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\section{Introduction}
Finite tight frames \cite{W18,BF03} are a natural generalization of orthonormal bases \cite{KC07, KC072}. These frames play a central role in quantum physics, as they are one-to-one related to Positive Operator Valued measure (POVM), the most general kind of measurements allowed by quantum mechanics. Tight frames also find applications in signal theory \cite{BYP04, PY97}, coding and communication \cite{SH03}, allowing to design polarimeters that maximize the signal-to-noise ratio and minimizing propagation of systematic errors \cite{T02,CK13}
. Optimal ways to implement quantum mechanical protocols typically require to consider sets of quantum measurements induced by vectors having a special geometrical distribution in Hilbert space, e.g. mutually unbiased bases (MUB) \cite{I81,WF89} and symmetric informationally complete (SIC)-POVM \cite{R04}. These constellations are relevant to quantum state estimation \cite{WF89,AS10}, entanglement detection \cite{SHBAH12}, quantum key distribution \cite{MDGGMPKPLF13}, and quantum quantum-nonlocality \cite{BCPSW14,TFRBK21}. On the other hand, sets of frames are relevant in quantum state tomography \cite{PBJD10}, quantum cloning \cite{JRD10} and quantum state discrimination \cite{RSHK11}.
Along the last years, the notion of mutually unbiasedness has be extended to mutually unbiased simplices \cite{FS20,S19}, mutually unbiased equiangular tight frames \cite{FM20}, and mutually unbiased POVM \cite{BBBCHT13}. Motivated by trying to understand the tree from exploring the forest, we introduce the most general notion of mutually unbiasedness for constellations composed by linearly independent vectors, namely \emph{mutually unbiased frames} (MUF). This notion allows to study some geometrical problems from a general perspective, revealing interesting properties that are not simple to see otherwise. For instance, we derive a series of analytic results for SIC-POVM, holding for any fiducial state in every finite dimension.\medskip
This work is organized as follows: in Section \ref{sec:preliminaries}, we introduce the basic notions required to understand the work, as well as fix the notation. In Section \ref{sec:MUF}, we introduce mutually unbiased frames (MUF) and some basic results. In Section \ref{sec:matrixMUF}, we show a matrix approach to MUF and study the particular case where these matrices are circulant. Here, both the mutually unbiased bases problem in prime dimensions and the SIC-POVM problem in every dimension can be studied as the problem to find $d$ MUF, encoded in $d$ circulant matrices. We also find the most general pair of MUF for a qubit system, showing that MUB and SIC-POVM arise as particular cases. In Section \ref{sec:sic}, we apply our formalism to the SIC-POVM problem. We show that Zauner's conjecture about the existence of Weyl-Heisenberg covariant SIC-POVM is equivalent to the most reasonable way to try to search $d$ MUF in dimension $d$, assuming that circulant matrices is the simplest class of commuting matrices to deal with. We also show that real fiducial states, with respect to the standard Weyl-Heisenberg group, do not exist in even dimensions. Furthermore, we demonstrate that fiducial states belong to an explicitly given $\lfloor(d-1)/2\rfloor+d-1$-dimensional set, in every finite dimension $d$. Finally, fiducial states are shown to be minimum uncertainty states with respect to a large number of subsets of $d+1$ bases, induced by Clifford unitaries. Proofs of our results can be found in Appendix \ref{app:proofs}.
\section{Preliminaries}\label{sec:preliminaries}
In this section, we introduce the notions required to understand the rest of the work, as well as we fix the notation. Let us start by noting that pure states $|\psi\rangle$ are normalized with respect to the norm induced by the scalar product in a $d$-dimensional Hilbert space $\mathcal{H}_d$. That is, $\|\psi\|^2=\langle\psi|\psi\rangle=\sum_{j=1}^d|\langle j|\psi\rangle|^2=1$, where $|j\rangle$ denotes the $j$th element of the canonical basis, also called computational basis in quantum information theory. Here, $|\psi\rangle$ denotes an element of $\mathcal{H}_d$, $\langle\phi|$ an element of the dual space $\mathcal{H}_d^*$, $\langle\phi|\psi\rangle$ and $|\phi\rangle\langle\psi|$ the inner and outer products, respectively, according to the notation. Note that $|\phi\rangle\langle\phi|$ denotes the rank-one projector associated to the direction $|\phi\rangle$. Let us now introduce the concept of frames \cite{DS52}.
\begin{defi}
A set of $n\geq d$ state vectors $\{|\phi_i\rangle\}_{i=1,\dots,n}$ in $\mathcal{H}_d$ define a frame if there exist constants $0<A\leq B<\infty$ such that
\begin{equation}\label{frame}
A\leq \sum_{i=0}^{n-1}|\langle x,\phi_i\rangle|^2\leq B,\qquad\mbox{for all }|x\rangle\in\mathcal{H}_d.
\end{equation}
\end{defi}
Here, $A$ and $B$ are called the lower and upper bounds of the frame, given by the minimal and maximal eigenvalues of the \emph{frame operator} $S=\sum_{i=0}^{n-1}|\phi_i\rangle \langle\phi_i|$, respectively \cite{CK13}. A frame is called \emph{tight} if $A$ equals $B$, having in such case that $A=B=n/d$. When $A=B=1$, we have Parseval tight Frames \cite{FEBS}, equivalent to orthonormal bases in finite dimensions. There is a simple way to detect whether a given set of vectors forms a frame: linearly independence of $d$ out of $n\geq d$ vectors is a necessary and sufficient condition to have a frame in $\mathcal{H}_d$. A gentle introduction to finite frames theory can be found in Ref. \cite{C13}.
A special kind of tight frames occurs when its $n$ vectors are equiangular, known as \emph{equiangular tight frames} (ETF) \cite{STDH07}, meaning that there is a constant $c>0$ such that $|\langle\phi_i|\phi_j\rangle|^2=c$, for every $i\not=j$ taken from the set $\{0,\dots,n-1\}$. Here, the number $\sqrt{c}$ is typically called \emph{coherence} but, for convenient reasons, along this work we alternatively refer to the constant $c$ as the \emph{overlap} of a MUF. See \cite{FM15} for a summary of the current state of the art of the ETF problem.
Among all constellations of vectors in Hilbert space, there is a specially distinguished one: mutually unbiased bases \cite{BSTW05}.
\begin{defi}
Two orthonormal bases $\{|\phi_j\rangle\}_{j=0,\dots,d-1}$ and $\{|\psi_k\rangle\}_{k=0,\dots,d-1}$ defined in $\mathcal{H}_d$ are mutually unbiased (MUB) if $|\langle\phi_j,\psi_k\rangle|^2=\frac{1}{d}$, for every $j,k=0,\dots,d-1$. Also, a set of $m$ orthonormal bases are MUB if they are pairwise MUB.
\end{defi}
For instance, the eigenvectors bases of the three Pauli matrices define $m=3$ MUB in $\mathcal{H}_2$. More generally, let \begin{equation}\label{XZ}
X=\sum_{k=0}^{d-1}|k+1\rangle\langle k| \mbox{ and } Z=\sum_{k=0}^{d-1}\omega^k|k\rangle\langle k|,
\end{equation}
where $\omega=e^{e\pi i/d}$, the \emph{shift} and \emph{clock} operators, respectively. Therefore, the essentially unique eigenvectors bases of operators $$X,Z,XZ,XZ^2,\dots,XZ^{d-1,}$$ form $m=d+1$ MUB in $\mathcal{H}_d$, whenever $d$ is a prime number. Here, and in the rest of the work, we assume addition modulo $d$ in kets, i.e. $|(d-1)+1\rangle=|0\rangle$.
MUB have been extensively studied along the last 40 years. A maximal set of $d+1$ MUB exists in every prime \cite{I81} and prime power \cite{WF89} dimension $d$. On the other hand, the maximal number of MUB in any other composite dimension $d$ is still open, even in the lowest dimensional case $d=6$, where at most triplets of MUB are known \cite{DEBZ10,RLE11,BBELTZ07,G13}. Another remarkable constellation are the SIC-POVM \cite{Z99,RBSC04,SG10}, defines as follows:
\begin{defi}
A set of $n=d^2$ state vectors $\{|\phi_j\rangle\}\subset\mathcal{H}_d$ form a SIC-POVM if $|\langle\phi_j|\phi_k\rangle|^2=\frac{1}{d+1}$, for every $j\not=k$.
\end{defi}
SIC-POVM are known to exist in dimension $d=2-53$ and in $63$ higher dimensions, the highest being 5779 \cite{G21}. Also, numerical solutions are known in dimensions $d=2-193$ and in some higher dimensions, being the highest 2208 \cite{S17,GS17}. In very low dimensions, fiducial states are simple to find. However, as long as the dimension increases, even the problem to find numerical solutions becomes hard. This fact motivated the search of additional symmetries in fiducial states. In short, every known Weyl-Heisenberg fiducial state is eigenvector of an order 3 Clifford operator \cite{A05}. More details about symmetries in fiducial states can be found here \cite{Z99,SG10,S17,A05,BW19}. There is a single exception to the above rule for fiducial states: the \emph{Hoggar lines} \cite{H98}. These solutions for 3-qubit systems are characterized by fiducial states that are covariant with respect to the tensor product of single qubit WH groups. The 240 fiducials existing in this case can be divided in two classes, distinguished by the fact that they have different amounts of entanglement \cite{CGZ18}.
The above mentioned symmetries allowed to construct every known fiducial state. However, as far as we know, there is \emph{no single proof} for the veracity of such symmetries, beyond its remarkable success. In Section \ref{sec:sic}, we provide the first cornerstone along this direction by revealing an elusive symmetry that holds for any existing fiducial state, with respect to the Weyl-Heisenberg group, in every finite dimension. This symmetry allows us to define fiducial states within a set composed by $\lfloor (d-1)/2\rfloor+d-1$ real free parameters, in every dimension $d$, representing a strong reduction with respect to the full number of free parameters $2(d-1)$.
To conclude this section, we highlight the interesting connection existing between the SIC-POVM problem and algebraic number theory, including a close relation to the 12$th$ Hilbert problem \cite{AFMY17}.
\section{Mutually unbiased frames}\label{sec:MUF}
The notion of mutually unbiasedness for constellations of vectors has been extensively used in quantum mechanics. An important reason to do that is because two Von Neumann observables having mutually unbiased eigenvectors bases are complementary, e.g. position and momentum or orthogonal directions of spin $1/2$ observables. Mutually unbiasedness has been extended to regular simplices \cite{FS20,S19}, equiangular tight frames \cite{FM20} and POVM in general \cite{BBBCHT13}. In this section, we introduce a notion of unbiasedness that includes all the above notions. The aim to introduce this generalization is to study the existence and construction of some inequivalent geometrical structures under the same framework, e.g. MUB and SIC-POVM. Despite our generalization goes beyond the set of quantum measurements, the new point of view reveals interesting properties when restricting the attention to POVM, as we will see in Section \ref{sec:sic}. Let us start by defining the central concept of our work.
\begin{defi}
Let $\{|\phi_j\rangle\}_{j=0,\dots,n_1-1}$ and $\{|\psi_k\rangle\}_{k=0,\dots,n_2-1}$ be two frames in $\mathcal{H}_d$. We call them Mutually Unbiased Frames (MUF) if there is a constant $c>0$ such that
\begin{equation}\label{MUF}
|\langle\phi_j|\psi_k\rangle|^2=c,
\end{equation}
holds for every $j=0,\dots,n_1-1$ and $j,k=0,\dots,n_2-1$. The constant value $c$ is called the overlap of the MUF.
\end{defi}
In particular, MUF reduce to mutually unbiased POVM \cite{BBBCHT13} when both frames are tight, for any $n_1$ and $n_2$. Additionally, if $n_1=n_2=d$, then they are MUB. Figure \ref{Fig1} illustrates the relation existing between different
notions of unbiasedness. Given that the sum of all sub-normalized rank-one projector associated to a tight frame sum up to the identity, it is simple to show that $c=1/d$ holds in (\ref{MUF}) when both frames are tight \cite{BBBCHT13}. However, if the frames are not tight then $c$ might take another value. Let us study the range of allowed values for the overlap $c$ in MUF.
\begin{prop}\label{prop:muf}
Let $\{|\phi^{(\ell)}_j\rangle\}$ be a set of $m$ mutually unbiased frames in $\mathcal{H}_d$ composed by $n$ elements each, with overlap $c$. Also, assume that each frame has lower and upper frame bounds $A_{\ell}$ and $B_{\ell}$, respectively, where $\ell=0,\dots,m-1$. Therefore,
$\max_{\ell} A_{\ell} \leq cn\leq \min_{\ell}B_{\ell}$.
\end{prop}
\begin{figure}[htp]
\centering
\includegraphics[width=0.7\textwidth]{Fig1.jpg}
\caption{Mutually unbiased frames (MUF), the main concept introduced in this work, represents the most general notion of unbiasedness for sets of linearly independent vectors. It generalizes the existing notions of unbiasedness for orthonormal bases (MUB) \cite{I81,WF89}, simplices (MUS) \cite{FS20,S19}, equiangular tight frames (MU-ETF) \cite{FM20} and POVM (MU-POVM) \cite{BBBCHT13}. Additionally, SIC-POVM \cite{Z99,RBSC04} can be seen as sets of $d$ MUF in dimension $d$, as we show in Section \ref{sec:sic}.}
\label{Fig1}
\end{figure}
For the case of orthonormal bases, Proposition \ref{prop:muf} reduces to mutually unbiased bases, where $c=1/d$. Indeed, the same overlap $c$ occurs when at least one of the frames is tight, thus forming a POVM.
\begin{corol}\label{corol:muf}
Let $\{|\phi^{(\ell)}_j\rangle\}$ be a set of $m$ mutually unbiased frames in $\mathcal{H}_d$, composed by $n$ elements each, such that at least one of them is tight. Therefore,
$c=1/d$.
\end{corol}
From Corollary \ref{corol:muf}, note that Parseval MUF are mutually unbiased bases in every finite dimension $d$. In Section \ref{sec:matrixMUF}, we show examples of MUF in $\mathcal{H}_d$ having overlap $c\not=1/d$.
\section{Matrix approach to mutually unbiased frames}\label{sec:matrixMUF}
In this section, we consider a matrix approach to the problem of mutually unbiased frames. In particular, we study the case of $m$ MUF composed by $d$ vectors each in $\mathcal{H}_d$. Entries in the computational basis of the $j$-th frame are arranged as columns of a square complex matrix of order $d$, for every $j=0,\dots,m-1$. Thus, we have $m$ square matrices $M_0,\dots,M_{m-1}$ satisfying
\begin{equation}\label{matrices_muf}
M_j^{\dag}M_k=K^{(jk)},
\end{equation}
where $|(K^{(jk)})_{st}|^2=c$, for every $s,t=0,\dots,d-1$, and $j\not=k$. Here, $(K^{(jk)})_{st}$ denotes the entry of matrix $K^{(jk)}$ located in row $s$ and column $t$. Also, if $j=k$, we have $|(K^{(jj)})_{ss}|=1$, as all columns vectors of $M_j$ are normalized, for every $j=0,\dots,m-1$, and $s=0,\dots,d-1$. As a general property, it is simple to show that $M_j$ and $K^{(jk)}$ are full rank matrices, as their columns have to be linearly independent in order to form a frame. However, the problem to find sets of $m$ matrices $\{M_j\}$ satisfying (\ref{matrices_muf}) is evidently hard and hopeless to us, if no further assumptions are taken. In order to simplify the problem, we assume some symmetries about matrices $M_j$. The first assumption considers that all matrices $M_j$ are normal, i.e. $[M^{\dag}_j,M_j]=0$, and that they do commute, i.e. $[M_j,M_k]=0$. These assumptions imply that $[M_{\ell},K^{(jk)}]=0$, for every $j,k,\ell=0,\dots,d-1$. In this way, the problem simplifies to finding special relations existing between the spectrum of matrices $M_j$ and $K^{(jk)}$.
At this stage, a fundamental aspect is the choice of the unitary transformation $U$ that diagonalizes all matrices $M_j$. Note that the kind of MUF to be found strongly depend on this choice. For instance, if all matrices $M_j$ are diagonal in the computational basis, then there is no pair of MUF. Therefore, an interesting question arises: \emph{for which choices of $U$ there is a solution to (\ref{matrices_muf})} for a given $m$? This question seems hard to solve, and we do not study it within the framework of the current research. In turns, we study the case where all matrices $M_j$ are diagonalized by the discrete Fourier transform, i.e. $M_0,\dots,M_{m-1}$ are circulant matrices. This choice is justified by simplicity, as several useful properties are well-known for this class of matrices \cite{D98}. A square matrix $M$ is called circulant if the $j+1$-th row is a cyclic permutation of the $j$-th row, for every $j=0,\dots-d-2$. That is,
\begin{equation}
M=\left(\begin{array}{ccccc}
m_0&m_1&m_2&\dots&m_{d-1}\\
m_{d-1}&m_0&m_1&\dots&m_{d-2}\\
\vdots&&&&\vdots\\
m_1&m_2&m_3&\dots&m_0
\end{array}\right),
\end{equation}
where $m_0,\dots,m_{d-1}\in\mathbb{C}$.
Let $\vec{\mu}(M)$ be the vector formed by all elements of the first row of $M$, i.e. $\vec{\mu}=(m_0,\dots,m_{d-1})^T$, where $T$ denotes transposition. Also, let $\vec{\lambda}(M)$ be a column vector containing all eigenvalues of $M$, sorted according to the decomposition
\begin{equation}\label{M}
M=F\mathrm{diag}[\vec{\lambda}(M)]F^{\dag},
\end{equation}
where $\mathrm{diag}[\vec{\lambda}(M)]$ is a diagonal matrix containing the vector $\vec{\lambda}(M)$ along its main diagonal, and $F$ is the discrete Fourier transform, having entries $F_{jk}=\frac{1}{\sqrt{d}}\omega^{jk}$, where $\omega=e^{2\pi i/d}$. From (\ref{M}), it is simple to show that $\vec{\lambda}(M)=\sqrt{d}F^{\dag}\vec{\mu}(M)$. On the other hand, the unitary operator $F^2$ transforms the first row of $M$, i.e. $\vec{\mu}(M)$, to the first column of $M$, which has unit norm. Therefore, the condition to have normalized columns in $M$ translates to the condition $\|\vec{\mu}(M)\|=1$, or equivalently $\|\vec{\lambda}(M)\|=\sqrt{d}$.
Similarly, any matrix $K$ from (\ref{matrices_muf}) is circulant, having the form
\begin{equation}
K=\sqrt{c}\left(\begin{array}{ccccc}
e^{i\alpha_0}&e^{i\alpha_1}&e^{i\alpha_2}&\dots&e^{i\alpha_{d-1}}\\
e^{i\alpha_{d-1}}&e^{i\alpha_0}&e^{i\alpha_1}&\dots&e^{i\alpha_{d-2}}\\
\vdots&&&&\vdots\\
e^{i\alpha_1}&e^{i\alpha_2}&e^{i\alpha_3}&\dots&e^{i\alpha_{0}}
\end{array}\right),
\end{equation}
where $\alpha_0,\dots,\alpha_{d-1}\in[0,2\pi)$ and $0<c<1$. Let us arrange the first row of $K$ as the vector $\vec{\mu}(K)=\sqrt{c}(e^{i\alpha_0},\dots,e^{i\alpha_{d-1}})^T$, and call $\vec{\lambda}(K)$ the vector containing all eigenvalues of $K$, sorted according to
\begin{equation}\label{K}
K=F\mathrm{diag}[\vec{\lambda}(K)]F^{\dag}.
\end{equation}
Here, we also have $\vec{\lambda}(K)=\sqrt{d}F^{\dag}\vec{\mu}(K)$, where $\|\vec{\mu}(K)\|=\sqrt{cd}$ and $\|\vec{\lambda}(K)\|=d\sqrt{c}$. Now we are in position to establish the first result of this section.
\begin{prop}\label{equi_M}
Let $M_0,\dots,M_{m-1}$ be $m$ circulant matrices forming $m$ MUF composed by $d$ vectors each, in dimension $d$. Then, their normalized eigenvectors vectors form $m$ equiangular lines in $C^d$, with overlap $c$.
\end{prop}
An expert reader might recognize here the evidence of a well-known property holding for fiducial states of SIC-POVM: the Fourier transform of a fiducial state is a fiducial state. From combining Prop. \ref{equi_M} with results from equiangular lines theory \cite{GKMS}, we obtain the following constraints for the existence of MUF:
\begin{prop}
Suppose $M_1,\dots,M_m$ are $m$ circulant matrices of order $d$ forming $m$ MUF. Then, we have $m\leq d^2$.
\end{prop}
Moreover, when eigenvalues of matrices $M_j$ are real we can have stronger bounds.
\begin{prop}
Suppose a set MUF composed of $m$ hermitian circulant matrices, with overlap $c$, exists. Therefore, the following properties hold:
\begin{enumerate}
\item $m\leq d(d+1)/2$ (see \cite{LS73})
\item If $m\geq2d$, then $1/\sqrt{c}$ is an integer number (see \cite{LS73}).
\item If $c\leq1/(d+2)$, then $m\leq d(1-c)/(1-dc)$ (see Lemma 6.1 here \cite{LS66}).
\item If $c\leq1/d^2$, then $m\leq d+1$.
\end{enumerate}
\end{prop}
Let us now study further properties of eigenvalues of matrices $M_j$. By using (\ref{matrices_muf}), we have
\begin{equation}
\vec{\lambda}^*(M_j)\circ\vec{\lambda}(M_k)=\vec{\lambda}(K),
\end{equation}
where $\vec{\lambda}(M_j)$ is the vector associated to $M_j$, the circle ($\circ$) denotes the entrywise (Hadamard) product and the asterisk (*) complex conjugation in the computational basis. Note that the inequality norm for a Hadamard product, $\|\vec{v}_1\circ\vec{v}_2\|\leq\|\vec{v}_1\|\|\vec{v}_2\|$, produces $c\leq1$. In principle, one might think that this trivial bound can be improved. However, note that two matrices $M_1$ and $M_2$, with normalized and linearly independent columns, can approach arbitrarily close to a rank one projector $P$, producing $c\rightarrow1$. Also, we can have $M_1\rightarrow P_1$ and $M_2\rightarrow P_2$, with orthogonal projectors $P_1$ and $P_2$ satisfying $P_1P_2=0$, thus having $c\rightarrow0$.
Let us illustrate the introduced notion by showing the most general pair of MUF, arising from circulant matrices of order 2. That is,
\begin{eqnarray}\label{M1M2_MUF}
M_{1}=
\begin{pmatrix}
\cos(\theta) & \sin(\theta)e^{i\alpha} \\
\sin(\theta)e^{i\alpha} & \cos(\theta)
\end{pmatrix}\quad\mbox{and}\quad M_{2}=
\begin{pmatrix}
\cos(\eta) & \sin(\eta)e^{i\beta} \\
\sin(\eta)e^{i\beta}& \cos(\eta)
\end{pmatrix},
\end{eqnarray}
satisfying $M^{\dag}_1M_2=K$, with $|K_{ij}|^2=c$, $i,j=0,\dots,d-1$. From a straight calculation, a sufficient condition to have a pair of MUF from (\ref{M1M2_MUF}) is obtained:
\begin{eqnarray}\label{sol_muf_d2}
\sin(\alpha)\sin(\beta)\tan(2\theta)\tan(2\eta)=-1.
\end{eqnarray}
Note that (\ref{sol_muf_d2}) has a solution if and only if the following properties hold:
\begin{equation*}
\alpha,\beta,\theta,\eta\not\in\{0,\pi/2,\pi,3\pi/2\},
\end{equation*}
\begin{equation*} -1\leq\sin^{-1}(\alpha)\tan^{-1}(2\theta)\tan^{-1}(2\eta)\leq1,
\end{equation*}
and
\begin{equation*}
-1\leq\sin^{-1}(\beta)\tan^{-1}(2\theta)\tan^{-1}(2\eta)\leq1,
\end{equation*}
where the negative power means inverse multiplicative. This generic solution determines a 3-parametric nonlinear family of MUF with overlap
\begin{equation}
c=\cos ^2(\eta-\theta)+2\sin(\eta)\cos(\eta)\sin (\theta)\cos(\theta)(\cos(\alpha-\beta)-1),
\end{equation}
which includes SIC-POVM when, for instance, $\alpha=\beta$ and $\theta-\eta=\arccos(1/\sqrt{3})$. Note that there is no real pair of MUF within this continuous family. Nonetheless, real pairs of MUF arise from (\ref{M1M2_MUF}) when $\theta\in\{0,\pi\}$ and $\eta\in\{\pi/2,3\pi/2\}$, also when $\theta\in\{\pi/2,3\pi/2\}$ and $\eta\in\{0,\pi\}$, and finally when $\alpha,\beta\in\{\pi/2,3\pi/2\}$. All these isolated solutions form pairs of real MUB. These isolated solutions, together with (\ref{sol_muf_d2}), complete the entire set of two MUF that arise from (\ref{M1M2_MUF}).
It is interesting to see that MUF based on circulant matrices allow us to study both SIC-POVM and MUB problem for a qubit system, under the same framework. Moreover, the same occurs for every prime dimension $d$. Indeed, the eigenvectors bases of operators $Z,XZ,\dots,XZ^{d-1}$, forming a maximal set of $d+1$ MUB, can be arranged as $d$ circulant unitary matrices \cite{BBRV01}. Also, the fact that a SIC-POVM can be seen as a MUF decomposed by $d$ circulant matrices is guaranteed by Zauner's conjecture \cite{Z99}. Let us show a further result.
\begin{prop}\label{2MUF}
Let $M$ be a circulant matrix of order $d$ with columns forming a normalized frame. Then, there is another matrix $\tilde{M}$ inducing a frame such that $M$ and $\tilde{M}$ form a pair of MUF, in every dimension $d$.
\end{prop}
Let us illustrate Prop. \ref{2MUF} with a 3-parametric family of MUF for a qubit system. Consider the vector $v=(1,i)^T$, which is MU to the pair $\{\mathbb{I},F\}$ in dimension $d=2$. Thus, taking $r_0=\sqrt{2}\cos(\nu)$ and $r_1=\sqrt{2}\sin(\nu)$, we have
\begin{eqnarray*}
M&=&Fdiag[\vec{\lambda}(M)]F^{\dag}\\
&=&\frac{1}{2}\left(\begin{array}{cc}
1&1\\
1&-1
\end{array}\right)
\left(\begin{array}{cc}
\sqrt{2}\cos(\nu)\omega^{\alpha_0}&0\\
0&\sqrt{2}\sin(\nu)\omega^{\alpha_1}
\end{array}\right)
\left(\begin{array}{cc}
1&1\\
1&-1
\end{array}\right)\\
&=&\frac{1}{\sqrt{2}}\left(\begin{array}{cc}
\cos(\nu)\omega^{\alpha_0}+\sin(\nu)\omega^{\alpha_1}&\cos(\nu)\omega^{\alpha_0}-\sin(\nu)\omega^{\alpha_1}\\
\cos(\nu)\omega^{\alpha_0}-\sin(\nu)\omega^{\alpha_1}&\cos(\nu)\omega^{\alpha_0}+\sin(\nu)\omega^{\alpha_1}
\end{array}\right),
\end{eqnarray*}
and
\begin{eqnarray*}
\tilde{M}&=&Fdiag[\vec{\lambda}(\tilde{M})]F^{\dag}\\
&=&\frac{1}{2}\left(\begin{array}{cc}
1&1\\
1&-1
\end{array}\right)
\left(\begin{array}{cc}
\frac{\sqrt{c}}{\cos(\nu)}\omega^{\alpha_0+1/4}&0\\
0&\frac{\sqrt{c}}{\sin(\nu)}\omega^{\alpha_1-1/4}
\end{array}\right)
\left(\begin{array}{cc}
1&1\\
1&-1
\end{array}\right)\\
&=&\frac{1}{2} \sqrt{c}\left(
\begin{array}{cc}
\sec(\nu ) w^{\alpha_0+\frac{1}{4}}+\csc(\nu )
w^{\alpha_1-\frac{1}{4}} & w^{\alpha_0+\frac{1}{4}}\sec(\nu)-w^{\alpha_1-\frac{1}{4}} \csc (\nu ) \\
w^{\alpha_0+\frac{1}{4}} \sec (\nu )-w^{\alpha_1-\frac{1}{4}} \csc (\nu )& \sec (\nu )
w^{\alpha_0+\frac{1}{4}}+\csc (\nu ) w^{\alpha_1-\frac{1}{4}}
\end{array}
\right),
\end{eqnarray*}
with overlap $c=2\cos^2(\nu)\sin^2(\nu)$. In particular, for $\nu=\pi/4$ and $\nu=-\arctan \left(\sqrt{\frac{3-\sqrt{3}}{3+\sqrt{3}}}\right)$, we have two MUB and a SIC-POVM, respectively.
\section{Application to the SIC-POVM problem}\label{sec:sic}
Along this section, we organize SIC-POVM as $d$ sets of mutually unbiased frames, composed by $d$ vectors each. This slightly different way of seeing the constellation carries out remarkably interesting consequences, as we will see below. Based on the matrix approach to MUF introduced in Section \ref{sec:matrixMUF}, we assume that the $d^2$ vectors of a SIC-POVM can be arranged as column vectors in $m=d$ circulant matrices of order $d$, $M_0,\dots,M_{d-1}$, each of them defining a frame. This assumption reduces the problem to finding a single suitable matrix of order $d$, called $\mathcal{M}$, that contains the normalized vectors $\vec{\lambda}(M_0)/\sqrt{d},\dots,\vec{\lambda}(M_{d-1})/\sqrt{d}$ along its columns. The problem to find a solution $\mathcal{M}$ is hard, as it has a quadratic number of free parameters as a function of the dimension $d$. In order to make the problem simpler, we adopt a further reasonable assumption: the matrix $\mathcal{M}$ is circulant. Thus, all the problem reduces to find a suitable first row of $\mathcal{M}$, that has $2(d-1)$ free parameters without loss of generality. After these intuitive -but drastic- assumptions, one might wonder whether there is a SIC-POVM satisfying the above symmetries. Below, we show that the answer is positive in every dimension $d$ where a SIC-POVM is known.
\begin{prop}\label{MZauner}
The assumption that matrices $M_0,\dots,M_{d-1}$ and $\mathcal{M}$ are circulant is equivalent to the assumption that a SIC-POVM can be defined as an orbit of a fiducial state under the Weyl-Heisenberg group.
\end{prop}
Note that normalized eigenvalues vectors $\vec{\lambda}(M_j)/\sqrt{d}$, $j=0,\dots,d-1$, are also fiducial states. This is simple to see, as the first row of $M_j$ is connected with the normalized vector $\vec{\lambda}(M_j)/\sqrt{d}$ via the Fourier transform $F$. Also, the first row of $M_j$ is connected with its first column via $F^2$. Given that $F$ belongs to the Clifford group, it transforms fiducial states into fiducial states. Note that this result is in agreement with Prop. \ref{equi_M}. If matrices $M_0,\dots,M_{d-1}$ have vanishing determinant we cannot see them as frames. However, in such case one can consider an equivalent SIC-POVM induced by the eigenvalues vector $\vec{\lambda}'=U\vec{\lambda}(M_0)$, where $U$ is an element of the Clifford group, e.g. the Fourier transform. This transformation produces a vector $\vec{\lambda}'$ having non-zero entries, that generates a set of $d$ invertible matrices $M'_0,\dots,M'_{d-1}$. That is, we have $d$ frames composed by $d$ vectors each. A vanishing determinant case occurs when the eigenvalues vector $\lambda(M_0)$ has a zero entry. As this vector is also a fiducial state, up to a rescaling factor, it is interesting to wonder whether a fiducial state can have a vanishing entry. As far as we know, there is a single evidence in the literature for such fiducial states, in dimension 3 \cite{Z99}. Nonetheless, some recent explorations in high dimensions reveal the existence of such fiducials for $d=26, 28, 62, 98$ and $228$ \cite{Markus_comment}. For an extended study about linearly dependencies of elements of a SIC-POVM see here \cite{DBBA}.
There is a further symmetry that one can impose in order to reduce the number of free parameters in SIC-POVM. One might assume that all matrices $M_0,\dots,M_{d-1}$ are hermitian, which holds if and only if $\vec{\lambda}(M_0)/\sqrt{d}$ is a real fiducial state. Along this direction, it has been conjectured that real fiducial states exist in odd dimensions of the form $d=4n^2+3$, $n\in\mathbb{Z}$ \cite{S17}. This conjecture is supported with 18 real fiducials existing in dimensions of such form, the highest dimensional case corresponding to $d=19603$ \cite{ABHGM21}. For instance, in $d=3$, there is a family of real fiducial states \cite{RBSC04}:
\begin{equation}
|\phi(r_0)\rangle=r_0|0\rangle-(r_0 /2 + \sqrt{2-3r_0^2}/2)|1\rangle- (r_0/2-\sqrt{2-3r_0^2}/2)|2\rangle,
\end{equation}
with $1/\sqrt{2}<r_0<\sqrt{2/3}$, inducing the following 1-parametric family of hermitian matrices:
\begin{eqnarray*}
M_j&=&\left(
\begin{array}{ccc}
0 & a_j & a_j^*\\
a_j^*&0&a_j\\
a_j&a_j^*&0
\end{array}
\right),\quad j=0,1,2,
\end{eqnarray*}
where
\begin{eqnarray*}
a_0&=&\frac{1}{2} \left(\sqrt{3} r_0+i \sqrt{2-3 r_0^2}\right)\nonumber\\ a_1&=&\frac{1}{12} \left(3-i \sqrt{3}\right) \left(\sqrt{6-9 r_0^2}-3 i r_0\right)\nonumber\\
a_2&=&-\frac{1}{12} \left(3+i \sqrt{3}\right) \left(\sqrt{6-9 r_0^2}-3 i r_0\right).
\end{eqnarray*}
At this stage, an interesting question arises: \emph{are there real fiducial states, covariant under the standard Weyl-Heisenberg group, in every dimension $d$?} Below, we provide a conclusive answer to this question.
\begin{prop}\label{real_fid}
Real fiducial states, covariant under the standard Weyl-Heisenberg group, do not exist in every even dimension.
\end{prop}
Now, let us reveal a hidden symmetry for the amplitudes in fiducial states.
\begin{prop}\label{prop:lambdas}
Any fiducial state $|\phi\rangle$, with respect to a Weyl-Heisenberg covariant SIC-POVM, satisfies
\begin{equation}\label{reduction}
\sum_{j=0}^{d-1}|\langle j|\phi\rangle|^2|j\rangle=F^{\dag}\left(\frac{1}{\sqrt{d}}|0\rangle+\frac{1}{\sqrt{d(d+1)}}\sum_{j=1}^{d-1}\omega^{\alpha_j}|j\rangle\right),
\end{equation}
in every finite dimension $d$. Here,
$\omega=e^{2\pi i/d}$, $F$ is the discrete Fourier transform of order $d$, and $\alpha_j\in[0,2\pi)$ satisfies $\alpha_{d-j}=-\alpha_j$, for every $j=1,\dots,d-1$. As a consequence, $|\phi\rangle$ depends on $\lfloor (d-1)/2\rfloor+d-1$ free parameters only, i.e., $\lfloor (d-1)/2\rfloor$ amplitudes given by (\ref{reduction}) and $d-1$ complex phases.
\end{prop}
After contemplating (\ref{reduction}), we note that Prop. \ref{prop:lambdas} can be equivalently obtained by applying the discrete version of the Wiener-Khintchine Theorem \cite{W30,K34} to the fiducial state $|\phi\rangle$. Nonetheless, our approach to MUF played a fundamental role to naturally arrive to the result, which might not be simple to realize otherwise. Indeed, Prop. \ref{prop:lambdas} seems to be a new observation, as far as we know. Let us stress that some numerical simulations made by Chris Fuchs in low dimensions suggest that the number of free parameters in fiducials might be $3(d-1)/2-1$ for $d$ odd and $3d/2-1$ for $d$ even, having an almost perfect match with our result, see page 1258 at Chris Fuchs's \emph{samizdat} \cite{marcus_comment,F14}.
\subsection{Minimum uncertainty states}
Fiducial states are minimum uncertainty states for a complete set of MUB in every prime dimension \cite{ADF14}. More precisely, let $\{|\varphi^j_k\rangle\}$, $k=0,\dots,d$, $j=0,\dots,d-1$, be a set of $d+1$ MUB in prime dimension $d$, $|\phi\rangle$ a pure quantum state and $p^j_k=|\langle\phi|\varphi^j_k\rangle|^2$. Further, let $H_j=-\log_2[\sum_{k=0}^{d-1}(p^j_k)^2]$ be the quadratic R\'enyi entropy \cite{R61} associated to the $j$th MUB. Therefore, it holds that \cite{BW07}:
\begin{equation}\label{minH}
\sum_{m=0}^dH_m\geq (d+1)\log_2\left[\frac{d+1}{2}\right],
\end{equation}
where the saturation of inequality (\ref{minH}) defines a minimum uncertainty states if and only if $\sum_{k=0}^{d-1}(p^j_k)^2=\frac{2}{d+1}$, for every $j=0,\dots,d$. It has been shown that this last property is satisfied by any fiducial state $|\phi\rangle$ in every prime dimension $d$ \cite{ADF14}. Nonetheless, there are further quantum states minimizing (\ref{minH}). An interesting question is whether there are further sets of $d+1$ orthonormal bases for which fiducial states are also minimum uncertainty states. The following result, based on Prop. \ref{prop:lambdas}, allows us to provide a positive answer to the question, in every dimension where a fiducial state exists.
\begin{prop}\label{product_prob}
Let $|\phi\rangle$ be a $d$-dimensional quantum state satisfying (\ref{reduction}), $|\xi_0\rangle,\dots,|\xi_{d-1}\rangle$ the columns of any unitary matrix belonging to the Clifford group, and $p_k=|\langle\phi|\xi_k\rangle|^2$. Therefore,
\begin{equation}\label{min_uncertainty}
\sum_{j=0}^{d-1}p_jp_{j+k}=\frac{1+\delta_{k,0}}{d+1}.
\end{equation}
In particular, (\ref{min_uncertainty}) holds for fiducial states $|\phi\rangle$, and the bases include maximal sets of $d+1$ MUB in prime dimension $d$.
\end{prop}
Proposition \ref{product_prob} has the following immediate consequence.
\begin{corol}\label{corol:mus}
Any $d$-dimensional quantum state $|\phi\rangle$ that satisfies (\ref{reduction}) is a minimum uncertainty state with respect to every subset of $d+1$ orthonormal bases formed by the columns of elements of the Clifford group. In particular, the result holds for any fiducial state in every dimension $d$.
\end{corol}\medskip
Let us now find a relation between some classes of fiducial states and mutually unbiased bases. Assume that $|\phi\rangle$ is a fiducial state, eigenvector of the Zauner's operator
\cite{Z99}
\begin{equation}\label{Zauner}
\mathcal{Z}=e^{i\pi(d-1)/12}FG,
\end{equation}
where $G$ is a diagonal unitary matrix having diagonal entries $G_{jj}=e^{i\pi(d+1)j^2/d}$, $j=0,\dots,d-1$. Thus, we obtain the following result.
\begin{prop}\label{triplet}
Any eigenvector of the Zauner operator (\ref{Zauner}) has identical probability distributions with respect to the triplet of MUB given by the columns of $\{\mathbb{I},\mathcal{Z},\mathcal{Z}^2\}$, in every dimension $d$. In particular, the result holds for fiducial states that are eigenvectors of $\mathcal{Z}$.
\end{prop}
For $d=2$, any eigenvector of $\mathcal{Z}$ is a \emph{MUB-balanced} state \cite{WS07}, in particular when it is a fiducial state. The MUB-balanced property means that a $d$-dimensional quantum state has identical probability distributions with respect to a maximal set of $d+1$ MUB.
\section{Conclusions}
We introduced the ultimate notion of unbiasedness for sets of linearly independent normalized vectors in $d$-dimensional Hilbert space: \emph{mutually unbiased frames} (MUF). These sets are characterized by a constant overlapping that can take any value between the absolute bounds 0 and 1, reduced to $1/d$ when at least one of the sets is a tight frame, equivalently a positive operator valued measure (POVM). We illustrated the introduced notion by finding the most general pair of MUF composed by two vectors each for a qubit system. The concept of MUF allowed us to think a Symmetric Informationally Complete (SIC)-POVM in dimension $d$ as a set of $d$ MUF composed by $d$ vectors each, thus having a common root with the mutually unbiased bases problem. This approach allowed us to reveal some remarkable properties about SIC-POVM, namely: real fiducial states do not exist in even dimensions (see Prop. \ref{real_fid}) and, any $d$-dimensional fiducial state depends on $\lfloor(d-1)/2\rfloor+d-1$ real parameters only (see Prop. \ref{prop:lambdas}). Within our scheme, Zauner's conjecture about the existence of fiducial states arose as the simplest way to study $d$ MUF in dimension $d$ (see Prop. \ref{MZauner}). Additionally, we have shown that some classes of quantum states $|\phi\rangle$, those satisfying (\ref{reduction}), are minimum uncertainty states with respect to $d+1$ bases formed by the columns of any subset of Clifford operations. In particular, the result holds for any fiducial state in every finite dimension (see Prop. \ref{product_prob} and Corol. \ref{corol:mus}). Finally, any eigenvector of the Zauner's operator `looks the same' for three MUB, in every dimension $d$ (see Prop. \ref{triplet}).
\vspace{1cm}
\textbf{Acknowledgements}
\medskip
We kindly acknowledge M. Appleby, I. Bengtsson and M. Grassl for providing valuable comments about SIC-POVM and the Clifford group. This work is supported by grant FONDECyT Iniciaci\'{o}n nro. 11180474, Chile. FCP and VGA belong to the first year of the PhD Program \emph{Doctorado en F\'isica, menci\'on F\'isica matem\'atica}, Universidad de Antofagasta, Antofagasta, Chile.\bigskip
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 5,180 |
{"url":"https:\/\/www.educator.com\/mathematics\/basic-math\/pyo\/adding-and-subtracting-decimals.php","text":"Mary Pyo\n\nSlide Duration:\n\nSection 1: Algebra and Decimals\nExpressions and Variables\n\n5m 57s\n\nIntro\n0:00\nVocabulary\n0:06\nVariable\n0:09\nExpression\n0:48\nNumerical Expression\n1:08\nAlgebraic Expression\n1:35\nWord Expression\n2:04\nExtra Example 1: Evaluate the Expression\n2:27\nExtra Example 2: Evaluate the Expression\n3:16\nExtra Example 3: Evaluate the Expression\n4:04\nExtra Example 4: Evaluate the Expression\n4:59\nExponents\n\n5m 34s\n\nIntro\n0:00\nWhat Exponents Mean\n0:07\nExample: Ten Squared\n0:08\nExtra Example 1: Exponents\n0:50\nExtra Example 2: Write in Exponent Form\n1:58\nExtra Example 3: Using Exponent and Base\n2:37\nExtra Example 4: Write the Equal Factors\n4:26\nOrder of Operations\n\n8m 40s\n\nIntro\n0:00\nPlease Excuse My Dear Aunt Sally\n0:07\nStep 1: Parenthesis\n1:16\nStep 2: Exponent\n1:25\nStep 3: Multiply and Divide\n1:30\n2:00\nExample: Please Excuse My Dear Aunt Sally\n2:26\nExtra Example 1: Evaluating Expression\n3:37\nExtra Example 2: Evaluating Expression\n4:59\nExtra Example 3: Evaluating Expression\n5:34\nExtra Example 4: Evaluating Expression\n6:25\nComparing and Ordering Decimals\n\n13m 37s\n\nIntro\n0:00\nPlace Value\n0:13\nExamples: 1,234,567.89\n0:19\nWhich is the Larger Value?\n1:33\nWhich is Larger: 10.5 or 100.5\n1:46\nWhich is Larger: 1.01 or 1.10\n2:24\nWhich is Larger: 44.40 or 44.4\n4:20\nWhich is Larger: 18.6 or 16.8\n5:18\nExtra Example 1: Order from Least to Greatest\n5:55\nExtra Example 2: Order from Least to Greatest\n7:56\nExtra Example 3: Order from Least to Greatest\n9:16\nExtra Example 4: Order from Least to Greatest\n10:42\nRounding Decimals\n\n12m 31s\n\nIntro\n0:00\nDecimal Place Value\n0:06\nExample: 12,3454.6789\n0:07\nHow to Round Decimals\n1:17\nExample: Rounding 1,234.567\n1:18\nExtra Example 1: Rounding Decimals\n3:47\nExtra Example 2: Rounding Decimals\n6:10\nExtra Example 3: Rounding Decimals\n7:45\nExtra Example 4: Rounding Decimals\n9:56\n\n11m 30s\n\nIntro\n0:00\n0:06\nAlign the Decimal Point First\n0:12\n0:47\nPlace the Decimal Point in the Same Place\n0:55\nCheck by Estimating\n1:09\nExamples\n1:28\nAdd: 3.45 + 7 + 0.835\n1:30\nFind the Difference: 351.4 - 65.25\n3:34\n5:32\nExtra Example 2: How Much Money?\n6:09\nExtra Example 3: Subtracting Decimals\n7:20\n9:32\nMultiplying Decimals\n\n10m 30s\n\nIntro\n0:00\nMultiply the Decimals\n0:05\nMethods for Multiplying Decimals\n0:06\nExample: 1.1 x 6\n0:38\nExtra Example 1: Multiplying Decimals\n1:51\nExtra Example 2: Work Money\n2:49\nExtra Example 3: Multiplying Decimals\n5:45\nExtra Example 4: Multiplying Decimals\n7:46\nDividing Decimals\n\n17m 49s\n\nIntro\n0:00\nWhen Dividing Decimals\n0:06\nMethods for Dividing Decimals\n0:07\nDivisor and Dividend\n0:37\nExample: 0.2 Divided by 10\n1:35\nExtra Example 1 : Dividing Decimals\n5:24\nExtra Example 2: How Much Does Each CD Cost?\n8:22\nExtra Example 3: Dividing Decimals\n10:59\nExtra Example 4: Dividing Decimals\n12:08\nSection 2: Number Relationships and Fractions\nPrime Factorization\n\n7m\n\nIntro\n0:00\nTerms to Review\n0:07\nPrime vs. Composite\n0:12\nFactor\n0:54\nProduct\n1:15\nFactor Tree\n1:39\nExample: Prime Factorization\n2:01\nExample: Prime Factorization\n2:43\nExtra Example 1: Prime Factorization\n4:08\nExtra Example 2: Prime Factorization\n5:05\nExtra Example 3: Prime Factorization\n5:33\nExtra Example 4: Prime Factorization\n6:13\nGreatest Common Factor\n\n12m 47s\n\nIntro\n0:00\nTerms to Review\n0:05\nFactor\n0:07\nExample: Factor of 20\n0:18\nTwo Methods\n0:59\nGreatest Common Factor\n1:00\nMethod 1: GCF of 15 and 30\n1:37\nMethod 2: GCF of 15 and 30\n2:58\nExtra Example 1: Find the GCF of 6 and 18\n5:16\nExtra Example 2: Find the GCF of 36 and 27\n7:43\nExtra Example 3: Find the GCF of 6 and 18\n9:18\nExtra Example 4: Find the GCF of 54 and 36\n10:30\nFraction Concepts and Simplest Form\n\n10m 3s\n\nIntro\n0:00\nFraction Concept\n0:10\nExample: Birthday Cake\n0:28\nExample: Chocolate Bar\n2:10\nSimples Form\n3:38\nExample: Simplifying 4 out of 8\n3:46\nExtra Example 1: Graphically Show 4 out of 10\n4:41\nExtra Example 2: Finding Fraction Shown by Illustration\n5:10\nExtra Example 3: Simplest Form of 5 over 25\n7:02\nExtra Example 4: Simplest Form of 14 over 49\n8:30\nLeast Common Multiple\n\n14m 16s\n\nIntro\n0:00\nTerm to Review\n0:06\nMultiple\n0:07\nExample: Multiples of 4\n0:15\nTwo Methods\n0:41\nLeast Common Multiples\n0:44\nMethod 1: LCM of 6 and 10\n1:09\nMethod 2: LCM of 6 and 10\n2:56\nExtra Example 1: LCM of 12 and 15\n5:09\nExtra Example 2: LCM of 16 and 20\n7:36\nExtra Example 3 : LCM of 15 and 25\n10:00\nExtra Example 4 : LCM of 12 and 18\n11:27\nComparing and Ordering Fractions\n\n13m 10s\n\nIntro\n0:00\nTerms Review\n0:14\nGreater Than\n0:16\nLess Than\n0:40\nCompare the Fractions\n1:00\nExample: Comparing 2\/4 and 3\/4\n1:08\nExample: Comparing 5\/8 and 2\/5\n2:04\nExtra Example 1: Compare the Fractions\n3:28\nExtra Example 2: Compare the Fractions\n6:06\nExtra Example 3: Compare the Fractions\n8:01\nExtra Example 4: Least to Greatest\n9:37\nMixed Numbers and Improper Fractions\n\n12m 49s\n\nIntro\n0:00\nFractions\n0:10\nMixed Number\n0:21\nProper Fraction\n0:47\nImproper Fraction\n1:30\nSwitching Between\n2:47\nMixed Number to Improper Fraction\n2:53\nImproper Fraction to Mixed Number\n4:41\nExamples: Switching Fractions\n6:37\nExtra Example 1: Mixed Number to Improper Fraction\n8:57\nExtra Example 2: Improper Fraction to Mixed Number\n9:37\nExtra Example 3: Improper Fraction to Mixed Number\n10:21\nExtra Example 4: Mixed Number to Improper Fraction\n11:31\nConnecting Decimals and Fractions\n\n15m 1s\n\nIntro\n0:00\nExamples: Decimals and Fractions\n0:06\nMore Examples: Decimals and Fractions\n2:48\nExtra Example 1: Converting Decimal to Fraction\n6:55\nExtra Example 2: Converting Fraction to Decimal\n8:45\nExtra Example 3: Converting Decimal to Fraction\n10:28\nExtra Example 4: Converting Fraction to Decimal\n11:42\nSection 3: Fractions and Their Operations\nAdding and Subtracting Fractions with Same Denominators\n\n5m 17s\n\nIntro\n0:00\nSame Denominator\n0:11\nNumerator and Denominator\n0:18\nExample: 2\/6 + 5\/6\n0:41\nExtra Example 1: Add or Subtract the Fractions\n2:02\nExtra Example 2: Add or Subtract the Fractions\n2:45\nExtra Example 3: Add or Subtract the Fractions\n3:17\nExtra Example 4: Add or Subtract the Fractions\n4:05\nAdding and Subtracting Fractions with Different Denominators\n\n23m 8s\n\nIntro\n0:00\nLeast Common Multiple\n0:12\nLCM of 6 and 4\n0:31\nFrom LCM to LCD\n2:25\n3:12\nExtra Example 1: Add or Subtract\n6:23\nExtra Example 2: Add or Subtract\n9:49\nExtra Example 3: Add or Subtract\n14:54\nExtra Example 4: Add or Subtract\n18:14\n\n19m 44s\n\nIntro\n0:00\nExample\n0:05\n0:17\nExtra Example 1: Adding Mixed Numbers\n1:57\nExtra Example 2: Subtracting Mixed Numbers\n8:13\nExtra Example 3: Adding Mixed Numbers\n12:01\nExtra Example 4: Subtracting Mixed Numbers\n14:54\nMultiplying Fractions and Mixed Numbers\n\n21m 32s\n\nIntro\n0:00\nMultiplying Fractions\n0:07\nStep 1: Change Mixed Numbers to Improper Fractions\n0:08\nStep2: Multiply the Numerators Together\n0:56\nStep3: Multiply the Denominators Together\n1:03\nExtra Example 1: Multiplying Fractions\n1:37\nExtra Example 2: Multiplying Fractions\n6:39\nExtra Example 3: Multiplying Fractions\n10:20\nExtra Example 4: Multiplying Fractions\n13:47\nDividing Fractions and Mixed Numbers\n\n18m\n\nIntro\n0:00\nDividing Fractions\n0:09\nStep 1: Change Mixed Numbers to Improper Fractions\n0:15\nStep 2: Flip the Second Fraction\n0:27\nStep 3: Multiply the Fractions\n0:52\nExtra Example 1: Dividing Fractions\n1:23\nExtra Example 2: Dividing Fractions\n5:06\nExtra Example 3: Dividing Fractions\n9:34\nExtra Example 4: Dividing Fractions\n12:06\nDistributive Property\n\n11m 5s\n\nIntro\n0:00\nDistributive Property\n0:06\nMethods of Distributive Property\n0:07\nExample: a(b)\n0:35\nExample: a(b+c)\n0:49\nExample: a(b+c+d)\n1:22\nExtra Example 1: Using Distributive Property\n1:56\nExtra Example 2: Using Distributive Property\n4:36\nExtra Example 3: Using Distributive Property\n6:39\nExtra Example 4: Using Distributive Property\n8:19\nUnits of Measure\n\n16m 36s\n\nIntro\n0:00\nLength\n0:05\nFeet, Inches, Yard, and Mile\n0:20\nMillimeters, Centimeters, and Meters\n0:43\nMass\n2:57\nPounds, Ounces, and Tons\n3:03\nGrams and Kilograms\n3:38\nLiquid\n4:11\nGallons, Quarts, Pints, and Cups\n4:14\nExtra Example 1: Converting Units\n7:02\nExtra Example 2: Converting Units\n9:31\nExtra Example 3: Converting Units\n12:21\nExtra Example 4: Converting Units\n14:05\nSection 4: Positive and Negative Numbers\nIntegers and the Number Line\n\n13m 24s\n\nIntro\n0:00\nWhat are Integers\n0:06\nIntegers are all Whole Numbers and Their Opposites\n0:09\nAbsolute Value\n2:35\nExtra Example 1: Compare the Integers\n4:36\nExtra Example 2: Writing Integers\n9:24\nExtra Example 3: Opposite Integer\n10:38\nExtra Example 4: Absolute Value\n11:27\n\n16m 5s\n\nIntro\n0:00\nUsing a Number Line\n0:04\nExample: 4 + (-2)\n0:14\nExample: 5 + (-8)\n1:50\n3:00\n3:10\n3:37\n4:44\nExtra Example 1: Add the Integers\n8:21\nExtra Example 2: Find the Sum\n10:33\nExtra Example 3: Find the Value\n11:37\nExtra Example 4: Add the Integers\n13:10\nSubtracting Integers\n\n15m 25s\n\nIntro\n0:00\nHow to Subtract Integers\n0:06\nTwo-dash Rule\n0:16\nExample: 3 - 5\n0:44\nExample: 3 - (-5)\n1:12\nExample: -3 - 5\n1:39\nExtra Example 1: Rewrite Subtraction to Addition\n4:43\nExtra Example 2: Find the Difference\n7:59\nExtra Example 3: Find the Difference\n9:08\nExtra Example 4: Evaluate\n10:38\nMultiplying Integers\n\n7m 33s\n\nIntro\n0:00\nWhen Multiplying Integers\n0:05\nIf One Number is Negative\n0:06\nIf Both Numbers are Negative\n0:18\nExamples: Multiplying Integers\n0:53\nExtra Example 1: Multiplying Integers\n1:27\nExtra Example 2: Multiplying Integers\n2:43\nExtra Example 3: Multiplying Integers\n3:13\nExtra Example 4: Multiplying Integers\n3:51\nDividing Integers\n\n6m 42s\n\nIntro\n0:00\nWhen Dividing Integers\n0:05\nRules for Dividing Integers\n0:41\nExtra Example 1: Dividing Integers\n1:01\nExtra Example 2: Dividing Integers\n1:51\nExtra Example 3: Dividing Integers\n2:21\nExtra Example 4: Dividing Integers\n3:18\nIntegers and Order of Operations\n\n11m 9s\n\nIntro\n0:00\nCombining Operations\n0:21\nSolve Using the Order of Operations\n0:22\nExtra Example 1: Evaluate\n1:18\nExtra Example 2: Evaluate\n4:20\nExtra Example 3: Evaluate\n6:33\nExtra Example 4: Evaluate\n8:13\nSection 5: Solving Equations\nWriting Expressions\n\n9m 15s\n\nIntro\n0:00\nOperation as Words\n0:05\nOperation as Words\n0:06\nExtra Example 1: Write Each as an Expression\n2:09\nExtra Example 2: Write Each as an Expression\n4:27\nExtra Example 3: Write Each Expression Using Words\n6:45\nWriting Equations\n\n18m 3s\n\nIntro\n0:00\nEquation\n0:05\nDefinition of Equation\n0:06\nExamples of Equation\n0:58\nOperations as Words\n1:39\nOperations as Words\n1:40\nExtra Example 1: Write Each as an Equation\n3:07\nExtra Example 2: Write Each as an Equation\n6:19\nExtra Example 3: Write Each as an Equation\n10:08\nExtra Example 4: Determine if the Equation is True or False\n13:38\n\n24m 53s\n\nIntro\n0:00\nSolving Equations\n0:08\ninverse Operation of Addition and Subtraction\n0:09\nExtra Example 1: Solve Each Equation Using Mental Math\n4:15\nExtra Example 2: Use Inverse Operations to Solve Each Equation\n5:44\nExtra Example 3: Solve Each Equation\n14:51\nExtra Example 4: Translate Each to an Equation and Solve\n19:57\nSolving Multiplication Equation\n\n19m 46s\n\nIntro\n0:00\nMultiplication Equations\n0:08\nInverse Operation of Multiplication\n0:09\nExtra Example 1: Use Mental Math to Solve Each Equation\n3:54\nExtra Example 2: Use Inverse Operations to Solve Each Equation\n5:55\nExtra Example 3: Is -2 a Solution of Each Equation?\n12:48\nExtra Example 4: Solve Each Equation\n15:42\nSolving Division Equation\n\n17m 58s\n\nIntro\n0:00\nDivision Equations\n0:05\nInverse Operation of Division\n0:06\nExtra Example 1: Use Mental Math to Solve Each Equation\n0:39\nExtra Example 2: Use Inverse Operations to Solve Each Equation\n2:14\nExtra Example 3: Is -6 a Solution of Each Equation?\n9:53\nExtra Example 4: Solve Each Equation\n11:50\nSection 6: Ratios and Proportions\nRatio\n\n40m 21s\n\nIntro\n0:00\nRatio\n0:05\nDefinition of Ratio\n0:06\nExamples of Ratio\n0:18\nRate\n2:19\nDefinition of Rate\n2:20\nUnit Rate\n3:38\nExample: $10 \/ 20 pieces 5:05 Converting Rates 6:46 Example: Converting Rates 6:47 Extra Example 1: Write in Simplest Form 16:22 Extra Example 2: Find the Ratio 20:53 Extra Example 3: Find the Unit Rate 22:56 Extra Example 4: Convert the Unit 26:34 Solving Proportions 17m 22s Intro 0:00 Proportions 0:05 An Equality of Two Ratios 0:06 Cross Products 1:00 Extra Example 1: Find Two Equivalent Ratios for Each 3:21 Extra Example 2: Use Mental Math to Solve the Proportion 5:52 Extra Example 3: Tell Whether the Two Ratios Form a Proportion 8:21 Extra Example 4: Solve the Proportion 13:26 Writing Proportions 22m 1s Intro 0:00 Writing Proportions 0:08 Introduction to Writing Proportions and Example 0:10 Extra Example 1: Write a Proportion and Solve 5:54 Extra Example 2: Write a Proportion and Solve 11:19 Extra Example 3: Write a Proportion for Word Problem 17:29 Similar Polygons 16m 31s Intro 0:00 Similar Polygons 0:05 Definition of Similar Polygons 0:06 Corresponding Sides are Proportional 2:14 Extra Example 1: Write a Proportion and Find the Value of Similar Triangles 4:26 Extra Example 2: Write a Proportional to Find the Value of x 7:04 Extra Example 3: Write a Proportion for the Similar Polygons and Solve 9:04 Extra Example 4: Word Problem and Similar Polygons 11:03 Scale Drawings 13m 43s Intro 0:00 Scale Drawing 0:05 Definition of a Scale Drawing 0:06 Example: Scale Drawings 1:00 Extra Example 1: Scale Drawing 4:50 Extra Example 2: Scale Drawing 7:02 Extra Example 3: Scale Drawing 9:34 Probability 11m 51s Intro 0:00 Probability 0:05 Introduction to Probability 0:06 Example: Probability 1:22 Extra Example 1: What is the Probability of Landing on Orange? 3:26 Extra Example 2: What is the Probability of Rolling a 5? 5:02 Extra Example 3: What is the Probability that the Marble will be Red? 7:40 Extra Example 4: What is the Probability that the Student will be a Girl? 9:43 Section 7: Percents Percents, Fractions, and Decimals 35m 5s Intro 0:00 Percents 0:06 Changing Percent to a Fraction 0:07 Changing Percent to a Decimal 1:54 Fractions 4:17 Changing Fraction to Decimal 4:18 Changing Fraction to Percent 7:50 Decimals 10:10 Changing Decimal to Fraction 10:11 Changing Decimal to Percent 12:07 Extra Example 1: Write Each Percent as a Fraction in Simplest Form 13:29 Extra Example 2: Write Each as a Decimal 17:09 Extra Example 3: Write Each Fraction as a Percent 22:45 Extra Example 4: Complete the Table 29:17 Finding a Percent of a Number 28m 18s Intro 0:00 Percent of a Number 0:06 Translate Sentence into an Equation 0:07 Example: 30% of 100 is What Number? 1:05 Extra Example 1: Finding a Percent of a Number 7:12 Extra Example 2: Finding a Percent of a Number 15:56 Extra Example 3: Finding a Percent of a Number 19:14 Extra Example 4: Finding a Percent of a Number 24:26 Solving Percent Problems 32m 31s Intro 0:00 Solving Percent Problems 0:06 Translate the Sentence into an Equation 0:07 Extra Example 1: Solving Percent Problems 0:56 Extra Example 2: Solving Percent Problems 14:49 Extra Example 3: Solving Percent Problems 23:44 Simple Interest 27m 9s Intro 0:00 Simple Interest 0:05 Principal 0:06 Interest & Interest Rate 0:41 Simple Interest 1:43 Simple Interest Formula 2:23 Simple Interest Formula: I = prt 2:24 Extra Example 1: Finding Simple Interest 3:53 Extra Example 2: Finding Simple Interest 8:08 Extra Example 3: Finding Simple Interest 12:02 Extra Example 4: Finding Simple Interest 17:46 Discount and Sales Tax 17m 15s Intro 0:00 Discount 0:19 Discount 0:20 Sale Price 1:22 Sales Tax 2:24 Sales Tax 2:25 Total Due 2:59 Extra Example 1: Finding the Discount 3:43 Extra Example 2: Finding the Sale Price 6:28 Extra Example 3: Finding the Sale Tax 11:14 Extra Example 4: Finding the Total Due 14:08 Section 8: Geometry in a Plane Intersecting Lines and Angle Measures 24m 17s Intro 0:00 Intersecting Lines 0:07 Properties of Lines 0:08 When Two Lines Cross Each Other 1:55 Angles 2:56 Properties of Angles: Sides, Vertex, and Measure 2:57 Classifying Angles 7:18 Acute Angle 7:19 Right Angle 7:54 Obtuse Angle 8:03 Angle Relationships 8:56 Vertical Angles 8:57 Adjacent Angles 10:38 Complementary Angles 11:52 Supplementary Angles 12:54 Extra Example 1: Lines 16:00 Extra Example 2: Angles 18:22 Extra Example 3: Angle Relationships 20:05 Extra Example 4: Name the Measure of Angles 21:11 Angles of a Triangle 13m 35s Intro 0:00 Angles of a Triangle 0:05 All Triangles Have Three Angles 0:06 Measure of Angles 2:16 Extra Example 1: Find the Missing Angle Measure 5:39 Extra Example 2: Angles of a Triangle 7:18 Extra Example 3: Angles of a Triangle 9:24 Classifying Triangles 15m 10s Intro 0:00 Types of Triangles by Angles 0:05 Acute Triangle 0:06 Right Triangle 1:14 Obtuse Triangle 2:22 Classifying Triangles by Sides 4:18 Equilateral Triangle 4:20 Isosceles Triangle 5:21 Scalene Triangle 5:53 Extra Example 1: Classify the Triangle by Its Angles and Sides 6:34 Extra Example 2: Sketch the Figures 8:10 Extra Example 3: Classify the Triangle by Its Angles and Sides 9:55 Extra Example 4: Classify the Triangle by Its Angles and Sides 11:35 Quadrilaterals 17m 41s Intro 0:00 Quadrilaterals 0:05 Definition of Quadrilaterals 0:06 Parallelogram 0:45 Rectangle 2:28 Rhombus 3:13 Square 3:53 Trapezoid 4:38 Parallelograms 5:33 Parallelogram, Rectangle, Rhombus, Trapezoid, and Square 5:35 Extra Example 1: Give the Most Exact Name for the Figure 11:37 Extra Example 2: Fill in the Blanks 13:31 Extra Example 3: Complete Each Statement with Always, Sometimes, or Never 14:37 Area of a Parallelogram 12m 44s Intro 0:00 Area 0:06 Definition of Area 0:07 Area of a Parallelogram 2:00 Area of a Parallelogram 2:01 Extra Example 1: Find the Area of the Rectangle 4:30 Extra Example 2: Find the Area of the Parallelogram 5:29 Extra Example 3: Find the Area of the Parallelogram 7:22 Extra Example 4: Find the Area of the Shaded Region 8:55 Area of a Triangle 11m 29s Intro 0:00 Area of a Triangle 0:05 Area of a Triangle: Equation and Example 0:06 Extra Example 1: Find the Area of the Triangles 1:31 Extra Example 2: Find the Area of the Figure 4:09 Extra Example 3: Find the Area of the Shaded Region 7:45 Circumference of a Circle 15m 4s Intro 0:00 Segments in Circles 0:05 Radius 0:06 Diameter 1:08 Chord 1:49 Circumference 2:53 Circumference of a Circle 2:54 Extra Example 1: Name the Given Parts of the Circle 6:26 Extra Example 2: Find the Circumference of the Circle 7:54 Extra Example 3: Find the Circumference of Each Circle with the Given Measure 11:04 Area of a Circle 14m 43s Intro 0:00 Area of a Circle 0:05 Area of a Circle: Equation and Example 0:06 Extra Example 1: Find the Area of the Circle 2:17 Extra Example 2: Find the Area of the Circle 5:47 Extra Example 3: Find the Area of the Shaded Region 9:24 Section 11: Geometry in Space Prisms and Cylinders 21m 49s Intro 0:00 Prisms 0:06 Polyhedron 0:07 Regular Prism, Bases, and Lateral Faces 1:44 Cylinders 9:37 Bases and Altitude 9:38 Extra Example 1: Classify Each Prism by the Shape of Its Bases 11:16 Extra Example 2: Name Two Different Edges, Faces, and Vertices of the Prism 15:44 Extra Example 3: Name the Solid of Each Object 17:58 Extra Example 4: Write True or False for Each Statement 19:47 Volume of a Rectangular Prism 8m 59s Intro 0:00 Volume of a Rectangular Prism 0:06 Volume of a Rectangular Prism: Formula 0:07 Volume of a Rectangular Prism: Example 1:46 Extra Example 1: Find the Volume of the Rectangular Prism 3:39 Extra Example 2: Find the Volume of the Cube 5:00 Extra Example 3: Find the Volume of the Solid 5:56 Volume of a Triangular Prism 16m 15s Intro 0:00 Volume of a Triangular Prism 0:06 Volume of a Triangular Prism: Formula 0:07 Extra Example 1: Find the Volume of the Triangular Prism 2:42 Extra Example 2: Find the Volume of the Triangular Prism 7:21 Extra Example 3: Find the Volume of the Solid 10:38 Volume of a Cylinder 15m 55s Intro 0:00 Volume of a Cylinder 0:05 Volume of a Cylinder: Formula 0:06 Extra Example 1: Find the Volume of the Cylinder 1:52 Extra Example 2: Find the Volume of the Cylinder 7:38 Extra Example 3: Find the Volume of the Cylinder 11:25 Surface Area of a Prism 23m 28s Intro 0:00 Surface Area of a Prism 0:06 Surface Area of a Prism 0:07 Lateral Area of a Prism 2:12 Lateral Area of a Prism 2:13 Extra Example 1: Find the Surface Area of the Rectangular Prism 7:08 Extra Example 2: Find the Lateral Area and the Surface Area of the Cube 12:05 Extra Example 3: Find the Surface Area of the Triangular Prism 17:13 Surface Area of a Cylinder 27m 41s Intro 0:00 Surface Area of a Cylinder 0:06 Introduction to Surface Area of a Cylinder 0:07 Surface Area of a Cylinder 1:33 Formula 1:34 Extra Example 1: Find the Surface Area of the Cylinder 5:51 Extra Example 2: Find the Surface Area of the Cylinder 13:51 Extra Example 3: Find the Surface Area of the Cylinder 20:57 Section 10: Data Analysis and Statistics Measures of Central Tendency 24m 32s Intro 0:00 Measures of Central Tendency 0:06 Mean 1:17 Median 2:42 Mode 5:41 Extra Example 1: Find the Mean, Median, and Mode for the Following Set of Data 6:24 Extra Example 2: Find the Mean, Median, and Mode for the Following Set of Data 11:14 Extra Example 3: Find the Mean, Median, and Mode for the Following Set of Data 15:13 Extra Example 4: Find the Three Measures of the Central Tendency 19:12 Histograms 19m 43s Intro 0:00 Histograms 0:05 Definition and Example 0:06 Extra Example 1: Draw a Histogram for the Frequency Table 6:14 Extra Example 2: Create a Histogram of the Data 8:48 Extra Example 3: Create a Histogram of the Following Test Scores 14:17 Box-and-Whisker Plot 17m 54s Intro 0:00 Box-and-Whisker Plot 0:05 Median, Lower & Upper Quartile, Lower & Upper Extreme 0:06 Extra Example 1: Name the Median, Lower & Upper Quartile, Lower & Upper Extreme 6:04 Extra Example 2: Draw a Box-and-Whisker Plot Given the Information 7:35 Extra Example 3: Find the Median, Lower & Upper Quartile, Lower & Upper Extreme 9:31 Extra Example 4: Draw a Box-and-Whiskers Plots for the Set of Data 12:50 Stem-and-Leaf Plots 17m 42s Intro 0:00 Stem-and-Leaf Plots 0:05 Stem-and-Leaf Plots 0:06 Extra Example 1: Use the Data to Create a Stem-and-Leaf Plot 2:28 Extra Example 2: List All the Numbers in the Stem-and-Leaf Plot in Order From Least to Greatest 7:02 Extra Example 3: Create a Stem-and-Leaf Plot of the Data & Find the Median and the Mode. 8:59 The Coordinate Plane 19m 59s Intro 0:00 The Coordinate System 0:05 The Coordinate Plane 0:06 Quadrants, Origin, and Ordered Pair 0:50 The Coordinate Plane 7:02 Write the Coordinates for Points A, B, and C 7:03 Extra Example 1: Graph Each Point on the Coordinate Plane 9:03 Extra Example 2: Write the Coordinate and Quadrant for Each Point 11:05 Extra Example 3: Name Two Points From Each of the Four Quadrants 13:13 Extra Example 4: Graph Each Point on the Same Coordinate Plane 17:47 Section 11: Probability and Discrete Mathematics Organizing Possible Outcomes 15m 35s Intro 0:00 Compound Events 0:08 Compound Events 0:09 Fundamental Counting Principle 3:35 Extra Example 1: Create a List of All the Possible Outcomes 4:47 Extra Example 2: Create a Tree Diagram For All the Possible Outcomes 6:34 Extra Example 3: Create a Tree Diagram For All the Possible Outcomes 10:00 Extra Example 4: Fundamental Counting Principle 12:41 Independent and Dependent Events 35m 19s Intro 0:00 Independent Events 0:11 Definition 0:12 Example 1: Independent Event 1:45 Example 2: Two Independent Events 4:48 Dependent Events 9:09 Definition 9:10 Example: Dependent Events 10:10 Extra Example 1: Determine If the Two Events are Independent or Dependent Events 13:38 Extra Example 2: Find the Probability of Each Pair of Events 18:11 Extra Example 3: Use the Spinner to Find Each Probability 21:42 Extra Example 4: Find the Probability of Each Pair of Events 25:49 Disjoint Events 12m 13s Intro 0:00 Disjoint Events 0:06 Definition and Example 0:07 Extra Example 1: Disjoint & Not Disjoint Events 3:08 Extra Example 2: Disjoint & Not Disjoint Events 4:23 Extra Example 3: Independent, Dependent, and Disjoint Events 6:30 Probability of an Event Not Occurring 20m 5s Intro 0:00 Event Not Occurring 0:07 Formula and Example 0:08 Extra Example 1: Use the Spinner to Find Each Probability 7:24 Extra Example 2: Probability of Event Not Occurring 11:21 Extra Example 3: Probability of Event Not Occurring 15:51 Loading... This is a quick preview of the lesson. For full access, please Log In or Sign up. For more information, please see full course syllabus of Basic Math Bookmark & Share Embed ## Share this knowledge with your friends! ## Copy & Paste this embed code into your website\u2019s HTML Please ensure that your website editor is in text mode when you paste the code. (In Wordpress, the mode button is on the top right corner.) \u00d7 \u2022 - Allow users to view the embedded video in full-size. Since this lesson is not free, only the preview will appear on your website. \u2022 ## Discussion \u2022 ## Answer Engine \u2022 ## Study Guides \u2022 ## Practice Questions \u2022 ## Download Lecture Slides \u2022 ## Table of Contents \u2022 ## Transcription \u2022 ## Related Books Lecture Comments (17) 0 answersPost by Jenny on November 10, 2020Thank you, this helped me alot 0 answersPost by Jianjun Ni on March 11, 2019this is ez 0 answersPost by Alex Liu on January 9, 2019I would like to comment on saying how would you try to subtract the problem (or add) and how do you do it more easily? 3 answersLast reply by: Mingyang CenWed Aug 15, 2018 7:56 PMPost by Tom Miller on January 16, 2017In one of the practice questions is states that John has$120.80 and running shoes are 45.65. But then the question is how much money will Sarah have after she buys the running shoes. 2 answersLast reply by: Mingyang CenWed Aug 15, 2018 7:54 PMPost by colton falgoust on May 19, 2014K I'm stuck find the deference 120 .25 - 50.55. K u start with 5-5=0 next I borrowed 1 from the zero and made the two a 12 and made the zero a ten 5-12=7 10-0 how do u get 9 out of 10-0 any help so I can move to the next problem 0 answersPost by yun wu on July 5, 2013Good lesson 0 answersPost by Fahad Chandia on June 20, 2013good lesson 2 answersLast reply by: Emily BaiFri Jul 13, 2018 8:22 AMPost by Ramez Hajelsawi on March 9, 2013GOOD LESSON I LOVE IT :) 0 answersPost by Nik Googooli on August 27, 2012perfect,, really teacher,,, a real teacher,,,,, simple,,,,, ### Adding and Subtracting Decimals #### Related Links \u2022 When adding and subtracting decimals, align the decimal points first \u2022 Then, add or subtract the digits \u2022 Place the decimal point in the same place ### Adding and Subtracting Decimals Add: 6.23 + 7 + 0.825 14.055 Find the difference: 462.9 \u2212 84.27 378.63 Add: 145.3 + 23.2 168.5 Add: 915.2 + 20.3 + 0.26 935.76 Add: 0.05 + 0.55 + 0.25 0.85 Find the difference: 120.25 \u2212 50.55 69.70 Sarah has 100. Running shoes cost $50.50. How much money will Sarah have left after buying the running shoes? \u2022$ 100 \u2212 $50.50$ 49.50\nJohn has $120.80. Running shoes cost$ 45.65. How much money will Sarah have left after buying the running shoes?\n\u2022 $120.80 \u2212$ 45.65\n\\$ 75.15\nSubtract:\n450.89 \u2212 20.85\n430.04\nSubtract:\n165.32 \u2212 10.385\n154.935\n\n*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.\n\nLecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.\n\n\u2022 Intro 0:00\n\u2022 When Adding and Subtracting 0:06\n\u2022 Align the Decimal Point First\n\u2022 Add or Subtract the Digits\n\u2022 Place the Decimal Point in the Same Place\n\u2022 Check by Estimating\n\u2022 Examples 1:28\n\u2022 Add: 3.45 + 7 + 0.835\n\u2022 Find the Difference: 351.4 - 65.25\n\u2022 Extra Example 1: Adding Decimals 5:32\n\u2022 Extra Example 2: How Much Money? 6:09\n\u2022 Extra Example 3: Subtracting Decimals 7:20\n\u2022 Extra Example 4: Adding Decimals 9:32\n\n### Transcription: Adding and Subtracting Decimals\n\nWelcome back to Educator.com; this lesson is on adding and subtracting decimals.0000\n\nWhen you add and subtract decimals, there are some rules to follow.0008\n\nThe first thing, the most important thing, is to align the decimal points.0013\n\nWhenever you have decimal numbers and you have to add or subtract them, make sure the decimal point is lined up.0018\n\nFor example, if you are going to add 4.1 with 3.2,0027\n\nyou have to make sure that the decimal points, this and this, line up.0036\n\nWe are adding these numbers together; this becomes 3; this is 7.0047\n\nAfter you add them, you have to make sure you place a decimal point in the same place.0055\n\nYou just align them straight down right here.0059\n\n4.1 plus 3.2 is going to be 7.3; check by estimating.0065\n\n4.1 is very close to 4; 4 is the whole number that it rounds to.0073\n\n4 plus 3; this 3.2 rounds to 3; so it is about 7.0077\n\nThat is what it means to check by estimating.0085\n\nLet's do a few problems; add 3.45 plus 7.835.0088\n\nAgain the first rule, the very important rule, is to line up the decimals.0099\n\nMake sure you align them; 3.45.0103\n\nThe next number, we don't see a decimal point here.0110\n\n7 doesn't have a decimal point; it actually does; we just can't see it.0114\n\nIf we have a whole number that doesn't show a decimal point, then the decimal point is right behind it.0118\n\nIt becomes 7.0; 7 is the same thing as 7.0.0126\n\nIf you have a whole number with no decimal point, that does not show a decimal point,0132\n\njust place the decimal point at the end of it, right behind it.0136\n\nI need to make sure that the decimal point goes right there; align them.0141\n\nThis one, 0.835; again 0.835, align the decimal.0149\n\nThat is all that matters; when you add them or subtract them, this is the main thing.0159\n\nYou don't line up the numbers; you line up the decimals.0165\n\nHere, whenever I have decimals, I can place 0s at the end of them.0170\n\nAs long as it is behind the decimal and it is at the end, we can add as many 0s as we want.0176\n\nHere it is behind the decimal point and it is at the end.0182\n\nI can put 0s here to fill in those blanks.0186\n\nIf I add straight down, this becomes 5; 5 plus 3 is 8.0191\n\n4 plus 8 is 12; 7 plus 3 plus 1 is 11.0197\n\nMy answer is 11.285 or 11 and 285.0206\n\nThe next problem, find the difference.0215\n\nThis is--write the first number--351 and 4 tenths, minus... be careful here.0217\n\nIf I write it like this, this would be wrong; this is wrong.0229\n\nI have to make sure not to line up the numbers but line up the decimals.0242\n\n65 and 25 hundredths; I have space right here; I have to put something there.0251\n\nIt is behind the decimal point and it is after the number so I can place a 0 there.0266\n\nI can only place 0s if it is behind the decimal point and it is at the very end.0272\n\n0 minus 5, remember I have to borrow; this becomes 3; this becomes a 10.0277\n\nGet 5; 3 minus 2 is 1; decimal point comes down.0284\n\nAgain I have to borrow here; this becomes 4; this becomes 11.0291\n\n11 minus 5 is 6; 4, again I have to borrow; this becomes a 2.0298\n\n14; this becomes 8 and then 2; this is the answer.0305\n\nMake sure your decimals are lined up.0314\n\nDon't forget again if you have any empty spaces, you have to place 0s in them0318\n\nbecause it is behind the decimal point and it is after the number so you can place 0s there.0325\n\nLet's do a few more examples.0331\n\nThe next one, we want to add these two numbers, 123.1 and then 140.0334\n\nWe have a whole number that does not show a decimal point.0343\n\nIn that case, we are just going to place it at the very end.0346\n\nIt is going to be 140.0; adding these numbers.0350\n\nIt is 1; bring the decimal point down; 3; 6; and 2.0358\n\nNext example, Sarah has 90 dollars and 75 cents.0371\n\nRunning shoes cost 55 dollars and 45 cents.0377\n\nHow much money will Sarah have left after buying the running shoes?0381\n\nIf she is going to buy something that costs 55 dollars and 45 cents, this is how much she has.0386\n\nWe have to see how much she will have left; we have to subtract the numbers.0392\n\nShe has 90 dollars and 75 cents; she spent 55 dollars and 45 cents.0401\n\nI am going to subtract them; this will be 0; 3.0412\n\nAgain I have to borrow; this is 8; this is 10.0421\n\n10 minus 5 is 5; this is 3.0424\n\nAfter buying the running shoes, Sarah will have 35 dollars and 30 cents left.0429\n\nThe next example, we are going to subtract these two numbers.0442\n\nJust write it out; 7 minus... be careful here.0449\n\nThe decimal place has to line up with this decimal place.0455\n\nI am going to write 9 under the 2; 91 and 386 thousandths.0458\n\nAgain I have an empty space right here that I have to fill in with a 0.0469\n\nDon't forget that; the reason why you have to place that 0 there.0476\n\nIf I don't have a 0... let's say I don't have a 0.0480\n\nWhen you subtract this, this seems like it would be a 6 right here.0486\n\nIt seems like you would write a 6 here.0491\n\nBut if you place a 0 there, then you know that you have to borrow.0494\n\nThis is actually 10 minus 6.0497\n\nIf I borrow this, this is going to become 6; 10; 4.0501\n\nBorrow again; 7; 16; there is 8.0507\n\n7 minus 3 is 4; bring down the decimal point.0514\n\nI am going to make this a 10; this becomes 11; this becomes 5.0520\n\n10 minus 1 is 9; 11 minus 9 is 2.0527\n\n5 minus 0 is 5; there is my answer.0533\n\nIf you just round it to the nearest hundreds, it is 600.0546\n\nHere, this is about 90 or maybe let's say 100.0550\n\n600 minus 100 is about 500; we have 500 something; it sounds right.0554\n\nIf we had let's say 50 something as our answer, we know that is wrong0562\n\nbecause we know that if we estimate, it should be around 500.0566\n\nThe last example, adding these decimals together.0573\n\nWe have 8 and 215 dollars and 49 cents and 75 cents.0578\n\nRemember the rule when we add or subtract decimals is to line up the decimal point.0586\n\nThat is very, very important.0591\n\nWhen we have a whole number that is not showing a decimal point,0593\n\nthen we can place a decimal point at the end of it.0597\n\nJust because it is not showing a decimal point does not mean it does not have one.0601\n\nThis is 8 point... and then I can add 0s to it.0607\n\n8 dollars is the same thing as 8.00.0610\n\nThen I am going to add 215 and 49 cents; line up the decimal point.0614\n\nIt is going to go 215.49; 75 cents here is going to be 0.75.0621\n\nGoing to add these all up together; 9 plus 5 is 14.0637\n\n1 plus 4 is 5; 5 plus 7 is 12.0646\n\nBring down the decimal point; come straight down.0655\n\n1 plus 8 is 9; plus 5 is 14; 1 plus 1, 2; and 2.0660\n\nThis is your answer--8 dollars plus 215 dollars and 49 cents plus 75 cents becomes 224 dollars and 24 cents.0674\n\nThat is it for this lesson on adding and subtracting decimals.0686\n\nThank you for watching Educator.com.0689\n\nOR\n\n### Start Learning Now\n\nOur free lessons will get you started (Adobe Flash\u00ae required).","date":"2023-02-05 04:33:59","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.5088266730308533, \"perplexity\": 4925.705878428347}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2023-06\/segments\/1674764500215.91\/warc\/CC-MAIN-20230205032040-20230205062040-00151.warc.gz\"}"} | null | null |
\section{Introduction}
The formation of dwarf spheroidal (dSph) galaxies in the Local Group remains an open question but
one of the most promising scenarios for their origin is via the tidal interaction of their disky
progenitors with more massive hosts like the Milky Way. The scenario,
proposed by Mayer et al. (2001) explains the morphology-density relation observed among the dwarfs of the Local Group
and accounts for the non-sphericity of the dSph objects.
The efficiency of the mechanism and its observational
predictions have been investigated in detail by Klimentowski et al. (2009), Kazantzidis et al. (2011) and
{\L}okas et al. (2011, 2012). These studies explored the dependence of the process on a large number of orbital
and structural parameters of the dwarf. The general picture that emerged from these studies is that a disky dwarf
progenitor, once accreted by a massive host, undergoes strong tidal stirring and mass stripping
if the orbit is tight enough.
Typically,
at the first pericenter passage, the disk transforms into a tidally induced bar (for a detailed description
of the properties of such a bar see {\L}okas et al. 2014a).
The bar becomes thicker and shorter in time leading in the end to the formation of a spheroidal stellar component.
The morphological transformation is accompanied by strong changes in the kinematics as quantified by the amount
of ordered to random motion. The latter starts to dominate at some point and at the end the galaxy is pressure
supported.
Among the parameters expected to have a strong impact on the evolution is the inclination between the angular
momentum of the dwarf galaxy disk and its orbital angular momentum. However, in the studies mentioned above
only a narrow range of inclinations was studied in detail, namely those with values $i = 0^\circ$,
$45^\circ$ and $90^\circ$. Because of this range, and the way the properties of the dwarf were measured, no
clear evidence for the dependence on this parameter was found. On the other hand, the difference between the
prograde and retrograde galaxy encounters has been recognized for a long time and known to lead to very different
outcomes (e.g. Holmberg 1941; H\'{e}non 1970; Kozlov et al. 1972; Toomre \& Toomre 1972;
Keenan \& Innanen 1975).
The issue has been recently addressed again by D'Onghia et al. (2009, 2010)
using an improved version of the impulse approximation applied to rotating systems.
Although this approximation is not directly applicable to our simulations, we attempt a comparison between these results
and the numerical ones.
\begin{table*}
\begin{center}
\caption{Initial conditions for the simulations}
\begin{tabular}{lrrrcl}
\hline
\hline
Simulation & $L_{\rm X}$ & $L_{\rm Y}$ & $L_{\rm Z}$ & Inclination & Line color/type \\
& & & & (deg) & \\
\hline
\ \ \ \ I0 & 0.0 & 0.0 & 1.0 &\ \ 0 & red/solid \\
\ \ \ \ I90 & 0.0 & $-1.0$ & 0.0 &\ 90 & blue/short-dashed \\
\ \ \ \ I180 & 0.0 & 0.0 & $-1.0$ & 180 & green/dotted \\
\ \ \ \ I270 & 0.0 & 1.0 & 0.0 & 270 & cyan/long-dashed \\
\hline
\label{initial}
\end{tabular}
\end{center}
\end{table*}
In this paper we aim at clarifying the issue of the dependence of the results
of tidal encounters between dwarfs and their hosts on the inclination of the dwarf's disk. For this
purpose we performed four simulations of tidal evolution of a dwarf galaxy orbiting a Milky Way-like host
with disk inclinations $i = 0^\circ$, $90^\circ$, $180^\circ$ and $270^\circ$. The angles of $0^\circ$ and
$180^\circ$ correspond to exactly prograde and exactly retrograde orientations of the dwarf's disk. We also measured
the properties of the dwarf galaxy in a different way that enables clear comparisons between different runs.
Preliminary results of this study, using lower resolution simulations, were discussed in {\L}okas \& Semczuk (2014).
The paper is organized as follows. In section 2 we present the simulations used in this study. In section 3 we discuss
the properties of the dwarf galaxies as they evolve in time, focusing on their kinematics, morphology and density
profiles.
Section 4 compares the results of the simulations to the
predictions of semi-analytic models. The conclusions follow in section 5.
\hspace{0.2in}
\section{The simulations}
The simulations used in this study were similar to those described in detail in {\L}okas et al. (2014a). Here we
therefore provide only a short summary.
The initial conditions for the simulations consisted of $N$-body realizations of two galaxies:
the Milky Way-like host and the dwarf galaxy, generated via procedures described in Widrow \& Dubinski (2005)
and Widrow et al. (2008). Both galaxies contained exponential disks embedded in
NFW (Navarro et al. 1997) dark matter haloes, each made of
$10^6$ particles ($4 \times 10^6$ total). We note that the results presented here differ only slightly (are less noisy)
from those of lower resolution simulations in {\L}okas \& Semczuk (2014) where a smaller number of particles
was used (by a factor of five). We are therefore confident that our present resolution is sufficient
to grasp all the essential features of the evolution.
The dwarf galaxy model had a dark
halo of mass $M_{\rm h} = 10^9$ M$_{\odot}$ and concentration $c=20$. Its disk had a
mass $M_{\rm d} = 2 \times 10^7$ M$_{\odot}$, an exponential scale-length $R_{\rm d} = 0.41$ kpc and thickness
$z_{\rm d}/R_{\rm d} = 0.2$. The model is stable against formation of the bar in isolation for the
time scales of interest here.
The host galaxy was similar to the model MWb of Widrow \& Dubinski (2005). It had a dark matter halo of mass
$M_{\rm H} = 7.7 \times 10^{11}$ M$_{\odot}$ and concentration $c=27$. The disk of the host had a mass $M_{\rm D}
= 3.4 \times 10^{10}$ M$_{\odot}$, the scale-length $R_{\rm D} = 2.82$ kpc and thickness $z_{\rm D} = 0.44$ kpc.
The disk was also stable against bar formation to avoid strong variations of the host potential in time. The disk
of the Milky Way was coplanar with the orbit of the dwarf. Although this may seem contrary to observational
constraints where most of satellite orbits are found to be polar (e.g. Pawlowski \& Kroupa 2014),
the choice was motivated by the necessity to
avoid any additional variability which may be due to the passages through the plane of the Milky Way disk. However,
we have performed an additional simulation to verify that for the orbits used here the evolution of the dwarf depends
very weakly on the orientation of the orbit with respect to the Milky Way disk.
The dwarf galaxy was initially placed at an apocenter of a typical, eccentric orbit around the Milky Way
with apo- to pericenter distance ratio of $r_{\rm apo}/r_{\rm peri} = 120/25$ kpc. The initial position was
at the coordinates $(X,Y,Z) = (-120,0,0)$ kpc of the simulation box and the velocity vector of the dwarf was toward the
negative $Y$ direction. We performed four simulations
with different dwarf disk orientations with respect to the orbit: two coplanar with the orbit (prograde and retrograde)
and two perpendicular to the orbit with angular momenta in the same and opposite direction to the dwarf's orbital
velocity. The different initial conditions, in particular the components of the unit angular momentum vector, are
listed in Table~\ref{initial}. We will refer to the simulations by names indicating the initial inclination of the
disk, I0, I90, I180 and I270, where the inclination is measured as the rotation angle around the $X$ axis of the
simulation box.
The evolution of the system in each simulation was followed for 10 Gyr using the GADGET-2 $N$-body code
(Springel et al. 2001; Springel 2005) with outputs saved every 0.05 Gyr.
The adopted softening scales were $\epsilon_{\rm d} = 0.02$ kpc and
$\epsilon_{\rm h} = 0.06$ kpc for the disk and halo of the dwarf while $\epsilon_{\rm D} = 0.05$ kpc and
$\epsilon_{\rm H} = 2$ kpc for the disk and halo of the host, respectively.
\hspace{0.2in}
\section{Evolution of the dwarfs}
In this section we look at the inner properties of the dwarf galaxies as they are transformed by the tidal forces from
the Milky Way. All measurements discussed below were made for stars (and dark matter particles) within the radius
of 0.5 kpc from the center of the dwarf.
\subsection{Mass content}
We begin the analysis by measuring the mass inside this radius.
Figure~\ref{compmass} compares the mass of stars (upper panel) and dark matter (lower panel) in the four simulations.
As expected, the dark mass content decreases systematically, most significantly at pericenters that occur at
$t = 1.2$, 3.3, 5.5, 7.6, 9.7 Gyr from the start of the simulation and there is no dependence on the inclination
because the halos were spherical and isotropic in all cases.
On the other hand, for the stellar component (upper panel)
there is a significant difference between dwarfs with varying initial inclination. However, the dependence may seem
surprising because we expect the prograde disk to be more stripped, while the opposite is seen in the Figure: the
stellar mass for the I0 simulation is even increased after the first pericenter passage, while it is decreased in the
remaining cases. While the tidal stripping is indeed stronger in the outer parts, in the inner region we probe by
this measurement, the stellar content increases as a result of the significant change in the structure
of the stellar component due to the formation of a tidally induced bar.
\begin{figure}
\begin{center}
\leavevmode
\epsfxsize=8cm
\epsfbox[0 0 186 190]{compmass.eps}
\end{center}
\caption{The evolution of the stellar (upper panel) and dark (lower panel) mass of the dwarf galaxy
enclosed within the radius of 0.5 kpc. Different lines correspond to different initial inclination
of the dwarf galaxy disk.}
\label{compmass}
\end{figure}
\subsection{Kinematics}
The main criteria usually applied in order to verify if a dwarf galaxy transformed from a disky object to a
dSph are based on kinematics and shape of the stellar component. A dSph galaxy is supposed to be characterized
by the dominance of random motions of the stars over the amount of rotation and its shape should be
sufficiently close to spherical. In this and the following subsection we look in detail at the evolution of
these quantities.
In order to measure these properties, for each simulation output we determine the directions of the principal
axes of the stellar component from stars within the radius of 0.5 kpc using the inertia tensor and rotate
these stars so that the new coordinate system is aligned with the principal axes (the $x$ axis is along the longest,
the $y$ axis along the intermediate and the $z$ axis along the shortest axis of the stellar component).
We then introduce a standard spherical coordinate system such that $\phi$ measures the angle around the $z$
axis and $\theta$ the angle from the $z$ axis towards the $xy$ plane.
The kinematic properties were estimated using these coordinates. In the top panel of Figure~\ref{compvsig}
we plot the rotation around the shortest axis $V=V_\phi$ as a function of time. We note that there is no other
significant streaming motion along the other spherical coordinates or around the two other principal axes (see
{\L}okas et al. 2014b for a brief discussion of this issue). Clearly, the rotation is decreasing most strongly
for the exactly prograde inclination of the disk (I0), a significantly smaller decrease is seen for the perpendicular
orientations (I90 and I270) and the effect is the weakest for the retrograde case (I180). Note that the decrease of
rotation is not steadily monotonic even in the exactly prograde case (I0). This is due to the tidal torques acting
on the bar at pericenter passages that can speed up or slow down the bar depending on its particular orientation at
this moment (see {\L}okas et al. 2014a for details).
\begin{figure}
\begin{center}
\leavevmode
\epsfxsize=8cm
\epsfbox[0 0 186 279]{compvsig.eps}
\end{center}
\caption{The evolution of the mean rotation velocity (upper panel), the velocity dispersion (middle panel)
and the ratio of the two (lower panel) as a function of time.}
\label{compvsig}
\end{figure}
The behavior of the velocity dispersion is exactly the opposite. In the middle panel of Figure~\ref{compvsig}
we show the 1D velocity dispersion calculated as
$\sigma = [(\sigma_r^2 + \sigma_{\theta}^2 + \sigma_{\phi}^2)/3]^{1/2}$. The increase of $\sigma$ at the first
pericenter passage is strongest for the I0 case, intermediate for I90 and I270 and for I180 $\sigma$ remains
constant in time or even slightly decreases due to mass loss. The ratio $V/\sigma$ shown in the lower panel of
Figure~\ref{compvsig} decreases for all simulations, but reaches a value significantly below unity only for the
prograde case. For the three remaining cases a substantial amount of rotation is retained, although the hierarchy
of lower $V/\sigma$ for more prograde cases is preserved.
In Figure~\ref{compsigma} we show the evolution of the different velocity dispersions $\sigma_r$ (upper panel),
$\sigma_\theta$ (second panel) and $\sigma_\phi$ (third panel) as a function of time. In all cases the increase
of a given dispersion at the first pericenter passage and its level at later times is highest for the I0 case,
intermediate for I90 and I270 and non-existent for I180. In addition, this increase is the most abrupt for the
prograde dwarf, while for the intermediate inclinations I90 and I270 the increase occurs much more slowly in
time and takes about half the orbital period between the first and second pericenter passage. Interestingly,
significant increase is seen in all dispersions in spite of the fact that, due to the formation of the bar, one
could expect the radial $\sigma_r$ to increase much more significantly.
\begin{figure}
\begin{center}
\leavevmode
\epsfxsize=8cm
\epsfbox[0 0 186 358]{compsigma.eps}
\end{center}
\caption{The evolution of the velocity dispersions of the stars in spherical coordinates $\sigma_r$ (upper panel),
$\sigma_\theta$ (second panel) and $\sigma_\phi$ (third panel). The lower panel shows the evolution of the
anisotropy parameter $\beta$.}
\label{compsigma}
\end{figure}
\begin{figure}
\begin{center}
\leavevmode
\epsfxsize=8cm
\epsfbox[0 0 186 358]{compshape.eps}
\end{center}
\caption{The evolution of the shape of the stellar component in time. The panels from top show the axis ratios
$b/a$, $c/a$, the triaxiality parameter $T$ and the bar mode $A_2$.}
\label{compshape}
\end{figure}
We end the analysis of the kinematics by plotting in the lower panel of Figure~\ref{compsigma} the value of the
anisotropy parameter $\beta = 1 - \sigma_t^2/(2 \sigma_r^2)$ where $\sigma_t^2 = \sigma_\theta^2 + \sigma_\phi^2
+V_\phi^2$ is the tangential second moment including rotation. A systematic dependence of this parameter on the
inclination of the disk is also present: the stellar orbits are most radial for the I0 case with $\beta$ almost
constantly at the level of 0.5, corresponding to mildly radial orbits of the stars, characteristic of the bar.
For the intermediate cases I90 and I270 the $\beta$ values stay between 0 and 0.4,
while for the retrograde case I180 $\beta$ remains negative due to the dominant presence of rotation.
Interestingly, at the end of the evolution all three non-prograde cases I90, I180 and I270 have almost isotropic
orbits ($\beta=0$) although they reach this special value via different evolutionary paths.
\subsection{Shapes}
Figure~\ref{compshape} illustrates the evolution of the shape of the stellar component of the dwarfs in time.
In the first and second panels from top we plot the axis ratios $b/a$ (intermediate to longest) and $c/a$
(shortest to longest). The thickening of the dwarf, as quantified by the increasing value of $c/a$ is
similar in all cases, although for the prograde case $c/a$ decreases after the first pericenter due to the formation of
the bar. Much more significant differences are seen in the evolution of $b/a$. This value decreases most
strongly for the prograde I0 case signifying the prolate shape characteristic of the bar. The intermediate
inclinations I90 and I270 lead to less prolate shapes, while I180 remains disky for the whole evolution, as
indicated by $b/a$ remaining constantly close to unity.
The shape can be also quantified in terms of the triaxiality parameter $T = [1-(b/a)^2]/[1-(c/a)^2]$ which is shown
in the third panel of Figure~\ref{compshape}. The values of
the parameter $T>2/3$ at all times confirm the prolate shape due to the bar in the case of I0. For the intermediate
cases I90 and I270 we get $1/3 < T < 2/3$ indicating a triaxial shape. For the retrograde case I180 the parameter
remains low, $T < 0.2$, at all times as is characteristic of a disk.
\begin{figure}
\begin{center}
\leavevmode
\epsfxsize=4.1cm
\epsfbox[0 0 185 200]{surdenrotI0xz.eps}
\leavevmode
\epsfxsize=4.1cm
\epsfbox[0 0 185 200]{surdenrotI90xz.eps}
\leavevmode
\epsfxsize=4.1cm
\epsfbox[0 0 185 200]{surdenrotI180xz.eps}
\leavevmode
\epsfxsize=4.1cm
\epsfbox[0 0 185 200]{surdenrotI270xz.eps}
\end{center}
\caption{Surface density distributions of the stars in the dwarfs at the last apocenter
($t=8.65$ Gyr) seen along the intermediate ($y$)
axis of the stellar component. The surface density
measurements were normalized to the maximum value $\Sigma_{\rm max} = 4.8 \times 10^5$ stars kpc$^{-2}$ occurring
for I180. Contours are equally spaced in $\log \Sigma$ with $\Delta \log \Sigma = 0.05$.}
\label{surdenrot}
\end{figure}
The presence of a bar is usually detected by measuring the bar mode
$A_2$ of the Fourier decomposition of the stars projected along the shortest axis of the stellar distribution
(see a more detailed discussion in {\L}okas et al. 2014a).
Usually, $A_2 > 0.3$ is considered as high enough to be interpreted as a bar. As we can see in the lower panel
of Figure~\ref{compshape} this is always the case for simulation I0 after the first pericenter passage and also
for some significant periods of time for the intermediate cases I90 and I270, which means that the bar also forms
there, but it is much weaker. The exactly retrograde disk does not
form a bar as its $A_2 < 0.06$ at all times. Slight temporary increases of this value are due to stretching of
the dwarf at the pericenters.
These measurements are confirmed by the maps of the surface density distribution of the stars in the four dwarfs
plotted in Figure~\ref{surdenrot}. The distributions are shown in projection along the intermediate axis of the
stellar component so in their most non-spherical appearance. The snapshots were selected for the time $t=8.65$ Gyr
after the start of the simulations, corresponding to the last apocenter passage. In all cases, except for the
retrograde one I180, the remnant of the bar formed after the first pericenter passage is still visible in the
inner parts. In addition, the distribution of the stars in the prograde case I0 is much less diffuse in the outer
parts of the maps. In spite
of the fact that this dwarf evolves most strongly, its neighborhood is not uniformly filled with stripped debris
because the lost stars form well-defined, narrow tidal streams.
\subsection{Density profiles}
\begin{figure}
\begin{center}
\leavevmode
\epsfxsize=7cm
\epsfbox[0 0 174 174]{compprofiles.eps}
\end{center}
\caption{Comparison of stellar density profiles of the dwarfs
at the fourth apocenter (6.6 Gyr after the start of the simulations, colored lines).
For completeness we also show the dark matter profile (black line) which is similar for all simulations.}
\label{compprofiles}
\end{figure}
The dependence of the properties of the dwarfs on the initial inclination of the disk also manifests itself in
the evolved density profiles. In Figure~\ref{compprofiles} we show examples of the stellar density profiles
(different colors) and the dark matter
density profile (black line) for different simulations considered here, measured at the fourth apocenter.
The transition from the bound component to the tidal tails is visible as the break in the slope of the density
profiles. This transition is however only well defined
for the exactly prograde case I0 (and the dark matter profile). In this case the transition from the steeper
to the shallower profile (where the slope is around
$r^{-4}$) occurs at around 3 kpc. For other simulations no such clear break radius is seen.
As discussed in {\L}okas et al. (2013) using similar (but only mildly prograde) simulation setups, the break radii
can be interpreted as the tidal radii. In this case, the dependence on the orbit of the star within the satellite
is expected (Keenan \& Innanen 1975; Read et al. 2006) and we will attempt a detailed comparison in a
follow-up paper. Since the
stellar density profiles of our simulated dwarfs do not show clear signatures of the
break radius (except for the exactly prograde case) here we propose a comparison in terms of density.
The stellar profiles shown in Figure~\ref{compprofiles} demonstrate clear hierarchy:
at the outer radii (larger than 1 kpc) the stellar density profile of I180 (green line) is above all the other
profiles, the one of I0 (red lines) is the lowest, and the ones of I90 and I270 fall exactly on top of each other
and between the other two. This means that we can quantify the amount of tidal stripping in these different cases by
measuring the density of the stars, rather than the break radius.
To do so, we calculated the mean density of stars in the shells of radii 2 kpc $< r <$ 4 kpc
as a function of time for different simulations. The results are shown in the upper panel of Figure~\ref{comptidal}.
In the lower panel we plot the radius $r_d$ at which the density of stars drops below $10^4 $M$_{\odot}/$kpc$^3$
as a function of time. In both plots there is a clear systematic difference between the measurements for different
initial disk orientations: the stars are stripped more effectively on prograde orbits as demonstrated by the lower
outer densities and smaller radii where the density drops to a fixed value.
Although the values for the measurements were chosen in an arbitrary way, we expect the results to be similar if
these parameters are slightly modified. They clearly confirm that the amount of tidal stripping depends
very strongly on the initial inclination of the dwarf's disk.
\begin{figure}
\begin{center}
\leavevmode
\epsfxsize=8cm
\epsfbox[0 0 186 178]{comptidal1.eps}
\end{center}
\caption{Upper panel: the density of stars in the shell of radii 2 kpc $< r <$ 4 kpc
as a function of time. Lower panel: the radius at which the density of stars drops below $10^4 $M$_{\odot}/$kpc$^3$
as a function of time.}
\label{comptidal}
\end{figure}
\section{Comparison with semi-analytic predictions}
To describe the encounter between a disky dwarf galaxy and the Milky Way we use the impulse approximation
as discussed in Binney \& Tremaine (1987) and D'Onghia et al. (2010). According to this approximation the
$k$th component of the acceleration of each dwarf's star with respect to the center of mass is given by
\begin{equation} \label{acceleration}
\dot{v}_k=-\sum_j \frac{\partial^2\psi}{\partial x_k \partial x_j}\at[\bigg]{\boldsymbol{x}=0}x_j,
\end{equation}
where $\psi$ is the Milky Way's (i.e. perturber's) potential and $x_i$ are Cartesian coordinates. We integrate equation
(\ref{acceleration}) over a finite time period to obtain velocity increments that can be compared with
increments measured from simulations
\begin{equation} \label{deltav}
\Delta v_k=\int\limits_0^{\Delta t}\dot{v}_k\; {\rm d} t.
\end{equation}
\begin{figure*}
\begin{center}
\leavevmode
\epsfxsize=17cm
\epsfbox[0 0 610 650]{panel3.eps}
\end{center}
\caption{The components of velocity increments along the $x$, $y$ and $z$ axis for simulations with different
initial inclination of the dwarf's disk. The blue
dots indicate the values measured for a subsample of stars in the simulation. Red dots show the corresponding
semi-analytic predictions.}
\label{velocities}
\end{figure*}
We work in a frame centered on the center of the dwarf's mass, the dwarf's disk lies in $xy$ plane and
position of the Milky Way at the pericenter is $(b_x,0,b_z)$.
In this frame the trajectory of each star originating from the dwarf is
\begin{equation} \label{trajectory}
\boldsymbol{x}=[r\cos(\Omega t +\phi_0), r\sin(\Omega t +\phi_0), 0],
\end{equation}
where $r$ is the radius, $\Omega$ is the angular velocity and $\phi_0$ is the initial azimuthal angle
of the star. The trajectory of Milky Way is given by
\begin{equation} \label{trajectoryMW}
\boldsymbol{X}=\boldsymbol{V} t+\boldsymbol{X_0},
\end{equation}
where $\boldsymbol{V}$ and $\boldsymbol{X_0}$ are constant vectors, fitted to mimic the perturber's trajectory
from simulations as a straight line during a given time period.
Note that $\boldsymbol{V}$ and $\boldsymbol{X_0}$ are different for each of our simulations as
they depend on the inclination $i$. This dependence can be found by rotating the
trajectory for the prograde case with matrix $\hat{A}$ defined by Euler angles, in order to obtain
trajectories in other cases. One of the Euler angles is the inclination $i$ and other two depend
on the perturber's orbit.
To approximate the gravitational potential of the Milky Way we sum the potential from its stars and
the dark matter halo.
The first part is represented as a point-mass potential, while the second is given by the NFW profile
\begin{equation} \label{potentialMW}
\psi=-\frac{G M_{\rm D}}{|\boldsymbol{x}-\boldsymbol{X}|}-g\frac{G M_{\rm H}\ln(1+
c |\boldsymbol{x}-\boldsymbol{X}|/r_{\rm{v}})}{|\boldsymbol{x}-\boldsymbol{X}|},
\end{equation}
where $M_{\rm D}$ is the mass of the Milky Way disk, $M_{\rm H}$ is the virial mass of its halo, $r_{\rm v}$ is
the virial radius, $c$ is the concentration parameter and $g=[\ln(1+c)-c/(1+c)]^{-1}$ (see {\L}okas \& Mamon 2001).
Substituting equations (\ref{acceleration}), (\ref{trajectory}), (\ref{trajectoryMW}) and (\ref{potentialMW})
into (\ref{deltav}) we obtain formulae which can be numerically integrated to get
velocity increments.
\begin{figure*}
\begin{center}
\leavevmode
\epsfxsize=7cm
\epsfbox[0 0 380 400]{map0.eps}
\leavevmode
\epsfxsize=7cm
\epsfbox[0 0 380 400]{map90.eps}
\leavevmode
\epsfxsize=7cm
\epsfbox[0 0 380 400]{map180.eps}
\leavevmode
\epsfxsize=7cm
\epsfbox[0 0 380 400]{map270.eps}
\end{center}
\caption{Surface density distributions of the stars in the dwarfs at the first pericenter
($t=1.15$ Gyr) in projection onto the initial disk plane. Arrows indicate the
velocities of the randomly selected subsample of 100 stars outside the radius of 2 kpc.}
\label{pericenter}
\end{figure*}
In order to calculate the velocity increments described above we estimate the angular velocity in the dwarf as
$\Omega=|\boldsymbol{v}|/r$. We calculate integrals of equation (\ref{deltav}) over a small period of time,
so that the assumptions concerning the trajectories are valid. In our simulations the outputs were saved every 0.05 Gyr
and we choose to integrate over this time to compare our predictions with velocity increments occurring in
the simulations when the tidal force is the strongest, i.e. between the output preceding the first pericenter
and the one as close as possible to this pericenter. We check how the distribution of increments changes
with the distance from the center of the dwarf. The results for all simulations are summarized in
Figure~\ref{velocities}.
The red points in Figure~\ref{velocities} represent the values predicted by the impulse approximation,
while the blue points correspond
to the values measured from the simulations. At radii smaller than 2 kpc the semi-analytical predictions do not
reproduce the distribution of velocity increments because at these radii the velocity changes are dominated by the
gravitational potential of the dwarf which was not included in the predictions.
However, for radii greater than 2 kpc the agreement between the full simulations and semi-analytic predictions
for this short time period is very good. In particular, in the upper left panel of Figure~\ref{velocities}
for simulation I0 we find two very well-defined
branches corresponding to stars on different sides of the dwarf. Branches from simulations
are not exactly symmetric with respect to zero while the branches from theoretical predictions are.
The difference is due to the fact that the analytic predictions only take into account the lowest order terms.
For simulations I270 the increments are almost identical as for I90, as expected due to symmetry of the
two configurations with respect to the orbital plane.
In some of the panels in Figure~\ref{velocities} the velocity increments for radii
larger than 2 kpc are approximately zero. However,
the negligible values are consistently obtained both from the simulations and the semi-analytic calculations.
We further illustrate these results in Figure~\ref{pericenter} where we plot the surface density maps of
the stellar component
and the velocity vectors for a random sample of a hundred stars at radii larger than 2 kpc.
The plots show the dwarfs at the first pericenter passage, i.e. after they have been affected by a tidal impulse
from the Milky Way for the first time. The coordinate system is as defined above, with $xy$ coordinates in the plane
of the dwarf's disk.
The comparison of the upper left panel of Figure~\ref{pericenter}, corresponding to simulation I0, to the other three
confirms that for the prograde encounter the effect of
the tidal force in the strongest: the dwarf galaxy disk is already strongly distorted toward a bar-like shape and
two tidal arms are formed. There are significant increments of velocity along the $x$ axis.
Small increments of velocities are also visible along the $y$ axis in the right panels of Figure~\ref{pericenter}
corresponding to simulations I90 and I270.
The effect of the tidal force is weakest for the retrograde case (lower left panel of Figure~\ref{pericenter})
where the dwarf's initial disk is affected very little.
As discussed by D'Onghia et al. (2010) strongest tidal interactions occur when the intrinsic angular velocities of
stars in the dwarf's disk are comparable to the angular velocity of the satellite on its orbit.
This condition can be written as
\begin{equation} \label{resonance}
\Omega_{\rm disk} \simeq \Omega_{\rm orb}
\end{equation}
and we can define the resonance parameter
\begin{equation} \label{resparameter}
\alpha=\frac{|\Omega|}{\Omega_{\rm orb}},
\end{equation}
that should be of the order of unity for the strongest, resonant response.
To demonstrate the resonant nature of the tidal effects in our simulations we measured $\alpha (r)$ for each
simulation output and found radii from the center of the dwarf at which $\alpha =1$. The time dependence of
this radius for simulation I0 is shown in Figure~\ref{alpha}. We can see that the variability
of this radius reflects the varying orbital velocity $\Omega_{\rm orb}$
of the dwarf resulting in rather large values at apocenters
and much smaller ones at pericenters. The slow variation over time scales much larger than the orbital period is caused
by the mass loss and decreasing $\Omega$.
Whenever the dwarf
reaches the pericenter of its orbit around the Milky Way, this characteristic radius drops below 2 kpc, and even
down to 1 kpc at the later pericenters. Note that these radii are of the order of 2-3 half-light radii of the
stellar component which means that a significant fraction of stars is affected.
Comparing with Figure~\ref{velocities} showing the velocities at the first pericenter we confirm that this radius
(equal to 1.7 kpc at this time) is exactly where the tidal effects start to prevail
over the dwarf's potential.
\section{Conclusions}
In this work we extended previous studies of the efficiency of the tidal stirring mechanism to include the
dependence on the initial inclination of the dwarf galaxy disk with respect to its orbit around the Milky Way.
Our simulation setups involved a dwarf galaxy placed on a typical, eccentric orbit around a Milky Way-like host and
its evolution was followed for 10 Gyr. We considered four configurations, an exactly prograde, an exactly retrograde and
two intermediate orientations of the disk. We found the efficiency of tidal stirring to be very strongly dependent on this
inclination.
\begin{figure}
\begin{center}
\leavevmode
\epsfxsize=8cm
\epsfbox[0 0 186 94]{alpha1.eps}
\end{center}
\caption{The evolution of the radius at which the resonance parameter $\alpha=1$ in time. Only results for
simulation I0 are shown because the measurements yield similar values for the remaining simulations.}
\label{alpha}
\end{figure}
The effect of the tidal interaction turns out to be the strongest for the exactly prograde orientation of the
dwarf's disk (I0). In this case the disk transforms into a strong bar ($A_2 = 0.4$)
at the first pericenter passage and a remnant bar is
retained until the end of the evolution. Although the bar becomes weaker with time, the shape of the stellar component
is consistently prolate, but tending to a spherical toward the end of the evolution. This morphological
transformation is mirrored in the kinematics by the gradual decrease of the dwarf's rotation velocity. Already at the
first pericenter passage the rotation drops significantly and the velocity dispersion increases, mostly in the
radial direction due to the formation of the bar. During subsequent pericenter passages the rotation is further
decreased down to rather low values at the end of the simulation where $V/\sigma =0.3$.
Thus the streaming motions are
almost completely replaced by random motions of the stars. Interestingly, the radial velocity dispersion dominates
the whole time, which manifests itself in the anisotropy parameter close to $\beta = 0.5$ even at the end.
In the two cases of perpendicular orientations of the dwarf's disk with respect to the orbit (I90 and I270) the
evolution is to some extent similar to the I0 case. In these configurations the bar also forms after the first
pericenter passage, but more slowly (over a time scale of about 1 Gyr corresponding to half the orbital period)
and is significantly weaker
($A_2 = 0.3$), also in the subsequent evolution. The overall shape of the stellar component can be characterized
more as triaxial than decidedly prolate. The transition from the streaming to random motions of the stars also
happens less efficiently with $V/\sigma$ only slightly below unity at the final outputs. The anisotropy parameter
is close to zero due to the contribution of the still significant rotation.
In the exactly retrograde case (I180) no strong evolution is present: the dwarf's stellar
component does not form a bar and remains disky. The only signatures of tidal evolution in this case are the mass
loss (similar as in other cases, mostly in dark matter), small decrease of the rotation velocity,
slight evolution of the anisotropy parameter from negative toward isotropic and a non-negligible thickening of the disk.
The difference between this and the other cases is also visible in the stellar density profiles which are less
affected and do not show any clear transition from the bound component to the tidal tails.
We have interpreted these changes in the context of the resonant stripping mechanism recently discussed by
D'Onghia et al. (2009, 2010). In particular, we calculated the velocity increments the dwarf's stars should
experience at the first pericenter and compared them with the direct measurements from simulations. We find a
very good agreement between the two, confirming the interpretation that the evolution we see in the full
$N$-body treatment is indeed due to the orientation of the dwarf's disk. The resonant nature of the phenomenon
is further confirmed by the behavior of the ratio between the angular velocity of the stars in the dwarf and
the angular velocity of its orbital motion. This ratio turns out to be of the order of unity only near the
pericenters and this is indeed when the tidal effects are the strongest.
The results presented here suggest that the most important mechanism underlying the tidal evolution of
disky dwarfs orbiting a bigger galaxy is indeed of resonant nature. We propose to refer to the processes of
morphological and dynamical evolution of the dwarfs we described as
`resonant stirring' in analogy to the `resonant stripping' mechanism found by D'Onghia et al. (2009, 2010)
to increase the mass loss in similar configurations. As discussed by D'Onghia et al. (2010), the resonance
is broad, hence the name `quasi-resonant stirring' would be more appropriate. In physical terms, this resonance can be
traced to the fact that the stars with $\alpha \approx 1$ remain for an extended
period of time on the line joining the dwarf galaxy and the perturber. For these stars
the tidal force (which is strongest along this line) has the longest time to
operate which results in the largest velocity increments and the largest stirring. In the context of the tidal
radius calculations, the difference between the prograde and retrograde cases comes from the change of sign of the
Coriolis force. The relation between the two approaches remains to be investigated and we plan to address this
issue in our future work.
\section*{Acknowledgments}
This work was supported in part by PL-Grid Infrastructure, the Polish National Science Centre under
grant 2013/10/A/ST9/00023, by US National Science Foundation Grant No. PHYS-1066293 and the hospitality of
the Aspen Center for Physics. ED gratefully acknowledges the
support of the Alfred P. Sloan Foundation and of the NSF Grant No. AST-1211258 and ATP-NASA Grant No. NNX14AP53G.
MS benefited from the summer student program of the Copernicus Center. We thank L.
Widrow for providing procedures to generate $N$-body models of galaxies for initial conditions.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 7,044 |
{"url":"https:\/\/mathsmadeeasy.co.uk\/gcse-maths-revision\/data-sampling-and-questionnaires-gcse-revision-and-worksheets\/","text":"# Data Sampling and Questionnaires Worksheets and Revision\n\nGCSE 4 - 5AQAEdexcelOCRWJEC\n\n## Data Sampling and Questionnaires\n\nSampling is the process of looking at a small selection of people\/objects from a wider population in order to learn new information about that population. There are many ways of sampling data, but the main methods we look at are random sampling and questionnaires.\n\nMake sure you are happy with the following topics before continuing.\n\nGCSE\n\n## Data Sampling\n\nData sampling is when you use a smaller sample of\u00a0 a large group to in order to represent the whole sample.\n\nExample: Suppose we wanted to know the reading habits of people living in the UK.\n\nIn reality, it would be better to ask every single member of the population \u2013 this would be VERY reliable, but VERY expensive, and time consuming.\n\nA positive of data sampling is that it is less time-consuming,\n\nA negative of data sampling is that it might be less reliable, and possibly biased. For example, if you only asked people who were leaving a library, then all your results may be much higher than otherwise.\n\nThe aim is to make a sample as representative of the whole population as possible (i.e. reduce the amount of bias).\n\nLevel 1-3 GCSE\n\n## Random Sampling\n\nRandom sampling is the best way to avoid any sort of bias.\n\nFor instance, in biological studies where biologists want to determine the number of plant species in a large wide open area, they select random spaces to sample by plotting the area on a grid and using a random number generator to select coordinates, which tells them which area to sample.\n\nLevel 1-3 GCSE\nLevel 1-3 GCSE\n\n## Questionnaires\n\nThe purpose of a questionnaire is to help extract the information you require, whilst avoiding bias and giving people the opportunity to give a whole range of possible answers. Constructing questionnaires in the correct way and being able to spot poor questionnaires is what you will be asked questions on.\n\nLevel 1-3 GCSE\nLevel 4-5 GCSE\n\n## Example: Questionnaires\n\nPhillipa has designed a questionnaire\u00a0to learn about TV-watching habits. She intends to put this questionnaire to $30$ of her classmates.\n\na) Find $2$ problems with this questionnaire and explain your answer.\n\nb) Redesign the questionnaire so that it is more likely to collect meaningful data.\n\n[4 marks]\n\na) There are more than just two problems with this, so we\u2019ll go through them all.\n\nTime frame\u00a0\u2013 The question must include a time frame, per day, per week, per month?\n\nOptions \u2013 There are no options for someone who watches no TV at all, or someone who watches more than $20$ hours of TV.\n\nOverlapping options \u2013 There is a crossover between the options \u2013 if I watch $5$ hours of TV, then should I tick the first box or the second?\n\nBias \u2013 Only people from one class are being asked, this doesn\u2019t represent the wider population and ultimately may result in biased data.\n\nClarity \u2013 It isn\u2019t clear whether the hours spent watching TV includes time spent watching online streaming services on other types of devices such as phones and computers.\n\nb) Redesigning the questionnaire will mean addressing all of the points listed above.\u00a0The following questionnaire is just an example.\n\nAdditionally, there are a couple of other important things to consider when writing questionnaires.\n\nLeading questions \u2013 e.g. \u201chow amazing was the last Star Wars film?\u201d \u2013 this could pressure the person into answering positively, adding Bias into the answer.\n\nPersonal questions \u2013 e.g \u201cdo you have a criminal record?\u201d might make the person feel uncomfortable and not answer truthfully.\n\nLevel 4-5GCSE\n\n## Example Questions\n\nThe first criticism is that Tilly\u2019s question is a leading question \u2013 she leads people into agreeing with her opinion that the new government will be a disaster.\n\nThe second criticism is that there are not enough options \u2013 somebody might have no opinion on the matter, or they could be neutral about it.\n\nHere is an example of an improved question:\n\nThe first criticism is that there are crossovers between the options \u2013 if I spend $\u00a330$ on food each week I won\u2019t know whether to tick the first or the second box.\n\nThe second criticism is that there are not enough options \u2013 there is no suitable box to tick for someone who spends more than $\u00a3120$ on food every week.\n\nBecause he is only asking people at the end of his street, Saru will probably get answers which are lower than if he asked this question at the end of a different street where the houses were bigger\/more expensive, therefore his survey will probably be biased.\n\n(a)\n\n\u2022 although it does specify the time frame the results will depend on the time of day the person is completing the questionnaire.\n\u2022 overlapping response boxes. Someone who has taken $5000$ steps could tick two of the responses.\n\n(b)\n\nHere the question is framed better as it is independent of when the form is completed and asks for a value people are more likely to know the answer to on the spot. Also the response boxes no longer overlap. Other improvements could include a response box for those who do not know.\n\nThere needs to be some time frame referenced in the question otherwise people will answer over varying time frames. Someone could respond with the amount the receive per week and someone else could give their pocket money amount over the course over year.\n\nIncluding a option for zero and having no overlapping response boxes is also important.\n\nThe new questionnaire should look something like:\n\nThe original questionnaire has very subjective answer that are qualitative rather than quantitative.\n\nA better questionnaire has the options with a specified number of average visits over a certain time frame. This will gather much more useful data for the manager of the cinema.\n\nLevel 1-3GCSE\n\nLevel 4-5GCSE\n\n## Worksheet and Example Questions\n\n### (NEW) Data and Questionnaires Exam Style Questions - MME\n\nLevel 4-5NewOfficial MME\n\nLevel 4-5\n\nLevel 4-5\n\n## You May Also Like...\n\n### GCSE Maths Revision Cards\n\nRevise for your GCSE maths exam using the most comprehensive maths revision cards available. These GCSE Maths revision cards are relevant for all major exam boards including AQA, OCR, Edexcel and WJEC.\n\n\u00a38.99\n\n### GCSE Maths Revision Guide\n\nThe MME GCSE maths revision guide covers the entire GCSE maths course with easy to understand examples, explanations and plenty of exam style questions. We also provide a separate answer book to make checking your answers easier!\n\nFrom: \u00a314.99","date":"2022-07-06 01:38:43","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 8, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.34024471044540405, \"perplexity\": 1203.9912555382957}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-27\/segments\/1656104655865.86\/warc\/CC-MAIN-20220705235755-20220706025755-00259.warc.gz\"}"} | null | null |
\section{Analysis}
We gather U.S. mortality rates from www.cdc.gov$^2$ and aggregate search data of U.S. citizens from www.google.com$^3$. The mortality data is annual, and Google search data is weekly, so in order to make Google search data annual, we take the yearly average. Then, we want to measure the change in mortality correlated with change in average search volume, because we want to know if next years mortality rates will increase or decrease, and by how much. We then take logs of the differenced data since it is a monotone transformation and reduces variance in the data (the purpose of this paper is to show if correlations exist - further methods can be employed to find how strong they are). \\
\begin{figure}[t]
\centering
\includegraphics[width=3in]{figure1.pdf}
\caption{Shows cross correlation function plotted at different lags for all races both sexes and the word hate. The blue lines are significance intervals at $\alpha=0.1$. The line outside of the significant band at $\Delta t=-1$ shows that there is a significant correlation at lag $\Delta t=-1$.}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[width=3in]{figure2.pdf}
\caption{The significant correlation at $\Delta t=-1$ from figure 1 suggests a correlation between how mortality moves and how hate moves one year in the future. The time series plot checks out with figure 1 and suggests that hate should increase at time 6 to 7, since mortality rate increased from time 5 to 6. }
\end{figure}
Table 1 shows where we found significant cross-correlations and at what lag. An x denotes that there were no significant lags. The significant lags at $\Delta t=-1$ for all people and whites for foreclosure and hate suggests that mortality rates can predict how much people search for those words. The black female data had very strange results, which may suggest that their mortality data in the period that we have is an outlier compared to how their mortality rates normally behave. The only positive lag correlation was with black females and "ship," which, in theory, could suggest that when more people are searching for "ship," more black females are dying, but since the black female data produced strange results, it is most likely that their data is unreliable. More data should be needed to help with this claim. \\
Unfortunately there are no other significant lags with a positive value, suggesting that none of these words can help predict mortality rates. On the other hand, since for both "foreclosure" and "hate" we found significance at lag $\Delta t=-1$, it may suggest psychological or sociological evidence into how humans react to financial failure and death. Perhaps more people search for "hate" when white people die, but a black person's death has no effect on how often the word "hate" is searched for.
\begin{table}[t]
\centering
\begin{tabular}{|l||c|c|c|c|c|c|c|c|c|}
\hline
Word & All & \male & \female & White & Black & White \male & White \female & Black \male & Black \female \\
\hline
Marijuana & x & x & x & x & x & x & x & x & x\\
\hline
Alcohol & x & x & x & x & x & x & x & x & x\\
\hline
Debt & x & x & x & x & x & x & x & x & x\\
\hline
Foreclosure & $-1^-$ & $-1^-$ & $-1^-$ & $-1^-$ & $-1^-$ & x & x & x & x\\
\hline
Bunnies & x & x & x & x & x & x & x & x & $-3^-$\\
\hline
Stress & x & x & x & x & x & x & x & x & x\\
\hline
Hate & $-1^+$ & $-1^+$ & $-1^+$ & $-1^+$ & $-1^+$ & $-1^+$ & x & x & $-3^-$\\
\hline
Ship & x & x & x & x & x & x & x & $0^-$ & $1^+$\\
\hline
Obesity & x & x & x & x & x & x & x & x & x\\
\hline
\end{tabular}
\caption{Table indicating which lags produced statistically significant results. \male = male, and \female = female. Where applicable, the superscripts indicate the sign of the correlation.}
\end{table}
\section{Limitations and Suggested Improvements}
The data was very limited which caused many problems. Mortality data was annual, and the latest data point was for 2010. Google search data only went back to 2004, so we had 7 points of data, and differencing these gave us only 6 changes. We are also doing multiple testing at once, so we should account for some sort of test correction to handle the probability of type I error, such as the Bonferroni test correction. There are essentially 81 tests being performed, however, so in order to get some preliminary results we forego these test corrections, as the purpose of this paper is to demonstrate how this method can be used to find correlations and show that there may be significance. In addition, reducing the number of tests is also possible, since we divided up the analysis into several groups races and sexes. \\
This method could be repeated removing outlier data to possibly improve the result. For example, the black women data acted strangely, which may suggest the data we had for this time period did not follow normal mortality trends. If we remove black women from the overall mortality data, we may get more accurate conclusions. \\
In addition, in the future our analytical methods are more practical. For example, 10 years from now we will have more than double the data points we had previously, causing for a huge improvement in accuracy and potential results. In addition, if more detailed mortality data exists, we can test for correlations between certain words and age groups. As an example, we could find positive correlation between people searching for "alcohol" and deaths for people in the age range of 15-25. \\
Although it will probably not be possible, if mortality data was available weekly, it would be tremendous for data analysis, since we will have 52 times as many data points.
\section{Conclusion}
We did find significant evidence at negative lags even though our data provided no significant conclusions for positive lag cross-correlations, This opens two new doors. The first being that maybe we haven't found the right word, and given the correct word we can predict mortality rates. The second being that we did find significant evidence for negative lags, providing psychological or sociological insight as to how mortality rates affect our searching on Google. Further work can definitely be done to find correlations with mortality, but even aside from that it provides more evidence as to how our lives affects Google search data and how Google search data affects our lives.
\section{References}
\begin{enumerate}[1.]
\item Preis, T., D. Reith, and H. E. Stanley. "Complex Dynamics of Our Economic Life on Different Scales: Insights from Search Engine Query Data." \textit{Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences} 368.1933 (2010): 5707-719. Print.
\item \textit{Centers for Disease Control and Prevention.} Centers for Disease Control and Prevention. Web. 27 Apr. 2012. <http://www.cdc.gov/>.
\item "Google Insights for Search." \textit{Google.} Web. 27 Apr. 2012.\\ <http://www.google.com/insights/search/>.
\end{enumerate}
\end{document}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 4,194 |
<?php
/*
* This file is part of Mandango.
*
* (c) Pablo Díez <pablodip@gmail.com>
*
* This source file is subject to the MIT license that is bundled
* with this source code in the file LICENSE.
*/
namespace Mandango\Tests\Cache;
use Mandango\Cache\ArrayCache;
class ArrayCacheTest extends CacheTestCase
{
protected function getCacheDriver()
{
return new ArrayCache();
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 8,092 |
Hans Jesper Helsø (born July 9, 1948), is a former Danish Chief of Defence.
Helsø served his conscription in 1968 and initially became sergeant and officer of the reserve. In 1974 he becomes lieutenant and served the following four years at King's Artillery Regiment. In 1982-83 he is - as a captain - at the General Staff Course in the United States. In 1990 he was appointed Lieutenant Colonel and come back to the artillery as Head of Department for a few years. After a short staff service at the Army Operational Command, he became Head of the King's Artillery Regiment between 1994 and 1996. In 2000-02, he is Chief of the Army Operational Command, and then Chief of Operations and Planning Staffs until his appointment as Chief of Defence. In August 2008 Helsø reached the army's age-limit and was replaced by Tim Sloth Jørgensen.
His service has been supplemented with a few international missions; first in Cyprus (UNFICYP) in 1979 and then UNPROFOR in the Balkans in 1995.
In 2002 - a few months before planned - Helsø had to step in as Chief of Defence as a result of Christian Hvidt's early resignation. His period as Chief of Defence is marked by the Defence moving from being a mobilization defense to be a conscription based professional defense, in which overseas missions form a significant part of the effort. The planning of this change occurred in the Defence's own management and is described in the episode K-notatet in the TV series Magtens Billeder.
After his time as Chief of Defense Helsø engaged in organizations related to the military, including as a member of Soldier Scholarship Award Committee and on the board of the YMCA Soldiers Mission.
Helsø is married and has four children.
Awards and decorations
References
General Hans Jesper Helsø's CV
Living people
1948 births
Danish generals
Helso, H | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 4,029 |
Q: Why UserPrincipal.Enabled returns different values? I am trying to determine if a user account in AD is enabled. For this I use the following code:
string domain = "my domain";
string group = "my security group";
string ou = "my OU";
//init context
using (var cnt= new PrincipalContext(ContextType.Domain, domain))
{
//find the necessary security group
using (GroupPrincipal mainGroup
= GroupPrincipal.FindByIdentity(cnt, IdentityType.Guid, group))
{
if (mainGroup != null)
{
//get the group's members
foreach (var user in mainGroup.GetMembers()
.OfType<UserPrincipal>()
.Where(u => u.DistinguishedName.Contains(ou)))
{
//ensure that all the info about the account is loaded
//by using FindByIdentity as opposed to GetMembers
var tmpUser= UserPrincipal.FindByIdentity(cnt,
user.SamAccountName);
//actually I could use `user` variable,
//as it gave the same result as `tmpUser`.
//print the account info
Console.WriteLine(tmpUser.Name + "\t" +
tmpUser.Enabled.HasValue + "\t" +
tmpUser.Enabled.Value);
}
}
}
}
The problem is, when I run this code under an administrative account, I get the real result, while when I run it under a non-priviledged account, user.Enabled returns false for some of the accounts, while it should be true.
The only similar q&a I managed to find are
*
*UserPrincipal.Enabled returns False for accounts that are in fact enabled?
*Everything in Active Directory via C#.NET 3.5 (Using System.DirectoryServices.AccountManagement)
which do not help here.
Why is that so? What are my options to get this info under a non-priviledged account?
Here is another approach: How to determine if user account is enabled or disabled:
private bool IsActive(DirectoryEntry de)
{
if (de.NativeGuid == null)
return false;
int flags = (int)de.Properties["userAccountControl"].Value;
if (!Convert.ToBoolean(flags & 0x0002))
return true;
else
return false;
}
Same approach is described in Active Directory Objects and C#.
However when running under an unpriviledged user account, userAccountControl attribute is null and it's not possible to determine the state of the account.
The workaround here is to use PrincipalContext Constructor, specifying the credentials of a user with enough priviledges to access AD.
It stays unclear to me, why the unpriviledged user had access to AD at all, and couldn't get values of some certain account attributes. Probably this has nothing to do with C#, and should be configured in AD...
A: You'll need to delegate permissions in Active Directory for the accounts that will be performing the AD queries. This is what I had to do for my applications to work (though we are performing other administrative tasks on user accounts).
Check Here for instructions on how to delegate permissions(or see blockquote below).
You may referred the following procedure to run the delegation:
*
*Start the delegation of control wizard by performing the following steps:
*
*Open Active Directory Users and Computers.
*In the console tree, double click the domain node.
*In the details menu, right click the organizational unit, click delegate control, and click next.
*Select the users or group to which you want to delegate common administrative tasks. To do so, perform the following steps:
*On the Users or Groups page, click Add.
*In the select Users, computers or Groups, write the names of the users and groups to which you have to delegate control of the organizational unit, click OK. And click next.
*Assign common tasks to delegate. To do so perform the following common tasks.
*On the tasks to delgate page, click delegate the following common tasks.
*On the tasks to delegate page, select the tasks you want to delegate, and click OK. Click Finish
For Example: To delegate administrator to move user/computer objects, you can use advance mode in AD User and Computer and run delegation. It should have write privilege in both OU for the object moving. For writing new values, the administrators account should have delegated values on the user account (Full privilege in specific OU as well.
Something else worth looking into is if the accounts have the userAccountControl attribute. I've heard that accounts missing this attribute may not report correctly. In most scenarios this attribute should be set to NormalAccount.
| {
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} | 107 |
Q: How to best display 30+ bpp graphics on a 24 bpp display? What would be the best way to display the most faithfully graphics that have more than 8 bits per channel on a regular 24 bpp display?
A: The best solution I can think of is based on a random dithering that changes every frame. This combines the advantage of dithering with not having a fixed dithering pattern, and since a given pixel changes values many times a second what you perceive is closer to the average of those various values, which is closer to the original "deep color" value than any given 24 bpp value.
How it looks
A gradient of green, undithered, dithered (10 frames are shown), then both enhanced in the same way for visibility:
The dithering
The dithering is achieved by adding the gamma-compressed deep color value for each channel with a random value, then rounding to the nearest 8-bit value. It would seem natural to use random numbers with a uniform distribution between -0.5 and 0.5 (I'm talking in units that are equivalent to 1 in 8-bit gamma-compressed values, like the difference between 0 and 1 or 254 and 255), however this would result in a sort of banding artifact where the values of a gradient close to an 8-bit value would have little noise whereas values the furthest from any 8-bit value would show a lot more noise. A Gaussian noise is much more suitable as it gives a much smoother noise level. I chose a sigma of 1.0, but for less noise a sigma of 0.8 might do.
You can create a Gaussian PRNG by taking two random numbers, n1 and n2, fitting them each in the [-1 , 1] range, and if they represent a point within the unit circle (if the sum sum of their squares is inferior or equal to 1, otherwise start again) return sqrt(-2. * log(sum) / sum) * n1.
Practical implementation
I chose to implement this by converting a 15 bit per channel linear RGB framebuffer into an 8 bit per channel sRGB framebuffer. The linear to sRGB part is just a detail, I use a lookup table to transform the linear values into gamma-compressed values (I chose to make those intermediate values use 13 bits, you can see it as an 8.5 fixed point notation for sRGB values).
It should go without saying that you're not going to generate a new random Gaussian number for each pixel, you'll want to precalculate a bunch of them and put them in a circular buffer. I chose to make 16384 of them, yes, only 16384, I avoid any repeating patterns by choosing a random entry point in this buffer, a random length to go through (between 100 and 1123, this is pretty arbitrary), and when I reach the end of the length I chose a new random starting point and a new random length. This way I get pretty random non-repeating patterns out of a relatively small buffer of numbers. The numbers in the buffer are stored in 2.5 fixed point format, this way they are all between -4.0 and 4.0 which covers for the range of Gaussian random numbers I want to have. Just make sure to add 0.5 to your random numbers as this will take care of the rounding to the nearest integer later.
Here's basically how it works for each pixel and each channel:
15-bit linear value --via LUT--> 13-bit (8.5 fixed point) gamma-compressed value then ADD 2.5 fixed point random number then SHIFT 5 bits to the right.
Now you get an integer value between -4 and 260, you can use if()s to limit those, but it's much faster to use a 264 element LUT that returns 0 for negative numbers (you can use negative numbers as the index by allocating your buffer then doing buffer = &buffer[4], saves you an addition I guess) and that returns 255 for numbers above 255. Also I use the same random number for each of the three color channels, this avoids chromatic noise, though arguably the result might look somewhat less noisy if those three use independent numbers.
For a single pixel's red channel my code looks like this:
sfb[i].r = bytecheck_l.lutb[lsrgb_l.lutint[fb[i].r] + dither_l.lutint[id] >> 5];
sfb being the sRGB 24 bpp buffer, fb being the 45 bpp linear RGB buffer, lsrgb_l.lutint[] being the linear to gamma-compressed LUT, dither_l.lutint[] the LUT containing the random Gaussian numbers in 2.5 fixed point format and bytecheck_l.lutb[] returning values clipped to [0 , 255].
Performance
I get over 50 FPS in a 1400x820 SDL window with my test gradient using just one core of a 2.4 GHz Core 2 Quad Q6600 and dual channel 800 MHz DDR2 memory, a somewhat mediocre machine by current standards, so this solution seems definitely suitable for modern computers.
Please let me know if any of my explanations require clarifications.
| {
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} | 2,515 |
Recombinant Influenza A H1N1 (A/Beijing/22808/2009) Hemagglutinin / HA Protein (His Tag)
Recombinant Influenza A H1N1 (A/Beijing/22808/2009) Hemagglutinin / HA Protein (His Tag) Antigens The extracellular domain of influenza A virus hemagglutinin (A/Beijing/22808/2009 (H1N1)) (ADD64203.1) (Met 1-Gln 529) was fused with a polyhistidine tag at the C-terminus. The secreted recombinant hemagglutinin of H1N1(A/Beijing/22808/2009)HA (A/Beijing/22808/2009 (H1N1)) comprises 523a.a and has a predicted molecular weight of 59 kDa. As a result of glycosylation, it migrates as an approximately 72 kDa band in SDS-PAGE under reducing conditions.
Formulation: The secreted recombinant hemagglutinin of H1N1(A/Beijing/22808/2009)HA (A/Beijing/22808/2009 (H1N1)) comprises 523a.a and has a predicted molecular weight of 59 kDa. As a result of glycosylation, it migrates as an approximately 72 kDa band in SDS-PAGE under reducing conditions.
Usage: Our products are for research use only. This product is not intended or approved for human, diagnostics or veterinary use.
Molecular Weight: The secreted recombinant hemagglutinin of H1N1(A/Beijing/22808/2009)HA (A/Beijing/22808/2009 (H1N1)) comprises 523a.a and has a predicted molecular weight of 59 kDa. As a result of glycosylation, it migrates as an approximately 72 kDa band in SDS-PAGE under reducing conditions.
Formulation: Recombinant Influenza A H1N1 (A/Beijing/22808/2009) Hemagglutinin / HA Protein was lyophilized from sterile PBS, pH 7.41.
The secreted recombinant hemagglutinin of H1N1(A/Beijing/22808/2009)HA (A/Beijing/22808/2009 (H1N1)) comprises 523a.a and has a predicted molecular weight of 59 kDa. As a result of glycosylation, it migrates as an approximately 72 kDa band in SDS-PAGE under reducing conditions.
Met 1-Gln 529
< 1.0 EU per μg of the protein as determined by the LAL method
Recombinant Influenza A H1N1 (A/Beijing/22808/2009) Hemagglutinin / HA Protein was lyophilized from sterile PBS, pH 7.41.
The extracellular domain of influenza A virus hemagglutinin (A/Beijing/22808/2009 (H1N1)) (ADD64203.1) (Met 1-Gln 529) was fused with a polyhistidine tag at the C-terminus.
The influenza viral Hemagglutinin (HA) protein is a homo trimer with a receptor binding pocket on the globular head of each monomer.HA has at least 18 different antigens. These subtypes are named H1 through H18.HA has two functions. Firstly, it allows the recognition of target vertebrate cells, accomplished through the binding to these cells' sialic acid-containing receptors. Secondly, once bound it facilitates the entry of the viral genome into the target cells by causing the fusion of host endosomal membrane with the viral membrane.The influenza virus Hemagglutinin (HA) protein is translated in cells as a single protein, HA, or hemagglutinin precursor protein. For viral activation, hemagglutinin precursor protein (HA) must be cleaved by a trypsin-like serine endoprotease at a specific site, normally coded for by a single basic amino acid (usually arginine) between the HA1 and HA2 domains of the protein. After cleavage, the two disulfide-bonded protein domains produce the mature form of the protein subunits as a prerequisite for the conformational change necessary for fusion and hence viral infectivity.
Greater than 95% as determined by SDS-PAGE
Asp 18
Our products are for research use only. This product is not intended or approved for human, diagnostics or veterinary use. | {
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Las Elecciones del Distrito Federal de 2000 se llevaron a cabo el domingo 2 de julio del 2000, simultáneamente con las Elecciones federales y en ellas fueron renovados los titulares de los siguientes cargos de elección popular del Distrito Federal:
Jefe de Gobierno del Distrito Federal. Titular del Gobierno del Distrito Federal, con funciones intermedias entre un Alcalde y el Gobernador de un Estado, electo por primera vez para un periodo de seis años no reelegibles en ningún caso. El candidato electo fue Andrés Manuel López Obrador.
16 Jefes Delegacionales. Titulares de cada una de las delegaciones políticas, equivalentes a los Municipios en el Distrito Federal.
66 Diputados a la Asamblea Legislativa. 40 elegidos por mayoría relativa en cada uno de los Distritos Electorales y 26 electos por el principio de representación proporcional mediante un sistema de listas.
Resultados electorales
Presidente de la república
Jefe de Gobierno
Jefes delegacionales
Azcapotzalco
Margarita Saldaña Hernández
Coyoacán
María Rojo
Gustavo A. Madero
Joel Ortega Cuevas
Diputados
Diputados Electos por el principio de Mayoría Relativa
Diputados Electos por el principio de Representación Proporcional
Véase también
Elecciones estatales de México
Elecciones federales de México de 2000
Referencias
Distrito Federal
2000 | {
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} | 8,638 |
«Черинг Кросс Роуд, 84» () — британо-американский кинофильм 1987 года. Экранизация пьесы Джеймса Руза-Эванса, поставленной по мемуарам Хелен Ханфф. Премия BAFTA.
Сюжет
Действие фильма происходит в Лондоне и Нью-Йорке и начинается в 1949 году. Элен Ханфф — американская писательница, увлекающаяся коллекционированием старых книг. Она просматривает объявления о продаже старых изданий и однажды откликается на объявление о продаже раритетной книги в магазине, который находится в буквальном смысле за океаном, а именно в Лондоне. Так начинается двадцатилетний роман в письмах между американкой и британцем, заведующим книжным магазином.
Многолетняя переписка переходит в глубокие и тёплые взаимные чувства. Однако, увидеть друг друга им так и не было суждено. В 1968 году Фрэнк Доэл умирает. В 1971 году Элен приезжает в Лондон и навещает опустевший книжный магазин Фрэнка.
Актёры
Энн Бэнкрофт — Элен Ханфф
Энтони Хопкинс — Фрэнк Доэл
Джуди Денч — Нора Доэл
Морис Денем — Джордж Мартин
Элеанор Дэвид — Сесили Фарр
Мерседес Рул — Кей
Даниил Джеролл — Брайан
Венди Морган — Меган Уэллс
Иэн Макнис — Билл Хамфрис
Дж. Смит-Камерон — Джинни
Премии и номинации
1988 премия BAFTA:
лучшая актриса (Энн Бэнкрофт)
Номинации
Лучшая актриса второго плана, лучший адаптированный сценарий
1987 фильм участник конкурсного показа ММКФ:
приз лучшему актёру (Энтони Хопкинс)
Ссылки
Примечания
Экранизации пьес
Фильмы-мелодрамы Великобритании
Фильмы-мелодрамы США
Фильмы-биографии США
Фильмы о писателях
Фильмы-биографии Великобритании
Фильмы Columbia Pictures
Фильмы Brooksfilms
Фильмы на английском языке
Фильмы Дэвида Хью Джонса | {
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Il tolperisone è un derivato della piperidina. Il composto agisce come muscolo rilassante ad azione centrale. Viene commercializzato, tra gli altri, con i marchi di Mydocalm o Muscodol in diversi paesi asiatici e europei nella forma farmaceutica di compresse contenenti 50 mg oppure 150 mg di principio attivo.
Farmacodinamica
Il tolperisone è un composto che svolge un'azione muscolo rilassante centrale agendo a livello della formazione reticolare del tronco cerebrale, bloccando i canali del sodio e del calcio ad alto voltaggio. Tolperisone causa un'incidenza relativamente bassa di sedazione rispetto ad altri miorilassanti ad azione centrale. Il suo esatto meccanismo d'azione rimane tuttavia non completamente compreso.
Farmacocinetica
A seguito di somministrazione per via orale la molecola viene assorbita in modo rapido e pressoché completo dal tratto gastroenterico. La concentrazione plasmatica di picco (Cmax) viene raggiunta a distanza di 1,5 ore circa dall'assunzione. Una volta assorbito il composto è ampiamente metabolizzato a livello della ghiandola epatica e del rene. L'eliminazione avviene prevalentemente attraverso l'emuntorio renale in due fasi: la prima con un'emivita di circa 2 ore e la seconda che si caratterizza per un'emivita di circa 12 ore.
Usi clinici
Il farmaco trova indicazione nel trattamento di tutte le patologie che si caratterizzano per un aumento patologico del tono muscolare, generalmente a seguito di malattie neurologiche (lesioni del tratto piramidale, sclerosi multipla, mielopatie, encefaliti, stroke), di paralisi spastica e di altre forme di encefalopatia che si associano a distonia.
Viene anche utilizzato per il trattamento degli spasmi muscolari, particolarmente in soggetti che soffrono di disturbi dolorosi di origine muscolo-scheletrica, da patologie a carico della colonna vertebrale (spondilosi, spondiloartrosi, sindromi cervicali e lombari) oppure delle grandi articolazioni (periartrite scapolo-omerale, artrosi e artriti, mialgie in genere come la sindrome fibromialgica o la cefalea muscolotensiva).
Effetti collaterali e indesiderati
Il tolperisone è generalmente ben tollerato. In corso di terapia sono stati comunque segnalati alcuni effetti collaterali quali astenia e debolezza muscolare, cefalea, ipotensione arteriosa, dispepsia, sensazione di bocca secca, nausea, vomito, diarrea, eccessiva sonnolenza diurna. Molti di questi effetti indesiderati scompaiono con la semplice riduzione del dosaggio di farmaco. Da segnalare anche alcune rare reazioni da ipersensibilità che includono rash cutaneo, prurito, edema di Quincke e, molto rararamente, shock anafilattico.
Controindicazioni
Il tolperisone è controindicato nei soggetti con ipersensibilità nota al principio attivo, a molecole chimicamente correlate oppure ad uno qualsiasi degli eccipienti contenuti nella formulazione farmaceutica. Un'ulteriore controindicazione è rappresentata dagli individui affetti da miastenia grave (patologia che già di per sé tende a determinare una marcata debolezza muscolare). Il composto non è stato adeguatamente studiato per l'assunzione in età pediatrica. Pertanto la somministrazione a bambini con meno di 15 anni di età è indicata solo sotto stretta sorveglianza medica e quando i benefici attesi superino i rischi potenziali.
Dosi terapeutiche
Nei soggetti adulti il dosaggio consigliato di tolperisone è pari a 300–450 mg al giorno, suddivisi in 2-3 somministrazioni. In casi particolari e a seconda della risposta clinica il medico può ritenere di ricorrere a una dose più elevata (600 mg/die in 4 somministrazioni). Nei soggetti anziani in genere si ricorre prudenzialmente a dosaggi inferiori rispetto a quelli usuali.
Gravidanza e allattamento
Gli studi sperimentali di riproduzione disponibili eseguiti sugli animali non suggeriscono alcun effetto teratogeno reale o potenziale attribuibile a tolperisone. Non si dispone, tuttavia, di studi specifici effettuati su donne incinte. La molecola non è perciò indicata per il trattamento in gravidanza (in particolare nel primo trimestre) o in donne che potrebbero essere in stato gravido. Allo stato attuale delle conoscenze non è noto se il composto venga secreto nel latte materno. Per questo motivo il farmaco non deve essere usato nelle donne che allattano al seno ed in quelle in cui è necessario somministrare eperisone l'allattamento deve essere interrotto.
Sovradosaggio
In caso di overdose, volontaria o accidentale esiste il rischio potenziale di ipotonia muscolare, che può interessare anche la muscolatura respiratoria. Nei bambini sono stati notati anche segni di eccitazione e agitazione psicomotoria.
Note
Altri progetti
Chetoni aromatici
Miorilassanti
Piperidine | {
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No two days at the markets are ever the same. With a plethora of vendors, shoppers and merchandise the dynamics are constantly changing and evolving. By expecting the unexpected, you can be pleasantly surprised to find that a day at the flea markets is always filled with fun!
a.) Find a deal: shopping at the markets, you never know what you're going to find but you're sure to find a deal. The markets are an unconventional way of shopping that allows room for the art of negotiation. Don't be afraid to offer up a fair price and you'll probably find a good deal.
b.) Find a treasure: a long lost childhood toy, a vintage dress that looks just like your grandma's on her wedding day, an authentic piece of art by your favorite artist. At the markets, there's always a treasure worth finding if you're willing to make the trip. So why not go and explore the markets today?
c.) Find a lifelong friend: ever run into the same people on the subway platform each morning? You're both waiting for the subway to go to different places but at the same time. At the markets you'll run into some of the same people shopping at the markets, at the same time, for different things. You never know if one of these encounters will spark a new friendship. Also, seeing your favorite regular vendors you'll find that they too become like old friends that you see each weekend.
Stay tuned for more Shop Like a Pro Tips! Have a great tip that you'd like to submit? Email us at info@hellskitchenfleamarket.com and share it with us! | {
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\section{Introduction}
\import{Sections/}{Intro.tex}
\section{Preliminaries}
\import{Sections/}{Preliminaries.tex}
\section{Exotic aromatic trees and forests}
\import{Sections/}{Definition.tex}
\section{Analysis of invariant measure order conditions using exotic aromatic forests}
\label{section:using_eat}
In this section, we show how the framework of Section \ref{section:exotic_aromatic_forests} applies for the study of order conditions for the invariant measure of numerical integrators.
\subsection{Weak Taylor expansion using exotic aromatic forests}
\import{Sections/}{Expectation.tex}
\subsection{Integration by parts of the exotic aromatic forests}
\import{Sections/}{IPP.tex}
\subsection{Order conditions using exotic aromatic forests}
\import{Sections/}{IPP_Bseries.tex}
\section{Construction of high order integrators}
\label{section:construction_high_order_integrators}
\subsection{Improvement of a method order via a modified equation}
\import{Sections/}{High_order_integrators.tex}
\subsection{Order conditions for stochastic Runge-Kutta schemes}
\import{Sections/}{Runge-Kutta.tex}
\subsection{Order conditions for postprocessed integrators}
\import{Sections/}{Postprocessed_integrators.tex}
\subsection{Order conditions for partitioned methods}
\import{Sections/}{Partitioned_methods.tex}
\subsection{Non-reversible perturbation}
\import{Sections/}{Antisymmetric_perturbation.tex}
\bigskip
\noindent \textbf{Acknowledgements.}\
The authors would like to thank Hans Munthe-Kaas and Olivier Verdier for helpful discussions about an earlier version of this work.
This work was partially supported by the Swiss National Science Foundation, grants No. 200020\_144313/1 and 200021\_162404.
\bibliographystyle{abbrv}
\subsection*{Isometric equivariance of $\Phi(f)=\Id+f'(\Delta f)$}
We saw the $\Or_d(\R)\ltimes \R^d$-equivariance of $\Phi$ is the same as the $\Or_d(\R)\times \{0\}$-equivariance of $\Phi_0(f)=f'(\Delta f)$.
We take $A=(a_{i,j})_{i,j}\in \GL_d(\R)$, then
$$f'(\Delta f)=\sum_i \sum_j \partial_{x_j,x_j} f_i. \partial_{x_i} f.$$
We develop $\Phi_0 (A*f)$.
\begin{align*}
\Phi_0 (A*f)_{j_0}(x)&=\sum_i \sum_j \partial_{x_j,x_j} (A*f)_i . \partial_{x_i} (A*f)_{j_0}\\
&=\sum_i \sum_j \left( \sum_{k'} \sum_{q_1,q_2} a_{i,k'} a_{j,q_1} a_{j,q_2} \partial_{x_{q_1},x_{q_2}} f_{k'} (A^{-1}x) \right)\\
&\quad \times \left( \sum_k \sum_p a_{j_0,k} a_{i,p} \partial_{x_p} f_k (A^{-1}x)\right)\\
&=\sum_p \sum_{q_1,q_2} \sum_{k'} \left(\sum_i a_{i,k'} a_{i,p}\right)\times \left(\sum_j a_{j,q_1} a_{j,q_2}\right)\\
&\quad \times \left(\partial_{x_{q_1},x_{q_2}} f_{k'} (A^{-1}x) \sum_k a_{j_0,k} \partial_{x_p} f_k (A^{-1}x)\right)\\
&=\sum_p \sum_{q_1} \partial_{x_{q_1},x_{q_1}} f_{p} (A^{-1}x) \sum_k a_{j_0,k} \partial_{x_p} f_k (A^{-1}x)\\
&=(A\Phi_0 f)_{j_0}(A^{-1}x)
\end{align*}
Thus $\Phi_0$ is isometric equivariant.
\subsection{Aromatic trees and forests}
\label{section:aromatic_forests}
We first consider directed graphs $\gamma=(V,E)$ with $V$ a finite set of nodes and $E\subset V\times V$ the set of directed edges. If $(v,w)\in E$, we say that the edge is going from $v$ to $w$, and $v$ is called a predecessor of $w$.
Two directed graphs $(V_1,E_1)$ and $(V_2,E_2)$ are equivalent if there exists a bijection $\varphi: V_1\rightarrow V_2$ with $(\varphi\times\varphi) (E_1)=E_2$.
For brevity of notation, to avoid drawing arrows on the forests, an edge linking two nodes goes from the top node to the bottom one. If there is an eventual cycle, the arrows on it are going in the clockwise direction. For example, $$\includegraphics[scale=0.5]{Other_trees/cycles.eps}=\includegraphics[scale=0.5]{Other_trees/cycles2.eps}.$$
We call aromatic forests the equivalence classes of directed graphs where each node has at most one outgoing edge.
The connected components making an aromatic forest are called aromatic trees.
According to the above definition, there are two types of trees:
\begin{itemize}
\item \emph{aromas} are aromatic trees\footnote{Such graphs with one cycle are not strictly speaking ``trees'', they are however called aromatic trees in the literature as an analogy with carbon chemistry.} with exactly one cycle: $\etree 1 2 0 1$, $\etree 2 2 0 1$, $\etree 5 2 0 1$, $\etree 5 2 0 2$, \dots
\item \emph{rooted trees} do not have a cycle ; they have a unique node that has no outgoing edge and that is called the root, graphically represented at the bottom: $\etree 1 1 0 1$, $\etree 2 1 0 1$, $\etree 3 2 0 1$, $\etree 3 1 0 1$, \dots
\end{itemize}
Thus, an aromatic forest is a collection of aromas in addition to at most one rooted tree. We call $\AA\TT$ the set of aromatic forests containing exactly one rooted tree, and we name its elements the aromatic rooted forests.
\begin{definition}[Elementary differentials]
\label{Elementary_differentials}
Let $\gamma=(V,E)\in \AA\TT$, $f: \R^d \rightarrow \R^d$ a smooth function. We denote $\pi(v)=\{w\in V, (w,v)\in E\}$ the set of all predecessors of the node $v\in V$ and $r$ the root of $\gamma$. We also call $V^0=V\smallsetminus\{r\}=\{v_1,\dots ,v_m\}$ the other nodes of $\gamma$.
Finally we introduce the notations $I_{\pi(v)}=(i_{q_1},\dots ,i_{q_s})$ where the $q_k$ are the predecessors of $v$, and $$\partial_{I_{\pi(v)}} f=\frac{\partial^s f}{\partial x_{i_{q_1}}\dots \partial x_{i_{q_s}}}.$$
Then $F(\gamma)$ is defined as $$F(\gamma)(f)=\sum_{i_{v_1},\dots ,i_{v_m}=1}^d \left(\prod_{v\in V^0} \partial_{I_{\pi(v)}} f_{i_v}\right)\partial_{I_{\pi(r)}} f.$$
\end{definition}
\begin{ex*}
Let $\gamma=\includegraphics[scale=0.5]{Other_trees/exemplediff.eps}$ and $\widetilde{\gamma}=\includegraphics[scale=0.5]{Other_trees/exemplediff2.eps}$
where we added indices to apply the formula of Definition \ref{Elementary_differentials}. Note that there is no index for the root.
Then the associated differentials are respectively
$F(\gamma)(f)=\sum_{i,j,k,l,m=1}^d \partial_{m} f_{m} f_i \partial_{i} f_{j} f_k \partial_{j,k}f_l \partial_l f=\Div(f)\times f'f''(f'f,f)$ and $F(\widetilde{\gamma})(f)=\sum_{i,j,k,l,m=1}^d \partial_{l} f_{m} \partial_{m,k} f_{l} f_k f_i \partial_{i} f_{j} \partial_{j}f=\sum_{m=1}^d f'_m((\partial_m f)'(f)) \times f'f'f$.
\end{ex*}
\subsection{Exotic aromatic trees and forests}
\label{section:eat}
We now introduce a new kind of edge, called lianas, for the aromatic forests. The corresponding generalization is called exotic aromatic forests.
Let $(V,E)$ be an aromatic forest and $L$ be a finite list of pairs of elements of $V$ (possibly with duplicates), then $\gamma=(V,E,L)$ is an exotic aromatic forest.
The elements of $L$ are called lianas and correspond to non-oriented edges between any two nodes of the forest. We graphically represent them with a dashed edge linking the two given nodes. As we authorize duplicates, there can be several lianas between two given nodes. Also lianas can link a node to itself.
For a node $v$, $\Gamma(v)$ denotes the list of the lianas (also with possible duplicates) linked to $v$. The predecessors of $v$ only take in account the edges of $E$.
An exotic aromatic tree of an exotic aromatic forest $\gamma=(V,E,L)$ is a connected component of the associated aromatic forest $(V,E)$.
We call $\EE\AA\TT$ the set of exotic aromatic forests with exactly one rooted tree, and name its elements exotic aromatic rooted forests.
\begin{ex*}
The lianas can link different trees of an aromatic forest and thus yield an exotic aromatic forest.
For instance, linking the aroma $\etree 1 2 0 1$ and the rooted tree $\etree 5 3 0 1$ gives $\includegraphics[scale=0.5]{Other_trees/exempleeat.eps}$.
\end{ex*}
The definition of elementary differentials is extended as follows.
\begin{definition}
\label{elementary_differential}
Let $\gamma=(V,E,L)\in \EE\AA\TT$, $f: \R^d \rightarrow \R^d$ a smooth function. We name $r$ the root of $\gamma$ and $V^0=V\smallsetminus\{r\}=\{v_1,\dots ,v_m\}$ the other nodes of $\gamma$. We denote $l_1$,\dots ,$l_s$ the elements of $L$ and for $v\in V$, $J_{\Gamma(v)}$ the multiindex $(j_{l_{x_1}},\dots ,j_{l_{x_t}})$ where $\Gamma(v)=\{l_{x_1},\dots ,l_{x_t}\}$.
Then $F(\gamma)$ is defined as $$F(\gamma)(f)=\sum_{i_{v_1},\dots ,i_{v_m}=1}^d \sum_{j_{l_1},\dots ,j_{l_s}=1}^d \left(\prod_{v\in V^0} \partial_{I_{\pi(v)}} \partial_{J_{\Gamma(v)}} f_{i_v}\right) \partial_{I_{\pi(r)}} \partial_{J_{\Gamma(r)}} f.$$
\end{definition}
\begin{exs*}
The differential that corresponds to the rooted tree $\etree 1 1 1 1$ with a single node and a single liana is $F(\etree 1 1 1 1)(f)=\Delta f$.
We can also represent as exotic aromatic forest more complicated derivatives. For instance, let $\gamma=\includegraphics[scale=0.5]{Other_trees/exempleeat2.eps}$,
then
$$F(\gamma)(f)=\sum_{i,j,k=1}^d \Div(\partial_i f)\times f'((\partial_{kl} f)'(f''(\partial_{ijj} f,\partial_{kl} f))).$$
\end{exs*}
\subsection{Grafted exotic aromatic trees}
For the study of the order for the invariant measure of numerical integrators, we introduce an extension of exotic aromatic forests. The root now symbolizes a test function $\phi$, and it has leafs (nodes without predecessors) that represent a random standard normal vector $\xi$. Note that these new trees can be seen as bi-coloured trees in the context of P-series (see \cite[Chap.\thinspace 3]{Hairer06gni}), where the nodes represented with crosses cannot have predecessors.
\begin{definition}
\label{definition:grafted_trees}
A grafted node is a new type of node graphically represented by a cross.
Let $V$ be a set of nodes whose subset of grafted nodes is $V_g$, $E$ a set of edges such that each node in $V_g$ has exactly one outgoing edge and no ingoing edge, and $L$ a set of lianas that link nodes in $V\smallsetminus V_g$, then $\gamma=(V,E,L)$ is a grafted exotic aromatic forest.
We define as before the grafted exotic aromatic trees and grafted exotic aromatic rooted forests, that we denote $\EE\AA\TT_g$.
If $\gamma=(V,E,L)$ is a grafted exotic aromatic rooted forest, $\phi: \R^d\rightarrow \R$ a smooth function, and $\xi$ a random vector of $\R^d$ whose components are independent and follow a standard normal law, the associated elementary differential of $\gamma$ is, with the same notations as Definition \ref{elementary_differential} and $V^0=V\smallsetminus (V_g\cup \{r\})$,
$$F(\gamma)(f,\phi,\xi)=\sum_{i_{v_1},\dots ,i_{v_m}=1}^d \sum_{j_{l_1},\dots ,j_{l_s}=1}^d \left(\prod_{v\in V^0} \partial_{I_{\pi(v)}} \partial_{J_{\Gamma(v)}} f_{i_v}\right)
\left(\prod_{v\in V_g} \xi_{i_v}\right) \partial_{I_{\pi(r)}} \partial_{J_{\Gamma(r)}} \phi.$$
\end{definition}
\begin{ex*}
The differential associated to the forest $\ctree 2 1 2 1$ is $F(\ctree 2 1 2 1)(f,\phi,\xi)=\phi'(f''(\xi,\xi))$.
\end{ex*}
If $\gamma$ is such that $V_g$ is empty, we recover the exotic aromatic forests of Definition \ref{elementary_differential}, where $\phi$ is replaced by $f$. For the rest of the paper (except Section \ref{section:isometric_equivariance}), we update the definition of the elementary differential of an exotic aromatic forests so that the root is associated to the function $\phi$.
This definition can be straightforwardly extended on non-rooted exotic aromatic forests.
For brevity of notations, we also write $F(\gamma)(\phi)$ instead of $F(\gamma)(f,\phi,\xi)$. We note that $\phi \to F(\gamma)(\phi)$ is a linear differential operator (dependent of $f$ and $\xi$).
\subsection{Grafted exotic aromatic B-series}
In this section, we adapt the formalism of aromatic B-series of \cite{MuntheKaas16abs} to grafted exotic aromatic forests, in order to use it as a numerical tool for weak Taylor expansions in the next sections.
We define the order $\abs{\gamma}$ of a tree $\gamma \in \EE\AA\TT_g$.
We denote $N(\gamma)$ the number of nodes, $N_l(\gamma)$ the number of lianas, $N_c(\gamma)$ the number of grafted nodes and $N_v(\gamma)=N(\gamma)-N_c(\gamma)-1$ the number of nodes that are non grafted and different from the root, then
$$\abs{\gamma}=N_v(\gamma)+N_l(\gamma)+\frac{N_c(\gamma)}{2}.$$
\begin{definition}
Let $a: \EE\AA\TT_g\to \R$ a map, $f: \R^d \rightarrow \R^d$ and $\phi: \R^d \rightarrow \R$ two smooth functions, then the grafted exotic aromatic B-series $B(a)(\phi)$ is a formal series indexed over $\EE\AA\TT_g$ defined by
$$B(a)(\phi)=\sum_{\gamma \in \EE\AA\TT_g} h^{\abs{\gamma}} a(\gamma) F(\gamma)(\phi).$$
We extend the definition of $F$ on $\Vect(\EE\AA\TT_g)$ by writing
$$F\bigg(\sum_{\gamma \in \EE\AA\TT_g} h^{\abs{\gamma}}a(\gamma) \gamma\bigg)(\phi)=B(a)(\phi).$$
\end{definition}
The variable $h$ is formal and thus can be chosen to be equal to 1.
If the series is indexed only on (exotic) aromatic rooted forests, then it is called an (exotic) aromatic B-series. In Section \ref{section:isometric_equivariance}, we shall focus on exotic aromatic B-series.
\begin{remark}
The coefficients $a(\gamma)$ of standard B-series are sometimes renormalized as $\frac{a(\gamma)}{\rho(\gamma)}$ where $\rho$ is a function determined by the symmetries of the associated forest. If $\rho$ is appropriately chosen, it simplifies greatly the composition laws of (aromatic) B-series (see \cite{Hairer06gni,Chartier10aso,Bogfjellmo15aso}).
Finding the best definition of $\rho$ for this exotic extension of B-series is out of the scope of this paper.
\end{remark}
\subsection{Isometric equivariance of exotic aromatic rooted forests}
\label{section:isometric_equivariance}
In this subsection, we show that the exotic aromatic B-series satisfy an isometric equivariance property in the spirit of \cite{MuntheKaas16abs, McLachlan16bsm}.
We consider exotic aromatic rooted forests $\gamma$ where the differential associated to the root is $f$. As the function $f$ is no longer fixed, we denote the associated differential $F(\gamma)(f)$. Also we adapt the definition of exotic aromatic B-series to this change.
First we add a new tree: the empty tree $\varnothing$.
The function $F$ is then extended on $\EE\AA\TT_g\cup\{\varnothing\}$ by $F(\varnothing)(f)=\Id_{\R^d}$.
Then, for a function $a:\EE\AA\TT\cup\{\varnothing\}\rightarrow \R$, the associated exotic aromatic B-series is
$$B(a)(f)=\sum_{\gamma \in \EE\AA\TT\cup\{\varnothing\}} a(\gamma) F(\gamma)(f).$$
We study (exotic) aromatic B-series $B(a)$ with $a(\varnothing)=1$. We call these (exotic) aromatic B-series methods.
Let $G$ be a subgroup of $\GL_d(\R)\ltimes \R^d$, the action of an element $(A,b)\in G$ on $\R^d$ is
$x\mapsto Ax+b$,
and the action on a vector field $f:\R^d \to \R^d$ is
$$((A,b)*f)(x):=Af(A^{-1}(x-b)).$$
We simplify the notations by writing $A*f:=(A,0)*f$.
We recall the definition of equivariance from \cite{MuntheKaas16abs}. The property of equivariance means the method is unchanged by an affine coordinate transformation.
Let $\Phi$ be a differential operator and $G$ a subgroup of $\GL_d(\R)\ltimes \R^d$, then $\Phi$ is called $G$-equivariant if
$$\forall (A,b)\in G, \forall f\in \CC^\infty(\R^d,\R^d), \Phi((A,b)*f)=(A,b)\circ\Phi(f)\circ (A,b)^{-1}.$$
In particular, $\Phi$ is said affine equivariant if $G=\GL_d(\R)\ltimes \R^d$ and isometric equivariant if $G=\Or_d(\R)\ltimes \R^d$.
\begin{theorem}
\label{theorem:equivariance}
Consider an exotic aromatic B-series method $B(a)$ (with $a(\varnothing)=1$), then $B(a)$ is isometric equivariant.
\end{theorem}
\begin{remark}
It is proved in \cite{MuntheKaas16abs} that standard B-series methods are
exactly the affine equivariant methods.
Analogously, it would be interesting to characterize the isometric equivariant maps.
\end{remark}
For the sake of brevity, we omit the proof of Theorem \ref{theorem:equivariance}.
The proof can be made in
the spirit of the result \cite[Prop.\thinspace 2.1]{MuntheKaas16abs} for affine equivariant B-series.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 23 |
Turkish lira: Commerzbank analyst asks 'Who'd want to invest in currency with valuation based on fear?'
Veteran populist campaigner Erdogan has been rousing the crowds by saying Turkey will go after rogue bankers he blames for lira woes. He is pictured at a March 25 campaign rally in Agri, eastern Turkey.
By bne IntelliNews March 25, 2019
"Who would want to invest in a currency that's valuation is only based on fear?" a senior Commerzbank analyst reportedly remarked on May 25 as the markets digested Turkey's move to place JPMorgan Chase & Co and other unnamed banks under investigation following the latest slump in the value of the Turkish lira.
On March 24, the day after Turkish watchdogs announced the probes, Turkish President Recep Tayyip Erdogan, facing landmark local elections widely seen as a referendum on his rule on March 31, followed up by warning bankers that there would be "a heavy price" to pay after the polls. Lashing out at bankers whom he sees as driving up demand for hard currency and making misleading predictions on foreign exchange rates, Erdogan warned at an Istanbul election rally: "I am calling on those who are engaging in such activities ahead of the elections. We know the identities of all of you. We know what you are doing."
Ulrich Leuchtmann, head of FX & EM Research at Commerzbank in Frankfurt, was also quoted by Reuters as saying: "With archaic measures of this kind he [Erdogan] will scare away even the last courageous Turkey investor."
March 22 saw the Turkish lira (TRY) weaken 5.42% d/d against the USD to close last week's trading at 5.76. It was the first major weakening of the currency since the lira crisis last summer. However, by around 19:00 on March 25, the TRY had strengthened by 3.33% to 5.57. Whether or not populist Erdogan's menacing words had delivered most of that recovery was certainly on bankers' minds.
Turkey CDS jump 27 bp
March 25 also saw the cost of insuring exposure to Turkey's sovereign debt soaring to a six-month high. Turkey 5-year credit default swaps (CDS) leapt by 27 basis points to 426 bp, IHS Markit data showed. Such a level was last recorded last September after August's big lira selloff.
Turkish money market rates also jumped for a second successive session, Reuters reported. Rates for borrowing 1-month funding in 9-month's time and 3-month funding in 9-month's time hit more than 29%.
Turkey's main interest rate is currently 24%.
Traders on March 25 also took note that Turkey's dollar-denominated bonds were again on the slide. The biggest fall hit the country's 2034 bonds which dropped 1.3 cents in the dollar while other maturities slid between 0.2 and 1 cents according to Tradeweb data.
The Turkish Central Bank started the new trading week by saying that it intended to use all monetary policy and liquidity management instruments to maintain price stability if deemed necessary. It said it was closely monitoring fluctuations and unhealthy price formations following the lira plunge three days ago.
The national lender said recent fluctuations in gross reserves were driven by ordinary transactions and periodic factors. There were no unforeseen incidents, it said, adding that it was decisive about its policy towards reinforcing its reserves. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 965 |
Loop: Fade Out
Artist: Loop
Title: Fade Out
Label: Reactor
A remastered and repackaged re-release of Loop's Fade Out. The album in it's entirety on CD 1, with the bonus tracks (including relevant Peel session material from the time) on CD 2, the bonus material is available on CD formats only. Fade Out was originally released on Chapter 22 Records in late 1988, the Peel session was recorded at that time too: Loop were loosely influenced by bands such as The Velvet Underground, The Stooges, The MC5, but retaining an avant-garde and experimental edge from Can, Faust, Neu!, Rhys Chatham, Glenn Branca and minimalist systems music, to name but a few. As with Heavens End, Fade Out was remastered by Robert Hampson with Kevin Metcalfe at Soundmasters, with the original album on CD 1, and 4 unreleased demo versions, 3 peel session tracks, and the 5 guitar loops(!!!) on CD 2.
1.1 Black Sun
1.2 This Is Where You End
1.3 Fever Knife
1.4 Torched
1.5 Fade Out
1.6 Pulse
1.7 A Vision Stain
1.8 Got to Get It Over
2.1 Black Sun (Feedback)
2.2 Torched (Orig. Mix)
2.3 Got to Get It Over (Orig. Mix)
2.4 This Is Where You End (Demo)
2.5 Pulse (Peel Session)
2.6 This Is Where You End (Peel)
2.7 Collision (Peel Session)
2.8 Fade Out Guitar Loops
2.9 I, II, III, IV, V | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 5,748 |
Home » Music » Concerts » The New York Pops: "Life is a Cabaret: The Songs of Kander and Ebb"
The New York Pops: "Life is a Cabaret: The Songs of Kander and Ebb"
A triumphant celebration of the legendary song writing team with the orchestra's superb musicianship as well as guest artists Tony Yazbeck and Caissie Levy.
Posted on March 14, 2017 by Darryl Reilly in Concerts, Music
Guest artists Tony Yazbeck and Caissie Levy with music director Steven Reineke and The New York Pops presents "Life is a Cabaret: The Songs of Kander and Ebb" (March 10, 2017) (Photo credit: Richard Termine)
Darryl Reilly, Critic
Laughter and applause were contagious as The New York Pops played the dynamic opening notes of the overture from Chicago. This raucous composition was followed by a symphonic medley of songs from that show. It was the delightful opening of the orchestra's triumphant tribute concert, Life is a Cabaret: The Songs of Kander and Ebb.
"I've wanted to put this program together for some time, and now is the perfect occasion," remarked the beaming Steven Reineke, The New York Pops' music director and conductor.
Composer John Kander and lyricist Fred Ebb's legendary musical theater writing collaboration began in 1952 and ended with Mr. Ebb's death in 2004, at the age of 76. The "perfect occasion" is Mr. Kander's 90th birthday that is on March 18th. He currently has a new show, Kid Victory, running Off-Broadway and another new musical in the works, The Beast in The Jungle, adapted from the Henry James novella.
"It looks like when you got your Kennedy Center Honor!" exclaimed Mr. Reineke, as a spotlight shone on John Kander, who was attending the concert from a first tier box at Carnegie Hall. He grinned to a thunderous reaction. Sitting with him, was Susan Stroman, who has directed several Kander and Ebb productions. Near the end of the show, at Reineke's instigation, the house lights went up, and the orchestra and the audience joined in for "Happy Birthday" to Kander.
Guest artist Tony Yazbeck with music director Steven Reineke and The New York Pops presents "Life is a Cabaret: The Songs of Kander and Ebb" (March 10, 2017) (Photo credit: Richard Termine)
The 78-piece orchestra also magnificently played "Hot Honey Rag" from Chicago ,"Gimme Love" from Kiss of The Spiderwoman, and a glorious piece of Americana, "Minstrel March" from The Scottsboro Boys.
Joining The New York Pops as guest artists were the vocalists, Caissie Levy and Tony Yazbeck.
The youthfully suave Mr. Yazbeck has performed on Broadway in Gypsy, Finding Neverland and was nominated for a Tony Award for On The Town. He has played Billy Flynn in the Broadway revival of Chicago, and will soon return to the role. His spirited renditions of "Razzle Dazzle" and "All I Care About" demonstrated that he is ideal for the part.
Yazbeck also scored with compelling versions of the less frequently performed songs "Sometimes a Day Goes By" from Woman of The Year and "You, You, You," from The Visit. He also shone with the gems "City Lights" from The Act, where he superbly tap danced, "Coffee in a Cardboard Cup" from 70, Girls, 70, and "But The World Goes 'Round" from the film New York, New York.
The sleek, lovely and appealing Ms. Levy has appeared on Broadway in Ghost, Hair, Wicked, and Les Misérables. She had the golden opportunity to perform many of Kander and Ebb's great songs for female characters.
Guest artist Caissie Levy with music director Steven Reineke and The New York Pops presents "Life is a Cabaret: The Songs of Kander and Ebb" (March 10, 2017) (Photo credit: Richard Termine)
Levy powerfully closed the show with "Maybe This Time" from Cabaret. She also wonderfully sang the other Sally Bowles' classics "Mein Herr," and "Cabaret," joined by Yazbeck. They appeared together in "Money, Money" with the delirious dance moves.
Her first number was a joyous "Sing Happy" from Flora The Red Menace. There was a sly "Roxie" from Chicago, a stirring "Colored Lights" from The Rink, and the wistful "Everybody's Girl" from Steel Pier.
The rousing encore was, of course, "New York, New York," done as a duet by Yazbeck and Levy, once again accompanied by The New York Pops.
The next New York Pops concert at Carnegie Hall, "You've Got a Friend: A Celebration of Singers and Songwriters," a tribute to the soundtrack of a generation, inspired by the music of James Taylor, Carole King, and more, will take place on Friday, April 21 at 8 PM.
The New York Pops: "Life is a Cabaret: The Songs of Kander and Ebb" (March 10, 2017)
Carnegie Hall, Stern Auditorium, 881 7th Avenue at 57th Street, in Manhattan
For tickets, call 212-247-7800 or visit http://www.carnegiehall.org
For information on The New York Pops, visit http://www.newyorkpops.org
Running time: two hours including one intermission
Caissie Levy
Fred Ebb
John Kander
Life is a Cabaret: The Songs of Kander and Ebb
Richard Termine
Steven Reineke
The New York Pops
Tony Yazbeck
About Darryl Reilly (773 Articles)
A native New Yorker, Darryl Reilly graduated from NYU with a BFA in Cinema Studies. For the Broadway League, (formerly The League of American Theatres and Producers) he developed, and for five years conducted their Broadway Open House Tours, which took visitors through The Theatre District and into several Broadway theaters. He contributed to Broadway Musicals Show by Show: Sixth Edition (Applause Books). Since 2013, he has reviewed theater, cabaret, and concerts for Theaterscene.net.
Charming romantic comedy with a dose of serious problems treated lightly.
The New York Pops: "Find Your Dream: The Songs of Rodgers and Hammerstein"
Glorious New York Pops concert of the ever-popular and melodic songs of Broadway's most successful team sung by guest artists Laura Michelle Kelly and Max von Essen.
The Judas Kiss
British film star Rupert Everett gives a bravura performance as Oscar Wilde in a revival of a talky play by David Hare.
Lady in the Dark
Victoria Clark stars as Liza Elliott, the fashion editor who undergoes psychoanalysis, in the world premiere of a new adaptation by MasterVoices in semi-staged concert.
The Mad Ones
"On the Road" is a touchstone for a girl graduating high school and figuring out her life in this musical that might be best appreciated by young adults. | {
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} | 4,371 |
\section{Introduction}
Reinforcement Learning (RL), originally introduced in the artificial intelligence community \citep{barto}, has recently seen a resurgence in optimal control of dynamical systems through a variety of papers such as \citet{vrabie1,lewis_mag,jiang1,nonlin2,nonlin1,V17} using solution techniques such as adaptive dynamic programming (ADP), actor-critic methods, Q-learning, etc. Curse of dimensionality, however, continues to be an ongoing debate for all of these RL-based control designs. Depending on the size and complexity of the plant, it may take an unacceptably long amount of time to even start the initialization step of RL, let alone control. {Our goal in this paper is to counteract this problem by exploiting certain physical characteristics of the plant dynamics that allow for model reduction so that learning only a reduced-dimensional controller is sufficient for stabilizing the full-dimensional plant. The specific property that we study is singular perturbation (SP). We consider plants whose dynamics are separated into two time-scales. Traditionally, SP theory has been used for model reduction (\citet{SPreduction, chow1985}), and control (\citet{chowslowfast}) of large-scale systems, but only by using full knowledge of the original plant model. Its extension to model-free control using RL has not been addressed. To bridge this gap, we present several sets of RL-based control designs where we exploit the underlying SP property of the plant to learn a controller for only its dominant slow time-scale dynamics, thereby saving significant amount of learning time. We provide sub-optimality and stability results for the resulting closed-loop system.}
The main contributions are as follows. Three distinct RL control designs for singularly perturbed systems are presented. The first design assumes that the slow state variable is either directly measurable, or can be constructed from the measurements of the full state vector. Using this assumption, we develop a modified ADP algorithm which learns a reduced-dimensional RL controller using only feedback from the slow state variables. The controller is shown to guarantee closed-loop stability of the full-dimensional system if the fast dynamics are stable. The second design extends this algorithm to output feedback control using a neuro-adaptive state estimator \citep{obs}. The estimation of full-dimensional states is essential for our design to extract the slow states, in contrast to the time-shifted discrete-time output-feedback designs like \cite{obs_discrete} that uses a combination of inputs and outputs in the control law. The third design shows the relevance of these two designs to SP models of multi-agent consensus networks where time-scale separation arises due to clustering of the network nodes. Along with a centralized design, a variant is proposed that imposes a block-diagonal structure on the RL controller to facilitate its implementation. Numerical results show that our approach saves significant amount of learning time than the conventional RL while still maintaining a modest closed-loop performance. All the designs are described by implementable algorithms together with theoretical guarantees.
