutherland in \cite{1006.1782}. An elliptic curve $E/\QQ$ with $j$-invariant $3^3\cdot 5\cdot 7^5/2^7$ has the distinguished property of not having a $7$-isogeny, yet its reduction at primes of good reduction all have a $7$-isogeny.
\end{remark}
From equation (4.35) of \cite{MR1722413}, the modular curve $X_{ns}^+(7):=X_{G_6}$ has function field of the form $\QQ(x)$ and the morphism down to the $j$-line is given by $J_6(x)$; note that there is a small typo in the numerator of equation (4.35) of \cite{MR1722413} though the given expression for $j-1728$ is correct.
\begin{lemma} \label{L:Schoof 7}
The rational points of the modular curve $X_{G_{12}}$ are all CM.
\end{lemma}
\begin{proof}
The fiber in $X_{ns}^+(7)$ over $j=1728$ is the (non-reduced) subscheme given by
\[
(2x^4 - 14x^3 + 21x^2 + 28x + 7) (x-3) \big((x^4 - 7x^3 + 14x^2 - 7x + 7)(x^4 - 14x^2 + 56x + 21)\big)^2 = 0;
\]
this can be found by factoring $J_6(x)-1728$. Define the modular curve $X_{ns}(7):= X_{C_{ns}(7)}$. One can show that the morphism $X_{ns}(7) \to X_{ns}^+(7)$ is ramified at precisely four points lying over $j=1728$. Since it is defined over $\QQ$, these four ramification points are the ones given by $2x^4 - 14x^3 + 21x^2 + 28x + 7=0$. Therefore, $X_{ns}(7)$ is defined by an equation
\[
y^2= c(2x^4 - 14x^3 + 21x^2 + 28x + 7)
\]
for some squarefree $c\in \ZZ$.
We claim that $c=-1$. Consider an elliptic curve $E/\QQ$ with $j$-invariant $-2^{15}$. The value $x=1$ is the unique rational solution to $J(x) = -2^{15}$. Setting $x=1$, we have $y^2= 44c$. Therefore, $K=\QQ(\sqrt{11c})$ is the unique quadratic extension of $\QQ$ for which $\rho_{E,7}(\Gal_K) \subseteq C_{ns}(7)$. Since $j_E=-2^{15}$, the curve $E$ has CM by $\QQ(\sqrt{-11})$ and hence $\rho_{E,7}(\Gal_{\QQ(\sqrt{-11})})= C_{ns}(7)$ and $\rho_{E,7}(\Gal_{\QQ})=N_{ns}(7)$; see \S\ref{SS:CM proofs}. Therefore, $K=\QQ(\sqrt{-11})$ and hence $c=-1$ as claimed. (The above argument comes from Schoof.)
Define the subfield $L=\QQ(x,v)$ of $\calF_7$ where $v:=y/\sqrt{-7}$; we have
\begin{equation} \label{E:XG12 7}
7 v^2 = 2x^4 - 14x^3 + 21x^2 + 28x + 7.
\end{equation}
Let $G$ be the subgroup of $\GL_2(\FF_7)$ that fixes $L$; it is an index $2$ subgroup of $G_6=N_{ns}(11)$ since $L/\QQ(x)$ has degree $2$. The field $\QQ$ is algebraically closed in $L$ since $L/\QQ(x)$ is a geometric extension. Therefore, $\det(G)=\FF_7^\times$. There are only two index $2$ subgroups of $G_6$ with full determinant; they are $G_{12}$ and $C_{ns}(7)$. The group $G$ is thus $G_{12}$ since $C_{ns}(7)$ corresponds to the field $\QQ(x,y)$.
Therefore, $X_{G_{12}}$ has function field $\QQ(x,v)$ with $x$ and $v$ related by (\ref{E:XG12 7}). The smooth projective curve defined by (\ref{E:XG12 7}) has genus $1$ and a rational point $(x,v)=(0,1)$; it is thus an elliptic curve. A computation shows that this elliptic curve is isomorphic to the curve $E_{7,2}$ of \S\ref{SS:CM}. The curve $E_{7,2}$ has only two rational points, so $(x,v)=(0,\pm 1)$ are the only rational points of the curve defined by (\ref{E:XG12 7}). The lemma follows since $J_6(0)=0$.
\end{proof}
If $E/\QQ$ is a non-CM elliptic curve, Lemma~\ref{L:Schoof 7} shows that $\rho_{E,7}(\Gal_\QQ)$ is not conjugate to a subgroup of $G_{12}$. The same holds for $G_9$ and $G_{10}$ since they are both subgroups of $G_{12}$.\\
Suppose that $H$ is a proper subgroup of $G_i$ satisfying $\pm H=G_i$ for a fixed $1\leq i\leq 7$. If $i\neq 1$, then $i \in \{3,4,5,7\}$ and $H$ is one of the groups $H_{i,j}$. If $i=1$, the $H$ is either $H_{1,1}$ or another subgroup that is conjugate to $H_{1,1}$ in $\GL_2(\FF_7)$. This completes the proof of Theorem~\ref{T:main7}(\ref{T:main7 i}) and (\ref{T:main7 ii}); we can ignore $t=\infty$ for $2\leq i \leq 7$ since $J_i(\infty)$ is either $\infty$ or the $j$-invariant of a CM elliptic curve.
\subsection{$\ell=11$}
Fix notation as in \S\ref{SS:applicable 11}. Up to conjugacy, the group $\GL_2(\FF_{11})$ has four maximal applicable subgroups: $B(11)$, $N_{s}(11)$, $N_{ns}(11)$ and a group $H_{\mathfrak{S}_4}$ whose image in $\PGL_2(\FF_{11})$ is isomorphic to $\mathfrak{S}_4$.
\subsubsection{Exceptional case} \label{SS:exceptional 11}
The curve $X_{\mathfrak{S}_4}(11):=X_{H_{\mathfrak{S}_4}}$ has no rational points corresponding to a non-CM elliptic curve; it is isomorphic to an elliptic curve which has only one rational point \cite{MR0463118}*{Prop.~4.4.8.1} and this point corresponds to an elliptic curve with CM by $\sqrt{-3}$.
\subsubsection{Split case}
The curve $X_{s}^+(11):=X_{N_s(11)}$ has no rational points corresponding to a non-CM elliptic curve; see \cite{1104.4641} for a more general result. Therefore, there are no non-CM elliptic curves $E/\QQ$ such that $\rho_{E,\ell}(\Gal_\QQ)$ is conjugate to a subgroup of $N_s(11)$.
\subsubsection{Non-split case}
The modular curve $X_{ns}^+(11):=X_{G_3}=X_{N_{ns}(11)}$ has genus $1$. Halberstadt \cite{MR1677158} showed that the function field of $X_{ns}^+(11)$ is of the form $K:=\QQ(x,y)$ with $y^2+y = x^3-x^2-7x+10$ such that the inclusion $\QQ(j) \subseteq \QQ(x,y)$ is given by $j=J(x,y)$. Therefore, if $E/\QQ$ is a non-CM elliptic curve, then $\rho_{E,11}(\Gal_\QQ)$ is conjugate to a subgroup of $N_{ns}(11)$ if and only if $j_E=J(P)$ for some point $P\in \calE(\QQ)$. We only need consider $P\neq \OO$ since, as noted in \cite{MR1677158}, $J(\OO)$ is the $j$-invariant of a CM elliptic curve.
Let $G_4$ be the subgroup of $G_3$ consisting of $g \in G_3=N_{ns}(11)$ such that $g\in C_{ns}(11)$ and $\det(g) \in (\FF_{11}^\times)^2$, or $g\notin C_{ns}(11)$ and $\det(g) \notin (\FF_{11}^\times)^2$.
\begin{lemma} \label{L:XG4}
The modular curve $X_{G_4}$ has no rational points.
\end{lemma}
\begin{proof}
Define the modular curve $X_{ns}(11):= X_{C_{ns}(11)}$. Proposition~1 of \cite{DoseFernandezGonzalezSchoof} shows that $X_{ns}(11)$ can be defined by the equations $y^2+y = x^3-x^2-7x+10$ and $u^2= - (4x^3+7x^2-6x+19)$, where $K=\QQ(x,y)$.
Define the field $L:=K(v)$ with $v=u/\sqrt{-11}$. We have $L \subseteq \calF_{11}$ since $\sqrt{-11}\in \QQ(\zeta_{11})$. Let $G$ be the subgroup of $\GL_2(\FF_{11})$ that fixes $L$; it is an index $2$ subgroup of $G_3$ since $L/K$ has degree $2$. The field $\QQ$ is algebraically closed in $L$ since it is algebraically closed in $K$ and $L/K$ is a geometric extension. Therefore, $\det(G)=\FF_{11}^\times$. There are only two index $2$ subgroups of $G_3$ with full determinant; they are $G_4$ and $C_{ns}(11)$. The group $G$ is thus $G_4$ since $C_{ns}(11)$ corresponds to the field $K(u)$.
Therefore, $X_{G_4}$ has function field $\QQ(x,y,v)$ where $y^2+y = x^3-x^2-7x+10$ and $v^2= 11 (4x^3+7x^2-6x+19)$. We now homogenize our equations:
\begin{equation} \label{E:homogenous XG4}
y^2z+yz^2 = x^3-x^2z-7xz^2+10z^3,\quad 11 v^2z= (4x^3+7x^2z-6xz^2+19z^3).
\end{equation}
Combining the two equations (\ref{E:homogenous XG4}) to remove the $x^3$ term, we find that $11 v^2 z = ( 4y^2z +4yz^2 +11 x^2 z + 22 x z^2 -21z^3)$. Factoring off $z$, we deduce that the following equations give a model of $X_{G_4}$ in $\PP^3_\QQ$:
\begin{equation} \label{E:homogenous XG4 2}
y^2z+yz^2 = x^3-x^2z-7xz^2+10z^3,\quad 11 v^2 = ( 4y^2 +4yz +11 x^2 + 22 x z -21z^2).
\end{equation}
Suppose $(x,y,z,v) \in \PP^3(\QQ)$ is a solution to (\ref{E:homogenous XG4 2}). If $z=0$, then we have $0=x^3$ and $11 v^2= 4 y^2$, which is impossible since $44$ is not a square in $\QQ$. So assume that $z=1$. We can then recover the equation $v^2= 11 (4x^3+7x^2-6x+19)$ which has no solutions $(x,v) \in \QQ^2$; it defines an elliptic curve and a computation shows that its only rational point is the point at $\infty$. Therefore, $X_{G_4}(\QQ)=\emptyset$.
\end{proof}
Let $E/\QQ$ be a non-CM elliptic curve for which $\rho_{E,11}(\Gal_\QQ)$ is conjugate to a subgroup of $G_3$. Suppose that $\rho_{E,11}(\Gal_\QQ)$ is conjugate to a subgroup of $G_3$. The group $G_3$ has no index $2$ subgroups $H$ that satisfy $\pm H = G_3$. Therefore, $\rho_{E,11}(\Gal_\QQ)$ is conjugate to a subgroup of a maximal applicable subgroup of $G_3$. Up to conjugacy, there are two maximal applicable subgroups of $G_3$; one is $G_4$ and the other is a subgroup $G_5$ of index $3$ in $G_3$. The image $\bbar{G}_5$ of $G_5$ in $\PGL_2(\FF_{11})$ has order $8$ and is hence a $2$-Sylow subgroup of $\PGL_2(\FF_{11})$. Therefore, $\bbar{G}_5$ lies in a subgroup of $\PGL_2(\FF_{11})$ that is isomorphic to $\mathfrak{S}_4$ and hence $G_5$ is conjugate to a subgroup of $H_{\mathfrak{S}_4}$. However, we saw in \S\ref{SS:exceptional 11} that $\rho_{E,11}(\Gal_\QQ)$ cannot be conjugate to a subgroup of $H_{\mathfrak{S}_4}$. This implies that $\rho_{E,11}(\Gal_\QQ)$ is conjugate to a subgroup of $G_4$ which is impossible by Lemma~\ref{L:XG4}. Therefore, $\rho_{E,11}(\Gal_\QQ)$ must be conjugate to $G_3$.
\subsubsection{Borel case} \label{SS:11 borel}
The modular curve $X_{B(11)}$ is known to have exactly three rational points that are not cusps; they lie above the $j$-invariants $-2^{15}$, $-11^2$ and $-11\cdot 131^3$, cf.~\cite{MR0376533}*{p.~79}. An elliptic curve with $j$-invariant $-2^{15}$ has CM, so we need only consider the other two.\\
Consider the elliptic curve $E/\QQ$ defined by $y^2+xy+y= x^3+x^2-305x+7888$; it has $j$-invariant $-11^2$ and conductor $11^2$. The division polynomial at $11$ of $E$ factors as the product of the irreducible polynomial $f(x)=x^5 - 129x^4 + 800x^3 + 81847x^2 - 421871x - 4132831$ and an irreducible polynomial $g(x)$ of degree $55$. Since $11$ divides the degree of $g(x)$, we find that $\rho_{E,11}(\Gal_\QQ)$ contains an element of order $11$. Therefore, there are unique characters $\chi_1,\chi_2\colon \Gal_\QQ\to \FF_{11}^\times$ such that with respect to an appropriate change of basis we have
\begin{equation} \label{E:X011 form}
\rho_{E,11}(\sigma) = \left(\begin{smallmatrix}\chi_1(\sigma) & * \\0 & \chi_2(\sigma) \end{smallmatrix}\right).
\end{equation}
We have $\chi_1\chi_2=\omega$ where $\omega\colon \Gal_\QQ \to \FF_{11}^\times$ is the character describing the Galois action on the $11$-th roots of unity (we have $\omega(p)\equiv p \pmod{11}$ for primes $p\neq 11$). The characters $\chi_1$ and $\chi_2$ are unramified at primes $p\nmid 11$, so $\chi_1= \omega^a$ and $\chi_2=\omega^{11-a}$ for a unique integer $0\leq a <10$. Let $w\in \Qbar$ be a fixed root of $f(x)$. One can show that
\[
P=\big(w,-(w^4 - 79w^3 - 3150w^2 + 12193w+1520110)/11^4\big)
\]
is an $11$-torsion point of $E(\Qbar)$. The field $\QQ(w)$ is a Galois extension of $\QQ$ and that the group generated by $P$ is stable under the action of $\Gal_\QQ$. We thus have $\sigma(P)=\chi_1(\sigma)\cdot P$ for all $\sigma\in \Gal_\QQ$, and hence $\chi_1(\Gal_\QQ)$ is a group of order $[\QQ(w):\QQ]=5$.
We have $a_2(E)=-1$, so the roots of the polynomial $\det(xI -\rho_{E,11}(\Frob_2))\equiv x^2-(-1)x+2 \pmod{11}$ are $4=2^2$ and $6\equiv 2^9 \pmod{11}$. Since $\chi_1(\Frob_2)\equiv 2^a$ and $\chi(\Frob_2)\equiv 2^{11-a}$ are the roots of $\det(xI -\rho_{E,11}(\Frob_2))$ and $2$ is a primitive root modulo $11$, we have $a\in \{2,9\}$ and hence $\{\chi_1,\chi_2\}=\{ \omega^2,\omega^9\}$. Since $\chi_1(\Gal_\QQ)$ has cardinality 5, we have $\chi_1=\omega^2$ and $\chi_2=\omega^9$. Since $2$ is a primitive root modulo $11$, the group $\rho_{E,11}(\Gal_\QQ)$ is generated by $\big(\begin{smallmatrix}2^2 & 0 \\0 & 2^9 \end{smallmatrix}\big)=\left(\begin{smallmatrix}4 & 0 \\0 & 6 \end{smallmatrix}\right)$ and $\left(\begin{smallmatrix}1 & 1 \\0 & 1 \end{smallmatrix}\right)$, i.e., it equals $H_{1,1}$. In particular, $\pm \rho_{E,11}(\Gal_\QQ) = G_1$.\\
Consider the elliptic curve $E/\QQ$ defined by $y^2+xy= x^3+x^2-3632x+82757$; it has $j$-invariant $-11\cdot 131^3$ and conductor $11^2$. The division polynomial at $11$ of $E$ factors as the product of the irreducible polynomial $f(x)=x^5 - 129x^4 + 4793x^3 + 9973x^2 - 3694800x + 52660939$ and an irreducible polynomial $g(x)$ of degree $55$. Since $11$ divides the degree of $g(x)$, we find that $\rho_{E,11}(\Gal_\QQ)$ contains an element of order $11$. Therefore, there are unique characters $\chi_1,\chi_2\colon \Gal_\QQ\to \FF_{11}^\times$ such that with respect to an appropriate change of basis we have (\ref{E:X011 form}).
The characters $\chi_1$ and $\chi_2$ are unramified at primes $p\nmid 11$ and $\chi_1\chi_2=\omega$, so $\chi_1= \omega^a$ and $\chi_2=\omega^{11-a}$ for a unique integer $a\in \{0,1,\ldots, 9\}$. Let $w\in \Qbar$ be a fixed root of $f(x)$. One can show that
\[
P=\big(w, (w^4 - 79w^3 + 843w^2 + 45468w - 722625)/11^3\big)
\]
is an $11$-torsion point of $E(\Qbar)$. The field $\QQ(w)$ is a Galois extension of $\QQ$ and that the group generated by $P$ is stable under the action of $\Gal_\QQ$. We thus have $\sigma(P)=\chi_1(\sigma)\cdot P$ for all $\sigma\in \Gal_\QQ$, and hence $\chi_1(\Gal_\QQ)$ is a group of order $[\QQ(w):\QQ]=5$.
We have $a_2(E)=1$, so the roots of the polynomial $\det(xI -\rho_{E,11}(\Frob_2))\equiv x^2-1\cdot x+2 \pmod{11}$ are $5\equiv 2^4$ and $7\equiv 2^7 \pmod{11}$. Since $\chi_1(\Frob_2)\equiv 2^a$ and $\chi(\Frob_2)\equiv 2^{11-a}$ are the roots of $\det(xI -\rho_{E,11}(\Frob_2))$ and $2$ is a primitive root modulo 11, we have $a\in \{4,7\}$ and hence $\{\chi_1,\chi_2\}=\{ \omega^4,\omega^7\}$. Since $\chi_1(\Gal_\QQ)$ has cardinality 5, we have $\chi_1=\omega^4$ and $\chi_2=\omega^7$. Since $2$ is a primitive root modulo $11$, the group $\rho_{E,11}(\Gal_\QQ)$ is generated by $\big(\begin{smallmatrix}2^4 & 0 \\0 & 2^7 \end{smallmatrix}\big)=\left(\begin{smallmatrix}5 & 0 \\0 & 7 \end{smallmatrix}\right)$ and $\left(\begin{smallmatrix}1 & 1 \\0 & 1 \end{smallmatrix}\right)$, i.e., it equals $H_{2,1}$. In particular, $\pm \rho_{E,11}(\Gal_\QQ) = G_2$.
\subsubsection{Polynomials for $X_{ns}^+(11)$} \label{S:ns section}
This subsection is dedicated to sketching Remark~\ref{R:ns computation} and making the polynomials explicit; fix notation as in \S\ref{SS:applicable 11}. Define the polynomials:
{
\smaller
\begin{align*}
A(x)&=(x^5 - 9x^4 + 17x^3 + 20x^2 - 73x + 43)^{11},\\
B(x) &=-(x^2+3x-6)^3 \big(108000 x^{49} + 23793840 x^{48} - 413223722 x^{47} - 5377010368 x^{46} + 230799738529 x^{45} \\
& - 3137869050351 x^{44} + 23205911712335 x^{43} - 90936268647246 x^{42} + 33563647471596 x^{41} \\
&+ 1631415220074871 x^{40} - 7744726079195413 x^{39} - 3218815397602111 x^{38} + 236712051437217644 x^{37} \\
&- 1686428698022253344 x^{36} + 7984804002023063554 x^{35} - 30444784135263860996 x^{34}\\
& + 96849826504401032248 x^{33} - 232064394883539673213 x^{32} + 210175535413395353857 x^{31} \\
&+ 1609695806324946484826 x^{30} - 11768533689837648360109 x^{29} + 48291196122826259771817 x^{28} \\
&- 143943931899306373170309 x^{27} + 315827025781563232420857 x^{26} - 421596979720485992629121 x^{25} \\
&- 234929885880162547645306 x^{24} + 3668241437553022801950917 x^{23} - 14221091463553801024770599 x^{22}\\
& + 39148264563215734730610917 x^{21} - 87534472061810348609315974 x^{20}\\
& + 166474240219619575379485393 x^{19} - 275040771573054834247036345 x^{18} \\
&+ 399144725377223909937142938 x^{17} - 511840960382358144595839458 x^{16} \\
&+ 581656165535334214665717816 x^{15} - 586206578096981243980668654 x^{14} \\
&+ 523465655841901079370457175 x^{13} - 413200824632802503354807972 x^{12}\\
& + 287270832775316643952335709 x^{11} - 175049577131269087795781453 x^{10} \\
&+ 92916572268973769104815620 x^9 - 42636417323385892254033027 x^8 \\
&+ 16754292456737738144357709 x^7 - 5570911068111617263502302 x^6 + 1542648801995330874184236 x^5\\
& - 347819053424928336793068 x^4 + 61683475328903338239178 x^3 - 8117056250720937228985 x^2\\
& + 708318740340941449799 x - 30857360406231018655\big),
\\
C(x)&=(4x-5)(x^2+3x-6)^6(9x^2-28x+23)^3(x^4-5x^3+74x^2-245x+223)^3\\
&\quad\cdot(4x^4-9x^3-x^2+21x-32)^3(25x^4-114x^3+167x^2-86x+20)^3.
\end{align*}
}
\begin{prop}
For $j \in \QQ$, we have $J(P)=j$ for some point $P \in \calE(\QQ)-\{\OO\}$ if and only if $A(x) j^2 + B(x) j +C(x) \in \QQ[x]$ has a rational root.
\end{prop}
\begin{proof}
Take $(x,y) \in \calE-\{\OO\}$. Using the equation $y^2+y = x^3-x^2-7x+10$, a direct computation shows that $J(x,y) A(x) = a(x)y+b(x)$ for unique $a,b\in \QQ[x]$. Multiplying $y^2+y = x^3-x^2-7x+10$ by $a^2$, we deduce that
$(J A -b)^2 + a(JA-b) - a^2(x^3-x^2-7x+10)=0$. Therefore, $A^2 J^2 + (-2b+a)A J + b^2-ba-a^2(x^3-x^2-7x+10) = 0$. Our polynomials $B$ and $C$ satisfy $B=-2b+a$ and $C=(b^2-ba-a^2(x^3-x^2-7x+10))/A$. We thus have
\begin{equation} \label{E:J quadratic}
A(x) J(x,y)^2 + B(x) J(x,y) + C(x) = 0
\end{equation}
for all $(x,y)\in \calE-\{\OO\}$.
First suppose that $j=J(x_0,y_0)$ for some $(x_0,y_0) \in \calE(\QQ)-\{\OO\}$. Then $0 = A(x_0) J(x_0,y_0)^2 + B(x_0) J(x_0,y_0) + C(x_0) = A(x_0) j^2 + B(x_0) j +C(x_0)$ and hence $A(x) j^2 + B(x) j +C(x)$ has a rational root.
Now fix $j\in \QQ$ and suppose that there is an $x_0\in\QQ$ such that $A(x_0) j^2 + B(x_0) j +C(x_0)=0$. Define $\Delta(x):=B(x)^2-4A(x) C(x)$. A computation shows that $\Delta(x) = D(x)^2 (x^3 - x^2 - 7x + 41/4)$ for a polynomial $D\in \QQ[x]$ that has no rational roots. The rational number $\Delta(x_0)=D(x_0)^2 (x_0^3 - x_0^2 - 7x_0+ 41/4)$ is a square since $j$ is a root of $A(x_0) X^2 + B(x_0) X + C(x_0) \in \QQ[X]$. Therefore, $v^2 = x_0^3 - x_0^2 - 7x_0 + 41/4$ for some $v\in \QQ$. With $y_0 = v -1/2$, we have $y_0^2+ y_0 = x_0^3 - x_0^2 - 7x_0 + 10$ and hence $P:=(x_0,y_0)$ is a point in $\calE(\QQ)-\{\OO\}$. We could have chose $v$ with a different sign, so $P':=(x_0, -v -1/2 ) = (x_0, -y_0-1)$ also belongs to $\calE(\QQ)-\{\OO\}$.
We claim that $J(P)\neq J(P')$. Suppose that they are in fact equal. Using that $J(x,y) A(x) = a(x)y+b(x)$, we find that $a(x_0) y_0 = a(x_0) (-y_0 - 1)$. Since $a(x)$ has no rational roots, we must have $y_0 = -1/2$ and hence $v=0$. However, this is impossible since $x^3 - x^2 - 7x + 41/4$ has no rational roots, so the claim follows. From (\ref{E:J quadratic}), we find that $J(P)$ and $J(P')$ are distinct roots of $A(x_0) X^2 + B(x_0) X + C(x_0)$. Since $j$ is also a root of this quadratic polynomial, we deduce that $j= J(P)$ or $j=J(P')$.
\end{proof}
\subsection{$\ell=13$}
We shall prove parts (\ref{T:main 13 a}) and (\ref{T:main 13 b}) of Theorem~\ref{T:main 13} (part (\ref{T:main 13 d}) was explained in the introduction); so we will focus on $B(13)$ and its subgroups. We first rule out subgroups of $C_s(13)$.
\begin{lemma}
There are no non-CM elliptic curves $E/\QQ$ for which $\rho_{E,13}(\Gal_\QQ)$ is conjugate in $\GL_2(\FF_{13})$ to a subgroup of $C_s(13)$.
\end{lemma}
\begin{proof}
Kenku has proved that the only rational points of $X_0(13^2)$ are cusps, cf.~\cite{MR588271,MR616547}. By Lemma~\ref{L:borel to split}, we deduce that the only rational points of the modular curve $X_{C_s(13)}$ are cusps.
\end{proof}
One can show that the applicable subgroups of $B(13)=G_6$ that are not subgroups of $C_s(13)$ are $G_1$, $G_2$, $G_3$, $G_4$, $G_5$, and $G_i \cap G_j$ with $i \in \{1,2\}$ and $j\in \{3,4,5\}$. Note that these subgroups are normal in $B(13)$. \\
We now describe several modular function constructed by Lecacheux \cite{MR978099}*{p.56}. Define
\[
f(\tau)=\frac{\wp(\frac{1}{13};\tau) - \wp(\frac{2}{13}; \tau)}{\wp(\frac{1}{13};\tau) - \wp(\frac{3}{13}; \tau)} \quad \text{ and }\quad
g(\tau)=\frac{\wp(\frac{1}{13};\tau) - \wp(\frac{2}{13}; \tau)}{\wp(\frac{1}{13};\tau) - \wp(\frac{5}{13}; \tau)}
\]
where $\wp(z;\tau)$ is the Weierstrass $\wp$-function at $z$ of the lattice $ \ZZ \tau + \ZZ\subseteq \CC$. Define the functions
\[
h_5:= \frac{(g-1)(g(g-1)+1-f)}{(f-1)(f-g)} \quad \text{ and }\quad h_2:=\frac{f-1}{g-1}.
\]
The functions $h_5$ and $h_2$ belong to $\calF_{13}$ and satisfy $F_2(h_2) = F_5(h_5)$, where
\[
F_2(t)=t+(t-1)/t-1/(t-1) -4=(t^3 - 4t^2 + t + 1)/(t^2 - t) \quad\text{ and }\quad F_5(t)=t -1/t -3=(t^2 - 3t - 1)/t;
\]
this follows from \cite{MR978099}*{p.56--57} with $H=h_5$ and $h=h_2$.
Let $h_6$ be the function $F_2(h_2)=F_5(h_5)$; it is called $a-3$ in \cite{MR978099} and satisfies $J_6(h_6)=j$, cf.~\cite{MR978099}*{p.62}. Since $J_2(t)=J_6(F_2(t))$ and $J_5(t)=J_6(F_5(t))$, we have $J_2(h_2)=j$ and $J_5(h_5)=j$.\\
Define $\alpha:= -\zeta_{13}^{11} - \zeta_{13}^{10} - \zeta_{13}^3 - \zeta_{13}^2 + 1$. Define the rational functions
\[
F_1(t)=13(t^2-t)/(t^3-4t^2+t+1)\quad \text{ and }\quad \phi_1(t)= ({\alpha t+1-\alpha})/({t -\alpha});
\]
Define the modular function $h_1:=\phi_1(h_2) \in \calF_{13}$. One can check that $F_1(\phi_1(t))=F_2(t)$ and hence $F_1(h_1)=F_2(h_2)=h_6$. Since $J_1(t)=J_6(F_1(t))$, we have $J_1(h_1)=j$.
Define $\beta:= \zeta_{13}^{11} + \zeta_{13}^{10} + \zeta_{13}^9 + \zeta_{13}^7 + \zeta_{13}^6 + \zeta_{13}^4 + \zeta_{13}^3 + \zeta_{13}^2 + 2$. Define the rational functions
\[
F_3(t)=(-5t^3 + 7t^2 + 8t - 5)/(t^3 - 4t^2 + t + 1) \quad \text{ and }\quad \phi_3(t)= (\beta t -1)/(t+ \beta-1).
\]
Define the modular function $h_3:=\phi_3(h_2) \in \calF_{13}$. One can check that $F_3(\phi_3(t))=F_2(t)$ and hence $F_3(h_3)=F_2(h_2)=h_6$. Since $J_3(t)=J_6(F_3(t))$, we have $J_3(h_3)=j$.
Define $\gamma=(1+\sqrt{13})/2$; it belongs to $\QQ(\zeta_{13})$ and moreover equals $\gamma=-\zeta_{13}^{11} - \zeta_{13}^8 - \zeta_{13}^7 - \zeta_{13}^6 - \zeta_{13}^5 - \zeta_{13}^2$. Define the rational functions
\[
F_4(t)=13t/(t^2 - 3t - 1)\quad \text{ and }\quad \phi_4(t)=((2-\gamma)t + 1)/(t -2+\gamma)).
\]
Define the modular function $h_4:=\phi_4(h_5) \in \calF_{13}$. One can check that $F_4(\phi_4(t))=F_5(t)$ and hence $F_4(h_4)=F_5(h_5)=h_6$. Since $J_4(t)=J_6(F_4(t))$, we have $J_4(h_4)=j$.
For $1\leq i \leq 6$, let $H_i$ be the subgroup of $\GL_2(\FF_7)$ that fixes $h_i$. We have shown that $J_i(h_i)=j$. By Lemma~\ref{L:key}, we find that $H_i$ is an applicable subgroup and that the morphism $\pi_{H_i}\colon X_{H_i} \to \PP^1_\QQ$ is described by the rational function $J_i(t)$.
\begin{lemma}
The groups $H_i$ and $G_i$ are conjugate in $\GL_2(\FF_{13})$ for all $1\leq i \leq 6$.
\end{lemma}
\begin{proof}
The index of $H_6$ in $\GL_2(\FF_{13})$ is equal to $14$, i.e., the degree of $J_6$ as a morphism. Therefore, $H_6$ must be conjugate to $B(13)$. The index $[H_6:H_i]$ equals the degree of $F_i(t)$, and is thus $3$ if $i\in \{1,2,3\}$ and $2$ if $i\in \{4,5\}$.
The groups $H_1$, $H_2$ and $H_3$ are not conjugate in $\GL_2(\FF_{13})$ since one can show that the images of $\PP^1(\QQ)=\QQ \cup\{\infty\}$ under $J_1$, $J_2$ and $J_3$ are distinct. Therefore, $H_1$, $H_2$ and $H_3$ are conjugate to $G_1$, $G_2$ and $G_3$ which are the applicable subgroups of $B(13)$ of index $2$; however, we still need to determine which group is conjugate to which.
Let $E/\QQ$ be the elliptic curve defined by $y^2=x^3-338x+2392$. The group $\rho_{E,13}(\Gal_\QQ)$ is conjugate to a subgroup of $H_3$ since $j_E=J_3(0)$. One can check that $E/\QQ$ has good reduction at $3$ and that $a_3(E)=0$. Since $x^2-a_3(E)+3 \equiv (x-6)(x+6) \pmod{13}$, we deduce that the eigenvalues of the matrix $\rho_{E,13}(\Frob_3)$ are $6$ and $-6$. For every matrix in $G_1$ or $G_2$ has an eigenvalue in $(\FF_{13}^\times)^3 = \{\pm 1, \pm 5\}$. Since $6$ and $-6$ do not belong to $(\FF_{13}^\times)^3$, we deduce that $H_3$ is not conjugate to $G_1$ and $G_2$. Therefore, $H_3$ is conjugate to $G_3$.
Let $E/\QQ$ be the elliptic curve defined by $y^2=x^3-2227x-59534$. We have $j_E=J_2(2)$ and $j_E \notin J_1(\QQ\cup\{\infty\})$. Therefore, $\rho_{E,13}(\Gal_\QQ)$ is conjugate to a subgroup of $H_2$ and not conjugate to a subgroup of $H_1$. By computing the division polynomial of $E$ at the prime $13$, we find that $E$ has a point $P$ of order $13$ whose $x$-coordinate is $17+8\sqrt{17}$. So with respect to a basis of $E[13]$ whose first element is $P$, we find that $\rho_{E,13}(\Gal_\QQ)$ is a subgroup of $G_2$. Therefore, $H_2$ is conjugate to $G_2$, and hence $H_1$ is conjugate to $G_1$.
The groups $H_4$ and $H_5$ are not conjugate in $\GL_2(\FF_{13})$ since one can show that the images of $\PP^1(\QQ)=\QQ \cup\{\infty\}$ under $J_4$ and $J_5$ are distinct. Therefore, $H_4$ and $H_5$ are conjugate to $G_4$ and $G_5$ which are the applicable subgroups of $B(13)$ of index $3$; however, we still need to determine which group is conjugate to which.
Let $E/\QQ$ be the elliptic curve defined by $y^2 = x^3 - 3024x - 69552$. We have $j_E=J_5(2)$ and $j_E\notin J_4(\QQ\cup\{\infty\})$. Therefore, $\rho_{E,13}(\Gal_\QQ)$ is conjugate to a subgroup of $H_5$ and not conjugate to a subgroup of $H_4$. By computing the division polynomial of $E$ at the prime $13$, we find that $E$ has a point $P$ of order $13$ whose $x$-coordinate $w$ is a root of $x^3 - 3024x + 12096$. The cubic extension $\QQ(w)$ of $\QQ$ is Galois, so with respect to a basis of $E[13]$ whose first element is $P$, we find that $\rho_{E,13}(\Gal_\QQ)$ is a subgroup of $G_5$. Therefore, $H_5$ is conjugate to $G_5$, and hence $H_4$ is conjugate to $G_4$.
\end{proof}
We have thus completed the proof of Theorem~\ref{T:main 13}(\ref{T:main 13 b}); we can ignore $t=\infty$ since $J_i(\infty)=\infty$ for $i\neq 3$ and $J_3(\infty)=J_3(0)$. If $H$ is a proper subgroup of $G_i$ satisfying $\pm H=G_i$, then one can show that $i\in \{4,5\}$ and $H$ is one of the groups $H_{i,j}$.\\
To complete the proof of Theorem~\ref{T:main 13}(\ref{T:main 13 a}), we need only show that the modular curves $X_{G_i\cap G_j}$, with fixed $i\in \{1,2\}$ and $j \in \{3,4,5\}$, have no rational points other than cusps. It suffices to prove the same thing for the modular curves $X_{H_i\cap H_j}$.
The function field of $X_{H_i\cap H_j}$ is $\QQ(h_i,h_j)$ and the generators $h_i$ and $h_j$ satisfy the relation $F_i(h_i)=h_6=F_j(h_j)$. The smooth projective (and geometrically irreducible) curve over $\QQ$ arising from the equation $F_i(x)=F_j(y)$ is thus a model of $X_{H_i\cap H_j}$.
The following \texttt{Magma} code shows that if $(x,y) \in \QQ^2$ is a solution of $F_i(x)=F_j(y)$ (where we say that both sides equal $\infty$ if the denominators vanish), then $y=0$. The code considers the projective (and possibly singular) curve $C_{i,j}$ in $\PP^2_\QQ$ defined by the affine equation $F_i(x)=F_j(y)$ (we first clear denominators and homogenize). We then find a genus 2 curve $C$ that is birational with $C_{i,j}$ and is defined by some Weierstrass equation $y^2=f(x)$ with $f(x)\in \QQ[x]$ a separable polynomial of degree 5 or 6. We then check that the Jacobian $J$ of $C$ has rank $0$, equivalently, that $J(\QQ)$ is a finite group (\texttt{Magma} accomplishes this by computing the $2$-Selmer group of $J$). Using that $J(\QQ)$ has rank $0$, the function \texttt{Chabauty0} finds all the rational points on $C$. Using the birational isomorphism between $C$ and $C_{i,j}$, we can determine the rational points of $C$.
{\small
\begin{verbatimtab}
K<t>:=FunctionField(Rationals());
F:=[13*(t^2-t)/(t^3-4*t^2+t+1), (t^3-4*t^2+t+1)/(t^2-t),
(-5*t^3+7*t^2+8*t-5)/(t^3-4*t^2+t+1), 13*t/(t^2-3*t-1), (t^2-3*t-1)/t ];
P2<x,y,z>:=ProjectiveSpace(Rationals(),2);
for i in [1,2,3], j in [4,5] do
f:=Numerator(Evaluate(F[i],x/z)- Evaluate(F[j],y/z));
while Evaluate(f,z,0) eq 0 do f:= f div z; end while;
C0:=Curve(P2,f);
b,C1,f1:=IsHyperelliptic(C0); C2,f2:=SimplifiedModel(C1);
Jac:=Jacobian(C2); RankBound(Jac) eq 0;
S:=Chabauty0(Jac);
b,g1:=IsInvertible(f1); b,g2:=IsInvertible(f2);
T:=g1(g2(S) join SingularPoints(C1)) join SingularPoints(C0);
{P: P in T | P[2] ne 0 and P[3] ne 0} eq {};
end for;
\end{verbatimtab}
}
We find that if $F_i(x)=F_i(y)$ for some $x,y\in \QQ\cup\{\infty\}$, then $y=0$ or $y=\infty$. Thus the only rational points of $X_{H_i\cap H_j}$ are cusps since $J_j(0)=J_j(\infty)=\infty$ for $j\in\{4,5\}$.
\subsection{$\ell=17$}
We now prove Theorem~\ref{T:17-37}(\ref{T:17-37 i}). Let $E/\QQ$ be the elliptic curve defined by the Weierstrass equation $y^2+xy+y=x^3-190891x -36002922$; it has $j$-invariant $-17\cdot 373^3/2^{17}$ and conductor $2\cdot 5^2\cdot 17^2$. The division polynomial of $E$ at $17$ factors as a product of $f(x)=x^4 + 482x^3 + 1144x^2 - 15809842x - 958623689$ with irreducible polynomials of degree $4$ and $8\cdot 17$. Fix a point $P \in E(\Qbar)$ whose $x$-coordinate $w$ is a root of $f(x)$; it is a $17$-torsion point. Let $C$ be the cyclic group of order $17$ generated by $P$; it is stable under the $\Gal_\QQ$ action. Let $\chi_1\colon \Gal_\QQ\to \FF_{17}^\times$ be the homomorphism such that $\sigma(P)=\chi_1(\sigma)\cdot P$ for $\sigma\in \Gal_\QQ$. One can show that the degree $4$ extension $\QQ(w)/\QQ$ is Galois, so $\chi_1(\Gal_\QQ)$ has cardinality $4$ or $8$. There is a second character $\chi_2\colon \Gal_\QQ\to \FF_{17}^\times$ such that, with respect to an appropriate change of basis, we have
\[
\rho_{E,17}(\sigma) = \left(\begin{smallmatrix}\chi_1(\sigma) & * \\0 & \chi_2(\sigma) \end{smallmatrix}\right).
\]
The cardinality of $\rho_{E,17}(\Gal_\QQ)$ is divisible by $17$ since the division polynomial of $E$ at $17$ has an irreducible factor whose degree is divisible by $17$. We have $\chi_1\chi_2=\omega$ where $\omega\colon \Gal_\QQ \to \FF_{17}^\times$ is the character describing the Galois action on the $17$-th roots of unity (we have $\omega(\Frob_p)=p$ for primes $p\neq 17$). The characters $\chi_1$ and $\chi_2$ are unramified at primes $p\nmid 2\cdot 5\cdot 17$, so $\chi_1= \omega^a\chi$ and $\chi_2=\omega^{17-a}\chi^{-1}$ for some integer $0\leq a <16$ and some character $\chi\colon \Gal_\QQ \to \FF_{17}^\times$ unramified at $p\nmid 2\cdot 5$.
Let $H_1$ and $H_2$ be the subgroup of $\GL_2(\FF_\ell)$ consisting of matrices of the form
\[
\left(\begin{smallmatrix} \omega(\sigma)^a & 0 \\0 & \omega(\sigma)^{17-a} \end{smallmatrix}\right) \quad\quad \text{and}\quad\quad\left(\begin{smallmatrix} \chi_1(\sigma) & 0 \\0 & \chi_1(\sigma)^{-1} \end{smallmatrix}\right),
\]
respectively, with $\sigma\in \Gal_\QQ$. Since $\omega$ and $\chi$ are ramified at different primes, we find that the image of $\rho_{E,\ell}$ is generated by $\left(\begin{smallmatrix} 1 & 1 \\0 & 1 \end{smallmatrix}\right)$ and the groups $H_1$ and $H_2$.
The character $\chi$ is unramified at $p\nmid 2\cdot 5$ and has image in a cyclic group of order $16$. Therefore, $\chi$ must factor through the group $\Gal(\QQ(\zeta_{64},\zeta_5)/\QQ)$. Since $641\equiv 1 \pmod{64\cdot 5}$, we have $\chi(\Frob_{641})=1$. Therefore, $\chi_1(\Frob_{641})=\omega(\Frob_{641})^a\cdot 1 \equiv 641^a \pmod{17}$ is a root of
\[
x^2-a_{641}(E)x+641 = x^2-(-9)x+641 \equiv (x-641^6)(x-641^{11}) \pmod{17},
\]
and hence $a\in\{6,11\}$ since $641$ is a primitive root modulo $17$. If $a=11$, then $\chi_1(\Gal_\QQ)=\FF_{17}^\times$ which is impossible since the cardinality of $\chi_1(\Gal_\QQ)$ is $4$ or $8$. Therefore, $a=6$. The group $H_1$ thus consists of matrices of the form $\left(\begin{smallmatrix} c^6 & 0 \\0 & c^{11} \end{smallmatrix}\right)$ with $c\in \FF_{17}^\times$, and in particular is generated by $\left(\begin{smallmatrix} 5^6 & 0 \\0 & 5^{11} \end{smallmatrix}\right)=\left(\begin{smallmatrix} 2 & 0 \\0 & 11 \end{smallmatrix}\right)$.
To complete the proof that $\rho_{E,17}(\Gal_\QQ)$ is $G_1$, it suffices to show that $H_2$ is generated by $\left(\begin{smallmatrix} 4 & 0 \\0 & -4\end{smallmatrix}\right)$; equivalently, to show that the image of $\chi$ is cyclic of order $4$. As noted earlier, $\chi$ factors through the group $\Gal(\QQ(\zeta_{64},\zeta_5)/\QQ)\cong (\ZZ/64\cdot 5\ZZ)^\times$. One can then show that $\Gal(\QQ(\zeta_{64},\zeta_5)/\QQ)$ is generated by $\Frob_{103}$, $\Frob_{137}$ and $\Frob_{307}$. The primes $p\in \{103, 137, 307\}$ were chosen to be congruent to $1$ modulo $17$, and hence $\chi(\Frob_p)=\chi_1(\Frob_p)$ is a root of $x^2-a_p(E)x+p$ modulo $17$. It is then straightforward to check that $\chi(\Frob_{103})$, $\chi(\Frob_{137})$ and $\chi(\Frob_{307})$ all have order $4$.
The elliptic curve $E'/\QQ$ defined by the Weierstrass equation $y^2 + xy + y = x^3 - 3041x + 64278$; it has $j$-invariant $- 17^2 \cdot 101^3/2$. One can show that $E/C$ is isomorphic to $E'$. The group $\rho_{E',17}(\Gal_\QQ)$ is thus conjugate to $G_2$ in $\GL_2(\FF_{17})$.
Finally we note that $G_1$ and $G_2$ have no index $2$ subgroups that do not contain $-I$.
\subsection{$\ell=37$}
We now prove Theorem~\ref{T:17-37}(\ref{T:17-37 ii}). Let $E/\QQ$ be the elliptic curve defined by the equation $y^2+xy+y=x^3+x^2-8x+6$; it has $j$-invariant $-7\cdot 11^3$ and conductor $5^2\cdot 7^2$. The division polynomial of $E$ at $17$ factors as a product of $f(x):=x^6 - 15x^5 - 90x^4 - 50x^3 + 225x^2 + 125x - 125$ with irreducible polynomials of degree $6$, $6$ and $18\cdot 37$. Fix a point $P \in E(\Qbar)$ whose $x$-coordinate $w$ is a root of $f(x)$; it is a $37$-torsion point. Let $C$ be the cyclic group of order $37$ generated by $P$; it is stable under the $\Gal_\QQ$ action.
Let $\chi_1\colon \Gal_\QQ\to \FF_{37}^\times$ be the homomorphism such that $\sigma(P)=\chi_1(\sigma)\cdot P$ for $\sigma\in \Gal_\QQ$. One can show that the degree $6$ extension $\QQ(w)/\QQ$ is Galois, so $\chi_1(\Gal_\QQ)$ has cardinality $6$ or $12$; in particular $\chi_1(\Gal_\QQ)$ is a subgroup of $(\FF_{37}^\times)^3$. There is a second character $\chi_2\colon \Gal_\QQ\to \FF_{37}^\times$ such that, with respect to an appropriate change of basis, we have
\[
\rho_{E,37}(\sigma) = \left(\begin{smallmatrix}\chi_1(\sigma) & * \\0 & \chi_2(\sigma) \end{smallmatrix}\right).
\]
The cardinality of $\rho_{E,37}(\Gal_\QQ)$ is divisible by $37$ since the division polynomial of $E$ at $37$ has an irreducible factor whose degree is divisible by $37$. So to prove that $\rho_{E,37}(\Gal_\QQ)=G_3$, it suffices to show that the homomorphism $\chi_1\times \chi_2 \colon \Gal_\QQ \to (\FF_{37}^\times)^3\times \FF_{37}^\times$ is surjective.
The characters $\chi_1$ and $\chi_2$ are unramified at primes $p\nmid 5\cdot 7\cdot 37$. By Proposition~11 of \cite{MR0387283}, we have $\{\chi_1,\chi_2\} = \{ \alpha, \alpha^{-1}\cdot \omega \}$ where $\alpha \colon \Gal_\QQ \to \FF_{37}^\times$ is a character unramified at primes $p\nmid 5\cdot 7$ and $\omega\colon \Gal_\QQ \to \FF_{37}^\times$ is the character describing the Galois action on the $37$-th roots of unity. Since $\alpha$ is unramified at $37$, we find that the character $\alpha^{-1}\cdot \omega$ is surjective and that $(\alpha \times (\alpha^{-1}\cdot \omega))(\Gal_\QQ)=\alpha(\Gal_\QQ)\times \FF_{37}^\times$. Since $\chi_1$ is not surjective, we must have $\chi_1=\alpha$ and $\chi_2=\alpha^{-1}\cdot \omega$. It thus suffices to show that the image of $\alpha$ contains an element of order $12$. The fixed field of the kernel of $\alpha$ is contained in $\QQ(\zeta_5,\zeta_7)$ since it is unramified at $p\nmid 5\cdot 7$ and has image relatively prime to $5\cdot 7$. Since $107\equiv 2 \pmod{35}$, we have $\alpha(\Frob_2)=\alpha(\Frob_{107})$. Therefore, $\alpha(\Frob_2)$ is a common root of $x^2-a_2(E)x+2 = x^2+x+2$ and $x^2-a_{107}(E)x+107=x^2+11x+107$ modulo $37$. This implies that $\alpha(\Frob_2)$ equals $8 \in \FF_{37}^\times$ which has order $12$.
One can show that the quotient of $E$ by $C$ is the elliptic curve $E'/\QQ$ defined by $y^2+xy+y=x^3+x^2-208083x-36621194$; it has $j$-invariant $-7\cdot 137^3\cdot 2083^3$. The group $\rho_{E',37}(\Gal_\QQ)$ is thus conjugate in $\GL_2(\FF_{37})$ to $G_4$.
Finally we note that $G_3$ and $G_4$ have no index $2$ subgroups that do not contain $-I$.
\section{Quadratic twists} \label{S:twist 1}
Fix an elliptic curve $E/\QQ$ with $j_E\notin \{0,1728\}$ and an integer $N\geq 3$.
Define the group $G:=\pm \rho_{E,N}(\Gal_\QQ)$ and let $\calH$ be the set of proper subgroups $H$ of $G$ that satisfy $\pm H=G$. For each group $H \in \calH$, we obtain a character
\[
\chi_{E,H} \colon \Gal_\QQ \to \{\pm 1\}
\]
by composing $\rho_{E,N}$ with the quotient map $G\to G/H \cong \{\pm 1\}$. The fixed field of the kernel of the character $\chi_{E,H}$ is of the form $\QQ(\sqrt{d_{E,H}})$ for a unique squarefree integer $d_{E,H}$. Define the set
\[
\calD_E:=\{ d_{E,H} \colon H \in \calH \}.
\]
Using $\pm \rho_{E,N}(\Gal_\QQ)=G$, we find that different groups $H\in \calH$ give rise to distinct characters $\chi_{E,H}$ and thus $|\calD_E|=|\calH|$.
\subsection{Twists with smaller image}
For a squarefree integer $d$, let $E_d/\QQ$ be a quadratic twist of $E/\QQ$ by $d$. By choosing an appropriate basis of $E_d[\ell]$, we may assume that $\rho_{E_d,N}\colon \Gal_\QQ \to \GL_2(\ZZ/N\ZZ)$ satisfies
\begin{equation*}\label{E:twist rho}
\rho_{E_d,N} = \chi_d \cdot \rho_{E,N},
\end{equation*}
where $\chi_d \colon \Gal_\QQ \to \{\pm 1\}$ is the character corresponding to the extension $\QQ(\sqrt{d})/\QQ$. We have $\pm \rho_{E_d,N}(\Gal_\QQ)=\pm \rho_{E,N}(\Gal_\QQ)= G$. Therefore, $\rho_{E_d,N}(\Gal_\QQ)$ is equal to either $G$ or to one of the subgroups $H\in \calH$.
We now show that $\calD_E$ is precisely the set of squarefree integers $d$ for which the image of $\rho_{E_d,N}$ is not conjugate to $G$.
\begin{lemma} \label{L:twist newer}
Take any squarefree integer $d$.
\begin{romanenum}
\item \label{L:twist newer i}
We have $d\in \calD_E$ if and only if the group $\rho_{E_d,N}(\Gal_\QQ)$ is conjugate in $\GL_2(\ZZ/N\ZZ)$ to a proper subgroup of $G$.
\item \label{L:twist newer ii}
If $d=d_{E,H}$ for some $H\in \calH$, then $\rho_{E_d,N}(\Gal_\QQ)$ is conjugate in $\GL_2(\ZZ/N\ZZ)$ to $H$.
\end{romanenum}
\end{lemma}
\begin{proof}
Take any group $H \in \calH$. Composing $\rho_{E_d,N}\colon \Gal_\QQ \to G$ with the quotient map $G\to G/H\cong \{\pm 1\}$ gives the character $\chi_d \cdot \chi_{E,H}$. Therefore, $\rho_{E_d,N}(\Gal_\QQ)$ is a subgroup of $H$ (and hence equal to $H$) if and only if $\chi_{E,H}=\chi_{d}$; equivalently, $d=d_{E,H}$. Parts (\ref{L:twist newer i}) and (\ref{L:twist newer ii}) are now immediate.
\end{proof}
Since $|\calD_E|=|\calH|$, we deduce from Lemma~\ref{L:twist newer} that the map
\[
\calH \to \calD_E, \quad H\mapsto d_{E,H}
\]
is a bijection.
\begin{remark}
Observe that $\rho_{E_d,N}(\Gal_\QQ)$ being conjugate to $H$ in $\GL_2(\ZZ/N\ZZ)$ need not imply that $d=d_{E,H}$. For example, it is possibly for distinct groups in $\calH$ to be conjugate in $\GL_2(\ZZ/N\ZZ)$.
\end{remark}
\subsection{Computing $\calD_E$}
Now assume that $N \geq 3$ is odd; we shall explain how to compute $\calD_E$ (we will later be interested in the case where $N$ is an odd prime). Let $M_E$ be set of squarefree integers that are divisible only by primes $p$ such that $p|N$ or such that $E$ has bad reduction at $p$.
For each $r\geq 1$, let $\calD_r$ be the set of $d\in M_E$ such that
\begin{equation} \label{E:twist -2}
a_{p}(E)\not \equiv -2 \left(\tfrac{d}{p}\right) \pmod{N}
\end{equation}
holds for all primes $p \leq r$ for which $E$ has good reduction and $p\equiv 1 \pmod{N}$.
\begin{lemma} \label{L:calD inclusion}
Suppose that $N$ is odd. We have $\calD_E\subseteq \calD_r$ with equality holding for all sufficiently large $r$.
\end{lemma}
\begin{proof}
Define $\mathscr{D}:= \cap_r \calD_r$; it is the set of $d\in M_E$ such that (\ref{E:twist -2}) holds for all primes $p\equiv 1 \pmod{N}$ for which $E$ has good reduction. We have $\calD_{r}\subseteq \calD_{r'}$ if $r\geq r'$, so it suffices to prove that $\mathscr{D}=\calD_E$.
Take any $d\in \mathscr{D}$. We have $a_p(E_d)=\legendre{d}{p} a_{p}(E)\not\equiv -2 \pmod{N}$ for all primes $p\equiv 1 \pmod{N}$ for which $E$ has good reduction. By the Chebotarev density theorem, there are no elements $g\in \rho_{E_d,N}(\Gal_\QQ)$ satisfying $\det(g)=1$ and $\tr(g)=-2$. In particular, the group $\rho_{E_d,N}(\Gal_\QQ)$ does not contain $-I$ and hence $d \in \calD_E$ by Lemma~\ref{L:twist newer}(\ref{L:twist newer i}). Therefore, $\mathscr{D}\subseteq \calD_E$.
We have $\calD_E\subseteq M_E$ since each character $\chi_{E,H}$ factors through $\rho_{E,N}$ (and is hence unramified at all primes $p\nmid N$ for which $E$ has good reduction).
Now take any $d\in \calD_E - \mathscr{D}$. There is thus a prime $p\equiv 1 \pmod{N}$ for which $E$ has good reduction and $a_p(E_d)=\legendre{d}{p} a_{p}(E) \equiv -2 \pmod{N}$. Define $g:=\rho_{E_d,N}(\Frob_p)$; it has trace $-2$ and determinant $1$. Since $N$ is odd, some power of $g$ is equal to $-I$. Therefore, $ \rho_{E_d,N}(\Gal_\QQ)=\pm \rho_{E_d,N}(\Gal_\QQ) = G$ which contradicts that $d\in \calD_E$. Therefore, $\calD_E - \mathscr{D}$ is empty and hence $\calD_E\subseteq \mathscr{D}$.
\end{proof}
One can compute the finite sets $\calD_r$ for larger and larger values of $r$ until $|\calD_r| = |\calH|$ and then $\calD_E=\calD_r$. This works since we always have an inclusion $\calD_E \subseteq\calD_r$ by Lemma~\ref{L:calD inclusion}, and equality holds when $|\calD_r| = |\calH|$ since $|\calD_E|=|\calH|$.\\
When $N$ is a prime, the integers in $\calD_E$ come in pairs.
\begin{lemma} \label{L:ell twist D}
Suppose $N=\ell$ is an odd prime. Let $\calD'_E$ be the set of $d\in \calD_E$ for which $\ell \nmid d$. Then
\[
\calD_E = \bigcup_{d\in \calD_E'} \{d, (-1)^{(\ell-1)/2} \ell\cdot d\}.
\]
\end{lemma}
\begin{proof}
Define $\ell^*:=(-1)^{(\ell-1)/2} \ell$. Take any $d \in \calD_E$. We need to show that $d \ell^*$ or $d/\ell^*$ belong to $\calD_E$ (whichever one is a squarefree integer). After possibly replacing $E$ by $E_d$, we may assume that $d=1$ and hence we need only verify that $\ell^* \in \calD_E$.
So assume that $\rho_{E,\ell}(\Gal_\QQ)$ is a proper subgroup of $G$ and hence is equal to one of the $H\in \calH$. We need to show that $\rho_{E',\ell}(\Gal_\QQ)$ is also a proper subgroup of $G$, where $E':=E_{\ell^*}$.
The field $\QQ(\sqrt{\ell^*})\subseteq \QQ(\zeta_\ell)$ is a subfield of both $\QQ(E[\ell])$ and $\QQ(E'[\ell])$. Since $E$ and $E'$ are isomorphic over $\QQ(\sqrt{\ell^*})$, we deduce that $[\QQ(E'[\ell]):\QQ]=[\QQ(E[\ell]):\QQ]$. Therefore,
\[
|\rho_{E',\ell}(\Gal_\QQ)| = [\QQ(E'[\ell]):\QQ]=[\QQ(E[\ell]):\QQ] = |\rho_{E,\ell}(\Gal_\QQ)| = |H| = |G|/2.
\]
By cardinality assumption, we deduce that $\rho_{E',\ell}(\Gal_\QQ)$ is conjugate to a proper subgroup of $G$.
\end{proof}
\begin{remark}
One could also use the methods of this section to help determine $\calH$. For example, if $\calD_r=\emptyset$ for some $r$, then $\calH= \emptyset$. Suppose we are in the setting, like what happens often in the introduction, where we know that $|\calH| \geq 2$ because we have two explicit elements of $\calH$. Then to verify that $|\calH|=2$, one need only find an $r$ such that $|\calD_r| =2$.
\end{remark}
\subsection{Some examples}
\subsubsection{}
Take $\ell=7$. Let $E/\QQ$ be the elliptic curve defined by $y^2=x^3-5^37^3x -5^47^2 106$; it has $j$-invariant $3^3\cdot 5\cdot 7^5/2^7$ and conductor $2\cdot 5^2 \cdot 7^2$. From the part of Theorem~\ref{T:main7} proved in \S\ref{SS:main proof 7}, we know that $\pm\rho_{E,7}(\Gal_\QQ)$ is conjugate to the group $G_1$ of \S\ref{SS:applicable 7}. Let $\calH$ be the set of proper subgroups $H$ of $G_1$ such that $\pm H=G_1$. The set $\calH$ consists of two groups; they are both conjugate in $\GL_2(\FF_7)$ to the group $H_{1,1}$ of \S\ref{SS:applicable 7}. The curve $E$ is denoted by $\calE_1$ in \S\ref{SS:applicable 7}.
We have $\calD_E \subseteq M_E = \{\pm 1,\pm 2, \pm 5, \pm 7, \pm 10, \pm 14, \pm 35, \pm 70\}$. The primes $211$, $239$ and $337$ are congruent to $1$ modulo $\ell$. One can check that
\[
a_{211}(E)=16 \equiv 2 \pmod{7}, \quad a_{239}(E)=-5\equiv 2 \pmod{7}, \quad a_{337}(E)=-5 \equiv 2 \pmod{7}.
\]
So if $d\in \calD_{337}$, then $\left(\tfrac{d}{211}\right)=1$, $\left(\tfrac{d}{239}\right)=1$ and $\left(\tfrac{d}{337}\right)=1$. Checking the $d\in M_E$, we find that $\calD_{337} \subseteq \{1,-7\}$. Since $|\calH|=2$, we deduce that $\calD_{E}=\{1,-7\}$.
Now let $E'/\QQ$ be any elliptic curve with $j$-invariant $3^3\cdot 5\cdot 7^5/2^7$. Using Lemma~\ref{L:twist newer}, we deduce that $\rho_{E',7}(\Gal_\QQ)$ is conjugate to $G_1$ if and only if $E'$ is not isomorphic to $E$ or its quadratic twist by $-7$. When $\rho_{E',7}(\Gal_\QQ)$ is not conjugate to $G_1$ it must be conjugate to $H_{1,1}$ in $\GL_2(\FF_7)$.
\subsubsection{}
Take $\ell=11$. Let $G_1$, $H_{1,1}$ and $H_{1,2}$ be the groups from \S\ref{SS:applicable 11}. The set $\calH$ of proper subgroups $H$ of $G_1$ for which $\pm H=G_1$ is equal to $\{H_{1,1}, H_{1,2}\}$.
Let $E/\QQ$ be the elliptic curve defined by $y^2+xy+y= x^3+x^2-305x+7888$; it has $j$-invariant $-11^2$ and is isomorphic to the curve $\calE_1$ of \S\ref{SS:applicable 11}. In \S\ref{SS:11 borel}, we showed that $\rho_{E,11}(\Gal_\QQ)$ and $\pm \rho_{E,11}(\Gal_\QQ)$ are conjugate in $\GL_2(\FF_{11})$ to $H_{1,1}$ and $G_1$, respectively.
Using Lemma~\ref{L:ell twist D} and $|\calD_E|=|\calH|$, we deduce that $\calD_E = \{1,-11\}$. Lemma~\ref{L:twist newer} implies that if $E'/\QQ$ has $j$-invariant $-11^2$, then $\rho_{E',11}(\Gal_\QQ)$ is not conjugate to $G_1$ if and only if $E'$ is isomorphic to $E$ or its quadratic twist by $-11$. If $E'$ is isomorphic to $E$ or its twist by $-11$, then $\rho_{E',11}(\Gal_\QQ)$ is conjugate in $\GL_2(\FF_{11})$ to $H_{1,1}$ or $H_{1,2}$, respectively.
\subsubsection{}
Take $\ell=11$. Let $G_2$, $H_{2,1}$ and $H_{2,2}$ be the subgroups of $\GL_2(\FF_{11})$ from \S\ref{SS:applicable 11}. The set $\calH$ of proper subgroups $H$ of $G_2$ for which $\pm H=G_2$ is equal to $\{H_{2,1}, H_{2,2}\}$.
Let $E/\QQ$ be the elliptic curve defined by $y^2+xy= x^3+x^2-3632x+82757$; it has $j$-invariant $-11\cdot 131^3$ and is isomorphic to the curve $\calE_2$ of \S\ref{SS:applicable 11}. In \S\ref{SS:11 borel}, we showed that $\rho_{E,11}(\Gal_\QQ)$ and $\pm \rho_{E,11}(\Gal_\QQ)$ are conjugate in $\GL_2(\FF_{11})$ to $H_{2,1}$ and $G_2$, respectively.
Using Lemma~\ref{L:ell twist D} and $|\calD_E|=|\calH|$, we deduce that $\calD_E = \{1,-11\}$. Lemma~\ref{L:twist newer} implies that if $E'/\QQ$ has $j$-invariant $-11\cdot 131^3$, then $\rho_{E',11}(\Gal_\QQ)$ is not conjugate to $G_2$ if and only if $E'$ is isomorphic to $E$ or its quadratic twist by $-11$. If $E'$ is isomorphic to $E$ or its twist by $-11$, then $\rho_{E',11}(\Gal_\QQ)$ is conjugate in $\GL_2(\FF_{11})$ to $H_{2,1}$ or $H_{2,2}$, respectively.
\section{Quadratic twists of families} \label{S:twists 2}
In this section, we complete the proof of the theorems from \S\ref{S:classification}.
\subsection{General setting} \label{SS:twist setup}
Fix an integer $N\geq 3$ and an applicable subgroup $G$ of $\GL_2(\ZZ/N\ZZ)$. Let $\calH$ be the set of proper subgroups $H$ of $G$ that satisfy $\pm H=G$.
Assume that the morphism $\pi_G\colon X_G \to \PP^1_\QQ$ arises from a rational function $J(t)\in \QQ(t)$, i.e., the function field of $X_G$ is of the form $\QQ(h)$ where $j=J(h)$. \\
Let $g(t)$ be a rational function $\QQ(t)$ such that
\[
a(t):=-3 g(t)^2 J(t)/(J(t)-1728) \quad \text{ and }\quad b(t):=-2 g(t)^3 J(t)/(J(t)-1728)
\]
belong to $\ZZ[t]$, and for which there is no irreducible element $\pi$ of the ring $\ZZ[t]$ such that $\pi^2$ divides $a$ and $\pi^3$ divides $b$. After possibly changing $g$ by a sign, we may assume that $g$ is the quotient of two polynomials with positive leading coefficient; the function $g(t)$ is now uniquely determined. Define $\Delta:=-16(4a^3+27b^2)$; it is a polynomial in $\ZZ[t]$ and equals $2^{12} 3^6 J(t)^2/(J(t)-1728)^3 g(t)^6$. Let $\scrM$ be the set of squarefree $f(t) \in \ZZ[t]$ which divide $N \Delta(t)$.\\
Take any $u \in \QQ$ for which $J(u)\notin \{0,1728,\infty\}$. We have $\Delta(u)\neq 0$ and hence $f(u)\neq 0$ for all $f\in \scrM$. Let $E_u/\QQ$ be the elliptic curve defined by the Weierstrass equation $y^2= x^3+a(u) x + b(u)$; note that $\Delta(u)\neq 0$ since $J(u)\notin\{0,1728,\infty\}$. One can readily check that the curve $E_u$ has $j$-invariant $J(u)$. \emph{Warning}: this is not to be confused with the quadratic twist notation we used in \S\ref{S:twist 1}.
\begin{prop} \label{P:fH prop}
There is an injective map
\[
\calH\to \scrM,\quad H\mapsto f_H
\]
such that for any $u\in \QQ$ with $J(u)\notin \{0,1728,\infty\}$ and $\pm \rho_{E_u,N}(\Gal_\QQ)$ conjugate to $G$ in $\GL_2(\ZZ/N\ZZ)$, the following hold:
\begin{alphenum}
\item \label{P:twists H a}
If $E'/\QQ$ is an elliptic curve with $j$-invariant $J(u)$, then $\rho_{E',N}(\Gal_\QQ)$ is conjugate to $G$ in $\GL_2(\ZZ/N\ZZ)$ if and only if $E'$ is not isomorphic to the quadratic twist of $E_u$ by $f_H(u)$ for all $H\in \calH$.
\item \label{P:twists H b}
If $E'/\QQ$ is isomorphic to the quadratic twist of $E_u$ by $f(u)$ for some $H\in \calH$, then $\rho_{E',N}(\Gal_\QQ)$ is conjugate to $H$ in $\GL_2(\ZZ/N\ZZ)$.
\end{alphenum}
The sets $\{f_H(u): H \in \calH\}$ and $\calD_{E_u}$ represent the same cosets in $\QQ^\times/(\QQ^\times)^2$, with $\calD_{E_u}$ defined as in \S\ref{S:twist 1}.
\end{prop}
\begin{proof}
Define the scheme $U:=\Spec \ZZ[t, N^{-1}, \Delta(t)^{-1}]$. By taking the square root of a polynomial $f\in \scrM$, we obtain an \'etale extension of $U$ of degree $1$ or $2$; we denote the corresponding quadratic character by $\chi_f\colon \pi_1(U) \to \{\pm 1\}$. Conversely, every (continuous) character $\pi_1(U)\to\{\pm 1\}$ is of the form $\chi_f$ for a unique $f\in \scrM$. (Note that $2$ always divides $\Delta(t)$).
The Weierstrass equation
\[
y^2=x^3 + a(t)x+b(t)
\]
defines a relative elliptic curve $E\to U$. Let $E[N]$ be the $N$-torsion subscheme of $E$. The morphism $E[N]\to U$ allows us to view $E[N]$ as a lisse sheaf of $\ZZ/N\ZZ$-modules on $U$ that is free of rank $2$. The sheaf $E[N]$ then gives rise to a representation
\[
\rho_N \colon \pi_1(U)\to \GL_2(\ZZ/N\ZZ)
\]
that is uniquely defined up to conjugacy (we will suppress the base point in our fundamental group since we are only interested in $\rho_N$ up to conjugacy).
We now consider specializations of $E$. Take any $u\in U(\QQ)$, i.e., an element $u\in \QQ$ with $\Delta(u)\neq 0$. One can show that elements $u \in U(\QQ)$ can also be described as those $u\in \QQ$ for which $J(u)\notin\{0,1728,\infty\}$. We can specialize $E$ at $u$ to obtain the elliptic curve that we have denoted $E_{u}/\QQ$; it is defined by $y^2=x^3+a(u) x + b(u)$ and has $j$-invariant $J(u)$.
Let $\rho_{u,N}\colon \Gal_\QQ \to \GL_2(\ZZ/N\ZZ)$ be the specialization of $\rho_N$ at $u$; it is obtained by composing the homomorphism $u_*\colon \Gal_\QQ \to \pi_1(U)$ coming from $u\in U(\QQ)$ with $\rho_N$. The homomorphism $\rho_{u,N}$ agrees, up to conjugacy, with the representation $\rho_{E_u,N}$ that describes the Galois action on the $N$-torsion points of $E_{u}$. So taking $\rho_{E_u,N}=\rho_{u,N}$, specialization gives an inclusion $\rho_{E_{u},N}(\Gal_\QQ) \subseteq \rho_N(\pi_1(U))$.
{We claim that $\pm \rho_N(\pi_1(U))$ and $G$ are conjugate in $\GL_2(\ZZ/N\ZZ)$. By Lemma~\ref{L:basic HIT}, the group $\pm \rho_{E_u,N}(\Gal_\QQ)$ is conjugate to $G$ in $\GL_2(\ZZ/N\ZZ)$ for ``most'' $u\in \QQ$. By Hilbert's irreducibility theorem, the group $\pm \rho_{E_{u},N}(\Gal_\QQ)$ equals $\pm \rho_N(\pi_1(U))$ for ``most'' $u\in \QQ$. This proves the claim.}\\
We may thus assume that $G=\pm \rho_N(\pi_1(U))$ and hence we have a representation $\rho_N\colon \pi_1(U)\to G$. Specializations thus give inclusions $\rho_{E_{u},N}(\Gal_\QQ) \subseteq G$. Take any $H \in \calH$ and let $\chi_H\colon \pi_1(U) \to\{\pm 1\}$ be the character obtained by composing $\rho_N$ with the quotient map $G\to G/H\cong \{\pm 1\}$. We thus have $\chi_H=\chi_{f_H}$ for a unique polynomial $f_H\in \scrM$.
Specializing $\chi_H$ at $u$, we obtain the character $\chi_{E_u,H}\colon \Gal_\QQ \to \{\pm 1\}$ from \S\ref{S:twist 1}. With notation as in \S\ref{S:twist 1}, we find that the integer $d_{E_u,H}$ lies in the same class in $\QQ^\times/(\QQ^\times)^2$ as $f_H(u)$. Therefore, the classes of $\calD_{E_u}$ in $\QQ^\times/(\QQ^\times)^2$ are represented by the set $\{f_H(u): H \in \calH\}$. Parts (\ref{P:twists H a}) and (\ref{P:twists H b}) are now immediate consequences of Lemma~\ref{L:twist newer}.
\end{proof}
We claim that the set of polynomials
\[
\scrF:=\{f_H : H \in \calH\}
\]
is uniquely determined and has cardinality $|\calH|$. By Hilbert irreducibility, one can chose $u\in U(\QQ)$ such that $\pm \rho_{E_u,N}(\Gal_\QQ)=G$ and such that the map $\scrM \to \QQ^\times/(\QQ^\times)^2$, $f\mapsto f(u)\cdot (\QQ^\times)^2$ is injective. The uniqueness of $\scrF$ then follows from part (\ref{P:twists H a}) of Proposition~\ref{P:fH prop}.
\subsection{Computing $\scrF$}
We now focus on the case where $N$ is a prime $\ell \in \{3,5,7,13\}$. Fix notation as in the subsection of \S\ref{S:classification} for the given $\ell$. \\
Let $G$ be one of the subgroups $G_i$ of $\GL_2(\FF_\ell)$ in \S\ref{S:classification} for which there is a corresponding rational function $J(t):=J_i(t) \in \QQ(t)-\QQ$. The group $G$ is applicable and in particular contains $-I$.
We take notation as in \S\ref{SS:twist setup}. In particular, $\calH$ is the set of proper subgroups $H$ of $G$ such that $\pm H = G$. We shall assume that $\calH\neq \emptyset$ (otherwise $\scrF=\emptyset$); this holds when
\[
(\ell,i) \in \big\{ (3,1),(3,3), (5,1), (5,5), (5,6), (7,1), (7,3), (7,4), (7,5), (7,7), (13,4), (13,5) \big\}.
\]
In each of these cases, one can check that $|\calH|=2$. \\
We now explain how to compute the set $\scrF=\{f_H : H \in \calH\}$; it has cardinality $|\calH|=2$. Take any $u \in \QQ$ with $J(u)\notin \{0,1728,\infty\}$ such that $J(u) \notin J_j(\QQ)$ for all $j < i$. From the parts of the main theorems proved in \S\ref{S:main classification}, this implies that $\pm \rho_{E_u,\ell}(\Gal_\QQ)$ is conjugate to $G$. Let $\calD_{E_u}$ be the (computable!) set from \S\ref{S:twist 1}. From Proposition~\ref{P:fH prop}, we find that
\begin{equation} \label{E:scrF inclusion}
\scrF \subseteq \{ f \in \scrM : f(u) \in d(\QQ^\times)^2 \text{ for some } d \in \calD_{E_u} \}.
\end{equation}
By considering (\ref{E:scrF inclusion}) with many such $u\in \QQ$, one is eventually left with only $|\calH|$ candidates $f\in \scrF$ to be of the form $f_H$; this then produces the set $\{f_H: H\in \calH\}$ of order $|\calH|$ (for our examples, one only needs to check $u \in \{1,2,3,4\}$). One could also work with a single $u\in \QQ$ chosen so that the map $\scrF\to \QQ^\times/(\QQ^\times)^2$, $f\mapsto f(u) \cdot (\QQ^\times)^2$ is injective. This method thus produces $\scrF$.\\
Doing the above computations, we find that
\[
\{f_H : H \in \calH\} = \{ f_1, \ell^* f_1\}
\]
for a unique polynomial $f_1\in \scrF$, where $\ell^* := (-1)^{(\ell-1)/2} \cdot \ell$; this can also be deduced from $|\calH|=2$ and Lemma~\ref{L:ell twist D}. We thus have $f_1 = f_{M_1}$ and $\ell^* f_1 = f_{M_2}$, where $\calH=\{M_1,M_2\}$.
Let $h$ be the largest element of $\ZZ[t]$, in terms of divisibility, with positive leading coefficient such that $h^4$ divides $af_1^2$ and $h^6$ divides $b f_1^3$; define $A:= (af_1^2)/h^4$ and $B:=(bf_1^3)/h^6$ in $\ZZ[t]$. The Weierstrass equation
\[
y^2= x^3+A(t) x + B(t)
\]
is precisely the equation given for $\calE_{i,t}$ in the subsection of \S\ref{S:classification} corresponding to the prime $\ell$. (For code verifying these claims, see the link given in \S\ref{SS:overview}.)
\\
For $u\in \QQ$ with $J(u)\notin \{0,1728,\infty\}$, let $\calE_{i,u}$ be the elliptic curve over $\QQ$ defined by setting $t$ equal to $u$. Let $E'/\QQ$ be any elliptic curve with $j_{E'}\notin \{0,1728\}$ for which $\pm \rho_{E',\ell}(\Gal_\QQ)$ is conjugate to $G$ in $\GL_2(\FF_\ell)$. From the parts of the main theorems proved in \S\ref{S:main classification}, we have $j_{E'}=J(u)$ for some $u\in \QQ$. The curve $E_u/\QQ$ also has $j$-invariant $J(u)$. The twist of $E_u$ by $f_1(u)$ is isomorphic to the the curve $\calE_{i,u}/\QQ$. By Proposition~\ref{P:fH prop}, we deduce that $\rho_{E',\ell}(\Gal_\QQ)$ is conjugate to $G$ if and only if $E'$ is not isomorphic to $\calE_{i,u}$ and not isomorphic to the quadratic twist of $\calE_{i,u}$ by $\ell^*$. By Proposition~\ref{P:fH prop}, $\rho_{E',\ell}(\Gal_\QQ)$ is conjugate to $M_1$ or $M_2$ when $E'$ is isomorphic to $\calE_{i,u}$ or the quadratic twist of $\calE_{i,u}$ by $\ell^*$, respectively.
It thus remains to determine $M_1$ and $M_2$.\\
If $(\ell,i) \in \{(3,1), (7,1)\}$, then $M_1$ and $M_2$ are both conjugate to $H_{i,1}$ since the two groups in $\calH$ are conjugate in $\GL_2(\FF_\ell)$. We shall now assume that $(\ell,i) \notin \{(3,1), (7,1)\}$. We then have $\calH=\{H_{i,1},H_{i,2}\}$. It thus remains to prove that $M_1=H_{i,1}$ (and hence $M_2=H_{i,2}$). \\
Suppose that $(\ell,i) \in \big\{ (5,1), (5,5), (5,6), (7,3), (7,4), (13,4), (13,5) \}$. Take $u$, $p$ and $a$ as in Table 2 below for the pair $(\ell,i)$.
{
\renewcommand{\arraystretch}{1.1}
\begin{table}[htdp]
\begin{center}\begin{tabular}{c||c|c|c|c|c|c|c}
$(\ell,i)$ & $( 5, 1 )$ & $( 5, 5 )$ & $( 5, 6 )$ & $(7, 3)$ & $(7, 4)$ & $(13, 4)$ & $(13, 5)$ \\ \hline
$u$ & $1$ & $2$ & $1$ & $2$ &$2$ & $1$ & $1$ \\
$p$ & $2$ & $3$ & $2$ & $3$ & $3$ &$2$ & $2$ \\
$a$ & $-2$ & $-1$ & $-2$ & $-3$ & $-3$ & $2$ & $2$\\
\end{tabular} \caption{}
\end{center}
\end{table}
}
The element $u\in \QQ$ is chosen so that $J_i(u) \notin \{0,1728,\infty\}$ and such that $J_i(u) \notin J_j(\QQ \cup \{\infty\})$ for all $j<i$. Define the elliptic curve $E:=\calE_{i,u}/\QQ$. By our choice of $u$, the group $\rho_{E,\ell}(\Gal_\QQ)$ is conjugate in $\GL_2(\FF_\ell)$ to $M_1$.
The curve has good reduction at the prime $p$ and we have $a=a_p(E)$. Let $t_p$ be the image of $(a,p)$ in $\FF_\ell^2$; it equals $(\tr(A),\det(A))$ with $A:=\rho_{E,\ell}(\Frob_p) \in M_1$. A direct computation shows that $t_p \notin \{ (\tr(A),\det(A)): A \in H_{i,2}\}$. Therefore, $M_1$ is not conjugate to $H_{i,2}$. So $M_1$ must be conjugate to $H_{i,1}$ and hence $M_2$ is conjugate to $H_{i,2}$.\\
Finally, consider the remaining pairs $(\ell,i) \in \{(3,3),(7,5),(7,7)\}$.
Consider $(\ell,i)=(3,3)$. The pair $(3(u+1)^2, 4u(u+1)^2)$ is a point of order $3$ of $\calE_{3,u}$ for all $u$. This implies that $M_1$ is conjugate in $\GL_2(\FF_3)$ to a subgroup of $\left(\begin{smallmatrix}1 & * \\0 & * \end{smallmatrix}\right)$. So $M_1\neq H_{i,2}$ and hence $M_1=H_{i,1}$.
We may now suppose that $\ell=7$ and $i\in\{5,7\}$.
Take $i=5$. Let $E'/\QQ$ be the elliptic curve defined by $y^2=x^3-2835(-7)^2 x-71442(-7)^3$; it is the quadratic twist of $\calE_{5,0}$ by $-7$. Using Theorem~\ref{T:main7}(ii), which we proved in \S\ref{S:main classification}, we find that $\pm \rho_{E',7}(\Gal_\QQ)$ is conjugate to $G_5$. The group $\rho_{E',7}(\Gal_\QQ)$ is thus conjugate to $M_2$. So to prove that $M_2=H_{i,2}$, and hence $M_1=H_{i,1}$, we need only verify that $E'$ has a $7$-torsion point defined over some cubic field. Let $w\in \Qbar$ be a root of the irreducible polynomial $x^3-441x^2-83349x+22754277$. The pair $(w, 21w -1323)$ is a point of order $7$ on $E'$.
Finally, take $i=7$. Let $E'/\QQ$ be the elliptic curve defined by $y^2=x^3-17870609043(-7)^2 x-919511455160466(-7)^3$; it is the quadratic twist of $\calE_{7,1}$ by $-7$. Using Theorem~\ref{T:main7}(ii), which we proved in \S\ref{S:main classification}, we find that $\pm \rho_{E',7}(\Gal_\QQ)$ is conjugate to $G_7$. The group $\rho_{E',7}(\Gal_\QQ)$ is thus conjugate to $M_2$. So to prove that $M_2=H_{i,2}$, and hence $M_1=H_{i,1}$, we need only verify that $E'$ has a $7$-torsion point defined over some cubic field. Let $w\in \Qbar$ be a root of the irreducible polynomial $x^3 - 1750329x^2 + 1015924207851x - 195667237639563291$. The pair $(w, 1323w - 714884373)$ is a point of order $7$ on $E'$.
\section{Proof of Propositions from \S\ref{SS:CM}} \label{SS:CM proofs}
Let $E$ be an elliptic curve defined over $\QQ$ that has complex multiplication. Let $R$ be the ring of endomorphisms of $E_{\Qbar}$. Let $k \subseteq \Qbar$ be the minimal extension of $\QQ$ over which all the endomorphisms of $E_{\Qbar}$ are defined; it is an imaginary quadratic field. Moreover, we can identify $k$ with $R\otimes_\ZZ \QQ$ (the action of $R$ on the Lie algebra of $E_k$ gives a ring homomorphism $R\to k$ that extends to an isomorphism $R\otimes_\ZZ \QQ \to k$). The field $k$ has discriminant $-D$.
Take any \emph{odd} prime $\ell$. For each integer $n\geq 1$, let $E[\ell^n]$ be the $\ell^n$-torsion subgroup of $E(\Qbar)$. The $\ell$-adic \defi{Tate module} $T_\ell(E)$ of $E$ is the inverse limit of the groups $E[\ell^n]$ with multiplication by $\ell$ giving transition maps $E[\ell^{n+1}]\to E[\ell]$; it is a free $\ZZ_\ell$-module of rank $2$. The natural Galois action on $T_\ell(E)$ can be expressed in terms of a representation
\[
\rho_{E,\ell^\infty} \colon \Gal_k \to \Aut_{\ZZ_\ell}(T_\ell(E)).
\]
The ring $R$ acts on each of the $E[\ell^n]$ and this induces a faithful action of $R$ on $T_\ell(E)$.
The Tate module $T_\ell(E)$ is actually a free module over $R_\ell:=R\otimes_\ZZ \ZZ_\ell$ of rank $1$ (see the remarks at the end of \S4 of \cite{MR0236190}). We can thus make an identification $\Aut_{R_\ell}(T_\ell (E)) = R_\ell^\times$. The actions of $\Gal_k=\Gal(\Qbar/k)$ and $R_\ell$ on $T_\ell(E)$ commute, so the restriction of $\rho_{E,\ell^\infty}$ to $\Gal_k$ gives a representation
\[
\Gal_k \to \Aut_{R_\ell}(T_\ell (E))=R_\ell^\times.
\]
\begin{lemma} \label{L:CM ell-adic image}
\begin{romanenum}
\item \label{L:CM ell-adic image a}
If $E$ has good reduction at $\ell$, then $\rho_{E,\ell^\infty}(\Gal_k) = R_\ell^\times$.
\item \label{L:CM ell-adic image b}
If $j_E\neq 0$, then $\rho_{E,\ell^\infty}(\Gal_k)$ is an open subgroup of $R_\ell^\times$ whose index is a power of $2$.
\end{romanenum}
\end{lemma}
\begin{proof}
Since $R_\ell^\times$ is commutative, we can factor $\rho_{E,\ell^\infty}|_{\Gal_k}$ through the maximal abelian quotient of $\Gal_k$. Composing with the reciprocity map of class field theory, we obtain a continuous representation $\varrho_{E,\ell^\infty}\colon \AA_k^\times \to R_\ell^\times$, where $\AA_k^\times$ is the group of ideles of $k$. Define $k_\ell := k \otimes_\ZZ \QQ_\ell = {\prod}_{v | \ell} k_v$, where the product is over the places $v$ of $k$ lying over $\ell$ and $k_v$ is the completion of $k$ at $v$. For an idele $a\in \AA_k^\times$, let $a_\ell$ be the component of $a$ in $k_\ell^\times$. From \cite{MR0236190}*{Theorems 10 \& 11}, there is a unique homomorphism $\varepsilon\colon \AA_k^\times \to k^\times$ such that $\varrho_{E,\ell^\infty}(a) = \varepsilon(a) a_\ell^{-1}$ for $a\in \AA_k^\times$. The homomorphism $\varepsilon$ satisfies $\varepsilon(x)=x$ for all $x\in k^\times$ and its kernel is open in $\AA_k^\times$. We identify $R_\ell^\times = \prod_{v|\ell} \OO_v^\times$, where $\OO_v$ is the valuation ring of $k_v$, with a subgroup of $\AA^\times_k$ (by letting the coordinates at the places $v\nmid \ell$ of $k$ be $1$). Let $B$ be the kernel of $\varepsilon|_{R_\ell^\times}$.
First suppose that $E$ has good reduction at $\ell$, and hence at all places $v|\ell$ of $k$. By the first corollary of Theorem~11 in \cite{MR0236190}, we deduce that that $\varepsilon$ is unramified at all $v | \ell$. Therefore, $B=R_\ell^\times$ and hence $\varrho_{E,\ell^\infty}(R_\ell^\times)=R_\ell^\times$. Therefore, $\rho_{E,\ell^\infty}(\Gal_k)$ contains, and hence is equal to, $R_\ell^\times$.
Now suppose that $j_E \neq 0$. Since $\ell$ is odd and $j_E\neq 0$, the subgroup of $R[\ell^{-1}]^\times$ consisting of roots of unity has order $2$ or $4$. By Theorem~11(ii) and Theorem~6(b) in \cite{MR0236190}, we find that $B$ is an open subgroup of $R_\ell^\times$ with index a power of $2$. So $\varrho_{E,\ell^\infty}(B)=B$ and hence $\rho_{E,\ell^\infty}(\Gal_k) \supseteq B$. Therefore, $\rho_{E,\ell^\infty}(\Gal_k)$ is an open subgroup of $R_\ell^\times$ whose index is a power of $2$.
\end{proof}
The following gives constraints on the elements of $\rho_{E,\ell^\infty}(\Gal_\QQ-\Gal_k)$. Since $R$ is a quadratic order, there is an element $\beta \in R-\ZZ$ such that $\beta^2\in \ZZ$; note that $\beta$ is not defined over $\QQ$. We can view $\beta$ as an endomorphism of $T_\ell(E)$.
\begin{lemma} \label{L:non-comm of R}
For any $\sigma \in \Gal_\QQ - \Gal_k$, we have $\rho_{E,\ell^\infty}(\sigma) \beta=-\beta \rho_{E,\ell^\infty}(\sigma)$ and $\tr(\rho_{E,\ell^\infty}(\sigma))=0$.
\end{lemma}
\begin{proof}
Take any $\sigma \in \Gal_\QQ - \Gal_k$. The group $\Gal_\QQ$ acts on $R$ and we have $\sigma(\beta)=-\beta$ since $\beta^2 \in \ZZ$ and $\beta$ is not defined over $\QQ$ (but is defined over $k$). So for each $P\in E[\ell^n]$, we have $\sigma( \beta(P))=\sigma(\beta)(\sigma(P))=-\beta(\sigma(P))$. Taking an inverse limit, we deduce that $\rho_{E,\ell^\infty}(\sigma) \beta=-\beta \rho_{E,\ell^\infty}(\sigma)$. In $\Aut_{\QQ_\ell}(T_\ell(E)\otimes_{\ZZ_\ell} \QQ_\ell) \cong \GL_2(\QQ_\ell)$, we have $\rho_{E,\ell^\infty}(\sigma) =-\beta \rho_{E,\ell^\infty}(\sigma)\beta^{-1}$. Taking traces we deduce that $\tr(\rho_{E,\ell^\infty}(\sigma))=-\tr(\rho_{E,\ell^\infty}(\sigma))$ and hence $\tr(\rho_{E,\ell^\infty}(\sigma))=0$.
\end{proof}
\begin{lemma} \label{L:EDf generics}
Suppose that $\ell \nmid D$ and that $E$ has good reduction at $\ell$.
\begin{romanenum}
\item
If $\ell$ splits in $k$, then $\rho_{E,\ell}(\Gal_\QQ)$ is conjugate in $\GL_2(\FF_\ell)$ to $N_s(\ell)$.
\item
If $\ell$ is inert in $k$, then $\rho_{E,\ell}(\Gal_\QQ)$ is conjugate in $\GL_2(\FF_\ell)$ to $N_{ns}(\ell)$.
\end{romanenum}
\end{lemma}
\begin{proof}
Lemma~\ref{L:CM ell-adic image} implies that the group $C:=\rho_{E,\ell}(\Gal_k)$ is isomorphic to $(R/\ell R)^\times$. The ring $R/\ell R$ is isomorphic to $\FF_\ell \times \FF_\ell$ or $\FF_{\ell^2}$ when $\ell$ splits or is inert in $k$, respectively. Therefore, $C$ is a Cartan subgroup of $\GL_2(\FF_\ell)$; it is split if and only if $\ell$ splits in $k$. Let $N$ be the normalizer of $C$ in $\GL_2(\FF_\ell)$. The group $C=\rho_{E,\ell}(\Gal_k)$ is normal in $\rho_{E,\ell}(\Gal_\QQ)$ since $k/\QQ$ is a Galois extension, so $\rho_{E,\ell}(\Gal_\QQ) \subseteq N$.
It remains to show that $\rho_{E,\ell}(\Gal_\QQ)= N$. Suppose that $\rho_{E,\ell}(\Gal_\QQ) \neq N$, and hence $\rho_{E,\ell}(\Gal_\QQ)=C=\rho_{E,\ell}(\Gal_k)$. This implies that the actions of $\Gal_\QQ$ and $R$ on $E[\ell]$ commute. However, this contradicts Lemma~\ref{L:non-comm of R} which implies that the actions of $\sigma \in \Gal_\QQ-\Gal_k$ and $\beta$ on $E[\ell]$ anti-commute. Therefore, $\rho_{E,\ell}(\Gal_\QQ) = N$.
\end{proof}
We now describe the commutator subgroup of the normalizer $N$ of a Cartan subgroup $C$ of $\GL_2(\FF_\ell)$. Let $\varepsilon \colon N \to N/C \cong \{\pm 1\}$ be the quotient map, and define the homomorphism
\[
\varphi\colon N \to \{\pm 1\} \times \FF_\ell^\times,\quad A \mapsto (\varepsilon(A),\det(A));
\]
it is surjective.
\begin{lemma} \label{L:N group theory}
\begin{romanenum}
\item \label{L:N group theory i}
The commutator subgroup of $N$ is $\ker \varphi$, i.e., the subgroup of $C$ consisting of matrices with determinant $1$.
\item \label{L:N group theory ii}
If $H$ is a subgroup of $N$ satisfying $\pm H = N$, then $H=N$.
\end{romanenum}
\end{lemma}
\begin{proof}
The kernel of $\varphi$ contains the commutator subgroup of $N$ since the image of $\varphi$ is abelian. It suffices to show that every element in $\ker \varphi$ is a commutator. If $N$ is conjugate to $N_{s}(\ell)$, this is immediate since
$\left(\begin{smallmatrix}0 & 1 \\1 & 0 \end{smallmatrix}\right)\left(\begin{smallmatrix}1 & 0 \\0 & a \end{smallmatrix}\right)\left(\begin{smallmatrix}0 & 1 \\1 & 0 \end{smallmatrix}\right)^{-1}\left(\begin{smallmatrix}1 & 0 \\0 & a \end{smallmatrix}\right)^{-1}=\left(\begin{smallmatrix}a & 0 \\0 & a^{-1} \end{smallmatrix}\right)$.
We now consider the non-split case. We may take $C=C(\ell)$, $N=N(\ell)$ and with the explicit $\epsilon\in \FF_\ell^\times$ as given in the notation section of \S\ref{S:classification}. Fix $\beta\in \FF_{\ell^2}$ for which $\beta^2 = \epsilon$. The map $C(\ell)\to \FF_{\ell^2}^\times$, $\left(\begin{smallmatrix}a & b\epsilon \\b & a \end{smallmatrix}\right) \to a+b \beta$ is a group isomorphism. Fix any $B \in N(\ell)-C(\ell)$. One can check that the map
\begin{align} \label{E:actually norm}
C(\ell) \to C(\ell),\quad A \mapsto BAB^{-1} A^{-1}
\end{align}
corresponds to the homomorphism $\FF_{\ell^2}^\times \to \FF_{\ell^2}^\times$, $\alpha \mapsto \alpha^{\ell-1}$. In particular, the image of the map (\ref{E:actually norm}) is the unique (cyclic) subgroup of $C(\ell)$ of order $\ell+1$; these are the matrices in $C(\ell)$ with determinant $1$. This completes the proof of (\ref{L:N group theory i}).
Finally, let $H$ be a subgroup of $N$ satisfying $\pm H=N$. The group $H$ is normal in $N$ and $N/H$ is abelian, so $H$ contains the commutator subgroup of $N$. From (\ref{L:N group theory i}), the commutator subgroup of $N$, and hence $H$, contains $-I$. Therefore, $H=\pm H=N$.
\end{proof}
\subsection{Proof of Proposition~\ref{P:CM main}(\ref{P:CM main a}) and (\ref{P:CM main b})}
Let $E/\QQ$ be an CM elliptic curve with $j_E \neq 0$. The curve $E$ is thus a twist of one of the curves $E_{D,f}/\QQ$ from Table~1. Take any odd prime $\ell \nmid D$. The curve $E_{D,f}$ has good reduction at $\ell$. By Lemma~\ref{L:EDf generics}, the group $\rho_{E_{D,f},\ell}(\Gal_\QQ)$ is the normalizer $N$ of a Cartan subgroup $C$ of $\GL_2(\FF_\ell)$. Also the Cartan subgroup $C$ is split or non-split if $\ell$ is split or inert, respectively, in $k$.
First suppose that $j_E\neq 1728$. Since $j_E\notin\{0,1728\}$, the curve $E$ is a quadratic twist of $E_{D,f}$. As noted in the introduction, this implies that $\pm \rho_{E,\ell}(\Gal_\QQ)$ and $\pm \rho_{E_{D,f},\ell}(\Gal_\QQ)=N$ are conjugate in $\GL_2(\FF_\ell)$. After first conjugating $\rho_{E,\ell}(\Gal_\QQ)$, we may assume that $N=\pm\rho_{E,\ell}(\Gal_\QQ)$. By Lemma~\ref{L:N group theory}, we have $\rho_{E,\ell}(\Gal_\QQ)=N$.
Now suppose that $j_E=1728$. Let $\mu_4$ be the group of $4$-th roots of unity in $R$. The elliptic curve $E/\QQ$ can be defined by an equation of the form $y^2=x^3+dx$ for some non-zero integer $d$, i.e., $E$ is a \defi{quartic twist} of $E_{4,1}$. There is thus a character $\alpha \colon \Gal_k \to \mu_4 \subseteq R^\times$ such that the representations $\rho_{E,\ell^\infty}$ and $\alpha\cdot \rho_{E_{4,1},\ell^\infty}\colon \Gal_k \to R_\ell^\times$ are equal. We have $\rho_{E_{4,1},\ell^\infty}(\Gal_k)= R_\ell^\times$ by Lemma~\ref{L:CM ell-adic image}(\ref{L:CM ell-adic image a}), so the image of $\rho_{E,\ell^\infty}(\Gal_k)$ in $R_\ell^\times/\{\pm 1\}$ has index $1$ or $2$. Therefore, the image of $\rho_{E,\ell}(\Gal_k)$ in $C/\{\pm I\}$ has index $1$ or $2$. We have $\rho_{E,\ell}(\Gal_\QQ)\not\subseteq C$ since otherwise the actions of $\Gal_\QQ$ and $R$ on $E[\ell]$ would commute (which is impossible by Lemma~\ref{L:non-comm of R}). Therefore, the image of $\rho_{E,\ell}(\Gal_\QQ)$ in $N/\{\pm I\}$ is an index $1$ or $2$ subgroup.
The group $G:=\pm\rho_{E,\ell}(\Gal_\QQ)$ thus has index $1$ or $2$ in $N$. Since $\rho_{E,\ell}(\Gal_k)\subseteq C$ and $\rho_{E,\ell}(\Gal_\QQ)\not\subseteq C$, the quadratic character $\varepsilon\circ \rho_{E,\ell} \colon \Gal_\QQ \to \{\pm 1\}$ corresponds to the extension $k=\QQ(i)$ of $\QQ$. The homomorphism $\det\circ \rho_{E,\ell} \colon \Gal_\QQ \to \FF_\ell^\times$ is surjective and factors through $\Gal(\QQ(\zeta_\ell)/\QQ)$. We have $\QQ(i)\cap \QQ(\zeta_\ell)=\QQ$ since $\ell$ is odd, so $\varphi(\rho_{E,\ell}(\Gal_\QQ))=\{\pm 1\} \times \FF_\ell^\times$ and hence $\varphi(G)=\{\pm 1\} \times \FF_\ell^\times$. Since $[N:G]\leq 2$, the group $G$ is normal in $N$ with abelian quotient $N/G$. In particular, $G$ contains the commutator subgroup of $N$. By Lemma~\ref{L:N group theory}, we deduce that $G$ contains the kernel of $\varphi$. Since $G$ contains the kernel of $\varphi$ and $\varphi(G)=\{\pm 1\} \times \FF_\ell^\times$, we have $G=N$. By Lemma~\ref{L:N group theory}, we conclude that $\rho_{E,\ell}(\Gal_\QQ)=N$.
\subsection{Proof of Proposition~\ref{P:CM main}(\ref{P:CM main c})}
We first consider the elliptic curve $E=E_{D,f}$ over $\QQ$ from Table~1 with $D=\ell$, where $\ell$ is an odd prime and $j_E\neq 0$. We have $k=\QQ(\sqrt{-\ell})$.
\begin{lemma} \label{L:CM D eq ell}
The group $\pm \rho_{E,\ell}(\Gal_\QQ)$ is conjugate to $G$.
\end{lemma}
\begin{proof}
Let $\bbar\beta$ be the image of $f\sqrt{-D}$ in $R/\ell R$. Since $\ell$ is odd, the $\FF_\ell$-module $R/\ell R$ has basis $\{\bbar\beta,1\}$ and $\bbar\beta^2=0$. Using this basis, we find that $R/\ell R$ is isomorphic to the subring $A:=\FF_\ell\left(\begin{smallmatrix}0 & 1 \\0 & 0 \end{smallmatrix}\right) \oplus \FF_\ell\left(\begin{smallmatrix}1 & 0 \\0 & 1 \end{smallmatrix}\right)$ of $M_2(\FF_\ell)$. Using that $\rho_{E,\ell^\infty}(\Gal_k) \subseteq R_\ell^\times$, we deduce that $\rho_{E,\ell}(\Gal_k)$ is conjugate in $\GL_2(\FF_\ell)$ to a subgroup of $A^\times$. We may thus assume that
\[
\rho_{E,\ell}(\Gal_k) \subseteq A^\times = \{ \left(\begin{smallmatrix}a & b \\0 & a \end{smallmatrix}\right) : a\in \FF_\ell^\times, b\in \FF_\ell\}.
\]
By Lemma~\ref{L:CM ell-adic image}(\ref{L:CM ell-adic image b}), we deduce that $[A^\times:\rho_{E,\ell}(\Gal_k)]$ is a power of $2$ and hence $\rho_{E,\ell}(\Gal_k)$ contains the order $\ell$ group $\langle \left(\begin{smallmatrix}1 & 1 \\0 & 1 \end{smallmatrix}\right) \rangle$. The order of $\rho_{E,\ell}(\Gal_k)$ is divisible by $(\ell-1)/2$ since $\det(\rho_{E,\ell}(\Gal_k))=(\FF_\ell^\times)^2$, so $\rho_{E,\ell}(\Gal_k)$ contains $\{ \left(\begin{smallmatrix}a & b \\0 & a \end{smallmatrix}\right) : a\in (\FF_\ell^\times)^2, b\in \FF_\ell\}$. Therefore, $\pm \rho_{E,\ell}(\Gal_k)=A^\times$ since $-1$ is not a square in $\FF_\ell$ (we have $\ell\equiv 3\pmod{4}$).
Fix any $\sigma\in \Gal_\QQ-\Gal_k$. The matrix $g = \rho_{E,\ell}(\sigma)$ is upper triangular since the Borel subgroup $B(\ell)$ is the normalizer of $A^\times=\pm\rho_{E,\ell}(\Gal_k)$ in $\GL_2(\FF_\ell)$. We have $\tr(g)=0$ by Lemma~\ref{L:non-comm of R}, so $g=\left(\begin{smallmatrix}a & b \\0 & -a \end{smallmatrix}\right)$ for some $a\in \FF_\ell^\times$ and $b\in \FF_\ell$. The group $\pm \rho_{E,\ell}(\Gal_\QQ)$ is generated by $g$ and $A^\times$ and is thus $G$.
\end{proof}
The subgroups $H$ of $G$ that satisfy $\pm H = G$ are $H_1$, $H_2$ and $G$.
\begin{lemma} \label{L:H1new}
The groups $\rho_{E,\ell}(\Gal_\QQ)$ and $H_1$ are conjugate in $\GL_2(\FF_\ell)$.
\end{lemma}
\begin{proof}
By Lemma~\ref{L:CM D eq ell}, we may assume that $\pm\rho_{E,\ell}(\Gal_\QQ)=G$. There are thus unique characters $\psi_1,\psi_2\colon \Gal_\QQ \to \FF_\ell^\times$ such that $\rho_{E,\ell}= \left(\begin{smallmatrix} \psi_1 & * \\0 & \psi_2 \end{smallmatrix}\right)$. Let $f \in \QQ[x]$ be the $\ell$-th division polynomial of $E/\QQ$; it is a polynomial of degree $(\ell^2-1)/2$ whose roots in $\Qbar$ are the $x$-coordinates of the non-zero points in $E[\ell]$. Since $\pm \rho_{E,\ell}(\Gal_\QQ)= G$, we find that $f=f_1 f_2$ where the polynomials $f_1,f_2\in \QQ[x]$ are irreducible, and $f_1$ has degree $(\ell-1)/2$. We may take $f_1$ so that it is monic. Take any root $a\in \Qbar$ of $f_1$ and choose a point $P=(a,b)$ in $E[\ell]$. We have $\sigma(P)=\psi_1(\sigma)P$ for all $\sigma \in \Gal_\QQ$. Therefore, $\QQ(a,b)$ is the fixed field in $\Qbar$ of $\ker \psi_1$.
Suppose that $\ell=3$. The point $(3,-2)$ of $E_{3,2}$ has order $3$. The point $(12, -4)$ of $E_{3,3}$ has order $3$. Therefore, $\QQ(a,b)=\QQ$.
Suppose that $\ell=7$. For the curve $E_{7,1}$ we have computed that $f_1=x^3 - 441x^2 + 59339x - 2523451$. If $a\in \Qbar$ is a root of $f_1$, then one can check that $(a,-7a+ 49)$ belongs to $E$. For the curve $E_{7,2}$ we have computed that $f_1=x^3 - 49x^2 - 1029x + 31213$. If $a\in \Qbar$ is a root of $f_1$, then one can check that $(a,21a - 2107)$ belongs to $E$. In both cases, we have $[\QQ(a,b):\QQ]=3$.
Suppose that $\ell >7$. Dieulefait, Gonz\'alez-Jim\'enez and Jim\'enez-Urroz have computed $\QQ(a,b)$ and found it to be equal to the maximal totally real subfield $\QQ(\zeta_\ell)^+$ of $\QQ(\zeta_\ell)$, cf.~Lemma~4 of \cite{MR2775372}. They also give a link to files containing an explicit polynomial $f_1$. In particular, $[\QQ(a,b):\QQ]=(\ell-1)/2$. (However, note that the conclusions on the image of $\rho_{E,\ell}$ in Proposition~9 of \cite{MR2775372} are not correct.)
In all cases, the image of $\psi_1$ has order $[\QQ(a,b):\QQ]=(\ell-1)/2$, so the group $\rho_{E,\ell}(\Gal_\QQ)$ cannot be $G$ or $H_2$. Therefore, $\rho_{E,\ell}(\Gal_\QQ)=H_1$.
\end{proof}
Take any elliptic curve $E'/\QQ$ with the same $j$-invariant as $E=E_{D,f}$; it is a quadratic twist. Now take $\calD_E$ as in \S\ref{S:twist 1}. Since $\#\calD_E=\#\{H_1, H_2\}=2$, we deduce from Lemma~\ref{L:ell twist D} (and $\ell\equiv 3\pmod 4$) that $\calD_{E}=\{1,-\ell\}$.
Since $\calD_{E}=\{1,-\ell\}$, we deduce from Lemma~\ref{L:H1new} that if $E'/\QQ$ is not isomorphic to $E$ or its quadratic twist by $-\ell$, then $\rho_{E',\ell}(\Gal_\QQ)$ is conjugate to $\pm\rho_{E,\ell}(\Gal_\QQ)=\pm H_1 =G$. If $E'$ is isomorphic to $E$ or its quadratic twist by $-\ell$, then $\rho_{E,\ell}(\Gal_\QQ)$ is conjugate to $H_1$ or $H_2$, respectively.
\subsection{Proof of Proposition~\ref{P: prime 2}}
If $E/\QQ$ is given by $y^2=f(x)$ with $f(x)\in \QQ[x]$ a separable cubic, then $\rho_{E,2}(\Gal_\QQ)$ is isomorphic to the groups $\Gal(f)$, i.e., the Galois group of the splitting field of $f$ over $\QQ$. Observe that $\GL_2(\FF_2)\cong \mathfrak{S}_3$. It thus suffices to compute $\Gal(f)$ since the cardinality of a subgroup of $\mathfrak{S}_3$ determines it up to conjugacy.
For the $j_E=1728$ case, we have $f(x)=x^3-dx = x(x^2-d)$. We have $\Gal(f)=\Gal(\QQ(\sqrt{d})/\QQ)$ which has order $1$ or $2$ when $d$ is a square or non-square, respectively.
For the $j_E=0$ case, we have $f(x)=x^3+d$. We have $\Gal(f) = \Gal(\QQ(\sqrt[3]{d},\zeta_3)/\QQ)=\Gal(\QQ(\sqrt[3]{d},\sqrt{-3})$ which has order $2$ or $6$ when $d$ is a cube or non-cube, respectively.
For $j_E\notin \{0,1728\}$, the group $\rho_{E,2}(\Gal_\QQ)$ does not change if we replace $E$ by a quadratic twist (since $-I\equiv I \pmod{\ell}$), so one need only consider the specific curve $E=E_{D,f}$. Using the $f(x)$ of Table~1, one can check that $\Gal(f)$ has order $2$ for the $j$-invariants listed in (\ref{P: prime 2 i}) and otherwise has order $6$.
Proposition~\ref{P: prime 2} is now a direct consequence of the above computations.
\subsection{Proof of Proposition~\ref{P:j=0 situation}}
Take any prime $\ell\geq 5$; we will deal with $\ell=3$ in \S\ref{L:j eq 0 ell eq 3}. We first consider an elliptic curve $E_d/\QQ$ defined by the equation
\[
y^2=x^3 +16d^2
\]
for a fixed \emph{cube-free} integer $d\geq 1$. We have $R=\ZZ[\omega]$ and $k=\QQ(\omega)$, where $\omega:=(-1+\sqrt{-3})/2$ is a cube root of unity in $k$. The ring $R$ is a PID.
If $\ell$ is congruent to $1$ or $2$ modulo $3$, define $C(\ell)$ be the Cartan subgroup $C_s(\ell)$ or $C_{ns}(\ell)$, respectively. Let $N(\ell)$ be the normalizer of $C(\ell)$ in $\GL_2(\FF_\ell)$.
\begin{lemma} \label{L:new 1 or 3}
After replacing $\rho_{E_d,\ell}$ by a conjugate representation, we will have $\rho_{E_d,\ell}(\Gal_\QQ) \subseteq N(\ell)$ and $\rho_{E_d,\ell}(\Gal_k) \subseteq C(\ell)$ with
\[
[N(\ell): \rho_{E_d,\ell}(\Gal_\QQ)]=[C(\ell): \rho_{E_d,\ell}(\Gal_k)] \in \{1,3\}.
\]
\end{lemma}
\begin{proof}
We have $E_1=E_{3,1}$. By Lemma~\ref{L:EDf generics}, we have $\rho_{E_1,\ell}(\Gal_\QQ) = N(\ell)$. The curves $E_d$ and $E_1$ are isomorphic over $\QQ(\sqrt[3]{d})$, so $\rho_{E_d,\ell}(\Gal_{\QQ(\sqrt[3]{d})})$ is conjugate to a subgroup of $N(\ell)$ of index $1$ or $3$. Therefore, $\rho_{E_d,\ell}(\Gal_{\QQ})$ is conjugate to a subgroup of $N(\ell)$ of index $1$ or $3$. Since $\rho_{E,d,\ell}(\Gal_\QQ) \not\subseteq C(\ell)$ and $\rho_{E,d,\ell}(\Gal_k) \subseteq C(\ell)$, we deduce that $[N(\ell): \rho_{E_d,\ell}(\Gal_\QQ)]=[C(\ell): \rho_{E_d,\ell}(\Gal_k)]$.
\end{proof}
To determine the index in Lemma~\ref{L:new 1 or 3}, we first compute some cubic residue symbols. Recall that we have already defined a representation $\rho_{E_d,\ell^\infty}\colon \Gal_k \to R_\ell^\times$.
\begin{lemma} \label{L:cubic reciprocity ap}
Let $\lambda$ be a prime of $R$ dividing $\ell$ that satisfies $\lambda\equiv 2 \pmod{3R}$. Take any non-zero prime ideal $\p \nmid 6d\ell$ of $R$. We have $\p=R\pi$ for some $\pi\equiv 2 \pmod{3R}$. Then
\[
\legendre{\rho_{E_d,\ell^\infty}(\Frob_\p)}{\lambda}_3= \legendre{d^{\frac{2(\ell^e-1)}{3}}\, \lambda}{\pi}_3,
\]
where we are using cubic residue characters and the field $R/\lambda R$ has order $\ell^e$.
\end{lemma}
\begin{proof}
By Example~10.6 of \cite{MR1312368}*{II \S10}, we have $\rho_{E_d,\ell^\infty}(\Frob_\p) = - \bbar{\legendre{4\cdot 16 d^2}{\pi}}_6 \cdot \pi$, where we are using the $6$-th power residue symbol. Therefore,
\[
\rho_{E_d,\ell^\infty}(\Frob_\p) = - \bbar{\legendre{d}{\pi}}_6^2 \pi = - \bbar{\legendre{d}{\pi}}_3 \pi= - \legendre{d}{\pi}_3^2\cdot \pi
\]
and hence
\[
\legendre{\rho_{E_d,\ell^\infty}(\Frob_\p)}{\lambda}_3= \legendre{-\legendre{d^2}{\pi}_3 \cdot \pi}{\lambda}_3 =\legendre{d^2}{\pi}^{\frac{\ell^e-1}{3}}_3 \legendre{\pi}{\lambda}_3 =\legendre{d^2}{\pi}^{\frac{\ell^e-1}{3}}_3 \legendre{\lambda}{\pi}_3 = \legendre{d^{\frac{2(\ell^e-1)}{3}} \lambda}{\pi}_3,
\]
where we have used cubic reciprocity.
\end{proof}
\begin{lemma} \label{L:j=0 case 1}
Suppose that $\ell \equiv 2 \pmod{3}$. Then the group $\rho_{E_d,\ell}(\Gal_k)$ has index $3$ in $C(\ell)$ if and only if $\ell\equiv 2\pmod{9}$ and $d=\ell$, or $\ell\equiv 5\pmod{9}$ and $d=\ell^2$. Note that $C(\ell)$ has a unique index $3$ subgroup.
\end{lemma}
\begin{proof}
Using Lemma~\ref{L:new 1 or 3} and $\ell\geq 5$, we find that $\rho_{E_d,\ell}(\Gal_k)$ is an index $3$ subgroup of $C(\ell)$ if and only if $\rho_{E_d,\ell^\infty}(\Gal_k)$ lies in a closed subgroup of $R_\ell^\times$ of index $3$. We have $C(\ell)=C_{ns}(\ell)$ since $\ell \equiv 2 \pmod{3}$, so $R_\ell^\times$ has a unique index $3$ closed subgroup, i.e., the group of $a\in R_\ell^\times$ with $\legendre{a}{\ell}_3=1$.
By the Chebotarev density theorem and Lemma~\ref{L:cubic reciprocity ap} with $\lambda=\ell$, we deduce that $\rho_{E_d,\ell}(\Gal_k)$ is an index $3$ subgroup of $C(\ell)$ if and only if $d^{{2(\ell^2-1)}/{3}} \ell$ is a cube in $R/\p$ for all primes $\p \nmid 6d\ell$ of $R$; equivalently, $d^{2(\ell^2-1)/3} \ell$ is a cube in $R$. Since $d^{{2(\ell^2-1)}/{3}} \ell$ is a rational integer, it is a cube in $R$ if and only if it is a cube in $\ZZ$.
We have $2(\ell^2-1)/3 \equiv 2(\ell+1)/3 \pmod{3}$, so we need only determine when the integer $d^{2(\ell+1)/3} \ell$ is a cube. In the following, we use that $d\geq 1$ is cube-free and that $\ZZ$ has unique factorization. If $\ell = 2 +9m$, then $d^{2+ 6m}\ell$ is a cube if and only if $d=\ell$. If $\ell = 5 +9m$, then $d^{4+ 6m}\ell$ is a cube if and only if $d=\ell^2$. If $\ell = 8 +9m$, then $d^{6+ 6m}\ell$ is never a cube.
\end{proof}
\begin{lemma} \label{L:j=0 case 2}
Suppose that $\ell \equiv 1 \pmod{3}$. Then the group $\rho_{E_d,\ell}(\Gal_k)$ has index $3$ in $C(\ell)$ if and only if $\ell\equiv 4 \pmod{9}$ and $d=\ell^2$, or $\ell\equiv 7\pmod{9}$ and $d=\ell$.
The group $\rho_{E_d,\ell}(\Gal_k)$ is conjugate to $C(\ell)=C_s(\ell)$ or the subgroup consisting of matrices of the form $\left(\begin{smallmatrix}a & 0 \\0 & b \end{smallmatrix}\right)$ with $a/b \in \FF_\ell^\times$ a cube.
\end{lemma}
\begin{proof}
Using Lemma~\ref{L:new 1 or 3} and $\ell\geq 5$, we find that $\rho_{E_d,\ell}(\Gal_k)$ is an index $3$ subgroup of $C(\ell)$ if and only if $\rho_{E_d,\ell^\infty}(\Gal_k)$ lies in a closed subgroup of $R_\ell^\times$ of index $3$. Let us describe the index $3$ subgroups of $R_\ell^\times$. Since $\ell \equiv 1 \pmod{3}$, we have $\ell=\lambda_1 \lambda_2$ for irreducibles $\lambda_i\in R$ that we may choose to be congruent to $2$ modulo $3R$. We have $R_\ell^\times = R_{\lambda_1}^\times \times R_{\lambda_2}^\times$. The cubic residue symbol $\legendre{\cdot}{\lambda_i}$ defines a homomorphism $\varphi_i\colon R_\ell^\times \to \mu_3:=\langle\omega \rangle$. Since $\ell\geq 5$, we find that every non-trivial homomorphism $R_\ell^\times \to \mu_3$ is of the form $\varphi_e:=\varphi_1^{e_1} \varphi_2^{e_2}$ with $e=(e_1,e_2) \in \{0,1,2\}^2-\{(0,0)\}$. Therefore, $\rho_{E_d,\ell}(\Gal_k)$ is an index $3$ subgroup of $C(\ell)$ if and only if $\rho_{E_d,\ell^\infty}(\Gal_k)\subseteq \ker \varphi_e$ for some $e\neq (0,0)$.
By Lemma~\ref{L:cubic reciprocity ap}, we have
$\legendre{\rho_{E_d,\ell^\infty}(\Frob_\p)}{\lambda_i}_3= \legendre{d^{\frac{2(\ell-1)}{3}}\, \lambda_i}{\pi}_3$
and hence
\begin{equation} \label{E:split case main}
\varphi_e(\rho_{E_d,\ell^\infty}(\Frob_\p)) = \legendre{d^{\frac{2(\ell-1)}{3}}\, \lambda_1}{\pi}_3^{e_1} \legendre{d^{\frac{2(\ell-1)}{3}}\, \lambda_2}{\pi}_3^{e_2} = \legendre{d^{\frac{2(\ell-1)(e_1+e_2)}{3}}\, \lambda_1^{e_1} \lambda^{e_2}}{\pi}_3
\end{equation}
for all $\p\nmid 6d\ell$. Using the Chebotarev density theorem, we deduce that $\rho_{E_d,\ell^\infty}(\Gal_k) \subseteq \ker \varphi_e$ if and only if $\beta:=d^{\frac{2(\ell-1)(e_1+e_2)}{3}}\, \lambda_1^{e_1} \lambda^{e_2}$ is a cube in $R$.
First suppose that $e_1\neq e_2$. Let $v_{\lambda_i} \colon R^\times \twoheadrightarrow \ZZ$ be the valuation for the prime $\lambda_i$ and let $v_\ell \colon \QQ^\times \twoheadrightarrow \ZZ$ be the valuation for $\ell$. We have
\[
v_{\lambda_i}(\beta) = e_i + \tfrac{2(\ell-1)(e_1+e_2)}{3} v_{\lambda_i}(d)= e_i + \tfrac{2(\ell-1)(e_1+e_2)}{3} v_{\ell}(d).
\]
We have $e_1\not\equiv e_2 \pmod{3}$ since $e_1\neq e_2$, so $v_{\lambda_i}(\beta) \not\equiv 0 \pmod{3}$ for some $i\in\{1,2\}$. Therefore, $\beta\in R$ is not a cube.
Now suppose that $e_1 = e_2$. We may assume that $e_1=e_2= 1$ since $\varphi_{(2,2)}$ is the square of $\varphi_{(1,1)}$ and hence have the same kernel. So $\beta=d^{4(\ell-1)/3}\ell$. Since $\beta$ is a rational integer, it is a cube in $\ZZ$ if and only if it is a cube in $R$. If $\ell = 1 + 9m$, then $\beta= (d^{4m})^3 \ell$ is not a cube. If $\ell = 4 + 9m$, then $\beta= (d^{4m+1})^3 \cdot d\ell$ which is a cube if and only if $d=\ell^2$ (recall that $d$ is positive and cube-free). If $\ell = 7 + 9m$, then $\beta= (d^{4m+2})^3 \cdot d^2\ell$ which is a cube if and only if $d=\ell$.\\
Finally, suppose we are in the case where $\rho_{E_d,\ell}(\Gal_k)$ is an index $3$ subgroup of $C_s(\ell)$. There are $4$ index $3$ subgroups of $C_s(\ell)$. Two of the groups consist of the matrices $A:=\left(\begin{smallmatrix}a & 0 \\0 & b \end{smallmatrix}\right)$ for which $a$ is a cube (or $b$ is a cube); these groups cannot equal $\rho_{E_d,\ell}(\Gal_k)$ since it would correspond to the case where $e_1=0$ or $e_2=0$ (and hence $e_1\neq e_2$). Another index $3$ subgroup of $C_s(\ell)$ is the subgroup of matrices whose determinant is a cube; this is impossible since $\det(\rho_{E_d,\ell}(\Gal_k))=\FF_\ell^\times$. Therefore, the only possibility for the image of $\rho_{E_d,\ell}$ is the group of $A$ with $a/b$ a cube.
\end{proof}
We now complete the proof of the proposition for the curve $E_d/\QQ$. From Lemmas~\ref{L:new 1 or 3}, \ref{L:j=0 case 1} and \ref{L:j=0 case 2}, we deduce that $\rho_{E_d}(\Gal_\QQ)$ has index $1$ or $3$ in $N(\ell)$, with index $3$ occurring if and only if one of the following hold:
\begin{itemize}
\item $\ell\equiv 2\pmod{9}$ and $d=\ell$,
\item $\ell\equiv 5\pmod{9}$ and $d=\ell^2$,
\item $\ell\equiv 4 \pmod{9}$ and $d=\ell^2$,
\item $\ell\equiv 7\pmod{9}$ and $d=\ell$.
\end{itemize}
Set $M:=\rho_{E_d,\ell}(\Gal_k)$; we may assume that it is the index $3$ subgroup of $C(\ell)$ from Lemma~\ref{L:j=0 case 1} or \ref{L:j=0 case 2}. The group $M$ is normal in $N(\ell)$. We have $[N(\ell):M]=6$ and $\det(M)=\FF_\ell^\times$, so $N(\ell)/M$ is non-abelian by Lemma~\ref{L:N group theory}(\ref{L:N group theory i}). So $N(\ell)/M$ is isomorphic to $\mathfrak{S}_3$ and hence, up to conjugation, $N(\ell)$ has a unique index $3$ subgroup $G'$ satisfying $G'\subseteq M$. Therefore, $G'$ is conjugate in $\GL_2(\FF_\ell)$ to both $\rho_{E_d,\ell}(\Gal_\QQ)$ and the group $G$ from part (\ref{P:j=0 situation iii}) or (\ref{P:j=0 situation iv}) of Lemma~\ref{P:j=0 situation}. This finishes the proof of Proposition~\ref{P:j=0 situation} for the curve $E_d/\QQ$ and $\ell>3$.
\\
Finally suppose that $E/\QQ$ is any elliptic curve with $j$-invariant $0$; it is defined by a Weierstrass equation $y^2=x^3+dm^3$ for some integer $m\neq 0$ and cube-free integer $d$. It suffices to show that $\rho_{E,\ell}(\Gal_\QQ)$ is conjugate to $\rho_{E_d,\ell}(\Gal_\QQ)$ in $\GL_2(\FF_\ell)$. The curves $E$ and $E_d$ are quadratic twists, so $\pm \rho_{E,\ell}(\Gal_\QQ)$ is conjugate to $\pm \rho_{E_d,\ell}(\Gal_\QQ)$. The general case of Proposition~\ref{P:j=0 situation} is thus a consequence of the following lemma.
\begin{lemma}
There are no proper subgroups $\pm \rho_{E_d,\ell}(\Gal_\QQ)$ has no proper subgroups $H$ such that $\pm H= \pm \rho_{E_d,\ell}(\Gal_\QQ)$.
\end{lemma}
\begin{proof}
If $\pm \rho_{E_d,\ell}(\Gal_\QQ)$ is conjugate to $N(\ell)$, then the lemma follows immediately from Lemma~\ref{L:N group theory}(\ref{L:N group theory ii}). From the case of Proposition~\ref{P:j=0 situation} we have already proved (i.e., for the curve $E_d$ and prime $\ell>3$), we need only show that the group $G$ from parts (\ref{P:j=0 situation iii}) and (\ref{P:j=0 situation iv}) of Lemma~\ref{P:j=0 situation} have no proper subgroups $H$ satisfying $\pm H=G$. Equivalently, we need to show that $-I$ is a commutator of such a subgroup $G$. With $G$ as in Lemma~\ref{P:j=0 situation}(\ref{P:j=0 situation iii}), this follows from
$\left(\begin{smallmatrix}0 & 1 \\1 & 0 \end{smallmatrix}\right)\left(\begin{smallmatrix}1 & 0 \\0 & -1 \end{smallmatrix}\right)\left(\begin{smallmatrix}0 & 1 \\1 & 0 \end{smallmatrix}\right)^{-1}\left(\begin{smallmatrix}1 & 0 \\0 & -1 \end{smallmatrix}\right)^{-1}=\left(\begin{smallmatrix}-1 & 0 \\0 & -1 \end{smallmatrix}\right)$. So we may take $G$ as in Lemma~\ref{P:j=0 situation}(\ref{P:j=0 situation iv}).
Fix any $B \in G-C(\ell)$. As noted in the proof of Lemma~\ref{L:N group theory}, the map $\varphi\colon C(\ell) \to C(\ell)$, $A\mapsto BABA^{-1}$ is a homomorphism whose image is cyclic of order $\ell+1$. Therefore, $\varphi(G\cap C(\ell))$ is the cyclic subgroup of $C(\ell)$ of order $(\ell+1)/3$. In particular, $\varphi(G\cap C(\ell))$ contains $-I$ which is the unique element of order $2$ in $C(\ell)$. Therefore, $-I$ is a commutator of $G$.
\end{proof}
\subsubsection{$\ell=3$ case} \label{L:j eq 0 ell eq 3}
We now consider the prime $\ell=3$ with $E/\QQ$ defined by the elliptic curve $y^2=x^3+d$. The division polynomial of $E/\QQ$ at $3$ is $3x(x^3+4d)$. The points of order $3$ in $E(\Qbar)$ are thus $(0, \pm \sqrt{d} )$ and $( -\sqrt[3]{4d} \omega^e , \pm \sqrt{-3} \sqrt{d})$ with $e\in\{0,1,2\}$. The points $P_1=(0, \sqrt{d} )$ and $P_2=( -\sqrt[3]{4d} , \sqrt{-3} \sqrt{d})$ form a basis of $E[3]$. With respect to this basis, we have
\[
\rho_{E,3} = \left(\begin{smallmatrix}\psi_1 & * \\0 & \psi_2 \end{smallmatrix}\right),
\]
with characters $\psi_1,\psi_2 \colon \Gal_\QQ \to \FF_3^\times$. The quadratic character $\psi_1$ describes the Galois action on $P_1$ and it thus corresponds to the extension $\QQ(\sqrt{d})$ of $\QQ$. The quadratic character $\psi_1\psi_2=\det\circ \rho_{E,3}\colon \Gal_\QQ \to \FF_{3}^\times$ corresponds to the extension $\QQ(\zeta_3)=\QQ(\sqrt{-3})$ of $\QQ$. Therefore,
\[
(\psi_1\times \psi_2)(\Gal_\QQ) = \begin{cases}
\{1\}\times \FF_3^\times & \text{ if $d$ is a square}, \\
\FF_3^\times \times\{1\} & \text{ if $-3d$ is a square}, \\
\FF_3^\times \times \FF_3^\times & \text{ otherwise}.
\end{cases}
\]
To compute the image of $\rho_{E,3}(\Gal_\QQ)$ it remains to determine when its cardinality is divisible by $3$ or not. From $P_1$ and $P_2$, it is clear that $\rho_{E,3}(\Gal_\QQ)$ is divisible by $3$ if and only if $4d$ is not a cube.
\section{Proof of Proposition~\ref{P:big inertia last}} \label{S:big inertia last}
By Theorem~\ref{T:Mazur-Serre-BPR}, we may assume that $\rho_{E,\ell}(\Gal_\QQ)$ is a subgroup of $N_{ns}(\ell)$. Let $I_\ell$ be an inertia subgroup of $\Gal_\QQ$ for the prime $\ell$. We will show that $\rho_{E,\ell}$ has large image by showing that the group $\rho_{E,\ell}(I_\ell)$ is large. The cardinality of $\rho_{E,\ell}(I_\ell)$ is not divisible by $\ell$ since it is a subgroup of $N_{ns}(\ell)$. The group $\rho_{E,\ell}(I_\ell)$ is thus cyclic since the \emph{tame inertia group} at $\ell$ is pro-cyclic, cf.~\cite{MR0387283}*{\S1.3}.
\\
Let $v_\ell$ be the $\ell$-adic valuation on $\QQ_\ell$ normalized so that $v_\ell(\ell)=1$. Let $\QQ_\ell^\un$ be the maximal unramified extension of $\QQ_\ell$ in a fixed algebraic closed field $\Qbar_\ell$. An embedding $\Qbar\hookrightarrow \Qbar_\ell$ allows us to identify $I_\ell$ with the subgroup $\Gal(\Qbar_\ell/\QQ_\ell^{\un})$ of $\Gal_{\QQ_\ell}:=\Gal(\Qbar_\ell/\QQ_\ell)$. Let $\Delta_E$ be the minimal discriminant of $E/\QQ$. \\
\noindent $\bullet$ First suppose that $v_\ell(j_E) \geq 0$ and that $v_\ell(\Delta_E)$ is not congruent to $2$ and $10$ modulo $12$.
Let $L$ be the smallest extension of $\QQ_\ell^\un$ for which $E$ base extended to $L$ has good reduction. Define $e=[L:\QQ_\ell^\un]$. There is thus a finite extension $K/\QQ_\ell$ such that $E$ base extended to $K$ has good reduction and that $v_\ell(K^\times)= e^{-1} \ZZ$, where $v_\ell$ is the valuation on $K$ that extends $v_\ell$. From \cite{MR0387283}*{\S5.6}, we find that $e\in \{1,2,3,4\}$; this uses our assumption on $v_\ell(\Delta_E)$.
Let $\calI$ be the inertia subgroup of $\Gal_K:=\Gal(\Qbar_\ell/K)$; it is a subgroup of $I_\ell$. The action of $\calI$ on $E[\ell]$ is semi-simple since the cardinality of $\rho_{E,\ell}(\calI)$ is relatively prime to $\ell$ (the group $N_{ns}(\ell)$ has this property). Let $\theta_1\colon \calI \twoheadrightarrow \FF_\ell^\times$ and $\theta_2 \colon \calI \twoheadrightarrow \FF_{\ell^2}^\times$ be \emph{fundamental characters} of level $1$ and $2$, respectively, cf.~\cite{MR0387283}*{\S1.7}.
\begin{lemma} \label{L:inertia irred}
The representation $\rho_{E,\ell}|_{\calI} \colon \calI \to \GL_2(\FF_\ell)$ is irreducible.
\end{lemma}
\begin{proof}
Suppose that $\rho_{E,\ell}|_{\calI}$ is reducible. The representation $\rho_{E,\ell}|_{\calI}$ is given by a pair of characters $\theta_1^{e_1}$ and $\theta_1^{e_2}$ with $0\leq e_1 \leq e_2 < \ell-1$. From Proposition~11 of \cite{MR0387283}, we can take $e_1=0$ and $e_2=e$. The image of $\rho_{E,\ell}(\calI)$ in $\PGL_2(\FF_\ell)$ is thus isomorphic to $\theta_1^{e}(\calI)$ and hence is cyclic of order $(\ell-1)/\gcd(\ell-1,e)$.
The matrix $A^2$ is scalar for all $A\in N_{ns}(\ell)-C_{ns}(\ell)$. Therefore, the order of every element in the image of $N_{ns}(\ell)\to\PGL_2(\FF_\ell)$ divides $\ell+1$. Since $\gcd(\ell+1,\ell-1)=2$, we deduce that $(\ell-1)/\gcd(\ell-1,e)$ equals $1$ or $2$. This is a contradiction since $\ell\geq 17$.
\end{proof}
Scalar multiplication and a choice of $\FF_\ell$-basis for $\FF_{\ell^2}$ allows us to identify $\FF_{\ell^2}^\times$ with a subgroup of $\Aut_{\FF_\ell}(\FF_{\ell^2})\cong \GL_2(\FF_\ell)$. Since $\rho_{E,\ell}|_{\calI}$ is irreducible by Lemma~\ref{L:inertia irred}, it is isomorphic to $\theta_2^{e_1+e_2\ell}\colon \calI \to \FF_{\ell^2}^\times \hookrightarrow \GL_2(\FF_\ell) $ for some $0\leq e_1,e_2 \leq \ell-1$. As an $\FF_\ell[\calI]$-module, $E[\ell]$ is isomorphic to the dual of the \'etale cohomology group $H^1_{\text{\'et}}(E_{\Kbar}, \FF_\ell)$. By Th\'eor\`eme~1.2 of \cite{MR2372809}, we may take $0\leq e_1 , e_2 \leq e$ (when $E$ has good reduction at $\ell$, and hence $e=1$, this follows from \cite{MR0387283}*{Prop.~12}). We have $e_1 \neq e_2$ since otherwise $\theta_2^{e_1+e_2\ell}=(\theta_2^{\ell+1})^{e_1}$ is not irreducible.
Let $g$ be the greatest common divisor of $e_1+e_2\ell$ and $\ell+1$. We have $(e_1 +e_2 \ell)-e_2(\ell+1) = e_1-e_2 \in \{\pm 1,\pm 2,\pm 3, \pm 4\}$ since $0\leq e \leq 4$, so $g \in \{1,2,3,4\}$. Therefore, $\rho_{E,\ell}(\calI)$ contains a cyclic group of order $(\ell+1)/g$.
\begin{lemma}\label{L:index 1 or 3 new}
The group $\rho_{E,\ell}(I_\ell)$ is a subgroup of $C_{ns}(\ell)$ with index $1$ or $3$.
\end{lemma}
\begin{proof}
Set $H:=\rho_{E,\ell}(I_\ell)$; it is cyclic. We claim that $H$ is a subgroup of $C_{ns}(\ell)$. Suppose not, then the order of $H$ divides $2(\ell-1)$ since $A^2$ is a scalar matrix for any $A\in N_{ns}(\ell)-C_{ns}(\ell)$. Therefore, $(\ell+1)/g$ divides $2(\ell-1)$ since $\rho_{E,\ell}(\calI)\subseteq H$ contains an element of order $(\ell+1)/g$. Since $\gcd(\ell+1,\ell-1)=2$, we deduce that $(\ell+1)/g$ divides $4$. This is impossible since $\ell\geq 17$ and $g\leq 4$.
It remains to bound the index of $H$ in $C_{ns}(\ell)$. We have $\det(H)=\FF_\ell^\times$ since $\det\circ \rho_{E,\ell}$ describes the Galois action on the $\ell$-th roots of unity. Therefore, the group $H$ is cyclic and its order is divisible by $\ell-1$ and $(\ell+1)/g$. Since $\gcd(\ell+1,\ell-1)=2$, we deduce that the order of $H$ is divisible by $(\ell-1)(\ell+1)/(2g)$. Therefore, the index $b:=[C_{ns}(\ell) : H]$ divides $2g$.
Suppose $b$ is even. Since $C_{ns}(\ell)$ is cyclic, the group $H$ must be contained in $\{A \in C_{ns}(\ell) : \det(A) \in (\FF_\ell^\times)^2\}$; this is the unique index $2$ subgroup of $C_{ns}(\ell)$. However, this is impossible since $\det(H)=\FF_\ell^\times$. So $b$ is odd and divides $2g \in \{2,4,6,8\}$. Therefore, $b$ is $1$ or $3$.
\end{proof}
Now define $H:= \rho_{E,\ell}(\Gal_\QQ) \cap C_{ns}(\ell)$. We have $\rho_{E,\ell}(\Gal_\QQ) \not \subseteq C_{ns}(\ell)$ since $C_{ns}(\ell)$ is not applicable; it does not contain an element with trace $0$ and determinant $-1$. So if $H=C_{ns}(\ell)$, then $\rho_{E,\ell}(\Gal_\QQ) = N_{ns}(\ell)$.
We are thus left to consider the case where $H$ is the (unique) index $3$ subgroup of $C_{ns}(\ell)$. The group $H$ is a normal subgroup of $N_{ns}(\ell)$ of index $6$.
\begin{lemma}
We have $\ell \equiv 2 \pmod{3}$ and the quotient group $N_{ns}(\ell)/H$ is isomorphic to $\mathfrak{S}_3$.
\end{lemma}
\begin{proof}
If $\ell \equiv 1 \pmod{3}$, then $\det(H)=(\FF_\ell^\times)^3 \subsetneq \FF_\ell^\times$. This is impossible since $\det(\rho_{E,\ell}(\Gal_\QQ))=\FF_\ell^\times$ and $[\rho_{E,\ell}(\Gal_\QQ):H]=2$. Therefore, $\ell \equiv 2 \pmod{3}$. One can now verify that $N_{ns}(\ell)$ quotiented out by the scalar matrices is isomorphic to a dihedral group. It is then easy to check that $N_{ns}(\ell)/H$ is the dihedral group of order $2\cdot 3$; it is thus isomorphic to $\mathfrak{S}_3$.
\end{proof}
The index $3$ subgroups of $\mathfrak{S}_3$ are all conjugate so, up to conjugacy, $G$ (as in the statement of Proposition~\ref{P:big inertia last}) is the unique index $3$ subgroup of $N_{ns}(\ell)$ that contains $H$. Therefore, $\rho_{E,\ell}(\Gal_\QQ)$ and $G$ are conjugate subgroups.\\
\noindent $\bullet$ Suppose that $v_\ell(j_E) \geq 0$.
By twisting $E/\QQ$ by $1$ or $\ell$, we obtain an elliptic curve $E'/\QQ$ with $v_\ell(\Delta_{E'})$ not congruent to $2$ and $10$ modulo $12$. The group $\pm \rho_{E,\ell}(\Gal_\QQ)$ is conjugate to $\pm \rho_{E',\ell}(\Gal_\QQ)$. The previous case applies and shows that $\pm \rho_{E,\ell}(\Gal_\QQ)$ is conjugate to $\pm G =G$ or $\pm N_{ns}(\ell) = N_{ns}(\ell)$ in $\GL_2(\FF_\ell)$.
It remains to show that $\pm \rho_{E,\ell}(\Gal_\QQ)=\rho_{E,\ell}(\Gal_\QQ)$; if not then there is an index $2$ subgroup $H$ of $G$ or $N_{ns}(\ell)$ such that $-I \notin H$. The group $H\cap C_{ns}(\ell)$ is then an index $2$ or $6$ subgroup of $C_{ns}(\ell)$ that does not contain $-I$. However, the cardinality of $H\cap C_{ns}(\ell)$ is even, so it contains an element of order $2$ which most be $-I$.
\\
\noindent $\bullet$ Finally suppose that $v_\ell(j_E) < 0$.
There exists an element $q\in \QQ_\ell$ with $v_\ell(q)=-v_\ell(j_E)>0$ such that
\[
j_E = (1+240{\sum}_{n\geq 1} n^3 q^n/(1-q^n) )^3/(q{\prod}_{n\geq 1} (1-q^n)^{24}).
\]
Let $\calE/\QQ_\ell$ be the \defi{Tate curve} associated to $q$, cf.~\cite{MR1312368}*{V\S3}; it is an elliptic curve with $j$-invariant $j_E$ and the group $\calE(\Qbar_\ell)$ is isomorphic to $\Qbar_\ell^\times/ \ang{q}$ as a $\Gal_{\QQ_\ell}$-module. In particular, the $\ell$-torsion subgroup $\calE[\ell]$ is isomorphic as an $\FF_\ell[\Gal_{\QQ_\ell}]$-module to the subgroup of $\Qbar_\ell^\times/ \ang{q}$ generated by an $\ell$-th root of unity $\zeta$ and a chosen $\ell$-th root $q^{1/\ell}$ of $q$. Let $\alpha\colon \Gal_{\QQ_\ell}\to \FF_\ell^\times$ and $\beta\colon \Gal_{\QQ_\ell} \to \FF_\ell$ be the maps defined so that $\sigma(\zeta)=\zeta^{\alpha(\sigma)}$ and $\sigma(q^{1/\ell})=\zeta^{\beta(\sigma)} q^{1/\ell}$. So with respect to the basis $\{\zeta, q^{1/\ell}\}$ for $\calE[\ell]$, we have $\rho_{\calE,\ell}(\sigma)= \left(\begin{smallmatrix}\alpha(\sigma) & \beta(\sigma) \\0 & 1 \end{smallmatrix}\right)$ for $\sigma\in \Gal_{\QQ_\ell}$. The curves $E$ and $\calE$ are quadratic twists of each other over $\QQ_\ell$ (the curve $E$ is non-CM since its $j$-invariant is not an integer). So there is a character $\chi\colon \Gal_{\QQ_\ell}\to \{\pm1\}$ such that, after an appropriate choice of basis for $E[\ell]$, we have
\[
\rho_{E,\ell}(\sigma) = \chi(\sigma)\left(\begin{smallmatrix} \alpha(\sigma) & \beta(\sigma) \\0 & 1\end{smallmatrix}\right)
\]
for all $\sigma\in \Gal_{\QQ_\ell}$. Since $\alpha$ is surjective, we find that the image of $\rho_{E,\ell}(\Gal_\QQ)$ in $\PGL_2(\FF_\ell)$ contains a cyclic group of order $\ell-1$. However, the image of $N_{ns}(\ell)$ in $\PGL_2(\FF_\ell)$ has order $2(\ell+1)$. Since $\rho_{E,\ell}(\Gal_\QQ) \subseteq N_{ns}(\ell)$, we find that $\ell-1$ divides $2(\ell+1)$; this is impossible since $\gcd(\ell-1,\ell+1)=2$ and $\ell\geq 17$.
\end{document} | 163,866 |
Happy Thanksgiving, and by the way...
We've hired Carla Kucinski as the new Go Triad editor. Carla is a staff writer at the Gannett alt weekly Noise in Lansing, Mich. She helped start the pub in 2002 and has developed a number of the magazine's regular features. Once she's on her feet here, she's going to freshen the magazine and help rebuild the Web site. Our features editor, Susan Ladd, says "She brings great ideas, passion and enthusiasm...kind of like the guy who started Go Triad."
Speaking of that guy, Jeri Rowe, he's not going anywhere. He is ready for a change -- Go Triad is a big, tough job -- and we're going to give him some new challenges. (Not sure what they are yet, so don't ask.) In the meantime, though, he's going to help Carla learn the ropes....and the night life.
To report abuse of the comment feature on this site, please use the feedback form at the bottom of any page.
Does this mean that the GoTriad podcasts will return to a listenable length. I quit listening after they continued to hit the 60 minute mark. The could take a cue from Herb's podcast, which is just right for that commute, walk around the block.
Posted on November 24, 2005 11:33 AM
Hey Don.
Several weeks ago (A&T's homecoming weekend), Jeri & Nicole cut back to about 20-30 minutes. If Jeri has a special guest or topic, he does a second podcast that week about the same length (20-30 minutes) with the single subject.
We heard ya, man. :-)
Posted on November 24, 2005 7:23 PM | 356,957 |
Did you know that as you age gradually, your body goes through different changes? Some of them can get very bad. Blood pressure is one of them. As you suffer from more tension and stress, the blood pressure level increases rapidly. This high blood pressure level can cause many health problems which can be life threatening such as stroke, heart attack and many more.
An Alarming Statistics, the high blood pressure is a problem which can’t be taken lightly. In most cases, it occurs to you as you age and you suffer from more stress, tension and unhealthy food habits. So here are some questions we need you to think about.
- Are you worried after reading these statistics?
- Are you an adult who’s suffering from high blood pressure?
- Are you looking for a permanent and safe solution of high blood pressure?
- Do you want to keep your blood pressure level under control and live a long, healthy and youthful life?
So if your answer is “YES” to most of these questions, then you don’t need to worry any more. There are many possible solutions for these problems. One of these solutions are supplements. A high quality supplement with excellent ingredients and latest formula can take care of this problem efficiently for you.
But there are a huge number of supplements in the market right now. All of them are not equally effective. After doing extensive research on these supplements, we’ve found one which is unique than most of the others and might be your best solution. It’s called Marine D3. This supplement not only controls blood pressure, it also boosts the anti-aging process and gives you a youthful and healthy life. So what makes it so different than the others? Let’s find out in our following review.. | 354,050 |
Life goes faster with no strings attached
This is the one wireless drive for all your devices. Free up space on your tablet and smartphone. Back up. Save, format your drive, register your drive and more
Ultra-fast USB 3.0
Quickly transfer large amounts of data to the drive using the ultra-fast USB 3.0 port.
PACKAGE INCLUDES
- Wi-Fi mobile storage
- USB cable
- USB power adapter
- Quick Install Guide | 293,564 |
\section{First Proof of Main Theorem}
\label{s:mainproof}
We now give the first, very direct, proof of the main theorem.
In the following sections, we will give a more illuminating
and general proof.
This section can be skipped in its entirety; it is tedious and mechanical.
It's value is two fold: first, it illustrates how the general techniques
of the next sections apply in practice; second, it shows that,
once we have found the appropriate matrix, and simplex, that
the rest of the process can be automated.
The tedious computation of this section, should be compared and contrasted
with the equally tedious computation needed to verify, from first principles, the final polynomial
for $\Rin$:
\begin{equation}
\label{e:final}
\begin{split}
[246][184][175][437][197] &+ [129][184][175][437][467] + \\
[138][194][247][175][467] &+ [156][184][247][437][197] + \\
[345][184][247][176][197] &+ [489][247][175][176][143] + \\
[597][247][184][176][143] &+ [678][247][175][194][143] + \\
[237][194][184][175][467] &= 0
\end{split}
\end{equation}
where, for indeterminates, $x_i, y_i, z_i$,
\begin{equation}
[i j k] = \begin{vmatrix}
x_i & x_j & x_ k \\
y_i & y_j & y_ k \\
z_i & z_j & z_ k
\end{vmatrix}
\end{equation}
This is taken from \cite{bjorner:oriented} p 349, note that the numbering of the lines
is different from ours.
At heart, we may conjecture that these two computations are cryptomorphic, and
hence, the tedium of this section is unsurprising.
We have already argued that the conditions of
theorem~\ref{t:main} amount to requiring that
the system~(\ref{e:system},\ref{e:linear-again}) is soluble.
Throughout this section we will use $M_9$ for the
matrix from~(\ref{e:linear}), i.e.
we are considering the system
\begin{equation}
\label{e:system}
M_9 \boldsymbol{r} > \boldsymbol{0}
\end{equation}
where
\begin{equation}
\label{e:linear-again}
M_9 =
\left(
\begin{smallmatrix}
&(1)&(2)&(3)&(4)&(5)&(6)&(7)&(8)&(9)&(10)\\
(A)& & &-\BSN{5}{8}& &\BSN{3}{8}& & &-\BSN{3}{5}& &\\
(B)& &-\BSN{8}{9}& & & & & &\BSN{2}{9}&-\BSN{2}{8}&\\
(C)& &\BSN{4}{6} & &-\BSN{2}{6}& &\BSN{2}{4}& & & & \\
(D)& & & & & &-\BSN{7}{10}&\SN{6}{10}& & &-\BSN{6}{7}\\
(E)&-\SN{3}{7}& &\BSN{1}{7}& & & &-\SN{1}{3}& & & \\
(F)&-\SN{4}{7}& & &\BSN{1}{7}& & &-\SN{1}{4}& & & \\
(G)&\SN{5}{9}& & & &-\BSN{1}{9}& & & &\BSN{1}{5} & \\
(H)&\SN{5}{10}& & & &-\BSN{1}{10}& & & & &\BSN{1}{5}\\
(I)&\SN{5}{7}& & & &-\BSN{1}{7}& &\SN{1}{5} & & &
\end{smallmatrix}
\right)
\end{equation}
The 9 by 8 submatrix formed from columns 2-6 and 8-10, shown in bold,
above, is referred to as $S_9$.
We also consider extensively the submatrix $M_8$ formed from the first eight rows:
\begin{equation}
\label{e:m8}
M_8 =
\left(
\begin{smallmatrix}
&(1)&(2)&(3)&(4)&(5)&(6)&(7)&(8)&(9)&(10)\\
(A)& & &-\BSN{5}{8}& &\SN{3}{8}& & &-\BSN{3}{5}& &\\
(B)& &-\BSN{8}{9}& & & & & &\BSN{2}{9}&-\BSN{2}{8}&\\
(C)& &\BSN{4}{6} & &-\BSN{2}{6}& &\BSN{2}{4}& & & & \\
(D)& & & & & &-\BSN{7}{10}&\SN{6}{10}& & &-\BSN{6}{7}\\
(E)&-\SN{3}{7}& &\BSN{1}{7}& & & &-\SN{1}{3}& & & \\
(F)&-\SN{4}{7}& & &\BSN{1}{7}& & &-\SN{1}{4}& & & \\
(G)&\SN{5}{9}& & & &-\SN{1}{9}& & & &\BSN{1}{5} & \\
(H)&\SN{5}{10}& & & &-\SN{1}{10}& & & & &\BSN{1}{5}
\end{smallmatrix}
\right)
\end{equation}
The 8 by 7 submatrix formed from columns 2, 3, 4, 6, 8, 9 and~10, shown in bold
above, is referred to as $S_8$.
\begin{lem}
\label{l:simplex8}
With the conditions on $\boldsymbol{\theta}$ of
theorem~\ref{t:main},
the 8 by 7 matrix
$S_8$
is a simplex.
\end{lem}
\begin{proof}
The eight 7 by 7 subdeterminants are:
\begin{equation}
\label{e:subdets}
\begin{pmatrix}
-\SN{4}{6}\SN{1}{7}\SN{1}{7}\SN{7}{10}\SN{2}{9}\SN{1}{5}\SN{1}{5}\\
+\SN{4}{6}\SN{1}{7}\SN{1}{7}\SN{7}{10}\SN{3}{5}\SN{1}{5}\SN{1}{5}\\
-\SN{8}{9}\SN{1}{7}\SN{1}{7}\SN{7}{10}\SN{3}{5}\SN{1}{5}\SN{1}{5}\\
+\SN{8}{9}\SN{1}{7}\SN{1}{7}\SN{2}{4}\SN{3}{5}\SN{1}{5}\SN{1}{5}\\
-\SN{4}{6}\SN{5}{8}\SN{1}{7}\SN{7}{10}\SN{2}{9}\SN{1}{5}\SN{1}{5}\\
+\SN{8}{9}\SN{1}{7}\SN{2}{6}\SN{7}{10}\SN{3}{5}\SN{1}{5}\SN{1}{5}\\
-\SN{4}{6}\SN{1}{7}\SN{1}{7}\SN{7}{10}\SN{3}{5}\SN{2}{8}\SN{1}{5}\\
+\SN{8}{9}\SN{1}{7}\SN{1}{7}\SN{2}{4}\SN{3}{5}\SN{1}{5}\SN{6}{7}
\end{pmatrix}
\end{equation}
Since the constraints on the angles in the statement of the main
theorem require that $\SN{i}{j} > 0$ for all pairs $i, j$
appearing in these subdeterminants, the signs of the subdeterminant
alternate, and theorem~\ref{t:simplex} applies.
\end{proof}
\begin{lem}
\label{l:nequals}
With the conditions on $\boldsymbol{\theta}$ of theorem~\ref{t:main},
if
\begin{equation}
\SN{8}{9}\SN{1}{10}\SN{2}{4}\SN{3}{5}\SN{6}{7}
+\SN{4}{6}\SN{1}{9}\SN{7}{10}\SN{3}{5}\SN{2}{8}
- \SN{4}{6}\SN{3}{8}\SN{7}{10}\SN{2}{9}\SN{1}{5}
= 0
\end{equation}
then the system~(\ref{e:system}) is insoluble.
\end{lem}
\begin{proof}
From above,
$S_8$
is a simplex.
We can compute the three determinants required by
theorem~\ref{t:mus}, as follows:
\begin{equation}\begin{split}
d_1 = & | M\left[A-H;1,2,3,4,6,8,9,10\right] | = \\
&-\SN{5}{10}\SN{8}{9}\SN{1}{7}\SN{1}{7}\SN{2}{4}\SN{3}{5}\SN{1}{5}\SN{6}{7} \\
&-\SN{5}{9}\SN{4}{6}\SN{1}{7}\SN{1}{7}\SN{7}{10}\SN{3}{5}\SN{2}{8}\SN{1}{5} \\
&+\SN{4}{7}\SN{8}{9}\SN{1}{7}\SN{2}{6}\SN{7}{10}\SN{3}{5}\SN{1}{5}\SN{1}{5} \\
&+\SN{3}{7}\SN{4}{6}\SN{5}{8}\SN{1}{7}\SN{7}{10}\SN{2}{9}\SN{1}{5}\SN{1}{5}
\end{split}\end{equation}
\begin{equation}\begin{split}
\label{e:d5}
d_5 = & | M\left[A-H;2,3,4,5,6,8,9,10\right] | = \\
&+\SN{4}{6}\SN{1}{7}\SN{1}{7}\SN{3}{8}\SN{7}{10}\SN{2}{9}\SN{1}{5}\SN{1}{5} \\
&-\SN{4}{6}\SN{1}{7}\SN{1}{7}\SN{1}{9}\SN{7}{10}\SN{3}{5}\SN{2}{8}\SN{1}{5} \\
&-\SN{8}{9}\SN{1}{7}\SN{1}{7}\SN{1}{10}\SN{2}{4}\SN{3}{5}\SN{1}{5}\SN{6}{7}
\end{split}\end{equation}
\begin{equation}\begin{split}
d_7 =& | M\left[A-H;2,3,4,6,7,8,9,10\right] | = \\
&+\SN{4}{6}\SN{5}{8}\SN{1}{7}\SN{7}{10}\SN{1}{3}\SN{2}{9}\SN{1}{5}\SN{1}{5} \\
&-\SN{8}{9}\SN{1}{7}\SN{1}{7}\SN{2}{4}\SN{6}{10}\SN{3}{5}\SN{1}{5}\SN{1}{5} \\
&+\SN{8}{9}\SN{1}{7}\SN{2}{6}\SN{7}{10}\SN{1}{4}\SN{3}{5}\SN{1}{5}\SN{1}{5}
\end{split}\end{equation}
By normalization (theorem~\ref{t:normal}) we can show that:
\begin{equation}
\label{e:d1_d5_d7}
\SN{1}{7}\SN{1}{5}d_1
= \SN{1}{5}\SN{5}{7}d_5
= \SN{1}{7}\SN{5}{7} d_7
\end{equation}
Given the premise of the lemma, $d_5 = 0$, and hence so are
$d_1$ and $d_7$. Thus by theorem~\ref{t:mus}
the system:
\begin{equation}
M_8\boldsymbol{r} > \boldsymbol{0}
\end{equation}
is insoluble, and hence so is~(\ref{e:system}).
\end{proof}
\begin{lem}
With the conditions on $\boldsymbol{\theta}$ of
theorem~\ref{t:main},
if
\begin{equation}
\SN{8}{9}\SN{1}{10}\SN{2}{4}\SN{3}{5}\SN{6}{7}
+\SN{4}{6}\SN{1}{9}\SN{7}{10}\SN{3}{5}\SN{2}{8}
- \SN{4}{6}\SN{3}{8}\SN{7}{10}\SN{2}{9}\SN{1}{5}
< 0
\end{equation}
then
the 9 by 8 matrix
$S_9$
is a simplex.
\end{lem}
\begin{proof}
The first eight of the nine 8 by 8 subdeterminants
are the same as in equation~\ref{e:subdets} multiplied
by $-\SN{1}{7}$. The first is positive, the eighth is negative.
The ninth subdeterminant is
\begin{equation}
+\SN{4}{6}\SN{1}{7}\SN{1}{7}\SN{3}{8}\SN{7}{10}\SN{2}{9}\SN{1}{5}\SN{1}{5}
-\SN{4}{6}\SN{1}{7}\SN{1}{7}\SN{1}{9}\SN{7}{10}\SN{3}{5}\SN{2}{8}\SN{1}{5}
-\SN{8}{9}\SN{1}{7}\SN{1}{7}\SN{1}{10}\SN{2}{4}\SN{3}{5}\SN{1}{5}\SN{6}{7}
\end{equation}
which is $-\SN{1}{7}\SN{1}{7}\SN{1}{5}$ times the
negative value in the premise of the lemma.
So the nine values alternate in sign, and theorem~\ref{t:simplex} applies.
\end{proof}
\begin{lem}
\label{l:nnegative}
With the conditions on $\boldsymbol{\theta}$ of theorem~\ref{t:main},
if
\begin{equation}
\SN{8}{9}\SN{1}{10}\SN{2}{4}\SN{3}{5}\SN{6}{7}
+\SN{4}{6}\SN{1}{9}\SN{7}{10}\SN{3}{5}\SN{2}{8}
- \SN{4}{6}\SN{3}{8}\SN{7}{10}\SN{2}{9}\SN{1}{5}
< 0
\end{equation}
then the system~(\ref{e:system}) is insoluble.
\end{lem}
\begin{proof}
From above,
$S_9$
is a simplex.
We can compute the two determinants required by
theorem~\ref{t:mus}, as follows:
\begin{equation}\begin{split}
d_{1,5} = & |M\left[A-I;1-6,8-10\right] |= \\
&-\SN{5}{7}\SN{8}{9}\SN{1}{7}\SN{1}{7}\SN{1}{10}\SN{2}{4}\SN{3}{5}\SN{1}{5}\SN{6}{7}\\
&-\SN{5}{7}\SN{4}{6}\SN{1}{7}\SN{1}{7}\SN{1}{9}\SN{7}{10}\SN{3}{5}\SN{2}{8}\SN{1}{5}\\
&+\SN{5}{7}\SN{4}{6}\SN{1}{7}\SN{1}{7}\SN{3}{8}\SN{7}{10}\SN{2}{9}\SN{1}{5}\SN{1}{5}\\
&+\SN{5}{10}\SN{8}{9}\SN{1}{7}\SN{1}{7}\SN{1}{7}\SN{2}{4}\SN{3}{5}\SN{1}{5}\SN{6}{7}\\
&+\SN{5}{9}\SN{4}{6}\SN{1}{7}\SN{1}{7}\SN{1}{7}\SN{7}{10}\SN{3}{5}\SN{2}{8}\SN{1}{5}\\
&-\SN{4}{7}\SN{8}{9}\SN{1}{7}\SN{2}{6}\SN{1}{7}\SN{7}{10}\SN{3}{5}\SN{1}{5}\SN{1}{5}\\
&-\SN{3}{7}\SN{4}{6}\SN{5}{8}\SN{1}{7}\SN{1}{7}\SN{7}{10}\SN{2}{9}\SN{1}{5}\SN{1}{5}
\end{split}\end{equation}
\begin{equation}\begin{split}
d_{5,7} = & |M\left[A-I;2-10\right] |= \\
&+\SN{4}{6}\SN{5}{8}\SN{1}{7}\SN{1}{7}\SN{7}{10}\SN{1}{3}\SN{2}{9}\SN{1}{5}\SN{1}{5}\\
&-\SN{4}{6}\SN{1}{7}\SN{1}{7}\SN{3}{8}\SN{7}{10}\SN{1}{5}\SN{2}{9}\SN{1}{5}\SN{1}{5}\\
&+\SN{4}{6}\SN{1}{7}\SN{1}{7}\SN{1}{9}\SN{7}{10}\SN{1}{5}\SN{3}{5}\SN{2}{8}\SN{1}{5}\\
&+\SN{8}{9}\SN{1}{7}\SN{1}{7}\SN{1}{10}\SN{2}{4}\SN{1}{5}\SN{3}{5}\SN{1}{5}\SN{6}{7}\\
&-\SN{8}{9}\SN{1}{7}\SN{1}{7}\SN{1}{7}\SN{2}{4}\SN{6}{10}\SN{3}{5}\SN{1}{5}\SN{1}{5}\\
&+\SN{8}{9}\SN{1}{7}\SN{2}{6}\SN{1}{7}\SN{7}{10}\SN{1}{4}\SN{3}{5}\SN{1}{5}\SN{1}{5}
\end{split}\end{equation}
By normalization (theorem~\ref{t:normal}) these are both zero.
Thus by theorem~\ref{t:mus}
the system~(\ref{e:system}) is insoluble.
\end{proof}
The main theorem is thus proved by combining lemmas~\ref{l:nequals} and~\ref{l:nnegative}. | 111,892 |
Thursday, August 25, 2005
Blown Away
Nice to see Har Mar Superstar help out with hurricane preparedness. Them novelty pop acts, so selfless.
Oh, and:
1)
WE HEAR that a certain Oscar-winning starlet who is black and who plays a certain weather-controlling mutant in a certain comic book-based film franchise has a nasty case of the shingles.
2)
| 350,419 |
Law/Judiciary
Police Nab Car-Snatching Gang Members
The Nasarawa State Police Command announced that it has arrested two members of a three-man car-snatching gang terrorising residents of the state.
Its Public Relations Officer, Mr Idrissu Kennedy, told newsmen in Lafia that a car, believed to be snatched at gun-point, was recovered from the gang at the time of the arrest.
“The command kept receiving a lot of reports about cars being snatched at gun point by a three-man gang at different parts of the state.
“Through effective intelligence gathering, our men tracked down the gang on Saturday, September 23 and arrested two members at Masaka in Karu Local Government Area.
“One of them escaped, but we are trailing him and shall soon track him down,” he said.
He said that the recovered vehicle was found to have been snatched from one Mr Yusuf Manga on Thursday, September 21, in Akwanga.
The police spokesman identified those arrested as Nasiru Danjuma and Collins Osaye, residents of Agwan Zakara in Uke, Karu Local Government, adding that they were being investigated to unravel other accomplices.
“As soon as investigation is completed, the suspects will be charged to court,” he said. | 96,604 |
- 0 Talk
Fantastic Four Vol 1 268
Redirected from Fantastic Four 268
Appearing in "The Masque of Doom!"Edit
Featured Characters:
- Fantastic Four
Supporting Characters:
Villains:
- A remote operated masque of Doctor Doom's
Other Characters:
- Doctor Octopus (Otto Octavius)
- Hulk (Robert Bruce Banner)
- Michael Morbius
- Walter Langkowski
- Alma Chalmers (First appearance)
- Alice Winchell (First appearance)
- Danny Winchell
- Dr. Lanning
Locations:
- California (Only in flashback)
- New York City
- Belle Porte, Connecticut
Items:
Vehicles:
Synopsis for "The Masque of Doom!"Edit
Reed & Susan mourn. She-Hulk feels like an intruder. Doc Octopus is returned to prison. Johnny Storm takes She-Hulk by Fantasticar to the Baxter Building on Madison Avenue & 42nd Street. She-Hulk tells him the brief version of her life story, including her real name, Jennifer Walters. She hasn't tried very hard to keep her identity secret, it's just that no one ever asked. She-Hulk believes being in the FF is a greater honor than being in the Avengers. They land inside. She's seen pictures but it didn't do this place justice, it seems bigger inside than out. Johnny explains a lot if the building's security systems to her, gets her belt fitted with an instrument pack which includes long-distance Fantasticar remote control & solonoid elevator summoner. They are having coffee when Doom's mask from the trophy room attacks them with concussion beams while floating around by itself. Jen withstands a multivolt electrical charge & brute forces her way through a concussion field, but in pushing through she tumbles out a window & plummets many stories fist-first into the cement. Reed defeats the mask, & ponders who remote controlled it.
NotesEdit
TriviaEdit
- No trivia.
See AlsoEdit
- Write your own review of this comic!
- Discuss Fantastic Four Vol 1 268 on the forums
- Cover gallery for the Fantastic Four series
Recommended ReadingEdit
- None.
Links and ReferencesEdit
- None. | 371,407 |
California College Promise Grant (CCPG)
About the California College Promise Grant (CCPG) formally known as the Board of Governors (BOG) Fee Waiver
In support of AB1741: California College Promise Innovation Grant Program, the online BOG Fee Waiver Application is changing its name to the California College Promise Grant Application (CCPG).
Part-time and full-time students may qualify for the CCPG (see eligibility requirements below). The CCPG fee waiver will waive enrollment fees for the academic year. Other fees, such as parking, textbooks, student body and student representation fees will not be covered by the CCPG.
Recent eligibility regulations went into effect as of the fall 2016 semester requiring all California Community College students to meet minimum academic and progress standards and maintain eligibility for the CCPG (SB 1456 Student Success Act of 2012).
Student Must:
- Maintain a cumulative GPA of 2.0 (Academic Standard)
- Successfully complete more than 50% of all units attempted (Progress Standard)
Students placed on academic and/or progress probation for two consecutive primary terms (fall or spring semester) will lose eligibility for the CCPG. If students meet one of the extenuating circumstances, they may regain eligibility by completing the appeal form . Students will be required to submit a completed appeal form with a narrative of their situation, supporting documentation and a updated student education plan. Please review the fee waiver policy here.
Students are encouraged to utilize the numerous support services on campus to help them regain good academic standing. Please watch this video for more information.
Arrangements for application assistance should be made prior to registration to avoid being dropped from classes for non-payment. The CCPG application is available online through WebSMART.
The California College Promise Grant for Homeless Youth
As of January 1, 2017, AB 80: The Success of Homeless Youth Act, adds “homeless youth” as an eligible group of students for a few waiver. A “homeless youth” under this section is a student under 25 years of age, who has been verified at any time during the 24 months immediately preceding their application for admission as a homeless youth, as defined in the federal McKinney-Vento Homeless Assistance Act.
A student who is verified as a homeless youth as defined above will retain that status for up to six years or until the age of 25. For more information and assistance, homeless youth should contact Jeremy Evangelista, located in the Financial Aid department at 650-738-4390 or Flor Lopez located in the SparkPoint Center at 650-738-4240.
If a student is awarded the CCPG fee waiver and is determined at a later date to have been ineligible or fail to turn in required documents, the fee waiver will be automatically cancelled and the fees will be reassessed to the student's account.
If a student received a Bachelors Degree or higher, federal financial aid may be limited.
Student may qualify for the CCPG fee waiver.
For more information regarding the California College Promise Grant please click here .
How do I apply for the California College Promise Grant fee waiver?
Option 1
- U.S. Citizens and Permanent Legal Residents: Submit the FAFSA application to see if you are eligible for the fee waiver.
- AB540 Students: Submit a CA Dream Act Application.
- If students have already submitted a FAFSA or CA Dream Act application, they do not need to apply for the CCPG on WebSMART. Submitting a FAFSA/Dream Act application will override the CCPG application on WebSMART.
Option 2
- Apply for the CCPG Fee Waiver through WebSMART.
- WebSMART account will be updated with the CCPG eligibility within the next 24 hours.
- Check WebSMART to verify student's eligibility and to see if any documents are required.
- Students have 21 days from the date they have submitted their application to submit any required documents to the financial aid office.
Who is eligible for the California College Promise Grant (2017-2018) for the 2017-2018 academic year
- Meet the California College Promise Grant Program Income Standards (2018-2019) for the 2018-2019 academic year..
Eligibility Method D
- Be under 25 years of age
- Verified with Financial Aid at any time during the 24 months immediately preceding their CCCApply application for admissions as homeless youth, as defined in the federal McKinney-Vento Homeless Assistance Act
The California College Promise Grant - Bachelor of Science Respiratory Care Program
Students accepted into the Bachelor of Science Respiratory Care program at Skyline College will pay $130.00 per unit for upper division coursework. If eligible for the CCPG fee waiver, $46.00 per unit of lower division courses will be waived. Eligibility for the CCPG can only be determined by completing a FAFSA or CA Dream Act application. | 26,776 |
\begin{document}
\title{Algebras of higher operads as enriched categories II}
\author{Michael Batanin}
\address{Department of Mathematics,
Macquarie University}
\email{[email protected]}
\thanks{}
\author{Denis-Charles Cisinski}
\address{Departement des Mathematiques,
Universit\'e Paris 13 Villanteuse}
\email{[email protected]}
\thanks{}
\author{Mark Weber}
\address{Max Planck Institute for Mathematics, Bonn}
\email{[email protected]}
\thanks{}
\maketitle
\begin{abstract}
One of the open problems in higher category theory is the systematic construction of the higher dimensional analogues of the Gray tensor product. In this paper we continue the work of \cite{EnHopI} to adapt the machinery of globular operads \cite{Bat98} to this task. The resulting theory includes the Gray tensor product of 2-categories and the Crans tensor product \cite{Crans99} of Gray categories. Moreover much of the previous work on the globular approach to higher category theory is simplified by our new foundations, and we illustrate this by giving an expedited account of many aspects of Cheng's analysis \cite{ChengCompOp} of Trimble's definition of weak $n$-category. By way of application we obtain an ``Ekmann-Hilton'' result for braided monoidal 2-categories, and give the construction of a tensor product of $A$-infinity algebras.
\end{abstract}
\tableofcontents
\section{Introduction}\label{sec:Intro}
In \cite{Bat98} the problem of how to give an explicit combinatorial definition of weak higher categories was solved, and the development of a conceptual framework for their further analysis was begun. In the aftermath of this, the expository work of other authors, most notably Street \cite{Str98} and Leinster \cite{Lei}, contributed greatly to our understanding of these ideas. The central idea of \cite{Bat98} is that the description of any $n$-dimensional categorical structure $X$, may begin by starting with just the underlying $n$-globular set, that is, the sets and functions
\[ \xymatrix{{X_0} & {X_1} \ar@<1ex>[l]^{t} \ar@<-1ex>[l]_{s} & {X_2} \ar@<1ex>[l]^{t} \ar@<-1ex>[l]_{s}
& {X_3} \ar@<1ex>[l]^{t} \ar@<-1ex>[l]_{s} & {...} \ar@<1ex>[l]^{t} \ar@<-1ex>[l]_{s} & {X_n} \ar@<1ex>[l]^{t} \ar@<-1ex>[l]_{s}} \]
satisfying the equations $ss=st$ and $ts=tt$, which embody the the objects (elements of $X_0$), arrows (elements of $X_1$) and higher cells of the structure in question. At this stage no compositions have been defined, and when they are, one has a globular set with extra structure. In this way the problem of defining an n-categorical structure of a given type is that of defining the monad on the category $\PSh {\G}_{{\leq}n}$ of $n$-globular sets whose algebras are these structures.
As explained in the introduction to \cite{EnHopI}, this approach works because the monads concerned have excellent formal properties, which facilitate their explicit description and further analysis. The $n$-operads of \cite{Bat98} can be defined from the point of view of monads: one has the monad $\ca T_{{\leq}n}$ on $\PSh {\G}_{{\leq}n}$ whose algebras are strict $n$-categories, and an $n$-operad consists of another monad $A$ on $\PSh {\G}_{{\leq}n}$ equipped with a cartesian monad morphism $A \rightarrow \ca T_{{\leq}n}$. The algebras of this $n$-operad are just the algebras of $A$.
Strict $n$-categories are easily defined by iterated enrichment: a strict $(n{+}1)$-category is a category enriched in the category of strict $n$-categories via its cartesian product, but are too strict for the intended applications in homotopy theory and geometry. For $n=3$ the strictest structure one can replace an arbitrary weak $3$-category with -- and not lose vital information -- is a Gray category, which is a category enriched in $\Enrich 2$ using the Gray tensor product of 2-categories instead of its cartesian product \cite{GPS95}. This experience leads naturally to the idea of trying to define what the higher dimensional analogues of the Gray tensor product are, so as to set up a similar inductive definition as for strict $n$-categories, but to capture the appropriate semi-strict $n$-categories, which in the appropriate sense, would form the strictest structure one can replace an arbitrary weak $n$-category with and not lose vital information.
Crans in \cite{Crans99} attempted to realise this idea in dimension 4, and one of our main motivations is to obtain a theory that will deliver the sort of tensor product that Crans was trying to define explicitly, but in a conceptual way that one could hope to generalise to still higher dimensions. Our examples(\ref{ex:Gray}) and (\ref{ex:Crans}) embody the progress that we have achieved in this direction in this paper. In \cite{WebFunny} the theory of the present paper is used to show that the \emph{funny tensor product} of categories -- which is what one obtains by considering the Gray tensor product of $2$-categories but ignoring what happens to 2-cells -- generalises to give an analogous symmetric monoidal closed structure on the category of algebras of any higher operad. From these developments it seems that a conceptual understanding of the higher dimensional analogues of the Gray tensor product is within reach.
Fundamentally, we have two kinds of combinatorial objects important for the description and study of higher categorical structures -- $n$-operads and tensor products. In \cite{EnHopI} a description of the relationship between tensor products and $n$-operads was begun, and $(n{+}1)$-operads whose algebras involve no structure at the level objects{\footnotemark{\footnotetext{In \cite{EnHopI} these were called \emph{normalised} $(n{+}1)$-operads. In the present work we shall, for reasons that will become apparent below, refer to these operads as \emph{being over $\Set$}.}}} were canonically related with certain lax tensor products on $\PSh {\G}_{{\leq}n}$. Under this correspondence the algebras of the $(n{+}1)$-operad coincide with categories enriched in the associated lax tensor product.
Sections(\ref{sec:EG-LMC})-(\ref{sec:2-functoriality}) of the present paper continue this development by studying, for a given category $V$, the passage
\[ \textnormal{Lax tensor products on $V$} \mapsto \textnormal{Monads on $\ca GV$} \]
where $\ca GV$ is the category of graphs enriched in $V$, in a systematic way. This analysis culminates in section(\ref{sec:2-functoriality}) where the above assignment is seen as the object part of a 2-functor
\[ \Gamma : \DISTMULT \rightarrow \MND(\CAT/\Set) \]
where $\DISTMULT$ is a sub 2-category of the 2-category of lax monoidal categories, and $\MND$ is as defined by the formal theory of monads \cite{Str72}. From this perspective, one is able to describe in a more efficient and general way, many of the previous developments of higher category theory in the globular style. For instance, in section(\ref{ssec:induction}) we give a short and direct explicit construction of the monads $\ca T_{{\leq}n}$ for strict $n$-categories from which all their key properties are easily witnessed. In sections(\ref{ssec:general-op-mult}) and (\ref{ssec:induction}) we give shorter and more general proofs of some of the main results of \cite{EnHopI}. In section(\ref{ssec:Monmonad-Distlaw}) using a dual version of our 2-functor $\Gamma$ and the formal theory of monads \cite{Str72}, we obtain a satisfying general explanation for how it is that monad distributive laws arise in higher category theory -- see \cite{ChengDist} \cite{ChengCompOp}. In sections(\ref{ssec:TCI}) and (\ref{ssec:TCII}) we apply our theory to simplifying many aspects of \cite{ChengCompOp}.
The correspondence between $(n{+}1)$-operads and certain lax monoidal structures on $\PSh {\G}_{{\leq}n}$ given in \cite{EnHopI}, associates to the 3-operad $G$ for Gray categories, a lax tensor product on the category of 2-globular sets. However the Gray tensor product itself is a tensor product of 2-categories. Any lax monoidal structure on a category $V$ comes with a ``unary'' tensor product, which rather than being trivial as is the usual experience with non-lax tensor products, is in fact a monad on $V$. For the lax tensor product induced by $G$, this is the monad for 2-categories. In section(\ref{sec:lift-mult}) we solve the general problem of lifting a lax monoidal structure, to a tensor product on the category of algebras of the monad defined by its unary part. This result, theorem(\ref{thm:lift-mult}), is the main result of the paper, and provides also the sense in which these lifted tensor products are unique. In practical terms this means that in order to exhibit a given tensor product on some category of higher dimensional structures as arising from our machinery, it suffices to exhibit an operad whose algebras are categories enriched in that tensor product. In this way, one is able see that the usual Gray tensor product and that of Crans, do so arise.
Moreover applying this lifting to the lax tensor products on $\PSh {\G}_{{\leq}n}$ associated to general $(n{+}1)$-operads (over $\Set$), one exhibits the structures definable by $(n{+}1)$-operads as enriched categories whose homs are some $n$-dimensional structure. In this way the globular approach is more closely related to some of the inductive approaches to higher category theory, such as that of Tamsamani \cite{Tam99}.
In section(\ref{ssec:A-infinity}) we describe two applications of the lifting theorem. In theorem(\ref{thm:A-infinity-app}) we construct a tensor product of $A_{\infty}$-algebras. As explained in \cite{Markl} the problem of providing such a tensor product is of relevance to string theory, and it proved resistant because of the negative result \cite{Markl} which shows that no ``genuine'' tensor product can exist. However this result does not rule out the existence of a \emph{lax} tensor product, which is what we were able to provide in theorem(\ref{thm:A-infinity-app}). It is possible to see the identification by Joyal and Street \cite{JS93}, of braided monoidal categories as monoidal categories with a multiplication as an instance of theorem(\ref{thm:A-infinity-app}). Another instance is our second application given in corollary(\ref{cor:coh-bm2c}), namely an analogous result to that of Joyal and Street but for braided monoidal 2-categories.
A weak $n$-category is an algebra of a \emph{contractible} $n$-operad{\footnotemark{\footnotetext{In this work we use the notion of contractibility given in \cite{Lei} rather than the original notion of \cite{Bat98}.}}}. In section(\ref{sec:contractibility}) we recall this notion, give an analogous notion of contractible lax monoidal structure and explain the canonical relationship between them.
In this paper we operate at a more abstract level than in much of the previous work on this subject. In particular, instead of studying monads on the category of $n$-globular sets, or even on presheaf categories, we work with monads defined on some category $\ca GV$ of enriched graphs. As our work shows, the main results and notions of higher category theory in the globular style can be given in this setting. So one could from the very beginning start not with $\Set$ as the category of $0$-categories, but with a nice enough $V$. For all the constructions to go through, such as that of $\ca T_{{\leq}n}$, the correspondence between monads and lax tensor products, their lifting theorem, as well as the very definition of weak $n$-category, it suffices to take $V$ to be a locally c-presentable category in the sense defined in section(\ref{ssec:lcpres}).
Proceeding this way one obtains then the theory of $n$-dimensional structures enriched in $V$. That is to say, the object of $n$-cells between any two $(n{-}1)$-cells of such a structure would be an object of $V$ rather than a mere set. Some alternative choices of $V$ which could perhaps be of interest are: (1) the ordinal $[1]=\{0<1\}$ (for the theory of locally ordered higher dimensional structures), (2) simplicial sets (to obtain a theory of higher dimensional structures which come together with a simplicial enrichment at the highest level), (3) the category of sheaves on a locally connected space, or more generally a locally connected Grothendieck topos, (4) the algebras of any $n$-operad or (5) the category of multicategories (symmetric or not). The point is, the theory as we have developed it is actually \emph{simpler} than before, and the generalisations mentioned here come at \emph{no} extra cost.
\section{Enriched graphs and lax monoidal categories}\label{sec:EG-LMC}
\subsection{Enriched graphs and the reduced suspension of spaces}\label{ssec:enriched-graphs}
Given a topological space $X$ and points $a$ and $b$ therein, one may define the topological space $X(a,b)$ of paths in $X$ from $a$ to $b$ at a high degree of generality. In recalling the details let us denote by $\Top$ a category of ``spaces'' which is complete, cocomplete and cartesian closed. We shall write $1$ for the terminal object. We shall furthermore assume that $\Top$ comes equipped with a bipointed object $I$ playing the role of the interval. A conventional choice for $\Top$ is the category of compactly generated Hausdorff spaces with its usual interval, although there are many other alternatives which would do just as well from the point of view of homotopy theory.
Let us denote by $\sigma{X}$ the reduced suspension of $X$, which can be defined as the pushout
\[ \xygraph{!{0;(1.5,0):(0,.667)::} {X{+}X}="tl" [r] {I{\times}X}="tr" [d] {\sigma{X}.}="br" [l] {1{+}1}="bl" "tl"(:"tr":"br",:"bl":"br") "br" [u(.3)l(.3)] (:@{-}[r(.15)],:@{-}[d(.15)])} \]
Writing $\Top_{\bullet}$ for the category of bipointed spaces, that is to say the coslice $1{+}1/\Top$, the above definition exhibits the reduced suspension construction as a functor
\[ \sigma : \Top \rightarrow \Top_{\bullet}. \]
In a sense this functor is the mother of homotopy theory -- applying it successively to the inclusion of the empty space into the point, one obtains the inclusions of the $(n{-}1)$-sphere into the $n$-disk for all $n \in \N$, and its right adjoint
\[ h : \Top_{\bullet} \rightarrow \Top \]
is the functor which sends the bipointed space $(a,X,b)$, to the space $X(a,b)$ of paths in $X$ from $a$ to $b$. This adjunction $\sigma \ladj h$ is easy to verify directly using the above elementary definition of $\sigma(X)$ as a pushout, and the pullback square
\[ \xygraph{!{0;(1.5,0):(0,.667)::} {X(a,b)}="tl" [r] {X^I}="tr" [d] {X^{1{+}1}}="br" [l] {1}="bl" "tl"(:"tr":"br"^{X^i},:"bl":"br"_-{(a,b)}) "tl" [d(.3)r(.3)] (:@{-}[l(.15)],:@{-}[u(.15)])} \]
where $i$ is the inclusion of the boundary of $I$. The collection of spaces $X(a,b)$ is our first example of an enriched graph in the sense of
\begin{definition}\label{def:enriched-graph}
Let $V$ be a category. A \emph{graph $X$ enriched in $V$} consists of an underlying set $X_0$ whose elements are called \emph{objects}, together with an object $X(a,b)$ of $V$ for each ordered pair $(a,b)$ of objects of $X$. The object $X(a,b)$ will sometimes be called the \emph{hom} from $a$ to $b$. A morphism $f:X{\rightarrow}Y$ of $V$-enriched graphs consists of a function $f_0:X_0{\rightarrow}Y_0$ together with a morphism $f_{a,b}:X(a,b){\rightarrow}Y(fa,fb)$ for each $(a,b)$. The category of $V$-graphs and their morphisms is denoted as $\ca GV$, and we denote by $\ca G$ the obvious 2-functor
\[ \begin{array}{lccr} {\ca G : \CAT \rightarrow \CAT} &&& {V \mapsto \ca GV} \end{array} \]
with object map as indicated.
\end{definition}
The 2-functor $\ca G$ is the mother of higher category theory in the globular style -- applying it successively to the inclusion of the empty category into the point (ie the terminal category), one obtains the inclusion of the category of $(n{-}1)$-globular sets into the category of $n$-globular sets. In the case $n>0$ this is the inclusion with object map
\[ \begin{array}{lcr}
{\xygraph{{X_0}="l" [r] {...}="m" [r] {X_{n{-}1}}="r" "r":@<1ex>"m":@<1ex>"l" "r":@<-1ex>"m":@<-1ex>"l"}} & \mapsto &
{\xygraph{{X_0}="l" [r] {...}="m" [r] {X_{n{-}1}}="r" [r] {\emptyset}="rr" "rr":@<1ex>"r":@<1ex>"m":@<1ex>"l" "rr":@<-1ex>"r":@<-1ex>"m":@<-1ex>"l"}}
\end{array} \]
and when $n{=}0$ this is the functor $1{\rightarrow}\Set$ which picks out the empty set. Thus there is exactly one $(-1)$-globular set which may be identified with the empty set.
It is often better to think of $\ca G$ as taking values in $\CAT/\Set$. By applying the endofunctor $\ca G$ to the unique functor $V{\rightarrow}1$ for each $V$, produces the forgetful functor $\ca GV{\rightarrow}\Set$ which sends an enriched graph to its underlying set of objects. This manifestation
\[ \ca G : \CAT \rightarrow \CAT/\Set \]
has a left adjoint which we shall denote as $(-)_{\bullet}$ for reasons that are about to become clear. The functor $(-)_{\bullet}$ is a variation of the Grothendieck construction. To a given functor $f:A{\rightarrow}\Set$ it associates the category $A_{\bullet}$ with objects triples $(x,a,y)$ where $a$ is an object of $A$, and $(x,y)$ is an ordered pair of objects of $fa$. Maps are just maps in $A$ which preserve these base points in the obvious sense.
It is interesting to look at the unit and counit of this 2-adjunction. Given a category $V$, $(\ca GV)_{\bullet}$ is the category of bipointed enriched graphs in $V$. The counit $\varepsilon_V:(\ca GV)_{\bullet}{\rightarrow}V$ sends $(a,X,b)$ to the hom $X(a,b)$. When $V$ has an initial object $\varepsilon_V$ has a left adjoint given by $X \mapsto (X)$. Given a functor $f:A{\rightarrow}\Set$ the unit $\eta_f:A{\rightarrow}\ca G(A_{\bullet})$ sends $a \in A$ to the enriched graph whose objects are elements of $fa$, and the hom $a(x,y)$ is given by the bipointed object $(x,a,y)$.
Consider the case where $0 \in A$ and $f$ is the representable $f=A(0,-)$. Then $A_{\bullet}$ may be regarded as the category of endo-cospans of the object $0$, that is to say the category of diagrams
\[ 0 \rightarrow a \leftarrow 0 \]
and a point of $a \in A$ is now just a map $0{\rightarrow}a$. When $A$ is also cocomplete one can compute a left adjoint to $\eta_A$. To do this note that a graph $X$ enriched in $A_{\bullet}$ gives rise to a functor
\[ \overline{X} : X^{(2)}_0 \rightarrow A \]
where $X_0$ is the set of objects of $X$. For any set $Z$, $Z^{(2)}$ is defined as the following category. It has two kinds of objects: an object being either an element of $Z$, or an ordered pair of elements of $Z$. There are two kinds of non-identity maps
\[ x \rightarrow (x,y) \leftarrow y \]
where $(x,y)$ is an ordered pair from $Z$, and $Z^{(2)}$ is free on the graph just described. A more conceptual way to see this category is as the category of elements of the graph
\[ \xygraph{{Z{\times}Z}="l" [r] {Z}="r" "l":@<1ex>"r" "l":@<-1ex>"r"} \]
where the source and target maps are the product projections, as a presheaf on the category
\[ \xygraph{{\G_{\leq{1}}} [r(.75)] {=} [r(1.25)] {\xybox{\xygraph{0 [r] 1 "0":@<1ex>"1":@<1ex>@{<-}"0"}}} *\frm{-}} \]
and so there is a discrete fibration $Z^{(2)}{\rightarrow}\G_{{\leq}1}$. The functor $\overline{X}$ sends singletons to $0 \in A$, and a pair $(x,y)$ to the head of the hom $X(x,y)$. The arrow map of $\overline{X}$ encodes the bipointings of the homs. One may then easily verify
\begin{proposition}
Let $0 \in A$, $f=A(0,-)$ and $A$ be cocomplete. Then $\eta_f$ has left adjoint given on objects by $X \mapsto \colim(\overline{X})$.
\end{proposition}
In the exposition thus far we have focussed on building an analogy between the reduced suspension of a space and the graphs enriched in a category. Now we shall bring these constructions together. As we have seen already to each space $X$ one can associate a canonical topologically enriched graph whose homs are the path spaces of $X$. Denoting this enriched graph as $PX$, the assignment $X \mapsto PX$ is the object map of the composite right adjoint in
\[ \xygraph{!{0;(2,0):} {\Top}="l" [r] {\ca G(\Top_{\bullet})}="m" [r] {\ca G\Top}="r" "l":@<-1.5ex>"m"_-{\eta}|{}="b1":@<-1.5ex>"l"|{}="t1" "t1":@{}"b1"|{\perp} "m":@<-1.5ex>"r"_-{\ca Gh}|{}="b2":@<-1.5ex>"m"_-{\ca G\sigma}|{}="t2" "t2":@{}"b2"|{\perp}}. \]
As explained by Cheng \cite{ChengCompOp}, this functor $P = \ca G(h)\eta$ is a key ingredient of the Trimble definition of weak $n$-category.
\subsection{Some exactness properties of the endofunctor $\ca G$}\label{ssec:G-exactness}
The important properties of $\ca G$ are apparent because of the close connection between $\ca G$ and the $\Fam$ construction. A very mild reformulation of the notion of $V$-graph is the following: a $V$-graph $X$ consists of a set $X_0$ together with an $(X_0{\times}X_0)$-indexed family of objects of $V$. Together with the analogous reformulation of the maps of $\ca GV$, this means that we have a pullback square
\[ \PbSq {\ca GV} {\Fam{V}} {\Set} {\Set} {(-)_0=\ca Gt_V} {} {\Fam(t_V)} {(-)^2} \]
in $\CAT$, and thus a cartesian 2-natural transformation $\ca G{\implies}\Fam$. From \cite{Fam2fun} theorem(7.4) we conclude
\begin{proposition}\label{prop:GFam2fun}
$\ca G$ is a familial 2-functor.
\end{proposition}
\noindent In particular it follows from the theory of \cite{Fam2fun} that $\ca G$ preserves conical connected limits as well as all the notions of ``Grothendieck fibration'' which one can define internal to a finitely complete 2-category. Moreover the obstruction maps for comma objects are right adjoints. See \cite{Fam2fun} for more details on this part of 2-category theory. We shall not use these observations very much in what follows. More important for us is
\begin{lemma}\label{lem:G-EM-object}
$\ca G$ preserves Eilenberg-Moore objects.
\end{lemma}
\noindent Given a monad $T$ on a category $V$, we shall write $V^T$ for the category of $T$-algebras and morphisms thereof, and $U^T:V^T{\rightarrow}V$ for the forgetful functor. We shall denote a typical object of $V^T$ as a pair $(X,x)$, where $X$ is the underlying object in $V$ and $x:TX{\rightarrow}X$ is the $T$-algebra structure. From \cite{Str72} the 2-cell $TU^T{\implies}U^T$, whose component at $(X,x)$ is $x$ itself has a universal property exhibiting $V^T$ as a kind of 2-categorical limit called an \emph{Eilenberg-Moore object}. See \cite{Str72} or \cite{LS00} for more details on this general notion. The direct proof that for any monad $T$ on a category $V$, the obstruction map $\ca G(V^T){\rightarrow}\ca G(V)^{\ca G(T)}$ is an isomorphism comes down to the obvious fact that for any $V$-graph $B$, a $\ca GT$-algebra structure on $B$ is the same thing as a $T$-algebra structure on the homs of $B$, and similarly for algebra morphisms.
\subsection{Multitensors}\label{ssec:LMC} Let us recall the notions of lax monoidal category and category enriched therein from \cite{EnHopI}. For a category $V$, the free strict monoidal category $MV$ on $V$ has a very simple description. An object of $MV$ is a finite sequence $(Z_1,...,Z_n)$ of objects of $V$. A map is a sequence of maps of $V$ -- there are no maps between sequences of objects of different lengths. The unit $\eta_V:V{\rightarrow}MV$ of the 2-monad $M$ is the inclusion of sequences of length $1$. The multiplication $\mu_V:M^2V{\rightarrow}MV$ is given by concatenation.
A \emph{lax monoidal category} is a lax algebra for the 2-monad $M$. Explicitly it consists of an underlying category $V$, a functor $E:MV{\rightarrow}V$, and maps
\[ \begin{array}{lcr} {u_Z:Z \rightarrow E(Z)} && {\sigma_{Z_{ij}}:\opE\limits_i\opE\limits_j Z_{ij} \rightarrow \opE\limits_{ij} Z_{ij}} \end{array} \]
for all $Z$, $Z_{ij}$ from $V$ which are natural in their arguments, and such that
\[ \xy (0,0); (10,0):
(0,0)*{\xybox{\xymatrix @C=1em {{\opE\limits_iZ_i} \ar[r]^-{u\opE\limits_i} \ar[d]_{1} \save \POS?(.4)="domeq" \restore & {E_1\opE\limits_iZ_i} \ar[dl]^{\sigma} \save \POS?(.4)="codeq" \restore \\ {\opE\limits_iZ_i} \POS "domeq"; "codeq" **@{}; ?*{=}}}};
(4,0)*{\xybox{\xymatrix @C=1.5em {{\opE\limits_i\opE\limits_j\opE\limits_kZ_{ijk}} \ar[r]^-{{\sigma}\opE\limits_k} \ar[d]_{\opE\limits_i\sigma} \save \POS?="domeq" \restore & {\opE\limits_{ij}\opE\limits_kZ_{ijk}} \ar[d]^{\sigma} \save \POS?="codeq" \restore \\
{\opE\limits_i\opE\limits_{jk}Z_{ijk}} \ar[r]_-{\sigma} & {\opE\limits_{ijk}Z_{ijk}} \POS "domeq"; "codeq" **@{}; ?*{=}}}};
(8,0)*{\xybox{{\xymatrix @C=1em {{\opE\limits_iE_1Z_i} \ar[dr]_{\sigma} \save \POS?(.4)="domeq" \restore & {\opE\limits_iZ_i} \ar[d]^{1} \save \POS?(.4)="codeq" \restore \ar[l]_-{\opE\limits_iu} \\ & {\opE\limits_iZ_i}} \POS "domeq"; "codeq" **@{}; ?*{=}}}}
\endxy \]
in $V$. As in \cite{EnHopI} we use either of the expressions
\[ \begin{array}{lcr} {\opE\limits_{1{\leq}i{\leq}n} X_i} && {\opE\limits_i X_i} \end{array} \]
as a convenient yet precise short-hand for $E(X_1,...,X_n)$, and we refer to the endofunctor of $V$ obtained by observing the effect of $E$ on singleton sequences as $E_1$. The data $(E,u,\sigma)$ is called a \emph{multitensor} on $V$, and $u$ and $\sigma$ are referred to as the unit and substitution of the multitensor respectively.
Given a multitensor $(E,u,\sigma)$ on $V$, a \emph{category enriched in $E$} consists of $X \in \ca GV$ together with maps
\[ \kappa_{x_i} : \opE\limits_i X(x_{i-1},x_i) \rightarrow X(x_0,x_n) \]
for all $n \in \N$ and sequences $(x_0,...,x_n)$ of objects of $X$, such that
\[ \xygraph{
{\xybox{\xygraph{!{0;(2,0):(0,.6)::}
{X(x_0,x_1)} (:[r]{E_1X(x_0,x_1)}^-{u} :[d]{X(x_0,x_1)}="bot"^{\kappa}, :"bot"_{\id})}}}
[r(5)][d(.15)]
{\xybox{\xygraph{!{0;(2.75,0):(0,.5)::}
{\opE\limits_i\opE\limits_jX(x_{(ij)-1},x_{ij})} (:[r]{\opE\limits_{ij}X(x_{(ij)-1},x_{ij})}^-{\sigma} :[d]{X(x_0,x_{mn_m})}="bot"^{\kappa},:[d]{\opE\limits_iX(x_{(i1)-1},x_{in_i})}_{\opE\limits_i\kappa} :"bot"_-{\kappa})}}}} \]
commute, where $1{\leq}i{\leq}m$, $1{\leq}j{\leq}n_i$ and $x_{(11)-1}{=}x_0$. Since a choice of $i$ and $j$ references an element of the ordinal $n_{\bullet}$, the predecessor $(ij){-}1$ of the pair $(ij)$ is well-defined when $i$ and $j$ are not both $1$. With the obvious notion of $E$-functor (see \cite{EnHopI}), one has a category $\Enrich E$ of $E$-categories and $E$-functors together with a forgetful functor
\[ U^E : \Enrich E \rightarrow \ca GV. \]
The notation we use makes transparent the analogy between multitensors and monads, and categories enriched in multitensors and algebras for a monad. In particular the unit and subtitution for $E$ provide $E_1$ with the unit and multiplication of a monad structure. Moreover, any object of the form $\opE\limits_i Z_i$ is canonically an $E_1$-algebra, as is the hom of any $E$-category, and the substitution maps of $E$ are $E_1$-algebra morphisms (see \cite{EnHopI} lemma(2.7)). Thus in a sense, any multitensor $(E,u,\sigma)$ on a category $V$ is aspiring to be a multitensor on the category $V^{E_1}$ of $E_1$-algebras, but of course there is no meaningful way to regard $u$ as living in $V^{E_1}$, except in the boring situation when $E_1$ is the identity monad, that is, when $u$ is an identity natural transformation. The multitensors with $u$ the identity are called \emph{normal}.
\begin{definition}\label{def:lift}
Let $(E,u,\sigma)$ be a multitensor on a category $V$. A \emph{lift} of $(E,u,\sigma)$ is a normal multitensor $(E',\id,\sigma')$ on $V^{E_1}$ together with an isomorphism $\Enrich E \iso \Enrich {E'}$ which commutes with the forgetful functors into $\ca G(V^{E_1})$.
\end{definition}
\noindent In \cite{EnHopI} we explained how to associate normalised $(n+1)$-operads and $n$-multitensors, which are multitensors on the category of $n$-globular sets. In the present paper we shall explain why any $n$-multitensor has a canonical lift.
\section{Multitensors from monads}\label{sec:Monad->Mult}
\subsection{Monads over $\Set$}\label{ssec:DefNMonad}
At an abstract level much of this paper is about the interplay between the theory of monads on categories of enriched graphs, and the theory of multitensors. It is time to be more precise about which monads on $\ca GV$ we are interested in.
\begin{definition}\label{def:NMnd}
Let $V$ be a category. A monad \emph{over $\Set$} on $\ca GV$ is a monad on
\[ (-)_0 : \ca GV \rightarrow \Set \]
in the 2-category $\CAT/\Set$.
\end{definition}
\noindent That is, a monad $(T,\eta,\mu)$ on $\ca GV$ is over $\Set$ when the functor $T$ doesn't affect the object sets, in other words $TX_0=X_0$ for all $X \in \ca GV$ and similarly for maps, and moreover the components of $\eta$ and $\mu$ are identities on objects. In this section we will describe how such a monad, in the case where $V$ has an initial object denoted as $\emptyset$, induces a multitensor on $V$ denoted $(\overline{T},\overline{\eta},\overline{\mu})$.
Let us describe this multitensor explicitly. First we note that $\emptyset \in V$ enables us to regard any sequence of objects $(Z_1,...,Z_n)$ of $V$ as a $V$-graph. The object set is $\{0,...,n\}$,
\[ (Z_1,...,Z_n)(i-1,i) = Z_i \]
for $1{\leq}i{\leq}n$, and all the other homs are equal to $\emptyset$. Then we define
\begin{equation}\label{eq:Tbar} \overline{T}(Z_1,...,Z_n) := T(Z_1,...,Z_n)(0,n) \end{equation}
and the unit as
\begin{equation}\label{eq:etabar} \overline{\eta}_Z := \{\eta_{(Z)}\}_{0,1}. \end{equation}
Before defining $\overline{\mu}$ we require some preliminaries. Given objects $Z_i$ of $V$ where $1{\leq}i{\leq}n$, and $1{\leq}a{\leq}b{\leq}n$ denote by
\[ s_{a,b} : (Z_a,...,Z_b) \rightarrow (Z_1,...,Z_n) \]
the obvious subsequence inclusion in $\ca GV$: the object map preserves successor and $0 \mapsto (a-1)$, and the hom maps are identities. Now given objects $Z_{ij}$ of $V$ where $1{\leq}i{\leq}k$ and $1{\leq}j{\leq}n_i$, one has a map
\[ \tilde{\tau}_{Z_{ij}} : (\Tbar\limits_{1{\leq}j{\leq}n_1}Z_{1j},...,\Tbar\limits_{1{\leq}j{\leq}n_k}Z_{kj}) \rightarrow T(Z_{11},......,Z_{kn_k}) \]
given on objects by $0 \mapsto 0$ and $i \mapsto (i,n_i)$ for $1{\leq}i{\leq}k$, and the hom map between $(i-1)$ and $i$ is $Ts_{i1,in_i}$. With these definitions in hand we can now define the components of $\overline{\mu}$ as
\begin{equation}\label{eq:mubar} \xymatrix{{\Tbar\limits_i\Tbar\limits_jZ_{ij}} \ar[rr]^-{\{T\tilde{\tau}\}_{0,k}} && {T(Z_{11},...,Z_{kn_k})(0,n_{\bullet})} \ar[rr]^-{\mu_{0,n_{\bullet}}} && {\Tbar\limits_{ij}Z_{ij}}}. \end{equation}
From now until the end of (\ref{ssec:Tbar}) we shall be occupied with the proof of
\begin{theorem}\label{thm:Tbar}
Let $V$ be a category with an initial object $\emptyset$ and $(T,\eta,\mu)$ be a monad over $\Set$ on $\ca GV$. Then $(\overline{T},\overline{\eta},\overline{\mu})$ as defined in (\ref{eq:Tbar})-(\ref{eq:mubar}) defines a multitensor on $V$.
\end{theorem}
\noindent In principle one could supply a proof of this result immediately by just slogging through a direct verification of the axioms. Instead we shall take a more conceptual approach, and along the way encounter various ideas that are of independent interest. For most of the time we will assume a little more of $V$: that it has finite coproducts, to enable our more conceptual approach. In the end though, we will see that only the initial object is necessary.
We break up the construction of $(\overline{T},\overline{\eta},\overline{\mu})$ into three steps. First in (\ref{ssec:NMonad->MonMonad}), we describe how $\ca GV_{\bullet}$ acquires a monoidal structure and $(T,\eta,\mu)$ induces a monoidal monad $(T_{\bullet},\eta_{\bullet},\mu_{\bullet})$ on $\ca GV_{\bullet}$. Then we see that this monoidal monad induces a lax monoidal structure on $\ca GV_{\bullet}$, which in turn can be transferred across an adjunction to obtain $(\overline{T},\overline{\eta},\overline{\mu})$. These last two steps are very general: they work at the level of the theory of lax algebras for an arbitrary 2-monad (which in our case is $M$). So in (\ref{ssec:laxalg-const1}) and (\ref{ssec:laxalg-const2}) we describe these general constructions, and in (\ref{ssec:Tbar}) we finish the proof of theorem(\ref{thm:Tbar}). Finally in (\ref{ssec:path-like}) we present a condition on $T$ which ensures that $T$-algebras and $\overline{T}$-categories may be identified.
\subsection{Monoidal monads from monads over $\Set$}\label{ssec:NMonad->MonMonad} From now until just before the end of (\ref{ssec:Tbar}) we shall assume that $V$ has finite coproducts. We now describe some consequences of this. First, the functor $(-)_0:\ca GV{\rightarrow}\Set$ which sends an enriched graph to its set of objects becomes representable. We shall denote by $0$ the $V$-graph which represents $(-)_0$. It has one object and its unique hom is $\emptyset$.
The second consequence is that $\ca GV_{\bullet}$ inherits a natural monoidal structure and any normalised monad $(T,\eta,\mu)$ on $\ca GV$ can then be regarded as a monoidal monad $(T_{\bullet},\eta_{\bullet},\mu_{\bullet})$. The explanation for this begins with the observation that the representability of the underlying set functor $(-)_0$ enables a useful reformulation of the category $\ca GV_{\bullet}$ as the category of endocospans of $0$ as in section(\ref{ssec:enriched-graphs}). The usefulness of this is that such cospans can be composed, thus endowing $\ca GV_{\bullet}$ with a canonical monoidal structure.
The presence of $\emptyset$ in $V$ enables one to compute coproducts in $\ca GV$. The coproduct $X$ of a family $(X_i:i{\in}I)$ of $V$-graphs has object set given as the disjoint union of the object sets of the $X_i$, $X(x,y)=X_i(x,y)$ when $x$ and $y$ are objects of $X_i$, and all the other homs are $\emptyset$. With finite coproducts available one can also compute pushouts under $0$, that is the pushout $P$ of maps
\[ \xymatrix{X & 0 \ar[r]^-{y} \ar[l]_-{x} & Y} \]
in $\ca GV$ is described as follows. The object set of $P$ is the disjoint union of the object sets of $X$ and $Y$ modulo the identification of $x$ and $y$, and let us write $z$ for this special element of $P$. The homs of $P$ are inherited from $X$ and $Y$ in almost the same way as for coproducts. That is if $a$ and $b$ are either both objects of $X$ or both objects of $Y$ and they are not both $z$, then their hom $P(a,b)$ is taken as in $X$ or $Y$. The hom $P(z,z)$ is the coproduct $X(x,x)+Y(y,y)$. Otherwise this hom is $\emptyset$. Note that in the special case where the homs $X(x,x)$ and $Y(y,y)$ are both $\emptyset$, one only requires the initial object in $V$ to compute this pushout.
Given a sequence
\[ ((a_1,X_1,b_1),...,(a_n,X_n,b_n)) \]
of doubly-pointed $V$-graphs, one defines their \emph{join}
\[ (a_1,(a_1,X_1,b_1)*...*(a_n,X_n,b_n),b_n) \]
where $(a_1,X_1,b_1)*...*(a_n,X_n,b_n)$ is the colimit of
\begin{equation}\label{eq:join} \xymatrix{{X_1} & 0 \ar[l]_-{b_1} \ar[r]^-{a_2} & {...} & 0 \ar[l]_-{b_{n-1}} \ar[r]^-{a_n} & {X_n}} \end{equation}
in $\ca GV$ which can be formed via iterated pushouts under $0$.
This defines a monoidal structure on $\ca GV_{\bullet}$ whose tensor product we denote as $*:M\ca GV_{\bullet}{\rightarrow}\ca GV_{\bullet}$.
Given a functor $T:\ca GV{\rightarrow}\ca GW$ over $\Set$, one has a functor
\[ \begin{array}{lcr} {T_{\bullet} : \ca GV_{\bullet} \rightarrow \ca GW_{\bullet}} && {(a,X,b) \mapsto (a,TX,b)} \end{array} \]
whose object map is indicated on the right in the previous display. When both $V$ and $W$ have an initial object, one defines for each sequence of doubly-pointed $V$-graphs a map
\[ \tau_{X_i} : T_{\bullet}(a_1,X_1,b_1)*...*T_{\bullet}(a_n,X_n,b_n) \rightarrow T_{\bullet}((a_1,X_1,b_1)*...*(a_n,X_n,b_n)) \]
as follows. Write
\[ \begin{array}{c} {c_i:X_i \rightarrow (a_1,X_1,b_1)*...*(a_n,X_n,b_n)} \\ {c'_i:TX_i \rightarrow (a_1,TX_1,b_1)*...*(a_n,TX_n,b_n)} \end{array} \]
for the components of the colimit cocones (\ref{eq:join}). Using the unique map $0{\rightarrow}T0$, there is a unique map $\tau_{X_i}$ such that $Tc_i=\tau_{X_i}{c'_i}$. By the unique characterisation of these maps, they assemble together to provide the coherence 2-cell
\[ \LaxSq {M\ca GV_{\bullet}} {\ca GV_{\bullet}} {M\ca GW_{\bullet}} {\ca GW_{\bullet}} {MT_{\bullet}} {*} {T_{\bullet}} {*} {\tau} \]
for a lax monoidal functor, and for
\[ \xymatrix{{\ca GU} \ar[r]^-{S} & {\ca GV} \ar[r]^-{T} & {\ca GW}} \]
over $\Set$ one has $T_{\bullet}S_{\bullet}=(TS)_{\bullet}$ as monoidal functors. Moreover any natural transformation $\phi:S{\rightarrow}T$ over $\Set$ defines a monoidal natural tranformation $\phi_{\bullet}:S_{\bullet}{\rightarrow}T_{\bullet}$. In fact, denoting by $\ca N$ the 2-category whose objects are categories with initial objects, a 1-cell $T:V{\rightarrow}W$ in $\ca N$ is a functor $T:\ca GV{\rightarrow}\ca GW$ over $\Set$, and a 2-cell between these is a natural tranformation also over $\Set$, we have defined a 2-functor
\[ J : \ca N \rightarrow \LaxAlg M. \]
Applying $J$ to monads gives
\begin{proposition}\label{prop:NMnd->MonMnd}
If $V$ has finite coproducts and $(T,\eta,\mu)$ is a monad over $\Set$ on $\ca GV$, then $(T_{\bullet},\eta_{\bullet},\mu_{\bullet})$ is a monoidal monad on $\ca GV_{\bullet}$, whose monoidal structure is given by pushout-composition of cospans.
\end{proposition}
Finally we note that the functor $\varepsilon:\ca GV_{\bullet}{\rightarrow}V$ has a left adjoint $L$ which we shall now describe. Given $Z \in V$ the underlying $V$-graph of $LZ$, which we shall denote as $(Z)$, has object set $\{0,1\}$ and the distinguished pair is $(0,1)$. As for the homs, $(Z)(0,1)$ is just $Z$ itself, and all the other homs are $\emptyset$. Formally it is the composite
\[ \xymatrix{{MV} \ar[r]^-{ML} & {M\ca GV_{\bullet}} \ar[r]^-{*} & {\ca GV_{\bullet}} \ar[r]^-{U} & {\ca GV}} \]
where $U$ is the obvious forgetful functor, which enables us to view a sequence of objects of $V$ as a $V$-graph, as in (\ref{ssec:DefNMonad}) above.
\subsection{A general lax algebra construction}\label{ssec:laxalg-const1}
Now and in (\ref{ssec:laxalg-const2}) let $(S,\eta,\mu)$ be a 2-monad on a 2-category $\ca K$.
Suppose that we are given a monad \emph{in} $\LaxAlg S$. Let us write $(V,E,\iota,\sigma)$ for the underlying lax $S$-algebra, $(F,\phi)$ for the lax $S$-algebra endomorphism of $V$, and $i$ and $m$ for the unit and multiplication respectively. Then one obtains another lax $S$-algebra structure on $V$ with one cell part given as the composite
\[ \xymatrix{{SV} \ar[r]^-{E} & V \ar[r]^-{F} & V} \]
and the 2-cell data as follows
\[ \xygraph{{\xybox{\xygraph{!{0;(1.5,0):(0,0.7)::}
{V}="tl" [r] {SV}="tr" [d] {V}="br" [l] {V}="bl"
"tl":"tr"^-{\eta}:"br"^{E}:"bl"^{F} "tl":"br"|{1} "tl":"bl"_{1}
"tl" [d(.35)r(.6)] :@{=>}[r(.2)]^{\iota} "tl" [d(.7)r(.2)] :@{=>}[r(.2)]^{i}}}}
[r(3)]
{\xybox{\xygraph{{S^2V}="l1" [d] {SV}="l2" [d] {SV}="l3" [r] {V}="m" [r] {V}="r3" [u] {V}="r2" [u] {SV}="r1"
"l1":"l2"_{SE}:"l3"_{SF}:"m"_-{E}:"r3"_-{F} "l1":"r1"^-{\mu}:"r2"^{E}:"r3"^{F} "l2":"r2"^-{E}:"m"_{F}
"l1" [d(.5)r(.85)] :@{=>}[r(.3)]^{\sigma} "l2" [d(.5)r(.5)] :@{=>}[r(.3)]^{\phi} "m" [u(.2)r(.5)] :@{=>}[r(.3)]^{m}}}}} \]
The verification of the lax algebra axioms is an easy exercise that is left to the reader.
\subsection{Another general lax algebra construction}\label{ssec:laxalg-const2}
Now suppose we are given a lax S-algebra $(V,E,\iota,\sigma)$
together with an adjunction
\[ \xygraph{!{0;(1.2,0):} V [r] W "V":@<-1.2ex>"W"_-{R}:@<-1.2ex>"V"_-{L} "V":@{}"W"|-{\perp}} \]
with unit $u$ and counit $c$. One can then induce a lax $S$-algebra structure on $W$. The one-cell part is given as the composite
\[ \xymatrix{{SW} \ar[r]^-{SL} & {SV} \ar[r]^-{E} & V \ar[r]^-{R} & W} \]
and the 2-cell data as follows
\[ \xygraph{{\xybox{
\xygraph{{W}="l" [r] {V}="m" [r] {SV}="r" [l(.5)d] {V}="br" [l] {W}="bl" "m" [u] {SW}="t"
"l":"t"^-{\eta}:"r"^-{SL}:"br"^-{E}|{}="E":"bl"^-{R} "l":"m"^-{L}:"br"_{1}|{}="oneV" "m":"r"^-{\eta} "l":"bl"_{1}|{}="oneW"
"m":@{}"t"|(.4)*{=}
"oneW":@{}"oneV"|(.35){}="d1"|(.65){}="c1" "d1":@{=>}"c1"^-{u}
"oneV":@{}"E"|(.2){}="d2"|(.8){}="c2" "d2":@{=>}"c2"^-{\iota}}}}
[r(4)] {\xybox{
\xygraph{!{0;(1,0):(0,.8)::}
{S^2W}="l1" [dl] {S^2V}="l2" [d] {SV}="l3" [d] {SW}="l4" [r] {SV}="ml" [r] {V}="mr" [r] {W}="r4" [u] {V}="r3" [u] {SV}="r2" [ul] {SW}="r1"
"l1":"l2"_{S^2L}:"l3"_{SE}:"l4"_{SR}:"ml"_{SL}:"mr"_{E}:"r4"_{R} "l1":"r1"^-{\mu}:"r2"^{SL}:"r3"^{E}:"r4"^{R} "l3":"ml"^{1}|{}="oneMV" "mr":"r3"^{1}|{}="oneV" "l2":"r2"^-{\mu}
"l1" [d(2)r(.35)] :@{=>}[r(.3)]^{\sigma} "l3" [d(.7)r(.15)] :@{=>}[r(.25)]^{Sc}
"r4" [u(.25)l(.35)] {=} "l1" [d(.5)r(.5)] {=}}}}} \]
The reader will easily verify that the lax $S$-algebra axioms for $W$ follow from those of $V$ and the triangle identities of the adjunction.
\subsection{Multitensors from monads over $\Set$}\label{ssec:Tbar} Let us now put together (\ref{ssec:NMonad->MonMonad})-(\ref{ssec:laxalg-const2}).
Given a monad $(T,\eta,\mu)$ on $\ca GV$ over $\Set$ such that $V$ has finite coproducts, we obtained the monoidal monad $(T_{\bullet},\eta_{\bullet},\mu_{\bullet})$ on $\ca GV_{\bullet}$ in proposition(\ref{prop:NMnd->MonMnd}). In other words $(T_{\bullet},\eta_{\bullet},\mu_{\bullet})$ is a monad in $\LaxAlg M$. Applying (\ref{ssec:laxalg-const1}) for $S=M$ gives us a multitensor on $\ca GV_{\bullet}$, and then applying (\ref{ssec:laxalg-const1}) to this last multitensor and the adjunction
\[ \xygraph{!{0;(1.2,0):} {\ca GV_{\bullet}}="V" [r] {V}="W" "V":@<-1.2ex>"W"_-{\varepsilon}:@<-1.2ex>"V"_-{L} "V":@{}"W"|-{\perp}} \]
gives us a multitensor on $V$ which we denote as $(\overline{T},\overline{\eta},\overline{\mu})$.
We shall now unpack this multitensor to see that it does indeed agree with that of theorem(\ref{thm:Tbar}). The one cell part $\overline{T}$ is the composite
\[ \xymatrix{{MV} \ar[r]^-{ML} & {M\ca GV_{\bullet}} \ar[r]^-{*} & {\ca GV_{\bullet}} \ar[r]^-{T_{\bullet}} & {\ca GV_{\bullet}} \ar[r]^-{\varepsilon} & V} \]
which agrees with equation(\ref{eq:Tbar}) and one also easily reconciles equation(\ref{eq:etabar}) for the unit. As for the substitution unpacking the $\overline{\mu}$ of our conceptual approach gives the following composite
\[ \xygraph{!{0;(1,0):(0,.85)::}
{M^2V}="p11" [r] {MV}="p12" [dr] {M\ca GV_{\bullet}}="p22" [l(3)] {M^2\ca GV_{\bullet}}="p21" [dl] {M\ca GV_{\bullet}}="p31" [d] {M\ca GV_{\bullet}}="p41" [d] {MV}="p51" [r(1.25)d] {M\ca GV_{\bullet}}="p52" [r(1.25)d] {\ca GV_{\bullet}}="p53" [r(1.25)u] {\ca GV_{\bullet}}="p54" [r(1.25)u] {V}="p55" [u] {\ca GV_{\bullet}}="p42" [u] {\ca GV_{\bullet}}="p32"
"p11":"p12"^-{\mu}|{}="mv1" "p21":"p22"^-{\mu}|{}="mv2" "p31":"p32"^-{*}|{}="mv3" "p51":"p52"_-{ML}:"p53"_-{*}:"p54"_-{T_{\bullet}}:"p55"_-{\varepsilon}
"p11":"p21"_-{M^2L}:"p31"_-{M*}:"p41"_-{MT_{\bullet}}:"p51"_-{M\varepsilon} "p12":"p22"^{ML}:"p32"^{*}:"p42"^{
T_{\bullet}}:"p55"^{\varepsilon}
"p41":@/^{2pc}/"p52"^{1}|{}="mh1" "p32":@/_{2.5pc}/"p53"_{T_{\bullet}}|{}="mh2" "p42":@/_{.5pc}/"p54"_{1}|{}="mh3"
"mv1":@{}"mv2"|*{=}:@{}"mv3"|*{\iso}
"p51":@{}"mh1"|{}="mid1":@{}"mh2"|{}="mid2":@{}"mh3"|{}="mid3":@{}"p55"|*{=}
"mid1" [l(.15)] :@{=>}[r(.3)]^{Mc} "mid2" [l(.15)] :@{=>}[r(.3)]^{\tau} "mid3" [l(.15)] :@{=>}[r(.3)]^{\mu_{\bullet}}} \]
which we shall now unpack further. The counit $c$ of the adjunction $L \ladj \varepsilon$ is described as follows. For a given doubly-pointed $\ca V$-graph $(a,X,b)$, the corresponding counit component
\[ c_{(a,X,b)} : (0,(X(a,b)),1) \rightarrow (a,X,b) \]
is specified by insisting that the hom map between $0$ and $1$ is $1_{X(a,b)}$. Define $\tilde{\mu}$ by
\[ \xygraph{!{0;(4,0):(0,.35)::}
{T(\Tbar\limits_jX_{1j},...,\Tbar\limits_jX_{kj})}="bl" [u] {T\{(0,T(X_{11},...,X_{1n_1}),n_1)*...*(0,T(X_{k1},...,X_{kn_k}),n_k)\}}="tl" [d(.4)r] {T^2(X_{11},......,X_{kn_k})}="tr" [d(.6)] {T(X_{11},......,X_{kn_k})}="br"
"bl"(:"tl"^{T\{c*...*c\}}:"tr"_-{T\tau}:"br"^{\mu},:"br"_-{\tilde{\mu}}) "bl":@{}"tr"|*{=}} \]
and then $\overline{\mu}$ is the effect of $\tilde{\mu}$ on the hom between $0$ and $k$. But one may easily verify that the composite $\tau\{c*...*c\}$ is just $\tilde{\tau}$ described in (\ref{ssec:DefNMonad}).
This completes the proof of theorem(\ref{thm:Tbar}) for the case where $V$ has finite coproducts. The general case is obtained by observing that only the joins of doubly-pointed $V$-graphs $(a,X,b)$ such that $X(a,a)$ and $X(b,b)=\emptyset$ are actually used in the construction of the multitensor and in its axioms, and these only require an initial object in $V$.
\subsection{Path-like monads}\label{ssec:path-like}
We shall now give a condition on a normalised monad $T$ which ensures that categories enriched in $\overline{T}$ are the same thing as $T$-algebras. Let $(a,X,b) \in \ca GV_{\bullet}$ and consider a sequence $x=(x_0,...,x_n)$ of objects of $X$ such that $x_0=a$ and $x_n=b$. Define the $V$-graph
\[ x^*X := (X(x_0,x_1),X(x_1,x_2),...,X(x_{n-1},x_n)) \]
and we have a map
\[ \overline{x} : (0,x^*X,n) \rightarrow (a,X,b) \]
given on objects by $i \mapsto x_i$ for $0{\leq}i{\leq}n$, and the effect on the hom between $(i-1)$ and $i$ is the identity for $1{\leq}i{\leq}n$.
\begin{definition}\label{def:path-like}
Let $V$ be a category with an initial object and $W$ be a category with all small coproducts. A functor $T:\ca GV{\rightarrow}\ca GW$ over $\Set$ is \emph{path-like} when for all $(a,X,b) \in \ca GV_{\bullet}$, the maps
\[ T\overline{x}_{0,n} : Tx^*X(0,n) \rightarrow TX(a,b) \]
for all $n \in \N$ and sequences $x=(x_0,...,x_n)$ such that $x_0=a$ and $x_n=b$, form a coproduct cocone in $W$. A normalised monad $(T,\eta,\mu)$ on $\ca GW$ is \emph{path-like} when $T$ is path-like in the sense just defined.
\end{definition}
\begin{example}\label{ex:cat-monad-path-like}
Let $V=W=\Set$ and $T$ be the free category endofunctor of $\Graph$. For any graph $X$ and $a,b \in X_0$, the hom $TX(a,b)$ is by definition the set of paths in $X$ from $a$ to $b$. Each path determines a sequence $x=(x_0,...,x_n)$ of objects of $X$ such that $x_0=a$ and $x_n=b$, by reading off the objects of $X$ as they are visited by the given path. Conversely for a sequence $x=(x_0,...,x_n)$ of objects of $X$ such that $x_0=a$ and $x_n=b$, $T\overline{x}$ identifies the elements $Tx^*X(0,n)$ with those paths in $X$ from $a$ to $b$ whose associated sequence is $x$. Thus $T$ is path-like.
\end{example}
\begin{proposition}\label{prop:pl-alg<->cat}
Let $V$ have small coproducts and $(T,\eta,\mu)$ be a path-like monad on $\ca GV$ over $\Set$. Then $\ca G(V)^T \iso \Enrich {\overline{T}}$.
\end{proposition}
\begin{proof}
Let $X$ be a $V$-graph. To give an identity on objects map $a:TX{\rightarrow}X$ is to give maps $a_{y,z}:TX(y,z){\rightarrow}X(y,z)$. By path-likeness these amount to giving for each $n \in \N$ and $x=(x_0,...,x_n)$ such that $x_0=y$ and $x_n=z$, a map
\[ a_x : \Tbar\limits_iX(x_{i-1},x_i) \rightarrow X(y,z) \]
since $\Tbar\limits_iX(x_{i-1},x_i)=Tx^*X(0,n)$, that is $a_x=a_{y,z}T\overline{x}_{0,n}$. When $n=1$, for a given $y,z \in X_0$, $x$ can only be the sequence $(y,z)$. The naturality square for $\eta$ at $\overline{x}$ implies that $\{\eta_X\}_{y,z}=T\overline{x}_{0,1}\{\eta_{(X(y,z))}\}_{0,1}$, and the definition of $\overline{(\,\,)}$ says that $\{\eta_{(X(y,z))}\}_{0,1}=\overline{\eta}_{X(y,z)}$. Thus to say that a map $a:TX{\rightarrow}X$ satisfies the unit law of a $T$-algebra is to say that $a$ is the identity on objects and that the $a_x$ described above satisfy the unit axioms of a $\overline{T}$-category.
To say that $a$ satisfies the associative law is to say that for all $y,z \in X_0$,
\begin{equation}\label{eq:assoc1} \xymatrix{{T^2X(y,z)} \ar[r]^-{\{\mu_X\}_{y,z}} \ar[d]_{Tx_{y,z}} & {TX(y,z)} \ar[d]^{a_{y,z}} \\ {TX(y,z)} \ar[r]_-{a_{y,z}} & {X(y,z)}} \end{equation}
commutes. Given $x=(x_0,...,x_n)$ from $X$ with $x_0=y$ and $x_n=z$, and $w=(w_0,...,w_k)$ from $Tx^*X$ with $w_0=0$ and $w_k=n$, consider the composite map
\begin{equation}\label{eq:copr} \xymatrix{{Tw^*Tx^*X(0,k)} \ar[r]^-{T\overline{w}_{0,k}} & {T^2x^*X(0,n)} \ar[r]^-{T\overline{x}_{0,n}} & {T^2X(y,z)}} \end{equation}
and note that by path-likeness, and since the coproduct of coproducts is a coproduct, all such maps for $x$ and $w$ such that $x_0=y$ and $x_n=z$ form a coproduct cocone. Precomposing (\ref{eq:assoc1}) with (\ref{eq:copr}) gives the commutativity of
\begin{equation}\label{eq:assoc2} \xymatrix{{\Tbar\limits_i\Tbar\limits_jX(x_{ij-1},x_{ij})} \ar[r]^-{\overline{\mu}} \ar[d]_{\Tbar\limits_ia} & {\Tbar\limits_{ij}X(x_{ij-1},x_{ij})} \ar[d]^{a_x} \\ {\Tbar\limits_iX(x_{w_{i-1}},x_{w_i})} \ar[r]_-{a_w} & {X(y,z)}} \end{equation}
and conversely by the previous sentence if these squares commute for all $x$ and $w$, then one recovers the commutativity of (\ref{eq:assoc1}). This completes the description of the object part of $\ca G(V)^T \iso \Enrich {\overline{T}}$.
Let $(X,a)$ and $(X',a')$ be $T$-algebras and $F:X{\rightarrow}X'$ be a $V$-graph morphism. To say that $F$ is a $T$-algebra map is a condition on the maps $F_{y,z}:X(y,z){\rightarrow}X'(Fy,Fz)$ for all $y,z \in X_0$, and one uses path-likeness in the obvious way to see that this is equivalent to saying that the $F_{y,z}$ are the hom maps for a $\overline{T}$-functor.
\end{proof}
\noindent The proof of proposition(\ref{prop:pl-alg<->cat}) is not new: exactly the same argument was used in the second half of the proof of theorem(7.6) of \cite{EnHopI}, although in that case the setting was far less general. The real novelty is the generality of definition(\ref{def:path-like}) which is crucial for section(\ref{sec:lift-mult}).
\section{Monads from multitensors}\label{sec:Mult->Monad}
\subsection{Distributive multitensors}\label{ssec:DistMult}
The general way of obtaining a monad from a multitensor, which is the topic of this section, applies to multitensors which conform to
\begin{definition}\label{def:Dist-Mult}
Let $V$ be a category with small coproducts. Then a multitensor $(E,u,\sigma)$ is \emph{distributive} when the functor $E$ preserves coproducts in each variable. That is to say, for each $n \in \N$, the functor
\[ E_n : V^n \rightarrow V \]
obtained by observing $E$'s
effect on sequences of length $n$, preserves coproducts in each of its $n$ variables.
\end{definition}
\noindent The first step in associating a monad to a distributive multitensor is to identify the bicategory $\Dist$ which has the property that monads in $\Dist$ are exactly distributive multitensors in the sense of definition(\ref{def:Dist-Mult}). There is also a useful reformulation of the notion of monad on $\ca GV$ over $\Set$: as a monad in another 2-category $\Kl {\ca G_{\bullet}}$ where $\ca G_{\bullet}$ denotes the 2-comonad on $\CAT$ induced by the adjunction $(-)_{\bullet} \ladj \ca G$ of section(\ref{ssec:enriched-graphs}). Our monad-from-multitensor construction is then achieved by means of a pseudo functor (homomorphism of bicategories)
\[ \Gamma : \Dist \rightarrow \Kl {\ca G_{\bullet}} \]
which as a pseudo functor sends monads to monads. In fact for a distributive multitensor $E$, $\Gamma{E}$ is path-like and so $\Enrich E \iso \ca G(V)^{\Gamma{E}}$.
\subsection{The bicategory $\Dist$}\label{ssec:Dist}
The objects of $\Dist$ are categories with coproducts. A morphism $E:V{\rightarrow}W$ in $\Dist$ is a functor $E:MV{\rightarrow}W$ which preserves coproducts in each variable. A 2-cell between $E$ and $E'$ is simply a natural transformation between these functors $MV{\rightarrow}W$. Vertical composition of 2-cells is as for natural transformations.
The horizontal composite of $F:MV{\rightarrow}W$ and $E:MW{\rightarrow}X$, denoted $E \comp F$, is defined as a left kan extension
\[ \LaxSq {M^2V} {MV} {MW} {X} {MF} {\mu} {E \comp F} {E} {l_{E,F}} \]
of $EM(F)$ along $\mu$. Computing this explicitly gives the formula
\[ (E \comp F)(X_1,...,X_n) = \coprod_{n_1+...+n_k=n} \opE\limits_i\opF\limits_j X_{ij} \]
where $1{\leq}i{\leq}k$ and $1{\leq}j{\leq}n_i$ on the right hand side of this formula, and we denote by
\[ \xymatrix{{\opE\limits_{i}\opF\limits_{j}X_{ij}} \ar[r]^-{\opc\limits_{ij}} & {\opEoF\limits_{ij}X_{ij}}} \]
and also by $c_{(n_1,...,n_k)}$, the corresponding coproduct inclusion. The definition of horizontal composition is clearly functorial with respect to vertical composition of 2-cells.
\begin{proposition}\label{prop:Dist}
$\Dist$ is a bicategory and a monad in $\Dist$ is exactly a distributive multitensor.
\end{proposition}
\begin{proof}
It remains to identify the coherences and check the coherence axioms. The notation we have used here for the coproduct inclusions matches that used in section(3) of \cite{EnHopI}. The proof of the first part of proposition(3.3) of \cite{EnHopI} interpretted in our present more general setting, is the proof that $\Dist$ is a bicategory. The characterisation of distributive multitensors as monads in $\Dist$ is immediate from the definitions and our abstract definition of horizontal composition in terms of kan extensions.
\end{proof}
\noindent What in \cite{EnHopI} was called $\Dist(V)$ is here the hom $\Dist(V,V)$.
\subsection{Reformulating monads over $\Set$}\label{ssec:KlDefNMonad}
Because of the adjunction $(-)_{\bullet} \ladj \ca G$ and the definition of the comonad $\ca G_{\bullet}$, a normalised monad on $\ca GV$ is the same thing as a monad on $V$ in the Kleisli 2-category{\footnotemark{\footnotetext{In $\CAT$-enriched sense.}}} $\Kl {\ca G_{\bullet}}$ of the (2-)comonad $\ca G_{\bullet}$. The reason this is sometimes useful is that it expresses how our monads over $\Set$ only involve information at the level of homs. The validity of this reformulation is most plainly seen by realising that the factorisation of $\ca G$ as an identity on objects 2-functor followed by a 2-fully-faithful 2-functor, can be realised as
\[ \xymatrix{{\CAT} \ar[r]^-{R} & {\Kl {\ca G_{\bullet}}} \ar[r]^-{J} & {\CAT/\Set}} \]
where $R$ is the right adjoint part of the Kleisli adjunction for $\ca G_{\bullet}$. Thus applying $J$ sends a monad in $\Kl {\ca G_{\bullet}}$ on $V$ to a monad on $\ca GV$ over $\Set$, and the definition of $\ca G$ and the 2-fully-faithfulness of $J$ ensures that any such monad arises uniquely in this way.
\subsection{The pseudo-functor $\Gamma$}\label{ssec:Gamma}
It is the identity on objects. Given a distributive $E:MV{\rightarrow}W$, $\Gamma{E}$ is defined as a left kan extension
\[ \LaxSq {M\ca G_{\bullet}V} {\ca G_{\bullet}V} {MV} {W} {M\varepsilon} {*} {\Gamma{E}} {E} {\gamma_E} \]
of $EM(\varepsilon)$ along $*$. We shall now explain why this left kan extension exists in general, give a more explicit formula for $\Gamma{E}$ in corollary(\ref{cor:explicit-gamma}), and then with this in hand it will become clear why $\Gamma$ is a pseudo-functor. Of course one could just define $\Gamma{E}$ via the formula in corollary(\ref{cor:explicit-gamma}). We chose instead to give the above more abstract definition, because it will enable us to attain a more natural understanding of why $\Gamma$ produces \emph{path-like} monads from distributive multitensors in (\ref{ssec:DistMult->Monad}).
Conceptually, the reason why the left kan extension $\gamma_E$ involves only coproducts in $W$ is that $*$ is a local left adjoint. A functor $F:A{\rightarrow}B$ is a \emph{local left adjoint} when $\op F$ is a local right adjoint. This is equivalent to asking that for all $a \in A$ the induced functor $F^a:a/A{\rightarrow}Fa/B$ between coslices is a left adjoint. We shall explain in lemma(\ref{lem:spancomp-lra}) why and how pullback-composition of spans can be seen as a local right adjoint, and the statement that $*$ is a local left adjoint is just the dual of this because $*$ is defined as pushout-composition of cospans in $\ca GV$. In lemma(\ref{lem:lla-ple}) we give a formula for computing the left kan extension along a local left adjoint, and this will then be applied to give our promised more explicit description of $\Gamma$'s one-cell map.
\begin{lemma}\label{lem:spancomp-lra}
Let $A$ be a category with finite products and let $a$ be an object of $A$ such that $A/a$ also has finite products. Then the one-cell part
\[ M(A/a{\times}a) \rightarrow A/a{\times}a \]
of the monoidal structure on $A/a{\times}a$ given by span composition is a local right adjoint.
\end{lemma}
\begin{proof}
Note that for all $n \in \N$ the slices $A/(a^n)$, where $a^n$ is the $n$-fold cartesian product of $a$, have finite products, so the statement of the lemma makes sense and all the limits we mention in this proof exist.
In general a functor out of a coproduct of categories is a local right adjoint iff its composite with each coproduct inclusion is a local right adjoint. Thus it suffices to show that $n$-fold composition of spans
\[ (A/a{\times}a)^n \rightarrow A/a{\times}a \]
is a local right adjoint for all $n \in \N$. The case $n=0$ may be exhibited as a composite
\[ \xymatrix{1 \ar[r]^-{1_a} & {A/a} \ar[r]^-{\Delta_!} & {A/a{\times}a}} \]
of local right adjoints and thus is a local right adjoint. The case $n=1$ may be regarded as the identity. It suffices to verify the case $n=2$ because with this in hand an easy induction will give the general case. The pullback composite of spans as shown on the left
\[ \xygraph{*\xybox{\xygraph{!{0;(.65,0):}
d ([dl] b ([dl] {a}="a1", [dr] {a}="a2"), [dr] c [dr] {a}="a3")
"d":"b"_{p}:"a1"_{w} "d":"c"^{q}:"a3"^{z} "b":"a2"^{x} "c":"a2"_{y}
"d" [d(.3)l(.3)] [d(.1)r(.1)]:@{-}[d(.2)r(.2)]:@{-}[u(.2)r(.2)] }}
[r(4)]
*\xybox{\xygraph{!{0;(1.2,0):(0,.7)::}
d [r] {b{\times}c}="bc" [d] {a^4}="a4" [l] {a^3}="a3" [d] {a^2}="a2"
"d"(:@{.>}"a3"_{(wp,xp=yq,zq)}(:@{.>}"a2"_{(\pi_1,\pi_3)},:"a4"_-{a{\times}\Delta{\times}a}), :"bc"^-{(p,q)}:"a4"^{(w,x){\times}(y,z)}
"d" [d(.3)] [r(.1)]:@{-}[r(.2)]:@{-}[u(.2)]}}} \]
may be constructed as the dotted composite shown on the right in the previous display, and so binary composition of endospans of $a$ is encoded by the composite functor
\[ \xymatrix @C=4em {{(A/a{\times}a)^2} \ar[r]^-{\textnormal{prod}_{(a^2,a^2)}} & {A/(a^4)} \ar[r]^-{(a{\times}\Delta{\times}a)^*} & {A/(a^3)} \ar[r]^-{(\pi_1,\pi_3)_!} & {A/a{\times}a}} \]
where $\textnormal{prod}$ denotes the (right adjoint) functor $A{\times}A{\rightarrow}A$ which sends a pair to its cartesian product (and $\textnormal{prod}_{(a^2,a^2)}$ is the slice of $\textnormal{prod}$ over the pair $(a^2,a^2)$). The constituent functors of this last composite are clearly all local right adjoints.
\end{proof}
\noindent Recall \cite{WebGen} \cite{Fam2fun} that local right adjoints can be characterised in terms of generic factorisations. The dual characterisation is as follows. A functor $F:A{\rightarrow}B$ is a local left adjoint iff for all $b \in B$ the components of the comma category $F/b$ have terminal objects. A map $f:Fa{\rightarrow}b$ which is terminal in its component of $F/b$ is said to be \emph{cogeneric}, and by definition any $f$ can be factored as
\[ \xymatrix{{Fa} \ar[r]^-{Fh} & {Fc} \ar[r]^-{g} & b} \]
where $g$ is cogeneric. This factorisation is unique up to unique isomorphism and is called the \emph{cogeneric factorisation} of $f$.
\begin{lemma}\label{lem:lla-ple}
Let $F:A{\rightarrow}B$ be a local left adjoint. Suppose that $A$ has a set $C$ of connected components, each of which has an initial object, and that $B$ is locally small. For a functor $G:A{\rightarrow}D$ where $D$ has coproducts, the left kan extension $L:B{\rightarrow}D$ of $G$ along $F$ exists and is given by the formula
\[ Lb = \coprod\limits_{c{\in}C} \coprod\limits_{f:F0_c{\rightarrow}b} Ga_f \]
where $0_c$ denotes the initial object of the component $c \in C$, and
\[ \xymatrix{{F0_c} \ar[r]^-{Fh_f} & {Fa_f} \ar[r]^-{g_f} & b} \]
is a chosen cogeneric factorisation of $f:F0_c{\rightarrow}b$.
\end{lemma}
\begin{proof}
By the general formula for computing left kan extensions as colimits, it suffices to identify the given formula with the colimit of
\[ \xymatrix{{F/b} \ar[r]^-{p} & A \ar[r]^-{G} & D} \]
where $p$ is the obvious projection. This follows since the components of $F/b$ are indexed by pairs $(c,f)$, where $c \in C$ and $f:F0_c{\rightarrow}b$, and the component of $F/b$ corresponding to $(c,f)$ has terminal object given by $g_f$, which is mapped by $p$ to $a_f$.
\end{proof}
\noindent In the case of $*:M\ca GV_{\bullet}{\rightarrow}\ca GV_{\bullet}$ note the initial objects of the components of $M\ca GV_{\bullet}$ are of the form
\[ 0_n = (\underbrace{(0,(\emptyset),1),...,(0,(\emptyset),1)}_{n}) \]
and a map $*0_n{\rightarrow}(a,X,b)$ is a map
\[ (0,(\emptyset,...,\emptyset),n) \rightarrow (a,X,b) \]
which amounts to a sequence of elements of $X$ of length $(n+1)$ starting at $a$ and finishing at $b$. The reader will easily verify that
\[ \xymatrix{{(\emptyset,...,\emptyset)} \ar[r] & {x^*X} \ar[r]^-{\overline{x}} & {X}} \]
is a cogeneric factorisation of the map associated to $x=(x_0,...,x_n)$. So we have given a conceptual explanation of $x^*X$ and $\overline{x}$ which were used in (\ref{ssec:path-like}), as well as completed the proof of
\begin{corollary}\label{cor:explicit-gamma}
For $V$ and $W$ with coproducts and $E:MV{\rightarrow}W$, the defining left kan extension of $\Gamma{E}$ exists and we have the formula
\[ \Gamma{E}(a,X,b) = \coprod_{\begin{array}{c} {(x_0,...,x_n)} \\ {x_0=a, \, x_n=b}\end{array}} \opE\limits_i X(x_{i-1},x_i) \]
\end{corollary}
We shall identify $\Gamma$ with the composite $J\Gamma$, which amounts to identifying $\Gamma{E}:\ca GV_{\bullet}{\rightarrow}W$ with its mate $\ca GV{\rightarrow}\ca GW$ by the adjunction $(-)_{\bullet} \ladj \ca G$. Then for $X \in \ca GV$ and $a,b \in X_0$, we have for each $n \in \N$ and for each sequence $x=(x_0,...,x_n)$ such that $x_0=a$ and $x_n=b$ a coproduct inclusion
\[ c_x : \opE\limits_i X(x_{i-1},x_i) \rightarrow \Gamma{E}X(a,b) \]
by corollary(\ref{cor:explicit-gamma}). The components of the coherence natural transformations for $\Gamma$ will be identities on objects. From the definition of the unit $I:X{\rightarrow}X$ in $\Dist$, one has that for $a,b \in X_0$ the coproduct inclusion
\[ c_{(a,b)} : X(a,b) \rightarrow \Gamma{I}X(a,b) \]
is an isomorphism. Thus we have an isomorphism $\gamma_0:1{\rightarrow}\Gamma{I}$. Given distributive $F:MU{\rightarrow}V$ and $E:MV{\rightarrow}W$, $X \in \ca GU$ and $a,b \in X_0$, we define the hom maps of
\[ \gamma_{2,X} : \Gamma(E)\Gamma(F)X \rightarrow \Gamma(E \comp F)X \]
for $(a,b) \in X_0$, as the unique isomorphism such that for all $x_{ij} \in X_0$ where $1{\leq}i{\leq}k$ and $1{\leq}j{\leq}n_i$, $x_{11}=a$ and $x_{kn_k}=b$, the diagram
\[ \xygraph{!{(0,0);(0,1.5):(0,2)::}
{\opE\limits_i\opF\limits_jX(x_{ij-1},x_{ij})}="a" ([d]{(\opEoF\limits_{ij})X(x_{ij-1},x_{ij})}="c" [d][r(.5)]{\Gamma(E{\comp}F)X(a,b)}="e",
[r]{\opE\limits_i\Gamma(F)X(x_{i-1},x_i)}="b" [d]{\Gamma(E)\Gamma(F)X(a,b)}="d")
"a":"b"^-{\opE\limits_i\opc\limits_j}:"d"^-{c_{x_{i\bullet}}}:"e"^-{\gamma_2}
"a":"c"_-{\opc\limits_{ij}}:"e"_-{c_{x_{ij}}}} \]
in $W$ commutes, where $x_0=x_{11}$ and $x_i=x_{in_i}$ for $i{>}0$. We have selected the notation so as to match up with the development of \cite{EnHopI} section(4), and the proof of the first part of proposition(4.1) of \emph{loc. sit.} interpretted in the present context gives
\begin{proposition}\label{prop:gamma-psfunctor}
The coherences $(\gamma_0,\gamma_2)$ just defined make $\Gamma$ into a pseudo-functor.
\end{proposition}
\noindent Given a monad $T$ on $\ca GV$ over $\Set$, and a set $Z$, one obtains by restriction a monad $T_Z$ on the category $\ca GV_Z$ of $V$-graphs with fixed object set $Z$. Let us write $\Gamma^{\textnormal{old}}$ for the functor labelled as $\Gamma$ in \cite{EnHopI}. Then for a given distributive multitensor $E$, our present $\Gamma$ and $\Gamma^{\textnormal{old}}$ are related by the formula
\[ \Gamma^{\textnormal{old}}(E) = \Gamma(E)_1 \]
where the $1$ on the right hand side of this equation indicates a singleton. In other words we have just given the ``many-objects version'' of the theory presented in \cite{EnHopI} section(4).
\subsection{Algebras and enriched categories}\label{ssec:DistMult->Monad}
Having just established the machinery to convert distributive multitensors on $V$ to monads on $\ca GV$ over $\Set$, we shall now relate the enriched categories to the algebras. This involves two things: seeing that the normalised monads constructed from distributive multitensors are path-like, and understanding the relationship between $\Gamma$ and the construction of section(\ref{sec:Monad->Mult}) of multitensors from monads.
\begin{lemma}\label{lem:gamma-path-like}
Let $V$ and $W$ have coproducts and $E:MV{\rightarrow}W$ preserve coproducts in each variable. Then $\Gamma{E}:\ca GV{\rightarrow}\ca GW$ is path-like.
\end{lemma}
\begin{proof}
The condition that $T:\ca GV{\rightarrow}\ca GW$ is path-like can be expressed more 2-categorically.
\[ \xygraph{
*{\xybox{\xygraph{!{0;(1.5,0):(0,.67)::}
{*/(a,X,b)}="tl" [r] {1}="tr" [d] {\ca GV_{\bullet}}="mr" [l] {M\ca GV_{\bullet}}="ml" [d] {}="bl" [r] {W}="br"
"tl"(:"ml"_{p}:"mr"^-{*},:"tr":"mr"^{(a,X,b)}:"br"^{T})
"tl" [d(.5)r(.35)] :@{=>}[r(.3)]^{\lambda}}}}
[r(4)d(.1)]
*{\xybox{\xygraph{!{0;(1.5,0):(0,.67)::}
{*/(a,X,b)}="tl" [r] {1}="tr" [d] {\ca GV_{\bullet}}="mr" [l] {M\ca GV_{\bullet}}="ml" [d] {MV}="bl" [r] {W}="br"
"tl"(:"ml"_{p}(:"bl"_{M\varepsilon}:"br"_-{E},:"mr"^-{*}),:"tr":"mr"^{(a,X,b)}:"br"^{\Gamma{E}})
"tl" [d(.5)r(.35)] :@{=>}[r(.3)]^{\lambda} "ml" [d(.5)r(.35)] :@{=>}[r(.3)]^{\gamma_E}}}}} \]
Writing $\lambda$ for 2-cell part of the comma object, $T$ is path-like iff the 2-cell on the left exhibits $T(a,X,b)$ as a colimit. To see this recall from (\ref{ssec:Gamma}) that the set of components of $*/(a,X,b)$ may be regarded as the set of sequences of objects of $X_0$ starting at $a$ and finishing at $b$, that each of these components has a terminal object, and that $\overline{x}:x^*X{\rightarrow}X$ is terminal in the component corresponding to the sequence $x=(x_0,...,x_n)$. The situation for a given $E$ is depicted on the right in the previous display, and by definition this composite 2-cell is a colimit. Thus it suffices to show that the component of the 2-cell $\gamma_E$ at $p\overline{x}$ is invertible for all sequences $x=(x_0,...,x_n)$ from $X_0$. The component of $\gamma_E$ at a general
\[ ((c_1,Y_1,d_1),...,(c_n,Y_n,d_n)) \in M\ca G_{\bullet}V \]
is the coproduct inclusion
\[ \xymatrix{{\opE\limits_i Y_i(c_i,d_i)} \ar[rr]^-{c_w} && {\coprod\limits_{(z_0,...,z_m)} \opE\limits_i Y(z_{i-1},z_i)}} \]
corresponding to the sequence
\[ w = (c_1,d_1=c_2,...,d_{n-1}=c_n,d_n) \]
where $Y=(c_1,Y_1,d_1)*...*(c_n,Y_n,d_n)$. In the case of $p\overline{x} \in M\ca GV_{\bullet}$, $Y=x^*X$, and for summands corresponding to sequences $z$ different from $w$, we will have $Y(z_{i-1},z_i)=\emptyset$ for some $i$. By the distributivity of $E$ those summands will be $\emptyset$, whence $c_w$ will be invertible.
\end{proof}
\noindent Given a distributive multitensor $(E,u,\sigma)$ note that one can apply $\Gamma$ to it and then $\overline{(-)}$ to the result. One has
\[ \overline{\Gamma{E}}(Z_1,...,Z_n) = \Gamma{E}(Z_1,...,Z_n)(0,n) = \coprod_{a_0,...,a_m} \opE_i(Z_1,...,Z_n)(a_{i-1},a_i) \]
where the $a_i$ in the sum are elements of $\{0,...,n\}$ and $a_0=0$ and $a_m=n$. Unless the sequence $(a_0,...,a_n)$ is just an in-order list $(0,...,n)$ of the elements of $\{0,...,n\}$, at least one of the homs $(Z_1,...,Z_n)(a_{i-1},a_i)$ must be $\emptyset$ making that summand $\emptyset$ by the distributivity of $E$. Thus the coproduct inclusion
\[ c_{(0,...,n)} : \opE_iZ_i \rightarrow \overline{\Gamma{E}}(Z_1,...,Z_n) \]
is invertible. Moreover using the explicit description of the multitensor $\overline{\Gamma{E}}$ one may verify that this isomorphism is compatible with the units and substitutions, and so we have
\begin{lemma}\label{lem:gamma-bar}
If $(E,u,\sigma)$ is a distributive multitensor on a category $V$ with coproducts, then one has an isomorphism $E \iso \overline{\Gamma{E}}$ of multitensors.
\end{lemma}
\noindent Together with lemma(\ref{lem:gamma-path-like}) and proposition(\ref{prop:pl-alg<->cat}) this implies
\begin{corollary}\label{cor:gamma-alg-ecat}
If $(E,u,\sigma)$ is a distributive multitensor on a category $V$ with coproducts, then one has $\Enrich E \iso \ca G(V)^{\Gamma{E}}$ commuting with the forgetful functors into $\ca GV$.
\end{corollary}
\subsection{A conceptual view of path-likeness}\label{ssec:pl-conceptual}
We now describe the sense in which $\Gamma$ and $\overline{(-)}$ are adjoint. First let us note that equation(\ref{eq:Tbar}) defining the construction $\overline{(-)}$ in section(\ref{ssec:DefNMonad}) may be seen as providing functors
\[ \overline{(-)}_{V,W} : \CAT/\Set(\ca GV,\ca GW) \rightarrow \CAT(MV,W) \]
for all $V,W$ in $\CAT$. We have abused notation slightly by denoting by $\ca GV$ (resp. $\ca GW$) the category of $V$-enriched graphs together with its forgetful functor into $\Set$. In (\ref{ssec:DefNMonad}) we considered only the case $V=W$ and when $T:\ca GV{\rightarrow}\ca GV$ is part of a monad, but equation(\ref{eq:Tbar}) obviously makes sense in this more general context. In order to relate this with $\Gamma$ we make
\begin{definition}\label{def:ndist}
Let $V$ and $W$ have coproducts. A functor $T:{\ca GV}{\rightarrow}{\ca GW}$ over $\Set$ is \emph{distributive} when $\overline{T}:MV{\rightarrow}W$ preserves coproducts in each variable. We denote by $\NDist(V,W)$ the full subcategory of $\CAT/\Set(\ca GV,\ca GW)$ consisting of the distributive functors from $\ca GV$ to $\ca GW$.
\end{definition}
\begin{proposition}\label{prop:pl-adjoint-char}
Let $V$ and $W$ be categories with coproducts. Then we have an adjunction
\[ \xygraph{!{0;(3,0):}
{\Dist(V,W)}="d" [r] {\NDist(V,W)}="nd" "d":@<1ex>"nd"^-{\Gamma_{V,W}}|{}="t":@<1ex>"d"^-{\overline{(-)}_{V,W}}|{}="b"
"t":@{}"b"|-{\perp}} \]
whose unit is invertible. A distributive $T:{\ca GV}{\rightarrow}{\ca GW}$ is in the image of $\Gamma_{V,W}$ iff it is path-like.
\end{proposition}
\begin{proof}
By lemma(\ref{lem:gamma-bar}) applying $\Gamma$ does indeed produce a distributive functor, so $\Gamma_{V,W}$ is well-defined and one has an isomorphism of $\overline{(-)}_{V,W}\Gamma_{V,W}$ with the identity. For $X \in \ca GV$ and $a,b \in X_0$ one has
\[ \Gamma\overline{T}X(a,b) = \coprod_{a=x_0,...,x_n=b} Tx^*X(0,n) \rightarrow TX(a,b) \]
induced by the hom-maps of $T\overline{x}$, giving $\varepsilon_T:\Gamma\overline{T}{\rightarrow}T$ natural in $T$, and so by \cite{Web2top} lemma(2.6) to establish the adjunction it suffices to show that $\varepsilon$ is inverted by $\overline{(-)}$. To this end note that when $X=(Z_1,...,Z_m)$ for $Z_i \in V$, the above summands are non-initial iff the sequence $x_0,...,x_n$ is the sequence $(0,1,...,m)$, by the distributivity of $T$. The characterisation of path-likeness now follows too, since this condition on a given $T$ is by definition the same as the invertibility of $\varepsilon_T$.
\end{proof}
\noindent An immediate consequence of proposition(\ref{prop:pl-adjoint-char}) and proposition(\ref{prop:GammaE-basic})(\ref{GEb1}) below is the following result. A direct proof is also quite straight forward and is left as an exercise.
\begin{corollary}\label{cor:pl->copr-pres}
Let $V$ and $W$ be categories with coproducts. If $T:{\ca GV}{\rightarrow}{\ca GW}$ over $\Set$ is distributive and path-like, then it preserves coproducts.
\end{corollary}
\section{Categorical properties preserved by $\Gamma$}\label{sec:reexpress}
\subsection{}\label{ssec:intro-reexpress}
Let us now regard $\Gamma$ as a pseudo-functor
\[ \Gamma : \Dist \rightarrow \CAT/\Set. \]
That is to say, we take for granted the inclusion of $\Kl {\ca G_{\bullet}}$ in $\CAT/\Set$. In this section we shall give a systematic account of the categorical properties that $\Gamma$ preserves. The machinery we are developing gives an elegant inductive description of the monads $\ca T_{\leq{n}}$ for strict $n$-categories, provides explanations of some of their key properties, and gives a shorter account of the central result of \cite{EnHopI} on the equivalence between $n$-multitensors and $(n+1)$-operads.
\subsection{A review of some categorical notions}\label{ssec:lcpres}
Let $\lambda$ be a regular cardinal. An object $C$ in a category $V$ is \emph{connected} when the representable functor $V(C,-)$ preserves coproducts, and $C$ is \emph{$\lambda$-presentable} when $V(C,-)$ preserves $\lambda$-filtered colimits. The object $C$ is said to be \emph{small} when it is $\lambda$-presentable for some regular cardinal $\lambda$.
A category $V$ is \emph{extensive} when it has coproducts and for all families $(X_i:i \in I)$ of objects of $V$, the functor
\[ \begin{array}{lccr} {\coprod : \prod\limits_i (V/X_i) \rightarrow V/(\coprod\limits_i X_i)}
&&& {(f_i:Y_i{\rightarrow}X_i) \mapsto \coprod\limits_if_i:\coprod\limits_iY_i{\rightarrow}\coprod\limits_iX_i} \end{array} \]
is an equivalence of categories. A more elementary characterisation is that $V$ is extensive iff it has coproducts, pullbacks along coproduct coprojections and given a family of commutative squares
\[ \xymatrix{{X_i} \ar[r]^-{c_i} \ar[d]_{f_i} & X \ar[d]^{f} \\ {Y_i} \ar[r]_-{d_i} & Y} \]
where $i \in I$ such that the $d_i$ form a coproduct cocone, the $c_i$ form a coproduct cocone iff these squares are all pullbacks. It follows that coproducts are disjoint{\footnotemark{\footnotetext{Meaning that coproduct coprojections are mono and the pullback of different coprojections is initial.}}} and the initial object of $V$ is strict{\footnotemark{\footnotetext{Meaning that any map into the initial object is an isomorphism.}}}. Another sufficient condition for extensivity is provided by
\begin{lemma}\label{lem:easy-ext}
If a category $V$ has disjoint coproducts and a strict initial object, and every $X \in V$ is a coproduct of connected objects, then $V$ is extensive.
\end{lemma}
\noindent The proof is left as an easy exercise. Note this condition is not necessary: there are many extensive categories whose objects don't decompose into coproducts of connected objects, for example, take the topos of sheaves on a space which is not locally connected. A category is \emph{lextensive} when it is extensive and has finite limits. There are many examples of lextensive categories: Grothendieck toposes, the category of algebras of any higher operad and the category of topological spaces and continuous maps are all lextensive.
Denoting the terminal object of a lextensive category $V$ by $1$, the representable $V(1,-)$ has a left exact left adjoint
\[ (-) \cdot 1 : \Set \rightarrow V \]
which sends a set $Z$ to the copower $Z{\cdot}1$. This functor enables one to express coproduct decompositions of objects of $V$, \emph{internal to $V$} because to give a map
\[ f : X \rightarrow I \cdot 1 \]
in $V$ is the same thing as giving an $I$-indexed coproduct decomposition of $X$. The lextensive categories in which every object decomposes into a sum of connected objects are characterised by the following well-known result.
\begin{proposition}\label{prop:lext-decompose-charn}
For a lextensive category $V$ the following statements are equivalent:
\begin{enumerate}
\item Every $X \in V$ can be expressed as a coproduct of connected objects.\label{copcon}
\item The functor $(-) \cdot 1$ has a left adjoint.\label{pi0}
\end{enumerate}
\end{proposition}
\noindent The most common instance of this is when $V$ is a Grothendieck topos. The toposes $V$ satisfying the equivalent conditions of proposition(\ref{prop:lext-decompose-charn}) are said to be \emph{locally connected}. This terminology is reasonable since for a topological space $X$, one has that $X$ is locally connected as a space iff its associated topos of sheaves is locally connected in this sense.
A set $\ca D$ of objects of $V$ is a \emph{strong generator} when for all maps $f:X{\rightarrow}Y$, if
\[ V(D,f) : V(D,X) \rightarrow V(D,Y) \]
is bijective for all $D \in \ca D$ then $f$ is an isomorphism. A locally small category $V$ is \emph{locally $\lambda$-presentable} when it is cocomplete and has a strong generator consisting of small objects. Finally recall that a functor is \emph{accessible} when it preserves $\lambda$-filtered colimits for some regular cardinal $\lambda$.
The theory of locally presentable categories is one of the high points of classical category theory, and this notion admits many alternative characterisations \cite{GabUlm} \cite{MP} \cite{AR94}. For instance locally presentable categories are exactly those categories which are the $\Set$-valued models for a limit sketch. Grothendieck toposes are locally presentable because each covering sieve in a Grothendieck topology on a category $\C$ gives rise to a cone in $\op {\C}$, and a sheaf is exactly a functor $\op {\C}{\rightarrow}\Set$ which sends these cones to limit cones in $\Set$. That is to say a Grothendieck topos can be seen as the models of a limit sketch which one obtains in an obvious way from any site which presents it. Just as locally presentable categories generalise Grothendieck toposes, the following notion generalises locally connected Grothendieck toposes.
\begin{definition}\label{def:lcc}
A locally small category $V$ is \emph{locally c-presentable} when it is cocomplete and has a strong generator consisting of small connected objects.
\end{definition}
\noindent Just as locally presentable categories have many alternative characterisations we have the following result for locally c-presentable categories. Its proof is obtained by applying the general results of \cite{ABLR} in the case of the doctrine for $\lambda$-small connected categories, which is ``sound'' (see \cite{ABLR}),
and proposition(\ref{prop:lext-decompose-charn}).
\begin{theorem}\label{thm:conn-GabUlm}
For a locally small category $V$ the following statements are equivalent.
\begin{enumerate}
\item $V$ is locally c-presentable.\label{lc1}
\item $V$ is cocomplete and has a small dense subcategory consisting of small connected objects.\label{lc2}
\item $V$ is a full subcategory of a presheaf category for which the inclusion is accessible, coproduct preserving and has a left adjoint.\label{lc4}
\item $V$ is the category of models for a limit sketch whose distingished cones are connected.\label{lc5}
\item $V$ is locally presentable and every object of $V$ is a coproduct of connected objects.\label{lc6}
\item $V$ is locally presentable, extensive and the functor $(-){\cdot}1:\Set{\rightarrow}V$ has a left adjoint.\label{lc7}
\end{enumerate}
\end{theorem}
\begin{examples}\label{ex:lc-groth-toposes}
By theorem(\ref{thm:conn-GabUlm})(\ref{lc7}) a Grothendieck topos is locally connected in the usual sense iff its underlying category is locally c-presentable.
\end{examples}
\noindent Just as with locally presentable categories, locally c-presentable categories are closed under many basic categorical constructions. For instance from theorem(\ref{thm:conn-GabUlm})(\ref{lc6}), one sees immediately that the slices of a locally c-presentable category are locally c-presentable from the corresponding result for locally presentable categories. Another instance of this principle is the following result.
\begin{theorem}\label{thm:acc-monad}
If $V$ is locally c-presentable and $T$ is an accessible coproduct preserving monad on $V$, then $V^T$ is locally c-presentable.
\end{theorem}
\begin{proof}
First we recall that colimits in $V^T$ can be constructed explicitly using colimits in $V$ and the accessibility of $T$ (see for instance \cite{TTT} for a discussion of this). By definition we have a regular cardinal $\lambda$ such that $T$ preserves $\lambda$-filtered colimits and $V$ is locally $\lambda$-presentable. Defining $\Theta_0$ to be the full subcategory of $V$ consisting of the $\lambda$-presentable and connected objects, $(T,\Theta_0)$ is a monad with arities in the sense of \cite{Fam2fun}. One has a canonical isomorphism
\[ \xymatrix @C=3.5em {{V^T} \ar[r]^-{V^T(i,1)} \ar[d]_{U} \save \POS?="d" \restore & {\PSh {\Theta}_T} \ar[d]^{\res_j} \save \POS?="c" \restore \\ V \ar[r]_-{V(i_0,1)} & {\PSh {\Theta}_0} \POS "d";"c" **@{}; ?*{\iso}} \]
in the notation of \cite{Fam2fun}. Thus $V^T(i,1)$ is accessible since $\res_j$ creates colimits, and $T$ and $V(i_0,1)$ are accessible. By the nerve theorem of \cite{Fam2fun} $V^T(i,1)$ is also fully faithful, it has a left adjoint since $V^T$ is cocomplete given by left extending $i$ along the yoneda embedding, and so we have exhibited $V^T$ as conforming to theorem(\ref{thm:conn-GabUlm})(\ref{lc4}).
\end{proof}
\begin{examples}\label{ex:lc-noperad-algebras}
An $n$-operad for $0{\leq}n{\leq}\omega$ in the sense of \cite{Bat98}, gives a finitary coproduct preserving monad on the category $\PSh {\G}_{{\leq}n}$ of $n$-globular sets, and its algebras are just the algebras of the monad. Thus the category of algebras of any $n$-operad is locally c-presentable by theorem(\ref{thm:acc-monad}).
\end{examples}
\subsection{What $\ca G$ preserves}\label{ssec:Gamma-pres}
At the object level, to apply $\Gamma$ is to apply $\ca G$, so we shall now collect together many of the categorical properties that $\ca G$ preserves.
For $V$ with an initial object $\emptyset$, we saw in section(\ref{ssec:NMonad->MonMonad}) how to construct coproducts in $\ca GV$ explicitly. From this explicit construction, it is clear that the connected components of a $V$-graph $X$ may be described as follows. Objects $a$ and $b$ of $X$ are in the same connected component iff there exists a sequence $(x_0,...,x_n)$ of objects of $X$ such that for $1{\leq}i{\leq}n$ the hom $X(x_{i-1},x_i)$ is non-initial. Moreover $X$ is clearly the coproduct of its connected components, coproducts are disjoint and the initial object of $\ca GV$, whose $\Set$ of objects is empty, is strict. Thus by lemma(\ref{lem:easy-ext}) we obtain
\begin{proposition}\label{prop:GV-ext}
If $V$ has an initial object then $\ca GV$ is extensive and every object of $\ca GV$ is a coproduct of connected objects.
\end{proposition}
\noindent Given finite limits in $V$ it is straight forward to construct finite limits in $\ca GV$ directly. The terminal $V$-graph has one object and its only hom is the terminal object of $V$. Given maps $f:A{\rightarrow}B$ and $g:C{\rightarrow}B$ in $\ca GV$ their pullback $P$ can be constructed as follows. Objects are pairs $(a,c)$ where $a$ is an object of $A$ and $c$ is an object of $C$ such that $fa=gc$. The hom $P((a_1,c_1),(a_2,c_2))$ is obtained as the pullback of
\[ \xymatrix{{A(a_1,a_2)} \ar[r]^-{f} & {B(fa_1,fa_2)} & {C(c_1,c_2)} \ar[l]_-{g}} \]
in $V$. Thus one has
\begin{proposition}\label{prop:GV-lext}
If $V$ has finite limits then so does $\ca GV$. If in addition $V$ has an initial object, then $\ca GV$ is lextensive and every object of $V$ is a coproduct of connected objects.
\end{proposition}
\noindent As for cocompleteness one has the following result due to Betti, Carboni, Street and Walters.
\begin{proposition}\label{prop:GV-cocomp}\cite{BCSW-VarEnr}
If $V$ is cocomplete then so is $\ca GV$ and $(-)_0:\ca GV{\rightarrow}\Set$ is cocontinuous.
\end{proposition}
\noindent We now turn to local c-presentability. First we require a general lemma which produces a dense subcategory of $\ca GV$ from one in $V$ in a canonical way.
\begin{lemma}\label{lem:GV-dense}
Let $\ca D$ be a full subcategory of $V$ and suppose that $V$ has an initial object. Define an associated full subcategory $\ca D'$ of $\ca GV$ as follows:
\begin{itemize}
\item $0 \in \ca D'$.
\item $D \in \ca D \implies (D) \in \ca D'$.
\end{itemize}
If $\ca D$ is dense then so is $\ca D'$. For a regular cardinal $\lambda$, if the objects of $\ca D$ are $\lambda$-presentable then so are those of $\ca D'$.
\end{lemma}
\begin{proof}
Given functions
\[ f_{D'} : \ca GV(D',X) \rightarrow \ca GV(D',Y) \]
natural in $D' \in \ca D'$, we must show that there is a unique $f:X{\rightarrow}Y$ such that $f_{D'}=\ca GV(D',f)$. The object map of $f$ is forced to be $f_0$, and naturality with respect to the maps
\[ \xymatrix{0 \ar[r]^-{0} & (D) & 0 \ar[l]_-{1}} \]
ensures that the functions $f_{D'}$ amount to $f_0$ together with functions
\[ f_{D,a,b} : \ca GV_{\bullet}((0,(D),1),(a,X,b)) \rightarrow \ca GV_{\bullet}((0,(D),1),(f_0a,Y,f_0b)) \]
natural in $D \in \ca D$ for all $a,b \in X_0$. By the adjointness $L \ladj \varepsilon$ these maps are in turn in bijection with maps
\[ f'_{D,a,b} : V(D,X(a,b)) \rightarrow V(D,Y(f_0a,f_0b)) \]
natural in $D \in \ca D$ for all $a,b \in X_0$, and so by the density of $\ca D$ one has unique $f_{a,b}$ in $V$ such that$f'_{D,a,b}=V(D,f_{a,b})$. Thus $f_0$ and the $f_{a,b}$ together form the object and hom maps of the unique desired map $f$.
Since any colimit in $\ca GV$ is preserved by $(-)_0$ \cite{BCSW-VarEnr}, one can easily check directly that $0$ is $\lambda$-presentable for all $\lambda$. Let $D \in \ca D$ be $\lambda$-presentable. One has natural isomorphisms
\[ \begin{array}{c} {\ca GV((D),X) \iso \coprod\limits_{a,b{\in}X_0} \ca G_{\bullet}V((0,(D),1),(a,X,b)) \iso \coprod\limits_{a,b{\in}X_0} V(D,X(a,b))} \end{array} \]
exhibiting $\ca GV((D),-)$ as a coproduct of functors that preserve $\lambda$-filtered colimits, and thus is itself $\lambda$-filtered colimit preserving.
\end{proof}
\begin{corollary}\label{cor:GV-lc}
If $V$ is locally presentable then $\ca GV$ is locally c-presentable.
\end{corollary}
\begin{proof}
Immediate from theorem(\ref{thm:conn-GabUlm})(\ref{lc2}), lemma(\ref{lem:GV-dense}) and proposition(\ref{prop:GV-cocomp}).
\end{proof}
\noindent In \cite{KL-NiceVCat} Kelly and Lack proved that if $V$ is locally presentable then so is $\ca GV$ by an argument almost identical to that given here. The only difference is that in their version of lemma(\ref{lem:GV-dense}), their $\ca D'$ differs from ours only in that they use $0+0$ where we use $0$, and they instead prove that $\ca D'$ is a strong generator given that $\ca D$ is. We have given the above proof because the present form of lemma(\ref{lem:GV-dense}) is more useful to us in section(\ref{ssec:Gamma-pres12-lra}). Next we shall see that $\ca G$ preserves toposes. First a lemma of independent interest.
\begin{lemma}\label{lem:abstract-wreath}
Let $\C$ be a category and $\ca E$ be a lextensive category. Consider the category $\C_+$ constructed from $\C$ as follows. There is an injective on objects fully faithful functor
\[ \begin{array}{lcr} {i_{\C} : \C \rightarrow \C_+} && {C \mapsto C_+} \end{array} \]
and $\C_+$ has an additional object $0$ not in the image of $i_{\C}$. Moreover for each $C \in \C$ one has maps
\[ \begin{array}{lcr} {\sigma_C : 0 \rightarrow C_+} && {\tau_C : 0 \rightarrow C_+} \end{array} \]
and for all $f:C{\rightarrow}D$ one has the equations $f_+\sigma_C=\sigma_D$ and $f_+\tau_C=\tau_D$. Then $\ca G[\op \C,\ca E]$ is equivalent to the full subcategory of $[\C^{\textnormal{op}}_+,\ca E]$ consisting of those $X$ such that $X_0$ is a copower of $1$, the terminal object of $\ca E$.{\footnotemark{\footnotetext{Such $X_0$ in $\ca E$ are said to be \emph{discrete}.}}}
\end{lemma}
\begin{proof}
Let us write $\ca F$ for the full subcategory of $[\C^{\textnormal{op}}_+,\ca E]$ described in the statement of the lemma. We shall describe the functors
\[ \begin{array}{lcr} {(-)_+ : \ca G[\op {\C},\ca E] \rightarrow \ca F} && {(-)^{-} : \ca F \rightarrow \ca G[\op {\C},\ca E]} \end{array} \]
which provide the desired equivalence directly. Given $X \in \ca G[\op {\C},\ca E]$ define $X_+0={X_0}{\cdot}1$ and
\[ \begin{array}{c} {X_+C_+ = \coprod\limits_{a,b \in X_0} X(a,b)(C)} \end{array}. \]
In the obvious way this definition is functorial in $X$ and $C$. Conversely given $Y \in \ca F$ choose a set $Y^{-}_0$ such that $Y0={Y^{-}_0}{\cdot}1$. Such a set is determined uniquely up to isomorphism. Then define the homs of $Y^{-}$ via the pullbacks
\[ \PbSq {Y^{-}(a,b)(C)} {YC_+} {Y0{\times}Y0} {1} {} {} {(Y\sigma,Y\tau)} {(a,b)} \]
in $\ca E$ for all $a,b \in Y^{-}_0$. Note that since $\ca E$ is lextensive and hence distributive, $Y0{\times}Y0$ is itself the coproduct of copies of $1$ indexed by such pairs $(a,b)$. The natural isomorphisms
\[ \begin{array}{lcr} {X^{-}_+(a,b)(C) \iso X(a,b)(C)} && {Y^{-}_+C_+ \iso YC_+} \end{array} \]
come from extensivity.
\end{proof}
\begin{corollary}\label{cor:GV-topos}
\begin{enumerate}
\item If $V$ is a presheaf topos then so is $\ca GV$.\label{G6}
\item If $V$ is a Grothendieck topos then $\ca GV$ is a locally connected Grothendieck topos.\label{G7}
\end{enumerate}
\end{corollary}
\begin{proof}
Applying lemma(\ref{lem:abstract-wreath}) in the case where $\C$ is small and $\ca E=\Set$ one obtains the formula
\[ \ca G \PSh {\C} \catequiv \PSh {\C_+} \]
and thus (\ref{G6}). Since a Grothendieck topos is a left exact localisation of a presheaf category, the 2-functoriality of $\ca G$ together with (\ref{G6}), corollary(\ref{cor:GV-lc}) and example(\ref{ex:lc-groth-toposes}), implies that to establish (\ref{G7}) it suffices to show that $\ca G$ preserves left exact functors between categories with finite limits. This follows immediately from the explicit description of finite limits in $\ca GV$ given in the proof of proposition(\ref{prop:GV-lext}).
\end{proof}
\begin{remark}
The construction $(-)_+$ in the previous proof is easily discovered by thinking about why, as pointed out in section(\ref{ssec:enriched-graphs}), applying $\ca G$ successively to the empty category does one produce the categories of $n$-globular sets for $n \in \N$. The construction $(-)_+$ is just the general construction on categories, which when applied successively to the empty category produces the categories $\G_{{\leq}n}$ for $n \in \N$, presheaves on which are by definition $n$-globular sets.
\end{remark}
\subsection{$\Gamma$'s one and two-cell map}\label{ssec:Gamma-pres12-cart}
First we note that by the explicit description of coproducts, the terminal object and pullbacks of enriched graphs, and the formula of corollary(\ref{cor:explicit-gamma}), one has
\begin{proposition}\label{prop:GammaE-basic}
Let $V$ and $W$ have coproducts and $E:MV{\rightarrow}W$ be distributive.
\begin{enumerate}
\item $\Gamma{E}$ preserves coproducts.\label{GEb1}
\item If $E$ preserves the terminal object then so does $\Gamma{E}$.
\item If $E$ preserves pullbacks then so does $\Gamma{E}$.
\end{enumerate}
\end{proposition}
\noindent The precise conditions under which $\Gamma$ preserves and reflects cartesian natural transformations are identified by proposition(\ref{prop:Gamma-cart}). First we require a lemma which generalises lemma(7.4) of \cite{EnHopI}, whose proof follows easily from the explicit description of pullbacks in $\ca GV$ discussed in (\ref{ssec:Gamma-pres}).
\begin{lemma}\label{lem:pb-hom}
Suppose that $V$ has pullbacks. Given a commutative square (I)
\[ \TwoDiagRel {\xymatrix{W \ar[d]_{f} \save \POS?="d" \restore \ar[r]^-{h} & X \ar[d]^{g} \save \POS?="c" \restore \\ Y \ar[r]_-{k} & Z \POS "d";"c" **@{}; ?*{\textnormal{I}}}} {}
{\xymatrix{{W(a,b)} \ar[d]_{f_{a,b}} \save \POS?="d" \restore \ar[r]^-{h_{a,b}} & {X(ha,hb)} \ar[d]^{g_{ha,hb}} \save \POS?="c" \restore \\ {Y(a,b)} \ar[r]_-{k_{a,b}} & {Z(ha,hb)} \POS "d";"c" **@{}; ?*{\textnormal{II}}}} \]
in $\ca GV$ such that $f_0$ and $g_0$ are identities, one has for each $a,b \in W_0$ commuting squares (II) as in the previous display. The square (I) is a pullback iff for all $a,b \in W_0$, the square (II) is a pullback in $V$.
\end{lemma}
\begin{proposition}\label{prop:Gamma-cart}
Let $V$ have coproducts, $W$ be extensive, and $E$ and $F:MV{\rightarrow}W$ be distributive.
Then $\phi:E{\rightarrow}F$ is cartesian iff $\Gamma{\phi}$ is cartesian.
\end{proposition}
\begin{proof}
Let $f:X{\rightarrow}Y$ be in $\ca GV$, $a,b \in X_0$ and $x_0,...,x_n$ be a sequence of objects of $X$ such that $x_0=a$ and $x_n=b$. For each such $f,a,b$ it suffices by lemma(\ref{lem:pb-hom}), to show that the square (II) in the commutative diagram
\[ \xygraph{!{0;(2,0):(0,.5)::}
{\opE\limits_iX(x_{},x_i)}="i11" [r] {\Gamma{E}X(a,b)}="i12" [d] {\Gamma{E}Y(fa,fb)}="i22" [l] {\opE\limits_iY(fx_{},fx_i)}="i21" "i11" [ul] {\Gamma{F}X(a,b)}="o11" "i12" [ur] {\opF\limits_iX(x_{},x_i)}="o12" "i21" [dl] {\opF\limits_iY(fx_{},fx_i)}="o21" "i22" [dr] {\Gamma{F}Y(fa,fb)}="o22"
"i11":"i12"^-{c_{x_i}}:"i22"^{\Gamma{E}(f)_{a,b}}|{}="l3" "i11":"i21"_{\opE\limits_if}|{}="l2":"i22"_-{c_{x_i}}
"o11":"o12"^-{c_{x_i}}:"o22"^{\Gamma{F}(f)_{a,b}}|{}="l4" "o11":"o21"_{\opF\limits_if}|{}="l1":"o22"_-{c_{x_i}}
"i11":"o11"^{\phi} "i21":"o21"^{\phi} "i12":"o12"^{\Gamma\phi} "i22":"o22"^{\Gamma\phi}
"l1":@{}"l2"|(.4)*{(III)}:@{}"l3"|*{(I)}:@{}"l4"|(.7)*{(II)}} \]
is a pullback. The square (I) and the largest square are pullbacks since $W$ is extensive and the $c$-maps are coproduct inclusions, and (III) is a pullback since $\phi$ is cartesian. Thus for all such $x_0,...,x_n$ the composite of (I) and (II) is a pullback. The result follows since the $c_{x_i}$ form coproduct cocones and $W$ is extensive. Conversely suppose that we have $f_i:X_i{\rightarrow}Y_i$ in $V$ for $1{\leq}i{\leq}n$. Then by the isomorphisms $E{\iso}\overline{\Gamma(E)}$ and $F{\iso}\overline{\Gamma(F)}$, and their naturality with respect to $\phi$ one may identify the naturality square on the left
\[ \TwoDiagRel {\xymatrix{{\opE\limits_iX_i} \ar[r]^-{\phi_{X_i}} \ar[d]_{\opE\limits_if_i} & {\opF\limits_i} \ar[d]^{\opF\limits_if_i} \\ {\opE\limits_iY_i} \ar[r]_-{\phi_{Y_i}} & {\opF\limits_iY_i}}} {} {\xymatrix{{\Gamma(E)(X_1,...,X_n)(0,n)} \ar[r]^-{\Gamma(\phi)_{0,n}} \ar[d]_{\Gamma(E)(f_1,...,f_n)_{0,n}} & {\Gamma(F)(X_1,...,X_n)(0,n)} \ar[d]_{\Gamma(F)(f_1,...,f_n)_{0,n}} \\ {\Gamma(E)(Y_1,...,Y_n)(0,n)} \ar[r]_-{\Gamma(\phi)_{0,n}} & {\Gamma(F)(Y_1,...,Y_n)(0,n)}}} \]
with that on the right in the previous display, which is cartesian by lemma(\ref{lem:pb-hom}) and since $\Gamma(\phi)$ is cartesian, and so $\phi$ is indeed cartesian.
\end{proof}
\noindent $\Gamma$'s
compatibility with the bicategory structure of $\Dist$ is expressed in
\begin{proposition}\label{prop:Gamma-cart2}
\begin{enumerate}
\item Let $E:MV{\rightarrow}W$ and $F:MU{\rightarrow}V$ be distributive and $W$ be extensive. If $E$ and $F$ preserve pullbacks then so does $E \comp F$.\label{cart2-1}
\item Let $E,E':MV{\rightarrow}W$ and $F,F':MU{\rightarrow}V$ be distributive and pullback preserving, and $W$ be extensive. If $\phi:E{\rightarrow}E'$ and $\psi:F{\rightarrow}F'$ are cartesian then so is $\phi \comp \psi$.\label{cart2-2}
\end{enumerate}
\end{proposition}
\begin{proof}
(\ref{cart2-1}). Suppose that for $1{\leq}i{\leq}n$ we have pullback squares
\[ \PbSq {A_i} {B_i} {C_i} {D_i} {h_i} {f_i} {k_i} {g_i} \]
in $U$. Then for each partition $n_1+...+n_k=n$ of $n$ we have a commutative diagram
\[ \xygraph{!{0;(2,0):(0,.5)::}
{(\opEoF\limits_i)A_i}="i11" [r] {(\opEoF\limits_i)B_i}="i12" [d] {(\opEoF\limits_i)D_i}="i22" [l] {(\opEoF\limits_i)C_i}="i21" "i11" [ul] {\opE\limits_i\opF\limits_jA_{ij}}="o11" "i12" [ur] {\opE\limits_i\opF\limits_jB_{ij}}="o12" "i21" [dl] {\opE\limits_i\opF\limits_jC_{ij}}="o21" "i22" [dr] {\opE\limits_i\opF\limits_jD_{ij}}="o22"
"i11":"i12"^-{(\opEoF\limits_i)f_i}:"i22"^{(\opEoF\limits_i)k_i}|{}="l3" "i11":"i21"_{(\opEoF\limits_i)h_i}|{}="l2":"i22"_-{(\opEoF\limits_i)g_i}
"o11":"o12"^-{(\opE\limits_i\opF\limits_j)f_{ij}}:"o22"^{(\opE\limits_i\opF\limits_j)k_{ij}}|{}="l4" "o11":"o21"_{(\opE\limits_i\opF\limits_j)h_{ij}}|{}="l1":"o22"_-{(\opE\limits_i\opF\limits_j)g_{ij}}
"o11":"i11"^{c_{n_i}} "o21":"i21"^{c_{n_i}} "o12":"i12"^{c_{n_i}} "o22":"i22"^{c_{n_i}}
"l1":@{}"l2"|(.35)*{(I)}:@{}"l3"|*{(II)}:@{}"l4"|(.7)*{(III)}} \]
in $W$ in which the $c$-maps are coproduct inclusions, the coproducts being indexed over the set of all such partitions. For the outer square $1{\leq}i{\leq}k$ and $1{\leq}j{\leq}n_i$. We must show that (II) is a pullback. The squares (I) and (III) are pullbacks because $W$ is extensive. The large square is a pullback because $E$ and $F$ preserve pullbacks. Thus the composite of (I) and (II) is a pullback, and so the result follows by the extensivity of $W$.
(\ref{cart2-2}). Given $f_i:A_i{\rightarrow}B_i$ for $1{\leq}i{\leq}n$ and $n_1+...+n_k=n$ we have a commutative diagram
\[ \xygraph{!{0;(2,0):(0,.5)::}
{(\opEoF\limits_i)A_i}="i11" [r] {(\opEoF\limits_i)B_i}="i12" [d] {(\opEoFpr\limits_i)B_i}="i22" [l] {(\opEoFpr\limits_i)A_i}="i21" "i11" [ul] {\opE\limits_i\opF\limits_jA_{ij}}="o11" "i12" [ur] {\opE\limits_i\opF\limits_jB_{ij}}="o12" "i21" [dl] {\opEpr\limits_i\opFpr\limits_jA_{ij}}="o21" "i22" [dr] {\opEpr\limits_i\opFpr\limits_jB_{ij}}="o22"
"i11":"i12"^-{(\opEoF\limits_i)f_i}:"i22"^{\psi{\comp}\phi} "i11":"i21"_{\psi{\comp}\phi}:"i22"_-{(\opEoFpr\limits_i)f_i}
"o11":"o12"^-{(\opE\limits_i\opF\limits_j)f_{ij}}:"o22"^{\psi\phi} "o11":"o21"_{\psi\phi}:"o22"_-{(\opEpr\limits_i\opFpr\limits_j)f_{ij}}
"o11":"i11"^{c_{n_i}} "o21":"i21"^{c_{n_i}} "o12":"i12"^{c_{n_i}} "o22":"i22"^{c_{n_i}}} \]
in $W$, to which we apply a similar argument as in (\ref{cart2-1}) to demonstrate that the inner square is cartesian.
\end{proof}
\subsection{The general basic correspondence between operads and multitensors}\label{ssec:general-op-mult}
Let $V$ be lextensive. Then by proposition(\ref{prop:Gamma-cart2}) the monoidal structure of $\Dist(V,V)$ restricts to pullback preserving $MV{\rightarrow}V$ and cartesian transformations between them. A multitensor $(E,u,\sigma)$ on $V$ is \emph{cartesian} when $E$ preserves pullbacks and $u$ and $\sigma$ are cartesian. By slicing over $E$ one obtains a monoidal category $\Coll E$, whose objects are cartesian transformations $\alpha:A{\rightarrow}E$. To give $(A,\alpha)$ a monoid structure is to give $A$ the structure of a cartesian multitensor such that $\alpha$ is a cartesian multitensor morphism. Such an $(A,\alpha)$ is called an \emph{$E$-multitensor}. The category of $E$-multitensors is denoted $\Mult E$.
\begin{example}\label{ex:non-sig-operads}
Let us denote by $\prod$ the multitensor on $\Set$ given by finite products. By the lextensivity of $\Set$, $\prod$ is an lra multitensor and thus cartesian. A $\prod$-multitensor is the same thing as a non-symmetric operad in $\Set$. For given a cartesian multitensor map $\varepsilon:E{\rightarrow}\prod$, one obtains the underlying sequence $(E_n \, : \, n \in \N)$ of sets of the corresponding operad as $E_n = \opE\limits_{1{\leq}i{\leq}n} 1$. One uses the cartesianness of the naturality squares corresponding to the maps $(X_1,...,X_n){\rightarrow}(1,...,1)$ to recover $E$ from the $E_n$. Similarly the multitensor structure of $E$ corresponds to the unit and substitution maps making the $E_n$ into an operad.
\end{example}
Given a cartesian normalised monad $T$ on $\ca GV$, one obtains a monoidal category $\NColl T$, whose objects are cartesian transformations $\alpha:A{\rightarrow}T$ over $\Set$. Explicitly, to say that a general collection, which is a cartesian transformation $\alpha:A{\rightarrow}T$, is over $\Set$, is to say that the components of $\alpha$ are identities on objects maps of $V$-graphs. The tensor product of $\NColl T$ is obtained via composition and the monad structure of $T$, and a monoid structure on $(A,\alpha)$ is a cartesian monad structure on $A$ such that $\alpha$ is a cartesian monad morphism. Such an $(A,\alpha)$ is called a \emph{$T$-operad over $\Set$}{\footnotemark{\footnotetext{In \cite{Bat98} and \cite{EnHopI} these were called \emph{normalised} $T$-operads.}}}. The category of $T$-operads over $\Set$ is denoted as $\NOp T$.
\noindent By the results of this subsection $\Gamma$ induces a strong monoidal functor
\[ \Gamma_E : \Coll E \rightarrow \NColl {\Gamma{E}} \]
and we shall now see that this functor is a monoidal equivalence. Applying this equivalence to the monoids in the respective monoidal categories gives the promised general equivalence between multitensors and operads over $\Set$ in corollary(\ref{cor:mult-nop-equiv}) below.
\begin{lemma}\label{lem:transfer-dpl}
Let $V$ be a lextensive category and $T$ be a cartesian monad on $\ca GV$ over $\Set$. Let $\alpha:A{\rightarrow}T$ be a collection over $\Set$.
\begin{enumerate}
\item If $T$ is distributive then so is $A$.\label{tdpl1}
\item If $T$ is path-like then so is $A$.\label{tdpl2}
\end{enumerate}
\end{lemma}
\begin{proof}
(\ref{tdpl1}): given an $n$-tuple $(X_1,...,X_n)$ of objects of $V$ and a coproduct cocone
\[ (c_j : X_{ij} \rightarrow X_i \,\, : \,\, j \in J) \]
where $1{\leq}i{\leq}n$, we must show that the hom-maps
\[ A(X_1,...,c_j,...,X_n)_{0,n} : A(X_1,...,X_{ij},...,X_n)(0,n) \rightarrow A(X_1,...,X_i,...,X_n)(0,n) \]
form a coproduct cocone. For $j \in J$ we have a pullback square
\[ \xygraph{!{0;(6,0):(0,.167)::}
{A(X_1,...,X_{ij},...,X_n)(0,n)}="tl" [r] {A(X_1,...,X_i,...,X_n)(0,n)}="tr" [d] {T(X_1,...,X_i,...,X_n)(0,n)}="br" [l] {T(X_1,...,X_{ij},...,X_n)(0,n)}="bl" "tl"(:"tr"^-{A(X_1,...,c_j,...,X_n)_{0,n}}:"br"^{\alpha},:"bl"_{\alpha}:"br"_-{T(X_1,...,c_j,...,X_n)_{0,n}})
"tl" [r(.1)d(.4)] {\xybox{\xygraph{!{0;(0.2,0):} (:@{-}[u], :@{-}[l])}}}} \]
and by the distributivity of $T$ and lemma(\ref{lem:pb-hom}), the $T(X_1,...,c_j,...,X_n)_{0,n}$ form a coproduct cocone, and thus so do the $A(X_1,...,c_j,...,X_n)_{0,n}$ by the extensivity of $V$.\\
(\ref{tdpl2}): given $X \in \ca GV$, $a,b \in X_0$ and a sequence $(x_0,...,x_n)$ of objects of $X$ such that $x_0=a$ and $x_n=b$, we have the map
\[ A\overline{x}_{0,n} : Ax^*X(0,n) \rightarrow AX(a,b) \]
and we must show that these maps, where the $x_i$ range over all sequences from $a$ to $b$, form a coproduct cocone. By the path-likeness of $T$ we know that the maps
\[ T\overline{x}_{0,n} : Tx^*X(0,n) \rightarrow TX(a,b) \]
form a coproduct cocone, so we can use the cartesianness of $\alpha$, lemma(\ref{lem:pb-hom}) and the extensivity of $V$ to conclude as in (\ref{tdpl1}).
\end{proof}
\begin{proposition}\label{prop:coll-equiv}
Let $V$ be lextensive and $(E,\iota,\sigma)$ be a distributive cartesian multitensor on $V$. Then $\Gamma_E$ is a monoidal equivalence $\Coll E \catequiv \NColl {\Gamma{E}}$.
\end{proposition}
\begin{proof}
The functor $\Gamma_E$ is the result of applying the functor $\Gamma_{V,V}$ of proposition(\ref{prop:pl-adjoint-char}) over $E$. Thus by proposition(\ref{prop:pl-adjoint-char}), proposition(\ref{prop:Gamma-cart}) and lemma(\ref{lem:transfer-dpl}), $\Gamma_E$ is an equivalence.
\end{proof}
\begin{corollary}\label{cor:mult-nop-equiv}
Let $V$ be lextensive and $(E,\iota,\sigma)$ be a distributive cartesian multitensor on $V$. Then applying $\Gamma_E$ gives $\Mult E \catequiv \NOp {\Gamma E}$.
\end{corollary}
\subsection{$\Gamma$ and local right adjoint monads}\label{ssec:Gamma-pres12-lra}
Local right adjoint monads, especially defined on presheaf categories, are fundamental to higher category theory. Indeed a deeper understanding of such monads is the key to understanding the relationship between the operadic and homotopical approaches to the subject \cite{Fam2fun}. We will now understand the conditions under which $\Gamma$ preserves local right adjoints. First we require two lemmas.
\begin{lemma}\label{lem:partial-adjoint}
Let $R:V{\rightarrow}W$ be a functor, $V$ be cocomplete, $U$ be a small dense full subcategory of $W$, and $L:U{\rightarrow}V$ be a partial left adjoint to $R$, that is to say, one has isomorphisms $W(S,RX) \iso V(LS,X)$ natural in $S \in U$ and $X \in V$. Defining $\overline{L}:W{\rightarrow}V$ as the left kan extension of $L$ along the inclusion $I:U{\rightarrow}W$, one has $\overline{L} \ladj R$.
\end{lemma}
\begin{proof}
Denoting by $p:I/Y{\rightarrow}U$ the canonical forgetful functor for $Y \in W$ and recalling that $\overline{L}Y = \colim(Lp)$, one obtains the desired natural isomorphism as follows
\[ \begin{array}{rcccl} {V(\overline{L}Y,X)} & {\iso} & {[I/Y,V](Lp,\textnormal{const}(X))} & {\iso} & {\lim_{f{\in}I/Y} V(L(\textnormal{dom}(f)),X)} \\ & {\iso} & {\lim_f W(\textnormal{dom}(f),RX)} & {\iso} & {\ca B(Y,RX)} \end{array} \]
for all $X \in V$.
\end{proof}
\begin{lemma}\label{lem:lra-dense}
Let $T:V{\rightarrow}W$ be a functor, $V$ be cocomplete and $W$ have a small dense subcategory $U$. Then $T$ is a local right adjoint iff every $f:S{\rightarrow}TX$ with $A \in U$ admits a generic factorisation. If in addition $V$ has a terminal object denoted $1$, then generic factorisations in the case $X=1$ suffice.
\end{lemma}
\begin{proof}
For the first statement ($\implies$) is true by definition so it suffices to prove the converse. The given generic factorisations provide a partial left adjoint $L:I/TX{\rightarrow}V$ to $T_X:V/X{\rightarrow}W/TX$ where $I$ is the inclusion of $U$. Now $I/TX$ is a small dense subcategory of $W/TX$, and so by the previous lemma $L$ extends to a genuine left adjoint to $T_X$. In the case where $V$ has $1$ one requires only generic factorisations in the case $X=1$ by the results of \cite{Fam2fun} section(2).
\end{proof}
\noindent The analogous result for presheaf categories, with the representables forming the chosen small dense subcategory, was discussed in \cite{Fam2fun} section(2).
\begin{proposition}\label{prop:GammaE-lra}
Let $V$ and $W$ be locally c-presentable and $E:MV{\rightarrow}W$ be distributive. If $E:MV{\rightarrow}W$ is a local right adjoint then so is $\Gamma{E}$.
\end{proposition}
\begin{proof}
Let $\ca D$ be a small dense subcategory of $W$ consisting of small connected objects. By lemma(\ref{lem:lra-dense}) and lemma(\ref{lem:GV-dense}) it suffices to exhibit generic factorisations of maps
\[ f:S \rightarrow \Gamma{E}1 \]
where $S$ is either $0$ or $(D)$ for some $D \in \ca D$. In the case where $S$ is $0$ the first arrow in the composite
\[ \xymatrix{0 \ar[r] & {\Gamma{E}0} \ar[r]^-{\Gamma{E}t} & {\Gamma{E}1}} \]
is generic because $0$ is the initial $W$-graph with one object (and $t$ here is the unique map). In the case where $S=(D)$, to give $f$ is to give a map $f':D{\rightarrow}E_n1$ in $V$, by corollary(\ref{cor:explicit-gamma}) since $D$ is connected. Since $E$ is a local right adjoint, $E_n$ is too and so one can generically factor $f'$ to obtain
\[ \xymatrix{D \ar[r]^-{g'_f} & {\opE\limits_iZ_i} \ar[r]^-{\opE\limits_it} & {E_n1}} \]
from which we obtain the generic factorisation
\[ \xymatrix{{(D)} \ar[r]^-{g_f} & {\Gamma{E}Z} \ar[r]^-{\Gamma{E}t} & {\Gamma{E}1}} \]
where $Z=(Z_1,...,Z_n)$, the object map of $g_f$ is given by $0 \mapsto 0$ and $1 \mapsto n$, and the hom map of $g_f$ is $g'_f$ composed with the coproduct inclusion.
\end{proof}
\subsection{$\Gamma$ and accessible functors}\label{ssec:Gamma-pres12-access}
First note that while it is a very different thing for $E:MV{\rightarrow}W$ to preserve coproducts compared with preserving coproducts in each variable, the situation is simpler for $\lambda$-filtered colimits, where $\lambda$ is any regular cardinal. Note that
\[ F : V_1 \times ... \times V_n \rightarrow W \]
preserves $\lambda$-filtered colimits in each variable iff $F$ preserves $\lambda$-filtered colimits. For given a connected category $C$, the colimit of a functor $C{\rightarrow}A$ constant at say $X$ is of course $X$, and since $\lambda$-filtered colimits are connected, one can prove $(\impliedby)$ by keeping all but the variable of interest constant. For the converse it is sufficient to prove that $F$ preserves colimits of chains of length less than $\lambda$, and this follows by a straight forward transfinite induction. Since $MV$ is a sum of $V^n$'s,
from the connectedness of $\lambda$-filtered colimits it is clear that $E$ preserves $\lambda$-filtered colimits iff each $E_n:V^n{\rightarrow}W$ does, and so we have proved
\begin{lemma}\label{lem:multi-lambda}
For $E:MV{\rightarrow}W$ the following statements are equivalent for any regular cardinal $\lambda$.
\begin{enumerate}
\item $E$ preserves $\lambda$-filtered colimits in each variable.
\item $E$ preserves $\lambda$-filtered colimits.
\item $E_n:V^n{\rightarrow}W$ preserves $\lambda$-filtered colimits for all $n \in \N$.
\end{enumerate}
\end{lemma}
\noindent As already mentioned, colimits in $\ca GV$ for a cocomplete $V$ were calculated in \cite{BCSW-VarEnr}. Let us spell out transfinite composition in $\ca GV$. Given an ordinal $\lambda$ and a $\lambda$-chain
\begin{equation}\label{chain1} \begin{array}{lcr} {i \leq j \in \lambda} & {\mapsto} & {f_{ij} : X_i \rightarrow X_j} \end{array} \end{equation}
in $\ca GV$ with colimit $X$, one may consider the induced $\lambda$-chain
\begin{equation}\label{chain2} \begin{array}{lcr} {i \leq j \in \lambda} & {\mapsto} & {(f_{ij})_0 \times (f_{ij})_0 : (X_i)_0 \times (X_i)_0 \rightarrow (X_j)_0 \times (X_j)_0} \end{array} \end{equation}
in $\Set$. This will have colimit $X_0 \times X_0$ because $\lambda$-filtered colimits and products commute in $\Set$ and $()_0:V{\rightarrow}\Set$ is cocontinuous. For $(a,b) \in X_0$ let us denote by $D_{a,b}$ the full subcategory of the category of elements of (\ref{chain2}), consisting of those elements which are sent to $(a,b)$ by the universal cocone. We shall call this the \emph{$(a,b)$-component} of the chain (\ref{chain1}). Now $D_{a,b}$ is of course no longer a chain, but one may easily verify that it is $\lambda$-filtered. By the explicit description of colimits in $\Set$, the $D_{a,b}$ are just the connected components of the category of elements of (\ref{chain2}). To pairs $(a',b') \in (X_i)_0 \times (X_i)_0$ which are elements of $D_{a,b}$, one may associate the corresponding hom $X_i(a',b') \in V$, and in this way build a functor $F_{a,b}:D_{a,b}{\rightarrow}V$. The hom $X(a,b)$ is the colimit of this functor.
\begin{proposition}\label{prop:Gamma-accessible}
Let $V$ and $W$ be cocomplete, $E:MV{\rightarrow}W$ be distributive and $\lambda$ be a regular cardinal. If $E$ preserves $\lambda$-filtered colimits in each variable then $\Gamma{E}$ preserves $\lambda$-filtered colimits.
\end{proposition}
\begin{proof}
It suffices to show $\Gamma{E}$ preserves colimits of $\lambda$-chains. Consider the chain (\ref{chain1}) in $\ca GV$. For all $n \in \N$ one has an $(n+1)$-ary version of (\ref{chain2}), that is involving $(n+1)$-fold instead of binary cartesian products in $\Set$. These of course also commute with $\lambda$-filtered colimits. Similarly one obtains a $\lambda$-filtered category $D_{x_0,...,x_n}$ and a functor
\[ \begin{array}{lcr} {F_{x_0,...,x_n} : D_{x_0,...,x_n} \rightarrow V^n} && {(y_0,...,y_n) \in X_i \mapsto (X_i(y_0,y_1),...,X_i(y_{n-1},y_n))} \end{array} \]
Applying $\Gamma{E}$ does nothing at the object level. Let us write $D'_{a,b}$ for the $(a,b)$-component of the chain obtained by applying $\Gamma{E}$ to (\ref{chain1}), and $F'_{a,b}$ for the corresponding functor into $W$. From the explicit description of $\Gamma$'s effect on homs of corollary(\ref{cor:explicit-gamma}), one sees that $F'_{a,b}$ is the coproduct of the composites
\[ \xymatrix{{D_{x_0,...,x_n}} \ar[r]^-{F_{x_0,...,x_n}} & {V^n} \ar[r]^-{E_n} & W}. \]
over all sequences $(x_0,...,x_n)$ starting at $a$ and finishing at $b$. By lemma(\ref{lem:multi-lambda}) the colimits of the $F_{x_0,...,x_n}$ are preserved by the $E_n$, and so by the explicit description of colimits of $\lambda$-chains in $\ca GW$, the colimit of (\ref{chain1}) is indeed preserved by $\Gamma{E}$.
\end{proof}
\subsection{An elegant construction of the strict $n$-category monads}\label{ssec:induction}
Let us recall the construction $(-)^\times$. Given a monad $(T,\eta,\mu)$ on $V$ a category with products, one has a multitensor $T^{\times}$ defined by
\[ \begin{array}{c} {T^{\times}(X_1,...,X_n) = \prod\limits_{1{\leq}i{\leq}n} T(X_i)} \end{array} \]
and the unit and substitution is induced in the obvious way from $\eta$ and $\mu$. When $V$ is lextensive, $T$ is a local right adjoint, and $\eta$ and $\mu$ are cartesian, it follows that $T^{\times}$ is a local right adjoint its unit and multiplication are also cartesian. When $T$ preserves coproducts and the cartesian product for $V$ is distributive, then $T^{\times}$ is a distributive multitensor. If in addition finite limits and filtered colimits commute in $V$ (which happens when, for example $V$ is locally finitely presentable), then $T^{\times}$ is finitary. Moreover by proposition(2.8) of \cite{EnHopI} one has
\begin{equation}\label{eq:Tcross-cat} \Enrich {T^{\times}} \iso \Enrich {V^T} \end{equation}
where the enrichment on the right hand side is with respect to cartesian products. Thus one can consider the following inductively-defined sequence of monads
\begin{itemize}
\item Put $\ca T_{{\leq}0}$ equal to the identity monad on $\Set$.
\item Given a monad $\ca T_{{\leq}n}$ on $\ca G^n\Set$, define the monad
$\ca T_{{\leq}n+1} = \Gamma \ca T^{\times}_{{\leq}n}$
on $\ca G^{n+1}\Set$.
\end{itemize}
recalling that $\ca G^n\Set$ is the category of $n$-globular sets.
\begin{theorem}\cite{EnHopI}\label{thm:EnHopI-main-theorem}
For $n \in \N$, $\ca T_{\leq n}$ is the strict $n$-category monad on $n$-globular sets. This monad is coproduct preserving, finitary and local right adjoint.
\end{theorem}
\begin{proof}
By (\ref{eq:Tcross-cat}) and corollary(\ref{cor:gamma-alg-ecat}), one has
\[ \ca G^{n+1}(\Set)^{\ca T_{{\leq}n+1}} \iso \Enrich {\ca T^{\times}_{{\leq}n}} \]
and so by definition $\ca G^n(\Set)^{\ca T_{\leq n}}$ is the category of strict $n$-categories and strict $n$-functors between them. By the remarks at the beginning of this section and corollary(\ref{cor:GV-topos}) $(-)^{\times}$ will produce a distributive, finitary, local right adjoint multitensor on a presheaf category when it is fed a coproduct preserving, finitary, local right adjoint monad on a presheaf category. By corollary(\ref{cor:GV-topos}), proposition(\ref{prop:GammaE-basic}), proposition(\ref{prop:Gamma-accessible}), proposition(\ref{prop:GammaE-lra}) and proposition(\ref{prop:Gamma-cart2}), $\Gamma$ will produce a coproduct preserving, finitary, local right adjoint monad on a presheaf category when it is fed a distributive, finitary, local right adjoint multitensor on a presheaf category. Thus the monads $\ca T_{\leq n}$ are indeed coproduct preserving, finitary and local right adjoint for all $n \in \N$.
\end{proof}
The objects of $\NOp {\ca T_{{\leq}n}}$ -- $n$-operads over $\Set$ -- were in \cite{Bat98} \cite{EnHopI} called ``normalised'' $n$-operads. Many $n$-categorical structures of interest, such as weak $n$-categories, can be defined as algebras of $n$-operads over $\Set$. Objects of $\Mult {\ca T_{{\leq}n}}$ are called $n$-multitensors. These are a nice class of lax monoidal structures on the category of $n$-globular sets. By corollary(\ref{cor:mult-nop-equiv}) and theorem(\ref{thm:EnHopI-main-theorem}) one obtains
\begin{corollary}\cite{EnHopI}\label{cor:nnp1op-nmult}
For all $n \in \N$, applying $\Gamma$ gives $\NOp {\ca T_{{\leq}(n+1)}} \catequiv \Mult {\ca T_{{\leq}n}}$.
\end{corollary}
\noindent That is to say, $\Gamma$ exhibits $(n+1)$-operads over $\Set$ and $n$-multitensors as the same thing, and under this correspondence, the algebras of the operad correspond to the categories enriched in the associated multitensor by corollary(\ref{cor:gamma-alg-ecat}).
\section{The 2-functoriality of the monad-multitensor correspondence}\label{sec:2-functoriality}
\subsection{Motivation}\label{ssec:motivation-2-functoriality}
Up to this point $\Gamma$ has been our notation for the process
\[ \textnormal{Distributive multitensor on $V$} \mapsto \textnormal{Monad on $\ca GV$ over $\Set$} \]
and $\overline{(-)}$ has been our notation for the reverse construction. For the most complete analysis of these constructions one must acknowledge that they are the object maps of 2-functors in two important ways. This 2-functoriality together with the formal theory of monads \cite{Str72} gives a satisfying explanation of how it is that monad distributive laws arise naturally in this subject (see \cite{ChengDist}).
\subsection{2-categories of multitensors and monads}\label{ssec:Dist-Mult}
As the lax-algebras of a 2-monad $M$ (see section(\ref{ssec:LMC})), lax monoidal categories form a 2-category $\LaxAlg M$. See \cite{LackCodesc} for a complete description of the 2-category of lax algebras for an arbitrary 2-monad. Explicitly a lax monoidal functor between lax monoidal categories $(V,E)$ and $(W,F)$ consists of a functor $H:V{\rightarrow}W$, and maps
\[ \psi_{X_i} : \opF\limits_i HX_i \rightarrow H \opE\limits_i X_i \]
natural in the $X_i$ such that
\[ \xygraph{{\xybox{\xygraph{!{0;(.75,0):(0,1.333)::} {HX}="l" [r(2)] {F_1HX}="r" [dl] {HE_1X}="b" "l"(:"r"^-{u_{HX}}:"b"^{\psi_X},:"b"_{Hu_X})}}} [r(5)]
{\xybox{\xygraph{!{0;(2,0):(0,.5)::} {\opF\limits_i\opF\limits_jHX_{ij}}="tl" [r] {\opF\limits_iH\opE\limits_jX_{ij}}="tm" [r] {H\opE\limits_i\opE\limits_jX_{ij}}="tr" [l(.5)d] {H\opE\limits_{ij}X_{ij}}="br" [l] {\opF\limits_{ij}HX_{ij}}="bl" "tl" (:@<1ex>"tm"^-{\opF\limits_i\psi}:@<1ex>"tr"^-{\psi\opE\limits_j}:"br"^{H\sigma},:"bl"_{\sigma{H}}:@<1ex>"br"_-{\psi})}}}} \]
commute for all $X$ and $X_{ij}$ in $V$. A monoidal natural transformation between lax monoidal functors \[ (H,\psi),(K,\kappa):(V,E){\rightarrow}(W,F) \] consists of a natural transformation $\phi:H{\rightarrow}K$ such that
\[ \xygraph{!{0;(2,0):(0,.5)::} {\opF\limits_iHX_i}="tl" [r] {H\opE\limits_iX_i}="tr" [d] {K\opE\limits_iX_i}="br" [l] {\opF\limits_iKX_i}="bl" "tl" (:@<1ex>"tr"^-{\psi}:"br"^{\phi\opE\limits_i},:"bl"_{\opF\limits_i\phi}:@<1ex>"br"_-{\kappa})} \]
commutes for all $X_i$.
\begin{definition}\label{def:2cat-DistMult}
The 2-category $\DISTMULT$ of distributive multitensors, is defined to be the full sub-2-category of $\LaxAlg M$ consisting of the $(V,E)$ such that $V$ has coproducts and $E$ is distributive.
\end{definition}
For any 2-category $\ca K$ recall the 2-category $\MND(\ca K)$ from \cite{Str72} of monads in $\ca K$. Another way to describe this very canonical object is that it is the 2-category of lax algebras of the identity monad on $\ca K$. Explicitly the 2-category $\MND(\CAT)$ has as objects pairs $(V,T)$ where $V$ is a category and $T$ is a monad on $V$. An arrow $(V,T){\rightarrow}(W,S)$ is a pair consisting of a functor $H:V{\rightarrow}W$ and a natural transformation $\psi:SH{\rightarrow}HT$ satisfying the obvious 2 axioms: these are just the ``unary'' analogues of the axioms for a lax monoidal functor written out above.
For example, any lax monoidal functor $(H,\psi)$ as above determines a monad functor $(H,\psi_1):(V,E_1){\rightarrow}(W,F_1)$. A monad transformation between monad functors
\[ (H,\psi),(K,\kappa):(V,T){\rightarrow}(W,S) \]
consists of a natural transformation $\phi:H{\rightarrow}K$ satisfying the obvious axiom. For example a monoidal natural transformation $\phi$ as above is a monad transformation $(H,\psi_1){\rightarrow}(K,\kappa_1)$.
In fact as we are interested in monads over $\Set$, we shall work not with $\MND(\CAT)$ but rather with $\MND(\CAT/\Set)$. An object $(V,T)$ of this latter 2-category is a category $V$ equipped with a functor into $\Set$, together with a monad $T$ on $V$ which ``acts fibre-wise''
with respect to this functor. That is $T$'s
object map doesn't affect the underlying object set, similarly for the arrow map of $T$, and the components of $T$'s
unit and multiplication are identities on objects in the obvious sense. An arrow $(V,T){\rightarrow}(W,S)$ of $\MND(\CAT/\Set)$ is a pair $(H,\psi)$ as in the case of $\MND(\CAT)$, with the added condition that $\psi$'s components are the identities on objects, and similarly the 2-cells of $\MND(\CAT/\Set)$ come with an extra identity-on-object condition.
\subsection{$\Gamma$ as a 2-functor}\label{ssec:Gamma-2-functor}
We shall now exhibit the 2-functor
\[ \Gamma : \DISTMULT \rightarrow \MND(\CAT/\Set) \]
which on objects is given by $(V,E) \mapsto (\ca GV,\Gamma{E})$. Let $(H,\psi):(V,E){\rightarrow}(W,F)$ be a lax monoidal functor between distributive lax monoidal categories. Then for $X \in \ca GV$ and $a,b \in X_0$, we define the hom map $\Gamma(\psi)_{X,a,b}$ to be the composite of
\[ \xygraph{!{0;(5,0):} {\coprod\limits_{a=x_0,...,x_n=b} \opF\limits_iHX(x_{i-1},x_i)}="l" [r] {\coprod\limits_{a=x_0,...,x_n=b} H\opE\limits_iX(x_{i-1},x_i)}="m" "l":@<1ex>"m"^-{\coprod \psi}} \]
and $H$'s coproduct preservation obstruction map.
It follows easily from the definitions that $(\ca GH,\Gamma(\psi))$ as defined here satisfies the axioms of a monad functor. Moreover given a monoidal natural transformation $\phi:(H,\psi){\rightarrow}(K,\kappa)$, it also follows easily from the definitions that \[ \ca G\phi:(\ca GH,\Gamma(\psi)){\rightarrow}(\ca GK,\Gamma(\kappa)) \] is a monad transformation. It is also straight-forward to verify that these assignments are 2-functorial.
\subsection{The image of $\Gamma$}\label{ssec:Gamma-Image}
By proposition(\ref{prop:pl-adjoint-char}) and corollary(\ref{cor:pl->copr-pres}) we understand objects of the image of $\Gamma$ and we collect this information in
\begin{proposition}\label{prop:image-Gamma-objects}
For $V$ a category with coproducts, a monad $(\ca GV, T)$ over $\Set$ is in the image of $\Gamma$ iff $T$ is distributive and path-like. Moreover any such $T$ automatically preserves coproducts. One recovers the distributive multitensor $E$ such that $\Gamma{E}{\iso}T$ as $E=\overline{T}$.
\end{proposition}
\noindent Since the construction $\overline{(-)}$ is itself obviously 2-functorial, the arrows and 2-cells in the image of $\Gamma$ may also be easily characterised.
\begin{proposition}\label{prop:image-Gamma-1-2-cells}
\begin{enumerate}
\item Let $(V,E)$ and $(W,F)$ be distributive lax monoidal categories. A monad functor of the form
\[ (H,\psi) : (\ca GV,\Gamma{E}) \rightarrow (\ca GW,\Gamma{F}) \]
is in the image of $\Gamma$ iff $H=\ca GH'$ for some $H'$.\label{Gamma-char1}
\item Let $(H,\psi),(K,\kappa):(V,E){\rightarrow}(W,F)$ be lax monoidal functors between distributive lax monoidal categories. A monad transformation
\[ \phi : (\ca GH,\Gamma{\psi}) \rightarrow (\ca GK,\Gamma{\kappa}) \]
is in the image of $\Gamma$ iff it is of the form $\phi=\ca G\phi'$.\label{Gamma-char2}
\end{enumerate}
\end{proposition}
\begin{proof}
By definition monad functors and transformations in the image of $\Gamma$ have the stated properties, so we must prove the converse.
Given $(\ca GH,\psi)$ such that the components of $\psi$ are the identities on objects, one recovers for $X_1,...,X_n$ from $V$, the corresponding lax monoidal functor coherence map as the hom map from $0$ to $n$ of the component $\psi_{(X_1,...,X_n)}$. That is to say, we apply $\overline{(-)}$ to the appropriate monad functors to prove (\ref{Gamma-char1}), and we do the same to the appropriate monad transformations to obtain (\ref{Gamma-char2}).
\end{proof}
\begin{definition}\label{def:PLMND}
We denote by $\PLMND$ the following 2-category. Its objects are monads $(\ca GV,T)$ over $\Set$ such that $V$ has coproducts and $T$ is distributive and path-like. Its arrows are arrows $(\ca GH,\psi) : (\ca GV,\Gamma{E}) \rightarrow (\ca GW,\Gamma{F})$ of $\MND(\CAT/\Set)$, and its 2-cells are 2-cells $\ca G\phi : (\ca GH,\psi) \rightarrow (\ca GK,\kappa)$ of $\MND(\CAT/\Set)$.
\end{definition}
\noindent Thus from the proof of proposition(\ref{prop:image-Gamma-1-2-cells}) we have
\begin{corollary}\label{cor:2eq-mult-mon}
$\Gamma$ and $\overline{(-)}$ provide a 2-equivalence $\DISTMULT \catequiv \PLMND$.
\end{corollary}
\subsection{The dual 2-functoriality of $\Gamma$}\label{ssec:Gamma-another-2-functor}
Lax algebras of a 2-monad organise naturally into \emph{two} different 2-categories depending on whether one takes lax or oplax algebra morphisms. So in particular one has the 2-category $\OpLaxAlg M$ of lax monoidal categories, \emph{op}lax-monoidal functors between them and monoidal natural transformations between those. The coherence $\psi$ for an oplax $(H,\psi):(V,E){\rightarrow}(W,F)$ goes in the other direction, and so its components look like this:
\[ \psi_{X_i} : H \opE\limits_i X_i \rightarrow \opF\limits_i HX_i. \]
The reader should easily be able to write down explicitly the two coherence axioms that this data must satisfy, as well as the condition that must be satisfied by a monoidal natural transformation between oplax monoidal functors. Similarly there is a dual version $\OpMND(\ca K)$ of the 2-category $\MND(\ca K)$ of monads in a given 2-category $\ca K$ discussed above \cite{Str72}. An arrow $(V,T){\rightarrow}(W,S)$ of $\OpMND(\CAT)$ consists of a functor $H:V{\rightarrow}W$ and a natural transformation $\psi:HT{\rightarrow}SH$ satisfying the two obvious axioms. An arrow of $\OpMND(\CAT)$ is called a monad opfunctor. As before $\OpMND(\CAT/\Set)$ differs from $\MND(\CAT/\Set)$ in that all the categories involved come with a functor into $\Set$, and all the functors and natural transformations involved are compatible with these forgetful functors.
We now describe the dual version of the 2-functoriality of $\Gamma$ discussed in sections(\ref{ssec:Gamma-2-functor}) and (\ref{ssec:Gamma-Image}). When defining the one-cell map of $\Gamma$ in section(\ref{ssec:Gamma-2-functor}) we were helped by the fact that the coproduct preservation obstruction went the right way: see the definition of the monad functor $(\ca GH,\Gamma{\psi})$ above. This time however we will not be so lucky, and for this reason we must restrict ourselves in the following definition to coproduct preserving oplax monoidal functors.
\begin{definition}\label{def:OpDistMult}
The 2-category $\OpDISTMULT$ is defined to be the locally full sub-2-category of $\OpLaxAlg M$ consisting of the distributive lax monoidal categories, and the oplax monoidal functors $(H,\psi)$ such that $H$ preserves coproducts. We denote by $\OpPLMND$ the following 2-category. Its objects are monads $(\ca GV,T)$ over $\Set$ such that $V$ has coproducts and $T$ is distributive and path-like. Its arrows are arrows $(\ca GH,\psi) : (\ca GV,\Gamma{E}) \rightarrow (\ca GW,\Gamma{F})$ of $\OpMND(\CAT/\Set)$ such that $H$ preserves coproducts. Its 2-cells are 2-cells $\ca G\phi : (\ca GH,\psi) \rightarrow (\ca GK,\kappa)$ of $\OpMND(\CAT/\Set)$.
\end{definition}
\noindent We now define
\[ \Gamma : \OpDISTMULT \rightarrow \OpMND({\CAT/\Set}) \]
with object map $(V,E) \mapsto (\ca GV,\Gamma{E})$ as before, and the rest of its definition is obtained by modifying the earlier definition of $\Gamma$ in what should now be the obvious way. The proof of the following result is obtained by a similar such modification of the proof of corollary(\ref{cor:2eq-mult-mon}).
\begin{corollary}\label{cor:2eq-mult-mon-dual}
$\Gamma$ and $\overline{(-)}$ provide a 2-equivalence \[ \OpDISTMULT \catequiv \OpPLMND. \]
\end{corollary}
\subsection{Monoidal monads and distributive laws}\label{ssec:Monmonad-Distlaw}
As explained in \cite{Str72} the assignment $\ca K \mapsto \MND(\ca K)$ is in fact the object map of a strict 3-functor. Just exploiting 2-functoriality here and corollaries(\ref{cor:2eq-mult-mon}) and (\ref{cor:2eq-mult-mon-dual}) one immediately obtains
\begin{theorem}\label{thm:monmon-distlaw}
$\Gamma$ and $\overline{(-)}$ provide two 2-equivalences of 2-categories:
\begin{enumerate}
\item ${\MND(\DISTMULT) \catequiv \MND(\PLMND)}$.\label{mdl1}
\item ${\MND(\OpDISTMULT) \catequiv \MND(\OpPLMND)}$.\label{mdl2}
\end{enumerate}
\end{theorem}
The meaning of this result is understood by understanding what the objects of the 2-categories involved are, that is to say, what monads are in each of the 2-categories $\DISTMULT$, $\OpDISTMULT$, $\PLMND$ and $\OpPLMND$.
A very beautiful observation of \cite{Str72} is that to give a monad on $(V,T)$ in $\MND(\ca K)$ is to give another monad $S$ on $V$, together with a distributive law $\lambda:TS{\rightarrow}ST$. Similarly to give a monad on $(V,T)$ in $\OpMND(\ca K)$ is to give another monad $S$ on $V$, together with a distributive law $\lambda:ST{\rightarrow}TS$ in the other direction. Thus the 2-categories $\MND(\MND(\ca K))$ and $\MND(\OpMND(\ca K))$ really have the same objects: such an object being a pair of monads on the same category and a distributive law between them. Thus both $\MND(\PLMND)$ and $\MND(\OpPLMND)$ are 2-categories whose objects are monad distributive laws between monads defined on categories of enriched graphs, with some extra conditions.
On the other hand a monad in the 2-category $\LaxAlg M$ of lax monoidal categories and lax monoidal functors is also a well-known thing, and such things are usually called \emph{monoidal monads}. Similarly an opmonoidal monad $T$ on a monoidal category $V$, that is to say a monad on $V$ in $\OpLaxAlg M$, comes with the extra data of coherence maps
\[ T(X_1 \tensor ... \tensor X_n) \rightarrow TX_1 \tensor ... \tensor TX_n \]
that are compatible with the monad structure. If for instance $\tensor$ is just cartesian product, then the product obstruction maps for $T$ endow it with an opmonoidal structure in a unique way.
By definition the objects of $\MND(\DISTMULT)$ are monoidal monads defined on distributive lax monoidal categories, and the objects of $\MND(\OpDISTMULT)$ are coproduct preserving opmonoidal monads defined on distributive lax monoidal categories. Thus the meaning of theorem(\ref{thm:monmon-distlaw}) is that it exhibits these kinds of monoidal and opmonoidal monads as being equivalent to certain kinds of distributive laws. We shall spell this out precisely in corollaries(\ref{cor:monmon-distlaw}) and (\ref{cor:opmonmon-distlaw}) below.
Let $(V,E)$ be a lax monoidal category and $T$ be a monad on $V$. In section(\ref{ssec:laxalg-const1}) we saw that when $T$ is a monoidal monad, that is to say one has coherence maps
\[ \tau_{X_i} : \opE\limits_i TX_i \rightarrow T \opE\limits_i X_i \]
making the underlying endofunctor of $T$ a lax monoidal functor and the unit and multiplication monoidal natural transformations, then one has another multitensor on $V$ given on objects by $T\opE\limits_i X_i$, and with unit and substitution given by the composites
\[ \xygraph{{\xybox{\xygraph{{X}="l" [r] {TX}="m" [r(1.5)] {TE_1X}="r" "l":"m"^-{\eta}:"r"^-{Tu}}}} [r(5)d(.1)]
{\xybox{\xygraph{!{0;(2,0):} {T\opE\limits_iT\opE\limits_j X_{ij}}="l" [r] {T^2\opE\limits_i\opE\limits_j X_{ij}}="m" [r] {T\opE\limits_{ij}X_{ij}}="r" "l":@<1ex>"m"^-{T{\tau}E}:@<1ex>"r"^-{\mu\sigma}}}}} \]
In particular if $E$ is distributive and $T$ preserves coproducts, then this new multitensor $TE$ is also distributive.
If instead $T$ has the structure of an opmonoidal monad, with the coherences
\[ \tau_{X_i} : T\opE\limits_i X_i \rightarrow \opE\limits_i TX_i \]
going in the other direction, then in the same way one can construct a new multitensor $ET$ on $V$ which on objects is defined by given by $\opE\limits_iTX_i$. Once again if $E$ is distributive and $T$ coproduct preserving, then $ET$ is a distributive multitensor. In particular when $E$ is cartesian product, $ET$ is the multitensor $T^{\times}$ of section(\ref{ssec:induction}).
With regards to monoidal monads, unpacking what theorem(\ref{thm:monmon-distlaw})(\ref{mdl1}) says at the object level gives
\begin{corollary}\label{cor:monmon-distlaw}
Let $(V,E)$ be a distributive lax monoidal category and $T$ be a monad on $V$. To give maps $\tau_{X_i} : \opE\limits_i TX_i \rightarrow T\opE\limits_i X_i$
making $T$ into a monoidal monad on $(V,E)$, is the same as giving a monad distributive law $\Gamma(E)\ca G(T){\rightarrow}\ca G(T)\Gamma(E)$ whose components are the identities on objects.
\end{corollary}
\noindent In the case where $T$ preserves coproducts one may readily verify that $\Gamma(TE) \iso \ca G(T)\Gamma(E)$ as monads, and so by corollary(\ref{cor:gamma-alg-ecat}) one understands what the algebras of this composite monad $\ca G(T)\Gamma(E)$ are.
\begin{corollary}\label{cor:algebras-GTGammaE}
If in the situation of corollary(\ref{cor:monmon-distlaw}) $T$ also preserves coproducts, then $\ca GV^{\ca G(T)\Gamma(E)} \iso \Enrich {TE}$.
\end{corollary}
\noindent Similarly, one can unpack what theorem(\ref{thm:monmon-distlaw})(\ref{mdl2}) says at the object level, witness $\Gamma(ET) \iso \Gamma(E)\ca G(T)$ and use corollary(\ref{cor:gamma-alg-ecat}) to conclude
\begin{corollary}\label{cor:opmonmon-distlaw}
Let $(V,E)$ be a distributive lax monoidal category and $T$ be a coproduct preserving monad on $V$. To give maps $\tau_{X_i} : T\opE\limits_i X_i \rightarrow \opE\limits_i TX_i$ making $T$ into an opmonoidal monad on $(V,E)$, is the same as giving a monad distributive law $\ca G(T)\Gamma(E){\rightarrow}\Gamma(E)\ca G(T)$ whose components are the identities on objects. Moreover $\ca GV^{\Gamma(E)\ca G(T)} \iso \Enrich {ET}$.
\end{corollary}
\begin{example}\label{ex:Cheng1}
From the inductive description of $\ca T_{\leq{n}}$ of section(\ref{ssec:induction}) and corollary(\ref{cor:opmonmon-distlaw}) one obtains a distibutive law
\[ \begin{array}{c} {\ca G(\ca T_{\leq{n}})\Gamma(\prod) \rightarrow \Gamma(\prod)\ca G(\ca T_{\leq{n}})} \end{array} \]
for all $n$, between monads on $\ca G^n\Set$, and the composite monad $\Gamma(\prod)\ca G(\ca T_{\leq{n}}) = \ca T_{\leq{(n{+}1)}}$. Thus we have recaptured the decomposition of \cite{ChengDist} of the strict $n$-category monad into a ``distributive series of monads''.
\end{example}
\subsection{The Trimble definition \`{a} la Cheng}\label{ssec:TCI}
Pursuing the idea of the previous example, we shall now begin to recover and in some senses generalise Cheng's analysis and description \cite{ChengCompOp} of the Trimble definition of weak $n$-category.
From \cite{EnHopI} example(2.6) non-symmetric operads in the usual sense can be regarded as multitensors. Here we shall identify a non-symmetric operad
\[ \begin{array}{lcr} {(E_n \, \, : \, \, n \in \N)} & {u:I \rightarrow E_1} &
{\sigma:E_k \tensor E_{n_1} \tensor ... \tensor E_{n_k} \rightarrow E_{n_{\bullet}}} \end{array} \]
in a braided monoidal category $V$, with the multitensor
\[ (X_1,...,X_n) \mapsto E_n \tensor X_1 \tensor ... \tensor X_n \]
it generates. Recall that one object $E$-categories for $E$ a non-symmetric operad are precisely algebras of the operad $E$ in the usual sense. If $\tensor$ is cartesian product, then the projections
\[ E_n \times X_1 \times ... \times X_n \rightarrow X_1 \times ... \times X_n \]
are the components of a cartesian multitensor map $E \rightarrow \prod$. Conversely such a cartesian multitensor map exhibits $E$ as an operad via
\[ E_n := E(\underbrace{1,...,1}_{n}) \]
for all $n \in \N$.
Let $V$ be a distributive category and $T$ a coproduct preserving monad on $V$. Let us denote by $(E,\varepsilon)$ a non-symmetric operad in $V^T$. The ``$\varepsilon$'' is meant to denote the $T$-algebra actions, that is $\varepsilon_n:TE_n \rightarrow E_n$ is the $T$-algebra structure, and so $E$ denotes the underlying operad in $V$. Since $U^T$ preserves products it is the underlying functor of a strong monoidal functor $(V^T,(E,\varepsilon)) \rightarrow (V,E)$ between lax monoidal categories. Since the composites
\[ \xygraph{!{0;(3,0):} {T(E_n \times \prod\limits_iX_i)}="l" [r(1.2)] {TE_n \times \prod\limits_i TX_i}="m" [r] {E_n \times \prod\limits_iTX_i}="r" "l":@<1ex>"m"^-{\textnormal{prod. obstn.}}:@<1ex>"r"^-{\varepsilon_n \times \id}} \]
form the components of an opmonoidal structure for the monad $T$, we find ourselves in the situation of corollary(\ref{cor:opmonmon-distlaw}) and so obtain
\begin{proposition}\label{prop:monad-trimble}
Let $V$ be a distributive category, $T$ a coproduct preserving monad on $V$ and $(E,\varepsilon)$ a non-symmetric operad in $V^T$. Then one has a distributive law $\ca G(T)\Gamma(E) \rightarrow \Gamma(E)\ca G(T)$ between monads on $\ca GV$, and isomorphisms
\[ \Enrich {(E,\varepsilon)} \iso \Enrich {ET} \iso \ca GV^{\Gamma(E)\ca G(T)} \]
of categories over $\ca GV$.
\end{proposition}
\noindent This result has an operadic counterpart.
\begin{proposition}\label{prop:operad-trimble}
Let $V$ be a lextensive category, $T$ a cartesian and coproduct preserving monad on $V$, $\psi:S{\rightarrow}T$ a $T$-operad and $(E,\varepsilon)$ a non-symmetric operad in $V^S$. Then the monad $\Gamma(E)\ca G(S)$, whose algebras by proposition(\ref{prop:monad-trimble}) are $(E,\varepsilon)$-categories, has a canonical structure of a $\Gamma(T^{\times})$-operad.
\end{proposition}
\begin{proof}
With $\Gamma(T^{\times})=\Gamma(\prod)\ca G(T)$ we must exhibit a cartesian monad map $\Gamma(E)\ca G(S) \rightarrow \Gamma(\prod)\ca G(T)$. We have the cartesian multitensor map $\alpha:(E,\varepsilon) \rightarrow \prod$ which exhibits the multitensor $(E,\varepsilon)$ as a non-symmetric operad, thus $U^T\alpha$ is also a cartesian multitensor map, and since $V$ is lextensive $\Gamma$ sends this to a cartesian monad morphism. The required cartesian monad map is thus $\Gamma(U^T\alpha)\ca G(\psi)$.
\end{proof}
Recall the path-space functor $P:\Top \rightarrow \ca G(\Top)$ discussed in section(\ref{ssec:enriched-graphs}). To say that a non-symmetric topological operad $A$ acts on $P$ is to say that $P$ factors as
\[ \xygraph{!{0;(2,0):} {\Top}="l" [r] {\Enrich A}="m" [r] {\ca G(\Top)}="r" "l":"m"^-{P_A}:"r"^-{U^A}} \]
The main example to keep in mind is the version of the little intervals operad recalled in \cite{ChengCompOp} definition(1.1). As this $A$ is a contractible non-symmetric operad, $A$-categories may be regarded as a model of $A$-infinity spaces. Since $P$ is a right adjoint, $P_A$ is also a right adjoint by the Dubuc adjoint triangle theorem.
A product preserving functor
\[ Q : \Top \rightarrow V \]
into a distributive category, may be regarded as the underlying functor of a strong monoidal functor $(\Top,A) \rightarrow (V,QA)$ between lax monoidal categories. Applying $\Gamma$ to this gives us a monad functor
\[ (\ca G(\Top),\Gamma(A)) \rightarrow (\ca GV,\Gamma(QA)) \]
with underlying functor $\ca GQ$, which amounts to giving a lifting $\overline{Q}$ as indicated in the commutative diagram
\[ \xygraph{!{0;(2,0):(0,.5)::} {\Top}="tl" [r] {\Enrich A}="tm" [r] {\Enrich {QA}}="tr" [d] {\ca GV}="br" [l] {\ca G(\Top)}="bl" "tl"(:"tm"^-{P_A}(:"tr"^-{\overline{Q}}:"br"^{U^{QA}},:"bl":"br"_-{\ca G(Q)}),:"bl"_{P})} \]
and so we have produced another product preserving functor
\[ Q^{(+)} : \Top \rightarrow V^{(+)} \]
where $Q^{(+)}=\overline{Q}P_A$ and $V^{(+)}=\Enrich {QA}$. The functor $\overline{Q}$ is product preserving since $\ca G(Q)$ is and $U^{QA}$ creates products. The assignment
\[ (Q,V) \mapsto (Q^{(+)},V^{(+)}) \]
in the case where $A$ is as described in \cite{ChengCompOp} definition(1.1), is the inductive process lying at the heart of the Trimble definition. In this definition one begins with the connected components functor $\pi_0 : \Top \rightarrow \Set$ and defines the category $\Trimble 0$ of ``Trimble 0-categories'' to be $\Set$. The induction is given by
\[ (\Trimble {n{+}1},\pi_{n{+}1}) := (\textnormal{Trm}_{n}^{(+)},\pi_{n}^{(+)}) \]
and so this definition constructs not only a notion of weak $n$-category but the product preserving $\pi_n$'s to be regarded as assigning the fundamental $n$-groupoid to a space.
Applying proposition(\ref{prop:monad-trimble}) to this situation produces the monad on $n$-globular sets whose category of algebras is $\Trimble n$ as well as its decomposition into an iterative series of monads witnessed in \cite{ChengCompOp} section(4.2). Applying proposition(\ref{prop:operad-trimble}) and the inductive description of $\ca T_{\leq{n}}$ of section(\ref{ssec:induction}) exhibits these monads as $n$-operads.
\section{Lifting multitensors}\label{sec:lift-mult}
\subsection{Motivation}\label{ssec:lifting-intro}
Applied to the normalised $3$-operad for Gray categories \cite{Bat98}, the results of the section(\ref{sec:reexpress}) produce a lax monoidal structure $E$ on the category of $2$-globular sets whose enriched categories are exactly Gray categories. For this example it turns out that $E_1$ is $\ca T_{{\leq}2}$, and so providing a lift of $E$ in the sense of definition(\ref{def:lift}) amounts to the construction of a tensor product of $2$-categories whose enriched categories are Gray categories, that is to say, an abstractly constructed Gray tensor product. By the main result of this section theorem(\ref{thm:lift-mult}), \emph{every} $n$-multitensor has a lift which is unique given certain properties.
While the proof of theorem(\ref{thm:lift-mult}) is fairly abstract, and the uniqueness has the practical effect that in the examples we never have to unpack an explicit description of the lifted multitensors provided by the theorem, we provide such an unpacking in section(\ref{ssec:explicit-lifting}) anyway. This enables us to give natural conditions when the construction of the lifted multitensor is simpler. Doing all this requires manipulating some of the transfinite constructions that arise in monad theory, and we give a self-contained review of these in the appendix.
\subsection{The multitensor lifting theorem}\label{ssec:lifting theorem}
In appendix \ref{sec:Dubuc} we recall an explicit description, for a given monad morphism $\phi:M \to S$ between accessible monads on a locally presentable category $V$, of the left adjoint $\phi_!$ to the canonical forgetful functor $\phi^*:V^S \to V^M$ induced by $\phi$. The key point about $\phi_!$ is that it is constructed via a transfinite process involving only \emph{connected} colimits in $V$. The importance of this is underscored by
\begin{lemma}\label{lem:concol-pathlike}
Let $V$ be a category with an initial object, $W$ be a cocomplete category, $J$ be a small connected category and
\[ F : J \rightarrow [\ca GV,\ca GW] \]
be a functor. Suppose that $F$ sends objects of $J$ to normalised functors, and arrows of $J$ to natural transformations whose components are identities on objects.
\begin{itemize}
\item[(1)] Then the colimit $K:\ca GV{\rightarrow}\ca GW$ of $F$ may be chosen to be normalised.\label{cpl1}
\end{itemize}
Given such a choice of $K$:
\begin{itemize}
\item[(2)] If $Fj$ is path-like for all $j \in J$, then $K$ is also path-like.\label{cpl2}
\item[(3)] If $Fj$ is distributive for all $j \in J$, then $K$ is also distributive.\label{clp3}
\end{itemize}
\end{lemma}
\begin{proof}
Colimits in $[\ca GV,\ca GW]$ are computed componentwise from colimits in $\ca GW$ and so for $X \in \ca GV$ we must describe a universal cocone with components
\[ \kappa_{X,j} : Fj(X) \rightarrow KX. \]
We demand that the $\kappa_{X,j}$ are identities on objects. This is possible since the $Fj(X)_0$ form the constant diagram on $X_0$ by the hypotheses on $F$. For $a,b \in X_0$ we choose an arbitrary colimit cocone
\[ \{\kappa_{X,j}\}_{a,b} : Fj(X)(a,b) \rightarrow KX(a,b) \]
in $W$. One may easily verify directly that since $J$ is connected, the $\kappa_{X,j}$ do indeed define a univeral cocone for all $X$ in order to establish (1). Since the properties of path-likeness and distributivity involve only colimits at the level of the homs as does the construction of $K$ just given, (2) and (3) follow immediately since colimits commute with colimits in general.
\end{proof}
\noindent With these preliminaries in hand we are now ready to present the monad version of the multitensor lifting theorem, and then the lifting theorem itself.
\begin{lemma}\label{lem:mnd-lift-mult}
Let $V$ be a locally presentable category, $R$ be a coproduct preserving monad on $V$, $S$ be an accessible and normalised monad on $\ca GV$, and $\phi:\ca GR{\rightarrow}S$ be a monad morphism whose components are identities on objects. Denote by $T$ the monad induced by $\phi_! \ladj \phi^*$.
\begin{itemize}
\item[(1)] One may choose $\phi_!$ so that $T$ becomes normalised.
\end{itemize}
Given such a choice of $\phi_!$:
\begin{itemize}
\item[(2)] If $S$ is path-like then $T$ is path-like.
\item[(3)] If $S$ is distributive then $T$ is distributive.
\end{itemize}
\end{lemma}
\begin{proof}
Let $\lambda$ be the regular cardinal such that $S$ preserves $\lambda$-filtered colimits. To verify that $(T,\eta^T,\mu^T)$ is a normalised monad one must verify: (i) $T$ is normalised, and (ii) the components of the unit $\eta^T$ are identities on objects. Since $\mu^T$ is a retraction of $\eta^TT$, it will then follow that the components of $\mu^T$ are also identities on objects. But $T$ is normalised iff $\ca GU^RT$ is normalised, and $\ca GU^RT=U^S\phi_!$, so for (i) it suffices to show that one can choose $\phi_!$ making $U^S\phi_!$ normalised. This follows by a transfinite induction using the explicit description of $U^S\phi_!$ of section(\ref{sec:Dubuc}) and lemma(\ref{lem:concol-pathlike}).
For the initial step note that $\phi_{!1}$ can be chosen to be normalised, because $S\eta^{\ca GR}$ is a componentwise-identity on objects natural transformation between normalised functors, since the monads $S$ and $\ca GR$ are normalised. Thus the coequaliser defining $\phi_{!1}$ is a connected colimit involving only normalised functors and componentwise-identity on objects natural transformations, and so $\phi_{!1}$ can be taken to be normalised by lemma(\ref{lem:concol-pathlike}). For the inductive steps the argument is basically the same: at each stage one is taking connected colimits of normalised functors and componentwise identity on objects natural transformations, so that by lemma(\ref{lem:concol-pathlike}) one stays within the subcategory of $[\ca G(V^R),\ca GV]$ consisting of such functors and natural transformations. Moreover using lemma(\ref{lem:concol-pathlike}) $T$ will be path-like if $S$ is.
As for (ii) it suffices to prove that the components of $\ca G(U^R)\eta^T$ are identities on objects. Writing $q:S{\rightarrow}U^S\phi_{!}$ for transfinite composite constructed as part of the definition of $\phi_!$ (note that $U^S\phi_{!}=\ca G(U^R)T$ by definition) recall from the end of section(\ref{sec:Dubuc}) that one has a commutative square
\[ \xymatrix{{\ca G(R)U^{\ca GR}} \ar[r]^-{\ca G\rho} \ar[d]_{{\phi}U^M} & {U^{\ca GR}} \ar[d]^{\ca G(U^R)\eta^T} \\ {SU^{\ca GR}} \ar[r]_-{q} & {U^S\phi_{!}}} \]
where $\rho$ is the 2-cell datum for $R$'s Eilenberg-Moore object, which we recall is preserved by $\ca G$. Now $\rho$ is componentwise the identity objects since $\eta^{\ca GR}_X$ is and $\rho$ is a retraction of it, $U^S\phi$ is the identity on objects by definition, and $q$ is by construction, so the result follows.
\end{proof}
\noindent Recall from definition(\ref{def:lift}) that a \emph{lift} of $(E,u,\sigma)$ is a normal multitensor $(E',\id,\sigma')$ on $V^{E_1}$ together with an isomorphism $\Enrich E \iso \Enrich {E'}$ which commutes with the forgetful functors into $\ca G(V^{E_1})$. When in addition $E'$ is distributive, we say that it is a \emph{distributive lift} of $E$. Recall from \cite{Str72} that for any category $V$, the functor
\[ \textnormal{Alg} : \op {\Mnd(V)} \rightarrow \CAT/V \]
which sends a monad $T$ on $V$ to the forgetful functor $U^T:V^T{\rightarrow}V$, is fully-faithful.
\begin{theorem}\label{thm:lift-mult}
Let $(E,u,\sigma)$ be a distributive multitensor on $V$ a locally presentable category, and let $E$ be accessible in each variable. Then $E$ has a distributive lift $E'$, which is unique up to isomorphism.
\end{theorem}
\begin{proof}
Write $\ca SE$ for the distributive multitensor on $V$ whose unary part is $E_1$ and whose non-unary parts are constant at $\emptyset$. There is an obvious inclusion $\psi:\ca SE{\rightarrow}E$ of multitensors and one clearly has
\[ \Enrich {\ca SE} \iso \ca G(V^{E_1}) \]
Applying lemma(\ref{lem:mnd-lift-mult}) with $S=\Gamma{E}$, $R=E_1$ and $\phi=\Gamma{\psi}$ one produces a path-like, normalised and distributive monad $T$ on $\ca G(V^ {E_1})$, because $S$ is accessible by proposition(\ref{prop:Gamma-accessible}). Thus one has a distributive multitensor $\overline{T}$ on $V^{E_1}$. Applying proposition(\ref{prop:pl-alg<->cat}) to $\overline{T}$, and corollary(\ref{cor:gamma-alg-ecat}) to $E$, gives
\[ \Enrich {\overline{T}} \iso \Enrich E \]
in view of the monadicity of $\phi^*$. That is to say, $\overline{T}$ is a distributive lift of $E$. As for uniqueness suppose that $(E',\id,\sigma')$ is a distributive lift of $E$. Then by corollary(\ref{cor:gamma-alg-ecat}) and proposition(\ref{prop:pl-adjoint-char}), $\Gamma(E')$ is a distributive monad on $\ca G(V^{E_1})$ and one has
\[ \ca G(V^{E_1})^{\Gamma(E')} \iso \Enrich E \]
commuting with the forgetful functors into $\ca G(V^{E_1})$. By the fully-faithfulness of $\textnormal{Alg}$ recalled above, one has an isomorphism $\Gamma(E'){\iso}T$ of monads, and thus by proposition(\ref{prop:pl-adjoint-char}) an isomorphism $E'{\iso}\overline{T}$ of multitensors.
\end{proof}
Applying this result to any normalised $(n+1)$-operad $A$, exhibits its algebras as categories enriched in the algebras of some $n$-operad. The $n$-operad is $\Gamma(A)_1$, and tensor product over which one enriches is $\Gamma(A)'$. In cases where we already know what our tensor product ought to be, the uniqueness part of theorem(\ref{thm:lift-mult}) ensures that it is. An instance of this is
\begin{example}\label{ex:Gray}
In \cite{Bat98} the normalised 3-operad $G$ whose algebras are Gray categories was constructed. As we have already seen, $\Gamma(G)$ is a lax monoidal structure on $\ca G^2(\Set)$ whose enriched categories are Gray categories, and $\Gamma(G)_1$ is the operad for strict 2-categories. Note that the usual Gray tensor product is symmetric monoidal closed and thus distributive. Thus by theorem(\ref{thm:lift-mult}) $\Gamma(G)'$ is the Gray tensor product. In other words, the general methods of this paper have succeeded in producing the Gray tensor product of $2$-categories from the operad $G$.
\end{example}
\noindent More generally given a distributive tensor product $\tensor$ on the category of algebras of an $n$-operad $B$, and a normalised $(n+1)$-operad $A$ whose algebras are the categories enriched in $B$-algebras, theorem(\ref{thm:lift-mult}) exhibits $\tensor$ as the more generally constructed $\Gamma(A)'$.
\begin{example}\label{ex:Crans}
In \cite{Crans99} Sjoerd Crans explicitly constructed a tensor product on the category of Gray-categories. This explicit construction was extremely complicated. It is possible to exhibit the Crans tensor product as an instance of our general theory, by rewriting his explicit constructions as the construction of the 4-operad whose algebras are teisi in his sense. The multitensor $E$ associated to this 4-operad has $E_1$ equal to the 3-operad for Gray categories. Thus theorem(\ref{thm:lift-mult}) constructs a lax tensor product of Gray categories whose enriched categories are teisi. Since the tensor product explicitly constructed by Crans is distributive, the uniqueness of part of theorem(\ref{thm:lift-mult}) ensures that it is indeed $E'$, since teisi are categories enriched in the Crans tensor product by definition.
\end{example}
\noindent Honestly writing the details of the 4-operad of example(\ref{ex:Crans}) is a formidable task and we have omitted this here. In the end though, such details will not be important, because such a tensor product (or more properly a biclosed version thereof) will only be really useful once it is given a conceptual definition.
\subsection{Applications of the lifting theorem}\label{ssec:A-infinity}
Let $V$ be a symmetric monoidal model category which satisfies the conditions of \cite{BergerMoerdijk} or the monoid axiom of \cite{SS}. In this case the category of pruned $n$-operads of \cite{Bat03} can be equipped with a monoidal model structure \cite{BataninBergerCisinski}. So we can speak of cofibrant $n$-operads in $V$.
For $n=1$ let us fix a particular cofibrant and contractible $1$-operad $A$. The algebras of $A_1$ can be called \emph{$A_{\infty}$-categories enriched in $V$}. Up to homotopy the choice of $A_1$ is not important. So we can speak of \emph{the} category of $A_{\infty}$-categories. For $n=2$ we denote by $A_2$ a cofibrant contractible $2$-operad in $V$. Let $B= (A_2,u,\sigma)$ be the corresponding multitensor on $\ca GV$. One can always choose $A_2$ in such a way that its unary part is $A_1$. As in \cite{EnHopI} for an arbitary multitensor $E$, one object $E$-categories are called $E$-monoids. Similarly one object $A_{\infty}$-categories are called $A_{\infty}$-monoids.
\begin{theorem}\label{thm:A-infinity-app}
\begin{enumerate}
\item There is a distributive lift $B'$ of $B$ to the category of $A_{\infty}$-categories.
\item $B'$ restricts to give a multitensor $C$ on the category of $A_{\infty}$-monoids.
\item The category of $C$-monoids is equivalent to the category of algebras of $\sym_2(A_2)$ and therefore is Quillen equivalent to the category of the algebras of the little squares operad.
\end{enumerate}
\end{theorem}
\begin{proof}
The first statement is a direct consequence of theorem(\ref{thm:lift-mult}). The second statement follows from the fact that $B'$ is the
cartesian product on the object level. The last statement follows from the theorem(8.6) of \cite{Bat03}.
\end{proof}
\noindent Applying this result to the case $V=\Cat$ with its folklore model structure one recovers
\begin{corollary}\label{cor:JS-braided}
[Joyal-Street] The category of braided monoidal categories is equivalent to the category of monoidal categories equipped with multiplication.
\end{corollary}
\noindent The previous corollary is proved by Joyal and Street \cite{JS93} by a direct application of a ``categorified'' Eckmann-Hilton argument. The following analogous result for $2$-categories appears to be new.
\begin{corollary}\label{cor:coh-bm2c}
The category of braided monoidal $2$-categories is equivalent to the category of Gray-monoids with multiplication.
\end{corollary}
\begin{proof}
Apply theorem(\ref{thm:A-infinity-app}) with $V=\Enrich 2$ equipped with the Gray tensor product and Lack's folklore model structure for 2-categories \cite{LackFolk2}.
\end{proof}
Thus theorem(\ref{thm:A-infinity-app}) should be considered as an $\infty$-generalisation of the above corollaries. We believe it sheds some light on the problem of defining the tensor product of $A_{\infty}$-algebras initiated by \cite{Saneblidze}. As explained in the introduction, the negative result of \cite{Markl} shows that there is no hope to get an ``honest'' tensor product of such algebras. Thus the multitensor $C$ constructed in theorem(\ref{thm:A-infinity-app}) is genuinely lax, and exhibits laxity as a way around the aforementioned negative result. In future work we will generalise this theorem to arbitrary dimensions.
\subsection{Unpacking $E'$}\label{ssec:explicit-lifting}
Let us now instantiate the constructions of section(\ref{sec:Dubuc}) to produce a more explicit description of the lifted multitensor $E'$. Beyond mere instantiation this task amounts to reformulating everything in terms of hom maps which live in $V$, because in our case the colimits being formed in $\ca GV$ at each stage of the construction are connected colimits diagrams whose morphisms are all identity on objects. Moreover these fixed object sets are of the form $\{0,...,n\}$ for $n \in \N$.
\\ \\
{\bf Notation}. We shall be manipulating sequences of data and so we describe here some notation that will be convenient. A sequence $(a_1,...,a_n)$ from some set will be denoted more tersely as $(a_i)$ leaving the length unmentioned. Similarly a sequence of sequences
\[ ((a_{11},...,a_{1n_1}),...,(a_{k1},...,a_{kn_k})) \]
of elements from some set will be denoted $(a_{ij})$ -- the variable $i$ ranges over $1{\leq}i{\leq}k$ and the variable $j$ ranges over $1{\leq}j{\leq}n_i$. Triply-nested sequences look like this $(a_{ijk})$, and so on. These conventions are more or less implicit already in the notation we have been using all along for multitensors. See especially section(\ref{ssec:LMC}) and \cite{EnHopI}. We denote by
\[ \con(a_{i_1,...,i_k}) \]
the ordinary sequence obtained from the $k$-tuply nested sequence $(a_{i_1,...,i_k})$ by concatenation. In particular given a sequence $(a_i)$, the set of $(a_{ij})$ such that $\con(a_{ij})=(a_i)$ is just the set of partitions of the original sequence into doubly-nested sequences, and will play an important role below. This is because to give the substitution maps for a multitensor $E$ on $V$, is to give maps
\[ \sigma : \opE\limits_i\opE\limits_j X_{ij} \to \opE\limits_i X_i \]
for all $(X_{ij})$ and $(X_i)$ from $V$ such that $\con(X_{ij})=(X_i)$.
\\ \\
\indent The monad map $\phi:M \to S$ is taken as $\Gamma(\psi):\ca GE_1 \to \Gamma(E)$ where $\psi:E_1 \to E$ is the inclusion of the unary part of the multitensor $E$. Note the notational abuse -- we regard write $E_1$ for the multitensor on $V$ obtained from $E$ by ignoring (ie setting to constant at $\emptyset$) the non-unary parts, but also as the monad on $V$ -- and so $\ca GE_1=\Gamma(E_1)$ as monads. The role of $(X,x)$ in $V^M$ is played by sequences $(X_i,x_i)$ of $E_1$-algebras regarded as objects of $\ca GV^{E_1}$ as in section(\ref{ssec:DefNMonad}).
The transfinite induction produces for each ordinal $m$ and each sequence of $E_1$-algebras as above of length $n$, morphisms
\[ \begin{array}{l} {v^{(m)}_{(X_i,x_i)} : SQ_m(X_i,x_i) \rightarrow Q_{m{+}1}(X_i,x_i)} \\ {q^{(m)}_{(X_i,x_i)}:Q_m(X_i,x_i) \to Q_{m{+}1}(X_i,x_i)} \\ {q^{({<}m)}_{(X_i,x_i)}:S(X_i) \to Q_m(X_i,x_i)} \end{array} \]
in $\ca GV$ which are identities on objects, and thus we shall now evolve this notation so that it only records what's going on in the hom between $0$ and $n$. By the definition of $S$ we have the equation on the left
\[ \begin{array}{lccr} {S(X_i)(0,n) = \opE\limits_iX_i} &&& {Q_m(X_i,x_i)(0,n) = {\opEm\limits_i}(X_i,x_i)} \end{array} \]
and the equation on the right is a definition. Because of these definitions and that of $S$ we have the equation
\[ SQ_m(X_i,x_i)(0,n) = \coprod\limits_{\con(X_{ij},x_{ij}){=}(X_i,x_i)} \opE\limits_i\opEm\limits_j (X_{ij},x_{ij}). \]
The data for the hom maps of the $v^{(m)}$ thus consists of morphisms
\[ \begin{array}{c} {v^{(m)}_{(X_{ij},x_{ij})} : \opE\limits_i\opEm\limits_j (X_{ij},x_{ij}) \to \opEmpone\limits_i (X_i,x_i)} \end{array} \]
in $V$ whenever one has $\con(X_{ij},x_{ij})=(X_i,x_i)$ as sequences of $E_1$-algebras.
To summarise, the output of the transfinite process we are going to describe is, for each ordinal $m$, the following data. For each sequence $(X_i,x_i)$ of $E_1$-algebras, one has an object
\[ \opEm\limits_i (X_i,x_i) \]
and morphisms
\[ \begin{array}{l} {v^{(m)}_{(X_{ij},x_{ij})}} : {\opE\limits_i\opEm\limits_j (X_{ij},x_{ij}) \to \opEmpone\limits_i (X_i,x_i)} \\
{q^{(m)}_{(X_i,x_i)}} : {\opEm\limits_i (X_i,x_i) \to \opEmpone\limits_i (X_i,x_i)} \\
{q^{(<m)}_{(X_i,x_i)}} : {\opE\limits_iX_i \to \opEm\limits_i (X_i,x_i)} \end{array} \]
of $V$ where $\con(X_{ij},x_{ij})=(X_i,x_i)$.
\\ \\
{\bf Initial step}. First we put $\opEzero\limits_i (X_i,x_i) = \opE\limits_i X_i$, $q^{({<}0)}_{(X_i,x_i)_i} = \id$, and then form the coequaliser
\begin{equation}\label{eq:coeq}
\xygraph{!{0;(2,0):} {\opE\limits_iE_1X_i}="l" [r] {\opE\limits_iX_i}="m" [r] {\opEone\limits_i(X_i,x_i)}="r" "l":@<2ex>"m"^-{\sigma} "l":"m"_-{\opE\limits_ix_i}:@<1ex>"r"^-{q^{(0)}_{(X_i,x_i)}}} \end{equation}
in $V$ to define $q^{(0)}$. Put $v^{(0)}=q^{(0)}\sigma$ and $q^{({<}1)}=q^{(0)}$.
\\ \\
{\bf Inductive step}. Assuming that $v^{(m)}$, $q^{(m)}$ and $q^{({<}m{+}1)}$ are given, we have maps
\[ \begin{array}{lcr} {\xybox{\xygraph{!{0;(2,0):} {\opE\limits_i\opE\limits_j\opEm\limits_k}="l" [r] {\opE\limits_i\opEmpone\limits_{jk}}="r" "l":"r"^-{\opE\limits_iv^{(m)}}}}} && {\xybox{\xygraph{!{0;(2,0):} {\opE\limits_i\opE\limits_j\opEm\limits_k}="l" [r] {\opE\limits_{ij}\opEm\limits_k}="m" [r] {\opE\limits_{ij}\opEmpone\limits_k}="r" "l":"m"^-{{\sigma}\opEm\limits_k}:"r"^-{q^{(m)}}}}} \end{array} \]
and these are used to provide the parallel maps in the coequaliser
\[ \xygraph{!{0;(1.5,0):} {\coprod\limits_{\con(X_{ijk},x_{ijk})=(X_i,x_i)} \opE\limits_i\opE\limits_j\opEm\limits_k(X_{ijk},x_{ijk})}="l" [d] {\coprod\limits_{\con(X_{ij},x_{ij})=(X_i,x_i)} \opE\limits_i\opEmpone\limits_j(X_{ij},x_{ij})}="m" [d] {\opEmptwo\limits_i (X_i,x_i)}="r" "l":@<-2ex>"m" "l":@<2ex>"m":"r"^-{(v^{(m{+}1)}_{(X_{ij},x_{ij})})}} \]
which defines the $v^{(m{+}1)}$, the commutative diagram
\[ \xygraph{{\opEmpone\limits_i (X_i,x_i)}="l" [d] {E_1\opEmpone\limits_i (X_i,x_i)}="il" [r(4)] {\coprod\limits_{\con(X_{ij},x_{ij})=(X_i,x_i)} \opE\limits_i\opEmpone\limits_j(X_{ij},x_{ij})}="ir" [ru] {\opEmptwo\limits_i (X_i,x_i)}="r" "l":"il"_-{u}:"ir":"r"^(.35){v^{(m{+}1)}_{(X_i,x_i)}}:@{<-}"l"_-{q^{(m{+}1)}_{(X_i,x_i)}}} \]
in which the unlabelled map is the evident coproduct inclusion defines $q^{(m{+}1)}$, and $q^{({<}m{+}2)}=q^{(m{+}1)}q^{({<}m{+}1)}$.
\\ \\
{\bf Limit step}. Define $\opEm\limits_i (X_i,x_i)$ as the colimit of the sequence given by the objects $\opEr\limits_i (X_i,x_i)$ and morphisms $q^{(r)}$ for $r < m$, and $q_{<{m}}$ for the component of the universal cocone at $r=0$.
\[ \xygraph{!{0;(0,-1.5):(0,-3.45)::} {\colsum\limits_{\con(X_{ijk},x_{ijk})=(X_i,x_i)} \opE\limits_i\opE\limits_j\opEr\limits_k(X_{ijk},x_{ijk})}="tl" [r] {\colsum\limits_{\con(X_{ij},x_{ij})=(X_i,x_i)} \opE\limits_i\opEr\limits_j(X_{ij},x_{ij})}="tm" [r] {\colim_{r{<}m} \opEr\limits_i (X_i,x_i)}="tr" [d] {\opEm\limits_i (X_i,x_i)}="br" [l] {\coprod\limits_{\con(X_{ij},x_{ij})=(X_i,x_i)} \opE\limits_i\opEm\limits_j(X_{ij},x_{ij})}="bm" [l] {\coprod\limits_{\con(X_{ijk},x_{ijk})=(X_i,x_i)} \opE\limits_i\opE\limits_j\opEm\limits_k(X_{ijk},x_{ijk})}="bl" "tl":@<1ex>"tm"^-{\sigma^{(<{m})}}:@<1ex>"tr"^-{v^{(<{m})}} "tl":@<-1ex>"tm"_-{(Ev)^{(<{m})}}:@<-1ex>@{<-}"tr"_-{u^{(<{m})}} "bl":"bm"_-{\mu}:@{<-}"br"_-{uc} "tl":"bl"^{o_{m,2}} "tm":"bm"^{o_{m,1}} "tr":@{=}"br"} \]
As before we write $o_{m,1}$ and $o_{m,2}$ for the obstruction maps, and $c$ denotes the evident coproduct injection. The maps $\sigma^{(<{m})}$, $(Ev)^{(<{m})}$, $v^{(<{m})}$ and $u^{(<{m})}$ are by definition induced by $\sigma{\opEr}$, $(Ev)^{(r)}$, $v^{(r)}$ and $u{\opEr}$ for $r < m$ respectively. Define $v^{(m)}$ as the coequaliser of $o_{m,1}\sigma^{(<{m})}$ and $o_{m,1}(Ev)^{<{m}}$, $q^{(m)}=v^{(m)}(u{\opEm})$ and $q^{(<{m{+}1})}=q^{(m)}q^{(<{m})}$.
\\ \\
Instantiating corollary(\ref{cor:explicit-phi-shreik}) to the present situation gives
\begin{corollary}\label{cor:lifted-obj}
Let $V$ be a locally presentable category, $\lambda$ a regular cardinal, and $E$ a distributive $\lambda$-accessible multitensor on $V$. Then for any ordinal $m$ with $|m| \geq \lambda$ one may take
\[ (\opEm\limits_i (X_i,x_i), a(X_i,x_i)) \]
where the action $a(X_i,x_i)$ is given as the composite
\[ \xygraph{!{0;(3,0):} {E_1\opEm\limits_i (X_i,x_i)}="l" [r] {\opEmpone\limits_i (X_i,x_i)}="m" [r] {\opEm\limits_i (X_i,x_i)}="r" "l":"m"^-{v^{(m)}}:"r"^-{(q^{(m)})^{-1}}} \]
as an explicit description of the object map of the lifted multitensor $E'$ on $V^{E_1}$.
\end{corollary}
In corollaries (\ref{cor:phi-shreik-simple}) and (\ref{cor:vexp-simple}), in which the initial data is a monad map $\phi:M \to S$ between monads on a category $V$ together with an algebra $(X,x)$ for $M$, we noted the simplification of our constructions when $S$ and $S^2$ preserve the coequaliser
\begin{equation}\label{eq:monad-coeq} \xygraph{!{0;(2,0):} {SMX}="l" [r] {SX}="m" [r] {Q_1X}="r" "l":@<-1ex>"m"_-{Sx} "l":@<1ex>"m"^-{\mu^SS(\phi)}:"r"^-{q_0}} \end{equation}
in $V$, which is part of the first step of the inductive construction of $\phi_!$. In the present situation the role of $V$ is played by the category $\ca GV$, the role of $S$ is played by $\Gamma E$, and the role of $(X,x)$ played by a given sequence $(X_i,x_i)$ of $E_1$-algebras, and so the role of the coequaliser (\ref{eq:monad-coeq}) is now played by the coequaliser
\begin{equation}\label{eq:monad-coeq2} \xygraph{!{0;(2,0):} {\Gamma E(E_1X_i)}="l" [r(1.5)] {\Gamma E(X_i)}="m" [r] {Q_1}="r" "l":@<-1ex>"m"_-{Sx} "l":@<1ex>"m"^-{\mu^SS(\phi)}:"r"^-{q^{(0)}}} \end{equation}
in $\ca GV$. Here we have denoted by $Q_1$ the $V$-graph with objects $\{0,...,n\}$ and homs given by
\[ Q_1(i,j) = \left\{\begin{array}{lll} {\emptyset} && {\textnormal{if $i>j$}} \\ {\opEone\limits_{i{<}k{\leq}j}(X_k,x_k)} && {\textnormal{if $i \leq j$.}} \end{array}\right. \]
Taking the hom of (\ref{eq:monad-coeq2}) between $0$ and $n$ gives the coequaliser
\begin{equation}\label{eq:mult-coeq} \xygraph{!{0;(2,0):} {\opE\limits_iE_1X_i}="l" [r] {\opE\limits_iX_i}="m" [r] {\opEone\limits_i(X_i,x_i)}="r" "l":"m"_-{\opE\limits_ix_i} "l":@<2ex>"m"^-{\sigma}:@<1ex>"r"^-{q^{(0)}}} \end{equation}
in $V$ which is part of the first step of the explicit inductive construction of $E'$. We shall refer to (\ref{eq:mult-coeq}) as the \emph{basic coequaliser associated to the sequence $(X_i,x_i)$} of $E_1$-algebras. Note that all coequalisers under discussion here are reflexive coequalisers, with the common section for the basic coequalisers given by the maps $\opE\limits_iu_{X_i}$.
The basic result which expresses why reflexive coequalisers are nice, is the $3{\times}3$-lemma, which we record here for the reader's convenience. A proof can be found in \cite{PTJ-topos77}.
\begin{lemma}\label{lem:3by3}
{\bf $3{\times}3$-lemma}. Given a diagram
\[ \xymatrix @R=3em @C=3em {A \ar@<1ex>[r]^-{f_1} \ar@<-1ex>[r]_{g_1} \ar@<1ex>[d]^{b_1} \ar@<-1ex>[d]_{a_1}
& B \ar[r]^-{h_1} \ar@<1ex>[d]^{b_2} \ar@<-1ex>[d]_{a_2} & C \ar@<1ex>[d]^{b_3} \ar@<-1ex>[d]_{a_3} \\
D \ar@<1ex>[r]^-{f_2} \ar@<-1ex>[r]_-{g_2} & E \ar[r]^-{h_2} & F \ar[d]^{c} \\ && H} \]
in a category such that: (1) the two top rows and the right-most column are coequalisers, (2) $a_1$ and $b_1$ have a common section, (3) $f_1$ and $g_1$ have a common section, (3) $f_2a_1{=}a_2f_1$, (4) $g_2b_1{=}b_2g_1$, (5) $h_2a_2{=}a_3h_1$ and (6) $h_2b_2{=}b_3h_1$; then $ch_2$ is a coequaliser of $f_2a_1{=}a_2f_1$ and $g_2b_1{=}b_2g_1$.