The first design has been presented as a preliminary result in our recent conference paper \citet{sayak_cdc}. The second design, however, is completely new. The multi-agent RL controllers, which were presented only for scalar dynamics in \citet{sayak_cdc, sayak_acc}, are now extended to vector-dimensional states. Moreover, unlike prior results, the consensus model here is more generic as we allow each node to have self dynamics. The simulation examples presented in Section $7$ are much larger-dimensional than in \citet{sayak_cdc} to demonstrate the numerical benefits of the designs.
The rest of the paper is organized as follows. The state-feedback and output-feedback RL design problems are formulated in Section $2$, followed by their respective solutions and stability analyses in Sections 3 and 4. Section $5$ and $6$ interprets these designs to multi-agent consensus networks with node clustering, presenting both centralized and block-decentralized RL. Numerical simulations are shown in Section $7$. Concluding remarks are provided in Section $8$. Proofs of theorems and lemmas are presented in the Appendix.
\par
\textbf{Notations:} $\mathbb{RH}_{\infty}$ is the set of all proper, real and rational stable transfer matrices; $\otimes$ denotes Kronecker product; $diag(m)$ is a diagonal matrix with vector $m$ on its principal diagonal; $\bf{1_n}$ denotes a column vector of size $n$ with all ones; $\cup$ denotes union operation of sets; $blkdiag(m_1,\dots,m_n)$ denotes a block-diagonal matrix with $m_1,\dots,m_n$ as its block diagonal elements; $|M|$ denotes the cardinality of set $M$; $\norm{.}$ denotes Euclidean norm of a vector and Frobenius norm of a matrix unless mentioned otherwise.
\vspace{-.4 cm}
\section{Problem Formulation}
Consider a linear time-invariant (LTI) system
\begin{align}
\label{eq:statecompact1}
& \dot{x} = Ax + Bu, \;\; x(0)=x_0,\;
q=\mathcal C x,
\end{align}
where, $x \in \mathbb{R}^n$ is the state, $u \in \mathbb{R}^m$ is the control input, and $q \in \mathbb{R}^p $ is the output. We assume that the matrices $A$ and $B$ are unknown, although the values of $n$, $m$ and $p$ are known. The following assumption is made.
\par
\noindent \textit{Assumption 1:} The system \eqref{eq:statecompact1} exhibits a singular perturbation property, i.e., there exist a small parameter $1 \gg \epsilon > 0$ and a similarity transform $\mathcal{T} = [T^T \; G^T]^T$ such that by defining $y \in \mathbb{R}^r$ and $z \in \mathbb{R}^{n-r}$ as
\begin{align}\label{similarity}
&\begin{bmatrix}
y \\ z
\end{bmatrix} = \mathcal{T}x = \begin{bmatrix}
T \\ G
\end{bmatrix}x,
\end{align}
the state-variable model \eqref{eq:statecompact1} can be rewritten as
\begin{subequations}
\label{eq:SP}
\begin{align}
&\dot{y} = A_{11}y + A_{12}z + B_1u, \; \;y(0)=Tx_0=y_0, \\
& \epsilon\dot{z} = A_{21}y + A_{22}z + B_2u,\; z(0)=Gx_0=z_0,\\
& q= \mathcal C \mathcal T^{-1}\begin{bmatrix}
y \\ z
\end{bmatrix} = C\begin{bmatrix}
y \\ z
\end{bmatrix}.
\end{align}
\end{subequations} \par
In the transformed model \eqref{eq:SP}, $y(t)$ represents the {\it slow} states and $z(t)$ represents the {\it fast} states. Since $A$ and $B$ are unknown, the matrices $A_{11}, A_{12}, A_{21}, A_{22}$, $B_1$ and $B_2$ are unknown as well.
\vspace{-.2 cm}
\subsection{Problem Statement for State-Feedback RL}
\textbf{P1.} \textit{Learn} a control gain $K \in \mathbb{R}^{m \times r}$ for the singularly perturbed system \eqref{eq:SP} without knowing the model using online measurements of $u(t)$ and $y(t)$ such that
\begin{equation}
u(t) = -Ky(t) = -KTx(t) \label{spu}
\end{equation} minimizes
\begin{align}\label{state_J} &J(y(0);u)= \int_0^{\infty} (y^T Q y + u^T R u )dt, \\
&\mbox{s.t.} \;\; A- BKT \in \mathbb{RH}_{\infty}.
\end{align}
We assume $(A,B)$ to be stabilizable. We consider $y(t)$ to be directly measurable, or $x(t)$ to be measurable (i.e. $\mathcal C=I$) and $T$ to be known so that $y(t)$ can be computed at all time $t$. This is not a restrictive assumption as in many SP systems the identity of the slow and fast states are often known a priori \citep{Khalilcontrol} even if the model is unknown. If the system is explicitly represented in form (2), then $ \mathcal{T} = I$, and we assume that the slow variable $y(t)$ is available.
The benefit of using $y(t)$ as the feedback variable is that one has to learn only a $(m\times r)$ matrix instead of a $(m \times n)$ matrix if full state feedback $x(t)$ was used. This will improve the learning time, especially if $r \ll n$. Before proceeding with the control design, we make the following assumption.
\par
\noindent \textit{Assumption 2:} $A_{22}$ in (3b) is Hurwitz. \par
This assumption means that the fast dynamics of \eqref{eq:SP} are stable, which allows us to skip feeding back $z(t)$ in \eqref{spu}.
\vspace{-.2 cm}
\subsection{Problem Statement for Output Feedback RL}
\textbf{P2.} Considering that $q(t)$ is measured and $\mathcal C$ is known, but $A$ and $B$ are both unknown in \eqref{eq:statecompact1}, \textit{estimate} the states $\hat{y}(t),\hat{z}(t)$ (or, equivalently estimate $\hat{x}(t)$ and compute $\hat{y}(t)=T\hat{x}(t)$ assuming that $T$ is known), \textit{learn} a controller $K \in \mathbb{R}^{m\times r}$ using online measurements of $q(t)$ and $u(t)$ such that
\begin{equation}
u = -K\hat{y} = -KT\hat{x}
\end{equation}
minimizes
\color{black}
\begin{align}\label{output_J}
J(y(0);u)= \int_0^{\infty} (y^T Q y + u^T R u )dt.
\end{align}
\color{black}
We assume $(A,B)$ to be stabilizable, and $(A,C)$ to be detectable. Our approach would be to estimate the slow states $\hat{y}(t)$ without knowing $(A,\,B)$ using an observer employing a neural structure that does not require exact information of the state dynamics, and then using $u(t)$ and $\hat{y}(t)$ to learn the controller $K$ using adaptive dynamic programming.
\par
We present the solutions for $\bf{P1}$ and $\bf{P2}$ with associated stability proofs in the following two respective sections.
\vspace{-.25 cm}
\section{Reduced-dimensional State Feedback RL}
Following \citet{khalil}, the \textit{reduced} slow subsystem of \eqref{eq:SP} can be defined by substituting $\epsilon=0$, resulting in
\begin{align}
\label{eq:slowsubsystem}
&\dot{y}_s = A_s y_s + B_s u_s, \;\; y_s(0)= y(0), \;\;
u=u_s +u_f,
\end{align}
where $A_s = A_{11} - A_{12}A_{22}^{-1}A_{21}$ and $B_s = B_1 - A_{12}A_{22}^{-1}B_{2}$.
Since our intent is to only use the slow variable for feedback, we substitute the fast control input $u_f=0$, and the slow control input $u_s=u$. If the controller were to use $y_s(t)$ for feedback then it would find $u = -\bar{K}y_s(t)$ to solve:
\begin{align}\label{J_red}
\text{minimize} \;\; &\bar{J}(y_s(0);u)= \int_0^{\infty} (y_s^T Q y_s + u^T R u )dt, \\
&\mbox{s.t.} \;\; A_s - B_s\bar{K} \in \mathbb{RH}_{\infty}.
\end{align}
The optimal solution for the above problem is given by the following algebraic Riccati equation (ARE):
\begin{align}
&A_s^T\bar{P} + \bar{P}A_s + Q -\bar{P}B_sR^{-1} B_s^T\bar{P} = 0,
\bar{K} = R^{-1}B_s^T\bar{P},\nonumber
\end{align}
\par
where $\bar{P} = \bar{P}^T \succ 0$. If $A_s$ and $B_s$ are unknown, then the RL controller $\bar{K}$ can be learned using measurements of $y_s(t)$ and of an exploration input $u(t)=u_0(t)$ by the ADP algorithm presented in \cite{jiang_book}, which is an iterative version of Kleinman's algorithm \citet{kleinman}. The control policy $u_0(t)$ must be persistently exciting, and can be chosen arbitrarily as long as the system states remain bounded. For example, one choice of $u_0$ is a sum of sinusoidal signals.
In reality, however, $y_s$ is not accessible as $\epsilon \neq 0$. We, therefore, recall the following theorem from \citet{chowslowfast}, which will allow us to replace $y_s(t)$ with $y(t)$ in the learning algorithm.
\par
\noindent \textbf{Theorem 1 \citep{chowslowfast,khalil}:}
\textit{Consider the two systems \eqref{eq:SP} and \eqref{eq:slowsubsystem}. There exists $0<\epsilon^* \ll 1$ such that for all $0< \epsilon \leq \epsilon^*$, the trajectories $y(t)$ and $y_s(t)$ satisfy uniformly for $t \in [0,t_1]$}
\begin{align}
y(t) = y_s (t) + O(\epsilon).
\end{align}
Algorithm 1 shows how the controller $K$ is learned using $y$ and $u_0$, based on \citet{jiang1}.
\begin{algorithm}[t]
\footnotesize
\caption{SP-RL using slow dynamics}
\textbf{Input:} Measurements of $y(t)$ and $u_0(t)$\\
\textbf{Step 1 - Data storage:}
\textit{Store} data (i.e., $y(t)$ and $u_0(t)$) for sufficiently large uniformly sampled time instants $(t_1,t_2,\cdots,t_l)$, and \textit{construct} the following matrices:
\vspace{-.47 cm}
\begin{align}
& \hspace{-.3 cm} \delta_{yy} = \begin{bmatrix}
y \otimes y |_{t_1}^{t_1+T} ,& \cdots &, y \otimes y |_{t_l}^{t_l+T}
\end{bmatrix}^T,\\
& \hspace{-.3 cm} I_{yy} = \begin{bmatrix}
\int_{t_1}^{t_1+T}(y \otimes y) d\tau ,& \cdots &, \int_{t_l}^{t_l+T} (y \otimes y) d\tau \\
\end{bmatrix} ^T,\\
& \hspace{-.3 cm} I_{yu_0} = \begin{bmatrix}
\int_{t_1}^{t_1+T}(y \otimes u_0) d\tau ,& \cdots & ,\int_{t_l}^{t_l+T} (y \otimes u_0) d\tau \\
\end{bmatrix} ^T,
\end{align}
such that rank($I_{yy} \;\; I_{yu_0}) = r(r+1)/2 + rm$ satisfies.
\textbf{Step 2 - Controller update:}
Starting with a stabilizing $K_0$, \textit{solve} for $K$ iteratively ($k=0,1,\cdots$) following the update equation:
\vspace{-.6 cm}
\begin{align}\label{eq:updateA1}
\underbrace{\begin{bmatrix}
\delta_{yy} -2I_{yy}(I_r \otimes K_k^TR) -2I_{yu_0}(I_r \otimes R)
\end{bmatrix}}_{\Theta_k}\begin{bmatrix}
vec(P_k) \\ vec(K_{k+1})
\end{bmatrix} =\underbrace{-I_{yy}vec(Q_k)}_{\Phi_k}.
\end{align}
The stopping criterion for this update is $\norm{P_k - P_{k-1}} < \gamma$, where $\gamma$ is a chosen small positive threshold. \\
\textbf{Step 3 - Applying control:} After $P$ and $K$ converge, remove $u_0$ and apply $u=-Ky$.
\end{algorithm}
\normalsize
The condition rank($\Theta_k$) = $r(r+1)/2 + rm$ can be satisfied, for example, by utilizing data from at least twice as many sampling intervals as the number of unknowns. We next provide the analytical guarantees of Algorithm $1$ related to the SP-based approximations.
\vspace{-.3 cm}
\subsection{Sub-optimality and Stability Analysis}
The optimal controller parameters $P,K$ can be written as
$
P=\bar{P} + \Delta P ,
K=\bar{K} + \Delta K,
$
where $\bar{P},\bar{K}$ are the optimal solutions if $y_s(t)$ were available for design, and $\Delta P, \Delta K$ are matrix perturbations resulting from the fact that $\epsilon \neq 0$. The following theorem establishes the sub-optimality of the learned controller using $y(t)$.
\noindent \textbf{Theorem 2:} \textit{Assuming $||y_s(t)||$ and $||u_0(t)||$ are bounded for a finite time $t \in [0,t_1]$, the solutions of Algorithm 1 are given by $P=\bar{P} + O(\epsilon)$, $K=\bar{K} + O(\epsilon)$, and $ J =\bar{J} + O(\epsilon)$.}
\noindent \textit{Proof:} See theorems $2$ and $3$ in \citet{sayak_cdc}.\par
Theorem $2$ shows that the controller obtained from Algorithm 1 is $O(\epsilon)$ close to that obtained from the ideal design using the actual slow variables. Next, we analyze how this perturbation affects the optimal objective. The next theorem provides a sufficient condition that is required to achieve asymptotic stability for the $(k+1)^{th}$ iteration of Algorithm 1 assuming that the control policy at the $k^{th}$ iteration stabilizes \eqref{eq:SP}.
\noindent \textbf{Theorem 3:}
\textit{Assume that the control policy $u = -K_ky$ at the $k^{th}$ iteration asymptotically stabilizes \eqref{eq:SP}. Consider $R \succ 0$ and $Q \succ 0$ with $ \lambda_{min}(Q)$ sufficiently large. Then the control policy at the $(k+1)^{th}$ iteration given by $u =- K_{k+1}y$ is asymptotically stabilizing for \eqref{eq:SP}.} \hfill \ensuremath{\square}
\noindent \textit{Proof:} Please see Theorem $4$ in \citet{sayak_cdc}.
\noindent \textbf{Remark 1: (Design trade-off)} The proof of Theorem 3 is based on Lyapunov function based stability analysis, where $Q$ compensates for the error due to $O(\epsilon)$ approximation of the fast dynamics such that $Q - O(\epsilon) \succ 0$. This translates to the requirement of a sufficiently large $\lambda_{min}(Q)$. In practice, one can start the off-policy RL iteration in a computing platform after gathering sufficient data with a considerable $Q \succ 0$, and if that is found to be not stabilizing then tune $Q$ until the states are bounded.
\vspace{-.4 cm}
\section{Reduced-Dimensional Output Feedback RL}
We next address the RL design when the full state information is not available. We start by considering the generic system \eqref{eq:SP}, and then design an observer to estimate the state $x$ as $\hat{x}(t) = [\hat{y}(t);\hat{z}(t)]$. As $T$ is known, the slow state can be estimated as $\hat{y}(t) = T\hat{x}(t)$. The idea then is to simply replace $y(t)$ by $\hat{y}(t)$ in Algorithm 1.
Algorithm $2$ shows the steps for this output feedback RL-based design.
In Section $4.2$ we will present one such observer which can estimate $x$ without having a proper knowledge about the model \eqref{eq:SP}. Before that, we first analyze the stability properties of the output feedback design.
\begin{algorithm}
\footnotesize
\caption{Output feedback ADP/RL}
\textbf{Input:} Measurements of $\hat{y}(t)$ and $u_0(t)$\\
\textbf{Step 1 - Data storage:} Construct the matrices $\delta_{\hat{y}\hat{y}},I_{\hat{y}\hat{y}},I_{\hat{y}u_0}$ with similar structures as $\delta_{yy},I_{yy},I_{yu_0}$ respectively but with $y(t)$ replaced by $\hat{y}(t)$.\\
\textbf{Step 2 - Controller update:}
Following Step $2$ of Algorithm $1$, \textit{update} the control gains as:
\begin{align}
&\underbrace{\begin{bmatrix}
\delta_{\hat{y}\hat{y}} -2I_{\hat{y}\hat{y}}(I_r \otimes K_k^TR) -2I_{\hat{y}u_0}(I_r \otimes R)
\end{bmatrix}}_{\hat{\Theta}_k}\begin{bmatrix}
vec(P_k) \\ vec(K_{k+1})
\end{bmatrix} = \underbrace{-I_{\hat{y}\hat{y}}vec(Q_k)}_{\hat{\Phi}_k}.
\end{align}
The stopping criterion for this update is $\norm{P_k - P_{k-1}} < \gamma_1$, where $\gamma_1$ is a chosen small positive threshold.\\
\textbf{Step 3 - Applying control:} Remove $u_0$ and apply $\tilde{u} = -K\hat{y}$.
\end{algorithm}
\vspace{-.35 cm}
\subsection{Sub-optimality and Stability Analysis}
\textbf{Lemma 1:} Define $e(t)=x(t)-\hat{x}(t)$. If $e$ is uniformly ultimately bounded (UUB) with a bound $b$ for all $t \geq t_0 +T_1$ for some initial time $t_0$, then there exists positive constants $\epsilon^{*}$ and $k$ such that for all $0 < \epsilon \leq \epsilon^{*}$
\vspace{-.3 cm}
\begin{align}
||\hat{y}(t) - y_s(t) || \leq \bar{k}|\epsilon| + b := c(\epsilon,b)
\end{align}
holds uniformly for $t \in [t_2,t_1]$.\\
\textit{Proof:}
Since $e(t)$ is UUB, there exists positive constants $b$ and $\hat{b}$, independent of $t_0 \geq 0,$ and for every $a \in (0,\hat{b})$, there exists $T_1=T_1(a,b)$, independent of $t_0$, such that $||\hat{y}(t_0) - y(t_0)|| \leq a$, which implies that
\begin{equation} ||\hat{y}(t) - y(t)|| \leq b, \; \forall t \geq t_0 + T_1 :=t_2. \label{int1}
\end{equation}
From Theorem $1$, it follows that there exist positive constants $k$ and $p$ such that,
\begin{align}
\hspace{-.4 cm}||y(t) - y_s(t)|| \leq \bar{k}|\epsilon | \;\;\;\; \forall t \in [t_0,t_1], t_1>t_2, \forall |\epsilon| < p. \label{int2}
\end{align}
Combining \eqref{int1} and \eqref{int2}, for $ t \in [t_2, t_1]$ we have
\begin{align}
& ||\hat{y}(t) - y_s(t) || \leq \bar{k}|\epsilon| + b := c(\epsilon,b).
\end{align}
This completes the proof.\hfill \ensuremath{\square} \\
\textbf{Corollary 2:} If $e(t) = O(\epsilon)$ for $t \in [t_2,t_1]$, then $\hat{y}(t) = y_s(t) + O(\epsilon).$
\noindent \textit{Proof:} The proof directly follows from Lemma $1$. \hfill \ensuremath{\square} \\
We know that if $y_s(t)$ were available for feedback then $\bar{P},\bar{K}$ would be the optimal solutions. However, due to the state estimation error bound $b$ and the singular perturbation error $O(\epsilon)$, the actual solutions are given as $P=\bar{P} +\Delta P$, $ K=\bar{K} + \Delta K$,
where $\Delta P$ and $\Delta K$ are matrix perturbations resulting from non-ideal feedback.
\noindent \textbf{Proposition 1: } Perturbations $\Delta P, \Delta K$ are bounded, i.e., there exist two positive constants $\rho,\, \rho_1$, dependent on $b$ and $\epsilon$, such that $\norm{\Delta P}\leq \rho, \norm{\Delta K} \leq \rho_1$. Moreover, if $e(t) = O(\epsilon)$ for $t \in [t_2,t_1]$, then we will recover $P= \bar{P} + O(\epsilon), K=\bar{K} + O(\epsilon)$.
\noindent\textit{Proof:}
Please see Appendix A.
\color{black}
If $e(t) $ can be made sufficiently small by proper tuning of the observer gain then we would recover the design characteristics of Algorithm 1. To this end, we present the following stability result.
\noindent \textbf{Theorem 4:} Assume that the control policy $u = -K_k \hat{y}$ is asymptotically stabilizing for the $k^{th}$ iteration in Step 2 of Algorithm 2. Then, there exist sufficiently small $b^{*},$ and $0<\epsilon^* \ll 1$ such that for $b \leq b^*, 0 < \epsilon \leq \epsilon^*$, with $Q \succ 0, R \succ 0$, $u=-K_{k+1}\hat{y}$ will asymptotically stabilize \eqref{eq:SP} at the $(k+1)^{th}$ iteration.\par
\noindent \textit{Proof:} Please see Appendix B.\\
As shown in Appendix B, the estimation error enters the closed-loop system as an exogenous disturbance. Since $K_{k+1}$ is stabilizing, the states converge to a neighborhood of the origin for sufficiently small $b^*$ and $\epsilon^*$. Note that the designer does not need the explicit knowledge of $\epsilon^*$, and can simply assume a strong time-scale separation in the plant dynamics resulting in a small enough $\epsilon$.
\noindent \textbf{Remark 2:} The convergence of the observer dynamics and that of the RL iterations are handled sequentially. The observer is used to gather sufficient amount of data samples to meet the rank condition $\mbox{rank}(\hat{\Theta}_k) =r(r+1)/2 + rm$, after which the control gain is computed iteratively. $\hat{\Theta}_k$ has same structure as $\Theta_k$ but with $y(t)$ replaced by $\hat{y}(t)$. The designer may start gathering data samples after a few initial time-steps over which the observer may have converged close to its steady-state. The observer is designed to achieve fast convergence, as discussed next. The state estimation error that may be present in the observer output has been taken into consideration in the sub-optimality and the stability analysis, as discussed in Proposition 1 and Theorem 4.
\vspace{-.3 cm}
\subsection{Neuro-adaptive Observer}
A candidate observer to estimate $\hat{y}(t)$ without knowing $(A,\,B)$ is the neuro-adaptive observer proposed in \citet{obs}. The observer employs a neural network structure to account for the lack of dynamic model information. This observer guarantees boundedness of $e(t)$, which, with proper tuning, can also be made arbitrarily small. We next recall the mechanism of this observer. We rewrite \eqref{eq:statecompact1} as
\begin{align}\label{obs1}
&\dot{x} = \hat{A}x + \underbrace{(Ax - \hat{A}x) + Bu}_{g(x,u) },
\; q=\mathcal{C}x,
\end{align}
where $\hat{A}$ is a Hurwitz matrix, and $(\mathcal{C},\hat{A})$ is observable. We do not have proper knowledge about $g(x,u)$, and a neural network (NN) with sufficiently large number of neurons can approximate $g(x,u)$, as $ g(x,u) = W\sigma (V\bar{x}) + \eta(x)$. Here, $\bar{x} = [x,u]$, while $\sigma(.)$ and $\eta(x)$ are the activation function and the bounded NN approximation error, respectively. $W$ and $V$ are the ideal fixed NN weights. We choose $G$ such that $ \hat{A}-G\mathcal{C}$ is Hurwitz. The observer dynamics follow as
\begin{align}\label{observer}
& \dot{\hat{x}} = \hat{A} \hat{x} + \underbrace{g(\hat{x},u)}_{ = \hat{W}\sigma (\hat{V} \hat{\bar{x}})} + G(q - \mathcal{C}\hat{x}),\;
\hat{q} = \mathcal{C}\hat{x},
\vspace{-.4 cm}
\end{align}
where $\hat{W},\,\hat{V}$ are neural network weights when driven by $\hat{x}$, and are updated
based on the modified Back Propagation (BP) algorithm. The observer \eqref{observer} requires the knowledge of $\mathcal{C}$. Accordingly, we define the output error as $\tilde{q} = q - \mathcal{C}\hat{x}$. The objective function is to minimize
$
J = \frac{1}{2}(\tilde{q}^T\tilde{q}).
$
Following \citet{obs}, the update law follows from gradient descent as:
\begin{align}\label{updateW}
&\dot{\hat{W}} = - \eta_1 (\tilde{q}^T \mathcal{C} A_c^{-1})^T(\sigma(\hat{V}\hat{\bar{x}}))^T - \rho_1 ||\tilde{q}||\hat{W},\\
&\dot{\hat{V}} = - \eta_2 (\tilde{q}^T \mathcal{C} A_c^{-1} \hat{W}(I - \Lambda(\hat{V}\hat{\bar{x}})))^T \mbox{sgn}(\hat{\bar{x}})^T - \rho_2 ||\tilde{q}||\hat{V},\nonumber
\end{align}
where, $\eta_1,\, \eta_2 > 0$ are learning rates and $\rho_1, \, \rho_2$ are small positive numbers. Considering $k$ neurons we have $ \Lambda(\hat{V}\hat{\bar{x}})) = diag(\sigma_i^2(\hat{V}_i \hat{\bar{x}})), i=1,2,..,k$, where sgn(.) is the sign function. The update law \eqref{updateW} depends on the knowledge of $\mathcal{C}$. This observer guarantees the following boundedness property.\\
\textbf{Theorem 5 \citep[Theorem 1]{obs}:} With the update law described as \eqref{updateW}, the state estimation error $\tilde{x} = x-\hat{x}$ and weight estimation errors $\tilde{W} = W-\hat{W}, \tilde{V}= V-\hat{V}$ are uniformly ultimately bounded (UUB).
The size of the estimation error bound can be made arbitrarily small by properly selecting the parameters and learning rates as shown in \citet{obs}. Selecting $\hat{A}$ to have fast eigenvalues will also keep the state estimation error small.
\section{Applying to Clustered Multi-Agent Networks}
We next describe how SP-based RL designs can be applied for the control of clustered multi-agent consensus networks. Example of such networks abound in practice including power systems, robotic swarms, and biological networks. The LTI model of these networks can be brought into the standard SP form \eqref{eq:statecompact1} by exploiting the time-scale separation in its dynamics arising from the clustering of nodes.
\vspace{-.35 cm}
\subsection{SP representation of clustered networks}
Consider a network of $n$ agents, where the dynamics of the $i^{th}$ agent is given by
\begin{align}
\label{eq:network}
\dot{x}_i = Fx_i+ \sum_{j \in \mathcal{N}_i} a_{ij} (x_{j} - x_{i}) + b_iu_i,
\end{align}
where $x_i \in \mathbb{R}^{s}$ is the state, $u_i \in \mathbb{R}^{p}$ is the input, and $\mathcal{N}_i$ denotes the set of agents that are connected to agent $i$, for $i =1,\dots n$. The connection graph between agents is assumed to be connected and time-invariant. The constants $a_{ij} = a_{ji} > 0$ denote the coupling strengths of the interaction between agents $i$ and $j$, and vice versa. The matrix $F \in \mathbb{R}^{s \times s}$ models the self-feedback of each node. The overall network model is written as
\begin{align}
\label{eq:statecompact}
\dot{x} = Ax + Bu, \;\; x(0)=x_0,
\end{align}
where, $x \in \mathbb{R}^{ns}$ is the vector of all agent states, $u \in \mathbb{R}^{ns}$ is the control input, $B= diag(b_1,\dots,b_n)$, $A = I_n\otimes F + L\otimes I_s$, $L \in \mathbb{R}^{n \times n}$ being the weighted network Laplacian matrix satisfying $L\bf{1}_n = \bf{0}$.
\noindent \textit{Assumption 3:} $F$ is marginally stable.
\textcolor{black} {Let the agents be divided into $r$ non-empty, non-overlapping, distinct groups ${\mathcal{I}_1,\dots,\mathcal{I}_r}$ such that agents inside each group are {\it strongly} connected while the groups themselves are {\it weakly} connected. In other words, $a_{ij} \gg a_{pq}$ for any two agents $i$ and $j$ inside a group and any other two agents $p$ and $q$ in two different groups. This type of clustering has been shown to induce a two-time scale behavior in the network dynamics of \eqref{eq:network}. Please see \cite{chow1985} for details. Fig. \ref{arch_central} shows an example of such a clustered dynamic network.} The clustered nature of the network helps decompose $L$ as $L = L^I + \epsilon L^E$, where $L^I$ is a block-diagonal matrix that represents the internal connections within each area, $L^E$ is a sparse matrix that represents the external connections, and $\epsilon$ is the singular perturbation parameter arising from the worst-case ratio of the coupling weights inside a cluster to that between the clusters. The slow and fast variables are defined as
\begin{align}\label{similarity}
&\begin{bmatrix}
y \\ z
\end{bmatrix} = \begin{bmatrix}
T \\ G
\end{bmatrix}x,\;\;
x = (U \;\; G^\dagger)\begin{bmatrix}
y \\ z
\end{bmatrix},
\end{align}
where, $T =T_1 \otimes I_s, G=G_1 \otimes I_s$. The definitions of $T_1 \in \mathbb{R}^{r\times n}$ and $G_1 \in \mathbb{R}^{(n-r)\times n}$ can be found in \cite{chow1985}. Applying this transformation to \eqref{eq:statecompact}, and redefining the time-scale as $t_s = \epsilon t$, the following SP form is obtained:
\begin{subequations}\label{eq:sp_stan}
\label{eq:SP2}
\begin{align}
&\frac{dy}{dt_s} = A_{11}y + A_{12}z + B_1u, \\
& \epsilon\frac{dz}{dt_s} = A_{21}y + A_{22}z + B_2u,
\end{align}
\end{subequations}
\vspace{-.4 cm}
\begin{align*}
& A_{11} = T(L^E \otimes I_s)U + (I_r \otimes F)/\epsilon,
A_{12}= T(L^E \otimes I_s)G^{\dagger}, \\
&
A_{21} = G(L^E \otimes I_s)U, A_{22} = G(L^I \otimes I_s)G^\dagger + (I_{n-r} \otimes F) + \\ & \epsilon G(L^E \otimes I_s)G^\dagger,
B_1 = TB/\epsilon,B_2= GB.
\end{align*}
The detailed derivation is shown Appendix C.
\color{black}
All six matrices are assumed to be unknown. Following Assumption 2, we assume that $A_{22}$ is Hurwitz.
\vspace{-.35 cm}
\subsection{Projection of control to agents}
One important distinction between controlling the multi-agent system \eqref{eq:SP2} and a generic SP system \eqref{eq:SP} is that the control input $u$ for the former has a physical meaning in terms of each agent. Therefore, even if $u$ is designed using a reduced-dimensional controller, it must be actuated in its actual dimension. \textcolor{black}{One way to design $u(t)$ can be to use $u = M\tilde{u}$ where $\tilde{u} \in \mathbb{R}^{(rp)\times (rs)}$ is the actual control signal learned using ADP, and the matrix $M$ is a projection matrix of the form
$
M = blkdiag(M^{1},\dots,M^{r}),M^{i}=\bar{M}^i \otimes I_s, \bar{M}^i = \bf{1}_{|\mathcal{I}_i|},
$
which projects the reduced-dimensional controller to the full-dimensional plant. The projection matrix $M$ is constructed by the designer with the assumption that the designer knows the cluster identity of each agent.}
We assume $(A,BM)$ to be stabilizable. The same back-projection concept can be used for output feedback RL.
\vspace{-.3 cm}
\section{Block-decentralized Multi-agent RL}
The controllers learned in Section $3$ and $4$ need to be computed in a centralized way. In this section we show that for the clustered consensus model \eqref{eq:SP2} the clustered nature of the system can also aid in learning a cluster-wise decentralized RL controller. Figs. \ref{arch_central},\ref{arch_decen} describe the centralized and block-decentralized architectures.
\vspace{-.35 cm}
\begin{figure*}
\begin{subfigure}[b]{1\textwidth}
\centering
\includegraphics[width=.8\linewidth, trim = 4 4 4 4,clip]{figures25/cluster5v2.pdf}
\caption{\small{Centralized control architecture for the clustered network}}
\label{arch_central}
\end{subfigure}
\qquad
\begin{subfigure}[b]{1\textwidth}
\centering
\includegraphics[width=.8\linewidth, trim = 4 4 4 4,clip]{figures25/cluster5decen.pdf}
\caption{\small{Control architecture for the area-wise decentralized design}}
\label{arch_decen}
\end{subfigure}
\caption{\small{Centralized and block-decentralized control architectures }}
\label{fig:coherent}
\vspace{- .4 cm}
\end{figure*}
\subsection{Cluster-wise representation}
Let the states of the agents in cluster $\alpha$ be denoted as $(x_{1}^\alpha,\,x_{2}^\alpha,\dots,x_{n_{\alpha}}^\alpha) \in \mathbb{R}^{n_\alpha s}.$ Following \cite{chow1985}, the transformation matrix $T$ in \eqref{similarity} is an averaging operation on the states of agents inside a cluster,
which implies that the slow variable for the cluster $\alpha$ is
\vspace{-.45 cm}
\begin{align}\label{eq:y}
& y^{\alpha} = \frac{1}{n_{\alpha}}(x_{1}^{\alpha}+x_{2}^{\alpha}+\dots+x_{n_{\alpha}}^{\alpha}),\; \alpha=1,\dots,r,\\
& y=[y^1;y^2;\dots;y^r].\label{eq:y1}
\end{align}
For the cluster-wise decentralized design, the starting point is to consider the scenario if all clusters were decoupled from each other. We denote the states in cluster $\alpha$ in that scenario as $x_{d1}^\alpha,\,x_{d2}^\alpha,\dots,x_{dn_{\alpha}}^\alpha \in \mathbb{R}^{n_\alpha s} $, and the concatenated state vector considering all the clusters are denoted as $x_d$. For this decoupled scenario, $y_d^\alpha$ and $y_d$ are similarly defined following \eqref{eq:y} and \eqref{eq:y1}. Then we will have,
\vspace{-.35 cm}
\begin{align}\label{yd}
&\dot{x}_d = (I_n \otimes F + L^I \otimes I_s)x_d + Bu,\\
&\dot{y}_d = T\dot{x}_d= (T_1 \otimes I_s)(I_n \otimes F + L^I \otimes I_s)x_d + \tilde{B}_1u, \nonumber
\end{align}
where $\tilde{B}_1 = TB$. As $x_d = U y_d + G^\dagger z_d$, \eqref{yd} is reduced to
\begin{align}
\dot{y}_d = (I_r \otimes F)y_d + \tilde{B}_1 u. \label{dec}
\end{align}
The controller can be represented cluster-wise as
$
u =[u^1;\,u^2;\,\dots,\,u^r].
$
Using the projected controller discussed in Section 5.2, we can design $u^\alpha(t)$ as
\begin{align}
& u^{\alpha} = M^{\alpha}\tilde{u}^{\alpha},\;
M^{\alpha}=\bar{M}^\alpha \otimes I_s, \bar{M}^i = \bf{1}_{|\mathcal{I}_i|_c},
\end{align}
where $\tilde{u}^{\alpha} $ is the controller learned in cluster $\alpha$, $\alpha = 1,\cdots,r$.
Taking a hint from the cluster-wise decentralized structure of $y_d$-dynamics in \eqref{dec}, we next state our design problem as follows.
\noindent \textbf{P3.} Consider the multi-agent consensus model \eqref{eq:statecompact} where $A$ and $B$ are unknown. \textit{Learn} a control gain $K^{\alpha}$ for every area $\alpha$, $\alpha = 1,\dots,r$, using $y^\alpha(t)$ and $\tilde{u}^\alpha (t)$ such that $u^{\alpha} = M^{\alpha}\tilde{u}^{\alpha} = -M^{\alpha} K^\alpha y^\alpha $ stabilizes the closed-loop system and minimizes the following individual cluster-wise objectives
\vspace{-.35 cm}
\begin{align}
J^\alpha(y^\alpha (0);\tilde{u}^\alpha)= & \int_0^{\infty} (y^{\alpha T} Q^\alpha y^\alpha + \tilde{u}^{\alpha T} R^\alpha \tilde{u}^\alpha )dt,
\end{align}
for $\alpha = 1,\dots,r$. We assume that $(A,BM)$ is stabilizable.
\vspace{-.35 cm}
\subsection{RL Algorithm}
We exploit a different $O(\epsilon)$ separation existing between the trajectories of the actual average variable of an area and the same variable when the areas are decoupled. We start by providing a lemma proving how the actual average variable $y^\alpha$ is related to the decoupled average variable $y_d^\alpha$ for an area $\alpha$.
\par
\noindent \textbf{Lemma 2:} \textit{The cluster-wise average variable $y^\alpha(t)$ and the decoupled average variable $y_d^\alpha(t)$ are related as,}
\vspace{-.35 cm}
\begin{align}
y^\alpha(t) = y_d^\alpha(t) + O(\epsilon), \forall t \in [0,t_1].
\end{align}
\noindent \textit{Proof:}
The proof is shown in Appendix D.
\color{black}
\par
We first consider the scenario when the clusters are decoupled. The average operation can be considered accordingly in $T$. The decoupled slow dynamics is given in \eqref{yd}.
The controller for area $\alpha$ uses the $y_{d}^{\alpha}(t)$ feedback and implements $\tilde{u}^\alpha = -\bar{K}^{\alpha}y_{d}^{\alpha}(t)$ so that the decoupled dynamics are stabilized and the following objective is minimized for area $\alpha$ with the ARE solution $\bar{P}^\alpha \succ 0$ and the optimal control gain $\bar{K}^\alpha$:
\vspace{-.35 cm}
\begin{align}\label{Jbar}
& \hspace{-.3 cm} \bar{J}^{\alpha}(y_{d}^\alpha(0);\tilde{u}^\alpha(0))= \int_0^{\infty} (y_d^{\alpha T} Q^\alpha y_d^\alpha + \tilde{u}^{\alpha T} R^\alpha \tilde{u}^\alpha )dt.
\end{align}
\vspace{-.45 cm}
As the decoupled system is fictitious, based on Lemma 2, it is plausible to replace $y_d^\alpha(t)$ with $y^\alpha(t)$ in the learning algorithm and then follow the same procedure as the Kleinman's algorithm. The resulting algorithm is given in Algorithm $3$.
\footnotesize
\begin{algorithm}[]
\footnotesize
\caption{ Cluster-wise Decentralized ADP}
\label{alg1}
\textbf{For} area $\alpha = 1,2,\dots,r$ \\
\textbf{Step 1:} Construct matrices $\delta_{y^\alpha y^\alpha },I_{y^\alpha y^\alpha },I_{y^\alpha u_0^\alpha }$ having similar structures as $\delta_{yy},I_{yy},I_{yu_0}$ but with $y(t)$ replaced by $y^\alpha (t)$.\\
\textbf{Step 2:}
Starting with a stabilizing $K_0^\alpha $, \textit{Solve} for $K^\alpha $ iteratively ($k=0,1,\dots$) once matrices $\delta_{y^\alpha y^\alpha },I_{y^\alpha y^\alpha },I_{y^\alpha u_0^\alpha }$ are constructed and iterative equation can be written for each small learning steps as,
\begin{align}\label{eq:update}
\hspace{-.3 cm} \underbrace{\begin{bmatrix}
\delta_{y^\alpha y^\alpha} & -2I_{y^\alpha y^\alpha}( I_s \otimes K_k^{\alpha T}R^\alpha) -2I_{y^\alpha u_0^\alpha }(I_s \otimes R^\alpha)
\end{bmatrix}}_{\Theta_k^\alpha } \times \begin{bmatrix}
vec(P_k^\alpha ) \\ vec(K_{k+1}^\alpha)
\end{bmatrix}
=\underbrace{-I_{y^\alpha y^\alpha }vec(Q_k^\alpha )}_{\Phi_k^\alpha }.
\end{align}
The stopping criterion for this update is $\norm{P_k^\alpha - P_{k-1}^\alpha} < \gamma_2$, where $\gamma_2$ is a chosen small positive threshold.\\
\textbf{Step 3:} Next $\tilde{u}^\alpha=-K^\alpha y^\alpha $ is applied and $u_0^\alpha$ source is removed.\\
\textbf{End For}
\end{algorithm}
\normalsize
\vspace{-.35 cm}
\subsubsection{Analysis and Stability for the Decentralized design}
In this section we analyze the sub-optimality and stability aspects of the area-wise decentralized controller learned from Algorithm $2$. The learned controller $K^{\alpha} \in \mathbb{R}$ for all the areas will be perturbed from the controller computed using $y_d^\alpha$, i.e.,
\begin{align}
P^\alpha=\bar{P}^\alpha + \Delta P^\alpha ,
K^\alpha=\bar{K}^\alpha + \Delta K^\alpha,
\end{align}
where $\bar{P}^\alpha,\bar{K}^\alpha$ are the optimal solutions if the clusters were decoupled and $y_d^\alpha(t)$ were available for design, and $\Delta P^\alpha, \Delta K^\alpha$ are matrix perturbations.
The following theorem shows that the matrix perturbations are $O(\epsilon)$ small. \\
\textbf{Theorem 6:} \textit{Assuming $||y_d^\alpha(t)||$ and $||u_0^\alpha(t)||$ are bounded, the area-wise decentralized solutions satisfy for $\alpha = 1,\dots,r$
\vspace{-.4 cm}
\begin{align}\label{decen_sub}
\hspace{-.4 cm} P^\alpha=\bar{P}^\alpha + O(\epsilon),
K^\alpha=\bar{K}^\alpha + O(\epsilon) ,\; J^\alpha = \bar{J}^\alpha + O(\epsilon).
\vspace{-.35 cm}
\end{align}}
\noindent \textit{Proof:}
This proof directly follows from the analysis performed for Theorem $2$. Here the time-scale separation exists between the decoupled average variable $y_d^\alpha$ and the actual average variable $y^\alpha$. Using Lemma $2$, these variables are $O(\epsilon)$ apart, which leads to \eqref{decen_sub} following the analysis of Theorem 2, and Corollary $1$. \hfill \ensuremath{\square}
Next we analyze the closed-loop stability conditions for the block-decentralized design.
\par
\noindent \textbf{Theorem 7:}
\textit{Assume that the control policy $u^\alpha = -M^\alpha K_k^\alpha y^\alpha$ for area $\alpha$ at the $k^{th}$ iteration is asymptotically stable. Then the control policy at the $(k+1)^{th}$ iteration given by $u^\alpha =- M^\alpha K_{k+1}^\alpha y^\alpha$ is asymptotically stable with $R^\alpha \succ 0$ and $Q^\alpha \succ 0$, if $\epsilon $ is sufficiently small.}\hfill \ensuremath{\square}
\noindent \textit{Proof:} The proof is given in Appendix E.
\vspace{-.4 cm}
\section{Numerical Simulations}
\subsection{Centralized State Feedback Design}
A singularly perturbed system in the form of~\eqref{eq:SP} is considered with two fast and two slow states. We choose $\epsilon =0.01$, $Q=10I_2, R=I$, the initial conditions as $[1,\,2,\,1,\,0]$, and the learning time-step as $0.01$ seconds. The model matrices are taken from \citet{chowslowfast} as
\begin{align*}
& A_{11}=\begin{bmatrix}0 & 0.4 \\ 0 & 0 \end{bmatrix},\; A_{12}=\begin{bmatrix}0 & 0 \\ 0.345 & 0 \end{bmatrix},\;A_{21}=\begin{bmatrix}0 & -0.524 \\ 0 & 0 \end{bmatrix}\\
& A_{22}=\begin{bmatrix}-0.465 & 0.262 \\ 0 & -1 \end{bmatrix},\; B_1 = B_2 =\begin{bmatrix}1\\ 1 \end{bmatrix}.
\end{align*}
\normalsize
The system is persistently excited by exploration noise following \citet{jiang_book}. The control gain is learned as $K = [3.80 \;\; 1.38]$, producing a closed-loop objective $J=7.72$ units. The convergence plots for $P$ and $K$ are shown in Fig.~\ref{fig:SP2}.
We next compare the closed-loop responses learned by ADP for the ideal reduced slow system ($\epsilon = 0$) versus the full-order system ($\epsilon \neq 0$) in Fig.~\ref{fig:SP3}. For the ideal slow system, the following controller is learned: $\bar{K} = [3.1623 \;\; 1.9962], \bar{J}= 7.2950$ units. The top panel of Fig.~\ref{fig:SP3} shows this comparison for $\epsilon=0.01$, while the bottom panel shows this for $\epsilon=0.001$. It can be seen that the responses of the ideal and non-ideal reduced-dimensional systems get closer to each other over time as $\epsilon$ decreases.\par
We next consider a clustered multi-agent network with $25$ agents, divided into $5$ clusters. Each agent has a scalar state with $F=0$. Therefore the network has $4$ slow eigenvalues, one zero eigenvalue and the rest are the fast eigenvalues. The slow eigenvalues are $-0.128,-0.195,-0.196,$ and $-0.2638$. The control architecture is shown in Fig.~\ref{arch_central}. Each cluster is assumed to have a local coordinator that averages the states from inside the cluster, and transmits the average state to a central controller, which learns the reduced-dimensional control input $\tilde{u}(t) \in \mathbb R^5$ and subsequently back-projects it to individual agents.
\begin{figure*}
\begin{minipage}{.33\linewidth}
{ \includegraphics[width=.9\linewidth, height= 2.3 cm, trim = 4 4 4 4,clip]{ADP_SP/P_it.eps}}
\includegraphics[ width=.9\linewidth, height= 2.3 cm, trim = 4 4 4 4,clip]{ADP_SP/K_it.eps}
\caption{\small{Convergence of $P$ \protect\\ and $K$ for the standard \protect\\ SP system \protect\\ }}
\label{fig:SP2}
\end{minipage}%
\begin{minipage}{.33\linewidth}
\includegraphics[width=.9\linewidth, height= 2.3 cm,, trim = 4 4 4 4,clip]{ADP_SP/state1_com01.eps}
\includegraphics[width=.9\linewidth, height=2.3 cm,, trim = 4 4 4 4,clip]{ADP_SP/state1_comeps001.eps}
\caption{\small{Comparison of slow state $1$ with $\epsilon = 0.01,0.001$ and reduced slow subsystem \protect\\}}
\label{fig:SP3}
\end{minipage}
\begin{minipage}{.33\linewidth}
\includegraphics[width=.9\linewidth, height= 2.3 cm,, trim = 4 4 4 4,clip]{figures25/reducedQ10.eps}
\includegraphics[width=.9\linewidth, height= 2.3 cm,, trim = 4 4 4 4,clip]{figures25/reducedQ1000.eps}
\caption{\small{Improved dynamics for the clustered network (Top panel - $Q=10I_5$, Bottom panel - $Q=1000I_5$)}}
\label{fig:cluster}
\end{minipage}
\bigskip
\begin{minipage}{.33\linewidth}
{ \includegraphics[width=.9\linewidth, height= 2.3 cm, trim = 4 4 4 4,clip]{fig_decen25/iso_area1.eps}}
\centering
\caption{\small{ Decentralized\protect\\ design for ideal decoupled clusters\protect\\ \protect\\}}
\label{fig:iso}
\end{minipage}%
\begin{minipage}{.33\linewidth}
\includegraphics[width=.9\linewidth, height= 2.3 cm,, trim = 4 4 4 4,clip]{fig_decen25/itK.eps}
\caption{\small{Convergence of $K$ and $P$ for the cluster-wise decentralized design \protect\\}}
\label{fig:KP}
\end{minipage}
\begin{minipage}{.33\linewidth}
\includegraphics[width=.9\linewidth, height= 2.3 cm,, trim = 4 4 4 4,clip]{fig_decen25/statesQvary.eps}
\caption{\small{Dynamic performance with cluster-wise decentralized design (Top panel - $Q = 10I_5$ for all areas, Bottom panel - varying $Q$ }}
\label{fig:decen}
\end{minipage}
\vspace{-.5 cm}
\end{figure*}
\begin{figure}[H]
\centering
\includegraphics[width=.7\linewidth, trim = 4 4 4 4,clip]{figures25/full25.eps}
\caption{\small{Learned controller with full-state feedback}}
\label{fig:full}
\vspace{-.3 cm}
\end{figure}
Fig.~\ref{fig:full} shows the learning of the full-dimensional optimal LQR controller. It takes at least $18.75$ seconds to learn $K \in \mathbb{R}^{25\times 25}$. The exploration signal here is a sum of sinusoidal signals with different frequencies. With $r=5$, the reduced-dimensional controller, on the other hand, requires only $r^2 + 2r^2 = 75$ samples for learning. It dominantly affects the slow poles, and with $Q=10I_5$, the closed-loop slow poles are placed at $-3.14,-3.18,-3.17,-3.15,$ and $-3.16$. {Dynamic performance is improved with increase in the weights of $Q$ as shown in Fig.~\ref{fig:cluster}. }
A comparison between the full and the reduced-dimensional design in terms of minimum learning and CPU run times is given in Table $1$.
\begin{table}[H]
\centering
\caption{\normalsize{Reduction in learning and CPU run times for the slow state feedback-based design with $25$ agents}}
\label{label1}
\begin{tabular}{l|l|l|}
\cline{2-3}
& \begin{tabular}[c]{@{}l@{}}\scriptsize{Ideal min. learning} \\ \scriptsize{time (T=0.01 s)}\end{tabular} & \begin{tabular}[c]{@{}l@{}}\scriptsize{CPU run} \\ \scriptsize{times}\end{tabular} \\ \hline
\multicolumn{1}{|l|}{\begin{tabular}[c]{@{}l@{}} \scriptsize{Full-state feedback}\end{tabular}} & \scriptsize{18.75 s} & \scriptsize{72.19 s} \\ \hline
\multicolumn{1}{|l|}{\begin{tabular}[c]{@{}l@{}}\scriptsize{Reduced-dim state feedback}\end{tabular}} & \scriptsize{0.75 s} & \scriptsize{1.34 s} \\ \hline
\end{tabular}
\end{table}
\vspace{-.4 cm}
\subsection{Cluster-wise decentralized state feedback design}
Considering the same multi-agent example, we first perform the ADP-based learning of the controller when the clusters are fully decoupled (i.e., the ideal decentralized scenario). Each area is equipped with an aggregator.
Note that the average of all the cluster states represents the decoupled slow state $y_d^\alpha$ for cluster $\alpha$. The state evolution of two representative areas are shown in Fig.~\ref{fig:iso}. We consider similar coupling strengths between the agents inside all the clusters with $Q=10,\, R=1$ but with different initial conditions. The computed scalar control gain for each area is $K=3.1623$, and the corresponding objective values are $\bar{J}^1 = 1.317,\,\bar{J}^2 = 0.745,\, \bar{J}^3 = 1.765,\, \bar{J}^4 = 0.8451$ and $\bar{J}^5 = 0.5244$.
\par
Thereafter, the decentralized ADP computation is performed on the actual system following Algorithm $2$. The average states from each cluster is used as the feedback signal for the ADP computation block as shown in Fig.~\ref{arch_decen}. Fig.~\ref{fig:KP} shows the fast convergence of the ADP iterations.
With $Q=10,R=1$ for all the areas, the cluster-wise decentralized control gains are computed as $K^1=3.139,\,K^2=3.195, \, K^3=3.130, \, K^4 = 3.187, \, K^5=3.173,$ with the objective values as $J^1=1.308,\, J^2= 0.754,\, J^3= 1.7478, \, J^4=0.8524$ and $J^5=0.5261$. In Fig.~\ref{fig:decen}, we can see that with the increasing value of $Q^\alpha,\alpha = 1,\dots,5$, the dynamic performance of the agent states increases. The dynamic performance of different cluster states can be controlled independently using different $Q$ for the different areas. The learning time is also decreased because of the reduced number of feedback variables. The exploration is performed for only $0.2$ seconds.
\vspace{-.3 cm}
\subsection{Output feedback RL (OFRL) design}
We first consider the singularly perturbed system as in Section $7.1$ with $\epsilon =0.01$, initial condition $[1,\,2,\,1,\,0]$. We consider $C=[1,1,0,0;0,0,1,1]$. The learning time step is $0.01$ seconds. Data is gathered for $0.7$ s with the system being persistently excited with exploration noise. Fig.~\ref{fig:iteration} shows the convergence of $P$ and $K$ during the ADP-based computations using the estimated states. Fig. \ref{fig:spslow} and Fig.~\ref{fig:spfast} show the actual versus estimated state trajectories using the NN observer.
For the design of the NN observer, the Hurwitz matrix $\hat{A}$ is considered to be of SP structure but different than the original state matrix. We can see from Figs. \ref{fig:spslow}-\ref{fig:spfast} that the estimation error is small, and the ADP controller using these estimates maintains closed-loop stability. Also, Fig.~\ref{fig:spcomp} compares the output feedback control responses with the ideal ($\epsilon =0$) state feedback responses.
\par
We next consider the $5$-cluster, $25$-agent clustered consensus network.
We consider a slightly different set of couplings with similar structure as considered for the state feedback design. The slow eigenvalues are $-0.127,-0.192,-0.191$ and $-0.258$.
For the estimator design, the Hurwitz matrix $\hat{A}$ is taken to be of similar structure as $A$ but the coupling between the agents in a same cluster is $20 \% $ off from the original, while the inter-cluster strengths are $50 \%$ off from the original. For the full-order system, Fig.~\ref{fig:clusterfull} shows few examples of the state estimation, where the learning takes approximately $20$ s. In the reduced-dimensional design, using the NN observer estimates the aggregator generates the average states for each cluster. These average states and inputs are used for the reduced-dimensional ADP iterations. Fig.~\ref{fig:clusterred} shows that the reduced-dimensional design using the NN observer requires approximately $1$ s of exploration. The comparison of learning and CPU run-times between the full-dimensional observer-based design and the reduced-dimensional observer-based design is presented in Table \ref{table2}.
\begin{table}[H]
\centering
\caption{\normalsize{Reduction in learning and CPU run times for the output feedback based reduced order design with $25$ agents}}
\label{table2}
\begin{tabular}{l|l|l|}
\cline{2-3}
& \begin{tabular}[c]{@{}l@{}} \scriptsize{Ideal min. learning}\\ \scriptsize{time (T=0.01 s)}\end{tabular} & \begin{tabular}[c]{@{}l@{}}\scriptsize{CPU run}\\ \scriptsize{ times} \end{tabular} \\ \hline
\multicolumn{1}{|l|}{\begin{tabular}[c]{@{}l@{}} \scriptsize{Full-dim output feedback} \end{tabular}} & \scriptsize{18.75 s} & \scriptsize{298 s} \\ \hline
\multicolumn{1}{|l|}{\begin{tabular}[c]{@{}l@{}} \scriptsize{Reduced-dim output feedback}\end{tabular}} & \scriptsize{0.75 s} & \scriptsize{13.82 s} \\ \hline
\end{tabular}
\end{table}
\vspace{-.6 cm}
\begin{figure}[t]
\begin{minipage}{.33\linewidth}
\includegraphics[width=.9\linewidth, height= 2.3 cm,, trim = 4 4 4 4,clip]{fig_spnnadp/K_it.eps}
\includegraphics[width=.9\linewidth, height= 2.3 cm, trim = 4 4 4 4,clip]{fig_spnnadp/P_it.eps}
\caption{\small{Convergence of K and P for the standard SP system (OFRL)}}
\label{fig:iteration}
\end{minipage}
\begin{minipage}{.33\linewidth}
\includegraphics[width=.9\linewidth, height= 2.3 cm,, trim = 4 4 4 4,clip]{fig_spnnadp/sp_01actesty1.eps}
\includegraphics[width=.9\linewidth, height= 2.3 cm,, trim = 4 4 4 4,clip]{fig_spnnadp/sp_01actesty2.eps}
\caption{\small{Slow states for the \protect\\ standard SP system (OFRL)}}
\label{fig:spslow}
\end{minipage}%
\begin{minipage}{.33\linewidth}
\includegraphics[width=.9\linewidth, height= 2.3 cm,, trim = 4 4 4 4,clip]{fig_spnnadp/sp_01actestz1.eps}
\includegraphics[width=.9\linewidth, height= 2.3 cm,, trim = 4 4 4 4,clip]{fig_spnnadp/sp_01actestz2.eps}
\caption{\small{Fast state trajectories for the standard SP system (OFRL)}}
\label{fig:spfast}
\end{minipage}
\bigskip
\begin{minipage}{.33\linewidth}
\includegraphics[width=.9\linewidth, height= 2.3 cm,, trim = 4 4 4 4,clip]{fig_spnnadp/compep_01slow1.eps}
\includegraphics[width=.9\linewidth, height= 2.3 cm,, trim = 4 4 4 4,clip]{fig_spnnadp/compep_01slow2.eps}
\caption{\small{Comparison with \protect\\state feedback for the $\epsilon =0$ \protect\\ system (OFRL) }}
\label{fig:spcomp}
\end{minipage}
\begin{minipage}{.33\linewidth}
\includegraphics[width=.9\linewidth, height= 2.3 cm,, trim = 4 4 4 4,clip]{fig_obs25/fullx1.eps}
\includegraphics[width=.9\linewidth, height= 2.3 cm,, trim = 4 4 4 4,clip]{fig_obs25/fullx6.eps}
\caption{\small{Learning with full \protect\\ state estimates for the \protect\\clustered network}}
\label{fig:clusterfull}
\end{minipage}
\begin{minipage}{.33\linewidth}
\includegraphics[width=.92\linewidth, height= 2.3 cm,, trim = 4 4 4 4,clip]{fig_obs25/red_comp.eps}
\includegraphics[width=.9\linewidth, height= 2.3 cm,, trim = 4 4 4 4,clip]{fig_obs25/red_estimates.eps}
\caption{\small{Learning with slow state estimates for the clustered network}}
\label{fig:clusterred}
\end{minipage}
\vspace{-1 cm}
\end{figure}
\section{Conclusion}
The paper presented RL based optimal control designs incorporating ideas from model reduction following from time-scale separation properties in LTI systems. Both state feedback and output feedback RL designs are reported. The designs are extended to clustered multi-agent networks for which an additional cluster-wise block-decentralized RL control is also discussed. Sub-optimality and stability analyses for each design are performed using SP approximation theorems. For the state feedback designs only the SP approximation error affects the sub-optimality, whereas for the output feedback designs the state estimation error adds to it. Results are validated using multiple simulation case studies.
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<bill session="115" type="h" number="2699" updated="2017-09-28T18:30:13Z">
<state datetime="2017-05-25">REFERRED</state>
<status>
<introduced datetime="2017-05-25"/>
</status>
<introduced datetime="2017-05-25"/>
<titles>
<title type="official" as="introduced">To direct the Secretary of Defense to more effectively provide mental health resources for members of the Armed Forces at high risk of suicide, and for other purposes.</title>
<title type="display">To direct the Secretary of Defense to more effectively provide mental health resources for members of the Armed Forces at high risk of suicide, and for other purposes.</title>
</titles>
<sponsor bioguide_id="K000380"/>
<cosponsors>
<cosponsor bioguide_id="M001196" joined="2017-05-25"/>
</cosponsors>
<actions>
<action datetime="2017-05-25">
<text>Introduced in House</text>
</action>
<action datetime="2017-05-25" state="REFERRED">
<text>Referred to the Committee on Armed Services, and in addition to the Committee on Veterans' Affairs, for a period to be subsequently determined by the Speaker, in each case for consideration of such provisions as fall within the jurisdiction of the committee concerned.</text>
</action>
<action datetime="2017-05-25">
<text>Referred to House Armed Services</text>
</action>
<action datetime="2017-05-25">
<text>Referred to House Veterans' Affairs</text>
</action>
<action datetime="2017-05-25">
<text>Referred to the Subcommittee on Health.</text>
</action>
<action datetime="2017-07-17">
<text>Referred to the Subcommittee on Military Personnel.</text>
</action>
</actions>
<committees>
<committee subcommittee="" code="HSAS" name="House Armed Services" activity="Referral"/>
<committee subcommittee="Military Personnel" code="HSAS02" name="House Armed Services" activity="Referral"/>
<committee subcommittee="" code="HSVR" name="House Veterans' Affairs" activity="Referral"/>
<committee subcommittee="Health" code="HSVR03" name="House Veterans' Affairs" activity="Referral"/>
</committees>
<relatedbills/>
<subjects>
<term name="Armed forces and national security"/>
</subjects>
<amendments/>
<summary date="2017-05-25T04:00:00Z" status="Introduced in House">This bill directs the Department of Defense (DOD) to develop a methodology to:
identify units of the Armed Forces that have a disproportionately high rate of suicide and suicide attempts; provide additional mental health resources for members of the Armed Forces deployed with such units; identify the circumstances of deployments associated with increased vulnerability to suicide, including the length and area of deployment and the nature and extent of contact with enemy forces; and provide additional preventative mental health care to units deployed under similar circumstances. DOD shall: (1) develop a methodology to assess the rate of suicide and suicide attempts of members of the Armed Forces of units that have been deployed in support of a contingency operation after September 11, 2001, and (2) provide outreach regarding the available mental health resources for veterans enrolled in the Department of Veterans Affairs (VA) health care system who were deployed with such units.
Information obtained pursuant to this bill may be used by DOD and VA officers, employees, and contractors only for the purposes of carrying it out.</summary>
<committee-reports/>
</bill>
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Ljungskile est une localité de la municipalité de Uddevalla, dans le comté de Västra Götaland en Suède. En 2010, y vivaient.
Références
Localité dans le comté de Västra Götaland | {
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{"url":"https:\/\/www.physicsforums.com\/threads\/optic-mirage.74837\/","text":"Optic Mirage\n\n1. May 8, 2005\n\n2. May 8, 2005\n\nStaff: Mentor\n\nThis is a clever toy consisting of two concave parabolic mirrors, one on top of the other, facing each other. The mirrors are made so that the focal point of one mirror is right at the surface of the other. You place a small object on the surface of the bottom mirror (inside the thing, where you can't see it directly). Since light from the object is at the focal point of the top mirror, the light reflects off the top mirror as parallel light, which then reflects off the bottom mirror to focus at its focal point: which is the the surface of the top mirror. Of course, a hole is cut out of the top mirror so that the real image can be seen. The real image looks like the object is really there.\n\nLast edited: May 8, 2005\n3. May 8, 2005\n\nuser123\n\nFirst of all, thanks for the help.\n\nI'm still a bit confused though; it might be just some semantic problems. When you say \"virtual image\" do you really mean virtual image as in the image that's behind the mirror? If that were so, how we see a projection of it? Also, if the image is at the focal point, doesn't that reduce the size of the image?\n\nLast edited: May 8, 2005\n4. May 8, 2005\n\nStaff: Mentor\n\nthe image is real\n\nOops! I meant to say real image!\n\n5. May 8, 2005\n\nuser123\n\nStill, doesn't shouldn't the image be only half the size of the orignial object if the light converges at the focal length?\n\n6. May 9, 2005\n\nStaff: Mentor\n\nWhat makes you think that the image will be half the size?\n\nNote that light from the object isn't exactly at the focal point, since it's an object not a point source. According to my calculations, if the object is small enough (compared to the focal length) the image will be the same size as the object.\n\n7. May 14, 2005\n\nuser123\n\nNever mind, I figured it out. Thanks a lot for your help.","date":"2017-02-22 09:16:33","metadata":"{\"extraction_info\": {\"found_math\": false, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8428441286087036, \"perplexity\": 457.995397368689}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 5, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2017-09\/segments\/1487501170925.44\/warc\/CC-MAIN-20170219104610-00529-ip-10-171-10-108.ec2.internal.warc.gz\"}"} | null | null |
Tycoon Richard Li & Miss Hong Kong's Karmen Kwok Split Because Of This M'sian Actress?
Source: Ming Pao
The internet is currently buzzing with the latest break up between Richard Li Tzar Kai (李澤楷) and Miss Hong Kong 2015 second runner-up Karmen Kwok (郭嘉文).
The pair started dating back in 2016 when they were spotted in Japan during Valentine's Day. It was reported that the 29-year-old beauty got a penthouse unit at One Mayfair, Beacon Hill in Kowloon Tong in 2017 thanks to her super-rich beau. Karmen terminated her contract with TVB that same year.
Source: Oriental Daily
Richard and Karmen made their first public debut as a couple during a public function in 2019. Since then, they've attended many events together. Rumours began circulating that the Hong Kong tycoon (who also happens to be the son of Hong Kong's richest man Li Ka Shing) recently ended his 5-year relationship with the Miss Hong Kong finalist.
Serene Lim Returns To Malaysia After Parting Ways With TVB
Malaysian Beauty Queen Serene Lim To Star Alongside Kristal Tin In 1st TVB Drama
Karmen herself seemed to confirm their split when she left a cryptic message on her Instagram account. In her IG Story, she wrote, "Thank you for sharing the story from your heart. I also hope that this is a place for you to share your happiness, express your sorrow, and gain positive energy, or a little comforting place. As for those negative voices, if you change the angle, they are also considered a nutrient in life, because what doesn't kill you makes you stronger/No worries."
A post shared by Karmen Kwok 郭嘉文 (@karmen_kkm)
Netizens also noticed that Karmen has started to become active again on her socials. In addition to posting more photos online, she's also been promoting several products and brands. It wouldn't come as a surprise that she's now building up her portfolio as a KOL (Key Opinion Leader) since she has also included her work contact in her IG bio.
So what caused the break up?
Word on the street has it that Richard might be seeing Serene Lim Shyi Yee (林宣妤). The Malaysian actress (who just turned 25 last weekend) hails from Perak and was the first artist to sign with Richard's entertainment company, ViuTV. Guess who introduced them? It was fellow Malaysian icon Tan Sri Michelle Yeoh!
A post shared by Serene Lim 宣妤 (@serene_lsy)
Since winning the Miss Astro Chinese International Pageant in 2016, Serene has gone on to star in "The Garden Of Evening Mists" and other notable projects after making her acting debut in Chiu Keng Guan's "Think Big Big". However, she parted ways with TVB last year after her contract ended and returned to Malaysia during the pandemic.
It remains to be seen if Serene and Richard are an item. Prior to Karmen, the 55-year-old Hong Kong businessman was previously dating actress-singer Isabella Leong, with whom they have 3 sons together.
Sources: DimSum, 38JieJie.
Tags: Karmen KwokRichard Li Tzar KaiSerene LimSerene Lim Shyi Yee李澤楷林宣妤郭嘉文
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Włodzimierz Połoszynowicz (ukr. Володимир Полошинович, ur. w 1865, zm. 9 października 1914 w Szczawnem) – ksiądz greckokatolicki, członek Rady Powiatowej i Wydziału Powiatowego powiatu sanockiego.
Wyświęcony w 1889. Żonaty. W latach 1889–1894 administrator parafii w Kamiannej, w latach 1894–1896 proboszcz tamże, od 1896 do śmierci proboszcz parafii w Szczawnem.
Bibliografia
Dmytro Błażejowśkyj - "Historical Sematism of the Eparchy of Peremysl", Lviv 1995
Galicyjscy duchowni greckokatoliccy
Duchowni związani z Sanokiem
Członkowie Rady Powiatowej Sanockiej (autonomia galicyjska)
Członkowie Wydziałów Powiatowych Galicji
Urodzeni w 1865
Zmarli w 1914 | {
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<p>Dr. David Wright Crawford is a staple in the playwright industry hailing from rural East Texas. His love for simplicity bred by the great outdoors and playful adventures life brings is evident in his scripts. Described as "a new Southern Gentleman
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\section{Introduction}
Few years after the conception of Generative Adversarial Networks (GANs)~\cite{goodfellow2014generative}, we have witnessed an impressive progress on generating illusory plausible images. From the early low-resolution and hazy results, the quality of the artificially generated images has been notably enhanced. We are now able to synthetically produce high-resolution~\cite{brock2018large} artificial images that are indiscernible from real ones to the human observer~\cite{karras2019style}.
In the original GAN architecture, inputs were randomly sampled from a latent space, so that it was hard to control which kind of images were being generated. The conception of conditional Generative Adversarial Networks (cGANs)~\cite{mirza2014conditional} led to an important improvement. By allowing to condition the generative process on an input class label, the networks were then able to produce images from different given types~\cite{choi2018stargan}. However, such classes had to be predefined beforehand during the training stage and thus, it was impossible to produce images from other unseen classes during inference.
\begingroup
\setlength{\tabcolsep}{8pt}
\begin{figure}[t!]
\centering
\begin{tabular}{ccccc}
\includegraphics[width=0.15\linewidth,height=.8cm]{images/real_iam_imgs/173_d06-020-00-04.png}&
\includegraphics[width=0.15\linewidth,height=.8cm]{images/in_te/180-21_anyone-anyone.png}&
\includegraphics[width=0.15\linewidth,height=.8cm]{images/oo_te/173-72_push-push.png}&
\includegraphics[width=0.15\linewidth,height=.8cm]{images/in_tr/136-92_easily-easily.png}&
\includegraphics[width=0.15\linewidth,height=.8cm]{images/oo_tr/000-30_reviews-reviems.png}\\
\includegraphics[width=0.15\linewidth,height=.8cm]{images/in_tr/024-68_mouth-month.png}&
\includegraphics[width=0.15\linewidth,height=.8cm]{images/real_iam_imgs/170_c04-150-01-05.png}&
\includegraphics[width=0.15\linewidth,height=.8cm]{images/in_te/198-40_Having-Having.png}&
\includegraphics[width=0.15\linewidth,height=.8cm]{images/real_iam_imgs/169_c04-139-00-03.png}&
\includegraphics[width=0.15\linewidth,height=.8cm]{images/in_te/168-19_sound-sound.png}\\
\includegraphics[width=0.15\linewidth,height=.8cm]{images/oo_tr/134-78_monkey-monkey.png}&
\includegraphics[width=0.15\linewidth,height=.8cm]{images/real_iam_imgs/168_c04-116-01-18.png}&
\includegraphics[width=0.15\linewidth,height=.8cm]{images/oo_te/545-65_Flower-Flower.png}&
\includegraphics[width=0.15\linewidth,height=.8cm]{images/real_iam_imgs/168_c04-116-01-00.png}&
\includegraphics[width=0.15\linewidth,height=.8cm]{images/in_tr/013-39_twice-trice.png}\\
\end{tabular}
\caption{Turing's test. Just five of the above words are real. Try to distinguish them from the artificially generated samples\protect\footnotemark[1].}
\label{fig:turinggame}
\end{figure}
\endgroup
But generative networks have not exclusively been used to produce synthetic images. The generation of data that is sequential in nature has also been largely explored in the literature. Generative methods have been proposed to produce audio signals~\cite{dong2018musegan}, natural language excerpts~\cite{yu2017seqgan}, video streams~\cite{tulyakov2018mocogan} or stroke sequences~\cite{graves2013generating,ha2017neural,ganin2018synthesizing,zheng2019strokenet} able to trace sketches, drawings or handwritten text. In all of those approaches, in order to generate sequential data, the use of Recurrent Neural Networks (RNNs) has been adopted.
Yet, for the specific case of generating handwritten text\footnotetext[1]{\rotatebox[origin=c]{180}{The real words are: \texttt{"that"}, \texttt{"vision"}, \texttt{"asked"}, \texttt{"hits"} and \texttt{"writer"}.}}, one could also envisage the option of directly producing the final images instead of generating the stroke sequences needed to pencil a particular word. Such non-recurrent approach presents several benefits. First, the training procedure is more efficient since recurrencies are avoided and the inherent parallelism nature of convolutional networks is leveraged. Second, since the output is generated all at once, we avoid the difficulties of learning long-range dependencies as well as vanishing gradient problems. Finally, online training data (pen-tip location sequences), which is hard to obtain, is no longer needed.
Nevertheless, the different attempts to directly generate raw word images present an important drawback. Similarly to the case with cGANs, most of the proposed approaches are just able to condition the word image generation to a predefined set of words, limiting its practical use. For example~\cite{gregor2015draw} is specifically designed to generate isolated digits, while~\cite{chang2018generating} is restricted to a handful of Chinese characters. To our best knowledge, the only exception to that is the approach by Alonso \emph{et al.}~\cite{alonso2019adversarial}. In their work they propose a non-recurrent generative architecture conditioned to input content strings. By having such design, the generative process is not restricted to a particular predefined vocabulary, and could potentially generate any word. However, the produced results are not realistic, still exhibiting a poor quality, sometimes producing barely legible word images. Their proposed approach also suffers from the mode collapse problem, tending to produce images with a unique writing style. In this paper we present a non-recurrent generative architecture conditioned to textual content sequences, that is specially tailored to produce realistic handwritten word images, indistinguishable to humans. Real and generated images are actually difficult to tell apart, as shown in Fig.~\ref{fig:turinggame}. In order to produce diverse styled word images, we propose to condition the generative process not only with textual content, but also with a specific writing style, defined by a latent set of calligraphic attributes.
Therefore, our approach\footnote{Our code is available at \url{https://github.com/omni-us/research-GANwriting}} is able to artificially render realistic handwritten word images that match a certain textual content and that mimic some style features (text skew, slant, roundness, stroke width, ligatures, etc.) from an exemplar writer. To this end, we guide the learning process by three different learning objectives~\cite{odena2017conditional}. First, an adversarial discriminator ensures that the images are realistic and that its visual appearance is as closest as possible to real handwritten word images. Second, a style classifier guarantees that the provided calligraphic attributes, characterizing a particular handwriting style, are properly transferred to the generated word instances. Finally, a state-of-the-art sequence-to-sequence handwritten word recognizer~\cite{michael2019evaluating} controls that the textual contents have been properly conveyed during the image generation.
To summarize, the main contributions of the paper are the following:
\begin{itemize}
\item Our model conditions the handwritten word generative process both with calligraphic style features and textual content, producing varied samples indistinguishable by humans, surpassing the quality of the current state-of-the-art approaches.
\item We introduce the use of three complementary learning objectives to guide different aspects of the generative process.
\item We propose a character-based content conditioning that allows to generate any word, without being restricted to a specific vocabulary.
\item We put forward a few-shot calligraphic style conditioning to avoid the mode collapse problem.
\end{itemize}
\section{Related Work}
The generation of realistic synthetic handwritten word images is a challenging task. To this day, the most convincing approaches involved an expensive manual intervention aimed at clipping individual characters or glyphs~\cite{wang2005combining,konidaris2007keyword,lin2007style,thomas2009synthetic,haines2016my}. When such approaches were combined with appropriate rendering techniques including ligatures among strokes, textures and background blending, the obtained results were indeed impressive. Haines \emph{et al.}~\cite{haines2016my} illustrated how such approaches could artificially generate indistinguishable manuscript excerpts as if they were written by Sir Arthur Conan Doyle, Abraham Lincoln or Frida Kahlo. Of course such manual intervention is extremely expensive, and in order to produce large volumes of manufactured images the use of truetype electronic fonts has also been explored~\cite{krishnan2016generating,kang2020unsupervised}. Although such approaches benefit from a greater scalability, the realism of the generated images clearly deteriorates.
With the advent of deep learning, the generation of handwritten text was approached differently. As shown in the seminal work by Alex Graves~\cite{graves2013generating}, given a reasonable amount of training data, an RNN could learn meaningful latent spaces that encode realistic writing styles and their variations, and then generate stroke sequences that trace a certain text string. However, such sequential approaches~\cite{graves2013generating,ha2017neural,ganin2018synthesizing,zheng2019strokenet} need temporal data, obtained by recording with a digital stylus pen real handwritten samples, stroke-by-stroke, in vector form.
Contrary to sequential approaches, non-recurrent generative methods have been proposed to directly produce images. Both variational auto-encoders~\cite{kingma2013auto} and GANs~\cite{goodfellow2014generative} were able to learn the MNIST manifold and generate artificial handwritten digit images in the original publications. With the emergence of cGANs~\cite{mirza2014conditional}, able to condition the generative process on an input image rather than a random noise vector, the adversarial-guided image-to-image translation problem started to rise. Image-to-image translation has since been applied to many different style transfer applications, as demonstrated in~\cite{isola2017image} with the \emph{pix2pix} network. Since then, image translation approaches have been acquiring the ability to disentangle style attributes from the contents of the input images, producing better style transfer results~\cite{taigman2016unsupervised,pondenkandath2019historical}. Geometry-aware synthesizing methods~\cite{zhan2019spatial,zhan2019ga} have been successfully applied on scene text images, but cursive words are not considered.
Concerning the generation of handwritten text, such approaches have been mainly used for synthesising Chinese ideograms~\cite{lyu2017auto,tian2017zi2zi,chang2018generating,jiang2018w,wu2020calligan} and glyphs~\cite{azadi2018multi}. However, they are restricted to a predefined set of content classes. The incapability to generate out of vocabulary (OOV) text limits its practical application. Few works can actually deal with OOV words. First, in the work by Alonso \emph{et al.}~\cite{alonso2019adversarial}, the generation of handwritten word samples is conditioned by character sequences, but it suffers from the mode collapse problem, hindering the diversity of the generated images. Second, Fogel~\emph{et al.}~\cite{fogel2020scrabblegan} generate handwritten word by assembling the images generated by its characters, but the generated characters have the same receptive field width, which can make the generated words look unrealistic. Third, Mayr~\emph{et al.}~\cite{mayr2020spatio} propose a conversion model to approximate online handwriting from offline data and then apply style transfer method to online data, so that offline handwritten text images could be generated by leveraging online handwriting synthesizer. However, this method highly depends on the performance of the conversion model and needs online data to train. Techniques like FUNIT~\cite{liu2019few}, able to transfer unseen target styles to the content generated images could be beneficial for this limitation. In particular, the use of Adaptive Instance Normalization (AdaIN) layers, proposed in~\cite{huang2017arbitrary}, shall allow to align both textual content and style attributes within the generative process.
Summarizing, state-of-the-art generative methods are still unable to produce plausible yet diverse images of whatever handwritten word automatically. In this paper we propose to condition a generative model for handwritten words with unconstrained text sequences and stylistic typographic attributes, so that we are able to generate any word with a great diversity over the produced results.
\section{Conditioned Handwritten Generation}
\subsection{Problem Formulation}
Let $\{\mathcal{X},\mathcal{Y},\mathcal{W}\}$ be a multi-writer handwritten word dataset, containing grayscale word images $\mathcal{X}$, their corresponding transcription strings $\mathcal{Y}$ and their writer identifiers $\mathcal{W}=\{w_i\}_{i=1}^N$. Let $X_i = \{x_{w_i,j}\}_{j=1}^K \subset \mathcal{X}$ be a subset of $K$ randomly sampled handwritten word images from the same given writer $w_i \in \mathcal{W}$. Let $\mathcal{A}$ be the alphabet containing the allowed characters (letters, digits, punctuation signs, etc.), $\mathcal{A}^l$ being all the possible text strings with length $l$. Given a set of images $X_i$ as a few-shot example of the calligraphic style attributes for writer $w_i$ on the one hand, and given a textual content provided by any text string $t \in \mathcal{A}^l$ on the other hand; the proposed generative model has the ability to combine both sources of information. It has the objective to yield a handwritten word image having textual content equal to $t$ and sharing calligraphic style attributes with writer $w_i$. Following this formulation, the generative model $H$ is defined as
\begin{equation}
\bar{x} = H\left(t, X_i\right) = H\left(t, \left\{x_1, \ldots, x_K\right\}\right),
\end{equation}
where $\bar{x}$ is the artificially generated handwritten word image with the desired properties. Moreover, we denote $\bar{\mathcal{X}}$ as the output distribution of the generative network $H$.
The proposed architecture is divided in two main components. The generative network produces human-readable images conditioned to the combination of calligraphic style and textual content information. The second component are the learning objectives which guide the generative process towards producing images that look realistic; exhibiting a particular calligraphic style attributes; and having a specific textual content. Fig.~\ref{fig:arch} gives an overview of our model.
\begin{figure}[t!]
\centering
\includegraphics[width=\linewidth]{images/overview.pdf}
\caption{Architecture of the proposed handwriting generation model.}
\label{fig:arch}
\end{figure}
\subsection{Generative Network}
The proposed generative architecture $H$ consists of a calligraphic style encoder $S$, a textual content encoder $C$ and a conditioned image generator $G$. The overall calligraphic style of input images $X_i$ is disentangled from their individual textual contents, whereas the string $t$ provides the desired content.
\myparagraph{Calligraphic style encoding.} Given the set $X_i \subset \mathcal{X}$ of $K=15$ word images from the same writer $w_i$, the style encoder aims at extracting the calligraphic style attributes, \emph{i.e.}~slant, glyph shapes, stroke width, character roundness, ligatures etc. from the provided input samples. Specifically, our proposed network $S$
learns a style latent space mapping, in which the obtained style representations $F_s = S(X_i)$ are disentangled from the actual textual contents of the images $X_i$. The VGG-19-BN~\cite{simonyan2014very} architecture is used as the backbone of $S$. In order to process the input image set $X_i$, all the images are resized to have the same height $h$, padded to meet a maximum width $w$ and concatenated channel-wise to end up with a single tensor $h\times w\times K$. If we ask a human to write the same word several times, slight involuntary variations appear. In order to imitate this phenomenon, randomly choosing permutations of the subset $X_i$ will already produce such characteristic fluctuations. In addition, an additive noise $Z \sim \mathcal{N}(0, 1)$ is applied to the output latent space to obtain a subtly distorted feature representation $\hat{F_s} = F_s + Z$.
\myparagraph{Textual content encoding.} The textual content network $C$ is devoted to produce an encoding of the given text string $t$ that we want to artificially write. The proposed architecture outputs content features at two different levels. Low-level features encode the different characters that form a word and their spatial position within the string. A subsequent broader representation aims at guiding the whole word consistency. Formally, let $t\in \mathcal{A}^l$ be the input text string, character sequences shorter than $l$ are padded with the empty symbol $\varepsilon$. Let us define a character-wise embedding function $e \colon \mathcal{A} \to \mathbb{R}^n$. The first step of the content encoding stage embeds with a linear layer each character $c\in t$, represented by a one-hot vector, into a character-wise latent space. Then, the architecture is divided into two branches.
\emph{Character-wise encoding:} Let $g_1 \colon \mathbb{R}^n \to \mathbb{R}^m$ be a Multi-Layer Perceptron (MLP). Each embedded character $e(c)$ is processed individually by $g_1$ and their results are later stacked together.
In order to combine such representation with style features, we have to ensure that the content feature map meets the shape of $\hat{F_s}$. Each character embedding is repeated multiple times horizontally to coarsely align the content features with the visual ones extracted from the style network, and the tensor is finally vertically expanded. The two feature representations are concatenated to be fed to the generator $F = [\hat{F_s} \parallel F_c]$. Such a character-wise encoding enables the network to produce OOV words, \emph{i.e.} words that have never been seen during training.
\emph{Global string encoding:} Let $g_2 \colon \mathbb{R}^{l\cdot n} \to \mathbb{R}^{2p\cdot q}$ be another MLP aimed at obtaining a much broader and global string representation. The character embeddings $e(c)$ are concatenated into a large one-dimensional vector of size $l \cdot n$ that is then processed by $g_2$. Such global representation vector $f_c$ will be then injected into the generator splitted into $p$ pairs of parameters.
Both functions $g_1(\cdot)$ and $g_2(\cdot)$ make use of three fully-connected layers with ReLU activation functions and batch normalization~\cite{ioffe2016batchnorm}.
\myparagraph{Generator.} Let $F$ be the combination of the calligraphic style attributes and the textual content information character-wise; and $f_c$ the global textual encoding. The generator $G$ is composed of two residual blocks~\cite{huang2018multimodal} using the AdaIN as the normalization layer. Then, four convolutional modules with nearest neighbor up-sampling and a final $\tanh$ activation layer generates the output image $\bar{x}$. AdaIN is formally defined as
\begin{equation}
\operatorname{AdaIN}\left(z, \alpha, \beta\right) = \alpha \left(\frac{z-\mu\left(z\right)}{\sigma\left(z\right)}\right) + \beta,
\end{equation}
where $z \in F$, $\mu$ and $\sigma$ are the channel-wise mean and standard deviations. The global content information is injected four times ($p=4$) during the generative process by the AdaIN layers. Their parameters $\alpha$ and $\beta$ are obtained by splitting $f_c$ in four pairs. Hence, the generative network is defined as
\begin{equation}
\bar{x} = H\left(t,X_i\right) = G\left( C\left(t\right), S\left(X_i\right)\right) = G\left( g_1\left(\hat{t}\right), g_2\left(\hat{t}\right), S\left(X_i\right)\right),
\end{equation}
where $\hat{t} = \left[e(c) ; \forall c\in t\right]$ is the encoding of the string $t$ character by character.
\subsection{Learning Objectives}
We propose to combine three complementary learning objectives: a discriminative loss, a style classification loss and a textual content loss. Each one of these losses aim at enforcing different properties of the desired generated image $\bar{x}$.
\myparagraph{Discriminative Loss.} Following the paradigm of GANs~\cite{goodfellow2014generative}, we make use of a discriminative model $D$ to estimate the probability that samples come from a real source, \emph{i.e.} training data $\mathcal{X}$, or belong to the artificially generated distribution $\bar{\mathcal{X}}$. Taking the generative network $H$ and the discriminator $D$, this setting corresponds to a $\min \max$ optimization problem. The proposed discriminator $D$ starts with a convolutional layer, followed by six residual blocks with LeakyReLU activations and average poolings. A final binary classification layer is used to discern between fake and real images. Thus, the discriminative loss only controls that the general visual appearance of the generated image looks realistic. However, it does not take into consideration neither the calligraphic styles nor the textual contents. This loss is formally defined as
\begin{equation}
\mathcal{L}_d\left(H,D\right) = \mathbb{E}_{x \sim \mathcal{X}} \left[ \log\left(D\left(x\right)\right) \right] + \mathbb{E}_{\bar{x} \sim \bar{\mathcal{X}}}\left[\log\left(1-D\left(\bar{x}\right)\right)\right].
\label{e:discriminator}
\end{equation}
\myparagraph{Style Loss.} When generating realistic handwritten word images, encoding information related to calligraphic styles not only provides diversity on the generated samples, but also prevents the mode collapse problem. Calligraphy is a strong identifier of different writers. In that sense, the proposed style loss guides the generative network $H$ to generate samples conditioned to a particular writing style by means of a writer classifier $W$. Given a handwritten word image, $W$ tries to identify the writer $w_i \in \mathcal{W}$ who produced it. The writer classifier $W$ follows the same architecture of the discriminator $D$ with a final classification MLP with the amount of writers in our training dataset. The classifier $W$ is only optimized with real samples drawn from $\mathcal{X}$, but it is used to guide the generation of the synthetic ones. We use the cross entropy loss, formally defined as
\begin{equation}
\mathcal{L}_w\left(H, W\right) = - \mathbb{E}_{x \sim \left\{ \mathcal{X},\bar{\mathcal{X}} \right\}} \left[ \sum_{i=1}^{\left|\mathcal{W}\right|} w_i \log\left(\hat{w}_i\right) \right],
\label{e:style}
\end{equation}
where $\hat{w} = W(x)$ is the predicted probability distribution over writers in $\mathcal{W}$ and $w_i$ the real writer distribution. Generated samples should be classified as the writer $w_i$ used to construct the input style conditioning image set $X_i$.
\begin{figure}[t!]
\centering
\includegraphics[width=.7\linewidth]{images/recognizer.pdf}
\caption{Architecture of the attention-based sequence-to-sequence handwritten word recognizer $R$.}
\label{fig:arch2}
\end{figure}
\myparagraph{Content Loss.} A final handwritten word recognizer network $R$ is used to guide our generator towards producing synthetic word images with a specific textual content. We implemented a state-of-the-art sequence-to-sequence model~\cite{michael2019evaluating} for handwritten word recognition to examine whether the produced images $\bar{x}$ are actually decoded as the string $t$. The recognizer, depicted in Fig.~\ref{fig:arch2}, consists of an encoder and a decoder coupled with an attention mechanism. Handwritten word images are processed by the encoder and high-level feature representations are obtained. A VGG-19-BN~\cite{simonyan2014very} architecture followed by a two-layered Bi-directional Gated Recurrent Unit (B-GRU)~\cite{chung2014empirical} is used as the encoder network. The decoder is a one-directional RNN that outputs character by character predictions at each time step. The attention mechanism dynamically aligns context features from each time step of the decoder with high-level features from the encoder, hopefully corresponding to the next character to decode. The Kullback-Leibler divergence loss is used as the recognition loss at each time step. This is formally defined as
\begin{equation}
\mathcal{L}_r\left(H, R\right) = - \mathbb{E}_{x \sim \left\{ \mathcal{X},\bar{\mathcal{X}} \right\}} \left[ \sum_{i=0}^{l} \sum_{j=0}^{\left|\mathcal{A}\right|} t_{i,j} \log\left(\frac{t_{i,j}}{\hat{t}_{i,j}}\right) \right],
\label{e:content}
\end{equation}
where $\hat{t} = R(x)$; $\hat{t}_{i}$ being the $i$-th decoded character probability distribution by the word recognizer, $\hat{t}_{i,j}$being the probability of $j$-th symbol in $\mathcal{A}$ for $\hat{t}_{i}$, and $t_{i,j}$ being the real probability corresponding to $\hat{t}_{i,j}$. The empty symbol $\varepsilon$ is ignored in the loss computation; $t_i$ denotes the $i$-th character on the input text $t$.
\subsection{End-to-end Training}
Overall, the whole architecture is trained end to end with the combination of the three proposed loss functions
\begin{equation}
\mathcal{L}(H, D, W, R) = \mathcal{L}_d(H, D) + \mathcal{L}_w(H, W) + \mathcal{L}_r(H, R),
\label{e:final}
\end{equation}
\begin{equation}
\min_{H,W,R} \max_D \mathcal{L}(H, D, W, R).
\end{equation}
Algorithm~\ref{alg:train} presents the training strategy that has been followed in this work. $\Gamma(\cdot)$ denotes the optimizer function. Note that the parameter optimization is performed in two steps. First, the discriminative loss is computed using both real and generated samples (line~\ref{alg:line:D}). The style and content losses are computed by just providing real data (line~\ref{alg:line:wr}).
Even though $W$ and $D$ are optimized using only real data and, therefore, they could be pre-trained independently from the generative network $H$, we obtained better results by initializing all the networks from scratch and jointly training them altogether. The network parameters $\Theta_D$ are optimized by gradient ascent following the GAN paradigm whereas the parameters $\Theta_W$ and $\Theta_R$ are optimized by gradient descent. Finally, the overall generator loss is computed following Equation~\ref{e:final} where only the generator parameters $\Theta_H$ are optimized (line~\ref{alg:line:H}).
\begin{algorithm}[t!]
\hspace*{\algorithmicindent} \textbf{Input:} Input data $\{\mathcal{X}, \mathcal{Y}, \mathcal{W}\}$; alphabet $\mathcal{A}$; max training iterations $T$ \\
\hspace*{\algorithmicindent} \textbf{Output: } Networks parameters $\{\Theta_{H}, \Theta_{D}, \Theta_{W}, \Theta_{R}\}$.
\begin{algorithmic}[1]
\Repeat
\State Get style and content mini-batches $\{X_i, w_i\}_{i=1}^{N_B}$ and $\{t^i\}_{i=1}^{N_B}$
\State $\mathcal{L}_d \leftarrow $ Eq.~\ref{e:discriminator} \label{alg:line:D} \algorithmiccomment{Real and generated samples $x \sim \{\mathcal{X}, \bar{\mathcal{X}}\}$}
\State $\mathcal{L}_{w,r} \leftarrow $ Eq.~\ref{e:style} + Eq.~\ref{e:content} \label{alg:line:wr} \algorithmiccomment{Real samples $x \sim \mathcal{X}$}
\State $\Theta_D \leftarrow \Theta_D + \Gamma(\nabla_{\Theta_D}\mathcal{L}_d)$
\State $\Theta_{W,R} \leftarrow \Theta_{W,R} - \Gamma(\nabla_{\Theta_{W,R}}\mathcal{L}_{w,d})$
\State $\mathcal{L} \leftarrow $ Eq.~\ref{e:final} \algorithmiccomment{Generated samples $x \sim \bar{\mathcal{X}}$}
\State $\Theta_H \leftarrow \Theta_H - \Gamma(\nabla_{\Theta_{H}}\mathcal{L})$ \label{alg:line:H}
\Until{Max training iterations $T$}
\end{algorithmic}
\caption{Training algorithm for the proposed model.} \label{alg:train}
\end{algorithm}
\section{Experiments}
To carry out the different experiments, we have used a subset of the IAM corpus~\cite{marti2002iam} as our multi-writer handwritten dataset $\{\mathcal{X},\mathcal{Y},\mathcal{W}\}$. It consists of $62,857$ handwritten word snippets, written by 500 different individuals. Each word image has its associated writer and transcription metadata. A test subset of 160 writers has been kept apart during training to check whether the generative model is able to cope with unseen calligraphic styles. We have also used a subset of $22,500$ unique English words from the Brown~\cite{bird2009natural} corpus as the source of strings for the content input. A test set of 400 unique words, disjoint from the IAM transcriptions has been used to test the performance when producing OOV words. To quantitatively measure the image quality, diversity and the ability to transfer style attributes of the proposed approach we will use the Fr{\'e}chet Inception Distance (FID)~\cite{heusel2017gans,borji2019pros}, measuring the distance between the Inception-v3 activation distributions for generated $\bar{\mathcal{X}}$ and real samples $\mathcal{X}$ for each writer $w_i$ separately, and finally averaging them. Inception features, trained over ImageNet data, have not been designed to discern between different handwriting images. Although this measure might not be ideal to evaluate our specific case, it will still serve as an indication of the similarity between generated and real images.
\begingroup
\setlength{\tabcolsep}{1pt}
\renewcommand{\arraystretch}{1.25}
\begin{figure}[t!]
\centering
\begin{tabular}{l@{\hskip 6pt}ccccccc}
a) \tiny{IV-S}&\includegraphics[width=0.12\linewidth, valign=c]{images/in_tr/000-5_whom-whom.png}&
\includegraphics[width=0.12\linewidth, valign=c]{images/in_tr/013-39_twice-trice.png}&
\includegraphics[width=0.12\linewidth, valign=c]{images/in_tr/022-19_sound-sound.png}&
\includegraphics[width=0.12\linewidth, valign=c]{images/in_tr/023-38_blonde-blonde.png}&
\includegraphics[width=0.12\linewidth, valign=c]{images/in_tr/024-68_mouth-month.png}&
\includegraphics[width=0.12\linewidth, valign=c]{images/in_tr/025-21_anyone-anyone.png}&
\includegraphics[width=0.12\linewidth, valign=c]{images/in_tr/025-36_posse-pose.png}\\
b) \tiny{IV-U}&\includegraphics[width=0.12\linewidth, valign=c]{images/in_te/168-19_sound-sound.png}&
\includegraphics[width=0.12\linewidth, valign=c]{images/in_te/180-21_anyone-anyone.png}&
\includegraphics[width=0.12\linewidth, valign=c]{images/in_te/181-25_fever-fever.png}&
\includegraphics[width=0.12\linewidth, valign=c]{images/in_te/181-29_humans-himans.png}&
\includegraphics[width=0.12\linewidth, valign=c]{images/in_te/183-44_virtual-virtual.png}&
\includegraphics[width=0.12\linewidth, valign=c]{images/in_te/198-40_Having-Having.png}&
\includegraphics[width=0.12\linewidth, valign=c]{images/in_te/304-31_beyond-beyond.png}\\
c) \tiny{OOV-S}&\includegraphics[width=0.12\linewidth, valign=c]{images/oo_tr/000-30_reviews-reviems.png}&
\includegraphics[width=0.12\linewidth, valign=c]{images/oo_tr/032-57_Similar-Similar.png}&
\includegraphics[width=0.12\linewidth, valign=c]{images/oo_tr/034-20_grades-grades.png}&
\includegraphics[width=0.12\linewidth, valign=c]{images/oo_tr/098-72_push-push.png}&
\includegraphics[width=0.12\linewidth, valign=c]{images/oo_tr/098-74_skiing-stiing.png}&
\includegraphics[width=0.12\linewidth, valign=c]{images/oo_tr/100-24_mood-mod.png}&
\includegraphics[width=0.12\linewidth, valign=c]{images/oo_tr/134-78_monkey-monkey.png}\\
d) \tiny{OOV-U}&\includegraphics[width=0.12\linewidth, valign=c]{images/oo_te/168-42_squares-squares.png}&
\includegraphics[width=0.12\linewidth, valign=c]{images/oo_te/170-1_planets-planets.png}&
\includegraphics[width=0.12\linewidth, valign=c]{images/oo_te/173-72_push-push.png}&
\includegraphics[width=0.12\linewidth, valign=c]{images/oo_te/177-30_reviews-reviews.png}&
\includegraphics[width=0.12\linewidth, valign=c]{images/oo_te/179-23_foul-foul.png}&
\includegraphics[width=0.12\linewidth, valign=c]{images/oo_te/543-28_nodding-nodiing.png}&
\includegraphics[width=0.12\linewidth, valign=c]{images/oo_te/545-65_Flower-Flower.png}\\
\end{tabular}
\caption{Word image generation. a) In-Vocabulary (IV) words and seen (S) styles; b) In-Vocabulary (IV) words and unseen (U) styles; c) Out-of-Vocabulary (OOV) words and seen (S) styles and d) Out-of-Vocabulary (OOV) words and unseen (U) styles.}
\label{fig:sample}
\end{figure}
\endgroup
\subsection{Generating Handwritten Word Images}
We present in Fig.~\ref{fig:sample} an illustrative selection of generated handwritten words. We appreciate the realistic and diverse aspect of the produced images. Qualitatively, we observe that the proposed approach is able to yield satisfactory results even when dealing with both words and calligraphic styles never seen during training. But, when analyzing the different experimental settings in Table~\ref{tab:4_scenario}, we appreciate that the FID measure slightly degrades when either we input an OOV word or a style never seen during training. Nevertheless, the reached FID measures in all four settings satisfactorily compare with the baseline achieved by real data.
\begin{table}[t!]
\caption{FID between generated images and real images of corresponding set.}
\label{tab:4_scenario}
\centering
\begin{tabular}{lccccc}
\toprule
&Real images & IV-S & IV-U & OOV-S & OOV-U\\
\midrule
FID & $90.43$ & $120.07$ & $124.30$& $125.87$ &$130.68$\\
\bottomrule
\end{tabular}
\end{table}
\begin{figure}[h!t]
\centering
\includegraphics[width=\linewidth]{images/TSNE_ALL_LR_15_PER_25_ITE_10000.png}
\caption{t-SNE embedding visualization of $2.500$ generated instances of the word \texttt{"deep"}.}
\label{fig:tsne}
\end{figure}
In order to show the ability of the proposed method to produce a diverse set of generated images, we present in Fig.~\ref{fig:tsne} a t-SNE~\cite{maaten2008visualizing} visualization of different instances produced with a fixed textual content while varying the calligraphic style inputs. Different clusters corresponding to particular slants, stroke widths, character roundnesses, ligatures and cursive writings are observed.
\begin{figure}[t!]
\centering
\begin{tabular}{c@{\hskip 8pt} ccccccc}
\toprule
&& \multicolumn{6}{c}{\textbf{Style Images}}\\
\cmidrule{3-8}
&&
\includegraphics[width=.12\linewidth, valign=c]{images/vs_funit1/190_d04-062-03-01.png} &
\includegraphics[width=.12\linewidth, valign=c]{images/vs_funit1/203_d06-003-01-03.png} &
\includegraphics[width=.12\linewidth, valign=c]{images/vs_funit1/279_f04-053-01-10.png} &
\includegraphics[width=.12\linewidth,, valign=c]{images/vs_funit1/281_f04-068-01-01.png} &
\includegraphics[width=.12\linewidth, valign=c]{images/vs_funit1/583_n04-048-00-07.png} &
\includegraphics[width=.12\linewidth, valign=c]{images/vs_funit1/616_p03-027-00-08.png} \\
\midrule
\multirow{12}{*}{\raisebox{-8pt}{\rotatebox[origin=r]{90}{\textbf{Textual Content}}}} & \textbf{FUNIT} &
\multirow{2}{*}{\includegraphics[width=.12\linewidth, valign=c]{images/vs_funit1/content_207_d06_037_02_00_style_190.jpg}} &
\multirow{2}{*}{\includegraphics[width=.12\linewidth, valign=c]{images/vs_funit1/content_207_d06_037_02_00_style_203.jpg}} &
\multirow{2}{*}{\includegraphics[width=.12\linewidth, valign=c]{images/vs_funit1/content_207_d06_037_02_00_style_279.jpg}} &
\multirow{2}{*}{\includegraphics[width=.12\linewidth, valign=c]{images/vs_funit1/content_207_d06_037_02_00_style_281.jpg}} &
\multirow{2}{*}{\includegraphics[width=.12\linewidth, valign=c]{images/vs_funit1/content_207_d06_037_02_00_style_583.jpg}} &
\multirow{2}{*}{\includegraphics[width=.12\linewidth, valign=c]{images/vs_funit1/content_207_d06_037_02_00_style_616.jpg}} \\
& \includegraphics[height=0.3cm, valign=c]{images/vs_funit1/d06-037-02-00.png} &&&&&\\
\cmidrule{3-8}
& \textbf{ours} &
\multirow{2}{*}{\includegraphics[width=.12\linewidth, valign=c]{images/vs_funit1/190-1_which-which.png}} &
\multirow{2}{*}{\includegraphics[width=.12\linewidth, valign=c]{images/vs_funit1/203-1_which-which.png}} &
\multirow{2}{*}{\includegraphics[width=.12\linewidth, valign=c]{images/vs_funit1/279-1_which-which.png}} &
\multirow{2}{*}{\includegraphics[width=.12\linewidth, valign=c]{images/vs_funit1/281-1_which-which.png}} &
\multirow{2}{*}{\includegraphics[width=.12\linewidth, valign=c]{images/vs_funit1/583-1_which-which.png}} &
\multirow{2}{*}{\includegraphics[width=.12\linewidth, valign=c]{images/vs_funit1/616-1_which-which.png}} \\
& \texttt{"which"} &&&&&\\
\cmidrule{2-8}
& \textbf{FUNIT} &
\multirow{2}{*}{\includegraphics[width=.12\linewidth, valign=c]{images/vs_funit2/content_550_m04-231-02-05_style_190.jpg}} &
\multirow{2}{*}{\includegraphics[width=.12\linewidth, valign=c]{images/vs_funit2/content_550_m04-231-02-05_style_203.jpg}} &
\multirow{2}{*}{\includegraphics[width=.12\linewidth, valign=c]{images/vs_funit2/content_550_m04-231-02-05_style_279.jpg}} &
\multirow{2}{*}{\includegraphics[width=.12\linewidth, valign=c]{images/vs_funit2/content_550_m04-231-02-05_style_281.jpg}} &
\multirow{2}{*}{\includegraphics[width=.12\linewidth, valign=c]{images/vs_funit2/content_550_m04-231-02-05_style_583.jpg}} &
\multirow{2}{*}{\includegraphics[width=.12\linewidth, valign=c]{images/vs_funit2/content_550_m04-231-02-05_style_616.jpg}} \\
& \includegraphics[height=0.3cm, valign=c]{images/vs_funit2/m04-231-02-05.png} &&&&&\\
\cmidrule{3-8}
& \textbf{ours} &
\multirow{2}{*}{\includegraphics[width=.12\linewidth, valign=c]{images/vs_funit2/190-1_Thank-Thank.png}} &
\multirow{2}{*}{\includegraphics[width=.12\linewidth, valign=c]{images/vs_funit2/203-1_Thank-Thank.png}} &
\multirow{2}{*}{\includegraphics[width=.12\linewidth, valign=c]{images/vs_funit2/279-1_Thank-Thank.png}} &
\multirow{2}{*}{\includegraphics[width=.12\linewidth, valign=c]{images/vs_funit2/281-1_Thank-Thank.png}} &
\multirow{2}{*}{\includegraphics[width=.12\linewidth, valign=c]{images/vs_funit2/583-1_Thank-Thank.png}} &
\multirow{2}{*}{\includegraphics[width=.12\linewidth, valign=c]{images/vs_funit2/616-1_Thank-Thank.png}} \\
& \texttt{"Thank"} &&&&&\\
\cmidrule{2-8}
& \textbf{FUNIT} &
\multirow{2}{*}{\includegraphics[width=.12\linewidth, valign=c]{images/vs_funit3/content_598_p02-069-02-02_style_190.jpg}} &
\multirow{2}{*}{\includegraphics[width=.12\linewidth, valign=c]{images/vs_funit3/content_598_p02-069-02-02_style_203.jpg}} &
\multirow{2}{*}{\includegraphics[width=.12\linewidth, valign=c]{images/vs_funit3/content_598_p02-069-02-02_style_279.jpg}} &
\multirow{2}{*}{\includegraphics[width=.12\linewidth, valign=c]{images/vs_funit3/content_598_p02-069-02-02_style_281.jpg}} &
\multirow{2}{*}{\includegraphics[width=.12\linewidth, valign=c]{images/vs_funit3/content_598_p02-069-02-02_style_583.jpg}} &
\multirow{2}{*}{\includegraphics[width=.12\linewidth, valign=c]{images/vs_funit3/content_598_p02-069-02-02_style_616.jpg}} \\
& \includegraphics[height=0.3cm, valign=c]{images/vs_funit3/p02-069-02-02.png} &&&&&\\
\cmidrule{3-8}
& \textbf{ours} &
\multirow{2}{*}{\includegraphics[width=.12\linewidth, valign=c]{images/vs_funit3/190-1_inside-inside.png}} &
\multirow{2}{*}{\includegraphics[width=.12\linewidth, valign=c]{images/vs_funit3/203-1_inside-inside.png}} &
\multirow{2}{*}{\includegraphics[width=.12\linewidth, valign=c]{images/vs_funit3/279-1_inside-inside.png}} &
\multirow{2}{*}{\includegraphics[width=.12\linewidth, valign=c]{images/vs_funit3/281-1_inside-inside.png}} &
\multirow{2}{*}{\includegraphics[width=.12\linewidth, valign=c]{images/vs_funit3/583-1_inside-inside.png}} &
\multirow{2}{*}{\includegraphics[width=.12\linewidth, valign=c]{images/vs_funit3/616-1_inside-inside.png}} \\
& \texttt{"inside"} &&&&&\\
\bottomrule
\end{tabular}
\caption{Comparison of handwritten word generation with FUNIT~\cite{liu2019few}.}
\label{tab:funit}
\end{figure}
To further demonstrate the ability of the proposed approach to coalesce content and style information into the generated handwritten word images, we compare in Fig.~\ref{tab:funit} our produced results with the outcomes of the state-of-the-art approach FUNIT~\cite{liu2019few}. Being an image-to-image translation method, FUNIT starts with a content image and then injects the style attributes derived from a second sample image. Although FUNIT performs well for natural scene images, it is clear that such kind of approaches do not apply well for the specific case of handwritten words. Starting with a content image instead of a text string confines the generative process to the shapes of the initial drawing. When infusing the style features, the FUNIT method is only able to deform the stroke textures, often resulting in extremely distorted words. Conversely, our proposed generative process is able to produce realistic and diverse word samples given a content text string and a calligraphic style example. We observe how for the different produced versions of the same word, the proposed approach is able to
change style attributes as stroke width or slant, to produce both cursive words, where all characters are connected through ligatures as well as disconnected writings, and even render the same characters differently, \emph{e.g.} note the characters \texttt{n} or \texttt{s} in \texttt{"Thank"} or \texttt{"inside"} respectively.
\subsection{Latent Space Interpolations}
The generator network $G$ learns to map feature points $F$ in the latent space to synthetic handwritten word images. Such latent space presents a structure worth exploring. We first interpolate in Fig.~\ref{fig:interpolation} between two different points $F_s^A$ and $F_s^B$ corresponding to two different calligraphic styles $w_A$ and $w_B$ while keeping the textual contents $t$ fixed. We observe how the generated images smoothly adjust from one style to another. Again note how individual characters evolve from one typography to another, \emph{e.g.} the \texttt{l} from \texttt{"also"}, or the \texttt{f} from \texttt{"final"}.
\begin{figure}[t!]
\centering
\resizebox{\linewidth}{!}{
\begin{tabular}{rccccccccccc}
& $w_A$ & & & & & & & & & & $w_B$\\ \midrule
\tiny{Real} & \includegraphics[width=0.07\linewidth,valign=c]{images/interpolation/also/real_1.png} &
& & & & & & & & & \includegraphics[width=0.07\linewidth,valign=c]{images/interpolation/also/real_0.png} \\
\tiny{Generated}&\includegraphics[width=0.07\linewidth,valign=c]{images/interpolation/also/fake_10.png} &
\includegraphics[width=0.07\linewidth,valign=c]{images/interpolation/also/fake_9.png} &
\includegraphics[width=0.07\linewidth,valign=c]{images/interpolation/also/fake_8.png} &
\includegraphics[width=0.07\linewidth,valign=c]{images/interpolation/also/fake_7.png} &
\includegraphics[width=0.07\linewidth,valign=c]{images/interpolation/also/fake_6.png} &
\includegraphics[width=0.07\linewidth,valign=c]{images/interpolation/also/fake_5.png} &
\includegraphics[width=0.07\linewidth,valign=c]{images/interpolation/also/fake_4.png} &
\includegraphics[width=0.07\linewidth,valign=c]{images/interpolation/also/fake_3.png} &
\includegraphics[width=0.07\linewidth,valign=c]{images/interpolation/also/fake_2.png} &
\includegraphics[width=0.07\linewidth,valign=c]{images/interpolation/also/fake_1.png} &
\includegraphics[width=0.07\linewidth,valign=c]{images/interpolation/also/fake_0.png} \\
\midrule
\tiny{Real} & \includegraphics[width=0.07\linewidth,valign=c]{images/interpolation/final/real_0.png} &
& & & & & & & & & \includegraphics[width=0.07\linewidth,valign=c]{images/interpolation/final/real_1.png} \\
\tiny{Generated}&\includegraphics[width=0.07\linewidth,valign=c]{images/interpolation/final/fake_0.png} &
\includegraphics[width=0.07\linewidth,valign=c]{images/interpolation/final/fake_1.png} &
\includegraphics[width=0.07\linewidth,valign=c]{images/interpolation/final/fake_2.png} &
\includegraphics[width=0.07\linewidth,valign=c]{images/interpolation/final/fake_3.png} &
\includegraphics[width=0.07\linewidth,valign=c]{images/interpolation/final/fake_4.png} &
\includegraphics[width=0.07\linewidth,valign=c]{images/interpolation/final/fake_5.png} &
\includegraphics[width=0.07\linewidth,valign=c]{images/interpolation/final/fake_6.png} &
\includegraphics[width=0.07\linewidth,valign=c]{images/interpolation/final/fake_7.png} &
\includegraphics[width=0.07\linewidth,valign=c]{images/interpolation/final/fake_8.png} &
\includegraphics[width=0.07\linewidth,valign=c]{images/interpolation/final/fake_9.png} &
\includegraphics[width=0.07\linewidth,valign=c]{images/interpolation/final/fake_10.png} \\
\midrule
\tiny{Real} & \includegraphics[width=0.07\linewidth,valign=c]{images/interpolation/which/real_0.png} &
& & & & & & & & & \includegraphics[width=0.07\linewidth,valign=c]{images/interpolation/which/real_1.png} \\
\tiny{Generated}&\includegraphics[width=0.07\linewidth,valign=c]{images/interpolation/which/fake_0.png} &
\includegraphics[width=0.07\linewidth,valign=c]{images/interpolation/which/fake_1.png} &
\includegraphics[width=0.07\linewidth,valign=c]{images/interpolation/which/fake_2.png} &
\includegraphics[width=0.07\linewidth,valign=c]{images/interpolation/which/fake_3.png} &
\includegraphics[width=0.07\linewidth,valign=c]{images/interpolation/which/fake_4.png} &
\includegraphics[width=0.07\linewidth,valign=c]{images/interpolation/which/fake_5.png} &
\includegraphics[width=0.07\linewidth,valign=c]{images/interpolation/which/fake_6.png} &
\includegraphics[width=0.07\linewidth,valign=c]{images/interpolation/which/fake_7.png} &
\includegraphics[width=0.07\linewidth,valign=c]{images/interpolation/which/fake_8.png} &
\includegraphics[width=0.07\linewidth,valign=c]{images/interpolation/which/fake_9.png} &
\includegraphics[width=0.07\linewidth,valign=c]{images/interpolation/which/fake_10.png} \\
\midrule
\tiny{Real} & \includegraphics[width=0.07\linewidth,valign=c]{images/interpolation/that/real_0.png} &
& & & & & & & & & \includegraphics[width=0.07\linewidth,valign=c]{images/interpolation/that/real_1.png} \\
\tiny{Generated}&\includegraphics[width=0.07\linewidth,valign=c]{images/interpolation/that/fake_0.png} &
\includegraphics[width=0.07\linewidth,valign=c]{images/interpolation/that/fake_1.png} &
\includegraphics[width=0.07\linewidth,valign=c]{images/interpolation/that/fake_2.png} &
\includegraphics[width=0.07\linewidth,valign=c]{images/interpolation/that/fake_3.png} &
\includegraphics[width=0.07\linewidth,valign=c]{images/interpolation/that/fake_4.png} &
\includegraphics[width=0.07\linewidth,valign=c]{images/interpolation/that/fake_5.png} &
\includegraphics[width=0.07\linewidth,valign=c]{images/interpolation/that/fake_6.png} &
\includegraphics[width=0.07\linewidth,valign=c]{images/interpolation/that/fake_7.png} &
\includegraphics[width=0.07\linewidth,valign=c]{images/interpolation/that/fake_8.png} &
\includegraphics[width=0.07\linewidth,valign=c]{images/interpolation/that/fake_9.png} &
\includegraphics[width=0.07\linewidth,valign=c]{images/interpolation/that/fake_10.png} \\
\midrule
\tiny{Real} & \includegraphics[width=0.07\linewidth,valign=c]{images/interpolation/point/real_0.png} &
& & & & & & & & & \includegraphics[width=0.07\linewidth,valign=c]{images/interpolation/point/real_1.png} \\
\tiny{Generated}&\includegraphics[width=0.07\linewidth,valign=c]{images/interpolation/point/fake_0.png} &
\includegraphics[width=0.07\linewidth,valign=c]{images/interpolation/point/fake_1.png} &
\includegraphics[width=0.07\linewidth,valign=c]{images/interpolation/point/fake_2.png} &
\includegraphics[width=0.07\linewidth,valign=c]{images/interpolation/point/fake_3.png} &
\includegraphics[width=0.07\linewidth,valign=c]{images/interpolation/point/fake_4.png} &
\includegraphics[width=0.07\linewidth,valign=c]{images/interpolation/point/fake_5.png} &
\includegraphics[width=0.07\linewidth,valign=c]{images/interpolation/point/fake_6.png} &
\includegraphics[width=0.07\linewidth,valign=c]{images/interpolation/point/fake_7.png} &
\includegraphics[width=0.07\linewidth,valign=c]{images/interpolation/point/fake_8.png} &
\includegraphics[width=0.07\linewidth,valign=c]{images/interpolation/point/fake_9.png} &
\includegraphics[width=0.07\linewidth,valign=c]{images/interpolation/point/fake_10.png} \\
\midrule
\tiny{Real} & \includegraphics[width=0.07\linewidth,valign=c]{images/interpolation/because/real_0.png} &
& & & & & & & & & \includegraphics[width=0.07\linewidth,valign=c]{images/interpolation/because/real_1.png} \\
\tiny{Generated}&\includegraphics[width=0.07\linewidth,valign=c]{images/interpolation/because/fake_0.png} &
\includegraphics[width=0.07\linewidth,valign=c]{images/interpolation/because/fake_1.png} &
\includegraphics[width=0.07\linewidth,valign=c]{images/interpolation/because/fake_2.png} &
\includegraphics[width=0.07\linewidth,valign=c]{images/interpolation/because/fake_3.png} &
\includegraphics[width=0.07\linewidth,valign=c]{images/interpolation/because/fake_4.png} &
\includegraphics[width=0.07\linewidth,valign=c]{images/interpolation/because/fake_5.png} &
\includegraphics[width=0.07\linewidth,valign=c]{images/interpolation/because/fake_6.png} &
\includegraphics[width=0.07\linewidth,valign=c]{images/interpolation/because/fake_7.png} &
\includegraphics[width=0.07\linewidth,valign=c]{images/interpolation/because/fake_8.png} &
\includegraphics[width=0.07\linewidth,valign=c]{images/interpolation/because/fake_9.png} &
\includegraphics[width=0.07\linewidth,valign=c]{images/interpolation/because/fake_10.png} \\
\bottomrule
\end{tabular}}
\caption{Latent space interpolation between two calligraphic styles for different words while keeping contents fixed.}
\label{fig:interpolation}
\end{figure}
Contrary to the continuous nature of the style latent space, the original content space is discrete in nature. Instead of computing point-wise interpolations, we present in Fig.~\ref{fig:ladder} the obtained word images for different styles when following a ``word ladder'' puzzle game, \emph{i.e.} going from one word to another, one character difference at a time. Here we observe how different contents influence stylistic aspects. Usually \texttt{s} and \texttt{i} are disconnected when rendering the word \texttt{"sired"} but often appear with a ligature when jumping to the word \texttt{"fired"}.
\begin{figure}[t!]
\centering
\begin{tabular}{cccccccccccc}
\texttt{\scriptsize{"three"}}&
\texttt{\scriptsize{"threw"}}&
\texttt{\scriptsize{"shrew"}}&
\texttt{\scriptsize{"shred"}}&
\texttt{\scriptsize{"sired"}}&
\texttt{\scriptsize{"fired"}}&
\texttt{\scriptsize{"fined"}}&
\texttt{\scriptsize{"firer"}}&
\texttt{\scriptsize{"fiver"}}&
\texttt{\scriptsize{"fever"}}&
\texttt{\scriptsize{"sever"}}&
\texttt{\scriptsize{"seven"}}\\
\midrule
\includegraphics[width=0.07\linewidth]{images/word_ladder/1/168-1_three-three.png}&
\includegraphics[width=0.07\linewidth]{images/word_ladder/1/168-2_threw-threm.png}&
\includegraphics[width=0.07\linewidth]{images/word_ladder/1/168-3_shrew-shren.png}&
\includegraphics[width=0.07\linewidth]{images/word_ladder/1/168-4_shred-shred.png}&
\includegraphics[width=0.07\linewidth]{images/word_ladder/1/168-5_sired-sired.png}&
\includegraphics[width=0.07\linewidth]{images/word_ladder/1/168-6_fired-fired.png}&
\includegraphics[width=0.07\linewidth]{images/word_ladder/1/168-7_fined-fined.png}&
\includegraphics[width=0.07\linewidth]{images/word_ladder/1/168-8_finer-finer.png}&
\includegraphics[width=0.07\linewidth]{images/word_ladder/1/168-9_fiver-fiver.png}&
\includegraphics[width=0.07\linewidth]{images/word_ladder/1/168-10_fever-fever.png}&
\includegraphics[width=0.07\linewidth]{images/word_ladder/1/168-11_sever-sever.png}&
\includegraphics[width=0.07\linewidth]{images/word_ladder/1/168-12_seven-seven.png}\\
\includegraphics[width=0.07\linewidth]{images/word_ladder/2/204-1_three-three.png}&
\includegraphics[width=0.07\linewidth]{images/word_ladder/2/204-2_threw-threw.png}&
\includegraphics[width=0.07\linewidth]{images/word_ladder/2/204-3_shrew-shrew.png}&
\includegraphics[width=0.07\linewidth]{images/word_ladder/2/204-4_shred-shred.png}&
\includegraphics[width=0.07\linewidth]{images/word_ladder/2/204-5_sired-sired.png}&
\includegraphics[width=0.07\linewidth]{images/word_ladder/2/204-6_fired-fired.png}&
\includegraphics[width=0.07\linewidth]{images/word_ladder/2/204-7_fined-fined.png}&
\includegraphics[width=0.07\linewidth]{images/word_ladder/2/204-8_finer-finer.png}&
\includegraphics[width=0.07\linewidth]{images/word_ladder/2/204-9_fiver-fiver.png}&
\includegraphics[width=0.07\linewidth]{images/word_ladder/2/204-10_fever-fever.png}&
\includegraphics[width=0.07\linewidth]{images/word_ladder/2/204-11_sever-sever.png}&
\includegraphics[width=0.07\linewidth]{images/word_ladder/2/204-12_seven-seven.png}\\
\includegraphics[width=0.07\linewidth]{images/word_ladder/3/304-1_three-three.png}&
\includegraphics[width=0.07\linewidth]{images/word_ladder/3/304-2_threw-threw.png}&
\includegraphics[width=0.07\linewidth]{images/word_ladder/3/304-3_shrew-shren.png}&
\includegraphics[width=0.07\linewidth]{images/word_ladder/3/304-4_shred-shred.png}&
\includegraphics[width=0.07\linewidth]{images/word_ladder/3/304-5_sired-sired.png}&
\includegraphics[width=0.07\linewidth]{images/word_ladder/3/304-6_fired-fired.png}&
\includegraphics[width=0.07\linewidth]{images/word_ladder/3/304-7_fined-fined.png}&
\includegraphics[width=0.07\linewidth]{images/word_ladder/3/304-8_finer-finer.png}&
\includegraphics[width=0.07\linewidth]{images/word_ladder/3/304-9_fiver-fiver.png}&
\includegraphics[width=0.07\linewidth]{images/word_ladder/3/304-10_fever-fever.png}&
\includegraphics[width=0.07\linewidth]{images/word_ladder/3/304-11_sever-sever.png}&
\includegraphics[width=0.07\linewidth]{images/word_ladder/3/304-12_seven-seven.png}\\
\includegraphics[width=0.07\linewidth]{images/word_ladder/4/281-1_three-three.png}&
\includegraphics[width=0.07\linewidth]{images/word_ladder/4/281-2_threw-threm.png}&
\includegraphics[width=0.07\linewidth]{images/word_ladder/4/281-3_shrew-shrem.png}&
\includegraphics[width=0.07\linewidth]{images/word_ladder/4/281-4_shred-shred.png}&
\includegraphics[width=0.07\linewidth]{images/word_ladder/4/281-5_sired-sired.png}&
\includegraphics[width=0.07\linewidth]{images/word_ladder/4/281-6_fired-fired.png}&
\includegraphics[width=0.07\linewidth]{images/word_ladder/4/281-7_fined-fined.png}&
\includegraphics[width=0.07\linewidth]{images/word_ladder/4/281-8_finer-finer.png}&
\includegraphics[width=0.07\linewidth]{images/word_ladder/4/281-9_fiver-fiver.png}&
\includegraphics[width=0.07\linewidth]{images/word_ladder/4/281-10_fever-fever.png}&
\includegraphics[width=0.07\linewidth]{images/word_ladder/4/281-11_sever-sever.png}&
\includegraphics[width=0.07\linewidth]{images/word_ladder/4/281-12_seven-seven.png}\\
\includegraphics[width=0.07\linewidth]{images/word_ladder/5/592-1_three-three.png}&
\includegraphics[width=0.07\linewidth]{images/word_ladder/5/592-2_threw-theew.png}&
\includegraphics[width=0.07\linewidth]{images/word_ladder/5/592-3_shrew-shrew.png}&
\includegraphics[width=0.07\linewidth]{images/word_ladder/5/592-4_shred-shred.png}&
\includegraphics[width=0.07\linewidth]{images/word_ladder/5/592-5_sired-sired.png}&
\includegraphics[width=0.07\linewidth]{images/word_ladder/5/592-6_fired-fired.png}&
\includegraphics[width=0.07\linewidth]{images/word_ladder/5/592-7_fined-fined.png}&
\includegraphics[width=0.07\linewidth]{images/word_ladder/5/592-8_finer-finer.png}&
\includegraphics[width=0.07\linewidth]{images/word_ladder/5/592-9_fiver-fiver.png}&
\includegraphics[width=0.07\linewidth]{images/word_ladder/5/592-10_fever-fiver.png}&
\includegraphics[width=0.07\linewidth]{images/word_ladder/5/592-11_sever-sever.png}&
\includegraphics[width=0.07\linewidth]{images/word_ladder/5/592-12_seven-seven.png}\\
\bottomrule
\end{tabular}
\caption{Word ladder. From \texttt{"three"} to \texttt{"seven"} changing one character at a time, generated for five different calligraphic styles.}
\label{fig:ladder}
\end{figure}
\subsection{Impact of the Learning Objectives}
Along this paper, we have proposed to guide the generation process by three complementary goals. The discriminator loss $\mathcal{L}_d$ controlling the genuine appearance of the generated images $\bar{x}$. The writer classification loss $\mathcal{L}_w$ forcing $\bar{x}$ to mimic the calligraphic style of input images $X_i$. The recognition loss $\mathcal{L}_r$ guaranteeing that $\bar{x}$ is readable and conveys the exact text information $t$. We analyze in Table~\ref{tab:abla_loss} the effect of each learning objective.
\begingroup
\setlength{\tabcolsep}{4pt}
\begin{table}[t!]
\centering
\caption{Effect of each different learning objectives when generating the content $t=\texttt{"vision"}$ for different styles.}
\label{tab:abla_loss}
\begin{tabular}{c@{\hskip 4pt}c@{\hskip 4pt}c @{\hskip 8pt}ccccc}
\toprule
\multirow{2}{*}{$\mathcal{L}_d$} & \multirow{2}{*}{$\mathcal{L}_w$} & \multirow{2}{*}{$\mathcal{L}_r$} &
\multirow{2}{*}{FID} & \multicolumn{4}{c}{\textbf{Style Images}} \\
\cmidrule{5-8}
&&&&\includegraphics[height=0.6cm,valign=c]{images/sample/190_d04-062-03-01.png} &
\includegraphics[height=0.6cm,valign=c]{images/sample/203_d06-003-01-03.png} &
\includegraphics[height=0.6cm,valign=c]{images/sample/281_f04-068-01-01.png} &
\includegraphics[height=0.6cm,valign=c]{images/sample/616_p03-027-00-08.png}\\
\midrule
\checkmark & - & - & 364.10 & \includegraphics[height=0.6cm,valign=c]{images/sample/gansolo190-3_great-MMMKffff.png} &
\includegraphics[height=0.6cm,valign=c]{images/sample/gansolo203-3_great-huMMMJff.png} &
\includegraphics[height=0.6cm,valign=c]{images/sample/gansolo281-3_great-gegelWXf.png} &
\includegraphics[height=0.6cm,valign=c]{images/sample/gansolo616-3_great-MMff.png}\\
\checkmark & \checkmark & - & 207.47 &
\includegraphics[height=0.6cm,valign=c]{images/sample/gan_wr190-vision-dQrwwDee.png} &
\includegraphics[height=0.6cm,valign=c]{images/sample/ganwr203-vision-wwwwIwrd.png} &
\includegraphics[height=0.6cm,valign=c]{images/sample/ganwr281-vision-dwwwwwww.png} &
\includegraphics[height=0.6cm,valign=c]{images/sample/ganwr616-vision-mwwwwwww.png}\\
\checkmark & - & \checkmark & 138.80 &
\includegraphics[height=0.6cm,valign=c]{images/sample/gan_rec190-vision-vision.png} &
\includegraphics[height=0.6cm,valign=c]{images/sample/ganrec203-vision-vision.png} &
\includegraphics[height=0.6cm,valign=c]{images/sample/ganrec281-vision-vision.png} &
\includegraphics[height=0.6cm,valign=c]{images/sample/ganrec616-vision-vision.png}\\
\checkmark & \checkmark & \checkmark & \textbf{130.68} &
\includegraphics[height=0.6cm,valign=c]{images/sample/ours190-2vision-vision.png} &
\includegraphics[height=0.6cm,valign=c]{images/sample/ours203-2_vision-vision.png} &
\includegraphics[height=0.6cm,valign=c]{images/sample/ours281-2_vision-vision.png} &
\includegraphics[height=0.6cm,valign=c]{images/sample/our616-vision-vision.png}\\
\bottomrule
\end{tabular}
\end{table}
\endgroup
The sole use of the $\mathcal{L}_d$ leads to constantly generating an image that is able to fool the discriminator. Although the generated image looks like handwritten strokes, the content and style inputs are ignored. When combining the discriminator with the auxiliary task of writer classification $\mathcal{L}_w$, the produced results are more encouraging, but the input text is still ignored, always tending to generate the word \texttt{"the"}, since it is the most common word seen during training. When combining the discriminator with the word recognizer loss $\mathcal{L}_r$, the desired word is rendered. However, as in~\cite{alonso2019adversarial}, we suffer from the mode collapse problem, always producing unvarying word instances. When combining the three learning objectives we appreciate that we are able to correctly render the appropriate textual content while mimicking the input styles, producing diverse results. We appreciate that the FID measure also decreases for each successive combination.
\subsection{Human Evaluation}
Finally, we also tested whether the generated images were actually indistinguishable from real ones by human judgments. We have conducted a human evaluation study as follows: we have asked $200$ human examiners to assess whether a set of images were written by a human or artificially generated. Appraisers were presented a total of sixty images, one at a time, and they had to choose if each of them was real of fake. We chose thirty real words from the IAM test partition from ten different writers. We then generated thirty artificial samples by using OOV textual contents and by randomly taking the previous writers as the sources for the calligraphic styles. Such sets were not curated, so the only filter was that the generated samples had to be correctly transcribed by the word recognizer network $R$. In total we collected $12,000$ responses. In Table~\ref{tab:human_study} we present the confusion matrix of the human assessments, with Accuracy (ACC), Precision (P), Recall (R), False Positive Rate (FPR) and False Omission Rate (FOR) values. The study revealed that our generative model was clearly perceived as plausible, since a great portion of the generated samples were deemed genuine. Only a $49.3\%$ of the images were properly identified, which shows a similar performance than a random binary classifier. Accuracies over different examiners were normally distributed. We also observe that the synthetically generated word images were judged more often as being real than correctly identified as fraudulent, with a final FPR of $55.4\%$.
\begin{table}[t!]
\caption{Human evaluation plausibility experiment.}
\label{tab:human_study}
\centering
\begin{tabular}{c @{\hskip 32pt} c}
\begin{tabular}{c @{\hskip 12pt}c @{\hskip 8pt} c @{\hskip 8pt} c}
\toprule
\multirow{2}{*}{Actual} & \multicolumn{2}{c}{Predicted}\\
\cmidrule{2-3}
& Real & Fake&\\
\midrule
Genuine & $27.01$ & $22.99$& R: $ 54.1 $\\
Generated & $27.69$ & $22.31$& FPR: $55.4$\\
& P: $49.4$& FOR: $50.8$& ACC: $49.3$\\
\bottomrule
\end{tabular}&
\includegraphics[width=0.365\linewidth,valign=c]{images/histogram.pdf}\\
a) Confusion matrix ($\%$) &b) Accuracy distribution\\
\end{tabular}
\end{table}
\section{Conclusion}
We have presented a novel image generation architecture that produces realistic and varied artificially rendered samples of handwritten words. Our pipeline can yield credible word images by conditioning the generative process with both calligraphic style features and textual content. Furthermore, by jointly guiding our generator with three different cues: a discriminator, a style classifier and a content recognizer, our model is able to render any input word, not depending on any predefined vocabulary, while incorporating calligraphic styles in a few-shot setup. Experimental results demonstrate that the proposed method yields images with such a great realistic quality that are indistinguishable by humans.
\section*{Acknowledgements}
This work was supported by EU H2020 SME Instrument project 849628, the Spanish projects TIN2017-89779-P and RTI2018-095645-B-C21, and grants 2016-DI-087, FPU15/06264 and RYC-2014-16831. Titan GPU was donated by NVIDIA.
\bibliographystyle{splncs04}
\section{Video Interpolation}
To better showcase the meaningfulness of the learned stylistic embedding space, find attached a video where we animate a much finer interpolation than the one pictured in the paper, between different calligraphic styles of several words, composing the first sentence of Ernest Hemingway's \emph{``The Old Man and The Sea''}. We appreciate how the generator is able to provide a smooth transition between different writing styles for a given static content. We provide some screenshots of such video in Figure~\ref{tab:screenshots}.
\begin{figure}[ht!]
\centering
\begin{tabular}{c}
\frame{\includegraphics[width=0.9\linewidth,valign=c]{images/video_screenshots/screen01.png}}\\
\frame{\includegraphics[width=0.9\linewidth,valign=c]{images/video_screenshots/screen02.png}}\\
\end{tabular}
\caption{Sample frames of the interpolation video.}
\label{tab:screenshots}
\end{figure}
\section{Limitation of the proposed method when dealing with calligraphic styles}
We evidence in Figure~\ref{tab:limits} the limitations of the proposed approach on imitating calligraphic styles. Unlike in~\cite{haines2016my}, where characteristic glyphs from a given writer were manually cropped to perfectly compose a fraudulent text excerpt as if it was written by a certain person, our approach is not able to produce such levels of mimicking. When the model, trained with the IAM dataset, is fed with an unconventional calligraphic style, the proposed approach is not able to convey such stylistic aspects to the generated word samples. In Figure~\ref{tab:limits}, we injected word samples written by Mary Shelley, and, the reader will appreciate how the rendered results are not able to imitate the visual aspect of such handwriting. However, the proposed generative method is still able to correctly render the textual contents, regardless of the provided calligraphic style.
\begin{figure}[ht!]
\centering
\resizebox{\textwidth}{!}{
\begin{tabular}{c@{\hskip 6pt}ccc}
\toprule
\multirow{4}{*}{\includegraphics[width=0.42\linewidth,valign=c]{images/limits/shelley.png}}&
\includegraphics[width=0.18\linewidth,valign=c]{images/limits/Shelley-20_grades-grades.png}&
\includegraphics[width=0.18\linewidth,valign=c]{images/limits/Shelley-21_heroic-heroic.png}&
\includegraphics[width=0.18\linewidth,valign=c]{images/limits/Shelley-22_Clever-Clever.png} \\
&\includegraphics[width=0.18\linewidth,valign=c]{images/limits/Shelley-23_foul-foul.png}&
\includegraphics[width=0.18\linewidth,valign=c]{images/limits/Shelley-24_mood-mod.png}&
\includegraphics[width=0.18\linewidth,valign=c]{images/limits/Shelley-25_warrior-warion.png} \\
&\includegraphics[width=0.18\linewidth,valign=c]{images/limits/Shelley-26_Morning-Morning.png}&
\includegraphics[width=0.18\linewidth,valign=c]{images/limits/Shelley-27_poetic-potiic.png}&
\includegraphics[width=0.18\linewidth,valign=c]{images/limits/Shelley-28_nodding-nodding.png}\\
&\includegraphics[width=0.18\linewidth,valign=c]{images/limits/Shelley-29_certify-certify.png}&
\includegraphics[width=0.18\linewidth,valign=c]{images/limits/Shelley-30_reviews-revieus.png}&
\includegraphics[width=0.18\linewidth,valign=c]{images/limits/Shelley-31_mosaics-mosaics.png}\\
Original manuscript & \multicolumn{3}{c}{Generated samples}\\
\bottomrule
\end{tabular}
}
\caption{Limitations of the proposed approach when mimicking Mary Shelley's handwriting style.}
\label{tab:limits}
\end{figure}
\section{Qualitative comparison with Alonso \emph{et al.}~\cite{alonso2019adversarial}}
We present in Table~\ref{tab:alonso}, a qualitative comparison with the work of Alonso \emph{et al.}~\cite{alonso2019adversarial}. We can appreciate how our proposed method clearly produces much credible generated images while being able to render the same content word with different calligraphic styles. Whereas~\cite{alonso2019adversarial} suffers from the mode collapse problem, always tending towards producing similar glyphs, our proposed method is able to yield different stylistic instances of the same textual content.
\newlength{\myheight}
\setlength{\myheight}{5ex}
\begin{table}[ht!]
\caption{Qualitative comparison with Alonso \emph{et al.}. Images reprinted from~\cite{alonso2019adversarial}.}
\label{tab:alonso}
\centering
\begin{tabular}{c@{\hskip 8pt}c cc c c@{\hskip 8pt} c}
\toprule
\multirow{2}{*}{Content} && \multirow{2}{*}{Alonso \emph{et al.}~\cite{alonso2019adversarial}} & & \multicolumn{3}{c}{Ours}\\
\cmidrule{5-7}
&&&& Style $A$ & Style $B$ & Style $C$\\
\cmidrule[\lightrulewidth]{1-1} \cmidrule[\lightrulewidth]{3-3} \cmidrule[\lightrulewidth]{5-7}
\texttt{"olibus"}&&
\includegraphics[height=5ex,valign=c]{images/alonso/Alonso01.png} & &
\includegraphics[height=\myheight,valign=c]{images/alonso/583-1_olibus-olious.png} &
\includegraphics[height=\myheight,valign=c]{images/alonso/281-1_olibus-olitus.png} &
\includegraphics[height=\myheight,valign=c]{images/alonso/lot_13_01258_L-1_olibus-olieus.png} \\
\midrule
\texttt{"reparer"}&&
\includegraphics[height=\myheight,valign=c]{images/alonso/Alonso04.png} & &
\includegraphics[height=\myheight,valign=c]{images/alonso/583-2_reparer-reparer.png} &
\includegraphics[height=\myheight,valign=c]{images/alonso/281-2_reparer-reparer.png} &
\includegraphics[height=\myheight,valign=c]{images/alonso/lot_13_01258_L-2_reparer-reparer.png} \\
\midrule
\texttt{"bonjour"}&&
\includegraphics[height=\myheight,valign=c]{images/alonso/Alonso05.png} & &
\includegraphics[height=\myheight,valign=c]{images/alonso/583-3_bonjour-bonjor.png} &
\includegraphics[height=\myheight,valign=c]{images/alonso/281-3_bonjour-bonjor.png} &
\includegraphics[height=\myheight,valign=c]{images/alonso/lot_13_01258_L-3_bonjour-bonjou.png} \\
\midrule
\texttt{"famille"}&&
\includegraphics[height=\myheight,valign=c]{images/alonso/Alonso06.png} & &
\includegraphics[height=\myheight,valign=c]{images/alonso/583-4_famille-famile.png} &
\includegraphics[height=\myheight,valign=c]{images/alonso/281-4_famille-famille.png} &
\includegraphics[height=\myheight,valign=c]{images/alonso/lot_13_01258_L-4_famille-famile.png} \\
\midrule
\texttt{"gorille"}&&
\includegraphics[height=\myheight,valign=c]{images/alonso/Alonso08.png} & &
\includegraphics[height=\myheight,valign=c]{images/alonso/583-5_gorille-grille.png} &
\includegraphics[height=\myheight,valign=c]{images/alonso/281-5_gorille-goille.png} &
\includegraphics[height=\myheight,valign=c]{images/alonso/lot_13_01258_L-5_gorille-grille.png} \\
\midrule
\texttt{"malade"}&&
\includegraphics[height=\myheight,valign=c]{images/alonso/Alonso09.png} & &
\includegraphics[height=\myheight,valign=c]{images/alonso/583-6_malade-malade.png} &
\includegraphics[height=\myheight,valign=c]{images/alonso/281-6_malade-malade.png} &
\includegraphics[height=\myheight,valign=c]{images/alonso/lot_13_01258_L-6_malade-malade.png} \\
\midrule
\texttt{"certes"}&&
\includegraphics[height=\myheight,valign=c]{images/alonso/Alonso10.png} & &
\includegraphics[height=\myheight,valign=c]{images/alonso/583-7_certes-certes.png} &
\includegraphics[height=\myheight,valign=c]{images/alonso/281-7_certes-certes.png} &
\includegraphics[height=\myheight,valign=c]{images/alonso/lot_13_01258_L-7_certes-certes.png} \\
\midrule
\texttt{"golf"}&&
\includegraphics[height=\myheight,valign=c]{images/alonso/Alonso11.png} & &
\includegraphics[height=\myheight,valign=c]{images/alonso/583-8_golf-golf.png} &
\includegraphics[height=\myheight,valign=c]{images/alonso/281-8_golf-golf.png} &
\includegraphics[height=\myheight,valign=c]{images/alonso/lot_13_01258_L-8_golf-golf.png} \\
\midrule
\texttt{"des"}&&
\includegraphics[height=\myheight,valign=c]{images/alonso/Alonso12.png} & &
\includegraphics[height=\myheight,valign=c]{images/alonso/583-9_des-des.png} &
\includegraphics[height=\myheight,valign=c]{images/alonso/281-9_des-des.png} &
\includegraphics[height=\myheight,valign=c]{images/alonso/lot_13_01258_L-9_des-des.png} \\
\midrule
\texttt{"ski"}&&
\includegraphics[height=\myheight,valign=c]{images/alonso/Alonso13.png} & &
\includegraphics[height=\myheight,valign=c]{images/alonso/583-10_ski-shi.png} &
\includegraphics[height=\myheight,valign=c]{images/alonso/281-10_ski-ski.png} &
\includegraphics[height=\myheight,valign=c]{images/alonso/lot_13_01258_L-10_ski-ski.png} \\
\midrule
\texttt{"le"}&&
\includegraphics[height=\myheight,valign=c]{images/alonso/Alonso14.png} & &
\includegraphics[height=\myheight,valign=c]{images/alonso/583-11_le-le.png} &
\includegraphics[height=\myheight,valign=c]{images/alonso/281-11_le-be.png} &
\includegraphics[height=\myheight,valign=c]{images/alonso/lot_13_01258_L-11_le-le.png} \\
\bottomrule
\end{tabular}
\end{table}
\section{t-SNE Embedding visualizations}
Due to space constrains, we are aware that the t-SNE plot presented in the paper in Figure 5 is shown at a quite small scale. This difficult its inspection. We provide here in Figures~\ref{fig:tsne0},~\ref{fig:tsne1},~\ref{fig:tsne2} and~\ref{fig:tsne3}, four different t-SNE plots for images generated with the same textual content and for various calligraphic styles.
\begin{figure}[ht!]
\centering
\includegraphics[angle=90,height=.93\textheight]{images/TSNE_ALL_LR_15_PER_25_ITE_10000.png}
\caption{t-SNE embedding visualization of $2.500$ generated instances of the word \texttt{"deep"}.}
\label{fig:tsne0}
\end{figure}
\begin{figure}[ht!]
\centering
\includegraphics[angle=90,height=.93\textheight]{images/vision_TSNE_ALL_LR_15_PER_25_ITE_10000.png}
\caption{t-SNE embedding visualization of $2.500$ generated instances of the word \texttt{"vision"}.}
\label{fig:tsne1}
\end{figure}
\begin{figure}[ht!]
\centering
\includegraphics[angle=90,height=.93\textheight]{images/hello_TSNE_ALL_LR_15_PER_25_ITE_10000.png}
\caption{t-SNE embedding visualization of $2.500$ generated instances of the word \texttt{"hello"}.}
\label{fig:tsne2}
\end{figure}
\begin{figure}[ht!]
\centering
\includegraphics[angle=90,height=.93\textheight]{images/world_TSNE_ALL_LR_15_PER_25_ITE_10000.png}
\caption{t-SNE embedding visualization of $2.500$ generated instances of the word \texttt{"world"}.}
\label{fig:tsne3}
\end{figure}
\bibliographystyle{splncs04}
| {
"redpajama_set_name": "RedPajamaArXiv"
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{"url":"https:\/\/it.wenda123.org\/question\/csstack\/33852\/counting-nodes-in-a-trie","text":"# \u8ba1\u6570Trie\u4e2d\u7684\u8282\u70b9 -- data-structures \u9886\u57df \u548c trees \u9886\u57df \u548c probability-theory \u9886\u57df \u548c average-case \u9886\u57df cs \u76f8\u5173 \u7684\u95ee\u9898\n\n## Counting nodes in a trie\n\n3\n\n### \u95ee\u9898\n\nIf you don't have any degree-1 nodes in your trie (which is a tree) than you have more leaves than interior nodes. So in this case you have $I\\le n$.\n\nIt depends a bit how you define the trie whether you can have many interior degree-1 nodes. If you study a compressed trie the all the path of degree-1 nodes are merges to an edge, so you are done. For an uncompressed trie, I am afraid, you can have many degree-1 nodes. Say you have one letter $a_i$ that is very common and has a high probability of $1-\\varepsilon$ and $\\varepsilon\\in \\Omega(1\/n^2)$. Then your trie contains many long degree-1 paths with high probability. In this case the you can have more than $O(n)$ interior nodes. Please do the computations by yourself (you might choose a smaller $\\varepsilon$ if you like).\n\n## \u76f8\u5173\u95ee\u9898\n\n0\u00a0 \u5c06\u9879\u76ee\u9644\u52a0\u5230\u6570\u7ec4\u7684\u5e73\u5747\u8fd0\u884c\u65f6\u95f4\u662f\u591a\u5c11\uff1f\u00a0\u00a0(\u00a0What is the average runtime of appending items to arrays\u00a0)\n\n0\u00a0 \u641c\u7d22\u6570\u7ec4\u7684\u5e73\u5747\u65f6\u95f4\u590d\u6742\u6027[\u5173\u95ed]\u00a0\u00a0(\u00a0Average time complexity of searching an array\u00a0)\n\n3\u00a0 \u4e3a\u4ec0\u4e48\u7ebf\u6027\u641c\u7d22\u5e73\u5747\u662f$frac {n} {2}$\u6bd4\u8f83\uff1f\u00a0\u00a0(\u00a0Why does linear search have fracn2 comparisons on average\u00a0)\n\n2\u00a0 \u5982\u4f55\u5b8c\u6210\u5de5\u4f5c\u7684\u5e73\u5747\u6848\u4f8b\u8fd0\u884c\u65f6\u95f4\uff08\u548c\u5176\u4ed6\u7b97\u6cd5\uff09\uff1f\u00a0\u00a0(\u00a0How to go about working the average case run time of this trivial algorithm and\u00a0)\n\n5\u00a0 \u8bc1\u660e\u4e8c\u8fdb\u5236\u641c\u7d22\u7684\u5e73\u5747\u6848\u4f8b\u590d\u6742\u6027\u662fO\uff08log n\uff09\u00a0\u00a0(\u00a0Proving that the average case complexity of binary search is olog n\u00a0)\n\n2\u00a0 \u5e73\u5747\u6848\u4f8b\u5206\u6790\u5e2e\u52a9\u00a0\u00a0(\u00a0Average case analysis help\u00a0)\n\n0\u00a0 Quicksort\u7684\u5e73\u5747\u6848\u4f8b\u8fd0\u884c\u65f6\u95f4\u5206\u6790\u5f88\u597d\u53c2\u8003\u00a0\u00a0(\u00a0Good reference for average case runtime analysis of quicksort\u00a0)\n\n15\u00a0 Knuth\uff0cDe Bruijn\u548cRice\uff081972\uff09\u7684\u201c\u79cd\u690d\u5e73\u9762\u6811\u7684\u5e73\u5747\u9ad8\u5ea6\u201d\u00a0\u00a0(\u00a0On the average height of planted plane trees by knuth de bruijn and rice 197\u00a0)\n\n1\u00a0 \u4e00\u79cd\u7b80\u5355\u7684\u6700\u5927\u53d1\u73b0\u7b97\u6cd5\u7684\u5e73\u5747\u5206\u6790\u00a0\u00a0(\u00a0Average case analysis of a simple max finding algorithm\u00a0)\n\n6\u00a0 \u8fd9\u79cd\u53ef\u6076\u52a3\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u7a0b\u5ea6\u662f\u591a\u5c11\uff1f\u00a0\u00a0(\u00a0What is the time complexity of this atrocious algorithm\u00a0)\n\n3\u00a0 \u5728\u901a\u7528\u5e03\u5c14\u516c\u5f0f\u4e2d\u627e\u5230\u771f\u5b9e\u6216\u9519\u8bef\u5206\u914d\u7684\u786c\u5ea6\uff1f\u00a0\u00a0(\u00a0Hardness of finding a true or a false assignment into a generic boolean formula\u00a0)\n\n0\u00a0 \u627e\u5230\u6700\u5927\u7b97\u6cd5\u7684\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\u00a0\u00a0(\u00a0Finding the average time complexity for a max algorithm\u00a0)\n\n4\u00a0 \u7528\u4e8e\u6267\u884c\u63d2\u5165\u7684\u5355\u4e2a\u94fe\u63a5\u5217\u8868\u7684\u5e73\u5747\u65f6\u95f4\u590d\u6742\u6027\u662f\u591a\u5c11\uff1f\u00a0\u00a0(\u00a0What is the average time complexity for a single linked list for performing an\u00a0)\n\n2\u00a0 \u95ee\u9898\u53d1\u73b0PESIN MAX\u7b97\u6cd5\u7684\u5e73\u5747\u6848\u4f8b\u00a0\u00a0(\u00a0Trouble finding average case of a find max algorithm\u00a0)\n\n2\u00a0 \u5faa\u73af\u7684\u65f6\u95f4\u590d\u6742\u6027\u00a0\u00a0(\u00a0Time complexity of this while loop\u00a0)\n\n14\u00a0 \u6ce1\u6cab\u6392\u5e8f\u4e2d\u7684\u6389\u6b21\u6570\u00a0\u00a0(\u00a0Expected number of swaps in bubble sort\u00a0)\n\n1\u00a0 \u60a8\u5982\u4f55\u8868\u8fbe\u5173\u4e8e\u4e0d\u6210\u529f\u641c\u7d22\u7684\u5b9a\u7406\u58f0\u660e\u5e73\u5747 - \u4e0e\u91cf\u5b50\u6563\u5217\u4e2d\u7684\u4e0d\u6210\u529f\u641c\u7d22\uff1f\u00a0\u00a0(\u00a0How do you express the theorem statement about unsuccessful search on average ca\u00a0)\n\n11\u00a0 \u8bc4\u4f30\u7ed9\u5b9a\u7684Bubblesort\u7b97\u6cd5\u7684\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\u3002\u00a0\u00a0(\u00a0Evaluating the average time complexity of a given bubblesort algorithm\u00a0)\n\n1\u00a0 \u9274\u4e8e\u7b97\u6cd5\uff0c\u5176\u8fd0\u884c\u65f6\u6848\u4f8b\u7684\u6982\u7387\u662f\u4ec0\u4e48\uff1f\u00a0\u00a0(\u00a0Given an algorithm what are the probabilities for its run time cases\u00a0)\n\n3\u00a0 \u9884\u671f\u6210\u672c\u548c\u7b97\u6cd5\u5e73\u5747\u6210\u672c\u4e4b\u95f4\u6709\u4ec0\u4e48\u533a\u522b\uff1f\u00a0\u00a0(\u00a0What is the difference between expected cost and average cost of an algorithm\u00a0)\n\n7\u00a0 \u5728\u6d41\u4e2d\u4fdd\u6301k $\u6700\u5c0f\u7684\u6574\u6570\u7684\u590d\u6742\u6027 ( Complexity of keeping track of k smallest integers in a stream ) \u6211\u9700\u8981\u5206\u6790\u5728\u7ebf\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u6027\uff0c\u4ee5\u8ddf\u8e2a\u4ece$ $numb\u7684\u6d41\u4e2d\u8ffd\u8e2a\u6700\u4f4e$ k $\u53f7\u7801\u3002\u8be5\u7b97\u6cd5\u662f \u5047\u8bbe\u6d41\u4e2d\u7684$ i $th\u53f7\u7801\u662f$ s_i $\u3002 \u4fdd\u6301\u5c3a\u5bf8\u7684\u6700\u5927\u5806$ k $\u3002 \u5982\u679c\u5806\u5305\u542b\u5c11\u4e8e$ k $\u5143\u7d20\uff0c\u8bf7\u4e3a\u5806\u6dfb\u52a0$ s_i $\u3002 \u5426\u5219\uff1a\u5982\u679c$ s_i $\u5c0f\u4e8e\u5806\u4e2d\u7684\u6700\u5927\u5143\u7d20\uff08\u5373\u5806\u7684\u6839\uff09\uff0c\u8bf7\u7528$ s_i $\u66ff\u6362\u5806\u7684... 6 \u4e8c\u8fdb\u5236\u641c\u7d22\u6811\u548cAVL\u6811\u7684\u5e73\u5747\u6df1\u5ea6 ( Average depth of a binary search tree and avl tree ) \u6211\u7684\u6559\u6388\u6700\u8fd1\u63d0\u5230\u4e8c\u8fdb\u5236\u641c\u7d22\u6811\u4e2d\u7684\u8282\u70b9\u7684\u5e73\u5747\u6df1\u5ea6\u5c06\u662f$ o\uff08log\uff08n\uff09\uff09$why$ n $\u662f\u6811\u4e2d\u7684\u8282\u70b9\u91cf\u3002\u6211\u6700\u7ec8\u7ed8\u5236\u4e86\u4e00\u5806\u4e8c\u8fdb\u5236\u641c\u7d22\u6811\uff0c\u6211\u4e0d\u8ba4\u4e3a\u6211\u662f\u6b63\u786e\u4e86\u89e3\u8fd9\u4e2a\u6982\u5ff5\u3002\u4f8b\u5982\uff0c\u5982\u679c$ n = 4 \\$ the tree\uff0c\u5219\u5c06\u5177\u6709\u6700\u59273\u7f8e\u5143\u6216\u8282\u70b9\u7684\u6700\u5927\u6df1\u5ea6\u7684\u8282\u70b9\uff0c\u6700\u5927\u6df1\u5ea6\u4e3a2\u7f8e\u5143\u3002\u5728\u6700\u5927\u503c\u4e3a3\u7f8e\u5143\u7684\u60c5\u51b5\u4e0b\uff0c\u6211\u4eec\u5c06\u67090\u7f8e\u5143...","date":"2021-11-28 07:55:33","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.5440747737884521, \"perplexity\": 1040.9394564307518}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-49\/segments\/1637964358480.10\/warc\/CC-MAIN-20211128073830-20211128103830-00097.warc.gz\"}"} | null | null |
**Physical Characteristics of the Welsh Terrier**
(from the American Kennel Club breed standard)
**Head:** Rectangular.
**Eyes:** Small, dark brown and almond-shaped, well set in the skull.
**Ears:** V-shaped, small...fold is just above the topline of the skull.
**Foreface:** Strong with powerful, punishing jaws.
**Muzzle:** Strong and squared off.
**Nose:** Black and squared off.
**Lips:** Black and tight.
**Teeth:** Large and strong, set in powerful, vise-like jaws.
**Backskull:** Of equal length to the foreface.
**Neck:** Of moderate length and thickness, slightly arched and sloping gracefully into the shoulders.
**Forequarters:** The front is straight. The shoulders are long, sloping and well laid back. The legs are straight and muscular with upright and powerful pasterns.
**Coat:** Hard, wiry, and dense with a close-fitting thick jacket.
**Topline:** Level.
**Loin:** Strong and moderately short.
**Body:** Shows good substance and is well ribbed up.
**Tail:** Docked to a length approximately level (on an imaginary line) with the occiput.
**Hindquarters:** Strong and muscular with well-developed second thighs and the stifles well bent. The hocks are moderately straight, parallel and short from joint to ground.
**Color:** The jacket is black, spreading up onto the neck, down onto the tail and into the upper thighs. The tan is a deep reddish color.
**Height:** Males are about 15 inches at the withers, with an acceptable range between 15 and 15.5. Bitches may be proportionally smaller.
**Weight:** Twenty pounds is considered an average weight.
**Feet:** Small, round, and catlike.
Contents
**History of the Welsh Terrier**
Enter the proverbial fox hole and follow this fascinating Celtic earthdog through the tunnels of its exciting history. Find out why the breed is called Welsh Terrier and meet the breeders responsible for promoting the dog in Wales, including the smitten royals! Trace the fancy of the Welsh to the US and various countries on the Continent.
**Characteristics of the Welsh Terrier**
Do you have to learn to speak Welsh to communicate with this overly bright terrier? Is the Welsh as difficult to train as other terriers? What imperfections exist in this delightful terrier? Find out the answers to these and other pressing questions that prospective owners should ask about the breed.
**Breed Standard for the Welsh Terrier**
Learn the requirements of a well-bred Welsh Terrier by studying the description of the breed as set forth in the American Kennel Club's breed standard. Both show dogs and pets must possess key characteristics as outlined in the breed standard.
**Your Puppy Welsh Terrier**
Find out about how to locate a well-bred Welsh Terrier puppy. Discover which questions to ask the breeder and what to expect when visiting the litter. Prepare for your puppy-accessory shopping spree. Also discussed are home safety, the first trip to the vet, socialization and acclimating puppy to his new home.
**Proper Care of Your Welsh Terrier**
Cover the specifics of taking care of your Welsh Terrier every day: feeding for the puppy, adult and senior dog; grooming, including coat care, ears, eyes, nails and bathing; and exercise needs for your dog. Also discussed are the essentials of dog identification.
**Training Your Welsh Terrier**
Begin with the basics of training the puppy and adult dog. Learn the principles of house-training the Welsh Terrier, including the use of crates and basic scent instincts. Get started by introducing the pup to his collar and leash and progress to the basic commands. Find out about obedience classes and other activities.
**Healthcare of Your Welsh Terrier**
_By Lowell Ackerman DVM, DACVD_
Become your dog's healthcare advocate and a well-educated canine keeper. Select a skilled and able veterinarian. Discuss pet insurance, vaccinations and infectious diseases, the neuter/spay decision and a sensible, effective plan for parasite control, including fleas, ticks and worms.
**Your Senior Welsh Terrier**
Know when to consider your Welsh Terrier a senior and what special needs he will have. Learn to recognize the signs of aging in terms of physical and behavioral traits and what your vet can do to optimize your dog's golden years.
**Showing Your Welsh Terrier**
Step into the center ring and find out about the world of showing pure-bred dogs. Acquaint yourself with the basics of AKC conformation shows and take a leap into the realms of obedience trials, agility, earthdog events, natural hunts and tracking tests.
**Behavior of Your Welsh Terrier**
Analyze the canine mind to understand what makes your Welsh Terrier tick. Among the potential problems addressed are different types of aggression, separation anxiety, sex-related behavior issues, chewing, digging and barking.
KENNEL CLUB BOOKS® **W ELSH TERRIER**
**ISBN 13: 978-1-59378-294-8**
**eISBN 13: 978-1-62187-003-6**
Copyright © 2005 • Kennel Club Books® • An Imprint of I-5 Press™ • A Division of I-5 Publishing, LLC™
3 Burroughs, Irvine, CA 92618 USA
Cover Design Patented: US 6,435,559 B2 • Printed in South Korea
All rights reserved. No part of this book may be reproduced in any form, by photostat, scanner, microfilm, xerography or any other means, or incorporated into any information retrieval system, electronic or mechanical, without the written permission of the copyright owner.
10 9 8 7 6 5 4 3 2 1
Photography by Carol Ann Johnson and Michael Trafford with additional photographs by:
Ashbey Photography, Paulette Braun, Linda Brisbin, T.J. Calhoun, Alan and Sandy Carey, Isabelle Français, Chris Halvorson, Carol Ann Johnson, Bill Jonas, Bardi McLennan, Melia Photography, Hiroshi Saito, Chuck Tatham "The Standard Image" and Top Dog Photos.
**Illustrations by Patricia Peters.**
The publisher wishes to thank all of the owners whose dogs are featured in this book, including Ms. Wendy Allen, Judy Averis, Mereda Cornick, Gillian Griffiths, Mrs. M. Kelles, Mr. Juha Korhonen, Sari Mäkelä, Anne Maughan, Cathleen Saito, Dave Scawthorn, Kim Skillman and Linda Taranto.
**Glansevin Coquette and Eng. Ch. Glansevin Coda, in a painting from 1907, illustrate quality Welsh Terrier type of that time.**
**THE WELSH TERRIER ORIGINAL**
There are no written pedigrees before the 1800s for the small black and tan terriers bred by Celtic farmers, but clear references to them, including their monetary value (three curt pence), can be found in writings from as early as the 10th century. The dogs, like their owners, worked in the soil, hence the name "terrier," taken from the Latin _terra,_ meaning "earth." These small dogs helped rid the farms of all forms of vermin from mice to martens and provided an occasional rabbit for the dinner table. They were true workers, and when their job was done they had earned the right to relax by the fire with the family. Thus a sensible temperament has always been intrinsic to the breed: sensible and single-minded working in the fields or alone underground; agreeable working with other dogs; steady and reliable in the home.
When it was found that fox hunting benefited from the addition of terriers to the packs of hounds in order to bolt the fox, the terriers' worth was raised a notch. However, it was due to the rapid expansion of dog shows that the Welsh Terrier gained popularity and prestige in the Terrier Group and became the handsome dog we know today. An opposing view holds that dog shows have put the cart before the horse by placing the emphasis on conformation and looks rather than on intelligence and working ability. Fortunately, this is not true in all countries.
In the beginning, the Welsh Terrier was little known outside his Welsh environs, which, at one time, went far beyond the boundaries of Wales as we know it today, encompassing a portion of the Continent and stretching into present-day Scotland and England. It is based on this historical fact that many in the breed feel so strongly that the breed eventually named the Welsh Terrier was indeed the progenitor of the other black and tan terriers inhabiting the British Isles.
Around 1450, a Welsh thank-you note ( _Englyn diolgarwch_ ) was written, acknowledging the gift of a terrier. It reads in part, "And a good black and red terrier bitch to choke the brown polecat and to tear up the red fox." It is quite possibly the earliest written description of the Welsh Terrier—a dog that was black and tan (the same "red" as a fox) and a worker of note.
**Welsh Terriers from the renowned Brynhir Kennels of leading breed advocate Walter Glynn in the early 1900s.**
**DOGS AS LIFE**
The old Celtic religion used dogs as symbols of healing, death and rebirth. Apparently the Celtic canine was sufficiently regarded to cover all aspects of life.
The breed was known to have been used with the Glansevin Welsh Hounds from the early 1600s, and records of 1760 show that these terriers had also been used for several generations by the Jones family with their Ynysfor Otterhounds.
**WHAT'S IN A NAME?**
In the first dog shows where Welsh Terriers were exhibited, the classes were all-inclusive; for example, "Working Terriers" or "Any Variety Terriers." The individual naming of each breed occurred with the increase in dog shows and the establishment of The Kennel Club of England in 1873. In the case of the Welsh Terrier, The Kennel Club was to become the arena for conflicting national canine contentions to do battle!
The Welsh people had always considered the black and tans as their very own, referred to as _daeargi_ , of course, not as Welsh Terriers. The English now laid claim to their version of the black and tan terrier as the taproot of the breed under several elongated names such as the Old English Broken-Haired Black and Tan Terrier or the Old English Wire-Haired Black and Tan Terrier. So it was that in 1885 The Kennel Club had an international crisis of sorts on its hands.
A decisive event had taken place a year before in Pwllheli, North Wales: the first show with separate classes for Welsh Terriers, a specialty, if you will. On August 28, 1884, at the Lleyn & Eifionydd Agricultural Show, two highly regarded elderly Welshmen, who had bred these terriers for many years, were asked to judge an entry of 90 dogs, divided into three classes. Mr. Griffith Owen and Mr. Humphrey Griffith spent the entire day going over the dogs one at a time, each dog being judged on its own merits, not compared to any other dog. It is said that all were pleased with the results, which today in itself might be considered something of a miracle!
_**CANIS LUPUS**_
"Grandma, what big teeth you have!" The gray wolf, a familiar figure in fairy tales and legends, has had its reputation tarnished and its population pummeled over the centuries. Yet it is the descendants of this much-feared creature to which we open our homes and hearts. Our beloved dog, _Canis domesticus_ , derives directly from the gray wolf, a highly social canine that lives in elaborately structured packs. In the wild, the gray wolf can range from 60 to 175 pounds, standing between 25 and 40 inches in height.
**ACTION!**
In these early years, Welsh Terriers were most numerous in North Wales, and it was said that Cledwyn Owen, William Jones and Price O. Pughe knew all the Welsh Terriers in the area as well as their pedigrees. By 1885, Welsh terriermen (including Owen, Jones and Pughe) had had enough of the name nonsense and, with nine others, formed the Welsh Terrier Club. They were recognized by The Kennel Club in 1886, where-upon, in what can only be described as an act of diplomacy, The Kennel Club provided classes for both the Welsh Terriers and the Old English Broken-Haired Black and Tan Terriers (OEBHBTT). However, that was not to be the end of the story. Things began to fall apart for the Old English supporters, who never were able to put together an organization to back them. Nor did their dogs help matters.
For two years, 1885 and 1886, no OEBHBTT shown was from the mating of an OEBHBTT sire and dam. All were first-generation cross-breds. The dogs were said, in general, to be somewhat more handsome than the Welsh Terrier, but since they were manufactured and could not reproduce in kind, this alone would seem to give credibility to the Welsh Terrier as the true black and tan. It should be noted that in those days there was no Kennel Club ruling against cross-breeding, but serious breeders in the new sport of dog showing wanted pure-bred stock.
**STAND BACK, PATAGONIA!**
It is now fashionable, especially among the young, to speak Welsh in Patagonia, an area of Argentina originally settled by the Welsh in the 1800s. Teachers have been sent there by the Welsh Office in Britain with an added plum for the students—six annual scholarships to study Welsh in Wales.
The results during those two years bordered on breed chaos, for many dogs were shown—and declared winners—in both classes. For example, one dog named Crib was shown as an OEBHBTT, but had been sired by a well-known Smooth Fox Terrier out of a solid black rough-coated bitch. He was also known to be deaf, a fault that the judges of the day apparently chose to overlook.
A dog named Dick Turpin was also famous for being caught in the middle of the dispute. The dog was Welsh-bred, but changed owners four times. Each owner was apparently unsure where to enter the dog, so Dick was entered in both OEBHBTT and Welsh Terrier classes. The show results proved his heritage, because he won a first place only when shown as a Welsh Terrier.
**AND THE WINNER IS...**
The Kennel Club ended the battle, and on April 5, 1887 dropped the Old English, etc., leaving the Welsh Terrier as the only recognized breed. One has to wonder if the long, clumsy OEBHBTT name may have influenced The Kennel Club's decision.
It should be noted that the Welsh Terriers entered in these early shows were, for the most part, a far cry from the beautiful dogs we see in the ring today. They were still primarily working terriers and meant to look the part. Cropped ears, for example, were allowed, since terriers' ears were often ripped by prey. The breed was not yet consistent in size or type. Many dogs had unattractive broad heads, drooping ears and white feet. The looks of the scruffy working terrier improved when popular opinion demanded it. The fashionable Wire Fox Terrier was what the public admired, and no doubt crosses with the Welsh were made to the Wire Fox. Judging from existing photographs, quite a few of these early dogs would be considered satisfactory breed specimens today.
**EARLY SHOW DOGS**
At the Bangor show in 1887, a bitch appropriately named Bangor Dau Lliw (Bangor Two Colors) became the first Welsh Terrier bitch champion. She was bred and owned by Mr. Dew. At this same show Walter Glynn, who was to become the leading advocate of the breed, exhibited his first Welsh Terrier, a male puppy named Dim Saesonaeg (No English). Bred by Mr. Pughe, the dog was made up a champion in 1889. He was highly regarded in his day. However, his son Eng. Ch. Cymro-o-Gymru (The Welshman from Wales) became the leading light of the breed. Bred by J. Mitchell out of Mitchell's bitch Blink Bonny in 1891, Cymro won 27 Challenge Certificates (awards at Kennel Club shows) and for many years was thought to be the embodiment of something close to perfection in the breed.
**WELSH ROOTS AT BULLDOG UNIVERSITY**
Elihu Yale, founder of Yale University in New Haven, Connecticut, had close ties to Wales. His family was from Plas-yn-Ial and although Elihu was born in America (where the spelling of the family name was changed from Ial to Yale), he retained his Welsh roots to the end and is buried in St. Giles Churchyard in Wrexham, Wales. As an acknowledgment of this association, a replica of the St. Giles tower stands at the university.
The first dog champion was made up in 1887. Mawddwy Nonsuch, sired by Fernyhurst Crab, was said to be a dog with an excellent head, albeit with cropped ears. Nonsuch was purchased by Edmund Buckley (Master of Otterhounds in Merioneth) for what was, at the time, the huge sum of 200 guineas. Possibly due in part to this extravagant expenditure, or the fact that the dog was said to lack type, doubts spread regarding the authenticity of the dog's dam. The gossip may well have been true, since apparently he was never used at stud. Eng. Ch. Bob Bethesda, a dog of Buckley's own breeding, was made up a champion soon after and was highly acclaimed as a show dog and for his excellent temperament.
**Eng. Ch. Bangor Dau Lliw was a record-breaking champion in the 1880s. In 1887, she became the first Welsh Terrier bitch champion.**
**Eng. Ch. Vaynor Again, a male Welsh Terrier born in 1928.**
**Eng. Ch. What's Wanted caused a sensation when she appeared on the show scene as a puppy during the 1919–1920 season. Her debut was followed by a brilliant show career.**
The breed was gaining much attention throughout Wales, as well it might, for this was also a period of intense Welsh national pride. As the number of dog shows increased and the dogs proved themselves winners in competition, the popularity of the breed increased at a rapid rate throughout the British Isles. Welsh Terriers were well represented in large English shows such as Crystal Palace, Crufts and Birmingham. By 1896, Welsh Terrier breeding and show stock was being exported in increasing numbers to Germany, Belgium, South Africa, India and the US. Judges from the UK were much in demand to critique the progress of foreign breeding.
Two years after Walter Glynn purchased his first Welsh Terrier in 1887, he began judging classes at home and abroad and became a member of The Kennel Club. Dogs with his Brynhir affix became the foundation of many kennels worldwide. When Mr. Glynn died in 1933, he had owned and bred more Welsh Terrier champions than anyone else in the previous half-century.
Another breeder of note on both sides of the Atlantic was T. H. Harris of Sennybridge, whose Eng. Ch. Resiant was made up in 1895. The first champion he actually owned and made up was Nell Gwynne in 1897. His Senny affix became synonymous with top-quality Welsh Terriers.
**The Welsh Terrier as it appeared in an 1887 publication.**
The first woman to award Challenge Certificates to Welsh Terriers was Mrs. H. L. Aylmer in 1907 at the Bristol show. It is historically significant that she was chosen for this honor since her affix, Glansevin, came from her family ties with the Glansevin Welsh Hound Pack, noted for the Welsh Terriers that ran with it in the 1700s.
**WELSH GRAMMAR LESSON**
The Welsh alphabet does not include the letters K, J, Q, V, X or Z, but it does have six others to make up for it. It goes like this: A, B, C, CH, D, E, F, FF, G, H, I, (J only for words borrowed from English), L, LL, M, N, O, P, PH, R, RH, S, T, TH, U, W and Y.
It should be noted that many Welsh Terriers in 1900 were still working terriers. By no means had they given up their day jobs for stardom! The ownership of all dogs as pets had only become acceptable and popular due to the example set by Queen Victoria. Prior to that time, royalty carried about various toy breeds, but commoners could not afford to feed a non-working animal. Dog shows took matters a step further. Those who could not afford to keep race horses, or horses and hounds for the hunt, could and did manage kennels of show dogs. Additionally, showing dogs held more prestige than showing livestock, although most of the early breeders had their roots in breeding sheep, rabbits or chickens to exhibit at agricultural shows. It may have been the idea that one could now take pride in selling a top show dog to a member of the aristocracy, whereas one could not hold one's head as high on the similar sale of a prize chicken!
**Galen Rexus, bred and owned by Mr. J. S. Gilbert, was born in September 1930.**
**Kynan O' Gaint was Mr. A. T. Morris's dog, born in March 1930.**
In 1899, Princess Adolphus of Teck bought a bitch, Brynafon Nellie, and promptly became a breeder, exhibitor and ardent supporter of the Welsh Terrier Club. Then, in 1911, a group of Welsh Terrier fanciers in North Wales raised the necessary money and bought a dog for HRH The Prince of Wales. The rest of that story reads like pure fiction. The bitch was Queen Llechwedd (called Gwen), sired by Dewi Sant (St. David, patron saint of Wales). The following year (1912) His Royal Highness registered two pups whelped on March 1, St. David's Day.
**DOGGIE HUBBARD LIVES ON**
There is a room at the library of the University of Wales in Aberystwyth dedicated to Clifford ("Doggie") Hubbard's superb collection of books on dogs, said to be the finest and most extensive in the world.
**Gochel Fi, owned by Mrs. O. Jones, was born in 1926. This fine Welsh Terrier beat three breed champions in the show ring.**
Serendipity continued to be on the side of the Welsh, although until the 1920s the controversy continued with articles referring to the "so-called Welsh Terrier" and comparisons of the breed to the "beautiful" Fox Terrier. Not to worry. A new world of dogs was on the rise and pets were becoming better fed, better housed and better cared for both at home and through outstanding advances in veterinary medicine.
**Eng. Ch. Delswood Welcome, born in September 1931, was bred and owned by Mr. A. H. Symonds.**
**Eng. Ch. Lady Gwen, born in August 1925, earned her champion title at the National Terrier and Kennel Club Championship Shows. She was exported to Germany.**
**THE WELSH GAINS FAME**
Joe Hitchings's Aman kennels in the Rhondda Valley were a dominant force in the breed after World War I and largely contributed to the area's being called "the whelping box of the Welsh Terrier." Hitchings handled many breeds but is best remembered for putting a modern stamp on the Welsh Terrier.
**Eng. Ch. Senny Rex, owned by Mr. T. H. Harris, was born in 1925.**
Hitchings, along with Sam Warburton in England and George Steadman Thomas (who was in effect a trans-Atlantic commuter) were responsible for sending a steady supply of quality Welsh Terriers to American kennels. With dogs being lined up on one side of the ocean and Welsh Terrier enthusiasts eagerly awaiting their arrival on the other, the stage was set for the breed's solid future in the modern world.
There were numerous true terriermen at this time. Hitchings continued breeding Welsh Terriers for 42 years. Arthur Harris (Ronvale), T. H. Harris (Senny) and A. E. Harris (Penhill) also remained in the breed for close to half a century. Harold Snow's Felstead kennels were continued by his son Emlyn and grandson Lyn: three generations of dedication to the breed.
After World War II, the breed's popularity skyrocketed and names such as Mervin Pickering (Groveview), Dai Rees (Ebbw Swell), Cyril Williams (Caiach) and Phil Thomas (Sandstorm) came to the fore-front. There were two prestigious wins in this period to boost the breed dramatically. The Crufts Best in Show winner in 1951 was Ch. Twynstar Dyma Fi; in 1959, Ch. Sandstorm Saracen repeated the feat, handled by his breeder, Mr. Thomas. It wasn't until 1994 that another Welsh Terrier was to claim the same honor, this time Ch. Purston Hit and Miss From Brocolitia, bred by Michael Collings and owned by Mrs. Anne J. Maughan. The next Crufts Best in Show achievement followed just four years later, by Ch. Saredon Forever Young, bred by Judy Averis and David Scawthorn.
Lord Atlee, former Prime Minister, chose his two Welsh Terriers as supporters for his coat of arms and the words _Labor omnia vincit_ or "Work conquers all"—a fitting tribute to both the man and the dogs. The Welsh fashion designer Laura Ashley brought a touch of fame to the breed when her Welsh Terrier, Clem, became the subject of a series of popular children's books.
In the 20th century, registration numbers in the UK were somewhat static until a high in 1927 of 288. By 1951, due in part to the fame of Twynstar Dyma Fi, those numbers escalated to 359. They remained in the 200–300 range, but new heights of popularity are expected in this 21st century. In the US, the registrations average 700 annually.
**A PINT FOR BEST OF BREED?**
Some of the first gatherings of dog owners, the precursors of today's dog shows, took place in pubs. Everyone in the pub became judge, exhibitor and spectator, all going over the dogs and giving their opinions.
There is always a threat to a breed such as the Welsh Terrier that inherited defects will become magnified in their small gene pool or that the overall quality will diminish. Fortunately for the breed, as import-export laws have been relaxed and the use of frozen semen increased, sound Welsh Terrier breeding stock is more easily obtainable worldwide.
**ABOUT THE WTA**
In 1923, the Welsh Terrier Association (WTA) was founded in England, becoming the second club for the breed. The year prior, an older club, the South Wales Welsh Terrier Breeders' Association, which had been the breed's second club, joined up with the Welsh Terrier Club (WTC), the original club for the breed. A good portion of British breed fanciers today belong to both clubs. In 1970, the WTA began a most informative and well-organized yearbook, reflecting the Welsh Terriers in Britain and in foreign lands. On June 1, 1980, this club held its first Open Show at the home of George and Olive Jackson (Jokyl), with 39 dogs in 88 entries. Judged by Beryl Blower (Turith), Best in Show was awarded to Mr. Jenkins and Miss Nock's Ch. Bowers Princess, handled by Ray Davies.
**Eng. Ch. Hold Up, owned by A. E. Harris, was born in 1925 and won many Challenge Certificates at important shows.**
**Ch. Bardwyn Bronze Bertram, bred by author Bardi McLennan. He was a top producing sire with 25 champions to his credit.**
In 1981, when the Welsh Terrier Club was granted permission to award Challenge Certificates, club president Mrs. Margaret Thomas was asked to judge the premier event. It is difficult to imagine why it took 95 years for The Kennel Club to acknowledge this stalwart club's standing; perhaps they never asked. The top Welsh Terrier in Britain that year and the following was Mr. Jenkins and Miss Nock's Ch. Puzzle of Kenstaff, bred by R. Ogles. This lovely bitch won Best of Breed at Crufts and Best in Show at the National Terrier Specialty.
**MADOC AT BAY**
One of perhaps the wildest of Celtic claims concerned a Prince Madoc, who was said to have crossed the Atlantic Ocean and landed in what is now Mobile Bay, Alabama in 1170. The legend was endorsed by Queen Elizabeth in 1580, no doubt due in some large part to pressure from her science advisor and magician, John Dee, who was—of course—a Welshman!
**CHARLIE KENNEDY**
Caroline Kennedy's Welsh Terrier, Charlie, swam in the White House pool with President John F. Kennedy. Unfortunately, the dog was no diplomat—he lifted his leg indiscriminately, so was confined to his quarters much of the time.
**Ch. Kirkwood Brazen Overture, bred and owned by Ann Baumgardner and Judith Ford Anspach, winning Best of Breed at Westminster Kennel Club in 2004.**
**MADE UP IN AMERICA**
Not surprisingly, the largest population of Welsh Terriers today is in the United States. The Welsh's rise in popularity began when a young dog named Nigwood Nailer won a 30-guinea Challenge Cup in the UK for Best Welsh, Irish or Fox Terrier in Show in 1899 and was immediately bought up by Major Carnochan and brought to America. The following year, the Welsh Terrier Club of America was formed, with Major Carnochan as treasurer. Nigwood Nailer went on to become the first Welsh Terrier to become an American Kennel Club champion, which he did in 1903.
The Misses Beatrice and Gertrude de Coppet were to be the backbone of the breed club from 1900 until 1960. The sisters' Windermere kennels were among the first to be devoted solely to the Welsh Terrier, based on dogs imported from T. H. Harris in 1890. The ladies were always attired in hats and white kennel coats while handling their own dogs, and they could be quite intimidating in the show ring. Among numerous contributions to the club, Beatrice de Coppet designed the club logo.
**Ch. Hapitails Hit Parade, bred by co-owner Elizabeth Leaman and Richard Powell, co-owned with Jill and Peter See, was number-one Welsh in 2003 with multiple Bests in Show.**
Over the ensuing years in the US, many people and dogs established the Welsh Terrier as a worthy challenger in the Terrier Group. From the 1940s through the 1970s, kennel affixes such as Halcyon, Strathglass, Twin Ponds, Coltan, Pool Forge, Licken Run, Penzance and Tujays became staples of the breed. In more recent years, such names as Anasazi, Sunspryte, Hapitails, Kirkwood, Cisseldale and Czar came to the forefront. That is not to say that the imported dogs are lagging behind. Imports keep arriving and doing a fair share of winning.
**A team from Russia's zo Strelki kennel, winning Best Breeders Group at the Novgorod, Russia show. Novgorod is home to several Welsh Terrier breeders, and interest in the breed is high in that area.**
Of special note are three Welsh Terriers bred by Michael and Nancy O'Neal of New Mexico. Each of them has earned a place in the record books. Ch. Anasazi Annie Oakley took her place as top-winning Welsh Terrier bitch with 40 Bests in Show and 106 Terrier Group wins. Ch. Anasazi Trail Boss topped the stud record with 60 champion get and Ch. Anasazi Billy The Kid retired in 1999 after breaking all Welsh Terrier records with 100 Bests in Show and over 150 Group Firsts. An amazing feat from a small kennel!
**WALES FOREVER!**
In the nation's capital, Washington, DC, halfway up the tall, cylindrical Washington Monument is inscribed _Cymru am bryth_ —"Wales forever."
**THE WORLD OF THE WELSH**
In many countries on the Continent, dogs must still prove that they can perform the tasks for which they were originally bred. As late as 1990 at the World Dog Show in the former Czechoslovakia, Welsh Terriers were a novelty as show dogs to people from Poland, Russia, East Germany and what is now the Czech Republic and Slovakia, but the breeders were charmed by the dogs' presence in the show ring, and the breed is making great strides in these countries. In Russia, a team of seven Welsh Terriers from zo Strelki kennels won the Breeders Group at a show near Novgorod—and very nice specimens they were. Welsh Terriers are still primarily working earthdogs in Russia, and in Poland they are actually the breed of choice for hunting.
**Shaireab's Honor Among Thieves, known to friends as "Stuart," is owned by Sharon Abmeyer.**
**Crufts winner the successful Eng. Ch. Purston Hit and Miss from Brocolitia.**
**G ERMANY**
The first classes for Welsh Terriers in Germany were at the Berlin show in 1896. Today the breed is in good shape with several dedicated breeders such as Mr. Axel Mohrke, who has exported his Bismarckquelle dogs worldwide. Mrs. Irmatraud Becker (v. Ganseliesel) is another breeder with top-winning dogs.
**D ENMARK**
Both the Borchorst and the El-Fri-Ba kennels are at the top of the breed in Denmark. Their Welsh Terrier stock has traveled as far as America and Australia.