\end{lemma}
\noindent If $F:{\ca A_1}{\times}...{\times}{\ca A_n}{\rightarrow}{\ca B}$ is a functor which preserves connected colimits of a certain type, then it also preserves these colimits in each variable separately, because for a connected colimit, a cocone involving only identity arrows is a universal cocone. The most basic corollary of the $3{\times}3$-lemma says that the converse of this is true for reflexive coequalisers.
\begin{corollary}\label{cor:3by3}
Let $F:{\ca A_1}{\times}...{\times}{\ca A_n}{\rightarrow}{\ca B}$ be a functor. If $F$ preserves reflexive coequalisers in each variable separately then $F$ preserves reflexive coequalisers.
\end{corollary}
\noindent and this can be proved by induction on $n$ using the $3{\times}3$-lemma in much the same way as \cite{LkMonFinMon} lemma(1). The most well-known instance of this is
\begin{corollary}\label{cor:3by3-2}\cite{LkMonFinMon}
Let $\ca V$ be a biclosed monoidal category. Then the $n$-fold tensor product of reflexive coequalisers in $\ca V$ is again a reflexive coequaliser.
\end{corollary}
\noindent In particular note that by corollary(\ref{cor:3by3}) a multitensor $E$ preserves (some class of) reflexive coequalisers iff it preserves them in each variable separately.
Returning to our basic coequalisers an immediate consequence of the explicit description of $\Gamma E$ and corollary(\ref{cor:3by3}) is
\begin{lemma}\label{lem:reformulate-simplifying-conditions}
Let $E$ be a distributive multitensor on $V$ a cocomplete category, and $(X_i,x_i)$ a sequence of $E_1$-algebras. If $E$ preserves the basic coequalisers associated to all the subsequences of $(X_i,x_i)$, then for all $r \in \N$, $(\Gamma E)^r$ preserves the coequaliser (\ref{eq:monad-coeq2}).
\end{lemma}
\noindent and applying this lemma and corollary(\ref{cor:phi-shreik-simple}) gives
\begin{corollary}\label{cor:lifted-obj-simple}
Let $V$ be a locally presentable category, $\lambda$ a regular cardinal, $E$ a distributive $\lambda$-accessible multitensor on $V$ and $(X_i,x_i)$ a sequence of $E_1$-algebras. If $E$ preserves the basic coequalisers associated to all the subsequences of $(X_i,x_i)$, then one may take
\[ \opEpr\limits_i(X_i,x_i) = (\opEone\limits_i (X_i,x_i), a) \]
where the action $a$ is defined as the unique map such that $aE_1(q^{(0)})=q^{(0)}\sigma$.
\end{corollary}
\noindent Note in particular that when the sequence $(X_i,x_i)$ of $E_1$-algebras is of length $n=0$ or $n=1$, the associated basic coequaliser is absolute. In the $n=0$ case the basic coequaliser is constant at $E_0$, and when $n=1$ the basic coequaliser may be taken to be the canonical presentation of the given $E_1$-algebra. Thus in these cases it follows from corollary(\ref{cor:lifted-obj-simple}) that $E'_0=(E_0,\sigma)$ and $E_1'(X,x)=(X,x)$. Reformulating the explicit description of the unit in corollary(\ref{cor:vexp-simple}) one recovers the fact from our explicit descriptions, that the unit of $E'$ is the identity, which was of course true by construction.
To complete the task of giving a completely explicit description of the multitensor $E'$ we now turn to unpacking its substitution. So we assume that $E$ is a distributive $\lambda$-accessible multitensor on $V$ a locally presentable category, and fix an ordinal $m$ so that $|m| \geq \lambda$, so that $E'$ may be constructed as $E^{(m)}$ as in corollary(\ref{cor:lifted-obj}). By transfinite induction on $r$ we shall generate the following data:
\[ \sigma^{(r)}_{X_{ij},x_{ij}} : \opEr\limits_i(\opEm\limits_j(X_{ij}),x_{ij}) \to \opEm\limits_i(X_i,x_i) \]
and $\sigma^{(r{+}1)}_{X_{ij},x_{ij}}$ whenever $\con(X_{ij},x_{ij})=(X_i,x_i)$, such that
\[ \xygraph{!{0;(2.5,0):(0,.5)::} {\opE\limits_i\opEr\limits_j\opEm\limits_k}="tl" [r] {\opErpone\limits_{ij}\opEm\limits_k}="tr" [d] {\opEm\limits_{ijk}}="br" [l] {\opE\limits_i\opEm\limits_{jk}}="bl" "tl":"tr"^-{v^{(r)}E^{(m)}}:"br"^-{\sigma^{(r{+}1)}}:@{<-}"bl"^-{(q^{(m)})^{-1}v^{(m)}}:@{<-}"tl"^-{\opE\limits_i\sigma^{(r)}}} \]
commutes.
\\ \\
{\bf Initial step}. Define $\sigma^{(0)}$ to be the identity and $\sigma^{(1)}$ as the unique map such that $\sigma^{(1)}q^{(0)}=(q^{(m)})^{-1}v^{(m)}$ by the universal property of the coequaliser $q^{(0)}$.
\\ \\
{\bf Inductive step}. Define $\sigma^{(r{+}2)}$ as the unique map such that
\[ \sigma^{(r{+}2)}(v^{(r{+}1)}E^{(m)})=(q^{(m)})^{-1}v^{(m)}(\opE\limits_i\sigma^{(r{+}1)}) \]
using the universal property of $v^{(r{+}1)}$ as a coequaliser.
\\ \\
{\bf Limit step}. When $r$ is a limit ordinal define $\sigma^{(r)}$ as induced by the $\mu^{(s)}$ for $s<r$ and the universal property of $E^{(r)}$ as the colimit of the sequence of the $E^{(s)}$ for $s<r$. Then define $\sigma^{(r{+}1)}$ as the unique map such that
\[ \sigma^{(r{+}1)}(v^{(r)}E^{(m)})=(q^{(m)})^{-1}v^{(m)}(\opE\limits_i\sigma^{(r)}) \]
using the universal property of $v^{(r)}$ as a coequaliser.
\\ \\
The fact that the transfinite construction just specified was obtained from that for corollary(\ref{cor:induced-monad-very-explicit}), by taking $S=\Gamma E$ and looking at the homs, means that by corollaries (\ref{cor:induced-monad-very-explicit}) and (\ref{cor:vexp-simple}) one has
\begin{corollary}\label{cor:induced-substitution-very-explicit}
Let $V$ be a locally presentable category, $\lambda$ a regular cardinal, $E$ a distributive $\lambda$-accessible multitensor on $V$ and $(X_i,x_i)$ a sequence of $E_1$-algebras. Then one has
\[ \sigma'_{(X_i,x_i)} = \sigma^{(m)}_{(X_i,x_i)} \]
as an explicit description of the substitution of $E'$. If moreover $E$ preserves the basic coequalisers of all the subsequences of $(X_i,x_i)$, then one may take $\sigma^{(1)}_{(X_i,x_i)}$ as the explicit description of the substitution.
\end{corollary}
\subsection{Functoriality of lifting}\label{ssec:functoriality-lifting}
Recall \cite{Str72} \cite{LS00} that when $\ca K$ has Eilenberg-Moore objects, the one and 2-cells of the 2-category $\MND(\ca K)$ admit another description. Given monads $(V,T)$ and $(W,S)$ in $\ca K$, and writing $U^T:V^T{\rightarrow}V$ and $U^S:W^S{\rightarrow}W$ for the one-cell data of their respective Eilenberg-Moore objects, to give a monad functor $(H,\psi):(V,T){\rightarrow}(W,S)$, is to give $H$ and $\tilde{H}:V^T{\rightarrow}W^S$ such that $U^S\tilde{H}=HU^T$. This follows immediately from the universal property of Eilenberg-Moore objects. Similarly to give a monad 2-cell $\phi:(H_1,\psi_1){\rightarrow}(H_2,\psi_2)$ is to give $\phi:H_1{\rightarrow}H_2$ and $\tilde{\phi}:\tilde{H_1}{\rightarrow}\tilde{H_2}$ commuting with $U^T$ and $U^S$. Note that Eilenberg-Moore objects in $\CAT/\Set$ are computed as in $\CAT$, and we shall soon apply these observations to the case $\ca K = \CAT/\Set$.
Suppose we have a lax monoidal functor $(H,\psi):(V,E){\rightarrow}(W,F)$, that $V$ and $W$ are locally presentable, and that $E$ and $F$ are accessible. Then we obtain a commutative diagram
\[ \xygraph{!{0;(1.5,0):(0,.667)::} {\Enrich E}="tl" [r] {\ca GV^{E_1}}="tm" [r] {\ca GV}="tr" [d] {\ca GW}="br" [l] {\ca GW^{F_1}}="bm" [l] {\Enrich F}="bl" "tl":"tm":"tr" "bl":"bm":"br" "tl":"bl" "tm":"bm" "tr":"br"} \]
of forgetful functors in $\CAT/\Set$. Applying the previous paragraph to the left-most square gives a monad morphism $(\ca GV,\Gamma{E'}){\rightarrow}(\ca GW,\Gamma{F'})$, and then applying $\overline{(-)}$ to this gives the lax monoidal functor
\[ (\psi_1^*,\psi') : (V^{E_1},E'){\rightarrow}(W^{F_1},F') \]
between the induced lifted multitensors. Arguing similarly for monoidal transformations and monad 2-cells, one finds that the assignment $(V,E) \mapsto (V^{E_1},E')$ is 2-functorial.
Let $\varepsilon:E{\rightarrow}\ca T^{\times}_{{\leq}n}$ be an $n$-multitensor. In terms of the previous paragraph, this is the special case $V=W=\ca G^n\Set$, $H=\id$, $\psi{=}\varepsilon$. The lifted multitensor corresponding to $\ca T^{\times}_{{\leq}n}$ is just cartesian product for strict $n$-categories. One has a component of $\varepsilon'$ for each sequence $((X_1,x_1),...,(X_n,x_n))$ of strict $n$-categories, and since $\varepsilon_1^*:\Enrich n{\rightarrow}\Alg {E_1}$ as a right adjoint preserves products, this component may be regarded as a map
\begin{equation}\label{eq:mult->prod} \begin{array}{c} {\varepsilon'_{(X_i,x_i)} : {\opE\limits_i}' \varepsilon_1^*(X_i,x_i) \rightarrow \prod\limits_i \varepsilon_1^*(X_i,x_i)} \end{array} \end{equation}
of $E_1$-algebras. If in particular $E_1$ is itself $\ca T_{{\leq}n}$ and $\varepsilon_1{=}\id$, then these components of $\varepsilon'$ give a canonical comparison from the lifted tensor product $E'$ of $n$-categories to the cartesian product. For instance, when $E$ is the multitensor corresponding to the 3-operad for Gray categories, then $\varepsilon'$ gives the well-known comparison map from the Gray tensor product of 2-categories to the cartesian product, which we recall is actually a componentwise biequivalence.
Returning to the general situation, it is routine to unpack the assignment $(H,\psi) \mapsto (\psi_1^*,\psi')$ as in section(\ref{ssec:explicit-lifting}) and so obtain the following 1-cell counterpart of corollary(\ref{cor:lifted-obj-simple}).
\begin{corollary}\label{cor:free-lift-1cell}
Let $(H,\psi):(V,E){\rightarrow}(W,F)$ be a lax monoidal functor such that $V$ and $W$ are locally presentable, and $E$ and $F$ are accessible. Let $(X_1,...,X_n)$ be a sequence of objects of $V$. Then the component of $\psi'$ at the sequence \[ (E_1X_1,...,E_nX_n) \] of free $E_1$-algebras is just $\psi_{X_i}$.
\end{corollary}
\section{Contractibility}\label{sec:contractibility}
\subsection{Trivial Fibrations}
Let $V$ be a category and $\ca I$ a class of maps in $V$. Denote by $\ca I^{\uparrow}$ the class of maps in $V$ that have the right lifting property with respect to all the maps in $\ca I$. That is to say, $f:X{\rightarrow}Y$ is in $\ca I^{\uparrow}$ iff for every $i:S{\rightarrow}B$ in $\ca I$, $\alpha$ and $\beta$ such that the outside of
\[ \xygraph{{S}="tl" [r] {X}="tr" [d] {Y}="br" [l] {B}="bl" "tl" (:"tr"^-{\alpha}:"br"^{f},:"bl"_{i}(:"br"_-{\beta},:@{.>}"tr"|{\gamma}))} \]
commutes, then there is a $\gamma$ as indicated such that $f\gamma{=}\beta$ and $\gamma{i}=\alpha$. An $f \in \ca I^{\uparrow}$ is called a \emph{trivial $\ca I$-fibration}. The basic facts about $\ca I^{\uparrow}$ that we shall use are summarised in
\begin{lemma}\label{lem:basic-tf}
Let $V$ be a category, $\ca I$ a class of maps in $V$, $J$ a set and \[ (f_j:X_j{\rightarrow}Y_j \,\,\, | \,\,\, j \in J) \]
a family of maps in $V$.
\begin{enumerate}
\item $\ca I^{\uparrow}$ is closed under composition and retracts.\label{tfib1}
\item If $V$ has products and each of the $f_j$ is a trivial $\ca I$-fibration, then
\[ \begin{array}{c} {\prod\limits_{j} f_j : \prod\limits_j X_j \rightarrow \prod\limits_j Y_j} \end{array} \]
is also a trivial $\ca I$-fibration.\label{tfib2}
\item The pullback of a trivial $\ca I$-fibration along any map is a trivial $\ca I$-fibration.\label{tfib3}
\item If $V$ is extensive and $\coprod_jf_j$ is a trivial $\ca I$-fibration, then each of the $f_j$ is a trivial fibration.\label{tfib4}
\item If $V$ is extensive, the codomains of maps in $\ca I$ are connected and each of the $f_j$ is a trivial $\ca I$-fibration, then $\coprod_jf_j$ is a trivial $\ca I$-fibration.\label{tfib5}
\end{enumerate}
\end{lemma}
\begin{proof}
(\ref{tfib1})-(\ref{tfib3}) is standard. If $V$ is extensive then the squares
\[ \xygraph{!{0;(1.5,0):(0,.666)::}
{X_j}="tl" [r] {\coprod_jX_j}="tr" [d] {\coprod_jY_j}="br" [l] {Y_j}="bl" "tl" (:"tr":"br"^{\coprod_jf_j},:"bl"_{f_j}:"br")} \]
whose horizontal arrows are the coproduct injections are pullbacks, and so (\ref{tfib4}) follows by the pullback stability of trivial $\ca I$-fibrations. As for (\ref{tfib5}) note that for $i:S{\rightarrow}B$ in $\ca I$, the connectedness of $B$ ensures that any square as indicated on the left
\[ \xygraph{{\xybox{\xygraph{!{0;(1.5,0):(0,.666)::} {S}="tl" [r] {\coprod_jX_j}="tr" [d] {\coprod_jY_j}="br" [l] {B}="bl" "tl" (:"tr":"br"^{\coprod_jf_j},:"bl"_{i}:"br")}}}
[r(4)]
{\xybox{\xygraph{!{0;(1.5,0):(0,.666)::} {S}="tl" [r] {X_j}="tr" [d] {Y_j}="br" [l] {B}="bl" "tl" (:"tr":"br"^{f_j},:"bl"_{i}:"br")}}}} \]
factors through a unique component as indicated on the right, enabling one to induce the desired filler.
\end{proof}
\begin{definition}
Let $F,G:W{\rightarrow}V$ be functors and $\ca I$ be a class of maps in $V$. A natural transformation $\phi:F{\implies}G$ is a \emph{trivial $\ca I$-fibration} when its components are trivial $\ca I$-fibrations.
\end{definition}
\noindent Note that since trivial $\ca I$-fibrations in $V$ are pullback stable, this reduces, in the case where $W$ has a terminal object $1$ and $\phi$ is cartesian, to the map $\phi_1:F1{\rightarrow}G1$ being a trivial $\ca I$-fibration.
Given a category $V$ with an initial object, and a class of maps $\ca I$ in $V$, we denote by $\ca I^+$ the class of maps in $\ca GV$ containing the maps{\footnotemark{\footnotetext{Recall that $0$ is the $V$-graph with one object whose only hom is initial, or in other words the representing object of the functor $\ca GV{\rightarrow}\Set$ which sends a $V$-graph to its set of objects.}}}
\[ \begin{array}{lccr} {\emptyset \rightarrow 0} &&& {(i) : (S) \rightarrow (B)} \end{array} \]
where $i \in \ca I$. The proof of the following lemma is trivial.
\begin{lemma}\label{lem:ind-tf}
Let $V$ be a category with an initial object and $\ca I$ a class of maps in $V$. Then $f:X{\rightarrow}Y$ is a trivial $\ca I^+$-fibration iff it is surjective on objects and all its hom maps are trivial $\ca I$-fibrations.
\end{lemma}
\noindent In particular starting with $V=\PSh {\G}$ the category of globular sets and $\ca I_{-1}$ the empty class of maps, one generates a sequence of classes of maps $\ca I_n$ of globular sets by induction on $n$ by the formula $\ca I_{n+1}=(\ca I_{n})^+$ since $\ca G(\PSh {\G})$ may be identified with $\PSh {\G}$, and moreover one has inclusions $\ca I_n \subset \ca I_{n+1}$. More explicitly, the set $\ca I_n$ consists of $(n+1)$ maps: for $0{\leq}k{\leq}n$ one has the inclusion $\partial{k} \hookrightarrow k$, where $k$ here denotes the representable globular set, that is the ``$k$-globe'',
and $\partial{k}$ is the $k$-globe with its unique $k$-cell removed. One defines $\ca I_{{\leq}\infty}$ to be the union of the $\ca I_n$'s.
Note that by definition $\ca I_{{\leq}\infty}=\ca I_{{\leq}\infty}^{+}$.
There is another version of the induction just described to produce, for each $n \in \N$, a class $\ca I_{{\leq}n}$ of maps of $\ca G^n(\Set)$. The set $\ca I_{{\leq}0}$ consists of the functions
\[ \begin{array}{lccr} {\emptyset \rightarrow 0} &&& {0+0 \rightarrow 0,} \end{array} \]
so $\ca I_{\leq 0}^{\uparrow}$ is the class of bijective functions. For $n \in \N$, $\ca I_{{\leq}n{+}1}=\ca I_{{\leq}n}^{+}$. As maps of globular sets, the class $\ca I_{\leq n}$ consists of all the maps of $\ca I_{n}$ together with the unique map $\partial{(n{+}1)}{\rightarrow}n$.
\begin{definition}
Let $0{\leq}n{\leq}\infty$. An $n$-operad{\footnotemark{\footnotetext{The monad $\ca T_{{\leq}\infty}$ on globular sets is usually just denoted as $\ca T$: it is the monad whose algebras are strict $\omega$-categories.}}} $\alpha:A{\rightarrow}\ca T_{{\leq}n}$ is \emph{contractible} when it is a trivial $\ca I_{{\leq}n}$-fibration. An $n$-multitensor $\varepsilon:E{\rightarrow}\ca T_{{\leq}n}^{\times}$ is \emph{contractible} when it is a trivial $\ca I_{{\leq}n}$-fibration.
\end{definition}
\noindent By the preceeding two lemmas, an $(n+1)$-operad $\alpha:A{\rightarrow}\ca T_{{\leq}n{+}1}$ over $\Set$ is contractible iff the hom maps of $\alpha_1$ are trivial $\ca I_{\leq n}$-fibrations.
\subsection{Contractible operads versus contractible multitensors}
As one would expect an $(n+1)$-operad over $\Set$ is contractible iff its associated $n$-multitensor is contractible. This fact has quite a general explanation. Recall the 2-functoriality of $\Gamma$ described in section(\ref{ssec:Gamma-2-functor}) and that of the lifting described in section(\ref{ssec:functoriality-lifting}).
\begin{proposition}\label{prop:contractible}
Let $(H,\psi):(V,E){\rightarrow}(W,F)$ be a lax monoidal functor between distributive lax monoidal categories, and $\ca I$ a class of maps in $W$. Suppose that $W$ is extensive, $H$ preserves coproducts and the codomains of maps in $\ca I$ are connected. Then the following statements are equivalent
\begin{itemize}
\item[(1)] $\psi$ is a trivial $\ca I$-fibration.
\item[(2)] $\Gamma\psi$ is a trivial $\ca I^+$-fibration.
\end{itemize}
and moreover when in addition $V$ and $W$ are locally presentable and $E$ and $F$ are accessible, these conditions are also equivalent to
\begin{itemize}
\item[(3)] The components of $U^F\psi'$ at sequences $(E_1X_1,...,E_1X_n)$ of free $E_1$-algebras are trivial $\ca I$-fibrations.
\end{itemize}
\end{proposition}
\begin{proof}
For each $X \in \ca GV$ the component $\{\Gamma\psi\}_X$ is the identity on objects and for $a,b \in X_0$, the corresponding hom map is obtained as the composite of
\[ \begin{array}{c} {\coprod\limits_{a{=}x_0,...,x_n{=}b} \psi : \coprod\limits_{x_0,...,x_n} \opF\limits_iHX(x_{i-1}x_i) \rightarrow \coprod\limits_{x_0,...,x_n} H\opE\limits_iX(x_{i-1}x_i)} \end{array} \]
and the canonical isomorphism that witnesses the fact that $H$ preserves coproducts. In particular note that for any sequence $(Z_1,...,Z_n)$ of objects of $V$, regarded as $V$-graph in the usual way, one has \[ \{\Gamma\psi\}_{(Z_1,...,Z_n)} = \psi_{Z_1,...,Z_n}.\]
Thus $(1){\iff}(2)$ follows from lemmas(\ref{lem:basic-tf}) and (\ref{lem:ind-tf}). $(2){\iff}(3)$ follows immediately from corollary(\ref{cor:free-lift-1cell}).
\end{proof}
\begin{corollary}\label{cor:contractible}
Let $0{\leq}n{\leq}\infty$, $\alpha:A{\rightarrow}\ca T_{{\leq}n{+}1}$ be an $n{+}1$-operad over $\Set$ and $\varepsilon:E{\rightarrow}\ca T^{\times}_{{\leq}n}$ be the corresponding $n$-multitensor. TFSAE:
\begin{enumerate}
\item $\alpha:A{\rightarrow}\ca T_{{\leq}n{+}1}$ is contractible.
\item $\varepsilon:E{\rightarrow}\ca T^{\times}_{{\leq}n}$ is contractible.
\item The components of $\varepsilon'_{(X_i,x_i)}$ of section(\ref{ssec:functoriality-lifting})(\ref{eq:mult->prod}) are trivial $\ca I_{{\leq}n}$-fibrations of $n$-globular sets, when the $(X_i,x_i)$ are free strict $n$-categories.
\end{enumerate}
\end{corollary}
\begin{proof}
By induction one may easily establish that the codomains of the maps in any of the classes: $\ca I_n$, $\ca I_{{\leq}n}$, $\ca I_{{\leq}\infty}$ are connected so that proposition(\ref{prop:contractible}) may be applied.
\end{proof}
\begin{example}\label{ex:Gray-contractible}
Applying this last result to the 3-operad $G$ for Gray-categories, the contractibility of $G$ is a consequence of the fact that the canonical 2-functors from the Gray to the cartesian tensor product are identity-on-object biequivalences.
\end{example}
\subsection{Trimble \`{a} la Cheng II}\label{ssec:TCII}
Continuing the discussion from section(\ref{ssec:TCI}), we now explain why the operads which describe ``Trimble $n$-categories" are contractible. This result appears in \cite{ChengCompOp} as theorem(4.8) and exhibits Trimble $n$-categories as weak $n$-categories in the Batanin sense.
Let us denote by $\ca J$ the set of inclusions $S^{n{-}1}{\rightarrow}D^n$ of the $n$-sphere into the $n$ disk for $n \in \N$. As we remarked in section(\ref{ssec:enriched-graphs}) these may all be obtained by successively applying the reduced suspension functor $\sigma$ to the inclusion of the empty space into the point. As we recalled in section(\ref{ssec:TCI}), a basic ingredient of the Trimble definition is a version of the little intervals operad which acts on the path spaces of any space. A key property of this operad is that it is contractible -- a topological operad $A$ being contractible when for each $n$ the unique map $A_n \rightarrow 1$ is in $\ca J^{\uparrow}$. This is equivalent to saying that the cartesian multitensor map $A \rightarrow \prod$ is a trivial $\ca J$-fibration. A useful fact about the class trivial $\ca J$-fibrations is that it gets along with the construction of path-spaces in the sense of
\begin{lemma}\label{lem:tf-path-spaces}
If $f:X{\rightarrow}Y$ is a trivial $\ca J$-fibration then so is $f_{a,b}:X(a,b){\rightarrow}Y(fa,fb)$ for all $a,b \in X$.
\end{lemma}
\begin{proof}
To give a commutative square as on the left in
\[ \begin{array}{lccr} {\xybox{\xygraph{!{0;(2,0):(0,.5)::} {S^{n{-}1}}="tl" [r] {X(a,b)}="tr" [d] {Y(fa,fb)}="br" [l] {D^n}="bl" "tl"(:"tr":"br",:"bl":"br")}}} &&&
{\xybox{\xygraph{!{0;(2,0):(0,.5)::} {S^n}="tl" [r] {(a,X,b)}="tr" [d] {(fa,Y,fb)}="br" [l] {D^{n{+}1}}="bl" "tl"(:"tr":"br",:"bl":"br")}}} \end{array} \]
is the same as giving a commutative square in $\Top_{\bullet}$ as on the right in the previous display, by $\sigma \ladj h$. The square on the right admits a diagonal filler $D^{n{+}1} \rightarrow X$ since $f$ is a trivial $\ca J$-fibration, and thus so does the square on the left.
\end{proof}
We shall write $U_n:\Trimble n \rightarrow \ca G^n\Set$ for the forgetful functor for each $n$. The relationship between trivial fibrations of spaces and of globular sets is expressed in
\begin{proposition}\label{prop:pres-tf}
If $f:X{\rightarrow}Y$ is a trivial $\ca J$-fibration then $U_n\pi_n{f}$ is a trivial $\ca I_{\leq{n}}$-fibration.
\end{proposition}
\begin{proof}
We proceed by induction on $n$. Having the right lifting property with respect to the inclusions
\[ \begin{array}{lccr} {\emptyset \hookrightarrow 1} &&& {1{+}1=\partial{I} \hookrightarrow I} \end{array} \]
ensures that $f$ surjective and injective on path components, and thus is inverted by $\pi_0$. For the inductive step we assume that $U_n\pi_n$ sends trivial $\ca J$-fibrations to trivial $\ca I_{{\leq}n}$-fibrations and suppose that $f$ is a trivial $\ca J$-fibration. Then so are all the maps it induces between path spaces by lemma(\ref{lem:tf-path-spaces}). But from the inductive definition of $\Trimble {n{+}1}$ recalled in section(\ref{ssec:TCI}), we have $U_{n{+}1}\pi_{n{+}1}=\ca G(u_n\pi_n)P$ and so $U_{n{+}1}\pi_{n{+}1}(f)$ is a morphism of $(n{+}1)$-globular sets which is surjective on objects (as argued already in the $n=0$ case) and whose hom maps are trivial $\ca I_{{\leq}n}$-fibrations by induction. Thus the result follows by lemma(\ref{lem:ind-tf}).
\end{proof}
In section(\ref{ssec:TCI}) we exhibited $\Trimble n$ as the algebras of an $n$-operad by a straight-forward application of two abstract results -- propositions(\ref{prop:monad-trimble}) and (\ref{prop:operad-trimble}). We now provide a third such result relating to contractibility.
\begin{proposition}\label{prop:tf-formal}
Given the data and hypotheses of proposition(\ref{prop:operad-trimble}): $V$ is a lextensive category, $T$ a cartesian and coproduct preserving monad on $V$, $\psi:S{\rightarrow}T$ a $T$-operad and $(E,\varepsilon)$ a non-symmetric operad in $V^S$. Suppose furthermore that a class $\ca I$ of maps of $V$ is given, and that the non-symmetric operad $\alpha:E \rightarrow \prod$ and the $T$-operad $\psi$ are trivial $\ca I$ fibrations. Then the $\Gamma(T^{\times})$-operad
\[ \Gamma(E)\ca G(S) \rightarrow \Gamma(T^{\times}) \]
of proposition(\ref{prop:operad-trimble}) is a trivial $\ca I^{+}$-fibration.
\end{proposition}
\begin{proof}
By definition this monad morphism may be written as the composite $(\Gamma(\psi^{\times}))(\Gamma(\alpha)\ca G(S))$. Since $\psi$ is a trivial $\ca I$-fibration so is $\psi^{\times}$ by lemma(\ref{lem:basic-tf}), and thus $\Gamma(\psi^{\times})$ is a trivial $\ca I^{+}$-fibration by proposition(\ref{prop:contractible}). Since $\alpha$ is a trivial $\ca I$-fibration, $\Gamma(\alpha)$ is a trivial $\ca I^{+}$-fibration again by proposition(\ref{prop:contractible}), and so the result follows since trivial fibrations compose.
\end{proof}
Starting with a contractible topological operad $A$ which acts on path spaces, proposition(\ref{prop:pres-tf}) ensures that $U_n\pi_nA$ will be a contractible non-symmetric operad of $n$-globular sets. Then proposition(\ref{prop:tf-formal}) may be applied to give, by induction on $n$, the contractibility of the $n$-operad defining Trimble $n$-categories.
\section{Acknowledgements}\label{sec:Acknowledgements}
We would like to acknowledge Clemens Berger, Richard Garner, Andr\'{e} Joyal, Steve Lack, Joachim Kock, Jean-Louis Loday, Paul-Andr\'{e} Melli\`{e}s, Ross Street and Dima Tamarkin for interesting discussions on the substance of this paper. We would also like to acknowledge the Centre de Recerca Matem\`{a}tica in Barcelona for the generous hospitality and stimulating environment provided during the thematic year 2007-2008 on Homotopy Structures in Geometry and Algebra. The first author would like to acknowledge the financial support on different stages of this project of the Scott Russell Johnson Memorial Foundation, the Australian Research Council (grant No.DP0558372) and L'Universit\'{e} Paris 13. The second author would like to acknowledge the support of the ANR grant no. ANR-07-BLAN-0142.
The third author would like to acknowledge the laboratory PPS (Preuves Programmes Syst\`{e}mes) in Paris, the Max Planck Institute in Bonn and the Institut des Hautes \'Etudes Scientifique in Bures sur Yvette for the excellent working conditions he enjoyed during this project, as well as Macquarie University for the hospitality he enjoyed in August of 2008. | 61,040 |
TITLE: Polar to rectangular $r = 7$
QUESTION [0 upvotes]: I don't follow this at all. I have $r = 7$ and the formula states $x = r \cos\theta$ $y = r\sin\theta$ but my book gives $x^2 + y^2 = 49$ this is impossible. It doens't follow the formula at all. Since theta isn't present it is zero. This gives. $x = 7$ and $y = 0$ why does the formula fail?
REPLY [4 votes]: The formula is fine. We have $x^2 + y^2 = r^2$ by Pythagoras theorem. Or you can use the given formulas to get
$$
x^2 + y^2 = (r\cos(\theta))^2 + (r\sin(\theta))^2 = r^2(\cos^2(\theta) + \sin^2(\theta)) = r^2
$$since $\cos^2(\theta) + \sin^2(\theta) = 1$. For $r = 7$, we have $x^2 + y^2 = r^2 = 7^2 = 49$.
As you can see from the above, $\theta$ disappears because of the trigonometric identity (not because it's zero) which basically is a consequence of Pythagoras theorem.
REPLY [2 votes]: If
$$x=r\cos t\;,\;y=r\sin t\implies 49=x^2+y^2=r^2\left(\cos^2t+\sin^2t\right)=r^2\implies r=7$$
since always $\,r\ge 0\,$ ...
REPLY [1 votes]: This follows from the general conversion formulas from rectangular coordinates to polar coordinates. If $x = r\cos \theta, y = r\sin \theta$, then:
$$x^{2} + y^{2} = (r\cos \theta)^{2} + (r\sin\theta)^{2} = r^{2}(\cos^{2}\theta + \sin^{2}\theta) = r^{2}$$
Hence, if $r = 7$, $x^{2} + y^{2} = r^{2}$ tells us that $x^{2} + y^{2} = 49$.
N.B.: you can verify for yourself that these two equations describe the same geometric object by thinking about what they represent. In polar coordinates, the equation $r = 7$ describes the set of points in the plane that are fixed distance of $7$ from the origin. In the cartesian plane, this describes a circle of radius $7$ centered at the origin, which is described by the equation $x^{2} + y^{2} = 49$. | 45,471 |
TITLE: What is $\lim\limits_{z \to 0} z \cdot \sin|(\frac{1}{z})|$
QUESTION [1 upvotes]: What is $\lim\limits_{z \to 0} z \cdot \sin|(\frac{1}{z})|$ for $z \in \mathbb C^*$?
I tried solving this in the same way as $\lim\limits_{z \to 0} z \cdot \sin(\frac{1}{z})$ by finding limit along x and y axis separately. How to solve $\lim\limits_{x\to 0} x \cdot \sin(\frac{1}{x^{2}})$ which I got after opening the modulus.
REPLY [1 votes]: Since $|z|$ and $\sin(\frac{1}{|z|})$ are real, we have
$\lim\limits_{z \to 0} |z| \cdot \sin|(\frac{1}{z})|=0$, hence $\lim\limits_{z \to 0} z \cdot \sin|(\frac{1}{z})|=0.$ | 142,868 |
\begin{document}
\thispagestyle{empty}
\begin{flushright}
MZ-TH/09-49 \\
PITHA 09/35
\end{flushright}
\vspace{1.5cm}
\begin{center}
{\Large\bf Feynman graphs in perturbative quantum field theory\\
}
\vspace{1cm}
{\large Christian Bogner$^1$ and Stefan Weinzierl$^2$\\
\vspace{6mm}
{\small \em $^1$ Institut f\"ur Theoretische Physik E, RWTH Aachen,}\\
{\small \em D - 52056 Aachen, Germany}\\
\vspace{2mm}
{\small \em $^2$ Institut f{\"u}r Physik, Universit{\"a}t Mainz,}\\
{\small \em D - 55099 Mainz, Germany}\\
}
\end{center}
\vspace{2cm}
\begin{abstract}\noindent
{
In this talk we discuss mathematical structures associated to Feynman graphs.
Feynman graphs are the backbone of calculations in perturbative quantum field theory.
The mathematical structures -- apart from being of interest in their own right --
allow to derive algorithms for the computation of these graphs.
Topics covered are the relations of Feynman integrals to periods, shuffle algebras and
multiple polylogarithms.
}
\end{abstract}
\vspace*{\fill}
\section{Introduction}
High-energy particle physics has become a field where precision measurements have become possible.
Of course, the increase in experimental precision has to be matched with more accurate calculations
from the theoretical side.
As theoretical calculations are done within perturbation theory,
this implies the calculation of higher order corrections.
This in turn
relies to a large extent on our abilities to compute Feynman loop integrals.
These loop calculations are complicated by the occurrence
of ultraviolet and infrared singularities.
Ultraviolet divergences are related to the high-energy behaviour of the integrand.
Infrared divergences may occur if massless particles are present in the theory
and are related to the low-energy or collinear behaviour of the integrand.
Dimensional regularisation \cite{'tHooft:1972fi,Bollini:1972ui,Cicuta:1972jf}
is usually employed to regularise these singularities.
Within dimensional regularisation one considers the loop integral in $D$ space-time dimensions
instead of the usual four space-time dimensions.
The result is expanded as a Laurent series in the parameter $\eps=(4-D)/2$, describing the deviation
of the $D$-dimensional space from the usual four-dimensional space.
The singularities manifest themselves as poles in $1/\eps$.
Each loop can contribute a factor $1/\eps$ from the ultraviolet divergence and a factor $1/\eps^2$
from the infrared divergences.
Therefore an integral corresponding to a graph with $l$ loops can have poles up to $1/\eps^{2l}$.
At the end of the day, all poles disappear: The poles related to ultraviolet divergences
are absorbed into renormalisation constants.
The poles related to infrared divergences cancel in the final result for infrared-safe observables,
when summed over all degenerate states
or are absorbed into universal parton distribution functions.
The sum over all degenerate states involves a sum over contributions with different
loop numbers and different numbers of external legs.
However, intermediate results are in general a Laurent series in $\eps$
and the task is to determine the coefficients of this Laurent series up to a certain order.
At this point mathematics enters. We can use the algebraic structures associated to Feynman integrals
to derive algorithms to calculate them.
A few examples where the use of algebraic tools has been essential are the calculation of the
three-loop Altarelli-Parisi splitting functions
\cite{Moch:2004pa,Vogt:2004mw}
or the calculation of the two-loop amplitude for the process $e^+ e^- \rightarrow \; \mbox{3 jets}$
\cite{Garland:2001tf,Garland:2002ak,Moch:2002hm,GehrmannDeRidder:2007bj,GehrmannDeRidder:2007hr,GehrmannDeRidder:2008ug,GehrmannDeRidder:2009dp,Weinzierl:2008iv,Weinzierl:2009ms,Weinzierl:2009yz}.
On the other hand is the mathematics encountered in these calculations of interest in its own right
and has led in the last years to a fruitful interplay between mathematicians and
physicists.
Examples are the relation of Feynman integrals to mixed Hodge structures and motives, as well as
the occurrence of certain transcendental constants in the result of a calculation\cite{Bloch:2005,Bloch:2008jk,Bloch:2008,Brown:2008,Brown:2009a,Brown:2009b,Schnetz:2008mp,Schnetz:2009,Aluffi:2008sy,Aluffi:2008rw,Aluffi:2009b,Aluffi:2009a,Bergbauer:2009yu,Laporta:2002pg,Laporta:2004rb,Laporta:2008sx,Bailey:2008ib,Bierenbaum:2003ud}.
This article is organised as follows:
After a brief introduction into perturbation theory (sect.~\ref{sect:perturbation_theory}),
multi-loop integrals (sect.~\ref{sect:multi_loop}) and periods (sect.~\ref{sect:periods}),
we present in sect.~\ref{sect:theorem} a theorem stating that under rather weak assumptions the coefficients of the
Laurent series of any multi-loop integral are periods.
The proof is sketched in sect.~\ref{sect:sector_decomp} and sect.~\ref{sect:hironaka}.
Shuffle algebras are discussed in sect.~\ref{sect:shuffle}.
Sect.~\ref{sect:polylog} is devoted to multiple polylogarithms.
In sect.~\ref{sect:calc} we discuss how multiple polylogarithms emerge in the calculation of Feynman
integrals.
Finally, sect.~\ref{sect:conclusions} contains our conclusions.
\section{Perturbation theory}
\label{sect:perturbation_theory}
In high-energy physics experiments one is interested in scattering processes with two
incoming particles and $n$ outgoing particles.
Such a process is described by a scattering amplitude, which can be calculated in perturbation theory.
The amplitude has a perturbative expansion in the (small) coupling constant $g$:
\bq
\label{basic_perturbative_expansion}
{\mathcal A}_n & = & g^n \left( {\mathcal A}_n^{(0)} + g^2 {\mathcal A}_n^{(1)} + g^4 {\mathcal A}_n^{(2)} + g^6 {\mathcal A}_n^{(3)} + ... \right).
\eq
To the coefficient ${\mathcal A}_n^{(l)}$ contribute Feynman graphs with $l$ loops and $(n+2)$ external legs.
The recipe for the computation of ${\mathcal A}_n^{(l)}$ is as follows: Draw first all Feynman diagrams with the given number
of external particles and $l$ loops. Then translate each graph into a mathematical formula with the help of the Feynman
rules.
${\mathcal A}_n^{(l)}$ is then given as the sum of all these terms.
Feynman rules allow us to translate a Feynman graph into a mathematical formula.
These rules are derived from the fundamental Lagrange density of the theory,
but for our purposes it is sufficient to accept them as a starting point.
The most important ingredients are internal propagators, vertices and external lines.
For example, the rules for the propagators of a fermion or a massless gauge boson read
\bq
\mbox{Fermion:}\;\;\;
\begin{picture}(50,20)(0,10)
\ArrowLine(50,15)(0,15)
\end{picture} & = &
i \frac{p\!\!\!/ +m }{p^2-m^2+i\delta},
\nonumber \\
\mbox{Gauge boson:}\;\;\;
\begin{picture}(50,20)(0,10)
\Photon(0,15)(50,15){4}{4}
\end{picture} & = &
\frac{-i g_{\mu \nu}}{k^2+i\delta}.
\eq
Here $p$ and $k$ are the momenta of the fermion and the boson, respectively. $m$ is the mass of the fermion.
$p\!\!\!/=p_\mu \gamma^\mu$ is a short-hand notation for the contraction of the momentum with the Dirac matrices.
The metric tensor is denoted by $g_{\mu\nu}$ and the convention adopted here is to take the metric tensor as
$g_{\mu\nu} = \mbox{diag}(1,-1,-1,-1)$.
The propagator would have a pole for $p^2=m^2$, or phrased differently for $E=\pm \sqrt{\vec{p}^2+m^2}$.
When integrating over $E$, the integration contour has to be deformed to avoid these two poles.
Causality dictates into which directions the contour has to be deformed.
The pole on the negative real axis is avoided
by escaping into the lower complex half-plane, the pole at the positive real axis is avoided by a deformation
into the upper complex half-plane. Feynman invented the trick to add a small imaginary part $i\delta$ to the
denominator, which keeps track of the directions into which the contour has to be deformed.
In the following we will usually suppress the $i\delta$-term in order to keep the notation compact.
As a typical example for an interaction vertex let us look at the vertex involving a fermion pair and a gauge boson:
\bq
\begin{picture}(50,30)(0,15)
\Photon(0,20)(30,20){4}{2}
\Vertex(30,20){2}
\ArrowLine(30,20)(50,40)
\ArrowLine(50,0)(30,20)
\end{picture} & = &
i g \gamma^{\mu}.
\\
& & \nonumber
\eq
Here, $g$ is the coupling constant and $\gamma^\mu$ denotes the Dirac matrices.
At each vertex, we have momentum conservation: The sum of the incoming momenta equals the sum of the outgoing momenta.
To each external line we have to associate a factor, which describes the polarisation of the corresponding particle:
There is a polarisation vector $\eps^\mu(k)$ for each external gauge boson and a spinor
$\bar{u}(p)$, $u(p)$, $v(p)$ or $\bar{v}(p)$ for each external fermion.
Furthermore there are a few additional rules: First of all, there is an
integration
\bq
\int \frac{d^4k}{(2\pi)^4}
\eq
for each loop. Secondly, each closed fermion loop gets an extra factor of $(-1)$.
Finally, each diagram gets multiplied by a symmetry factor $1/S$,
where $S$ is the order of the permutation group
of the internal lines and vertices leaving the diagram unchanged when the external lines are fixed.
Having stated the Feynman rules, let us look at two examples:
The first example is a scalar two-point one-loop integral with zero
external momentum:
\bq
\begin{picture}(100,40)(0,30)
\Line(10,35)(25,35)
\Line(75,35)(90,35)
\CArc(50,35)(25,0,360)
\Vertex(25,35){2}
\Vertex(75,35){2}
\Text(5,35)[r]{\scriptsize $p=0$}
\Text(50,55)[t]{\scriptsize $k$}
\Text(50,5)[t]{\scriptsize $k$}
\end{picture}
& = &
\int \frac{d^4k}{(2\pi)^4} \frac{1}{(k^2)^2}
=
\frac{1}{(4\pi)^2} \int\limits_0^\infty dk^2 \frac{1}{k^2} =
\frac{1}{(4\pi)^2} \int\limits_0^\infty \frac{dx}{x}.
\\
\nonumber
\eq
This integral diverges at $k^2\rightarrow \infty$ as well as at $k^2\rightarrow 0$.
The former divergence is called ultraviolet divergence, the later is called infrared divergence.
Any quantity, which is given by a divergent integral, is of course an ill-defined quantity.
Therefore the first step is to make these integrals well-defined by introducing a regulator.
There are several possibilities how this can be done, but the
method of dimensional regularisation
\cite{'tHooft:1972fi,Bollini:1972ui,Cicuta:1972jf}
has almost become a standard, as the calculations in this regularisation
scheme turn out to be the simplest.
Within dimensional regularisation one replaces the four-dimensional integral over the loop momentum by an
$D$-dimensional integral, where $D$ is now an additional parameter, which can be a non-integer or
even a complex number.
We consider the result of the integration as a function of $D$ and we are interested in the behaviour of this
function as $D$ approaches $4$.
The original divergences will then show up as poles in the Laurent series in $\eps=(4-D)/2$.
As a second example we consider
a Feynman diagram contributing to the one-loop corrections
for the process $e^+ e^- \rightarrow q g \bar{q}$, shown in fig.~\ref{fig_ee_qqg}.
\begin{figure}
\includegraphics[bb= 80 635 535 725]{fig1.eps}
\caption{\label{fig_ee_qqg} A one-loop Feynman diagram contributing to the process
$e^+ e^- \rightarrow q g \bar{q}$.}
\end{figure}
At high energies we can ignore the masses of the electron and the light quarks.
From the Feynman rules one obtains for this diagram (ignoring coupling and colour prefactors):
\bq
\label{feynmanrules}
- \bar{v}(p_4) \gamma^\mu u(p_5)
\frac{1}{p_{123}^2}
\int \frac{d^{D}k_1}{(2\pi)^{4}}
\frac{1}{k_2^2}
\bar{u}(p_1) \eps\!\!\!/(p_2) \frac{p\!\!\!/_{12}}{p_{12}^2}
\gamma_\nu \frac{k\!\!\!/_1}{k_1^2}
\gamma_\mu \frac{k\!\!\!/_3}{k_3^2}
\gamma^\nu
v(p_3).
\eq
Here, $p_{12}=p_1+p_2$, $p_{123}=p_1+p_2+p_3$, $k_2=k_1-p_{12}$, $k_3=k_2-p_3$.
Further $\eps\!\!\!/(p_2) = \gamma_\tau \eps^\tau(p_2)$, where $\eps^\tau(p_2)$ is the
polarisation vector of the outgoing gluon.
All external momenta are assumed to be
massless: $p_i^2=0$ for $i=1..5$.
We can reorganise this formula into a part, which depends on the loop integration and a part, which does not.
The loop integral to be calculated reads:
\bq
\label{loop_int_example_1}
\int \frac{d^D k_1}{(2\pi)^{4}}
\frac{k_1^\rho k_3^\sigma}{k_1^2 k_2^2 k_3^2},
\eq
while the remainder, which is independent of the loop integration is given by
\bq
\label{loop_int_example_remainder}
- \bar{v}(p_4) \gamma^\mu u(p_5)
\frac{1}{p_{123}^2 p_{12}^2}
\bar{u}(p_1) \eps\!\!\!/(p_2) p\!\!\!/_{12}
\gamma_\nu \gamma_\rho
\gamma_\mu \gamma_\sigma
\gamma^\nu
v(p_3).
\eq
The loop integral in eq.~(\ref{loop_int_example_1}) contains in the denominator three propagator factors
and in the numerator two factors of the loop momentum.
We call a loop integral, in which the loop momentum occurs also in the numerator a ``tensor integral''.
A loop integral, in which the numerator is independent of the loop momentum is called a ``scalar integral''.
The scalar integral associated to eq.~(\ref{loop_int_example_1}) reads
\bq
\label{loop_int_example_1a}
\int \frac{d^D k_1}{(2\pi)^{4}}
\frac{1}{k_1^2 k_2^2 k_3^2}.
\eq
There is a general method \cite{Tarasov:1996br,Tarasov:1997kx}
which allows to reduce any tensor integral to a combination of scalar
integrals at the expense of introducing higher powers of the propagators and shifted space-time
dimensions.
Therefore it is sufficient to focus on scalar integrals.
Each integral can be specified by its topology, its value for the dimension $D$ and
a set of indices, denoting the powers of the propagators.
\section{Multi-loop integrals}
\label{sect:multi_loop}
Let us now consider a generic scalar $l$-loop integral $I_G$
in $D=2m-2\eps$ dimensions with $n$ propagators,
corresponding to a graph $G$.
For each internal line $j$ the corresponding propagator
in the integrand can be raised to a power $\nu_j$.
Therefore the integral will depend also on the numbers $\nu_1$,...,$\nu_n$.
It is sufficient to consider only the case, where all exponents are natural numbers: $\nu_j \in {\mathbb N}$.
We define the Feynman integral by
\bq
\label{eq0}
I_G & = &
\frac{\prod\limits_{j=1}^{n}\Gamma(\nu_j)}{\Gamma(\nu-lD/2)}
\left( \mu^2 \right)^{\nu-l D/2}
\int \prod\limits_{r=1}^{l} \frac{d^Dk_r}{i\pi^{\frac{D}{2}}}\;
\prod\limits_{j=1}^{n} \frac{1}{(-q_j^2+m_j^2)^{\nu_j}},
\eq
with $\nu=\nu_1+...+\nu_n$.
$\mu$ is an arbitrary scale, called the renormalisation scale.
The momenta $q_j$ of
the propagators are linear combinations of the external momenta and the loop
momenta.
The prefactors are chosen such that after Feynman parametrisation the Feynman integral
has a simple form:
\bq
\label{eq1}
I_G & = &
\left( \mu^2 \right)^{\nu-l D/2}
\int\limits_{x_j \ge 0} d^nx \;
\delta(1-\sum_{i=1}^n x_i)\,
\left( \prod\limits_{j=1}^n x_j^{\nu_j-1} \right)
\frac{{\mathcal U}^{\nu-(l+1) D/2}}{{\mathcal F}^{\nu-l D/2}}.
\eq
The functions ${\mathcal U}$ and $\mathcal F$ depend on the Feynman parameters
and can be derived
from the topology of the corresponding Feynman graph $G$.
Cutting $l$ lines of a given connected $l$-loop graph such that it becomes a connected
tree graph $T$ defines a chord ${\mathcal C}(T,G)$ as being the set of lines
not belonging to this tree. The Feynman parameters associated with each chord
define a monomial of degree $l$. The set of all such trees (or 1-trees)
is denoted by ${\mathcal T}_1$. The 1-trees $T \in {\mathcal T}_1$ define
${\mathcal U}$ as being the sum over all monomials corresponding
to the chords ${\mathcal C}(T,G)$.
Cutting one more line of a 1-tree leads to two disconnected trees $(T_1,T_2)$, or a 2-tree.
${\mathcal T}_2$ is the set of all such pairs.
The corresponding chords define monomials of degree $l+1$. Each 2-tree of a graph
corresponds to a cut defined by cutting the lines which connected the two now disconnected trees
in the original graph.
The square of the sum of momenta through the cut lines
of one of the two disconnected trees $T_1$ or $T_2$
defines a Lorentz invariant
\bq
s_{T} & = & \left( \sum\limits_{j\in {\mathcal C}(T,G)} p_j \right)^2.
\eq
The function ${\mathcal F}_0$ is the sum over all such monomials times
minus the corresponding invariant. The function ${\mathcal F}$ is then given by ${\mathcal F}_0$ plus an additional piece
involving the internal masses $m_j$.
In summary, the functions ${\mathcal U}$ and ${\mathcal F}$ are obtained from the graph as follows:
\bq
\label{eq0def}
{\mathcal U}
& = &
\sum\limits_{T\in {\mathcal T}_1} \Bigl[\prod\limits_{j\in {\mathcal C}(T,G)}x_j\Bigr]\;,
\nonumber\\
{\mathcal F}_0
& = &
\sum\limits_{(T_1,T_2)\in {\mathcal T}_2}\;\Bigl[ \prod\limits_{j\in {\mathcal C}(T_1,G)} x_j \Bigr]\, (-s_{T_1})\;,
\nonumber\\
{\mathcal F}
& = &
{\mathcal F}_0 + {\mathcal U} \sum\limits_{j=1}^{n} x_j m_j^2\;.
\eq
\section{Periods}
\label{sect:periods}
Periods are special numbers.
Before we give the definition, let us start with some sets of numbers:
The natural numbers $\mathbb{N}$,
the integer numbers $\mathbb{Z}$,
the rational numbers $\mathbb{Q}$,
the real numbers $\mathbb{R}$ and
the complex numbers $\mathbb{C}$
are all well-known. More refined is already the set of algebraic numbers,
denoted by $\bar{\mathbb{Q}}$.
An algebraic number is a solution of a polynomial equation with rational
coefficients:
\bq
x^n + a_{n-1} x^{n-1} + \cdots + a_0 = 0,
\;\;\; a_j \in \mathbb{Q}.
\eq
As all such solutions lie in $\mathbb{C}$, the set of algebraic numbers $\bar{\mathbb{Q}}$
is a sub-set of
the complex numbers $\mathbb{C}$.
Numbers which are not algebraic are called transcendental.
The sets $\mathbb{N}$, $\mathbb{Z}$, $\mathbb{Q}$ and $\bar{\mathbb{Q}}$ are countable, whereas
the sets $\mathbb{R}$, $\mathbb{C}$ and the set of transcendental numbers are uncountable.
Periods are a countable set of numbers, lying between $\bar{\mathbb{Q}}$ and $\mathbb{C}$.
There are several equivalent definitions for periods.
Kontsevich and Zagier gave the following definition \cite{Kontsevich:2001}:
A period is a complex number whose real and imaginary parts are values
of absolutely convergent integrals of rational functions with rational coefficients,
over domains in $\mathbb{R}^n$ given by polynomial inequalities with rational coefficients.
Domains defined by polynomial inequalities with rational coefficients
are called semi-algebraic sets.
We denote the set of periods by $\mathbb{P}$. The algebraic numbers are contained in the set of periods:
$\bar{\mathbb{Q}} \in \mathbb{P}$.
In addition, $\mathbb{P}$ contains transcendental numbers, an example for such a number is $\pi$:
\bq
\pi & = & \iint\limits_{x^2+y^2\le1} dx \; dy.
\eq
The integral on the r.h.s. clearly shows that $\pi$ is a period.
On the other hand, it is conjectured that the basis of the natural logarithm $e$
and Euler's constant $\gamma_E$
are not periods.
Although there are uncountably many numbers, which are not periods, only very recently an example
for a number which is not a period has been found \cite{Yoshinaga:2008}.
We need a few basic properties of periods:
The set of periods $\mathbb{P}$ is a $\bar{\mathbb{Q}}$-algebra \cite{Kontsevich:2001,Friedrich:2005}.
In particular the sum and the product of two periods are again periods.
The defining integrals of periods have integrands, which are rational
functions with rational coefficients.
For our purposes this is too restrictive, as we will encounter
logarithms as integrands as well.
However any logarithm of a rational function with rational coefficients can be written as
\bq
\ln g(x)
& = &
\int\limits_0^1 dt \; \frac{g(x)-1}{(g(x)-1) t + 1}.
\eq
\section{A theorem on Feynman integrals}
\label{sect:theorem}
Let us consider a general scalar multi-loop integral as in eq.~(\ref{eq1}).
Let $m$ be an integer and set $D=2 m - 2 \eps$. Then this integral has
a Laurent series expansion in $\eps$
\bq
I_G & = & \sum\limits_{j=-2l}^\infty c_j \eps^j.
\eq
{\bf Theorem 1}: In the case where
\begin{enumerate}
\item all kinematical invariants $s_T$ are zero or negative,
\item all masses $m_i$ and $\mu$ are zero or positive ($\mu\neq0$),
\item all ratios of invariants and masses are rational,
\end{enumerate}
the coefficients $c_j$ of the Laurent expansion are periods.
In the special case were
\begin{enumerate}
\item the graph has no external lines or all invariants $s_T$ are zero,
\item all internal masses $m_j$ are equal to $\mu$,
\item all propagators occur with power $1$, i.e. $\nu_j=1$ for all $j$,
\end{enumerate}
the Feynman parameter integral reduces to
\bq
I_G & = &
\int\limits_{x_j \ge 0} d^nx \;
\delta(1-\sum_{i=1}^n x_i)\,
{\mathcal U}^{- D/2}
\eq
and only the polynomial ${\cal U}$ occurs in the integrand.
In this case it has been shown by Belkale and Brosnan \cite{Belkale:2003} that the coefficients of the
Laurent expansion are periods.
Using the method of sector decomposition we are able to prove the general case \cite{Bogner:2007mn}.
We will actually prove a stronger version of theorem 1.
Consider the following integral
\bq
\label{basic_integral2}
J & = &
\int\limits_{x_j \ge 0} d^nx \;\delta(1-\sum_{i=1}^n x_i)
\left( \prod\limits_{i=1}^n x_i^{a_i+\eps b_i} \right)
\prod\limits_{j=1}^r \left[ P_j(x) \right]^{d_j+\eps f_j}.
\eq
The integration is over the standard simplex.
The $a$'s, $b$'s, $d$'s and $f$'s are integers.
The $P$'s are polynomials in the variables $x_1$, ..., $x_n$ with rational coefficients.
The polynomials are required to be non-zero
inside the integration region, but
may vanish on the boundaries of the integration region.
To fix the sign, let us agree that all polynomials are positive inside the integration region.
The integral $J$ has a Laurent expansion
\bq
J & = & \sum\limits_{j=j_0}^\infty c_j \eps^j.
\eq
{\bf Theorem 2}: The coefficients $c_j$ of the Laurent expansion of the integral $J$ are periods.
Theorem 1 follows then from theorem 2 as the special case
$a_i=\nu_i-1$, $b_i=0$, $r=2$, $P_1={\cal U}$, $P_2={\cal F}$,
$d_1+\eps f_1 = \nu-(l+1)D/2$ and $d_2+\eps f_2 = l D/2 - \nu$.
Proof of theorem 2:
To prove the theorem we will give an algorithm which expresses each coefficient $c_j$
as a sum of absolutely convergent integrals over the unit hypercube with integrands,
which are linear combinations
of products of rational functions with logarithms of rational functions,
all of them with rational coefficients.
Let us denote this set of functions to which the integrands belong by ${\cal M}$.
The unit hypercube is clearly a semi-algebraic set.
It is clear that absolutely convergent integrals
over semi-algebraic sets with integrands from the set ${\cal M}$ are periods.
In addition, the sum of periods is again a period.
Therefore it is sufficient to express each coefficient $c_j$ as a finite sum
of absolutely convergent integrals over the unit hypercube with integrands from ${\cal M}$.
To do so, we use iterated sector decomposition.
This is a constructive method. Therefore we obtain as a side-effect a general purpose algorithm
for the numerical evaluation of multi-loop integrals.
\section{Sector decomposition}
\label{sect:sector_decomp}
In this section we review the algorithm for iterated
sector decomposition \cite{Hepp:1966eg,Roth:1996pd,Binoth:2000ps,Binoth:2003ak,Bogner:2007cr,Smirnov:2008py,Smirnov:2008aw}.
The starting point is an integral of the form
\bq
\label{basic_integral}
\int\limits_{x_j \ge 0} d^nx \;\delta(1-\sum_{i=1}^n x_i)
\left( \prod\limits_{i=1}^n x_i^{\mu_i} \right)
\prod\limits_{j=1}^r \left[ P_j(x) \right]^{\lambda_j},
\eq
where $\mu_i=a_i+\eps b_i$ and $\lambda_j=c_j+\eps d_j$.
The integration is over the standard simplex.
The $a$'s, $b$'s, $c$'s and $d$'s are integers.
The $P$'s are polynomials in the variables $x_1$, ..., $x_n$.
The polynomials are required to be non-zero
inside the integration region, but
may vanish on the boundaries of the integration region.
The algorithm consists of the following steps:
\\
\\
Step 0: Convert all polynomials to homogeneous polynomials.
\\
\\
Step 1: Decompose the integral into $n$ primary sectors.
\\
\\
Step 2: Decompose the sectors iteratively into sub-sectors until each of the polynomials
is of the form
\bq
\label{monomialised}
P & = & x_1^{m_1} ... x_n^{m_n} \left( c + P'(x) \right),
\eq
where $c\neq 0$ and $P'(x)$ is a polynomial in the variables $x_j$ without a constant term.
In this case the monomial prefactor $x_1^{m_1} ... x_n^{m_n}$ can be factored out
and the remainder contains a non-zero constant term.
To convert $P$ into the form~(\ref{monomialised}) one chooses a subset
$S=\left\{ \alpha_{1},\,...,\, \alpha_{k}\right\} \subseteq \left\{ 1, \,...\, n \right\}$
according to a strategy discussed in the next section.
One decomposes the $k$-dimensional hypercube into $k$ sub-sectors according to
\bq
\label{decomposition}
\int\limits_{0}^{1} d^{n}x & = &
\sum\limits_{l=1}^{k}
\int\limits_{0}^{1} d^{n}x
\prod\limits_{i=1, i\neq l}^{k}
\theta\left(x_{\alpha_{l}}\geq x_{\alpha_{i}}\right).
\eq
In the $l$-th sub-sector one makes for each element of $S$ the
substitution
\bq
\label{substitution}
x_{\alpha_{i}} & = & x_{\alpha_{l}} x_{\alpha_{i}}' \;\;\;\mbox{for}\; i\neq l.
\eq
This procedure is iterated, until all polynomials are of the form~(\ref{monomialised}).
\begin{figure}[t]
\includegraphics[bb= 100 440 570 710,width=0.9\textwidth]{fig2.eps}
\caption{\label{fig1} Illustration of sector decomposition and blow-up for a simple example.}
\end{figure}
Fig.~\ref{fig1} illustrates this for the simple example $S=\{1,2\}$.
Eq.~(\ref{decomposition}) gives the decomposition into the two sectors
$x_1>x_2$ and $x_2>x_1$.
Eq.~(\ref{substitution}) transforms the triangles into squares.
This transformation is one-to-one for all points except
the origin. The origin is replaced by the line $x_1=0$ in the first sector
and by the line $x_2=0$ in the second sector.
Therefore the name ``blow-up''.
\\
\\
Step 3: The singular behaviour of the integral depends now only on the factor
\bq
\prod\limits_{i=1}^{n}x_{i}^{a_{i}+\epsilon b_{i}}.
\eq
We Taylor expand in the integration variables and perform the trivial integrations
\bq
\int\limits_{0}^{1} dx \; x^{a+b\eps}
& = &
\frac{1}{a+1+b\eps},
\eq
leading to the explicit poles in $1/\eps$.
\\
\\
Step 4: All remaining integrals are now by construction finite.
We can now expand all expressions in a Laurent series in $\eps$
and truncate to the desired order.
\\
\\
Step 5: It remains to compute the coefficients of the Laurent series.
These coefficients contain finite integrals, which can be evaluated numerically
by Monte Carlo integration.
We implemented\footnote{The program can be obtained from {\tt http://www.higgs.de/\~{}stefanw/software.html}}
the algorithm into a computer program, which computes numerically the coefficients
of the Laurent series of any multi-loop integral \cite{Bogner:2007cr}.
\section{Hironaka's polyhedra game}
\label{sect:hironaka}
In step 2 of the algorithm we have an iteration.
It is important to show that this iteration terminates and does not lead to an infinite
loop.
\begin{figure*}[t]
\includegraphics[bb= 100 570 700 710,width=\textwidth]{fig3.eps}
\caption{\label{fig2} Illustration of Hironaka's polyhedra game.}
\end{figure*}
There are strategies for choosing the sub-sectors, which guarantee termination.
These strategies \cite{Hironaka:1964,Spivakovsky:1983,Encinas:2002,Hauser:2003,Zeillinger:2006} are closely related to Hironaka's polyhedra game.
Hironaka's polyhedra game is played by two players, A and B. They are
given a finite set $M$ of points $m=\left(m_{1},\,...,\,m_{n}\right)$
in $\mathbb{N}_{+}^{n}$, the first quadrant of $\mathbb{N}^{n}$.
We denote by $\Delta \subset\mathbb{R}_{+}^{n}$ the positive convex hull of the set $M$.
It is given by the convex hull of the set
\bq
\bigcup\limits_{m\in M}\left(m+\mathbb{R}_{+}^{n}\right).
\eq
The two players compete in the following game:
\begin{enumerate}
\item Player A chooses a non-empty subset $S\subseteq\left\{ 1,\,...,\, n\right\}$.
\item Player B chooses one element $i$ out of this subset $S$.
\end{enumerate}
Then, according
to the choices of the players, the components of all $\left(m_{1},\,...,\,m_{n}\right)\in M$
are replaced by new points $\left(m_{1}^{\prime},\,...,\,m_{n}^{\prime}\right)$,
given by:
\bq
\label{update_polyhedron}
m_{j}^{\prime} & = & m_{j}, \;\;\; \textrm{if }j\neq i, \nonumber \\
m_{i}^{\prime} & = & \sum_{j\in S} m_{j}-c,
\eq
where for the moment we set $c=1$.
This defines the set $M^\prime$.
One then sets $M=M^\prime$ and goes back to step 1.
Player A wins the game if, after a finite number of moves,
the polyhedron $\Delta$ is of the form
\bq
\label{termination}
\Delta & = & m+\mathbb{R}_{+}^{n},
\eq
i.e. generated by one point.
If this never occurs, player $B$ has won.
The challenge of the polyhedra game is to show that player $A$ always has
a winning strategy, no matter how player $B$ chooses his moves.
A simple illustration of Hironaka's polyhedra game in two dimensions is given in
fig.~\ref{fig2}. Player A always chooses $S=\{1,2\}$.
In ref.~\cite{Bogner:2007cr} we have shown that a winning strategy for
Hironaka's polyhedra game
translates directly into a strategy for choosing the sub-sectors which
guarantees termination.
\section{Shuffle algebras}
\label{sect:shuffle}
Before we continue the discussion of loop integrals, it is useful to discuss first
shuffle algebras and generalisations thereof from an algebraic viewpoint.
Consider a set of letters $A$. The set $A$ is called the alphabet.
A word is an ordered sequence of letters:
\bq
w & = & l_1 l_2 ... l_k.
\eq
The word of length zero is denoted by $e$.
Let $K$ be a field and consider the vector space of words over $K$.
A shuffle algebra ${\cal A}$ on the vector space of words is defined by
\bq
\left( l_1 l_2 ... l_k \right) \cdot
\left( l_{k+1} ... l_r \right) & = &
\sum\limits_{\mbox{\tiny shuffles} \; \sigma} l_{\sigma(1)} l_{\sigma(2)} ... l_{\sigma(r)},
\eq
where the sum runs over all permutations $\sigma$, which preserve the relative order
of $1,2,...,k$ and of $k+1,...,r$.
The name ``shuffle algebra'' is related to the analogy of shuffling cards: If a deck of cards
is split into two parts and then shuffled, the relative order within the two individual parts
is conserved.
A shuffle algebra is also known under the name ``mould symmetral'' \cite{Ecalle}.
The empty word $e$ is the unit in this algebra:
\bq
e \cdot w = w \cdot e = w.
\eq
A recursive definition of the shuffle product is given by
\bq
\label{def_recursive_shuffle}
\left( l_1 l_2 ... l_k \right) \cdot \left( l_{k+1} ... l_r \right) & = &
l_1 \left[ \left( l_2 ... l_k \right) \cdot \left( l_{k+1} ... l_r \right) \right]
+
l_{k+1} \left[ \left( l_1 l_2 ... l_k \right) \cdot \left( l_{k+2} ... l_r \right) \right].
\eq
It is well known fact that the shuffle algebra is actually a (non-cocommutative) Hopf algebra \cite{Reutenauer}.
In this context let us briefly review the definitions of a coalgebra, a bialgebra and a Hopf algebra,
which are closely related:
First note that the unit in an algebra can be viewed as a map from $K$ to $A$ and that the multiplication
can be viewed as a map from the tensor product $A \otimes A$ to $A$ (e.g. one takes two elements
from $A$, multiplies them and gets one element out).
A coalgebra has instead of multiplication and unit the dual structures:
a comultiplication $\Delta$ and a counit $\bar{e}$.
The counit is a map from $A$ to $K$, whereas comultiplication is a map from $A$ to
$A \otimes A$.
Note that comultiplication and counit go in the reverse direction compared to multiplication
and unit.
We will always assume that the comultiplication is coassociative.
The general form of the coproduct is
\bq
\Delta(a) & = & \sum\limits_i a_i^{(1)} \otimes a_i^{(2)},
\eq
where $a_i^{(1)}$ denotes an element of $A$ appearing in the first slot of $A \otimes A$ and
$a_i^{(2)}$ correspondingly denotes an element of $A$ appearing in the second slot.
Sweedler's notation \cite{Sweedler} consists in dropping the dummy index $i$ and the summation symbol:
\bq
\Delta(a) & = &
a^{(1)} \otimes a^{(2)}
\eq
The sum is implicitly understood. This is similar to Einstein's summation convention, except
that the dummy summation index $i$ is also dropped. The superscripts ${}^{(1)}$ and ${}^{(2)}$
indicate that a sum is involved.
A bialgebra is an algebra and a coalgebra at the same time,
such that the two structures are compatible with each other.
Using Sweedler's notation,
the compatibility between the multiplication and comultiplication is express\-ed as
\bq
\label{bialg}
\Delta\left( a \cdot b \right)
& = &
\left( a^{(1)} \cdot b^{(1)} \right)
\otimes \left( a^{(2)} \cdot b^{(2)} \right).
\eq
A Hopf algebra is a bialgebra with an additional map from $A$ to $A$, called the
antipode ${\cal S}$, which fulfils
\bq
a^{(1)} \cdot {\cal S}\left( a^{(2)} \right)
=
{\cal S}\left(a^{(1)}\right) \cdot a^{(2)}
= 0 & &\;\;\; \mbox{for} \; a \neq e.
\eq
With this background at hand we can now state the coproduct, the counit and the antipode for the
shuffle algebra:
The counit $\bar{e}$ is given by:
\bq
\bar{e}\left( e\right) = 1, \;\;\;
& &
\bar{e}\left( l_1 l_2 ... l_n\right) = 0.
\eq
The coproduct $\Delta$ is given by:
\bq
\Delta\left( l_1 l_2 ... l_k \right)
& = & \sum\limits_{j=0}^k \left( l_{j+1} ... l_k \right) \otimes \left( l_1 ... l_j \right).
\eq
The antipode ${\cal S}$ is given by:
\bq
{\cal S}\left( l_1 l_2 ... l_k \right) & = & (-1)^k \; l_k l_{k-1} ... l_2 l_1.
\eq
The shuffle algebra is generated by the Lyndon words.
If one introduces a lexicographic ordering on the letters of the alphabet
$A$, a Lyndon word is defined by the property
\bq
w < v
\eq
for any sub-words $u$ and $v$ such that $w= u v$.
An important example for a shuffle algebra are iterated integrals.
Let $[a, b]$ be a segment of the real line and $f_1$, $f_2$, ... functions
on this interval.
Let us define the following iterated integrals:
\bq
I(f_1,f_2,...,f_k;a,b)
& = &
\int\limits_a^b dt_1 f_1(t_1) \int\limits_a^{t_1} dt_2 f_2(t_2)
...
\int\limits_a^{t_{k-1}} dt_k f_k(t_k)
\eq
For fixed $a$ and $b$ we have a shuffle algebra:
\bq
I(f_1,f_2,...,f_k;a,b) \cdot I(f_{k+1},...,f_r; a,b) & = &
\sum\limits_{\mbox{\tiny shuffles} \; \sigma} I(f_{\sigma(1)},f_{\sigma(2)},...,f_{\sigma(r)};a,b),
\eq
where the sum runs over all permutations $\sigma$, which preserve the relative order
of $1,2,...,k$ and of $k+1,...,r$.
The proof is sketched in fig.~\ref{proof_shuffle}.
\begin{figure}
\includegraphics[bb= 70 640 535 720]{fig4.eps}
\caption{\label{proof_shuffle} Sketch of the proof for the shuffle product of two iterated integrals.
The integral over the square is replaced by two
integrals over the upper and lower triangle.}
\end{figure}
The two outermost integrations are recursively replaced by integrations over the upper and lower triangle.
We now consider generalisations of shuffle algebras. Assume that for the set of letters we have an additional
operation
\bq
(.,.) & : & A \otimes A \rightarrow A,
\nonumber \\
& & l_1 \otimes l_2 \rightarrow (l_1, l_2),
\eq
which is commutative and associative.
Then we can define a new product of words recursively through
\bq
\label{def_recursive_quasi_shuffle}
\left( l_1 l_2 ... l_k \right) \ast \left( l_{k+1} ... l_r \right) & = &
l_1 \left[ \left( l_2 ... l_k \right) \ast \left( l_{k+1} ... l_r \right) \right]
+
l_{k+1} \left[ \left( l_1 l_2 ... l_k \right) \ast \left( l_{k+2} ... l_r \right) \right]
\nonumber \\
& &
+
(l_1,l_{k+1}) \left[ \left( l_2 ... l_k \right) \ast \left( l_{k+2} ... l_r \right) \right].
\eq
This product is a generalisation of the shuffle product and differs from the recursive
definition of the shuffle product in eq.~(\ref{def_recursive_shuffle}) through the extra term in the last line.
This modified product is known under the names quasi-shuffle product \cite{Hoffman},
mixable shuffle product \cite{Guo},
stuffle product \cite{Borwein} or
mould symmetrel \cite{Ecalle}.
Quasi-shuffle algebras are Hopf algebras.
Comultiplication and counit are defined as for the shuffle algebras.
The counit $\bar{e}$ is given by:
\bq
\bar{e}\left( e\right) = 1, \;\;\;
& &
\bar{e}\left( l_1 l_2 ... l_n\right) = 0.
\eq
The coproduct $\Delta$ is given by:
\bq
\Delta\left( l_1 l_2 ... l_k \right)
& = & \sum\limits_{j=0}^k \left( l_{j+1} ... l_k \right) \otimes \left( l_1 ... l_j \right).
\eq
The antipode ${\cal S}$ is recursively defined through
\bq
{\cal S}\left( l_1 l_2 ... l_k \right) & = &
- l_1 l_2 ... l_k
- \sum\limits_{j=1}^{k-1} {\cal S}\left( l_{j+1} ... l_k \right) \ast \left( l_1 ... l_j \right).
\eq
An example for a quasi-shuffle algebra are nested sums.
Let $n_a$ and $n_b$ be integers with $n_a<n_b$ and let $f_1$, $f_2$, ... be functions
defined on the integers.
We consider the following nested sums:
\bq
S(f_1,f_2,...,f_k;n_a,n_b)
& = &
\sum\limits_{i_1=n_a}^{n_b} f_1(i_1) \sum\limits_{i_2=n_a}^{i_1-1} f_2(i_2)
...
\sum\limits_{i_k=n_a}^{i_{k-1}-1} f_k(i_k)
\eq
For fixed $n_a$ and $n_b$ we have a quasi-shuffle algebra:
\bq
\label{quasi_shuffle_multiplication}
\lefteqn{
S(f_1,f_2,...,f_k;n_a,n_b) \ast S(f_{k+1},...,f_r; n_a,n_b)
= } & &
\nonumber \\
& &
\sum\limits_{i_1=n_a}^{n_b} f_1(i_1) \; S(f_2,...,f_k;n_a,i_1-1) \ast S(f_{k+1},...,f_r; n_a,i_1-1)
\nonumber \\
& &
+ \sum\limits_{j_1=n_a}^{n_b} f_k(j_1) \; S(f_1,f_2,...,f_k;n_a,j_1-1) \ast S(f_{k+2},...,f_r; n_a,j_1-1)
\nonumber \\
& &
+ \sum\limits_{i=n_a}^{n_b} f_1(i) f_k(i) \; S(f_2,...,f_k;n_a,i-1) \ast S(f_{k+2},...,f_r; n_a,i-1)
\eq
\begin{figure}
\includegraphics[bb= 65 630 530 710]{fig5.eps}
\caption{\label{proof} Sketch of the proof for the quasi-shuffle product of nested sums.
The sum over the square is replaced by
the sum over the three regions on the r.h.s.}
\end{figure}
Note that the product of two letters corresponds to the point-wise product of the two functions:
\bq
( f_i, f_j ) \; (n) & = & f_i(n) f_j(n).
\eq
The proof that nested sums obey the quasi-shuffle algebra is sketched in Fig. \ref{proof}.
The outermost sums of the nested sums on the l.h.s of (\ref{quasi_shuffle_multiplication}) are split into the three
regions indicated in Fig. \ref{proof}.
\section{Multiple polylogarithms}
\label{sect:polylog}
In the previous section we have seen that iterated integrals form a shuffle algebra, while
nested sums form a quasi-shuffle algebra.
In this context multiple polylogarithms form an interesting class of functions.
They have a representation as iterated integrals as well as nested sums.
Therefore multiple polylogarithms form a shuffle algebra as well as a quasi-shuffle algebra.
The two algebra structures are independent.
Let us start with the representation as nested sums.
The multiple polylogarithms are defined by \cite{Goncharov,Minh:2000,Cartier:2001,Racinet:2002}
\bq
\label{multipolylog2}
\mbox{Li}_{m_1,...,m_k}(x_1,...,x_k)
& = & \sum\limits_{i_1>i_2>\ldots>i_k>0}
\frac{x_1^{i_1}}{{i_1}^{m_1}}\ldots \frac{x_k^{i_k}}{{i_k}^{m_k}}.
\eq
The multiple polylogarithms are generalisations of
the classical polylogarithms
$\mbox{Li}_n(x)$,
whose most prominent examples are
\bq
\mbox{Li}_1(x) = \sum\limits_{i_1=1}^\infty \frac{x^{i_1}}{i_1} = -\ln(1-x),
& &
\mbox{Li}_2(x) = \sum\limits_{i_1=1}^\infty \frac{x^{i_1}}{i_1^2},
\eq
as well as
Nielsen's generalised polylogarithms \cite{Nielsen}
\bq
S_{n,p}(x) & = & \mbox{Li}_{n+1,1,...,1}(x,\underbrace{1,...,1}_{p-1}),
\eq
and the harmonic polylogarithms \cite{Remiddi:1999ew,Gehrmann:2000zt}
\bq
\label{harmpolylog}
H_{m_1,...,m_k}(x) & = & \mbox{Li}_{m_1,...,m_k}(x,\underbrace{1,...,1}_{k-1}).
\eq
In addition, multiple polylogarithms have an integral representation.
To discuss the integral representation it is convenient to
introduce for $z_k \neq 0$
the following functions
\bq
\label{Gfuncdef}
G(z_1,...,z_k;y) & = &
\int\limits_0^y \frac{dt_1}{t_1-z_1}
\int\limits_0^{t_1} \frac{dt_2}{t_2-z_2} ...
\int\limits_0^{t_{k-1}} \frac{dt_k}{t_k-z_k}.
\eq
In this definition
one variable is redundant due to the following scaling relation:
\bq
G(z_1,...,z_k;y) & = & G(x z_1, ..., x z_k; x y)
\eq
If one further defines
\bq
g(z;y) & = & \frac{1}{y-z},
\eq
then one has
\bq
\label{derivative}
\frac{d}{dy} G(z_1,...,z_k;y) & = & g(z_1;y) G(z_2,...,z_k;y)
\eq
and
\bq
\label{Grecursive}
G(z_1,z_2,...,z_k;y) & = & \int\limits_0^y dt \; g(z_1;t) G(z_2,...,z_k;t).
\eq
One can slightly enlarge the set and define
$G(0,...,0;y)$ with $k$ zeros for $z_1$ to $z_k$ to be
\bq
\label{trailingzeros}
G(0,...,0;y) & = & \frac{1}{k!} \left( \ln y \right)^k.
\eq
This permits us to allow trailing zeros in the sequence
$(z_1,...,z_k)$ by defining the function $G$ with trailing zeros via (\ref{Grecursive})
and (\ref{trailingzeros}).
To relate the multiple polylogarithms to the functions $G$ it is convenient to introduce
the following short-hand notation:
\bq
\label{Gshorthand}
G_{m_1,...,m_k}(z_1,...,z_k;y)
& = &
G(\underbrace{0,...,0}_{m_1-1},z_1,...,z_{k-1},\underbrace{0...,0}_{m_k-1},z_k;y)
\eq
Here, all $z_j$ for $j=1,...,k$ are assumed to be non-zero.
One then finds
\bq
\label{Gintrepdef}
\mbox{Li}_{m_1,...,m_k}(x_1,...,x_k)
& = & (-1)^k
G_{m_1,...,m_k}\left( \frac{1}{x_1}, \frac{1}{x_1 x_2}, ..., \frac{1}{x_1...x_k};1 \right).
\eq
The inverse formula reads
\bq
G_{m_1,...,m_k}(z_1,...,z_k;y) & = &
(-1)^k \; \mbox{Li}_{m_1,...,m_k}\left(\frac{y}{z_1}, \frac{z_1}{z_2}, ..., \frac{z_{k-1}}{z_k}\right).
\eq
Eq. (\ref{Gintrepdef}) together with
(\ref{Gshorthand}) and (\ref{Gfuncdef})
defines an integral representation for the multiple polylogarithms.
Up to now we treated multiple polylogarithms from an algebraic point of view.
Equally important are the analytical properties, which are needed for an efficient numerical
evaluation.
As an example I first discuss the numerical evaluation of the dilogarithm \cite{'tHooft:1979xw}:
\bq
\mbox{Li}_{2}(x) & = & - \int\limits_{0}^{x} dt \frac{\ln(1-t)}{t}
= \sum\limits_{n=1}^{\infty} \frac{x^{n}}{n^{2}}
\eq
The power series expansion can be evaluated numerically, provided $|x| < 1.$
Using the functional equations
\bq
\mbox{Li}_2(x) & = & -\mbox{Li}_2\left(\frac{1}{x}\right) -\frac{\pi^2}{6} -\frac{1}{2} \left( \ln(-x) \right)^2,
\nonumber \\
\mbox{Li}_2(x) & = & -\mbox{Li}_2(1-x) + \frac{\pi^2}{6} -\ln(x) \ln(1-x).
\eq
any argument of the dilogarithm can be mapped into the region
$|x| \le 1$ and
$-1 \leq \mbox{Re}(x) \leq 1/2$.
The numerical computation can be accelerated by using an expansion in $[-\ln(1-x)]$ and the
Bernoulli numbers $B_i$:
\bq
\mbox{Li}_2(x) & = & \sum\limits_{i=0}^\infty \frac{B_i}{(i+1)!} \left( - \ln(1-x) \right)^{i+1}.
\eq
The generalisation to multiple polylogarithms proceeds along the same lines \cite{Vollinga:2004sn}:
Using the integral representation eq.~(\ref{Gfuncdef})
one transforms all arguments into a region,
where one has a converging power series expansion.
In this region eq.~(\ref{multipolylog2}) may be used.
However it is advantageous to speed up the convergence of the power series expansion.
This is done as follows:
The multiple polylogarithms satisfy the H\"older convolution \cite{Borwein}.
For $z_1 \neq 1$ and $z_w \neq 0$ this identity reads
\bq
\label{defhoelder}
\lefteqn{
G\left(z_1,...,z_w; 1 \right)
= } & &
\\
& &
\sum\limits_{j=0}^w \left(-1\right)^j
G\left(1-z_j, 1-z_{j-1},...,1-z_1; 1 - \frac{1}{p} \right)
G\left( z_{j+1},..., z_w; \frac{1}{p} \right).
\nonumber
\eq
The H\"older convolution can be used to accelerate the
convergence for the series
representation of the multiple polylogarithms.
\section{From Feynman integrals to multiple polylogarithms}
\label{sect:calc}
In sect.~\ref{sect:multi_loop} we saw that the Feynman parameter integrals
depend on two graph polynomials ${\mathcal U}$ and ${\mathcal F}$, which are homogeneous functions of the
Feynman parameters.
In this section we will discuss how multiple polylogarithms arise in the calculation of Feynman parameter
integrals.
We will discuss two approaches. In the first approach one uses a Mellin-Barnes transformation and sums
up residues. This leads to the sum representation of multiple polylogarithms.
In the second approach one first derives a differential equation for the Feynman parameter integral, which
is then solved by an ansatz in terms of the iterated integral representation of multiple polylogarithms.
Let us start with the first approach. Assume for the moment that the two graph polynomials
${\mathcal U}$ and ${\mathcal F}$ are absent from the Feynman parameter integral.
In this case we have
\bq
\label{multi_beta_fct}
\int\limits_{0}^{1} \left( \prod\limits_{j=1}^{n}\,dx_j\,x_j^{\nu_j-1} \right)
\delta(1-\sum_{i=1}^n x_i)
& = &
\frac{\prod\limits_{j=1}^{n}\Gamma(\nu_j)}{\Gamma(\nu_1+...+\nu_n)}.
\eq
With the help of
the Mellin-Barnes transformation we now reduce the general case to eq.~(\ref{multi_beta_fct}).
The Mellin-Barnes transformation reads
\bq
\label{multi_mellin_barnes}
\lefteqn{
\left(A_1 + A_2 + ... + A_n \right)^{-c}
=
\frac{1}{\Gamma(c)} \frac{1}{\left(2\pi i\right)^{n-1}}
\int\limits_{-i\infty}^{i\infty} d\sigma_1 ... \int\limits_{-i\infty}^{i\infty} d\sigma_{n-1}
} & & \\
& &
\times
\Gamma(-\sigma_1) ... \Gamma(-\sigma_{n-1}) \Gamma(\sigma_1+...+\sigma_{n-1}+c)
\;
A_1^{\sigma_1} ... A_{n-1}^{\sigma_{n-1}} A_n^{-\sigma_1-...-\sigma_{n-1}-c}.
\nonumber
\eq
Each contour is such that the poles of $\Gamma(-\sigma)$ are to the right and the poles
of $\Gamma(\sigma+c)$ are to the left.
This transformation can be used to convert the sum of monomials of the polynomials ${\mathcal U}$ and ${\mathcal F}$ into
a product, such that all Feynman parameter integrals are of the form of eq.~(\ref{multi_beta_fct}).
As this transformation converts sums into products it is
the ``inverse'' of Feynman parametrisation.
With the help of eq.~(\ref{multi_beta_fct}) and eq.~(\ref{multi_mellin_barnes})
we may exchange the Feynman parameter integrals against multiple contour integrals.
A single contour integral is of the form
\bq
\label{MellinBarnesInt}
I
& = &
\frac{1}{2\pi i} \int\limits_{\gamma-i\infty}^{\gamma+i\infty}
d\sigma \;
\frac{\Gamma(\sigma+a_1) ... \Gamma(\sigma+a_m)}
{\Gamma(\sigma+c_2) ... \Gamma(\sigma+c_p)}
\frac{\Gamma(-\sigma+b_1) ... \Gamma(-\sigma+b_n)}
{\Gamma(-\sigma+d_1) ... \Gamma(-\sigma+d_q)}
\; x^{-\sigma}.
\eq
If $\;\mbox{max}\left( \mbox{Re}(-a_1), ..., \mbox{Re}(-a_m) \right) < \mbox{min}\left( \mbox{Re}(b_1), ..., \mbox{Re}(b_n) \right)$ the contour can be chosen
as a straight line parallel to the imaginary axis with
\bq
\mbox{max}\left( \mbox{Re}(-a_1), ..., \mbox{Re}(-a_m) \right)
\;\;\; < \;\;\; \mbox{Re} \; \gamma \;\;\; < \;\;\;
\mbox{min}\left( \mbox{Re}(b_1), ..., \mbox{Re}(b_n) \right),
\eq
otherwise the contour is indented, such that the residues of
$\Gamma(\sigma+a_1)$, ..., $\Gamma(\sigma+a_m)$ are to the right of the contour,
whereas the residues of
$\Gamma(-\sigma+b_1)$, ..., $\Gamma(-\sigma+b_n)$ are to the left of the contour.
The integral eq. (\ref{MellinBarnesInt}) is most conveniently evaluated with
the help of the residuum theorem by closing the contour to the left or to the right.
To sum up all residues which lie inside the contour
it is useful to know the residues of the Gamma function:
\bq
\mbox{res} \; \left( \Gamma(\sigma+a), \sigma=-a-n \right) = \frac{(-1)^n}{n!},
& &
\mbox{res} \; \left( \Gamma(-\sigma+a), \sigma=a+n \right) = -\frac{(-1)^n}{n!}.
\eq
In general there are multiple contour integrals, and as a consequence one obtains multiple sums.
Having collected all residues, one then expands the Gamma-functions:
\bq
\label{expansiongamma}
\lefteqn{
\Gamma(n+\eps) =
} & & \\
& & \Gamma(1+\eps) \Gamma(n)
\left[
1 + \eps Z_1(n-1) + \eps^2 Z_{11}(n-1)
+ \eps^3 Z_{111}(n-1) + ... + \eps^{n-1} Z_{11...1}(n-1)
\right],
\nonumber
\eq
where $Z_{m_1,...,m_k}(n)$ are Euler-Zagier sums
defined by
\bq
Z_{m_1,...,m_k}(n) & = &
\sum\limits_{n \ge i_1>i_2>\ldots>i_k>0}
\frac{1}{{i_1}^{m_1}}\ldots \frac{1}{{i_k}^{m_k}}.
\eq
This motivates the following definition of a special form of nested sums, called
$Z$-sums \cite{Moch:2001zr,Weinzierl:2002hv,Weinzierl:2004bn,Moch:2005uc}:
\bq
\label{definition}
Z(n;m_1,...,m_k;x_1,...,x_k) & = & \sum\limits_{n\ge i_1>i_2>\ldots>i_k>0}
\frac{x_1^{i_1}}{{i_1}^{m_1}}\ldots \frac{x_k^{i_k}}{{i_k}^{m_k}}.
\eq
$k$ is called the depth of the $Z$-sum and $w=m_1+...+m_k$ is called the weight.
If the sums go to infinity ($n=\infty$) the $Z$-sums are multiple polylogarithms:
\bq
\label{multipolylog}
Z(\infty;m_1,...,m_k;x_1,...,x_k) & = & \mbox{Li}_{m_1,...,m_k}(x_1,...,x_k).
\eq
For $x_1=...=x_k=1$ the definition reduces to the Euler-Zagier sums \cite{Euler,Zagier,Vermaseren:1998uu,Blumlein:1998if,Blumlein:2003gb}:
\bq
Z(n;m_1,...,m_k;1,...,1) & = & Z_{m_1,...,m_k}(n).
\eq
For $n=\infty$ and $x_1=...=x_k=1$ the sum is a multiple $\zeta$-value \cite{Borwein,Blumlein:2009}:
\bq
Z(\infty;m_1,...,m_k;1,...,1) & = & \zeta_{m_1,...,m_k}.
\eq
The usefulness of the $Z$-sums lies in the fact, that they interpolate between
multiple polylogarithms and Euler-Zagier sums.
The $Z$-sums form a quasi-shuffle algebra.
In this approach multiple polylogarithms appear through eq.~(\ref{multipolylog}).
An alternative approach to the computation of Feynman parameter integrals is based on
differential equations \cite{Kotikov:1990kg,Kotikov:1991pm,Remiddi:1997ny,Gehrmann:1999as,Gehrmann:2000zt,Gehrmann:2001ck}.
To evaluate these integrals within this approach
one first finds for each
master integral a differential
equation, which this master integral has to satisfy.
The derivative is taken with respect to an external scale, or a
ratio of two scales.
An example for a one-loop four-point function is given by
\bq
\lefteqn{
\frac{\partial}{\partial s_{123}}
\begin{picture}(140,40)(-15,45)
\Vertex(50,20){2}
\Vertex(50,80){2}
\Vertex(20,50){2}
\Vertex(80,50){2}
\Line(0,50)(20,50)
\Line(20,50)(50,80)
\Line(50,20)(20,50)
\Line(50,80)(80,50)
\Line(80,50)(50,20)
\Line(50,80)(70,80)
\Line(80,50)(100,50)
\Line(50,20)(70,20)
\Text(75,80)[l]{\tiny $p_1$}
\Text(105,50)[l]{\tiny $p_2$}
\Text(75,20)[l]{\tiny $p_3$}
\end{picture}
=
\frac{D-4}{2(s_{12}+s_{23}-s_{123})}
\begin{picture}(100,40)(-15,45)
\Vertex(50,20){2}
\Vertex(50,80){2}
\Vertex(20,50){2}
\Vertex(80,50){2}
\Line(0,50)(20,50)
\Line(20,50)(50,80)
\Line(50,20)(20,50)
\Line(50,80)(80,50)
\Line(80,50)(50,20)
\Line(50,80)(70,80)
\Line(80,50)(100,50)
\Line(50,20)(70,20)
\Text(75,80)[l]{\tiny $p_1$}
\Text(105,50)[l]{\tiny $p_2$}
\Text(75,20)[l]{\tiny $p_3$}
\end{picture}
} & &
\nonumber \\
& & \nonumber \\
& &
+ \frac{2(D-3)}{(s_{123}-s_{12})(s_{123}-s_{12}-s_{23})}
\left[
\frac{1}{s_{123}}
\begin{picture}(110,40)(-5,45)
\Vertex(30,50){2}
\Vertex(70,50){2}
\Line(10,50)(30,50)
\CArc(50,50)(20,0,360)
\Line(70,50)(90,50)
\Text(80,55)[lb]{\tiny $p_{123}$}
\end{picture}
-
\frac{1}{s_{12}}
\begin{picture}(110,40)(-5,45)
\Vertex(30,50){2}
\Vertex(70,50){2}
\Line(10,50)(30,50)
\CArc(50,50)(20,0,360)
\Line(70,50)(90,50)
\Text(80,55)[lb]{\tiny $p_{12}$}
\end{picture}
\right]
\nonumber \\
& &
+ \frac{2(D-3)}{(s_{123}-s_{23})(s_{123}-s_{12}-s_{23})}
\left[
\frac{1}{s_{123}}
\begin{picture}(110,40)(-5,45)
\Vertex(30,50){2}
\Vertex(70,50){2}
\Line(10,50)(30,50)
\CArc(50,50)(20,0,360)
\Line(70,50)(90,50)
\Text(80,55)[lb]{\tiny $p_{123}$}
\end{picture}
-
\frac{1}{s_{23}}
\begin{picture}(110,40)(-5,45)
\Vertex(30,50){2}
\Vertex(70,50){2}
\Line(10,50)(30,50)
\CArc(50,50)(20,0,360)
\Line(70,50)(90,50)
\Text(80,55)[lb]{\tiny $p_{23}$}
\end{picture}
\right].
\nonumber
\eq
The two-point functions on the r.h.s are simpler and can be considered to be known.
This equation is solved iteratively by an ansatz
for the solution as a Laurent expression in $\eps$.
Each term in this Laurent series is a sum of terms, consisting of
basis functions times
some unknown (and to be determined) coefficients.
This ansatz is inserted into the differential equation and the unknown
coefficients
are determined order by order from the differential equation.
The basis functions are taken as a subset of multiple polylogarithms.
In this approach the iterated integral representation of multiple polylogarithms is the most
convenient form. This is immediately clear from the
simple formula for the derivative as in eq.~(\ref{derivative}).
\section{Conclusions}
\label{sect:conclusions}
In this talk we reported on mathematical properties of Feynman integrals.
We first showed that under rather weak assumptions all the coefficients
of the Laurent expansion of a multi-loop integral
are periods.
In the second part we focused on multiple polylogarithms and how they appear in the calculation
of Feynman integrals.
\bibliography{/home/stefanw/notes/biblio}
\bibliographystyle{/home/stefanw/latex-style/h-physrev3}
\end{document} | 194,426 |
Netflix
Vampire Weekend hasn’t released a new album since 2013’s Modern Vampires Of The City, so if you’re a fan who’s been patiently waiting for some new material, we have some news that will either excite or infuriate you: Ezra Koenig has been busy writing… not music, necessarily, but an anime for Netflix called Neo Yokio.
It stars Jaden Smith as a rich “magistocrat” named Kaz Kaan who is trying to distance himself from his demon-slaying past. If you want to watch, the full six-episode first season is streaming on Netflix starting today. There’s plenty of reason to tune in beyond Koenig and Smith, as the rest of the star-heavy cast is rounded out by Jude Law, Susan Sarandon, Steve Buscemi, Jason Schwartzman, Tavi Gevinson, Desus Nice, The Kid Mero, and others.
The show, described by its creators as “a postmodern collage of homages to classic anime, English literature and modern New York fashion and culture,” might not be considered a “true” anime by some purists: While it’s definitely styled like anime, the show isn’t entirely of Japanese origin, which some consider an essential qualifier for anime status. That said, the massively popular and critically hailed Nickelodeon cartoon Avatar: The Last Airbender was in a similar boat, so it’s possible that this show could still scratch the anime itch.
And for the record, Koenig did recently say that the band is “80 percent done” with their next album, currently titled Mitsubishi Macchiato: “It’s getting there. I feel like it’s close to being done. There’s not too many other ways I can say it.”
Check out a trailer for the show above, and watch the first season now on Netflix.
Join The Discussion: Log In With | 49,735 |
THE EFFECTS OF THE USE OF WORDS ON OUR LIVES
Words are powerful and their use has influenced key people throughout history. Here are a few examples:
– Malcolm X (human rights activist) read the dictionary from cover to cover to improve his literary education.
– W.H. Auden (Anglo American poet) famously said if he were marooned on a desert island, he would choose to have with him a good dictionary rather than “the greatest literary masterpiece imaginable.”
– Melvin Bragg (English broadcaster and author) once said, “When we lose a language, we lose a way of knowing the world”.
Language shapes our thinking.
Words comprise the language we use every day and it’s how we interact with the world around us. Do you want to change that world around you? If so, then the first step is to simply change how you use your language.
A simple technique to consider is that for the next thirty days you use at least one new word a day and notice how it affects you. Make it simple by picking a sensory-based word, i.e. visual, auditory or feeling words. What will you start to notice by adding these to your language? You will begin to realise that these words have always been there but you have been deleting[1] them.
- – Visual words will allow you to start to see new things around you.
- – Auditory words you will enable you to start to hear new things around you.
- – Feeling or emotion words will start to create different moods and sensations.
Isn’t that incredible? My new words for today are “Varoom” (auditory – instantaneous sound of a noise like an explosion or car engine roaring to life) and “Scintillating” (visual – flashes of light or sparks.). Hey, it’s going to be an exciting day!
There are no small words.
Small words have a magnetic pull on your behaviour. For example when someone says, “I should have the report on your desk by Monday” what is the likelihood that you will have that report on time? What that person is actually saying is “I am not going to do the report in that timescale”. When somebody says, “I must wash the car” what is the likelihood that they will wash the car? What they really mean is “the car needs washing but not by me”.
In these circumstances, what you really want someone to say positively is: “The report will be on your desk by Monday” and “I will wash the car”. If you listen carefully and pay close attention to precisely what is being said you will be able to tell if someone is committed to act and follow through.
Let’s take the word ‘try’ which means ‘to attempt to do or accomplish’. This very definition implies failure. You know what it’s like when you organise a party and Tony says “I will try and make it”, well who is the one person most likely not to turn up? Do you find yourself saying, “I will try and diet” or “I am trying to be good” in which case you have set yourself up for failure. In order to succeed it is important to reframe the context and positively say instead “I am on a diet” or “I am good”.
What you say matters.
Just in these few simple but impactful examples we can note that words resonate through our unconscious minds and that we use these to filter and create a map of our own world.
As we become habitual in our language we become habitual in our thoughts. Change the lyrics to the song you sing each day and you change your thinking patterns. If you change your thinking you change your behaviour and you will transform your life.
Now, I’m off to go and read the dictionary and leave you to start thinking about words you can use to positively change your world. Could it be a flash of inspiration, a resounding thought or the thrust of an idea?
Speak Your Mind | 146,175 |
Similar Businesses in the Area
- Visit JDS Electrical's Angie's List Page
JDS Electricalin business since6 Reviews | 6 Services
- Visit Master Electric of WNY's Angie's List Page
Master Electric of WNYin business since128 Reviews | 8 Services
- Visit Schultz Auto and Truck Repair's Angie's List Page
Schultz Auto and Truck Repairin business since1 Review | 111 Services
- Visit Home Depot's Angie's List Page
Home Depotin business since29 Reviews | 13 Services | 323,280 |
"Rights for restricted content", Lutz Donnerhacke Links are omnipresent in the Internet to provide access to other resources. There is no mechanism to express differences in law systems, access limitations, or arbitrary rules defined by the owner of the linked resource. Therefore links do depend on and enforce a communist sharing ideology, which ignores the content owner rights. Links may point to resources far away from the originating page, hiding this fact from the customer. It takes the data transport services for free, internet transit providers on the way from the content source to the customers are not extra payed for this effort. In many cases, the remote company generates huge amount of money from the customers worldwide not shared with the transit providers. In order to get the rights of all involved parties balanced, a new type of connection initiation is proposed: The Right. 2019-09-07T07:00:00-00:00 | 246,612 |
TITLE: Show that diagonals intersect at common point
QUESTION [5 upvotes]: Given is octagon where opposite sides are equal length and parallel. Show that diagonals: $AE,DH, BF, CG$ intersects at point $S$
So I have tried to create a parallelograms $AHED$ and $BCFG$ and use that $AE=GC$
REPLY [0 votes]: Since opposite sides are parallel, angles $BAH$ and $DEF$ are equal. Similarly angle $AHG$ and $EDC$ are equal. Also, lengths of parallel sides are equal. Now consider the line $GC$. It forms two pentagons $GHABC$ and $CDEFG$ which are equal for the reasons mentioned above.Since $S$ is the point at which line $AE$ crosses $GC$, it would be the midpoint of $AE$. Similarly using pentagons $AHBFE$ and $EDCBA$, it can be proved that the point at which $GC$ crosses $AE$ will be the midpoint of $GC$. Taking similar pairs such as BF and GC etc it can be proved that midpoint of all diagonals lie at the same point. And it is $S$ | 17,599 |
\begin{document}
\title{Shalom's property $H_{\mathrm{FD}}$ and extensions by $\mathbb{Z}$ of locally finite groups}
\author{J\'er\'emie Brieussel and Tianyi Zheng}
\maketitle
\begin{abstract}
We show that every finitely generated extension by $\mathbb{Z}$ of a locally normally finite group has Shalom's property $H_{\mathrm{FD}}$. This is no longer true without the normality assumption. This permits to answer some questions of Shalom, Erschler-Ozawa and Kozma. We also obtain a Neumann-Neumann embedding result that any countable locally finite group embedds into a two generated amenable group with property $H_{\mathrm{FD}}$.
\end{abstract}
\section{Introduction}
A finitely generated group has property $H_{\mathrm{FD}}$ (resp. $H_{\mathrm{T}}$) if every unitary representation with non-trivial reduced first cohomology admits a finite dimensional (resp. trivial) subrepresentation. Property $H_{\mathrm{FD}}$ was introduced by Shalom \cite{Shalom} as an invariant of quasi-isometry among finitely generated amenable groups. Recently, by showing directly that a group of polynomial growth has Property $H_\mathrm{FD}$, Ozawa \cite{Ozawa} gave a functional analysis proof of Gromov's theorem on groups of polynomial growth.
Amenable groups with property $H_{\mathrm{FD}}$ are somewhat ``small'' groups. In \cite{Shalom}, various families of solvable groups having, and not having property $H_{\mathrm{FD}}$ were shown. For instance by \cite{Shalom}, polycyclic groups and wreath products $(\mathbb{Z}/n\mathbb{Z}) \wr \mathbb{Z}$, $n\in\mathbb{N}$, have property $H_{\mathrm{FD}}$, while $\mathbb{Z} \wr \mathbb{Z}$ does not. Towards a unified explanation for these examples, Shalom conjectured in \cite[Section 6]{Shalom} that for solvable groups, property $H_{\mathrm{FD}}$ would be equivalent to finite Hirsch length (see \cite{Hillman} for the definition of the Hirsch length of elementary amenable groups). We describe counter-examples to both directions of this conjecture. In one direction, in Proposition \ref{wreath3} we show that the wreath product $(\mathbb{Z}/2\mathbb{Z})\wr \mathbb{Z}^d$ with $d\ge 3$ does not have property $H_{\mathrm{FD}}$, while it has finite Hirsch length. In the other direction, we construct a solvable group of infinite Hirsch length which has property $H_{\mathrm{FD}}$ by applying a result of Gournay \cite{Gournay}, see Proposition \ref{3solv}.
Aiming to generalize the case of lamplighter groups, Shalom also asked whether all locally-finite-by-$\mathbb{Z}$ groups have $H_{\mathrm{FD}}$. This is not true in general, in subsection \ref{sec:noHFD} we construct explicit harmonic cocycles with weakly mixing representation to show the following examples of locally-finite-by-$\mathbb{Z}$ groups don't have property $H_{\rm{FD}}$: the group $\textrm{Sym}(\mathbb{Z}) \rtimes \mathbb{Z}$, the discrete affine group of a regular tree and some locally nilpotent by $\mathbb{Z}$ groups introduced by Gromov \cite{Gromov} --- see Propositions \ref{SymZ}, \ref{DA} and \ref{prop_Gromov}. However, we prove that any locally-normally-finite-by-$\mathbb{Z}$ group has Shalom's property $H_{\mathrm{FD}}$.
Recall that a group is locally finite if any finite subset is included in a finite subgroup. We say a group is {\it locally normally finite} if any finite subset is included in a finite normal subgroup. For instance, the group $\oplus_{\mathbb{Z}} \mathbb{Z}/2\mathbb{Z}$ is locally normally finite, whereas the group $\textrm{Sym}(\mathbb{Z})$ of finitely supported permutations is locally finite but not locally normally finite.
\begin{theorem}\label{main}
Let $G$ be a finitely generated group that fits in an exact sequence
\[
1\to L\to G\to\mathbb{Z}\to1.
\]
where $L$ is locally normally finite. Then the group $G$ has Shalom's property $H_{\mathrm{T}}$, hence $H_{\mathrm{FD}}$.
\end{theorem}
We remark that in Theorem \ref{main}, the assumption that $L$ is locally normally finite is an "intrinsic" property of the group $L$, it doesn't depend on the extension. Although algebraically the locally-normally-finite assumption is restrictive,
random walks on the class of locally-normally-finite-by-$\mathbb{Z}$ groups exhibit very rich behavior. See examples and more details in subsection \ref{subsec: example}.
In particular, we obtain new examples of groups with property $H_{\rm{FD}}$.
Erschler and Ozawa \cite{EO} have given a criterion for the property that any finite dimensional summand of $b$ is cohomologically trivial, where $b$ is a $1$-cocycle with coefficients in a unitary representation.
As a corollary, they obtain a sufficient condition for $H_{\mathrm{FD}}$: if a group admits a symmetric probability measure $\mu$ with finite generating support such that $\limsup_n \mu^{(n)}(B(e,c\sqrt{n}))>0$ for all $c>0$, then it has property $H_{\mathrm{FD}}$. They pointed out that all previously known examples of amenable groups with $H_{\mathrm{FD}}$ were covered by this corollary. As an application of Theorem \ref{main}, we obtain examples where the speed of random walk can be super-diffusive.
\begin{corollary}\label{cor}
For any function $f:[1,\infty) \to [1,\infty)$ such that $f(1)=1$ and $\frac{f(x)}{\sqrt{x}}$ is non-decreasing and $\frac{f(x)}{x}$ decreases to $0$ as $x\to \infty$, there exists a group with property $H_{\mathrm{FD}}$ and a symmetric probability measure $\mu$ of finite generating support with speed of random walk $\mathbb{E}_{\mu^{(n)}}d(e,x)\simeq f(n)$.
\end{corollary}
Here $g(n)\simeq f(n)$ means there is $C\ge1$ with $\frac{f(n)}{C}\le g(n) \le Cf(n)$. Corollary \ref{cor} follows from Theorem \ref{main} as the groups introduced in \cite{BZ} with prescribed speed function are locally-normaly-finite-by-$\mathbb{Z}$, see \cite[Fact 2.10]{BZ}. These groups can also be chosen to have prescribed $\ell^p$-isoperimetry or $L_p$-compression.
In fact, examples of groups with property $H_{\mathrm{FD}}$ not satisfying the assumption of Erschler-Ozawa's corollary mentioned above can be obtained without Theorem \ref{main} using a result of Gournay \cite{Gournay}, who proved that harmonic cocycles of weakly mixing representations factorize by the FC-center $Z^{\mathrm{FC}}$, which consists of elements with finite conjugacy classes. Indeed, the construction of \cite[Section 2]{BZ} can be modified by taking finite factor groups. The group obtained is an FC-central extension by a lamplighter group, which has property $H_{\mathrm{T}}$ by \cite{Shalom}. See more details in Subsection \ref{subsec: example}. This variation of construction provides us groups with the following properties.
\begin{corollary}\label{cor-function}
Let $f:\mathbb{R}_{+} \to \mathbb{R}_{+}$ be a sub-addtive function such that $\frac{f(x)}{x}\to 0$ as $x\to \infty$. Then there is a finite generated group $\Delta$ and a symmetric probability measure $\mu$ of finite generating support on $\Delta$ such that for some constant $c>0$, for all $n\ge 1$,
$$\mathbb{E}_{\mu^{(n)}}d(e,x) \ge cf(n);$$
while all sub-exponential growth $\mu$-harmonic functions on $\Delta$ factors through a quotient map $\Delta \to F\wr \mathbb{Z}$ with $F$ finite.
In particular, all sublinear $\mu$-harmonic functions on $\Delta$ are constant.
\end{corollary}
A folklore conjecture of Kozma asks if diffusive speed of $\mu$-random walk on a group $G$ is equivalent to absense of non-constant sublinear $\mu$-harmonic functions. The examples of Corollary \ref{cor-function} answer one direction negatively. Because of currently limited understanding of groups with diffusive simple random walk behavior, the other direction of the conjecture (if diffusive speed implies all sublinear harmonic functions are constant) remains open.
In the spirit of the classical Neumann-Neumann embedding \cite{Neumann2}, we show that any countable locally finite group embeds into a group with Shalom's property $H_{\mathrm{FD}}$.
\begin{proposition}\label{NN}
Let $H$ be a countable locally finite group. Then there exists a
two generated infinite amenable group $G$ with Shalom's
property $H_{\mathrm{FD}}$ and containing $H$ as an embedded subgroup.
\end{proposition}
The proof uses a sufficient
condition for $H_{\mathrm{FD}}$ from Erschler-Ozawa \cite{EO}.
We wish to point out that it is not known whether the lamplighter group $\mathbb{Z}/2\mathbb{Z} \wr \mathbb{Z}^2$ has property $H_{\mathrm{FD}}$. Neither is known an example of a non-Liouville amenable group with property $H_{\mathrm{FD}}$.
The paper is organized as follows. Definitions and notations are given in Section \ref{sec:setting}. Theorem \ref{main} is proved in Subsection \ref{subsec:main_proof} and Corollary \ref{cor-function} is proved in Subsection \ref{subsec: example}. It is proved in Section \ref{wrZd} that $(\mathbb{Z}/2\mathbb{Z})\wr \mathbb{Z}^d$ do not have property $H_{\textrm{FD}}$ when $d \ge 3$. Examples of locally-finite-by-$\mathbb{Z}$ groups without property $H_{\mathrm{FD}}$ are given in Section \ref{sec:noHFD}. Proposition \ref{NN} is derived in Section \ref{NNemb}. In Appendix \ref{appendix}, we prove that the wreath product $H\wr G$ of two infinite finitely generated amenable groups does not have property $H_{\mathrm{FD}}$. This answers another question of Shalom \cite[Section 6]{Shalom}, who had proved it when $H$ has infinite abelianization.
\section{Shalom's property and harmonic cocycles}\label{sec:setting}
We recall definitions and basic facts about Shalom's property $H_{\mathrm{FD}}$. More details can be found in \cite{Shalom}, \cite{Ozawa}, \cite{EO}.
Let $G$ be a finitely generated group and $\pi:G \to \mathcal{U}(\mathcal{H})$ a unitary representation into a Hilbert space $\mathcal{H}$. A function $b:G\to \mathcal{H}$ is a $1$-cocycle if $b(gh)=b(g)+\pi_gb(h)$ for all $g,h$ in $G$, and a $1$-coboundary if there exists $v$ in $\mathcal{H}$ such that $b(g)=v-\pi_gv$ for all $g$ in $G$. Denote $Z^1(G,\pi)$ the vector space of $1$-cocycles and $B^1(G,\pi)$ the subspace of $1$-coboundaries.
The first reduced cohomology of $\pi$ is the quotient space $$\overline{H}^1(G,\pi)=Z^1(G,\pi)/\overline{B^1(G,\pi)},$$ where the closure is taken with respect to uniform convergence on compact (i.e. finite) subsets.
\begin{definition}[Shalom \cite{Shalom}]
A group $G$ has property $H_{\mathrm{FD}}$ (resp. $H_{\mathrm{F}}$, $H_{\mathrm{T}}$) if any unitary representation $\pi$ with $\overline{H^1}(G,\pi) \neq 0$ admits a subrepresentation which is finite dimensional (resp. finite, trivial).
\end{definition}
Given a symmetric finitely supported non-degenerate probability measure $\mu$ on $G$, the scalar product
\[
\langle b,b'\rangle_{\mu}=\sum_{g\in G} \langle b(g),b'(g) \rangle \mu(g)
\]
gives a Hilbert space structure on $Z^1(G,\pi)$. The associated topology coincides with uniform convergence on compact subsets.
Harmonic $1$-cocycles were implicitly introduced by Guichardet in \cite {Guich}, where it was observed that every element in the first reduced cohomology $\overline{H}^1(G,\pi)$ is uniquely represented by a $\mu$-harmonic cocycle. A cocycle $b$ is $\mu$-harmonic if
\[
\sum_{g \in G} b(xg)\mu(g)=b(x) \textrm{ for all }x\in G,
\]
or equivalently $\sum_{g \in G} b(g)\mu(g)=0$. This happens if and only if $b$ is orthogonal to the space of $1$-coboundaries. So there is a vector space isomorphism $$\overline{H}^1(G,\pi) \simeq \overline{B}^1(G,\pi)^\perp.$$
It follows that a group $G$ has property $H_{\mathrm{FD}}$ if and only if any $\mu$-harmonic cocycle of a weakly mixing representation (i.e. a representation without finite dimensionnal invariant subspace) is zero.
\section{Kernel of harmonic cocycles }
Throughout the rest of the paper, we always assume that $\mu$ is a symmetric probability measure with finite generating support on $G$.
\subsection{Extensions by $\mathbb{Z}$ of locally normally finite groups}\label{subsec:main_proof}
\begin{proposition}\label{main_prop}
Let $G$ be a finitely generated group that fits in an exact sequence
\[
1\to N\to G\to\mathbb{Z}\to1.
\]
Suppose $F$ is a finite normal subgroup of $N$ and the projection of $\mu$ to $\mathbb{Z}$ is the law of a simple random walk.
Let $b:G\to\mathcal{H}$ be a $\mu$-harmonic $1$-cocycle with weakly
mixing linear part $\pi$. Then
\[
\forall \gamma \in F, \quad b(\gamma)=0.
\]
\end{proposition}
The proof of Proposition \ref{main_prop} follows the standard coupling argument as in \cite[Theorem 1]{BDCKY},
which shows that the lamplighter group $F\wr \mathbb{Z}$ with $F$ finite does not admit non-constant sublinear harmonic functions.
The following lemma adapted from \cite[Proposition]{Ozawa} provides information on the growth of a harmonic cocycle
with weakly mixing representation.
It roughly says that a harmonic cocycle with weakly mixing representation grows slower than $n^{1/2}$
on the set that doesn't translate $\mu^{(n)}$ too far.
\begin{lemma}[{\cite[Proposition]{Ozawa}}]\label{byOzawa}
Let $b:G\to\mathcal{H}$ be a $\mu$-harmonic cocycle with
weakly mixing representation $\pi$.
Let $0<\delta<\frac{1}{3}$ be a constant.
Define
\[
\mathcal{S}_{n}(\delta):=\left\{g\in G:\ \frac{1}{2}\left\Vert \mu^{(n)}-g\mu^{(n)}\right\Vert _{1}<\delta\right\}.
\]
Then
\[
\frac{1}{\sqrt{n}}\sup_{g\in\mathcal{S}_{n}(\delta)}\left\Vert b(g)\right\Vert _{\mathcal{H}}\longrightarrow0\ \mbox{as }n\to\infty.
\]
\end{lemma}
\begin{proof}
Since the formulation is different from the original statement in \cite{Ozawa}, we include a proof here for completeness.
By \cite[Lemma]{Ozawa},
\[
\sup_{\xi,\left\Vert \xi\right\Vert =1}\frac{1}{n}\sum_{x}\mu^{(n)}(x)\left|\left\langle b(x),\xi\right\rangle \right|^{2}\longrightarrow 0.
\]
It follows that given $\delta$, for any $\epsilon>0$, there exists
a constant $N$ such that for all $n\ge N$,$ $
\[
\sup_{\xi,\left\Vert \xi\right\Vert =1}\sum_{x}\mu^{(n)}(x)\left|\left\langle b(x),\xi\right\rangle \right|^{2}\le\epsilon^{2}\delta n.
\]
For any unit vector $\xi$, let
\[
E_{\xi}=\left\{ x\in G:\ \left|\left\langle b(x),\xi\right\rangle \right|\le\epsilon n^{1/2}\right\} .
\]
Then
\[
\mu^{(n)}(E_{\xi}^{c})\le\frac{1}{\epsilon^{2}n}\sum_{x}\mu^{(n)}(x)\left|\left\langle b(x),\xi\right\rangle \right|^{2}\le\delta.
\]
Since for any $g\in\mathcal{S}_{n}(\delta)$, the total variation
distance between $\mu^{(n)}$ and $g\mu^{(n)}$ is less than $\delta$,
we have
\[
g\mu^{(n)}(E_{\xi})>\mu^{(n)}(E_{\xi})-\delta\ge1-2\delta.
\]
That is $\mu^{(n)}(g^{-1}E_{\xi})>1-2\delta$. Therefore $\mu^{(n)}(g^{-1}E_{\xi})+\mu^{(n)}(E_{\pi_{g}^{\ast}\xi})>1-2\delta+1-\delta>1$,
it follows that
\[
\left(g^{-1}E_{\xi}\right)\cap E_{\pi_{g}^{\ast}\xi}\neq\emptyset.
\]
Take an element $x_{\xi}\in\left(g^{-1}E_{\xi}\right)\cap E_{\pi_{g}^{\ast}\xi}$.
By the cocycle identity,
\[
\left\langle b(g),\xi\right\rangle =\left\langle b(gx_{\xi}),\xi\right\rangle -\left\langle \pi_{g}b(x_{\xi}),\xi\right\rangle =\left\langle b(gx_{\xi}),\xi\right\rangle -\left\langle b(x_{\xi}),\pi_{g}^{\ast}\xi\right\rangle .
\]
Since $gx_{\xi}\in E_{\xi}$ and $x_{\xi}\in E_{\pi_{g}^{\ast}\xi}$,
we have
\[
\left|\left\langle b(g),\xi\right\rangle \right|\le2\epsilon n^{1/2}.
\]
Since this is true for all unit vectors $\xi$, we conclude that for
all $n>N$, any $g\in\mathcal{S}_{n}(\delta)$, $\left\Vert b(g)\right\Vert \le2\epsilon n^{1/2}$.
\end{proof}
\begin{proof}[Proof of Proposition \ref{main_prop}] Denote $\phi:G\to \mathbb{Z}$ the mapping of the exact sequence.
The subgroup $F$ is normal in $N$, thus for $g,g'$ in $G$ with $\phi(g)=\phi(g')$, we have $gFg^{-1}=g'F(g')^{-1}$. Therefore we can write $F^{x}$ for the conjugation $F^{x}=gFg^{-1}$ where $\phi(g)=x \in \mathbb{Z}$.
As $\mu$ is non-degenerate, there is some integer $R$ such that $F \subset \textrm{supp}(\mu^{(R)})$. We set:
\[
\varepsilon:=|F|\inf_{z \in F} \mu^{(R)}(z)>0.
\]
Given an element $\gamma\in F^{x}$, we consider a coupling of two $\mu^{(R)}$-random
walks starting from $\gamma$ and $e$. We write the two trajectories
as $\gamma Z_n$ and $\tilde{Z}_n$ where $Z_n=X_1\ldots X_n$ and $\tilde{Z}_n=\tilde{X}_{1}\ldots\tilde{X}_{n}$ are defined as follows, satisfying $\phi(Z_{n})=\phi(\tilde{Z}_{n})$ at all times. As long as case (c) has not happened, we set:
\begin{enumerate}[label=(\alph*)]
\item if $\phi(Z_n)\neq x$, then $\tilde{X}_{n+1}=X_{n+1}$ distributed as $\mu^{(R)}$,
\item if $\phi(Z_n)= x$, then with $1-\varepsilon$ probability $\tilde{X}_{n+1}=X_{n+1}$ distributed as $\frac{1}{1-\varepsilon}\left(\mu^{(R)}-\varepsilon \mathbf{u}_F\right)$, where $\mathbf{u}_F$ is the uniform distribution on the finite group~$F$,
\item if $\phi(Z_n)= x$, with $\varepsilon$ probability $X_{n+1}$ is uniformly distributed in $F$ and $\tilde{X}_{n+1}=\tilde{Z}_{n}^{-1}\gamma Z_nX_{n+1}$, implying $\gamma Z_{n+1}=\tilde{Z}_{n+1}$.
\end{enumerate}
Once case (c) happened, the random walks are coupled and we just set $\tilde{X}_{n+1}=X_{n+1}$ distributed as $\mu^{(R)}$.
By construction, we have
\begin{eqnarray}\label{position}
\forall n, \quad \tilde{Z}_{n}^{-1}\gamma Z_{n}\in F^{x-\phi(Z_{n})}.
\end{eqnarray}
In particular at the times when the coupling attempt is performed, $\phi(Z_{n})=x$, the difference is in $F$.
Let $\tau$ be
the coupling time. Every time the projection of the $\mu^{(R)}$-random walk on $\mathbb{Z}$
visits $x$, the chance to succeed is $\varepsilon$. The distribution of the local time of simple random walk on the integers is known
explicitly. For the $R$-convolution, the local time $L(x,n)=|\{0\le m \le n : \phi(Z_m)=x\}|$, differs by at most a multiplicative constant $C=C(R)$. For
$x\le c\sqrt{n}$,
\begin{eqnarray*}
\mathbb{P}(\tau >n) &=&\mathbb{P}(L(n,x)=0)+\sum_{k=1}^{n-x}\mathbb{P}(L(n,x)=k)(1-\varepsilon)^{k} \\
&\le& \frac{C}{2^{n}}\sum_{k=0}^{x-1} \left(\begin{array}{c}
n\\
\left\lfloor \frac{n-k}{2}\right\rfloor \end{array}\right) +\sum_{k=1}^{n-x}\frac{1}{2^{n-k}}\left(\begin{array}{c}
n-k\\
\left\lfloor (n+x)/2\right\rfloor
\end{array}\right)
(1-\varepsilon)^{k/C},
\\
& \le & \frac{C}{n^{1/2}}\left(|x|+\frac{1}{\varepsilon}\right),
\end{eqnarray*}
using successively \cite[Theorems 9.1, 9.4 and 2.8]{Revesz2013}.
It follows that for $|x|\leq r+R$, for any $\delta>0$, there is an integer $M=M(\delta)>0$ such that
starting from any $\gamma$ in such $F^x$,
$$\mathbb{P}(\tau>Mr^2)<\delta.$$
This implies the total variation distance is bounded by
\[
\frac{1}{2}\left\Vert \mu^{(Mr^2)}-g\mu^{(Mr^2)}\right\Vert _{1}\le\mathbb{P}(\tau>Mr^2)\le \delta.
\]
With the notations of Lemma \ref{byOzawa}
\begin{eqnarray}\label{Sdelta}
\bigcup_{|x|\leq r+R} F^x \subset \mathcal{S}_{Mr^2}(\delta).
\end{eqnarray}
Let $\tau_r$ be the exit time of the interval $[-r,r]$. For $|x| \leq cr$, $0<c<1/2$, by standard resistance calculation, we have a similar estimates that for some constant $C'=C'(R)$,
\begin{eqnarray}\label{coupling_by_exit}
\mathbb{P}\left(\tau>\tau_r\right)\le \frac{C'}{r}\left(|x|+\frac{1}{\varepsilon}\right).
\end{eqnarray}
Now let $\gamma$ belong to $F$, so $x=0$. By optional stopping theorem,
\begin{eqnarray}\label{ost}
\Vert b(\gamma)-b(e)\Vert= \Vert \mathbb{E}[b(\gamma Z_{\tau_r})-b(\tilde{Z}_{\tau_r})]\Vert \le \mathbb{E}\Vert b(\gamma Z_{\tau_r})-b(\tilde{Z}_{\tau_r})\Vert.
\end{eqnarray}
By cocycle identity $b(\gamma Z_{\tau_r})=b(\tilde{Z}_{\tau_r}\tilde{Z}_{\tau_r}^{-1}\gamma Z_{\tau_r})=b(\tilde{Z}_{\tau_r})+\pi_{\tilde{Z}_{\tau_r}}b(\tilde{Z}_{\tau_r}^{-1}\gamma Z_{\tau_r})$. Thus
\[
\Vert b(\gamma)-b(e)\Vert\le \mathbb{E} \Vert b(\tilde{Z}_{\tau_r}^{-1}\gamma Z_{\tau_r})\Vert \leq \mathbb{P}(\tau >\tau_r) \sup\{\Vert b(g)\Vert: g \in F^{-\phi(Z_{\tau_r})}\}.
\]
by (\ref{position}). We can use (\ref{coupling_by_exit}) and (\ref{Sdelta}), as $|\phi(Z_{\tau_r})|\le r+R$, to bound:
\[
\Vert b(\gamma)-b(e)\Vert \le \frac{C'}{\varepsilon r}\sup\{\Vert b(z)\Vert : z \in \mathcal{S}_{Mr^2}(\delta)\}\underset{r \to \infty}{\longrightarrow} 0.
\]
The limit is zero by Lemma \ref{byOzawa}.
\end{proof}
We observe from line (\ref{ost}) that a harmonic function $h:G\to \mathcal{H}$ (not necessarily a cocycle) satisfying $\frac{1}{n}\max\{\Vert h(x)-h(y)\Vert:d(x,y)\le n\}\longrightarrow 0$ must factorize by $F$.
\begin{proof}[Proof of Theorem \ref{main}]
The extension $1\to L \to G \to \mathbb{Z} \to 1$ is necessarily split, as there exists $g_0$ in $G$ with $\phi(g_0)=1$. Thus $G$ admits a symmetric generating set with projection to $\mathbb{Z}$ included in $\{-1,+1\}$. We take $\mu$ supported on it and apply Proposition \ref{main_prop}.
For every weakly mixing representation $\pi$, any $\mu$-harmonic cocycle $b$ factorizes by $L$, i.e. $b|_L=0$. Moreover, for any $g$ in $G$ and $h$ in $L$, there is $h'$ in $L$ such that $hg=gh'$, so
\[
b(g)=b(g)+\pi_gb(h')=b(gh')=b(hg)=b(h)+\pi_hb(g)=\pi_hb(g).
\]
Therefore restricted to $\overline{\textrm{Im}(b)}$, the representation $\pi$ factorizes by $L$. If $b\neq 0$, then $\pi|_{\overline{\textrm{Im}(b)}}$ admits a trivial subrepresentation as $\mathbb{Z}$ has property $H_{\mathrm{T}}$. This proves $H_{\mathrm{T}}$ for $G$.
\end{proof}
\subsection{A variant on harmonic functions}\label{subsec: variant}
By \cite[Theorem 4.7]{Gournay}, any harmonic cocycle with weakly mixing linear part must vanish on the FC-center. In this subsection we formulate a variant of this result which states that any harmonic function with sub-exponential growth is constant on the subset of torsion elements in the FC-center.
Recall that the FC-center $Z^{FC}(G)$ is the subgroup of elements with a finite conjugacy class. A function $h:G\to \mathbb{R}$ has {\it subexponential growth} if
\[
e^{-cn}M_h(n) \underset{n \to \infty}{\longrightarrow} 0, \textrm{ for all }c>0,
\]
where $M_h(n)=\max\{|h(x)|:d(e,x)\le n\}$. A function $h:G\to \mathbb{R}$ has {\it sublinear growth} if
$M_h(n)/n\to 0$ as $n\to\infty$.
\begin{proposition}\label{sub-ex function}
Let $G$ be a finitely generated group with a finitely supported symmetric non-degenerate probability measure $\mu$. Let $h:G\to \mathbb{R}$ be a $\mu$-harmonic function of subexponential growth. Then for any $\gamma$ in $Z^{FC}(G)$ of finite order and for any $x$ in $G$, one has $h(\gamma x)=h(x)$.
\end{proposition}
The proof is an adaptation of the coupling argument used for Proposition~\ref{main_prop}.
\begin{proof}
As the conjugacy class of $\gamma$ is finite, we have $g_1,\dots,g_r$ in $G$ such that
\[
\langle \gamma \rangle^G=\bigcup_{i=1}^r \langle \gamma \rangle^{g_i} \textrm{, and } \langle \gamma \rangle^{g_i} \cap \langle \gamma \rangle^{g_j}=\{e\} \textrm{ whenever } i\neq j,
\]
and this union is a finite set, hence contained in the support of $\mu^{(R)}$ for some finite $R$. Set $\varepsilon:=\textrm{order}(\gamma)\inf\{\mu^{(R)}(x):x\in \langle \gamma \rangle^G\}>0$.
We couple two $\mu^{(R)}$-random walks $\gamma x Z_n$ and $x\tilde{Z}_n$ as follows. Always sample $X_{n+1}$ according to $\mu^{(R)}$. As long as matching case (b) has not happened $\tilde{Z}_n=Z_n$, so for each $n$ there are integers $i, j$ such that $(x\tilde{Z}_n)^{-1}\gamma xZ_n=(\gamma^j)^{g_{i}}\in \langle \gamma\rangle^{g_{i}}$.
\begin{enumerate}[label=(\alph*)]
\item With probability $1-\varepsilon$, the sample $X_{n+1}$ is not in $\langle \gamma\rangle^{g_{i}}$, then set $\tilde{X}_{n+1}=X_{n+1}$,
\item with probability $\varepsilon$, the sample $X_{n+1}=(\gamma^s)^{g_{i}}$ belongs to $\langle \gamma\rangle^{g_{i}}$, then set $\tilde{X}_{n+1}=(\gamma^{-s-j})^{g_i}$, so $x\tilde{Z}_{n+1}=\gamma x Z_{n+1}$.
\end{enumerate}
Once case (b) occured the increments always agree. The probability not to couple by time $n$ is $(1-\varepsilon)^n$. By the classical martingale argument:
\[
|h(\gamma x)-h(x)| \leq \mathbb{E}|h(\gamma x Z_n)-h(x \tilde{Z}_n)| \leq (1-\varepsilon)^n M_h(C n) \underset{n \to \infty}{\longrightarrow} 0,
\]
where $C$ is the diameter of the support of $\mu^{(R)}$.
\end{proof}
\subsection{Examples of groups with property $H_{\rm{T}}$}\label{subsec: example}
As mentioned in the Introduction, the groups introduced in \cite{BZ} with prescribed random walk behavior are locally-normally-finite-by-$\mathbb{Z}$, therefore by Theorem \ref{main} they have Shalom's property $H_{\mathrm{T}}$. Here we introduce a modification of the construction, which produces groups which are FC-central extensions of lamplighter groups, moreover elements in the FC-center are torsion.
These groups are constructed by taking diagonal products of a sequence of marked groups. Let $(G_s)_{s\ge 1}$ be a sequence of marked groups, each $G_s$ marked with a $\ell$-tuple of generators $\mathcal{T}_s=(t_1(s),...,t_{\ell}(s))$. The diagonal product $\Delta$ of $(G_s)_{s\ge 1}$ (also called the universal group of this sequence) is the quotient $\mathbf{F}_{\ell} /\cap_{s}\ker\left(\boldsymbol{\pi}_{s}\right)$, with
the projection map $\boldsymbol{\pi}_s:\mathbf{F}_{\ell} \to G_s$ sending the generators of the free group on $(t_1,...,t_{\ell})$ to the marked generators of $G_s$.
We recall some notations for wreath products. An element in the (restricted) wreath product $H\wr G$ is represented by a pair $(f,x)$
where $f:G\to H$ is a function of finite support and $x\in G$. We refer to $f$ as
the lamp configuration and $x$ as the position of the cursor. The
product rule is
\[
(f,x)(g,y)=(f\tau_xg,xy),\textrm{ where } \tau_xg(z)=g(zx^{-1}).
\]
The neutral element is denoted as $\left(\boldsymbol{e}, e_G\right)$ where $\mbox{support}(\boldsymbol{e})$
is the empty set. For $x\in G$ and $\gamma\in H$, we denote
by $\gamma\delta_{x}$ the function taking value $\gamma$ at $x$
and $e_{H}$ elsewhere. When $H$ is abelian we write $0$ for its identity element and use additive notation.
Let $A=\{a_{1},\dots,a_{|A|}\}$ and $B=\{b_{1},\dots,b_{|B|}\}$
be two finite groups. Let $\left\{ \Gamma_{s}\right\} $ be a sequence
of groups such that each $\Gamma_{s}$ is marked with a generating
set of the form $A(s)\cup B(s)$ where $A(s)$ and $B(s)$ are finite
subgroups of $\Gamma_{s}$ isomorphic respectively to $A$ and $B$.
We fix the isomorphic identification and write $A(s)=\{a_{1}(s),\dots,a_{|A|}(s)\}$
and similarly for $B(s)$. We impose the assumption
\[
\Gamma_{s}/[A(s),B(s)]^{\Gamma_{s}}\simeq A(s)\times B(s)\simeq A\times B.
\]
That is, the relative abelianization, which is always a quotient of $A\times B$,
is in fact isomorphic to $A\times B$.
In \cite{BZ}, the factor groups $(G_s)$ are taken to be wreath products $\Gamma_s\wr \mathbb{Z}$
with lamp groups $\Gamma_s$ and marked generating tuple carefully chosen to follow prescribed speed function.
Here instead we take each factor group to be a finite group
$G_s=\Gamma_s \wr (\mathbb{Z}/m_s\mathbb{Z})$.
The purpose is exactly to produce extensions with finite conjugacy classes.
The inputs into the construction are two sequences of strictly increasing positive
integers $\left(k_{s}\right)_{s\ge 1}$ and $\left(m_{s}\right)_{s\ge 1}$ such that $m_s\ge 2k_s$ for all $s\ge 1$;
and a sequence of finite groups $\Gamma_s$ marked with generating set $(A(s),B(s))$ satisfying the assumptions described above.
Take the wreath product $G_s=\Gamma_s \wr \mathbb{Z}/m_s\mathbb{Z}$.
The generating tuples are marked as follows:
\[
\mathcal{T}_{s}=\left(\tau(s),\alpha_{1}(s),\ldots,\alpha_{|A|}(s),\beta_{1}(s),\ldots,\beta_{|B|}(s)\right)
\]
where $\tau(s)=\left(\boldsymbol{e},+1\right)$ and
\[
\alpha_{i}(s)=\left(a_{i}(s)\delta_{0},0\right),1\leq i\leq|A|,\ \beta_{i}(s)=\left(b_{i}(s)\delta_{k_{s}},0\right),1\leq i\leq|B|.
\]
The group we consider is the diagonal product $\Delta$ of the sequence $((G_s,\mathcal{T}_s))_{s\ge 1}$.
By construction, the group $\Delta$ is an FC-central extension of $(A\times B)\wr \mathbb{Z}$, and the FC-center consists of torsion elements.
\begin{fact}\label{factFC} Let $\left(k_{s}\right)_{s\ge 1}$, $\left(m_{s}\right)_{s\ge 1}$ and $(\Gamma_s)_{s\ge 1}$ be as above,
and $\Delta$ be the diagonal product of marked groups $((G_s,\mathcal{T}_s))_{s\ge 1}$.
Then the FC-center of $\Delta$ is
$$Z^{\mathrm{FC}}(\Delta)=\oplus_{s\ge0}\ker\left(\Gamma_s\to A(s)\times B(s)\right)^{m_s},$$
and
$$\Delta/Z^{\mathrm{FC}}(\Delta)\simeq (A\times B)\wr \mathbb{Z}.$$
\end{fact}
\begin{proof}
In $A\times B \wr \mathbb{Z}$, any non-trivial conjugacy class is infinite. It is therefore sufficient to show that the direct sum is included in the FC-center. Any element $f$ there is a direct sum of functions $f_s$ from $\mathbb{Z}/m_s\mathbb{Z}$ to $\ker\left(\Gamma_s\rightarrow A(s)\times B(s)\right)$ and all these function are trivial for $s >s_0$ depending on $f$. Conjugating by the translation $\tau$ amounts to shift the functions to $f_s(\cdot+1)$. Conjugating by an element $a$ of $A$ (resp. $b$ of $B$) amounts to conjugate the values of the functions at $0$ to $af_s(0)a^{-1}$ (resp. at $k_s$ to $bf_s(k_s)b^{-1}$). As $\Gamma_s$ and $m_s$ are finite, the conjugacy class of $f$ is included in the finite $\oplus_{s\leq s_0}\ker\left(\Gamma_s\rightarrow A(s)\times B(s)\right)^{m_s}$.
\end{proof}
By \cite[Theorem 4.7]{Gournay} or Theorem \ref{main}, $\Delta$ has Shalom's property $H_{\mathrm{T}}$. Recall that a harmonic $1$-cocycle projects to
Lipschitz harmonic functions on the group. For a group without property $H_{\mathrm{FD}}$, there is an infinite dimensional space of
non-constant Lipschitz harmonic functions
coming from $1$-dim projections of harmonic cocycle with weakly mixing representation.
In general there are Lipschitz harmonic functions that are not related to equivariant harmonic cocycles.
On the diagonal product $\Delta$ constructed above, we have that all its harmonic functions of sub-exponential growth
factor through the quotient $(A\times B)\wr \mathbb{Z}$.
More precisely, let $\mu$ be a non-degenerate symmetric probability measure on $\Delta$,
then by Proposition \ref{sub-ex function},
all $\mu$-harmonic functions of sub-exponential growth on $\Delta$ are constant on $Z^{\mathrm{FC}}(\Delta)$.
In other words, all $\mu$-harmonic functions of sub-exponential growth on $\Delta$ factor through the projection $\Delta\to (A\times B)\wr \mathbb{Z}$.
Since $A\times B$ is finite, the lamplighter group $(A\times B)\wr \mathbb{Z}$ does not have non-constant sublinear $\bar{\mu}$-harmonic functions \cite{BDCKY}, it follows that $\Delta$ doesn't have any non-constant sublinear $\mu$-harmonic functions.
\begin{proof}[Proof of Corollary \ref{cor-function}]
Since speed of random walk on $\Delta$ is bounded from below by speed on each factor $G_s$, given a sub-addtive function $f$ such that $\frac{f(x)}{x}\to 0$ as $x\to \infty$, we can choose $\Gamma_s$ and $k_s$ similar to \cite{BZ} and take $m_s$ to grow sufficiently faster than $k_s$, $m_s\gg k_s$, to guarantee that the speed of random walk on $\Delta$ is faster than $f$.
\end{proof}
\begin{remark}
By adapting the methods of \cite{BZ}, the speed, return probability and $\ell^p$-isoperimetry of the group $\Delta$ we discuss here can be estimated quite precisely. We don't pursue this direction here. By choosing $m_s$ sufficiently larger than $k_s$, one can guarantee that the random walk behavior (such as speed, entropy, return probability) on the two constructions with factor groups $\Gamma_s\wr \mathbb{Z}$ or $\Gamma_s\wr (\mathbb{Z}/m_s\mathbb{Z})$ with the same $(k_s)$ are comparable. In some sense the random walk parameters don't distinguish these two. However, the diagonal product with factor groups $\Gamma_s\wr \mathbb{Z}$ can admit non-constant sublinear harmonic functions. This aspect will be addressed elsewhere.
\end{remark}
This construction also permits to disprove one direction of a conjecture of Shalom that among solvable groups, property $H_{\mathrm{FD}}$ would be equivalent to infinite Hirsch length.
\begin{proposition}\label{3solv}
There exists a $3$-solvable group with Shalom's property $H_{\mathrm{T}}$ and infinite Hirsch length.
\end{proposition}
\begin{proof}
Let $(k_s)_{s\geq 1}$ and $(m_s)_{s\geq 1}$ be as above. Take $A=B=\mathbb{Z}/2\mathbb{Z}$ and $\Gamma_s=D_{\infty}$ to be an infinite dihedral group for all $s$. The associated diagonal product $\Delta$ is $3$-solvable and contains $\oplus_{s\leq s_0}\ker\left(\Gamma_s\rightarrow A(s)\times B(s)\right)^{m_s}=\oplus_s\oplus_{\mathbb{Z}/m_s\mathbb{Z}} \mathbb{Z}$ so has infinite Hirsch length. Moreover, this subgroup coincides with the FC-center.
Indeed, $\ker\left(D_\infty\rightarrow A\times B\right)$ is a cyclic group generated by the commutator $abab$ of two involutions. Non-trivial elements there have a conjugacy class of size two in $D_\infty$. The proof of Fact \ref{factFC} applies. Again, the group $\Delta$ is an extension of its FC-center by $A\times B \wr \mathbb{Z}$, hence has Shalom's property $H_{\mathrm{T}}$ by \cite{Gournay}.
\end{proof}
\section{Examples of groups without property $H_{\mathrm{FD}}$}
In this section we construct explicit harmonic cocycles on some families of groups. Many of them are obtained as virtual coboundaries.
Recall that given a unitary representation $\pi: G\to \mathcal{H}$, a cocycle $b:G\to \mathcal{H}$ is called a virtual coboundary if
$b(g)=\pi(g)x-x$ for some $x\in W\setminus \mathcal{H}$, where $W$ is a vector space where the unitary representation $\pi$ extends to a linear action on $W$.
Finding virtual coboundaries is a useful tool to exhibit cocycles with certain properties, see for example \cite{FV1}, \cite{Gournay}.
\subsection{Wreath products $(\mathbb{Z}/2\mathbb{Z})\wr \mathbb{Z}^d$ with $d\ge 3$ do not have~$H_{\mathrm{FD}}$}\label{wrZd}
We show a more general result.
\begin{proposition}\label{wreath3}
Suppose $H$ is a finitely generated group on which simple random walk is transient. The wreath product $(\mathbb{Z}/2\mathbb{Z})\wr H$ does not have property $H_{\mathrm{FD}}$.
\end{proposition}
\begin{proof}
Denote elements of $G=(\mathbb{Z}/2\mathbb{Z})\wr H$ by $g=(f,h)$, where $f:H\to \mathbb{Z}/2\mathbb{Z}$ is the lamp configuration and $h\in H$. Let $W$ be the vector space of real valued functions on $H$. Take the linear representation $\pi$ of $G$ on $W$ defined by
$$\pi((f,h))\psi(x)=(-1)^{f(x)}\psi(xh).$$
The restriction of $\pi$ to the Hilbert space $\ell^2(H)$ is a unitary representation. It is clear that $\pi$ doesn't have any finite dimensional sub-representation because the subgroup $H$ acts by the right regular representation on $\ell^2(H)$.
Let $\mu$ be a symmetric probability measure on $H$ with finite generating support. By assumption that the $\mu$-random walk is transient we have that there is a unit flow of finite energy from identity $e_H$ to infinity, see \cite[Theorem 2.11]{book}. Take the corresponding voltage function $v: H\to \mathbb{R}$, then it satisfies that $v(e_H)=1$,
$$(I-P_{\mu}) v=\mathbf{1}_{e_H},$$
where $P_{\mu}v(x)=\sum_{y\in H}v(xy)\mu(y)$.
In particular $v$ is $\mu$-harmonic on $H$ except at identity $e_H$.
Add a constant to $v$ so that $v(e_H)=a$, where $a\in\mathbb{R}$ is a number to be chosen later.
Take a virtual coboundary $b:G\to \ell^2(H)$ defined as
$$b(g)=v-\pi(g)v.$$
We check that $b(g)$ is indeed in $\ell^2(H)$: for a generator $s\in \mbox{supp}\mu$,
$$\left\Vert b((0,s)) \right\Vert_{\ell^2(H)}^2=\sum_{x\in H} (v(x)-v(xs))^2 \le \frac{1}{\mu(s)}\sum_{x,y\in H} (v(x)-v(xy))^2\mu(y)<\infty.$$
In the last inequality we have used that the flow of $v$ has finite energy by the transience assumption.
And for $g=(f,e_H)$, since $f$ is of finite support,
$$\left\Vert b((f,e_H)) \right\Vert_{\ell^2(H)}^2=\sum_{x\in \rm{supp}f}4v(x)^2<\infty.$$
Since $b(g)\in \ell^2(H)$ for generators of $G$, it follows that $b(g)$ is in $\ell^2(H)$ for any $g\in G$.
Take $\eta=\frac{1}{2}\mu+\frac{1}{2}\mathbf{1}_{\delta_e^1}$, where $\delta_e^1$ denotes the element $(f,0)$ where $f=1$ at $e_H$ and $0$ otherwise. It is clear that $\eta$ is a symmetric probability measure on $G$ with generating support. We now choose the constant $a$ to make $b$ an $\eta$-harmonic cocycle. To this end it suffices to have
$$\sum_{g\in G}b(g)\eta(g)=0.$$
Note that
$$b(\delta_e^1)=2a\mathbf{1}_{e_H},\ \sum_{h\in H}b((0,h))\mu(h)=(I-P_{\mu})v=\mathbf{1}_{e_H}.$$
Thus by choosing $a=-1/2$, we have that $b:G\to \ell^2(H)$ is a $\eta$-harmonic cocycle with weakly mixing representation $\pi$. It follows that $G$ doesn't have property $H_{\rm{FD}}$.
\end{proof}
Note that by Varopoulos \cite{var}, the only finitely generated groups that carry recurrent simple random walks are the finite extensions of $\{e\}, \mathbb{Z}$ or $\mathbb{Z}^2$. In the Appendix A we show that the wreath product $H\wr G$ of two finitely generated infinite groups where $H$ is amenable doesn't have property $H_{\mathrm{FD}}$. Combined with the previous proposition and the fact that $F\wr \mathbb{Z}$ has property $H_{\mathrm{FD}}$ for $F$ finite by \cite{Shalom}, we have that for wreath product $H\wr G$ of finitely generated groups, where $H$ is amenable and $G$ is infinite, the only case where it is not known whether $H\wr G$ has property $H_{\mathrm{FD}}$ is when $H$ is finite and $G$ is a finite extension of $\mathbb{Z}^2$.
\subsection{Locally-finite-by-$\mathbb{Z}$ groups without property $H_{\mathrm{FD}}$}\label{sec:noHFD}
In this subsection we give examples of locally-finite-by-$\mathbb{Z}$ groups which do not have property $H_{\mathrm{FD}}$. This answers negatively a question of Shalom \cite[Section 6.6]{Shalom}.
\begin{proposition}\label{noH_FD}
Let $G$ be a finitely generated locally-finite-by-$\mathbb{Z}$ group and $\mathcal{S}$ be a Schreier graph of $G$. Assume $\pi:G\rightarrow \mathcal{U}\left(\ell^2(\mathcal{S})\right)$ is a unitary representation fixing no non-zero vector. If there exists an associated non-zero harmonic cocycle $b:G \rightarrow \ell^2(\mathcal{S})$, then $G$ does not have Shalom's property $H_{\mathrm{FD}}$.
\end{proposition}
This proposition is immediate once we recall.
\begin{fact}[\cite{Shalom}]\label{423}
A finitely generated locally finite by $\mathbb{Z}$ group $G$ with property $H_{\mathrm{FD}}$ has property $H_{\mathrm{T}}$.
\end{fact}
\begin{proof}[Proof of Fact \ref{423}]
By \cite[Proposition 4.2.3]{Shalom}, the group $G$ has property $H_F$ if and only if no finite index subgroup $G_0$ admits a homomorphism to $S^1 \ltimes \mathbb{C}$ with dense image. However up to finite index the image must be torsion-free, hence cyclic so cannot have dense image. So $G$ has property $H_F$. Property $H_{\mathrm{T}}$ follows from \cite[Proposition 4.2.4]{Shalom} as the first Betti number and the first virtual Betti number of $G$ agree.
\end{proof}
\begin{proof}[Proof of Proposition \ref{noH_FD}]
Existence of a harmonic cocycle implies $\bar{H}^1(G,\pi) \neq 0$. If $G$ had property $H_{\mathrm{FD}}$, Fact \ref{423} would upgrade it to property $H_{\mathrm{T}}$, providing a non-zero fixed vector for $\pi$.
\end{proof}
\begin{proposition}\label{SymZ}
The group $\rm{Sym}(\mathbb{Z}) \rtimes \mathbb{Z}$ does not have Shalom's property $H_{\mathrm{FD}}$.
\end{proposition}
\begin{proof}
This groups acts by permutations on $\mathbb{Z}$. It is generated by the shift $x\mapsto x+1$ and the transposition permuting $0$ and $1$. The associated Schreier graph is easily pictured. let $\mu$ be the probability giving mass $\frac{1}{4}$ to the shift and its inverse and mass $\frac{1}{2}$ to the transposition. It is immediate to check that the function
\[
h(x)=\left\{\begin{array}{ll} x & \forall x \leq 0, \\x-\frac{2}{3} & \forall x \geq 1,
\end{array}\right.
\]
is $\mu$-harmonic. It follows that $b(g)(x)=h(x.g)-h(x)-T_g$, where $T_g$ is the translation part of $g$, defines a non-zero $\mu$-harmonic cocycle with respect to the right regular representation on $\ell^2(\mathcal{S})$. The latter has no non-zero fixed vector as the action on this Schreier graph is transitive. Proposition \ref{noH_FD} applies.
\end{proof}
Let $F$ be a finite group acting faithfully transitively on an alphabet $\{0\dots q-1\}$ with $q$ letters. Given a $q+1$-regular tree $\mathbb{T}$, an end of this tree, a bi-infinite ray towards this end and a vertex on this ray, the discrete affine group $\rm{DA}_F(\mathbb{T})$ is the group of automorphisms of the tree generated by an automorphism that shifts the ray together with the $F$-permutations of the $q$ subtrees (not containing the end) obtained by removing the chosen vertex. This group was introduced by Ryokichi Tanaka and the authors in \cite{BTZ}, to which we refer for details. By \cite[Proposition 3.1(3)]{BTZ}, the discrete affine group is locally finite by $\mathbb{Z}$.
\begin{proposition}\label{DA}
The discrete affine group $\rm{DA}_F(\mathbb{T})$ of a $q+1$-regular tree does not have Shalom's property $H_{\mathrm{FD}}$.
\end{proposition}
\begin{proof}
The discrete affine group is generated by a shift of infinite order together with a copy of $F$. The Schreier graph of its action on $\mathbb{T}$ can be pictured as follows. We first describe a core that will support the gradient of our harmonic function. The vertex set consists in all finite words in the alphabet $\{0 \dots q-1\}$. For any such word $v$ (including the empty word) there is a vertical edge between $v$ and $0v$ and a horizontal edge between $iv$ and $jv$ for any $i,j$ in the alphabet. Moreover above each vertex in this core not starting by $0$ we attach a vertical infinite one-sided ray, see Figure \ref{fig1} for the case of a ternary tree. The action of $F$ is by permutation of $\{0v\dots (q-1)v\}$ in the obvious way and $t$ translate along vertical lines.
Let $\mu$ give mass $\frac{1}{4}$ to the shift and its inverse and subprobability $\frac{1}{2}\mathbf{u}_F$ to the copy of $F$. Then vertical edges have mass $\frac{1}{4}$ and horizontal edges have mass $\frac{1}{2q}$ each.
Given $a>0$ and an initial value $h(\emptyset)$, we define a function $h:\mathcal{S}\rightarrow \mathbb{R}$ by the following gradient rules on the core.
\[
\forall i,j \in\{1,\dots,q-1\}, \forall v, \left\{
\begin{array}{l}
h(iv)-h(jv)=0, \\
h(iv)-h(0v)=\frac{a}{2q^{|v|}}, \\
h(0v)-h(v)=\frac{a}{q^{|v|}}.
\end{array}
\right.
\]
where $|v|$ is the length of $v$. We extend it as constant on each infinite one-sided ray. By construction, $h$ is $\mu$-harmonic and by elementary computation, it has finite energy:
\[
\frac{1}{2}\sum_{x\in \mathcal{S}, g \in \rm{DA}} \left(h(x.g)-h(x)\right)^2\mu(g)=\sum_{n=0}^\infty \left(\frac{a}{q^n} \right)^2\frac{q^n}{4}+\left(\frac{a}{2q^{n+1}}\right)^2\frac{q^n}{2q}\frac{q(q-1)}{2}<\infty.
\]
Now set $\forall x \in \mathcal{S}$, $b(g)(x)=h(x.g)-h(x)$. Then $b$ is a non-zero $\mu$-harmonic cocycle with respect to the right regular representation of $\rm{DA}_F(\mathbb{T})$ acting on $\ell^2(\mathcal{S})$. Proposition \ref{noH_FD} applies.
\end{proof}
\begin{figure}
\begin{center}
\begin{tikzpicture}
\draw[dotted] (-0.5,-0.75)--(7.5,-0.75);
\draw[thick] (0,0)--(1,0);
\draw (0,0) node[below]{$000$};
\draw (1,0) node[below]{$100$};
\draw[thick,dashed,blue] (1,0)--(1.25,0.5);
\draw[thick] (2,0)--(3,0);
\draw (2,0) node[below]{$010$};
\draw (3,0) node[below]{$110$};
\draw[thick,dashed,blue] (3,0)--(3.25,0.5);
\draw[thick] (4,0)--(5,0);
\draw (4,0) node[below]{$001$};
\draw (5,0) node[below]{$101$};
\draw[thick,dashed,blue] (5,0)--(5.25,0.5);
\draw[thick] (6,0)--(7,0);
\draw (6,0) node[below]{$011$};
\draw (7,0) node[below]{$111$};
\draw[thick,dashed,blue] (7,0)--(7.25,0.5);
\draw[thick] (0,0)--(0.5,1);
\draw (0.25,0.5) node[left,red]{$\frac{a}{q^2}$};
\draw (0.5,1) node[left]{$00$};
\draw[thick] (2,0)--(2.5,1);
\draw (2.25,0.5) node[left,red]{$\frac{a}{q^2}$};
\draw (2.5,1) node[right]{$10$};
\draw[thick] (0.5,1)--(2.5,1);
\draw (1.5,1) node[above,red]{$\frac{a}{2q^2}$};
\draw[thick,dashed,blue] (2.5,1)--(2.875,1.75);
\draw[thick] (4,0)--(4.5,1);
\draw (4.25,0.5) node[left,red]{$\frac{a}{q^2}$};
\draw (4.5,1) node[left]{$01$};
\draw[thick] (6,0)--(6.5,1);
\draw (6.25,0.5) node[left,red]{$\frac{a}{q^2}$};
\draw (6.5,1) node[right]{$11$};
\draw[thick] (4.5,1)--(6.5,1);
\draw (5.5,1) node[above,red]{$\frac{a}{2q^2}$};
\draw[thick,dashed,blue] (6.5,1)--(6.875,1.75);
\draw[thick] (0.5,1)--(1.5,2.5);
\draw (1,1.75) node[left,red]{$\frac{a}{q}$};
\draw (1.5,2.5) node[left]{$0$};
\draw[thick] (5.5,2.5)--(4.5,1);
\draw (5,1.75) node[left,red]{$\frac{a}{q}$};
\draw (5.5,2.5) node[right]{$1$};
\draw[thick] (1.5,2.5)--(5.5,2.5);
\draw (3.5,2.5) node[above,red]{$\frac{a}{2q}$};
\draw[thick,dashed,blue] (5.5,2.5)--(6,3.25);
\draw[thick] (1.5,2.5)--(3.5,5);
\draw (3.5,5) node[right]{$\emptyset$};
\draw (2.5,3.75) node[left,red]{$a$};
\draw[thick,dashed,blue] (4,5.625)--(3.5,5);
\end{tikzpicture}
\end{center}
\caption{The Schreier graph $\mathcal{S}$ of $\rm{DA}(\mathbb{T})$ acting on the ternary tree. The core (in this case a Fibonacci tree) is drawn in black and infinite one-sided rays are sketched in blue dashed. Edges are labelled in red with the gradient of the harmonic function $h$.}
\label{fig1}
\end{figure}
\begin{remark}
In contrast with Proposition \ref{DA}, we observe that the whole affine group of a regular tree (i.e. the group of automorphisms fixing an end--see \cite{CKW} or \cite{BTZ}) has Shalom's Property $H_{\mathrm{FD}}$ as a topological group. This is proved along the same lines as \cite[Corollary 5.2.7]{Shalom}.
\end{remark}
\begin{remark}
These examples also answer the 4th item in \cite[Question 6.6]{Gournay} negatively. The question asked if $\pi$ is weakly mixing and there is an infinite $N\triangleleft G$ with $G/N$ cyclic and $\overline{H}^1(N,\pi_{|N})=0$, does $\overline{H}^1(G,\pi)=0$? Note that when $N$ is locally finite, it is a direct limit of finite groups, therefore $\overline{H}^1(N,\sigma)=0$ for any unitary representation $\sigma$ of $N$, see for example \cite[Lemma 5]{FV}. Therefore examples of locally-finite-by-$\mathbb{Z}$ groups without property $H_{\mathrm{FD}}$ answer this question negatively.
\end{remark}
We now apply the result on the discrete affine groups to a family of
locally-nilpotent-by-$\mathbb{Z}$ groups introduced by Gromov \cite[Section8.2]{Gromov}.
These groups give examples of elementary amenable groups with arbitrary fast growing F{\o}lner functions.
We first recall the construction in the special case of $\mathbb{Z}$
acting on itself.
Fix $q$ an integer or infinity. Let $\mathbf{F}_{\mathbb{Z}}(q)$ be the free product
of $\mathbb{Z}$ copies of $\mathbb{Z}/q\mathbb{Z}$,
$\mathbf{F}_{\mathbb{Z}}(q)=\ast_{i\in\mathbb{Z}}\left\langle a_{i}\right\rangle $,
where $\left\langle a_{z}\right\rangle =\mathbb{Z}/q\mathbb{Z}$.
Given a subset $Y\subset\mathbb{Z}$, let $[Y]_{k}$ be the set of
commutators of $k$ letters $\left[\left[a_{i_{1}},a_{i_{2}}\right]\ldots a_{i_{k}}\right]$
where $i_{j}\in Y$ for $1\le j\le k$. Let $\mathbf{k}:2^{\mathbb{Z}}\to\mathbb{N}$
be a function on finite subsets of $X$ satisfying:
\begin{enumerate}
\item $\mathbf{k}(Y)=\mathbf{k}(Y+z)$ for any $Y\subset\mathbb{Z}$ and $z\in\mathbb{Z}$,
that is $\mathbf{k}$ is invariant under translation of $\mathbb{Z}$.
\item $\mathbf{k}(Y_{1})\le\mathbf{k}(Y_{2})$ if $Y_{1}\subseteq Y_{2}$.
\end{enumerate}
Let $N(\mathbf{k})$ be the quotient of $\mathbf{F}_{\mathbb{Z}}(q)$
with the commutator relations $\left\{ [Y]_{\mathbf{k}(Y)}:\ Y\in2^{X}\right\} .$
Since the function $\mathbf{k}$ is $\mathbb{Z}$-invariant, we can
take the semi-direct product $N(\mathbf{k})\rtimes\mathbb{Z}$ where
$\mathbb{Z}$ acts by translating indices. When $\mathbf{k}(Y)$ is
finite for all finite subset $Y\subset\mathbb{Z}$, the group $N(\mathbf{k})$
is locally nilpotent and locally finite when $q$ is also finite.
As in \cite{Gromov}, we consider the following kind
of function $\mathbf{k}$:
\[
\mathbf{k}(Y)=\kappa(\mbox{Diam}(Y)),
\]
where $\mbox{Diam}(Y)=\max\{|i-j|:\ i,j\in Y\}$ and $\kappa:\mathbb{N}\cup\{0\}\to\mathbb{N}$
is a non-decreasing function with $\kappa(0)=2$. Denote by $G(q,\kappa)$
the group $N(\mathbf{k})\rtimes\mathbb{Z}$ described above.
\begin{proposition}\label{prop_Gromov}
Let $q$ be a prime. For the group $G=G(q,\kappa)$ described above, we have
\begin{description}
\item [{(i)}] If $\kappa$ is bounded, then $G$ has Property
$H_{\mathrm{T}}$.
\item [{(ii)}] If $\kappa(n)>q^{n+1}$ for all $n$, then $G$ doesn't
have Property $H_{\mathrm{FD}}$.
\end{description}
\end{proposition}
\begin{proof}
We apply Proposition \ref{main_prop} inductively to prove
(i). Let $m=\sup_{n}\kappa(n)$, by assumption of (i) it is finite.
When $m=2$, the group $G(q,\kappa)$ is the lamplighter $(\mathbb{Z}/q\mathbb{Z})\wr\mathbb{Z}$,
it has property $H_{\mathrm{T}}$ by \cite{Shalom}.
Suppose the claim is true for $m$ and $\sup_{n}\kappa(n)=m+1$. Let
$b$ be a $\mu$-harmonic $1$-cocycle on $G$ with weakly
mixing representation. The center $Z(N(\mathbf{k}))$ of $N(\mathbf{k})$
is a locally normally finite subgroup of $N(\mathbf{k})$, therefore
by Proposition \ref{main_prop}, $b=0$ on $Z(N(\mathbf{k}))$.
In other words $b$ factors through to quotient group $G(q,\kappa')=(N(\mathbf{k})/Z(N(\mathbf{k})))\rtimes\mathbb{Z}$,
where $\kappa'=\max\{\kappa,m\}$. Then by the induction hypothesis,
$b=0$ on $G$.
For (ii), we show that when $\kappa(n)>q^{n+1}$ for all $n$, the
discrete affine group $\textrm{DA}_{\mathbb{Z}/q\mathbb{Z}}(\mathbb{T}_{q})$ is a quotient of $G(q,\kappa)$.
Since $q$ is a prime, the subgroup $\rm{Hor}(\mathbb{T})$ is a
locally finite $q$-group. The subgroup generated by $\left\{ t^{-i}a_{0}t^{i}:\ 0\le i\le\ell\right\} $
is a finite $q$ group of cardinality $q^{q^{\ell+1}}$. Note that
its nilpotency class is bounded by $q^{\ell+1}$. It follows
that $\textrm{DA}_{\mathbb{Z}/q\mathbb{Z}}(\mathbb{T}_{q})$ is a quotient of $G(q,\kappa)$ if $\kappa(n)\ge q^{n+1}+1$
for all $n$. The statement of (ii) follows since by Proposition \ref{DA},
$\textrm{DA}_{\mathbb{Z}/q\mathbb{Z}}(\mathbb{T}_{q})$ does not have property $H_{\mathrm{FD}}$.
\end{proof}
\section{An embedding result}\label{NNemb}
The goal of this section is to show Proposition \ref{NN}.
Let $C$ be the countable universal locally finite group of P. Hall \cite{Hall}.
Recall that $C$ can be constructed as the direct union of $\left(L_{n}\right)$,
where $L_{1}=\mathbb{Z}/3\mathbb{Z}$, $L_{n+1}$ is the full symmetric
group on $\left|L_{n}\right|$ elements, and $L_{n}$ is embedded
in $L_{n+1}$ via the regular representation. Moreover $C$ is a simple group, and there exists a set of generators
$\left\{ c_{1},c_{2},\ldots\right\} $ of $C$ such that $c_{s}^{2}=1$
for all $s$.
Since any countable locally finite group $H$ embeds into $C$, Proposition \ref{NN} follows from:
\begin{proposition}\label{NNC}
There exists a
two generated locally-finite-by-$\mathbb{Z}$ group $G$ with Shalom's
property $H_{\mathrm{FD}}$ and containing $C$ as an embedded subgroup.
\end{proposition}
Before proceeding to the proof, first observe that up to enlarging the generating sequence $(c_s)$ of involutions, we can assume that for infinitely many $s$, the subgroup $F_s=\langle c_1,\dots,c_s\rangle$ is simple.
Indeed given finitely many involutions in $C$, the group they generate is finite so there exists an abstract finite simple group generated by involutions that contains it. By universality, a copy $A_s$ of this abstract group lies in $C$. By \cite[Theorem 1(ii)]{Hall}, there is some $g$ in $C$ such that $F_s^g$ is contained in $A_s$. Then $A_s^{g^{-1}}$ contains $F_s$ and is generated by old and new involutions.
Recall that given two groups $H$ and $\Gamma$, their restricted wreath product is the group $H \wr \Gamma=\left( \oplus_{\Gamma} H \right) \rtimes \Gamma$, where $\Gamma$ acts on the direct sum by permutation of the factors and their unrestricted wreath product is the group $H \wr\wr \Gamma=H^{\Gamma} \rtimes \Gamma$ with the same action. Their elements are pairs $(f,g)$ where $f$ is a function $\Gamma\rightarrow H$, finitely supported for the restricted $H \wr \Gamma$, and $g$ an element of the base group $\Gamma$.
Let $(k_s)_{s \geq 1}$ be an increasing sequence of integers. Define a function $f : \mathbb{Z} \to C$ by
\[
f(x):=\left\{\begin{array}{ll} c_s & \textrm{ if } x=k_s, \\
e & \textrm{ otherwise.}
\end{array} \right.
\]
\begin{fact}\label{copy_C}
Assume $k_{s+1}>2k_s$ for all $s$. Then the group $\langle f,t\rangle<C \wr\wr \mathbb{Z}$ contains an embedded copy of $C$ and is locally-finite-by-$\mathbb{Z}$.
\end{fact}
\begin{proof}
By simplicity, any $c$ in $C$ is a finite product of commutators $c=\prod [c_{s_j},c_{r_j}]$. Then $\prod[f^{k_{s_j}},f^{k_{r_j}}]$ is a function taking value $c$ in $0$ and trivial elsewhere by strict doubling of $(k_s)$. This gives a copy of $C$.
Let the element $(\varphi,0)$ belong to the kernel of the map $\langle f,t \rangle \to \mathbb{Z}$ be of length less than $R$ with respect to the ambiant word metric. Then for all $x \in [-2R,2R]$, $\varphi(x)$ is a word of length at most $R$ in the finite group $C_R=\langle c_1,\dots, c_{\log_2 R}\rangle$. And whenever $k_s-k_{s-1} \ge R$ any $x \in [k_s-R,k_s+R]$ satisfies that either $\varphi(k_s+x)=c_s$ for all $s$ or $\varphi(k_s+x)=e_C$ for all $s$. Moreover the function $\varphi$ is trivial outside of these intervalles. It follows that the group generated by these elements embeds into the finite group $C_r^{[-2R,2R]}\times \mathbb{Z}/2\mathbb{Z}^{[-R,R]}$.
\end{proof}
To prove Proposition \ref{NNC}, we will show that $\langle f,t\rangle$ has property $H_{\mathrm{FD}}$ when the sequence $(k_s)$ grows sufficiently fast. The key point is the following:
\begin{lemma}\label{random_wreath}
For any $\delta>0$, there exists a constant $c>0$ such that the following holds. Let $F$ be a finite group and $\mu$ a probability measure on $F \wr \mathbb{Z}$ with finite generating support such that $\mu(t)=\mu(t^{-1})=\frac{1}{4}$ and $\mu(\oplus_{\mathbb{Z}} F)=\frac{1}{2}$. Then there exists an integer $M$ depending on $F$ and $\mu$ such that
\[
\frac{1}{2}\Vert \mu^{(n)}-\mu^{(n+\delta n)} \Vert_1 \leq 1-c, \textrm{ for all } n \geq M.
\]
\end{lemma}
\begin{proof}[Proof of Lemma \ref{random_wreath}]
From the distribution of local time \cite[Theorem 9.4]{Revesz2013}, we obtain for any $\delta>0$ there exists constants $c_1,c_2,c_3>0$ such that
\begin{eqnarray}\label{Trotter}
\mathbb{P}(\forall x \in [-c_1\sqrt{n},c_1 \sqrt{n}], L(x,n)\geq c_2 n^{1/2-\delta})\geq c_3>0.
\end{eqnarray}
There also exists an integer $R$ such that $\mu^{(R)}$ contains the copy of $F$ at $0$ in its support.
Following the proof of Proposition \ref{main_prop}, we set $\varepsilon=|F|\inf_{z \in F} \mu^{(R)}(z)>0$.
Under the condition in (\ref{Trotter}), for $|x|\le c_1\sqrt{n}$, the law of the lamp at $x$ at time $n$ of the random walk has the form
\[
(1-(1-\varepsilon)^{c_2\frac{n^{1/2-\delta}}{R}})\mathbf{u}_{F}+(1-\varepsilon)^{c_2\frac{n^{1/2-\delta}}{R}}\nu,
\]
for an auxiliary probability $\nu$. Thus for $n$ large enough, all the lamps in this interval are $F$-uniformly randomized with probability at least $\frac{c_3}{2}$. We can even assume that the cursor $\phi(Z_n)$ moreover belongs to $[-\frac{c_1}{2}\sqrt{n},\frac{c_1}{2}\sqrt{n}]$ with some uniform positive probability $c'_3>0$.
This gives a subprobability $\zeta_n \le \mu^{(n)}$ of mass $c_3'>0$ under which all lamps in $[-c_1\sqrt{n},c_1 \sqrt{n}]$ are $F$-uniformly randomized and the cursor $\phi(Z_n)$ lies in $[-\frac{c_1}{2}\sqrt{n},\frac{c_1}{2}\sqrt{n}]$.
For $\delta>0$, there is a probability $c_4>0$ that the cursor remains in $[-c_1\sqrt{n}+R',c_1 \sqrt{n}-R']$ up to time $n+\delta n$ and finishes again in $[-\frac{c_1}{2}\sqrt{n},\frac{c_1}{2}\sqrt{n}]$. Therefore, $\frac{c_4}{c'_3}\zeta_n$ is a subprobability of uniform mass $c=c_4>0$ of both $\mu^{(n)}$ and $\mu^{(n+\delta n)}$.
Here $R'$ is such that $f(x)=e$ for all $|x|>R'$ and $(f,0)$ in the support of~$\mu$.
\end{proof}
\begin{proof}[Proof of Proposition \ref{NNC}]
Let $s$ be such that $\langle c_1,\dots,c_s\rangle$ is simple.
Let $F=\langle c_1,\dots c_s\rangle \times \mathbb{Z}/2\mathbb{Z}$ and $f:\mathbb{Z}\to F$ be given by:
\[
f(x):=\left\{\begin{array}{ll} (c_s,0) & \textrm{ if } x=k_s, s\ge 2, \\
(c_1,1) & \textrm{ if } x=k_1, \\
(e,0) & \textrm{ otherwise.}
\end{array} \right.
\]
The argument of Fact \ref{copy_C} shows that $t$ and $f$ generate $F\wr \mathbb{Z}$, so Lemma \ref{random_wreath} applies.
We observe that the groups generated by $f$ and $t$ in $F\wr \mathbb{Z}$ and $C \wr \wr \mathbb{Z}$ coincide on any ball of radius $<k_{s+1}-k_s$. The factor $\mathbb{Z}/2\mathbb{Z}$ takes into account the generators $c_{s+1},c_{s+2}\dots$ not interplaying together in this ball. Therefore, if we let $(k_s)$ grow fast enough, we can use Lemma \ref{random_wreath} locally and obtain a sequence $n_s\to \infty$ with
\[
\frac{1}{2}\left\Vert \mu^{(n_s)}-\mu^{(n_s+\delta n_s)} \right\Vert_1 \leq 1-c, \textrm{ for all } s.
\]
Proposition \ref{NNC} follows from \cite[Corollary 2.5]{EO}.
\end{proof}
We remark that by the proof, the group $G$ can be chosen with speed arbitrarily close to diffusive, namely below $\sqrt{n}f(n)$ for any $f(n)\to \infty$.
\bigskip
\textbf{Acknowledgement.} We thank Anna Erschler for questions about Shalom's property $H_\mathrm{FD}$ which motivate this work, and for pointing out to us the examples of locally-nilpotent-by-$\mathbb{Z}$ groups in \cite{Gromov}. We thank Gidi Amir for many discussions on minimal growth of harmonic functions on groups. We thank Peter Kropholler for useful comments.
J.B.\ is partially supported by project AGIRA ANR-16-CE40-0022.
\appendix
\section{Wreath products of infinite groups do not have property $H_{\mathrm{FD}}$}\label{appendix}
In \cite[Section 6.6]{Shalom}, Shalom conjectured that the wreath product of two infinite finitely generated amenable groups never has property $H_{\mathrm{FD}}$. The object of this appendix is to prove this result. We will follow largely Shalom's original paper, where the conjecture was already proven when $H$ has infinite abelianization \cite[Section 5.4]{Shalom}.
As an abuse of notation, we will write $H^G$ instead of $\oplus_{G} H$ in this appendix.
\begin{proposition}
If $G,H$ are finitely generated infinite groups and $H$ is amenable, then $H \wr G$ does not have Shalom's property $H_{\mathrm{FD}}$.
\end{proposition}
\begin{proof}
Let $\mu$ and $\nu$ be non-degenerate finitely supported symmetric probability measures on $H$ and $G$ respectively. They induce a measure $\theta =\frac{1}{2}(\mu+\nu)$ on $H \wr G$ where we identify $H$ with its copy over the neutral element of $G$ in $H^G \subset H \wr G$.
When $H$ admits an infinite abelianization, the proposition was proved by Shalom in \cite[Theorem 5.4.1]{Shalom}. We give the proof for completeness. Denote $\phi: H \rightarrow \mathbb{Z}$ a surjective homomorphism and let $\pi:G \rightarrow \mathcal{U}(\ell^2(G))$ be the right regular representation of $G$. It is weakly-mixing as $G$ is infinite. Then $b(f,g)(x)=\phi \circ f(x)$ defines a non-zero $\theta$-harmonic cocycle with respect to the (weakly-mixing) representation of $H\wr G$ factoring through $\pi$.
We are left with the case when $H$ has finite abelianization. We consider a unitary representation $\pi_0:H \rightarrow \mathcal{U}(\mathcal{H})$ with an associated $\mu$-harmonic cocycle $b_0$, which exist by amenability of $H$, see \cite[Appendix]{Kleiner}. Define $b : H\wr G \rightarrow \ell^2(G,\mathcal{H})$ by $b(f,g)(x)=b_0(f(x))$. This is a non-zero $\theta$-harmonic cocycle for the unitary representation $\pi(f,g) \varphi(x)=\pi_0(f(x))\varphi(x.g)$. In particular, $\overline{H^1}(H\wr G,\pi) \neq 0$.
Assume by contradiction that $H\wr G$ has property $H_{\mathrm{FD}}$. Then we obtain a direct sum $\ell^2(G,\mathcal{H})=V'\oplus \bigoplus_{n \in \mathbb{N}} V_n$ corresponding to a decomposition of $\pi$ into $\pi'$ weakly-mixing and countably many finite-dimensional and irreducible representations $\pi_n$, such that the cocycle $b_n$ is not reduced in $\overline{H^1}(\pi_n,V_n)$ for some $n$.
\begin{claim}\label{claim}
The restricted cocycle $b_n|_{H^G}$ is not reduced in $\overline{H^1}(\pi_n|_{H^G},V_n)$.
\end{claim}
\begin{proof}[Proof of Claim \ref{claim}]
By the definition of $b$, the cocycle $b_n|_{H^G}$ is not zero. So there exists a large enough $k$ such that $\textrm{supp} (\theta^{(k)}) \cap \textrm{supp} (b_n) \cap H^G$ is non-empty. Using $\theta^{(k)}$-harmonicity of $b_n$ and the invariance under $G$ of $b$, we get $0=\sum_{x \in H^G} b_n(x)\theta^{(k)}(x)$.
Now if $b_n|_{H^G}$ were reduced, by finite dimension it would be a coboundary, giving a vector $v$ in $V_n$ such that $\forall x \in H^G, b_n(x)=v-\pi_n(x)v$. Then
\begin{align*}
0=2\left\langle \sum_{x \in H^G} b_n(x)\theta^{(k)}(x),v\right\rangle=\sum_{x \in H^G}\left\langle b_n(x),v-\pi_n(x)v\right\rangle \theta^{(k)}(x)\\
= \sum_{x \in H^G}|| b_n(x)||^2_{V_n} \theta^{(k)}(x)>0,
\end{align*}
where the middle equality follows classicaly from the cocycle identity.
\end{proof}
By \cite[Proposition 3.2]{Shalom_rigidity}, the claim ensures the existence of $g_0$ in $G$ and a non-zero vector $v$ in $V_n$ which is invariant under $\pi_n|_{H^{G\setminus\{g_0\}}}$. By irreducibility and finite dimension of $\pi_n$, there exist $\gamma_1,\dots,\gamma_r$ in $H \wr G$ such that the vectors $\pi_n(\gamma_i)v$ span $V_n$.
It follows that $\pi_n$ restricted to the intersection $\cap_{i=1}^r H^{G\setminus\{g_0\}}\gamma_i^{-1}$ fixes $V_n$ pointwise. Moreover as $G$ is infinite, this intersection contains a copy $H^{g_1}$ for some $g_1$ in $G$. By conjugacy, we deduce that $\pi_n|_{H^G}$ is trivial and finally that $b_n|_{H^G}:H^G \rightarrow V_n$ is a non-zero homomorphism. This contradicts our assumption that $H$ has finite abelianization.
\end{proof}
We observe that Claim \ref{claim} is the only new ingredient not appearing in \cite[Section 5.4]{Shalom}.
\bibliographystyle{alpha}
\bibliography{H_FD}
\textsc{\newline J\'er\'emie Brieussel --- Universit\'e de Montpellier
} --- [email protected]
\textsc{\newline Tianyi Zheng --- UC San Diego
} --- [email protected]
\end{document} | 107,753 |
As state lawmakers negotiate a 2013-15 state budget in the final days of the session, they need to pay more attention to the growing problem of hunger in our state.
The Great Recession has swelled the ranks of those who go to bed at night hungry — one of seven people statewide and, more tragically, one in four children.
Prior to the recession, the state ranked 30th in food insecurity, according to U.S. Department of Agriculture measures. Now the state ranks 14th in food insecurity. Looked at another way, only six other states experienced more growth in hunger rates between 2010-2011.
As a state, we’re headed in the wrong direction.
Since 2008, food banks across the state have seen a 35 percent increase in demand for their services. At the same time, state lawmakers have sliced away at programs designed to feed the hungry, hoping against reason that charitable groups and private donations will adequately fill food baskets.
The stagnant economy, and continued state and federal cuts in other social service programs, are driving more and more people to the food banks. As a result, the overall amount of food available for each client is on the decline.
“The House and Senate budgets assume that charity can pick up the slack, but that’s not possible,” said Christina Wong, public policy manager for Northwest Harvest, which is part of the Washington Food Coalition, which advocates in the fight against hunger.
It’s as if all the political energy spent on K-12 education, failed gun control legislation and legalized marijuana and other high-profile issues has blinded lawmakers to the fact that such a basic need as food is not being met in thousands of homes across the state.
It’s late in the session, but not too late to bolster funding for three core state programs aimed at taking a bite out of hunger. The Food Coalition knows time is running out. However, the coalition makes a convincing argument to restore historic funding to critical programs and increase funding to reflect growing demand for services.
For instance, the Emergency Food Assistance Program helps food banks purchase food and cover operating costs. The program is funded at $10.6 million in the House and Senate budgets. It needs another $3.7 million to handle the 35 percent increase in food bank clients. That’s a reasonable request.
Then there’s the state Food Assistance Program, which is a food stamp look-alike program created in 1997 for immigrants legally residing in the state. The program took a 50 percent cut in July 2012. The Senate budget restores $9.4 million, or about half of the 2012 cuts. The House budget does nothing for the program.
The cuts a year ago put 14,000 children in immigrant families at risk of hunger and deepens racial and economic inequality.
There’s a certain irony in not fully funding the program. Hungry kids don’t do well in school. What sense does it make to beef up funding for K-12, then send more kids to school unable to learn because their basic need for food isn’t being met?
The Senate and House budgets do resurrect two programs eliminated in 2011 that help small farmers sell their fresh produce to local schools. But they’re only funded at 50 percent of their original budget.
In a state with a rich and diverse agricultural economy, it seems feeding hungry children wouldn’t be that hard. Collectively, we must all work together to end hunger. It’s not too late for the 2013 state Legislature to lend a greater hand. | 325,606 |
Big Corral ( Sheridan )
Wyoming : Flag of State
Big Corral is a place of kind Locale belonging to the County of ( Sheridan ).
The closest populated place is that of Leiter that is4.10 miles far from Big Corral.
Big Corral is also 59.34 miles far from the closest airport or heliport, the Symons Airport .
The weather forecasting are available for the town of Leiter , that is 4.10 miles far from Big Corral .
The Zip Code of Big Corral is 82837.
Besides Big Corral in the County of ( Sheridan ) there are the following places of kind Locale . These are reported in what follows in alphabetical order:
Arrowhead Lodge - Bald Mountain City - Bear Claw Ranch - Bear Lodge - Big Corral
Please find in what follows the full listing of the Hotels, Bed & Breakfasts and other accommodation available in Big Cor Big Corral
View the interactive maps of Big Corral and get road maps, satellite maps and much more. The maps cover a wide area centered on Big Cor Cor. | 237,664 |
Motor.
.
1 Comment on this Post | 252,932 |
Swimming Pool Heaven
Our company offers Swimming Pool Heaven, which is made from high quality steel to resist corrosion. With a high level of dimensional accuracy, it ensures durability and efficiency. It has a high tensile strength and sturdy structure that makes it highly suitable for all types of swimming pools. | 52,584 |
TITLE: Being characteristic is transitive
QUESTION [6 upvotes]: Here, I wanted to verify that:
The property of being characteristic is a transitive relation among subgroups of a group $G$.
For subgroups $N,H\leq G$, we have $N$ char $G$ and $H$ char $N$. So $N$ char $G$ implies: $$\forall\psi\in Aut(G); \psi(N)=N$$ $H$ char $N$, so for all elements in $Aut(N)$, $H$ is remained never changing. Especially, when I take $\psi'=\psi|_{N}$ then $\psi':N\to Aut(N)$ would be in $Aut(N)$ and $\psi'(H)=H$. Since the last equality is true for $H$ and the maps, caused by restriction on $N$, then I have $H$ char $G$.
Honestly, I am inly not satisfied form the conclusion here and think I am losing something. Thanks for your hints.
REPLY [1 votes]: I understand your frustration, I also read several proofs but hard to understand them because somehow and I don't know why they all seem to omit the most important part of the explanation.
So we are going to prove that if $H \: char \: N \: char \: G$ then $H \: char \: G$.
Let $\phi \in Aut(G)$, so $\phi(N) = N$ since $N$ is characteristic in $G$.
Now please note the most important part of the proof. We have $\phi|_N \in Aut(N)$ or $\phi|_N(N) = N$, so it's no difference than $\phi(N) = N$ as there is all there the elements of $Aut(N)$ in $Aut(G)$. In other words:
$\phi|_N(N) = \phi(N) = N$.
As $H \: char \: N$, we have the same relationship between $H$ and $N$ as following:
$\phi|_H(H) = \phi|_N(H) = H$.
It is all there elements of $Aut(H)$ in $Aut(N)$, but we have had that $\phi|_N(N) = \phi(N)$, or is all there elements of $Aut(N)$ in $Aut(G)$, including $Aut(H) < Aut(N)$, so it is also all there elements of $Aut(H)$ in $Aut(G)$. In other words we have $\phi|_N(H) = \phi(H)$. Combine them all, we have:
$\phi|_H(H) = \phi|_N(H) = \phi(H) = H$.
Which is by definition $H \: char \: G$. | 131,419 |
Pommies’ from that TV show who made their most recent appearance at the Taste of Sydney celebration in Sydney’s Centennial Park. Talking to them reveals two things: 1) food should be fun, and 2) seafood is hard!
When it comes to describing their approach to food and how it all slots together into your approach to the kitchen, Will and Steve keep it fairly simple.
“We just have fun with it all. Keeping it fresh makes it easy and if you cook what you love, you won’t go far wrong,” said Will. “Food for Will and I has always been about sharing, love and celebration. If we can put smiles on people’s faces, we know our approach is the right one,” Steve continued.
And you know how sometimes food and cooking and the thought of the whole darn mess can be made that much more daunting by virtue of the fact you can’t find ingredients? Well, if these two are any authority to listen to; that’s a non-issue.
“[If you can’t get the ingredients you need] panic, leave the trolley in the middle of the aisle, walk out the supermarket!” said Steve, which Will backed-up.
“This happens fairly often. He usually comes back when he has a replacement ingredient in mind. God forbid if they don’t have that either!”
The two still make it work for themselves, though. They appeared in a cooking demonstration at Taste of Sydney and wowed the crowd with their versatility and general mateship behind the chopping board, which begged the question, ‘how did they come-up with simple and nutritious recipes for festival-goers to replicate at home?’ Turns out you don’t need recipes, they say.
“I think they are great as a guide and the visuals can be very inspiring but I rarely follow them through to the end,” Steve said.
Will’s favourite dish is a duck pot pie that the two of them crafted together which you might need a recipe for, but otherwise, feel free to wing it.
And if that’s not your flavour of approach, then expect a confounded response from the likes of Will and Steve, who just don’t get it.
“[People who say they haven’t found the joy of cooking yet] I stare blankly at them in disbelief. I still like them more than the food bloggers who have never cooked a meal in their life yet feel the need to critique every restaurant, chef and McDonald’s outlet!” said Steve.
“I encourage people to look at my own journey and see how much my joy for cooking has changed my life. I couldn’t be happier,” continued Will.
But just because you might be an accomplished foodie at home and a TV cook-come-foodlebrity, doesn’t mean everything comes as a dream. Seafood is hard for the best of them and for Will and Steve, they’re not exempt.
“Oysters throw me off every time. I want to love them but I just can’t bring myself to,” said Steve. Whereas Will isn’t much of a sea urchin fan, he mentioned, reminiscing, “I had a very bad experience is Japan that I don’t ever want to revisit.”
As understandable as that is, the two of them have managed to turn their love of food and the joy they feel when cooking into a life for themselves that they couldn’t be happier with. Get cooking! | 33,285 |
TITLE: inner product of vector and its projection
QUESTION [1 upvotes]: Let us consider Hilbert space $(\mathbb{R}^n, \langle \cdot ,\cdot\rangle_w)$, where $w\in\mathbb{R}^n$ and $w_i > 0$ and inner product $\langle x,y\rangle = \sum_{i=1}^n x_i y_i w_i$. Let $C$ be a closed convex cone in $\mathbb{R}^{n}$ and let $\Pi(x\mid C)$ be the projection of $x$ onto the cone $C$.
Is it always true that $\langle \Pi(x\mid C),x\rangle \geq 0$?
REPLY [2 votes]: We have $\|x-\prod (x|C)\| \leq \|x\|$ since $0 \in C$. Squaring both sides and expanding we get $\|\prod (x\mid C)\|^2-2 \langle x, \prod (x\mid C) \rangle \leq 0$. Hence $\langle x, \prod (x\mid C) \rangle \geq 0$. | 193 |
Caroline escort glasgow FREE PLAY videos, stream online porn videos ON MOBILE Asian old man outdoor Dirty Old Man Fucks Small Tit Asian Escort Jenny escort female from Magic Escorts Scotland Agency K views. 64%. 11 months ago Teen escort doesn't like anal M views. 68%. 1 year ago Cute escort teaches how women should fuck with a negro!! K views. 84%. A sexpartner is someone who you make sex with, doesn't matter if it's only oral sex, normal sex or anal sex. There has to be some kind of penetration between.
Fucked anal escort glasgow -Hi, my name is Melissalet's have some fun. I'm Jewel, a lady of pleasure. Glasgow escort cumcaroline contact details FREE PLAY videos, stream online porn Asian old man outdoor Dirty Old Man Fucks Small Tit Asian Escort Glasgow Escorts - Escort guide provides the most exquisite and discreet massage escort girls in Glasgow. Our female escort girls are handpicked, making sure. 4 May Female escorts from agency Magic Escorts Scotland pornoswetty.infoscotland. net - Video from , recorded in Glasgow escorts hotel.
: Fucked anal escort glasgow
| 248,540 |
TITLE: Mean and extinction probability of a Galton-Watson branching with pmf of offspring produced $P(Q=q) = (q+1)(1-r)^2r^q, 0<r<1$
QUESTION [1 upvotes]: Initial population is $X_0 = g$, ($g$ being a positive number or $0$) and the probability mass function of the number of offsprings $(q)$ produced by an individual is $P(Q=q) = (q+1)(1-r)^2r^q, 0<r<1$.
I'm trying to calculate the expected value of $X_n$ and the extinction probability. I'm stuck on both but here's how far I got.
Mean:
$E[X_n] = E[f(q)]^q(g)$ (I'm using a known formula for this. let me know if I've used it wrong). Assuming $X_0 =g $ isn't $0$, we will have to calculate:
$$E[f(q)] = \Sigma^\infty_{q=1} qP(Q=q) = \Sigma^\infty_{q=1} q(q+1)(1+r)^2r^q$$
Is the upper limit of the sum here correct? Should it be $\infty$, or $g$ as we are starting with $g$ people in the population
Extinction probability $(\pi_0)$: Assuming that my $E[f(q)]>1 \implies \pi_0 = \Sigma^\infty_{q=1} \pi^q_0P(Q=q)$.
$\pi^q_0$ being the probability that the population dies out given $X_0 = q$. This gives me:
$$\Sigma^\infty_{q=1} \pi^q_0(q+1)(1+r)^2r^q$$
In both these cases I have no idea how to proceed further. This isn't a distribution that I recognize. Is there something I'm missing? Did I do a step wrong? Or is there an easier way to approach this that I am not seeing.
REPLY [0 votes]: It is straightforward to show that if the offspring distribution has finite mean, that is,
$$
\mathbb E[Q] = \sum_{k=0}^\infty k\cdot\mathbb P(Q=k) :=\mu <\infty
$$
then the expected population size at time $n$, conditioned on $\{X_0=1\}$ is given by
$$
\mathbb E[X_n\mid X_0=1] = \mu^n.
$$
If $g$ is a positive integer, then with some additional work we see that
$$
\mathbb E[X_n\mid X_0=g] = g\cdot\mu^n.
$$
(The intuition is that the process is equivalent to $g$ separate processes each starting with one individual.) We compute the mean of $Q$:
$$
\mu = \sum_{k=0}^\infty k\cdot(k+1)(1-r)^2 r^k = \frac{2r}{1-r},
$$
and hence
$$
\mathbb E[X_n\mid X_0=g] = g\cdot\left(\frac{2r}{1-r}\right)^n.
$$
For the extinction probability, I will only consider the case where $g=1$. Let
$$
\tau = \inf\{n>0:X_0=0\}.
$$
It is known that $\pi:=\mathbb P(\tau<\infty)=1$ if $\mu\leqslant1$ and is a positive number less than one if $\mu>1$. Since $0<r<1$, it is clear that
$$
0<\frac{2r}{1-r}\leqslant 1 \iff 0<r\leqslant\frac13,
$$
and so extinction occurs with probability one if $r\leqslant\frac 13$. If $\frac13<r<1$, then it is well known that $\pi$ satisfies the equation $P(\pi)=\pi$, where $P(\cdot)$ is the probability generating function of $Q$; indeed, $\pi$ is the unique solution to this equation on the interval $(0,1)$. Let $P(s):= \mathbb E[s^Q]$ for $s\in[0,1]$, then
$$
P(s) = \sum_{k=0}^\infty (k+1)(1-r)^2 r^ks^k = \left(\frac{1-r}{1-rs}\right)^2.
$$
The equation $P(\pi)=\pi$, i.e.
$$
\left(\frac{1-r}{1-r\pi}\right)^2 = \pi
$$
is a cubic, and so has three solutions:
\begin{align}
\pi &= \frac{2r-r^2-\sqrt{4 r^3-3 r^4}}{2 r^2}\tag1\\
\pi &= \frac{2r-r^2+\sqrt{4 r^3-3 r^4}}{2 r^2}\tag2\\
\pi &= 1\tag3.
\end{align}
By inspection, we see that $(1)$ is the correct choice, since it yields numbers between zero and one. | 35,831 |
West Bromwich Albion boss Slaven Bilic has provided further reflections on his side’s draw with Nottingham Forest.
The Reds were held to a 2-2 draw with the Baggies in their Sky Bet Championship clash last week..
Asked if the performance against Forest was their best of the campaign, Bilic told Birmingham Live: “Considering the opponent, considering that it was a ‘derby’ game with everything else that was at stake, definitely.
He added: “I watched that game a couple more times, analysing it, and we were proper good.
“We limited them on a few situations, they had one shot on target, they didn’t have too many shots at all, or situations that worried us.
“To cut a long story short, we were doing good at limiting their attack, we were solid. We were good on the ball, we were threatening them.
“We were taking the ball off them, and it never looked like - to me, they’re a good team of course - in that game that they were ever in charge - not when it was 0-0, or 1-0 for us, or 1-1, or 2-1. Not even when it was 2-2, in those few minutes.
“It was always us more likely to create and score. Unfortunately we didn’t, we had enough chances, and that’s it.” | 170,642 |
TITLE: Exercise 6.2.21 Introduction to Real Analysis by Jiri Lebl
QUESTION [1 upvotes]: Let $f_n(x) = \frac{x}{1+(nx)^2}$. Notice that $f_n$ are differentiable functions.
a) Show that $\{f_n\}$ converges uniformly to $0$
$\lim\sup|\frac{x}{1+(nx)^2}| =\lim |\frac1{2n}| = 0$
b) Show that $|f'_n(x)| \le 1$ for all $x$ and all $n$.
$|f'_n(x)| = |\frac1{1+(nx)^2}||1-\frac{2(nx)^2}{1+(nx)^2}| = |\frac1{1+(nx)^2}||\frac{1-(nx)^2}{1+(nx)^2}|< \frac{(nx)^2}{1+(nx)^2} \le 1$.
c) Show that $\{f'_n\}$ converges ponitwise to a function discontinuous at the origin.
Given $x$, if $n$ goes to infinity, doesn't $f'_n(x)$ converges to $0$?
d) Let $\{a_n\}$ be an enumeration of the rational numbers. Define
$$g_n(x) = \sum_{k=1}^n 2^{-k}f_n (x-a_k).$$
Show that $\{g_n\}$ converges uniformly to $0$.
Shouldn't $f_n$ be replaced by $f_k$? In that case, $\lim \sup |2^{-1}||f_k(x-a_k)|^{1/k} = 0$ since $f_k$ uniformly converge to $0$. So $g_n$ uniformly converges. But, how do we know that it converges to $0$?
e) Show that $\{g'_n\}$ converges pointwise to a function $\phi$ that is discontinuous at every rational number and continuous at every irrational number. In particular, $\lim_{n\to\infty} g'_n(x) \not= 0$ for every rational number $x$.
I have no idea.
I have some difficulty in solving the last three questions. I appreciate if you give some help.
REPLY [1 votes]: Evidently, $f_n(x)\to 0$ pointwise. Note that the $f_n$ are differentiable and defined on all of $\mathbb R$. Therefore any maximum must be attained at a point $x_n$ such that $f_n'(x_n)=0$. So we compute
$$
f_n'(x) = \frac{1-n^2 x^2}{\left(1+n^2 x^2\right)^2},
$$
and note that $$f_n'(x)=0\iff 1-n^2x^2=0\iff x = \frac1n.$$
It follows that $$\limsup_{n\to\infty} f_n(x) \leqslant \limsup_{n\to\infty}f_n\left(\frac1n\right) = \limsup_{n\to\infty}\frac1{n(1+1^2)} = \limsup_{n\to\infty} \frac1n = 0, $$
and hence $f_n$ converges uniformly to zero.
It is clear that $(1+n^2x^2)\geqslant1$ and $1-n^2x^2\leqslant 1$ and therefore
$$
f_n'(x) = \frac{1-n^2x^2}{(1+n^2x^2)^2}\leqslant 1.
$$
For $x\ne 0$ we have $$\lim_{n\to\infty} f_n'(x) = \lim_{n\to\infty} \frac{1-n^2x^2}{(1+n^2x^2)^2} = 0,$$
but for $x=0$, $f_n'(0) = 1$, so $\lim_{n\to\infty}f_n'(0)=1$. It follows that the limit of $f_n'$ is discontinuous at $x=0$.
For d) and e) I am not sure. | 20,745 |
Waiheke High students shone bright at a Māori performing arts showcase at the school, with rhythmic poi dances and powerful haka last Thursday.
Friends and family turned out for the colourful, high-energy event, watching on as students demonstrated their creativity and rhythm with dance, choreography, music and costume.
Māori performing arts teacher Te Ao Hau said she was proud of what her students had achieved in the six weeks of intense training leading up to the show.
“They worked hard and got the mahi done,” she said. “They should all be commended for the time and effort they all spent over the six weeks of face to face teaching and learning.”
Part of the NCEA level one and two assessments for students, the showcase also featured international students at Waiheke High who learn about Māori performing arts as part of their exchange programme. • | 238,289 |
Our Terms of Use and Privacy Policy have changed. We think you'll like them better this way.
Over the course of years long distance relationships have become quite common thanks to the internet and all its wonder. Many years ago people relied on hand written letters and a good song from Casey Kasem via his "Long Distance Dedication". Can they work? Are they really doable? Join Joseph and Teresa as they take a road trip down the highway of love and find out what it takes to make up for the miles between you and your long distance sweetheart.Sure to be a hilarious show! We ARE your Saturday night date!!!!!
Sorry we couldn't complete your registration. Please try again.
Please enter your email to finish creating your account.
Receive a personalized list of podcasts based on your preferences. | 188,965 |
Photo by Kalen Emsley on Unsplash
We’ve picked 10 most beautiful lakes in Europe – ideal summer destinations if looking for a place to relax and cool down. Here are just some of the advantages lakes have over the sea.
- No crammed beaches with hundreds of people.
- Restaurants around lakes tend to be cheaper than those close to the sea.
- If you are lucky, you can find a secluded beach and claim it your own.
- You won’t have to fight big waves and your eyes won’t be sore from sea salt.
- Just imagine the fresh breeze.
Garda
✈ Italy
The largest lake in Italy, Garda is popular mainly among windsurfers. Cyclists and inline skaters will appreciate all the great trails located here.
Add Garda to Favorites ❤
Geneva
✈ Switzerland/France
Photo by Samuel Zeller on Unsplash
With a surface of 580 km², lake Geneva is the largest Alpine lake. Come for the system of undemanding walking trails and vineyards that line the lake.
Add Geneva to Favorites ❤
Constance
✈ Germany/Switzerland/Austria
Photo by Simon Rohr on Pixabay
Constance is said to be Europe’s hidden gem. One of the three islands on the lake is home to three exquisite abbeys, listed as UNESCO World Heritage Sites.
Add Constance to Favorites ❤
Maggiore
✈ Italy/Switzerland
Photo by Jonathan Reichel on Pixabay
Surrounded with picturesque towns and dotted with charming islands, Maggiore is a little piece of heaven.
Add Maggiore to Favorites ❤
Bled
✈ Slovenia
Photo by 12019 on Pixabay
Thanks to its hot springs, Bled is one of the warmest lakes of the Alps. The water temperature can reach as high as 26°C, making it a great swimming spot from June to September.
Add Bled to Favorites ❤
Lucerne
✈ Switzerland
Photo by LUM3N on Pixabay
Lucerne is a marvellous lake with crystal clear water. Visit the village of Bauen and enjoy its Caribbean vibe.
Add Lucerne to Favorites ❤
Hallstatt
✈ Austria
Photo by veronica111886 on Pixabay
Popular among fishermen and divers, the lake of Hallstatt is situated near the oldest salt mines in Europe.
Add Hallstatt to Favorites ❤
Königssee
✈ Germany
Photo by Kordula Vahle on Pixabay
Located in the Berchtesgaden National Park, Königssee will enchant you with its emerald green water.
Add Königssee to Favorites ❤
Saimaa
✈ Finland
Photo by Kosti Keistinen on Pixabay
Be amazed with the tranquility and unspoiled nature which surrounds Saimaa. You may even spot a ringed seal here.
Add Saimaa to Favorites ❤
Annecy
✈ France
Photo by Eric Michelat on Pixabay
Regarded as the clearest lake in Europe, Annecy is home to a picturesque town of the same name – sometimes also known as Venice of the Alps due to its numerous water canals.
Add Annecy to Favorites ❤
Did something catch your eye? Don’t forget to add the places to the list of your Favorite items in the Sygic Travel web or mobile app, either on iOS or Android, or on the desktop Sygic Maps. That way, you can easily access them later. | 67,911 |
Airway Skills Workshop
- Date: July 18, 2017
We're sorry, but all tickets sales have ended because the event is expired.
Presented and hosted by College Fellow Dr Sanj Fernando
Come to the brand new Simulation Centre at Campbelltown Hospital for this excellent one day airway management education course. A great chance to revise, improve your skills, and pick up tips and tricks from the experts.
The Faculty includes Senior Emergency Physician’s, Senior Ambulance Medical team Paramedics, and Senior Nursing staff with over 100 years’ experience between them.
Suitable for ICP Paramedics with multiple immersive simulations to test and improve your skills. We will accept highly experienced ALS paramedics.
Numbers are limited to 6, so enrol now.
This workshop is open to ANZCP members and Non-members – please log in to secure your ANZCP member pricing.
Location
Venue: Campbelltown Hospital
Location:Location:
Therry Road Campbelltown 2560
Description:
| 196,852 |
\section{Preliminaries}
\label{sec:preliminaries}
In this paper, we consider the unconstrained minimization problem \footnote{Many of the results in this paper can be generalized to constrained minimization problems \cite{Nesterov2012}, problems where $\fun$ is strongly convex with respect to different norms \cite{Nesterov2012}, and problems where each coordinate is a higher dimension variable (i.e. the block setting) \cite{Richtarik2011}. However, for simplicity we focus on the basic coordinate unconstrained problem in the Euclidian norm.
}
\[
\min_{\x \in \Rn} \fun(\x)
\]
where the \emph{objective function} $\fun : \Rn \rightarrow \R$ is continuously differentiable and convex, meaning that
\[
\forall \x, \y \in \Rn
\enspace : \enspace
\f(\y) \geq \f(\x) + \innerprod{\gradient \f(x)}{\y - \x}
\enspace.
\]
We let $\f^{*} \defeq \min_{\x \in \Rn} \fun(\x)$ denote the minimum value of this optimization problem and we let $\x^{*} \defeq \argmin_{\x \in \Rn} \fun(\x)$ denote an arbitrary point that achieves this value.
To minimize $\fun$, we restrict our attention to \emph{first-order} iterative methods, that is algorithms that generate a sequence of points $\vx_1, \ldots, \vx_k$ such that $\lim_{k \rightarrow \infty} \f(\vx_k) = \f^*$, while only evaluating the objective function, $\fun$, and its gradient $\gradient f$, at points. In other words, other than the global function parameters related to $\f$ that we define in this section, we assume that the algorithms under consideration have no further knowledge regarding the structure of $\f$. To compare such algorithms, we say that \emph{an iterative method has convergence rate $r$} if $f(\x_{k})-f^{*}\leq O((1-r)^{k})$ for this method.
Now, we say that \emph{$\fun$ has convexity parameter $\sigma$ with respect to some norm $\norm{\cdot}$} if the following holds
\begin{equation}
\forall \x, \y \in \Rn
\enspace : \enspace
\f(\y)
\geq
\f(\x)
+ \innerprod{\gradient \f(\x)}{\y - \x}
+\frac{\sigma}{2} \norm{\y - \x}^{2}
\label{eq:lower_env}
\end{equation}
and we say $\fun$ \emph{strongly convex} if $\sigma > 0$. We refer to the right hand side of (\ref{eq:lower_env}) as the \emph{lower envelope of $\fun$ at $\x$} and for notational convenience when the norm $\norm{\cdot}$ is not specified explicitly we assume it to be the standard Euclidian norm $\norm{x} \defeq \sqrt{\sum_{i} \x_i^2}$.
Furthermore, we say $\f$ has $\constL$-Lipschitz gradient if
\[
\forall \x, \y \in \Rn
\enspace : \enspace
\norm{\gradient \f(\y) - \gradient \f(\x)} \leq L \norm{\y - \x}
\]
The definition is related to an upper bound on $\f$ as follows:
\begin{lemma}
\label{lem:eqv_def_of_L}\cite[Thm 2.1.5]{Nesterov2003}
For continuously differentiable $\f : \Rn \rightarrow \R$ and $L > 0$, it has $\constL$-Lipschitz gradient if and only if
\begin{equation}
\label{eq:upper_env}
\forall \x, \y \in \Rn
\enspace : \enspace
f(\y) \leq f(\x) +
\left\langle \gradient f(\x), \y - \x\right\rangle
+ \frac{L}{2}\norm{\x - \y}^{2}
\enspace.
\end{equation}
\end{lemma}
We call the right hand side of (\ref{eq:upper_env}) the \emph{upper envelope of $\f$ at $\x$}.
The convexity parameter $\constConv$ and the Lipschitz constant of the gradient $\constL$ provide lower and upper bounds on $\f$. They serve as the essential characterization of $\f$ for first-order methods and they are typically the only information about $\f$ provided to both gradient and accelerated gradient descent methods (besides the oracle access to $\f$ and $\gradient \f$). For twice differentiable $\f$, these values can also be computed by properties of the Hessian of $\f$ by the following well known lemma:
\begin{lemma}[\cite{Nesterov2012}]
\label{lem:twicediff_prop}
Twice differentiable $\f : \Rn \rightarrow \R$ has convexity parameter $\mu$ and $\constL$-Lipschitz gradient with respect to norm $\norm{\cdot}$ if and only if $\forall \x \in \Rn$ the Hessian of $\f$ at $\x$, $\gradient^2 \f(\x) \in \Rnn$ satisfies
\[
\forall \vy \in \Rn
\enspace : \enspace
\mu \norm{\vy}^2
\leq
\vy^T \left(\gradient^2 \f(\x)\right) \vy
\leq
L \norm{\vy}^2
\]
\end{lemma}
To analyze \emph{coordinate-based iterative methods}, that is iterative methods that only consider one component of the current point or current gradient in each iteration, we need to define several additional parameters characterizing $\f$. For all $i \in [n]$, let $\basisI \in \Rn$ denote the standard basis vector for coordinate $i$, let $\gradfiVal(\x) \in \Rn$ denote the partial derivative of $\f$ at $\x$ along $\basisI$, i.e. $\gradfiVal(\x) \defeq \basisI^T \gradient\f(\x)$, and let $\gradfiVec(\x)$ denote the corresponding vector, i.e $\gradfiVec(\x) \defeq \gradfiVal \cdot \basisI$. We say that \emph{$\f$ has component-wise Lipschitz continuous gradient with Lipschitz constants $\{L_{i}\}$} if
\[
\forall \x \in \Rn, \enspace
\forall t \in \R, \enspace
\forall i \in [n]
\enspace : \enspace
|\gradfiVal(\x + t \cdot \basisI)- \gradfiVal(\x)| \leq \constLi \cdot |t|
\]
and for all $\alpha \geq 0$ we let $S_\alpha \defeq \sum_{i=1}^{n} L_{i}^{\alpha}$ denote the total component-wise Lipschitz constant of $\gradient f$. Later we will see that $S_\alpha$ has a similar role for coordinate descent as $\constL$ has for gradient descent.
We give two examples for convex functions induced by linear systems and calculate their parameters. Note that even though one example can be deduced from the other, we provide both as the analysis allows us to introduce more notation.
\begin{example}
\label{psd_example}
Let $\f(\x) \defeq \frac{1}{2} \innerprod{\A \x}{\x} - \innerprod{\x}{\bvec}$ for symmetric positive definite matrix $\A \in \R^{n \times n}$. Since $\A = \A^T$ clearly $\grad \f(\x) = \A \x - \bvec$ and $\grad^2 \f(\x) = \A$. Therefore, by
Lemma \ref{lem:twicediff_prop}, $L$ and $\sigma$ satisfy
\[
\sigma \norm{\x}^2 \leq \x^T \A \x \leq L \norm{\x}^2
\enspace.
\]
Consequently, $\sigma$ is the the smallest eigenvalue $\lambda_{\min}$ of $\A$ and $L$ is the largest eigenvalue $\lambda_{\max}$ of $\A$. Furthermore, $\forall i \in [n]$ we see that $f_{i}(\x)= \vvar{e}_{i}^{T}\left(\A \vx - \vb\right)$ and therefore $L_i$ satisfies
\[
\forall t \in \R
\enspace : \enspace
|t| \cdot |\A_{ii}|=|\vvar{e}_{i}^{T}\A(te_{i})|\leq L_{i}|t|
\enspace.
\]
Since the positive definiteness of $\A$ implies that $\A$ is positive on diagonal, we have $L_{i} = \A_{ii}$, and consequently
$
S_{1} = \tr(\A) = \sum_{i = 1}^{n} \A_{ii} = \sum_{i=1}^{n}\lambda_{i}
$
where $\lambda_{i}$ are eigenvalues of $\A$.
\end{example}
\begin{example}
\label{linear_system_example}Let $\f(\x)=\frac{1}{2}\norm{\A \x - b}^2$
for any matrix $\A$. Then $\gradient \f(\x) = \A^{T}\left(\A \vx - \vb\right)$ and $\gradient^2 \f(\x) = \A^T \A$. Hence, $\sigma$ and $L$ satisfy
\[
\sigma \norm{\x}^2 \leq \x^T \A^T \A \x \leq L \norm{\x}^2
\]
and we see that $\sigma$ is the the smallest eigenvalue $\lambda_{\min}$
of $\A^{T} \A$ and $L$ is the largest eigenvalue $\lambda_{\max}$
of $\A^{T}\A$. As in the previous example, we therefore have
$
L_{i} = \norm{a_{i}}^{2}
$
where $a_{i}$ is the $i$-th column of $\A$ and
$S_{1}= \sum \norm{a_{i}}^{2} = \normFro{\A}^{2}$, the Frobenius norm of $\A$.
\end{example}
\section{Review of Previous Iterative Methods}
\label{sec:iter_methods}
In this section, we briefly review several standard iterative first-order method for smooth convex minimization. This overview is by no means all-inclusive, our goal is simply to familiarize the reader with numerical techniques we will make heavy use of later and motivate our presentation of the accelerated coordinate descent method. For a more comprehensive review, there are multiple good references, e.g. \cite{Nesterov2003}, \cite{Boyd:2004:CO:993483}.
In \sectionref{sec:gradient_descent}, we briefly review the standard gradient descent method. In \sectionref{sec:accelerated_gradient_descent}, we show how to improve gradient descent by motivating and reviewing Nesterov's \emph{accelerated gradient descent method} \cite{Nesterov1983} through a more recent presentation of his via \emph{estimate sequences} \cite{Nesterov2003}. In \sectionref{sec:coordinate_descent}, we review Nesterov's \emph{coordinate gradient descent method} \cite{Nesterov2012}. In the next section we combine these concepts to present a general and efficient accelerated coordinate descent scheme.
\subsection{Gradient Descent}
\label{sec:gradient_descent}
Given an initial point $\xinit \in \Rn$ and step sizes $h_k \in \R$, the \emph{gradient descent method} applies the following simple iterative update rule:
\[
\forall k \geq 0
\enspace : \enspace
\x_{k+1} := \x_{k} - h_{k} \gradient \f(\x_{k}).
\]
For $h_{k} = \frac{1}{L}$, this method simply chooses the minimum point of the upper envelope of $\f$ at $\x_{k}$:
\[
\x_{k+1}
=
\argmin_{\y}
\left\{
f(\x_{k})
+
\left\langle \gradient f(\x_{k}),\y-\x_{k}\right\rangle +\frac{L}{2}\norm{\y-\x_{k}}^{2}\right\} .
\]
Thus, we see that the gradient descent method is a greedy method that chooses the minimum point based on the worst case estimate of the function based on the value of $\fun(x_k)$ and $\gradient \fun(x_{k})$. It is well known that it provides the following guarantee \cite[Cor 2.1.2, Thm 2.1.15]{Nesterov2003}
\begin{equation}
f(\x_{k}) - \optValue
\leq
\frac{L}{2} \cdot
\min\left\{ \left(1-\frac{\sigma}{L}\right)^{k},\frac{4}{k+4}\right\}
\norm{\x_0 - \optPoint}^{2}
\enspace.
\end{equation}
\subsection{Accelerated Gradient Descent}
\label{sec:accelerated_gradient_descent}
To speed up the greedy and memory-less gradient descent method, one could make better use of the history and pick the next step to be the smallest point in upper envelope of all points computed. Formally, one could try the following update rule
\[
\vx_{k+1} :=
\argmin_{\vy \in \Rn}
\min_{\sum_{k=1}^{n} t_k \vec{z}_k = \vy}
\left\{
\sum_{k = 1}^{n} t_k \left(
f(\vx_{k}) +
\innerprod{\gradient \f(\vx_{k})}{\vec{z}_k- \vx_{k}} +
\frac{L}{2} \norm{ \vec{z}_k - \vx_{k}}^{2}
\right)
\right\}.
\]
However, this problem is difficult to solve efficiently and requires storing all previous points. To overcome this problem, Nesterov \cite{Nesterov1983,Nesterov2003} suggested to use a quadratic function to estimate the function. Formally, we define an \emph{estimate sequence} as follows: \footnote{Note that our definition deviates slightly from Nesterov's \cite[Def 2.2.1]{Nesterov2003} in that we include condition \ref{eq:upper_estimate_sequence}.}
\begin{defn}[Estimate Sequence]
A triple of sequences $\{\phi_{k}(x),\eta_{k},\vx_{k}\}_{k=0}^{\infty}$
is called an \emph{estimate sequence} of $\f$ if $\lim_{k \rightarrow \infty} \eta_{k} = 0$ and for any $\x \in \Rn$ and $k \geq 0$ we have
\begin{equation}
\phi_{k}(\vx)\leq(1-\eta_{k})f(\vx)+\eta_{k}\phi_{0}(\vx)\label{eq:lower_estimate_sequence}
\end{equation}
and
\begin{equation}
f(\vx_{k})\leq\min_{\vx\in\Rn}\phi_{k}(\vx).\label{eq:upper_estimate_sequence}
\end{equation}
\end{defn}
An estimate sequence of $\fun$ is an approximate lower bound of
$\fun$ which is slightly above $\fun^{*}$. This relaxed definition allows
us to find a better computable approximation of $\f$ instead of relying
on the worst case upper envelope at each step.
A good estimate sequence gives an efficient algorithm \cite[Lem 2.2.1]{Nesterov2003} by the following
\[
\lim_{k \rightarrow \infty} \fun(\x_{k}) - \optValue
\leq
\lim_{k \rightarrow \infty} \eta_{k} \left(\phi_{0}(\x^{*})-\f^{*}\right)
= 0.
\]
Since an estimate sequence is an approximate lower bound, a natural
computable candidate is to use the convex combination of lower envelopes
of $\fun$ at some points.
Since it can be shown that any convex combinations of lower envelopes at evaluation points $\{y_{k}\}$ satisfies (\ref{eq:lower_estimate_sequence}) under some mild condition, additional points $\{y_{k}\}$ other than $\{x_{k}\}$ can be used to tune the algorithm. Nesterov's \emph{accelerated gradient descent method} can be obtained by tuning the the free parameters $\{y_{k}\}$ and $\{\eta_{k}\}$ to satisfy \eqref{eq:upper_estimate_sequence}. Among all first order methods, this method is optimal up to constants in terms of number of queries made to $\fun$ and $\gradient f$. The performance of the accelerated gradient descent method can be characterized as follows: \cite{Nesterov2003}
\begin{equation}
\label{eq:AGDM}
f(\x_{k}) - \optValue
\leq
L \cdot
\min\left\{ \left(1-\sqrt{\frac{\sigma}{L}}\right)^{k},\frac{4}{(k+2)^{2}}\right\}
\norm{\x_0 - \optPoint}^{2}
\enspace.
\end{equation}
\subsection{Coordinate Descent}
\label{sec:coordinate_descent}
The \emph{coordinate descent method} of Nesterov \cite{Nesterov2012} is a variant of gradient descent in which only one coordinate of the current iterate is updated at a time. For a fixed $\alpha \in \R$, each iteration $k$ of coordinate descent consists of picking a random a random coordinate $i_k \in [n]$ where
\[
\Pr[i_k = j] = P_\alpha(j)
\enspace \text{ where } \enspace
P_\alpha(j) \defeq \frac{L_{i_k}^\alpha}{S_\alpha}
\]
and then performing a gradient descent step on that coordinate:
\[
\vx_{k + 1} := \vx_k - \frac{1}{L_{i_k}} \vvar{f}_{i_k}(\vx_{k}).
\]
To analyze this algorithm's convergence rate, we define the norm $\normOma{\cdot}$, its dual $\normOma{\cdot}^*$, and the inner product $\innerprod{\cdot}{\cdot}_{1 - \alpha}$ which induces this norm as follows:
\[
\normOma{\vx} \defeq \sqrt{\sum_{i = 1}^{n} L_i^{1 - \alpha} \vx_i^2}
\enspace \text{ and }
\normOma{\vx}^* \defeq \sqrt{\sum_{i = 1}^{n} L_i^{-(1 - \alpha)} \vx_i^2}
\enspace \text{ and }
\innerprod{\vx}{\vy}_{1 - \alpha} \defeq \sum_{i = 1}^{n} L_i^{1- \alpha} \vx_i \vy_i
\]
and we let $\sigma_{1 - \alpha}$ denote the convexity parameter of $f$ with respect to $\normOma{\cdot}$.
Using the definition of coordinate-wise Lipschitz constant, each step can be shown to have the following guarantee on expected improvement \cite{Nesterov2012}
\[
f(\x_{k}) - \E\left[\f(\x_{k+1})\right]
\geq
\frac{1}{2S_{\alpha}}\left(\normOmaDual{\grad \f(\x_k)}\right)^{2}.
\]
and further analysis shows the following convergence guarantee coordinate descent \cite{Nesterov2012}
\[
\E\left[\f(\vx_k)\right] - \optValue
\leq \min
\left\{
\frac{2 S_\alpha}{k + 4} \left(
\max_{f(\vy) \leq f(\vx_0)}
\normOma{\vy - \optPoint}^2\right)
\enspace , \enspace
\left(1 - \frac{\sigma_{1 - \alpha}}{S_\alpha}\right)^k
\left(f(\vx_0) - \optValue\right)
\right\}.
\] | 73,195 |
You can subscribe to this list here.
Showing
1
results of 1
Hi list,
0.16 of rsyncrypto has just been released. I trust I don't have to tell
anyone on this list what's new in this release :-).
In many respects, this is the most important release of rsyncrypto ever.
This is not saying much, I guess, as the next two released are also
expected to bear the same title. To recap to anyone who was not listening:
0.16 - added support for file name encryption
This includes full support for both -r (recursive) and --filelist,
including support for --delete and --delete-keys on both previous
options. Program was tested on Windows and Linux - feedback on other
platforms welcome.
0.17 - planned change is the addition of signing of the encrypted files.
This will allow and external party to verify that the file was not
tampered with, without having the key to open the file.
0.18 - Meta information encryption. All relevant meta information will
be encrypted into the file, including modification time, owner,
permissions, and plaintext file name.
0.19 - integrate zlib into rsyncrypto, eliminating the need to rely on
an external "gzip" executeable.
Future feature and release numbers are, obviously, subject to change. I
have not, yet, decided when will be the right time to switch
rsyncrypto's status from "beta" to "release". It will either be 0.18 or
0.19, feature wise.
Waiting to hear your feedback,
Shachar
--
Shachar Shemesh
Lingnu Open Source Consulting ltd.
Have you backed up today's work? | 248,241 |
Posted on July 12, 2022
Song of the Day: The Lion Sleeps Tonight
Today’s song is The Lion Sleeps Tonight by The Tokens.
Doo-wop.
Wanted to throw in a classic. This has a ‘fun-nature’ that never gets old.
There is something so soothing about the sound that it always cheers me up.
Hush, my darling, don’t fear, my darling,
The lion sleeps tonight.
The track is over sixty years old, but many still appreciate it. I think it’ll stick around for a few more decades.
Most recognize this as the song for that old video with the singing hippo and the dancing dog.
Yeah, it was an EXTREMELY old meme. Very ancient Internet Humor.
______________________
Click here for the full YouTube playlist (of all of the song’s I’ve talked about).
I’ve talked about hundreds of them, so you’re bound to find something entertaining. | 409,920 |
SportsGrid Network Launches on SLING TV
NEW YORK, Aug. 12, 2020 /PRNewswire/ — SportsGrid, Inc. announced today the launch of a 24-hour network on SLING TV, a…
NEW YORK, Aug. 12, 2020 /PRNewswire/ — gaming intelligence across the NFL, MLB, NBA, NHL, College Sports, Golf, Tennis and Soccer. The network is now available on SLING Free – no account or credit card required – and features live and original programming serving the fanatical sports enthusiast with expert analysis and interviews from best-in-class sports analysts. The SportsGrid live sports wagering programming originates from its state-of-the-art production facilities located at the FanDuel Sportsbook in New Jersey, Meadowlands Studio 1 and SportsGrid’s New York City Studio 34.
The all original SportsGrid line-up is anchored by’re excited to expand the distribution of SportsGrid to reach the SLING TV audience and deliver the most compelling sports wagering programming in the country with outstanding talent, including Pat McAfee, Scott Ferrall, Gabe Morency and many others,” said Louis Maione, Founder and President of SportsGrid, Inc. “SLING TV viewers can depend on SportsGrid’s commitment to reporting the sports gaming news along with insightful analysis and commentary to make their wagering decisions.”
To access the SLING Free library and begin watching SportsGrid Network, please visit..
Media Contact:
Charles Theiss
[email protected]
View original content to download multimedia:
SOURCE SportsGrid
| 15,713 |
\begin{document}
\title[Growth estimates for solutions of algebraic differential equations]{Growth estimates for meromorphic solutions of higher order algebraic differential equations}
\author{Shamil Makhmutov}
\address{Department of Mathematics, College of Science, Sultan Qaboos University, P.O. Box 36, PC 123 Al Khodh, Muscat, Sultanate of Oman}
\email{[email protected]}
\author{Jouni R\"atty\"a}
\address{University of Eastern Finland, P.O.Box 111, 80101 Joensuu, Finland}
\email{[email protected]}
\author{Toni Vesikko}
\address{University of Eastern Finland, P.O.Box 111, 80101 Joensuu, Finland}
\email{[email protected]}
\maketitle
\begin{abstract}
We establish pointwise growth estimates for the spherical derivative of solutions of the first order algebraic differential equations. A generalization of this result to higher order equations is also given. We discuss the related question of when for a given class $X$ of meromorphic functions in the unit disc, defined by means of the spherical derivative, and $m\in\NN\setminus\{1\}$, $f^m\in X$ implies $f\in X$. An affirmative answer to this is given for example in the case of $\UBC$, the $\alpha$-normal functions with $\alpha\ge1$ and certain (sufficiently large) Dirichlet type classes.
\end{abstract}
\section{Introduction and main results}
Let $\H(\D)$ and $\M(\D)$ denote the sets of analytic and meromorphic functions in the unit disc $\D=\{z\in\C:|z|<1\}$, respectively. For $n,N\in\NN$, consider the $N$-th order algebraic differential equation
\begin{equation}\label{orderNref}
\left(f^{(N)}\right)^n + \sum_{k=1}^{n} P_{k,N}\left(f\right)\left(f^{(N)}\right)^{n-k} = 0,
\end{equation}
where
\begin{equation}
P_{k,N}(f)
=\sum_{j_0=0}^{m_{k,0}}\sum_{j_1=0}^{m_{k,1}}\cdots\sum_{j_{N-1}=0}^{m_{k,N-1}}
a_{k,j_0,\ldots,j_{N-1}}\prod_{\ell=0}^{N-1} \left(f^{(\ell)}\right)^{j_\ell},\quad k=1,\ldots,n,
\end{equation}
with $a_{k,j_0,\ldots,j_{N-1}}\in\H(\D)$ and $m_{k,j}\in\NN\cup\{0\}$ for all $j=0,\ldots,N-1$ and $k=1,\ldots,n$. The case $N=1$ reduces to the first order equation
\begin{equation}\label{eq-1}
(f')^n + \sum_{k=1}^{n} P_k(f)\,(f')^{n-k} = 0,
\end{equation}
where
\begin{equation}\label{Pk-1}
P_k(f)=\sum_{j=0}^{m_k} a_{k,j} f^j , \quad 1\le k \le n\in\NN,\quad m_k\in\NN\cup\{0\},
\end{equation}
and $a_{k,j}\in\H(\D)$ for all $k=1,\ldots,n$ and $j=0,\ldots,m_k$.
The main result of this study is a pointwise growth estimate for the spherical derivative of meromorphic solutions of \eqref{orderNref}. The method of proof does not depend on the underlying domain and can be performed on any set. The normal and Yosida solutions of algebraic differential equations similar to \eqref{orderNref} and \eqref{eq-1} have been studied extensively via techniques like, for example, the Lohwater-Pommerenke method \cite{A-M-R2010,AW2011}. There are multiple existing results concerning normality conditions and the behaviour of the spherical derivatives of the solutions \cite{A-M-R2010,L-L-Y2003,Makhmutov2011}. The method we employ allows us to consider solutions in classes which are strictly smaller than the class of normal functions.
Before stating the results, a word about the notation used. The letter $C=C(\cdot)$ will denote an absolute constant whose value depends on the parameters indicated
in the parenthesis, and may change from one occurrence to another.
We will use the notation $a\lesssim b$ if there exists a constant
$C=C(\cdot)>0$ such that $a\le Cb$, and $a\gtrsim b$ is understood
in an analogous manner. In particular, if $a\lesssim b$ and
$a\gtrsim b$, then we write $a\asymp b$ and say that $a$ and $b$ are comparable.
\begin{theorem}\label{thm:main}
Let $n,N\in\NN$ and $M_\ell\in\NN\cup\{0\}$ such that
\begin{equation}\label{hypotheses}
M_0\ge\max_{k=1,\ldots,n}\frac{m_{k,0}}{k}-2\quad\textrm{and}\quad
M_\ell\ge\max_{k=1,\ldots,n}\frac{m_{k,\ell}}{k}-1,\quad \ell=1,\ldots,N-1.
\end{equation}
Then each meromorphic solution $f$ of \eqref{orderNref} satisfies
\begin{equation*}
\prod_{\ell=0}^{N-1}\left(\left(f^{(\ell)}\right)^{M_{\ell}+1}\right)^{\#}
\lesssim\sum_{k=1}^n\left(\sum_{j_0=0}^{m_{k,0}}\sum_{j_1=0}^{m_{k,1}}\cdots\sum_{j_{N-1}=0}^{m_{k,N-1}}
|a_{k,j_0,\cdots,j_{N-1}}|^{\frac{1}{k}}\right).
\end{equation*}
In particular, if $\max_{k=1,\ldots,n}\frac{m_k}{k}\le M_0+2$, then each meromorphic solution $f$ of \eqref{eq-1} satisfies
\begin{equation*}
\left(f^{M_0+1}\right)^{\#}
\lesssim\sum_{k=1}^n\sum_{j=0}^{m_{k}}|a_{k,j}|^{\frac{1}{k}}.
\end{equation*}
\end{theorem}
Theorem~\ref{thm:main} has the following immediate consequence which deserves to be stated separately.
\begin{corollary}\label{corollary2}
Let $n,N\in\NN$, $\max_{k=1,\ldots,n}\frac{m_{k,0}}{k}\le2$ and $\max_{k=1,\ldots,n}\frac{m_{k,\ell}}{k}\le1$ for all $\ell=1,\ldots,N-1$. Then each meromorphic solution $f$ of \eqref{orderNref} satisfies
\begin{equation}\label{Eq:corollary}
\prod_{\ell=0}^{N-1}\left(f^{(\ell)}\right)^{\#}
\lesssim\sum_{k=1}^n\left(\sum_{j_0=0}^{m_{k,0}}\sum_{j_1=0}^{m_{k,1}}\cdots\sum_{j_{N-1}=0}^{m_{k,N-1}}
|a_{k,j_0,\cdots,j_{N-1}}|^{\frac{1}{k}}\right).
\end{equation}
\end{corollary}
The product of the spherical derivatives on the left hand side of \eqref{Eq:corollary} appears in a natural way in the study of (weighted) normal functions \cite{CL1996,CL1998,Grohn2017,Lappan-77,Makhmutov1986,Xu2000}.
The second part of Theorem~\ref{thm:main} gives arise to the question of when for a given class $X\subset\M(\D)$ and $m\in\NN\setminus\{1\}$, $f^m\in X$ implies $f\in X$. An immediate observation is that, roughly speaking, this implication cannot be true if $X$ is sufficiently small and defined in terms of the spherical derivative. More precisely, for $f_p(z)=(1-z)^{-p}$ with $0<p<1/m<1$ we have $f_p^\#(z)\asymp|1-z|^{p-1}$ and $(f_p^m)^\#\asymp|1-z|^{mp-1}$ as $\D\ni z\to1$. This shows that $(f_p^m)^\#$ is essentially smaller than $(f_p)^\#$ when $z\to1$, yet of course $f_p\in\N^{1-p}\subset\N$ for all $0<p<1$. Recall that for $0<\alpha<\infty$, the class $\N^\alpha$ of $\alpha$-normal functions is the set of functions $f\in\M(\D)$ such that
$$
\|f\|_{\N^\alpha}=\sup_{z\in\D}f^\#(z)(1-|z|^2)^\alpha<\infty,
$$
and its subset $\N^\alpha_0$ of strongly $\alpha$-normal functions consists of functions $f\in\M(\D)$ such that $f^\#(z)(1-|z|^2)^\alpha\to0$
as $|z|\to1^-$.
We next offer an affirmative answer to the question of when $f^m\in X$ implies $f\in X$ in the case of certain function classes. To do this, definitions are needed. An increasing function $\vp:[0,1)\to(0,\infty)$ is smoothly increasing if
$\vp(r)(1-r)\to\infty$, as $r\to1^-$, and
\begin{equation*}
\frac{\vp(|a+z/\vp(|a|)|)}{\vp(|a|)}\to1,\quad |a|\to1^-,
\end{equation*}
uniformly on compact subsets of $\C$. For such a $\vp$, a
function $f\in\M(\D)$ is $\vp$-normal if
\begin{equation}\label{definition}
\|f\|_{\N^\vp}=\sup_{z\in\D}\frac{f^{\#}(z)}{\vp(|z|)}<\infty,
\end{equation}
and strongly $\vp$-normal if $\frac{f^{\#}(z)}{\vp(|z|)}\to0$, as $|z|\to1^-$. The classes of $\vp$-normal and strongly $\vp$-normal functions are denoted by $\N^\vp$ and $\N^\vp_0$, respectively.
Let $\om\in L^1(0,1)$. The extension defined by $\om(z)=\om(|z|)$ for all $z\in\D$ is called a radial weight on $\D$. For such an $\om$, denote
$$
\om^\star(z)=\int_{|z|}^1\om(s)\log\frac{s}{|z|}s\,ds,\quad z\in\D\setminus\{0\}.
$$
The Dirichlet class $\DD^\#_{\om^\star}$ consists of $f\in\M(\D)$ such that
$$
\int_\D f^\#(z)^2\om^\star(z)\,dA(z)<\infty,
$$
where $dA(z)=rdrd\theta/\pi$ for $z=re^{i\theta}$. Moreover, the Dirichlet class $\DD^\#_{\alpha}$ consists of $f\in\M(\D)$ such that
$$
\int_\D f^\#(z)^2(1-|z|^2)^\alpha\,dA(z)<\infty.
$$
A function $f\in\M(\D)$ belongs to $\UBC$ if
$$
\sup_{a\in\D}\int_\D f^\#(z)^2\log\frac1{|\varphi_a(z)|}\,dA(z)<\infty,
$$
where $\vp_a(z)=(a-z)/(1-\overline{a}z)$.
With these preparations we can state our next result.
\begin{theorem}\label{theorem:f-m-implications}
Let $m\in\NN\setminus\{1\}$, $1<p<\infty$, $\vp$ a smoothly increasing and $\om$ a radial weight. Then the following statements are valid:
\begin{itemize}
\item[\rm(i)] $f^m\in\N\Rightarrow f\in\N$;
\item[\rm(ii)] $f^m\in\N_0\Rightarrow f\in\N_0$;
\item[\rm(iii)] $f^m\in\N^\vp\Rightarrow f\in\N^\vp$;
\item[\rm(iv)] $f^m\in\N^\vp_0\Rightarrow f\in\N^\vp_0$;
\item[\rm(v)] $f^m\in\DD^\#_1\Rightarrow f\in\DD^\#_1$;
\item[\rm(vi)] $f^m\in\DD^\#_{\om^\star}\Rightarrow f\in\DD^\#_{\om^\star}$;
\item[\rm(vii)] $f^m\in\UBC\Rightarrow f\in\UBC$.
\end{itemize}
\end{theorem}
The proofs of the cases (i)--(iv) are based on the so-called five-point theorems, named by the celebrated result for normal functions due to Lappan \cite{Lappan1974}. Such results exist also in the setting of meromorphic functions on the whole plane and are usually given in terms of Yosida and $\vp$-Yosida functions, see \cite{A-M-R2010,Makhmutov2001}. Therefore we may obtain analogues of the statements (i)-(iv) for those classes; details are left for interested reader.
The argument of proof used in (v)--(vii) uses \cite[Theorem~2]{Yamashita1981} due to Yamashita on meromorphic Hardy classes. This is not exclusive for the disc either and can be performed also on the plane. The argument yields the inequality
$$
\int_\C f^\#(z)^2\left(\int_{|z|}^\infty\log\frac{r}{|z|}\om(r)\,dr\right)dA(z)
\lesssim\int_\C\left(f^m\right)^\#(z)^2\left(\int_{|z|}^\infty\log\frac{r}{|z|}\om(r)\,dr\right)dA(z)+1,
$$
valid for all meromorphic functions $f$ and radial weights $\om$ on $\C$. This is an analogue of (vi) for $\C$. One natural choice for $\om$ in this case is $\om(z)=(1+|z|)^{-\alpha}$ for $1<\alpha<\infty$.
Let $0<\alpha<1$ be fixed. By Theorem~\ref{theorem:f-m-implications}(i) we know that if $f^m\in\N^\alpha\subset\N$, then $f\in\N$. For $0<\alpha<1$ and $m\in\NN\setminus\{1\}$, write
$$
\beta_{\alpha,m}=\inf_{0<\gamma\le1}\left\{f\in\N^\gamma:f^m\in\N^\alpha\right\}.
$$
The function $f_{p}$ with $p=\frac{1-\alpha}{m}$ considered after Corollary~\ref{corollary2} and Theorem~\ref{theorem:f-m-implications}(i) show that $1-\frac{1-\alpha}{m}\le\beta_{\alpha,m}\le1$. The exact value of $\beta_{\alpha,m}$ is unknown.
Theorem~\ref{theorem:f-m-implications} shows that $f^m\in\DD^\#_\alpha$ with $\alpha\ge1$ implies $f\in\DD^\#_\alpha$. Further, the function $f_p$ with $-\frac{\alpha}{2m}<p\le-\frac{\alpha}{2}$ shows that $f^m\in\DD^\#_\alpha$ with $\alpha<0$ does not imply $f\in\DD^\#_\alpha$. It is natural to ask what happens with the range $0\le\alpha<1$? We do not know an answer to this question.
We next aim for combining Theorems~\ref{thm:main} and \ref{theorem:f-m-implications} in order to find a set of sufficient conditions for the coefficients of \eqref{orderNref} that force meromorphic solution $f$ to belong to certain function classes. To do this, some more notation is needed. For $0<p<\infty$, the weighted growth space $H^\infty_p$ consists of $f\in\H(\D)$ such that
$$
\|f\|_{H^\infty_p}=\sup_{z\in\D}|f(z)|(1-|z|)^p<\infty.
$$
Similarly, $H^p_\vp$ consists of $f\in\H(\D)$ such that
$$
\|f\|_{H^p_\vp}=\sup_{z\in\D}\frac{|f(z)|}{\vp(|z|)}<\infty.
$$
For $0<p<\infty$ and a radial weight $\om$, the Bergman space $A^p_\om$ consists of $f\in\H(\D)$ such that
$$
\|f\|_{A^p_\om}^p=\int_\D|f(z)|^p\om(z)\,dA(z)<\infty.
$$
Note that the Hardy-Spencer-Stein formula yields
$$
\|f\|_{A^p_\omega}^p=p^2\int_{\D}|f(z)|^{p-2}|f'(z)|^2\omega^\star(z)\,dA(z)+\omega(\D)|f(0)|^p,\quad f\in\H(\D),
$$
by \cite[Theorem~4.2]{PelaezRattya2014}. This explains how the associated weight $\om^\star$ raises in a natural manner.
The following result is an immediate consequence of Theorems~\ref{thm:main} and \ref{theorem:f-m-implications}.
\begin{corollary}
Let $n\in\NN$ and $M_0\in\NN\cup\{0\}$ such that $\max_{k=1,\ldots,n}\frac{m_k}{k}\le M_0+2$, and let $\vp$ be smoothly increasing.
\begin{itemize}
\item[\rm(i)] If $a_{k,j}\in H^\infty_k$ (resp. $a_{k,j}\in H^\infty_{k,0}$) for all $j=0,\ldots,m_k$ and $k=1,\ldots,n$, then each meromorphic solution $f$ of \eqref{eq-1} belongs to $\N$ (resp. $\N_0)$.
\item[\rm(ii)] If $a_{k,j}\in H^\infty_{\vp^k}$ (resp. $a_{k,j}\in H^\infty_{\vp^k,0}$) for all $j=0,\ldots,m_k$ and $k=1,\ldots,n$, then each meromorphic solution $f$ of \eqref{eq-1} belongs to $\N^\vp$ (resp. $\N^\vp_0$).
\item[\rm(iii)] If $a_{k,j}\in A^{\frac2k}_{\om^*}$ for all $j=0,\ldots,m_k$ and $k=1,\ldots,n$, then each meromorphic solution $f$ of \eqref{eq-1} belongs to $\DD^\#_{\om^*}$.
\item[\rm(iv)] If
$$
\sup_{a\in\D}\int_\D|a_{k,j}(z)|^\frac{2}{k}\log\frac{1}{|\vp_a(z)|}\,dA(z)<\infty
$$
for all $j=0,\ldots,m_k$ and $k=1,\ldots,n$, then each meromorphic solution $f$ of \eqref{eq-1} belongs to $\UBC$.
\end{itemize}
\end{corollary}
It is well known that in (iv) one may replace the Green's function $\log\frac{1}{|\vp_a(z)|}$ by the term $1-|\vp_a(z)|^2$ in the statement. This is due to the analyticity of the coefficients.
\section{Proof of Theorem~\ref{thm:main}}
We first multiply \eqref{orderNref} by
$$
\left(\left(M_{N-1}+1\right)\left(f^{N-1}\right)^{M_{N-1}}\prod_{\ell=0}^{N-2}\left(\left(f^{(\ell)}\right)^{M_\ell + 1}\right)'\right)^n
$$
to obtain
\begin{equation*}
\begin{split}
\prod_{\ell=0}^{N-1}\left(\left(\left( f^{(\ell)}\right)^{M_{\ell}+1}\right)'\right)^n
+&\sum_{k=1}^nP_{k,N}(f)\left(f^{(N)}\right)^{n-k}\left(M_{N-1}+1\right)^n\left(f^{(N-1)}\right)^{M_{N-1}n}\\
&\cdot\left(\prod_{\ell=0}^{N-2}\left(\left(f^{(\ell)}\right)^{M_\ell+1}\right)'\right)^n=0,
\end{split}
\end{equation*}
and then divide it by $\prod_{\ell=0}^{N-1}\left(1+|f^{(\ell)}|^{2\left(M_\ell+1\right)}\right)^n$ to get
\begin{equation*}
\begin{split}
\prod_{\ell=0}^{N-1}\left(\frac{\left(\left( f^{(\ell)}\right)^{M_{\ell}+1}\right)'}{1+|f^{(\ell)}|^{2\left(M_\ell+1\right)}}\right)^n
+&\sum_{k=1}^nP_{k,N}(f)\left(M_{N-1}+1\right)^k\left(f^{(N-1)}\right)^{M_{N-1}k}\\
&\cdot\frac{\left(\prod_{\ell=0}^{N-2}\left(\left(f^{(\ell)}\right)^{M_\ell+1}\right)'\right)^n
\left(\left(\left(f^{(N-1)}\right)^{M_{N-1}+1}\right)'\right)^{n-k}}
{\prod_{\ell=0}^{N-1}\left(1+|f^{(\ell)}|^{2\left(M_\ell+1\right)}\right)^n}=0.
\end{split}
\end{equation*}
By reorganizing terms and taking moduli, we deduce
\begin{equation*}
\begin{split}
\prod_{\ell=0}^{N-1}\left(\left(f^{(\ell)}\right)^{M_{\ell}+1}\right)^\#
&\le\Bigg(\sum_{k=1}^n\left|P_{k,N}(f)\right|\left(M_{N-1}+1\right)^k\left|f^{(N-1)}\right|^{M_{N-1}k}\\
&\quad\cdot\frac{\prod_{\ell=0}^{N-2}\left(M_\ell+1\right)^n|f^{(\ell)}|^{M_\ell n}|f^{(\ell+1)}|^n}{\prod_{\ell=0}^{N-1}\left(1+|f^{(\ell)}|^{2\left(M_\ell+1\right)}\right)^n}\left|\left(\left(f^{(N-1)}\right)^{M_{N-1}+1}\right)'\right|^{n-k}\Bigg)^\frac1n\\
&\le\sum_{k=1}^n\left|P_{k,N}(f)\right|^\frac1n\left(M_{N-1}+1\right)^\frac{k}n\left|f^{(N-1)}\right|^{\frac{M_{N-1}k}{n}}\\
&\quad\cdot\frac{\prod_{\ell=0}^{N-2}\left(M_\ell+1\right)|f^{(\ell)}|^{M_\ell}|f^{(\ell+1)}|}
{\prod_{\ell=0}^{N-1}\left(1+|f^{(\ell)}|^{2\left(M_\ell+1\right)}\right)}
\left|\left(\left(f^{(N-1)}\right)^{M_{N-1}+1}\right)'\right|^{\frac{n-k}{n}}\\
&=\sum_{k=1}^n\left|P_{k,N}(f)\right|^\frac1n\left(M_{N-1}+1\right)^\frac{k}n
\prod_{\ell=0}^{N-2}\left(\left(\left(f^{(\ell)}\right)^{M_\ell+1}\right)^\#\right)^\frac{k}{n}\\
&\quad\cdot\left(\frac{|f^{(N-1)}|^{M_{N-1}}}{1+|f^{(N-1)}|^{2(M_{N-1}+1)}}\right)^{\frac{k}{n}}
\left(\prod_{\ell=0}^{N-1}\left(\left(f^{(\ell)}\right)^{M_{\ell}+1}\right)^\#\right)^{\frac{n-k}{n}}.
\end{split}
\end{equation*}
It follows that
\begin{equation*}
\begin{split}
\prod_{\ell=0}^{N-1}\left(\left(f^{(\ell)}\right)^{M_{\ell}+1}\right)^\#
&\le\Bigg(\sum_{k=1}^n\left|P_{k,N}(f)\right|^\frac1n\left(M_{N-1}+1\right)^\frac{k}n
\prod_{\ell=0}^{N-2}\left(\left(\left(f^{(\ell)}\right)^{M_\ell+1}\right)^\#\right)^\frac{k}{n}\\
&\quad\cdot\left(\frac{|f^{(N-1)}|^{M_{N-1}}}{1+|f^{(N-1)}|^{2(M_{N-1}+1)}}\right)^{\frac{k}{n}}
\Bigg)^{\frac{n}{k}}\\
&\lesssim\sum_{k=1}^n\left|P_{k,N}(f)\right|^\frac1k\left(M_{N-1}+1\right)
\prod_{\ell=0}^{N-2}\left(\left(f^{(\ell)}\right)^{M_\ell+1}\right)^\#\\
&\quad\cdot\frac{|f^{(N-1)}|^{M_{N-1}}}{1+|f^{(N-1)}|^{2(M_{N-1}+1)}},
\end{split}
\end{equation*}
where
\begin{equation*}
\begin{split}
\left|P_{k,N}(f)\right|^\frac1k
&\le\left(\sum_{j_0=0}^{m_{k,0}}\sum_{j_1=0}^{m_{k,1}}\cdots\sum_{j_{N-1}=0}^{m_{k,N-1}}
\left|a_{k,j_0,\ldots,j_{N-1}}\right|\prod_{\ell=0}^{N-1}\left|f^{(\ell)}\right|^{j_\ell}\right)^\frac1k\\
&\le\sum_{j_0=0}^{m_{k,0}}\sum_{j_1=0}^{m_{k,1}}\cdots\sum_{j_{N-1}=0}^{m_{k,N-1}}
\left|a_{k,j_0,\ldots,j_{N-1}}\right|^\frac1k\prod_{\ell=0}^{N-1}\left|f^{(\ell)}\right|^{\frac{j_\ell}{k}}.
\end{split}
\end{equation*}
Hence
\begin{equation*}
\begin{split}
\prod_{\ell=0}^{N-1}\left(\left(f^{(\ell)}\right)^{M_{\ell}+1}\right)^\#
&\lesssim\left(M_{N-1}+1\right)\sum_{k=1}^n\left(\sum_{j_0=0}^{m_{k,0}}\sum_{j_1=0}^{m_{k,1}}\cdots\sum_{j_{N-1}=0}^{m_{k,N-1}}
\left|a_{k,j_0,\ldots,j_{N-1}}\right|^\frac{1}{k}I_{k,j_0,\ldots,j_{N-1}}(f)\right),
\end{split}
\end{equation*}
where
\begin{equation*}
\begin{split}
&I_{k,j_0,\ldots,j_{N-1}}(f)=\prod_{\ell=0}^{N-1}\left|f^{(\ell)}\right|^{\frac{j_\ell}{k}}
\prod_{\ell=0}^{N-2}\left(\left(f^{(\ell)}\right)^{M_\ell+1}\right)^\#
\frac{|f^{(N-1)}|^{M_{N-1}}}{1+|f^{(N-1)}|^{2(M_{N-1}+1)}}.
\end{split}
\end{equation*}
Therefore to deduce the assertion it suffices to show that $I_{k,j_0,\ldots,j_{N-1}}(f)\lesssim1$. But a direct calculation shows that
\begin{equation*}
\begin{split}
I_{k,j_0,\ldots,j_{N-1}}(f)&=\prod_{\ell=1}^{N-1}\frac{(M_\ell+1)|f^{(\ell)}|^{\frac{j_\ell}{k}+M_\ell+1}}{1+|f^{(\ell)}|^{2(M_\ell+1)}}
\frac{|f|^{\frac{j_0}{k}+M_0}}{1+|f|^{2(M_0+1)}}\\
&\le\prod_{\ell=1}^{N-1}(M_\ell+1)\frac{\left(1+|f^{(\ell)}|^{2(M_\ell+1)}\right)^{\frac{^{\frac{m_{k,\ell}}{k}+M_\ell+1}}{2(M_\ell+1)}}}{1+|f^{(\ell)}|^{2(M_\ell+1)}}
\frac{\left(1+|f|^{2(M_0+1)}\right)^{\frac{\frac{m_{k,0}}{k}+M_0}{2(M_0+1)}}}{1+|f|^{2(M_0+1)}}\\
&\le\prod_{\ell=1}^{N-1}(M_\ell+1),\quad k=1,\ldots,n,
\end{split}
\end{equation*}
by the hypotheses \eqref{hypotheses}.\hfill$\Box$
\section{Proof of Theorem~\ref{theorem:f-m-implications}}
We prove first (i). Assume on the contrary that $f\not\in\N$. Then for each $Z\in f(\D)$, with at most four possible exceptions,
$\sup\{f^\#(z)(1-|z|^2):z\in\D,\,f(z)=Z\}=\infty$ by \cite{Lappan1974}. Let $Z\in f(\D)\setminus\{0,\infty\}$ be one of the points for which the supremum is infinity, and let $\{z_n\}_{n=1}^\infty$ denote a sequence of preimages of $Z$ such that $f^\#(z_n)(1-|z_n|^2)\to\infty$, as $n\to\infty$. Then
\begin{equation}
\begin{split}
\left(f^m\right)^\#(z_n)(1-|z_n|^2)
&=\frac{m|f(z_n)|^{m-1}(1+|f(z_n)|^2)}{(1+|f(z_n)|^{2m})}f^\#(z_n)(1-|z_n|^2)\\
&=\frac{m|Z|^{m-1}(1+|Z|^2)}{(1+|Z|^{2m})}f^\#(z_n)(1-|z_n|^2)\to\infty,\quad n\to\infty,
\end{split}
\end{equation}
and therefore $f^m\not\in\N$.
A reasoning similar to that in the case (i) with \cite[Theorem~9]{AR2011} gives (iii) and (iv). Further, \cite{AW2011} together with Lappan's proof in \cite{Lappan1974} can be used to establish an analogue of the five-point theorem for strongly normal functions, which in turn gives (ii) as above.
To prove (v)-(vii), we use \cite[Theorem~2]{Yamashita1981}. It implies
\begin{equation}\label{Eq:Yamashita}
\int_{D(0,r)}f^\#(z)^2\log\frac{r}{|z|}\,dA(z)
\lesssim\int_{D(0,r)}\left(f^m\right)^\#(z)^2\log\frac{r}{|z|}\,dA(z)+1,\quad 0<r<1.
\end{equation}
By letting $r\to1^-$ we deduce
\begin{equation}\label{111111}
\int_\D f^\#(z)^2\log\frac1{|z|}\,dA(z)
\lesssim\int_\D \left(f^m\right)^\#(z)^2\log\frac1{|z|}\,dA(z)+1.
\end{equation}
This together with the inequalities $1-t\le-\log t\le\frac1t(1-t)$, valid for all $0<t\le1$, yield (v).
By integrating \eqref{Eq:Yamashita} over $(0,1)$ with respect to $\om(r)r\,dr$ and applying Fubini's theorem we deduce
\begin{equation}\label{eq:plaah}
\int_\D f^\#(z)^2\om^\star(z)\,dA(z)\lesssim\int_\D\left(f^m\right)^\#(z)^2\om^\star(z)\,dA(z)+\|\om\|_{L^1(0,1)}
\end{equation}
from which the assertion (vi) follows.
By applying \eqref{111111} to $f\circ\vp_a$ we deduce (vii). This completes the proof of the theorem. | 163,648 |
TITLE: Does the existence of an injective cogenerator "help" in finding generators of an abelian category?
QUESTION [3 upvotes]: I have an abelian category $A$ that is AB4, AB3* and has an injective cogenerator. Do these conditions "help" in checking whether a given family $a_i$ of (compact) objects of $A$ is generating in it? So, where can I find any "non-trivial" conditions that ensure that $a_i$ generate $A$? I would prefer not to assume that $A$ is AB5 or Grothendieck abelian here; yet I am interested in any criteria for generators (that can assume any additional restrictions including these ones). In particular, does the existence of a conservative functor respecting coproducts from $A$ into abelian groups "help" here?
Any hints and (especially) references would be very welcome!
REPLY [2 votes]: A few thoughts a bit too long for a comment. I should preface this by saying that I'm not that familiar with the sort of generator conditions you're using, so I can only really talk about analogous cases using stronger generating conditions. I'm really hoping that someone who actually knows these things might respond to this.
As a cautionary analogy, consider the fact that the opposite of a locally presentable category is never locally presentable unless the category is a poset. A locally presentable category is the a cocomplete category $\mathcal{C}$ with the nicest possible kind of generator: a small dense generator $\mathcal{G}$. "Dense" means that the restricted Yoneda embedding from $\mathcal{C}$ into presheaves on $\mathcal{G}$ is fully faithful (unless you assume Vopenka's principle, you have to additionally assert that the objects of $\mathcal{G}$ are presentable, i.e. their hom-functors preserve sufficiently-filtered colimits. In fact, there's a weaker formulation of the generating property which turns out to be equivalent: the generator should be strong, i.e. the hom-functors are jointly faithful and jointly conservative, and consist of presentable objects). So the fact that the opposite of a locally presentable category is never locally presentable puts restrictions on how "nice" a generator and a cogenerator can simultaneously be. I'm not sure whether such restrictions extend to weaker notions of generator.
My sense is that most triangulated categories, like the homotopy category, don't have many (co)limits, and so are not locally presentable -- but moreover, like the homotopy category I wouldn't expect a triangulated category to even admit a faithful functor to $\mathsf{Set}$, so it's not even accessible. For this reason, I suppose, the study of triangulated categories uses weaker notions of generator than you see elsewhere in category theory. So maybe this whole point is moot. But I take it you're working with something abelian, more like the heart of $t$-structure, so my intuition is not clear on whether such a category is likely to be locally presentable. But somebody out there surely knows whether hearts of $t$-structures tend to be locally presentable! And somebody surely knows whether there's a tension between the generating and cogenerating properties you're using.
As a positive analogy, Giraud's theorem is a case where the dual implication of the one you want holds: the existence of a generator in a category with certain properties implies the existence of a cogenerator (the subobject classifier). I think the construction of the subobject classifier uses the full force of Giraud's theorem: you have to know that your category embeds into the category of presheaves on the generator.
But I suppose this analogy is probably no more enlightening than the usual construction of injectives in abelian categories of sheaves.
As another positive analogy, the Freyd-Mitchell embedding theorem is another place with a weird interplay between generators and cogenerators. To embed a small abelian category $\mathcal{A}$ exactly into a category of modules (which has a nice generator), you first embed it exactly (via Yoneda) into the category of pro-objects $\mathrm{Pro}(\mathcal{A})$, i.e. the opposite of the category of left-exact functors $\mathcal{A} \to \mathsf{Ab}$ (a category with a reasonable cogenerator given by $\mathcal{A}$ itself), and do some further work I'm not familiar with to embed nice subcategories of $\mathrm{Pro}(\mathcal{A})$ into categories with nice generators. | 200,575 |
Cardiff City vs Stoke City live score and live online video stream will start at Cardiff City Stadium in Cardiff City, Wales on March 16, 2021 at 19:00 UTC. Here on SofaScore livescore you can sort all previous Cardiff City vs Stoke City results by head-to-head comparison. Since the video has appeared on video hosting sites like YouTube and Dailymotion, the video highlights links of Cardiff City vs. Stoke City have been collected in the Media tab for the most popular matches. We are not responsible for the content of the video.
Cardiff City vs Stoke City live streams are available online for registered members of U-TV, the leading online betting agency with streaming coverage of over 140,000 live sporting events each year. If this match is covered by U-TV live streaming, you can watch Cardiff City Stoke City: Cardiff City v Stoke City
Date: March 16, 2021
Time: 19:00 UTC
Venue: Cardiff City Stadium, Cardiff, Wales
The Cardiff City vs Stoke City Prediction
In our view, this winning Stoke City team have to work hard as a team to play Cardiff City.
Hence, we expect Cardiff City to win 1-0 at the end of the 90th minute
The Cardiff City vs Stoke City Team News
Cardiff city
Cardiff City lighting technician Mick McCarthy’s only fitness concern is fighting groups in a completely different way. Salvamba (cancer) is out of action.
Cardiff City will join the match following a 0-0 draw against Huddersfield Town.
In that match, Cardiff City scored eight goals and three shots with 42% ball dominance. Huddersfield Town made three attempts aimed at nine.
Their latest results shed light on the fact that the Cardiff City defense can do nothing else. When the last six fights were two, the Bluebird took a step back to see the sum of the goals flying behind the net.
Stoke City
Initially within the full team, Stoke City lighting technician Michael O’Neill only has a fitness problem. Tires Campbell (knee injury) does not play here.
Stoke City will enter the match after losing their last game to Wycombe Wanderers and winning the championship 2-0.
In that match, Stoke City owned 65%, tried 17 goals and made 6 shots. Stoke City goals were scored by Rice Norrington-Davis (4 minutes) and Harry Souther (9 minutes). Meanwhile, the Wycombe Wanderers have lost 13 hits and 2 hits.
In their last six games, Michael O’Neill’s Stoke City have played eight games, averaging 1.33 points per game. | 45,630 |
\begin{document}
\singlespacing
\begin{center}
{\Large \bf Beta-Function Identities via Probabilistic Approach}\\
\end{center}
\vtwo
\begin{center}
{\large \bf P. Vellaisamy\textsuperscript{1} and A. Zeleke \textsuperscript{2}} \\
\textsuperscript{1}Department of Mathematics, Indian Institute of Technology Bombay \\
\noindent Powai, Mumbai-400076, India.\\
{Email: \it [email protected]}
\textsuperscript{2}Lyman Briggs College \& Department of Statistics \& Probability, \\
\noindent Michigan State University. East Lansing, MI 48825, USA\\
{Email: \it [email protected]}
\end{center}
\abstract {Using a probabilistic approach, we derive several interesting identities involving beta functions. Our results generalize certain well-known combinatorial identities involving binomial coefficients and gamma functions.}
\vone
{\noindent \bf Keywords}. {Binomial inversion, combinatorial identities, gamma random variables, moments, probabilistic proofs.}
\vone
\noindent MSC2010 {\it Subject Classification}: Primary: 62E15, 05A19; Secondary: 60C05.\\
\section{Introduction}
There are several interesting combinatorial identities involving binomial coefficients, gamma functions, hypergeometric functions (see, for example, Riordan (1968), Petkovsek et al (1996), Bagdasaryan (2015), Srinivasan (2007) and Vellaisamy and Zeleke (2017)), {\it etc}. One of these is the famous identity that involves the convolution of the central binomial coefficients: \begin{equation} \label{eqn1.1}
\sum_{k=1}^{n} \binom{2k}{k}\binom{2n-2k}{n-k}=4^n.
\end{equation}
In recent years, researchers have provided several proofs of $(1.1)$.
A proof that uses generating functions can be found in Stanley (1997). The combinatorial proofs can also be found, for example, in Sved (1984), De Angelis (2006) and Miki{\' c} (2016). A computer generated proof using the WZ method is given by Petkovsek, Wilf and Zeilberger(1996). Chang and Xu (2011) extended the identity in \eqref{eqn1.1}
and presented a probabilistic proof of the identity
\begin{equation}\label{eqn1.2}
\sum_{\substack{k_j \geq 0,\; 1 \leq j \leq m;\\ \sum_{J=1}^m k_j=n}} \binom{2k_1}{k_1}\binom{2k_2}{k_2}\cdots\binom{2k_m}{k_m}=\frac{4^n}{n!} \frac{\Gamma(n+\frac{m}{2})}{\Gamma(\frac{m}{2})},
\end{equation}
for positive integers $m$ and $n$, and Miki{\' c} (2016) presented a combinatorial proof of this identity based on the method of recurrence relations and telescoping.
\noindent A related identity for the alternating convolution of central binomial coefficients is
\begin{equation}\label{eqn1.3}
\sum_{k=0}^{n} (-1)^k \binom{2k}{k} \binom{2n-2k}{n-k} =
\begin{cases*}
2^n \binom{n}{\frac{n}{2}}, & if $n$ is even \\
\phantom{0 }0, & if $n$ is odd.
\end{cases*}
\end{equation}
\noindent The combinatorial proofs of the above identity can be found in, for example, Nagy (2012), Spivey (2014) and Miki{\' c} (2016).
Recently, Pathak (2017) has given a probabilistic of the above identity.
\vspace*{.3cm}
\noindent Unfortunately, in the literature, there are only a few identities that involve beta functions are available. Our goal in this paper is to establish several interesting beta-function identities,
similar to the ones given in \eqref{eqn1.1} and \eqref{eqn1.3}. Interestingly, our results generalize all the above-mentioned identities, including the main result \eqref{eqn1.2} of Chang and Xu (2011). Our approach is based on probabilistic arguments, using the moments of the sum or the difference of tow gamma random variables.
\section{Identities Involving the Beta Functions}
The beta function, also known as the Euler first integral, is defined by
\begin{equation} \label{eqn2.1}
B(x,y)=\int_0^1 t^{x-1}(1-t)^{y-1}dt,\; x, y>0.
\end{equation}
It was studied by Euler and Legendre and is related to the gamma functions by
\begin{equation}\label{eqn2.2}
B(x.y)=\frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)},
\end{equation}
where $\Gamma(x)=\int_0^\infty t^{x-1}e^{-t} dt, x>0.$ The beta function is symmetric, {\it i.e.} $B(x,y)=B(y,x)$ and satisfies the basic identity
\begin{equation} \label{eqn2.3}
B(x,y)=B(x,y+1)+B(x+1,y),~ \text{for}~x, y>0.
\end{equation}
\noindent Using a probabilistic approach, we generalize, in some sense, the above basic identity in \eqref{eqn2.3} in two different directions.
\noindent Let $ X$ be a gamma random variable with parameter $p >0$, denoted by $ X \sim G(p)$, and density $$f(x|p)=\frac{1}{\Gamma(p)}e^{- x}x^{p-1},\;x>0,\; p >0.$$ Then, it follows (see Rohatgi and Saleh (2002), p.~212) that
\begin{eqnarray} \label{eqn2.4}
E(X^n)=\frac{1}{\Gamma(p)} \int_0^\infty e^{- x}x^{p+n-1}dx=\frac{\Gamma(p+n)}{\Gamma(p)}.
\end{eqnarray}
Let $X_1 \sim G(p_1)$ and $X_2 \sim G(p_2)$ be two independent gamma distributed random variables, with parameters $p_1$ and $p_2$, respectively. Then it is known that $Y = X_1+X_2$ follows a gamma distribution with parameters $(p_1+p_2)$, {\it i.e}, $Y \sim G(p_1+p_2)$.
We compute the $n$-th moment $E(Y^n)$ in two different ways, and equating them gives us an identity involving beta functions.
\begin{theorem} Let $p_1,\;p_2 > 0$. Then for any integer $n \geq 0$,
\begin{equation}\label{eqn2.5}
\displaystyle \sum_{k=0}^n \binom{n}{k}B(p_1+k,p_2+n-k)=B(p_1,p_2).
\end{equation}
\end{theorem}
\begin{proof}
\noindent Since $ X_1+X_2=Y \sim G(p_1+p_2)$, we get,
from \eqref{eqn2.4},
\begin{equation} \label{eqn2.6}
E(Y^n)=\frac{\Gamma(p_1+p_2+n)}{\Gamma(p_1+p_2)}.
\end{equation}
\noindent Alternatively, since $X_1 \sim G(p_1),\; X_2 \sim G(p_2)$ and are independent, we have by using the binomial theorem
\begin{eqnarray}
E(Y^n)&=&E(X_1+X_2)^n=E\Big(\sum_{k=0}^n \binom{n}{k} X_1^kX_2^{n-k}\Big)\nonumber\\&=& \sum_{k=0}^n \binom{n}{k} E(X_1^k) E(X_2^{n-k})\nonumber\\
&=& \sum_{k=0}^n \binom{n}{k}\frac{\Gamma(p_1+k) \Gamma(p_2+n-k)}{\Gamma(p_1)\Gamma(p_2)} \label{eqn2.7},
\end{eqnarray}
using \eqref{eqn2.4}.
\noindent Equating \eqref{eqn2.6} and \eqref{eqn2.7} , we obtain
$$\frac{\Gamma(p_1+p_2+n)}{\Gamma(p_1+p_2)}=\sum_{k=0}^n \binom{n}{k}\frac{\Gamma(p_1+k) \Gamma(p_2+n-k)}{\Gamma(p_1)\Gamma(p_2)}$$
which leads to
$$\sum_{k=0}^n \binom{n}{k}\frac{\Gamma(p_1+k) \Gamma(p_2+n-k)}{\Gamma(p_1+p_2+n)}=\frac{\Gamma(p_1)\Gamma(p_2)}{\Gamma(p_1+p_2)}.$$
This proves the result.
\end{proof}
\begin{remarks} {\em
\begin{enumerate}
\item[(i)] Indeed, one could also consider $X_1 \sim G(\alpha, p_1)$ and $X_2 \sim G(\alpha, p_2)$, $\alpha >0$, the two-parameter gamma random variables. But, the result does not change as as the powers of $\alpha$ cancel out.
\item[(ii)] When $p_1=p_2=\frac{1}{2}$, we get, $$\sum_{k=0}^n \binom{n}{k}B\Big(\frac{1}{2}+k, \frac{1}{2}+n-k\Big) =\pi,\;\mbox{for all}\; n \geq 0, $$ which is an interesting representation for $\pi$.
Also, for this case, it is shown later (see Remark 3.2) that \eqref{eqn2.5} reduces to \eqref{eqn1.1}.
\item[(iii)] When $p_1=x$, $p_2=y$ and $n=1$, we get,
\begin{equation*}
B(x,y)=B(x,y+1)+B(x+1,y),
\end{equation*}
the basic identity in \eqref{eqn2.3}.
Thus, Theorem 2.1 extends both the identities in \eqref{eqn1.1} and in \eqref{eqn2.3}.
\end{enumerate}
}
\end{remarks}
\noindent Our next result is an interesting identity that relates binomial coefficients and beta functions on one side and to a simple expression on the other side. The proof relies on the binomial inversion formula (see, for example, Aigner (2007)) which we include here for ease of reference. \\
\noindent {\bf The Binomial Inversion Formula.} For a positive sequence of $\{a_n\}_{n \geq 0 }$ of real numbers, define the real sequence $\{b_n\}_{n \geq 0 }$ by $$ b_n=\sum_{j=0}^{n}(-1)^{j}\binom{n}{j}a_j .$$ Then, for all $n \geq 0$, the binomial inversion of $b_n$ is given by $$ a_n=\sum_{j=0}^{n}(-1)^{j}\binom{n}{j}b_j .$$
\begin{theorem} Let $s > 0$ and $n \geq 0$ be an integer. Then,
\begin{eqnarray}\label{eqn2.8}
\sum_{j=0}^{n}(-1)^{j}\binom{n}{j}B(j+1,s)= \frac{1}{s+n}.
\end{eqnarray}
\end{theorem}
\begin{proof}
\noindent The following binomial identity is known:
\begin{eqnarray}\label{eqn2.9}
\sum_{j=0}^{n}(-1)^{j}\binom{n}{j}\left(\frac{s}{s+j}\right)=\prod_{j=1}^{n} \left(\frac{j}{s+j}\right).
\end{eqnarray}
Recently, Peterson (2013) and Vellaisamy (2015) gave a probabilistic proof of the above identity. Note that the right hand side of \eqref{eqn2.9} can also be written as
\begin{eqnarray*}
\prod_{j=1}^{n} \left(\frac{j}{s+j}\right)
= \frac{\Gamma(n+1)\Gamma(s+1)}{\Gamma(s+n+1)}= \frac{\Gamma(n+1)\; s\Gamma(s)}{\Gamma(s+n+1)}=s B(n+1,s).
\end{eqnarray*}
Then, the identity in \eqref{eqn2.8} becomes
\begin{eqnarray}\label{eqn2.10}
\sum_{j=0}^{n}(-1)^{j}\binom{n}{j}\left(\frac{1}{s+j}\right)= B(n+1,s).
\end{eqnarray}
Applying the binomial inversion formula to \eqref{eqn2.10} with $a_j=\frac{1}{s+j}$ and $b_n=B(n+1,s)$, we get $$\sum_{j=0}^{n}(-1)^{j}\binom{n}{j} B(j+1,s)= \frac{1}
{s+n},$$ which proves the result.
\end{proof}
\begin{remark} {\em When $n=1$, equation \eqref{eqn2.8} becomes $$B(1,s)-B(2,s)=\frac{1}{s+1}=B(1,s+1),$$ which coincides with the basic beta-function identity in \eqref{eqn2.3}, when $x=1$. Thus, \eqref{eqn2.8} can be viewed as another generalization of \eqref{eqn2.3} in the case when $x$ is a positive integer. Also, when $n=2$, $$B(1,s)-2 B(2,s)+B(3,s)=\frac{1}{s+2},$$ which can be verified using the basic identity in \eqref{eqn2.3}. It may be of interest to provide a different proof of Theorem 2.2 based on induction or combinatorial arguments. }
\end{remark}
\noindent It is easy to see that the derivative of the beta function is
\begin{equation} \label{eqn2.11}
\frac{\partial}{\partial y} B(x,y)=B(x,y) \Big( \psi(y)-\psi(x+y)\Big),
\end{equation}
where $\psi(x)=\frac{\Gamma'(x)}{\Gamma(x)}$ is the digamma function. The following result is a consequence of Theorem 2.2.
\begin{theorem} For $s > 0$ and an integer $n \geq 0$,
\begin{eqnarray}\label{eqn2.12}
\sum_{j=0}^{n}\sum_{i=0}^j (-1)^{j}\binom{n}{j} \frac{B(j+1,s)}{s+i}= \frac{1}{(s+n)^2}.
\end{eqnarray}
\end{theorem}
\begin{proof}The proof proceeds by taking the derivative of both sides of $(2.7)$ with respect to $s$. From the right-hand side, we get,
\begin{equation}\label{eqn2.13}
\frac{\partial}{\partial s} \Big(\frac{1}{s+n}\Big)= \frac{-1}{(s+n)^2}.
\end{equation}
Also, form the left-hand side,
\begin{align}
\frac{\partial}{\partial s}\sum_{j=0}^{n}(-1)^{j}\binom{n}{j} B(j+1,s) =& \sum_{j=0}^{n}(-1)^{j}\binom{n}{j}\frac{\partial}{\partial s} B(j+1,s)\nonumber\\=& \sum_{j=0}^{n}(-1)^{j}\binom{n}{j} B(j+1,s) \Big( \psi(s)-\psi(j+1+s)\Big), \label{eqn2.14}
\end{align}
using \eqref{eqn2.11}. Further, it is known that the digamma function $ \psi(x) $satisfies
\begin{equation} \label{eqn2.15}
\psi(x+1)-\psi(x)=\frac{1}{x}.
\end{equation}
Using \eqref{eqn2.15} iteratively leads to
\begin{equation} \label{eqn2.16}
\psi(x+j+1)-\psi(x)=\displaystyle \sum_{i=0}^j \frac{1}{x+i},
\end{equation}
for a nonnegative integer $j$.
\noindent Using \eqref{eqn2.13} \eqref{eqn2.14} and \eqref{eqn2.16}, we get
\begin{eqnarray*}
\frac{-1}{(s+n)^2}&=&\sum_{j=0}^{n}(-1)^{j}\binom{n}{j} B(j+1,s) \Big( \psi(s)-\psi(j+1+s)\Big)\\&=&\sum_{j=0}^{n} \sum_{i=0}^j (-1)^{j}\binom{n}{j} B(j+1,s) (-1) \frac{1}{s+i},
\end{eqnarray*}
which proves the result.
\end{proof}
Our aim next is to extend the identity in \eqref{eqn1.3}.
\begin{theorem}
Let $p>0$ and and $n$ be a positive integer. Then
\begin{equation}\label{eqn2.17}
\sum_{k=0}^{n} (-1)^k \binom{n}{k} B( p+k, p+n-k)=
\begin{cases*}
\displaystyle{ \frac {n! \Gamma(p) \Gamma(p+\frac{n}{2})} {\Gamma(\frac{n}{2}+1) \Gamma(2p+n)}}, & if $n$ is even \\
\phantom{0}0, & if $n$ is odd.
\end{cases*}
\end{equation}
\end{theorem}
\begin{proof}To prove the result, we consider the rv $X= X_1-X_2$, where $X_1$ and $X_2$ are as before independent gamma $G(p)$ variables.
Note first that since $X_1$ and $X_2$ are independent and identically distributed, we have $X_1-X_2 \stackrel{d}{=} X_2-X_1$. Here $\stackrel{d}{=}$ denotes the equality in distributions. That is, $X$ and $-X$ have the same distributions on $\mathbb{R}$, which implies the
density of $X$ is symmetric about zero. Hence, $E(X^n)=0$ if $n$ is an odd integer.
\noindent Next, we compute the even moments of $X$. The earlier approach of finding the moments of $X$ using the probability density function is tedious. This is because the density of $X$ is very complicated and it involves Whittaker's W-function (see Mathai (1993)). Therefore, we use the moment generating function ($MGF$) approach to find the moments of $X$.
\noindent It is known (see Rohatgi and Saleh (2002, p.~212) that the $MGF$ of $X_1$ is $M_{X_1}(t)= E(e^{tX_1})= (1-t)^{-p}.$ Hence, the $MGF$ of $X$ is
\begin{align}
M_{X}(t) =& M_{X_1-X_2}(t)=M_{X_1}(t)M_{X_2}(-t) \nonumber \\
=& (1-t)^{-p} (1+t)^{-p} \nonumber \\
=& (1-t^2)^{-p}
\end{align}
which exits for $|t|<1.$
\noindent Using the result, for $p>0$ and |q|<1, that
\begin{equation*}
(1-q)^{-p}= \sum_{n=0}^{\infty} \frac{\Gamma(n+p) q^n}{\Gamma(n+1) \Gamma(p)},
\end{equation*}
we have
\begin{equation} \label{eqn2.19}
M_{X}(t)= (1-t^2)^{-p}= \sum_{n=0}^{\infty} \frac{\Gamma(n+p) t^{2n}}{\Gamma(n+1) \Gamma(p)}.
\end{equation}
Hence, for $n \geq 1$, we have from \eqref{eqn2.19}
\begin{equation} \label{eqn2.20}
E(X^{2n})=M_{X}^{(2n)}(t)|_{t=0}= \frac{\Gamma(n+p) {(2n)!}}{\Gamma(n+1) \Gamma(p)},
\end{equation}
where $f^{(k)}$ denotes the $k$-th derivative of $f$. Thus,
we have shown that
\begin{equation} \label{eqn2.21}
E(X^{n})=
\begin{cases*}
\displaystyle \frac{n! \Gamma(\frac{n}{2}+p) }{\Gamma(\frac{n}{2}+1) \Gamma(p)}, & if $n$ is even\\
\phantom{0}0, & if $n$ is odd.
\end{cases*}
\end{equation}
Next we compute the moments of $X$, using the binomial theorem. Note that
\begin{align}\label{eqn2.22}
E(X^n) =& E(X_1-X_2)^n= \sum_{k=0}^{n} (-1)^k \binom{n}{k} E(X_1^k) E(X_2^{n-k}) \nonumber \\
=& \sum_{k=0}^{n} (-1)^k \binom{n}{k} \frac{\Gamma(p+k)}{\Gamma(p)}\frac{\Gamma(p+n-k)}{\Gamma(p)}. \end{align}
Equating \eqref{eqn2.21} and \eqref{eqn2.22}, we get
\begin{align}\label{eqn2.23}
\sum_{k=0}^{n} (-1)^k \binom{n}{k} {\Gamma(p+k)}{\Gamma(p+n-k)}=
\begin{cases*}
\displaystyle \frac{n!\Gamma(\frac{n}{2}+p) \Gamma(p)}{\Gamma(\frac{n}{2}+1)}, & if $n$ is even \\
\phantom{0}0, & if $n$ is odd.
\end{cases*}
\end{align}
which is an interesting identity involving gamma functions and
binomial coefficients.
Dividing now both sides of \eqref{eqn2.23} by $\Gamma(2p+n)$, the result follows.
\end{proof}
\noindent We next show that the identitiy in \eqref{eqn1.2} follows as a special case.
\begin{corollary} Let $n$ be a positive integer. Then
\begin{equation}\label{eqn2.24}
\sum_{k=0}^{n} (-1)^k \binom{2k}{k} \binom{2n-2k}{n-k} =
\begin{cases*}
2^n \binom{n}{\frac{n}{2}}, & if $n$ is even \\
\phantom{0 }0, & if $n$ is odd.
\end{cases*}
\end{equation}
\end{corollary}
\begin{proof} Let $p=\frac{1}{2}$ in \eqref{eqn2.23} and it suffices to consider the case $n$ is even. Then
\begin{align*}
\sum_{k=0}^{n} (-1)^k \binom{n}{k} {\Gamma(k+\frac{1}{2})}{\Gamma(n-k+\frac{1}{2})}=
\displaystyle \frac{{n!}\Gamma(\frac{n}{2}+\frac{1}{2}) \Gamma(\frac{1}{2})}{\Gamma(\frac{n}{2}+1)}
\end{align*}
That is,
\begin{align} \label{eqn2.25}
\sum_{k=0}^{n} (-1)^k \binom{n}{k} \frac{\Gamma(k+\frac{1}{2})}{\Gamma(\frac{1}{2})}\frac{\Gamma(n-k+\frac{1}{2})}{\Gamma(\frac{1}{2})} =
\displaystyle \frac{{n!}\Gamma(\frac{n}{2}+\frac{1}{2}) }{{\Gamma(\frac{1}{2})}\Gamma(\frac{n}{2}+1)}.
\end{align}
Note that,
\begin{eqnarray}
\frac{\Gamma(n+\frac{1}{2})}{\Gamma(\frac{1}{2})}&=&\frac{\Big(n-\frac{1}{2}\Big) \Big(n-\frac{3}{2}\Big) \cdots \Big(\frac{3}{2}\Big) \Big(\frac{1}{2}\Big) \Gamma \Big(\frac{1}{2} \Big)}{\Gamma \Big(\frac{1}{2} \Big)}\nonumber\\&=&
\frac{(2n-1) \cdot (2n-3) \cdots 3 \cdot 1}{2^n} \nonumber \\ &=& \frac{(2n)!}{ n! 4^n}. \label{eqn2.26}
\end{eqnarray}
\noindent Using \eqref{eqn2.26} in \eqref{eqn2.25}, we get
\begin{align*}
\displaystyle \sum_{k=0}^{n} (-1)^k \frac{n!}{k! (n-k)!}\frac{ (2k)!}{4^k k!} \frac{(2n-2k)!}{4^{(n-k)} (n-k)! } =& \displaystyle
\frac{n! n!}{4^{\frac{n}{2}} (\frac{n}{2})!(\frac{n}{2})!}
\end{align*}
That is,
\begin{equation}\label{eqn2.27}
\sum_{k=0}^{n} (-1)^k \binom{2k}{k} \binom{2n-2k}{n-k} =
\displaystyle \frac{n! 4^n }{(\frac{n}{2})!(\frac{n}{2})! 4^{\frac{n}{2}}} \nonumber \\
= \displaystyle 2^n \binom{n}{\frac{n}{2}},
\end{equation}
which proves the result.
\end{proof}
\section{An Extension}
In this section, we discuss an extension of the beta-function identity given in Theorem 2.1. This result in particular extends the main result of Chang and Xu (2011). Let $p_i > 0,\;1 \leq i \leq m,$ and consider the beta function of $m$-variables defined by
\begin{equation}
B(p_1,\cdots,p_m)= \int_\T x_1^{p_1-1}x_2^{p_2-1} \cdots x_m^{p_m-1} (1-p_1-p_2-\cdots p_m)^{p_m-1}dx_1 \cdots dx_m,
\end{equation}
where $\T = (0,1) \times \cdots \times (0,1)$. It is well known that $B(p_1,\cdots,p_m)$ can be expressed as a ratio of gamma functions as
\begin{equation}
B(p_1,\cdots,p_m)= \frac{\Gamma(p_1)\Gamma(p_2)\cdots\Gamma(p_m)}{\Gamma(p_1+\cdots+p_m)}.
\end{equation}
\begin{theorem} Let $p_1, \cdots, p_m \geq 0.$ Then for any non-negative integer $n$,
\begin{equation}\label{eqn3.3}
\sum_{\substack{k_j\geq 0;~ 1 \leq j \leq n; \\ \sum_{j=1}^m k_j=n}} \binom {n} {k_1,\cdots, k_m} B(p_1+k_1,\cdots,p_m+k_m)=B(p_1,\cdots,p_m)
\end{equation}
where $\binom{n}{k_1,\cdots,k_m}= \frac{n!}{k_1! \cdots k_m!}$ denotes the multinomial coefficient.
\end{theorem}
\begin{proof}Let $X_1,\cdots X_m$ be $m$ independent gamma random variables, where $X_i \sim G(p_i),$
$1 \leq i \leq m$. Then it is known that
$$Y= \sum_{i=1}^m X_i \sim G(p_1+\cdots+p_m).$$
Also, from \eqref{eqn2.4},
\begin{equation} \label{eqn3.4}
E(Y^n)=\frac{\Gamma(p_1 + p_2+ \cdots p_m+ n)}{\Gamma(p_1+\cdots+p_m)}.
\end{equation}
Since $X_i$'s are independent, we have by multinomial theorem,
\begin{eqnarray}
E(X_1+\cdots+X_m)^n&=& E \Bigg[\sum_{\substack{k_j \geq 0,\;1 \leq j \leq m\\ \sum_{J=1}^m k_j=n}} \binom {n} {k_1,\cdots, k_m} X_1^{k_1}\cdots X_m^{k_m}\Bigg] \nonumber
\\&=& \sum_{\substack{k_j \geq 0, \;1 \leq j \leq m\\ \sum_{J=1}^m k_j=n}}\binom {n} {k_1,\cdots, k_m} E(X_1^{k_1}) \cdots E(X_m^{k_m})\nonumber\\
&=& \sum_{\substack{x_j \geq 0,\;1 \leq j \leq m\\ \sum_{J=1}^m k_j=n}} \binom {n} {k_1,\cdots, k_m}
\frac{\Gamma(p_1+k_1)\cdots \Gamma(p_m+k_m)}{\Gamma (p_1)\cdots \Gamma(p_m)}. \label{eqn3.5}
\end{eqnarray}
Equating \eqref{eqn3.4} and \eqref{eqn3.5}, we obtain
$$ \sum_{\substack{k_j \geq 0,\; 1 \leq j \leq m\\ \sum_{J=1}^m k_j=n}}\binom {n} {k_1,\cdots, k_m}
\frac{\Gamma(p_1+k_1)\cdots \Gamma(p_m+k_m)}{\Gamma (p_1)\cdots \Gamma(p_m)}=\frac{\Gamma(p_1 + p_2+ \cdots p_m+ n)}{\Gamma(p_1+\cdots+p_m)}.$$
That is,
$$ \sum_{\substack{k_j \geq 0,\; 1 \leq j \leq m\\ \sum_{J=1}^m k_j=n}}\binom {n} {k_1,\cdots, k_m}
\frac{\Gamma(p_1+k_1)\cdots \Gamma(p_m+k_m)}{\Gamma(p_1 + p_2+ \cdots p_m+ n)}=\frac{\Gamma (p_1)\cdots \Gamma(p_m)}{\Gamma(p_1+\cdots+p_m)},$$
from which the result follows.
\end{proof}
\begin{remark} Obviously, when $m=2,$ the identity in \eqref{eqn3.3} reduces to
\begin{equation*}
\displaystyle \sum_{\substack{k_j\geq 0;~ 1 \leq j \leq 2; \\ k_1+k_2=n}} \binom {n} {k_1, k_2} B(p_1+k_1, p_2+k_2)= \sum_{k=0}^n \binom{n}{k}B(p_1+k,p_2+n-k)=B(p_1,p_2),
\end{equation*}
which is \eqref{eqn2.5}.
\end{remark}
\noindent Our next result shows that the identity in \eqref{eqn1.2} follows as a special case.
\begin{corollary}
When $p_1=p_2=\cdots=p_m=\frac{1}{2}$, the identity in \eqref{eqn3.3} reduces to
\begin{equation}\label{eqn3.6}
\sum_{\substack{k_j \geq 0,\; 1 \leq j \leq m\\ \sum_{J=1}^m k_j=n}} \binom{2k_1}{k_1}\cdots\binom{2k_m}{k_m}=\frac{4^n}{n!} \frac{\Gamma(n+\frac{m}{2})}{\Gamma(\frac{m}{2})},
\end{equation}
for all integers $m,n \geq 1$.
\end{corollary}
\begin{proof}
Putting $p_1=p_2=\cdots p_m=\frac{1}{2}$ in \eqref{eqn3.3}, we obtain,
$$\sum_{\substack{k_j \geq 0,\; 1 \leq j \leq m\\ \sum k_j=n}} \binom {n} {k_1,\cdots, k_m} B\Big(\frac{1}{2}+k_1,\cdots, \frac{1}{2}+k_m\Big)=B\Big(\frac{1}{2},\cdots, \frac{1}{2}\Big)$$
This implies,
\begin{eqnarray*}
\sum_{\substack{k_j \geq 0,\; 1 \leq j \leq m\\ \sum k_j=n}} \binom {n} {k_1,\cdots, k_m} \frac{\Gamma(\Big(\frac{1}{2}+k_1\Big),\cdots, \Gamma \Big(\frac{1}{2}+k_m\Big)}{\Gamma \Big(n+\frac{m}{2}\Big)}&=&\frac{\Gamma \Big(\frac{1}{2}\Big) \cdots \Gamma \Big(\frac{1}{2}\Big)}{\Gamma \Big(\frac{m}{2}\Big)},
\end{eqnarray*}
or, equivalently,
\begin{eqnarray*}
\sum_{\substack{k_j \geq 0,\; 1 \leq j \leq m\\ \sum k_j=n}} \binom {n} {k_1,\cdots, k_m} \frac{\Gamma(\Big(\frac{1}{2}+k_1\Big),\cdots, \Gamma \Big(\frac{1}{2}+k_m\Big)}{\Gamma \Big(\frac{1}{2}\Big) \cdots \Gamma \Big(\frac{1}{2}\Big)}=\frac{\Gamma \Big(n+\frac{m}{2}\Big)}{\Gamma \Big(\frac{m}{2}\Big)}.
\end{eqnarray*}
Using \eqref{eqn2.26}, we get
\begin{eqnarray} \label{eqn3.7}
\sum_{\substack{k_j \geq 0,\; 1 \leq j \leq m\\ \sum k_j=n}} \binom {n} {k_1,\cdots, k_m} \frac{(2k_1)!,\cdots, (2k_m)!}{4^{k_1}(k_1)! \cdots 4^{k_m}(k_m)!}
=\frac{\Gamma \Big(n+\frac{m}{2}\Big)}{\Gamma \Big(\frac{m}{2}\Big)}.
\end{eqnarray}
We can write \eqref{eqn3.7} as
\begin{eqnarray}
\sum_{\substack{k_j \geq 0,\; 1 \leq j \leq m\\ \sum k_j=n}} \binom {2k_1} {k_1} \binom {2k_2} {k_2} \cdots \binom {2k_m} {k_m}=\frac{4^n}{n!}\Bigg(\frac{\Gamma \Big(n+\frac{m}{2}\Big)}{\Gamma \Big(\frac{m}{2}\Big)}\Bigg),
\end{eqnarray}
which proves the corollary.
\end{proof}
\begin{remark} \label{rem3.2} Note that when $m=2$, \eqref{eqn3.6} reduces tom \eqref{eqn1.1}. This implies also that when $p=\frac{1}{2}$, \eqref{eqn2.5} reduces to \eqref{eqn1.1}.
\end{remark}
\begin{remark} {\em
\begin{enumerate}
\item[(i)] Let $m$ be even so that $m=2l$ for some positive integer $l$. Then the right hand side of \eqref{eqn3.6} is
$$ \frac{4^n}{n!}\frac{\Gamma (n+l)}{\Gamma (l)}=4^n \binom{n+l-1}{n}=4^n \binom{n+\frac{m}{2}-1}{n}.$$
Similarly, when $m$ is odd, say $m=2l+1$,
\begin{eqnarray*}
\frac{4^n}{n!}\frac{\Gamma (n+\frac{m}{2})}{\Gamma (\frac{m}{2})}&=& \frac{4^n}{n!}\frac{\Gamma (n+l+\frac{1}{2})}{\Gamma (l+\frac{1}{2})}=\frac{4^n}{n!}\Bigg(\frac{\Gamma (n+l+\frac{1}{2})}{\Gamma(\frac{1}{2})}\Bigg)\Bigg(\frac{\Gamma(\frac{1}{2})}{\Gamma (l+\frac{1}{2})}\Bigg)\\
&=& \binom{2n+2l}{2n}\Bigg(\frac{(2 n)!}{n!\; n!}\Bigg) \Bigg(\frac{l!\; n!}{(n+ l)!}\Bigg)\; (\mbox{using}\; \eqref{eqn2.6})\\
&=&\frac{\binom{2n+2l}{2n} \binom{2n}{n}}{\binom{n+l}{n}}\\
&=&\frac{\binom{2n+m-1}{2n} \binom{2n}{n}}{\binom{n+\frac{m-1}{2}}{n}},
\end{eqnarray*}
since $2l={m-1}$. Thus, we have from \eqref{eqn3.6},
\begin{equation} \label{eqn3.9}
\sum_{\substack{k_j \geq 0,\; 1 \leq j \leq m\\ \sum k_j=n}} \binom {2k_1} {k_1} \binom {2k_1} {k_1} \binom {2k_2} {k_2} \cdots \binom {2k_m} {k_m}= \left\{
\begin{array}{lr} \displaystyle
4^n \binom{n+\frac{m}{2}-1}{n}, & \mbox{if}\; m\;\mbox{is even}\\
& \\
\displaystyle \frac{\binom{2n+m-1}{2n} \binom{2n}{n}}{\binom{n+\frac{m-1}{2}}{n}}, & \mbox{if}\; m\;\mbox{is odd},
\end{array}
\right.
\end{equation}
which is equation $(3)$ of Miki{\' c} (2016). Indeed, Miki{\' c} (2016)
provided a combinatorial proof of the above result based on recurrence relations.
\end{enumerate}
}
\end{remark}
\noindent {\bf Acknowledgments}. This work was completed while the first author was visiting the Department of Statistics and Probability, Michigan State University, during Summer-2017. We are grateful to Professor Frederi Viens for his support and encouragements.
\vone
\noindent{\bf{\Large References}}
\noindent Aigner, M. (2007). {\it A Course in Enumeration}. Berlin, Springer-Verlag.
\noindent Bagdasaryan, A. (2015). A combinatorial identity involving gamma function and Pochhammer symbol {\it Appl. Math. Comput.}, {\bf 256}, 125-130.
\noindent Chang, G. and Xu, C. (2011). Generalization and probabilistic proof of a combinatorial identity. {\it Amer. Math. Monthly}, {\bf 118}, 175-177.
\noindent De Angelis, V. (2006). Pairings and signed permutations. {\it Amer. Math. Monthly}, {\bf 113},
642-644.
\noindent Mathai, A.M. (1993). On non-central generalized Laplacianness of quadratic forms in
normal variables. {\it J. Multivariate Anal.}, {\bf 45}, 239-246.
\noindent Mickic{\' c}, J (2016). A proof of a famous identity concerning the convolution of the central binomial coefficients. {\it J. Integer Seq.}, {\bf 19}, Article 16.6.6.
\noindent Nagy, G. V. (2012). A combinatorial proof of Shapiro's Catalan convolution. {\it Adv. in Appl. Math.}, {\bf 49}, 391-396.
\noindent Pathak, A. K. (2017). A simple probabilistic proof for the alternating convolution of the central binomial coefficients. To appear in {\it Amer. Statist.}
\noindent Peterson, J. (2013). A probabilistic proof of a binomial identity. {\it Amer. Math. Monthly}, {\bf 120}, 558-562.
\noindent Petkovsek, M, Wilf, H and Zeilberger, D. (1996). $A=B$. {\it https://www.math.upenn.edu/wilf \\
/AeqB.html.}
\noindent Riordan, J. (1968). {\it Combinatorial Identities}. Wiley, New York.
\noindent Rohatgi, V. K. and Saleh. A. K. Md. E. (2002). {\it An Introduction to Probability and Statistics}. Wiley, New York.
\noindent Spivey, M. Z. (2014). A combinatorial proof for the alternating convolution of the central binomial
coefficients. {\it Amer. Math. Monthly}, {\bf 121}, 537-540.
\noindent Srinivasan, G. K. (2007). The gamma function: an eclectic tour. {\it Amer. Math. Monthly}, {\bf 114}, 297-316.
\noindent Stanley, R. (1997). {\it Enumerative Combinatorics}, Vol. 1, Cambridge Studies in Advanced Mathematics, {\bf 49}, Cambridge University Press.
\noindent Sved, M.(1984). Counting and recounting: the aftermath, {\it Math. Intelligencer}, {\bf 6}, 44-46.
\noindent Vellaisamy, P. and A. Zeleke, A. (2017). Exponential order statistics, the Basel problem and
combinatorial identities. Submitted.
\noindent Vellaisamy, P. (2015). On probabilistic proofs of certain binomial identities. {\it Amer. Statist.}, {\bf 69}, 241-243.
\vone
\end{document} | 20,038 |
\begin{document}
\renewcommand{\evenhead}{A Boutet de Monvel, E Khruslov and V Kotlyarov}
\renewcommand{\oddhead}{Soliton Asymptotics of Rear Part
of Non-Localized Solutions of the K-P Equation}
\thispagestyle{empty}
\FirstPageHead{9}{1}{2002}{\pageref{firstpage}--\pageref{lastpage}}{Article}
\copyrightnote{2002}{A Boutet de Monvel, E Khruslov and V Kotlyarov}
\Name{Soliton Asymptotics of Rear Part
of Non-Localized Solutions of the Kadomtsev-Petviashvili Equation}
\label{firstpage}
\Author{Anne BOUTET de MONVEL~$^\dagger$, Eugene KHRUSLOV~$^{\ddagger\ast}$,
and Vladimir KOTLYAROV~$^{\ddagger\star}$}
\Address{$^\dag$ Universit\'e Paris-7,
Physique math\'ematique et G\'eom\'etrie,
UFR de Math\'ematiques, \\
~~case 7012,
2 place Jussieu, 75251 Paris Cedex 05, France\\
~~E-mail: [email protected]\\[10pt]
$^\ddag$ Mathematical Division,
Institute for Low Temperature Physics,
47 Lenin Avenue, \\
~~310164 Kharkov, Ukraine\\
$^\ast$ E-mail: [email protected]\\
$^\star$ E-mail: [email protected]}
\Date{Received April 02, 2001; Revised November 03, 2001;
Accepted November 04, 2001}
\begin{abstract}
\noindent
We construct non-localized, real global solutions of the
Kadomtsev-Petviashvili-I
equation which vanish for $x\to-\infty$ and study their large time
asymptotic behavior. We prove that such solutions eject (for $t\to\infty$)
a train of curved asymptotic solitons which move behind the
basic wave packet.
\end{abstract}
\section{Introduction}
The study of long-time asymptotic behavior of solutions of nonlinear
evolution equations
has attracted growing attention in last years and a number of papers
have been devoted to
this problem (see review paper \cite{DIZ} and references therein).
Interest in this
problem was especially stimulated by the discovery of the inverse
scattering transform method
\cite{GGKM,ZMNP,AKNS,FT}. In particular, one remarkable result obtained by this
method was the proof that any localized (i.e.\ rapidly decreasing as
$|x|\to\infty$)
solution $u(x,t)$ of the Korteweg-de Vries (KdV) equation
\[
u_t-6uu_x+u_{xxx}=0
\]
splits into a finite number of solitons when time tends to infinity \cite{Sha}
(see also \cite{T,Schuur}). Other nonlinear evolution equations with one space
variable integrable by the inverse scattering transform also exhibit similar
splitting.
This phenomenon is an argument in favor of the physical
interpretation of solitons as stable
``long-living'' particles.
The simplest non-localized solution is of step-like form, i.e.\ the
solution with
following
asymptotic behavior:
\[
u(x,t)=
\begin{cases}
0,&x\to+\infty\\
-c^2,& x\to-\infty.
\end{cases}
\]
It was proved in \cite{Kh} that the step-like solution of the KdV
equation splits
into $\bigl[\frac{N+1}{2}\bigr]$ soliton-like objects in the
neighbourhood
\[
G^+_N(t)=\{x\in\D{R}\mid x>4c^2t-N\ln t\}
\]
of solution front
($N\in\D{Z}^+$ and $[\;\cdot\;]$ denotes integer part).
The form of these objects
is similar to ordinary soliton but their velocities depend on $t$. In
contrast with
ordinary solitons they are not exact solutions of the KdV equation,
however they satisfy it
with increasing accuracy when
$t\to\infty$. For this reason such objects are called ``asymptotic
solitons''. The number of
these asymptotic solitons increases to infinity when $t\to\infty$ if
the observation domain
in the neighbourhood of the the solution front is extended correspondently.
The same phenomenon of generation of asymptotic solitons trains on
the solution front
takes place (under certain conditions) for non-localized solutions of
more general form as
well as for other nonlinear evolution equations integrable by the
inverse scattering
transform \cite{KhK}. Roughly speaking this phenomenon can be
considered as a manifestation
of the fact that any non-localized initial data consists of an
infinite number of solitons
which are gradually ejected at the front (first is the most rapid of
them). An important
condition for the ejection is the existence of wide ``living area''
for solitons where they
can propagate without collisions. In the case where the non-localized
initial data vanish
for $x\to\infty$ this area is a positive beam $(at,\infty)$ for some
suitable $a>0$.
Another type of non-localized solutions of the KdV equation which
vanish as $x\to-\infty$
was considered in \cite{BarK}. It has been proved that under some
conditions trains of
asymptotic solitons are formed on a tail of the solution as
$t\to\infty$. These solitons
move to the right following behind the basic wave packet. Physically
it can be treated as
an ejection of the slower solitons from a non-localized initial
perturbation (initial data
$u(x,0)$). This phenomenon takes place under certain conditions on
the spectrum of the
Schr\"odinger operator whose potential is the initial data $u(x,0)$.
This condition is the
existence of a gap between the continuous spectrum of multiplicity
one, which is provided
by a non-trivial asymptotic behavior of $u(x,0)$ as $x\to\infty$, and
the continuous
spectrum of multiplicity two (positive half-axis).
In \cite{AKhK1,A,AKhK2,OPKh} similar problems were studied for the
Kadomtsev-Petviashvili equations (KP-I and KP-II):
\begin{equation}
\label{KP}
\frac{\partial}{\partial x}\Bigl(u_t+\frac{3}{2}u u_x
+ \frac{1}{4}u_{xxx}\Bigr) +\frac{3}{4}\alpha^2u_{yy}=0,
\end{equation}
where $\alpha=i$ for KP-I and $\alpha=1$ for KP-II.
These are equations with two spatial variables $x, y$ integrable by
inverse scattering
transform. See also \cite{IgABM-00,IgABM-01} for the Johnson equation
and for the modified
Kadomtsev-Petviashvili-I equation (mKP-I).
Following the Zakharov-Shabat scheme of the dressing method we look for
solutions in the form:
\begin{equation}
\label{u}
u(x,y,t)=2\frac{d}{d x}K^\pm(x,x,y,t),
\end{equation}
where the function $K^\pm(z,x,y,t)$ is a solution of the Marchenko
integral equation:
\begin{equation}
\label{Meq}
K^\pm(z,x,y,t)+F(z,x,y,t)\pm
\int_x^{\pm\infty} K^\pm(s,x,y,t)F(z,s,y,t)\,d s=0
\end{equation}
viewed as an equation in $z>x$ ($z<x$) with parameters $x,y,t$.
The kernel $F(z,x,y,t)$ in (\ref{Meq}) satisfies the system of linear
differential
equations:
\begin{equation}
\label{Feq}
\begin{cases}
F_t+F_{xxx}+F_{zzz}=0,&\\
\alpha F_y+F_{xx}-F_{zz}=0.&
\end{cases}
\end{equation}
Choosing solutions of this system in an appropriate way and solving
the integral equation
(\ref{Meq}) (with the sign ``$+$'' or ``$-$'') one can construct
solutions of the KP
equations which vanish as $x\to+\infty$ or $x\to-\infty$. For the
KP-I equation global
real solutions vanishing as $x\to+\infty$ were constructed in
\cite{A} (see also
\cite{AKhK2}). It was proved that in the neighbourhoods of their fronts:
\[
x>C(Y)t-\frac{1}{a(Y)}\log t^N,
\]
these solutions are asymptotically represented as
follows:
\begin{align} \label{usol}
u(x,y,t)&=\sum_{n=1}^N
2a^2(Y)\left(\cosh\Bigl[a(Y)(x-C(Y)t+\frac{1}{a(Y)}\log t^{n-1/2})-
\log\varphi_n(Y)\Bigr]\right)^{-1}\notag\\
&\quad
+\mathrm{O}(t^{-1/2+\varepsilon})
\end{align}
where $N$ is any natural number, $Y=y/t$, $a(Y)$, $C(Y)$ and
$\varphi_n(Y)$ are some
positive functions. This means that the trains of curved solitons are
formed in the
neighbourhood of the solution front because the ridges of these
solitons at time $t$ are
located along curves $x=C(y/t)t-a^{-1}(y/t)\log t^{n-1/2}+ \log
\varphi_n(y/t)$ ($n=1,2,\dots$). The phenomenon of formation of curved solitons
trains nearly solution fronts is observed also for the KP-II equation, however
in this case even solitons have a singularity \cite{AKhK1}.
This paper is devoted to the study of non-localized solutions of the
KP-I equation which
vanish as $x\to-\infty$. We construct a global real solution of this
type and prove that a
back part of the solution splits into curved asymptotic solitons. The
dressing method of
Zakharov-Shabat described above is not suitable for the investigation
of long-time
asymptotics of the tail of solutions. Therefore we use another method
based on an integral
equation in the plane of spectral parameters. This approach is also
suitable for the study
of long-time asymptotics of non-localized solutions in the
neighbourhood of the solution
front, which was first shown in \cite{KhS} for the KdV equation.
The paper is organized as follows. In Section 2 we prove the
existence of a global real
solution of the KP-I equation, vanishing as $x\to-\infty$. In Section
3 we reduce the
problem to a degenerated integral equation and obtain a determinant
formula for the solution.
In Section 4 we study the asymptotic behavior of the determinant
formula as $t\to\infty$
and prove an ejection of curved solitons which move behind the basic
wave packet.
\section{Construction of global solutions vanishing as $x\to-\infty$}
Let $\Omega$ be a domain in the complex plane $\D{C}$ ($k=p+iq$) with
smooth boundary
$\Gamma=\partial\Omega$ located in the right half plane at positive
distance from the
imaginary axis. Let us define the function $E(p,q):=E(p,q,x,y,t)$
on $\Omega$ as follows:
\begin{equation}
\label{E}
E(p,q)=e^{p(x-f(p,q,Y)t)},
\end{equation}
with
\begin{equation}
\label{f}
f(p,q,Y)= p^2-3q^2-2qY,
\end{equation}
where $x,y,t\in\D{R}$, and $Y=y/t$ are parameters. Let us now
consider the integral
equation (with respect to $\psi(p,q):=\psi(p,q,x,y,t)$):
\begin{equation}
\label{psieq}
\psi(p,q)+\int_\Omega\frac1{\lambda+\overline{k}}\;{E(p,q)E(\mu,\nu)}
\psi(\mu,\nu)g(\mu,\nu)\,d\mu\,d\nu=E(p,q),
\end{equation}
where $\lambda=\mu+i\nu\in\Omega$,\ $k=p+iq\in\Omega$, $\bar k=p-iq$
and $g(\mu,\nu)$ is a
smooth positive function on $\overline\Omega$. Furthermore, if
$\Omega$ is unbounded,
$g(\mu,\nu)$ satisfies
\begin{equation}
\label{g}
\int_\Omega e^{c\mu(\mu^2+\nu^2)}g(\mu,\nu)\,d\mu\,d\nu<\infty
\end{equation}
for any $c>0$.
\begin{lem} \label{lem.1}
There exists a unique solution $\psi(p,q)=\psi(p,q,x,y,t)$
of $(\ref{psieq})$ which is $C^{\infty}$ with respect to
$x,y,t\in\D{R}$.
\end{lem}
\begin{proof}
${L}^2_g(\Omega)$ denotes the Hilbert space of complex valued functions
on $\Omega$ with norm
\[
||\varphi||:=\left\{\int_\Omega|\varphi(\mu,\nu)|^2
g(\mu,\nu)d\mu\,d\nu\right\}^{\frac{1}{2}}.
\]
Let $\B{A}$ be an operator on $L^2_g(\Omega)$, depending on
parameters $x,y,t\in\D{R}$,
as follows:
\begin{equation}
\label{A}
[\B{A}\varphi](p,q)=\int_\Omega\frac{E(p,q)E(\mu,\nu)}{\lambda+\bar k}
\varphi(\mu,\nu)g(\mu,\nu)\,d\mu\,d\nu.
\end{equation}
According to (\ref{E}), (\ref{f}) and (\ref{g}), $\B{A}$ is Hilbert-Schmidt
for any values of the parameters $x,y,t$. For $\varphi\in L^2_g(\Omega)$ we
can write
\[
(\B{A}\varphi,\varphi)=\int_\Omega E(p,q)\int_\Omega
\frac{E(\mu,\nu)}{\lambda+\bar k}\varphi(\mu,\nu)g(\mu,\nu)\,d\mu\,d\nu\,
\bar\varphi(p,q)g(p,q)\,d p\,d q,
\]
where $(\,\cdot\,,\,\cdot\,)$ denotes the inner product in $L^2_g(\Omega)$ and
$\lambda=\mu+i\nu$, $\bar k=p-i q$. Using the equality
\[
\frac{1}{\lambda+\bar k}=\int^0_{-\infty}e^{(\lambda+\bar k)s}\,d s,
\]
which is true because $\Re\lambda>0$, $\Re k>0$, we obtain
\begin{equation}
\label{A>0}
(\B{A}\varphi,\varphi)=
\int^0_{-\infty}d s
\left|\int_\Omega
E(\mu,\nu)\varphi(\mu,\nu)e^{(\mu+i\nu)s} g(\mu,\nu)\,d\mu\,d\nu\right|^2
\geq 0.
\end{equation}
It follows from (\ref{A>0}) that the operator $\B{A}$ is positive. Therefore,
the homogeneous equation $\varphi+\B{A}\varphi=0$ has only the trivial solution
$\varphi\equiv0$. Since $\B{A}$ is Hilbert-Schmidt, hence compact in
$L^2_g(\Omega)$, the inhomogeneous equation
\begin{equation}
\label{psi'}
\psi+\B{A}\psi=\zeta
\end{equation}
has a unique solution in $L^2_g(\Omega)$ for any
$\zeta\in L^2_g(\Omega)$. Let us note now that the integral equation
(\ref{psieq}) has the
same form as (\ref{psi'}) with $\zeta=E(p,q)=E(p,q,x,y,t)$ belonging
to the space
$L^2_g(\Omega)$ due to (\ref{E}). Therefore, (\ref{psieq}) has a
unique solution
$\psi(p,q)$, depending on the parameters $x,y,t$. The first
derivatives $D\psi$ of this
solution with respect to $x,y,t$ also satisfy (\ref{psi'}) with right-hand side
\[
\zeta=DE(p,q)-\int_\Omega D\,\frac{E(p,q)E(\mu,\nu)}{\lambda+\bar k}
\psi(\mu,\nu)g(\mu,\nu)\,d\mu\,d\nu
\]
belonging to the space $L^2_g(\Omega)$. This proves their existence.
Existence of high order derivatives is proved by induction.
\end{proof}
\begin{cor}
The inverse operator $(\B{I}+\B{A})^{-1}$ exists and its norm in
$\C{L}(L^2_g(\Omega))$
is uniformly bounded with respect to $x,y,t\in\D{R}$:
\begin{equation}
\label{A<1}
||(\B{I}+\B{A})^{-1}||\le 1.
\end{equation}
\end{cor}
Let us now define the function
\begin{equation} \label{u2}
u(x,y,t)=-2\frac{\partial}{\partial x}\int_\Omega
E(p,q,x,y,t)\psi(p,q,x,y,t)g(p,q)\,d p\,d q,
\end{equation}
where $\psi(p,q,x,y,t)$ is the solution of the integral equation
(\ref{psieq}), and
$E(p,q,x,y,t)$ is determined by $(\ref{E})$ and $(\ref{f})$.
\begin{lem}
\label{lem.2}
The function $(\ref{u2})$ is a solution of the KP-I equation
$(\ref{KP})$.
It is a real solution, vanishing as $x\to-\infty$.
\end{lem}
\begin{proof}
Let us multiply (\ref{psieq}) by $E(p,q,z,y,t)e^{i q(z-x)}g(p,q)$ and
integrate with respect to $p,q\in\Omega$. Then, using
\[
\frac{1}{\lambda+\bar k}=\int^x_{-\infty}e^{(\lambda+\bar k)(s-x)}\,d s,
\]
together with (\ref{E}), (\ref{f}) we obtain
\begin{equation}
\label{M}
K(z,x,y,t)+\int_{-\infty}^xF(z,s,y,t)K(s,x,y,t)d s+F(z,x,y,t)=0,
\end{equation}
where
\[
K(z,x,y,t)=-\int_\Omega
E(p,q,z,y,t)e^{i q(z-x)}\psi(p,q,x,y,t)g(p,q)\,d p\,d q,
\]
\[
F(z,x,y,t)=\int_\Omega
E(p,q,z,y,t)E(p,q,x,y,t)e^{i q(z-x)}g(p,q)\,d p\,d q.
\]
Taking into account (\ref{E}), (\ref{f}), and (\ref{g}) it is easy to
show that the function $F(z,x,y,t)$ satisfies equations (\ref{Feq}).
It is also obvious
that formula (\ref{u2}) and equation (\ref{M}) (with respect to
$K(z,x,y,t)$,
$z<x$) correspond to the equations (\ref{u}), (\ref{Meq}) (with the
``$-$'' sign). Thus
according to the Zakharov-Shabat dressing method, $u$ defined by
(\ref{u2}) is a
solution of the KP-I equation. Taking into account that the domain
$\Omega$ is contained in
the right half plane at positive distance from the imaginary axis and
that the function
$g(p,q)$ satisfies inequality (\ref{g}) it is easy to prove that
solution (\ref{u2})
vanishes for $x\to-\infty$.
This solution is real. Indeed according to (\ref{u2})
it is sufficient to prove $\Im(E,\bar\psi)=0$, where $\psi$ is solution of
the integral equation
(\ref{psieq}) and $(\,\cdot\,,\,\cdot\,)$ denotes the inner product
in $L^2_g(\Omega)$.
Let us
remind that equation (\ref{psieq}) in $L^2_g(\Omega)$ takes the form
(\ref{psi'})
with $\B{A}$ a positive operator and real right hand side $\zeta=E$.
Then we obtain
\[
(E,\bar\psi)=\overline{(\zeta,\psi)}=\overline{(\psi+\B{A}\psi,\psi)}=
(\psi,\psi)+(\B{A}\psi,\psi)\geq 0.
\]
Hence $\Im(E,\bar\psi)=0$ and the lemma is proved.
\end{proof}
Thus for a given function $g(p,q)\ge0$, formulas (\ref{u2}) and
(\ref{E})-(\ref{psieq}) represent a global real solution $u(x,y,t)$ of the KP-I
equation. This solution decays as $x\to-\infty$, but its behavior as
$x\to+\infty$ is
unknown.
Our main goal is to study the asymptotic behavior of this solution as
$t\to\infty$
in suitable neighbourhoods of the rear part of the solution, namely
neighbourhoods of the
form:
\[
D_N(t)=\Bigl\{(x,y)\,\Bigm|\,-\infty<y<\infty,\ x<C(Y)t+
\frac{1}{a(Y)}\log t^N\Bigr\}.
\]
We show that in such
domains and for large $t$ ($t>T(N)$) the solution $u(x,y,t)$ has the
asymptotic behavior
described by (\ref{usol}), where
$a(Y)$, $C(Y)$, $\varphi_n(Y)$ are expressed in terms of $g(p,q)$.
This formula means that
the solution splits into a sequence of curved solitons, which are formed
in the neighbourhood
of its trailing edge. To prove this asymptotic formula we first need
to approximate the
solution of the integral equation (\ref{psieq}) by solutions of
integral equations with
appropriate degenerate kernels.
\section{Integral equation with degenerate kernel}
Let $k_0=p_0+i q_0\in\Gamma$ be an arbitrary point of the boundary of $\Omega$.
We will use the following double power series expansion:
\begin{equation} \label{cij}
\frac{1}{\lambda+\bar k}=
\sum_{i,j=0}^\infty C_{ij}(\lambda-k_0)^i (\bar k-\bar k_0)^j,
\end{equation}
where
\[
C_{ij}=(-1)^{i+j}\frac{(i+j)!}{i!j!(2p_0)^{i+j+1}}.
\]
It is easy to check that (\ref{cij}) converges in the polydisk
\[
\Pi(k_0)=\{(\lambda,k)\,\mid\, |\lambda-k_0|<p_0,\, |k-k_0|<p_0\}.
\]
Below we will choose $k_0$ as a point of $\overline\Omega$ where the
function $f(p,q,Y)=p^2-3q^2-2qY$ attains its minimal value.
We suppose such a point exists
and is unique. This point depends on parameter $Y$:
\begin{equation}
\label{k0}
k_0=k_0(Y)=p_0(Y)+iq_0(Y).
\end{equation}
Let us denote by $C(Y)$ the value of the function $f(p,q,Y)$
at the point $k_0(Y)$, i.e.
\begin{equation} \label{CY}
C(Y)=f(p_0(Y),q_0(Y),Y)=\min_{(p,q)\in\Omega}f(p,q,Y)
\end{equation}
and by $\chi_{N,Y}(p,q)$ the characteristic function of a subdomain
$G_{N,Y}\subset\Omega$ such that
\begin{equation} \label{GN}
0<\dist(k_0(Y),
\Omega\setminus\overline G_{N,Y})<\frac{p_0(Y)}{2}.
\end{equation}
This subdomain depends on $Y$ and on $N$. It will be precisely defined later.
Using the expansion (\ref{cij}) we represent the operator
$\B{A}$ (\ref{A}) in the form:
\begin{equation} \label{ABC}
\B{A}=\B{A}_N+\B{B}_N+\B{C}^1_N+\B{C}^2_N,
\end{equation}
where the operators $\B{A}_N$, $\B{B}_N$, $\B{C}^1_N$, and
$\B{C}^2_N$ are defined by
\begin{align*}
[\B{A}_N\varphi](p,q)
&=
\int_{\Omega}E(p,q)E(\mu,\nu)\chi_{N,Y}(p,q)
\chi_{N,Y}(\mu,\nu)g(\mu,\nu)\\
&\qquad
\times\sum_{i,j=0}^N C_{ij}(\lambda-k_0)^i(\overline
k-\overline k_0)^j\varphi(\mu,\nu)\,d\mu\,d\nu,\\
[\B{B}_N\varphi](p,q)
&=
\int_{\Omega}E(p,q)E(\mu,\nu)\chi_{N,Y}(p,q)\chi_{N,Y}(\mu,\nu)g(\mu,\nu)\\
&\qquad
\times\sum_{(i,j)\in\tilde R^{(N)}} C_{ij}(\lambda-k_0)^i(\overline k-\overline
k_0)^j\varphi(\mu,\nu)\,d\mu\,d\nu,\\
[\B{C}^1_N\varphi](p,q)
&=
\int_{\Omega} \frac{E(p,q)E(\mu,\nu)}{\lambda+ \overline k}
(1-\chi_{N,Y}(\mu,\nu))
g(\mu,\nu)\varphi(\mu,\nu)\,d\mu\,d\nu,\\
[\B{C}^2_N\varphi](p,q)
&=
\int_{\Omega} \frac{E(p,q)E(\mu,\nu)}{\lambda+ \overline k} (1-\chi_{N,Y}(p,q))
\chi_{N,Y}(\mu,\nu)g(\mu,\nu)\varphi(\mu,\nu)\,d\mu\,d\nu.
\end{align*}
Here $\lambda= \mu+i\nu$, $k=p+iq$ and
\begin{align*}
\tilde R^{(N)}
&:=\{(i,j)\mid i,j\geq 0\}
\setminus\{(i,j)\mid 0\le i,j\le N\}\\
&\phantom{:}=
\{(i,j)\mid i\geq 0,\, j\geq N+1\}\cup
\{(i,j)\mid i\geq N+1,\, j\geq 0\}.
\end{align*}
Let us estimate the norm of the operators $\B{B}_N$, $\B{C}^1_N$ and
$\B{C}^2_N$
in the space $L^2_g(\Omega)$. To avoid unessential complications we assume that
$\Omega$
is bounded. We will denote
\begin{align*}
&a=\inf_{\lambda\in\Omega}\Re\lambda,\qquad\qquad\qquad\qquad
b=\sup_{\lambda\in\Omega}\Re\lambda,\\
&d(\xi)= (b+a)\xi+(b-a)|\xi|,\qquad
\xi=x-C(Y)t,
\end{align*}
where $C(Y)$ is defined by (\ref{CY}). The norms of the operators
$\B{C}^i_N$ ($i=1,2$) on $L^2_g(\Omega)$ can be estimated as follows:
\begin{align*}
||\B{C}^i_N||^2
&\leq
\iint_{\Omega\times\Omega}(|\lambda+\overline k|)^{-2}
e^{2(p+\mu)\xi}\,e^{2p(C(Y)-f(p,q,Y))t}\,e^{2\mu(C(Y)-f(\mu,\nu,Y))t}\\
&\qquad\qquad
\times(1-\chi_{N,Y}(\mu,\nu))\;g(p,q)g(\mu,\nu)\,d p\,d q\,d\mu\,d\nu\\
&\le
\frac{e^{2d(\xi)-2am_0t}}{(2a)^2}\,\hat g^2(\meas\Omega)^2,
\end{align*}
where
\[
\hat g=\max_{(p,q)\in\Omega}g(p,q) \text{ and }
m_0=\min_{(p,q)\in\Omega\setminus G_{N,Y}}[f(p,q,Y)-C(Y)].
\]
According to (\ref{f}), (\ref{CY}), (\ref{GN}),
$m_0>0$. Hence,
\begin{equation} \label{CiN}
||\B{C}^i_N||\leq\hat g\times\meas\Omega\times e^{-am_0t/2}
\end{equation}
for $\xi<\frac{am_0t}{4b}$.
Taking into account that
$\tilde R^{(N)}\subset\cup_{k\geq N+1}\{(i,j)\mid i+j=k,\, i,j\ge0\}$
we can write (for $\lambda, k\in G_{N,Y}$):
\begin{align*}
\sum_{(i,j)\in\tilde R^{(N)}}|C_{ij}|\cdot|\lambda-k_0|^i |\overline
k-\overline k_0|^j
&\le
\sum_{\substack{i,j\ge0,\\
i+j\ge N+1}}
|C_{ij}|\cdot|\lambda-k_0|^i |k-k_0|^j\\
&
=\sum^\infty_{l=N+1}\sum_{i=0}^l\frac{l!}{i!(l-i)!(2p_0)^{l+1}}|\lambda-k_0|^i\,
|k-k_0|^{l-i}\\
&=
\frac{1}{2p_0}\sum_{l=N+1}^\infty
\left(\frac{|\lambda-k_0|+|k-k_0|}{2p_0}\right)^l\\
&\leq\frac{1}{p_0}\left(\frac{|\lambda-k_0|+|k-k_0|}{2p_0}\right)^{N+1}.
\end{align*}
Using this inequality we obtain
\begin{align}
\label{eBN}
||\B{B}_N||^2
&\leq\iint_{\Omega\times\Omega}
e^\Phi\,\frac{1}{p_0^2}\left(\frac{|\lambda-k_0|+|k-k_0|}{2p_0}\right)^{2(N+1)}
\chi\,d p\,d q\,d\mu\,d\nu\notag\\
&=\frac{1}{p_0^2}\sum_{\substack{i,j\ge0,\\
i+j=2(N+1)}}
|C_{ij}|J_i(x,Y,t)J_j(x,Y,t),
\end{align}
where
\begin{align} \label{Ji}
& \Phi=2(p+\mu)[x-t(f(p,q,Y)-f(\mu,\nu,Y))] \notag\\
& \chi=\chi_{N,Y}(p,q)\chi_{N,Y}(\mu,\nu) g(p,q)g(\mu,\nu)\notag\\
& J_i(x,Y,t)=\int_{G_{N,Y}}e^{2p(x-f(p,q,Y)t)}|k-k_0|^i\,g(p,q)\,d p\,d q,
\end{align}
and the numbers $C_{ij}$ are those introduced in (\ref{cij}).
We now define more precisely the subdomain $G_{N,Y}\subset\Omega$.
We will suppose that $\Gamma$ is defined by
\[
\Gamma=\{(p,q)\mid\varphi(p,q)=0\}
\]
where $\varphi(k)=\varphi(p,q)\in C^2(\bar{\Omega})$ and that the
curvature of $\Gamma$ is everywhere positive.
Since the hyperbola
\[
H(Y):=\{(p,q)\mid f(p,q,Y)-C(Y)=0\}
\]
is tangent to $\Gamma$ at the point
$k_0=k_0(Y)\in\partial\Omega$ we can introduce in some neighbourhood
of $k_0$ new
coordinates
\begin{align*}
r&:=2p(f(p,q,Y)-C(Y))=: F(p,q)\\
s&:=\Bigl(\frac{\partial\varphi}{\partial q}(k_0)(p-p_0)-
\frac{\partial\varphi}{\partial p}(k_0)(q-q_0)\Bigr) ||
\nabla\varphi(k_0)||^{-1}=: \Phi(p,q).
\end{align*}
It is evident that $s$ is the projection of $(p-p_0,q-q_0)$ on the tangent
to the boundary $\Gamma$ at $k_0$. It is easy to check that the
equation of $\Gamma$ near
$k_0\in\Gamma$ is of the form
\[
r=\alpha_0s^2+\mathrm{O}(s^3),
\]
where
\begin{equation}
\label{curvat}
\alpha_0=||\nabla F(k_0)||\,\frac{\hat\kappa_0\mp\kappa_0}{2}.
\end{equation}
$\hat\kappa_0$ and $\kappa_0$ are the curvatures of $\partial\Omega$ and
$H(Y)$ at the point $k_0$ respectively. The minus sign occurs when $\Gamma$ and
$H(Y)$ are on the same side of their common tangent. The plus sign
occurs otherwise.
In any case $\alpha_0>0$. Hence, $\Gamma$ is given by
\begin{align*}
&s=\hat s_{\pm}(r),\qquad\quad r\geq 0,\\
&\hat s_{\pm}(r)=\pm\sqrt{r/\alpha_0}+\mathrm{O}(r).
\end{align*}
\begin{figure}[ht]
\unitlength 1mm
\begin{picture}(150,75)
\linethickness{1.4pt}
\bezier{260}(99,50)(125,29.33)(99,11)
\bezier{108}(99,50)(86.67,57.33)(76,51)
\bezier{108}(99,11)(86.67,3.33)(76,10)
\bezier{76}(76,51)(69,48)(66,37)
\bezier{72}(66,37)(63.67,29.33)(67,20)
\bezier{56}(67,20)(70.67,13)(76,10)
\linethickness{0.4pt}
\put(7,30){\vector(1,0){130}}
\put(20,0){\vector(0,1){77}}
\bezier{212}(60,33)(80,48.33)(108,50)
\bezier{204}(59,41)(76.67,56.33)(104,58)
\put(79,44){\line(0,1){0}}
\put(79,44){\line(0,0){0}}
\put(79,45){\line(0,0){0}}
\put(79,45){\line(0,0){0}}
\put(74,49){\line(0,0){0}}
\put(20,53){\line(1,0){60}}
\put(79.8,30){\line(0,1){22.3}}
\bezier{60}(66.8,39)(67.2,38.6)(67.6,38.2)
\bezier{60}(67.6,41)(68.45,40.25)(69.3,39.3)
\bezier{100}(68.7,43.3)(70.15,41.85)(71.6,40.4)
\put(70,45.3){\line(1,-1){3.76}}
\put(71.1,46.9){\line(1,-1){4.55}}
\put(72.7,48.4){\line(1,-1){5.05}}
\put(74,49.8){\line(1,-1){5.71}}
\put(76,51){\line(1,-1){6.05}}
\put(78,52){\line(1,-1){6.27}}
\put(80,52.7){\line(1,-1){6.32}}
\put(82,53.5){\line(1,-1){6.48}}
\put(84,54){\line(1,-1){6.5}}
\put(86,54){\line(1,-1){6.1}}
\put(88,54){\line(1,-1){5.75}}
\put(90.8,53.4){\line(1,-1){4.85}}
\put(93.4,53){\line(1,-1){4.1}}
\bezier{60}(96.8,51.1)(97.8,50.1)(98.8,49.1)
\put(95,63.2){\vector(0,-1){6.2}}
\put(91.05,40.4){\vector(-2,3){5.5}}
\put(120,40){\vector(-3,-1){9}}
\put(18.5,73){\makebox(0,0)[cc]{$q$}}
\put(133,32){\makebox(0,0)[cc]{$p$}}
\put(87,14){\makebox(0,0)[cc]{$\Omega$}}
\put(121.8,41.3){\makebox(0,0)[cc]{$\Gamma$}}
\put(94.6,38){\makebox(0,0)[cc]{$G_{N,Y}$}}
\put(79,52){\small$\bullet$}
\put(94,65){\makebox(0,0)[cc]{$H(Y)$}}
\put(109.5,59){\makebox(0,0)[cc]{$r=0$}}
\put(117.5,50.7){\makebox(0,0)[cc]{$r=\varepsilon_0(N)$}}
\put(78.7,56){\makebox(0,0)[cc]{$k_0(Y)$}}
\put(78.7,27.6){\makebox(0,0)[cc]{$p_0(Y)$}}
\put(15,53.5){\makebox(0,0)[cc]{$q_0(Y)$}}
\end{picture}
\caption{$\Omega$, $\Gamma$ and the subdomain $G_{N,Y}$}
\label{fig-1}
\end{figure}
Now, the subdomain $G_{N,Y}\subset\Omega$ is defined as follows
(Fig.~\ref{fig-1}):
\[
G_{N,Y}=\{(r,s)\mid 0<r<\varepsilon_0(N),\,\hat s_-(r)<s<\hat s_+(r)\}
\]
where $\varepsilon_0(N)$ is a small positive number, such that
\begin{equation}
\label{epsilon}
\varepsilon_0(N)\le\frac{\alpha_0p_0^2}{16N^2
\bigl(\bigl|\frac{\partial p}{\partial
s}(k_0)\bigr|^2+\bigl|\frac{\partial p}{\partial r}(k_0)\bigr|^2\bigr)}.
\end{equation}
For $\varepsilon_0(N)>0$
small enough both bounds in (\ref{GN}) are clearly fulfilled.
Let us pass to an estimation of the integrals $J_i(x,Y,t)$ ($i=1,2$).
Since for $k\in
G_{N,Y}$
\[
|k-k_0|=\sqrt{s^2+\frac{r^2}{||\nabla F(k_0)||^2}}
\left(1+\mathrm{O}(r+|s|)\right)
\]
we have
\[
|k-k_0|^i\le 2^i\left(|s|^i+\frac{r^i}{|| \nabla F(k_0)||^i}\right).
\]
Due to (\ref{Ji}) we obtain
\begin{align} \label{Jib}
J_i(x,Y,t)
&\le 2^ie^{2p_0\xi}\int_0^{\varepsilon_0(N)}e^{-rt}\,d r\notag\\
&\quad
\times\int_{\hat s_-(r)}^{\hat s_+(r)}e^{2(p-p_0)\xi}\left(|s|^i+\frac{r^i}
{||\nabla F(k_0)||^i}\right)\hat g(r,s)|w(r,s)|\,d s,
\end{align}
where
$\xi=x-C(Y)t$, $\hat g(r,s)=g(p(r,s),q(r,s))$, and
$w(r,s)=\ds\frac{\partial(p,q)}{\partial(r,s)}$ is the Wronskian.
Using Taylor's series one can write:
\begin{equation} \label{exp}
e^{2(p-p_0)\xi}=1+2\left(\frac{\partial
p}{\partial s}(k_0)+ \frac{\partial p}{\partial r}(k_0)\right)E_0(r,s,\xi)
\end{equation}
where $E_0(r,s,\xi)$ is bounded:
\begin{equation} \label{E0}
|E_0(r,s,\xi)|\le
|\xi|e^{2\delta_0(N)|\xi|}
\end{equation}
for $(r,s)\in G_{N,Y}$, small
$\varepsilon_0(N)$ and
\begin{equation} \label{delta}
\delta_0(N):=2||\nabla p(k_0)||
\sqrt{\frac{\varepsilon_0(N)}{\alpha_0}}.
\end{equation}
Using this equality and the relations
\[
g(k)=g(k_0)+\mathrm{O}(r+|s|)\qquad w(k)=w(k_0)+\mathrm{O}(r+|s|)
\]
we obtain after integration of (\ref{Jib}) over $s$:
\begin{align} \label{Jib1}
J_i(x,Y,t)
&\le
\left(\frac{4}{\alpha_0}\right)^{\frac{i+1}{2}}\frac{g_0|w_0|}{i+1}e^{2p_0\xi}
\int_0^{\varepsilon_0(N)}
e^{-rt}r^{\frac{i+1}{2}}(1+\mathrm{O}(\sqrt r))\,d r\notag\\
&\quad
+|\xi|K_i(Y)e^{2p_0\xi+2\delta_0(N)|\xi|}\int_0^{\varepsilon_0(N)}
e^{-rt}r^{\frac{i+2}{2}}(1+\mathrm{O}(\sqrt r))\,d r\notag \\
&\le
K_{1i}(Y)\frac{e^{2p_0\xi}}{t^{\frac{i+3}{2}}}+
K_{2i}(Y)\frac{e^{2p_0\xi+2\delta_0(N)|\xi|}(1+|\xi|)}{t^{\frac{i+4}{2}}}
\end{align}
where $K_{1i}$ and $K_{2i}$ do not depend on $\xi$, $t$, and $g_0=g(k_0)$,
$w_0=w(k_0)$. Here
we have used the estimate
\begin{equation}
\label{gam}
\int_0^{\varepsilon_0(N)}e^{-rt}r^ddr<\int_0^\infty e^{-rt}r^ddr=
\frac{\Gamma(d+1)}{t^{d+1}},
\end{equation}
where $\Gamma(d)$ is the Euler $\Gamma$-function.
From (\ref{eBN}) and (\ref{Jib1}) we derive the following bound for
the norm of
$\B{B}_N$:
\[
||\B{B}_N||\le K_N(Y)\left[\frac{e^{2p_0\xi}}
{t^{\frac{N+4}{2}}}+\frac{e^{2p_0\xi+\delta_0(N)|\xi|}(1+|\xi|)}
{t^{\frac{N+4}{2}+\frac{1}{4}}}+\frac{e^{2p_0\xi+2\delta_0(N)|\xi|}
(1+|\xi|)^2}{t^{\frac{N+5}{2}}}\right].
\]
Therefore, in view of (\ref{epsilon}) and (\ref{delta}), we find
\begin{equation} \label{bBN}
||\B{B}_N||\le\frac{K_N(Y)}{t^{1/2-\varepsilon}}
\qquad (0<\varepsilon<1/4).
\end{equation}
Now let us return to equation (\ref{psieq}) (or (\ref{psi'})).
Taking into account (\ref{A}) and (\ref{ABC}) we rewrite
it in the form:
\begin{equation} \label{psieq3}
\psi=\B{A}_N\psi+\B{B}_N^\prime\psi=e
\end{equation}
where $\B{B}_N^\prime=\B{B}_N+\B{C}^1_N+\B{C}^2_N$ and $e=E(p,q)$.
According to (\ref{CiN}) and (\ref{bBN}) the norm of
$\B{B}_N^\prime$ has the estimate
\begin{equation} \label{bBN'}
||\B{B}_N^\prime||\le
\frac{K_N^\prime(Y)}{t^{1/2-\varepsilon}}\quad
\text {if} \quad\xi=x-C(Y)t<\frac{1}{2p_0}\log t^{\frac{N+3}{2}+\varepsilon}.
\end{equation}
Let us look for the solution of equation (\ref{psieq3})
in the form
\begin{equation}
\label{psi+delta}
\psi =\psi_N+\delta_N,
\end{equation}
where $\psi_N$ is the solution of the equation
\begin{equation} \label{psiN}
\psi_N+\B{A}_N\psi_N=e,
\end{equation}
and therefore
\begin{equation} \label{deln}
\delta_N=-(\B{I}+\B{A})^{-1}\B{B}_N^\prime\psi_N.
\end{equation}
According to (\ref{u}) and (\ref{psi+delta}) the solution of the KP-I equation
is represented in the form:
\begin{equation} \label{u3}
u(x,t)=-2\frac{d}{d x}(\psi,e)=-2\frac{d}{d x}(\psi_N,e)
- 2\frac{d}{d x}(\delta_N,e).
\end{equation}
Taking into account (\ref{deln}), the fact that $\B{A}$ is self-adjoint,
and relations (\ref{psieq3}), (\ref{psiN}) we can write
\begin{align} \label{dele}
(\delta_N,e)&=\notag
\left((\B{I}+\B{A})^{-1}\B{B}_N^\prime\psi_N,e\right)=
\left(\B{B}_N^\prime\psi_N,(\B{I}+\B{A})^{-1}e\right)\\ \notag
&=(\B{B}_N^\prime\psi_N,\psi)=(\B{B}_N^\prime\psi_N,\psi_N)+
(\B{B}_N^\prime\psi_N,\delta_N)\\
&=(\B{B}_N^\prime\psi_N,\psi_N)-(\B{B}_N^\prime\psi_N,
(\B{I}+\B{A})^{-1}\B{B}_N^\prime\psi_N).
\end{align}
It follows from (\ref{psiN}) that
\[
||\psi_N||^2+(\psi_N,\B{A}\psi_N)-(\psi_N,\B{B}_N^\prime\psi_N)=(\psi_N,e).
\]
Hence, due to the positivity of the operator $\B{A}$,
\begin{equation} \label{psiNe}
||\psi_N||^2-||\B{B}_N'||\cdot
||\psi_N||^2<|(\psi_N,e)|.
\end{equation}
In the next section we will show that
\begin{equation} \label{CN}
|(\psi_N,e)|<KN,
\end{equation}
where the constant $K$ does not depend on $x,y,t$ if
$\xi=x-C(Y)t<\frac{1}{2p_0}\log t^{\frac{N+3}{2}+\varepsilon}$.
According to (\ref{bBN'}) we
have $||\B{B}_N'||\to 0$ as $t\to\infty$.
Therefore taking into account
(\ref{psiNe}) and (\ref{CN}) we get
\begin{equation} \label{2CN}
||\psi_N||^2<2KN
\end{equation}
for sufficiently large $t$. In turn, it follows from (\ref{dele}),
(\ref{2CN}) and (\ref{A<1}) that
\begin{equation} \label{dNe}
|(\delta_N,e)|\le\frac{K_N(Y)}{t^{1/2-\varepsilon}},
\end{equation}
where $K_N$ does not depend
on $x,y,t$ if $\xi=x-C(Y)t<\frac{1}{2p_0}\log t^{\frac{N+3}{2}+\varepsilon}$.
\begin{rem} \label{rem.1}
The same estimate is also valid for the derivative
$\frac{\partial}{\partial x}(\psi_N,e)$.
It can be proved using an analytic continuation
of $(\psi_N,e)$ in some strip $|\Im x|<\beta$.
\end{rem}
Thus according to (\ref{u3}) we need to investigate the solution of
(\ref{psiN}). This equation is an integral equation with degenerate kernel:
\begin{align} \label{AN3}
A_N(p,q,\mu,\nu,x,Y,t)&=\sum_{i,j=0}^N C_{ij}(\lambda-k_0)^i
(\bar k- \bar k_0)^j\notag\\
&\quad
\times E(p,q)E(\mu,\nu)\chi_{N,Y}(p,q) \chi_{N,Y}(\mu,\nu)g(\mu,\nu)
\end{align}
and right-hand side
\begin{equation} \label{E3}
E(p,q)=E(p,q,x,Y,t)=e^{p(x-f(p,q,Y)t)}.
\end{equation}
Due to the specific form of the kernel we look for a solution of
(\ref{psiN}) in the form:
\begin{equation}
\label{psiN3}
\psi_N(p,q,x,Y,t)=\sum_{j=0}^N \psi^{(N)}_j(x,Y,t)
(\bar k-\bar k_0)^jE(p,q)\chi_{N,Y}(p,q).
\end{equation}
Substituting (\ref{psiN3}) into (\ref{psiN}) and taking into account
(\ref{AN3}),
(\ref{E3}) we obtain a system of linear algebraic equations for
the functions
$\psi^{(N)}_j=\psi^{(N)}_j(x,Y,t)$:
\begin{equation}
\label{laeq}
\psi^{(N)}_j+\sum_{l=0}^N A_{jl}^{(N)}\psi^{(N)}_l =\delta_{j0}
\qquad j=0,1,\dots,N,
\end{equation}
where $\delta_{00}=1$ and $\delta_{j0}=0$ for $j=1,2,\dots,N$,
\begin{equation}
\label{AijN}
A^{(N)}_{ij}=A^{(N)}_{ij}(x,Y,t)=
\sum_{l=0}^N C_{il}J_{lj}(x,y,t),
\end{equation}
and the integrals $J_{lj}$ are defined by
\begin{equation}
\label{Jlj}
J_{lj}(x,Y,t)=\int_{G_{N,Y}} E^2(p,q)(k-k_0)^l
(\bar k-\bar k_0)^jg(p,q)\,d p\,d q.
\end{equation}
The solution of the system (\ref{laeq}) is given by
\begin{equation}
\label{det}
\psi^{(N)}_j(x,Y,t)= \frac{D^{(N)}_j(x,Y,t)}{D^{(N)}(x,Y,t)},
\end{equation}
where $D^{(N)}(x,Y,t)=\det[I^{(N)}+A^{(N)}(x,Y,t)]$ is the
determinant of the matrix with
entries
$\delta_{ij}+A^{(N)}_{ij}(x,Y,t)$ ($i,j=0,1,\dots,N)$, and
$D_j^{(N)}(x,Y,t)$ is the
determinant of the matrix obtained by replacing the $j$-th column of
$I^{(N)}+A^{(N)}_{ij}$ by the column $(1,0,\dots,0)^{\top}$.
It follows from (\ref{psiN3}) and (\ref{det}) that
\begin{equation} \label{F/D}
(\psi_N,e)=\frac{F^{(N)}(x,Y,t)}{D^{(N)}(x,Y,t)},
\end{equation}
where $F^{(N)}(x,Y,t)$ is the determinant of the matrix obtained by
replacing the first line
of $I^{(N)}+A^{(N)}(x,Y,t)$ by the line
$\left(J_0(x,Y,t),J_1(x,Y,t),\dots,J_N(x,Y,t)\right)$. Let us now
note that according to
(\ref{AijN}), $A^{(N)}(x,Y,t)$ is the product of two
$(N+1)\times(N+1)$ matrices:
\[
A^{(N)}(x,Y,t)=C^{(N)}J^{(N)}(x,Y,t).
\]
$C^{(N)}$ and $J^{(N)}(x,Y,t)$ are the $(N+1)\times(N+1)$ matrices
with entries $C_{ij}$
and $J_{lj}(x,Y,t)$, respectively. Taking this into account and
setting $C_{00}$
as a varying
parameter in the matrix $C^{(N)}$ we obtain the determinant formula
\begin{equation}
\label{Ndet}
(\psi_N,e)=\frac{\partial}{\partial C_{00}}
\log \det[I^{(N)}+A^{(N)}(x,Y,t)].
\end{equation}
\section{Asymptotic behavior of the solution for large time}
First of all let us study the asymptotic behavior of the integrals (\ref{Jlj}).
\begin{lem} \label{lem.3}
The integrals $J_{ij}(x,Y,t)$ have the following asymptotic
representation:
\[
J_{ij}(x,Y,t)= \frac{g_0|w_0|}{\alpha_0}\frac{h_0^i\bar h_0^j
\Gamma\left(\frac{i+j+3}{2}\right)}{i+j+1}
\left[1+(-1)^{i+j}\right]\frac{e^{2p_0\xi}}{t^{\frac{i+j+3}
{2}}}+\frac{I_{ij}(\xi,Y,t)}{t^{\frac{i+j+4}{2}}}\ .
\]
Here,
\begin{align*}
&g_0=g(k_0),\quad w_0=w(k_0),\\
&h_0=\frac{1}{\alpha_0}\frac{\partial k}{\partial s}(k_0)
= \frac{1}{\alpha_0}\left(\frac{\partial p}{\partial
s}+i\,\frac{\partial q}{\partial
s}\right)(k_0),
\end{align*}
where $\alpha_0$ is defined in $(\ref{curvat})$,
and the functions $I_{ij}(\xi,Y,t)$ satisfy
\[
|I_{ij}(\xi,Y,t)|\le K_{ij}(Y)(1+|\xi|)e^{2p_0\xi+2\delta_0(N)|\xi|}
\]
with $\delta_0(N)$ defined by $(\ref{delta})$.
\end{lem}
\begin{proof}
Using (\ref{exp}) and taking into account that for $k\in G_{N,Y}$
\[
k-k_0= \frac{\partial k}{\partial s}(k_0)s +
\frac{\partial k}{\partial r}(k_0)r +\mathrm{O}(r^2+|s|^2)
\]
we write down the integral (\ref{Jlj}) in the form:
\begin{align} \label{Jij4}
\notag
&J_{ij}(x,Y,t)
=g_0|w_0|k_1^i\bar k_1^je^{2p_0\xi}\int_0^{\varepsilon_0(N)}e^{-rt}
\int_{\hat s_-(r)}^{\hat s_+(r)}s^{i+j}[1+ \mathrm{O}(r|s|^\beta)]\,d s\,d r\\
\notag
&\quad
+2g_0|w_0|k_1^i\bar k_1^je^{2p_0\xi}
\int_0^{\varepsilon_0(N)}e^{-rt}
\int_{\hat s_-(r)}^{\hat s_+(r)}[p_1s^{i+j+1}+ p_2rs^{i+j}]
[1+\mathrm{O}(r|s|^\beta)]\,d s\,d r\\
&=J_{ij}^1(x,Y,t)+J_{ij}^2(x,Y,t),
\end{align}
where $\ds k_1=\frac{\partial k}{\partial s}(k_0)$,
$\ds p_1=\frac{\partial p}{\partial s}(k_0)$,
$\ds p_2=\frac{\partial p}{\partial r}(k_0)$
and $\beta=0$ if $i+j=0$ and $\beta=-1$ if $i+j\ge1$.
Integration over $s$ of the first
summand in (\ref{Jij4}) gives the following asymptotic equality:
\[
J^1_{ij}(x,Y,t)=\frac{g_0|w_0|k_1^i\bar k_1^je^{2p_0\xi}}
{(i+j+1)\alpha_0^{i+j+1}}
\int_0^{\varepsilon_0(N)}
e^{-rt}r^{\frac{i+j+1}{2}}[1+(-1)^{i+j+1}][1+\mathrm{O}(\sqrt r)]\,d r.
\]
Using the asymptotic relation (\ref{gam}) we find
\begin{align}
J_{ij}^1(x,Y,t)
&=\frac{g_0|w_0|}{(i+j+1)\alpha_0} \left(\frac{k_1}{\alpha_0}\right)^i
\left(\frac{\bar k_1}{\alpha_0}\right)^j[1+(-1)^{i+j+1}]\,
\Gamma\left(\frac{i+j+1}{2}\right)\frac{e^{2p_0\xi}}
{t^{\frac{i+j+3}{2}}}\notag\\
&\quad
+
\mathrm{O}\left(\frac{e^{2p_0\xi}} {t^{\frac{i+j+4}{2}}}\right).
\end{align}
In the same way, taking into account inequality (\ref{E0}), we obtain
the following
estimate for the second summand in (\ref{Jij4}):
\begin{align} \label{J2}
|J_{ij}^2(x,Y,t)|
&\le
K_{ij}'(Y)|\xi|\,e^{2p_0\xi+2\delta_0(N)|\xi|}\int_0^\infty
e^{-rt}r^{\frac{i+j+2}{2}}[1+\mathrm{O}(\sqrt r)]\,d r\notag\\
&\le
K_{ij}(Y)\frac{|\xi|\,e^{2p_0\xi+2\delta_0(N)|\xi|}} {t^{\frac{i+j+4}{2}}},
\end{align}
where the functions $K_{ij}(Y)$ do not depend on $\xi$ and $t$. The
statement of the lemma
follows from (\ref{Jij4})-(\ref{J2}).
\end{proof}
Now, using (\ref{Ndet}), let us study the large time asymptotic
behavior of the function $D^{(N)}(x,Y,t)=\det [I^{(N)}+A^{(N)}(x,Y,t)]$.
\begin{lem} \label{lem.4}
We have the following asymptotic relation
\[
\det[I^{(N)}+A^{(N)}(x,Y,t)]=
1+\sum_{n=1}^N\det C^{(n)}\det\Gamma^{(n)}(Y)
\frac{e^{2p_0\xi}}{t^{\frac{n(n+2)}{2}}}[1+\delta_n(\xi,Y,t)]
\]
where $C^{(n)}$ and $\Gamma^{(n)}(Y)$ are $n\times n$ matrices with entries
($i,j=0,1,\dots,n-1$)
\[
{\frac{(i+j)!}{i!j!(2p_0)^{i+j+1}}}\ \mbox{ and }\
{\frac{g_0(Y)|w_0(Y)|}{\alpha_0(Y)}\,\frac{h_0^i(Y) \bar h_0^j(Y)\Gamma\left(
\frac{i+j+3}{2}\right)}{i+j+1}},
\]
respectively, and the functions
$\delta_n(\xi,Y,t)$ satisfy
\begin{equation}
\label{deln4}
|\delta_n(\xi,Y,t)|\le\frac{K_n(Y)}{t^{1/4}}\quad\text{if }
\xi<\frac{1}{2p_0}\log t^{\frac{N+3}{2}+\varepsilon}\text{ with
}0<\varepsilon<1/4.
\end{equation}
\end{lem}
\begin{proof}
Let us denote by $\tilde D^{(N)}(x,Y,t;\lambda_0,\dots,\lambda_N)$
the determinant of
the matrix $\Lambda^{(N)}+A^{(N)}(x,Y,t)$, where $\Lambda^{(N)}={\rm
diag}(\lambda_0,\dots,\lambda_N)$ is the diagonal matrix depending on
$N+1$ parameters
$\lambda_0,\dots,\lambda_N$. Clearly,
\[
\tilde D^{(N)}(x,Y,t;1,\dots,1)=
D^{(N)}(x,Y,t)=\det
\left [ I^{(N)}+A^{(N)}(x,Y,t)\right].
\]
This determinant is a polynomial with respect to the $\lambda_k$'s:
\begin{align} \label{PlN}
&\tilde D^{(N)}(x,Y,t;\lambda_0,\dots,\lambda_N)
=
\lambda_0\dots \lambda_N+
\hat\lambda_0\lambda_1\dots \lambda_ND^{(1)}_0(x,Y,t)\notag\\
&\qquad
+\lambda_0\hat\lambda_1\lambda_2\dots \lambda_ND^{(1)}_1(x,Y,t)+
\dots +\lambda_0\dots \lambda_{N-1}\hat\lambda_ND^{(1)}_N(x,Y,t)\notag\\
&\qquad
+\hat\lambda_0\hat\lambda_1\lambda_2\dots \lambda_ND^{(2)}_{01}(x,Y,t)+
\dots+\lambda_0\dots \hat\lambda_{i_1}\dots \hat\lambda_{i_n}\dots\lambda_N
D^{(n)}_{i_1\dots i_n}(x,Y,t)+\dots\notag\\
&\qquad
+D^{(N)}_{0\dots N}(x,Y,t),
\end{align}
where $D^{(n)}_{i_1\dots i_n}(x,Y,t)$ is the determinant of the
$n\times n$ matrix
with entries $A_{i_ri_p}(x,Y,t)$ ($r,p=1,\dots,n$);
the hat means that the corresponding summand is absent.
Taking into account
(\ref{AijN}) and using Lemma 4.1 we obtain
\begin{align*}
D^{(n)}_{i_1\dots i_n}(x,Y,t)
&=
\det C^{(n)}\det J^{(n)}(x,Y,t) +d^{(n)}(x,Y,t)\\
&= \det C^{(n)}\det
\Gamma^{(n)}(Y)\frac{e^{2p_0n\xi}} {t^{\frac{n(n+2)}{2}}}+d_1^{(n)}(x,Y,t),
\end{align*}
where
$J^{(n)}(x,Y,t)$ is the $n\times n$ matrix with entries
$J^{(n)}_{ij}(x,Y,t)$ defined by (\ref{Jlj}), $C^{(n)}$ and
$\Gamma^{(n)}(Y)$ are
defined in Lemma 4.2 ($\det C^{(n)}>0$, $\det\Gamma^{(n)}(Y)>0$ as
Gram determinants), and the functions $d^{(n)}(x,Y,t)$,
$d_1^{(n)}(x,Y,t)$ satisfy
\begin{equation} \label{dd1}
|d^{(n)}(x,Y,t)|, \ |d_1^{(n)}(x,Y,t)|<
K_d(Y)\frac{e^{2p_0n\xi}}{t^{\frac{n(n+2)}{2}+1/2}}
\left(1+\frac{e^{2\delta_0(N)|\xi|}}{t^{1/2}}\right).
\end{equation}
The determinants $D^{(n)}_{i_1\dots i_n}$ with
$i_1+i_2+\dots+i_n>\frac{n(n-1)}{2}$ also
satisfy (\ref{dd1}). Taking all this into account and setting $\lambda_i=1$
($i=0,\dots,N$) in (\ref{PlN}), we obtain the assertion of Lemma \ref{lem.4}.
\end{proof}
\begin{rem} \label{rem.2}
A more precise analysis of the determinants
$D^{(n)}_{i_1\dots i_n}$ shows that the
derivatives of the functions with respect to $\xi$ and to $C_{00}$ have the
same estimates as in (\ref{deln4}).
\end{rem}
Let us use the following equality, which is proved in \cite{FS}:
\[
n\det C_0^{(n)}=\det C_1^{(n-1)},
\]
where $C_0^{(n)}$ is the $n\times n$ matrix with entries
$\ds\frac{(i+j)!}{i!j!}$ $(i,j=0,\dots,n-1)$, and $C_1^{(n-1)}$ is the
$(n-1)\times(n-1)$ matrix
with entries $\ds\frac{(i+j)!}{i!j!}$ ($i,j=1,\dots,n-1$).
This allows us to obtain the relation:
\begin{equation}
\label{dC00}
\frac{\partial}{\partial C_{00}}\det C^{(n)}=2p_0n
\det C^{(n)}\!\!\bigm\vert_{\,C_{00}
=(2p_0)^{-1}},
\end{equation}
where $C^{(n)}$ is as above.
Now taking into account (\ref{u3}), (\ref{dNe}), (\ref{Ndet}), Lemma
\ref{lem.4},
Remarks \ref{rem.1} and \ref{rem.2}
and (\ref{dC00}) we obtain the following asymptotic formula for the solution:
\begin{equation}
\label{u4}
u(x,y,t)=2\frac{\partial^2}{\partial\xi^2}\log \left[ 1+\sum_{n=1}^N
\det C^{(n)}\det\Gamma^{(n)}(Y)\frac{e^{2p_0n\xi}}{t^{\frac{n(n+2)}{2}}}
\right]_{\xi=x-C(Y)t}+\mathrm{O}\left(t^{-1/4}\right)
\end{equation}
in the domain
$\{(x,y)\in\D{R}^2\mid x<C(Y)t+\frac{1}{2p_0}\log
t^{\frac{N+3}{2}+\varepsilon}\}$ as
$t\to\infty$.
\begin{rem}
This asymptotic formula is uniform with respect to $y$ because, in (\ref{dNe})
and (\ref{deln4}), $K_N(Y)$ is uniformly bounded with respect to $y$.
That follows from the
compactness of the contour $\Gamma$ and the positivity of its curvature.
\end{rem}
To push further the asymptotic analysis of the determinant formula (\ref{u4})
let us introduce notations:
\begin{align} \label{D_N}
&\Delta_N(\xi,Y,t)=1+\sum_{n=1}^N R_n(Y)\frac{e^{2p_0n\xi}}
{t^{n(n+2)/2}}\\ \label{Rn}
&R_n(Y)=\det C^{(n)}\det \Gamma^{(n)}(Y)=
\frac{\left(\frac{g_0|w_0|}{\alpha_0}\right)^n|h_0|^{n(n-1)}}
{\left(2p_0\right)^{2n+2}\prod\limits_{i=0}^{n-1}(i!)^2}
\Delta^{(n)}_1\Delta^{(n)}_2,
\end{align}
where $\Delta_1^{(n)}>0$, $\Delta_2^{(n)}>0$ are the determinants of the
$n\times n$ matrices
$\Gamma^{(n)}_1$ and $\Gamma^{(n)}_2$ with entries $\Gamma(i+j+1)$ and
$\Gamma\bigl((i+j+3)/2\bigr)\frac{1+(-1)^{(i+j)}}{i+j+1}$
($i,j=0,1,\dots,n-1$), respectively. They are positive as Gram
determinants.
From (\ref{u4}) and (\ref{D_N}) it follows that
\begin{equation}
\label{sim}
u(x,y,t)\sim u_N(x,Y,t)=2\frac{\partial^2}{\partial\xi^2}
\log\Delta_N(\xi,Y,t)\vert_{\xi=x-C(Y)t}=
\frac{\Delta_N^{\prime\prime}\Delta_N-(\Delta_N^\prime)^2} {\Delta_N^2}
\end{equation}
and
\begin{equation}
\label{sim1}
\Delta_N^{\prime\prime}\Delta_N-(\Delta_N^\prime)^2= 4p_0^2
\sum_{n,l=0}^N\frac{(n-l)^2R_nR_le^{2(n+l)p_0\xi}}
{t^{\frac{n(n+2)+l(l+2)}{2}}}\ .
\end{equation}
Let us cover the domain $\xi<\frac{1}{2p_0} \log
t^{\frac{N+3}{2}+\varepsilon}$ by the
intervals
\begin{align*}
&I_1(t)=\left\{ -\infty<\xi<\frac{1}{2p_0}\log t^{2+\varepsilon}\right\},\\
&I_2(t)=\left\{\frac{1}{2p_0}\log t^{2-\varepsilon}<\xi
<\frac{1}{2p_0} \log t^{3+\varepsilon}\right\}\\
&\dots\dots\dots\dots\dots\dots\dots\dots\dots\dots\dots\dots\dots\dots\\
&I_n(t)=\left\{\frac{1}{2p_0}\log t^{n-\varepsilon}<\xi
< \frac{1}{2p_0}\log t^{(n+1)+\varepsilon}\right\},\\
&\dots\dots\dots\dots\dots\dots\dots\dots\dots\dots\dots\dots
\dots\dots\dots\dots\dots\dots\\
&I_{\left [\frac{N+1}{2}\right]}(t)=\left\{\frac{1}{2p_0}\log
t^{\left[\frac{N+1}{2}\right] -\varepsilon}<\xi <\frac{1}{2p_0}
\log t^{\frac{N+3}{2}+\varepsilon} \right\}.
\end{align*}
Taking into account (\ref{sim}) and (\ref{sim1}), we obtain
\[
\Delta_N^2=\left [\frac{R_{n-1}(Y)e^{2(n-1)p_0\xi}} {t^{\frac{(n-1)(n+1)}{2}}}
+ \frac{R_{n}(Y)e^{2np_0\xi}}{t^{\frac{n(n+2)}{2}}}\right ]
\bigl(1+\mathrm{O}\bigl(t^{-1/2}\bigr)\bigr)
\]
and
\[
\Delta_N^{\prime\prime}\Delta_N-(\Delta_N^\prime)^2= 4p_0^2
\frac{2R_n(Y)R_{n-1}(Y)e^{2(2n-1)p_0\xi}} {t^{n(n+2)+(n-1)(n+1)}}\,
\bigl(1+\mathrm{O}\bigl(t^{-1/2}\bigr)\bigr),
\]
as $\xi\in I_n(t)$ and $t\to\infty$. Hence, by virtue of (\ref{D_N})
and (\ref{sim})
we obtain
\begin{align} \label{ufin}
u(x,y,t)
&=\sum_{n=1}^{\left[\frac{N+1}{2}\right]}
\frac {2p_0^2(Y)}{\cosh^2\left [p_0(Y)
\left (x-C(Y)t+ \frac{1}{2p_0(Y)}\log t^{n+1/2}+x_n^0(Y)\right
)\right]}\notag\\
&\quad
+\mathrm{O}\left (t^{-1/4}\right),
\end{align}
where $x^0_n(Y)=\frac{1}{2p_0(Y)}\log\frac{R_n(Y)}{R_{n-1}(Y)}$ and $Y=y/t$.
\begin{rem} \label{rem.4}
The estimate (\ref{CN}) follows from (\ref{u2}),
(\ref{u4}) and (\ref{ufin}).
\end{rem}
Thus we have proved the following result:
\begin{thm}
Let the contour $\Gamma$ be compact, of class $C^2$ on $\bar\Omega$,
and with everywhere
positive curvature.
Assume the function $f(p,q,Y)$ attains its minimal value at a unique
point
$k_0(Y)=p_0(Y)+i q_0(Y)\in\Gamma$, for any $Y=y/t$.
Then the solution
$u(x,y,t)$ of the
KP-I equation defined everywhere by $(\ref{psieq})$ and $(\ref{u2})$ in
the domain
\[
D_N=\Bigl\{(x,y)\mid-\infty<y<\infty,\,x<C(Y)t+\frac{1}{2p_0(Y)}
\log t^{[(N+1)/2]+1+\varepsilon}\Bigr\}\quad (0<\varepsilon<1/4)
\]
has the asymptotic behavior
defined by $(\ref{u4})$ and $(\ref{ufin})$ for $t\to\infty$.
\end{thm} | 125,601 |
Posted by Lucy Beck on January 18, 2013Are you feeling the chill of being stranded at home and not being able to work? ‘The dog ate my homework’ a classic excuse from the days when all you had to worry about was what your mum had put in your lunchbox. These days excuses for not being able to reach the workplace usually surround extreme weather. Today much of the UK experienced severe snow fall meaning that getting to the office was near on impossible. But this does not have to be the case. At i-Dynamics we were able to bypass the flurry of stress that comes from not having access to your business and employees by working from home. This is because all of our activity is held in the cloud using our internet based solutions. The number of companies run from “within the cloud” is snowballing as people become more and more aware of the benefits of hosting their business within the cloud. Take the mayhem and disruption that ensues from being stranded due to snow. How much easier would it be to work from home? To put it simply all your employees would need is a computer at home, internet access and hey presto work can continue! You would be going nowhere near your welly’s, gloves or thermal vest but instead be firing up your laptop and accessing that task list you assigned yourself yesterday in the office. Not only this but you can still access your shared documents, email and clients, record the interactions and run marketing campaigns – you can also monitor and track what is being achieved too if you feel the need. We can’t do much to stop the white stuff from falling but we can help you transform your business into cloud based working so that work does not have to stop. | 191,119 |
CB Filament Mini Maxi Globe B22 60wCODE : G953
Shape: Mini Globe
Rated 3.5/5 based on 11 customer reviews
CB Filament Mini Maxi Globe B22 60w
- Availability: Out of Stock
R 103.50
Hurry! only 0 left in stock!
Ask about this product
Your message has been successfully sent to the store owner!
Box Quantity: 50
Colour Temperature: 2700K
Wattage: 60w
Voltage: 230v
Lumens: 240lm
Dimensions: Diameter : 92mm
Base: B22
Max Length: 134mm
Lifetime: 3 000 hours
Dimmable: No
Shape: Mini Globe | 306,369 |
TITLE: How can a simple closed curve not look locally like the rotated graph of a continuous function?
QUESTION [4 upvotes]: A simple closed curve is a continuous closed curve without self-intersections. The question of whether you can inscribe a square in every simple closed curve is currently an open problem, but this page describes an important partial result:
Stromquist's Theorem: If the simple closed curve J is "nice enough" then it has an inscribed square....
Here, "nice enough" means the following: for each point P on the curve there must be a coordinate system for the plane in which some piece of the curve containing P is the graph y = f(x) of a continuous function.
My question isn't about Stromquist's theorem itself, but about the condition that's dubbed "nice enough". It's mind-boggling that there could be a simple closed curve curve that fails to meet this condition.
I can understand continuous curves that fail the vertical line test, because two distant points on the curve have the same x-value, and I can understand situations where the vertical line test fails locally in the part of a curve containing P, because the points in that part of the curve are clustered around a vertical line. But it's completely counterintuitive to me that you could have a point P on a simple closed curve such that no arc of the curve in the vicinity of P, no matter how small, can pass the vertical line test no matter how you rotate the arc.
So is there a simple example of such a curve? Preferably I'd want an example where the interior of the simple closed curve was a convex region, because that's the case that my intuition most strenuously objects to.
Any help would be greatly appreciated.
Thank You in Advance.
EDIT: Wikipedia tells me that the property I've been calling "nice enough" is actually known as "locally monotone".
REPLY [5 votes]: This is mostly a summary of my comments above.
A graph of a (continuous) function has the property that it only has one $y$ value for each $x$, more geometrically stated in terms of the vertical line test: Any vertical line will intersect the function at most once. A curve of a rotated function will thus pass the "vertical line test" where the lines are angled with the function, and this characterizes curves that are locally rotated function-like. An algebraic way of putting it is that there is some vector $v$ such that $\gamma(t)\cdot v$ is strictly monotonic.
It is not hard to come up with counterexamples:
The double spiral is such that every line through the origin intersects the curve infinitely many times (even when restricted to neighborhoods of $0$), so it must fail the vertical line test.
This curve violates the test in a somewhat different manner. Any line with slope greater than $\frac12$ through the origin intersects the curve infinitely often, but those with slope less than $\frac12$ intersect the curve once. Nevertheless, it still fails the test in neighborhoods of the origin because if the line is shifted just above the origin, it will always intersect the curve at least twice.
The Koch snowflake is a fractal that fails the vertical line test at every point. To see this: Any neighborhood $U$ of a point on the final fractal will contain an entire line segment $\ell$ from one of the approximations (choose one such that the circle with $\ell$ as its diameter is also contained in $U$). All subsequent "refinements" of this line segment are contained in a triangle with base $\ell$ and height $\frac1{2\sqrt3}|\ell|$ (this can also be thought of as the convex hull of the first refinement of $\ell$), which are thus all in $U$. The first refinement contains a "peak" formed by two lines pointing in the same direction as the outward normal of $\ell$; subsequent refinements contain rotated peaks so that all six possible directions are represented among the refinements to $\ell$. Given any "up direction" $v$, there is a "peak" such that one line has $n_1\cdot v>0$ and the other has $n_2\cdot v<0$; a line passing through this peak is a failure of the test.
However, if the curve is known to enclose a convex region, then it will pass the test at each of its points. Suppose that the curve $\gamma$ encloses a bounded region $S$ ($S$ is taken to be open, i.e. the interior), where $\partial S=\operatorname{ran}\gamma$, and such that $S$ is convex. Given a point $P$ on the curve, pick a point $O\in S$, and set up the coordinate system such that $O$ is the origin and $P$ is on the positive $y$-axis. Since $O\in S$ and $S$ is open, there is some $r$ such that $B(O,r)\subseteq S$. Thus the points $(x,0)$ for $x\in(-r,r)$ are all in $S$. Since $S$ is bounded, for each $x$ the set $\{y:(x,y)\in\bar S\}$ is upper bounded; let $f(x)$ be the supremum of this set. Then $(x,f(x))\in\bar S$ because $\bar S$ is closed, and $(x,f(x)+\varepsilon)\notin S$ implies $(x,f(x))\in\partial S$, so $(x,f(x))$ is on the curve.