**F INLAND**
Registrations of the breed in Finland were on a downturn until quite recently. Due in great part to the success of Sari Mäkëla's Vicway dogs, things are looking up for the breed. Ch. Vicway Live Free Or Die, a son of Multi-Ch. Vicway Live and Let Die (a dog sired by Ch. High Flyer's Welsh Baron), was a World Winner in 1998. Participation in agility has a slight edge over participation in conformation shows.
**S WEDEN**
Sweden has been involved with the Welsh Terrier since dog shows began in that country. The late Per Thorsen (Snowdonia) played a leading role in establishing the popularity of the breed. Lars Adenheimer (Aden) is yet another Welsh Terrier breeder whose kennel affix is recognized everywhere.
The Swedish law against tail-docking not only has hurt the importing of dogs for the show ring but also has prevented Welsh Terriers with docked tails from competing in dog shows as well as agility, obedience or other such events. A recent World Show held in Sweden drew much criticism from owners of dogs that were disqualified for this reason, when the docking of tails is still legal (but not mandatory) in the breed's land of origin.
**F RANCE**
Mesdames Remy and Bernaudin of France owned the Best in Show Felstead Formulate and later purchased Ch. Solentine Sugar Ray, a dog bred by Wendy Gatto-Ronchieri that was top stud dog in Britain in 1995.
**H OLLAND**
Holland is fortunate to have one of the world's top breeders, Jan Albers, whose High Flyer Welsh Terriers have a definite stamp, a unique look. Many High Flyer dogs are top winners and producers and have been the foundation of other kennels on the Continent and overseas.
**A junior handler winning a high award at an International Championship Show in Holland, under an American judge.**
**B RITAIN**
Among the kennels of note today in the UK are Felstead, Philtown, Serenfach, Alokin, Davannadot, Saredon, Glyncastle and Wigmore. The breed is in excellent form in its native land with a steady growth of new breeders and exhibitors smitten by the Welsh Terrier.
**The late Frank Kellett handled Ch. Purston Hit and Miss from Brocolitia to Best in Show at Crufts in 1994. He is shown with owner Anne Maughan.**
**PURE-BRED PURPOSE**
Given the vast range of the world's 400 or so pure breeds of dog, it's fair to say that domestic dogs are the most versatile animal in the kingdom. From the tiny 1-pound lap dog to the 200-pound guard dog, dogs have adapted to every need and whim of their human masters. Humans have selectively bred dogs to alter physical attributes like size, color, leg length, mass and skull diameter in order to suit our own needs and fancies. Dogs serve humans not only as companions and guardians but also as hunters, exterminators, shepherds, rescuers, messengers, warriors, babysitters and more!
**WHY THE WELSH TERRIER?**
Sometimes one has to wonder if perchance the Welsh Terrier speaks only Welsh, for the dog has an uncanny knack of ignoring directions and commands given in the owner's native tongue. This is often mistaken for stubbornness, but that's not quite true. The Welsh Terrier is easily distracted and therefore may not be paying attention to you, or, more accurately, is paying strict attention to something else. That's the contradictory nature of the Welsh Terrier—easily distracted or intensely focused, which is, after all, how an earthdog must function. Take your eye off the target (be it rat, fox or badger) and you've lost the "game"!
A similar scenario mistaken for obstinacy occurs when the dog is asked to obey a command that he has demonstrated over and over again that he can perform perfectly. Being a sensible dog, the Welsh Terrier sees nothing to be gained by pointless repetition.
The breed is intelligent and, as everyone knows, it isn't always easy to cope with intelligence. He is not a canine robot, but instead will show you (without having been asked) just how many different ways he can execute your request. It may be amusing to watch his mental wheels go round, and no harm is done so long as you remain amiably in control. The Welsh Terrier may have coined the phrase "equal opportunity employer," for he will seize every opportunity to become your equal, or better. If you drop your role as leader, rest assured that your Welsh friend will retrieve it instantly. The Welsh Terrier is an intelligent, alert dog and great fun to teach basic obedience and home rules, even if a bit of a challenge.
**TRAINABILITY**
Begin as you mean to continue. Training begins from the moment the puppy (or adult) steps across the threshold of your household. Undeniably the best training method is bribery and coercion. Well, at least bribery! Later in this book, it is more politely referred to as "positive reinforcement," which means whenever the dog does as he's told, you hand out tiny food rewards. When he does not, you accept the fact that you did not properly explain what you wanted him to do, and you begin again. An occasional "No!" is permissible, with an exaggerated frown to signify your total displeasure. Physical punishment is definitely not acceptable and might even encourage reciprocation in kind. Welsh Terriers do not have strong jaws and large teeth for naught! It is wise not to become involved in trading smacks for bites.
**All terriers are curious, active and easily distracted—are your family and home ready for a Welsh?**
House-training a Welsh Terrier is seldom a problem when a consistent schedule is followed, ample praise is given for relieving himself where he should and the dog is confined when no one is free to keep an eye on him.
**WITH YOUNGSTERS**
A question often asked is how the breed gets along with children. There are two answers. The Welsh Terrier is very good with slightly older children; he is ready to obey them and ready for almost any game they want to play, even agreeing to be dressed up! However, a Welsh puppy is not a suitable new pet to consider for families with babies or children under the age of five years. The puppy will treat these little ones as littermates, and if you've ever watched a litter of pups in action, you know that needle-sharp puppy teeth are invariably involved in play. Small children who have never had a puppy can't be expected to understand.
Speaking of babies, it should be pointed out that the Welsh Terrier, regardless of age, is not a baby and should never be treated like one. He is a dog, knows he's a dog and, what's more, is proud to be one. He's also a terrier, which makes him a bit more of a dog if that's possible!
**THE BEST HOME**
Given a choice, Welsh Terriers would no doubt prefer to live in the country, but will settle down contentedly if an apartment in the city is where his family will be. In any home with any type of yard or outdoor area, a fence is essential for the dog's safety in today's world of busy streets and traffic, even in residential areas. The breed is not given to excessive or senseless barking, which is a blessing to both owners and neighbors, no matter where you live.
**A well-behaved child and an equally well-behaved Welsh Terrier make a wonderful pair.**
The hunting instincts of the breed make walks more than a mere stroll down the lane. A brisk 30-minute walk with frequent stops for sniffing, exploring, tracking and greeting passers-by (human and canine) is ideal. Twice a day would be lovely. Once will suffice if augmented by periods of vigorous play, such as games of fetch.
The Welsh Terrier is a calm housedog, not given to boisterous behavior when adequately exercised. Most will alert you to a car or pedestrian coming up the path, but, to be honest, they are more likely to sound the alarm at an invasion by the neighbor's cat. Welsh Terriers raised with cats are generally tolerant of them, although one cannot always say the same for the cats! Introducing a cat into the home of an older Welsh who is unfamiliar with felines is another matter entirely. Proceed with utmost caution.
**NOT QUITE PERFECT**
As a Welsh Terrier owner, you may run into a behavioral problem based on something no one warned you about. It is called the Welsh Terrier Code of Ownership: "What's mine is mine and if I have any part of it (or anything else) in my mouth, that's mine, too!" It's rather like dealing with a child in the "terrible twos" stage! I don't mean to make light of it, however, for it can develop into aggressive behavior in an otherwise very compliant dog. It is not a game; you are dealing with a true terrier.
Never try to snatch anything away from the Welsh. You could be bitten in his attempt merely to hang onto his prize. Nor should you ever attempt to crawl under a bed or table to pull him out in order to retrieve your socks. Not only will you meet those jaws again, but you are confronting an earthdog. When he retreats into a small dark place with something he caught (well, "stole" is more accurate), he is in his totally natural element and will breach no mortal meddling in his domain.
The most you might achieve is a terrier battle of wits with the unpleasant sound of growling and the unacceptable appearance of curled lips and menacing teeth. The dog's owner needs to realize that he is actually causing the dog to react in this way. Let's say it's unacceptable behavior on the part of the human.
How to avoid such confrontation? Easily, by the simple means of prevention. The day your Welsh Terrier enters your home, begin to teach "drop it" or "give it" by offering a tiny treat in your left hand while holding out your right hand to accept the surrendered toy. Food is always more desirable than a mere toy or even a stolen object. As with all training, gradually diminish the use of treats, but do keep up a verbal "Good dog." This clever trick could save the dog's life when he picks up a poisonous object (or makes off with your leather purse). Lure him out of hiding with a treat worth his while—a bit of cheese or sausage, for example.
Aggressive behavior in any dog is dangerous. In the Welsh Terrier, as in any other terrier, it is compounded by the dog's natural speed and the strength of his jaws. Luckily for us (and for the dogs), this turn of events is easily preventable in the normally good-natured Welsh Terrier.
Remember that Welsh Terriers are, above all, intelligent. They study and understand our body language more clearly than our words. Therefore the exaggerated frown is a big help in relaying your message of disapproval. Welsh Terriers are also gluttons for treats and can be persuaded by food to do almost anything. If you ever hear a growl or a snarl, say "No" with a big frown. Then quickly give the dog a familiar command like "Sit," to which you can say "Good dog" and give a reward. Gradually the food can be eliminated and verbal praise alone will be effective.
**THE LASTING LOVE OF A WELSH**
The Welsh Terrier is eager and able to take on whatever lifestyle is asked of him. He will be equally adept as a lap dog, foot warmer or companion to an elderly person, or, for the more active, as a hiking, hunting, swimming or boating partner. An interesting phenomenon about the Welsh Terrier is how faithful people are to the breed. Adults who grew up with Welsh Terriers invariably want the same breed for their children, and another when those children leave home and still another for their retirement years.
**HEART-HEALTHY**
In this modern age of ever-improving cardio-care, no doctor or scientist can dispute the advantages of owning a dog to lower a person's risk of heart disease. Studies have proven that petting a dog, walking a dog and grooming a dog all show positive results toward lowering your blood pressure. The simple routine of exercising your dog—going outside with the dog and walking, jogging or playing catch—is heart-healthy in and of itself. If you are normally less active than your physician thinks you should be, adopting a dog may be a smart option to improve your own quality of life as well as that of another creature.
These small black and tan terriers have a unique way of fitting into each phase of our lives with charm, personality and sensible companionship that elicit an extreme loyalty in their owners. Airedale owners who, in their later years, can no longer cope with the size and strength of that breed, switch in great numbers to the Welsh. Then they often have to respond to that persistently annoying query, "Is that a miniature Airedale?" with a somewhat defiant, "No, sir (or ma'am). It is a Welsh Terrier!"
**GOOD HEALTH**
A genetically sound Welsh Terrier fed a good canine diet and given sufficient exercise and routine visits to the vet will live 12 to 15 years in good health. There are no health problems that are breed-specific, but certain problems that can be seen in all dogs, pure-bred and mixed, do occur now and then in the Welsh. Buying from a breeder with a good reputation for sound stock is the best way to avoid such genetic disorders as glaucoma, lens luxation, epilepsy, hypothyroidism and skin allergies, which can affect the Welsh.
There is no way to guarantee a lifetime free of all illness for any individual dog, but usually Welsh Terriers are a healthy, hardy lot. The explanation for this good fortune may lie in the fact that the breed has never become overly popular. Working with small numbers, dedicated breeders are quickly aware of any genetically transmitted disease and thus are able to remove affected animals from their breeding programs.
**ADDITIONAL ACCOMPLISHMENTS**
While the reader has been taken through many of the charming ways in which a Welsh Terrier can usurp authority, it is not always a battle of wits. Here's a look at other uses for both his terrier tenacity and common sense.
**T HERAPY DOGS**
As therapy dogs, Welsh Terriers are quite astonishing to behold. They quickly sense what is expected of them and become calm, almost serene, moving slowly and confidently among the ill and aged. They are unafraid of wheelchairs or walkers, and are willing to be petted by unsteady hands.
**A GILITY TRIALS**
Agility must have been made for terriers! All those obstacles, jumps and tunnels are second nature to the Welsh Terrier's physical stamina, sense of adventure and natural terrier instincts. To be realistic, however, getting him to cover the course in the required order is another matter entirely. Halfway through the tunnel, he may decide to wait for a fox to appear! However, do not despair. Keep your sense of humor and enjoy his amusing behavior.
**T HE EARTHDOG**
The reason these dogs were put on earth is—the _earth_! That is, to go to ground, and how they know it! Nothing can compare with the total body expression of a Welsh Terrier—eyes, ears, neck, tail—fired up by the smells of the earth, the natural hunting instinct put to the test. It makes little difference if the hunt is a natural one in the fields or along riverbanks, or in one of the artificial earths used in many earthdog tests with protected vermin (usually rats) to assess and maintain the working capabilities of the terriers. The Welsh Terrier is in his element and relishes the activity. Yet, on returning home, your friend will curl up near you by the fire, completely content.
**Terrier means "earth dog"...and any Welsh welcomes the opportunity to put his paws to work!**
**THE DELIGHTFUL DOG OF WALES**
It's difficult to sum up these delightful Welsh dogs, because they can be so diverse in personality and in the roles they play in our individual lives. I've tried to give you the bad along with the good, lest you think these black and tans belong on a pedestal. Indeed, they have their feet firmly on the ground with a heavenly, very high opinion of themselves.
It is useful to keep in mind that these dogs come from Wales, a country known for a folk tradition of argument or debate; perhaps the Welsh Terrier is carrying on that tradition. The dog is strong willed (anything weaker would be no match for his natural prey), but with enough common sense to know when it is wiser to follow the rules. For that reason alone, the owner of a Welsh Terrier must be something of a Welsh Terrier himself, able to understand the debate but also able always to remain in charge.
One initial obstacle for dog historians was the breeders' reluctance to use originality in naming their dogs. Literally hundreds of Welsh Terrier bitches named either Fan or Nell were further identified only occasionally by the owner's name. And how many Joneses would you guess there are in Wales? Since dogs changed owners from one show to the next, it was almost impossible to keep track of all the Fans and Nells. Major P. F. Brine took on the task of putting together a stud book for Britain's Welsh Terrier Club with records going back to 1854. He completed this herculean work in 1903.
Once founded, the Welsh Terrier Club (WTC) immediately set about drawing up a breed standard, which it completed in 1895. The first dog-show breeders and judges were stockmen and horse-men whose great knowledge was based on the working aspects of their animals. "Form follows function" was the rule and the reason why the original standard did not include the obvious. It remained unchanged until 1948, when the breed height was raised from 15 to 15.5 inches, a decision arrived at jointly by the WTC and the Welsh Terrier Association (WTA). The format in which the English standard appears today was approved by The Kennel Club in 1994.
The Kennel Club standard was used in the US until 1984, when parts of it were rewritten in an effort to translate it into "American English." To assist new breeders unfamiliar with breed vernacular, the Welsh Terrier Club of America publishes an annotated standard, "The Welsh Terrier in Profile," which explains the standard in detail and with sketches of each portion of the standard.
The standard is the blueprint or written description of the perfect dog of that breed, and thus serves to train the breeder's eye. It serves as a quick reference sheet for the show judge. Although one cannot quarrel with obvious deviations from it, nevertheless every breed standard is open to subjective interpretation. One person may wish to forgive a slightly gay tail and focus instead on the lovely head, while another observer sees that gay tail as a major defect, taking precedence over the nice head.
In judging the Welsh Terrier, whether by qualified conformation judges, breeders or ringside spectators, the emphasis must be on the working aspect of the dog and thus on soundness, not mere beauty. A dog with a weak front or hindquarters, or one with too short a back, or with a quarrelsome temperament, could not perform a productive day's work.
The lovely alert eye and ear expression of a Welsh Terrier, combined with good ears and seton of tail, are the beauty aspects of the dog, referred to as "type." Soundness and type must be considered jointly to be judged against that unattainable perfect specimen as described in the standard.
Mr. Walter Glynn's description is as valid today as when he wrote it 100 years ago: "The Welsh Terrier is built on the lines of a powerful, short-legged, short-backed hunter. He is best with a jet black back and neck, and deep tan head, ears, legs and tail; ears a shade deeper than elsewhere." You will note that soundness is foremost.
The most notable change in the Welsh Terrier seen from pictures of the early show dogs to those of the present is the acquisition of face and leg furnishings (the profusion of fuzzy hair on those parts). Since no farmer or miner of the day would have bothered to pluck out the hair in these areas, it is apparent that the breed originally had little or no excess hair on the legs or muzzle. The furnishings came about with the beauty aspect of the dog shows. It wasn't until the late 1920s and '30s that these were considered essential parts of the Welsh Terrier's show coat. Welsh Terriers in the US have their facial furnishings trimmed and shaped for the show ring. In Britain and on the Continent, the Welsh Terrier is shown in a somewhat more natural, or workmanlike, state.
**THE AMERICAN KENNEL CLUB BREED STANDARD FOR THE WELSH TERRIER**
**General Appearance:** The Welsh Terrier is a sturdy, compact, rugged dog of medium size with a coarse wire-textured coat. The legs, underbody and head are tan; the jacket black (or occasionally grizzle). The tail is docked to length meant to complete the image of a "square dog" approximately as high as he is long. The movement is a terrier trot typical of the long-legged terrier. It is effortless, with good reach and drive. The Welsh Terrier is friendly, outgoing to people and other dogs, showing spirit and courage. The "Welsh Terrier expression" comes from the set, color and position of the eyes combined with the use of the ears.
**A head study illustrating correct type.**
**A male Welsh Terrier of correct type and balance.**
**Size, Proportion, Substance:** Males are about 15 inches at the withers, with an acceptable range between 15 and 15.5. Bitches may be proportionally smaller. Twenty pounds is considered an average weight, varying a few pounds depending on the height of the dog and the density of bone. Both dog and bitch appear solid and of good substance.
**Head:** The entire head is rectangular. The _eyes_ are small, dark brown and almond-shaped, well set in the skull. They are placed fairly far apart. The size, shape, color and position of the eyes give the steady, confident but alert expression that is typical of the Welsh Terrier. The _ears_ are V-shaped, small, but not too thin. The fold is just above the topline of the skull. The ears are carried forward close to the cheek with the tips falling to, or toward, the outside corners of the eyes when the dog is at rest. The ears move slightly up and forward when at attention. _Skull_ —The foreface is strong with powerful, punishing jaws. It is only slightly narrower than the backskull. There is a slight stop. The backskull is of equal length to the foreface. They are on parallel planes in profile. The backskull is smooth and flat (not domed) between the ears. There are no wrinkles between the ears. The cheeks are flat and clean (not bulging). The _muzzle_ is one-half the length of the entire head from tip of nose to occiput. The foreface in front of the eyes is well made up. The furnishings on the foreface are trimmed to complete without exaggeration the total rectangular outline. The muzzle is strong and squared off, never snipy. The nose is black and squared off. The lips are black and tight. A scissors bite is preferred, but a level bite is acceptable. Either one has complete dentition. The teeth are large and strong, set in powerful, vise-like jaws.
**Faults: Upright shoulders; dip in back; high in rear.**
**Faults: Short, thick neck; upright shoulders; low on leg.**
**Faults: Thick neck and shoulders; lack of angulation in the rear; marginal tail set.**
**Faults: General lack of substance; ewe-neck; gay tail; long back.**
**Comparing type in the Welsh and Wire-haired Fox Terrier**
**The Welsh is stockier, with more substance. The Fox Terrier has a longer, narrower head, with smaller, high-set ears.**
**Neck, Topline, Body:** The neck is of moderate length and thickness, slightly arched and sloping gracefully into the shoulders. The throat is clean with no excess of skin. The topline is level. The body shows good substance and is well ribbed up. There is good depth of brisket and moderate width of chest. The loin is strong and moderately short. The tail is docked to a length approximately level (on an imaginary line) with the occiput, to complete the square image of the whole dog. The root of the tail is set well up on the back. It is carried upright.
**Forequarters:** The front is straight. The shoulders are long, sloping and well laid back. The legs are straight and muscular with upright and powerful pasterns. The feet are small, round, and catlike. The pads are thick and black. The nails are strong and black; any dewclaws are removed.
**Hindquarters:** The hindquarters are strong and muscular with well-developed second thighs and the stifles well bent. The hocks are moderately straight, parallel and short from joint to ground. The feet should be the same as in the forequarters.
**Coat:** The coat is hard, wiry, and dense with a close-fitting thick jacket. There is a short, soft undercoat. Furnishings on muzzle, legs, and quarters are dense and wiry.
**Color:** The jacket is black, spreading up onto the neck, down onto the tail and into the upper thighs. The legs, quarters, and head are clear tan. The tan is a deep reddish color, with slightly lighter shades acceptable. A grizzle jacket is also acceptable.
**Ch. Shaireab's Sam I Am, owned by Sharon Abmeyer, finished his championship from the puppy classes. Sam is producing good bone, excellent movement and lovely headpieces.**
**Gait:** The movement is straight, free and effortless, with good reach in front, strong drive behind, with feet naturally tending to converge toward a median line of travel as speed increases.
**Temperament:** The Welsh Terrier is a game dog—alert, aware, spirited, but at the same time, is friendly and shows self control. Intelligence and desire to please are evident in his attitude. A specimen exhibiting an overly aggressive attitude, or shyness, should be penalized.
**Faults:** Any deviation from the foregoing should be considered a fault; the seriousness of the fault depending upon the extent of the deviation.
**Approved August 10, 1993**
**Effective September 29, 1993**
**Am-Int. Ch. Kirkwood Top Brass, owned by Frank Stevens, was a Montgomery County and Westminster Best of Breed winner. He is the sire of Group and specialty winners.**
**SIGNS OF A HEALTHY PUPPY**
Healthy puppies are robust little fellows who are alert and active, sporting shiny coats and supple skin. They should not appear lethargic, bloated or pot-bellied, nor should they have flaky skin or runny or crusted eyes or noses. Their stools should be firm and well formed, with no evidence of blood or mucus.
**SELECTING A PUPPY**
You'll notice a few things when taking your first look at a Welsh litter. If you see the Welsh pups at three or four weeks of age, they are likely to be almost entirely black. However, the tan increases in all the right places as the pups mature. Also, don't worry if the dam is a bit protective of her brood; that's normal.
Nine to ten weeks is the ideal age to bring a Welsh puppy into your life. Each puppy needs that much time to learn that he is a dog, how to behave as a dog and—so very importantly—how to read the body language of other dogs. A misunderstanding in the latter is usually the spark that sets off a fight. The Welsh can best learn all of this from his dam and littermates. The physical, and sometimes vocal, activity among littermates is part of the essential learning experience.
Unless you have previously owned a Welsh Terrier, know something about the breed and consider yourself to be a "terrier person," I would not advise you to select the most active pup in the litter. In some breeds, that would be exactly the pup to choose, but not in the Welsh Terrier. The one that wants to continue to play energetically when the others stop is not going to be an easy pup to convince that he must take his orders from you! He will need a pleasant but very firm, very consistent hand to tone down his natural exuberance, which otherwise will turn into dominance.
**FINDING A QUALIFIED BREEDER**
Before you begin your puppy search, ask for references from the breed club, your veterinarian and other breeders to refer you to someone they believe is reputable. Responsible breeders usually raise only one or two breeds of dog. Avoid any breeder who has several different breeds or has several litters at the same time. Dedicated breeders are usually involved with a breed or other dog club. Many participate in some sport or activity related to their breed. Just as you want to be assured of the breeder's qualifications, the breeder wants to be assured that you will make a worthy owner. Expect the breeder to interview you, asking questions about your goals for the pup, your experience with dogs and what kind of home you will provide.
There is also the pup that looks you straight in the eye with a look that says, "See me? I'm the best!" That one may be destined for the show ring, or could be a nice pet with a good attitude. A calm, somewhat quiet puppy may be sizing you up as a potential partner. At the other end of the temperament scale is the shy pup. True shyness is almost unheard of in Welsh Terriers, so beware the pup that shuns your hand or creeps away to sit by himself. That one may require the help of a professional behaviorist before too long.
**Bring the family along when visiting the litter. Everyone should take part in selecting the new puppy.**
**CREATE A SCHEDULE**
Puppies thrive on sameness and routine. Offer meals at the same time each day, take him out at regular times for potty trips and do the same for play periods and outdoor activity. Make note of when your puppy naps and when he is most lively and energetic, and try to plan his day around those times. Once he is house-trained and more predictable in his habits, he will be better able to tolerate changes in his schedule.
Choose any one of the happy, friendly, normal Welsh pups wanting to kiss your hands, nibble your fingers or just have your attention. The chances are good that you won't even be confronted with either of the two extremes.
**PUPPY NEEDS AND DEVELOPMENT**
Your Welsh Terrier puppy will need all-day everyday attention for about six weeks. The routine of feeding, house-training and exercise is broken only by frequent naps (yours coinciding with the pup's). If you are not there to teach the puppy what he can do (as well as where and when) and what he cannot ever do, he will instantly teach himself. If he gets away with something naughty because nobody was there to tell him otherwise, the puppy will have taught himself that it was the right thing to do. And he'll do it again. The owner is the teacher, guidance counselor and corrections officer!
At any time between five and seven months, a Welsh Terrier may go through what is commonly called a fear phase, when the normally outgoing, happy pup may suddenly seem shy or fearful. The phase can last a day or a few weeks and the best way to live through it is not to give in to it. Keep the pup with you (on lead, if necessary) and go about your business, using a normal tone of voice and lots of cheerful chat. Ask your guests to pay no attention until the pup makes the first overture. All of this is nature's way of cautioning the rapidly maturing puppy to slow down. Fear of thunderstorms is quite different. That is nature's way of telling all animals to seek shelter. Don't fall into the trap of cuddling your puppy and saying "It's all okay" because it most decidedly is not so in the pup's mind. Use the cheerful chat distraction routine, add some toys and turn up the TV.
**PEDIGREE VS. REGISTRATION CERTIFICATE**
Too often new owners are confused between these two important documents. Your puppy's pedigree, essentially a family tree, is a written record of a dog's genealogy of three generations or more. The pedigree will show you the names as well as performance titles of all dogs in your pup's background. Your breeder must provide you with a registration application, with his part properly filled out. You must complete the application and send it to the AKC with the proper fee. Every puppy must come from a litter that has been AKC-registered by the breeder, born in the US and from a sire and dam that are also registered with the AKC.
The seller must provide you with complete records to identify the puppy. The AKC requires that the seller provide the buyer with the following: breed; sex, color and markings; date of birth; litter number (when available); names and registration numbers of the parents; breeder's name; and date sold or delivered.
After giving you all of this advice, however, the vast majority of Welsh Terriers do not go through this phase for more than five minutes. It may be due to their conviction that they will all grow up to be Airedales! Although the Welsh reaches full height by ten months of age, he is not fully mature and in fact will probably begin a period of teenage nonsense. By the age of two years, he will be a self-confident adult.
**INDOOR/OUTDOOR**
A Welsh Terrier is not a dog to be left outdoors despite his harsh coat and hardy physique. He thrives on social interaction with his family and craves the creature comforts of his home. If you don't want a dog to "help" make the beds, rearrange the flowers or read a book tucked up close to you, forget the Welsh. He won't be underfoot, just close by because he is certain that you'll need his assistance at any minute.
**Outdoors on a rainy day is no place for the family dog. Your Welsh is a rugged outdoorsman, but he thrives on the companionship of his family and the comforts of home.**
**NEW RELEASES**
Most Welsh Terrier breeders release their puppies between nine and ten weeks of age. A breeder who allows puppies to leave the litter at five or six weeks of age may be more concerned with profit than with the puppies' welfare. However, some breeders of show or working breeds may hold one or more top-quality puppies longer, occasionally until three or four months of age, in order to evaluate the puppies' career or show potential and decide which one(s) they will keep for themselves.
**IF YOU'D PREFER AN ADULT**
If you do not have the time needed to teach a puppy all he must learn, there are several ways to obtain an adult Welsh Terrier. Breeders often have dogs or bitches that have finished their show or breeding careers—or perhaps never lived up to their potential. There are also dogs in need of adoption for any number of reasons, such as the death of a previous owner or an irresponsible, impulsive owner's abandonment. Contact the Welsh Terrier club nearest you and follow their advice. Welsh Terrier rescuers also take in abandoned dogs and dogs who end up in shelters and work hard to find loving homes for them.
The only caveat is that the adult was trained, rightly or wrongly, by someone else. The adult Welsh coming into a new home will have all the basic things to learn, such as your house rules, the family members, your home routine, the words you use, the sounds, smells and sights. He will also have an equal amount to unlearn, including habits that you may not permit in your home. Welsh Terriers are not one-man dogs, although they are skilled at making each of their owners think otherwise. There will be an adjustment period and at some point the dog may indicate that it has been pleasant visiting with you, but now he'd like to go home. On the other hand, he may think he's landed in Heaven and never look back. Either way, as soon as he settles in, the new owner will become his best friend. As I've said, the Welsh Terrier is a very sensible dog.
**DOG OR BITCH?**
The prevalent view is that all males are aggressive watchdogs and all bitches will be sweet and stay close to home. This is pure fabrication! The differences in Welsh Terrier temperament and character are almost non-existent. Males are slightly larger and, properly raised, are complete gentlemen. Bitches can be very sweet and feminine—or not. Go with the individual pup that attracts you and, in either case, have your pet neutered or spayed by six months of age. It is a fallacy that such surgery will keep either sex closer to home or make your pet fat and lazy. It will restrain the hormones, however, and offers important health benefits. Only future show or breeding dogs should be kept sexually intact.
**A CONFIDENT NEW OWNER**
Like most of the other terriers, Welsh Terriers do best with terrier-like people. These are people who are themselves alert, ready to go and inquisitive, but calm, self-assured and sensible. This is not the breed for the meek and mild or indecisive.
By now you should understand what makes the Welsh Terrier a most unique and special dog, one that may fit nicely into your family and lifestyle. If you have contacted the Welsh Terrier Club of America, asked for breeder referrals and researched breeders, you should be able to recognize a knowledgeable and responsible Welsh Terrier breeder who cares not only about his pups but also about what kind of owner you will be. The Welsh Terrier Club of America's website (<http://clubs.akc.org/wtca>) holds a wealth of information on the breed, including breeder information, health, living with a Welsh, etc. WTCA member breeders are obliged to follow a strict code of ethics in their breeding programs.
**WTCA RESCUE**
WTCARES, a branch of the parent club, rescues and re-homes Welsh Terriers throughout the United States. On average, 55 to 60 dogs go through the system annually. About 25 WTCA members serve as rescue volunteers. They locate and collect the dogs, groom them, take them to a veterinarian and evaluate temperament and training, in order to place each dog with a suitable new owner. The breed adjusts easily, and WTCARES usually has a waiting list of potential owners, carefully screened by the chairperson.
**Visiting the breeder's facilities says a lot about the breeder and how he cares for his dogs. View the litter, meet the other dogs on the premises, see where the dogs are kept and exercised, etc.**
**GETTING ACQUAINTED**
When visiting a litter, ask the breeder for suggestions on how best to interact with the puppies. If possible, get right into the middle of the pack and sit down with them. Observe which pups climb into your lap and which ones shy away. Toss a toy for them to chase and bring back to you. It's easy to fall in love with the puppy who picks you, but keep your future objectives in mind before you make your final decision.
If you have completed the final step in your journey, you have found a litter, or possibly two, of quality Welsh Terrier pups. A visit with the puppies and their breeder should be an education in itself. Breed research, breeder selection and puppy visitation are very important aspects of finding the puppy of your dreams. Beyond that, these things also lay the foundation for a successful future with your pup. Puppy personalities within each litter vary, from the shy and easygoing puppy to the one who is dominant and assertive, with most pups falling somewhere in between. By spending time with the puppies, you will be able to recognize certain behaviors and what these behaviors indicate about each pup's temperament. Which type of pup will complement your family dynamics is best determined by observing the puppies in action within their "pack." Your breeder's expertise and recommendations are also valuable. Although you may fall in love with a bold and brassy male, the breeder may suggest that another pup would be best for you. The breeder's experience in rearing Welsh Terrier pups and matching their temperaments with appropriate humans offers the best assurance that your pup will meet your needs and expectations. The type of puppy that you select is just as important as your decision that the Welsh Terrier is the breed for you.
The decision to live with a Welsh Terrier is a serious commitment and not one to be taken lightly. This puppy is a living sentient being that will be dependent on you for basic survival for his entire life. Beyond the basics of survival—food, water, shelter and protection—he needs much, much more. The new pup needs love, nurturing and a proper canine education to mold him into a responsible, well-behaved canine citizen. Your Welsh Terrier's health and good manners will need consistent monitoring and regular "tuneups," so your job as a responsible dog owner will be ongoing throughout every stage of his life. If you are not prepared to accept these responsibilities and commit to them for the next 12 or more years, then you are not prepared to own a dog of any breed.
**SELECTING FROM THE LITTER**
Before you visit a litter of puppies, promise yourself that you won't fall for the first pretty face you see! Decide on your goals for your puppy—show prospect, hunting dog, obedience competitor, family companion—and then look for a puppy who displays the appropriate qualities. In most litters, there is an Alpha pup (the bossy puppy), and occasionally a shy fellow who is less confident, with the rest of the litter falling somewhere in the middle. "Middle-of-the-roaders" are safe bets for most families and novice competitors.
**Chow time! Sturdy bowls and good-quality food are among the items you will need for your Welsh Terrier.**
Although the responsibilities of owning a dog may at times tax your patience, the joy of living with your Welsh Terrier far outweighs the workload, and a well-mannered adult dog is worth your time and effort. Before your very eyes, your new charge will grow up to be your most loyal friend, devoted to you unconditionally.
**YOUR WELSH TERRIER SHOPPING LIST**
Just as expectant parents prepare a nursery for their baby, so should you ready your home for the arrival of your Welsh Terrier pup. If you have the necessary puppy supplies purchased and in place before he comes home, it will ease the puppy's transition from the warmth and familiarity of his mom and littermates to the brand-new environment of his new home and human family. You will be too busy to stock up and prepare your house after your pup comes home, that's for sure! Imagine how a pup must feel upon being transported to a strange new place. It's up to you to comfort him and to let your little pup know that he is going to be happy with you.
**Durable stainless steel bowls are recommended as they will withstand the wear-and-tear of terrier teeth.**
**F OOD AND WATER BOWLS**
Your puppy will need separate bowls for his food and water. Stainless steel pans are generally preferred over plastic bowls since they sterilize better and pups are less inclined to chew on the metal. Heavy-duty ceramic bowls are popular, but consider how often you will have to pick up those heavy bowls. Buy adultsized pans, as your puppy will grow into them quickly.
**T HE DOG CRATE**
If you think that crates are tools of punishment and confinement for when a dog has misbehaved, think again. Most breeders and almost all trainers recommend a crate as the preferred house-training aid as well as for all-around puppy training and safety. Because dogs are natural den creatures that prefer cave-like environments, the benefits of crate use are many. The crate provides the puppy with his very own "safe house," a cozy place to sleep, take a break or seek comfort with a favorite toy; a travel aid to house your dog when on the road, at motels or at the vet's office; a training aid to help teach your puppy proper toileting habits; a place of solitude when non-dog people happen to drop by and don't want a lively puppy—or even a well-behaved adult dog—saying hello or begging for their attention.
**COST OF OWNERSHIP**
The purchase price of your puppy is merely the first expense in the typical dog budget. Quality dog food, veterinary care (sickness and health maintenance), dog supplies and grooming costs will add up to big bucks every year. Can you adequately afford to support a canine addition to the family?
Crates come in several types, although the wire crate and the fiberglass airline-type crate are the most popular. Both are safe and your puppy will adjust to either one, so the choice is up to you. The wire crates offer better visibility for the pup as well as better ventilation. Many of the wire crates collapse for easy transport. The fiberglass crates, similar to those used by the airlines for animal transport, are sturdier and more den-like. However, the fiber-glass crates do not collapse and are less ventilated than a wire crate, which can be problematic in hot weather. Some of the newer crates are made of heavy plastic mesh; they are very lightweight and fold up into slim-line suitcases. However, a mesh crate might not be suitable for a pup with manic chewing habits.
The size of the crate is another thing to consider, but for the Welsh Terrier you will only need to do so once. Buy an adultsized crate from the outset, as the pup will soon grow into it. A crate approximately 24 inches long by 20 inches wide by 21 inches high will last the Welsh's lifetime.
The crate begins as his special puppy place to be, and as his bed overnight with the door closed. With the door left open during the day, the crate is his den where he can put his toys and be by himself for a nap. A special feature of the crate is the travel aspect, not just for safety in the car, but when you go off visiting friends and relatives, the crate is the dog's home away from home, making him feel "at home" even when he's not. The crate is the "den" of the earthdog.
**B EDDING AND CRATE PADS**
Your puppy will enjoy some type of soft bedding in his "room" (the crate), something he can snuggle into to feel cozy and secure. Old towels or blankets are good choices for a young pup, since he may (and probably will) have a toileting accident or two in the crate or decide to chew on the bedding material. Once he is fully trained and out of the early chewing stage, you can replace the puppy bedding with a permanent crate pad if you prefer. Crate pads and other dog beds run the gamut from inexpensive to high-end doggie-designer styles, but don't splurge on the good stuff until you are sure that your puppy is reliable and won't tear it up or make a mess on it.
**The most common crate types: mesh on the left, wire on the right and fiberglass on top.**
**P UPPY TOYS**
Just as infants and older children require objects to stimulate their minds and bodies, puppies need toys to entertain their curious brains, wiggly paws and achy teeth. A fun array of safe doggie toys will help satisfy your puppy's chewing instincts and distract him from gnawing on the leg of your antique chair or your new leather sofa. Most puppy toys are cute and look as if they would be a lot of fun, but not all are necessarily safe or good for your puppy, so use caution when you go puppy-toy shopping.
**In addition to a crate, your Welsh will appreciate a cozy dog bed. Be aware, though, that a wicker bed can be destroyed in short order by a chewing pup and the pieces he chews off could be harmful.**
**CRATE EXPECTATIONS**
To make the crate more inviting to your puppy, you can offer him a cookie inside the crate, always keeping the crate door open so that he does not feel confined. Keep a favorite toy or two in the crate for him to play with while inside. You can also cover the crate at night with a lightweight sheet to make it more den-like and remove the stimuli of household activity. Never put him into his crate as punishment or as you are scolding him, since he will then associate his crate with negative situations and avoid going there.
Plush squeaky toys are the earthdogs' favorites! What else is there that is soft and squashy and squeals like a cornered rat? All Welsh puppies enjoy them. Indulge your pup, but monitor all toys and get rid of any that have been chewed to divulge the stuffing or squeaker or have any small parts (eyes, etc.) in danger of becoming detached and swallowed. Some Welsh Terriers will play with a squeaky toy for years, even after it no longer squeaks. Admittedly, others will destroy the entire toy in half an hour. If your dog is one of the latter, you'll have to forgo the squeaky variety and stick with heavy-duty rubber toys or knotted rope toys and large knucklebones.
You can make an excellent puppy teething toy out of a knotted piece of towel, dampened and put in the fridge for an hour or so. Chewing into the cold toweling relieves itchy gums caused by the new teeth's erupting and provides good exercise for the jaws. An inexpensive, beneficial toy!
The best "chewcifiers" are sturdy nylon and hard rubber bones, which are safe to gnaw on and come in sizes appropriate for all age groups and breeds. Be especially careful of natural bones, which can splinter or develop dangerous sharp edges; pups can easily swallow or choke on those bone splinters. Veterinarians often tell of surgical nightmares involving bits of splintered bone, because in addition to the danger of choking, the sharp pieces can damage the intestinal tract.
Similarly, rawhide chews, while a favorite of most dogs and puppies, can be equally dangerous. Pieces of rawhide are easily swallowed after they get soft and gummy from chewing, and dogs have been known to choke on large pieces of ingested rawhide. Rawhide chews should be offered only when you can supervise the puppy.
**TEETHING TIME**
All puppies chew. It's normal canine behavior. Chewing just plain feels good to a puppy, especially during the three- to five-month teething period when the adult teeth are breaking through the gums. Rather than attempting to eliminate such a strong natural chewing instinct, you will be more successful if you redirect it and teach your puppy what he may or may not chew. Correct inappropriate chewing with a sharp "No!" and offer him a chew toy, praising him when he takes it. Don't become discouraged. Chewing usually decreases after the adult teeth have come in.
If you believe that your pup has ingested a piece of one of his toys, check his stools for the next couple of days to see if he passes the item when he defecates. At the same time, also watch for signs of intestinal distress. A call to your veterinarian might be in order to get his advice and be on the safe side.
**TOYS 'R SAFE**
The vast array of tantalizing puppy toys is staggering. Stroll through any pet shop or pet-supply outlet and you will see that the choices can be overwhelming. However, not all dog toys are safe or sensible. Most very young puppies enjoy soft woolly toys that they can snuggle with and carry around. (You know they have outgrown them when they shred them up!) Avoid toys that have buttons, tabs or other enhancements that can be chewed off and swallowed. Soft toys that squeak are fun, but make sure your puppy does not disembowel the toy and remove (and swallow) the squeaker. Toys that rattle or make noise can excite a puppy, but they present the same danger as the squeaky kind and so require supervision. Hard rubber toys that bounce can also entertain a pup, but make sure that the toy is too big for your pup to swallow.
An all-time favorite toy for puppies (young and old!) is the empty gallon milk jug. Hard plastic juice containers—46 ounces or more—are also excellent. Such containers make lots of noise when they are batted about, and puppies go crazy with delight as they play with them. However, they don't often last very long, so be sure to remove and replace them when they get chewed up.
A word of caution about homemade toys: be careful with your choices of non-traditional play objects. Never use old shoes or socks, since a puppy cannot distinguish between the old ones on which he's allowed to chew and the new ones in your closet that are strictly off-limits. That principle applies to anything that resembles something that you don't want your puppy to chew.
**C OLLARS**
A lightweight nylon collar is the best choice for a very young pup. Quick-clip collars are easy to put on and remove, and they can be adjusted as the puppy grows. Introduce him to his collar as soon as he comes home to get him accustomed to wearing it. He'll get used to it quickly and won't mind a bit. Make sure that it is snug enough that it won't slip off, yet loose enough to be comfortable for the pup. You should be able to slip two fingers between the collar and his neck. Check the collar often, as puppies grow in spurts, and his collar can become too tight almost overnight.
**C OLLARING OUR CANINES**
The standard flat collar with a buckle or a snap, in leather, nylon or cotton, is widely regarded as the everyday all-purpose collar. If the collar fits correctly, you should be able to fit two fingers between the collar and the dog's neck.
The martingale, Greyhound or limited-slip collar is preferred by many dog owners and trainers. It is fixed with an extra loop that tightens when pressure is applied to the leash. The martingale collar gets tighter but does not "choke" the dog. The limited-slip collar should only be used for walking and training, not for free play or interaction with another dog. These types of collar should never be left on the dog, as the extra loop can lead to accidents.
Choke collars, usually made of stainless steel, are made for training purposes, though are not recommended for small or heavily coated dogs, and certain other breeds, including the Welsh Terrier. Thin nylon choke leads are commonly used on show dogs while in the ring, though they are not practical for everyday use.
The harness, with two or three straps that attach over the dog's shoulders and around his torso, is a humane and safe alternative to the conventional collar. By and large, a well-made harness is virtually escape-proof. Harnesses are available in nylon and mesh and can be outfitted on most dogs, with chest girths ranging from 10 to 30 inches.
A head collar, composed of a nylon strap that goes around the dog's muzzle and a second strap that wraps around his neck, offers the owner better control over his dog. This device is recommended for problem-solving with dogs (including jumping up, pulling and aggressive behaviors), but must be used with care.
A training halter, including a flat collar and two straps, made of nylon and webbing, is designed for walking. There are several on the market; some are more difficult to put on the dog than others. The halter harness, with two small slip rings at each end, is recommended for ease of use.
**This Welsh pup is becoming accustomed to his nylon lead, attached to a sturdy, light collar.**
Welsh puppies seldom object to any collar for more than a few minutes. The martingale collar (which is a double loop) prevents the dog from backing out as well as allowing you to make gentle corrections when he lunges ahead. It can be used for training as well. For the more athletic types, or those Welsh Terriers resistant to class instruction, a head collar is the answer. The head collar is an attention-getting device that allows you to turn the dog's head toward you, giving you better control over those persistent terrier distractions. Choke collars should never be used; the head collar is more conducive to mutual understanding.
**L EASHES**
A 4-foot nylon lead is an excellent choice for a young puppy. It is lightweight and not as tempting to chew as a leather lead. For initial puppy walks and house-training purposes, the shorter 4-foot length will give you more control over the puppy. At first you don't want him wandering too far away from you and, when taking him out for toileting, you will want to keep him in the specific area chosen for his potty spot. As he becomes house-trained and polite on lead, you can progress to a 6-foot lead for walks.
The best lead for training is a 6-foot cotton lead. It is gentle on your hands and won't slip as easily as nylon. Welsh Terriers are generally cooperative about this "attachment for safety" program of ours. As an adult, when he won't be so likely to chew it to bits, your Welsh Terrier will look quite handsome with a matching leather collar and lead.
Once the puppy is heel-trained with a traditional leash, you can consider purchasing a retractable lead. This type of lead is excellent for walking adult dogs that are already leash-wise. The retractable lead allows the dog to roam farther away from you and explore a wider area when out walking, and also retracts when you need to keep him close.
**HOME SAFETY FOR YOUR PUPPY**
The importance of puppy-proofing cannot be overstated. In addition to making your house comfortable for your Welsh Terrier's arrival, you also must make sure that your house is safe for your puppy before you bring him home. There are countless hazards in the owner's personal living environment that a pup can sniff, chew, swallow or destroy. Many are obvious; others are not. Do a thorough advance house check to remove or rearrange those things that could hurt your puppy, keeping any potentially dangerous items out of areas to which he will have access.
Electrical cords are especially dangerous, since puppies view them as irresistible chew toys. Unplug and remove all exposed cords or fasten them beneath a baseboard where the puppy cannot reach them. Veterinarians and firefighters can tell you horror stories about electrical burns and house fires that resulted from puppy-chewed electrical cords. Consider this a most serious precaution for your puppy and the rest of your family.
Scout your home for tiny objects that might be seen at a pup's eye level. Keep medication bottles and cleaning supplies well out of reach, and do the same with waste baskets and other trash containers. It goes without saying that you should not use rodent poison or other toxic chemicals in any puppy area and that you must keep such containers safely locked up. You will be amazed at how many places a curious puppy can discover!
Once your house has cleared inspection, check your yard. A sturdy fence, well embedded into the ground, will give your dog a safe place to play and potty. Welsh Terriers are athletic dogs, so a 6-foot-high fence will be required to contain an agile youngster or adult. "Well-embedded into the ground" is especially important with terriers, who were born to dig. Check the fence periodically for necessary repairs. If there is a weak link or space to squeeze through or under, you can be sure a Welsh Terrier will discover it. A very determined pup may return to the same spot to "work on it" until he is able to get through. Also be very careful about doors that open into unfenced areas. Each family member must be on guard lest the Welsh Terrier slip out unnoticed.
**TOXIC PLANTS**
Plants are natural puppy magnets, but many can be harmful, even fatal, if ingested by a puppy or adult dog. Scout your yard and home interior and remove any plants, bushes or flowers that could be even mildly dangerous. It could save your puppy's life. You can obtain a complete list of toxic plants from your veterinarian, at the public library or by looking online.
**A D OG-SAFE HOME**
The dog-safety police are taking you and your new puppy on a house tour. Let's go room by room and see how safe your own home is for your new pup. The following items are doggie dangers, so either they must be removed or the dog should be monitored or not have access to these areas.
**PUPPY PARASITES**
Parasites are nasty little critters that live in or on your dog or puppy. Most puppies are born with ascarid roundworms, which are acquired from dormant ascarids residing in the dam. Other parasites can be acquired through contact with infected fecal matter. Take a stool sample to your vet for testing. He will prescribe a safe wormer to treat any parasites found in your puppy's stool. Always have a fecal test performed at your puppy's annual veterinary exam.
The garage and shed can be hazardous places for a dog, as things like fertilizers, chemicals and tools are usually kept there. It's best to keep these areas off-limits to your Welsh. Antifreeze is especially dangerous to dogs, as they find the taste appealing and it takes only a few licks from the driveway to kill a dog, puppy or adult, small breed or large.
**VISITING THE VETERINARIAN**
A good veterinarian is your Welsh Terrier puppy's best health-insurance policy. If you do not already have a vet, ask friends and experienced dog people in your area for recommendations so that you can select a vet before you bring your Welsh Terrier puppy home. Also arrange for your puppy's first veterinary examination beforehand, since many vets do not have appointments available immediately and your puppy should visit the vet within a day or so of coming home.
It's important to make sure your puppy's first visit to the vet is a pleasant and positive one. The vet should take great care to befriend the pup and handle him gently to make their first meeting a positive experience. The vet will give the pup a thorough physical examination and set up a schedule for vaccinations and other necessary wellness visits. Be sure to show your vet any health and inoculation records, which you should have received from your breeder. Your vet is a great source of canine health information, so be sure to ask questions and take notes. Creating a health journal for your puppy will make a handy reference for his wellness and any future health problems that may arise.
**Your Welsh Terrier puppy will be very curious about his new home and surroundings. Be sure that the yard is securely enclosed and that there are no dangerous plants or toxic chemicals in your landscaping.**
**MEETING THE FAMILY**
Your Welsh Terrier's homecoming is an exciting time for all members of the family, and it's only natural that everyone will be eager to meet him, pet him and play with him. However, for the puppy's sake, it's best to make these initial family meetings as uneventful as possible so that the pup is not overwhelmed with too much too soon. Remember, he has just left his dam and his littermates and is away from the breeder's home for the first time. Despite his fuzzy wagging tail, he is still apprehensive and wondering where he is and who all these strange humans are. It's best to let him explore on his own and meet the family members as he feels comfortable. Let him investigate all the new smells, sights and sounds at his own pace. Children should be especially careful to not get overly excited, use loud voices or hug the pup too tightly. Be calm, gentle and affectionate, and be ready to comfort him if he appears frightened or uneasy.
Be sure to show your puppy his new crate during this first day home. Toss a treat or two inside the crate; if he associates the crate with food, he will associate the crate with good things. Do not feed your puppy inside the crate, as this can lead to food-aggressive behavior. Leave the door ajar so he can wander in and out as he chooses.
**THE WORRIES OF MANGE**
Sometimes called "puppy mange," demodectic mange is passed to the puppy through the mother's milk. The microscopic mites that cause the condition take up residence in the puppy's hair follicles and sebaceous glands. Stress can cause the mites to multiply, causing bare patches on the face, neck and front legs. If neglected, it can lead to secondary bacterial infections, but if diagnosed and treated early, demodectic mange can be localized and controlled. Most pups recover without complications.
**FIRST NIGHT IN HIS NEW HOME**
So much has happened in your Welsh Terrier puppy's first day away from the breeder. He's had his first car ride to his new home. He's met his new human family and perhaps the other family pets. He has explored his new house and yard, at least those places where he is to be allowed during his first weeks at home. He may have visited his new veterinarian. He has eaten his first meal or two away from his dam and littermates. Surely that's enough to tire out a nine-week-old Welsh Terrier pup...or so you hope!
It's bedtime. During the day, the pup investigated his crate, which is his new den and sleeping space, so it is not entirely strange to him. Line the crate with a soft towel or blanket that he can snuggle into and gently place him into the crate for the night. Some breeders send home a piece of bedding from where the pup slept with his littermates, and those familiar scents are a great comfort for the puppy on his first night without his siblings.
He will probably whine or cry. The puppy is objecting to the confinement and the fact that he is alone for the first time. This can be a stressful time for you as well as for the pup. It's important that you remain strong and don't let the puppy out of his crate to comfort him. He will fall asleep eventually. If you release him, the puppy will learn that crying means "out" and will continue that habit. You are laying the groundwork for future habits. Some breeders find that soft music can soothe a crying pup and help him get to sleep.
**SOCIALIZING YOUR PUPPY**
The first 20 weeks of your Welsh Terrier puppy's life are the most important of his entire lifetime. A properly socialized puppy will grow up to be a confident and stable adult who will be a pleasure to live with and a welcome addition to the neighborhood.
The importance of socialization cannot be overemphasized. Research on canine behavior has proven that puppies who are not exposed to new sights, sounds, people and animals during their first 20 weeks of life will grow up to be timid and fearful, even aggressive, and unable to flourish outside of their home environment.
Socializing your puppy is not difficult and, in fact, will be a fun time for you both. Lead training goes hand in hand with socialization, so your puppy will be learning how to walk on a lead at the same time that he's meeting the neighborhood. Because the Welsh Terrier is such a terrific breed, everyone will enjoy meeting "the new kid on the block." Take him for short walks, to the park and to other dog-friendly places where he will encounter new people, especially children. Puppies automatically recognize children as "little people" and are drawn to play with them. Just make sure that you supervise these meetings and that the children do not get too rough or encourage him to play too hard. An overzealous pup can often nip too hard, frightening the child and in turn making the puppy overly excited. A bad experience in puppyhood can impact a dog for life, so a pup that has a negative experience with a child may grow up to be shy or even aggressive around children.
**Consider your pup's safety both indoors and out, by puppy-proofing and supervising. A pup doesn't know the difference between an electrical cord and a chew toy, which could lead to danger.**
Take your puppy along on your daily errands. Puppies are natural "people magnets," and most people who see your pup will want to pet him. All of these encounters will help to mold him into a confident adult dog. Likewise, you will soon feel like a confident, responsible dog owner, rightly proud of your handsome Welsh Terrier.
**THE FAMILY FELINE**
A resident cat has feline squatter's rights. The cat will treat the newcomer (your puppy) as she sees fit, regardless of what you do or say. So it's best to let the two of them work things out on their own terms. Cats have a height advantage and will generally leap to higher ground to avoid direct contact with a rambunctious pup. Some will hiss and boldly swat at a pup who passes by or tries to reach the cat. Keep the puppy under control in the presence of the cat and they will eventually become accustomed to each other.
Here's a hint: move the cat's litter box where the puppy can't get into it! It's best to do so well before the pup comes home so the cat is used to the new location.
Be especially careful of your puppy's encounters and experiences during the eight-to-ten-week-old period, which is also called the "fear period," if you have him for part of this time. This is a serious imprinting period, and all contact during this time should be gentle and positive. A frightening or negative event could leave a permanent impression that could affect his future behavior if a similar situation arises.
Also make sure that your puppy has received his first and second rounds of vaccinations before you expose him to other dogs or bring him to places that other dogs may frequent. Avoid dog parks and other strange-dog areas until your vet assures you that your puppy is fully immunized and resistant to the diseases that can be passed between canines. Discuss socialization with your breeder, as some breeders recommend socializing the puppy even before he has received all of his inoculations, depending on how outgoing the puppy may be.
**LEADER OF THE PUPPY'S PACK**
Like other canines, your puppy needs an authority figure, someone he can look up to and regard as the leader of his "pack." His first pack leader was his dam, who taught him to be polite and not chew too hard on her ears or nip at her muzzle. He learned those same lessons from his littermates. If he played too rough, they cried in pain and stopped the game, which sent an important message to the rowdy puppy.
As puppies play together, they are also struggling to determine who will be the boss. Being pack animals, dogs need someone to be in charge. If a litter of puppies remained together beyond puppy-hood, one of the pups would emerge as the strongest one, the one who calls the shots.
Once your puppy leaves the pack, he will look intuitively for a new leader. If he does not recognize you as that leader, he will try to assume that position for himself. Of course, it is hard to imagine your adorable Welsh Terrier puppy trying to be in charge when he is so small and seemingly helpless. You must remember that these are natural canine instincts. Do not cave in and allow your pup to get the upper "paw"!
Just as socialization is so important during these first 20 weeks, so too is your puppy's early education. He was born without any bad habits. He does not know what is good or bad behavior. If he does things like nipping and digging, it's because he is having fun and doesn't know that humans consider these things as "bad." It's your job to teach him proper puppy manners, and this is the best time to accomplish that...before he has developed bad habits, since it is much more difficult to "unlearn" or correct unacceptable learned behavior than to teach good behavior from the start.
Make sure that all members of the family understand the importance of being consistent when training their new puppy. If you tell the puppy to stay off the sofa and your daughter allows him to cuddle on the couch to watch her favorite television show, your pup will be confused about what he is and is not allowed to do. Have a family conference before your pup comes home so that everyone understands the basic principles of puppy training and the rules you have set forth for the pup, and agrees to follow them.
**Socialization begins at the breeder's home as the pups interact with their dam, their littermates and the people that live there.**
The old adage that "an ounce of prevention is worth a pound of cure" is especially true when it comes to puppies. It is much easier to prevent inappropriate behavior than it is to change it. It's also easier and less stressful for the pup, since it will keep discipline to a minimum and create a more positive learning environment for him. That, in turn, will also be easier on you!
**CHEWING AND NIPPING**
Nipping at fingers and toes is normal puppy behavior. Chewing is also the way that puppies investigate their surroundings. However, you will have to teach your puppy that chewing anything other than his toys is not acceptable. That won't happen overnight and at times puppy teeth will test your patience. However, if you allow nipping and chewing to continue, just think about the damage that a mature Welsh Terrier can do with a full set of adult terrier teeth.
Whenever your puppy nips your hand or fingers, cry out "Ouch!" in a loud voice, which should startle your puppy and stop him from nipping, even if only for a moment. Immediately distract him by offering a small treat or an appropriate toy for him to chew instead (which means having chew toys and puppy treats handy or in your pockets at all times). Praise him when he takes the toy and tell him what a good fellow he is. Praise is even more important in puppy training than discipline and correction.
Puppies also tend to nip at children more often than adults, since they perceive little ones to be more vulnerable and more similar to their littermates. Teach your children appropriate responses to nipping behavior. If they are unable to handle it themselves, you may have to intervene. Puppy nips can be quite painful and a child's frightened reaction will only encourage a puppy to nip harder, which is a natural canine response. As with all other puppy situations, interaction between your Welsh Terrier puppy and children should be supervised.
**CONFINEMENT**
It is wise to keep your puppy confined to a small "puppy-proofed" area of the house for his first few weeks at home. Gate or block off a space near the door he will use for outdoor potty trips. Expandable baby gates are useful to create puppy's designated area. If he is allowed to roam through the entire house or even only several rooms, it will be more difficult to house-train him.
**FIRST CAR RIDE**
The ride to your home from the breeder will no doubt be your puppy's first automobile experience, and you should make every effort to keep him comfortable and secure. Bring a large towel or small blanket for the puppy to lie on during the trip and an extra towel in case the pup gets carsick or has a potty accident. It's best to put the puppy on the towel in a crate. Most puppies will fall fast asleep from the rolling motion of the car.
If the ride is lengthy, you may have to stop so that the puppy can relieve himself, so be sure to bring a leash and collar for those stops. Avoid rest areas for potty trips, since those are frequented by many dogs, who may carry parasites or disease. It's better to stop at grassy areas near gas stations or shopping centers to prevent unhealthy exposure for your pup.
Chewing on objects, not just family members' fingers and ankles, is also normal canine behavior that can be especially tedious (for the owner, not the pup) during the teething period when the puppy's adult teeth are coming in. At this stage, chewing just plain feels good. Furniture legs and cabinet corners are common puppy favorites. Shoes and other personal items also taste pretty good to a pup.
The best solution is, once again, prevention. If you value something, keep it tucked away and out of reach. You can't hide your dining-room table in a closet, but you can try to deflect the chewing by applying a bitter product made just to deter dogs from chewing. Available in a spray or cream, this substance is vile-tasting, although safe for dogs, and most puppies will avoid the forbidden object after one tiny taste. You also can apply the product to your leather leash if the puppy tries to chew on his lead during leash-training sessions.
Keep a ready supply of safe chews handy to offer your Welsh Terrier as a distraction when he starts to chew on something that's a "no-no." Remember, at this tender age he does not yet know what is permitted or forbidden, so you have to be "on call" every minute he's awake and on the prowl.
You may lose a treasure or two during puppy's growing-up period, and the furniture could sustain a nasty nick or two. These can be trying times, so be prepared for those inevitable accidents and comfort yourself in knowing that this too shall pass.
Adding a Welsh Terrier to your household means adding a new family member who will need your care each and every day. When your Welsh Terrier pup first comes home, you will start a routine with him so that, as he grows up, your dog will have a daily schedule just as you do. The aspects of your dog's daily care will likewise become regular parts of your day, so you'll both have a new schedule. Dogs learn by consistency and thrive on routine: regular times for meals, exercise, grooming and potty trips are just as important for your dog as they are to you! Your dog's schedule will depend much on your family's daily routine, but remember that you now have a new member of the family who is part of your day every day.
**NOT HUNGRY?**
No dog in his right mind would turn down his dinner, would he? If you notice that your dog has lost interest in his food, there could be any number of causes. Dental problems are a common cause of appetite loss, one that is often overlooked. If your dog has a toothache, a loose tooth or sore gums from infection, chances are it doesn't feel so good to chew. Think about when you've had a toothache! If your dog does not approach the food bowl with his usual enthusiasm, look inside his mouth for signs of a problem. Whatever the cause, you'll want to consult your vet so that your chow hound can get back to his happy, hungry self as soon as possible.
**FEEDING**
Feeding your dog the best diet is based on various factors, including age, activity level, overall condition and size of breed. When you visit the breeder, he will share with you his advice about the proper diet for your dog based on his experience with the breed and the foods with which he has had success. Likewise, your vet will be a helpful source of advice throughout the dog's life and will aid you in planning a diet for optimal health.
**F EEDING THE PUPPY**
Of course, your pup's very first food will be his dam's milk. There may be special situations in which pups fail to nurse, necessitating that the breeder hand-feed them with a formula, but for the most part pups spend the first weeks of life nursing from their dam. The breeder weans the pups by gradually introducing solid foods and decreasing the milk meals. Pups may even start themselves off on the weaning process, albeit inadvertently, if they snatch bites from their mom's food bowl.
By the time the pups are ready for new homes, they are fully weaned and eating a good puppy food. As a new owner, you may be thinking, "Great! The breeder has taken care of the hard part." Not so fast.
A puppy's first year of life is the time when all or most of his growth and development takes place. This is a delicate time, and diet plays a huge role in proper skeletal and muscular formation. Improper diet and exercise habits can lead to damaging problems that will compromise the dog's health and movement for his entire life. That being said, new owners should not worry needlessly. With the myriad types of food formulated specifically for growing pups of different-sized breeds, dog-food manufacturers have taken much of the guesswork out of feeding your puppy well. Since growth-food formulas are designed to provide the nutrition that a growing puppy needs, it is unnecessary and, in fact, can prove harmful to add supplements to the diet. Research has shown that too much of certain vitamin supplements and minerals predispose a dog to skeletal problems. It's by no means a case of "if a little is good, a lot is better." At every stage of your dog's life, too much or too little in the way of nutrients can be harmful, which is why a manufactured complete food is the easiest way to know that your dog is getting what he needs.
**Ch. Vicway Modesty Blaise is a perfect example of a well-maintained Welsh Terrier in top condition. Diet, exercise and grooming are important parts of the dog's care and overall well-being.**
Because of a young pup's small body and accordingly small digestive system, his daily portion will be divided up into small meals throughout the day. This can mean starting off with three or more meals a day and decreasing the number of meals as the pup matures. Eventually you can feed only one meal a day, although it is generally thought that dividing the day's food into two meals on a morning/evening schedule is healthier for the dog's digestion.
Regarding the feeding schedule, feeding the pup at the same times and in the same place each day is important for both house-breaking purposes and establishing the dog's everyday routine. As for the amount to feed, growing puppies generally need proportionately more food per body weight than their adult counterparts, but a pup should never be allowed to gain excess weight. Dogs of all ages should be kept in proper body condition, but extra weight can strain a pup's developing frame, causing skeletal problems.
**SWITCHING FOODS**
There are certain times in a dog's life when it becomes necessary to switch his food; for example, from puppy to adult food and then from adult to senior-dog food. Additionally, you may decide to feed your pup a different type of food from what he received from the breeder, and there may be "emergency" situations in which you can't find your dog's normal brand and have to offer something else temporarily. Anytime a change is made, for whatever reason, the switch must be done gradually. You don't want to upset the dog's stomach or end up with a picky eater who refuses to eat something new. A tried-and-true approach is, over the course of about a week, to mix a little of the new food in with the old, increasing the proportion of new to old as the days progress. At the end of the week, you'll be feeding his regular portions of the new food, and he will barely notice the change.
Watch your pup's weight as he grows and, if the recommended amounts seem to be too much or too little for your pup, consult the vet about appropriate dietary changes. Keep in mind that treats, although small, can quickly add up throughout the day, contributing unnecessary calories. Treats are fine when used prudently; opt for dog treats specially formulated to be healthy or for nutritious snacks like small pieces of cheese or cooked chicken.
**F EEDING THE ADULT DOG**
For the adult (meaning physically mature) dog, feeding properly is about maintenance, not growth. Again, correct weight is a concern. Your dog should appear fit and should have an evident "waist." His ribs should not be protruding (a sign of being underweight), but they should be covered by only a slight layer of fat. Under normal circumstances, an adult dog can be maintained fairly easily with a high-quality nutritionally complete adult-formula food.
Factor treats into your dog's overall daily caloric intake, and avoid offering table scraps. Not only are certain "people foods," like chocolate, onions, grapes, raisins and nuts, toxic to dogs, but feeding from the table encourages begging and overeating. Over-weight dogs are more prone to health problems. Research has even shown that obesity takes years off a dog's life. With that in mind, resist the urge to overfeed and over-treat. Don't make unnecessary additions to your dog's diet, whether with tidbits or with extra vitamins and minerals.
The amount of food needed for proper maintenance will vary depending on the individual dog's activity level, but you will be able to tell whether the daily portions are keeping him in good shape. With the wide variety of good complete foods available, choosing what to feed is largely a matter of personal preference. Just as with the puppy, the adult dog should have consistency in his mealtimes and feeding place. In addition to a consistent routine, regular mealtimes also allow the owner to see how much his dog is eating. If the dog seems never to be satisfied or, likewise, becomes uninterested in his food, the owner will know right away that something is wrong and can consult the vet.
**DIET DON'TS**
• Got milk? Don't give it to your dog! Dogs cannot tolerate large quantities of cows' milk, as they do not have the enzymes to digest lactose.
• You may have heard of dog owners' who add raw eggs to their dogs' food for a shiny coat or to make the food more palatable, but consumption of raw eggs too often can cause a deficiency of the vitamin biotin.
• Avoid feeding table scraps, as they will upset the balance of the dog's complete food. Additionally, fatty or highly seasoned foods can cause upset canine stomachs.
• Do not offer raw meat to your dog. Raw meat can contain parasites; it also is high in fat.
• Vitamin A toxicity in dogs can be caused by too much raw liver, especially if the dog already gets enough vitamin A in his balanced diet, which should be the case.
• Bones like chicken, pork chop and other soft bones are not suitable, as they easily splinter.
**Welsh Terriers are enthusiastic "chow hounds" who approach mealtimes with vigor!**
**D IETS FOR THE AGING DOG**
A good rule of thumb is that once a dog has reached 75% of his expected lifespan, he has reached "senior citizen" or geriatric status. Your Welsh Terrier will be considered a senior at about 8 or 9 years of age; he has a projected lifespan of at least 12 years. Terriers in general are relatively long-lived dogs.
What does aging have to do with your dog's diet? No, he won't get a discount at the local diner's early-bird special. Yes, he will require some dietary changes to accommodate the changes that come along with increased age. One change is that the older dog's dietary needs become more similar to that of a puppy. Specifically, dogs can metabolize more protein as youngsters and seniors than in the adult-maintenance stage. Discuss with your vet whether you need to switch to a higher-protein or senior-formulated food or whether your current adult-dog food contains sufficient nutrition for the senior.
Watching the dog's weight remains essential, even more so in the senior stage. Older dogs are already more vulnerable to illness, and obesity only contributes to their susceptibility to problems. As the older dog becomes less active and thus exercises less, his regular portions may cause him to gain weight. At this point, you may consider decreasing his daily food intake or switching to a reduced-calorie food. As with other changes, you should consult your vet for advice.
**D ON'T FORGET THE WATER!**
For a dog, it's always time for a drink! Regardless of what type of food he eats, there's no doubt that he needs plenty of water. Fresh cold water, in a clean bowl, should be freely available to your dog at all times. There are special circumstances, such as during puppy housebreaking, when you will want to monitor your pup's water intake so that you will be able to predict when he will need to relieve himself, but water must be available to him nonetheless. Water is essential for hydration and proper body function just as it is in humans.
You will get to know how much your dog typically drinks in a day. Of course, in the heat or if exercising vigorously, he will be more thirsty and will drink more. However, if he begins to drink noticeably more water for no apparent reason, this could signal any of various problems, and you are advised to consult your vet.
Water is the best drink for dogs. Some owners are tempted to give milk from time to time or to moisten dry food with milk, but dogs do not have the enzymes necessary to digest the lactose in milk, which is much different from the milk that nursing puppies receive. Therefore, stick with clean fresh water to quench your dog's thirst, and always have it readily available to him.
**PUPPY STEPS**
Puppies are brimming with activity and enthusiasm. It seems that they can play all day and night without tiring, but don't overdo your puppy's exercise regimen. Easy does it for the puppy's first six to nine months. Keep walks brief and don't let the puppy engage in stressful jumping games. The puppy frame is delicate, and too much exercise during those critical growing months can cause injury to his bone structure, ligaments and musculature. Save his first jog for his first birthday!
**A water bowl should be kept outside so that your Welsh can quench his thirst during outdoor play and exercise.**
**EXERCISE**
The Welsh Terrier, like all other terrier breeds, is an active dog that welcomes the chance to exercise. Two vigorous walks daily are ideal for the adult Welsh. Do not begin brisk walks with your Welsh until he is at least four months of age. As the dog reaches adulthood, the speed and distance of the walks can be increased as long as they are both kept reasonable and comfortable for both of you. A good walk will stimulate the youngster's heart rate as well as promote development of musculature.
Most importantly, your Welsh Terrier looks for structured time to spend with his owner in an active pursuit of fun. Play sessions and letting the dog run free in the fenced yard also are great exercise for the Welsh Terrier. Keep an eye on your Welsh's yard time to make sure he's staying safe and out of mischief. Fetching games can be played indoors or out; these are excellent for giving your dog active play that he will enjoy. Chasing things that move comes naturally to dogs of all breeds, and the Welsh has strong instincts for catching things on the run. If you choose to play games outdoors, you must have a securely fenced-in yard and/or have the dog attached to at least a 25-foot light line for security. You want your Welsh Terrier to run, but not run away!
**A Welsh's body should appear fit and athletic.**
Bear in mind that an over-weight dog should never be suddenly over-exercised; instead, he should be encouraged to increase exercise slowly. Also remember that not only is exercise essential to keep the dog's body fit but it also is essential to his mental well-being. A bored dog will find something to do, which often manifests itself in some type of destructive behavior. In this sense, exercise is essential for the owner's mental well-being as well!
**GROOMING**
**C OAT CARE**
The Welsh Terrier has a wiry outer coat and a soft, somewhat woolly, undercoat. Neither one actually casts out or sheds; that is, the hair does not reach a certain stage of growth and fall out in profusion all over the furniture. Dead hairs are adequately removed with a good weekly brushing and combing. If the coat is not properly trimmed, the adorable puppy will become a shaggy, unattractive woolly-bully within a year, with matted lumps harboring unwanted parasites.
Don't expect a Welsh Terrier puppy to cooperate with being groomed for more than five or ten minutes. Begin with short daily sessions and lengthen them as the puppy learns to tolerate the brushing and handling. A worthwhile investment is a small fold-up grooming table with an adjustable noose at one end to keep the dog's head up and facing the right direction. You'll get good use out of it for 12 to 14 years. It's important that the dog feel perfectly safe, so if you use something else, such as an ordinary table, for grooming, be sure it is steady and has a non-slip surface. Never leave any dog—puppy or adult—unsupervised on the table. A fall could cause serious injury.
**Ask your breeder for advice about what type of equipment you'll need to care for your terrier's wiry coat.**
The only grooming required for the first few weeks is gentle brushing and combing, because the primary purpose is to accustom your puppy to being handled for the real grooming to come. A thorough brushing and combing will precede every trimming session.
There are two ways to keep the Welsh looking as neat and handsome as he should. The preferred method is called plucking, or stripping, and for anyone not familiar with the process, it is best undertaken with the instruction of someone who knows exactly how to do it and can show you. It is not difficult, but it is time-consuming. Done correctly, stripping will not hurt the dog. A few hairs at a time are methodically lifted and pulled (in the direction the hair grows) using the fingers or a stripping knife.
**WATER SHORTAGE**
No matter how well behaved your dog is, bathing is always a project! Nothing can substitute for a good warm bath, but owners do have the option of giving their dogs "dry" baths. Pet shops sell excellent products, in both powder and spray forms, designed for spot-cleaning your dog. These dry shampoos are convenient for touch-up jobs when you don't have the time to bathe your dog in the traditional way.
Muddy feet, messy behinds and smelly coats can be spot-cleaned and deodorized with a "wet-nap"-style cleaner. On those days when your dog insists on rolling in fresh goose droppings and there's no time for a bath, a spot bath can save the day. These pre-moistened wipes are also handy for other grooming needs like wiping faces, ears and eyes and freshening tails and behinds.
All of the trimming is done to follow the lines of the dog. The Welsh Terrier has no frills or skirts or other enhancements to his outline. Short eyebrows are left to protect his small, deep-set eyes from nettles and twigs. All hair on the inside of the ears is removed so they fold properly for protection and are more easily kept clean. The only parts left somewhat long are the furnishings, or whiskers, on the muzzle and on the legs, and that's in part because they can take months to grow back in. The front legs are trimmed as columns, the face is styled as a rectangle and the hindquarters furnishings follow the angulation. Use photographs of show dogs for guidance.
**Billy poses on a grooming table. The lead is attached overhead to the table's metal arm.**
The alternative method is to use an electric clipper, following the same pattern. A few lessons in "clipper control" would be a good idea, since it is only easy when you've got the hang of it. Holding a noisy machine in one hand and getting the Welsh to stand still when you put this object on his head is not as easy at it looks when watching a long-time terrierman or other professional! Clipping is quicker, but it has a downside. When the hairs of both coats (wire and soft) are cut, rather than just dead hair removed, the coat often loses its deep color. Plucking or stripping allows the strong-colored tips of new hairs to be seen.
Of course, there is a third method—that is to pack your Welsh into the car, head to a professional groomer and pay to have someone else do it, either by stripping or clipping. No matter which method you choose, this coat work only needs to be undertaken about every three months.
For weekly grooming, you'll need two brushes. One is a terrier palm pad (also called a dollingup pad) and the other a slicker (made with bent wires) or a stiff bristle brush. The pad is used on the furnishings, brushing against and then with the way the hair grows. The slicker or bristle brush is used on the rest of the dog. Use both gently, getting down to the skin, but not digging into it. You'll need a metal comb and a pair of scissors for trimming between the pads of the feet and around the edges of the feet, ears and so on.
**The palm pad is used on the furnishings, both against and with the lie of the hair.**
All of this brushing promotes good healthy skin and removes dead hair as well as dirt and debris caught in the coat. It will also leave you with a Welsh Terrier that is handsome to look at and nice to have around the house.
**The slicker brush is used to thoroughly brush the rest of the body.**
**A metal comb is helpful for detangling and removing debris from the coat.**
**The excess hair growing on the bottom of the feet, between the pads, should be carefully scissored.**
Many Welsh Terriers are bathed only two or three times in their lives. Their coats shed dirt and with it any doggy odor. Most Welsh Terriers love the rain, which is a good thing, considering their country of origin, but after coming in from the rain, a good toweling and a brushing are all that's needed. The wire Welsh coat is akin to a duck's back! However, if your dog has rolled in muck or mud, a bath will be in order. Use a dog shampoo (people shampoos contain ingredients harmful to the dog's skin and coat) and rinse thoroughly several times. Muddy paws, and face furnishings caught up in the dinner dish, only need rinsing off as needed. Towel-dry, brush and comb all hair into place.
**N AIL CLIPPING**
Having his nails trimmed is not on many dogs' lists of favorite things to do. With this in mind, you will need to accustom your puppy to the procedure at a young age so that he will sit still (well, as still as he can) for his pedicures. Long nails can cause the dog's feet to spread, which is not good for him; likewise, long nails can hurt if they unintentionally scratch, not good for you!
Some dogs' nails are worn down naturally by regular walking on hard surfaces, so the frequency with which you clip depends on your individual dog. Look at his nails from time to time and clip as needed; a good way to know when it's time for a trim is if you hear your dog clicking as he walks across the floor.
There are several types of nail clippers and even electric nail-grinding tools made for dogs; first we'll discuss using the clipper. To start, have your clipper ready and some doggie treats on hand. You want your pup to view his nail-clipping sessions in a positive light, and what better way to convince him than with food? You may want to enlist the help of an assistant to comfort the pup and offer treats as you concentrate on the clipping itself. The guillotine-type clipper is thought of by many as the easiest type to use; the nail tip is inserted into the opening, and blades on the top and bottom snip it off in one clip.
**THE EARS KNOW**
Examining and cleaning your puppy's ears helps ensure good internal health. The ears are the eyes to the dog's innards! Begin handling your puppy's ears when he's still young so that he doesn't protest every time you lift a flap or touch his ears. Yeast and bacteria are two of the culprits that you can detect by examining the ear. You will notice a strong, often foul, odor, debris, redness or some kind of discharge. All of these point to health problems that can worsen over time. Additionally, you are on the lookout for wax accumulation, ear mites and other tiny bothersome parasites and their even tinier droppings. You may have to pluck hair with tweezers in order to have a better view into the dog's ears, but this is painless if done carefully. Healthy ears should be gently cleaned once a week, using an ear-cleaning formula and a soft wipe or pad.
**Never probe into the ear with a cotton swab; only clean that which is visible. It is safer to use a soft cotton wipe. Ear cleanser for dogs is available at pet shops or through your vet.**
**Check your dog's teeth regularly to ensure that plaque is not accumulating on the teeth and gums.**
**Initiate a home dental-care regimen. Use toothbrushes and toothpaste made especially for dogs.**
**You will appreciate the time you spent acclimating the pup to his pedicures when you have a politely behaved adult who stands still while you clip.**
Start by grasping the pup's paw; a little pressure on the foot pad causes the nail to extend, making it easier to clip. Clip off a little at a time. If you can see the "quick," which is a blood vessel that runs through each nail, you will know how much to trim, as you do not want to cut into the quick. On that note, if you do cut the quick, which will cause bleeding, you can stem the flow of blood with a styptic pencil or other clotting agent. If you mistakenly nip the quick, do not panic or fuss, as this will cause the pup to be afraid. Simply reassure the pup, stop the bleeding and move on to the next nail. Don't be discouraged; you will become a professional canine pedicurist with practice.
**SCOOTING HIS BOTTOM**
Here's a doggy problem that many owners tend to neglect. If your dog is scooting his rear end around the carpet, he probably is experiencing anal-sac impaction or blockage. The anal sacs are the two grape-sized glands on either side of the dog's vent. The dog cannot empty these glands, which become filled with a foul-smelling material. The dog may attempt to lick the area to relieve the pressure. He may also rub his anus on your walls, furniture or floors.
Don't neglect your dog's rear end during grooming sessions. By squeezing both sides of the anus with a soft cloth, you can express some of the material in the sacs. If the material is pasty and thick, you likely will need the assistance of a veterinarian. Vets know how to express the glands and can show you how to do it correctly without hurting the dog or spraying yourself with the contents.
You may or may not be able to see the quick, so it's best to just clip off a small bit at a time. If you see a dark dot in the center of the nail, this is the quick and your cue to stop clipping. Tell the puppy he's a "good boy" and offer a piece of treat with each nail. You can also use nail-clipping time to examine the foot-pads, making sure that they are not dry and cracked and that nothing has become embedded in them.
The nail grinder, the other choice, is many owners' first choice. Accustoming the puppy to the sound of the grinder and sensation of the buzz presents fewer challenges than the clipper, and there's no chance of cutting through the quick. Use the grinder on a low setting and always talk soothingly to your dog. He won't mind his salon visit, and he'll have nicely polished nails as well.
**E YE CARE**
During grooming sessions, pay extra attention to the condition of your dog's eyes. If the area around the eyes is soiled or if tear staining has occurred, there are various cleaning agents made especially for this purpose. Look at the dog's eyes to make sure no debris has entered; dogs with large eyes and those who spend time outdoors are especially prone to this.
The signs of an eye infection are obvious: mucus, redness, puffiness, scabs or other signs of irritation. If your dog's eyes become infected, the vet will likely prescribe an antibiotic ointment for treatment. If you notice signs of more serious problems, such as opacities in the eye, which usually indicate cataracts, consult the vet at once. Taking time to pay attention to your dog's eyes will alert you in the early stages of any problem so that you can get your dog treatment as soon as possible. You could save your dog's sight!
**ID FOR YOUR DOG**
You love your Welsh Terrier and want to keep him safe. Of course, you take every precaution to prevent his escaping from the yard or becoming lost or stolen. You have a sturdy high fence and you always keep your dog on-lead when out and about in public places. If your dog is not properly identified, however, you are overlooking a major aspect of his safety. We hope to never be in a situation where our dog is missing, but we should practice prevention in the unfortunate case that this happens; identification greatly increases the chances of your dog's being returned to you
**PET OR STRAY?**
Besides the obvious benefit of providing your contact information to whoever finds your lost dog, an ID tag makes your dog more approachable and more likely to be recovered. A strange dog wandering the neighborhood without a collar and tags will look like a stray, while the collar and tags indicate that the dog is someone's pet. Even if the ID tags become detached from the collar, the collar alone will make a person more likely to pick up the dog.
There are several ways to identify your dog. First, the traditional dog tag should be a staple in your dog's wardrobe, attached to his everyday collar. Tags can be made of sturdy plastic and various metals and should include your contact information so that a person who finds the dog can get in touch with you right away to arrange his return. Many people today enjoy the wide range of decorative tags available, so have fun and create a tag to match your dog's personality. Of course, it is important that the tag stays on the collar, so have a secure "O" ring attachment; you also can explore the type of tag that slides right onto the collar.
**In a crate is the safest way for your Welsh to travel.**
**CAR CAUTION**
You may like to bring your canine companion along on the daily errands, but if you will be running in and out from place to place and can't bring him indoors with you, leave him at home. Your dog should never be left alone in the car, not even for a minute—never! A car heats up very quickly, and even a cracked-open window will not help. In fact, leaving the window cracked will be dangerous if the dog becomes uncomfortable and tries to escape. When in doubt, leave your dog home, where you know he will be safe.
In addition to the ID tag, which every dog should wear even if identified by another method, two other forms of identification have become popular: microchipping and tattooing. In microchipping, a tiny scannable chip is painlessly inserted under the dog's skin. The number is registered to you so that, if your lost dog turns up at a clinic or shelter, the chip can be scanned to retrieve your contact information.
The advantage of the microchip is that it is a permanent form of ID, but there are some factors to consider. Several different companies make microchips, and not all are compatible with the others' scanning devices. It's best to find a company with a universal microchip that can be read by scanners made by other companies as well. It won't do any good to have the dog chipped if the information cannot be retrieved. Also, not every humane society, shelter and clinic is equipped with a scanner, although more and more facilities are equipping themselves. In fact, many shelters microchip dogs that they adopt out to new homes.
Because the microchip is not visible to the eye, the dog must wear a tag that states that he is microchipped so that whoever picks him up will know to have him scanned. He of course also should have a tag with contact information in case his chip cannot be read. Humane societies and veterinary clinics offer micro-chipping service, which is usually very affordable.
Though less popular than microchipping, tattooing is another permanent method of ID for dogs. Most vets perform this service, and there are also clinics that perform dog tattooing. This is also an affordable procedure and one that will not cause much discomfort for the dog. It is best to put the tattoo in a visible area, such as the ear, to deter theft. It is sad to say that there are cases of dogs' being stolen and sold to research laboratories, but such laboratories will not accept tattooed dogs.
**Your Welsh Terrier's ID tag must be securely fastened to his everyday collar.**
To ensure that the tattoo is effective in aiding your dog's return to you, the tattoo number must be registered with a national organization. That way, when someone finds a tattooed dog, a phone call to the registry will quickly match the dog with his owner.
**BASIC TRAINING PRINCIPLES: PUPPY VS. ADULT**
There's a big difference between training an adult dog and training a young puppy. With a young puppy, everything is new. When your pup comes home with you, he will be experiencing many things, and he has nothing with which to compare these experiences. Up to this point, he has been with his dam and littermates, not one-on-one with people except in his interactions with his breeder and visitors to the litter.
**LEADER OF THE PACK**
Canines are pack animals. They live according to pack rules, and every pack has only one leader. Guess what? That's you! To establish your position of authority, lay down the rules and be fair and good-natured in all your dealings with your dog. He will consider young children as his littermates, but the one who trains him, who feeds him, who grooms him, who expects him to come into line, that's his leader. And he who leads must be obeyed.
When you first bring the puppy home, he is eager to please you. This means that he accepts doing things your way. During the next couple of months, he will absorb the basis of everything he needs to know for the rest of his life. This early age is even referred to as the "sponge" stage. After that, for the next 18 months, it's up to you to reinforce good manners by building on the foundation that you've established. Once your puppy is reliable in basic commands and behavior and has reached the appropriate age, you may gradually introduce him to some of the interesting sports, games and activities available to pet owners and their dogs.
Raising your puppy is a family affair. Each member of the family must know what rules to set forth for the puppy and how to use the same one-word commands to mean exactly the same thing every time. Even if yours is a large family, one person will soon be considered by the pup to be the leader, the Alpha person in his pack, the "boss" who must be obeyed. Often that highly regarded person turns out to be the one who feeds the puppy. Food ranks very high on the puppy's list of important things! That's why your puppy is rewarded with small treats along with verbal praise when he responds to you correctly. As the puppy learns to do what you want him to do, the food rewards are gradually eliminated and only the praise remains. If you were to keep up with the food treats, you could have two problems on your hands—an obese dog and a beggar.
**OUR CANINE KIDS**
"Everything I learned about parenting, I learned from my dog." How often adults recognize that their parenting skills are mere extensions of the education they acquired while caring for their dogs. Many owners refer to their dogs as their "kids" and treat their canine companions like real members of the family. Surveys indicate that a majority of dog owners talk to their dogs regularly, celebrate their dogs' birthdays and purchase Christmas gifts for their dogs. Another survey shows that dog owners take their dogs to the veterinarian more frequently than they visit their own physicians.
**Everyone in the family should take part in the Welsh Terrier's training so that the dog will respect and obey all members of his "pack."**
Training begins the minute your Welsh Terrier puppy steps through the doorway of your home, so don't make the mistake of putting the puppy on the floor and telling him by your actions to "Go for it! Run wild!" Even if this is your first puppy, you must act as if you know what you're doing: be the boss. An uncertain pup may be terrified to move, while a bold one will be ready to take you at your word and start plotting to destroy the house! Before you collected your puppy, you decided where his own special place would be, and that's where to put him when you first arrive home. Give him a house tour after he has investigated his area and had a nap and a bathroom "pit stop."
**Welsh Terriers on a winter walk in Finland with Jaana Matto.**
It's worth mentioning here that if you've adopted an adult dog that is completely trained to your liking, lucky you! You're off the hook! However, if that dog spent his life up to this point in a kennel, or even in a good home but without any real training, be prepared to tackle the job ahead. A dog three years of age or older with no previous training cannot be blamed for not knowing what he was never taught. While the dog is trying to understand and learn your rules, at the same time he has to unlearn many of his previously self-taught habits and general view of the world.
Working with a professional trainer will speed up your progress with an adopted adult dog. You'll need patience, too. Some new rules may be close to impossible for the dog to accept. After all, he's been successful so far by doing everything his way! (Patience again.) He may agree with your instruction for a few days and then slip back into his old ways, so you must be just as consistent and understanding in your teaching as you would be with a puppy. (More patience needed yet again!) Your dog has to learn to pay attention to your voice, your family, the daily routine, new smells, new sounds and, in some cases, even a new climate.
One of the most important things to find out about a newly adopted adult dog is his reaction to children (yours and others), strangers and your friends, and how he acts upon meeting other dogs. If he was not socialized with dogs as a puppy, this could be a major problem. This does not mean that he's a "bad" dog, a vicious dog or an aggressive dog; rather, it means that he has no idea how to read another dog's body language. There's no way for him to tell whether the other dog is a friend or foe. Survival instinct takes over, telling him to attack first and ask questions later. This definitely calls for professional help and, even then, may not be a behavior that can be corrected 100% reliably (or even at all). If you have a puppy, this is why it is so very important to introduce your young puppy properly to other puppies and "dog-friendly" adult dogs.
**BASIC PRINCIPLES OF DOG TRAINING**
1. Start training early. A young puppy is ready, willing and able.
2. Timing is your all-important tool. Praise at the exact time that the dog responds correctly. Pay close attention.
3. Patience is almost as important as timing!
4. Repeat! The same word has to mean the same thing every time.
5. In the beginning, praise all correct behavior verbally, along with treats and petting.
**HOUSE-TRAINING YOUR WELSH TERRIER**
Dogs are tactility-oriented when it comes to house-training. In other words, they respond to the surface on which they are given approval to eliminate. The choice is yours (the dog's version is in parentheses): The lawn (including the neighbors' lawns)? A bare patch of earth under a tree (where people like to sit and relax in the summertime)? Concrete steps or patio (all sidewalks, garages and basement floors)? The curbside (watch out for cars)? A small area of crushed stone in a corner of the yard (mine!)? The latter is the best choice if you can manage it, because it will remain strictly for the dog's use and is easy to keep clean.
**Your pup's mealtimes have a direct effect on the house-training schedule. What goes in must come out; with a pup, that's usually sooner, not later!**
You can start out with paper-training indoors and switch over to an outdoor surface as the puppy matures and gains control over his need to eliminate. For the nay-sayers, don't worry—this won't mean that the dog will soil on every piece of newspaper lying around the house. You are training him to go outside, remember? Starting out by paper-training often is the only choice for a city dog.
**W HEN YOUR PUPPY'S "GOT TO GO"**
Your puppy's need to relieve himself is seemingly non-stop, but signs of improvement will be seen each week. From 9 to 10 weeks old, the puppy will have to be taken outside every time he wakes up, about 10–15 minutes after every meal and after every period of play—all day long, from first thing in the morning until his bedtime! That's a total of ten or more trips per day to teach the puppy where it's okay to relieve himself. With that schedule in mind, you can see that house-training a young puppy is not a part-time job. It requires someone to be home all day.
**A fenced yard makes the task of house-training much easier, but the pup must learn the difference between potty time and exploring time.**
If that seems overwhelming or impossible, do a little planning. For example, plan to pick up your puppy at the start of a vacation period. If you can't get home in the middle of the day, plan to hire a dog-sitter or ask a neighbor to come over to take the pup outside, feed him his lunch and then take him out again about ten or so minutes after he's eaten. Also make arrangements with that or another person to be your "emergency" contact if you have to stay late on the job. Remind yourself—repeatedly—that this hectic schedule improves as the puppy gets older.
**H OME WITHIN A HOME**
Your Welsh Terrier puppy needs to be confined to one secure, puppy-proof area when no one is able to watch his every move. Generally, the kitchen is the place of choice because the floor is washable. Likewise, it's a busy family area that will accustom the pup to a variety of noises, everything from pots and pans to the telephone, blender and dishwasher. He will also be enchanted by the smell of your cooking (and will never be critical when you burn something). An exercise pen (also called an "expen," a puppy version of a playpen) within the room of choice is an excellent means of confinement for a young pup. He can see out and has a certain amount of space in which to run about, but he is safe from dangerous things like electrical cords, heating units, trash baskets or open kitchen-supply cabinets. Place the pen where the puppy will not get a blast of heat or air conditioning.
**DAILY SCHEDULE**
How many relief trips does your puppy need per day? A puppy up to the age of 14 weeks will need to go outside about 8 to 12 times per day! You will have to take the pup out any time he starts sniffing around the floor or turning in small circles, as well as after naps, meals, games and lessons or whenever he's released from his crate. Once the puppy is 14 to 22 weeks of age, he will require only 6 to 8 relief trips. At the ages of 22 to 32 weeks, the puppy will require about 5 to 7 trips. Adult dogs typically require 4 relief trips per day, in the morning, afternoon, evening and late at night.
**LEASH TRAINING**
House-training and leash training go hand in hand, literally. When taking your puppy outside to do his business, lead him there on his leash. Unless an emergency potty run is called for, do not whisk the puppy up into your arms and take him outside. If you have a fenced yard, you have the advantage of letting the puppy loose to go out, but it's better to put the dog on the leash and take him to his designated place in the yard until he is reliably house-trained. Taking the puppy for a walk is the best way to house-train a dog. The dog will associate the walk with his time to relieve himself, and the exercise of walking stimulates the dog's bowels and bladder. Dogs that are not trained to relieve themselves on a walk may hold it until they get back home, which of course defeats half the purpose of the walk.
In the pen, you can put a few toys, his bed (which can be his crate if the dimensions of pen and crate are compatible) and a few layers of newspaper in one small corner, just in case. A water bowl can be hung at a convenient height on the side of the ex-pen so it won't become a splashing pool for an innovative puppy. His food dish can go on the floor, near but not under the water bowl.
Crates are something that pet owners are at last getting used to for their dogs. Wild or domestic canines have always preferred to sleep in den-like safe spots, and that is exactly what the crate provides. How often have you seen adult dogs that choose to sleep under a table or chair even though they have full run of the house? It's the den connection.
**SOMEBODY TO BLAME**
House-training a puppy can be frustrating for the puppy and the owner alike. The puppy does not instinctively understand the difference between defecating on the pavement outside and on the ceramic tile in the kitchen. He is confused and frightened by his human's exuberant reactions to his natural urges. The owner, arguably the more intelligent of the duo, is also frustrated that he cannot convince his puppy to obey his commands and instructions.
In frustration, the owner may struggle with the temptation to discipline the puppy, scold him or even strike him on the rear end. These harsh corrections are unsuitable and unnecessary, and will defeat your purpose in gaining your puppy's trust and respect. Don't blame your nine-week-old puppy. Blame yourself for not being 100% consistent in the puppy's lessons and routine. The lesson here is simple: try harder and your puppy will succeed.
In your "happy" voice, use the word "Crate" every time you put the pup into his den. If he's new to a crate, toss in a small biscuit for him to chase the first few times. At night, after he's gone outside to potty, he should sleep in his crate. The crate may be kept in his designated area at night or, if you want to be sure to hear those wake-up yips in the morning, put the crate in a corner of your bedroom. However, don't make any response whatsoever to whining or crying. If he's completely ignored, he'll settle down and get to sleep.
Good bedding for a young puppy is an old folded bath towel or an old blanket, something that is easily washable and disposable if necessary ("accidents" will happen!). Never put newspaper in the puppy's crate. Also, those old ideas about adding a clock to replace his mother's heartbeat, or a hot-water bottle to replace her warmth, are just that—old ideas. The clock could drive the puppy nuts, and the hot-water bottle could end up as a very soggy waterbed! An extremely good breeder would have introduced your puppy to the crate by letting two pups sleep together for a couple of nights, followed by several nights alone. How thankful you will be if you found that breeder!
**If a fenced area is not available, you will have to be diligent in taking your dog out on his lead, at the same times each day, for him to relieve himself.**
Safe toys in the pup's crate or area will keep him occupied, but monitor their condition closely. Discard any toys that show signs of being chewed to bits. Squeaky parts, bits of stuffing or plastic or any other small pieces can cause intestinal blockage or possibly choking if swallowed.
**P ROGRESSING WITH POTTY-TRAINING**
After you've taken your puppy out and he has relieved himself in the area you've selected, he can have some free time with the family as long as there is someone responsible for watching him. That doesn't mean just someone in the same room who is watching TV or busy on the computer, but one person who is doing nothing other than keeping an eye on the pup, playing with him on the floor and helping him understand his position in the pack.
**A small crate like this is fine for a pup, but he will outgrow it quickly. It is wise to buy a crate from the outset that is large enough for a fully grown Welsh.**
**EXTRA! EXTRA!**
The headlines read: "Puppy Piddles Here!" Breeders commonly use newspapers to line their whelping pens, so puppies learn to associate newspapers with relieving themselves. Do not use newspapers to line your pup's crate, as this will signal to your puppy that it is OK to urinate in his crate. If you choose to paper-train your puppy, you will layer newspapers on a section of the floor near the door he uses to go outside. You should encourage the puppy to use the papers to relieve himself, and bring him there whenever you see him getting ready to go. Little by little, you will reduce the size of the newspaper-covered area so that the puppy will learn to relieve himself "on the other side of the door."
This first taste of freedom will let you begin to set the house rules. If you don't want the dog on the furniture, now is the time to prevent his first attempts to jump up onto the couch. The word to use in this case is "Off," not "Down." "Down" is the word you will use to teach the down position, which is something entirely different.
Most corrections at this stage come in the form of simply distracting the puppy. Instead of telling him "No" for "Don't chew the carpet," distract the chomping puppy with a toy and he'll forget about the carpet.
As you are playing with the pup, do not forget to watch him closely and pay attention to his body language. Whenever you see him begin to circle or sniff, take the puppy outside to relieve himself. If you are paper-training, put him back into his confined area on the newspapers. In either case, praise him as he eliminates while he actually is _in the act_ of relieving himself. Three seconds after he has finished is too late! You'll be praising him for running toward you, picking up a toy or whatever he may be doing at that moment, and that's not what you want to be praising him for. Timing is a vital tool in all dog training. Use it.
Remove soiled newspapers immediately and replace them with clean ones. You may want to take a small piece of soiled paper and place it in the middle of the new clean papers, as the scent will attract him to that spot when it's time to go again. That scent attraction is why it's so important to clean up any messes made in the house by using a product specially made to eliminate the odor of dog urine and droppings. Regular household cleansers won't do the trick. Pet shops sell the best pet deodorizers. Invest in the largest container you can find.
Scent attraction eventually will lead your pup to his chosen spot outdoors; this is the basis of outdoor training. When you take your puppy outside to relieve himself, use a one-word command such as "Outside" or "Go-potty" (that's one word to the puppy!) as you pick him up and attach his leash. Then put him down in his area. If for any reason you can't carry him, snap the leash on quickly and lead him to his spot. Now comes the hard part—hard for you, that is. Just stand there until he urinates and defecates. Move him a few feet in one direction or another if he's just sitting there looking at you, but remember that this is neither playtime nor time for a walk. This is strictly a business trip! Then, as he circles and squats (remember your timing!), give him a quiet "Good dog" as praise. If you start to jump for joy, ecstatic over his performance, he'll do one of two things: either he will stop midstream, as it were, or he'll do it again for you—in the house—and expect you to be just as delighted!
**POTTY COMMAND**
Most dogs love to please their masters; there are no bounds to what dogs will do to make their owners happy. The potty command is a good example of this theory. If toileting on command makes the master happy, then more power to him. Puppies will obligingly piddle if it really makes their keepers smile. Some owners can be creative about which word they will use to command their dogs to relieve themselves. Some popular choices are "Potty," "Tinkle," "Piddle," "Let's go," "Hurry up" and "Toilet." Give the command every time your puppy goes into position and the puppy will begin to associate his business with the command.
**BE UPSTANDING!**
You are the dog's leader. During training, stand up straight so your dog looks up at you, and therefore up to you. Say the command words distinctly, in a clear, declarative tone of voice. (No barking!) Give rewards only as the correct response takes place (remember your timing!). Praise, smiles and treats are "rewards" used to positively reinforce correct responses. Don't repeat a mistake. Just change to another exercise—you will soon find success!
Give him five minutes or so and, if he doesn't go in that time, take him back indoors to his confined area and try again in another ten minutes, or immediately if you see him sniffing and circling. By careful observation, you'll soon work out a successful schedule.
Accidents, by the way, are just that—accidents. Clean them up quickly and thoroughly, without comment, after the puppy has been taken outside to finish his business and then put back into his area or crate. If you witness an accident in progress, say "No!" in a stern voice and get the pup outdoors immediately. No punishment is needed. You and your puppy are just learning each other's language, and sometimes it's easy to miss a puppy's message. Chalk it up to experience and watch more closely from now on.
**KEEPING THE PACK ORDERLY**
Discipline is a form of training that brings order to life. For example, military discipline is what allows the soldiers in an army to work as one. Discipline is a form of teaching and, in dogs, is the basis of how the successful pack operates. Each member knows his place in the pack and all respect the leader, or Alpha dog. It is essential for your puppy that you establish this type of relationship, with you as the Alpha, or leader. It is a form of social coexistence that all canines recognize and accept. Discipline, therefore, is never to be confused with punishment. When you teach your puppy how you want him to behave, and he behaves properly and you praise him for it, you are disciplining him with a form of positive reinforcement.
For a dog, rewards come in the form of praise, a smile, a cheerful tone of voice, a few friendly pats or a rub of the ears. Rewards are also small food treats. Obviously, that does not mean bits of regular dog food. Instead, treats are very small bits of special things like cheese or pieces of soft dog treats. The idea is to reward the dog with something very small that he can taste and swallow, providing instant positive reinforcement. If he has to take time to chew the treat, he will have forgotten what he did to earn it by the time he is finished.
Your puppy should never be physically punished. The displeasure shown on your face and in your voice is sufficient to signal to the pup that he has done something wrong. He wants to please everyone higher up on the social ladder, especially his leader, so a scowl and harsh voice will take care of the error. Growling out the word "Shame!" when the pup is caught in the act of doing something wrong is better than the repetitive "No." Some dogs hear "No" so often that they begin to think it's their name! By the way, do not use the dog's name when you're correcting him. His name is reserved to get his attention for something pleasant about to take place.
**Welsh Terriers are a food-motivated bunch, something that owners can use to their advantage in training.**
There are punishments that have nothing to do with you. For example, your dog may think that chasing cats is one reason for his existence. You can try to stop it as much as you like but without success, because it's such fun for the dog. But one good hissing, spitting, swipe of a cat's claws across the dog's nose will put an end to the game forever. Intervene only when your dog's eyeball is seriously at risk. Cat scratches can cause permanent damage to an innocent but annoying puppy.
**A sturdy leash and collar, a good attitude and a willing Welsh are key ingredients in training success.**
**PUPPY KINDERGARTEN**
**C OLLAR AND LEASH**
Before you begin your Welsh Terrier puppy's education, he must be used to his collar and leash. Choose a collar for your puppy that is secure, but not heavy or bulky. He won't enjoy training if he's uncomfortable. A flat buckle collar is fine for everyday wear and for initial puppy training. For older dogs, there are several types of training collars such as the martingale, which is a double loop that tightens slightly around the neck, or the head collar, which is similar to a horse's halter. Do not use a chain choke collar with your Welsh Terrier. It is neither necessary nor effective.
A lightweight 6-foot woven cotton training leash is preferred by most trainers because it is easy to fold up in your hand and comfortable to hold because there is a certain amount of give to it. There are lessons where the dog will start off 6 feet away from you at the end of the leash. The leash used to take the puppy outside to relieve himself is shorter because you don't want him to roam away from his area. The shorter leash will also be the one to use when you walk the puppy.
If you've been wise enough to enroll in a Puppy Kindergarten training class, suggestions will be made as to the best collar and leash for your young puppy. I say "wise" because your puppy will be in a class with puppies in his age range (up to five months old) of all breeds and sizes. It's the perfect way for him to learn the right way (and the wrong way) to interact with other dogs as well as their people. You cannot teach your puppy how to interpret another dog's sign language. For a first-time puppy owner, these socialization classes are invaluable. For experienced dog owners, they are a real boon to further training.
**SMILE WHEN YOU ORDER ME AROUND!**
While trainers recommend practicing with your dog every day, it's perfectly acceptable to take a "mental health day" off. It's better not to train the dog on days when you're in a sour mood. Your bad attitude or lack of interest will be sensed by your dog, and he will respond accordingly. Studies show that dogs are well tuned in to their humans' emotions. Be conscious of how you use your voice when talking to your dog. Raising your voice or shouting will only erode your dog's trust in you as his trainer and master.
**A TTENTION**
You've been using the dog's name since the minute you collected him from the breeder, so you should be able to get his attention by saying his name—with a big smile and in an excited tone of voice. His response will be the puppy equivalent of "Here I am! What are we going to do?" Your immediate response (if you haven't guessed by now) is "Good dog." Rewarding him at the moment he pays attention to you teaches him the proper way to respond when he hears his name.
**EXERCISES FOR A BASIC CANINE EDUCATION**
**T HE SIT EXERCISE**
There are several ways to teach the puppy to sit. The first one is to catch him whenever he is about to sit and, as his backside nears the floor, say "Sit, good dog!" That's positive reinforcement and, if your timing is sharp, he will learn that what he's doing at that second is connected to your saying "Sit" and that you think he's clever for doing it!
Another method is to start with the puppy on his leash in front of you. Show him a treat in the palm of your right hand. Bring your hand up under his nose and, almost in slow motion, move your hand up and back so his nose goes up in the air and his head tilts back as he follows the treat in your hand. At that point, he will have to either sit or fall over, so as his back legs buckle under, say "Sit, good dog," and then give him the treat and lots of praise. You may have to begin with your hand lightly running up his chest, actually lifting his chin up until he sits. Some (usually older) dogs require gentle pressure on their hindquarters with the left hand, in which case the dog should be on your left side. Puppies generally do not appreciate this physical dominance.
**Once your pup is comfortable with his collar and lead and you're ready to start with basic commands, the sit exercise is the first you will teach.**
After a few times, you should be able to show the dog a treat in the open palm of your hand, raise your hand waist-high as you say "Sit" and have him sit. You will thereby have taught him two things at the same time. Both the verbal command and the motion of the hand are signals for the sit. Your puppy is watching you almost more than he is listening to you, so what you do is just as important as what you say.
Don't save any of these drills only for training sessions. Use them as much as possible at odd times during a normal day. The dog should always sit before being given his food dish. He should sit to let you go through a doorway first, when the doorbell rings or when you stop to speak to someone on the street.
**T HE DOWN EXERCISE**
Before beginning to teach the down command, you must consider how the dog feels about this exercise. To him, "down" is a submissive position. Being flat on the floor with you standing over him is not his idea of fun. It's up to you to let him know that, while it may not be fun, the reward of your approval is worth his effort.
Start with the puppy on your left side in a sit position. Hold the leash right above his collar in your left hand. Have an extra-special treat, such as a small piece of cooked chicken or hot dog, in your right hand. Place it at the end of the pup's nose and steadily move your hand down and forward along the ground. Hold the leash to prevent a sudden lunge for the food. As the puppy goes into the down position, say "Down" very gently.
**A SIMPLE "SIT"**
When you command your dog to sit, use the word "Sit." Do not say "Sit down," as your dog will not know whether you mean "Sit" or "Down," or maybe you mean both. Be clear in your instructions to your dog; use one-word commands and always be consistent.
The difficulty with this exercise is twofold: it's both the submissive aspect and the fact that most people say the word "Down" as if they were a drill sergeant in charge of recruits! So issue the command sweetly, give him the treat and have the pup maintain the down position for several seconds. If he tries to get up immediately, place your hands on his shoulders and press down gently, giving him a very quiet "Good dog." As you progress with this lesson, increase the "down time" until he will hold it until you say "Okay" (his cue for release). Practice this one in the house at various times throughout the day.
By increasing the length of time during which the dog must maintain the down position, you'll find many uses for it. For example, he can lie at your feet in the vet's office or anywhere that both of you have to wait, when you are on the phone, while the family is eating and so forth. If you progress to training for competitive obedience, he'll already be all set for the exercise called the "long down."
**T HE STAY EXERCISE**
You can teach your Welsh Terrier to stay in the sit, down and stand positions. To teach the sit/stay, have the dog sit on your left side. Hold the leash at waist level in your left hand and let the dog know that you have a treat in your closed right hand. Step forward on your right foot as you say "Stay." Immediately turn and stand directly in front of the dog, keeping your right hand up high so he'll keep his eye on the treat hand and maintain the sit position for a count of five. Return to your original position and offer the reward.
**Before progressing to the down/stay, the dog must be comfortable with the down command.**
Increase the length of the sit/stay each time until the dog can hold it for at least 30 seconds without moving. After about a week of success, move out on your right foot and take two steps before turning to face the dog. Give the "Stay" hand signal (left palm back toward the dog's head) as you leave. He gets the treat when you return and he holds the sit/stay. Increase the distance that you walk away from him before turning until you reach the length of your training leash. But don't rush it! Go back to the beginning if he moves before he should. No matter what lesson you are teaching, never be upset by having to back up for a few days. The repetition and practice are what will make your dog reliable in these commands. It won't do any good to move on to something more difficult if the command is not mastered at the easier levels. Above all, even if you do get frustrated, never let your puppy know! Always keep a positive, upbeat attitude during training, which will transmit to your dog for positive results.
**TIPS FOR TRAINING AND SAFETY**
1. Whether on- or off-leash, practice only in a fenced area.
2. Remove the training collar when the training session is over.
3. Don't try to break up a dogfight.
4. "Come," "Leave it" and "Wait" are safety commands.
5. The dog belongs in a crate or behind a barrier when riding in the car.
6. Don't ignore the dog's first sign of aggression. Aggression only gets worse, so take it seriously.
7. Keep the faces of children and dogs separated.
8. Pay attention to what the dog is chewing.
9. Keep the vet's number near your phone.
10. "Okay" is a useful release command.
The down/stay is taught in the same way once the dog is completely reliable and steady with the down command. Again, don't rush it. With the dog in the down position on your left side, step out on your right foot as you say "Stay." Return by walking around in back of the dog and into your original position. While you are training, it's okay to murmur something like "Hold on" to encourage him to stay put. When the dog will stay without moving when you are at a distance of 3 or 4 feet, begin to increase the length of time before you return. Be sure he holds the down on your return until you say "Okay." At that point, he gets his treat—just so he'll remember for next time that it's not over until it's over.
**T HE COME EXERCISE**
No command is more important to the safety of your Welsh Terrier than "Come." It is what you should say every single time you see the puppy running toward you: "Binky, come! Good dog." During playtime, run a few feet away from the puppy and then turn and tell him to "Come" as he is already running to you. You can go so far as to teach your puppy two things at once if you squat down and hold out your arms. As the pup gets close to you and you're saying "Good dog," bring your right arm in about waist high. Now he's also learning the hand signal, an excellent device should you be on the phone when you need to get him to come to you! You'll also both be one step ahead when you enter obedience classes.
**OKAY!**
This is the signal that tells your dog that he can quit whatever he was doing. Use "Okay" to end a session on a correct response to a command. (Never end on an incorrect response.) Lots of praise follows. People use "Okay" a lot and it has other uses for dogs, too. Your dog is barking. You say, "Okay! Come!" "Okay" signals him to stop the barking activity and "Come" allows him to come to you for a "Good dog."
**The trainer uses both verbal commands and hand signals in teaching the stay. Distance and time are increased gradually as the dog learns.**
**LET'S GO!**
Many people use "Let's go" instead of "Heel" when teaching their dogs to behave on lead. It sounds more like fun! When beginning to teach the heel, whatever command you use, always step off on your left foot. That's the one next to the dog, who is on your left side, in case you've forgotten. Keep a loose leash. When the dog pulls ahead, stop, bring him back and begin again. Use treats to guide him around turns.
When the puppy responds to your well-timed "Come," try it with the puppy on the training leash. This time, catch him off guard, while he's sniffing a leaf or watching a bird: "Binky, come!" You may have to pause for a split second after his name to be sure you have his attention. If the puppy shows any sign of confusion, give the leash a mild jerk and take a couple of steps backward. Do not repeat the command. In this case, you should say "Good come" as he reaches you.
That's the number-one rule of training. Each command word is given just once. Anything more is nagging. You'll also notice that all commands are one word only. Even when they are actually two words, you say them as one.
Never call the dog to come to you—with or without his name—if you are angry or intend to correct him for some misbehavior. When correcting the pup, you go to him. Your dog must always connect "Come" with something pleasant and with your approval; then you can rely on his response. Always greet his coming to you with plenty of happy praise.
Puppies, like children, have notoriously short attention spans, so don't overdo it with any of the training. Keep each lesson short. Break it up with a quick run around the yard or a ball toss, repeat the lesson and quit as soon as the pup gets it right. That way, you will always end with a "Good dog."
Life isn't perfect and neither are puppies. A time will come, often around ten months of age, when he'll become "selectively deaf" or choose to "forget" his name. He may respond by wagging his tail (and even seeming to smile at you) with a look that says "Make me!" Laugh, throw his favorite toy and skip the lesson you had planned. Pups will be pups!
**T HE HEEL EXERCISE**
The second most important command to teach, after the come, is the heel. When you are walking your growing puppy, you need to be in control. Besides, it looks terrible to be pulled and yanked down the street, and it's not much fun either. Your nine- to ten-week-old puppy will probably follow you everywhere, but that's his natural instinct, not your control over the situation. However, any time he does follow you, you can say "Heel" and be ahead of the game, as he will learn to associate this command with the action of following you before you even begin teaching him to heel.
There is a very precise, almost military, procedure for teaching your dog to heel. As with all other obedience training, begin with the dog on your left side. He will be in a very nice sit and you will have the training leash across your chest. Hold the loop and folded leash in your right hand. Pick up the slack leash above the dog in your left hand and hold it loosely at your side. Step out on your left foot as you say "Heel." If the puppy does not move, give a gentle tug or pat your left leg to get him started. If he surges ahead of you, stop and pull him back gently until he is at your side. Tell him to sit and begin again.
**A tasty tidbit for a job well done! Verbal praise is even more important, so don't forget to also tell your Welsh what a good dog he is.**
Walk a few steps and stop while the puppy is correctly beside you. Tell him to sit and give mild verbal praise. (More enthusiastic praise will encourage him to think the lesson is over.) Repeat the lesson, increasing the number of steps you take only as long as the dog is heeling nicely beside you. When you end the lesson, have him hold the sit and then give him the "Okay" to let him know that this is the end of the lesson. Praise him so that he knows he did a good job.
**The first Welsh to be awarded the Master Agility Champion (MACH) is Webster, MACH Cisseldale's Double Trouble CGC, owned by Linda and Brad Brisbin and bred by Barbara Cissel. Webster was the top Agility Welsh for 2001–2004.**
The cure for excessive pulling (a common problem) is to stop when the dog is no more than 2 or 3 feet ahead of you. Guide him back into position and begin again. With a really determined puller, try switching to a head collar. This will automatically turn the pup's head toward you so you can bring him back easily to the heel position. Give quiet, reassuring praise every time the leash goes slack and he's staying with you.
Staying and heeling can take a lot out of a dog, so provide playtime and free-running exercise to shake off the stress when the lessons are over. You don't want him to associate training with all work and no fun.
**OBEDIENCE CLASSES**
The advantages of an obedience class are that your dog will have to learn amid the distractions of other people and dogs and that your mistakes will be quickly corrected by the trainer. Teaching your dog along with a qualified instructor and other handlers who may have more dog experience than you is another plus of the class environment. The instructor and other handlers can help you to find the most efficient way of teaching your dog a command or exercise. It's often easier to learn from other people's mistakes than your own. You will also learn all of the requirements for competitive obedience trials, in which you can earn titles and go on to advanced jumping and retrieving exercises, which are fun for many dogs. Obedience classes build the foundation needed for many other canine activities (in which we humans are allowed to participate, too!).
**NO MORE TREATS!**
When your dog is responding promptly and correctly to commands, it's time to eliminate treats. Begin by alternating a treat reward with a verbal-praise-only reward. Gradually eliminate all treats while increasing the frequency of praise. Overlook pleading eyes and expectant expressions, but if he's still watching your treat hand, you're on your way to using hand signals.
**TRAINING FOR OTHER ACTIVITIES**
Once your dog has basic obedience under his collar and is 12 months of age, you can enter the world of agility training. Dogs think agility is pure fun, like being turned loose in an amusement park full of obstacles! In addition to agility, your Welsh may enjoy participating in go-to-ground events for terriers and/or tracking, which is open to all "nosey" dogs (which would include all dogs!). For those who like to volunteer, there is the wonderful feeling of owning a therapy dog and visiting hospices, nursing homes and veterans' homes to bring smiles, comfort and companionship to those who live there.
**We thought that the Welsh Terrier was an earthdog, not a water dog! Every Welsh has individual likes and aptitudes, so find out what your dog enjoys and have a great time together.**
Around the house, your Welsh Terrier can be taught to do some simple chores. You might teach him to carry a small basket or to fetch the morning newspaper. The kids can teach the dog all kinds of tricks, from playing hide-and-seek to balancing a biscuit on his nose. The Welsh will enjoy joining you on outings and hikes; a long lead will let him explore safely and keep him from going off "on the hunt." A family dog is what rounds out the family. Everything he does, including sitting in your lap or gazing lovingly at you, represents the bonus of owning a dog.
**HEALTHCARE FOR A LIFETIME**
When you own a dog, you become his healthcare advocate over his entire lifespan, as well as being the one to shoulder the financial burden of such care. Accordingly, it is worthwhile to focus on prevention rather than treatment, as you and your pet will both be happier.
Of course, the best place to have begun your program of preventive healthcare is with the initial purchase or adoption of your dog. There is no way of guaranteeing that your new furry friend is free of medical problems, but there are some things you can do to improve your odds. You certainly should have done adequate research into the Welsh Terrier and have selected your puppy carefully rather than buying on impulse. Health issues aside, a large number of pet abandonment and relinquishment cases arise from a mismatch between pet needs and owner expectations. This is entirely preventable with appropriate planning and finding a good breeder.
Regarding healthcare issues specifically, it is very difficult to make blanket statements about where to acquire a problem-free pet, but, again, a reputable breeder is your best bet. In an ideal situation, you have the opportunity to see both parents, get references from other owners of the breeder's pups and see genetic-testing documentation for several generations of the litter's ancestors. At the very least, you must thoroughly investigate the Welsh Terrier and the problems inherent in the breed, as well as the genetic testing available to screen for those problems. Genetic testing offers some important benefits, but testing is available for only a few disorders in a relatively small number of breeds and is not available for some of the most common genetic diseases, such as hip dysplasia, cataracts, epilepsy, cardiomyopathy, etc. This area of research is indeed exciting and increasingly important, and advances will continue to be made each year. In fact, recent research has shown that there is an equivalent dog gene for 75% of known human genes, so research done in either species is likely to benefit the other.
We've also discussed that evaluating the behavioral nature of your Welsh Terrier and that of his immediate family members is an important part of the selection process that cannot be underestimated or overemphasized. It is sometimes difficult to evaluate temperament in puppies because certain behavioral tendencies, such as some forms of aggression, may not be immediately evident. More dogs are euthanized each year for behavioral reasons than for all medical conditions combined, so it is critical to take temperament issues seriously. Start with a well-balanced, friendly companion and put the time and effort into proper socialization, and you will both be rewarded with a lifelong valued relationship.
Assuming that you have started off with a pup from healthy, sound stock, you then become responsible for helping your veterinarian keep your pet healthy. Some crucial things happen before you even bring your puppy home. Parasite control typically begins at two weeks of age, and vaccinations typically begin at six to eight weeks of age. A pre-pubertal evaluation is typically scheduled for about six months of age. At this time, a dental evaluation is done (since the adult teeth are now in), heartworm prevention is started and neutering or spaying is most commonly done.
**DOGGIE DENTAL DON'TS**
A veterinary dental exam is necessary if you notice one or any combination of the following in your dog:
• Broken, loose or missing teeth
• Loss of appetite (which could be due to mouth pain or illness caused by infection)
• Gum abnormalities, including redness, swelling and bleeding
• Drooling, with or without blood
• Yellowing of the teeth or gumline, indicating tartar
• Bad breath
It is critical to commence regular dental care at home if you have not already done so. It may not sound very important, but most dogs have active periodontal disease by four years of age if they don't have their teeth cleaned regularly at home, not just at their veterinary exams. Dental problems lead to more than just bad "doggie breath." Gum disease can have very serious medical consequences. If you start brushing your dog's teeth and using antiseptic rinses from a young age, your dog will be accustomed to it and will not resist. The results will be healthy dentition, which your pet will need to enjoy a long, healthy life.
Most dogs are considered adults at a year of age, although the Welsh Terrier continues to mature up to about the age of two. Even individual dogs within each breed have different healthcare requirements, so work with your veterinarian to determine what will be needed and what your role should be. This doctor-client relationship is important, because as vaccination guidelines change, there may not be an annual "vaccine visit" scheduled. You must make sure that you see your veterinarian at least annually, even if no vaccines are due, because this is the best opportunity to coordinate healthcare activities and to make sure that no medical issues creep by unaddressed.
When your Welsh Terrier reaches three-quarters of his anticipated lifespan, he is considered a "senior" and likely requires some special care. In general, if you've been taking great care of your canine companion throughout his formative and adult years, the transition to senior status should be a smooth one. Age is not a disease, and as long as everything is functioning as it should, there is no reason why most of late adulthood should not be rewarding for both you and your pet. This is especially true if you have tended to the details, such as regular veterinary visits, proper dental care, excellent nutrition and management of bone and joint issues.
At this stage in your Welsh Terrier's life, your veterinarian may want to schedule visits twice yearly, instead of once, to run some laboratory screenings, electrocardiograms and the like, and to change the diet to something more digestible. Catching problems early is the best way to manage them effectively. Treating the early stages of heart disease is so much easier than trying to intervene when there is more significant damage to the heart muscle. Similarly, managing the beginning of kidney problems is fairly routine if there is no significant kidney damage. Other problems, like cognitive dysfunction (similar to senility and Alzheimer's disease), cancer, diabetes and arthritis, are more common in older dogs, but all can be treated to help the dog live as many happy, comfortable years as possible. Just as in people, medical management is more effective (and less expensive) when you catch things early.
**TAKING YOUR DOG'S TEMPERATURE**
It is important to know how to take your dog's temperature at times when you think he may be ill. It's not the most enjoyable task, but it can be done without too much difficulty. It's easier with a helper, preferably someone with whom the dog is friendly, so that one of you can hold the dog while the other inserts the thermometer.
Before inserting the thermometer, coat the end with petroleum jelly. Insert the thermometer slowly and gently into the dog's rectum about one inch. Wait for the reading—digital thermometers will register in less than a minute. Be sure to remove the thermometer carefully and clean it thoroughly after each use.
A dog's normal body temperature is between 100.5 and 102.5 degrees F. Immediate veterinary attention is required if the dog's temperature is below 99 or above 104 degrees F.
**YOUR DOG NEEDS TO VISIT THE VET IF:**
• He has ingested a toxin such as antifreeze or a toxic plant; in these cases, administer first aid and call the vet right away
• His teeth are discolored, loose or missing or he has sores or other signs of infection or abnormality in the mouth
• He has been vomiting, has had diarrhea or has been constipated for over 24 hours; call immediately if you notice blood
• He has refused food for over 24 hours
• His eating habits, water intake or toilet habits have noticeably changed; if you have noticed weight gain or weight loss
• He shows symptoms of bloat, which requires immediate attention
• He is salivating excessively
• He has a lump in his throat
• He has a lump or bumps anywhere on the body
• He is very lethargic
• He appears to be in pain or otherwise has trouble chewing or swallowing
• His skin loses elasticity Of course there will be other instances in which a visit to the vet is necessary; these are just some of the signs that could be indicative of serious problems that need to be caught as early as possible.
**SELECTING A VETERINARIAN**
There is probably no more important decision that you will make regarding your pet's healthcare than the selection of his doctor. Your pet's veterinarian will be a pediatrician, family-practice physician and gerontologist, depending on the dog's life stage, and will be the individual who makes recommendations regarding issues such as when specialists need to be consulted, when diagnostic testing and/or therapeutic intervention is needed and when you will need to seek outside emergency and critical-care services. Your vet will act as your advocate and liaison throughout these processes.
Everyone has his own idea about what to look for in a vet, an individual who will play a big role in his dog's (and, of course, his own) life for many years to come. For some, it is the compassionate caregiver with whom they hope to develop a professional relationship to span the lifetime of their dogs and even their future pets. For others, they are seeking a clinician with keen diagnostic and therapeutic insight who can deliver state-of-the-art healthcare. Still others need a veterinary facility that is open evenings and weekends, is in close proximity or provides mobile veterinary services to accommodate their schedules; these people may not much mind that their dogs might see different veterinarians on each visit. Just as we have different reasons for selecting our own healthcare professionals (e.g., covered by insurance plan, expert in field, convenient location, etc.), we should not expect that there is a one-size-fits-all recommendation for selecting a veterinarian and veterinary practice. The best advice is to be honest in your assessment of what you expect from a veterinary practice and to conscientiously research the options in your area. You will quickly appreciate that not all veterinary practices are the same, and you will be happiest with one that truly meets your needs.
**AIRBORNE ALLERGIES**
Just as humans have hay fever, rose fever and other fevers from which they suffer during the pollinating season, many dogs suffer from the same allergies, and these commonly plague some Welsh Terriers. When the pollen count is high, your dog might suffer, but don't expect him to sneeze and have a runny nose like a human would. Dogs react to pollen allergies the same way they react to fleas—they scratch and bite themselves. Dogs, like humans, can be tested for allergens. Discuss the testing with your veterinary dermatologist.
There is another point to be considered in the selection of veterinary services. Not that long ago, a single veterinarian would attempt to manage all medical and surgical issues as they arose. That was often problematic, because veterinarians are trained in many species and many diseases, and it was just impossible for general veterinary practitioners to be experts in every species, every breed, every field and every ailment. However, just as in the human healthcare fields, specialization has allowed general practitioners to concentrate on primary healthcare delivery, especially wellness and the prevention of infectious diseases, and to utilize a network of specialists to assist in the management of conditions that require specific expertise and experience. Thus there are now many types of veterinary specialists, including dermatologists, cardiologists, ophthalmologists, surgeons, internists, oncologists, neurologists, behaviorists, criticalists and others to help primary-care veterinarians deal with complicated medical challenges. In most cases, specialists see cases referred by primary-care veterinarians, make diagnoses and set up management plans. From there, the animals' ongoing care is returned to their primary-care veterinarians. This important team approach to your pet's medical-care needs has provided opportunities for advanced care and an unparalleled level of quality to be delivered.
With all of the opportunities for your Welsh Terrier to receive high-quality veterinary medical care, there is another topic that needs to be addressed at the same time—cost. It's been said that you can have excellent healthcare or inexpensive health-care, but never both; this is as true in veterinary medicine as it is in human medicine. While veterinary costs are a fraction of what the same services cost in the human healthcare arena, it is still difficult to deal with unanticipated medical costs, especially since they can easily creep into hundreds or even thousands of dollars if specialists or emergency services become involved. However, there are ways of managing these risks. The easiest is to buy pet health insurance and realize that its foremost purpose is not to cover routine healthcare visits but rather to serve as an umbrella for those rainy days when your pet needs medical care and you don't want to worry about whether or not you can afford that care.
**PROBLEM: AND THAT STARTS WITH "P"**
Urinary tract problems more commonly affect female dogs, especially those who have been spayed. The first sign that a urinary tract problem exists usually is a strong odor from the urine or an unusual color. Blood in the urine, known as hematuria, is another sign of an infection, related to cystitis, a bladder infection, bladder cancer or a blood-clotting disorder. Urinary tract problems can also be signaled by the dog's straining while urinating, experiencing pain during urination and genital discharge as well as excessive water intake and urination.
Excessive drinking, in and of itself, does not indicate a urinary tract problem. A dog who is drinking more than normal may have a kidney or liver problem, a hormonal disorder or diabetes mellitus. Behaviorists report a disorder known as psychogenic polydipsia, which manifests itself in excessive drinking and urination. If you notice your dog drinking much more than normal, take him to the vet.
Pet insurance policies are very cost-effective (and very inexpensive by human health-insurance standards), but make sure that you buy the policy long before you intend to use it (preferably starting in puppy-hood, because coverage will exclude pre-existing conditions) and that you are actually buying an indemnity insurance plan from an insurance company that is regulated by your state or province. Many insurance policy look-alikes are actually discount clubs that are redeemable only at specific locations and for specific services. An indemnity plan covers your pet at almost all veterinary, specialty and emergency practices and is an excellent way to manage your pet's ongoing healthcare needs.
**VACCINATIONS AND INFECTIOUS DISEASES**
There has never been an easier time to prevent a variety of infectious diseases in your dog, but the advances we've made in veterinary medicine come with a price—choice. Now while it may seem that choice is a good thing (and it is), it has never been more difficult for the pet owner (or the veterinarian) to make an informed decision about the best way to protect pets through vaccination.
Years ago, it was just accepted that puppies got a starter series of vaccinations and then annual "boosters" throughout their lives to keep them protected. As more and more vaccines became available, consumers wanted the convenience of having all of that protection in a single injection. The result was "multivalent" vaccines that crammed a lot of protection into a single syringe. The manufacturers' recommendations were to give the vaccines annually, and this was a simple enough protocol to follow. However, as veterinary medicine has become more sophisticated and we have started looking more at healthcare quandaries rather than convenience, it became necessary to reevaluate the situation and deal with some tough questions. It is important to realize that whether or not to use a particular vaccine depends on the risk of contracting the disease against which it protects, the severity of the disease if it is contracted, the duration of immunity provided by the vaccine, the safety of the product and the needs of the individual animal. In a very general sense, rabies, distemper, hepatitis and parvovirus are considered core vaccine needs, while parainfluenza, _Bordetella bronchiseptica_ , leptospirosis, coronavirus and borreliosis (Lyme disease) are considered non-core needs and best reserved for animals that demonstrate reasonable risk of contracting the diseases.
**C OMMON INFECTIOUS DISEASES**
Let's discuss some of the diseases that create the need for vaccination in the first place. Following are the major canine infectious diseases and a simple explanation of each.
**Rabies:** A devastating viral disease that can be fatal in dogs and people. In fact, vaccination of dogs and cats is an important public-health measure to create a resistant animal buffer population to protect people from contracting the disease. Vaccination schedules are determined on a government level and are not optional for pet owners; rabies vaccination is required by law in all 50 states.
**Parvovirus:** A severe, potentially life-threatening disease that is easily transmitted between dogs. There are four strains of the virus, but it is believed that there is significant "cross-protection" between strains that may be included in individual vaccines.
**Distemper:** A potentially severe and life-threatening disease with a relatively high risk of exposure, especially in certain regions. In very high-risk distemper environments, young pups may be vaccinated with human measles vaccine, a related virus that offers cross-protection when administered at four to ten weeks of age.
**Hepatitis** : Caused by canine adenovirus type 1 (CAV-1), but since vaccination with the causative virus has a higher rate of adverse effects, cross-protection is derived from the use of adenovirus type 2 (CAV-2), a cause of respiratory disease and one of the potential causes of canine cough. Vaccination with CAV-2 provides long-term immunity against hepatitis, but relatively less protection against respiratory infection.
**Canine cough:** Also called tracheobronchitis, actually a fairly complicated result of viral and bacterial offenders; therefore, even with vaccination, protection is incomplete. Wherever dogs congregate, canine cough will likely be spread among them. Intranasal vaccination with _Bordetella_ and parainfluenza is the best safeguard, but the duration of immunity does not appear to be very long, typically a year at most. These are non-core vaccines, but vaccination is sometimes mandated by boarding kennels, obedience classes, dog shows and other places where dogs congregate to try to minimize spread of infection.
**Leptospirosis:** A potentially fatal disease that is more common in some geographic regions. It is capable of being spread to humans. The disease varies with the individual "serovar," or strain, of _Leptospira_ involved. Since there does not appear to be much cross-protection between serovars, protection is only as good as the likelihood that the serovar in the vaccine is the same as the one in the pet's local environment. Problems with _Leptospira_ vaccines are that protection does not last very long, side effects are not uncommon and a large percentage of dogs (perhaps 30%) may not respond to vaccination.
**_Borrelia burgdorferi_ :** The cause of Lyme disease, the risk of which varies with the geographic area in which the pet lives and travels. Lyme disease is spread by deer ticks in the eastern US and western black-legged ticks in the western part of the country, and the risk of exposure is high in some regions. Lameness, fever and inappetence are most commonly seen in affected dogs. The extent of protection from the vaccine has not been conclusively demonstrated.
**Coronavirus:** This disease has a high risk of exposure, especially in areas where dogs congregate, but it typically causes only mild to moderate digestive upset (diarrhea, vomiting, etc.). Vaccines are available, but the duration of protection is believed to be relatively short and the effectiveness of the vaccine in preventing infection is considered low.
There are many other vaccinations available, including those for _Giardia_ and canine adenovirus-1. While there may be some specific indications for their use, and local risk factors to be considered, they are not widely recommended for most dogs.
**NEUTERING/SPAYING**
Sterilization procedures (neutering for males/spaying for females) are meant to accomplish several purposes. While the underlying premise is to address the risk of pet overpopulation, there are also some medical and behavioral benefits to the surgeries as well. For females, spaying prior to the first estrus (heat cycle) leads to a marked reduction in the risk of mammary cancer. There also will be no manifestations of "heat" to attract male dogs and no bleeding in the house. For males, there is prevention of testicular cancer and a reduction in the risk of prostate problems. In both sexes, there may be some limited reduction in aggressive behaviors toward other dogs, and some diminishing of urine marking, roaming and mounting.
While neutering and spaying do indeed prevent animals from contributing to pet overpopulation, even no-cost and low-cost neutering options have not eliminated the problem. Perhaps one of the main reasons for this is that individuals that intentionally breed their dogs and those that allow their animals to run at large are the main causes of unwanted offspring. Also, animals in shelters are often there because they were abandoned or relinquished, not because they came from unplanned matings. Neutering/ spaying is important, but it should be considered in the context of the real causes of animals' ending up in shelters and eventually being euthanized.
**HIT ME WITH A HOT SPOT**
What is a hot spot? Technically known as pyotraumatic dermatitis, a hot spot is an infection on the dog's coat, usually by the rear end, under the tail or on a leg, which the dog inflicts upon himself. The dog licks and bites the itchy spot until it becomes inflamed and infected. The hot spot can range in size from the circumference of a grape to the circumference of an apple. Provided that the hot spot is not related to a deeper bacterial infection, it can be treated topically by clipping the area, cleaning the sore and giving prednisone. For bacterial infections, antibiotics are required. In some cases, an Elizabethan collar is required to keep the dog from further irritating the hot spot. The itching can intensify and the pain becomes worse. Medicated shampoos and cool compresses, drying agents and topical steroids may be prescribed by your vet as well.
Hot spots can be caused by fleas, an allergy, an ear infection, anal sac problems, mange or a foreign irritant. Likewise, they can be linked to psychoses. The underlying problem must be addressed in addition to the hot spot itself.
One of the important considerations regarding neutering is that it is a surgical procedure. This sometimes gets lost in discussions of low-cost procedures and commoditization of the process. In females, spaying is specifically referred to as an ovariohysterectomy. In this procedure, a midline incision is made in the abdomen and the entire uterus and both ovaries are surgically removed. While this is a major invasive surgical procedure, it usually has few complications, because it is typically performed on healthy young animals. However, it is major surgery, as any woman who has had a hysterectomy will attest.
In males, neutering has traditionally referred to castration, which involves the surgical removal of both testicles. While still a significant piece of surgery, there is not the abdominal exposure that is required in the female surgery. In addition, there is now a chemical sterilization option, in which a solution is injected into each testicle, leading to atrophy of the sperm-producing cells. This can typically be done under sedation rather than full anesthesia. This is a relatively new approach, and there are no long-term clinical studies yet available.
Neutering/spaying is typically done around six months of age at most veterinary hospitals, although techniques have been pioneered to perform the procedures in animals as young as eight weeks of age. In general, the surgeries on the very young animals are done for the specific reason of sterilizing them before they go to their new homes. This is done in some shelter hospitals for assurance that the animals will definitely not produce any pups. Otherwise, these organizations need to rely on owners to comply with their wishes to have the animals "altered" at a later date, something that does not always happen.
**FOOD ALLERGY**
Severe itching, leading to bald patches and open sores on the feet, face, ears, armpits and groin, could be caused by a food allergy. Studies indicate that up to 10% of dogs suffer from food allergies, which develop slowly over time without a change in diet. Dogs who suffer from chronic ear problems may actually have a food allergy. Unfortunately, there are no tests available to determine whether your dog definitely suffers from a food allergy. The dog will be miserable and you will be frustrated and stressed.
Take the problem into your own hands and kitchen. Select a type of meat that your dog is not getting from his existing diet, perhaps white fish, lamb or venison, and prepare a home-cooked food. The food should consist of two parts carbohydrate (rice, pasta or potatoes) and one part protein (the chosen meat). It's better not to start with soy as the protein source unless all of the meats cause a reaction.
Monitor your dog's intake carefully. He must eat only your prepared meal without any treats or side-trips to the garbage can. All family members (and visiting friends) must be informed of the plan. After four or five weeks on the new diet, you will reintroduce a portion of his original diet to determine whether this food is the cause of the skin irritation (or other reactions). Once the dog reacts to the change in diet, resume the new diet. Make dietary modifications every two weeks and keep careful records of any reactions the dog has to the diet.
There are some exciting immunocontraceptive "vaccines" currently under development, and there may be a time when contraception in pets will not require surgical procedures. We anxiously await these developments.
**EXTERNAL PARASITES**
**F LEAS**
Fleas have been around for millions of years and, while we have better tools now for controlling them than at any time in the past, there still is little chance that they will end up on an endangered species list. Actually, they are very well adapted to living on our pets, and they continue to adapt as we make advances.
The female flea can consume 15 times her weight in blood during active reproduction and can lay as many as 40 eggs a day. These eggs are very resistant to the effects of insecticides. They hatch into larvae, which then mature and spin cocoons. The immature fleas reside in this pupal stage until the time is right for feeding. This pupal stage is also very resistant to the effects of insecticides, and pupae can last in the environment without feeding for many months. Newly emergent fleas are attracted to animals by the warmth of the animals' bodies, movement and exhaled carbon dioxide. However, when they first emerge from their cocoons, they orient towards light; thus when an animal passes between a flea and the light source, casting a shadow, the flea pounces and starts to feed. If the animal turns out to be a dog or cat, the reproductive cycle continues. If the flea lands on another type of animal, including a person, the flea will bite but will then look for a more appropriate host. An emerging adult flea can survive without feeding for up to 12 months but, once it tastes blood, it can survive off its host for only three to four days.
**FLEA PREVENTION FOR YOUR DOG**
• Discuss with your veterinarian the safest product to protect your dog, likely in the form of a monthly tablet or a liquid preparation placed on the back of the dog's neck.
• For dogs suffering from flea-bite dermatitis, a shampoo or topical insecticide treatment is required.
• Your lawn and property should be sprayed with an insecticide designed to kill fleas and ticks that lurk outdoors.
• Using a flea comb, check the dog's coat regularly for any signs of parasites.
• Practice good housekeeping. Vacuum floors, carpets and furniture regularly, especially in the areas that the dog frequents, and wash the dog's bedding weekly.
• Follow up house-cleaning with carpet shampoos and sprays to rid the house of fleas at all stages of development. Insect growth regulators are the safest option.
It was once thought that fleas spend most of their lives in the environment, but we now know that fleas won't willingly jump off a dog unless leaping to another dog or when physically removed by brushing, bathing or other manipulation. Flea eggs, on the other hand, are shiny and smooth, and they roll off the animal and into the environment. The eggs, larvae and pupae then exist in the environment, but once the adult finds a susceptible animal, it's home sweet home until the flea is forced to seek refuge elsewhere.
**KILL FLEAS THE NATURAL WAY**
If you choose not to go the route of conventional medication, there are some natural ways to ward off fleas:
• Dust your dog with a natural flea powder, composed of such herbal goodies as rosemary, wormwood, pennyroyal, citronella, rue, tobacco powder and eucalyptus.
• Apply diatomaceous earth, the fossilized remains of single-cell algae, to your carpets, furniture and pet's bedding. Even though it's not good for dogs, it's even worse for fleas, which will dry up swiftly and die.
• Brush your dog frequently, give him adequate exercise and let him fast occasionally. All of these activities strengthen the dog's system and make him more resistant to disease and parasites.
• Bathe your Welsh Terrier with a capful of pennyroyal or eucalyptus oil.
• Feed a natural diet, free of additives and preservatives. Add some fresh garlic and brewer's yeast to the dog's morning portion, as these items have flea-repelling properties.
Since adult fleas live on the animal and immature forms survive in the environment, a successful treatment plan must address all stages of the flea life cycle. There are now several safe and effective flea-control products that can be applied on a monthly basis. These include fipronil, imidacloprid, selamectin and permethrin (found in several formulations). Most of these products have significant flea-killing rates within 24 hours. However, none of them will control the immature forms in the environment. To accomplish this, there are a variety of insect growth regulators that can be sprayed into the environment (e.g., pyriproxyfen, methoprene, fenoxycarb) as well as insect development inhibitors such as lufenuron that can be administered. These compounds have no effect on adult fleas, but they stop immature forms from developing into adults. In years gone by, we relied heavily on toxic insecticides (such as organophosphates, organochlorines and carbamates) to manage the flea problem, but today's options are not only much safer to use on our furry friends but also safer for the environment.
**T ICKS**
Ticks are members of the spider class (arachnids) and are blood-sucking parasites capable of transmitting a variety of diseases, including Lyme disease, ehrlichiosis, babesiosis and Rocky Mountain spotted fever. It's easy to see ticks on your own skin, but it is more of a challenge when your furry companion is affected. Whenever you happen to be planning a stroll in a tick-infested area (especially forests, grassy or wooded areas or parks) be prepared to do a thorough inspection of your dog afterward to search for ticks. Ticks can be tricky, so make sure you spend time looking in the ears, between the toes and everywhere else where a tick might hide. Ticks need to be attached for 24–72 hours before they transmit most of the diseases that they carry, so you do have a window of opportunity for some preventive intervention.
Female ticks live to eat and breed. They can lay between 4,000 and 5,000 eggs and die soon after. Males, on the other hand, live only to mate and continue the process as long as they are able. Most ticks live on multiple hosts before parasitizing dogs. The immature forms typically reside on grass and shrubs, waiting for susceptible animals to walk by. The larvae and nymph stages typically feed on wildlife.
**A TICKING BOMB**
There is nothing good about a tick's harpooning his nose into your dog's skin. Among the diseases caused by ticks are Rocky Mountain spotted fever, canine ehrlichiosis, canine babesiosis, canine hepatozoonosis and Lyme disease. If a dog is allergic to the saliva of a female wood tick, he can develop tick paralysis.
If only a few ticks are present on a dog, they can be plucked out, but it is important to remove the entire head and mouthparts, which may be deeply embedded in the skin. This is best accomplished with forceps designed especially for this purpose; fingers can be used but should be protected with rubber gloves, plastic wrap or at least a paper towel. The tick should be grasped as closely as possible to the animal's skin and should be pulled upward with steady, even pressure. It is important not to squeeze, crush or puncture the body of the tick or you risk exposure to any disease carried by that tick. Once the ticks have been removed, the sites of attachment should be disinfected. Your hands should then be thoroughly washed with soap and water to further minimize risk of contagion. The tick should be disposed of in a container of alcohol or household bleach.
**TICK CONTROL**
Removal of underbrush and leaf litter and the thinning of trees in areas where tick control is desired are recommended. These actions remove the cover and food sources for small animals that serve as hosts for ticks. With continued mowing of grasses in these areas, the probability of ticks' surviving is further reduced. A variety of insecticide ingredients (e.g., resmethrin, carbaryl, permethrin, chlorpyrifos, dioxathion and allethrin) are registered for tick control around the home.
Some of the newer flea products on the market, specifically those with fipronil, selamectin and permethrin, have proven to be effective against some, but not all, species of tick. Flea collars containing appropriate pesticides (e.g., propoxur, chlorfenvinphos) can aid in tick control. In most areas, such collars should be placed on animals in March, at the beginning of the tick season, and changed regularly. Leaving the collar on when the pesticide level is waning invites the development of resistance. Amitraz collars are also good for tick control, and the active ingredient does not interfere with other flea-control products. The ingredient helps prevent the attachment of ticks to the skin and will cause those ticks already attached on the skin to detach themselves.
**M ITES**
Mites are tiny arachnid parasites that parasitize the skin of dogs. Skin diseases caused by mites are referred to as "mange," and there are many different forms seen in dogs. These forms are very different from one another, each one warranting an individual description.
Sarcoptic mange, or scabies, is one of the itchiest conditions that affects dogs. The microscopic _Sarcoptes_ mites burrow into the superficial layers of the skin and can drive dogs crazy with itchiness. They are also communicable to people, although they can't complete their reproductive cycle on people. In addition to being tiny, the mites also are often difficult to find when trying to make a diagnosis. Skin scrapings from multiple areas are examined microscopically but, even then, sometimes the mites cannot be found.
Fortunately, scabies is relatively easy to treat, and there are a variety of products that will successfully kill the mites. Since the mites can't live in the environment for very long without feeding, a complete cure is usually possible within four to eight weeks.
Cheyletiellosis is caused by a relatively large mite, which sometimes can be seen even without a microscope. Often referred to as "walking dandruff," this also causes itching, but not usually as profound as with scabies. While _Cheyletiella_ mites can survive longer in the environment than scabies mites, they too are easy to treat, being responsive to not only the medications used to treat scabies but also often to flea-control products.
_Otodectes cynotis_ is the canine ear mite and is one of the more common causes of mange, especially in young dogs in shelters or pet stores. That's because the mites are typically present in large numbers and are quickly spread to nearby animals. The mites rarely do much harm but can be difficult to eradicate if the treatment regimen is not comprehensive. While many try to treat the condition with ear drops only, this is the most common cause of treatment failure. Ear drops cause the mites to simply move out of the ears and as far away as possible (usually to the base of the tail) until the insecticide levels in the ears drop to an acceptable level—then it's back to business as usual! The successful treatment of ear mites requires treating all animals in the household with a systemic insecticide, such as selamectin, or a combination of miticidal ear drops combined with whole-body flea-control preparations.
Demodicosis, or red mange, can be one of the most difficult forms of mange to treat. Part of the problem has to do with the fact that the mites live in the hair follicles and they are relatively well shielded from topical and systemic products. The main issue, however, is that demodectic mange typically results only when there is some underlying process interfering with the dog's immune system.
Since _Demodex_ mites are normal residents of the skin of mammals, including humans, there is usually a mite population explosion only when the immune system fails to keep the number of mites in check. In young animals, the immune deficit may be transient or may reflect an actual inherited immune problem. In older animals, demodicosis is usually seen only when there is another disease hampering the immune system, such as diabetes, cancer, thyroid problems or the use of immune-suppressing drugs. Accordingly, treatment involves not only trying to kill the mange mites but also discerning what is interfering with immune function and correcting it if possible.
Chiggers represent several different species of mite that don't parasitize dogs specifically, but do latch on to passersby and can cause irritation. The problem is most prevalent in wooded areas in the late summer and fall. Treatment is not difficult, as the mites do not complete their life cycle on dogs and are susceptible to a variety of miticidal products.
**M OSQUITOES**
Mosquitoes have long been known to transmit a variety of diseases to people, as well as just being biting pests during warm weather. They also pose a real risk to pets. Not only do they carry deadly heartworms but recently there also has been much concern over their involvement with West Nile virus. While we can avoid heartworm with the use of preventive medications, there are no such preventives for West Nile virus. The only method of prevention in endemic areas is active mosquito control. Fortunately, most dogs that have been exposed to the virus only developed flu-like symptoms and, to date, there have not been the large number of reported deaths in canines as seen in other species.
**MOSQUITO REPELLENT**
Low concentrations of DEET (less than 10%), found in many human mosquito repellents, have been safely used in dogs but, in these concentrations, probably give only about two hours of protection. DEET may be safe in these small concentrations, but since it is not licensed for use on dogs, there is no research proving its safety for dogs. Products containing permethrin give the longest-lasting protection, perhaps two to four weeks. As DEET is not licensed for use on dogs, and both DEET and permethrin can be quite toxic to cats, appropriate care should be exercised. Other products, such as those containing oil of citronella, also have some mosquito-repellent activity, but typically have a relatively short duration of action.
**INTERNAL PARASITES: WORMS**
**A SCARIDS**
Ascarids are intestinal roundworms that rarely cause severe disease in dogs. Nonetheless, they are of major public health significance because they can be transferred to people. Sadly, it is children who are most commonly affected by the parasite, probably from inadvertently ingesting ascarid-contaminated soil. In fact, many yards and children's sand-boxes contain appreciable numbers of ascarid eggs. So, while ascarids don't bite dogs or latch onto their intestines to suck blood, they do cause some nasty medical conditions in children and are best eradicated from our furry friends. Because pups can start passing ascarid eggs by three weeks of age, most parasite-control programs begin at two weeks of age and are repeated every two weeks until pups are eight weeks old. It is important to realize that bitches can pass ascarids to their pups even if they test negative prior to whelping. Accordingly, bitches are best treated at the same time as the pups.
**H OOKWORMS**
Unlike ascarids, hookworms do latch onto a dog's intestinal tract and can cause significant loss of blood and protein. Similar to ascarids, hookworms can be transmitted to humans, where they cause a condition known as cutaneous larval migrans. Dogs can become infected either by consuming the infective larvae or by the larvae's penetrating the skin directly. People most often get infected when they are lying on the ground (such as on a beach) and the larvae penetrate the skin. Yes, the larvae can penetrate through a beach blanket. Hookworms are typically susceptible to the same medications used to treat ascarids.
**W HIPWORMS**
Whipworms latch onto the lower aspects of the dog's colon and can cause cramping and diarrhea. Eggs do not start to appear in the dog's feces until about three months after the dog was infected. This worm has a peculiar life cycle, which makes it more difficult to control than ascarids or hook-worms. The good thing is that whipworms rarely are transferred to people.
Some of the medications used to treat ascarids and hookworms are also effective against whipworms, but, in general, a separate treatment protocol is needed. Since most of the medications are effective against the adults but not the eggs or larvae, treatment is typically repeated in three weeks, and then often in three months as well. Unfortunately, since dogs don't develop resistance to whipworms, it is difficult to prevent them from getting rein-fected if they visit soil contaminated with whipworm eggs.
**ASCARID DANGERS**
The most commonly encountered worms in dogs are roundworms known as ascarids. Toxascaris leonine and Toxocara canis are the two species that infect dogs. Subsisting in the dog's stomach and intestines, adult roundworms can grow to 7 inches in length and adult females can lay in excess of 200,000 eggs in a single day.
In humans, visceral larval migrans affects people who have ingested eggs of Toxocara canis, which frequently contaminates children's sandboxes, beaches and park grounds. The roundworms reside in the human's stomach and intestines, as they would in a dog's, but do not mature. Instead, they find their way to the liver, lungs and skin, or even to the heart or kidneys in severe cases. Deworming puppies is critical in preventing the infection in humans, and young children should never handle nursing pups who have not been dewormed.
**T APEWORMS**
There are many different species of tapeworm that affect dogs, but _Dipylidium caninum_ is probably the most common and is spread by fleas. Flea larvae feed on organic debris and tapeworm eggs in the environment and, when a dog chews at himself and manages to ingest fleas, he might get a dose of tapeworm at the same time. The tapeworm then develops further in the intestine of the dog.
**HOOKED ON _ANCYLOSTOMA_**
Adult dogs can become infected by the bloodsucking nematodes we commonly call hookworms via ingesting larvae from the ground or via the larvae penetrating the dog's skin. It is not uncommon for infected dogs to show no symptoms of hookworm infestation. Sometimes symptoms occur within ten days of exposure. These symptoms can include bloody diarrhea, anemia, loss of weight and general weakness. Dogs pass the hook-worm eggs in their stools, which serves as the vet's method of identifying the infestation. The hookworm larvae can encyst themselves in the dog's tissues and be released when the dog is experiencing stress.
Caused by an _Ancylostoma_ species whose common host is the dog, cutaneous larval migrans affects humans, causing itching and lumps and streaks beneath the surface of the skin.
The tapeworm itself, which is a parasitic flatworm that latches onto the intestinal wall, is composed of numerous segments. When the segments break off into the intestine (as proglottids), they may accumulate around the rectum, like grains of rice. While this tapeworm is disgusting in its behavior, it is not directly communicable to humans (although humans can also get infected by swallowing fleas).
A much more dangerous flat-worm is _Echinococcus multilocularis_ , which is typically found in foxes, coyotes and wolves. The eggs are passed in the feces and infect rodents, and, when dogs eat the rodents, the dogs can be infected by thousands of adult tapeworms. While the parasites don't cause many problems in dogs, this is considered the most lethal worm infection that people can get. Take appropriate precautions if you live in an area in which these tapeworms are found. Do not use mulch that may contain feces of dogs, cats or wildlife, and discourage your pets from hunting wildlife. Treat these tapeworm infections aggressively in pets, because if humans get infected, approximately half die.
**H EARTWORMS**
Heartworm disease is caused by the parasite _Dirofilaria immitis_ and is seen in dogs around the world. A member of the roundworm group, it is spread between dogs by the bite of an infected mosquito. The mosquito injects infective larvae into the dog's skin with its bite, and these larvae develop under the skin for a period of time before making their way to the heart. There they develop into adults, which grow and create blockages of the heart, lungs and major blood vessels there. They also start producing offspring (microfilariae) and these microfilariae circulate in the bloodstream, waiting to hitch a ride when the next mosquito bites. Once in the mosquito, the microfilariae develop into infective larvae and the entire process is repeated.
When dogs get infected with heartworm, over time they tend to develop symptoms associated with heart disease, such as coughing, exercise intolerance and potentially many other manifestations. Diagnosis is confirmed by either seeing the microfilariae themselves in blood samples or using immunologic tests (antigen testing) to identify the presence of adult heart-worms. Since antigen tests measure the presence of adult heartworms and microfilarial tests measure offspring produced by adults, neither are positive until six to seven months after the initial infection. However, the beginning of damage can occur by fifth-stage larvae as early as three months after infection. Thus it is possible for dogs to be harboring problem-causing larvae for up to three months before either type of test would identify an infection.
**WORM-CONTROL GUIDELINES**
• Practice sanitary habits with your dog and home.
• Clean up after your dog and don't let him sniff or eat other dogs' droppings.
• Control insects and fleas in the dog's environment. Fleas, lice, cockroaches, beetles, mice and rats can act as hosts for various worms.
• Prevent dogs from eating uncooked meat, raw poultry and dead animals.
• Keep dogs and children from playing in sand and soil.
• Kennel dogs on cement or gravel; avoid dirt runs.
• Administer heartworm preventives regularly.
• Have your vet examine your dog's stools at your annual visits.
• Select a boarding kennel carefully so as to avoid contamination from other dogs or an unsanitary environment.
• Prevent dogs from roaming. Obey local leash laws.
The good news is that there are great protocols available for preventing heartworm in dogs. Testing is critical in the process, and it is important to understand the benefits as well as the limitations of such testing. All dogs six months of age or older that have not been on continuous heartworm-preventive medication should be screened with microfilarial or antigen tests. For dogs receiving preventive medication, periodic antigen testing helps assess the effectiveness of the preventives. The American Heartworm Society guidelines suggest that annual retesting may not be necessary when owners have absolutely provided continuous heartworm prevention. Retesting on a two- to three-year interval may be sufficient in these cases. However, your veterinarian will likely have specific guidelines under which heartworm preventives will be prescribed, and many prefer to err on the side of safety and retest annually.
It is indeed fortunate that heartworm is relatively easy to prevent, because treatments can be as life-threatening as the disease itself. Treatment requires a two-step process that kills the adult heartworms first and then the microfilariae. Prevention is obviously preferable; this involves a once-monthly oral or topical treatment. The most common oral preventives include ivermectin (not suitable for some breeds), moxidectin and milbemycin oxime; the once-a-month topical drug selamectin provides heartworm protection in addition to flea, tick and other parasite controls.
**Abrasions**
Clean wound with running water or 3% hydrogen peroxide. Pat dry with gauze and spray with antibiotic. Do not cover.
**Animal Bites**
Clean area with soap and saline solution or water. Apply pressure to any bleeding area. Apply antibiotic ointment.
**Antifreeze Poisoning**
Induce vomiting and take dog to the vet.
**Bee Sting**
Remove stinger and apply soothing lotion or cold compress; give antihistamine in proper dosage.
**Bleeding**
Apply pressure directly to wound with gauze or towel for five to ten minutes. If wound does not stop bleeding, wrap wound with gauze and adhesive tape.
**Bloat/Gastric Torsion**
Immediately take the dog to the vet or emergency clinic; phone from car. No time to waste.
**Burns**
**Chemical:** Bathe dog with water and pet shampoo. Rinse in saline solution. Apply antibiotic ointment.
**Acid:** Rinse with water. Apply one part baking soda, two parts water to affected area.
**Alkali:** Rinse with water. Apply one part vinegar, four parts water to affected area.
**Electrical:** Apply antibiotic ointment. Seek veterinary assistance immediately.
**Choking**
If the dog is on the verge of collapsing, wedge a solid object, such as the handle of a screwdriver, between molars on one side of mouth to keep mouth open. Pull tongue out. Use long-nosed pliers or fingers to remove foreign object. Do not push the object down the dog's throat. For small or medium dogs, hold dog upside down by hind legs and shake firmly to dislodge foreign object.
**Chlorine Ingestion**
With clean water, rinse the mouth and eyes. Give dog water to drink; contact the vet.
**Constipation**
Feed dog 2 tablespoons bran flakes with each meal. Encourage drinking water. Mix ¼teaspoon mineral oil in dog's food.
**Diarrhea**
Withhold food for 12 to 24 hours. Feed dog anti-diarrheal with eyedropper. When feeding resumes, feed one part boiled hamburger, one part plain cooked rice, ¼- to ¾-cup four times daily.
**Dog Bite**
Snip away hair around puncture wound; clean with 3% hydrogen peroxide; apply tincture of iodine. If wound appears deep, take the dog to the vet.
**Frostbite**
Wrap the dog in a heavy blanket. Warm affected area with a warm bath for ten minutes. Red color to skin will return with circulation; if tissues are pale after 20 minutes, contact the vet.
**Heat Stroke**
Submerge the dog (up to his muzzle) in cold water; if no response within ten minutes, contact the vet.
**Hot Spots**
Mix 2 packets Domeboro® with 2 cups water. Saturate cloth with mixture and apply to hot spots for 15–30 minutes. Apply antibiotic ointment. Repeat every six to eight hours.
**Poisonous Plants**
Wash affected area with soap and water. Cleanse with alcohol. For foxtail/grass, apply antibiotic ointment.
**Rat Poison Ingestion**
Induce vomiting. Keep dog calm, maintain dog's normal body temperature (use blanket or heating pad). Get to the vet for antidote.
**Shock**
Keep the dog calm and warm; call for veterinary assistance.
**Snake Bite**
If possible, bandage the area and apply pressure. If the area is not conducive to bandaging, use ice to control bleeding. Get immediate help from the vet.
**Tick Removal**
Apply flea and tick spray directly on tick. Wait one minute. Using tweezers or wearing plastic gloves, grasp the tick's body firmly. Apply antibiotic ointment.
**Vomiting**
Restrict dog's water intake; offer a few ice cubes. Withhold food for next meal. Contact vet if vomiting persists longer than 24 hours.
_Use a portable, durable container large enough to contain all items_
**DOG OWNER'S FIRST-AID KIT**
**Gauze bandages/swabs**
**Adhesive and non-adhesive bandages**
**Antibiotic powder**
**Antiseptic wash**
**Hydrogen peroxide 3%**
**Antibiotic ointment**
**Lubricating jelly**
**Rectal thermometer**
**Nylon muzzle**
**Scissors and forceps**
**Eyedropper**
**Syringe**
**Anti-bacterial/fungal solution**
**Saline solution**
**Antihistamine**
**Cotton balls**
**Nail clippers**
**Screwdriver/pen knife**
**Flashlight**
**Emergency phone numbers**
**Number-One Killer Disease in Dogs: CANCER**
In every age, there is a word associated with a disease or plague that causes humans to shudder. In the 21st century, that word is "cancer." Just as cancer is the leading cause of death in humans, it claims nearly half the lives of dogs that die from a natural disease as well as half the dogs that die over the age of ten years.
Described as a genetic disease, cancer becomes a greater risk as the dog ages. Vets and dog owners have become increasingly aware of the threat of cancer to dogs. Statistics reveal that one dog in every five will develop cancer, the most common of which is skin cancer. Many cancers, including prostate, ovarian and breast cancer, can be avoided by spaying and neutering our dogs by the age of six months.
Early detection of cancer can save or extend a dog's life, so it is absolutely vital for owners to have their dogs examined by a qualified vet or oncologist immediately upon detection of any abnormality. Certain dietary guidelines have also proven to reduce the onset and spread of cancer. Foods based on fish rather than beef, due to the presence of Omega-3 fatty acids, are recommended. Other amino acids such as glutamine have significant benefits for canines, particularly those breeds that show a greater susceptibility to cancer.
Cancer management and treatments promise hope for future generations of canines. Since the disease is genetic, breeders should never breed a dog whose parents, grandparents and any related siblings have developed cancer. It is difficult to know whether to exclude an otherwise healthy dog from a breeding program, as the disease does not manifest itself until the dog's senior years.
**RECOGNIZE CANCER WARNING SIGNS**
Since early detection can possibly rescue your dog from becoming a cancer statistic, it is essential for owners to recognize the possible signs and seek the assistance of a qualified professional.
• Abnormal bumps or lumps that continue to grow
• Bleeding or discharge from any body cavity
• Persistent stiffness or lameness
• Recurrent sores or sores that do not heal
• Inappetence
• Breathing difficulties
• Weight loss
• Bad breath or odors
• General malaise and fatigue
• Eating and swallowing problems
• Difficulty urinating and defecating
In general, pure-bred dogs are considered to have achieved senior status when they reach 75% of their breed's average lifespan, with lifespan being based on breed size along with breed-specific factors. Fortunately, the Welsh Terrier, like most of his terrier brethren, is a long-lived dog that can live in good health for 12 to 15 years (or even longer) and is considered a senior at around 8 or 9 years of age.
Obviously, the old "seven dog years to one human year" theory is not exact. In puppyhood, a dog's year is actually comparable to more than seven human years, considering the puppy's rapid growth during his first year. Then, in adulthood, the ratio decreases. Small breeds tend to live longer than larger breeds, and terriers, in general, are a hardy lot. Of course, lifespan varies among individual dogs, with many living longer than expected, which we hope is the case!
By the time your dog has reached his senior years, you will know him very well, so the physical and behavioral changes that accompany aging should be noticeable to you. Humans and dogs share the most obvious physical sign of aging: gray hair! Graying often occurs first on the muzzle and face, around the eyes. Other telltale signs are the dog's overall decrease in activity. Your older dog might be more content to nap and rest, and he may not show the same old enthusiasm when it's time to play in the yard or go for a walk. Other physical signs include significant weight loss or gain; more labored movement; skin and coat problems, possibly hair loss; sight and/or hearing problems; changes in toileting habits, perhaps seeming "unhousebroken" at times; tooth decay, bad breath or other mouth problems.
There are behavioral changes that go along with aging, too. There are numerous causes for behavioral changes. Sometimes a dog's apparent confusion results from a physical change like diminished sight or hearing. If his confusion causes him to be afraid, he may act aggressively or defensively. He may sleep more frequently because his daily walks, though shorter now, tire him out. He may begin to experience separation anxiety or, conversely, become less interested in petting and attention.
There also are clinical conditions that cause behavioral changes in older dogs. One such condition is known as cognitive dysfunction (familiarly known as "old-dog" syndrome). It can be frustrating for an owner whose dog is affected with cognitive dysfunction, as it can result in behavioral changes of all types, most seemingly unexplainable. Common changes include the dog's forgetting aspects of the daily routine, such as times to eat, go out for walks, relieve himself and the like. Along the same lines, you may take your dog out at the regular time for a potty trip and he may have no idea why he is there. Sometimes a placid dog will begin to show aggressive or possessive tendencies or, conversely, a hyperactive dog will start to "mellow out."
Disease also can be the cause of behavioral changes in senior dogs. Hormonal problems (Cushing's disease is common in older dogs), diabetes and thyroid disease can cause increased appetite, which can lead to aggression related to food guarding. It's better to be proactive with your senior dog, making more frequent trips to the vet if necessary and having bloodwork done to test for the diseases that can commonly befall older dogs.
The aforementioned changes are discussed to alert owners to what may happen in a dog's senior years. Many hardy dogs remain active and alert well into old age. However, it can be frustrating and heartbreaking for owners to see their beloved dogs change physically and temperamentally. Just know that it's the same Welsh Terrier under there, and that he still loves you and appreciates your care, which he needs now more than ever.
**ADAPTING TO AGE**
As dogs age and their once-keen senses begin to deteriorate, they can experience stress and confusion. However, dogs are very adaptable, and most can adjust to deficiencies in their sight and hearing. As these processes often deteriorate gradually, the dog makes adjustments gradually, too. Because dogs become so familiar with the layout of their homes and yards, and with their daily routines, they are able to get around even if they cannot see or hear as well. Help your senior dog by keeping things consistent around the house. Keep up with your regular times for walking and potty trips, and do not relocate his crate or rearrange the furniture. Your dog is a very adaptable creature and can make compensation for his diminished ability, but you want to help him along the way and not make changes that will cause him confusion.
Even if he shows no outward signs of aging, your dog should begin a senior-care program once he reaches the determined age. By providing him with extra attention to his veterinary care at this age, you will be practicing good preventive medicine, ensuring that the rest of your dog's life will be as long, active, happy and healthy as possible. If you do notice indications of aging, such as graying and/or changes in sleeping, eating or toileting habits, this is a sign to set up a senior-care visit with your vet right away to make sure that these changes are not related to any health problems.
To start, senior dogs should visit the vet twice yearly for exams, routine tests and overall evaluations. Many veterinarians have special screening programs especially for senior dogs that can include a thorough physical exam; blood test to determine complete blood count; serum biochemistry test, which screens for liver, kidney and blood problems as well as cancer; urinalysis; and dental exams. With these tests, it can be determined whether your dog has any health problems; the results also establish a baseline for your pet against which future test results can be compared.
In addition to these tests, your vet may suggest additional testing, including an EKG, tests for glaucoma and other problems of the eye, chest x-rays, screening for tumors, blood pressure test, test for thyroid function and screening for parasites and reassessment of his preventive program. Your vet also will ask you questions about your dog's diet and activity level, what you feed and the amounts that you feed. This information, along with his evaluation of the dog's overall condition, will enable him to suggest proper dietary changes, if needed.
This may seem like quite a work-up for your pet, but veterinarians advise that older dogs need more frequent attention so that any health problems can be detected as early as possible. Serious conditions like kidney disease, heart disease and cancer may not present outward symptoms, or the problem may go undetected if the symptoms are mistaken by owners as just part of the aging process.
Cognitive dysfunction shares much in common with senility and Alzheimer's disease, and dogs are not immune. Dogs can become confused and/or disoriented, lose their house-training, have abnormal sleep-wake cycles and interact differently with their owners. There is good evidence that continued stimulation in the form of games, play, training and exercise can help to maintain cognitive function. There are also medications (such as seligiline) and antioxidant-fortified senior diets that have been shown to be beneficial.
Cancer is also a condition more common in the elderly. Almost all of the cancers seen in people are also seen in pets. If pets are getting regular physical examinations, cancers are often detected early. There are a variety of cancer therapies available today, and many pets continue to live happy lives with appropriate treatment.
Degenerative joint disease, often referred to as arthritis, is another malady common to both elderly dogs and humans. A lifetime of wear and tear on joints and running around at play eventually takes its toll and results in stiffness and difficulty in getting around. As dogs live longer and healthier lives, it is natural that they should eventually feel some of the effects of aging. Once again, if your Welsh has always received regular veterinary care, he should not have been carrying extra pounds all those years and wearing those joints out before their time. If your pet was unfortunate enough to inherit hip dysplasia, osteochondritis dissecans or any of the other developmental orthopedic diseases, battling the onset of degenerative joint disease was probably a longstanding goal. In any case, there are now many effective remedies for managing degenerative joint disease and a number of remarkable surgeries as well.
**WEATHER WORRIES**
Older pets are less tolerant of extremes in weather, both heat and cold. Your older dog should not spend extended periods in the sun; when outdoors in the warm weather, make sure he does not become overheated. In chilly weather, consider a sweater for your dog when outdoors and limit time spent outside. Whether or not his coat is thinning, he will need provisions to keep him warm when the weather is cold. You may even place his bed by a heating duct in your living room or bedroom.
Aside from the extra veterinary care, there is much you can do at home to keep your older dog in good condition. The dog's diet is an important factor. If your dog's appetite decreases, he will not be getting the nutrients he needs. He also will lose weight, which is unhealthy for a dog at a proper weight. Conversely, an older dog's metabolism is slower and he usually exercises less, but he should not be allowed to become obese. Obesity in an older dog is especially risky, because extra pounds mean extra stress on the body, increasing his vulnerability to heart disease. Additionally, the extra pounds make it harder for the dog to move about. You should discuss age-related feeding changes with your vet.
As for exercise, the senior dog should not be allowed to become a "couch potato" despite his old age. He may not be able to handle the morning run, long walks and vigorous games of fetch, but he still needs to get up and get moving. Keep up with your daily walks, but keep the distances shorter and let your dog set the pace. If he gets to the point where he's not up for walks, let him stroll around the yard. On the other hand, many dogs remain very active in their senior years, so base changes to the exercise program on your own individual dog and what he's capable of. Don't worry, your Welsh Terrier will let you know when it's time to rest.
Keep up with your grooming routine as you always have. Be extra diligent about checking the skin and coat for problems. Older dogs can experience thinning coats as a normal aging process, but they can also lose hair as a result of medical problems. Some thinning is normal, but patches of baldness or the loss of significant amounts of hair is not.
Hopefully, you've been regular with brushing your dog's teeth throughout his life. Healthy teeth directly affect overall good health. We already know that bacteria from gum infections can enter the dog's body through the damaged gums and travel to the organs. At a stage in life when his organs don't function as well as they used to, you don't want anything to put additional strain on them. Clean teeth also contribute to a healthy immune system. Offering the dental-type chews in addition to toothbrushing can help, as they remove plaque and tartar as the dog chews.
Along with the same good care you've given him all of his life, pay a little extra attention to your dog in his senior years and keep up with twice-yearly trips to the vet. The sooner a problem is uncovered, the greater the chances of a full recovery.
**RUBDOWN REMEDY**
A good remedy for an aching dog is to give him a gentle massage each day, or even a few times a day if possible. This can be especially beneficial before your dog gets out of his bed in the morning. Just as in humans, massage can decrease pain in dogs, whether the dog is arthritic or just afflicted by the stiffness that accompanies old age. Gently massage his joints and limbs, as well as petting him on his entire body. This can help his circulation and flexibility and ease any joint or muscle aches. Massaging your dog has benefits for you, too; in fact, just petting our dogs can cause reduced levels of stress and lower our blood pressure. Massage and petting also help you find any previously undetected lumps, bumps or abnormalities. Often these are not visible and only turn up by being felt.
**Ch. Shaireab's On Your Honor, bred and owned by Sharon Abmeyer, is known as "Opie." He won his championship at the Montgomery County show in 2003.**
**AKC CONFORMATION SHOWING**
Is dog showing in your blood? Are you excited by the idea of gaiting your handsome Welsh Terrier around the ring to the thunderous applause of an enthusiastic audience? Are you certain that your beloved Welsh Terrier is flawless? You are not alone! Every loving owner thinks that his dog has no faults, or too few to mention. No matter how many times an owner reads the breed standard, he cannot find any faults in his aristocratic companion dog. If this sounds like you, and if you are considering entering your Welsh Terrier in a dog show, here are some basic questions to ask yourself:
• Did you purchase a "show-quality" puppy from the breeder?
• Is your puppy at least six months of age?
• Does the puppy exhibit correct show type for his breed?
• Does your puppy have any disqualifying faults?
• Is your Welsh Terrier registered with the American Kennel Club?
• How much time do you have to devote to training, grooming, conditioning and exhibiting your dog?
• Do you understand the rules and regulations of a dog show?
• Do you have time to learn how to show your dog properly?
• Do you have the financial resources to invest in showing your dog?
• Will you show the dog yourself or hire a professional handler?
• Do you have a vehicle that can accommodate your weekend trips to the dog shows?
Success in the show ring requires more than a pretty face, a waggy tail and a pocketful of liver. Even though dog shows can be exciting and enjoyable, the sport of conformation makes great demands on the exhibitors and the dogs. Winning exhibitors live for their dogs, devoting time and money to their dogs' presentation, conditioning and training. Very few novices, even those with good dogs, will find themselves in the winners' circle, though it does happen. Don't be disheartened, though. Every exhibitor began as a novice and worked his way up to the Group ring. It's the "working your way up" part that you must keep in mind.
**Here she is world: it's "Rose"! The number-one Welshie for 2003 and 2004, Ch. Tothwood's American Beauty, owned by breeder Arthur Toth, Jr., Dr. Isaac Wood and Marge McClung.**
Visiting a dog show as a spectator is a great place to start. Pick up the show catalog to find out what time your breed is being shown, who is judging the breed and in which ring the classes will be held. To start, Welsh Terriers compete against other Welsh Terriers, and the winner is selected as Best of Breed by the judge. This is the procedure for each breed. At a group show, all of the Best of Breed winners go on to compete for Group One in their respective groups. For example, all Best of Breed winners in a given group compete against each other; this is done for all seven groups. Finally, all seven group winners go head to head in the ring for the Best in Show award.
What most spectators don't understand is the basic idea of conformation. A dog show is often referred as a "conformation" show. This means that the judge should decide how each dog stacks up (conforms) to the breed standard for his given breed: how well does this Welsh Terrier conform to the ideal representative detailed in the standard? Ideally, this is what happens. In reality, however, this ideal often gets slighted as the judge compares Welsh Terrier #1 to Welsh Terrier #2. Again, the ideal is that each dog is judged based on his merits in comparison to his breed standard, not in comparison to the other dogs in the ring. It is easier for judges to compare dogs of the same breed to decide which they think is the better specimen; in the Group and Best in Show ring, however, it is very difficult to compare one breed to another, like apples to oranges. Thus the dog's conformation to the breed standard—not to mention advertising dollars and good handling—is essential to success in conformation shows. The dog described in the standard (the standard for each AKC breed is written and approved by the breed's national parent club and then submitted to the AKC for approval) is the perfect dog of that breed, and breeders keep their eye on the standard when they choose which dogs to breed, hoping to get closer and closer to the ideal with each litter.
Another good first step for the novice is to join a dog club. You will be astonished by the many and different kinds of dog clubs in the country, with about 5,000 clubs holding events every year. Dog clubs may specialize in a single breed, like a local or regional Welsh Terrier club, or in a specific pursuit, such as obedience, tracking or hunting tests. There are all-breed clubs for all dog enthusiasts; they sponsor special training days, seminars on topics like grooming or handling or lectures on breeding or canine genetics. There are also clubs that specialize in certain types of dogs, like terriers, herding dogs, companion dogs, etc.
**FOR MORE INFORMATION....**
For reliable up-to-date information about registration, dog shows and other canine competitions, contact one of the national registries by mail or via the Internet.
American Kennel Club
5580 Centerview Dr., Raleigh, NC 27606-3390
www.akc.org
United Kennel Club
100 E. Kilgore Road, Kalamazoo, MI 49002
www.ukcdogs.com
Canadian Kennel Club
89 Skyway Ave., Suite 100, Etobicoke, Ontario M9W 6R4 Canada
www.ckc.ca
The Kennel Club
1-5 Clarges St., Piccadilly, London W1Y 8AB, UK
www.the-kennel-club.org.uk
A parent club is the national organization, sanctioned by the AKC, which promotes and safeguards its breed in the country. The Welsh Terrier Club of America was formed in 1900 and can be contacted on the Internet at <http://clubs.akc.org/wtca>. The parent club holds an annual national specialty show, usually in a different city each year, in which many of the country's top dogs, handlers and breeders gather to compete. At a specialty show, only members of a single breed are invited to participate. There are also group specialties, in which all members of a group are invited. For more information about dog clubs in your area, contact the AKC at www.akc.org on the Internet or write them at their Raleigh, NC address.
**OBEDIENCE TRIALS**
Mrs. Helen Whitehouse Walker, a Standard Poodle fancier, can be credited with introducing obedience trials to the United States. In the 1930s, she designed a series of exercises based on those of the Associated Sheep, Police, Army Dog Society of Great Britain. These exercises were intended to evaluate the working relationship between dog and owner. Since those early days of the sport in the US, obedience trials have grown more and more popular, and now more than 2,000 trials each year attract over 100,000 dogs and their owners. Any dog registered with the AKC, regardless of neutering or other disqualifications that would preclude entry in conformation competition, can participate in obedience trials.
There are three levels of difficulty in obedience competition. The first (and easiest) level is the Novice, in which dogs can earn the Companion Dog (CD) title. The intermediate level is the Open level, in which the Companion Dog Excellent (CDX) title is awarded. The advanced level is the Utility level, in which dogs compete for the Utility Dog (UD) title. Classes at each level are further divided into "A" and "B," with "A" for beginners and "B" for those with more experience. In order to win a title at a given level, a dog must earn three "legs." A "leg" is accomplished when a dog scores 170 or higher (200 is a perfect score). The scoring system gets a little trickier when you understand that a dog must score more than 50% of the points available for each exercise in order to actually earn the points. Available points for each exercise range between 20 and 40.
**A tunnel is hardly an obstacle for this athletic Welsh Terrier in an agility trial.**
Once he's earned the UD title, a dog can go on to win the prestigious title of Utility Dog Excellent (UDX) by winning "legs" in ten shows. Additionally, Utility Dogs who win "legs" in Open B and Utility B earn points toward the lofty title of Obedience Trial Champion (OTCh.). Established in 1977 by the AKC, this title requires a dog to earn 100 points as well as three first places in a combination of Open B and Utility B classes under three different judges. The "brass ring" of obedience competition is the AKC's National Obedience Invitational. This is an exclusive competition for only the cream of the obedience crop. In order to qualify for the invitational, a dog must be ranked in either the top 25 all-breeds in obedience or in the top three for his breed in obedience. The title at stake here is that of National Obedience Champion (NOC).
**Flying high! This agile Welsh clears a jump gracefully with room to spare.**
**AGILITY TRIALS**
Agility trials became sanctioned by the AKC in August 1994, when the first licensed agility trials were held. Since that time, agility certainly has grown in popularity by leaps and bounds, literally! The AKC allows all registered breeds (including Miscellaneous Class breeds) to participate, providing the dog is 12 months of age or older. Agility is designed so that the handler demonstrates how well the dog can work at his side. The handler directs his dog through, over, under and around an obstacle course that includes jumps, tires, the dog walk, weave poles, pipe tunnels, collapsed tunnels and more. While working his way through the course, the dog must keep one eye and ear on the handler and the rest of his body on the course. The handler runs along with the dog, giving verbal and hand signals to guide the dog through the course.
The first organization to promote agility trials in the US was the United States Dog Agility Association, Inc. (USDAA). Established in 1986, the USDAA sparked the formation of many member clubs around the country. To participate in USDAA trials, dogs must be at least 18 months of age. The USDAA and AKC both offer titles to winning dogs, although the exercises and requirements of the two organizations differ.
Agility trials are a great way to keep your dog active, and they will keep you running, too! You should join a local agility club to learn more about the sport. These clubs offer sessions in which you can introduce your dog to the various obstacles as well as training classes to prepare him for competition. In no time, your dog will be climbing A-frames, crossing the dog walk and flying over hurdles, all with you right beside him. Your heart will leap every time your dog jumps through the hoop—and you'll be having just as much (if not more) fun!
**TRACKING**
Tracking tests are exciting ways to test your Welsh Terrier's instinctive scenting ability on a competitive level. All dogs have noses, and all breeds are welcome in tracking tests. The first AKC-licensed tracking test took place in 1937 as part of the Utility level at an obedience trial, and thus competitive tracking was officially begun. The first title, Tracking Dog (TD), was offered in 1947, ten years after the first official tracking test. It was not until 1980 that the AKC added the title Tracking Dog Excellent (TDX), which was followed by the title Versatile Surface Tracking (VST) in 1995. Champion Tracker (CT) is awarded to a dog who has earned all three of those titles.
**Ch. Rubicon's Sugar Bear, ME, CD, OA, OAJ, CG, CGC, TDI, hunting vermin in a woodpile. Owned by Hiroshi Saito.**
**EARTHDOG EVENTS**
Earthdog tests are held for breeds that were developed to "go to ground" into badger, fox, woodchuck and gopher holes and bring out the quarry. The Welsh and other terriers of like size, or smaller, as well as Dachshunds are eligible to participate in the AKC and AWTA (American Working Terrier Association) trials. These trials test the ability of the dogs to follow the scent underground right up to the quarry. A trench, 9 inches square, is dug and a wooden liner is used to cover the sides and top of the tunnel, which in turn is covered with earth so only the entrance is visible to the dog. The earth floor is rat-scented for the dog to follow up to the caged rat. The dog must then show he can "work" the quarry by digging, growling or otherwise trying to get the rat, which in fact he cannot touch or harm in any way due to the protective cage.
There are four levels in AKC earthdog trials. The first, Introduction to Quarry, is for beginners and uses a 10-foot tunnel. No title is awarded at this level. The Junior Earthdog (JE) title is awarded at the next level, which uses a 30-foot tunnel with three 90-degree turns. Two qualifying JE runs are required for a dog to earn the title. The next level, Senior Earthdog (SE), uses the same length tunnel and number of turns as in the JE level, but also has a false den and exit and requires the dog to come out of the tunnel when called. To try for the SE title, a dog must have at least his JE; the SE title requires three qualifying runs at this level. The most difficult of the earthdog tests, Master Earthdog (ME), again uses the 30-foot tunnel with three 90-degree turns, with a false entrance, exit and den. The dog is required to enter in the right place and, in this test, honor another working dog. The ME title requires four qualifying runs, and a dog must have earned his SE title to attempt the ME level. The first terrier to earn a Senior Earthdog title was a Welsh.
**NATURAL HUNTS**
Completely natural hunting is not only instinctive to the Welsh but also allows the breed to use all its working terrier capabilities to hunt by sight, scent, sound–and digging. In the field, while following scent, Welsh Terriers can often be seen to rise up on their hind legs, or even jump up, to sight prey or to flush it. Going to ground to follow the rabbit, fox, woodchuck or other game is the sequence of the hunt. The terrier follows the prey into the earth and either engages it while the accompanying people can dig it out, kills it or flushes it out of the den–at which point, the chase continues.
Welsh Terrier owners continue to emphasize their dogs' natural instincts by honing them through hunts and trials. Farmers are grateful for these hard working terriers that help rid their fields of destructive vermin.
**CANINE GOOD CITIZEN ® PROGRAM**
Have you ever considered getting your dog "certified"? The AKC's Canine Good Citizen® Program affords your dog just that opportunity. Your dog shows that he is a well-behaved canine citizen, using the basic training and good manners you have taught him, by taking a series of ten tests that illustrate that he can behave properly at home, in a public place and around other dogs. The tests are administered by participating dog clubs, colleges, 4-H clubs, Scouts and other community groups and are open to all pure-bred and mixed-breed dogs. Upon passing the ten tests, the suffix CGC is then applied to your dog's name.
The ten tests are: 1. Accepting a friendly stranger; 2. Sitting politely for petting; 3. Appearance and grooming; 4. Walking on a lead; 5. Walking through a group of people; 6. Sit, down and stay on command; 7. Coming when called; 8. Meeting another dog; 9. Calm reaction to distractions; 10. Separation from owner.
The AWTA is the precursor of the AKC earthdog events and was started in 1972 by Patricia Lent. The AWTA rules do not extend to the degree of difficulty of those offered by the AKC, but their trials are held nationwide. There is also an emphasis on natural hunting in which many members and their dogs participate. Welsh Terriers have proven their ability in both versions of earthdog trials and in natural hunts.
**Here's an accomplished Welsh entering the tunnel at an earthdog trial. Owner, Hiroshi Saito.**
**UNDERSTANDING THE CANINE MINDSET**
For starters, you and your dog are on different wavelengths. Your dog is similar to a toddler in that both live in the present tense only. A dog's view of life is based primarily on cause and effect, which is similar to the old saying, "Nothing teaches a youngster to hang on like falling off the swing." If your dog stumbles down a flight of three steps, the next time he hopefully will be more careful, or he may just avoid the steps altogether.
Your Welsh makes connections based on the fact that he lives in the present, so when he is doing something and you interrupt to dispense praise or a correction, a connection, positive or negative, is made. To the dog, that's like one plus one equals two! In the same sense, it's also easy to see that when your timing is off, you will cause an incorrect connection. The one-plus-one way of thinking is why you must never scold a dog for behavior that took place an hour, 15 minutes or even 5 seconds ago. But it is also why, when your timing is perfect, you can teach him to do all kinds of wonderful things—as soon as he has made that essential connection. What helps the process is his desire to please you and to have your approval.
**THE TOP-DOG TUG**
When puppies play tug-of-war, the dominant pup wins. Children also play this kind of game but, for their own safety, must be prevented from ever engaging in this type of play with their dogs. Playing tug-of-war games can result in a dog's developing aggressive behavior. Don't be the cause of such behavior.
There are behaviors we admire in dogs, such as friendliness and obedience, as well as those behaviors that cause problems to a varying degree. The dog owner who encounters minor behavioral problems is wise to solve them promptly or get professional help. Bad behaviors are not corrected by repeatedly shouting "No" or getting angry with the dog. Only the giving of praise and approval for good behavior lets your dog understand right from wrong. The longer a bad behavior is allowed to continue, the harder it is to overcome. A responsible breeder is often able to help. Each dog is unique, so try not to compare your dog's behavior with your neighbor's dog or the one you had as a child.
Many things, such as environment and inherited traits, form the basic behavior of a dog, just as in humans. You also must factor into his temperament the purpose for which your dog was originally bred. The major obstacle lies in the dog's inability to explain his behavior to us in a way that we understand. The one thing you should not do is to give up and abandon your dog. Somewhere a misunderstanding has occurred, but, with help and patient understanding on your part, you should be able to work out the majority of bothersome behaviors.
**GET A WHIFF OF HIM!**
Dogs sniff each others' rears as their way of saying "hi" as well as to find out who the other dog is and how he's doing. That's normal behavior between canines, but it can, annoyingly, extend to people. The command for all unwanted sniffing is "Leave it!" Give the command in a no-nonsense voice and move on.
**AGGRESSION**
"Aggression" is a word that is often misunderstood and is sometimes even used to describe what is actually normal canine behavior. For example, it's normal for puppies to growl when playing tug-of-war. It's puppy talk. There are different forms of dog aggression, but all are degrees of dominance, indicating that the dog, not his master, is (or thinks he is) in control. When the dog feels that he (or his control of the situation) is threatened, he will respond. The extent of the aggressive behavior varies with individual dogs. It is not at all pleasant to see bared teeth or to hear your dog growl or snarl, but these are signs of behavior that, if left uncorrected, can become extremely dangerous. A word of warning here: never challenge an aggressive dog. He is unpredictable and therefore unreliable to approach.
Nothing gets a "hello" from strangers on the street quicker than walking a puppy, but people should ask permission before petting your dog so you can tell him to sit in order to receive the admiring pats. If a hand comes down over the dog's head and he shrinks back, ask the person to bring their hand up, underneath the pup's chin. Now you're correcting strangers, too! But if you don't, it could make your dog afraid of strangers, which in turn can lead to fear-biting. Socialization prevents much aggression before it rears its ugly head.
The body language of an aggressive dog about to attack is clear. The dog will have a hard, steady stare. He will try to look as big as possible by standing stiff-legged, pushing out his chest, keeping his ears up and holding his tail up and steady. The hackles on his back will rise so that a ridge of hairs stands up. This posture may include the curled lip, snarl and/or growl, or he may be silent. He looks, and definitely is, very dangerous.
Uncontrolled aggression, sometimes called "irritable aggression," is not something for the pet owner to try to solve. If you cannot solve your dog's dangerous behavior with professional help, and you (quite rightly) do not wish to keep a canine time-bomb in your home, you will have some important decisions to make. Aggressive dogs often cannot be rehomed successfully, as they are dangerous and unreliable in their behavior. An aggressive dog should be dealt with only by someone who knows exactly the situation that he is getting into and has the experience, dedication and ideal living environment to attempt rehabilitating the dog, which often is not possible. In these cases, the dog ends up having to be humanely put down. Making a decision about euthanasia is not an easy undertaking for anyone, for any reason, but you cannot pass on to another home a dog that you know could cause harm.
**FEAR BITING**
The remedy for the fear biter is in the hands of a professional trainer or behaviorist. This is not a behavior that the average pet owner should attempt to correct. However, there are things you should not do. Don't sympathize with him, don't pet him and don't, whatever you do, pick him up—you could be bitten in the process, which is even scarier if you bring him up near your face.
A milder form of aggression is the dog's guarding anything that he perceives to be his—his food dish, his toys, his bed and/or his crate. This can be prevented if you take firm control from the start. The young puppy can and should be taught that his leader will share, but that certain rules apply. Guarding is mild aggression only in the beginning stages, and it will worsen and become dangerous if you let it.
Don't try to snatch anything away from your puppy. Bargain for the item in question so that you can positively reinforce him when he gives it up. Punishment only results in worsening any aggressive behavior.
We've mentioned that many Welsh Terriers extend their guarding impulse toward items they've stolen. The dog figures, "If I have it, it's mine!" (Some ill-behaved kids have similar tendencies.) An angry confrontation will only increase the dog's aggression. (Have you ever watched a child have a tantrum?) Try a simple distraction first, such as tossing a toy or picking up his leash for a walk. If that doesn't work, the best way to handle the situation is with basic obedience. Show the dog a treat, followed by calm, almost slow-motion commands: "Come. Sit. Drop it. Good dog," and then hand over the cheese! That's one example of positive-reinforcement training.
Children can be bitten when they try to retrieve a stolen shoe or toy, so they need to know how to handle the dog or to let an adult do it. They may also be bitten as they run away from a dog, in either fear or play. The dog sees the child's running as reason for pursuit, and even a friendly young puppy will nip at the heels of a runaway. Teach the kids not to run away from a strange dog and when to stop overly exciting play with their own puppy.
**Properly socialized dogs should show neither fear nor aggression toward other dogs that they meet. This Welsh Terrier and German Shepherd dog are pleased to make each other's acquaintance.**
**SEPARATION ANXIETY**
Any behaviorist will tell you that separation anxiety is the most common problem about which pet owners complain. It is also one of the easiest to prevent. Unfortunately, a behaviorist usually is not consulted until the dog is a stressed-out, neurotic mess. At that stage, it is indeed a problem that requires the help of a professional.
Training the puppy to the fact that people in the house come and go is essential in order to avoid this anxiety. Leaving the puppy in his crate or a confined area while family members go in and out, and stay out for longer and longer periods of time, is the basic way to desensitize the pup to the family's frequent departures. If you are at home most of every day, make it a point to go out for at least an hour or two whenever possible.
How you leave is vital to the dog's reaction. Your dog is no fool. He knows the difference between sweats and business suits, jeans and dresses. He sees you pat your pocket to check for your wallet, open your briefcase, check that you have your cell phone or pick up the car keys. He knows from the hurry of the kids in the morning that they're off to school until afternoon. Lipstick? Aftershave lotion? Lunch boxes? Every move you make registers in his sensory perception and memory. Your puppy knows more about your departures than the FBI. You can't get away with a thing!
**JEALOUS PETS**
In households with more than one pet, one pet must be dominant. This means that one pet gets more attention, sits closest to you, goes out the door first, takes up more room on the bed and in hundreds of other tiny ways exerts his dominance. The pets will occasionally squabble over your unintended partiality, but it's best not to interfere.
Before you got dressed, you checked the dog's water bowl, made sure he has a supply of safe toys and turned the radio on low. You will leave him in what he considers his "safe" area, not with total freedom of the house. If you've invested in pet gates, you can be reasonably sure that he'll remain in the designated area. Don't give him access to a window where he can watch you leave the house. If you're leaving for an hour or two, just put him into his crate with a safe toy.
Now comes the test! You are ready to walk out the door. Do not give your Welsh Terrier a big hug and a fond farewell. Do not drag out a long goodbye. Those are the very things that jump-start separation anxiety. Toss a biscuit into the dog's area, call out "So long, pooch" and close the door. You're gone. The chances are that the dog may bark a couple of times, or maybe whine once or twice, and then settle down to enjoy his biscuit and take a lovely nap, especially if you took him for a nice long walk after breakfast. As he grows up, the barks and whines will stop because it's an old routine, so why should he make the effort?
When you first brought home the puppy, the come-and-go routine was intermittent and constant. He was put into his crate with a tiny treat. You left (silently) and returned in 3 minutes, then 5, then 10, then 15, then half an hour, until finally you could leave without a problem and be gone for 2 or 3 hours. If, at any time in the future, there's a "separation" problem, refresh his memory by going back to that basic training.
**CHASE INSTINCT**
Chasing small animals is in the blood of many dogs, perhaps most, and definitely terriers. They think that this is a fun recreational activity (although some are more likely to bring you an undesirable "gift" as a result of the hunt). The good old "Leave it" command works to deter your dog from taking off in pursuit of "prey," but only if taught with the dog on leash for control.
Chasing cars or bikes is dangerous for all parties concerned: dogs, drivers and cyclists. Something about those wheels going around fascinates dogs, but that fascination can end in disastrous results. Corrections for your dog's chasing behavior must be immediate and firm. Tell him "Leave it!" and then give him either a sit or a down command. Get kids on bikes to help saturate your dog with spinning wheels while he politely practices his sits and downs.
**"Are you home yet?" A dog left home alone all day will often anxiously await his owner's return.**
Now comes the next most important part—your return. Do not make a big production of coming home. "Hi, poochie" is as grand a greeting as he needs. When you've taken off your hat and coat, tossed your briefcase on the hall table and glanced at the mail, and the dog has settled down from the excitement of seeing you "in person" from his confined area, then go and give him a warm, friendly greeting. A potty trip is needed and a walk would be appreciated, since he's been such a good dog.
**CHEWING**
All puppies chew. All dogs chew. Terriers are built to chew! This is a fact of life for canines, and sometimes you may think it's what your dog does best! A pup starts chewing when his first set of teeth erupts and continues throughout the teething period. Chewing gives the pup relief from itchy gums and incoming teeth and, from that time on, he gets great satisfaction out of this normal, somewhat idle, canine activity. Providing safe chew toys is the best way to direct this behavior in an appropriate manner. Chew toys are available in all sizes, textures and flavors, but you must monitor the wear-and-tear inflicted on your pup's toys to be sure that the ones you've chosen are safe and remain in good condition.
**Always keep an eye on what your pup puts in his mouth. This pup has chosen a stick over his array of plush toys, which could prove dangerous.**
Puppies cannot distinguish between a rawhide toy and a nice leather shoe or wallet. It's up to you to keep your possessions away from the dog and to keep your eye on the dog. There's a form of destruction caused by chewing that is not the dog's fault. Let's say you allow him on the sofa. One day he takes a rawhide bone up on the sofa and, in the course of chewing on the bone, takes up a bit of fabric. He continues to chew. Disaster! Now you've learned the lesson: dogs with chew toys have to be either kept off furniture and carpets, carefully supervised or put into their confined areas for chew time.
The wooden legs of furniture are favorite objects for chewing. The first time, tell the dog "Leave it!" (or "No!") and offer him a chew toy as a substitute. But your clever dog may be hiding under the chair and doing some silent destruction, which you may not notice until it's too late. In this case, it's time to try one of the foul-tasting products, made specifically to prevent destructive chewing, that is sprayed on the objects of your dog's chewing attention. These products also work to keep the dog away from plants, trash, etc. It's even a good way to stop the dog from "mouthing" or chewing on your hands or the leg of your pants. (Be sure to wash your hands after the mouthing lesson!) A little spray goes a long way.
**THE MACHO DOG**
The Venus/Mars differences are found in dogs, too. Males have distinct behaviors that, while seemingly sex-related, are more closely connected to the role of the male as leader. Marking territory by urinating on it is one means that male dogs use to establish their presence. Doing so merely says, "I've been here." Small dogs often attempt to lift their legs higher on the tree than the previous male. While this is natural behavior outdoors on items like telephone poles, fence posts, fire hydrants and most other upright objects, marking indoors is totally unacceptable. Treat it as you would a house-training accident and clean thoroughly to eradicate the scent. Another behavior often seen in the macho male, mounting is a dominance display. Neutering the dog before six months of age helps to deter this behavior. You can discourage him from mounting by catching the dog as he's about to mount you, stepping quickly aside and saying "Off!"
**DIGGING**
Digging, which is seen as a destructive behavior to humans, is actually quite a natural behavior in dogs. If ever a dog was meant to dig, the Welsh Terrier was bred to do the job. However, he is not an offender unless moles or voles live in your yard, in which case the Welsh may actually prove beneficial to your lawn and thus have your forgiveness for any divots incurred in the process.
As earthdogs, Welsh Terriers are not the indiscriminate diggers one might expect. Some individuals will go at it like Welsh miners, but by and large they save their digging prowess purely for going after vermin. Although terriers are most associated with digging, any dog's desire to dig can be irrepressible and most frustrating to his owners.
**FOUR ON THE FLOOR**
You must discourage your dog from jumping up to get attention or for any other reason. To do so, turn away from the dog as he attempts to jump up. Do not bump him in the chest, as this can cause injury to the dog. "Four on the floor" requires praise. Once the dog sits on command, prevent him from attempting to jump again by asking him to sit/stay before petting him. Back away if he breaks the sit.
Domesticated dogs also dig to escape, and that's a lot more dangerous than it is destructive. A dog that digs under the fence is the one that is hit by a car or becomes lost. A good fence to protect a digger should be set at least 12 inches below ground level, and every fence needs to be routinely checked for even the smallest openings that can become possible escape routes.
Catching your dog in the act of digging is the easiest way to stop it, because your dog will make the "one-plus-one" connection, but digging is too often a solitary occupation, something the lonely dog does out of boredom. Catch your young puppy in the act and put a stop to it before you have a yard full of craters. It is more difficult to stop if your dog sees you gardening. If you can dig, why can't he? Because you say so, that's why! Some dogs are excavation experts, and some dogs never dig. However, when it comes to any of these instinctive canine behaviors, never say "never."
**BARKING**
Here's a big, noisy problem! Telling a dog he must never bark is like telling a child not to speak! Consider how confusing it must be to your dog that you are using your voice (which is your form of barking) to teach him when to bark and when not to! That is precisely the reason not to "bark back" when the dog's barking is annoying you (or your neighbors). Try to understand the scenario from the dog's viewpoint. He barks. You bark. He barks again, you bark again. This "conversation"can go on forever!
**You won't believe where your Welsh will stick his nose in search of a tasty reward. Stop a potential thief by removing temptation–don't leave food within your Welsh's reach.**
**STOP, THIEF!**
The easiest way to prevent a dog from stealing food is to stop this behavior before it starts by never leaving food out where he can reach it. However, if it is too late and your dog has already made a steal, you must stop your furry felon from becoming a repeat offender. Once Sneaky Pete has successfully stolen food, place a bit of food where he can reach it. Place an empty soda can with some pebbles in it on top of the food. Leave the room and watch what happens. As the dog grabs the tasty morsel, the can comes with it. The noise of the tumbling pebble-filled can makes its own correction, and you don't have to say a word.
The first time your adorable little puppy said "Yip" or "Yap, you were ecstatic. His first word! You smiled, you told him how smart he was—and you allowed him to do it. So there's that one-plus-one thing again, because he will understand by your happy reaction that "Mr. Alpha loves it when I talk." Ignore his barking in the beginning, and allow it, but don't encourage barking during play. Instead, use the "put a toy in it" method to tone it down. Add a very soft "Quiet" as you hand off the toy. If the barking continues, stand up straight, fold your arms and turn your back on the dog. If he barks, you won't play, and you should follow the same rule for all undesirable behavior during play.
Dogs bark in reaction to sounds and sights. Another dog's bark, a person passing by or even just rustling leaves can set off a barker. If someone coming up your driveway or to your door provokes a barking frenzy, use the saturation method to stop it. Have several friends come and go every three or four minutes over as long a period of time as they can spare (it could take a couple of hours). Attach about a foot of rope to the dog's collar and have very small treats handy. Each time a car pulls up or a person approaches, let the dog bark once (grab the rope if you need to physically restrain him), say "Okay, good dog," give him a treat and make him sit. "Okay" is the release command. It lets the dog know that he has alerted you and tells him that you are now in charge. That person leaves and the next arrives, and so on and so on until everyone—especially the dog—is bored and the barking has stopped. Don't forget to thank your friends. Your neighbors, by the way, may be more than willing to assist you in this parlor game.
Excessive barking outdoors is more difficult to keep in check because, when it happens, he is outside and you are probably inside. A few warning barks are fine, but use the same method to tell him when enough is enough. You will have to stay outside with him for that bit of training.
**"What's for lunch?" Has your Welsh taken a place at the head of the table? If so, you need to stand your ground in enforcing the "no-begging" laws.**
There is one more kind of vocalizing which is called "idiot barking" (from _idiopathic_ , meaning "of unknown cause"). It is usually rhythmic or a timed series of barks. Put a stop to it immediately by calling the dog to come. This form of barking can drive neighbors crazy and commonly occurs when a dog is left outside at night or for long periods of time during the day. He is completely and thoroughly bored! A change of scenery may help, such as relocating him to a room indoors when he is used to being outside. A few new toys or different dog biscuits might be the solution. If he is left alone and no one can get home during the day, a noontime walk with a local dog-sitter would be the perfect solution.
| {
"redpajama_set_name": "RedPajamaBook"
} | 248 |
\section{INTRODUCTION}
The system of $N$ identical fermions interacting resonantly with a
distinguishable atom exhibits a rich and interesting physics, including
universal phenomena and the celebrated Efimov physics.
For a recent review see, e.g., Ref.~\cite{NaiEnd16}.
An important parameter here is the ratio of the impurity mass $m$ and the
identical fermions mass $M$.
In the ultracold limit the interaction between identical fermions can
be neglected, and therefore in the heavy impurity case $m \gg M$ the problem
is decoupled to $N$ independent fermions interacting with a static impurity.
The opposite limit, where $m \ll M$, corresponds to a
dynamic impurity which induces interaction between the identical fermions.
The simplest non trivial example is the $(2+1)$ system, composed of two
identical fermions of mass $M$ and a distinguishable atom of mass $m$,
where different particles have zero-range resonant interaction
while identical particles do not interact.
Efimov has shown that when the mass ratio $\alpha=M/m$ is larger than the
critical value $\alpha_c=13.607$, an infinite tower of trimers with
angular momentum and parity $L^\pi=1^-$ is produced \cite{Efimov1973}.
The $n$-th trimer energy is $E_n=E_0e^{-2\pi n/|s|}$, where $E_0$ is the
trimer ground-state energy. The scale factor $s=s(\alpha)$
is a function of the mass ratio and vanishes at the Efimov threshold
$s(\alpha_c)=0$.
In the non-Efimovian regime $\alpha<\alpha_c$ the scale
factor characterizes the short-distance (and large momenta)
behavior of a universal trimer, which exists for $8.173<\alpha<\alpha_c$
for finite positive scattering length \cite{KarMal07}.
The physical interpretation of the scale factor can be understood
from the adiabatic hyperspherical formalism \cite{Mac68}.
To see that, one rearranges the relative coordinates into the hyperradius
$\rho$, the only coordinate with a dimension of length,
and $3N-1$ hyperangles.
Here $\rho \propto \sqrt{m r^2+M\sum_{i=1}^{N}R_i^2}$,
where ${\bf r}$ (${\bf R_i}$) is the position of the
distinguishable (identical) atom in the center-of-mass frame.
At small $\rho$, where $E$ and $1/a$ can be neglected, the hyperradial motion
separates from hyperangular degrees of freedom and is governed by
\begin{equation}\label{Schr}
\left[-\frac{\partial^2}{\partial \rho^2}-\frac{3N-1}{\rho}
\frac{\partial}{\partial \rho}+\frac{s^2-(3N/2-1)^2}{\rho^2}\right]
\Psi(\rho)=0,
\end{equation}
where $s^2$ is the hyperangular eigenvalue.
The general solution of Eq.~(\ref{Schr}) is a linear combination of
$\Psi_+(\rho)\propto \rho^{-3N/2+1+s}$ and $\Psi_-(\rho)\propto \rho^{-3N/2+1-s}$.
The case $s^2<0$ ($s=is_0$) corresponds to the Efimovian regime, where this
linear combination is an oscillating function, and a three-body parameter
is required to fix the relative phase of $\Psi_+$ and $\Psi_-$.
The non-Efimovian regime appears for $s^2>0$ ($s>0$) where,
far from few-body resonances, $\Psi(\rho)$ is dominated by $\Psi_+(\rho)$.
Interestingly, the same factor determines the energy of the trapped
system at unitarity \cite {Tan04,WerCas06}, namely,
\begin{equation}\label{trap}
E=\hbar \omega (s+2n+1),
\end{equation}
where $\omega$ is the trapping frequency, taken to be identical for all
particles, $n$ is a non-negative integer and the center-of-mass zero-point
energy is omitted.
This is because the trapping potential is involved only in the hyperradial
equation, while $s$ is determined by the hyperangular equation which
is identical in free space and in a trap.
For a recent review of the trapped few-body problem, see Ref.~\cite{Blu12}.
Following Efimov, the mass-imbalanced (2+1) system has attracted wide attention
(see, e.g., Refs.~\cite{Efimov1973,KarMal07,Pet03,Fon79,PetSalShl04,NisSonTan08,
LevTieWal09, RitMehGre10,MatParHus11,HelHam11,EndNaiUed11,LevPet11,CasTig11,
Safavi2013,KarMal16,EndCas16}).
The scale factor of the (2+1) system was first calculated for the equal-mass
case to be $s(1) = 1.7727$ for the $1^-$ ground state and
$s(1) = 2.1662$ for the $0^+$ excited state \cite{PetSalShl04}.
Later, the method was generalized to include any angular momentum and
mass ratio \cite{RitMehGre10}.
The $1^-$ trimer energy crosses the dimer+atom energy in a trap at
$\alpha=8.6186$ \cite{Pet03}.
An ultracold mixture of $^6$Li and $^{40}$K
($\alpha \approx 6.4$) was realized experimentally, and
a strong atom-dimer attraction was observed.
This attraction was interpreted as $p$-wave interaction
between two heavy particles induced by the light atom \cite{Rudi14}.
The trend of moving from a non-Efimovian universal state to an Efimovian state
with the same symmetry as the mass ratio increases was discovered also in the
$(3+1)$ and $(4+1)$ systems \cite{CasMorPri10,Blu12b,BazPet17}.
The mass-imbalanced $(3+1)$ system has been the subject of a few studies
\cite{CasMorPri10,Blu12b,BazPet17,BluDai10,EndCas16}.
Here a tower of $1^+$ Efimovian tetramers exists above $\alpha_c=13.384$
\cite{CasMorPri10}, and a universal non-Efimovian $1^+$ tetramer is bound in
free space for $8.862<\alpha<\alpha_c$ \cite{Blu12b,BazPet17}.
The scale factor of the tetramer ground state has been calculated for a few
mass ratios \cite{BluDai10}, while that of excited states is known only for
the equal-mass case \cite{RakDaiBlu12}.
The tetramer energy crosses the trimer+atom energy in a trap at
$\alpha=8.918$ \cite{BazPet17}.
The mass-imbalanced $(4+1)$ system was studied in
Refs.~\cite{BluDai10,BazPet17}.
A tower of $0^-$ Efimovian pentamers exists above $\alpha_c=13.279$,
while a universal $0^-$ pentamer is bound in free space
for $9.672<\alpha<\alpha_c$ \cite{BazPet17}.
Here the scale factor is known for
equal mass \cite{RakDaiBlu12}, when the pentamer is bound in free space
\cite{BazPet17} and for few other mass ratios \cite{BluDai10}.
The pentamer energy crosses the tetramer+atom energy in a trap at $\alpha=9.41$
\cite{BazPet17}.
The ground-state properties of the $(N+1)$ systems are summarized in
Table~\ref{tbl:Thresholds}.
\begin{table}
\begin{center}
\caption{The ground-state properties of mass-imbalanced $(N+1)$ fermionic
mixtures, for $N\le5$.
Shown are the angular momentum and parity of the state,
the mass ratio where it crosses the threshold of the system with one
particle less in free space and in a harmonic trap,
and the mass ratio where Efimov physics emerges.
See text for references.
\label{tbl:Thresholds}}
\vspace{0.3cm}
{\renewcommand{\arraystretch}{1.25}%
\begin{tabular}
{c@{\hspace{5mm}} c@{\hspace{5mm}} c@{\hspace{5mm}} c@{\hspace{5mm}} c}
\hline\hline
System & $L^\pi$ & Free crossing & Trap crossing & Efimov \\
\hline
2+1 & $1^-$ & 8.173 & 8.619 & 13.607 \\
3+1 & $1^+$ & 8.862 & 8.918 & 13.384 \\
4+1 & $0^-$ & 9.672 & 9.41 & 13.279 \\
5+1 & $0^-$ \\
\hline\hline
\end{tabular}}
\end{center}
\end{table}
Very little is known about the $(5+1)$ system.
A simplified model explains the similar trends in the
$(2+1)$, $(3+1)$, and $(4+1)$ systems as populating a $p$ shell atom
by atom. The $(5+1)$ system, therefore, should be different, since the
$p$ shell is now full and the additional atom has to open a new shell
\cite{BazPet17}.
Intriguing open questions are thus the following:
is there a non-Efimovian universal bound hexamer
and does the six-body Efimov effect exist?
The extrapolation toward the case of fermionic polaron,
corresponding to the $N \gg 1$ case, is of special interest.
As a step in this direction the shell structure of the few-body
systems is studied here.
In contrast to the static heavy-impurity case, it is shown
that non perturbative physics arise in the dynamic
light-impurity case.
The goal of this work is to study the scale factor, or equivalently the
energy in a trap, of the $(N+1)$ ($N \le 5$) fermionic mixtures
few lowest states, and to identify their properties.
Calculation are done for a wide range of mass ratios,
from the static-impurity limit $m \gg M$ to the
dynamic-impurity limit $m \ll M$.
A convenient way to describe the system is the
Skorniakov and Ter-Martirosian (STM) integral equation
\cite{STM,MorCasPri11}, which deals directly with zero-range interaction by
applying the Bethe-Peierls boundary condition when two different particles
approach each other.
One has to solve an integral equation in $3(N-1)$ dimensions,
but utilizing the system symmetries the number of dimensions
can be reduced further.
For $N=2$, the STM equation for the scale factor is reduced to a
transcendental equation which can be easily solved.
For $N=3$, it can be reduced to two dimensions,
allowing the solution on a grid \cite{CasMorPri10}.
For $N=4$, however, a five-dimensional equation
makes a grid-based approach challenging if possible at all.
A method based on a Monte-Carlo process to solve the STM equation was
developed for this case in Ref.~\cite{BazPet17}.
However, this method is limited to bound systems and therefore
cannot be used to calculate the scale factor for all mass ratios.
In addition, as a fermionic Monte-Carlo method it might suffer from
a sign problem if the wave function has radial nodes.
Thus we take here another approach.
We solve the Schr\"{o}dinger equation for
the trapped system with \emph{finite}-range interspecies potential
and then extrapolate to the zero-range limit.
A similar method was applied in Refs.~\cite{BluDai10,RakDaiBlu12}.
Using this method we calculate the scale factor for $0 \le \alpha \le 12$
for the ground state, as well as for a few lowest excited states, of the
$(N+1)$ fermionic system up to $N \le 5$.
We set a simple model to understand the shell structure for the static-impurity
case, and explore the effects of the dynamic impurity as the mass ratio
increases.
We find that no $(5+1)$ Efimov states exist for $\alpha \le 12$.
As the mass ratio increases, finite-range corrections become
significant and the extrapolation to the zero-range limit cannot be trusted
anymore. A further study is therefore needed to explore such states for
larger mass ratios, $12 < \alpha < 13.279$.
\section{METHODS}
As we have explained, the zero-range limit is not directly used here; instead,
a series of calculations with a finite-range potential with decreasing range
is used to extrapolate the zero-range limit.
The Hamiltonian of the $(N+1)$ system is
\begin{equation}
H = T + U + V,
\end{equation}
where $T$ is the internal kinetic energy
and $U$ is the confining harmonic potential.
Here, $V$ is the interspecies attractive interaction,
taken of the form
\begin{equation}
V=-V_0 \sum_{i=1}^N \exp\left(-\frac{({\bf r}-{\bf R}_i)^2}{2R_0^2}\right),
\end{equation}
where $V_0>0$ is the potential strength and $R_0$ is its range.
We seek the limit of $R_0 \longrightarrow 0$
while $V_0$ is tuned to keep the two-body system at unitarity.
To solve the few-body problem, we use the stochastic variational method (SVM)
\cite{SuzVar98}. The wave function is expanded in an over-complete basis
of correlated Gaussians, where the basis functions are chosen in a stochastic
way utilizing the variational principle. The energies and the corresponding wave
functions can be found then by solving a generalized eigenvalue problem.
The basis functions are chosen to have the necessary permutational symmetry,
parity $\pi$, and angular momentum $L$ and its projection $M$,
\begin{equation}
\phi^\pi_{LM}(A,u;\eta) = \hat {\mathcal A} e^{-\frac{1}{2}\eta^T A \eta}\,
\theta^\pi_{LM}(u;\eta)
\end{equation}
where $\eta \equiv \{\ensuremath{\boldsymbol{\eta}}_1,\ldots,\ensuremath{\boldsymbol{\eta}}_N\}$ is a set of $N$ Jacobi
coordinates, $\hat {\mathcal A}$ is the appropriate anti-symmetrization
operator,
$A$ is an $N \times N$ real, symmetric, and positive definite matrix,
and $\theta^\pi_{LM}(u;\eta)$ is the angular part.
The $N(N+1)/2$ real numbers defining $A$ are optimized in a stochastic way
such as the energy is minimized.
Spin and isospin functions can be introduced but are not needed here.
The angular part is characterized by the global vector
representation \cite{VarSuz95,SuzUsu00}.
For a natural parity $\pi=(-1)^L$ it is
\begin{equation}
\theta^\pi_{LM}(u;\eta) = \mathcal Y_{LM}(\mathbf v),
\end{equation}
where $\mathcal Y_{LM}$ is the regular solid harmonic
and $\mathbf{v}=u^T\eta$ is a global vector, whose
elements are also optimized in a stochastic way.
To get the unnatural parity $\pi=(-1)^{L+1}$ for $L>0$ one has to couple two
global vectors,
\begin{equation}
\theta^\pi_{LM}(u;\eta) =
\left[\mathcal Y_L(\mathbf v_1) \otimes \mathcal Y_1(\mathbf v_2)\right]_{LM},
\end{equation}
while three global vectors are needed to get the $0^-$ symmetry,
\begin{equation}
\theta^-_{00}(u;\eta) =
\left[\left[\mathcal Y_1(\mathbf v_1) \otimes \mathcal Y_1(\mathbf v_2)\right]_1
\otimes \mathcal Y_1(\mathbf v_3)\right]_{00}.
\end{equation}
The overlap of such basis functions, as well as the matrix elements of the
Hamiltonian, are known analytically
\cite{VarSuz95,SuzVar98,SuzUsu00,RakDaiBlu12,BazEliKol16}.
For a given number of particles, angular momentum, and parity,
the ground-state energy is calculated for various potential ranges. From
these energies, the zero-range limit is extrapolated.
Typical results for the (2+1) $1^-$ ground state are shown in
Fig.~\ref{Fig:Convergence}, where results calculated from finite-range
potentials are compared to the zero-range results.
The radius of convergence for the extrapolation is shown to be much larger
for $\alpha=4$ than for $\alpha=12$.
In the latter case, close to the Efimovian limit,
the extrapolated value will be completely off if one uses, say,
results with $R_0>0.03 \sqrt{\hbar^2/m \omega}$ \cite{BluDai10}.
\begin{figure}
\vskip 0 pt \includegraphics[clip,width=1\columnwidth]{Convergence.pdf}
\caption{
Convergence of finite-range potentials toward the zero-range limit
$R_0 \longrightarrow 0$ for the $(2+1)$ ground state.
(a) $\alpha=4$, away from the Efimovian limit.
(b) $\alpha=12$, near the Efimovian limit.
The zero-range result (red square) is the exact solution of Eq. (\ref{ZR}).
}
\label{Fig:Convergence}
\end{figure}
To estimate the extrapolation uncertainty, we fit the results with
a few shortest $R_0$ with linear and parabolic curves
and account for their differences. The error due to the finite basis set
becomes significant for $N>3$ and is also considered.
Taking the potential range to be smaller, the numerical calculation
becomes harder.
Therefore close to the Efimovian limit, where finite-range corrections become
significant, the extrapolations can not be trusted anymore.
To correctly treat this region one should use a method dealing with the
zero-range limit directly. For example, one would like to solve the
STM equation using a diffusion Monte-Carlo (DMC)-like approach \cite{BazPet17}.
This task is left for future work.
\section{RESULTS}
\subsection{The $\alpha = 0$ limit}
We start to analyze the $\alpha = 0$ limit, where the impurity
is infinitely heavy and therefore static.
This case reduces to the problem of $N$
trapped fermions scattering on a zero-range potential at
the trap center.
The analytic solution for the two-body problem is known \cite{BusEngRza98},
giving at unitarity an energy shift of $-\,\hbar\omega$ for the $s$ shell
with respect to the non interacting case.
The quantum numbers characterizing a shell are the radial number $n$ and
the angular momentum $l$ and its projection; its energy is given by
\begin{equation}
E_{nl}=\hbar \omega (2n+l-\delta_{l,0}+3/2),
\end{equation}
and the energy of the $(N+1)$ system is just a sum of $N$ single-particle
energies.
To ease comparison between clusters with different particle numbers,
the zero-point energy $\hbar \omega \,3N/2$ is subtracted.
Energy is measured in units of $\hbar \omega$
and with respect to the dimer energy, i.e.
\begin{equation}
\epsilon = E/\hbar \omega-3N/2+1.
\end{equation}
Only interacting states, i.e., those states which have
an atom in an $s$ shell, are considered.
Applying the fermionic symmetry, the spectrum and properties of the $(N+1)$
systems can be calculated.
Table~\ref{tbl:GroundState} summarizes
the ground-state properties of the $(N+1)$ systems.
For completeness, the properties of the two lowest excited states
are also tabulated in the Appendix. Here and thereafter we ignore the
trivial $2L+1$ degeneracy due to different total angular momentum projections.
\begin{table}
\begin{center}
\caption{The ground-state properties in the static-impurity limit,
$\alpha=0$.
Shown are the energy, the angular momentum, the parity, and the shell
configuration for the $(N+1)$ mixtures.
\label{tbl:GroundState}}
\vspace{0.3cm}
{\renewcommand{\arraystretch}{1.25}%
\begin{tabular}
{c@{\hspace{5mm}} c@{\hspace{5mm}} c@{\hspace{5mm}} c@{\hspace{5mm}} c}
\hline\hline
System & $\epsilon$ & $\pi$ & $L$ & Configuration \\
\hline
1+1 & 0 & + & 0 & $0s$ \\
\hline
\multirow{2}{*}{2+1} & \multirow{2}{*}{1} & + & 0 & $0s\,1s$ \\
\cline{3-5}
& & -- & 1 & $0s\,0p$ \\
\hline
\multirow{2}{*}{3+1} & \multirow{2}{*}{2} & + & 1 & $0s\,0p^2$ \\
\cline{3-5}
& & -- & 1 & $0s\,1s\,0p$ \\
\hline
\multirow{2}{*}{4+1} &\multirow{2}{*}{3} & + & 1 & $0s\,1s\,0p^2$ \\
\cline{3-5}
& & -- & 0 & $0s\,0p^3$ \\
\hline
5+1 & 4 & -- & 0 & $0s\,1s\,0p^3$ \\
\hline\hline
\end{tabular}}
\end{center}
\end{table}
\subsection {The (2+1) case}
We move now to the general mass-imbalanced case and start with
two identical fermions interacting with a distinguishable atom.
For the natural parity case, the scale factor $s$ corresponding to
a total angular momentum $L$ is
the solution of a transcendental equation,
\begin{equation}\label{ZR}
\frac{2}{\Gamma(a-1/2)\Gamma(b-1/2)}+
\frac{\left(-\gamma\right)^L}{\sqrt \pi \Gamma (c)}
\,_2F_1\left(a,b;c;\gamma^2\right)=0
\end{equation}
where $a=1+(L-s)/2$, $b=1+(L+s)/2$, $c=L+3/2$,
$_2F_1$ is the hypergeometric function, and $\gamma=\alpha/(\alpha+1)$
\cite{RitMehGre10}.
Unnatural parity means here that both identical fermions
are excited to $l>0$ shell, resulting in a non interacting case that will be
ignored here.
For $\alpha=0$ the ground state has two degenerate states, $1^-$ and $0^+$,
where in the first case the additional atom populates a $p$ shell while in the
latter it sits in an excited $s$ shell.
The energy degeneracy is lifted for $\alpha>0$, where the dynamic impurity
induces interaction between the identical fermions,
which is attractive (repulsive) for an odd (even) angular momentum.
Hence, the $1^-$ state becomes the ground state.
This behavior can be understood in the Born-Oppenheimer (BO)
approximation, which holds for $\alpha \gg 1$ \cite{Fon79}.
Utilizing the mass difference,
the distance between heavy particles $\ensuremath{\boldsymbol{R}}=\ensuremath{\boldsymbol{R}}_1-\ensuremath{\boldsymbol{R}}_2$
can be treated as a parameter in the light-particle equation,
which becomes simply the double-well potential problem,
with the known eigenvalues $\epsilon_\pm(R)$.
In the heavy-particle equation, $\epsilon_\pm(R)$ has the meaning of an
effective potential and is attractive or repulsive, depending on the parity.
Applying the fermionic symmetry for heavy particles' permutation, the
effective potential for odd-$L$ states is found to be attractive and
goes like $-1/mR^2$ for $R \ll a$, while the effective potential for
even-$L$ states is repulsive.
For the attractive channel, the mass ratio governs the competition between
the centrifugal barrier $\propto L(L+1)/MR^2$ and the effective attraction.
Increasing $\alpha$ tips the scales in favor of the attraction;
hence the trimer energy decreases.
In a trap the trimer energy crosses the dimer+atom energy
($\epsilon=0$ in our conventions)
for $\alpha$ slightly larger than needed in free space.
Increasing $\alpha$ further the effective interaction becomes purely
attractive and the system becomes Efimovian.
In the $(2+1)$ system, the $1^-$ symmetry is the only symmetry where this
phenomenon occurs.
To benchmark our method, we calculate the unitary $(2+1)$ trapped system
energy by extrapolating finite-range results to the zero-range limit.
The scale factor can be easily calculated from Eq. (\ref{ZR}) and is connected
to the energy in a trap by Eq. (\ref{trap}), giving here (for $n=0$)
$s=\epsilon+1$.
Hence, the Efimovian limit $s=0$ corresponds here to $\epsilon=-1$.
Our results are plotted in Fig.~\ref{Fig:Trimer}, showing
a nice agreement with the solutions of Eq. (\ref{ZR}).
The limit of $\alpha=0$ from Tables \ref{tbl:GroundState},
\ref{tbl:ExcitedState} and \ref{tbl:2ExcitedState} is also reproduced.
\begin{center}
\begin{figure}
\vskip 0 pt \includegraphics[clip,width=1\columnwidth]{Trimer.pdf}
\caption{
The energy of the unitary $(2+1)$ trapped system is shown as a function of
the mass ratio for a few lowest states.
Symbols are the zero-range extrapolation from finite-range potentials,
and dashed curves are the zero-range results calculated from Eq. (\ref{ZR}).
The Efimovian limit $s=0$ is the dotted horizontal line, which the lowest
$1^-$ curve hits at $M/m=13.607$.
}
\label{Fig:Trimer}
\end{figure}
\par\end{center}
Note that in a trap, each solution of Eq. (\ref{ZR}) starts a ladder of
solutions, corresponding to hyperradial excitations and giving an additional
$2\hbar\omega$ for each hyperradial node.
The first excited state of the $1^-$ symmetry is also shown
in Fig.~\ref{Fig:Trimer}.
\subsection {The (3+1) case}
We now add another identical particle and move to the $(3+1)$ system.
For $\alpha=0$, the ground state has two degenerate states, $1^+$ and $1^-$,
both with $\epsilon=2$.
These states have different atomic configurations: while in the $1^+$
state the additional atom sits in a $p$ shell, the $1^-$ state
corresponds to atom-trimer $s$-wave scattering. $d$-wave atom-trimer scattering
states, corresponding to $1^-$, $2^-$, and $3^-$ symmetries,
have higher energy in this limit, $\epsilon=3$.
The energy degeneracy is lifted for $\alpha>0$, where
the $1^+$ state energy becomes lower than the $1^-$ state energy, in qualitative
agreement with the BO picture where the interaction induced by the impurity is
attractive in a $p$ wave and repulsive in an $s$ wave.
For a larger mass ratio, the $1^+$ state becomes bound in free space,
then crosses the trimer+atom threshold in a trap,
and eventually reaches the Efimov threshold, corresponding here
to $\epsilon=-2.5$.
States of other symmetries, nevertheless, does not reach the Efimov limit for
any mass ratio smaller than the $(2+1)$ Efimov threshold \cite{CasMorPri10}.
The $1^+$ ground-state scale factor has been calculated in Ref.~\cite{BazPet17}
using a grid-based method, similar to that of Ref.~\cite{CasMorPri10}.
That method is more accurate than our current method and can be used up to,
and even beyond, the Efimov limit.
For a benchmark, we compare in Fig.~\ref{Fig:Tetramer} the results of
both methods, which are in nice agreement.
The $\alpha=0$ limit from Table~\ref{tbl:GroundState} is also reproduced.
For this symmetry the calculations for $\alpha>10$ become sensitive,
signing a non universal resonance, identified in Ref.~\cite{BluDai10} to occur
at $\alpha=10.4(2)$ for a Gaussian interaction.
\begin{figure}
\vskip 0 pt \includegraphics[clip,width=1\columnwidth]{Tetramer.pdf}
\caption{
The energy of the unitary $(3+1)$ trapped system is shown as a function of
the mass ratio for a few lowest states.
Symbols are the zero-range extrapolation from finite-range potentials,
and the dashed curve is the zero-range result of Ref.~\cite{BazPet17}.
The results of Refs.~\cite{BluDai10,RakDaiBlu12} are shown as purple
triangles.
The Efimovian limit $s=0$ is the dotted horizontal line, which the $1^+$
curve approaches at $M/m=13.384$.
}
\label{Fig:Tetramer}
\end{figure}
The scale factor of the $1^-$ lowest excited state has been calculated
for an equal-mass system only \cite{RakDaiBlu12}.
Our results are tabulated in Table~\ref{tbl:TetramerX} and shown in
Fig.~\ref{Fig:Tetramer}, agreeing well with the $\alpha=0$ limit and with
the $\alpha=1$ result of Ref.~\cite{RakDaiBlu12}.
The bending in the $1^-$ energy around $\alpha=2$ is to be understood as
level repulsion with an excited $1^-$ state.
To make this point clear, the energies of a few lowest $1^-$ states are shown
in Fig.~\ref{Fig:TetramerX}.
The atomic configurations for $\alpha=0$ are the following.
The state with $\epsilon=2$ corresponds to
the configuration $0s\,0p\,1s$, i.e. an atom-trimer $s$-wave state, while
for $\epsilon=3$ it is $0s\,0p\,0d$, i.e. an atom-trimer $d$-wave state.
A clear avoided crossing between these states is seen around $\alpha=2$.
Note, however, that the crossing of levels with different quantum numbers is
allowed.
States with different hyperradial quantum number $n$ can therefore cross,
and are also shown in Fig.~\ref{Fig:TetramerX}.
\begin{table}
\begin{center}
\caption{The energies of the trapped tetramer lowest $1^-$ state.
\label{tbl:TetramerX}}
\vspace{0.3cm}
{\renewcommand{\arraystretch}{1.25}%
\begin{tabular}
{c@{\hspace{5mm}} c@{\hspace{5mm}} c@{\hspace{5mm}} c@{\hspace{5mm}} c}
\hline\hline
$M/m$ & This work & Ref.~\cite{RakDaiBlu12} & $M/m$ & This work \\
\hline
0 & 2 & & 6 & 1.613(1) \\
1 & 2.183(2) & 2.177(4) & 7 & 1.428(1) \\
2 & 2.221(2) & & 8 & 1.232(1) \\
3 & 2.115(2) & & 9 & 1.024(1) \\
4 & 1.959(1) & & 10 & 0.805(2) \\
5 & 1.791(1) & & 11 & 0.569(3) \\
\hline\hline
\end{tabular}}
\end{center}
\end{table}
\begin{figure}
\vskip 0 pt \includegraphics[clip,width=1\columnwidth]{TetramerX.pdf}
\caption{
The energy of the unitary $(3+1)$ trapped system is shown as a function of
the mass ratio, for a few lowest $1^-$ states.
}
\label{Fig:TetramerX}
\end{figure}
The next state, with $3^-$ symmetry, is also shown in Fig.~\ref{Fig:Tetramer}.
It moves closer to the $1^-$ state as the mass ratio increases.
Since the lowest $1^-$ for large $\alpha$ is dominated by a $d$-wave
atom-trimer state, like the $3^-$ state, this similarity makes sense.
As we show later, this phenomena also exists,
and is even stronger, for larger $N$.
\subsection {The (4+1) case}
Adding another identical particle, we now consider the $(4+1)$ system.
For $\alpha=0$, two states are degenerate at $\epsilon=3$, with symmetries
$0^-$ and $1^+$.
In the $0^-$ state the additional atom populates the last place in
the $p$ shell,
while the $1^+$ state corresponds to atom-tetramer $s$-wave scattering.
The degeneracy is lifted for $\alpha>0$, where
the $0^-$ state energy becomes lower than the $1^+$ energy.
For larger mass ratios, the $0^-$ state
crosses the tetramer+atom energy in a trap,
becomes bound in free space,
and eventually reaches the Efimov threshold, corresponding here to
$\epsilon=-4$ \cite{BazPet17}.
The $0^-$ scale factor has been calculated for a few mass ratios using
finite-range models \cite{BluDai10}.
For $\alpha>9.672$, when the pentamer is bound in free space, it was
calculated by fitting the wave-function high-momentum tail to
$F(Q)\propto Q^{-3N/2+1-s}$, where $Q$ is the hypermomentum conjugate to the
the hyperradius $\rho$
and $F$ is the momentum-space wave-function calculated in the STM-DMC method
\cite{BazPet17}.
Our results are tabulated in Table~\ref{tbl:Pentamer} and shown
in Fig.~\ref{Fig:Pentamer}.
\begin{table}
\begin{center}
\caption{The energies of the trapped pentamer $0^-$ state for various
mass ratios.
\label{tbl:Pentamer}}
\vspace{0.3cm}
{\renewcommand{\arraystretch}{1.25}%
\begin{tabular}
{c@{\hspace{2mm}} c@{\hspace{4mm}} c@{\hspace{3mm}} c@{\hspace{2mm}} c@{\hspace{4mm}} c@{\hspace{4mm}} c}
\hline\hline
$M/m$ & This work & Ref.~\cite{BluDai10} & $M/m$ & This work & Ref.~\cite{BazPet17} \\
\hline
0 & 3 & & 6 & 1.01(1) \\
1 & 2.42(1) & 2.45 & 7 & 0.77(1) \\
2 & 2.11(1) & 2.15 & 8 & 0.44(1) \\
3 & 1.83(1) & & 9 & 0.26(3) \\
4 & 1.57(1) & 1.68 & 10 & -0.2(1) & -0.41(1) \\
5 & 1.28(1) & & 11 & -0.5(1) & -0.90(1) \\
\hline\hline
\end{tabular}}
\end{center}
\end{table}
\begin{figure}
\vskip 0 pt \includegraphics[clip,width=1\columnwidth]{Pentamer.pdf}
\caption{
The energy of the unitary $(4+1)$ trapped system is shown as a function of
the mass ratio for a few lowest states.
Symbols are the zero-range extrapolation from finite-range potentials,
and the dashed curve is the zero-range result of Ref.~\cite{BazPet17}.
The results of Refs.~\cite{BluDai10,RakDaiBlu12} are shown as purple
triangles.
The Efimovian limit $s=0$ is the dotted horizontal line, which the $0^-$
curve approaches at $M/m=13.279$.
}
\label{Fig:Pentamer}
\end{figure}
The $1^+$ scale factor has been calculated only for the equal-mass
case \cite{RakDaiBlu12}.
Our results are tabulated in Table~\ref{tbl:PentamerX} and shown
in Fig.~\ref{Fig:Pentamer}.
Since for large mass ratio the zero-range extrapolation is not conclusive,
we cannot work close to the Efimov threshold. However, no sign
for an Efimov state with any symmetry other than $0^-$ is found in the
explored mass ratios.
\begin{table}
\begin{center}
\caption{The energies of the trapped pentamer $1^+$ state for various
mass ratios.
\label{tbl:PentamerX}}
\vspace{0.3cm}
{\renewcommand{\arraystretch}{1.25}%
\begin{tabular}
{c@{\hspace{5mm}} c@{\hspace{5mm}} c@{\hspace{5mm}} c@{\hspace{5mm}} c}
\hline\hline
$M/m$ & This work & Ref.~\cite{RakDaiBlu12} & $M/m$ & This work\\
\hline
0 & 3 & & 6 & 2.01(2) \\
1 & 3.19(1) & 3.155 & 7 & 1.77(1) \\
2 & 3.05(1) & & 8 & 1.56(3) \\
3 & 2.85(1) & & 9 & 1.19(4) \\
4 & 2.56(1) & & 10 & 0.99(1) \\
5 & 2.31(1) & & \\
\hline\hline
\end{tabular}}
\end{center}
\end{table}
Similar to the $(3+1)$ case, the bending in the $1^+$ energy results from
avoided crossing around $\alpha=1$ with another $1^+$ state (not shown).
The latter state has $\epsilon=4$ in the $\alpha=0$ limit and corresponds to the
$d$-wave atom-tetramer state. The same is true for the $2^+$ and $3^+$ states,
also shown in Fig.~\ref{Fig:Pentamer},
and indeed the energies of these state are close apart from the avoided
crossing region.
\subsection {The (5+1) case}
Adding another atom, we now move to the $(5+1)$ system.
Since no room is left in the $p$ shell, the additional
atom can populate an excited $s$ shell, keeping the $0^-$ symmetry of
the $(4+1)$ core, or a $d$ shell, resulting in a $2^-$ state.
The energies of these states in a trap are tabulated in Table~\ref{tbl:Hexamer}
and plotted in Fig.~\ref{Fig:Hexamer}.
\begin{center}
\begin{figure}
\vskip 0 pt \includegraphics[clip,width=1\columnwidth]{Hexamer.pdf}
\caption{
The energy of the unitary $(5+1)$ trapped system is shown as a function of
the mass ratio for a few lowest states.
Symbols are the zero-range extrapolation from finite-range potentials.
The Efimovian limit $s=0$ is the dotted horizontal line. For the mass ratios
explored here, the scale factors do not cross this limit and therefore
no $(5+1)$ Efimov effect exists.
}
\label{Fig:Hexamer}
\end{figure}
\par\end{center}
\begin{table}
\begin{center}
\caption{The energies of the two lowest $(5+1)$ hexamer states in a trap,
with $0^-$ and $2^-$ symmetries, for various mass ratios.
\label{tbl:Hexamer}}
\vspace{0.3cm}
{\renewcommand{\arraystretch}{1.25}%
\begin{tabular}
{c@{\hspace{5mm}} c@{\hspace{5mm}} c@{\hspace{5mm}} c@{\hspace{5mm}} c@{\hspace{5mm}} c}
\hline\hline
$M/m$ & $0^-$ & $2^-$ & $M/m$ & $0^-$ & $2^-$ \\
\hline
0 & 4 & 5 & 6 & 2.7(1) & 2.73(4) \\
1 & 4.23(1) & 4.34(1) & 7 & 2.3(1) & 2.44(6) \\
2 & 3.89(3) & 3.96(2) & 8 & 2.4(1) & 2.20(3) \\
3 & 3.52(3) & 3.63(2) & 9 & 1.8(1) & 1.8(1) \\
4 & 3.19(3) & 3.31(2) & 10 & 1.8(3) & 1.5(1) \\
5 & 2.87(4) & 2.99(3) & 11 & 1.3(3) & 1.2(2) \\
\hline\hline
\end{tabular}}
\end{center}
\end{table}
As the mass ratio becomes larger, the $0^-$ and $2^-$ states becomes degenerate
within our error bars.
The Efimov limit corresponds here to $\epsilon=-5.5$.
Our results show no sign for a $(5+1)$ Efimov state for any symmetry
up to $\alpha \le 12$.
As was have claimed, a different method would be probably needed to extend this
conclusion up the the $(4+1)$ Efimovian threshold.
\section{CONCLUSION}
We study mass-imbalanced mixtures of $N$ identical fermions interacting
resonantly with a distinguishable atom.
The scale factor, or the energy
of the unitary system in a harmonic trap, was calculated for a few lowest
states of the $N \le 5$ systems. We solve the trapped few-body system
with finite-range inter-species potentials using the
stochastic variational method. The zero-range limit is then extrapolated.
The shell structure of the system is explored and the effect of level
repulsion is shown, revealing the significant change from the static-impurity
case to the dynamic-impurity case.
A series of Efimov states with $N=2,3$, and $4$
exist for large enough mass ratio. Nevertheless, no sign for
the existence of a $(5+1)$ Efimov effect is shown in the mass ratios
explored here, $\alpha\le 12$. Further studies that
would deal directly with the zero-range limit should be carried out to
check the validity of this statement for mass ratios up to the $(4+1)$
Efimovian threshold.
\section*{ACKNOWLEDGMENT}
I would like to thank Dmitry Petrov, Nir Barnea, Kalman Varga,
Johannes Kirscher, Ronen Weiss, and Yvan Castin
for useful discussions and communications.
This research was supported by the Pazi Fund.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 2,487 |
\section{Introduction}
The notion of magnetic charge has intrigued physicists since Dirac \cite{dirac}
showed that it was consistent with quantum mechanics provided a suitable
quantization condition was satisfied: For a monopole of magnetic charge $g$ in
the presence of an electric charge $e$, that quantization condition is
(in this paper we use rationalized units)
\begin{equation}
{eg\over4\pi}={n\over2}\hbar c,
\label{quant}
\end{equation}
where $n$ is an integer. For a pair of dyons, that is, particles carrying
both electric and magnetic charge, the quantization condition is replaced
by \cite{schwinger}
\begin{equation}
{e_1g_2-e_2g_1\over4\pi}={n\over 2}\hbar c,
\label{squant}
\end{equation}
where $(e_1,g_1)$ and $(e_2,g_2)$ are the charges of the two dyons.
With the advent of ``more unified'' non-Abelian theories, classical composite
monopole solutions were discovered \cite{nonabel}. The mass of these monopoles
would be of the order of the relevant gauge-symmetry breaking scale,
which for grand unified theories is of order
$10^{16}$ GeV or higher. But there are models where the electroweak symmetry
breaking can give rise to monopoles of mass $\sim 10$ TeV \cite{ewmono}.
Even the latter are not yet accessible to accelerator experiments,
so limits on heavy monopoles depend either on cosmological considerations
\cite{cosmo}, or detection of cosmologically produced (relic) monopoles
impinging upon the earth or moon \cite{relic}.
However, {\it a priori}, there is no reason that Dirac/Schwinger monopoles
or dyons of arbitrary mass might not exist: It is important to set limits
below the 1 TeV scale.
Such an experiment is currently in progress at the University of Oklahoma
\cite{ou}, where we expect to be able to set limits on {\it direct\/} monopole
production at Fermilab up to several hundred GeV. This will be a substantial
improvement over previous limits \cite{prev}. But {\it indirect\/}
searches have
been proposed and carried out as well. De R\'ujula \cite{DeRujula} proposed
looking at the three-photon decay of the $Z$ boson, where the process proceeds
through a virtual monopole loop. If we use his formula \cite{DeRujula} for the
branching ratio for the $Z\to3\gamma$ process, compared to the current
experimental upper limit \cite{exp}
for the branching ratio of $10^{-5}$, we can rule out
monopole masses lower than about 400 GeV, rather than the 600 GeV quoted in
Ref.~\cite{DeRujula}.
Similarly, Ginzburg and Panfil \cite{ginz1} and very recently Ginzburg and
Schiller \cite{ginz2} considered the production of two photons with
high transverse momenta by the collision of two photons produced either
from $e^+e^-$ or quark-(anti-)quark collisions. Again the final photons are
produced through a virtual monopole loop. Based on this theoretical scheme,
an experimental limit has
appeared by the D0 collaboration \cite{d0}, which sets the following bounds
on the monopole mass $M$:
\begin{equation}
{M\over n}>\left\{\begin{array}{cc}
610 \mbox{ GeV}&\mbox{ for } S=0\\
870 \mbox{ GeV}&\mbox{ for } S=1/2\\
1580 \mbox{ GeV}&\mbox{ for } S=1 \end{array}\right.,
\end{equation}
where $S$ is the spin of the monopole.
It is worth noting that a mass limit of 120 GeV for a Dirac monopole has been
set by Graf, Sch\"afer, and Greiner \cite{graf}, based on the monopole
contribution to the vacuum polarization correction to the muon anomalous
magnetic moment. (Actually, we believe that the correct limit, obtained
from the well-known textbook formula \cite{js1} for the $g$-factor correction
due to a massive Dirac particle is 60 GeV.)
The purpose of this paper is to critique the theory of Refs.~\cite{DeRujula},
\cite{ginz1}, \cite{ginz2}, and \cite{graf}. We will show that it is based on
a naive application of electromagnetic duality; the resulting
cross section cannot be valid because
unitarity is violated for monopole masses as low as the quoted limits, and
the process is subject to enormous, uncontrollable radiative corrections.
It is not correct, in any sense, as Refs. \cite{ginz2} and \cite{d0}
state, that the effective expansion parameter is $g\omega/M$, where $\omega$
is some external photon energy; rather, the factors of $\omega/M$ emerge
kinematically from the requirements of gauge invariance at the one-loop
level. If, in fact, a correct calculation introduced such
additional factors of $\omega/M$, arising from the complicated coupling
of magnetic charge to photons, we argue that no limit could be deduced
for monopole masses from the current experiments. It may even be the
case, based on preliminary field-theoretic calculations, that processes
involving the production of real photons vanish.
\section{Duality and the Euler-Heisenberg Lagrangian}
Let us concentrate on the process contemplated in Refs.~\cite{ginz2}
and \cite{d0}, that is
\begin{equation}
\left(\begin{array}{ccc}
qq&\to& qq\\
\bar qq&\to&\bar qq
\end{array}\right)+\gamma\gamma,\quad \gamma\gamma\to\gamma\gamma,
\end{equation}
where the photon scattering process is given by the one-loop light-by-light
scattering graph shown in Fig.\ \ref{fig1}. If the particle in the loop
is an ordinary electrically charged electron, this process is
well-known \cite{js,js1,ll}. If, further, the photons involved are of very low
momentum compared the the mass of the electron, then the result may be
simply derived from the well-known Euler-Heisenberg Lagrangian \cite{eh}, which
for a spin 1/2 charged-particle loop in the presence of homogeneous
electric and magnetic fields is\footnote{We emphasize that
Eq.~(\ref{ehlagrangian}) is only valid when $\partial_\alpha F_{\mu\nu}=0$.}
\begin{equation}
{\cal L}=-{\cal F}-{1\over8\pi^2}\int_0^\infty {ds\over s^3}e^{-m^2 s}
\left[(es)^2{\cal G}{\mbox{Re}\cosh esX\over\mbox{Im}\cosh esX}-1-{2\over3}
(es)^2{\cal F}\right].
\label{ehlagrangian}
\end{equation}
Here the invariant field strength combinations are
\begin{equation}
{\cal F}={1\over 4}F^2={1\over2}({\bf H}^2-{\bf E}^2),\quad
{\cal G}={1\over 4}F \tilde F={\bf E\cdot H},
\end{equation}
$\tilde F_{\mu\nu}={1\over2}\epsilon_{\mu\nu\alpha\beta}F^{\alpha\beta}$ being
the dual field strength tensor, and the argument of the hyperbolic cosine
in Eq.~(\ref{ehlagrangian}) is given in terms of
\begin{equation}
X=[2({\cal F}+i{\cal G})]^{1/2}=[({\bf H}+i{\bf E})^2]^{1/2}.
\end{equation}
If we pick out those terms quadratic, quartic and
sextic in the field strengths,
we obtain\footnote{Incidentally, note that the coefficient of the last term is
36 times larger than that given in Ref.~\cite{DeRujula}.}
\begin{eqnarray}
{\cal L}&=&-{1\over4}F^2+{\alpha^2\over360}{1\over m^4}
[4(F^2)^2+7(F \tilde F)^2]\nonumber\\
&&\mbox{}-{\pi\alpha^3\over630}{1\over m^8}F^2[8(F^2)^2+13 (F {}^*F)^2]+\dots.
\label{ehlag}
\end{eqnarray}
The Lagrangian for a spin-0 and spin-1 charged particle in the loop
is given by similar formulas which are derived in Ref.~\cite{js,js1}
and (implicitly) in Ref.~\cite{spin1}, respectively.
Given this homogeneous-field effective Lagrangian, it is a simple matter to
derive the cross section for the $\gamma\gamma\to\gamma\gamma$ process in the
low energy limit. (These results can, of course, be directly calculated
from the corresponding one-loop Feynman graph with on-mass-shell photons.
See Refs.~\cite{js1,ll}.)
Explicit results for the differential cross section are given
by Ref.~\cite{ll}:
\begin{equation}
{d\sigma\over d\Omega}={139\over32400\pi^2}\alpha^4{\omega^6\over m^8}
(3+\cos^2\theta)^2,
\end{equation}
and the total cross section for a spin-1/2 charged particle in the loop
is\footnote{The numerical coefficient in the total cross section for a spin-0
and spin-1 charged particle in the loop is $119/20250\pi$ and $2751/250\pi$,
respectively. Numerically the coefficients are $0.00187$, $0.0306$, and
$3.50$ for spin 0, spin 1/2, and spin 1, respectively.}
\begin{equation}
\sigma={973\over10125\pi}\alpha^4{\omega^6\over m^8}.
\label{llcs}
\end{equation}
Here, $\omega$ is the energy of the photon in the center of mass frame,
$s=4\omega^2$. This result is valid provided $\omega/m\ll 1$.
The dependence on $m$ and $\omega$ is evident from the
Lagrangian (\ref{ehlag}),
the $\omega$ dependence coming from the field strength tensor.
Further note that perturbative quantum corrections are small,
because they are of relative order $3\alpha\sim10^{-2}$ \cite{dicus}.
Processes in which
four final-state photons are produced, which may be easily calculated from
the last displayed term in Eq.~(\ref{ehlag}), are even smaller, being
of relative order $\sim\alpha^2 (\omega/m)^8$. So light-by-light scattering,
which has been indirectly observed through its contribution to the
anomalous magnetic moment of the electron \cite{anmm}, is completely under
control for electron loops.
How is this applicable to photon scattering through a monopole loop? At first
blush this calculation seems formidable. The interaction of a magnetically
charged particle with a photon involves a ``string,'' that is, an arbitrary
vector function $f_\mu(x-x')$ that satisfies
\begin{equation}
\partial_\mu f^\mu(x-x')=\delta(x-x'),
\end{equation}
which can be realized by a line integral, for example, the semi-infinite
one
\begin{equation}
f_\mu(x)=\int_0^\infty d\xi_\mu\,\delta(x-\xi),
\end{equation}
where the $\xi$ integration follows some path from the origin to infinity.
For the case of a straight line with direction $n_\mu$, this can be
written in the form
\begin{equation}
f_\mu(x)={n_\mu\over i}\int{(dq)\over(2\pi)^4}{e^{iqx}\over n\cdot q
-i\epsilon}.
\label{string}
\end{equation}
The interaction between a magnetic current $J_m^\mu$
and the electromagnetic field is given by
\begin{equation}
W_{\rm int}=\int (dx)(dx')\tilde F_{\mu\nu}(x')f^\nu(x'-x)J_m^\mu(x),
\label{mmint}
\end{equation}
where the magnetic current must be conserved, $\partial_\mu J_m^\mu=0$.
Here, the string-dependent field strength tensor \cite{dirac,schwinger}
is
\begin{equation}
F_{\mu\nu}(x)=\partial_\mu A_\nu-\partial_\nu A_\mu+\epsilon_{\mu\nu\sigma\tau}
\int(dy)f^\sigma(x-y)J^\tau_m(y).
\end{equation}
The interaction (\ref{mmint}) corresponds to a coupling between electric
and magnetic currents of
\begin{equation}
W^{(eg)}=
-\epsilon_{\mu\nu\sigma\tau}\int(dx)(dx')(dx'')J_e^\mu(x)\partial^\sigma
D_+(x-x')f^\tau(x'-x'')J_m^\nu(x'').
\label{weg}
\end{equation}
From Eqs.~(\ref{string})--(\ref{weg})
one obtains the relevant string-dependent
monopole-photon coupling vertex in momentum space,
\begin{equation}
\Gamma_\mu(q)=ig{\epsilon_{\mu\nu\sigma\tau} n^\nu q^\sigma\gamma^\tau\over
n\cdot q-i\epsilon}.
\label{mmvertex}
\end{equation}
The choice of the string is arbitrary; reorienting the string is a kind
of gauge transformation. In fact, it is this requirement that leads to
the quantization conditions (\ref{quant}) and (\ref{squant}). The consistency
of magnetic charge has been demonstrated in quantum mechanics (for
example, see Refs.~\cite{dyondyon} and \cite{dm}),
but never completely in quantum field theory.\footnote{Arguments have been
given to demonstrate the relativistic invariance of the theory, and the
string independence of the action for classical particle currents
\cite{schwinger}. See also
Ref.~\cite{brandt}. This consistency is a consequence of the quantization
condition (\ref{quant}) or (\ref{squant}). We should also bear in mind
Schwinger's warning, in the first reference in Ref.~\cite{schwinger}:
``Relativistic invariance will appear to be violated
in any treatment based on a perturbation expansion. Field theory is
more than a set of `Feynman's rules.'~'' } The use of the string-dependent
vertex (\ref{mmvertex}) directly is not meaningful. In this regard,
the ``remedy'' proposed by Deans \cite{deans} and cited as a solution
to the gauge-string dependence of Drell-Yan processes is implausible.
The authors of Refs.~\cite{DeRujula}, \cite{ginz1}, and \cite{ginz2} do not
attempt a calculation of the ``box'' diagram with the
interaction (\ref{mmint}).
Rather, they (explicitly or implicitly) appeal to duality, that is, the
symmetry that the introduction of magnetic charge brings to Maxwell's
equations:
\begin{equation}
{\bf E}\to {\bf H}, \quad {\bf H}\to -{\bf E},
\label{duality}
\end{equation}
and similarly for charges and currents. Thus the argument is that
for low energy photon processes
it suffices to compute the fermion loop graph in the presence of
zero-energy photons, that is, in the presence of static, constant fields.
The box diagram shown in Fig.~\ref{fig1} with a spin-1/2
monopole running around the loop in the presence of a homogeneous $\bf E, H$
field is then obtained from that analogous process with an electron in the
loop in the presence of a homogeneous $\bf H, -E$ field, with the substitution
$e\to g$. Since the Euler-Heisenberg Lagrangian (\ref{ehlag}) is invariant
under the substitution (\ref{duality}) on the fields alone,
this means we obtain the low energy
cross section $\sigma_{\gamma\gamma\to\gamma\gamma}$ through the monopole
loop from Eq.~(\ref{llcs}) by the substitution $e\to g$, or
\begin{equation}
\alpha\to\alpha_g={137\over 4}n^2,\quad n=1,2,3,\dots.
\label{subs}
\end{equation}
\section{Inconsistency of the Duality Approximation}
It is critical to emphasize that the Euler-Heisenberg Lagrangian is
an effective
Lagrangian for calculations at the {\it one fermion loop level\/} for low
energy, i.e., $\omega/M\ll1$. It is commonly asserted that the Euler-Heisenberg
Lagrangian is an {\it effective Lagrangian\/} in the sense used in chiral
perturbation theory \cite{weinberg,halter}. This is not true. The QED
expansion generates derivative terms which do not arise in the effective
Lagrangian expansion of the Euler-Heisenberg Lagrangian \cite{dicus}.
One can only say that the Euler-Heisenberg Lagrangian is a good approximation
for light-by-light scattering (without monopoles)
at low energy because radiative corrections are down by factors of
$\alpha$. However, it becomes unreliable if radiative corrections are
large.
In this regard, both the Ginzburg \cite{ginz1,ginz2} and the De R\'ujula
\cite{DeRujula} articles, particularly Ref.~\cite{ginz2}, are rather misleading
as to the validity of the approximation sketched in the previous section.
They state that the expansion parameter is not $g$ but $g\omega/M$, $M$ being
the monopole mass, so that the perturbation expansion may be valid for large
$g$ if $\omega$ is small enough. But this is an invalid argument.
It is only when external photon lines are attached that
extra factors of $\omega/M$ occur, due to the appearance of the field
strength tensor in the Euler-Heisenberg Lagrangian. Moreover, the powers
of $g$ and $\omega/M$ are the same only for the $F^4$ process.
The expansion parameter
is $\alpha_g$, which is huge. Instead of radiative corrections being of the
order of $\alpha$ for the electron-loop process, these corrections will
be of order $\alpha_g$, which implies an uncontrollable sequence of
corrections. For example, the internal radiative correction to the box
diagram in Fig.~\ref{fig1} have been computed by Ritus \cite{ritus} and by
Reuter, Schmidt, and Schubert \cite{reuter} in QED. In the $O(\alpha^2)$
term in Eq.~(\ref{ehlag}) the coefficients of the $(F^2)^2$ and the
$(F\tilde F)^2$ terms are multiplied by $\left(1+{40\over9}{\alpha\over\pi}
+O(\alpha^2)\right)$ and $\left(1+{1315\over252}{\alpha\over\pi}+O(\alpha^2)
\right)$,
respectively. The corrections become meaningless when we {\it replace\/}
$\alpha\to\alpha_g$.
This would seem to be a
devastating objection to the results quoted in Ref.~\cite{ginz2} and used
in Ref.~\cite{d0}. But even if one closes one's eyes to higher order effects,
it seems clear that the mass limits quoted are inconsistent.
If we take the cross section given by Eq.~(\ref{llcs}) and make the
substitution (\ref{subs}), we obtain for the low energy light-by-light
scattering cross section in the presence of a monopole loop
\begin{equation}
\sigma_{\gamma\gamma\to\gamma\gamma}\approx {973\over2592000\pi}
{n^8\over\alpha^4}{\omega^6\over M^8}=4.2\times 10^4 \,n^8{1\over M^2}\left(
\omega\over M\right)^6.
\label{monocs}
\end{equation}
If the cross section were dominated by a single partial wave of angular
momentum $J$, the cross section would be bounded by
\begin{equation}
\sigma\le{\pi(2J+1)\over s}\sim {3\pi\over s},
\end{equation}
if we take $J=1$ as a typical partial wave. Comparing this with the
cross section given in Eq.~(\ref{monocs}), we obtain the following
inequality for the cross section to be consistent with unitarity,
\begin{equation}
{M\over\omega}\gtrsim 3 n.
\label{unitbd}
\end{equation}
But the limits quoted for the monopole mass are less than this:
\begin{equation}
{M\over n}>870 \mbox{ Gev}, \quad \mbox{spin } 1/2,
\end{equation}
because, at best, a minimum $\langle\omega\rangle\sim 300$ GeV;
the theory cannot sensibly be applied below a monopole mass of about 1 TeV.
(Note that changing the
value of $J$ in the unitarity limits has very little effect on the bound
(\ref{unitbd}) since an 8th root is taken: replacing $J$ by 50 reduces
the limit (\ref{unitbd}) only by 50\%.)
Similar remarks can be directed toward the De R\'ujula limits \cite{DeRujula}.
That author, however, notes the ``perilous use of a perturbative expansion
in $g$.'' However, he fails to use the correct vertex, Eq.~(\ref{mmvertex}),
instead appealing to duality, and even so he admittedly omits
enormous radiative corrections of $O(\alpha_g)$ without any justification
other than what we believe is a specious reference to the use of effective
Lagrangian techniques for these processes.
\section{Proposed Remedies}
Apparently, then, the formal small $\omega$ result obtained from the
Euler-Heisenberg Lagrangian cannot be valid beyond a photon energy
$\omega/M\gtrsim0.1$. The reader might ask why one cannot use
duality to convert the monopole coupling with an arbitrary photon to
the ordinary vector coupling. The answer is that little is thereby
gained, because the coupling of the photon to ordinary charged particles
is then converted into a complicated form analogous to Eq.~(\ref{mmint}).
This point is stated and then ignored in Ref.~\cite{DeRujula} in the
calculation of $Z\to3\gamma$.
There is, in general, no way of avoiding the complication of including
the string.
We are currently undertaking realistic calculations of virtual (monopole
loop) and real (monopole production) magnetic monopole processes \cite{gm}.
These calculations are, as the reader may infer, somewhat difficult
and involve subtle issues of principle involving the string, and it will
be some time before we have results to present. Therefore, here we wish
to offer plausible qualitative considerations, which we believe
suggest bounds that call into question
the results of Ginzburg et al. \cite{ginz1,ginz2}.
Our point is very simple. The interaction (\ref{mmint}) couples the magnetic
current to the dual field strength. This corresponds to
the velocity suppression in the interaction
of magnetic fields with electrically charged particles,
or to the velocity suppression in the interaction
of electric fields with magnetically charged
particles, as most simply seen in the magnetic analog of the Lorentz force,
\begin{equation}
{\bf F}=g({\bf B}-{{\bf v}\over c}\times {\bf E}).
\end{equation}
That is, the force between an electric charge $e$ and magnetic charge $g$,
moving with relative velocity $\bf v$ and with relative separation $\bf r$
is
\begin{equation}
{\bf F}=-{eg\over c}{{\bf v\times r}\over4\pi r^3}.
\end{equation}
This velocity suppression is reflected in nonrelativistic calculations.
For example, the energy loss in matter
of a magnetically charge particle is approximately obtained from that
of a particle with charge $Ze$ by the substitution \cite{ce}
\begin{equation}
{Ze\over v}\to{g\over c}.
\end{equation}
And the classical nonrelativistic dyon-dyon scattering cross section near
the forward direction is \cite{dyondyon}
\begin{equation}
{d\sigma\over d\Omega}\approx{1\over(2\mu v)^2}\left[\left(e_1g_2-e_2g_1
\over4\pi c\right)^2+\left(e_1e_2+g_1g_2\over4\pi v\right)^2\right]
{1\over(\theta/2)^4},\quad \theta\ll1,
\end{equation}
the expected generalization of the Rutherford scattering cross section at
small angles.
Of course, the true structure of the magnetic interaction and the resulting
scattering cross section is much more complicated. For example, classical
electron-monopole or dyon-dyon scattering exhibits rainbows and glories,
and the quantum scattering exhibits
a complicated oscillatory behavior in the backward direction \cite{dyondyon}.
These reflect the complexities of the magnetic interaction between
electrically and magnetically charged particles, which can be represented as a
kind of angular momentum \cite{dm,gold}. Nevertheless, for the purpose of
extracting qualitative information, the naive substitution,
\begin{equation}
e\to {v\over c}g,
\end{equation}
seems a reasonable first step.\footnote{This, and the extension of this
idea to virtual processes, leaves aside the troublesome issue
of radiative corrections. The hope is that an effective Lagrangian
can be found by approximately integrating over the fermions
which incorporates these effects.}
Indeed, such a substitution was used in the
proposal \cite{ou} to estimate production rates of monopoles at Fermilab.
The situation is somewhat less clear for the virtual processes
considered here. Nevertheless, the interaction (\ref{mmint}) suggests
there should, in general, be a softening of the vertex. In the current
absence of a valid calculational scheme, we will merely suggest two
plausible alternatives to the mere replacement procedure adopted in
Refs.~\cite{DeRujula,ginz1,ginz2,graf}.
We first suggest, as seemingly Ref.~\cite{ginz2} does,
that the approximate effective vertex incorporates
an additional factor of $\omega/M$.
Thus we propose the following estimate for the $\gamma\gamma$ cross
section in place of Eq.~(\ref{monocs}),
\begin{equation}
\sigma_{\gamma\gamma\to\gamma\gamma}\sim10^4 n^8{1\over M^2}\left(\omega\over
M\right)^{14},
\label{goodcs}
\end{equation}
since there are four suppression factors in the amplitude.
Now a considerably larger value of $\omega$ is consistent with unitarity,
\begin{equation}
{M\over\omega}\gtrsim\sqrt{3n},
\end{equation}
if we take $J=1$ again. We now must re-examine the $\sigma_{pp\to\gamma
\gamma X}$ cross section.
In the model
given in Ref.~\cite{ginz2}, where the photon energy distribution is
given in terms of the functions $f(y)$, $y=\omega/E$,
the physical cross section is given by
\begin{equation}
\sigma_{pp\to\gamma\gamma X}=\left(\alpha\over\pi\right)^2\int{dy_1\over y_1}
{dy_2\over y_2}f(y_1)f(y_2)\sigma_{\gamma\gamma\to\gamma\gamma}=
\int dy_1 dy_2{d\sigma\over dy_1\,dy_2},
\end{equation}
where now ({\it cf.} Eq.~(25) of Ref.~\cite{ginz2})
\begin{equation}
{d\sigma\over dy_1\,dy_2}=\left(\alpha\over\pi\right)^2R E^6\left(E\over M
\right)^8y_1^6f(y_1)y_2^6f(y_2),
\label{newdsigma}
\end{equation}
where, for spin 1/2, (up to factors of order unity)
\begin{equation}
R\sim{10^{-4}\over\alpha^4}\left( n\over M\right)^8.
\end{equation}
The result in (\ref{newdsigma}) differs from that in Ref.~\cite{ginz2}
by a factor of $(E/M)^8y_1^4y_2^4$. The photon distribution
function $y^2f(y)$ used is rather strongly
peaked at $y\sim 0.3$. (This peaking is necessary to have any chance
of satisfying the low-frequency criterion.) When we multiply by $y^4$, the
amplitude is greatly reduced and the peak is shifted above $y=1/2$, violating
even the naive criterion for the validity of perturbation theory.
Nevertheless, the integral of the distribution function is reduced by two
orders of magnitude, that is,
\begin{equation}
{\int_0^1 dy\,y^6f(y)\over\int_0^1dy\,y^2 f(y)}\sim 10^{-2}.
\end{equation}
This reduces the mass limit quoted in \cite{d0} by a factor of $1/\sqrt{3}$,
to about 500 GeV, where $\langle\omega\rangle/M\approx 0.9$. This
dubious result makes us conclude that
it is impossible to derive any limit for the monopole mass from the present
data.
As for the De R\'ujula limit\footnote{We note that De R\'ujula also considers
the monopole vacuum polarization correction to $g_V/g_A$, $g_A$, and $m_W/m_Z$,
proportional to $(m_Z/M)^2$ in each case, once again ignoring both the
string and the radiative correction problem. He assumes that the
monopole is a heavy vector-like fermion, and obtains a limit of
$M/n>8m_Z$. Our ansatz changes $(m_Z/M)^2$ to $(m_Z/M)^4$, so that
$M/\sqrt{n}>\sqrt{8}m_Z\approx250$ GeV, a substantial reduction.}
from the $Z\to3\gamma$ process, if we
insert a suppression factor of $\omega/M$ at each vertex and integrate over
the final state photon distributions, given by Eq.~(18)
of Ref.~\cite{DeRujula},
the mass limit is reduced to $M/\sqrt{n}\gtrsim1.4m_Z\sim120$
GeV, again grossly violating the low
energy criterion. And the limit deduced from the vacuum polarization
correction to the anomalous magnetic moment of the muon due to virtual
monopole pairs \cite{graf} is reduced to 2 GeV.
The reader might object that this $\omega/M$ softening of the vertex has
little field-theoretic basis. Therefore, we propose a second possibility that
does have such a basis. The vertex (\ref{mmvertex}) suggests, and detailed
calculation supports (based on the tensor structure of the photon
amplitudes\footnote{For example, the naive monopole loop contribution to vacuum
polarization differs from that of an electron loop (apart from charge and
mass replacements) entirely by the replacement in the latter of
$(g_{\mu\nu}-q_\mu q_\nu/q^2)\to({\bf q}^2/q_0^2)(\delta_{ij}-q_iq_j/{\bf q}^2)
$, when $n^\mu$ points in the time direction.
Apart from this different tensor structure, the vacuum polarization
is given by exactly the usual formula, found, for example in Ref.~\cite{js1}.
Details of this and related
calculations will be given in Ref.~\cite{gm}.}) the introduction of the
string-dependent factor $\sqrt{q^2/(n\cdot q)^2}$ at each vertex, where $q$ is
the photon momentum. Such a factor is devastating to the indirect monopole
searches---for any process involving a real photon, such as that of the D0
experiment \cite{d0} or for $Z\to3\gamma$ discussed in \cite{DeRujula}, the
amplitude vanishes. Because such factors can and do appear in full monopole
calculations, it is clearly premature to claim any limits based on virtual
processes involving real final-state photons.
\section{Conclusions}
We do not take our reduced limits on monopole masses very seriously. Rather,
we believe they demonstrate our point that given the dual difficulties
of theoretically treating monopoles, that is, incorporating the string
and dealing with enormously strong coupling,\footnote{None of the papers
dealing with virtual monopole effects, \cite{DeRujula,ginz1,ginz2,graf},
in fact, incorporate these nontrivial effects.} it is premature to attempt
to set any limits on monopole masses based on virtual effects. A direct
search stills seems much less problematic.
\section* {ACKNOWLEDGMENTS}
We thank Igor Solovtsov for helpful conversations and we are
grateful to the U.S.~Department of Energy for financial support.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 9,225 |
Q: Unable to resolve domain names from PHP in Apache. Works from command line we just migrated our production server to a new host (dedicated servers). And now we are unable resolve domain names from PHP pages when loaded from Apache. However, it works fine when we execute the php page from the command line.
For example, the following PHP code returns the IP address when executed from command line but fails when page is loaded in browser thru Apache.
$ip = gethostbyname('www.google.com');
echo $ip;
Any thoughts? We are running Apache 2.4 on Windows Server 2012.
Thanks for your help.
| {
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} | 782 |
Q: Placing text on top of a image I've been looking on several threads on stackoverflow, but cant seem to make it work. What i've found out is that i need to apply relative position on the parent div and then absolute on the child text, but this is not working? what am i doing wrong`
.the-image {
position: relative;
border: 1px solid;
width: auto;
}
.the-h3 {
z-index:100;
position:absolute;
color:white;
font-size:24px;
font-weight:bold;
left:150px;
top:350px;
}
.the-h3 span {
color: #ffffff;
letter-spacing: -1px;
background: rgb(0, 0, 0); /* fallback color */
background: rgba(0, 0, 0, 0.7);
padding: 10px;
}
<div class="the-image">
<img style="height: 200px" src="http://i.imgur.com/w15Db.jpg"></img>
<h3 class="the-h3"><span>TEST</span></h3>
</div>
A: You are giving the h3 a top property which is more than the image is high.
Simply lower that value to something more fitting:
.the-image {
position: relative;
border: 1px solid;
width: auto;
}
.the-h3 {
z-index:100;
position:absolute;
color:white;
font-size:24px;
font-weight:bold;
left:150px;
top:10px;
}
.the-h3 span {
color: #ffffff;
letter-spacing: -1px;
background: rgb(0, 0, 0); /* fallback color */
background: rgba(0, 0, 0, 0.7);
padding: 10px;
}
<div class="the-image">
<img style="height: 200px" src="http://i.imgur.com/w15Db.jpg"></img>
<h3 class="the-h3"><span>TEST</span></h3>
</div>
| {
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} | 7,966 |
\section{Introduction and results}\label{intro}
\subsection{The equation}
Consider the stochastic heat equation with Neumann boundary
conditions:
\begin{equation}\label{she}
\left\{
\begin{array}{rcl}
\displaystyle\partial_t u(t,x) &=&
\partial_{xx}u(t,x) + b(u(t,x)) + \sigma(u(t,x))\dot W(t,x),
\quad t\geq 0,\; x\in [0,1],\\
\displaystyle u(0,x)&=&u_0(x) , \quad x\in[0,1],\\
\partial_x u(t,0) &=&\partial_x u(t,1)=0,\quad t> 0.
\end{array}
\right. \hskip-0.3cm
\end{equation}
Here $b,\sigma: {\mathbb{R}}\mapsto {\mathbb{R}}$ are the drift and diffusion coefficients
and $u_0:[0,1]\mapsto {\mathbb{R}}$ is the initial condition. We write formally
$W(dt,dx)=\dot W(t,x)dtdx$, for $W(dt,dx)$ a white noise on
$[0,\infty)\times [0,1]$ based on
$dtdx$, see Walsh \cite{w}.
We will always assume in this paper that $b,\sigma$ are Lipschitz-continuous,
that is for some $C$,
\renewcommand{\theequation}{${{\mathcal H}}$}
\begin{equation}
\hbox{for all $r,z \in {\mathbb{R}}$, } \quad
|b(r)-b(z)|+|\sigma(r)-\sigma(z)| \leq C |r-z|.
\end{equation}
\renewcommand{\theequation}{\arabic{equation}}
\addtocounter{equation}{-1}
Our goals in this paper are the following:
$\bullet$ prove a strong existence and (partial)
uniqueness result when the initial condition
$u_0$ only belongs to $L^1([0,1])$ and some stability results of the
solution with respect to such an initial condition;
$\bullet$ study the uniqueness of invariant measures and the asymptotic
confluence of solutions.
\vskip0.2cm
We will investigate these two points by using some {\it a priori}
estimates on the difference between two solutions
$u,v$, obtained as a martingale dissipation of the
$L^1([0,1])$-norm of $u(t)-v(t)$.
\vskip0.2cm
Let us mention that our results extend without difficulty
to the case of Dirichlet boundary conditions and to the case of
the unbounded domain ${\mathbb{R}}$ (with $u_0 \in L^1({\mathbb{R}})$).
\vskip0.2cm
This equation has been much investigated, in particular
since the work of Walsh \cite{w}. In \cite{w}, one can find definitions of
weak solutions, existence and uniqueness results, as well as
proofs that solutions are H\"older-continuous, enjoy a Markov property, etc.
Let us mention for example
the works of Bally-Gyongy-Pardoux \cite{bgp} (existence of solutions
when the drift is only measurable), Gatarek-Goldys \cite{gg} (existence
of solutions in law), Donati-Pardoux (comparison results and reflection
problems), Bally-Pardoux (smoothness of the law of the solution),
Bally-Millet-Sanz \cite{bms} (support theorem), etc.
Sowers \cite{s}, Mueller \cite{m} and Cerrai \cite{c} have obtained some results
on the invariant distributions and convergence to equilibrium.
\subsection{Weak solutions}
We will consider two types of {\it weak} solutions, which we now precisely
define, following the ideas of Walsh \cite{w}. When we refer to predictability,
this is with respect to the filtration $({{\mathcal F}}_t)_{t\geq 0}$
generated by $W$, that is ${{\mathcal F}}_t=\sigma(W(A),A\in {{\mathcal B}}([0,t]\times[0,1]))$.
\vskip0.2cm
We denote by $L^p([0,1])$ the set of all measurable functions
$f:[0,1]\mapsto {\mathbb{R}}$ such that $||f||_{L^p([0,1])}=
(\int_0^1 |f(x)|^pdx)^{1/p}<\infty$.
\vskip0.2cm
Finally, we denote by $G_t(x,y)$ the Green kernel associated with the heat
equation $\partial_t u = \partial_{xx}u$ on ${\mathbb{R}}_+\times [0,1]$
with Neumann boundary conditions, whose explicit form can be found in
Walsh \cite{w}.
Here we will only use that for some $C_T$, for all $x,y\in [0,1]$,
all $t\in [0,T]$, see \cite{w},
\begin{equation}\label{ineqgt}
0\leq G_t(x,y)\leq \frac{C_T}{\sqrt t}e^{-|x-y|^2/4t}.
\end{equation}
\begin{defi} Assume $({{\mathcal H}})$, and consider a ${\mathbb{R}}$-valued
predictable process $u=(u(t,x))_{t\geq 0, x\in [0,1]}$.
(i) For $u_0\in L^1([0,1])$, $u$
is said to be a {\bf weak} solution to (\ref{she}) starting from
$u_0$ if a.s.,
\begin{equation}\label{bd1}
\hbox{for all } T>0, \quad
\sup_{[0,T]} ||u(t)||_{L^1([0,1])}
+ \int_0^T ||\sigma(u(t))||_{L^2([0,1])}^2 dt<\infty
\end{equation}
and if for all $\varphi \in C^2_b([0,1])$ such
that $\varphi'(0)=\varphi'(1)=0$, for all $t\geq 0$,
a.s.,
\begin{eqnarray}\label{sheweak}
\int_0^1 u(t,x)\varphi(x) dx &=& \int_0^1 u_0(x)\varphi(x) dx
+ \int_0^t \int_0^1 \sigma(u(s,x))\varphi(x) W(ds,dx) \\
&&+ \int_0^t \int_0^1 [u(s,x)\varphi''(x) + b(u(s,x))\varphi(x)]dxds.
\nonumber
\end{eqnarray}
(ii) For $u_0$ bounded-measurable, $u$
is said to be a {\bf mild} solution to (\ref{she}) starting from
$u_0$ if a.s.,
\begin{equation}\label{bd2}
\hbox{for all } T>0, \quad
\sup_{[0,T]\times [0,1]} |u(t,x)|<\infty
\end{equation}
and if for all $t\geq 0$, all $x\in [0,1]$, a.s.,
\begin{equation}\label{shemild}
u(t,x)= \int_0^1 G_t(x,y)u_0(y)dy + \ds\int_0^t \int_0^1 G_{t-s}(x,y)
[\sigma(u(s,y)) W(ds,dy)+ b(u(s,y)) dyds].
\end{equation}
\end{defi}
Let us make a few comments.
Recall that for $(H(s,y))_{s\geq 0, y\in [0,1]}$ a ${\mathbb{R}}$-valued
predictable process,
the stochastic integral $\int_0^t \int_0^1 H(s,y)W(ds,dy)$ is well-defined
if and only if $\int_0^t \int_0^1 H^2(s,y) dyds <\infty$ a.s.
\vskip0.2cm
$\bullet$ Thus (\ref{bd1}) implies that all the terms in
(\ref{sheweak})
are well-defined. Clearly, condition (\ref{bd1}) is not far from minimal.
\vskip0.2cm
$\bullet$ Next, (\ref{bd2}) and (\ref{ineqgt}) imply that all the terms
in (\ref{shemild})
are well-defined, but here (\ref{bd2}) is clearly far from optimal.
\vskip0.2cm
When $u_0$ only belongs to $L^1([0,1])$, we will only be able to prove that
(\ref{bd1}) holds.
\vskip0.2cm
Let us finally recall that Walsh \cite{w} proved, under $({{\mathcal H}})$, that for any
bounded-measurable initial condition $u_0$, there exists a unique
mild solution $u$ to (\ref{she}),
which is also a weak solution and which furthermore
satisfies, for all $p\geq 1$, all $T>0$,
${\mathbb{E}}[\sup_{[0,T]\times[0,1]} |u(t,x)|^p ]<\infty$.
\subsection{Existence and stability in $L^1([0,1])$}
Our first goal is to extend the existence theory to more
general initial conditions.
\begin{theo}\label{exstab}
Assume $({{\mathcal H}})$.
(i) For $u_0 \in L^1([0,1])$, there exists
a weak solution $u$
to (\ref{she}) starting from $u_0$.
(ii) This solution is unique in the following sense: for any sequence
of bounded-measurable functions $u_0^n:[0,1]\mapsto {\mathbb{R}}$ such that
$\lim_n ||u_0^n-u_0||_{L^1([0,1])}=0$, the sequence
$\sup_{[0,T]} ||u^n(t)-u(t)||_{L^1([0,1])}$ tends to $0$ in probability for
any $T$. Here $u^n$ is the unique mild solution to (\ref{she})
starting from $u_0^n$.
(iii) For $u_0,v_0 \in L^1([0,1])$, consider the two weak solutions
$u$ and $v$ to (\ref{she}) starting from $u_0$ and $v_0$ built in (i).
For all $\gamma \in (0,1)$, all $T\geq 0$, we have
$$
{\mathbb{E}}\left[\sup_{[0,T]}||u(t)-v(t)||_{L^1([0,1])}^\gamma +
\left(\int_0^T ||\sigma(u(t))-\sigma(v(t))||_{L^2([0,1])}^2dt \right)^{\gamma/2}
\right] \leq C_{b,\gamma,T} ||u_0-v_0||_{L^1([0,1])}^\gamma,
$$
where $C_{b,\gamma,T}$ depends only on $b,\gamma,T$.
(iv) Assume now that $b$ is non-increasing.
For $u_0,v_0 \in L^1([0,1])$, let $u,v$ be the two weak solutions
to (\ref{she}) starting from $u_0$ and $v_0$ built in (i).
For all $\gamma \in (0,1)$, we have
\begin{eqnarray*}
{\mathbb{E}}\Big[\sup_{[0,\infty)}||u(t)-v(t)||_{L^1([0,1])}^\gamma +
\left(\int_0^\infty ||b(u(t))-b(v(t)) ||_{L^1([0,1])} dt \right)^\gamma &&\\
+\left(\int_0^\infty
||\sigma(u(t))-\sigma(v(t))||_{L^2([0,1])}^2dt \right)^{\gamma/2}
\Big] &\leq& C_{\gamma} ||u_0-v_0||_{L^1([0,1])}^\gamma,
\end{eqnarray*}
where $C_{\gamma}$ depends only on $\gamma$.
\end{theo}
Observe that this result contains a regularization property.
For example if $\sigma(z)=z$,
even if $u_0$ does not belong to $L^2([0,1])$, the weak solution
satisfies (\ref{bd1}) and in particular $\sigma(u(t))=u(t)\in L^2([0,1])$
for a.e. $t>0$. For the same reasons, the stability result (iii)
provides a better estimate for a.e. $t>0$ than for $t=0$.
\vskip0.2cm
To our knowledge, Theorem \ref{exstab} is the first result concerning
$L^1([0,1])$ initial conditions. Many works concern bounded-measurable
(or continuous)
initial conditions, see Walsh \cite{w}, Bally-Gyongy-Pardoux \cite{bgp},
Cerrai \cite{c}. Another abundant literature deals with the Hilbert
case (initial conditions in $L^2([0,1])$), see Pardoux \cite{p},
Da Prato-Zabczyk
\cite{dz}, Gatarek-Goldys \cite{gg}.
\vskip0.2cm
The present well-posedness result is quite satisfying, since the requirement
that $u_0 \in L^1([0,1])$ is very weak and seems necessary for
(\ref{sheweak}) to make sense.
\subsection{Large time behavior}
We now wish to study the uniqueness of invariant measures.
\begin{defi}\label{dinv}
A probability measure $Q$ on $L^1([0,1])$ is said to be an invariant
distribution for (\ref{she}) if, for $u_0$ a $L^1([0,1])$-valued
random variable with law $Q$ independent of $W$, for $u$
the weak solution to (\ref{she}) starting from $u_0$ built
in Theorem \ref{exstab}, ${{\mathcal L}}(u(t))=Q$ for all $t\geq 0$.
\end{defi}
We have the following result.
\begin{theo}\label{uniinv}
Assume $({{\mathcal H}})$, that $b$ is non-increasing and that
$(\sigma,b):{\mathbb{R}}\mapsto{\mathbb{R}}^2$ is injective. Then
(\ref{she}) admits at most one invariant distribution.
\end{theo}
To prove the asymptotic confluence of solutions,
we need to strengthen the injectivity assumption.
\renewcommand{\theequation}{${{\mathcal I}}$}
\begin{equation}
\left\{
\begin{array}{l}
\hbox{There is a strictly increasing convex function }
\rho:{\mathbb{R}}_+\mapsto {\mathbb{R}}_+ \hbox{ with } \rho(0)=0 \hbox{ such that }\\
\hbox{for all $r,z\in {\mathbb{R}}$, } \quad
|b(r)-b(z)|+|\sigma(r)-\sigma(z)|^2 \geq \rho(|r-z|).
\end{array}
\right.
\end{equation}
\renewcommand{\theequation}{\arabic{equation}}
\addtocounter{equation}{-1}
\begin{theo}\label{confl}
Assume $({{\mathcal H}})$, that $b$ is non-increasing and $({{\mathcal I}})$.
(i) The following asymptotic
confluence property holds: for $u_0,v_0\in L^1([0,1])$,
for $u,v$
the weak solutions to (\ref{she}) starting from $u_0$ and $v_0$
built in Theorem \ref{exstab},
$$
\hbox{a.s., } \quad \lim_{t\to \infty} ||u(t)-v(t)||_{L^1([0,1])} =0.
$$
(ii) Assume additionally that (\ref{she}) admits an invariant
distribution $Q$. Then for $u_0\in L^1([0,1])$, for
$u$ the corresponding weak solution
to (\ref{she}), $u(t)$ goes in law to $Q$ as $t\to\infty$.
\end{theo}
Clearly, $({{\mathcal I}})$ holds if $b$ is $C^1$ with $b'\leq -\epsilon<0$
(choose $\rho(z)=\epsilon z$)
or if $\sigma$ is $C^1$ with $|\sigma'|\geq \epsilon>0$ (choose $\rho(z)=(\epsilon z)^2$).
One may also combine conditions on $b$ and $\sigma$.
But $({{\mathcal I}})$ also holds if $b$ is $C^1$ and if $b'\leq 0$ vanishes reasonably.
For example if $b(z)=-{\rm{sg}}(z)\min(|z|,|z|^p)$ for some $p\geq 1$,
choose $\rho= \epsilon \rho_p$ with $\epsilon$ small enough and
$\rho_p(z)= z^p$ for $z\in [0,1]$
and $\rho_p(z)= p z -p+ 1$ for $z\geq 1$. If
$b(z)=-z-\sin z$, choose $\rho=\epsilon \rho_3$ with $\epsilon$ small enough.
One may also consider the case where
$\sigma$ is monotonous with $\sigma'$ vanishing
reasonably.
\vskip0.2cm
Let us now compare Theorems \ref{uniinv} and \ref{confl} with known results.
The works cited below sometimes
concern different boundary conditions, but we believe this is not
important.
\vskip0.2cm
$\bullet$ Sowers \cite{s} has proved the existence of an invariant
distribution supported by $C([0,1])$, assuming $({{\mathcal H}})$,
that $\sigma$ is bounded
and that $b$ is of the form $b(z)=- \alpha z + f(z)$, for some bounded $f$
and some $\alpha>0$.
He obtained uniqueness of this invariant distribution
when $\sigma$ is sufficiently small and bounded from below.
\vskip0.2cm
$\bullet$ Mueller \cite{m} has obtained some surprising coupling results,
implying in particular the uniqueness of
an invariant distribution as well as a the trend to equilibrium.
He assumes
$({{\mathcal H}})$, that $\sigma$ is bounded from above and from below and that
$b$ is non-increasing, with $|b(z)-b(r)|\geq \alpha |z-r|$
for some $\alpha>0$.
\vskip0.2cm
$\bullet$ Cerrai \cite{c} assumed that $\sigma$ is strictly
monotonous (it may vanish, but only at one point).
(i) She obtained an asymptotic
confluence result which we do not recall here and concerns,
roughly, the case
$b(z)\simeq -{\rm{sg}}(z) |z|^{m}$ as $z\to \pm \infty$, for some $m>1$.
(ii) Assuming $({{\mathcal H}})$, she proved uniqueness of the invariant
distribution as well as an asymptotic confluence property, under the conditions
that for all $r\leq z$, $b(z)-b(r)\leq \lambda (z-r)$,
and $|\sigma(z)-\sigma(r)| \geq \mu |z-r|$, for some $\mu>0$
and some $\lambda < \mu^2/2$
(if $b$ is non-increasing, choose $\lambda=0$).
\vskip0.2cm
Thus the main advantages of the present paper are that
the uniqueness of the invariant measure requires very few
conditions, and we allow $\sigma$
to vanish (it may be compactly supported).
\vskip0.2cm
{\it Example 1.} Assume $({{\mathcal H}})$ and that $b$ strictly decreasing.
Then there exists at most one invariant distribution.
If $b(z)=-z$ or $b(z)=-z-\sin z$ or $b(z)=-{\rm{sg}}(z)
\min(|z|,|z|^p)$ for some $p>1$,
then we have asymptotic confluence of solutions.
Here to apply \cite{s,m} one needs to assume additionally that
$\sigma$ is bounded from above and from below, while to apply
\cite{c}, one has to suppose that $\sigma$ is strictly monotonous.
\vskip0.2cm
{\it Example 2.} Assume $({{\mathcal H}})$, that $b$ is non-increasing and that
$\sigma$ is strictly monotonous.
Then there exists at most one invariant distribution.
If furthermore $\sigma$ is $C^1$ with $0< c< \sigma'<C$,
then we get asymptotic
confluence of solutions using \cite{c} or Theorem \ref{confl}
(here \cite{s,m} cannot apply, since $\sigma$ vanishes).
But now if $\sigma'\geq 0$ reasonably vanishes
then Theorem \ref{confl} applies, which is not the case of \cite{c}:
take e.g. $\sigma(z)= {\rm{sg}}(z) \min(|z|,|z|^p)$ for some $p>1$,
or $\sigma(z)=z+\sin z$.
\vskip0.2cm
{\it Example 3.} Consider the compactly supported coefficient
$\sigma(z)=(1-z^2){\bf 1}_{\{|z|\leq 1\}}$.
Assume that $b$ is $C^1$, non-increasing, with $b'(z) \leq -\epsilon<0$ for
$z\in (-\infty,-1)\cup\{0\}\cup(1,+\infty)$.
Then Theorems \ref{uniinv} and \ref{confl} apply, while \cite{s,m,c} do not.
Observe here that if $b(z_0)=0$ for some $z_0\notin (-1,1)$, then
$u(t)\equiv z_0$ is the (unique) stationary solution.
If now $b(-1)>0$ and $b(1)<0$,
then the invariant measure $Q$
(that exists due to Sowers \cite{s}) is unique and one may show,
using the comparison
Theorem of Donati-Pardoux \cite{dp}, that
$Q$ is supported by $[-1,1]$-valued continuous functions on $[0,1]$.
\vskip0.2cm
However, there are some cases where \cite{s,c} provide some better
results than ours.
\vskip0.2cm
{\it Example 4.} If $\sigma(z)=\mu z$ and $b(z)=\lambda z$, then
$u(t)\equiv 0$ is an obvious stationary solution.
Theorems \ref{uniinv} and \ref{confl} apply if
$\lambda \leq 0$ and $|\lambda|+|\mu|>0$. Cerrai \cite{c} was
able to treat the case $\lambda>0$ provided $\mu^2/2>\lambda$.
\vskip0.2cm
{\it Example 5.} If $\sigma$ is small enough and bounded from below
and if $b(z)=-\alpha z + h(z)$, with $\alpha>0$ and $h$ bounded,
then Sowers \cite{s} obtains the uniqueness of the invariant distribution
even if $b$ is not non-increasing.
\subsection{Plan of the paper}
In the next section, we prove some inequalities concerning
the $L^1([0,1])$-norm of the difference between any pair of {\it mild} solutions
to (\ref{she}). Section \ref{exist} is dedicated to the proof
of our existence result Theorem \ref{exstab}.
Theorems \ref{uniinv} and \ref{confl} are checked in Section \ref{ergo}.
We briefly discuss the multi-dimensional equation in Section \ref{dim2}
and conclude the paper with an appendix containing technical results.
\section{On the $L^1([0,1])$-norm of the difference
between two mild solutions}\label{basic}
All our study is based on the following result. We set
${\rm{sg}}(z)=1$ for $z\geq 0$ and ${\rm{sg}}(z)=-1$ for $z< 0$.
\begin{pro}\label{eql1}
Assume $({{\mathcal H}})$.
For two bounded-measurable initial conditions
$u_0,v_0$, let $u,v$ be the corresponding mild
solutions to (\ref{she}). Then, enlarging the probability space if necessary,
there is a
Brownian motion $(B_t)_{t\geq 0}$ such that a.s., for all
$t\geq 0$,
\begin{eqnarray}\label{ineqfc}
||u(t)-v(t)||_{L^1([0,1])}&\leq&||u_0-v_0||_{L^1([0,1])}
+ \ds\int_0^t || \sigma(u(s))-\sigma(v(s)) ||_{L^2([0,1])} dB_s \\
&&+\ds\int_0^t \int_0^1 {\rm{sg}}(u(s,x)-v(s,x))(b(u(s,x))-b(v(s,x)))dxds.\nonumber
\end{eqnarray}
\end{pro}
\begin{proof} We divide the proof into several steps, following closely
the ideas of Donati-Pardoux \cite[Theorem 2.1]{dp}, to which we refer
for technical details.
\vskip0.2cm
{\it Step 1.} Consider an orthonormal basis $(e_k)_{k\geq 1}$ of
$L^2([0,1])$. For $k\geq 1$, we
set $B^k_t=\int_0^t \int_0^1 e_k(x) W(ds,dx)$. Then
$(B^k)_{k\geq 1}$ is a family of independent Brownian motions.
For $n\geq 1$, consider the unique adapted solution
$u^n\in L^2(\Omega\times[0,T],V)$, where $V=\{f\in H^1([0,1]), f'(0)=f'(1)=0\}$,
to
$$
u^n(t,x)=u_0(x)+\ds\int_0^t \left[\partial_{xx}u^n(s,x)ds + b(u^n(s,x))\right]ds
+ \sum_{k=1}^n \int_0^t \sigma(u^n(s,x))e_k(x)dB^k_s.
$$
We refer to Pardoux \cite{p} for existence, uniqueness and properties
of this solution. We also consider the solution $v^n$ to the same equation
starting from $v_0$. Then, as shown in \cite{dp},
\begin{equation}\label{cacv}
\lim_n \sup_{[0,T]\times[0,1]}{\mathbb{E}}[|u^n(t,x)-u(t,x)|^2 +|v^n(t,x)-v(t,x)|^2 ]=0.
\end{equation}
{\it Step 2.} For $\epsilon>0$, we introduce a nonnegative
$C^2$ function $\phi_\epsilon$ such that $\phi_\epsilon(z)= |z|$ for $|z|\geq \epsilon$,
with $|\phi_\epsilon'(z)| \leq 1$ and $0\leq \phi_\epsilon''(z)
\leq 2\epsilon^{-1}{\bf 1}_{|z|<\epsilon}$.
When applying the It\^o formula (see \cite{dp} for details), we get
\begin{eqnarray}\label{pbfc}
&&\hskip-1.5cm\int_0^1 \phi_\epsilon(u^n(t,x)-v^n(t,x))dx
= \int_0^1 \phi_\epsilon(u_0(x)-v_0(x))dx \\
&&+ \ds\int_0^t \int_0^1 \phi_\epsilon'(u^n(s,x)-v^n(s,x))
\partial_{xx}[u^n(s,x)-v^n(s,x)]dxds \nonumber \\
&&+ \ds\int_0^t \int_0^1 \phi_\epsilon'(u^n(s,x)-v^n(s,x))[b(u^n(s,x))-b(v^n(s,x))]dxds\nonumber \\
&&+ \sum_{k=1}^n \int_0^t \int_0^1 \phi_\epsilon'(u^n(s,x)-v^n(s,x))
[\sigma(u^n(s,x))-\sigma(v^n(s,x)) ] e_k(x)dx dB^k_s\nonumber \\
&&+ \frac{1}{2} \sum_{k=1}^n\ds\int_0^t \int_0^1 \phi_\epsilon''(u^n(s,x)-v^n(s,x))
[\sigma(u^n(s,x))-\sigma(v^n(s,x)) ]^2 e^2_k(x)dx ds\nonumber \\
&&=:I_\epsilon^1+I_\epsilon^2(t)+I_\epsilon^3(t)+I_\epsilon^4(t)+I_\epsilon^5(t).\nonumber
\end{eqnarray}
Since $|z|\leq \phi_\epsilon(z)\leq |z|+\epsilon$ for all $z$, we easily get, a.s.,
$$
\lim_{\epsilon\to 0}\int_0^1 \phi_\epsilon(u^n(t,x)-v^n(t,x))dx =||u^n(t)-v^n(t)||_{L^1([0,1])}
\quad \hbox{ and } \quad
\lim_{\epsilon\to 0} I^1_\epsilon =||u_0-v_0||_{L^1([0,1])}.
$$
An integration by parts, using that $\partial_x[u^n(t,0)-v^n(t,0)]=
\partial_x[u^n(t,1)-v^n(t,1)]=0$ shows that
$$
I^2_\epsilon(t)=-\ds\int_0^t \int_0^1 \phi_\epsilon''(u^n(s,x)-v^n(s,x))
[\partial_{x}(u^n(s,x)-v^n(s,x))]^2\leq 0.
$$
Since $\phi_\epsilon''(z-r)(\sigma(z)-\sigma(r))^2\leq C
\epsilon^{-1} {\bf 1}_{|z-r|\leq \epsilon} |z-r|^2 \leq C \epsilon$ by $({{\mathcal H}})$,
we have $I^5_\epsilon(t) \leq Cnt\epsilon $, whence
$$
\lim_{\epsilon\to 0} I_\epsilon^5(t)=0 \hbox{ a.s.}
$$
Using that $|\phi_\epsilon'(z)- {\rm{sg}}(z)|\leq {\bf 1}_{\{|z|\leq \epsilon\}}$ and $({{\mathcal H}})$,
one obtains a.s.
\begin{eqnarray*}
&&\lim_{\epsilon\to 0}\left| I^3_\epsilon(t) - \int_0^t\int_0^1 {\rm{sg}}(u^n(s,x)-v^n(s,x))
(b(u^n(s,x))-b(v^n(s,x))) dxds \right| \\
&\leq & \lim_{\epsilon\to 0} \int_0^t \int_0^1 {\bf 1}_{|u^n(s,x)-v^n(s,x)|\leq \epsilon}
|b(u^n(s,x))-b(v^n(s,x))| dxds \leq \lim_{\epsilon\to 0} Ct \epsilon =0.
\end{eqnarray*}
Similarly,
\begin{eqnarray*}
&&\lim_{\epsilon\to 0} {\mathbb{E}}\left[\left(I^4_\epsilon(t)-
\sum_{k=1}^n \int_0^t \int_0^1 {\rm{sg}}(u^n(s,x)-v^n(s,x))
[\sigma(u^n(s,x))-\sigma(v^n(s,x)) ] e_k(x)dx dB^k_s \right)^2 \right]=0.
\end{eqnarray*}
Thus we can pass to the limit as $\epsilon\to 0$ in (\ref{pbfc}) and get, a.s.,
\begin{eqnarray}\label{pbfcf}
||u^n(t)-v^n(t)||_{L^1([0,1])}&\leq&||u_0-v_0||_{L^1([0,1])} \nonumber \\
&&+\ds\int_0^t \int_0^1 {\rm{sg}}(u^n(s,x)-v^n(s,x))[b(u^n(s,x))-b(v^n(s,x))]dxds
\nonumber\\
&&+ \sum_{k=1}^n \ds\int_0^t \int_0^1
{\rm{sg}}(u^n(s,x)-v^n(s,x)) [\sigma(u^n(s,x))-\sigma(v^n(s,x))]
e_k(x)dx dB^k_s .
\end{eqnarray}
{\it Step 3.} Using $({{\mathcal H}})$, there holds, for all $r_1,z_1,r_2,z_2$ in
${\mathbb{R}}$,
\begin{eqnarray}\label{sglip}
\big|{\rm{sg}}(r_1-z_1)[\sigma(r_1)-\sigma(z_1)]
-{\rm{sg}}(r_2-z_2)[\sigma(r_2)-\sigma(z_2)] \big|
\leq C(|r_1-r_2|+|z_1-z_2|),\\
\big|{\rm{sg}}(r_1-z_1)[b(r_1)-b(z_1)]
-{\rm{sg}}(r_2-z_2)[b(r_2)-b(z_2)] \big|
\leq C(|r_1-r_2|+|z_1-z_2|).
\end{eqnarray}
Indeed, it suffices, by symmetry, to check that
$\big|{\rm{sg}}(r_1-z_1)[\sigma(r_1)-\sigma(z_1)]
-{\rm{sg}}(r_2-z_1)[\sigma(r_2)-\sigma(z_1)] \big|
\leq C|r_1-r_2|$. If ${\rm{sg}}(r_1-z_1)={\rm{sg}}(r_2-z_2)$, this is obvious.
If now $r_1\leq z_1 \leq r_2$ (or $r_1\geq z_1 \geq r_2$)
we get the upper-bound
$|\sigma(r_1)+\sigma(r_2)-2\sigma(z_1)| \leq C(|r_1-z_1|+|r_2-z_1|)
= C |r_1-r_2|$.
Using (\ref{cacv}), it is thus routine
to make $n$ tend to infinity in (\ref{pbfcf}) and to obtain, a.s.,
\begin{eqnarray}\label{coucou}
||u(t)-v(t)||_{L^1([0,1])}&\leq&||u_0-v_0||_{L^1([0,1])}
+\ds\int_0^t \int_0^1 {\rm{sg}}(u(s,x)-v(s,x))[b(u(s,x))-b(v(s,x))]dxds \nonumber \\
&&+ \sum_{k=1}^\infty \ds\int_0^t \int_0^1
{\rm{sg}}(u(s,x)-v(s,x)) [\sigma(u(s,x))-\sigma(v(s,x))]
e_k(x)dx dB^k_s.
\end{eqnarray}
For the last term, we used that, by the Plancherel identity,
setting for simplicity
\begin{eqnarray*}
\alpha_n(s,x)&=&{\rm{sg}}(u^n(s,x)-v^n(s,x)) [\sigma(u^n(s,x))-\sigma(v^n(s,x))],\\
\alpha(s,x)&=&{\rm{sg}}(u(s,x)-v(s,x)) [\sigma(u(s,x))-\sigma(v(s,x))],
\end{eqnarray*}
there holds
\begin{eqnarray*}
&&{\mathbb{E}}\Big[\Big( \sum_{k=1}^n \ds\int_0^t \int_0^1
\alpha_n(s,x) e_k(x) dB^k_s - \sum_{k=1}^\infty \ds\int_0^t \int_0^1
\alpha(s,x) e_k(x) dB^k_s\Big)^2 \Big]\nonumber \\
&\leq& \ds\int_0^t {\mathbb{E}}\Big[ \sum_{k\geq 1} \Big(\int_0^1\Big\{
\alpha_n(s,x)-\alpha(s,x)\Big\}e_k(x)dx \Big)^2 \Big]ds
+ \sum_{k\geq n+1} \ds\int_0^t {\mathbb{E}}\Big[ \Big(\int_0^1 \alpha(s,x)e_k(x)dx \Big)^2
\Big]ds\\
&\leq& \ds\int_0^t {\mathbb{E}}\Big[ || \alpha_n(s)-\alpha(s)||_{L^2([0,1])}^2\Big]ds
+ \sum_{k\geq n+1} \ds\int_0^t {\mathbb{E}}\Big[ \Big(\int_0^1
\alpha(s,x) e_k(x)dx \Big)^2 \Big]ds=:I_n(t)+J_n(t).
\end{eqnarray*}
Using (\ref{sglip}) and then (\ref{cacv}), $I_n(t)\leq
C \int_0^t \int_0^1 {\mathbb{E}}[|u^n(s,x)-u(s,x)|^2+|v^n(s,x)-v(s,x)|^2 ]dxds$
tends to $0$ as $n\to \infty$. Finally, $J_n(t)$ tends to $0$
because $\sum_{k\geq 1} \int_0^t {\mathbb{E}} [(\int_0^1
\alpha(s,x) e_k(x)dx )^2 ] ds = \int_0^t {\mathbb{E}}[||\alpha(s)||_{L^2([0,1])}^2] ds
\leq C\int_0^t \int_0^1 {\mathbb{E}}(|u(s,x)-v(s,x)|^2) dxds <\infty$.
\vskip0.2cm
{\it Step 4.}
A standard representation argument (see e.g. Revuz-Yor
\cite[Proposition 3.8 and Theorem 3.9 p 202-203]{ry})
concludes the proof, because the last term on the RHS of
(\ref{coucou}) is a continuous
local martingale with bracket
\begin{equation*}
\ds\int_0^t \sum_{k=1}^\infty \left(\int_0^1
{\rm{sg}}(u(s,x)-v(s,x))[\sigma(u(s,x))-\sigma(v(s,x))]e_k(x)dx \right)^2 ds
=\ds\int_0^t ||\sigma(u(s))-\sigma(v(s))||_{L^2([0,1])}^2 ds.
\end{equation*}
We used here again that $(e_k)_{k\geq 1}$ is an orthonormal
basis of $L^2([0,1])$.
\end{proof}
\begin{cor}\label{ineql1g}
Adopt the notation and assumptions of Proposition \ref{eql1}.
For all $\gamma \in (0,1)$, all $T\geq 0$,
$$
{\mathbb{E}}\left[\sup_{[0,T]}||u(t)-v(t)||_{L^1([0,1])}^\gamma +
\left(\int_0^T ||\sigma(u(t))-\sigma(v(t))||_{L^2([0,1])}^2dt \right)^{\gamma/2}
\right] \leq C_{b,\gamma,T} ||u_0-v_0||_{L^1([0,1])}^\gamma,
$$
where $C_{b,\gamma,T}$ depends only on $b,\gamma,T$.
\end{cor}
\begin{proof}
Let $C$ be the Lipschitz constant of $b$. Denote by $L_t$ the RHS of
(\ref{ineqfc}). The It\^o formula yields
\begin{eqnarray*}
||u(t)-v(t)||_{L^1([0,1])}e^{-Ct}&\leq& L_te^{-Ct} \\
&=& ||u_0-v_0||_{L^1([0,1])}- C \ds\int_0^t
e^{-Cs }L_s ds\\
&&+ \ds\int_0^t || \sigma(u(s))-\sigma(v(s)) ||_{L^2([0,1])}e^{-Cs} dB_s \\
&&+\ds\int_0^t \int_0^1 e^{-Cs }{\rm{sg}}(u(s,x)-v(s,x))(b(u(s,x))-b(v(s,x)))dxds.
\end{eqnarray*}
But $\int_0^1 {\rm{sg}}(u(s,x)-v(s,x))(b(u(s,x))-b(v(s,x)))dx \leq
C ||u(s)-v(s)||_{L^1([0,1])} \leq C L_s$.
Hence
\begin{eqnarray*}
||u(t)-v(t)||_{L^1([0,1])}e^{-Ct}&\leq& ||u_0-v_0||_{L^1([0,1])}
+\ds\int_0^t || \sigma(u(s))-\sigma(v(s)) ||_{L^2([0,1])}e^{-Cs} dB_s=:M_t.
\end{eqnarray*}
Hence $M_t$ is a nonnegative local martingale with bracket
$\left< M \right> _t= \int_0^t || \sigma(u(s))-\sigma(v(s)) ||_{L^2([0,1])}^2
e^{-2Cs} ds$. Applying Lemma \ref{martingales}, we immediately get,
for $\gamma \in (0,1)$,
\begin{eqnarray*}
&&{\mathbb{E}}\left[\sup_{[0,\infty)}||u(t)-v(t)||_{L^1([0,1])}^\gamma e^{-C\gamma t}
+ \left( \int_0^\infty || \sigma(u(s))-\sigma(v(s)) ||_{L^2([0,1])}^2
e^{-2Cs} ds \right)^{\gamma/2} \right] \\
&\leq& C_\gamma ||u_0-v_0||_{L^1([0,1])}^\gamma.
\end{eqnarray*}
The result easily follows.
\end{proof}
Finally, one can say a little more when $b$ is non-increasing.
\begin{cor}\label{ineql1d}
Adopt the notation and assumptions of Proposition \ref{eql1} and assume
that $b$ is non-increasing. Then for all $\gamma \in (0,1)$,
\begin{eqnarray*}
{\mathbb{E}}\Big[\sup_{[0,\infty)}||u(t)-v(t)||_{L^1([0,1])}^\gamma +
\left(\int_0^\infty ||b(u(t))-b(v(t))||_{L^1([0,1])}dt \right)^{\gamma} &&\\
+\left(\int_0^\infty
||\sigma(u(t))-\sigma(v(t))||_{L^2([0,1])}^2dt \right)^{\gamma/2}
\Big] &\leq& C_{\gamma}||u_0-v_0||_{L^1([0,1])}^\gamma,\nonumber
\end{eqnarray*}
where $C_{\gamma}$ depends only on $\gamma$.
\end{cor}
\begin{proof}
Since $b$ is non-increasing, Proposition \ref{eql1} yields
\begin{eqnarray*}
&&||u(t)-v(t)||_{L^1([0,1])}+ \ds\int_0^t ||b(u(s))-b(v(s))||_{L^1([0,1])}ds \\
&\leq& ||u_0-v_0||_{L^1([0,1])}
+ \ds\int_0^t || \sigma(u(s))-\sigma(v(s)) ||_{L^2([0,1])} dB_s=:M_t,
\end{eqnarray*}
which is thus a nonnegative martingale with bracket
$\left< M \right> _t= \int_0^t || \sigma(u(s))-\sigma(v(s)) ||_{L^2([0,1])}^2 ds$.
Lemma \ref{martingales} allows us to conclude.
\end{proof}
\section{Existence theory in $L^1([0,1])$}\label{exist}
The goal of this section is to give the
\vskip0.2cm
\begin{preuve} {\it of Theorem \ref{exstab}.} We start with point (i).
Let thus $u_0\in L^1([0,1])$ and consider a sequence of bounded-measurable
initial conditions $(u_0^n)_{n\geq 1}$
such that $||u_0^n-u_0||_{L^1([0,1])}\leq 2^{-n}$. For each $n\geq 1$,
denote by $u^n$ the mild solution
to (\ref{she}) starting from $u_0^n$. Using Corollary \ref{ineql1g}
(with $\gamma=1/2$), we deduce that a.s.,
$$
\sum_{n\geq 1} \left[\sup_{[0,T]}||u^{n+1}(t)-u^n(t)||_{L^1([0,1])}^{1/2}
+ \left(\int_0^T ||\sigma(u^{n+1}(t))-\sigma(u^n(t))||_{L^2([0,1])}^2 dt
\right)^{1/4} \right]< \infty,
$$
which implies that
$$
\sum_{n\geq 1} \left[\sup_{[0,T]}||u^{n+1}(t)-u^n(t)||_{L^1([0,1])}
+ ||\sigma(u^{n+1})-\sigma(u^n)||_{L^2([0,T]\times[0,1])} \right]< \infty.
$$
Using some completeness arguments, we deduce that there are some
(predictable) processes $u$ and $S$ such that a.s., for all $T>0$,
$\sup_{[0,T]}||u(t)||_{L^1([0,1])}+\int_0^T ||S(t)||_{L^2([0,1])}^2dt <\infty$ and
$$
\lim_n \sup_{[0,T]}||u(t)-u^n(t)||_{L^1([0,1])} = 0, \quad
\lim_n ||S-\sigma(u^n)||_{L^2([0,T]\times[0,1])} = 0.
$$
Since $\sigma$ is Lipschitz-continuous, we deduce from
the first equality that $\lim_n ||\sigma(u)-\sigma(u^n)||_{L^1([0,T]\times[0,1])}
= 0$,
while from the second one,
$\lim_n ||S-\sigma(u^n)||_{L^1([0,T]\times[0,1])}= 0$. Consequently, $S=\sigma(u)$
a.e. and we finally conclude that a.s.,
\begin{equation}\label{ettac}
\hbox{ for all } T>0,\quad
\lim_n \left(\sup_{[0,T]}||u(t)-u^n(t)||_{L^1([0,1])} +\int_0^T ||\sigma(u(t))
-\sigma(u^n(t))||_{L^2([0,1])}^2 dt\right)= 0.
\end{equation}
It remains to prove that $u$ is a weak solution to (\ref{she}).
We have already seen that $u$ satisfies (\ref{bd1}). Next,
for $\varphi\in C^2_b([0,1])$ with $\varphi'(0)=\varphi'(1)=0$, for
$t\geq 0$, we know that a.s., $A^{n,\varphi}_t=B^{n,\varphi}_t$ for all
$n\geq 1$, where
\begin{eqnarray*}
A^{n,\varphi}_t&:=&\int_0^1 \varphi(x)u^n(t,x)dx
- \int_0^1 \varphi(x)u^n_0(x)dx
- \ds\int_0^t \int_0^1 [u^n(s,x)\varphi''(x) + b(u^n(s,x))\varphi(x)] dxds
\\
B^{n,\varphi}_t&:=& \ds\int_0^t \int_0^1 \sigma(u^n(s,x))\varphi(x) W(ds,dx).
\end{eqnarray*}
It directly follows from (\ref{ettac}) and $({{\mathcal H}})$ that a.s.,
$$
\lim_{n\to \infty} A^{n,\varphi}_t=\int_0^1 \varphi(x)u(t,x)dx
- \int_0^1 \varphi(x)u_0(x)dx
- \ds\int_0^t \int_0^1 [u(s,x)\varphi''(x) + b(u(s,x))\varphi(x)] dxds.
$$
We deduce that $B^{\varphi}_t:=\lim_n B^{n,\varphi}_t$ exists a.s.,
and it only remains to check that $B^\varphi_t=C^\varphi_t$ a.s., where
$C^\varphi_t:=\int_0^t \int_0^1
\sigma(u(s,x))\varphi(x) W(ds,dx)$.
To this end, consider, for $M>0$, the stopping time
$$
\tau_M = \inf\left\{ r \geq 0, \; \int_0^r ||\sigma(u(s))||_{L^2([0,1])}^2ds
+ \sup_n \int_0^r ||\sigma(u^n(s))||_{L^2([0,1])}^2ds \geq M\right\}.
$$
Using (\ref{ettac}) and the dominated convergence Theorem,
we see that for each $M>0$,
$$
\lim_n {\mathbb{E}}[|B^{n,\varphi}_{t\land \tau_M}- C^{\varphi}_{t\land \tau_M} |^2]
= \lim_n {\mathbb{E}}\left[\int_0^{t\land \tau_M}
||(\sigma(u(s))-\sigma(u^n(s))\varphi ||_{L^2([0,1])}^2 ds \right] = 0.
$$
But we also deduce from (\ref{ettac}) that a.s.,
$\sup_n \int_0^T ||\sigma(u^n(s))||_{L^2([0,1])}^2ds <\infty$ for all $T>0$,
whence $\lim_{M \to \infty} \tau_M = \infty$ a.s.
We easily conclude that $B^{n,\varphi}_t$ tends to $C^\varphi_t$ in probability,
whence $B^\varphi_t=C^\varphi_t$ a.s.
\vskip0.2cm
Point (ii) is easily checked: let $({\tilde u}_0^n)_{n\geq 1}$
be another sequence of bounded-measurable
initial conditions converging to $u_0$ and let
$({\tilde u}^n)_{n\geq 1}$ be the corresponding sequence of mild solutions
to (\ref{she}). Then necessarily, $||u_0^n - {\tilde u}_0^n||_{L^1([0,1])}$
tends to $0$,
whence, by Corollary \ref{ineql1g}, $\sup_{[0,T]}||u^n(t)-{\tilde u}^n(t)||_{L^1([0,1])}$
tends also to $0$, in probability. Using (\ref{ettac}),
we conclude that $\sup_{[0,T]}||u(t)-\tilde u^n(t)||_{L^1([0,1])}$ tends to $0$
in probability.
\vskip0.2cm
We now prove point (iii). For $u_0$ and $v_0$ in $L^1([0,1])$,
we consider $u_0^n$ and $v_0^n$ bounded-measurable with
$||u_0^n-u_0||_{L^1([0,1])}
+||v_0^n-v_0||_{L^1([0,1])} \leq 2^{-n}$. We denote by $u,v,u^n,v^n$ the
corresponding
weak solutions to (\ref{she}). In the proof of (i), we have seen that
a.s., $\lim_n \sup_{[0,T]} [||u^n(t)-u(t)||_{L^1([0,1])}
+||v^n(t)-v(t)||_{L^1([0,1])} ]=0$ and
$\lim_n \int_0^T [||\sigma(u^n(t))-\sigma(u(t))||_{L^2([0,1])}^2
+||\sigma(v^n(t))-\sigma(v(t))||_{L^2([0,1])}^2 ]dt =0$.
Using the Fatou Lemma and Corollary \ref{ineql1g}, we thus get
\begin{eqnarray*}
&&{\mathbb{E}}\left[\sup_{[0,T]}||u(t)-v(t)||_{L^1([0,1])}^\gamma
+ \left(\int_0^T ||\sigma(u(t))-\sigma(v(t))||_{L^2([0,1])}^2 dt
\right)^{\gamma/2} \right] \nonumber \\
&\leq& \liminf_n {\mathbb{E}}\left[\sup_{[0,T]}||u^{n}(t)-v^n(t)||_{L^1([0,1])}^\gamma
+ \left(\int_0^T ||\sigma(u^{n}(t))-\sigma(v^n(t))||_{L^2([0,1])}^2 dt
\right)^{\gamma/2} \right] \\
&\leq& \liminf_n C_{\gamma,T} ||u_0^n-v_0^n||_{L^1([0,1])}^\gamma
= C_{\gamma,T} ||u_0-v_0||_{L^1([0,1])}^\gamma.
\end{eqnarray*}
Point (iv) is checked similarly.
\end{preuve}
\section{Large time behavior}\label{ergo}
We now prove the uniqueness of the invariant measure.
\vskip0.2cm
\begin{preuve} {\it of Theorem \ref{uniinv}.}
Consider two invariant distributions $Q$ and ${\tilde Q}$ for (\ref{she}),
see Definition \ref{dinv}.
Let $u_0$ be $Q$-distributed and ${\tilde u}_0$ be ${\tilde Q}$-distributed.
Consider the corresponding (stationary) weak solutions $u,{\tilde u}$ to (\ref{she}).
Applying Theorem \ref{exstab}-(iv) and the Cauchy-Schwarz inequality,
$\int_0^\infty K_sds <\infty$ a.s., where
\begin{eqnarray*}
K_s:=K(u(s),{\tilde u}(s))=
||b(u(s))-b({\tilde u}(s))||_{L^1([0,1])} +
||\sigma(u(s))-\sigma({\tilde u}(s))||_{L^1([0,1])}^2.
\end{eqnarray*}
Using Lemma \ref{conc}, there is a sequence $(t_n)_{n\geq 1}$ such that
$K_{t_n}$ tends to $0$ in probability. Consider the function
$\phi(r)=r/(1+r)$ on ${\mathbb{R}}_+$, and define
$\Psi:L^1([0,1])\times L^1([0,1]) \mapsto {\mathbb{R}}_+$ as $\Psi(f,g)=\phi(K(f,g))$.
Then $\lim_n {\mathbb{E}}[\Psi(u(t_n),v(t_n))]=\lim_n {\mathbb{E}}[\phi(K_{t_n})]=0$.
We now apply Lemma \ref{cou}. The space $L^1([0,1])$ is Polish and
for each $n\geq 1$, ${{\mathcal L}}(u(t_n))=Q$ and ${{\mathcal L}}({\tilde u}(t_n))={\tilde Q}$.
The function $\Psi$ is clearly continuous on $L^1([0,1])\times L^1([0,1])$,
(because $\sigma,b$ are Lipschitz-continuous).
Finally, $\Psi(f,g)>0$ for all $f\ne g$ (because
$\Psi(f,g)=0$ implies that $b\circ f=b\circ g$ and
$\sigma \circ f=\sigma \circ g$
a.e., whence $f=g$ a.e. since $(\sigma,b)$ is injective).
Lemma \ref{cou} thus yields $Q={\tilde Q}$.
\end{preuve}
Finally, we give the
\vskip0.2cm
\begin{preuve} {\it of Theorem \ref{confl}.}
Point (ii) is immediately deduced from point (i). Let thus
$u_0,v_0\in L^1([0,1])$
be fixed and let $u,v$ be the corresponding weak solutions
to (\ref{she}). We know from $({{\mathcal I}})$, the Jensen inequality
and Theorem \ref{exstab}-(iv) that a.s.,
\begin{eqnarray*}
\int_0^\infty \rho( ||u(t)-v(t)||_{L^1([0,1])}) dt &\leq&
\int_0^\infty || \rho(|u(t)-v(t)|)||_{L^1([0,1])} dt \\
&\leq&
\int_0^\infty \big|\big| |b(u(t))-b(v(t))|+ |\sigma(u(t))-\sigma(v(t))|^2
\big|\big|_{L^1([0,1])} dt
<\infty.
\end{eqnarray*}
Using Lemma \ref{conc}, one may thus find an increasing
sequence $(t_n)_{n\geq 1}$
such that $\rho( ||u(t_n)-v(t_n)||_{L^1([0,1])})$ tends to $0$ in probability,
so that $||u(t_n)-v(t_n)||_{L^1([0,1])}$ also tends to $0$ in probability
(because due to ${{\mathcal I}}$, $\rho$ is strictly increasing and vanishes only at
$0$).
Next, we use Theorem \ref{exstab}-(iv) with e.g. $\gamma=1/2$ to get,
setting $\Delta_t=\sup_{[t,\infty)}||u(s)-v(s)||_{L^1([0,1])}$,
$$
{\mathbb{E}}\left[\left. \Delta_{t_n}^{1/2} \right\vert {{\mathcal F}}_{t_n}\right]
\leq C ||u(t_n)-v(t_n)||_{L^1([0,1])}^{1/2} \to 0 \hbox{ in probability}.
$$
We used here that conditionally on ${{\mathcal F}}_{t_n}$,
$(u(t_n+t,x))_{t\geq 0, x\in [0,1]}$
is a weak solution to (\ref{she}), starting from $u(t_n)$
(with a translated white noise).
Thus for any $\epsilon>0$, using the Markov inequality
\begin{equation*}
P\left[\Delta_{t_n}>\epsilon \right]
= {\mathbb{E}}\left[ P\left( \left. \Delta_{t_n}>\epsilon
\right\vert {{\mathcal F}}_{t_n} \right) \right]
\leq {\mathbb{E}}\left[ \min\left(1, \epsilon^{-1/2}
{\mathbb{E}}\left[\left. \Delta_{t_n}^{1/2} \right\vert {{\mathcal F}}_{t_n}\right]\right)\right],
\end{equation*}
which tends to $0$ as $n\to \infty$ by dominated convergence.
Consequently, as $n$ tends to infinity,
\begin{equation}\label{cvtz}
\Delta_{t_n}
\hbox{ tends to $0$ in probability.}
\end{equation}
But a.s. $s\mapsto \Delta_s=\sup_{[s,\infty)} ||u(t)-v(t)||_{L^1([0,1])}$ is
non-increasing,
and thus admits a limit as $s\to \infty$, which can be only $0$
due to (\ref{cvtz}).
\end{preuve}
\section{Toward the multi-dimensional case?}\label{dim2}
Consider now a bounded smooth domain $D \subset
{\mathbb{R}}^d$, for some $d\geq 2$. Consider the (scalar) equation
\begin{equation}\label{sherd}
\partial_t u(t,x)=\Delta u(t,x) + b(u(t,x))+ \sigma(u(t,x))\dot W(t,x), \quad
t\geq 0,\; x\in D,
\end{equation}
with some Neumann boundary condition.
Here $W(dt,dx)=\dot W (t,x)dtdx$ is a white noise
on $[0,\infty)\times D$ based on $dtdx$.
We assume that
$\sigma,b:{\mathbb{R}} \mapsto {\mathbb{R}}$ are Lipschitz-continuous.
\vskip0.2cm
It is well known that the mild equation makes no sense in such a case,
since even if $\sigma(u)$ is bounded, $G_{t-s}(x,y)\sigma(u(s,y))$ does not
belong to $L^2([0,t]\times D)$. The existence of solutions
is thus still an open problem.
See however Walsh \cite{w} when $\sigma\equiv 1$,
$b(u)=\alpha u$
and Nualart-Rozovskii
\cite{nr} when $\sigma(u)=u$, $b(u)=\alpha u$. In these works,
the authors manage to define some {\it ad-hoc} notion of solutions,
using that the equations can be solved more or less
explicitly. In the literature, one almost always
considers the simpler case where the noise $W$ is colored, see Da Prato-Zabczyk
\cite{dz}.
\vskip0.2cm
However the weak form makes sense: a predictable process
$u=(u(t,x))_{t\geq 0,x\in D}$ is a weak solution if a.s.,
\begin{equation}\label{crd}
\hbox{for all } T>0, \quad
\sup_{[0,T]}||u(t)||_{L^1(D)}+ \int_0^T ||\sigma(u(t))||^2_{L^2(D)}dt <\infty
\end{equation}
and if for all function $\varphi\in C^2_b(D)$
(with Neumann conditions on $\partial D$), all $t\geq 0$, a.s.,
\begin{equation*}\label{weakrd}
\int_{D}u(t,x)\varphi(x)dx=\int_{D}u_0(x)\varphi(x)dx
+ \int_0^t\int_{D} [\{u(s,x)\Delta\varphi(x)+b(u(s,x))\}dxds+
\sigma(u(s,x))\varphi(x)W(ds,dx)].
\end{equation*}
Assume now that $\sigma(0)=b(0)=0$. Then $v\equiv 0$ is a
weak solution. Furthermore, the estimate of Theorem \ref{exstab}-(iii)
{\it a priori} holds. Choosing $u_0\in L^1(D)$ and $v_0=0$, this would
imply (\ref{crd}).
Unfortunately, we are not able to make this {\it a priori}
estimate rigorous.
\vskip0.2cm
But following the proof of Proposition \ref{eql1}
and Corollary \ref{ineql1g}, one can easily check rigorously
the following result.
For $(e_k)_{k\geq 1}$ an orthonormal basis
of $L^2(D)$, set $B^k_t= \int_0^t \int_{D} e_k(x) W(ds,dx)$. For
$u_0 \in L^\infty(D)$
and $n\geq 1$, consider the solution (see Pardoux \cite{p}) to
$$
u^n(t,x)=u_0(x)+\ds\int_0^t [\partial_{xx}u^n(s,x) + b(u^n(s,x))]ds
+ \sum_{k=1}^n \int_0^t \sigma(u^n(s,x))e_k(x)dB^k_s.
$$
Then if $\sigma(0)=b(0)=0$,
for any $\gamma\in (0,1)$, any $T>0$,
\begin{equation}\label{uin}
{\mathbb{E}}\left[ \sup_{[0,T]} ||u^n(t)||_{L^1(D)}^\gamma + \left\{
\int_0^T \sum_{k=1}^n \left(
\int_{{\mathbb{R}}^d} \sigma(u^n(t,x))e_k(x)dx \right)^2 ds \right\}^\gamma \right]
\leq C_{b,\gamma,T} ||u_0||_{L^1(D)}^\gamma,
\end{equation}
where the constant $C_{b,\gamma,T}$ depends only on $\gamma,T,b$
(the important fact is that it does not depend on $n$).
Passing to the limit formally in (\ref{uin}) would yield
(\ref{crd}).
Unfortunately, (\ref{uin}) is not sufficient to ensure
that the sequence $u^n$ is compact and tends, up to extraction of
a subsequence, to a
weak solution $u$ to (\ref{sherd}).
But this suggests
that, when $\sigma(0)=b(0)=0$,
weak solutions to (\ref{sherd}) do exist and satisfy (\ref{crd}).
\section{Appendix}
First, we recall the following results on continuous local
martingales.
\begin{lem}\label{martingales}
Let $(M_t)_{t\geq 0}$ be a nonnegative continuous local martingale
starting from $m \in (0,\infty)$.
For all $\gamma \in (0,1)$, there exists a constant $C_\gamma$ (depending
only on $\gamma$) such that
$$
{\mathbb{E}}\left[\sup_{[0,\infty)} M_t^\gamma +
\left< M \right> _\infty^{\gamma/2} \right] \leq C_\gamma m^\gamma.
$$
\end{lem}
\begin{proof}
Classically (see e.g. Revuz-Yor \cite[Theorems 1.6 and 1.7 p 181-182]{ry} ),
enlarging the probability space if necessary,
there is a
standard Brownian motion $\beta$ such that $M_t=m+\beta_{\left< M \right> _t}$.
Denote now by $\tau_a=\inf\{t\geq 0;\; \beta_t=a\}$. Since $M$
is nonnegative, we deduce that
$$
\left< M\right> _\infty \leq \tau_{-m} \hbox{ and } \sup_{[0,\infty)} M_t
\leq m+\sup_{[0,\tau_{-m})} \beta_s.
$$
Thus we just have to prove that ${\mathbb{E}}[\tau_{-m}^{\gamma/2} ]
+{\mathbb{E}}[S_m^\gamma ] \leq C_\gamma m^\gamma$, where $S_m=\sup_{[0,\tau_{-m})} \beta_s$.
\vskip0.2cm
First, for $x \geq 0$, $P[S_m\geq x]=P[\tau_{x}\leq \tau_{-m}]=m/(m+x)$.
As a consequence, since $\gamma\in (0,1)$,
\begin{equation*}
{\mathbb{E}}[S_m^\gamma]= \int_0^\infty P[S_m^\gamma\geq x] dx
=\int_0^\infty \frac{m}{m+x^{1/\gamma}} dx
=m^\gamma \int_0^\infty \frac{1}{1+y^{1/\gamma}} dy = C_\gamma m^\gamma.
\end{equation*}
Next, for $t\geq 0$, $P[\tau_{-m}\geq t]=P[\inf_{[0,t]}\beta_s > -m]$.
Recalling that $\inf_{[0,t]}\beta_s$ has the same law as $-\sqrt t |\beta_1|$,
we get $P[\tau_{-m}\geq t]=P[|\beta_1|< m/\sqrt t ]$. Hence
\begin{equation*}
{\mathbb{E}}[\tau_{-m}^{\gamma/2}]= \int_0^\infty P [\tau_{-m}^{\gamma/2}\geq t ]dt=
\int_0^\infty P[|\beta_1|< m/ t^{1/\gamma}] dt
= \int_0^\infty P[(m/|\beta_1|)^\gamma>t ] dt = m^\gamma
{\mathbb{E}}\left[|\beta_1|^{-\gamma}\right].
\end{equation*}
This concludes the proof, since
${\mathbb{E}}\left[|\beta_1|^{-\gamma}\right]<\infty$ for $\gamma \in (0,1)$.
\end{proof}
Next, we state a technical result on a.s. converging integrals.
\begin{lem}\label{conc}
Let $(K_t)_{t\geq 0}$ be a nonnegative process. Assume that
$A_\infty= \int_0^\infty K_t dt <\infty$. Then one may find
a sequence $(t_n)_{n\geq 1}$ increasing to infinity such that
$K_{t_n}$ tends to $0$ in probability as $n\to \infty$.
\end{lem}
\begin{proof}
Consider a strictly increasing continuous concave function
$\phi:{\mathbb{R}}_+\mapsto [0,1]$ such that $\phi(0)=0$.
Using the Jensen inequality, we deduce that
$$
\frac{1}{T}\int_0^T {\mathbb{E}}[\phi(K_s)]ds= {\mathbb{E}} \left[ \frac{1}{T}\int_0^T \phi(K_s)ds
\right]\leq {\mathbb{E}} \left[\phi\left(\frac{1}{T}\int_0^T K_s ds\right)\right]
\leq {\mathbb{E}} \left[\phi\left(\frac{A_\infty}{T}\right)\right],
$$
which tends to $0$ as $T\to \infty$
by the dominated convergence Theorem.
As a consequence, we may find a sequence $(t_n)_{n\geq 1}$ such that
$\lim_n {\mathbb{E}} [\phi(K_{t_n})]=0$. The conclusion follows.
\end{proof}
Finally, we prove a technical result on coupling.
\begin{lem}\label{cou}
Consider two probability measures $\mu,\nu$ on a Polish space ${{\mathcal X}}$.
Let $\Psi:{{\mathcal X}}\times{{\mathcal X}} \mapsto {\mathbb{R}}_+$ be continuous and
assume that $\Psi(x,y)> 0$ for all $x\ne y$. If there is a sequence
of ${{\mathcal X}}\times{{\mathcal X}}$-valued random variables $(X_n,Y_n)_{n\geq 1}$
such that for all $n\geq 1$, ${{\mathcal L}}(X_n)=\mu$ and ${{\mathcal L}}(Y_n)=\nu$ and
if $\lim_n {\mathbb{E}}[\Psi(X_n,Y_n)]=0$, then $\mu=\nu$.
\end{lem}
\begin{proof}
The sequence of probability measures
$({{\mathcal L}}(X_n,Y_n))_{n\geq 1}$ is obviously tight,
so up to extraction of a subsequence, we may assume that
$(X_n,Y_n)$ converges in law, to some $(X,Y)$. Of course,
${{\mathcal L}}(X)=\mu$ and ${{\mathcal L}}(Y)=\nu$. Since $\Psi\land 1$ is continuous and bounded,
we deduce that ${\mathbb{E}}[\Psi(X,Y)\land 1]=\lim_n {\mathbb{E}}[\Psi(X_n,Y_n)\land 1]=0$,
whence $\Psi(X,Y)=0$ a.s. By
assumption, this implies that $X=Y$ a.s., so that $\mu=\nu$.
\end{proof}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 2,935 |
London: Change City
London Discounts
When The Crows Visit
23 October - 30 November 2019  From: £12.00
The Kiln Theatre (formerly The Tricycle Theatre), London
When The Crows Visit - Kiln Theatre presents WHEN THE CROWS VISIT by Anupama Chandrasekhar
…and all the sins of his father and his forefathers came out of his body, through the pores of his skin, in the form of crows.
When a son returns home after being accused of a violent crime, a mother is forced to confront the ghosts of her past when the crows visit.
Inspired by true events in modern-day India, Anupama Chandrasekhar explores the themes of Ibsen's Ghosts and the cyclical nature of oppression in a dark and thrilling world premiere.
Inspired by true events in modern-day India, Anupama Chandrasekhar explores the themes of Ibsen's Ghosts and the cyclical nature of oppression in a dark and thrilling world premiere. More Details
WonTix Privacy | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 4,299 |
Longobarditidae is a family of ceratitd ammonoids known from the early Triassic, included in the Danubitaceae. Longobarditidae includes genera formerly placed in Hungaritidae by the American Treatise on Invertebrate Paleontology, Part L, 1957 as well as genera that have been described since.
Taxonomy
Longobarditidae includes 17 genera, 16 distributed among 4 subfamilies plus one unassigned. Six genera were named prior to the first publication of part L of the Treatise on Invertebrate Paleontology in 1957, the remaining 11 since. Arctohungerites, Groenlandites, Longobardites, and Noetingites were previously included in the Hungeritidae, Czekanowskites in the Meekoceritidae, and Pearylandites in the Siberitidae
Fm. Longobarditidae
Gen Azarianites
Subfm. Czekanowskitinae
Gen. Arctohungerites
Gen. Czekanowskites
Gen. Stannakhites
Gen. Tetsaoceras
Subfm. Groenlanditinae
Gen. Groenlandites
Gen. Lenotropites
Gen. Pearylandites
Subfm. Longobarditinae
Gen. Grambergia
Gen. Intornites
Gen. Longobardites
Gen. Longobarditoides
Gen. Oxylongobardites
Gen. Subhungarites
Subfm. Noetlingitinae
Gen. Noetlingites
Gen. Pronoetlingites
Gen. Silberlingeria
References
Classification of E. T. Tozer 1981
E. T. Tozer. 1981. Triassic Ammonoidea: Classification, evolution and relationship with Permian and Jurassic Forms. The Ammonoidea: The evolution classification, mode of life and geological usefulness of a major fossil group 66-100
Classification of E. T. Tozer 1994
E. T. Tozer. 1994. Canadian Triassic Ammonoid Faunas. Geological Survey of Canada Bulletin 467:1-663
Treatise on Invertebrate Paleontology, Part L, Ammonoidea. R. C. Moore (ed) Geol Soc of America and Univ of Kansas press, 1957.
Danubitaceae
Ceratitida families
Early Triassic first appearances
Early Triassic extinctions | {
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package org.apache.drill.common.logical.data;
import org.apache.drill.common.JSONOptions;
import org.apache.drill.common.logical.data.visitors.LogicalVisitor;
import com.fasterxml.jackson.annotation.JsonCreator;
import com.fasterxml.jackson.annotation.JsonProperty;
import com.fasterxml.jackson.annotation.JsonTypeName;
import com.google.common.base.Preconditions;
@JsonTypeName("constant")
public class Constant extends SourceOperator {
private final JSONOptions content;
@JsonCreator
public Constant(@JsonProperty("content") JSONOptions content){
super();
this.content = content;
Preconditions.checkNotNull(content, "content attribute is required for source operator 'constant'.");
}
public JSONOptions getContent() {
return content;
}
@Override
public <T, X, E extends Throwable> T accept(LogicalVisitor<T, X, E> logicalVisitor, X value) throws E {
return logicalVisitor.visitConstant(this, value);
}
}
| {
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107 Landing Drive in Kill Devil Hills is for sale.
Headline High End Home - GREAT Price!
If the seller of this property located at 107 Landing Drive in Kill Devil Hills has provided the expenses, you will see them below, as provided by the listing agent. If there are no expenses listed, give Shore Realty of the OBX a call and let us do the research for you. It helps to know! Local 252-441-3416 or 800-647-1868. OBX Local experts here to help you!
This section will only apply to homes already in a some type of rental program. If the property located at 107 Landing Drive is used as a vacation rental home or as a year round rental, you will see the income listed below if provided by the seller's agent. A lack of rental numbers does not necessarily mean the property is not rented out. For more call 800-647-1868 and we can get you the actual rental numbers.
$$PRICE REDUCED$$....Discerning Buyers look for a certain home....well, THIS IS IT! The quality throughout speaks to ones that seek out an Outer Banks Beach Home that rings has strengths like construction techniques, interesting details and higher level finishes & decor. Located in the heart of the WONDERFUL OBX this 5 BR/5.5BA home offers comfort with quality. There is an office/sitting room as well as a game room in addition to the expansive Great Room, too. The immediate reaction when one sees this is WOW! From the high end tile work to the Amish Crafted furnishing it more than satisfies. The outdoor space has green space that whispers....."come and relax". There are times you know what you want but most times you are unsure until you see it! This is that time! The searching for the right home...whether you like strong rental income or seek out a 2nd home for your personal use, THIS IS IT! Contact an OBX AGENT today and find out more about this wonderful opportunity.
The data relating to real estate for sale on this web site comes from the Broker Reciprocity Program of the Outer Banks Association of REALTORS. This property located at 107 Landing Drive in Kill Devil Hills is provided courtesy of Mike Bishal of Great Escapes Realty. Information is believed to be accurate but is not warranted. Copyrighted by Outer Banks Association of REALTORS. All Rights Reserved. IDX information is provided by Shore Realty of the OBX exclusively for consumers, personal, non-commercial use and may not be used for any purpose other than to identify prospective properties consumers may be interested in purchasing. If this page disappears from this site it means the property at 107 Landing Drive in Kill Devil Hills is no longer available for sale. If you are interested in this property or have any questions about buying or selling real estate on the Outer Banks call Shore Realty. Local 252-441-3416 or 800-647-1868. | {
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{"url":"https:\/\/tex.stackexchange.com\/questions\/348680\/tikz-spy-library-resets-line-properties-e-g-line-join-line-width-etc","text":"# TikZ spy library resets line properties, e.g. line join, line width, etc\n\nWhen using the TikZ spy library in a tikzpicture it \u201cforgets\u201d the line properties, e.g. line join, set previously.\n\nI have looked into the spy library code (at pgf\/pgf\/generic\/pgf\/frontendlayer\/tikz\/libraries\/tikzlibraryspy.code.tex) and noted these lines:\n\n\\tikzset{\ntikz@lib@reset@gs\/.style={black,thin,solid,opaque,line cap=butt,line join=miter}\n}\n\n\nA first quick fix was to use\n\n\\tikzset{\nspy scope\/.append style={\nline width=5pt,\nline join=round,\n},\n}\n\n\nwhich fixes it for all the image parts that are not magnified, but the lines in the magnified part are still drawn with properties reset.\n\nSetting the property directly to the drawn path preserves it for both parts of the picture.\n\n\\documentclass{article}\n\\usepackage{tikz}\n\\usetikzlibrary{spy}\n\n\\tikzset{\nthick round path\/.append style={\nline width=5pt,\nline join=round,\n},\nevery picture\/.append style={\nthick round path\n},\n}\n\n\\begin{document}\n\n\\section{No \\texttt{spy} library}\n\\begin{tikzpicture}\n\\draw (0,.2) -- (1,0) -- (0,-.2);\n\\end{tikzpicture}\n\n\\section{\\texttt{spy scope} set}\n\\begin{tikzpicture}[spy scope]\n\\draw (0,.2) -- (1,0) -- (0,-.2);\n\\end{tikzpicture}\n\n\\tikzset{\nspy scope\/.append style={\nthick round path\n},\n}\n\n\\section{\\texttt{spy scope} set but \\texttt{spy scope\/.append style} used}\n\\begin{tikzpicture}[spy scope]\n\\draw (0,.2) -- (1,0) -- (0,-.2);\n\\end{tikzpicture}\n\n\\section{Actually magnifying}\n\\begin{tikzpicture}[spy using outlines={circle, size=1cm, magnification=2}]\n\\draw (0,.2) -- (1,0) -- (0,-.2);\n\\spy on (1,0) in node at (2,0);\n\\end{tikzpicture}\n\n\\section{Setting property directly on drawn line}\n\\begin{tikzpicture}[spy using outlines={circle, size=1cm, magnification=2}]\n\\draw[thick round path] (0,.2) -- (1,0) -- (0,-.2);\n\\spy on (1,0) in node at (2,0);\n\\end{tikzpicture}\n\n\\end{document}\n\n\nI would like to prevent having to specify a property to every line I draw since that can be many. Is there a different way to achieve what I want?\n\n\u2022 Why not every path\/.style={thick round path} \u2013\u00a0Salim Bou Jan 14 '17 at 15:35\n\u2022 Well that is embarrassing. I didn't know about every path. Totally works. Do you want to make an answer out of your comment? \u2013\u00a0Artemis Jan 16 '17 at 9:07\n\u2022 A small follow up: The every path solution prevents clip from working. The solution for this can be found here: tex.stackexchange.com\/a\/88222 \u2013\u00a0Artemis Jan 16 '17 at 9:27\n\n\\tikzset{","date":"2019-10-23 10:41:54","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.6221824884414673, \"perplexity\": 9280.464013911882}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-43\/segments\/1570987833089.90\/warc\/CC-MAIN-20191023094558-20191023122058-00520.warc.gz\"}"} | null | null |
This directory includes 8 Palm Springs Hospice service providers. It is the goal of hospice care professionals to help seniors, who are dying, to feel peace, comfort and dignity. These services may also provide comfort to the families in this difficult time. Hospice care ranges from basic pain control to spa treatments, assistance with bucket list items, and other special service to help comfort the dying. Many nursing homes and assisted living facilities have a hospice program.
Results Limited to 8 for download speed reasons. See More Palm Springs hospice care providers.
555 Tachevah 3E-101Palm Springs, CA 92262 Hospice of the Desert Communities offers -Hospice care .
61675 29 Palms HighwayJoshua Tree, CA 92252 Hospice of Morongo Basin offers -Hospice care . | {
"redpajama_set_name": "RedPajamaC4"
} | 4,983 |
Q: How can I use TypeScript to clone a node that contains an Angular selector? I have an Angular app, and in one component, the HTML contains a selector of another component, like this:
<div id="header">
<selector>
Text Content
</selector>
</div>
In my Typescript, I'm trying to clone this div using the following:
var headerClone = document.getElementById('header').cloneNode(true);
I want the headerClone that gets created to look exactly like the div above, so the selector from the other component can work properly, but instead, here is what the clone looks like (from console.log):
<div _ngcontent-ikv-c2 class id="header">Text Content</div>
It isn't copying the <selector> tag, which causes the HTML/CSS of the selector component to not load. How can I get the clone to include the <selector> tag properly?
A: What about getting the innerHtml and creating a new div?
var headerClone = document.createElement('div);
headerClone.setAttribute("id", "header");
headerClone.innerHtml = document.getElementById('header').innerHtml;
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 6,485 |
\section{Introduction}
The main part of this introduction is divided into three subsections discussing three main topics considered in this paper: the symmetric subgroup $K_k$ of a reductive group $G_k$; the quantum Frobenius splitting of an $\imath$quantum group; the Frobenius splitting of $K_k$-orbit closures on the flag variety $\CB_k$ of $G_k$.
\subsection{The symmetric subgroup $K_k$}
\subsubsection{Quantum groups and coordinate rings}
Let $G_{\mathbb{C}}$ be a connected reductive algebraic group over $\BC$. We denote by $\RO_\BC$ the coordinate ring of $G_{\mathbb{C}}$. In his famous papers \cites{Ch55, Ch95}, Chevalley constructed an integral form $\RO_\BZ$ of $\RO_\BC$ such that $\RO_\BC = \RO_\BZ \otimes_{\BZ} \BC$. The integral form defines the {\it Chevalley group scheme} $\G=\G_\BZ$ over $\BZ$ such that the geometric fiber $G_k\footnote{We identify an algebraic variety over $k$ with its set of $k$-rational points.} = \G_{\BZ} \times_{{Sp\, \BZ}} {Sp\, } k$ is the linear algebraic group associated with the given root datum for any algebraically closed field $k$.
Chevalley's approach depends on the choice of representations of $G_{\mathbb{C}}$. Kostant \cite{Ko66} identified (without proof) $\RO_\BZ$ intrinsically as the restricted dual of the Kostant's $\BZ$-form of the enveloping algebra of the Lie algebra of $G_{\mathbb{C}}$. Lusztig \cite{Lu07} reformulated (and proved) Kostant's construction using his theory of canonical bases on the quantum group $\U$. The coordinate ring $\RO_\BZ$ is identified as a dual subspace of the modified quantum group (at $q=1$) spanned by the dual canonical basis.
\subsubsection{The symmetric subgroups} Let $G_k$ be a connected reductive algebraic group over an algebraically closed field $k$ of characteristic $\neq 2$. Let $\theta_k$ be an involution of $G_k$. We denote the fixed point subgroup by $K_k$. It was shown by Steinberg \cite{Ste68} that $K_k$ is reductive. We remark that $K_k$ may not be connected.
It was shown by Springer \cite{Spr87} that involutions of $G_k$ are classified in terms of Satake diagrams (when $G_k$ is simple) independent of the characteristic of $k$, provided $\neq 2$. We reformulate Springer's classification using {\em $\imath$root datum} in \S\ref{sec:class}.
We assume $K_k$ or $(G_k, \theta_k)$ is of quasi-split type, that is, there exists a Borel subgroup $B_k\subset G_k$ such that $B_k \cap \theta_k(B_k) =T_k$ is a maximal torus of $G_k$. Such a Borel subgroup $B_k$ is called $\theta_k$-anisotropic. In terms of Satake diagrams, this means no black dots in the Satake diagram. We also call the relevant $\imath$root datum quasi-split in this case.
We only consider quasi-split cases in this paper. We fix a quasi-split $\imath$root datum for the rest of this introduction. Hence, for any algebraically closed field $k$ of characteristic $\neq 2$, we obtain $G_k$, $\theta_k$, $K_k$, $\theta_k$-anisotropic $B_k$.
\subsubsection{Quantum symmetric pairs}
Associated to the $\imath$root datum, we can construct a quantum symmetric pair $(\U,\U^\imath)$\footnote{The superscript $\imath$ stands for invariant or involution. Since $\U^\imath$ is not the fixed point of any natural involution $\theta$ of $\U$ for generic $q$, one can not denote it by $\U^\theta$.}, where $\U^\imath \subset \U$ is a coideal subalgebra of the quantum group associated to the root datum of $G_k$. The quantum symmetric pair $(\U,\U^\imath)$ is a quantization of the pair of enveloping algebras of the symmetric pair $(\mathfrak{g}_k, \mathfrak{g}_k^{\theta_k})$. Here $\mathfrak{g}_k$ denotes the Lie algebra of $G_k$, and $\mathfrak{g}_k^{\theta_k}$ denotes the Lie algebra of $K_k$ (cf. \cite{Bo91}*{\S9.4}). We often call $\U^\imath$ the {\em $\imath$quantum group}.
Quantum symmetric pairs were originally introduced by Letzter \cite{Le99}, generalized by Kolb \cite{Ko14} to Kac-Moody cases. The first named author and Wang \cite{BW18} initiated the theory of canonical bases arising from quantum symmetric pairs. We refer to the survey \cite{WangICM} for recent developments in quantum symmetric pairs.
Let ${}_\A\dot{\U}^\imath$ be the $\A = \BZ[q,q^{-1}]$-form of the modified $\imath$quantum group. This is a free $\A$-subalgebra of the modified $\imath$quantum group $\dot{\U}^\imath$ with basis $\dot{\B}^\imath$. We call $\dot{\B}^\imath$ the $\imath$canonical basis of ${}_\A\dot{\U}^\imath$.
For any commutative ring $R$ and ring homomorphism $\A \rightarrow R$, we write ${}_R\dot{\U}^\imath= R \otimes_{\A} {}_\A\dot{\U}^\imath$ if there is no ambiguity. We abuse notations and denote by $\dot{\B}^\imath$ the basis of ${}_R\dot{\U}^\imath$ after base change.
\subsubsection{Symmetric subgroup schemes}
We consider the ring homomorphism $\A \rightarrow \BZ$ mapping $q$ to $1$ and the algebra ${}_\BZ\dot{\U}^\imath= \BZ \otimes_{\A} {}_\A\dot{\U}^\imath$. Let $\RO^\imath_{\BZ}$ be the subspace of $\text{Hom}_{ {\BZ}}( {}_\BZ\dot{{\U}}^\imath, \BZ)$ spanned by the dual basis of $\dot{\B}^\imath$. We state the first main theorem of this paper.
\begin{thmintro}[Theorem~\ref{thm:Hopfi} \& Proposition~\ref{prop:GAi} \& \S\ref{sec:pro} (a) \& Theorem~\ref{thm:Oik}]\label{thm:1} \phantom{x}
\begin{enumerate}
\item The subspace $\RO^\imath_{{\BZ}}$ is naturally a commutative and cocommutative Hopf algebra.
\item The algebra $\RO^\imath_{{\BZ}}$ defines a closed subgroup scheme of the Chevalley group scheme $\G$, denoted by $\G^\imath \subset \G$.
\item Assume $R$ is an integral domain with characteristic not 2. Then $\RO^\imath_{R} = \RO^\imath_{{\BZ}} \otimes_\BZ R$ is reduced.
\item Let $k$ be any algebraically closed field $k$ of characteristic $\neq 2$. We identify $\G^\imath_k$ with its set of $k$-rational points, denoted by $G^\imath_k$.
We have $G^\imath_k = K_k \subset G_k$. In particular, $\RO^\imath_{k}$ is the coordinate ring of $K_k$.
\end{enumerate}
\end{thmintro}
We call $\G^\imath$ the {\em symmetric subgroup scheme} over $\BZ$.
The theorem generalizes Lusztig's construction \cite{Lu07} of the Chevalley group scheme $\G$ using (dual) canonical bases arising from quantum groups. While $\G$ parameterizes reductive groups associated to a given root datum, the pair $(\G, \G^\imath)$ parameterizes symmetric pairs associated to a given $\imath$root datum.
\subsubsection{Future works}
We expect to generalize Theorem~\ref{thm:1} to arbitrary (quantum) symmetric pairs in our future works. Our current results rely on the (strong) compatibility of $\imath$canonical bases with respect to the projective system introduced by the first named author and Wang in \cite{BW18a}*{\S6.2} (see Theorem~\ref{thm:stab}). Such compatibility was conjectured by the first named author and Wang in \cite{BW18a}*{Remark~6.18} and established by Watanabe \cite{Wa21} for quasi-split cases (as well as all real rank one cases in \cite{Wa22}) using the crystal theory for $\imath$quantum groups.
While the crystal theory for $\imath$quantum groups is an interesting research direction on its own, only a weaker version is required in our project.
We expect that $\G^\imath$ is the fixed point subscheme of $\G$ (cf. Fogarty \cite{Fo73}). It would be interesting to see the connection with the recent preprint \cite{ALRR22} by Achar-Louren{\c c}o-Richarz-Riche.
\subsection{Quantum Frobenius splittings}
Let $k$ be an algebraically closed field of characteristic $p > 2$.
\subsubsection{Definitions}
The concept of Frobenius splitting was introduced by Mehta-Ramanathan \cite{MR85}, followed by Ramanan-Ramanathan \cite{RR85}, in their study of Schubert varieties. Mehta-Ramanathan \cite{MR85} and Ramanan-Ramanathan \cite{RR85} used Frobenius splitting method to prove the vanishing of higher cohomologies of ample line bundles on Schubert variety, as well as the normality of Schubert varieties (which implies the Demazure character formula). Such results apply to characteristic $0$ as well by the semi-continuity theorem \cite{Ha77}*{Chapter III, Theorem 12.8}.
A variety $\mathcal{X}$ over $k$ is called Frobenius split if the natural morphism $\mathcal{O}_\mathcal{X} \rightarrow F_* \mathcal{O}_\mathcal{X}$ admits a splitting. Here $F: \mathcal{X} \rightarrow \mathcal{X}$ denotes the absolute Frobenius morphism, which fixes all the points and raises the functions to their $p$-th powers.
Let $\mathcal{Y} \subset \mathcal{X}$ be a closed subvariety of $\mathcal{X}$. A splitting of $F: \mathcal{X} \rightarrow \mathcal{X}$ is said to compatibly split $\mathcal{Y}$ if it induces a splitting of $F: \mathcal{Y} \rightarrow \mathcal{Y}$. This can be equivalently defined as a splitting preserving the ideal sheaf of $\mathcal{Y}$ as $\mathcal{O}_\mathcal{X}$-modules.
We refer to the book \cite{BK05} by Kumar-Brion for details on Frobenius splittings.
\subsubsection{Quantum Frobenius splittings of $\U$} We assume $p > 2$ (and $p > 3$ if $G_k$ has a $G_2$ component). Let $\A'$ be the quotient of $\A$ by the $p$-th cyclotomic polynomial, that is, $q$ is a $p$-th primitive root of $1$ in $\A'$. We are interested in two ring homomorphisms from $\A$ to $\A'$, mapping $q$ to $1$ and $p$-th root of $1$, respectively. We then obtain the two $\A'$-algebras ${}_{\A'} \dot{\mathfrak{U}}$ (mapping $q$ to $1$) and ${}_{\A'} \dot{\U}$ (mapping $q$ to $p$-th root of $1$) via base changes from the $\A$-form ${}_{\A} \dot{\U}$ of the modified quantum group.
Lusztig \cite{Lu90} defined a quantum Frobenius morphism $\ofr :{}_{\A'} \dot{\U} \rightarrow {}_{\A'} \dot{\mathfrak{U}}$ mapping divided powers $E^{(n)}_i 1_\zeta $ of Chevalley generators to $E^{(n/p)}_i 1_{\zeta/p}$ (and $0$ if not divisible). Lusztig \cite{Lu93} also defined a (more mysterious) splitting $\fr: {}_{\A'} \dot{\mathfrak{U}} \rightarrow {}_{\A'} \dot{\U}$ of the quantum Frobenius morphism, such that $\ofr \circ \fr = {\rm id}$.
\subsubsection{A Frobenius splitting of $G_k$} Note that ${}_{k}\dot{\U} = k \otimes_{\A'}{}_{\A'} \dot{\mathfrak{U}} = k \otimes_{\A'}{}_{\A'}\dot{\U}$, when $k$ is a field of characteristic $p$.
By taking duals, and combining with the Frobenius automorphis of $k$ (see \S\ref{sec:kFr} for more details), both ${}_k\ofr^*$ and ${}_k\fr^{, *}$ become additive endomorphisms of the coordinate ring $\RO_k$ of $G_k$. The following theorem seems never appear explicitly in literature, which we would like to highlight.
\begin{thmintro}[Theorem~\ref{thm:splitO}] \label{introthm:2}
The induced morphims ${}_k\ofr^{*}: \RO_k \rightarrow \RO_k$ is the $p$-th power map. The induced map ${}_k\fr^{, *}:\RO_k \rightarrow \RO_k$ gives a Frobenius splitting of the algebraic group $G_k$.
\end{thmintro}
Since $G_k$ is a smooth affine variety, it is always Frobenius split. The quantum Frobenius splitting ${}_k\fr^{, *}$ induces a concrete splitting on $G_k$. By construction, the splitting compatibly splits the chosen Borel subgroup $B_k$ and the maximal torus $T_k \subset B_k$.
\subsubsection{Quantum Frobenius splittings of $\U^\imath$}We now consider the two $\A'$-algebras ${}_{\A'} \dot{\mathfrak{U}}^\imath$ and ${}_{\A'} \dot{\U}^\imath$ associated to the $\imath$quantum group. They are obtained via base changes from ${}_{\A} \dot{\U}^\imath$ and ring homomorphisms $\A \rightarrow \A'$, where $q$ maps $1$ for ${}_{\A'} \dot{\mathfrak{U}}^\imath$ and $p$-th root of $1$ for ${}_{\A'} \dot{\U}^\imath$. Here we use different notations to distinguish the two $\A'$-algebras when considered together.
It was shown by the first named author and Sale \cite{BS21} that Lusztig's quantum Frobenius morphism restricts to a quantum Frobenius morphism $\iofr: {}_{\A'} \dot{\U}^\imath \rightarrow {}_{\A'} \dot{\mathfrak{U}}^\imath$. On the other hand, one can not expect Lusztig's quantum Frobenius splitting restricts to a splitting of the $\imath$quantum group. This is more transparent geometrically, that is, the splitting ${}_k\fr^{ ,*}$ of $G_k$ induced by $\fr$ does not compatibly split the symmetric subgroup $K_k$ in general.
The following theorem generalizes Lusztig's quantum Frobenius splitting.
\begin{thmintro}[Theorem~\ref{thm:iFr}]
We construct an $\A'$-algebra homomorphism $\ifr : {}_{\A'} \dot{\mathfrak{U}}^\imath \rightarrow {}_{\A'} \dot{\U}^\imath $ that splits the quantum Frobenius morphism $\iofr: {}_{\A'} \dot{\U}^\imath \rightarrow {}_{\A'} \dot{\mathfrak{U}}^\imath $, that is $\iofr \circ \ifr = {\rm id}$.
\end{thmintro}
The theorem is established by direct computation. Our computation uses crucially the explicit formula of $\imath$divided powers by Berman-Wang \cite{BerW18}, as well as higher $\imath$Serre relations by Chen-Lu-Wang \cite{CLW18}.
\subsubsection{A Frobenius splitting of $K_k$}
Recall ${}_k \dot{\mathfrak{U}}^\imath = {}_k \dot{\U}^\imath$ as $k$-algebras. So ${}_k\iofr^*$ and ${}_k\ifr^{ ,*}$ (again combined with the Frobenius automorphism of $k$) become additive endomorphisms of the coordinate ring $\RO_k^\imath$ of $K_k$.
\begin{thmintro}[Theorem~\ref{thm:splitO}]\label{introthm:4}
The map $_k\iofr^*: \RO_k^\imath \rightarrow \RO_k^\imath $ is the $p$-th power map. The map $_k\ifr^{, *}: \RO_k^\imath \rightarrow \RO_k^\imath $ gives a Frobenius splitting of the symmetric subgroup $K_k$.
\end{thmintro}
\subsubsection{Future works}
It is shown by Benito-Muller-Rajchgot-Smith \cite{BMRS15} that any upper cluster algebra admits a canonical Frobenius splitting. We believe the splittings constructed here are compatible with the cluster structure on the reductive groups.
We expect the splitting of $G_k$ in Theorem~\ref{introthm:2} compatibly splits Bruhat cells. It is also interesting to determine the compatibly split subvarieties of $K_k$ induced by the splitting in Theorem~\ref{introthm:4}.
Recall that the splitting in Theorem~\ref{introthm:2} does not compatibly split $K_k$ in general. Since $K_k \subset G_k$ is a smooth subvariety of $G_k$, we know abstractly there exists a Frobenius splitting of $G_k$ that compatibly splits $K_k$ \cite{BK05}*{Proposition~1.1.6}. It remains an open question to construct concretely an algebraic Frobenius splitting of $G_k$ that compatibly splits $K_k$.
We would like to extend our results to general symmetric pairs (including Kac-Moody types). The main difficulty is that the $\imath$divided powers beyond quasi-split types are much more involved and not well-understood. We hope results here can motivate further research in this direction as well.
\subsection{$K_k$-orbits on flag varieties}Let $k$ be an algebraically closed field of characteristic $\neq 2$.
\subsubsection{Algebraic Frobenius splittings}
Mehta-Ramanathan's splitting on the flag variety $\CB_k$ has been translated by Kummar-Littelmann \cite{KL2} to the algebraic setting using Lusztig's quantum Frobenius splittings.
The algebraic splitting method has also been applied to the setting of cotangent bundles of flag varieties by Hague \cite{Ha13}. This simplifies the (geometric) Frobenius splitting on cotangent bundles of flag varieties by Kumar-Lauritzen-Thomsen \cite{KLT99}.
\subsubsection{$K_k$-orbits on flag varieties} Let $\CB_k = G_k/B_k$ be the flag variety of $G_k$. It follows by Springer \cite{Spr85} that $K_k$ has only finitely many orbits on $\CB_k$.
Since the Borel subgroup $B_k$ is $\theta_k$-anisotropic, the $K_k$-orbit of $B_k/B_k$ is the unique open dense orbit. The geometry of $K_k$-orbits is a classical subject of study, with intrinsic connections with real groups, number theory, etc.
We show the poset of $K_k$-orbits on $\CB_k$ can be parameterized independently of the characteristic of $k$ (provided not $2$) in Proposition~\ref{prop:Korbits}.
Using the symmetric subgroup scheme $\G^\imath_{\BZ[2^{-1}]}$ over ${\BZ[2^{-1}]}$, we further show the closure of orbits can be defined over a open subset of $ \BZ[2^{-1}]$.
\begin{propintro}[Proposition~\ref{prop:KorbitsZ}]\label{prop:5}Let $\CO_k$ be a $K_k$-orbit on $\CB_k$.
There exists a closed subscheme $\CZ$ of the flag scheme $\CB_{\BZ[2^{-1}]}$, such that
(1) $\CZ\rightarrow Sp\,\BZ[2^{-1}]$ is flat;
(2) there is a nonempty open subset $U$ of $Sp\,\BZ[2^{-1}]$ such that for any algebraically closed field $k'$ and a morphism $Sp\,k'\rightarrow U$, the base change $\CZ_{k'} = \CZ \times_{Sp\, \BZ[2^{-1}]} Sp\, k'$ gives the closure $\overline{\mathcal{O}_{k'}}$ of the $K_{k'}$-orbit on $\mathcal{B}_{k'}$ of the same type. In particular, $\CZ_{k'}$ is reduced.
\end{propintro}
The existence of such closed sub-schemes of $\CB_{\BZ[2^{-1}]}$ enables us to deduce characteristic $0$ results from the characteristic $p$ results.
\subsubsection{Codimensional one orbits} The involution $\theta_k$ of $G_k$ induces an involution $\tau$ on the Dynkin diagram of $G_k$. Let $\I$ be the set of simple roots of $G_k$. The codimensional one $K_k$-orbits on $\CB_k$ can be described as $\{\mathcal{O}_i, i \in \I \vert \tau(i) \neq i\} \cup \{\mathcal{O}^{\pm}_i, i \in \I \vert \tau(i) = i\}$. We might have $\mathcal{O}^{+}_i = \mathcal{O}^{-}_i$, e.g., when $G_k = PGL_{2,k}$.
The closures of the orbits $\{\mathcal{O}_i, i \in \I \vert \tau(i) \neq i\}$ are multiplicity-free divisors on $\CB_k$ in the sense of Brion \cites{Br01a, Br01b}, while the closures of the orbits $\{\mathcal{O}^{\pm}_i, i \in \I \vert \tau(i) = i\}$ are often of multiplicity-two.
Various nice geometric properties (e.g. normality) of multiplicity-free subvarieties have been established by Brion \cite{Br01a}.
\subsubsection{Frobenius splitting of $K_k$-orbits} Assume the characteristic of $k$ is positive now. It is desirable to find a Frobenius splitting of the flag variety $\CB_k$ that compatibly splits $K_k$-orbit closures.
However, one can not expect splitting of all $K_k$-orbit closures, cf. \cite{Br01b}*{Introduction}. So the splitting of $K_k$-orbit closures is much more complicated than the case of Schubert varieties.
\begin{thmintro}[Theorem~\ref{thm:spl1} \& Theorem~\ref{thm:spl2}]\label{thmintro:split}
We parameterize the codimensional one $K_k$-orbits on $\CB_k$ as $\{\mathcal{O}_i, i \in \I \vert \tau(i) \neq i\} \cup \{\mathcal{O}^{\pm}_i, i \in \I \vert \tau(i) = i\}$.
\begin{enumerate}
\item The quantum Frobenius splitting $\ifr$ induces a Frobenius splitting of $\CB_k$ that compatibly splits all the closures of $\{\mathcal{O}_i, i \in \I \vert \tau(i) \neq i\} $.
\item Let $J \subset \I$ satisfying conditions in \ref{sec:propJ}. The quantum Frobenius splitting $\ifr$ induces a Frobenius splitting of $\CB_k$ that compatibly splits all the closures of $\{\mathcal{O}^{\pm}_j \vert j \in J\} $.
\item In particular, when $G_k$ is simply-laced, there exists a Frobenius splitting of $\CB_k$ that compatibly splits any given codimensional one $K_k$-orbit.
\end{enumerate}
\end{thmintro}
The splittings constructed in (1) and (2) of the theorem are different. The splitting of multiplicity-free divisors is known by Knutson \cite{Kn09}. See also the work by He-Thomsen \cite{HT12} for some splittings relevant to symmetric subgroups. Our algebraic construction using $\imath$quantum groups is new. The splitting of non-multiplicity-free divisors is new.
\subsubsection{Geometric consequences}
As standard consequences of the splittings in Theorem~\ref{thmintro:split}, we obtain cohomological vanishing of ample line bundles, reducedness of scheme-theoretical intersections, etc.
The splittings apply to partial flag varieties as well as some orbits of smaller dimensions. We further show vanishing of higher cohomology of certain semi-ample line bundles in \S\ref{sec:geo}. These results hold even for characteristic $0$ by semicontinuity, thanks to Proposition~\ref{prop:5}.
Normality of orbit closures can sometime be deduced from Frobenius splittings. However, this would require detailed information on the Bruhat order of $K_k$-orbits on $\CB_k$. We demonstrate for the symmetric pair $(GL_{n,k}, GL_{\lceil n/2\rceil,k}\times GL_{\lfloor n/2 \rfloor,k} )$ in \S\ref{sec:AIII}. Here the Bruhat order has been obtained by Wyser \cite{Wy16}.
\subsubsection{Future works} Frobenius splittings of the flag varieties can be classified by elements in the tensor product of Steinberg modules \cite{BK05}*{\S2.3}. We expect that our splittings correspond to (dual) $\imath$canonical basis elements in such a tensor product (with respect to various choice of parameters).
We expect $K_k$-orbit closures can be defined over $\BZ[2^{-1}]$, that is, $U = Sp\, \BZ[2^{-1}]$ in Proposition~\ref{prop:5}. Similar results for Schubert varieties were first established by Seshadri \cite{Se83}.
\subsection{Organization of this paper}
Let us discuss the organization of this paper in more details.
We give a rather comprehensive review on the preliminaries in \S\ref{sec:pre}. Results are essentially known (except Lemma~\ref{le:nop}). Various formulations are new, including Proposition~\ref{prop:cft} and Proposition~\ref{prop:Korbits}.
We construct the symmetric subgroup scheme $\G^\imath$ in \S\ref{sec:SymGpSch}. We show $\G^\imath$ is a closed subgroup scheme of the Chevalley group scheme $\G$. We show it is reduced over any integral domain with characteristic not 2. As consequences, we construct the coordinate ring of $K_k$ using dual $\imath$canonical basis in Theorem~\ref{thm:Oik} and relate $K_k$-modules with ${}_k\dot{\U}^\imath$-modules in Proposition~\ref{prop:KvsUi}.
We construct the quantum Frobenius splitting of the $\imath$quantum group in \S\ref{sec:qFri}. The first part of this section is devoted to study the $\imath$quantum group over a localization of the ring $\CA= \BZ[q,q^{-1}]$ (essentially inverting the quantum $2$). The second part of this section is devoted to the proof of Theorem~\ref{thm:iFr}. We consider the induced splittings on $\U^\imath$-modules in \S\ref{sec:splitmod}.