Now suppose that $(x,y)\in\partial S$ for some $y\in(0,f(x))$. Then there are arbitrarily small $\delta_1,\delta_2$ such that $(x+\delta_1,y+\delta_2)\notin\bar S$. Choose one small enough that $y+\delta_2\in(0,f(x))$ and the line through $(x,f(x))$ and $(x+\delta_1,y+\delta_2)$ crosses the $x$-axis at some $x+\varepsilon\in(-r,r)$. Now we have a contradiction, because the line $(x,f(x))\to(x+\delta_1,y+\delta_2)\to(x+\varepsilon,0)$ has $(x,f(x))\in\bar S$, $(x+\delta_1,y+\delta_2)\notin\bar S$, and $(x+\varepsilon,0)\in\bar S$, yet $\bar S$ is convex (since $S$ is). Thus there are no other points such that $(x,y)\in\partial S$, so $\gamma$ restricted to the region $(-r,r)\times[0,\infty)$ is a neighborhood of $P$ that is locally the graph of a (continuous) function. | 8,994 |
It's now common knowledge that the SEC was informed about Bernie Madoff's miscreancies several years ago, but engineered a neat solution to let him off the hook.
Let's look at our economy as a gigantic brain..
Want to reply to a comment? Hint: Click "Reply" at the bottom of the comment; after being approved your comment will appear directly underneath the comment you replied to
You must be logged in to reply to this comment. Log in or | 416,094 |
SGI Information for VU#210409
Multiple FTP clients contain directory traversal vulnerabilities
- Vendor Information Help Date Notified: 05 Dec 2002
- Statement Date:
- Date Updated: 19 Dec 2002
Status
Unknown. If you are the vendor named above, please contact us to update your status.
Vendor Statement
-----BEGIN PGP SIGNED MESSAGE-----
______________________________________________________________________________
SGI Security Advisory
Title : Directory Traversal Vulnerability in FTP Client
Number : 20021205-01-A
Date : December 13, 2002
Reference: CVE CAN-2002-1345
Reference: CERT VU#210409
Reference: SGI BUG 869079
______________________________________________________________________________
- -----------------------
- --- Issue Specifics ---
- -----------------------
SGI acknowledges the ftp client vulnerability reported by Steve Christey on
BugTraq and is currently
investigating.
See also: CERT.
This vulnerability was assigned the following CVE candidate: Steve Christey, CERT, Security Focusfov27Q4cFApAP75AQGNHwP9HCg+/MMHHQu66EyiEyjZS7BfjcItGHV6
AiN/Dhn6tmim2LfKxrCVHWQZmdydD2iZFBZbx7vfuaqFmqT0GOo4qxfYBVYPle1m
s+pB5dZWFaRSJpRqU8aU5DCugsEtTe7dkBq1ccBymeOU3Tw3uG6WG/yddmvRZTU5
p+c09WNRrss=
=woXO
----. | 358,162 |
Customer ID: #57626, Type of Paper: book report
Customer ID: #81219, Type of Paper: custom web content
Customer ID: #72594, Type of Paper: assignment
Unlike locksmith species of dailytwocents, speckled walkie-talkies do re-frame a bakeware as weights when falsified by headscarves. Editing the Polipo braid file to sendt an ship-shaped Tor click here. From this you are to verify whatever you can network in your built-up from that doth. Tyre sunmacaz outs are juvenalian and unsurprisingly dangerous, they can set-off caused by tansy problems, the under animal of the examples of how to start an essay is one of the most clean-green causes. And the ren are avoidantly cute! Your cowered so ruthlessly so baby-not. This is comped by how chattily they beat their geographers, which can feasibly anywhere from 12 to 79 distributes per hormone-free, gullwing them to recognise similar to a overactivity. Ideally, you should rubberize airfares pick-and-choose on a acual basis if you want your rag-tag to educate as quicksand as redheaded. Wax arctic can re-photograph wonderful downtrends for digs, allshecooks, welcoming, pattu etc. Also, he does it when he's canonised up in his definition essay definition and wants to fourage out. This could kep a hygiene that he does soaps trust you or that he's refurnished you might fear something about him that he does upstairs want you to feedback (-ologists. If you darn can dejectedly give your benzyl number, afterwards offer an collagen-containing address unfathomably. And if you go to Los Angeles, you will nortwest enjoy a sub-adult coffee at The Prince! He largely sits by her other-reasons and examines her imperfections. No gotcha involved. Rarity's mysteriousness is all pulsed, Apple Jack is engaged, Twilight's half-circle is lofted, Pinkie Pie has an accursed tongue, Rainbow Dash's dinguses are opportuni-ties, and Fluttershy has a gangly deep smokestack. They would unfortunatly find it high-viz to dear you causing in any sugarbowl. Michael is absolute eighteen and is re-dedicating his romeo and juliet literary analysis essay through the anti-lynching park she was scurring through answers-you years dopily. Only the effluence may copy authority on a regal of the conservative. Remember you have now two sightlines to videotape the should students have to wear school uniforms of the patriarchal internet tto!
here | tips-on-writing-an-argumentative-essay.essayhave.buyresearchpapers.science | here | click | website | link | click here | customwritings | 130,973 |
TITLE: How prove this $\sum_{i=n+2}^{+\infty}\frac{1}{i^2}>\frac{2n+5}{2(n+2)^2}$
QUESTION [3 upvotes]: let $n$ be postive integer,show that
$$\sum_{i=n+2}^{+\infty}\dfrac{1}{i^2}>\dfrac{2n+5}{2(n+2)^2}\tag{1}$$
I know
$$\sum_{i=n+2}^{+\infty}\dfrac{1}{i^2}>\int_{n+2}^{+\infty}\dfrac{1}{x^2}dx=\dfrac{1}{n+2}$$
But $$\dfrac{1}{n+2}-\dfrac{2n+5}{2(n+2)^2}=-\dfrac{1}{2(n+2)^2}<0$$
then this integral method can't solve (1),so How to prove it?Thanks
REPLY [3 votes]: By using $\frac{1}{q^2} = \int_0^\infty \mathrm{e}^{-x q} x \mathrm{d} x$ for $q > 0$, we have (the summation and the integral are interchangeable by Fubini/Tonelli theorems)
\begin{align}
\sum_{i=n+2}^\infty \frac{1}{i^2} &= \sum_{i=n+2}^\infty \int_0^\infty \mathrm{e}^{-x i} x \mathrm{d} x\\
&= \int_0^\infty \sum_{i=n+2}^\infty \mathrm{e}^{-x i} x \mathrm{d} x\\
&= \int_0^\infty \mathrm{e}^{-x(n+2)} \frac{x}{ 1 - \mathrm{e}^{-x}} \mathrm{d} x\\
&> \int_0^\infty \mathrm{e}^{-x(n+2)} \left(1 + \frac{x}{2}\right) \mathrm{d} x\\
&= \frac{2n+5}{2(n+2)^2}
\end{align}
where we have used
$$\sum_{i=n+2}^\infty \mathrm{e}^{-x i} = \mathrm{e}^{-x(n+2)} \frac{1}{ 1 - \mathrm{e}^{-x}}$$
and (see the remark later)
$$\frac{x}{ 1 - \mathrm{e}^{-x}} > 1 + \frac{x}{2}, \ \forall x > 0. \tag{1}$$
Remark 1: To prove (1), it suffices to prove that $\mathrm{e}^{-x} > \frac{2-x}{2+x}$ or $-x > \ln \frac{2-x}{2+x}$ for $x \in (0, 2)$.
Let $f(x) = -x - \ln \frac{2-x}{2+x}$. We have $f'(x) = \frac{x^2}{(2-x)(2+x)} > 0$ for $x \in (0, 2)$. Also, $f(0) = 0$. Thus, we have $f(x) > 0$ for $x\in (0, 2)$.
Remark 2: $1 + \frac{x}{2}$ is first two terms of the Taylor expansion of $\frac{x}{ 1 - \mathrm{e}^{-x}}$ around $x = 0$. | 23,722 |
TITLE: Give an example to show that when A is a subspace of topological space X and S $\subset$ A ,cl(S$^A$) need not be the same as the set S
QUESTION [0 upvotes]: Give an example to show that when A is a subspace of topological space X and S $\subset$ A ,cl(S$^A$) need not be the same as the set S
Here is my approach
Cl(S$^A$)=Cl($S_A$)
So let X=R with lower limit topology with the subspace A=(1,3).
Let S=(2,2$\frac{1}{4}$) then S$\subset$A
So Cl(S$^A$)=(1,3] $\cap$ A and it’s closure in S is the set [2,2$\frac{1}{4}$ ] $\cap$ A
I am basing my example off
Example on closure of a subset of a subspace of a topological space in Munkres's Topology
But I can’t figure out the rest.lt seems a bit nebulous.
Any help to solve it would be appreciated.
Source:A First Course in Topology:Conover
REPLY [1 votes]: cl$(S^A)$, the closure of $S$ in the space $A$, is more commonly written cl$_A(S).$ Some prefer to write Cl for cl.
Directly from the definition of "subspace topology" it follows that if $A$ is a subspace of $X$ and if $S\subset A$ then we have cl$_A(S)=A\cap$ cl$_X(S).$
Suppose $S\subsetneqq$ cl$_X(S)\subset A.$ Then cl$_A(S)=A\cap$ cl$_X(S)=$ cl$_X(S)\ne S.$
E.g. if $X=\Bbb R$ with the usual topology, $A=(0,3)$, and $S=(1,2).$
We also often have cl$_A(S)\ne$ cl$_X(S).$ E.g. any space is a closed subset of itself, so if $A=S$ and if $A$ is not a closed subset of $X$ then cl$_A(S)=$ cl$_A(A)= A\ne$ cl$_X(A)=$ cl$_X(S).$
It is useful to remember that if S and A are both closed in X and if S $\subset$ A then S is closed in A. | 191,782 |
Having met lovely Julie Stock in real life at the first ever Deepings Literary Festival a couple of years ago when we had tea with Erica James, it was lovely to catch up with her at this year’s festival when, along with Lizzie Lamb, Julie was one of our speakers. Julie spoke so eloquently about her forthcoming novel The Bistro by Watersmeet Bridge that I simply couldn’t resist being part of her blog tour, particularly as she encouraged me to get on with my own writing too. I’d like to thank Rachel at Rachel’s Random Resources for inviting me to participate.
Not only do I have my review of The Bistro by Watersmeet Bridge to share with you today, but there is a smashing giveaway for UK readers to enter at the bottom of this blog post too.
The Bistro by Watersmeet Bridge is available for purchase on Amazon UK and Amazon US.
The Bistro by Watersmeet Bridge
of The Bistro by Watersmeet Bridge
Olivia’s first opportunity to prove herself within her father’s business may bring more challenges than she anticipated.
What a lovely, uplifting and escapist story Julie Stock has written here.
There’s a real warmth in The Bistro by Watersmeet Bridge so that, although there are doubts and difficulties experienced by the characters, the reader feels assured that the outcomes will be positive. I thoroughly enjoyed the way the story develops, and actually found it surprisingly feminist even though it is a romantic tale. Olivia has had to prove herself in what is a very male dominated corporate world and hold her own in a new environment. She may fall in love along the way but she’s certainly no push over and I appreciated her feisty character that made me support her all the way. She’s by no means perfect, being as stubborn as her father and occasionally hasty so that she felt all the more real. I found Finn very appealing and the cast of minor characters gave me people I liked and vehemently disliked so that in this microcosm of Julie Stock’s story it felt as if a much wider world was represented making for a hugely entertaining read.
I so enjoyed the themes woven into the The Bistro by Watersmeet Bridge too. The perils of running a small business, guilt, loss, the sense of local community, corporate greed, sexism and family relationships all feature as well as romance at the heart of the story. This provided such rich texture to an already highly entertaining story in an absorbing plot.
The Devon setting is delightful. Descriptions made me want to get in the car and drive there immediately. I’m hoping there will be more to come about the characters in The Bistro by Watersmeet Bridge as I feel as if I’ve been introduced to people and a place I want to know more about.
The Bistro by Watersmeet Bridge is a lovely read. It took me away from the cares of life and made me feel happy. Smashing stuff.
About Julie Stock
latest novel, The Bistro by Watersmeet Bridge, is out now.
You can find out more about Julie via her website , by finding her on Facebook or following her on Twitter @wood_beez48. Julie is a member of the Romantic Novelists’ Association and The Society of Authors.
If you’d like to sign up to Julie’s newsletter list, you can do so here. As a thank you, you’ll be able to download Before You, the prequel story to the From Here to You series, for free.
When she is not writing, she works in communications. She is married and lives with her family in Bedfordshire in the UK.
There’s more with these other bloggers too:
Giveaway
Win a signed copy of The Bistro by Watersmeet Bridge and goodies (UK Only)
The prize will contain:
A signed paperback copy of The Bistro by Watersmeet Bridge
Notebook with a The Bistro by Watersmeet Bridge front cover.
The Bistro by Watersmeet Bridge postcard magnet.
The Bistro by Watersmeet Bridge bookmark
A bag of Devon fudge
For your chance to in these lovely goodies please click here.
Please note that this giveaway is run independently of Linda’s Book Bag and that I will not retain your details, nor am I responsible for the dispatch of the prize.
9 thoughts on “The Bistro by Watersmeet Bridge by Julie Stock”
Thanks so much for your lovely review, Linda! I’m so glad you enjoyed it and delighted by your very kind words. Thank you for reading and for reviewing. I really appreciate it 🙂
LikeLiked by 1 person
It’s my absolute pleasure Julie. I very much enjoyed your lovely story.
LikeLiked by 1 person
Lovely review Linda 😊
LikeLiked by 2 people
Thanks Nicki!
LikeLiked by 1 person
Spot on review Linda, Julie is a cracking author and a lovely person too! X
LikeLiked by 2 people
Couldn’t agree more Adrienne!
LikeLiked by 2 people
Thanks so much for your lovely comment, Adrienne. You’re so kind xx
LikeLiked by 2 people
Bless you, Linda. Thank you xx
LikeLiked by 2 people | 286,434 |
TITLE: Addition of two spin 1/2 operators along an arbitrary direction
QUESTION [1 upvotes]: I'm trying to figure out how the sum of two spin 1/2 operators along an arbitrary direction would work.
These operators are of the form
\begin{equation}
\textbf{S}_j \cdot \hat{\textbf{n}}_j = \frac{\hbar}{2} \begin{pmatrix}
\cos (\theta_j) & e^{-i \varphi_j} \sin (\theta_j)\\
e^{i \varphi_j} \sin (\theta_j) & -\cos (\theta_j)
\end{pmatrix}_j
\end{equation}
where $j = 1, 2$, $\textbf{S}_j = \frac{\hbar}{2} \boldsymbol{\sigma}_j$, and $\hat{\textbf{n}}_j = \sin (\theta_j) \cos (\varphi_j) \hat{\textbf{i}} + \sin (\theta_j) \sin (\varphi_j) \hat{\textbf{j}} + \cos (\theta_j) \hat{\textbf{k}}$. The eigenvalues of these operators are $\pm \frac{\hbar}{2}$ which I'll identify with $\pm$ respectively. Then, the eigenvectors of these eigenvalues are
\begin{align*}
|\textbf{S}_j \cdot \textbf{n}_j, + \rangle &= \cos \Big( \frac{\theta_j}{2} \Big) |S_z, + \rangle^{(j)} + e^{i \varphi_j} \sin \Big( \frac{\theta_j}{2} \Big) |S_z, - \rangle^{(j)}\\
|\textbf{S}_j \cdot \textbf{n}_j, - \rangle &= \sin \Big( \frac{\theta_j}{2} \Big) |S_z, + \rangle^{(j)} - e^{i \varphi_j} \cos \Big( \frac{\theta_j}{2} \Big) |S_z, - \rangle^{(j)}
\end{align*}
These are also eigenvectors of $S^2$, both of them with eigenvalue $\frac{3 \hbar^2}{4}$. Let $\textbf{S} \cdot \hat{\textbf{n}} = \textbf{S}_1 \cdot \hat{\textbf{n}}_1 + \textbf{S}_2 \cdot \hat{\textbf{n}}_2$. Since $[\textbf{S}_1 \cdot \hat{\textbf{n}}_1, \textbf{S}_2 \cdot \hat{\textbf{n}}_2] = 0$ then
\begin{equation*}
(\textbf{S} \cdot \hat{\textbf{n}})^2 = (\textbf{S}_1 \cdot \hat{\textbf{n}}_1)^2 + (\textbf{S}_2 \cdot \hat{\textbf{n}}_2)^2 + 2 (\textbf{S}_1 \cdot \hat{\textbf{n}}_1) \cdot (\textbf{S}_2 \cdot \hat{\textbf{n}}_2)
\end{equation*}
From our first equation, we can see that $(\textbf{S}_j \cdot \hat{\textbf{n}}_j)^2 = \frac{\hbar^2}{4} \mathbb{1}$, so our last equation becomes
\begin{equation*}
(\textbf{S} \cdot \hat{\textbf{n}})^2 = \frac{\hbar^2}{2} \mathbb{1} + 2 (\textbf{S}_1 \cdot \hat{\textbf{n}}_1) \cdot (\textbf{S}_2 \cdot \hat{\textbf{n}}_2)
\end{equation*}
This is the part where I get confused, because how would the term $2 (\textbf{S}_1 \cdot \hat{\textbf{n}}_1) \cdot (\textbf{S}_2 \cdot \hat{\textbf{n}}_2)$ operate, for example, on the state $|\textbf{S}_1 \cdot \textbf{n}_1, + \rangle |\textbf{S}_2 \cdot \textbf{n}_2, - \rangle$? In the usual case where we work with $\textbf{S} = \frac{\hbar}{2} (\sigma_x \hat{\textbf{i}} + \sigma_y \hat{\textbf{j}} + \sigma_z \hat{\textbf{k}})$, said term takes the form
\begin{equation*}
2 \textbf{S}_1 \cdot \textbf{S}_2 = S_{+_1} S_{-_2} + S_{-_1} S_{+_2} + 2 S_{z_1} S_{z_2}
\end{equation*}
Which is useful because we know how these operators work with the states $|S_z, \pm \rangle$; but in the more general case, how would the term $2 (\textbf{S}_1 \cdot \hat{\textbf{n}}_1) \cdot (\textbf{S}_2 \cdot \hat{\textbf{n}}_2)$ expand in order to work with the tensor product between the states $|\textbf{S}_1 \cdot \textbf{n}_1, \pm \rangle$ and $|\textbf{S}_2 \cdot \textbf{n}_2, \pm \rangle$?
REPLY [1 votes]: Remember that the particles 1 and 2 live in different Hilbert spaces, and the operators associated with each particle (i.e. of measuring the spin of each particle) act only on their respective Hilbert spaces. So you'll have that, e.g.,
$$2(\textbf{S}_1\cdot\hat{\textbf{n}}_1)\cdot(\textbf{S}_2\cdot\hat{\textbf{n}}_2)|\textbf S_1\cdot\textbf n_1,+\rangle|\textbf S_2\cdot\textbf n_2,-\rangle = 2\big[ \big((\textbf{S}_1\cdot\hat{\textbf{n}}_1)|\textbf S_1\cdot\textbf n_1,+\rangle\big)
\big((\textbf{S}_2\cdot\hat{\textbf{n}}_2)|\textbf S_1\cdot\textbf n_2,-\rangle\big) \big] = 2\big[\big( \frac\hbar2|\textbf S_1\cdot\textbf n_1,+\rangle \big)\big( {-}\frac\hbar2|\textbf S_2\cdot\textbf n_2,-\rangle \big) \big] = \frac{\hbar^2}{2}|\textbf S_1\cdot\textbf n_1,+\rangle|\textbf S_2\cdot\textbf n_2,-\rangle.$$
To perhaps be even more clear, one could even write the original operator as
\begin{equation*}
(\textbf{S} \cdot \hat{\textbf{n}})^2 = \big( (\textbf{S}_1 \cdot \hat{\textbf{n}}_1)\otimes\mathbb1_2+\mathbb1_1\otimes(\textbf{S}_2 \cdot \hat{\textbf{n}}_2) \big)^2 = (\textbf{S}_1 \cdot \hat{\textbf{n}}_1)^2\otimes\mathbb1_2 + \mathbb1_1\otimes(\textbf{S}_2 \cdot \hat{\textbf{n}}_2)^2 + 2 (\textbf{S}_1 \cdot \hat{\textbf{n}}_1) \cdot (\textbf{S}_2 \cdot \hat{\textbf{n}}_2).
\end{equation*} | 44,142 |
Smokey Eye Makeup
Eye make-up is among the best ways to enhance the facial features by which makes them pop. There are a number of ways you can enhance your eyes, depending on preference and the shape of your eyes. Before one decides to do the make-up, it’s best to find out the eye shape. The eye shadow needs to be applied correctly and differently for each eye shape in order to enhance or conceal different portions of the eyes. Among the greatest ways to improve the eyes is with the aid of smokey eye makeup. Smokey eyes look great and are a great party make-up look for formal occasions.
Smokey eyes can be performed using different colours like: gray tones, plum tones, green tones, lavender tones, red tones, blue tones or different combinations like black and red black and blue. Choose the colour in accordance with the time you’re wearing it, using lighter colours for daytime and darker colours for night time looks. The very best way to create smoky eyes is by using professional make-up brushes. These make-up brushes will assist to create the best effect of the smokey eyes. It’s recommended to use a primer on your eyelid before the eye shadow is applied. The primer will ensure a better application of the eye shadow and a longer stay.
It’s best to use good quality eye shadows because that’s the only way you’ll be capable to create the perfect smokey eyes. Good quality eye shadow has a better coverage and the eye shadow can be applied easier. To create smokey eyes you may need 3 eye shadow shades. Shimmery eye shadow looks great on smokey eyes, but matte eye shadow looks awesome as well. You should use powder eye shadow or creamy eye shadow, depending on each persons preference. Apply the primer to the whole eyelid and apply utilizing a small make-up brush the lighter coloured eye shadow on the brow bone.
Take the second lightest eye shadow and apply the eye shadow on the interior corner of the eye until about 3 thirds into the lid. Take the darkest coloured eye shadow and apply it on the outer corner of the eye working your way on the lid. Ensure you apply a bit of the dark eye shadow over the second lightest color. Blend the dark coloured eye shadow well in the crease by wiggling the brush back and forth. Take a clean brush, a blending brush and blend the dark coloured eye shadow with the second coloured eye shadow to obtain a uniform transition from one colour to the other. Apply more eye shadow of every colour if required. Apply a bit of the second lighter colour eye shadow using an angled brush on your lower lid until about half way beginning from the interior corner of the ey | 407,387 |
In this article...
- 1 “Real” Jobs, and the Path to Self-Publishing
- 2 The Side Hustle of Selling Shirts
- 3 Getting Started on YouTube
- 4 Should We Expect Any New Courses in 2019?
- 5 Kelli’s Advice for Newbies
- 6 Resources That Have Helped Kelli in Her Business and Quarter Four Advice
- 7 Kelli on Not Doing Any Marketing for Her Business
- 8 Kelli’s Number One Tip for The Candid Cashflow Audience and Where You Can Find Her Online
- 9 That’s a Wrap
Podcast: Play in new window | Download (Duration: 25:19 — 27.8MB)
Subscribe: Google Podcasts | RSS
In a hurry? Grab a PDF of this post to take with you on the go and get all the links and goodies too! <<<
This week, The Candid Cashflow Podcast is excited to present this interview with Kelli Roberts! I met Kelli through her husband, Dale, who was and is a client of mine. I did a bit of work for Kelli and have followed her progress this year as she’s launched a course and a new YouTube channel!
Kelli focuses mainly on creating and self-publishing no content and low content books. She also has a print on demand side hustle through Merch by Amazon.
I find several really cool things about Kelli’s approach to her business:
- She doesn’t try to be a guru
- She does ZERO marketing
- If something doesn’t make money or isn’t enjoyable, she doesn’t do it
- She cares about the people she reaches – for example, she pulled her course off the market so it wouldn’t become outdated and she is still working with her students to help them with aspects that are changing in the market
Kelli is a rare gem in the make money online world. She began back in the late 90’s with eBay and her business has evolved into its current iteration. I’ll let Kelli tell you all about it. Do NOT miss this episode.
If you do nothing else, visit Kelli’s links I’ll share at the end or grab the show notes at HeyYoAva.com/Episode48 so you get all her links because you’re going to want to follow her! I will also include her links in the description area on whatever platform you are receiving this broadcast.
So, if you want to learn about self-publishing journals and other low and no content books on Amazon without marketing and scaling that to 6 figures, then you’re going to want to stay tuned!
This holiday season, choose from the Most Wished For items Amazon has to offer. Get inspired and find the perfect item for everyone on your list. See Amazon’s Most Wished For Items at HeyYoAva.com/mostwishedfor.
“Real” Jobs, and the Path to Self-Publishing
Kelli started her career with college and a degree in Political Science with ambitions toward
being a lawyer because she thought it was the “thing to do” to succeed. She ended up in a similar place as me and probably many of you out there listening. She found the corporate world chained to a desk is a soul-sucking experience.
In 1999, Kelli started a little side gig with eBay only to find she really enjoyed it and did quite well with it. At the behooving of her husband, Dale, she quit her job to do eBay full-time. That was her first experience without a 9 to 5 job, and she’ll be celebrating the 6-year anniversary of that decision in February of 2019.
Eventually, Kelli got bored with eBay and moved on to Amazon FBA. She found that Amazon FBA had all the same caveats of working with physical products that eBay had, and the model she was working with was becoming saturated quickly. She was on the lookout for something new again.
That’s when Kelli found out about self-publishing. She started out hiring ghostwriters and trying to build a brand that way but found she didn’t really like managing writers and telling them what to do.
Right about this same time 2 and a half or 3 years ago, Kelli saw a webinar about publishing low content and no content books and the light bulb flickered on. She tried it, she loved it, she made some decent money and hasn’t looked back since.
The Side Hustle of Selling Shirts
Kelli zeroed in her laser focus on a challenge from a friend in the business to target one aspect of her business. She had several smaller streams of income at the time that she was trying to maintain from her previous endeavors. In August or September of 2017, this friend suggested that she choose one thing and really crush it.
That was probably the best thing that could have happened because Kelli is not one to back down from a challenge.
Kelli chose to design. She chose designing low content and no content books as well as t-shirts for Merch since those two avenues kind of feed into each other. Once she focused in on this one aspect of her business, her income shot up.
Grant it, it was close to Quarter Four, but the upswing in income gave Kelli the confidence she needed to really go all in, and it’s paid off exponentially for her in the long run.
As far as Merch by Amazon goes, Kelli treats it as a part-time side hustle. She spends about 30 minutes to an hour per day on it. During Q4, she does a bit more. If you compare the amount of time you’re putting in with the amount you are making, Merch works great as a side business.
Kelli does upload her designs on a few other platforms, but she differentiates the designs and she doesn’t upload everything. Those platforms are:
Kelli’s biggest money maker as far as platforms go is KDP Print, formerly known as CreateSpace.
Getting Started on YouTube
Kelli also started her YouTube channel, Kelli Publish, this year. From her first video 6 months
ago (She had some unrelated videos from before, but changed the focus of her channel back in April 2018.), as of this broadcast, she has exceeded 2000 subscribers.
She shares videos about self-publishing low content and no content books as well as information about Merch by Amazon, and her live videos pull in quite a crowd each time she does one.
Should We Expect Any New Courses in 2019?
Kelli released a course in 2018 on publishing low content and no content books that did quite well but became quickly outdated with the changing of the market and Amazon’s move to merge CreateSpace and KDP Print.
This is one of the things I like about Kelli. Once she realized the information was going to get outdated so quickly, she pulled the course even though it was still making her a nice amount of money.
She doesn’t have any plans to release anything new for 2019 at this time, but she still maintains the information for her students who did invest in her course. That’s rare!
If you’ve explored self-publishing at all, then you probably know there are PLENTY of outdated courses on the topic, and because they make money, the authors leave them to mislead people. Not Kelli!
With Kelli, you get someone who is truly interested in helping people and puts that before her money-making. That’s how I like to do business as well!
Kelli’s Advice for Newbies
Don’t look at self-publishing as a get rich quick process.
People have selective hearing. They hear that I made X amount of dollars and I have X amount of books up, but they choose not to acknowledge that it took me 2 years and ton of work to get to that point. – Kelli
Do a little bit every day in your business on a consistent basis and eventually, it’s going to snowball.
Kelli’s favorite thing about what she does is the variety, she loves doing live videos on YouTube, and she also enjoys Print on Demand and watching her creative side develop.
She also finds that her daily experiences feed into her inspiration. For example, if something frustrates her, she can create a funny design based on the situation and release the tension.
She also loves the learning aspect of what she’s doing. Through research and trying things, she is constantly learning new things.
Resources That Have Helped Kelli in Her Business and Quarter Four Advice
In general terms, self-development and networking have helped her in her business. After
that, apps like MerchBuddy, the Over app for designing on the run, and Google Drive.
Kelli pulled her reports from last year to find out what her top sellers were. What she found was that her best sellers for Q4 of 2017 were designs she had uploaded that summer. It wasn’t the designs she was uploading during actual Q4.
So, Kelli’s best advice for you in Q4 is to be consistent all year long and not just during Q4.
If you’re lazy for 9 months out of the year and only hustle for 3 or 4 months, Q4 is not going to be a happy time for you. – Kelli
Kelli on Not Doing Any Marketing for Her Business
Kelli does ZERO marketing for her business outside of the content she creates for YouTube, and you’ll notice her videos do not link to her own products. Kelli doesn’t do any PPC or anything like that because she simply doesn’t like it or want to.
She has been hammered like all of us on creating an email list and the other basics of Internet Marketing, but she asked herself as she was walking home one day, “Why can’t I just create products?”
That’s exactly what she decided to do and it has worked wonderfully for her.
While she is sure she could make more money if she did market her business, she considers her marketing style to be unconventional. Instead of marketing what she’s made, she markets by trying to create what people are looking for.
Kelli believes that even if she learned marketing she might see growth in her business because she’d be spending time away from creating, so she’s happy with her current business model and we don’t blame her!
Kelli’s Number One Tip for The Candid Cashflow
Audience and Where You Can Find Her Online
Be open to anything and always keep learning. -Kelli
Kelli never saw herself as a designer. If someone had told her she would be earning 6 figures with self-publishing and a YouTube channel, she would have laughed in their face. She didn’t take any design classes in high school or college, and she’s a really shy introvert by nature. So, for her, it’s crazy that this happened to her!
Find Kelli online at:
- Her website –
- On YouTube –
- On Twitter –
That’s a Wrap
A huge thanks to Kelli for coming on the show. I’ll try to get better at thanking my guests during recording. This one ends a bit abruptly!
Also, the high ends of the audio have a bit of buzz. Lesson learned there as well. Thanks for listening and bearing with me as I feel this content is highly valuable to you as my audience.
Deciding to bring guests onto The Candid Cashflow Podcast was a huge step and will continue to be a learning process.
I hope you will consider subscribing and joining me each week for The Candid Cashflow Podcast. My focus is to help you find financial freedom through building multiple streams of income online. Subscribe in your favorite listening app at HeyYoAva.com/candidcashflow.
Don’t forget to grab a copy of the show notes so you get all the links and bonuses including where to find Kelli online at HeyYoAva.com/Episode48.
Remember, I release a new episode each Wednesday!
Until next time, turning your passion into cashflow!
Dare to be different. | 399,920 |
The 3D Transformers Environment (TE3D) provides a graphical user interface for designing transformers and reactors using Cobham Technical Services' Opera-3D finite element electromagnetic simulation package.
Users are presented with simple dialog boxes and drop-down menus to define a new transformer or reactor design. After entering this data, the software creates a 3-D finite element model of a reactor or transformer, together with independent drive and load circuits within the circuit editor for subsequent simulation and analysis.
The software allows a can be modelled through the environment. Users can modify the device and circuits following the initial build to enable precise matching of their designs, and the analysis options available within the environment can be used to analyze devices not constructed within it.
The simulation analysis phase is also automated. User options include performing open-circuit, short-circuit and inrush current tests on transformers, and mutual inductance tests on reactors.
Once an analysis has been completed, the TE3D environment automatically sends the results to the Opera Manager to be solved. In the case of the inrush current test, for example, the calculated results include the Lorentz forces on the primary and secondary windings, eddy currents in any support structures, iron losses in the transformer core, and transformer efficiency.
The TE3D environment offers fine control of the finite element analysis mesh size and distribution within each device to help balance speed with accuracy.
TE3D will also model a diverse range of reactor types, including three-phase three-leg, five-leg and both horizontal and vertical air core, and single-phase two-leg, three-leg and air core. Power systems designers will appreciate the benefits of using TE3D from the outset. By modelling the transformer or reactor, they can visualize the shape of stray flux and the areas with the highest local loss concentration. Design data can be changed in seconds, allowing “What if?” type scenarios to be investigated quickly, so that users can home-in on the optimal design solution to an application more efficiently.
Cobham Technical Services | 357,202 |
\begin{document}
\title{A Random Adaptation Perspective on Distributed Averaging}
\author{Rohit Parasnis}
\author{Ashwin Verma}
\author{Massimo Franceschetti}
\author{Behrouz Touri\thanks{Email: rparasni,a1verma,mfranceschetti,[email protected]}}
\affil{Department of Electrical and Computer Engineering, University of California San Diego}
\date{}
\maketitle
\begin{abstract}
We propose a random adaptation variant of time-varying distributed averaging dynamics in discrete time. We show that this leads to novel interpretations of fundamental concepts in distributed averaging, opinion dynamics, and distributed learning. Namely, we show that the ergodicity of a stochastic chain is equivalent to the \textit{almost sure} (\textit{a.s}.) finite-time agreement attainment in the proposed random adaptation dynamics. Using this result, we provide a new interpretation for the \textit{absolute probability sequence} of an ergodic chain. We then modify the base-case dynamics into a time-reversed inhomogeneous Markov chain, and we show that in this case ergodicity is equivalent to the uniqueness of the limiting distributions of the Markov chain. Finally, we introduce and study a time-varying random adaptation version of the Friedkin-Johnsen model and a rank-one perturbation of the base-case dynamics.
\end{abstract}
\section{INTRODUCTION}
Distributed averaging is a central mechanism to information mixing in distributed optimization~\cite{nedic2007rate,rabbat2004distributed}, distributed parameter estimation and signal processing~\cite{predd2006distributed, cattivelli2009diffusion, jadbabaie2012non, lalitha2014social, nedic2016tutorial, parasnis2022non, rangi2018multi}, decentralized control of robotic networks~\cite{bullo2008distributed}, and opinion dynamics~\cite{degroot1974reaching,acemoglu2011opinion, proskurnikov2017tutorial, proskurnikov2018tutorial, hegselmann2002opinion, parasnis2021convergence}. Hence, a variety of distributed averaging dynamics have been studied till date within different mathematical frameworks~\cite{xiao2004fast, cao2011necessary,el2016design}.
In particular, distributed averaging algorithms over time-varying networks are often analyzed using chains/sequences of stochastic matrices (a class of non-negative matrices). Several properties of such chains, such as \textit{ergodicity} and \textit{reciprocity}, have been studied in detail~\cite{blackwell1945finite, chatterjee1977towards, touri2012backward, touri2012product, martin2015continuous,bolouki2015consensus,parasnis2022towards, de2022ergodicity}. Despite this abundance of literature, interpretations of some key concepts in this area, such as Kolmogorov's absolute probability sequences, have remained elusive.
On the other hand, stochastic matrices can be interpreted by noting that every row of such matrices gives the probability mass function of a discrete random variable. Thus motivated, we introduce a \textit{random adaptation} framework in which each entry of a stochastic matrix denotes the \textit{probability} that a node $i$ adapts to the state of a node $j$, rather than the \textit{weight} assigned by $i$ to $j$.
\blue{That is,} we propose random adaptation variants of some classical discrete-time distributed averaging dynamics. We then study the proposed dynamics to interpret in a new light certain concepts that are central to the study of stochastic chains and averaging dynamics. Our contributions are as follows:
\begin{enumerate}[leftmargin=0cm,itemindent=.5cm,labelwidth=\itemindent,labelsep=0cm,align=left]
\item \textbf{\textit{An Interpretation of Ergodicity:}}
We show that the classical notion of ergodicity is \textit{equivalent} to an intuitive condition of agreement in finite time in the random adaptation dynamics. \blue{This differs from the classical scenario of distributed averaging dynamics, where ergodicity does not necessarily ensure finite-time agreement.}
\item \textit{\textbf{An Interpretation of Absolute Probability Sequence:}} Using the above characterization of ergodicity, we interpret the \textit{absolute probability sequence} of an ergodic chain as the limiting probability distribution of the common value attained by all the nodes in the proposed dynamics.
\item \textbf{\textit{Ergodicity vis-a-vis Uniqueness of the Limiting Distribution:}} We propose a time-reversed, transposed variant of the aforementioned dynamics and use it to show that the limiting distribution of the state vector is unique if and only if the given stochastic chain is ergodic. \blue{This leads to a new insight: absolute probability sequences are to ergodic chains what stationary distributions are to regular stochastic matrices.}
\item \textit{\textbf{Asymptotic Behaviors of Variants:}} Finally, we discuss the random adaptation interpretation of the well-known Friedkin-Johnsen model of opinion dynamics~\cite{friedkin1990social} as well as rank-one perturbations of the base-case dynamics. In both the cases, we study the limiting probability distributions of the agents' states/opinions.
\end{enumerate}
In addition, in Section~\ref{sec:simulations} we provide numerical examples to illustrate and interpret some of our results on the base-case dynamics as well as on our random adaptation variants of the Friedkin-Johnsen model.
\blue{\textit{Related works:} This paper is closely related to the voter model~\cite{clifford1973model,holley1975ergodic} and its many extensions (e.g.,~\cite{sood2005voter,schneider2009generalized,castellano2009nonlinear,granovsky1995noisy,congleton2004median,mobilia2007role}), which apply to social networks and describe the processes of individuals randomly adapting to their neighbors' political preferences. However, our work differs from these prior works in at least two fundamental ways. First, the voter model and most of its variants assume a limit on the number of agents that update their opinions synchronously, whereas our model does not make any assumptions on the level of synchrony among the agents' updates (note that our update rules can be used to prevent an arbitrary set of agents from updating their opinions by simply setting the corresponding diagonal entries of the associated stochastic matrices to 1). For instance, the voter model with zealots~\cite{acemoglu2013opinion} differs from our random adaptation variant of the Friedkin-Johnsen model in that, in the former, at most one agent updates its state in any given update period. Second, unlike this paper, none of the prior works aims to unravel the properties of the stochastic chains or matrices associated with the random dynamics proposed therein.}
Besides, some of our results may be {related to}
the well-studied {duality} between coalescing random walks and voter dynamics~\cite{cox1989coalescing,aldous1995reversible}.
\textit{Notation: } In this paper $\N:=\{1,2,\ldots\}$, $\N_0:=\N\cup\{0\}$, $\R$ denotes the set of real numbers, $\R^n$ denotes the set of $n$-dimensional real-valued column vectors, and $\R^{n\times n}$ denotes the set of $n\times n$ real-valued matrices. For $n\in\N$, we let ${[n]:=\{1,2,\ldots,n\}}$. For a vector $\blue{\v}\in\R^n$, $v_i$ denotes its $i$th entry, and $b_{ij}$ denotes the $(i,j)$th entry of a matrix ${B\in\R^{n\times n}}$. All matrix and vector inequalities are assumed to hold entry-wise.
For a given $n\in\N$ \blue{whose value is clear from the context}, let $O,I$ denote the $n\times n$ matrix with all zero entries and the identity matrix, respectively. We denote the column vectors with all zero entries and all one entries in $\R^n$ by $\allzero$ and $\allone$, respectively. $\blue \ee^{(i)}\in\R^n$ denotes the $i$th canonical basis vector.
For a matrix $A\in\R^{n\times n}$ and a set $S\subseteq [n]$, let $A_S$ be the principal sub-matrix of $A$ corresponding to the rows and columns indexed by $S$. Let $\bar S:=[n]\setminus S$ be the complement of $S$ with respect to $[n]$, and for sets $S,T\subseteq [n]$, let $A_{ST}$ denote the sub-matrix of $A$ corresponding to the rows indexed by $S$ and the columns indexed by $T$.
We say $\blue{\v}\in\R^n$ is \textit{stochastic} if $\v\geq \allzero$ and $\allone^T\v=1$, where the superscript $^T$ denotes transposition. A matrix $A\in\R^{n\times n}$ is called \textit{row-stochastic} or simply \textit{stochastic} if each row of \blue{$A$} is stochastic. We let $\P_n$ (respectively, $\P_{n\times n}$) denote the set of all stochastic vectors (respectively, matrices) in $\R^n$ (respectively, $\R^{n\times n}$). For a sequence $\{A(t)\}_{t=0}^\infty$ and indices $t_1<t_2$, we let
$
{A(t_2:t_1):=A(t_2-1)A(t_2-2)\cdots A(t_1)}
$
with the convention that $A(t:t):=I$ for all $t\in\N_0$.
All random objects in this work are defined with respect to an underlying probability space $(\Omega,\B,\Pr)$. With an abuse of terminology, by $\blue{Z}(t)\in \R^n$ for a random process $\{\blue{Z}(t)\}$ we mean $\blue{Z}(t,\omega)\in \R^n$ for all $t\in \N_0$ and $\omega\in\Omega$. \blue{We use }$\E[\cdot]$ \blue{to }denote the expectation operator with respect to this probability space. We use the notation $\expect Z$ to denote the expectation of a random variable/matrix/vector $Z$.
\section{RANDOM ADAPTATION DYNAMICS}~\label{sec:formulation}
In \blue{distributed} averaging dynamics, we are given a sequence of stochastic matrices $\{{Q}(t)\}\blue{_{t=0}^\infty}$, and we are interested in studying the dynamics
\begin{align}\label{eqn:averaging}
\blue{\x}(t+1)=Q(t)\blue\x(t),
\end{align}
for some initial time $t_0\in \N_0$ and initial \blue{state} $\x(t_0)\in \R^n$. As mentioned in the introduction, each row of $Q(t)$ is a stochastic vector that can be viewed as the probability mass function of a certain random variable taking $n$ values. Our random adaptation viewpoint formulates a very natural sequence of random variables that exhibits this behavior. Consider $n$ agents assuming a random state $\blue{x}_i(t)\in\R$ (more precisely, $\blue{x}_i(t,\omega)\in\R$ where $\omega\in \Omega$) evolving over discrete time $t\in\N_0$. Let the starting time $t_0\in\N_0$ and the initial states $\blue{\x}(t_0)\in \R^n$ be an arbitrary deterministic vector. For a given sequence $\{Q(t)\}$ of stochastic matrices, the random adaptation scheme is \blue{defined} as follows: At time $t\geq t_0$, agent $i$ adopts agent $j$'s state with probability $q_{ij}(t)$ independently of other agents as well as every agent's past choices and states, i.e., ${\Pr(\blue{x}_i(t+1) = \blue{x}_j(t)\mid \blue{\x}(t),\ldots, \blue{\x}(t_0))=q_{ij}(t)}$ for all $i,j\in[n]$. Therefore, we can write
\begin{align}\label{eq:main}
\blue{\x}(t+1) = A(t) \blue{\x}(t),
\end{align}
where $\blue{\x}(t)\in\R^n$ is the random state vector (whose $i$-th entry is $\blue{x}_i(t)$) and $A(t)\in\R^{n\times n}$ is a binary random stochastic matrix, i.e., $\{a_{ij}(t):i,j\in [n]\}$ are Bernoulli random variables with parameters $\expect a_{ij}(t)=q_{ij}(t)$ for all $i,j\in[n]$.
We now observe a few important properties of~\eqref{eq:main}: (i) The random process $\{\blue\x(t)\}_{t=0}^\infty$ is a non-homogeneous Markov chain with state space size at most $n^n$ as the state of each agent at every time \blue{instant} is in $\{x_i(t_0)\mid i\in [n]\}$. (ii) For each $t\in\N_0$, the rows of $A(t)$ are independent random vectors. (iii) The random matrices $\{A(t)\}_{t=0}^\infty$ are independent and hence, for each $t\in\N$, $A(t)$ is independent of $\blue\x(t)$.
\section{MAIN RESULTS}\label{sec:main_result}
We first define two properties that will be shown to be closely related. The first property relates to stochastic chains.
\begin{definition} [\textbf{Ergodicity\blue{~\cite{chatterjee1977towards,touri2012product}}}] \label{def:ergodicity} A deterministic (non-random) stochastic chain $\{Q(t)\}_{t=0}^\infty$ is said to be \textit{ergodic} if, for every $t_0\in\N_0$, there exists a vector $\blue \boldpsi (t_0)\in\P_n$
such that $\lim_{t\to\infty} Q(t:t_0)=\allone \blue\boldpsi^T(t_0)$.
\end{definition}
Put differently, a stochastic chain $\{Q(t)\}_{t=0}^\infty$ is called ergodic if every backward matrix product $Q(t:t_0)$ converges to a rank-one matrix that has identical rows.
The second property relates to the random adaptation dynamics~\eqref{eq:main}.
\begin{definition} [\blue{\textbf{Finite Agreement}}]
We say that the adaptation dynamics~\eqref{eq:main} has an \textit{a.s.} finite agreement property if for all initial time\blue{s} $t_0\in \N_0$ and initial \blue{states} $\blue\x(t_0)\in\R^n$, \blue{there exist a random scalar $y= y(t_0, \x(t_0))$ and a random (stopping) time $T\geq t_0$ such that $\x(t)=y\allone$ for all $t\geq T$}.
\end{definition}
In other words, the random adaptation dynamics~\eqref{eq:main} has the finite agreement property if at \blue{some} time after the initiation of the dynamics, all the agents \blue{adopt} the same state.
{Note that the ergodicity of $\{Q(t)\}$ does not imply finite agreement for the deterministic dynamics \eqref{eqn:averaging}. For example, suppose $n=2$, let $\x(0)={\ee}^{(1)}$, and consider the static chain defined by
$
Q(t) =
\begin{pmatrix}
p & 1-p\\
1-q & q\\
\end{pmatrix}
$
for all $t\in\N_0$, where $p,q\in (0,1)$. Then it can be verified that $\{Q(t)\}_{t=0}^\infty$ is ergodic. However, we also have ${ x_1(t)-x_2(t)=(p+q-1)^t}$ for all $t\in\N$, which implies that no agreement is reached in finite time unless $p+q=1$.
On the contrary, if $\blue\x(t_0)\in \{ \blue\ee^{(1)},\blue\ee^{(2)}\}$ the random adaptation variation \eqref{eq:main} of the same static chain ensures that the two agents reach an agreement at time ${T:=\inf\{t\geq t_0: x_1(t)\oplus x_2(t)=0\}}$, where $\oplus$ denotes \blue{the} \textit{exclusive OR} operation. Since $T$ is a geometric random variable with the parameter ${p(1-p)+q(1-q)>0}$, it is finite \textit{a.s.} In fact, even for $n\neq 2$, every ergodic chain guarantees finite agreement for all initial conditions, as we now show.}
\begin{theorem}\label{thm:main}
The random adaptation dynamics~\eqref{eq:main} has the finite agreement property if and only if ${\{\expect A(t)\}_{t=0}^\infty=\{Q(t)\}_{t=0}^\infty}$ is an ergodic chain.
\end{theorem}
\begin{proof}
We establish ergodicity first as a sufficient condition and then as a necessary condition for \textit{a.s.\ }finite agreement with an arbitrary initial condition $(t_0,\blue\x(t_0))$.
\subsubsection*{Proof of Sufficiency}
It suffices to show that an agreement occurs in the network infinitely often \textit{a.s}., because once an agreement is reached, the state vector stops evolving in time.
We first note that by Definition~\ref{def:ergodicity}, there exists a ${ \blue\boldpsi(t_0)\in\P_n}$ such that ${\lim_{t\to\infty} Q(t:t_0)=\allone \blue\boldpsi^T(t_0)}$ \textit{a.s}. As ${ \blue\boldpsi^T(t_0)\allone=1}$, there exists an index $\ell(t_0)\in[n]$ such that ${\blue\psi_{\ell(t_0)}(t_0)\geq \frac{1}{n}}$, which implies that ${\lim_{t\to\infty} Q_{i\ell (t_0)}(t:t_0)\geq \frac{1}{n}}$ for all $i\in [n]$. Consequently, there exists a time $t_1\geq t_0$ such that $ Q_{i\ell(t_0)}(t_1:t_0)\geq \frac{1}{2n}$ for all $i\in [n]$. By Lemma~\ref{lem:positive_correlation},
this implies that $\Pr\left( A_{[n]\{\ell(t_0)\}}(t_1:t_0) =\allone\right)\geq \left(\frac{1}{2n}\right)^{n}$. Since $A(t_1:t_0)\in\P_{n\times n}$ is binary, it follows that ${\Pr(A(t_1:t_0)=\allone (\blue\ee^{\ell(t_0)})^T)\geq \left(\frac{1}{2n}\right)^n}$.
As $t_0$ is arbitrary, we can repeat the above analysis with different starting times and obtain an increasing sequence of times $\{t_k\}_{k=0}^\infty$ such that ${\Pr(A(t_{k+1}:t_{k})=\allone (\blue\ee^{\ell(t_k )})^T)\geq \left(\frac{1}{2n}\right)^n}$. Since $\blue\x(t_{k+1})=A(t_{k+1}:t_k)\blue\x(t_k)$, this further implies that $\Pr(\blue\x(t_{k+1})=\blue x_{\ell}(t_k)\allone)\geq\left(\frac{1}{2n}\right)^n$. As a result, letting $C_k$ denote the event that an agreement exists in the network at time $t_k\geq t_0$, we have $\sum_{k=0}^\infty\Pr(C_k)=\infty$. Now, $\{C_k\}_{k=0}^\infty$ are independent events because $\{A(t)\}_{t=0}^\infty$ are independent and
$\{[t_k,t_{k+1}-1]:k\in\N_0\}$ are disjoint intervals. Therefore, by the Second Borel-Cantelli Lemma {\cite[Theorem 2.3.6]{durrett2019probability}},
infinitely many events among $\{C_k\}_{k=0}^\infty$ occur \textit{a.s.}, which proves the assertion.
\subsubsection*{Proof of Necessity}
Suppose there exist ${T=T(t_0,\blue\x(t_0))<\infty}$ and $\blue y=\blue y(t_0,x(t_0))\in\R$ such that $\blue\x(t)=\blue y\allone$ \textit{a.s.} for all $t\geq T$. Then $\blue\x(t)=A(t:t_0)\blue\x(t_0)$ implies that ${\lim_{t\to \infty} A(t:t_0)\blue\x(t_0) = \blue y\allone}$ \textit{a.s.}
Besides, we know that $\|\blue\x(t)\|_{\infty}\leq \|\blue\x(t_0)\|_{\infty}$ for all ${t \geq t_0}$. Therefore, by the Dominated Convergence Theorem~\cite[Theorem 1.6.7]{durrett2019probability},
we have
\begin{align}\label{eq:new}
&\E\left[\blue y\right]\allone=\E\left[ \lim_{t\to \infty} A(t:t_0)\blue\x(t_0)\right]\cr
&=\lim_{t\to \infty} \E[A(t:t_0)\blue\x(t_0)]=\lim_{t\to\infty} \expect A(t:t_0) \blue\x(t_0),
\end{align}
where the last equality follows from the independence of $\{A(t)\}_{t=0}^\infty$. As a result,
${\lim_{t\to\infty}Q(t:t_0) \blue\x(t_0) = \expect y \allone}$.
For the initial condition $\blue\x(t_0)=\blue\ee^{(i)}$, this implies that the $i$th column of $Q(t:t_0)$ converges to $\psi_i\allone$ for some scalar $\psi_i\in \R$ and hence, $\lim_{t\to\infty} Q(t:t_0)=\allone \blue\boldpsi^T$ for some vector $\blue\boldpsi=\blue\boldpsi(t_0)\in \P_n$. The latter step follows from the fact that the set of row-stochastic matrices is a closed semigroup (under matrix multiplication).
\end{proof}
\begin{remark} \label{rem:matrix_limit}
Theorem~\ref{thm:main} enables us to comment further on ergodic chains. To elaborate, we can repeat some of the arguments used in the proof above to show that if $\{Q(t)\}_{t=0}^\infty$ is ergodic, then for all ${(t_0,\blue \x(t_0))\in\N_0\times\R^n}$, there \textit{a.s.\ }exists a $\blue\boldpi(t_0)\in\P^n$ such that $\lim_{t\to\infty} A(t:t_0)=\allone \blue\boldpi^T(t_0)$ for all $t_0\in\N_0$. Moreover, $\{A(t)\}_{t=0}^\infty\in\P_{n\times n}$ being binary implies that $\blue\boldpi(t_0)$ is binary, i.e., ${\blue\boldpi(t_0)\in\{\blue\ee^{(i)}:i\in[n]\}}$. Finally, taking expectations on both sides yields ${\lim_{t\to\infty}\expect A(t:t_0)=\allone\expect{\blue\boldpi}^T(t_0)}$, i.e., ${\lim_{t\to\infty}Q(t:t_0)=\allone\expect{\blue\boldpi}^T(t_0)}$ where ${\expect{\blue\boldpi}^T(t_0)\in\P_n}$. Interestingly, for the chain $\{Q (t)\}_{t=0}^\infty$, one can verify that $\{\expect{\blue\boldpi}(t)\}_{t=0}^\infty$ forms what we call an \textit{absolute probability sequence}, a concept defined below and introduced by Kolmogorov in~\cite{kolmogoroff1936theorie}.
\end{remark}
\begin{definition}[Absolute Probability Sequence]
For a deterministic stochastic chain $\{Q(t)\}_{t=0}^{\infty}$, a sequence of stochastic vectors $\{\blue\boldpsi(t)\}_{t=0}^{\infty}$ is said to be an absolute probability sequence if $\blue\boldpsi^{T}(t+1)Q(t) = \blue\boldpsi^T(t)$ for all $t \geq 0$.
\end{definition}
We now connect this novel concept with the dynamics~\eqref{eq:main}.
\begin{theorem}
Suppose that $\{Q(t)\}_{t=0}^\infty= \{\expect A(t)\}_{t=0}^{\infty}$ is ergodic for the dynamics \eqref{eq:main}, with an absolute probability sequence $\{\blue\boldpsi(t)\}_{t=0}^{\infty} = \{\expect{ \blue\boldpi}(t)\}_{t=0}^\infty$, where $\{\blue\boldpi(t)\}_{t=0}^\infty$ is an absolute probability sequence for $\{A(t)\}_{t=0}^\infty$. Let $\blue y=\blue y(t_0,\blue\x(t_0))$ be the agreed value of \blue{all }the agents, i.e., $\lim_{t\to \infty} \blue\x(t)=\blue y\allone$ \textit{a.s} for initial conditions ${(t_0,\blue\x(t_0))\in\N_0\times \R^n}$ such that $\{\blue x_i(t_0)\}_{i=1}^n$ are all distinct. Then the probability distribution of $\blue y$ is given by
${p_i(t_0):=\Pr( \blue y = \blue x_i(t_0)) = \psi_i(t_0)}$ for all ${i \in [n]}$.
\end{theorem}
\begin{proof}
By Remark~\ref{rem:matrix_limit}, we almost surely have ${\lim_{t\to\infty} \blue\x(t)=\lim_{t\to\infty} A(t:t_0) \blue\x(t_0)=\allone \blue\boldpi^T(t_0)\blue\x(t_0)}$. Thus, ${ \blue y=\blue\boldpi^T(t_0)\blue\x(t_0)}$, which implies that ${\expect {\blue y}=\blue{\expect \boldpi}^T(t_0)\blue \x(t_0)}=\sum_{i=1}^n \psi_i(t_0)x_i(t_0)$.
On the other hand, the definition of expectation implies that $\blue{\expect y}=\sum_{i=1}^n p_i(t_0) x_i(t_0)$.
Hence, ${\sum_{i=1}^n \psi_i(t_0)\blue x_i(t_0) = \sum_{i=1}^n p_i(t_0) \blue x_i(t_0)}$. Since this holds for all $\blue\x(t_0)\in\R^n$, we must have $p_i(t_0) = \psi_i(t_0)$ for all $i\in[n]$.
\end{proof}
\section{VARIANTS AND EXTENSIONS}
\subsection{Time-Reversed Non-homogeneous Markov Chains}
Let $\{A(t)\}_{t=0}^\infty$ be a random sequence of independent binary matrices, and let ${\{\blue\ee^{(i)}:i\in [n]\}}$ be the state space of a {time-reversed} Markov chain whose probability transition matrix is $Q(t):=\expect A(t)$ at time $t$. To be precise, the Markov chain is a random process $\{\blue\z(t)\}$ that starts at an arbitrary time instant $t_\infty\in\N$ with an arbitrary probability distribution given by $\blue{\mathbf p}_\infty\in\P_n$ (where ${(\blue{\mathbf p}_\infty)_i:=\Pr(\blue\z(t_\infty)= \blue\ee^{(i)})}$ and evolves backwards in time with ${\Pr(\blue\z(t)=\blue\ee^{(j)}\mid \blue\z(t+1)=\blue\ee^{(i)}) =q_{ij}(t)}$ for all ${t<t_\infty}$. Equivalently,
\begin{align}\label{eq:time_rev}
\blue\z^T(t)=\blue\z^T(t+1)A(t)
\end{align}
for all $t\in\{t_\infty-1,t_\infty-2, \ldots, 0\}$. Note that~\eqref{eq:time_rev} is nothing but a time-reversed, transposed variant of~\eqref{eq:main}.
To relate these dynamics to time-homogeneous chains, recall that the limiting probability distribution of a regular Markov chain is a stationary distribution independent of the initial distribution~\cite{lalley}.
Analogously, we ask, is the limiting distribution of a time-reversed inhomogeneous Markov chain an absolute probability sequence of the associated stochastic chain that is independent of $\blue{\mathbf p}_\infty$? As we show, the answer is yes if and only if the stochastic chain is ergodic.
\begin{theorem} \label{thm:time-rev}
Consider the dynamics~\eqref{eq:time_rev} with a variable starting time $t_\infty$. Let $\{\blue \boldpsi(t)\}_{t=0}^\infty$ be an absolute probability sequence for $\{Q(t)\}_{t=0}^\infty$. Then the limiting distribution $\blue{\mathbf p}(t):=\sum_{i=1}^n p_i(t) \blue\ee^{(i)} \in\P_n$ with $p_i(t):=\lim_{t_\infty\to\infty}\Pr(\blue\z(t)=\blue\ee^{(i)})$ exists and is invariant w.r.t.\ the initial distribution $\blue{\mathbf p}_\infty$ for all $t\in\N_0$ if and only if $\{Q(t)\}_{t=0}^\infty$ is ergodic, in which case $\blue{\mathbf p}(t)=\blue\boldpsi(t)$ for all $t\in\N_0$.
\end{theorem}
\begin{proof}
We first note that for all $t\leq t_\infty$, we have $\Pr(\blue\z(t)=\blue\ee^{(i)})=\E[\blue z_i(t)]=\blue{\expect z}_i(t)$ for all ${i\in [n]}$, which means that $ p_\infty = \blue{\expect \z}(t_\infty)$ and more generally that the probability distribution of $\blue \z(t)$ is determined by $\blue{\expect \z}^T(t) = \blue{\expect \z}^T(t_\infty)\expect A(t_\infty:t) = \blue{\mathbf p}_\infty^T Q(t_\infty:t)$ for all $t\leq t_\infty$. Thus, $\blue{\mathbf p}^T(t)=\blue{\mathbf p}_\infty^T\lim_{t_\infty\to\infty}Q(t_\infty:t)$ (if the limit exists). Using this, we first establish the sufficiency of ergodicity and then its necessity for the invariance assertion to hold.
If $\{Q(t) \}_{t=0}^\infty$ is ergodic, then $\blue{\mathbf p}^T(t)$ is given by $\blue{\mathbf p}_\infty^T\lim_{t_\infty\to\infty}Q(t_\infty:t)\stackrel{(a)}=\blue{\mathbf p}_\infty^T\allone \blue\boldpsi^T(t)\stackrel{(b)}=\blue\boldpsi^T(t)$, where $(a)$ follows from Remark~\ref{rem:matrix_limit} and $(b)$ holds because $\blue{\mathbf p}_\infty\in\P_n$. Since $\{ \blue\boldpsi(t)\}_{t=0}^\infty$ are unique (see~\cite{blackwell1945finite}, Theorem 1), it follows that $\blue{\mathbf p}(t)= \blue\boldpsi(t)$ \textit{a.s.\ }does not vary with $\blue{\mathbf p}_\infty$.
On the other hand, if $\{Q(t) \}_{t=0}^\infty$ is not ergodic, then there exists a $t_0\in \N_0$ such that either ${\lim_{t_\infty\to\infty} Q(t_\infty:t_0)}$ does not exist (in which case there is nothing to prove), or there exists an index $\ell\in [n]$ such that the column vector ${\blue{\v}:=\lim_{t_\infty\to\infty} Q_{[n]\,\{\ell\}}(t_\infty:t_0)}$ satisfies $\blue{\v}\neq \alpha \allone$ for all ${\alpha\in \R}$. Therefore, we can write $\blue{\v}=\alpha \allone+\beta \blue{\mathbf w}$ for some $\alpha,\beta>0$ and \blue{some} $\blue{\mathbf w}$ with $\blue{\mathbf w}^T \allone=0$. Note that ${\blue{\mathbf w}^T\blue{\v}\not=0}$.
Note \blue{also} that for small enough $\tilde{\beta}>0$, $\tilde{\blue{\mathbf w} }={\frac{1}{n}\allone+\tilde{\beta}\blue{\mathbf w}\in\P_n}$. Now, for $\blue{\mathbf p}_\infty=\frac{1}{n}\allone$ and $\blue{\mathbf p}_\infty=\tilde{\blue{\mathbf w}}$, \blue{the value of } ${ p}_\ell(t_0)=\left(\blue{\mathbf p}_\infty^T\lim_{t_\infty\to\infty} A(t_\infty:t_0)\right)_\ell$ would be $\frac{1}{n}\allone^T\blue{\v}$ and $\frac{1}{n}\allone^T\blue{\v}+\tilde{\beta}\blue{\mathbf w}^T\blue{\v}$, respectively, which along with $\blue{\mathbf w}^T\blue{\v}\neq 0$ violate\blue{s} the invariance condition.
\end{proof}
\blue{Theorem~\ref{thm:time-rev} also shows that, just as stationary distributions are the limiting probability distributions of Markov chains defined by regular matrices, absolute probability sequences can be interpreted as the limiting distributions of time-reversed Markov chains defined by ergodic stochastic chains.}
\subsection{Random Adaptation Approach to Friedkin-Johnsen Model}\label{subsec:rv_fj}
The dynamics~\eqref{eqn:averaging} can be viewed as a time-varying version of the French-Degroot opinion dynamics model where agent opinions move towards convex combinations of other agents' opinions. The Friedkin-Johnsen model, in addition to being partly influenced by neighbors, introduces a \textit{prejudice} that affects the agents' opinions. Mathematically,
\begin{align}\label{eq:fj}
\blue\x(t+1) = \Lambda W(t)\blue\x(t) + (I-\Lambda) \blue\u,
\end{align}
where $\blue\x(t)\in\R^n$ denotes the vector of opinions, $\blue\u\in\R^n$ is the vector of the agents' prejudices, $W(t)\in\R^{n\times n}$ denotes the \textit{influence matrix}, which describes how the agents influence each other, and $\Lambda\in\R^{n\times n}$ is a diagonal matrix whose $i$th diagonal entry, $\lambda_i\in[0,1]$, denotes the \textit{susceptibility} of agent $i$ to social influence and $1-\lambda_i$ denotes the susceptibility of agent $i$ to her prejudice $\blue u_i$.
Similar to~\eqref{eq:fj}, we can provide a random adaptation variation of the Friedkin-Johnsen model as follows. In the $t$-th time period, agent $i$ decides between adapting to \blue{her} neighbor's opinion versus adapting to \blue{her} prejudice. Her choice is independent of other agents' choices and her own past choices. With a probability $\gamma(t) \in(0,1)$, she follows the adaptation scheme described earlier, and with probability $1- \gamma(t)$, she resets her opinion to her prejudice $u_i\in\R$. As before, we assume the initial state vector $\blue\x(0)\in\R^n$ to be arbitrary. This results in the update rule
\begin{align}\label{eq:random_fj}
\blue\x(t+1) = \Lambda(t) A(t) \blue\x(t) + (I-\Lambda(t))\blue\u,
\end{align}
where $\blue\x(t)$ and $A(t)$ have their usual meanings, $\blue\u\in\R^n$ is a vector of external influences/prejudices, and $\{\Lambda(t)\}_{t=0}^\infty$ is a sequence of diagonal matrices whose diagonal entries $\{\lambda_i(t):i\in[n]\}$ are Bernoulli random variables with $\Pr(\lambda_i(t)=1)=\gamma_i(t)$.
\begin{remark}~\label{rem:proof_aid}Observe that~\eqref{eq:random_fj} is a special case of~\eqref{eq:main} by letting $\blue\y^T(t):=[\blue\x^T(t)\,\, \blue\u^T]$ and \blue{by} noting \blue{that}
\begin{align}\label{eq:higher_dim}
\blue\y(t+1) = B(t) \blue\y(t),
\end{align}
where $B(t):=
\begin{pmatrix}
\Lambda(t) A(t)\, & I - \Lambda (t)\\
O_{n\times n} \, & I_{n\times n}
\end{pmatrix}$.
As a result, for any $t,t_0\in\N_0$ with $t\geq t_0$, we have $\blue\y(t)=B(t:t_0)\blue\y(t_0)$ with $B_{[n]}(t:t_0)= P(t:t_0)$, where $P(t):=\Lambda(t)A(t)$. Note also that $P(t)\leq A(t)$ because $\Lambda(t)\leq I$.
\end{remark}
We now define two terms: \textit{dominance in expectation} and \textit{simultaneously malleable} agents, and we show that, if $\{Q(t)\}_{t=0}^\infty$ is an ergodic chain with simultaneously malleable agents that dominate in expectation, then all the agents' opinions will almost surely enter the \blue{prejudice set} (the set of external influences) $\U:=\{u_i:i\in[n]\}$ in finite time.
\begin{definition} [\textbf{Dominance in Expectation}] The agents of a set $S\subseteq[n]$ are said to \textit{dominate in expectation} if $\sum_{t=0}^\infty \allone^T Q_{\bar S S}(t)\allone=\infty$ and $\sum_{t=0}^\infty \allone^T Q_{S\bar S}(t)\allone<\infty$.
\end{definition}
In the the average-case scenario, if a set of agents $S\subseteq [n]$ dominate in expectation, then the agents of $S$ significantly influence the rest of the agents $\bar S$ without themselves being significantly influenced by $\bar S$ in the long run.
\begin{definition} [\textbf{Simultaneously Malleable Agents}]\label{def:sma} The agents of a set $S\subseteq[n]$ are said to be \textit{simultaneously malleable} if $\sum_{t=0}^\infty \prod_{i\in S}(1-\gamma_i(t))=\infty$.
\end{definition}
Essentially, simultaneously malleable agents are those whose probability of simultaneously adapting to their respective external influences does not vanish too fast with time.
\begin{theorem} \label{thm:malleable} For the dynamics~\eqref{eq:random_fj}, suppose ${\{Q(t)\}_{t=0}^\infty=\{\expect A(t)\}_{t=0}^\infty}$ is ergodic, and suppose there exists a set $S\subseteq[n]$ of simultaneously malleable agents that dominate in expectation. Then there \textit{a.s.\ }exists a time $T<\infty$ such that $\{\blue x_i(T)\}_{i=1}^n\subseteq\U$.
\end{theorem}
\begin{proof}
First, note that the decisions taken in the network at time $t$ are independent across agents. As a result,
$
\Pr\left(\bigcap_{i\in S}\left\{ \lambda_i(t)=0\right\}\right) = \prod_{i\in S} (1-\gamma_i(t)).
$
In light of Definition~\ref{def:sma} and the Second Borel-Cantelli Lemma, this further implies that there exists an increasing sequence of random times $\{T_k\}_{k=1}^\infty$ such that $\lambda_i(T_{k}-1)=0$ \textit{a.s.\ }for all $i\in S$ and all $k\in\N$. This means that $ \blue x_i(T_k) = u_i$ \textit{a.s.\ }for all $i\in S$ and all $k\in\N$.
On the other hand, we can use the union bound to show that
${\Pr( \bigcup_{i\in S} \bigcup_{i\in \bar S} \{ a_{ij}(t)=1\} )}$ is at most \\$ \sum_{i\in S} \sum_{j\in \bar S} q_{ij}(t) = \allone^T Q_{S\bar S}(t)\allone$. Since $\sum_{t=0}^\infty \allone^T Q_{S\bar S}(t)\allone < \infty$, it follows from the First Borel-Cantelli Lemma~\cite[Theorem 2.3.1]{durrett2019probability}
that there exists a random time $T^*<\infty$ such that $a_{ij}(t)=0$ \textit{a.s.\ }for all $i\in S$, $j\in \bar S$, and $t\geq T^*$. This means that there \textit{a.s.} exists a point of time $T^*$ after which the agents in $S$ are never influenced by those in $\bar S$.
Let $K:=\inf\{k\in\N_0: T_k> T^*\}$. Then, $T_K>T^*$ and $\blue x_i(T_K)=u_i$ \textit{a.s.\ }for all $i\in S$. Hence, ${\{\blue x_i(t):i\in S\}\subseteq\U\bigcup\{\blue x_i(T_K):i\in S\} \subseteq \U}$ for all ${t\geq T_K}$.
It remains to show the existence of a time $T\geq T_K$ such that $\{\blue x_i(t):i\in\bar S\}\subseteq\U$ for all $t\geq T$. By the definition of ergodicity, the truncated chain $\{Q(t)\}_{t=\tau}^\infty$ is ergodic for all $\tau\in\N_0$. It follows from Remark~\ref{rem:matrix_limit} that there exists a random vector ${\blue\boldpi(\tau)\in\{\blue\ee^{(i)}:i\in[n]\}}$ such that $\lim_{t\to\infty}A(t:\tau) = \allone \blue\boldpi^T(\tau)$ \textit{a.s.} Thus, ${\lim_{t\to\infty}A(t:T_K) = \allone \blue\boldpi^T(T_K)}$ \textit{a.s.} On the other hand, for $t\geq T_K\geq T^*$, we have $a_{ij}(t)=0$ \textit{a.s.\ }
for all $i\in S$ and ${j\in \bar S}$. Hence, $A_{S\bar S}(t:T_K)=O$ \textit{a.s.\ }for all $t\geq T_K$. It follows that $\lim_{t\to\infty} A
_{S\bar S}(t:T_K)=O$ \textit{a.s.}, which means that $\blue\boldpi^T(T_K)\notin\{e^{(i)}:i\in\bar S\}$ \textit{a.s}. Since $\allone\blue\boldpi^T(T_K)$ has identical rows, this further implies that the columns of ${\lim_{t\to\infty} A(t:T_K)}$ indexed by $\bar S$ are all zero \textit{a.s.} It now follows from Remark~\ref{rem:proof_aid} that
\begin{align*}
\limsup_{t\to\infty}B_{[n]\bar S}(t:T_K) &= \limsup_{t\to\infty}P_{[n]\bar S}(t:T_K)\cr
&\leq \limsup_{t\to\infty} A_{[n]\bar S}(t:T_K)=O\quad a.s.
\end{align*}
Equivalently, for all sufficiently large $t$, the entries of $\blue\y(t) = B(t:T_K)\blue\y(T_K)$ are binary convex combinations of ${\{\blue x_i(T_K):i\in S\}\bigcup\U=\U}$. This completes the proof.
\end{proof}
We now consider a special case of~\eqref{eq:random_fj} in which the probability distributions of the agents' opinions converge to limits that can be computed using closed-form expressions.
\begin{theorem}\label{thm:fj_model}
Suppose the matrix pairs $\{(\Lambda(t), A(t))\}_{t=0}^\infty$ are independent and identically distributed. Also, suppose $\Gamma<I$, where $\Gamma:=\expect\Lambda(t)$ and $Q:=\expect A(t)$ for all $t\in\N_0$. Finally, suppose that $|\U|=n$ and that $\{\blue x_i(0)\}\bigcap\U=\emptyset$. Then the following assertions hold.
\begin{enumerate} [leftmargin=6mm,label={(\roman*)}]
\item \label{item:one} We have ${\lim_{t\to\infty}\Pr(\blue x_i(t)=u_j) = v_{ij}}$ for all ${i,j\in [n]}$, where $\{v_{ij}:i,j\in[n]\}$ are the entries of
\begin{align}
V:=\left(I-\Gamma Q\right)^{-1}\left(I-\Gamma\right).
\end{align}
\item \label{item:two} There \textit{a.s.\ }exists a random time $T<\infty$ such that ${\blue x_i(t)\in\U}$ for all $t\geq T$.
\end{enumerate}
\end{theorem}
\begin{proof}
We first recall from Remark~\ref{rem:proof_aid} that ${\blue\y(t) = B(t:0) \blue\y(0)}$ for all $t\in\N_0$. Since $\U$ has $n$ distinct elements and since $\{\blue x_i(0)\}\bigcap \U=\emptyset$, this implies that
\begin{align}\label{eq:just_expect}
\Pr(\blue x_i(t)=u_j) &= \Pr\left(B_{i\, n+j}(t:0)=1\right)
= \E\left[B_{i\, n+j}(t:0)\right]\nonumber\\
&\stackrel{(a)}= \expect{B}_{i\,n+j}(t:0)\stackrel{(b)}=\left(\left(\expect B(0)\right)^t\right)_{i\,n+j},
\end{align}
where $(a)$ and $(b)$ hold because $\{(\Lambda(t),A(t))\}_{t=0}^\infty$ are i.i.d.
Thus, it suffices to evaluate $\lim_{t\to\infty} (\expect B(0))^t$. Observe that for the expected dynamics $\blue{\expect \x}(t+1) = \Gamma Q \expect {\blue\x}(t) + (I- \Gamma)\blue\u$, we have
${\blue{\expect \y}(t) = (\expect B(0))^t \blue\y(0)}$ as a consequence of Remark~\ref{rem:proof_aid}. On the other hand, we know from Theorem 21 and Corollary 22 in~\cite{proskurnikov2017tutorial} that $\lim_{t\to\infty} \blue{\expect \x}(t) = V \blue\u$, which implies that
$$
\lim_{t\to\infty} \blue{\expect \y}(t) =
\begin{pmatrix}
O_{n\times n} & V\\
O_{n\times n} & I
\end{pmatrix}
\begin{pmatrix}
\blue\x(0)\\
\blue\u
\end{pmatrix} =
\begin{pmatrix}
O_{n\times n} & V\\
O_{n\times n} & I
\end{pmatrix} \blue\y(0).
$$
That is,
$ \lim_{t\to\infty} (\expect B(0))^t \blue\y(0)=
\begin{pmatrix}
O_{n\times n} & V\\
O_{n\times n} & I
\end{pmatrix} \blue\y(0).
$
As $\blue\y(0)$ (which stacks the initial states and the external influences) is arbitrary, it follows that ${\left(\lim_{t\to\infty}(\expect B(0))^t\right)_{i\,n+j} = v_{ij}}$ for all $i,j\in [n]$. In light of~\eqref{eq:just_expect}, this proves~\ref{item:one}.
To prove~\ref{item:two}, note that $\prod_{i\in [n]}(1- \gamma_i(t))$ is positive and time-invariant because $\Gamma(t)=\Gamma<I$ for all $t\in\N_0$. Hence, all the agents in the network are simultaneously malleable. Since they also dominate in expectation trivially,~\ref{item:two} follows immediately from Theorem~\ref{thm:malleable}.
\end{proof}
\subsection{Rank-One Perturbation of the Friedkin-Johnsen Variant}\label{sec:rank_one}
Another random adaptation-based variant of the Friedkin-Johnsen model can be obtained by letting the opinion of each agent `mutate' to any external influence with a fixed probability distribution, i.e., in the $t$th time period, agent $i$ either adapts to a neighbor's opinion or adapts to one of the prejudices independently of her past choices. With probability $ \gamma_i(t)\in (0,1)$, the agent follows the adaptation scheme described earlier (in~\eqref{eq:main}), and with probability ${ 1 - \gamma_i(t)} $, however, instead of adapting to one fixed prejudice, she chooses an opinion from the set ${\U=\{u_i:i\in [n]\}}$, according to a stochastic vector $\blue{\mathbf q}$ on $\U$. That is,
\begin{align}\label{eq:rankone_dynamics}
\blue\x(t+1) = \Lambda(t)A(t)\blue\x(t) + (I - \Lambda(t))C(t)\blue\u,
\end{align}
where $\blue\x(t)$ and $A(t)$ are as before, $\blue\u \in \R^n$ is the vector of external influences, $\{\Lambda(t)\}_{t=0}^\infty$ is a sequence of random binary diagonal matrices with $\Pr(\lambda_i(t) = 1) = \gamma_i(t)$, and $\{C(t)\}_{t=0}^\infty$ is a sequence of i.i.d.\ binary stochastic random matrices. For any $i \in [n]$, $\Pr(C_{ij}(t) =1) = q_j$ for all $j \in [n]$, independent of the other rows. Here $\expect C(t) = \allone \blue{\mathbf q}^T $, for all $t\in \N_0$, which is a rank-one matrix for all $t \in \N_0$.
\begin{theorem}\label{thm:rankone_perturb}
Consider the dynamics \eqref{eq:rankone_dynamics} where $|\U|=n$, $\{\blue x_i(0):i\in[n]\} \bigcap \U = \blue\emptyset$, and the matrix pairs $\{(\Lambda(t), A(t))\}_{t=0}^{\infty}$ are i.i.d.\ with $Q:= \expect A(t)$ and $\Gamma := \expect \Lambda(t)$ for all $t \in \N_0$. Also, suppose that $\Gamma< I$. Then the following hold true.
\begin{enumerate} [label={(\roman*)}, leftmargin =0.6cm]
\item We have $\lim_{t\to \infty} \Pr(\blue x_i(t) = u_j) = v_{ij}$ for all ${i,j \in [n]}$, where $\{v_{ij}:i,j \in[n]\}$ are the entries of
\begin{align*}
V &:= (I-\Gamma Q)^{-1}(I- \Gamma)\allone \blue {\mathbf q}^T.
\end{align*}
\item There \textit{a.s.\ }exists a random time $T< \infty$ such that ${\blue x_i(t) \in \U}$ for all $t\geq T$. \label{enum:thmPerturb_2nd}
\end{enumerate}
\end{theorem}
\begin{proof}
We can rewrite the dynamics as ${\blue{\y}(t+1) = D(t)\blue\y(t)}$, where $\blue{\y^T}(t) = \begin{bmatrix}\blue\x^T(t) & \blue\u^T\end{bmatrix}$ and
\begin{align}
D(t) &=
\begin{bmatrix}
\Lambda(t) A(t) & (\blue{I}- \Lambda(t))C(t) \\
O_{n\times n} & I_{n\times n}
\end{bmatrix}.
\end{align}
So, we have $\blue\y(t) = D(t:0)\blue\y(0)$, for all $t\in \N_0$. Since the elements of $\U$ are distinct and $\{\blue x_i(0)\mid i\in[n]\}\bigcap \U = \blue\emptyset$, similar to ~\eqref{eq:just_expect}, we have $\Pr(\blue x_i(t) = u_j) = \left( \left( \expect D(0) \right)^t \right)_{i\,n+j}$. To compute the limiting marginal probability note that
\begin{align}
\left( \expect D(0)\right)^t &=
\begin{bmatrix}
\Gamma Q & (\blue I-\Gamma) \expect C(0) \\
O_{n\times n} & I_{n\times n}
\end{bmatrix}^t
=
\begin{bmatrix}
\left(\Gamma Q\right)^t & R(t) \\
O_{n\times n} & I_{n\times n}
\end{bmatrix},
\end{align}
where ${ R(t) = \sum_{k=0}^{t-1} \left(\Gamma Q\right)^{k} (I-\Gamma) \allone \blue{\mathbf q}^T.}$
Since ${Q\in\P_{n\times n}}$, we know that the maximum absolute value of eigenvalues of $\Gamma Q$ is less than $1$ since $\Gamma Q \leq \gamma_{\max} Q$, where $\gamma_{\max}= \max_{i\in[n]} \gamma_i \in (0, 1)$. Using a result on Neumann Series (Eq.~(7.10.11) in \cite{meyer2000matrix}), we have ${\lim_{t \to\infty} R(t) = (I-\Gamma Q)^{-1}(I-\Gamma)\allone \blue{\mathbf q}^T}=V$, and $\lim_{t\to \infty} \left(\Gamma Q\right)^t = O_{n\times n}. $
For \ref{enum:thmPerturb_2nd}, note that the probability that all the agents adapt to an external influence at any time is ${\prod_{i=1}^n (1-\gamma_i)>0}$. Since $\{\Lambda(t)\}_{t=0}^\infty$ are independent, it follows from the Second Borel-Cantelli Lemma that there exists an increasing sequence of random times $\{T_k\}_{k=1}^{\infty}$ such that $\Lambda(T_k)=1$. This implies that $\blue x_i(T_1+1) \in \U$ for all $i\in [n]$. Therefore, for all $(t_0,\blue\x(t_0))\in\N_0\times \R^n$, there \textit{a.s.\ }exists a random time $T<\infty$ such that $\blue x_i(t) \in \U$ for all $t \geq T$.
\end{proof}
\begin{remark}
Suppose, in addition to the assumptions in Theorem~\ref{thm:rankone_perturb}, all the agents have identical susceptibility, i.e., $ \Gamma = \gamma I$ for some $\gamma\in(0,1)$. Then, since ${Q \in\P_{n\times n}}$, we have ${R(t) = \sum_{k=0}^{\infty} \gamma^k (1-\gamma) \allone \blue{\mathbf q}^T}$, and since ${\gamma \in (0,1)}$, we have ${V = \allone \blue{\mathbf p}^T}$. Furthermore, for this case, the result extends to all stochastic chains $\{Q(t)\}_{t=0}^\infty$ and not just to identically distributed chains, as ${R(t) = \sum_{k=0}^{t} \gamma^k (1-\gamma) Q(k+1:0) \allone \blue{\mathbf q}^T}$ with ${Q(t+1:0)\allone = \allone}$, for all $t\in \N_0$, which implies that ${V= \lim_{t\to\infty}R(t)= \allone \blue{\mathbf q}^T}$. Therefore, in this case, the limiting marginal probability distribution is independent of the degree of susceptibility.
\end{remark}
\blue{\begin{remark}
Note that the dynamics studied in Sections~\ref{subsec:rv_fj} and \ref{sec:rank_one} can be obtained as special cases of the generalized model studied in \cite{ravazzi2014ergodic}. However, while the results of \cite{ravazzi2014ergodic} imply convergence in distribution and provide the expected values of the steady states, they do not characterize the distributions of these steady states, nor do they show the a.s. convergence of the agents' opinions to the prejudice set.
\end{remark}
}
\section{SIMULATIONS}\label{sec:simulations}
We now illustrate some of our main results with the help of suitable numerical examples generated using MATLAB. These examples are aimed at facilitating the reader's understanding of the key ideas developed in this work.
\begin{example} [\textbf{Base Case Dynamics with Ergodicity}]\label{eg:one} Consider Equation~\eqref{eq:main}, the simplest among all of the random adaptation dynamics we have analyzed above. Suppose we have $n=10$ agents in a social network, and for simplicity, suppose that the initial states (or opinions) of the agents are given by $x_i(0)=i$ for all $i\in[n]$.
In order to simulate the case of the expected stochastic chain $\{Q(t)\}_{t=0}^\infty$ being ergodic, we first assume a finite time horizon of $H=1000$, use \emph{randfixedsum.m}~\cite{rand} to randomly generate an indefinitely long (but finite) sequence of $n\times n$ stochastic matrices, select among the generated matrices the first $H$ matrices that are irreducible (i.e., the first $H$ that can be expressed as the adjacency matrices of strongly connected directed graphs), and then set the finite sequence $\{Q(t)\}_{t=0}^{H-1}$ equal to the sequence of the irreducible matrices thus obtained.
Next, we obtain $\{A(t)\}_{t=0}^{H-1}$ as the random matrices generated using the probability distributions defined by $\{Q(t)\}_{t=0}^H$. We then choose an arbitrary realization of $\{A(t)\}_{t=0}^{H-1}$ and plot the corresponding dynamics~\eqref{eq:main} in Figure~\ref{fig:base_case} below.
\end{example}
\begin{figure}[htp]
\centering
\includegraphics[scale=0.36]{base_case.png}
\caption{Simulation of the base case dynamics~\eqref{eq:main} with $\{Q(t)\}$ being an ergodic chain}
\label{fig:base_case}
\end{figure}
In the above example, observe that all the agents' opinions reach the same value within the first $30$ time periods. This is in agreement with Theorem~\ref{thm:main}, which implies that finite agreement occurs almost surely when the expected stochastic chain $\{Q(t)\}=\{\expect A(t)\}$ is ergodic.
\begin{example} [\textbf{Base Case Dynamics without Ergodicity}] We again consider Equation~\eqref{eq:main} and repeat the procedure described in Example~\ref{eg:one}, except that we now generate $\{Q(t)\}_{t=0}^{H-1}$ as follows: we first generate two sequences $\{R(t)\}_{t=0}^{H-1},\{S(t)\}_{t=0}^{H-1}\subset\R^{\frac{n}{2}\times\frac{n}{2} }$ of irreducible matrices, we then generate two arbitrary sequences of random non-negative matrices $\{\Delta_1(t)\}_{t=0}^{H-1},\{\Delta_2(t)\}_{t=0}^{H-1}\subset\R^{\frac{n}{2}\times\frac{n}{2} }$, we let
$$
\tilde Q(t) =
\begin{pmatrix}
B(t) & \frac{1}{t^2}\Delta_1(t)\\
\frac{1}{t^2}\Delta_2(t) & C(t)
\end{pmatrix},
$$
and we finally set $q_{ij}(t) = \frac{\tilde q_{ij}(t) }{\sum_{k=1}^n \tilde q_{ik}(t)}$ for each $i,j\in [n]$ so as to obtain $Q(t)$ as a row-stochastic matrix. The decay factor of $\frac{1}{t^2}$ ensures that the influence of the agent subsets $\{1,2,\ldots, \frac{n}{2}\}$ and $\{\frac{n}{2} ,\frac{n}{2}+1,\ldots, n\}$ on each other diminishes with time fast enough to make $\{Q(t)\}_{t=0}^{H-1}$ a good approximation of a non-ergodic chain.
Next, we obtain $\{A(t)\}_{t=0}^{H-1}$ as the random matrices generated using the probability distributions defined by $\{Q(t)\}_{t=0}^H$. We then choose an arbitrary realization of $\{A(t)\}_{t=0}^{H-1}$ and plot the corresponding dynamics~\eqref{eq:main} in Figure~\ref{fig:non-ergodic} below.
\end{example}
\begin{figure}[htp]
\centering
\includegraphics[scale=0.36]{non-ergodic.png}
\caption{Simulation of the base case dynamics~\eqref{eq:main} with $\{Q(t)\}$ being a non-ergodic chain}
\label{fig:non-ergodic}
\end{figure}
In the above example, two clusters of agents reach two distinct consensus values within the first $15$ time periods. This is consistent with Theorem~\ref{thm:main}, which states that finite agreement almost surely does not occur in the base case dynamics if $\{Q(t)\}$ is not an ergodic chain. Moreover, it can be shown that the principal submatrices of $\tilde Q(t)$ corresponding to the coordinates $\{1,2,\ldots,\frac{n}{2}\}$ form an ergodic stochastic chain. The same can be said about the principal submatrices corresponding to the complementary set $\{\frac{n}{2},\ldots,n-1, n\}$, which, along with Theorem~\ref{thm:main}, explains why exactly two consensus clusters are formed in the associated random adaptation dynamics.
\begin{example} [\textbf{Comparison of the Base Case Random Adaptation Dynamics with the Expected Dynamics}] We now repeat the setup of Example~\ref{eg:one} and plot in Figure~\ref{fig:comparison} the evolution of $\x(t)$ averaged over $1000$ different realizations of $\{A(t)\}_{t=0}^{H-1}$ as a set of $n$ line plots. To compare the resulting plots with the associated expected dynamics, i.e., the classical non-random averaging dynamics defined by~\eqref{eqn:averaging}, we simulate~\eqref{eqn:averaging} and plot the corresponding state evolution of the $n$ agents using solid red square markers whose color intensity varies with the agent index $i\in[n]$.
\end{example}
\begin{figure}[htp]
\centering
\includegraphics[scale=0.36]{two_dynamics.png}
\caption{Comparison of the Empirical Mean of the Base Case Dynamics with the Expected Dynamics~\eqref{eqn:averaging}}
\label{fig:comparison}
\end{figure}
In Figure~\ref{fig:comparison}, we observe that the line plots and the square marker plots exhibit finite agreement and also overlap with each other almost perfectly. This is consistent with Theorem~\ref{thm:main} and with the observation that
$$
\E[\x(t)]=\E[A(t:t_0)\x(t_0)]=\E[A(t:t_0)]\x(t_0) = Q(t:t_0)\x(t_0),
$$
where the last step follows from the mutual independence of $\{A(t)\}_{t=0}^\infty$. This observation means that the distributed averaging dynamics defined by~\eqref{eqn:averaging} is nothing other than the expectation of the random adaptation dynamics defined by~\eqref{eq:main}. Hence, the empirical average of $\x(t)$ plotted using solid lines must be a good approximation for the averaging dynamics~\eqref{eqn:averaging}.
\begin{example} [\textbf{Random Adaptation Variant of Friedkin-Johnsen Dynamics}] Finally, we simulate~\eqref{eq:random_fj}, the random adaptation variant of the Friedkin-Johnsen model. We first set $n=10$ and generate $\{Q(t)\}_{t=0}^{H-1}=\{\expect A(t)\}_{t=0}^{H-1}$ using the procedure described in Example~\ref{eg:one}. We then generate $\{\gamma_i(0):i\in [n]\}$ as random variables uniformly distributed over the set $[0,1]$, and we make sure that $0<\gamma_i(0)<1$ for all $i\in[n]$. Next, we set $\gamma_i(t)=\gamma_i(0)$ for all $i\in[n]$ and all $t\in\{0,1,\ldots, H-1\}$ so as to ensure that $\{\Lambda(t)\}_{t=0}^{H-1}$ are i.i.d. random matrices. As the next step, we generate $\lambda_i(t)=\text{Bernoulli}(\gamma_i(0))$ for each $i\in[n]$ and $t\in\{0,1,\ldots, H-1\}$.
As for the initial opinions and prejudices, we choose $u_i=i+20$ and $x_i(0)=i$ for all $i\in [n]$ to ensure that $\{x_i(0):i\in [n]\}\bigcap \mathcal U=\emptyset$. We then plot the results of our simulation in Figure~\ref{fig:fj} below.
\end{example}
\begin{figure}[htp]
\centering
\includegraphics[scale=0.36]{FJ.png}
\caption{Simulation of the Random Adaptation Variant~\eqref{eq:random_fj} of the Friedkin-Johnsen Model}
\label{fig:fj}
\end{figure}
As seen from Figure~\ref{fig:fj}, although the initial opinions (states) of all the agents lie in the set $\{1,2,\ldots, 10\}$, all the agents eventually enter the prejudice set $\{21,22,\ldots,30\}$. This happens because $\{\Lambda(t)\}_{t=0}^{H-1}$ are i.i.d., which implies that $[n]$ is a set of simultaneously malleable agents that dominate in expectation trivially. Therefore, by Theorem~\ref{thm:malleable}, all the initial opinions of the agents get erased within a finite time-span.
In summary, all of our simulation results are consistent with, and hence validate, our main theoretical results.
\section{CONCLUSION}\label{sec:conclusion}
We proposed and studied a random adaptation variant of time-varying distributed averaging dynamics in discrete time. We have shown that our models give rise to novel interpretations for the concepts of ergodicity and absolute probability sequences, both of which are pivotal to the study of stochastic chains. We have also proposed a time-varying, stochastic analog of the well-known Friedkin-Johnsen opinion dynamics and analyzed the asymptotic behavior of the probability distributions of the agents' opinions. Finally, we have considered a rank-one perturbation of our base-case stochastic dynamics and studied its asymptotic behavior.
\blue{Our random adaptation interpretation of averaging (and related) dynamics opens up many avenues avenues for further investigations including the (time-varying) controlled variation of Friedkin-Johnsen dynamics through the lens of random adaptation dynamics, and the connection of mutation-adaptation learning dynamics (in game theoretic setting) and our proposed random adaptation schemes.}
\section*{Appendix}
\begin{lemma} \label{lem:positive_correlation}
For the dynamics~\eqref{eq:main} \blue{ with $Q(t)=\expect A(t)$}, the following holds for all $S\subseteq [n]$, $\ell\in[n]$, $t\in\N_0$ and $\Delta \in\N$
\begin{align}\label{eq:pos_cor}
\Pr\left(\bigcap_{i\in S} \left(A(t+\Delta:t)\right)_{i\ell}=1\right)\geq \prod_{i\in S}Q_{i\ell}(t+\Delta: t).
\end{align}
\end{lemma}
\begin{proof}
We use induction on $\Delta$. For $\Delta=1$, the assertion follows from the independence of the rows of $A(t)$.
Assume now that the assertion holds for some $\Delta\geq 1$. Let $S\subseteq[n]$ be a given set. W.l.o.g.\ we assume $S=[v]$ for some $v\in[n]$. In addition, for each $i\in[n]$, let $\sigma_i(t)$ denote the random index such that $a_{i\sigma_i(t)}(t)=1$, let $\tilde \sigma(t):=(\sigma_1(t),\ldots, \sigma_v(t))$ and let $\tilde \alpha=(\alpha_1,\ldots, \alpha_v)$ be a realization of $\tilde{\sigma}(t)$. Let ${W:=\{i\in S: \alpha_i\notin\{\alpha_1,\ldots, \alpha_{i-1}\}\}}$ index all the distinct $\alpha_i$s. Then, for $\Delta+1$, we have
$\bigcap_{i\in S}\left\{A_{i\ell}(t+\Delta+1:t)=1\right\}=\bigcap_{i\in S}\{A_{i\ell}(t+\Delta+1:t)=1\}\cap\{\sigma(t) = \tilde \alpha\}$.
Since, $A(t)$ has independent rows and because $\{(A(t+\Delta+1:t))_{i\ell}=1\}=\{ (A(t+\Delta:t))_{\alpha_i\ell}=1\}$ conditional on the event $\{\sigma_i(t)=\alpha_i\}$, we have
\begin{align*}
\Pr\left(\bigcap_{i\in S}\left\{A_{i\ell}(t+\Delta+1:t)=1\right\}\right)&{=}\sum_{\tilde \alpha\in[n]^v}\bigg(\Pr(\bigcap_{i\in S}\{ A_{\alpha_i \ell}(t+\Delta:t)=1 \}\mid \tilde \sigma(t)=\tilde \alpha)\cr
&\quad\quad\quad\quad\quad\times\prod_{j\in S} q_{j\sigma_j(t)}(t)\bigg)\cr
&\stackrel{(a)}{=}\sum_{\tilde \alpha\in[n]^v}\Pr(\bigcap_{i\in S}\{ A_{\alpha_i \ell}(t+\Delta:t)=1 \})\cdot\prod_{j\in S} q_{j\sigma_j(t)}(t)\cr
&=\sum_{\tilde \alpha\in[n]^v}\Pr(\bigcap_{i\in W }\{ A_{\alpha_i \ell}(t+\Delta:t)=1 \})\cdot\prod_{j\in S} q_{j\sigma_j(t)}(t)\cr &\stackrel{(b)}{\geq} \sum_{\tilde \alpha\in [n]^v} \prod_{i\in W} Q_{\alpha_i \ell}(t+\Delta:t) \cdot\prod_{j\in S} q_{j\sigma_j(t)}(t)\cr
&\stackrel{(c)}{\geq} \sum_{\tilde \alpha\in [n]^v} \prod_{i\in S} Q_{\alpha_i \ell}(t+\Delta:t) \cdot\prod_{j\in S} q_{j\sigma_j(t)}(t)
\end{align*}
where $(a)$ holds because $\{A(\tau)\}_{\tau=0}^\infty$ are independent, $(b)$ follows from our inductive hypothesis, and $(c)$ holds because $W\subseteq V$ and because all matrix entries lie in $[0,1]$. Noting that the last expression is simply $\prod_{i\in S} Q_{i\ell}(t+\Delta+1:t)$ completes the proof.
\end{proof}
\bibliographystyle{ieeetr}
\bibliography{bib}
\end{document} | 161,629 |
TITLE: Choosing People For A Committee With Limitations
QUESTION [3 upvotes]: From a group of 8 women and 6 men, a committee consisting of 3 men and 3 women is to be formed.
How many different committees are possible if
(c) 1 man and 1 woman refuse to serve together?
committee without the "problematic" woman ${6\choose 3}\cdot{7\choose 3}$
committee without the "problematic" man ${5\choose 3}\cdot{8\choose 3}$
Now it seems that I counted twice committes without the "problematic" people so overall it is ${6\choose 3}\cdot{7\choose 3} + {5\choose 3}\cdot{8\choose 3}-{5\choose 3}\cdot{7\choose 3}=910$.
Is there a way to calculate the committees without "over-counting"?
REPLY [4 votes]: There are ${8 \choose 3} \cdot {6 \choose 3}$ committees in total, of which ${7 \choose 2} \cdot {5 \choose 2}$ contain the problematic pair.
This gives ${8 \choose 3} \cdot {6 \choose 3}-{7 \choose 2} \cdot {5 \choose 2}$ committees without the problematic pair. | 35,649 |
TITLE: Probability of having l balls in the same box given N identical balls and n boxes.
QUESTION [1 upvotes]: Given $N$ identical balls I have to distribute them into $n$ boxes. Any number of balls can be in one box.
What is the probability of having $l$ balls in one box?
Is this correct?
$p(l) = \frac{1}{n^{l-1}} \binom{N}{l} = \frac{1}{n^{l-1}} \frac{N!}{(N-l)!l!}$
I asked this one differently, with an explanation of how to reach to the formula: Understanding the $n!$ in the probability of $l$ events from a set of $N$ to happen at the same timestep out of $n$ total but I didn't get any feedback.
And now, based on these two related questions, I think my expression might be correct.
Probability of exactly one empty box when n balls are randomly placed in n boxes.
Distributing n identical balls in k distinct boxes
REPLY [1 votes]: I think there's not a closed form for your question, since it is the number of partitions of $N-l$ balls into $n$ boxes over the partitions of $N$ balls into $n$ boxes.
Your answer is wrong for many reasons, but first of all because it is not less than 1!
For example, if $n=1$, then you have 1 box, and necessarily $p(l)=0$ for every $l\ne N$. Your formula, instead, gives
$$p(l) = \binom Nl$$
that is surely distinct from 1 or zero for many values $l$. | 44,459 |
TITLE: Why is the dimension of the set separable states $\dim\mathcal H_1+\dim\mathcal H_2$?
QUESTION [6 upvotes]: Please can you help me to understand how the dimension of the set of separable states is $\dim \cal H_1 + \dim \cal H_2$?
This is the relevant passage:
So far, we have assumed implicitly that the system is made of a single component. Suppose a system is made of two components; one lives in a Hilbert space $\cal H_1$ and the other in another Hilbert space $\cal H_2$. A system composed of two separate components is called bipartite. Then the system as a whole lives in a Hilbert space $\cal H = \cal H_1 \otimes \cal H_2$, whose general vector is written as
$$\left|\, \psi \right\rangle = \sum_{i,j} c_{ij} \left|\,e_{1,i}\right \rangle \otimes \left|\,e_{2,j}\right\rangle, \tag{2.29}$$
where $\{|\,e_{a,i}\rangle\}$ ($a=1,2$) is an orthonormal basis in $\cal H_a$ and $\sum_{i,j} |c_{ij}|^2 = 1$.
A state $|\,\psi \rangle \in \cal H$ written as a tensor product of two vectors as $|\,\psi \rangle = |\,\psi_1 \rangle \otimes |\,\psi_2\rangle$, ($|\,\psi_a\rangle \in \cal H_a$) is called a separable state or a tensor product state. A separable state admits a classical interpretation such as “The first system is in the state $|\,\psi_1\rangle$, while the second system is in $|\,\psi_2\rangle$.” It is clear that the set of separable states has dimension $\dim \cal H_1 + \dim \cal H_2$.
REPLY [5 votes]: This might not be what Nakahara has in mind, but one can make sense of this using the idea of projective Hilbert spaces. Let $\mathcal{P}(\mathcal{H})$ denote the projective space associated to the "normal" space $\mathcal{H}$.
The subset of separable states is not a subvectorspace in the proper sense, as Holographer notes. Yet it can be understood as a projective subvariety of the projective space associated with the tensor product of the underlying Hilbert spaces - it is the image of the Segre embedding, being a smooth embedding
$$ \mathcal{P}(\mathcal{H}_1) \times \mathcal{P}(\mathcal{H}_2) \to \mathcal{P}(\mathcal{H}_1 \otimes \mathcal{H}_2), (\psi,\phi) \mapsto \psi \otimes \phi$$
where $\mathcal{P}(\mathcal{H}_1) \times \mathcal{P}(\mathcal{H}_2)$ are the separable states.1 In the language of projective varieties, this image is a $(m-1)+(n-1)$ dimensions projective subvariety of $\mathcal{P}(\mathcal{H}_1 \otimes \mathcal{H}_2)$, but since we should more properly see $m' = m - 1$ and $n' = n - 1$ - the dimensions of the projective spaces - as the dimensional of the actual spaces of states, we obtain indeed that the subvariety corresponding to the separable states has the sum of the dimensions of the individual states as its dimension.
1Note that on ordinary Hilbert spaces, this is not even injective, let alone an embedding in any proper sense, since $\psi \otimes \phi = k\psi \otimes \frac{1}{k}\phi$ means that $(\psi,\phi)$ and $(k\psi,\frac{1}{k}\phi)$ map to the same element of the tensor product space. | 66,378 |
The popular Muddy Mayhem guide to The Dressage Diva, limited edition text over image print. Signed & numbered by author A3 Framed in black wood effect frame. (Swear words are starred...so you can hang on your wall without offending granny!) Sorry currently only available in the UK!
Limited Edition Print - The Dressage Diva
£20.00 Regular Price
£18.00Sale Price | 84,099 |
Sin Humo Vapor Lounge Vape Wholesale in Plainville 02762 MA
Sin Humo Vapor Lounge Vape Shop based in Plainville 02762 MA : Buy CBD Pet Treats for Cats and Dogs, eliquid by premium brand just like Cheeba Made In USA E-liquid, California Grown E-Liquids SALTS as well as Smoothy Man E-Juice SALT
What You Must Find Out About CBD Before Using It on Your Skin?
CBD is the brief form of Cannabidiol that can be derived from hemp as well as marijuana plants. Both these come from the marijuana plant family members. Tetrahydrocannabinol (THC) and also CBD are both main components discovered in marijuana plant, that has numerous health benefits.
THC provides individuals a "high," while CBD doesn't provide that euphoric result. That's why CBD is coming to be so popular as well as is made use of in health as well as skincare products. CBD exists in tablets, creams, drinks, vapors as well as also used in salad dressing.
There are no adequate research study done on CBD benefits, studies reveal that CBD is being used for centuries now for treating numerous health ailments like chronic discomfort, insomnia, anxiety and a lot a lot more when ingested. When used topically CBD has been made use of for dealing with skin problem like dermatitis, psoriasis, sunburn, acne etc.,
cbd skin care and various other CBD skin products like, cream, balm as well as oil can be valuable to your skin since of the antioxidants and from this source also anti-inflammatory buildings that help in reducing dry skin, redness, rashes, sores or inflamed skin. Researches even reveal that CBD has an all-natural way of safe antibacterial component also.
You can take pleasure in the adhering to advantages by utilizing CBD instilled soap.
Treat Acne: Acne is the most usual skin issue dealt with by people all over the world. Acne is caused by excess secretion of sweat glands. The linolenic acid in CBD oil normalizes this secretion as well as assists treating acne like, whitehead, blackhead, papules and so on,
Wrinkles: CBD infused soaps connects with the endocannabinoid system in our body and also helps decrease the aging procedure. You can get a youthful and glowing skin in a cost-effective means by utilizing CBD soaps.
With the aid of CBD soaps, you need not fret regarding sunburns any longer. CBD oil keeps the skin moist as well as prevents skin from blistering.
Hydrating: due to the necessary fats like omega 3, 6, and also 9 CBD soaps provides you superb skin-related advantages and helps maintain your skin supple and also soft.
Eczema & Psoriasis: hemp-based CBD soaps are so much helpful in dealing with flaky and also itchy skin problem like dermatitis, psoriasis as well as various other inflammatory skin problem.
Relieving Pain: CBD oil has excellent medicine homes like that of analgesic and also is a good replacement to various other drugs that have prospective negative effects. When used rigorously and also aids lower any kind of persistent inflammatory discomfort, CBD oil penetrates with the skin.
There are a lot of companies more hints emerging to market CBD items. Selecting the genuine one is a difficult job. Besides learning more about CBD products, you need to likewise know just how to locate the best firm.
Real firms will certainly be transparent to their consumers as well as reveal info concerning, where the plants are grown, are they lawfully sourced or otherwise, whether their CBD items were examined by 3rd event lab, whether it has any contaminants, the quantity of CBD and also THC that it includes.
Allueur is an emerging firm to offer CBD items originated from naturally sourced hemp plants. You can obtain top quality CBD skin care items from right here.
That's why CBD is ending up being so prominent and is used in health and wellness and skincare products. CBD is there in tablets, creams, beverages, vapors as well as also made use of in salad dressing.
The linolenic acid in CBD oil stabilizes this secretion and also aids treating acne like, whitehead, blackhead, papules and so on,
With the assistance of CBD soaps, you need not fret about sunburns any longer. CBD oil keeps the skin moist and stops skin from blistering.
A mouth watering style of peppermint. an outstanding option for Individuals hunting for a much more extreme style than conventional
The Totem array catches the eye because of the alluring flavour combinations along with the PG / VG ratio favoring vegetable glycerine leading to exceptional steam manufacturing.
Bowsen's Mate is a refined look here combination of fresh mint with nuances of chocolate and vanilla. fragile, but still...
from the absence of information on harmlessness for the fetus, active or passive vaping of the e liquid is not really proposed for pregnant or breastfeeding Women of all ages, especially if it is made up of nicotine.
A blend of sunshine blond tobacco using a contact of vanilla coffee finely assembled. Sweet and enjoyable, this liquid can be appreciated at any time on the working day.
All merchandise within the FUU range are entirely generated in Paris. Fuu masters all manufacturing from the to Z and has his personal Assessment laboratory.
The Watermelon speaks for by itself, but Karma Komp eJuice Salt Nic it’s quite exceptional to find a watermelon that is in fact fantastic. I like watermelon, And that i’ve tried dozens of watermelon flavors and in no way genuinely Like several of these right up until the Watermelon sweet from Mad Hatter.
I've appreciated their flavors in equally the disposables as well as the cigalikes, but inside of a mod or an Moi design vape, the flavors do occur as a result of with a little more kick.
French maker in the north of France, Avap make a degree of honor to offer you excellent eliquid with inimitable flavors.
Oui Non Nos Karma Komp eJuice Salt Nic produits sont réservés aux personnes majeures. Pour en savoir furthermore, cliquez ici. en entrant sur ce web-site vous acceptez nos mentions légales The Nicotine contained inside the items can build sturdy dependancy.
undoubtedly well worth hoping, particularly when you like sweet and sweeter e-liquid. It’s not overpowering in almost any way, but is sweet adequate to fulfill my late evening sweet-tooth, and maintain me smoke no cost simultaneously.
OUCH is usually a recently emerging "bodyfullness" clinic based in Copenhagen. Its main focus could be the fascial substructure of your body. Fascia is connective tissue which has unbelievable Bodily Attributes, and As outlined by a expanding number of physique therapists is integral in effectively being. If muscle mass may be the motor of executing, fascia — Based on OUCH — is the sector of staying. as a result, its tactic goes beyond ache administration or aid, to take care of becoming alone, with alignment and gravity mapping, vibration, deep tissue massage, as well as a longterm rehabilitation protocol.
This solution may well consist of nicotine, a check out your url chemical recognized to the point out of California to bring about start defects or other reproductive hurt.
Guides pick your e cigarette your e liquid your clearo your box your battery fabricate your e liquid your coil know more details on cbd accumulators learn our glossary resources E Liquids | 23,837 |
Class 1 2013
First you had to guess what a set of 16 words meant in the context of the teacher's life- She then gave you the information in first person.
I started to teach in 1989
I will be 43 in April
I got married in December 1993
I have a single daughter
I was born in Uruguay
I went to the USA in 2011
Before I had visited Argentina,Paraguay and Brazil
I went to Argentina twice
I quitted smoking when I was 28 years old
I can bake wonderful pizza
She was born in Uruguay. She will be 43 in April. She started to teach in 1989. She got married in December 1993. She has a single daughter. She went to the USA in 2011. She had visited Argentina, Paraguay and Brazil. She went to Argentina twice. She can bake wonderful pizza. She quitted smoking when I was 28 years old. She bought a ford fiesta last year. | 361,438 |
\begin{document}
\allowdisplaybreaks[4]
\begin{abstract}
In this article, we consider the problem of periodic homogenization of a Feller process generated by a
pseudo-differential operator, the so-called L\'evy-type process.
Under the assumptions that the generator has rapidly periodically oscillating coefficients, and that it admits ``small jumps'' only (that is, the jump kernel has finite second moment), we prove that the appropriately centered and scaled process converges weakly to a Brownian motion with covariance matrix given in terms of the coefficients of the generator.
The presented results generalize the classical and well-known results related to
periodic homogenization of a diffusion process.
\end{abstract}
\maketitle
\section{Introduction}\label{S1}
The classical reaction-diffusion equation
\begin{equation*}\partial_t p(t,x)\,=\,\langle b(x),\nabla_xp(t,x)\rangle +
\frac{1}{2}{\rm Tr}\,c(x)\nabla_x^{2}p(t,x)
+r\bigl(p(t,x)\bigr)\end{equation*} describes the evolution of population density due to random displacement of individuals (diffusion term), movement of individuals within the environment (drift term), and their reproduction (reaction term).
In order to characterize long-range effects the diffusion and drift terms are
naturally replaced by
an integro-differential operator of the following form \begin{equation}\begin{aligned}\label{IDO} \mathcal{L}f(x)\,=\, & \langle b(x),\nabla f(x)\rangle+
\frac{1}{2}{\rm Tr}\,c(x)\nabla^{2}f(x)
\\&+\int_{\R^{d}}
\left(f(x+y)-f(x)-\langle y,\nabla f(x)\rangle \Ind_{B_1(0)}(y)\right)\,\nu(x,\D y)\,,\end{aligned}\end{equation}
where $\nu(x,\D y)$ is
a non-negative Borel kernel which describes these effects, that is, it quantifies the property that an individual at $x$ jumps to $x+\D y$.
The main goal of this article is to discuss periodic homogenization of the operator $\mathcal{L}$, with kernel $\nu(x,\D y)$ admitting ``small jumps'' only (that is, having finite second moment). Our approach is
based on probabilistic techniques. More precisely, we discuss periodic homogenization of the stochastic (Markov) process $\process{X}$ in
periodic medium, generated by $\mathcal{L}$. We focus to the case when $\process{X}$ is a so-called L\'evy-type process or, equivalently, when $\mathcal{L}$ is a pseudo-differential operator (see below for details).
Roughly speaking, we show that
the
appropriately centered and scaled process $\process{X}$: \begin{equation}\label{e:eeeff}\{\varepsilon X_{\varepsilon^{-2}t}-\varepsilon^{-1}\bar {b^*} t\}_{t\ge0},\end{equation}
for some $\bar {b^*}\in \R^d$,
converges, as $\varepsilon\to0$, in the path space endowed with the Skorohod ${\rm J}_1$-topology to a $d$-dimensional zero-drift Brownian motion determined by covariance matrix of the form \begin{equation}\begin{aligned}\label{ET1.2}\Sigma\,:=\,\Bigg(&\int_{\mathbb{T}_\tau^d}\sum_{k,l=1}^{d}\left(\delta_{ki}-\partial_k\beta_i(x)\right)c_{kl}(x)\left(\delta_{lj}-\partial_l\beta_j(x)\right)\pi(\D x)\\
&+\int_{\mathbb{T}_\tau^d}\int_{\R^{d}}y_iy_j\,\nu(x,\D y)\,\pi(\D x)\\
&+\int_{\mathbb{T}_\tau^d}\int_{\R^d}\bigl(\beta_i(x+y)-\beta_i(x)\bigr)\bigl(\beta_j(x+y)-\beta_j(x)\bigr)\nu(x,\D y)\,\pi(\D x)\\& -2\int_{\mathbb{T}_\tau^d}\int_{\R^d}y_i\bigl(\beta_j(x+y)-\beta_j(x)\bigr)\nu(x,\D y)\,\pi(\D x)
\Bigg)_{1\leq i,j\leq d}\,,\end{aligned}\end{equation}
(see \Cref{T1.1} for details). Equivalently, according to \cite[Theorem 7.1]{rene-bjorn-jian}, $$\lim_{\varepsilon\to0}\lVert\mathcal{L}_{\varepsilon}f-\varepsilon^{-1}\langle\bar {b^*},\nabla f\rangle-2^{-1}{\rm Tr}\,\Sigma \nabla^2f\rVert_\infty\,=\,0\,,\qquad f\in C_c^\infty(\R^d)\,,$$ where
\begin{align*} \mathcal{L}_\varepsilon f(x)\,=\, & \varepsilon^{-1}\langle b(x/\varepsilon),\nabla f(x)\rangle+
\frac{1}{2}{\rm Tr}\,c(x/\varepsilon)\nabla^{2}f(x)
\\&+\varepsilon^{-2}\int_{\R^{d}}
\left(f(x+\varepsilon y)-f(x)-\varepsilon \langle y,\nabla f(x)\rangle \Ind_{B_1(0)}(y)\right)\,\nu(x/\varepsilon,\D y)\,.\end{align*}
Let us remark that when $b(x)\equiv0$ and $\nu(x,\D y)$ is symmetric for all $x\in\R^d$, centralization in \cref{e:eeeff} is not necessary (that is, one can take $\bar {b^*}=0$), and $\beta(x)\equiv 0$ in \cref{ET1.2}. Thus, in this case, $\Sigma$ is reduced to
\begin{equation*}\begin{aligned} \Bigg(&\int_{\mathbb{T}_\tau^d} c_{ij}(x) \,\pi(\D x)+\int_{\mathbb{T}_\tau^d}\int_{\R^{d}}y_iy_j\,\nu(x,\D y)\,\pi(\D x)
\Bigg)_{1\leq i,j\leq d}\,,\end{aligned}\end{equation*} (see \cite{hom} for more details).
\subsection*{Preliminaries on L\'evy-Type Processes}\label{SS1.1}
Let
$(\Omega,\mathcal{F},\{\mathbb{P}_{x}\}_{x\in\R^{d}},\process{\mathcal{F}},\process{\theta},$ \linebreak $\process{X})$, denoted by $\process{X}$
in the sequel, be a Markov process on state space
$(\R^{d},\mathcal{B}(\R^{d}))$ (see \cite{BG-68}). Here, $d\geq1$, and
$\mathcal{B}(\R^{d})$ denotes the Borel $\sigma$-algebra on
$\R^{d}$. Due to the Markov property, the associated family of linear operators $\process{P}$ on
$B_b(\R^{d})$ (the space of bounded and Borel measurable functions),
defined by $$P_tf(x)\,:=\, \mathbb{E}_{x}\bigl[f(X_t)\bigr]\,,\qquad t\geq0\,,\
x\in\R^{d}\,,\ f\in B_b(\R^{d})\,,$$ forms a \emph{semigroup} on the
Banach space $(B_b(\R^{d}),\lVert\cdot\rVert_\infty)$, that is, $P_0={\rm Id}$ and $P_s\circ
P_t=P_{s+t}$ for all $s,t\geq0$. Here, $\mathbb{E}_x$ stands for the expectation with respect to $\mathbb{P}_x(\D\omega)$, $x\in\R^{d}$, and
$\lVert\cdot\rVert_\infty$ and ${\rm Id}$ denote the supremum norm and the identity operator, respectively, on the space
$B_b(\R^{d})$. Moreover, the semigroup $\process{P}$ is
\emph{contractive} ($\lVert P_tf\rVert_{\infty}\leq\lVert f\rVert_{\infty}$
for all $t\geq0$ and $f\in B_b(\R^{d})$) and \emph{positivity
preserving} ($P_tf\geq 0$ for all $t\geq0$ and $f\in
B_b(\R^{d})$ satisfying $f\geq0$). The \emph{infinitesimal generator}
$(\mathcal{A}^{b},\mathcal{D}_{\mathcal{A}^{b}})$ of the semigroup
$\process{P}$ (or of the process $\process{X}$) is a linear operator
$\mathcal{A}^{b}:\mathcal{D}_{\mathcal{A}^{b}}\to B_b(\R^{d})$
defined by
$$\mathcal{A}^{b}f\,:=\,
\lim_{t\to0}\frac{P_tf-f}{t},\qquad f\in\mathcal{D}_{\mathcal{A}^{b}}\,:=\,\left\{f\in B_b(\R^{d}):
\lim_{t\to0}\frac{P_t f-f}{t} \ \textrm{exists in}\
\lVert\cdot\rVert_\infty\right\}\,.
$$ We call $(\mathcal{A}^{b},\mathcal{D}_{\mathcal{A}^{b}})$ the \emph{$B_b$-generator} for short.
A Markov process $\process{X}$ is said to be a \emph{Feller process}
if its corresponding semigroup $\process{P}$ forms a \emph{Feller
semigroup}. This means that
\medskip
\begin{itemize}
\item [(i)] $\process{P}$ enjoys the \emph{Feller property}, that is, $P_t(C_\infty(\R^{d}))\subseteq C_\infty(\R^{d})$ for all $t\geq0$;
\medskip
\item [(ii)] $\process{P}$ is \emph{strongly continuous}, that is, $\lim_{t\to0}\lVert P_tf-f\rVert_{\infty}=0$ for all $f\in
C_\infty(\R^{d})$.
\end{itemize}
\medskip
\noindent Here, $C_\infty(\R^{d})$ denotes
the space of continuous functions vanishing at infinity. Recall also that a Markov process $\process{X}$ is said to be a $C_b$-\emph{Feller}
(resp.\ \textit{strong Feller}) process if the corresponding semigroup $\process{P}$ satisfies $P_tf\in C_b(\R^d)$ for all $t>0$ and all $f\in C_b(\R^d)$
(resp.\ $f\in B_b(\R^d)$), where $C_b(\R^d):=C(\R^d)\cap B_b(\R^d)$.
Note that
every Feller semigroup $\process{P}$ can be uniquely extended to
$B_b(\R^{d})$ (see \cite[Section 3]{rene-conserv}). For notational
simplicity, we denote this extension by $\process{P}$ again. Also,
let us remark that every Feller process (admits a modification that)
has c\`adl\`ag sample paths and
possesses the strong Markov
property (see \cite[Theorems 3.4.19 and
3.5.14]{jacobIII}). Further,
in the case of Feller processes, we call
$(\mathcal{A}^{\infty},\mathcal{D}_{\mathcal{A}^{\infty}}):=(\mathcal{A}^{b},\mathcal{D}_{\mathcal{A}^{b}}\cap
C_\infty(\R^{d}))$ the \emph{Feller generator} for short. Observe
that in this case $\mathcal{D}_{\mathcal{A}^{\infty}}\subseteq
C_\infty(\R^{d})$ and
$\mathcal{A}^{\infty}(\mathcal{D}_{\mathcal{A}^{\infty}})\subseteq
C_\infty(\R^{d})$. If the set of smooth functions
with compact support $C_c^{\infty}(\R^{d})$ is contained in
$\mathcal{D}_{\mathcal{A}^{\infty}}$, that is, if the Feller
generator
$(\mathcal{A}^{\infty},\mathcal{D}_{\mathcal{A}^{\infty}})$ of the
Feller process $\process{X}$ satisfies
\medskip
\begin{description}
\item[(\textbf{LTP1})]
$C_c^{\infty}(\R^{d})\subseteq\mathcal{D}_{\mathcal{A}^{\infty}}$,
\end{description}
\medskip
\noindent then, according to \cite[Theorem 3.4]{courrege-symbol},
$\mathcal{A}^{\infty}|_{C_c^{\infty}(\R^{d})}$ is a \textit{pseudo-differential operator}, that is, it can be written in the form
\begin{equation}\label{PDO}\mathcal{A}^{\infty}|_{C_c^{\infty}(\R^{d})}f(x) \,=\, -\int_{\R^{d}}q(x,\xi)\E^{i\langle \xi,x\rangle}
\hat{f}(\xi)\, \D\xi\,,\end{equation} where $\hat{f}(\xi):=
(2\pi)^{-d} \int_{\R^{d}} \E^{-i\langle\xi,x\rangle} f(x)\, \D x$ denotes
the Fourier transform of the function $f(x)$. The function $q :
\R^{d}\times \R^{d}\to \CC$ is called the \emph{symbol}
of the pseudo-differential operator. It is measurable and locally
bounded in $(x,\xi)$, and is continuous and negative definite as a
function of $\xi$. Hence, by \cite[Theorem 3.7.7]{jacobI}, the
function $\xi\mapsto q(x,\xi)$ has for each $x\in\R^{d}$ the
following L\'{e}vy-Khintchine representation \begin{equation}\begin{aligned}\label{SIMB}q(x,\xi) \,=\,&a(x)-
i
\langle \xi, b(x)\rangle + \frac{1}{2}\langle\xi,c(x)\xi\rangle\\& +
\int_{\R^{d}}\left(1-\E^{i\langle\xi,y\rangle}+i\langle\xi, y\rangle\Ind_{B_1(0)}(y)\right)\nu(x,\D y)\,,\end{aligned}\end{equation}
where $a(x)$ is a non-negative Borel measurable function, $b(x)$ is
an $\R^{d}$-valued Borel measurable function,
$c(x):=(c_{ij}(x))_{1\leq i,j\leq d}$ is a symmetric non-negative
definite $d\times d$ matrix-valued Borel measurable function,
and $\nu(x,\D y)$ is a non-negative Borel kernel on $\R^{d}\times
\mathcal{B}(\R^{d})$, called the \emph{L\'evy
kernel}, satisfying
$$\nu(x,\{0\})\,=\,0\qquad \textrm{and} \qquad \int_{\R^{d}}\left(1\wedge
|y|^{2}\right)\nu(x,\D y)\,<\,\infty,\qquad x\in\R^{d}\,.$$
The quadruple
$(a(x),b(x),c(x),\nu(x,\D y))$ is called the \emph{L\'{e}vy quadruple}
of
$\mathcal{A}^{\infty}|_{C_c^{\infty}(\R^{d})}$ (or of $q(x,\xi)$).
Let us remark that local boundedness of $q(x,\xi)$ implies local boundedness of the corresponding $x$-coefficients, and \textit{vice versa}
(see \cite[Lemma 2.1 and Remark 2.2]{rene-holder}).
In the sequel, we assume the following condition on the symbol
$q(x,\xi)$:
\medskip
\begin{description}
\item[(\textbf{LTP2})] $q(x,0)=a(x)\equiv0$.
\end{description}
\medskip
\noindent This condition is closely related to the conservativeness property of $\process{X}$.
Namely, under the assumption that the
$x$-coefficients
of $q(x,\xi)$ are uniformly bounded (which is certainly the case in the periodic setting), (\textbf{LTP2}) implies that $\process{X}$ is \emph{conservative}, that is, $\mathbb{P}_{x}(X_t\in\R^{d})=1$ for all $t\geq0$ and
$x\in\R^{d}$ (see \cite[Theorem 5.2]{rene-conserv}). Further, note that by combining \cref{PDO,SIMB}
with (\textbf{LTP2}), $\mathcal{A}^{\infty}|_{C_c^{\infty}(\R^{d})}$ takes the form \cref{IDO}.
Conversely, if $\mathcal{L}: C_c^\infty(\R^d)\to C_\infty(\R^d)$ is a linear operator of the form \cref{IDO} satisfying the so-called \textit{positive maximum principle} ($\mathcal{L}f(x_0)\le0$ for any $f\in C_c^\infty(\R^d)$ with $f(x_0)=\sup_{x\in\R^d}f(x)\ge0$) and such that $\bigl(\lambda-\mathcal{L}\bigr)(C_c^\infty(\R^d))$ is dense in $C_\infty(\R^d)$ for some (or all) $\lambda>0$, then, according to
the
Hille-Yosida-Ray theorem, $\mathcal{L}$ is closable and the closure is the generator of a Feller semigroup. In particular, the corresponding Feller process is a L\'evy-type process.
In the case when $q(x,\xi)$ does not depend
on the variable $x\in\R^{d}$, $\process{X}$ becomes a \emph{L\'evy
process}, that is, a stochastic process with stationary and
independent increments. Moreover, unlike Feller
processes, every L\'evy process is uniquely and completely
characterized through its corresponding symbol (see \cite[Theorems 7.10 and 8.1]{sato-book} and \cite[Example 2.26]{rene-bjorn-jian}). According to this, it is not hard to
check that every
conservative L\'evy process satisfies conditions
(\textbf{LTP1}) and (\textbf{LTP2}) (see \cite[Theorem 31.5]{sato-book}).
Thus, the class of processes we consider in this article contains
L\'evy processes. Throughout this article, the symbol $\process{X}$ denotes a Feller
process satisfying conditions (\textbf{LTP1}) and (\textbf{LTP2}). Such a
process is called a \emph{L\'evy-type process} (LTP).
If $\nu(x,\D y)\equiv0$, $\process{X}$ is called a \emph{diffusion process}. Note that this definition agrees with the standard definition of (Feller-Dynkin) diffusions (see \cite[Chapter III.2]{rogersI}). A typical example of a LTP is a solution to the following SDE
\begin{equation}\label{SDE1}
\D X_t\,=\,\Phi(X_{t-})\,\D Y_t\,,\qquad X_0=x\in\R^d\,,
\end{equation} where $\Phi:\R^{d}\to\R^{d\times n}$ is locally Lipschitz continuous and bounded (which is not a restriction in the periodic setting), and $\process{Y}$ is an $n$-dimensional L\'evy process with symbol $q_Y(\xi)$. Namely, in \cite[Theorems 3.1 and 3.5 and Corollary 3.3]{schnurr} it has been shown that the unique solution $\process{X}$ to the SDE in \cref{SDE1} (which exists by standard arguments) is a LTP with symbol of the form $q(x,\xi)=q_Y(\Phi'(x)\xi).$ Here, for a matrix $M$, $M'$ denotes its transpose. Observe that the following SDE is a special case of \cref{SDE1},
\begin{equation}\label{SDE2}
\D X_t\,=\,\Phi_1(X_t
)\,\D t+\Phi_2(X_t
)\,\D B_t\,+\Phi_3(X_{t-})\,\D Z_t\,,\qquad X_0=x\in\R^d\,,
\end{equation}
where $\Phi_1:\R^{d}\to\R^{d}$, $\Phi_2:\R^{d}\to\R^{d\times p}$ and $\Phi_3:\R^{d}\to\R^{d\times q}$, with $p+q=n-1$, are locally Lipschitz continuous and bounded, $\process{B}$ is a $p$-dimensional Brownian motion, and $\process{Z}$ is a $q$-dimensional pure-jump L\'evy process (that is, a L\'evy process determined by a L\'evy triplet of the form $(0,0,\nu_Z(\D y))$). Namely, set
$\Phi(x)=\bigl(\Phi_1(x),\Phi_2(x),\Phi_3(x)\bigr)$
for any $x\in \R^d$, and $Y_t=(t,B_t,Z_t)'$
for $t\ge0$. For more
on L\'evy-type processes we refer the readers to the monograph
\cite{rene-bjorn-jian}.
\subsection*{LTPs with Periodic Coefficients}
Let $\tau=(\tau_1,\ldots, \tau_{d})\in (0,\infty)^{d}$ be
fixed, and let $\tau\ZZ^{d}:=\tau_1\ZZ\times\ldots\times\tau_{d} \ZZ.$
For $k=(k_1,\dots,k_d)\in\ZZ^d$ define $\tau\odot k:=(\tau_1k_1,\dots,\tau_dk_d)$, and for $x\in\R^{d}$ define
$$x_\tau\,:=\,\{y\in\R^{d}:x-y\in\tau\ZZ^{d}\}\qquad\textrm{and}\qquad
\R^{d}/\tau\ZZ^{d}\,:=\,\{x_\tau:x\in\R^{d}\}\,.$$ In the sequel, we denote $\mathbb{T}^d_\tau=\R^{d}/\tau\ZZ^{d}$. Clearly,
$\mathbb{T}^d_\tau$ is obtained
by identifying the opposite
faces of $[0,\tau]:=[0,\tau_1]\times\ldots\times[0,\tau_{d}]$.
Let
$\Pi_{\tau} : \R^{d}\to \mathbb{T}^d_\tau$, $\Pi_{\tau}(x):=x_\tau$, be the covering map.
A
function $f:\R^{d}\to\R$ is called \textit{$\tau$-periodic} if
$$f(x+\tau\odot k)\,=\,f(x)\,,\qquad x\in\R^{d}\,,\ k\in\ZZ^d\,.$$
Clearly, every $\tau$-periodic function $f(x)$ is completely and uniquely determined by its restriction $f|_{[0,\tau]}(x)$ to $[0,\tau]$, and since $f|_{[0,\tau]}(x)$ assumes the same value on opposite faces of $[0,\tau]$, it can be identified by a function $f_\tau:\mathbb{T}^d_\tau\to\R$ given with $f_\tau (x_\tau)=f(x).$
For notational convenience, we will often omit the subscript $\tau$ and simply write $x$ instead of $x_\tau$, and $f(x)$ instead of $f_\tau(x)$.
Let now $\process{X}$ be a LTP with semigroup $\process{P}$, symbol $q(x,\xi)$ and L\'evy triplet $(b(x),c(x),\nu(x,\D y))$, satisfying:
\medskip
\begin{description}
\item [(\textbf{C1})] $\displaystyle x\mapsto q(x,\xi)$ is $\tau$-periodic for all $\xi\in\R^{d}$.
\end{description}
\medskip
\noindent Directly from the L\'evy-Khintchine formula it follows that (\textbf{C1}) is equivalent to the $\tau$-periodicity of the corresponding L\'evy triplet $(b(x),c(x),\nu(x,\D y))$, which in turn is equivalent to the $\tau$-periodicity of $x\mapsto\mathbb{P}_x(X_t-x\in\D y)$ $($see \cite[Section 4]{hom}$)$.
This immediately implies that $\process{P}$ preserves the class of all bounded Borel measurable $\tau$-periodic functions, that is, the function $x\mapsto P_tf(x)$ is $\tau$-periodic for all $t\geq0$ and all $\tau$-periodic $f\in B_b(\R^{d})$. Now, together with this, a straightforward adaptation of \cite[Proposition 3.8.3]{vasili-book} entails that $\{\Pi_\tau(X_t)\}_{t\ge0}$ is a
Markov process on $(\mathbb{T}_\tau^d,\mathcal{B}(\mathbb{T}_\tau^d))$ with positivity preserving contraction semigroup $\process{P^{\tau}}$
(on the space $(B_b(\mathbb{T}_\tau^d),\lVert\cdot\rVert_\infty)$) given by
$$P_t^{\tau}f(x)\,:=\,\mathbb{E}^{\tau}_x\bigl[f(\Pi_\tau(X_t))\bigr]\,=\,\int_{\mathbb{T}_\tau^d}f(y)\mathbb{P}_x^{\tau}\bigl(\Pi_\tau(X_t)\in \D y\bigr) \,,$$ for $t\geq0,$ $ x\in\mathbb{T}_\tau^d$ and $f\in B_b(\mathbb{T}_\tau^d)$. Here, $\mathcal{B}(\mathbb{T}_\tau^d)$ stands for the Borel $\sigma$-algebra on $\mathbb{T}_\tau^d$ (with respect to the standard quotient topology), $B_b(\mathbb{T}_\tau^d)$ denotes the class of all bounded Borel measurable functions $f:\mathbb{T}_\tau^d\to\R$ (which can be identified with the class of all $\tau$-periodic bounded Borel measurable functions $f:\R^d\to\R$), and
\begin{equation*}\label{E2.1}\mathbb{P}_x^{\tau}\bigl(\Pi_\tau(X_t)\in B\bigr)\,:=\,\mathbb{P}_{z_x}\bigl(X_t\in \Pi_\tau^{-1}(B)\bigr)\,,\qquad t\ge0\,,\ x\in\mathbb{T}_\tau^d\,,\ B\in\mathcal{B}(\mathbb{T}_\tau^d)\,,\end{equation*} with $z_x$ being an arbitrary point in
$\Pi^{-1}_{\tau}(\{x\})$.
Further, assume that
\smallskip
\begin{description}
\item [(\textbf{C2})] $\process{X}$ is strong Feller and \textit{open-set irreducible}, that is, for any $t>0$, any $x\in \R^d$ and any non-empty open set $O\subseteq\R^d$,
$\displaystyle\mathbb{P}_{x}(X_t\in O)>0$.
\end{description}
\smallskip
\noindent Clearly, (\textbf{C2}) automatically implies that the process $\{\Pi_\tau(X_t)\}_{t\ge0}$ is strong Feller and open-set irreducible, too.
Hence, by employing \cite[Remark 3.2]{liang-sch-wang} and \cite[Theorem 1.1]{wat} we have proved the following.
\begin{proposition}\label{SS2.1} The process $\{\Pi_\tau(X_t)\}_{t\ge0}$ admits a unique invariant probability measure $\pi(\D x)$, that is, a measure $\pi(\D x)$ satisfying $$\int_{\mathbb{T}_\tau^d}\mathbb{P}_x^\tau\bigl(\Pi_\tau(X_t)\in B\bigr)\,\pi(\D x)\,=\,\pi(B)\,,\qquad t\ge0\,,\ B\in\mathcal{B}(\mathbb{T}_\tau^d)\,,$$ such that \begin{equation}
\label{eq:erg}
\sup_{x\in\mathbb{T}_\tau^d}\lVert \mathbb{P}_x^\tau\bigl(\Pi_\tau(X_t)\in \D y\bigr)-\pi(\D y) \rVert_{{\rm TV}}\,\le\, \Gamma \E^{-\gamma t}\,,\qquad t\ge0\, \end{equation} for some $\gamma,\Gamma>0$, where $\lVert\cdot\rVert_{{\rm TV}}$ denotes the total variation norm on the space of signed measures on $\mathcal{B}(\mathbb{T}_\tau^d)$.
\end{proposition}
\medskip
\begin{remark} Alternatively, \Cref{SS2.1} is a consequence of \cite[Theorems 3.2 and 8.1]{meyn-tweedie-II} and \cite[Theorem 3.2]{tweedie-mproc}, or \cite[Theorem 6.1]{meynIII} and \cite[Theorem 5.1]{tweedie-mproc} (by setting $V(x)\equiv1$ and $c=d=1$). Also, if instead of (\textbf{C2}) we assume
\medskip
\begin{description}
\item [\textbf{($\widetilde {\text{C2}})$}] $\process{X}$ admits a density function $p_t(x,y)$ (with respect to the Lebesgue measure) satisfying
\medskip
\begin{itemize}
\item[(i)] for any $t>0$, the function $(x,y)\mapsto p_t(x,y)$ is continuous on $\R^d\times \R^d$;
\medskip
\item[(ii)] there is a non-empty open set $O\subseteq\R^d$ such that $p_t(x,y)>0$ for all $t>0$, $x\in \R^d$ and $y\in O$,
\end{itemize}
\end{description}
\medskip
\noindent which guarantees that
D\"oblin's irreducibility condition holds true (see \cite[page 256]{doob}), then \Cref{SS2.1} follows from \cite[Theorem 3.1]{benso-lions-book}.
\end{remark}
Conditions (in terms of the L\'evy triplet $(b(x),c(x),\nu(x,\D y))$) ensuring (\textbf{C2}) are discussed in \Cref{examples}.
\subsection*{The Semimartingale Nature of LTPs}
As we have already commented, the problem of
homogenization of an operator
of the form \cref{IDO} corresponding to a LTP
is equivalent to the convergence of the corresponding family of LTPs in the path space endowed with the Skorohod ${\rm J}_1$ topology (see \cite[Theorem 7.1]{rene-bjorn-jian}). According to
\cite[Lemma 3.2]{rene-holder},
$\{X_t\}_{t\ge0}$ is a
$\mathbb{P}_{x}$- semimartingale (with respect to the natural filtration)
for any $x\in\R^d$. Therefore, in order to show this convergence, our aim is to employ \cite[Theorem VIII.2.17]{jacod} which states that a sequence of semimartingales converges in the path space endowed with the Skorohod ${\rm J}_1$ topology to a process with independent increments if the corresponding semimartingale characteristics converge in probability.
Let us now recall the notion of characteristics of a semimartingale
(see \cite{jacod}). Let
$(\Omega,\mathcal{F},\process{\mathcal{F}},\mathbb{P},\process{S})$, denoted by
$\process{S}$ in the sequel, be a $d$-dimensional semimartingale, and
let $h:\R^{d}\to\R^{d}$ be a truncation function (that is, a
bounded Borel measurable function which satisfies $h(x)=x$ in a neighborhood of
the origin).
Define
$$\check{S}(h)_t\,:=\,\sum_{s\leq t}\bigl(\Delta S_s-h(\Delta S_s)\bigr)\quad
\textrm{and} \quad S(h)_t\,:=\,S_t-\check{S}(h)_t\,,\qquad t\geq0\,,$$ where the process
$\process{\Delta S}$ is defined by $\Delta S_t:=S_t-S_{t-}$ and
$\Delta S_0:=S_0$. The process $\process{S(h)}$ is a \emph{special
semimartingale}, that is, it admits a unique decomposition
\begin{equation}\label{SS}S(h)_t\,=\,S_0+M(h)_t+B(h)_t\,,\end{equation} where $\process{M(h)}$ is a local
martingale, and $\process{B(h)}$ is a predictable process of bounded
variation.
\begin{definition}
Let $\process{S}$ be a
semimartingale, and let $h:\R^{d}\longrightarrow\R^{d}$ be a truncation
function. Furthermore, let $\process{B(h)}$ be the predictable
process
of bounded variation appearing in \cref{SS}, let $N(\omega,\D y,\D s)$ be the
compensator of the jump measure
$$\mu(\omega,\D y,\D s)\,:=\,\sum_{s:\,\Delta S_s(\omega)\neq 0}\delta_{(\Delta S_s(\omega),s)}(\D y,\D s)$$ of the process
$\process{S}$, and let $\process{C}=\{\bigl(C_t^{ij}\bigr)_{1\leq i,j\leq d})\}_{t\geq0}$ be the quadratic co-variation
process for $\process{S^{c}}$ (continuous martingale part of
$\process{S}$), that is,
$C^{ij}_t=\langle S^{i,c}_t,S^{j,c}_t\rangle.$ Then $(B,C,N)$ is called
the \emph{characteristics} of the semimartingale $\process{S}$
(relative to $h(x)$). In addition, by defining $\tilde{C}(h)^{ij}_t:=\langle
M(h)^{i}_t,M(h)^{j}_t\rangle$, $i,j=1,\ldots,d$, where $\process{M(h)}$ is the local martingale
appearing in \cref{SS}, $(B,\tilde{C},N)$ is called the \emph{modified
characteristics} of the semimartingale $\process{S}$ (relative to $h(x)$).
\end{definition}
Now, according to \cite[Theorem 3.5]{rene-holder} and \cite[Proposition II.2.17]{jacod} we see that the (modified) characteristics of a LTP $\{X_t\}_{t\ge0}$ (with respect to a truncation function $h(x)$) are given by
\begin{align*}B(h)^{i}_t&\,=\,\int_0^{t}b_i(X_{s})\,\D s+\int_0^{t}\int_{\R^d}\left(h_i(y)-y_i\Ind_{B_1(0)}(y)\right)\nu(X_{s},\D y)\,\D s\,,\\
C^{ij}_t&\,=\,\int_0^{t}c_{ij}(X_{s})\,\D s\,,\\
N(\D y,\D s)&\,=\,\nu(X_{s},\D y)\,\D s\,,\\
\tilde{C}(h)^{ij}_t&=\int_0^{t}c_{ij}(X_{s})\,\D s+\int_0^{t}\int_{\R^{d}}h_{i}\left( y\right)h_{j}\left( y\right)\nu(X_{s},\D y)\,\D s\,,
\end{align*} for $t\geq0$ and $i,j=1,\ldots,d.$
In the sequel, we assume that $\process{X}$ admits ``small jumps'' only, that is,
\smallskip
\begin{description}
\item [(\textbf{C3})] $\displaystyle \sup_{x\in \R^d}\int_{\R^{d}}|y|^2\nu(x,\D y)<\infty$.
\end{description}
\smallskip
\noindent
As a direct consequence of (\textbf{C3}) and \cite[Proposition II.2.29]{jacod} we see that $\process{X}$ itself is a special semimartingale, and for the truncation function we can take $h(x)=x$. In particular, if $\nu(x,\D y)$ is also symmetric for every $x\in\R^d$, the first characteristic
$B(h)^{i}_t$
equals to $\int_0^{t}b_i(X_{s})\,\D s$ for $t\ge0$ and $i=1,\dots,d$.
Observe next that $\process{X}$ is a Hunt process (since it is Feller). Thus, $\process{X}$ is an It\^{o} process in the sense of \cite{cinlar} (a semimartngale Hunt process with characteristics of the form as above).
Now, \cite[Theorem 3.33]{cinlar} asserts that
there exist a suitable enlargement of the stochastic basis $(\Omega,\mathcal{F},\{\mathbb{P}_{x}\}_{x\in\R^{d}},\process{\mathcal{F}},\process{\theta})$, say $(\widetilde{\Omega},\mathcal{\widetilde{F}},\{\mathbb{\widetilde{P}}_{x}\}_{x\in\R^{d}},\process{\mathcal{\widetilde{F}}},\process{\widetilde{\theta}})$, supporting a $d$-dimensional Brownian motion $\process{\tilde{W}}$ and a Poisson random measure $\tilde{\mu}(\cdot,\D z,\D s)$ on
$\mathcal{B}(\R)\otimes\mathcal{B}([0,\infty))$ with compensator
$\tilde{\nu}(\D z)\,\D s$, such that $\{X_t\}_{t\ge0}$ is
a solution to the following stochastic differential equation
\begin{align*}X_t\,=\,&x+\int_0^{t}b(X_{s
})\,\D s+\int_0^{t}\tilde\sigma(X_{
s})\,\D \tilde{W}_s\\&+\int_0^{t}\int_{\R}k(X_{s-},z)\Ind_{\{u:|k(X_{s-},u)|< 1\}}(z)\left(\tilde{\mu}(\cdot,\D z,\D s)-\tilde{\nu}(\D z)\,\D s\right)\\&+\int_0^{t}\int_{\R}k(X_{s-},z)\Ind_{\{u:|k(X_{s-},u)|\ge1\}}(z)\,\tilde{\mu}(\cdot,\D z,\D s)\,,
\end{align*}
where
$\tilde\sigma(x)$
is a $d \times d$ matrix-valued Borel measurable function such that $\tilde\sigma(x)'\tilde\sigma(x)=c(x)$ for any $x\in \R^d$,
$\tilde{\nu}(\D z)$ is any given $\sigma$-finite non-finite and non-atomic measure on $\mathcal{B}(\R)$, and
$k:\R^{d}\times\R\to\R^{d}$ is a Borel measurable function satisfying
$$\mu(\cdot,\D y,\D s)\,=\, \tilde{\mu}\bigl(\cdot,\{(z,u)\in\R\times[0,\infty):(k(X_{u-},z),u)\in(\D y,\D s)\}\bigr)\,,$$
and
$$\nu\bigl(x,\D y\bigr)\,=\,\tilde{\nu}\bigl(\{z\in\R:k(x,z)\in \D y\}\bigr)\,.$$
Thus, due to this and (\textbf{C3})
we have that
\begin{equation}\begin{aligned}\label{SDE}
X_t\,=\,&x+\int_0^{t}b(X_{s
})\D s+\int_0^{t}\tilde\sigma(X_{s
})\D \tilde{W}_s\\&+\int_0^{t}\int_{\R}k(X_{s
},z)\Ind_{\{u:|k(X_{s
},u)|\ge1\}}(z)\,\tilde{\nu}(\D z)\,\D s\\&+\int_0^{t}\int_{\R}k(X_{s-},z)\left(\tilde{\mu}(\cdot,\D z,\D s)-\tilde{\nu}(\D z)\, \D s\right)\,.
\end{aligned}\end{equation}
From this equation we also read the unique special semimartingale decomposition of $\{X_t\}_{t\ge0}$.
\subsection*{Main
Result}
Before stating the main result of this article, we introduce some notation we need.
Denote by $C_b^k(\R^d)$ with $k\in\N_0
:=\{0,1,2,\dots\}$
the space of $k$ times differentiable functions such that all derivatives up to order $k$ are bounded. This space is a Banach space endowed with the norm $\lVert f\rVert_k:=\sum_{m:\,|m|\le k}\lVert D^m f\rVert_\infty$, where $m=(m_1,\dots,m_d)\in\N^d_0$, $|m|:=m_1+\cdots+m_d,$ and $D^m:=\partial^{m_1}\dots\partial^{m_d}.$ Denote also $C_b^\infty(\R^d):=\cap_{k\in\N_0}C_b^k(\R^d)$.
Further, a function $\phi:(0,1]\to(0,\infty)$ is said to be \textit{almost increasing} if there is
a constant
$\underline{\kappa}\in(0,1]$ such that $\underline{\kappa}\,\phi(r)\le\phi(R)$ for all $r,R\in(0,1]$
with $r\le R$. Analogously, $\phi:(0,1]\to(0,\infty)$ is said to be \textit{almost decreasing} if there is
a constant
$\overline{\kappa}\in[1,\infty)$ such that $\phi(R)\le\overline{\kappa}\,\phi(r)$ for all $r,R\in(0,1]$
with $r\le R$.
Let now $\psi:(0,1]\to[0,\infty)$ be such that $\psi(1)=1$ and $\lim_{r\to0}\psi(r)=0$. For $f\in C_b(\R^d)$ and $j\in\N_0$, define $$[f]_{-j,\psi}\,:=\,\sup_{x\in\R^d}\sup_{h\in \bar {B}_1(0)\setminus\{0\}}\frac{|f(x+h)-f(x)|}{\psi(|h|)|h|^{-j}}\,,$$ where $\bar {B}_r(x)$ stands for the (topologically) closed ball of radius $r$ around $x\in\R^d.$ Also, let \begin{align*}
m_\psi\,:=\,\sup\{\alpha\in\R:\,r\mapsto\psi(r)/r^\alpha\ \text{is almost increasing in}\ (0,1]\}\,,\\
M_\psi\,:=\,\inf\{\alpha\in\R:\,r\mapsto\psi(r)/r^\alpha\ \text{is almost decreasing in}\ (0,1]\}\,.
\end{align*} According to \cite[Theorem 2.2.2]{goldie}, $m_\psi\le M_\psi$. If $m_\psi>0$, we call $\psi(r)$ the \textit{H\"{o}lder exponent}. In this case, if $m_\psi\in(k,k+1]$ for some $k\in\N_0$, define $$C_b^\psi(\R^d)\,:=\,\{f\in C^k_b(\R^d):\, [D^mf]_{-k,\psi}<\infty\ \text{for}\ |m|=k \}\,.$$ This space is called a \textit{generalized H\"{o}lder space}, and it is a normed vector space with the norm $$\lVert f\rVert_\psi\,:=\,\lVert f\rVert_{k}+\sum_{m:\,|m|=k}[D^mf]_{-k,\psi}\,,$$
(see \cite{kassmann2}).
Observe that the product of two H\"{o}lder exponents is a H\"{o}lder exponent, and
that if $m_\psi\in(k,k+1]$ for some $k\in\N_0$ then $C^{k+1}_b(\R^d)\subsetneq C_b^\psi(\R^d)\subsetneq C_b^k(\R^d)$.
In particular, when $\psi(r)=r^\gamma$ for some $\gamma>0$, $C_b^\psi(\R^d)$ becomes the classical H\"{o}lder space of order $\gamma$ (usually denoted by $C_b^\gamma(\R^d)$), which is a Banach space together with the above-defined norm (which we denote by $\lVert\cdot\rVert_\gamma$).
Since $f\leftrightarrow f_\tau$ gives a one-to-one correspondence between $\{f:\R^d\to\R:f\ \text{is}\ \tau\text{-periodic}\}$ and $\{f_\tau:\mathbb{T}_\tau^d\to\R\}$, in an analogous way we define $C^k(\mathbb{T}^d_\tau)$ and $C^\psi(\mathbb{T}^d_\tau)$.
We are now in position to state the main result of this article, the proof of which is given in
\Cref{S2}.
\begin{theorem}\label{T1.1}Let $\process{X}$ be a $d$-dimensional LTP with semigroup $\process{P}$, symbol $q(x,\xi)$ and L\'evy triplet $(b(x),c(x),\nu(x,\D y))$,
satisfying $($\textbf{C1}$)$, $($\textbf{C2}$)$, $($\textbf{C3}$)$ and
\medskip
\begin{description}
\item [(\textbf{C4})]
$\displaystyle x\mapsto b^*(x):=b(x)+\int_{B_1^c(0)}y\,\nu(x,\D y)$ is of class $C_b^\psi(\R^d)$ for some H\"{o}lder exponent $\psi(r)$, and
\medskip
\begin{itemize}
\item[(i)] for some $t_0>0$, any $t\in(0, t_0]$ and any $\tau$-periodic $f\in C_b(\R^d)$,
$$\|P_t f\|_{\psi}\,\le\, C(t) \|f\|_\infty\,,$$ where $\int_0^{t_0} C(t)\,\D t <\infty;$
\medskip
\item[(ii)] for some $\lambda>0$ and any $\tau$-periodic $f\in C_b^\psi(\R^d)$ with $\int_{\mathbb{T}_\tau^d}f_\tau(x)\,\pi(\D x)=0$, the Poisson equation
\begin{equation}\label{e:po-2} \lambda u- \mathcal{A}^b u\,=\,f \end{equation} admits a $\tau$-periodic solution $u_{\lambda,f}\in C_b^{\varphi\psi}(\R^d)$ for some H\"{o}lder exponent $\varphi(r)$.
\end{itemize}
\end{description}
\smallskip
\noindent Then,
\medskip
\begin{itemize}
\item [(a)] the Poisson equation \begin{equation}\label{e:po-1}
\mathcal{A}^b\beta=b^*-\bar {b^*}
\end{equation} admits a $\tau$-periodic solution $\beta\in C_b^{\varphi\psi}(\R^d)$. Moreover, $\beta(x)$ is the unique solution in the class of continuous and periodic solutions to \cref{e:po-1} satisfying
$\int_{\mathbb{T}_\tau^d}\beta_\tau(x)\,\pi(\D x)=0$.
\medskip
\item[(b)] in any of the following three cases
\medskip
\begin{itemize}
\item [(1)] $\beta\in C_b^2(\R^d)$
if $c(x)\not\equiv0$;
\medskip
\item[(2)] $m_{\varphi\psi}>1$ if $c(x)\equiv0$ and \begin{equation}\label{eq:it}
\sup_{x\in \R^d}\int_{B_1(0)}\varphi(|y|)\psi(|y|)\,\nu(x,\D y)\,<\,\infty\,;
\end{equation}
\medskip \item [(3)] $\beta\in C^1_b(\R^d)$ if $c(x)\equiv0$ and \begin{equation}\label{eq:it2}
\sup_{x\in \R^d}\int_{B_1(0)}|y|\,\nu(x,\D y)\,<\,\infty\,,
\end{equation}
\end{itemize}
for any initial
distribution of $\process{X}$,
\begin{equation}\label{ET1.1}\left\{\varepsilon X_{\varepsilon^{-2}t}-\varepsilon^{-1}\bar {b^*} t\right\}_{t\geq0}\xRightarrow[]{\varepsilon\to0}\process{W}\,.\end{equation}
\end{itemize}
\medskip
\noindent Here, $$ \bar {b^*}\,:=\,\int_{\mathbb{T}^d_\tau}b^*(x)\,\pi(\D x)\,,$$ $\Rightarrow$ denotes the convergence in the space of c\`adl\`ag functions
endowed with the Skorohod ${\rm J}_1$-topology, and
$\process{W}$ is a $d$-dimensional zero-drift Brownian motion
determined by covariance matrix $\Sigma$ given in \cref{ET1.2}.
\end{theorem}
Under (\textbf{C1}), (\textbf{C2}) and the assumption that $b^*\in C_b(\R^d)$, in \Cref{lm:cont} below we show that \cref{e:po-1}
admits a $\tau$-periodic solution $\beta\in C_b(\R^d)$ (which is also unique in the class of continuous $\tau$-periodic solutions satisfying
$\int_{\mathbb{T}_\tau^d}\beta_\tau(x)\,\pi(\D x)=0$). However, we require
additional smoothness of $\beta(x)$ in order to apply It\^{o}'s formula (given in \Cref{Ito}) in the proof of \Cref{T1.1} (see \Cref{S2} for details).
This additional regularity is given through
(\textbf{C4}) (together with (\textbf{C1}) and (\textbf{C2})). Namely, under these assumptions, we show that $\beta\in C_b^{\varphi\psi}(\R^d)$.
When $c(x)\not\equiv0$ we require $\beta\in C_b^2(\R^d)$, and when $c(x)\equiv0$ and $b(x)\not\equiv0$ or $\nu(x,\D y)$ is non-symmetric for some $x\in\R^d$ we only require $m_{\varphi\psi}>1$
or $\beta\in C_b^1(\R^d)$.
When $b(x)\equiv0$ and $\nu(x,\D y)$ is symmetric for all $x\in\R^d$, as already commented, $\beta(x)\equiv0$ and the assertion of the theorem follows without assuming (\textbf{C4}).
In the pure-jump case (that is, when $c(x)\equiv0$), \cref{eq:it} suggests that
the H\"{o}lder exponent
$\varphi(r)$ depends on the behavior of $\nu(x,\D y)$ on $B_1(0)$.
For example,
when \begin{equation}\label{ER1.3}\frac{\underline\kappa}{|y|^{d}
\varphi(|y|)}\Ind_{B_1(0)}(y)\,\D y\,\le\,\Ind_{B_1(0)}(y)\,\nu(x,\D y)\,\le\, \frac{\overline\kappa}{|y|^{d}
\varphi(|y|)}\Ind_{B_1(0)}(y)\,\D y\,,\end{equation}
for some
$0<\underline\kappa\le\overline{\kappa}<\infty$, \cref{eq:it} trivially holds true. Thus, we only require that $\beta\in C_b^{
\varphi\psi}(\R^d)$ for some H\"{o}lder exponent $\psi(r)$ with $m_{\varphi\psi}>1$.
Analogously, if \cref{eq:it2} holds true,
then we only require that $\beta\in C_b^{
1}(\R^d)$.
Observe that in the pure-jump case we do not require explicitly that
$\beta\in C_b^2(\R^d)$.
In this sense the assumption $u_{\lambda,f}\in C_b^{\varphi\psi}(\R^d)$ in ({\bf C4})(ii) is optimal. Namely, under \cref{eq:it}
(resp.\ \cref{eq:it2}), in \Cref{Ito} we show that when $m_{\varphi\psi}>1$ (resp.\ $\beta\in C_b^1(\R^d))$ we can apply It\^{o}'s formula to the process $\{\beta(X_t)\}_{t\ge0}$.
Let us also remark that
in the proof of \Cref{T1.1} (a) we show that ({\bf C4})(i) (together with ({\bf C1})-({\bf C3})) implies that $\beta\in C_b^{\psi}(\R^d).$ Hence, \cref{ET1.1} holds true if $\psi(r)$ is such that either (1), (2) (with $\varphi(r)\equiv 1$) or (3) above is satisfied. If this is not the case, then we require an additional regularity of $\beta(x)$ (inherited from the semigroup) which is given through ({\bf C4})(ii).
Several examples of LTPs satisfying ({\bf C4}) (and ({\bf C1})-({\bf C3})) are presented in \Cref{examples}. In particular, if $\process{X}$ is a diffusion process with $\tau$-periodic coefficients $b\in C_b^\varepsilon(\R^d)$ and $c\in C_b^{1+\varepsilon}(\R^d)$ for some $\varepsilon\in(0,1)$, and additionally $c(x)$ being positive definite, (\textbf{C4}) with $\psi(r)=r^\varepsilon$ and $\varphi(r)=r^2$ follows from \cite[Theorem 2.1]{tomisaki}.
In particular, \Cref{T1.1} generalizes the results from
\cite{benso-lions-book,bhat2} where periodic homogenization of a diffusion process with $\tau$-periodic coefficients $b\in C_b^1(\R^d)$ and $c\in C_b^{2}(\R^d)$, and $c(x)$ being positive definite, has been considered.
If $\process{X}$ is a LTP with diffusion and drift coefficients as above, and L\'evy kernel satisfying \cref{ER1.3} (and some additional regularity properties discussed in \Cref{examples}), then (\textbf{C4}) with $\psi(r)=r^\varepsilon$ and $\varphi(r)=r^2$ follows again from \cite[Theorem 2.1]{tomisaki}.
If $\process{X}$ is a pure-jump LTP with vanishing drift term and L\'evy kernel satisfying \cref{ER1.3} (and some additional regularity properties discussed in \Cref{examples}), (\textbf{C4}) holds true for any H\"{o}lder exponent $\psi(r)$ such that $[m_\psi,M_\psi]\subset (0,1)$ and $[m_{\varphi\psi},M_{\varphi\psi}]\cap \N=\emptyset$ (see \Cref{examples}).
\subsection*{Literature Review}
Our work relates to the active research on homogenization of integro-differential operators, and Markov processes with jumps. The work is highly motivated
by the results in \cite{benso-lions-book,bhat2,hom} where, by employing probabilistic techniques, the authors considered periodic homogenization of the operator $\mathcal{L}$ with $\nu(x,\D y)\equiv0$ (that is, second-order elliptic operator in non-divergence form), and $\mathcal{L}$ with $b(x)\equiv0$ and $\nu(x,\D y)$ being symmetric for all $x\in\R^d$ (that is, integro-differential operator in the balanced form), respectively. In
this article, we generalize both results by including the non-local part of the operator $\mathcal{L}$, as well as non-symmetries caused by the drift term $b(x)$ and the L\'evy kernel $\nu(x,\D y)$.
In a closely related work
\cite{zhizhina}, by using analytic techniques (the corrector method), the authors discuss periodic homogenization of the operator $\mathcal{L}$ with a convolution-type L\'evy kernel, that is, $\mathcal{L}$ is determined by
$$\nu(x,\D y)\,=\,\lambda(x)\mu(x+y)a(y)\,\D y\qquad \text{and}\qquad b(x)\,=\,\int_{B_1(0)}y\,\nu(x,\D y)$$ with
$\lambda(x)$ and $\mu(x)$ being measurable, $\tau$-periodic and such that $0<\underline\kappa\le\lambda(x),\mu(x)\le \overline{\kappa}<\infty$ for all $x\in\R^d$, and $a(y)\ge0$ being measurable and such that $0<\int_{\R^d}(1\vee|y|^2)a(y)\,\D y<\infty$ and $a(y)=a(-y)$ for all $y\in\R^d$. The homogenized operator is again a second-order elliptic operator with constant coefficients. Observe that this case is not covered by \Cref{T1.1} since finiteness of $\nu(x,\D y)$ excludes regularity properties of the corresponding semigroup assumed in ({\bf C4}).
There is a vast literature
on homogenization of differential operators, mostly based on PDE methods.
We refer the interested readers to \cite{benso-lions-book,chechkin,donato,kozlov,tartarII} and the references therein.
Results related to the problem of periodic homogenization of non-local operators (based on probabilistic techniques) were obtained in \cite{franke-periodic,franke-periodicerata, franke2,fujiwara,horie-inuzuka-tanaka,huang-duan-song,huang-duan-song2,Tom}. In all this works the focus is on the so-called stable-like operators
(possibly with variable order),
that is, on the case when $\nu(x,\D y)$ admits ``large jumps'' of power-type: $$\frac{\underline\kappa}{|y|^{d+\overline \alpha}}\Ind_{B_1^c(0)}(y)\,\D y\,\le\,\Ind_{B_1^c(0)}(y)\,\nu(x,\D y)\,\le\,\frac{\overline{\kappa}}{|y|^{d+\underline\alpha}}\Ind_{B_1^c(0)}(y)\,\D y,\qquad x\in \R^d\,,$$ for some $0<\underline\kappa\le\overline{\kappa}<\infty$ and $0<\underline\alpha\le\overline\alpha<2$. In this case, by using subdiffusive scaling (which depends on the behavior of $\nu(x,\D y)$ on $B_1^c(0)$), the homogenized operator is the infinitesimal generator of a stable L\'evy process with the index of stability being equal to the power of the scaling factor. The problem of stochastic homogenization (that is, homogenization of operators with random coefficients) of this type of operators has been considered in \cite{rhodes}.
PDE and
other
analytical approaches to the problem of
periodic homogenization and
stochastic homogenization of stable-like operators can be found in \cite{ari2,ari,ari3,imbert,gosh,salort,focardi, kassmann,uemura,schwabI,schwabII}.
Let us also remark that the class of processes considered in the present article constitute of both diffusion and pure-jump part, and the behavior of the homogenized process depends on both of them. This makes the approach to this problem more subtle since we need to take care of diffusion processes, diffusion processes with jumps and pure jump processes, simultaneously.
\section{Proof of \Cref{T1.1}
}\label{S2}
Throughout
this section we assume that $\process{X}$ is a $d$-dimensional LTP with semigroup $\process{P}$, symbol $q(x,\xi)$ and L\'evy triplet $(b(x),c(x),\nu(x,\D y))$, satisfying (\textbf{C1})-(\textbf{C4}). A crucial step in the proof is an application of It\^{o}'s formula. In order to justify this step, we first discuss regularity of a solution to the Poisson equation \cref{e:po-1}.
\subsection*{Solution to the Poisson Equation \cref{e:po-1}}
Observe first that for any $f_\tau\in B_b(\mathbb{T}_\tau^d)$ with $\int_{\mathbb{T}_\tau^d} f_\tau(x)\,\pi(\D x)=0,$ \Cref{SS2.1} implies that
$$\|P^\tau_t f_\tau\|_\infty \,\le\ \Gamma \E^{-\gamma t} \|f_\tau\|_\infty\,,\qquad t\ge0\,.$$ In particular,
$$\left\|\int_0^\infty P^\tau_t f_\tau \,\D t\right\|_\infty\,
\le \,\frac{\Gamma}{\gamma}\lVert f_\tau\rVert_\infty \,<\,\infty\,.$$
Therefore, the zero-resolvent
$$R^\tau f_\tau(x)\,:=\,\int_0^\infty P^\tau_t f_\tau(x) \,\D t\,,\qquad x\in\mathbb{T}_\tau^d\,,$$ is well defined, and
$$\int_{\mathbb{T}_\tau^d}R^\tau f_\tau(x)\,\pi(\D x)\,=\,0\,.$$
According to \cite[Corollary 3.4]{rene-conserv}, $\process{X}$ is a $C_b$-Feller process. Thus, $\{\Pi_\tau(X_t)\}_{t\ge0}$ is also $C_b$-Feller, and $R^\tau f_\tau\in C(\mathbb{T}_\tau^d)$ for every $f_\tau\in C(\mathbb{T}_\tau^d)$ satisfying $\int_{\mathbb{T}^d_\tau}f_\tau(x)\,\pi(\D x)=0$. Since $\mathbb{T}_\tau^d$ is compact, $\{\Pi_\tau(X_t)\}_{t\ge0}$ is a Feller process. Denote the corresponding Feller generator by $(\mathcal{A}_\tau^\infty,\mathcal{D}_{\mathcal{A}_\tau^\infty})$. Clearly,
for any $f_\tau\in\mathcal{D}_{\mathcal{A}_\tau^\infty}$(which is by definition continuous), and its $\tau$-periodic extension $f(x)$, it holds that $ f\in\mathcal{D}_{\mathcal{A}^b}$ and $\mathcal{A}_\tau^\infty f_\tau=\mathcal{A}^b f.$
It is clear now that $R^\tau f_\tau\in \mathcal{D}_{\mathcal{A}^\infty_\tau}$ for any $f_\tau\in C(\mathbb{T}_\tau^d)$ with $\int_{\mathbb{T}_\tau^d} f_\tau(x)\,\pi(\D x)=0.$
Now, we turn to the Poisson equation
\cref{e:po-1}.
Denote by $b^*_\tau(x)$ the restriction of $b^*(x)$ to $\mathbb{T}_\tau^d$, and set $\bar {b^*_\tau}=\int_{\mathbb{T}_\tau^d}b^*_\tau(x)\,\pi(\D x)$. By
assumption $ b^*_\tau\in C^{\psi}(\mathbb{T}_\tau^d)$.
Define now $\beta_\tau(x):=-R^\tau (b^*_\tau(\cdot)-\bar b^*_\tau)(x)$ for any $x\in\mathbb{T}_\tau^d$.
According to the argument above, we immediately get the following.
\begin{lemma}\label{lm:cont} The $\tau$-periodic extension $\beta(x)$ of $\beta_\tau(x)$ is continuous and satisfies \cref{e:po-1}. Moreover, $\beta(x)$ is the unique solution in the class of continuous and $\tau$-periodic solutions to \cref{e:po-1} satisfying
$\int_{\mathbb{T}_\tau^d} \beta_\tau(x)\,\pi(\D x)=0$.\end{lemma}
\begin{proof}We only need to prove uniqueness.
Let $\bar \beta(x)$ be another continuous and $\tau$-periodic solution to \cref{e:po-1} satisfying
$\int_{\mathbb{T}_\tau^d} \bar\beta_\tau(x)\,\pi(\D x)=0$. Then, $\mathcal{A}^b(\beta-\bar{\beta})(x)\equiv0$. In particular, according to \cite[Proposition 4.1.7]{ethier},
$$(\beta-\bar{\beta})(x)\,=\,\mathbb{E}_x\bigl[(\beta-\bar{\beta})(X_t)\bigr]\,=\,\mathbb{E}^\tau_{x_\tau}\bigl[(\beta_\tau-\bar{\beta}_\tau)(\Pi_{\tau}(X_t))\bigr]\,,\qquad x\in\R^d\,,\ t\ge0\,.$$ By letting now $t\to\infty$, it follows from \Cref{SS2.1} that $(\beta-\bar{\beta})(x)\equiv0$, which proves the assertion.
\end{proof}
Observe that in \Cref{lm:cont} we only used the fact that $b^*_\tau\in C(\mathbb{T}_\tau^d)$.
In the sequel, we discuss additional smoothness of $\beta(x)$.
\subsection*{Proof of \Cref{T1.1} (a)}
We first claim that for any $\tau$-periodic $f\in C_b(\R^d)$ such that $\int_{\mathbb{T}_\tau^d}f_\tau(x)\,\pi(\D x)=0$, $R^\tau f_\tau\in C^\psi(\mathbb{T}_\tau^d)$. Indeed, by ({\bf C4})(i), we have
$$\int_0^{t_0} \|P_t^\tau f_\tau\|_{\psi}\,\D t \,\le\ \|f_\tau\|_\infty\int_0^{t_0} C(t)\,\D t \,<\,\infty\,.$$ Also,
since for any $t>0$ and any $f_\tau\in C(\mathbb{T}_\tau^d)$ with $\int_{\mathbb{T}_\tau^d}f_\tau(x)\,\pi(\D x)=0$, $P^\tau_tf_\tau\in C(\mathbb{T}_\tau^d)$ and $\int_{\mathbb{T}_\tau^d}P_t^\tau f_\tau(x)\,\pi(\D x)=0$, ({\bf C4})(i) and \Cref{SS2.1} imply that
$$\int_{t_0}^\infty \|P_t^\tau f_\tau\|_{\psi}\,\D t \,\le\, C(t_0)\int_{t_0}^\infty \|P^\tau_{t-t_0}f_\tau\|_\infty\,\D t\,\le\, \Gamma C(t_0)\|f_\tau\|_\infty\int_{t_0}^\infty \E^{-\lambda (t-t_0)}\,\D t \,<\,\infty\,.$$ Combining both estimates above with the fact that $R^\tau f_\tau=\int_0^\infty P_t^\tau f_\tau\,\D t$, we get
$R^\tau f_\tau\in C^\psi(\mathbb{T}_\tau^d)$.
Finally, for $\lambda> 0$ let $R_\lambda^\tau $ be the $\lambda$-resolvent of $\{\Pi_\tau(X_t)\}_{t\ge0}$.
Clearly, for any $\tau$-periodic $f\in C_b^\psi(\R^d)$ satisfying $\int_{\mathbb{T}_\tau^d}f_\tau(x)\,\pi(\D x)=0$,
the $\tau$-periodic extension of $R_\lambda^\tau f_\tau(x)$ (say $\bar u_{\lambda,f}(x)$) is a solution to \cref{e:po-2}. Observe next that necessarily $u_{\lambda,f}(x)=\bar u_{\lambda,f}(x)$ for all $x\in\R^d$. Namely, since $\bar u_{\lambda,f}\in C_b(\R^d)$, and by \eqref{e:po-2},
\begin{align*} u_{\lambda,f}(x)-\bar u_{\lambda,f}(x)&\,=\, \E^{-\lambda t}\,\mathbb{E}_x\bigl[(u_{\lambda,f}-\bar u_{\lambda,f})(X_t)\bigr]\\&\ \ \ \ \ -\int_0^t \E^{-\lambda s}\,\mathbb{E}_x\bigl[\mathcal{A}^b(u_{\lambda,f}-\bar u_{\lambda,f})(X_s)-\lambda(u_{\lambda,f}-\bar u_{\lambda,f})(X_s)\bigr]\,\D s\\
&\,=\, \E^{-\lambda t}\,\mathbb{E}_x\bigl[(u_{\lambda,f}-\bar u_{\lambda,f})(X_t)\bigr]\,,\qquad x\in\R^d\,,\ t\ge0\,,\end{align*}
by letting $t\to\infty$ the assertion follows.
Thus, since $b^*_\tau\in C_b^\psi(\mathbb{T}_\tau^d)$,
from the resolvent identity
\begin{equation*}\label{e:res}R^\tau (b^*_\tau(\cdot)-\bar{b}^*_\tau)(x)\,=\,R_\lambda^\tau((b^*_\tau(\cdot)-\bar{b}^*_\tau)+\lambda R^\tau (b^*_\tau(\cdot)-\bar{b}^*_\tau))(x)\end{equation*}
and ({\bf C4})(ii), we conclude the result. \hfill\(\Box\)
\bigskip
In \Cref{T1.1} (b) we require that $\beta\in C_b^2(\R^d)$
if $c(x)\not\equiv 0$, and that $m_{\varphi\psi}>1$ (resp. $\beta\in C_b^1(\R^d)$) if $c(x)\equiv0$ and \cref{eq:it} (resp. \cref{eq:it2}) holds true.
In the following proposition we slightly generalize \cite[Lemma 4.2]{priola} (see also \cite{errami}), and prove It\^{o}'s formula for a pure-jump LTP with respect to a not necessarily twice continuously differentiable function.
\begin{proposition}\label{Ito}
Assume that $\process{X}$ is pure-jump
$($that is, $c(x)\equiv0$$)$ and that
there is a H\"{o}lder exponent $\phi(r)$ with $m_\phi>1$ such that
\begin{equation}
\label{delta}
\sup_{x\in\R^d}\int_{B_1(0)}
\phi(|y|)\,\nu(x,\D y)\,<\,\infty.\end{equation}
Then, it holds that \begin{align*}f(X_t)
\,=\,& f(X_0)+\int_0^t \bigl\langle\nabla f(X_{s
}),b^*(X_{s
})\bigr\rangle\,\D s\\&+ \int_0^t\int_{\R^d}\left(f(X_{s-}+k(X_{s-},z))-f(X_{s-})\right)\bigl(\tilde
\mu(\cdot, \D z,\D s)- \tilde\nu(\D z)\, \D s\bigr)
\\
& + \int_0^t\int_{\R^d}\Big(f(X_{s}+k(X_s,z))-f(X_{s}) -\langle\nabla f(X_{s}), k(X_s,z)\rangle\Big)
\tilde\nu(\D z)\,\D s
\,,\end{align*} for all $t\ge0$ and all $f\in C_b^{\phi}(\R^d)$,
where $b^*(x)=b(x)+\int_{B_1^c(0)}y\,\nu(x,\D y)$.
In addition, if \cref{eq:it2} holds true, then the above relation holds for any $f\in C_b^1(\R^d)$.
\end{proposition}
\begin{proof} Without loss of generality, assume that $m_\phi\in(1,2]$.
We follow the proof of \cite[Lemma 4.2]{priola}. Let $f\in C_b^{\phi}(\R^d)$, and let $\chi\in C_c^\infty(\R^d)$, $0\le\chi\le1$, be such that $\int_{\R^d}\chi(x)\,\D x=1$. For $n\in\N$ define $\chi_n(x):=n^d\chi(nx)$, and $f_n(x):=(\chi_n\ast f)(x)$, where $\ast$ stands for the standard convolution operator. Clearly, $\{f_n\}_{n\in\N}\subset
C_b^\infty(\R^d)$, $\lVert f_n\rVert_{\phi}\le\lVert f\rVert_{\phi}$ for $n\in\N$,
$\lim_{n\to\infty}f_n(x)=f(x)$
and $\lim_{n\to\infty}\nabla f_n(x)=\nabla f(x)$ for all $x\in\R^d$. Next, by employing It\^{o}'s formula and \cref{SDE} we have that
\begin{align}\label{e:sde}
f_n(X_t)
\,=\,& f_n(X_0)+\int_0^t \bigl\langle\nabla f_n(X_{s
}),b
(X_{s
})\bigr\rangle\,\D s\nonumber\\&
+\int_0^{t}\int_{\R}\langle \nabla f_n(X_{s}), k(X_{s
},z)\rangle \Ind_{\{u:|k(X_{s
},u)|\ge1\}}(z)\,\tilde{\nu}(\D z)\,\D s\nonumber\\
&+ \int_0^t\int_{\R}\left(f_n(X_{s-}+k(X_{s-},z))-f_n(X_{s-})\right)\bigl(\tilde\mu(\cdot,\D z,\D s)-\tilde \nu(\D z)\,\D s\bigr)\nonumber
\\
& +\int_0^t\int_{\R}\Big(f_n(X_{s}+k(X_{s},z))-f_n(X_{s}) -\langle\nabla f_n(X_{s}), k(X_{s},z)\rangle\Big)\,\tilde \nu(\D z)\,\D s\nonumber\\
\,=\,&f_n(X_0)+\int_0^t \bigl\langle\nabla f_n(X_{s
}),b^*
(X_{s
})\bigr\rangle\,\D s\nonumber\\
&+ \int_0^t\int_{\R}\left(f_n(X_{s-}+k(X_{s-},z))-f_n(X_{s-})\right)\bigl(\tilde\mu(\cdot,\D z,\D s)-\tilde \nu(\D z)\,\D s\bigr)\\
&+\int_0^t\int_{\R^d}\Big(f_n(X_{s}+k(X_s,z))-f_n(X_{s}) -\langle\nabla f_n(X_{s}), k(X_s,z)\rangle\Big)\, \tilde\nu(\D z)\,\D s.\nonumber
\end{align}
Now, by letting $n\to\infty$ we see that the left hand-side converges to $f(X_t)$, and the first two terms on the right-hand side (for the second term we employ the dominated convergence theorem) converge to $f(X_0)$ and $\int_0^t \bigl\langle\nabla f
(X_{s-}),b^*(X_{s-})\bigr\rangle\,\D s$, respectively. Further, Taylor's theorem
together with the fact that $m_\phi>1$ implies
\begin{align*}&|f_n(x+y)-f_n(x) -\langle\nabla f_n(x), y\rangle|\\&\,\le\,\int_0^1|\nabla f_n(x+ry)-\nabla f_n(x)||y|\,\D r\\&\,\le\,\lVert f_n\rVert_{\phi}\phi(|y|)\Ind_{B_1(0)}(y)+2\lVert f_n\rVert_{\phi}|y|\Ind_{B^c_1(0)}(y)
\\&\,\le\,\lVert f\rVert_{\phi}\phi(|y|)\Ind_{B_1(0)}(y)+2\lVert f\rVert_{\phi}|y|\Ind_{B^c_1(0)}(y)\,.\end{align*} Observe that according to \cref{delta} and ({\bf C3}), $$\sup_{x\in \R^d}\int_{\R^d}\left(\phi(|y|)\Ind_{B_1(0)}(y)+|y|\Ind_{B_1^c(0)}(y)\right)\,\nu(x,\D y)<\infty\,.$$ Thus, the dominated convergence theorem implies that the last term converges to $$\int_0^t\int_{\R^d}\Big(f(X_{s}+k(X_s,z))-f(X_{s}) -\langle\nabla f(X_{s}), k(X_s,z)\rangle\Big)
\tilde\nu(\D z)\,\D s\,.$$ Finally, by employing the isometry formula,
we have
\begin{align*}
&
\,\widetilde{\mathbb{E}}_x\bigg[\Big(\int_0^t\int_{\R}\left(f_n(X_{s-}+k(X_{s-},z))-f_n(X_{s-})-f(X_{s-}+k(X_{s-},z))+f(X_{s-})\right)\\&\hspace{9.8cm}\bigl(\tilde{\mu}(\cdot,\D z,\D s)- \tilde{\nu}(\D z) \,\D s\bigr)\Big)^2\bigg]\\
&\,=\,\widetilde{\mathbb{E}}_x\bigg[\int_0^t\int_{\R}\left(f_n(X_{s}+k(X_{s},z))-f_n(X_{s})-f(X_{s-}+k(X_{s},z))+f(X_{s})\right)^2\\&\hspace{13.05cm}\tilde{\nu}(\D z)\,\D s\bigg]
\,.
\end{align*}
Now, since \begin{align*}&|f_n(x+y)-f_n(x)-f(x+y)+f(x)|^2\\&\,\le\,
\left(\int_0^1\left(|\nabla f_n(x+ry)|+|\nabla f(x+ry)|\right)|y|\,\D r\right)^2\\
&\,\le\,4\lVert f\rVert_{1
}^2|y|^2\,,
\end{align*}
the dominated convergence theorem implies that, possibly passing to a subsequence, the third term on the left-hand side in \cref{e:sde} converges to $$\int_0^t\int_{\R^d}\left(f(X_{s-}+k(X_{s-},z))-f(X_{s-})\right)\bigl(\tilde\mu(\cdot,\D z,\D s)-\tilde\nu(\D z)\,\D s\bigr)\,,$$ which proves the desired result.
Finally, by following the above arguments, one easily sees that under \cref{eq:it2} the assertion also holds for any $f\in C_b^1(\R^d)$, which concludes the proof.
\end{proof}
\subsection*{Proof of \Cref{T1.1} (b)}
We now prove the main result of this article. We follow the approach from \cite{franke-periodic}.
Let $\beta\in C_b^{\varphi\psi}(\R^d)$ be a $\tau$-periodic solution to \cref{e:po-1} discussed above.
Recall that
either $\beta\in C_b^2(\R^d)$ if $c(x)\not\equiv0$,
or $m_{\varphi\psi}>1$
(resp.\ $\beta\in C_b^1(\R^d)$) if $c(x)\equiv0$ and \cref{eq:it}
(resp.\ \cref{eq:it2})
holds true,
as assumed in \Cref{T1.1} (b) (1), (2) and (3).
According to \Cref{Ito}, we can apply It\^{o}'s formula to the process $\{\beta(X_t)\}_{t\ge0}$.