We apply our quantum Frobenius splitting to construct a Frobenius splitting of the coordinate ring of $G^\imath_k= K_k$ in \S\ref{sec:iFrO}. We construct a Frobenius splitting of the coordinate ring of $G_k$ using Lusztig's Frobenius splitting in Theorem~\ref{thm:splitO}.
We apply our quantum Frobenius splitting to study the geometry of $K_k$-orbits on the flag variety in \S\ref{sec:Frflag}. We construct two families of splittings $\Psi_k$ and $\Psi^J_k$ of $\CB_k$. The splitting $\Psi_k$ (\S\ref{sec:Phi}) compatibly splits multiplicity-free codimensional one $K$-orbit closures simultaneously. For certain subset $J \subset \I$, we construct the splitting $\Psi^J_k$ (\S\ref{sec:PhiJ}) compatibly splits multiplicity-two codimensional one $K_k$-orbit closures relevant to $J$. We obtain geometric consequences of our splittings in \S\ref{sec:geo}. Finally, we study in details in \S\ref{sec:AIII} the symmetric pair $( GL_{n,k}, GL_{\lceil n/2\rceil,k}\times GL_{\lfloor n/2 \rfloor,k})$ to indicate how to deduce normality from our results.
\vspace{.2cm}
\noindent {\bf Acknowledgement: } HB is supported by MOE grants A-0004586-00-00 and A-0004586-01-00.
\section{Preliminaries} \label{sec:pre}
\subsection{Quantum groups}
\subsubsection{Root data}
A \emph{root datum} $(\I,Y,X,A,(\alpha_i)_{i\in\I},(\alpha_i^\vee)_{i\in\I})$ consists of
\begin{itemize}
\item a finite set $\I$;
\item two finitely generated free abelian groups $X$, $Y$ and a perfect pairing $\langle\,,\,\rangle:Y\times X\rightarrow \mathbb{Z}$;
\item a (symmetrizable) generalized Cartan matrix $A=(a_{ij})_{i,j\in\I}$;
\item an embedding $\I\subset X$ ($i\mapsto \alpha_i$) and an embedding $\I\subset Y$ ($i\mapsto \coroot_i$) such that $\langle \coroot_i,\alpha_j\rangle=a_{ij}$.
\end{itemize}
Let $D=diag(\epsilon_i)_{i\in\I}$ be the (unique) diagonal matrix such that $DA$ is a symmetric matrix, with $\epsilon_i\in\mathbb{\BZ}_{>0}$ and $\epsilon_i=1$ for some $i\in\I$. The number $\epsilon_i$ is called the \emph{root length} of $i$. A root datum is called of \emph{finite type} if the matrix $A$ is a Cartan matrix, namely, $DA$ is positive definite.
In this paper, we will assume that the root datum defined above is \emph{X-regular} and \emph{Y-regular}, namely, elements in $\{\alpha_i\mid i\in\I\}$ are linearly independent in $X$, and elements in $\{\coroot_i\mid i\in\I\}$ are linearly independent in $Y$. The isomorphism between two root data is defined in an evident manner.
Let $X^+=\{\mu\in X\mid\langle\alpha_i^\vee,\mu\rangle\geqslant 0,\forall i\in\I\}$ be the set of \emph{dominant weights}, and let $X^{++}=\{\mu\in X\mid\langle\alpha_i^\vee,\mu\rangle>0,\forall i\in\I\}$ be the set of \emph{regular dominant weights}.
\begin{example}
Let $k$ be an algebraically closed field. For any triple $(G,T,B)$, where $G$ is a connected reductive group, $T$ is a maximal torus of $G$, and $B$ is a Borel subgroup of $G$ containing $T$, one can associate a root datum (of finite type) in the following way.
Let $X=\text{Hom}(T,k^\times)$, and $Y=\text{Hom}(k^\times,T)$ be the character lattice and cocharacter lattice. Then there is a natural pairing between $Y$ and $X$. Let $(\alpha_i)_{i\in\I}\subset X$ (resp. $(\coroot_i)_{i\in\I}\subset Y$) be the simple roots (resp. simple coroots) associated to the Borel subgroup $B$, where $\I$ is a finite index set. Write $a_{ij}=\langle \alpha_i^\vee,\alpha_j\rangle$, for any $i,j$ in $\I$. Then $(\I,Y,X,A,(\alpha_i)_{i\in\I},(\coroot_i)_{i\in\I}))$ is a root datum of finite type.
For fixed $G$, by choosing different pair $(T,B)$, one will get isomorphic root data. We call any root datum in this isomorphism class a \emph{root data associated to $G$}. There is a bijection between isomorphism classes of connected reductive groups over algebraically closed field, and isomorphism classes of root data of finite type.
\end{example}
\subsubsection{Definitions}
Fix a root datum $(\I,Y,X,A,(\alpha_i)_{i\in\I},(\coroot_i)_{i\in\I})$. Let $q$ be an indeterminate. We write $\A=\mathbb{Z}[q,q^{-1}]$ be the subring of $\mathbb{Q}(q)$. Write $q_i=q^{\epsilon_i}$ for all $i$ in $\I$. For $m,n,d\in \mathbb{Z}$ with $m\geqslant 0$, define
\begin{equation*}
[n]=\frac{q^n-q^{-n}}{q-q^{-1}} \quad \text{ and } \quad [m]!=[1][2]\cdots[m].
\end{equation*}
These are called \emph{q-integers} and \emph{q-factorials}. Also define \emph{q-binomial coefficients}:
\begin{equation*}
\qbinom{n}{d}=\left\{\begin{array}{cc}
\frac{[n][n-1]\cdots[n-d+1]}{[d]!}, & \text{ if }d\geqslant 0 \\
0, & \text{ if }d<0
\end{array}
\right.
\end{equation*}
Similarly define $[n]_i$, $[m]_i!$ and $\qbinom{n}{d}_i$ with $q$ replaced by $q_i$. Note that these are all elements in $\A$.
The quantum group $\U$ associated to the given root datum is the associated $\mathbb{Q}(q)$-algebra (with 1) with generators
\begin{equation*}
E_i \quad (i\in\I),\qquad F_i \quad (i\in\I), \qquad K_\mu \quad ( \mu\in Y),
\end{equation*}
subject to the relations
\begin{align*}
&K_0=1,\quad K_\mu K_{\mu'}=K_{\mu+\mu'}, \text{ for all }\mu,\,u'\in Y;\\
&K_\mu E_i=q^{\langle \mu,\alpha_i\rangle}E_i K_\mu, \text{ for all }i\in\I,\, \mu\in Y;\\
&K_\mu F_i=q^{-\langle \mu,\alpha_i\rangle}F_i K_\mu, \text{ for all }i\in\I,\, \mu\in Y;\\
&E_iF_j-F_jE_i=\delta_{ij}\frac{\Tilde{K}_i-\Tilde{K}_{i}^{-1}}{q_i-q_i^{-1}}, \text{ for all }i,j\in\I, \text{ where }\Tilde{K}_i=K_{\coroot_i}^{\epsilon_i} ;\\
&\sum_{n=0}^{1-a_{ij}}(-1)^nE_i^{(n)}E_jE_i^{(1-a_{ij}-n)}=0,\qquad
\sum_{n=0}^{1-a_{ij}}(-1)^nF_i^{(n)}F_jF_i^{(1-a_{ij}-n)}=0
\end{align*}
Here $E_i^{(n)}=E_i^n/[n]_i!$ and $F_i^{(n)}= F_i^n/[n]_i!$ are the divided powers.
Let $\dot{\U}$ be the modified quantum group \cite{Lu93}*{23.1} and $\dot{\RB}$ be its canonical basis (\cite{Lu93}*{25.2.1}). Let $\AU$ be the $\A$-form of $\dot{\U}$, which is an $\A$-subalgebra of $\dot{\U}$ generated by $E_i^{(n)} 1_\lambda$ and $F_i^{(n)} 1_\lambda$ for various $i \in I$, $n \ge 0$ and $\lambda \in X$. Then $\AU$ is a free $\A$-module with basis $\dot{\RB}$. We denote by $\AU^{>0}$ (resp. $\AU^{<0}$) the $\A$-subalgebra generated by $E_i^{(n)} 1_\lambda$ (resp. $F_i^{(n)} 1_\lambda$) for various $i \in I$, $n \ge 0$ and $\lambda \in X$.
\subsubsection{Modules} \label{sec:qanniR}
For any $\lambda\in X^+$, let $L(\lambda)$ be the highest weight $\U$-module with highest weight $\lambda$, and $^\omega L(\lambda)$ be the $\U$-module where the action is twisted by the involution $\omega$ (\cite{Lu93}*{3.1.3}). We denote by $\RB(\lambda)$ and ${}^\omega\RB(\lambda)$ the canonical bases of $L(\lambda)$ and $^\omega L(\lambda)$, respectively. Write $v_\lambda^+$ and $v_{-\lambda}^-$ to denote the highest weight vector and lowest weight vector in $\RB(\lambda)$ and ${}^\omega\RB(\lambda)$, respectively. $L(\lambda)$ also admits a natural $\dot{\U}$-action. We write $_\A L(\lambda)={_\A\dot{\U}\cdot v_\lambda^+}$ to be the $\A$-form of $L(\lambda)$.
For any $\A$-algebra $R$ and $\lambda \in X^+$, we write $_R\dot{\U}=R\otimes_{\A} {_\A\dot{\U}}$ and $_R L(\lambda)=R\otimes_{\A} {_\A L(\lambda)}$.
For any $\lambda_1, \lambda_2$ in $X^+$, and any $\A$-algebra $R$, the action map
$
{}_R\dot{\U}\rightarrow {_R^\omega L(\lambda_1)}\otimes{_R L(\lambda_2)}$, $ x\mapsto x\cdot (v_{\lambda_1}^-\otimes v_{\lambda_2}^+)$ has kernel
\[
_R P(\lambda_1,\lambda_2) = \!\!\!\sum_{\lambda\neq\lambda_2-\lambda_1} \!\!\! {_R\dot{\U}\one_\lambda}
+
\!\!\!\sum_{i\in\I,\,a_i>\langle \alpha_i^\vee,\lambda_1\rangle}\!\!\! {_R\dot{\U}E_i^{(a_i)}}\one_{\lambda_2-\lambda_1}
+
\!\!\!\sum_{i\in\I,\, b_i>\langle \coroot_i,\lambda_2\rangle}\!\!\! {_R\dot{\U}}F_i^{(b_i)}\one_{\lambda_2-\lambda_1}.
\]
The claim follows from \cite{Lu93}*{Proposition 23.3.6} for the case $R=\A$. The general case follows from the fact that $_\A P(\lambda_1,\lambda_2)$ is a free $\A-$submodule of $_\A\dot{\U}$ spanned by canonical basis elements \cite{Lu93}*{Theorem 25.2.1}. In particular, $_R P(\lambda_1,\lambda_2)$ is a free $R$-module.
\subsubsection{Quantum Frobenius homomorphism}\label{sec:qFr}
Fix an odd number $l>1$. We require that $l$ is relatively prime to all the root length $\epsilon_i$. Let $f_l\in \mathcal{A}$ be the $l$-th cyclotomic polynomial. Set $\mathcal{A}'=\mathcal{A}/(f_l)$, and $\mathbf{F}$ be the quotient ring of $\mathcal{A}'$. Let $\phi:\mathcal{A}\rightarrow\mathcal{A}'$ be the natural quotient map. Then $\phi$ extends to a ring homomorphism from the local ring $\mathcal{A}_{(f_l)}$ to $\mathbf{F}$.
Let $c:\mathcal{A}\rightarrow\mathcal{A'}$ be the ring homomorphism sending $q$ to 1. We will write $\A'_\phi$ and $\A'_c$ to distinguish two different $\A$-algebra structures. Define
\begin{align*}
_{\mathcal{A}'}\dot{\mathfrak{U}}={_{\A'_c}\dot{\U}}={\mathcal{A}'_c}\otimes_\A ({_{\mathcal{A}}\dot{\mathrm{U}}}),\qquad
_{\mathcal{A}'}\dot{\mathrm{U}}={_{\A'_\phi}\dot{\U}}=\mathcal{A}'_\phi\otimes_\A (_\mathcal{A}\dot{\mathrm{U}}).
\end{align*}
For any $i\in\I$, $\zeta\in X$, we use $\mathfrak{e}_i^{(n)}\one_\zeta$ (resp. $\mathrm{E}^{(n)}_i\one_\zeta$) to denote the image of $E_i^{(n)}\one_\zeta$ in $\cdotU$ (resp. $\pdotU$). Similar notations are used for $\mathfrak{f}_i^{(n)}\one_\zeta$ and $\mathrm{F}_i^{(n)}\one_\zeta$.
\begin{theorem}(\cite{Lu93}*{35.1.9},\cite{Mc07}*{Proposition~3.4})\label{thm:ClaFr}
(a) There is a unique $\A'$-algebra homomorphism \ofr: $\pdotU \rightarrow\cdotU$, such that for all $i\in\I$, $n\in\mathbb{N}$ and $\zeta\in X$, we have:
$\ofr(\mathrm{E}_i^{(n)}\one_\zeta)$ equals $\mathfrak{e}_i^{(n/l)}\one_{\zeta/l}$, if $l$ divides $n$ and $\zeta\in lX$, and equals 0, if otherwise;
$\ofr(\mathrm{F}_i^{(n)}\one_\zeta)$ equals $\mathfrak{f}_i^{(n/l)}\one_{\zeta/l}$, if $l$ divides $n$ and $\zeta\in lX$, and equals 0, if otherwise.
(b) There is a unique $\A'$-algebra homomorphism $\fr$: $\cdotU\rightarrow\pdotU$, such that for all $i\in\I$, $n\in\mathbb{N}$ and $\zeta\in X$, we have $\fr(\mathfrak{e}_i^{(n)}\one_\zeta)=\mathrm{E}_i^{(nl)}\one_{l\zeta}$, and $\fr(\mathfrak{f}_i^{(n)}\one_\zeta)=\mathrm{F}_i^{(nl)}\one_{l\zeta}$. In particular, we have $\ofr\circ\fr=$id.
\end{theorem}
For any $\lambda\in X^+$, set ${_{\A'}L}(\lambda)=\A'_{\phi} \otimes_\A ({_\A L(\lambda)})$, and ${_{\A'}\mathfrak{L}}(\lambda)=\A'_c \otimes_\A ({_\A L(\lambda)})$. We will abuse the notations, and still use $v_\lambda^+$ to denote the highest weight vector in these modules.
We also have the following proposition by Theorem \ref{thm:ClaFr}{(a)} and \S\ref{sec:qanniR}.
\begin{proposition}\label{prop:ga}
For any $\mu\in X^+$, there exists a unique $_{\A'}\dot{\U}$-module homomorphism
\[
\gamma_\mu: {_{\A'}L}(l\mu)\longrightarrow \big({_{\A'}\mathfrak{L}}(\mu)\big)^\ofr, \quad v_{l\mu}^+ \mapsto v_\mu^+.
\]
Here $\big({_{\A'}\mathfrak{L}}(\mu)\big)^\ofr$ stands for the $_{\A'}\dot{\U}$-module which is isomorphic to ${_{\A'}\mathfrak{L}}(\mu)$ as $\A'$-modules, and the action is twisted by $\ofr$.
\end{proposition}
\subsection{Chevalley group schemes}
In this subsection, we shall assume that our root datum $(\I,Y,X,A,(\alpha_i)_{i\in\I},(\alpha_i^\vee)_{i\in\I})$ is of finite type. We recall Lusztig's construction of the group scheme $\G$ in \cite{Lu07}.
\subsubsection{Lusztig's construction}\label{sec:nocp}
Let $A$ be a commutative ring with 1, which is viewed as an $\A$-algebra, where $q$ acts by 1. For any element $a\in\A$, we will abuse the notation and still write $a$ to denote its image in $A$.
Let ${}_A\dot{\U} = A \otimes_{\A} {}{_\A}\dot{\U}$ and ${}_A\dot{\RB}=\{1\otimes b\mid b\in\dot{\RB}\}$. The set ${}_A\dot{\RB}$ is called the canonical basis of $_A\dot{\U}$. If there is no confusion, we will write ${}_A\dot{\RB}=\dot{\RB}$, and use $b$ to denote the image $1\otimes b$.
Let $_A\widehat{\mathrm{U}}$ be the $A$-module consisting of all the formal linear combinations
\[
\sum_{a\in\dot{\RB}} n_a a, \quad n_a\in A.
\]
By \cite{Lu07}*{1.11}, $_A\widehat{\U}$ has a structure of $A$-algebra compatible with the embedding $_A\dot{\U} \subset {}_A\widehat{\U}$.
Let $_A\widehat{\mathrm{U}}^{(2)}$ be the $A$-module consisting of all the formal linear combinations
\[
\sum_{(a,a')\in \dot{\RB}\times \dot{\RB}} n_{a,a'}a\otimes a', \quad n_{a,a'}\in A.
\]
Then ${}_A\widehat{\U}^{(2)}$ also has an $A$-algebra structure by \cite{Lu07}*{1.11}. We have an $A$-algebra homomorphism $\WD:{_A\widehat{\mathrm{U}}}\longrightarrow {_A\widehat{\mathrm{U}}}^{(2)}$, compatible with the coproduct on $_A\dot{\U} $ by \cite{Lu07}*{1.17}.
Let $S:{_A\dot{\U}}\rightarrow {_A\dot{\U}}$ be the antipode map. It extends to $\widehat{S}:{_A\widehat{\U}}\rightarrow {_A\widehat{\U}}$.
Let $\RO_A$ be the $A$-submodule of $_A\dot{\U}^*=\text{Hom}_A({_A\dot{\U}},A)$, spanned by $\{b^*\mid b\in \dot{\RB}\}$, where $b^* \in {}_A\dot{\U}^*$ is the \emph{dual canonical basis} element which sends $b$ to 1, and sends other canonical basis element to 0. Then for any element $f$ in $\RO_A$, it extends to an $A$-linear form on the completion $_A\widehat{\U}$. We write $\hat{f}$ to denote this extension.
Lusztig defined an $A$-Hopf algebra structure on $\RO_A$, using structure constants for the algebra $_A\dot{\U}$ (\cite{Lu07}*{3.1}). Let $\delta:\RO_A\rightarrow\RO_A\otimes\RO_A$ to be the coproduct, $\sigma:\RO_A\rightarrow\RO_A$ to be the antipode, and $\epsilon:\RO_A\rightarrow A$ to be the counit.
For any $f$, $g \in \RO_A$, it follows from \cite{Lu07}*{3.1} that the following diagrams commute
\begin{equation*}
\begin{tikzcd}
& _A\widehat{\U} \arrow[r,"\WD"] \arrow[rd,"\widehat{fg}"'] & {_A\widehat{\U}}^{(2)} \arrow[d,"f\otimes g"] \\ & & A
\end{tikzcd}
\begin{tikzcd}
& {_A\widehat{\U}}\otimes{_A\widehat{\U}} \arrow[r,"m"] \arrow[rd,"\delta(f)"'] & {_A\widehat{\U}} \arrow[d,"\hat{f}"] \\ & & A
\end{tikzcd}
\quad
\begin{tikzcd}
_A\widehat{\U} \arrow[r,"\widehat{S}"] \arrow[rd,"\widehat{\sigma(f)}"'] & {_A\widehat{\U}} \arrow[d,"\hat{f}"] \\ & A
\end{tikzcd}.
\end{equation*}
Here $f\otimes g: {_A\widehat{\U}^{(2)}}\rightarrow A$ sends formal linear combination $\sum_{(a,a')\in \dot{\RB}\times\dot{\RB}}n_{a,a'}a\otimes a'$ to $\sum_{(a,a')\in \dot{\RB}\times\dot{\RB}}n_{a,a'}f(a) g(a')$, where the second sum is finite, thanks to the definition of $\RO_A$. Similarly $\delta(f)\in \RO_A\otimes\RO_A$ induces a well-defined $A$-linear form on $_A\widehat{\U}\otimes{_A\widehat{\U}}$. We still use $\delta(f)$ to denote this map.
Since $\RO_A$ is a commutative Hopf algebra over $A$, the set $\text{Hom}_{A-\text{alg}}(\RO_A, A)$, consisting of $A$-algebra homomorphisms $\RO_A\rightarrow A$ preserving 1, has a group structure.
By definition, $\text{Hom}_{A-\text{alg}}(\RO_A,A)$ is an $A$-submodule of $\RO_A^{*}$, the $A$-linear duals of $\RO_A$. There is an $A$-linear bijective map from $\RO_A^{*}$ to $_A\widehat{\U}$, given by $\phi\mapsto \sum_{b\in\dot{\RB}}\phi(b^*)b$. Then under this bijection, the subset $\text{Hom}_{A-\text{alg}}(\RO_A,A)$ is sent to
\[
G_A=\{\xi=\sum_{b\in\dot{\RB}}n_b b\in {_A\widehat{\mathrm{U}}}\mid \WD(\xi)=\xi\otimes\xi,\,n_{1_0}=1\}.
\]
Here for any $\xi=\sum_{b\in \dot{\RB}}n_bb$ and $\xi'=\sum_{b'\in\dot{\RB}}n'_{b'}b'$ in $_A\widehat{\U}$, the element $\xi\otimes\xi'$ in $_A\widehat{\U}^{(2)}$ is defined to be the formal linear combination
\[
\xi\otimes\xi'=\sum_{(b,b')\in \dot{\RB}\times \dot{\RB}} n_{b}n'_{b'}b\otimes b'.
\]
Then the subset $G_A$ is closed taking products. It moreover admits a group structure, where the unit is $\sum_{\lambda\in X}\one_\lambda$, and the inverse is given by the restriction of the antipode $\widehat{S}:{_A\widehat{\U}}\rightarrow{_A\widehat{\U}}$. The bijection $\text{Hom}_{A-\text{alg}}(\RO_A,A)\xrightarrow{\sim}G_A$ is a group isomorphism.
Define the $\BZ$-group scheme $\G$ by setting $\G(A)=G_A$ \footnote{Here we identify a $\BZ$-group scheme with the associated $\BZ$-group functor following \cite{Jan03}} for any (unital) commutative ring $A$. Then $\G\cong Sp\,\RO_\BZ$ is the Chevalley group scheme associated to the given root datum.
\subsubsection{The reductive group $G_k$}\label{sec:regpq}
Let $k$ be an algebraically closed field. Set $G_k$ be the subset of $_k\widehat{\U}$ defined as above by letting $A=k$.
Then $G_k$ is a connected reductive group over $k$ with the coordinate ring $\RO_k$ by \cite{Lu07}*{Theorem~4.11}.
Let $T_k$ be the subset of $G_k$ consisting of elements of the form $\sum_{\lambda\in X}n_\lambda\one_\lambda$, such that $n_\lambda n_{\lambda'}=n_{\lambda+\lambda'}$. Let $_k\widehat{\U}^{>0}$ be the $k$-subspace of $_k\widehat{\U}$ consisting of all elements of the form $ \sum_{b\in \dot{\RB} \cap {}_k\dot{\U}^{>0},\lambda\in X}n_b(b\one_\lambda)$ with $n_b\in k$. Set $G_k^{>0}=G_k\cap {_k\widehat{\U}^{>0}}$ and $B_k=G_k^{>0}T_k$. It follows from \cite{Lu07}*{4.11} that $T_k$ is a maximal torus of $G_k$ and $B_k$ is a Borel subgroup of $G_k$.
We have the following isomorphisms of free abelian groups
\begin{align*}
X&\xrightarrow{\sim} \text{Hom}(T_k,k^\times),\qquad &Y\xrightarrow{\sim} &\text{Hom}(k^\times,T_k)\\
\lambda & \longmapsto \big(\sum_{\lambda\in X}n_\lambda\one_\lambda\mapsto n_\lambda\big),\qquad &\gamma\longmapsto &\big(a\mapsto \sum_{\lambda\in X}a^{\langle \gamma,\lambda\rangle}\one_\lambda\big).
\end{align*}
These two maps give an isomorphism between the root data defining the quantum group with the root data associated to $(G_k,T_k,B_k)$.
For any $i\in\I$ and $\xi\in k$, define
\[
x_i(\xi)=\sum_{c\in\BN,\;\lambda\in X}\xi^cE_i^{(c)}\one_\lambda,\qquad y_i(\xi)=\sum_{c\in\BN,\;\lambda\in X}\xi^cF_i^{(c)}\one_\lambda.
\]
Then $\{x_i,y_i\mid i\in\I\}$ is a {\em pinning} for the triple $(G_k,T_k,B_k)$, that is, the morphism
\[
\begin{pmatrix}
1 & a \\ & 1
\end{pmatrix}\mapsto x_i(a)\quad
\begin{pmatrix}
1 & \\ b &1
\end{pmatrix}\mapsto y_i(b)\quad
\begin{pmatrix}
t & \\ & t^{-1}
\end{pmatrix}\mapsto \coroot_i(t)
\]
for any $a, b$ in $k$, and $t$ in $k^*$, give a group morphism from $\mathrm{SL}_{2,k}$ to $G$.
\subsubsection{Lie algebras and algebras of distributions}\label{sec:liealg}
For any linear algebraic group $H_k$ over $k$, we use the notation $\lie(H_k)$ and $\dist(H_k)$ to denote the Lie algebra and the algebra of distributions on $H_k$. They are subalgebras of $k[H_k]^*$, the linear dual of the coordinate ring of $H_k$.
Recall $\RO_k$ is the coordinated ring the connected reductive group $G_k$. We identify $\RO_k^*$ with $_k\widehat{\U}$, via $\mu\mapsto\sum_{b\in\dot{\RB}}\mu(b^*)b$. Then its Lie algebra and the distribution algebra are identified with subalgebras of $_k\widehat{\U}$.
For any $i\in\I$, $n\in\BN$, and $\mu\in Y$, define the following elements in $_k\widehat{\U}$:
\[
e_i^{(n)}=\sum_{\lambda\in X}E_i^{(n)}\one_\lambda,\qquad f_i^{(n)}=\sum_{\lambda\in X}F_i^{(n)}\one_\lambda,\qquad \binom{h_\mu}{n}=\sum_{\lambda\in X}\binom{\langle \mu,\lambda\rangle}{n}\one_\lambda.
\]
Set $e_i=e_i^{(1)}$, $f_i=f_i^{(1)}$, and $h_\mu=\binom{h_\mu}{1}$. Here we use the same notations to denote the elements in the algebra after base change.
The Lie algebra $\lie(G_k)$ is identified as the Lie subalgebra of $_k\widehat{\U}$ generated by $e_i$ ($i\in\I$), $f_i$ ($i\in\I$), and $h_\mu$ ($\mu\in Y$). The distribution algebra of $G_k$ is identified as the subalgebra of $_k\widehat{\U}$ generated by $e_i^{(n)}$ ($i\in\I$, $n\in\BN$), $f_i^{(n)}$ ($i\in\I$, $n\in\BN$), and $\binom{h_\mu}{n}$ ($\mu\in Y$, $n\in \BN$).
\subsection{Quantum symmetric pairs}
\subsubsection{$\imath$root datum}
\begin{definition}
A quasi-split \emph{$\imath$root datum} consists of the following
\begin{itemize}
\item a root datum $(\I,Y,X,A,(\alpha_i)_{i\in\I},(\coroot_i)_{i\in\I}))$;
\item an involution $\tau$ of the index set $\I$, such that $a_{ij}=a_{\tau i,\tau j}$, for all $i, j\in\I$;
\item an involution $\theta_X$ on $X$, and an involution $\theta_Y$ on $Y$, such that $\langle \theta_Y(\lambda),\theta_X(\mu)\rangle=\langle \lambda,\mu\rangle$, for $\lambda\in Y$, $\mu\in X$, and $\theta_X(\alpha_i)=-\alpha_{\tau i}$, $\theta_Y(\coroot_i)=-\coroot_{\tau i}$, for any $i\in\I$.
\end{itemize}
\end{definition}
If there is no ambiguity, we will use $\theta$ to denote both the involution on $X$ and on $Y$. We write $(\I,Y,X,A,(\alpha_i)_{i\in\I},(\coroot_i)_{i\in\I}),\tau,\theta)$ to denote an $\imath$root datum. We call an $\imath$root datum \emph{of finite type} if the associated Cartan datum is of finite type. Morphisms and isomorphisms between $\imath$root data are defined in an evident manner.
For an $\imath$root datum $(\I,Y,X,A,(\alpha_i)_{i\in\I},(\coroot_i)_{i\in\I}),\tau,\theta)$, set
\begin{equation*}
X_\imath=X/\langle \lambda-\theta\lambda\mid\lambda\in X\rangle\qquad \text{and}\qquad Y^\imath=\{\mu\in Y\mid \theta\mu=\mu\}.
\end{equation*}
We call $X_\imath$ the \emph{$\imath$weight lattice}, and call $Y^\imath$ the \emph{$\imath$coweight lattice}. There is a natural pairing $Y^\imath\times X_\imath\rightarrow\mathbb{Z}$ inherited from the pairing between $Y$ and $X$. Note that this pairing is not necessarily perfect. For any $\lambda\in X$, we write $\overline{\lambda}$ its image in $X_\imath$.
\begin{remark}
One can define $\imath$root data of more general (not necessarily quasi-split) type. In this paper we shall only consider quasi-split $\imath$root data, and we will just call them $\imath$root datum for brevity.
\end{remark}
\begin{lemma}\label{le:nop}
The abelian group $X_\imath$ has no odd torsion. Namely, for any $\overline{\lambda} \in X_\imath$ and any odd $n\in\BN$ with $n\overline{\lambda}=0$, we have $\overline{\lambda}=0$.
\end{lemma}
\begin{proof}
Let $\BZ[2^{-1}] \subset \BQ$ be the localization of $\BZ$ inverting $2$. Let $\Breve{X} = \langle \lambda-\theta\lambda\mid\lambda\in X\rangle$. By definition, we have the short exact sequence
\[
0\rightarrow \Breve{X}\rightarrow X\rightarrow X_\imath\rightarrow 0.
\]
Tensoring with the flat $\BZ$-module $\BZ[2^{-1}]$, we get
\[
0\rightarrow \Breve{X}\otimes \BZ[2^{-1}]\rightarrow X\otimes \BZ[2^{-1}]\rightarrow X_\imath\otimes \BZ[2^{-1}]\rightarrow 0.
\]
Note that the $\BZ[2^{-1}]$-module homomorphism
\[
X\otimes\BZ[2^{-1}]\longrightarrow \Breve{X}\otimes \BZ[2^{-1}]
\]
sending $\lambda\otimes 1$ to $(\lambda-\theta\lambda)\otimes 1/2$ gives a splitting for the above short exact sequence. Hence $X_\imath\otimes \BZ[2^{-1}]$ can be viewed as a submodule of $X\otimes \BZ[2^{-1}]$, which is torsion-free. We deduce that $X_\imath\otimes \BZ[2^{-1}]$ is torsion-free. Then by the structure theory of finitely generated abelian groups, we see that $X_\imath$ has no odd torsion.
\end{proof}
\subsubsection{$\imath$quantum groups}\label{sec:iqp}
For each $i\in\I$, we fix a parameter $\varsigma_i\in q^{\mathbb{Z}}$, such that
\begin{equation*}
\varsigma_{\tau i}=q_i^{-a_{i,\tau i}}{\varsigma}_i^{-1},\quad
\varsigma_{\tau i}=\varsigma_i\text{ if }a_{i,\tau i}=0,\quad
\varsigma_i=q_i^{-1} \text{ if }\tau i=i.
\end{equation*}
Then the associated {\em $\imath$quantum group} $\U^\imath=\U^\imath_{\mathbf{\varsigma}}$ is defined to be the $\mathbb{Q}(q)$-subalgebra of $\U$ generated by elements
\begin{equation*}
B_i=F_i+\varsigma_iE_{\tau i}\Tilde{K}_i^{-1}\quad (i\in\I),\qquad K_\mu\quad (\mu\in Y^\imath).
\end{equation*}
The pair $(\U, \U^\imath)$ is called a {\em quantum symmetric pair}. We refer to \cite{BW21}*{\S3} for a more general definition for $\imath$quantum groups.
Let $\dot{\U}^\imath$ be the modified algebra of $\U^\imath$ and let $\dot{\RB}^\imath$ be the canonical basis of $\dot{\U}^\imath$ defined in \cite{BW18a}*{\S3.7, \S6.4}. The algebra $\dot{\U}$ is naturally a $(\dot{\U}^\imath, \dot{\U}^\imath)$-bimodule by \cite{BW18a}*{\S3.7}.
Let $_\A\dot{\U}^\imath$ be the $\A$-form of $\dot{\U}^\imath$ \cite{BW18a}*{Definition~3.19}. For any $\A$-algebra $A$, set ${}_A\dot{\U}^\imath=A\otimes_{\A} {_\A\dot{\U}^\imath}$.
(a) {\it For any $\A$-algebra $A$, we have an $A$-algebra embedding ${}_A\dot{\U}^\imath \rightarrow {_A\widehat{\mathrm{U}}}$, $x \mapsto \sum_{\lambda \in X} x \one_\lambda$. Here $x \one_\lambda \in \dot{\U}$ is via the ${}_A\dot{\U}^\imath$-action on ${}_A\dot{\U}$.}
Since ${}_\A\dot{\U}^\imath$ is a free $\A$-module by \cite{BW18a}*{Theorem~6.17}, it suffices to prove the statement for $A = \A$. The claim for $A = \A$ follows from \cite{BW18a}*{\S3.7} and \cite{Lu07}*{\S1.11}.
We denote by $\iota_\lambda: {_\A\dot{\U}^\imath\one_{\overline{\lambda}}} \rightarrow {_\A\dot{\U}\one_\lambda}$ the map $ x \mapsto x \one_\lambda$. We define the composition
\begin{equation}\label{eq:pim}
p_{\imath,\lambda}=\pi_\lambda\circ\iota_\lambda:{_\A\dot{\U}^\imath\one_{\overline{\lambda}}}\xrightarrow{\iota_\lambda}{_\A\dot{\U}\one_\lambda}\xrightarrow{\pi_\lambda}{_\A\U^-\one_\lambda},
\end{equation}
where $\pi_\lambda:{_\A\dot{\U}\one_\lambda}\rightarrow {_\A\dot{\U}\one_\lambda}/{_\A\dot{\U}\U^+\one_\lambda}\xrightarrow{\sim} {_A\U^-}\one_\lambda$.
It follows from \cite[Corollary 6.20]{BW18a} that $p_{\imath,\lambda}$ is an isomorphism of $\A$-modules.
Let $A$ be any $\A$-algebra. After tensoring with $A$, we deduce that the composition
\begin{equation}\label{eq:pimA}
p_{\imath,\lambda}=\pi_\lambda\circ\iota_\lambda:{_A\dot{\U}^\imath\one_{\overline{\lambda}}}\xrightarrow{\iota_\lambda}{_A\dot{\U}\one_\lambda}\xrightarrow{\pi_\lambda} {_A\U^-\one_\lambda}
\end{equation}
is an isomorphism between $A$-modules.
We recall the construction for \emph{$\imath$divided powers}.
For $i\in\I$ with $\tau i\neq i$, and $m\in\mathbb{N}$, set
\begin{equation*}
B_i^{(m)}=\frac{B_i^m}{[m]_i!}.
\end{equation*}
For $i\in \I$ with $\tau i=i$, and $m\in\mathbb{N}$, set
\begin{eqnarray*}
&&B_{i,\odd}^{(m)} =\frac{1}{[m]_i^!}\left\{ \begin{array}{ccccc} B_i\prod_{j=1}^k (B_i^2-[2j-1]_i^2 ), & \text{if }m=2k+1;\\
\prod_{j=1}^k (B_i^2-[2j-1]_i^2), &\text{if }m=2k; \end{array}\right.
\\
&&B_{i,\ev}^{(m)} = \frac{1}{[m]_i^!}\left\{ \begin{array}{ccccc} B_i\prod_{j=1}^k (B_i^2-[2j]_i^2 ), & \text{if }m=2k+1;\\
\prod_{j=1}^{k} (B_i^2-[2j-2]_i^2), &\text{if }m=2k. \end{array}\right.
\end{eqnarray*}
We also recall the {\em modified $\imath$divided powers}.
For $\tau i\neq i$, $\zeta\in X_\imath$, and $m\in\mathbb{N}$, we define
$B_{i,\zeta}^{(m)}=B_i^{(m)}\one_\zeta$.
For $\tau i=i$, $\zeta\in X_\imath$, and $m\in\mathbb{N}$, we define $B_{i,\zeta}^{(m)}=B_{i,\ev}^{(m)}\one_\zeta$ if $\langle \alpha^\vee_i,\zeta\rangle$ is even;
$B_{i,\zeta}^{(m)}=B_{i,\odd}^{(m)}\one_\zeta$ if $\langle \alpha^\vee_i,\zeta\rangle$ is odd.
The algebra ${_\A}\dot{\U}^\imath$ is generated by $B_{i,\zeta}^{(n)}$, for various $i\in\I$, $\zeta\in X_\imath$, and $n\in\mathbb{N}$.
\subsubsection{Stability}
For the remaining part of this section, we assume root datum is of finite type. We also fix parameters as following: $\varsigma_i=q^{-1}_i$, if $\tau i=i$; $\varsigma_i=1$, if $a_{i,\tau i}=0$; and $\{\varsigma_i,\varsigma_{\tau i}\}=\{1,q_i^{-1}\}$, if $a_{i,\tau i}=-1$.
For any $\mu\in X^+$, let $\RB[\lambda]^\imath$ be the $\imath$canonical basis for the irreducible module $L(\lambda)$ (\cite{BW18a}*{Theorem 5.7}).
Let $\pi_\lambda^\imath:\dot{\U}^\imath\one_{\overline{\lambda}}\rightarrow L(\lambda)$ be the $\U^\imath$-module homomorphism sending $u$ to $u\cdot v_\lambda^+$. Note that $\dot{\U}^\imath\one_{\overline{\lambda}}$ is spanned by the $\imath$canonical basis.
\begin{theorem}\label{thm:stab}(\cite{Wa21}*{Theorem 6.2.8})
For any $\lambda\in X^+$, the map $\pi_\lambda^\imath$ sends $\imath$canonical basis element to an $\imath$canonical basis element or zero. And the kernel of $\pi_\lambda^\imath$ is a subspace of $\dot{\U}^\imath\one_{\overline{\lambda}}$ spanned by $\imath$canonical basis elements. In particular, there are only finitely many $\imath$canonical basis element $b$ in $\dot{\RB}^\imath$, such that $b\cdot v_\lambda^+\neq 0$.
\end{theorem}
\subsection{Symmetric subgroups}
Let $k$ be an algebraically closed field with characteristic not 2.
\subsubsection{Definitions} Let $G_k$ be a connected reductive algebraic group defined over $k$. A \emph{symmetric pair} $(G_k,\theta_k)$ consists of a connected reductive group $G_k$, and an involution $\theta_k$ of $G_k$. Two symmetric pairs $(G_k,\theta_k)$ and $(G'_k,\theta_k')$ are called isomorphic if there exists an isomorphism between algebraic groups $f:G_k\rightarrow G'_k$, such that $f\circ\theta_k=\theta_k'\circ f$.
\begin{definition}\cite{Spr87}*{3.4}
An involution $\theta_k$ of $G_k$ as algebraic groups is called \emph{quasi-split} if there exists some Borel subgroup $B_k$, such that $\theta_k(B_k)\cap B_k$ is a maximal torus of $G_k$. Such a Borel subgroup $B_k$ is called \emph{$\theta_k$-anisotropic}.
We call a symmetric pair $(G_k,\theta_k)$ \emph{quasi-split} if the involution $\theta_k$ is quasi-split.
\end{definition}
In this paper, we shall only consider quasi-split symmetric pairs. In the sequel, symmetric pairs are always assumed to be quasi-split.
For any {\em anisotropic triple} $(G_k,\theta_k,B_k)$, where $(G_k,\theta_k)$ is a symmetric pair and $B_k$ is a $\theta_k$-anisotropic Borel subgroup of $G_k$, one can associate an $\imath$root datum in the following way.
Let $T_k=B_k\cap\theta_k(B_k)$ be the maximal torus. Let $(\I,Y,X,(\alpha_i)_{i\in\I},(\coroot_i)_{i\in\I}))$ be the root datum associated to $(G_k,T_k,B_k)$. It is easy to see that $\theta_k$ induces involutions on $X$ and $Y$, which respects the pairing, and $\theta_k$ restricts to an involution on the set of simple roots, which induces a graph involution $\tau$ on the Cartan datum $(\I,\cdot)$. Therefore, $(\I,Y,X,(\alpha_i)_{i\in\I},(\coroot_i)_{i\in\I},\tau,\theta_k)$ forms a quasi-split $\imath$root datum of finite type.
For a fixed symmetric pair $(G_k,\theta_k)$, it is direct to see that by choosing different $\theta_k$-anisotropic Borel subgroups $B_k$, the triple $(G_k,\theta_k,B_k)$ gives isomorphic $\imath$root data. We call any $\imath$root datum in this isomorphism class the \emph{$\imath$root datum associated to $(G_k,\theta_k)$.}
\begin{lemma}\label{lem:pinning}
Let $(G_k,\theta_k,B_k)$ be a triple as above, and write $T_k=B_k\cap\theta_k(B_k)$. Then there is a pinning $\{x_i,y_i;i\in\I\}$ of $(G_k,T_k,B_k)$, such that $\theta_k(x_i(\xi))=y_{\tau i}(\xi)$, for any $i\in\I$ and $\xi\in k$.
\end{lemma}
\begin{proof}
By \cite{Spr87}*{\S1.5}, there exists a pinning $\{x_i,y_i;i\in\I\}$ such that $\theta_k(x_i(\xi))=y_{\tau i}(-\xi)$, for any $i\in\I$ and $\xi\in k$. This is because $n$ in \cite{Spr87}*{\S1.5} is a representative of the longest element of $W$ for quasi-split cases. Now the lemma is immediate by rescaling.
\end{proof}
We will call such pinning an \emph{anisotropic pinning} of $(G_k,\theta_k,B_k)$. It is easy to see that anisotropic pinning is not unique.
\subsubsection{Classifications}\label{sec:class}
Take an $\imath$root datum $(\I,Y,X,(\alpha_i)_{i\in\I},(\coroot_i)_{i\in\I},\tau,\theta)$ of finite type. We follow the same notations as before. There is a well-defined $\A$-algebra automorphism ${_\A\dot{\U}}\rightarrow{_\A\dot{\U}}$, sending $E_i^{(n)}\one_\lambda$ to $F_{\tau i}\one_{\theta\lambda}$, and $F_i^{(n)}\one_\lambda$ to $E_{\tau i}\one_{\theta\lambda}$, for any $i\in\I$, $n\in\BN$, and $\lambda\in Y$. Let $A$ be any commutative ring with unit. It is direct to see that after base change, we have the induced $A$-algebra involution
\begin{equation}\label{eq:thetaA}
\theta_A:{_A\widehat{\U}}\rightarrow{_A\widehat{\U}}, \text{ which restricts to a group homomorphism } \theta_A:G_A\rightarrow G_A.
\end{equation}
In particular, $\theta_k:G_k\rightarrow G_k$ defines a quasi-split involution for the reductive group $G_k$, with $B_k$ an anisotropic Borel subgroup (\S \ref{sec:regpq}). Then the $\imath$root datum associated to the pair $(G_k,\theta_k)$ is isomorphic to
the one we start with.
By Lemma~\ref{lem:pinning} and \cite{Spr87}*{\S1.6}, we have the following proposition.
\begin{prop}\label{prop:cft}
We have a canonical bijection:
\[
\{\text{iso. classes of symmetric pairs $(G_k,\theta_k)$}\}\leftrightarrow\{\text{iso. classes of finite type $\imath$root data}\}.
\]
\end{prop}
In particular, the isomorphic classes of symmetric pairs $(G_k,\theta_k)$ is independent of the field $k$ (provided char $k$ $\neq 2$).
\subsubsection{$K_k$-orbits on the flag variety}\label{sec:Korbits}
We fix an anisotropic triple $(G_k,\theta_k,B_k)$ and set $K_k=G_k^{\theta_k}$ be the closed subgroup of $G_k$ consisting of $\theta_k$-fixed elements. It is called the \emph{symmetric subgroup} of $G_k$ associated to $\theta_k$. Let $\{x_i,y_i;i\in\I\}$ be an anisotropic pinning of $(G_k,\theta_k,B_k)$. Let $T_k=B_k\cap \theta_k(B_k)$ be the maximal torus, and $W=N(T_k)/T_k$. For any $i\in\I$, we set $n_i=x_i(1)y_i(-1)x_i(1)$.
Let $\CB_k=G_k/B_k$ be the flag variety of $G_k$. Then $K_k$ acts on $\CB_k$ with finitely many orbits by \cite{Spr85}*{Corollary~4.3}. The $K_k$-orbits on $\CB_k$ can be parameterized equivalently using $B_k$-orbits on $G_k/K_k$. The results are essentially in \cite{Spr85}. Since our choice of $(\theta_k, B_k)$ is different than Springer, who considers $B_k$ to be $\theta_k$-stable, we reformulate the results here.
Let $\phi: G_k/K_k \rightarrow G_k$, $gK_k \mapsto g \theta(g)^{-1}$ be the $G_k$-equivariant embedding \cite{Spr85}*{Proposition~2.2}, where the $G_k$-action on $G_k$ is given by $h \ast g = hg\theta(h)^{-1}$. Let $S_k = \phi (G_k/K_k)$. Since $\Big(B_k \times \theta_k(B_k)\Big)\text{-double cosets of $G_k$}$ are parameterized by the Weyl group $W$, we can define the composition
\[
\Pi: \{B_k\text{-orbits on } G_k/K_k\} \rightarrow \{\Big(B_k \times \theta_k(B_k)\Big)\text{-orbits on } G_k\} \rightarrow W.
\]
One sees that $K_kB_k/B_k$ is the unique open $K_k$-orbit on $\CB_k$ by \cite{Spr85} or \cite{Spr87}*{\S1.3}.
We next construct $K_k$-orbits inductive from the open orbit following \cite{Spr85}*{Theorem~6.5}. We consider $B_k$-orbits on $S_k \cong G_k/K_k$. Let $\Omega \subset S_k$ be a $B_k$-orbit via the $\ast$-action. We denote by $P_i$ the standard parabolic subgroup of $G_k$ associated to $i \in I$. We describe $P_i \ast \Omega$ following \cite{Spr85}*{\S6.7}. Let $P_i \ast \Omega = \Omega \cup \Omega'$, where
\[
\Omega' = \{u n_i x \theta(un_i)^{-1} \vert x \in \Omega, u \in U_i\}
\]
Let $w=\Pi(\Omega)\in W$. We divided into several cases.
\begin{enumerate}
\item[(a)$'$] If $\ell(s_i w \theta(s_i)) = \ell(w) -2$, then $\Omega'$ is the unique open dense $B_k$-orbit in $\overline{P_i \ast \Omega}$. We have $\Omega' =B_kn_i \ast \Omega$. We have $\dim \Omega' = \dim \Omega +1$.
\item[(a)$''$] If $\ell(s_i w \theta(s_i)) = \ell(w) + 2$, then $\Omega'$ is a single $B_k$-orbit in $P_i \ast \Omega$. We have $\Omega' =B_kn_i \ast \Omega$. We have $\dim \Omega' = \dim \Omega - 1$.
\end{enumerate}
Now we consider the cases where $s_i w \theta(s_i) = w$. We have an involution $\psi$ on the subgroup $G_i$ generated by $ x_i(k), y_i(k) $ defined by $\psi(g) = n \theta(g) n^{-1}$, where $n$ is any representative of $w$ in $\Omega$. We are reduced to rank one computations.
\begin{enumerate}
\item[(b)] If $\psi =\rm{id}$, then $\Omega' = \Omega$.
\item[ (c)] We assume $\psi(x_i(a)) = x_i(-a)$, $\psi(y_i(a)) = y_i(-a)$, $\psi(n_i) = n_i^{-1}$. Note that $ n_i = y_i(-1) x_i(1/2) \psi( y_i(-1) x_i(1/2))^{-1}$ in this case.
We write $\Omega_1 = B_k \ast n_i \Omega= B_ky_i(-1) x_i(1/2) \ast \Omega$ and $\Omega_2 = B_k \ast n_i \Omega \theta(n_i)^{-1}= B_k n_i \ast \Omega$. We then have $\Omega' = \Omega_1 \cup \Omega_2 $. Note that $\Omega = \Omega_2$ if $G_i \cong PGL_{2,k}$.
\item[(d)] We assume $\psi(x_i(a)) = y_i(a)$, $\psi(y_i(a)) = x_i(a)$, $\psi(n_i) = n_i^{-1}$. Note that $x_i(1/2) y_i(-1) \psi(x_i(1/2) y_i(-1))^{-1} = n_i$.
Then we write $\Omega_1 = B_k \ast n_i \Omega= B_k x_i(1/2) y_i(-1) \ast \Omega$ and $\Omega_2= B_k \ast \alpha^\vee_i(-1) n_i \Omega_1= B_k x_i(-1/2) y_i(1) * \Omega $. Hence we have
$\Omega' = \Omega_1 \cup \Omega_2$. Note that $\Omega_1 = \Omega_2$ if $G_i \cong PGL_{2,k}$.
\end{enumerate}
\begin{prop}\label{prop:Korbits}
(1) For any $K_k$-orbit on $\CB_k$, one can take a representative $vB_k$, which can be written as products of $n_i^{- 1}$, $ y_{i}(1)x_i(-1/2)$ and $ y_i(-1) x_{i}(1/2) $, for various $i \in \I$, and $v^{-1}\theta_k(v)\in N(T_k)$.
(2) (for quasi-split types) Codimension one $K_k$-orbits are of the form:
\begin{itemize}
\item $\mO_i=K_k n_iB_k/B_k$, for $i\in \I$, with $\tau i\neq i$; (Note that these orbits may not be distinct.)
\item $\mO_i^+=K_ky_i(1)B_k/B_k$ and $\mO_i^-=K_ky_i(-1)B_k/B_k$, for $i\in \I$, with $\tau i=i$. (Note that $\mO_i^+$ and $\mO_i^-$ may not be distinct.)
\end{itemize}
(3) The poset of $K_k$-orbits on $\mathcal{B}_k$ only depends on the $\imath$root datum associated to $(G_k,\theta_k)$. (Hence it is independent of the underlying field $k$.) We write $\CO_k = \CO$ whenever it is necessary to emphasize the field.
\end{prop}
\begin{proof}
Part (1) and part (2) is immediate from the above discussion.
We prove part (3). The claim on partial orders follows by \cite{RS90}*{Theorem~4.6 or Theorem~7.11}, since the case analysis above for $P_i \ast \Omega$ is independent of the characteristic of $k$. It suffices to show that the set of $K_k$-orbits on $\mathcal{B}_k$ is independent of the characteristic of $k$.
The description for the representatives in (1) only depends on the Weyl group $W$ and the involution on $W$ induced by $\theta_k$, which can be read from the $\imath$root datum. Two different representatives in (1) may represents the same orbit. It will suffice to show this phenomenon can be verified independent of $k$.
Let $v$, $v'$ be two different representatives in (1). Write $n=v^{-1}\theta_k(v)$, and $n'=v'^{-1}\theta_k(v')$. Then $n$ and $n'$ belong to $N(T_k)$. Then by the above discussion and (uniqueness part of) Bruhat's lemma, we have: $K_kvB_k=K_kv'B_k$ $\Leftrightarrow$ $n'\in B_k\ast n$ $\Leftrightarrow$ $n'\in T_k \ast n$ $\Leftrightarrow$ $n$ and $n'$ represent the same element in $W$, say $w$, and
\begin{equation}\label{eq:tsol}
\text{$\exists\; t\in T_k$, such that }w^{-1}(t)\theta_k(t)=n^{-1}n' = t_0\in T_k.
\end{equation}
Recall $Y=\text{Hom}(k^\times, T_k)$, and there is a canonical isomorphism $T_k\cong k\otimes_\BZ Y\cong k\otimes_{\BZ[2^{-1}]}Y_2$, where $Y_2=\BZ[2^{-1}]\otimes_\BZ Y$. Under our construction, $t_0=1\otimes t_0'$, for some $t_0'\in Y_2$, independent of $k$. And $\theta_k$ is obtained from $\theta:Y\rightarrow Y$. Since $\theta_k(n)=n^{-1}$, we have $(\theta w)^2=id:Y\rightarrow Y$. By the same proof of Lemma \ref{le:nop}, the cokernel of the map $id-\theta w:Y\rightarrow Y$ has no odd torsion. Hence condition \eqref{eq:tsol} is equivalent to $\theta(t_0')\in \text{im}(id-\theta w)\subseteq Y_2$. Here we extend $\theta$ and $w$ to $Y_2$. This is clearly independent of the field $k$. We complete the proof.
\end{proof}
\subsection{Frobenius splittings} \label{sec:algFr}
Let $k$ be an algebraically closed field of positive characteristic $p$ (possibly $2$ in this section).
\subsubsection{Definitions}
By \emph{schemes}, we mean separated schemes of finite type over $k$.
\begin{defi}
Let $ \CX$ be a scheme; then the \emph{absolute Frobenius morphism}
\[
F:\CX\longrightarrow \CX
\]
is the identity on the underlying space of $\CX$, and the $p$-th power map on the structure sheaf $\mathcal{O}_\CX$.
\end{defi}
Following \cite[\S1.1.3]{BK05}, a scheme $\CX$ is \emph{Frobenius split} (or simply \emph{split}) if the $\mathcal{O}_\CX$-linear map $F^\#:\mathcal{O}_\CX\longrightarrow F_*\mathcal{O}_\CX$ splits. Namely, there exists a $\mathcal{O}_\CX$-linear map $\varphi\in\text{Hom }(F_*\mathcal{O}_\CX,\mathcal{O}_\CX)$ such that $\varphi\circ F^\#={\rm id}$. Such a map $\varphi$ is called a \emph{splitting} of $X$. Then clearly any $\mathcal{O}_\CX$-linear map $\varphi:F_*\mathcal{O}_\CX\longrightarrow \mathcal{O}_\CX$ is a splitting if and only if $\varphi(1)=1$.
Let $\CY\subset \CX$ be a closed subscheme of $\CX$, then we say $\CX$ is \emph{split compatibly with $\CY$} (or \emph{$\CY$ is compatibly split}) if there is a splitting $\varphi$ of $\CX$
such that $\varphi(F_*\mathcal{I}_\CY)\subset \mathcal{I}_\CY$, where $\mathcal{I}_\CY$ is the ideal sheaf of $\CY$.
We collect the following consequences on the Frobenius splitting schemes by \cite{BK05}*{\S1.2}.
(a) {\em Frobenius split schemes are reduced, and weakly normal.}
(b) {\em Let $\CX$ be a scheme, $\phi$ be a Frobenius splitting of $X$. Then the set of closed subschemes of $\CX$ which are compatibly split under $\phi$ is closed under taking irreducible components, and taking scheme-theoretic intersection. In particular, the scheme-theoretic intersection of two compatibly split subschemes is reduced.}
(c) {\em Let $\CX$ be a proper scheme and $\CY$ be a closed subscheme of $\CX$. Suppose $\CX$ is Frobenius split compatibly with $\CY$. Let $\mathcal{L}$ be an ample line bundle on $\CX$. Then $H^i(\CX,\mathcal{L})=0$, for all $i>0$, the restriction map $H^0(\CX,\mathcal{L})\rightarrow H^0(\CY,\mathcal{L})$ is surjective, and $H^i(\CY,\mathcal{L})=0$, for all $i>0$.}
One can often apply the positive characteristic techniques of Frobenius splitting to certain schemes in characteristic zero as well, e.g., Schubert varieties in characteristic zero. We refer to \cite{BK05}*{\S1.6} for details.
\subsubsection{Flag varieties}
Let $\CB_k=G_k/B_k$ be the flag variety of a connected reductive group $G_k$. For any $\lambda\in X$, let $k_\lambda$ be the one-dimensional representation of the group $B_k=U_kT_k$, where $U_k$ is the unipotent radical of $B_k$ and $T_k$ is the maximal torus, on which $T_k$ acts via the isomorphism $X\xrightarrow{\sim} \text{Hom}(T_k,k^\times)$ and $U_k$ acts trivially. Let
\[
\mathcal{L}_\lambda=G_k\times^{B_k} k_{\lambda}\longrightarrow \CB_k
\]
be the associated line bundle over the flag variety $\CB_k$. By definition, the total space $G_k\times ^{B_k} k_{\lambda}$ is the quotient space $(G_k\times k)/B_k$, where $B_k$ acts by $(g,t)\cdot b=(gb^{-1},\lambda(b)t)$, for $g\in G_k$, $b\in B_k$, and $t\in k$. The line bundle $\mathcal{L}_\lambda$ is (very) ample if and only if $\lambda \in X^{++}$.
The space of global sections $H^0(\lambda)=H^0(\CB_k,\mathcal{L}_\lambda)$ admits a natural (rational) $G_k$-action. It is well-known that $H^0(\lambda)$ is nonzero if and only if $\lambda\in X^+$, in which case $H^0(\lambda) \cong V_k(\lambda)^*$. Here $V_k(\lambda)$ denotes the Weyl module of $G_k$ of highest weight $\lambda$.
Let $\mathcal{L}_\lambda$ be an ample line bundle on $\CB_k$. Define a graded $k$-algebra
\[
R_{\mathcal{L}_\lambda}=\bigoplus_{n\geqslant 0} H^0(\mathcal{B}_k,\mathcal{L}_\lambda^n) =\bigoplus_{n\geqslant 0} H^0(\mathcal{B}_k,\mathcal{L}_{n\lambda}).
\]
Let $\mathcal{Y}\subset \CB_k$ be a closed subvariety, and $I_{\mathcal{Y},{\mathcal{L}_\lambda^n}}$ be the kernel of the restriction map $H^0(\CB_k,{\mathcal{L}_\lambda^n})\rightarrow H^0(\mathcal{Y},{\mathcal{L}_\lambda^n})$, for any $n\geqslant 0$. Set
\[
I_{\mathcal{Y},{\mathcal{L}_\lambda}}=\bigoplus_{n\geqslant 0}I_{\mathcal{Y},{\mathcal{L}_\lambda^n}}.
\]
It is an ideal of $R_{\mathcal{L}_\lambda}$. The following lemma follows from \cite{KL2}*{Theorem~6.4 \& 6.7}.
\begin{lem}\label{le:algFrfl}
Let $\phi : R_{\mathcal{L}_\lambda} \rightarrow R_{\mathcal{L}_\lambda}$ be a graded additive endomorphism such that (a) $\phi(f^pg)=f\phi(g)$, for any $f,g$ in $R_{\mathcal{L}_\lambda}$; (b) $\phi(1)=1$; (c) $\phi ( I_{\mathcal{Y},{\mathcal{L}_\lambda}}) \subset I_{\mathcal{Y},{\mathcal{L}_\lambda}}$. Then
the $\mathcal{B}_k$ is Frobenius split compatibly with $\mathcal{Y}$.
\end{lem}
\section{Symmetric subgroup schemes}\label{sec:SymGpSch}
Let $(\I,Y,X,A,(\alpha_i)_{i\in\I},(\coroot_i)_{i\in\I},\tau,\theta)$ be a quasi-split $\imath$root datum of finite type. Let $(\U, \U^\imath)$ be the quantum symmetric pair associated to this datum. We also fix parameters of $\U^\imath$ as follows: $\varsigma_i=q^{-1}_i$, if $\tau i=i$; $\varsigma_i=1$, if $a_{i,\tau i}=0$; and $\{\varsigma_i,\varsigma_{\tau i}\}=\{1,q_i^{-1}\}$, if $a_{i,\tau i}=-1$.
\subsection{The finiteness conditions}\label{sec:fin}
Recall the isomorphism of $\A$-modules from \eqref{eq:pim}:
\[
p_{\imath,\lambda}=\pi_\lambda\circ\iota_\lambda:{_\A\dot{\U}^\imath\one_{\overline{\lambda}}}\xrightarrow{\iota_\lambda}{_\A\dot{\U}\one_\lambda}\xrightarrow{\pi_\lambda}{_\A\dot{\U}^-\one_\lambda}.
\]
Set $s_\lambda=p_{\imath,\lambda}^{-1}\circ\pi_\lambda:{_\A\dot{\U}\one_\lambda}\longrightarrow{_\A\dot{\U}^\imath}\one_{\overline{\lambda}}.$ Then $s_\lambda\circ \iota_\lambda=id$.
Recall the canonical basis $\dot{\RB}$ of $\dot{\U}$ and the $\imath$canonical basis $\dot{\RB}^\imath$ of $\dot{\U}^\imath$.
For any $\lambda\in X^+$, let $\dot{\RB}^\imath_{\overline{\lambda}} =\dot{\RB}^\imath\cap{\dot{\U}^\imath\one_{\overline{\lambda}}}$
and $\dot{\RB}_\lambda=\dot{\RB}\cap{\dot{\U}\one_\lambda}$.
Then ${\dot{\U}^\imath\one_{\overline{\lambda}}}$ has basis $\dot{\RB}^\imath_{\overline{\lambda}}$ and ${\dot{\U}\one_\lambda}$ has basis $\dot{\RB}_\lambda$.
We write $$\iota_\lambda(b)=\sum_{b'\in\dot{\RB}_\lambda}\iota_{\lambda,b,b'}b', \quad \text{for } b\in\dot{\RB}^\imath_{\overline{\lambda}}, \iota_{\lambda,b,b'} \in \A; $$
$$s_\lambda(b)=\sum_{b'\in\dot{\RB}_{\overline{\lambda}}}s_{\lambda,b,b'}b',\quad \text{for } b\in\dot{\RB}_\lambda, s_{\lambda,b,b'} \in \A.$$
We view $_\A\dot{\U}^\imath$ as an $\A$-subalgebra of $_\A\widehat{\U}$ via the embedding $x \mapsto \sum_{\lambda \in X}\iota_\lambda(x) $ as in \S \ref{sec:iqp}. We write
\[
b=\sum_{b'\in\dot{\RB}}\iota_{b,b'}b' \in {}_\A\widehat{\U}, \quad \text{for } b\in\dot{\RB}^\imath, \iota_{b,b'} \in \A.
\]
\begin{lemma}\label{le:finit}
Let $\mu\in X$. Then for any $b'\in \dot{\RB}_\mu$, there are only finitely many $b \in \dot{\RB}^\imath_{\overline{\mu}}$, such that $\iota_{\mu,b,b'}\neq 0$. For any $b'\in\dot{\RB}^\imath_{\overline{\mu}}$, there are only finitely many $b\in\dot{\RB}_\mu$, such that $s_{\mu,b,b'}\neq 0$.
In particular, for any $b'\in \dot{\RB}$, there are only finitely many $b\in\dot{\RB}^\imath$, such that $\iota_{b,b'}\neq 0$.
\end{lemma}
\begin{proof}
Suppose there exists some $b'\in\dot{\RB}_\mu$, such that there are infinitely many $b\in\dot{\RB}^\imath_{\overline{\mu}}$, such that $\iota_{\mu,b,b'}\neq 0$.
Take $\mu_1, \mu_2\in X^+$, such that $b'\cdot v_{-\mu_1}^-\otimes v_{\mu_2}^+\neq 0$ in $^\omega L(\mu_1)\otimes L(\mu_2)$. Then by the assumption, we get infinitely many $b\in\dot{\RB}^\imath_{\overline{\mu}}$, such that $b\cdot v_{-\mu_1}\otimes v_{\mu_2}\neq 0$.
Recall from \cite{BW18a}*{Theorem 4.18} that we have $\U^\imath$-module isomorphism $$\mathcal{T}:{ L(\tau\mu_1)}\rightarrow {^\omega L(\mu_1)}$$ sending $v_{\tau\mu_1}^+$ to $v_{-\mu_1}^-$. Then we have $\U^\imath$-module isomorphism $$\mathcal{T}^{-1}\otimes id:{^\omega L(\mu_1)}\otimes L(\mu_2)\rightarrow L(\tau\mu_1)\otimes L(\mu_2)$$ sending $v_{-\mu_1}^-\otimes v_{\mu_2}^+$ to $v_{\tau \mu_1}^+\otimes v_{\mu_2}^+$. Hence via this isomorphism, we get infinitely many $\imath$canonical basis elements $b$, with $b\cdot v_{\tau \mu_1}^+\otimes v_{\mu_2}^+\neq 0$. Via the $\U$-module embedding $L(\tau \mu_1+\mu_2)\rightarrow L(\tau \mu_1)\otimes L(\mu_2)$, sending $v_{\tau\mu_1+\mu_2}^+$ to $v_{\tau \mu_1}^+\otimes v_{\mu_2}^+$, we get infinitely many $\imath$canonical basis elements $b$, such that $b\cdot v_{\tau\mu_1+\mu_2}^+\neq 0$. This contradicts Theorem \ref{thm:stab}. We proved the first claim.
For the second claim, suppose there exists some $b'\in \dot{\RB}^\imath_{\overline{\mu}}$, such that there are infinitely many $b\in\dot{\RB}_\mu$, with $s_{\mu,b,b'}\neq 0$. We take $\mu\in X^+$, such that $b'\cdot v_{\mu}^+\neq 0$ in $L(\mu)$. Then for any $b\in \dot{\RB}_\mu$, since $b\cdot v_{\mu}^+=s_\mu(b)\cdot v_{\mu}^+$, it follows that $b\cdot v_{\mu}^+\neq 0$, whenever $s_{\mu,b,b'}\neq0$. Therefore we get infinitely many canonical basis element $b$, with $b\cdot v_{\mu}^+\neq 0$, which is a contradiction. This proves the second claim.
\end{proof}
\begin{corollary}\label{cor:fimu}
For any $b',b''\in\dot{\RB}^\imath$, write $b'\cdot b''=\sum_{b\in\dot{\RB}^\imath}m_{b',b''}^b b$. Then for any $b\in\dot{\RB}^\imath$, there are only finitely many pair $(b',b'')$ in $\dot{\RB}^\imath$, such that $m_{b',b''}^b\neq 0$.
\end{corollary}
\begin{proof}
For any $b\in\dot{\RB}^\imath$, we choose $\mu\in X^+$, such that $b\cdot v_\mu^+\neq 0$ in $L(\mu)$. Suppose $m_{b',b''}^b\neq 0$, for some $b',b''$ in $\dot{\RB}^\imath$. Then thanks to Theorem \ref{thm:stab}, we have $(b'\cdot b'')\cdot v_\mu^+\neq 0$. In particular, $b'\mid_{L(\mu)}\not\equiv 0$, and $b''\mid_{L(\mu)}\not\equiv 0$. Finally, note that for all but finitely many canonical basis element $c$, we have $c\mid_{L(\mu)}\equiv 0$. Then thanks to Lemma \ref{le:finit}, we only have finitely many such pair $(b',b'')$.
\end{proof}
\begin{remark}
Theorem~\ref{thm:stab} has been established by Watanabe \cite{Wa22} for all real rank one cases as well. Hence Lemma~\ref{le:finit} and Corollary~\ref{cor:fimu} hold for real rank one cases by similar proofs. One can also show all results in \S\ref{sec:SymGpSch} remain valid for real rank one cases.
All results in \S\ref{sec:SymGpSch} hold if the strong compatibility conjecture in \cite{BW18a}*{Remark~6.18} is true.
\end{remark}
\subsection{The group scheme $\G^\imath$}
Let $A$ be a commutative ring (with unit) viewed as an $\A$-algebra via $q\mapsto 1$. We follow the same notations and conventions as in \S \ref{sec:nocp}. We write $_A\dot{\U}^\imath=A\otimes_\A\big({_\A\dot{\U}^\imath}\big)$. We will abuse the notations, and use $\dot{\RB}^\imath$ to denote the basis of $_A\dot{\U}^\imath$, consisting of the image of the $\imath$canonical basis elements after base change. This should not cause any confusion.
\subsubsection{}\label{sec:Ui1}
Let $_A\widehat{\mathrm{U}}^\imath$ be the $A$-module consisting of formal linear combinations
\[ \sum_{b\in\dot{\RB}^\imath} n_b b, \quad n_b\in A.
\]
Then thanks to Lemma \ref{le:finit}, $_A\widehat{\mathrm{U}}^\imath$ can be naturally embedded into $_A\widehat{\mathrm{U}}$. By the Corollary \ref{cor:fimu}, the product structure on $_A\widehat{\U}^\imath$ is well defined, and the embedding $_A\widehat{\U}^\imath\hookrightarrow{_A\widehat{\U}}$ is compatible with products.
Define $_A\widehat{\U}^{\imath,1}$ be the $A$-module consisting of all the formal linear combinations
\[
\sum_{(a,a')\in \dot{\RB}^\imath\times \dot{\RB}} n_{a,a'}a\otimes a', \quad n_{a,a'}\in A
\]
Again, by Lemma \ref{le:finit}, we have natural embedding $_A\widehat{\U}^{\imath,1}\hookrightarrow{_A\widehat{\U}^{(2)}}$. Thanks to the Corollary \ref{cor:fimu} and \cite{Lu07}*{Lemma 1.8}, one can define a product structure on $_A\widehat{\U}^{\imath,1}$ in an evident way. Then the embedding is moreover compatible with products.
Since $\U^\imath$ is a right coideal subalgebra of $\U$, it follows that $\WD:{_A\widehat{\U}}\rightarrow{_A\widehat{\U}}$ will restrict to an $A$-algebra homomorphism
\begin{equation*}
\WD: {_A\dot{\U}^\imath}\longrightarrow {_A\widehat{\U}^{\imath,1}}.
\end{equation*}
\subsubsection{}\label{sec:pro}
Define $\RO_A^\imath$ be the $A$-submodule of $(_A\dot{\U}^\imath)^*=\text{Hom}_A({_A\dot{\U}^\imath},A)$, spanned by the \emph{dual $\imath$canonical basis} $\{b^*\mid b\in \dot{\RB}^\imath\}$, where $b^*$ stands for the $A$-linear maps sending $b'$ to $\delta_{b', b}$, for any $b'\in\dot{\RB}^\imath$.
Recall \S\ref{sec:nocp} that $\RO_A$ is an $A$-submodule of $_A\dot{\U}^*$, spanned by the dual canonical basis. Define the $A$-linear map
\[
r:\RO_A\rightarrow\RO_A^\imath, \quad b^* \mapsto \sum_{b_1\in\dot{\RB}^\imath}\iota_{b_1,b}b_1^*, \quad \text{for }b\in\dot{\RB}.
\]
The summation is finite thanks to Lemma \ref{le:finit}. Then it is clear that for any $f\in \RO_A$, we have $r(f)=\hat{f}\mid_{_A\dot{\U}^\imath}$. Here $\hat{f}$ is the extension of $f$ to $_A\widehat{\U}$.
For any $\mu\in X$, let $\RO_{A,\mu}$ be the $A$-submodule of $\RO_A$, spanned by all $b^*$, such that $b\in \dot{\RB}\cap {_A\dot{\U}\one_\mu}$. Then $\RO_A=\bigoplus_{\mu\in X}\RO_{A,\mu}$. Moreover, we have $\RO_{A,\mu}\cdot\RO_{A,\mu'}\subseteq \RO_{A,\mu+\mu'}$. Via the restriction, we have a natural map $\RO_A\rightarrow(_A\dot{\U}\one_\mu)^*$. Under this restriction, the submodule $\RO_{A,\mu}$ is sent injectively to an $A$-submodule of $(_A\dot{\U}\one_\mu)^*$. We identify $\RO_{A,\mu}$ with its image in $(_A\dot{\U}\one_\mu)^*$.
Similarly, for any $\zeta\in X_\imath$, define $\RO_{A,\zeta}^\imath$ be the $A$-submodule of $\RO_A^\imath$, spanned by all the elements $b^*$, such that $b\in \dot{\RB}^\imath\cap {_A\dot{\U}^\imath\one_\zeta}$. Then $\RO_A^\imath=\bigoplus_{\zeta\in X_\imath}\RO_{A,\zeta}^\imath$. Similarly, the submodule $\RO_{A,\zeta}^\imath$ can be identified with an $A$-submodule of $(_A\dot{\U}^\imath\one_\zeta)^*$ via restriction.
By \S\ref{sec:fin} we have $A$-linear maps
\begin{equation*}
\begin{tikzcd}
{_A\dot{\U}\one_\mu}\arrow[r,bend right,"s_\mu"']& {_A\dot{\U}^\imath\one_{\overline{\mu}}} \arrow[l,"\iota_\mu"'],
\end{tikzcd}, \quad \text{ such that }s_\mu\circ\iota_\mu=id, \quad \text{for any } \mu\in X.
\end{equation*}
Taking linear dual of these maps, and using Lemma \ref{le:finit}, we have
\begin{equation*}
\begin{tikzcd}
\RO_{A,\mu}\arrow[r,"\iota_\mu^*"] & \RO_{A,\overline{\mu}}^\imath \arrow[l,bend left,"s_\mu^*"]
\end{tikzcd}, \quad \text{such that }\iota_\mu^*\circ s_\mu^*=id.
\end{equation*}
In particular, we deduce that $\iota_\mu^*$ is surjective, and $s_\mu^*$ is injective.
It follows from the definition that $\iota_\mu^*(f)=r(f)$, for any $f\in\RO_{A,\mu}$. Hence we deduce
(a) {\em the map $r:\RO_A\rightarrow\RO_A^\imath$ is surjective.}
\subsubsection{}\label{sec:HoOi}
Recall from \S \ref{sec:nocp} that $(\RO_A,\delta, \sigma,\epsilon)$ is a commutative $A$-Hopf algebra.
\begin{theorem}\label{thm:Hopfi}
The $A$-module $\RO_A^\imath$ has a structure of a commutative $A$-Hopf algebra, such that the surjection $r:\RO_A\rightarrow\RO_A^\imath$ is a Hopf algebra homomorphism.
\end{theorem}
\begin{proof}
Note that the kernel $I_A$ of $r$ consists of linear forms $f$, such that $\hat{f}\mid_{{{}_A\dot{\U}^\imath}} = 0$. We firstly show that $I_A$ is an ideal. Take any $f\in I_A$, and $g\in \RO_A$. For any $x\in{_A\dot{\U}^\imath}$, we have $\widehat{fg}(x)=f\otimes g\circ(\widehat{\Delta}(x))=0$. The last equality follows from the fact $\widehat{\Delta}({_A\dot{\U}^\imath})\subseteq {_A\widehat{\U}^{\imath,1}}$. Hence $fg$ belongs to $I_A$. Since $\RO_A$ is commutative, $I_A$ is a two-sided ideal.
Since the embedding $_A\dot{\U}^\imath\hookrightarrow{_A\widehat{\U}}$ is compatible with products, it follows that the comultiplication $\delta:\RO_A\rightarrow\RO_A\otimes\RO_A$ induces $\RO_A/I_A\rightarrow \RO_A/I_A\otimes\RO_A/I_A$. Finally, it is direct to check that $\sigma(I_A)\subseteq I_A$, and $\epsilon(I_A)=0$. Hence $\RO_A/I_A$ admits a commutative $A$-Hopf algebra structure. We then endow $\RO_A^\imath$ with the commutative $A$-Hopf algebra structure via the isomorphism $\RO_A^\imath\cong \RO_A/I_A$ and the theorem is proved.
\end{proof}
Since $\RO_A^\imath$ is a commutative Hopf algebra over $A$, the set $\text{Hom}_{A-\text{alg}}(\RO_A^\imath, A)$ of $A$-algebra homomorphisms $\RO_A^\imath\rightarrow A$ preserving 1, has a group structure.
By definition, $\text{Hom}_{A-\text{alg}}(\RO_A^\imath,A)$ is an $A$-submodule of $\RO_A^{\imath,*}$, the $A$-linear duals of $\RO_A^\imath$. There is an ($A$-linear) bijective map from $\RO_A^{\imath,*}$ to $_A\widehat{\U}^\imath$, given by $\phi\mapsto \sum_{b\in\dot{\RB}^\imath}\phi(b^*)b$. We write $G_A^\imath$ to be the image of $\text{Hom}_{A-\text{alg}}(\RO_A^\imath,A)$ under this bijection. Via the embedding $_A\widehat{\U}^\imath\hookrightarrow{_A\widehat{\U}}$, we view $G_A^\imath$ as a subset of ${_A\widehat{\U}}$.
Recall the construction for the group $G_A$ from \S \ref{sec:regpq}. The following claim is immediate.
(a) {\em As subsets of $_A\widehat{\U}$, we have $G_A^\imath={_A\widehat{\U}^\imath}\cap G_A$, which is a subgroup of $G_A$. And the bijection $\text{Hom}_{A-\text{alg}}(\RO_A^\imath,A)\xrightarrow{\sim} G_A^\imath$ is moreover a group isomorphism. }
\begin{definition}
We define $\G^\imath$ as the $\BZ$-group scheme, by setting $\G^\imath(A)=G_A^\imath$, for any $\BZ$-algebra A. We have $\G^\imath \cong Sp\, \RO_{\BZ}^\imath $ as affine group schemes.
\end{definition}
By \S\ref{sec:pro} (a), $\G^\imath$ is a closed subsgroup scheme of $\G$, where $\G$ denotes the Chevalley group scheme over $\BZ$ associated to (the root data) of $G$.
\subsubsection{}
We next show that $\G^\imath_{\BZ[2^{-1}]}\rightarrow Sp\,\BZ[2^{-1}]$ has reduced geometric fibres.
\begin{proposition}\label{prop:GAi}
Suppose $A$ is an integral domain with characteristic not 2. Then $\RO_A^\imath$ is a reduced $A$-algebra.
\end{proposition}
\begin{proof}
For any $\mu,\mu'\in X$, we have the following commutative diagram by definitions:
\begin{equation*}
\begin{tikzcd}
{_A\dot{\U}^\imath}\one_{\overline{\mu+\mu'}} \arrow[r,"\iota_{\mu+\mu'}"] \arrow[d,"\Delta_{\overline{\mu},\overline{\mu'}}"'] & {_A\dot{\U}\one_{\mu+\mu'}} \arrow[r,"\pi_{\mu+\mu'}"] \arrow[d,"\Delta_{\mu,\mu'}"'] & {_A\U^-\one_{\mu+\mu'}} \arrow[d,"\Delta_{\mu,\mu'}"] \\ {_A\dot{\U}^\imath\one_{\overline{\mu}}}\otimes{}{_A\dot{\U}^\imath\one_{\overline{\mu'}}} \arrow[r,"\iota_\mu\otimes\iota_{\mu'}"] & {_A\dot{\U}\one_\mu}\otimes{} {_A\dot{\U}\one_{\mu'}} \arrow[r,"\pi_\mu\otimes\pi_{\mu'}"] & {_A\U^-\one_\mu}\otimes{} {_A\U^-\one_{\mu'}}.
\end{tikzcd}
\end{equation*}
Here $\Delta_{\mu,\mu'}$ (resp. $\Delta_{\overline{\mu},\overline{\mu'}}$) stands for the comultiplication restricting to the corresponding weight spaces (resp. $\imath$weight spaces). Then we have the commutative diagram:
\begin{equation}\label{dia:As}
\begin{tikzcd}
{_A\dot{\U}\one_{\mu+\mu'}} \arrow[r,"s_{\mu+\mu'}"] \arrow[d,"\Delta_{\mu,\mu'}"'] & {_A\dot{\U}^\imath\one_{\overline{\mu+\mu'}}} \arrow[d,"\Delta_{\overline{\mu},\overline{\mu'}}"]\\ {_A\dot{\U}\one_\mu}\otimes{}{_A\dot{\U}\one_{\mu'}} \arrow[r,"s_\mu\otimes s_{\mu'}"] & {_A\dot{\U}^\imath\one_{\overline{\mu}}}\otimes {_A\dot{\U}^\imath\one_{\overline{\mu'}}}.
\end{tikzcd}
\end{equation}
By taking the linear dual of the diagram \eqref{dia:As} and restricting to the proper subspaces, we get the commutative diagram:
\begin{equation}\label{dia:OmA}
\begin{tikzcd}
\RO_{A,\mu+\mu'} & \RO_{A,\overline{\mu+\mu'}}^\imath \arrow[l,"s_{\mu+\mu'}^*"'] \\ \RO_{A,\mu}\otimes \RO_{A,\mu'} \arrow[u] & \RO_{A,\overline{\mu}^\imath}\otimes\RO_{A,\overline{\mu'}}^\imath \arrow[u] \arrow[l,"s_\mu^*\otimes s_{\mu'}^*"'].
\end{tikzcd}
\end{equation}
Since $\RO_A^\imath$ is a flat $A$-module, it can be viewed as a subring of $\RO_K^\imath$, where $K$ is the fractional field of $A$. Hence we may assume that $A$ is a field. Set $p=\text{char }A$. Then by our assumption, $p\neq 2$.
Firstly assume $p>0$. Note that for any $\zeta,\zeta'\in X_\imath$, we have $\RO_{A,\zeta}^\imath\cdot\RO_{A,\zeta'}^\imath\subset\RO_{A,\zeta+\zeta'}^\imath$. So $\RO_A^\imath=\bigoplus_{\zeta\in X_\imath}\RO_{A,\zeta}^\imath$ is a $X_\imath$-graded algebra. We firstly prove that a homogeneous element cannot be nilpotent.
Take $f\in \RO_{A,\zeta}^\imath$, $g\in \RO_{A,\zeta'}^\imath$, such that $f\cdot g=0$. Choose $\mu,\mu'\in X$, such that $\overline{\mu}=\zeta$, $\overline{\mu'}=\zeta'$. By the commuting diagram \eqref{dia:OmA}, we deduce that $s_\mu^*(f)\cdot s_{\mu'}^*(g)=0$. Since $\RO_A$ is a integral domain, and $s_\mu^*$, $s_{\mu'}^*$ are injective, we deduce that either $f$ or $g$ is 0. Now suppose $f^n=0$, for some homogeneous element $f$, using the argument inductively, we deduce that $f=0$.
Suppose $f=\sum_{\zeta\in X_\imath} f_\zeta\in \RO_A^\imath$ be a nilpotent element, where $f_\zeta\in\RO_{A,\zeta}^\imath$. Then there exists $r\in \BN$, such that $f^{p^r}=0$. Note that $f^{p^r}=\sum_{\zeta\in X_\imath} f_\zeta^{p^r}$. Element $f_\zeta^{p^r}$ has degree $p^r\zeta$. Since $X_\imath$ has no $p$-torsion by Lemma \ref{le:nop}, we deduce that $f_\zeta^{p^r}$ and $f_{\zeta'}^{p^r}$ have different degrees whenever $\zeta\neq \zeta'$. Hence $f_\zeta^{p^r}=0$, for any $\zeta\in X_\imath$, which implies $f_\zeta=0$, thanks to the above argument. Therefore $f=0$.
The case when $p=0$ either follows from \cite{BK05}*{1.6.5}, or follows from a well-known result by Cartier (which asserts that any finitely generated Hopf algebra over a field with characteristic 0 is reduced).
\end{proof}
\begin{remarks}
The assumption that $\textrm{char}(A)\neq 2$ can not be dropped in general. In the case of of rank 1, one can show that $\RO_A^\imath\cong A[u,v]/(u^2-v^2-1)$, which is non-reduced if $A$ has characteristic 2.
\end{remarks}
\subsection{$G_A^\imath$ as a symmetric subgroup}
Let $A$ be an integral domain with characteristic not 2. Let $k$ be an algebraic closure of the quotient field of $A$. Then characteristic of $k$ in not 2.
\subsubsection{}
Recall \eqref{eq:thetaA} the group involution $\theta_A:G_A\rightarrow G_A$. Set $K_A=G_A^{\theta_A}$ be the subgroup of $G_A$ consisting of $\theta_A$ fixed points. It is called the \emph{symmetric subgroup} of $G_A$ corresponding to the involution $\theta_A$.
Since $B_k$ is a $\theta_k$-anisotropic Borel subgroup of $G_k$, the maximal torus $T_k = B_k \cap \theta_k(B_k)$ contains a maximal $\theta_k$-anisotropic torus. We denote by $K_k^\circ$ the identity component of $K_k$. Then by \cite{Vu74}*{Proposition~7} (cf. \cite{Ri82}*{\S2.9}), we have \begin{equation}\label{eq:TK}
K_k = T_k^{\theta_k} K^\circ_k.
\end{equation}
\begin{theorem}\label{thm:Oik}
As closed subgroups of $G_k$, we have $G_k^\imath=K_k$. In particular, the coordinate ring of the symmetric subgroup $K_k$ is isomorphic to $\RO_k^\imath$.
\end{theorem}
\begin{proof}
We first show $G_k^\imath\subseteq K_k$. It suffices to show the subalgebra $_k\dot{\U}^\imath \subset {}_k\widehat{\U}$ is fixed by $\theta_k$ pointwise. Since $_k\dot{\U}^\imath\cong k\otimes_\BZ {}_\BZ\dot{\U}^\imath$, and $_\BZ\dot{\U}^\imath$ is a subalgebra of $_\BC\dot{\U}^\imath$, we may assume $k=\BC$. Since $_\BC\dot{\U}^\imath$ is generated by $B_{i,\zeta}$ ($i\in\I$, $\zeta\in X_\imath$), we only need to check $\theta_\BC$ fixes these generators. This is direct, and we leave it for readers.
Next we compare the Lie algebras of $G_k^\imath$ and $K_k$. Let $\mathfrak{g}$ be the Lie algebra of $G$. Recall from \S \ref{sec:liealg} that we have a canonical embedding $\mathfrak{g}\hookrightarrow{}_k\widehat{\U}$. Also recall that $\mathfrak{g}$ has generators (as a Lie algebra)
\[
e_i=\sum_{\lambda\in X}E_i\one_\lambda,\qquad f_i=\sum_{\lambda\in X}F_i\one_\lambda,\qquad h_\mu=\sum_{\lambda\in X}\langle \mu,\lambda\rangle\one_\lambda,
\]
for any $i\in\I$, and $\mu\in Y$. Here we are abusing notations by using the same notations to denote the elements after base change.
The involution $\theta_k$ on $_k\widehat{\U}$ also restricts a (Lie algebra) involution on $\mathfrak{g}$, which is the same as the differentiation of the involution on the group $G_k$. We still write $\theta_k$ to denote the involution on $\mathfrak{g}$.
Let $\mathfrak{k}$ be the Lie algebra of $K_k$. By \cite{Bo91}*{\S9.4}, we have $\mathfrak{k}=\mathfrak{g}^{\theta_k}$. By the triangular decomposition of $\mathfrak{g}$ (cf. \cite{Ko14}*{Lemma~2.7}), it is easy to see that $\mathfrak{k}=\mathfrak{g}^{\theta_k}$ is the subalgebra of $\mathfrak{g}$ generated by $f_i+e_{\tau i}$ ($i\in\I$) and $h_\mu$ ($\mu\in Y^\imath$).
We write $\mathfrak{g}^\imath\subseteq \mathfrak{g}$ to be the Lie algebra of $G^\imath_k$. Then it follows from the definition that $\mathfrak{g}^\imath=\mathfrak{g}\cap{_k\widehat{\U}^\imath}$, as Lie subalgebras of $_k\widehat{\U}$. For any $i\in\I$, $\mu\in Y^\imath$, we have
\[
f_i+e_{\tau i}=\sum_{\zeta\in X_\imath}B_{i,\zeta},\qquad h_\mu=\sum_{\zeta\in X_\imath}\langle \mu,\zeta\rangle \one_\zeta.
\]
Again, we abuse notations here. Then we deduce that $\mathfrak{k}$ is contained in $\mathfrak{g}^\imath$. Combined with $G_k^\imath\subseteq K_k$, it follows that $K_k^\circ$ is contained in $G_k^\imath$.
Thanks to \eqref{eq:TK}, it remains to show that $T_k^{\theta_k}$ is contained in $G_k^\imath$.
Take any $\xi=\sum_{\lambda\in X}n_\lambda\one_\lambda$ in $T_k^{\theta_k}$. Then $n_\lambda\in k^\times$ and $n_{\lambda'}n_{\lambda''}=n_{\lambda'+\lambda''}$, for any $\lambda',\lambda''\in X$. Moreover we also have $n_\mu=n_{\theta\mu}$, for any $\mu\in X$. Therefore $n_{\mu-\theta\mu}=1$. Hence $n_{\lambda'}=n_{\lambda''}$, whenever $\overline{\lambda'}=\overline{\lambda''}$ in $X_\imath$. For any $\zeta\in X_\imath$, we define $n_\zeta=n_\lambda$, for any $\lambda\in X$ with $\overline{\lambda}=\zeta$. Then we can write $\xi=\sum_{\zeta\in X_\imath}n_\zeta\one_\zeta \in {}_k\widehat{\U}^\imath$.
We finish the proof now.
\end{proof}
\begin{corollary}
As subgroups of $G_A$, we have $G_A^\imath=K_A$.
\end{corollary}
\begin{proof}
The proof for $G_A^\imath\subseteq K_A$ is the same as the first part of the proof of Theorem \ref{thm:Oik}. For the other side inclusion, we embed all the objects into the ambient space $_k\widehat{\U}$. It follows from definitions that
\[
K_A\subseteq {_A\widehat{\U}}\cap K_k={_A\widehat{\U}}\cap G_k^\imath={_A\widehat{\U}}\cap{_k\widehat{\U}^\imath}\cap G_k={_A\widehat{\U}^\imath}\cap G_A=G_A^\imath.
\]
We completed the proof.
\end{proof}
\subsubsection{}
For any $\mu\in X^+$, write $V_k(\mu)=k\otimes_\A\big({_\A L(\mu)}\big)$ as before. Then $V_k(\mu)$ admits a $_k\widehat{\U}$-action, and hence a (rational) $G_k$-action by restriction. Via the embedding $_k\dot{\U}^\imath\hookrightarrow{_k\widehat{\U}}$ the space $V_k(\mu)$ also admits a $_k\dot{\U}^\imath$-action. Since there are only finitely many $\imath$canonical basis elements acting non-trivially on $V_k(\mu)$, $V_k(\mu)$ admits a natural $_k\widehat{\U}^\imath$-action.
\begin{prop}\label{prop:KvsUi}
Let $M$ be a $k$-subspace of $V_k(\mu)$. Then $M$ is stable under the $K_k$-action if and only if it is stable under the $_k\dot{\U}^\imath$-action.
\end{prop}
\begin{proof}
Suppose $M$ is $_k\dot{\U}^\imath$-stable. Since we $K_k = G^\imath_k=G_k\cap{_k\widehat{\U}^\imath}$, it is clear $M$ is $K_k$-stable.
Next suppose $M$ is $G_k^\imath$-stable. Take any $f$ in $_k\dot{\U}^\imath$ and $m$ in $M$. Then $f$ naturally induces a linear form on $\RO_k^\imath$, which we denote by $\tilde{f}$. It is direct to check that $f\cdot m=(id\otimes\tilde{f})\circ\Delta_M(m)$, where $\Delta_M:M\rightarrow M\otimes \RO_k^\imath$ is the comodule morphism corresponding to the $G_k^\imath$ action on $M$ (cf. \cite{Jan03}*{\S2.8}). It follows that $f\cdot m$ belongs to $M$.
\end{proof}
\section{Quantum Frobenius splittings} \label{sec:qFri}
In this section, we assume given an $\imath$root datum $(\I,Y,X,A,(\alpha_i)_{i\in\I},(\coroot_i)_{i\in\I},\tau,\theta)$ of arbitrary (quasi-split) type.
\subsection{$\imath$Quantum groups over $\mathcal{A}_2$}
Set $\mathcal{A}_2=S^{-1}\mathcal{A}$, where $S\subseteq\mathcal{A}$ is the multiplicative system generated by $q^a+q^{-a}$, for all $a\in\mathbb{Z}$. Then $\mathcal{A}_2 \subset \mathbb{Q}(q)$ is a subring containing $\mathcal{A}$.
This subsection is devoted to study the $\imath$quantum group over the ring $\A_2$.
\subsubsection{} We define the following \emph{q-double binomial coefficients} in $\Qq$:
\begin{align*}
\LR{m}{2k} &=\frac{[m][m-2]\cdots[m-2k+2]}{[2][4]\cdots[2k]} =\frac{\prod_{s=0}^{k-1} (q^{m-2s} - q^{-m+2s})}{\prod_{s=1}^{k}(q^{2s} - q^{-2s}) }, \text{for } m\in\mathbb{Z}, k\in\mathbb{Z}_{>0};\\
\LR{m}{0}&=1, \text{for } m\in\mathbb{Z}; \qquad \LR{m}{2k}=0, \text{for }k<0.
\end{align*}
We denote by $\LR{m}{2k}_{q^a}$ if $q$ is replaced by $q^a$ for some $a \in \BZ_{>0}$.
We collect some basic properties the $q$-double binomial coefficients from the definition ($m,k\in\mathbb{Z}$).
\begin{equation}\label{eq:mk1}
\LR{m+2}{2k}=q^{-2k}\LR{m}{2k}+q^{m-2k+2}\LR{m}{2k-2},
\end{equation}
\begin{equation}\label{eq:mk2}
\LR{m}{2k}=(-1)^k\LR{-m+2k-2}{2k},
\end{equation}
\begin{equation}\label{eq:mk3}
\LR{2m}{2k}=\qbinom{m}{k}_{q^2}.
\end{equation}
\begin{lem}\label{lem:DoubleInA2}
(a) Let $m,k\in\mathbb{Z}$. We have $\LR{m}{2k}\in\mathcal{A}_2$.
(b) Let $m', m'',k\in\mathbb{Z}$. We have
\[
\LR{m'+m''}{2k}=\sum_{k'+k''=k} q^{e(m'k''-m''k')}\LR{m'}{2k'}\LR{m''}{2k''}, \quad \text{for } e = \pm 1.
\]
\end{lem}
\begin{proof}
We show (a). It suffices to consider the case when $k>0$ and $m$ is odd. Thanks to \eqref{eq:mk1} and \eqref{eq:mk2}, it suffices to show $\LR{-1}{2k} \in \CA_2$. Note that $[2n] = [n] (q^n + q^{-n})$ for $n \in \BZ_{>0}$. We have
\begin{align*}
\LR{-1}{2k}&= (-1)^k \LR{2k-1}{2k} = (-1)^k\frac{[2k-1][2k-3]\cdots[1]}{[2k][2k-2]\cdots [2]}\\&=(-1)^k\frac{[2k]!}{[2]^2[4]^2\cdots[2k]^2}=(-1)^k\qbinom{2k}{k}\prod_{i=1}^k\frac{1}{(q^i+q^{-i})^2} \in \CA_2.
\end{align*}
We prove (b). When both $m'$, $m''$ are even, the claim is clear by \eqref{eq:mk3} and \cite[\S1.3.1 (e)]{Lu93}.
For fixed $k$, we may regard (b) as an identity of rational functions in three variables: $q, q^{m'}, q^{m''}$. Since the identity holds for all even $m' $, $m'' $, it must hold as a formal identity in the three variables. This proves the proposition.
\end{proof}
\subsubsection{} \label{sec:qsetup}
We recall the setup in \S\ref{sec:qFr}.
Fix an odd number $l>1$. We require that $l$ is relatively prime to all the root length $\epsilon_i$. Let $f_l\in \mathcal{A}$ be the $l$-th cyclotomic polynomial. Set $\mathcal{A}'=\mathcal{A}/(f_l)$, and let $\mathbf{F}$ be the quotient ring of $\mathcal{A}'$. Let $\phi:\mathcal{A}\rightarrow\mathcal{A}'$ be the natural quotient map. Then $\phi$ extends to a ring homomorphism from the local ring $\mathcal{A}_{(f_l)}$ to $\mathbf{F}$. Let $c:\mathcal{A}\rightarrow\mathcal{A'}$ be the ring homomorphism sending $q$ to 1. Then $c$ also extends to $\A_{(f_l)}$.
Write $\bq_i = \phi (q_i)\in\A '$, for any $i\in\I$. For $m,k\in\mathbb{Z}$, set
\begin{equation*}
\lr{m}{2k} =c(\LR{m}{2k})=\frac{m\cdot (m-2)\cdots (m-2k+2)}{2\cdot 4\cdots (2k)}\in \mathbb{Z}[2^{-1}].
\end{equation*}
The following lemma is an analogue of \cite[Lemma 34.1.2 (c)]{Lu93}.
\begin{lemma}\label{le:qBinomAtUnity}
For $n,k\in\mathbb{Z}$, write $n=n_0+n_1l$, with $n_0,n_1\in\mathbb{Z}$ such that $n_0\in\{0,2,\cdots,2l-2\}$, and $k=k_0+k_1l$, with $k_0,k_1\in\mathbb{Z}$ such that $k_0\in\{0,1,\cdots, l-1\}$. We have
\begin{equation*}
\phi\left(\LR{n}{2k}\right)=\lr{n_1}{2k_1}\phi\left(\LR{n_0}{2k_0}\right).
\end{equation*}
\begin{proof}
Let $n$ be even. Then by Lemma \ref{lem:DoubleInA2}, and \cite[Lemma 34.1.2 (c)]{Lu93}, we have
\begin{align*}
\phi\left(\LR{n}{2k}\right)=\phi\left(\qbinom{n/2}{k}_{q^2}\right)=\phi\left(\qbinom{n_0/2}{k_0}_{q^2}\right)\binom{n_1/2}{k_1}=\phi\left(\LR{n_0}{2k_0}\right)\lr{n_1}{2k_1}.
\end{align*}
Now suppose $n$ is odd. The lemma holds when $n=l$, since
\[
\phi\left(\LR{l}{2k}\right) =
\begin{cases}
0, & \text{if } l \nmid k;\\
\lr{1}{2k/l}, & \text{otherwise}.
\end{cases}
\]
Note that $n-l$ is even. By the Lemma \ref{lem:DoubleInA2}, we have
\begin{align*}
\phi\left(\LR{n}{2k}\right)&=\sum_{k'+k''=k}\bq^{nk''}\phi\left(\LR{n-l}{2k'}\LR{l}{2k''}\right)\\&=\sum_{k'+lk''=k}\phi\left(\LR{n-l}{2k'}\right)\lr{1}{2k''}\\&=\phi\left(\LR{n_0}{2k_0}\right)\sum_{0\leqslant k''\leqslant k_1}\lr{n_1-1}{2k_1-2k''}\lr{1}{2k''}\\&=\phi\left(\LR{n_0}{2k_0}\right)\lr{n_1}{2k_1}.
\end{align*}
Hence the lemma is proved.
\end{proof}
\end{lemma}
\subsubsection{}\label{sec:baid}
Recall the construction for \emph{$\imath$divided powers} in \S\ref{sec:iqp}.
Let $i \in \I$ with $\tau i=i$. We define the \emph{balanced $\imath$divided powers} for $n \ge 0$:
\[
B_i^{(n)} = B_{i,\overline{n+1}}^{(n)} = \frac{(B_i- [-n+1]_i )(B_i- [-n+3]_i) \cdots (B_i- [n-3]_i)(B_i- [n-1]_i)}{[n]_i^!}.
\]
Here $B_i^{(0)} = 1$ by definition.
\begin{lem} \label{lem:ff}
For $i\in\I$ with $\tau i=i$, and any $n\in \BZ_{\ge 0}$, we have
\begin{equation}\label{eq:ff}
B_i^{(n)}=\sum_{t\geqslant 0}\LR{-1}{2t}_i B_{i,\overline{n}}^{(n-2t)}, \qquad
B_{i,\overline{n}}^{(n)} =\sum_{t\geqslant 0}\LR{1}{2t}_i B_i^{(n-2t)}.
\end{equation}
\end{lem}
\begin{proof}
We prove the first equation by induction on $n$. The second equation can be proved similarly. The claim is trivial for $n = 0, 1$. Note that
\begin{align*}
B_i^2B_i^{(n)} &=[n+1]_i[n+2]_iB_i^{(n+2)}+[n+1]_i^2B_i^{(n)},\\
B_i^2B_{i,\overline{n}}^{(n-2t)} &=[n-2t]_i^2B_{i,\overline{n}}^{(n-2t)} +[n-2t+1]_i[n-2t+2]_iB_{i,\overline{n}}^{(n-2t+2)} .
\end{align*}
Suppose the equation holds for $n$. Multiplying $B_i^2$ on both sides of the first equation in \eqref{eq:ff}, we obtain
\begin{align*}
&[n+1]_i[n+2]_iB_i^{(n+2)}\\
= & -[n+1]_i^2 \sum_{t\geqslant 0} \LR{-1}{2t}_i B^{(n-2t)}_{i,\overline{n}}+ \sum_{t\geqslant 0} \LR{-1}{2t}_i [n-2t]_i^2 B^{(n-2t)}_{i,\overline{n}} \\
& + \sum_{t\geqslant 0} \LR{-1}{2t}_i [n-2t+1]_i[n-2t+2]_i B^{(n-2t+2)}_{i,\overline{n}} \\
= & [n+1]_i[n+2]_iB^{(n+2)}_{i,\overline{n}} \\
+ & \sum_{t\geqslant 0} \LR{-1}{2t+2}_i \Big( ([n-2t]_i^2-[n+1]_i^2)\frac{[2t+2]_i}{[-1-2t]_i}+[n-2t-1]_i[n-2t]_i \Big) B^{(n-2t)}_{i,\overline{n}}\\
= & [n+1]_i[n+2]_iB^{(n+2)}_{i,\overline{n}} + \sum_{t\geqslant 0} \LR{-1}{2t+2}_i \Big([n+1]_i[n+2]_i \Big) B^{(n-2t)}_{i,\overline{n}}\\
= & [n+1]_i[n+2]_i\sum_{t\geqslant 0} \LR{-1}{2t}_i B^{(n+2-2t)}_{i,\overline{n}}.
\end{align*}
This finishes the proof.
\end{proof}
\begin{lem}\label{lem:balanced}
For $i\in\I$ with $\tau i=i$, and $a,k\in \mathbb{Z}_{\ge 0}$, we have
\begin{align*}
B_i^{(a)}B_i^{(k)}&=\sum_{t\geqslant 0}\qbinom{a+k}{a}_i\prod_{m=1}^t\frac{[a-2m+2]_i[k-2m+2]_i}{[a+k-2m+1]_i[2m]_i}B_i^{(a+k-2t)}\\
& =\sum_{t\geqslant 0}\qbinom{a+k}{a}_i\frac{\LR{a}{2t}_i \LR{k}{2t}_i}{\LR{a+k-1}{2t}_i}B_i^{(a+k-2t)}
\end{align*}
In particular, we have $B_iB_i^{(n)}=\sum_{t\geqslant 0}[n+1]_i\LR{1}{2t}_iB_i^{(n+1-2t)}$.
\end{lem}
\begin{proof}
We prove by induction on $a + k$. The base cases are $a+k \le 2$, which can be checked directly. We write $c_{a,k;t} = \qbinom{a+k}{a}_i\displaystyle \prod_{m=1}^t\frac{[a-2m+2]_i[k-2m+2]_i}{[a+k-2m+1]_i[2m]_i}$.
Multiplying $B_i^2$ on both sides of the equation, we obtain, by the induction hypothesis, that
\begin{align*}
&([a+1]_i[a+2]_iB_i^{(a+2)}+[a+1]_i^2B_i^{(a)})B_i^{(k)}\\&=\sum_{t\geqslant 0}c_{a,k;t}\Big([a+k+1-2t]_i [a+k+2-2t]_i B_i^{(a+k+2-2t)}+[a+k+1-2t]_i ^2 B_i^{(a+k-2t)}\Big).
\end{align*}
Therefore we have $B_i^{(a+2)} B_i^{(k)} = \sum_{t\geqslant 0}c'_{a+2,k;t}B_i^{(a+2+k - 2t)} $ such that
\[
[a+1]_i[a+2]_i c'_{a+2,k;t} = [a+k-2t+1]_i[a+k-2t+2]_ic_{a,k;t}+[2a+k-2t+4]_i[k-2t+2]_ic_{a,k;t-1}
\]
The right hand side is
\begin{align*}
&[a+k-2t+1]_i[a+k-2t+2]_i\qbinom{a+k}{a}_i \prod_{m=1}^t\frac{[a-2m+2]_i[k-2m+2]_i}{[a+k-2m+1]_i[2m]_i}\\&+[2a+k-2t+4]_i[k-2t+2]_i\qbinom{a+k}{a}_i \prod_{m=1}^{t-1}\frac{[a-2m+2]_i[k-2m+2]_i}{[a+k-2m+1]_i[2m]_i}\\
&=[a+1]_i[a+2]_ic_{a+2,k;t} \Big( \frac{ [a+k-2t+2]_i [a-2t+2]_i}{ [a+k+2]_i[a+2]_i } + \frac{[2a+k-2t+4]_i [2t]_i}{[a+k+2]_i [a+2]_i }\Big) \\
&=[a+1]_i[a+2]_ic_{a+2,k;t}.
\end{align*}
Therefore $c_{a+2,k;t} = c'_{a+2,k;t}$. This completes the proof.
\end{proof}
\begin{remark}
This formula is motivated by the formulas for the structure constants for $\imath$divided powers in \cite{CW22}.
\end{remark}
\subsubsection{}
Define $_{\A_2}\dot{\U}^\imath$ be the $\A_2$-span of the free $\A$-submodule of $_\A\dot{\U}^\imath$ inside $\dot{\U}^\imath$. It is then a free $\A_2$-submodule, as well as an $\A_2$-subalgebra.
For $i\in\I$ with $\tau i=i$, and $\zeta\in X_\imath$, define $'B_{i,\zeta}^{(n)}=B_{i}^{(n)}\one_\zeta\in\dot{\U}^\imath$. By Lemma~\ref{lem:ff}, we have the following claim.
(a) {\em
the algebra $_{\A_2}\dot{\U}^\imath$ is generated by $'B_{i,\zeta}^{(n)}$ ($\tau i=i$, $\zeta \in X_\imath$, $n\in\BN$), and $B_{i,\zeta}^{(n)}$ ($\tau i\neq i$, $\zeta \in X_\imath$, $n\in\BN$) as an $\A_2$-algebra. }
\subsubsection{}
Fix $i\neq j\in\I$ with $\tau i=i$ in this subsection. For any $m,n\in\mathbb{Z}$, with $n>0$ and $e=\pm 1$, we define elements $y_{n,m,e}=y_{i,j;n,m,e}$ in $\Ui$ inductively as follows: for $m<0$, set $y_{n,m,e}=0$, and set $y_{n,0,e}=B_j^{(n)}$; for $m\geqslant0$, $y_{n,m,e}$ are determined by the following formula:
\begin{align}\label{fo:recursion}
q_i^{-e(2m+na_{ij})}&B_iy_{n,m,e}-y_{n,m,e}B_i\\&=-[m+1]_iy_{n,m+1,e}+[m+na_{ij}-1]_iq_i^{-e(2m+na_{ij}-1)}y_{n,m-1,e}\notag.
\end{align}
The elements are analogues of Lusztig's higher relations for $\imath$quantum groups. They were first studied in \cite{CLW21}, where explicit formulas were obtained in \cite[\S6.1]{CLW21}. We give a simpler expression in the $\CA_2$-form.
\begin{prop}\label{prop:HigherSerreRe}
For $m,n\in\mathbb{N}$, with $n>0$, and $e=\pm 1$, we have
\begin{equation}\label{eq:HigherSerreRe}
y_{i,j;n,m,e}=\sum_{\substack{r+s+2t=m\\t\geqslant 0}}(-1)^rq_i^{-e(m+na_{ij}-1)(r+t)}\LR{m+na_{ij}}{2t}_iB_i^{(r)}B_j^{(n)}B_i^{(s)}.
\end{equation}
\end{prop}
\begin{proof}
We prove by induction on $m$. When $m= 0, 1$, one can check this by direct computation. We check the right hand side of \eqref{eq:HigherSerreRe} satisfies the recursion formula \eqref{fo:recursion}.
By Lemma \ref{lem:balanced}, we have $
B_iB_i^{(n)}=\sum_{u\geqslant 0}[n+1]_i\LR{1}{2u}B_i^{(n+1-2u)}$. Hence by induction hypothesis, we have
\begin{align*}
&B_iy_{n,m,e}\\
= &\sum_{\substack{r+s+2t=m\\t\geqslant 0}}\sum_{u\geqslant 0}(-1)^rq_i^{-e(m+na_{ij}-1)(r+t)}[r+1]_i\LR{m+na_{ij}}{2t}_i\LR{1}{2u}_iB_i^{(r+1-2u)}B_j^{(n)}B_i^{(s)}\\
= &\sum_{\substack{r+s+2t=m+1\\t\geqslant 0}}\sum_{u\geqslant 0}(-1)^{r+1}q_i^{-e(m+na_{ij}-1)(r+t+u-1)}[r+2u]_i\LR{m+na_{ij}}{2t-2u}_i\LR{1}{2u}_iB_i^{(r)}B_j^{(n)}B_i^{(s)}.
\end{align*}
We write $J_{r,s}$ to denote the coefficient before $B_i^{(r)}B_j^{(n)}B_i^{(s)}$. Then
\begin{align*}
J_{r,s} =&\sum_{u\geqslant 0}(-1)^{r+1}q_i^{-e(m+na_{ij}-1)(r+t+u-1)}(q_i^{er}[2u]_i+q_i^{-2eu}[r]_i)\LR{m+na_{ij}}{2t-2u}_i\LR{1}{2u}_i\\
=&(-1)^{r+1}q_i^{-e(m+na_{ij}-1)(r+t-1)}[r]_i\sum_{u\geqslant 0}q_i^{-e(m+na_{ij}+1)u}\LR{m+na_{ij}}{2t-2u}_i\LR{1}{2u}_i\\
&+(-1)^{r+1}q_i^{-e(m+na_{ij}-1)(r+t-1)+er}\sum_{u\geqslant 0}q_i^{-e(m+na_{ij}-1)u}[2u]_i\LR{m+na_{ij}}{2t-2u}_i\LR{1}{2u}_i\\
\stackrel{(\heartsuit 1)}{=}&(-1)^{r+1}q_i^{-e(m+na_{ij}-1)(r+t-1)-et}[r]_i\LR{m+na_{ij}+1}{2t}_i \\
&+(-1)^{r+1}q_i^{-e(m+na_{ij}-1)(r+t-1)+er}\sum_{u\geqslant 0}q_i^{-e(m+na_{ij}-1)u}\LR{m+na_{ij}}{2t-2u}_i\LR{-1}{2u-2}_i\\
\stackrel{(\heartsuit 2)}{=}&(-1)^{r+1}q_i^{-e(m+na_{ij}-1)(r+t-1)-et}[r]_i\LR{m+na_{ij}+1}{2t}_i\\
&+(-1)^{r+1}q_i^{-e(m+na_{ij}-1)(r+t-1)+e(r+t-m-na_{ij})}\LR{m+na_{ij}-1}{2t-2}_i,
\end{align*}
where both $(\heartsuit 1)$ and $(\heartsuit 2)$ are by Lemma~\ref{lem:DoubleInA2}.
Hence
\begin{align*}
B_iy_{n,m,e}=&\sum_{\substack{r+s+2t=m+1\\t\geqslant 0}}(-1)^{r+1}q_i^{-e(m+na_{ij}-1)(r+t-1)}\\&\cdot\left(q_i^{-et}[r]_i\LR{m+na_{ij}+1}{2t}_i+q_i^{e(r+t-m-na_{ij})}\LR{m+na_{ij}-1}{2t-2}_i\right)B_i^{(r)}B_j^{(n)}B_i^{(s)}.
\end{align*}
Similarly, one can compute
\begin{align*}
y_{n,m,e}B_i=&\sum_{\substack{r+s+2t=m+1\\t\geqslant 0}}(-1)^rq_i^{-e(m+na_{ij}-1)(r+t)}\\&\cdot\left(q_i^{et}[s]_i\LR{m+na_{ij}+1}{2t}_i+q_i^{e(m+na_{ij}-t-s)}\LR{m+na_{ij}-1}{2t-2}_i\right)B_i^{(r)}B_j^{(n)}B_i^{(s)}.
\end{align*}
Hence one has
\begin{align*}
q_i^{-e(2m+na_{ij})}B_iy_{n,m,e}-y_{n,m,e}B_i=\sum_{\substack{r+s+2t=m+1\\t\geqslant 0}}(-1)^{r+1}q_i^{-e(m+na_{ij})(r+t)}a_{r,s}B_i^{(r)}B_j^{(n)}B_i^{(s)},
\end{align*}
where
\begin{align*} a_{r,s}=&(q_i^{-e(s+2t)}[r]_i+q_i^{e(2t+r)}[s]_i)\LR{m+na_{ij}+1}{2t}_i\\&+(q_i^{-e(m+na_{ij}+s-r)}+q_i^{e(m+na_{ij}-s+r)})\LR{m+na_{ij}-1}{2t-2}_i.
\end{align*}
On the other hand, by definition, we have
\begin{align*}
-[m+1]_iy_{n,m+1,e}+&[m+na_{ij}-1]_iq_i^{-e(2m+na_{ij}-1)}y_{n,m-1,e}\\&=\sum_{\substack{r+s+2t=m+1\\t\geqslant 0}}(-1)^{r+1}q_i^{-e(m+na_{ij})(r+t)}b_{r,s}B_i^{(r)}B_j^{(n)}B_i^{(s)},
\end{align*}
where
$ b_{r,s}=[m+1]_i\LR{m+na_{ij}+1}{2t}_i-q_i^{e(r-s)}[m+na_{ij}-1]_i\LR{m+na_{ij}-1}{2t-2}_i$.
It remains to verify $a_{r,s} = b_{r,s}$. Since $m+1=r+s+2t$, we have
$$[m+1]_i=q_i^{e(r+2t)}[s]_i+q_i^{-e(s+2t)}[r]_i+q_i^{e(r-s)}[2t]_i.$$
Hence
\begin{align*}
a_{r,s}-b_{r,s}=&q_i^{e(r-s)}(q_i^{-e(m+na_{ij})}+q_i^{e(m+na_{ij})}+[m+na_{ij}-1]_i)\LR{m+na_{ij}-1}{2t-2}_i\\&-q_i^{e(r-s)}[2t]_i\LR{m+na_{ij}+1}{2t}_i\\=&q_i^{e(r-s)}[m+na_{ij}+1]_i\LR{m+na_{ij}-1}{2t-2}_i-q_i^{e(r-s)}[2t]_i\LR{m+na_{ij}+1}{2t}_i\\=&0.
\end{align*}
The proposition is proved.
\end{proof}
It follows from \cite[Theorem 6.3]{CLW21} that $y_{i,j;n,m,e}=0$, whenever $m>-na_{ij}$, $e=\pm 1$. Letting $n=1$, $m=1-a_{ij}$, we have
\begin{equation}\label{eq:iSerreNew}
\sum_{\substack{r+s+2t=1-a_{ij}\\t\geqslant 0}}(-1)^r\LR{1}{2t}_iB_i^{(r)}B_jB_i^{(s)}=0.
\end{equation}
\subsubsection{} \label{subsec:rel}
Let $k$ be any field of characteristic zero. We view $k$ as an $\A$-module on which $q \mapsto 1$.
Write $\udoti=k\otimes_\A (_{\A}\dot{\U}^\imath)$. For any $i\in \I$, $\zeta\in X_\imath$ and $n\in\mathbb{N}$, we denote by $b_{i,\zeta}^{(n)}$ the image of $B_{i,\zeta}^{(n)}$. It is clear that all $b_{i,\zeta}^{(n)}$ ($i\in \I$, $\zeta\in X_\imath$, $n\geqslant 1$) is generated by $b_{i,\zeta}$ ($i\in\I$, $\zeta\in X_\imath$). Hence the elements $b_{i,\zeta}$ ($i\in\I$, $\zeta\in X_\imath$), $\one_\zeta$ ($\zeta\in X_\imath$) generate $\udoti$.
The algebra $\udoti$ is generated by elements $\one_\zeta$ ($\zeta\in X_\imath$), $b_{i,\zeta}$ ($i\in\I$, $\zeta\in X_\imath$) which subject to the following relations ($i\neq j\in\I$, $\zeta,\zeta'\in X_\imath$):
\begin{align}
&\quad b_{i,\zeta}\one_{\zeta'}=\delta_{\zeta,\zeta'}b_{i,\zeta}, \qquad \one_{\zeta'}b_{i,\zeta}=\delta_{\zeta-\overline{\alpha_i} \zeta'}b_{i,\zeta}, \quad \one_{\zeta'} \one_{\zeta} = \delta_{\zeta, \zeta'} \one_{\zeta}, \label{re:wt}\\
&\sum_{r+s=1-a_{ij}}(-1)^rb_{i,\zeta-s\overline{\alpha_i}-\overline{\alpha_j}}^{(r)}b_{j,\zeta-s\overline{\alpha_i}}b_{i,\zeta}^{(s)}=0, \quad \text{if } j \neq \tau i \neq i, \label{re:iSerre}\\
&\!\!\!\sum_{r+s=1-a_{i,\tau i}}\!\!\!(-1)^rb_{i,\zeta-s\overline{\alpha_i}-\overline{\alpha_{\tau i}}}^{(r)}b_{\tau i,\zeta-s\overline{\alpha_i}}b_{i,\zeta}^{(s)}=\left\{\begin{array}{ll}
\langle \coroot_i-\coroot_{\tau i}, \zeta\rangle\one_\zeta, & \text{if }a_{i,\tau i}=0,\\
-2b_{i,\zeta}, & \text{if }a_{i,\tau i}=-1,\\
0, & \text{if }a_{i,\tau i}\leqslant -2,
\end{array}\right. \label{re:iwithtaui} \text {if }\tau i\neq i, \\
&\sum_{\substack{r+s+2t=1-a_{ij}\\t\geqslant 0}}(-1)^r\lr{1}{2t}{'b_{i,\zeta-s\overline{\alpha_i}-\overline{\alpha_j}}^{(r)}}b_{j,\zeta-s\overline{\alpha_i}}{'b_{i,\zeta}^{(s)}}=0, \quad \text{if } \tau i = i.\label{re:iSerre2}
\end{align}
The presentation was obtained in \cite[Theorem~3.1]{CLW18}. Passing to the modified form is straightforward (cf. \cite{Lu93}*{\S31.1.3}). We replace the relation \cite[Theorem~3.1(3.9)]{CLW18} by the equivalent relation \eqref{re:iSerre2} thanks to \eqref{eq:iSerreNew}.
\subsection{Main theorem}\label{sec:iFr} Retain the setup in \S \ref{sec:qsetup}. Let $\CA'_2 = \CA_2 / (f_l) \supset \CA'$. Define
\begin{align*}
_{\mathcal{A}'}\dot{\mathfrak{U}}^{\imath}&={\mathcal{A}'}\otimes_c ({_{\mathcal{A}}\dot{\mathrm{U}}^\imath}),\quad
_{\mathcal{A}'}\dot{\mathrm{U}}^\imath=\mathcal{A}'\otimes_\phi (_\mathcal{A}\dot{\mathrm{U}}^\imath),\\
_{\mathcal{A}_2'}\dot{\mathfrak{U}}^{\imath}&={\mathcal{A}_2'}\otimes_c ({_{\mathcal{A}}\dot{\mathrm{U}}^\imath}),\quad
_{\mathcal{A}'_2}\dot{\mathrm{U}}^\imath=\mathcal{A}_2'\otimes_\phi (_\mathcal{A}\dot{\mathrm{U}}^\imath),\\
_{\mathbf{F}} \dot{\mathfrak{U}}^{\imath}&={{\mathbf{F}}}\otimes_c ({_{\mathcal{A}}\dot{\mathrm{U}}^\imath}),\quad
{_{\mathbf{F}}}\dot{\mathrm{U}}^\imath={{\mathbf{F}}}\otimes_\phi (_\mathcal{A}\dot{\mathrm{U}}^\imath).
\end{align*}
Then we have $_{\mathcal{A}'}\dot{\mathfrak{U}}^{\imath}\subseteq {_{\mathcal{A}_2'}\dot{\mathfrak{U}}^{\imath}}\subseteq {_{\mathbf{F}}\dot{\mathfrak{U}}^{\imath}}$, and $_{\mathcal{A}'}\dot{\U}^{\imath}\subseteq {_{\mathcal{A}_2'}\dot{\U}^{\imath}}\subseteq {_{\mathbf{F}}\dot{\U}^{\imath}}$.
For any $i\in\I$, $n\in\mathbb{N}$, and $\zeta\in X_\imath$, we use $\mathfrak{b}_{i,\zeta}^{(n)}$ (resp. $\mathrm{B}_{i,\zeta}^{(n)}$) to denote the image of $B_{i,\zeta}^{(n)}$ in $_{\mathcal{A}'}\dot{\mathfrak{U}}^{\imath}$ (resp. $_{\mathcal{A}'}\dot{\mathrm{U}}^\imath$). For $i=\tau i$, set $'\mathfrak{b}_{i,\zeta}^{(n)}$ (resp. $'\mathrm{B}_{i,\zeta}^{(n)}$) be the image of $'B_{i,\zeta}^{(n)}$ in $_{\mathcal{A}'_2}\dot{\mathfrak{U}}^\imath$ (resp. $_{\mathcal{A}'_2}\dot{\mathrm{U}}^\imath$). We state the main theorem of this section.
\begin{theorem}\label{thm:iFr}
There exists a unique $\mathcal{A}_2'$-algebra homomorphim
$\ifr: {_{\mathcal{A}_2'}\dot{\mathfrak{U}}^{\imath}}\longrightarrow {_{\mathcal{A}_2'}\dot{\mathrm{U}}^\imath},$
such that: for $\tau i\neq i$, $n\in\mathbb{N}$, we have
$\ifr(\mathfrak{b}_{i,\zeta}^{(n)})=\mathrm{B}_{i,l\zeta}^{(nl)}$;
for $\tau i=i$, $n\in\mathbb{N}$, we have
$\ifr('\mathfrak{b}_{i,\zeta}^{(n)})={'\mathrm{B}_{i,\zeta}^{(nl)}}.$
\end{theorem}
The uniqueness is clear. It suffices to prove the existence over the field of fraction $\mathbf{F}$. This theorem will be proved in \S\ref{subsec:pf}. Let us assume this theorem for now.
\begin{corollary}\label{cor:iFr}
For $\tau i=i$, we have
\[
\ifr(\mathfrak{b}_{i,\zeta}^{(n)}) =
\begin{cases} \mathrm{B}_{i,l\zeta}^{(ln)}, &\text{if } \langle \coroot_i,\zeta\rangle \neq n \text{ {\rm mod }}2;\\
\sum_{t=0}^{(l-1)/2}\phi\left(\qbinom{(l-1)/2}{t}_{q_i^2}\right)\mathrm{B}_{i,l\zeta}^{(nl-2t)}, &\text{if } \langle \coroot_i,\zeta\rangle = n \text{ {\rm mod }}2.
\end{cases}
\]
In particular, the morphism $\ifr$ restricts to an $\A'$-algebra homomorphism $\ifr: {_{\mathcal{A}'}\dot{\mathfrak{U}}^{\imath}}\longrightarrow {_{\mathcal{A}'}\dot{\mathrm{U}}^\imath}$.
\end{corollary}
\begin{remarks}
Even though we do not need to consider the localization $\CA'_2$ in light of Corollary~\ref{cor:iFr}, it seems impossible to obtain the formulas without Theorem~\ref{thm:iFr}. For applications of Frobenius splittings, we only consider fields of characteristic not $2$. So we believe the localization in Theorem~\ref{thm:iFr} is conceptual.
\end{remarks}
\begin{proof}
If $\langle \coroot_i,\zeta\rangle \neq n$ {\rm mod} $2$, then $\mathfrak{b}_{i,\zeta}^{(n)}={'\mathfrak{b}_{i,\zeta}^{(n)}}$ and $\mathrm{B}_{i,\zeta}^{(n)}={'\mathrm{B}_{i,\zeta}^{(n)}}$. Then the identity follows directly.
Suppose $\langle \coroot_i,\zeta\rangle = n$ {\rm mod} $2$. By Lemma \ref{lem:ff} and Theorem \ref{thm:iFr}, we have
\begin{align*}
\ifr(\mathfrak{b}_{i,\zeta}^{(n)})&=\ifr(\sum_{t\geqslant 0}\lr{1}{2t}{'\mathfrak{b}_{i,\zeta}^{(n-2t)}})\\
&=\sum_{t \geqslant 0}\lr{1}{2t}{'\mathrm{B}_{i,l\zeta}^{(nl-2tl)}}\\
&=\sum_{t\geqslant 0}\lr{1}{2t}\sum_{s\geqslant 0}\phi\left(\LR{-1}{2s}_i\right)\mathrm{B}_{i,l\zeta}^{(nl-2tl-2s)}\\
&=\sum_{k\geqslant 0}\sum_{\substack{tl+s=k\\ t\geqslant 0, s\geqslant 0}}\lr{1}{2t}\phi\left(\LR{-1}{2s}_i\right)\mathrm{B}_{i,l\zeta}^{(nl-2k)}.
\end{align*}
By Lemma \ref{le:qBinomAtUnity}, $\phi\left(\LR{l}{2k}_i\right)=\lr{1}{2k/l}$ if $l \mid k$, and $\phi\left(\LR{l}{2k}_i\right) = 0$ if $l \nmid k$. Hence we have
\begin{equation*}
\sum_{\substack{tl+s=k\\ t\geqslant 0, s\geqslant 0}}\lr{1}{2t}\phi\left(\LR{-1}{2s}_i\right)=\sum_{t'+s=k}\bq_i^{t'}\phi\left(\LR{l}{2t'}_i\LR{-1}{2s}_i\right)=\phi\left(\LR{l-1}{2k}_i\right).
\end{equation*}
Note that $\phi\left(\LR{l-1}{2k}_i\right)= 0$, unless $0\leqslant k\leqslant (l-1)/2$. Hence we have
\begin{equation*}
\ifr(\mathfrak{b}_{i,\zeta}^{(n)})=\sum_{k=0}^{(l-1)/2}\phi\left(\LR{l-1}{2k}_i\right)\mathrm{B}_{i,l\zeta}^{(nl-2k)}=\sum_{k=0}^{(l-1)/2}\phi\left(\qbinom{(l-1)/2}{k}_{q_i^2}\right)\mathrm{B}_{i,l\zeta}^{(nl-2k)}.
\end{equation*}
We finish the proof.
\end{proof}
\subsection{Proof of Theorem \ref{thm:iFr}}\label{subsec:pf}
For $i\in\I$, $\zeta\in X_\imath$, we define
\begin{align*}
\ifr: \cAtUi &\rightarrow \pAtUi,\\
'\fb_{i,\zeta} &\mapsto {'\B_{i,l\zeta}^{(l)}},\quad \text{if } \tau i =i;\\
\fb_{i,\zeta} &\mapsto {\B_{i,l\zeta}^{(l)}}, \quad \text{if } \tau i \neq i.
\end{align*}
We show $\ifr$ is a well-defined algebra homomorphism satisfying Theorem~\ref{thm:iFr}. This proof is organized as follows:
\begin{itemize}
\item we check the relation \eqref{re:wt} is preserved under $\ifr$ (this is trivial and will be skipped);
\item we show $\ifr(\fb_{i,\zeta}^{(a)})=\B_{i,l\zeta}^{(al)}$ if $\tau i \neq i$ in \S\ref{subsec:pf1};
\item we show $\ifr('\fb_{i,\zeta}^{(a)})={}'\B_{i,l\zeta}^{(al)}$ if $\tau i = i$ in \S\ref{subsec:pf2}
\item we check the relation \eqref{re:iwithtaui} is preserved under $\ifr$ in \S\ref{subsec:pf3};
\item we check the relation \eqref{re:iSerre} is preserved under $\ifr$ in \S\ref{subsec:pf4};
\item we check the relation \eqref{re:iSerre2} is preserved under $\ifr$ in \S\ref{subsec:pf5};
\end{itemize}
\subsubsection{} \label{subsec:pf1}
Let $\zeta\in X_\imath$ and $ i\in \I$ be such that $\tau i\neq i$. We have
\begin{equation*}
\ifr(\fb_{i,\zeta}^{(a)})=\frac{(\B_i^{(l)})^a}{a!}\one_{l\zeta}=\frac{\phi\left({[al]_i!}/{([l]_i!)^a}\right)}{a!}\B_i^{(al)}\one_{l\zeta}=\B_{i,l\zeta}^{(al)}.
\end{equation*}
\subsubsection{} \label{subsec:pf2}
Let $\zeta\in X_\imath$ and $ i\in \I$ be such that $\tau i = i$. We proceed by induction on $a$. Thanks to Lemma \ref{lem:balanced}, we have
\begin{equation*}
'\fb_{i,\zeta-a \overline{\alpha_i}}{'\fb_{i,\zeta}}^{(a)}=(a+1)\sum_{u\geqslant 0}\lr{1}{2u}{'\fb_{i,\zeta}}^{(a+1-2u)},
\end{equation*}
and
\begin{equation*}
'\B_{i,l\zeta-al \overline{\alpha_i}}^{(l)}{'\B}_{i,l\zeta}^{(al)}=\phi\left(\qbinom{(a+1)l}{l}_i\prod_{m=1}^u\frac{[al-2m+2]_i[l-2m+2]_i}{[al+l-2m+1]_i[2m]_i}\right){'\B}_{i,l\zeta}^{(al+l-2u)}.
\end{equation*}
It follows by direct computation that
\begin{equation*}
\phi\left(\qbinom{(a+1)l}{l}_i\prod_{m=1}^u\frac{[al-2m+2]_i[l-2m+2]_i}{[al+l-2m+1]_i[2m]_i}\right)
=
\begin{cases}
(a+1)\lr{1}{2u/l}, &\text{if } l\mid u;\\
0, &\text{otherwise}.
\end{cases}
\end{equation*}
Therefore $\ifr('\fb_{i,\zeta}^{(a)})={}'\B_{i,l\zeta}^{(al)}$.
\subsubsection{} \label{subsec:pf3}
Let $\zeta\in X_\imath$, $i\in\I$ with $\tau i\neq i$. We write $\alpha=-a_{i,\tau i}=-a_{\tau i,i}$.
\begin{lemma}\label{lem:rel3}
For $a,b\geqslant 0$, with $0\leqslant a+b<l$, we have the following identity in $_{\A'}\U^-$:
\begin{equation*}
\sum_{\substack{r+s=1+\alpha\\r,s\geqslant 0}}(-1)^rF_i^{(rl-a)}F_{\tau i}^{(l-a-b)}F_i^{(sl-b)}=0.
\end{equation*}
\end{lemma}
\begin{proof}
The identity is trivial for $\alpha=0$. If $a=b=0$, the identity follows by \cite[35.2.3]{Lu93}.
We assume $\alpha \ge 1$ and $a > 0$ (the case when $ b > 0$ is similar). Then we have $\alpha l-b>\alpha(l-a-b)$. Then by the higher Serre relation \cite[\S7.1.1\&Proposition~7.1.5]{Lu93}, we have
\[
\sum_{u+v=\alpha l-b}(-1)^u\bq_i^{(\alpha(a+b) - b -1)u}F_i^{(u)}F_{\tau i}^{(l-a-b)}F_i^{(v)}=0.
\]
For any $ u\ge 0$, we write $l-a+u = u_a + u_1l$ for some $u_1 \in \BZ$ and $0\le u_a \le l-1$. Since $0 < l-a < l$, by \cite[Lemma~34.1.2]{Lu93}, we have
\[
F^{(l-a)}_i F^{(u)}_i =\phi\left( \qbinom{l-a+u}{l-a}_i \right)F^{(l-a+u)}_i = \binom{u_a}{l-a} F^{(l-a+u)}_i.
\]
Therefore, we have
\begin{align*}
0 = &F^{(l-a)}_i \left( \sum_{u+v=\alpha l-b}(-1)^u\bq_i^{(\alpha(a+b) - b -1)u}F_i^{(u)}F_{\tau i}^{(l-a-b)}F_i^{(v)}\right) \\
= &\sum_{u+v=\alpha l-b}(-1)^u\bq_i^{(\alpha(a+b) - b -1)u} \binom{u_a}{l-a} F_i^{(l-a+u)}F_{\tau i}^{(l-a-b)}F_i^{(v)}\\
= & \sum_{u+v=\alpha l-b, u_a = l-1}(-1)^u\bq_i^{(\alpha(a+b) - b -1)u} \binom{l-1}{l-a} F_i^{(l-a+u)}F_{\tau i}^{(l-a-b)}F_i^{(v)} \\
& + \sum_{u+v=\alpha l-b, u_a = l-2}(-1)^u\bq_i^{(\alpha(a+b) - b -1)u} \binom{l-2}{l-a} F_i^{(l-a+u)}F_{\tau i}^{(l-a-b)}F_i^{(v)} \\
& + \cdots \cdots \\
= & \sum_{u+v=\alpha l-b, u_a = l-1}(-1)^u\bq_i^{(\alpha(a+b) - b -1)u} \binom{l-1}{l-a} F_i^{(rl-1)}F_{\tau i}^{(l-a-b)}F_i^{(sl -b-a+1)} \\
& + \sum_{u+v=\alpha l-b, u_a = l-2}(-1)^u\bq_i^{(\alpha(a+b) - b -1)u} \binom{l-2}{l-a} F_i^{(rl-2)}F_{\tau i}^{(l-a-b)}F_i^{(sl-b-a+2)} \\
& + \cdots \cdots .
\end{align*}
Note that $ \binom{u_a}{l-a} = 0$ if $u_a < l-a$. The lemma follows by induction on $a$ now.
\end{proof}
We show the following equalities hold in $\fpdotU^\imath$:
\begin{equation}
\sum_{r+s=1+\alpha}(-1)^r\B_{i,*}^{(rl)}\B_{\tau i,*}^{(l)}\B_{i,l\zeta}^{(sl)}=\left\{\begin{array}{ll}
\langle \coroot_i-\coroot_{\tau i}, \zeta\rangle\one_{l\zeta} & \text{ if }\alpha=0\\
-2\B_{i,l\zeta}^{(l)} & \text{ if }\alpha=1\\
0 & \text{ if }\alpha\geqslant 2
\end{array}\right. \label{eq:checkiwithtaui}
\end{equation}
Here $*$ on the subscripts stand for the appropriate elements in $X_{\imath}$.
Take $\lambda\in X$ such that $\bar{\lambda}=\zeta$. Recall \eqref{eq:pimA} the $\A$-module isomorphism $p_\imath:{_\A\dot{\U}^\imath}\one_{l\zeta}\rightarrow {_\A\dot{\U}^-}\one_{l\lambda}$. Hence it induces an isomorphism $p_\imath': {_{\A'}\dot{\U}^\imath}\one_{l\zeta}\rightarrow {_{\A'}\dot{\U}^-}\one_{l\lambda}$.
For any $n\in\BN$, we have the following identity in $\U$:
\begin{equation}\label{eq:spandi}
B_i^{(a)}=\sum_{t=0}^aq_i^{t(a-t)-(t(t-1)/2)\cdot \alpha}\varsigma_i^tF_i^{(a-t)}E_{\tau i}^{(t)}\Tilde{K}_i^{-t}
\end{equation}
For any $a,b\in\BN$, and $\mu\in X$, we have the following identity in $_\A\dot{\U}$ by \cite{Lu93}*{23.1.3}:
\begin{equation}\label{eq:quoUp}
\begin{split}
E_i^{(a)}F_i^{(b)}\one_\mu &=\sum_{t\geqslant 0}\qbinom{a-b+\langle \coroot_i,\mu\rangle}{t}_iF_i^{(b-t)}E_i^{(a-t)}\one_\mu\\ &\equiv \qbinom{a-b+\langle \coroot_i,\mu\rangle}{a}_iF_i^{(b-a)}\one_\mu\quad \text{mod }{_\A\dot{\U}}{_\A\U}^+.
\end{split}
\end{equation}
As usual, we understand $F_i^{(n)}=0$ if $n<0$.
Hence in $_\A\U^-\one_\lambda$ (which is viewed as the quotient ${_\A\dot{\U}\one_\lambda}/{_\A\dot{\U}}{_\A\U}^+\one_\lambda$), we have:
\begin{align*}
&p_\imath(\sum_{r+s=1+\alpha}(-1)^rB_{i,*}^{(rl)}B_{\tau i,*}^{(l)}B_{i,l\zeta}^{(sl)})=\sum_{r+s=1+\alpha}(-1)^rB_{i}^{(rl)}B_{\tau i}^{(l)}F_{i}^{(sl)}\one_{l\lambda}\\
&\stackrel{(\heartsuit 1)}{=}\sum_{r+s=1+\alpha}(-1)^rB_{i}^{(rl)}\sum_{b=0}^lq_i^{b(l-b)-b(b-1)/2\cdot\alpha}\varsigma_{\tau i}^bF_{\tau i}^{(l-b)}E_i^{(b)}\tilde{K}_{\tau i}^{-b}F_{i}^{(sl)}\one_{l\lambda}\\
&\stackrel{(\heartsuit 2)}{=}\sum_{r+s=1+\alpha}(-1)^rB_{i}^{(rl)}\sum_{b=0}^lq_i^{b(l-b)-b(b-1)/2\cdot\alpha-al(\langle \coroot_{\tau i},\lambda\rangle-s\alpha)}\varsigma_{\tau i}^b\\
&\cdot \qbinom{b-sl+\langle \coroot_i,l\lambda\rangle}{b}_iF_{\tau i}^{(l-b)}F_i^{(sl-b)}\one_{l\lambda}\\
&\stackrel{(\heartsuit 3)}{=}\sum_{r+s=1+\alpha}(-1)^r\sum_{a=0}^{rl}q_i^{a(rl-a)-a(a-1)/2\cdot \alpha}\varsigma_i^aF_i^{(rl-a)}E_{\tau i}^{(a)}\tilde{K}_i^{-a}\\
&\cdot \sum_{b=0}^lq_i^{b(l-b)-b(b-1)/2\cdot\alpha-al(\langle \coroot_{\tau i},\lambda\rangle-s\alpha)}\varsigma_{\tau i}^b\qbinom{b-sl+\langle \coroot_i,l\lambda\rangle}{b}_iF_{\tau i}^{(l-b)}F_i^{(sl-b)}\one_{l\lambda}\\
&\stackrel{(\heartsuit 4)}{=}\sum_{\substack{0\leqslant a+b\leqslant l\\a,b\geqslant 0}}\sum_{\substack{r+s=1+\alpha\\r,s\geqslant 0}}(-1)^rq_i^{-(a+b)^2+\alpha(a+b)(a+b-1)/2+lN} \\&\cdot \varsigma_i^a\varsigma_{\tau i}^b\qbinom{a+b-l+(sl-b)\alpha+\langle \coroot_{\tau i},l\lambda\rangle}{a}_i\qbinom{b-sl+\langle \coroot_i,l\lambda\rangle}{b}_i\notag
\\&\cdot F_i^{(rl-a)}F_{\tau i}^{(l-a-b)}F_i^{(sl-b)}\one_{l\lambda}.\notag
\end{align*}
Here $N$ is some integer. The equality $(\heartsuit 1)$ and $(\heartsuit 3)$ follow from \eqref{eq:spandi} The equality $(\heartsuit 2)$ and $(\heartsuit 4)$ follow from \eqref{eq:quoUp}.
Therefore in $_{\A'}\U^-\one_{l\lambda}$, we have
\begin{align*}
&p_\imath'(\sum_{r+s=1+\alpha}(-1)^r\B_{i,*}^{(rl)}\B_{\tau i,*}^{(l)}\B_{i,l\zeta}^{(sl)})
=\sum_{\substack{0\leqslant a+b\leqslant l\\a,b\geqslant 0}}\sum_{\substack{r+s=1+\alpha\\r,s\geqslant 0}}(-1)^r\bq_i^{-(a+b)^2+\alpha(a+b)(a+b-1)/2} \\&\cdot \phi\left(\varsigma_i^a\varsigma_{\tau i}^b\qbinom{a+b-l+(sl-b)\alpha+\langle \alpha^\vee_{\tau i},l\lambda\rangle}{a}_i\qbinom{b-sl+\langle \alpha^\vee_i,l\lambda\rangle}{b}_i\right)\notag
\\&\cdot F_i^{(rl-a)}F_{\tau i}^{(l-a-b)}F_i^{(sl-b)}\one_{l\lambda}.\notag
\end{align*}
The coefficients $\phi\left(\varsigma_i^a\varsigma_{\tau i}^b\qbinom{a+b-l+(sl-b)\alpha+\langle \alpha^\vee_{\tau i},l\lambda\rangle}{a}_i\qbinom{b-sl+\langle \alpha^\vee_i,l\lambda\rangle}{b}_i\right)$ are independent of $s$ and $r$ by \cite[Lemma~34.1.2]{Lu93}.
Hence by Lemma~\ref{lem:rel3}, it suffices to consider the summands when $a+b = l$. Hence we have
\begin{align*}
&p_\imath'(\sum_{r+s=1+\alpha}(-1)^r\B_{i,*}^{(rl)}\B_{\tau i,*}^{(l)}\B_{i,l\zeta}^{(sl)})\notag\\
=&\sum_{a=0}^l\sum_{\substack{r+s=1+\alpha\\r,s\geqslant 0}}(-1)^r\phi\left(\varsigma_i^a\varsigma_{\tau i}^{-a}\qbinom{(sl-l+a)\alpha+\langle \coroot_{\tau i},l\lambda\rangle}{a}_i\qbinom{l-a-sl+\langle \coroot_i,l\lambda\rangle}{l-a}_i\right)\\
&\cdot F_i^{(rl-a)}F_i^{(sl-l+a)}\one_{l\lambda}\notag\\
=&\sum_{r=0}^\alpha(-1)^r(r-\alpha+\langle \coroot_i,\lambda\rangle)\binom{\alpha}{r}F_i^{(\alpha l)}\one_{l\lambda} +\sum_{s=0}^\alpha(-1)^{1+\alpha-s}(s\alpha+\langle \coroot_{\tau i},\lambda\rangle)\binom{\alpha}{s}F_i^{(\alpha l)}\one_{l\lambda}
\end{align*}
Note that
\[
F_i^{(rl-a)}F_i^{(sl-l+a)} =
\begin{cases}
\binom{\alpha}{s}F_i^{(\alpha l)}, &\text{if } a =l;\\
\binom{\alpha}{r}F_i^{(\alpha l)}, &\text{if } a =0;\\
0, &\text{otherwise}.
\end{cases}
\]
By the identities
\begin{equation*}
\sum_{t=0}^n(-1)^t\binom{n}{t}=0\quad \text{ if }n>0,\qquad \sum_{t=0}^n(-1)^tt\binom{n}{t}=0\quad\text{ if }n>1,
\end{equation*}
we deduce that
\[
p_\imath'(\sum_{r+s=1+\alpha}(-1)^r\B_{i,*}^{(rl)}\B_{\tau i,*}^{(l)}\B_{i,l\zeta}^{(sl)})
= \begin{cases}
0, &\text{if } \alpha\ge 2;\\
\langle \coroot_i-\coroot_{\tau i}, \lambda\rangle \one_{l\lambda}, &\text{if } \alpha=0;\\
-2F_i^{(l)}\one_{l\lambda}, &\text{if } \alpha=1.
\end{cases}
\]
Now \eqref{eq:checkiwithtaui} follows by the isomorphism $p_\imath': {_{\A'}\dot{\U}}^\imath\one_{l\zeta}\rightarrow {_{\A'}\dot{\U}}^-\one_{l\lambda}$.
\subsubsection{}\label{subsec:pf4} Let $\zeta\in X_\imath$ and $i \in \I$ with $\tau i \neq i$. We write $\alpha=-a_{ij}$. We need to show the following equality in $_{\mathbf{F}}\dot{\U}^\imath$:
\[
\sum_{r+s=1+\alpha}(-1)^r\B_{i,*}^{(rl)}{'\B_{j,*}^{(l)}}\B_{i,l\zeta}^{(sl)}=0.
\]
Here $'\B_{j,*}^{(l)}$ stands for the standard $\imath$divided power $\B_{j,*}^{(l)}$ if $\tau j\neq j$. As before $*$ denotes appropriate elements in $X_\imath$.
Let $\lambda\in X$ such that $\bar{\lambda}=\zeta$ and let $p'_\imath=p'_{\imath,l\lambda}$ be the isomorphism in \eqref{eq:pimA}. Then we have
\begin{equation}\label{eq:r}
p_\imath'(\sum_{r+s=1+\alpha}(-1)^r\B_{i,*}^{(rl)}{'\B_{j,*}^{(l)}}\B_{i,l\zeta}^{(sl)})=\sum_{t=0}^{(l-1)/2}c_t\sum_{r+s=1+\alpha}(-1)^rF_i^{(rl)}F_j^{(l-2t)}F_i^{(sl)}\one_{l\lambda}
\end{equation}
where $c_t\in\A_2'$.
The following claim can be proved similar to \cite[\S35.2.3]{Lu93}.
(a) {\it For $0\leqslant a \leqslant l$, we have $\sum_{r+s=1+\alpha}(-1)^rF_i^{(rl)}F_j^{(a)}F_i^{(sl)}=0$ in $_{\A'}\U^-$.}
By \eqref{eq:r}, we complete the proof in this case.
\subsubsection{}\label{subsec:pf5}
Let $i \neq j\in\I$ with $\tau i=i$. We write $\alpha=-a_{ij}$. We show the following equality in $_{\mathbf{F}}\dot{\U}^\imath$:
\[
\displaystyle \sum_{\substack{r+s+2t=1+\alpha\\t\geqslant 0}}(-1)^r\lr{1}{2t}{'\B_{i,*}^{(rl)}}{'\B_{j,*}^{(l)}}{'\B_{i,l\zeta}^{(sl)}}=0.
\]
Here $'\B_{j,*}^{(l)}$ stands for the standard $\imath$divided power $\B_{j,*}^{(l)}$ if $\tau j\neq j$, and $*$ denotes appropriate elements in $X_\imath$.
For any $n,a\in\mathbb{N}$, $e=\pm 1$, set
\begin{equation*}
D_{i;a,e}^{(n)}=\sum_{u\geqslant 0}q_i^{-e(a-1)u}\LR{a}{2u}_iB_i^{(n-2u)}\in \U^\imath.
\end{equation*}
In particular, $D_{i;0,e}^{(n)}=B_i^{(n)}$. We write $D_i^{(n)}=D_{i; 1,\pm 1}^{(n)}$. We have $B_iB_i^{(a)}=[a+1]_iD_i^{(a+1)}$ by Lemma ~\ref{lem:balanced} for any $a\in\mathbb{N}$.
Then for $n,m\in\mathbb{N}$, thanks to Proposition~ \ref{prop:HigherSerreRe}, we have
\begin{equation*}
y_{n,m}=y_{i,j;n,m}=\sum_{r=0}^m(-1)^rq_i^{-(m-n\alpha-1)r}B_i^{(r)}B_j^{(n)}D_{i;m-n\alpha,1}^{(m-r)}.
\end{equation*}
Let $\U_i^\imath\subseteq \U^\imath$ be the subalgebra (with 1) generated by $B_i$. It is a polynomial ring with one variable over $\mathbb{Q}(q)$. Elements $\{B_i^{(n)}\mid n\in\mathbb{N}\}$ form a $\mathbb{Q}(q)$-basis for $\U_i^\imath$. Define linear operators $\delta=\delta_i$, $E_e=E_{i,e}$ ($e=\pm 1$) on $\U_i^\imath$, such that $\delta(B_i^{(n+1)})=B_i^{(n)}$, $E_e(B_i^{(n)})=q_i^{en}D^{(n)}$, for any $n\in\mathbb{N}$. We understand $B_i^{(n)}$ as $0$ if $ n < 0$.
By a proof similar to Lemma~\ref{lem:balanced}, we obtain
\begin{align}
B_i^{(a-1)}D_i^{(n)} &=\sum_{t\geqslant 0}\qbinom{a+n-1}{n}_i\prod_{m=1}^t\frac{[a-2m+2]_i[n-2m+2]_i}{[a+n-2m+1]_i[2m]_i}B_i^{(a+n-2t-1)} \notag\\
& = \sum_{t\geqslant 0}\qbinom{a+n-1}{n}_i \frac{\LR{a}{2t}_i \LR{n}{2t}_i}{\LR{a+n-1}{2t}_i}B_i^{(a+n-2t-1)}. \label{eq:BD}
\end{align}
\begin{proposition}\label{prop:Rank1Operators}
(a) For any $n,a \in\mathbb{N}$, we have $E_e(D_{i;a,e}^{(n)})=q_i^{en}D_{i;a+1,e}^{(n)}$.
(b) As operators on $\U_i^\imath$, we have $\delta E_e=q_i^e E_e\delta$.
(c) For any $f,g\in\U^\imath_i$, $k\in\mathbb{N}$, we have
\begin{equation*}
\delta^k(fg)=\sum_{s=0}^k\qbinom{k}{s}_i(E_e^{k-s}\delta^sf)(E_{-e}^s\delta^{k-s}g).
\end{equation*}
(d) For $k,N\in\mathbb{N}$, we have
\begin{equation*}
\delta_i^N(D_i^{(k)}B_i^{(N)})=q_i^{Nk}\sum_{s=0}^N\qbinom{N}{s}_iq_i^{-(N+k)s}D_{i;N-s+1,1}^{(k-s)}D_{i;s,-1}^{(s)}.
\end{equation*}
\end{proposition}
\begin{proof}
(a) By direct computation, we have
\begin{align*}
E_{i,e}(D_{i;a,e})&=E_{i,e}(\sum_{u\geqslant 0}q_i^{-e(a-1)u}\LR{a}{2}_iB_i^{(n-2u)})\\
&=\sum_{u\geqslant 0}q_i^{-e(a-1)u}\LR{a}{2u}_iq_i^{e(n-2u)}D_i^{(n-2u)}\\
&=\sum_{u\geqslant 0}q_i^{-e(a-1)u+e(n-2u)}\LR{a}{2u}_i\sum_{t\geqslant 0}\LR{1}{2t}_iB_i^{(n-2u-2t)}\\
&=\sum_{m\geqslant 0}\sum_{u+t=m}q_i^{en-e(a+1)u}\LR{a}{2u}_i\LR{1}{2t}_iB_i^{(n-2m)}\\
&=q_i^{en}\sum_{m\geqslant 0}q_i^{-eam}\LR{a+1}{2m}_iB_i^{(n-2m)}\\&=q_i^{en}D_{i;a+1,e}^{(n)}.
\end{align*}
(b) For any $a\in\mathbb{N}$, we have
\begin{align*}
\delta_i E_{i,e}(B_i^{(a)})=q_i^{ae}\delta_i(\sum_{u\geqslant 0}\LR{1}{2u}_iB_i^{(a-2u)})=q_i^{ae}\sum_{u\geqslant 0}\LR{1}{2u}_iB_i^{(a-2u-1)}.
\end{align*}
Also, we have
\begin{equation*}
E_{i,e}\delta_i(B_i^{(a)})=E_{i,e}(B_i^{(a-1)})=q_i^{(a-1)e}\sum_{u\geqslant 0}\LR{1}{2u}_iB_i^{(a-1-2u)}=q_i^{-e}\delta_iE_{i,e}(B_i^{(n)}).
\end{equation*}
(c) We may assume $k\geqslant 1$. We firstly prove the formula for $k=1$. It will suffice to check that for $a,c\in\mathbb{N}$:
\begin{equation*}
\begin{split}
\delta_i(B_i^{(a)}B_i^{(c)})&= E_{e} ( B_i^{(a)} ) \delta_i ( B_i^{(c)}) + \delta_i ( B_i^{(a)} ) E_{-e}( B_i^{(c)})\\& = q_i^{ea}D_i^{(a)}B_i^{(c-1)}+q_i^{-ec}B_i^{(a-1)}D_i^{(c)}.
\end{split}
\end{equation*}
Note that
\begin{align*}
q_i^{ea}D_i^{(a)}B_i^{(c-1)}+q_i^{-ec}B_i^{(a-1)}D_i^{(c)}&=q_i^{ea}\frac{B_iB_i^{(a-1)}B_i^{(c-1)}}{[a]_i}+q_i^{-ec}B_i^{(a-1)}D_i^{(c)}\\
&=\left(q_i^{ea}\frac{[c]_i}{[a]_i}+q_i^{-ec}\right)B_i^{(a-1)}D_i^{(c)}\\
&=\frac{[a+c]_i}{[a]_i}B_i^{(a-1)}D_i^{(c)}.
\end{align*}
By Lemma \ref{lem:balanced} and \eqref{eq:BD}, we have
\begin{equation*}
\delta_i(B_i^{(a)}B_i^{(c)})=\sum_{t\geqslant 0}\qbinom{a+c}{a}_i\prod_{m=1}^t\frac{[a-2m+2]_i[c-2m+2]_i}{[a+c-2m+1]_i[2m]_i}B_i^{(a+c-2t-1)} = B_i^{(a-1)}D_i^{(c)}.
\end{equation*}
Now suppose the formula holds for $k$. Then
\begin{align*}
\delta_i^{k+1}(fg)&=\sum_{s=0}^k\qbinom{k}{s}_i\delta_i((E_{i,e}^{k-s}\delta_i^sf)(E_{i,-e}^s\delta_i^{k-s}g))\\&=\sum_{s=0}^k\qbinom{k}{s}_i((\delta_iE_{i,e}^{k-s}\delta_i^sf)(E_{i,-e}^{s+1}\delta_i^{k-s}g)+(E_{i,e}^{k+1-s}\delta_i^sf)(\delta_iE_{i,-e}^s\delta_i^{k-s}g))\\
&=\sum_{s=0}^k\qbinom{k}{s}_i((q_i^{e(k-s)}E_{i,e}^{k-s}\delta_i^{s+1}f)(E_{i,-e}^{s+1}\delta_i^{k-s}g)+(E_{i,e}^{k+1-s}\delta_i^sf)(q_i^{-es}E_{i,-e}^s\delta_i^{k+1-s}g))\\
&=\sum_{s=0}^{k+1}\left(\qbinom{k}{s-1}_iq_i^{e(k-s+1)}+q_i^{-es}\qbinom{k}{s}_i\right)(E_{i,e}^{k+1-s}\delta_i^sf)(E_{i,-e}^s\delta_i^{k+1-s}g)\\
&=\sum_{s=0}^{k+1}\qbinom{k+1}{s}_i(E_{i,e}^{k+1-s}\delta_i^sf)(E_{i,-e}^s\delta_i^{k+1-s}g).
\end{align*}
Hence the formula holds for $k+1$. We complete the proof by induction.
(d) We have
\begin{align*}
\delta_i^N(D_i^{(k)}B_i^{(N)})&=\sum_{s=0}^N\qbinom{N}{s}_i(E_{i,1}^{N-s}\delta_i^sD_i^{(k)})(E_{i,-1}^s\delta_i^{N-s}B_i^{(N)})\\
&=q_i^{-k}\sum_{s=0}^N\qbinom{N}{s}_i(E_{i,1}^{N-s}\delta_i^sE_{i,1}B_i^{(k)})(E_{i,-1}^sB_i^{(s)})\\
&=q_i^{-k}\sum_{s=0}^Nq_i^{s}\qbinom{N}{s}_i(E_{i,1}^{N-s+1}\delta_i^sB_i^{(k)})(q_i^{-s^2}D_{i,s,-1}^{(s)})\\
&=q_i^{Nk}\sum_{s=0}^N\qbinom{N}{s}_iq_i^{-(N+k)s}D_{i,N-s+1,1}^{(k-s)}D_{i,s,-1}^{(s)}.
\end{align*}
This finishes the proof.
\end{proof}
For $0\leqslant k<l$, we have $y_{l,(1+\alpha)l-k} = 0$ by higher Serre relations. We have
\begin{align}
0&= \sum_{k=0}^{l-1}(-1)^kq_i^{k} y_{l,(1+\alpha)l-k} D_{i;k,-1}^{(k)} \notag \\
&=\sum_{k=0}^{l-1}(-1)^kq_i^{k} \Big( \sum_{r=0}^{(1+\alpha)l-k}(-1)^rq_i^{-(l-k-1)r}B_i^{(r)}B_j^{(l)}D_{i;l-k,1}^{((1+\alpha)l-k-r)}\Big) D_{i;k,-1}^{(k)}\notag\\
&=\sum_{r=0}^{(1+\alpha)l}(-1)^rq_i^{-(l-1)r}B_i^{(r)}B_j^{(l)}J_r,\label{eq:S}
\end{align}
where
$J_r=\sum_{k=0}^{l-1}(-1)^kq_i^{k(r+1)}D_{i;l-k,1}^{((1+\alpha)l-k-r)}D_{i;k,-1}^{(k)}\in \U^\imath_i$.
Let $_{\A_2}\U^\imath_i$ be the $\A_2$ submodule of $\U^\imath_i$ spanned by $B_i^{(n)}$ for $n\in\mathbb{N}$. Then it is also an $\A_2$-subalgebra. Note that elements $D_{i;a,e}^{(n)}$ ($a,n\in\mathbb{N}$, $e=\pm 1$) belong to this subalgebra. And set $_{\A_2'}\U_i^\imath={\A_2'}\otimes_{\A_2}(_{\A_2}\U^\imath_i)$ where $\A_2\rightarrow\A_2'$ is the canonical quotient. Note that operators $\delta_i$, $E_{i,e}$ preserve $_{\A_2}\U^\imath_i$, so they induce linear operators on $_{\A_2'}\U_i^\imath$. We use the same notation to denote the corresponding operator after base change.
By Proposition~\ref{prop:Rank1Operators}, we have
\begin{align}\label{eq:J}
&\delta_i^{l-1}(D_i^{((1+\alpha)l-r)}B_i^{(l-1)}) \notag \\
&= \bq_i^{(l-1)((1+\alpha)l-r)}\sum_{s=0}^{l-1}\phi\left(\qbinom{l-1}{s}_i\right)\bq_i^{-((2+\alpha)l-r-1)s}D_{i;l-s,1}^{((1+\alpha)l-r-s)}D_{i;s,-1}^{(s)}\notag \\
&\stackrel{\heartsuit}{=}\bq_i^r\sum_{s=0}^{l-1}(-1)^s\bq_i^{(r+1)s}D_{i;l-s,1}^{((1+\alpha)l-r-s)}D_{i;s,-1}^{(s)} =\bq_i^rJ_r \quad \text{in } _{\A_2'}\U_i^\imath
\end{align}
Here $\heartsuit$ follows by $\phi\left(\qbinom{l-1}{s}_i\right)=(-1)^s$, for $0\leqslant s<l$. By the proof of Proposition \ref{prop:Rank1Operators} (c), we have
\begin{equation}\label{eq:l}
[(2+\alpha)l-r]_iD_i^{((1+\alpha)l-r)}B_i^{(l-1)}=[l]_i\delta_i(B_i^{(l)}B_i^{((1+\alpha)l-r)}) \quad \text{in } _{\A_2}\U^\imath_i
\end{equation}
Note that $\phi([l]_i)=0$ in $\A'_2$. Suppose $l\nmid r$, then $\phi([(2+\alpha)l-r]_i)\neq 0$. Hence
\begin{align*}
J_r &= \bq_i^{-r} \delta_i^{l-1}(D_i^{((1+\alpha)l-r)}B_i^{(l-1)}) \\
&= \bq_i^{-r}\phi([(2+\alpha)l-r]_i)^{-1}\phi([l]_i) \delta_i^{l}(B_i^{(l)}B_i^{((1+\alpha)l-r)}) =0 \quad \text{in } _{\A'_2}\U^\imath_i.
\end{align*}
Suppose $r=lr'$, with $0\leqslant r'\leqslant 1+\alpha$. Then by \eqref{eq:l} and \S\ref{subsec:pf2}, we have
\begin{align*}
&(2+\alpha-r')D_i^{((1+\alpha-r')l)}B_i^{(l-1)} \\
& =\delta_i(B_i^{(l)}B_i^{((1+\alpha-r')l)}) \\
& = (2+\alpha-r')\sum_{u\geqslant 0}\lr{1}{2u}B_i^{((2+\alpha-r'-2u)l-1)}
\quad \text{in } _{\A'_2}\U^\imath_i.
\end{align*}
Hence by \eqref{eq:J}, we deduce
\begin{equation*}
J_r=\sum_{u\geqslant 0}\lr{1}{2u}B_i^{((1+\alpha-r'-2u)l)} \quad \text{in } _{\A_2'}\U^\imath_i.
\end{equation*}
For any $\zeta\in X_\imath$, combining with \eqref{eq:S}, we have the following equality in $_{\A_2'}\dot{\U}^\imath$
\begin{align*}
0&=\sum_{r'=0}^{1+\alpha}(-1)^{r'}B_i^{(r'l)}B_j^{(l)}\sum_{u\geqslant 0}\lr{1}{2u}B_i^{((1+\alpha-r'-2u)l)}\one_{l\zeta}\\
&=\sum_{\substack{r+s+2u=1+\alpha\\u\geqslant 0}}(-1)^r\lr{1}{2u}B_i^{(rl)}B_j^{(l)}B_i^{(sl)}\one_{l\zeta}\\
&=\sum_{\substack{r+s+2u=1+\alpha\\u\geqslant 0}}(-1)^r\lr{1}{2u}{'\B_{i,*}^{(rl)}}{'\B_{j,*}^{(l)}}{'\B_{i,l\zeta}^{(sl)}}.
\end{align*}
We finish the proof in this case.
\subsection{The splitting on modules}\label{sec:splitmod}
\subsubsection{}
For any $\lambda\in X^+$, recall \S\ref{sec:qanniR} that $L(\lambda)$ denotes the irreducible $\mathrm{U}$-module of highest weight $\lambda$. It follows from \cite[3.8]{BW18a} that the $\mathcal{A}$-linear map $_\mathcal{A}\dot{\mathrm{U}}^\imath\rightarrow {_\mathcal{A}L(\lambda)}$ sending $u$ to $u\cdot v_\lambda ^+$ is surjective.
\begin{lem}\label{lem:annil}
The map $\pi:{_\mathcal{A}\dot{\mathrm{U}}^\imath}\rightarrow {_\mathcal{A}L(\lambda)}$ sending $u$ to $u\cdot v_\lambda ^+$ has kernel
\[
J(\lambda)=\sum_{\zeta\in X_\imath,\;\zeta\neq\bar{\lambda}}{_\mathcal{A}\dot{\mathrm{U}}^\imath}\one_\zeta+\sum_{i\in\I,\;n>\langle \coroot_i,\lambda\rangle}{_\mathcal{A}\dot{\mathrm{U}}^\imath}B_{i,\bar{\lambda}}^{(n)}.
\]
\end{lem}
\begin{proof}
Note that ${_\mathcal{A}\dot{\mathrm{U}}^\imath}=\bigoplus_{\zeta\in X_\imath}{_\mathcal{A}\dot{\mathrm{U}}^\imath}\one_\zeta$, and $\one_\zeta\cdot v_\lambda^+=0$ unless $\zeta=\bar{\lambda}$. Hence it will suffice to show the restricting map $\pi:{_\mathcal{A}\dot{\mathrm{U}}^\imath}\one_{\bar{\lambda}} \rightarrow {_\mathcal{A}L(\lambda)}$ has kernel
$$
Q(\lambda)=\sum_{i\in\I,\;n>a_i}{_\mathcal{A}\dot{\mathrm{U}}^\imath}B_{i,\bar{\lambda}}^{(n)}.
$$
The action map factors through the ${_\mathcal{A}\dot{\mathrm{U}}^\imath}$-module isomorphism \eqref{eq:pimA} $\psi: {_\mathcal{A}\dot{\mathrm{U}}^\imath}\one_{\bar{\lambda}}\xrightarrow{\sim}{_\mathcal{A}\dot{\mathrm{U}}^-}\one_\lambda \xrightarrow{\sim} {}_\CA M(\lambda)$. Here ${}_\CA M(\lambda)$ denotes the $\A$-form of the Verma module $M(\lambda)$ with highest weight $\lambda$.
It follows from \cite[23.3.3 \& 23.3.6]{Lu93} that the $\mathcal{A}$-linear map $ {}_\CA M(\lambda) \rightarrow {_\mathcal{A}L(\lambda)}$ sending the highest weight vector to $v_\lambda^+$ has kernel
\[
P(\lambda)=\sum_{i\in\I,\; n>a_i}{_\mathcal{A}\dot{\mathrm{U}}}\cdot F_i^{(n)}v^+_\lambda =\sum_{i\in\I,\; n>a_i}{_\mathcal{A}\mathrm{U}}^- F_i^{(n)}v^+_\lambda.
\]
Therefore $Q(\lambda) = \psi^{-1} (P(\lambda) ) = \sum_{i\in\I,\;n>a_i}{_\mathcal{A}\dot{\mathrm{U}}^\imath} \psi^{-1}(F_i^{(n)}v^+_\lambda)$. If $\tau i \neq i$, then we have $\psi^{-1}(F_i^{(n)}v^+_\lambda) = B_{i,\bar{\lambda}}^{(n)} $ by direct computation. If $\tau i = i$, then the $\CA$-span of $\{B_{i,\bar{\lambda}}^{(n)}\}_{n > a_i}$ is the same as the $\CA$-span of $\{\psi^{-1}(F_i^{(n)}v^+_\lambda)\}_{n > a_i}$ by \cite[Theorem~2.10\&3.6]{BerW18}. This finishes the proof.
\end{proof}
For any $\mathcal{A}$-algebra $R$ and $\lambda\in X^+$, recall $_R L(\lambda)=R\otimes_{\A} {_\mathcal{A}L(\lambda)}$. Since ${_\mathcal{A}L(\lambda)}$ is a free (hence flat) $\CA$-module, we obtain the following corollary by base change.
\begin{cor}\label{cor:annlR}
The $R$-linear map $_R\dot{\mathrm{U}}^\imath \rightarrow {_RL(\lambda)}$ given by $r\otimes u\mapsto (r\otimes u)(1\otimes v_\lambda^+)$ is surjective and the kernel is the left ideal of $_R\dot{\mathrm{U}}^\imath$ generated by $1\otimes\one_\zeta$ ($\zeta\neq\bar{\lambda}$), $1\otimes B_{i,\bar{\lambda}}^{(n)}$ for various $i\in\I,\, n>\langle \coroot_i,\lambda \rangle$.
\end{cor}
\subsubsection{}\label{se:L}
Recall \S\ref{sec:qFr}.
\begin{proposition}\label{prop:fa}
For any $\lambda\in X^+$, there exists a unique $_{\A'}\dot{\mU}^\imath$-module homomorphism
\[
\psi_\lambda:{_{\A'}\mathfrak{L}}(\lambda)\longrightarrow \big({_{\A'}L} (l\lambda)\big)^\ifr, \quad v_{\lambda}^+ \mapsto v_{l\lambda}^+.
\]
Here $\big({_{\A'}L} (l\lambda)\big)^{\ifr}$ stands for the $_{\A'}\dot{\mU}^\imath$-module, which is the same as ${_{\A'}L} (l\lambda)$ as $\A'$-modules, and the action of $_{\A'}\dot{\mU}^\imath$ is twisted by $\ifr$.
\end{proposition}
\begin{proof}
Since ${_{\A'}\mathfrak{L}}(\lambda)$ is generated by $v_\lambda^+$ as $\cAUi$-module, the uniqueness is clear. Thanks to Corollary \ref{cor:annlR}, in order to show such map is well-defined, it will suffice to show that $\ifr (\fb_{i,\bar{\lambda}}^{(n)})v_{l\lambda}^+=0$, for any $i\in\I$ and $n>\langle \alpha^\vee_i,\lambda\rangle$.
Suppose $\tau i\neq i$. Then $n>\langle \alpha^\vee_i,\lambda\rangle$ implies $nl>\langle \alpha^\vee_i,l\lambda\rangle$. Thanks to Corollary \ref{cor:annlR} and Theorem~\ref{thm:iFr}, we have
$\ifr (\fb_{i,\bar{\lambda}}^{(n)})v_{l\lambda}^+=\RB_{i,\overline{l\lambda}}^{(nl)}v_{l\lambda}^+=0$.
Suppose $\tau i=i$. By Corollary \ref{cor:iFr}, we have
$$
\ifr(\fb_{i,\bar{\lambda}}^{(n)}) = \sum_{t=0}^{(l-1)/2} c_t \RB_{i,\overline{l\lambda}}^{(ln-2t)}, \quad c_t \in \CA'.
$$
Since $n>\langle \alpha^\vee_i,\lambda\rangle$, we have $ln\geqslant \langle \alpha^\vee_i,l\lambda\rangle + l$. Therefore $ln-2t\geqslant \langle \alpha^\vee_i,l\lambda\rangle +1$ for $t = 0, \dots, (l-1)/2$. By Corollary \ref{cor:annlR} again, we deduce that $\ifr(\fb_{i,\bar{\lambda}}^{(n)})v_{l\lambda}^+=0$.
The proposition is proved.
\end{proof}
\subsubsection{}\label{sec:propJ}
For a subset $J$ of the index set $\I$, we call $J$ has \emph{property (*)} if:
(a) $\tau j=j$, for any $j\in J$; (b) $\langle \alpha^\vee_i,\alpha_j\rangle \geqslant -1$, for any $j\in J$, and $i\in \I$; (c) for any fixed $i\in \I$, there are at most one $j\in J$, with $\langle \alpha^\vee_i, \alpha_j\rangle \neq 0$.
For a subset $J\subseteq \I$ with property (*), we define
\begin{equation*}
\RB_{J,\overline{\lambda}}^{(l-1)} =\prod_{j\in J}{\RB}_j^{(l-1)}\one_{\bar{\lambda}} \in \pAUi, \quad \text{for any }\lambda\in X.
\end{equation*}
We see that the product is independent of the order by condition (c). Note that $\bq_j^k+\bq_j^{-k}$ is invertible in $\A'$, whenever $l$ does not divide $k$. Hence by Lemma \ref{lem:ff}, the element $\RB_{J,\overline{\lambda}}^{(l-1)}$ indeed belongs to $\pAUi$.
\begin{proposition}\label{prop:faJ}
For any $\lambda\in X^+$, and a subset $J$ of $\I$ which has property (*), there exists a unique $\cAUi$-module homomorphism
\[
\psi^J_\lambda: {_{\A'}\mathfrak{L}}(\lambda)\longrightarrow \big({_{\A'}L}(l\lambda)\big)^\ifr, \quad v_\lambda^+ \mapsto \RB_{J,\overline{\lambda}}^{(l-1)}v_{l\lambda}^+\, \text{ (the untwisted action}).
\]
Here $\big({_{\A'}L} (l\lambda)\big)^\ifr$ stands for the $_{\A'}\dot{\mU}^\imath$-module, which is the same as ${_{\A'}L} (l\lambda)$ as $\A'$-modules, and the action of $_{\A'}\dot{\mU}^\imath$ is twisted by $\ifr$.
\end{proposition}
\begin{proof}
It will suffice to prove that $\ifr(\fb_{i,\bar{\lambda}}^{(n)})\RB_{J,\overline{l\lambda}}^{(l-1)}\cdot v_{l\lambda}^+=0$, for any $i\in \I$ and $n>\langle \alpha^\vee_i, \lambda\rangle$ by Lemma~\ref{lem:annil}. We divide the proof into two cases.
Case (a): $i \not \in J$.
If $\langle \alpha^\vee_i,\alpha_j\rangle=0$ for all $j\in J$, then $\ifr(\fb_{i,\bar{\lambda}}^{(n)})$ commutes with $\RB_{J,\overline{l\lambda}}^{(l-1)}$. By the proof of Proposition \ref{prop:fa}, we have $\ifr(\fb_{i,\bar{\lambda}}^{(n)})v_{l\lambda}^+=0$ if $n>\langle \alpha^\vee_i,\lambda\rangle$. Hence $\ifr(\fb_{i,\bar{\lambda}}^{(n)})\RB_{J,\overline{l\lambda}}^{(l-1)} v_{l\lambda}^+=0$
Otherwise we can find a unique $j'\in J$ such that $\langle \alpha^\vee_{i},\alpha_{j'}\rangle =-1$. Then $\ifr(\fb_{i,\bar{\lambda}}^{(n)})$ commutes with $\RB_{j,\overline{l\lambda}}^{(l-1)}$, for any $j' \neq j\in J$. Hence
\[
\ifr(\fb_{i,\bar{\lambda}}^{(n)})\RB_{J,\overline{l\lambda}}^{(l-1)} v_{l\lambda}^+=\RB_{J\backslash\{j'\},*}^{(l-1)}\ifr(\fb_{i,\bar{\lambda}}^{(n)}){'\RB_{j',\overline{l\lambda}}^{(l-1)}}v_{l\lambda}^+.
\]
We claim $\ifr(\fb_{i,\bar{\lambda}}^{(n)}){'\RB_{j',\overline{l\lambda}}^{(l-1)}}v_{l\lambda}^+=0$. Since $'\RB_{j',\overline{l\lambda}}^{(l-1)}v_{l\lambda}^+$ is an $\A'$-linear combinations of $ \RF_{j'}^{(t)}v_{l\lambda}^+$ with $ 0 \leqslant t \leqslant l-1$, it suffices to show $\ifr(\fb_{i,\bar{\lambda}}^{(n)})\RF_{j'}^{(t)}v_{l\lambda}^+=0$ for any $ 0 \leqslant t \leqslant l-1$. Note that $\RF_{j'}^{(t)}v_{l\lambda}^+$ is a highest weight vector of highest weight $\langle \alpha^\vee_i, l\lambda-t\alpha_{j'}\rangle < nl$, with respect to the rank one quantum group associated to $i \in \I$. By Corollary~\ref{cor:iFr}, we have
\[
\ifr(\mathfrak{b}_{i,\overline{\lambda}}^{(n)}) =
\begin{cases}
\RB_{i,\overline{l\lambda}}^{(ln)}, &\text{if } \tau i \neq i;\\
\mathrm{B}_{i,\overline{l\lambda}}^{(ln)}, &\text{if } \tau i =i \text{ and } n = \langle \coroot_i,\overline{\lambda}\rangle + 1;\\
\sum_{t=0}^{(l-1)/2}c_t\mathrm{B}_{i,\overline{l\lambda}}^{(nl-2t)} (c_t \in \A'), &\text{if } \tau i =i \text{ and } n > \langle \coroot_i,\overline{\lambda}\rangle +1.
\end{cases}
\]
The claim follows by the proof of Lemma~\ref{lem:annil} or \cite{BerW18}*{Theorem~2.10\&3.6}. We hence finish Case (a).
Case (b): $i \in J$.
Then by the definition of the property (*), $\ifr(\fb_{i,\bar{\lambda}}^{(n)})$ commutes with $\RB_{J,\overline{l\lambda}}^{(l-1)}$. Then by similar computations, we have
\[
\ifr(\fb_{i,\bar{\lambda}}^{(n)})\RB_{J,\overline{l\lambda}}^{(l-1)} v_{l\lambda}^+=\RB_{J,\overline{l\lambda}}^{(l-1)}\ifr(\fb_{i,\bar{\lambda}}^{(n)}) v_{l\lambda}^+=0, \quad \text{if }n>\langle \alpha^\vee_i,\lambda\rangle.
\]
We finish the proof.
\end{proof}
\section{Frobenius splittings of algebraic groups}\label{sec:iFrO}
In this section, we assume given an $\imath$root datum $(\I,Y,X,A,(\alpha_i)_{i\in\I},(\coroot_i)_{i\in\I},\tau,\theta)$ of finite (quasi-split) type. Retain the settings in \S\ref{sec:qFr}.
\subsection{The maps $\ofr$ and $\iofr$}
\subsubsection{}
Recall \S \ref{sec:iqp} the embedding $_\A\dot{\U}^\imath\hookrightarrow{_\A\widehat{\U}}$. We have induced embeddings $_{\A'}\dot{\U}^\imath\hookrightarrow{_{\A'}\widehat{\U}}$, and $_{\A'}\dot{\mathfrak{U}}^\imath\hookrightarrow{_{\A'}\widehat{\mathfrak{U}}}$. It follows from \cite{BS21}\footnote{In \cite{BS21}, the authors use a different completion, but the same proof applies in our setting.} that $\ofr:{_{\A'}\widehat{\U}}\rightarrow {_{\A'}\widehat{\mU}}$ restricts to $\iofr:{_{\A'}\dot{\U}^\imath}\rightarrow {_{\A'}\dot{\mU}^\imath}$.
For any canonical basis element $b$ of $_{\A'}\dot{\U}$, we write
\[
\ofr(b)=\sum_{b'\in\dot{\RB}}x_{b,b'}b', \quad \text{for } x_{b,b'}\in \A', b' \in {}_{\A'}\dot{\mU}.
\]
For any $\imath$canonical basis element $b$ of $_{\A'}\dot{\U}^\imath$, we write
\[
\iofr(b)=\sum_{b'\in\dot{\RB}^\imath}x^\imath_{b,b'}b', \quad \text{for } x^\imath_{b,b'}\in\A', b' \in {}_{\A'}\dot{\mathfrak{\U}}^\imath.
\]
\begin{lemma}\label{le:fiFr}
(1) Let $b'$ be a canonical basis element of $_{\A'}\dot{\mU}$. There are only finitely many canonical basis element $b$ of $_{\A'}\dot{\U}$, such that $x_{b,b'}\neq 0$.
(2) Let $b'$ be a $\imath$canonical basis element of $_{\A'}\dot{\mU}^\imath$. There are only finitely many $\imath$canonical basis element $b$ of $_{\A'}\dot{\U}^\imath$, such that $x^\imath_{b,b'}\neq 0$.
\end{lemma}
\begin{proof}
We show part (1). Choose some $\mu_1,\mu_2\in X^+$, such that $b'\cdot (v_{-\mu_1}^-\otimes v_{\mu_2}^+)\neq 0$ in $^\omega {_{\A'}\mathfrak{L}}(\mu_1)\otimes_{\A'} {_{\A'}\mathfrak{L}}(\mu_2)$.
By \S\ref{sec:qanniR}, it is direct to check that we have the $_{\A'}\dot{\U}$-module homomorphism:
\[
^\omega {_{\A'}L}(l\mu_1)\otimes_{\A'} {_{\A'}L}(l\mu_2)\longrightarrow \left({^\omega {_{\A'}\mathfrak{L}}(\mu_1)\otimes_{\A'} {_{\A'}\mathfrak{L}}(\mu_2)}\right)^\ofr,\quad v_{-l\mu_1}^-\otimes v_{l\mu_2}^+ \mapsto v_{-\mu_1}^-\otimes v_{\mu_2}^+.
\]
Here $\left({^\omega {_{\A'}\mathfrak{L}}(\mu_1)\otimes_{\A'} {_{\A'}\mathfrak{L}}(\mu_2)}\right)^\ofr$ is defined similar to that in Proposition~\ref{prop:ga}.
Suppose $b$ is some canonical basis element of $_{\A'}\dot{\U}$, such that $x_{b,b'}\neq 0$. Then thanks to the map above, we deduce that $b\cdot( v_{-l\mu_1}^-\otimes v_{l\mu_2}^+)\neq 0$. There are only finitely many such $b$ by \cite{Lu93}*{\S25.2}.
We now show part (2). For any $\mu\in X^+$, it follows from Corollary \ref{cor:annlR} that there is an $_{\A'}\dot{\U}^\imath$-module homomorphism :
\[
{_{\A'}L}(l\mu)\rightarrow {_{\A'}\mathfrak{L}}(\mu)^\iofr, \quad v_{l\mu}^+ \mapsto v_\mu^+.
\]
Here ${_{\A'}\mathfrak{L}}(\mu)^\iofr$ is defined similar to that in Proposition~\ref{prop:ga}.
For any fixed $\imath$canonical basis element $b'$ of $_{\A'}\dot{\mU}^\imath$, we can find $\mu\in X^+$, such that $b'\cdot v_\mu^+\neq 0$. Here $b'\cdot v_\mu^+$ stands for the standard action (not twisted by $\iofr$).
Suppose $b$ is some $\imath$canonical basis element of $_{\A'}\dot{\U}^\imath$, such that $x_{b,b'}^\imath\neq 0$. Then thanks to the map above and Theorem \ref{thm:stab}, we deduce that $b\cdot v_{l\mu}^+\neq 0$. By Theorem~\ref{thm:stab} again, there are only finitely many such $b$.
\end{proof}
\subsubsection{}\label{sec:oFrbach}
Let $l=p$ be a prime number, which is relatively prime to all the root length.
Let $\mathbb{F}_p$ be the finite field consisting of $p$ elements. Note that the $p$-th cyclotomic polynomial $f_p(x)$ is $1+x+\cdots+x^{p-1}$. So $f_p(1)=0$ modulo $p$. Hence there is a ring homomorphism $v:\A'\rightarrow \BF_p$, sending $\bq$ to 1. It follows that the compositions
\begin{equation*}
\begin{tikzcd}
\mathcal{A} \arrow[r,"c\text{,}\,\phi"] & \mathcal{A}' \arrow[r,"v"] & \mathbb{F}_p.
\end{tikzcd}
\end{equation*}
are given by $q\mapsto 1$, regardless of the first map. We shall view $\BF_p$ as an $\A$-algebra in either way without ambiguity.
The we have canonical isomorphisms
\[
\BF_p\otimes_{\A'} {_{\A'}\dot{\mU}}\cong\BF_p\otimes_{\A'} {_{\A'}\dot{\U}}\cong {_{\BF_p}\!\dot{\U}}\quad\text{and}\quad\BF_p\otimes_{\A'} {_{\A'}\dot{\mU}^\imath}\cong\BF_p\otimes_{\A'} {_{\A'}\dot{\U}^\imath}\cong {_{\BF_p}\!\dot{\U}^\imath}.
\]
Therefore the $\A'$-algebra homomorphisms
\[
\ofr: {_{\A'}\dot{\U}}\longrightarrow{_{\A'}\dot{\mU}}\quad \text{ and } \quad \iofr:{_{\A'}\dot{\U}^\imath}\longrightarrow{_{\A'}\dot{\mU}^\imath}
\]
induce $\BF_p$-algebra homomorphisms
\[
\ofr:{_{\BF_p}\!\dot{\U}}\longrightarrow{_{\BF_p}\!\dot{\U}}\quad \text{ and }\quad \iofr:{_{\BF_p}\!\dot{\U}^\imath}\rightarrow{_{\BF_p}\!\dot{\U}^\imath}.
\]
Thanks to Lemma \ref{le:fiFr}, we further have well-defined $\BF_p$-algebra homomorphisms
\[
\ofr:{}_{\BF_p}\!\widehat{\U} \longrightarrow {}_{\BF_p}\!\widehat{\U} \quad \text{ and }\quad \iofr:{}_{\BF_p}\!\widehat{\U}^\imath \rightarrow {}_{\BF_p}\!\widehat{\U}^\imath.
\]
For any $i\in\I$, $n\in\BN$, $\lambda\in X$, we write $e_i^{(n)}\one_\lambda$ and $f_i^{(n)}\one_\lambda$ to denote the images of $E_i^{(n)}\one_\lambda$ (or $\mathfrak{e}_i^{(n)}\one_\lambda$) and $F_i^{(n)}\one_\lambda$ (or $\mathfrak{f}_i^{(n)}\one_\lambda$) in ${}_{\BF_p}\!\dot{\U}$, respectively. Also, for any $i\in\I$, $n\in\BN$, $\zeta\in X_\imath$, we write $b_{i,\zeta}^{(n)}$ and $'b_{i,\zeta}^{(n)}$ to denote the image of $B_{i,\zeta}^{(n)}$ (or $\mathfrak{b}_{i,\zeta}^{(n)}$) and $'B_{i,\zeta}^{(n)}$ (or $'\mathfrak{b}_{i,\zeta}^{(n)}$) in ${}_{\BF_p}\!\dot{\U}^\imath$.
\subsubsection{}
Recall the commutative Hopf algebras $\RO_{\BF_p}$ and $\RO^\imath_{\BF_p}$ in \S\ref{sec:nocp} and \S\ref{sec:pro}, respectively.
Thanks to Lemma \ref{le:fiFr}, the maps $\ofr$ and $\iofr$ induce $\BF_p$-linear maps
\[
{\ofr}^*:\RO_{\BF_p}\longrightarrow\RO_{\BF_p}\quad \text{ and } \quad {\iofr}^*:\RO_{\BF_p}^\imath\longrightarrow\RO_{\BF_p}^\imath.
\]
It follows by definitions that $r\circ{\ofr^*}={\iofr^*}\circ r$, where $r:\RO_{\BF_p}\rightarrow\RO_{\BF_p}^\imath$ denotes the restriction defined in \ref{sec:pro}.
\begin{proposition}\label{prop:pthFr}
(1) For any $f\in \RO_{\BF_p}$, we have ${\ofr^*}(f)=f^p$.
(2) For any $f\in \RO_{\BF_p}^\imath$, we have ${\iofr^*}(f)=f^p$.
\end{proposition}
\begin{proof}
We prove part (1). Part (2) follows from the identity $r\circ{\ofr^*}={\iofr^*}\circ r$ and the fact that $r$ is a surjective algebra homomorphism.
Let ${}_{\BF_p}\!\widehat{\U}^{(p)}$ be the $\BF_p$-linear space consisting of all the formal $\BF_p$-linear combinations
\[
\sum_{b_i\in\dot{\RB};\;1\leqslant i\leqslant p }n_{b_1,\cdots,b_p}b_1\otimes\cdots\otimes b_p, \quad \text{for }n_{b_1,\cdots,b_p}\in \BF_p.
\]
Then ${}_{\BF_p}\!\widehat{\U}^{(p)}$ is naturally endowed with an structure of $\BF_p$-algebra. By applying comultiplication $p$ times, we obtain an $\BF_p$-algebra homomorphism $\widehat{\Delta}^p:{{}_{\BF_p}\!\widehat{\U}}\rightarrow {{}_{\BF_p}\!\widehat{\U}^{(p)}}$. For any $f\in\RO_A$, let $f^{\otimes p}:{{}_{\BF_p}\!\widehat{\U}^{(p)}}\rightarrow \BF_p$ be the $\BF_p$-linear map defined in the obvious way.
Then it suffices to check that the following diagram commutes:
\begin{equation}\label{dia:pthFr}
\begin{tikzcd}
{}_{\BF_p}\!\dot{\U} \arrow[r,"\ofr"] \arrow[d,"\widehat{\Delta}^p"'] & {}_{\BF_p}\!\widehat{\U} \arrow[d,"f"] \\ {}_{\BF_p}\!\widehat{\U}^{(p)} \arrow[r,"f^{\otimes p}"] & \BF_p
\end{tikzcd}
\end{equation}
Let $x=E_{i_1}^{(a_1)}\cdots E_{i_s}^{(a_s)}F_{j_1}^{(b_1)}\cdots F_{j_t}^{(b_t)}\one_\lambda\in {{}_{\BF_p}\!\dot{\U}}$. We have
\begin{align*}
\widehat{\Delta}^{p}(x)=&\sum_{\substack {a_{n,1}+a_{n,2}+\cdots a_{n,p}=a_n \\ 1\leq n \leq s \\ b_{m,1}+b_{m,2}+\cdots+b_{m,p}=b_m \\ 1\leq m\leq t\\ \lambda_1+\lambda_2+\cdots \lambda_p=\lambda}} E_{i_1}^{(a_{1,1})}\cdots E_{i_s}^{(a_{s,1})}F_{j_1}^{(b_{1,1})}\cdots F_{j_t}^{(b_{t,1})}\one_{\lambda_1} \otimes \cdots\\& \otimes E_{i_1}^{(a_{1,p})}\cdots E_{i_s}^{(a_{s,p})}F_{j_1}^{(b_{1,p})}\cdots F_{j_t}^{(b_{t,p})}\one_{\lambda_p}
\end{align*}
The symmetric group $S_p$ acts on ${}_{\BF_p}\!\widehat{\U} $ by permutation. The element $\widehat{\Delta}^{p}(x)$ is $S_p$-invariant. If a summand $\widehat{\Delta}^{p}(x)$ is not fixed by $S_p$, then the number of elements in its $S_p$-orbit will be divisible by $p$. Therefore the evaluation of $f^{\otimes p}$ over non-trivial $S_p$-orbits is $0$.
Recall $z^p = z$ for any $z \in \BF_p$. Hence we have
\begin{align*}
f^{\otimes p}\circ \widehat{\Delta}^p(x)=f\left(E_{i_1}^{(a_1/p)}\cdots E_{i_s}^{(a_s/p)}F_{j_1}^{(b_1/p)}\cdots F_{j_t}^{(b_t/p)}\one_{\lambda/p}\right)
\end{align*}
if $p$ divides $a_n$ ($1\leq n\leq s$), $p$ divides $b_m$ ($1\leq m\leq t$), and $\lambda\in p X$;
and
\[
f^{\otimes p}\circ \widehat{\Delta}^p(x)=0, \quad \text{otherwise}.
\]
Finally, by the definition of $\ofr$, we conclude that $f^{\otimes p}\circ \widehat{\Delta}^p(x)=f\circ{\ofr}(x)$. Hence the diagram \eqref{dia:pthFr} commutes.
\end{proof}
\subsection{The maps $\fr$ and $\ifr$}
\subsubsection{}
Recall the $\A'$-algebra homomorphisms $\fr:{_{\A'}\dot{\mU}}\rightarrow {_{\A'}\dot{\U}}$ in \S\ref{sec:qFr} and $\ifr:{_{\A'}\dot{\mU}^\imath}\rightarrow{_{\A'}\dot{\U}^\imath}$ in \S\ref{sec:iFr}.
For any canonical basis element $b$ of $_{\A'}\dot{\mU}$, we write
\[
\fr (b) =\sum_{b'\in \dot{\RB}}y_{b,b'}b', \quad \text{for }y_{b,b'}\in \A', b'\in{_{\A'}\dot{\U}}.
\]
Similarly, for any $\imath$canonical basis element $b$ of $_{\A'}\dot{\U}^\imath$, we write
\[
\ifr (b) =\sum_{b'\in \dot{\RB}^\imath}y^\imath_{b,b'}b', \quad \text{for }y^\imath_{b,b'}\in \A', b'\in{_{\A'}\dot{\U}^\imath}.
\]
\begin{lemma}\label{le:fiFrs}
(1) Let $b'$ be a canonical basis element of ${_{A'}\dot{\U}}$. Then there are only finitely many canonical basis element $b$ of $_{\A'}\dot{\mU}$, such that $y_{b,b'}\neq 0$.
(2) Let $b'$ be an $\imath$canonical basis element of ${_{A'}\dot{\U}^\imath}$. Then there are only finitely many $\imath$canonical basis element $b$ of $_{\A'}\dot{\mU}^\imath$, such that $y^\imath_{b,b'}\neq 0$.
\end{lemma}
\begin{proof}
We show part (1). Let $\mu_1,\mu_2\in X^+$ be such that $b'\cdot( v_{-l\mu_1}^-\otimes v_{l\mu_2}^+)\neq 0$ in ${^\omega _{\A'}L}(l\mu_1)\otimes_{\A'} {_{\A'}L}(l\mu_2)$. Thanks to \S\ref{sec:qanniR}, we have an $_{\A'}\dot{\mU}$-module homomorphism:
\[
{^\omega _{\A'}\mathfrak{L}}(\mu_1)\otimes_{\A'} {_{\A'}\mathfrak{L}}(\mu_2)\longrightarrow \left( {^\omega _{\A'}L}(l\mu_1)\otimes_{\A'} {_{\A'}L}(l\mu_2)\right)^\fr,\quad v_{-\mu_1}^-\otimes v_{\mu_2}^+ \mapsto v_{-l\mu_1}^-\otimes v_{l\mu_2}^+.
\]
Suppose $b$ is some canonical basis element of $_{\A'}\dot{\mU}$, such that $y'_{b,b'}\neq 0$. Then we deduce that $\fr(b)\cdot (v_{-l\mu_1}^-\otimes v_{l\mu_2}^+)\neq 0$. Hence $b\cdot (v_{-\mu_1}^-\otimes v_{\mu_2}^+)\neq 0$. There are only finitely many such $b$ by \cite{Lu93}*{\S25.2}.
We now show part (2). Let $\mu\in X^+$ be such that $b'\cdot v_{l\mu}^+\neq 0$. By Proposition \ref{prop:fa}, we have the $_{\A'}\dot{\mU}^\imath$-module homomorphism:
\[
{_{\A'}\mathfrak{L}}(\mu)\longrightarrow \left({_{\A'}L}(l\mu)\right)^\ifr, \quad v_\mu^+ \mapsto v_{l\mu}^+.
\]
Suppose $b$ is some $\imath$canonical basis element of $_{\A'}\dot{\mU}^\imath$, such that $y^\imath_{b,b'}\neq 0$. Then we deduce that $\ifr(b)\cdot v_{l\mu}^+\neq 0$. Hence $b\cdot v_\mu^+\neq 0$. There are only finitely many such $b$ by Theorem~\ref{thm:stab}.
\end{proof}
As consequences, we obtain well-defined $\A'$-algebra homomorphisms
\begin{equation*
\fr:{_{\A'}\widehat{\mU}}\rightarrow {_{\A'}\widehat{\U}}\quad \text{and} \quad \ifr:{_{\A'}\widehat{\mU}^\imath}\rightarrow {_{\A'}\widehat{\U}^\imath}.
\end{equation*}
\subsubsection{}
Recall \S\ref{sec:nocp} and \S\ref{sec:qFr}. Write ${_{\A'}\widehat{\U}^{(2)}}={_{\A'_\phi}\widehat{\U}^{(2)}}$ and ${_{\A'}\widehat{\mU}^{(2)}}={_{\A'_c}\widehat{\U}^{(2)}}$. Let ${_{\A'}\widehat{\U}}\widehat{\otimes}{_{\A'}\widehat{\mU}}$ be the $\A'$-module consisting of formal $\A'$-linear combinations
\[
\sum_{(b,b')\in\dot{\RB}\times\dot{\RB}} n_{b,b'}b\otimes b', \quad \text{for }n_{b,b'} \in \A'.
\]
This is moreover an $\A'$-algebra.
Thanks to Lemma \ref{le:fiFr} and Lemma \ref{le:fiFrs}, we have well-defined $\A'$-algebra homomorphisms
\[
id\otimes \ofr:{_{\A'}\widehat{\U}^{(2)}}\rightarrow {_{\A'}\widehat{\U}}\widehat{\otimes}{_{\A'}\widehat{\mU}} \quad \text{ and } \quad \fr\otimes id:{_{\A'}\widehat{\mU}^{(2)}}\rightarrow{_{\A'}\widehat{\U}}\widehat{\otimes}{_{\A'}\widehat{\mU}}.
\]
\begin{prop}
The following diagram commutes:
\begin{equation}\label{dia:frspl}
\begin{tikzcd}
_{\mathcal{A}'}\dot{\mathfrak{U}} \arrow[r,"\fr"] \arrow[d,"\widehat{\Delta}"'] & _{\mathcal{A}'}\dot{\mathrm{U}} \arrow[r,"\widehat{\Delta}"] & _{\mathcal{A}'}\widehat{\mathrm{U}}^{(2)} \arrow[d,"id\otimes\ofr"] \\
_{\mathcal{A}'}\widehat{\mathfrak{U}}^{(2)}
\arrow[rr, "\fr\otimes id"] & & _{\mathcal{A}'}\widehat{\mathrm{U}}\widehat{\otimes}{_{\A'}\widehat{\mU}}.
\end{tikzcd}
\end{equation}
\end{prop}
\begin{proof}
It suffices to check the commutative diagram over the field of fractions $\mathbf{F}$. Since $_{\mathbf{F}}\dot{\mU}$ is generated by $\one_\lambda$ ($\lambda\in X$), $\mathfrak{e}_{i}\one_\lambda$ ($i\in\I$, $\lambda\in X$) and $\mathfrak{f}_{i}\one_\lambda$ ($i\in\I$, $\lambda\in X$), it will suffice to check the identity
\begin{equation*}
(id\otimes\ofr)\circ\widehat{\Delta}\circ\fr(u)=(\fr\otimes id)\circ \widehat{\Delta}(u), \quad \text{for any such generator } u \in {}_{\mathbf{F}}\dot{\mU}.
\end{equation*}
By direct computations, one has
\[
(id\otimes\ofr)\circ\widehat{\Delta}\circ\fr(\one_\lambda)=\sum_{\lambda_1+\lambda_2=l\lambda}\one_{\lambda_1}\otimes \ofr(\one_{\lambda_2})=\sum_{\lambda_1'+\lambda_2'=\lambda}\one_{l\lambda_1'}\otimes\one_{\lambda_2'}=(\fr\otimes id)\circ \widehat{\Delta}(\one_\lambda)
\]
By \cite{Lu93}*{\S3.1.5}, for any $i\in\I$, $n\in\BN$, and $\lambda\in X$, we have
\begin{align*}
&\widehat{\Delta}(E_i^{(n)}\one_\lambda)=\sum_{\substack{t+s=n\\ \mu+\mu'=\lambda}}q_i^{(t+\langle \alpha_i^\vee,\mu\rangle)s}E_i^{(t)}\one_\mu\otimes E_i^{(s)}\one_{\mu''},\\ &\widehat{\Delta}(F_i^{(n)}\one_{\lambda})=\sum_{\substack{t+s=n\\\mu+\mu'=\lambda}}q_i^{(s-\langle \alpha_i^\vee,\mu'\rangle)t} F_i^{(t)}\one_\mu\otimes F_i^{(s)}\one_{\mu'}.
\end{align*}
Therefore
\begin{align*}
&\widehat{\Delta}\circ\fr(\mathfrak{e}_i\one_\lambda)=\widehat{\Delta}(\RE_i^{(l)}\one_{l\lambda})=\sum_{\substack{t+s=l\\ \mu+\mu'=l\lambda}}\bq_i^{(t+\langle \alpha_i^\vee,\mu\rangle)s}\RE_i^{(t)}\one_\mu\otimes \RE_i^{(s)}\one_{\mu'},\\
&\widehat{\Delta}\circ\fr(\mathfrak{f}_i\one_\lambda)=\widehat{\Delta}(\RF_i^{(l)}\one_{l\lambda})=\sum_{\substack{t+s=l\\\mu+\mu'=l\lambda}}\bq_i^{(s-\langle \alpha_i^\vee,\mu'\rangle)t} \RF_i^{(t)}\one_\mu\otimes \RF_i^{(s)}\one_{\mu'}
\end{align*}
Composing with $id\otimes \ofr$, the summands on the right hand sides are 0, unless $\{t,s\}=\{0,l\}$ and $\mu,\mu' \in l X$. Hence we obtain
\begin{align*}
(id\otimes \ofr)\circ\widehat{\Delta}\circ\fr(\mathfrak{e}_i\one_\lambda)=\sum_{\mu+\mu'=\lambda}(\one_{l\mu}\otimes\mathfrak{e}_i\one_{\mu'}+\RE_i^{(l)}\one_{l\mu}\otimes\one_{\mu'})=(\fr\otimes id)\circ \widehat{\Delta}(\mathfrak{e}_i\one_\lambda),\\
(id\otimes \ofr)\circ\widehat{\Delta}\circ\fr(\mathfrak{f}_i\one_\lambda)=\sum_{\mu+\mu'=\lambda}(\one_{l\mu}\otimes\mathfrak{f}_i\one_{\mu'}+\RF_i^{(l)}\one_{l\mu}\otimes\one_{\mu'})=(\fr\otimes id)\circ \widehat{\Delta}(\mathfrak{f}_i\one_\lambda).
\end{align*}
This completes the proof.
\end{proof}
\subsubsection{}
For any $i\in\I$ with $\tau i=i$, and any $n\in\BN$, recall \S \ref{sec:baid} the definition of the elements $B_i^{(n)}$ in $\U^\imath$. Also recall the elements $'\RB_{i,\zeta}^{(n)}$ ($\zeta\in X_\imath$, $n\in\BN$) in $_{\A_2'}\dot{\U}^\imath$ in \S \ref{sec:iFr}.
\begin{lemma}\label{le:conew}
For any $i\in\I$ with $\tau i=i$, and any $n\in\BN$, we have
\[
\Delta(B_i^{(n)})=\sum_{r=0}^n B_i^{(n-r)}\otimes M_{n,r}
\]
where $M_{n,r}\in \U$, such that for any $\lambda\in X$, we have
\[
M_{n,r}\one_\lambda=\sum_{\substack{a+a'+2c=r\\c\geqslant 0}}q_i^{s_i(a,a',\lambda,n,c)}\LR{a'-a-\lambda}{2c}_iE_i^{(a)}F_i^{(a')} \one_\lambda
\]
where ($\lambda_i=\langle\coroot_i,\lambda\rangle$)
\[
s_i(a,a',\lambda,n,c)=(a-a'+n+\lambda_i)a'-n(\lambda_i+a)+(1+a-a'+\lambda_i)c.
\]
\end{lemma}
\begin{proof}
In the algebra $\U^\imath$, by the comultiplication formulas \cite{CW22}*{Theorem 4.2 \& Theorem 5.1}, we have
\[
\Delta(B_i^{(n)})=\Delta(B_{i,\overline{n+1}}^{(n)})=\sum_{r=0}^n B_{i,\overline{n+1}}^{(n-r)}\otimes S_{n,r},
\]
with
\[
S_{n,r}=\sum_{\substack{a+a'+2c=r\\c\geqslant 0}}q_i^{\binom{2c+1}{2}+(a+a')(r-n)-aa'}\Check{E_i}^{(a)}\qbinom{h;-\lfloor \frac{r-1}{2}\rfloor}{c}_iK_i^{r-n}F_i^{(a')},
\]
where
\[
\check{E}_i^{(a)}=\frac{(q_i^{-1}E_iK_i^{-1})^a}{[a]_i!}, \qquad \qbinom{h;a}{n}_i=\prod_{s=1}^n\frac{q_i^{4a+4s-4}K_i^{-2}-1}{q_i^{4s}-1}.
\]
In particular, we have
\[
\qbinom{h;a}{n}_i\one_\lambda =q_i^{(2a-2-\lambda_i)n}\LR{2a+2n-\lambda_i-2}{2n}_i\one_\lambda, \quad \text{where }\lambda_i=\langle \coroot_i,\lambda\rangle.
\]
Then it is direct to compute
\[
S_{n,r}\one_\lambda=
\begin{cases}
\displaystyle\sum_{\substack{a+a'+2c=r\\c\geqslant 0}}q_i^{s_i(a,a',\lambda,n,c)}\LR{a'-a-\lambda_i}{2c}_iE_i^{(a)}F_i^{(a')}\one_\lambda, &\text{if $r$ is even};\\
\displaystyle\sum_{\substack{a+a'+2c=r\\c\geqslant 0}}q_i^{s_i(a,a',\lambda,n,c)-c}\LR{a'-a-\lambda_i-1}{2c}_iE_i^{(a)}F_i^{(a')}\one_\lambda, &\text{if $r$ is odd}.
\end{cases}
\]
Hence by Lemma \ref{lem:ff} , we have
\begin{align*}
&\Delta(B_{i}^{(n)})= \sum_{\substack{0\leqslant r\leqslant n\\r\,\text{even}}}\left( {B_{i}^{(n-r)}}\otimes \sum_{\substack{a+a'+2c=r\\c\geqslant 0}}q_i^{s_i(a,a',\lambda,n,c)}\LR{a'-a-\lambda_i}{2c}_iE_i^{(a)}F_i^{(a')}\one_\lambda\right)+ \\
& \sum_{\substack{0\leqslant r\leqslant n\\r\,\text{odd}}} \left(\left(\sum_{t\geqslant 0}\LR{1}{2t}_i{B_{i}}^{(n-r-2t)}\right)\otimes \sum_{\substack{a+a'+2c=r\\c\geqslant 0}}q_i^{s_i(a,a',\lambda,n,c)-c}\LR{a'-a- \lambda_i-1}{2c}_iE_i^{(a)}F_i^{(a')}\one_\lambda\right).
\end{align*}
It remains to compute the coefficient of $B_{i}^{(n-r)}\otimes E_i^{(a)}F_i^{(a')}\one_\lambda$ when both $r$ and $a+a'$ are odd. The coefficient equals
\begin{align*}
&\sum_{c+t=(r-a-a')/2}q_i^{s_i(a,a',\lambda,n,c)-c}\LR{1}{2t}_i\LR{a'-a-\lambda-1}{2c}_i\\
=&q_i^{(a-a'+n+\lambda)a'-n(\lambda+a)}\sum_{c+t=(r-a-a')/2}q_i^{(a-a'+\lambda)c}\LR{1}{2t}_i\LR{a'-a-\lambda-1}{2c}_i\\
=&q_i^{s_i(a,a',\lambda,n,(r-a-a')/2)}\LR{a'-a-\lambda_i}{r-a-a'}_i. \qquad \text{(By Lemma~\ref{lem:DoubleInA2}.)}
\end{align*}
This completes the proof.
\end{proof}
Following the notations in \S\ref{sec:nocp} and \S\ref{sec:Ui1}, we write ${_{\A'}\widehat{\U}^{\imath,1}}={_{\A'_\phi}\widehat{\U}^{\imath,1}}$ and ${_{\A'}\widehat{\mU}^{\imath,1}}={_{\A'_c}\widehat{\U}^{\imath,1}}$. Let ${_{\A'}\widehat{\U}^\imath}\widehat{\otimes}{_{\A'}\widehat{\mU}}$ be the $\A'$-module consisting of formal $\A'$-linear combinations
\[
\sum_{(b,b')\in\dot{\RB}_\phi\times\dot{\RB}_c}n_{b,b'}b\otimes b', \quad \text{for }n_{b,b'} \in \A'
\]
This is moreover an $\A'$-algebra. By Lemma \ref{le:fiFr} and Lemma \ref{le:fiFrs} we have well-defined $\A'$-algebra homomorphisms
\[
id\otimes \ofr:{_{\A'}\widehat{\U}^{\imath,1}}\rightarrow {_{\A'}\widehat{\U}^\imath}\widehat{\otimes}{_{\A'}\widehat{\mU}} \quad \text{ and } \quad \ifr\otimes id:{_{\A'}\widehat{\mU}^{\imath,1}}\rightarrow{_{\A'}\widehat{\U}^\imath}\widehat{\otimes}{_{\A'}\widehat{\mU}}.
\]
\begin{prop}\label{prop:cosp}
The following diagram commutes:
\begin{equation}\label{dia:ifrspl}
\begin{tikzcd}
_{\mathcal{A}'}\dot{\mathfrak{U}}^\imath \arrow[r,"\ifr"] \arrow[d,"\widehat{\Delta}"'] & _{\mathcal{A}'}\dot{\mathrm{U}}^\imath \arrow[r,"\widehat{\Delta}"] & _{\mathcal{A}'}\widehat{\mathrm{U}}^{\imath,1} \arrow[d,"id\otimes\ofr"] \\
_{\mathcal{A}'}\widehat{\mathfrak{U}}^{\imath,1}
\arrow[rr, "\ifr\otimes id"] & & {_{\A'}\widehat{\U}^\imath}\widehat{\otimes}{_{\A'}\widehat{\mU}}
\end{tikzcd}
\end{equation}
\end{prop}
\begin{proof}
It suffices to check
\[
(id\otimes \ofr) \circ \widehat{\Delta}\circ \ifr(u)=(\ifr\otimes id)\circ \widehat{\Delta}(u)
\]
for $u=\one_\zeta$ ($\zeta\in X_\imath$), and $u=\mathfrak{b}_{i,\zeta}$ ($i\in\I$, $\zeta\in X_\imath$). The first case is straightforward.
Assume $u=\mathfrak{b}_{i,\zeta}$. Then by definitions, we have
\[
(\ifr\otimes id)\circ \widehat{\Delta}(\mathfrak{b}_{i,\zeta})=\sum_{\substack{\zeta'+\overline{\lambda}=\zeta\\\zeta'\in X_\imath,\lambda\in X}}\left('\RB_{i,l\zeta'}^{(l)}\otimes\one_\lambda+\one_{l\zeta'}\otimes(\fe_i\one_\lambda+\ff_i\one_\lambda)\right).
\]
We then compute $(id\otimes \ofr) \circ \widehat{\Delta}\circ \ifr(\fb_{i,\zeta})$, for any $i\in\I$ and $\zeta\in X_\imath$. We divide into two cases.
Case (a): $\tau i=i$.
Recall Theorem~\ref{thm:ClaFr} that
\[
\ofr(\RE_i^{(a)}\RF_i^{(a')}\one_{l\lambda}) =
\begin{cases}
\fe_i^{(a/l)}\ff_i^{(a'/l)}\one_\lambda, &\text{if } l\mid a \text{ and }l\mid a';\\
0, &\text{otherwise.}
\end{cases}
\]
Therefore by the Lemma \ref{le:conew}, we have
\begin{align*}
&(id\otimes \ofr) \circ \widehat{\Delta}\circ \ifr(\mathfrak{b}_{i,\zeta}) = (id\otimes \ofr) \circ \widehat{\Delta}('\B_{i,l\zeta}^{(l)}) \\
=\, &\bq_i^{s_i(0,0,l\lambda,l,0)}{'\RB_{i,\zeta'}^{(l)}}\otimes \one_{\lambda}+\bq_i^{s_i(l,0,l\lambda,l,0)}\one_{\zeta'}\otimes \fe_i\one_\lambda+\bq_i^{s_i(0,l,l\lambda,l,0)}\one_{\zeta'}\otimes \ff_i\one_\lambda.
\end{align*}
It is direct to check that $s_i(0,0,l\lambda,l,0)$, $s_i(l,0,l\lambda,l,0)$, and $s_i(0,l,l\lambda,l,0)$ are all divisible by $l$. We deduce that
\begin{align*}
(id\otimes \ofr) \circ \widehat{\Delta}\circ \ifr(\mathfrak{b}_{i,\zeta}) &=\sum_{\substack{\zeta'+\overline{\lambda}=\zeta\\\zeta'\in X_\imath,\lambda\in X}}\left('\RB_{i,l\zeta'}^{(l)}\otimes\one_\lambda+\one_{l\zeta'}\otimes(\fe_i\one_\lambda+\ff_i\one_\lambda)\right) \\
&=(\ifr\otimes id)\circ \widehat{\Delta}(\mathfrak{b}_{i,\zeta}).
\end{align*}
Case (b): $\tau i\neq i$.
By direct computation in $\mathrm{U}^\imath$, we have
\[
\Delta(B_i)=B_i\otimes \Tilde{K}_i^{-1}+1\otimes F_i+\Tilde{K}_i^{-1}\Tilde{K}_{\tau i}\otimes \varsigma_i E_{\tau i}\Tilde{K}_i^{-1}.
\]
Therefore we have
\begin{equation*}
\begin{split}
&\Delta(B_i^{(l)})=(B_i\otimes \Tilde{K}_i^{-1}+1\otimes F_i+\Tilde{K}_i^{-1}\Tilde{K}_{\tau i}\otimes \varsigma_i E_{\tau i}\Tilde{K}_i^{-1})^l/[l]_i^!\\
&=B_i^{(l)}\otimes \Tilde{K}_i^{-l}+1\otimes F_i^{(l)}+(\Tilde{K}_i^{-l}\Tilde{K}_{\tau i}^l)\otimes \varsigma_i^l(E_{\tau i}\Tilde{K}_i^{-1})^{l}/[l]_i^!+\text{other terms},
\end{split}
\end{equation*}
where the "other terms" stands for the terms, such that the degree of the second factor does not belong to $l\BZ[\I]$ (recall that $\U$ is naturally $\BZ[\I]$-graded). Hence they are in the kernel of $id \otimes \ofr$.
Therefore
\begin{align*}
&(id\otimes \ofr) \circ \widehat{\Delta}\circ \ifr(\mathfrak{b}_{i,\zeta})= (id\otimes \ofr) \circ \widehat{\Delta}(\B_{i,l\zeta}^{(l)})\\
=\, &(id\otimes \ofr) \Big( \sum_{\substack{\zeta'+\overline{\lambda}=\zeta\\\zeta'\in X_\imath,\lambda\in X}}\big(\B_{i,l\zeta'}^{(l)}\otimes \one_{l\lambda}+\one_{l\zeta'}\otimes \RF_i^{(l)}\one_{l\lambda}+\one_{l\zeta'}\otimes \RE_{\tau i}^{(l)}\one_{l\lambda}+\text{other terms}\big) \Big)\\
=&\sum_{\substack{\zeta'+\overline{\lambda}=\zeta\\\zeta'\in X_\imath,\lambda\in X}}\left('\RB_{i,l\zeta'}^{(l)}\otimes\one_\lambda+\one_{l\zeta'}\otimes(\fe_i\one_\lambda+\ff_i\one_\lambda)\right)\\
=\, &(\ifr\otimes id)\circ \widehat{\Delta}(\mathfrak{b}_{i,\zeta}).
\end{align*}
This finishes the proof.
\end{proof}
\subsection{Frobenius splittings of algebraic groups}\label{sec:Frbach}
We assume $l=p$ is an odd prime number, which is relatively prime to all the root lengths. We follow the notations in \S \ref{sec:oFrbach}.
\subsubsection{}
Tensoring the $\A'$-algebra homomorphisms $\fr: {_{\A'}\dot{\mU}}\longrightarrow{_{\A'}\dot{\U}}$ and $\ifr:{_{\A'}\dot{\mU}^\imath}\longrightarrow{_{\A'}\dot{\U}^\imath}$ with the $\A'$-algebra $\BF_p$,
we obtain $\BF_p$-algebra homomorphisms
\[
\fr:{{}_{\BF_p}\!\dot{\U}}\longrightarrow{{}_{\BF_p}\!\dot{\U}}\quad \text{ and } \quad \ifr:{{}_{\BF_p}\!\dot{\U}^\imath}\longrightarrow{{}_{\BF_p}\!\dot{\U}^\imath}.
\]
Thanks to Lemma \ref{le:fiFrs}, we also have $\BF_p$-algebra homomrophisms
\[
\fr:{{}_{\BF_p}\!\widehat{\U}}\longrightarrow{{}_{\BF_p}\!\widehat{\U}}\quad \text{ and } \quad \ifr:{{}_{\BF_p}\!\widehat{\U}^\imath}\longrightarrow{{}_{\BF_p}\!\widehat{\U}^\imath}.
\]
Recall the ${\BF_p}$-Hopf algebras $\RO_{\BF_p}$ in \S\ref{sec:nocp} and $\RO_{\BF_p}^\imath$ in \S\ref{sec:pro}. We hence have the induced well-defined $\BF_p$-linear maps
\[
{ \fr}^{,*}:\RO_{\BF_p}\longrightarrow \RO_{\BF_p}\quad \text{ and } \quad { \ifr}^{,*}:\RO_{\BF_p}^\imath\longrightarrow\RO_{\BF_p}^\imath.
\]
\begin{prop}\label{prop:splp}
(1) The map $\fr^{,*}$ is a Frobenius splitting of $\RO_{\BF_p}$, that is, we have $\fr^{,*}(fg^p)={ \fr^{,*}}(f)\cdot g$ for any $f$, $g$ in $\RO_{\BF_p}$, and $\fr^{,*}(1)=1$.
(2) The map $\ifr^{,*}$ is a Frobenius splitting of $\RO_{\BF_p}^\imath$. That is, we have $\ifr^{,*}(fg^p)={\ifr^{,*}}(f)\cdot g$ for any $f$, $g$ in $\RO_{\BF_p}^\imath$, and $\ifr^{,*}(1)=1$.
\end{prop}
\begin{proof}
We show part (1). It follows from the commuting diagram \eqref{dia:frspl} that the following diagram commutes
\begin{equation*}
\begin{tikzcd}
{}_{\BF_p}\!\dot{\U} \arrow[r,"\fr"] \arrow[d,"\widehat{\Delta}"'] & {}_{\BF_p}\!\widehat{\U} \arrow[r,"\widehat{\Delta}"] & {}_{\BF_p}\!\widehat{\U}^{(2)} \arrow[d,"id\otimes\ofr"] \\
{}_{\BF_p}\!\widehat{\U}^{(2)}
\arrow[rr, "\fr\otimes id"] & & {}_{\BF_p}\!\widehat{\U}^{(2)}
\end{tikzcd}
\end{equation*}
Taking ${\BF_p}$-linear duals, we obtain the commutative diagram
\begin{equation}\label{dia:opsplit}
\begin{tikzcd}
\RO_{\BF_p} & \RO_{\BF_p} \arrow[l,"\fr^{,*}"'] & \RO_{\BF_p}\otimes\RO_{\BF_p} \arrow[l,"m"'] \\ \RO_{\BF_p} \otimes\RO_{\BF_p} \arrow[u,"m"] & & \RO_{\BF_p}\otimes\RO_{\BF_p} \arrow[ll,"\fr^{,*}\otimes id"'] \arrow[u,"id\otimes\ofr^{*}"']
\end{tikzcd}
\end{equation}
We have seen that $\ofr^*$ is the $p$-th power map. The diagram \eqref{dia:opsplit} implies that $\fr^{,*}(fg^p)=\fr^{,*}(f)g$, for any $f,g\in \RO_{\BF_p}$. Since $\ofr\circ\fr=id$, we deduce that $\fr^{,*}(f^p)=f$, for any $f\in \RO_{\BF_p}$. Hence $\fr^{,*}$ is a Frobenius splitting of $\RO_{\BF_p}$.
Part (2) follows similarly using the commutative diagram \eqref{dia:ifrspl}.
\end{proof}
\subsubsection{}\label{sec:kFr}
Let $k$ be an algebraically closed field with characteristic $p$. Recall $p$ is an odd prime number which is relatively prime to all the root lengths.
The algebra $\RO_k\cong k\otimes_{\BF_p}\RO_{\BF_p}$ is the coordinate ring of the algebraic group $G_k$ associated to the root datum by \S\ref{sec:nocp}. The algebra $\RO_k^\imath\cong k\otimes_{\BF_p}\RO_{\BF_p}^\imath$ is coordinate ring of the symmetric subgroup $G_k^\theta$ associated to the $\imath$-root datum by Theorem~\ref{thm:Oik}.
Let $(\;\cdot\;)^p:k\rightarrow k$ be the $\BF_p$-linear map sending any $t$ in $k$ to $t^p$. We define
\begin{align*}
_k\ofr^*& =(\;\cdot\;)^p\otimes_{\BF_p} { \ofr^*}: \RO_k\longrightarrow\RO_k\\ _k\iofr^*&=(\;\cdot\;)^p\otimes_{\BF_p}{\iofr^*}:\RO^\imath_k\longrightarrow\RO_k^\imath.
\end{align*}
Here we add a subscript $_k({\cdot})$ to indicate the non-trivial automorphism $(\;\cdot\;)^p$ on $k$.
Let $(\;\cdot\;)^{1/p}:k\rightarrow k$ be the $\BF_p$-linear map sending any $t$ in $k$ to $t^{1/p}$, the unique $p$-th root of $t$. We define
\begin{align*}
_k\fr^{,*}& =(\;\cdot\;)^{1/p}\otimes_{\BF_p}{\fr^{,*}}:\RO_k\longrightarrow\RO_k,\\
_k\ifr^{,*}& =(\;\cdot\;)^{1/p}\otimes_{\BF_p}{\ifr^{,*}}:\RO^\imath_k\longrightarrow\RO_k^\imath.
\end{align*}
We again add a subscript $_k({\cdot})$ here to indicate the non-trivial automorphism $(\;\cdot\;)^{1/p}$ on $k$.
The following theorem is immediate by Proposition~\ref{prop:pthFr} and Proposition \ref{prop:splp}.
\begin{thm}\label{thm:splitO}
(1) We have
\begin{align*}
{_k\ofr^*}(f)&=f^p, \quad \text{for any } f\in \RO_k,\\
{_k\iofr^*}(f)&=f^p, \quad \text{for any } f\in \RO_k^\imath.
\end{align*}
(2) The maps $_k\fr^{,*}$ and $_k\ifr^{,*}$ provide explicit Frobenius splittings for algebraic groups $G_k$ and $G^\imath_k$, respectively.
\end{thm}
\begin{remark}
(1) The existence of such splittings is guaranteed by \cite{BK05}*{Proposition~1.1.6}. Our construction is explicit and uniform for all positive characteristics (not $2$ for $G^\imath_k$ of course).
(2) By \cite{BK05}*{Proposition~1.1.6}, there exists a splitting of $G_k$ that compatibly splits $G_k^\imath$. We expect our method can be used to construct explicitly such an Frobenius splitting. This will be addressed in future works.
\end{remark}
\section{Frobenius splittings of flag varieties}\label{sec:Frflag}
In this section, we assume given an $\imath$root datum $(\I,Y,X,A,(\alpha_i)_{i\in\I},(\coroot_i)_{i\in\I}),\tau,\theta)$ of finite (quasi-split) type. Let $p$ be an odd prime number, and relatively prime to all the root lengths (so $p>3$ if the underlying root datum has a $G_2$ factor). We fix an algebraically closed field $k$ of characteristic $p$.
Let $(G_k, \theta_k,B_k)$ be an anisotropic triple assocaited to the $\imath$root datum following Proposition \ref{prop:cft}. We identify this triple as we constructed in \S \ref{sec:class} (using quantum groups). Let $\{x_i,y_i;i\in\I\}$ be the corresponding anisotropic pinning, and $\mathcal{B}_k=G_k/B_k$ be the flag variety.
\subsection{The algebra $R_{\BF_p}$ and $R_k$}
We follow the notations in \S \ref{sec:oFrbach} and \S \ref{sec:Frbach}.
For any $\lambda\in X^+$, we write
\[
\baV(\lambda) =\BF_p\otimes_{\A}{_\A L(\lambda)}.
\]
It is a ${}_{\BF_p}\!\dot{\U}$-module with maximal vector $v_\lambda^+=1\otimes v_\lambda^+$ (again we abuse notations here).
Set
\[
R_{\BF_p}=\bigoplus_{\lambda\in X^+}\baV(\lambda)^*.
\]
It has a $\BF_p$-algebra structure defined in the following way. For any $\lambda_1,\lambda_2\in X^+$, there is a unique ${}_{\BF_p}\!\dot{\U}$-module homomorphism
\[
\baV(\lambda_1+\lambda_2)\longrightarrow \baV(\lambda_1)\otimes \baV(\lambda_2)
\]
sending $v_{\lambda_1+\lambda_2}^+$ to $v_{\lambda_1}^+\otimes v_{\lambda_2}^+$.
Taking dual of this map, we get $\BF_p$-linear maps
\[
\baV(\lambda_1)^*\otimes \baV(\lambda_2)^*\longrightarrow \baV(\lambda_1+\lambda_2)^*,
\]
which defines the multiplication in $R_{\BF_p}$. The unit of $R_{\BF_p}$ is the linear dual of $\baV(0)=\BF_p\cdot v_0^+$, sending $v_0^+$ to 1.
Set $R_k = k\otimes_{\BF_p} R_{\BF_p} \cong \bigoplus_{\lambda\in X^+}V_k(\lambda)^*$.
For any $\lambda\in X^+$, recall the construction for the $k$-algebra $R_{\mathcal{L}_\lambda}$ in \S \ref{sec:algFr}. By definitions, $R_{\mathcal{L}_\lambda}$ is a $k$-subalgebra of $R_k$.
\subsection{The $p$-th power maps}
\subsubsection{}
Retain the notations in \S \ref{sec:oFrbach} and \S \ref{sec:Frbach}.
For any $\lambda \in X^+$, recall \S\ref{sec:qFr} the modules ${_{\A'}\mathfrak{L}}(\mu)$ and ${_{\A'}L}(\mu)$.
By the arguments in \S \ref{sec:oFrbach}, we have canonical isomorphisms (as ${}_{\BF_p}\!\dot{\U}$-modules)
\[
\BF_p\otimes_{\A'}{{_{\A'}\mathfrak{L}}(\mu)}\cong \BF_p\otimes_{\A'}{{_{\A'}L}(\mu)}\cong \baV(\mu).
\]
Hence by Proposition \ref{prop:ga}, we have a ${}_{\BF_p}\!\dot{\U}$-module homomorphism
\[
\gamma_\mu: \baV(p\mu)\longrightarrow \baV(\mu)^{\ofr}, \quad v_{p\mu}^+ \mapsto v_\mu^+.
\]
Define $\BF_p$-linear map $\Gamma: R_{\BF_p}\rightarrow R_{\BF_p}$, such that $\Gamma(f)=\gamma_\mu^*(f)$, for $f\in \baV(\mu)^*$. Here $\gamma_\mu^*: (\baV(\mu)^{\ofr})^* \rightarrow \baV(p\mu)^*$ denotes the ${\BF_p}$-linear dual of $\gamma_\mu$.
Recall \S\ref{sec:kFr} the map $(\;\cdot \;)^p:k\rightarrow k$, mapping $ t$ to $t^p$. We further define
\[
\Gamma_k =(\;\cdot\;)^p\otimes \Gamma: R_k\longrightarrow R_k,
\]
\begin{proposition}[\cite{KL1}*{Theorem~1}] \label{prop:Ga}
(1) The map $\Gamma: R_{\BF_p}\rightarrow R_{\BF_p}$ is the $p$-th power map, that is, $\Gamma(f)=f^p$, for any $f\in R_{\BF_p}$.
(2) The map $\Gamma_k: R_{k}\rightarrow R_{k}$ is the $p$-th power map, that is, $\Gamma(f)=f^p$, for any $f\in R_{k}$.
\end{proposition}
\begin{proof}
It suffices to show part (1). Let $f\in \baV(\mu)^*$ for some $\mu\in X^+$. By definition, the equality $\Gamma(f)=f^p$ is equivalent to the commutativity of the following diagram:
\begin{equation}\label{dia:Fr}
\begin{tikzcd}
\baV(p\mu) \arrow[r,"\gamma_\mu"] \arrow[d] & \baV(\mu)^{\ofr} \arrow[d,"f"]\\
\baV(\mu)^{\otimes p} \arrow[r,"f^{\otimes p}"] & \mathbb{F}_p.
\end{tikzcd}
\end{equation}
Note that $\baV(\mu)^{\ofr}= \baV(\mu)$ as $\BF_p$-vector spaces.
Let $I_\mu\subset \baV(\mu)^{\otimes p}$ be the subspace spanned by elements $v_1\otimes \cdots \otimes V_{\BF_p}-v_{\sigma(1)}\otimes \cdots \otimes v_{\sigma(p)}$, for all $v_i\in V_{\BF_p}(\mu)$ and $\sigma\in S_n$. Then $I_\mu$ is moreover a ${}_{\BF_p}\!\dot{\U}$-submodule. Set $S^p\baV(\mu)=\baV(\mu)^{\otimes p}/I_\mu$ be the quotient module.
Note that the $\BF_p$-linear map $f^{\otimes p}$ factors through $S^p\baV(\mu)$, denoted by $f^p: S^p\baV(\mu)\longrightarrow \mathbb{F}_p$. Define a $\BF_p$-linear map
\[
\vartheta: \baV(\mu)\longrightarrow \baV(\mu)^{\otimes p} \longrightarrow S^p\baV(\mu), \quad v \mapsto v^{\otimes p} \mapsto v^p=\overline{v^{\otimes p}}.
\]
Since $(f(v) )^p = f(v)$ for any $v \in \baV(\mu)$, we have the following commuting diagram
\begin{equation*}
\begin{tikzcd}
\big(\baV(\mu)^{\ofr}=\big)\baV(\mu) \ar[r, "\vartheta"] \ar[dr, "f"]& S^p\baV(\mu) \ar[d, "f^p"]\\
& \BF_p
\end{tikzcd}
\end{equation*}
Hence in order to check the commutativity of the diagram \eqref{dia:Fr}, it will suffice to show the commutativity of the following diagram:
\begin{equation}\label{eq:prop:Ga}
\begin{tikzcd}
\baV(p\mu) \arrow[r,"\gamma_\mu"] \arrow[d] & \baV(\mu)^{\ofr} \arrow[d,"\vartheta"] \\
\baV(\mu)^{\otimes p} \arrow[r] & S^p\baV(\mu).
\end{tikzcd}
\end{equation}
We first show that $\vartheta: \baV(\mu)^{\ofr}\rightarrow S^p\baV(\mu)$ is ${}_{\BF_p}\!\dot{\U}$-equivariant, that is,
\begin{equation}\label{id:the}
\vartheta({ \ofr(u)}\cdot v)=u\cdot \vartheta(v), \quad \text{for any } u\in{{}_{\BF_p}\!\dot{\U}}, v\in \baV(\mu).
\end{equation}
We may assume $v$ is homogeneous of weight $\gamma$, and $u$ equals $e_i^{(a)}\one_\lambda$, or $f_i^{(a)}\one_\lambda$, for some $a\in\BN$ and $\lambda\in X$. If $\lambda\neq p\gamma$, then it is clear that both sides of \eqref{id:the} are 0. If $ \lambda = p \gamma$, then entirely similar to the proof of Proposition~\ref{prop:pthFr}, we have
\[
e_i^{(a)}\one_{p\gamma}\cdot v^p=
\begin{cases} (e_i^{(a/p)}v)^p, &\text{if } p \mid a;\\
0, &\text{otherwise}.
\end{cases}
\]
Hence $ \vartheta (\ofr(e_i^{(a)}\one_{p\gamma}) \cdot v)= e_i^{(a)}\one_{p\gamma}\cdot v^p $, whence the identity \eqref{id:the}. Similar computation holds when $u=f_i^{(a)}\one_{p\gamma}$.
Now all the maps in the diagram \eqref{eq:prop:Ga} are ${{}_{\BF_p}\!\dot{\U}}$-equivariant. Since $\baV(p\mu)$ is generated by $v^+_{p\mu}$ as ${{}_{\BF_p}\!\dot{\U}}$-module and $v^+_{p\mu} \in \baV(p\mu)$ maps to the same element via the two maps, we conclude the diagram is commutative.
\end{proof}
\subsection{The splitting $\Psi_k$}\label{sec:Phi}
\subsubsection{}
For any $\mu\in X^+$, recall \S \ref{se:L} the $\A'$-linear maps $\psi_\mu:{_{\A'}\mathfrak{L}}(\mu)\rightarrow ({_{\A'}L}(p\mu))^{\ifr}$. Applying base change to $\BF_p$, we obtain the ${}_{\BF_p}\!\dot{\U}^\imath$-module homomorphisms:
\[
\psi_\mu: \baV(\mu)\longrightarrow \baV(p\mu)^{\ifr}, \quad v_\mu^+ \mapsto v_{p\mu}^+.
\]
Define the $\mathbb{F}_p$-linear map $\Psi: R_{\BF_p} \longrightarrow R_{\BF_p} $ by
\[
\Psi(f)=
\begin{cases}
\psi_\mu^*(f), &\text{if } f\in \baV(p\mu)^*;\\
0, &\text{if }f\in \baV(\mu)^* \text{ with }\mu\not\in pX.
\end{cases}
\]
\begin{prop}\label{prop:Psi}
The map $\Psi$ is a Frobenius splitting for the algebra $R_{\BF_p}$. Namely, it is an additive map satisfying:
(a) $\Psi(f^pg)=f\Psi(g),$ for $f,g\in R$;
(b) $\Psi(1)=1$.
\end{prop}
\begin{proof}
Condition (b) follows easily from the definition. Now we check the condition (a).
Let $f\in \baV(\lambda)^*$ and $g\in \baV(p\mu)^*$. By definition, the equality $\Psi(f^pg)=f\Psi(g)$
holds if and only if the following diagram commutes
\begin{equation*}
\begin{tikzcd}
\baV(\mu+\lambda) \arrow[r] \arrow[d,"\psi_{\mu+\lambda}"'] & \baV(\mu)\otimes \baV(\lambda) \arrow[r,"\psi_\mu\otimes id"] & \baV(p\mu)^{\ifr}\otimes \baV(\lambda) \arrow[d,"g\otimes f"] \\
\baV(p\mu+p\lambda)^{\ifr} \arrow[r] & \big(\baV(p\mu)\otimes \baV(\lambda)^{\otimes p}\big)^{\ifr} \arrow[r,"g\otimes f^{\otimes p}"] & \mathbb{F}_p.
\end{tikzcd}
\end{equation*}
Note that $\baV(p\mu)^{\ifr} = \baV(p\mu)$ as $\BF_p$-vector spaces.
Following the proof of Proposition \ref{prop:Ga}, we define $S^p\baV(\lambda)$, and the $\BF_p$-linear map $\vartheta: \baV(\lambda)\rightarrow S^p\baV(\lambda)$ sending $v$ to $v^p=\overline{v^{\otimes p}}$. Note that the map $g\otimes f^{\otimes p}$ factors through $\baV(p\mu)\otimes S^p\baV(\lambda)$. Moreover, we have the following commuting diagram by Proposition~\ref{prop:Ga}:
\begin{equation*}
\begin{tikzcd}
\baV(p\mu)\otimes S^p\baV(\lambda) \arrow[rd,"g\otimes f^p"] & \baV(p\mu)\otimes \baV(\lambda) \arrow[l,"id \otimes \vartheta"'] \arrow[d,"g\otimes f"] \\ \baV(p\mu)\otimes \baV(\lambda)^{\otimes p} \arrow[u] \arrow[r,"g\otimes f^{\otimes p}"'] & \mathbb{F}_p.
\end{tikzcd}
\end{equation*}
Hence it will suffice to check the following diagram commutes:
\begin{equation}\label{dia:v}
\begin{tikzcd}
\baV(\mu+\lambda) \arrow[r] \arrow[d,"\psi_{\mu+\lambda}"'] & \baV(\mu)\otimes \baV(\lambda) \arrow[r,"\psi_\mu\otimes id"] & \baV(p\mu)^{\ifr}\otimes \baV(\lambda) \arrow[d,"id\otimes \vartheta"] \\
\baV(p\mu+p\lambda)^{\ifr} \arrow[r] & (\baV(p\mu)\otimes \baV(\lambda)^{\otimes p})^{\ifr} \arrow[r] & (\baV(p\mu)\otimes S^p\baV(\lambda))^{\ifr}.
\end{tikzcd}
\end{equation}
Note that each space of the diagram admits a ${}_{\BF_p}\!\dot{\U}^\imath$-action, ${}_{\BF_p}\!\widehat{\U}^\imath$-action, as well as a ${}_{\BF_p}\!\widehat{\U}$-action.
We claim that each morphism is ${}_{\BF_p}\!\widehat{\U}^\imath$-equivariant. We will only check this claim for the map
\[
id\otimes\vartheta: \baV(p\mu)^{ \ifr}\otimes \baV(\lambda)\longrightarrow (\baV(p\mu)\otimes S^pV(\lambda))^{ \ifr}.
\]
The checkings for other maps are trivial.
Let $v\in \baV(p\mu)$, $w\in \baV(\lambda)$ and $u\in{{}_{\BF_p}\!\widehat{\U}^\imath}$. We show
\begin{equation}\label{id:id}
(id\otimes \vartheta) ({\ifr} \otimes id \circ \widehat{\Delta}(u) \cdot( v\otimes w))={\ifr}(u)\cdot (v\otimes w^p).
\end{equation}
It follows from Proposition~\ref{prop:cosp} that
\[
\big({ \ifr\otimes id}\big)\circ\widehat{\Delta}(u)=\big(id\otimes{ \ofr}\big)\circ \widehat{\Delta}\circ{ \ifr}(u).
\]
Recall \eqref{eq:prop:Ga} that
\[
\vartheta \big(( \ofr (x)\cdot v) \big)= x \cdot\vartheta (v), \text{ for } v\in \baV(\lambda)^{ \ofr}, x\in {{}_{\BF_p}\!\dot{\U}}.
\]
Hence we have
\begin{align*}
&(id\otimes \vartheta) ({\ifr} \otimes id \circ \widehat{\Delta}(u) \cdot( v\otimes w))\\
&=\big(id\otimes \vartheta\big)\big((id\otimes {\ofr})\circ \widehat{\Delta}\circ {\ifr(u)}\cdot (v\otimes w)\big)\\
&=\big(\widehat{\Delta}\circ {\ifr(u)}\big)\cdot \big(id\otimes \vartheta\big)\big(v\otimes w \big)\\
&={\ifr(u)}\cdot (v\otimes w ^p).
\end{align*}
This proves the identity \eqref{id:id}.
The highest weight vector $v_{\mu+\lambda}^+$ maps to the same vector $v_{p\mu}^+\otimes (v_\lambda^{+})^p$ via both compositions by the following computation
\begin{equation*}
\begin{tikzcd}
v_{\mu+\lambda}^+ \arrow[r,mapsto] \arrow[d,mapsto] & v_\mu^+\otimes v_\lambda^+ \arrow[r,mapsto] & v_{p\mu}^+\otimes v_\lambda^+ \arrow[d,mapsto] \\ v_{p\mu+p\lambda}^+ \arrow[r,mapsto] & v_{p\mu}^+\otimes (v_\lambda^{+})^{\otimes p} \arrow[r,mapsto] & v_{p\mu}^+\otimes (v_\lambda^{+})^p.
\end{tikzcd}
\end{equation*}
Since all morphisms in the diagram \eqref{dia:v} are ${}_{\BF_p}\!\dot{\U}^\imath$-equivariant, and $\baV(\mu+\lambda)$ is generated by $v_{\mu+\lambda}^+$ as a ${}_{\BF_p}\!\widehat{\U}^\imath$-module, the proof follows now.
\end{proof}
\subsubsection{}\label{sec:sptfl}
Recall \S\ref{sec:kFr} the map $(\;\cdot\;)^{1/p}:k\rightarrow k$ sends $t$ to $t^{1/p}$. Define
\[
\Psi_k =(\;\cdot\;)^{1/p}\otimes \Psi: k \otimes_{\BF_p} R_{\BF_p}= R_k\longrightarrow R_k.
\]
For any $\lambda\in X^{++}$, it is clear that the subalgebra $k$-subalgebra $R_{\mathcal{L}_\lambda}$ is preserved by $\Psi_k$. Thanks to Proposition \ref{prop:Psi} and Lemma~\ref{le:algFrfl}, we make the following conclusion.
(a) {\em The map $\Psi_k$ induces a Frobenius splitting of the flag variety $\CB_k$. }
The splitting is independent of the choice of $\lambda\in X^{++}$ by a similar argument as \cite{KL2}*{Lemma~6.3}.
\subsubsection{}
Recall Proposition~\ref{prop:Korbits} the construction of codimension-one $K_k$-orbits on $\CB_k$.
For any $\mu\in X^{++}$, and $i\in I$ with $\tau i\neq i$, we consider the restriction map
$
r_{\mu,i}:H^0(\mu)=H^0(\CB_k,\mathcal{L}_\mu)\longrightarrow H^0(\overline{\mO_i},\mathcal{L}_\mu)$. Let $I_{\mu,i}$ be the kernel.
\begin{lemma}\label{lem:Iui}
Under the isomorphism $V_k(\mu)^*\cong H^0(\mu)$, the subspace $I_{\mu,i}$ consists of linear forms vanishing on $_k\dot{\U}^\imath\cdot n_iv_\mu^+$, which equals the $_k\dot{\U}^\imath$-submodule of $V_k(\mu)^*$ generated by the extremal vector of weight $s_i\mu$, where $s_i \in W$ is the simple reflection associated to $i$.
\end{lemma}
\begin{proof}
For any $f\in V_k(\mu)^*$, we identify $f$ with a section in $H^0(\mu)$. Then $r_{\mu,i}(f)=0$ $\Leftrightarrow$ $f\mid_{\mathcal{O}_i} = 0$ $\Leftrightarrow$ $f(kn_i\cdot v_\mu^+)=0$, for all $k\in K$ $\Leftrightarrow$ $f$ vanishes on the $K$-submodule generated by the extremal vector $n_iv_\mu^+$, which is exactly $_k\dot{\U}^\imath\cdot n_i v_\mu^+$, thanks to Proposition \ref{prop:KvsUi}. Hence the lemma is proved.
\end{proof}
\begin{theorem}\label{thm:spl1}
The map $\Psi_k: R_k \rightarrow R_k$ provides a Frobenius splitting of $\CB_k$, which compatibly splits subvarieties $\overline{\mO_i}$, for all $i\in \I$ with $\tau i\neq i$.
\end{theorem}
\begin{proof}
Recall \S\ref{sec:Phi}. The ${}_{\BF_p}\!\dot{\U}^\imath$-module homomorphism
\[
\psi_\mu: \baV(\mu)\longrightarrow \baV(p\mu)^{\ifr}
\]
sends ${}_{\BF_p}\!\dot{\U}^\imath$-submodules to ${}_{\BF_p}\!\dot{\U}^\imath$-submodules. We compute
\[
\psi_\mu(f^{(\mu_i)}_i\one_\mu\cdot v_\mu^+)=\psi_\mu(b_{i,\bar{\mu}}^{(\mu_i)}\cdot v_\mu^+)=b_{i,\overline{p\mu}}^{(p\mu_i)}\cdot v_{p\mu}^+=f_i^{(p\mu_i)}\cdot v_{p\mu}^+, \quad \text{where }\mu_i=\langle \coroot_i,\mu\rangle.
\]
Hence $\psi_\mu$ sends the weight space $\baV(\mu)_{s_i\mu}$ to the weight space $\baV(p\mu)_{s_i(p\mu)}$.
Thus by definition, we have \[
\Psi_k\mid_{H^0(p\mu)}:H^0(p\mu)\longrightarrow H^0(\mu), \quad \text{sending }I_{p\mu,i} \mapsto I_{\mu,i}.
\]
The theorem now follows from \S\ref{sec:sptfl} and Lemma~\ref{le:algFrfl}.
\end{proof}
\subsection{The splitting $\Psi^J_k$} \label{sec:PhiJ}
Recall \S \ref{sec:propJ} and the notations in \S\ref{sec:oFrbach}. We fix a subset $J \subset \I$ with the property (*). We denote by $b_{J,\bar{\lambda}}^{(p-1)}$ the image of $\RB_{J,\bar{\lambda}}^{(p-1)}$ in ${}_{\BF_p}\!\dot{\U}^\imath$ under the canonical isomorphism $\BF_p\otimes_{\A'}{_{\A'}\dot{\U}^\imath}\cong{{}_{\BF_p}\!\dot{\U}^\imath}$.
\subsubsection{}
For any $\mu\in X^+$, recall the map $\psi_\mu^J:{_{\A'}\mathfrak{L}}(\mu)\rightarrow ({_{\A'}L}(p\mu))^{\ifr}$ in Proposition \ref{prop:faJ}. Applying base change, we obtain the ${}_{\BF_p}\!\dot{\U}^\imath$-module homomorphism:
\[
\psi_\mu^J:\baV(\mu)\longrightarrow \baV(p\mu)^{\ifr}, \quad v_\mu^+ \mapsto b_{J,\overline{p\mu}}^{(p-1)}\cdot v_{p\mu}^+.
\]
We denote its $\BF_p$-linear dual by $(\psi_\mu^J)^*$. Define the $\BF_p$-linear map $\Psi^J: R\longrightarrow R$ by
\[
\Psi^J(f) =
\begin{cases}
(\psi_\mu^J)^*(f), &\text{if }f\in \baV(p\mu)^*;\\
0, &\text{if }f\in \baV(\mu)^* \text{ with }\mu\not\in pX.
\end{cases}
\]
\begin{prop}\label{prop:PsiJ}
The map $\Psi^J$ is a Frobenius splitting for the algebra $R_{\BF_p}$.
\end{prop}
\begin{proof}
The proof is similar to the proof of the Proposition \ref{prop:Psi}. It suffices to check the commutativity of the diagram
\begin{equation}\label{dia:vp}
\begin{tikzcd}
\baV(\mu+\lambda) \arrow[r] \arrow[d,"\psi_{\mu+\lambda}^J"'] & \baV(\mu)\otimes \baV(\lambda) \arrow[r,"\psi_{\mu}^J\otimes id"] & \baV(p\mu)^{\ifr}\otimes \baV(\lambda) \arrow[d,"id\otimes \vartheta"] \\
\baV(p\mu+p\lambda)^{\ifr} \arrow[r] & (\baV(p\mu)\otimes \baV(\lambda)^{\otimes p})^{\ifr} \arrow[r] & (\baV(p\mu)\otimes S^p\baV(\lambda))^{\ifr}.
\end{tikzcd}
\end{equation}
One can show that all the maps in the above diagram is ${}_{\BF_p}\!\widehat{\U}^\imath$-equivariant. Hence it will suffice to keep track of the maximal vector:
\begin{equation*}
\begin{tikzcd}
v^+_{\mu+\lambda} \arrow[r,mapsto] \arrow[d,mapsto] & v^+_\mu\otimes v^+_\lambda \arrow[r,mapsto] & \left(b_{J,\overline{p\mu}}^{(p-1)}v_{p\mu}^+\right)\otimes v_\lambda^+ \arrow[d,mapsto]\\ b_{J,\overline{p\mu+p\lambda}}^{(p-1)}v_{p\mu+p\lambda}^+ \arrow[r,mapsto] & b_{J,\overline{p\mu+p\lambda}}^{(p-1)}\cdot \left(v_{p\mu}^+\otimes (v_\lambda^+)^{\otimes p}\right) \arrow[r,mapsto,"(\heartsuit)"] & \left(b_{J,\overline{p\mu}}^{(p-1)}v_{p\mu}^+\right)\otimes (v_\lambda^+)^p.
\end{tikzcd}
\end{equation*}
The $(\heartsuit)$ follows from the fact that
$f_i^{(s)}\cdot (v_\lambda^+)^{\otimes p}\in I_\lambda$,
for any $0<s<p$ and $i\in\I$ (cf. the proof of Proposition~\ref{prop:Ga}). Here $I_\lambda\subset V(\lambda)^{\otimes p}$ stands for the kernel of the quotient map $V(\lambda)^{\otimes p}\rightarrow S^p\baV(\lambda)$.
Hence the diagram \eqref{dia:vp} commutes. This completes the proof for the proposition.
\end{proof}
\subsubsection{}
For $\lambda\in X^+$, $i\in \I$ with $\tau i=i$, and $\epsilon=\pm 1$,
we define
\[
v_{\lambda,i,\epsilon} =\sum_{a\geqslant 0}\epsilon^{\langle \coroot_i,\lambda\rangle+a}f_i^{(a)}v_\lambda^+ \in \baV(\lambda).
\]
By \cite[(2.16) \& (2.17) \& (3.9)\& (3.8)]{BerW18}, we have
\[
b_{i,\bar{\lambda}}^{(\langle \coroot_i,\lambda\rangle )}v_\lambda^+=\sum_{t\geqslant 0}f_i^{(\langle \coroot_i,\lambda\rangle -2t)}v_\lambda^+,\qquad
b_{i,\bar{\lambda}}^{(\langle \coroot_i,\lambda\rangle -1)}v_\lambda^+=\sum_{t\geqslant 0}f_i^{(\langle \coroot_i,\lambda\rangle-2t-1)}v_\lambda^+.
\]
Hence we conclude
\begin{equation}\label{eq:bv}
v_{\lambda,i,1}=(b_{i,\bar{\lambda}}^{(\langle \coroot_i,\lambda\rangle )}+b_{i,\bar{\lambda}}^{(\langle \coroot_i,\lambda\rangle -1)})v_\lambda^+, \quad v_{\lambda,i,-1}=(b_{i,\bar{\lambda}}^{(\langle \coroot_i,\lambda\rangle )}-b_{i,\bar{\lambda}}^{(\langle \coroot_i,\lambda\rangle -1)})v_\lambda^+.
\end{equation}
\begin{lemma}\label{le:bb}
For $i\in \I$ with $\tau i=i$, $\zeta\in X_\imath$, and $n\in\mathbb{N}$, we have
\[
'b_{i,\zeta}^{(np)}\cdot {'b_{i,\zeta}^{(p-1)}}={'b_{i,\zeta}^{(np+p-1)}} \quad \text{in } {{}_{\BF_p}\!\dot{\U}^\imath}.
\]
\end{lemma}
\begin{proof}
By Lemma~\ref{lem:balanced} in $\mathrm{U}^\imath$, we have
\[
B_i^{(np)}B_i^{(p-1)}=\sum_{t\geqslant 0}\qbinom{np+p-1}{np}_i\prod_{m=1}^t\frac{[np-2m+2]_i[p-2m+1]_i}{[np+p-2m]_i[2m]_i}B_i^{(np+p-1-2t)}.
\]
Note that
\[
\qbinom{np+p-1}{np}_i\prod_{m=1}^t\frac{[np-2m+2]_i[p-2m+1]_i}{[np+p-2m]_i[2m]_i} = 0, \quad \text{unless } 0\leqslant t\leqslant (p-1)/2.
\]
Assume $0\leqslant t\leqslant (p-1)/2$, then the above element belongs to $\mathcal{A}_{f_p}$, i.e., the denominator is not divisible by $f_p$.
Then since $\phi([np]_i)=0$, we have
\[
\phi\left(\qbinom{np+p-1}{np}_i\prod_{m=1}^t\frac{[np-2m+2]_i[p-2m+1]_i}{[np+p-2m]_i[2m]_i}\right)=0, \quad \text{if } t \neq 0.
\]
Finally, when $t=0$, we obtain $\phi\left(\qbinom{np+p-1}{np}_i\right)=1$. This proved the lemma.
\end{proof}
\begin{lem}\label{lem:pj}
For any $j\in J$, the element $\psi_\mu^J(v_{\mu,j,\epsilon})$ belongs to the ${}_{\BF_p}\dot{\U}^\imath$-submodule of $\baV(p\mu)$ generated by the vector $v_{\mu,j,\epsilon}$.
\end{lem}
\begin{proof}
We write $\mu_j = \langle \coroot_j, \mu \rangle $. Thanks to \eqref{eq:bv}, it suffices to prove that
\[
\psi_\mu^J(b_{j,\bar{\mu}}^{(\mu_j)}v_\mu^+)\in {{}_{\BF_p}\!\dot{\U}^\imath}\cdot b_{j,\overline{p\mu}}^{(p\mu_j)}v_{p\mu}^+ \quad\text{and}\quad\psi_\mu^J(b_{j,\bar{\mu}}^{(\mu_j-1)}v_\mu^+)\in {{}_{\BF_p}\!\dot{\U}^\imath}\cdot b_{j,\overline{p\mu}}^{(p\mu_j-1)}v_{p\mu}^+
.\]
By definition,
\begin{equation}\label{id:jm}
\psi_\mu^J(b_{j,\bar{\mu}}^{(\mu_j)}v_\mu^+)=\ifr (b_{j,\bar{\mu}}^{(\mu_j)})b_{J,\overline{p\mu}}^{(p-1)}v_{p\mu}^+=b_{J\backslash \{j\},\overline{p\mu}}^{(p-1)}\cdot \ifr(b_{j,\bar{\mu}}^{(\mu_j)})'b_{j,\overline{p\mu}}^{(p-1)}v_{p\mu}^+.
\end{equation}
By the proof of Corollary \ref{cor:iFr} and Lemma \ref{le:bb}, we have
\begin{equation*}
\begin{split}
\ifr(b_{j,\overline{\mu}}^{(\mu_j)})\cdot {'b_{i,\overline{p\mu}}^{(p-1)}}&=\sum_{t\geqslant 0}\lr{1}{2t}{'b_{j,\overline{p\mu}}^{(p\mu_j-2tp)}}\cdot {'b_{i,\overline{p\mu}}^{(p-1)}}\\&=\sum_{t\geqslant 0}\lr{1}{2t} {'b}_{i,\overline{p\mu}}^{(p\mu_j-2tp+p-1)}\\
&=\sum_{t\geqslant 0}\sum_{s\geqslant 0}\lr{1}{2t} \lr{-1}{2s} {b}_{i,\overline{p\mu}}^{(p\mu_j-2tp+p-1-2s)}\\
&=\sum_{k\geqslant 0}\sum_{t\geqslant 0}\lr{1}{2t}\lr{-1}{2k-1+p-2tp}b_{i,\overline{p\mu}}^{(p\mu_j-2k)}.
\end{split}
\end{equation*}
For any $k\geqslant 0$, let $k_0\in [0,p)$, such that $k_0\equiv k+(p-1)/2\; \text{mod }p$. By the Lemma \ref{le:qBinomAtUnity}, we have
\begin{align*}
\lr{-1}{2k-1+p-2tp} & = \lr{p-1}{2k_0}\lr{-p}{2k-2k_0-1+p-2tp}\\
&=\lr{p-1}{2k_0}\lr{-1}{(2k-2k_0-1)/p+1-2t}.
\end{align*}
Hence, for any $k \geqslant 0$, we have
\begin{align*}
&\sum_{t\geqslant 0}\lr{1}{2t}\lr{-1}{2k-1+p-2tp}\\
&=\lr{p-1}{2k_0}\sum_{t\geqslant 0}\lr{1}{2t}\lr{-1}{(2k-2k_0-1)/p+1-2t}\\&=\lr{p-1}{2k_0}\delta_{0,k+(p-1)/2-k_0}. \quad (\text{By Lemma~\ref{lem:DoubleInA2}})
\end{align*}
When $k+(p-1)/2-k_0= 0$, we have
\[
\lr{p-1}{2k_0}=\binom{(p-1)/2}{k_0}=\binom{(p-1)/2}{(p-1)/2+k} = \delta_{0,k}.
\]
Hence we conclude that
\[
\sum_{t\geqslant 0}\lr{1}{2t}\lr{-1}{2k-1+p-2tp}=\delta_{0,k},
\]
and hence
$\ifr(b_{j,\overline{\mu}}^{(\mu_j)})\cdot {'b_{i,\overline{p\mu}}^{(p-1)}}=b_{j,\overline{p\mu}}^{(p\mu_j)}$.
Now \eqref{id:jm} implies
\[
\psi_\mu^J(b_{j,\bar{\mu}}^{(\mu_j)}v_\mu^+)=b_{J\backslash \{j\},\overline{p\mu}}^{(p-1)}\cdot b_{j,\overline{p\mu}}^{(p\mu_j)}v_{p\mu}^+\in {{}_{\BF_p}\!\dot{\U}}^\imath\cdot b_{j,\overline{p\mu}}^{(p\mu_j)}v_{p\mu}^+.
\]
For another relation, still by (the easier version of) Corollary \ref{cor:iFr} and Lemma \ref{le:bb}, we have
\begin{align*}
\psi_\mu^J(b_{j,\overline{\mu}}^{(\mu_j-1)}v_{p\mu}^+)&=\ifr(b_{j,\overline{\mu}}^{(\mu_j-1)})b_{J,\overline{p\mu}}^{(p-1)}v_{p\mu}^+=b_{J\backslash \{j\},\overline{p\mu}}^{(p-1)}\cdot {'b_{j,\overline{p\mu}}^{(p\mu_j-p)}}{'b_{j,\overline{p\mu}}^{(p-1)}}v_{p\mu}^+\\
&=b_{J\backslash \{j\},\overline{p\mu}}^{(p-1)}\cdot {'b_{j,\overline{p\mu}}^{(p\mu_j-1)}}v_{p\mu}^+ \in {{}_{\BF_p}\!\dot{\U}^\imath}\cdot b_{j,\overline{p\mu}}^{(p\mu_j)} v_{p\mu}^+.
\end{align*}
Hence we complete the proof for the lemma.
\end{proof}
\subsubsection{}\label{sec:spltJk}
Recall \S\ref{sec:kFr} the map $(\;\cdot\;)^{1/p}: k \rightarrow k$ mapping $t$ to $t^{1/p}$. Define
\[
\Psi_k^J =(\;\cdot\;)^{1/p}\otimes \Psi^J:R_k\longrightarrow R_k.
\]
For any $\lambda\in X^{++}$, it is clear that the subalgebra $k$-subalgebra $R_{\mathcal{L}_\lambda}$ is preserved by $\Psi_k^J $. Thanks to Proposition \ref{prop:PsiJ} and Lemma~\ref{le:algFrfl}, we make the following conclusion.
(a) {\em The map $\Psi_k^J $ induces a Frobenius splitting of the flag variety $\CB_k$.}
The splitting is independent of the choice of $\lambda\in X^{++}$ by a similar argument as \cite{KL2}*{Lemma~6.3}.
\subsubsection{}
Recall that $\mO_j^\pm=K_ky_j(\pm 1)B_k/B_k$ are codimensional one $K_k$-orbits on the flag variety $\CB_k$ by Proposition~\ref{prop:Korbits}, for any $ j \in J$.
For any $\mu\in X^+$ and $j \in J$, let $I_{\mu,j}^+$ be the kernel of the restriction map
\[
r_{\mu,i}:H^0(\mu)=H^0(\CB_k,\mathcal{L}_\mu)\longrightarrow H^0(\overline{\mO_j^+},\mathcal{L}_\mu).
\]
Similarly define the subspace $I_{\mu,j}^-$ of $H^0(\mu)$. Similar to Lemma~\ref{lem:Iui}, we see that
the subspace $I_{\mu,j}^+$ (resp. $I_{\mu,j}^-$) consists of linear forms vanishing on $_k\dot{\U}^\imath\cdot y_j(1)v_\mu^+$ (resp. ${}_k\dot{\U}^\imath\cdot y_j(-1)v_\mu^+$), under the isomorphism $V_k(\mu)^*\cong H^0(\mu)$.
\begin{theorem}\label{thm:spl2}
The map $\Psi^J_k: R_k \rightarrow R_k$ induces a Frobenius splitting of $\CB_k$, which compatibly splits subvarieties $\overline{\mO_j^\pm}$, for all $j\in J$.
\end{theorem}
\begin{proof}Let $j \in J$.
Note that $y_j(1)v_\mu^+=\sum_{a\geqslant 0}f_j^{(a)}\one_\mu\cdot v_\mu^+$ and similarly $y_j(-1)v_\mu^-=\sum_{a\geqslant 0}(-1)^af_j^{(a)}\one_\mu\cdot v_\mu^+$. Hence by Lemma~\ref{lem:pj}, the map
\[
\Psi_k^J\mid_{H^0(p\mu)}: H^0(p\mu)\longrightarrow H^0(\mu), \quad \text{ mapping } I_{p\mu,j}^\pm \text{ to }I_{\mu,j}^\pm.
\]
Take $\lambda\in X^{++}$. We deduce that the map $\Psi_k^J$ induces a Frobenius splitting for the algebra $R_{\mathcal{L}_\lambda}$ which preserves the ideals $I_{\mathcal{L}_\lambda,\overline{\mO_i^\pm}}$, for any $i\in J$. The theorem follows from \S\ref{sec:spltJk} and Lemma \ref{le:algFrfl}.
\end{proof}
\subsection{Results on partial flags}
Let $P = P_S$ be a standard parabolic subgroup of $G_k$ (recall we fixed a pinning of $G_k$), corresponding to $S\subset\I$. Let $\pi=\pi_{P}:G_k/B_k\rightarrow G_k/P$ be the projection. Let $\mu\in X^+$ be such that $\langle \alpha_j^\vee,\mu\rangle=0$ for any $j\in S$. Then $\mu$ extends to a character of $P$. Let
\[
\mathcal{L}^P_\mu =G_k\times^{P}k_\mu\longrightarrow G_k/P
\]
be the associated line bundle of $G_k/P$. Then $\pi_P^*(\mathcal{L}^P_\mu) \cong \mathcal{L}_\mu$, as line bundles on $G_k/B_k$. Hence we have
\[
H^0(G_k/P,\mathcal{L}^P_\mu)\cong H^0(G_k/B_k,\mathcal{L}_\mu)\cong V_k(\mu)^*.
\]
The $k$-algebra $R_{\mathcal{L}^P_\mu} =\bigoplus_{n\geq 0}H^0(G_k/P,\mathcal{L}^P_{n\mu})$ is isomorphic to a subalgebra of $R_k$. It is clear that the splitting maps defined in \ref{sec:sptfl} and \ref{sec:spltJk} restrict to $R_{\mathcal{L}^P_\mu}$.
\begin{corollary}\label{cor:parab}
The map $\Psi_k$ induces a Frobenius splitting of $G_k/P$, which compatibly splits $\pi(\overline{\mathcal{O}_i})$, for all $i\in\I$ with $\tau i\neq i$.
For any $J\subset\I$ with the property (*), the map $\Psi_k^J$ induces a Frobenius splitting of $G_k/P$, which compatibly splits $\pi\big(\overline{\mathcal{O}_j^\pm}\big)$ for all $j\in J$.
\end{corollary}
\subsection{Geometric consequences} \label{sec:geo}
Theorem~\ref{thm:spl1}, Theorem~\ref{thm:spl2}, and Corollary~\ref{cor:parab} imply standard consequences on $K_k$-orbit closures following \S\ref{sec:algFr}.
We futher obtain the following stronger vanishing result on semi-ample line bundles.
\begin{theorem}
Let $i\in\I$ be such that $\langle\coroot_j,\alpha_i\rangle\geqslant -1$, for any $j\in\I$, and be such that $\langle\coroot_i,\mu\rangle >0$ and $\langle\coroot_{\tau i},\mu\rangle >0$, for any $\mu\in X^+$. We have
(a) the restriction $H^0(G_k/B_k,\mathcal{L}_\mu)\rightarrow H^0(\overline{\mathcal{O}_i^\pm},\mathcal{L}_\mu)$ is surjective;
(b) $H^n(\overline{\mathcal{O}_i^\pm},\mathcal{L}_\mu)=0$, for $n>0$.
Here we write $\mathcal{O}_i^\pm=\mathcal{O}_i$, if $\tau i\neq i$.
\end{theorem}
\begin{proof}
Set $S=\{j\in\I:\langle \alpha_j^\vee,\mu\rangle=0\}$. Let $P =P_S$ be the parabolic subgroup associated to the subset $S$, and $\pi:G_k/B_k\rightarrow G_k/P$ be the projection. It can be deduced from the case study in \S \ref{sec:Korbits} that $\overline{\mathcal{O}_i^\pm}=\pi^{-1}\circ\pi(\overline{\mathcal{O}_i^\pm})$. Hence the restriction $\pi_{i}^\pm:\overline{\mathcal{O}_i^\pm}\rightarrow\pi(\overline{\mathcal{O}_i^\pm})$ is a locally trivial fibration with fibre isomorphic to $P/B_k$. By the projection formula, we have isomorphisms (for any $n$)
\[
\pi^*\!:\!H^n(G_k/P,\mathcal{L}^{P}_\mu)\xrightarrow{\sim}H^n(G_k/B_k,\mathcal{L}_\mu), \,\, \pi_{i}^{\pm,*}\!:\!H^n(\pi\big(\overline{\mathcal{O}_i^\pm}\big),\mathcal{L}_\mu^{P})\xrightarrow{\sim}H^n(\overline{\mathcal{O}_i^\pm},\mathcal{L}_\mu).
\]
We also have the commutative diagram
\[
\begin{tikzcd}
H^0(G_k/P,\mathcal{L}_\mu^P) \arrow[r] \isoarrow{d} & H^0(\pi_P(\overline{\mathcal{O}_i^\pm}),\mathcal{L}_\mu^P) \isoarrow{d}\\ H^0(G_k/B_k,\mathcal{L}_\mu) \arrow[r] & H^0(\overline{\mathcal{O}_i^\pm},\mathcal{L}_\mu).
\end{tikzcd}
\]
It follows from Corollary \ref{cor:parab} that $G_k/P$ is Frobenius split compatibly with $\pi_P(\overline{\mathcal{O}_i^\pm})$. Note that the assumption on $i$ guarantees $\{i\}$ is a subset satisfying property (*), if $\tau i=i$. Since $\mathcal{L}_\mu^P$ is a very ample line bundle on $G/P$, by Corollary \S \ref{sec:algFr} (c), we deduce that the restriction $H^0(G_k/P,\mathcal{L}_\mu^P)\rightarrow H^0(\pi_P(\overline{\mathcal{O}_i^\pm}),\mathcal{L}_\mu^P)$ is surjective, and $H^n(\pi_P(\overline{\mathcal{O}_i^\pm}),\mathcal{L}_\mu^P)=0$ for $n>0$. Hence the theorem follows.
\end{proof}
\begin{remark}
If $\tau i\neq i$ (or, $\tau i = i$ with $\mathcal{O}_i^+\neq\mathcal{O}_i^-$), the orbit closures $\overline{\mathcal{O}_i^\pm}$ are multiplicity-free. In these cases, a stronger statement is obtained by Brion \cite{Br01b}*{Corollary 8}, in which he showed the results for any dominant weights $\mu$.
Our result is new in the case $\tau i=i$ with $\mathcal{O}_i^+=\mathcal{O}_i^-$, which corresponds to $\overline{\mathcal{O}_i^\pm}$ being multiplicity-two. The assumption on $\mu$ cannot be dropped by \cite{Br01b}*{Proposition 10}.
\end{remark}
Let $\CO$ be a $K_k$-orbit on the flag variety $\CB_k=G_k/B_k$. Recall from Proposition \ref{prop:Korbits} (3) that the
parameterization of such orbits is independent of the base filed. We denote by $\CB_{\BZ[2^{-1}]}$ the flag scheme over $\BZ[2^{-1}]$ (cf. \cite{Jan03}*{Part II, \S1}).
\begin{prop}\label{prop:KorbitsZ}
There exists a closed subscheme $\CZ=\CZ (\mathcal{O})$ of $\CB_{\BZ[2^{-1}]}$ (over $\BZ[2^{-1}]$), such that,
(1) $\CZ\rightarrow Sp\,\BZ[2^{-1}]$ is flat;
(2) there is a nonempty open subset $U$ of $Sp\,\BZ[2^{-1}]$ such that for any algebraically closed field $k'$ and a morphism $Sp\,k'\rightarrow U$, the base change $\CZ_{k'} = \CZ \times_{Sp\, \BZ[2^{-1}]} Sp\, k'$ gives the closure of the corresponding $K_{k'}$-orbit on $\mathcal{B}_{k'}$. In particular, $\CZ_{k'}$ is reduced.
\end{prop}
\begin{proof}
Thanks to Proposition~\ref{prop:Korbits}, a representative of $\CO$ can be chosen in the $\BZ[2^{-1}]$-points of $\CB_{\BZ[2^{-1}]}$. Let $\G^\imath_{\BZ[2^{-1}]} \rightarrow \CB_{\BZ[2^{-1}]}$ be the orbit map. Let $\CZ = \CZ (\mathcal{O})$ be the scheme-theoretical image (cf. \cite{Ha77}*{Exercise II 3.11(d)}). The rest follows from similar arguments as in the proof of \cite{MR85}*{Lemma 3}.
\end{proof}
\begin{remark}
It is an open question if one can take $U=Sp\,\BZ[2^{-1}]$ for part (2). Similar question for Schubert varieties is addressed in \cite{Se83}*{Theorem 2 (ii)}.
\end{remark}
Thanks to the above lemma and semiconuity, all the results stated in this subsection remain true over algebraically closed fields with characteristic 0 (cf. \cite{BK05}*{\S 1.6}).
\subsection{Normality: type $\textrm{AIII}$}\label{sec:AIII}
Let $k$ be an algebraically closed field of characteristic $>2$. We consider the type AIII symmetric pairs in this subsection. We illustrate how to deduce normality from our splittings, which requires detailed information of the Bruhat order of $K_k$-orbits on $\CB_k$.
\subsubsection{}
Let $G_k=GL_{n,k}$ and $a=\begin{pmatrix}
I_{\lceil n/2\rceil} & 0 \\ 0 & -I_{\lfloor n/2 \rfloor}
\end{pmatrix} \in G_k$.
Let $\theta_k: G_k \rightarrow G_k$ be the conjugation by $a$. Then $K_k=G_k^{\theta_k}=GL_{\lceil n/2\rceil,k}\times GL_{\lfloor n/2 \rfloor,k}$. The symmetric pair $(G_k,\theta_k)$ is quasi-split. The underlying Satake diagram is of type $\textrm{AIII}_{n-1}$.
Let $B^{st}_k$ be the subgroup of $G_k$ consisting of upper-triangular matrices, and $T^{st}_k$ be the subgroup of diagonal matrices. Then $B^{st}_k$ is a $\theta$-stable Borel subgroup. Set
\[
g=\begin{pmatrix}
1 & & & & & -1\\ & 1 & & & -1 & \\ & & \ddots & \reflectbox{$\ddots$} & & \\ & & \reflectbox{$\ddots$} & \ddots & & \\ & 1 & & & 1 & \\ 1 & & & & & 1
\end{pmatrix} \in G_k.
\]
The middle entry of $g$ can be either $1$ or $-1$ when $n$ is odd.
Let $B_k=gB_k^{st}g^{-1}$, and $T_k=gT^{st}_kg^{-1}$. Then $B_k$ is a $\theta$-anisotropic Borel subgroup and $T_k = B_k \cap \theta_k(B_k)$.
Let $W$ be the (absolute) Weyl group, and $\I=\{1,2,\dots,n-1\}$ be the index set for the simple roots. The graph automorphism $\tau$ induced by $\theta$ is given by $\tau(i)=n-i$, for any $i\in\I$. Let $\mathcal{B}_k$ denote the flag variety.
Let $\CV$ denote the set of $K_k$-orbits of $\mathcal{B}_k$, and let $\mathcal{O}_0\in \CV$ be the open orbit. The Weyl group $W$ acts on the set $\CV$ by \cite{RS90}*{\S 2} by identifying $W \cong N(T_k) / T_k$. Set $J=\{1,2,\dots,\lfloor n/2\rfloor-1\}$, and $J'=\tau(J)=\{n-1,n-2,\cdots,\lceil n/2\rceil+1\}$ be two subsets of $\I$. Let $W_J$ and $W_{J'}$ be the parabolic subgroup of $W$ associated to $J$ and $J'$, respectively. For any $j\in J\cup J'$, we have $\tau j\neq j$. Set $\CV'=W_{J'}\cdot\mathcal{O}_0\cup W_J\cdot\mathcal{O}_0$. Note that $W_{J'}\cdot\mathcal{O}_0=W_J\cdot\mathcal{O}_0$, when $n$ is even.
\begin{proposition}
The splitting defined in \S \ref{sec:sptfl} compatibly splits all the orbits in $\CV'$ simultaneously.
\end{proposition}
\begin{proof}
Let $\mathcal{O}\in \CV'$. Then $\mathcal{O}=K_knB_k/B_k$, for some $n\in N(T_k)$, representing $w\in W_J\cup W_{J'}$. Let $w=s_{i_1}s_{i_2}\cdots s_{i_r}$ be a reduced expression for $w$, with $i_t$ ($1\leqslant t\leqslant r$) either all belong to $J$ , or all belong to $J'$.
Let $\mu\in X^+$. Note that in the $\U$-module $L(\mu)$, we have
\[
B_{i_1}^{(n_1)}B_{i_2}^{(n_2)}\cdots B_{i_r}^{(n_r)}v_\mu^+=F_{i_1}^{(n_1)}F_{i_2}^{(n_2)}\cdots F_{i_r}^{(n_r)}v_\mu^+
\]
which is a nonzero vector of extremal weight $w\cdot\mu$, where $n_t=\langle \coroot_{i_t},\mu-n_r\alpha_{i_r}-\cdots-n_{t+1}\alpha_{i_{t+1}}\rangle$, for $1\leqslant t\leqslant r$. Then the proposition follows by a similar proof to Theorem \ref{thm:spl1}.
\end{proof}
\subsubsection{} \label{sec:AIIIa}
For any $i\in\I$, let $P_i$ be the minimal parabolic subgroup containing $B_k$ associated to the index $i$. Let $\pi_i:G_k/B_k\rightarrow G_k/P_i$ be the projection. Then for any $\mathcal{O}\in \CV$, the decomposition of $\pi_i^{-1} \circ \pi_i (\CO) = \tilde{\pi}_i (\CO)$ into $K_k$-orbits depends on the case analysis in \S\ref{sec:Korbits}. The following claim follows from \cite{Wy16}*{Theorem~1.2}. The proof can be obtained merely by translating our languages into clans, which we shall skip.
(a) {\em Let $\mathcal{O}\in \CV$ and $i \in I$ be such that $s_i \cdot \CO \neq \CO$.
Then for any $\mathcal{O}'\in \CV$ with $\overline{\mathcal{O}'}\supseteq \mathcal{O}$ and $\overline{\mathcal{O}'}\supseteq s_i\cdot\mathcal{O}$, we have $\overline{\mathcal{O}'}\supseteq\tilde{\pi}_i (\CO)$.}
\begin{lem}\label{lem:AIII}
Let $\CO' \in \CV$ and $\CO \in \CV$ the unique open dense $K_k$-orbit in $\tilde{\pi}_i(\CO')$ for some $ i \in \I$. Let $\widetilde{\mathcal{O}'}$ be the preimage of $\overline{\mathcal{O}'}$ via the projection $G_k \rightarrow G_k / B_k$. Then the following surjective morphism has connected fibers
\[
f: \widetilde{\mathcal{O}'}\times^{B_k} P_{i}/B_k\longrightarrow\overline{\mathcal{O}}.
\]
\end{lem}
\begin{proof}
Recall the closed embedding via the convolution product
\[
\widetilde{\mathcal{O}'}\times^B P_{i}/B_k \rightarrow \overline{\mathcal{O}'}\times \overline{\mathcal{O}}, \quad (g_1, g_2 B_k/B_k) \mapsto (g_1B_k/B_k, g_1g_2 B_k/B_k).
\]
Let $xB_k / B_k \in \overline{\mathcal{O}}$ for some $x \in G_k$. Then we have
\[
f^{-1} (xB_k / B_k) \cong \tilde{\pi}_i (xB_k / B_k) \cap \overline{\mathcal{O}'}.
\]
Since the fiber is stable under the action of $K \cap x P_i x^{-1}$, it remains to consider $H$-orbits on $\tilde{\pi}_i (xB_k / B_k) \cong P_i / B_k \cong \BP^1_k$, where $H$ denotes the image of $K \cap x P_i x^{-1}$ in $Aut(\BP^1_k) \cong PGL_{2,k}$. By construction, $H$ has an open dense orbit. We have four cases to consider by \cite{RS90}*{\S4}:
Case I: $H^\circ$ is unipotent. In this case, we have two orbits, one open dense orbit and one closed fixed point. The fibre is always connected.
Case II: $H = PGL_{2,k}$. There is only one orbit. The fibre is connected.
Case III: $H$ is a torus. There are three orbits, one open dense orbit and two fixed points. However, the fibre can not be the union of the two fixed points, by \S\ref{sec:AIIIa} (a). The fibre is always connected in this case.
Case IV: $H$ is the normalizer of a torus. Then $H^\circ$ is a torus. The open dense $H^\circ$-orbit is $H$-stable and $H$ permutes the two fixed points of $H^\circ$. However, this case can not happen in our setting by \cite{Br01b}*{Corollary~2 \& Page 18}
We now finish the proof.
\end{proof}
\begin{remark}
(1) We expect the claim \S\ref{sec:AIIIa} (a) holds for arbitrary symmetric pairs.
(2) It follows from the proof of Lemma~\ref{lem:AIII} that the connectedness of the fibres can be read from Brion's graphs in \cite{Br01b}. This is closely related to multiplicity-free subvarieties.
Brion's graph has only simple edges for the symmetric pair $(GL_{n,k}, GL_{\lceil n/2\rceil,k}\times GL_{\lfloor n/2 \rfloor,k})$, so that Lemma~\ref{lem:AIII} holds for all orbits in this case.
\end{remark}
\subsubsection{}
We now prove the main result of this section.
\begin{prop}
For any $\mathcal{O}\in \CV$, its closure $\overline{\mathcal{O}}$ is normal if it is Frobenius split. In particular, $\overline{\mathcal{O}}$ is normal for any $\mathcal{O}\in \CV'$.
\end{prop}
\begin{proof}
Let $\mathcal{O}\in \CV$. By \cite{RS90}*{Theorem 4.6}, there exists a closed orbit $\mathcal{O}_1\in \CV$ and a sequence $i_1, \dots, i_m \in \I$, such that $\tilde{\pi}_{i_m} \circ \cdots \circ \tilde{\pi}_{i_1}(\mathcal{O}_1)=\mathcal{O}$ and $\textrm{dim}(\mathcal{O})=\textrm{dim}(\mathcal{O}_1)+m$. Let $\widetilde{\mathcal{O}_1}\subset G$ be the preimage of $\mathcal{O}_1$ via the natural projection $G_k \rightarrow G_k/B_k$. Then there is a proper surjective morphism:
\[
f:\widetilde{\mathcal{O}_1}\times^{B_k}P_{i_1}\times^{B_k}\cdots\times^{B_k}P_{i_m}/B_k\longrightarrow \overline{\mathcal{O}}.
\]
We show the map $f$ has connected fibres by induction on $m$.
The base case is by Lemma~\ref{lem:AIII}. By the induction hypothesis, the map
\[
f_1:\widetilde{\mathcal{O}_1}\times^{B_k}P_{i_1}\times^{B_k}\cdots\times^{B_k}P_{i_{m-1}}\times^{B_k} P_{i_m}/B_k\longrightarrow \widetilde{\mathcal{O}'}\times ^{B_k} P_{i_m}/B_k
\]
has connected fibers. The projection $f_2: \widetilde{\mathcal{O}'}\times ^{B_k} P_{i_m}/B_k \rightarrow \overline{\mathcal{O}}$ also has connected fibres. Since $f_1$ is proper hence closed, we conclude that $f = f_2 \circ f_1$ has connected fibers as well.
The rest of the proposition follows by \cite{MR87}*{Lemma 1}.
\end{proof}
Thanks to Proposition \ref{prop:KorbitsZ}, any orbit closure over characteristic 0 can be viewed as the geometric generic fibre of the corresponding (flat) scheme over $\BZ[2^{-1}]$. Since geometric normality is a open condition over $Sp\,\BZ[2^{-1}]$ (\cite{EGA}*{Theorem 12.2.4 (iv)}), closures of orbits in $\mathcal{V}'$ are normal over characteristic 0.
\begin{remark}
Similar maps to $f$ were considered by Barbasch-Evens in \cite{BE94}*{6.3.7}. Barbasch-Evens' construction starts from ``distinguished orbits'', instead of closed ones, which guarantees the map is birational. In general, the map defined from closed orbits can be generically finite-to-one.
\end{remark}
\begin{remark}
In the langauage of \cite{Wy16}, the orbits in $\CV'$ correspond to clans which have at most one sign, and the first $\lfloor n/2\rfloor$ natural numbers are distinct.
\end{remark}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 2,291 |
\section{Introduction}
Degenerate elliptic operators on manifolds with boundary or corners arise naturally in many different problems in
partial differential equations, geometric analysis, mathematical physics and elsewhere. Over the past
several decades, many types of such equations have been studied, often by ad hoc methods but sometimes
through the development of a more systematic theory to handle various classes of operators. One particularly
fruitful direction concerns the elliptic operators associated to (complete or incomplete) iterated edge metrics
on smoothly stratified spaces. The simplest examples of such operators include the Laplace
operators for spaces with isolated conic singularities or with asymptotically cylindrical ends. Other
important special cases include nondegenerate elliptic operators on manifolds with boundaries
or the Laplacians on asymptotically hyperbolic (conformally compact) manifolds. The class of elliptic
operators on spaces with simple edge singularities includes both of these sets of examples. A final
example is the Laplacian (or any other elliptic operator) on a smooth manifold written in Fermi coordinates around
a smooth embedded submanifold.
To be more specific, let $M$ be a compact manifold with boundary, and suppose that $\partial M$ is the total
space of a fibration with base $B$ and fibre $F$. Choose coordinates $(x,y,z)$ near the boundary so
that $x = 0$ defines $\partial M$, $y$ is a set of coordinates on $B$ lifted to $\partial M$ and then extended
into $M$ and $z$ are independent functions which restrict to a coordinate system on each fibre $F_y$.
A differential operator of order $L$ is called an edge operator of order $m$ if it has the form
\[
L = \sum_{j + |\alpha| + |\beta| \leq m} a_{j \alpha \beta}(x,y,z) (x\partial_x)^j (x\partial_y)^\alpha \partial_z^\beta.
\]
We assume that the coefficients $a_{j \alpha \beta}$ are all $\mathcal C^\infty$ on the closed manifold $M$; these
can either be scalar or (if $L$ acts between sections of vector bundles) matrix-valued. We say that $L$
is edge elliptic if it is an elliptic combination of the constituent vector fields $x\partial_x$, $x\partial_{y_i}$
and $\partial_{z_j}$; an invariant definition is indicated in \S 2. The examples above
all fall into this class, or else are of the form $x^{-m}L$ where $L$ has this form; operators of this latter
sort are called `incomplete' edge operators since they include Laplacians of metrics with incomplete
edge singularities.
The present paper is the continuation of a now rather old paper by the first author \cite{M} which develops a
framework for the analysis of elliptic differential edge operators based on the methods of geometric
microlocal analysis. That paper establishes many fundamental results concerning the mapping properties of
these operators and the regularity properties of solutions, and those results have had very many applications,
both in analysis and geometry, in the intervening years. We review this theory in \S 2. The mapping properties
considered there were for an elliptic edge operator acting between weighted Sobolev and H\"older spaces. This
left open, however, any development of a more general theory of ``elliptic edge boundary value problems''.
The present paper finally addresses this aspect of elliptic edge theory.
An important generalization involves the study of elliptic operators with iterated edge singularities;
examples include Laplacians on $\mathcal C^\infty$ polyhedra or conifolds, see \cite{MM}, as well as on general
smoothly stratified spaces \cite{ALMP}, \cite{ALMP2}. As of yet, there is no complete elliptic theory
in this general setting, although many special cases and specific results have been obtained by a variety of authors.
Certainly the closest to what we do here is the work of Gil, Krainer and Mendoza \cite{GKM} and the recent and ongoing
work of Krainer-Mendoza \cite{KM}, \cite{KM2}. In various respects, these last papers go much farther than what
we do here. We mention also the theory developed by Schulze,
\cite{S1}, \cite{S2}, \cite{S3}. The emphasis in those papers and monographs is the development of a
hierarchy of algebras of pseudodifferential operators, structured as in \cite{BdM}, with emphasis on the
operator-symbol quantization associated to spaces with both simple and iterated edge singularities. Other notable
contributions include the work of Maz'ya and his collaborators, see \cite{KMR}, as well as Nistor \cite{Ni},
Ammann-Lauter-Nistor \cite{ALN} and Br\"uning-Seeley \cite{BS}. As noted above, Krainer and Mendoza \cite{KM}, \cite{KM2}
also treat edge boundary problems, and in fact do so for more general operators with variable indicial roots, but their methods
are somewhat different from the ones here. These last authors have very generously shared some important ideas
from their work before the appearance of \cite{KM}, described here in \S 3, which form a necessary part of our analysis.
There are many reasons for developing a theory of more general boundary conditions for elliptic edge operators, in
particular from the point of view developed here. Perhaps most significant is the importance of mixed or global
boundary conditions, either of local (Robin) or Atiyah-Patodi-Singer type, in the study of index theory for generalized
Dirac operators on spaces with simple edge singularities, all of which appear in many natural problems. Similarly, the
study of the eta invariant and analytic torsion for Dirac-type operators on spaces with various boundary conditions
on spaces with isolated conic singularities has proved to be quite interesting. All of these directions fall within
the scope of one or more of the other approaches cited above.
The geometric microlocal methods used here have a distinct advantage over other (e.g.\ more directly Fourier analytic)
approaches: our primary focus is on Schwartz kernels rather than abstract mapping properties or methods too
closely tied to more standard pseudodifferential theory, and because of this it is equally easy to obtain results
adapted to any standard types of function spaces that one might wish to use, e.g.\ weighted H\"older or $L^p$ spaces.
This transition between mapping properties on different types of spaces seems more difficult using those other
approaches, although having such properties available is quite important when studying nonlinear geometric
problems on spaces with edge singularities, cf.\ \cite{JMR} for a recent example.
One limitation of the current development, however, is that we do not treat the delicate regularity issues associated
with the possibility of smoothly varying indicial roots. As in \cite{M}, we make a standing assumption that all
operators considered here have constant indicial roots, at least in the critical weight-range $(\underline{\delta}, \overline{\delta})$.
We refer to \S 2 for a description of all of this.
Because their precise description requires a number of preliminary definitions, we defer to \S 3 a careful description
of our main results; however, we now state them briefly and somewhat
informally. The starting point is the basic statement that if $L$ is an elliptic edge operator, as above, then under appropriate
hypotheses on the indicial roots and assuming the unique continuation property for the reduced Bessel operator
$B(L)$, see \S 2 for these, the mapping
\[
L: x^\delta H^m_e(M) \longrightarrow x^\delta L^2(M)
\]
is essentially surjective, i.e.\ has closed range with finite dimensional cokernel, provided
$\delta \leq \underline{\delta}$ and $\delta$ is nonindicial, and is essentially injective, i.e.\ has
closed range with finite dimensional nullspace, if $\delta \geq \overline{\delta}$.
Suppose that $u \in x^{\underline{\delta}} H^m_e$ and $Lu = f \in x^{\overline{\delta}}$. Then it is proved in \cite{M} that
\[
u \sim \sum_{j=1}^N \sum_{\ell, p \in \mathbb N_0} u_{j p \ell}(y,z) x^{\gamma_j + \ell} (\log x)^p + \tilde{u},
\]
where the sum is over all indicial roots (see \S 2) of $L$ and indices $p, \ell$ such that $\gamma_j + \ell \in (\underline{\delta}, \overline{\delta})$
and $p$ is no greater than some integer $N _{j,\ell}$, and where $\tilde{u} \in x^{\overline{\delta}}H^m_e$. The
subcollection of leading coefficients $\{u_{j, N_{j 0}, 0}\}$ is called the Cauchy data of $u$ and denoted
$\mathcal C(u)$. A boundary condition for this edge problem consists of a finite collection $Q = \{Q_{kj}\}$
of pseudodifferential operators acting on these leading coefficients. (Since the precise formulation is somewhat
intricate, we defer this for now.) We then study the mappings:
\[
\begin{aligned}
& L u = f \in x^{\underline{\delta}}L^2 \\
& Q( \mathcal C(u)) = \phi.
\end{aligned}
\]
The main results here give conditions for when this mapping is Fredholm or semi-Fredholm acting on appropriate
weighted Sobolev spaces. We follow the methods due originally to Calderon, described particularly well in
the monograph of Chazarain and Piriou \cite{CP}, and later extended significantly by Boutet de Monvel and others; however, we
do not define the full Boutet de Monvel calculus in this edge setting. We also give the precise structure of the
Schwartz kernel of the generalized inverse of this mapping, and consequently can study this problem on other
function spaces. We do not treat any application of these results in this paper, but must rely on the reader's
knowledge of the centrality of elliptic boundary problems in the standard setting, and on his or her faith
that this extension of that theory will also have broad applicability.
\bigskip
{\bf Acknowledgements.} The first author's understanding of elliptic boundary problems reflects his long and fruitful
interactions with Richard Melrose, and also Pierre Albin, Charlie Epstein, Gerd Grubb, Thomas Krainer,
Gerardo Mendoza and Andras Vasy. The second author would like to express his gratitude to Bert-Wolfgang Schulze for
encouragement and many useful discussions concerning his alternate formulation of edge calculus. He also
wishes to acknowledge many useful discussions with Andras Vasy. The second author also thanks the
Department of Mathematics at Stanford University for its hospitality during a major part of the research and
writing which led to this paper. Both authors offer special thanks to Thomas Krainer and Gerardo Mendoza for many
useful discussions on this subject, and in particular for explaining their theory of trace bundles to us before the appearance of \cite{KM}.
We hope the reader will regard their work, as we do, as a good counterpart to ours, with somewhat different aims.
\section{A review of the edge calculus} \label{fundamentals}
We begin by recalling in more detail the geometric and analytic framework necessary to discuss the theory of
differential and pseudodifferential edge operators, and then review the main theorems from \cite{M} concerning
the semi-Fredholm theory and asymptotics of solutions. This section is meant as a brief review, and is not meant
to be self-contained. We refer the reader to \cite{M} for elaboration and proofs of all the definitions and facts
presented here.
\medskip
\noindent{\bf Edge structures}
As in the introduction, let $M$ be compact manifold with boundary, and suppose that $\partial M$ is the total space of a fibration
$\phi:\partial M \to B$ with fibre $F$. We set $b = \dim B$ and $f = \dim F$.
The fundamental object in this theory is the space $\mathcal V_{\mathrm{e}}$ of all smooth vector fields on $M$ which are unconstrained in
the interior and which are tangent to the fibres of $\phi$ at $\partial M$; clearly $\mathcal V_{\mathrm{e}}$ is
closed under Lie bracket. We shall routinely use local coordinate systems near the boundary of the following form:
$x$ is a defining function for the boundary (i.e.\ $\partial M = \{x = 0\}$), $y_1, \ldots, y_b$ is a set of local coordinates
on $B$ lifted to $\partial M$ and then extended into $M$, and $z_1, \ldots, z_f$ is a set of independent functions
which restricts to a coordinate system on each fibre $F_y$. In terms of these,
\begin{equation}
\mathcal V_{\mathrm{e}} = \mathrm{Span}_{\mathcal C^\infty} \, \{ x\partial_x, x\partial_{y_1}, \ldots, x \partial_{y_b}, \partial_{z_1}, \ldots, \partial_{z_f} \}.
\label{v-e}
\end{equation}
In other words, any $V \in \mathcal V_{\mathrm{e}}$ can be expressed locally as
\[
V = a x\partial_x + \sum b_i x \partial_{y_i} + \sum c_j \partial_{z_j}, \quad \mbox{where}\ a, b_i, c_j \in \mathcal C^\infty(\overline{M}).
\]
Any differential operator can be expressed locally as the sum of products of vector fields, and so we can define
interesting subclasses of operators by restricting the vector fields allowed in these decompositions. In particular,
define $\mathrm{Diff}_e^*(M)$ to consist of all differential operators which are locally finite sums of products of
elements in $\mathcal V_{\mathrm{e}}$. With the subscript corresponding to the usual order filtration, we have, in local coordinates,
\begin{equation}
\mathrm{Diff}_e^m(M) \ni L = \sum_{j+|\alpha|+|\beta| \leq m} a_{j \alpha \beta}(x,y,z) (x\partial_x)^j (x\partial_y)^\alpha \partial_z^\beta,
\label{diffem}
\end{equation}
with all $a_{j \alpha \beta} \in \mathcal C^\infty$.
Here and later we use standard multi-index notation to describe (differential) monomials. If $L$ acts between sections
of two bundles $E$ and $F$, then taking local trivializations of these bundles, the coefficients here are matrix-valued.
There is a natural edge tangent bundle ${}^eTM$ defined by the property that $\mathcal V_{\mathrm{e}}$ coincides with its {\it full} space
of $\mathcal C^\infty$ sections; its dual is the edge cotangent bundle ${}^eT^*M$, which has a local $\mathcal C^\infty$ basis of
sections consisting of the $1$-forms
\[
\frac{dx}{x}, \frac{dy_1}{x}, \ldots, \frac{dy_b}{x}, dz_1, \ldots, dz_f.
\]
Any $L \in \mathrm{Diff}_e^m(M)$ has symbol
\[
{}^e\sigma_m(L)(x,y,z, \xi, \eta, \zeta) = \sum_{j+|\alpha|+|\beta| = m} a_{j \alpha \beta} (x,y,z) \xi^j \eta^\eta \zeta^\beta,
\]
which is well-defined as a smooth function on ${}^eT^*M$ which is a homogenous polynomial of degree $m$ on
each fibre. If $L$ acts between sections of two vector bundles $E$ and $F$, then ${}^e\sigma_m(L)$ takes values
in $\mathrm{End}(\pi^*E, \pi^* F)$, where $\pi: {}^e T^*M \to M$. The operator $L$ is said to be elliptic (in the
edge sense) if this symbol is invertible when $(\xi,\eta,\zeta) \neq (0,0,0)$.
A (complete) edge metric is a smooth positive definite section of $\mathrm{Sym}^2 ({}^e T^*M)$. It is not hard to
check that if $g$ is any metric of this type, then its scalar Laplacian, Hodge Laplacian, and all other natural
elliptic geometric operators (e.g. the rough Laplacian, the Lichnerowicz Laplacian, twisted Dirac operators, etc.)
are all elliptic edge operators (N.B.; some of these operators are of this type only if expressed in terms of an
appropriate basis of sections of the bundles on which they act).
Similarly, an incomplete edge metric $g$ is one of the form $x^2 \tilde{g}$, where
$\tilde{g}$ is a complete edge metric. Its Laplacian is of the form $x^{-2}L$ where $L \in \mathrm{Diff}_e^2(M)$, and there
are analogous assertions for the other elliptic operators mentioned above. In practice one often restricts
to a smaller class of metrics (for example, requiring that $g$ does not contain the term $x^{-1}dxdy$, though
even even more rigid hypotheses arise naturally), see \cite{MV} for more on this.
\medskip
\noindent{\bf Model operators}
Let $L$ be an elliptic edge operator of order $m$, expressed as in \eqref{diffem}. The analysis of the mapping
properties of $L$ relies on a variety of associated model operators.
First, the principal edge symbol ${}^e\sigma_m(L)$ is a purely algebraic model for $L$ at any point; the microlocal
inversion of $L$, uniformly up to the boundary, relies on the invertibility of this object.
Next, associated to every point $y_0 \in B$ (taken as the origin in the $y$ coordinate system) is the {\it normal operator}
\begin{equation}
N_{y_0}(L) = \sum_{j+|\alpha| + |\beta| \leq m} a_{j \alpha \beta}(0, 0, z) (s\partial_s)^j (s\partial_u)^\alpha \partial_z^\beta;
\label{normalop}
\end{equation}
this acts on functions on the model space $\mathbb R^+_s \times \mathbb R^b_u \times F_{y_0}$, where $s$ and $u$ are global
{\it linear} variables on a half-space which can be regarded as being the part of the tangent bundle which is the
inward normal to $F_{y_0}$. This model space is naturally identified with the tangent cone with respect
to the family of dilations $(x,y,z) \mapsto (\lambda x, \lambda y, z)$ as $\lambda \nearrow \infty$.
Another operator which models the behaviour of $L$ near $F_{y_0}$ is the Bessel operator
\begin{equation}
B_{y_0, \hat{\eta}}(L) = \sum_{j + |\alpha| + |\beta| \leq m} a_{j \alpha \beta} (0,0,z) (t\partial_t)^j (-i t \hat{\eta})^\alpha \partial_z^\beta.
\label{Bessel}
\end{equation}
Here $\hat{\eta} = \eta/|\eta|$, where $\eta \in T_{y_0}^*B$ (i.e.\ $\hat{\eta}$ lies in the spherical conormal bundle
$S_{y_0}^*B$). This is obtained from $N(L)$ by first passing to the Fourier transform in $u$ (which transforms $s \partial_u$
to $-i s \eta$) and then rescaling by setting $t = s |\eta|$. These operations are reversible so the family $B_{y_0, \hat{\eta}}(L)$
is completely equivalent to $N_{y_0}(L)$ even though it appears to be simpler.
Note that the structure of $B(L)$ as $t \to \infty$ captures the behaviour as $|\eta| \to \infty$,
and hence corresponds to local behaviour for $N(L)$.
Finally, the indicial operator, which is an elliptic $b$-operator in the sense of \cite{Mel-APS},
\cite{M}, on $\mathbb R^+ \times F$, is defined by
\begin{equation}
I_{y_0}(L) = \sum_{j+|\beta| \leq m} a_{j 0 \beta} (0, 0, z) (t\partial_t)^j \partial_z^\beta.
\label{indicial}
\end{equation}
This is obtained from the Bessel-normal operator by dropping the terms which are lower order in the $b$-theory.
Thus $B_{y_0,\hat\eta}(L) = I_{y_0}(L) + E$, where $E$ is truly lower order on any finite interval $0 < t < t_0$. However,
as remarked above, the large $t$ behaviour of $B_{y_0,\hat\eta}(L)$ contains important information missing in the indicial operator.
\medskip
\noindent{\bf Indicial roots and the trace bundle} The indicial operator can be conjugated, via the Mellin transform,
to the {\it operator pencil},
\begin{equation}
I_{y_0}(L)(\zeta) := \sum_{j+|\beta| \leq m} a_{j 0 \beta} (0, 0, z)(-i \zeta)^j \partial_z^\beta,
\label{indfamily}
\end{equation}
which depends smoothly on $y_0 \in B$; this is often called the indicial family of $L$. (An operator pencil, an important
generalization of a resolvent family, is simply a polynomial family with operator coefficients.) Because the
coefficient of $\zeta^j$ has order $m-j$, and the coefficient of $\zeta^0$ is an elliptic operator on $F_{y_0}$
of order $m$. Hence the analytic Fredholm theorem may be applied, and this implies that this family is either never
invertible, for any $\zeta
\in \mathbb C$, or else that this inverse is meromorphic, and the Laurent coefficients at each pole are operators of finite
rank. It is certainly necessary to assume that $I_{y_0}(L)(0) = \sum_{|\beta| = m} a_{0 0 \beta}(z) \partial_z^\beta$ has
index zero, otherwise we are necessarily in the first, nowhere invertible, case. A standard condition to ensure
that the inverse exists at one point, and hence away from a discrete set, is that the resolvent of the (ordinary) symbol,
$(\sigma_m( I_{y_0}(L)(0)) - \lambda)^{-1}$, satisfy standard elliptic symbol-with-parameter estimates
in some open conic sector. In any case, we shall assume that we are in some setting which allows
us to conclude that the indicial family has meromorphic inverse.
\begin{defn}
The boundary spectrum of $L$, $\mbox{Spec}_b(L)$, is the set of locations of the poles of the meromorphic family
$I_{y_0}(L)^{-1}$ (this is also called the spectrum of the operator pencil); elements of $\mbox{Spec}_b(L)$ are called
the indicial roots of $L \ \textup{(at $y_0$)}$.
\end{defn}
We have tacitly suppressed the fact that these indicial roots may vary with $y_0$. The analysis of edge operators
with indicial roots depending nontrivially on $y_0$ is an interesting and difficult topic, and is discussed
in detail in the forthcoming work of Krainer and Mendoza. However, partly because this behaviour often does not
occur for the natural examples of edge operators, we choose to make the basic
\begin{assumption}[Constancy of indicial roots]
\label{discrete}
The spectrum of the indicial family is discrete and the location of the poles of $I_{y_0}(L)(\zeta)^{-1}$ does
not depend on $y_0 \in B$.
\label{assump-const-ind-rts}
\end{assumption}
It is necessary to make one further hypothesis:
\begin{assumption}
For each $(y_0, \hat\eta)$, the Bessel operator $B_{y_0,\hat\eta}(L)$ is injective on
$t^{\delta}L^2(dt dz)$ for $\delta \gg 0$, and is surjective when $\delta \ll 0$.
\label{assump-uniq-cont}
\end{assumption}
This holds in many interesting situations, see \cite{Ma-UC}.
We henceforth always work with operators satisfying both of these assumptions. In the following, we
fix two nonindicial values $\underline{\delta}$ and $\overline{\delta}$ such that $B(L)$ is injective on $t^{\overline{\delta}}L^2$ and surjective
on $t^{\overline{\delta}} L^2$. We then write $\mathfrak{S}(L)$ for the set of all indicial roots in this critical strip,
omitting their multiplicity:
\[
\mathfrak{S}(L):=\{\zeta_j \in \mathbb{C}: \underline{\delta} -1/2 < \Im\zeta_0 < \overline{\delta} - 1/2\ \mbox{and}\ \exists \, p \in \mathbb{N}\
\mbox{s.t.}\ (\zeta_j,p) \in \textup{Spec}_b(L)\}.
\]
The shift by $1/2$ appears because we are using the measure $dt dz$.
\begin{prop}
Let $\mathrm{\omega} \in t^{\underline{\delta}} L^2( dt dz)$ and $I_{y_0}(L) \mathrm{\omega} = f \in t^{\overline{\delta}}L^2(dt dz)$. Then
\[
\mathrm{\omega} = \sum_{\zeta_j \in \mathfrak{S}(L)} \, \sum_{p = 0}^{p_j} \mathrm{\omega}_{j, p}(z) t^{-i\zeta + \ell} (\log t)^p + \widetilde{\mathrm{\omega}},
\quad \mathrm{\omega}_{j,p} \in \mathcal C^\infty(F),\ \widetilde{\mathrm{\omega}} \in t^{\overline{\delta}}L^2(dt dz).
\]
Here $p_j + 1$ is equal to the order of the pole of $I_{y_0}(L)^{-1}$ at $\zeta_j$.
\label{explsoln}
\end{prop}
Dropping the subscript $y_0$, we pass to the Mellin transform of this equation (see \cite{M}),
which is $I(L)(\zeta) \mathrm{\omega}_M(\zeta, z) = f_M(\zeta, z)$. The Mellin transforms $\mathrm{\omega}_M(\zeta, z)$ and $f_M(\zeta,z)$ are
holomorphic in $\Im \zeta < \underline{\delta} - 1/2$ and $\Im \zeta < \overline{\delta} - 1/2$, respectively, both with values in $L^2(F; dz)$.
Thus $\mathrm{\omega}_M = - I(L)(\zeta)^{-1} f_M$ extends meromorphically to this larger half-plane. Taking the inverse Mellin
transform by integrating along the line $\Im \zeta = \delta < \underline{\delta} - 1/2$ and then shifting the contour across the
poles in this horizontal strip produces the expansion.
Using this result, we can define the trace of $\mathrm{\omega}$ to consist of the set of functions $\mathrm{\omega}_{j,p}$ over all $\zeta_j
\in \mathfrak{S}(L)$ and $p \leq p_j$. There is a subtlety here in that although we require that each $\zeta_j$
is independent of $y_0$, the same may not be true of the $p_j$, so we need to explain carefully the sense in which
this expansion depends smoothly on $y_0$.
To describe this we need make a small detour. First note that the algebraic multiplicity of each pole is well-defined.
As in the special case of the resolvent family of a non self-adjoint operator, this algebraic multiplicity is a
positive integer which measures the dimension of the space of generalized eigenvectors associated to that
indicial root. Since there are several different-looking (but equivalent) definitions of this quantity,
we provide a slightly longer description than strictly necessary, for the reader's convenience. For simplicity of notation,
omit the dependence on $y_0$ for the moment, and suppose that $\zeta_0$ is the indicial root in question.
The geometric eigenspace of $I(L)(\zeta_0)$ is the subspace of $\mathcal C^\infty(F)$ consisting of all $\phi_0$ such that
$I(L)(\zeta_0)\phi_0 = 0$, and its dimension is called the geometric multiplicity of the indicial root.
Suppose now that there exist additional functions $\phi_j \in \mathcal C^\infty(F)$, $j = 1, \ldots, k-1$ such that
\begin{equation}
\sum_{j=0}^{\ell} \frac{1}{j!} (\partial_\zeta^j I(L))(\zeta_0) \phi_j = 0, \ \ \ell = 1, \ldots, k-1;
\label{jordanchain}
\end{equation}
this sequence of equations is equivalent to the single condition
\[
I(L)(\zeta)( \phi_0 + (\zeta-\zeta_0) \phi_1 + \ldots (\zeta-\zeta_0)^{k-1} \phi_{k-1}) = \mathcal O(|\zeta-\zeta_0|^k).
\]
The ordered $k$-tuple $(\phi_0, \ldots, \phi_{k-1})$ is called a generalized eigenvector, and the maximal length of all such
chains beginning with $\phi_0$ is called the multiplicity of $\phi_0$ and denoted $m(\phi_0)$. In other words, $m(\phi_0)$
measures the order to which $\phi_0$ can be extended as a formal series solution of $I(L)(\zeta) \sum
(\zeta-\zeta_0)^j \phi_j \equiv 0$. We refer to $\phi_0(\zeta)= \sum (\zeta-\zeta_0)^j \phi_j$ as
the root function associated to the eigenvector $\phi_0$. Following \cite[\S 1.1]{KMR}, choose a basis
$\{\phi_{0,1}, \ldots, \phi_{0,N}\}$ for the geometric eigenspace of $I(L)(\zeta_0)$ so that $m(\phi_{0,1}) \leq
m(\phi_{0,2})\leq \cdots \leq m(\phi_{0,N})$, and then define the algebraic multiplicity of the pole to be the number
\[
m(\zeta_0) = \sum_{j=1}^N m(\phi_{0,j}).
\]
There is an alternate description, following \cite[Ch. XI]{GGK}, which gives a slightly different intuition for this number.
The first step is to write the holomorphic family $I(L)(\zeta)$ as the product $E(\zeta)D(\zeta) F(\zeta)$, locally
near $\zeta_0$. Here $E(\zeta)$ and $F(\zeta)$ are holomorphic and invertible for $|\zeta-\zeta_0| < \epsilon$ and
\[
D(\zeta) = P_0 + (\zeta- \zeta_0)^{\kappa_1} P_1 + \ldots + (\zeta- \zeta_0)^{\kappa_r} P_r,
\]
where the $P_j$ are mutually disjoint projectors (i.e.\ $P_i P_j =0$ if $i \neq j$), $P_0$ has infinite rank,
$\mbox{rank} \, P_j = 1$, $j > 0$, and $P_0 + \ldots + P_r = \mbox{Id}$. Clearly $\kappa_r$ is the order of
the pole of $I(L)(\zeta)^{-1}$ at $\zeta_0$, and a straightforward calculation shows that the algebraic multiplicity
$m(\zeta_0)$ is equal to $\kappa_1 + \ldots + \kappa_r$.
In any case, by an extension of the theorem of Keldy\u{s} \cite{Kel1},\cite{Kel2}, see Gohberg-Sigal \cite{GohSig} and
Menniken-M\"oller \cite{MM}, the generalized eigenvectors characterize the singular part of the Laurent expansion
of $I_{y_0}(L)(\zeta)^{-1}$ at $\zeta_0$, as follows. These sources prove that there exists a set of polynomials in $\zeta$,
$(\psi_1(\zeta), .., \psi_N(\zeta))$, taking values in $\mathcal D'(F)$ (distributions on $F$), and an operator-valued family $H(\zeta)$
which is holomorphic near $\zeta_0$, such that
\begin{align}\label{representation}
I_{y_0}(L)(\zeta)^{-1} = \sum_{j=1}^N (\zeta - \zeta_0)^{-k_j} \phi_{0,j}(\zeta) \otimes \psi_j(\zeta) + H(\zeta).
\end{align}
Here, for each $j$, $\phi_{0,j}(\zeta)$ is the root function corresponding to the element $\phi_{0,j}$ in the geometric
eigenspace and $k_j = m(\phi_{0,j})$. This implies that for any holomorphic function $u(\zeta)$ taking
values in $\mathcal C^\infty(F)$, each singular Laurent coefficient of $I_{y_0}(L) (\zeta)^{-1} u(\zeta)$ at $\zeta_0$
is a linear combinations of the coefficients $\phi_{\ell, j}$ of the root functions $\phi_{0,j}(\zeta) = \sum \phi_{\ell,j} (\zeta -
\zeta_0)^\ell$.
Going back to Proposition~\ref{explsoln}, and using the notation there, it is clear that each of the singular Laurent
coefficients of $\mathrm{\omega}_M$ at any pole $\zeta_0$ in the horizontal strip is linear combination of coefficients of the root
functions for $I(L)(\zeta_0)$, hence the coefficients of the terms $t^{-i\zeta_0} (\log t)^p$ in the partial expansion for $\mathrm{\omega}$
are constituted by these same functions.
We now finally come to the issue of smooth dependence on $y_0$. The key fact is that the algebraic multiplicity $m(\zeta_0)$
of each pole is invariant under small perturbations, and hence is locally independent of $y_0$. This follows from an
operator-valued version of Rouch\'e's theorem; we refer to \cite[\S 1.1.2]{KMR} for a more careful description,
and to \cite[Theorem 9.2]{GGK} for a proof.
In fact, slightly more is true: the direct sum of the coefficients of the root functions $\{\phi_{0,j}(\zeta), j \leq N\}$
form a vector space $\mathcal E_{ y_0}(\zeta_0)$ of dimension $m(\zeta_0)$, and as $y_0$ varies, these vector spaces fit
together as a smooth vector bundle $\mathcal E(\zeta_0) = \mathcal E(L; \zeta_0)$ over each connected component of $B$.
To define this bundle, write $\mathscr{M}(\zeta_0)$ and $\mathscr{H}(\zeta_0)$ for the spaces of germs of meromorphic
and holomorphic functions, respectively, at $\zeta_0$. Following the definitions above, if $u(\zeta)$ is a holomorphic
$\mathcal C^\infty(F)$-valued function defined near $\zeta_0$, then the Laurent coefficients of $I_{y_0}(L)(\zeta)^{-1} u(\zeta)$
at $\zeta_0$ lie in $\mathcal E_{y_0}(\zeta_0)$. We take this as our primary definition and hence let
\[
\mathcal E_{y_0}(\zeta_0) = \{ [u] \in \mathscr{M}(\zeta_0) / \mathscr{H}(\zeta_0) : [I_{y_0}(L)(\zeta)^{-1} u]=0\};
\]
equivalently, $\mathcal E_{ y_0}(\zeta_0)$ is identified with the kernel of $I_{y_0}(L)$ on the space of all finite combinations
$\sum_q a_{j, q}(z) t^{-i\zeta_0 }(\log t)^q$ with $a_{j, q} \in \mathcal C^\infty(F)$.
\begin{prop}[Krainer and Mendoza \cite{KM}]
\[
\mathcal E(L;\zeta_0) :=\coprod\limits_{y_0 \in B} \mathcal E_{y_0}(L;\zeta_0) \overset{\pi}\longrightarrow B
\]
is a smooth vector bundle of rank $m(\zeta_0)$.
\label{trace-bundle}
\end{prop}
We sketch some elements of the proof (recalling however that those authors work in the more general setting where the
$\zeta_j$ may also vary with $y$). For each $y_0 \in B$, and indicial root $\zeta_0 \in \mathfrak{S}(L)$, Krainer and Mendoza
construct an independent set of smooth functions $\{\phi_{y_0, j}\}_{j=1}^{m_0}$, $m_0=m(\zeta_0)$, in a neighbourhood $\mathcal U$ of
$y_0$, which form a basis of $\mathcal E_{y}(\zeta_0)$ for each $y\in \mathcal U$. They then show that if $\phi(t,y,z) \in t^{\underline{\delta}}L^2$
depends smoothly on $y \in \mathcal U \subset B$ and $z \in F$, and $I_y(L) \phi \equiv 0$, then there exist smooth functions
$f_j:\mathcal U\to \mathbb C$, $j=1,.., m_0$ such that
\[
\phi(t,y,z) = \sum_{j=1}^{m_0} f_j(y) \phi_{y,j}(t,z).
\]
It follows from this that $\mathcal E(L; \zeta_0)$ is a smooth vector bundle over $B$. A nonobvious
consequence is that the ranges of the various singular Laurent coefficients of $I_{y}(L)(\zeta)^{-1}$ remain
independent of one another as $y$ varies.
To conclude, let us remark that the full strength of Assumption~\ref{assump-const-ind-rts} is not needed: it is
only necessary that every indicial root with real part in the critical interval $[\underline{\delta}, \overline{\delta}]$ for any $y_0$ is independent
of $y_0$, so in particular, there are no indicial roots with imaginary parts crossing the levels $\overline{\delta}$, $\underline{\delta}$. We shall
phrase most results as if all indicial roots are constant, but remark at various points how
results change in this slightly more general setting.
\medskip
\noindent{\bf Mapping properties}
Each of the model operators described above plays an important role in determining the refined mapping properties of $L$.
The basic result, stated more carefully below, is that if both ${}^e\sigma_m(L)$ and $N(L)$ are invertible (as a bundle map and
as an operator between weighted Sobolev spaces, respectively); we encompass this pair of properties by saying
that $L$ is {\it fully elliptic} -- then $L$ itself is Fredholm between the analogous weighted Sobolev spaces. For this reason,
the pair $({}^e \sigma_m(L), N(L))$ should be regarded as the full symbol of $L$. This is the simplest nontrivial case of a symbol
hierarchy for iterated edge structures (as in Schulze's work).
We shall let $L$ act on weighted Sobolev and H\"older spaces. Fix a reference measure $dV = dx dy dz$ (more precisely,
$dV$ is a smooth, strictly positive multiple of Lebesgue measure). For any $k \in \mathbb N_0$, define
\[
H^k_e(M) = \{u: V_1 \ldots V_\ell u \in L^2(dV)\ \forall\, V_j \in \mathcal V_{\mathrm{e}}\ \mbox{and}\ \ell \leq k\}.
\]
Using interpolation and duality (or using edge pseudodifferential operators) one also defines $H^s_e(M)$ for any $s \in \mathbb R$.
We also define their weighted versions
\[
x^\delta H^s_e(M) = \{ u = x^\delta v: v \in H^s_e(M)\}.
\]
Note that these are the Sobolev spaces associated to any complete edge metric $g$ (though the measure $dV$ is equal
to $x^{b+1}$ times the Riemannian density for such a metric).
Similarly, we define the H\"older seminorm
\[
[ u ]_{e; 0,\alpha} = \sup_{ (x,y,z) \neq (x',y',z')} \frac{ |u(x,y,z) - u(x',y',z')|(x+x')^\alpha}{ |x-x'|^\alpha + |y-y'|^\alpha +
(x+x')^\alpha |z-z'|^\alpha}.
\]
This is simply the standard H\"older seminorm associated to the Riemannian distance associated to the complete metric $g$.
The edge H\"older space $\Lambda^{0,\alpha}_e(M)$ consists of functions $u$ such that $\sup |u| + [ u ] _{e, 0,\alpha} < \infty$.
We also define the weighted edge H\"older spaces
\[
x^\delta \Lambda^{k,\alpha}_e(M) = \{ u = x^\delta v: V_1 \ldots V_\ell v \in \Lambda^{0,\alpha}_e(M)\ \ell \leq k\ \mbox{and}\
V_j \in \mathcal V_{\mathrm{e}}\}.
\]
It is clear from the definitions that if $L \in \mathrm{Diff}_e^m(M)$, then
\begin{eqnarray}
& L: x^\delta H^s_e(M) &\longrightarrow x^\delta H^{s-m}(M) \label{Lsob} \\
& L: x^\delta \Lambda^{k+m,\alpha}_e(M) &\longrightarrow x^\delta \Lambda^k_e(M) \label{Lhold}
\end{eqnarray}
are bounded mappings for every $\delta, s \in \mathbb R$ and $k \in \mathbb N_0$. This is \cite[Cor. 3.23]{M}.
The basic and most important mapping property for elliptic edge operators is the following.
\begin{prop}(\cite[Thm. 6.1]{M}) Suppose that $L \in \mathrm{Diff}_e^m(M)$ is elliptic satisfying the
Assumption~\ref{assump-const-ind-rts}, and that $\delta \notin \mbox{Spec}_b(L)$. Suppose finally that
\[
B_{y_0, \widehat{\eta}}(L): t^\delta H^m_b( \mathbb R^+\times F; t^{-1} dt dz) \longrightarrow t^\delta L^2(\mathbb R^+ \times F; t^{-1} dt dz)
\]
is invertible for every $(y_0, \widehat{\eta})$. Then both \eqref{Lsob} and \eqref{Lhold} are Fredholm mappings. If
we only know that $B(L)$ is injective for all $(y_0, \widehat{\eta})$, then \eqref{Lsob} and \eqref{Lhold} are
semi-Fredholm and essentially injective; if $B(L)$ is surjective for every $(y_0, \widehat{\eta})$, then \eqref{Lsob} and \eqref{Lhold}
are semi-Fredholm and essentially surjective.
\label{semi-fred}
\end{prop}
\noindent{\bf Normalizations and conventions.} We first rewrite Assumption~\ref{assump-uniq-cont} in the
following form:
\begin{assumption}
There exists values $\underline{\delta} < \overline{\delta}$, $\underline{\delta}, \overline{\delta} \notin \mbox{Spec}_b(L)$, such that, for every $(y_0, \widehat{\eta})$,
\[
B_{y_0, \widehat{\eta}}(L): t^{\underline{\delta}}H^m_b \longrightarrow t^{\underline{\delta}} L^2
\]
is surjective, and
\[
B_{y_0, \widehat{\eta}}(L): t^{\overline{\delta}}H^m_b \longrightarrow t^{\overline{\delta}}L^2
\]
is injective.
\label{defdlb}
\end{assumption}
\begin{remark}
It is enough to assume that an `injectivity weight' $\overline{\delta}$ exists for both $B(L)$ and its adjoint $B(L)^*$ (taken with
respect to any fixed measure of the form $t^\gamma dt dz$). This holds simply because injectivity of $B(L)^*$
on some $t^\delta L^2$ is equivalent to surjectivity of $B(L)$ on another space $t^{\delta^*}L^2$, where $\delta^*$
is determined by $\delta$ and $\gamma$.
\end{remark}
Based on this, we see that Assumption~\ref{defdlb} will hold if both $B(L)$ and $B(L)^*$ satisfy the more basic
\begin{assumption}[Unique continuation property]
Any solution $u$ to $B(L) u = 0$ which vanishes to infinite order at $t = 0$ and which has subexponential growth as $t \to \infty$
is the trivial solution $u \equiv 0$.
\label{uniq-cont}
\end{assumption}
That this should always be true is quite believable, but has not been proved in general. It is known to hold in
the special case where $L$ is second order with diagonal principal part and $\dim F = 0$, see \cite{Ma-UC}.
\begin{remark}
Another observation which simplifies notation below is that the precise choice of measure $t^\gamma dt dz$ for
$B(L)$, or $x^\delta dx dy dz$ for $L$, (or other measures which differ from these by a smooth function $J$
which is uniformly bounded above and away from $0$) is irrelevant for these various mapping and regularity
properties. Obviously, the values of $\gamma$ and $J$ enter into the precise computations of adjoints, normalization
of weight parameters, etc., but do not in any way effect the nature of the any of the results below.
Thus we always assume that we are working with respect to the measure $dt dz$, or $dx dy dz$.
We also fix the two values $\underline{\delta}$ and $\overline{\delta}$ (and this choice of fixed measures) henceforth for the rest of the paper.
\label{fixmeas}
\end{remark}
One final remark: as noted earlier, we really only need to assume constancy of indicial roots with real part in the interval
$[\underline{\delta}, \overline{\delta}]$, though in that more general case, one has slightly weaker regularity statements (conormality
rather than complete polyhomogeneity).
\medskip
\noindent{\bf Generalized inverses}
Assume that $L\in \textup{Diff}^m_e(M)$ is elliptic and satisfies Assumptions~\ref{assump-const-ind-rts}
and \ref{uniq-cont}, and that $\underline{\delta}$ and $\overline{\delta}$ have been chosen as above. By Proposition~\ref{semi-fred}, the mapping
\eqref{Lsob} is semi-Fredholm whenever $\delta \geq \overline{\delta}$ or $\delta \leq \underline{\delta}$, and in either case, $\delta \notin \mbox{Spec}_b(L)$;
in other words, this mapping has closed range, and either finite dimensional nullspace or finite dimensional cokernel, respectively.
General functional analysis then gives, for each $s \in \mathbb R$, the existence of a generalized inverse $G$ for \eqref{Lsob}, which
is to say, there exists a bounded map
\begin{equation}
G:x^\delta H^s_e(M) \to x^\delta H^{s+m}_e(M)
\end{equation}
which satisfies $GL = I - P_1$, $LG = I - P_2$ where $P_1$ and $P_2$ are the orthogonal projectors onto the
nullspace of $L$ and orthogonal complement of the range of $L$, respectively. By the simplest form of elliptic
regularity in the edge setting, we obtain that $P_1$ and $P_2$ are both smoothing in the sense that
$P_1: x^\delta H^{s+m}_e \to x^\delta H^{t+m}_e$ and $P_2: x^\delta H^s_e \to x^\delta H^t_e$ are
bounded for any $t > s$.
Much more is true, and one of the strengths of the pseudodifferential edge theory is that it allows one
to give a fairly explicit description of the Schwartz kernels of these operators. Fix $\delta$ as above, and
set $s=0$ (to normalize the choice of projectors). Then Theorem 6.1 in \cite{M} asserts that $G$, $P_1$ and $P_2$
are all pseudodifferential edge operators. When $\delta > \overline{\delta}$, then $P_1$ has finite rank and maps into the space
of polyhomogeneous functions, while when $\delta < \underline{\delta}$, then $P_2$ has finite rank and maps into the space of
polyhomogeneous functions. We shall recall the definitions of these spaces of pseudodifferential operators in \S 4,
but for now point out that this description of their Schwartz kernels has a number of important ramifications.
For example, once one establishes a general boundedness
theorem for pseudodifferential edge operators on weighted edge H\"older spaces, then it is an immediate
consequence of this Sobolev semi-Fredholmness that one can then deduce that for this same value of $\delta$,
the mapping \eqref{Lhold} is also semi-Fredholm, and that $P_1$ and $P_2$ are the appropriate projectors
in that case too. Indeed, the equations $GL = I - P_1$ and $LG = I - P_2$ still hold, and all operators are
bounded on the appropriate spaces. Note in particular that if $\delta < \underline{\delta}$, for example, then the nullspace
of \eqref{Lhold} is infinite dimensional and in this case it does not follow from general theory that this
nullspace is complemented in $x^\delta \Lambda_e^{m+k}$. Nonetheless, since the infinite rank projectors
$P_1$ and $I - P_1$ are bounded, we see that this nullspace has a complement, as claimed.
\section{Outline and statement of the main result}\label{overview}
We are now in a position to provide a more careful statement of our main results and to
sketch the arguments to prove them.
There are several closely related conceptual frameworks for studying elliptic boundary problems;
the one we follow here is very close to the one developed by Boutet de Monvel \cite{BdM}, and used
in many other places since, including by Schulze \cite{S1, S2} for edge operators. This theory is
centered around the idea of extending the use of `interior' pseudodifferential edge operators
by introducing the associated spaces of trace and Poisson operators, as well as boundary
operators along the edge $B$.
What distinguishes our approach here is the focus on the geometric structures of the
Schwartz kernels of these various types of operators. As in \cite{M}, any one of
these operators has a Schwartz kernel which is a polyhomogeneous distribution
on a certain blown up space. Section~\ref{micro} describes all of this more carefully.
Amongst the tasks we must face is to show that the composition of an interior
edge operator and a trace operator (interior to boundary) is again a trace operator,
and similarly the composition of a Poisson operator (boundary to interior) with
an interior edge operator is again of Poisson type. These composition formul\ae\
are perhaps the most technically demanding part of this presentation.
The operators which arise in these elliptic boundary problems are of a somewhat
more special type, which we call representable. This is described in \S \ref{representable},
where we introduce these subclasses of interior, trace and Poisson edge operators
and examine their normal operators.
Following these more general `structural' definitions and results, we turn to the analysis
specific to elliptic differential edge operators. In the steps below, we first define each object
at the level of Bessel operators, where the issues are typically finite dimensional. We then
rescale and take inverse Fourier transforms and obtain the corresponding objects at the
level of normal operators. Although everything becomes infinite dimensional, it is still
completely equivalent to the finite dimensional problem. The last step is to extend each
object from the normal operator level to that of the actual operators, and this is where
the special class of representable operators becomes important.
The starting point is to identify the spaces on which the boundary trace map is well defined.
We define
\begin{equation}
\begin{split}
\mathcal H^{B(L)}_{\overline{\delta},\underline{\delta}} & = \{u \in t^{\underline{\delta}} H^m(\mathbb R^+ \times F_{y_0}; dt\, dz) \mid B_{y_0,\widehat{\eta}}(L) u \in t^{\overline{\delta}} L^2\} \\
\mathcal H^{N(L)}_{\overline{\delta},\underline{\delta}} & = \{u \in s^{\underline{\delta}} H^m(\mathbb R^+ \times \mathbb R^b \times F_{y_0}; ds\, dY \, dz) \mid N_{y_0}(L) u \in s^{\overline{\delta}} L^2\} \\
\mathcal H^{L}_{\overline{\delta},\underline{\delta}} & = \{u \in x^{\underline{\delta}} H^m(M; dx \, dy\, dz) \mid L u \in x^{\overline{\delta}} L^2 \},
\end{split}
\label{defHsp}
\end{equation}
where by implication the second inclusion is supposed to hold for all $y_0, \widehat{\eta}$. For simplicity we often denote
these simply as $\mathcal H^B$, $\mathcal H^N$ and $\mathcal H$, omitting the subscript $\overline{\delta},\underline{\delta}$. These are Hilbert spaces with respect
to the norms
\begin{equation}
\begin{split}
||u||_{\mathcal H^B} & = || u||_{t^{\underline{\delta}}L^2} + || B(L)u ||_{t^{\overline{\delta}}L^2} \\
||u||_{\mathcal H^N} & = || u||_{s^{\underline{\delta}}L^2} + || N(L)u ||_{s^{\overline{\delta}}L^2} \\
||u||_{\mathcal H} & = || u||_{x^{\underline{\delta}}L^2} + ||L u ||_{x^{\overline{\delta}}L^2}.
\end{split}
\end{equation}
In \S \ref{trace-pot} we construct successively the trace and Poisson operators associated to an elliptic edge operator $L$
by first constructing the corresponding operators for $B(L)$ and $N(L)$. Since $B(L)$ is Fredholm at all nonindicial weights,
most of the considerations for it are finite dimensional and we may formulate the analogue of the Calderon, or
Lopatinski-Schapiro conditions directly. The starting point is that $\mathcal H^B$ is the natural domain for the boundary trace
map for $B(L)$, and in fact for each $y_0 \in B$,
\[
\mathrm{Tr}_{B(L)}: \mathcal H^B \longrightarrow \mathcal E_{y_0} := \bigoplus \mathcal E _{y_0} (L, \zeta_j),
\]
where $\mathcal E_{y_0}(L, \zeta_j)$ is the fibre of the trace bundle \eqref{trace-bundle} associated to the indicial root $\zeta_j$ at $y_0$
and the direct sum is over all indicial roots with imaginary part in the interval $(\underline{\delta}-1/2, \overline{\delta}-1/2)$. The
corresponding trace map for the normal operator $N(L)$ is obtained by rescaling and taking the inverse Fourier transform, and
\[
\mathrm{Tr}_{N(L)}: \mathcal H^N \longrightarrow \bigoplus H^{-(\Im (\zeta_j) -\underline{\delta} +1/2)}(\mathbb R^b; \mathcal E_{y_0}).
\]
The trace map for $L$ itself is bounded as a map
\[
\mathrm{Tr}_{L}: \mathcal H \longrightarrow \bigoplus H^{-(\Im (\zeta_j) -\underline{\delta} +1/2)}(B; \mathcal E).
\]
In a similar way, we construct the Poisson edge operators $P_{B(L)}$, $P_{N(L)}$ and
\[
P_L: \bigoplus H^{-(\Im (\zeta_j) -\underline{\delta} +1/2)}(B, \mathcal E(\zeta_j)) \longrightarrow \ker L \cap x^{\underline{\delta}}H^{\infty}_e(M).
\]
By construction, $P_L \circ \mathrm{Tr}_L$ is the identity on $\ker L \cap \mathcal H$. The Calderon subspaces
\begin{multline*}
\mathcal C_{B(L)} = \mathrm{Tr}_{B(L)} ( \ker B(L) \cap \mathcal H^B), \quad \mathcal C_{N(L)} = \mathrm{Tr}_{N(L)} ( \ker N(L) \cap \mathcal H^N), \quad \mbox{and}\\
\hfill \mathcal C_{L} = \mathrm{Tr}_{L} ( \ker L \cap \mathcal H) \hfill
\end{multline*}
are of fundamental importance. For $B(L)$ this subspace depends smoothly on $(y_0,\widehat{\eta})$, and for $N(L)$ it
depends smoothly on $y_0$.
Let us now explain how to formulate a boundary problem for the edge operator $L$. Fix a vector bundle $W$ over $B$
and a pseudodifferential operator
\[
Q:\mathcal C^\infty (B, \mathcal E) \to \mathcal C^\infty (B,W).
\]
For many operators of interest, $W$ splits as a finite direct sum $\bigoplus W_k$, and of course $\mathcal E$ also splits
into the summands corresponding to each indicial root, so $Q$ has a matrix form $(Q_{jk})$ where the different components
may have different orders.
\begin{defn}
With all notation as above, an edge boundary value problem $(L,Q)$ is a system
\begin{align*}
Lu & = f\in x^{\overline{\delta}}L^2(M), \ u \in \mathcal H_{\underline{\delta},\overline{\delta}} \subset x^{\underline{\delta}}H^m_e(M), \\
Q (\mathrm{Tr}_L u) & =\phi \in \bigoplus\limits_{k=1}^M H^{\underline{\delta} -d_k-1/2}(B,W_k).
\end{align*}
\end{defn}
As in the classical theory on a manifold with boundary, the determinantion of whether this problem is Fredholm is formulated
using the (left or right) invertibility of the principal symbol of the boundary conditions restricted to the Calderon subspace:
\begin{defn}
The boundary conditions $Q$ of an edge boundary value problem $(L,Q)$ are
\begin{enumerate}
\item right-elliptic if $\sigma(Q)(y_0,\hat\eta)\restriction \mathcal C_{B(L)(y_0,\widehat{\eta})}: \mathcal C_{B(L)(y_0,\widehat{\eta})} \to \pi^* W_{y_0}$
is surjective,
\item left-elliptic if $\sigma(Q)(y_0,\hat{\eta})\restriction \mathcal C_{B(L)(y_0,\widehat{\eta})}: \mathcal C_{B(L)(y_0,\widehat{\eta})} \to \pi^* W_{y_0}$
is injective, and
\item elliptic if $\sigma(Q)(y_0,\hat{\eta})\restriction \mathcal C_{B(L)(y_0,\widehat{\eta})}:\mathcal C_{B(L)(y_0,\widehat{\eta})} \to \pi^* W_{y_0}$
is an isomorphism
\end{enumerate}
for all $(y_0,\hat\eta) \in S^*B$, where $\pi: S^*B \to B$ is the standard projection.
\label{typesofbcs}
\end{defn}
The final section, \S \ref{fredholm}, assembles the various types of operators considered earlier to construct parametrices
in each of these three cases. Our main result is the
\begin{thm}
Let $(L,Q)$ be right-elliptic. Let $G$ be the generalized inverse for $L$ on $x^{\underline{\delta}}L^2$. Then
\[
(L,Q): (\mathcal H,\|\cdot \|_\mathcal H)\to x^{\overline{\delta}}L^2(M)\oplus \left(\bigoplus\limits_{k=1}^MH^{\underline{\delta}-d_k-1/2}(B,W_k)\right),
\]
is semi-Fredholm with right parametrix
\begin{align*}
\mathcal{G}(f,\phi)= Gf + P_L[K(\phi - Q(\mathrm{Tr}_{L}Gf))].
\end{align*}
In particular, $(L,Q)$ has closed range of finite codimension.
\end{thm}
\begin{thm}
Let $(L,Q)$ be left-elliptic. Then
\[
(L,Q): (\mathcal H,\|\cdot \|_\mathcal H)\to x^{\overline{\delta}}L^2(M)\oplus \left(\bigoplus\limits_{k=1}^MH^{\underline{\delta}-d_k-1/2}(B,W_k)\right),
\]
is semi-Fredholm with left parametrix
\begin{align*}
\mathcal{G}(f,\phi)= Gf + P_L[K(\phi - Q(\mathrm{Tr}_{L}\, Gf))].
\end{align*}
In particular, $(L,Q)$ has a finite-dimensional kernel.
\end{thm}
These results together prove that an elliptic edge boundary problem gives a Fredholm mapping.
\section{Interior, trace and Poisson edge operators}\label{micro}
In this section we recall the space of pseudodifferential edge operators and introduce the corresponding spaces
of trace and Poisson operators. As explained earlier, our focus is on the Schwartz kernels of these operators, in
particular their structure as polyhomogeneous distributions. We keep the notation of the preceding sections.
The definitions below are phrased in the language of manifolds with corners and various spaces of conormal or
polyhomogeneous functions on them, so we review some of this now. A manifold with corners is a space locally
diffeomorphically modelled on neighbourhoods in the standard orthant $(\mathbb R^+)^\ell \times \mathbb R^{n-\ell}$. A standing
assumption is that every boundary face of a manifold with corners is embedded.
This implies, in particular, that if $H$ is a boundary hypersurface, then there is a globally defined boundary defining
function $\rho_H$ which vanishes precisely on $H$ and is strictly positive everywhere else, and is such that $d\rho_H \neq 0$ at $H$.
The most useful and natural classes of `smooth' functions on a manifold with corners $\mathfrak{W}$ are the conormal and
polyhomogeneous distributions. Let $\{(H_i,\rho_i)\}_{i=1}^N$ enumerate the boundary hypersurfaces and corresponding
defining functions of $\mathfrak{W}$. For any multi-index $b= (b_1,\ldots, b_N)\in \mathbb{C}^N$ set $\rho^b = \rho_1^{b_1} \ldots \rho_N^{b_N}$.
Similarly, for $p = (p_1, \ldots, p_N) \in \mathbb N_0^N$, we write $(\log \rho)^p = (\log \rho_1)^{p_1} \ldots (\log \rho_N)^{p_N}$.
Finally, let $\mathcal V_b(\mathfrak{W})$ be the space of all smooth vector fields on $\mathfrak{W}$ which are unconstrained
in the interior but which lie tangent to all boundary faces.
\begin{defn}\label{phg}
A distribution $u$ on $\mathfrak{W}$ is said to be conormal of order $b$ at the faces of $\mathfrak{W}$,
written $u \in \mathscr{A}^b(\mathfrak{W})$,
if $u\in \rho^b L^\infty(\mathfrak{W})$ for some $b\in \mathbb{C}^N$ and $V_1 \ldots V_\ell u \in \rho^b L^\infty(\mathfrak{W})$
for all $V_j \in \mathcal V_b(\mathfrak{W})$ and for every $\ell \geq 0$.
An index set $E$ is a collection of pairs $\{(\gamma,p)\} \subset \mathbb C \times \mathbb N_0\}$ satisfying the following hypotheses:
\begin{enumerate}
\item $\Re \gamma$ accumulates only at plus infinity, while the second index $p$ for a given $\gamma$ is bounded above
by a constant depending on $\gamma$, i.e.\ $p \leq P_\gamma < \infty$;
\item If $(\gamma,p) \in E$, then $(\gamma+j,p') \in E_i$ for all $j \in \mathbb N$ and $0 \leq p' \leq p$.
\end{enumerate}
An index family $\mathcal E = (E_1, \ldots, E_N)$ is an $N$-tuple of index sets associated to each of
the boundary hypersurfaces of $\mathfrak{W}$. In the rest of this paper, we typically let $k$ stand for
the simple index set $\{(k+\ell,0): \ell \in \mathbb N_0\}$.
A conormal distribution $u$ on $\mathfrak{W}$ is said to be polyhomogeneous with index family $\mathcal E$,
$u \in \mathscr{A}_{\mathrm{phg}}^\mathcal E(\mathfrak{W})$, if $u\in \mathscr{A}^*$, and if in addition, near each $H_i$,
\[
u \sim \sum_{(\gamma,p) \in E_i} a_{\gamma,p} \rho_i^{\gamma} (\log \rho_i)^p, \ \mbox{as} \ \rho_i\to 0,
\]
with coefficients $a_{\gamma,p}$ conormal on $H_i$, polyhomogeneous with index $E_j$ at any $H_i\cap H_j$.
We also require that $u$ have product type expansions at all corners of $\mathfrak{W}$.
\end{defn}
A $p$-submanifold in a manifold with corners $\mathfrak{W}$ is an embedded submanifold with the property that
if $p \in S$, then it is possible to choose coordinates $(x,y) \in (\mathbb R^+)^k \times \mathbb R^{n-k}$ for $\mathfrak{W}$
with $p = (0,0)$, and such that $S = \{(x,y): x'' = 0, y'' = 0\}$, where $x = (x',x'')$ and $y = (y', y'')$ are some
subdivisions of these sets of coordinates. In other words, $\mathfrak{W}$ has a product structure near $S$.
We may then define the new manifold with corners $[\mathfrak{W}, S]$ by blowing up $\mathfrak{W}$ around $S$.
This consists of taking the disjoint union $\mathfrak{W} \setminus S$ and the inward-pointing normal bundle
of $S$, and endowing this set with the structure of a smooth manifold with corners, with the unique minimal
differential structure so that smooth functions on $\mathfrak{W}$ and polar coordinates around $S$ all lift to
be smooth. This blown up space has a `front face', which is a new boundary hypersurface which projects
down to $S$ in the `blowdown'; it is the total space of a fibration over $S$ with fibre some spherical orthant.
\subsection{Pseudodifferential edge operators}\label{edge-pseudos}
Let $M^2_e$ denote the double edge space, which is obtained by blowing up the fibre diagonal of $(\partial M)^2$
in the product $M^2$, $M^2_e = [ M^2; \mathrm{fdiag}]$. In standard adapted local coordinates $(x,y,z)$
on $M$ near $\partial M$, with $(\widetilde{x},\widetilde{y},\widetilde{z})$ a copy of these coordinates on the other factor of $M$ in $M^2$,
the fibre diagonal $\mathrm{fdiag}$ is the submanifold $\{x = \widetilde{x} = 0, y = \widetilde{y}\}$; it is the total space of a fibration over
$\mbox{diag}\,(B \times B)$ with fibre $S^n_+ \times F \times F$. The space $M^2_e$ is a manifold with corners up to codimension
three; there are three boundary hypersurfaces, denoted $\mathrm{ff}$ (the front face), $\mathrm{lf}$ (the left face) and $\mathrm{rf}$ (the right face).
The front face is the one created by the blowup; it is the total space of a fibration over $\mathrm{fdiag}$ with each fibre a copy
of the quarter-sphere $\{\omega = (\omega_0, \omega', \omega_n) \in S^n: \omega_0, \omega_n \geq 0\}$.
It is often more convenient to use projective coordinates rather than polar coordinates. Thus away from $\mathrm{rf}$, we use
\begin{equation}
\label{proj-coord}
s=\frac{x}{\widetilde{x}}, \, Y=\frac{y-\widetilde{y}}{\widetilde{x}}, \, z, \, \widetilde{x}, \, \widetilde{y}, \, \widetilde{z},
\end{equation}
where $\widetilde{x}$ and $s$ are defining functions of $\mathrm{ff}$ and $\mathrm{rf}$, respectively. Note that in these coordinates, $\mathrm{ff}$
is the face where $\widetilde{x} = 0$. There are analogous coordinates valid away from $\mathrm{rf}$, obtained by interchanging the roles of $x$ and $\widetilde{x}$.
Figure 1 illustrates $M^2_e$
\begin{figure}[h]
\begin{center}
\begin{tikzpicture}
\draw (0,0) -- (2,1);
\draw (4,2) -- (6,3);
\draw (0,0) -- (4,0);
\draw (0,0) -- (0, 2.5);
\draw (2,1) .. controls (2.1,2.3) and (2.9,2.7) .. (4,2);
\draw (2,1) .. controls (3.8,0.6) and (4.2,1.4) .. (4,2);
\node at (3.1,1.6) {\large{ff}};
\node at (1.1,2.7) {\large{lf}};
\node at (5.2,0.7) {\large{rf}};
\end{tikzpicture}
\end{center}
\label{figure-edge}
\caption{The edge double space $M^2_e$.}
\end{figure}
This space has a distinguished submanifold, the edge diagonal $\mathrm{diag}_e$, which is the lift of
the diagonal to $M^2_e$. (Strictly speaking, it is the closure of the lift of the interior of the diagonal.)
A linear operator $A$ on $M$ is called a pseudodifferential edge operator of order $m$ and with index family
$\mathcal E$, $A \in \Psi_e^{m, \mathcal E}(M)$, if the lift of its Schwarz kernel $K_A$ to $M^2_e$ is polyhomogeneous
distribution on this space, where the index sets $\mathcal E = (E_{\mathrm{ff}}, E_{\mathrm{lf}}, E_{\mathrm{rf}})$ describe the expansions at the three faces. The superscript $-\infty$ indicates
the pseudodifferential order, hence the lifted Schwartz kernel is smooth along $\mbox{diag}_e$. The full space
of pseudodifferential edge operators, $\Psi^{*,\mathcal E}_e(M)$, consists of the space of sums $A + B$ where
$A$ is an operator of order $-\infty$ as above, and where the lift of the Schwartz kernel of $B$ to $M^2_e$
is supported near $\mbox{diag}_e$, has a classical conormal singularity along that submanifold, and is smoothly
extendible (after factoring out a certain singular density) across $\mathrm{ff}$. To understand the singular density
here, note that the identity operator has Schwartz kernel which lifts as
\[
\delta(x-\widetilde{x}) \delta(y-\widetilde{y}) \delta(z-\widetilde{z}) d\widetilde{x} d\widetilde{y} d\widetilde{z} = \delta(s-1) \delta(Y) \delta(z-\widetilde{z}) \, \widetilde{x}^{-b-1} d\widetilde{x} d\widetilde{y} d\widetilde{z}.
\]
This is smoothly extendible across the front face, which in these projective coordinates is where $\widetilde{x} = 0$,
provided we factor out the final singular measure. In the language above, $\mbox{Id} \in \Psi^{0, \varnothing}_e(M)$.
There is a distinguished subalgebra $\Psi_e^{*}(M)$, called the small calculus, which consists of operators
which vanish to infinite order at the left and right faces, $E_{\mathrm{lf}} = E_{\mathrm{rf}} = \varnothing$, and with
$E_{\mathrm{ff}} = 0$. The residual calculus $\Psi_e^{-\infty, 0, E_{\mathrm{lf}}, E_{\mathrm{rf}}}(M)$ consists of operators with no singularity
along the lifted diagonal and with standard index set $0$ at the front face.
Many details have been suppressed here, and we refer to \cite{M} where all of this is described more carefully.
\subsection{Edge trace operators}
Whereas the edge operators introduced in the previous subsection map functions on $M$ to functions on $M$, the
other two classes of operators we consider map functions on $M$ to functions on $\partial M$ (these are the edge
trace operators) or functions on $\partial M$ to functions on $M$ (these are the edge Poisson operators. We now
describe the former of these.
An edge trace operator $T$ is again described in terms of the lifting properties of its Schwartz kernel.
Initially this Schwartz kernel is a distribution on $\partial M \times M$; this space has the same distinguished
submanifold as before, namely the fibre diagonal of $(\partial M)^2$, $\mathrm{fdiag} = \{\widetilde{x} = 0, y = \widetilde{y}\}$.
We define the edge trace double space
\[
T^2_e = [\partial M \times M; \mathrm{fdiag}];
\]
note that this is nothing other than the right face $\mathrm{rf}$ of $M^2_e$. It has two boundary hypersurfaces,
the new front face of which, still denoted here by $\mathrm{ff}$, is simply one boundary face of the front face
of $M^2_e$, and hence a bundle of hemispheres $S^{n-1}_+$ over $\mathrm{fdiag}$. The lift of the original
face here is denoted $\mathrm{of}$, and still called the original face.
We can use the same projective coordinates as before, namely $(Y, z, \widetilde{x}, \widetilde{y}, \widetilde{z})$ with $Y = (y-\widetilde{y})/\widetilde{x}$.
Figure 2 illustrates this space.
\begin{figure}[h]
\begin{center}
\begin{tikzpicture}
\draw (0,0) -- (2,1);
\draw (4,2) -- (5.5,2.75);
\draw (0,0) -- (0, 2.5);
\draw (2,1) .. controls (2.1,2.3) and (3.2,3) .. (4,2);
\node at (3.1,1.8) {\large{ff}};
\node at (6.1,3.1) {\large{rf}};
\end{tikzpicture}
\end{center}
\caption{The trace blowup $T^2_e$.}
\label{fig-trace}
\end{figure}
\begin{defn}\label{trace-op}
The space $\Psi^{k, F_{\mathrm{rf}}}_e(M)$ of trace operators of order $k\in \mathbb{N}_0$
is the space of all operators $T$ with Schwartz kernels $K_T$ which are pushforwards
from polyhomogeneous conormal distributions $\kappa_T$ on the trace blowup space $T^2_e$ which
have index set $F_{\mathrm{of}}$ at the original face, and index set $F_{\mathrm{ff}}= (-1-b + k)+\mathbb{N}_0$ at the front face.
\end{defn}
\subsection{Edge Poisson operators}
The last class of operators we define are those which act from functions on $\partial M$ to functions on $M$.
The ones amongst these in which we are particularly interested are analogues of the classical Poisson operators,
and hence take functions on the boundary to functions in the interior which are solutions of an elliptic edge operator $L$.
However, it is advantageous to consider the full class of all operators with the relevant structure.
The Schwartz kernel of an edge Poisson operator $P$ is a distribution on $M \times \partial M$, and as usual,
we consider distributions which lift to be polyhomogeneous on the edge Poisson double space $P^2_e$,
obtained from $M \times \partial M$ by blowing up the same fibre diagonal $\mathrm{fdiag}$. The space
$P^2_e$ is naturally identified with the left face $\mathrm{lf}$ of $M^2_e$; it has two boundary hypersurfaces,
the front face $\mathrm{ff}$, which is `the other' boundary hypersurface of the front face of $M^2_e$, and the
original face $\mathrm{of}$. We often use projective coordinates $(x, z, Y, \widetilde{y}, \widetilde{z})$ with $Y=(y-\widetilde{y})/x$.
It is illustrated in Figure 3 (which is just the `transpose' of Figure 2).
\begin{figure}[h]
\begin{center}
\begin{tikzpicture}
\draw (0,0) -- (2,1);
\draw (4,2) -- (5.5,2.75);
\draw (0,0) -- (4,0);
\draw (2,1) .. controls (4,0.6) and (4.3,1.7) .. (4,2);
\node at (3.3,1.5) {\large{ff}};
\node at (6,3) {\large{lf}};
\end{tikzpicture}
\end{center}
\caption{The Poisson double space $P_e$.}
\label{fig-pot}
\end{figure}
\begin{defn}\label{potential-op}
The space $\Psi^{ k, J_{\mathrm{lf}}}_e(P^2_e)$ of edge Poisson operators of order $k\in \mathbb{N}_0$
is the space of all operators $P$ with Schwartz kernels $K_P$ which are pushforwards
from polyhomogeneous conormal distributions $\kappa_P$ on the edge Poisson double
space $P^2_e$, with index sets $J_{\mathrm{of}}$ at the original face, and $J_{\mathrm{ff}}= (-1-b+k)+\mathbb{N}_0$ at the front face.
\end{defn}
Comparing with the Boutet de Monvel calculus, one expects that we should also include operators mapping
functions on $\partial M$ to functions on $\partial M$. Indeed, a complete analogue of that calculus (as in the work
of Schulze) would indeed include these, but this is not necessary for our purposes here. Note that the operators
of this type we would need are not of any particularly standard type; their Schwartz kernels on $(\partial M)^2$
should be conormal at the fibre diagonal $\mathrm{fdiag}$, rather than the diagonal of the boundary.
These are, in some sense, lifts of pseudodifferential operators from $B^2$ to $(\partial M)^2$.
\subsection{Composition formul\ae}\label{triple}
The key fact which makes the definitions above useful is that these classes of operators are closed
under composition. This statement must be qualified to account for two issues. The first is the trivial
observation that one can only compose operators of the appropriate types, e.g.\ $T \circ A$ is defined
if $A$ is an interior edge operator and $T$ is an edge trace operator, and similarly, $A \circ P$ is defined
if $P$ is an edge Poisson operator and $A$ an interior edge operator, but of course $P \circ T$ is not defined, etc.
More seriously, however, even when composing two interior edge operators, the composition may not be defined because
of integrability issues. Thus if $A \in \Psi^{*,\mathcal E}_e$ and $A' \in \Psi^{*,\mathcal E'}_e$, then $A \circ A'$ is defined only
if $E_{\mathrm{rf}} + E'_{\mathrm{lf}} > -1$ (this lower bound depends on the choice of reference measure). The full composition
theorem for interior edge operators is proved in \cite{M}, and we prove here the analogous results
for compositions involving edge Poisson and trace operators. The main point in all of this is the more subtle fact
that if two operators have Schwartz kernels which lift to be polyhomogeneous on the appropriate blown-up space, then the
same is true for the composition. This can be verified `by hand', breaking up the
regions of integration into different neighbourhoods and using projective coordinate systems
to check the polyhomogeneity of these localized integrals. There is a much more elegant
and conceptual way, due to Melrose, and employed in \cite{M} (and many
other places), using the `pushforward theorem'. This states that under appropriate conditions on
a map $f: X \to X'$ between two manifolds with corners, the pushforward of a polyhomogeneous
distribution is polyhomogeneous. We review this result now and apply it to state the composition formul\ae.
First introduce some terminology. Let $X$ and $X'$ be two compact manifolds with corners, and $f: X \to X'$
a smooth map. Let $\{H_i\}$ and $\{H_j'\}$ be enumerations of the codimension one boundary faces of $X$ and $X'$,
respectively, and let $\rho_i$, $\rho_j'$ be global defining functions for $H_i$, resp.\ $H_j'$. We say that the map $f$
is a $b$-map if
\[
f^* \rho_i' = A_{ij} \prod_i \rho_j^{e(i,j)}, \quad 0 < A_{ij} \in \mathcal C^\infty(X),\ e(i,j) \in \mathbb{N} \cup \{0\};
\]
in other words, $f^* \rho_j'$ vanishes to constant order along each boundary face of $X$. In particular, this
means that if $f(H_i) \cap H_j' \neq \varnothing$, then $f(H_i) \subset H_j'$, and the order of vanishing of $f$ in the
direction normal to $H_i$ is constant along the entire face.
Next, $f$ is called a $b$-submersion if $f_*$ induces a surjective map between the $b$-tangent bundles of $X$ and $X'$. (The
$b$-tangent space at a point $p$ of $\partial X$ on a codimension $k$ corner is spanned locally by the
sections $x_1 \partial_{x_1}, \ldots, x_k \partial_{x_k}, \partial_{y_j}$, where $x_1, \ldots, x_k$ are the defining functions for the faces
meeting at $p$ and the $y_j$ are local coordinates on the corner through $p$.) Finally, if we require that $f$ is not
only a $b$-submersion, but that in addition, for each $j$ there is at most one $i$ such that $e(i,j) \neq 0$
(this condition simply means that each hypersurface face $H_i$ in $X$ gets mapped into \emph{at most one}
$H_j'$ in $X'$, or in other words, no hypersurface in $X$ gets mapped to a corner in $X'$), then $f$ is called a $b$-fibration.
Let $\nu_0$ be any smooth density on $X$ which is everywhere nonvanishing and smooth up to all boundary faces of $X$.
A smooth $b$-density $\nu_b$ is, by definition, any density of the form $\nu_b =\nu_0 (\Pi \rho_i)^{-1}$.
Fix smooth nonvanishing $b$-densities $\nu_b$ on $X$ and $\nu_b'$ on $X'$.
\begin{prop}[The Pushforward Theorem (Melrose)]
Let $u$ be a polyhomogeneous function on $X$ with index set $E_j$ at the face $H_j$, for all $j$. Suppose that if
$e(i,j) = 0$ for all $i$, i.e.\ $H_j$ is mapped to the interior of $X'$, then $\mbox{Re}\, z > 0$ for all $(z,p) \in E_j$.
In this case, the pushforward $f_* (u \nu_b)$ is well-defined and equals $h \nu_b'$ where $h$ is polyhomogeneous
on $X'$ and has an index family $f_b(\mathcal E)$ given by an explicit formula in terms of the index family $\mathcal E$ for $X$.
\end{prop}
We do not state the formula for the index set of the pushforward in generality, but give an informal description sufficient
for the present situation. If $H_{j_1}$ and $H_{j_2}$ are both mapped to a face $H_i'$, and if $H_{j_1} \cap H_{j_2} = \varnothing$,
then the pushforward has index set $E_{j_1} + E_{j_2}$ at $H_i'$. If they do intersect, then the contribution is the extended
union $H_{j_1} \overline{\cup} H_{j_2}$. For any two index sets $E,E'$ their \emph{extended union} $E\overline{\cup}E'$
is defined by
\begin{align}
\label{extended}
E \overline{\cup} E' = E \cup E' \cup \{((z, p + q + 1): \exists \, (z,p) \in E,\
\mbox{and}\ (z,q) \in E' \}.
\end{align}
After these generalities, we can now state the composition results between interior and Poisson operators and
between Poisson and trace operators. Note that the composition formula between trace and interior operators
is the adjoint of the interior-Poisson composition, so we do not state it separately.
\subsubsection{Interior $\circ$ Poisson}
Let $G$ be an interior edge operator and $P$ a Poisson edge operator and consider the (only possible) composition $A=G \circ P$.
To show that this is again a Poisson edge operator, we must verify that the Schwartz kernel of this composition
lifts to be polyhomogeneous on $P^2_e$ and has the stated index sets. This is accomplished by
constructing the interior-Poisson triple space $M^3_{i-p}$, obtained by a sequence of blowups from $M \times M \times \partial M$.
Recall the fibre diagonal $\mathrm{fdiag}$ which is blown up in the definitions of the interior edge and Poisson operators.
Here, using local coordinates $(x,y,z)$, $(x', y', z')$ and $(y'',z'')$ in the three factors, $\mathrm{fdiag}_{g} := \{ x = x' = 0,
y = y'\}$ is the fibre diagonal that needs to blown for polyhomogeneity of $G$, $\mathrm{fdiag}_{p} := \{x' = 0, y' = y''\}$
is the fibre diagonal that needs to blown for polyhomogeneity of $P$, and finally $\mathrm{fdiag}_{a} := \{x = 0, y = y''\}$
is the fibre diagonal that needs to blown for polyhomogeneity of $A$. All the three submanifolds intersect at
$\mathrm{fdiag}_{0} := \{x=x'=x''=0, y=y'=y''\}$. We define the triple space by
$$
M^3_{i-p}:=[[M \times M \times \partial M; \mathrm{fdiag}_{0} ]; \mathrm{fdiag}_{g}, \mathrm{fdiag}_{p}, \mathrm{fdiag}_{a}].
$$
Then there exist natural projections
\begin{align*}
&\pi_p: M^3_{i-p} \to P^2_e \times M_{(x,y,z)} \to P^2_e, \\
&\pi_a: M^3_{i-p} \to P^2_e \times M_{(x',y',z')} \to P^2_e, \\
&\pi_g: M^3_{i-p} \to M^2_e \times \partial M_{(y'',z'')} \to M^2_e.
\end{align*}
The maps $\pi_*$ are $b$-fibrations by construction of the triple space.
The triple space is also equipped with the natural blowdown map $\beta_3: M^3_{i-p} \to M \times M \times \partial M$.
We also consider the natural blowdown maps $\beta_2: M^2_e \to M^2$ and
$\beta_1: P^2_e \to M \times \partial M$.
The Schwartz kernel $K_G$ of an interior edge operator $G\in \Psi^{-\infty, \gamma, E_{\mathrm{lf}}, E_{\mathrm{rf}}}_e(M^2_e)$ lifts
to a polyhomogeneous conormal distribution $\kappa_G= \beta_2^* K_G$ on $M^2_e$ of leading order $(-1-b+\gamma)$
at the front face. The Schwartz kernel $K_P$ of an edge Poisson operator $P\in \Psi^{-\infty, \rho, J_{\mathrm{rf}}}_e(P^2_e)$ lifts
to a polyhomogeneous conormal distribution $\kappa_P= \beta_1^* K_P$ on $P^2_e$ of leading order $(-1-b+\rho)$
at the front face. The kernel of the composition $A=G \circ P$
can be expressed using pullbacks and pushforwards as
\[
\kappa_A:=\beta_1^*(K_{A})= (\pi_a)_*[\pi_g^*\kappa_G \cdot \pi_p^*\kappa_P].
\]
Applying the pushforward theorem we obtain the
\begin{thm}\label{triple-ep}
If $\Re E_{\mathrm{lf}} + \Re J_\mathrm{rf}> -1$ then
\[
\Psi^{-\infty, \gamma, E_{\mathrm{rf}}, E_{\mathrm{lf}}}_e(M^2_e) \circ \Psi^{-\infty, \rho, J_\mathrm{rf}}_e(P^2_e) \subset \Psi^{-\infty, \gamma+\rho, E_{\mathrm{rf}}}_e(P^2_e).
\]
\end{thm}
\begin{proof}
In view of the pushforward theorem, it remains to identify the leading order behaviour
at the various boundary faces. Denote the boundary defining functions of the boundary faces introduced by blowing up
$\mathrm{fdiag}_*$ by $\rho_*$, where $*\in \{0, g,p,a\}$. We write $\rho_\mathrm{ff}$ for the front
face defining functions in the double spaces $M^2_e$ and $P^2_e$. The defining functions of the boundary faces
$\{x=0\}$ and $\{x'=0\}$ in $M^3_{i-p}$ are denoted by $\rho_r$ and $\rho_l$, respectively.
The defining functions of the boundary faces
$\{x=0\}$ and $\{x'=0\}$ in either $M^2_e$ or $P^2_e$ are denoted by $\rho_\mathrm{rf}$ and $\rho_\mathrm{lf}$, respectively.
We then obtain
\begin{align*}
&\pi_g^*(\rho_\mathrm{ff}) = \rho_0 \rho_g, \ \pi_g^*(\rho_{\mathrm{rf}, \mathrm{lf}}) = \rho_{r,l}, \\
&\pi_p^*(\rho_\mathrm{ff}) = \rho_0 \rho_p, \ \pi_g^*(\rho_{\mathrm{lf}}) = \rho_{l}, \\
&\pi_a^*(\rho_\mathrm{ff}) = \rho_0 \rho_a, \ \pi_g^*(\rho_{\mathrm{rf}}) = \rho_{r}. \\
\end{align*}
We denote by $\nu_3$ a $b$-volume on $M^3_{i-p}$ and by $\nu_1$ a $b$-volume on $P^2_e$.
We compute
\begin{align*}
&\beta_3^*(dx\, dy\, dz\, dx'\, dy'\, dz'\, dy''\, dz'') = \rho_0^{2+2b} (\rho_g \rho_p \rho_r \rho_l)^{1+b} \nu_3, \\
&\beta_1^*(dx\, dy\, dz\, dy''\, dz'') = \rho_\mathrm{ff}^{1+b} \rho_\mathrm{rf}^{1+b} \nu_1.
\end{align*}
The leading order behaviour of $\kappa_A$ at the various boundary faces of $P^2_e$ follows
now from the following computation
\begin{align*}
&\kappa_A \cdot \beta_1^*(dx\, dy\, dz\, dy''\, dz'') =
\beta_1^*(K_{A} dx\, dy\, dz\, dy''\, dz'') \\ &=
(\pi_a)_*[\pi_g^*\kappa_G \cdot \pi_p^*\kappa_P \cdot \beta_3^*(dx\, dy\, dz\, dx'\, dy'\, dz'\, dy''\, dz'')] \\
&= (\pi_a)_*[ \rho_0^{\gamma+\rho} \rho_g^{\gamma} \rho_p^{\rho} \rho_r^{1+b+E_{\mathrm{rf}}} \rho_l^{1+b+E_{\mathrm{lf}}+J_\mathrm{rf}} \nu_3] \\
&=\rho_\mathrm{ff}^{\gamma+\rho} \rho_\mathrm{rf}^{1+b+E_{\mathrm{rf}}} \nu_1 = \rho_\mathrm{ff}^{-1-b+ \gamma+\rho} \rho_\mathrm{rf}^{E_{\mathrm{rf}}} \beta_1^*(dx\, dy\, dz\, dy''\, dz'').
\end{align*}
This proves the statement.
\end{proof}
\subsubsection{Poisson $\circ$ trace}
Let $P$ be a Poisson and $T$ a trace edge operator, and consider the (only possible) composition $G=P \circ T$.
To show that this is again an interior edge operator, we must verify that the Schwartz kernel of this composition
lifts to be polyhomogeneous conormal on $M^2_e$ and has the stated index sets. This is accomplished by
constructing the Poisson-trace triple space $M^3_{p-t}$, obtained by a sequence of blowups from $M \times \partial M \times M$.
Recall the fibre diagonal $\mathrm{fdiag}$ which is blown up in the definitions of the Poisson and the trace edge operators.
Here, using local coordinates $(x,y,z)$, $(y', z')$ and $(x'', y'',z'')$ in the three factors, $\mathrm{fdiag}_{p} := \{ x = 0,
y = y'\}$ is the fibre diagonal that needs to blown for polyhomogeneity of $P$, $\mathrm{fdiag}_{t} := \{x'' = 0, y' = y''\}$
is the fibre diagonal that needs to blown for polyhomogeneity of $T$, and finally $\mathrm{fdiag}_{g} := \{x = x''=0, y = y''\}$
is the fibre diagonal that needs to blown for polyhomogeneity of $G$. All the three submanifolds intersect at
$\mathrm{fdiag}_{0} := \{x=x''=0, y=y'=y''\}$. We define the triple space by
$$
M^3_{p-t}:=[[M\times \partial M \times M ; \mathrm{fdiag}_{0} ]; \mathrm{fdiag}_{g}, \mathrm{fdiag}_{p}, \mathrm{fdiag}_{t}].
$$
Then there exist natural projections
\begin{align*}
&\pi_p: M^3_{i-p} \to P^2_e \times M_{(x'',y'',z'')} \to P^2_e, \\
&\pi_t: M^3_{i-p} \to T^2_e \times M_{(x,y,z)} \to T^2_e, \\
&\pi_g: M^3_{i-p} \to M^2_e \times \partial M_{(y',z')} \to M^2_e.
\end{align*}
The maps $\pi_*$ are $b$-fibrations by construction of the triple space.
The triple space is also equipped with the natural blowdown map $\beta_3: M^3_{p-t} \to M\times \partial M \times M$.
We also consider the natural blowdown maps $\beta_g: M^2_e \to M^2$,
$\beta_p: P^2_e \to M \times \partial M$ and $\beta_t: T^2_e \to \partial M \times M$.
The Schwartz kernel $K_P$ of an edge Poisson operator $P\in \Psi^{-\infty, \rho, J_\mathrm{rf}}_e(P^2_e)$ lifts
to a polyhomogeneous conormal distribution $\kappa_P= \beta_p^* K_P$ on $P^2_e$ of leading order $(-1- b+\rho)$
at the front face. The Schwartz kernel $K_T$ of an edge trace operator $T\in \Psi^{-\infty, \tau, F_{\mathrm{lf}}}_e(T^2_e)$ lifts
to a polyhomogeneous conormal distribution $\kappa_T= \beta_t^* K_T$ on $T^2_e$ of leading order $(-1-b+\tau)$
at the front face. The kernel of the composition $G=P \circ T$
can be expressed using pullbacks and pushforwards as
\[
\kappa_G:=\beta_2^*(K_G)= (\pi_g)_*[\pi_p^*\kappa_P \cdot \pi_t^*\kappa_T].
\]
Applying the pushforward theorem we obtain the
\begin{thm}\label{triple-pt}
\[
\Psi^{-\infty, \rho, J_{\mathrm{rf}}}_e(P^2_e) \circ \Psi^{-\infty, \tau , F_{\mathrm{lf}}}_e(T^2_e) \subset \Psi^{-\infty, \rho+\tau , J_{\mathrm{rf}},F_{\mathrm{lf}}}_e(M^2_e).
\]
\end{thm}
\begin{proof}
In view of the pushforward theorem, it remains to identify the leading order behaviour
at the various boundary faces. Denote the boundary defining functions of the boundary faces introduced by blowing up
$\mathrm{fdiag}_*$ by $\rho_*$, where $*\in \{0, g,p,t\}$. We write $\rho_\mathrm{ff}$ for the front
face defining functions in the double spaces $M^2_e$ and $P^2_e, T^2_e$. The defining functions of the boundary faces
$\{x=0\}$ and $\{x'=0\}$ in $M^3_{i-p}$ are denoted by $\rho_r$ and $\rho_l$, respectively.
The defining functions of the boundary faces
$\{x=0\}$ and $\{x'=0\}$ in either $M^2_e$ or $P^2_e, T^2_e$ are denoted by $\rho_\mathrm{rf}$ and $\rho_\mathrm{lf}$, respectively.
We then obtain
\begin{align*}
&\pi_p^*(\rho_\mathrm{ff}) = \rho_0 \rho_p, \ \pi_p^*(\rho_{\mathrm{rf}}) = \rho_{r}, \\
&\pi_t^*(\rho_\mathrm{ff}) = \rho_0 \rho_t, \ \pi_t^*(\rho_{\mathrm{lf}}) = \rho_{l}, \\
&\pi_g^*(\rho_\mathrm{ff}) = \rho_0 \rho_g, \ \pi_g^*(\rho_{\mathrm{rf},\mathrm{lf}}) = \rho_{r,l}. \\
\end{align*}
We denote by $\nu_3$ a $b$-volume on $M^3_{i-p}$ and by $\nu_2$ a $b$-volume on $M^2_e$.
We compute
\begin{align*}
&\beta_3^*(dx\, dy\, dz\, dx'\, dy'\, dx'\, dy''\, dz'') = \rho_0^{2+2b} (\rho_p \rho_t \rho_r \rho_l)^{1+b} \nu_3, \\
&\beta_g^*(dx\, dy\, dz\, dx''\, dy''\, dz'') = \rho_\mathrm{ff}^{1+b} \rho_\mathrm{rf}^{1+b} \rho_\mathrm{lf}^{1+b} \nu_2.