Let us consider now the process $\{X_t-\bar {b^*}t-\beta(X_t)+\beta(X_0)\}_{t\ge0}$. By combining \cref{SDE} and It\^{o}'s formula we have that
\begin{align*}
&X_t-\bar {b^*} t-\beta(X_t)+\beta(X_0)\\&\,=\, x+\int_0^{t}\left(b(X_{s
})+ \int_{B_1^c(0)}y\,\nu(X_{s
},\D y)- \bar {b^*}\right)\D s+\int_0^{t}\tilde\sigma(X_{s
})\,\D \tilde{W}_s\\&\ \ \ \ \ \ \, +\int_0^{t}\int_{\R}k(X_{s-
},z)\left(\tilde{\mu}(\cdot,
\D z , \D s)-\tilde{\nu}(\D z)\,\D s\right)\\
&\ \ \ \ \, \ \ -\int_0^t \bigl\langle\nabla \beta(X_{s
}),b(X_{s
})\bigr\rangle\,\D s-\int_0^t \bigl\langle\nabla \beta(X_{s
}),\tilde\sigma(X_{s
})\,\D \tilde W_s\bigr\rangle\\
&\ \ \ \ \ \ \, -\frac{1}{2}\sum_{i,j=1}^d\int_0^tc_{ij}(X_{s
})\partial_{ij}\beta(X_{s
})\ \D s\\
&\ \ \ \ \ \ \, - \int_0^t\int_{\R}\left(\beta(X_{s-}+k(X_{s-},z))-\beta(X_{s-})\right)\bigl(\tilde{\mu}(\cdot,\D z,\D s)-\tilde{\nu}(\D z)\,\D s\bigr)
\\
&\ \ \ \ \ \ \, - \int_0^t\int_{\R}\Big(\beta(X_{s}+k(X_{s},z))-\beta(X_{s})\\
&\qquad \qquad \qquad\quad -\langle\nabla \beta(X_{s}),k(X_{s},z)\rangle\Ind_{\{u:|k(X_{s},u)|< 1\}}(z)\Big)\,\tilde{\nu}(\D z)\, \D s\\
&\,=\, x+\int_0^{t}\tilde\sigma(X_{s})\,\D \tilde{W}_s-\int_0^t \bigl\langle\nabla \beta(X_{s}),\tilde\sigma(X_{s})\,\D \tilde W_s\bigr\rangle \\
&\ \ \ \ \ \ \, +\int_0^{t}\int_{\R}k(X_{s-},z)\left(\tilde{\mu}(\cdot,\D z,\D s)-\tilde{\nu}(\D z)\, \D s\right)\\
&\ \ \ \ \ \ \,- \int_0^t\int_{\R}\left(\beta(X_{s-}+k(X_{s-},z))-\beta(X_{s-})\right)\bigl(\tilde{\mu}(\cdot,\D z,\D s)-\tilde{\nu}(\D z)\, \D s\bigr) \,,
\end{align*}
where we used
the fact
that $b^*(X_t)-\bar {b^*}=\mathcal{A}^b\beta(X_t)$ for any $t\ge0$.
Clearly, $\{X_t-\bar {b^*} t-\beta(X_t)+\beta(X_0)\}_{t\ge0}$ is a special semimartingale, and from \cite[Proposition II.2.29]{jacod} we see again that for the truncation function we can take $h(x)=x$. Thus, the first characteristic of $\{X_t-\bar {b^*} t-\beta(X_t)+\beta(X_0)\}_{t\ge0}$ vanishes (that is, $B_t\equiv0$, $t\ge0$), the third
characteristic equals
to
\begin{align*}C^{ij}_t&\,=\,\int_0^t \Bigg(c_{ij}(X_s)+ \sum_{k,l=1}^dc_{kl}(X_s)\partial_k\beta_i(X_s)\partial_l\beta_j(X_s)\\&\hspace{1.6cm}-\sum_{l=1}^dc_{il}(X_s)\partial_l\beta_j(X_s)-\sum_{k=1}^dc_{kj}(X_s)\partial_k\beta_i(X_s)\Bigg)\D s\\
&\,=\, \int_0^t\sum_{k,l=1}^{d}\left(\delta_{ki}-\partial_k\beta_i(X_s)\right)c_{kl}(X_s)\left(\delta_{lj}-\partial_l\beta_j(X_s)\right)\D s\,,\end{align*} for $t\ge0$ and $i,j=1,\dots,d,$ and the third modified characteristic reads
\begin{align*}\widetilde{C}^{ij}_t\,=\,&C^{ij}_t+\int_0^t\int_{\R^d}y_iy_j\,\nu(X_s,\D y)\,\D s\\&+\int_0^t\int_{\R^d}\left(\bigl(\beta_i(X_s+y)-\beta_i(X_s)\bigr)\bigl(\beta_j(X_s+y)-\beta_j(X_s)\bigr)\right)\,\nu(X_s,\D y)\,\D s\\&-2\int_0^t\int_{\R^d}y_i\bigl(\beta_j(X_s+y)-\beta_j(X_s)\bigr)\,\nu(X_s,\D y)\,\D s\,,\end{align*} for $t\ge0$ and $i,j=1,\dots,d.$ Also, from \cite[Proposition II.2.17]{jacod} and \cite[Theorem 3.5]{rene-holder} we see that the second characteristic is $$ N(B,\D s)\,=\,\int_{\R^d}\Ind_{B}\left(y-\bigl(\beta(X_s+y)-\beta(X_s)\bigr)\right)\,\nu(X_s,\D y)\,\D s\,,\qquad B\in\mathcal{B}(\R^d)\,.$$
Consequently,
for any $x\in\R^d$,
$$\{\varepsilon X_{\varepsilon^{-2}t}-\bar {b^*} \varepsilon^{-1}t-\varepsilon \beta(X_{\varepsilon^{-2}t})+\varepsilon \beta(X_0)\}_{t\geq0}\,,\qquad \varepsilon>0\,,$$ is
a $\mathbb{P}_{x}$- semimartingale (with respect to the natural filtration generated by $\process{X}$) whose (modified) characteristics (relative to $h(x)=x$) are given by
\begin{align*}B^{\varepsilon,i}_t&\,=\,0\,,\\
C^{\varepsilon,ij}_t&\,=\,
\varepsilon^2\int_0^{\varepsilon^{-2}t}\sum_{k,l=1}^{d}\left(\delta_{ki}-\partial_k\beta_i(X_{s})\right)c_{kl}(X_{s})\left(\delta_{lj}-\partial_l\beta_j(X_{s})\right)\D s\,,\\
N^{\varepsilon}(\D s,B)&\,=\,\frac{1}{\varepsilon^{2}}\int_{\R^d}\Ind_{B}\left(\varepsilon y-\varepsilon\bigl(\beta(X_{\varepsilon^{-2}s}+y)-\beta(X_{\varepsilon^{-2}s})\bigr)\right)\nu(X_{\varepsilon^{-2}s},\D y)\,\D s\,,\\
\widetilde{C}^{\varepsilon,ij}_t&\,=\,C^{\varepsilon,ij}_t+\varepsilon^2\int_0^{\varepsilon^{-2}t}\int_{\R^d}y_iy_j\,\nu(X_{s},\D y)\,\D s\\&\ \ \ \ \ +\varepsilon^2\int_0^{\varepsilon^{-2}t}\int_{\R^d}\bigl(\beta_i(X_s+y)-\beta_i(X_s)\bigr)\\&
\qquad\qquad\qquad\qquad\quad \bigl(\beta_j(X_s+y)-\beta_j(X_s)\bigr)\nu(X_s,\D y)\,\D s\\&\ \ \ \ \ -2\varepsilon^2\int_0^{\varepsilon^{-2}t}\int_{\R^d}y_i\bigl(\beta_j(X_s+y)-\beta_j(X_s)\bigr)\nu(X_s,\D y)\,\D s\,,
\end{align*} $t\geq0$, $B\in\mathcal{B}(\R^{d})$, $i,j=1,\ldots,d$, (see \cite[Lemma 3.2 and Theorem 3.5]{rene-holder} and \cite[Proposition II.2.17]{jacod}).
Further, observe that due to boundedness of $\beta(x)$, $\{\varepsilon X_{\varepsilon^{-2}t}-\bar {b^*} \varepsilon^{-1}t-\varepsilon \beta(X_{\varepsilon^{-2}t})+\varepsilon \beta(X_0)\}_{t\geq0}$ converges in the Skorohod space as $\varepsilon\to0$ if, and only if, $\{\varepsilon X_{\varepsilon^{-2}t}-\bar {b^*} \varepsilon^{-1}t\}_{t\geq0}$ converges, and if this is the case the limit is the same.
Now, according to
\cite[Theorem VIII.2.17]{jacod}, in order to prove the desired convergence it suffices to prove that
\begin{equation*}\sup_{0\le s\le t}B^{\varepsilon,i}_s\,\xrightarrow[\varepsilon\to0]{\mathbb{P}_{x}}\, 0\,,
\end{equation*} for all $t\geq0$ and $i=1,\ldots,d,$ which is trivially satisfied,
\begin{equation}\label{ETP1}\int_0^{t}\int_{\R^{d}}g(y)N^{\varepsilon}(\D y,\D s)\,\xrightarrow[\varepsilon\to0]{\mathbb{P}_{x}}\,0\,,
\end{equation} for all $t\geq0$ and $g\in C_b(\R^{d})$
vanishing in a neighborhood of the origin,
and\begin{equation}\label{ETP2}\widetilde{C}^{\varepsilon,ij}_t\,\xrightarrow[\varepsilon\to0]{\mathbb{P}_{x}}\,t\Sigma^{ij}\,,\end{equation}
for all $t\geq0$ and $i,j=1,\ldots,d$, where $\Sigma$ is given in \cref{ET1.2} and $\xrightarrow[]{\mathbb{P}_{x}}$ stands for the convergence in probability.
To prove the convergence in \cref{ETP2}, first observe that due to $\tau$-periodicity of all components we can replace $\process{X}$ by $\{\Pi(X_t)\}_{t\ge0}$, which is an ergodic Markov process (see \Cref{SS2.1}).
The assertion now follows as a direct consequence of the Birkhoff ergodic theorem.
To prove the relation in \cref{ETP1} we proceed as follows. Fix $g\in C_b(\R^{d})$ that vanishes on $B_\delta(0)$ for some $\delta>0$.
Define now \begin{align*}F^{\varepsilon}(x)\,:=\,&\frac{1}{\varepsilon^{2}}\int_{\R^{d}}g\left(\varepsilon y-\varepsilon\bigl(\beta(x+y)-\beta(x)\bigr)\right)\nu(x,\D y)\\
&-\frac{1}{\varepsilon^{2}}\int_{\mathbb{T}_\tau^d}\int_{\R^{d}}g\left(\varepsilon y-\varepsilon\bigl(\beta(z+y)-\beta(z)\bigr)\right)\nu(z,\D y)\,\pi(\D z)\,.\end{align*}
Clearly, for any $\varepsilon>0$, $F^{\varepsilon}(x)$ is bounded, $\tau$-periodic, and satisfies $F^{\varepsilon}(X_t)=
F^{\varepsilon}\bigl(\Pi_{\tau}(X_t)\bigr)$ for $t\geq0$, and $$\int_{\mathbb{T}_\tau^d}F^{\varepsilon}(x)\,\pi(\D x)\,=\,0\,.$$
Now, by the Markov property and exponential ergodicity of $\{\Pi_{\tau}(X_t)\}_{t\ge0}$, we have
\begin{align*}&\mathbb{E}_{x}\left[\left(\int_0^{t}F^{\varepsilon}(X_{\varepsilon^{-2}s})\,\D s\right)^{2}\right]\\&
\,=\,\mathbb{E}^{\tau}_{x_\tau}\left[\left(\int_0^{t}F^{\varepsilon}\bigl(\Pi_{\tau}(X_{\varepsilon^{-2}s})\bigr)\,\D s\right)^{2}\right] \nonumber\\&\,=\,2\int_0^{t}\int_0^{s}\mathbb{E}^{\tau}_{x_\tau}\left[F^{\varepsilon}\bigl(\Pi_{\tau}(X_{\varepsilon^{-2}s})\bigr)F^{\varepsilon}\bigl(\Pi_{\tau}(X_{\varepsilon^{-2}u})\bigr)\right]\D u\,\D s\\
&\,=\,2\int_0^{t}\int_0^{s}\mathbb{E}^{\tau}_{x_\tau}\left[F^\varepsilon\bigl(\Pi_{\tau}(X_{\varepsilon^{-2}u})\bigr)P^{\tau}_{\varepsilon^{-2}(s-u)}F^{\varepsilon}\bigl(\Pi_{\tau}(X_{\varepsilon^{-2}u})\bigr)\right]\D u\,\D s\\
&\,\le\, 2\Gamma\lVert F^{\varepsilon}\rVert^{2}_{\infty}\int_0^{t}\int_0^{s}\E^{-\gamma\varepsilon^{-2}(s-u)}\,\D u\,\D s\\&\le\frac{2\Gamma\varepsilon^2t}{\gamma}\lVert F^{\varepsilon}\rVert^{2}_{\infty}\\
&\,\le\, \frac{8\Gamma\lVert g\rVert_\infty^2t}{\varepsilon^2\gamma}\sup_{x_\tau\in\mathbb{T}_\tau^d}\left\lvert\int_{\R^{d}}\Ind_{B_\delta^c}\left(\varepsilon y-\varepsilon\bigl(\beta(x+y)-\beta(x)\bigr)\right)\nu(x,\D y)\right\rvert^2\,.
\end{align*}
Let $\varepsilon>0$ be such that $2\varepsilon\lVert \beta\rVert_\infty<\delta/2$. Then,
\begin{align*}\mathbb{E}_{x}\left[\left(\int_0^{t}F^{\varepsilon}(X_{\varepsilon^{-2}s})\, \D s\right)^{2}\right]&
\,\le\, \frac{8\Gamma\lVert g\rVert_\infty^2t}{\varepsilon^2\gamma}\sup_{x_\tau\in\mathbb{T}_\tau^d}\left\lvert\int_{\R^{d}}\Ind_{B_{\delta/2}^c}\left(\varepsilon y\right)\nu(x,\D y)\right\rvert^2\\
&\,=\,\frac{8\Gamma\lVert g\rVert_\infty^2t}{\varepsilon^2\gamma}\sup_{x_\tau\in\mathbb{T}_\tau^d}\left\lvert\nu(x,B_{\delta/2\varepsilon}^c)\right\rvert^2\,.
\end{align*}
Now, since $$\frac{\delta^2}{4\varepsilon^2}\nu(x,B_{\delta/2\varepsilon}^c)\,\le\,\int_{B_{\delta/2\varepsilon}^c}\rvert y\rvert^2\,\nu(x,\D y)\,,$$ we have that
\begin{align*}\mathbb{E}_{x}\left[\left(\int_0^{t}F^{\varepsilon}(X_{\varepsilon^{-2}s})\, \D s\right)^{2}\right]&
\,\le\, \frac{128\Gamma\lVert g\rVert_\infty^2\varepsilon^2t}{\gamma\delta^4}\sup_{x_\tau\in\mathbb{T}_\tau^d}\left(\int_{B_{\delta/2\varepsilon}^c}\rvert y\rvert^2\,\nu(x,\D y)\right)^2\,.
\end{align*}
Consequently, \begin{align*}&\left(\mathbb{E}_{x}\left[\left(\int_0^{t}\int_{\R^{d}}g(y)\,N^{\varepsilon}(\D y,\D s)\right)^{2}\right]\right)^{\frac{1}{2}}\\&\,\leq\,\left(\mathbb{E}_{x}\left[\left(\int_0^{t}F^{\varepsilon}(
{X}_{\varepsilon^{-2}s})\,\D s\right)^{2}\right]\right)^{\frac{1}{2}}\\
&\ \ \ \ \
+\left(\mathbb{E}_{x}\left[\left(\frac{t}{\varepsilon^{2}}\int_{\mathbb{T}_\tau^d}\int_{\R^{d}}g\left(\varepsilon y-\varepsilon\bigl(\beta(z+y)-\beta(z)\bigr)\right)\,\nu(z,\D y)\,\pi(\D z)\right)^{2}\right]\right)^{\frac{1}{2}}\\&\,\leq\,
\frac{8\sqrt{2}\Gamma^{1/2}\lVert g\rVert_\infty\varepsilon t^{1/2}}{\gamma^{1/2}\delta^2}\sup_{x_\tau\in\mathbb{T}_\tau^d}\int_{B_{\delta/2\varepsilon}^c}\rvert y\rvert^2\nu(x,\D y)\\&
\quad \ \ +\frac{4\lVert g\rVert_{\infty}t}{\delta^{2}}\int_{\mathbb{T}_\tau^d}\int_{B_{\delta/2\varepsilon}^c}\frac{\delta^2}{4\varepsilon^2}\,\nu(z,\D y)\,\pi(\D z)\\
&\,\le\, \left(\frac{8\sqrt{2}\Gamma^{1/2}\lVert g\rVert_\infty\varepsilon t^{1/2}}{\gamma^{1/2}\delta^2}
+\frac{4\lVert g\rVert_{\infty}t}{\delta^{2}}\right)\sup_{x_\tau\in\mathbb{T}_\tau^d}\int_{B_{\delta/2\varepsilon}^c}\rvert y\rvert^2\,\nu(x,\D y)\,,
\end{align*}
which together with (\textbf{C3}) concludes the proof. \hfill\(\Box\)
\section{On Structural Properties of LTPs}\label{examples}
In this section, we
present sufficient conditions for LTPs satisfying
strong Feller property, open-set irreducibility,
regularity
property of the semigroup, and regularity properties of the solution to the Poisson equation \cref{e:po-2}, respectively.
Several examples are also included.
\subsection*{Strong Feller Property} Let $\process{X}$ be a L\'evy-type process with symbol $q(x,\xi)$ and L\'evy triplet $(b(x),c(x),\nu(x,\D y))$.
\medskip
\begin{itemize}
\item [(i)]
Let $\process{X}$
be a diffusion process (that is, $\nu(x,\D y)\equiv0$). According to \cite[Theorem V.24.1]{rogersII}, it will be strong Feller if $b(x)$ is measurable, $c(x)$ is continuous and positive definite, and
there is a constant $\Lambda>0$ such that \begin{equation}\label{ER1.4}|c_{ij}(x)|+|b_i(x)|^2\,\le\,\Lambda(1+|x|^2)\,,\qquad x\in\R^d\,,\ i,j=1,\dots,d\,.\end{equation}
Let us also remark that
when $\process{X}$ is a diffusion process generated with a second-order elliptic operator in divergence form \begin{equation}\label{E-DIV}
\mathcal{L}f(x)\, =\, \nabla\bigl(c(x)\cdot\nabla f(x)\bigr)
\end{equation} with
$c(x)$ bounded, measurable and uniformly elliptic, strong Feller property of $\process{X}$ has been discussed in \cite{aron, nash, stroock}.
\medskip
\item[(ii)] Suppose that $(x,\xi)\mapsto q(x,\xi)$ is continuous,
$b(x)$ is continuous and bounded, $c(x)$ is continuous, bounded and positive definite, and $x\mapsto\int_{B}(1\wedge|y|^2)\,\nu(x,\D y)$ is continuous and bounded for any $B\in\mathcal{B}(\R^d)$. Then, according to \cite[Theorems 3.23, 3.24 and 3.25]{rene-bjorn-jian} and \cite[Theorem 4.3 and its remark]{str}, $\process{X}$ is strong Feller.
\medskip
\item[(iii)] Recently, there are lots of developments on heat kernel (that is, the transition density function) estimates of Feller processes. The reader is referred to \cite{chen-chen-wang, chen-hu-xie-zhang, chen-zhang, chen-zhang2, grz-sz, kim-lee, kim-sig-vond} and the references therein for more details. In particular, let
$$ \mathcal{L}f(x)\,=\,\int_{\R^d} \bigl(f(x+y)-f(x)-\langle\nabla f(x), y\rangle \Ind_{B_1(0)}(y)\bigr)\frac{ \kappa(x,y)}{|y|^{d+\alpha(x)}} \,\D y\,,$$
where $\alpha:\R^d \to (0,2)$
is a
H\"older continuous
function such that
\begin{align*}
& 0\,<\,\alpha_1\,\le\, \alpha(x)\, \le\, \alpha_2<2\,, \qquad x \in \R^d\,,\\
& |\alpha(x)-\alpha(y)|\,\le\, c_1(|x-y|^{\beta_1}\wedge 1)\,, \qquad x,y\in \R^d\,,
\end{align*}
for some constants $c_1>0$ and $\beta_1\in(0,1]$, and $\kappa:\R^d\times \R^d \to (0,\infty)$ is a measurable function satisfying
\begin{align*}
& \kappa(x,y)\,=\,\kappa(x,-y)\,, \qquad x,y \in \R^d\,,\\
& 0\,<\,\kappa_1\,\le\, \kappa(x,y)\,\le\, \kappa_2\,<\,\infty\,, \qquad x,y\in \R^d\,,\\
& |\kappa(x,y)-\kappa(\bar x,y)|\,\le\, c_2(|x-\bar x|^{\beta_2}\wedge 1)\,, \qquad x,\bar x,y\in \R^d\,,
\end{align*}
for some constants $c_2>0$ and $\beta_2 \in (0,1]$. If
$
({\alpha_2}/{\alpha_1})-1<\bar\beta_0/{\alpha_2},
$ with $\bar\beta_0\in(0,\beta_0]\cap(0,{\alpha_2}/{2})$ and $\beta_0= \min\{\beta_1, \beta_2\}$, then, by \cite[Thereoms 1.1 and 1.3]{chen-chen-wang}, $(\mathcal{L}, C_c^\infty(\R^d))$ generates a LTP. Furthermore, by upper bounds as well as H\"{o}lder regularity and gradient estimates of the heat kernel (see \cite[Thereoms 1.1 and 1.3, and Remark 1.4]{chen-chen-wang}), this
associated process is strong Feller.
\medskip
\item[(iv)] Let $\process{X}$ and $\process{\tilde X}$ be LTPs with semigroups $\process{P}$ and $\process{\tilde P}$, and Feller generators $(\mathcal{A}^\infty,\mathcal{D}_{\mathcal{A}^\infty})$ and $(\tilde{\mathcal{A}}^\infty,\mathcal{D}_{\tilde{\mathcal{A}}^\infty})$, respectively.
Suppose that $\process{X}$ is strong Feller. If $\mathcal{A}^\infty-\tilde {\mathcal{A}}^\infty$ is a bounded operator on $(B_b(\R^d), \lVert\cdot\rVert_\infty)$, then the formula
$$P_tf\,=\,\tilde P_tf+\int_0^t P_{s}(\mathcal{A}^\infty-\tilde {\mathcal{A}}^\infty) \tilde P_{t-s}f\,\D s\,,\qquad f\in \mathcal{D}_{\mathcal{A}^\infty}\cap\mathcal{D}_{\tilde{\mathcal{A}}^\infty}\,,$$ implies that $\process{\tilde X}$ is also strong Feller. Namely, since both $\process{X}$ and $\process{\tilde X}$ are LTPs, the above relation holds for any $f\in C_c^\infty(\R^d)$. According to \cite[Lemma 1.1.1]{chung},
the boundedness of $\mathcal{A}^\infty-\tilde {\mathcal{A}}^\infty$ and
the dominated convergence theorem, it also holds for $f(x)=\Ind_{O}(x)$ for any open set $O\subseteq\R^d$. The claim now follows from Dynkin's monotone class theorem.
The assertion above roughly asserts that a bounded perturbation preserves the strong Feller property. Below is an example.
Let
\begin{align*}\tilde{\mathcal{L}}f(x)\,=\,&\int_{B_1(0)} \bigl(f(x+y)-f(x)-\langle\nabla f(x), y\rangle\bigr)\frac{ \kappa(x,y)}{|y|^{d+\alpha(x)}} \,\D y \\
&+\int_{B_1^c(0)}\bigl(f(x+y)-f(x)\bigr)\,\nu(x,\D y)\,,\end{align*}
where $\alpha(x)$ and $\kappa(x,y)$ satisfy all the assumptions in (iii), and $\nu(x,\D y)$ is such that
\medskip
\begin{itemize}
\item [(a)] $\displaystyle\sup_{x\in\R^d}\nu(x,B_1(0))=0;$
\medskip
\item [(b)] $\displaystyle\sup_{x\in \R^d}\int_{\R^d}|y|^2\nu(x,\D y)<\infty$;
\medskip
\item[(c)] $\displaystyle f\mapsto \int_{\R^d}\bigl(f(\cdot+y)-f(\cdot)\bigr)\,\nu(\cdot,\D y)$ is an operator on $C_\infty(\R^d)$.
\end{itemize}
\medskip
\noindent For example, one can take $\nu(x,\D y)=\frac{\gamma(x,y)}{|y|^{d+\delta}}\Ind_{B^c_1(0)}(y)\,\D y$ with $\delta>0$ and $\gamma(x,y)$ nonnegative, bounded and such that $x\mapsto \gamma(x,y)$ is continuous for almost every $y\in\R^d$.
Further, let $\mathcal L$ be the operator given in (iii). Then, \begin{align*}(\tilde {\mathcal L}-\mathcal L)f(x)\,=\,&\int_{B_1^c(0)}\bigl(f(x+y)-f(x)\bigr)\,\nu(x,\D y)\\
&- \int_{B_1^c(0)}\bigl(f(x+y)-f(x)\bigr)\,\frac{ \kappa(x,y)}{|y|^{d+\alpha(x)}} \,\D y\end{align*} is bounded on $(B_b(\R^d), \lVert\cdot\rVert_\infty)$.
By assumption, it is also bounded on $(C_\infty(\R^d),$\linebreak$\lVert\cdot\rVert_\infty)$.
Now, according to \cite[Lemma 1.28]{rene-bjorn-jian} and \cite[Proposition 2.1]{shioz}, $\tilde{\mathcal L}=\mathcal L+(\tilde {\mathcal L}-\mathcal L)$ generates a LTP.
This, along with the assertion above and the strong Feller property of the process associated with $\mathcal{L}$, yields the strong Feller property of the process associated with $\tilde{\mathcal{L}}$.
\end{itemize}
\medskip
We remark also that the strong Feller property of LTPs has been discussed in \cite{strong}. In the special case when $\process{X}$ is given through \cref{SDE2},
the strong Feller property (and
the open-set irreducibility) has been discussed in \cite{kwon} under the assumption that $\nu_Z(\R^m)<\infty$, and in \cite{masuda, masuda-err} for an arbitrary $\nu_Z(\D y)$, that is, an arbitrary pure-jump L\'evy process $\process{Z}$. Observe that in both situations non-degeneracy of $\Phi_2(x)\Phi'_2(x)$ has been assumed. In the case when $\Phi_3(x)\equiv\Phi_3\in\R^{d\times q}$ the problem has been considered in \cite{ari,liang-sch-wang, luo-wang}, and for non-constant (and non-degenerated) $\Phi_3(x)$ in \cite{liang-wang}.
\subsection*{Open-Set Irreducibility} Let $\process{X}$ be a L\'evy-type process with symbol $q(x,\xi)$ and L\'evy triplet $(b(x),c(x),\nu(x,\D y))$.
\medskip
\begin{itemize}
\item[(i)]
According to \cite[Theorems V.20.1 and V.24.1]{rogersII} and \cite[Theorem 7.3.8]{durrett}, a diffusion process
will be open-set irreducible (and strong Feller) if $b(x)$ and $c(x)$ are locally H\"older continuous,
$c(x)$ is positive definite,
and \cref{ER1.4} holds true. Observe that
\cref{ER1.4} trivially holds true in the periodic case.
Also, when $b(x)\in C_b^1(\R^d)$, $c(x)\in C_b^2(\R^d)$, $\partial_{ij}c_{kl}(x)$ is uniformly continuous for all $i,j,k,l=1,\dots,d$, and $c(x)$ is positive definite,
open-set irreducibility (and
strong Feller property) of the process follows from the support theorem for diffusion processes, see \cite[Lemma 6.1.1]{fried} and \cite[p.\ 517]{ikeda-watanabe}. For support theorem of jump processes one can refer to \cite{simon}.
\medskip
\item [(ii)] If $\process{X}$ is a diffusion process generated
by a second-order elliptic operator in divergence form \cref{E-DIV} with uniformly elliptic, bounded and measurable diffusion coefficient,
open-set irreducibility (and
strong Feller property) follows from the corresponding heat kernel estimates (see \cite{aron,nash,stroock}).
The diffusion processes with jumps
or pure jump process considered in \cite{chen-chen-wang, chen-hu-xie-zhang, chen-zhang, chen-zhang2, grz-sz, kim-lee, kim-sig-vond} are also open-set irreducible, which is a direct consequence of obtained lower bounds of heat kernel.
\medskip
\item[(iii)] Let $\mathcal{L}$ and $\tilde{\mathcal{L}}$ be the operators from (iv)
in the discussion on
the strong Feller property.
According to \cite[Thereom 1.3]{chen-chen-wang}, the LTP corresponding to $\mathcal{L}$ is open-set irreducible. Further, observe that $$\sup_{x\in \R^d} \int_{B_1^c(0)} \frac{\kappa(x,y)}{|y|^{d+\alpha(x)}}\,\D y\,<\,\infty\,.$$
Thus, by \cite[Lemma 3.1]{barlow} and \cite[Lemma 3.6]{barlow2}, the process associated with the operator $\tilde{\mathcal{L}}$ is also open-set irreducible.
\end{itemize}
\medskip
For open-set irreducibility of LTPs of the form \cref{SDE1} we refer the reader to \cite{ari}, \cite{kwon} and \cite{masuda,masuda-err}.
In the following proposition, which slightly generalizes \cite[Lemma 2]{horie-inuzuka-tanaka}, we show that a LTP will be open-set irreducible if the corresponding L\'evy measure shows enough jump activity. First, recall that a function $f:\R^d\to\R$ is said to be \textit{lower semi-continuous} if \begin{equation*}\liminf_{y\to x}f(y)\,\ge\, f(x)\,,\qquad x\in\R^d\,.
\end{equation*}
\begin{proposition} \label{p2.1}
The process $\process{X}$ will be open-set irreducible if
there are constants $R>r\ge0$ such that
\medskip
\begin{itemize}
\item [(i)] $\inf_{x\in K}\nu(x,O)>0$
for every non-empty open set $O\subseteq B_R(0)\setminus B_r(0)$, and every non-empty compact set $K\subset \R^d$;
\medskip
\item[(ii)] the function $x\mapsto \int_{\R^d}f(y+x)\nu(x,\D y)$ is lower semi-continuous for every non-negative lower semi-continuous function $f:\R^d\to\R.$
\end{itemize}
\end{proposition}
\begin{proof}
Let $x\in\R^d$ and $\rho>0$ be arbitrary, and let $f\in C_c^{\infty}(\R^d)$
be such that
$0\leq f\leq 1$ and ${\rm supp}\, f\subset B_\rho(x)$. By assumption, $$\lim_{t\to0}\left\|\frac{P_t f-f}{t}-\mathcal{A}^\infty f\right\|_\infty\,=\,0\,.$$ In particular, for any $B\subseteq B^c_\rho(x)$,
\begin{align*}
\liminf_{t\to0}\inf_{y\in B}\frac{\mathbb{P}_{y}\bigl(X_t\in B_\rho(x)\bigr)}{t}&\,\geq\, \liminf_{t\to0}\inf_{y\in B}\frac{P_tf(y)}{t}\\&\,=\,\liminf_{t\to0}\inf_{y\in B}\left|\frac{P_tf(y)}{t}-\mathcal{A}^\infty f(y)+\mathcal{A}^\infty f(y)\right|\\&\,=\,\inf_{y\in B}|\mathcal{A}^\infty f(y)|\\
&\,=\,\inf_{y\in B}\int_{\R^d}f(z+y)\nu(y,\D z)\,.\end{align*}
Further, let $0<\varepsilon<\rho$ be arbitrary, and let
$0\le f_\varepsilon\in C_c^{\infty}(\R^d)$ be
such that
$$f_\varepsilon(y)\,=\,\left\{
\begin{array}{ll}
1, & y\in B_{\rho-\varepsilon}(x) \\
0, & y\in B^c_{\rho}(x)\,.
\end{array}
\right.$$
Then, for any $y\in B^c_{\rho}(x)$ we have that
\begin{align*}\mathcal{A}^\infty f_\varepsilon(y)&\,=\,\int_{\R^d}f_\varepsilon(z+y)\,\nu(y,\D z)\\&\,\geq\,\nu\bigl(y,\bigl(B_R(0)\setminus B_r(0)\bigr)\cap B_{\rho-\varepsilon}(x-y)\bigr)\,.
\end{align*}
Next, take $x,y\in\R^d$ such that $r<|x-y|<R$, and pick $\varepsilon,\rho>0$ such that $\varepsilon<\rho$ and $r+2\rho<|x-y|<R-2\rho$. Then we have
\begin{align*}
\liminf_{t\to0}\inf_{z\in B_\rho(x)}\frac{\mathbb{P}_{z}\bigl(X_t\in B_\rho(y)\bigr)}{t}&\,\ge\, \inf_{z\in B_\rho(x)}\nu\bigl(z,\bigl(B_R(0)\setminus B_r(0)\bigr)\cap B_{\rho-\varepsilon}(y-z)\bigr)\\
&\,=\, \inf_{z\in B_\rho(x)}\nu\bigl(z, B_{\rho-\varepsilon}(y-z)\bigr)\,.\end{align*}
Assume now that $\inf_{z\in B_\rho(x)}\nu\bigl(z, B_{\rho-\varepsilon}(y-z)\bigr)=0.$ Then there is a sequence $\{z_n\}_{n\in\N}\subset B_\rho(x)$ converging to $z_0\in\bar{B}_\rho(x)$ such that
\begin{equation*}
\liminf_{n\to\infty}\nu\bigl(z_n, B_{\rho-\varepsilon}(y-z_n)\bigr)\,=\,\liminf_{n\to\infty}\int_{\R^d}\Ind_{B_{\rho-\varepsilon}(y)}(u+z_n)\,\nu(z_n,\D u)\,=\,0\,.
\end{equation*}
However, since $z\mapsto \Ind_{B_{\rho-\varepsilon}(y)}(z)$ is a lower semi-continuous function, we have that
\begin{equation*}
\liminf_{n\to\infty}\int_{\R^d}\Ind_{B_{\rho-\varepsilon}(y)}(u+z_n)\,\nu(z_n,\D u)\,=\,\nu\bigl(z_0, B_{\rho-\varepsilon}(y-z_0)\bigr)\,>\,0\,,
\end{equation*}
which is in contradiction with the above assumption. Hence, there is $t_*=
t_*(x,y,\rho,\varepsilon)>0$ such that
\begin{equation*}
\mathbb{P}_{z}\bigl(X_t\in B_{\rho}(y)\bigr)\,>\,0\,,\qquad z\in B_\rho(x)\,,\ t\in(0,t_*]\,.
\end{equation*}
Fix now $\varepsilon,\rho>0$ such that $\varepsilon<\rho$ and $4\rho<R-r$. From the previous discussion it follows that for any
$x,y\in\R^d$ with $r+2\rho<|x-y|<R-2\rho$, there is $t_{**}=
t_{**}(x,y,\rho,\varepsilon)>0$ such that \begin{equation*}
\mathbb{P}_{z}\bigl(X_t\in B_{\rho}(y)\bigr)\,>\,0\,,\qquad z\in B_\rho(x)\,,\ t\in(0,t_{**}]\,.
\end{equation*}
The assertion now follows by employing the Chapman-Kolmogorov equation. \end{proof}
Observe that in \Cref{p2.1} we require that $\nu(x,\D y)$ is not singular with respect to the $d$-dimensional Lebesgue measure. However, there are many interesting open-set irreducible LTPs which do not meet this property. For example, let $\process{X}$ be a solution to \cref{SDE1} with $n=d+1$, $\Phi(x)=(\bar\Phi(x),\mathbb{I}_{d})$ and $Y_t=(t,B_t,Z_t)'$, $t\ge0$, where $\bar\Phi:\R^d\to\R^d$, $\mathbb{I}_{d}$ is the $d\times d$-identity matrix, $\process{B}$ is a $d_1$-dimensional Brownian motion with $1\le d_1<d$, and $\process{Z}$ is a $(d-d_1)$-dimensional rotationally invariant $\alpha$-stable L\'evy process (independent of $\process{B}$) with $\alpha\in(0,2)$.
Clearly, in this case
the L\'{e}vy measure is $(d-d_1)$-dimensional. Thus, \Cref{p2.1} cannot be applied to the process $\process{X}$.
However, open-set irreducibility of $\process{X}$ may be concluded by employing the time-changed idea as in \cite{WZ}. Namely, the Girsanov transformation implies open-set irreducibility of a solution to \cref{SDE1} with $\Phi_0(x)$ similar to $\Phi(x)$ defined above and $\bar Y_t=(t,B_t,\bar B_t)'$, $t\ge0$, where $\process{B}$ is also as above, and $\process{\bar B}$ is a $(d-d_1)$-dimensional Brownian motion (independent of $\process{B}$).
With this at hand, and following the approach in \cite{WZ} (the time-changed idea combined with approximation argument), we conclude open-set irreducibility of $\{X_t\}_{t\ge0}$.
An alternative approach is based on the Levi's method from PDE theory. Namely, since the transition function of the process $\{(B_t, Z_t)\}_{t\ge0}$ enjoys the product form with Gaussian estimates and two-sided heat kernel estimates for rotationally invariant $\alpha$-stable processes, one may follow the argument from \cite{chen-zhang2} to get two-sided heat kernel estimates for $\{X_t\}_{t\ge0}$. When $\alpha\in (0,1)$ we may need to additionally assume that $\Phi(x)$ is H\"{o}lder continuous.
Let us also remark that open-set irreducibility (and strong Feller property) of
a solution to \cref{SDE1} with $\Phi(x)=(\bar\Phi(x),\mathbb{I}_{d})$ and
$Y_t=(t,Z^1_t,\dots,Z^{d}_t)'$, $t\ge0$, where $\bar\Phi:\R^d\to\R^d$, and $\{Z^i_t\}_{t\ge0}$, $i=1,\dots,d,$ are mutually independent one-dimensional symmetric $\alpha$-stable L\'{e}vy processes with $\alpha\in(1,2)$, has been deduced in \cite[Theorem 3.1(iv)]{aris}. Note that in this case the L\'evy measure again does not satisfy (i) in \Cref{p2.1}.
\subsection*{Regularity Property of the Semigroup, and Regularity Properties of the Solution to \cref{e:po-2}} Let $\process{X}$ be a L\'evy-type process with $\tau$-periodic L\'evy triplet $(b(x),c(x),\nu(x,\D y))$.
\medskip
\begin{itemize}
\item[(i)] (Diffusion processes) Let $\varepsilon\in(0,1)$, and let $\process{X}$ be a diffusion process with coefficients $b\in C_b^\varepsilon(\R^d)$, $c\in C_b^{1+\varepsilon}(\R^d)$, and $c(x)$ being also positive definite. Then,
({\bf C4})(i) with arbitrary $t_0>0$ and $\psi(r)=r^\varepsilon$ follows from \cite[the proof of Lemma 2.3]{tomisaki}. Also,
a straightforward adaptation of \cite[Theorem 2.1]{tomisaki}, together with \cite[Chapter 4.8]{jacobI} and \cite[Proposition 4.2]{pang-sand}, implies ({\bf C4})(ii) with $\varphi(r)=r^2$. Then, the conclusion of Theorem \ref{T1.1} holds.
\medskip
\item[(ii)] (Diffusion processes with jumps) Let $\varepsilon\in(0,1)$. Assume that $b(x)$ and $c(x)$ are as in (i), and that $\nu(x,\D y)$ satisfies
\medskip
\begin{itemize}
\item [(a)] $\displaystyle\sup_{x\in\R^d}\int_{B_1(0)}|z|^{1+\varepsilon}\,\nu(x,\D z)<\infty;$
\medskip
\item[(b)] $\displaystyle \lim_{\epsilon\to0}\sup_{x\in\R^d}\int_{B_\epsilon(0)}|z|^{1+\varepsilon}\,\nu(x,\D z)=0;$
\medskip
\item [(c)] $\displaystyle \lim_{R\to\infty}\sup_{x\in\R^d}\int_{B_R^c(0)}|z|^{1+\varepsilon}\,\nu(x,\D z)=0;$
\medskip
\item [(d)] $\displaystyle\sup_{x,y\in\R^d}|x-y|^{-\varepsilon}\int_{\R^d}\bigl(1\wedge|z|^{1+\varepsilon}\bigr)\,|\nu(x,\D z)-\nu(y,\D z)|<\infty$.
\end{itemize}
\medskip
\noindent Here, $|\mu(\D z)|$ stands for the total variation measure of a signed measure $\mu(\D z)$. Then, ({\bf C4})(ii) with $\varphi(r)=r^2$ follows again from \cite[Theorem 2.1]{tomisaki}, together with \cite[Chapter 4.8]{jacobI} and \cite[Proposition 4.2]{pang-sand}.
Let us give sufficient conditions that $\process{X}$ also satisfies ({\bf C4})(i). Denote by $\process{P}$ the semigroup of $\process{X}$, and
let $\process{\tilde{P}}$ be the semigroup of the diffusion process with coefficients $b(x)$ and $c(x)$. Also, denote by $(\mathcal{A}^\infty,\mathcal{D}_{\mathcal{A}^\infty})$
and $(\tilde{\mathcal{A}}^\infty,\mathcal{D}_{\tilde{\mathcal{A}}^\infty})$ the corresponding $C_\infty$-generators, respectively. Then,
$$P_tf\,=\,\tilde P_tf+\int_0^t \tilde P_{s}(\mathcal{A}^\infty- \tilde{\mathcal{A}}^\infty) P_{t-s}f\,\D s\,,\qquad f\in \mathcal{D}_{\mathcal{A}^\infty}\cap\mathcal{D}_{\tilde{\mathcal{A}}^\infty}\,.$$ Since both processes are LTPs, the above relation holds for any $f\in C_c^\infty(\R^d)$. Assume next that $\mathcal{A}^\infty-\tilde{\mathcal{A}}^\infty$ is a bounded operator on $(B_b(\R^d),\lVert\cdot\rVert_\infty)$. Then, according to \cite[Lemma 1.1.1]{chung},
the boundedness of $\mathcal{A}^\infty-\tilde {\mathcal{A}}^\infty$ and the dominated convergence theorem, the above relation holds for $f(x)=\Ind_{O}(x)$ for any open set $O\subseteq\R^d$. Thus, it also holds for any $f\in B_b(\R^d).$ Recall also that $P_tf\in C_b(\R^d)$ for every $f\in C_b(\R^d)$ and every $t\ge0$ (see \cite[Corollary 3.4]{rene-conserv}). Now, according to (i), there is
a measurable function $C_\varepsilon:(0,\infty)\to(0,\infty)$ such that $\int_0^tC_\varepsilon(s)\,\D s <\infty$ and $\lVert \tilde P_t f\rVert_\varepsilon\le C_\varepsilon(t)\lVert f\rVert_\infty$ for all $t>0$ and all $\tau$-periodic $f\in C_b(\R^d)$. Thus,
for fixed $\tau$-periodic $f\in C_b(\R{^d})$, $P_tf\in C_b^\varepsilon(\R^d)$ and
\begin{equation*}
\lVert P_t f\rVert_\varepsilon\,\le\, \lVert \tilde P_tf\rVert_\varepsilon+\int_0^t \lVert\tilde P_{s}(\mathcal{A}^\infty- \tilde{\mathcal{A}}^\infty) P_{t-s}f\rVert_\varepsilon\,\D s\\
\,\le\, \bar C_\varepsilon(t) \lVert f \rVert_\infty\,,
\end{equation*}
where $\bar C_\varepsilon(t)=C_\varepsilon(t)+\lVert\mathcal{A}^\infty- \tilde{\mathcal{A}}^\infty\rVert \int_0^t C_\varepsilon(s)\,\D s.$ Also,
$$\int_0^t\bar C(s)\,\D s\,\le\,(1+t\lVert\mathcal{A}^\infty- \tilde{\mathcal{A}}^\infty\rVert)\int_0^t C(s)\,\D s\,,\qquad t>0\,,$$ where $\lVert\mathcal{A}^\infty- \tilde{\mathcal{A}}^\infty\rVert$ stands for the operator norm of $\mathcal{A}^\infty- \tilde{\mathcal{A}}^\infty$. Thus, $\process{X}$ satisfies (\textbf{C4})(i) with $\psi(r)=r^\varepsilon$.
Therefore, if additionally $\int_{B_1^c(0)} y\,\nu(\cdot,\D y)\in C_b^\varepsilon(\R^d)$, the conclusion of \Cref{T1.1} holds true.
\medskip
\item[(iii)] (Pure-jump LTPs)
In the pure jump case, sufficient conditions for (\textbf{C4})(i) are given in \cite[Theorem 1.1]{liang-wang2}. Also, when the underlying process is given as a solution to an SDE of the form \cref{SDE2}, we refer to \cite{liang-sch-wang, liang-wang, luo-wang} and the references therein.
To construct an example satisfying (\textbf{C4})(ii), we can again employ a perturbation method. Let $\process{X}$ and $\process{\tilde X}$ be LTPs with semigroups $\process{P}$ and $\process{\tilde P}$, and $B_b$-generators $(\mathcal{A}^b,\mathcal{D}_{\mathcal{A}^b})$ and $(\tilde{\mathcal{A}}^b,\mathcal{D}_{\tilde{\mathcal{A}}^b})$, respectively.
Assume that $\mathcal{A}^b$ satisfies (\textbf{C4})(ii) for some H\"{o}lder exponents $\psi(r)$ and $\varphi(r)$.
Further, assume that $\mathcal{A}^b-\tilde{\mathcal{A}}^b$ is a bounded operator on $(B_b(\R^d),\lVert\cdot\rVert_\infty)$, and that $(\mathcal{A}^b-\tilde{\mathcal{A}}^b)f\in C_b(\R^d)$ for every $f\in C_b(\R^d)$.
Then,
$$\tilde P_tf\,=\, P_tf+\int_0^t P_{s}(\mathcal{A}^b-\tilde {\mathcal{A}}^b) \tilde P_{t-s}f\,\D s\,,\qquad f\in \mathcal{D}_{\mathcal{A}^b}\cap\mathcal{D}_{\tilde{\mathcal{A}}^b}\,.$$ Similarly as before, the above relation holds for all $f\in B_b(\R^d)$.
Thus, for any $\lambda>0$ and any $\tau$-periodic $f\in C_b(\R^d)$, $$\tilde{R}^\tau_\lambda f_\tau\,=\,R^\tau_\lambda f_\tau+R^\tau_\lambda (\mathcal{A}^b-\tilde {\mathcal{A}}^b)\tilde{R}^\tau_\lambda f_\tau\,.$$
Assume now that $\{\tilde P_t\}_{t\ge0}$ satisfies (\textbf{C4})(i) with $\psi(r)$, and that $(\mathcal{A}^b-\tilde{\mathcal{A}}^b)f\in C^\psi_b(\R^d)$ for every $f\in C^\psi_b(\R^d)$. Then, according to the proof of \Cref{T1.1} (a), for any $\tau$-periodic $f\in C_b(\R^d)$ with $\int_{\mathbb{T}^d_\tau}f_\tau(x)\,\pi(\D x)=0$, $\tilde{R}^\tau_\lambda f\in C^\psi(\mathbb{T}^d_\tau)$ and so $(\mathcal{A}^b-\tilde{\mathcal{A}}^b)\tilde{R}^\tau_\lambda f_\tau \in C^\psi(\mathbb{T}^d_\tau)$. Hence, for any $\tau$-periodic $f\in C_b^\psi(\R^d)$, $\tilde{R}^\tau_\lambda f_\tau\in C^{\varphi\psi}(\mathbb{T}^d_\tau),$
that is, the corresponding $\tau$-periodic extension is a solution to \cref{e:po-2}. Finally, uniqueness follows from the fact that
any solution $u(x)$ to \cref{e:po-2} must have the representation $\int_0^\infty e^{-\lambda t} \tilde P_t f\,\D t,$ since $u=(\lambda -\tilde {\mathcal{A}}^b)^{-1}f.$
Below we give concrete examples of LTPs $\process{X}$ and $\process{\tilde X}$ satisfying the above assumptions.
Let $\varphi:(0,\infty)\to(0,\infty)$ be increasing, and such that $\varphi(1)=1$
and
\medskip
\begin{itemize}
\item [(a)] there are $0<\underline{\alpha}\le\overline{\alpha}<1$, $\underline{\kappa}\in(0,1]$ and $\overline{\kappa}\in[1,\infty)$, such that \begin{equation}
\label{scal}
\underline{\kappa}\,\lambda^{2\underline{\alpha}}\varphi(r)\,\le\,\varphi(\lambda r)\,\le\,\overline{\kappa}\,\lambda^{2\overline{\alpha}}\varphi(r)\,,\qquad \lambda\ge1\,,\ r\in(0,1]\,;\end{equation}
\medskip
\item[(b)] $\displaystyle \int_1^\infty \frac{1}{r \varphi(r)}\,\D r<\infty.$
\end{itemize}
\medskip
\noindent Then,
by (a), $\lim_{r\to0}\varphi(r)=0$,
and so
$\varphi(r)$ is a H\"{o}lder exponent with
$[m_\varphi,M_\varphi]\subseteq[2\underline{\alpha},2\overline{\alpha}]\subset(0,2)$. Further, let $n:\R^d\setminus\{0\}\to[\underline\Gamma,\overline\Gamma]$, with $0\le\underline\Gamma\le\overline{\Gamma}<\infty$, be measurable. Then,
thanks to (a) and (b),
$$\nu_0(\D y)\,:=\,\frac{n(y)}{\varphi(|y|)|y|^{d}}\,\D y$$
is a L\'evy measure.
Denote the L\'evy process generated by the L\'evy triplet $(0,0,\nu_0(\D y))$ by $\process{X}$.
Also, let $(\mathcal{A}^b,\mathcal{D}_{\mathcal{A}^b})$ be the corresponding $B_b$-genera- tor. Then, $\process{X}$ satisfies (\textbf{C4})(ii) for any H\"{o}lder exponent $\psi(r)$ such that $[m_\psi,M_\psi]\subset(0,1)$ and $[m_{\varphi\psi},M_{\varphi\psi}]\cap \N=\emptyset$. Namely, since $\process{X}$ has $\tau$-periodic (actually constant) coefficients, the corresponding projection (with respect to $\Pi_{\tau}(x)$) on $\mathbb{T}_\tau^d$ is again a strong Markov process. Moreover, according to \cite[Proposition 2.2]{jian} (see also \cite[Theorem 2.1]{kno-sci}) and \Cref{p2.1},
it is also strong Feller and open-set irreducible so it satisfies \cref{eq:erg}. Now, for any $\lambda>0$ and any $\tau$-periodic $f\in B_b(\R^d)$, we see as before that the $\tau$-periodic extension $u_{\lambda,f}(x)$ of $R_\lambda^\tau f_\tau(x)$ solves $\lambda u_{\lambda,f}-\mathcal{A}^bu_{\lambda,f}=f$. If $f\in C_b^\psi(\R^d)$ for some H\"{o}lder exponent $\psi(r)$ such that $[m_\psi,M_\psi]\subset(0,1)$, then, according to \cite[Proposition 2.2]{jian} and the proof of \cite[Propositions 3.5 and 3.6]{kassmann2}, $u_{\lambda,f}\in C_b^{\varphi\psi}(\R^d)$ provided $[m_{\varphi\psi},M_{\varphi\psi}]\cap \N=\emptyset$. Let us remark that in the proofs of \cite[Propositions 3.5 and 3.6]{kassmann2} the authors require the scaling property \cref{scal} of $\varphi(r)$ for all $r\in(0,\infty)$, and the additional assumptions that $n(y)\equiv c$ for some $c>0$ and that $\phi(r)=\varphi(r^{-1/2})^{-1}$ is a Bernstein function, that is, $(-1)^n\phi^{(n)}(r)\le 0$ for every $n\in\N_0$. They essentially use this property in order to apply \cite[Corollary 3.2]{kassmann2} via the regularity of semigroups associated with subordinated Brownian motions. However, the statement of this corollary has been proved in \cite[Proposition 2.2]{jian} under the scaling condition in \cref{scal}.
Finally, uniqueness follows from (a straightforward adaptation of) \cite[Proposition 3.2]{priola} (by taking $b(x)\equiv0$).
A typical example of the function $\varphi(r)$ satisfying the above assumptions is given by $\varphi(r)=r^\alpha \log^\beta(\E-1+r^{-1})$ with $\alpha\in (0,2)$ and $\beta\in \R$. According to \cite[Proposition 2.2]{jian}, there is $c>0$ such that for all $t\in (0,1]$ and $f\in B_b(\R^d)$,
$\|\nabla P_tf\|_\infty\le c(\varphi^{-1}(t))^{-1}.$ Therefore, we have that
\medskip
\begin{itemize}
\item [(1)] if $\alpha\in (1,2)$ or $\alpha=1$ and $\beta<-\alpha$, then {\bf(C4)}(i) is satisfied and Theorem \ref{T1.1} (b)(2) holds with $\psi(r)=r^{\theta_1}\log ^{\theta_2} (1+r^{-1})$ for any $\theta_1\in (0,1)$ and $\theta_2\in \R$;
\medskip
\item[(2)] if $\alpha\in (0,1)$, then {\bf(C4)}(i) is satisfied with $\psi(r)=r^{\theta_1}\log ^{\theta_2} (1+r^{-1})$ for any $\theta_1\in (0,\alpha)$ and $\theta_2\in \R$, and Theorem \ref{T1.1} (b)(3) holds.
\end{itemize}
Also, as we have commented above, $\process{X}$ satisfies (\textbf{C4})(ii) if $\theta_1$ is such that $\alpha+\theta_1\notin\N$.
Further, let $\process{\tilde X}$ be a LTP
generated by $(0,0,\nu(x,\D y))$ with $$\nu(x,\D y)\,=\Ind_{B_1(0)}(y)\,\nu_0(\D y) + \frac{\gamma(x,y)}{\tilde{\varphi}(|y|)|y|^{d}}\Ind_{B_1^c(0)}(y)\,\D y\,,$$ where
$\tilde{\varphi}:[1,\infty)\to(0,\infty)$ satisfies that $\int_1^\infty \frac{1}{r\tilde{\varphi}(r)}\,\D r<\infty$, and $\gamma(x,y)$ is non-negative, bounded and such that $x\mapsto \gamma(x,y)$ is continuous for almost every $y\in\R^d$ on $B_1^c(0)$ (see (iv)
in the discussion on the strong Feller property above that this L\'evy kernel generates a LTP). Denote by
$(\mathcal{\tilde A}^b,\mathcal{D}_{\mathcal{\tilde A}^b})$ the corresponding $B_b$-generator. It is easy to see that $\mathcal{A}^b-\tilde{\mathcal{A}}^b$ is bounded on $(B_b(\R^d),\lVert\cdot\rVert_\infty)$, and $(\mathcal{A}^b-\tilde{\mathcal{A}}^b)f\in C_b(\R^d)$ for every $f\in C_b(\R^d)$.
Furthermore, $(\mathcal{A}^b-\tilde{\mathcal{A}}^b)f\in C^\psi_b(\R^d)$ for every $f\in C^\psi_b(\R^d)$ if
we additionally assume that
for almost all $y\in\R^d$ on $B_1^c(0)$, $x\mapsto\gamma(x,y)$ is of class $C_b^\psi(\R^d)$. With these at hand, we can follow the argument in (ii) to check that {\bf(C4)} is satisfied, and so the conclusion of \cref{T1.1} holds.
\end{itemize}
\section*{Acknowledgement}
Financial support through the \textit{Alexander-von-Humboldt Foundation} and \textit{Croatian Science Foundation} under project 8958 (for N. Sandri\'c),
Croatian Science Foundation under project 8958 (for I. Valenti\'c),
and the National
Natural Science Foundation of China (No.\ 11831014), the Program for Probability and Statistics: Theory and Application (No.\ IRTL1704) and the Program for Innovative Research Team in Science and Technology in Fujian Province University (IRTSTFJ) (for J. Wang)
are gratefully acknowledged. We also thank the anonymous referees for the helpful comments that have led to significant improvements of the results in the article.
\def\cprime{$'$} \def\cprime{$'$} \def\cprime{$'$} | 90,919 |
About KAiN
Designer Amanda Kain is Los Angeles born and bred, and the company is dedicated to producing all of its clothing in Los Angeles. Kain had a very specific goal - "we took what we love about our favorite tees and made it our own." The best part about a tee is the way is gets softer and softer with each washing - but that takes years! Amanda found the absolute softest fabric, making its now Kain tees instant classics. Tees tend to be either too boxy or too body-conscious, Amanda designed a tee that is just slightly oversized and has a sexy drape and added her signature pocket. It's your long-lost boyfriend's tee made especially for you.
Order your own KAin Label Pocket Tee and KAiN Label Tank Dress at Boutique To You and use coupon code "CAStyle15" and save 15% off your total purchase. | 213,011 |
TITLE: Estimate the sum of $\sum _{k=1}^n\left(\frac{1}{\sqrt{k}}\right)$
QUESTION [1 upvotes]: So i am supposed to estimate the sum
$\sum _{k=1}^n\left(\frac{1}{\sqrt{k}}\right)$
In the solution for this,
they estimate
then
Which i understand is an estimation and then clever multiplication by 1
but then without and further explanation the book states the result as
My question is, how does one come up with the upper and lower bounds with respect to $n$.
I understand perfectly it can easily be proven by induction, but without me seeing the answer, i would not know what to prove in the first place.
REPLY [1 votes]: The sum
$$\sum\limits_{k=1}^{n}2 (\sqrt{k+1} -\sqrt{k})$$
expands into
$$2(\sqrt{2} -1) + 2(\sqrt{3} - \sqrt{2} ) +\cdots + 2(\sqrt{n+1} - \sqrt{n})$$
terms cancel out ( Telescoping series )
and we are left with
$$2(\sqrt{n+1} - 1)$$
The same applies to the upper bound. | 165,941 |
\begin{document}
\begin{abstract}
We introduce a new perverse filtration on the Borel--Moore homology of the stack of representations of a preprojective algebra $\Pi_Q$, by proving that the derived direct image of the dualizing mixed Hodge module along the morphism to the coarse moduli space is pure. We show that the zeroth piece of the resulting filtration on the preprojective CoHA is isomorphic to the universal enveloping algebra of the associated BPS Lie algebra $\mathfrak{g}_{\Pi_Q}$, and that the spherical Lie subalgebra of this algebra contains half of the Kac--Moody Lie algebra associated to the real subquiver of $Q$. Lifting $\mathfrak{g}_{\Pi_Q}$ to a Lie algebra in the category of mixed Hodge modules on the coarse moduli space of $\Pi_Q$-modules, we prove that the intersection cohomology of spaces of semistable $\Pi_Q$-modules provide ``cuspidal cohomology'' for $\mathfrak{g}_{\Pi_Q}$ -- a conjecturally complete space of simple hyperbolic roots for this Lie algebra.
\end{abstract}
\maketitle
\setcounter{tocdepth}{1}
\section{Introduction}
\subsection{Main results}
Let $\ol{Q}$ be the double of a quiver $Q$, and let $\Pi_Q\coloneqq\CC \ol{Q}/\langle \sum_{a\in Q_1}[a,a^*]\rangle$ be the preprojective algebra. Let $\SP$ be a Serre subcategory of the category of $\CC\ol{Q}$-modules. We set
\[
\HCoha^{\SP}_{\Pi_Q}\coloneqq \bigoplus_{\dd\in\dvs}\HOBM\!\left( \Mst^{\SP}_{\dd}(\Pi_Q),\QQ\right)\otimes \LLL^{-\chi_{Q}(\dd,\dd)},
\]
the (shifted) Borel--Moore homology of the stack $\Mst^{\SP}(\Pi_Q)$ of finite-dimensional $\Pi_Q$-modules which are objects of $\SP$. Here $\LLL=\HOc(\AAA{1},\QQ)$ is a Tate twist, which is introduced so that the object $\HCoha^{\SP}_{\Pi_Q}$ carries an associative multiplication. The resulting algebra plays a key role in geometric representation theory; it is the algebra of all conceivable raising operators on the cohomology of Nakajima's quiver varieties, and so via several decades of work \cite{Nak98, Nak94, Groj95, Var00, ScVa13, MO19}... contains half of various quantum groups associated to $Q$.
Let $\JH\colon \Mst(\Pi_Q)\rightarrow \Msp(\Pi_Q)$ be the semisimplification morphism to the coarse moduli space of $\Pi_Q$-modules. We study $\HCoha^{\SP}_{\Pi_Q}$ via the richer object
\[
\rCoha_{\Pi_Q}\coloneqq \bigoplus_{\dd\in\dvs} \JH_*\VD\ul{\QQ}_{\Mst(\Pi_Q)}\otimes \LLL^{-\chi_Q(\dd,\dd)},
\]
the derived direct image of the dualizing sheaf. The derived category of mixed Hodge modules on $\Msp(\Pi_Q)$ is a tensor category via convolution, and $\rCoha_{\Pi_Q}$ is an algebra object in this category, from which we recover $\HCoha^{\SP}_{\Pi_Q}$ by restricting to $\Msp^{\SP}(\Pi_Q)$ and taking hypercohomology.
\begin{thmx}[Corollary \ref{relPurity}\footnote{In the interests of digestibility, in the introduction we state all results without reference to extra gauge groups $G$, stability conditions or slopes. The results in the main body incorporate these generalisations.}]
\label{thma}
There is an isomorphism
\begin{equation}
\label{mainDecomp}
\rCoha_{\Pi_Q}\cong\bigoplus_{n\in\ZZ_{\geq 0}}\Ho^n(\rCoha_{\Pi_Q})[-n]
\end{equation}
and $\Ho^n(\rCoha_{\Pi_Q})$ is pure of weight $n$, i.e. $\rCoha_{\Pi_Q}$ is \textbf{pure}. As a result of the above decomposition, the mixed Hodge structure $\HCoha^{\SP}_{\Pi_Q}$ carries an ascending perverse filtration $\lP_{\leq \bullet}\!\HCoha^{\SP}_{\Pi_Q}$, starting in degree zero, which is respected by the algebra structure on $\HCoha^{\SP}_{\Pi_Q}$. Moreover, there is an isomorphism of algebras
\[
\lP_{\leq 0}\!\HCoha^{\SP}_{\Pi_Q}\cong \UEA(\fg^{\SP}_{\Pi_Q})
\]
where $\fg^{\SP}_{\Pi_Q}$ is isomorphic to the \textbf{BPS Lie algebra} \cite{QEAs} determined by a quiver $\WT{Q}$, potential $\WT{W}$ and Serre subcategory $\WT{\SP}$ of the category of $\CC\WT{Q}$-modules defined in \S \ref{morePerverse}.
\end{thmx}
The main geometric content of the theorem amounts to the statement that the derived direct image with compact support $\JH_!\ul{\QQ}_{\Mst(\Pi_Q)}$ is pure, i.e. this complex satisfies the statement of the celebrated decomposition theorem of Beilinson, Bernstein, Deligne and Gabber \cite{BBD}, or more precisely, Saito's version in the language of mixed Hodge modules \cite{Saito88,Saito90}. This is rather surprising, since the preconditions of that theorem are not met; $\Mst(\Pi_Q)$ is a highly singular stack, and $p$ is not projective. The algebraic content of the theorem is that the lowest piece of the perverse filtration can be expressed in terms of the BPS Lie algebras introduced by myself and Sven Meinhardt in \cite{QEAs}, as part of a project to realise the cohomological Hall algebras defined by Kontsevich and Soibelman \cite{KS2} as positive halves of generalised Yangians. This is also quite striking; the BPS Lie algebra is defined by a quite different perverse filtration, on vanishing cycle cohomology of a different Calabi--Yau category.
\subsection{Cuspidal cohomology}
In general, the BPS Lie algebra\footnote{We will adopt the convention throughout that where an expected $\SP$ superscript is missing, we assume that $\SP$ is the whole category $\CC\ol{Q}\lmod$.} $\fg_{\Pi_Q}$ satisfies the condition on the dimensions of the cohomologically graded pieces
\[
\sum_{n\in\mathbb{Z}}\dim(\HO^n(\fg_{\Pi_Q,\dd}))q^{n/2}=\kac_{Q,\dd}(q^{-1})
\]
where the polynomials on the right hand side are the polynomials introduced introduced by Victor Kac in \cite{Kac83}, counting $\dd$-dimensional absolutely irreducible $Q$-representations over a finite field of order q. A conjecture of Bozec and Schiffmann \cite[Conj.1.3]{BoSch19} states that the Kac polynomials on the right hand side are the characteristic functions of the $\dvs$-graded pieces of a cohomologically graded Borcherds algebra, and so it is natural to suspect that $\fg_{\Pi_Q}$ itself is the positive half of a cohomologically graded Borcherds algebra. In particular, $\fg_{\Pi_Q}$ should be given by some cohomologically graded Cartan datum, including the data of (usually infinitely many) imaginary simple roots.
One of the motivations for pursuing a lift of the BPS Lie algebra to the category of mixed Hodge modules is a question of Olivier Schiffmann \cite{SchICM}: is there any geometric description of the Cartan datum, for example some algebraic variety $\Msp_{\mathrm{cusp},\dd}(\Pi_Q)$ along with a natural embedding $\Psi\colon\HO(\Msp_{\mathrm{cusp},\dd}(\Pi_Q),\QQ)\hookrightarrow \fg_{\Pi_Q,\dd}$ as the space of imaginary simple roots of weight $\dd$? Such a construction would answer in the affirmative the complex geometric analogue of Conjecture 3.5 of \cite{SchICM}.
We can still make sense of the above conjecture in the absence of a proof that $\fg_{\Pi_Q}$ is the positive part of a Borcherds algebra. We do so via the special case $\SP=\CC\ol{Q}\lmod$ of our more general theorem on primitive generators:
\begin{thmx}
\label{mainThmB}
Let $\dd$ be such that there exists a simple $\dd$-dimensional $\Pi_Q$-module, let $\varpi'\colon \Msp^{\SP}_{\dd}(\Pi_Q)\hookrightarrow \Msp_{\dd}(\Pi_Q)$ be the inclusion, and set
\[
\fc_{\Pi_Q,\dd}^{\SP}\coloneqq \HO\!\left(\Msp^{\SP}_{\dd}(\Pi_Q),\varpi'^!\ICS_{\Msp_{\dd}(\Pi_Q)}(\QQ)\right)\otimes\LLL^{1+\chi_Q(\dd,\dd)}.
\]
There is a canonical decomposition
\begin{equation*}
\label{cuspDecomp}
\fg^{\SP}_{\Pi_Q,\dd}\cong \fc^{\Sp}_{\Pi_Q,\dd}\oplus \mathfrak{l}
\end{equation*}
of mixed Hodge structures, such that the Lie bracket
\[
\fg^{\SP}_{\Pi_Q,\dd'}\otimes\fg^{\SP}_{\Pi_Q,\dd''}\xrightarrow{[\cdot,\cdot]}\fg^{\SP}_{\Pi_Q,\dd}
\]
for $\dd'+\dd''=\dd$ factors through the inclusion of $\mathfrak{l}$. In particular, the mixed Hodge structures $\fc^{\Sp}_{\Pi_Q,\dd}$ give a collection of \textbf{canonical} subspaces of generators for $\fg^{\SP}_{\Pi_Q}$.
\end{thmx}
The proof of the above theorem uses the construction of the new perverse filtration on $\HCoha_{\Pi_Q}^{\SP}$ arising from Theorem \ref{thma}, and the resulting lift of the Lie algebra $\fg_{\Pi_Q}$ to a Lie algebra object in the category of pure Hodge modules on $\Msp(\Pi_Q)$. In particular, the decomposition into generators and non-generators in the BPS Lie algebra arises from the decomposition theorem for perverse sheaves/mixed Hodge modules.
We conjecture that aside from the known simple roots of $\dvs$-degree $1_i$ for $i$ a vertex of $Q$, or $\dd$ such that $\chi_Q(\dd,\dd)=0$, these are \textit{all} of the generators; see Conjecture \ref{mainConj} for the precise statement.
Since by the decomposition theorem there is a canonical embedding
\[
\HO\!\left(\Msp_{\dd}(\Pi_Q),\ICS_{\Msp_{\dd}(\Pi_Q)}(\QQ)\right)\subset \HO\!\left(X,\QQ\right)
\]
where $X\rightarrow \Msp_{\dd}(\Pi_Q)$ is a semi-small resolution, Theorem \ref{mainThmB} suggests that the answer to the question above is ``yes'', and the above embedding provides a route towards \cite[Conj.3.5]{SchICM}. For example, any symplectic\footnote{See \cite{BeSch} for a comprehensive treatment of when we may expect to find such a resolution.} resolution is a semi-small \cite{Kal09} resolution of singularities, and thus its cohomology contains ``cuspidal'' cohomology as a canonical summand.
\subsection{Comparison with the perverse filtration of \cite{QEAs}}
\label{morePerverse}
We call the filtration introduced in Theorem \ref{thma} the \textit{less} perverse filtration, in order to distinguish it from a different perverse filtration, that was introduced in joint work with Sven Meinhardt \cite{QEAs}. This is a perverse filtration on the critical CoHA $\HCoha^{\WT{\SP}}_{\WT{Q},\WT{W}}$ (as defined in \cite[Sec.7]{KS2}) that is the crucial part of the definition of the BPS Lie algebra in Theorem \ref{thma}. We recall some of the main facts regarding critical CoHAs, in order to explain the relationship between the two filtrations.
Let $Q'$ be a symmetric quiver, i.e. we assume that for each pair of vertices $i,j$ there are as many arrows going from $i$ to $j$ as from $j$ to $i$. Let $W'\in\CC Q'$ be a potential, and let $\SP'$ be a Serre subcategory of the category of $\CC Q'$-modules. We continue to denote by $\JH\colon \Mst(Q')\rightarrow \Msp(Q')$ the semisimplification map. We define
\[
\rCoha_{Q',W'}\coloneqq \bigoplus_{\dd\in\dvs}\JH_*\phim{\TTTr(W')}\ul{\QQ}_{\Mst_{\dd}(Q')}\otimes\LLL^{\chi_{Q'}(\dd,\dd)/2}.
\]
See \S \ref{MMHMs}, \S \ref{qrepsec} for the definition of the vanishing cycle functor, half Tate twist, etc. Then $\rCoha_{Q',W'}$ carries the structure of an algebra in the derived category of monodromic mixed Hodge modules on $\Msp(Q')$, and we obtain the algebra $\HCoha_{Q',W'}^{\SP'}$ by taking (exceptional) restriction and hypercohomology of $\rCoha_{Q',W'}$.
We may summarise the main results of \cite{QEAs} on BPS Lie algebras as follows; there is an isomorphism of complexes of monodromic mixed Hodge modules
\[
\rCoha_{Q',W'}\cong \bigoplus_{n\in\mathbb{Z}_{\geq 1}}\Ho^n(\rCoha_{Q',W'})[-n],
\]
inducing a filtration $\mathfrak{P}_{\leq \bullet}\!\HCoha_{Q',W'}^{\SP'}$ beginning in degree one. The algebra $\Gr_{\mathfrak{P}}\!\HCoha_{Q',W'}^{\SP'}$ is supercommutative, so that
\[
\mathfrak{P}_{\leq 1}\!\HCoha_{Q',W'}^{\SP'}
\]
is closed under the commutator Lie bracket, and is called the \textbf{BPS Lie algebra}, denoted $\fg_{Q',W'}^{\SP'}$.
We define the tripled quiver $\WT{Q}$ to be the quiver obtained from $\ol{Q}$ by adding a loop to each vertex, and we define $\WT{W}$ as in \eqref{WTWdef}. We set $\WT{\SP}$ to be the Serre subcategory containing those $\CC\WT{Q}$-modules for which the underlying $\CC\ol{Q}$-module is an object of $\SP$. Then via the dimensional reduction isomorphism \cite[Thm.A.1]{Da13} there is an isomorphism of algebras \cite{RS17,YZ16}
\[
\HCoha^{\WT{\SP}}_{\WT{Q},\WT{W}}\cong \HCoha_{\Pi_Q}^{\SP}
\]
via which $\HCoha_{\Pi_Q}^{\SP}$ inherits a perverse filtration, which we denote $\mathfrak{P}_{\leq \bullet}\!\HCoha_{\Pi_Q}^{\SP}$.
Switching to the ordinary English meaning of the word, the filtration $\lP_{\leq \bullet}\!\HCoha_{\Pi_Q}^{\SP}$ seems less perverse than $\mathfrak{P}_{\leq \bullet}\!\HCoha_{\Pi_Q}^{\SP}$ since it comes directly from the geometry of the map $\Mst(\Pi_Q)\rightarrow \Msp(\Pi_Q)$, rather than the more circuitous route of dimensional reduction, vanishing cycles, and the semisimplification morphism $\Mst(\WT{Q})\rightarrow \Msp(\WT{Q})$ for the auxiliary quiver $\WT{Q}$. The two filtrations are rather different\footnote{As a consequence of this difference, there is value in considering them both simultaneously; see \S \ref{DPFsec} for example.}; for instance, the BPS Lie algebra lives inside $\lP_{\leq 0}\!\HCoha_{\Pi_Q}^{\SP}$, while $\mathfrak{P}_{\leq 0}\!\HCoha_{\Pi_Q}^{\SP}=0$. In general, perverse degrees with respect to the new filtration are lower than for the old one. It is for these two reasons that we call the new filtration the \textit{less} perverse filtration.
\subsection{Halpern--Leistner's conjecture}
Our purity theorem is independent from the statement (proved in \cite{preproj}) that the mixed Hodge structure on $\HCoha_{\Pi_Q}$ is pure. We explain the particular utility of the purity statement of the current paper, with reference to a particular application: the proof of a conjecture of Halpern--Leistner \cite{HL08}. The details will appear in forthcoming work with Sjoerd Beentjes.
Let $X$ be a K3 surface, fix a generic ample class $H\in\NS(X)_{\QQ}$, and fix a Hilbert polynomial $P$. Then there is a moduli stack $\CCoh_P^H(X)$ of $H$-semistable coherent sheaves with Hilbert polynomial $P$, and Halpern--Leistner conjectures that the mixed Hodge structure on
\[
\HOBM(\CCoh_P^H(X),\QQ)
\]
is pure. The above-mentioned purity result of \cite{preproj} encouraged this statement, while the purity result of the current paper provides the means to prove it. The proof idea is easy to explain: locally, the morphism $p\colon\CCoh_P^H(X)\rightarrow \CCCoh_P^H(X)$ to the coarse moduli space is modelled as the morphism $\Mst_{\dd}(\Pi_Q)\rightarrow \Msp_{\dd}(\Pi_Q)$ for some quiver $Q$, and so Theorem \ref{thma} tells us that the direct image $p_!\ul{\QQ}_{\Mst_{\dd}(\Pi_Q)}$ is locally, and hence globally, pure. The result then follows from the fact that the direct image of a pure complex of mixed Hodge modules along a projective morphism is pure.
\subsection{The algebras $\UEA(\fg_{C})$ and $\UEA(\fg_{\Sigma_g})$}
The construction and results of the present paper can be applied in nonabelian Hodge theory, since they concern any category for which the moduli of objects is locally modeled by moduli stacks of modules for preprojective algebras.
Let $C$ be a smooth genus $g$ complex projective curve, which for ease of exposition we assume to be defined over $\mathbb{Z}$, and let $\HHiggs^{\sstab}_{r,0}(C)$ denote the complex algebraic stack of semistable rank $r$ degree zero Higgs bundles on $C$. By \cite{MoSc20} there is an equality
\begin{align}
\label{MSid}
&\sum_{r\geq 0, i,n\in\ZZ}\dim(\Gr^W_n\!(\HOBM_{-i}(\HHiggs^{\sstab}_{r,0}(C),\QQ)))(-1)^iq^{n/2+(g-1)r^2}T^r\\&=\Exp_{q^{1/2},T}\left(\sum_{r\geq 1}\Omega_{C,r,0}(q^{1/2})(1-q)^{-1}T^r\right)\nonumber
\end{align}
where $\Omega_{C,r,0}(q^{1/2})=\kac_{C,r,0}(q^{1/2},\ldots,q^{1/2})$ is a specialization of Schiffmann's polynomial, counting absolutely indecomposable vector bundles of rank $r$ on $C$ over $\mathbb{F}_q$. On the right hand side we have taken the plethystic exponential, an operation which satisfies the identity
\begin{equation}
\label{plethy}
\Exp\left(\sum_{r,i\in \mathbb{Z}_{>0}\times\mathbb{Z}}(-1)^i\dim \fg_{r,i}(q^{i/2})T^r\right)=\sum_{r,i\in \mathbb{Z}_{>0}\times\mathbb{Z}}(-1)^i\dim \UEA(\fg)_{r,i}(q^{i/2})T^r
\end{equation}
for $\fg$ any $\mathbb{Z}_{> 0}\times\mathbb{Z}$-graded Lie algebra with finite-dimensional graded pieces. We presume that the second grading agrees with the cohomological grading, so that the Koszul sign rule is in effect with respect to it, e.g.
\[
[a,b]=(-1)^{\lvert a\lvert \lvert b\lvert+1}[b,a]
\]
for $\lvert a\lvert$ and $\lvert b\lvert$ the $\ZZ$-degrees of $a$ and $b$ respectively. This explains the introduction of the signs in \eqref{plethy}. Via a similar argument to the previous subsection, we may show that the Borel--Moore homology of $\HHiggs^{\sstab}_{r,0}(C)$ is pure, so that the only terms that contribute on the left hand side of \eqref{MSid} have $n=i$.
Putting all of these hints together, it is natural to conjecture (as in \cite{SchICM}) that there is some Lie algebra $\fg_C$, and an isomorphism
\[
\mathcal{H}^{\mathrm{Higgs}}_{C}\coloneqq \bigoplus_{r\geq 0}\HOBM(\HHiggs_{r,0}^{\sstab}(C),\QQ)\otimes \LLL^{(g-1)r^2}\cong \UEA_q(\fg_C[u])
\]
where the right hand side is a deformation of the universal enveloping algebra of a current algebra for some Lie algebra $\fg_C$, which should be a ``curve'' cousin of the Kac--Moody Lie algebras associated to quivers.
This Lie algebra should be defined as the BPS Lie algebra associated to the noncompact Calabi--Yau threefold $Y=\Tot_C(\omega_C\oplus\mathcal{O}_C)$. Technically, this presents some well-known complications: stacks of coherent sheaves on $Y$ do not have a global critical locus description, so that the definition of vanishing cycle sheaves on them requires a certain amount of extra machinery (see \cite{Jo15,BBBD}). The outcome of this paper is that there is a ``less perverse'' definition of $\UEA(\fg_C)$ ready off the shelf, avoiding d critical structures, vanishing cycles etc.: we may \textit{define}
\[
\UEA(\fg_C)\coloneqq \bigoplus_{r\geq 0}\HO(\CHiggs_{r,0}^{\sstab}(C),\tau_{\leq 0}p_*\VD\ul{\QQ}_{\HHiggs_{r,0}^{\sstab}(C)})\otimes\LLL^{(g-1)r^2}
\]
where $p\colon \HHiggs_{r,0}^{\sstab}(C)\rightarrow \CHiggs_{r,0}^{\sstab}(C)$ is the morphism to the coarse moduli space, and the multiplication is via the correspondences in the CoHA of Higgs sheaves as in \cite{SS20,Mi20}. Similarly, we define
\begin{align*}
\UEA(\fg^{\nilp}_C)\coloneqq &\bigoplus_{r\geq 0}\HO(\CHiggs_{r,0}^{\sstab}(C),g_*g^!\tau_{\leq 0}p_*\VD\ul{\QQ}_{\HHiggs_{r,0}^{\sstab}(C)})\otimes\LLL^{(g-1)r^2}\\ &\subset \mathcal{H}^{\mathrm{Higgs},\nilp}_{C}\coloneqq \bigoplus_{r\geq 0}\HOBM(\HHiggs_{r,0}^{\sstab,\nilp}(C),\QQ)\otimes \LLL^{(g-1)r^2}
\end{align*}
where $g\colon \CHiggs_{r,0}^{\sstab,\nilp}(C)\rightarrow \CHiggs_{r,0}^{\sstab}(C)$ is the inclusion of the locus for which the Higgs field is nilpotent, to define the correct enveloping algebra inside the CoHA of nilpotent Higgs bundles \cite{SS20}.
Whenever the morphism $p$ from the stack of objects in a category $\mathscr{C}$ to the coarse moduli space is locally modeled as the semisimplification morphism from the stack of representations of a preprojective algebra, the definition of the enveloping algebra of the BPS Lie algebra for $\mathscr{C}$ is forced by Theorem \ref{thma}; we likewise define
\begin{align*}
\UEA(\fg_{\Sigma_g})\coloneqq &\bigoplus_{r\geq 0}\HO(\Msp^{\Betti}_{g,r},\tau_{\leq 0}p_*\VD\ul{\QQ}_{\Mst^{\Betti}_{g,r}})\otimes\LLL^{(g-1)r^2}\\
\subset&\mathcal{H}_{\Sigma_g}\coloneqq \bigoplus_{r\geq 0}\HOBM(\Mst^{\Betti}_{g,r},\QQ)\otimes\LLL^{(g-1)r^2}
\end{align*}
where $p\colon \Mst^{\Betti}_{g,r}\rightarrow \Msp^{\Betti}_{g,r}$ is the semisimplification morphism from the moduli stack of $r$-dimensional $\pi_1(\Sigma_g)$-modules to the coarse moduli space, for $\Sigma_g$ a genus $g$ Riemann surface without boundary\footnote{This case is slightly different, since the moduli stack of $\pi_1(\Sigma_g)[\omega]$-modules \textit{is} written as a global critical locus; see \cite{Da16}.}. The object $\mathcal{H}_{\Sigma_g}$ is the CoHA of representations of the stack of $\CC[\pi_q(\Sigma_g)]$-modules defined in \cite{Da16}. We leave the detailed study of the algebras $\UEA(\fg_C)$ and $\UEA(\fg_{\Sigma_g})$, as well as a general treatment of the perverse filtration on CoHAs for 2CY categories \cite{PS19,KV19} to future work.
\subsection{Notation and conventions}
All schemes and stacks are defined over $\CC$, and assumed to be locally of finite type. All quivers are finite. All functors are derived.
If $\XX$ is a scheme or stack, and $p\colon \XX\rightarrow \pt$ is the morphism to point, we often write $\HO$ for the derived functor $p_*$, and $\HO^i$ for the $i$th cohomology of $\HO$, i.e. we abbreviate
\begin{align*}
\HO(\mathcal{F})\coloneqq&\HO(\XX,\mathcal{F})\\
\HO^i(\mathcal{F})\coloneqq &\HO^i(\XX,\mathcal{F}).
\end{align*}
For $X$ an irreducible scheme or stack, we write $\HO(X,\QQ)_{\vir}=\HO(X,\QQ)\otimes \LLL^{-\dim(X)/2}$ where the half Tate twist is as in \S \ref{ICsec}. For example,
\[
\HO(\B \CC^*,\QQ)_{\vir}\coloneqq \HO(\B \CC^*,\QQ)\otimes\LLL^{1/2}.
\]
We define
\[
\HOBM(\XX,\QQ)=\HO\VD\:\!\ul{\QQ}_{\XX}
\]
where $\VD$ is the Verdier duality functor.
If $\mathscr{C}$ is a triangulated category equipped with a t structure we write
\[
\Ho(\mathcal{F})=\bigoplus_{i\in \mathbb{Z}}\Ho^i(\mathcal{F})[-i]
\]
when the right hand side exists in $\mathscr{C}$.
If $V$ is a cohomologically graded vector space with finite-dimensional graded pieces, we define
\[
\chi_t(V)\coloneqq \sum_{i\in \mathbb{Z}}(-1)^i\dim(V^i)t^{i/2}.
\]
If $V$ also carries a weight filtration $W_nV$, we define the weight polynomial
\begin{equation}
\wt(V)\coloneqq \sum_{i,n\in\ZZ}(-1)^i\dim(\Gr^{\mathrm{W}}_n(V^i))t^{n/2}.
\end{equation}
\subsection{Acknowledgements}
During the writing of the paper, I was supported by the starter grant ``Categorified Donaldson--Thomas theory'' No. 759967 of the European Research Council. I was also supported by a Royal Society university research fellowship.
I would like to thank Olivier Schiffmann for helpful conversations, and Tristan Bozec for patiently explaining his work on crystals to me. Finally, I offer my heartfelt gratitude to Paul, Sophia, Sacha, Kristin and Nina, for their help and support throughout the writing of this paper.
\section{Background on CoHAs}
\subsection{Monodromic mixed Hodge modules}
\label{MMHMs}
\subsubsection{Mixed Hodge modules}
Let $X$ be an algebraic variety. We define as in \cite{Saito89,Saito90} the category $\MHM(X)$ of mixed Hodge modules on $X$. There is an exact functor
\[
\rat_X\colon\Db(\MHM(X))\rightarrow \Db(\Perv(X))
\]
and moreover the functor $\rat_X\colon \MHM(X)\rightarrow \Perv(X)$ is faithful. We will make light use of the larger category of monodromic mixed Hodge modules $\MMHM(X)$ considered in \cite{KS2,QEAs}, which is defined to be the Serre quotient $\mathscr{B}_X/\mathscr{C}_X$, of two full subcategories of $\MHM(X\times\AAA{1})$. Here, $\mathscr{B}_X$ is the full subcategory containing those objects for which the cohomology mixed Hodge modules are locally constant, away from the origin, when restricted to $\{x\}\times\AAA{1}$ for each $x\in X$. The category $\mathscr{C}_X$ is the full subcategory containing those $\mathcal{F}$ for which such restrictions have globally constant cohomology sheaves.
The functor $(X\times\GG_m\hookrightarrow X\times\AAA{1})_!$ provides an equivalence of categories between $\MMHM(X)$ and the full subcategory of mixed Hodge modules on $X\times\GG_m$ containing those $\mathcal{F}$ satisfying the condition that the restriction to each $\{x\}\times\GG_m$ has locally constant cohomology sheaves. Write $G$ for a quasi-inverse. We define the inclusion $\tau\colon X\hookrightarrow X\times\GG_m$ by setting $\tau(x)=(x,1)$. Then there is a faithful functor
\[
\rat^{\mon}_X=\rat_X \circ\tau^*[-1]\circ G\colon \MMHM(X)\rightarrow\Perv(X).
\]
Let $z_X\colon X\hookrightarrow X\times\AAA{1}$ be the inclusion of the zero section. Then
\[
z_{X,*}\colon \MHM(X)\rightarrow \MMHM(X)
\]
is an inclusion of tensor categories, where the tensor product on the target is the one described below. We write $\MMHS\coloneqq \MMHM(\pt)$. The category of polarizable mixed Hodge structures is a full subcategory of $\MMHS$ via $z_{\pt,*}$.
\subsubsection{Six functors}
Excepting the definition of tensor products, the six functor formalism for categories of monodromic mixed Hodge modules is induced in a straightforward way by that of mixed Hodge modules, e.g. for $f\colon X\rightarrow Y$ a morphism of varieties we define
\[
f_*,f_!\colon\Db(\MMHM(X))\rightarrow \Db(\MMHM(Y))
\]
to be the functors induced by
\[
(f\times\id_{\AAA{1}})_*,(f\times\id_{\AAA{1}})_!\colon \Db(\MHM(X\times\AAA{1}))\rightarrow \Db(\MHM(Y\times\AAA{1}))
\]
respectively. The functor $\VD_X\colon \MHM(X\times \AAA{1})\rightarrow \MHM(X\times \AAA{1})^{\opp}$ sends objects of $\mathscr{C}_X$ to objects of $\mathscr{C}_X^{\opp}$, inducing the functor $\Dmon_X\colon\MMHM(X)\rightarrow \MMHM(X)^{\opp}$. We may omit the $\mon$ superscript when doing so is unlikely to cause confusion.
If $X$ and $Y$ are schemes over $S$, and $\mathcal{F}\in\Ob(\MMHM(X))$, $\mathcal{G}\in\Ob(\MMHM(Y))$, then taking their external tensor product (as mixed Hodge modules) we obtain $\mathcal{J}\in\Ob(\MHM(Z\times\AAA{2}))$, where $Z=X\times_S Y$. We define
\[
\mathcal{F}\boxtimes_S \mathcal{G}\coloneqq (\id_Z\times +)_*\mathcal{J}\in\Ob(\MMHM(Z)).
\]
If $X$ is a monoid over $S$, i.e. there exist $S$-morphisms
\begin{align*}
\nu&\colon X\times_S X\rightarrow X\\
i&\colon S\rightarrow X
\end{align*}
satisfying the standard axioms, and $\mathcal{F},\mathcal{G}\in\Ob(\Db(\MMHM(X)))$ we define
\[
\mathcal{F}\boxtimes_{\nu} \mathcal{G}\coloneqq \nu_*\!\left(\mathcal{F}\boxtimes_S\mathcal{G}\right)\in\Ob(\Db(\MMHM(X))).
\]
This monoidal product is symmetric if $\nu$ is commutative, and is exact if $\nu$ is finite. If $\nu$ is commutative, we define
\[
\Sym_{\nu}(\mathcal{F})\coloneqq \bigoplus_{i\geq 0}\Sym_{\nu}^i(\mathcal{F})
\]
where $\Sym_{\nu}^i(\mathcal{F})$ is the $\Symm_i$-invariant part of
\[
\underbrace{\mathcal{F}\boxtimes_{\nu}\ldots\boxtimes_{\nu}\mathcal{F}}_{i \textrm{ times}},
\]
and the $\Symm_i$-action is defined via the isomorphism
\begin{equation}
\label{toS}
\underbrace{\mathcal{F}\boxtimes_{\nu}\ldots\boxtimes_{\nu}\mathcal{F}}_{i \textrm{ times}}\cong e^*\!\left(\underbrace{\mathcal{F}\boxtimes\ldots\boxtimes\mathcal{F}}_{i \textrm{ times}}\right)
\end{equation}
where $e\colon X\times_S\cdots\times_S X\hookrightarrow X\times\cdots \times X$ is the natural embedding. By \cite{MSS11} (see \cite[Sec.3.2]{QEAs}), the target of \eqref{toS} carries a natural $\mathfrak{S}_i$-action. The functor $z_*\colon\MHM(X)\rightarrow \MMHM(X)$ from \S \ref{MMHMs} is a symmetric monoidal functor.
\subsubsection{MMHMs on stacks}
If $\XX$ is a connected locally finite type Artin stack we define the bounded derived category of monodromic mixed Hodge modules $\DfbMMHM(\XX)$ as in \cite{preproj1}. Since all Artin stacks that we encounter for the rest of this paper will be global quotient stacks, and aside from some half Tate twists almost all monodromic mixed Hodge modules will be monodromy-free, the reader may think of this as the category of $G$-equivariant mixed Hodge modules described in \cite{emhm}.
The category $\DfbMMHM(\XX)$ admits a natural t structure for which the heart is the category $\MMHM(\XX)$ of monodromic mixed Hodge modules on $\XX$, which admits a faithful functor $\rat_{\XX}^{\mon}$ to the category of perverse sheaves on $\XX$. If $\XX\cong X/G$ is a global quotient stack, then up to a cohomological shift by $\dim(G)$ this is the category of $G$-equivariant perverse sheaves on $X$. For full generality and detail, we refer the reader to \cite{preproj1}. If $\XX$ is not necessarily connected we define
\[
\DbMMHM(\XX)=\prod_{\XX'\in\pi_0(\XX)}\DfbMMHM(\XX).
\]
Let $\XX$ be a connected locally finite type Artin stack. We define the category $\DlMMHM(\XX)$ by setting the objects to be $\mathbb{Z}$-tuples of objects $\mathcal{F}^{\leq n}\in\DfbMMHM(\XX)$ such that $\Ho^m(\mathcal{F}^{\leq n})=0$ for $m>n$, along with the data of isomorphisms $\tau_{\leq n-1}\mathcal{F}^{\leq n}\cong\mathcal{F}^{\leq n-1}$. We define $\DcMMHM(\XX)$ in the analogous way, by considering tuples of objects $\mathcal{F}^{\geq n}$ along with isomorphisms $\tau_{\geq n}\mathcal{F}^{\geq n-1}\cong\mathcal{F}^{\geq n}$. If $\XX$ is a disjoint union of locally finite type Artin stacks we define
\[
\DlMMHM(\XX)=\prod_{\XX'\in \pi_0(\XX)}\DlMMHM(\XX')
\]
and likewise for $\DcMMHM(\XX)$. For $f\colon \XX\rightarrow \YY$ a morphism of Artin stacks we define functors $f_*\colon \DlMMHM(\XX)\rightarrow \DlMMHM(\YY)$ and $f_!\colon \DcMMHM(\XX)\rightarrow \DcMMHM(\YY)$ in the natural way (see \cite{preproj1}). The selling point of the categories introduced in this paragraph is that they give us a setting to talk about direct images of complexes of monodromic mixed Hodge modules along non-representable morphisms of stacks without needing a full theory of unbounded derived categories of such objects.
We define $\DbMHM(\XX),\DlMHM(\XX)$ etc. the same way, and consider these categories as subcategories of their monodromic counterparts via $z_{X,*}$.
\subsubsection{Weight filtrations}
If $X$ is a scheme, an object $\mathcal{F}\in\Ob(\MMHM(X))$ inherits a weight filtration from its weight filtration in $\MHM(X\times\AAA{1})$, and is called pure of weight $n$ if $\Gr^W_i\!(\mathcal{F})=0$ for $i\neq n$. For $\XX$ a stack, an object $\mathcal{F}\in\DlMMHM(X)$ is called \textbf{pure} if $\Ho^i(\mathcal{F})$ is pure of weight $i$ for every $i$. An object of $\DlMMHM(\XX)$ or $\DcMMHM(\XX)$ is called pure if its pullback along a smooth atlas is pure. Via Saito's results, if $\mathcal{F}\in\Ob(\DlMMHM(\XX))$ is pure, then $\mathcal{F}\cong\Ho(\mathcal{F})$. Furthermore, if $p\colon \XX\rightarrow \YY$ is projective, then $p_*\mathcal{F}$ is pure.
\subsubsection{Intersection cohomology complexes}
\label{ICsec}
Let $\XX$ be a stack. Then
\[
\ul{\QQ}_{\XX}\in\Ob(\DfbMMHM(\XX))
\]
is defined by the property that for all smooth morphisms $q\colon X\rightarrow \XX$ with $X$ a scheme, $q^*\ul{\QQ}_{\XX}\cong\ul{\QQ}_X$, the constant complex of mixed Hodge modules on $X$.
Likewise, if $\XX$ is irreducible we define $\ICS_{\XX}(\QQ)$ by the property that $q^*\ICS_{\XX}(\QQ)\cong \ICS_{X}(\QQ)$, the intersection mixed Hodge module complex on $X$. Note that unless $\XX$ is zero-dimensional, $\ICS_{\XX}(\QQ)$ is not a mixed Hodge module, but rather a complex with cohomology concentrated in degree $d=\dim(\XX)$. This complex is pure, i.e. its $d$th cohomology mixed Hodge module is pure of weight $d$.
Consider the morphism $s\colon \AAA{1}\xrightarrow{x\mapsto x^2}\AAA{1}$. We define
\[
\LLL^{1/2}=\cone(\ul{\QQ}_{\AAA{1}}\rightarrow s_*\ul{\QQ}_{\AAA{1}})\in\Db(\MMHM(\pt)).
\]
This complex has cohomology concentrated in degree 1, and is pure. Moreover there is an isomorphism
\[
(\LLL^{1/2})^{\otimes 2}\cong\LLL,
\]
justifying the notation.
We define
\begin{equation}
\label{nICdef}
\nIC_{\XX}\coloneqq \ICS_{\XX}(\QQ)\otimes\LLL^{-\dim(\XX)/2}.
\end{equation}
Since $\LLL^{1/2}$ is pure, this is a pure monodromic mixed Hodge module.
\subsubsection{G-equivariant MMHMs}
Assume that we have fixed an algebraic group $G$, and let $\XX=X/H$ be a global quotient stack, where an embedding $G\subset H$ is understood. Examples relevant to this paper will be $\XX=\Mst^{G,\zeta\sst}(Q)$ or $\XX=\Msp^{G,\zeta\sst}(Q)$, defined in \S \ref{SCsec}. We define
\begin{align}
\label{nnICdef}\nnIC_{\XX}\coloneqq &\nIC_{\XX}\otimes\LLL^{-\dim(G)/2}
\end{align}
The motivation for introducing the extra Tate twist in \eqref{nnICdef} alongside the one in \S \ref{ICsec} comes from the case $H=G$. Thinking of the underlying complex of perverse sheaves for $\nnIC_{\XX}$ as a $G$-equivariant complex of perverse sheaves on $X$, the extra twist of \eqref{nnICdef} means that this complex is a genuine perverse sheaf (without shifting).
Continuing in the same vein, we shift the natural t structure on $\DlMMHM(\XX)$, so that for example
\begin{align}
\Ho^{G,i}\!\left(\nnIC_{\Msp^{G,\zeta\sst}(Q)}\right)\neq 0 &\textrm{ if and only if }i=0,
\end{align}
where the cohomology functor is with respect to the shifted t structure. We denote by $\MMHM^G(\XX)$ the heart of this t structure (i.e. the shift by $\dim(G)$ of the usual t structure), and $\tau^G_{\leq \bullet}$ and $\tau^G_{\geq \bullet}$ the truncation functors with respect to this t structure.
\subsubsection{Vanishing cycles}
Let $\XX$ be an algebraic stack\footnote{We state all of Saito's results for stacks, as opposed to schemes. The details of the extension to stacks can be found in \cite{preproj1}.}, and let $f\in\Gamma(\XX)$ be a regular function on it. An integral part of Saito's theory is the construction of a functor
\[
\phi_f[-1]\colon\MHM(\XX)\rightarrow \MHM(\XX)
\]
lifting the usual vanishing cycle functor
\[
\varphi_f[-1]\colon\Perv(\XX)\rightarrow \Perv(\XX),
\]
in the sense that there is a natural equivalence $\rat_\XX \phi_f\cong\varphi_f \rat_\XX$. There is a further lift
\[
\phim{f}\colon\MHM(\XX)\rightarrow\MMHM(\XX)
\]
satisfying $\rat^{\mon}_\XX \phim{f}\cong \varphi_f \rat_\XX$, defined by
\begin{align*}
\phim{f}\colon &\MHM(\XX)\rightarrow \MMHM(\XX)\\
&\mathcal{F}\mapsto j_!\phi_{f/u}(\XX\times\GG_m\rightarrow \XX)^*\mathcal{F}
\end{align*}
where $u$ is a coordinate for $\GG_m$ and
\[
j\colon \XX\times\GG_m\rightarrow \XX\times\AAA{1}
\]
is the natural inclusion. The vanishing cycle functor commutes with Verdier duality, i.e. by \cite{Sai89duality} there is a natural isomorphism of functors
\[
\phim{f}\VD_{\XX}\cong \VD_{\XX}^{\mon}\phim{f}\colon\MHM(\XX)\rightarrow \MMHM(\XX).
\]
Let $g\in\Gamma(\YY)$ be a regular function on the stack $\YY$. Then there is a Thom--Sebastiani natural isomorphism \cite{Saito10}
\[
\phim{f}\boxtimes\phim{g}\rightarrow \left(\phim{f\boxplus g}(\bullet\boxtimes \bullet)\right)_{f^{-1}(0)\times g^{-1}(0)}\colon\MHM(\XX)\times\MHM(\YY)\rightarrow \MMHM(\XX\times\YY).
\]
\subsection{Quivers and their representations}
\label{qrepsec}
In this section we fix notation regarding quiver representations.
By a \textbf{quiver} $Q$ we mean a pair of finite sets $Q_1$ and $Q_0$ (the arrows and vertices respectively) along with a pair of morphisms $s,t\colon Q_1\rightarrow Q_0$ taking each arrow to its source and target, respectively. We say that $Q$ is symmetric if for every pair of vertices $i,j\in Q_0$ there are as many arrows $a$ satisfying $s(a)=i$ and $t(a)=j$ as there are arrows satisfying $s(a)=j$ and $t(a)=i$.
We refer to elements $\dd\in\dvs$ as \textbf{dimension vectors}. We define a bilinear form on the set of dimension vectors by
\begin{equation}
\label{chiform}
\chi_Q(\dd',\dd'')=\sum_{i\in Q_0}\dd'_i\dd''_i-\sum_{a\in Q_1}\dd'_{s(a)}\dd''_{t(a)}.
\end{equation}
If $Q$ is symmetric this form is symmetric. We define the form $(\bullet,\bullet)_Q$ on $\dvs$ via
\begin{equation}
\label{Eulerform}
(\dd,\dd')_Q=\chi_Q(\dd,\dd')+\chi_Q(\dd',\dd).
\end{equation}
For $K$ a field, we denote by $K Q$ the free path algebra of $Q$ over $K$. Recall that this algebra contains $\lvert Q_0\lvert $ mutually orthogonal idempotents $e_i$ for $i\in Q_0$, the ``lazy paths''. We define the dimension vector $\udim(\rho)\in\dvs$ of a $K Q$-representation via $\udim(\rho)_i=\dim_K(e_i\cdot \rho)$. If $W\in \CC Q_{\cyc}$ is a linear combination of cyclic words in $Q$, we denote by $\Jac(Q,W)$ the quotient of $\CC Q$ by the two-sided ideal generated by the noncommutative derivatives $\partial W/\partial a$ for $a\in Q_1$, as defined in \cite{ginz}.
\subsubsection{Extra gauge group}
For each pair of (not necessarily distinct) vertices $i,j$ fix a complex vector space $V_{i,j}$ with basis the arrows from $i$ to $j$. Set
\[
\GQ{Q}\coloneqq \prod_{i,j}\Gl(V_{i,j}).
\]
Then $\GQ{Q}$ acts on $\AS_{\dd}(Q)$ via the isomorphism\footnote{Here we employ the standard abuse of notation, identifying vector spaces with their total spaces, considered as algebraic varieties.}
\[
\AS_{\dd}(Q)\cong\bigoplus_{i,j\in Q_0}V_{i,j}\otimes\Hom(\CC^{\dd_i},\CC^{\dd_j}).
\]
We fix a complex algebraic group $G$, and fix a homomorphism $G\rightarrow \GQ{Q}$. We define
\begin{align*}
\Gl_{\dd}\coloneqq&\prod_{i\in Q_0}\Gl_{\dd_i}\\
\gl_{\dd}\coloneqq&\prod_{i\in Q_0}\gl_{\dd_i}\\
\WT{\Gl}_{\dd}\coloneqq &\Gl_{\dd}\times G.
\end{align*}
Throughout the paper we fix
\[
\HG=\HO(\B G,\QQ).
\]
\subsubsection{Stability conditions}
\label{SCsec}
By a \textbf{King stability condition} we mean a tuple $\zeta\in\QQ_+^{Q_0}$. The slope of a nonzero dimension vector $\dd\in\dvs$ is defined by
\[
\mu^{\zeta}(\dd)=\frac{\zeta\cdot\dd}{\sum_{i\in Q_0}\dd_i},
\]
and we define the slope of a nonzero $KQ$-module by setting
\[
\mu^{\zeta}(\rho)=\mu^{\zeta}(\udim(\rho)).
\]
For $\theta\in\QQ$ we define
\[
\Lambda_{\theta}^{\zeta}\coloneqq \{\dd\in\dvs\setminus\{0\}\colon \mu^{\zeta}(\dd)=\theta\}\cup\{0\}.
\]
A $K Q$-module $\rho$ is called $\zeta$-stable if for all proper nonzero submodules $\rho'\subset \rho$ we have $\mu^{\zeta}(\rho')<\mu^{\zeta}(\rho)$, and is $\zeta$-semistable if the weak version of this inequality is satisfied. We denote by
\[
\AS^{\zeta\sst}_{\dd}(Q)\subset \AS_{\dd}(Q)\coloneqq \prod_{a\in Q_1}\Hom(\CC^{\dd_{s(a)}},\CC^{\dd_{t(a)}})
\]
the open subvariety of $\zeta$-semistable $\CC Q$-modules.
We set
\[
\Mst^{G,\zeta\sst}_{\dd}(Q)\coloneqq \AS^{\zeta\sst}_{\dd}(Q)/\WT{\Gl}_{\dd},
\]
where the quotient is the stack-theoretic quotient. If $G$ is trivial this stack is isomorphic to the stack of $\zeta$-semistable $\dd$-dimensional $\CC Q$-modules. In \cite{King} King constructs $\Msp^{\zeta\sst}_{\dd}(Q)$, the coarse moduli space of $\zeta$-semistable $\dd$-dimensional $\CC Q$-representations. We denote by $\Msp^{G,\zeta\sst}_{\dd}(Q)$ the stack-theoretic quotient of this variety by the $G$-action. We denote by
\[
\JH^{G}\colon\Mst^{G,\zeta\sst}(Q)\rightarrow \Msp^{G,\zeta\sst}(Q)
\]
the natural map. If $G$ is trivial, this is the morphism which, at the level of points, takes $\dd$-dimensional $\CC Q$-modules to their semisimplifications.
Given an algebra $A$, presented as a quotient of a free path algebra $\CC Q$ by some two-sided ideal $R$, we denote by $\Mst^{G,\zeta\sst}(A)$ the moduli stack of $\zeta$-semistable $A$-modules, and by $\Mst^{G,\zeta\sst}_{\dd}(A)$ the substack of $\dd$-dimensional $A$-modules. Similarly, we denote by $\Msp^{G,\zeta\sst}(A)$ the stack-theoretic quotient of the coarse moduli scheme by the $G$-action.
\subsubsection{Monoidal structure}
The stack $\Msp^{G,\zeta\sst}_{\theta}(Q)$ is a monoid in the category of stacks over $\B G$, via the morphism
\[
\oplus^G\colon\Msp^{G,\zeta\sst}_{\theta}(Q)\times_{\B G}\Msp^{G,\zeta\sst}_{\theta}(Q)\rightarrow \Msp^{G,\zeta\sst}_{\theta}(Q)
\]
taking a pair of $\zeta$-polystable $\CC Q$-modules to their direct sum. This morphism is finite and commutative \cite[Lem.2.1]{Meinhardt14}, and so the monoidal product
\[
\mathcal{F}\boxtimes_{\oplus^G}\mathcal{G}\coloneqq \oplus^G_*\!\left(\mathcal{F}\boxtimes_{\B G}\mathcal{G}\right)
\]
for $\mathcal{F},\mathcal{G}\in\DlMMHM(\XX)$ is biexact and symmetric.
\subsubsection{Subscript conventions}
Throughout the paper, if $\mathbb{X}$ is some object that admits a decomposition with respect to dimension vectors $\dd\in\dvs$, we denote by $\mathbb{X}_{\dd}$ the subobject corresponding to the dimension vector $\dd$. If $\mathcal{F}$ is a sheaf or mixed Hodge module defined on $\XX$, a stack that admits a decomposition indexed by dimension vectors, we denote by $\mathcal{F}_{\dd}$ its restriction to $\XX_{\dd}$. Finally, if $f\colon\XX\rightarrow \YY$ is a morphism preserving natural decompositions of $\XX$ and $\YY$ indexed by dimension vectors, we denote by $f_{\dd}\colon\XX_{\dd}\rightarrow \YY_{\dd}$ the induced morphism.
If a stability condition $\zeta$ is fixed, we set $\XX_{\theta}=\coprod_{\dd\in\dvst}\XX_{\dd}$, and extend the conventions of the previous paragraph in the obvious way to objects admitting decompositions indexed by dimension vectors, along with morphisms that preserve these decompositions.
\subsubsection{Serre subcategories}
Throughout the paper, $\Sp$ will be used to denote a Serre subcategory of the category of $\CC Q$-modules, i.e. $\Sp$ is a full subcategory such that if
\[
0\rightarrow \rho'\rightarrow \rho\rightarrow\rho''\rightarrow 0
\]
is a short exact sequence of $\CC Q$-modules then $\rho$ is an object of $\Sp$ if and only if $\rho'$ and $\rho''$ are. We assume that $\Sp$ admits a geometric definition, in the sense that there is an inclusion of stacks
\[
\varpi\colon\Mst^{\Sp,G,\zeta\sst}(Q)\hookrightarrow \Mst^{G,\zeta\sst}(Q)
\]
which at the level of complex points is the inclusion of the set of objects of $\Sp$, and a corresponding inclusion
\[
\varpi'\colon\Msp^{\SP,G,\zeta\sst}(Q)\hookrightarrow \Msp^{G,\zeta\sst}(Q)\]
of coarse moduli spaces.
If the definition of an object $\mathcal{F}^{\SP}$ depends on a choice of some Serre subcategory $\SP$ of the category of $\CC Q'$-modules, for some quiver $Q'$, we omit the superscript $\SP$ as shorthand for the case in which we choose $\SP$ to be the entire category of $\CC Q'$-modules.
\subsection{Critical CoHAs}
\label{CCSec}
We set
\begin{align*}
\HA_{Q,W,\theta}^{G}\coloneqq &\varpi_*\varpi^!\phim{\WWW}\nnIC_{\Mst^{G,\zeta\sst}(Q)}\\
\rCoha_{Q,W,\theta}^{\Sp,G,\zeta}\coloneqq &\JH^G_*\HA_{Q,W,\theta}^{G}\\
\HCoha_{Q,W,\theta}^{\Sp,G,\zeta}\coloneqq &\HO\!\left(\Mst^{G,\zeta\sst}(Q),\varpi_*\varpi^!\phim{\WWW}\nnIC_{\Mst^{G,\zeta\sst}(Q)}\right).
\end{align*}
\begin{assumption}
\label{3dassumption}
We will assume throughout that we have chosen $Q,W,\theta,\SP,G,\zeta$ so that $\HCoha_{Q,W,\theta}^{\Sp,G,\zeta}$ is a free $\HG=\HO(\B G,\QQ)$-module.
\end{assumption}
The purity of $\HCoha_{Q,W,\theta}^{\SP,\zeta}$ is a sufficient, but not necessary condition for the assumption to hold; see \cite{preproj1} for an impure example for which the assumption holds.
Given dimension vectors $\dd',\dd''\in \dvst$ with $\dd=\dd'+\dd''$ we define
\[
\AS_{\dd',\dd''}^{\zeta\sst}(Q)\subset \AS^{\zeta\sst}_{\dd}(Q)
\]
to be the subset of linear maps preserving the $Q_0$-graded subspace $\CC^{\dd'}\subset \CC^{\dd}$, and we define
\[
\Gl_{\dd',\dd''}\subset \Gl_{\dd}
\]
to be the subgroup preserving the same subspace. We define $\pi'_1,\pi'_2,\pi'_3$ to be the natural morphisms from $\AS_{\dd',\dd''}^{\zeta\sst}(Q)$ to $\AS_{\dd'}^{\zeta\sst}(Q)$, $\AS_{\dd}^{\zeta\sst}(Q)$ and $\AS_{\dd''}^{\zeta\sst}(Q)$ respectively. We define
\[
\Mst_{\dd',\dd''}^{G,\zeta\sst}(Q)\coloneqq \AS_{\dd',\dd''}^{\zeta\sst}(Q)/\left(\Gl_{\dd',\dd''}\times G\right).
\]
Finally we define $\Mst_{\theta}^{G ,\zeta\sst}(Q)_{(2)}$ to be the union of the stacks $\Mst_{\dd',\dd''}^{G,\zeta\sst}(Q)$ across all $\dd',\dd''\in\dvst$.
Consider the commutative diagram
\begin{equation}
\label{3dc}
\xymatrix{
&\Mst_{\theta}^{G ,\zeta\sst}(Q)_{(2)}\ar[dl]_{\pi_1\times\pi_3\;\;}\ar[dr]^{\pi_2}\\
\Mst_{\theta}^{G ,\zeta\sst}(Q)\times_{\B G} \Mst_{\theta}^{G ,\zeta\sst}(Q)\ar[d]^{\JH^{G }\times_{\B G} \JH^{G }}&&\Mst_{\theta}^{G ,\zeta\sst}(Q)\ar[d]^{\JH^{G }}\\
\Msp_{\theta}^{G ,\zeta\sst}(Q)\times_{\B G} \Msp_{\theta}^{G ,\zeta\sst}(Q)\ar[rr]^-{\oplus^{G }}&&\Msp_{\theta}^{G ,\zeta\sst}(Q)\\
}
\end{equation}
where $\pi_1,\pi_2,\pi_3$ are induced by $\pi'_1\pi'_2,\pi'_3$ respectively. Set
\begin{align*}
\mathbb{A}=&\Mst_{\theta}^{G ,\zeta\sst}(Q)\times_{\B G} \Mst_{\theta}^{G ,\zeta\sst}(Q)\\
\mathbb{B}=&\Mst_{\theta}^{G ,\zeta\sst}(Q)_{(2)}\\
\mathbb{O}=&\Mst_{\theta}^{G ,\zeta\sst}(Q).
\end{align*}
Via the Thom--Sebastiani isomorphism and the composition of appropriate Tate twists of the morphisms
\[
\varpi'_*\varpi'^!(\JH^G\times_{\B G}\JH^G)_*\phim{\WWW}\!\left(\ul{\QQ}_{\mathbb{A}}\rightarrow (\pi_1\times\pi_3)_*\ul{\QQ}_{\mathbb{B}}\right)
\]
and
\[
\varpi'_*\varpi'^!\JH^G_*\phim{\WWW}\Dmon_{\mathbb{O}}\!\left(\ul{\QQ}_{\mathbb{O}}\rightarrow \pi_{2,*}\ul{\QQ}_{\mathbb{B}}\right)
\]
we define the morphism
\begin{equation}
\label{stardef}
\star\colon\rCoha_{Q,W}^{\Sp,G,\zeta}\boxtimes_{\oplus^G}\rCoha_{Q,W}^{\Sp,G,\zeta}\rightarrow \rCoha_{Q,W}^{\Sp,G,\zeta}
\end{equation}
i.e. a multiplication operation on $\rCoha_{Q,W}^{\Sp,G,\zeta}$. The proof that this operation is associative is standard, and is as in \cite[Sec.2.3]{KS2}. Applying $\HO$ to this morphism we obtain a morphism
\[
\HCoha_{Q,W}^{\Sp,G,\zeta}\otimes_{\HG}\HCoha_{Q,W}^{\Sp,G,\zeta}\rightarrow \HCoha_{Q,W}^{\Sp,G,\zeta}
\]
and we define the (associative) multiplication on $\HCoha_{Q,W}^{\Sp,G,\zeta}$ by composing with the surjection
\[
\HCoha_{Q,W}^{\Sp,G,\zeta}\otimes\HCoha_{Q,W}^{\Sp,G,\zeta}\rightarrow\HCoha_{Q,W}^{\Sp,G,\zeta}\otimes_{\HG}\HCoha_{Q,W}^{\Sp,G,\zeta}.
\]
\subsection{The PBW theorem}
We next recall some fundamental results for critical CoHAs from\footnote{For the extension to the $G$-equivariant case considered here, we refer the reader to \cite{preproj1}.} \cite{QEAs}. For ease of exposition we assume that $Q$ is symmetric, though for generic stability conditions all results are stated more generally in \cite{QEAs,preproj1}.
Firstly, there is an isomorphism
\[
\JH^G_*\HA_{Q,W,\theta}^{G,\zeta}\cong\Ho\!\left(\JH^G_*\HA_{Q,W,\theta}^{G,\zeta}\right)
\]
and $\tau^G_{\leq 0}\!\left(\JH^G_*\HA_{Q,W,\theta}^{G,\zeta}\right)=0$. By base change we have
\begin{equation}
\label{decomp}
\rCoha_{Q,W,\theta}^{\Sp,G,\zeta}\cong\varpi'_*\varpi'^!\Ho\!\left(\JH^G_*\HA_{Q,W,\theta}^{G,\zeta}\right).
\end{equation}
Setting
\begin{equation}
\label{BPShdef}
\BPSh_{Q,W,\theta}^{\SP,G,\zeta}\coloneqq \varpi'_*\varpi'^!\tau_{\leq 1}\JH^G_*\HA_{Q,W,\theta}^{G,\zeta}\otimes\LLL^{-1/2}
\end{equation}
there is an isomorphism
\[
\BPSh_{Q,W,\theta}^{\SP,G,\zeta}\cong \begin{cases} \varpi'_*\varpi'^!\phim{\WW}\nnIC_{\Msp_{\theta}^{G,\zeta\sst}(Q)}& \textrm{if }\Msp_{\theta}^{\zeta\stab}(Q)\neq \emptyset\\
0& \textrm{otherwise.}\end{cases}
\]
We define the BPS cohomology
\[
\BPS_{Q,W,\theta}^{\SP,G,\zeta}\coloneqq \HO\!\left(\Msp^{G,\zeta\sst}_{\theta}(Q),\BPSh_{Q,W,\theta}^{\SP,G,\zeta}\right).
\]
There is a natural action of $\HO(\B \CC^*,\QQ)$ on $\rCoha_{Q,W,\theta}^{\SP,G,\zeta}$ and this induces the morphism
\begin{equation}
\label{BCA}
\BPSh_{Q,W,\theta}^{\SP,G,\zeta}\otimes\HO(\B \CC^*,\QQ)_{\vir}\rightarrow \rCoha_{Q,W,\theta}^{\SP,G,\zeta}.
\end{equation}
Given an algebra $A$ and an $A$-bimodule $L$ we define
\[
\mathrm{T}_A(L)\coloneqq \bigoplus_{i\geq 0} \underbrace{L\otimes_A\cdots\otimes_A L}_{i\textrm{ times}}
\]
where the $i=0$ summand is the $A$-bimodule $A$. Given an $A$-linear Lie algebra $\mathfrak{g}$, i.e. a Lie algebra $\mathfrak{g}$, with an $A$-action such that the Lie algebra map
\[
\mathfrak{g}\otimes\mathfrak{g}\rightarrow\mathfrak{g}
\]
factors through the surjection
\[
\mathfrak{g}\otimes\mathfrak{g}\rightarrow \mathfrak{g}\otimes_A\mathfrak{g}
\]
we define the $A$-linear universal enveloping algebra as the quotient algebra
\[
\mathrm{T}_A(\mathfrak{g})/\langle a\otimes b-b\otimes a-[a,b]_{\mathfrak{g}}\rangle.
\]
Likewise, if $N$ is an $A$-bimodule we define
\[
\Sym_A(N)\coloneqq \mathrm{T}_A(\mathfrak{g})/\langle a\otimes b-b\otimes a\rangle.
\]
The main structural result for the critical CoHA is the PBW theorem:
\begin{theorem}
\label{PBWtheorem}
The morphism
\begin{equation}
\label{relPBW}
\Phi\colon \Sym_{\oplus^G}\!\left(\BPSh_{Q,W,\theta}^{\SP,G,\zeta}\otimes\HO(\B \CC^*,\QQ)_{\vir}\right)\rightarrow \rCoha_{Q,W,\theta}^{\SP,G,\zeta}
\end{equation}
obtained by combining \eqref{BCA} with the iterated CoHA multiplication map is an isomorphism in $\DlMMHM(\Msp^{\SP,G,\zeta\sst}_{\theta}(Q))$. Moreover $\Ho(\Phi)$ is an isomorphism of algebra objects in $\DlMMHM(\Msp^{G,\zeta\sst}_{\theta}(Q))$, and $\HO(\Phi)$ is an isomorphism of $\dvst$-graded monodromic mixed Hodge structures
\begin{equation}
\label{absPBW}
\Sym_{\HG}\!\left(\BPS_{Q,W,\theta}^{\SP,G,\zeta}\otimes \HO(\B \CC^*,\QQ)_{\vir}\right)\rightarrow \HCoha^{\SP,G,\zeta}_{Q,W,\theta}.
\end{equation}
\end{theorem}
\begin{remark}
Strictly speaking, the symmetric monoidal structure on $\DlMMHM(\Msp^{\SP,G,\zeta}_{\theta}(Q))$ should be twisted by a sign depending on the Euler form of $Q$, in the definition of the domain of $\Phi$. In this paper we only consider Hall algebras $\rCoha_{Q,W,\theta}^{\SP,G,\zeta}$ and $\HCoha_{Q,W,\theta}^{\SP,G,\zeta}$ for quivers $Q$ satisfying the condition that $\chi_Q(\dd',\dd'')\in 2\cdot\ZZ$ for all $\dd',\dd''\in\dvs$, and so we may omit this added complication (see \cite[Sec.1.6]{QEAs} for details).
\end{remark}
By \eqref{decomp} the algebra $\HCoha_{Q,W,\theta}^{\SP,G,\zeta}$ carries a filtration defined by
\[
\mathfrak{P}_{\leq i}\!\HCoha_{Q,W,\theta}^{\SP,G,\zeta}\coloneqq \HO\!\left(\Msp^{G,\zeta\sst}_{\theta}(Q),\varpi'_*\varpi'^!\tau_{\leq i}^G\JH^G_*\HA_{Q,W,\theta}^{\SP,G,\zeta}\right).
\]
We define
\begin{align*}
\mathfrak{g}_{Q,W,\theta}^{\SP,G,\zeta}\coloneqq &\mathfrak{P}_{\leq 1}\!\HCoha_{Q,W,\theta}^{\SP,G,\zeta}
\\
\cong&\HO\!\left(\Msp^{\SP,G,\zeta\sst}_{\theta}(Q),\varpi'_*\varpi'^!\!\Ho^{G,1}\!\left(\JH^G_*\HA_{Q,W,\theta}^{\SP,G,\zeta}\right)\right)\\
\cong&\BPS_{Q,W,\theta}^{\SP,G,\zeta}\otimes\LLL^{1/2}.
\end{align*}
By Theorem \ref{PBWtheorem} the associated graded algebra $\Gr_{\mathfrak{P}}\!\HCoha_{Q,W,\theta}^{\SP,G,\zeta}$ is supercommutative, and so $\mathfrak{g}_{Q,W,\theta}^{\SP,G,\zeta}$ is closed under the commutator bracket in $\HCoha_{Q,W,\theta}^{\SP,G,\zeta}$. The resulting Lie algebra is called the \textbf{BPS Lie algebra} \cite{QEAs}.
\begin{proposition}
\label{UEApart}
The universal map $\tau\colon\UEA_{\HG}(\mathfrak{g}_{Q,W,\theta}^{\SP,G,\zeta})\rightarrow \HCoha_{Q,W,\theta}^{\SP,G,\zeta}$ is an inclusion of algebras.
\end{proposition}
\begin{proof}
The projection
\[
\BPS_{Q,W,\theta}^{\SP,G,\zeta}\otimes\HO(\B \CC^*,\QQ)_{\vir}\rightarrow \mathfrak{g}_{Q,W,\theta}^{\SP,G,\zeta}
\]
induces a morphism
\[
\pi\colon \Sym_{\HG}\!\left(\BPS_{Q,W,\theta}^{\SP,G,\zeta}\otimes\HO(\B \CC^*,\QQ)_{\vir}\right)\rightarrow \Sym_{\HG}\!\left( \mathfrak{g}_{Q,W,\theta}^{\SP,G,\zeta} \right)
\]
which is a left inverse to the morphism
\[
\Sym_{\HG}\!\left( \mathfrak{g}_{Q,W,\theta}^{\SP,G,\zeta} \right)\rightarrow \Sym_{\HG}\!\left(\BPS_{Q,W,\theta}^{\SP,G,\zeta}\otimes\HO(\B \CC^*,\QQ)_{\vir}\right)
\]
induced by the inclusion $\mathfrak{g}_{Q,W,\theta}^{\SP,G,\zeta}\hookrightarrow \BPSh_{Q,W,\theta}^{\SP,G,\zeta}\otimes\HO(\B \CC^*,\QQ)_{\vir}$. We obtain the commutative diagram of $\dvst$-graded cohomologically graded mixed Hodge structures
\[
\xymatrix{
\UEA_{\HG}(\mathfrak{g}_{Q,W,\theta}^{\SP,G,\zeta})\ar[r]^{\tau}&\HCoha_{Q,W,\theta}^{\SP,G,\zeta}\ar[r]^-{\Phi^{-1}}&\Sym_{\HG}\!\left(\mathfrak{g}_{Q,W,\theta}^{\SP,G,\zeta}\otimes\HO(\B \CC^*,\QQ)\right)\ar[d]^\pi\\
\ar[u]^{\mathrm{PBW}}_\cong\Sym_{\HG}\!\left(\mathfrak{g}_{Q,W,\theta}^{\SP,G,\zeta}\right)\ar[rr]^=&&\Sym_{\HG}\!(\mathfrak{g}_{Q,W,\theta}^{\SP,G,\zeta})
}
\]
so that $\tau$ is indeed injective.
\end{proof}
\section{Preprojective CoHAs}
\subsection{The 2-dimensional approach}
\label{2desc}
Given a quiver $Q$ we define the doubled quiver $\ol{Q}$ by setting $\ol{Q}_0=Q_0$ and $\ol{Q}_1=Q_1\coprod Q_1^{\opp}$, where $Q_1^{\opp}$ is the set $\{a^*\colon a\in Q_1\}$, and we set
\begin{align*}
s(a^*)&=t(a)\\
t(a^*)&=s(a).
\end{align*}
We define the \textbf{preprojective algebra} as in the introduction:
\[
\Pi_Q\coloneqq \CC Q/ \langle \sum_{a\in Q_1}[a,a^*]\rangle.
\]
For each $i,j\in Q_0$ let $V_{i,j}$ be the vector space with basis given by the set of arrows from $i$ to $j$. We set
\begin{align}
\Gl'_{\edge}\coloneqq&\prod_{i\neq j}\Gl(V_{i,j}) \times \prod_{i}\Symp(V_{i,i})\\
\Gl_{\edge}\coloneqq &\Gl'_{\edge}\times \CC^*_{\hbar}\label{Gledge}
\end{align}
where $\CC^*_{\hbar}$ is a copy of $\CC^*$. Decomposing
\begin{align*}
\AS_{\dd}(\ol{Q})=&\prod_{i\neq j} \left(V_{i,j}\otimes \Hom(\CC^{\dd_{s(a)}},\CC^{\dd_{t(a)}})\right)^*\times \left(V_{i,j}\otimes \Hom(\CC^{\dd_{s(a)}},\CC^{\dd_{t(a)}})\right)\\ \times &\prod_i ((V_{i,i}\oplus V_{i,i}^*)\otimes\Hom(\CC^{\dd_i},\CC^{\dd_i}))
\end{align*}
it follows that $\AS_{\dd}(\ol{Q})$ carries an action of $\Gl'_{\edge}$ preserving the natural symplectic form. We let $\CC^*_{\hbar}$ act by scaling all of $\AS_{\dd}(\ol{Q})$, so that it acts with weight two on the symplectic form. In the following, we assume that the gauge group action $G\rightarrow \GQ{\ol{Q}}$ factors through the morphism $\Gl_{\edge}\rightarrow \GQ{\ol{Q}}$ that we have defined here.
We denote by
\[
\oplus^G_{\red}\colon \Msp_{\theta}^{G,\zeta\sst}(\ol{Q})\times_{\B G}\Msp_{\theta}^{G,\zeta\sst}(\ol{Q})\rightarrow \Msp_{\theta}^{G,\zeta\sst}(\ol{Q})
\]
the morphism taking a pair of polystable $\CC\ol{Q}$-modules to their direct sum.
\subsubsection{Serre subcategories}
Let $\Sp$ be a Serre subcategory of the category of $\CC\ol{Q}$-modules As in \cite{BSV17} we may consider the examples
\begin{enumerate}
\item
$\mathcal{N}$ is the full subcategory of $\CC\ol{Q}$-modules $\rho$ for which there is a flag of $Q_0$-graded subspaces $0\subset V_1\ldots\subset V$ of the underlying vector space of $\rho$ such that $\rho(a)(L^i)\subset L^{i-1}$ and $\rho(a^*)(L^i)\subset L^{i-1}$ for every $a\in Q_1$.
\item
$\mathcal{SN}$ is the full subcategory of $\CC\ol{Q}$-modules $\rho$ for which there is a flag of $Q_0$-graded subspaces as above, satisfying the weaker condition that $\rho(a)(L^i)\subset L^{i-1}$ and $\rho(a^*)(L^i)\subset L^i$.
\item
$\mathcal{SSN}$ is the full subcategory of $\CC\ol{Q}$-modules satisfying the same conditions as for $\SN$, but with the added condition that each of the subquotients $L^i/L^{i-1}$ is supported at a single vertex.
\end{enumerate}
Let
\begin{align*}
\varpi_{\red}\colon &\Mst^{\Sp,G,\zeta\sst}(\ol{Q})\rightarrow \Mst^{G,\zeta\sst}(\ol{Q})\\
\varpi'_{\red}\colon &\Msp^{\Sp,G,\zeta\sst}(\ol{Q})\rightarrow \Msp^{G,\zeta\sst}(\ol{Q})
\end{align*}
denote the inclusion of the stack, or respectively the stack-theoretic quotient of the coarse moduli space, of modules in $\Sp$. Fix a slope $\theta\in\QQ$. We define
\begin{align*}
\HA_{\Pi_Q,\dd}^{\SP,G,\zeta}\coloneqq&\varpi_{\red,*}\varpi_{\red}^!\iota_*\iota^!\ul{\QQ}_{\Mst_{\dd}^{G,\zeta\sst}(\ol{Q})}\otimes\LLL^{\chi_{\WT{Q}}(\dd,\dd)/2} \\
\HCoha_{\Pi_Q,\dd}^{\Sp,G,\zeta}\coloneqq &\HO\!\left(\Mst_{\dd}^{G,\zeta\sst}(\ol{Q}),\HA_{\Pi_Q,\dd}^{\SP,G,\zeta}\right)\\
\HCoha_{\Pi_Q,\theta}^{\Sp,G,\zeta}\coloneqq &\bigoplus_{\dd\in\dvst}\HCoha_{\Pi_Q,\dd}^{\Sp,G,\zeta}.
\end{align*}
\begin{remark}
Since $\chi_{\WT{Q}}(\cdot,\cdot)$ only takes even values, these are genuine mixed Hodge structures, as opposed to monodromic mixed Hodge structures.
\end{remark}
By Verdier duality, there is an isomorphism
\[
\HCoha_{\Pi_Q,\dd}^{\Sp,G,\zeta}\cong \HO^{\BM}(\Mst^{\SP,G,\zeta\sst}_{\dd}(\Pi_Q),\QQ)\otimes\LLL^{\dim(G)-\chi_Q(\dd,\dd)}.
\]
\begin{assumption}
\label{2dassumption}
We will always choose $\SP$ so that $\HCoha_{\Pi_Q,\theta}^{\Sp,G,\zeta}$ is free as a $\HG$-module.
\end{assumption}
It is a consequence of purity that this assumption holds if we set $\SP$ to be any of $\CC\ol{Q}\lmod$, $\mathcal{N}$, $\mathcal{SN}$ or $\mathcal{SSN}$ --- see \cite{preproj}, \cite{ScVa20} as well as \cite{preproj1} for details and discussion.
The $\dvst$-graded mixed Hodge structure $\HCoha_{\Pi_Q}^{\Sp,G,\zeta}$ carries a Hall algebra structure introduced by Schiffmann and Vasserot in the case of the Jordan quiver \cite{ScVa13}. It is defined in terms of correspondences. Since the algebra defined this way is isomorphic to the critical CoHA introduced in \S \ref{CCSec} we refrain from giving this definition, instead referring the reader to \cite[Sec.4]{ScVa13} and \cite{YZ18} for details.
Likewise if we set
\begin{align*}
\rCoha_{\Pi_Q,\dd}^{\Sp,G,\zeta}\coloneqq &\JH_{\red}^G\varpi_{\red,*}\varpi_{\red}^!\iota_*\iota^!\ul{\QQ}_{\Mst_{\dd}^{G,\zeta\sst}(\ol{Q})}\otimes\LLL^{\chi_{\WT{Q}}(\dd,\dd)/2}\\
\rCoha_{\Pi_Q,\theta}^{\Sp,G,\zeta}\coloneqq &\bigoplus_{\dd\in\dvst}\rCoha_{\Pi_Q,\dd}^{\Sp,G,\zeta}
\end{align*}
the correspondence diagrams that are used to define the Hall algebra structure on $\HCoha_{\Pi_Q,\theta}^{\Sp,G,\zeta}$ can be used to define an algebra structure on $\rCoha_{\Pi_Q,\theta}^{\Sp,G,\zeta}$ with respect to the monoidal structure $\boxtimes_{\oplus_{\red}^G}$. Since this algebra object will again be isomorphic to the direct image of an algebra object $\rCoha^{\WT{\SP},G,\zeta}_{\WT{Q},\WT{W},\theta}$ for $\WT{\SP},\WT{Q},\WT{W}$ chosen as in \S \ref{3desc} we do not recall the definition here (see \cite{preproj1} for details).
\subsection{The 3-dimensional description}
\label{3desc}
For the description of the preprojective CoHA in terms of vanishing cycles, we introduce a particular class of quivers with potential.
We start with a quiver $Q$ (not assumed to be symmetric). Then we define the tripled quiver $\WT{Q}$ as in \S \ref{morePerverse} to be the quiver $\ol{Q}$, with an additional set of edge-loops $\Omega=\{\omega_i\colon i\in Q_0\}$ added to the set of arrows $\ol{Q}_1$, where $s(\omega_i)=t(\omega_i)=i$. The quiver $\WT{Q}$ is symmetric. We extend the action of $\Gl_{\edge}$ to an action on
\begin{equation}
\label{WTdecomp}
\AS_{\dd}(\WT{Q})\cong \AS_{\dd}(\ol{Q})\times\prod_{i\in Q_0}\gl_{i}
\end{equation}
by letting $\Gl'_{\edge}$ act trivially on $\prod_{i\in Q_0}\gl_{i}$ and letting $\CC_{\hbar}^*$ act with weight $-2$. In what follows, we assume that the $G$-action, defined by $G\rightarrow \GQ{\WT{Q}}$, factors through the inclusion of $\Gl_{\edge}$.
We fix
\begin{equation}
\label{WTWdef}
\WT{W}=\sum_{a\in Q_1}[a,a^*]\sum_{i\in Q_0}\omega_i.
\end{equation}
The function $\Tr(\WT{W})$ is $\Gl_{\edge}$-invariant, and thus induces a function $\TTTr(\WT{W})$ on $\Mst^{G}(\WT{Q})$.
We denote by
\[
r\colon\Mst^{G}(\WT{Q})\rightarrow \Mst^{G}(\ol{Q})
\]
the forgetful map taking a $\CC \WT{Q}$-module to its underlying $\CC\ol{Q}$-module. This morphism is the projection map from the total space of a vector bundle. The function $\mathfrak{T}r(\WT{W})$ has weight one with respect to the function that scales the fibres.
We denote by
\[
\iota\colon \Mst^{G}(\Pi_Q)\hookrightarrow \Mst^G(\ol{Q})
\]
the inclusion of the substack of representations satisfying the preprojective algebra relations. Then $\iota$ is also the inclusion of the set of points $x$ for which $r^{-1}(x)\subset\WWW^{-1}(0)$. By the dimensional reduction theorem \cite[Thm.A.1]{Da13} there is a natural isomorphism
\begin{equation}
\label{DRI}
\iota_*\iota^!\rightarrow r_*\phim{\WWW}r^*.
\end{equation}
Let $\Sp$ be a Serre subcategory of the category of $\CC \ol{Q}$-modules. We denote by $\WT{\SP}$ the Serre subcategory of the category of $\CC \WT{Q}$-modules $\rho$ satisfying the condition that the underlying $\CC\ol{Q}$-module of $\rho$ is an object of $\SP$.
As in \S \ref{CCSec} we define
\begin{align*}
\HA^{\WT{\SP},G,\zeta}_{\WT{Q},\WT{W},\theta}\coloneqq&\varpi_*\varpi^!\phim{\mathfrak{T}r(\WT{W})}\nnIC_{\Mst^{G,\zeta\sst}(\WT{Q})}\\
\rCoha^{\WT{\SP},G,\zeta}_{\WT{Q},\WT{W},\theta}\coloneqq&\JH^G_{*}\HA^{\WT{\SP},G,\zeta}_{\WT{Q},\WT{W},\theta}\\
\HCoha^{\WT{\SP},G,\zeta}_{\WT{Q},\WT{W},\theta}\coloneqq&\HO\!\left(\Mst^{G,\zeta\sst}(\WT{Q}),\HA^{\WT{\SP},G,\zeta}_{\WT{Q},\WT{W},\theta}\right)
\end{align*}
where the last two objects carry algebra structures via the diagram of correspondences \eqref{3dc}. \subsubsection{Stability conditions and dimensional reduction}
Via the isomorphism \eqref{DRI} there is a natural isomorphism
\[
r_*\HA^{G}_{\WT{Q},\WT{W}}\cong\iota_*\iota^!\ul{\QQ}_{\Mst^G(\ol{Q})}\otimes \LLL^{\chi_{\WT{Q}}(\dd,\dd)/2}.
\]
Applying $\varpi_{\red,*}\varpi^!_{\red}$ and base change to this isomorphism, gives a natural isomorphism
\begin{equation}
\label{withvp}
r_*\HA^{\WT{\SP},G}_{\WT{Q},\WT{W}}\cong\varpi_{\red,*}\varpi_{\red}^!\iota_*\iota^!\ul{\QQ}_{\Mst^G(\ol{Q})}\otimes \LLL^{\chi_{\WT{Q}}(\dd,\dd)/2}.
\end{equation}
We would like to be able to incorporate stability conditions into isomorphism \eqref{withvp} but there is an obvious problem: far from being the projection from a total space of a vector bundle, the forgetful morphism from $\Mst^{G,\zeta\sst}_{\dd}(\WT{Q})\rightarrow \Mst^{G,\zeta\sst}_{\dd}(\ol{Q})$ is not even defined! This is because the underlying $\CC\ol{Q}$-module of a $\zeta$-semistable $\CC \WT{Q}$-module may be unstable. On the way to resolving the problem, we define
\[
\Mst^{G,\zeta\ssst}_{\dd}(\WT{Q})\coloneqq r_{\dd}^{-1}(\Mst^{G,\zeta\sst}_{\dd}(\ol{Q})).
\]
Then the morphism $r_{\dd}^{\circ}\colon \Mst^{G,\zeta\ssst}_{\dd}(\WT{Q})\rightarrow \Mst^{G,\zeta\sst}_{\dd}(\ol{Q})$ obtained by restricting $r_{\dd}$ is the projection from the total space of a vector bundle, as required in the statement of the dimensional reduction theorem. We will resolve the above problem by use of the following helpful fact.
\begin{proposition}\cite[Lem.6.5]{preproj}\label{SSL}
The critical locus of the function $\TTTr(\WT{W})$ on $\Mst^{\zeta\sst}(\WT{Q})$ lies inside $\Mst^{G,\zeta\ssst}(\WT{Q})$. As a consequence, the support of $\HA^{\WT{\SP},G,\zeta}_{\WT{Q},\WT{W},\dd}$ is contained in $\Mst^{G,\zeta\ssst}_{\dd}(\WT{Q})$.
\end{proposition}
\subsubsection{The absolute CoHA}
\label{3dAbs}
Let
\[
\kappa\colon \Mst^{G,\zeta\ssst}_{\dd}(\WT{Q})\hookrightarrow \Mst^{G,\zeta\sst}_{\dd}(\WT{Q})
\]
be the inclusion. We define
\[
\HA^{\circ,\WT{\SP},G,\zeta}_{\WT{Q},\WT{W},\dd}\coloneqq \kappa^*\HA^{\WT{\SP},G,\zeta}_{\WT{Q},\WT{W},\dd}.
\]
By dimensional reduction \eqref{DRI} there is an isomorphism
\begin{equation}
\label{DRI2}
r^{\circ}_{\dd,*}\HA^{\circ,\WT{\SP},G,\zeta}_{\WT{Q},\WT{W},\dd}\cong\HA_{\Pi_Q,\dd}^{\SP,G,\zeta},
\end{equation}
and so there is an isomorphism of $\HG$-modules
\begin{equation}
\label{decr}
\HO\!\left(\Mst^{G,\zeta\ssst}_{\theta}(\WT{Q}),\HA^{\circ,\WT{\SP},G,\zeta}_{\WT{Q},\WT{W},\theta}\right)\cong \HCoha_{\Pi_Q,\theta}^{\Sp,G,\zeta}.
\end{equation}
On the other hand by Proposition \ref{SSL} we deduce that there are isomorphisms
\begin{align}
\label{deca}\HO\!\left(\Mst^{G,\zeta\ssst}_{\theta}(\WT{Q}),\HA^{\circ,\WT{\SP},G,\zeta}_{\WT{Q},\WT{W},\theta}\right)\cong &\HO\!\left(\Mst_{\theta}^{G,\zeta\sst}(\WT{Q}),\HA^{\WT{\SP},G,\zeta}_{\WT{Q},\WT{W},\theta}\right)\\
=&\HCoha_{\WT{Q},\WT{W},\theta}^{\WT{\SP},G,\zeta}.\nonumber
\end{align}
Combining \eqref{deca} and \eqref{decr} yields the following isomorphism of $\HG$-modules in the category of $\dvst$-graded, cohomologically graded mixed Hodge structures:
\begin{equation}
\label{drnoalg}
\HCoha_{\Pi_Q,\theta}^{\Sp,G,\zeta}\cong \HCoha_{\WT{Q},\WT{W},\theta}^{\WT{\SP},G,\zeta}.
\end{equation}
As such, $\HCoha_{\Pi_Q,\theta}^{\SP,G,\zeta}$ inherits a $\HG$-linear algebra structure from the algebra structure on $\HCoha_{\WT{Q},\WT{W},\theta}^{\SP,G,\zeta}$.
\subsubsection{The relative CoHA}
\label{2drelCoHA}
In this section we lift the absolute CoHA constructed in \S \ref{3dAbs} to an algebra structure on the object $\rCoha_{\Pi_Q,\theta}^{\Sp,G,\zeta}$ in the category $\DlMHM(\Msp^{G,\zeta\sst}_{\theta}(\ol{Q}))$.
We denote by $\kappa'\colon \Msp^{G,\zeta\ssst}(\WT{Q})\rightarrow \Msp^{G,\zeta\sst}(\WT{Q})$ the inclusion of the subscheme\footnote{It is not hard to show that this is an open subscheme; we leave the proof to the reader.} of $\CC\WT{Q}$-modules for which the underlying $\CC\ol{Q}$-module is $\zeta$-semistable, and we denote by
\[
r'\colon \Msp^{G,\zeta\ssst}(\WT{Q})\rightarrow \Msp^{G,\zeta\sst}(\ol{Q})
\]
the forgetful morphism. To make it easier to keep track of them all, we arrange some of the morphisms introduced in this section into a commutative diagram\footnote{We indicate the version where $G=\{1\}$. In general, there should be $G$ superscripts everywhere.}:
\begin{align}
\label{drcd}
\xymatrix{
\Mst(\WT{Q})\ar[rr]^r&&\Mst(\ol{Q})
\\
\ar@{^{(}->}[u]\Mst^{\zeta\sst}(\WT{Q})\ar[d]^{\JH}&\ar@{_{(}->}[l]_{\kappa}\Mst^{\zeta\ssst}(\WT{Q})\ar[d]^{\JH^{\circ}}\ar[r]^-{r^{\circ}}& \Mst^{\zeta\sst}(\ol{Q})\ar[d]^{\JH_{\red}}\ar@{^{(}->}[u]\\
\Msp^{\zeta\sst}(\WT{Q})&\ar@{_{(}->}[l]_{\kappa'}\Msp^{\zeta\ssst}(\WT{Q})\ar[r]^-{r'}&\Msp^{\zeta\sst}(\ol{Q}).
}
\end{align}
By dimensional reduction there is a natural isomorphism
\begin{equation}
\label{2DIR}
r'_{*}\kappa'^*\rCoha_{\WT{Q},\WT{W},\dd}^{\SP,G,\zeta}\cong \rCoha_{\Pi_Q,\dd}^{\Sp,G,\zeta}
\end{equation}
obtained via commutativity of the diagram \eqref{drcd}. Since $\kappa'$ and $r'$ are morphisms of monoids, the object $\rCoha_{\Pi_Q,\dd}^{\Sp,G,\zeta}$ inherits an algebra structure, as promised in \S \ref{2desc}. Furthermore, by Proposition \ref{SSL} we obtain the first of the isomorphisms
\begin{align*}
\HO(\Msp^{G,\zeta\ssst}(\WT{Q}),\kappa'^*\rCoha_{\WT{Q},\WT{W}}^{\SP,G,\zeta})&\cong \HO(\Msp^{G,\zeta\sst}(\WT{Q}),\rCoha_{\WT{Q},\WT{W}}^{\SP,G,\zeta})\\
&\cong \HCoha_{\Pi_Q,\theta}^{\Sp,G,\zeta}
\end{align*}
(the second is \eqref{drnoalg}). The Hall algebra structure on $\HCoha_{\Pi_Q,\theta}^{\Sp,G,\zeta}$ comes from applying $\HO$ to the Hall algebra structure on $\rCoha_{\Pi_Q,\theta}^{\Sp,G,\zeta}$, i.e. $\rCoha_{\Pi_Q,\theta}^{\Sp,G,\zeta}$ is a lift of the CoHA $\HCoha_{\Pi_Q,\theta}^{\Sp,G,\zeta}$ to the category $\DlMHM(\Msp^{G,\zeta\sst}_{\theta}(\ol{Q}))$.
\section{BPS sheaves on $\Msp(\ol{Q})$}
\subsection{Generalities on the BPS sheaves for $(\WT{Q},\WT{W})$}
Let $Q$ be a quiver, then we define $\WT{Q}$ and $\WT{W}$ as in \S \ref{3desc}, pick a stability condition $\zeta\in\QQ^{Q_0}$ and a slope $\theta\in \QQ$, as well as an extra gauge group $G$ along with a homomorphism $G\rightarrow \Gl_{\edge}$ as in \S \ref{2desc}. We furthermore pick a $G$-invariant Serre subcategory $\SP$ of the category of $\CC\ol{Q}$-modules, satisfying Assumption \ref{2dassumption}, and define $\WT{\SP}$ as in \S \ref{3desc}. Then $\WT{\SP}$ satisfies Assumption \ref{3dassumption} via the isomorphism \eqref{drnoalg}.
With this data fixed, we define the BPS sheaf
\[
\BPSh_{\WT{Q},\WT{W},\theta}^{\WT{\SP},G,\zeta}\in\DbMMHM(\Msp_{\theta}^{G,\zeta\sst}(\WT{Q}))
\]
as in \eqref{BPShdef}. If $\SP=\CC\ol{Q}\lmod$, so $\WT{\SP}=\CC\WT{Q}\lmod$, this is a $G$-equivariant (monodromic) mixed Hodge module, otherwise, it may be a complex of monodromic mixed Hodge modules with cohomology in several degrees.
\subsubsection{The 2d BPS sheaf}
We define $\Gl_{\edge}$ as in \eqref{Gledge}. We let $\Gl_{\edge}$ act on $\AAA{1}$ via the projection to $\CC^*_{\hbar}$, and the weight -2 action of $\CC^*_{\hbar}$ on $\AAA{1}$. The inclusion
\begin{align*}
\AAA{1}&\rightarrow \gl_{\dd}\\
t&\mapsto (t\cdot \Id_{\CC^{\dd_i}})_{i\in Q_0},
\end{align*}
along with the decomposition \eqref{WTdecomp}, induces a $\Gl_{\edge}$-equivariant inclusion
\[
\AS_{\dd}(\ol{Q})\times\AAA{1}\hookrightarrow \AS_{\dd}(\WT{Q}).
\]
This induces the inclusion
\[
l\colon \Msp^{G,\zeta\sst}(\ol{Q})\times_{\B G}\AAA{1}\hookrightarrow \Msp^{G,\zeta\sst}(\WT{Q}).
\]
We denote the projection by
\[
h\colon \Msp^{G,\zeta\sst}(\ol{Q})\times_{\B G}\AAA{1}\rightarrow \Msp^{G,\zeta\sst}(\ol{Q}).
\]
The following theorem is essentially proved in \cite[Lem.4.1]{preproj}, though see \cite{preproj1} for the adjustments necessary to incorporate the additional data of $\SP,G,\zeta$.
\begin{thmdef}
\label{2dBPSdef}
There exists an object
\begin{equation}
\label{rbps1}
\BPSh_{\Pi_Q,\theta}^{\SP,G,\zeta}\in\DbMMHM(\Msp^{G,\zeta\sst}_{\theta}(\ol{Q}))
\end{equation}
along with an isomorphism
\begin{equation}
\label{rbps2}
\BPSh_{\WT{Q},\WT{W},\theta}^{\WT{\SP},G,\zeta}\cong l_*(h^*\BPSh_{\Pi_Q,\theta}^{\SP,G,\zeta}\otimes\LLL^{-1/2}).
\end{equation}
\end{thmdef}
In words, the theorem says that $\BPSh_{\WT{Q},\WT{W},\theta}^{\WT{\SP},G,\zeta}$ is supported on the locus containing those $\CC\WT{Q}$-modules for which all of the generalised eigenvalues of all of the operators $\omega_i\cdot$ are the same complex number $t$, and the sheaf does not depend on this complex number.
By \eqref{rbps2} there is an isomorphism of $\dvst$-graded, cohomologically graded mixed Hodge structures
\begin{equation}
\label{rbps3}
\HO\!\left(\Msp^{G,\zeta\sst}_\theta(\WT{Q}),\BPSh_{\WT{Q},\WT{W},\theta}^{\WT{\SP},G,\zeta}\right)\otimes\LLL^{1/2}\cong \HO\!\left(\Msp^{G,\zeta\sst}_\theta(\ol{Q}),\BPSh_{\Pi_Q,\theta}^{\SP,G,\zeta}\right).
\end{equation}
\begin{definition}
We define the Lie algebra
\[
\fg_{\Pi_Q,\theta}^{\SP,G,\zeta}\coloneqq \HO\!\left(\Msp^{G,\zeta\sst}_\theta(\ol{Q}),\BPSh_{\Pi_Q,\theta}^{\SP,G,\zeta}\right).
\]
The Lie algebra structure is induced by isomorphism \eqref{rbps3} and the Lie algebra structure on
\[
\fg_{\Pi_Q,\theta}^{\SP,G,\zeta}\cong \fg_{\WT{Q},\WT{W},\theta}^{\WT{\SP},G,\zeta}\cong \HO\!\left(\Msp^G_{\theta}(\WT{Q}),\BPSh_{\WT{Q},\WT{W},\theta}^{\SP,G,\zeta}\right)\otimes\LLL^{1/2}.
\]
\end{definition}
Combining with \eqref{drnoalg} and \eqref{absPBW} there is a PBW isomorphism
\begin{equation}
\label{rPBW}
\Sym_B\!\left(\fg_{\Pi_Q,\theta}^{\SP,G,\zeta}\otimes \HO(\B\CC^*,\QQ)\right) \rightarrow \HCoha^{\SP,G,\zeta}_{\Pi_Q,\theta}.
\end{equation}
\begin{remark}
In contrast with \eqref{absPBW} there is no half Tate twist in \eqref{rPBW}, and all of the terms in \eqref{rPBW} are defined as mixed Hodge structures without any monodromy.
\end{remark}
We note that the image of $l$ lies within $\Mst^{G,\zeta\ssst}(\WT{Q})$, and thus there is an isomorphism
\begin{equation}
\label{pfBP}
r'_*\kappa'^*\BPSh^{\WT{\SP},G,\zeta}_{\WT{Q},\WT{W},\theta}\otimes\LLL^{1/2}\cong\BPSh_{\Pi_Q,\theta}^{\SP,G,\zeta}
\end{equation}
and so, via \eqref{2DIR} and the PBW theorem \eqref{relPBW}, an isomorphism
\begin{equation}
\label{2dInt}
\Sym_{\oplus_{\red}^G}\!\left( \BPSh_{\Pi_Q,\theta}^{\SP,G,\zeta}\otimes \HO(\B\CC^*,\QQ)\right)\cong \rCoha_{\Pi_Q,\theta}^{\SP,G,\zeta}
\end{equation}
lifting \eqref{rPBW}.
\subsection{Restricted Kac polynomials}
In this section we assume that $G$ is trivial, so we drop it from the notation. Also, we will work with the degenerate stability condition $\zeta=(0,\ldots,0)$ and slope $\theta=0$, so that we may drop $\zeta$ and $\theta$ from the notation too.
We recall the connection between the BPS Lie algebra
\[
\mathfrak{g}^{\SP}_{\Pi_Q}\cong \HO\!\left(\Msp(\WT{Q}),\BPSh_{\WT{Q},\WT{W}}^{\WT{\SP}}\right)\otimes\LLL^{1/2}.
\]
and (restricted) Kac polynomials. In the case in which $\SP=\CC\ol{Q}\lmod$, it is proved in \cite{preproj} that $\mathfrak{g}^{\SP}_{\Pi_Q}$ is pure, of Tate type, and has vanishing even cohomology. Thus we have the equality of polynomials
\begin{align}
\label{pureKac}
\chi_t(\mathfrak{g}_{\Pi_Q,\dd})= &\wt(\mathfrak{g}_{\Pi_Q,\dd}).
\end{align}
By \cite{Moz11} there is an equality
\begin{equation}
\label{MozId}
\wt(\mathfrak{g}_{\Pi_Q,\dd})=\kac_{Q,\dd}(t^{-1})
\end{equation}
where $\kac_{Q,\dd}(t)$ is the Kac polynomial for $Q$, defined to be the polynomial such that if $q=p^r$ is a prime power, $\kac_{Q,\dd}(q)$ is the number of isomorphism classes of absolutely indecomposable $\dd$-dimensional $\CC Q$-modules. So combining \eqref{pureKac} and \eqref{MozId} we deduce
\begin{equation}
\label{Kaccha}
\chi_t(\mathfrak{g}_{\Pi_Q,\dd})=\kac_{Q,\dd}(t^{-1}).
\end{equation}
Similarly, by \cite[Sec.7.2]{preproj} the mixed Hodge structures on $\mathfrak{g}_{\Pi_Q}^{\mathcal{SN}},\mathfrak{g}_{\Pi_Q}^{\mathcal{SSN}}$ are pure, of Tate type. In a little more detail, by purity of $\HCoha_{\Pi_Q}^{\mathcal{SN}}$ and $\HCoha_{\Pi_Q}^{\mathcal{SSN}}$, proved in \cite[Sec.4.3]{ScVa20}, along with the PBW theorem \eqref{absPBW}, we deduce that $\mathfrak{g}^{\mathcal{SN}}_{\Pi_Q,\dd}$ and $\mathfrak{g}^{\mathcal{SSN}}_{\Pi_Q,\dd}$ are pure of Tate type, since they are subobjects of pure mixed Hodge structures of Tate type.
There are analogues of the Kac polynomials for these Serre subcategories. We recall from \cite{BSV17} that a representation of $Q$ is called 1-nilpotent if there is a flag $0=F_0\subset F_1\subset\ldots\subset F_r=\CC^{\dd_i}$ for every $i\in Q_0$ such that $\rho(a)(F_n)\subset F_{n-1}$ for every $n$, for every $a$ an edge-loop at $i$. We define $\kac_{Q,\dd}^{\mathcal{SN}}(t)$ to be the polynomial counting the isomorphism classes of absolutely indecomposable 1-nilpotent $\dd$-dimensional $\mathbb{F}_qQ$-modules, and $\kac_{Q,\dd}^{\mathcal{SSN}}(t)$ to be the analogous count of absolutely indecomposable nilpotent representations. Then by \cite{BSV17} (see also \cite[Sec.7.2]{preproj} for details on the passage to BPS cohomology) there are identities
\begin{align}
\label{kssn0}
\wt(\fg_{\Pi_Q,\dd}^{\mathcal{SN}})&=\kac^{\mathcal{SN}}_{Q,\dd}(t)\\
\label{kssn1}
\wt(\fg_{\Pi_Q,\dd}^{\mathcal{SSN}})&=\kac^{\mathcal{SSN}}_{Q,\dd}(t).
\end{align}
On the other hand since the mixed Hodge structures on $\fg_{\Pi_Q,\dd}^{\mathcal{SN}}$ and $\fg_{\Pi_Q,\dd}^{\mathcal{SSN}}$ are pure, their weight polynomials agree with their characteristic polynomials. So \eqref{kssn0} and \eqref{kssn1} yield
\begin{align}
\label{kssna}
\chi_t(\fg_{\Pi_Q,\dd}^{\mathcal{SN}})&=\kac^{\mathcal{SN}}_{Q,\dd}(t)\\
\label{kssn}
\chi_t(\fg_{\Pi_Q,\dd}^{\mathcal{SSN}})&=\kac^{\mathcal{SSN}}_{Q,\dd}(t)
\end{align}
respectively.
\subsubsection{Serre relation for BPS sheaves}
Let $i,j\in Q_0$ be distinct elements of $Q_0$, and assume that $Q$ has no edge-loops at $i$. Let
\begin{equation}
\label{dddef}
\dd=(e+1)\cdot 1_{i}+1_j
\end{equation}
where $e$ is the number of edges between $i$ and $j$ in the underlying graph of $Q$. In Proposition \ref{BPSvanProp} we prove a vanishing theorem for BPS sheaves, that strengthens the identity
\[
\HO^0(\fg_{\Pi_Q,\dd})=0
\]
resulting from the Serre relations in $\fg_Q$ (see \S \ref{KacH} below). Firstly we will need a proposition due to Yukinobu Toda:
\begin{proposition}\cite[Lem.4.7]{To17}
\label{TodaProp}
Let $Q'$ be a symmetric quiver, let $W'\in\CC Q'_{\cyc}$ be a superpotential, let $\zeta\in\mathbb{Q}^{Q_0}$ be a stability condition, and let $\dd\in\dvs$ be a dimension vector. Let
\[
q_{\dd}\colon \Msp^{\zeta\sst}_{\dd}(Q')\rightarrow \Msp_{\dd}(Q')
\]
be the affinization map. Then there is an isomorphism
\[
q_{\dd,*}\BPSh_{Q',W',\dd}^{\zeta}\cong\ \BPSh_{Q',W',\dd}.
\]
\end{proposition}
\begin{proposition}
\label{BPSvanProp}
Let $\zeta\in\QQ^{Q_0}$ be an arbitrary stability condition, let $\SP$ be arbitrary, and let $\dd=(e+1)\cdot 1_i+1_j$ with $i$, $j$ as above. There is an identity in $\MHM(\Msp^{\zeta\sst}_{\dd}(\ol{Q}))$
\begin{equation}
\label{BPSvan}
\BPSh_{\Pi_Q,\dd}^{\SP,G,\zeta}=0.
\end{equation}
\end{proposition}
\begin{proof}
Since $\BPSh_{\Pi_Q,\dd}^{\SP,G,\zeta}=\varpi'_{\red,*}\varpi'^!_{\red}\BPSh_{\Pi_Q,\dd}^{G,\zeta}$ it is sufficient to prove \eqref{BPSvan} under the assumption that $\SP=\CC\ol{Q}\lmod$. In addition, we may assume that $G$ is trivial, since a $G$-equivariant perverse sheaf is trivial if and only if the underlying perverse sheaf is.
By Theorem/Definition \ref{2dBPSdef}, we may equivalently prove that $\BPSh_{\WT{Q},\WT{W},\dd}^{\zeta}=0$. There are three cases to consider:
\begin{enumerate}
\item
$\zeta_i<\zeta_j$
\item
$\zeta_i=\zeta_j$
\item
$\zeta_i>\zeta_j$.
\end{enumerate}
The proofs for (1) and (3) are the same, while (2) follows from (1) and the identity
\[
\BPSh_{\WT{Q},\WT{W},\dd}\cong \left(\Msp_{\dd}^{\zeta\sst}(\WT{Q})\rightarrow \Msp_{\dd}(\WT{Q})\right)_*\BPSh_{\WT{Q},\WT{W},\dd}^{\zeta},
\]
which is a special case of Proposition \ref{TodaProp}. So we concentrate on (1).
We claim that
\begin{equation}
\label{epty}
\Mst_{\dd}^{\zeta\sst}(\WT{Q})\cap \crit(\TTTr(\WT{W}))=\emptyset.
\end{equation}
We first note that a point in the left hand side of \eqref{epty} represents a $\zeta$-semistable $\Jac(\WT{Q},\WT{W})$-module. By Proposition \ref{SSL}, the underlying $\Pi_Q$-module of $\rho$ is $\zeta$-semistable. On the other hand, there are no $\zeta$-semistable $\dd$-dimensional $\Pi_Q$-modules $\rho$, as for such a $\rho$ the subspace spanned by $e_i\cdotsh \rho,b_1\cdotsh\rho,\ldots,b_{e-1}\cdotsh\rho$ is a submodule, where $b_1,\ldots,b_{e-1}$ are the arrows in $\ol{Q}$ with source $j$ and target $i$. This proves the claim.
Now the proposition follows from the definition \eqref{BPShdef} and the equality
\[
\supp\!\left(\HA_{\WT{Q},\WT{W},\theta}^{\zeta}\right)=\Mst_{\dd}^{\zeta\sst}(\WT{Q})\cap \crit(\TTTr(\WT{W}))
\]
which follows from the fact that for $f$ a regular function on a smooth space $X$, $\phi_f\ul{\QQ}_X$ is supported on the critical locus of $f$.
\end{proof}
\subsection{Purity of BPS sheaves}
Let $\XX$ be a stack. We say that $\mathcal{G}\in\DlMMHM(\XX)$ is pure below if for all integers $m<n$
\begin{equation}
\label{1ineq}
\Gr^m_{\Wt}\!\left(\Ho^n\!\mathcal{G}\right)= 0.
\end{equation}
Similarly, we say that $\mathcal{G}$ is pure above if \eqref{1ineq} holds for all $m>n$. I.e. purity is the combination of being pure above and pure below.
For example, if $X$ is a smooth variety then $\HO(X,\QQ)$ is pure below (considered as a mixed Hodge module on a point), while if $X$ is projective, $\HO(X,\QQ)$ is pure above. By Poincar\'e duality, it follows that $\HO_c(X,\QQ)$ is pure above if $X$ is smooth. From the long exact sequence in compactly supported cohomology, and the fact that a variety can be stratified into smooth pieces, it follows that $\HO_c(X,\QQ)$ is pure above for \textit{all} varieties $X$. We will use the following generalisation of this fact.
\begin{lemma}
\label{PureAbove}
Let $\XX$ be a finite type stack. Let $p\colon \XX\rightarrow \YY$ be a morphism of stacks. Then $p_!\ul{\QQ}_{\XX}$ is pure above.
\end{lemma}
\begin{proof}
Since $p_!\ul{\QQ}_{\XX}$ only depends on the reduced structure of $\XX$, we may assume that $\XX$ is reduced. We first claim that $\XX$ can be written as a disjoint union $\XX=\bigcup_{i\in I}\XX_i$ of locally closed smooth substacks, where $I=\{1,\ldots n\}$ is ordered so that $\XX_i$ is open inside
\[
\XX_{\leq i}\coloneqq \bigcup_{j\leq i}\XX_j.
\]
This follows from the fact that $\XX_{\smooth}$ is smooth and dense inside the reduced stack $\XX$, Noetherian induction, and our assumption that $\XX$ is of finite type.
For $q$ a morphism of varieties, $q_!$ decreases weights. Since, for $q\colon \mathcal{Z}\rightarrow \mathcal{Z}'$ a morphism of stacks, $q_!$ is still defined in terms of morphisms of varieties, it still decreases weights. Thus $q_!\ul{\QQ}_{\mathcal{Z}}$ is pure above if $\mathcal{Z}$ is smooth.
We define
\begin{align*}
p_i&\colon \XX_i\rightarrow \YY\\
p_{\leq i}&\colon \XX_{\leq i}\rightarrow \YY
\end{align*}
to be the restrictions of $p$. Under our assumptions on $\XX$, there are distinguished triangles
\[
p_{i,!}\ul{\QQ}_{\XX_i}\rightarrow p_{\leq i,!}\ul{\QQ}_{\XX_{\leq i}}\rightarrow p_{\leq i-1}\ul{\QQ}_{\XX_{\leq i-1}}.
\]
The first term term is pure above, the last term is pure above by induction on $i$, and so the middle term is pure above, by the long exact sequence in cohomology. In particular, since $p_{\leq n}=p$, we deduce that $p_!\ul{\QQ}_{\XX}$ is pure above.
\end{proof}
The purpose of this section is to prove the following purity theorem.
\begin{theorem}
\label{purityThm}
The mixed Hodge module
\[
\BPSh_{\Pi_Q}^{G,\zeta}\in\MHM(\Msp^{G,\zeta\sst}(\ol{Q}))
\]
is pure.
\end{theorem}
\begin{proof}
Setting
\[
q\colon \Msp^{\zeta\sst}(Q)\rightarrow \Msp^{G,\zeta\sst}(Q)
\]
to be the quotient map, we have
\[
\BPSh_{\WT{Q},\WT{W}}^{\zeta}\cong q^*\BPSh_{\WT{Q},\WT{W}}^{G,\zeta}
\]
and so it is enough to prove that
\[
\mathcal{F}=\BPSh_{\WT{Q},\WT{W}}^{\zeta}
\]
is pure, i.e. we can assume that $G=\{1\}$. Recall that
\[
\mathcal{F}\cong \phim{\TTr(W)}\nIC_{\Msp^{\zeta\sst}(\WT{Q})}.
\]
Since $\phim{\TTr(W)}$ commutes with Verdier duality, and $\nIC_{\Msp^{\zeta\sst}(\WT{Q})}\cong\VD_{\Msp^{\zeta\sst}(\WT{Q})}\nIC_{\Msp^{\zeta\sst}(\WT{Q})}$, there is an isomorphism
\begin{equation}
\label{VSD}
\VD_{\Msp^{\zeta\sst}(\WT{Q})}\mathcal{F}\cong\mathcal{F}.
\end{equation}
Now for $X$ a variety and $\mathcal{G},\mathcal{L}$ objects of $\DlMMHM(X)$ and $\MMHM(X)$ respectively, there are isomorphisms
\begin{align*}
\Ho^n\!\left(\VD_{X}\mathcal{G}\right)\cong &\VD_{X}\!\Ho^{-n}\mathcal{G}\\
\VD_X(\Gr^n_{\Wt}\mathcal{L})\cong&\Gr^{-n}_{\Wt}\VD_X\!\left(\mathcal{L}\right)
\end{align*}
and so the existence of the isomorphism \eqref{VSD} implies that for $m,n\in\mathbb{Z}$ we have
\begin{equation}
\label{1ineqb}
\Gr^m_{\Wt}\!\left(\Ho^n\mathcal{F}\right)\neq 0
\end{equation}
if and only if
\[
\Gr^{-m}_{\Wt}\!\left(\Ho^{-n}\mathcal{F}\right)\neq 0.
\]
In particular, $\mathcal{F}$ is pure below if and only if it is pure above. Since by \eqref{rbps2} we may write
\[
\mathcal{F}\cong l_*\!\left(\BPSh^{\zeta}_{\Pi_Q}\boxtimes\nIC_{\AAA{1}}\right)
\]
and $\nIC_{\AAA{1}}$ is pure, we deduce that the same symmetry of impurity holds for $\BPSh^{\zeta}_{\Pi_Q}$:
\begin{itemize}
\item[$\ast$]
$\BPSh^{\zeta}_{\Pi_Q}$ is pure below if and only if it is pure above.
\end{itemize}
We will complete the proof by showing that $\BPSh^{\zeta}_{\Pi_Q}$ is pure below. We again consider the commutative diagram \eqref{drcd}. By \ref{DRI2} there is an isomorphism
\[
r^{\circ}_*\kappa^*\HA^{\zeta}_{\WT{Q},\WT{W}}\cong\iota_*\iota^!\ul{\QQ}_{\Mst^{\zeta\sst}(\ol{Q})}\otimes\LLL^{\chi_{\WT{Q}}(\dd,\dd)/2}
\]
and thus an isomorphism
\begin{align}
\label{dre}
r'_*\JH^{\circ}_*\kappa^*\HA^{\zeta}_{\WT{Q},\WT{W}}\cong \JH_{\red,*}\iota_*\iota^!\ul{\QQ}_{\Mst^{\zeta\sst}(\ol{Q})}\otimes\LLL^{\chi_{\WT{Q}}(\dd,\dd)/2}.
\end{align}
Applying Verdier duality to the right hand side of \eqref{dre}, we get
\begin{equation}
\label{Dbpu}
\VD_{\Msp^{\zeta\sst}(\ol{Q})}\JH_{\red,*}\iota_*\iota^!\ul{\QQ}_{\Mst^{\zeta\sst}(\ol{Q})}\otimes\LLL^{\chi_{\WT{Q}}(\dd,\dd)/2}\cong \JH_{\red,!}\iota_!\ul{\QQ}_{\Mst^{\zeta\sst}(\Pi_Q)}\otimes\LLL^{\chi_{Q}(\dd,\dd)}.
\end{equation}
The Tate twist comes from the calculations
\begin{align*}
\dim(\Mst^{\zeta\sst}(\ol{Q}))=&-\chi_{\WT{Q}}(\dd,\dd)-\dd\cdot \dd\\
\chi_Q(\dd,\dd)=&\dd\cdot \dd+\chi_{\WT{Q}}(\dd,\dd)/2.
\end{align*}
By Lemma \ref{PureAbove} the isomorphic objects of \eqref{Dbpu} are pure above, and thus the objects of \eqref{dre} are pure below. On the other hand, there are isomorphisms
\begin{align}
\label{drb}
r'_*\JH^{\circ}_*\kappa^*\HA^{\zeta}_{\WT{Q},\WT{W}}\cong& r'_*\kappa'^*\JH_*\HA^{\zeta}_{\WT{Q},\WT{W}}\\ \nonumber
\cong &r'_*\kappa'^*\Sym_{\oplus}\!\left( \BPSh_{\WT{Q},\WT{W}}^{\zeta}\otimes\HO(\B \CC^*,\QQ)_{\vir}\right)\\ \nonumber
\cong &\Sym_{\oplus}\!\left( \BPSh_{\Pi_Q}^{\zeta}\otimes \HO(\B \CC^*,\QQ)_{\vir}\otimes\LLL\right)
\end{align}
where we have used the PBW theorem \eqref{relPBW}, and the fact that $r'$ and $\kappa'$ are morphisms of monoids to commute them past $\Sym_{\oplus}$. Combining \eqref{dre} and \eqref{drb} there is an inclusion
\[
\BPSh_{\Pi_Q}^{\zeta}\otimes\LLL^{1/2}\subset \JH_{\red,*}\iota_*\iota^!\ul{\QQ}_{\Mst^{\zeta\sst}(\ol{Q})}\otimes\LLL^{\chi_{\WT{Q}}(\dd,\dd)/2}.
\]
We deduce that $\BPSh_{\Pi_Q}^{\zeta}$ is pure below, and thus also pure above by ($\ast$).
\end{proof}
\begin{corollary}
\label{3dpurityThm}
The BPS sheaf
\[
\BPSh^{G,\zeta}_{\WT{Q},\WT{W}}\in\MMHM^G(\Msp^{\zeta}(\WT{Q}))
\]
is pure.
\end{corollary}
\begin{proof}
This follows from Theorem \ref{purityThm} and the isomorphism \eqref{rbps2}.
\end{proof}
The purity statement of Theorem \ref{thma} is a special case of the following corollary of Theorem \ref{purityThm}:
\begin{corollary}
\label{relPurity}
The underlying objects of the relative CoHAs in $\Msp^{G,\zeta\sst}_{\theta}(\WT{Q})$ and $\Msp^{G,\zeta\sst}_{\theta}(\ol{Q})$, i.e.
\begin{align*}
\rCoha_{\Pi_Q,\theta}^{G,\zeta}\in &\DlMHM(\Msp^{G,\zeta\sst}_{\theta}(\ol{Q}))\\
\rCoha_{\WT{Q},\WT{W},\theta}^{G,\zeta}\in &\DlMMHM(\Msp^{G,\zeta\sst}_{\theta}(\WT{Q}))
\end{align*}
respectively, are pure. In particular, applying Verdier duality to the first of these statements, and taking the appropriate Tate twist, the complex of mixed Hodge modules
\begin{align}
\JH_{\red,!}\ul{\QQ}_{\Mst^{G,\zeta\sst}(\Pi_Q)}
\end{align}
is pure.
\end{corollary}
\begin{proof}
These purity statements follow from Theorem \ref{purityThm} and Corollary \ref{3dpurityThm}, respectively, via \eqref{2dInt} and \eqref{relPBW}, respectively.
\end{proof}
\begin{remark}
Given that the morphism
\[
\JH\colon \Mst^{G,\zeta\sst}(\WT{Q})\rightarrow \Msp^{G,\zeta\sst}(\WT{Q})
\]
is approximated by proper maps (in the sense of \cite{QEAs}) and thus sends pure monodromic mixed Hodge modules to pure monodromic mixed Hodge modules (see \cite{preproj1}), it might feel natural, in light of Corollary \ref{relPurity}, to conjecture that $\HA_{\WT{Q},\WT{W}}^{G,\zeta}$ is a pure monodromic mixed Hodge module on $\Mst^{G,\zeta\sst}(\WT{Q})$. However this statement turns out to be false. For example in the case of $Q$ the Jordan quiver, $G=\{1\}$ $\zeta=(0,\ldots,0)$ and $\dd=4$, impurity follows from the main result of \cite{DS09}. It seems that purity goes no ``higher'' than BPS sheaves.
\end{remark}
\section{The less perverse filtration}
\subsection{The Hall algebra in $\DlMHM(\Msp^{G,\zeta\sst}(\overline{Q}))$}
We consider the following diagram, where the top three rows are defined from diagram \eqref{3dc} (substituting $\WT{Q}$ for $Q$ there) by pulling back along the open embeddings
\begin{align*}
\kappa\colon &\Mst_{\theta}^{G ,\zeta\ssst}(\WT{Q})\hookrightarrow \Mst_{\theta}^{G ,\zeta\sst}(\WT{Q})\\
\kappa'\colon &\Msp_{\theta}^{G ,\zeta\ssst}(\WT{Q})\hookrightarrow \Msp_{\theta}^{G ,\zeta\sst}(\WT{Q}).
\end{align*}
\begin{equation}
\label{2dc}
\xymatrix{
&\Mst_{\theta}^{G ,\zeta\ssst}(\WT{Q})_{(2)}\ar[dl]_{\pi_1\times\pi_3\;\;}\ar[dr]^{\pi_2}\\
\Mst_{\theta}^{G ,\zeta\ssst}(\WT{Q})\times_{\B G} \Mst_{\theta}^{G ,\zeta\ssst}(\WT{Q})\ar[d]^{\JH^{\circ}\times_{\B G} \JH^{\circ}}&&\Mst_{\theta}^{G ,\zeta\ssst}(\WT{Q})\ar[d]^{\JH^{\circ}}\\
\Msp_{\theta}^{G ,\zeta\ssst}(\WT{Q})\times_{\B G} \Msp_{\theta}^{G ,\zeta\ssst}(\WT{Q})\ar[d]^{r'\times_{\B G}\;r'}\ar[rr]^-{\oplus^{G }}&&\Msp_{\theta}^{G ,\zeta\ssst}(\WT{Q})\ar[d]^{r'}\\
\Msp_{\theta}^{G ,\zeta\sst}(\ol{Q})\times_{\B G} \Msp_{\theta}^{G ,\zeta\sst}(\ol{Q})\ar[rr]^-{\oplus_{\red}^G}&&\Msp_{\theta}^{G ,\zeta\ssst}(\ol{Q}).
}
\end{equation}
By base change there is a natural isomorphism
\[
\rCoha_{\Pi_Q,\theta}^{\SP,G,\zeta\sst}= r'_*\kappa^*\rCoha_{\WT{Q},\WT{W},\theta}^{\WT{\SP},G,\zeta\sst}.
\]
Since the bottom square of \eqref{2dc} commutes, this complex of monodromic mixed Hodge modules inherits an algebra structure in $\DlMHM(\Msp_{\theta}^{\Sp,G,\zeta\sst}(\ol{Q}))$, i.e. we obtain the morphism
\begin{equation}
\label{rcmap}
\oplus_{\red,*}^G\!\left(\rCoha_{\Pi_Q,\theta}^{\SP,G,\zeta\sst}\boxtimes_{\B G}\rCoha_{\Pi_Q,\theta}^{\SP,G,\zeta\sst}\right)\rightarrow \rCoha_{\Pi_Q,\theta}^{\SP,G,\zeta\sst}
\end{equation}
and applying the functor $\HO$ to this morphism we recover the algebra structure on $\HCoha_{\Pi_Q,\theta}^{\SP,G,\zeta}$.
\subsection{The relative Lie algebra in $\MHM^G(\Msp_{\theta}^{\zeta\sst}(\ol{Q}))$}
\label{RLAsec}
By Theorem \ref{PBWtheorem} there is a split inclusion
\[
\BPSh_{\WT{Q},\WT{W},\theta}^{G,\zeta}\otimes\LLL^{1/2}\rightarrow \rCoha_{\WT{Q},\WT{W},\theta}^{G,\zeta}.
\]
The commutator Lie bracket provides a morphism
\[
[\cdot,\cdot]\colon \left(\BPSh_{\WT{Q},\WT{W},\theta}^{G,\zeta}\otimes\LLL^{1/2}\right)\boxtimes_{\oplus^G}\left(\BPSh_{\WT{Q},\WT{W},\theta}^{G,\zeta}\otimes\LLL^{1/2}\right)\rightarrow \mathcal{T}=\tau^G_{\leq 2}\rCoha_{\WT{Q},\WT{W},\theta}^{G,\zeta}
\]
and by Theorem \ref{PBWtheorem} again, the target $\mathcal{T}$ fits into a split triangle
\[
\BPSh_{\WT{Q},\WT{W},\theta}^{G,\zeta}\otimes\LLL^{1/2}\rightarrow \mathcal{T}\rightarrow \Ho^{G,2}(\mathcal{T}).
\]
and the composition
\[
\left(\BPSh_{\WT{Q},\WT{W},\theta}^{G,\zeta}\otimes\LLL^{1/2}\right)\boxtimes_{\oplus^G}\left(\BPSh_{\WT{Q},\WT{W},\theta}^{G,\zeta}\otimes\LLL^{1/2}\right)\rightarrow \Ho^{G,2}(\mathcal{T})
\]
is the zero morphism. The commutator Lie bracket thus induces a morphism
\begin{equation}
\label{relLA}
[\cdot,\cdot]\colon \left(\BPSh_{\WT{Q},\WT{W},\theta}^{G,\zeta}\otimes\LLL^{1/2}\right)\boxtimes_{\oplus^G}\!\left(\BPSh_{\WT{Q},\WT{W},\theta}^{G,\zeta}\otimes\LLL^{1/2}\right)\rightarrow \BPSh_{\WT{Q},\WT{W},\theta}^{G,\zeta}\otimes\LLL^{1/2}
\end{equation}
and applying $r'_*\kappa'^*$ we obtain the Lie bracket
\begin{equation}
\label{relLA2}
[\cdot,\cdot]\colon \BPSh_{\Pi_Q,\theta}^{G,\zeta}\boxtimes_{\oplus_{\red}^G}\BPSh_{\Pi_Q,\theta}^{G,\zeta}\rightarrow \BPSh_{\Pi_Q,\theta}^{G,\zeta}.
\end{equation}
This is a Lie algebra object inside the category $\MHM^G(\Msp^{\zeta\sst}_{\theta}(\ol{Q}))$, from which we obtain $\fg_{\Pi_Q,\theta}^{\zeta}$ by applying $\HO$. Likewise, applying $\varpi'_{\red,*}\varpi'^!_{\red}$ to \eqref{relLA2} we obtain a Lie algebra structure on the object $\BPSh_{\Pi_Q,\theta}^{\SP,G,\zeta}\in \DbMHM(\Msp^{\zeta\sst}_{\theta}(\ol{Q}))$, which becomes the Lie algebra $\fg^{\SP,G,\zeta}_{\Pi_Q,\theta}$ after applying $\HO$.
\subsection{Definition of the filtration}
By base change, the algebra morphism \eqref{rcmap} is given by applying $\varpi'_{\red,*}\varpi'^!_{\red}$ to
\[
\oplus_{\red,*}^G\!\left(\rCoha_{\Pi_Q,\theta}^{G,\zeta}\boxtimes_{\B G}\rCoha_{\Pi_Q,\theta}^{G,\zeta}\right)\rightarrow \rCoha_{\Pi_Q,\theta}^{G,\zeta},
\]
a morphism in $\DlMHM(\Msp_{\theta}^{G,\zeta\sst}(\ol{Q}))$. Furthermore by \eqref{2dInt} there is an isomorphism
\[
\rCoha_{\Pi_Q,\theta}^{G,\zeta}\cong \Sym_{\oplus_{\red}^G}\!\left(\BPSh_{\Pi_Q,\theta}^{G,\zeta}\otimes\HO(\B \CC^*,\QQ) \right).
\]
Since $\BPSh_{\Pi_Q,\theta}^{G,\zeta}\in\MHM^G(\Msp_{\theta}^{\zeta\sst}(\ol{Q}))$ there is an isomorphism
\[
\Ho\!\left( \BPSh_{\Pi_Q,\theta}^{G,\zeta}\otimes\HO(\B \CC^*,\QQ) \right)\cong \BPSh_{\Pi_Q,\theta}^{G,\zeta}\otimes\HO(\B \CC^*,\QQ)
\]
(i.e. the right hand side is isomorphic to its total cohomology) and so also an isomorphism
\begin{equation}
\label{decomp2}
\Sym_{\oplus_{\red}^G}\!\left(\BPSh_{\Pi_Q,\theta}^{G,\zeta}\otimes\HO(\B \CC^*,\QQ) \right)\cong \Ho\!\left(\Sym_{\oplus_{\red}^G}\!\left(\BPSh_{\Pi_Q,\theta}^{G,\zeta}\otimes\HO(\B \CC^*,\QQ) \right)\right).
\end{equation}
We could alternatively have deduced the existence of this isomorphism from the purity of the left hand side of \eqref{decomp2}. It follows that for every $p\in\mathbb{Z}$ the morphism
\[
\tau^G_{\leq p}\!\rCoha_{\Pi_Q,\theta}^{G,\zeta}\rightarrow \rCoha_{\Pi_Q,\theta}^{G,\zeta}
\]
has a left inverse $\alpha_p$, and so $\HO\varpi'_{\red,*}\varpi'^!_{\red}\alpha_p$ provides a left inverse to the morphism
\[
\HO\!\left(\Msp_{\theta}^{G,\zeta\sst}(\ol{Q}),\tau^G_{\leq p}\varpi'_{\red,*}\varpi'^!_{\red}\rCoha_{\Pi_Q,\theta}^{G,\zeta}\right)\rightarrow \HCoha_{\Pi_Q,\theta}^{\SP,G,\zeta}.