\end{align*}
The leading order behaviour of $\kappa_G$ at the various boundary faces of $M^2_e$ follows
now from the following computation
\begin{align*}
&\kappa_G \cdot \beta_g^*(dx\, dy\, dz\, dx''\, dy''\, dz'') =
\beta_g^*(K_G dx\, dy\, dz\, dx''\, dy''\, dz'') \\ &=
(\pi_g)_*[\pi_p^*\kappa_P \cdot \pi_t^*\kappa_T \cdot \beta_3^*(dx\, dy\, dz\, dy'\, dz'\, dx''\, dy''\, dz'')] \\
&= (\pi_g)_*[ \rho_0^{\rho+\tau} \rho_p^{\rho} \rho_t^{\tau} \rho_r^{1+b+ J_{\mathrm{rf}}} \rho_l^{1+b+F_{\mathrm{lf}}} \nu_3] \\
&=\rho_\mathrm{ff}^{\rho+\tau } \rho_\mathrm{rf}^{1+b+J_{\mathrm{rf}}} \rho_\mathrm{lf}^{1+b+F_{\mathrm{rf}}} \nu_2 = \rho_\mathrm{ff}^{-1-b+\rho+\tau}
\rho_\mathrm{rf}^{J_{\mathrm{rf}}} \rho_\mathrm{lf}^{F_{\mathrm{rf}}} \beta_g^*(dx\, dy\, dz\, dx''\, dy''\, dz'').
\end{align*}
This proves the statement.
\end{proof}
\section{Representable subclass of edge, trace and Poisson operators}\label{representable}
Within the more general classes of residual edge, Poisson and trace operators there are subclasses of operators
for which the restriction to the front face has a particular representation formula. We call these the subclasses
of representable operators. We introduce these now, and then show how the composition formul\ae\ specialize
in this setting, proving in particular that the composition of representable operators is again representable.
\subsection{Representable residual edge operators}\label{edge-op}
We may consider $\mathbb R^+ \times F$ as a manifold with boundary with a trivial edge structure, where the base $B$ reduces
to a single point. The corresponding edge double-space thus corresponds to the somewhat simpler $b$-double
space, from \cite{Mel-APS}, \cite{M}, and is denoted $(\mathbb R^+ \times F)^2_b$. This is a manifold with
corners, with three boundary faces, the left, right and the front face. Let $G_0(\widetilde{y}, \widehat{\eta}) \in \mathcal A_{\mathrm{phg}}^{\mathcal E'}((\mathbb R^+\times F)^2_b)$,
where $\mathcal E' = (E_{\mathrm{ff}}=\mathbb{N}_0, \mathcal E = (E_{\mathrm{lf}}, E_{\mathrm{rf}}))$, and the lf, rf index sets are constant (at least in the critical range) when varying
in smooth parameters $(\widetilde{y}, \widehat{\eta}) \in S^*B$.
\begin{defn}\label{edge-Bessel}
Let $G_0(t,z,\widetilde{t},\widetilde{z};\widetilde{y}, \widehat{\eta}) \in \mathcal A_{\mathrm{phg}}^{\mathcal E'}$ as above. Here $(\widetilde{y}, \widehat{\eta}) \in
S^*B$ are smooth parameters. We say that $G_0$ is edge Bessel operator, $G_0\in \Psi^{-\infty,\mathcal E'}_b((\mathbb R^+\times F)^2)$,
if it satisfies the following two conditions:
\begin{enumerate}
\item $G_0(t,z,\widetilde{t},\widetilde{z};\widetilde{y}, \widehat{\eta})$ decreases rapidly as $t\to \infty$, locally uniformly in $(z,\widetilde{t},\widetilde{z})$,
and as $\widetilde{t}\to \infty$, locally uniformly in $(t,z,\widetilde{z})$;
\item $G_0(t,z,\widetilde{t},\widetilde{z};\widetilde{y}, \widehat{\eta})$ admits a polyhomogeneous expansion as $t\to 0$, where the
coefficient functions decrease rapidly as $\widetilde{t}\to \infty$, uniformly in the other coordinates, and vice versa.
\end{enumerate}
\end{defn}
Following \cite[(5.18)]{M}, if $G_0$ is a edge Bessel operator and $k\in \mathbb{N}_0$, set
\begin{align}\label{N-edge}
N_k(G_0)= \int_{\mathbb{R}^b} e^{iY\eta} G_0(s|\eta|,z, |\eta|,\widetilde{z}; \widetilde{y}, \widehat{\eta})
|\eta|^{-k+1} {\mathchar'26\mkern-11mu\mathrm{d}} \eta.
\end{align}
The proof of \cite[Prop. 5.19]{M} shows that $N_k(G_0)$ is polyhomogeneous on the
front face $\mathrm{ff}$ of the edge double space $(\mathbb R^+ \times \mathbb R^b \times F)^2_e$.
It will be convenient below to use the homogeneity rescaling
\begin{align}
\kappa_\lambda u(x, \cdot ):=u(\lambda x, \cdot), \ x\in \mathbb{R}^+.
\end{align}
Consider a residual edge operator $\textup{Op}(G_0)\in \Psi^{-\infty, k, \mathcal E}_e(M^2_e)$.
By definition this acts on test functions $u$ supported near $\partial M$ by
\begin{multline*}
\left[\textup{Op}(G_0) u\right](x,y,z) \\ =\int e^{i(y-\widetilde{y})\eta} \kappa_{|\eta|}\circ
G_0(x,z,\widetilde{x},\widetilde{z}; y, \widehat{\eta}) \circ \kappa_{|\eta|}^{-1} u(\widetilde{x}, \widetilde{y},\widetilde{z})
|\eta|^{-k} \, {\mathchar'26\mkern-11mu\mathrm{d}} \eta \, d\widetilde{x} \, d\widetilde{y} \, d\widetilde{z}.
\end{multline*}
The Schwartz kernel is thus
\begin{equation}
\begin{split}
K_{\mathrm{Op}(G_0)}&(x,y,z, \widetilde{x}, \widetilde{y}, \widetilde{z}) = \int_{\mathbb{R}^b} e^{i(y-\widetilde{y})\eta}
G_0(x|\eta|,z,\widetilde{x} |\eta|,\widetilde{z}; y, \widehat{\eta}) |\eta|^{-k+1} \, {\mathchar'26\mkern-11mu\mathrm{d}} \eta \\
& = \widetilde{x}^{-1-b+k} \int_{\mathbb{R}^b} e^{iY\eta} G_0(s|\eta|,z, |\eta|,\widetilde{z}; \widetilde{y}+\widetilde{x} Y,
\widehat{\eta}) |\eta|^{-k+1} \, {\mathchar'26\mkern-11mu\mathrm{d}} \eta \\
& = \widetilde{x}^{-1-b+k} N_k(G_0) + \mathcal{O}(\widetilde{x}^{-b+k} ) .
\end{split}
\label{kernel-edge}
\end{equation}
The representable subcalculus of residual edge operators consists of those operators $G\in \Psi^{-\infty, k, \mathcal E}_e(M^2_e)$,
whose normal operator $N(G)$, defined as the restriction of $\rho_\mathrm{ff}^{1+b-k}\kappa_G$ to $\mathrm{ff}$, is given by
$N_k(G_0)$ for some $G_0\in \Psi^{-\infty, \mathcal E'}_b (\mathbb R^+\times F)$.
Recall that if $L$ is an elliptic edge operator, then for any nonindicial weight $\delta \in (\underline{\delta}, \overline{\delta})$,
there is a generalized inverse $G$ and projectors $P_1$ and $P_2$ onto the nullspace and cokernel.
By \cite[(4.22)]{M}, the lift of the Schwartz kernel of $P_1$ to $M^2_e$ is polyhomogeneous with index set
\begin{equation}
\label{P-index}
\begin{split}
E_{\mathrm{lf}} &=\{(\zeta,p)\in \textup{Spec}_b(L) \mid \Im \zeta >\delta -1/2\}, \\
E_{\mathrm{rf}} &= \{(\zeta,p) \in \mathbb{C}\times \mathbb{N}_0 \mid (\zeta +2\delta, p)\in E_{\mathrm{lf}}\}, \
E_{\mathrm{ff}} = \mathbb{N}_0.
\end{split}
\end{equation}
Furthermore, its normal operator $N(P_1)$ equals $N_0(P_{01})$
where $P_{01}\in \Psi^{-\infty, \mathcal E}_b((\mathbb R^+\times F)^2)$ is the
projector onto the nullspace for the Bessel operator $B(L)$. Similarly, the lift of the Schwartz kernel of
$P_2$ to $M^2_e$ is polyhomogeneous with index set
\begin{equation}
\begin{split}
F_{\mathrm{rf}} &=\{(\zeta,p) \in \mathbb{C}\times \mathbb{N}_0 \mid (-\zeta-2\delta -1, p) \in \textup{Spec}_b(L),
\Im \zeta >-\delta -1/2\}, \\
F_{\mathrm{lf}} &= \{(\zeta,p) \in \mathbb{C}\times \mathbb{N}_0 \mid (\zeta - 2\delta, p)\in F_{\mathrm{rf}}\}, \ F_{\mathrm{ff}} = \mathbb{N}_0,
\end{split}
\end{equation}
and has normal operator $N(P_2) = N_0(P_{02})$, where $P_{02}\in \Psi^{-\infty,
\mathcal{F}}_b((\mathbb{R}^+\times F)^2)$, $\mathcal F=(F_{\mathrm{ff}}, F_{\mathrm{lf}},F_{\mathrm{rf}})$. Note that
if $\delta > \overline{\delta}$ then $P_{01} = 0$ while if $\delta < \underline{\delta}$ then $P_{02} = 0$.
Finally, the lift of the Schwartz kernel of $G$ is polyhomogeneous on $M^2_e$ with index set
\[
H_{\mathrm{rf}} =E_{\mathrm{rf}}\overline{\cup}F_{\mathrm{rf}}, \
H_{\mathrm{lf}} =E_{\mathrm{lf}}\overline{\cup}F_{\mathrm{lf}}, \
H_{\mathrm{ff}} = \mathbb{N}_0,
\]
This has normal operator $N(G) = N_0(G_0)$ for $G_0\in \Psi^{-\infty, \mathcal{H}}_b((\mathbb{R}^+\times F)^2)$,
$\mathcal{H}=(H_{\mathrm{ff}},H_{\mathrm{lf}},H_{\mathrm{rf}})$.
\subsection{Representable trace operators}
We next introduce the Bessel trace kernels.
\begin{defn}\label{trace-Bessel}
Let $T_0(\widetilde{t},z,\widetilde{z};\widetilde{y}, \widehat{\eta})$ be polyhomogeneous on $F^2\times \mathbb R^+$,
smooth in the interior, and varying smoothly in $(\widetilde{y}, \widehat{\eta})\in S^*B$, with
index sets $\mathcal{F}=(F_{\mathrm{ff}},F_{\mathrm{rf}})$ constant (at least in the critical range) when varying
in $(\widetilde{y}, \widehat{\eta})$. Then $T_0$ is called a trace Bessel kernel, $T_0\in \Psi^{-\infty, \mathcal F}_b(F^2\times \mathbb{R}^+)$,
if it satisfies:
\begin{enumerate}
\item $T_0(\widetilde{t},z,\widetilde{z};\widetilde{y}, \widehat{\eta})$ is rapidly decreasing as $\widetilde{t}\to \infty$, locally uniformly in $(z,\widetilde{z})$;
\item $T_0(\widetilde{t},z,\widetilde{z};\widetilde{y}, \widehat{\eta})$ admits a polyhomogeneous
expansion as $\widetilde{t}\to 0$, uniformly in the other variables.
\end{enumerate}
\end{defn}
The class of representable trace operators consists of those operators $T\in \Psi^{-\infty, k, F_{\mathrm{rf}}}_e(T_e)$,
whose normal operator $N(T)$, defined as the restriction of $\rho_{\mathrm{ff}}^{1+b-k}\kappa_T$ to $\mathrm{ff}$, is given by
\begin{align}\label{N-trace}
N_k(T_0) := \int_{\mathbb{R}^b} e^{iY\eta} T_0(|\eta|, z,\widetilde{z}; \widetilde{y}, \widehat{\eta}) |\eta|^{-k+1} \, {\mathchar'26\mkern-11mu\mathrm{d}} \eta,
\end{align}
for some $T_0\in \Psi^{-\infty, \mathcal F}_b(F^2\times \mathbb{R}^+)$. An example is a trace operator
$\textup{Op}(T_0)\in \Psi^{-\infty, k, F_{\mathrm{rf}}}_e(T_e)$, defined on test functions $u$ supported near $\partial M$ by
\begin{align*}
\left[\textup{Op}(T_0) u \right] (y,z):=\int e^{i(y-\widetilde{y})\eta}
T_0(\widetilde{x},z,\widetilde{z}; y, \widehat{\eta}) \circ \kappa_{|\eta|}^{-1} u(\widetilde{x}, \widetilde{y},\widetilde{z})
|\eta|^{-k} {\mathchar'26\mkern-11mu\mathrm{d}} \eta \, d\widetilde{x} \, d\widetilde{y} \, d\widetilde{z}.
\end{align*}
and extended trivially away from the singular neighborhood. The corresponding operator kernel is given in local coordinates by
\begin{equation}
\begin{split}
K_{\textup{Op}(T_0)}(y,z, \widetilde{x}, \widetilde{y}, \widetilde{z}) &=
\int_{\mathbb{R}^b} e^{i(y-\widetilde{y})\eta} T_0(\widetilde{x}|\eta|,z,\widetilde{z}; y, \widehat{\eta}) |\eta|^{-k+1} {\mathchar'26\mkern-11mu\mathrm{d}} \eta \\
&= \widetilde{x}^{-1-b+k} \int_{\mathbb{R}^b} e^{iY\eta} T_0(|\eta|,z, \widetilde{z}; \widetilde{y}+\widetilde{x} Y, \widehat{\eta}) |\eta|^{-k+1} \, {\mathchar'26\mkern-11mu\mathrm{d}} \eta \\
&= \widetilde{x}^{-1-b+k} N_k(T_0) + \mathcal{O}(\widetilde{x}^{-b+k}).
\end{split}
\label{kernel-trace}
\end{equation}
\subsection{Representable Poisson operators}
Finally, we introduce the Bessel Poisson kernels.
\begin{defn}\label{potential-Bessel}
Let $P_0(t,z,\widetilde{z};\widetilde{y}, \widehat{\eta})$ be polyhomogeneous on $\mathbb{R}^+\times F^2$ with index set
$\mathcal{J}=(J_{\mathrm{lf}}, J_{\mathrm{ff}})$, parametrized and varying smoothly in $(\widetilde{y}, \widehat{\eta})\in S^*B$.
Then $P_0$ is called a Bessel Poisson operator, $P_0\in \Psi^{-\infty, \mathcal{J}}_b(\mathbb{R}^+\times F^2)$, if:
\begin{enumerate}
\item $P_0(t,z,\widetilde{z};\widetilde{y}, \widehat{\eta})$ is rapidly decreasing as $t\to \infty$,
locally uniformly in $(z,\widetilde{z})$;
\item $P_0(t,z,\widetilde{z};\widetilde{y}, \widehat{\eta})$ admits a polyhomogeneous expansion as $t\to 0$,
uniformly in the other variables.
\end{enumerate}
\end{defn}
The representable Poisson operators are operators $P\in \Psi^{-\infty, k, J_{\mathrm{lf}}}_e(P_e)$ with leading coefficient
at the front face, the normal operator $N(P)$, given by
\begin{align}\label{N-potential}
N_k(P_0) := \int_{\mathbb{R}^b} e^{iY\eta} P_0(|\eta|, z,\widetilde{z}; \widetilde{y}, \widehat{\eta}) |\eta|^{-k+1} \, {\mathchar'26\mkern-11mu\mathrm{d}} \eta
\end{align}
for some $P_0\in \Psi^{-\infty, J_{\mathrm{lf}}, J'_{\mathrm{ff}}}_b(\mathbb{R}^+ \times F)$. If $\textup{Op}(P_0)\in \Psi^{-\infty, k, J_{\mathrm{lf}}}_e(P_e)$
is defined near $\partial M$ by
\begin{align*}
\left[\textup{Op}(P_0) u \right] (x,y,z):=\int e^{i(y-\widetilde{y})\eta} \kappa_{|\eta|}
\circ P_0(x,z,\widetilde{z}; y, \widehat{\eta}) u(\widetilde{y},\widetilde{z}) |\eta|^{-k+1} \, {\mathchar'26\mkern-11mu\mathrm{d}} \eta \, d\widetilde{y} \, d\widetilde{z},
\end{align*}
then the Schwartz kernel is given locally by
\begin{equation}
\begin{split}
K_{\textup{Op}(P_0)}(x, y, z, \widetilde{y}, \widetilde{z}) &=
\int_{\mathbb{R}^b} e^{i(y-\widetilde{y})\eta} P_0(x|\eta|,z,\widetilde{z}; y, \widehat{\eta}) |\eta|^{-k+1} {\mathchar'26\mkern-11mu\mathrm{d}} \eta \\
&= x^{-1-b+k} \int_{\mathbb{R}^b} e^{iY\eta} P_0(|\eta|, z, \widetilde{z}; \widetilde{y}+xY, \widehat{\eta}) |\eta|^{-k+1} {\mathchar'26\mkern-11mu\mathrm{d}} \eta, \\
&= x^{-1-b+k} N_k(P_0) + \mathcal{O}(x^{-b+k}).
\end{split}
\label{kernel-potential}
\end{equation}
\subsection{Composition of representable operators}
We conclude this section by proving that the property of being representable is closed under composition.
\smallskip
\noindent{\bf Residual \ $\circ$\ Poisson: }
Let $\textup{Op}(G_0)$ and $\textup{Op}(P_0)$ be a residual edge and an edge Poisson operator associated to the
Bessel operator $G_0$ Bessel Poisson operator $P_0$, respectively. Using \eqref{kernel-edge} and \eqref{kernel-potential},
the composition $\textup{Op}(G_0) \circ \textup{Op}(P_0)$ is given by
\begin{equation}
\begin{split}
K_{\textup{Op}(G_0) \circ \textup{Op}(P_0)}&(x,y,z, \widetilde{y}, \widetilde{z}) \\
=\int\int &e^{i(y-y')\eta} G_0(x|\eta|,z,\widetilde{x} |\eta|, z'; y, \widehat{\eta}) |\eta|^{-g+1} \, {\mathchar'26\mkern-11mu\mathrm{d}} \eta \\
&e^{i(y'-\widetilde{y})\eta'} P_0(\widetilde{x}|\eta'|,z',\widetilde{z}; y', \widehat{\eta}') |\eta'|^{-p+1} \, {\mathchar'26\mkern-11mu\mathrm{d}} \eta' \, d\widetilde{x} \, dy' \, dz' \\
=\int\int &e^{i(Y-Y')\eta} G_0(|\eta|,z, t |\eta|, z'; \widetilde{y}+xY, \widehat{\eta}) x^{-1-b+g}|\eta|^{-g+1} \, {\mathchar'26\mkern-11mu\mathrm{d}} \eta \\
&e^{iY'\eta'} P_0(t|\eta'|,z',\widetilde{z}; \widetilde{y}+xY', \widehat{\eta}') x^{p} |\eta'|^{-p+1} \, {\mathchar'26\mkern-11mu\mathrm{d}} \eta' \, dt \, dY' \, dz',
\end{split}
\end{equation}
where we have substituted
\begin{align}
Y=\frac{y-\widetilde{y}}{x}, \ Y'=\frac{y'-\widetilde{y}}{x}, \ t=\frac{\widetilde{x}}{x}.
\end{align}
Replacing $Y'$ by $(-Y')$ we find for the leading $x^{-1-b+(p+g)}$ coefficient
\begin{equation}
\label{edge-potential}
\begin{split}
N(\textup{Op}& (G_0) \circ \textup{Op}(P_0)) = \int e^{iY\eta}G_0(|\eta|,z, t |\eta|, z'; \widetilde{y}, \widehat{\eta}) |\eta|^{-g+1} \\
& \times \int e^{i(\eta-\eta')Y'} P_0(t|\eta'|,z',\widetilde{z}; \widetilde{y}, \widehat{\eta}') |\eta'|^{-p+1} \, {\mathchar'26\mkern-11mu\mathrm{d}} \eta'\, dY' \, {\mathchar'26\mkern-11mu\mathrm{d}} \eta \, dt\, dz' \\
&=\int e^{iY\eta} (G_0\circ P_0)(|\eta|, z, \widetilde{z}; \widetilde{y}, \widehat{\eta})|\eta|^{-p-g+1} \, {\mathchar'26\mkern-11mu\mathrm{d}} \eta = N_g(G_0)\circ N_p(P_0).
\end{split}
\end{equation}
This proves that the normal operator of this composition is representable and has the form \eqref{N-potential}.
\smallskip
\noindent{\bf Poisson\ $\circ$\ trace:}
Now consider a Poisson operator $\textup{Op}(P_0)$ associated to the Bessel Poisson kernel $P_0$ and a trace operator
$\textup{Op}(T_0)$ associated to the Bessel trace kernel $T_0$. The composition in \eqref{kernel-potential} and
\eqref{kernel-trace} takes the form
\begin{equation}
\begin{split}
K_{\textup{Op}(P_0) \circ \textup{Op}(T_0)}&(x,y,z, \widetilde{x}, \widetilde{y}, \widetilde{z}) \\
=\int\int &e^{i(y-y')\eta} P_0(x|\eta|,z, z'; y, \widehat{\eta}) |\eta|^{-p+1} \, {\mathchar'26\mkern-11mu\mathrm{d}} \eta \\
&e^{i(y'-\widetilde{y})\eta'} T_0(\widetilde{x}|\eta'|,z',\widetilde{z}; y', \widehat{\eta}') |\eta'|^{-\tau+1} \, {\mathchar'26\mkern-11mu\mathrm{d}} \eta' \, dy' \, dz' \\
=\int\int &e^{i(Y-Y')\eta} P_0(s|\eta|,z, z'; \widetilde{y}+\widetilde{x} Y, \widehat{\eta}) x^{-1-b+p}|\eta|^{-p+1} \, {\mathchar'26\mkern-11mu\mathrm{d}} \eta \\
&e^{iY'\eta'} T_0(|\eta'|,z',\widetilde{z}; \widetilde{y}+\widetilde{x} Y', \widehat{\eta}') x^{\tau} |\eta'|^{-\tau+1} \, {\mathchar'26\mkern-11mu\mathrm{d}} \eta' \, dY' \, dz',
\end{split}
\end{equation}
where
\begin{align}
Y=\frac{y-\widetilde{y}}{\widetilde{x}}, \ Y'=\frac{y'-\widetilde{y}}{\widetilde{x}}, \ s=\frac{x}{\widetilde{x}}.
\end{align}
As before, substituting $Y'$ by $(-Y')$, we obtain
\begin{equation}
\label{potential-trace}
\begin{split}
N(\textup{Op}& (P_0) \circ \textup{Op}(T_0)) =\int e^{iY\eta}P_0(s|\eta|,z, z'; \widetilde{y}, \widehat{\eta}) |\eta|^{-p+1} \\
&\times \int e^{i(\eta-\eta')Y'} T_0(|\eta'|,z',\widetilde{z}; \widetilde{y}, \widehat{\eta}') |\eta'|^{-\tau+1} \, {\mathchar'26\mkern-11mu\mathrm{d}} \eta'\, dY' \, {\mathchar'26\mkern-11mu\mathrm{d}} \eta \, dz' \\
&=\int e^{iY\eta} (P_0\circ T_0)(s|\eta|, z, |\eta|, \widetilde{z}; \widetilde{y}, \widehat{\eta})|\eta|^{-p-\tau+1} \, {\mathchar'26\mkern-11mu\mathrm{d}} \eta = N_p(P_0)\circ N_{\tau}(T_0),
\end{split}
\end{equation}
so this composition is again representable.
\smallskip
\noindent{\bf Trace\ $\circ$\ Poisson:}
Finally, if $\textup{Op}(T_0)$ is a trace operator associated to the Bessel trace kernel $T_0$ and $\textup{Op}(P_0)$
is a Poisson operator associated to the Bessel Poisson kernel $P_0$, then \eqref{kernel-trace} and \eqref{kernel-potential}
becomes
\begin{equation}
\begin{split}
K_{\textup{Op}(T_0) \circ \textup{Op}(P_0)}&(y,z, \widetilde{y}, \widetilde{z}) \\
=\int\int &e^{i(y-y')\eta} T_0(x|\eta|,z, z'; y, \widehat{\eta}) |\eta|^{-\tau+1} \, {\mathchar'26\mkern-11mu\mathrm{d}} \eta \\
&e^{i(y'-\widetilde{y})\eta'} P_0(x|\eta'|,z',\widetilde{z}; y', \widehat{\eta}') |\eta'|^{-p+1} \, {\mathchar'26\mkern-11mu\mathrm{d}} \eta' \, dx\, dy' \, dz' \\
=\int\int &e^{i(Y-Y')\eta} T_0(t|\eta|,z, z'; \widetilde{y}+r Y, \widehat{\eta}) r^{-1-b+\tau}|\eta|^{-\tau+1} \, {\mathchar'26\mkern-11mu\mathrm{d}} \eta \\
&e^{iY'\eta'} P_0(t|\eta'|,z',\widetilde{z}; \widetilde{y}+r Y', \widehat{\eta}') r^{p} |\eta'|^{-p+1} \, {\mathchar'26\mkern-11mu\mathrm{d}} \eta' \, dt\, dY' \, dz',
\end{split}
\end{equation}
where
\begin{align}
Y=\frac{y-\widetilde{y}}{r}, \ Y'=\frac{y'-\widetilde{y}}{r}, \ t=\frac{x}{r}.
\end{align}
Substituting $Y'$ by $(-Y')$, and taking the limit $r\to 0$, we obtain the principal symbol of a
pseudodifferential operator on the closed manifold $B$ acting on sections of the trace bundle:
\begin{equation}
\label{trace-potential}
\begin{split}
N(\textup{Op}& (T_0) \circ \textup{Op}(P_0)) =\int e^{iY\eta}T_0(t|\eta|,z, z'; \widetilde{y}, \widehat{\eta}) |\eta|^{-\tau+1} \\
&\times \int e^{i(\eta-\eta')Y'} P_0(t|\eta'|,z',\widetilde{z}; \widetilde{y}, \widehat{\eta}') |\eta'|^{-p+1} \, {\mathchar'26\mkern-11mu\mathrm{d}} \eta'\, dY' \, {\mathchar'26\mkern-11mu\mathrm{d}} \eta \, dx\, dz' \\
&=\int e^{iY\eta} (T_0\circ P_0)(z, \widetilde{z}; \widetilde{y}, \widehat{\eta})|\eta|^{-p-\tau+1} \, {\mathchar'26\mkern-11mu\mathrm{d}} \eta = N_{\tau}(T_0)\circ N_p(P_0).
\end{split}
\end{equation}
\section{Trace and Poisson operators of an elliptic edge operator}\label{trace-pot}
Let $L\in \textup{Diff}^m_e(M)$ be an elliptic differential edge operator. We use all the same notation as above, and
assume, in particular, that $B(L)$ is injective on $t^{\overline{\delta}}L^2$ and surjective on $t^{\underline{\delta}}L^2$.
Define
\begin{equation}
\mathcal H_{\underline{\delta},\overline{\delta}}(L) = \{ u \in x^{\underline{\delta}}L^2: Lu \in x^{\overline{\delta}} L^2\}.
\label{calH}
\end{equation}
We often refer to this as $\mathcal H_{\underline{\delta},\overline{\delta}}$, or even just $\mathcal H$. This is a Hilbert space with respect the graph norm
\[
||u||_{\mathcal H} = ||u||_{ x^{\underline{\delta}} L^2} + ||Lu||_{ x^{\overline{\delta}} L^2}.
\]
In this section we define and study the trace map, which assigns to any $u \in \mathcal H_{\underline{\delta},\overline{\delta}}$ the set of
leading coefficients in its expansion with exponents between $\underline{\delta}$ and $\overline{\delta}$. We also construct the
Poisson operator for $L$, which assigns to an appropriate set of leading coefficients an element
of $\ker L \cap \mathcal H_{\underline{\delta},\overline{\delta}}$.
A subtlety in these definitions is that leading coefficients are sections of the trace bundle
\[
\mathcal E(L) = \bigoplus_{j=0}^N \mathcal E(L; \zeta_j)
\]
introduced in \S 2. A standing assumption in this paper is that the $\zeta_j$ are independent of $y \in B$, and because of
this, the different subbundles $\mathcal E(L; \zeta_j)$ do not interact with one another. Thus, to simplify the notation
in this section, we suppose that there is only a single indicial root $\zeta_0 \in \mathfrak{S}(L)$, and we
$\mathcal E(L) = \mathcal E(L; \zeta_0)$.
\subsection{The trace map for the model Bessel operator}
The model Bessel operator corresponding to $L$ is
\[
B_{\widetilde{y}, \widehat{\eta}}(L)=\sum\limits_{j+|\mathrm{\alpha}|+|\beta|\leq m}
a_{j,\mathrm{\alpha},\beta}(0,\widetilde{y},z)(t\partial_t)^j(it\widehat{\eta})^{\mathrm{\alpha}}\partial_z^{\beta},
\]
which acts (as an unbounded operator) on $t^{\underline{\delta}}H^m_{\mathrm{loc}}(\mathbb R^+ \times F; dt\, dz)$. Just as for $L$, however,
we are primarily interested in its restriction to
\[
\mathcal H_{\underline{\delta}, \overline{\delta}}^B = \{\mathrm{\omega}\in t^{\underline{\delta}}L^2(\mathbb{R}^+\times F; dt\, dz): \ B(L)\mathrm{\omega}\in t^{\overline{\delta}}L^2\}.
\]
If $\mathrm{\omega} \in \mathcal H^B_{\underline{\delta},\overline{\delta}}$, then we can follow the same strategy as in the proof of Proposition~\ref{explsoln}
to obtain the (strong) expansion
\[
\mathrm{\omega} \sim \sum_{\ell \geq 0} \sum_{p=0}^{p_0} t^{-i\zeta_0+\ell}(\log t)^p \, \mathrm{\omega}_{\ell,p}(\widetilde{y}, z) + \tilde{\mathrm{\omega}},
\quad \tilde{\mathrm{\omega}} \in \bigcap_{\epsilon > 0} t^{\overline{\delta} - \epsilon}L^2.
\]
Indeed, writing $B(L) = I(L) + E$, where $E$ contains all terms with `extra' powers of $t$, then $B(L) \mathrm{\omega} = f$
becomes $I(L) \mathrm{\omega} = f - E \mathrm{\omega}$. The term $E \mathrm{\omega}$ creates new `higher order' terms $t^{-i\zeta_0 + \ell}$ with
$\ell > 0$, but discarding these we obtain
\begin{equation}
I_{\widetilde{y}}(L) \left( \sum_{p=0}^{p_0} t^{-i\zeta_0}( \log t)^p \, \mathrm{\omega}_{0,p}(\widetilde{y}, z) \right) = 0.
\label{may}
\end{equation}
By definition of the fibres of the trace bundle, this expression in parentheses lies in $\mathcal E_{\widetilde{y}}(L; \zeta_0)$ for each $\widetilde{y}$.
Now consider how this expansion varies as a function of $\widetilde{y}$. Even if $f$ depends smoothly on $\widetilde{y}$,
the individual coefficients $\mathrm{\omega}_{0,p}$ may fail to be smooth (or even continuous) in $\widetilde{y}$ because the order
$p_0$ of the indicial root may vary. This is where the properties of the trace bundle from \cite{KM},
discussed above in \S 2, become crucial. As explained there, on any neighbourhood
$\mathcal U \subset B$ over which $\mathcal E$ is trivialized, there exist smooth functions $\phi_{\widetilde{y},k}(t,z)$,
$k \leq m_0 = m(\zeta_0)$, such that
\[
\sum_{p=0}^{p_0} t^{-i\zeta_0}(\log t)^p \, \mathrm{\omega}_{0,p}(\widetilde{y}, z) = \sum_{k=1}^{m_0} f_k(\widetilde{y}) \phi_{\widetilde{y},k}(t,z),
\]
where, somewhat remarkably, $f_k \in \mathcal C^\infty(\mathcal U)$ even though the number of terms in the sum on the left
may be discontinuous.
Using all of this, we can now state the
\begin{defn}
The Bessel trace map $\mathrm{Tr}_{B(L)}$ is the operator which assigns to each $\mathrm{\omega} \in \mathcal C^\infty(S^*B; \mathcal H_{\underline{\delta},\overline{\delta}}^B)$
a section of $\mathcal E(L; \zeta_0)$ which is represented in a neighbourhood $\mathcal U \subset B$
in which $\mathcal E(L; \zeta_0)$ is trivialized by the smooth basis of sections $\phi_{\widetilde{y},k}$ by the
$m_0$-tuple $\{f_1, \ldots, f_{m_0}\}$.
\end{defn}
Note that if $\mathrm{\omega}(\widetilde{y},\widehat{\eta}) \in t^{\overline{\delta}}L^2$ for each $(\widetilde{y}, \widehat{\eta})$, then $\mathrm{Tr}_{B(L)} \mathrm{\omega} =0$.
\begin{prop}
The operator $\mathrm{Tr}_{B(L)}$ is a representable Bessel trace kernel in the sense of Definition~\ref{trace-Bessel}.
\end{prop}
\begin{proof}
Recall the definition of the trace bundle in Proposition \ref{trace-bundle}.
Then for a solution $\mathrm{\omega}$, the singular part of its Mellin transform is a section of $\mathcal E(L)$.
Consider, following \cite{KM}, the Hilbert space adjoint $I_{\widetilde{y}}(L)(\zeta)^*$ of the indicial operator pencil and set
$I_{\widetilde{y}}(L)^*(\zeta):=I_{\widetilde{y}}(L)(\bar{\zeta})^*$. This depends smoothly on $y$ and is a holomorphic family of
Fredholm operators in $\zeta\in \mathbb{C}$. Its indicial roots are the complex conjugates of elements of
$\textup{Spec}_b(L)$. We denote its trace bundle by $\mathcal E^*(L)$. This suggestive notation is vindicated
by a central result in \cite[Theorem 5.3]{KM}, which asserts the nondegeneracy of the pairing
\begin{equation}
\begin{split}
&\mathcal E_{\widetilde{y}}(L)(\zeta_0) \times \mathcal E^*_{\widetilde{y}}(L)(\bar{\zeta_0}) \rightarrow \mathbb{C}, \\
&[\phi, \psi]:= \frac{1}{2\pi} \oint_{B_\epsilon(\zeta_0)} \phi(\zeta) I_{\widetilde{y}}(L)^*(\bar{\zeta}) \psi (\bar{\zeta}) \, d\zeta.
\end{split}
\end{equation}
for any sufficiently small $\epsilon >0$. Identifying $\mathcal E_{\widetilde{y}}(L, \zeta_0)$ with the kernel of $I_{\widetilde{y}}(L)$ on the space of
finite combinations $\sum a_q(z) t^{-i\zeta_0} \log^q (t), a_j \in C^\infty(F)$, we may assign to each basis element
$\phi_{\widetilde{y},j}$ its dual, $\phi^*_{\widetilde{y},j}$, with respect to this pairing. If $\chi\in \mathcal C^\infty_0(\mathbb R)$ is a cutoff function
which equals one near $0$, then the integral kernel of the Bessel trace map is
\begin{equation}
\begin{split}
\mathrm{Tr}_{B(L)}(t,\widetilde{z}; \widetilde{y}) &= \frac{1}{2\pi} \bigoplus_{j=1}^{m_0}
\oint_{B_\epsilon(\zeta_0)} t^{i\zeta -1} \chi(t)
I_{\widetilde{y}}(L)^*(\bar{\zeta}) \phi^*_{\widetilde{y},j} (\bar{\zeta}, \widetilde{z}) \, d\zeta \\
&:= \bigoplus_{j=1}^{m_0} \Phi^*_j(t,\widetilde{y},\widetilde{z}).
\end{split}
\end{equation}
This satisfies the conditions of Definition \ref{trace-Bessel}, and hence $\mathrm{Tr}_{B(L)}$ is a representable Bessel
trace kernel.
The absence of the variable $z$ in this formula is a result of the identification of the asymptotic coefficients
of $\mathrm{\omega}$ with local sections of the trace bundle, since this bundle is trivialized by the smooth basis $\{\phi_{\widetilde{y},j}\}$,
which has coefficients $\{f_1, ..., f_{m_0}\}$ depending only on $\widetilde{y}$.
\end{proof}
\subsection{Trace of solutions to the normal operator}
The next step is to carry out a similar analysis of the trace operator for the normal operator $N(L)$.
Recall that $N(L)$ is identified with the restriction of the lift $\beta^*L$ to the front face in $M^2_e$
with respect to the blowdown map $\beta:M^2_e \to M^2$, and in the projective coordinates $(s,Y,z)$
from \eqref{proj-coord} this takes the form \eqref{normalop} (with $Y$ replacing the variable $u$ there).
The normal operator is equivalent to the Bessel operator \eqref{Bessel} through Fourier transform (in $Y$)
and rescaling (setting $s=t/|\eta|$):
\[
\mathscr{F}\circ N_{\widetilde{y}}(L) \circ \mathscr{F}^{-1} \mid_{s=t/|\eta|} =
B_{\widetilde{y},\widehat{\eta}}(L).
\]
Thus if $\mathrm{\omega} \in s^{\underline{\delta}}L^2(ds\, dY\, dz)$ is such that $N(L)\mathrm{\omega}\in s^{\overline{\delta}}L^2$, then its Fourier transform $\widehat{\mathrm{\omega}}$
evaluated at $s = t/|\eta|$ is an element of $\mathcal H^B_{\underline{\delta},\overline{\delta}}$. As such, it can be written locally as
\[
\widehat{\mathrm{\omega}} (s, \eta, z) = \sum_{k=1}^{m_0} a_k(\widetilde{y}, \eta) \phi_{\widetilde{y},k}(s|\eta|, z).
\]
We define the trace map for $N_{\widetilde{y}}(L)$ as
\begin{align*}
\textup{Tr}_{N(L)} \mathrm{\omega} &:= \bigoplus_{j=1}^{m_0} \int e^{i(Y-\widetilde{Y})\eta} \Phi^*_j(s|\eta|,\widetilde{y}, \widetilde{z})
\, \mathrm{\omega} (s, \widetilde{Y}, \widetilde{z}) |\eta|^{-i\zeta_0+1}\, ds\, {\mathchar'26\mkern-11mu\mathrm{d}}\eta\, d\widetilde{Y}\, d\widetilde{z}\\
&= \bigoplus_{j=1}^{m_0} \int_{\mathbb{R}^b} e^{iY\eta} a_j(\widetilde{y}, \eta) |\eta|^{-i\zeta_0} \,{\mathchar'26\mkern-11mu\mathrm{d}} \eta\in H^{-(\Im\zeta_0 -\underline{\delta} +1/2)}(\mathbb R^b, dY) \otimes \mathcal E_{\widetilde{y}}(L;\zeta_0).
\end{align*}
where we used the regularity result \cite[Thm. 7.3]{M}.
From \eqref{N-trace} and since $\textup{Tr}_{B(L)}$
is a Bessel trace kernel, we infer that
\begin{align}\label{normal-trace}
\textup{Tr}_{N(L)}(Y, \widetilde{y}, \widetilde{z}) = \int_{\mathbb{R}^b} e^{iY\eta} \textup{Tr}_{B(L)} (|\eta|, \widetilde{y}, \widetilde{z}) |\eta|^{-i\zeta_0+1}\, {\mathchar'26\mkern-11mu\mathrm{d}} \eta,
\end{align}
is smooth on the front face of $T_e$ and polyhomogeneous at the boundaries of this face.
\subsection{The trace map of $L$}\label{trace-bounded}
The construction above determines a Schwartz kernel representation for a trace map of the operator $L$ itself.
Indeed, following \eqref{normal-trace}, define in local coordinates of the corner neighborhood in $M^2$
\[
\textup{Tr}_L(\widetilde{x}, y, \widetilde{y}, \widetilde{z}) := \int_{\mathbb{R}^b} e^{i\eta (y-\widetilde{y})} \textup{Tr}_{B(L)} (x|\eta|, \widetilde{y}, \widetilde{z}) |\eta|^{-i\zeta_0+1}\, {\mathchar'26\mkern-11mu\mathrm{d}} \eta,
\]
and extend smoothly to the interior. From the work above,
\[
\textup{Tr}_L : \mathcal H_{\underline{\delta}, \overline{\delta}} \to H^{-(\Im\zeta_0 -\underline{\delta} +1/2)}(B,\mathcal E(L;\zeta_0))
\]
is a bounded mapping, a representable trace operator. Note that this operator $\mathrm{Tr}_L$ is by no means unique.
\subsection{The edge Poisson operator}
We define the Bessel Poisson operator
\begin{align*}
P_0:\mathcal E(L,\zeta_0) \to \mathcal H^B_{\underline{\delta},\overline{\delta}}, \quad (f_1, ...,f_{m_0}) \mapsto \sum_{j=1}^{m_0} f_j \phi_{\widetilde{y}, j},
\end{align*}
with the integral kernel (as before $\widetilde{z}$ is absent)
\[
P_0(t,z;\widetilde{y}) = \bigoplus\limits_{j=1}^{m_0} \phi_{\widetilde{y}, j}(t,z).
\]
In particular, $P_0$ is representable in the sense of Definition \ref{potential-Bessel}, with the index set
$J_{\mathrm{lf}}$ determined by the asymptotic expansion of each $\phi_{\widetilde{y}, j}$. The associated normal operator
is given by
\[
N_{-i\zeta_0+1}(P_0) = \int_{\mathbb{R}^b} e^{iY\eta} P_0(|\eta|, z; \widetilde{y}) |\eta|^{i\zeta_0}\, {\mathchar'26\mkern-11mu\mathrm{d}} \eta,
\]
which we extend off the front face to define an edge Poisson operator
\begin{align*}
\textup{Op}(P_0): H^{-(\Im(\zeta_0) -\underline{\delta} +1/2)}(B, \mathcal E(\zeta_0))
\longrightarrow x^{\underline{\delta}}H^{\infty}_e(M,g).
\end{align*}
Consider the orthogonal projector (cf. \cite{M})
\begin{align}
P_1: x^{\underline{\delta}}L^2(M,g)\to \ker L\cap x^{\underline{\delta}}L^2(M,g),
\end{align}
which is a residual edge operator discussed in \S \ref{edge-pseudos},
with the corresponding edge Bessel kernel $P_{01}$, which is the
Schwartz kernel of the orthogonal projection of $t^{\underline{\delta}}L^2(dt\, dz)$
onto $\ker B(L)(\widetilde{y}, \widehat{\eta})\cap t^{\underline{\delta}}L^2(dt\, dz)$. We define
\begin{align}
P_L=P_1\circ \textup{Op}(P_0).
\end{align}
By the composition rule \eqref{edge-potential} we find
\begin{align*}
N(P_L)=N_0(P_1) \circ N_1(P_0)= \int e^{iY\eta} (P_{01}\circ P_0)(|\eta|, z) \, {\mathchar'26\mkern-11mu\mathrm{d}} \eta.
\end{align*}
The restriction of Bessel trace map $\textup{Tr}_{B(L)}$ to $\ker B(L) \cap t^{\underline{\delta}}L^2$ is injective,
since $B(L)$ is injective on $t^{\overline{\delta}}L^2$. Hence $\textup{Tr}_{B(L)}$ admits a left-inverse $\textup{Tr}_{B(L)}^{-1}$,
mapping $\mathcal E_{\widetilde{y}}(L,\zeta_0)$ to $\ker B(L) \cap t^{\underline{\delta}}L^2$, which is a true inverse when restricted to
$\textup{im}\, \textup{Tr}_{B(L)}(\ker B(L) \cap t^{\underline{\delta}}L^2)$.
\begin{lemma}\label{bessel-P-T}
$\textup{Tr}_{B(L)}^{-1} = P_{01} \circ P_0 \restriction \textup{im}\, \textup{Tr}_{B(L)}(\ker B(L) \cap t^{\underline{\delta}}L^2).$
\end{lemma}
\begin{proof}
Note that by \cite[(5.8)]{M} there exists a generalized inverse $G_0$ such that
$G_0B(L)=I-P_{01}$. Consequently, for any $\mathrm{\omega} \in \mathcal H^B_{\underline{\delta},\overline{\delta}}$ we find
$\mathrm{\omega}- P_{01}\mathrm{\omega} =G_0B(L)\mathrm{\omega} \in t^{\overline{\delta}}L^2$. Hence, $\textup{Tr}_{B(L)}\mathrm{\omega} = \textup{Tr}_{B(L)}P_{01} \mathrm{\omega}$. Thus
\[
\textup{Tr}_{B(L)} P_{01} \circ P_0 (\textup{Tr}_{B(L)}\mathrm{\omega}) = \textup{Tr}_{B(L)} P_0(\textup{Tr}_{B(L)}\mathrm{\omega}) = \textup{Tr}_{B(L)} \mathrm{\omega}.
\]
If $\mathrm{\omega}\in \ker B(L) \cap t^{\underline{\delta}}L^2$, then $\mathrm{\omega} = P_{01}\circ P_0 (\textup{Tr}_{B(L)} \mathrm{\omega})$ since $B(L)$ is
injective on $t^{\overline{\delta}}L^2$.
\end{proof}
\begin{prop} $(N(P_L)\circ \textup{Tr}_{N(L)}) \mathrm{\omega}=\mathrm{\omega}$, for $\mathrm{\omega}\in \ker N(L)\cap s^{\underline{\delta}}L^2$.
\end{prop}
\begin{proof}
We compute according to \eqref{potential-trace}
\begin{align*}
N(P_L)\circ \textup{Tr}_{N(L)}= \int e^{iY\eta}(P_{01} \circ P_0 \circ \textup{Tr}_{B(L)(\widetilde{y},\widehat{\eta})})
(s|\eta|, \widetilde{s}|\eta|, z, \widetilde{z}) \, {\mathchar'26\mkern-11mu\mathrm{d}} \eta.
\end{align*}
which is the normal operator of a residual edge operator. Consider $\mathrm{\omega}(s,Y,z)\in \ker N(L)\cap s^{\underline{\delta}}L^2$.
As before, $\widehat{\mathrm{\omega}}(t/|\eta|,\eta,z) \in \ker B(L)(\widetilde{y}, \widehat{\eta})\cap t^{\underline{\delta}}L^2$.
Thus we compute by Lemma \ref{bessel-P-T}
\begin{align*}
&N(P_L)\circ \textup{Tr}_{N(L)}\, \mathrm{\omega} \\ &=\int e^{i(Y-\widetilde{Y})\eta}
(P_{01} \circ P_0 \circ \textup{Tr}_{B(L)(\widetilde{y},\widehat{\eta})})
(s|\eta|, \widetilde{s}|\eta|, z, \widetilde{z}) \mathrm{\omega}(\widetilde{s}, \widetilde{Y}, \widetilde{z})\, {\mathchar'26\mkern-11mu\mathrm{d}} \eta\, d\widetilde{s}\, d\widetilde{Y}\, d\widetilde{z}\\
&=\int e^{iY\eta} (P_{01} \circ P_0 \circ \textup{Tr}_{B(L)(\widetilde{y},\widehat{\eta})})
(s|\eta|, \widetilde{s}|\eta|, z, \widetilde{z})\, \widehat{\mathrm{\omega}}
(\widetilde{s},\eta,\widetilde{z}) \, {\mathchar'26\mkern-11mu\mathrm{d}} \eta\, d\widetilde{s}\, d\widetilde{z} \\ &= \int e^{iY\eta}\, \widehat{\mathrm{\omega}}(s,\eta, z) \,
{\mathchar'26\mkern-11mu\mathrm{d}} \eta= \mathrm{\omega}(s,Y,z).
\end{align*}
\end{proof}
\section{Fredholm theory of elliptic edge boundary value problems}\label{fredholm}
We return now to the general situation, where $\mathfrak{S}(L) = \{\zeta_0,..,\zeta_N\}$. Fix a collection $E_1,..,E_M$ of finite
rank vector bundles over $B$ and set $E = \oplus_{k=1}^M E_k$. Now consider the collection of classical pseudodifferential operators
\begin{align*}
&Q_{kj} \in \Psi^{d_k - \Im (\zeta_j)}(B; \mathcal E(L; \zeta_j), E_k), \quad j=1,..,N, \ k=1,..,M, \\
&Q_{kj}: H^s(B, \mathcal E(L, \zeta_j)) \to H^{s-d_k+\Im (\zeta_j)}(B,E_k), \ s\in \mathbb{R}.
\end{align*}
Define the homogeneity rescalings
\begin{align*}
&\eta(L): \mathcal E \to \mathcal E, \
(u_1,..,u_N) \mapsto (|\eta|^{\Im\zeta_1}u_1,..,|\eta|^{\Im\zeta_N}u_N), \\
&\eta(Q): E\to E, \ (e_1,..,e_M)\mapsto (|\eta|^{d_1}e_1,..,|\eta|^{d_M}e_M).
\end{align*}
The matrix $(Q_{kj})$ defines the pseudodifferential system $Q$ where
\[
\sigma_0(Q)(\widetilde{y},\eta)=\eta(Q)\circ \sigma_0(Q)(\widetilde{y},\widehat{\eta})\circ \eta(L)^{-1}.
\]
(Note that $\eta$ appears on the left and $\widehat{\eta} = \eta/|\eta|$ on the right.)
We now recall the form of the general edge boundary value problem:
\begin{defn}
Let $L\in \textup{Diff}^m_e(M)$ be edge elliptic, and suppose that $Q = (Q_{kj})$ is
as above. Then the edge boundary value problem $(L,Q)$ is the set of equations
\begin{align*}
& Lu=f\in x^{\overline{\delta}}L^2(M), \\
& Q(\textup{Tr}_L\, u)=\phi \in \bigoplus\limits_{k=1}^M H^{\underline{\delta} -d_k-1/2}(B,E_k).
\end{align*}
for $u\in x^{\underline{\delta}}H^m_e(M)$.
\end{defn}
We have already stated, in Definition~\ref{typesofbcs}, the definitions of right-, left- and full ellipticity
of the boundary problem $(L,Q)$.
Clearly
\begin{equation}
\begin{split}
(L,Q): & \mathcal H \to x^{\overline{\delta}}L^2(M,g)\oplus \left(\bigoplus\limits_{k=1}^M H^{\underline{\delta}-d_k-1/2}(B,E_k)\right), \\
& u\mapsto (Lu, Q(\textup{Tr}_L\, u)).
\end{split}
\label{mainmap}
\end{equation}
is continuous. Our goal is to show that it is semi-Fredholm if $(L,Q)$ satisfies conditions i) or ii) of Definition~\ref{typesofbcs},
and Fredholm if $(L,Q)$ satisfies condition iii). This is proved by a parametrix construction.
\subsection{The right-elliptic case}
Consider a right-elliptic system $(L,Q)$. Since
\begin{align*}
\sigma(Q)\restriction \textup{im}\,
\textup{Tr}_{B(L)}:\textup{im}\, \textup{Tr}_{B(L)} \to E
\end{align*}
is surjective, there exists a right parametrix
\begin{align*}
K:\bigoplus\limits_{k=1}^MH^{\underline{\delta}-d_k-1/2}(B,E_k)\to \bigoplus\limits_{j=1}^N H^{\underline{\delta} -\Im (\zeta_j)-1/2}(B,\mathcal E(L; \zeta_j))
\end{align*}
for $Q$; this has principal symbol
\begin{equation}\label{K}
\begin{split}
&\sigma(K)(\widetilde{y}, \eta)=\eta(L)\circ \sigma(Q)^{-1}(\widetilde{y}, \widehat{\eta})\circ \eta(Q)^{-1}, \\
&\sigma(Q)^{-1}(\widetilde{y}, \widehat{\eta}): E_{\widetilde{y}} \to \textup{im}\, \textup{Tr}_{B(L)(\widetilde{y},\widehat{\eta})},
\end{split}
\end{equation}
where $\sigma(Q)^{-1}(\widetilde{y}, \widehat{\eta})$ is some choice of right-inverse for $\sigma(Q)(\widetilde{y},\widehat{\eta})
\restriction \textup{im}\, \textup{Tr}_{B(L)}$ which varies smoothly in $(\widetilde{y}, \widehat{\eta})$.
\begin{thm}\label{right}
If $(L,Q)$ is right-elliptic, then \eqref{mainmap} is semi-Fredholm, with closed range of finite codimension.
A right parametrix for it is given by
\begin{align*}
\mathcal{G}(f,\phi)= Gf + P_L[K(\phi - Q(\textup{Tr}_{L}Gf))],
\end{align*}
where $G$ is the generalized inverse for $L$ on $x^{\underline{\delta}}H^m_e(M)$.
\end{thm}
\begin{proof}
By definition, $LG = \mbox{Id} - P_2$, where $P_2$ is the orthogonal projection onto the finite-dimensional space
$\textup{coker}L\cap x^{\underline{\delta}}L^2$. Thus if $f\in x^{\overline{\delta}}L^2$, then
\begin{align*}
\|Gf\|_\mathcal H&=\|Gf\|_{x^{\underline{\delta}}H^m_e}+ \|LGf\|_{x^{\overline{\delta}}L^2}\\
&\leq \|Gf\|_{x^{\underline{\delta}}H^m_e}+ \|f\|_{x^{\overline{\delta}}L^2} + \|P_2f\|_{x^{\overline{\delta}}L^2}.
\end{align*}
Since $G$ is bounded on $x^{\underline{\delta}}L^2$ and $||f||_{x^{\underline{\delta}}L^2} \leq ||f||_{x^{\overline{\delta}}L^2}$, we have
$||Gf||_{\mathcal H} \leq C (||f||_{x^{\overline{\delta}}L^2} + ||P_2 f||_{x^{\underline{\delta}}L^2})$. Hence $G: \ker P_2 \cap x^{\overline{\delta}}L^2 \rightarrow \mathcal H$
is bounded. For simplicity below, we assume that $P_2 \equiv 0$; if this projector is nontrivial, it only changes
things by a finite dimensional amount, which does not affect any of the Fredholmness statements below.
Next, both
\begin{align}\label{bded2}
P_L: \bigoplus_{j=0}^N H^{-(\Im (\zeta_j) -\underline{\delta} +1/2)}(B, \mathcal E(L, \zeta_j)) \to \ker L \cap x^{\underline{\delta}}H^{\infty}_e(M) \subset \mathcal H
\end{align}
and
\begin{equation}\label{bded3}
\textup{Tr}_{L}: \mathcal H \to \bigoplus_{j=0}^N H^{-(\Im (\zeta_j) -\underline{\delta} +1/2)}(B, \mathcal E(L,\zeta_j)),
\end{equation}
are continuous, the latter by the discussion in \S \ref{trace-bounded}. All of this, together with continuity of the
pseudodifferential operators $Q$ and $K$ between appropriate Sobolev spaces over $B$, shows that the
parametrix $\mathcal G$ is a bounded mapping.
We now compute the error term $((L,Q)\mathcal{G} - \mbox{Id})(f,\phi)$.
Since $L P_L = 0$, and we are assuming that the cokernel of $L$ is trivial, we have $L \mathcal G (f,\phi) = LGf = f$.
Next,
\begin{align*}
Q\, \textup{Tr}_{L}\, \mathcal{G}(f,\phi)&= Q\left[\textup{Tr}_{L}\, Gf+
\textup{Tr}_{L}\, P_L(K(\phi-Q(\textup{Tr}_{L}\, Gf)))\right]\\
&= Q\textup{Tr}_{L}\, Gf+ (Q\circ \textup{Tr}_{L}\circ P_L\circ K)(\phi-Q(\textup{Tr}_{L}\, Gf))\\
&=\phi + (Q\circ \textup{Tr}_{L}\circ P_L\circ K-I)(\phi-Q(\textup{Tr}_{L}\, Gf)).
\end{align*}
It thus remains to prove that
\[
(Q\circ \textup{Tr}_{L}\circ P_L\circ K-I): \bigoplus\limits_{k=1}^M
H^{\underline{\delta}-d_k-1/2}(B,E_k) \to \bigoplus\limits_{k=1}^MH^{\underline{\delta}-d_k-1/2}(B,E_k)
\]
is compact. This is however simply a pseudo-differential operator over the closed manifold $B$, so it suffices to check that
its principal symbol vanishes. We compute, using \eqref{trace-potential}, that
\begin{align*}
&\sigma_0(Q\circ \textup{Tr}_{L}\circ P_L\circ K-I)(\widetilde{y},\eta)\\
&= \sigma(Q)(\widetilde{y}, \eta)\circ (\textup{Tr}_{B(L)}\circ P_{01} \circ P_0)\circ \sigma(K)(\widetilde{y}, \eta)-I.
\end{align*}
By definition, $\sigma(K)$ maps into $\mathcal C_{B(L)}$, so all terms cancel and this principal symbol vanishes.
This completes the proof.
\end{proof}
\subsection{Left-elliptic edge boundary value problem}
Now consider a set of boundary operators $Q$ which satisfy the left-elliptic conditions.
Since $\sigma_Q(\widetilde{y}, \widehat{\eta}) \restriction \mathcal C_{B(L)}$ is injective, there exists a matrix of pseudodifferential operators
\[
K:\bigoplus\limits_{k=1}^M H^{\underline{\delta}-d_k-1/2}(B,E_k)\to
\bigoplus\limits_{j=1}^N H^{\underline{\delta} -\Im (\zeta_j)-1/2}(B, \mathcal E(L; \zeta_j)),
\]
with principal symbol
\[
\sigma(K)(\widetilde{y}, \eta)=\eta(L)\circ \sigma(Q)^{-1}(\widetilde{y}, \widehat{\eta})\circ \eta(Q)^{-1}
\]
where
\[
\sigma(Q)^{-1}(\widetilde{y}, \widehat{\eta}): E_{\widetilde{y}} \to \mathcal C_{B(L)},
\]
is a left-inverse to $\sigma(Q)(\widetilde{y},\widehat{\eta})\restriction \mathcal C_{B(L)}$. Note that $K$ is
not necessarily a left-parametrix for $Q$, since $\sigma(Q)^{-1}(\widetilde{y}, \widehat{\eta})$ does not invert the full symbol
$\sigma(Q)(\widetilde{y},\widehat{\eta})$, but this is not required for our argument.
\begin{thm}\label{left}
If $(L,Q)$ is left-elliptic, then
\[
(L,Q): \mathcal H \to x^{\overline{\delta}}L^2(M)\oplus \left(\bigoplus\limits_{k=1}^MH^{\underline{\delta}-d_k-1/2}(B,E_k)\right),
\]
is semi-Fredholm with left parametrix
\begin{align*}
\mathcal G(f,\phi)= Gf + P_L[K(\phi - Q(\textup{Tr}_{L}\, Gf))].
\end{align*}
\end{thm}
\begin{proof}
As before, $\mathcal G$ is a bounded operator and we compute for any $u\in \mathcal H$,
\begin{align*}
\mathcal{G}(L,Q)u&= GLu + P_L[K(Q\, \textup{Tr}_{L}\, u-Q\textup{Tr}_{L}\, GLu)]\\
&=GLu + (P_L\circ K \circ Q \circ \textup{Tr}_{L})(u-GLu)\\
&=u + (P_L\circ K \circ Q \circ \textup{Tr}_{L}-I) P_1 u,
\end{align*}
where $P_1$ is the orthogonal projection onto the nullspace of $L$ in $x^{\underline{\delta}}L^2$. Hence we must show that
$(P_L\circ K \circ Q \circ \textup{Tr}_{L}-I)\circ P_1$ is compact on $\mathcal H$.
By the form of $||\cdot ||_{\mathcal H}$ and since $LP_L=0$ and $LP_1=0$, we need only check compactness of
\[
(P_L\circ K \circ Q \circ \textup{Tr}_{L}-I)\circ P_1: \mathcal H \longrightarrow \ker L \cap \mathcal H.
\]
By the composition results in \S \ref{triple}, $(P_L\circ K \circ Q \circ \textup{Tr}_{L})$ is an edge
operator of order $-\infty$, i.e.\ has no diagonal singularity, and has normal operator
\begin{multline*}
N(P_L\circ K \circ Q \circ \textup{Tr}_{L}) = \\ \int e^{iY\eta}
(P_{01} \circ P_0 \circ \sigma(K) \circ \sigma(Q) \circ \textup{Tr}_{B(L)})
(s|\eta|, |\eta|, z, \widetilde{z}; \widetilde{y}, \widehat{\eta}) \, {\mathchar'26\mkern-11mu\mathrm{d}} \eta,
\end{multline*}
whence
\begin{multline*}
N((P_L\circ K \circ Q \circ \textup{Tr}_{L}-I)\circ P_1) = \\
\int e^{iY\eta} (P_{01} \circ P_0 \circ \sigma(K) \circ \sigma(Q) \circ \textup{Tr}_{B(L)} \circ P_{01}
-P_{01})(s|\eta|, |\eta|, z, \widetilde{z}; \widetilde{y}, \widehat{\eta})\, |\eta| \, {\mathchar'26\mkern-11mu\mathrm{d}} \eta.
\end{multline*}
In this combination, $\sigma(Q)(\widetilde{y}, \widehat{\eta})$ acts on $\mathcal C_{B(L)}$, so that
$\sigma(Q)(\widetilde{y}, \widehat{\eta})$ and $\sigma(Q)^{-1}(\widetilde{y}, \widehat{\eta})$ cancel. After further obvious cancellations,
this normal operator reduces to
\[
\int e^{iY\eta} (P_{01}-P_{01})(s|\eta|, |\eta|, z, \widetilde{z}; \widetilde{y}, \widehat{\eta})
\, |\eta| \, {\mathchar'26\mkern-11mu\mathrm{d}} \eta=0.
\]
Finally, using the boundedness properties of $P_1$, $P_L$ and $\mathrm{Tr}_L$, we see that
\[
R:=(P_L\circ K \circ Q \circ \textup{Tr}_{L}-I)\circ P_1:
\mathcal H\longrightarrow \ker L \cap \, x^{\underline{\delta}} H^\infty_e(M) \hookrightarrow x^{\underline{\delta}}L^2
\]
is bounded as well. From the composition results in \S \ref{triple},
$R\in \Psi^{-\infty, 0, E_{\mathrm{lf}}, E_{\mathrm{rf}}}(M^2_e)$ and $N(R)=0$, so in fact $R\in \Psi^{-\infty, 1, E_{\mathrm{lf}}, E_{\mathrm{rf}}}(M^2_e)$,
with index sets
\begin{equation}
\begin{split}
E_{\mathrm{lf}} &=\{(\zeta,p)\in \textup{Spec}_b(L) \mid \Im \zeta) >\underline{\delta} -1/2\}, \\
E_{\mathrm{rf}} &= \{(\zeta,p) \in \mathbb{C}\times \mathbb{N}_0 \mid (\zeta +2\underline{\delta}, p)\in E_{\mathrm{lf}}\},
\end{split}
\end{equation}
see \eqref{P-index}. Its compactness is now a consequence of \cite[Prop. 3.29]{M}.
\end{proof}
From Theorems \ref{right} and \ref{left} we now conclude the
\begin{cor}
Let $(L,Q)$ be elliptic. Then
\[
(L,Q): \mathcal H_{\underline{\delta},\overline{\delta}} \to x^{\overline{\delta}}L^2(M)\oplus \left(\bigoplus\limits_{k=1}^MH^{\underline{\delta}-d_k-1/2}(B,E_k)\right),
\]
is Fredholm, with parametrix
\begin{align*}
\mathcal{G}(f,\phi)= Gf + P_L[K(\phi - Q(\textup{Tr}_{L}Gf))].
\end{align*}
\end{cor}
We conclude by presenting one simple application of this machinery.
\begin{prop}
Let $u\in x^{\underline{\delta}} L^2(M)$ and suppose that $Lu=0$ and $\mathrm{Tr}_L u = 0$. Then $u \in x^{\overline{\delta}}H^\infty_e(M)$.
\end{prop}
\begin{proof}
Choose any left elliptic boundary value problem $(L,Q)$, and let $\mathcal G$ be its left parametrix, as constructed above, so
that $\mathcal G \circ (L,Q) = \mbox{Id} - \mathcal R$. Then $\mathrm{Tr}_L u = 0$, so $u = \mathcal R u$. Since $N(\mathcal R)=0$, \cite[Thm. 3.25]{M} gives
that
\[
\mathcal R: x^{\delta}H^s_e(M)\to x^{\delta+\epsilon}H^{\infty}_e(M), \ s\geq 0
\]
is bounded for some $\epsilon >0$ which depends only on $\mathcal R$. Iterating this statement gives the result.
\end{proof}
\bibliographystyle{amsplain}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 76 |
Biografia
Nato a Catania da padre siciliano e da madre ligure, crebbe a Genova, dove s'era trasferito con la famiglia all'età di 5 anni. Laureato in psicologia, per il cinema ha interpretato ruoli di caratterista, dalla presenza corpulenta e con una voce inconfondibile, in numerosi film, soprattutto nei filoni della commedia sexy all'italiana e del poliziottesco, in cui interpretò spesso e volentieri il ruolo del bonario grassone. È stato anche attivo nel teatro: esordì nel 1972 allo Stabile di Genova. Sposato con Mariella, ha avuto una figlia di nome Giusy.
Tra le principali pellicole in cui ha lavorato: Sabato, domenica e venerdì, La casa stregata, Attila flagello di Dio, Il ragazzo di campagna, 7 chili in 7 giorni, La classe non è acqua e Innamorato pazzo.
Ebbe anche occasione di recitare, alquanto saltuariamente, in produzioni di carattere internazionale, come Il nome della rosa di Jean-Jacques Annaud, in cui interpretava un inviato papale, e soprattutto Fuga di mezzanotte di Alan Parker, in cui si distinse nel ruolo del losco avvocato difensore turco Yesil (in cui, per esigenze di copione, recita interamente in turco).
Negli ultimi anni di carriera, in cui aveva diradato le sue apparizioni al cinema anche a causa di problemi di salute, ha preso parte ad alcune fiction televisive. Nelle sue ultime interpretazioni risulta visibilmente dimagrito. È morto, all'età di 57 anni, per arresto cardio-circolatorio, all'ospedale San Martino di Genova, dopo un ricovero ospedaliero durato due settimane. È sepolto presso il cimitero di Staglieno, a Genova.
Filmografia
Anastasia mio fratello, regia di Steno (1973)
Teresa la ladra, regia di Carlo Di Palma (1974)
Maria Rosa la guardona, regia di Marino Girolami (1974)
Fatevi vivi, la polizia non interverrà, regia di Giovanni Fago (1974)
Il piatto piange, regia di Paolo Nuzzi (1974)
Il colonnello Buttiglione diventa generale, regia di Mino Guerrini (1974)
Buttiglione diventa capo del servizio segreto, regia di Mino Guerrini (1975)
Nude per l'assassino, regia di Andrea Bianchi (1975)
Morte sospetta di una minorenne, regia di Sergio Martino (1975)
La liceale, regia di Michele Massimo Tarantini (1975)
La collegiale, regia di Gianni Martucci (1975)
Il giustiziere di mezzogiorno, regia di Mario Amendola (1975)
40 gradi all'ombra del lenzuolo, regia di Sergio Martino (1976)
La madama, regia di Duccio Tessari (1976)
Signore e signori, buonanotte, registi vari (1976)
Roma, l'altra faccia della violenza, regia di Marino Girolami (1976)
I soliti ignoti colpiscono ancora - E una banca rapinammo per fatal combinazion (Ab morgen sind wir reich und ehrlich), regia di Franz Antel (1976)
Taxi Girl, regia di Michele Massimo Tarantini (1977)
Tentacoli, regia di Ovidio G. Assonitis (1977)
Fuga di mezzanotte (Midnight Express), regia di Alan Parker (1978)
Sabato, domenica e venerdì, regia di Castellano e Pipolo (1979)
I contrabbandieri di Santa Lucia, regia di Alfonso Brescia (1979)
Gardenia il giustiziere della mala, regia di Domenico Paolella (1979)
La poliziotta della squadra del buon costume, regia di Michele Massimo Tarantini (1979)
Supersexymarket, regia di Mario Landi (1979)
Il viziaccio, regia di Mario Landi (1980)
Delitto a Porta Romana, regia di Bruno Corbucci (1980)
L'insegnante al mare con tutta la classe, regia di Michele Massimo Tarantini (1980)
Trhauma, regia di Gianni Martucci (1980)
Il minestrone, regia di Sergio Citti (1981)
Innamorato pazzo, regia di Castellano e Pipolo (1981)
Attenti a quei P2, regia di Pier Francesco Pingitore (1982)
Grand Hotel Excelsior, regia di Castellano e Pipolo (1982)
Attila flagello di Dio, regia di Castellano e Pipolo (1982)
La casa stregata, regia di Bruno Corbucci (1982)
Il paramedico, regia di Sergio Nasca (1982)
Testa o croce, regia di Nanni Loy (1982)
Don Camillo, regia di Terence Hill (1983)
Stesso mare stessa spiaggia, regia di Angelo Pannacciò (1983)
Il ragazzo di campagna, regia di Castellano e Pipolo (1984)
Scemo di guerra, regia di Dino Risi (1985)
Killer contro killers, regia di Fernando Di Leo (1985)
Mare amore - Frammenti di storie d'amore, regia di Angelo Pann (Angelo Pannacciò) (1985)
7 chili in 7 giorni, regia di Luca Verdone (1986)
Il nome della rosa, regia di Jean-Jacques Annaud (1986)
Il burbero, regia di Castellano e Pipolo (1986)
Il lupo di mare, regia di Maurizio Lucidi (1987)
La dolce casa degli orrori, regia di Lucio Fulci (1989)
Occhio alla perestrojka, regia di Castellano e Pipolo (1990)
C'era un castello con 40 cani, regia di Duccio Tessari (1990)
Fuga da Kayenta, regia di Fabrizio De Angelis (1991)
Assolto per aver commesso il fatto, regia di Alberto Sordi (1992)
Ci hai rotto papà, regia di Castellano e Pipolo (1993)
La casa degli spiriti, regia di Billie August (1993)
Piccolo grande amore, regia di Carlo Vanzina (1993)
18000 giorni fa, regia di Gabriella Gabrielli (1993)
Poliziotti, regia di Giulio Base (1994)
La classe non è acqua, regia di Cecilia Calvi (1996)
La sindrome di Stendhal, regia di Dario Argento (1996)
Anni '60, regia di Carlo Vanzina (1999)
Un uomo perbene, regia di Maurizio Zaccaro (1999)
Boom, regia di Andrea Zaccariello (1999)
I banchieri di Dio - Il caso Calvi, regia di Giuseppe Ferrara (2002)
Doppiatori italiani
Manlio De Angelis in La casa stregata, Nude per l'assassino, Gardenia il giustiziere della mala, Delitto a Porta Romana
Sergio Fiorentini in La poliziotta della squadra del buon costume
Michele Gammino in Tentacoli
Rino Bolognesi in L'insegnante al mare con tutta la classe
Note
Altri progetti
Collegamenti esterni
Attori cinematografici italiani
Attori teatrali italiani
Sepolti nel cimitero monumentale di Staglieno | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 7,492 |
{"url":"https:\/\/codereview.stackexchange.com\/questions\/48552\/split-list-into-groups-of-n-in-haskell\/48564","text":"# Split list into groups of n in Haskell\n\nI need to split a list into equal sublists, e. g [1..9] split into groups of 3 will be [[1, 2, 3], [4, 5, 6], [7, 8, 9]]. I have accomplished this task in the following way:\n\nsplitInGroupsOf n = takeWhile ((n ==) . length)\n. map fst\n. drop 1\n. iterate (\\(res, list) -> splitAt n list)\n. (,) []\n\n\nwhere iterate creates list of tuples with first n elements and rest of list. This way I had to use (,) [] on argument to ensure correct type, and unwrap result afterwards. My questions are\n\n1. is there a better\/more elegant way of performing same task?\n2. is there some standard function I should make use of?\n\nP.S.: I'm not sure where to ask simple Haskell-related questions and will appreciate if someone points me a better place for this than SE.\n\nThere are Data.List.Split.chunksOf and Data.List.Grouping.splitEvery implementations of this routine in specialized packages (and a number included in other application packages: search by Int -> [a] -> [[a]] signature on Hayoo).\n\nI think splitEvery implementation is pretty elegant:\n\nsplitEvery :: Int -> [a] -> [[a]]\nsplitEvery _ [] = []\nsplitEvery n xs = as : splitEvery n bs\nwhere (as,bs) = splitAt n xs\n\n\u2022 strangely enough, Hoogle is unable to find these functions. I've got a second question: when I search Hayoo for Int -> [a] -> [[a]] then chunksOf is not at top, but when I search Int -> [e] -> [[e]], then chunksOf is first. How am I supposed to guess what letter to use as a type variable? \u2013\u00a0sukhmel Apr 30 '14 at 17:13\n\nThe ready made function chunksOf works very well. When tasked to create 3 elements in sublists with 11 elements in the source list, two elements will be in the last sublist of the result. The following function also includes trailers.\n\nmklsts n = takeWhile (not.null) . map (take n) . iterate (drop n)\n\n\nI use this as pairs with a 2 for n and no n parameter. Pairs rock.\n\nThe match up of iterate and splitOn is one made in hell. In the questioner above, placing splitOn in a lambda may have compounded the problems. It is possible to make splitOn work with iterate but you have to ditch the fst of the tuple produced. That defeats the entire purpose. It is way cleaner and easier to use drop n with iterate. The results are the same. That's what the preceding function does. Otherwise, it's the same idea.\n\nHere is a novel way of producing the identical results using tails imported from Data.List in a list comprehension. It picks up stragglers, too.\n\nts n ls = [take n l|l<-init$tails ls,odd (head l)] The parameters are size-of-sublist and list Edit 4\/17\/2018 Well, since I had some time at work a list comprehension version that does not use tails, a recursive version and a map version. ttx s ls=[take s.drop x$ls|x<-[0,s..s*1000]]\n\n\nRecursive\n\ntkn n []=[];tkn n xs=[take n xs]++(tkn n $drop n xs) Map tp n ls=takeWhile(not.null)$ map(take n.flip drop ls) [0,n..]\n\n\nThe list comprehension is virtually infinite. Change [0,s..s*200] to [0,s..] for true infinity. The recursive is, of course, inherently infinite and the map function uses a big takeWhile (not.null) to end itself.\n\n\u2022 \"The ready made function chunksOf\"? Which chunksOf do you speak of? Where do you address the code of the original poster? Did you intend to comment leventov's answer instead? \u2013\u00a0Zeta Apr 12 '18 at 7:03\n\u2022 The questions are two. 1 is there a better\/more elegant way of performing same task? 2. Is there some standard function I should make use of? It is implied, the code of the original works fine. What I suggest as an alternative is shorter and clear. More elegant? It is relative I thinks so. Am I wrong? i tried so hard to use splitAt` but this resulted instead out of frustration. \u2013\u00a0fp_mora Apr 12 '18 at 7:59\n\u2022 Sure, but on Code Review, we review code. We do not only provide an alternative solution without an explanation why its better. I can apply the same critcism on leventov's answer, by the way. \u2013\u00a0Zeta Apr 12 '18 at 8:14\n\u2022 Thank you so much for the clarification. Criticisms leveled at me are usually the opposite. I can be tedious. My interest in this function is not incidental nor frivolous. The value is information contained in the sublists exceed that of the source list so they can streamline code and lessen logic. There are few things I value more than criticism. Thank you so very much. \u2013\u00a0fp_mora Apr 12 '18 at 16:10","date":"2020-06-06 18:21:53","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.4343210756778717, \"perplexity\": 1982.5066781070984}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"absolute_threshold\": 20, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-24\/segments\/1590348517506.81\/warc\/CC-MAIN-20200606155701-20200606185701-00590.warc.gz\"}"} | null | null |
Glucocorticoid secretion, which is controlled by the suprachiasmatic nucleus, is highly tuned to circadian rhythmicity.
An extended-release dose of glucocorticoid replacement can improve circadian clock gene expression in patients with adrenal insufficiency, according to a report recently published in The Journal of Clinical Endocrinology & Metabolism.
Conventional glucocorticoid replacement has long been associated with a number of negative side effects. The single-blind DREAM trial reported significant effects of glucocorticoid replacement on immune function. An examination of patients' peripheral blood mononuclear cells showed higher levels of pro-inflammatory monocytes, as well as lower levels of CD16+CD14- and CD16+ natural killer cells. However, after switching to physiological, oral-release hydrocortisone once per day, the rate of infections dropped, suggesting that this method is a safer alternative to conventional glucocorticoid replacement.
Glucocorticoid secretion, which is controlled by the suprachiasmatic nucleus, is highly tuned to circadian rhythmicity. Combined with the large number of glucocorticoid receptors on peripheral cells/tissues and their absence on suprachiasmatic nucleus neurons, this makes glucocorticoid secretion a likely contender for a peripheral circadian oscillator. It directly regulates the circadian clock gene expression and performs intact cellular glucocorticoid signaling to entrain circadian oscillators in peripheral tissue.
Glucocorticoid plays an important role in regulating immune response. Administering glucocorticoid leads to entrainment of molecular clocks in human peripheral blood mononuclear cells by regulating BMAL-1 and PER2-3. | {
"redpajama_set_name": "RedPajamaC4"
} | 2,167 |
require 'thin-latency/rack/handler'
require 'thin-latency/connection'
require 'thin-latency/backends/tcp-server'
| {
"redpajama_set_name": "RedPajamaGithub"
} | 5,856 |
{"url":"https:\/\/stacks.math.columbia.edu\/tag\/01OE","text":"Lemma 27.21.5. Let $S$ be a scheme. Let $n \\geq 0$. Then $\\mathbf{P}^ n_ S$ is a projective bundle over $S$.\n\nProof. Note that\n\n$\\mathbf{P}^ n_{\\mathbf{Z}} = \\text{Proj}(\\mathbf{Z}[T_0, \\ldots , T_ n]) = \\underline{\\text{Proj}}_{\\mathop{\\mathrm{Spec}}(\\mathbf{Z})} \\left(\\widetilde{\\mathbf{Z}[T_0, \\ldots , T_ n]}\\right)$\n\nwhere the grading on the ring $\\mathbf{Z}[T_0, \\ldots , T_ n]$ is given by $\\deg (T_ i) = 1$ and the elements of $\\mathbf{Z}$ are in degree $0$. Recall that $\\mathbf{P}^ n_ S$ is defined as $\\mathbf{P}^ n_{\\mathbf{Z}} \\times _{\\mathop{\\mathrm{Spec}}(\\mathbf{Z})} S$. Moreover, forming the relative homogeneous spectrum commutes with base change, see Lemma 27.16.10. For any scheme $g : S \\to \\mathop{\\mathrm{Spec}}(\\mathbf{Z})$ we have $g^*\\mathcal{O}_{\\mathop{\\mathrm{Spec}}(\\mathbf{Z})}[T_0, \\ldots , T_ n] = \\mathcal{O}_ S[T_0, \\ldots , T_ n]$. Combining the above we see that\n\n$\\mathbf{P}^ n_ S = \\underline{\\text{Proj}}_ S(\\mathcal{O}_ S[T_0, \\ldots , T_ n]).$\n\nFinally, note that $\\mathcal{O}_ S[T_0, \\ldots , T_ n] = \\text{Sym}(\\mathcal{O}_ S^{\\oplus n + 1})$. Hence we see that $\\mathbf{P}^ n_ S$ is a projective bundle over $S$. $\\square$\n\nIn your comment you can use Markdown and LaTeX style mathematics (enclose it like $\\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).","date":"2022-07-02 08:52:07","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 2, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 2, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9899545907974243, \"perplexity\": 128.89408688096373}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-27\/segments\/1656103989282.58\/warc\/CC-MAIN-20220702071223-20220702101223-00030.warc.gz\"}"} | null | null |
Piper rionechianum är en pepparväxtart som beskrevs av Truman George Yuncker. Piper rionechianum ingår i släktet Piper och familjen pepparväxter. Inga underarter finns listade i Catalogue of Life.
Källor
Pepparväxter
rionechianum | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 1,155 |
require_relative '../test_helper'
module Decks
class SetTrackNumberScreenTest < ScreenTestCase
attr_reader :screen
def track
release[0]
end
def setup
super
@screen = SetTrackNumberScreen.new track, placeholder
end
def test_assigns_the_number
screen.set 5
assert_equal 5, track.number
end
def test_prints_a_useful_prompt
assert_equal 'Enter track # for this track', screen.prompt
end
end
end
| {
"redpajama_set_name": "RedPajamaGithub"
} | 2,301 |
Q: Filter ListView Items from Binding Source my app shows a ListView which uses Data Binding to show a list of several items. Some variables use TwoWay binding and thus changes are saved in the viewmodel instance which is the datacontext. This is my "main" instance of the viewmodel as it contains all items.
Now I want to filter this list (which contains all items) and only show a portion of the items. In the past, I created a new instance of the viewmodel in the code behind and copied all the items I wanted to show to from my main instance to the newly created instance of the viewmodel and then set this new viewmodel as the DataContext.
This worked fine, but when I changed Data it would only save to the newly created instance and not to the main instance. Thus, when I changed the filter the items would again be loaded from the main instance without any changes made in between.
Is there any way to filter my main instance of the view model? I would like to work on the main instance so that changes are saved automatically.
The ViewModel contains an ObservableCollection of my own class, I would like to only show some of the items for binding according to a filter.
A:
In the past, I created a new instance of the viewmodel in the code behind and copied all the items I wanted to show to from my main instance to the newly created instance of the viewmodel and then set this new viewmodel as the DataContext.
I think it is not necessary to create a new instance, you can directly manipulate your ObservableCollection for the source of ListView.
Just for example, I created a List which contains all the data, and add data to ObservableCollection from this List:
<Page.DataContext>
<local:MainPageViewModel x:Name="VM" />
</Page.DataContext>
<Grid>
<Grid.RowDefinitions>
<RowDefinition Height="4*" />
<RowDefinition Height="*" />
<RowDefinition Height="*" />
</Grid.RowDefinitions>
<ListView ItemsSource="{x:Bind VM.peopleCollection}">
<ListView.ItemTemplate>
<DataTemplate>
<StackPanel>
<TextBox Text="{Binding Name, Mode=TwoWay}" />
<TextBox Text="{Binding Company, Mode=TwoWay}" />
<TextBox Text="{Binding Age, Mode=TwoWay}" />
</StackPanel>
</DataTemplate>
</ListView.ItemTemplate>
</ListView>
<StackPanel Grid.Row="1">
<Button Content="Age From 20-29" Click="{x:Bind VM.Age_Filter}" />
<Button Content="Company AA" Click="{x:Bind VM.Company_Filter}" Margin="0,10" />
<Button Content="Name Peter" Click="{x:Bind VM.Name_Filter}" />
</StackPanel>
<Button Content="Show All" Click="{x:Bind VM.Show_All}" Grid.Row="2" />
</Grid>
code behind in the MainPageViewModel:
public class MainPageViewModel
{
public MainPageViewModel()
{
peopleList.Add(new Person { Name = "Jay", Company = "AA", Age = 25 });
peopleList.Add(new Person { Name = "Peter", Company = "BB", Age = 35 });
peopleList.Add(new Person { Name = "Jayden", Company = "AA", Age = 27 });
peopleList.Add(new Person { Name = "John", Company = "AAC", Age = 26 });
peopleList.Add(new Person { Name = "Alan", Company = "BB", Age = 45 });
peopleList.Add(new Person { Name = "Frank", Company = "BB", Age = 29 });
peopleList.Add(new Person { Name = "Ami", Company = "AA", Age = 24 });
peopleList.Add(new Person { Name = "Elvis", Company = "AA", Age = 30 });
peopleCollection.Clear();
foreach (var person in peopleList)
{
peopleCollection.Add(person);
}
}
private static List<Person> peopleList = new List<Person>();
public ObservableCollection<Person> peopleCollection = new ObservableCollection<Person>();
public void Age_Filter(object sender, RoutedEventArgs e)
{
foreach (var person in peopleList)
{
if (person.Age > 29 || person.Age < 20)
peopleCollection.Remove(person);
}
}
public void Company_Filter(object sender, RoutedEventArgs e)
{
foreach (var person in peopleList)
{
if (person.Company != "AA")
peopleCollection.Remove(person);
}
}
public void Name_Filter(object sender, RoutedEventArgs e)
{
foreach (var person in peopleList)
{
if (person.Name != "Peter")
peopleCollection.Remove(person);
}
}
public void Show_All(object sender, RoutedEventArgs e)
{
peopleCollection.Clear();
foreach (var person in peopleList)
{
peopleCollection.Add(person);
}
}
}
In the data model of class Person, there are only three properties: Name, Company and Age.
public class Person : INotifyPropertyChanged
{
private string _Name;
public string Name
{
get { return _Name; }
set
{
if (value != _Name)
{
_Name = value;
OnPropertyChanged();
}
}
}
private string _Company;
public string Company
{
get { return _Company; }
set
{
if (value != _Company)
{
_Company = value;
OnPropertyChanged();
}
}
}
private int _Age;
public int Age
{
get { return _Age; }
set
{
if (value != _Age)
{
_Age = value;
OnPropertyChanged();
}
}
}
public event PropertyChangedEventHandler PropertyChanged;
private void OnPropertyChanged([CallerMemberName]string propertyName = "")
{
if (this.PropertyChanged != null)
{
PropertyChanged(this, new PropertyChangedEventArgs(propertyName));
}
}
}
As you can see, I remove the item from ObservableCollection when filter, this will make the filtering based on the last filter result. If you want to filter based on all the items, you can for example code like this:
public void Company_Filter(object sender, RoutedEventArgs e)
{
peopleCollection.Clear();
foreach (var person in peopleList)
{
if (person.Company == "AA")
peopleCollection.Add(person);
}
}
A: Normally it's done like this:
ICollectionView dataView =
CollectionViewSource.GetDefaultView(IncList.ItemsSource);
dataView.Filter = o =>
{
EventData t = o as EventData;
return t.action == evGroup.action && t.objid == evGroup.objid;
};
dataView.Refresh();
This technique does filtering manipulation on ListView, not on ItemSource and it's convinient: You just use datasource as usual (add-remove elements) and ListView shows current View filtered.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 7,937 |
\section{Introduction} \label{introduction}
The limit order book (LOB) is used by financial exchanges to match buyers and sellers of a particular instrument and acts as an indicator of the supply and demand at a given point in time. It can be described as a self-evolving process with complex spatial and temporal structures revealing the price dynamics at the microstructural level. Market making, optimal execution and statistical arbitrage strategies, all require a good understanding of the LOB and its dynamics. Figure \ref{fig:lob} (A) shows a snapshot of the LOB with both the bid (buy) and ask (sell) order volumes accumulated at each price level. The mid-price is the average of the best (lowest) ask price and the best (highest) bid price and the difference between them is referred to as the bid-ask spread. The LOB gets updated continuously with order placements, cancellations and executions.
\begin{figure}[!htb]
\centering
\includegraphics[width=0.45\textwidth]{figures/LOB_illustration3.eps}
\caption{(\textbf{A}) A snapshot of the limit order book. (\textbf{B}) Workflow of a price forecasting task using LOB data with machine learning models.}
\label{fig:lob}
\end{figure}
\begin{figure*}[!tb]
\centering
\includegraphics[width=\textwidth]{figures/perturbation.eps}
\caption{(\textbf{A}) Original LOB data with 10 levels on ask and bid side without perturbation. (\textbf{B}) LOB data with 10 levels after data perturbation. Red blocks represent intentionally placed perturbation orders with order volume = 1. Compared with the original one, the new 10-level data representation has a much narrower vision on the market.}
\label{fig:perturbation}
\end{figure*}
The use of algorithmic trading strategies and the digitisation of exchange activities has made available a tremendous amount of LOB data for practitioners and researchers to study the market dynamics from data-driven approaches. This led to a surge in interest for big data applications in the financial markets and machine learning (including deep learning) models becoming a trend in the quantitative finance domain \cite{buehler2019deep}, \cite{wiese2020quant}. The LOB data come in different degrees of granularity with \emph{L1} data providing the best bid/ask prices and volumes, \emph{L2} data providing the same data across all price levels and \emph{L3} data containing the non-aggregated orders placed by market participants.
In our work, we focus on how LOB data is typically represented by taking a price forecasting task as an example. In practice, a vector representation of the raw limit order book information is needed for upcoming learning processes. This transformation from raw data to feature vectors is typically referred to as \emph{feature engineering}, which requires a good and comprehensive understanding of the domain knowledge to make sure the extracted features match the learning task. By contrast, \emph{representation learning}, also called \emph{feature learning}, is an automated approach to discover an optimal representation for the data. The major difference between feature engineering and representation learning is whether the representation is formed in a purely data-driven way. Also, it is common for a machine learning system to involve both feature engineering and representation learning with multiple levels of representation appearing at different stage of processing (see figure \ref{fig:lob} (B)).
The performance of machine learning models is heavily influenced by the data representation scheme \cite{bengio2013representation}. For neural networks, the representation learning and the prediction processes are combined within the network structure and are trained together towards the same target function. In this case, the original representation of LOB, \textit{i.e.} the input representation to neural networks, becomes the foundation of the entire model. Presently, the price level-based data representation scheme is used in almost all recent studies \cite{tsantekidis2017using,tsantekidis2017forecasting,tran2018temporal,zhang2019deeplob,mahfouz2019importance,sirignano2019deep,tsantekidis2020using,wallbridge2020transformers} applying deep learning models on LOB data. However, this representation scheme is rarely discussed or investigated towards its compatibility with machine learning especially deep learning models. In this paper, we propose a pioneer insight to challenge this level-based LOB representation for machine learning models, by showing potential risks under subtle perturbations and raising concerns regarding to its robustness.
\paragraph{Summary of contribution} This paper propose a perturbation paradigm to the LOB. By examining the performance change of LOB price forecasting machine learning models under perturbation, we examine the robustness of data representation. The experimental results confirm our concerns about the current level-based LOB representation as well as machine learning models designed based on this representation scheme. Furthermore, this paper present desiderata of LOB representations for guiding future research in this area.
\section{Problem Description} \label{problem_description}
We first introduce the commonly-used level-based LOB representation scheme found in benchmark and research datasets (e.g. \cite{ntakaris2018benchmark,huang2011lobster}). The spatial representation of this scheme, \textit{i.e.} an LOB snapshot, is a vector $s_{t} = \left\{p_{a}^{i}(t), v_{a}^{i}(t), p_{b}^{i}(t), v_{b}^{i}(t)\right\}_{i=1}^{L}$, where, $p_{a}^{i}(t)$, $p_{b}^{i}(t)$ are the ask and bid prices for price level $i$ and $v_{a}^{i}(t)$, $v_{b}^{i}(t)$ are the ask and bid volumes respectively. Temporally, the history of the LOB snapshots is stacked to reflect an evolution of the market, leading to the commonly-used limit order book data structure $S\in \mathbb{R}^{T\times 4L}$ where T is the history length and L is the amount of price levels considered for each side.
This level-based representation is efficient and convenient from the perspective of human understanding and how the matching engine in exchanges works. From the machine learning perspective, this representation has some particular characteristics. The most intuitive one is that the price and volume for each LOB level are tied together - any disentanglement or distortion to this would result in an invalid representation. In addition, the spatial structure across different levels is not homogeneous since there is no assumption for adjacent price levels to have fixed intervals. Note that, homogeneous spatial relationship is a basic assumption for convolutional neural networks (CNN) due to the parameter sharing mechanism. Thus, the heterogeneous spatial feature of level-based LOB data may reduce model robustness when learning with CNN models. From the temporal perspective, we also realise some instability of the representation due to occasional shifts of price levels - the previous best bid/ask data can suddenly shift to second best bid/ask channel if a new order is placed with a better price.
\section{Data Perturbation} \label{methods}
We present a simple data perturbation method to examine the robustness of the price level-based representation from the machine learning perspective. In some LOB data for equities, the price difference between adjacent price levels is sometimes larger than the tick size (the minimum price increment change allowed). This is especially prevalent in small-tick stocks and can result in the entire LOB shifting even if a small order of the minimum allowable size is placed at a price in between the existing price levels. The data perturbation method presented assumes that the data is perturbed by small size orders at empty price levels beyond the best ask/bid prices. This perturbation ensures no change is made to the mid-price before and after perturbation to make sure the prediction labels are not affected.
We illustrate this data perturbation with a synthetic LOB example as shown in Fig. \ref{fig:perturbation}. Fig. \ref{fig:perturbation} (A) shows the synthetic LOB snapshot with 10 price levels in both ask and bid sides of the LOB (marked as L1-L10) before any perturbation. We assume the tick size is 0.01 and the minimum order size present in our data is 1. In this LOB snapshot, the mid-price is 10.00 with bid-ask spread equal to 0.04. We can observe some price levels where no orders are placed, such as 10.03, 10.06 in the ask side and 9.96, 9.94 in the bid side. To perturb this LOB data, one can place orders with allowed minimum order size to fill these empty price levels. These minimum size orders may seem to be not influential since 1) they do not effect the mid price, 2) their volumes are tiny. c (see Fig. \ref{fig:perturbation} (B)). Approximately half of the original price level information is no longer visible after perturbation (e.g. ask-side L5 to L10 information is not included in representation after perturbation) and while the rest are preserved, they are shifted to different levels in the LOB representation (e.g., the ask-side L2 appears in ask-side L3 after perturbation).
Intuitively, this perturbation has two impacts from the machine learning point of view. Firstly, it shifts the 40-dimensional input space dramatically. For example, the Euclidean distance between these two 40-dimensional vectors before and after perturbation is 344.623 whereas actually the total volume of orders applied is only 10. This means that the level-based representation scheme does not bring local smoothness. Furthermore, it narrows the scope of vision of machine learning models to `observe' the market. As shown in the LOB data visualisation plot in Fig. \ref{fig:perturbation}, the gray areas are masked out for the model input after perturbation.
\section{Related Work} \label{related_work}
The study of the importance of robust data representation, the criteria for evaluating the quality of the representation, and the variety of methods for learning these representations is studied extensively in the machine learning literature with \cite{bengio2013representation} providing a survey of these methods. In our work, we focus on the representation of financial market microstructure data. \cite{bouchaud2018trades}, \cite{abergel2016limit} study the structure and empirical properties of limit order books and provide a set of statistical properties (referred to as \textit{stylized facts}) using NASDAQ exchange data. On the other hand, \cite{lehalle2018market} discusses the practical aspects and issues of market structure, design, price formation, discovery and the behaviour of different actors in limit order book markets. A significant amount of research in recent years focused on applying deep learning models on limit order book data for the purposes of price forecasting or price movements classification. Different model architectures were investigated including multi-layer perceptrons \cite{mahfouz2019importance}, recurrent neural networks \cite{sirignano2019universal}, convolutional neural networks \cite{zhang2019deeplob}, \cite{zhang2018bdlob} and self-attention transformer networks \cite{wallbridge2020transformers}.
\section{Experiments and Results} \label{results}
In this section, we implement a series of experiments to examine the robustness of the LOB representation under different data perturbations. We take price forecasting as the task paradigm and train various machine learning models to perform such task and examine their performance when encountering unexpected perturbation.
\subsection{Benchmark Dataset and Models}
We use the FI-2010 dataset \cite{ntakaris2018benchmark} as the benchmark dataset. The FI-2010 dataset consists of LOB data from 5 stocks in the Helsinki Stock Exchange during normal trading hours (no auctions) for 10 trading days. This dataset takes into account 10 price levels on the bid/ask sides of the limit order book, which are updated according to events such as order placement, executions and cancellations. In our experiments, we take into account the history of the LOB snapshots for future price movement prediction. Thus, each input data point is a short time series with input dimension $T \times 40$ where T is the total amount of historical snapshots. For experiments in this paper, we choose T = 10.
The prediction target is the micro-movement $l_t = \frac{m_+(t)-p_t}{p_t}$ where $m_+(t) = \frac{1}{k} \sum^k_{i=1}p_{t+i}$ is the smoothed mid-price with prediction horizon $k$. The movement is further categorised into three classes - 1:up ($l_t>0.002$), 2:stationary ($-0.002<l_t<0.002$), 3:up ($l_t<-0.002$). We choose the FI-2010 prediction labels with prediction horizon k-50 as the targets for model training and testing. The training set is relatively balanced with 34\%/ 32\%/ 34\% components of labels Up / Stationary / Down samples. By contrast, the testing set is an unbalanced set with 28\%/ 47\%/ 25\% samples for Up / Stationary / Down.
We choose 4 benchmark price forecasting models, \textit{i.e.} logistic regression, multi-layer perceptron (MLP), long short term memory (LSTM) and the DeepLOB model combining convolutional networks with LSTM \cite{zhang2019deeplob}. All these methods are trained with the same FI-2010 training dataset and then tested in 4 types of scenarios: the data is not perturbed (`None') and the data is perturbed by placing minimum-size orders to fill the ticks on the ask side only (`ask-side'), on the bid side only (`bid-side'), on both the ask and bid side.
\subsection{Model Performance under Perturbation}
Table \ref{table:performance} demonstrates the testing performance of these machine learning models in the price movement forecasting tasks. Since the testing set is unbalanced, we use 4 different metrics (scores) to evaluate and compare the performance - Accuracy (\%), Precision (\%), Recall (\%) and F-score (\%). Among these metrics, Accuracy (\%) is measured as the percentage of predictions of the test samples exactly matches the ground truth, which is the unbalanced accuracy score where as the rest metrics are all averaged across classes in an unweighted manner to eliminate the influence of data imbalance.
We observe a performance decay of all the machine learning models under the unexpected data perturbation introduced, especially these sophisticated models like LSTM and DeepLOB. In addition, the performance decline alters under different types of data perturbation. Compared with no perturbation scenario, ask-side and bid-side perturbations cause around 6\% accuracy decrease on MLP, 7\% on LSTM and 9\%-14\% on DeepLOB. When the perturbation is applied to both sides, the performance decrease becomes more severe - 11\% accuracy decrease on MLP, 12\% on LSTM and 30\% on DeepLOB. Similar trends can also be viewed for other evaluation metrics.
From the these performance decay results, we find that DeepLOB, the best performed model under normal condition as well as the most complicated one, is also the most vulnerable one under perturbation. Its predictive accuracy decreases to 47.5\% and the F-score is only 22.2\%, which even underperforms logistic regression. The reason behind this phenomenon may be a combination of various factors. On one hand, the complexity of model is related to overfitting, which may reduce the generalisation ability and become unstable under the perturbation. Also as we mentioned in earlier sections, CNN assumes homogeneous spatial relationship but the level-based LOB representation is obviously heterogeneous, which leads to a mismatch between the data representation and the network characteristics. Once the spatial relationship is further broken due to perturbation, the CNN descriptors may not be able to extract meaningful features and thus cause malfunction of the entire predictor.
\begin{table}[!tb]
\centering
\small
\begin{tabular}{c c c c c }
\toprule
\multirow{2}{*}{Data Perturbation} &
\multicolumn{4}{c}{Metrics (\%)} \\
\cmidrule(lr){2-5}
& Accuracy & Precision & Recall & F-score \\
\midrule
\multicolumn{5}{c}{Logistic Regression} \\
\midrule
None & 52.9 & 62.9 & 41.6 & 38.2\\
Ask & 49.7 & 50.4 & 42.9 & 42.4\\
Bid & 51.5 & 62.2 & 40.3 & 35.7\\
Both sides & 49.3 & 45.8 & 42.0 & 41.5\\
\midrule
\midrule
\multicolumn{5}{c}{MLP} \\
\midrule
None & 61.1 & 66.6 & 54.7 & 55.6\\
Ask & 55.5 & 61.4 & 49.2 & 48.5\\
Bid & 56.5 & 63.6 & 46.8 & 46.1\\
Both sides & 51.0 & 50.0 & 41.5 & 39.3\\
\midrule
\midrule
\multicolumn{5}{c}{LSTM} \\
\midrule
None & 70.4 & 68.4 & 68.0 & 68.2 \\
Ask & 63.6 & 62.1 & 62.0 & 61.5\\
Bid & 63.0 & 61.2 & 61.0 & 60.5\\
Both sides & 57.9 & 54.7 & 54.9 & 54.8\\
\midrule
\midrule
\multicolumn{5}{c}{DeepLOB \cite{zhang2019deeplob}} \\
\midrule
None & 77.2 & 76.2 & 74.3 & 75.1 \\
Ask & 68.0 & 70.1 & 60.8 & 62.3 \\
Bid & 63.2 & 68.6 & 54.6 & 55.5 \\
Both sides & 47.5 & 47.9 & 33.7 & 22.2 \\
\bottomrule
\end{tabular}
\caption{Price forecasting model performance under data perturbation. Each model is trained with a non-perturbed training set and when testing the model, we apply various data perturbation. None: no perturbation. Ask-side: perturbation only applied to the ask-side of data. Bid-side: perturbation only applied to the bid-side of the data. Both sides: perturbation applied to both ask and bid sides.}
\label{table:performance}
\end{table}
\begin{figure*}[!tb]
\centering
\includegraphics[width=0.95\textwidth]{figures/confusion.eps}
\caption{Confusion matrices for corresponding experimental results in Table. \ref{table:performance}}
\label{fig:cm}
\end{figure*}
Fig. \ref{fig:cm} further illustrates more details behind the numerical performance metrics in the form of a confusion matrix. The logistic regression model basically classify a majority of samples as `Stationary' no matter whether perturbation is applied. Similarly in MLP, about half of the `Up' and `Down' samples are misclassified as `Stationary' ones. Both LSTM and DeepLOB shows confusion matrices with obvious diagonal feature without perturbation - more than half of the samples from each class are classified the same as their true labels. When the perturbation is applied, LSTM shows performance decrease but the still near half of samples are correctly classified. DeepLOB, however, fails in the perturbation condition by misclassifying almost all the data to `Stationary' class (see DeepLOB+Both in Fig. \ref{fig:cm}).
\section{Conclusion and Future Work} \label{conclusion}
In this paper, we discussed the importance of data representations to machine learning models applied to LOB-related tasks. We designed data perturbation scenarios to test the robustness of commonly-used machine learning models with the level-based LOB representation scheme in price forecasting tasks. We show that, although the perturbation is subtle, it still shows a large impact to the level-based representation and thus leads to significant performance decay. In particular, this performance decay is more severe in sophisticated machine learning models.
Based on the findings in this paper and our understanding of representations, we would like to raise some desiderata for improving the robustness of LOB-related data representations and machine learning models designed on top of certain representations:
\begin{itemize}
\item \textbf{Smoothness}: LOB Data representations should not change dramatically under subtle perturbations.
\item \textbf{Efficiency}: LOB data representation should organise data in a efficient manner to reduce the \emph{curse of dimensionality}.
\item \textbf{Stochasticity}: LOB data representation should consider the stochasticity of the market, instead of treating the market process as a deterministic one.
\item \textbf{Validity}: Basic assumptions needs to be matched between data representations and learning models. If not, these models may contain unknown risks due to invalid fundamental settings.
\end{itemize}
Our future work would focus on feature engineering and representation learning for LOB data that can fulfill these desiderata and combining LOB representations with various machine learning tasks including forecasting, reinforcement learning and etc.
\section*{Acknowledgements}
The authors would like to acknowledge our colleagues Vacslav Gluckov, Rui Silva, Thomas Spooner and Jeremy Turiel and for their input and suggestions at various key stages of the research.
This paper was prepared for informational purposes by the Artificial Intelligence Research group of JPMorgan Chase \& Co and its affiliates ("J.P. Morgan"), and is not a product of the Research Department of J.P. Morgan. J.P. Morgan makes no representation and warranty whatsoever and disclaims all liability, for the completeness, accuracy or reliability of the information contained herein. This document is not intended as investment research or investment advice, or a recommendation, offer or solicitation for the purchase or sale of any security, financial instrument, financial product or service, or to be used in any way for evaluating the merits of participating in any transaction, and shall not constitute a solicitation under any jurisdiction or to any person, if such solicitation under such jurisdiction or to such person would be unlawful.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 4,856 |
If you choose "Bank Transfer" for your order, you can transfer the total of the order to our bank account. After you have confirmed the order, you will be sent further information regarding the payment by mail. Next, simply transfer the sum of your order to our bank account. As soon as we have received the transfer, your order will continue processing. For faster process you can send us confirmation of the payment and we'll handle your package the same day. Due to the transfer periods between banks, it may take a day or two for your payment to reach us. Please note that we can only send parcels once we have received your transfer payment.
You can also pay for your order with credit and debit card via paypal without needing to create an account at paypal. | {
"redpajama_set_name": "RedPajamaC4"
} | 7,818 |
THE LATE LIEUTENANT-COMMANDER ELWELL.
Lieutenant-Commander Charles Bingham Elwell was a son of the late Mr. P. B. Elwell, formerly electrical engineer to the New South Wales Railway and Tramway Department.
Mon 14 Sep 1914 - The Sydney Morning Herald (NSW : 1842 - 1954)
Page 8 - THE LATE LIEUTENANT-COMMANDER ELWELL.
THE LATE LIEUTENANT-COM
MANDEK ELWELL.
Lieutenant-Commander Charles Bingham El-
well was a son of the late Mr. P. B. Elwell,
formerly electrical engineer to tho New South
Wales Railway and Tramway Department.
The deceased officer was lent to the Royal
Australian navy by the Imperial authorities,
and returned to the Commonwealth as first
lieutenant of tho cruiser Melbourne last year.
He afterwards joined the gunnery tender
Pioneer, and at tho beginning of 1914 was
attached to the Royal Naval College at Gee-
long as skilled instructor. He then beenmo
Lieutenant-Commander. When the mixed
force was lately sent away from Sydney Lieu-
tenant-Commander Elwell was appointed to
tbo command of the Royal Naval RescrvlBts
who went with it. Ho was an officer who was
Intensely popular with the endets at Geelong
and with his brother officers. He had a great
charm of manner, and though quiet in speech
was full of energy In action-in work and
games. The college boat's crew owes its effi-
ciency to his coaching.
THE LATE LIEUTENANT-COMMANDER ELWELL. (1914, September 14). The Sydney Morning Herald (NSW : 1842 - 1954), p. 8. Retrieved January 21, 2021, from http://nla.gov.au/nla.news-article15551200
"THE LATE LIEUTENANT-COMMANDER ELWELL." The Sydney Morning Herald (NSW : 1842 - 1954) 14 September 1914: 8. Web. 21 Jan 2021 <http://nla.gov.au/nla.news-article15551200>.
1914 'THE LATE LIEUTENANT-COMMANDER ELWELL.', The Sydney Morning Herald (NSW : 1842 - 1954), 14 September, p. 8. , viewed 21 Jan 2021, http://nla.gov.au/nla.news-article15551200
{{cite news |url=http://nla.gov.au/nla.news-article15551200 |title=THE LATE LIEUTENANT-COMMANDER ELWELL. |newspaper=[[The Sydney Morning Herald]] |issue=23,925 |location=New South Wales, Australia |date=14 September 1914 |accessdate=21 January 2021 |page=8 |via=National Library of Australia}}
The Sydney Morning Herald (NSW : 1842 - 1954), Mon 14 Sep 1914, Page 8 - THE LATE LIEUTENANT-COMMANDER ELWELL. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 647 |
An enormous kite is lowering costs and emissions on cargo ships
Mobility News Technology
Image: Airseas
The image seems almost fictional: a giant kite, towing a five-gigaton commercial ship through the ocean. But a small Toulouse-based company is pulling it off, providing the shipping industry with a much-needed solution for its high costs and environmental impact.
Airseas, a spin-off of aerospace company Airbus, developed the Seawing, an autonomous parafoil kite to tow cargo ships. Via a simple on/off switch, the vessel's captain can launch or recover the autonomous kite, and harness the wind to continue its course with up to a 20% decrease in fuel use and associated emissions.
The Seawing is an important step towards reaching the aim of the International Maritime Organization to realize a carbon dioxide emissions cut of 40% by 2030, as well as reducing other emissions such as nitrogen oxide and particulate matter. The 28,000 ships navigating the seas that weigh more than 5,000 gigaton emit about 2.7% of global CO2 emissions. As fuel constitutes up to 50% of total operating costs, putting the wind to work is an appealing plan.
In June of 2019, the leading Japanese transport company K Line announced a pilot to equip one of their ships with a Seawing, with a potential 50 more in the future. A year earlier, Airbus was the first to order the kite for its cargo ship and from 2021, the Seawing will be available to large shipowners. Ultimately, Airseas' aim is to equip 15% of the world's fleet with the system.
Besides transport purposes, kites can also be used to generate electricity, as Kite Power Systems has proved. The Scotland-based engineering company has been developing its alternative to wind turbines since 2011, aiming at a solution that is cheaper and uses 85% less material.
Airseas
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"redpajama_set_name": "RedPajamaCommonCrawl"
} | 4,738 |
cp /tmp/tomcat/deployments/*.war /opt/tomcat/webapps
exec /usr/sbin/init | {
"redpajama_set_name": "RedPajamaGithub"
} | 4,822 |
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| {
"redpajama_set_name": "RedPajamaGithub"
} | 4,375 |
PRESS RELEASE June 6, 2018
Six Million Indonesians Will Gain Access to Water at Home
WASHINGTON D.C., June 6, 2018 – Six million Indonesians, including three million women, should have access to water through piped connections to their homes, due to a new US$100 million project approved today by the World Bank's Board of Executive Directors. The biggest beneficiaries are expected to be the three million women who are disproportionately, truly affected by lack of access to clean water at home.
The National Urban Water Supply Project seeks to support Indonesia's development through improved access to water sources and enhanced performance of water service providers in underserved urban areas. Today, nearly one out of two Indonesians lacks access to safe water, and more than 70 percent of the nation's 260 million people rely on potentially contaminated sources.
"Stunting is one of the most urgent public policy challenges in Indonesia today. Our experience around the world has taught us that a multisector response is necessary, including improvement in basic services such as water and sanitation," said Rodrigo A. Chaves, World Bank Country Director for Indonesia and Timor-Leste. "And with more than half of Indonesia's population living in urban areas, the potential for gaining the development benefits of urbanization can only be realized if the need for such basic public services is met."
The Government of Indonesia has set the goal of achieving universal access to water supply and sanitation by the end of 2019. This financing is part of a broader effort that will also combine national, provincial and local resources and the private sector's collaboration.
Despite robust economic growth and significant poverty reduction, Indonesia's inequality remains high. About one-third of this inequality can be traced back to inequality of opportunity, such as lack of access to clean water, leading to long-term consequences to human development such as stunting and malnutrition.
"The Bank's investment will contribute to the government's financing, targeting specific investments and technical assistance to directly increase water access and improve the efficiency of local water service providers," said Irma Magdalena Setiono, World Bank Water Supply and Sanitation Specialist in Indonesia.
A key component of this project will support the central government in providing investment support to at least 40 local governments and local government owned water supply enterprises. Some 200 local governments and local government owned water supply enterprises will benefit through better capacity and performance, as well as improved credit worthiness and climate resilience.
The World Bank's support to Indonesia's water and sanitation sector is an important component of the World Bank Group's Country Partnership Framework for Indonesia, which focuses on government priorities that have potentially transformational impact.
www.worldbank.org/Indonesia
PRESS RELEASE NO: ECR/180/EAP
Lestari Boediono
62-21-5299-3156
lboediono@worldbank.org
Marcela Sanchez-Bender
sanchezbender@worldbank.org | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 8,619 |
Q: windows phone 8.1 how to make a determinate circle progressbar In Windows Phone 8.1, how can I modify the progressbar like the circle below? The ring should be determinate.
A: You can use the ProgressRing. Take a look at this.
<ProgressRing x:Name="myProgressRing" IsActive="True" Height="90" Width="90" />
If you want to make it determinate look at this article.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 576 |
Milwaukee Area Researchers Have New, Faster Test for Sepsis
Researchers in the Milwaukee area have developed a new automated test to identify most leading causes of bacterial bloodstream infections 42 hours faster than conventional methods, potentially reducing medical bills by about $21,000 for patients suffering from sepsis.
The automated nucleic acid test, developed by researchers at Froedtert Hospital & The Medical College of Wisconsin and acompany called Nanosphere, identifies genetic information of bacteria and antibiotic resistance for 12 of the most common bacteria that cause sepsis.
Sepsis caused by bacterial bloodstream infections results in up to 500,000 hospitalizations each year and accounts for 11% of intensive care unit admissions in the United States, according to a study released Tuesday evaluating the effectiveness of the new test. It has a mortality rate of 25% to 80% in critically-ill patients. Gram-positive bacteria — which differ from gram-negative bacteria because of their thick cell walls — account for 52% to 77% of all bacterial sepsis.
Researchers for the study, published Tuesday in the journal PLOS Medicine, collected and tested 1,252 blood cultures containing gram-positive bacteria at five clinical centers across the country, including the Medical College of Wisconsin in Wauwatosa, between April 2011 and January 2012. An additional 387 contrived blood cultures with bacterial targets that are uncommon in the United States were included to further evaluate the performance of the test, named the Verigene Gram-Positive Blood Culture Test.
"We had looked at diagnosis of bloodstream infections and one of the major limitations ... was that results were taking a great deal of time to get to the patient," said the study's lead author, Nathan Ledeboer, an associate professor of pathology and medical director of microbiology and molecular diagnostics at Froedtert and the Medical College of Wisconsin. "What this panel allows us to do is look at a large percentage of those organisms and to report on a number of different resistance factors."
When compared with the commonly used reference culture method, the accuracy of the test — in terms of the number of true positives and negatives — ranged from 92.6% to 100% for all 12 of the gram-positive bacteria in the 1,157 collected cultures that contained a single target. About 7.5% of the cultures contained organisms that were not included on the panel.
Conventional methods take about three days to produce bacterial identification and antibiotic resistance results after the blood culture turns positive; the new test delivers the same results within 2.5 hours. Because of the long incubation period, patients with bloodstream infections are routinely treated with a broad spectrum of anti-microbials that are sometimes ineffective, leading to extended hospital stays.
While other techniques deliver results in as little as 30 to 60 minutes after blood culture positivity have been developed, most are limited to looking at just one milliliter of blood and thereby only one or a few specific target bacteria. The Verigene system can parse through 20 to 40 milliliters of blood and identify 12 gram-positive bacterial targets and three genetic markers of antibiotic resistance directly from the positive blood cultures, Ledeboer said.
"We're getting ready to use it," said Gary Procop, chair of molecular pathology at the Cleveland Clinic. "It gives a rapid identification of the organism and key resistant factors which is really important in tailoring therapy."
He added that the Verigene test is more user-friendly than the faster tests, because it's automated.
Froedtert has used the test for about 500 patients since the Food and Drug Administration approved it in September — and it's worked "exceedingly well," Ledeboer said.
The Verigene testing device itself costs between $10,000 and $20,000, and cartridges for individual blood cultures cost $75 each.
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UC Davis Developing Rapid Pathogen Tests for Food ... | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 5,484 |
Q: How do i count number of occurence i need to create a top 5 of interfaces that went up/down based on %LINK-3-UPDOWN from a log file.
and also need to count the amount of ICMP packets that are stopped based on amount of %SEC-6-IPACCESSLOGDP.
log file looks like this:
Sep 22 15:12:09 145.89.109.1 : %SEC-6-IPACCESSLOGP: list 120 denied tcp 80.82.77.33(0) -> 145.89.109.49(0), 1 packet
Sep 22 16:11:15 145.89.109.11 28w6d: %LINK-3-UPDOWN: Interface GigabitEthernet1/20, changed state to up
Sep 22 16:11:15 145.89.109.11 28w6d: %LINEPROTO-5-UPDOWN: Line protocol on Interface GigabitEthernet1/20, changed state to up
Sep 22 15:16:09 145.89.109.1 : %SEC-6-IPACCESSLOGP: list 120 denied tcp 216.158.238.186(0) -> 145.89.109.49(0), 1 packet
Sep 22 15:17:10 145.89.109.1 : %SEC-6-IPACCESSLOGP: list 120 denied tcp 184.105.139.98(0) -> 145.89.109.49(0), 1 packet
Sep 22 15:22:10 145.89.109.1 : %SEC-6-IPACCESSLOGS: list 78 denied 145.89.110.15 1 packet
Sep 22 16:20:46 145.89.109.11 28w6d: %LINEPROTO-5-UPDOWN: Line protocol on Interface GigabitEthernet1/20, changed state to down
Sep 22 16:20:46 145.89.109.11 28w6d: %LINK-3-UPDOWN: Interface GigabitEthernet1/20, changed state to down
My code is as followed but i am not getting the result i want:
infile = open("router1.log","r") #Open log bestand in "read modus"
dictionary = {} #Maak lege dictionary aan
for line in infile: #For-loop die elke regel afgaat in log-bestand
try:
naam = line.split(":")[3] #variable naam die regel split naar een lijst met index 3
naam2 = line.split(":")[4] #variable naam die regel split naar een lijst met index 4
if naam.strip()in dictionary.keys(): #"Als" naam zich bevindt in dictionary voer onderstaande uit:
dictionary[naam.strip()]+=1
else: #Anders voer onderstaan uit:
dictionary[naam.strip()]=0
except:
continue
A: If I had interpreted your question correctly, your problem here is that you are unable to obtain the correct count values.
To resolve that, you want set your first occurrence of a particular log issue to value 1 instead of 0.
So your else statement should be:
else:
dictionary[naam.strip()]=1
If you do not do that your counts will always be lesser by 1, hope that helps!
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 3,574 |
Q: NgxMatLuxonAdapter (Ngx-Mat-DateTimePicker) setting values on immutable object my projects used the Material DatePicker with the MatLuxonAdapter. Now we need a DateTIMEPicker and came across this: https://www.npmjs.com/package/@angular-material-components/datetime-picker
Since this only supports Moment and we use Luxon I wanted to extend on the Luxon Adapter for the DatePicker to make use of this DateTimePicker, but I seem to be stuck.
I extended and implemented the classes I wanted to use, I don't think this is a good practice, but Luxons DateAdapter is the basis I want to use.
export class MyDateAdapter extends LuxonDateAdapter implements NgxMatDateAdapter<DateTime> {}
I then implemented all methods of NgxMatDateAdapter. It's kinda working. I'm only having trouble setting a new time. There are three setter methods for hour, minute and second and I cannot make it work with Luxon since it's immutable.
setHour(date: DateTime, value: number): void {
date.set({hour:value}); // doesn't work since it will not change date
}
setHour(date: DateTime, value: number): void {
date = date.set({hour:value}); // also doesn't work
}
what can I do?
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 8,595 |
\section{Introduction}
Studying excitation energy transport (EET) has been of great interest across different fields bridging evolutionary biology to solar cell engineering for many years. Especially natural light-harvesting has been the subject of intense research. Pigment-protein complexes exhibit remarkable transport properties which facilitate highly efficient excitation energy transfer across long distances.\cite{caffarri2009a,baker2008a,kreisbeck2016a, amarnath2016a} Thus, identifying working principles that ultimately transform into blueprints for novel nature-inspired excitonic devices is an active research frontier.\cite{scholes2017a,scholes2011a}
Mechanistic studies reveal valuable insight into the microscopic details of EET. Prominent examples are given by studies probing
the impact of electronic coherence or non-trivial interactions between excitons and specific vibrational modes on transfer characteristics.\cite{blau2017a, kreisbeck2012a, chin2013a, christensson2012a, dean2016a, Romero2014a,desio2016a} However such investigations are tedious since they require sophisticated experimental setups,\cite{dean2016a, Romero2014a,desio2016a,collini2010a,engel2007a,brixner2005a} as well as computationally involved accurate simulations of open-quantum system dynamics.\cite{blau2017a,kreisbeck2012a,chin2013a,schulze2015a,hein2012a,suess2014a,nalbach2011a} Further, there are only a few fundamentally different natural light-harvesting complexes from which alone we cannot extract the relation between the structure of an excitonic system and its dynamics in full detail.
In order to relate the dynamics to the underlying structure, it is desirable to investigate a large number of artificially designed excitonic systems. This has been recently addressed in several theoretical works.\cite{Scholak2011, Mostarda2013, Baghbanzadeh2016, Baghbanzadeh2016lett} For example, analyzing perturbations on pigment geometries in the Fenna-Matthews-Olson (FMO) complex revealed that higher transport efficiencies tend to be realized by more compact structures.\cite{Knee2017} The drawback of these statistical approaches is that they need to run exciton dynamics calculations for ten thousands of randomly generated physically-plausible multi-chromophoric structures. Due to the sheer number of performed dynamics simulations, such an analysis becomes quickly computationally exhaustive, even though less sophisticated methods such as Lindblad equations are used.\cite{Knee2017}
\begin{figure*}[!ht]
\centering
\includegraphics[width = 0.95\textwidth]{fig_introduction}
\caption{Machine learning excitation energy transfer properties in open quantum systems. (A) Fenna-Matthews-Olson (FMO) pigment-protein complex with eight chlorophyll pigments in the conventional numbering scheme. Dominant energy transfer pathways from the donor pigment~8 (blue) to the acceptor pigment~3 (orange) are indicated. (B) Results for average transfer time $\langle t \rangle$ calculations for energy transfer in the FMO complex from the donor to the acceptor obtained from solving the hierarchical equations of motion (HEOM), the approximate secular Redfield formalism and predicted by multi-layer perceptrons (MLPs) designed in this study. Computational costs are reported for each method. (C) Illustration of the MLP architecture. MLPs accept Frenkel exciton Hamiltonians as input feature and predict average transfer times and efficiencies. The best network architectures were obtained through Bayesian optimization.}
\label{fig:fmo_illustration}
\end{figure*}
Here, we follow a novel path and leverage concepts from deep learning to bypass the computational demand of established techniques for exploring EET properties (see Fig.~\ref{fig:fmo_illustration}). Specifically, we train multi-layer perceptrons (MLPs), a class of fully connected feed-forward artificial neural networks to predict average exciton transfer times and overall transfer efficiencies. The input features to the MLPs are hereby given by the parameters of the corresponding Frenkel exciton Hamiltonians. For large scale screening of parameter space, only a fraction of all systems needs to be actually calculated to train the MLPs.
Once trained, our neural networks evaluate transfer times just within a few milliseconds and thus bypass the computational demand of established techniques for exploring EET properties, while maintaining sufficiently high prediction accuracy.
We demonstrate the potential of the MLPs by considering various artificial datasets which were generated by uniform sampling of pigment excitation energies and inter-pigment couplings in the vicinity of the energies and couplings of a set of relevant biological complexes: the FMO complex,\cite{Fenna1975} as well as the light-harvesting complexes CP43, CP47 and the reaction center (RC) of photosystem~II.\cite{muh2012a,raszewski2008a,raszewski2008b} We aim to predict average transfer times from an initially excited donor to a certain acceptor pigment. Fig.~(\ref{fig:fmo_illustration}) shows the situation for the FMO complex, which serves as an energy wire bridging the chlorosome and the reaction center in the photosynthetic apparatus of green sulfur bacteria and has become a standard system for comparing energy transfer properties.\cite{Mohseni2008} Initial excitation is assumed to be located at the donor pigment~8 since this pigment is in the proximity of the light-harvesting chlorosome antenna. Then, the excitation energy needs to be transferred to the target pigment~3 which couples to the reaction center where photochemical reactions are triggered. In the context of EET, the latter process is typically modeled as irreversible energy trapping.\cite{Kreisbeck2011,rebentrost2009a,caruso2009a,fassioli2010a}
The MLP models are trained based on transfer properties obtained with the hierarchically coupled equation of motion technique (HEOM),\cite{tanimura1989a,ishizaki2009c,tanimura2012a} which is a non-perturbative open quantum system approach taking into account non-Markovian effects. HEOM has become one of the standard tools in the field (a ready-to-run online package is available on nanohub.org)\cite{kreisbeck2013a} and serves in this manuscript as ground truth to quantify the error for the predictions made by the neural networks. The accuracy of the predictions critically depends on the choice of hyperparameters such as the number of neurons, number of hidden layers or the learning rate, which collectively define the specific architecture of the neural network. However, the best set of these parameters is \textit{a priori} unknown. Therefore, we determine the architectures for our MLP models from a Bayesian optimization on selected hyperparameters. This procedure is well-established in the machine learning community and was shown to outperform architectures built by domain experts.\cite{Snoek2012}
We assess the quality of our MLP predictions by comparing the relative error of our predicted transfer times to the relative error made by secular Redfield calculations. The latter is simple to implement and commonly used to avoid the numerical complexity of more accurate HEOM simulations.
Our findings demonstrate that MLPs provide a computationally significantly cheaper alternative to secular Redfield computations at comparable or, in most of our examples, even higher accuracy. Results for the FMO complex are summarized in Fig.~(\ref{fig:fmo_illustration}).
\section{Machine learning approach}\label{sec:machine_learning_approach}
A number of studies across many fields in recent years have demonstrated how machine learning models can be utilized to accelerate a variety of computations by several orders of magnitude at a reasonable level of accuracy. For example, Gaussian processes were used to predict formation of free energies for catalyst surface chemistry.\cite{Ulissi2017} Neural networks have been successfully employed for the construction of various forms of transferable and non-transferable atomistic potentials.\cite{Behler2007, Smith2017, Yao2017} Protein-ligand binding affinities were accurately predicted by atomic convolutional neural networks,\cite{Gomes2017} and multi-layer perceptrons were trained to predict excited state energies in the context of exciton dynamics,\cite{Hase2016} as well as other electronic properties of small molecules.\cite{Hansen2013, Montavon2013}
In the subsequent sections, we develop a machine learning framework based on multi-layer perceptrons which predict excitation energy transfer properties of excitonic systems rather than obtaining them from computationally expensive calculations. In future applications, this approach could facilitate large-scale screening such as the search for best-performing devices or studies on structure-function relationships in natural light-harvesting.
Overall, our procedure can be summarized as follows. Based on the Frenkel exciton Hamiltonian we leverage standard open quantum system approaches to generate a database comprising of average transfer times and efficiencies for EET from a donor to a target pigment for a random set of Frenkel exciton Hamiltonians. The complete dataset is split into a training set, on which we train each MLP model, as well as a validation and a test set. For training data selection we will compare two strategies: (i) random selection of data points and (ii) selection of training data based on a principal component analysis (PCA) which allows us to extract those data points covering the most information sampled in the dataset. As we show in Sec.~\ref{sec:results}, the latter strategy is of particular relevance if the space of transfer properties is not evenly sampled and many representatives in the training set exhibit redundant information. We run a Bayesian optimization procedure to identify the best architecture for our MLP models. The performance of each architecture is quantified by the average relative absolute error made when predicting transfer properties for the validation set. Finally, we run predictions on the test set to assess the ability of the optimized architecture to generalize to realizations that were neither employed for training nor for validation during the Bayesian optimization. The source code for exciton transfer property predictions along with all trained MLP models as well as the datasets generated in this study are made available on GitHub.\cite{githubRepo}
\subsection{Generating the excitation energy transfer database}\label{sec:dataset_generation}
To demonstrate the capabilities of our machine learning approaches, we investigate four datasets of randomly generated excitonic systems that are sampled around pigment-protein complexes found in natural light-harvesting. For future reference, the generated database can be downloaded from a GitHub repository.\cite{githubRepo}
For our first dataset, we sample Hamiltonians around the FMO complex (Fig.~\ref{fig:fmo_illustration}), which serves frequently as the prototype light-harvesting complex. We construct three additional datasets that are motivated by the photosystem~II of higher plants. For one set, we consider the eight pigments of the reaction center (RC) core, in which the primary step of charge separation is initiated through the electronically excited pigment Chl$_{\rm D1}$.\cite{holzwarth2006a,raszewski2008a} For the other two sets, the reaction center core is extended by including either light-harvesting complex CP47 or CP43 of photosystem~II into the exciton system. For simplicity, we refer to the dataset inspired by the CP43+RC (CP47+RC) complex as the CP43 (CP47) dataset from hereon. For each dataset, we generated 12000 exciton Hamiltonians by uniformly sampling excited state energies and inter-site couplings from a fixed range of values, as is summarized in Tab.~\ref{tab:hamiltonian_ranges}.
\begin{table}[!t]
\centering
\begin{tabular}{lccccc}
\toprule
Label & \#Sites & $\unit[\varepsilon_\text{low}]{[cm^{-1}]}$ & \quad$\unit[\varepsilon_\text{high}]{[cm^{-1}]}$ & \quad$\unit[V_\text{range}]{[cm^{-1}]}$ \\
\midrule
RC & 8 & 14800 & 15000 & -50 ... 50 \\
FMO & 8 & 12000 & 12800 & -100 ... 100 \\
CP43 & 21 & 14800 & 15100 & -60 ... 60 \\
CP47 & 24 & 14500 & 15300 & -100 ... 100 \\
\bottomrule
\end{tabular}
\caption{Lower and upper limits in between which excited state energies $\varepsilon$ and inter-site couplings $V$ were sampled uniformly to generate the four datasets of this study. Each dataset consists of 12000 Hamiltonians with excited state energies and inter-site couplings within the reported ranges. Note, that the labels CP43 (CP47) denote datasets which are inspired by the CP43+RC (CP47+RC) biological complexes.}
\label{tab:hamiltonian_ranges}
\end{table}
In the following, we are interested in transfer characteristics such as average transfer times from an initially excited pigment (donor) to a target pigment (acceptor). This model provides a simple description of the first step of photosynthesis, where energy is absorbed in the antenna pigments and subsequently transferred to the reaction center in which photochemical reactions are triggered. The energy transport in light-harvesting complexes is determined by coupled pigments which are embedded in a protein scaffold,\cite{May2004a, Cheng2009} and is typically modeled with an effective Frenkel exciton Hamiltonian.\cite{Leegwater1996, May2008} We include energy trapping in the acceptor pigment phenomenologically by introducing anti-Hermitian parts in the Hamiltonian. The exciton dynamics is expressed in terms of the reduced density matrix, which can be obtained from standard open quantum system approaches.
We compute exciton transfer times for all Hamiltonians in our datasets with the hierarchical equations of motion (HEOM)\cite{tanimura1989a,ishizaki2009c,tanimura2012a} method, implemented in the \emph{QMaster} software package, version 0.2.\cite{Kreisbeck2011, Kreisbeck2012, Kreisbeck2014} HEOM is a numerically exact method which accurately accounts for the reorganization process,\cite{Yan2004, Xu2005, Ishizaki2005, Ishizaki2009} in which the vibrational coordinates rearrange to their new equilibrium positions upon electronic transition from the ground to the excited potential energy surface. For all Hamiltonians we assumed identical Drude-Lorentz spectral densities $J(\omega)=2\lambda \frac{\omega \nu}{\omega^2+\nu^2}$, describing the exciton-phonon interaction. We do not use the parameters of the spectral density as input features for our neural networks. Extending our approach to predict transfer properties for various spectral densities goes beyond the present scope and is the aim of future work. More details on the Frenkel exciton Hamiltonian and the exciton dynamics methods, as well as the definition of the transfer time and transfer efficiencies, are given in the supplementary information Sec.~\ref{sec:modeling_exciton_transfer}.
\begin{figure}[!t]
\centering
\includegraphics[width = 0.98\columnwidth]{transfer_time_distributions.pdf}
\caption{Distributions of exciton transfer times computed for all 12000 generated exciton Hamiltonians for each dataset using the HEOM approach implemented in \textit{QMaster}. Vertical red lines indicate the transfer time of the exciton Hamiltonian corresponding to the biological complex. In all calculations we use a trapping rate of $\Gamma_\text{trap}^{-1}=1$~ps, an exciton life-times of $\Gamma_\text{loss}^{-1}=0.25$~ns, and a temperature of $T=300$~K. The parameters of the spectral density are set to $\lambda = \unit[35]{cm^{-1}}$, $\nu^{-1} = \unit[50]{fs}$.}
\label{fig:transfer_time_distributions}
\end{figure}
Distributions of transfer times for all exciton Hamiltonians of each dataset are depicted in Fig.~\ref{fig:transfer_time_distributions}. The transfer times for the Hamiltonians of the biological complexes are highlighted in every distribution. Excited states and inter-site couplings for the exciton Hamiltonians of the biological complexes are taken from literature,\cite{adolphs2006a,muh2012a,raszewski2008a,raszewski2008b} and are uploaded to the GitHub repository.\cite{githubRepo} All population dynamics simulations are initialized as a fully populated site~1, serving as a donor, while site~3 acts as acceptor that couples to an energy sink with trapping rate $\Gamma_\text{trap}$ (see supplementary information Sec.~\ref{sec:modeling_exciton_transfer}). Note that the labeling of the donor and acceptor state is without loss of generality as rows and columns of the Hamiltonian can be permuted in a suitable way, which effectively corresponds to a relabeling of the pigments.
We find large variations in the ranges of transfer times between the four datasets. The RC and CP43 datasets, both with relatively narrow ranges of excited state energies and site couplings, yield relatively small transfer times. In contrast, we observe a wider spread in transfer times for the FMO dataset and the CP47 dataset which is consistent with the broader range of excited state energies and site couplings that were sampled.
The transfer times of the actual biological complexes lie close to the mode of the distributions for all four datasets. This suggests that natural systems may not be specifically selected for extraordinary transfer properties, as they exhibit transport characteristics that are just likely to occur, even for a random choice of the exciton Hamiltonian. The conclusions of a recent evolutionary study for the FMO complex,\cite{Valleau2017} goes along a similar direction and suggests that the FMO complex has evolved towards stability to mutations rather than a selection of specific transfer characteristics. However, we note that we did not take into account structural considerations which could change the picture as many of our randomly generated artificial Hamiltonians may not be realizable under structural constraints.
\subsection{Principal component analysis for improved training data selection}\label{sec:training_data_selection}
We select the training sets for our MLP models following two methods for dataset splitting. In the simplest ansatz, we select the training set randomly from our created dataset. However, due to the nature of how we randomly sampled our Hamiltonians, the transfer characteristics are not distributed homogeneously and many representations of our Hamiltonians might be very similar and thus are expected to carry redundant information. As can be seen in Fig.~\ref{fig:transfer_time_distributions}, Hamiltonians yielding longer transfer time-scales are for example underrepresented in all four datasets.
Therefore, we follow a different path and carry out a more sophisticated selection process. The idea is to add those Hamiltonians to our training set which give the most information. We perform a principal component analysis (PCA) on the 8000 Hamiltonians containing dataset (after separating 2000 Hamiltonians each for validation and testing). We project each Hamiltonian onto a reduced space spanned by the most relevant principal components. The Hamiltonians for the training set are then selected such that they are maximally separated in the reduced space. This procedure guarantees that our training set constitutes the most diverse entities.
\subsection{Setup of the multi-layer perceptron architecture}\label{sec:machine_learning_aspects}
The architectures of our multi-layer perceptrons (MLPs) are designed for supervised learning of exciton energy transfer properties. All exciton Hamiltonians were reshaped into vectors and provided as input features to the MLPs, which were used to predict exciton transfer times and transfer efficiencies simultaneously. Since, the input features of neural networks need to be of fixed size, we construct separate MLPs for each dataset in order to treat the different dimensionalities of the exciton Hamiltonians. Details on the rescaling of the input features and predicted output, as well as on the training procedure are provided in the supplementary information (see Sec.~\ref{sec:mlp_comments}).
The 12000 Hamiltonians of each dataset were split into three sets: a training set of up to 6000 Hamiltonians for training MLP model instances with particular hyperparameters, a validation set of 2000 Hamiltonians used to evaluate the MLP architecture during optimization of the hyperparameters and a test set of 2000 Hamiltonians to probe out-of-sample prediction accuracies. All constructed MLP models were trained with stochastic gradient descent with 200 data points per batch and the ADAM optimizer,\cite{Kingma2014} until the average relative absolute error (see Eq.~\ref{eq:relative_time_deviation}) on the validation set increased over three full consecutive training epochs. Neuron saturation was avoided with L2 regularization on all weights of all neurons but the output neurons.
An essential component in developing accurate machine learning models consists in choosing proper values for the model hyperparameters. For this MLP framework, we consider a total of six hyperparameters. The initial learning rate $\mu$ for the ADAM optimizer and the regularization parameter $\lambda$. We also included the number of MLP layers and the number of neurons per layer, as well as the activation functions for neurons in each layer, for which we allowed five different options to choose from. The only exception is the last layer, for which we always use the softplus activation function to constrain our MLP models to the prediction of always positive transfer times and efficiencies. Lastly, we treat the number of training points as a hyperparameter in order to study the effect of the variations in the number of training samples on the prediction accuracy. The set of hyperparameters to be optimized and their allowed ranges are summarized in the supplementary information in Tab.~\ref{tab:bayes_opt_hyperparameter_selection}
We employ a Bayesian optimization algorithm,\cite{Dixon1978} in order to scan the space of hyperparameters for the most accurate model. The model accuracy was defined as the average relative absolute error (see Eq.~\ref{eq:relative_time_deviation}) in exciton transfer times predicted by the MLP and corresponding HEOM simulations for the validation set. All generated MLP models were constructed and trained with the same random seed. Bayesian optimization is a common tool in machine learning and balances exploration of parameter space and exploitation of previous information. The idea of this ansatz is to reduce the number of costly function evaluations under the assumption that the unknown function was sampled from a Gaussian process. In contrast to gradient or Hessian based optimization techniques, Bayesian optimization uses information of all previously evaluated points and can thus find a good approximation to the minimum of non-convex functions in relatively few iterations. We carried out the Bayesian optimization of MLP hyperparameters in the spearmint software package.\cite{Snoek2012} MLP models were generated and trained using the Tensorflow package, version 1.0.\cite{tensorflow2015-whitepaper}
\begin{table}[!t]
\centering
\begin{tabular}{llccc}
\toprule
Dataset & Model & \unit[$\Delta\tau_\text{train}$]{[\%]} & \unit[$\Delta\tau_\text{valid}$]{[\%]} & \unit[$\Delta\tau_\text{test}$]{[\%]} \\
\midrule
\multirow{2}{*}{FMO} & Network (PCA) & \textbf{4.53} & \textbf{4.38} & \textbf{7.41} \\
& Network & 10.53 & 10.75 & 11.56 \\
& Redfield & 9.70 & 9.96 & 9.60 \\
\midrule
\multirow{2}{*}{RC} & Network (PCA) & \textbf{2.71} & \textbf{2.73} & \textbf{3.35} \\
& Network & 3.61 & 3.58 & 3.76 \\
& Redfield & 8.62 & 8.67 & 8.60 \\
\midrule
\multirow{2}{*}{CP43} & Network (PCA) & \textbf{4.42} & \textbf{4.47} & \textbf{4.72} \\
& Network & 4.66 & 4.71 & 4.86 \\
& Redfield & 4.71 & 4.66 & 4.73 \\
\midrule
\multirow{2}{*}{CP47} & Network (PCA) & 12.36 & 12.32 & 12.59 \\
& Network & 13.36 & 13.34 & 13.59 \\
& Redfield & \textbf{10.48} & \textbf{10.47} & \textbf{10.51} \\
\bottomrule
\end{tabular}
\caption{Average relative absolute error $\Delta\tau$ (see Eq.~\ref{eq:relative_time_deviation}) of exciton transfer times computed with HEOM and either, predicted by the trained neural networks (with/without PCA selection) or computed with secular Redfield. For all four datasets, we show the results of the training, validation, and test set separately. Smallest errors for each dataset are printed in bold.}
\label{tab:mad_redfield_networks}
\end{table}
\section{Results: Prediction of transfer times with neural networks}\label{sec:results}
In the subsequent discussion, we demonstrate the capabilities of our trained MLP models by analyzing the average relative absolute error
\begin{equation}\label{eq:relative_time_deviation}
\Delta\tau= \Big\langle \frac{| t_\text{HEOM} - t_\text{model}|}{t_\text{HEOM}} \Big\rangle_\text{dataset},
\end{equation}
between predicted exciton transfer times and the ones obtained with the numerically exact HEOM calculations. Although we restrict our discussion to transfer times, we note that similar conclusions hold for the analysis of the transfer efficiencies since both characteristics are strongly correlated. Tab.~\ref{tab:mad_redfield_networks} summarizes the results for the predicted transfer times for our four generated datasets.
The predictions are carried out with the Bayesian optimized MLP architectures, which show slight variations in their best-performing hyperparameters depending on the dataset at hand. However, for all datasets, the neural networks tend to prefer shallow but broad architectures comprising of only a few layers with each layer containing a larger number of neurons. More details on the procedure and results for the hyperparameter optimization can be found in the supplementary information Sec.~\ref{sec:sup_bayesian_optimization}.
\subsection{Prediction accuracies of trained multi-layer perceptrons}
Our trained MLP models predict exciton transfer times for out-of-sample Hamiltonians at almost the same accuracy as for Hamiltonians on which MLP parameters and hyperparameters were optimized (see Tab.~\ref{tab:mad_redfield_networks}). This demonstrates the ability of our MLP models to generalize to previously unseen data and to provide accurate out-of-sample predictions. Noteworthy, there is no significant asymmetry in the distribution of the relative absolute errors for the individual Hamiltonians or the training/validation and test set (see Fig.~\ref{fig:out_of_sample_predictions}). Therefore, the architectures of the neural networks are well-balanced and neither in the regime of over-fitting, which would result in a large discrepancy in errors between the training and validation sets nor did we over-optimize the neural network architecture during Bayesian optimization.
Overall we find a high accuracy of our predictions and small average relative errors on the test sets which are in the range between \unit[3.35]{\%} for RC (PCA selected training set) and \unit[13.59]{\%} for the largest considered exciton system CP47 attached to RC (random selected training set). The CP47 dataset exhibits the most diverse transfer properties (see Fig.~\ref{fig:transfer_time_distributions}), which explains the larger average relative absolute errors in the predictions when compared to the other datasets.
The accuracy of the predictions can be enhanced by a more sophisticated PCA selection of the training set without the need of generating additional computationally expensive data points. The level of improvement of the PCA selection over a random selection of the training set differs for the four complexes. In general, we find that MLPs can be trained almost equally accurate with either selection method. The highest benefit of the PCA selected training set is obtained for the FMO and CP47 dataset, which are not only the most diverse ones out of our four datasets but are biased towards Hamiltonians showing fast transfer. As intuitively expected, selecting training points based on PCA is most advantageous for datasets with an extremely unevenly sampled feature space.
\begin{figure}[!t]
\centering
\includegraphics[width = 0.92\columnwidth]{comparison_training_test_violins.pdf}
\caption{Normalized distributions of the average relative absolute error of predicted exciton transfer times and exciton transfer times computed with HEOM. The left (blue) side of the plots illustrate the distributions of average relative absolute errors for predictions on the training and the validation set, while the right (orange) side of the plots illustrates the errors for predictions on the test set.}
\label{fig:out_of_sample_predictions}
\end{figure}
\subsection{Comparing multi-layer perceptron predictions to secular Redfield results}
Next, we provide a context for the observed MLP prediction accuracies by comparing them to the errors made by the frequently employed secular Redfield method, which is essentially derived from second order perturbation theory in the system-bath interaction in combination with a Markov approximation. Accuracies of the transfer times for both, the secular Redfield calculations and the MLP predictions are evaluated according to Eq.~(\ref{eq:relative_time_deviation}). Here, the HEOM calculations again serve as ground truth. For the datasets inspired by the smaller exciton systems FMO and RC, the trained MLPs outperform secular Redfield, even for out-of-sample predictions, whereas for the datasets around larger systems both approaches are similarly accurate.
For example in the case of the biological exciton Hamiltonian of the FMO complex, HEOM reveals a transfer time of \unit[7.95]{ps}. The trained MLP model predicts a transfer time of \unit[7.52]{ps} which is slightly more accurate than secular Redfield calculations that result in \unit[7.48]{ps}. Exciton transfer times obtained for all four biological complexes with all three approaches are reported in the supplementary information Tab.~\ref{tab:transfer_times_bio_complexes}. However, while the MLP prediction takes about 5~ms, secular Redfield calculations took about \unit[14.5]{min} on a single CPU (computation times are listed in the supplementary information in Tab.~\ref{tab:transfer_time_computing_times}). We conclude that our trained MLP predictions are competitive to secular Redfield calculations in terms of their accuracy, but (once trained) come at a significantly reduced computational cost.
\begin{figure}[!b]
\centering
\includegraphics[width = 0.95\columnwidth]{comparison_scatter_plot_all_four.pdf}
\caption{Relative errors in exciton transfer times computed with the hierarchical equations of motion (HEOM) approach and exciton transfer times computed with the secular Redfield approach and predicted by neural networks respectively. Displayed are relative deviations for all four datasets: the Fenna-Matthews-Olson (FMO) complex, the reaction center (RC) core, the RC with the CP43 complex and the RC with the CP47 complex. Regions in which the absolute of deviations of neural network predicted transfer times from HEOM computed transfer times are smaller than deviations for Redfield are shaded in green.}
\label{fig:scatter_comparisons_networks_redfield}
\end{figure}
Besides analyzing the accuracy in terms of averaging over all realizations in the datasets, we compare the relative errors in transfer time for secular Redfield and the MLP predictions in more detail on the level of individual Hamiltonians. Fig.~\ref{fig:scatter_comparisons_networks_redfield} depicts scatter plots where the horizontal axes measure the accuracy of secular Redfield calculations and the vertical axes reflect the accuracy of MLP predictions for MLPs trained on the PCA selected datasets. We do not distinguish between training, validation, and test set and show the complete dataset. Almost all the Hamiltonians show a $\Delta t_\text{Redfield}=(t_\text{HEOM} - t_\text{Redfield})/t_\text{HEOM}>0$, which demonstrates that secular Redfield systematically underestimates transfer time scales. On the other hand, the predictions under- as well as overestimate transfer time-scales yielding a more symmetrical distribution along the horizontal axis. For the RC (FMO) dataset, more than \unit[95]{\%} (\unit[80]{\%}) of the Hamiltonians fall into regions marked as green, for which the neural networks provide higher accuracy than secular Redfield. For all other datasets, secular Redfield and the MLP predictions are equally likely to give better results, with about \unit[59]{\%} (\unit[57]{\%}) of the Hamiltonians for CP43 (CP47) falling within the green shaded region. This is in agreement with our average relative absolute errors listed in Tab.~\ref{tab:mad_redfield_networks}. We did not observe any cases for which the MLPs show relative errors that significantly exceeded any of the secular Redfield ones. \\
\section*{Conclusion}
In this study, we have outlined how machine learning approaches can be employed to bypass computationally costly simulations of open quantum system dynamics in the context of excitation energy transfer. Overall we find that MLPs are capable of predicting transfer times for excitonic systems at higher or comparable accuracy than the frequently used secular Redfield approach albeit at much lower computational costs. Therefore we conclude that MLP models are a promising alternative for extracting excitation energy transfer properties when compared to frequently used rate equation methods.
The presented approach is of particular interest for large-scale analyses of the structure-transport relationship in excitonic systems. An area of great interest in excitonics is the study of the dynamics of charge dissociation at the interface present in bulk heterojunction photovoltaics.\cite{Jailaubekov2012, Vithanage2013} We believe a tool like this will help in the rapid screening of material properties in the mesoscale and therefore help the search for high-performance OPV systems.\cite{Hachmann2011}
Once trained, evaluations of MLP models come at almost no additional cost. Our four generated MLP architectures (each optimized for one of the four datasets) predict transfer times for an aggregated set of 48,000 exciton Hamiltonians just within a few seconds, while the corresponding quantum dynamics simulations take several GPU (CPU) years for the HEOM (secular Redfield) calculations. Our trained MLP models extend well to out-of-sample predictions for exciton Hamiltonians that are close to the sampled parameter regime. However, to employ MLPs on parameter regimes beyond those probed in the existing database requires running computationally expensive exciton dynamics for a few thousand Hamiltonians in order to extend our training set. To avoid this bottleneck a potential strategy could be to leverage already existing data, e.g. produced by a user community of existing software packages such as \textit{QMaster}. However such data can be quite diverse. To this end, future research needs to focus on novel more general neural network architectures that accurately predict transfer times for flexible spectral density parameters as well as for differently sized exciton systems.
\section*{Acknowledgments}
F.H. is supported by the Herchel Smith Graduate Fellowship. C.K. is supported by the National Science Foundation under award number CHE-1464862. A.A.-G. acknowledges support from the Center for Excitonics and Energy Frontier Research Center funded by the U.S. Department of Energy under award DE-SC0001088. All computations reported in this paper were completed on the Odyssey cluster supported by the FAS Division of Science, Research Computing Group at Harvard University.
\section*{Supplementary Information}
\subsection{Modeling of excitation energy transfer}\label{sec:modeling_exciton_transfer}
The energy transport in light-harvesting complexes is determined by coupled pigments which are embedded in a protein scaffold.\cite{May2004a, Cheng2009} The large number of degrees of freedom in the system renders \textit{ab initio} calculations on the atomistic level impossible. Therefore, the exciton transfer dynamics is typically modeled with an effective Frenkel exciton Hamiltonian.\cite{Leegwater1996, May2008} The exciton Hamiltonian for a system of $N$ sites for the single exciton manifold reads
\begin{align}
H_\text{system} = \sum\limits_{i = 1}^N \varepsilon_i |i\rangle\langle i| + \sum\limits_{i \neq j}^N V_{ij} |i\rangle \langle j|,
\end{align}
where $\varepsilon_i$ denotes the first excited state energy of the $i$-th pigment molecule and $V_{ij}$ denotes the Coulomb coupling between excited states at the $i$-th and $j$-th molecule. We assume that the exciton system couples linearly to the vibrational environment of each pigment, which is assumed to be given by a set of harmonic oscillators. The phonon mode dependent interaction strength is captured by the spectral density
\begin{equation}\label{eq:SpecDens}
J_i(\omega)=\pi\sum_k \hbar^2\omega_{i,k}^2 d_{i,k}^2\delta(\omega-\omega_{i,k}).
\end{equation}
Here, $d_{i,k}$ defines the coupling of the $k$-th phonon mode ($b^\dagger_{i,k}$) of the $i$-th pigment with frequency $\hbar \omega_{i,k}$. \\
In the first step of photosynthesis, energy is absorbed in the antenna pigments and subsequently transferred to the reaction center in which photochemical reactions are triggered. Within a simple picture, this process can be modeled by energy transfer from an initially excited pigment (donor) to a target state (acceptor). We model energy trapping in the acceptor state $|\text{acceptor} \rangle$ phenomenologically by introducing anti-Hermitian parts in the Hamiltonian
\begin{equation}
\mathcal{H}_\text{trap} = -i\hbar\Gamma_{\rm trap}/2\,|\mbox{acceptor}\rangle\langle\mbox{acceptor}|,
\end{equation}
where $\Gamma_\text{trap}$ defines the trapping rate. In a similar way, we model radiative or non-radiative decay to the electronic ground state as exciton losses
\begin{equation}
\mathcal{H}_\text{loss} = -i\hbar\Gamma_{\rm loss}/2\,\sum_i|i\rangle\langle i|.
\end{equation}
The rate $\Gamma_\text{loss}^{-1}$ defines the exciton lifetime. In this study we are interested in two different exciton propagation characteristics: the average transfer time
\begin{align}\label{eq:transfer_time_definition}
\langle t \rangle=\Gamma_\text{trap}/\eta \int_0^{t_{\rm max}} \mbox{d}t\, t \, \langle\mbox{acceptor}|\rho(t)|\mbox{acceptor}\rangle,
\end{align}
and the overall efficiency
\begin{align}\label{eq:transfer_efficiency}
\eta=\int_0^{t_{max}}\mbox{d}t\, \Gamma_\text{trap} \langle\mbox{acceptor}|\rho(t)|\mbox{acceptor}\rangle,
\end{align}
which corresponds to the accumulated trapped population during the transfer process. For numerical evaluations, we replace the upper integration limit by $t_{max}$ which is chosen such that the total population within the pigments has dropped below $0.0001$. \\
The exciton dynamics is expressed in terms of the reduced density matrix $\rho(t)$, which can be obtained from standard open quantum system approaches. Here we employ the hierarchical equations of motion (HEOM) approach which accounts for the reorganization process,\cite{Yan2004, Xu2005, Ishizaki2005, Ishizaki2009} in which the vibrational coordinates rearrange to their new equilibrium position upon electronic transition from the ground to the excited potential energy surface. The major drawback of the HEOM approach is the adverse computational scaling, which arises from the need to propagate a complete hierarchy of auxiliary matrices. Therefore, we employ a high-performance implementation of HEOM integrated into the \textit{QMaster} software package.\cite{Kreisbeck2011, Kreisbeck2012, Kreisbeck2014} As demonstrated in previous publications, \textit{QMaster} enables HEOM simulations for large systems comprising of up to hundred pigments,\cite{kreisbeck2016a} as well as to perform accurate calculations for highly structured spectral densities.\cite{blau2017a, kreisbeck2013, Kreisbeck2014, Kreisbeck2012} \\
A computationally much cheaper formalism, the Redfield approach, can be derived with the assumption of weak couplings between the system and the bath in combination with a Markov approximation.\cite{Breuer2002, May2008} The secular approximation simplifies the equation even further and allows to write the dynamics in the form of a Lindblad master equation. This drastically reduces the computational demand of this approach compared to exciton propagation under the HEOM, which explains the popularity of the secular Redfield equations. However, secular Redfield has been shown to underestimate the transfer times in certain light-harvesting complexes.\cite{Novoderezhkin2011}
\subsection{Exciton transfer properties of biological Hamiltonians}\label{sec:bio_transfer_times}
We report the exciton transfer times calculated for the four light-harvesting complexes (LHCs) FMO, RC, CP43+RC and CP47+RC.\cite{adolphs2006a, muh2012a, raszewski2008a, raszewski2008b} Exciton transfer times were obtained from HEOM calculations, secular Redfield calculations and MLP predictions. Results are listed in Tab.~\ref{tab:transfer_times_bio_complexes}. For all four biological complexes, we find that the MLP predictions are more accurate than the secular Redfield calculations.
\begin{table}[!ht]
\centering
\begin{tabular}{lcccccc}
\toprule
\multirow{2}{*}{LHC} & \multicolumn{3}{c}{~~~~~Exciton transfer time [ps]~~~~~} & \multicolumn{3}{c}{~~~~~Computation time~~~~~} \\
& ~~~HEOM~ & ~MLP~ & ~Secular Redfield~ & ~~~HEOM (GPU)~ & ~MLP (CPU)~ & Secular Redfield (CPU) \\
\midrule
FMO & 5.78 & 4.99 & 4.18 & \unit[2.1]{min} & \unit[5]{ms} & \unit[14.5]{min} \\
RC & 7.94 & 7.52 & 7.48 & \unit[3.9]{min} & \unit[3]{ms} & \unit[8.9]{min} \\
CP43+RC & 13.80 & 11.64 & 11.16 & \unit[7.4]{h} & \unit[5]{ms} & \unit[14.6]{h} \\
CP47+RC & 18.92 & 19.31 & 15.08 & \unit[31.2]{h} & \unit[4]{ms} & \unit[40.9]{h} \\
\bottomrule
\end{tabular}
\caption{Comparison of exciton transfer times for the light-harvesting complexes (LHCs) considered in this study computed with the hierarchical equations of motion (HEOM), multi-layer perceptrons (MLPs) and secular Redfield. We report the obtained exciton transfer times as well as the runtimes of the calculations. Note, that HEOM calculations were run on GPUs, while MLP predictions and secular Redfield calculations were run on CPUs. MLPs were trained on PCA selected training sets and neither of the LHCs was included in the training sets. }
\label{tab:transfer_times_bio_complexes}
\end{table}
\subsection{Computational cost of exciton dynamics calculations}\label{sec:transfer_time_results}
Established methods for computing the population dynamics of excitonic systems such as the hierarchical equations of motion (HEOM) approach suffer from adverse computational scaling. Because of this drawback less sophisticated techniques with lower computational demand such as the secular Redfield method are more popular. Both methods need to run a full population dynamics calculation for obtaining exciton transfer properties such as the average transfer time or transfer efficiency. \\
We report the runtimes of HEOM and secular Redfield calculations observed during the generation of the four datasets used in this study. Each dataset consists of 12000 randomly generated exciton Hamiltonians for which we computed average transfer times and transfer efficiencies with both methods (see Sec.~\ref{sec:dataset_generation} for details). HEOM calculations were carried out in the high-performance \textit{QMaster} package,\cite{Kreisbeck2011, Kreisbeck2012, Kreisbeck2014} which uses the architecture of GPUs for propagating a complete hierarchy of auxiliary matrices in parallel. Secular Redfield calculations were run on single core CPUs. Computation time for the individual Hamiltonians show some variations in computation time since depending on the excitation energy transfer times, we need to run the exciton dynamics for a larger or smaller number of time-steps. In Table~\ref{tab:transfer_time_computing_times} we show the average computation time for a single exciton Hamiltonian for each dataset (average over all 12,000 Hamiltonians), as well as the total computational cost to generate the complete datasets.\\
As a compromise between accuracy and computational costs we truncate the hierarchy at $N_\text{max} = 5$ for the datasets with fewer pigments (RC and FMO) and at $N_\text{max} = 4$ for the datasets with more pigments (CP43 and CP47). We tested that the accuracy is retained at these truncation levels by simulating a random selection of 10 exciton Hamiltonians drawn from each dataset at respective lower and higher levels of truncation. We observe an average \unit[1.6]{\%} deviation in the calculated transfer times between the higher and lower truncation levels for the FMO dataset. The RC and CP43 datasets show smaller deviations with \unit[1.1]{\%} for RC and \unit[0.1]{\%} for CP43. For the CP47 we observe a deviation of about \unit[2.4]{\%}.
Despite the small deviations induced by the earlier truncation of the hierarchy, we use the lower truncation level in our dataset generation to keep the computational costs at a reasonable level (see Tab.~\ref{tab:transfer_time_computing_times}). \\
While transfer properties in smaller systems, as they are modeled in the RC and FMO datasets, can be calculated within a few minutes, calculations on larger systems modeled in the CP43 and CP47 datasets show significantly higher computational demands with typical runtimes in the order of 6 to 18 hours. The significant computation times illustrate that large-scale simulations on artificial exciton systems can be computationally quite exhaustive. \\
Multi-layer perceptrons (MLPs), however, encode a set of matrix operations, which allows for a significantly faster calculation of the properties of interest. By construction, the time for computing the output of an MLP scales linearly with the number of neurons in the architecture. We find that our trained MLP models could predict exciton transfer times and transfer efficiencies of one complete dataset introduced in this study in less than 10 seconds on a single CPU. This estimate includes the time spent for loading all 12000 exciton Hamiltonian matrices into memory and running the matrix operations encoded in the MLP architecture as well as writing the results of the prediction to file. \\
\begin{table}[!ht]
\centering
\begin{tabular}{lcccccc}
\toprule
\multirow{2}{*}{Dataset~~~} & \multicolumn{2}{x{4cm}}{HEOM (GPU)~~} & \multicolumn{2}{x{4cm}}{MLPs (CPU)~~} & \multicolumn{2}{x{4cm}}{Secular Redfield (CPU)~~} \\
& $T_\text{single}$ & $T_\text{all}$ & $T_\text{single}$ & $T_\text{all}$ & $T_\text{single}$ & $T_\text{all}$ \\
\midrule
FMO & \unit[4.0]{min} & \unit[33.6]{days} & \unit[$<$0.1]{ms} & \unit[0.3]{s} & \unit[8.8]{min} & \unit[73.5]{days} \\
RC & \unit[2.2]{min} & \unit[18.3]{days} & \unit[0.1]{ms} & \unit[1.6]{s} & \unit[4.4]{min} & \unit[36.3]{days} \\
CP43 & \unit[5.8]{h} & \unit[7.9]{years} & \unit[0.6]{ms} & \unit[7.5]{s} & \unit[10.7]{h} & \unit[14.7]{years} \\
CP47 & \unit[18.2]{h} & \unit[24.9]{years} & \unit[0.1]{ms} & \unit[1.5]{} & \unit[36.6]{h} & \unit[41.9]{years} \\
\bottomrule
\end{tabular}
\caption{Runtimes $T$ for exciton transfer time computations carried out with the \emph{QMaster} package, version 0.2,\cite{Kreisbeck2011, Kreisbeck2012, Kreisbeck2014} for all four datasets. We report runtimes for the entire datasets, $T_\text{all}$, each comprising of 12000 exciton Hamiltonians, as well as the average runtime per exciton Hamiltonian, $T_\text{single}$. Hierarchical equations of motion (HEOM) calculations were carried out on a NVIDIA Tesla K80 GPU and secular Redfield calculations as well as multi-layer perceptron (MLP) predictions on Intel(R) Xeon(R) CPUs X5650 @ \unit[2.67]{GHz}.}
\label{tab:transfer_time_computing_times}
\end{table}
MLP models can be considered a valid alternative to secular Redfield calculations when running exciton dynamics calculations on a large number of excitonic systems due to the comparable accuracy at an orders of magnitude lower computational cost. \\
\subsection{Dataset preparations for multi-layer perceptron predictions}\label{sec:mlp_comments}
MLP models constructed in this study were designed to predict exciton transfer times and transfer efficiencies from the Frenkel exciton Hamiltonian of an excitonic system. The numerical values of excited state energies and inter-site couplings in the exciton Hamiltonian, as well as, the transfer times and transfer efficiencies depend on the chosen unit system. Further, the ranges of these properties can be very different (see for instance Tab.~\ref{tab:hamiltonian_ranges}). \\
Features and targets with significantly different numerical values are more challenging to learn for most machine learning models as the applied model first has to learn the general range of numerical values before it can learn more subtle differences. The training procedure of machine learning models in general and MLPs, in particular, can be accelerated by rescaling features and targets to similar numerical values. \\
Instead of providing the Frenkel exciton Hamiltonians as training features and the transfer times and efficiencies as training targets in a particular system of physical units we rescaled both features and targets based on the parameter ranges in the training set to facilitate faster and more stable MLP training. In particular, we subtracted the training set mean from all excited state energies in the exciton Hamiltonians and then mapped all feature elements $h_{ij}$ onto the interval $[-2, 2]$ (see Eq.~\ref{eq:feature_rescaling}), where $h_{ij}$ denotes a particular element of a particular exciton Hamiltonian and $H_\text{train}$ denotes the set of all exciton Hamiltonians in the training set. The rescaled features are denoted with $\tilde{h}_{ij}$. Rescaling the input features onto the $[-2, 2]$ interval ensures that all input values lie within sensitive regions of input layer activation functions, which avoids neuron saturation at the beginning of the MLP training. \\
\begin{align}\label{eq:feature_rescaling}
\tilde{h}_{ij} = 4 \frac{h_{ij} - h_\text{min}}{h_\text{max} - h_\text{min}} - 2, \qquad\qquad h_\text{min} = \min\limits_{h_{ij} \in H_\text{train}}(h_{ij}), \qquad h_\text{max} = \max\limits_{h_{ij} \in H_\text{train}}(h_{ij}).
\end{align}
Prediction targets need to be rescaled onto an interval, which lies within the codomain of the output layer activation function. The target properties in this study are exciton transfer times and efficiencies. Both of these properties are always positive, but while efficiencies also have an upper bound, transfer times could a priori be arbitrarily large. We, therefore, chose the softplus function for the activation function of the MLP output layer as it exhibits similar properties. To use most of the non-linear regime of the softplus function, we decided to map our training targets $t$ (the collective set of transfer times and efficiencies) onto the interval $(0, 4]$ based on the maximum transfer time and efficiency in the training set $T_\text{train}$ (see Eq.~\ref{eq:target_rescaling}). Exciton transfer times and transfer efficiencies were rescaled separately. Note, that both exciton transfer times and efficiencies are always positive which justifies the implicit lower bound of zero in the equation. \\
\begin{align}\label{eq:target_rescaling}
\tilde{t} = \frac{4t}{t_\text{max}},\qquad\qquad t_\text{max} = \max\limits_{t \in T_\text{train}}(t)
\end{align}
\subsection{Bayesian optimization for hyperparameter selection}\label{sec:sup_bayesian_optimization}
Multi-layer perceptrons (MLP) consist of a set of neurons organized in layers. Each neuron accepts an input, which is rescaled by a set of weights and biases intrinsic to the neuron, to calculate its output. The outputs of neurons in one layer are propagated through the MLP as the input for the subsequent layer. While weights and biases of each neuron are collectively referred to as the parameters of the MLP, an MLP model contains additional free parameters such as the number of layers and the number of neurons per layer. The latter are the hyperparameters of the model. MLP parameters are typically optimized on the training set with gradient based optimization techniques such as stochastic gradient descent. Hyperparameters are instead selected by optimizing the parameters of an MLP on the training set and evaluating the prediction accuracy on the validation set. \\
In this study, we decided to employ a Bayesian optimization approach to find hyperparameters for accurate MLP architectures. We chose a total of six hyperparameters to be optimized and set fixed ranges for each of them for the Bayesian optimization. Hyperparameters and ranges are reported in Tab.~\ref{tab:bayes_opt_hyperparameter_selection}. In particular, we also included the number of training points as a hyperparameter to investigate by how much the prediction accuracy of MLPs can increase when expanding the training set. \\
\begin{table}[!ht]
\centering
\begin{tabular}{lrr}
\toprule
Hyperparameter & low & high \\
\midrule
Training points & 4000 & 6000 \\
Network layers & 3 & 18 \\
Neurons per layer & 2 & 2000 \\
Learning rate & $10^{-7}$ & $10^{-1}$ \\
Regularization & $10^{-18}$ & $10^{6}$ \\
\midrule
\multirow{2}{*}{Activation function} & \multicolumn{2}{c}{sigmoid, relu, tanh} \\
& \multicolumn{2}{c}{softsign, softplus} \\
\bottomrule
\end{tabular}
\caption{Selection of hyperparameters for Bayesian optimization of the MLP architecture. Lower and upper bounds were applied to the search space for five of the six parameters. Five different options were provided to the Bayesian optimizer for choosing an activation function for all but the last network layer.}
\label{tab:bayes_opt_hyperparameter_selection}
\end{table}
The particular ranges for individual hyperparameters were chosen based on a few test training runs and chosen large enough that the Bayesian Optimizer is able to explore diverse MLP architectures. Although we found that MLPs perform more accurately when using more points in the training set, we restricted the number of training points to 6000 as a compromise between accuracy and computational demand. Especially for PCA sampled training sets we observed only a small advantage of larger training sets measured by the validation set error. Several activation functions were probed in the Bayesian optimization. \\
\subsubsection{Convergence of Bayesian optimization runs}
Bayesian optimization selects a particular set of hyperparameters from all the possible values for each of the hyperparameters (see Tab.~\ref{tab:bayes_opt_hyperparameter_selection}). The MLP corresponding to this set of hyperparameters is then constructed and trained on the training set. Prediction errors on the validation set are used to evaluate the prediction accuracy of each constructed MLP after it was trained. \\
The Bayesian optimization procedure was run for a total time period of seven days (walltime) on four GPUs (NVIDIA Tesla K80) for each dataset (FMO, RC, CP43, CP47) and each training set selection method (random, PCA). We extract the validation set error for all MLPs generated and trained in this process. During optimization, we keep track of validation error of the current optimal MLP architecture. Fig.~\ref{fig:bayesian_optimization_progress} to illustrate the progress of the hyperparameter optimization. After only a few iterations the prediction error for the validation set already has significantly dropped.
\begin{figure}[!ht]
\centering
\includegraphics[width = 0.6\columnwidth]{bayesian_optimization_progress.pdf}
\caption{Smallest relative validation error for the set of MLPs trained during the Bayesian Optimization over the number of MLP evaluations in the optimization.
}
\label{fig:bayesian_optimization_progress}
\end{figure}
In each of the Bayesian optimization procedures we did not see any decrease in the validation set errors for at least the last 200 proposed MLP architectures. Further, we observe that as opposed to a single best set of hyperparameters the Bayesian optimization instead reveals a number of MLP architectures with different hyperparameter values but similarly small validation set errors. Therefore, we conclude that the Bayesian optimization converged for all datasets and identified reasonably accurate MLP architectures.
\subsubsection{Bayesian optimization results}
We recorded the minimum validation set errors of all MLPs constructed and trained during the Bayesian optimization procedure to study the effect of particular hyperparameter choices on the prediction accuracy (see Fig.~\ref{fig:remaining_bayes_opt_networks}).
Optimal sets of hyperparameters resulting in the smallest validation set error for all four datasets with MLPs trained on PCA select (randomly drawn) training sets are reported in Tab.~\ref{tab:bayes_opted_hyperparameters} (Tab.~\ref{tab:bayes_opted_hyperparameters_random}). While MLPs trained on the RC and the FMO datasets, which consists of fewer excitonic sites, tend to prefer shallow but broad architectures we achieved the smallest relative validation set errors with more hidden layers and fewer neurons per hidden layer for the CP43 and the CP47 dataset with more excitonic sites. \\
\begin{table}[!ht]
\centering
\begin{tabular}{lcccc}
\toprule
Hyperparameter & RC & FMO & CP43 & CP47 \\
\midrule
Training points & 6000 & 6000 & 5480 & 6000 \\
Layers & 3 & 5 & 3 & 3 \\
Neurons per layer & 1729 & 1258 & 1899 & 749 \\
Learning rate & $10^{-3.83}$ & $10^{-3.44}$ & $10^{-3.24}$ & $10^{-3.19}$ \\
Activation & softsign & relu & tanh & softsign \\
Regularization & $10^{-14.37}$ & $10^{-18.0}$ & $10^{-7.80}$ & $10^{-2.91}$ \\
\bottomrule
\end{tabular}
\caption{Optimized multi-layer perceptron (MLP) hyperparameters for the four investigated datasets (PCA selected training sets) used in the main text. Hyperparameters were optimized in a Bayesian optimization procedure. }
\label{tab:bayes_opted_hyperparameters}
\end{table}
\begin{table}[!ht]
\centering
\begin{tabular}{lcccc}
\toprule
Hyperparameter & RC & FMO & CP43 & CP47 \\
\midrule
Training points & 6000 & 6000 & 6000 & 6000 \\
Layers & 3 & 3 & 3 & 7 \\
Neurons per layer & 901 & 1382 & 1913 & 962 \\
Learning rate & $10^{-2.84}$ & $10^{-3.21}$ & $10^{-4.23}$ & $10^{-4.08}$ \\
Activation & softsign & softsign & softsign & softsign \\
Regularization & $10^{-18.0}$ & $10^{-16.1}$ & $10^{-12.0}$ & $10^{-4.3}$ \\
\bottomrule
\end{tabular}
\caption{Optimized multi-layer perceptron (MLP) hyperparameters for the four investigated datasets (randomly selected training sets). Hyperparameters were optimized in a Bayesian optimization procedure. }
\label{tab:bayes_opted_hyperparameters_random}
\end{table}
\begin{figure}[!ht]
\centering
\begin{minipage}{0.45\textwidth}
\flushleft
A) FMO:\\
\centering
\includegraphics[width = 0.85\textwidth]{bayes_opt_networks_fmo_pca.pdf} \\
\end{minipage}
\begin{minipage}{0.45\textwidth}
\flushleft
B) RC:\\
\centering
\includegraphics[width = 0.85\textwidth]{bayes_opt_networks_rc_pca.pdf} \\
\end{minipage} \\
\begin{minipage}{0.45\textwidth}
\flushleft
C) CP43:\\
\centering
\includegraphics[width = 0.85\textwidth]{bayes_opt_networks_cp43_pca.pdf}
\end{minipage}
\begin{minipage}{0.45\textwidth}
\flushleft
D) CP47:\\
\centering
\includegraphics[width = 0.85\textwidth]{bayes_opt_networks_cp47_pca.pdf}
\end{minipage}
\caption{Scatter plot of the average absolute relative errors of multi-layer perceptrons (MLPs) constructed during the Bayesian optimization procedure and trained on the indicated data set in dependence of the particular choices of hyperparameters made by the Bayesian optimizer. Each point represents the hyperparameters for the best architecture for each optimization step. MLPs were trained on PCA selected training sets.}
\label{fig:remaining_bayes_opt_networks}
\end{figure}
\subsection{Comparison of exciton transfer times obtained from exciton dynamics calculations and multi-layer perceptron predictions}\label{sec:prediction_comparisons}
We trained MLP models to predict exciton transfer times and efficiencies from Frenkel exciton Hamiltonians to provide an alternative to computationally costly exciton dynamics calculations. In this section, we comment on the prediction accuracy of MLP models by comparing their predictions with results obtained from two popular exciton dynamics approaches, the secular Redfield method and the hierarchical equations of motion (HEOM) formalism (on which the MLPs were trained). While secular Redfield is known to underestimate exciton transfer times it is computationally cheaper than HEOM. \\
We provide a context for MLP prediction accuracies by comparing MLP predicted transfer times to transfer times obtained from secular Redfield and HEOM calculations on the level of individual Hamiltonians. Fig.~\ref{fig:transfer_times_scatter_plot} shows scatter plots of transfer times obtained from all three approaches. We plot transfer times predicted by MLPs and secular Redfield versus transfer times calculated with HEOM, which we consider as the ground truth for the purpose of this study. The green line in Fig.~\ref{fig:transfer_times_scatter_plot} indicates perfect agreement between HEOM predictions and predictions by either MLPs or secular Redfield. \\
\begin{figure}[!ht]
\centering
\includegraphics[width = 1.0\textwidth]{comparison_redfield_networks_all.pdf}
\caption{Exciton transfer times as computed with the hierarchical equations of motion (HEOM) approach compared to ecxiton transfer times calculated with the secular Redfield method or predicted from trained multi-layer perceptrons (MLPs). Panels show the exciton transfer times obtained for all 12000 exciton Hamiltonians in each of the four generated datasets. The griin line indicates perfect agreement between HEOM results and predictions by either MLPs or secular Redfield. }
\label{fig:transfer_times_scatter_plot}
\end{figure}
By comparing the exciton transfer time predictions of the secular Redfield approach to the exciton transfer times of HEOM we observe that secular Redfield almost always underestimates transfer times consistently in all four datasets. MLPs instead over- and underestimate exciton transfer times, which is due to the symmetric loss function employed during MLP training and hyperparameter optimization. \\
The absolute deviation between Redfield and HEOM transfer times generally increases with the value of the exciton transfer time, as observed in previous studies.\cite{Novoderezhkin2011} Also for the MLP predictions, we observe a larger deviation from the HEOM results for longer transfer times. However, in contrast to secular Redfield, the error is still distributed rather symmetrically around our ground truth. This observation is explained by the fact that MLPs were trained to minimize the relative deviation between predicted and HEOM transfer times. That is, for longer transfer times the predictions can afford larger deviations from HEOM which results in an opening funnel structure, especially seen in the scatter plot for the CP47 dataset. In addition, all of the presented datasets include fewer data points at larger transfer times (see Fig.~\ref{fig:transfer_time_distributions}), but MLPs are trained to minimize loss functions which take the unweighted average over all transfer times.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 7,238 |
Debate & Public Speaking Academy
Are you fascinated by the power of speech? We created the Debate & Public Speaking Academy especially for 15-18 year olds to help you develop your confidence and ability to persuade and captivate your audience. You'll also discover how you might use debating and public speaking skills in different professional contexts
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Welcome to the Debate & Public Speaking Academy. This two-week course is a hands-on, practical opportunity for you to explore and experiment with the power of speech in the contexts of Politics, International Relations, Law and Economics.
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I learnt eloquence in speech, debating skills and confidence. A fantastic experience.Jacques, France
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As part of the Debating & Public Speaking Academy, you'll take part in a series of nine classes. Eight classes are structured around the course topics below. Your final class is called "Class X", and will be based on your tutor's personal expertise, focusing on cutting-edge research in their field that they are really passionate about.
1. Introduction to Debate and Public Speaking
What speaking techniques are most effective? Work together to identify and evaluate techniques used by renowned public speakers to captivate their audience. Put what you learn into practice with a short speech of your own
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What is British Parliamentary (BP) Debating? We'll discuss the rules, styles and strategies of this format of debate and you'll be able to give it a go yourself in a practice debate
3. Schools Mace Debating
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Demonstrate your newly refined debating skills and international relations knowledge as you:
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Simranjit Kamal, LLB Hons
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Tom Corby
You'll have the opportunity to meet our Hero, Tom, who works as a Lawyer where he regularly engages in debate and public speaking.
Tom studied Modern History at New College, Oxford University, where he was a Scholar. He is now a barrister practicing at 20 Essex Street.
He specialises in commercial law which includes shipping, international sale of goods, conflicts of law and banking, and has acted for and against a variety of parties, including the Kingdom of Spain, a major Eastern European drinks brand, and an England cricketer.
As part of the Debate and Public Speaking Summer School, your classes will be held in the Blavatnik School of Government, an award-winning building which is part of the University of Oxford. Finished in 2015, the building is a modern addition to the University and is located right by the Oxford University Press building.
Debating and Public Speaking Summer School for High School Students – Dates and Prices | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 9,785 |
BioM: Orth, Jeanette M. (1948)
Surnames: Orth, Chamberland, Beschta, Kraus, Pudlas, Lukowicz, Frook, Luepke
----Source: Colby Phonograph (Colby, Clark County, Wis.) 07/08/1948
Orth, Jeanette M. (Marriage - 30 June 1948)
Jeanette M. Orth, daughter of Mr. and Mrs. Conrad Orth, became the bride of Edward J. Chamberland, son of Mr. and Mrs. Albert Chamberland of Spencer, at a service conducted at St. Mary's Catholic church Wednesday morning, June 30, Rev. N. B. Beschta officiating. The altars of the church were decorated with American beauty roses and peonies. "Ave Maria" was sung at offeratory by Miss Bernadette Kraus of Colby, Wisconsin.
The bride was attended by Donna Mae Pudlas of Forreston, Ill., niece of the bride, as maid of honor and Jane Lukowicz of Colby as bridesmaid. The groom was attended by Emory Orth of Tolusa, Ill., niece of the bride, as best man and Cpl. Eugene Chamberland of Anchorage, Alaska.
The bride was attired in a white slipper satin court entrained gown, all over lace bodice extending over top part of skirt, net fingertip veil with a lace Hollander headdress. Her cascade bouquet consisted of pink and yellow roses and lilies. She carried a sterling silver rosary, a gift of the groom. The maid of honor wore a yellow taffeta colonial styled gown with bouffant skirt, matching gauntlets, net sweetheart bonnet and wore a pearl necklace, a gift of the bride. Her colonial bouquet consisted of yellow and white carnations. The bridesmaid wore a dress of mint green, styled identical to that of the maid of honor's, and also wore a pearl necklace, a gift of the bride. Her bouquet also consisted of white and yellow carnations.
Dinner was served to the immediate families at the home of the bride which was decorated with mint green and yellow streamers and white wedding bells. In the evening, a dance was given at Walter's pavilion.
The couple left on a wedding trip to Northern Wisconsin and will be at home after July 4th at Marshfield where the groom is employed by the Roddis Co.
Out of town people here for the wedding were Mr. and Mrs. Walter Frook and two children of Fond du Lac, Mr. and Mrs. Conrad Orth, Jr., and Mr. and Mrs. John Pudlas and children of Forreston, Ill., Roger Orth of Racine, Mr. and Mrs. Vern Luepke and children and Mr. and Mrs. Joe Orth of Toluca, Ill., Norman Orth of Kenosha, Mr. and Mrs. Albert Chamberland and two sons of Spencer, Wis. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 3,131 |
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World Food Day – 16 October 2018
Today is World Food Day and the 73rd anniversary of the UN's Food and Agricultural Organisation. As one of the most celebrated days in the UN's calendar, events organised by FAO country offices, governments, local authorities and other partners in over 130 countries across the world will call for increased action to achieve Zero Hunger.
According to the UN, the official World Food Day (WFD) ceremony on 16 October at FAO headquarters in Rome will be an opportunity for leaders and key global players in the drive to achieve Zero Hunger and eliminate malnutrition. The day also serves as a reminder to the world that Zero Hunger by 2030 can still be possible.
The latest FAO 2018 State of Food Security and Nutrition in the World report shows that world hunger is once again on the rise, and over 820 million people are suffering from chronic undernourishment.
The same report shows that conflict, extreme weather events linked to climate change, economic slowdown and rapidly increasing overweight and obesity levels are reversing progress made in the fight against hunger and malnutrition.
World Food Day ties into the UN's second Sustainable Development Goal (SDG) of ending global hunger by 2030. The FAO says that World Food Day is a chance to show commitment to SDG 2 in the hopes of ending hunger and malnutrition.
Every year, a large number of events – from marathons and hunger marches, to exhibitions, cultural performances, contests and concerts – are organised in around 130 countries across the world to celebrate World Food Day.
World Food Week began on 15 October and a series of events including World Food Day will explore the actions needed to achieve Zero Hunger by 2030. World Food Week takes place in a global context where conflict, climate extremes and an increase in obesity are reversing progress made in the fight against hunger and malnutrition.
The AIDF Global Summit will return to Washington D.C, in 2019.
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"redpajama_set_name": "RedPajamaCommonCrawl"
} | 9,793 |
Q: Implicitlywait in selenium WebDriver I gave implicitly wait like following ------>
d.manage().timeouts().implicitlyWait(60,TimeUnit.SECONDS);
// And I wrote 3 locaters
d.findElement(By.id("element1")).click(); //assume it take to load 20 seconds
d.findElement(By.id("element2")).sendKeys(""); //assume it take to load 10 seconds
d.findElement(By.id("element3")).click();
// now my question is how much time webdriver wait for 3rd element3? , is it 60-20+10=30 seconds or full time 60 seconds?
Sorry, I changed my question for clarity.
A: Implicit wait "tells" the findElement() method to look for the element up to the specified amount of time or until the element exists in the DOM (not necessarily displayed). This occurs for every search separately, there aren't any dependencies between the searches.
A: Now the implicitly_wait( )(Python) or implicitlyWait() (Java) method tells the script , more precisely, it tells the Webdriver to poll the DOM for a certain amount of time , for 30 or 60 seconds or whatever the time you've specified, when trying to find an element or elements if they are not immediately available. Here by poll, we mean to check the DOM again and again.