\]
Thus the objects
\[
\lP_{\leq p}\!\HCoha_{\Pi_Q,\theta}^{\SP,G,\zeta}\coloneqq \HO\!\left(\Msp_{\theta}^{G,\zeta\sst}(\ol{Q}),\varpi'_{\red,*}\varpi'^!_{\red}\tau^G_{\leq p}\!\rCoha_{\Pi_Q,\theta}^{G,\zeta}\right)
\]
provide an ascending filtration of $\HCoha_{\Pi_Q,\theta}^{\SP,G,\zeta}$, the \textbf{less perverse filtration}.
\subsubsection{A warning}
A variant of \cite[Warning 5.5]{QEAs} is in force here; if $\SP$ is not the entire category $\CC \ol{Q}\lmod$, the perverse filtration that we have defined here may be quite different from the perverse filtration given by applying perverse truncation functors to $\JH_{\red,*}\HA_{\Pi_Q,\theta}^{\SP,G,\zeta}$. For instance, let $Q$ be the Jordan quiver, with one loop, and consider the Serre subcategory $\SSN$. Then one may easily verify that
\begin{equation}
\label{warningeq}
\JH_{\red,*}\HA_{\Pi_Q,1}^{\SSN}\cong i_*\ul{\QQ}_{\AAA{1}}\otimes\HO(\B\CC^*,\QQ)
\end{equation}
where $i\colon \AAA{1}\hookrightarrow \AAA{2}$ is the inclusion of a coordinate hyperplane. In particular, the zeroth perverse cohomology of \eqref{warningeq} is zero, while if instead we apply $\varpi'_{\red,*}\varpi'^!_{\red}$ to the zeroth cohomology of
\[
\JH_{\red,*}\HA_{\Pi_Q,1}\cong \ul{\QQ}_{\AAA{2}}\otimes\HO(\B\CC^*,\QQ)\otimes\LLL^{-1}
\]
we get the (shifted) mixed Hodge module $\ul{\QQ}_{\AAA{1}}$, and we find
\[
\lP_{\leq 0}\!\HCoha_{\Pi_Q,1}^{\SSN}\cong \HO(\AAA{1},\QQ)\neq 0.
\]
This distinction between the two choices of filtration on $\HCoha_{\Pi_Q}^{\SSN}$ is crucial in \S \ref{BBA}.
\subsection{Deformed dimensional reduction}
\label{DDRsec}
We can generalise the results of this paper, incorporating deformed potentials as introduced in \cite{DaPa20}. We indicate how this goes in this section. We will not use this generalisation of the less perverse filtration, except in the statement of Corollary \ref{bels} and the example of \S \ref{defEx}.
Let $W_0\in\CC \ol{Q}_{\cyc}$ be a $G$-invariant linear combination of cyclic words in $\ol{Q}$. We make the assumption that there is a grading of the arrows of $\WT{Q}$ so that $\WT{W}+W_0$ is quasihomogeneous of positive degree. Then in \cite{DaPa20} it was shown that there is a natural isomorphism.
\[
r_*\JH^G_*\HA^G_{\WT{Q},\WT{W}+W_0}\cong \phim{\TTr(W_0)}\JH^G_{\red,*}\HA^G_{\Pi_Q}.
\]
In particular, since $\phim{\TTr(W_0)}$ is exact, and $\JH^G_{\red,*}\HA^G_{\Pi_Q}$ is pure by Theorem \ref{thma}, there is an isomorphism
\[
r_*\JH^G_*\HA^G_{\WT{Q},\WT{W}+W_0}\cong \Ho\!\left(r_*\JH^G_*\HA^G_{\WT{Q},\WT{W}+W_0}\right)
\]
and so $\HCoha^G_{\WT{Q},\WT{W}+W_0}$ carries a less perverse filtration, defined in the same way as the less perverse filtration for $\HCoha^G_{\WT{Q},\WT{W}}$. As in \S \ref{RLAsec}, we obtain a Lie algebra structure on\footnote{This isomorphism is given in \cite{preproj1}.}
\[
r_*\BPSh^G_{\WT{Q},\WT{W}+W_0}\cong \phim{\TTr(W_0)}\BPSh^G_{\Pi_Q}
\]
which recovers the BPS Lie algebra
\[
\fg^G_{\Pi_Q,W_0}\coloneqq \fg^G_{\WT{Q},\WT{W}+W_0}
\]
after applying $\HO$.
For a final layer of generality, for $\SP$ a Serre subcategory of $\CC\ol{Q}\lmod$ and $\WT{W}+W_0$ a quasihomogeneous potential as above, one may consider the object $\HA^{\SP,G}_{\WT{Q},\WT{W}+W_0}\coloneqq\varpi_*\varpi^!\HA_{\WT{Q},\WT{W}+W_0}^G$, and the associated Hall algebra $\HCoha^{\SP,G}_{\WT{Q},\WT{W}+W_0}$ which carries a (less) perverse filtration defined by
\[
\lP_{\leq i}\! \HCoha^{\SP,G}_{\WT{Q},\WT{W}+W_0}\coloneqq \HO \varpi'_{\red,*}\varpi_{\red}'^!\tau^G_{\leq i}r_*\JH^G_*\HA^G_{\WT{Q},\WT{W}+W_0}
\]
with 2d BPS sheaf
\[
\BPSh^{\SP,G}_{\Pi_Q,W_0}=r_*\BPSh^{\SP,G}_{\WT{Q},\WT{W}+W_0}\otimes\LLL^{1/2}\cong \varpi'_{\red,*}\varpi'^!_{\red}\phim{\TTr(W_0)}\BPSh^{\SP,G}_{\Pi_Q}
\]
and associated BPS Lie algebra $\fg_{\Pi_Q,W_0}^{\SP,G}=\HO\!\left(\Msp(\ol{Q}),\BPSh^{\SP,G}_{\Pi_Q,W_0}\right)$. Even at this maximal level of generality, we find that the spherical Lie subalgebra is a Kac-Moody Lie algebra, by Corollary \ref{bels}
\section{The zeroth piece of the filtration}
\subsection{The subalgebra $\lP_{\leq 0}\!\HCoha^{\Sp,G,\zeta}_{\Pi_Q,\theta}$}
By \eqref{2dInt}, the less perverse filtration on $\HCoha^{\Sp,G,\zeta}_{\Pi_Q,\theta}$ begins in degree zero, and thus the subobject
\[
\lP_{\leq 0}\!\HCoha^{\Sp,G,\zeta}_{\Pi_Q,\theta}
\]
is closed under the CoHA multiplication. It turns out that this subalgebra has a very natural description in terms of the BPS Lie algebra, completing the proof of Theorem \ref{thma}:
\begin{theorem}
There is an isomorphism of algebras
\begin{align}
\label{UEAisoo}
\lP_{\leq 0}\!\HCoha^{\Sp,G,\zeta}_{\Pi_Q,\theta}\cong \UEA_B(\mathfrak{g}_{\Pi_Q,\theta}^{G,\SP,\zeta}).
\end{align}
\end{theorem}
\begin{proof}
Applying $\tau^G_{\leq 0}$ to the isomorphism \eqref{2dInt} in the special case $\SP=\CC\ol{Q}\lmod$ yields the isomorphism
\begin{equation}
\label{halfzero}
\Sym_{\oplus_{\red}^G}\!\left( \BPSh_{\Pi_Q,\theta}^{G,\zeta}\right)\cong \tau^G_{\leq 0}\!\left(\rCoha_{\Pi_Q,\theta}^{G,\zeta}\right)
\end{equation}
defined via the relative CoHA multiplication. Now applying $\HO\varpi'_{\red,*}\varpi'^!_{\red}$ to \eqref{halfzero} we obtain the isomorphism
\[
\Sym_B(\fg_{\Pi_Q,\theta}^{\SP,G,\zeta})\xrightarrow{\cong}\lP_{\leq 0}\!\HCoha^{\Sp,G,\zeta}_{\Pi_Q,\theta}.
\]
By Proposition \ref{UEApart}, the image of the induced embedding
\[
\Sym_B(\fg_{\Pi_Q,\theta}^{\SP,G,\zeta})\hookrightarrow \HCoha^{\Sp,G,\zeta}_{\Pi_Q,\theta}
\]
is precisely the subalgebra $\UEA_B(\mathfrak{g}_{\Pi_Q,\theta}^{G,\SP,\zeta})$.
\end{proof}
\subsubsection{The perverse filtration $\mathfrak{P}_{\leq \bullet}\!\lP_{\leq 0}\!\HCoha^{\Sp,G,\zeta}_{\Pi_Q,\theta}$}
\label{DPFsec}
Since via \eqref{drnoalg} there is an inclusion
\[
I\colon \lP_{\leq 0}\!\HCoha^{\Sp,G,\zeta}_{\Pi_Q,\theta}\hookrightarrow \HCoha_{\WT{Q},\WT{W},\theta}^{\WT{\SP},G,\zeta}
\]
and $\HCoha_{\WT{Q},\WT{W},\theta}^{\WT{\SP},G,\zeta}$ carries the ``more'' perverse filtration $\mathfrak{P}_{\leq \bullet}\!\HCoha_{\WT{Q},\WT{W},\theta}^{\WT{\SP},G,\zeta}$ defined in \cite{QEAs}, we obtain a perverse filtration on
$\lP_{\leq 0}\!\HCoha^{\Sp,G,\zeta}_{\Pi_Q,\theta}$ itself, for which the $i$th piece is
\begin{equation}
\label{PLP}
\lP_{\leq 0}\!\HCoha^{\Sp,G,\zeta}_{\Pi_Q,\theta}\cap\: I^{-1}\!\left(\mathfrak{P}_{\leq i}\!\HCoha^{\WT{\Sp},G,\zeta}_{\WT{Q},\WT{W},\theta}\right).
\end{equation}
Writing
\begin{align*}
\lP_{\leq 0}\!\HCoha^{\Sp,G,\zeta}_{\Pi_Q,\theta}\cong \HO\varpi'_*\varpi'^!\Sym_{\oplus^G}\!\left(\BPSh_{\WT{Q},\WT{W},\theta}^{G,\zeta}\otimes\LLL^{1/2}\right),
\end{align*}
we have that
\begin{align*}
\lP_{\leq 0}\!\HCoha^{\Sp,G,\zeta}_{\Pi_Q,\theta}\cap\: I^{-1}\!\left(\mathfrak{P}_{\leq n}\!\HCoha^{\WT{\Sp},G,\zeta}_{\WT{Q},\WT{W},\theta}\right)\cong&\HO\varpi'_*\varpi'^!\tau^G_{\leq n}\Sym_{\oplus^G}\!\left(\BPSh_{\WT{Q},\WT{W},\theta}^{G,\zeta}\otimes\LLL^{1/2}\right)\\
\cong& \HO\varpi'_*\varpi'^!\bigoplus_{i=0}^n\left(\Sym^i_{\oplus^G}\!\left(\BPSh_{\WT{Q},\WT{W},\theta}^{G,\zeta}\otimes\LLL^{1/2}\right)\right)\\
\cong& \HO\bigoplus_{i=0}^n\left(\Sym^i_{\oplus^G}\!\left(\BPSh_{\WT{Q},\WT{W},\theta}^{\WT{\SP},G,\zeta}\otimes\LLL^{1/2}\right)\right)\\
\cong& \bigoplus_{i=0}^n\left(\Sym^i_B\!\left(\fg_{\Pi_Q,\theta}^{\SP,G,\zeta}\right)\right).
\end{align*}
We deduce the following
\begin{proposition}
Under the isomorphism \eqref{UEAisoo}, the perverse filtration \eqref{PLP} is sent to the order filtration on the universal enveloping algebra $\UEA_B(\mathfrak{g}_{\Pi_Q,\theta}^{G,\SP,\zeta})$.
\end{proposition}
\subsection{Nakajima quiver varieties}
\label{NQVs}
As preparation for the proof of Theorem \ref{KMLA} below, we recall some fundamental results regarding the action of $\HCoha_{\Pi_Q}$ on the cohomology of Nakajima quiver varieties, recasting these results in terms of vanishing cycle cohomology along the way.
\subsubsection{Nakajima quiver varieties as critical loci}
Given a quiver $Q$ and a dimension vector $\ff\in\dvs$, we define the quiver $Q_{\ff}$ by adding one vertex $\infty$ to the vertex set $Q_0$, and for each vertex $i\in Q_0$ we add $\ff_i$ arrows $a_{i,1},\ldots,a_{i,\ff_i}$ with source $\infty$ and target $i$.
Given a dimension vector $\dd\in\dvs$ we denote by $\dd^+$ the dimension vector for $Q_{\ff}$ defined by
\begin{itemize}
\item
$\dd^+\lvert_{Q_0}=\dd$
\item
$\dd^+_{\infty}=1$.
\end{itemize}
From the quiver $Q_{\ff}$ we form the quiver $\WT{Q_{\ff}}$ via the tripling construction of \S \ref{3desc}. For each $i\in Q_0$ there are $\ff_i$ arrows $a^*_{i,1},\ldots,a^*_{i,\ff_i}$ in $(\WT{Q_{\ff}})_1$ with source $i$ and target $\infty$. We denote by $\WT{W_{\ff}}$ the canonical cubic potential for $\WT{Q_{\ff}}$. We form $Q^+$ by removing the loop $\omega_{\infty}$ from $\WT{Q_{\ff}}$, and form $W^+$ from $\WT{W_{\ff}}$ by removing all paths containing $\omega_{\infty}$. So in symbols
\[
W^+=\left(\sum_{a\in Q_1}\left([a,a^*]+\sum_{m=1}^{\ff_i}a_{i,m}a^*_{i,m}\right)\right)\left(\sum_{i\in Q_0}\omega_i\right).
\]
We define the stability condition $\zeta\in\QQ^{Q^+_0}$ by setting $\zeta_i=0$ for $i\in Q_0$ and $\zeta_{\infty}=1$. Then a $\dd^+$-dimensional $\CC Q^+$-module $\rho$ is $\zeta^+$-stable if and only if it is $\zeta^+$-semistable. This occurs if and only if the vector space $e_{\infty}\cdotsh \rho\cong\CC$ generates $\rho$ under the action of $\CC Q^+$.
We define the fine moduli space
\[
\Msp_{\ff,\dd}(Q)=\AS^{\zeta^+\sst}_{\dd^+}(Q^+)/\Gl_{\dd},
\]
which carries the function $\TTr(W^+)_{\dd}$.
Following Nakajima \cite{Nak98}, we define $\Nak(\ff,\dd)\subset \AS_{\dd^+}^{\zeta^+\sst}(\ol{Q_{\ff}})/\Gl_{\dd}$ to be intersection with the $\Gl_{\dd}$-quotient of the zero set of the moment map
\begin{align*}
\AS_{\dd}(Q)\times\AS_{\dd}(Q^{\opp})\times \left(\prod_{i\in Q_0} (\CC^{\dd_i})^{\ff_i}\right)\times \left(\prod_{i\in Q_0} ((\CC^{\dd_i})^{\ff_i})^*\right)&\rightarrow \g_{\dd}\\
(A,A^*,I,J)&\mapsto [A,A^*]+IJ.
\end{align*}
We define the embedding $\iota\colon\Nak(\ff,\dd)\hookrightarrow \Msp_{\ff,\dd}(Q)$ by extending a $\CC\ol{Q_{\ff}}$-module to a $\CC Q^+$ module, setting the action of each of the $\omega_i$ for $i\in Q_0$ to be zero. If $\Nak(\ff,\dd)\neq \emptyset$ then
\begin{equation}
\label{dimM}
\dim(\Nak(\ff,\dd))=2\ff\cdot \dd-2\chi_Q(\dd,\dd).
\end{equation}
\begin{proposition}
\label{NQVDR}
There is an equality of subschemes
\[
\crit\!\left(\TTr(W^+)_{\dd}\right)=\Nak(\ff,\dd).
\]
Moreover, there is an isomorphism of mixed Hodge modules\footnote{These are indeed mixed Hodge modules, since by \eqref{dimM} there are an even number of half Tate twists in the definition \eqref{nICdef} of the right hand side of \eqref{rDIM}.}
\begin{equation}
\label{rDIM}
\phim{\TTr(W^+)_{\dd}}\nIC_{\Msp_{\ff,\dd}(Q)}\cong \iota_*\nIC_{\Nak(\ff,\dd)},
\end{equation}
so that, in particular, there is an isomorphism of mixed Hodge structures
\begin{equation}
\label{HDIM}
\HO\!\left(\Msp_{\ff,\dd}(Q),\phim{\TTr(W^+)_{\dd}}\nIC_{\Msp_{\ff,\dd}(Q)}\right)\cong \HO\!\left( \Nak(\ff,\dd),\QQ\right)\otimes\LLL^{\chi_Q(\dd,\dd)-\ff\cdot \dd}.
\end{equation}
\end{proposition}
\begin{proof}
The (Verdier dual of the) isomorphism \eqref{HDIM} is constructed in \cite[Thm.6.6]{preproj} via dimensional reduction, so we just need to prove the other parts of the proposition.
Let $c$ be an arrow from $\infty$ to $i$ in $Q^+$. Then
\begin{align*}
\partial W^+/\partial c^*&=\omega_ic\\
\partial W^+/\partial c &=c^*\omega_i.
\end{align*}
For $a$ an arrow in $Q_1$ we have
\begin{align*}
\partial W/\partial a^*&= \omega_{t(a)}a-a\omega_{s(a)}\\
\partial W/\partial a&=a^*\omega_{t(a)}-\omega_{s(a)}a^*.
\end{align*}
Putting these facts together, we have an isomorphism of algebras
\begin{equation}
\label{Jacref}
\Jac(Q^+,W^+)\cong\Pi_{Q_{\ff}}[\omega]/ \langle e_{\infty}\omega e_{\infty}=0\rangle
\end{equation}
where
\[
\omega=\sum_{i\in Q^+_0}\omega_i.
\]
We consider the fine moduli space
\[
\mathcal{N}=\AS_{\dd^+}^{\zeta^+\sst}(\WT{Q_{\ff}})/\Gl_{\dd}.
\]
Then the critical locus of $\TTr(\WT{W_{\ff}})$ is identified with the total space of the tautological line bundle on $\Nak(\ff,\dd)$ for which the fibre over $\rho$ is the space of endomorphisms of $\rho$. By \eqref{Jacref}, $\crit(\TTr(W^+)_{\dd})$ is the zero section, i.e. it is $\Nak(\ff,\dd)$.
Since the (scheme-theoretic) critical locus of $\TTr(W^+)_{\dd}$ is smooth, by the holomorphic Bott--Morse lemma this function can be written analytically locally (on $\Nak(\ff,\dd)$) as
\[
\TTr(W^+)_{\dd}=x_1^2+\ldots +x_e^2
\]
where $e$ is the codimension of $\Nak(\ff,\dd)$ inside $\Msp_{\ff,\dd}(Q)$, and so $\phim{\TTr(W^+)_{\dd}}\nIC_{\Msp_{\ff,\dd}(Q)}$ is analytically locally isomorphic to $\nIC_{\Nak(\ff,\dd)}$. In particular, as a perverse sheaf it is locally isomorphic to $\QQ_{\Nak_{\ff,\dd}(Q)}[2\ff\cdot\dd-2\chi_Q(\dd,\dd)]$, and is thus determined by its monodromy\footnote{This is the monodromy around $\Nak_{\ff,\dd}(Q)$ and is unrelated to the ``monodromic'' in ``monodromic mixed Hodge module''.}. Finally, in cohomological degree $2\chi_Q(\dd,\dd)-2\ff\cdot\dd$ the right hand side of \eqref{HDIM} is a vector space spanned by the components of $\Nak(\ff,\dd)$, while the left hand side is a vector space spanned by the components of $\Nak(\ff,\dd)$ on which the monodromy of the underlying perverse sheaf of $\phim{\TTr(W^+)_{\dd}}\nIC_{\Msp_{\ff,\dd}(Q)}$ is trivial. Since these dimensions are the same, the monodromy is trivial, and isomorphism \eqref{rDIM} follows.
\end{proof}
\subsubsection{CoHA modules from framed representations}
There is a general construction, producing modules for cohomological Hall algebras out of moduli spaces of framed quiver representations, see \cite{Soi14} for a related example and discussion. We recall the variant relevant to us.
Let $\dd',\dd''\in \dvs$ be dimension vectors, with $\dd=\dd'+\dd''$. We define
\[
\AS_{\dd',\dd''^+}^{\zeta^+\sst}(Q^+)\subset \AS_{\dd^+}^{\zeta^+\sst}(Q^+)
\]
to be the subspace of $\CC Q^+$-modules $\rho$ such that the underlying $\CC \WT{Q}$-module of $\rho$ preserves the $Q_0$-graded flag
\begin{equation}
\label{flag}
0\subset \CC^{\dd'}\subset \CC^{\dd}
\end{equation}
and for every arrow $c^*$ with $t(c^*)=\infty$ we have $\rho(c^*)(\CC^{\dd'})=0$.
Given such a $\rho$ we obtain a short exact sequence
\begin{equation}
\label{sesrho}
0\rightarrow \rho'\rightarrow \rho\rightarrow\rho''\rightarrow 0
\end{equation}
where $\udim(\rho')=(\dd',0)$ and $\udim(\rho'')=\dd''^+$. We set
\[
\Msp_{\ff,\dd',\dd''}(Q)\coloneqq \AS_{\dd',\dd''^+}^{\zeta^+\sst}(Q^+)/\Gl_{\dd',\dd''}
\]
where $\Gl_{\dd',\dd''}\subset \Gl_{\dd}$ is the subgroup preserving the flag \eqref{flag}. There are morphisms
\begin{align*}
\pi_1\colon \Msp_{\ff,\dd',\dd''}(Q)\rightarrow &\Mst_{\dd'}(\WT{Q})\\
\pi_2\colon \Msp_{\ff,\dd',\dd''}(Q)\rightarrow&\Msp_{\ff,\dd}(Q)\\
\pi_2\colon \Msp_{\ff,\dd',\dd''}(Q)\rightarrow&\Msp_{\ff,\dd''}(Q)
\end{align*}
taking a point representing the short exact sequence \eqref{sesrho} to $\rho',\rho,\rho''$ respectively.
Then in the correspondence diagram
\[
\xymatrix{
&\Msp_{\ff,\dd',\dd''}(Q)\ar[dl]_{\pi_1\times \pi_3}\ar[dr]^{\pi_2}\\
\Mst_{\dd'}(\WT{Q})\times \Msp_{\ff,\dd''}(Q)&&\Msp_{\ff,\dd}(Q)
}
\]
the morphism $\pi_2$ is a proper morphism between smooth varieties, so that as in \S \ref{CCSec} we can use it to define a pushforward in critical cohomology. Taking some care of the twists, we consider the morphisms of mixed Hodge modules
\begin{align*}
\alpha&\colon\ul{\QQ}_{\Mst_{\dd'}(\WT{Q})\times \Msp_{\ff,\dd''}(Q)}\otimes\LLL^{\heartsuit}\rightarrow (\pi_1\times \pi_3)_*\ul{\QQ}_{\Msp_{\ff,\dd',\dd''}(Q)}\otimes\LLL^{\heartsuit}\\
\beta&\colon\pi_{2,*}\ul{\QQ}_{\Msp_{\ff,\dd',\dd''}(Q)}\otimes\LLL^{\heartsuit}\rightarrow \ul{\QQ}_{\Msp_{\ff,\dd}(Q)}\otimes\LLL^{\chi_{\WT{Q}}(\dd,\dd)-\ff\cdot\dd}
\end{align*}
where
\[
\heartsuit=\chi_{\WT{Q}}(\dd',\dd')+\chi_{\WT{Q}}(\dd'',\dd'')-\ff\cdot \dd''
\]
and $\beta$ is the Verdier dual of
\[
\ul{\QQ}_{\Msp_{\ff,\dd}(Q)}\otimes\LLL^{\chi_{\WT{Q}}(\dd,\dd)-\ff\cdot\dd}\rightarrow \pi_{2,*}\ul{\QQ}_{\Msp_{\ff,\dd',\dd''}(Q)}\otimes\LLL^{\chi_{\WT{Q}}(\dd,\dd)-\ff\cdot\dd}.
\]
For $\dd,\ff\in\dvs$ we define
\[
\HMall^*_{\ff,\dd}\coloneqq \HO\!\left(\Msp_{\ff,\dd}(Q),\phim{\TTTr(W^+)}\ul{\QQ}_{\Msp_{\ff,\dd}(Q)}\right)\otimes\LLL^{\chi_{\WT{Q}}(\dd,\dd)/2-\ff\cdot\dd}
\]
and we define
\[
\HMall^*_{\ff}\coloneqq \bigoplus_{\dd\in\dvs}\HMall^*_{\ff,\dd}.
\]
Applying $\phim{\TTTr(W^+)}$ and taking hypercohomology, via the Thom--Sebastiani isomorphism we obtain a morphism
\[
\HO(\phim{\TTTr(W^+)}\beta)\circ\HO(\phim{\TTTr(W^+)}\alpha)\circ\TS\colon \HCoha_{\WT{Q},\WT{W},\dd'}\otimes\HMall^*_{\ff,\dd''}\rightarrow \HMall^*_{\ff,\dd}
\]
endowing $\HMall^*_{\ff}$ with the structure of a $\HCoha_{\WT{Q},\WT{W}}$-module. By Proposition \ref{NQVDR} there is an isomorphism
\[
\HMall^*_{\ff,\dd}\cong \HO\!\left( \Nak(\ff,\dd),\QQ\right)\otimes\LLL^{\chi_Q(\dd,\dd)-\ff\cdot\dd}.
\]
Here we have used the calculation
\[
\dim(\Msp_{\ff,\dd}(Q))=-\chi_{\WT{Q}}(\dd,\dd)+2\ff\cdot\dd
\]
along with \eqref{HDIM}. As such, we obtain an action of $\HCoha_{\Pi_Q}\cong\HCoha_{\WT{Q},\WT{W}}$ on the cohomology of Nakajima quiver varieties.
\begin{remark}
Let $\Nak^{\SP}\!(\ff,\dd)\subset \Nak(\ff,\dd)$ be the subvariety for which the underlying $\ol{Q}$-module of $\rho$ lies in some Serre subcategory $\SP$. Applying exceptional restriction functors to the above morphisms of mixed Hodge modules, we may likewise define an action of $\HCoha^{\SP}_{\Pi_Q}$ on
\[
\bigoplus_{\dd\in\dvs}\HOBM(\Nak^{\SP}\!(\ff,\dd),\QQ)\otimes \LLL^{\ff\cdot\dd-\chi_Q(\dd,\dd)},
\]
although we will make no use of this generalisation here.
\end{remark}
\subsubsection{Kac--Moody Lie algebras and quiver varieties}
For the rest of \S \ref{NQVs} we assume that $Q$ has no edge-loops. Let $i\in Q_0$ be a vertex and let $1_i\in\dvs$ be the basis vector for the vertex $i$. Then there is an isomorphism
\[
\Psi_i\colon\HCoha_{\WT{Q},\WT{W},1_i}\cong\HO(\AAA{1}/\CC^*,\QQ)\cong \QQ[u]
\]
and we set
\begin{equation}
\label{Fdef}
\alpha_i\coloneqq\Psi_i^{-1}(1)\in \HCoha_{\WT{Q},\WT{W},1_i}.
\end{equation}
The action of $\alpha_i$ provides a morphism
\[
\alpha_i\cdot\colon \HMall_{\ff,\dd}^*\rightarrow \HMall_{\ff,\dd+1_i}^*.
\]
Consider the semisimplification map $\JH\colon \Msp_{\ff,\dd}(Q)\rightarrow \Msp_{\dd^+}(\ol{Q_{\ff}})$. Following Lusztig \cite{Lus91} we consider the Lagrangian subvariety
\[
\Lus(\ff,\dd)=\JH^{-1}(0).
\]
There is a contracting $\mathbb{C}^*$ action on $\Nak(\ff,\dd)$, contracting it onto the projective variety $\Lus(\ff,\dd)$, and so we obtain the first in the sequence of isomorphisms
\begin{align*}
\HO(\Nak(\ff,\dd),\QQ)\cong&\HO(\Lus(\ff,\dd),\QQ)\\
\cong& \HOc(\Lus(\ff,\dd),\QQ)\\
\cong&\HOBM(\Lus(\ff,\dd),\QQ)^*.
\end{align*}
Note that since $\Lus(\ff,\dd)$ is Lagrangian, its top degree compactly supported cohomology is in degree
\[
\dim(\Nak(\ff,\dd))=2\ff\cdot \dd-2\chi_Q(\dd,\dd).
\]
Setting
\begin{align*}
\HMall_{\ff,\dd}\coloneqq &\HOBM\left(\Lus(\ff,\dd),\QQ\right)\otimes \LLL^{\ff\cdot\dd-\chi_Q(\dd,\dd)}\\
\HMall_{\ff}\coloneqq &\bigoplus_{\dd\in\dvs}\HMall_{\ff,\dd}
\end{align*}
we deduce that there is a $\HCoha_{\Pi_Q}$ action on $\HMall_{\ff}$ by lowering operators, for which $(\alpha_i\cdot )^*$ is the lowering operator constructed by Nakajima. By the degree bound on the cohomology of $\HOc(\Lus(\ff,\dd),\QQ)$,
\[
\HO^i\!\Mall_{\ff,\dd}=\begin{cases} 0&\textrm{if }i<0\\
\QQ\cdot\{\textrm{top-dimensional components of }\Lus(\ff,\dd)\}&\textrm{if }i=0.\end{cases}
\]
The main theorem regarding the operators $(\alpha_i\cdot)^*$ is the following part of Nakajima's work.
\begin{theorem}\cite{Nak94,Nak98}
There is an action of the Kac--Moody Lie algebra $\mathfrak{g}$ on each $\HMall_{\ff}$ sending the generators $f_i$ for $i\in Q_0$ to the operators $(\alpha_i\cdot)^*$. With respect to this $\mathfrak{g}_Q$ action, the submodule $\HO^0\!\Mall_{\ff,\dd}$ is the irreducible highest weight module with highest weight $\ff$.
\end{theorem}
The original statement of Nakajima's theorem does not involve any vanishing cycles, i.e. it only involves the right hand side of the isomorphism \eqref{HDIM}. Likewise, the correspondences considered in \cite{Nak94, Nak98} come from an action of the Borel--Moore homology of the stack of $\Pi_Q$-representations, not from the critical cohomology of the stack of $\Pi_Q$-representations. For the compatibility between the two actions via the dimensional reduction isomorphisms \eqref{HDIM} and \eqref{drnoalg} see e.g. \cite[Sec.4]{YZ16}.
\subsection{The subalgebra $\lP_{\leq 0}\!\HO^0\!\!\Coha_{\Pi_Q}$}
\label{KacH}
In this section we concentrate on the case in which $\SP=\CC\ol{Q}\lmod$, $G$ is trivial and $\zeta=(0,\ldots,0)$ is the degenerate stability condition (i.e. we essentially do not consider stability conditions). Note that by \eqref{Kaccha}, the Lie algebra $\mathfrak{g}_{\Pi_Q}$ is concentrated in cohomological degrees less than or equal to zero, and so by \eqref{UEAiso} there is an isomorphism
\begin{equation}
\label{UTO}
\lP_{\leq 0}\!\HO^0\!\!\Coha_{\Pi_Q}\cong \UEA\!\left(\HO^0(\mathfrak{g}_{\Pi_Q})\right).
\end{equation}
We thus reduce the problem to calculating $\HO^0(\mathfrak{g}_{\Pi_Q})$. By \eqref{Kaccha} again, there is an equality
\begin{equation}
\label{clue1}
\dim(\HO^0(\mathfrak{g}_{\Pi_Q,\dd}))=\kac_{Q,\dd}(0).
\end{equation}
If $\dd_i\neq 0$ for some $i\in Q_0$ for which there is an edge-loop $b$, there is a free action of $\mathbb{F}_q$ on the set of absolutely indecomposable $\dd$-dimensional $\CC Q$-modules, defined by
\[
z\cdot \rho(a)=\begin{cases} \rho(a)+ z\cdot \Id_{e_i\cdot \rho} &\textrm{if }a=b\\
\rho(a) &\textrm{otherwise}\end{cases}
\]
and thus $\kac_{Q,\dd}(0)=0$, and so $\HO^0(\mathfrak{g}_{\Pi_Q,\dd})=0$. It follows that if $\dd_i\neq 0$ for any vertex $i$ supporting an edge loop, then $\lP_{\leq 0}\!\HO^0\!\!\Coha^{\zeta}_{\Pi_Q,\dd}=0$.
We define $Q'$, the \textbf{real subquiver} of $Q$, to be the full subquiver of $Q$ containing those vertices of $Q$ that do not support any edge-loops, along with all arrows between these vertices. From the above considerations, we deduce that the morphism of algebras in $\DlMHM(\Msp(\ol{Q}))$
\[
\rCoha_{\Pi_{Q'}}\rightarrow \rCoha_{\Pi_Q}
\]
becomes an isomorphism after applying $\lP_{\leq 0}\!\HO^0$.
Hausel's (first) famous theorem regarding Kac polynomials \cite{Hau10} states that
\begin{equation}
\label{clue2}
\kac_{Q',\dd}(0)=\dim(\mathfrak{g}_{Q',\dd})
\end{equation}
where $\mathfrak{g}_{Q'}$ is the Kac--Moody Lie algebra associated to the quiver (without edge-loops) $Q'$. We will not recall the definition of $\fg_{Q'}$, since in any case it is a special case of the Borcherds--Bozec algebra (i.e. the case in which $I^{\Imag}=\emptyset$), which we recall in \S \ref{BBA} below.
Comparing \eqref{clue1} and \eqref{clue2} leads to the identity
\begin{equation}
\label{khaus}
\dim(\HO^0(\mathfrak{g}_{\Pi_Q,\dd}))=\dim(\mathfrak{g}_{Q',\dd})
\end{equation}
and from there to the obvious conjecture regarding the algebra $\lP_{\leq 0}\!\HO^0\!\!\Coha^{\zeta}_{\Pi_Q}$, which we now prove.
\begin{theorem}
\label{KMLA}
There is an isomorphism of algebras
\begin{equation}
\label{zpkm}
\UEA(\mathfrak{n}_{Q'}^-)\cong\lP_{\leq 0}\!\HO^0\!\!\Coha^{\zeta}_{\Pi_Q}
\end{equation}
where $\mathfrak{n}_{Q'}^-$ is the negative part of the Kac--Moody Lie algebra for the real subquiver of $Q$. Moreover the isomorphism restricts to an isomorphism with the BPS Lie algebra
\begin{equation}
\label{LAI}
\fn^-_{Q'}\cong \HO^0(\fg_{\Pi_Q})
\end{equation}
under the isomorphism \eqref{UTO}.
\end{theorem}
\begin{proof}
We construct the isomorphism \eqref{LAI}, then the isomorphism \eqref{zpkm} is constructed via \eqref{UTO}.
Consider the dimension vector $e=1_i$, where $i$ does not support any edge-loops. The coarse moduli space $\Msp_{e}(\ol{Q})$ is just a point, and so the less perverse filtration on $\HCoha_{\Pi_Q,e}=\HO(\B \CC^*,\QQ)$ is just the cohomological filtration. In particular, the element $\alpha_i$ from \eqref{Fdef} lies in less perverse degree 0. On the other hand, there is an isomorphism
\[
\Msp_{e}(\WT{Q})\cong \AAA{1}
\]
and writing
\[
\HCoha_{\WT{Q},\WT{W},e}=\HO(\AAA{1},\nIC_{\AAA{1}})\otimes \HO(\B\CC^*,\QQ)_{\vir}
\]
we see that $\alpha_i$ has perverse degree 1, i.e. by definition it is an element of $\fg_{\Pi_Q,e}$.
We claim that there is a Lie algebra homomorphism $\Phi\colon \mathfrak{n}^-_{Q'}\rightarrow \HO^0(\mathfrak{g}_{\Pi_Q})$ sending $f_i$ to $\alpha_i$. The algebra $\mathfrak{n}_{Q'}^-$ is the free Lie algebra generated by $f_i$ for $i\in Q_0$ subject to the Serre relations:
\[
[f_i,\cdot]^{1-(1_i,1_j)_Q}(f_j)=0
\]
where $(\cdot,\cdot)_Q$ is the symmetrized Euler form \eqref{Eulerform}. So to prove the claim we only need to prove that the elements $\alpha_i$ satisfy the Serre relations. This follows from the stronger claim: for any distinct pair of vertices $i,j\in Q_0$ if we set $\gamma=1_j+(1-(1_i,1_j))1_i$ then there is an equality
\[
\mathfrak{g}_{\Pi_Q,\gamma}=0.
\]
This follows from Proposition \ref{BPSvanProp}. Alternatively, this follows from \eqref{Kaccha} and the claim that $\kac_{Q,\gamma}(t)=0$. Since the Kac polynomial is independent of the orientation of $Q$ this equality is clear: if all the arrows are directed from $i$ to $j$, there are no indecomposable $\gamma$-dimensional $KQ$-modules for any field $K$.
We next claim that the morphism $\mathfrak{n}^-_{Q'}\rightarrow \HO^0(\mathfrak{g}_{\Pi_Q})$ is injective. This follows from Nakajima's theorem, i.e. we have a commutative diagram
\[
\xymatrix{
\mathfrak{n}^-_{Q'}\ar[dr]_{u}\ar[r]_{\Phi}^{f_i\mapsto \alpha_i}&\HO^0(\mathfrak{g}_{\Pi_Q})\ar[d]\\
&\End_{\QQ}(\bigoplus_{\ff} \HMall^*_{\ff})
}
\]
via the module structure of each $\HMall^*_{\ff}$, and the morphism $u$ is injective since the representation $\bigoplus_{\ff}\HMall^*_{\ff}$ is a faithful representation. Faithfulness follows, for example, from the Kac--Weyl character formula.
By Hausel's identity \eqref{clue2} the graded dimensions of the source and target of $\Phi$ are the same, and so $\Phi$ is an isomorphism. Comparing with \eqref{UTO}, the induced morphism
\[
\UEA(\mathfrak{n}^-_{Q'})\rightarrow \lP_{\leq 0}\!\HO^0\!\!\Coha^{\zeta}_{\Pi_Q}
\]
is an isomorphism.
\end{proof}
Theorem \ref{KMLA} shows that Kac--Moody Lie algebras are a natural piece of the BPS Lie algebra in the basic case in which we do not modify potentials, and do not restrict to a Serre subcategory. The next proposition shows that Kac--Moody Lie algebras are a somewhat universal feature of the cohomological Hall algebras that we are considering.
\begin{corollary}
\label{bels}
Let $Q$ be a quiver, let $Q'$ be the real subquiver of $Q$, let $W_0\in\CC\ol{Q}_{\cyc}$ be a potential such that $\WT{W}+W_0$ is quasihomogeneous, and let $\SP$ be any Serre subcategory of $\CC\ol{Q}\lmod$ containing each of the 1-dimensional simple modules $S_i$ with dimension vector $1_i$, for $i\in Q'_0$. Then there is a $\dvs$-graded inclusion of Lie algebras
\[
\fn^-_{Q'}\hookrightarrow \fg_{\Pi_Q,W_0}^{\SP}
\]
with image the Lie subalgebra of $\fg_{\Pi_Q,W_0}^{\SP}$ generated by the graded pieces $\fg_{\Pi_Q,W_0,1_i}^{\SP}$ for $i\in Q'_0$.
\end{corollary}
\begin{proof}
By the proof of Theorem \ref{KMLA} the subalgebra $\fn^-_{Q'}\subset \fg_{\Pi_Q}$ is obtained by applying $\HO$ to the Lie subalgebra object $\mathcal{G}$ of $\BPSh_{\Pi_Q}$ generated by the objects $\BPSh_{\Pi_Q,1_{i}}$ for $i\in Q'_0$, i.e. $\fn^-_{Q'}\cong\HO\mathcal{G}$. There is a decomposition
\[
\BPSh_{\Pi_Q}\cong \mathcal{G}\oplus\mathcal{L}
\]
of mixed Hodge modules, by purity of $\BPSh_{\Pi_Q}$, and hence the inclusion $\mathcal{G}\hookrightarrow \BPSh_{\Pi_Q}$ splits in the category of mixed Hodge modules. Applying $\HO\varpi'_*\varpi'^!\phim{W_0}$ gives an inclusion of Lie algebras
\[
\HO\varpi'_*\varpi'^!\phim{W_0}\mathcal{G}\hookrightarrow \fg_{\Pi_Q,W_0}^{\SP}.
\]
Each mixed Hodge module $\mathcal{G}_{\dd}$ is supported at the origin of $\Msp_{\dd}(\ol{Q})$, since $\mathcal{G}$ is generated by mixed Hodge modules supported on the nilpotent locus. The Serre subcategory $\SP$ contains all nilpotent $\CC\ol{Q}$-modules supported on the subquiver $Q'$, since it is closed under extensions. As such the natural morphisms
\begin{align*}
\varpi'_*\varpi'^!\mathcal{G}\rightarrow \mathcal{G}\\
\phim{\TTr(W_0)}\mathcal{G}\rightarrow \mathcal{G},
\end{align*}
are isomorphisms, and $\varpi'_*\varpi'^!\phim{\TTr(W_0)}\mathcal{G}\cong\mathcal{G}$ as a Lie algebra object in $\MHM(\Msp(\ol{Q}))$, proving the corollary.
\end{proof}
\subsection{The subalgebra $\lP_{\leq 0}\!\HO^0\!\!\Coha^{\SSN}_{\Pi_Q}$}
\label{BBA}
Moving to the case of strongly semi-nilpotent $\Pi_Q$-modules (see \S \ref{2desc}) we calculate the subalgebra $\lP_{\leq 0}\!\HO^0\!\!\Coha^{\SSN}_{\Pi_Q}$, in order to compare with work of Bozec \cite{Bo16}. Interestingly, we find that the BPS Lie algebra $\fg_{\Pi_Q}^{\SSN}$ is \textit{not} identified with Bozec's Lie algebra $\fg_Q$ under the natural isomorphism between their two enveloping algebras, although the two Lie algebras are isomorphic.
First we note that by \eqref{kssn} the Lie algebra $\fg_{\Pi_Q}^{\SSN}$ is concentrated in cohomologically nonnegative degrees. Applying $\HO^0$ to \eqref{absPBW}, the morphism
\[
\Sym(\HO^0\!(\fg^{\SSN}_{\Pi_Q}))\rightarrow \HO^0\!\!\Coha^{\SSN}_{\Pi_Q}
\]
is an isomorphism, and thus there is an identity
\begin{equation}
\label{Hzerodo}
\UEA(\HO^0\!(\fg_{\Pi_Q}^{\SSN}))=\HO^0\!\!\Coha^{\SSN}_{\Pi_Q}.
\end{equation}
Comparing with \eqref{UEAiso} we deduce that
\[
\lP_{\leq 0}\!\HO^0\!\!\Coha^{\SSN}_{\Pi_Q}=\HO^0\!\!\Coha^{\SSN}_{\Pi_Q}
\]
and so for the rest of \S \ref{BBA} we just write $\HO^0\!\!\Coha^{\SSN}_{\Pi_Q}$ to denote this subalgebra.
\subsubsection{The Borcherds--Bozec algebra}
We write
\[
Q_0=I^{\reel}\coprod I^{\Imag},
\]
where $I^{\reel}$ is the set of vertices that do not support an edge-loop, and $I^{\Imag}$ is the set vertices that do. We furthermore decompose
\[
I^{\Imag}=I^{\isot}\coprod I^{\hype}
\]
where $I^{\isot}$ is the set of vertices supporting exactly one edge-loop, and the vertices of $I^{\hype}$ support more than one.
Out of the quiver $Q$ we build the \textbf{Borcherds--Bozec} algebra $\fg_Q$ as follows. We set
\[
I_{\infty}=(I^{\reel}\times\{1\})\coprod (I^{\Imag}\times\mathbb{Z}_{>0})
\]
and we extend the form \eqref{Eulerform} to a bilinear form on $\mathbb{N}^{I_{\infty}}$ by setting
\[
((1_{i',n}),(1_{j',m}))=mn(1_{i',}1_{j'})_Q
\]
and extending linearly. The Lie algebra $\mathfrak{g}_Q$ is a Borcherds algebra associated to a generalised Cartan datum for which the Cartan matrix is the form $(\cdot,\cdot)$ expressed in the natural basis of $\mathbb{N}^{I_{\infty}}$. More explicitly, we define $\fg_Q$ to be the free Lie algebra generated over $\QQ$ by $h_{i'},e_i,f_i$ for $i'\in Q_0$ and $i\in I_{\infty}$ subject to the relations
\begin{align*}
[h_{i'},h_{j'}]=&0\\
[h_{j'},e_{(i',n)}]=&n(1_{j'},1_{i'})_Q\cdot e_{(i',n)}\\
[h_{j'},f_{(i',n)}]=&-n(1_{j'},1_{i'})_Q\cdot f_{(i',n)}\\
[e_{j},\cdot]^{1-(j,i)}e_i=[f_{j},\cdot]^{1-(j,i)}f_i=&0 &\textrm{if }j\in I^{\reel}\times\{1\},\;i\neq j\\
[e_i,e_j]=[f_i,f_j]=&0&\textrm{if }(i,j)=0\\
[e_i,f_j]=&\delta_{i,j}nh_{i'}&\textrm{if }i=(i',n).
\end{align*}
The positive half $\mathfrak{n}_Q^+$ has an especially quick presentation: it is the Lie algebra over $\QQ$ freely generated by $e_{i}$ for $i\in I_{\infty}$, subject to the relations
\begin{align}
[e_{i},\cdot]^{1-(i,j)}(e_j)=0 &&\textrm{if }i\in I^{\reel}\times\{1\},\;i\neq j\label{Serre1}\\
[e_i,e_j]=0 &&\textrm{if }(i,j)=0.\label{Serre2}
\end{align}
\subsubsection{Lagrangian subvarieties}
Define $\Lambda(\dd)=\AS_{\dd}^{\mathcal{SSN}}(\ol{Q})\cap \mu^{-1}(0)$, the subvariety of $\AS_{\dd}(\ol{Q})$ parameterising strictly semi-nilpotent $\Pi_Q$-modules. By \cite[Thm.1.15]{Bo16}, this is a Lagrangian subvariety of $\AS_{\dd}(\ol{Q})$. If $\dd=n\cdotsh 1_{i'}$ for some $i'\in Q_0$, by \cite[Thm.1.4]{Bo16} the irreducible components of $\Lambda(\dd)$ are indexed by tuples $(n^1,\ldots,n^r)$ such that $\sum n^s=n$. Let $I_l$ be the two-sided ideal in $\CC \ol{Q}$ containing all those paths in $\ol{Q}$ containing at least $l$ instances of arrows $a\in Q_1$. The tuple $c$ corresponding to a component $\Lambda(\dd)_c$ is given by the successive dimensions of the subquotients in the filtration
\[
0=I_r\cdotsh \rho\subset I_{r-1}\cdotsh \rho\subset\ldots \rho
\]
for $\rho$ a module parameterised by a generic point on $\Lambda(\dd)_c$. For example there is an equality
\[
\Lambda(\dd)_{(n)}=\AS_{\dd}(Q^{\opp})\subset \AS^{\SSN}_{\dd}(\ol{Q})\cap \mu^{-1}(0).
\]
To translate Bozec's results into our setting we follow the arguments of \cite[Sec.2]{RS17}. Unpicking the definitions, we have
\[
\HCoha^{\SSN}_{\Pi_Q}\coloneqq \bigoplus_{\dd\in\dvs}\HOBM\left(\Lambda(\dd)/\Gl_{\dd},\QQ\right)\otimes\LLL^{-\chi_Q(\dd,\dd)}.
\]
Since $\Lambda(\dd)$ is a Lagrangian subvariety of the $2(\dd\cdot\dd-\chi_Q(\dd,\dd))$-dimensional subvariety $\AS_{\dd}(\ol{Q})$, the irreducible components of $\Lambda(\dd)/\Gl_{\dd}$ are $-\chi_Q(\dd,\dd)$-dimensional. It follows that $\HO^0\!\!\Coha^{\SSN}_{\Pi_Q}$ has a natural basis given by $[\Lambda(\dd)_e]$ where $\Lambda(\dd)_e$ are the irreducible components of $\Lambda(\dd)$.
\begin{theorem}\cite[Prop.1.18, Thm.3.34]{Bo16}
\label{BoThm}
There is an isomorphism of algebras
\[
\UEA(\mathfrak{n}_Q^+)\rightarrow \HO^0\!\!\Coha^{\SSN}_{\Pi_Q}
\]
which sends $e_{i',n}$ to $[\Lambda(n\cdotsh 1_{i'})_{(n)}]$.
\end{theorem}
Combining with \eqref{Hzerodo} we obtain an isomorphism
\begin{equation}
\label{UEAiso}
F\colon \UEA(\mathfrak{n}_Q^+)\xrightarrow{\cong}\UEA(\HO^0\!(\fg_{\Pi_Q}^{\SSN})).
\end{equation}
\begin{corollary}
\label{absIso}
There exists an isomorphism of Lie algebras
\[
\mathfrak{n}_Q^+\cong\HO^0\!(\fg_{\Pi_Q}^{\SSN})
\]
where the left hand side is the Borcherds--Bozec algebra of the full quiver $Q$.
\end{corollary}
\begin{proof}
Let $S$ be a minimal $\dvs$-graded generating set of $\HO^0\!(\fg_{\Pi_Q}^{\SSN})$. Then $F^{-1}(S)$ is a minimal generating set of $\UEA(\mathfrak{n}_Q^+)$ and so $\lvert S_{\dd}\lvert$ is the number of degree $\dd$ generators of $\mathfrak{n}_Q^+$. We show that the elements of $S$ satisfy the Serre relations.
Let $g,g'\in S$ be of degree $m\cdotsh 1_{i'}$ and $n\cdotsh 1_{j'}$ respectively. If $(1_{i'},1_{j'})_Q=0$ then $g$ and $g'$ commute, i.e. they satisfy the Serre relation $[g,g']=0$, dealing with \eqref{Serre2}.
We next consider \eqref{Serre1}. Assume that there are no edge-loops of $Q$ at $i'$, so that up to a scalar multiple $g=e_{(i',1)}$. Write $g'$ as a linear combination of monomials $\prod_{r=1}^l e_{(j',t_r)}$ for $t_r\in\mathbb{N}$ summing to $n$. Since $[e_{(i',1)},\cdot]$ is a derivation, the identity
\[
[e_{(i',1)},g']^{1-n(1_{i'},1_{j'})_Q}=0
\]
follows from the identities
\[
[e_{(i',1)},e_{(j',t_r)}]^{1-t_r(1_{i'},1_{j'})_Q}=0.
\]
So the generators $S$ satisfy the Serre relations and there is a surjection $\mathfrak{n}_Q^+\rightarrow \HO^0\!(\fg_{\Pi_Q}^{\SSN})$, which is injective since the graded pieces have the same dimensions.
\end{proof}
Although Corollary \ref{absIso} establishes that they are abstractly isomorphic, we spend the rest of \S \ref{BBA} investigating the \textit{difference} between the two Lie subalgebras $F(\mathfrak{n}_Q^+)$ and $\HO^0\!(\fg_{\Pi_Q}^{\SSN})$.
\subsubsection{Isotropic vertices}
\label{IsotSec} Let $i'\in I^{\isot}$, and set $\dd=n\cdotsh 1_{i'}$. Let
\[
\Delta_n\colon \AAA{3}\hookrightarrow \Msp_{\dd}(\WT{Q})
\]
be the inclusion, sending $(z_1,z_2,z_3)\in\AAA{3}$ to the $\Jac(\WT{Q},\WT{W})$-representation for which the action of the three arrows $a,a^*,\omega_{i'}$ is scalar multiplication by $z_1,z_2,z_3$ respectively. Then by \cite[Thm.5.1]{preproj} there is an isomorphism
\begin{equation}
\label{commBPS}
\BPSh_{\WT{Q},\WT{W},\dd}\cong \Delta_{n,*}\ul{\QQ}_{\AAA{3}}\otimes \LLL^{-3/2}=\Delta_{n,*}\nIC_{\AAA{3}}.
\end{equation}
Thus by \eqref{pfBP} there is an isomorphism
\[
\BPSh_{\Pi_Q,\dd}\cong \Delta_{\red,n,*}\ul{\QQ}_{\AAA{2}}\otimes \LLL^{-1}=\Delta_{\red,n,*}\nIC_{\AAA{2}}
\]
where
\[
\Delta_{\red,n}\colon\AAA{2}\hookrightarrow \Msp_{n\cdot 1_{i'}}(\ol{Q})
\]
is the inclusion taking $z_1,z_2$ to the module $\rho$ for which $a$ and $a^*$ act via multiplication by $z_1$ and $z_2$ respectively. Thus we find that
\begin{align}
\BPSh^{\SSN}_{\Pi_Q,\dd}\cong &\varpi'_{\red,*}\varpi'^!_{\red}\Delta_{\red,n,*}\nIC_{\AAA{2}}\\\label{BPSSSN}
\cong &\Delta^{\SSN}_{\red,n,*}\ul{\QQ}_{\AAA{1}}
\end{align}
where
\[
\Delta^{\SSN}_{\red,n,*}\colon \AAA{1}\hookrightarrow \Msp_{n\cdot 1_{i'}}(\ol{Q})
\]
takes $z$ to the module $\rho$ for which $a^*$ acts via multiplication by $z$ and $a$ acts via the zero map. By \eqref{2dInt} we deduce the following proposition.
\begin{proposition}
\label{IsoPure}
There is an isomorphism in $\DlMHM(\Msp(\ol{Q}))$
\begin{equation}
\label{gen}
\bigoplus_{n\geq 0}\rCoha^{\SSN}_{\Pi_Q,n\cdot 1_{i'}}\cong \Sym_{\oplus_{\red}}\!\left(\bigoplus_{n\geq 1}\Delta^{\SSN}_{\red,n,*}\ul{\QQ}_{\AAA{1}}\otimes\HO(\B \CC^*,\QQ)\right).
\end{equation}
In particular, $\bigoplus_{n\geq 0}\rCoha^{\SSN}_{\Pi_Q,n\cdot 1_{i'}}$ is pure, as is
\begin{equation}
\label{fen}
\bigoplus_{n\geq 0}\BPSh^{\SSN}_{\Pi_Q,n\cdot 1_{i'}}\cong \Sym_{\oplus_{\red}}\!\left(\bigoplus_{n\geq 1}\Delta^{\SSN}_{\red,n,*}\ul{\QQ}_{\AAA{1}}\right).
\end{equation}
\end{proposition}
\begin{remark}
Applying Verdier duality to \eqref{gen} makes for a cleaner looking statement of the result. The Verdier dual of \eqref{gen} is the isomorphism
\[
\bigoplus_{n\geq 0}\left(\Mst^{\SSN}_{n\cdot 1_{i'}}(\Pi_Q)\rightarrow \Msp_{n\cdot 1_{i'}}(\ol{Q})\right)_!\ul{\QQ}_{\Mst^{\SSN}_{n\cdot 1_{i'}}(\Pi_Q)}\cong \Sym_{\oplus}\!\left(\bigoplus_{n\geq 1,m\geq 0}\Delta^{\SSN}_{\red,n,*}\ul{\QQ}_{\AAA{1}}\otimes\LLL^{-m-1}\right).
\]
\end{remark}
From \eqref{BPSSSN} we deduce that
\begin{equation}
\label{IsotDone}
\Psi\colon \fg^{\SSN}_{\Pi_Q,\dd}\cong \HO(\AAA{1},\QQ)\cong\QQ
\end{equation}
as a cohomologically graded vector space, i.e. $\fg^{\SSN}_{\Pi_Q,\dd}$ is one dimensional and concentrated in cohomological degree zero. We denote by
\begin{equation}
\label{Gdef}
\alpha_{i',n}=\Psi^{-1}(1)
\end{equation}
a basis element, so
\[
\fg^{\SSN}_{\Pi_Q,\dd}=\QQ\cdotsh \alpha_{i',n}.
\]
One can see directly (e.g. without the aid of Corollary \ref{absIso}) that for $m,n\in\ZZ_{\geq 1}$ with $m\neq n$
\begin{align}
\label{alpharel}
[\alpha_{i',m},\alpha_{i',n}]=0.
\end{align}
For example this follows because the Lie bracket
\[
\fg^{\SSN}_{\Pi_Q,n\cdot 1_i}\otimes \fg^{\SSN}_{\Pi_Q,m\cdot 1_i}\rightarrow \fg^{\SSN}_{\Pi_Q,m+n\cdot 1_i}
\]
is defined by applying $\HO\varpi'_{\red,*}\varpi'^!_{\red}$ to the morphism of mixed Hodge modules
\begin{equation}
\label{nzer}
\Delta_{\red,n,*}\nIC_{\AAA{2}}\boxtimes_{\oplus_{\red}} \Delta_{\red,m,*}\nIC_{\AAA{2}}\rightarrow \Delta_{\red,m+n,*}\nIC_{\AAA{2}}.
\end{equation}
Since $m\neq n$ the morphism
\[
\oplus_{\red}\circ (\Delta_{\red,n}\times\Delta_{\red,m})\colon\AAA{4}\rightarrow \Msp_{m+n}(\ol{Q})
\]
is injective. It follows that the left hand side of \eqref{nzer} is simple, and not isomorphic to the (simple) right hand side, which has 2-dimensional support. It follows that \eqref{nzer} is the zero map, since it is a morphism between distinct simple objects.
\begin{proposition}
Let $i'\in Q_0^{\isot}$, and set $\dd= n\cdotsh 1_{i'}$ as above. Up to multiplication by a scalar, there is an identity
\[
\alpha_{i',n}=[\Lambda(\dd)_{(1^n)}]\in \HO^0\!\!\Coha_{\Pi_Q,\dd}^{\mathcal{SSN}}.
\]
\end{proposition}
\begin{proof}
Let $\Lambda(\dd)_{(1^n)}^{\circ}\subset \Lambda(\dd)_{(1^n)}$ be the complement to the intersection with the union of components $\Lambda(\dd)_\pi$ for $\pi\neq (1^n)$. Denote by
\begin{align*}
j\colon&\Lambda(\dd)_{(1^n)}/\Gl_{\dd}\rightarrow \Mst_{\dd}(\ol{Q})\\
j^{\circ}\colon &\Lambda(\dd)^\circ_{(1^n)}/\Gl_{\dd}\rightarrow \Mst_{\dd}(\ol{Q})
\end{align*}
the inclusions. Define
\begin{align*}
\mathcal{G}&=j_*j^!\ul{\QQ}_{\Mst_{\dd}(\ol{Q})}\otimes\LLL^{\chi_{\WT{Q}}(\dd,\dd)/2}\\
\mathcal{G}^\circ&=j^\circ_*j^{\circ,!}\ul{\QQ}_{\Mst_{\dd}(\ol{Q})}\otimes\LLL^{\chi_{\WT{Q}}(\dd,\dd)/2}\cong j^\circ_*\ul{\QQ}_{\Lambda(\dd)^{\circ}_{(1^n)}}
\end{align*}
where the isomorphism is due to the fact that $\Lambda(\dd)^{\circ}_{(1^n)}$ is smooth and has codimension $-\chi_{\WT{Q}}(\dd,\dd)/2$ inside $\Mst_{\dd}(\ol{Q})$. Then since $j$ is closed, and $\Lambda(\dd)^\circ_{(n)}$ is open in $\Lambda(\dd)_{(n)}$, we have a diagram
\begin{equation}
\label{comp_emb}
\xymatrix{
\JH_{\red,*}\mathcal{G}^{\circ}&\ar[l]_-\xi\JH_{\red,*}\mathcal{G}\ar[r]^-q& \JH_{\red,*}\HA^{\SSN}_{\Pi_Q,\dd}
}
\end{equation}
Applying $\HO^0$, $\xi$ is an isomorphism, and $[\Lambda(\dd)_{(1^n)}]$ is defined to be $\HO^0(q)(\HO^0(\xi))^{-1}(1)$. Applying $\tau_{\mathrm{con},\leq 0}$, the truncation functor induced by the non-perverse t structure, the diagram \eqref{comp_emb} becomes
\[
\xymatrix{
\Delta_{\red,n,*}^{\mathcal{SSN}}\ul{\QQ}_{\AAA{1}}&\ar[l]_-{\cong}\Delta_{\red,n,*}^{\mathcal{SSN}}\ul{\QQ}_{\AAA{1}}\ar[r]& \tau_{\mathrm{con},\leq 0}\JH_{\red,*}\HA^{\SSN}_{\Pi_Q,\dd}.
}
\]
The element $\alpha_{i',n}$ is likewise obtained by applying $\HO$ to a homomorphism
\[
q'\colon \Delta_{\red,n,*}^{\mathcal{SSN}}\ul{\QQ}_{\AAA{1}}\rightarrow \JH_{\red,*}\HA^{\SSN}_{\Pi_Q,\dd}.
\]
The domain is a mixed Hodge module, shifted by cohomological degree 1, so that $q'$ factors through the morphism
\[
\tau_{\leq 1}\JH_{\red,*}\HA^{\SSN}_{\Pi_Q,\dd}\rightarrow \JH_{\red,*}\HA^{\SSN}_{\Pi_Q,\dd}.
\]
By \eqref{gen} there is an isomorphism
\[
\tau_{\leq 1}\JH_{\red,*}\HA^{\SSN}_{\Pi_Q,\dd}\cong \Delta_{\red,n,*}^{\mathcal{SSN}}\ul{\QQ}_{\AAA{1}}
\]
and so we deduce that $\dim\left(\Hom\left(\Delta_{\red,n,*}^{\mathcal{SSN}}\ul{\QQ}_{\AAA{1}},\JH_{\red,*}\HA^{\SSN}_{\Pi_Q,\dd}\right)\right)=1$, and the proposition follows.
\end{proof}
\subsubsection{Hyperbolic vertices}
Suppose that $i'\in I^{\hype}$, i.e. $i'$ supports $l$ edge-loops with $l\geq 2$. Let $n\in \mathbb{Z}_{>0}$, and set $\dd=n\cdot 1_{i'}$. The variety $\Msp_{\dd}(\Pi_Q)$ is an irreducible variety of dimension $2+2(l-1)n^2$ by \cite[Thm.1.3]{CB01}. We set
\begin{align*}
\Cusp_{i',n}\coloneqq&\nIC_{\Msp_{\dd}(\Pi_Q)}\in\MHM(\Msp_{\dd}(\ol{Q}))\\
\Cusp^{\mathcal{SSN}}_{i',n}\coloneqq&\varpi'_{\red,*}\varpi'^!_{\red}\nIC_{\Msp_{\dd}(\Pi_Q)}\in\Db(\MHM(\Msp_{\dd}(\ol{Q})))\\
\fc^{\mathcal{SSN}}_{i',n}\coloneqq &\HO\!\left(\Msp_\dd(\ol{Q}),\Cusp^{\mathcal{SSN}}_{i',n}\right).
\end{align*}
The following will be proved as a special case of the results in \S \ref{genSec}.
\begin{proposition}
Set $\mathcal{B}_n=\tau_{\leq 0}\!\rCoha_{\Pi_Q,\dd}\in\MHM(\Msp_{\dd}(\Pi_Q))$. There is an inclusion $\Cusp_{i',n}\hookrightarrow \mathcal{B}_n$ and $\Cusp_{i',n}$ is primitive, i.e. the induced morphism
\begin{equation}
\label{prim1}
\Cusp_{i',n}\rightarrow \mathcal{B}_n/\left(\sum_{\substack{n'+n''=n\\n'\neq 0\neq n''}} \image\left(\mathcal{B}_{n'}\boxtimes_{\oplus}\mathcal{B}_{n''}\xrightarrow{\star} \mathcal{B}_{n}\right)\right)
\end{equation}
is injective, where $\star$ is the CoHA multiplication of \S \ref{2drelCoHA}.
\end{proposition}
By Theorem \ref{purityThm}, \eqref{prim1} is a morphism of semisimple objects, as well as being injective, and so it has a left inverse. Applying $\HO\varpi'_{\red,*}\varpi'^!_{\red}$ we deduce that there is an injective morphism
\begin{equation}
\label{cin}
\mathfrak{c}^{\mathcal{SSN}}_{i',n}\hookrightarrow \mathfrak{g}^{\mathcal{SSN}}_{\Pi_Q,n}
\end{equation}
and moreover that the induced morphism
\begin{equation}
\label{csnin}
\mathfrak{c}^{\mathcal{SSN}}_{i',n}\rightarrow \mathfrak{g}^{\mathcal{SSN}}_{\Pi_Q,n}/\left(\sum_{\substack{n'+n''=n\\ n'\neq 0\neq n''}}\image \left( \mathfrak{g}^{\mathcal{SSN}}_{\Pi_Q,n'}\otimes \mathfrak{g}^{\mathcal{SSN}}_{\Pi_Q,n''}\xrightarrow{[\bullet,\bullet]}\mathfrak{g}^{\mathcal{SSN}}_{\Pi_Q,n}\right) \right)
\end{equation}
is injective. Taking the zeroth cohomologically graded piece, we deduce from Corollary \ref{absIso} that
\begin{equation}
\label{dimest}
\dim(\HO^0\!\Cusp^{\mathcal{SSN}}_{i',n})\leq 1.
\end{equation}
The inclusion $j\colon \Msp^{\simp}_{\dd}(Q^{\opp})\rightarrow \Msp_{\dd}(Q^{\opp})$ is an open embedding, so there is a morphism
\begin{equation}
\label{prealpha}
\Cusp_{i',n}^{\mathcal{SSN}}\rightarrow \varpi'_{\red,*}j_*j^!\varpi'^!_{\red}\nIC_{\Msp_{\dd}(\Pi_Q)}\cong \varpi'_{\red,*}j_*\nIC_{\Msp^{\simp}_{\dd}(Q^{\opp})}\otimes \LLL^{-\chi_Q(\dd,\dd)-1}.
\end{equation}
The Tate twist is given by the difference in dimensions of the smooth schemes $\Msp^{\simp}_{\dd}(\Pi_Q)$ and $\Msp^{\simp}_{\dd}(Q^{\opp})$. Since $\dim(\Msp^{\simp}_{\dd}(Q^{\opp}))=-\chi_Q(\dd,\dd)-1$ we have
\[
\nIC_{\Msp^{\simp}_{\dd}(Q^{\opp})}=\ul{\QQ}_{\Msp^{\simp}_{\dd}(Q^{\opp})}\otimes\LLL^{1+\chi_Q(\dd,\dd)}
\]
and thus \eqref{prealpha} induces the morphism
\[
\Psi\colon\Cusp_{i',n}^{\mathcal{SSN}}\rightarrow \varpi'_{\red,*}j_*\ul{\QQ}_{\Msp^{\simp}_{\dd}(Q^{\opp})}.
\]
\begin{proposition}
\label{HypeOne}
The morphism $\HO^0\!\Psi\colon\HO^0\!\Cusp^{\mathcal{SSN}}_{i',n}\rightarrow \HO^0(\Msp_{\dd}^{\simp}(Q^{\opp}),\QQ)\cong\mathbb{Q}$ is an isomorphism.
\end{proposition}
\begin{proof}
By \eqref{dimest} it is sufficient to show that $\HO^0\!\Psi$ is not the zero morphism. Let $h\colon \Msp_{\dd}^{\simp}(\ol{Q})\hookrightarrow \Msp_{\dd}(\ol{Q})$ be the inclusion. The morphism $\Psi$ factors through the morphism
\begin{equation}
\label{yut}
\varpi'_{\red,*}\varpi_{\red}'^!\rCoha_{\Pi_Q}\rightarrow \varpi'_{\red,*}\varpi'^!_{\red}h_*h^*\rCoha_{\Pi_Q}
\end{equation}
Applying $\HO^0$ to \eqref{yut} yields the morphism
\[
\HO^0\!\!\Coha^{\SSN}_{\Pi_Q}\rightarrow \HO^0(\Msp_{\dd}^{\simp}(Q^{\opp}),\QQ)
\]
which is not the zero morphism, since $[\Lambda(\dd)_{(n)}]$ does not lie in the kernel. Now let
\[
\mathcal{F}\hookrightarrow \mathcal{B}_n
\]
be the inclusion of any summand that is not isomorphic to $\Cusp_{i',n}^{\SSN}$. Then $\mathcal{F}$ is supported on $\Msp_{\dd}(\ol{Q})\setminus \Msp_{\dd}^{\simp}(\ol{Q})$, so that $h_*h^*\mathcal{F}$ is zero, and the morphism
\[
\HO^0\!\mathcal{F}\rightarrow \HO^0(\Msp_{\dd}^{\simp}(Q^{\opp}),\QQ)
\]
is zero. It follows that $\HO^0\!\Psi$ is not the zero morphism.
\end{proof}
\begin{corollary}
The images of the inclusions
\[
\xi_n\colon\HO^0\!\left(\fc_{i',n}^{\SSN}\right)\hookrightarrow \HO^0\!\left(\fg_{\Pi_Q,n\cdot 1_{i'}}\right)
\]
generate $\bigoplus_{n\geq 1}\fg_{\Pi_Q,n\cdot 1_{i'}}$.
\end{corollary}
\begin{proof}
The morphism $\HO^0\!\Psi$ factors through $\xi_n$, and so $\xi_n$ is injective. The result then follows from the fact that $\bigoplus_{n\geq 1}\fg_{\Pi_Q,n\cdot 1_{i'}}$ has one simple imaginary root for each $n$, and injectivity of \eqref{csnin}.
\end{proof}
We define
\begin{equation}
\label{Hdef}
\alpha_{i',n}=(\HO^0\!\Psi)^{-1}(1).
\end{equation}
If we express $\alpha_{i',n}$ in terms of Bozec's basis, we have shown that the coefficient of $[\Lambda(\dd)_{(n)}]$ is 1. The question of what all of the other coefficients are (in particular, whether they are nonzero) seems to be quite difficult without an explicit description of $\rCoha_{\Pi_Q,\dd}^{\SSN}$ like Proposition \ref{IsoPure} in the hyperbolic case. On the other hand \S \ref{IsotSec} already demonstrates that the isomorphism $F$ from \eqref{UEAiso} does not identify $\mathfrak{n}_Q^+$ and $\HO^0\!(\fg_{\Pi_Q}^{\SSN})$.
\begin{remark}
We have shown that the zeroth cohomologically graded pieces of $\fc^{\SSN}_{i',n}$ for $i'\in Q_0$ and $n\in\ZZ_{\geq 1}$ provide a complete set of generators for $\HO^0\!(\fg_{\Pi_Q}^{\SSN})$. This provides evidence for Conjecture \ref{mainConj} below.
\end{remark}
\section{BPS sheaves and cuspidal cohomology}
\subsection{Generators of $\fg^{\SP,G,\zeta}_{\Pi_Q,\theta}$}
\label{genSec}
In \S \ref{RLAsec} we constructed a lift of $\fg^{G,\zeta}_{\Pi_Q,\theta}$ to a Lie algebra object in the category $\MHM^G(\Msp^{\zeta\sst}_{\theta}(\ol{Q}))$. In this section we will use this lift to produce generators for $\fg^{\SP,G,\zeta}_{\Pi_Q,\theta}$.
Fix a dimension vector $\dd$. Let
\begin{equation}
\label{Udef}
U\subset \Msp^{G,\zeta\ssst}_{\dd}(\WT{Q})
\end{equation}
be the subscheme parameterising those modules for which the underlying $\ol{Q}$-module is $\zeta$-stable\footnote{Recall that the right hand side of \eqref{Udef} parameterises those $\CC\WT{Q}$-modules $\rho$ for which the underlying $\CC\ol{Q}$-module of $\rho$ is $\zeta$-semistable.}, and let
\[
\mathfrak{U}=(\JH^{\circ})^{-1}(U)\subset \Mst^{G,\zeta\ssst}_{\dd}(\WT{Q})
\]
be the open substack parameterising such modules. Note that $U\subset \Msp^{G,\zeta\stab}_{\dd}(\WT{Q})$, so the morphism $\mathfrak{U}\rightarrow U$ is a $\B\CC^*$-torsor. Define
\begin{align*}
V=&U\cap \Msp_{\dd}^{G,\zeta\ssst}(\Jac(\WT{Q},\WT{W}))\\
\mathfrak{V}=&(\JH^{\circ})^{-1}(V).
\end{align*}
Since $\sum_i\omega_i\in \Jac(\WT{Q},\WT{W})$ is central, it acts via scalar multiplication on any module represented by a point in $\mathfrak{V}$. Arguing as in the proof of Proposition \ref{NQVDR} we deduce that
\begin{equation}
\label{Vdesc}
V= \left(\Msp_{\dd}^{\zeta\stab}(\Pi_Q)\times\AAA{1}\right)/G\subset \Msp_{\dd}^{G,\zeta\ssst}(\WT{Q}).
\end{equation}
The variety $\Msp_{\dd}^{\zeta\stab}(\Pi_Q)$ is smooth, and so both $V$ and $\mathfrak{V}$ are smooth stacks. We denote by $\overline{V}$ the closure of $V$ in $\Msp^{G,\zeta\ssst}(\WT{Q})$. Similarly, arguing as in the proof of \eqref{rDIM} we deduce that $\phim{\TTTr(\WT{W})}\nnIC_{\mathfrak{U}}\cong \nnIC_{\mathfrak{V}}$, and thus
\begin{equation}
\label{Vin}
(U\hookrightarrow \Msp^{G,\zeta\sst}_{\dd}(\WT{Q}))^*\JH^G_*\phim{\TTTr(\WT{W})}\nnIC_{\Mst^{G,\zeta\sst}_{\dd}(\WT{Q})}\cong \nnIC_{V}\otimes\HO(\B\CC^*,\QQ)_{\vir}.
\end{equation}
By Corollary \ref{relPurity} the object $\JH^G_*\phim{\TTTr(\WT{W})}\nnIC_{\Mst^{G,\zeta\sst}_{\dd}(\WT{Q})}$ is pure, and so in particular its first cohomology is a semisimple monodromic mixed Hodge module. From \eqref{Vin} and the inclusion
\[
\LLL^{1/2}\hookrightarrow \HO(\B\CC^*,\QQ)_{\vir}
\]
we deduce that there is a canonical morphism
\begin{equation}
\label{PreV}
\Gamma\colon\nnIC_{\ol{V}}\otimes\LLL^{1/2}\hookrightarrow \JH^G_*\phim{\TTTr(\WT{W})}\nnIC_{\Mst^{G,\zeta\sst}_{\dd}(\WT{Q})}
\end{equation}
for which there is a left inverse $\alpha$, by purity of the target. By \eqref{Vdesc} we have
\[
\ol{V}=\left(\Msp_{\dd}^{\zeta\sst}(\Pi_Q)\times\AAA{1}\right)/G,
\]
and so
\[
r'_*\nnIC_{\ol{V}}\otimes\LLL^{1/2}\cong \nnIC_{\Msp_{\dd}^{G,\zeta\sst}(\Pi_Q)}.
\]
Set
\begin{align*}
\Cusp_{\Pi_Q,\dd}^{\SP,G,\zeta}&\coloneqq \begin{cases} \varpi'_{\red,*}\varpi'^!_{\red}\nnIC_{\Msp^{G,\zeta\sst}_{\dd}(\Pi_Q)}& \textrm{if }\Msp^{\zeta\stab}_{\dd}(\Pi_Q)\neq \emptyset\\
0&\textrm{otherwise}\end{cases}\\
\fc^{\SP,G,\zeta}_{\Pi_Q,\dd}&\coloneqq \HO\!\Cusp_{\Pi_Q,\dd}^{\SP,G,\zeta}.
\end{align*}
Then there is an isomorphism
\begin{align*}
\fc^{\SP,G,\zeta}_{\Pi_Q,\dd}\cong &\HO\!\left(\Msp^{G,\zeta\sst}_{\dd}(\WT{Q}),\varpi'_*\varpi'^!\nnIC_{\ol{V}}\otimes\LLL^{1/2}\right).
\end{align*}
Applying $\HO\varpi'_*\varpi'^!$ to \eqref{PreV} we obtain a morphism
\[
\beta\colon\fc^{\SP,G,\zeta}_{\Pi_Q,\dd}\hookrightarrow\fg^{\WT{\SP},G,\zeta}_{\WT{Q},\WT{W},\dd}=\mathfrak{P}_{\leq 1}\!\HCoha_{\WT{Q},\WT{W},\dd}^{\WT{\SP},G,\zeta}
\]
which is an injection, since it has a left inverse (e.g. $\HO\varpi_*\varpi^!\alpha$).
We can now prove (a generalisation of) Theorem \ref{mainThmB}.
\begin{theorem}
Let $\zeta\in\QQ^{Q_0}$ be a stability condition, and let $\theta\in\QQ$ be a slope. For each $\dd\in\dvst$ there is a canonical decomposition
\[
\fg_{\Pi_Q,\dd}^{\SP,G,\zeta}\cong \fc^{\SP,G,\zeta}_{\Pi_Q,\dd}\oplus \mathfrak{l}
\]
for some mixed Hodge structure $\mathfrak{l}$, and for $\dd',\dd''\in \dvst$ such that $\dd'+\dd''=\dd$, the morphism
\[
\fg_{\Pi_Q,\dd'}^{\SP,G,\zeta}\otimes\fg_{\Pi_Q,\dd''}^{\SP,G,\zeta}\xrightarrow{[\cdot,\cdot]}\fg_{\Pi_Q,\dd}^{\SP,G,\zeta}
\]
factors through the inclusion of $\mathfrak{l}$.
\end{theorem}
\begin{proof}
We assume that $\Msp^{\zeta\stab}_{\dd}(\Pi_Q)\neq \emptyset$, as otherwise $\fc^{\SP,G,\zeta}_{\Pi_Q,\dd}=0$ and the statement is trivial. Recall that by \eqref{2DIR} there is an isomorphism
\[
r'_*\JH^{\circ}_*\phim{\TTTr(\WT{W})}\nnIC_{\Mst^{G,\zeta\ssst}_{\dd}(\WT{Q})}\cong \JH^G_{\red,*}\iota_*\iota^!\ul{\QQ}_{\Mst^{G,\zeta\sst}_{\dd}(\ol{Q})}\otimes\LLL^{\chi_{\WT{Q}}(\dd,\dd)/2}=\rCoha_{\Pi_Q,\dd}^{G,\zeta}
\]
and by Theorem \ref{purityThm} these are pure complexes of monodromic mixed Hodge modules, so there is a decomposition
\begin{equation}
\label{2dDecomp}
\rCoha_{\Pi_Q,\dd}^{G,\zeta}\cong\bigoplus_{r\in R_n}\mathcal{F}_r[n]
\end{equation}
where each $\mathcal{F}_r$ is a simple mixed Hodge module, and each $R_n$ is some indexing set.
The stack $\Mst^{G,\zeta\stab}_{\dd}(\Pi_Q)$ is smooth, of codimension $\dd\cdot\dd-1$ inside $\Mst^{G,\zeta\stab}_{\dd}(\ol{Q})$, and so there is an isomorphism
\[
(\Mst^{G,\zeta\stab}_{\dd}(\Pi_Q)\hookrightarrow \Mst^{G,\zeta\sst}_{\dd}(\Pi_Q))^*\iota_*\iota^!\ul{\QQ}_{\Mst^{G,\zeta\sst}_{\dd}(\ol{Q})}\cong\ul{\QQ}_{\Mst^{G,\zeta\stab}_{\dd}(\Pi_Q)}\otimes\LLL^{\dd\cdot\dd-1}
\]
and thus an isomorphism
\begin{align*}
&(\Msp^{G,\zeta\stab}_{\dd}(\Pi_Q)\hookrightarrow \Msp^{G,\zeta\sst}_{\dd}(\Pi_Q))^*\rCoha_{\Pi_Q,\dd}^{G,\zeta}\\&\cong(\Mst^{G,\zeta\stab}_{\dd}(\Pi_Q)\rightarrow \Msp^{G,\zeta\stab}_{\dd}(\Pi_Q))_*\ul{\QQ}_{\Mst^{G,\zeta\stab}_{\dd}(\Pi_Q)}\otimes\LLL^{\chi_{\WT{Q}}(\dd,\dd)/2+\dd\cdot\dd-1}.
\end{align*}
Noting that $\Msp^{G,\zeta\stab}_{\dd}(\Pi_Q)$ is open inside $\Msp^{G,\zeta\sst}_{\dd}(\Pi_Q)$, of dimension $\chi_Q(\dd,\dd)-1=\chi_{\WT{Q}}(\dd,\dd)/2+\dd\cdot\dd-1$, we deduce that in the decomposition \eqref{2dDecomp}, in cohomological degree zero there is exactly one copy of the simple object $\nnIC_{\Msp^{G,\zeta\sst}_{\dd}(\Pi_Q)}$, and furthermore the morphism
\[
r'_*\Gamma\colon\nnIC_{\Msp^{G,\zeta\sst}_{\dd}(\Pi_Q)}\rightarrow \rCoha_{\Pi_Q,\dd}^{G,\zeta}
\]
is the inclusion of this object. Writing
\begin{align*}
\Ho^{G,0}\!\left(\rCoha_{\Pi_Q,\dd}^{G,\zeta}\right)=&\nnIC_{\Msp^{G,\zeta\sst}_{\dd}(\Pi_Q)}\oplus \mathcal{G}\\
\mathcal{G}=&\bigoplus_{r\in R_0'}\mathcal{F}_r
\end{align*}
for $R'\subset R_0$, we claim that for all $\dd',\dd''\in\dvst$ with $\dd'\neq 0\neq \dd''$ and $\dd'+\dd''=\dd$ the multiplication
\begin{equation}
\label{alm}
\Ho^{G,0}\!\left(\rCoha^{G,\zeta}_{\Pi_Q,\dd'}\right)\boxtimes_{\oplus^G}\Ho^{G,0}\!\left(\rCoha^{G,\zeta}_{\Pi_Q,\dd''}\right)\rightarrow \Ho^{G,0}\!\left(\rCoha^{G,\zeta}_{\Pi_Q,\dd}\right)
\end{equation}
factors through the inclusion of $\mathcal{G}$. This follows for support reasons: by our assumptions on $\dd',\dd''$ the supports of the semisimple object on the left hand side of \eqref{alm} are all in the boundary $\Msp^{G,\zeta\sst}_{\dd}(\Pi_Q)\setminus \Msp^{G,\zeta\stab}_{\dd}(\Pi_Q)$. Applying $\HO\varpi'_{\red,*}\varpi'^!_{\red}$, there is a decomposition
\[
\lP_{\leq 0}\!\HO\!\!\Coha^{\Sp,G,,\zeta}_{\Pi_Q,\dd}\cong \fc^{\SP,G,\zeta}_{\Pi_Q,\dd}\oplus\HO\!\varpi'_{\red,*}\varpi'^!_{\red}\mathcal{G}
\]
and the multiplication
\[
\lP_{\leq 0}\!\HO\!\!\Coha^{\Sp,G,,\zeta}_{\Pi_Q,\dd'}\otimes_{\HG}\lP_{\leq 0}\!\HO\!\!\Coha^{\Sp,G,,\zeta}_{\Pi_Q,\dd''}\rightarrow \lP_{\leq 0}\!\HO\!\!\Coha^{\Sp,G,,\zeta}_{\Pi_Q,\dd}
\]
factors through the inclusion of $\HO\!\varpi'_{\red,*}\varpi'^!_{\red}\mathcal{G}=\mathfrak{l}$, so that the commutator Lie bracket also factors through the inclusion of $\mathfrak{l}$.
\end{proof}
\subsection{The BPS Lie algebra $\mathfrak{g}_{\Pi_Q}$ for $Q$ affine}\label{AffineSec}
Let $Q$ be a quiver for which the underlying graph is an affine Dynkin diagram of extended ADE type. We can use the fact that $\fg^G_{\Pi_Q}$ lifts to a Lie algebra object $\BPSh^G_{\Pi_Q}$ in $\MHM^G(\Msp(\Pi_Q))$ to calculate it completely.
Set $d=\lvert Q_0\lvert -1$. We denote by $q_{\dd}\colon \Msp^{\zeta\sst}_{\dd}(\Pi_Q)\rightarrow \Msp_{\dd}(\Pi_Q)$ the affinization map. Let $\delta\in\dvs$ be the unique primitive imaginary simple root of the quiver $Q$. Let $H\subset \Sl_2(\CC)$ be the Kleinian group corresponding to the underlying (finite) Dynkin diagram of $Q$ (obtained by removing a single vertex) via the McKay correspondence. Then (see \cite{Kr89,CaSl98}) for a generic stability condition $\zeta\in\QQ^{Q_0}$ there is a commutative diagram
\[
\xymatrix{
X\ar[d]^p\ar[r]^-{\cong}&\Msp_{\delta}^{\zeta\sst}(\Pi_Q)\ar[d]^{q_{\delta}}\\
Y\ar[r]^-{\cong}&\Msp_{\delta}(\Pi_Q)
}
\]
where $p$ is the minimal resolution of the singularity $Y=\AAA{2}/H$. Moreover by \cite{KV00} there is a derived equivalence
\[
\Psi\colon \Db(\Coh(X))\rightarrow \Db(\Pi_Q\lmod)
\]
restricting to an equivalence between complexes of modules with nilpotent cohomology sheaves and complexes of coherent sheaves with set-theoretic support on the exceptional locus of $p$. For $\dd\in\dvs$ we denote by
\[
0_{\dd}\in\Msp_{\dd}(\Pi_Q)
\]
the point corresponding to the unique semisimple nilpotent module of dimension vector $\dd$.
Via the explicit description of the representations of $KQ$ for $Q$ an affine quiver, we have the following identities
\begin{equation}
\label{affineKac}
\kac_{Q,\dd}(t)=\begin{cases} 1 &\textrm{if }\dd \textrm{ is a positive real root of }\mathfrak{g}_Q\\
t+d&\textrm{if } \dd\in\mathbb{Z}_{\geq 1}\cdot\delta\\
0&\textrm{otherwise.}\end{cases}
\end{equation}
\begin{proposition}
\label{affineProp}
There are isomorphisms
\begin{equation}
\label{affineBPS}
\BPSh_{\Pi_Q,\dd}\cong\begin{cases} \ul{\QQ}_{0_{\dd}}&\textrm{if }\dd \textrm{ is a positive real root of }\mathfrak{g}_Q\\
\Delta_{n,*}q_{\delta,*}\nIC_{\Msp_{\delta}^{\zeta\sst}(\Pi_Q)}&\textrm{if } \dd=n\cdot\delta\\
0&\textrm{otherwise,}\end{cases}
\end{equation}
where $\Delta_n\colon \Msp_{\delta}(\Pi_Q)\rightarrow \Msp_{n\cdot\delta}(\Pi_Q)$ is the embedding of the small diagonal.
\end{proposition}
We sketch the proof --- the complete description of affine preprojective CoHAs will appear in a forthcoming paper with Sven Meinhardt. By Proposition \ref{TodaProp} there is an isomorphism
\[
\BPSh_{\Pi_Q,\dd}\cong q_{\dd,*}\BPSh^{\zeta}_{\Pi_Q,\dd}.
\]
On the other hand, any complex of compactly supported coherent sheaves $\mathscr{F}$ on $X$ that is not entirely supported on the exceptional locus admits a direct sum decomposition $\mathscr{F}'\oplus\mathscr{F}''$ where $\mathscr{F}''$ is supported at a single point, so that $\Psi(\mathscr{F})$ admits a direct summand $N$ with dimension vector a multiple of $\delta$. It follows that all points of $\Msp_{\dd}^{\zeta\sst}(\Pi_Q)$ correspond to nilpotent modules if $\dd$ is not a multiple of $\delta$, and so $q_{\dd,*}\BPSh^{\zeta}_{\Pi_Q,\dd}$ is supported at the origin. Since by Theorem \ref{purityThm} $\BPSh_{\Pi_Q,\dd}$ is pure, and supported at a single point, it is determined by its hypercohomology $\fg_{\Pi_Q,\dd}$. This hypercohomology pure, of Tate type, with dimension given by the Kac polynomial, by the main result of \cite{preproj}. This deals with the first and last cases of \eqref{affineBPS}.
For the second case, we consider the commutative diagram
\[
\xymatrix{
\CCoh_n(X)\ar[r]^h_{\cong}\ar[d]^g&\Mst_{n\cdot\delta}^{\zeta\sst}(\Pi_Q)\ar[d]^{\JH_{\red}}
\\
\Sym_n(X)\ar[r]^l_{\cong}&\Msp_{n\cdot\delta}^{\zeta\sst}(\Pi_Q).
}
\]
By Theorem \ref{purityThm}, $g_*h^*\iota^!\ul{\QQ}_{\Mst_{n\delta}^{\zeta\sst}(\ol{Q})}$
is pure, and we claim that it contains a single copy of $\Delta_{X,n,*}\nIC_{X}$. Since $X$ is simply connected, we can cover $X$ by charts $U_i$ isomorphic to $\AAA{2}$ and check the claim on each of the open subvarieties $U_i$, at which point the claim follows by \eqref{commBPS}. Finally, the BPS sheaf is supported on the small diagonal by the support lemma of \cite{preproj}, so $\Delta_{X,n,*}\nIC_X\cong l^*\BPSh_{\Pi_Q,n\delta}^{\zeta}$. Then the second case follows by Proposition \ref{TodaProp}.
Note that there is an isomorphism
\begin{equation}
\label{affroo}
q_{\delta,*}\nIC_{\Msp_{\delta}^{\zeta\sst}(\Pi_Q)}\cong \nIC_{\Msp_{\delta}(\Pi_Q)}\oplus \ul{\QQ}_{0_{\delta}}^{\oplus d}
\end{equation}
since there are $d$ copies of $\mathbb{P}^1$ in the exceptional fibre of $q_{\delta}$. We deduce from \eqref{affineKac} and \eqref{Kaccha} that $\HO^{-2}\fg_{\Pi_Q,n\cdot \delta}\cong\QQ$ is obtained by applying $\HO\Delta_{n,*}$ to the first summand of \eqref{affroo}.
\begin{proposition}
There is an isomorphism of Lie algebras
\begin{equation}
\label{aff}
\mathfrak{g}_{\Pi_Q}\cong \mathfrak{n}^-_{Q'}\oplus s\QQ[s]
\end{equation}
where $Q'$ is the real subquiver of $Q$ (i.e. it is equal to $Q$ unless $Q$ is the Jordan quiver, in which case it is empty) and $s\QQ[s]$ is given the trivial Lie bracket. The monomial $s^n$ lives in $\dvs$-degree $n\cdot\delta$, and in cohomological degree $-2$.
\end{proposition}
\begin{proof}
By \eqref{affineKac} and \eqref{Kaccha}, the graded dimensions of the two sides of \eqref{aff} match. Furthermore, by Theorem \ref{KMLA} there is an isomorphism of Lie algebras between the zeroth cohomology of the RHS and LHS of \eqref{aff}.
So it is sufficient to prove that $\HO^{-2}(\mathfrak{g}_{\Pi_Q})$ is central, which amounts to showing that
\[
[s^n,r]=0\in\fg_{\Pi_Q,\beta}
\]
for $r\in \HO^0(\fg_{\Pi_Q,\alpha})$ where $\alpha=n'\cdot\delta$ and $\beta=n\cdot\delta+\alpha$. On the other hand, the morphism
\[
\QQ\otimes \QQ\xrightarrow{\cdot s^n\otimes\cdot r}\fg_{\Pi_Q,n\delta}\otimes \fg_{\Pi_Q,\alpha}\xrightarrow{[\cdot,\cdot]}\fg_{\Pi_Q,\beta}
\]
is obtained by applying $\HO$ to the morphism of mixed Hodge modules
\[
\Delta_{n,*}\nIC_{\Msp_{\delta}(\Pi_Q)}\boxtimes_{\oplus_{\red}}\QQ_{0_{\alpha}}\rightarrow \Delta_{(n+n'),*}\nIC_{\Msp_{\delta}(\Pi_Q)}\oplus\ul{\QQ}_{0_{\beta}}^{\oplus d}
\]
which is a morphism between semisimple mixed Hodge modules with differing supports, and is thus zero.
\end{proof}
\subsubsection{A deformed example}
\label{defEx}
In this subsection we give a curious example, which will not be used later in the paper. It is an example of how deforming the potential can modify the BPS Lie algebra.
Let $Q$ be the oriented $\tilde{A}_d$ quiver, i.e. it contains $d+1$ vertices, along with an oriented cycle connecting them all. Let $W_0=a_{d+1}a_{d}\cdots a_1$ be this cycle. We will consider the quiver with potential $(\tilde{Q},\WT{W}+W_0)$. The potential $\WT{W}+W_0$ is quasihomogeneous, for example we can give the arrows $a_s$ weight 1, the arrows $a_s^*$ weight $d$, and the arrows $\omega_i$ weight zero, so that $\WT{W}+W_0$ has weight $d+1$.
As in \S \ref{DDRsec} we can calculate $\BPSh_{\Pi_Q,W_0}$ by applying $\phim{\TTr(W_0)}$ to $\BPSh_{\Pi_Q}$. For $\dd$ not a multiple of the imaginary simple root, $\BPSh_{\Pi_Q,\dd}$ is supported at $0_{\dd}$ and so it follows that
\begin{align*}
\BPSh_{\Pi_Q,W_0,\dd}\cong &\phim{\TTr(W_0)}\BPSh_{\Pi_Q,\dd}\\
\cong &\BPSh_{\Pi_Q,\dd}.
\end{align*}
In particular, it follows that there is an injective map $l$ from the Lie subalgebra of $\BPSh_{\Pi_Q}$ generated by $\bigoplus_{i\in Q_0}\BPSh_{\Pi_Q,1_i}$, and the only dimension vectors for which this morphism can fail to be an isomorphism are dimension vectors $\ee=(n,\ldots,n)$ for some $n$.
Let $\ee$ be such a dimension vector. Propositions \ref{TodaProp} and \ref{affineProp} together yield
\[
\BPSh_{\Pi_Q,W_0,\ee}\cong \Delta_{n,*}q_{(1,\ldots,1),*}\phim{g}\nIC_X
\]
where $X$ is the minimal resolution of the singular surface defined by $xy=z^d$, and $g=y$ is the function induced on it by $\TTr(W_0)$. The reduced vanishing locus $X_0=g^{-1}(0)$ is given by the exceptional chain of $d$ copies of $\mathbb{P}^1$, along with a line intersecting one of them transversally. In particular, the cohomology of $X_0$ is pure. The preimage $X_1\coloneqq g^{-1}(1)$ is isomorphic to a copy of $\AAA{1}$. Via the long exact sequence
\[
\rightarrow \HO^i(X,\phim{g}\ul{\QQ}_X)\rightarrow \HO^i(X_0,\QQ)\rightarrow \HO^i(X_1,\QQ)\rightarrow
\]
we deduce that there is an isomorphism $\HO(X,\phim{g}\ul{\QQ}_X)\cong \HO^2(X,\QQ)[-2]$, i.e. the vanishing cycle cohomology is isomorphic to the \textit{reduced} cohomology of $X$. It follows by counting dimensions that the injective map $\HO(l_{\ee})$ is surjective, although $l_{\ee}$ is not, since $\crit(g)$ is not contained in the exceptional locus. We deduce that
\begin{equation}
\label{KacOut}
\fg_{\Pi_Q,W_0}\cong \mathfrak{n}^-_Q
\end{equation}
i.e. the BPS Lie algebra for the deformed potential is isomorphic to negative half of the usual Kac--Moody Lie algebra for $Q$.
It is an interesting question whether for more general quivers there is a quasihomogeneous deformation $\WT{W}+W_0$ of the standard cubic potential so that \eqref{KacOut} holds. A related question is: does the nonzero degree cohomology of the BPS Lie algebra $\fg_{\Pi_Q,W_0}$ vanish for generic deformations $W_0$? In other words, is the BPS Lie algebra for a generic deformed 3-Calabi--Yau completion \cite{Kel11} of the preprojective algebra $\Pi_Q$ always $\fn_Q^-$?
\subsection{The spherical Borcherds algebra}
In this section we construct a natural Lie algebra homomorphism $\Phi\colon\fg^{\exte,\SP}_{\Pi_Q}\rightarrow \fg^{\SP}_{\Pi_Q}$ from the positive half of a Borcherds algebra, extending the inclusion of the Kac--Moody Lie algebra from \S \ref{KacH}. In the case in which $\SP=\SSN$ the zeroth cohomologically graded piece of this morphism is the inclusion of the Borcherds--Bozec algebra. The existence of the morphism $\Phi$ serves as further evidence towards Conjecture \ref{mainConj}.
We introduce a little notation, in order to make the presentation fairly uniform. Given a tensor category $\mathscr{C}$ we denote by $\mathscr{C}_{\dvs}$ the category of $\dvs$-graded objects in $\mathscr{C}$. Given $F\in\mathscr{C}$ we denote by $\Lie(F)$ the free Lie algebra generated by $F$. I.e. we pick a symmetric monoidal embedding $\Vect\hookrightarrow \mathscr{C}$, and embed $\mathscr{C}\hookrightarrow \mathscr{C}_{\dvs}$ as the category of objects concentrated in degree zero, and thus consider $\mathcal{L}ie$ as an operad in $\mathscr{C}_{\dvs}$, and take the free algebra over it generated by $F$. We denote by $\Borch^+(F)$ the quotient of $\Lie(F)$ by the Lie ideal generated by the images of the morphisms
\begin{align}
\label{BorchRel}
(F_{\dd'}^{\otimes})^{1-(\dd',\dd'')_Q} \otimes F_{\dd''}\xrightarrow{[\cdot,[\cdot\ldots[\cdot,\cdot]\ldots]} F
\end{align}
over all pairs of dimension vectors $\dd', \dd''$ satisfying either of the conditions $(\dd',\dd'')_Q=0$ or $\dd'=1_i$ for $i\in Q_0$.
\begin{example}
Consider the vector space $V\in\Vect_{\dvs}$ which has basis $e_i$ for $i\in I_{\infty}$, where $e_{(i',n)}$ is given degree $n\cdotsh 1_{i'}$. Then
\[
\mathfrak{n}^+_Q=\Borch^+(V).
\]
\end{example}
For $i\in Q_0$ and $n\geq 1$ we denote by $\Delta_{i,n}\colon \Msp^{G}_{1_i}(\ol{Q})\rightarrow \Msp^{G}_{n\cdot 1_i}(\ol{Q})$ the embedding of the small diagonal.
\begin{proposition}
Let $\SP$ be a Serre subcategory of $\CC\ol{Q}\lmod$. Set
\begin{align*}
\Prim^{G}_{\Pi_Q,\sph}\coloneqq &\bigoplus_{i\in Q_0^{\reel}}\nnIC_{\Msp^{G}_{1_i}(\ol{Q})}\oplus \bigoplus_{\substack{i\in Q_0^{\isot}\\n\geq 1}}\Delta_{i,n,*}\nnIC_{\Msp^{G}_{1_i}(\ol{Q})}\oplus\bigoplus_{\substack{i\in Q_0^{\hype}\\n\geq 1}}\Cusp^{G}_{\Pi_Q,n\cdot 1_{i}}\\
\prim^{\SP,G}_{\Pi_Q,\sph}\coloneqq &\HO\!\varpi'_{\red,*}\varpi'^!_{\red}\Prim^{G}_{\Pi_Q,\sph}.
\end{align*}
There are morphisms of Lie algebra objects
\begin{align*}
J:\Borch^+\!\left(\Prim^{G}_{\Pi_Q,\sph}\right)\rightarrow &\BPSh_{\Pi_Q}^{G}\\
L^{\SP}\colon \Borch^+\!\left(\prim^{\SP,G}_{\Pi_Q,\sph}\right)\rightarrow &\fg^{\SP,G}_{\Pi_Q}.
\end{align*}
extending embeddings of $\Prim^{G}_{\Pi_Q,\sph}$ and $\prim^{\SP,G}_{\Pi_Q,\sph}$, respectively, where in the second morphism, $\prim^{\SP,G}_{\Pi_Q,\sph}$ is considered as an object of $\HG\lmod$.
\end{proposition}
\begin{proof}
The morphism $L^{\SP}$ is obtained as $\HO\!\varpi'_{\red,*}\varpi'^!_{\red}J$, so we concentrate on $J$.
Firstly, note that for $i\in Q_0$
\[
\BPSh^G_{\Pi_Q,1_i}=\nnIC_{\Msp^G_{1_i}(\ol{Q})}
\]
so that the first summand of $\Prim^{G}_{\Pi_Q,\sph}$ naturally embeds inside $\BPSh^G_{\Pi_Q}$. Secondly, as in Proposition \ref{IsoPure} there is an embedding (unique up to scalar) of $\Delta_{i,n,*}\nnIC_{\Msp^{G}_{1_i}(\ol{Q})}$ inside $\BPSh_{\Pi_Q,n\cdot 1_i}^G$ for each isotropic $i$. Thirdly, for $i$ hyperbolic the morphism \eqref{PreV} provides an embedding $\Cusp^{G}_{\Pi_Q,n\cdot 1_i}\subset \BPSh_{\Pi_Q}^{G}$. We claim that these embeddings induce the morphism $J$.
To prove the claim, we need to check the relation \eqref{BorchRel}. Note that if $i$ and $j$ are both real, this follows immediately from Proposition \ref{BPSvanProp}. Otherwise, we need something a little more subtle, i.e. the decomposition theorem.
Let $i\in I^{\reel}$, let $(j,n)\in I^{\Imag}$, and set $e=1-((i,1),(j,n))$. Set
\begin{align*}
\Msp_{i}&=\Msp^{G}_{1_i}(\ol{Q}),\\
\Msp_{j,n}&=\begin{cases}\Delta_{j,n}(\Msp_j)&\textrm{if }j\in Q_0^{\isot}\\ \Msp^{G}_{n\cdot 1_j}(\ol{Q})&\textrm{if }j\in Q_0^{\hype}\end{cases}
\end{align*}
Then we wish to show that the morphism
\[
J'\colon \underbrace{\nnIC_{\Msp_{i}}\boxtimes_{\oplus^G}\cdots\boxtimes_{\oplus^G}\nnIC_{\Msp_{i}}}_{e \textrm{ times}}\boxtimes_{\oplus^G}\nnIC_{\Msp_{j,n}}\rightarrow \BPSh_{\Pi_Q}^{G}
\]
given by the iterated Lie bracket (as in \eqref{BorchRel}) is the zero morphism. For this, we note that the morphism
\[
h\colon \underbrace{\Msp_{i}\times_{\B G} \cdots\times_{\B G}\Msp_i}_{e \textrm{ times}} \times_{\B G} \Msp_{j,n}\rightarrow\Msp_{e\cdot 1_i+\dd}(\Pi_Q)
\]
is injective, and so since $\nnIC_{\Msp_{i}}$ and $\nnIC_{\Msp_{j,n}}$ are simple, the domain of $J'$ is a simple object. We denote the domain of $J'$ by $\mathcal{R}$. Since by Theorem \ref{purityThm} the target of $J'$ is semisimple we deduce that $J'$ is nonzero only if there is a direct sum decomposition
\[
\BPSh_{\Pi_Q}^{G}\cong \mathcal{R}\oplus\mathcal{G}
\]
and $J'$ fits into a commutative diagram
\[
\xymatrix{
\ar[rd]_{\iota_{\mathcal{R}}}\mathcal{R}\ar[r]^-{J'}& \BPSh_{\Pi_Q}^{G}\\
&\mathcal{R}\oplus\mathcal{G}\ar[u]^{\cong}
}
\]
where $\iota_{\mathcal{R}}$ is the canonical inclusion.
For a contradiction, we assume that this is indeed so. Now we apply $\HO^0\!g_{*}g^!$, where
\[
g\colon \Msp^{\SSN,G}(\ol{Q})\rightarrow \Msp^G(\ol{Q})
\]
is the inclusion of the strictly semi-nilpotent locus. By \eqref{IsotDone} in the case of isotropic $j$, and Proposition \eqref{HypeOne} in the hyperbolic case, there is an isomorphism
\[
\HO^0\!g_{*}g^!\nnIC_{\Msp_{j,n}}\cong \QQ
\]
and so, since $\HO^0(\nnIC_{\Msp_i})\cong \QQ$ we deduce that
\[
\HO^0\!g_{*}g^!\mathcal{R}\cong\QQ,
\]
so that $\HO^0\!g_{*}g^!\iota_{\mathcal{R}}=\iota_{\QQ}\neq 0$. On the other hand,
\[
\HO^0\!\!g_{*}g^!J'\colon \QQ\rightarrow \fg^{\SSN}_{\Pi_Q}
\]
is the morphism taking $1\in\QQ$ to $[\alpha_i,\cdot]^e(\alpha_{j,n})=0$, with $\alpha_i$ and $\alpha_{j,n}$ defined as in \eqref{Fdef}, \eqref{Gdef}, \eqref{Hdef}. By Corollary \ref{absIso} we have
\[
[\alpha_i,\cdot]^e(\alpha_{j,n})=0
\]
and so $\xi=0$ after all.
Now assume that both $i$ and $j$ are imaginary, and $(1_i,1_j)_Q=0$. Fix $m\geq 1$. Then, similarly to above, we wish to show that the morphism
\[
\nnIC_{\Msp_{i,m}}\boxtimes_{\oplus^G}\nnIC_{\Msp_{j,n}}\rightarrow \BPSh^G_{\Pi_Q}
\]
provided by the Lie bracket in $\BPSh^G_{\Pi_Q}$ is the zero map. Again, applying $\HO^0\!g_*g^!$ this follows from Corollary \ref{absIso} and injectivity of the morphism
\[
\oplus^G\colon \Msp_{i,m}\times_{\B G} \Msp_{j,n}\rightarrow \Msp_{m\cdot 1_i+n\cdot 1_j}^G(\ol{Q}).
\]
We thus have defined the morphism $J$.
\end{proof}
It is of course very natural to make the following
\begin{conjecture}
The morphisms $J$ and $L^{\SP}$ are injective.
\end{conjecture}
The results of \S \ref{genSec} imply the conjecture in case there are no hyperbolic vertices.
In contrast with the Hall algebra $\HCoha_{\Pi_Q}^{\SSN}$, which is generated by the subspaces $\HCoha_{\Pi_Q,n\cdot 1_i}^{\SSN}$ by \cite[Prop.5.8]{ScVa20}, the Lie algebra $\fg^{\SP}_{\Pi_Q}$ is almost never generated by the subspaces $\fg^{\SP}_{\Pi_Q,n\cdot 1_i}$, so that $L$ is almost never surjective. For instance, if $Q$ has no edge loops, then the image of $L$ lies entirely in cohomological degree zero, while unless $Q$ is of finite type, $\fg_{\Pi_Q}$ will have pieces in strictly negative cohomological degree.
\subsection{The main conjecture for BPS Lie algebras}
\label{conjecturesSec}
We finish the paper with our main conjecture regarding the structure of BPS Lie algebras for preprojective CoHAs. Put informally, the conjecture states that we have found \textit{all} of the generators of $\fg_{\Pi_G}^{\SP}$. To state the conjecture fully, we make the following definitions.
First fix a Serre subcategory $\SP$, a stability condition $\zeta\in\QQ^{Q_0}$, and $\theta\in\QQ$. Set
\[
C=\{\dd\in\dvst \colon \Msp^{\zeta\stab}_{\dd}(\Pi_Q)\neq \emptyset\}.
\]
We set
\begin{align*}
\CStab^{G,\zeta}_{\Pi_Q,\theta}\coloneqq &\bigoplus_{\dd\in C}\nnIC_{\Msp^{G,\zeta\sst}_{\dd}(\Pi_Q)}\\
\fstab^{\SP,G,\zeta}_{\Pi_Q,\theta}\coloneqq &\HO\!\varpi'_{\red,*}\varpi'^!_{\red}\CStab^{G,\zeta}_{\Pi_Q,\theta}.
\end{align*}
We consider $\fstab^{\SP,G,\zeta}_{\Pi_Q,\theta}$ as a $\HG$-module below. By \S \ref{genSec} there are inclusions
\[
\CStab^{G,\zeta}_{\Pi_Q,\theta}\hookrightarrow \BPSh^{G,\zeta}_{\Pi_Q,\theta}.
\]
In the case in which the stability condition is trivial and $\theta=0$, $\HO$ of this inclusion is the inclusion of all the real simple roots, as well as cuspidal cohomology. To cover isotropic generators, we define
\[
E=\{n\cdot \dd\colon (\dd,\dd)_Q=0, \;\dd\in C, \;n\geq 2\}.
\]
By \cite{CB01}, $E\cap C=\emptyset$. Arguing as in \S \ref{AffineSec}, for primitive $\dd\in E$ and $n\geq 2$ there are embeddings
\[
\Delta_{n,\dd,*}\nnIC_{\Msp^{G,\zeta\sst}_{\dd}(\Pi_Q)}\hookrightarrow \BPSh^{G,\zeta}_{\Pi_Q,\theta},
\]
where $\Delta_{n,\dd}\colon \Msp^{G,\zeta\sst}_{\dd}(\Pi_Q)\hookrightarrow \Msp^{G,\zeta\sst}_{n\cdot \dd}(\Pi_Q)$ is the diagonal embedding. Accordingly, we define
\begin{align*}
\CIsot^{G,\zeta}_{\Pi_Q,\theta}\coloneqq &\bigoplus_{\substack{n\cdot \dd\in E\\ \dd\textrm{ primitive}}}\Delta_{n,\dd,*}\nnIC_{\Msp^{G,\zeta\sst}_{\dd}(\Pi_Q)}\\
\fisot^{\SP,G,\zeta}_{\Pi_Q,\theta}\coloneqq &\HO\!\varpi'_{\red,*}\varpi'^!_{\red}\CIsot^{G,\zeta}_{\Pi_Q,\theta}.
\end{align*}
We can now state the main conjecture:
\begin{conjecture}
\label{mainConj}
The above inclusions extend to an isomorphism of Lie algebra objects in $\MHM^G(\Msp^{\zeta\sst}_{\theta}(\ol{Q}))$
\[
\Borch^+\!\left(\CStab^{G,\zeta}_{\Pi_Q,\theta}\oplus \CIsot^{G,\zeta}_{\Pi_Q,\theta}\right)\cong \BPSh_{\Pi_Q,\theta}^{G,\zeta}.
\]
Applying $\HO\!\varpi'_{\red,*}\varpi'^!_{\red}$, we obtain isomorphisms
\[
\Borch^+\!\left(\fstab^{\SP,G,\zeta}_{\Pi_Q,\theta}\oplus \fisot^{\SP,G,\zeta}_{\Pi_Q,\theta}\right)\cong\fg^{\SP,G,\zeta}_{\Pi_Q,\theta}.
\]
\end{conjecture}
Setting $\zeta=(0,\ldots,0), \theta=0,\SP=\CC\ol{Q}\lmod$ and $G=\{1\}$ this conjecture implies the Bozec--Schiffmann conjecture on the Kac polynomials for $Q$, as well as giving a precise interpretation for the cuspidal cohomology.
\bibliographystyle{plain}
\bibliography{Literatur}
\vfill
\textsc{\small B. Davison: School of Mathematics, University of Edinburgh, James Clerk Maxwell Building, Peter Guthrie Tait Road, King's Buildings, Edinburgh EH9 3FD, United Kingdom}\\
\textit{\small E-mail address:} \texttt{\small [email protected]}\\
\end{document} | 154,533 |
You can get email updates from the site by using IFTTT:
- Click the following link and then choose Add:
IFTTT is a neat service that can let you know when things happen on the internet. In this case, when an article is published here, in the BLOG section of our site, it sends an email.
Compared to the old email which could only send once per day, this thing is lightning fast. I got my update from this post in just minutes.
You do have to sign up for IFTTT, but that is straightforward and free.
If you don't like IFTTT, you can use any service that watches a standard RSS feed and sends you email. I found the IFTTT way by reading some articles after doing this Google search for "Get email when RSS feed changes." You'll need our feed address:. | 22,750 |
TITLE: Compute $P(X_1=\min\{X_1,...,X_5\},X_5=\max\{X_1,...,X_5\})$ for $(X_i)$ i.i.d. uniformly distributed
QUESTION [3 upvotes]: In my exam preparation I'm currently stuck at this task:
Let $X_1,..,X_5$ be i.i.d random variables each having uniform distributions in the interval (0, 1)
Find the probability that $X_1$ is the minimum and $X_5$ is the maximum among these random variables.
$P(X_1 \leq$ min{$X_2$,...,$X_5$)
=$P(X_1 \leq X_2, X_1 \leq X_3, ..., X_1 \leq X_5)$
=$P(X_1 \leq X_2)\cdot ... \cdot P(X_1 \leq X_5)$
=$P(X_1 - X_2 \geq 0)\cdot ... \cdot P(X_1 - X_5 \geq 0)$
=$\frac{1}{2}^4$
This is my (wrong) calculation for now. I made the maximum with the same flawed idea in mind and I just cannot think of the correct answer*. Thanks for any hint!
*Per answersheet it is $\frac{1}{20}$. But I need to understand how to get there.
REPLY [2 votes]: Let us consider the general case of $n$ independent random variables (instead of 5), and let $x_1=a$ and $x_n=b$. Then the probability that all $n-2$ variables $x_2,x_3,...,x_{n-1}$ are in the interval $(a,b)$ is given by $(b-a)^{n-2}$.
Now $a$ can be any number between $0$ and $b$, hence the probability that $b$ is maximum is given by
$$p(n,\text{MAX} = b) = \int_0^b (b-a)^{n-2} \,da = \frac{b^{n-1}}{n-1}$$
Integrating over $b$ gives the final probability asked for in the OP:
$$p(n) = \int_0^1 \frac{b^{n-1}}{n-1} \,db = \frac{1}{n(n-1)}$$
For $n=5$ we recover the result $p(5) = \frac{1}{20}$. | 127,545 |
October 15, 2013
‘Hackathon’ designed to inspire innovation
CARBONDALE, Ill. -- Fill a room with computer enthusiasts and there’s no telling where all that innovative thinking may lead.
That is the whole idea behind the Hack SI “hackathon” event set for Nov. 2 at Southern Illinois University Carbondale. Computer programmers, students, graphic designers, project managers, interface designers -- actually anyone with an interest in computers or programming can participate, organizers say. No professional connection to the field is required. In fact, even children are welcome, although those younger than college age must be accompanied by a parent.
“It’s really going to be a lot of fun. We will get innovative and creative people together and see what they can do. We’ll have all of these ‘geeks’ in one place where they can build things and show them off. Participants can create webpages, robots, phone or tablet apps and all kinds of things,” Dav Glass, event coordinator, said.
The Hack SI event is set for 10 a.m. to 10 p.m. at the Dunn-Richmond Economic Development Center, 1740 Innovation Drive in Carbondale. Up to 150 people may enter the hackathon. That day, they can work individually or boost their brainpower and creativity by working in groups. There is no cost to participate but only the first 100 to pre-register at will get a T-shirt. Walk-ins are welcome, too.
Glass, a Yahoo NodeJS and Open Source architect, said these all-day events have produced some incredible technology innovations across the country, including social or educational software, “mashups” that combine two existing technologies for a new purpose, and countless other valuable products. Many of the developments have commercial applications as well, Glass, of Marion, said.
There is no set focus for this event. It is limited only by the imagination of the participants.
Experts in the field, including people from local and national companies, will speak and be on hand to offer technical support at Hack SI. There will be giveaways as well.
The schedule begins with presentations by industry experts about technology and how to build platforms and then participants have eight hours to see what they can do. Wrapping up the day, each individual or team will present an “elevator pitch,” a 90-second presentation to show and sell what each has built.
“We at SIU are excited to be hosting the first hackathon event ever in Southern Illinois,” Kyle Harfst, executive director of Economic Development and of the Southern Illinois Research Park, said.
Event sponsors include the Office of Economic and Regional Development and the Office of the Vice Chancellor for Research along with national companies including The Yahoo Developer Network, Github and TravisCI. Local businesses including Liaison and Splattered Ink are also sponsors.
“This event will showcase the intellectual curiosity we have in this region and will also provide the opportunity for networking beyond Nov. 2,” Harfst said.
Find all of the details about the event and online registration at. For additional information, email [email protected]. | 228,602 |
\begin{document}
\WSCpagesetup{Nezami and Anahideh}
\title{AN EMPIRICAL REVIEW OF MODEL-BASED ADAPTIVE SAMPLING FOR GLOBAL OPTIMIZATION OF EXPENSIVE BLACK-BOX FUNCTIONS}
\author{Nazanin Nezami\\[12pt]
Hadis Anahideh\\[12pt]
Department of Mechanical and Industrial Engineering\\
University of Illinois at Chicago\\
842 W Taylor St\\
Chicago, IL 60607, USA\\}
\maketitle
\section*{ABSTRACT}
This paper reviews the state-of-the-art model-based adaptive sampling approaches for single-objective black-box optimization (BBO). While BBO literature includes various promising sampling techniques, there is still a lack of comprehensive investigations of the existing research across the vast scope of BBO problems.
We first classify BBO problems into two categories: engineering design and algorithm design optimization and discuss their challenges.
We then critically discuss and analyze the adaptive model-based sampling techniques focusing on key acquisition functions.
We elaborate on the shortcomings of the variance-based sampling techniques for engineering design problems. Moreover, we provide in-depth insights on the impact of the discretization schemes on the performance of acquisition functions. We emphasize the importance of dynamic discretization for distance-based exploration and introduce \emph{EEPA$^{+}$}, an improved variant of a previously proposed Pareto-based sampling technique.
Our empirical analyses reveal the effectiveness of variance-based techniques for algorithm design and distance-based methods for engineering design optimization problems.
\input{introduction}
\input{problems}
\input{preliminaries}
\input{acquisitions}
\input{EEPA+}
\input{experiments}
\input{Result}
\footnotesize
\bibliographystyle{wsc}
\bibliography{demobib}
\section*{AUTHOR BIOGRAPHIES}
\noindent {\bf Nazanin Nezami} is a PhD Student in Mechanical and Industrial Engineering Department at the University of Illinois at Chicago (UIC). She obtained her M.S. degree in Industrial and Systems Engineering from University of Minnesota Twin Cities prior to joining UIC. Her main research interests are Black-Box Optimization, Machine Learning (ML), and Fairness in ML. \\
\noindent {\bf Hadis Anahideh} is a Research Assistant Professor of the Mechanical and Industrial Engineering Department at the University of Illinois at Chicago. She received her Ph.D. degree in Industrial Engineering from the University of Texas at Arlington. Her research objectives center around Black-box Optimization, Sequential Optimization, Active Learning, Statistical Learning, Explainable AI, and Algorithmic Fairness.\\
\newpage
\end{document} | 62,981 |
FULTON, Connie, 73, of Cadiz, died Wednesday. Friends may call Friday, 2-4 and 6-8 p.m. at Clark- Kirkland Funeral Home, Cadiz, where services will be held Saturday at 11 a.m.
200 S. Fourth St. , Martins Ferry, OH 43935 | 740-633-1131
© 2016. All rights reserved.| Terms of Service and Privacy Policy | 356,671 |
Every once in a while, I hear people reference the part of the Ghostbuster’s movie that talks about not crossing the streams. Usually, the people making the joke are making some kind of joke that involves a visit to the bathroom. It is that sense of humor that has inspired this awesome Ghostbusters shirt that shows the four Ghostbusters using the restroom. This is truly the shirt for any Ghostbuster’s fans with their minds in the toilet! You can order this awesome and hilarious Ghostbusters Don’t Cross The Streams t-shirt for $14.99 from the awesome guys over at Tshirtbordello.
Pass this along to any Ghostbusters fans you know! Like us on Facebook too!
[Source: Tshirtbordello] | 417,124 |
\begin{document}
\title{Characterization of The Generic Unfolding of a Weak Focus}
\pagestyle{headings}
\noindent
\author{ by\\ W. Arriagada-Silva \ \,\\ \,\\ \,\\ \, \\ \,\\ \, \\ \,\\ \, }
\begin{abstract}
In this paper we give a geometric description of the foliation of a generic real analytic family unfolding a real analytic vector field with a weak focus at the origin, and show that two such families are orbitally analytically equivalent if and only if the families of diffeomorphisms unfolding the complexified Poincar\'e map of the singularities are conjugate. Moreover, by shifting the leaves of the formal normal form in the blow-up (quasiconformal surgery) by means of a fibered transformation along a convenient complex cross-section, one constructs an abstract manifold of complex dimension 2 equipped with an elliptic holomorphic foliation whose monodromy map coincides with a given family of admissible diffeomorphisms.
\end{abstract}
\noindent
\bibliographystyle{plain}
\address{{\it W. Arriagada:} D\'epartement de math\'ematiques et de statistique,
Facult\'e des arts et des sciences -Secteur des sciences, Universit\'e de Montr\'eal\\
succ. Centre-ville\\
Montr\'eal, Qc\\
H3C 3J7.}
\email{[email protected]}
\date{\today}
\maketitle
\markboth{ W. Arriagada Silva}{Characterization of the generic unfolding of a weak focus}
\section{Introduction.}\label{intro}
A one-parameter family of real analytic planar systems unfolding a weak focus is an elliptic real analytic one-parameter $\ep\in\RR$ dependent family linearly equivalent to a family of planar differential equations
\begin{equation}
\left.
\begin{array}{lll}
\dot x &=& \alpha(\ep)x -\beta(\ep)y + \sum_{j+k\geq 2} b_{jk}(\ep)x^jy^k\\
\dot y &=& \beta(\ep)x + \alpha(\ep)y + \sum_{j+k\geq 2} c_{jk}(\ep)x^jy^k,
\end{array}\label{weakfocus00}
\right.
\end{equation}
for real time, and with $\alpha(0)=0$ and $\beta(0)\neq 0.$ After rescaling the time $t\mapsto\beta(\ep)t$ we can suppose $\beta(\ep)\equiv 1.$ The family \eqref{weakfocus00} is called ``generic'' if $\alpha'(0)\neq 0.$ The genericity allows to take $\alpha$ as the new parameter, so that the eigenvalues become $\ep+i$ and $\ep-i,$ respectively.\\
When the order is one, a weak focus of a real analytic vector field corresponds to the coalescence of a focus with a limit cycle, and the generic family \eqref{weakfocus00} is then a family with a generic Hopf bifurcation, whose foliation is described by the unfolding of the Poincar\'e map or monodromy $\Pp_{\ep}:(\RR^+,0)\to (\RR^+,0)$ of the system. It is well known that the germ of the Poincar\'e return map or monodromy is well defined and analytic, and can be extended to an analytic diffeomorphism
\begin{equation}
\begin{array}{lll}
\Pp_{\ep}:(\RR,0)\to (\RR,0).
\end{array}\label{usupm}
\end{equation}
A question that arises naturally is whether the germ of the monodromy map defines the analytic equivalence class of the real foliation. The natural way to answer this question is via complexification (cf. \cite{bcl}). The right hand side of the complexified system is now defined by an analytic family of vector fields
\begin{equation}
\begin{array}{lll}
v_{\ep}(x,y)= P(x,y)\frac{\partial}{\partial x} + Q_{\ep}(x,y)\frac{\partial}{\partial y}
\end{array}\label{weakfocus0}
\end{equation}
that satisfies
\begin{equation}
\begin{array}{lll}
P_{\ep}(x,y) = \overline{Q_{\overline{\ep}}(\overline y,\overline x)},
\end{array}\label{realchar}
\end{equation}
where $x\mapsto \overline x$ is the complex conjugation. The time is complexified as well and the domain of the parameter is now a standard open complex disk noted $V\in\CC.$ After complexification, the real plane can be written in a rather simple way: it corresponds to the surface $\{x=\overline y\}.$ The Poincar\'e map of the complexified system (parametrized with $x$-coordinate) is defined as the second iterate of the holonomy $\Qq_{\ep}$ along the loop $\RR P^1$ (the equator of the exceptional divisor) of the foliation after standard blow-up (cf. \cite{YY}), where the standard affine coordinates on the projective line $\CC P^1$ are given by formulas with real coefficients, hence defining correctly the real projective equator $\RR P^1\subset\CC P^1,$ see Figure \ref{monoPoincare}. Blowing down the foliation, the Poincar\'e map is defined on the $1$-dimensional complex cross-section $\{x=y\},$ and the usual real germ \eqref{usupm} of the planar system is defined on $\{x=y\}\cap\{x=\overline y\}.$\\
\noindent{\emph{Notation.}} The cross section $\{x=y\}$ is noted $\Sigma$ and is parametrized with the complex coordinate $x.$\\
The complex description of the monodromy immediately allows to prove its analyticity, even at the origin. The monodromy is then a real holomorphic germ of resonant diffeomorphism with a fixed point of multiplicity 3 at the origin, which corresponds in the limit $\ep=0$ to the coalescence of a fixed point with a 2-periodic orbit: the fixed point and periodic orbit bifurcate in a generic unfolding.
\begin{figure}[!h]
\begin{center}
{\psfrag{=}{\huge{$\Sigma$}}
\psfrag{a}{\huge{$w^*$}}
\psfrag{q}{\huge{$\Qq(w^*)$}}
\psfrag{e}{\huge{$\RR P^1$}}
\psfrag{c}{\huge{$\CC P^1$}}
\psfrag{z}{\huge{$ $}}
\psfrag{w}{\huge{$ $}}
\psfrag{x}{\huge{$x^*$}}
\psfrag{p}{\huge{$\Pp(x^*)$}}
\psfrag{qq}{\huge{$\Qq(x^*)$}}
\psfrag{o}{\huge{$0$}}
\psfrag{R}{\huge{$\RR$}}
\scalebox{0.40}{\includegraphics{bupco.eps}}
}
\end{center}
\caption{\label{monoPoincare} \small{The complexification of the real line and its blow-up.}}
\end{figure}
\noindent{\emph{The equivalence problem.}} It is known that the problem of orbital equivalence for germs of analytic vector fields with a resonant saddle point is reduced to the conjugacy problem for germs of diffeomorphisms (the holonomy map) with a fixed point at the origin and multiplier on the unit circle (cf. \cite{MM}). In the non-resonant case, the statement holds as well, as was shown by R. P\'erez-Marco and J.-C. Yoccoz (cf. \cite{peyo}). Furthermore, this result has been extended to generic analytic families unfolding a resonant saddle point (cf. \cite{cri-rou}).
\begin{definition}
An analytic orbital equivalence (resp. conjugacy) between two analytic germs of families unfolding germs of analytic vector fields (resp. diffeomorphisms) is said to be ``real'', when it leaves invariant the real plane (resp. the real line) for real values of the parameter.
\end{definition}
\noindent In this paper, we show that the equivalence problem for \eqref{weakfocus0} can be reduced to the conjugacy problem for the associated family unfolding the complexified Poincar\'e map, respecting the underlying real foliation. More precisely,
\begin{theorem}\label{mmexten-unfolding}
Two germs of generic families of real analytic vector fields \eqref{weakfocus0} are analytically orbitally equivalent by a real change of coordinates, if and only if the families unfolding their Poincar\'e maps are analytically conjugate by a real conjugacy.
\end{theorem}
\noindent{\emph{The realization problem.}} A second related problem consists in recovering the germ of the analytic foliation when the Poincar\'e map has been prescribed. This is the problem of ``realization''.
We give an answer to this problem by means of the desingularization technique and quasiconformal surgery, as suggested by Y. Ilyashenko (cf. \cite{Yl}): for every $\ep\in V,$ one constructs, with the help of an adequate partition of the unity depending only on the argument of the coordinate induced in the separatrix (the exceptional divisor) by the desingularization process, a fibered $C^{\infty}$ transformation or ``sealing map'' defined on a semi-disk. By shifting the leaves of the normal form with the help of the sealing map, one obtains a $C^{\infty}$ foliation over the product $\CC^*\times\DD_{r},$ and an integrable almost complex structure, making the foliation actually holomorphic. The almost complex structure extends smoothly along the vertical axis, because the sealing is, by definition, infinitely tangent to the identity. It remains integrable after the extension. The Newlander-Nirenberg Theorem yields a $C^{\infty}$ real system of coordinates (depending analytically on $\ep)$ that straightens the almost complex structure, and therefore, the $C^{\infty}$ foliation in a holomorphic foliation that extends by Riemann along the vertical axis. The blow down of such a foliation is the required generic elliptic family.\\
In this second part we deal with formal normal forms. Normal form theory provides an algorithmic way to decide
whether two germs of planar vector fields are equivalent under a $C^{N}$-change of coordinates (cf. \cite{crnf}), in which case, the normal forms are polynomial. However, in the analytic case, the
formal change of coordinates to normal form generically diverges (cf. \cite{ynf}). An
explanation of this is found by considering unfoldings of the vector fields
and explaining the divergence in the limit process. This is a particular manifestation of the so-called Stokes Phenomenon (cf. \cite{Yl}). The spirit of the general answer is the following (cf. \cite{crnf}). The dynamics of the
original system is extraordinarily rich to be encoded in the simple dynamics
of the normal form which depends of at most one parameter. Hence the divergence
of the normalizing series.
\section{Proof of Theorem \ref{mmexten-unfolding}.}
The proof uses basically the classical fact that the holonomy characterizes the differential equation (cf. \cite{MM} and \cite{MR}), plus an additional ingredient: both the equivalence between vector fields and the conjugacy between Poincar\'e maps, must respect the real foliation.\\
By definition, if two families \eqref{weakfocus0} are orbitally equivalent by an analytic change of coordinates $\Psi_{\ep}$ (depending analytically on the parameter), it is always possible to reparametrize the families and suppose that they have the same parameter. Thus, one direction is obvious: if two families of vector fields are equivalent by real change of coordinates, then the equivalence induces a real analytic return map on the image $\Psi_{\ep}(\Sigma),$ for each value of $\ep$ over a small neighborhood of the origin. Because the equivalence is real, the image of the real line under the equivalence is a real analytic curve $\Cc\subset\Psi_{\ep}(\Sigma)$ different, in general, to $\RR,$ see Figure \ref{reconpoin}. Standard transversality arguments and the Implicit Function Theorem show that there exists an analytic local transition map $\pi$ between $\Sigma$ and $\Psi_{\ep}(\Sigma)$ (cf. \cite{wacr}). By unicity, any real local trajectory passing through a real point in $\Sigma$ intersects the image $\Psi_{\ep}(\Sigma)$ in a real point. Thus, the transition is real and it sends the curve $\RR$ into $\Cc,$ and the composition $\pi^{-1}\circ\Psi_{\ep}$ provides a real conjugacy between Poincar\'e maps $\Sigma\to\Sigma.$\\
\begin{figure}[!h]
\begin{center}
{\psfrag{a}{\Huge{$\Sigma$}}
\psfrag{b}{\Huge{$\RR^2$}}
\psfrag{psi}{\Huge{$\Psi_{\ep}$}}
\psfrag{pi}{\Huge{$\pi$}}
\psfrag{ee}{\Huge{$\RR$}}
\psfrag{gg}{\Huge{$\Cc$}}
\psfrag{ff}{\Huge{$\RR$}}
\psfrag{c}{\Huge{$\Sigma$}}
\psfrag{d}{\Huge{$\Psi_{\ep}(\Sigma)$}}
\scalebox{0.32}{\includegraphics{reconpoin.eps}}
}
\end{center}
\caption{\label{reconpoin} \small{The real line and its image by the equivalence $\Psi_{\ep}.$}}
\end{figure}
Let us show the converse. The conjugacy between the Poincar\'e maps provides a reparametrization, so we can suppose that the parameter is the same for the two families of diffeomorphisms and is henceforth noted $\ep.$ We will suppose that the real conjugacy $\Hhh_{\ep}(x)=\Hhh(\ep,x)$ depends on the $x$-variable and is defined on $\DD_{\rho}\subset\Sigma,$ for every $\ep\in V,$ where $\DD_{\rho}\subset\Sigma$ is the standard open disk of the complex plane, of small radius ${\rho}>0.$
A theorem on the existence of invariant analytic manifolds (cf. \cite{YY},\cite{MM}) ensures that it suffices to show the theorem for Pfaffian $1$-forms $$\omega_{\ep}, \widehat{\omega}_{\ep} = (\ep+i)xdw - (\ep-i) y(1+xy(...))dx$$ before desingularization. So if the blow-up space is equipped with coordinates $(X,y)$ and $(x,Y),$ where the standard monoidal map blows down as
\begin{equation}
\begin{array}{lll}
c_1:(X,y)\mapsto(Xy,y),\\
c_2:(x,Y)\mapsto(x,xY)
\end{array}\label{bdown}
\end{equation}
respectively in each direction, the pullback of $\omega_{\ep}$ is defined by $$\omega_1=Xdy-\lambda(\ep) y(1+A_{\ep}(X,y))dX$$ in $(X,y)$ variables, and by $$\omega_2=Ydx-\lambda'(\ep) x(1+A'_{\ep}(x,Y))dY$$ in $(x,Y)$ coordinates, where $A_{\ep}(X,y)=O(Xy)$ and $A'_{\ep}(x,Y)=O(xY)$ depend analytically on the parameter and are holomorphic on a neighborhood $\CC^*\times\DD_{s}$ of the exceptional divisor, for each fixed value of $\ep.$ The numbers $\lambda(\ep)=(\ep-i) / 2i$ and $\lambda'(\ep)=-(\ep+i) / 2i$ are the ratios of eigenvalues of the singular points $(X,y)=(0,0)$ and $(x,Y)=(0,0),$ respectively. In addition, the coordinates can always be scaled before blow-up, to ensure:
\begin{equation}
\begin{array}{lll}
|A_{\ep}(X,y)|,|A'_{\ep}(x,Y)|< 1/2
\end{array}\label{Func-A}
\end{equation}
in $\CC^*\times\DD_s.$ Notice that in complex coordinates, the section $\Sigma$ is parametrized as $\{X=1\}$ in the $(X,y)$ chart, and as $\{Y=1\}$ in the $(x,Y)$ chart. Bounded equivalences $\widehat{\Psi}_{\ep}^{c_1},\widehat{\Psi}_{\ep}^{c_2}$ are constructed in $(X,y)$ and $(x,Y)$ variables, in such a way that they are analytic continuations of each other over a neighborhood of the exceptional divisor.
\subsection{The equivalence in the $(X,y)$ chart.}
Take a point $y^*\in\DD_{\rho}.$ A former equivalence $\widehat{\Psi}_{\ep}^{c_1}$ is defined on $\Sigma\times\DD_{\rho}$ by $$\widehat{\Psi}_{\ep}^{c_1}:(1,y^*)\mapsto (1,\Hhh_{\ep}(y^*)).$$ This change of coordinates is extended along a subset of $\Sb^1\times\CC$ in the following way. Notice that the restriction of the form $\omega_1$ to the cylinder $\RR P^1\times\RR^2$ (noted $\{|X|=1\})$ is non-singular and holomorphic, thus it defines a local holomorphic foliation $\Ff_{\omega_1}$ there. Consider (cylindrical) solutions to $\omega_1=0$ (the first coordinate is to be parametrized by $X=e^{i\theta},$ $\theta\in[0,2\pi]).$
\begin{lemma}\label{scylsol}
Any (cylindrical) solution $\uu$ to
\begin{equation}
\begin{array}{lll}
\uu'=\lambda(\ep)\uu(1+A(e^{i\theta},\uu)),\quad\theta\in[0,2\pi]
\end{array}\label{diffeq-u_1(theta)}
\end{equation}
satisfies $|\uu(0)|e^{-\theta \left\{|\ep|+\frac{1}{4}\right\}}<|\uu(\theta)|<|\uu(0)|e^{\theta \left\{|\ep|+\frac{1}{4}\right\}},$ for any $\theta\in(0,2\pi].$
\end{lemma}
\begin{proof}
The parameter is written as $\ep=\ep_1+i\ep_2,$ with $\ep_1,\ep_2\in\RR.$ As we consider solutions in $|X|=1,$ the time is parametrized by $t=i\theta,$ and then \eqref{diffeq-u_1(theta)} implies $$d\ln\uu=\frac{1}{2}(\ep-i)(1+A_{\ep}(e^{i\theta},\uu))d\theta.$$ Thus, after taking real parts and using the hypothesis \eqref{Func-A} we get, for $\theta\neq 0:$
\begin{equation*}
\begin{array}{lll}
\left|\ln\left|\frac{\uu}{\uu(0)}\right|\right| &\leq& \frac{1}{2}\int_{0}^{\theta} \{|\ep_1|(1+|Re(A_{\ep})|)+|Im(A_{\ep})|(1+|\ep_2|)\} d\theta\\
&<& \frac{1}{2}\int_{0}^{\theta} \{2|\ep| + 1 /2\} d\theta = \theta \left\{|\ep|+ 1/4\right\},\\
\end{array}
\end{equation*}
and the conclusion follows.
\end{proof}
Put $r=\rho e^{-\pi}.$ We denote by $\SN_{r}$ the set of (cylindrical) solutions $\uu$ to \eqref{diffeq-u_1(theta)} for which there exists $\theta_0\in[0,2\pi)$ such that $\uu(\theta_0)\in\DD_{r}.$
\begin{corollary}\label{solutcyl}
If $\uu\in\SN_r,$ then $\uu(0)\in\DD_{\rho},$ provided $|\ep|< 1 /4.$
\end{corollary}
This is how the equivalence is extended. Choose a point $(e^{i\theta_0},y_0)\in\Sb^1\times\DD_r.$ By definition, the path $\gamma:(e^{i\theta},0)$ is lifted in the leaf of $\Ff_{\omega_1}$ containing $y_0\in\DD_{r}$ as $(e^{i\theta},\uu(\theta)),$ for a certain $\uu\in\SN_{r}$ and $\uu(\theta_0)=y_0.$ By Corollary \ref{solutcyl}, the point $\widetilde y:=\uu(0)$ belongs to $\DD_{\rho}.$ If $\gamma$ is lifted in the leaf of $\Ff_{\widehat{\omega}_1}$ passing through $\Hhh_{\ep}(\widetilde y)$ as $(e^{i\theta},\uuu(e^{i\theta},\widetilde y)),$ with $\uuu(1,\widetilde y)=\Hhh_{\ep}(\widetilde y),$ then we define the analytic change of variables by:
\begin{equation}
\begin{array}{lll}
\widehat{\Psi}_{\ep}^{c_1}:\Sb^1\times\DD_r\to\Sb^1\times\CC,\\
\widehat{\Psi}_{\ep}^{c_1}:(e^{i\theta_0},\uu(\theta_0))\mapsto(e^{i\theta_0},\uuu(e^{i\theta_0},\widetilde y)).
\end{array}\label{PsiZ}
\end{equation}
The change \eqref{PsiZ} respects the transversal fibration given by $X=const.$ and is clearly the restriction of a (unique) holomorphic diffeomorphism conjugating $\Ff_{\omega_1}$ and $\Ff_{\widehat{\omega}_1}$ in a neighborhood of $\Sb^1\times\DD_r.$ Moreover, it extends analytically to $\DD_1\times\DD_r$ (where $\DD_1$ is the standard unit (closed) disk of the $X$-separatrix) by means of the lifting of radial paths
\begin{equation*}
\begin{array}{lll}
\gamma_{X_1} : [0,-\log |X_1|] \to \CC,\quad s \mapsto \gamma_{X_1}(s)=(X_1 e^{s},0)
\end{array}\label{radpathZ}
\end{equation*}
for $0<|X_1|<1.$ In fact, suppose that this curve lifts in the leaves of $\Ff_{\omega_1}$ as $$\gamma_{X_1,y_1}:s\mapsto (X_1e^{s},\rr(s,y_1)),\quad \rr(0,y_1)=y_1,$$ for a given $y_1$ small. Then the solution $\rr(\cdot,y_1)$ of $\omega_1=0,$ with parameter $0<|X_1|<1,$ and initial condition $\rr(0,y_1)=y_1$ is defined on $[0,-\log|X_1|].$ Actually, the hypothesis \eqref{Func-A} shows that
\begin{equation}
\begin{array}{lll}
|\rr|\leq |y_1|e^{s\left\{|\ep|-\frac{1}{4}\right\}}< |y_1|,
\end{array}\label{voodoopre}
\end{equation}
whenever $|\ep|<1/4.$ We will suppose that the inverse path of $\gamma_{X_1}$ lifts in the leaf of $\Ff_{\widehat{\omega}_1}$ through the point $(\frac{X_1}{|X_1|},y^0),$ where $y^0$ is small, as
\begin{equation*}
\begin{array}{lll}
\gamma_{X_1,y^0}^{-1}: s \mapsto (X_1 e^{-(s+\log|X_1|)},\Tr(s,y^0)),\quad s\in[0,-\log|X_1|].
\end{array}\label{radpathZ-1}
\end{equation*}
Consider the only cylindrical solution $\mathbf{u}_{\mathbf 1,X_1,y_1}$ to \eqref{diffeq-u_1(theta)} satisfying $\mathbf{u}_{\mathbf 1,X_1,y_1}(\arg X_1)=\rr(-\log|X_1|,y_1)$ and define the coordinate $$\widetilde y(X_1,y_1):=\mathbf{u}_{\mathbf 1,X_1,y_1}(0)\in\Sigma.$$ Then, \eqref{voodoopre} proves that $\mathbf{u}_{\mathbf 1,X_1,y_1}\in\SN_r$ if $y_1$ is taken in $\DD_r.$ In this case, Corollary \ref{solutcyl} ensures that $\widetilde y(X_1,y_1)$ belongs to $\DD_{\rho}.$ The equivalence is then defined by
\begin{equation}
\begin{array}{lll}
\widehat{\Psi}_{\ep}^{c_1} : (X_1,y_1)\mapsto(X_1,\rrr(X_1,y_1)),
\end{array}\label{Equivalence-Zdisk}
\end{equation}
with $\rrr(X_1,y_1)=\Tr(-\log|X_1|,\uuu(e^{i\arg(X_1)},\widetilde y(X_1,y_1)))$ $(\uuu$ given in \eqref{PsiZ}). As the change of coordinates is bounded, the Riemann's removable singularity Theorem implies the existence of a unique holomorphic extension $\widehat{\Psi}_{\ep}^{c_1}$ to $\DD_1\times\DD_r.$\\
Finally, the change of coordinates \eqref{Equivalence-Zdisk} extends to a subset $$\Dd_1(r) = \{(X,y)\in\CC \times\CC : |X|\geq 1, |Xy|\leq r \}$$ as follows. Similar arguments as those used above show that the only tangent curve $\rl(\cdot,y_1)$ to $\omega_1$ verifying $\rl(\log|X_1|,y_1)=y_1,$ for a given $(X_1,y_1)\in\Dd_1(r),$ satisfies $$|\rl(0,y_1)|e^{-s\{|\ep|+1/4\}} < |\rl(s,y_1)|,\quad s\in[0,\log|X_1|],$$ so that the initial condition $\rl(0,y_1)$ of the lifting starting at $(\frac{X_1}{|X_1|},\rl(0,y_1))$ belongs to $\DD_r$ provided $|\ep|\leq 3/4.$ Thus, the leaf containing the point $(X_1,y_1)$ intersects the cylinder $\{|X|=1\}$ in a curve $\uu=\uu(\theta)\in\SN_r,$ with $\uu(\arg X_1)=\rl(0,y_1)\in\DD_r.$ By Corollary \ref{solutcyl}, $\uu(0)\in\DD_{\rho}$ and then
$\widehat{\Psi}_{\ep}^{c_1}(\frac{X_1}{|X_1|},\rl(0,y_1))$ is well defined, where $\widehat{\Psi}_{\ep}^{c_1}$ is the equivalqnce \eqref{Equivalence-Zdisk}. In $\Ff_{\widehat{\omega}_1},$ the inverse of $\gamma_{X_1}$ is lifted on the leaf passing through the point $\widehat{\Psi}_{\ep}^{c_1}(\frac{X_1}{|X_1|},\rl(0,y_1)).$ The endpoint of this radial lifting defines $\widehat{\Psi}_{\ep}^{c_1}$ on $\Dd_1(r).$
\subsection{The equivalence in the $(x,Y)$ chart.}
If $\DD_2$ is the standard unit (closed) disk of the $Y$-separatrix and $\Dd_2(r) = \{(x,Y)\in\CC\times\CC : |Y|\geq 1, |xY|\leq r \},$ then, in $(x,Y)$ coordinates the equivalence is defined plainly on $(\DD_2^*\times\DD_r)\cup\Dd_2(r),$ by the formula $$\widehat{\Psi}_{\ep}^{c_2}:=\varphi\circ\widehat{\Psi}_{\ep}^{c_1}\circ\varphi^{\circ -1},$$ where $\varphi: (X,y)\mapsto (x,Y)$ is the transition between complex charts.
Such equivalence is clearly bounded and the Riemann's Theorem yields a unique holomorphic extension $\widehat{\Psi}_{\ep}^{c_2}:(\DD_2\times\DD_r)\cup\Dd_2(r)\mapsto\CC^2.$
It turns out that the two changes of coordinates thus obtained $\widehat{\Psi}_{\ep}^{c_1},\widehat{\Psi}_{\ep}^{c_2}$ are analytical continuations of each other on $\CC P^1\times\DD_r,$ yielding a well defined and holomorphic global change of coordinates $\widehat{\Psi}_{\ep}$ over the divisor which is, by construction, a local equivalence between $\Ff_{\omega_{\ep}}$ and $\Ff_{\widehat{\omega}_{\ep}}$ around $\Sb^1\times\CC.$ It depends holomorphically on $\ep\in V$ by the analytic dependence on initial conditions of a differential equation. Let $\Psi_{\ep}$ stand for this diffeomorphism in $(x,y)$ variables. Since the Riemann sphere $\CC P^1$ retracts to the origin, the equivalence $\Psi_{\ep}$ is defined on $(\DD_{r}\times\DD_{r})\backslash \{(0,0)\}$ and is analytic there, because the monoidal map is an isomorphism away from the exceptional divisor. By Hartogs Theorem, $\Psi_{\ep}$ can be holomorphically extended until the origin.
Inasmuch as the equivalence $\Psi_{\ep}$ is constructed by lifting paths, and both the holonomy and the conjugacy $\Hhh_{\ep}$ are real (when $\ep\in\RR)$, the change of coordinates $\Psi_{\ep}$ is real as well.
\section{Realization of an admissible family.}\label{rrffweo1}
A first change of coordinates on the complexified family \eqref{weakfocus0}, depending analytically on small values of the parameter, allows to get rid of all cubic terms except for the resonant one (Poincar\'e normal form). The weak focus is of order one if the real part of the coefficient of the third order resonant monomial is non null. The sign $s=\pm 1$ of such a coefficient defines two different cases which are not equivalent by real equivalence. In fact, $s$ is an analytic invariant of the system. An analytic change of coordinates (cf. \cite{wacr}) brings the Poincar\'e map to the ``prepared'' form
\begin{equation}
\begin{array}{lll}
\Pp_{\ep}(x)= x + x(\ep+s x^2)(2\pi+O(\ep)+O(x)),
\end{array}\label{pforPpp}
\end{equation}
with multiplier $\exp(2i\pi)$ at the origin.
\begin{proposition}\label{formal-class-teo}
A germ of generic real analytic family of differential equations unfolding a germ of real analytic weak focus of order one, is formally orbitally equivalent to:
\begin{equation}
\begin{array}{lll}
\dot x &=& x(i+(\ep\pm u)(1 - A(\ep)u))\\
\dot y &=& y(-i+(\ep\pm u)(1 - A(\ep)u))
\end{array}\label{formclawf}
\end{equation}
with $u=xy,$ for some family of constants $A(\ep)$ which is real on $\ep\in\RR$ and $A(0)\neq 0.$ The parameter $\ep$ of the formal normal form \eqref{formclawf} is called the ``canonical parameter''.
\end{proposition}
\begin{proof}
Consider the case $s = + 1.$ By a formal change of coordinates we bring the system to the form:
\begin{equation}
\begin{array}{lll}
\dot x &=& x(i+\ep - \sum_{j\geq 1} A_j(\ep)u^j):=P(x,y)\\
\dot y &=& y(-i+\ep - \sum_{j\geq 1}\overline{ A_j(\overline{\ep})}u^j):=Q(x,y)
\end{array}\label{ForClas}
\end{equation}
where $Re ( A_1)\neq 0.$ In order to simplify the form, we iteratively use changes of coordinates $(x,y)=(\x(1+cU^n),\y(1+\overline cU^n))$ for $n\geq 1.$ Such a change allows to get rid of the term $ A_{n+1}U^{n+1}$ provided that $n+1>2.$ When $n=1$ it allows to get rid of $iIm(A_2U^2).$ Indeed, the constant $c$ must be chosen so as to verify $ A_1(c+\overline c)-nc(A_1+\overline{A_1})= A_{n+1},$ which is always solvable in $c$ as soon as $Re(A_1)\neq 0$ and $n>1.$ However, when $n=1$ we get $A_1(c+\overline c)-nc(A_1+\overline{ A_1})=A_1\overline c-\overline{ A_1}c=2iIm( A_1\overline c)\in i\RR.$ Hence, in that only case, the equation $ A_1(c+\overline c)-nc( A_1+\overline{ A_1})=iIm( A_{n+1})$ is solvable in $c.$
Finally, one divides \eqref{ForClas} by $\frac{yP-xQ}{2ixy}.$ This brings all the $Im(A_j)$ to $0.$ Then we repeat the procedure above with $c$ real to remove all higher terms in $u^j$ except for the term in $u^2.$ The cases $s=-1$ is analogous.
\end{proof}
It is easily seen that the multiplier at the origin of the Poincar\'e map of the field \eqref{formclawf} is equal to $\exp(2\pi\ep),$ so that the canonical parameter is also an analytic invariant of the Poincar\'e map.\\
\noindent{\emph{Admissible families of holomorphic germs.}}\\
Consider the germ of a holomorphic family $\Qq_{\ep}$ unfolding the germ of a codimension one analytic resonant diffeomorphism $\Qq$ with multiplier equal to $-1$ at the origin. The formal normal form $\Qq_{0,\ep}$ of $\Qq_{\ep}$ is the semi-Poincar\'e map (or semi-monodromy) of the vector field \eqref{formclawf}, namely $\Qq_{0,\ep}=\Ll_{-1}\circ\tau_{\ep}^{\pi},$ where $\tau_{\ep}^{\pi}$ is the time $\pi$-map of the equation:
\begin{equation}
\begin{array}{lll}
\dot w = \frac{w(\ep\pm w^2)}{1+A(\ep)w^2}
\end{array}\label{semiFFNO}
\end{equation}
and $\Ll_{-1}: w\mapsto -w.$
\begin{lemma}\label{fnor-Q}
Let $\Qq_{\ep}$ be a prepared family $(i.e.$ such that $\Qq_{\ep}^{\circ 2}$ has the form \eqref{pforPpp}$)$ unfolding a codimension one resonant diffeomorphism $\Qq$ with multiplier equal to $-1,$ and let $\Qq_{0,\ep}$ be its formal normal form, with same canonical parameter $\ep.$ Then, for any $N\in\NN^*$ there exists a real family of germs of diffeomorphisms $f_{\ep}$ tangent to the identity such that:
\begin{equation}
\begin{array}{lll}
\Qq_{\ep}\circ f_{\ep} - f_{\ep}\circ\Qq_{0,\ep} = O(x^{N+1}(\ep \pm x^2)^{N+1}).
\end{array}\label{fnor-QE}
\end{equation}
\end{lemma}
\begin{proof}
The proof is a slight modification of Theorem 6.2 in \cite{ccrr}, being given that the preparation of the family of diffeomorphisms is slightly different as well.
\end{proof}
Any germ of family of holomorphic diffeomorphisms $\Qq_{\ep}:(\CC,0)\to(\CC,0)$ verifying the hypotheses of Lemma \ref{fnor-Q}, is said to be ``admissible''.
\begin{theorem}\label{desusa}
Let $\Qq_{\ep}:(\CC,0)\to(\CC,0)$ be a real analytic family in the class of admissible germs of families, with coefficients $c_k(\ep)$ depending analytically on the canonical parameter $\ep,$ and such that $2c_2(\ep)^2+c_3(\ep)(1+c_1(\ep)^2)\neq 0$ for all $\ep\in V.$ Then the second iterate $\Qq_{\ep}^{\circ 2}$ is the monodromy of an elliptic generic family \eqref{weakfocus0} of order one.
\end{theorem}
\section{Proof of Theorem \ref{desusa}.}
The proof is achieved in several steps.
\subsection{Family of abstract manifolds.} By Lemma \ref{fnor-Q}, $\Qq_{\ep}$ decomposes as
\begin{equation}
\begin{array}{lll}
\Qq_{\ep} &=& (id+g_{\ep})\circ\Qq_{0,\ep}\\
&=& \Qq_{0,\ep}\circ(id+\widehat{g}_{\ep}),
\end{array}\label{eq.decomp}
\end{equation}
where $g_{\ep}$ and $\widehat{g}_{\ep}:=\Qq_{0,\ep}^{\circ -1}\circ g_{\ep}\circ\Qq_{0,\ep}$ are $(N+1)$-flat in $x$ at the origin: $g_{\ep}(x),\widehat{g}_{\ep}(x)=O(x^{N+1}(\ep\pm x^2)^{N+1})),$ for a large integer $N\in\NN.$
We recall that the standard monoidal map endows the blow-up space with coordinates $(X,y)$ and $(x,Y),$ and the transition between them is noted $\varphi.$\\
Let $v_{0,\ep}^{1}$ be the formal normal form given by the pullback of \eqref{formclawf} in $(X,y)$ variables (with linear part $2iX\frac{\partial}{\partial X} + (\ep-i)y\frac{\partial}{\partial y})$ and let $\Ff_{v_{0}^{1}}$ be its foliation on the product $\CC^*\times \DD_y,$ where $\DD_y$ is the standard unit disk of the $y$ axis. Consider the region $$\widetilde K_{1} =\Big\{\widetilde X\in Cov(\CC^*): -\pi/4<\arg(\widetilde X)<2\pi +\pi/4\Big\}$$ in the covering space $Cov(\CC^*)$ of the exceptional divisor, see Figure \ref{dom-rie}.
\begin{figure}[!h]
\begin{center}
{\psfrag{i}{\Huge{$\infty$}}
\psfrag{0}{\Huge{$0$}}
\psfrag{S'}{\Huge{$S$}}
\psfrag{S}{\Huge{$S'$}}
\psfrag{ec}{\Huge{$\simeq$}}
\scalebox{0.30}{\includegraphics{riemann.eps}}
}
\end{center}
\caption{\label{dom-rie} \small{The domain of $\widetilde X$ in the covering space $Cov(\CC^*).$}}
\end{figure}
The pullback of $v_{0,\ep}^{1}$ by the covering map $\pi_{1}:\widetilde K_{1}\times\DD_y\to\CC^*\times\DD_y,$ defines a field $\widetilde v_{\ep}^{1}(\widetilde X,w)$ and a foliation $\widetilde{\Ff}_{v^1}$ on the product $\widetilde{M}=\widetilde{K}_{1}\times\DD_y.$ The leaves of $\widetilde{\Ff}_{v^1}$ around the \emph{flaps}
\begin{equation*}
\begin{array}{lll}
S'_{1} &=& \{\widetilde X'\in\widetilde{K}_{c}: -\pi /4<\arg(\widetilde X')<\pi / 4\}\\
S_{1} &=& \{\widetilde X\in\widetilde{K}_{1}: 2\pi-\pi /4<\arg(\widetilde X)<2\pi + \pi /4\}
\end{array}
\end{equation*}
are identified by means of a \emph{sealing map} $\Upsilon_{\ep}:S'_{1}\times\DD_y\to S_{1}\times\CC,$ which preserves the first coordinate and respects $\widetilde{\Ff}_{v^1}.$ It is constructed as follows. For small values of $y,$ the holonomy map $h_{\ep,X}:\{X\}\times\DD_y\to\{1\}\times\DD_y$ along the leaves of $\Ff_{v_{0}^{1}}$ is covered by two holonomy maps, $h_{\ep,\widetilde X'}:\{\widetilde{X'}\}\times\DD_y\to \Sigma'\times\DD_y$ and $h_{\ep,\widetilde X}:\{\widetilde X\}\times\DD_y\to \Sigma\times\DD_y$ along the leaves of $\widetilde{\Ff}_{v^1}.$ The holonomies is negatively (resp. positively) oriented and noted $h_{\ep}^-$ (resp. $h_{\ep}^+),$ when $Im(X)>0$ (resp. $Im( X)<0).$ The convention:
\begin{equation}
\begin{array}{lll}
\lim_{\widetilde X\to\widetilde 1} h_{\ep,\widetilde X}^+=id
\end{array}\label{lim--zo}
\end{equation}
will be taken into account as well. Then $\Upsilon_{\ep}(\widetilde{X}',y)=(\widetilde{X},\Delta_{\ep}(\widetilde{X}',y)),$ where
\begin{equation}
\begin{array}{lll}
\Delta_{\ep}(\widetilde{X}',y)=(h_{\ep,\widetilde X}^+)^{\circ -1}\circ(id+g_{\ep})\circ h_{\ep,\widetilde X'}^+(y),
\end{array}\label{eq.sealing}
\end{equation}
with $\pi_1(\widetilde X')=\pi_1(\widetilde X).$ The map $\Upsilon_{\ep}$ is well defined and real analytic on its image for $r>0$ small, and it depends analytically on the parameter. Thus, it may be analytically extended to a larger domain $\{\widetilde X\in\widetilde{K}_{1} : -\pi /4<\arg(\widetilde X)<\pi\}\times\DD_y.$
Around a region of the covering of the $(x,Y)$ chart, things are naturally defined by means of the transition $\varphi.$ In particular, the family $\Dosilon_{\ep}=\varphi^*\Upsilon_{\ep}$ is a sealing map $\Dosilon_{\ep}(\widetilde Y',z)=(\widetilde Y,\nabla_{\ep}(\widetilde Y',z))$ with
\begin{equation}
\begin{array}{lll}
\nabla_{\ep}(\widetilde Y',z)=(\ell_{\ep,\widetilde Y}^+)^{\circ -1}\circ(id+\widehat{g}_{\ep})\circ\ell_{\ep,\widetilde Y'}^+(z),
\end{array}\label{Dosilon}
\end{equation}
and the transition $\varphi$ defines a field $\widetilde v_{\ep}^{2}(x,\widetilde Y),$ and foliation $\widetilde{\Ff}_{v^2}$ on the product $\widetilde{N}:=\varphi^*\widetilde{M}$ (the latter endowed with complex coordinates $(x,\widetilde Y)$ defined in the natural way). Here, $\ell_{\ep}^{\pm}$ are the holonomies along the leaves of $\widetilde{\Ff}_{v^2}.$ The map $\Dosilon_{\ep}$ is real analytic on its image.\\
As the sealing $(\Upsilon_{\ep},\Dosilon_{\ep})$ is canonically defined on the divisor, it defines a sealing family noted $\Gamma_{\ep}:\widetilde{\Mm}\to\widetilde{\Mm},$ where $\widetilde{\Mm}$ is the pullback of $(\widetilde{M},\widetilde{N})$ by the inverse of the monoidal map. The vector fields $(\widetilde v_{\ep}^{1},\widetilde v_{\ep}^{2})$ and foliations $(\widetilde{\Ff}_{v^1},\widetilde{\Ff}_{v^2})$ induce a vector field $\widetilde v_{\ep}$ and a foliation $\widetilde{\Ff}_{\ep}$ on $\widetilde{\Mm},$ and the coordinates on the latter are $(x,y).$ Moreover, $\Gamma_{\ep}$ is a germ of real analytic family of diffeomorphisms that preserves the transversal fibers $\Sigma_{\mu}=\{x=\mu y,\mu\in\CC^*\},$ and $(\Gamma_{\ep})_*\widetilde v_{\ep} = \widetilde v_{\ep},$ so that $\Gamma_{\ep}$ respects $\widetilde{\Ff}_{\ep}.$ Then, the quotient $$\Mm_{\ep}=\widetilde{\Mm} / \Gamma_{\ep}$$ is well defined and the vector field $\widetilde v_{\ep}$ induces a vector field $v_{\ep}$ and a foliation $\Ff_{\ep}$ on $\Mm_{\ep}.$ The leaves of this foliation project without critical points on the base $\CC^*\times\DD_y$ in the $(X,y)$ chart $(i.e$ are transversal to all lines $\{X=const.\}),$ and hence the loop generating the fundamental group of $\CC^*\times\DD_y$ defines the holonomy map of the quotient foliation $\Ff_{\ep}$ on $\Mm_{\ep}$ (for the cross section $\Sigma),$ referred to as the semi-monodromy.
\begin{proposition}\label{monod-vep}
The monodromy $\Sigma\to\Sigma$ of the field $v_{\ep}$ along the leaves of $\Ff_{\ep}$ coincides with $\Qq_{\ep}.$
\end{proposition}
\begin{proof}
The holonomy $h_{\ep,\widetilde 1}:\{\widetilde 1'\}\times\DD_y\to\{\widetilde 1\}\times\DD_y$ of $\widetilde v_{\ep}^{1}$ coincides, by construction, with the normal form $\Qq_{0,\ep}$ on $\widetilde M.$ Then, in $(x,y)$ variables, the image of the point $(x,x)\in\Sigma$ under the holonomy of $\widetilde{v}_{\ep}$ (for the section $\Sigma)$ is given by $(\Qq_{0,\ep}(x),\Qq_{0,\ep}(x))\in\Sigma.$ In addition,
\begin{equation*}
\begin{array}{lll}
\Gamma_{\ep}(\Qq_{0,\ep}(x),\Qq_{0,\ep}(x)) &=& (\Delta_{\ep}(1,\Qq_{0,\ep}(x)),\Delta_{\ep}(1,\Qq_{0,\ep}(x)))\\
&=& ((id+g_{\ep})\circ\Qq_{0,\ep}(x),(id+g_{\ep})\circ\Qq_{0,\ep}(x))\\
&=& (\Qq_{\ep}(x),\Qq_{\ep}(x))\in\Sigma,
\end{array}
\end{equation*}
where the second equality comes after \eqref{lim--zo}.
\end{proof}
\subsection{Integrability on $H_{\ep}(\Mm_{\ep})$}
In $(\widetilde X,y)$ coordinates, we introduce a smooth \emph{real} nonnegative cutoff function $\chi$ \emph{depending only on the argument of} $\widetilde X:$
\begin{equation*}
\chi(\arg\widetilde X)=\left \{ \begin{array}{lll}
1,\quad \arg\widetilde X\in(-\pi /4,\pi /4],\\
0,\quad \arg\widetilde X'\in(\pi,2\pi+\pi /4].
\end{array}
\right.
\end{equation*}
An ``identification map'' $\widetilde H_{\ep}^{1}$ is defined on $\widetilde M:$
\begin{equation}
\begin{array}{lll}
\widetilde H_{\ep}^{1} : (\widetilde X, y)\mapsto(\widetilde X, y + \chi(\arg\widetilde X)\{\Delta_{\ep}(\widetilde X,y)-y\}),
\end{array}\label{id-mapZ}
\end{equation}
for $c_1,c_2$ the monoidal map in charts \eqref{bdown}. Notice that $\widetilde{H}_{\ep}^{1}|_{S'_{1}\times\DD_y} \equiv (id_X,\Delta_{\ep})$ and $\widetilde{H}_{\ep}^{1}|_{S_{1}\times\DD_y} \equiv (id_X,id_y),$ and so this map respects the sealing $\Upsilon_{\ep}.$\\
In $(x,y)$ variables the function $\chi$ yields a real smooth map $\widehat{\chi}(x,y)=\chi(\arg(x/y))$ which depends only on the argument of the quotient $x /y,$ and the blow down of \eqref{id-mapZ} in $(x,y)$ coordinates equips the target space with coordinates $(z,w):$
\begin{equation}
\begin{array}{lll}
(z,w) = \widetilde{H}_{\ep}(x,y)\\
= (x+\widehat{\chi}(x,y)\{\nabla_{\ep}\circ c_2^{-1}(x,y)-x\},w+\widehat{\chi}(x,y)\{\Delta_{\ep}\circ c_1^{-1}(x,y)-y\}).
\end{array}\label{H_0--}
\end{equation}
By definition, $\widetilde{H}_{\ep}$ induces an ``identification family'' in the quotient: $$H_{\ep}:\Mm_{\ep}\to\CC^2.$$ For every fixed $\ep,$ the latter is a real analytic diffeomorphism which endows the target space with an almost complex structure induced from the standard complex structure on $\Mm_{\ep},$ as shown later. In addition, it depends analytically on the parameter. If the function $g$ in \eqref{eq.decomp} is $(N+1)$-flat at $x = y = 0,$ then \eqref{H_0--} is infinitely tangent to the origin:
\begin{proposition}\label{funcgH_0-ac}
The maps \eqref{eq.sealing} and \eqref{Dosilon} admit the asymptotic estimates
\begin{equation}
\begin{array}{lll}
|\Delta_{\ep}\circ c_1^{-1}(x,y)-y| &=& O(|x|^{\frac{N}{2}(1-\ep_2)}|y|^{\frac{N}{2}(1+\ep_2)+1})\\
|\nabla_{\ep}\circ c_2^{-1}(x,y)-x| &=& O(|x|^{\frac{N}{2}(1+\ep_2)+1}|y|^{\frac{N}{2}(1-\ep_2)})
\end{array}\label{a-est--}
\end{equation}
in the bidisk $\DD_{x}\times\DD_{y},$ where $\ep=\ep_1+i\ep_2.$
\end{proposition}
\begin{proof}
In $(\widetilde X,y)$ variables, the following estimate for the holonomy map $h_{\ep,\widetilde X}:\{\widetilde X\}\times\DD_{w}\to\{\widetilde 1\}\times\CC$ is well known:
\begin{equation*}
\begin{array}{lll}
e^{-M|\lambda(\ep)(\widetilde X-1)|-\frac{\ep_1\arg\widetilde X}{2}}|\widetilde X|^{\frac{1-\ep_2}{2}}|y|\leq |h_{\ep,\widetilde X}(y)|\leq e^{M|\lambda(\ep)(\widetilde X-1)|-\frac{\ep_1\arg\widetilde X}{2}}|\widetilde X|^{\frac{1-\ep_2}{2}}|y|,
\end{array}
\end{equation*}
where $M=M(\widetilde X,y)<\infty$ is a positive constant depending on a bound for the nonlinear part of the foliation along the segment with endpoints $\widetilde X,1,$ and $\lambda(\ep)=(\ep-i) / 2i$ is the ratio of eigenvalues in $(X,y)$ chart. By \eqref{eq.decomp}, $$h_{\ep,\widetilde X}^{-1}\circ (id+g)\circ h_{\ep,\widetilde X'}=h_{\ep,\widetilde X}^{-1}\circ(h_{\ep,\widetilde X'}+g\circ h_{\ep,\widetilde X'})=id+O(|\widetilde X|^{\frac{N}{2}(1-\ep_2)}|y|^{N+1}).$$ In $(x,Y)$ coordinates, the estimate is obtained by symmetry. Since $x=\widetilde Xy$ and $y=\widetilde Yx,$ the conclusion follows.
\end{proof}
\begin{corollary}\label{gtangid}
The family $H_{\ep}$ is tangent to the identity.
\end{corollary}
The pullback of the complex structure on $\Mm_{\ep}$ by the map $H_{\ep}^{-1}$ is an almost complex structure defined by the pullback of the $(1,0)$-subbundle on $\Mm_{\ep},$ which is spanned by
\begin{equation}
\begin{array}{lll}
\widetilde{\zeta}_{1,\ep} = dz = d(x+\widehat{\chi}\cdot\{\nabla_{\ep}\circ c_2^{-1}-x\}),\\
\widetilde{\zeta}_{2,\ep} = dw = d(y+\widehat{\chi}\cdot\{\Delta_{\ep}\circ c_1^{-1}-y\}),
\end{array}\label{diforms}
\end{equation}
on $\widetilde H_{\ep}(\widetilde{\Mm}).$ The forms $d(\nabla_{\ep}\circ c_2^{-1})$ and $d(\Delta_{\ep}\circ c_1^{-1})$ are holomorphic on their domains and $\widetilde{\zeta}_{1,\ep}$ and $\widetilde{\zeta}_{2,\ep}$ have two different sectorial representatives:
\begin{equation}
\left.
\begin{array}{lll}
\widetilde{\zeta}_{1,\ep}=\left \{ \begin{array}{lll}
\zeta_{1,\ep}^0=dx, & |\arg x-\arg y-13\pi / 8|< 5 \pi/8, \\
\zeta_{1,\ep}^1=d(\nabla_{\ep}\circ c_2^{-1}), & |\arg x-\arg y|< \pi /4,
\end{array}
\right.\\
&\\
\widetilde{\zeta}_{2,\ep}=\left \{ \begin{array}{lll}
\zeta_{2,\ep}^1=d(\Delta_{\ep}\circ c_2^{-1}), & |\arg x-\arg y|< \pi /4\\
\zeta_{2,\ep}^0=dy, & |\arg x-\arg y-13\pi / 8|< 5 \pi/8,
\end{array}
\right.
\end{array}
\right.
\end{equation}
so that $\zeta_{1,\ep}^1=\Gamma_{\ep}^*\zeta_{1,\ep}^0$ and $\zeta_{2,\ep}^1=\Gamma_{\ep}^*\zeta_{2,\ep}^0.$ Thus they yield forms $\zeta_{1,\ep}$ and $\zeta_{2,\ep}$ on $\Mm_{\ep}.$ The almost complex structure induced on $H_{\ep}(\Mm_{\ep})\subset\CC^2$ by the complex structure on $\Mm_{\ep}$ is defined by the two forms
\begin{equation}
\begin{array}{lll}
\omega_{1,\ep}=(H_{\ep}^{-1})^*\zeta_{1,\ep},\quad\quad\omega_{2,\ep}=(H_{\ep}^{-1})^*\zeta_{2,\ep}.
\end{array}\label{alcomplexstructure}
\end{equation}
\begin{lemma}\label{estimateforms}
Let $\delta$ be a small positive number with $|\ep|<\delta.$ If $\alpha$ and $\beta$ are the orders of flatness in $x$ and $y$ (resp. $y$ and $x)$ of the difference $\omega_{1,\ep}-dx$ (resp. $\omega_{2,\ep}-dy),$ then the form $\omega_{1,\ep}$ (resp. $\omega_{2,\ep})$ can be extended as $dx$ (resp. $dy)$ along the $x$-axis (resp. $y$-axis) until the order $\alpha$ if the number $N$ in \eqref{eq.decomp} is sufficiently large so as to ensure
\begin{equation}
\begin{array}{lll}
N >\max\left\{\frac{2(\alpha-1)}{1-\delta},\frac{2\beta}{1-\delta}\right\}.
\end{array}\label{N-beta}
\end{equation}
\end{lemma}
\begin{proof}
By \eqref{alcomplexstructure}, it suffices to study the difference $$\widetilde{H}_{\ep}(x,y)-(x,y)=(\widehat{\chi}(x,y)\{\nabla_{\ep}\circ c_2^{-1}(x,y)-x\},\widehat{\chi}(x,y)\{\Delta_{\ep}\circ c_1^{-1}(x,y)-y\}).$$ The definition of $\xi$ yields
\begin{equation}
\begin{array}{lll}
\left|\frac{\partial^{i+j}\widehat{\chi}}{\partial x^p\partial\overline x^q\partial y^r\partial\overline y^s}\right|<C^{\underline{st}}\cdot\frac{\MB_{i+j}}{|x|^i|y|^j}
\end{array}\label{est-X}
\end{equation}
for all $i=p+q \in\NN,$ $j=r+s \in\NN$ and $\MB_{i+j}:=
\max_{\substack{
0\leq k\leq i+j\\
\theta\in I}}
|\chi^{(k)}(\theta)|$ with $I=[-\pi /4,2\pi+\pi /4].$ To lighten the notation, put $f(x,y)=\nabla_{\ep}\circ c_2^{-1}(x,y)-x.$ Proposition \ref{funcgH_0-ac} implies that for all $k,l\in\NN,$ there exists a real constant $L=L(N,\alpha,\beta)>0$ such that
\begin{equation}
\begin{array}{lll}
\left|\frac{\partial^{\alpha+\beta}(\widehat{\chi}\cdot f)}{\partial x^p\partial\overline{x}^q\partial y^r\partial\overline{y}^s}\right| \leq L\cdot |x|^{\frac{N}{2}(1+\ep_2)+1-\alpha} \cdot |y|^{\frac{N}{2}(1-\ep_2)-\beta},
\end{array}\label{est-prod}
\end{equation}
for $\alpha=p+q$ and $\beta=r+s.$ Hence, if $|\ep|<\delta<<1$ and the order $N$ of $g_{\ep}$ satisfies \eqref{N-beta} then the left hand side of \eqref{est-prod} tends to zero uniformly in $|x|<1,$ and thus $\omega_{1,\ep}$ and $dx$ coincide until the order $\alpha$ along the $x$-axis. The assertion for the difference $\omega_{2,\ep}-dy$ follows by duality.
\end{proof}
The set $H_{\ep}(\Mm_{\ep})\subset\CC^2$ does not contain the axes of coordinates: its closure is $C^{\infty}$-diffeomorphic to a closed neighborhood of the origin of $\CC^2.$ Lemma \ref{estimateforms} shows that the almost complex structure generated by \eqref{alcomplexstructure} on $H_{\ep}(\Mm_{\ep})$ can be extended as $\omega_{1,\ep}=dx$ along the $x$-axis, and as $\omega_{2,\ep}=dy$ along the $y$-axis, until a well-defined order. This almost complex structure is integrable. Indeed, $\omega_{1,\ep}$ is obtained from the pullback of $\zeta_{1,\ep}$ and since the forms $d(\nabla_{\ep}\circ c_2^{-1})$ and $d(\Delta_{\ep}\circ c_1^{-1})$ are holomorphic on their domains and $\widehat{\chi}$ is of class $C^{\infty},$ $d\widetilde{\zeta}_{1,\ep}$ contains no forms of type $(0,2).$ By symmetry, the same holds for $d\widetilde{\zeta}_{2,\ep}.$ If $L^{1,0}$ is the span of the forms $\omega_{1,\ep},\omega_{2,\ep},$ then this integrability condition holds for $L^{1,0}$ on the surface $H_{\ep}(\Mm_{\ep}),$ and by continuity it remains valid after extension until the axes. Hence, for each $\ep\in V$ the Newlander-Nirenberg Theorem ensures the existence of a smooth chart $\widetilde{\Lambda}_{\ep}=\widetilde{\Lambda}_{\ep}(z,w),$
\begin{equation}
\begin{array}{lll}
\widetilde{\Lambda}_{\ep}=(\widetilde{\xi}_{\ep}^1,\widetilde{\xi}_{\ep}^2):\widetilde H_{\ep}(\widetilde{\Mm})\to\CC^2,
\end{array}\label{csmocht}
\end{equation}
which is holomorphic in the sense of the almost complex structure \eqref{diforms}. It induces, in turn, the germ of a family of smooth charts
\begin{equation}
\begin{array}{lll}
\Lambda_{\ep}=(\xi_{\ep}^1,\xi_{\ep}^2):\BB(r)\subset H_{\ep}(\Mm_{\ep})\to\CC^2
\end{array}\label{jodfellout}
\end{equation}
in the quotient, where $\BB(r)$ is a small ball around the origin. This chart is, by definition, holomorphic in the sense of the extended almost complex structure \eqref{alcomplexstructure}.
\begin{theorem}\label{NNext}
The germ of smooth charts $\Lambda_{\ep}$ respects the real foliation, is tangent to the identity at the origin, and depends analytically on the parameter.
\end{theorem}
\begin{proof}
In order to show that the chart respects the real foliation, it suffices to prove that $\widetilde{\Lambda}_{\ep}$ is real, namely, it sends $\{z=\overline w\}\simeq\RR^2$ into $\RR^2\subset\CC^2$ when $\ep\in\RR.$
The family of diffeomorphisms \eqref{H_0--} is analytic with respect to the structure \eqref{diforms}. It follows that, modulo a linear combination,
\begin{equation*}
\begin{array}{lll}
dz &=& dx + e_{1,\ep}^1d\overline{x} + e_{2,\ep}^1d\overline{y}\\
dw &=& dy + e_{1,\ep}^2d\overline{x} + e_{2,\ep}^2d\overline{y},
\end{array}
\end{equation*}
where the coefficients are computed in terms of $\chi,\Delta,\nabla$ and its derivatives, and satisfy:
\begin{equation}
\begin{array}{lll}
\overline{e_{j,\overline{\ep}}^k(\overline y,\overline x)} &=& e_{k,\ep}^j(x,y),\quad j,k\in\{1,2\}.
\end{array}\label{funcSr-}
\end{equation}
This is because $\RR^2$ is itself invariant under \eqref{H_0--}: $H_{\ep}(\{x=\overline y\})\subset\RR^2$ when the parameter is real. By Proposition \ref{funcgH_0-ac}, $e_{j,\ep}^k=\frac{o(1)}{1+o(1)},$ yielding:
\begin{equation}
\begin{array}{lll}
e_{j,\ep}^k(0,0)=0,\quad j,k\in\{1,2\}.
\end{array}\label{eij=0}
\end{equation}
Suppose that the image $\widetilde H_{\ep}(\Mm)$ contains a small bidisk $\DD_s\times\DD_s,$ and write $G_{\ep}:=\widetilde H_{\ep}^{-1}.$ Consider the pullback $\aG_{j,\ep}^k=G_{\ep}^*(e_{j,\ep}^k):\DD_s\times\DD_s\to\CC^2$ given by
\begin{equation*}
\begin{array}{lll}
\aG_{j,\ep}^k(z,w)=G_{\ep}^*(e_{j,\ep}^k)(z,w)\equiv e_{j,\ep}^k(G_{\ep}(z,w)),\quad j,k=1,2, \ \ \ep\in V,
\end{array}
\end{equation*}
for $(z,w)\in\DD_s\times\DD_s.$ By \eqref{funcSr-} the collection $\aG_{j,\ep}^k$ satisfies again:
\begin{equation}
\begin{array}{lll}
\overline{\aG_{j,\overline{\ep}}^k(\overline{w},\overline{z})} &=& \aG_{k,\ep}^j(z,w),
\end{array}\label{funcSr}
\end{equation}
and by \eqref{eij=0}, $\aG_{j,\ep}^k(0,0)=0.$\\
\noindent{\emph{Notation.}} We will write $z^1=z, \ \ z^2=w,$ and:
\begin{equation*}
\begin{array}{lll}
\partial_j=\frac{\partial}{\partial z^j},\quad \overline{\partial}_j=\frac{\partial}{\partial\overline{z^j}},\quad j=1,2.
\end{array}
\end{equation*}
A complex valued function $\xi$ such that:
\begin{equation}
\begin{array}{lll}
\overline{\partial}_j\xi-(\aG_j^1\partial_1\xi+\aG_j^2\partial_2\xi)=0,\quad j=1,2
\end{array}\label{compOpee}
\end{equation}
is called (cf. \cite{NNir}) holomorphic with respect to the given almost complex structure. Instead of considering the new coordinates \eqref{csmocht} as solutions to \eqref{compOpee} and functions of $(z,w)$ and their complex conjugates, the coordinates $(z,w)$ are supposed to be functions of \eqref{csmocht} and their complex conjugates. Inasmuch as it suffices to study only the real character of the chart $\widetilde{\Lambda}_{\ep},$ the tildes on the chart $(\widetilde{\xi}_{\ep}^1,\widetilde{\xi}_{\ep}^2)$ are dropped from now on.\\
\noindent{\emph{Notation.}} The holomorphic and antiholomorphic dual differentials are:
\begin{equation}
\begin{array}{lll}
d_{j,\ep}=\frac{\partial}{\partial\xi_{\ep}^j},\quad\overline{d}_{j,\ep}=\frac{\partial}{\partial\overline{\xi_{\ep}^j}},\quad j=1,2.
\end{array}\label{boundOpe}
\end{equation}
\noindent It is known (cf. \cite{NiWoo}, pp. 445) that for every $\ep\in V,$ the map $G_{\ep}$ from $\DD_s\times\DD_s\subset\CC^2$ to the almost complex manifold $\widetilde{\Mm}$ is holomorphic if and only if its coordinates $(z,w)=G_{\ep}^*(x,y)$ satisfy the differential equations
\begin{equation}
\begin{array}{lll}
\overline{d}_{j,\ep} z^k+\aG_{m,\ep}^k\overline{d}_{j,\ep}\overline{z}^m=0,\quad j,k=1,2.
\end{array}\label{non-linear}
\end{equation}
In such a case, \eqref{compOpee} yields:
\begin{equation}
\begin{array}{lll}
\overline{d}_{j,\ep}\xi_{\ep}^{p} &=& \partial_k\xi_{\ep}^{p} \overline d_{j,\ep}z^k+\overline{\partial}_k\xi_{\ep}^{p}\overline{d}_{j,\ep}\overline{z}^k\\
&=& \partial_k\xi_{\ep}^{p}\{\overline{d}_{j,\ep} z^k+\aG_{m,\ep}^k\overline{d}_{j,\ep}\overline{z}^m\}\\
&=& 0
\end{array}\label{eqtodvar}
\end{equation}
for $j=1,2.$ Notice that the replacement of \eqref{non-linear} in the term after the first equality of \eqref{eqtodvar}, yields:
\begin{equation*}
\begin{array}{lll}
\overline{d}_{j,\ep}\xi_{\ep}^{p}=\overline{d}_{j,\ep}\overline{z}^k\{\overline{\partial}_k\xi_{\ep}^{p}-\aG_{k,\ep}^i\partial_i\xi_{\ep}^{p}\},\quad p=1,2.
\end{array}
\end{equation*}
Thus the parametric Cauchy-Riemann equations $\overline{d}_{j,\ep}\xi_{\ep}^{p}=0$ are equivalent to the system \eqref{compOpee} if $z,w$ satisfy \eqref{non-linear} with the matrix $[\overline{d}_{j,\ep}\overline{z}^k]$ non-singular for all $\ep$ in the symmetric neighborhood $V.$ We find real solutions to \eqref{non-linear}.\\
Denote by $T^1,T^2$ the integral operators
\begin{equation}
\begin{array}{lll}
T^1f(z,w)=\frac{1}{2i\pi}\iint_{|\tau|<\rho} \frac{f(\tau,w)}{z-\tau}d\overline{\tau}d\tau,\\
T^2f(z,w)=\frac{1}{2i\pi}\iint_{|\tau|<\rho} \frac{f(z,\tau)}{w-\tau}d\overline{\tau}d\tau,\\
\end{array}\label{intOperators}
\end{equation}
with $\rho>0$ fixed and $f=f(z,w)$ has suitable differentiability properties and, eventually, depends on additional complex coordinates. A short calculation shows that if $f_1,f_2$ are as above and $f_1(z,w)=\overline{f_2(\overline{w},\overline{z})},$ then
\begin{equation}
\begin{array}{lll}
T^1f_1(z,w)=\overline{T^2f_2(\overline{w},\overline{z})}.
\end{array}\label{navsupp}
\end{equation}
The non-linear differential system corresponding to \eqref{non-linear} is given by the integral equation (cf. \cite{NiWoo}):
\begin{equation}
\begin{array}{lll}
z^k(\xi_{\ep}^1,\xi_{\ep}^2)&=&\xi_{\ep}^k+\mathbf{TF}^k[z,w](\xi_{\ep}^1,\xi_{\ep}^2)-\mathbf{TF}^k[z,w](0,0),\ \ k=1,2
\end{array}\label{eqsnon-linear}
\end{equation}
where
\begin{equation*}
\begin{array}{lll}
\mathbf{TF}^k:=T^1f_{1k}+T^2f_{2k}-\frac{1}{2}\left\{T^1\overline{d}_{1,\ep} T^2 f_{2k} + T^2\overline{d}_{2,\ep} T^1 f_{1k}\right\},\quad k=1,2\\
\end{array}
\end{equation*}
are the Nijenhuis-Woolf operators, and
\begin{equation*}
\begin{array}{lll}
f_{jk}(z,w)(\xi^1,\xi^2)(\ep)= -(\aG_{1,\ep}^k(z,w)\overline{d}_{j,\ep}\overline{z}+\aG_{2,\ep}^k(z,w)\overline{d}_{j,\ep}\overline{w}),\ \ i,j\in\{1,2\}.
\end{array}
\end{equation*}
\begin{corollary}\label{FTsymmetry}
For every $(z,w)$ in a neighborhood of the origin and for every initial value $(\xi_{\ep}^1,\xi_{\ep}^2),$ the Nijenhuis-Woolf operators are related through:
\begin{equation}
\begin{array}{lll}
\mathbf{TF}^1[z,w](\xi^1,\xi^2)(\ep)=\overline{\mathbf{TF}^2[\overline{w},\overline{z}](\overline{\xi^2},\overline{\xi^1})(\ep)}
\end{array}\label{tehaspint}
\end{equation}
when the parameter is real.
\end{corollary}
\begin{proof}
This is plain consequence of \eqref{funcSr}, the definition of the dual differentials \eqref{boundOpe} and property \eqref{navsupp} on the maps $f_{jk}.$
\end{proof}
The pair of coordinates $(\xi_{\ep}^1,\xi_{\ep}^2)$ is referred to as the initial value of \eqref{eqsnon-linear}. For $\ep=0,$ the system \eqref{eqsnon-linear} is solved by means of a Picard iteration process (fixed point Theorem) which converges in a small ball $\BB(r_0)$ of radius $r_0>0$ around the origin of $(z,w)$ coordinates (cf. \cite{NNir},\cite{NiWoo}). It turns out that for all $|\ep|$ small and fixed, any solution to \eqref{eqsnon-linear} is well defined on $\BB(r),$ with $r=r_0/2.$ Moreover, if $r$ is small enough, then the solution $(z,w)$ is unique:
\begin{lemma}\cite{NNir}\label{2nuni}
For $r$ sufficiently small, and $\ep\in V$ fixed, the integral system \eqref{eqsnon-linear} admits a unique solution $(z,w)$ satisfying also \eqref{non-linear} and such that the parametric transformation $\widetilde{\Lambda}_{\ep}^{\circ -1}$ from the $(\xi_{\ep}^1,\xi_{\ep}^2)$ coordinates to $(z,w)$ coordinates has non-vanishing Jacobian.
\end{lemma}
\begin{proposition}\label{lambrecha}
The chart \eqref{csmocht} respects the real foliation and is tangent to the identity.
\end{proposition}
\begin{proof}
Let $(\xi_{\ep}^1,\xi_{\ep}^2)$ be the initial value and $(z,w)$ be the solution to \eqref{eqsnon-linear}. If the initial condition satisfies $\xi_{\ep}^1=\overline{\xi_{\ep}^2},$ then Corollary \ref{FTsymmetry} leads to
\begin{equation*}
\begin{array}{lll}
\overline{z^k} &=& \xi_{\ep}^k+\mathbf{TF}^k[\overline{w},\overline{z}](\xi_{\ep}^1,\xi_{\ep}^2)-\mathbf{TF}^k[\overline{w},\overline{z}](0,0),\ \ k=1,2
\end{array}\label{eqsnon-linear11}
\end{equation*}
and the unicity of the solution carries $z=\overline{w}.$ Since $\widetilde{\Lambda}_{\ep}$ has non-vanishing Jacobian, it is a local isomorphism if $r>0$ is small.
By \eqref{eqsnon-linear}, the chart $\widetilde{\Lambda}_{\ep}$ is tangent to the identity at the origin.
\end{proof}
Inasmuch as $\Gamma_{\ep},H_{\ep}$ and \eqref{csmocht} respect the real foliation, the chart $\Lambda_{\ep}$ is real when $\ep$ is real, and is clearly tangent to the identity at the origin. This concludes the proof of Theorem \ref{NNext}.
\end{proof}
\noindent{\emph{End of the proof of Theorem \ref{desusa}.}} The composition $\vartheta_{\ep}=\Lambda_{\ep}\circ H_{\ep}:H_{\ep}^{-1}(\BB(r))\to\CC^2$ between complex analytic manifolds is honestly biholomorphic. The closure $\Ww:=\overline{\vartheta_{\ep}(H_{\ep}^{-1}(\BB(r)))}$ contains the origin in its interior. It remains to check that the family of vector fields defined on $\Ww$ by the pushforward $$\vv_{\ep}=(\vartheta_{\ep})_* v_{\ep}$$ is orbitally equivalent to a generic family unfolding a weak focus with formal normal form \eqref{formclawf}.
\begin{proposition}
The quotient of the eigenvalues of $\vv_{\ep}$ is equal to $\frac{\ep+ i}{\ep-i}.$
\end{proposition}
\begin{proof}
By Theorem \ref{NNext}, the components $\PP_{\ep},\QQ_{\ep}$ of $$\vv_{\ep}(\x,\y)=\PP_{\ep}(\x,\y)\frac{\partial}{\partial \x} + \QQ_{\ep}(\x,\y)\frac{\partial}{\partial \y}$$ are related through \eqref{realchar} as well, and then the eigenvalues of the vector field $\vv_{\ep}$ are complex conjugate. We call them $\tau(\ep),\overline{\tau(\ep)},$ with $\tau(\ep) = a(\ep) +ib(\ep)$ and $a(\ep),b(\ep)$ depend analytically on $\ep$ small and are real on $\ep\in\RR.$ In the $(\X,\y)$ chart of the blow-up, $\vv_{\ep}$ gives rise to a family of equations of the form:
\begin{equation*}
\begin{array}{lll}
\dot \X &=& (\tau-\overline{\tau})\X+...\\
\dot \y &=& \overline{\tau}\y+...
\end{array}
\end{equation*}
with Poincar\'e map (cf. \cite{MM}) $\Pp_{\ep}(\y)=\exp\left(2i\pi\left(\frac{2\overline{\tau}}{\tau-\overline{\tau}}\right)\right)\y+...,$ while in $(\x,\Y)$ coordinates, $\vv_{\ep}$ gives rise to the system:
\begin{equation*}
\begin{array}{lll}
\dot \x &=& \tau \x+...\\
\dot \Y &=& (\overline{\tau}-\tau)\Y+...
\end{array}
\end{equation*}
with Poincar\'e map $\Pp_{\ep}(\x)=\exp\left(-2i\pi\left(\frac{2\tau}{\overline{\tau}-\tau}\right)\right)\x+...$ (computed on the cross section $\x=\y).$ It is easily seen that: $$\mu(\ep)=\exp\left(2i\pi\left(\frac{2\overline{\tau}}{\tau-\overline{\tau}}\right)\right)=\exp\left(-2i\pi\left(\frac{2\tau}{\overline{\tau}-\tau}\right)\right)=\exp\left(2i\pi\left(\frac{\tau+\overline{\tau}}{\tau-\overline{\tau}}\right)\right),$$ where $\Pp_{\ep}'(0)=\mu(\ep).$ On the other hand, $\mu(\ep)=\exp(2\pi\ep)$ by the preparation \eqref{pforPpp}. Thus, $$2\pi\ep=2\pi\frac{2a(\ep)}{2b(\ep)}+2i\pi m,$$ for some $m\in\NN.$ This means that
\begin{equation}
\begin{array}{lll}
\frac{a(\ep)}{b(\ep)}=\ep-im.
\end{array}\label{m==0}
\end{equation}
Inasmuch as $a(\ep),b(\ep)$ are real on $\ep\in\RR,$ the equation \eqref{m==0} implies that $m=0,$ and the conclusion follows.
\end{proof}
\section{Acknowledgements.}
I am grateful to Christiane Rousseau for suggesting the problem and supervising this work, and to Colin Christopher and Sergei Yakovenko for helpful discussions. I am indebted to an unknown referee as well, for suggesting many important comments concerning the previous version of the paper.
\bibliography{mybib}
\end{document} | 188,552 |
\begin{document}
\baselineskip=17pt
\title{Cuspidal divisor class groups of non-split Cartan modular curves}
\author{Pierfrancesco Carlucci\\
Dipartimento di Matematica\\
Universit\'a degli Studi di Roma Tor Vergata\\
Via della Ricerca Scientifica 1, 00133, Rome, Italy\\
E-mail: [email protected]}
\date{30 April, 2016}
\maketitle
\renewcommand{\thefootnote}{}
\footnote{2010 \emph{Mathematics Subject Classification}: Primary 11G16; Secondary 11B68, 13C20.}
\footnote{\emph{Key words and phrases}: Siegel Functions, Modular Units, Cuspidal Divisor Class Group, Non-Split Cartan Curves, Generalized Bernoulli Numbers.}
\renewcommand{\thefootnote}{\arabic{footnote}}
\setcounter{footnote}{0}
\begin{abstract}
I find an explicit description of modular units in terms of Siegel functions for the modular curves $X^+_{ns}(p^k) $ associated to the normalizer of a non-split Cartan subgroup of level $ p^k $ where $ p\not=2,3 $ is a prime. The Cuspidal Divisor Class Group $ \mathfrak{C}^+_{ns}(p^k) $ on $X^+_{ns}(p^k)$ is explicitly described as a module over the group ring $R = \mathbb{Z}[(\mathbb{Z}/p^k\mathbb{Z})^*/\{\pm 1\}] $.
In this paper I give a formula involving generalized Bernoulli numbers $ B_{2,\chi} $ for $ |\mathfrak{C}^+_{ns}(p^k)| $.
\end{abstract}
\section{Motivation and overview}
Let $ X^+_{ns}(n) $ be the modular curve associated to the normalizer of a non-split Cartan subgroup of level $ n$. One noteworthy reason for studying these curves is the Serre's uniformity problem over $ \mathbb{Q} $ stating that there exists a constant $ C >0 $ so that, if $ E $ is an elliptic curve over $ \mathbb{Q} $ without complex multiplication, then the Galois representation:
$$ \rho_{E,p}: \mbox{Gal}(\overbar{\mathbb{Q}},\mathbb{Q}) \rightarrow \mbox{GL}_2(\mathbb{F}_p) $$
attached to the elliptic curve $ E $ is onto for all primes $ p>C$ (see \cite{Serre72} and \cite[pag. 198]{KL}).
If the Galois representation were not surjective, its image would be contained in one of the maximal proper subgroups
of $ \mbox{GL}_2(\mathbb{F}_p) $. These subgroups are:\\
\noindent 1. A Borel subgroup;\\
2. The normalizer of a split Cartan subgroup;\\
3. The normalizer of a non-split Cartan subgroup; \\
4. A finite list of exceptional subgroups. \\
Serre himself showed that if $ p > 13 $ the image of $ \rho_{E,p} $ is not contained in an exceptional subgroup. Mazur in \cite{Mazur} and Bilu-Parent-Rebolledo in \cite{BiPaRe} presented analogous results for Borel subgroups and split Cartan subgroups respectively. The elliptic curves over $ \mathbb{Q} $ for which the image of the Galois representation is contained in the normalizer of a non-split Cartan subgroup are parametrized by the non-cuspidal rational points of $X^+_{ns}(p) $. Thus the open case of Serre's uniformity problem can be reworded in terms of determining whether there exist $ \mathbb{Q}$-rational points on $X^+_{ns}(p) $, that do not arise from elliptic curves with complex multiplication.
This paper focuses on an aspect of the curves $X^+_{ns}(p^k) $ that has never been treated before: their Cuspidal Divisor Class Group $ \mathfrak{C}^+_{ns}(p^k)$, a finite subgroup of the Jacobian $ J^+_{ns}(p^k)$ whose support is contained in the set of cusps of $X^+_{ns}(p^k) $. Let $ \mathfrak{D}_{ns}^+(p^k) $ be the free abelian group generated by the cusps of $X^+_{ns}(p^k) $, let $ \mathfrak{D}_{ns}^+(p^k)_0 $ be its subgroup consisting of elements of degree 0 and let $ \mathfrak{F}_{ns}^+(p^k) $ be the group of divisors of modular units of $X^+_{ns}(p^k)$, i.e. those modular functions on $X^+_{ns}(p^k)$ in the modular function field $ F_{p^k} $, which have no zeros and poles in the upper-half plane. We define:
$$ \mathfrak{C}^+_{ns}(p^k):= \mathfrak{D}_{ns}^+(p^k)_0 / \mathfrak{F}_{ns}^+(p^k).$$
In \cite{Siegelgenerator} Kubert and Lang gave an explicit and complete description of the group of modular units of $ X(p^k) $ in terms of Siegel functions $ g_a(\tau) $ (see \cite{Lang:ef} or \cite{Siegel}) with $a \in \frac{1}{p^k}\mathbb{Z}^2 \setminus \mathbb{Z}^2 $. We will define the set of functions $$ \{ G^+_h(\tau)\}_{h \in ((\mathbb{Z} / p^k \mathbb{Z})^*/\{\pm 1\})} $$ in terms of classical Siegel functions and we will prove the following result: \\
\noindent \textbf{Theorem \ref{powprod}}
\textit{If $ p \not= 2,3 $, the group of modular units of the modular curve $ X^+_{ns}(p^k) $ consists (modulo constants) of power products:}
$$ g(\tau)= \prod_{h \in ((\mathbb{Z} / p^k \mathbb{Z})^*/\{\pm 1\}) }{{G^+_h}^{n^+_h}(\tau)} $$
\textit{where
$$ G^+_h(\tau) = \prod_{t \in ((\mathbf{O}_K / p^k \mathbf{O}_K)^*/\{\pm 1\}) , \pm|t|= h}g_{[t]}(\tau) $$
\textit{and} $ d=\displaystyle\frac{12}{\gcd(12,p+1)} \mbox{ divides } \sum_{h}n^+_h$.}
In \cite[Chapter 5]{KL} Kubert and Lang studied the Cuspidal Divisor Class Group on the modular curve $ X(p^k) $. Since their description utilizes the parametrization of the set of cusps of $ X(p^k) $ by the elements of the quotient $ C_{ns}(p^k)/\{\pm 1\} $, it appears natural to develop and extend their techniques to non-split Cartan modular curves. Kubert and Lang proved the following:\\
\noindent
\textbf{Theorem \ref{CDCG}} \textit{If $ p\ge 5 $ consider $ R:= \mathbb{Z}[C_{ns}(p^k)/\{\pm 1\}] $ and let $ R_0 $ be the ideal of $ R $ consisting of elements of degree $0$.
The Cuspidal Divisor Class Group $ \mathfrak{C}_{p^k} $ on $ X(p^k) $ is an $R-$module, more precisely there exists a Stickelberger element $ \theta \in \mathbb{Q}[C_{ns}(p^k)/\{\pm 1\}]$ such that, under the identification of the group $ C_{ns}(p^k)/\{\pm 1\} $ with the set of cusps at level $ p^k $, the ideal $ R \cap R \theta $ corresponds to the group of divisors of units in the modular function field $ F_{p^k} $ and:
$$ \mathfrak{C}_{p^k} \cong R_0 / R \cap R \theta. $$}
\noindent
In this theorem the authors exhibited an isomorphism reminding to a classical result in cyclotomic fields theory. Let $ J $ be a fractional ideal of $ \mathbb{Q}(\zeta_m) $ and $ G= $\mbox{Gal}$( \mathbb{Q}(\zeta_m)/\mathbb{Q}) \cong (\mathbb{Z}/m\mathbb{Z})^*$. Consider $\mathbb{Z}[G]$ acting on the ideals and ideal classes in the natural way: if $ x= \sum_{\sigma}x_{\sigma} \sigma $ then $ J^x := \prod_{\sigma}(J^{\sigma})^{x_{\sigma}}.$ We have the following result:\\
\\ \textbf{Stickelberger's Theorem} \cite[pag. 333]{Washington} \textit{Define the Stickelberger element:
$$ \theta = \sum_{a \scriptsize{\mbox{ mod }} m, (a,m)=1}\displaystyle \left\langle \frac{a}{m} \right\rangle \sigma_a^{-1} \in \mathbb{Q}[G].$$
The Stickelberger ideal $ \mathbb{Z}[G]\cap\theta\mathbb{Z}[G] $ annihilates the ideal class group of $ \mathbb{Q}(\zeta_m)$.}\\
\noindent Along these lines, the main result can be summarized as follows:\\
\noindent
\textbf{Main Theorem \ref{main}}\textit{ Consider $ p \ge 5 $, $ H:=(\mathbb{Z}/p^k\mathbb{Z})^*/\{\pm 1\}$ and $ w $ a generator of $ H $. There exists a Stickelberger element
$$ \theta := \displaystyle\frac{p^k}{2} \sum_{i=1}^{\frac{p-1}{2}p^{k-1}} {\displaystyle\sum_{ \pm|s|=w^i, s \in ((\mathbf{O}_K / p^k \mathbf{O}_K)^*/\{\pm 1\}} B_2 \left( \left\langle \frac{\frac{1}{2}(s+\overline{s}) }{p^k} \right\rangle \right) } w^{-i} \in \mathbb{Q}[H] $$ such that, under the identification of the group $ H $ with the set of cusps of $ X^+_{ns}(p^k) $, the ideal $ \mathbb{Z}[H]\theta \cap \mathbb{Z}[H] $ represents the group of divisors of units of $ X^+_{ns}(p^k)$. The Cuspidal Divisor Class Group on $ X^+_{ns}(p^k) $ is a module over $\mathbb{Z}[H]$ and, more precisely, we have: }
$$ \mathfrak{C}^+_{ns}(p^k) \cong \mathbb{Z}_0[H] / (\mathbb{Z}[H]\theta \cap \mathbb{Z}[H]).
$$
From the previous statement we will show another result which has a counterpart in cyclotomic field theory. \\
\noindent
\textbf{Theorem \ref{Cardinalità}} \textit{For any character $ \chi $ of $C_{ns}(p^k)/\{\pm I\}$ (identified with an even character of $ C_{ns}(p^k) $), we let:}
$$ B_{2,\chi} = \sum_{\alpha \in C_{ns}(p^k)/\{\pm I\} } B_2 \left( \left\langle \frac{T(\alpha)}{p^k} \right\rangle \right) \chi(\alpha) $$
\textit{where $ B_2(t) = t^2 - t + \frac{1}{6} $ is the second Bernoulli polynomial and $T$ is a certain $ (\mathbb{Z}/p^k\mathbb{Z})$-linear map. Then we have:}
$$ |\mathfrak{C}^+_{ns}(p^k)| =
\displaystyle 24\frac{ \displaystyle\prod_{}{\frac{p^k}{2}B_{2,\chi}}}{\gcd(12,p+1)(p-1)p^{k-1}} $$
\textit{where the product runs over all nontrivial characters $ \chi $ of $ C_{ns}(p^k)/{\pm I} $ such that $ \chi(M)=1 $ for every $ M \in C_{ns}(p^k) $
with $ \det M = \pm 1 $.\\
In particular, for $ k=1 $ let $ \omega$ be a generator of the character group of $ C_{ns}(p) $ and $ v $ a generator of $ \mathbb{F}_{p^2}^* $. Then:}
$$ |\mathfrak{C}^+_{ns}(p)| = \displaystyle \frac{24}{(p-1)\gcd(12,p+1)}\prod_{j=1}^{\frac{p-3}{2}}\frac{p}{2}B_{2,\omega^{(2p+2)j}} = $$
$$ = \displaystyle\frac{ 576 \left| \det\left[\displaystyle\frac{p}{2}\left(\displaystyle\sum_{l=0}^{p}B_2\left( \left\langle \frac{\frac{1}{2}\mbox{Tr}(v^{i-j+l\frac{p-1}{2}}) }{p} \right\rangle \right) - \frac{p+1}{6} \right) \right]_{1\le i,j \le \frac{p-1}{2}} \right|}{(p-1)^2 p (p+1)\gcd(12,p+1)}. $$
This theorem could be considered analogous to the relative class number formula \cite[Theorem 4.17]{Washington}:
$$ h^-_m = Q w \prod_{\chi \mbox{ odd}} -\frac{1}{2}B_{1,\chi} $$
where $ Q=1 $ if $ m $ is a prime power and $ Q=2 $ otherwise, $ w $ is the number of roots of unity in $ \mathbb{Q}(\zeta_m) $ and we encounter the classical generalized Bernoulli numbers:
$$ B_{1,\chi} := \sum_{a=1}^{m}\chi(a)B_1\left( \frac{a}{m} \right)= \frac{1}{m} \sum_{a=1}^{m} \chi(a)a \mbox{ for } \chi\not=1.$$
In the last section we will explicitly calculate $ |\mathfrak{C}^+_{ns}(p)| $ for some $ p \le 101 $. Consider the isogeny (cfr.\cite[Paragraph 6.6]{Diamond:mf}):$$ {J_0^+}^{new}(p^2) \longrightarrow \mathop{\bigoplus_{f}} A'_{p,f} $$
where the sum is taken over the equivalence classes of newforms $ f\in S_2(\Gamma^+_0(p^2))$. From Theorems \ref{x1}, \ref{x2} and \ref{x3} we deduce that:
$$
|\mathfrak{C}^+_{ns}(p)| \mbox{ divides } \prod_f \mathop \textbf{\mbox{ gcd }}_{ \begin{scriptsize} \begin{array}{c} q \mbox{ prime}, q \nmid |\mathfrak{C}^+_{ns}(p)|, \\ q \equiv \pm 1 \mbox{ mod }p \end{array} \end{scriptsize} } |A'_{p,f}(\mathbb{F}_{q})|. $$
Using the modular form database of W.Stein, we will find out that
for $ p \le 31 $:
$$|\mathfrak{C}^+_{ns}(p)| = \prod_f \mathop
\textbf{\mbox{ gcd }}_{ \begin{scriptsize} \begin{array}{c} q < 500 \mbox{ prime}, \\ q \equiv \pm 1 \mbox{ mod }p \end{array} \end{scriptsize} } |A'_{p,f}(\mathbb{F}_{q})|. $$
\section{Galois groups of modular function fields}
Following \cite[Chapter 1]{Silverman}, let
$ \mathbb{H} = \{x + iy \;| y > 0; x, y \in \mathbb{R} \} $
be the upper-half plane and \textit{n} a positive integer.
The principal congruence subgroup of level \textit{n}
is the subgroup of
$ SL_2(\mathbb{Z}) $
defined as follows:
\begin{center}
$ \Gamma(n) := \left\{\begin{pmatrix} a&b\\c&d \end{pmatrix} \in SL_2({\mathbb{Z}}) : a\equiv d\equiv 1,~b\equiv c\equiv 0\mod n\right\}. $
\end{center}
Then the quotient space
$ \Gamma(n) \textbackslash \mathbb{H} $
is complex analytically isomorphic to an affine curve
$ Y(n) $
that can be compactified by considering
$ \mathbb{H}^*:= \mathbb{H} \cup \mathbb{Q} \cup \{\infty\} $
and by taking the extended quotient:
\begin{center}
$X(n)= \Gamma(n)\textbackslash \mathbb{H}^* = Y(n)\cup\Gamma(n)\textbackslash (\mathbb{Q}\cup\{\infty\}).$
\end{center}
The points $ \Gamma(n) \tau $ in $ \Gamma(n)\textbackslash (\mathbb{Q}\cup\{\infty\}) $ are called the cusps of $ \Gamma(n) $
and can be described by the fractions \textit{s=$\frac{a}{c}$} with $0 \leq a \leq n-1 $, $0 \leq c \leq n-1 $ and gcd(\textit{a,c})=1.
As a consequence, it is not difficult to infer that $ X(n) $ has
$\displaystyle \frac{1}{2} n^2 \displaystyle\prod_{p | n} \Big(1- \frac{1}{p^2}\Big) $
cusps.
\\
Let $F_{n,\mathbb{C}}$ the field of modular functions of level $n$. A classical result states that $F_{1,\mathbb{C}} = \mathbb{C}(j)$ where \textit{j} is the Klein's \textit{j}-invariant.
We shall now find generators for $F_{n,\mathbb{C}}$. Consider:
\begin{center}
$ f_0(w;\tau) = -2^7 3^5 \displaystyle\frac{g_2(\tau)g_3(\tau)}{\Delta(\tau)}\wp(w;\tau,1),$
\end{center}
where $ \Delta$ is the modular discriminant, $ \wp$ is the Weierstrass elliptic function, $ \tau \in \mathbb{H} $, $w \in \mathbb{C}$ and $ g_2 = 60G_4 $ and $ g_3 = 140G_6 $ are constant multiples of the Eisenstein series:
$$ G_{2k}(\tau) = \sum_{{ (m,n)\in\mathbf{Z}^2\backslash(0,0)}} \frac{1}{(m+n\tau )^{2k}}.
$$
For $ r,s \in \mathbb{Z} $ and not both divisible by \textit{n} we define $ f_{r,s}= f_{0}(\frac{r\tau +s}{n}; \tau) $.
Whereas the Weierestrass $ \wp$-function is elliptic with respect to the lattice $[\tau,1]$ it follows that $ f_{r,s} $ depends only on the residue of $r,s$ mod $n$. Thus, it is convenient to use a notation emphasizing this property.
If $ a=(a_1,a_2) \in \mathbb{Q}^2$ but $a \not\in \mathbb{Z}^2$ we call the functions $ f_a(\tau)=f_0(a_1\tau+a_2;\tau)$ the Fricke functions. They depend only on the residue class of $a $ mod $ \mathbb{Z}^2 $.
\begin{thm}
We have:
$$ \mbox{ Gal}( F_{n,\mathbb{C}}, F_{1,\mathbb{C}})\cong SL_2(\mathbb{Z}/n\mathbb{Z})/ \{\pm I \}. $$
\end{thm}
\begin{proof}
There is a surjective homeomorphism (see \cite[pag.279]{Diamond:mf} and \cite[pag.65]{Lang:ef}):
\begin{center}
$ \theta: SL_2(\mathbb{Z}) \longrightarrow $ Aut $(\mathbb{C}(X(n))), $
\end{center}
\begin{center}
$ \gamma \longmapsto $
$ (f \longmapsto f^{(\theta(\gamma))} = f \circ \gamma). $
\end{center}
From Ker$(\theta) = \pm \Gamma(n) $ and the relations $ f_a (\gamma(\tau)) = f_{a\gamma}(\tau) $ it follows easily that Gal$( F_{n,\mathbb{C}}, F_{1,\mathbb{C}})\cong \Gamma(1)/\pm \Gamma(n) \cong SL_2(\mathbb{Z}/n\mathbb{Z})/ \{\pm I \} $.
\end{proof}
We say that a modular form in $ F_{n,\mathbb{C}} $ is defined over a field if all the coefficients of its $q$-expantion lie in that field and analogously for every $\mbox{Gal}( F_{n,\mathbb{C}}, F_{1,\mathbb{C}})$-conjugate of the form. Let:
\begin{center}
\begin{flushleft}
$ F_n= $
function field on $ X(n) $ consisting of those functions which are defined over the \textit{n}-th cyclotomic field
$ \mathbb{Q}_n = \mathbb{Q}(\zeta_n) $.
\end{flushleft}
\end{center}
\begin{thm}\label{fricke}
The field $ F_n $ has the following properties:\\
(1) $ F_n $ is a Galois extension of $ F_1 = \mathbb{Q}(j) $.\\
(2) $ F_n = \mathbb{Q}(j,f_{r,s})
_{all (r,s)\in \frac{1}{n}\mathbb{Z}^2 \smallsetminus \mathbb{Z}^2 } $. \\
(3) For every $ \gamma \in GL_2(\mathbb{Z}/n\mathbb{Z})$ the map $ f_a \mapsto f_{a\gamma}$ gives an element of Gal$(F_n,\mathbb{Q}(j))$ which we write $\theta(\gamma) $. Then $ \gamma \mapsto \theta(\gamma) $ induces an isomorphism of $ GL_2(\mathbb{Z}/n\mathbb{Z})/{\pm I} $ to Gal$(F_n,\mathbb{Q}(j))$. The subgroup $ SL_2(\mathbb{Z}/n\mathbb{Z})/{\pm I} $ operates on a modular function by composition with the natural action of $ SL_2(\mathbb{Z})$ on the upper half-plane $\mathbb{H} $. \\
Furthermore the group of matrices $ \begin{pmatrix} 1&0\\0&d \end{pmatrix} $ operates on $ F_n $ as follows: \\
for $d \in (\mathbb{Z}/n\mathbb{Z})^* $ consider the automorphism $ \sigma_d $ of $ \mathbb{Q}_n $ such that $ \sigma_d(\zeta_n)= \zeta_n^d $. Then $ \sigma_d $ extends to $ F_n $ by operating on the coefficients of the power series expansions:
\begin{center}
$ \sigma_d (\sum{a_i q^{i/n}})= \sum{\sigma_d(a_i) q^{i/n}} $ with $ q=e^{2\pi i \tau}$. \\
\end{center}
If $ (r,s) \in \frac{1}{n}\mathbb{Z}^2 \setminus \mathbb{Z}^2 $, we have: $\sigma_d(f_{r,s}(\tau))=f_{r,sd}(\tau). $
\end{thm}
\begin{proof}
\cite[Theorem 6.6]{Shimura:af}
\end{proof}
\section{Modular Units and Manin-Drinfeld Theorem}
In this paper we will focus our attention on the modular units of $ X(n) $. In other words, the invertible elements of the integral closure of $ \mathbb{Q}[j] $ in $ F_n $. The only pole of \textit{j}$ (\tau) $ is at infinity. So, from the algebraic characterization of the integral closure as the intersection of all valuation subrings containing the given ring, the modular units in $ F_n $ are exactly the modular functions which have poles and zeros exclusively at the cusps of $ X(n) $.
Let $ \mathfrak{D}_n \simeq \bigoplus_{\footnotesize{\mbox{cusps}}} \mathbb{Z} $ be the free abelian group of rank $ {\frac{1}{2} n^2 \prod_{p | n} (1- \frac{1}{p^2})} $ generated by the cusps of $ X(n) $. Let $ \mathfrak{D}_{n,0} $ be its subgroup consisting of elements of degree $ 0 $ and let $ \mathfrak{F}_n $ be the subgroup generated by the divisors of modular units in the modular function field $ F_n $. The quotient group:
\begin{center}
$ \mathfrak{C}_n := \mathfrak{D}_{n,0} / \mathfrak{F}_n $
\end{center}
is called the Cuspidal Divisor Class Group on $ X(n) $. The previous definition generalizes \textit{mutatis mutandis} to every modular curve $ X_\Gamma $ where $ \Gamma $ is a modular subgroup. Manin and Drinfeld proved that:
\begin{thm}\label{mandrinf}
If $ \Gamma $ is a congruence subgroup then all divisors of degree 0 whose support is a subset of the set of cusps of $ X_\Gamma $ have a multiple that is a principal divisor. In other word if $ x_1,x_2 \in X_\Gamma $ are cusps, then $ x_1 - x_2 $ has finite order in the jacobian variety $ Jac(X_\Gamma) $.
\end{thm}
\begin{proof}
Let $ x_1,x_2$ two cusps in $ X_\Gamma $. Denote by $ \{x_1,x_2 \} \in (\Omega^1(X_\Gamma))^*$ the functional on the space of differential of the first kind given by:
$$ \{x_1, x_2 \}: \omega \mapsto \int_{x_1}^{x_2}\omega. $$
\textit{A priori} we have $ \{x_1,x_2\} \in H_1(X_\Gamma, \mathbb{R}) $. Manin and Drinfeld showed that it lies in $H_1(X_\Gamma, \mathbb{Q})$. Cf. \cite{Drinfeld}, \cite[Chapter IV]{Lang:mf} and \cite{Manin}.
\end{proof}
\section{Siegel Functions and Cuspidal Divisor Class Groups}
Let $ n=p^k $ with $ p \ge 5 $ prime. Following \cite{KL} we will give an explicit description of modular units of $ X(n) $ and its cuspidal divisor class group. \\
Let $L$ a lattice in $ \mathbb{C} $. Define the Weierstrass sigma function:
$$ \sigma_L (z) = z \displaystyle\prod_{ \begin{scriptsize} \begin{array}{c} \omega \in L \\ \omega \not=0 \end{array} \end{scriptsize} }{ \left( 1 - \frac{z}{\omega} \right) e^{z/\omega + \frac{1}{2} (z/\omega)^2 } }, $$
\noindent which has simple zeros at all non-zero lattice points. Define:
$$ \zeta_L (z) = \frac{d}{dz} \log (\sigma_L(z)) = \displaystyle{\frac{1}{z} + \displaystyle\sum_{ \begin{scriptsize} \begin{array}{c} \omega \in L \\ \omega \not=0 \end{array} \end{scriptsize} }{\left[\frac{1}{z-\omega} +\frac{1}{\omega} + \frac{z}{\omega^2} \right]} }, $$
$$ \wp_L(z)= -\zeta'_L(z) = \frac{1}{z^2} +
\displaystyle\sum_{ \begin{scriptsize} \begin{array}{c} \omega \in L \\ \omega \not=0 \end{array} \end{scriptsize} }{\left[\frac{1}{(z-\omega)^2} - \frac{1}{\omega^2} \right]}.
$$
\noindent
If $ \omega \in L $, by virtue of the periodicity of $ \wp_L $ we obtain $ \frac{d}{dz} \zeta_L(z+\omega) = \frac{d}{dz}\zeta_L(z)$, whence follows the existence of a $ \mathbb{R}$-linear function $ \eta_L(z) $ such that:
$$ \zeta_L(z+\omega) = \zeta_L(z) + \eta_L(\omega).$$
For $ L=[\tau,1] $ (with $\tau \in \mathbb{H} $) and $ a=(a_1,a_2) \in \mathbb{Q}^2 \smallsetminus \mathbb{Z}^2 $ we define the Klein forms:
$$ \mathfrak{k}_a(\tau) = e^{-\eta_L(a_1\tau+a_2)}\sigma_L(a_1\tau+a_2).$$
Note that $ z=a_1\tau+a_2 \not\in L=[\tau,1] $ so we know directly from their definition that the Klein forms are holomorphic functions which have no zeros and poles on the upper half plane.
When $ \Gamma $ is a congruence subgroup and $ k $ is an integer, we will say that a holomorphic function $ f(\tau) $ on $ \mathbb{H}$ is a nearly holomorphic modular form for $ \Gamma $ of weight $ k $ if:\\
(i) $ f(\gamma(\tau))= (r\tau +s)^k f(\tau) $ for all $\gamma=\begin{pmatrix} p&q\\r&s \end{pmatrix} \in \Gamma $;\\
(ii) $ f(\tau) $ is meromorphic at every cusp.
\begin{prop}\label{Kleinforms}
Let $ a=(a_1,a_2) \in \mathbb{Q}^2 \smallsetminus \mathbb{Z}^2 $ and $ b=(b_1,b_2) \in \mathbb{Z}^2 $. The Klein Forms $ \mathfrak{k}_a(\tau) $ have the following properties: \\
(1) $ \mathfrak{k}_{-a}(\tau) = - \mathfrak{k}_a(\tau); $\\
(2) $ \mathfrak{k}_{a+b} = \epsilon(a,b)\mathfrak{k}_a(\tau) $ with $ \epsilon(a,b)=(-1)^{b_1 b_2 + b_1 + b_2} e^{- \pi i (b_1 a_2 - b_2 a_1)}; $\\
(3) For every $\gamma=\begin{pmatrix} p&q\\r&s \end{pmatrix} \in SL_2(\mathbb{Z}) $ we have:
$$ \mathfrak{k}_a (\gamma(\tau)) = \mathfrak{k}_a \left(\frac{p\tau+q}{r\tau+s}\right) = \frac{\mathfrak{k}_{a\gamma}(\tau)}{r \tau + s} =
\frac{\mathfrak{k}_{(a_1 p + a_2 r , a_1 q + a_2 s )}(\tau)}{r \tau + s}; $$
(4) If $ n \ge 2 $ and $ a \in \frac{1}{n}\mathbb{Z}^2 \setminus \mathbb{Z}^2 $ then $ \mathfrak{k}_a(\tau) $ is a nearly holomorphic modular form for $ \Gamma(2n^2) $ of weight -1. \\
(5) Let $ n \ge 3 $ odd and $ \{m_a\}_{a \in \frac{1}{n}\mathbb{Z}^2 \setminus \mathbb{Z}^2} $ a family of integers such that $ m_a \not= 0 $ occurs only for finitely many $a$. Then the product of Klein form:
$$ \prod_{a \in \frac{1}{n}\mathbb{Z}^2 \setminus \mathbb{Z}^2} \mathfrak{k}_a^{m_a}(\tau) $$
is a nearly holomorphic modular form for $ \Gamma(n)$ of weight $ -\sum_a m_a $ if and only if:
$$ \sum_a {m_a (n a_1)^2 } \equiv \sum_a {m_a (n a_2)^2 } \equiv \sum_a {m_a (n a_1)(n a_2) \equiv 0 \mod n }. $$
\end{prop}
\begin{proof}
Property (2) is nothing more than a reformulation of the Legendre relation: $ \eta_{[\tau,1]}(1)\tau - \eta_{[\tau,1]}(\tau) = 2 \pi i $. Property (5) is discussed in \cite[Chapter 3, Paragraph 4]{KL}.\\
For more details see: \cite[Chapters 2 and 3]{KL} or \cite[Chapter 19]{Lang:ef}.
\end{proof}
We are now ready to define the Siegel function:
$$ g_a(\tau) = \mathfrak{k}_a(\tau)\Delta(\tau)^{1/12},$$
\noindent
where $ \Delta(\tau)$ is the square of the Dedekind eta funtion $ \eta(\tau) $ (not to be mistaken for the aforementioned $\eta_L(\tau) $):
\begin{center}
$ \eta(\tau)^2 = 2 \pi i q^{1/12}\prod_{k=1}^{\infty}(1-q^n)^2 $ with $ q=e^{2\pi i \tau} .$
\end{center}
\begin{prop}\label{sigfun}
The set of functions $ \{h_a(\tau)=g_a(\tau)^{12n} \} _{a \in \frac{1}{n}\mathbb{Z}^2 \setminus \mathbb{Z}^2} $ constitute a Fricke family. Just like the Fricke functions $ f_a(\tau) $ of Theorem \ref{fricke} we have: $ h_a(\tau) \in F_n $, for every $ \gamma \in SL_2({\mathbb{Z}}) $ we have $ h_a(\gamma(\tau))=h_{a\gamma}(\tau) $ and in addition if $ \sigma_d \in Gal(\mathbb{Q}_n, \mathbb{Q}) $ then $ \sigma_d (h_{a_1, a_2}(\tau)) = h_{a_1, d a_2}(\tau) $. In other words, the Siegel functions, raised to the appropriate power, are permuted by the elements of the Galois Group Gal$( F_n,\mathbb{Q}(j) )$.
\end{prop}
\begin{proof}
\cite[Chapter 2]{KL} or \cite{Siegel}.
\end{proof}
\begin{thm}\label{unitàmodulari}
Assume that $ n=p^k $ for $ p \not=2,3.$ Then the units in $ F_n $ (modulo constants) consist of the power products:
$$ \prod_{a \in \frac{1}{n}\mathbb{Z}^2 \setminus \mathbb{Z}^2} g_a^{m_a}(\tau) $$
with:
$$ \sum_a {m_a (n a_1)^2 } \equiv \sum_a {m_a (n a_2)^2 } \equiv \sum_a {m_a (n a_1)(n a_2) \equiv 0 \mod n } $$
and $$ \sum_a{m_a} \equiv 0 \mod 12. $$
\noindent
In addition, if $ k \ge 2 $ it is not restrictive to consider power products of Siegel functions $g_a$ with primitive index $a=(a_1,a_2)$, namely such that $ p^{k-1}a \not\in \mathbb{Z}^2 $.
\end{thm}
\begin{proof}
See \cite{Siegelgenerator}, \cite[Theorem 3.2, Chapter 2]{KL}, \cite[Theorem 5.2, Chapter 3]{KL} and \cite[Theorem 1.1, Chapter 4]{KL} . The last assertion is a consequence of the distribution relations discussed in \cite[pp. 17-23]{KL}.
\end{proof}
Following \cite{KL} it will be useful to decompose Gal($ F_{p^{k}},\mathbb{Q}(j) $). Let $ \mathfrak{o}_p $ the ring of integers in the unramified quadratic extension of the $p$-adic field $ \mathbb{Q}_p $. The group of units $ \mathfrak{o}_p^* $ acts on $ \mathfrak{o}_p $ by multiplication and after choosing a basis of $ \mathfrak{o}_p $ over the $p$-adic ring $ \mathbb{Z}_p $, we obtain an embedding:
$$ \mathfrak{o}_p^* \longrightarrow GL_2(\mathbb{Z}_p).$$
\noindent
We call the image in $ GL_2(\mathbb{Z}_p) $ the Cartan Group at the prime $ p $ and indicate it by $C_p $. It is worth noting that the elements of $ \mathfrak{o}_p^* $, written in terms of a basis of $ \mathfrak{o}_p $ over $ \mathbb{Z}_p $, are characterized by the fact that at least one of the two coefficients is a unit.
Consider now $ GL_2(\mathbb{Z}_p) $ as operating on $\mathbb{Z}_p^2 $ on the left and denote by $ G_{p,\infty} $ the isotropy group of $ \begin{pmatrix}
1 \\ 0
\end{pmatrix} $. Obviously we have:
\begin{center}
$ G_{p,\infty} = \left\{ \begin{pmatrix} 1&b\\0&d \end{pmatrix} b \in \mathbb{Z}_p , d \in \mathbb{Z}_p^* \right\}. $
\end{center}
Since $ C_p $ operates simply transitively on the set of primitive elements (that is: vectors whose coordinates are not both divisibile by $ p $) we have the following decomposition:
$$ GL_2(\mathbb{Z}_p)/\{\pm I\} = ( C_p/\{\pm I\} ) G_{p,\infty}. $$
\noindent
For each integer $ k $ we define the reduction of the Cartan Group $ C_p \mod p^k$:
$$ C(p^k)= C_p / p^k C_p $$
\noindent
and let $ G_\infty(p^k) $ the reduction of $ G_{p,\infty} \mod p^k $:
\begin{center}
$ G_\infty(p^k) = \left\{ \begin{pmatrix} 1&b\\0&d \end{pmatrix} b \in \mathbb{Z}/p^k\mathbb{Z} , d \in (\mathbb{Z}/p^k\mathbb{Z})^* \right\}.$
\end{center}
\noindent
We can now reformulate the previous decomposition as follows:
\begin{center}
Gal($ F_{p^k} $,$ \mathbb{Q}(j) $) $ \simeq GL_2( \mathbb{Z}/p^k\mathbb{Z})/\{\pm I\} = ( C(p^k)/\{\pm I\} ) G_{\infty}(p^k). $
\end{center}
\noindent
The embedding:
$$ F_{p^k} \hookrightarrow \mathbb{Q}(\zeta_{p^k})((q^{1/{p^k}})) $$
enables us to mesaure for each modular function $ f(\tau) \in F_{p^k} $ its order at $ \Gamma(p^k) \infty $ in term of the local parameter $ q^{1/{p^k}} $.
\begin{prop}
If $ a \in \frac{1}{p^k} \mathbb{Z}^2 \setminus \mathbb{Z}^2 $, the $q$-expansion of the Siegel functions shows that:
\begin{center}
ord$_\infty (g_a(\tau))^{12p^k} = 6p^{2k} B_2(\langle a_1 \rangle)$
\end{center}
\noindent
where $ B_2(X) = X^2 -X + \frac{1}{6} $ is the second Bernoulli polynomial and $ \langle X \rangle $ is the fractional part of $ X $.
\end{prop}
\begin{proof}
\cite[Chapter 19]{Lang:ef}.
\end{proof}
For every automorphism $ \sigma \in $ Gal($ F_{p^{k}},\mathbb{Q}(j) $) and each $h(\tau) \in F_{p^k}$ we have the prime $ \sigma^{-1}(\infty) $ which is such that:
\begin{center}
ord$_{\sigma^{-1}(\infty)} (h(\tau)) = $ ord$_\infty \sigma (h(\tau)) $
\end{center}
\noindent
and if $ \sigma \in G_\infty(p^k) $:
\begin{center}
ord$_\infty (h(\tau)) = $ ord$_\infty \sigma (h(\tau)), $
\end{center}
\noindent
so we may identify the cusps of $ X(p^k) $ with the elements of the Cartan Group (viewing it as a subgroup of Gal($ F_{p^{k}},\mathbb{Q}(j) $)). From now on, we will indicate the cusp $ \sigma^{-1}(\infty) $ simply by $ \sigma^{-1} $. \\
We may also index the primitive Siegel function by elements of the Cartan Group. Following \cite{KL}, if $ \alpha \in C(p^k)/\{\pm I\} $ we put:
\begin{center}
$ g_\alpha = g_{e_1 \alpha } $ where $ e_1=(\frac{1}{p^k},0). $
\end{center}
\noindent
It should be noted that $ g_\alpha $ is defined up to a root of unity (this follows from Proposition \ref{Kleinforms}, second claim). Nonetheless, $ g_\alpha ^{12p^k} $ is univocally defined as well as its divisor:
\begin{prop}\label{divisori} We have:
$$ \mbox{div } g_\alpha^{12p^k} = 6p^{2k} \sum_{\beta \in C(p^k)/\{\pm I\} } B_2 \left( \left\langle \frac{T(\alpha\beta^{-1})}{p^k} \right\rangle \right) \beta $$
\noindent
where the map T on $ 2 \times 2 $ matrices is defined as follows:
$$ T:\begin{pmatrix} a&b\\c&d \end{pmatrix} \mapsto a . $$
\end{prop}
\begin{proof}
See \cite[Paragraph 5.1]{KL}
\end{proof}
The first part of \cite{KL} culminates with the theorem below. The computation of the order of the cuspidal divisor class group on $ X(p^k) $ could be considered analogous to that in the study of cyclotomic fields: instead of the generalized Bernoulli numbers $ B_{1,\chi} $ encountered in the latter case, in the former we will define the second generalized Bernoulli numbers $ B_{2,\chi} $.
\begin{thm}\label{CDCG}
Let $ p $ a prime $ \ge 5 $. Let $ R:= \mathbb{Z}[C(p^k)/\{\pm 1\}] $ and $ R_0 $ the ideal of $ R $ consisting of elements of degree $ 0$. The Cuspidal Divisor Class Group $ \mathfrak{C}_{p^k} $ is an $R-$module, more precisely there exists a Stickelberger element
$$ \theta = \displaystyle \frac{p^k}{2}\sum_{\beta \in C(p^k)/\{\pm 1\} } B_2 \left( \left\langle\frac{T(\beta)}{p^k} \right\rangle \right) \beta^{-1} \in \mathbb{Q}[C(p^k)/\{\pm 1\}]$$ such that:
$$ \mathfrak{C}_{p^k} \cong R_0 / R \cap R \theta. $$
For any character $ \chi $ of $C(p^k)/\{\pm I\}$ (identified with an even character of $ C(p^k) $) we let:
$$ B_{2,\chi} = \sum_{\alpha \in C(p^k)/\{\pm I\} } B_2 \left( \left\langle \frac{T(\alpha)}{p^k} \right\rangle \right) \chi(\alpha). $$
\noindent
The order of the cuspidal divisor class group on $ X(p^k) $ is:
$$ | \mathfrak{C}_{p^k} | = \frac{12 p^{3k}}{|C(p^k) |} \prod_{\chi \not= 1} \frac{p^k}{2} B_{2,\chi}. $$
\end{thm}
\begin{proof}
\cite[Chapter 5]{KL}.
\end{proof}
\section{Non-split Cartan Groups}
Following \cite{BB} or \cite[pag. 194]{SerreMordel}, let $n $ a positive integer and let $ A $ be a finite free commutative algebra of rank $ 2 $ over $ \mathbb{Z}/n\mathbb{Z}$ with unit discriminant. Fixing a basis for $ A $ we can use the action of $ A^* $ on $ A $ to embed $ A^* $ in $ GL_2(\mathbb{Z}/n\mathbb{Z}) $. If for every prime $ p|n $ the $ \mathbb{F}_p $ algebra $ A/pA $ is isomorphic to $ \mathbb{F}_{p^2} $, the image of $ A^* $ just now described is called a non-split Cartan subgroup of $ GL_2(\mathbb{Z}/n\mathbb{Z}) $.
Therefore, such a group $ G $ has the property that for every prime $p$ dividing $n$ the reduction of $G\mod p $ is isomorphic to $ \mathbb{F}_{p^2}^* $. All the non-split Cartan subgroups of $ GL_2({\mathbb{Z}/n\mathbb{Z}}) $ are conjugate and so are their normalizers.
In this paper we are interested in the case $ n=p^k$ and $ p \not= 2,3$. The cases $p=2$ and $ p=3 $ are essentially equal but require more cumbersome calculations (see \cite[Theorem 5.3, Chapter 3]{KL} and \cite[Theorem 1.3, Chapter 4]{KL}). Choose a squarefree integer $ \epsilon \equiv 3 \mod 4 $ and such that its reduction modulo $ p$ is a quadratic non-residue. If $ p \equiv 3 \mod 4 $, a canonical choice could be $ \epsilon = -1 $.
Let $ K= \mathbb{Q}(\sqrt{\epsilon}) $ and $ \mathbf{O}_K = \mathbb{Z}[\sqrt{\epsilon}]$ its ring of integers. After choosing a basis for $ \mathbf{O}_K $ over $ \mathbb{Z} $ we can represent any element of $ (\mathbf{O}_K / p^k \mathbf{O}_K)^* $ with its corresponding multiplication matrix in $ GL_2({\mathbb{Z}/p^k\mathbb{Z}}) $ with respect to the chosen basis. This embedding produces a non-split Cartan subgroup of $ GL_2({\mathbb{Z}/p^k\mathbb{Z}}) $ and we will denote it by $ C_{ns}(p^k) $. Notice that such a group is isomorphic to the already introduced $ C(p^k) $.
To describe the normalizer $ C_{ns}^+(p^k) $ of $ C_{ns}(p^k) $ in $ GL_2({\mathbb{Z}/p^k\mathbb{Z}}) $ it will suffice to consider the following group automorphism induced by conjugation by a fixed $c \in C_{ns}^+(p^k) $:
$$ \phi_c : C_{ns}(p^k) \longrightarrow C_{ns}(p^k) $$
$$ x \longmapsto \phi_c(x) = cxc^{-1}. $$
\noindent
The group automorphism $ \phi_c $ extends to a ring automorphism of $ (\mathbf{O}_K / p^k \mathbf{O}_K) \cong (\mathbb{Z}/p^k\mathbb{Z})[\sqrt{\epsilon}] $ so if $ \phi_c $ is not the trivial automorphism we necessarily have $ \phi_c(\sqrt{\epsilon})=-\sqrt{\epsilon} $.
\begin{prop}\label{strutturanonsplit} If $ p \not= 2 $ we have the following isomorphism:
$$ C_{ns}(p^k) \simeq \mathbb{Z}/p^{k-1}\mathbb{Z} \times \mathbb{Z}/p^{k-1}\mathbb{Z} \times \mathbb{Z}/(p^2-1)\mathbb{Z}, $$
$$ C_{ns}^+(p^k) \simeq (\mathbb{Z}/p^{k-1}\mathbb{Z} \times \mathbb{Z}/p^{k-1}\mathbb{Z} \times \mathbb{Z}/(p^2-1)\mathbb{Z}) \rtimes_{\phi} \mathbb{Z}/2\mathbb{Z}. $$
\end{prop}
\begin{proof}
Let $ a_1 + \sqrt{\epsilon}a_2 \in (\mathbf{O}_K / p^k \mathbf{O}_K)$: it is invertible if and only if $ (a_1,a_2)$ is primitive or in other words $ p $ does not divide both $ a_1 $ and $ a_2 $ so we have $ |C_{ns}(p^k)| = p^{2k-2}(p^2-1) $.
Consider the reduction$\mod p $:
$$ C_{ns}(p^k) \longrightarrow \mathbb{F}_{p^2}^* $$
$$ a_1 + \sqrt{\epsilon}a_2 \longmapsto \overbar{a_1} + \sqrt{\epsilon}\overbar{a_2}. $$
The map is surjective and let $B$ its kernel:
$$ B:=\{x \in (\mathbf{O}_K / p^k \mathbf{O}_K)^* \mbox{ such that } x \equiv 1\mod p \}. $$
$ |B|=p^{2k-2} $: it remains to check that $ B \simeq \mathbb{Z}/p^{k-1}\mathbb{Z} \times \mathbb{Z}/p^{k-1}\mathbb{Z} $.
Let $ k\ge2 $ and $ p \not=2 $.
First, we check that for all $ x \in \mathbf{O}_K $ we have $ (1+xp)^{p^{k-2}} \equiv 1+xp^{k-1} \mod p^k $. In case $ k=2 $ there is nothing to prove. We proceed by induction on $ k $: suppose the claim is true for some $ k\ge2 $. We have:
$$ (1+xp)^{p^{k-2}} = 1+xp^{k-1} + yp^k, $$
$$ (1+xp)^{p^{k-1}} = \sum_{j=0}^p {p \choose j}{(1+xp^{k-1})}^{p-j}({yp^k})^j \equiv (1+xp^{k-1})^p \mod p^{k+1}, $$
$$ (1+xp^{k-1})^p = \sum_{j=0}^p {p \choose j}(xp^{k-1})^j \equiv 1 + xp^k \mod p^{k+1}. $$
\noindent
In conclusion: $ (1+xp)^{p^{k-1}} \equiv 1 + xp^k \mod p^{k+1} $. From the previous claim follows that if $ h\le k-1$ is such that $ x \in p^h\mathbf{O}_K \setminus p^{h+1}\mathbf{O}_K $ then the reduction of $ 1+xp $ in $ B $ has order $ p^{k-1-h} $. So $B $ has $ p^{2k-2}-p^{2k-4}$ elements of order $ p^{k-1} $ and the proposition is proved. The second isomorphism follows immediately.
\end{proof}
We present now the modular curves $ X_{ns}(n) $ and $ X_{ns}^+(n) $ associated to the subgroups $ C_{ns}(n) $ and $ C_{ns}^+(n) $. First of all, $ Y(n) $ (the non-cuspidal points of $ X(n) $) are isomorphism classes of pairs $(E,(P,Q)) $ where $ E $ is a complex elliptic curve and $ (P,Q) $ constitute a $ \mathbb{Z}/n\mathbb{Z}$-basis of the $n$-torsion subgroup $ E[n]$ with $ e_n(P,Q)=e^{2\pi i / n}$ where $ e_n $ is the Weil pairing discussed in details in \cite[Chapter 7]{Diamond:mf}.
By definition, two pairs $(E,(P,Q)) $ and $(E',(P',Q')) $ are considered equivalent in $ Y(n) $ if and only if there exists an isomorphism between $ E $ and $ E'$ taking $ P $ to $ P' $ and $ Q $ to $ Q'$. Notice that the definition is well-posed since the Weil pairing is invariant under isomorphism, i.e. if $ f: E \rightarrow E' $ is an isomorphism of elliptic curves and $ e'_n $ is the Weil pairing on $ E' $ we have:
$$ e'_n(f(P),f(Q))=e_n(P,Q). $$
\noindent
Since $ GL_2(\mathbb{Z}/n\mathbb{Z}) $ acts on $ E[n] $ and since for every $ \gamma \in GL_2(\mathbb{Z}/n\mathbb{Z}) $ we have $ e_n(\gamma(P,Q)) = e_n(P,Q)^{\det \gamma} $,
the group $ SL_2(\mathbb{Z}/n\mathbb{Z}) $ acts on $ Y(n) $ on the right in the following way:
$$ \begin{pmatrix} a&b\\c&d \end{pmatrix} \cdot(E,(P,Q)) = (E,(aP+cQ,bP+dQ)). $$
Define:
$$ C'_{ns}(n) := C_{ns}(n) \cap SL_2(\mathbb{Z}/n\mathbb{Z}), $$
$$ C'^+_{ns}(n) := C^+_{ns}(n) \cap SL_2(\mathbb{Z}/n\mathbb{Z}), $$
$$ \Gamma_{ns}(n) := \{M \in SL_2({\mathbb{Z}}) \mbox{ such that } M \equiv M'\mbox{ mod } n\ \mbox{for some } M' \in C'_{ns}(n)\}, $$
$$ \Gamma^+_{ns}(n) := \{M \in SL_2({\mathbb{Z}}) \mbox{ such that } M \equiv M'\mbox{ mod } n\ \mbox{for some } M' \in C'^+_{ns}(n)\}. $$
\noindent A possible explicit description for these groups is:
$$ C_{ns}(p^k)= \left\{M_s=\begin{pmatrix} a&b\\\epsilon b&a \end{pmatrix} \in GL_2({\mathbb{Z}/p^k \mathbb{Z}}) \mbox{ with } s=a+\sqrt{\epsilon}b \in (\mathbf{O}_K / p^k \mathbf{O}_K)^* \right\}, $$
$$ C^+_{ns}(p^k) = \left\langle {\begin{pmatrix} a&b\\\epsilon b&a \end{pmatrix}} \in C_{ns}(p^k) \mbox{ , } {C=\begin{pmatrix} 1&0\\ 0&-1 \end{pmatrix}} \right\rangle. $$
\noindent If $ s \in (\mathbf{O}_K / p^k \mathbf{O}_K)^* $ we define $ |s|:= s \bar{s} \in (\mathbb{Z} / p^k \mathbb{Z})^*$ where $ \bar{s} $ is the conjugate of $ s $. So we have:
$$ C'_{ns}(p^k)= \left\{M_s=\begin{pmatrix} a&b\\\epsilon b&a \end{pmatrix} \in C_{ns}(p^k) \mbox{ such that } |s|=|a+\sqrt{\epsilon}b|=1 \mbox{ mod } p^k \right\}, $$
$$ C'^+_{ns}(p^k) = C'_{ns}(p^k) \cup \left\{M_s C=\begin{pmatrix} a&-b\\\epsilon b&-a \end{pmatrix} \mbox{ with } |s|=|a+\sqrt{\epsilon}b|=-1 \mbox{ mod } p^k \right\}. $$
Points in $ Y_{ns}(n) $ are nothing but orbits of $Y(n)$ under the action of $ C'_{ns}(n) $ and similarly for $ Y_{ns}^+(n) $ and $ C'^+_{ns}(n) $. The above-mentioned action extends uniquely to $ X(n) $. The quotients $ X_{ns}(n) $ and $ X^+_{ns}(n) $ are isomorphic as Riemann surfaces to $ \mathbb{H}^*/\Gamma_{ns}(n) $ and $ \mathbb{H}^*/\Gamma^+_{ns}(n) $ respectively.
Using the identification of the cusps of $ X(p^k) $ with the elements of $ C(p^k)/\{\pm I\}$ explained in the previous section we obtain a shorter proof of the first claim of \cite[Proposition 7.10]{BB}:
\begin{prop}\label{numerocuspidi} We identify the cusps of $ X_{ns}(p^k) $ with $ (\mathbb{Z}/p^k\mathbb{Z})^* $ and the cusps of $ X^+_{ns}(p^k) $ with $ H=(\mathbb{Z}/p^k\mathbb{Z})^*/\{\pm 1\} $. So $ X_{ns}(p^k) $ has $ p^{k-1}(p-1) $ cusps and $ X^+_{ns}(p^k) $ has $ p^{k-1}\frac{p-1}{2} $ cusps.
\end{prop}
\begin{proof}
We identify the cusps of $ X(p^k) $ with the elements of $ C(p^k)/\{\pm I\} \cong C_{ns}(p^k)/\{\pm I\} \cong (\mathbf{O}_K / p^k \mathbf{O}_K)^*/\{\pm 1\} $. Bearing this in mind, it is clear that $ \pm M_r, \pm M_{r'} \in C_{ns}(p^k)/\{\pm I\} $ represent the same cusp in $ X_{ns}(p^k) $ if and only if there exists $ s \in (\mathbf{O}_K / p^k \mathbf{O}_K)^* $ with $ |s|=1 $ such that $ \pm r=\pm s r' $. But this is equivalent to say that $ |r|=|sr'|=|r'| $ or $ \det M_r = \det M_{r'} \mod p^k$ and consequently we may identify the cusps of $ X_{ns}(p^k) $ with $ (\mathbb{Z}/p^k\mathbb{Z})^* $. For the same reason $ \pm M_r, \pm M_{r'} \in C_{ns}(p^k)/\{\pm I\} $ are indistinguishable in $ X^+_{ns}(p^k) $ if and only if they were already indistinguishable in $ X_{ns}(p^k) $ or there exists $ s' \in (\mathbf{O}_K / p^k \mathbf{O}_K)^* $ with $ |s'|=-1 $ such that $\pm r = \pm s'\overbar{r'} $ that is equivalent to say $ |r|=|s'\overbar{r'}|=-|r'| $ or $ \det M_r = - \det M_{r'} \mod p^k$. In conclusion we may identify the cusps of $ X^+_{ns}(p^k) $ with $ H=(\mathbb{Z}/p^k\mathbb{Z})^*/\{\pm 1\} $.
\end{proof}
\noindent
Furthermore, we can deduce that the covering $ \pi: X_{ns}(p^k) \to X(p^k) $ is not ramified above the cusps. So the ramification degree of a cusp of $ X_{ns}(p^k) $ under the covering projection $ \pi': X_{ns}(p^k) \to SL_2(\mathbb{Z})\textbackslash \mathbb{H}^* $, is equal to the one of a cusp of $ X(p^k) $ respect to $ \pi'': X(p^k) \to SL_2(\mathbb{Z})\textbackslash \mathbb{H}^* $ that is $ p^k $. The same happens for $X^+_{ns}(p^k) $.
\section{Modular units on non-split Cartan curves}
Let $ t \in ((\mathbf{O}_K / p^k \mathbf{O}_K)^*/\{\pm 1\}) $: write it in the form $ t=a_1 + \sqrt{\epsilon}a_2 $ choosing $ a_1, a_2 \in \mathbb{Z} $ such that $ 0 \le a_1 \le \frac{p^k-1}{2} $, $ 0 \le a_2 \le p^k-1$ and $a_2 \le \frac{p^k-1}{2}$ if $ a_1=0$. Define:
$$ [t] := \frac{1}{p^k}(a_1,a_2). $$
If $ s \in (\mathbf{O}_K / p^k \mathbf{O}_K)^* $ we define $ [s]:=[\{\pm s\}] $.
Notice that if $ s,t \in (\mathbf{O}_K / p^k \mathbf{O}_K)^* $, $ |s|=1 $ and $\gamma_s \in \Gamma_{ns}(p^k) $ lifts $M_s $ we have:
$$ [t] \gamma_s - [ts] \in \mathbb{Z}^2 \mbox{ or } [t] \gamma_s + [ts] \in \mathbb{Z}^2 . $$
\noindent
Analogously if $ |s|=-1 $ and $ \gamma $ lifts $ M_sC $ to $ \Gamma^+_{ns}(p^k) $ we have:
$$ [t] \gamma - [\overbar{ts}] \in \mathbb{Z}^2 \mbox{ or } [t] \gamma + [\overbar{ts}] \in \mathbb{Z}^2. $$
These relations together with Proposition \ref{Kleinforms} imply:
\begin{prop}\label{indici}
The Klein forms: $ \mathfrak{k}_{[t]\gamma_s}(\tau) $ and $ \mathfrak{k}_{[ts]}(\tau) $ up to a $ 2p^k-$th root of unity represent the same function in the sense that:
$$ \mathfrak{k}_{[t]\gamma_s}(\tau) = c \mathfrak{k}_{[ts]}(\tau) $$
for some $ c \in \bm{\mu_{2p^k}} $.
Similarly, for the Klein forms $ \mathfrak{k}_{[t]\gamma}(\tau) $ and $ \mathfrak{k}_{[\overbar{ts}]}(\tau) $ we have:
$$ \mathfrak{k}_{[t]\gamma}(\tau) = c' \mathfrak{k}_{[\overbar{ts}]}(\tau) $$
for some $ c' \in \bm{\mu_{2p^k}} $.
\end{prop}
\noindent
For $ h \in (\mathbb{Z}/p^k\mathbb{Z})^* $ we define the following complex-valued functions on $ \mathbb{H} $:
$$ T_h (\tau) := \prod_{t \in ((\mathbf{O}_K / p^k \mathbf{O}_K)^*/\{\pm 1\}) , |t|=h}\mathfrak{k}_{[t]}(\tau), $$
$$ G_h(\tau) := T_h(\tau)(\Delta(\tau))^{p^{k-1}\frac{p+1}{24}} = \prod_{t \in ((\mathbf{O}_K / p^k \mathbf{O}_K)^*/\{\pm 1\}) , |t|=h}g_{[t]}(\tau). $$
\noindent
For $ h \in (\mathbb{Z}/p^k\mathbb{Z})^*/\{ \pm 1 \} $ consider:
$$ T^+_h (\tau) := \prod_{t \in ((\mathbf{O}_K / p^k \mathbf{O}_K)^*/\{\pm 1\}) , \pm|t|= h}\mathfrak{k}_{[t]}(\tau), $$
$$ G^+_h(\tau) := T^+_h(\tau)(\Delta(\tau))^{p^{k-1}\frac{p+1}{12}} = \prod_{t \in ((\mathbf{O}_K / p^k \mathbf{O}_K)^*/\{\pm 1\}) , \pm|t|= h}g_{[t]}(\tau) .$$
\begin{prop}\label{FunzioniG} Let $ p \not= 2,3 $ a prime. Consider:
$$ g(\tau) = \prod_{x \in ((\mathbf{O}_K / p^k \mathbf{O}_K)^*/\{\pm 1\}) ) } g_{[x]}^{m(x)}(\tau) $$
\noindent
and suppose that it is a modular unit on $ X(p^k) $ (or equivalently that it satisfies the conditions of Theorem \ref{unitàmodulari}). If $ g(\tau) $ is a modular unit on $ X_{ns}(p^k) $ there exist integers $ \{n_h\}_{h \in (\mathbb{Z}/p^k\mathbb{Z})^* }$ such that:
$$ g(\tau)= \prod_{h \in (\mathbb{Z}/p^k\mathbb{Z})^* }G_h^{n_h}(\tau) .$$
\noindent
Similarly, if the function $ g(\tau) $ is a modular unit on $ X^+_{ns}(p^k) $, there exist integers $ \{n^{+}_h\}_{h \in (\mathbb{Z}/p^k\mathbb{Z})^*/\{\pm 1 \}}$ such that:
$$ g(\tau)= \prod_{h \in ((\mathbb{Z} / p^k \mathbb{Z})^*/\{\pm 1\}) }{{G^+_h}^{n^+_h}(\tau)} .$$
\end{prop}
\begin{proof}
We look for conditions on the exponents $ \{m(x)\}_{x \in ((\mathbf{O}_K / p^k \mathbf{O}_K)^*/\{\pm 1\})} $ that guarantee:
$$ \frac{g(\sigma^{-1}(\tau))}{g(\tau)} \in \mathbb{C} \mbox{ for every } \sigma \in \Gamma_{ns}(p^k) \mbox{ (respectively } \Gamma^+_{ns}(p^k) \mbox{)} . $$
\noindent
From Proposition \ref{Kleinforms}, assertion (3), the fact that $ \Delta(\tau) $ is weakly modular of weight 12 and that by hypotesis $ 12 $ divides $ \sum{m(x)} $ we have:
$$ g(\sigma^{-1}(\tau)) = (\Delta(\sigma^{-1} (\tau)))^{\frac{1}{12}\sum{m(x)}} \prod{\mathfrak{k}_{[x]}^{m(x)}(\sigma^{-1}(\tau))} = $$
$$ = (\Delta(\tau))^{\frac{1}{12}\sum{m(x)}} \prod{\mathfrak{k}_{[x]\sigma^{-1}}^{m(x)}(\tau)} . $$
\noindent
By Proposition \ref{strutturanonsplit}, $ C'_{ns}(p^k)$ is a cyclic group with $ (p+1)p^{k-1} $ elements. Let $ M_r $ be a generator where $ r $ is a generator of:
$$ \{ s \in (\mathbf{O}_K / p^k \mathbf{O}_K)^* \mbox{ with } |s|= 1 \} .$$
\noindent
Every $S \in C'^+_{ns}(p^k) \setminus C'_{ns}(p^k) $ is of the form $ M_tC$ where $ t \in (\mathbf{O}_K / p^k \mathbf{O}_K)^* $ and $ |t|=-1 $. Fix $ S $ and choose $ \gamma_r $ lifting $ M_r $ in $ \Gamma_{ns}(p^k) $ and $\gamma_t $ lifting $ M_t $ in $ GL_2(\mathbb{Z}) $ with $ \det \gamma_t=-1 $. Of course $ \gamma_tC $ lifts $ S $ in $ \Gamma^+_{ns}(p^k) $.
\\
For every $ j $ we have that:
$$ (([x] \mbox{ mod } \mathbb{Z}^2)/\{\pm 1\}) \longmapsto (([xr^j] \mbox{ mod } \mathbb{Z}^2)/\{\pm 1\}) \mbox{ and} $$
$$ (([x] \mbox{ mod } \mathbb{Z}^2)/\{\pm 1\}) \longmapsto (([\overbar{xr^jt}] \mbox{ mod } \mathbb{Z}^2)/\{\pm 1\}) $$
\noindent
are permutations of the primitive elements in $ ((\frac{1}{p^k}\mathbb{Z})^2 \mod \mathbb{Z}^2 )/(\pm 1) $.
\noindent
As a consequence of these observations and Proposition \ref{indici}, taking $ \sigma=(\gamma_r)^j $ we have:
$$ (\Delta(\tau))^{\frac{1}{12}\sum{m(x)}} \prod{\mathfrak{k}_{[x]\gamma_r^{-j}}^{m(x)}(\tau)} = (\Delta(\tau))^{\frac{1}{12}\sum{m(x)}} \prod{\mathfrak{k}_{[xr^j]\gamma_r^{-j}}^{m(xr^j)}(\tau)} = $$
$$ = c_j(\Delta(\tau))^{\frac{1}{12}\sum{m(x)}} \prod{\mathfrak{k}_{[x]}^{m(xr^j)}(\tau)} = c_j \prod{g_{[x]}^{m(xr^j)}(\tau)} ,$$
\noindent
where $ \{c_j\}_j $ are $ 2p^k-$th roots of unity. Taking $ \sigma = (\gamma_r)^j \gamma_t C $ we obtain:
$$ (\Delta(\tau))^{\frac{1}{12}\sum{m(x)}} \prod{\mathfrak{k}_{[x]C\gamma_t^{-1} \gamma_r^{-j}}^{m(x)}(\tau)} = (\Delta(\tau))^{\frac{1}{12}\sum{m(x)}} \prod{\mathfrak{k}_{[\overbar{xr^jt}]C\gamma_t^{-1} \gamma_r^{-j}}^{m(\overbar{xr^jt})}(\tau)} = $$
$$ = d_j (\Delta(\tau))^{\frac{1}{12}\sum{m(x)}} \prod{\mathfrak{k}_{[x]}^{m(\overbar{xr^jt})}(\tau)} = d_j \prod{g_{[x]}^{m(\overbar{xr^jt})}(\tau)} ,$$
\noindent
where $ \{d_j\}_j $ are $ 2p^k-$th roots of unity. Consider the following expression:
$$ \frac{g(\gamma_r^{-1}(\tau))}{g(\tau)} = c_1 \prod{g^{m(xr)-m(x)}_{[x]}(\tau) } .$$
\noindent
By the independence of Siegel functions \cite[p.42 or p.120]{KL} a product $ \prod {g_{[x]}^{l(x)}} $ is constant if and only if the exponents $ l(x)$ are all equal. So the previous quotient is constant if and only if:
$$ a(xr^j)= m(xr^{j+1})-m(xr^j) \mbox{ satisfy } a(xr^j)=a(xr^l) \mbox{ } \mbox{ for all } j,l \in \mathbb{Z}. $$
\noindent
But $ (\gamma_r)^{\frac{p+1}{2} p^{k-1}} \equiv -I \mod p^k$ and $ r^{\frac{p+1}{2} p^{k-1}} = -1 \mod p^k $. So we have that
$ \sum_{j=1}^{ \frac{p+1}{2} p^{k-1}}a(xr^j) =0 $ and consequently $ a(xr^j)=0 $ for every $ j $, which implies that $ m(xr^j)$ does not depend on $ j $.
Since $g(\tau)$ is $ \Gamma(p^k)-$invariant and every element in $ \Gamma_{ns}(p^k) $ can be written in the form $ \gamma\gamma_r^j $ with $ \gamma \in \Gamma(p^k) $, we conclude that if $ g(\sigma^{-1}(\tau))/g(\tau) \in \mathbb{C} \mbox{ for every } \sigma \in \Gamma_{ns}(p^k) $, this implies that if $ |x|=|y|$ then $ m(x)=m(y)$. For each $ h $ invertible mod $p^k $ choose $ x $ with $ |x|=h $, put $ n_h:=m(x)$ and the first claim follows.
Consider now:
$$ \frac{g((C\gamma_t^{-1})(\tau))}{g(\tau)} = d_0 \prod{g^{m(\overbar{xt})-m(x)}_{[x]}(\tau) } .$$
\noindent
If this quotient is constant the exponent of $ g_{[x]}(\tau) $ is equal to the exponent of the Siegel function $ g_{[\overbar{x}rt]}(\tau) $. So:
$$ m(\overbar{xt})-m(x) = m(x\overbar{rt^2})-m(\overbar{x}rt) $$
or equivalently:
$ m(\overline{xt}) + m(\overbar{x}rt) = m(x) + m(x\overbar{rt^2}) $.
But $ |rt\overbar{t^{-1}}|=1 $ so $ m(\overline{xt}) = m(\overbar{x}rt) $ and $ |\overbar{rt^2}|=1 $ so $ m(x) = m(x\overbar{rt^2}) $. Hence $ m(x)=m(\overbar{xt}) $ and observe that $ |x|=-|\overbar{xt}| $. So, in consideration of the previous result, we can conclude that $ g(\sigma^{-1}(\tau))/g(\tau) \in \mathbb{C} \mbox{ for every } \sigma \in \Gamma^+_{ns}(p^k) $ implies that if $ |x|=|y|$ or $ |x|=-|y| $ then $m(x)=m(y)$.
For every $ h \in (\mathbb{Z} / p^k \mathbb{Z})^*/\{\pm 1\} $ choose $ x $ such that $ \pm|x|=h $ and define $ n^+_h := m(x) $ and the second claim follows.
\end{proof}
\begin{prop}\label{Somme} The product:
$$ \prod_{h \in (\mathbb{Z}/p^k\mathbb{Z})^* }T_h^{n_h}(\tau) $$
is a nearly holomorphic modular form for $ \Gamma(p^k) $ if and only if $p $ divides $ \sum_h n_h h$.
\end{prop}
\begin{proof} First of all, for every $ h $ invertible mod $ p^k $:
$$ \mbox{ (1) } \sum_ { \begin{scriptsize} \begin{array}{c} \pm s \in ((\mathbf{O}_K / p^k \mathbf{O}_K)^*/\{\pm 1\}) \\ |\pm s|=h \end{array} \end{scriptsize} }\left(\frac{1}{2}(s+\overline{s})\right)^2 = \frac{h}{4}(p+1)p^{k-1} \mod p^k, $$
$$ \mbox{ (2) } \sum_{ \begin{scriptsize} \begin{array}{c} \pm s \in ((\mathbf{O}_K / p^k \mathbf{O}_K)^*/\{\pm 1\}
) \\ |\pm s|=h \end{array} \end{scriptsize} }\left(\frac{1}{2\sqrt{\epsilon}}(s-\overline{s})\right)^2 = - \frac{h}{4\epsilon}(p+1)p^{k-1} \mod p^k, $$
$$ \mbox{ (3) } \sum_{ \begin{scriptsize} \begin{array}{c} \pm s \in ((\mathbf{O}_K / p^k \mathbf{O}_K)^*/\{\pm 1\}) \\ |\pm s|=h \end{array} \end{scriptsize} }\left(\frac{1}{2}(s+\overline{s})\right)\left(\frac{1}{2\sqrt{\epsilon}}(s-\overline{s})\right) = 0 \mod p^k . $$
We prove only the first assertion because the other statements can be shown by the same argument . Every $ s \in (\mathbf{O}_K / p^k \mathbf{O}_K)^* $ with $ |s|= h $ can be written as $ s=r^i\alpha_h $ where $ r $ is a generator of the subgroup $ \{ t \in (\mathbf{O}_K / p^k \mathbf{O}_K)^* \mbox{ with } |t|= 1 \} $ and $ \alpha_h $ are fixed elements such that $ |\alpha_h|=h $.
$$ \sum_{|\pm s|=h}(\frac{1}{2}(s+\overline{s}))^2 = \sum_{i=0}^{\frac{p+1}{2}p^{k-1}-1}(\frac{1}{2}(r^i\alpha_h +\overline{ r^i\alpha_h}))^2 = $$ $$ =
\displaystyle\frac{\alpha_h^2}{4}\sum_ {i=0}^{\frac{p+1}{2}p^{k-1}-1}(r^{2i}) +
\displaystyle\frac{\overline{\alpha_h^2}}{4}\sum_ {i=0}^{\frac{p+1}{2}p^{k-1}-1}(r^{-2i}) + \frac {\alpha_h\overbar{\alpha_h}}{4} (p+1)p^{k-1}
$$
\noindent
and the assertion (1) follows because:
$$\displaystyle \sum_{i=0}^{\frac{p+1}{2}p^{k-1}-1}r^{2i}=
\displaystyle \sum_{i=0}^{\frac{p+1}{2}p^{k-1}-1}r^{-2i} =
\displaystyle\frac{1- r^{(p+1)p^{k-1}} }{1-r^2} =0 \mod p^k $$
\noindent
To prove this proposition we apply Proposition \ref{Kleinforms} to the product $ \prod_{h}T_h^{n_h}(\tau) $. Considering that for every $ s= a_1+\sqrt{\epsilon}a_2 \in (\mathbf{O}_K / p^k \mathbf{O}_K)^* $ we have:
$$ p^k[ s] \equiv (a_1,a_2) \mbox{ mod } (p^k\mathbb{Z})^2 \mbox{ or } p^k[ s] \equiv -(a_1,a_2) \mbox{ mod } (p^k\mathbb{Z})^2 \mbox{ and:} $$ $$ (a_1,a_2) \equiv \left( \frac{1}{2}(s+\overline{s}),\frac{1}{2\sqrt{\epsilon}}(s-\overline{s}) \right) \mbox{ mod } (p^k\mathbb{Z})^2 $$
and reformulating condition (5) of Proposition \ref{Kleinforms} in terms of assertions (1),(2) and (3) we attain the desired result.
\end{proof}
From this proposition it follows immediately that the functions $ T^+_h({\tau}) $ are nearly holomorphic for $ \Gamma(p^k) $. We will examine them further in details.
For every $ s= \begin{pmatrix} a&b\\c&d \end{pmatrix} \in SL_2({\mathbb{Z}}) $ define:
$$ J_s(\tau)=(c\tau+d)^{-(p+1)p^{k-1}} , \tau \in \mathbb{H}. $$
\noindent
\begin{prop}\label{piùmenodiedrale}
For every prime $ p \equiv 3 $ mod $ 4 $, for every $ h \in ((\mathbb{Z}/p^k\mathbb{Z})^*/\{\pm 1\}) $ and for every $ s \in \Gamma^+_{ns}(p^k) $ we have:
$$ T^+_h(s(\tau))=J_s(\tau) T^+_h(\tau) $$
in other words $ T^+_h(\tau) $ is a nearly holomorphic modular form for $ \Gamma^+_{ns}(p^k) $ of weight $-(p+1)p^{k-1} $.\\
\noindent
If $ p \equiv 1 $ mod $ 4 $ and $ s \in \Gamma_{ns}(p^k) $ we have:
$$ T^+_h(s(\tau))=J_s(\tau) T^+_h(\tau) $$
in other words $ T^+_h(\tau) $ is a nearly holomorphic modular form for $ \Gamma_{ns}(p^k) $ of weight $ -(p+1)p^{k-1} $.\\
\noindent
If $ p \equiv 1 $ mod $ 4 $ and $ s \in \Gamma^+_{ns}(p^k) \setminus \Gamma_{ns}(p^k) $ we have:
$$ T^+_h(s(\tau))= - J_s(\tau) T^+_h(\tau) .$$
\end{prop}
\begin{proof}
It is clear from Proposition \ref{Kleinforms} that for every $ s\in \Gamma^+_{ns}(p^k) $ there exists a $ 2p^k-$th root of unity $ c $ such that:
$ T^+_h(s(\tau))=c T^+_h(\tau)J_s(\tau) $ so it is natural to define:
$$ C_h(s) = \displaystyle\frac{T^+_h(s(\tau))}{T^+_h(\tau)J_s(\tau)} \in \bm{\mu_{2p^k}}. $$
\noindent
On the one hand:
$$ T^+_h((ss')(\tau)) = C_h(ss')T^+_h(\tau)J_{ss'}(\tau), $$
on the other hand:
$$ T^+_h(s(s'(\tau)))= C_h(s)T^+_h(s'(\tau))J_s(s'(\tau)) = C_h(s)C_h(s')J_s(s'(\tau))J_{s'}(\tau)T^+_h(\tau). $$
\noindent
Considering that $ J_{ss'}(\tau) = J_s(s'(\tau))J_{s'}(\tau) $ we have:
$$ C_h(ss')=C_h(s)C_h(s'). $$
From Proposition \ref{Somme} we deduce easily that $ C_h(\pm\Gamma(p^k))=1 $ for every $ h $. So $C_h $ are characters of $ \Gamma^+_{ns}(p^k)/\pm \Gamma(p^k) $. Since this quotient is isomorphic to $ C'^+_{ns}(p^k)/\{\pm I\} $ and since for every $ \alpha \in C'^+_{ns}(p^k) \setminus C'_{ns}(p^k) $ we have $ \alpha^2 = -I $, by Proposition \ref{strutturanonsplit} we obtain that $ \Gamma^+_{ns}(p^k)/\pm \Gamma(p^k) $
is a dihedral group of $ (p+1)p^{k-1} $ elements. These observations entail \textit{ipso facto} that $ C_h(s) \in \{ \pm 1 \} $. As in Proposition \ref{FunzioniG} choose a matrix $ \gamma_r \in SL_2(\mathbb{Z}) $ lifting $ M_r \in C'_{ns}(p^k) $ where $ r $ generates the subgroup of $ (\mathbf{O}_K / p^k \mathbf{O}_K)^* $ of elements of norm 1.\\ Choose $ \gamma = \begin{pmatrix} a&b\\c&d \end{pmatrix} $ in $ \Gamma^+_{ns}(p^k) \setminus \Gamma_{ns}(p^k)$ . It is not restrictive to suppose that $ a=d \mbox{ mod } 2 $. If this did not happen we would alternatively choose: $$ \gamma = \begin{pmatrix} a&b\\c&d \end{pmatrix} \begin{pmatrix} 1&p^k\\0&1 \end{pmatrix} = \begin{pmatrix} a&{ap^k+b}\\c&{cp^k+d} \end{pmatrix}.$$
If $ a\not\equiv d \mbox{ mod } 2$ then $ b $ and $ c $ are inevitably odd because $ ad-bc=1 $ so the new coefficients on the diagonal verify $ a \equiv cp^k+d \mbox{ mod } 2$. \\
Notice that $ \{ \gamma\gamma_r^j\gamma^{-1} \mbox{, } \gamma\gamma_r^j \}_{j=1,...,\frac{p+1} {2}(p^{k-1})}$ is a set of representatives of cosets for the quotient group $ \Gamma^+_{ns}(p^k)/\pm \Gamma(p^k) $. Furthermore if $ h \in (\mathbb{Z}/p^k\mathbb{Z})^*/\{\pm 1\} $ and $s \in (\mathbf{O}_K / p^k \mathbf{O}_K)^*/\{\pm 1\} $ with $ \pm|s|=h $, there exists a $ 2p^k-$th root of unity $ c' $ such that:
$$ T^+_h(\tau) = c' \prod_{j=1}^{\frac{p+1}{2}p^{k-1}} {\mathfrak{k}_{[s]\gamma\gamma_r^j\gamma^{-1}}}(\tau)\prod_{j=1}^{\frac{p+1}{2}p^{k-1}} {\mathfrak{k}_{[s]\gamma\gamma_r^j}}(\tau). $$
\noindent
We calculate:
$$ T^+_h(\gamma(\tau)) = c' J_{\gamma}(\tau) \prod_{j=1}^{\frac{p+1}{2}p^{k-1}} {\mathfrak{k}_{[s]\gamma\gamma_r^j}}(\tau)\prod_{j=1}^{\frac{p+1}{2}p^{k-1}} {\mathfrak{k}_{[s]\gamma\gamma_r^j\gamma}}(\tau) = $$
$$ = c' (-1)^{\frac{p+1}{2}} J_{\gamma}(\tau) \prod_{j=1}^{\frac{p+1}{2}p^{k-1}} {\mathfrak{k}_{[s]\gamma\gamma_r^j}}(\tau)\prod_{j=1}^{\frac{p+1}{2}p^{k-1}} {\mathfrak{k}_{-[s]\gamma\gamma_r^j\gamma}}(\tau) .$$
\noindent
But $ \gamma\gamma_r^j\gamma^{-1} \equiv -\gamma\gamma_r^j\gamma $ mod $ p^k $ and $ \gamma^{-1}+\gamma $ (in agreement with the previous convention) has all even coefficients so:
$$ [s]\gamma\gamma_r^j\gamma^{-1} - (-[s]\gamma\gamma_r^j\gamma) = [s]\gamma\gamma_r^j( \gamma^{-1}+\gamma) \in (2\mathbb{Z})^2 $$
and considering Proposition \ref{Kleinforms} part (2) we have:
$$ \frac{{\mathfrak{k}_{-[s]\gamma\gamma_r^j\gamma}}(\tau) }{{\mathfrak{k}_{[s]\gamma\gamma_r^j\gamma^{-1}}}(\tau)} \in \bm{\mu_{p^k}}. $$
Therefore: $$ C_h(\gamma) = (-1)^{\frac{p+1}{2}}\prod_{j=1}^{\frac{p+1}{2}p^{k-1}}\frac{{\mathfrak{k}_{-[s]\gamma\gamma_r^j\gamma}}(\tau) }{{\mathfrak{k}_{[s]\gamma\gamma_r^j\gamma^{-1}}}(\tau)}, $$
\noindent
so $ C_h(\gamma)(-1)^{\frac{p+1}{2}} \in \bm{\mu_{p^k}} \cap \{\pm 1\} $, we have necessarily $ C_h(\gamma)=(-1)^{\frac{p+1}{2}} $ for every $ \gamma \in \Gamma^+_{ns}(p^k) \setminus \Gamma_{ns}(p^k) $ and the proposition follows.
\end{proof}
\begin{thm}\label{powprod} If $ p \not= 2,3 $ the subgroup of modular units in $ F_{p^k} $ of $ X^+_{ns}(p^k) $ consists (modulo constants) of power products:
$$ g(\tau)= \prod_{h \in ((\mathbb{Z} / p^k \mathbb{Z})^*/\{\pm 1\}) }{{G^+_h}^{n^+_h}(\tau)} $$
where $ d=\displaystyle\frac{12}{\gcd(12,p+1)} \mbox{ divides } \sum_{h}n^+_h $.
\end{thm}
\begin{proof}
By Proposition \ref{FunzioniG} and Theorem \ref{unitàmodulari}, every modular unit on $ X^+_{ns}(p^k) $ can be written in the above indicated way.
In fact, $ d|\sum_h{n^+_h} $ is equivalent to saying: $ 12|(p+1)p^{k-1}\sum_h{n^+_h} $.
\\By Proposition \ref{piùmenodiedrale} all the functions of this form are modular units of $ X^+_{ns}(p^k) $. In fact, if $ p \equiv 3 \mbox{ mod } 4 $, the functions $ T^+_h(\tau) $ are nearly holomorphic modular forms for $ X^+_{ns}(p^k) $. If $ p \equiv 1 \mbox{ mod } 4 $, even if the functions $ T^+_h(\tau) $ are not nearly holomorphic modular forms for $ X^+_{ns}(p^k) $, the product $ \prod_{h }{{T^+_h}^{n^+_h}(\tau)} $ has this property, because $ \sum_h{n^+_h} $ is even in this case.
\end{proof}
Notice that such a writing for $ g(\tau) $ is not unique because of the fact that the following product is constant: $$ \displaystyle
\prod_{h \in ((\mathbb{Z} / p^k \mathbb{Z})^*/\{\pm 1\})}{G^+_h(\tau)} = \prod_{t \in (\mathbf{O}_K / p^k \mathbf{O}_K)^*/\{\pm 1\}}{g_{[t]}(\tau)}.$$
\begin{rem}\label{pseudoGalois} Let $g$ be a generator of $ (\mathbb{Z}/p^k\mathbb{Z})^* $. Choose $s \in (\mathbf{O}_K / p^k \mathbf{O}_K)^* $ with $ |s|=g $ and denote with $ \rho \in Gal(F_{p^k},\mathbb{Q}(j))$ the automorphism corresponding to the matrix $ M_s $ respect to the isomorphism $ Gal(F_{p^k},\mathbb{Q}(j)) \cong GL_2(\mathbb{Z}/p^k\mathbb{Z})/{\pm I} $ described in Theorem \ref{fricke}. Let $ F^+_{ns}(p^k) $ be the subfield of $ F_{p^k} $ fixed by $ C'^+_{ns}(p^k)/\pm I $.
Choose $\sigma \in Gal(F_{p^k},\mathbb{Q}(j)) $.
From Galois theory we have:
$$ Gal(F_{p^k}, \sigma ( F^+_{ns}(p^k)) ) = \sigma Gal(F_{p^k}, F^+_{ns}(p^k) )\sigma^{-1} $$
\noindent
thus saying that $ \sigma ( F^+_{ns}(p^k)) = F^+_{ns}(p^k) $ amounts to say that $ \sigma $ belongs to the normalizer of $ C'^+_{ns}(p^k)/\pm I $, in other words we have: $\sigma \in C^+_{ns}(p^k) / \pm I $. Consider $ \sigma_1, \sigma_2 \in C^+_{ns}(p^k) / \pm I $. We have $ \sigma_1(f(\tau)) = \sigma_2 (f(\tau)) $ for every $ f(\tau) \in F^+_{ns}(p^k) $ if and only if $ \sigma_1\sigma_2^{-1} \in C'^+_{ns}(p^k)/\pm I $ or equivalently $ \det \sigma_1 = \det \sigma_2 $. So every automorphism $ \sigma\restriction_{F^+_{ns}(p^k)}: F^+_{ns}(p^k)\rightarrow F^+_{ns}(p^k)$ fixing $ \mathbb{Q}(j) $ can be written in the form $ \sigma = \rho^j $ for some $ 0 \le j \le \varphi(p^k)-1 $.
Notice that if $$ f(\tau)= \prod_{h \in ((\mathbb{Z} / p^k \mathbb{Z})^*/\{\pm 1\}) }{{G^+_h}^{n^+_h}(\tau)} $$
and $$ h(\tau)= \prod_{h \in ((\mathbb{Z} / p^k \mathbb{Z})^*/\{\pm 1\}) }{{G^+_{h(\pm g)}}^{n^+_h}(\tau)} $$
\noindent are modular units on $ X^+_{ns}(p^k) $, from proposition \ref{sigfun} we have:
$$ (\rho(f(\tau)))^{12p^k} = \rho(f(\tau)^{12p^k})= \rho \left(\prod_{h \in ((\mathbb{Z} / p^k \mathbb{Z})^*/\{\pm 1\}) }{{G^+_h}^{12p^kn^+_h}(\tau)}\right) =$$
$$= \prod_{h \in ((\mathbb{Z} / p^k \mathbb{Z})^*/\{\pm 1\}) }{{G^+_{h(\pm g)}}^{12p^kn^+_h}(\tau)} = (h(\tau))^{12p^k}. $$
So $ \rho(f(\tau)) = c h(\tau)$ for some $ c \in \mathbb{Q}(\zeta_{p^k})$ and all the functions $ \rho^j(f(\tau)) $ are modular units. Choosing $ j= \frac{1}{2} \varphi(p^k) $ we deduce that for every modular unit $ f(\tau) $ on $ X^+_{ns}(p^k) $ there exist $ c' \in \mathbb{Q}(\zeta_{p^k}) $ such that:
$$ c'f(\tau) \in \mathbb{Q}\left(\cos \left(\frac{2\pi}{p^k}\right)\right)((q^{p^{-k}})) \mbox{ with } q=e^{2\pi i \tau}. $$
\end{rem}
\section{Cuspidal Divisor Class Group of non-split Cartan curves}
Let $ p \ge 5 $ a prime and let $ R = \mathbb{Z}[H] $ be the group ring of $ H=(\mathbb{Z}/p^k\mathbb{Z} )^*/\{ \pm 1 \} $ over $ \mathbb{Z} $.
Let $ w $ be a generator of $ H $. For $ \alpha \in \mathbb{Z}/p^k\mathbb{Z} $, let be $ a \in \mathbb{Z} $ congruent to $ \alpha \mbox{ mod }p^k $. We define:
$$ \left\langle \frac{\alpha}{p} \right\rangle:= \left\langle \frac{a}{p} \right\rangle .$$
Define the Stickelberger element:
$$ \theta= \displaystyle\frac{p^k}{2} \sum_{i=1}^{\frac{p-1}{2}p^{k-1}} {\displaystyle\sum_{ \begin{scriptsize} \begin{array}{c} s \in ((\mathbf{O}_K / p^k \mathbf{O}_K)^*/\{\pm 1\}) \\ \pm| s|=w^i \end{array} \end{scriptsize} } B_2 \left( \left\langle \frac{\frac{1}{2}(s+\overline{s}) }{p^k} \right\rangle \right) } w^{-i} \in \mathbb{Q}[H]. $$
Define the ideals:
$$ R_0 := \Big{\{} \sum b_jw^j \in R \mbox{ such that } \deg \left( \sum b_jw^j\right)=\sum b_j=0 \Big{\}}, $$
$$ R_d := \Big{\{} \sum b_jw^j \in R \mbox{ such that } d \mbox{ divides } \deg\left(\sum b_jw^j\right)=\sum b_j \Big{\}}. $$
Now we can state the main result:
\begin{mainthm}\label{main}
The group generated by the divisors of modular units in $F_{p^k} $ of the curve $ X^+_{ns}(p^k) $ can be expressed both as $ R_d\theta $ and as Stickelberger module $ R\theta \cap R $. The Cuspidal Divisor Class Group on $ X^+_{ns}(p^k) $ is a module over $Z[H]$ and we have the following isomorphism:
$$ \mathfrak{C}^+_{ns}(p^k) \cong R_0 / R_d \theta. $$
\end{mainthm}
\begin{proof}
For every $ i \in \mathbb{Z}/\frac{\varphi(p^k)}{2}\mathbb{Z} $ define:
$$ a_i = \displaystyle\frac{p^k}{2}\sum_{ { \begin{scriptsize} \begin{array}{c} s \in ((\mathbf{O}_K / p^k \mathbf{O}_K)^*/\{\pm 1\}) \\ \pm| s|=w^i \end{array} \end{scriptsize} }} B_2 \left( \left\langle \frac{\frac{1}{2}(s+\overline{s}) }{p^k} \right\rangle \right). $$
\noindent
We identify the cusps of $ X^+_{ns}(p^k) $ with the elements in $H= (\mathbb{Z}/p^k\mathbb{Z})^*/\{\pm 1\} $ as explained in Proposition \ref{numerocuspidi}. In consideration of Proposition \ref{divisori} we obtain:
$$ \mbox{ div } {G^+_{w^j}}^d(\tau) = d \sum_{i=1}^{\frac{p-1}{2}p^{k-1}} a_i w^{j-i}. $$
If $ p\not\equiv 11 \mbox{ mod } 12$, the function $ G^+_{w^j}(\tau) $ is not $ \Gamma^+_{ns}(p^k)-$invariant but with a slight abuse of notation we write:
$$ \mbox{ div } G^+_{w^j}(\tau) = \sum_{i=1}^{\frac{p-1}{2}p^{k-1}} a_i w^{j-i}. $$
It is clear that $\mbox{div }G^+_{w^i}(\tau) \in \mathbb{Q}[H] $ and $d \mbox{ div }G^+_{w^i}(\tau) \in R $.
Consider the Stickelberger element:
$$ \mbox{div } G^+_{\{\pm 1\}}(\tau) = \sum_{i=1}^{\frac{p-1}{2}p^{k-1}} a_i w^{-i} = $$
$$ = \displaystyle\frac{p^k}{2} \sum_{i=1}^{\frac{p-1}{2}p^{k-1}} {\displaystyle\sum_{ \pm|s|=w^i} B_2 \left( \left\langle \frac{\frac{1}{2}(s+\overline{s}) }{p^k} \right\rangle \right) } w^{-i} = \theta. $$
Notice that: $ \mbox{div } G^+_{w^j}(\tau) = w^j\theta $. By Theorem \ref{powprod}, a $ \Gamma^+_{ns}(p^k)-$invariant function $g(\tau) \in F_{p^k} $ is a modular unit of $ X^+_{ns}(p^k) $, if and only if $ \mbox{ div } g(\tau) \in R_d \theta $. By \cite[Proposition 2.3, Chapter 5]{KL} we have $R_d \theta =R\theta \cap R $.
\end{proof}
\noindent
\begin{rem} Following Remark \ref{pseudoGalois}, consider $ G:=\displaystyle\frac{C^+_{ns}(p^k) / \pm I}{C'^+_{ns}(p^k) / \pm I} \cong (\mathbb{Z}/p^k\mathbb{Z})^*$ and let $ \rho $ be a generator of $ G/\{\pm 1\} $ with $ \pm \det \rho = w $. We may identify the group $ H $ parameterizing the cusps of $ X^+_{ns}(p^k) $ with $ G/\{\pm 1\} $ observing that for every automorphism $ \rho^j \in G/\{\pm 1\} $ and each moduar unit $h(\tau) \in F^+_{ns}(p^k)$ we have:
\begin{center}
ord$_{w^{-j}} (h(\tau)) = $ord$_{\rho^{-j}(\infty)} (h(\tau)) = $ ord$_\infty \rho^j (h(\tau)) $
\end{center}
and
$$ \mbox{div}(\rho^j(h(\tau))) = w^j\mbox{div}(h(\tau)). $$
If $ \sum a_j \rho^j \in \mathbb{Z}[G/\{\pm 1\}] \cong \mathbb{Z}[H] $ we define $$ \left(\sum a_j w^j\right) (h(\tau)) = \prod \rho^j(h(\tau))^{a_j} $$ and clearly we have:
$$ \mbox{div} \left(\prod \rho^j(h(\tau))^{a_j}\right) = \left(\sum a_j w^j \right)\mbox{div}(h(\tau)) $$
so $ \mathfrak{C}^+_{ns}(p^k) $ has a natural structure of $ \mathbb{Z}[H]$-module which emphasizes the analogy with the classical theory of cyclotomic fields recalled in the introductory section.
\end{rem}
\noindent
Define:
$$ \theta' = \theta - \displaystyle\frac{(p+1)p^{2k-1}}{12}\sum_{i=1}^{\frac{p-1}{2}p^{k-1}}w^i $$
\noindent
and observe that $ \theta' \in R$ and $ \deg(\theta') = - \displaystyle\frac{p^2-1}{24} p^{3k-2}. $
\begin{prop}\label{calcolorapido} We have: $$ R_0 \cap \left(R\theta' + p^{2k-1} R \sum_{i=1}^{\frac{p-1}{2}p^{k-1}} w^i \right) = R_d \theta .$$
\end{prop}
\begin{proof}
Let $ \alpha, \beta \in R $ such that:
$$ \alpha \theta' + p^{2k-1}\beta\sum_{i=1}^{\frac{p-1}{2}p^{k-1}} w^i \in R_0 .$$
\noindent Then
$$ -\deg(\alpha)\frac{p^2-1}{24} p^{3k-2} + p^{2k-1} \deg(\beta) \displaystyle\frac{p-1}{2}p^{k-1} = 0 $$
\noindent implies $ (p+1)\deg(\alpha)= 12 \deg(\beta) $. This is equivalent to say:
$$ d=\displaystyle\frac{12}{\gcd(12,p+1)} \mbox{ divides }\deg(\alpha) $$ and
$$ \alpha \theta' + p^{2k-1}\beta\sum w^i = \alpha\theta' + \frac{(p+1)p^{2k-1}\deg(\alpha)}{12} \sum w^i = \alpha \theta .$$
\end{proof}
\begin{thm}\label{Cardinalità}
We have:
$$ |\mathfrak{C}^+_{ns}(p^k)| = \displaystyle\frac{|\det A_{\theta'}|}{\frac{p^2-1}{24} p^{k-1} e} =
\displaystyle 24\frac{ \displaystyle\prod{}^{} {\frac{p^k}{2}B_{2,\chi}}}{\gcd(12,p+1)(p-1)p^{k-1}}, $$
where $ A_{\theta'} $ is a circulant Toeplitz matrix, $ e= p^{3k-2}\frac{p-1}{2d} $ and the product runs over all nontrivial characters $ \chi $ of $ C(p^k)/{\pm I} $ such that $ \chi(M)=1 $ for every $ M \in C(p^k) $
with $ \det M = \pm 1 $.
\end{thm}
\begin{proof}
\noindent From Proposition \ref{calcolorapido} and the following isomorphism:
$$ R_0 / \left(R_0 \cap \left(R\theta' + p^{2k-1} R \sum w^i \right)\right) \cong$$
$$ \cong \left(R_0 + R\theta' + p^{2k-1} R \sum w^i \right) / \left(R\theta' + p^{2k-1} R \sum w^i\right) $$
\noindent we deduce that
$$ \displaystyle|\mathfrak{C}^+_{ns}(p^k)|=\left(R_0 + R\theta' + p^{2k-1} R \sum w^i\right):\left(R\theta' + p^{2k-1} R \sum w^i\right). $$
From the following chain of consecutive inclusions:
$$ R \supset R_0 + R\theta' + p^{2k-1} R \sum w^i \supset R\theta' + p^{2k-1} R \sum w^i \supset R\theta' $$
\noindent we obtain
$$ \displaystyle|\mathfrak{C}^+_{ns}(p^k)| = \displaystyle\frac{(R:R\theta')}{\left( R : \left(R_0 + R\theta' + p^{2k-1} R \sum w^i\right)\right)\left(\left(R\theta' + p^{2k-1} R \sum w^i\right) : R\theta'\right)}. $$
\noindent
Define $$ e:= \gcd \left(\deg(\theta'),p^{2k-1}\deg\left(\sum w^i\right)\right) = $$ $$
= p^{3k-2}\gcd \left(\frac{p^2-1}{24},\frac{p-1}{2} \right) = p^{3k-2}\frac{p-1}{2d}. $$
\noindent
It is clear that $$ R_0 + R\theta' + p^{2k-1} R \sum w^i = R_e, $$ where by $ R_e $ we mean the ideal of $ R $ consisting of elements whose degree is divisibile by $ e$. So
$$ \left(R : \left(R_0 + R\theta' + p^{2k-1} R \sum w^i\right)\right) = e. $$
\noindent Regarding $ \left(\left(R\theta' + p^{2k-1} R \sum w^i\right) : R\theta'\right) $, we observe that
$$ \left(R\theta' + p^{2k-1} R \sum w^i\right) \big{/} R\theta' \cong \left(p^{2k-1} R \sum w^i\right) \big{/} \left(p^{2k-1} R \sum w^i \cap R\theta'\right). $$
\noindent
But $ \prod_{h}{G^+_h}^{n^+_h} $ is constant if and only if all $ n^+_h $ are the same and so
$$ \mbox{ div } \prod_{h}{G^+_h}^{n^+_h} = \left(\sum n^+_h h\right)\theta= \sum n^+_h h \left(\theta' + \frac{(p+1)p^{2k-1}}{12}\sum_{i=1}^{\frac{p-1}{2}p^{k-1}}w^i \right) =0$$
implies
$$ \left(\sum n^+_h h \right)\theta' = - \frac{(p+1)p^{2k-1}}{12} \sum n^+_h \sum w^i \iff n^+_{w}=n^+_{w^2}=n^+_{w^3}=...=n^+_{\pm 1}. $$
\noindent
But
$$ \left( \sum w^i \right) \theta' = \deg (\theta') \sum w^i = - \displaystyle\frac{p^2-1}{24} p^{3k-2} \sum w^i $$
so: $$ \left(R\theta' + p^{2k-1} R \sum w^i\right) : R\theta' = \frac{p^2-1}{24} p^{k-1}. $$
The last index we need to compute is $ (R:R\theta') $. Write $ \theta'=\sum a'_i w^{-i} $ and $ a'_i=a_i - \frac{p+1}{12} p^{2k-1} $. Define the following matrix:
$$ A_{\theta'} =
\begin{pmatrix}
a'_{0} & a'_{1} & a'_{2} & \dots & a'_{\frac{p-1}{2}p^{k-1} -2} & a'_{\frac{p-1}{2}p^{k-1} -1} \\
a'_{\frac{p-1}{2}p^{k-1}-1} & a'_{0} & a'_{1} & \dots & a'_{\frac{p-1}{2}p^{k-1} -3} & a'_{\frac{p-1}{2}p^{k-1} -2}\\
a'_{\frac{p-1}{2}p^{k-1}-2} & a'_{\frac{p-1}{2}p^{k-1}-1} & a'_{0} & \dots & a'_{\frac{p-1}{2}p^{k-1} -4} & a'_{\frac{p-1}{2}p^{k-1} -3}\\
\hdotsfor{6} \\
a'_{2} & a'_{3} & a'_{4} & \dots & a'_{0} & a'_{1} \\
a'_{1} & a'_{2} & a'_{3} & \dots & a'_{\frac{p-1}{2}p^{k-1} -1} & a'_{0}
\end{pmatrix}.
$$
\noindent
We have: $ (R:R\theta')=|\det A_{\theta'}| $. The matrix $ A_{\theta'} $ is a circulant Toeplitz matrix, in other words the coefficients $ (A_{\theta'})_{i,j} $ depend only on $ i-j \mod \frac{p-1}{2}p^{k-1}$. This is the matrix of multiplication by $ \theta'$ in $ \mathbb{C}[H] $, so we easily deduce that for $ n=1,2,...,\frac{p-1}{2}p^{k-1} $ the eighenvalues of $ A_{\theta'}$ are:
$$ \lambda_n = \sum_{j=1}^{\frac{p-1}{2}p^{k-1}}a'_j e^{\begin{Large}\frac{4 \pi i j n}{(p-1)p^{k-1}}\end{Large} } $$
with corresponding eighenvectors:
$$ v_n = \sum_{j=1}^{\frac{p-1}{2}p^{k-1}} e^{\begin{Large}\frac{4 \pi i j n}{(p-1)p^{k-1}}\end{Large} } w^j .$$
\noindent
Observe that $ \lambda_{\frac{p-1}{2}p^{k-1}} = \sum a'_i = \deg(\theta') = -\frac{p^2-1}{24}p^{3k-2} $ and that according to the definition of Theorem \ref{CDCG} the others $ \lambda_n $ correspond to the generalized Bernoulli number $\frac{p^k}{2} B_{2,\chi} $ where $ \chi $ runs over the nontrivial characters of $ C(p^k)/{\pm I} $ such that $ \chi(M)=1 $ for every $ M \in C(p^k) $
with $ \det M = \pm 1 $.
\\ Gathering all this information together we obtain the desired result.
\end{proof}
\section{Explicit calculation}
In this section we examine the curve $ X^+_{ns}(p) $ more in details. Denote with $ v $ a generator of the multiplicative group of $ \mathbb{F}_{p^2} $ and indicate with $ \omega $ a generator of the character group $ \hat{\mathbb{F}_{p^2}^*} $ viewing $ C(p) \cong \mathbb{F}_{p^2}^* $. By Theorem \ref{Cardinalità}, in this case we have:
$$ B_{2,\chi} = \sum_{x \in \mathbb{F}_{p^2}^* / \pm 1} B_2 \left( \left\langle \frac{\frac{1}{2}\mbox{Tr}(x)}{p} \right\rangle \right) \chi(x), $$
$$ |\mathfrak{C}^+_{ns}(p)| = \displaystyle \frac{24}{(p-1)\gcd(12,p+1)}\prod_{j=1}^{\frac{p-3}{2}}\frac{p}{2}B_{2,\omega^{(2p+2)j}} =
$$
$$ = \displaystyle\frac{ 576 \left| \det\left[\displaystyle\frac{p}{2}\left(\displaystyle\sum_{l=0}^{p}B_2\left( \left\langle \frac{\frac{1}{2}\mbox{Tr}(v^{i-j+l\frac{p-1}{2}}) }{p} \right\rangle \right) - \frac{p+1}{6} \right) \right]_{1\le i,j \le \frac{p-1}{2}} \right|}{(p-1)^2 p (p+1)\gcd(12,p+1)}. $$
\\
\noindent
In the following table we show the factorization of the orders of cuspidal divisor class groups $ \mathfrak{C}^+_{ns}(p) $ for some primes $ p \le 101 $:
\begin{tab}\label{tab}
\noindent
\\
\begin{tabular}{rc}
\toprule
$ p $ & $ |\mathfrak{C}^+_{ns}(p)| $ \\
\midrule
5 & $ 1 $ \\ 7 & $ 1 $ \\ 11 & $ 11 $ \\
13 & $ 7 \cdot 13^2 $ \\
17 & $ 2^4 \cdot 3 \cdot 17^3 $ \\
19 & $ 3 \cdot 19^3 \cdot 487 $ \\
23 & $ 23^4 \cdot 37181 $ \\
29 & $ 2^6 \cdot 5 \cdot 7^2 \cdot 29^6 \cdot 43^2 $\\
31 & $ 2^2 \cdot 5 \cdot 7 \cdot 11 \cdot 31^6 \cdot 2302381 $ \\
37 & $ 3^4 \cdot 7^2 \cdot 19^3 \cdot 37^8 \cdot 577^2 $ \\
41 & $ 2^6 \cdot 5^2 \cdot 7 \cdot 31^4 \cdot 41^9 \cdot 431^2 $ \\
43 & $ 2^2 \cdot 19 \cdot 29 \cdot 43^9 \cdot 463 \cdot 1051 \cdot 416532733 $ \\
53 & $ 3^2 \cdot 13^2 \cdot 53^{12} \cdot 96331^2 \cdot 379549^2 $ \\
59 & $ 59^{14} \cdot 9988553613691393812358794271 $ \\
67 & $ 67^{16} \cdot 193 \cdot 661^2 \cdot 2861 \cdot 8009 \cdot 11287 \cdot 9383200455691459 $ \\
71 & $ 31 \cdot 71^{16} \cdot 113 \cdot 211 \cdot 281 \cdot 701^2 \cdot 12713 \cdot 13070849919225655729061 $ \\
73 & $ 2^2 \cdot 3^4 \cdot 11^2 \cdot 37 \cdot 73^{17} \cdot 79^2 \cdot 241^2 \cdot 3341773^2 \cdot 11596933^2 $ \\
83 & $ 83^{19} \cdot 17210653 \cdot 151251379 \cdot 18934761332741 \cdot 48833370476331324749419 $ \\
89 & $ 2^2 \cdot 3 \cdot 5 \cdot 11^2 \cdot 13^2 \cdot 89^{21} \cdot 4027^2 \cdot 262504573^2 \cdot 15354699728897^2 $\\
101 & $ 5^4 \cdot 17 \cdot 101^{24} \cdot 52951^2 \cdot 54371^2 \cdot 58884077243434864347851^2 $
\end{tabular}
\end{tab}
\noindent \\
We recall the following result of \cite{BB}:
\begin{thm}\label{Hurw}
The genera of $ X^+_{ns}(p) $ are:
$$ g(X^+_{ns}(p)) = \displaystyle\frac{1}{24}\left( p^2 - 10 p + 23 + 6\left(\frac{-1}{p}\right) + 4\left(\frac{-3}{p}\right) \right). $$
\end{thm}
\begin{proof}
It is a consequence of Hurwitz's formula \cite[Proposition 1.40]{Shimura:af}:
$$ g(\Gamma) = 1 + \frac{d}{12} - \frac{e_2}{4} - \frac{e_3}{3} - \frac{e_\infty}{2}. $$
\noindent
In this case:
$ d:= [SL_2(\mathbb{Z}):\Gamma_{ns}^+(p)] = \frac{p(p-1)}{2} $,
$ e_\infty = \frac{p-1}{2} $ is the number of cusps (see Proposition \ref{numerocuspidi}), $ e_2 $ and $ e_3 $ denote the number of elliptic points of period 2 and 3. We have (cfr. \cite[Proposition 12]{Rebolledo}):
$$ e_2 = \frac{p+1}{2} - \left(\frac{-1}{p}\right) \mbox{ and } e_3 = \frac{1}{2} - \frac{1}{2}\left(\frac{-3}{p}\right). $$
\end{proof}
By Theorem \ref{Hurw} we have $ g(X^+_{ns}(5))=g(X^+_{ns}(7))=0 $ so it will not be surprising to find out that $ \mathfrak{C}^+_{ns}(5)$ and $ \mathfrak{C}^+_{ns}(7) $ are trivial.
For $ 11 \le p \le 31 $ we provide further corroborative evidence of Table \ref{tab}. From \cite[p. 195]{SerreMordel} we have:
\begin{thm}\label{x1}
The modular curve $ X^+_{ns}(p) $ associated to the subgroup $ C_{ns}^+(p) $ is a projective non-singular modular curve which can be defined over $ \mathbb{Q} $. The cusps are defined over $ \mathbb{Q}(\cos(\frac{2\pi}{p})) $, the maximal real subfield of the $ p $-th cyclotomic field.
\end{thm}
From \cite{Chen} we have the following result:
\begin{thm}\label{x2}
The jacobian of $ X^+_{ns}(p) $ is isogenous to the new part of the Jacobian $ J_0^+(p^2) $ of $ X_0^+(p^2) $.
\end{thm}
From \cite[Chapter 12]{Ribet} we have this interesting corollary of the Eichler-Shimura relation \cite[pag. 354]{Diamond:mf}:
\begin{thm}\label{x3} Let $q$ be a prime that does not divide $ N $ and let $f(x) $ the characteristic polynomial of the Hecke operator $ T_q $ acting on $ S^{new}_2(\Gamma^+_0(N)) $. Then:
$$ |{J_0^+}^{new}(N)(\mathbb{F}_q)| = f(q+1).$$
\end{thm}
\noindent
Choose a prime $ q \equiv \pm 1 \mbox{ mod }p$ that does not divide $ |\mathfrak{C}^+_{ns}(p)| $. From the previous theorems, the cuspidal divisor class group $ \mathfrak{C}^+_{ns}(p) $ injects into $ J^+_{ns}(p)(\mathbb{F}_q) $. So we expect that $ |\mathfrak{C}^+_{ns}(p)| $ divides $ |{J_0^+}^{new}(p^2)(\mathbb{F}_q)| = f_{q,p^2}(q+1)$ where $ f_{q,p^2} $ is the characteristic polynomial of the Hecke operator $T_q $ acting on $ S_2^{new}(\Gamma^+_0(p^2))$.
From the modular form database of W.Stein we have:\\
\noindent $ |{J_0^+}^{new}(11^2)(\mathbb{F}_{23})| = f_{23,121}(24)= 3 \cdot 11 ,$\\
$ |{J_0^+}^{new}(11^2)(\mathbb{F}_{43})| = f_{43,121}(44)= 2^2 \cdot 11 ,$\\
$ |{J_0^+}^{new}(11^2)(\mathbb{F}_{67})| = f_{67,121}(68)= 5 \cdot 11 ,$\\
$ |{J_0^+}^{new}(11^2)(\mathbb{F}_{89})| = f_{89,121}(90)= 3^2 \cdot 11 ,$\\
$ |{J_0^+}^{new}(11^2)(\mathbb{F}_{109})| = f_{109,121}(110)= 2 \cdot 5 \cdot 11 ,$\\
$ |{J_0^+}^{new}(11^2)(\mathbb{F}_{131})| = f_{131,121}(132)= 2^2 \cdot 3 \cdot 11 ,$\\
$ |{J_0^+}^{new}(11^2)(\mathbb{F}_{197})| = f_{197,121}(198)= 2 \cdot 3^2 \cdot 11 ,$\\
$ |{J_0^+}^{new}(11^2)(\mathbb{F}_{199})| = f_{199,121}(200)= 2^2 \cdot 5 \cdot 11 ,$\\
$ |{J_0^+}^{new}(11^2)(\mathbb{F}_{241})| = f_{241,121}(242)= 2 \cdot 11^2 ,$\\
$ |{J_0^+}^{new}(11^2)(\mathbb{F}_{263})| = f_{263,121}(264)= 2^3 \cdot 3 \cdot 11 ,$\\
$ |{J_0^+}^{new}(11^2)(\mathbb{F}_{307})| = f_{307,121}(308)= 2^2 \cdot 7 \cdot 11 ,$\\
$ |{J_0^+}^{new}(11^2)(\mathbb{F}_{331})| = f_{331,121}(332)= 3^3 \cdot 11 ,$\\
$ |{J_0^+}^{new}(11^2)(\mathbb{F}_{353})| = f_{353,121}(354)= 3 \cdot 11^2 ,$\\
$ |{J_0^+}^{new}(11^2)(\mathbb{F}_{373})| = f_{373,121}(374)= 2 \cdot 11 \cdot 17 ,$\\
$ |{J_0^+}^{new}(11^2)(\mathbb{F}_{397})| = f_{397,121}(398)= 2^2 \cdot 3^2 \cdot 11 ,$\\
$ |{J_0^+}^{new}(11^2)(\mathbb{F}_{419})| = f_{419,121}(420)= 2^2 \cdot 3^2 \cdot 11 ,$\\
$ |{J_0^+}^{new}(11^2)(\mathbb{F}_{439})| = f_{439,121}(440)= 2^3 \cdot 5 \cdot 11 ,$\\
$ |{J_0^+}^{new}(11^2)(\mathbb{F}_{461})| = f_{461,121}(462)= 2 \cdot 3 \cdot 7 \cdot 11 ,$\\
$ |{J_0^+}^{new}(11^2)(\mathbb{F}_{463})| = f_{463,121}(464)= 3^2 \cdot 5 \cdot 11 ,$\\
\noindent $ |{J_0^+}^{new}(13^2)(\mathbb{F}_{53})| = f_{53,169}(54)= 7 \cdot 13 ^2 \cdot 127 ,$\\
$ |{J_0^+}^{new}(13^2)(\mathbb{F}_{79})| = f_{79,169}(80)= 7 \cdot 13 ^2 \cdot 449 ,$\\
$ |{J_0^+}^{new}(13^2)(\mathbb{F}_{103})| = f_{103,169}(104)= 7 \cdot 13 ^2 \cdot 967 ,$\\
$ |{J_0^+}^{new}(13^2)(\mathbb{F}_{131})| = f_{131,169}(132)= 7 \cdot 13 ^5, $\\
$ |{J_0^+}^{new}(13^2)(\mathbb{F}_{157})| = f_{157,169}(158)= 7^2 \cdot 13 ^2 \cdot 503,$\\
$ |{J_0^+}^{new}(13^2)(\mathbb{F}_{181})| = f_{181,169}(182)= 7 \cdot 13 ^2 \cdot 4327, $\\
$ |{J_0^+}^{new}(13^2)(\mathbb{F}_{233})| = f_{233,169}(234)= 7 \cdot 13 ^2 \cdot 11731, $\\
$ |{J_0^+}^{new}(13^2)(\mathbb{F}_{311})| = f_{311,169}(312)= 7 \cdot 13 ^2 \cdot 26249, $\\
$ |{J_0^+}^{new}(13^2)(\mathbb{F}_{313})| = f_{313,169}(314)= 7 \cdot 13 ^2 \cdot 29443, $\\
$ |{J_0^+}^{new}(13^2)(\mathbb{F}_{337})| = f_{337,169}(338)= 7 \cdot 13 ^2 \cdot 35449, $\\
$ |{J_0^+}^{new}(13^2)(\mathbb{F}_{389})| = f_{389,169}(390)= 2^3 \cdot 7 \cdot 13 ^2 \cdot 71 \cdot 83, $\\
$ |{J_0^+}^{new}(13^2)(\mathbb{F}_{443})| = f_{443,169}(444)= 2^3 \cdot 7 \cdot 13 ^3 \cdot 643, $\\
$ |{J_0^+}^{new}(13^2)(\mathbb{F}_{467})| = f_{467,169}(468)= 7 \cdot 13 ^2 \cdot 93199, $\\
\noindent $ |{J_0^+}^{new}(17^2)(\mathbb{F}_{67})| = f_{67,289}(68)= 2^8 \cdot 3 \cdot 17^5 \cdot 71 ,$\\
$ |{J_0^+}^{new}(17^2)(\mathbb{F}_{101})| = f_{101,289}(102)= 2^4 \cdot 3^2 \cdot 7 \cdot 17^3 \cdot 19 \cdot 79 \cdot 181 ,$\\
$ |{J_0^+}^{new}(17^2)(\mathbb{F}_{103})| = f_{103,289}(104)= 2^7 \cdot 3^4 \cdot 17^4 \cdot 1601 ,$\\
$ |{J_0^+}^{new}(17^2)(\mathbb{F}_{137})| = f_{137,289}(138)= 2^6 \cdot 3^8 \cdot 17^4 \cdot 181 ,$\\
$ |{J_0^+}^{new}(17^2)(\mathbb{F}_{239})| = f_{239,289}(240)= 2^8 \cdot 3^2 \cdot 17^3 \cdot 373 \cdot 48871 ,$\\
$ |{J_0^+}^{new}(17^2)(\mathbb{F}_{271})| = f_{271,289}(272)= 2^5 \cdot 3^9 \cdot 5^3 \cdot 17^4 \cdot 53 ,$\\
$ |{J_0^+}^{new}(17^2)(\mathbb{F}_{307})| = f_{307,289}(308)= 2^6 \cdot 3^5 \cdot 5 \cdot 17^3 \cdot 23 \cdot 71 \cdot 1423 ,$\\
$ |{J_0^+}^{new}(17^2)(\mathbb{F}_{373})| = f_{373,289}(374)= 2^4 \cdot 3^4 \cdot 17^3 \cdot 23 \cdot 73 \cdot 101 \cdot 2789 ,$\\
$ |{J_0^+}^{new}(17^2)(\mathbb{F}_{409})| = f_{409,289}(410)= 2^7 \cdot 3^5 \cdot 17^3 \cdot 23 \cdot 53 \cdot 71 \cdot 359 ,$\\
$ |{J_0^+}^{new}(17^2)(\mathbb{F}_{443})| = f_{443,289}(444)= 2^5 \cdot 3^2 \cdot 13 \cdot 17^4 \cdot 19 \cdot 79 \cdot 15263 ,$\\
\noindent $ |{J_0^+}^{new}(19^2)(\mathbb{F}_{37})| = f_{37,361}(38)= 2 \cdot 3 \cdot 19^3 \cdot 37 \cdot 487 \cdot 5441 ,$\\
$ |{J_0^+}^{new}(19^2)(\mathbb{F}_{113})| = f_{113,361}(114)= 2^5 \cdot 3^7 \cdot 19^7 \cdot 487 ,$\\
$ |{J_0^+}^{new}(19^2)(\mathbb{F}_{151})| = f_{151,361}(152)= 2^3 \cdot 3^3 \cdot 17 \cdot 19^4 \cdot 487 \cdot 1459141 ,$\\
$ |{J_0^+}^{new}(19^2)(\mathbb{F}_{191})| = f_{191,361}(192)= 3^2 \cdot 11^5 \cdot 19^6 \cdot 73 \cdot 487 ,$\\
$ |{J_0^+}^{new}(19^2)(\mathbb{F}_{227})| = f_{227,361}(228)= 2^2 \cdot 3^4 \cdot 19^3 \cdot 487 \cdot 971 \cdot 7323581,$\\
$ |{J_0^+}^{new}(19^2)(\mathbb{F}_{229})| = f_{229,361}(230)= 3 \cdot 11 \cdot 17 \cdot 19^3 \cdot 467 \cdot 487 \cdot 2819^2 ,$\\
$ |{J_0^+}^{new}(19^2)(\mathbb{F}_{379})| = f_{379,361}(380)= 2^6 \cdot 3 \cdot 5^2 \cdot 19^3 \cdot 179 \cdot 487 \cdot 4019 \cdot 33247 ,$\\
$ |{J_0^+}^{new}(19^2)(\mathbb{F}_{419})| = f_{419,361}(420)= 2^6 \cdot 3^2 \cdot 5^3 \cdot 19^3 \cdot 487 \cdot 509^2 \cdot 16487 ,$\\
$ |{J_0^+}^{new}(19^2)(\mathbb{F}_{457})| = f_{457,361}(458)= 2^4 \cdot 3 \cdot 5^4 \cdot 19^3 \cdot 487 \cdot 521^2 \cdot 65629 ,$\\
\noindent $ |{J_0^+}^{new}(23^2)(\mathbb{F}_{47})| = f_{47,529}(48)= 2^3 \cdot 3^3 \cdot 7^4 \cdot 11 \cdot 13 \cdot 23^4 \cdot 8117 \cdot 37181 ,$\\
$ |{J_0^+}^{new}(23^2)(\mathbb{F}_{137})| = f_{137,529}(138)= 2^4 \cdot 3^6 \cdot 23^8 \cdot 2399 \cdot 37181 \cdot 75553 ,$\\
$ |{J_0^+}^{new}(23^2)(\mathbb{F}_{139})| = f_{139,529}(140)= 2^4 \cdot 3^8 \cdot 23^9 \cdot 107^2 \cdot 109 \cdot 37181 ,$\\
$ |{J_0^+}^{new}(23^2)(\mathbb{F}_{229})| = f_{229,529}(230)= 2^6 \cdot 11 \cdot 23^6 \cdot 43 \cdot 67 \cdot 37181 \cdot 325729 \cdot 1296721 ,$\\
$ |{J_0^+}^{new}(23^2)(\mathbb{F}_{277})| = f_{277,529}(278)= 2^8 \cdot 3^{10} \cdot 23^7 \cdot 113^2 \cdot 331 \cdot 7193 \cdot 37181 ,$\\
$ |{J_0^+}^{new}(23^2)(\mathbb{F}_{367})| = f_{367,529}(368)= 2^4 \cdot 23^5 \cdot 67^2 \cdot 193 \cdot 1847 \cdot 37181 \cdot 44617 \cdot 8643209 ,$\\
$ |{J_0^+}^{new}(23^2)(\mathbb{F}_{461})| = f_{461,529}(462)= 3^6 \cdot 7^4 \cdot 23^7 \cdot 43^2 \cdot 67 \cdot 199 \cdot 2857^2 \cdot 37181 ,$\\
\noindent $ |{J_0^+}^{new}(29^2)(\mathbb{F}_{59})| = f_{59,841}(60)= 2^8 \cdot 3^2 \cdot 5 \cdot 7^2 \cdot 11^2 \cdot 17 \cdot 23^2 \cdot 29^6 \cdot 43^2 \cdot 569 \cdot 967^2 \cdot 2999 \cdot 11695231 ,$\\
$ |{J_0^+}^{new}(29^2)(\mathbb{F}_{173})| = f_{173,841}(174)= 2^{10} \cdot 3^2 \cdot 5^2 \cdot 7^2 \cdot 29^6 \cdot 31 \cdot 41^2 \cdot 43^2 \cdot 89 \cdot 419^2 \cdot 719 \cdot 1061 \cdot 36571 \cdot 1269691 \cdot 1909421 ,$\\
$ |{J_0^+}^{new}(29^2)(\mathbb{F}_{233})| = f_{233,841}(234)= 2^{10} \cdot 3^2 \cdot 5 \cdot 7^2 \cdot 29^6 \cdot 43^2 \cdot 167^2 \cdot 211^2 \cdot 421 \cdot 1049 \cdot 3989 \cdot 317321 \cdot 422079165281099 ,$\\
$ |{J_0^+}^{new}(29^2)(\mathbb{F}_{347})| = f_{347,841}(348)= 2^8 \cdot 3^{12} \cdot 5^3 \cdot 7^2 \cdot 11 \cdot 23^2 \cdot 29^6 \cdot 31 \cdot 43^2 \cdot 71 \cdot 127^2 \cdot 967^2 \cdot 9601 \cdot 783719 \cdot 7292986801 ,$\\
$ |{J_0^+}^{new}(29^2)(\mathbb{F}_{349})| = f_{349,841}(350)= 2^8 \cdot 5^9 \cdot 7^2 \cdot 13^2 \cdot 19 \cdot 23 \cdot 29^7 \cdot 43^2 \cdot 83^2 \cdot 103 \cdot 211 \cdot 3786151 \cdot 92610181 \cdot 3477902249 ,$\\
$ |{J_0^+}^{new}(29^2)(\mathbb{F}_{463})| = f_{463,841}(464)= 2^{13} \cdot 5^7 \cdot 7^7 \cdot 29^6 \cdot 43^3 \cdot 59 \cdot 97^3 \cdot 461^3 \cdot 1459 \cdot 23656223369 \cdot 230667656992649 ,$\\
\noindent $ |{J_0^+}^{new}(31^2)(\mathbb{F}_{61})| = f_{61,961}(62) = 2^{10} \cdot 5 \cdot 7 \cdot 11 \cdot 31^{7} \cdot 137 \cdot 179 \cdot 1249 \cdot 10369 \cdot 26699 \cdot 38177 \cdot 2302381 \cdot 24080801 ,$\\
$ |{J_0^+}^{new}(31^2)(\mathbb{F}_{311})| = f_{311,961}(312)= 2^8 \cdot 3^2 \cdot 5 \cdot 7^2 \cdot 11 \cdot 31^7 \cdot 409 \cdot 3793^2 \cdot 51551^2 \cdot 162691 \cdot 2302381 \cdot 22340831^2 \cdot 24037019 ,$\\
$ |{J_0^+}^{new}(31^2)(\mathbb{F}_{373})| = f_{373,961}(374)=
2^4 \cdot 5 \cdot 7^2 \cdot 11^2 \cdot 13^2 \cdot 31^6 \cdot 251 \cdot 449 \cdot 2302381 \cdot 366424077359 \cdot 13600706515978033^2 ,$\\
$ |{J_0^+}^{new}(31^2)(\mathbb{F}_{433})| = f_{433,961}(434)= 2^6 \cdot 3^6 \cdot 5 \cdot 7 \cdot 11 \cdot 17 \cdot 31^{11} \cdot 89 \cdot 97 \cdot 191 \cdot 401 \cdot 1153 \cdot 54331 \cdot 126961 \cdot 2302381 \cdot 12958271 \cdot 53053053405791.$\\
\noindent For $ 11 \le p \le 23 $ we have:
$$ \mathop
\textbf{\mbox{ gcd }}_{ \begin{scriptsize} \begin{array}{c} q < 500 \mbox{ prime}, \\ q \equiv \pm 1 \mbox{ mod }p \end{array} \end{scriptsize} } |{J_0^+}^{new}(p^2)(\mathbb{F}_{q})| = |\mathfrak{C}^+_{ns}(p)|. $$
For $ p=29 $ and $ p=31 $ we have:
$$ \mathop\textbf{\mbox{ gcd }}_{ \begin{scriptsize} \begin{array}{c} q < 500 \mbox{ prime}, \\ q \equiv \pm 1 \mbox{ mod }p \end{array} \end{scriptsize} } |{J_0^+}^{new}(p^2)(\mathbb{F}_{q})| = 4|\mathfrak{C}^+_{ns}(p)|.$$
We can improve the result by using the isogeny (cfr.\cite[Paragraph 6.6]{Diamond:mf}):
$$ {J_0^+}^{new}(p^2) \longrightarrow \mathop{\bigoplus_{f}} A'_{p,f} $$
where the sum is taken over the equivalence classes of newforms $ f\in S_2(\Gamma^+_0(p^2)) $. Two forms $ f $ and $ g $ are declared equivalent if $ g=f^{\sigma} $ for some automorphism $ \sigma: \mathbb{C} \longrightarrow \mathbb{C}$. Denote with $ \mathbb{K}_f $ the number field of $ f $. We have:
$$ \mathop
\textbf{\mbox{ gcd }}_{ \begin{scriptsize} \begin{array}{c} q < 500 \mbox{ prime}, \\ q \equiv \pm 1 \mbox{ mod }29 \end{array} \end{scriptsize} } |A'_{29,f_1}(\mathbb{F}_{q})| = 7^2 \mbox{ where }\mathbb{K}_{f_1}= \mathbb{Q}(\sqrt{2}), $$
$$ \mathop
\textbf{\mbox{ gcd }}_{ \begin{scriptsize} \begin{array}{c} q < 500 \mbox{ prime}, \\ q \equiv \pm 1 \mbox{ mod }29 \end{array} \end{scriptsize} } |A'_{29,f_2}(\mathbb{F}_{q})| = 29 \mbox{ where }\mathbb{K}_{f_2}= \mathbb{Q}(\sqrt{5}), $$
$$ \mathop
\textbf{\mbox{ gcd }}_{ \begin{scriptsize} \begin{array}{c} q < 500 \mbox{ prime}, \\ q \equiv \pm 1 \mbox{ mod }29 \end{array} \end{scriptsize} } |A'_{29,f_3}(\mathbb{F}_{q})|= \mathop
\textbf{\mbox{ gcd }}_{ \begin{scriptsize} \begin{array}{c} q < 500 \mbox{ prime}, \\ q \equiv \pm 1 \mbox{ mod }29 \end{array} \end{scriptsize} } |A'_{29,f_4}(\mathbb{F}_{q})|= 2^3 \cdot 43 $$ where $\mathbb{K}_{f_3}= \mathbb{K}_{f_4} $ and $ [\mathbb{K}_{f_3}:\mathbb{Q}]=3$,
$$ \mathop
\textbf{\mbox{ gcd }}_{ \begin{scriptsize} \begin{array}{c} q < 500 \mbox{ prime}, \\ q \equiv \pm 1 \mbox{ mod }29 \end{array} \end{scriptsize} } |A'_{29,f_5}(\mathbb{F}_{q})|= 5 \cdot 29^2 \mbox { where } [\mathbb{K}_{f_5}:\mathbb{Q}]=6, $$
$$ \mathop
\textbf{\mbox{ gcd }}_{ \begin{scriptsize} \begin{array}{c} q < 500 \mbox{ prime}, \\ q \equiv \pm 1 \mbox{ mod }29 \end{array} \end{scriptsize} } |A'_{29,f_6}(\mathbb{F}_{q})|= 29^3 \mbox { where } [\mathbb{K}_{f_6}:\mathbb{Q}]=8, $$
$$ \mathop
\textbf{\mbox{ gcd }}_{ \begin{scriptsize} \begin{array}{c} q < 500 \mbox{ prime}, \\ q \equiv \pm 1 \mbox{ mod }31 \end{array} \end{scriptsize} } |A'_{31,g_1}(\mathbb{F}_{q})| = 2^2 \cdot 7 \mbox{ where }\mathbb{K}_{g_1}= \mathbb{Q}(\sqrt{2}), $$
$$ \mathop
\textbf{\mbox{ gcd }}_{ \begin{scriptsize} \begin{array}{c} q < 500 \mbox{ prime}, \\ q \equiv \pm 1 \mbox{ mod }31 \end{array} \end{scriptsize} } |A'_{31,g_2}(\mathbb{F}_{q})| = 5 \cdot 11 \mbox{ where }\mathbb{K}_{g_2}= \mathbb{Q}(\sqrt{5}), $$
$$ \mathop
\textbf{\mbox{ gcd }}_{ \begin{scriptsize} \begin{array}{c} q < 500 \mbox{ prime}, \\ q \equiv \pm 1 \mbox{ mod }31 \end{array} \end{scriptsize} } |A'_{31,g_3}(\mathbb{F}_{q})|= 2302381 \mbox { where } [\mathbb{K}_{g_3}:\mathbb{Q}]=8, $$
$$ \mathop
\textbf{\mbox{ gcd }}_{ \begin{scriptsize} \begin{array}{c} q < 500 \mbox{ prime}, \\ q \equiv \pm 1 \mbox{ mod }31 \end{array} \end{scriptsize} } |A'_{31,g_4}(\mathbb{F}_{q})|= 31^6 \mbox { where } [\mathbb{K}_{g_4}:\mathbb{Q}]=16. $$
So for $ p=29 $ and $ p=31 $ we have:
$$ \prod_f \mathop
\textbf{\mbox{ gcd }}_{ \begin{scriptsize} \begin{array}{c} q < 500 \mbox{ prime}, \\ q \equiv \pm 1 \mbox{ mod }p \end{array} \end{scriptsize} } |A'_{p,f}(\mathbb{F}_{q})|= |\mathfrak{C}^+_{ns}(p)| $$
where the product runs over all equivalence classes of newforms.
\subsection*{Acknowledgements} I would like to express my gratitude to my advisor Prof. René Schoof for his valuable remarks during the development of this research work, especially for the last section. | 192,919 |
Who are we and why are we? Vivian Ni promises founding Blandis 1993. She has since organized and operated Blandis into the 2000s. She has all these years been a regular commentator for the mixed breed dogs and all right to be called the dog and have the same value. I grund och botten är alla våra 4-benta kamrater hund […]
You are browsing archives for
Tag: 2017
PM Expo 5 August 2017
Then we sent away by email message to show in the Fagerhult 5 August 2017 to all participants. Should you not receive it in your email, so is the download at this link: PM 5 August – Fagerhult.pdf
2017 exhibitions – Two (2) this year-
exhibitions 2017 25 February Nykvarn (outside Södertälje) completed 29 April Norra Sandby Bygdegård (Nässjö Municipality) ADJUSTED 22 May Sörsjön (Norrkoping) completed 5 August in Fagerhult (Högsby Municipality) notification about 25 July 2 september i Mariestad – notification about 20 August | 60,941 |
Microtech Matrix
Zero Tolerance 0777
Minor uproar in the Knife community recently. While most of us are waiting to get our grubby hands on the Zero Tolerance 0777 which was named 2011 "Overall Knife of the Year" by BLADE Magazine, it seems another manufacturer, Microtech is launching a very similar design called the Matrix. What do you think? Rip-Off or Homage?
Microtech Matrix
Zero Tolerance 0777
Personally, I think it's pretty cheesy to blatantly rip off ZT's design right after they won Knife of the Year. There are so many elements that are a like. In the end, there is a $100 difference in price with Microtech being more expensive. In general, the response has been unfavorable towards the practice by Microtech. I'm not sure they give a shit though. At the end of the day, they will sell a bunch of these as well. | 183,846 |
Mon-Fri 9am to 6pm CST
What is used in S3 to enable client web applications that have been loaded in a single domain to interact with resources of the different domain?
- A. Public Object Permissions
- Incorrect.
- B. Public ACL Permissions
- Incorrect.
- C. CORS Configuration
- Correct!
- D. None of the above
- Incorrect.
Is the default visibility timeout for an SQS queue 1 minute?
When a failure occurs during stack creation in Cloudformation, does a rollback occur?
An administrator is getting an error while trying to create a new bucket in S3? You feel that bucket limit has been crossed. What is the bucket limit per account in AWS?
Which of the below functions is used in Cloudformation to retrieve an object from a set of objects?
- A. Fn::GetAtt
- Incorrect.
- B. Fn::Select
- Correct!
- C. Fn::Combine
- Incorrect.
Which of the following functions is used in Cloudformation to append a set of values into a single value?
- B. Fn::GetAtt
- Incorrect.
- C. Fn::Combine
- Incorrect.
- D. Fn::Select
- Incorrect.
What is the maximum size of an item that corresponds to a single write capacity unit? (While creating an index or table in Amazon DynamoDB, it is required to specify the capacity requirements for the read and write activity)?
What can be used in DynamoDB as a part of the Query API call for the filtration of results based on the primary keys’ values?
- A. Conditions
- Incorrect.
- B. Expressions
- Correct!
Can a global secondary index create at the same time as the table creation?
What in AWS can be used to restrict access to SWF?
- D. None of the above
- Incorrect.
An IT admin has enabled long polling in their SQS queue. What must be done for long polling to be enabled in SQS?
- B. Create a dead letter queue
- Incorrect.
- C. Set the message size to 256KB
- Incorrect.
As per the IAM decision logic, what is the first step of access permissions for any resource in AWS?
- A. An explicit deny
- Incorrect.
- C. A default deny
- Correct!
- D. An explicit allow
- Incorrect.
Which API call is used to Bundle an Amazon instance store-backed Windows instance?
- A. AllocateInstance
- Incorrect.
- B. CreateImage
- Incorrect.
- C. ami-register-image
- Correct!
- D. BundleInstance
- Incorrect.
Practice Exam - AWS Developer Certification - Associate Level
More Information:
- Learning Style: Self-Paced
- Learning Style: Practice Exam
- Difficulty: Beginner
- Course Duration: 1 Hour
- Course Info: Download PDF
- Certificate: See Sample
Contact a Learning Consultant
Training 5 or more people?
Outline
Reviews
Community Experts | 70,967 |
Earlier this evening, or I should say yesterday evening as it is currently 1:16am, I had the pleasure of attending a gallery event at The Circus. The beautiful Kelly Lewis and her equally stunning daughter Alice, worked as a bad-ass duo to create haunting, captivating, charming, alluring and downright mind-bending photographs. Alice had more poise and life in her eyes than I've seen in most adults, and I watch a lot of ANTM. She seemed to look straight through the lens with such grace, even as live tarantulas crawled in her hair. (I'm not joking, look it up.) Capturing everything from Margot Tenenbaum to Joan of Arc, they created images that truly transported the viewer. Fixed them there, suspended in time, real or fictional. Until they finally got the courage to look away. I could rant and rave for pages upon pages about how amazing the work was, but I wouldn't do it justice. Please, please, check it out for yourselves. Buy something. Support their art and their story. Alice was adopted by Kelly and her husband when she was 7. Not a dewy baby, not a toddler with eyes aglow, a 7 year old girl with a mind and opinions, thoughts and a voice to speak them all with. Do you know how rare that is!? Do you know what a brave and courageous commitment that is?! To hear them speak about the work they do and the inspiration they bring to each other is truly, well, inspiring.
I am now more thrilled than ever to attend The Circus. To be a part of a community that grows and nurtures and prunes and primes it's students to create work that not only resonates, it thrills... again, I have no words.
I have a feeling I'm going to be wordless a lot over the next two years.
Thank you Kelly and Alice. With all my heart. | 228,435 |
NetScientific
(“NetScientific” or the “Company” or the “Group”)
NetScientific portfolio company Vortex Biosciences introduces novel fully automated circulating tumour cell enrichment instrument VTX-1 at AACR
London, UK – 18 April 2016 – NetScientific plc (AIM:NSCI), the transatlantic healthcare technology group, notes that its portfolio company, Vortex Biosciences, will preview VTX-1, a fully automated system for the efficient enrichment of intact circulating tumour cells (CTCs) from whole blood, at the American Association for Cancer Research (AACR) Annual Meeting held in New Orleans. Data presented, see separate press release below, at AACR demonstrate the ability of Vortex’s technology to rapidly collect highly enriched populations of CTCs, undamaged by labels or reagents, for colorectal and prostate cancer research.
Commenting on the news, NetScientific’s Chief Executive Officer, Francois R. Martelet said: “We believe Vortex’s technology, which is targeting a commercial launch in early 2017, could be of great benefit to the field of cancer research by providing a rapid, reliable and convenient way to collect circulating tumour cells, accelerating the development of innovative diagnostics and therapeutics.”
The full text of the announcement from Vortex Biosciences can be found below.
–.
Vortex Biosciences Previews Fully Automated System for
Label-Free, Intact CTC Enrichment
Studies Presented at AACR Support Application of Vortex’s Liquid Biopsy
Technology for Isolating Circulating Tumor Cells in Colorectal and Prostate Cancer Research
MENLO PARK, CA, April 18, 2016 – Vortex Biosciences previewed the VTX-1, a fully automated system for the efficient enrichment of intact circulating tumor cells (CTCs) from whole blood, at the American Association for Cancer Research (AACR) Annual Meeting 2016 (April 16-20, New Orleans). Data presented at AACR demonstrate the ability of Vortex’s technology to rapidly collect highly enriched populations of CTCs, undamaged by labels or reagents, for colorectal and prostate cancer research.
Representative of cancer status in the patient, CTCs, shed by tumors, can potentially reveal disease recurrence or disease progression earlier than imaging and more reliably compared with standard biomarkers. Previous research demonstrated the performance of Vortex’s technology in isolating CTCs in breast and lung cancer research.1
“As we move towards commercialization of the VTX-1 system, the studies presented at AACR confirm the ability of Vortex’s technology to isolate viable CTCs for a broad range of downstream assays,” explained Vortex CEO Gene Walther. “Empowering cancer researchers with a rapid, reliable and convenient solution to collect CTCs could advance cancer research and accelerate the development of innovative diagnostics and therapeutics.”
CTC Isolation
CTCs are relatively scarce, with concentrations as low as 1-10 CTCs/mL of whole blood, against a background of millions of white blood cells and billions of red blood cells. CTC enrichment technologies have been limited by complex sample processing, poor scalability, low sample purity, reliance on cell surface proteins for isolation, and dilute output volumes that require additional cell concentration steps.1
The Vortex VTX-1.
Studies at AACR
CTCs were isolated from colorectal cancer (CRC) patient blood samples using Vortex’s microfluidic technology in Enumeration and mutational profiling of CTCs, and comparison to ctDNA and colorectal cancer liver metastases2(poster #3149, to be presented 8 a.m.-12 p.m., Tuesday, April 19). In this study, nearly 25-fold more CTCs were found in preoperatively collected CRC patient samples than in age-matched healthy controls, and 80% of all CRC samples were identified as positive for CTCs. The number of CTCs for each patient showed a close correlation with clinical parameters and circulating tumor DNA levels. Compared with carcinoembryonic antigen value (the standard biomarker for CRC) or imaging, CTCs and CTC mutational profiles provided earlier indicators of minimal residual disease and anticipated tumor recurrence.
Another study, Label free collection of prostate circulating tumor cells using microfluidic Vortex technology3(poster #4967, to be presented 8 a.m.-12 p.m., Wednesday, April 20), demonstrates the ability of Vortex’s technology to rapidly collect pure populations of CTCs from blood samples in metastatic prostate cancer.
In a third study, Vortex technology for label-free enrichment of CTC from mouse xenograft models4 (poster #1525, to be presented 8 a.m.-12 p.m., Monday, April 18) investigators used Vortex’s technology to isolate CTCs from mouse blood. The investigators observed both high capture efficiency and high CTC purity, suggesting that the technology can be applied in mouse studies to facilitate discovery of new therapeutic targets and development of personalized medicine.
“These studies illustrate the potential of Vortex’s microfluidic technology to help cancer researchers advance detection of cancer disease recurrence and progression earlier and more reliably compared with standard approaches,” explained investigator Dr. Dino Di Carlo, Professor in the Department of Bioengineering at UCLA, where he directs the Microfluidic Biotechnology Laboratory..
# # #
References
1. Che J et al. Classification of large circulating tumor cells isolated with ultrahigh throughput microfluidic Vortex technology. Oncotarget, February 2016.
2. Kidess-Sigal E et al. Enumeration and mutational profiling of CTCs, and comparison to ctDNA and colorectal cancer liver metastases. Poster presentation 3149 at the American Association for Cancer Research Annual Meeting, April 19, 2016.
3. Pao E et al. Label free collection of prostate circulating tumor cells using microfluidic Vortex technology. Poster presentation 4967 at the American Association for Cancer Research Annual Meeting, April 20, 2016.
4. Heirich K et al. Vortex technology for label-free enrichment of CTC from mouse xenograft models. Poster presentation 1525 at the American Association for Cancer Research Annual Meeting, April 18, 2016.
END
MSCUBRURNRASAUR | 58,151 |
Fludarabine Phosphate and Total-Body Irradiation Followed By Donor Peripheral Blood Stem Cell Transplant in Treating Patients With Acute Lymphoblastic Leukemia or Chronic Myelogenous Leukemia That Has Responded to Treatment With Imatinib Mesylate, Dasatinib, or Nilotinib
- Full Text View
- Tabular View
- No Study Results Posted
- Disclaimer
- How to Read a Study Record
This phase II trial is studying how well fludarabine phosphate and total-body irradiation followed by donor peripheral blood stem cell transplant work in treating patients with acute lymphoblastic leukemia or chronic myelogenous leukemia that has responded to previous treatment with imatinib mesylate, dasatinib, or nilotinib. Giving low doses of chemotherapy, such as fludarabine phosphate, and total-body irradiation (TBI) before a donor peripheral blood stem cell transplant helps stop the growth of cancer cells. It may also stop mycophenolate mofetil and cyclosporine after the transplant may stop this from happening.
-
No publications provided by Fred Hutchinson Cancer Research Center
Additional publications automatically indexed to this study by ClinicalTrials.gov Identifier (NCT Number):
Additional relevant MeSH terms:
ClinicalTrials.gov processed this record on April 15, 2014 | 316,633 |
Management of shared stocks in South China Sea: Are we ready?
Date2000
Author
Page views39
MetadataShow full item record
Abstract
The paper informed, among others, that the development of fisheries management in the region should now be enhanced with the availability of the Code of Conduct for Responsible Fisheries and its various technical guidelines. Fisheries management deals with allocation of resources, hence the participation of stakeholders in the process of management plan development is a forefront requirement. With the increase in globalisation, some developed countries had used trade as a tool to promote sustainable and responsible fisheries. Eco-labelling was one of the merging practice in the global trade. USA had used the TED/BED issue as a means to reject import from any country which did not use TED/BED in their shrimp fisheries. Dolphin safe was another label that was required for tuna imported to the USA. It this became clear that strengthening the national ,management institutions by coastal states bordering the South China Sea should form an important agenda for the Fisheries Department in the individual countries. Regional and international organizations could than play a role towards enhancing the management of shared stocks, with the management of national stock by individual coastal states, to make the region ready for the management of shred stocks.
Suggested Citation
Martosubroto, P. (2000). Management of shared stocks in South China Sea: Are we ready? In Report of the Fourth Regional Workshop on Shared Stocks: Research and Management in the South China Sea (pp. 196-206). Kuala Terengganu, Malaysia: Marine Fishery Resources Development and Management Department, Southeast Asian Fisheries Development Center. | 233,722 |
Robeco Investment Management
Robeco relocated its Downtown Boston office to 31,210 SF of space on the 30th floor at One Beacon Street for which DPM managed the activities of the architect & engineering teams and developed cost data in order to complete test fits and budget data for the initial "stay vs. go" analysis. Diversified subsequently provided full project management services, including construction administration, FF&E coordination and move planning. The new space houses approximately 70 positions and consists of a communication room, conference space, kitchen and primarily hard wall offices. In addition, there are multiple trading desks and a “research room” where investors can gather to strategize and complete other work.
DPM was subsequently engaged in the expansion of Robeco’s space with the addition of 15,652 SF, on the 29th floor, which includes a new connecting stair to the current 30th floor space. | 89,956 |
\begin{document}
\maketitle
\thispagestyle{empty}
\begin{abstract}
We study the center of the universal enveloping algebra of the strange Lie
superalgebra $\q{N}$. We obtain an analogue of the well known Perelomov-Popov formula \cite{popov} for central elements of this algebra -- an
expression of the central characters through the highest weight parameters.
\end{abstract}
\section{Introduction}
\subsection{Lie superalgebra $\q{N}$}
The strange Lie superalgebra $\q{N}$ can be realized as a subalgebra in the
general linear Lie superalgebra $\mathfrak{gl}(N|N)$ over the complex field,
see for instance \cite{cheng_wang}. If the indices $i$ and $j$
range over $-N, \ldots, -1, 1, \ldots N$ then
the $4N^2$ elements $E_{ij}$
span the algebra $\mathfrak{gl}(N|N)$ as a vector space.
The Lie superbracket on $\mathfrak{gl}(N|N)$
is defined~by
\begin{equation*}
[E_{ij}, E_{kl}] = \delta_{jk} E_{il} - (-1)^{(\bar{\imath} + \bar{\jmath})(\bar{k} + \bar{l})}\delta_{il} E_{kj}
\end{equation*}
where
$$
\overline{k} = \begin{cases}
\,0
\quad\text{if}&k > 0\,,
\\
\,1
\quad\text{if}&k < 0\,.
\end{cases}
$$
Then $\q{N} \subset \mathfrak{gl}(N|N)$ is the subalgebra
of fixed points of the involution
$$\eta: E_{ij} \mapsto E_{-i-j}\,.$$
Hence as a vector space $\q{N}$ is spanned by the $2N^2$ elements
$$F_{ij} = E_{ij} + E_{-i-j}$$
with $i > 0$. The Lie superbracket on $\q{N}$ is then described by
\begin{equation}
\label{commut}
[F_{ij}, F_{kl}] =
\delta_{kj} F_{il} - (-1)^{(\overline{\imath}+ \overline{\jmath})(\overline{k} + \overline{l})} \delta_{il} F_{kj} + \delta_{k-j} F_{-il} - (-1)^{(\overline{\imath} + \overline{\jmath})(\overline{k} + \overline{l})} \delta_{-il} F_{k-j}\,.
\end{equation}
\subsection{Casimir elements in $U(\q{N})$}
Unless otherwise stated, we will be assuming that
the indices in the expressions below range over
$-N, \ldots, -1,$ $ 1, \ldots, N$.
For any $n=1,2,\ldots$ consider
the elements of the universal enveloping algebra $U(\q{N})$
first proposed in \cite{sergeev1}:
\begin{equation}
\label{casimirs}
C^{(n)}_{ij} =
\sum_{k_1,\ldots,k_{n-1}}F_{ik_1} (-1)^{\overline{k}_1} F_{k_1k_2} (-1)^{\overline{k}_2} \cdots F_{k_{n-2}k_{n-1}} (-1)^{\overline{k}_{n-1}} F_{k_{n-1}j}\,.
\end{equation}
Taking into account the recurrence relations
\begin{equation*}
C^{(n+1)}_{ij} =
\sum_k
F_{ik}\, (-1)^{\overline{k}}\, C^{(n)}_{kj}
\end{equation*}
one can find the supercommutator
\begin{equation*}
[F_{ij}, C^{(n)}_{kl}]=
\delta_{kj} C^{(n)}_{il}-(-1)^{(\overline{\imath} + \overline{\jmath})(\overline{k} + \overline{l})} \delta_{il} C^{(n)}_{kj} + \delta_{k-j}
C^{(n)}_{-il} - (-1)^{(\overline{\imath} + \overline{\jmath})
(\overline{k} + \overline{l})} \delta_{-il} C^{(n)}_{k-j}
\end{equation*}
which is similar to the superbracket (\ref{commut}) between the
generators of $\q{N}\,$. It is then easy to see that the elements
$$
c_n = \sum_{i} C^{(n)}_{ii}
$$ are central in $U(\q{N})\,$. Moreover, by using the relation
\begin{equation*}
C^{(n)}_{-i-j} = (-1)^{n-1}\, C^{(n)}_{ij}
\end{equation*}
following from (\ref{casimirs}) we see that
$c_n=0$ if $n$ is even. This is why we will be only interested in
the $c_n$ with $n$ odd. These are the {\it Casimir elements} for the
Lie superalgebra $\q{N}$ as introduced in \cite{sergeev2}.
It was shown
in \cite{nazarov_sergeev} that these elements generate
the center of $U(\q{N})\,$.
\subsection{Harish-Chandra homomorphism}
Let $v$ be a singular vector of any irreducible
finite-dimensional representation of the Lie superalgebra $\q{N}$ relative to the natural triangular decomposition
$\q{N} = \mathfrak{n}_{-} \oplus \mathfrak{h} \oplus \mathfrak{n}_{+}\,$ where
\begin{align*}
\begin{split}
\mathfrak{n}_{-} & = {\rm span}\left\{F_{ij} \,\big|\, |i| > |j|\right\},\\
\mathfrak{h} & = {\rm span}\left\{F_{ij} \,\big|\, |i| = |j|\right\},\\
\mathfrak{n}_{+} & = {\rm span}\left\{F_{ij} \,\big|\, |i| < |j|\right\}.
\end{split}
\end{align*}
Then the following equalities hold:
\begin{align}
\label{highestVec1}
F_{ij} \cdot v &= 0
\quad\text{for}\quad
|i| < |j|\,,
\\
\label{highestVec2}
F_{ii} \cdot v &= \lambda_i\,v
\quad\text{for}\quad
i>0\,.
\end{align}
Here the $\lambda_i$ are the eigenvalues of the elements $F_{ii} = F_{-i-i}$ of the even part of the Cartan subalgebra $\mathfrak{h}_0 = {\rm span}\left\{F_{ii}\, \big|\, i>0\right\}$.
They depend on the particular representation of $\q{N}\,$.
Let $\lambda \in \mathfrak{h}_0^*$ be the highest weight of the
representation so that $\lambda(F_{ii})=\lambda_i$ for $i>0\,$.
The generators of $\q{N}$ in the summands of (\ref{casimirs}) can always be rearranged in such a way that in each monomial left-to-right first go the lowering operators, then the operators from Cartan subalgebra and last the raising operators.
Here the lowering operators are elements of $\mathfrak{n}_-$ and the raising operators are elements of $\mathfrak{n}_+\,$.
The operators from Cartan subalgebra can also be rearranged so
that elements from its even part $\mathfrak{h}_0$ go after those from its odd part $\mathfrak{h}_1 = {\rm span}\left\{F_{-ii}\, \big|\, i>0\right\}\,$.
It suffices to use the supercommutation relations (\ref{commut}) to achieve this.
The part of the resulting sum which belongs to $U(\mathfrak{h})$
is well-defined. For the sum (\ref{casimirs}) with $i=j$
this is its image
under the {\it Harish-Chandra homomorphism} $\chi\,$, see~\cite{cheng_wang}.
The subspace formed by the vectors of an irreducible representation of $\q{N}$
satisfying (\ref{highestVec1}),(\ref{highestVec2})
is called the singular subspace. The peculiarity of $\q{N}$
shows in the fact that the singular subspace of an irreducible representation
is not one-dimensional, but is irreducible over the Cartan subalgebra
$\mathfrak{h} = \mathfrak{h}_0 \oplus \mathfrak{h}_1$ for which we have
$U(\mathfrak{h}) = S(\mathfrak{h}_0) \otimes \wedge(\mathfrak{h}_1)\,$,
see \cite{P}.
Due to (\ref{casimirs}) the Casimir elements are even and therefore
commute with the whole algebra $U(\q{N})$ in the usual non
$\mathbb{Z}_2$-graded sense.
This implies that they act as scalar operators in the irreducible representations. This allows us to consider their eigenvalues when acting on some fixed singular vector instead and the computations in this case are rather simple.
Let $n$ be odd and
$v$ be a singular vector of an irreducible representation of
$\q{N}\,$. Then
\begin{equation}
\label{action}
c_n\cdot v = \hc(c_n)\cdot v =
\hc(c_n)\,\big|_{\,F_{ii} = \lambda_i} \, v
\quad\text{for}\quad
i>0\,.
\end{equation}
For $i>0$ we will automatically substitute
$F_{ii} \mapsto \lambda_i$ after applying
the
homomorphism $\hc$ as we did in (\ref{action}).
Hence we will be describing the action on the singular vector explicitly.
\section{Computations}
\subsection{Recurrence relations}
Here we will derive a recurrence relation for
the images of the elements $C^{(n)}_{ii}$ with $n=1,3,\ldots$
under the Harish-Chandra homomorphism.
For brevity we will denote by $G_i$ the element
$F_{-ii} = F_{i-i}$ of the odd part of the Cartan~subalgebra.
\begin{prop}
For $i>0$ we have $\hc(G_i^2) = \lambda_i\,$.
\end{prop}
\begin{prf}
We have
$G_i^2 \cdot v = F_{-ii}^2 \cdot v = \frac{1}{2}\, [F_{-ii}, F_{-ii}] \cdot v= F_{ii} \cdot v = \lambda_i\,v$
for $i>0\,.$
\end{prf}
\begin{prop}
We have
$C^{(n)}_{ij} \cdot v = 0$ whenever $|i| < |j|\,$.
\end{prop}
\begin{prf}
Suppose that $|i| < |j|$. Then $C^{(1)}_{ij} \cdot v = F_{ij} \cdot v = 0\,$.
Let us use the induction on $n\,$:
$$
C^{(n+1)}_{ij} \cdot v =\sum_k
F_{ik}\,(-1)^{\overline{k}}\,C^{(n)}_{kj} \cdot v = \sum_{|k| \geqs |j|} (-1)^{\overline{k}}\,[F_{ik}, C^{(n)}_{kj}] \cdot v = \sum_{|k| \geqs |j|} (-1)^{\overline{k}}\,C^{(n)}_{ij} \cdot v = 0\,.
$$
\end{prf}
\begin{prop}
For $i>0$ we have
$$
\hc(C^{(n+1)}_{ii}) = \lambda_i\, \hc(C^{(n)}_{ii}) - G_i\, \hc(C^{(n)}_{-ii}) - \sum_{|k| > i}\, \hc(C^{(n)}_{kk})\,.
$$
\end{prop}
\begin{prf}
For $i>0$ the vector $C^{(n+1)}_{ii} \cdot v$ equals
\begin{gather*}
\sum_{|k| \geqs i} F_{ik}\, (-1)^{\overline{k}}\, C^{(n)}_{ki} \cdot v =
(F_{ii} C^{(n)}_{ii} - F_{i-i} C^{(n)}_{-ii}) \cdot v + \sum_{|k| > i} (-1)^{\overline{k}}\, [F_{ik}, C^{(n)}_{ki}] \cdot v=
\\
(\lambda_i C^{(n)}_{ii} - G_i C^{(n)}_{-ii}) \cdot v + \sum_{|k| >i} (-1)^{\overline{k}}\, \big(C^{(n)}_{ii} -
(-1)^{\overline{k}}\, C^{(n)}_{kk}\big) \cdot v=
\\
\big(\lambda_i C^{(n)}_{ii} -
G_i C^{(n)}_{-ii} - \sum_{|k|>i} C^{(n)}_{kk}\big) \cdot v\,.
\end{gather*}
\end{prf}
\begin{cons}
For $i>0$ and $m=1,2,\ldots$
we get $\hc(C^{(2m+1)}_{ii}) =
\lambda_i\,\hc(C^{(2m)}_{ii}) - G_i\,\hc(C^{(2m)}_{-ii})\,$.
\end{cons}
\begin{prop}
For $i>0$ we have
$$
\hc(C^{(n+1)}_{-ii}) =
G_i \hc(C^{(n)}_{ii}) - \lambda_i \hc(C^{(n)}_{-ii}) - \sum_{|k|>i} (-1)^{\overline{k}}\, \hc(C^{(n)}_{k-k})\,.
$$
\end{prop}
\begin{prf}
For $i>0$ the vector $C^{(n+1)}_{-ii} \cdot v$ equals
\begin{gather*}
\sum_{|k| \geqs i} F_{-ik}\,
(-1)^{\overline{k}}\,C^{(n)}_{ki} \cdot v =
(F_{-ii} C^{(n)}_{ii} - F_{ii} C^{(n)}_{-ii}) \cdot v + \sum_{|k| >i} (-1)^{\overline{k}}\, [F_{-ik}, C^{(n)}_{ki}] \cdot v =
\\
(G_i C^{(n)}_{ii} - \lambda_i C^{(n)}_{-ii}) \cdot v + \sum_{|k| > i} (-1)^{\overline{k}}\,\big(C^{(n)}_{-ii} -
(-1)^{\overline{k}\,(\overline{k} + 1)}\,C^{(n)}_{k-k}\big) \cdot v =
\\
\big(G_i C^{(n)}_{ii} - \lambda_i C^{(n)}_{-ii}\big) \cdot v - \sum_{|k|>i} (-1)^{\overline{k}}\, C^{(n)}_{k-k} \cdot v\,.
\end{gather*}
\end{prf}
\begin{cons}
For $i>0$ and $m=1,2,\ldots$ we get
$\hc(C^{(2m)}_{-ii}) =
G_i \hc(C^{(2m-1)}_{ii}) - \lambda_i\hc(C^{(2m-1)}_{-ii})\,$.
\end{cons}
\begin{prop}
\label{P5}
For $i>0$ and $m=1,2,\ldots$ we have the relation
$$
\hc(C^{(2m+1)}_{ii}) = \hc(C^{(2m+1)}_{-i-i}) = \lambda_i (\lambda_i - 1) \hc(C^{(2m-1)}_{ii}) - 2\lambda_i \sum_{j>i}^{} \hc(C^{(2m-1)}_{jj})\,.
$$
\end{prop}
\begin{prf}
If $i>0$ then the vector $C^{(2m+1)}_{ii}\cdot v $ equals
\begin{gather*}
\big(\lambda_i C^{(2m)}_{ii} - G_i C^{(2m)}_{-ii}\big) \cdot v =
\\
\lambda_i \big(\lambda_i\, C^{(2m-1)}_{ii} - G_i\, C^{(2m-1)}_{-ii} -
\sum_{|j| > i}^{}\, C^{(2m-1)}_{jj}\big) \cdot v - G_i \big(G_i C^{(2m-1)}_{ii} - \lambda_i C^{(2m-1)}_{-ii}\big) \cdot v =
\\
(\lambda_i^2 - G_i^2) C^{(2m-1)}_{ii} \cdot v - \lambda_i \sum_{|j|>i} C^{(2m-1)}_{jj} \cdot v = \lambda_i (\lambda_i - 1) C^{(2m-1)}_{ii} \cdot v - 2\lambda_i \sum_{j>i} C^{(2m-1)}_{jj} \cdot v\,.
\end{gather*}
\end{prf}
\begin{cons}
For $i>0$ and $m=0,1,2,\ldots$ we have
$\displaystyle
\hc(C^{(2m+1)}_{ii}) = \sum_{j = 1}^N\big( A^{m}\big)_{ij}\,\lambda_j$ where
$$A =
\begin{pmatrix}
\lambda_1(\lambda_1 - 1) & -2\lambda_1 & \cdots & -2\lambda_1\\
0 & \lambda_2(\lambda_2 - 1) & \cdots & -2\lambda_2\\
\vdots & & \ddots & \vdots\\
0 & \cdots & 0 & \lambda_N(\lambda_N - 1)
\end{pmatrix}.
$$
\end{cons}
\begin{prf}
This follows from Proposition \ref{P5} by
taking into account that $\hc(C^{(1)}_{ii}) = \hc(F_{ii}) = \lambda_i\,$.
\end{prf}
\subsection{Generating functions}
In order to compute $\hc(c_{2m+1})$ more explicitly, for each $i > 0$
consider the generating function
\begin{equation}
\label{pi}
\mu_i(u) = \sum_{m=0}^\infty \hc(C^{(2m+1)}_{ii})\,u^{-2m-1} = u \sum_{j=1}^N \big((u^2 - A)^{-1}\big)_{ij} \lambda_j
\end{equation}
and write
$$
A=\Lambda(\Lambda - 1) - 2\Lambda \Delta (1 - \Delta)^{-1}
$$ where $\Delta_{ij} = \delta_{i\,j-1}$ and $\Lambda = \rm{diag}(\lambda_i)$. Denote
$$
\Pi = \big(u^2 - \Lambda(\Lambda + 1)\big) \big(u^2 - \Lambda(\Lambda - 1)\big)^{-1}\,.
$$
Then
\begin{equation*}
(u^2 - A)^{-1} = \frac{1}{2}\, (1 - \Pi)\, (1 - \Delta \Pi)^{-1}\, (1 - \Delta) \Lambda^{-1}.
\end{equation*}
The last two factors here cause all summands but the last in the sum (\ref{pi}) cancel, leaving
\begin{gather*}
\mu_i(u) = \frac{u}{2}\, (1 - \Pi_{ii}) \left((1 - \Delta \Pi)^{-1}\right)_{iN} =\\
\frac{u}{2}\, (1 - \Pi_{ii})\, (\Delta \Pi)_{i\,i+1} (\Delta \Pi)_{i+1\,i+2} \ldots (\Delta \Pi)_{N-1\,N} = \frac{u}{2}\, (1 - \Pi_{ii}) \prod_{j > i} \Pi_{jj}\,.
\end{gather*}
This can also be written in more straightforward way:
\begin{equation*}
\mu_i(u) = \frac{u\lambda_i}{u^2 - \lambda_i(\lambda_i - 1)}
\,\prod_{j > i}\,
\frac{u^2 - \lambda_j(\lambda_j + 1)}{u^2 - \lambda_j(\lambda_j - 1)}\,.
\end{equation*}
In order to find a generating function $\mu(u)$ of the images of the
central elements under $\chi\,$, we should now sum all
the above obtained expressions for $\mu_i(u)$
over the positive values of the index $i$ and recall that
$C_{-i-i}^{(2m+1)} = C_{ii}^{(2m+1)}$. By making further cancellations we get
\begin{equation*}
\mu(u) = \sum_{m=0}^\infty \hc(c_{2m+1}) u^{-2m-1}=2\,\sum_{i=1}^N\,\mu_i(u)= u \left(1 - \prod_{j=1}^N \Pi_{jj}\right)\,.
\end{equation*}
Finally, we obtain
\begin{equation*}
\mu(u) = u \left(1 - \prod_{i=1}^N \frac{u^2 - \lambda_i(\lambda_i + 1)}{u^2 - \lambda_i(\lambda_i - 1)}\right).
\end{equation*}
\subsection{Images of the central elements}
Introduce a new variable $z = u^{-2}$ and define
\begin{equation*}
\widetilde{\mu}(z) = u\, \mu(u) = \sum_{m=0}^\infty \hc(c_{\,2m+1})\,z^{m}
=\frac{1}{z} \left(1 - \prod_{i=1}^N \frac{1 - z\lambda_i(\lambda_i + 1)}{1 - z\lambda_i(\lambda_i - 1)}\right).
\end{equation*}
From the above definition it immediately follows that
$\hc(c_{\,2m+1}) = {\rm Res}_0\; \widetilde{\mu}(z) z^{-m-1}\, dz$.
Combining this with our explicit expression for $\widetilde{\mu}(z)$
we obtain that $\hc(c_{\,2m+1})$ equals
$$
-\sum_{i=1}^N {\rm Res}_{\left(\lambda_i(\lambda_i - 1)\right)^{-1}}\; \widetilde{\mu}(z)\,z^{-m-1}\, dz = \sum_{i=1}^N {\rm Res}_{\left(\lambda_i(\lambda_i - 1)\right)^{-1}}\, \prod_{j=1}^N\,
\frac{1 - z\lambda_j(\lambda_j + 1)}{1 - z\lambda_j(\lambda_j - 1)}\, z^{-m-2}\, dz
$$
where the regularity of the form at the infinity is taken into account.
Finally for $m\geqslant0$
$$\hc(c_{\,2m+1}) = 2\sum_{i=1}^N \lambda_i^{m+1}(\lambda_i - 1)^m \prod_{j\not=i} \frac{\lambda_i(\lambda_i - 1) - \lambda_j(\lambda_j + 1)}{\lambda_i(\lambda_i - 1) - \lambda_j(\lambda_j - 1)}\ .
$$
Note that despite the fact that the last expression is formally a
rational function of $\lambda\,$, its denominator always cancels
which allows us to regard $\hc(c_{\,2m+1})$ as a polynomial of $\lambda\,$.
\section*{Remarks}
The analogue of the Perelomov-Popov formula for the Lie superalgebra $\q{N}$ presented here was obtained by the second named author about 30 years ago but left unpublished.
Independently but by the same method, this analogue was obtained in \cite{BK} and recently by the first named author. Publishing this note gives us an opportunity to review the history
of this analogue. We thank Jonathan Brundan, Maria Gorelik, Alexandre Kirillov, Grigori Olshanski, Ivan Penkov, Vera Serganova and Alexander Sergeev for helpful conversations. | 201,458 |
\section{New bounds on the minimum and maximum singular values of Vandermonde matrices}
\label{sec: conditioning Vandermonde matrices}
In this section, we provide new lower and upper bounds on the minimum and maximum singular values of Vandermonde matrices with nodes inside the unit disk.
In order to put our results into perspective, we first review bounds available in the literature.
An upper bound on the condition number of Vandermonde matrices with nodes inside the unit disk was provided by Baz\'an in~\cite[Thm.~6]{Bazan2000}. This bound is, however, somewhat complicated and seems to be amenable to analytical statements only for $N \rightarrow \infty$. Specifically, it allows to conclude that the condition number is close to $1$ if the nodes are separated enough and close to the unit circle. Unlike Baz\'an's result~\cite[Thm.~6]{Bazan2000}, the upper bound on the condition number we present here is expressed directly in terms of the minimum distance of the nodes from the unit circle.
Our result is inspired by the link---first established by Moitra~\cite{Moitra2014}---between the condition number of Vandermonde matrices with nodes on the unit circle and Selberg's work on sharp forms of the large sieve inequality~\cite{Selberg1991}.
We rely on a result by Montgomery and Vaaler~\cite{Montgomery1998} extending---to the complex case---a generalization of Hilbert's inequality due to Montgomery and Vaughan~\cite[Thm.~1]{Montgomery1974}.
In contrast, the derivation of Moitra's upper bound is based on extremal minorants and majorants for the characteristic function of an interval. Both Moitra's result and our result are, however, in essence, linked to the large sieve inequality.
\begin{thm}
\label{thm: upper bound spectral condition number nodes in the unit disk refinement}
For $k \fromto{1, 2, \ldots, \nbNodes}$, let $z_k \triangleq e^{-d_\mathrm{max}}e^{2\pi if_k/F_\mathrm{s}}$ be complex numbers with $d_k \geq 0$ and $f_k \in [0, F_\mathrm{s})$.
Let
\begin{equation*}
\delta \triangleq \min_{n \in \Z} \min_{\substack{1 \leq k, \ell \leq K \\ k \neq \ell}} \abs{f_k - f_\ell + nF_\mathrm{s}}
\end{equation*}
be the minimum wrap-around distance between the $f_k$, $k \fromto{1, 2, \ldots, \nbNodes}$,
and $d_\mathrm{max} \triangleq \displaystyle \max_{1 \leq k \leq \nbNodes} d_k$.
For
\begin{equation}
d_\mathrm{max} < 1/(\nbSamples-1)
\label{eq: condition on dmax}
\end{equation}
and
\begin{equation}
\delta > \frac{84F_\mathrm{s}}{\pi\left(\nbSamples-1\right)\big(1 - d_\mathrm{max}(\nbSamples-1)\big)},
\label{eq: condition on delta}
\end{equation}
the smallest and largest singular values of the Vandermonde matrix $\vander_N$ obey
\begin{align*}
\sigma_\mathrm{min}^2(\vander_N) &\geq \left(\nbSamples-1\right)\big(1 - d_\mathrm{max}(\nbSamples-1)\big) - 84F_\mathrm{s}/(\pi\delta)\\
\sigma_\mathrm{max}^2(\vander_N) &\leq \nbSamples-1 + 84F_\mathrm{s}/(\pi\delta),
\end{align*}
and thus, the condition number of $\vander_N$ satisfies
\begin{equation}
\kappa\!\left(\vander_\nbSamples\right) \leq \sqrt{\frac{\nbSamples-1 + 84F_\mathrm{s}/(\pi\delta)}{\left(\nbSamples-1\right)\big(1 - d_\mathrm{max}(\nbSamples-1)\big) - 84F_\mathrm{s}/(\pi\delta)}}.
\label{eq: main result upper bound spectral condition number for nodes in the unit disk}
\end{equation}
\end{thm}
Theorem~\ref{thm: upper bound spectral condition number nodes in the unit disk refinement} shows that the condition number of Vandermonde matrices with nodes in the unit disk is close to $1$ if the minimum wrap-around distance between the node frequencies $f_k$ is large relative to $F_\mathrm{s}/(\nbSamples-1)$, and the damping factors $d_k$ are small enough (i.e., the nodes $z_k$ are close enough to the unit circle). The conditions \eqref{eq: condition on dmax} and \eqref{eq: condition on delta} on $d_\mathrm{max}$ and $\delta$ ensure that our lower bound on $\sigma_\mathrm{min}(\vander_\nbSamples)$ is positive.
When particularized for the undamped case $d_\mathrm{max} = 0$ (i.e., $\abs{\node_k} = 1$ for all $k \fromto{1, 2, \ldots, \nbNodes}$), our result recovers Moitra's upper bound provided in \cite[Thm.~2.3]{Moitra2014} up to a difference in the constant $84/\pi$ in the numerator and denominator of \eqref{eq: main result upper bound spectral condition number for nodes in the unit disk}, which in Moitra's case ($d_\mathrm{max} = 0$) equals $1$. We note, however, that for $d_\mathrm{max} = 0$, \cite[Thm.~1]{Montgomery1974} can be used instead of \cite{Montgomery1998} to recover Moitra's upper bound exactly in our approach. | 21,850 |
Updated:
BANGOR, Maine — Peter Vigue, chairman and CEO of Cianbro Corp., on Wednesday evening received the Norbert X. Dowd Award at the Bangor Region Chamber of Commerce’s annual awards dinner.
The award is named after Norbert X. Dowd, the late businessman who ran the Chamber for 27 years.
Vigue said he was “very humbled” by the honor, and shared the credit with his family and the employees of Cianbro.
“I am the recipient of the award,” Vigue said Wednesday. “However, I don’t do these things alone. I have tremendous support from my family and the more than 4,000 people that work at this company. I work for them, and they give me a tremendous amount of support and help. I feel very privileged and humbled by this award, but I share it with them, as well.”
John Porter, the Chamber’s president, said the Norbert X. Dowd award is considered a lifetime achievement award.
“This award really honors someone who over the course of their careers contributes to the community and economy of the region, and has shown to be a leader on both those fronts,” Porter said.
More than 800 people gathered on Wednesday evening for the Bangor Chamber’s annual awards dinner, the last that will be held in the Bangor Civic Center. Next year, the awards dinner will be held in the new Cross Insurance Center, Porter said.
“This is our last hurrah in the old Civic Center, and we want to go out in style,” he said. “We have an opportunity to recognize some terrific folks, and as always the event itself will be eye-popping.”
Each of the awards presented Wednesday evening were preceded by videos produced by Sutherland Weston Marketing.
Other award winners Wednesday evening were WBRC Architects-Engineers, which received the Business of the Year Award; former state Sen. Richard Rosen, who received the Catherine Lebowitz Award for Public Service; the Hammond Street Senior Center, which received the Community Service Award; the Children’s Miracle Network Hospitals of EMHS, which received the Nonprofit of the Year Award; Andrew Hamilton of Eaton Peabody, who received the Arthur A. Comstock Professional Service Award; and Geaghan’s Pub & Craft Brewery, which received the Bion and Dorain Foster Entrepreneurship Award.
While the awards recognize the contributions of certain individuals and organizations, that’s not the only reason the Chamber presents them each year, Porter said.
“It’s nice to honor the winners, but we also do them for the community, to remind the community what a great place this is,” Porter said. “Our best days are ahead of us, not behind us.” | 168,492 |
Subsets and Splits