Once defined, implicit wait will be defined for the whole life time of a Webdriver object instance, until it is changed. So once defined in a script, it will be active for the lifetime of a script, until modified. It will wait the same amount i.e. 60 seconds, in your case, for element1, element2 or element3. If the element is found within the stipulated time, then next command is executed. However, if not time, WebDriver raises a TimeoutException exception.
| {
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} | 6,435 |
package org.supler
import org.json4s.JValue
import org.json4s.JsonAST.{JField, JObject, JString}
case class FormMeta(meta: Map[String, String]) {
def apply(key: String): String = {
meta(key)
}
def +(key: String, value: String) = this.copy(meta = meta + (key -> value))
def toJSON = JField(FormMeta.JsonMetaKey, JObject(meta.toList.map {case (key, value) => JField(key, JString(value))}))
def isEmpty = meta.isEmpty
}
object FormMeta {
val JsonMetaKey = "supler_meta"
def fromJSON(json: JValue) = {
json match {
case JObject(fields) => fields.toMap.get(JsonMetaKey) match {
case Some(JObject(entries)) => FormMeta(entries.toMap.collect{case (key: String, value: JString) => key -> value.s})
case Some(_) => throw new IllegalArgumentException("Form meta is not well formed")
case None => FormMeta(Map())
}
case _ => FormMeta(Map())
}
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 5,703 |
Britain to shrug off wintry economic chill
Published: 09:48 pm Jan 10, 2010
LONDON: The British economy, forecast to emerge soon from recession, will likely shrug off the most brutal winter in decades as many Britons simply delay purchases and work from home to beat the big freeze.
The Confederation of British Industry (CBI), the nation's biggest employers' organisation, admitted that the cold weather was causing "massive disruption" for companies already suffering from weak demand in the downturn.
The CBI also argued, however, that the economic impact will be mitigated by the growing adoption of high-speed Internet services that allow many to shop or work from the comfort of their own homes.
Economist Howard Archer, who covers Britain and Europe for IHS Global Insight, played down the effects of heavy snowfall and freezing temperatures.
"These things tend not to have as much impact as often feared," Archer told AFP.
"Obviously, the longer it persists the more it will hit retail spending and affect some business activity but these things tend to be made up once conditions return to normal.
"For example, people tend to delay their retail spending rather than cancel it," he added.
With icy conditions making it almost impossible to travel, many people are choosing instead to sit on their sofas, flick on the heating and power up their home computers and laptops.
Consumers will also transfer their spending to other items, said Collin Ellis, economist at Daiwa Capital Markets Europe.
"I would not expect (the bad weather) to have a big impact on economic growth," Ellis said. "Obviously if people are struggling to get to work, that means it may take longer to fill orders.
"But I suspect the most likely outcome may be a further transfer between different types of consumption -- more meals at home versus eating out."
Britain's big freeze will slash around 1.0 billion pounds from the nation's daily economic output, according to forecasts from the Centre for Economics and Business Research consultancy.
However, Keith Pilbeam, economics professor at London's City University, said the true impact on the economy was impossible to quantify.
"Although the true cost of the current wintry weather on our economy cannot be calculated precisely, it will create a number of economic issues," Pilbeam told AFP.
"The full effects of the weather will depend on the severity and duration of the wintry conditions we are currently experiencing."
Profits and sales will be damaged but some businesses will also benefit as more people stay at home.
"For businesses, the reduction in revenues coupled with paying staff who cannot make it to work -- and so are not producing -- will be very damaging to their profitability," Pilbeam said.
"However, the absence of staff from the workplace will benefit some companies, such as utilities, as people stay at home, switch on the television and turn up the heating.
"The widespread wintry weather will mean an increased demand for commodities such as oil and gas, which will lead to higher prices that may prove long lasting," he added.
British annual inflation jumped to 1.9 percent in November because of rising fuel prices, recent official data showed.
CEBR head Douglas Williams warned that some businesses could go to the wall in the bad weather but agreed that the overall impact was limited.
"Don't exaggerate (the) economic impact of the freeze -- much of the lost GDP (gross domestic product) will be made up in the coming weeks -- but some cash-strapped businesses might be pushed over the edge," Williams said.
"But all the past research shows that the impact of extreme weather on GDP is surprisingly small," he said, adding that many businesses were already operating below full capacity because of the recession.
Most analysts agree that the effects of wintry weather will simply be reversed when the big freeze thaws out.
"The wintry spell is likely to have an adverse impact on industrial production as well as a generally disruptive effect on the service sector as three million workers are reckoned to have missed work in the worst affected areas," said VTB Capital economist Neil MacKinnon.
"Obviously, these are temporary effects -- as better weather would simply see a sharp rebound in output and activity.
"The underlying picture of the UK economy is one where there are tentative signs of a gradual move out of recession," MacKinnon added.
#Britain to shrug off wintry economic chill
Slow capital spending irks FinMin Paudel | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 23 |
\section{Introduction}
During the outbursts of transient black hole candidates (BHCs), in addition to the large changes in X-ray luminosity, marked variations are observed in the properties of the timing and the energy spectrum
often on very short time scales (see e.g. \citealt{belloni2005}).
We still do not have a detailed understanding of all the mechanisms that lead to changes in the X-ray emission properties, but the physics involves the structure of the accretion flow around the black hole as well as the connection between the accretion disc and the steady or impulsive jets that can be emitted from these systems. The main cause of the changes in the X-ray emission properties is the variation of the mass accretion rate onto the black hole; however some phenomena indicate that other parameters are also important (e.g. \citealt{Homan01}).
From the observational point of view, the emission properties of accreting black holes are often classified in terms of observed spectral and timing parameters. From their combination, a number of source states have been identified (\citealt{belloni2005,belloni2009}, for an alternative definition, see \citealt{remmc}).
The high-energy spectra can be described as the combination of a soft thermal component together with a hard power law component. The latter component often shows a cutoff at high energies~\citep{Tanaka}. This decomposition is the simplest phenomenological model. However for the hard component complex models can be used as for example the Comptonization models (see e. g. \citealt{Tit, PP}){\it . The} simplest decomposition is less model dependent and provides at least a qualitative measurement of the behaviour of the source.
The power density spectra, used to characterise the fast variability, show a combination of power-law and Lorentzian components. When the Lorentzians are zero-centered, they are referred to as ``band-limited noise'', while if they are narrow they are called {\it Quasi-Periodic Oscillations} (QPO, see e.g. \citealt{bpk}). Another important observable that can be used to trace source states is the total amount of variability in the 0.1-64 Hz band, expressed in terms of fractional integrated rms (see e.g. \citealt{belloni2005,hombel,belloni2009}).
A BHC spends its time mostly in a quiescent state at low flux level ($<$ 10$^{32}$--10$^{33}$ ergs s$^{-1}$, e.g. \citealt{Camp}). When the outburst begins, the luminosity of the source increases and the X-ray spectrum is dominated by a power law component with a hard photon index of $\sim$ 1.4-1.5 and a high-energy cutoff around 100 keV (low/hard state, hereafter LHS). The radio emission in this state indicates the presence of steady jets, while the power spectrum is dominated by a strong band limited noise ($>$30\% fractional rms).
Then the outburst evolves as the source increases its luminosity and its spectrum starts to change: the soft thermal component appears and becomes increasingly important, the energy peak of the emission softens and the photon index of the hard component steepens ($\sim$ 2.0-2.5). Two different states with these spectral characteristics have been defined: the hard intermediate state (HIMS) and the soft intermediate state (SIMS). The characteristics of these two states are quite complex: the changes can be established mostly by the timing properties \citep{hombel} and also by the ejection of relativistic jets associated to the transition from HIMS to SIMS.
After the SIMS, the source enters a state where the X-ray spectrum is dominated by the emission of the soft thermal component (high/soft state, hereafter HSS). A non-thermal power law tail is also present without any detectable cutoff, while the power spectrum is characterised by a low-level (1-2\% fractional rms) variability.
Then the flux starts to decrease, most likely following a parallel decrease in accretion rate. At some point, a reverse transition is started and the path is followed backwards all the way to the LHS and then to quiescence. As mentioned above, the luminosity level of this back-transition is always lower than that of the corresponding forward-transition.
The description above is the basic general pattern (the so-called "{\it q-track}" pattern in the hardness-intensity diagram (see Figure~\ref{figtesi}); several examples of "{\it q-track}" patterns are reported in \citealt{hombel} or in Dunn et al. 2009 in prep.), which has been modeled after repeated outbursts of GX 339-4 \citep{fbg2004,belloni2005,hombel,mela, belloni2009}. During the HSS, minor transitions to SIMS and HIMS have been observed also in GX 339-4 (e.g. \citealt{Pg,belloni2005,mela2}). Other sources behave in a more complicated way, but the general classification into four states holds for all of them (see e.g. \citealt{mike2000,frontera2001,chaty2003,hynes2003}).
Interestingly, until now all black-hole transients have shown two types of behaviour: after the initial LHS, most sources show a transition to the HIMS at a luminosity level which is always different and might be related to the previous history of the transient \citep{yu2007}. If this transition takes place, the source always reached the HSS. However, a few sources (both NS and BHC X-ray binaries) never left the LHS at all as reported for example by \citet{broc} for V404 Cyg, A1524-62, 4U1543-475, GRO J0422+32, GRO J1719-24, GRS1737-21 and GS 1354-64, by \citet{Stu} for XTE1550-564 and by \citet{Rod} for Aql X-1. The only possible exception to this dichotomy is represented by SAX~J1711.6--38 (see \citealt{wm02}), a faint transient X-ray binary classified as BHC~\citep{Liu}.
In this paper we present the results of the {\it RXTE}, {\it Swift } and INTEGRAL data analysis of the last outburst of the recurrent transient H~1743--322 during which only two states were sampled: the LHS and the HIMS. It is the first time that this kind of outburst is analysed in detail showing that the evolution of the BHC outbursts, through the different spectral states, is still difficult to predict.
\subsection{Short History of H~1743--322}
On 2003 March 21 {\it INTEGRAL} (MJD=52719) detected a relatively bright source ($\sim$60 mCrab at 15-40 keV) named IGR J17464--3213. The source was localised at R.A.\ (2000) $=17^{\rm h}46.3^{\rm m}$, Dec.\ $=-32\degr
14.4^\prime$, with an error box of 1.6 arcmin (90\% confidence) and then associated with H~1743--322 \citep{mark,rev}, a bright BHC observed by {\it HEAO-1} in 1977 with an intensity of 700 mCrab in the 2-10 keV~\citep{dox}. The outburst evolution of 2003 was followed by {\it RXTE } and {\it INTEGRAL} reporting strong flux and spectral variability \citep{parm,homan,cap2005,joinet}.
H~1743--322, after the first and brightest outburst, also underwent two fainter outbursts on September 2004 and September 2005~\citep{atel301,atel575,cap2006}.
On January 2008 a third outburst was detected by {\it RXTE}/ASM~\citep{kal}. Then a {\it Swift } ToO was granted in order to follow the source evolution~\citep{CapATel2,CapATel1}.
Seven months later (on September 23, MJD=54732), another outburst was detected by INTEGRAL during the Galactic bulge monitoring~\citep{kuul} showing that the source was in a hard state with an increasing flux. {\it Swift}, {\it RXTE } and INTEGRAL followed the outburst evolution.
Furthermore, on October 23 (MJD=54762) an {\it RXTE } observation of H~1743--322 indicated that the source had undergone a state transition from the LHS to the HIMS~\citep{belATel}. The study reported in this paper is focused on the last part of this outburst. Some preliminary results of these observations were already reported in various ATels (see e.g. \citealt{Ricci} and reference therein), while results on the early phase of the outburst are presented in \citet{pr}.
\begin{figure}
\centering
\includegraphics[angle=0, scale=0.5]{qtrak.ps}
\caption{Sketch of the "canonical q-track pattern" through all the different spectral states (for more details see ~\citealt{hombel} and ~\citealt{belloni2009}).}
\label{figtesi}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[angle=90, scale=0.37]{fig0.ps}
\caption{ASM 1.5-12 keV one-day averaged light curve of H~1743--322 from 2003 until the end of 2008.}
\label{asmtot}
\end{center}
\end{figure}
\section{Observation and analysis}
The {\it RXTE } campaign of pointed observations covers the period of the outburst starting from MJD=54740 (2008 October 10) to MJD=54775 (2009 November 5) for a total of 37 pointings with a total exposure time of 65.5 Ks. The PCA and HEXTE data analysis was performed with the standard {\it RXTE } software within {\tt HEASOFT V6.6} following the standard extraction procedure. For spectral analysis only PCA2 for PCA and cluster B for HEXTE were used. A systematic error of 0.6\% ~\citep{Wilms} was added to the PCA spectra. For the fitting, the energy ranges 3-20 keV and 20-130 keV were used for PCA and HEXTE respectively.
For the timing analysis of the PCA data, for each of the observations, we produced power spectra from 64-s stretches accumulated in the channel band 0-35 (2-15 keV) with a time resolution of 1/1024 s. The resulting power spectra were then averaged, resulting in one power spectrum per observation. The power spectra were normalised according to the description by \citealt{Leahy} and converted
to squared fractional rms~\citep{BelHas,Miy}. The contribution due to Poissonian statistics was subtracted (see \citealt{Zhang}). The timing analysis was performed with custom software.
During the 2008 October outburst a public INTEGRAL ToO campaign was carried out (\citealt{Ricci} and references therein).
Three observations were performed respectively in 2008 October 10 (65 ks, MJD=54749), October 22 (86 ks,MJD=54761) and October 28 (80 ks,MJD=54767
) with a total of 136 science windows (SCW)\footnote{ A SCW is a unit of INTEGRAL continuous observing time that is at the base of the INTEGRAL data processing.} of about 2500 s each (INTEGRAL revolutions 732, 734 and 736). In order to obtain a wider energy range for the spectra, our analysis was focused on ISGRI~\citep{Lebr}, the low energy detector of the $\gamma$-ray telescope IBIS~\citep{ube}. The data of the INTEGRAL spectrometer SPI were not used because of the low angular resolution of the instrument (for details see \citealt{Ver}). For the ISGRI data analysis we used the latest release of the standard Offline Scientific Analysis (OSA) version 7. The ISGRI energy range considered for the fitting is 20-200 keV with a systematic error of 2\% as usual in the INTEGRAL spectral analysis (see also \citealt{Jourdain}).
The ISGRI light curve was obtained by extracting the source count rate from the images in the 40-100 keV band for each SCW.
A {\it Swift } ToO was also performed with three pointings. We extracted the XRT spectra of the Window-Timing (WT) mode pointings ( less affected by pile-up) in order to better constrain the equivalent hydrogen column. The data were processed with the standard {\it Swift } tools:
XRT software version 0.12.2 and FTOOLS version 6.6.2
Only the second WT mode {\it Swift}/XRT observation was performed simultaneously with {\it RXTE } and INTEGRAL and thus it was used to obtain a joint spectrum with the other instruments (see the bottom panel of Figure~\ref{counts2}).
The {\it Swift }/BAT transient monitor light curve was provided by the {\it Swift }/BAT team\footnote{http://swift.gsfc.nasa.gov/docs/swift/results/transients/index.html}.
\section{Results}
\subsection{Outburst Evolution}
The 2008 October outburst of H 1743-322 was quite short, and fainter than the previous ones (see Figure~\ref{asmtot}). In Figure~\ref{bat_vs_asm} we show the temporal behaviour of H 1743-322 in different energy ranges.
For a direct comparison with the previous outbursts of the source, we show in the figure, as an example, also the light curves of the January 2008 outburst. During the January 2008 outburst the source reached the HSS~\citep{kal} showing the standard {\it q-track} behaviour as for the previous outbursts of H1743-322 (see for example \citealt{cap2005,cap2006,Mc}).
The top panel and the middle panel represent respectively the 4-45 keV ({\it RXTE}/PCA) and the 20-100 keV ({\it RXTE}/HEXTE) light curves. Both curves are binned to a single observation. The bottom panel shows the INTEGRAL IBIS/ISGRI 40-100 keV light curve, binned to a SCW~\footnote{ The data are taken from both the public archive of the INTEGRAL Galactic Bulge Monitoring., http://isdc.unige.ch/Science/BULGE/ and from the 2008 public INTEGRAL ToO.}.
The first outburst, was observed by {\it RXTE } only during the return to the quiescent state. The monitoring of the most recent outburst (blue curve) was more complete even though {\it RXTE } missed the rising phase of the outburst and its coverage ended before the full return to quiescence.
The INTEGRAL monitoring started at the end of the January outburst; a good coverage was only achieved during the October outburst (bottom panel of Figure~\ref{bat_vs_asm}).
Figure~\ref{bat_mon} shows the {\it Swift }/BAT daily averaged light curve (15-50 keV) of the two 2008 outbursts of H1743-322.
\begin{figure}
\centering
\includegraphics[angle=90, scale=0.34]{hexte.ps}
\caption{ Light curves of both 2008 January and October H~1743--322 outbursts in different energy ranges: in red the October outburst, in blue the January one. Top panel: {\it RXTE }/PCA 4-45 keV light curve (for the different symbols of the blue curve see Figure~\ref{licu}). Middle panel: {\it RXTE }/HEXTE 20-100 keV light curve. Bottom panel: IBIS/ISGRI 40-100 keV light curve binned to a SCW; the data are taken from the {\it INTEGRAL Galactic Bulge Monitoring}. The black circles represent the three ToO observations analysed in this paper.}
\label{bat_vs_asm}
\end{figure}
\begin{figure}
\centering
\hspace{-1.4cm}
\includegraphics[angle=90, scale=0.35]{BAT.ps}
\caption{H1743-322 BAT daily averaged light curve, 15-50 keV}
\label{bat_mon}
\end{figure}
\begin{figure}
\centering
\hspace{-1.4cm}
\includegraphics[angle=90, scale=0.34]{fig3.ps}
\caption{ {\it RXTE }/PCU2 hardness-intensity diagram (hardness is defined as the ratio of the count rates in the 11-20 keV and 4-10 keV bands) of the two 2008 outbursts of H 1743-322. The different symbols of the blue curve are associated with their averaged spectrum of Figure~\ref{spectra} and Table~\ref{spec_res} as follow: filled diamonds for {\it (a)}, asterisks for {\it (b)}, open diamonds for {\it (c)}, triangles for {\it (d)}, squares for {\it (e)}, crosses for {\it (f)}, circles for {\it (g)} and ellipses for {\it (h)}.}
\label{licu}
\end{figure}
The HIDs of the two outbursts are compared in Figure~\ref{licu}: for the first outburst (red curve), the PCA caught the source at the end of the {\it q-track} diagram (see \citealt{belloni2005,belloni2009}), when the energy spectrum of H~1743--322 was hardening again through the HIMS, then moving vertically downward along the LHS branch returning to quiescence.
For the second outburst, after the initial LHS rise (missed by {\it RXTE }) the source moves horizontally to the left, softening, then jumps to a much softer location, from which it slowly returns to the hard track along a diagonal path. Clearly, the final hard state is also missed. As Figure~\ref{licu} shows, the softest points of the second outburst reach only intermediate values of the hardness ($\sim$0.5) that correspond in the previous outburst to the HIMS.
This fact is confirmed also by the HEXTE light curve of the two outbursts (see middle panel of Figure~\ref{bat_vs_asm} red and blue curves): the ratio between the PCA and HEXTE fluxes shows that the October outburst is clearly harder than the January one.
\subsection{Timing Analysis}
Figure \ref{fig:rms} shows the Hardness-rms diagram for both 2008 outbursts. Also from this figure it is evident that the October outburst saw the source remaining at a high level of variability, with integrated fractional rms always above 10\%. In contrast, in January the sampling of the final part of the outburst, started at a low hardness and little variability (around 1\%).
Together with the HID, this figure suggests that in October the source never left the HIMS.
Inspection of the power-density spectra of the October outburst confirms this hypothesis. Band-limited noise is seen in all cases. All observations show a type-C QPO (see ~\citealt{rem02}) with the exception of that of October 25 (MJD=54764), the softest of the sample. However, the observations before and after, on October 23 (MJD=54762) and 27 (MJD=54766), show a QPO evolving from 5.6 Hz to 6.7 Hz, with an rms decreasing from 5.5\% to 3.1\%. The 3$\sigma$ upper limit for a 6 Hz QPO with the same FWHM (around 0.4 Hz) for October 25 is 3.3\%, which makes the non-detection compatible with neighboring observations.
From timing analysis, we can conclude that all observations of the October outburst indicate H 1743-322 being in the HIMS, a state characterised by the presence of a type-C QPO and intermediate hardness.
\begin{figure}
\centering
\includegraphics[angle=0, scale=0.41]{rms.eps}
\caption{Hardness-rms diagram of the two 2008 outbursts of H 1743-322. The blue asterisks indicate the January outburst, red circles the October one. Note that 1.7 days is the average distance between two subsequent observations (maximum gap 4.5 days) }
\label{fig:rms}
\end{figure}
\subsection{Spectral Analysis}
We analysed all the {\it RXTE } pointed observations in order to study the source spectral behaviour. The data were fitted with a simple model consisting of an absorbed disc blackbody plus a cutoff power-law component. From the analysis of the Swift/XRT data an equivalent hydrogen column value of N$_{H}$=(1.6$\pm$0.1)$\times$10$^{22}$ atoms cm$^{-2}$ was derived and in all the other fits the N$_{H}$ was fixed to the value derived from the {\it Swift } data analysis.
To account for cross-calibration problems between the three different instruments (PCA, HEXTE, IBIS/ISGRI), multiplicative constants were added to the fits. An emission line with centroid fixed at 6.4 keV was needed to obtain good fits.
The relative change of the disc inner radius is derived from the square root of the disk black body component normalisation constant (for details see ~\citealt{bb}).
After a detailed study of each pointing, we averaged the spectra of contiguous observations with consistent spectral parameters. Table~\ref{spec_res} summarises our results, while the unfolded spectra of different groups of observations are presented in Figure~\ref{spectra}. Each spectrum was rebinned with {\tt HEASOFT V6.4} tool {\tt GRPPHA} in order to get an adequate signal to noise ratio.
Concerning the first group of spectra, {\it (a)} in Table~\ref{spec_res}, the best fit model is described by a disk black body with an internal temperature of about 1 keV plus a high-energy power law component with a photon index of about 1.3 and a cutoff of 75 keV. The second group of spectra, {\it (b)}, is characterised by a softening of the photon index together with the decrease of the 0.1-500 keV flux (see Table~\ref{spec_res}).
\begin{figure}
\centering
\includegraphics[angle=90, scale=0.35]{fig5.ps}
\caption{{\it RXTE }/PCU and INTEGRAL IBIS/ISGRI spectral evolution of H 1743-322 during the 2008 October outburst: see Table~\ref{spec_res} for the spectral parameters value of each spectrum. See Figure~\ref{licu} for the position of each spectrum in the HID.}
\label{spectra}
\end{figure}
Between the observations {\it(b)} and {\it (c)}, the spectrum changes fast: the cutoff reaches a value of about 100 keV (red curve in Figure~\ref{spectra}). Two days after (spectrum {\it (d)}) the cutoff is no longer detectable; at the same time the photon index becomes softer.
The disc black body inner radius increases its value by about 70\% during the softening while the inner temperature remains contant (see Table~\ref{spec_res}).
After October 30 the source spectra {\it (e)} and {\it (f)} harden again: in accordance with the HID, the cutoff is again detectable at about 109 keV. Both the inner radius and the photon index approach values similar to those observed in the first two groups of observations (see Table~\ref{spec_res}).
We show in Figure~\ref{counts2} (bottom panel) the (f) spectrum fitted jointly with the simultaneous {\it Swift}/XRT window timing observations. This spectrum confirms the presence of the disk black body component and the fit parameters are all consistent within the errors with the RXTE spectrum (see Table~\ref{spec_res}). While the top and the middle panels show respectively the {\it (a)} and {\it (d)} spectra.
\begin{figure}
\centering
\includegraphics[angle=-90, scale=0.22]{ldatadel1.ps}
\includegraphics[angle=-90, scale=0.22]{ldatadel_d.ps}
\includegraphics[angle=-90, scale=0.22]{ldatadel_swift_new.ps}
\caption{Spectra {\it (a)} (Top panel), {\it (d)} (middle panel) and {\it (f)} (bottom panel) plotted in counts (see Table~\ref{spec_res} for spectral parameters).}
\label{counts2}
\end{figure}
At the end of the outburst (spectra {\it (g)} and {\it (h)}) the flux slightly continues to decrease, the inner temperature of the disk still remains unvaried, while it is no more possible to constrain the cutoff. Note, however that the last INTEGRAL observation, that permits to extend the spectrum up to 200 keV, was performed in the 2008 October 23 (MJD=54762) (spectrum (c)). Thus, the spectra taken after October 23 cover only energy range from 3 to 130 keV. This fact limits the possibility to constrain the cutoff for observations after that date.
\begin{table*}
\begin{minipage}{220mm}
\begin{tabular}{lccccccccccc}
\hline
ID & date & date & T$_{in}$ & R$^{*}_{in}$ &$\Gamma$ & E$_{c}$ & FLUX$_{(0.1-500)}$ & FLUX$_{(2-10)}$ & FLUX$_{(10-100)}$ & $\chi^{2}_{red.}$ & d.o.f. \\
\hline
- & mm/dd & MJD & keV & Km & - & keV & (erg s$^{-1}$cm$^{-2}$) & { \small (erg s$^{-1}$cm$^{-2}$)}& {\small (erg s$^{-1}$cm$^{-2}$)} & - &- \\
- & (2008) & - & - & - & - & - & $\times$10$^{-9}$ & $\times$ 10$^{-9}$ & $\times$ 10$^{-9}$ &-&-\\
\hline
{\it (a)}& 10/03-10/16 &{\bf 54742-54755} & 1.1$^{+0.1}_{-0.1}$ &4 $^{+3}_{-1}$ &1.23$^{+0.03}_{-0.03}$ & 64$^{+3}_{-3}$ & 8.9 &1.7 &5.0 & 1.1 & 96 \\
{\it (b)}&10/17-10/19 &{\bf 54756-54758} & 1.0$^{+0.1}_{-0.1}$ &5$^{+2}_{-1}$ &1.45$^{+0.03}_{-0.03}$ &84$^{+6}_{-6}$ &8.1 &1.7 &4.1 &1.0 & 104\\
{\it (c)}&10/23 &{\bf 54762} & 0.83$^{+0.03}_{-0.03}$ &16$^{+2}_{-1}$ &1.9$^{+0.1}_{-0.1}$ &109$^{+27}_{-19}$&10 &2.8 &2.0 & 1.1 & 92\\
{\it (d)}&10/25-10/27&54764-54767 & 0.79$^{+0.03}_{-0.02}$ & 20$^{+2}_{-2}$ & 2.10$^{+0.03}_{-0.03}$& -- & 12 &2.9 &2.1 &1.0 & 61\\
{\it (e)}&10/28-10/30 &54767-54769 & 0.72$^{+0.04}_{-0.04}$ &15$^{+4}_{-2}$ &1.95$^{+0.03}_{-0.03}$ & -- & 7.1 & 1.8 &2.2 &1.0 & 65\\
{\it (f)\footnote{ {\it Swift}/XRT, {\it RXTE}/PCA and {\it RXTE}/HEXTE joint spectrum} } &10/31-11/04&54770-54774 & 0.82$^{+0.03}_{-0.03}$ &6$^{+4}_{-2}$ &1.60$^{+0.04}_{-0.05}$ &104$^{+16}_{-26}$& 5.1 &1.2 &2.1 &0.9 & 215\\
{\it (g)}&11/07-11/16 &54776-54786 & 0.80$^{+0.1}_{-0.1}$ &5$^{+2}_{-1}$ &1.71$^{+0.02}_{-0.02}$ & -- &6.7 &0.9 &2.1 & 1.0 & 68\\
{\it (h)}&11/18-11/19 &54788-54789 & 1.0$^{+0.1}_{-0.1}$ & 3$^{+2}_{-1}$ &1.60$^{+0.03}_{-0.04}$ &-- & 5.3 &0.6 &1.7 & 1.1 & 51\\
\hline
\end{tabular}
\end{minipage}
\caption{Best fit spectral parameters of the seven groups of spectra. INTEGRAL and
{\it RXTE } commonly fitted spectra are represented by the MJD date in bolt.
Note: all the errors are at 90\% confidence level. T$_{in}$: the inner temperature of the disk;
R$^{*}_{in}$=(R$_{in}$/D$_{10}$)$\times\sqrt[]{cos(i)}$:
where R$_{in}$ is the inner radius of the disk in km; {\it i} is the
inclination angle of the system and D$_{10}$ is the distance of the
source in unit of 10kpc; $\Gamma$: power law photon index; E$_{c}$:
high energy cutoff; FLUX$_{(0.1-500)}$: unabsorbed flux extraploted
from the models in the 0.1-500 keV energy range; Flux$_{(2-10)}$:
unabsorbed flux between 2-10 keV; Flux$_{(10-100)}$: unabsorbed flux
between 10-100 keV; $\chi^{2}_{red}$: reduced $\chi$ square.}
\label{spec_res}
\end{table*}
\section{discussion}
At first inspection the shape of the HID of H 1743-322 suggests a normal evolution of the October outburst: (missed) hard state, followed by a HIMS, a fast jump to the soft state (with or without a sampling of the SIMS), then a return path at lower flux.
However, the softest points are only at an intermediate hardness, which, in the previous outburst, for example, corresponded to the HIMS. The hardness is only a rough indication of the spectral shape and similar states have been seen to correspond to slightly different hardness values even in different outbursts of the same source. Anyway the lack of soft states is an important fact and possibly suggests that the soft state was never reached. This fact, only supposed by the study of the HID, is confirmed by the results of the timing analysis.
The spectral analysis, in accordance with the HID, sampled the softening of the source: the flux of the black body component increases and the R$_{in}$ decreases. This means that the disk approaches the last stable orbit, while, curiously, the T$_{in}$ remains substantially unvaried being also quite high for a HIMS.
As far as we know, this is the first time that an outburst, that left the LHS and does not reach the HSS, has ever been studied in detail. Searching for similar cases in the literature, we have found only one outburst comparable with our results, the one of SAX J1711.6--3808 in 2001. This outburst was not covered very densely, but its softest observation showed a power spectrum and a total fractional rms typical of the HIMS, while the slope of the hard part of the energy spectrum hardly reached 2.4~\citep{wm02}. Also in this case we have an HIMS with an relatively high inner-disk temperature as 0.86$\pm$0.04 keV~\citep{wm02} and a luminosity consistent with the softest state of H 1743-322.
We conclude that we observed from H 1743-322 a failed outburst. In fact the source, even if softens, it never reaches the HSS during the 2008 October outburst.
The range in luminosities of hard-to-soft transitions in black-hole binaries observed with {\it RXTE } is 0.2-1 L$_{Edd}$~\citep{Chen}. Interestingly, the lowest transitions, at 0.2 L$_{Edd}$, are those observed from Cyg X-1, which also does not reach very soft spectral hardnesses (see \citealt{Wilms,belloni2009}). Similarly, we found that the luminosity of the softest state of H 1743-322 outburst (spectrum (d) in Table~\ref{spec_res}), computed in unit of Eddington luminosity, is L$\sim$0.1 L$_{Edd}$ ( considering a mass of 10 M$_{\odot}$ and
a distance of 10 kpc as estimated for H 1743-322 by \citealt{Mc}).
Although the mass accretion rate is a very important parameter involved in the transition, another parameter seems to prime the transition out from the hard state. This second parameter, whose nature is still not clear~\citep{Es,Homan01}, drove the October 2008 transition from the LHS to HIMS. Then probably an accretion rate decrease did not permit to continue the canonical transition pattern to the HSS. This means that the source did not pass the jet-line in the HID (see \citealt{hombel}) and probably there was not any jet major ejection ~\citep{fbg2004}. This despite the fact that the inner disk radius was seen to decrease and according to the ~\citet{fbg2004} model the acceleration of the jet to higher Lorentz factor had already started.
The results reported in this paper demonstrate that the processes starting at the beginning of the outburst can be reversible even after the transition to HIMS.
Other cases previously presented in literature as failed outbursts of transient X-ray binaries, are LHS-only outbursts without any sign of state transitions at all (see e.g. \citealt{broc} and \citealt{Stu}). Conversely, the data presented here show that the full pattern (LHS, HIMS, SIMS, HSS) and LHS-only pattern are not the only two possibilities for the temporal evolution of a BHC outburst.
The October 2008 outburst of H 1743-322, showing only LHS and HIMS, takes place at low luminosity (L$\sim$0.1 L$_{Edd}$) and the lack of soft-state transitions is probably connected to a premature decrease of the mass accretion rate. Again, this bring us back to one of the major problems for the interpretation of the spectra/timing evolution of the outbursts: what physical parameter determines the transitions starting from the low hard state.
\section*{Acknowledgements}
This work has been supported by the Italian Space Agency through grants I/008/07/0 and I/088/06/0.
TMB acknowledges support from the International Space Science Institute.
We acknowledge the use of public data from the {\it Swift } data archive and all the {\it Swift } team for its support. Our particular thanks goes to Dr. J. M. Miller and his colleagues who immediately returned to the scientific community their INTEGRAL ToO data.
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\section*{Acknowledgement}
We thank Nikhil Rao for providing the code for block signal recovery with the Latent Group Lasso approach.
\bibliographystyle{IEEEtran}
\section*{Appendix}
\appendices
\section{Dynamical programming for solving \eqref{eq:GWMC} for loopless pairwise overlapping groups}
\label{sec:dp_lp}
Here, we give the proof of Prop.~\ref{prop:DP2}.
The proof of Prop.~\ref{prop:DP1} follows along similar lines.
\begin{proof}
The proof consists in describing the algorithm and showing it is polynomial time.
\eqref{eq:GWMC} can be equivalently described by the following problem: given a signal ${\bf x} \in \mathbb R^N$ and a group structure $\GG$ consisting of $M$ groups defined over the index set $\mathcal{N} = \{1, \ldots, N\}$, with each index having an associated (non-negative) weight (i.e., $x_i^2,~\forall~i \in \mathcal{N}$), find the optimal selection of at most $K$ indices, to maximize the sum of their weights, such that the indices are contained in a union of at most $G$ groups.
We highlight the optimal substructure inherent in this problem, which allows us to solve it using a dynamic programming approach.
The optimal substructure is somewhat involved: we represent it below by two properties.
\begin{enumerate}
\item Suppose we know the $G$ groups that constitute the optimal solution.
Then the optimal choice of elements corresponds to picking the $K$ largest weight elements belonging to the union of the $G$ groups.
\item Suppose we know the groups and elements selected in the optimal solution under a $G$-groups and $K$-elements constraint.
Partition the set of chosen groups into two sets, $\mathcal{S}_1$ and $\mathcal{S}_2$, consisting of $g_1$ and $g_2$ groups respectively ($g_1 + g_2 = G$).
Suppose $\mathcal{S}_1$ contains $k_1$ of the elements in the optimal solution, and suppose $\mathcal{S}_2$ contains additional $k_2$ elements {\em excluding elements covered by} $\mathcal{S}_1$ ($k_1 + k_2 = K$).
Then, given the selection of groups and elements in $\mathcal{S}_1$, $\mathcal{S}_2$ represents the optimal selection of at most $k_2$ elements from at most $g_2$ groups in $\mathcal{S}_1^c$ (i.e. $\GG \setminus \mathcal{S}_1$), after the elements in $\mathcal{S}_1$ have been removed from the groups in $\mathcal{S}_1^c$.
\end{enumerate}
These properties lead us to a dynamic programming based method for obtaining the solution. The underlying graph has as nodes groups in $\GG$.
The algorithm gradually explores every node in the group-graph, storing the optimal solution among the visited nodes and it is defined by two rules: the {\bf Node Picking Rule} controls how the graph must be explored in order to minimize the number of values to store; the {\bf Value Update Rule} describes how the stored values are updated when a new node is considered.
Due to the looplessness constraint, the graph can be represented as a tree or a forest.
Choose an arbitrary node and call it the root node.
Let $\GG$ be the set of all nodes and let $\mathcal{S} \subseteq \GG$ be the set of currently explored nodes with $|\mathcal{S}| = m$.
Define $3$-valued indicator variables, $I_1, I_2, \ldots, I_M$ for each of the $M$ nodes.
$I_j = 1$ indicates that the $j$th node is selected, $I_j = 0$ shows that it is forbidden, while $I_j = 2$ represents a ``don't care'' state: there is no restriction on the $j$th node being either chosen or not.
For unexplored nodes, the indicator variables are always in the ``don't care'' state.
At every step of the algorithm, we store the optimal value for choosing at most $k$ elements, from at most $g$ nodes, $1 \leq k \leq K$ and $1 \leq g \leq G$ from the currently explored set of nodes, $\mathcal{S}$.
These optimal values are stored in the function, $F(\mathcal{S}, g, k, I_1, I_2, \ldots, I_M)$.
We define an additional function $H(\mathcal{S},k)$ to represent the optimal value for choosing $k$ elements from a set $\mathcal{S}$.
The set $\mathcal{S}$ could be a single group, a union of groups, or any well-defined collection of elements.
As noted earlier, the optimal selection corresponds to simply picking the $k$ largest weight elements in $\mathcal{S}$.
Our aim is to obtain the value: $F(\GG , G, K, I_1=2, I_2=2, \ldots, I_M=2)$.
All indicator variables are set to $2$, as we do not care about any particular group being selected or rejected in the final selection.
Notice that by definition,
\begin{align}
F(\mathcal{S}, g, k, I_1=i_1,\ldots, I_j=2, \ldots, I_M=i_M) = \max\lbrace &F(\mathcal{S}, g, k, I_1=i_1,\ldots, I_j=0, \ldots, I_M=i_M), \nonumber \\ &F(\mathcal{S}, g, k, I_1=i_1,\ldots, I_j=1, \ldots, I_M=i_M)\rbrace, \nonumber
\end{align} i.e., the optimal value when we do not care about a particular group being selected, is simply the maximum over the two cases of the group being selected and being rejected, \emph{ceteris paribus}.
Note that the function $F$ has an input space which is exponential in $M$ (since the indicator variables combined can take exponentially many values).
Therefore, if we tried to determine the values of $F$ at all possible points, we would need an exponential amount of space and time.
However, we shall see that our algorithm needs to work with only a small set of values of $F$, and hence runs in polynomial time.
This happens because the values of the indicator variables will be important only for certain specific nodes, called boundary nodes.
We define a {\bf boundary node} as an explored node adjacent to an unexplored node.
Hence the groups defined by the boundary node and the adjacent unexplored node overlap.
{\em Base Case.} We start by taking $\mathcal{S} = \emptyset$. For this case, all values of $F$ will be set to $0$:
$F(\emptyset , g, k, I_1=i_1, I_2=i_2,\ldots, I_M=i_M) = 0~\forall g, k, i_1, \ldots, i_M$.
Assume that we have ordered the nodes from $1$ to $M$ according to some criteria.
We shall now explore the nodes in this order and use the following value-update rules.
{\bf Value Update Rule.}
Suppose at a particular step, we have explored the first $m$ nodes.
We assume that we have stored the values of $F$ for each $g$ and $k$.
Further, we assume that we have stored the values of $F$ separately for each value of the indicator variable for each boundary nodes.
Denote by $\mathcal{S}_i$ the set of the first $i$ nodes.
Denote by $\mathcal{B}_m$ the set of boundary nodes when $m$ nodes have been explored.
Denote by $\mathcal{X}_m$ the $m$th group.
We assume that the following values are available at this step:
$F(\mathcal{S}_m,g,k,I_1 = i_1, \ldots, I_M = i_M)$ for all $1 \leq g \leq G$ and $1 \leq k \leq K$ and all $i_1, \ldots, i_M$ such that $i_j \in \{0,1\}$ for $j \in \mathcal{B}_m$ and $i_j = 2$ for $j \in \{1, \ldots, M\} \setminus \mathcal{B}_m$. Note that the indicator variables for all non-boundary nodes, as well as the unexplored nodes are set equal to $2$.
Thus the total number of values that have to be stored equals $GK2^{|\mathcal{B}_M|}$
The value update rule is divided into $3$ cases and a final condensation step.
\begin{enumerate}
\item {\em The new node is rejected}.
All optimal values for all $k$ and all $g$ remain the same as for $m$ nodes.
The added node is treated as a new boundary node and the stored values are associated to the new node being rejected.
\begin{align*}
& F(\mathcal{S}_{m+1}, g, k, I_1 = i_1, \ldots, I_m = i_m, {\bf I_{m+1} = 0}, I_{m+2} = 2, \ldots, I_M = 2)\\
& = F(\mathcal{S}_{m}, g, k, I_1 = i_1, \ldots, I_m = i_m, {\bf I_{m+1} = 2}, \ldots, I_M = 2)
\end{align*}
for all $1 \leq g \leq G$ and $1 \leq k \leq K$ and all $i_1, \ldots, i_M$ such that $i_j \in \{0,1\}$ for $j \in \mathcal{B}_{m}$ and $i_j = 2$ for $j \in \{1, \ldots, m \} \setminus \mathcal{B}_{m}$.
\item {\em The new node is accepted (no overlap with any explored node)}.
Since the new node is selected, we can choose at most $g-1$ explored nodes.
We first compute the sum of the optimal value for choosing the best $\ell$ elements from the new node and the optimal value for choosing $k - \ell$ elements from the $g-1$ explored nodes, for any $\ell$ such that $1 \leq \ell \leq k$.
Then, the new optimal value for each $g$ and $k$ is given by taking the maximum of these sums over $\ell$.
\begin{align*}
& F(\mathcal{S}_{m+1}, g, k, I_1 = i_1, \ldots, I_m = i_m, {\bf I_{m+1} = 1}, I_{m+2} = 2, \ldots, I_M = 2)\\
& = \max_{1 \leq \ell \leq k} \left\{ F(\mathcal{S}_{m}, {\bf g-1}, \boldsymbol{ k-\ell}, I_1 = i_1, \ldots, I_m = i_m, {\bf I_{m+1} = 0}, \ldots, I_M = 2) + H(\mathcal{X}_{m+1},\ell) \right\}
\end{align*}
for all $1 \leq g \leq G$ and $1 \leq k \leq K$ and all $i_1, \ldots, i_M$ such that $i_j \in \{0,1\}$ for $j \in \mathcal{B}_{m}$ and $i_j = 2$ for $j \in \{1, \ldots, m\} \setminus \mathcal{B}_{m}$.
\item {\em The new node is accepted (overlaps with some explored nodes)}.
The update rule is the same as for case 2, but the elements in the region of overlap between the new node and the selected explored nodes must not be considered as being part of the new node.
In other words, the new node must be `cleaned' of the overlapping region before updating.
For this step, we need to know exactly which nodes have been chosen while computing an optimal value.
This is the reason why we need to store separate values for each boundary node.
We further assume that the cleaning operation can be done in $O(1)$ time, leading to a total complexity of $O(GK^2 2^{|\mathcal{B}_{m}|})$.
\begin{align*}
& F(\mathcal{S}_{m+1}, g, k, I_1 = i_1, \ldots, I_m = i_m, {\bf I_{m+1} = 1}, I_{m+2} = 2, \ldots, I_M = 2)\\
& = \max_{1 \leq \ell \leq k} \left\{ F(\mathcal{S}_{m}, {\bf g-1}, \boldsymbol{ k-\ell}, I_1 = i_1, \ldots, I_m = i_m, {\bf I_{m+1} = 0}, \ldots, I_M = 2) + H(\mathcal{X}'_{m+1},\ell) \right \}
\end{align*}
for all $1 \leq g \leq G$ and $1 \leq k \leq K$ and all $i_1, \ldots, i_M$ such that $i_j \in \{0,1\}$ for $j \in \mathcal{B}_{m}$ and $i_j = 2$ for $j \in \{1, \ldots, m\} \setminus \mathcal{B}_{m}$. We also define $\mathcal{X}'_{m+1} = \mathcal{X}_{m+1} \setminus \bigcup_{j \in \mathcal{B}} \mathcal{X}_j$, with $\mathcal{B} = \{j \in \mathcal{B}_{m} : i_j = 1\}$. That is we ``clean'' $\mathcal{X}_{m}$ of the overlap with the currently selected boundary nodes.
\item {\em Condensation.}
After these steps, the number of stored values will be (at most) doubled.
We can reduce them: for each boundary node which has fallen into the interior of the explored nodes, we combine the optimal values for it being picked or unpicked, into a single value by taking the larger of the two values.
Each such operation reduces the number of stored values by half and we perform it after each value update
\begin{align*}
F(\mathcal{S}_{m+1}, g, k, I_1 = i_1, \ldots, {\bf I_j = 2}, \ldots, I_M = i_M) = \max \lbrace &F(\mathcal{S}_{m}, g,k, I_1 = i_1, \ldots, {\bf I_j = 0}, \ldots, I_M = i_M), \nonumber \\ &F(\mathcal{S}_{m}, g,k, I_1 = i_1, \ldots, {\bf I_j = 1}, \ldots, I_M = i_M) \rbrace
\end{align*}
for all $j \in (\mathcal{B}_{m} \cup X_{m+1}) \setminus \mathcal{B}_{m+1}$ and for all $1 \leq g \leq G$ and $1 \leq k \leq K$ and for all $i_1, \ldots, i_M$.
\end{enumerate}
{\bf Time Complexity.}
Let $B$ be the maximum number of boundary nodes encountered by the algorithm,
then the number of steps is bounded by $O(2^BK^2GM)$.
We now give an algorithm to explore the graph so that $B$ is logarithmic in $M$, establishing polynomial complexity.
{\bf Node Picking Rule.}
In order to minimize the number of boundary nodes encountered by the algorithm, we must explore the graph in a particular fashion.
The order with which the nodes are picked is determined by a value associated to each subtree of the graph, which we call the $D$-value.
In the following we describe how it is computed, how it depends logarithmically on the number of nodes in the graph and how the number of boundary nodes is bounded by the $D$-value.
The Node Rule Picking rule is defined as follows.
We first order all rooted subtrees with respect to the the $D$-value, so that $D_1 \geq \ldots \geq D_R$ for subtrees $T_1, T_2, \ldots, T_R$.
We then pick the subtrees in the order $\{T_1,\text{root}, T_2, \ldots, T_R\}$ and recurse until the explored subtree has only one node, see Fig.~\ref{fig:node_pick}.
\begin{figure}
\centering
\includegraphics[width=0.7\textwidth]{node_picking_rule.pdf}
\caption{\label{fig:node_pick} Node Picking Rule: explore nodes in the order $\mathcal{T}_1, \text{root}, \mathcal{T}_2, \mathcal{T}_3$ where $D_1 \geq D_2 \geq D_3$. For the subtree $\mathcal{T}_1$, the node connected to $\texttt{root}$ should be considered the root of $\mathcal{T}_1$, which we denote by $R_1$; similarly for the other subtrees.}
\end{figure}
The procedure for computing the $D$-values is also recursive.
If the tree has only one node, $D = 1$.
Now, assume the subtrees at a node $Q$ have values $D_1 \geq \ldots \geq D_R$.
Then, $D(Q) = \max( D_1, D_2+1 )$.
In case there is no $2^\text{nd}$ subtree, $D_2 = 0$.
We then have the following bound on the $D$-values.
\begin{lemma}
\label{lem:dvalue}
The $D$-value of a rooted tree graph is logarithmic in the number of nodes, i.e. $D(G) \leq \log_2(M) +1$.
\end{lemma}
\begin{proof}
Let $D$ be a positive integer and $N(D)$ be the minimum number of nodes that a rooted tree must have in order to have $D$-value of D.
We prove by induction that
\begin{equation}
\label{eq:dvalue}
N(D) \geq 2^{D-1}.
\end{equation}
{\em Base case: $D = 1$}. A tree with only one node will have a $D$-value of 1. Hence \eqref{eq:dvalue} is satisfied.
{\em Inductive case: $D > 0$}. Let $\mathcal{T}$ be the smallest rooted tree graph whose $D$-value is equal to $D$.
Spread out $\mathcal{T}$ in the form of root and subtrees.
Let the subtrees be $\mathcal{T}_1, \mathcal{T}_2, \ldots, \mathcal{T}_k$; with corresponding $D$-values $D_1, D_2, \ldots, D_k$.
Without loss of generality, assume that $D_1 \geq D_2 \geq \ldots \geq D_k$.
By definition, $D(\mathcal{T}) = \max(D_1, D_2+1)$.
By our assumption, $\mathcal{T}$ is the smallest graph with $D$-value equal to $D$, hence we cannot have $D_1 = D(G) = D$, since that would give us a smaller rooted tree graph ($\mathcal{T}_1$) with a $D$-value of $D$.
This means that $D_1 < D$, and hence $D_2 + 1 = D$, i.e. $D_2 = D - 1$.
Since $D_1 \geq D_2= D-1$ and $D_1 < D$, then $D_1 = D_2 = D-1$.
Thus, the graph $\mathcal{T}$ has 2 subtrees ($\mathcal{T}_1$ and $\mathcal{T}_2$), with $D$-values of $D-1$ each.
By definition, any rooted subtree with a $D$-value of $D-1$ must have at least $N(D-1)$ nodes.
By our induction hypothesis, $N(D-1) \geq 2^{D-2}$ .
Therefore, $\mathcal{T}$ has at least $2\times 2^{D-2} = 2^{D-1}$ nodes.
But since $\mathcal{T}$ was the smallest rooted tree graph with $D$-value of $D$, this means that $N(D) \geq 2^{D-1}$, as required.
\end{proof}
We now link the number of boundary nodes visited by the algorithm to the $D$-value of the group graph.
\begin{lemma}
\label{lem:boundary}
The total number of boundary nodes encountered by the node-picking algorithm cannot exceed the $D$-value of the graph.
\end{lemma}
\begin{proof}
Let $\mathcal{T}$ be the given rooted tree graph, with $M$ nodes.
We shall consider the number of boundary nodes when there is a {\em ghost node} connected to the root node.
This implies that the root, when explored, will always be counted as a boundary node.
The ghost node captures the fact when we are running the algorithm recursively on a subtree, there will be an additional (potentially unexplored) node connected to the root of the subtree, which may lead to the root being counted as a boundary node.
Let $B^*(\mathcal{T})$ denote the maximum number of boundary nodes encountered on $\mathcal{T}$ when we pick nodes according to our algorithm, and let $B^*_G(\mathcal{T})$ represent the same when we also have the ghost node.
Clearly, $B^*_G(\mathcal{T}) \geq B^*(\mathcal{T})$, hence it is enough to prove the following:
\begin{equation}
\label{eq:boundary}
B_G^*(\mathcal{T}) \leq D(\mathcal{T}).
\end{equation}
We prove this by induction on $M$.
{\em Base Case.} Suppose the rooted tree graph $\mathcal{T}$ has only $1$ node.
Then the maximum number of boundary nodes encountered is obviously $1$, which is equal to the $D$-value of the graph (by definition).
Hence $B^*_G(\mathcal{T}) \leq D(\mathcal{T})$.
{\em Inductive Case.} When the graph $\mathcal{T}$ consists of $M$ nodes, $M > 1$, consider the graph to be spread out in the form of root and subtrees.
Compute the $D$-values for each subtree, where w.l.o.g., $D_1 \geq D_2 \geq \ldots D_k$.
Let $\mathcal{T}_1, \mathcal{T}_2, \ldots, \mathcal{T}_k$ be the corresponding subtrees.
Our algorithm explores nodes in the sequence: $\mathcal{T}_1, \text{root}, \mathcal{T}_2, \mathcal{T}_3, \ldots \mathcal{T}_k$.
Since each subtree has strictly fewer than $M$ nodes, each subtree satisfies \eqref{eq:boundary} by the induction hypothesis.
Also, notice that when exploring the subtree $\mathcal{T}_1$ of $\mathcal{T}$, the number of boundary nodes encountered is less than or equal to the number of boundary nodes encountered when exploring $\mathcal{T}_1$ as a standalone rooted-tree-graph, with a ghost node connected to its root.
By construction, this is exactly equal to $B^*_G(\mathcal{T}_1)$, which by our induction hypothesis is bounded by $D_1$. Therefore, the number of boundary nodes encountered while exploring $\mathcal{T}_1$ in $\mathcal{T}$ cannot exceed $D_1$. Once we are finished with $\mathcal{T}_1$, we pick the root, so the total number of boundary nodes is $1$.
We now proceed to pick $\mathcal{T}_2$.
By a similar argument, the maximum number of boundary nodes in $\mathcal{T}_2$ at any point cannot exceed the number of boundary nodes encountered while exploring $\mathcal{T}_2$ as a standalone graph with attached ghost node. In addition, the root of $\mathcal{T}$ can contribute at most 1 additional boundary node (In fact, the ghost node for $\mathcal{T}$ ensures that the root, once picked, will always contribute an additional boundary node).
Therefore, the total number of boundary nodes in $\mathcal{T}$ while exploring $\mathcal{T}_2$ is at most $D_2+1$.
Similar arguments hold for all other subtrees --- the maximum number of boundary nodes while exploring the $k$-th subtree will be at most $D_k+1$, which is at most $D_2+1$.
Therefore, the maximum number of boundary nodes encountered at any step while exploring $\mathcal{T}$ is
$B_G^*(\mathcal{T}) \leq \max(D_1,D_2+1)$.
By definition, $D(\mathcal{T}) = \max(D_1,D_2+1)$.
Therefore $B_G^*(\mathcal{T}) \leq D(\mathcal{T})$.
\end{proof}
Combining Lemmas \ref{lem:dvalue} and \ref{lem:boundary}, we have the following result.
\begin{prop}
\label{prop:boundary}
The maximum number of boundary nodes at any step of the algorithm is logarithmic in the number of nodes, i.e., $B \leq \log_2(M) + 1$.
\end{prop}
The previous proposition establishes the polynomial time complexity of the dynamic program for solving the generalized integer problem \eqref{eq:GWMC}.
\begin{theorem}
The proposed dynamic program solves, in polynomial time, the problem of Weighted Maximum Cover with an additional constraint on element sparsity for loopless pairwise overlapping groups.
In particular, its time complexity is $O(M^2GK^2 )$, where $M$ is the number of groups, $G$ is the group sparsity constraint and $K$ is the element sparsity budget.
\end{theorem}
\end{proof}
\section{Dynamical programming for solving the hierarchical signal approximation problem \eqref{eq:hier}}
\label{sec:dp_hier}
Here, we give the proof of Prop.~\ref{prop:DP_hier}.
We start describing the dynamic program and then prove that its running time is polynomial.
\begin{proof}
Problem \eqref{eq:hier} can be equivalently rephrased as the following optimization problem.
Given a rooted tree $\mathcal{T}$ with each node having at most $D$ children, a non-negative real number (weight) assigned to every node and a positive integer $K$, choose a subset of its nodes forming a rooted-connected subtree that maximizes the sum of weights of the chosen elements, such that the number of selected nodes does not exceed $K$.
In our case, \eqref{eq:hier}, the weight of a node is the square of the value of the component of the signal associated to that node.
The proposed algorithm leverages the optimal substructure of the problem.
{\bf Nested Sub-problems.} Suppose that a particular node X belongs to the optimal $K$-node rooted-connected subtree.
Consider the subtree $\mathcal{T}_{X,d}$ obtained by choosing X, $d$ of its children ($1\leq d \leq D$) and all descendants of these children.
Consider the set of nodes $\mathcal{S}$ consisting of all the nodes of $\mathcal{T}_{X,d}$ which are also present in the optimal $K$-node rooted-connected subtree.
Suppose there are $L$ nodes in $\mathcal{S}$.
Then the nodes in $\mathcal{S}$ form the optimal $L$-node rooted-connected subtree at X, for the subgraph $\mathcal{T}_{X,d}$.
See Fig.~\ref{fig:DP_hier_subproblem} for an example.
\begin{figure}
\centering
\includegraphics[width=0.7\textwidth]{DP_hier_subproblem}
\caption{Example of a nested subproblem in hierarchical groups model}
\label{fig:DP_hier_subproblem}
\end{figure}
{\bf Dynamic Programming method.} For every node X, we store the weight of the optimal $k$-node rooted-connected subtree at X, using only the nodes in the $d$ rightmost children of X and their descendants, for each $k$ and $d$ such that $1 \leq k \leq K$ and $1 \leq d \leq D$.
We define a function $F(X,k,d)$, to store these optimal values.
We start from the leaf nodes and move upwards, for each node assessing all its subtrees from right to left, eventually covering the entire tree.
At the end, the optimal value will be given by $F(\text{root},K,D)$, that is the value of the best K-node rooted connected subtree of the root considering all its descendants.
{\it Base Case.} For every leaf node X and for all $1\leq k \leq K$ and $ 1\leq d \leq D$, we set $F(X,k,d) = \text{Weight}(X)$.
{\it Inductive Case.} By induction, for every non-leaf node X, all the F-values are known for the descendants of X.
Let $X_1, X_2, \ldots X_d$ be the $d$ children of X in the right-to-left order, where $1 \leq d \leq D$.
Then, we compute the F-values of X using the following value update rules.
{\bf Value Update Rules.}
\begin{enumerate}
\item For all $1\leq k \leq K$
$$
F(X,k,1) = \text{Weight}(X)+ F(X_1,k-1,D) \; .
$$
The optimal value for choosing a $k$-node subtree rooted at $X$, when only the rightmost child $X_1$ is allowed, equals the weight of $X$ itself (since $X$ must be chosen), plus the optimal value for choosing a rooted connected subtree with $k-1$ nodes from the rightmost child $X_1$.
\item For all $1 \leq k \leq K$ and $1 \leq i \leq d$
$$
F(X,k,i) = \max_{1 \leq \ell \leq k} \left \{ F(X,\ell,i-1) + F(X_d ,k-\ell,D) \right \} \; .
$$
For choosing the best $k$-node rooted connected subtree from the rightmost $d$ children, choose a positive integer $\ell \leq k$, pick the best $\ell$-node subtree at $X$ by including the rightmost $d-1$ children and pick the remaining $k-\ell$ nodes from the subtree of the $d$th child.
We then take the maximum over all $\ell$, $1 \leq \ell \leq k$ (since at least $1$ node must be chosen from the rightmost $d-1$ nodes, this node will be the root).
\item For all $1\leq k \leq K$ and $d \leq i \leq D$
$$
F(X,k,i)=F(X,k,d) \; .
$$
For convenience, when a node has only $d$ children, where $d$ is strictly less than $D$, we set F-values for cases involving more than $d$ children equal to the value for $d$ children.
\end{enumerate}
\iffalse
The algorithm leverages the optimal substructure of the problem:
given a node in the optimal $K$-node rooted and connected subtree, $K'$ of its descendants will also be present in the optimal selection.
Therefore, this subtree at that node will itself be an optimal rooted and connected subtree consisting of $K' < K$ nodes.
We denote by $D$ the maximum number of subtrees that any node in the tree can have.
The algorithm explores every node in the tree, starting from the leaves, storing the best possible value for choosing a $J$-node rooted and connected subtree, for each $J$ from $1$ to $K$.
For the leaf nodes, the best value for choosing a $J$-node tree is the weight of that node, because we are interesting in finding the rooted and connected subtree with {\em at most} $K$ nodes.
We then consider a non-leaf node.
We need to use the optimal values computed for its subtrees to obtain the optimal value for choosing a $J$-node subtree at the current node, for each $J$ from $1$ to $K$.
Clearly, when choosing a rooted subtree at a particular node, we must choose the node itself, hence we need to distribute the remaining $J-1$ nodes among its subtrees (which are at most $D$).
Suppose all these $J-1$ nodes were chosen from the rightmost child's subtree, then the optimal values obtained are simply the values stored in that node for picking a $J-1$ node subtree, for $J$ from $1$ to $K$.
We now extend these optimal values to the other subtrees on the left, one subtree at a time.
Thus the optimal value for choosing $J-1$ nodes from the rightmost two subtrees equals the sum of the optimal values for choosing an $L$-node subtree from the rightmost child and a $J-L-1$-node subtree from the second rightmost child, maximized over all $L$ from $0$ to $J-1$.
We repeat the procedure considering one subtree at a time and re-using the values computed for the current selection of children.
\fi
\begin{theorem}
The time complexity of the dynamic program for hierarchical structures is polynomial in the number of nodes.
\end{theorem}
\begin{proof}
Given the description of the algorithm above, we observe that we need to calculate at most $NDK$ F-values, and for calculating each value, we need to evaluate at most $K$ sums. Therefore the time required will be $O(NDK^2)$.
\end{proof}
\end{proof}
\section{Conclusions}
\label{sec:conclusions}
Many applications benefit from group sparse representations. Unfortunately, our result in this paper shows that finding a group-based interpretation of a signal is an integer optimization problem, which is in general NP-hard. To this end, we characterize group structures for which a dynamical programming algorithm can find a solution in polynomial time and also delineate discrete relaxations for special structures (i.e., totally unimodular constraints) that can obtain correct solutions in special circumstances.
Our examples and numerical simulations show the deficiencies of convex relaxations.
We observe that such methods only recover group-covers that lie in the convex hull of the Pareto frontier determined by the solutions of the original integer problem for different values of the group budget $G$ (and sparsity budget $K$ for the generalized model).
This, in turn, implies that convex and non-convex relaxations might miss some important groups or include spurious ones in the group-sparse model selection. We summarize our findings in Fig.~\ref{fig:diagram}.
\begin{figure}
\centering
\includegraphics[width=0.6\textwidth]{diagram_bw}
\caption{\label{fig:diagram} Characterization of tractability for group-based interpretations.}
\end{figure}
\section{Convex relaxations}
\label{sec:convex_relax}
For tractability and analysis, convex proxies to the group $\ell_0$-norm have been proposed (e.g., \cite{jacob2009group}) for finding group-sparse approximations of signals.
Given a group structure $\GG$, an example generalization is defined as
\begin{equation}
\label{eq:atomic_norm}
\|{\bf x}\|_{\GG,\{1,p\}} := \inf\limits_{{\scriptsize \begin{array}{c} \mathbf{v}^1, \ldots, \mathbf{v}^M \in \mathbb R^N\\\forall j, \supp(\mathbf{v}^j) = \G_j \end{array}}} \left \{ \sum_{j = 1}^M d_j \|\mathbf{v}^j\|_p: \sum_{j=1}^M \mathbf{v}^j = {\bf x} \right \} ,
\end{equation}
where $\|{\bf x}\|_p = \left ( \sum_{i=1}^N x_i^p \right )^{1/p}$ is the $\ell_p$-norm, and $d_j$ are positive weights that can be designed to favor certain groups over others \cite{obozinski2011group}.
This norm can be seen a weighted instance of the atomic norm described in \cite{rao2012signal}, where the authors leverage convex optimization for signal recovery, but not for model selection.
One can in general use \eqref{eq:atomic_norm} to find a group-sparse approximation under the chosen group norm
\begin{equation}
\label{eq:latent_gl}
\hat{\bf x} \in \mathrm{argmin}\limits_{{\bf z} \in \mathbb R^N} \left \{ \|{\bf x} - {\bf z}\|_2^2 : \|{\bf z}\|_{\GG,\{1,p\}} \leq \lambda\right \}, \;
\end{equation}
where $\lambda > 0$ controls the trade-off between approximation accuracy and group-sparsity.
However, solving \eqref{eq:latent_gl} does not yield a group-support for $\hat{\bf x}$:
even though we can recover one through the decomposition $\{ \mathbf{v}^j\}$ used to compute $\|\hat{\bf x}\|_{\GG, \{1,p\}}$, it may not be unique as observed in \cite{obozinski2011group} for $p = 2$.
In order to characterize the group-support for ${\bf x}$ induced by \eqref{eq:atomic_norm}, in \cite{obozinski2011group} the authors define two group-supports for $p = 2$.
The {\em strong group-support} $\breve{\mathcal{S}}({\bf x})$ contains the groups that constitute the supports of each decomposition used for computing \eqref{eq:atomic_norm}.
The {\em weak group-support} $\mathcal{S}({\bf x})$ is defined using a dual-characterisation of the group norm \eqref{eq:atomic_norm}.
If $\breve{\mathcal{S}}({\bf x}) = \mathcal{S}({\bf x})$, the group-support is uniquely defined.
However, \cite{obozinski2011group} observed that for some group structures and signals ${\bf x}$, even when $\breve{\mathcal{S}}({\bf x}) = \mathcal{S}({\bf x})$, the group-support does not capture the minimal group-cover of ${\bf x}$. Hence, the equivalence of $\ell_0$ and $\ell_1$ minimization \cite{donoho2006compressed, candes2006compressive} in the standard compressive sensing setting does not hold in the group-based sparsity setting.
\section{Case study: discrete vs.\ convex interpretability}
\label{sec:discrete_vs_convex}
The following stylized example illustrates situations that can potentially be encountered in practice.
In these cases, the group-support obtained by the convex relaxation will not coincide with the discrete definition of group-cover, while the dynamical programming algorithm of Prop.~\ref{prop:DP1} is able to recover the correct group-cover.
Let $\mathcal{N} = \{1, \ldots, 11\}$ and let $\GG = \{\G_1 = \{1, \ldots, 5\},~\G_2 = \{4, \ldots, 8\},~\G_3 = \{7, \ldots, 11\}\}$ be the loopless pairwise overlapping groups structure with $3$ groups of equal cardinality. Its group graph is represented in Fig.~\ref{fig:example}.
Consider the $2$-group sparse signal ${\bf x} = [0~0~1~1~1~0~1~1~1~0~0]^\top$, with minimal group-cover $\mathcal{M}({\bf x}) = \{\G_1, \G_3\}$.
\tikzstyle{place}=[circle,draw=black,fill=white,thick, minimum size=6pt, inner sep=0pt]
\begin{figure}
\centering
\begin{tikzpicture}[-,>=stealth',shorten >=1pt,auto,node distance=1.5cm, semithick]
\node[place] (n1) at (0,0) [label=below:$\G_1$] {}
\node[place] (n2) at (1.5,1) [label=above:$\G_2$] {}
\node[place] (n3) at (3,0) [label=below:$\G_3$] {}
\draw (n1) to node {$\{4,5\}$} (n2);
\draw (n2) to node {} (n1);
\draw (n2) to node {$\{7,8\}$} (n3);
\draw (n3) to node {} (n2);
\end{tikzpicture}
\caption{\label{fig:example} The group-graph for the example in Section~\ref{sec:discrete_vs_convex}}
\end{figure}
The dynamic program of Prop. \ref{prop:DP1}, with group budget $G = 2$, correctly identifies the groups $\G_1$ and $\G_3$.
The TU linear program \eqref{eq:PR}, with $0 < \lambda \leq 2$, also yields the correct group-cover.
Conversely, the decomposition obtained via \eqref{eq:atomic_norm} with unitary weights is unique, but is not group sparse. In fact, we have $\mathcal{S}({\bf x}) = \breve{\mathcal{S}}({\bf x}) = \GG$.
We can only obtain the correct group-cover if we use the weights $[1~d~1]$ with $d > \frac{2}{\sqrt{3}}$, that is knowing beforehand that $\G_2$ is irrelevant.
\begin{rem}
Indeed, if the convex relaxation always recovered the correct minimal group-cover, it would be possible to solve the discrete NP-hard problem in polynomial time.
\end{rem}
\section{\label{sec:basics} Basic Definitions}
Let $ {\bf x} \in \mathbb R^\dim$ be a vector and $\mathcal{N} = \{1, \ldots, \dim\}$ be the ground set of its indices.
We use $|\mathcal{S}|$ to denote the cardinality of an index set $\mathcal{S}$.
We use $\mathbb{B}^N$ to represent the space of $N$-dimensional binary vectors and define $\iota: \mathbb R^N \to \mathbb{B}^N$ to be the indicator function of the nonzero components of a vector in $\mathbb R^N$, i.e., $\iota({\bf x})_i = 1$ if $x_i \neq 0$ and $\iota({\bf x})_i = 0$, otherwise.
We let $\mathbf{1}_N$ to be the $N$-dimensional vector of all ones and $\mathbf{I}_N$ the $N \times N$ identity matrix.
The support of ${\bf x}$ is defined by the set-valued function $\supp({\bf x}) = \{i \in \mathcal{N} : x_i \neq 0 \}$.
Note that we use bold lowercase letters to indicate vectors and bold uppercase letters to indicate matrices.
\begin{definition}
A {\bf group structure} $\mathfrak{G} = \{\G_1, \ldots, \G_M\}$ is a collection of index sets, named {\em groups}, with $\G_j \subseteq \mathcal{N}$ and $|\G_j| = g_j$ for $ 1 \leq j \leq M$ and $\bigcup_{\G \in \mathfrak{G}} \G = \mathcal{N}$.
\end{definition}
We can represent a group structure $\GG$ as a bipartite graph, where on one side we have the $N$ variables nodes and on the other the $M$ group nodes.
An edge connects a variable node $i$ to a group node $j$ if $i \in \G_j$.
Fig.~\ref{fig:var_groups} shows an example.
The bi-adjacency matrix $\mathbf{A}^\mathfrak{G} \in \mathbb{B}^{N \times M}$ of the bipartite graph encodes the group structure,
$$
\label{eq:group_structure}
A^\mathfrak{G}_{ij} = \bigg \{ \begin{array}{lc} 1, & \text{if}~i \in \G_j; \\ 0, & \text{otherwise.} \end{array} \;
$$
\tikzstyle{vnode}=[circle,draw=black,fill=white,thick, minimum size=6pt, inner sep=0pt]
\tikzstyle{vnodeholder}=[circle,draw=black,fill=white,thick, minimum size=1pt, inner sep=0pt]
\tikzstyle{gnode}=[rectangle,draw=black,fill=white,thick, minimum size=6pt, inner sep=0pt]
\tikzstyle{gnodeholder}=[rectangle,draw=black,fill=white,thick, minimum size=1pt, inner sep=0pt]
\begin{figure}
\centering
\begin{tikzpicture}[-,>=stealth',shorten >=1pt,auto,node distance=1.5cm, semithick]
\node[vnode] (v1) at (0,0) [label=above:$1$] [label=left:variables] {};
\node[vnode] (v2) at (1,0) [label=above:$2$] {};
\node[vnode] (v3) at (2,0) [label=above:$3$] {};
\node[vnode] (v4) at (3,0) [label=above:$4$] {};
\node[vnode] (v5) at (4,0) [label=above:$5$] {};
\node[vnode] (v6) at (5,0) [label=above:$6$] {};
\node[vnode] (v7) at (6,0) [label=above:$7$] {};
\node[vnode] (v8) at (7,0) [label=above:$8$] {};
\node[gnode] (g1) at (0,-1.8) [label=below:$\G_1$] [label=left:groups] {};
\node[gnode] (g2) at (1.4,-1.8) [label=below:$\G_2$] {};
\node[gnode] (g3) at (2.8,-1.8) [label=below:$\G_3$] {};
\node[gnode] (g4) at (4.2,-1.8) [label=below:$\G_4$] {};
\node[gnode] (g5) at (5.6,-1.8) [label=below:$\G_5$] {};
\node[gnode] (g6) at (7,-1.8) [label=below:$\G_6$] {};
\draw (v1) to (g1);\draw (g1) to (v1);\draw (v2) to (g2);\draw (g2) to (v2);
\draw (v1) to (g3);\draw (v2) to (g3);\draw (v3) to (g3);\draw (v4) to (g3);\draw (v5) to (g3);\draw (g3) to (v1);\draw (g3) to (v2);\draw (g3) to (v3);\draw (g3) to (v4);\draw (g3) to (v5);
\draw (v3) to (g5); \draw (g5) to (v3);
\draw (v4) to (g4);\draw (v6) to (g4);\draw (g4) to (v4);\draw (g4) to (v6);
\draw (v5) to (g5);\draw (v7) to (g5);\draw (g5) to (v5);\draw (g5) to (v7);
\draw (v6) to (g6);\draw (v7) to (g6);\draw (v8) to (g6);\draw (g6) to (v6);\draw (g6) to (v7);\draw (g6) to (v8);
\end{tikzpicture}
\caption{\label{fig:var_groups} Example of bipartite graph between variables and groups induced by the group structure $\GG^1$, see text for details.
\end{figure}
Another useful representation of a group structure is via a {\em group graph} $(\mathcal{V}, \mathcal{E})$ where the nodes $\mathcal{V}$ are the groups $\G \in \mathfrak{G}$ and the edge set $\mathcal{E}$ contains $e_{ij}$ if $\G_i \cap \G_j \neq \emptyset$, that is an edge connects two groups that {\em overlap}.
A sequence of connected nodes $v_1, v_2, \ldots, v_n$, is a {\em loop} if $v_1 = v_n$.
In order to illustrate these concepts, consider the group structure $\GG^1$ defined by the following groups, $\G_1 = \{1\}$, $\G_2 = \{2\}$, $\G_3 = \{1, 2, 3, 4, 5\}$, $\G_4 = \{4,6\}$, $\G_5 = \{3, 5, 7\}$ and $\G_6 = \{6, 7, 8\}$.
$\GG^1$ can be represented by the variables-groups bipartite graph of Fig.~\ref{fig:var_groups} or by the group graph of Fig.~\ref{fig:bipartite}, which is bipartite and contains loops.
\tikzstyle{place_black}=[circle,draw=black,fill=black,thick, minimum size=6pt, inner sep=0pt]
\tikzstyle{place_red}=[circle,draw=black,fill=white,thick, minimum size=6pt, inner sep=0pt]
\begin{figure}
\centering
\begin{tikzpicture}[-,>=stealth',shorten >=1pt,auto,node distance=2.8cm, semithick]
\tikzstyle{every state}=[fill=blue!20,draw=none,text=black]
\node[place_black] (n1) at (0,0) [label=below:$\G_1$] {}
\node[place_black] (n2) at (0,1.5) [label=below:$\G_2$] {}
\node[place_red] (n3) at (2.25,0.75) [label=below:$\G_3$] {}
\node[place_black] (n4) at (4.5,0) [label=below:$\G_4$] {}
\node[place_black] (n5) at (4.5,1.5) [label=below:$\G_5$] {}
\node[place_red] (n6) at (6.75,0.75) [label=below:$\G_6$] {}
\path (n1) edge node {} (n3);
\path (n2) edge node {$\{2\}$} (n3);
\path (n3) edge node {} (n4);
\path (n3) edge node {$\{3, 5\}$} (n5);
\path (n4) edge node {} (n6);
\path (n5) edge node {$\{7\}$} (n6);
\path (n3) edge node {$\{1\}$} (n1);
\path (n3) edge node {} (n2);
\path (n4) edge node {$\{4\}$} (n3);
\path (n5) edge node {} (n3);
\path (n6) edge node {$\{6\}$} (n4);
\path (n6) edge node {} (n5);
\end{tikzpicture}
\caption{\label{fig:bipartite} Bipartite group graph with loops induced by the group structure $\GG^1$, where on each edge we report the elements of the intersection.}
\end{figure}
An important group structure is given by {\em loopless pairwise overlapping groups}.
This group structure consists of groups such that each element of the ground set occurs in at most two groups and the induced graph does not contain loops.
Therefore the group graph for these structures is actually a tree or a forest and hence bipartite.
For example, consider $\G_1 = \{1, 2, 3\}$, $\G_2 = \{3, 4, 5\}$, $\G_3 = \{5, 6, 7\}$, which can be represented by the graph in Fig.~\ref{fig:loopless}(Left).
If $\G_3$ were to include an element from $\G_1$, for example $\{2\}$, we would have the loopy graph of Fig.~\ref{fig:loopless}(Right).
Note that $\GG^1$ is pairwise overlapping, but not loopless, since $\G_3, \G_4, \G_5$ and $\G_6$ form a loop.
\tikzstyle{place}=[circle,draw=black,fill=white,thick, minimum size=6pt, inner sep=0pt]
\begin{figure}
\centering
\begin{tikzpicture}[-,>=stealth',shorten >=1pt,auto,node distance=1.5cm, semithick]
\node[place] (n1) at (0,0) [label=below:$\G_1$] {}
\node[place] (n2) at (1.5,1) [label=above:$\G_2$] {}
\node[place] (n3) at (3,0) [label=below:$\G_3$] {}
\draw (n1) to node {$\{3\}$} (n2);
\draw (n2) to node {} (n1);
\draw (n2) to node {$\{5\}$} (n3);
\draw (n3) to node {} (n2);
\node[place] (n4) at (4.5,0) [label=below:$\G_1$] {}
\node[place] (n5) at (6,1) [label=above:$\G_2$] {}
\node[place] (n6) at (7.5,0) [label=below:$\G_3$] {}
\draw (n4) to node {$\{3\}$} (n5);
\draw (n5) to node {$\{5\}$} (n6);
\draw (n6) to node [label=below:$\{2\}$] {} (n4);
\draw (n5) to node {} (n4);
\draw (n6) to node {} (n5);
\draw (n4) to node {} (n6);
\end{tikzpicture}
\caption{\label{fig:loopless}(Left) Loopless pairwise overlapping groups. (Right) By adding one element from $\G_1$ into $\G_3$, we introduce a loop in the graph.}
\end{figure}
We anchor our analysis of the tractability of interpretability via selection of groups on covering arguments.
\begin{definition}
A {\bf group cover} $\mathcal{S}({\bf x})$ for a signal ${\bf x} \in \mathbb R^N$ is a collection of groups such that $\supp({\bf x}) \subseteq \bigcup_{\G \in \mathcal{S}({\bf x})} \G$. An alternative equivalent definition is given by
$$
\mathcal{S}({\bf x}) = \{\G_j \in \mathfrak{G} : \boldsymbol{\omega} \in \mathbb{B}^M,~ \omega_j = 1,~ \mathbf{A}^\mathfrak{G}\boldsymbol{\omega} \geq \iota({\bf x})\} \; .
$$
\end{definition}
The binary vector $\boldsymbol{\omega}$ indicates which groups are active and the constraint $\mathbf{A}^\mathfrak{G}\boldsymbol\omega \geq \iota({\bf x})$ makes sure that, for every non-zero component of ${\bf x}$, there is at least one active group that covers it.
We also say that $\mathcal{S}({\bf x})$ {\em covers} ${\bf x}$.
Note that the group cover is often not unique and $\mathcal{S}({\bf x}) = \mathfrak{G}$ is a group cover for any signal ${\bf x}$. This observation leads us to consider more restrictive definitions of group cover.
\begin{definition}
A {\bf $G$-group cover} $\mathcal{S}^G({\bf x}) \subseteq \mathfrak{G}$ is a group cover for ${\bf x}$ with at most $G$ elements,
$$
\label{eq:G_cover}
\mathcal{S}^G({\bf x}) = \{\G_j \in \mathfrak{G} : \boldsymbol{\omega} \in \mathbb{B}^M,~\omega_j = 1,~\mathbf{A}^\mathfrak{G}\boldsymbol{\omega} \geq \iota({\bf x}),~\sum_{j=1}^M \omega_j \leq G\} \; .
$$
\end{definition}
It is not guaranteed that a $G$-group cover always exists for any value of $G$.
Finding the smallest $G$-group cover lead to the following definitions.
\begin{definition}
The {\bf group $\ell_0$-``norm"} is defined as
\begin{equation}
\label{eq:group_0_norm}
\|{\bf x}\|_{\GG,0} := \min\limits_{\boldsymbol{\omega} \in \mathbb{B}^M} \left \{ \sum_{j=1}^M \omega_j : \mathbf{A}^\mathfrak{G}\boldsymbol{\omega} \geq \iota({\bf x}) \right \} \; .
\end{equation}
\end{definition}
\begin{definition
A {\bf minimal group cover} for a signal ${\bf x} \in \mathbb R^N$ is defined as $\mathcal{M}({\bf x}) = \{\G_j \in \GG : \hat{\omega}({\bf x})_j =1\}$, where
$\hat{\boldsymbol{\omega}}$ is a minimizer for \eqref{eq:group_0_norm},
$$
\hat{\boldsymbol{\omega}}({\bf x}) \in \mathrm{argmin}\limits_{\boldsymbol{\omega} \in \mathbb{B}^M} \left \{ \sum_{j=1}^M \omega_j : \mathbf{A}^\mathfrak{G}\boldsymbol{\omega} \geq \iota({\bf x}) \right \} \; .
$$
\end{definition}
A {\em minimal group cover} $\mathcal{M}({\bf x})$ is a group cover for ${\bf x}$ with minimal cardinality.
Note that there exist pathological cases where for the same group $\ell_0$-``norm", we have different minimal group cover models.
\begin{definition}
A signal ${\bf x}$ is {\bf $G$-group sparse} with respect to a group structure $\mathfrak{G}$ if $\|{\bf x}\|_{\GG,0} \leq G$.
\end{definition}
In other words, a signal is {\em $G$-group sparse} if its support is contained in the union of at most $G$ groups from $\GG$ .
\section{\label{sec:exp}Pareto Frontier Example}
The purpose of this numerical simulation is to illustrate the limitations of relaxations for correctly estimating the $G$-group cover of an approximation.
We consider the problem of finding a $K$-sparse approximation of a signal imposing hierarchical constraints.
We generate a piecewise constant signal of length $N = 64$, to which we apply the Haar wavelet transformation, yielding a $25$-sparse vector of coefficients ${\bf x}$ that satisfies hierarchical constraints on a binary tree of depth $5$, see Fig.~\ref{fig:exp}(Left).
We compare the proposed dynamic program (DP) to the regularized totally unimodular linear program approach and two convex relaxations that use group-based norms.
The first \cite{rao2012signal} uses the Latent Group Lasso penalty \eqref{eq:latent_gl} with groups defined as all father-child relations in the tree.
This formulation will not enforce all hierarchical constraints to be satisfied, but will only `favor' them.
Therefore, we also report the number of hierarchical constraint violations.
The second\cite{jenatton2011proximal} considers a hierarchy of groups where $\G_j$ contains node $j$ and all its descendants.
Hierarchical constraints are enforced by the group lasso penalty $\Omega_{GL}({\bf x}) = \sum_{\G \in \GG} \|{\bf x}_\G\|_2$,
where ${\bf x}_\G$ is the restriction of ${\bf x}$ to $\G$. We call this method Hierarchical Group Lasso.
Once we determine the support of the solution, we assign to the components in the support the values of the corresponding components of the original signal.
In Fig.~\ref{fig:exp}(Right), we show the approximation error $\|{\bf x} - \hat{\bf x}\|_2^2$ as a function of the solution sparsity $K$ for the methods.
The values of the DP solutions form the discrete Pareto frontier of the optimization problem controlled by the parameter $K$. Note that there are points in the Pareto frontier that do not lie on its convex hull, hence these solutions are not achievable by the TU linear relaxation.
We observe that the Hierarchical Group Lasso\footnote{We used the code provided in \url{http://spams-devel.gforge.inria.fr/.}} also misses the solutions for $K = 21$ and $K=23$, while the Latent Group Lasso\footnote{We used the duplication of variables approach and solved the resulting Group Lasso problem using SpaRSA: \url{http://www.lx.it.pt/~mtf/SpaRSA/}} approach achieves more levels of sparsity (but still missing the solutions for $K=2, 13$ and $15$), although at the price of violating some of the hierarchical constraints.
These observations lead us to conclude that, in some cases, relaxations of the original discrete problem might not be able to find the correct group-based interpretation of a signal.
\begin{figure}
\begin{tabular}{cc}
\includegraphics[width=0.5\textwidth]{block_signals} &
\includegraphics[width=0.5\textwidth]{hierarchical_pareto}
\end{tabular}
\caption{\label{fig:exp} (Left) (a) Original piecewise constant signal. (b) Haar wavelet representation. (Right) Signal approximation on a binary tree. The original signal is $25$-sparse and satisfies hierarchical constraints. The numbers next to the Latent Group Lasso solutions indicate the number of constraint violations.}
\end{figure}
\section{Generalizations}
\label{sec:generalizations}
In this section, we first present a generalization of the discrete approximation problem \eqref{eq:WMC} by introducing an additional overall sparsity constraint.
Secondly, we show how this generalization encompasses approximation with hierarchical constraints that can be solved exactly via dynamic programming.
Finally, we show that the generalized problem can be relaxed into a linear binary problem and that hierarchical constraints lead to totally unimodular matrices for which there exists efficient polynomial time solvers.
\subsection{Sparsity within groups}
In many applications, for example genome-wide association studies \cite{zhou2010association}, it is desirable to find approximations that are not only group-sparse, but also sparse in the usual sense (see \cite{simon2012sparse} for an extension of the group lasso).
To this end, we generalize our original problem \eqref{eq:WMC} by introducing a sparsity constraint $K$ and allowing to individually select variables within a group.
The generalized integer problem then becomes
\begin{equation}
\label{eq:GWMC}
\max\limits_{\boldsymbol{\omega} \in \mathbb{B}^M,~{\bf y} \in \mathbb{B}^\dim} \left \{ \sum_{i=1}^N y_i x_i^2 : \mathbf{A}^\mathfrak{G} \boldsymbol{\omega} \geq {\bf y}, \sum_{i=1}^\dim y_i \leq K , \sum_{j=1}^M \omega_j \leq G \right \} \; .
\end{equation}
This problem is in general NP-hard too, but it turns out that it can be solved in polynomial time for the same group structures that allow to solve \eqref{eq:WMC}.
\begin{prop}
\label{prop:DP2}
Given a loopless pairwise overlapping groups structure $\mathfrak{G}$, there exists a polynomial time dynamic programming algorithm that solves \eqref{eq:GWMC}.
\end{prop}
\begin{proof}
The dynamic program is described in Appendix \ref{sec:dp_lp} alongside the proof that it has a polynomial running time.
\end{proof}
\subsection{Hierarchical constraints}
The generalized model allows to deal with hierarchical structures, such as regular trees, frequently encountered in image processing (e.g. denoising using wavelet trees).
In such cases, we often require to find $K$-sparse approximations such that the selected variables form a rooted connected subtree of the original tree, see Fig.~\ref{fig:hier}.
Given a tree $\mathcal{T}$, the rooted-connected approximation can be cast as the solution of the following discrete problem
\begin{equation}
\label{eq:hier}
\max_{{\bf y} \in \BB^\dim} \left \{ \sum_{i=1}^N y_ix_i^2: {\bf y} \in \mathcal{T_K} \right \} \; ,
\end{equation}
where $\mathcal{T}_K$ denotes all rooted and connected subtrees of the given tree $\mathcal{T}$ with at most $K$ nodes.
\tikzstyle{notsel}=[circle,draw=black,fill=white,thick, minimum size=6pt, inner sep=0pt]
\tikzstyle{sel}=[circle,draw=black,fill=black,thick, minimum size=6pt, inner sep=0pt]
\begin{figure}
\centering
\begin{tabular}{cc}
\begin{tikzpicture}[level distance=10mm]
\tikzstyle{level 1}=[sibling distance=15mm]
\tikzstyle{level 2}=[sibling distance=10mm]
\tikzstyle{level 3}=[sibling distance=5mm]
\node[sel] {}
child {node[sel] {}
child{node[notsel] {}}
child{node[sel] {}
child{node[sel] {}}
child{node[sel] {}}}}
child {node[notsel] {}
child{node[notsel] {}}
child{node[notsel] {}}};
\end{tikzpicture}
& \hskip 1cm
\begin{tikzpicture}[level distance=10mm]
\tikzstyle{level 1}=[sibling distance=15mm]
\tikzstyle{level 2}=[sibling distance=10mm]
\tikzstyle{level 3}=[sibling distance=5mm]
\node[sel] {}
child {node[notsel] {}
child{node[notsel] {}}
child{node[sel] {}
child{node[sel] {}}
child{node[sel] {}}}}
child {node[notsel] {}
child{node[notsel] {}}
child{node[notsel] {}}};
\end{tikzpicture}
\end{tabular}
\caption{\label{fig:hier} Hierarchical constraints. Each node represent a variable. (Left) A valid selection of nodes. (Right) An {\em invalid} selection of nodes.}
\end{figure}
This type of constraint can be represented by a group structure, where for each node in the tree we define a group consisting of that node and all its ancestors.
When a group is selected, we require that all its elements are selected as well.
We impose an overall sparsity constraint $K$, while discarding the group constraint $G$.
For this particular problem, for which convex approximations have been proposed \cite{jenatton2011proximal}, we present an exact dynamic program that runs in polynomial time.
\begin{prop}
\label{prop:DP_hier}
Given a hierarchical group structure $\mathfrak{G}$, there exists a polynomial time dynamic programming algorithm that solves \eqref{eq:hier}.
\end{prop}
\begin{proof}
The description of the algorithm and the proof of its polynomial running time can be found in Appendix \ref{sec:dp_hier}.
\end{proof}
\subsection{Additional discrete relaxations}
By relaxing both the group budget and the sparsity budget in \eqref{eq:GWMC} into regularization terms, we obtain the following binary linear program
\begin{equation}
\label{eq:LP_hier}
(\boldsymbol{\omega}^\lambda, {\bf y}^\lambda) \in \argmax\limits_{\boldsymbol{\omega} \in \BB^M, {\bf y} \in \BB^\dim} \left \{ \mathbf{w}^\top \mathbf{u} : \mathbf{u}^\top = [{\bf y}^\top~\boldsymbol{\omega}^\top~{\bf y}^\top],~\mathbf{Cu} \leq 0 \right \}
\end{equation}
where $\mathbf{w}^\top = [x_1^2, \ldots, x_N^2,-\lambda_G\mathbf{1}_M^\top,-\lambda_K\mathbf{1}_N^\top]$ and $\mathbf{C} = [\mathbf{I}_N,~-\mathbf{A}^\GG,~\mathbf{0}_N]$ and $\lambda_G, \lambda_K > 0$ are two regularization parameters that indirectly control the number of active groups and the number of selected elements.
\eqref{eq:LP_hier} can be solved in polynomial time if the constraint matrix $\mathbf{C}$ is totally unimodular.
Due to its structure, $\mathbf{C}$ is totally unimodular if $\mathbf{A}^\GG$ is totally unimodular \cite{wolsey1999integer}.
The next results proves that the constraint matrix of hierarchical group structures is totally unimodular.
\begin{prop}
Hierarchical group structures lead to totally unimodular constraints.
\end{prop}
\begin{proof}
We use the fact that a binary matrix is totally unimodular if there exists a permutation of its columns such that in each row the $1$s appear consecutively \cite{wolsey1999integer}.
For hierarchical group structures, such permutation is given by a depth-first ordering of the groups.
In fact, a variable is included in the group that has it as the leaf and in all the groups that contain its descendants.
Given a depth-first ordering of the groups, the groups that contain the descendants of a given node will be consecutive.
\end{proof}
\section{Introduction}
\IEEEPARstart{I}{nformation} in many natural and man-made signals can be exactly represented or well approximated by a sparse set of nonzero coefficients in an appropriate basis \cite{mallat1999wavelet}. Compressive sensing (CS) exploits this fact to recover signals from their compressive samples, which are dimensionality reducing, non-adaptive random measurements. According to the CS theory, the number of measurements for stable recovery is proportional to the signal sparsity, rather than to its Fourier bandwidth as dictated by the Shannon/Nyquist theorem \cite{donoho2006compressed, candes2006compressive, baraniuk2007compressive}. Unsurprisingly, the utility of sparse representations also goes well-beyond CS and permeates a lot of fundamental problems in signal processing, machine learning, and theoretical computer science.
Recent results in CS extend the simple sparsity idea to consider more sophisticated {\em structured} sparsity models, which describe the interdependency between the nonzero coefficients \cite{eldar2009robust, blumensath2009sampling, baraniuk2010model, rao2012signal}. There are several compelling reasons for such extensions:
The structured sparsity models allow to significantly reduce the number of required measurements for perfect recovery in the noiseless case and be more stable in the presence of noise. Furthermore, they facilitate the interpretation of the signals in terms of the chosen structures, revealing information that could be used to better understand their properties.
An important class of structured sparsity models is based on groups of variables that should either be selected or discarded together \cite{baraniuk2010low, jenatton2011structured, obozinski2011group, rao2012signal, rao2011convex}.
These structures naturally arise in applications such as neuroimaging \cite{gramfort2009improving,jenatton2011multiscale}, gene expression data \cite{subramanian2005gene,obozinski2011group}, bioinformatics \cite{rapaport2008classification, zhou2010association} and computer vision \cite{cevher2009sparse,baraniuk2010model}.
For example, in cancer research, the groups might represent genetic pathways that constitute cellular processes.
Identifying which processes lead to the development of a tumor can allow biologists to directly target certain groups of genes instead of others \cite{subramanian2005gene}.
Incorrect identification of the active/inactive groups can thus have a rather dramatic effect on the speed at which cancer therapies are developed.
As a result, in this paper, we consider {\em group-based} sparsity models, denoted as $\GG$. These structured sparsity models feature collections of groups of variables that could overlap arbitrarily, that is $\GG = \{\G_1, \ldots, \G_M\}$ where each $\G_j$ is a subset of the index set $\{1, \ldots, N\}$, with $N$ being the dimensionality of the signal that we model. Arbitrary overlaps mean that we do not restrict the intersection between any two sets $\G_j$ and $\G_\ell$ from $\GG$.
In this paper, we address the {\em signal approximation} problem based on a known group structures $\GG$. That is, given a signal ${\bf x} \in \mathbb R^N$, we seek an $\hat{\bf x}$ closest to it in the Euclidean sense, whose {\em support} (i.e., the index set of its non-zero coefficients) consists of the union of at most $G$ groups from $\GG$, where $G > 0$ is a user-defined group budget:
$$
\hat{\bf x} \in \mathrm{argmin}\limits_{{\bf z} \in \mathbb R^N} \left \{ \|{\bf x} - {\bf z}\|_2^2 : \supp({\bf z}) \subseteq \bigcup_{\G \in \mathcal{S}} \G, \mathcal{S} \subseteq \GG, |\mathcal{S}| \leq G \right \},
$$
where $\supp({\bf z})$ is the support of the vector ${\bf z}$. We call such an approximation as {\em G-group-sparse} or in short {\em group-sparse}.
More importantly, we seek to identify the {\em G-group-support} of the approximation $\hat{{\bf x}}$, that is the $G$ groups that constitute its support.
We call this the group-sparse {\em model selection} problem.
The G-group-support of $\hat{{\bf x}}$ allows us to ``interpret'' the original signal and discover its properties so that we can, for example, target specific groups of genes instead of others \cite{subramanian2005gene} or focus more precise imaging techniques on certain brain regions only \cite{michel2011total}.
As a result, we study under which circumstances we can correctly and tractably identify the group-support of a given signal.
{\bf Previous work.}
Recent works in compressive sensing and machine learning with group sparsity have mainly focused on leveraging the group structures for lowering the number of samples required for recovering signals \cite{stojnic2009reconstruction, eldar2009robust, blumensath2009sampling, baraniuk2010model, rao2012signal, huang2011learning,jacob2009group, obozinski2011group}. While these results have established the importance of group structures, many of these works have not fully addressed the relevant issue of model selection.
For the special case of non-overlapping groups, dubbed as the block-sparsity model, the problem of model selection does not present computational difficulties and features a well-understood theory \cite{stojnic2009reconstruction}. The first convex relaxations for group-sparse approximation \cite{yuan2006model} considered only non-overlapping groups. Its extension to overlapping groups \cite{zhao2009composite} has the drawback of selecting supports defined as the complement of a union of groups (see also \cite{jenatton2011structured}).
For overlapping groups, on the other hand, Eldar et al.\ \cite{eldar2009robust} consider the union of subspaces framework and cast the model selection problem as a block-sparse model selection one by duplicating the variables that belong to overlaps between the groups.
Their uniqueness condition \cite{eldar2009robust}[Prop.~1], however, is infeasible for any group structure with overlaps, because it requires that the subspaces intersect only at the origin, while two subspaces defined by two overlapping groups of variables intersect on a subspace of dimension equal to the number of elements in the overlap.
The recently proposed convex relaxations \cite{jacob2009group, obozinski2011group} for group-sparse approximations select group-supports that consist of union of groups.
However, the group-support recovery conditions in \cite{jacob2009group, obozinski2011group} should be taken with care, because they are defined with respect to a particular subset of group-supports and are not general. As we numerically demonstrate in this paper, the group-supports recovered with these methods might be incorrect.
Furthermore, the required consistency conditions in \cite{jacob2009group, obozinski2011group} are unverifiable {\em a priori}. For instance, they require tuning parameters to be known beforehand to obtain the correct group-support.
{\bf Contributions.}
This paper is an extended version of a prior submission to the IEEE International Symposium on Information Theory (ISIT), 2013.
This version contains all the proofs that were previously omitted due to lack of space, refined explanations of the concepts, and provides the full description of the proposed dynamic programming algorithms.
In stark contrast to the existing literature, we take an explicitly discrete approach to identifying group-supports of signals given a budget constraint on the number of groups. This fresh perspective enables us to show that the group-sparse model selection problem is NP-hard: if we can solve the group model selection problem in general, then we can solve any weighted maximum coverage (WMC) problem instance in polynomial time. However, WMC is known to be NP-Hard.
Given this connection, we can only hope to characterize a subset of instances which are tractable or find guaranteed and tractable approximations.
We then present characterizations of group structures that lead to computationally tractable problems via dynamic programming. We do so by leveraging a graph-based representation of the groups and exploiting properties of the induced graph. We present and describe a novel dynamic program that solves the WMC problem for a specific class of group structures and could be of interest by itself. We also identify tractable discrete relaxations of the group-sparse model selection problem that lead to efficient algorithms. Specifically, we relax the constraint on the number of groups into a penalty term and show that if the remaining group constraints satisfy a property related to the concept of total unimodularity \cite{wolsey1999integer}, then the relaxed problem can be efficiently solved using linear program solvers. We also extend the discrete model to incorporate an overall sparsity constraint and allowing to select individual elements from each group, leading to within-group sparsity. Furthermore, we discuss how this extension can be used to model hierarchical relationships between variables. We present a novel dynamic program that solves the hierarchical model selection problem exactly and discuss a tractable discrete relaxation.
We also interpret the implications of our results in the context of other group-based recovery frameworks. For instance, the convex approaches proposed in \cite{eldar2009robust, jacob2009group, obozinski2011group} also relax the discrete constraint on the cardinality of the group support. However, they first need to decompose the approximation into vector components whose support consists only of one group and then penalize the norms of these components.
It has been observed \cite{obozinski2011group} that these relaxations produce approximations that are group-sparse, but their group-support might include irrelevant groups. We concretely illustrate these cases via a Pareto frontier example.
{\bf Paper structure.}
The paper is organized as follows. In Section 2, we present definitions of group-sparsity and related concepts, while in Section \ref{sec:tract}, we formally define the approximation and model-selection problems and connect them to the WMC problem. We present and analyze discrete relaxations of the WMC in Section \ref{sec:discrete_relax} and consider convex relaxations in Section \ref{sec:convex_relax}.
In Section \ref{sec:discrete_vs_convex}, we illustrate via a simple example the differences between the original problem and the relaxations.
The generalized model is introduced and analyzed in Section \ref{sec:generalizations}, while numerical simulations are presented in Section \ref{sec:exp}.
We conclude the paper with some remarks in Section \ref{sec:conclusions}. The appendices contain the detailed descriptions of the dynamic programs.
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\section{\label{sec:tract}Tractability of interpretations}
A group-based interpretation of a signal consists in identifying the groups that constitute the support of its approximation.
In this section, we establish the hardness of finding group-based interpretations of signals in general and characterize a class of group structures that lead to tractable interpretations.
In particular, we present a polynomial time algorithm that finds the correct $G$-group-support of the $G$-group-sparse approximation of ${\bf x}$, given a positive integer $G$ and the group structure $\mathfrak{G}$.
We first define the $G$-group sparse approximation $\hat{{\bf x}}$ and then show that it can be easily obtained from its $G$-group cover $\mathcal{S} ^G(\hat{\bf x})$, which is the solution of the model selection problem.
We then reformulate the model selection problem as the weighted maximum coverage problem.
Finally, we present our main result, the polynomial time dynamic program for loopless pairwise overlapping group structures.
\begin{prob}[Signal approximation]
Given a signal ${\bf x} \in \mathbb R^N$, a best $G$-group sparse approximation $\hat{\bf x}$ is given by
\begin{equation}
\label{eq:approx}
\hat{\bf x} \in \mathrm{argmin}\limits_{{\bf z} \in \mathbb R^N} \left \{ \|{\bf x} - {\bf z}\|_2^2 : \|{\bf z}\|_{\GG,0} \leq G \right \}.
\end{equation}
\end{prob}
If we already know the $G$-group cover of the approximation $\mathcal{S}^G(\hat{\bf x})$, we can obtain $\hat{\bf x}$ as $\hat{\bf x}_\mathcal{I} = {\bf x}_\mathcal{I}$ and $\hat{\bf x}_{\mathcal{I}^c} = 0$, where $\mathcal{I} = \bigcup_{\G \in \mathcal{S}^G(\hat{\bf x})} \G$ and $\mathcal{I}^c = \mathcal{N} \setminus \mathcal{I}$.
Therefore, we can solve Problem 1 by solving the following discrete problem.
\begin{prob}[Model selection]
Given a signal ${\bf x} \in \mathbb R^N$, a $G$-group cover model for its $G$-group sparse approximation is expressed as follows
\begin{equation}
\label{eq:model_sel}
\mathcal{S}^G(\hat{\bf x}) \in \argmax\limits_{\scriptsize\begin{array}{c}\mathcal{S} \subseteq \GG \\|\mathcal{S}| \leq G \end{array}} \left \{ \sum\limits_{i \in \mathcal{I}} x_i^2 : \mathcal{I} = \bigcup_{\G \in \mathcal{S}} \G \right \}.
\end{equation}
\end{prob}
To show the connection between the two problems, we first reformulate Problem 1 as
$$
\min\limits_{{\bf z} \in \mathbb R^N} \left \{ \|{\bf x} - {\bf z}\|_2^2 : \supp({\bf z}) = \mathcal{I}, \mathcal{I} = \bigcup_{\G \in \mathcal{S}} \G, \mathcal{S} \subseteq \GG, |\mathcal{S}| \leq G \right \},
$$
which can be rewritten as
$$
\min\limits_{\scriptsize \begin{array}{c} \mathcal{S} \subseteq \GG\\ |\mathcal{S}| \leq G\\ \mathcal{I} = \bigcup_{\G \in \mathcal{S}} \G\end{array}} \min\limits_{\scriptsize \begin{array}{c} {\bf z} \in \mathbb R^N\\ \supp({\bf z}) = \mathcal{I}\end{array}} \|{\bf x} - {\bf z}\|_2^2 \; .
$$
The optimal solution is not changed if we introduce a constant, change sign of the objective and consider maximization instead of minimization
$$
\max\limits_{\scriptsize \begin{array}{c} \mathcal{S} \subseteq \GG\\ |\mathcal{S}| \leq G\\ \mathcal{I} = \bigcup_{\G \in \mathcal{S}} \G\end{array}} \max\limits_{\scriptsize \begin{array}{c} {\bf z} \in \mathbb R^N\\ \supp({\bf z}) = \mathcal{I}\end{array}} \bigg \{ \|{\bf x}\|_2^2 - \|{\bf x} - {\bf z}\|_2^2 \bigg \} \; .
$$
The internal maximization is achieved for $\hat{\bf x}$ as $\hat{\bf x}_\mathcal{I} = {\bf x}_\mathcal{I}$ and $\hat{\bf x}_{\mathcal{I}^c} = 0$, so that we have, as desired,
$$
\mathcal{S}^G(\hat{{\bf x}}) \in \argmax\limits_{\scriptsize \begin{array}{c} \mathcal{S} \subseteq \GG\\ |\mathcal{S}| \leq G\\\mathcal{I} = \bigcup_{\G \in \mathcal{S}} \G\end{array}} \|{\bf x}_\mathcal{I}\|_2^2 \; .
$$
The following reformulation of Problem 2 as a binary problem allows us to characterize its tractability.
\begin{lemma}
Given ${\bf x} \in \mathbb R^N$ and a group structure $\GG$, we have that $\mathcal{S}^G(\hat{\bf x}) = \{ \G_j \in \GG : \omega^G_j = 1 \}$, where $(\boldsymbol{\omega}^G, {\bf y}^G)$ is an optimal solution of
\begin{equation}
\label{eq:WMC}
\max\limits_{\boldsymbol{\omega} \in \mathbb{B}^M,~{\bf y} \in \mathbb{B}^\dim} \left \{ \sum_{i=1}^N y_i x_i^2 : \mathbf{A}^\mathfrak{G} \boldsymbol{\omega} \geq {\bf y}, \sum_{j=1}^M \omega_j \leq G \right \}.
\end{equation}
\end{lemma}
\begin{proof}
The proof follows along the same lines as the proof in \cite{kyrillidis2012combinatorial}. Note that in \eqref{eq:WMC}, $\boldsymbol{\omega}$ and $\bf y$ are binary variables that specify which groups and which variables are selected, respectively. The constraint $\mathbf{A}^\mathfrak{G} \boldsymbol{\omega} \geq {\bf y}$ makes sure that for every selected variable at least one group is selected to cover it, while the constraint $\sum_{j=1}^M \omega_j \leq G$ restricts choosing at most $G$ groups.
\end{proof}
Problem \eqref{eq:WMC} can produce all the instances of the weighted maximum coverage problem (WMC), where the weights for each element are given by $x_i^2$ ($1 \leq i \leq \dim$) and the index sets are given by the groups $\G_j \in \GG$ ($1 \leq j \leq M$).
Since WMC is in general NP-hard and given Lemma 1, the tractability of \eqref{eq:model_sel} directly depends on the hardness of \eqref{eq:WMC}, which leads to the following result.
\begin{prop}
The model selection problem \eqref{eq:model_sel} is in general NP-hard.
\end{prop}
It is possible to approximate the solution of \eqref{eq:WMC} using the greedy WMC algorithm \cite{nemhauser1978analysis}.
At each iteration, the algorithm selects the group that covers new variables with maximum combined weight until $G$ groups have been selected.
However, we show next that for certain group structures we can find an exact solution.
Our main result is an algorithm for solving \eqref{eq:WMC} for loopless pairwise overlapping groups structures.
The proof is given in Appendix \ref{sec:dp_lp}.
\begin{prop}
\label{prop:DP1}
Given a loopless pairwise overlapping group structure $\mathfrak{G}$, there exists a polynomial time dynamic programming algorithm that solves \eqref{eq:WMC}.
\end{prop}
\section{Discrete relaxations}
\label{sec:discrete_relax}
Relaxations are useful techniques that allow to obtain approximate, or even sometimes exact, solutions while being computationally less demanding.
In our case, by relaxing the constraint on the number of groups in \eqref{eq:WMC} into a regularization term with parameter $\lambda > 0$, we obtain the following binary linear program
\begin{equation}
(\boldsymbol{\omega}^\lambda, {\bf y}^\lambda) \in \argmax\limits_{\boldsymbol{\omega} \in \BB^M, {\bf y} \in \BB^\dim} \left \{ \sum_{i=1}^N y_i x_i^2 - \lambda\sum_{j=1}^M \omega_j: \mathbf{A}^\mathfrak{G} \boldsymbol{\omega} \geq {\bf y} \right \} \;
\label{eq:PR}
\end{equation}
We can rewrite the previous program in standard form. Let $\mathbf{u}^\top = [{\bf y}^\top~\boldsymbol{\omega}^\top] \in \BB^{\dim+M}$, $\mathbf{w}^\top = [x_1^2, \ldots, x_N^2,-\lambda\mathbf{1}_M^\top]$ and $\mathbf{C} = [\mathbf{I}_N,~-\mathbf{A}^\GG]$. We then have that \eqref{eq:PR} is equivalent to
\begin{equation}
\mathbf{u}^\lambda \in \argmax\limits_{\mathbf{u} \in \BB^{\dim+M}} \left \{ \mathbf{w}^\top \mathbf{u} : \mathbf{Cu} \leq 0 \right \}
\label{eq:PR_std}
\end{equation}
In general, \eqref{eq:PR_std} is NP-hard, however, it is well known \cite{wolsey1999integer} that if the constraint matrix $\mathbf{C}$ is Totally Unimodular (TU), then it can be solved in polynomial-time.
While the concatenation of two TU matrices is not TU in general, the concatenation of the identity matrix with a TU matrix results in a TU matrix. Thus, due to its structure, $\mathbf{C}$ is TU if $\mathbf{A}^\mathfrak{G}$ is TU \cite{wolsey1999integer}.
Group structures that can be represented by a bipartite graph, such as the one in Fig.~\ref{fig:bipartite}, lead to constraint matrices $\mathbf{A}^\mathfrak{G}$ that are TU \cite{wolsey1999integer}.
\begin{lemma}
Loopless pairwise overlapping groups lead to totally unimodular constraints.
\end{lemma}
\begin{proof}
We first use a result that establishes that if a matrix is TU, then its transpose is also TU \cite{wolsey1999integer}[Prop.2.1].
We then apply \cite{wolsey1999integer}[Corollary 2.8] to $\mathbf{A}^\mathfrak{G}$, swapping the roles of rows and columns.
Given a binary matrix whose columns can be partitioned into two disjoint sets and with no more than two nonzero elements in each row, this result provides two sufficient conditions for it being totally unimodular.
In our case, the columns of $\mathbf{A}^\mathfrak{G}$ can be partitioned in two sets, $\mathcal{S}_1$ and $\mathcal{S}_2$ because the group graph for loopless pairwise overlapping groups is bipartite. The two sets represents groups which have no common overlap. Furthermore, each row of $\mathbf{A}^\mathfrak{G}$ contains at most two nonzero entries due to the pairwise overlap.
We can now easily check that the two conditions on $\mathbf{A}^\mathfrak{G}$ are satisfied:
\begin{itemize}
\item If two nonzero entries in a row have the same sign, then the column of one is in $\mathcal{S}_1$ and the other is in $\mathcal{S}_2$: indeed if an element belongs to two groups, these groups must lie in two different sets;
\item If two nonzero entries in a row have opposite signs, then their columns are both in $\mathcal{S}_1$ or both in $\mathcal{S}_2$: there are no such rows in our case.
\end{itemize}
\end{proof}
Even though for this group structure we can use the dynamic program of Prop. \ref{prop:DP1},
for very large problems it may be computationally faster to solve the binary linear program.
The next proposition establishes when the regularized solution coincides with the solution of \eqref{eq:WMC}.
\begin{lemma}
If the value of the regularization parameter $\lambda$ is such that the solution $(\boldsymbol{\omega}^\lambda, {\bf y}^\lambda)$ of \eqref{eq:PR} satisfies $\sum_j \omega_j^\lambda = G$, then $(\boldsymbol{\omega}^\lambda, {\bf y}^\lambda)$ is also a solution for \eqref{eq:WMC}.
\end{lemma}
\begin{proof}
This lemma is a direct consequence of Prop. \ref{prop:pareto_optimal} below.
\end{proof}
However, as we numerically show in Section \ref{sec:exp}, given a value of $G$ it is not always possible to find a value of $\lambda$ such that the solution of \eqref{eq:PR} is also a solution for \eqref{eq:WMC}.
Let the set of points $\mathcal{P} = \{G, (f(G))\}_{G=1}^M$, where $f(G) = \sum_{i=1}^N y^G_i x_i^2$, be the Pareto frontier of \eqref{eq:WMC}.
We then have the following characterization of the solutions of the discrete relaxation.
\begin{prop}
\label{prop:pareto_optimal}
The discrete relaxation \eqref{eq:PR} yields only the solutions that lie on the intersection between the Pareto frontier of \eqref{eq:WMC}, $\mathcal{P}, $ and the boundary of the convex hull of $\mathcal{P}$.
\end{prop}
\begin{proof}
On the one hand, the solutions of \eqref{eq:WMC} for all possible values of $G$ are the Pareto optimal solutions of the following vector-valued minimization problem with respect to the positive orthant of $\mathbb R^2$, which we denote by $\mathbb R^2_+$,
\begin{equation}
\label{eq:vv_wmc}
\begin{array}{cc}
\min\limits_{\boldsymbol{\omega} \in \mathbb{B}^M,~{\bf y} \in \mathbb{B}^\dim} & f(\boldsymbol{\omega},{\bf y}) \\
\text{subject to} & \mathbf{A}^\mathfrak{G} \boldsymbol{\omega} \geq {\bf y}
\end{array}
\end{equation}
where $f(\boldsymbol{\omega},{\bf y}) = \left (\|x\|^2 - \sum_{i=1}^N y_i x_i^2, \sum_{j=1}^M \omega_j \right ) \in \mathbb R^2_+$.
On the other hand, the scalarization of \eqref{eq:vv_wmc} yields the following discrete problem, with $\lambda > 0$
\begin{equation}
\label{eq:scalar_wmc}
\begin{array}{cc}
\min\limits_{\boldsymbol{\omega} \in \mathbb{B}^M,~{\bf y} \in \mathbb{B}^\dim} & \|{\bf x}\|^2 - \sum_{i=1}^N y_i x_i^2 + \lambda \sum_{j=1}^M \omega_j \\
\text{subject~to} & \mathbf{A}^\mathfrak{G} \boldsymbol{\omega} \geq {\bf y}
\end{array}
\end{equation}
whose solutions are the same as for \eqref{eq:PR}.
Therefore, the relationship between the solutions of \eqref{eq:WMC} and \eqref{eq:PR} can be inferred by the relationship between the solutions of \eqref{eq:vv_wmc} and \eqref{eq:scalar_wmc}.
It is known that the solutions of \eqref{eq:scalar_wmc} are also Pareto optimal solutions of \eqref{eq:vv_wmc}, but only the Pareto optimal solutions of \eqref{eq:vv_wmc} that admit a supporting hyperplane for the feasible objective values of \eqref{eq:vv_wmc} are also solutions of \eqref{eq:scalar_wmc} \cite{boyd2004convex}.
In other words, the solutions obtainable via scalarization belong to the intersection of the Pareto optimal solution set and the boundary of its convex hull.
\end{proof} | {
"redpajama_set_name": "RedPajamaArXiv"
} | 3,898 |
Q: How to wipe RAID superblock? wipefs: error: /dev/sda: probing initialization failed I want to format my external USB.
lsblk shows
sda 8:0 1 57.6G 0 disk
├─sda1 8:1 1 2.9G 0 part /media/miki/Ubuntu 20.04.3 LTS amd64
├─sda2 8:2 1 3.9M 0 part
└─sda3 8:3 1 54.8G 0 part /media/miki/writable
I tried standard procedures
sudo mdadm --zero-superblock /dev/sda
mdadm: Couldn't open /dev/sda for write - not zeroing
And wipefs
sudo wipefs -a /dev/sda
wipefs: error: /dev/sda: probing initialization failed: Device or resource busy
What should I try now? What is the root cause of the problem?
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 733 |
Biografia
Discepolo di Wolf Caspar von Klengel, nel 1691 venne nominato sovrintendente generale all'edilizia di Dresda e dal 1704 fu architetto di corte di Augusto II il Forte, re di Sassonia.
Usò inizialmente modelli barocchi grevi, ma dopo i suoi viaggi a Vienna, Roma e in Francia del 1710, durante i quali ebbe modo di confrontarsi con le opere dei più grandi architetti del tempo, elaborò uno stile nuovo, in cui a prevalere erano le forme più semplici e chiare. Alla fine della sua carriera seppe creare effetti architettonici con sapiente cambio ritmico nei tetti degli edifici ed un moderato uso delle ornamentazioni.
Nel 1718 venne nominato Oberlandbaumeister da Augusto II. Sotto la sua guida Dresda, la capitale del regno, assunse un nuovo volto barocco: realizzò il grande giardino dello Zwinger, il palazzo della Cancelleria e le residenze reali di Dresda e Pillnitz, il Castello di Moritzburg, ecc.
Altri progetti
Collegamenti esterni
Architetti barocchi | {
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Omori (Stiliserat som OMORI) är ett datorrollspel utvecklat av indie-studion Omocat. Spelet är baserat på regissörens serie Omori (ひきこもり, hikikomori) och släpptes i december 2020. Spelet har framträdande koncept som ångest och depression, och har psykologiska skräckelement. I spel-berättelsen styr spelaren en hikkomori-pojke som heter Sunny och hans alter-ego, drömvärlden Omori. Figurerna utforskar både den verkliga världen och den surrealistiska drömvärlden för att övervinna sina rädslor och hemligheter. Hur de interagerar beror på val som görs av spelaren, vilket resulterar i ett av flera slut. Spelets turbaserade stridssystem inkluderar okonventionella statuseffekter baserade på figurens känslor.
Efter en framgångsrik Kickstarter-kampanj försenades spelet flera gånger och fick under flera tillfällen utvecklingssvårigheter. Spelet släpptes slutligen för Microsoft Windows och MacOS sex år efter den första finansieringen, och utvecklarna avslöjade också planer på en japansk översättning samt portar till Nintendo Switch, PlayStation 4 och Xbox One. Omori hyllades av kritiker, som hyllade det för dess grafik, berättande element, soundtrack och skildring av ångest och depression, och jämförde det positivt med Earthbound och Yume Nikki. Spelet blev nominerat till flera priser och vann DreamHacks kategori "Daringly Dramatic" 2021.
Spelupplägg
Omoris spelupplägg är inspirerat av traditionella japanska rollspel. Spelaren styr ett sällskap med fyra figurer: Omori, Aubrey, Kel och Hero, var och en med sina egna färdigheter i strid.
När man utforskar övervärlden spelas spelet från ett "top-down"-perspektiv. Spelet innehåller flera sidouppdrag och pussel för spelaren att lösa, vilket gör att man kan få belöningar och färdigheter. Vapen och föremål som gynnar sällskapet kan erhållas under hela spelet, inklusive genom att köpa dem med spelets valuta, Clams. När man inte är i strid kan figurerna läka och spara genom att gå till en picknickfilt, där Omoris äldre syster Mari befinner sig.
Striderna spelas turbaserat, där varje figur utför ett drag. Efter att en figur attackerat kan den och en annan figur tillsammans utföra en "follow up"-attack. Figurer och fiender har ett hjärta, som fungerar som hälsopoäng; om man tar skada, minskar hälsopoängen, och om hälsopoängen når noll, besegras figuren förvandlas till rostat bröd. "Juicemätaren" används för att utföra färdigheter och speciella förmågor som hjälper till i strid.
Till skillnad från de flesta rollspel är statuseffekter baserade på ett triangulärt känslosystem. En figur eller fiendes känslor kan förändras under loppet av en strid, vanligtvis på grund av rörelser från en annan figur eller fiende. Neutral är baslinjen och har inga effekter, Angry ökar attacker men sänker försvaret, Sad ökar försvaret men sänker hastigheten och Happy ökar hastigheten men sänker precisionen. Känslor är antingen starka eller svaga mot varandra - Happy slår Angry, Angry slår Sad och Sad slår Happy. Dessutom finns det högre intensitetsvarianter av varje känsla.
Handling
Huvudpersonen Omori, vaknar i "The White Space", ett litet vitt rum han har bott i "så länge [han] kan minnas". Han öppnar en dörr och går in i den livfulla världen "Headspace", där han träffar sin äldre syster Mari och hans vänner Aubrey, Kel, Hero och Basil. Vännerna tittar igenom delade minnen i Basils fotoalbum och de bestämmer sig för att bege sig till hans hus, men Mari väljer att stanna kvar. Längs vägen skadas albumet när Kel och Aubrey bråkar. När Basil ser ett obekant foto falla ur albumet, får Basil panik, och Omori teleporteras plötsligt tillbaka till "The White Space" ensam. Han hugger sig själv med en kniv och det visar sig att de tidigare händelserna är drömmarna från tonårspojken, Sunny.
När Sunny vaknar märker han att de ska flytta ut om tre dagar, och går ner för att äta ett mellanmål vid midnatt. Han blir konfronterad av en mardrömslik hallucination som symboliserar hans rädsla, han skingrar den genom att lugna ner sig och går tillbaka till sängen med en kniv. När Omori sen vaknar upp i "The White Space" igen, återförenas han med Aubrey, Kel och Hero, de fyra får dock veta att Basil är försvunnen. De bestämmer sig för att rädda honom och reser till olika delar av "Headspace" för att söka efter honom, med Mari som hjälper till på vägen. Gruppen blir distraherad av olika situationer de möter, och deras minnen av Basil och deras mål att rädda honom försvinner sakta.
Under tiden, i den vakna världen, visar det sig att Mari hade begått självmord för fyra år sedan, vilket ledde till att vängruppen splittrades. Även om Kel och Hero lyckades återhämta sig något känslomässigt, blev Sunny helt avskärmad och Aubrey lämnade efter att ha känt sig förrådd av gruppens uppenbara likgiltighet inför Maris död. Basil blev neurotisk och paranoid. Kel knackar på Sunnys dörr i ett sista försök att återansluta. Spelaren kan antingen ignorera Kel eller svara på dörren; att välja annorlunda kan utlösa olika vägar, om spelaren väljer den första stannar Sunny inne under de återstående tre dagarna, gör sysslor och fokuserar på sina drömmar istället för att försonas.
Om det sista alternativet väljs upptäcker Sunny och Kel att Aubrey attackerar Basil. De upptäcker att hon stal Basils fotoalbum, för att hindra honom från att vandalisera det. Efter att spelaren ha slagits mot Aubrey och hämtat albumet, lämnar de tillbaka det till Basil med några foton som saknas, men han låter Sunny behålla det. Medan de äter middag tillsammans blir Basil plötsligt stött över att få veta om Sunnys avgång, och avslöjar att han har liknande hallucinationer. I ett annat slagsmål nästa dag, puttar Aubrey av misstag ner Basil i en sjö. Sunny försöker rädda honom, och båda räddas från att drunkna av Hero. I drömvärlden återvänder Omori och hans vänner till Basils nu förfallna hus, och han transporteras till det mer oroande "Black Space". Basil dyker upp flera gånger, och försöker upprepade gånger prata med honom om något innan han dör. I det sista rummet dödar Omori Basil och placerar sig på toppen av en tron av massiva, röda händer.
Dagen före Sunnys avgång försonas de andra med Aubrey och hittar de saknade bilderna. För att komma i rätta med Maris död, bestämmer de sig för att tillbringa sin sista natt tillsammans i Basils hus, trots att han vägrar att lämna sitt rum. Den natten konfronterar Sunny sanningen i sina drömmar: under ett gräl dödade han Mari genom att av misstag putta ner henne för trappan. Basil förnekade att Sunny gjorde det, och hjälpte till att framställa Maris död som ett självmord genom att hänga hennes lik. Efteråt tittade de tillbaka på liket och såg ett öppet öga stirra tillbaka på dem vilket formade deras efterföljande hallucinationer. Medan Basil förtärdes av skuld och självförakt, fick Sunnys självmordsdepression honom att skapa "Headspace" och sin drömpersona Omori för att maskera sitt trauma. För att dölja sanningen återställde Omori "Headspace" varje gång deras minnen flydde från "Black Space". Sunny vaknar mitt i natten; spelaren kan antingen välja att gå in i Basils rum för att konfrontera honom om Maris död eller somna om.
Slut
Om spelaren svarar Kel vid dörren, konfronterar Basil under natten på den sista dagen, då attackerar och slåss Sunny och Basil mot varandra, där Sunny mitt i striden blir knivhuggen i ögat av Basil med sin trädgårdssax vilket resulterar i att båda pojkarna svimmar. När han är medvetslös, minns Sunny sina minnen med Mari och hans vänner och möter Omori. Omori som vägrar att dö, besegrar Sunny och spelaren får en "game over"-skärm.
Om spelaren väljer att försöka igen, reser sig Sunny upp och spelar duetten med Mari som var planerad för deras konsert. Omori kramar honom och försvinner. I den verkliga världen vaknar Sunny på sjukhuset som han och Basil skickades till och beger sig till den Basils säng. Medan de är omgivna av sina vänner, antyds det att Sunny berättar sanningen för dem om Maris död. Dessutom, om spelaren vattnade Basils trädgård dagligen i "Headspace", kommer en scen att visa Basil vakna upp på sjukhuset. Han och Sunny ler mot varandra, och hallucinationerna försvinner från båda pojkarna.
Skulle spelaren välja att inte fortsätta försvinner Sunny snarare än Omori. När han vaknar upp på sjukhuset begår han självmord genom att hoppa från balkongen.
Alternativt, om spelaren ignorerar Basil under natten till den sista dagen, kommer Sunny och hans vänner att vakna upp och upptäcka att Basil har begått självmord. Beroende på spelarens val kan Sunny sedan antingen ta livet av sig med sin kniv eller gå därifrån med sin skuld fortfarande oförminskad när sirener från ambulanser ringer i fjärran. Om spelaren till en början väljer att stanna inne och undvika Kel, är endast en variant av detta slut tillgänglig.
Noter
Datorspel 2020
Indiespel
Windows-spel
Macintosh-spel
Nintendo Switch-spel
Playstation 4-spel
Xbox One-spel | {
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Welcome to Structured Wealth Advisors. We are a fee based independent financial advisor associate firm in Duluth, MN. Since 1994, we have helped clients from the United States all the way to Australia take control of their financial future.
Our investment approach is completely different from traditional Wall Street brokers, financial planners, financial advisors or wealth managers.. Rather than fight the market, we work with the market. Through our partnership with Dimensional Fund Advisors, we let the science of capital markets guide the way.
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We look forward to working with you through all stages of your life, please call our Duluth office at 218-720-6989. | {
"redpajama_set_name": "RedPajamaC4"
} | 4,326 |
Grapevine research solves Chardonnay clonal mystery
Aug 4, 2020 | AWRI, News, Research stories
For many years, the origin of the popular Western Australian Chardonnay clone known as Gingin has been hotly debated. Now new genomic research by the Australian Wine Research Institute has solved the mystery.
A study recently published in the Australian Journal of Grape and Wine Research has used newly identified clonal markers to investigate the heritage of a specific Chardonnay clone for the first time.
Gingin is the most commonly planted clone of Chardonnay in Margaret River. It is known both for its tendency to produce loose grape bunches with berries of different sizes and for making complex and elegant wines, but its history has been unclear. Introduced into Western Australia in 1957 via University of California, Davis, Gingin was believed by some to be derived from the same source material as a clone known as Old Foundation Block (OF) Chardonnay, but was also commonly thought of as being the same as another clone with similar traits called Mendoza.
The new genomics research has revealed that all three clones have a shared heritage, in an old Californian source block at UC Davis. They are, however, quite distinct from each other. In particular, Gingin and Mendoza are as different from each other as they are from any of the other clonal selections of Chardonnay, despite their shared origins.
Discussing the recently published findings, AWRI Research Manager, Dr Simon Schmidt said, 'It has been very satisfying to apply our research on grapevine clonal markers in such a tangible way. Using this new science to address a longstanding industry question has been a great way to demonstrate the capabilities of whole genome sequencing. There is now huge potential to apply clonal marker mapping to other varieties – with Pinot Noir our next target, as part of support for Adelaide Hills vineyards damaged by the recent bushfires and requiring replanting with well characterised grapevines.'
This work has not only solved a decades-old mystery – it represents a significant leap forward for grapevine genomics and demonstrates that a great deal more is possible beyond simply confirming grape variety. The panel of markers identified for Chardonnay will be useful for verification of planting material by nurseries and vineyard owners, for targeted importation of clones not currently available in Australia. The work also forms a foundation for identifying clonal markers in other grape varieties and potentially other crops
Read the research paper | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 7,450 |
\section{Conclusions and Future Work}
\label{sec:conclusions}
In recent years there has been considerable interest in satisfiability
procedures (aka solvers) for theories over string equations, length,
and string-number conversions in the verification and security
communities~\cite{z3strcav, Liang2014}. These theories are also of
great interest to logicians, since there are many open problems
related to their decidability and complexity. We showed that a
first-order many-sorted quantifier-free theory $T_{s,n}$ of string
equations, linear arithmetic over length function, and string-number
conversion predicates, variations of which have been implemented in
solvers such as Z3str2 and CVC4, is undecidable. We establish expressibility
results for $numstr$ predicate that suggest that this predicate is far
more complex than appears at first glance. Finally, we also provide a
consistent axiomatization $\Gamma$ for the symbols of $T_{s,n}$, and
show that the theory $T_{\Gamma}$ is incomplete.
There are many decidability, complexity and efficient encoding
questions related to fragments of $T_{s,n}$ that remain open. For
example, it is not known whether the theory of word equations and
arithmetic over length functions is decidable~\cite{Matiyasevich}. The
satisfiability problem for the quantifier-free theory of string
equations by itself is known to be in PSPACE; however, it is not known
whether it is PSPACE-complete~\cite{plandowski2006}. Yet another open
question concerns efficient encoding of functions such as ``Replace''
that are heavily used in many programming languages, and predicates
such as string comparison. More generally, efficient encoding of
common programming language string-intensive functions and predicates
in terms of $T_{s,n}$-functions and predicates can be of great value
to practitioners, and remains a challenging practical problem.
\section{Result 2: Expressibility}
\label{sec:definability}
In this section we establish that the $\pi(p,x,y)$ and $numstr$
predicates are expressible in terms of each other.
We define a new theory $T_\pi$ (different from $T_p$), which is the same
as $T_{s,n}$ except that $numstr$ is removed and replaced by the
$\pi(p,x,y)$ predicate. From the previous section it is clear that
any formula involving the $\pi(p,x,y)$ predicate can be reduced to some
formula in $T_{s,n}$ using some Boolean combination of $numstr$
predicate, string equations, and length function.
This shows us that a reduction exists from $T_\pi$ to $T_{s,n}$.
We now show that a reduction in the opposite direction exists; that is,
the $numstr$ predicate can be expressed in terms of quantified
formulas over the $\pi(p,x,y)$ predicate, word equations, and length
function.
The value of these two recursive reductions is that it suggests that
the $\pi$ predicate is expressible using string equations and length
function iff $numstr$ is. Expressibility results are very useful tools
in constructing reductions, distinguishing the expressive powers of
various theories, and establishing (un)-decidability
results. Additionally, our expressibility results suggest that the
$numstr$ predicate is much more complex, both from a theoretical and a
practical point of view, than it seems at first glance.
\begin{Definition}
A predicate $P$ is \textbf{expressible} in some theory $T$ having
language $L_{T}$ if there exists an $L_{T}$-formula $\phi(x_1,
\hdots, x_n)$ such that for all interpretations $m_1, \hdots, m_n$
of $x_1, \hdots, x_n$ allowed by $T$ and such that $\phi(m_1,
\hdots, m_n)$ is well-sorted, we have that $P(m_1, \hdots, m_n)$ is
true iff $\phi(m_1, \hdots, m_n)$ is true.
\end{Definition}
The fact that $\pi(p,x,y)$ is expressible in terms of $numstr(i,s)$ in
the theory $T_{s,n}$ follows immediately from the reduction from $T_p$
to $T_{s,n}$ used to establish the undecidability theorem
in the previous section. We only have to show the reverse direction,
i.e., that $numstr(i,s)$ is expressible in terms of $\pi(p,x,y)$.
\footnote{Note that we do not present a reduction from $T_{s,n}$ to
$T_p$. However, we conjecture that one exists, due to the
possibility of mapping the countably infinite set of string
constants onto the countably infinite set of natural numbers and
then constructing string functions and predicates as operators over
natural numbers.}
\begin{Theorem}
$numstr(i,s)$ is expressible in terms of $\pi(p,x,y)$ in $T_\pi$.
\end{Theorem}
\begin{proof}
We represent $numstr(i,s)$ as a formula that asserts the
non-existence of a witness for one of two kinds of error in the
conversion. The first kind of error relates to the maximum possible
value of $i$. Suppose $s$ is a binary string of length $n$. Then
$s$ cannot represent a natural number greater than or equal to
$2^n$. The second error is a discrepancy between the binary
representation of $i$ and the binary string $s$. To check bit $t$
of the number $i$, decompose $i$ into $h2^{t+1} + x2^{t} + l$ where
$x$ is the $t$-th bit of $i$ and so $x = 0 \lor x = 1$, and $l$ is
the numeric representation of bits $t-1$ through 0 and so $l <
2^{t}$. Then if $x = 0$ and $s[len(s) - 1 - t] = ``1"$, or if $x =
1$ and $s[len(s) - 1 - t] = ``0"$, there is an error. This gives
us the following sentence:
\begin{align*}
numstr(i, s) \iff & \forall n\, p\, t\, h\, p_{h}\, x\, p_{x}\, l\, l_{u}\, s_h\, s_x\, s_l: \\
& \lnot (len(s) = n \land \pi(p, 1, n) \land i \ge p) \\
& \land \lnot (\pi(p_{h}, h, t+1) \land \pi(p_{x}, x, t) \\
& \land i = p_{h} + p_{x} + l \land \pi(l_{u}, 1, t) \land l < l_u \\
& \land s = s_h \cdot s_x \cdot s_l \land len(s_l) = t \land len(s_x) = 1 \\
& \land ((x = 0 \land s_x = ``1") \lor (x = 1 \land s_x = ``0")))
\end{align*}
We can apply this rule recursively to the input formula, along with similar rules to the ones presented previously,
to obtain a reduction from $T_{s,n}$ to $T_{\pi}$.
\end{proof}
\section{\bf Introduction}
The satisfiability problem for theories over finite-length strings
(aka words) has long been studied by mathematicians such as
Quine~\cite{Quine}, Post, Markov and Matiyasevich~\cite{Matiyasevich},
Makanin~\cite{makanin}, and Plandowski~\cite{KarhumakiPM97,
plandowski99, plandowski2006}. Post, Markov, and Quine were
motivated by the connections between theories over word
equations\footnote{In this paper, we interchangeably use the terms
\textit{word equations} and \textit{string equations}. The term
``word equations'' is the convention among logicians, while formal
verification researchers tend to use the term ``string equations''.}
and Peano arithmetic, while Matiyasevich's motivation for studying
them was their connection to Diophantine
equations~\cite{Matiyasevich}.
More recently there has been considerable interest in efficient
solvers for theories over string equations in the formal verification,
software engineering, and security research communities. Examples of
such solvers include Z3str2~\cite{z3strcav} and CVC4~\cite{Liang2014},
both of which support the quantifier-free (QF) first-order many-sorted
theory $T_{s,n}$ of string equations, length, and string-integer
conversions. This theory is expressive enough that many string-related
library functions and programming constructs from languages such as C,
C++, Java, PHP, and JavaScript can be easily encoded in terms of its
functions and predicates. The expressive power of $T_{s,n}$ and
efficient practical string solvers have enabled many applications in
program analysis and verification~\cite{z3strcav, prateek,
rupak}. Examples include dynamic symbolic execution aimed at
automated bug-finding~\cite{hampi, prateek}, and analysis of
database/web applications~\cite{emmiMS2007, rupak, WassermannSu2007}.
Given the fundamental nature of the theory $T_{s,n}$ and its fragments
(e.g., note that word equations essentially form a free semigroup
studied intensively by mathematicians over the last several
decades~\cite{lothaire}), it is no surprise that there is strong
motivation from logicians to study their decidability and
complexity. In the 1940's, Post and Markov conjectured that the
fully-quantified first-order theory of string equations (i.e.,
quantified sentences over Boolean combination of string equations)
must be undecidable. In his 1946 paper, Quine \cite{Quine} showed that
this theory is indeed undecidable. In 1977, Makanin famously proved
that the satisfiability problem for the quantifier-free theory of
string equations is decidable~\cite{makanin}. This result is often
considered as one of the most complex proofs in theoretical computer
science. In recent years, Plandowski and others considerably improved
Makanin's results and showed that satisfiability problem for string
equations is in PSPACE~\cite{plandowski2006}. In 2012, Ganesh et
al. showed that $\forall\exists$-fragment of positive string equations
is undecidable, strengthening Quine's result and
establishing the boundary between decidability and undecidability for
string equations~\cite{ganesh2012}. Additionally, Ganesh et al. also
proved conditional decidability results for the quantifier-free theory
of string equations and linear arithmetic over the string length
function~\cite{ganesh2012}.
As automated reasoning tools and algorithms for the satisfiability
problem for the theory $T_{s,n}$ continue to be intensively researched
and developed, it is a natural question to ask whether the theory is
indeed decidable. This question has been open for at least over the 15
years since interest in string solvers dramatically increased in the
formal methods community, and is the primary focus of this paper.
\subsection{Problem Statement}
We answer the following three questions in this paper:
\begin{enumerate}
\item Is the satisfiability problem for the quantifier-free fragment
of a first-order two-sorted theory $T_{s,n}$ of finite-length
strings over a finite alphabet decidable, whose functions and
predicates are as follows: concatenation function and the equality
predicate over string terms, string to natural number conversion
predicate, length function from string terms to natural numbers, and
linear arithmetic over natural numbers and length function.
An answer to this question may give us clues to decidability
questions relating to certain fragments of $T_{s,n}$ that remain
open. For example, it is not known whether the quantifier-free
theory of word equations and equality over the length function is
decidable, and this problem has been open for at least 5
decades~\cite{Matiyasevich}. Furthermore, as discussed above, the answer
impacts practical string solvers such as
CVC4 and Z3str2 which in currently implement incomplete algorithms
to decide the satisfiability problem for the theory $T_{s,n}$. We
show that the satisfiability problem for the theory $T_{s,n}$ is
undecidable.
\item Is the string-numeric conversion predicate expressible in terms
of string equations and length function? This question is important
from a theoretical point of view because if such an expressibility result exists,
then this immediately settles the open question
regarding the satisfiability problem for the quantifier-free theory
of string equations and length.
\item What is a consistent (possibly minimal) axiomatization $\Gamma$
for the functions and predicates of $T_{s,n}$? Is the first-order
many-sorted fully-quantified theory $T_{\Gamma}$ obtained as a
logical closure of the axiom set $\Gamma$ complete? (Note that the
existential closure of the quantifier-free first-order many-sorted
theory $T_{s,n}$ is a subset of $T_{\Gamma}$.)
\end{enumerate}
\subsection{Contributions in Detail}
In greater detail, the contributions of this paper are as follows:
\begin{enumerate}
\item We prove that the satisfiability problem for the quantifier-free
theory of string equations, linear arithmetic over string length,
and string-number conversion is undecidable. This problem has been
open for some time, and is of great interest to formal verification
researchers~\footnote{Note that the theory $T_{s,n}$ is stronger
than the quantifier-free theory of string equations and linear
arithmetic over string length function, since $T_{s,n}$
additionally has the string-number conversion predicate.}. The
ability to model string concatenation, equality, linear arithmetic
over length, and string-natural number conversions is particularly
useful in identifying security vulnerabilities in applications
developed using many modern programming languages, including
JavaScript web
applications~\cite{OISSTA14,saxena10kudzu}. (Section~\ref{sec:undecidability})
\item We also show that the $\pi$ predicate from the power arithmetic
theory $T_p$, which asserts the equality $z = x * 2^y$,
is expressible in terms of the $numstr$ predicate from
$T_{s,n}$. More precisely, we encode $\pi$ using only the $numstr$
predicate, string equations, and string length function. In the
above-mentioned undecidability theorem, we establish that
$numstr$ can be encoded using only the $\pi$ predicate, string
equations, and string length function. These two reductions put
together suggest that the $\pi$ predicate is expressible using string
equations and length function iff $numstr$ is. Expressibility results
are very useful tools in constructing reductions, distinguishing the
expressive powers of various theories, and in establishing
(un)-decidability results. Additionally, our expressibility results
suggest that the $numstr$ predicate is much more complex, both from
a theoretical and a practical point-of-view, than it seems at first
glance. (Section~\ref{sec:definability})
\item We establish a consistent finite axiomatization $\Gamma$ for the
functions and predicates in the language $L$ of
$T_{s,n}$. Additionally, we show that the first-order many-sorted
$L$-theory $T_{\Gamma}$, that is the closure of the axioms $\Gamma$,
is not a complete theory. That is, there are $L$-sentences $\phi$
such that $T_{\Gamma}$ does not entail either $\phi$ or its
negation. (Section~\ref{sec:soundness})
\end{enumerate}
The paper is organized as follows: In Section~\ref{sec:prelim} we
provide the syntax and semantics of the theory $T_{s,n}$. In
Section~\ref{sec:undecidability} we prove the undecidability of the
satisfiability problem of $T_{s,n}$. In
Section~\ref{sec:definability}, we show a reduction from the power
arithmetic theory to $T_{s,n}$. In Section~\ref{sec:soundness}, we
discuss the consistency of an axiom system $\Gamma$ for the language of $T_{s,n}$, and
in Section~\ref{sec:incompleteness} we establish
that the theory $T_{\Gamma}$ is incomplete. In
Section~\ref{sec:relwork} we provide a comprehensive overview of the
decidability/undecidability results for theories of strings over the
last several decades, and the practical relevance of this theory
in the context of verification and security. Finally, we conclude in
Section~\ref{sec:conclusions}, and provide a list of open problems
related to various extensions and fragments of the theory $T_{s,n}$
some of which have been open for many decades now.
\section{Preliminaries}
\label{sec:prelim}
In this section, we define the syntax and semantics of the
first-order, many-sorted, language $L$ of string (aka word) equations
with concatenation, length function over string terms, linear
arithmetic over natural numbers and the length function, and
string-number conversion predicate.
In Section~\ref{sec:soundness}, we will present an axiom system $\Gamma$ for this language
and prove that it is consistent.
\subsection{The Language $L$: Syntax for Theories over String Equations, Length, and String-Number Conversion}
We first define the countable language $L$ below, i.e., its sorts, and
constant, function, and predicate symbols.
\begin{enumerate}
\item {\bf Sorts:} The language is many-sorted, with a string sort
$str$ and a natural number sort $num$. The Boolean sort $Bool$ is
standard. When necessary, we write the sort of an $L$-term $t$
explicitly as $t:sort$.
\item {\bf Finite Alphabet:} We fix a finite alphabet $\Sigma =
\{0,1\}$ over which all strings are defined. As necessary, we may
subscript characters of $\Sigma$ with an $s$ to indicate that their
sort is str.
\item {\bf String and Natural Number Constants:} We fix a two-sorted
set of constants $Con = Con_{str} \cup Con_{num}$. The set
$Con_{str}$ is a subset of $\Sigma^*$, the set of all finite-length
string constants over the finite alphabet $\Sigma$. Elements of
$Con_{str}$ will be referred to as {\it string constants} or simply
{\it strings}. The empty string is represented by
$\epsilon$. Elements of $Con_{num}$ are the {\emph natural numbers}
starting from 0. As necessary, we may subscript numbers by $n$ to
indicate that their sort is num.
\item {\bf String and Numeric Variables:} We fix a disjoint two-sorted
set of variables $var = var_{str} \cup var_{num}$; $var_{str}$
consists of string variables, denoted $X,Y,S, \ldots$ that range
over string constants, and $var_{num}$ consists of numeric
variables, denoted $m,n,\ldots$ that range over the natural numbers.
\item {\bf String Function Symbols:} The string function symbols
include the concatenation operator $\cdot: str \times str \rightarrow str$
that take as argument two string terms and outputs a string term,
and the length function $len: str \rightarrow num$ that takes as
argument a string and outputs a natural number.
\item {\bf Linear Arithmetic Function Symbols:} The natural number
(aka numeric) function symbols include the addition symbol $+: num \times
num \rightarrow num$, that takes as argument two numeric terms and
outputs a numeric term. (Following standard practice in mathematical
logic literature, we allow multiplication by constants as a
shorthand.)
\item {\bf String Predicate Symbols:} The predicate symbols over
string terms include the equality symbol $=_s: str \times str \rightarrow
Bool$ that takes as argument two string terms and evaluates to a
Boolean value, and the string-number conversion predicate
$numstr:num \times string \rightarrow Bool$.
\item {\bf Natural Number Predicate Symbols:} The predicate symbols
over natural number terms include the equality symbol $=_n: num \times
num \rightarrow Bool$, and the inequality predicate $\leq: num \times num
\rightarrow Bool$.
\end{enumerate}
\subsection{Terms and Formulas in the Language $L$}
\noindent{\bf Terms:} $L$-terms may be of string or numeric sort. A
string term ($t_{str}$ in Figure~\ref{fig:syntax}) is inductively
defined as either an element of $var_{str}$, an element of
$Con_{str}$, or a concatenation of string terms (denoted by the
function $concat$ or interchangeably by the $\cdot$ operator). A numeric or
natural number term ($t_{num}$ in Figure~\ref{fig:syntax}) is an
element of $var_{num}$, an element of $Con_{num}$, the length function
applied to a string term, a constant multiple of a length term, or a
sum of length terms. (Note that for convenience we may write
concatenation and addition as $n$-ary functions, even though we define
them as binary operators.)
\noindent{\bf Atomic Formulas:} There are five types of atomic
formulas as given in Figure~\ref{fig:syntax}: (1) word equations
($A_{wordeqn}$), (2) linear arithmetic predicates over natural numbers
and length constraints ($A_{num}$), and (3) string-numeric conversion
predicates ($A_{numstr}$).
\noindent{\bf Quantifier-free Formulas:} Boolean combination of atomic
formulas. The term ``quantifier-free'' formulas means
that each free variable is implicitly existentially quantified and no
explicit quantifiers may be written in the formula.
\noindent{\bf Formulas and Prenex-normal Form:} $L$-Formulas are
defined inductively over atomic formulas (see
Figure~\ref{fig:syntax}). The symbol Qx refers to a block of
quantifiers over a set $x$ of variables. We assume that formulas are
always represented in prenex-normal form (a block of quantifiers
followed by a quantifier-free formula).
\noindent{\bf Free and Bound Variables, and Sentences:} We say that a
variable under a quantifier in a formula $\phi$ is bound. Otherwise we
refer to variables as free. A formula with no free variables is called
a sentence.
\begin{figure}[t!]
\[
\begin{array}{llll}
F & \Coloneqq & Atomic \hspace{3mm} | \hspace{3mm} F \wedge F \hspace{3mm} | \hspace{3mm} F \vee F \hspace{3mm} | \hspace{3mm} \neg F \hspace{3mm} | \hspace{3mm} Qx. F\\
Atomic & \Coloneqq & A_{wordeqn} \hspace{3mm} | \hspace{3mm} A_{num} \hspace{3mm} | \hspace{3mm} A_{numstr} & \\
A_{wordeqn} & \Coloneqq & t_{str} = t_{str} & \\
A_{num} & \Coloneqq & t_{num} = t_{num} \hspace{3mm} | \hspace{3mm} t_{num} < t_{num} & \\
A_{numstr} & \Coloneqq & numstr(n, s) \\
& & \text{where } n \in t_{num}, s \in t_{str} \\
t_{str} & \Coloneqq & a \hspace{3mm} | \hspace{3mm} X \hspace{3mm} | \hspace{3mm} concat(t_{str},...,t_{str}) \\
& & \text{where} \hspace{1mm} a \in Con_{str} \hspace{1mm} \& \hspace{1mm} X \in var_{str}\\
t_{num} & \Coloneqq & m \hspace{3mm} | \hspace{3mm} v \hspace{3mm} | \hspace{3mm} len(t_{str}) \hspace{3mm} | \hspace{3mm} t_{num} + t_{num} \\
& & \text{where} \hspace{1mm} m \in Con_{num} \hspace{1mm} \& \hspace{1mm} v \in var_{num}\\
\end{array}
\]
\caption{\label{fig:syntax} The syntax of $L$-formulas.}
\end{figure}
\subsection{Signature of the Theory $T_{s,n}$}
We define the signature of $T_{s,n} = \left< \Sigma^{*}, \mathbb{N},
0_s, 1_s, \cdot, 0_n, 1_n, +, len, numstr, =_{s}, =_{n}, <_{n}
\right>$, where $\Sigma^{*}$ is the set of all string constants over a
finite alphabet $\Sigma$, $\mathbb{N}$ is the set of natural numbers,
$\cdot$ is the two-operand string concatenation function, $+$ is the
two-operand addition function for natural numbers, $len$ is a
function that takes a string and returns its length as a natural
number, $=_{s}$ is the equality predicate over strings, $=_{n}$ and
$<_{n}$ are the equality and less-than predicates over natural
numbers, and $numstr$ is a two-argument predicate such that $numstr(i,
s)$ is true for natural number $i$ and string $s$ if and only if $s$
is a valid binary representation of the natural number $i$. By a
``valid binary representation'' we mean that $s$ does not contain any
characters other than `0' and `1', and interpreting the characters of
$s$ as the digits of a numeral in base 2, where the last character of
$s$ is the least significant digit, produces a natural number that is
equal to $i$. (Hence we require that the alphabet $\Sigma$ contain
characters `0' and `1'.)
Note that the signatures of all theories
considered in this paper are countable.
\subsection{$L$-Semantics and the Canonical Model $\mathbb{M}$}\
\label{sec:canonicalmodel}
In this section, we provide semantics for the symbols in the language
$L$ via what we call a canonical model $\mathbb{M}$. We take the
finite alphabet $\Sigma$ to be the set $\{0,1\}$. The results
presented here can be easily extended to other finite alphabets. We
assume standard definitions for the terms \textit{interpretation of
symbols} and \textit{model}~\cite{HodgesModelTheory}.
\noindent{\bf Universe of Discourse for symbols in $L$:} The universe
of discourse over which all symbols are interpreted is two-sorted
disjoint sets. The first set $\Sigma^*$, of sort str, is the set of all
finite-length strings over the alphabet $\Sigma = \{0,1\}$ including
the empty string (represented by $\epsilon$), and the second set $\mathbb{N}$, of sort
num, is the set of natural numbers starting from $0$.
\noindent{\bf Interpretation of Natural Number Variables, Constants,
Functions and Predicates:} Variables of num sort range over the set
$\mathbb{N}$ of natural numbers, and constants represent corresponding
natural numbers. Note that all natural number constants are
represented as binary numbers, unless otherwise specified. The
function $+$ and the predicates $=_n, \leq$ have the standard
interpretations. (Multiplication by constant is also treated in the
standard way as a shorthand for repeated addition.)
\noindent{\bf Interpretation of String Variables, Constants,
Functions, and Predicates:} String constants are interpreted as
a finite concatenation of letters $0$ and $1$ and correspond to appropriate
strings in $\Sigma^*$, and string variables range over values from
$\Sigma^*$. The string concatenation function is inductively defined
over elements of $\Sigma^*$ in the natural way.
\noindent{\bf What is meant by the Length of a String:} For a string
or a word, $w$, $len(w)$ denotes the length of $w$, or equivalently,
the (natural) number of characters from $\Sigma$ in the interpretation
of $w$ under a given assignment.
\noindent{\bf The Meaning of $numstr$ Predicate:} The $numstr$
predicate asserts that the interpretation of its string argument is a
valid binary representation of the natural number represented by its
numeric argument. A string $s$ is a valid binary representation of a
natural number $i$ iff the following properties hold:
\begin{enumerate}
\item $s$ does not contain any characters in $\Sigma$ other than `0'
and `1'.
\item Let $s[n]$ denote the $n$th character in $s$, where $n$ is a
natural number between 0 and $len(s) - 1$ inclusive. Let $s'[n]$
denote the numeric value of $s[n]$, where $s'[n] = 1$ if $s[n]$ is
`1', and $s'[n] = 0$ if $s[n]$ is `0'. Then it must be the case
that $\sum_{n=0}^{len(s) - 1} s'[n] 2^{len(s) - n - 1} = i$.
(Here we expand the characters of $s$ into a binary representation
of $i$.)
\end{enumerate}
\noindent{\bf The Meaning of Equality between String Terms:} For a
word equation of the form $t_1 = t_2$, we refer to $t_1$ as the left
hand side (LHS), and $t_2$ as the right hand side (RHS). Two string
terms are considered equal if their interpretations have the same
characters appearing in the same order, i.e., the LHS and RHS evaluate
to the same string in $\Sigma^*$ under the appropriate interpretation
for variables and constants in the LHS and RHS of the given equality.
\noindent{\bf The Canonical Model:} This interpretation
of $L$-symbols along with the universe of discourse defines the
canonical $L$-model. (An interpretation of a set of symbols in a
language $L$ along with universe of discourse is called an $L$-model.)
\subsection{Standard Logic Definitions}
Here we give some standard definitions such as assignment,
satisfiability, validity, consistency of an axiom system, and
completeness of a theory.
\noindent{\bf Assignments, Satisfiability, Validity, and Equisatisfiability:} Given an
$L$-formula $\theta$, an {\it assignment} for $\theta$ (with respect
to $\Sigma)$ is a map from the set of free variables in $\theta$ to
$\Sigma^* \cup \mathbb{N}$ (where string variables are mapped to
strings and natural number variables are mapped to numbers). Given
such an assignment, $\theta$ can be interpreted as an assertion about
$\Sigma^*$ and $\mathbb{N}$. If this assertion is true, then we say
that $\theta$ itself is {\it true} under the assignment. If there is
some assignment which makes $\theta$ true, then $\theta$ is called
{\it satisfiable}. An $L$-formula with no satisfying assignment is
called an {\it unsatisfiable} formula. We say two formulas $\theta, \phi$ are {\it equisatisfiable} if $\theta$ is satisfiable iff $\phi$
is satisfiable. Note that this is a broad definition: equisatisfiable
formulas may have different numbers of assignments and, in fact, need
not even be from the same language. We say a formula is valid if it is
true under all possible assignments.
\noindent{\bf The Satisfiability Problem:} The {\it satisfiability
problem} for a set $S$ of formulas is the problem of deciding
whether any given formula in $S$ is satisfiable or not. We say that
the satisfiability problem for a set $S$ of formulas is decidable if
there exists an algorithm (or \textit{satisfiability procedure}) that
solves its satisfiability problem. Satisfiability procedures must have
three properties: soundness, completeness, and termination. Soundness
and completeness guarantee that the procedure returns ``satisfiable"
if and only if the input formula is indeed satisfiable. Termination
means that the procedure halts on all inputs. In a practical
implementation, some of these requirements may be relaxed for the sake
of improved typical performance. Analogous to the definition of the
satisfiability problem for formulas, we can define the notion of the
\textit{validity problem} (aka decision problem) for a set $Q$ of
sentences in a language $L$. The validity problem for a set $Q$ of
sentences is the problem of determining whether a given sentence in
$Q$ is true under all assignments.
\noindent{\bf Logical Entailment:} We say that a set of sentences $C$
entails a sentence $\phi$, written as $C \models \phi$, if any
model $A$ of $C$ is also a model of $\phi$. We say a model $A$ is a
model of a set of sentences $C$, if all sentences of $C$ are true under
some assignments in $A$, written as $A \models C$.
\noindent{\bf Consistency of an Axiom System:} A set of $L$-sentences
may be designated as axioms. We say that an axiom system $A$ is
consistent if for any $L$-formula $\phi$, the axiom system $A$ does
not logically imply both a formula $\phi$ and its negation $\neg \phi$.
\noindent{\bf Theory, Closure of an Axiom System, Completeness of a
Theory:} A set of $L$-sentences is referred to as a theory. The
closure $C$ of an axiom system $A$ is the set of sentences that are
logically implied by $A$, i.e., every model of $A$ is a model of the
set $C$. We say that a theory $T$ is complete if for every
$L$-sentence $\phi$, $T$ logically entails either $\phi$ or its
negation.
\section{Related Work}
\label{sec:relwork}
We provide a relatively comprehensive overview of both theoretical and
practical work done by researchers in the context of theories over
strings.
\subsection{Theoretical Results over Theories of Strings}
In his original 1946 paper, Quine \cite{Quine} showed that the
first-order theory of string equations (i.e., quantified sentences
over Boolean combination of word equations) is undecidable. Due to
the expressibility of many key reliability and verification questions
within this theory, this work has been extended in many ways.
One line of research studies fragments and modifications of this base
theory which are decidable. Notably, in 1977, Makanin proved that the
satisfiability problem for the quantifier-free theory of word
equations is decidable \cite{makanin}. In a sequence of papers,
Plandowski and co-authors showed that the complexity of this problem
is in PSPACE \cite{plandowski2006}. Stronger results have been found
where equations are restricted to those where each variable occurs at
most twice\cite{robsondiekert} or in which there are at most two
variables \cite{CharaPach, IliePland, dabrowski2002weo}. In the first
case, satisfiability is shown to be NP-hard; in the second, polynomial
(which was improved further in the case of single variable word
equations). Concurrently, many researchers have looked for the exact
boundary between decidability and undecidability. Durnev \cite{durnev}
and Marchenkov \cite{marchenkov} both showed that $\forall\exists$
sentences over word equations is undecidable. Despite decades of
effort, however, the satisfiability problem for the quantifier-free
theory of word equations and numeric length remains
open~\cite{makanin,plandowski2006,ganesh2012,Matiyasevich}. More
recently, Artur J\"{e}z presents a technique called recompression that
gives more efficient algorithms for many fragments of theory of word
equations~\cite{jez}.
A related result was shown by
Furia~\cite{DBLP:journals/corr/abs-1001-2100}, wherein he proved that
the quantifier-free theory of integer sequences is decidable. The
framework he establishes in that paper is closely related to the
theory of concatenation and word equations, but weaker than either
strings plus numeric length or the theory of arrays due to the
inability of the theory of sequences
to express facts relating indices directly to elements.
Word equations augmented with additional predicates yield richer
structures which are relevant to many applications, as we have
considered here. In the 1970s, Matiyasevich formulated a connection
between string equations augmented with integer coefficients whose
integers are taken from the Fibonacci sequence and Diophantine
equations~\cite{MatiyasevichHilbertPub,Matiyasevich}. In particular, he showed that proving
undecidability for the satisfiability problem of this theory would
suffice to solve Hilbert's Tenth Problem in a novel way.
Schulz \cite{schulz} extended Makanin's satisfiability algorithm to
the class of formulas where each variable in the equations is
specified to lie in a given regular set (i.e. a set defined by a regular language).
This is a strict generalization of the solution sets of word equations. Further work
in~\cite{KarhumakiPM97} shows that the class of sets expressible
through word equations is incomparable to that of regular sets.
Matiyasevich extends Schulz's result to decision problems
involving trace monoids and free partially commutative monoids~\cite{matiyasevichExtra1,matiyasevichExtra2,matiyasevichExtra3}.
M\"oller~\cite{Moller} studies word equations and related theories as
motivated by questions from hardware verification. More specifically,
M\"oller proves the undecidability of the existential fragment of a
theory of fixed-length bit-vectors, with a special finite but parameterized
concatenation operation, extraction of substrings, and
equality predicate. Although this theory is related to the word
equations that we study, it is more powerful because of the
finite but possibly arbitrary concatenation.
The question of whether the satisfiability problem for the
quantifier-free theory of word equations and length constraints is
decidable has remained open for several decades. Our decidability
results are a partial and conditional
solution. Matiyasevich~\cite{matiyasevich2008} observed the relevance
of this question to a novel resolution of Hilbert's Tenth Problem. In
particular, he showed that if the satisfiability problem for the
quantifier-free theory of word equations and length constraints is
undecidable, then it gives us a new way to prove Matiyasevich's
Theorem (which resolved the famous problem)~\cite{Matiyasevich,
matiyasevich2008}.
B\"{u}chi et al.~\cite{Buchi90} consider extensions of the
quantifier-free theory of word equations with various length
predicates. They find that a predicate $Elg$ that asserts that two
strings have equal length is not existentially definable in this
theory, and that by introducing two stronger functions,
$Lg_{1}$ and $Lg_{2}$ which count the number of occurrences of the
characters `1' and `2' respectively, the resulting theory is
undecidable.
The source of undecidability, as the authors identify, is the
ability for these functions to match the number of occurrences of
certain subsequences, which allows them to encode addition and
multiplication. Our result is similar to
this one; B\"{u}chi proposes an encoding of arithmetic into word
equations, while we assume an extension of word equations that already
contains the $len$ function and natural number arithmetic (as well as
$numstr$), and encode an arithmetic operation into operations on
strings.
\subsection{String Solvers and their application in Program Analysis, Bug-finding, and Verification}
Formulas over strings became important in the context of automated
bug-finding~\cite{hampi, prateek} and analysis of database/web
applications~\cite{emmiMS2007, rupak, WassermannSu2007}. These program
analysis and bug-finding tools read string-manipulating programs and
generate formulas expressing their outputs. These formulas contain
equations over string constants and variables, membership queries over
regular expressions, inequalities between string lengths, and in some
cases the string-integer conversion predicate/functions. In practice,
formulas of this form have been solved by off-the-shelf solvers such
as HAMPI~\cite{hampi2, hampi}, Z3str2~\cite{z3strcav},
CVC4~\cite{Liang2014}, or Kaluza~\cite{prateek}. All these solvers are
based on sound algorithms, but are incomplete in different ways.
Zheng et al. \cite{z3strcav} present the Z3str2 solver for the
quantifier-free many-sorted theory $T_{wlr}$ over word equations,
membership predicate over regular expressions, and length function,
which consists of the string (str) and numeric (num) sorts.
S3~\cite{s3} is another tool that supports word equations, length
function, and regular expression membership predicate. S3 internally
uses a version of Z3str2 to handle word equations and length
functions.
CVC4 \cite{Liang2014} handles constraints over the theory of unbounded
strings with length and RE membership. Solving is based on
multi-theory reasoning backed by the DPLL($T$) architecture combined
with existing SMT theories. The Kleene star operator in RE formulas
is dealt with via unrolling as in Z3str2.
In a separate paper, Liang et al.~\cite{Liang2015} give a decision
procedure for regular language membership and numeric length
constraints over unbounded strings. However, their decision procedure
does not consider word equations, and hence is many ways weaker than
the theory $T_{s,n}$ we consider in this paper. Hence the algorithm
they propose, while useful in some contexts, is weaker than the full
theory of strings, and their result does not yet resolve the question
of whether the quantifier-free theory of strings and numeric length
constraints is decidable.
It must be stressed that all the solvers (including Z3str2, CVC4, and
S3) that purportedly solve the satisfiability problem for the theory
$T_{s,n}$ or the word equation and length function fragment of
$T_{s,n}$ are incomplete. Solvers such as HAMPI are limited by the
fact that they reason only over a bounded string domain, where the
bound is given as part of the input.
Pex \cite{Tillmann2008} is a parameterized unit testing tool for .NET
that observes program behaviour and uses a constraint solver in order
to produce test inputs which exercise new program behaviour. It
integrates a specialized string solver in order to generate string
inputs that satisfy the desired branch conditions.
\section{Result 3: Axiomatization $\Gamma$ of the Language $L$}
\label{sec:soundness}
In this section, we present a consistent
axiomatization $\Gamma$ for the functions and predicates of the
language $L$ (as presented earlier in Section~\ref{sec:prelim}).
\subsection{The Axiomatization $\Gamma$}
\label{sec:axioms}
We introduce an axiom system $\Gamma$ for the language $L$.
For the sake of readability, we choose not to specify the
sorts of various terms if they are clear from context.
\subsubsection{Axioms of Linear Arithmetic over the Natural Numbers}
The following axioms follow from the ones for Presburger
arithmetic. Note that both Presburger arithmetic and the linear
arithmetic as part of $\Gamma$ include only the addition symbol, and
do not have full multiplication. (Multiplication by constants is
simply a short-hand for repeated addition up to a known constant
bound.)
\begin{enumerate}
\item $0 \neq 1$
\item $\forall x : \lnot (0 = x + 1)$
\item $\forall x \exists y : x \ne 0 \to : y + 1 = x$
\item $\forall x \, y : \lnot (x < y \land y < x + 1)$
\item $\forall x\, y : x + y = y + x$
\item $\forall x\, y\, z : (x + y = x + z) \to (y = z)$
\item $\forall x\, y : x + 1 = y + 1 \to x = y$
\item $\forall x : x + 0 = x$
\item $\forall x\, y : x + (y + 1) = (x + y) + 1$
\item $\forall x\, y \exists c : x < y \to \lnot (c = 0) \land x + c = y$
\item $\exists c \forall x \, y: \lnot (c = 0) \land x + c = y \to x < y $
\end{enumerate}
\subsubsection{Axioms of Equality for Strings and Natural Numbers}
It is assumed that the equality predicate for both string and numeric
sorts is reflexive, symmetric, and transitive. In addition, we have
the following axiom recursively defined over string terms. Below we
present the axiom for string constants over the alphabet $\Sigma$.
\begin{enumerate}[resume]
\item $\forall A \, B : A = B \to len(A) = len(B)$
\end{enumerate}
\subsubsection{Axioms of Concatenation}
Concatenation is associative, but not commutative.
\begin{enumerate}[resume]
\item $\forall x : x \cdot \epsilon = \epsilon \cdot x = x$
\item $\forall xyz : x \cdot (y \cdot z) = (x \cdot y) \cdot z$
\end{enumerate}
\subsubsection{Axioms of the $len$ Function}
\begin{enumerate}[resume]
\item $\forall x : len(x) = 0 \iff x = \epsilon$
\item $\forall x : len(x) = 1 \to \bigvee_{c \in \Sigma} x = c$
\item $\forall x\, y : len(x \cdot y) = len(x) + len(y)$
\item $\forall c \in \Sigma : len(c) = 1$
\end{enumerate}
\subsubsection{Axioms of $numstr$}
The axioms for the $numstr$ predicate essentially allow us to define a
natural mapping between natural numbers, represented in binary, and
strings over $\Sigma$.
\begin{enumerate}[resume]
\item $\forall i : \lnot numstr(i, \epsilon)$
\item $numstr(0, ``0")$
\item $numstr(1, ``1")$
\item $\forall s\, i : len(s) = 1 \land s \ne ``0" \land s \ne ``1" \to \lnot numstr(i, s)$
\item $\forall i\, x\, z : numstr(i, x) \land ``0"z = z``0" \to numstr(i, zx)$
\item $\forall i\, x\, z : numstr(i, zx) \land ``0"z = z``0" \land z \ne \epsilon \land x \ne \epsilon \to numstr(i, x)$
\item $\forall x\, y\, z : (\exists u\, v : numstr(u, y) \land numstr(v, z)) \to (numstr(x, yz) \iff x = u_b v_b)$, where $u_b$ and $v_b$ are the binary
digits of $u$ and $v$ respectively. (This describes distribution of $numstr$ over a concatenation.)
\item $\forall x\, y\, z \exists u \, v \, w : numstr(x + y, z) \to : len(u) = x \land len(v) = y \land w = uv \land numstr(len(w), z)$
\item $\exists u\, v\, w \forall x \, y \, z: len(u) = x \land len(v) = y \land w = uv \land numstr(len(w), z) \to numstr(x + y, z)$
\end{enumerate}
\subsection{Relationship between $T_{\Gamma}$ and $T_{s,n}$}
We refer to the set of sentences logically entailed by the axiom
system $\Gamma$ as the theory $T_{\Gamma}$. Note that this set
contains sentences with arbitrary quantifiers in them. We assume that
sentences are always written in prenex normal form.
The set $T_{s,n}$ is a set of quantifier-free $L$-formulas. As discussed
before, when we use the term ``quantifier-free'' formulas, we mean
that each free variable is implicitly existentially quantified and
there are no other explicit quantifiers in the formula. When the
formulas in $T_{s,n}$ are existentially quantified, we get the same set
of sentences implied by $\Gamma$ that have a single set of existential
quantifiers in prenex normal form. We also call this the existential
fragment of $T_{\Gamma}$.
\subsection{Consistency of $\Gamma$}
\begin{Theorem}
The axiom system $\Gamma$ presented in Section~\ref{sec:axioms} is consistent.
\end{Theorem}
\begin{proof}
It is well known that a theory or axiom system is consistent if it
has a model~\cite{HodgesModelTheory}. We prove consistency by
showing that the structure established in
Section~\ref{sec:canonicalmodel} is in fact a model of $\Gamma$. The
remainder of the proof is structured in sections corresponding to
those in the description of $\Gamma$.
\begin{enumerate}
\item \textbf{Axioms of arithmetic over natural numbers:} These are
standard axioms for natural number arithmetic. Since we choose
$\mathbb{N}$ to model numeric terms, it follows that these axioms
are true over the natural numbers.
\item \textbf{Axioms of equality for strings and natural numbers:}
This axiom states that if two strings $A$ and $B$ are equal, then
$A$ and $B$ have the same length, in addition to the standard
axioms of equality. Our model of string terms states that two
strings are equal if they have the same characters appearing in the
same order, and that the length of a string is the natural number
of characters in that string. It follows that if two strings are
equal, then they have the same characters, and therefore have the
same length.
\item \textbf{Axioms of concatenation:} The first axiom states that
concatenating any string with the empty string, on either side,
produces a result equal to the original string. Our model
represents the result of concatenating $A$ and $B$ as a string
having all of $A$'s characters (in the same order) followed by all
of $B$'s characters (also in the same order). If one of $A$ or $B$
is empty, it follows that the resulting string has the same
characters and in the same order as the other string, and therefore
the two are equal.
The second axiom states that string concatenation is associative.
Suppose strings $X, Y, Z$ are composed of characters $x_1 \hdots
x_u$, $y_1 \hdots y_v$, $z_1 \hdots z_w$ respectively. Then by
definition of concatenation in our model, we have:
\begin{align*}
y \cdot z & = y_1 \hdots y_v z_1 \hdots z_w \\
x \cdot (y \cdot z) & = x_1 \hdots x_u y_1 \hdots y_v z_1 \hdots z_w \\
x \cdot y & = x_1 \hdots x_u y_1 \hdots y_v \\
(x \cdot y) \cdot z & = x_1 \hdots x_u y_1 \hdots y_v z_1 \hdots z_w \\
x \cdot (y \cdot z) & = (x \cdot y) \cdot z
\end{align*}
Therefore the axiom holds in this model.
\item \textbf{Axioms of the Length Function:} The first axiom states
that the only string having length 0 is the empty string $\epsilon$,
which follows trivially from the definition of the set of string
constants $\Sigma^{*}$.
The second axiom states that the length of the concatenation of $A$
and $B$ is equal to the sum of the lengths of $A$ and $B$ taken
separately. Our model represents the result of concatenating $A$
and $B$ as a string having all of $A$'s characters (in the same
order) followed by all of $B$'s characters (also in the same
order). Since characters are conserved by this process, it follows
that the resulting string has length equal to the sum of the lengths
of $A$ and $B$.
The third axiom states that all single-character strings have length
1, which holds trivially.
\item \textbf{Axioms of $numstr$ string-numeric conversion predicate:}
The first four axioms state some basic properties of string-number
conversion: $\epsilon$ is not the binary representation of any
number, ``0'' is the binary representation of 0, ``1'' is the binary
representation of 1, and single-character strings that are not ``0''
or ``1'' are not the binary representation of any number. These
axioms are true by inspection.
The fifth and sixth axioms show that leading zeroes can be added to and removed from
a string without changing its value.
We can show that this is true in our model by demonstrating that if $y$ is a binary string and $z$ is a string of all zeroes,
the binary expansions of $y$ and $zy$, denoted $y_b$ and $(zy)_{b}$ respectively, both represent the same natural number:
\begin{align*}
y_{b} & = y[0] 2^{length(y)-1} + y[1] 2^{length(y) - 2} + \hdots \\
& + y[length(y) - 2] 2^{1} + y[length(y)-1] 2^{0} \\
(zy)_{b} & = (zy)[0] 2^{length(zy) - 1} + (zy)[1] 2^{length(zy) - 2} + \hdots \\
& + (zy)[length(z)-1] 2^{length(zy) - length(z) - 2} \\
& + (zy)[length(z)] 2^{length(zy) - length(z) - 1} + \hdots \\
& + (zy)[length(zy) - 1] 2^{length(zy) - length(zy)} \\
(zy)[0] & = 0 \\
(zy)[1] & = 0 \\
\vdots & \\
(zy)[length(z)-1] & = 0 \\
(zy)_{b} & = 0 + 0 + \hdots + 0 \\
& + (zy)[length(z)] 2^{length(zy) - length(z) - 1} + \hdots \\
& + (zy)[length(zy) - 1] 2^{length(zy) - length(zy)} \\
(zy)[length(z)] & = y[0] \\
(zy)[length(z) + 1] & = y[1] \\
\vdots & \\
(zy)[length(z) + length(y) - 1] & = y[length(y) - 1] \\
length(zy) - length(z) & = length(y) \\
(zy)_{b} & = y[0] 2^{length(y) - 1} + \hdots + y[length(y) - 1] 2^{0} \\
& = y_{b}
\end{align*}
Hence adding or deleting leading zeroes has no effect on what number is represented by a given binary string, and so these axioms hold.
The seventh axiom holds if we assume that all numbers are written in binary;
concatenating the binary digits of two numbers is equivalent to
concatenating the string representations of those numbers.
The eighth axiom illustrates how to perform string-number conversion
on an addition term $x + y$. It suffices to show
that $len(w) = x + y$:
\begin{align*}
w & = uv \\
len(w) & = len(u) + len(v) \\
& = x + y
\end{align*}
\end{enumerate}
This completes the proof.
\end{proof}
\section{Result 4: Incompleteness of the Theory $T_{\Gamma}$}
\label{sec:incompleteness}
We first state a number of useful definitions and theorems related to
completeness of first-order theories from the standard model theory
literature~\cite{HodgesModelTheory}.
\begin{Definition}
A first-order theory $T$ in language $L$ is \textbf{complete} if for
all $L$-formulas $\phi$, exactly one of $\phi$ and $\lnot \phi$ is a
consequence of $T$.
\end{Definition}
\begin{Definition}
Two models $A, B$ of a first-order theory are \textbf{elementarily
equivalent} if for all first order $L$-sentences $\phi$, $A \vDash
\phi \iff B \vDash \phi$.
\end{Definition}
\begin{Theorem}
\label{thm:completeIffEquivalent}
A first-order theory $T$ is complete iff all of its models
are elementarily equivalent~\cite{HodgesModelTheory}.
\end{Theorem}
\noindent{We are now in a position to prove the following result.}
\begin{Theorem}
$T_{\Gamma}$ is incomplete.
\end{Theorem}
\begin{proof}
Consider two models $A, B$ of the theory $T_{\Gamma}$, defined as
follows: $A$ is the canonical model given in
Section~\ref{sec:canonicalmodel}, and $B$ is a restricted version of
$A$ where the only string constants that are allowed
are nonempty string constants with no leading zeroes. (In other
words, the only string constant in $B$ that starts with `0' is
``0''.) It is easy to see that both of these are models of
$T_{\Gamma}$.
Now consider the first-order sentence $J$ which states ``the
$numstr$ predicate describes a bijection between strings and natural
numbers''.
\footnote{Note that as long as the alphabet $\Sigma$ is finite and
string constants are concatenations of a finite number of
characters, in general there exists a bijection between strings and
natural numbers. This follows from the fact that the set $\Sigma^*$
of strings is countably infinite. The argument made in the proof
above deals with a very particular bijection as defined by
$numstr$.}
We state this sentence $J$ formally as follows:
\begin{align*}
& \forall_{num} \, i : \exists_{str} \, s : \left( numstr(i,s) \land \forall_{str} \, t : numstr(i,t) \to s = t \right) \\
\land & \forall_{str} \, s : \exists_{num} \, i : \left( numstr(i,s) \land \forall_{num} \, j : numstr(j,s) \to i = j \right)
\end{align*}
It follows that due to the restrictions on string constants imposed
in $B$, $numstr$ clearly defines a bijection between strings and
natural numbers, where each integer is mapped to the unique string
that is its minimal binary representation, and so $J$ is true in the
model $B$. However, in the model $A$, $numstr$ does not define a
bijection, as by counterexample, $numstr(3, ``11")$ and $numstr(3,
``0011")$ are both true. Therefore $J$ is false in the model $A$.
From this we conclude that $J$ is able to distinguish between $A$
and $B$, and hence $A$ is not elementarily equivalent to $B$; by
Theorem~\ref{thm:completeIffEquivalent}, $T_{\Gamma}$ is
incomplete.
\end{proof}
\section{Result 1: The Undecidability of the Satisfiability Problem for $T_{s,n}$}
\label{sec:undecidability}
In this section we prove that the satisfiability problem for the
first-order many-sorted quantifier-free theory $T_{s,n}$ over string
equations and linear arithmetic over natural numbers extended with
string length and a string-number conversion predicate is undecidable.
\subsection{The Theory of Power Arithmetic $T_{p}$, and B\"{u}chi's Results}
In this subsection, we present the syntax and semantics of the power
arithmetic theory $T_{p}$, and discuss B\"{u}chi's results for this
theory.
\subsubsection{Syntax, Semantics, and the Signature of Theory $T_{p}$}
We define the theory $T_{p}$ to have the signature $\left< \mathbb{N}, 0, 1, +, \pi,
<_{n}, =_{n} \right>$, where $\mathbb{N}$ is the set of natural
numbers, 0 and 1 are constants, $+$ is the two-operand addition
function, $<_{n}$ and $=_{n}$ are the two-operand less-than and
equality predicates, and $\pi$ is a three-operand predicate
\footnote{Representation of $\pi$ as a predicate is somewhat more
natural given that string-number conversion is also represented as a
predicate.} defined as $\pi(p, x, y) \iff p = x \times 2^{y}$. Note
that we only consider the satisfiability problem over the
quantifier-free fragment of $T_p$ (equivalently the existential
closure over quantifier-free formulas).
\subsubsection{B\"{u}chi's Undecidability Result}
\label{sec:buchiresult}
Below we briefly present the necessary context for B\"{u}chi's
undecidability result for theory $T_p$.
We note that Lemmas~\ref{lem:robinson} and~\ref{lem:buchi1},
as well as the statement of Theorem~\ref{thm:TpUndecidable},
are adapted from~\cite{Buchi90} where they were originally presented.
\begin{Lemma}{\bf (Julia Robinson's divisibility lemma)} \label{lem:robinson}
If $m \le n, l > 2n^2,$ and $l+m, l-m | l^2 - n$, then $m^2 = n$.
(Refer to Lemma~5 in~\cite{Buchi90}.)
\end{Lemma}
\begin{Lemma}{\bf (B\"{u}chi's Lemma)} \label{lem:buchi1}
In $T_p = \left< \mathbb{N}, 0, 1, +, \pi \right>$ we can
existentially define addition and multiplication
on $\mathbb{N}$. (Refer to Lemma~6 in~\cite{Buchi90}.)
\end{Lemma}
\begin{Theorem}{\bf (B\"{u}chi's Undecidability Theorem)} \label{thm:TpUndecidable}
The existential theory of $T_p = \left< \mathbb{N}, 0, 1, +, \pi
\right>$ is undecidable. (Corollary~5 in~\cite{Buchi90}.)
\end{Theorem}
\subsection{Proof Idea}
We present a sound, complete, and terminating (recursive) reduction
from the satisfiability problem for the theory of power arithmetic,
$T_{p}$, which is an extension of arithmetic over natural numbers with
a three-argument $\pi$ predicate defined as $\pi(p, x, y) \iff p = x *
2^{y}$, to the satisfiability problem of the theory $T_{s,n}$. This
theory $T_p$ (and its associated satisfiability problem) was shown by
B\"{u}chi to be undecidable~(in \cite{Buchi90}, as outlined in
Section~\ref{sec:buchiresult}).
As the theory $T_{s,n}$ already has arithmetic over natural numbers,
the only detail that is missing is an encoding of the $\pi$ predicate
into $T_{s,n}$. Recall that in
bit-vector arithmetic, an unsigned left shift corresponds to
multiplication by a power of 2. Therefore, if we have a binary string
that represents the natural number $x$ and we concatenate this string
with a string of all zeroes of a given length $y$, the resulting
string will be the binary representation of $x * 2^{y}$. Once this
encoding is provided, then it is easy to see that any quantifier-free
formula in $T_p$ can be reduced equisatisfiably to a quantifier-free
formula in $T_{s,n}$.
\subsection{The Undecidability Theorem}
\begin{Theorem}
The satisfiability problem for the theory $T_{s,n}$ is undecidable.
\end{Theorem}
\begin{proof}
We prove this result via a recursive reduction from the theory $T_p$
(B\"{u}chi's power arithmetic) to theory $T_{s,n}$, i.e., any
quantifier-free formula in $T_p$ can be equisatisfiably reduced to a
quantifier-free formula in $T_{s,n}$. Thus, if the satisfiability
problem for $T_{s,n}$ is decidable then so is the satisfiability
problem for $T_p$. By B\"{u}chi's theorem~\cite{Buchi90} the
satisfiability problem for $T_p$ is undecidable, and hence so is the
satisfiability problem for $T_{s,n}$.
\noindent{\bf The Reduction from $T_p$ to $T_{s,n}$.} We reduce each
constant, function, predicate, and atomic formula of $T_{p}$ to
$T_{s,n}$ by applying the following rules recursively over the input formula:
\begin{enumerate}
\item Each natural number in $\mathbb{N}$ is represented directly as a
constant in $T_{s,n}$.
\item Variables in $T_{p}$ are represented directly as variables of
numeric sort in $T_{s,n}$.
\item Addition of two terms $t_1 + t_2$ is represented directly as
addition over natural numbers, $t_1 + t_2$, in $T_{s,n}$.
\item Equality of terms in $T_{p}$ is represented directly via a
recursive reduction as equality $t_1 =_{n} t_2$ of terms of numeric
sort.
\item The less-than predicate in $T_{p}$ is represented directly as
comparison of natural numbers, $t_1 <_{n} t_2$.
\item The predicate $\pi(p, x, y)$ is expressible as follows: $\exists
z:str, \exists x_{s}:str : \, (``0" \cdot z = z \cdot ``0" \land
len(z) = y \land numstr(p, x_{s} \cdot z) \land numstr(x, x_{s})
)$. The interpretation of the $\pi$ predicate is $p = x\times
2^y$. The variables $z$ and $x_{s}$ are string variables, and $z$ is
a string of the ``0'' character of length equal to y. The $x_s$
variable is the string binary representation of the natural number
$x$. The concatenation of $x_s$ followed by $z$ is a binary
representation of $p$. It is easy to verify that the given formula
over free numeric variables $x,y,p$ is satisfiable iff $\pi(p,x,y)$
is satisfiable.
\end{enumerate}
\noindent{The reduction can easily be extended to arbitrary
quantifier-free formulas in $T_p$. It is easy to verify that the
reduction is sound, complete, and terminating for all
inputs.}
\end{proof}
\subsection{Discussion}
Recall that the satisfiability problem for the theory of
quantifier free string equations with string length remains
open. Knowing whether that theory is decidable would be of value in
many program analysis applications. The theory $T_{s,n}$ we consider
here is arguably more directly relevant to program analysis since many
state-of-art solvers implement exactly this theory, as the extension
of string-number conversion allows it to model similar operations
which are present in almost all programming languages that have a data
structure for strings. Examples of programming language
operations/functions that could be modelled with the string-numeric
conversion predicate include JavaScript's \texttt{parseInt} and
\texttt{toNumber} methods, which perform integer-string and
string-integer conversion.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 5,502 |
{"url":"https:\/\/edge.docs.solana.com\/es\/wallet-guide\/hardware-wallets","text":"Saltar al contenido principal\n\n# Using Hardware Wallets on the Solana CLI\n\nSigning a transaction requires a private key, but storing a private key on your personal computer or phone leaves it subject to theft. Adding a password to your key adds security, but many people prefer to take it a step further and move their private keys to a separate physical device called a hardware wallet. A hardware wallet is a small handheld device that stores private keys and provides some interface for signing transactions.\n\nThe Solana CLI has first class support for hardware wallets. Anywhere you use a keypair filepath (denoted as <KEYPAIR> in usage docs), you can pass a keypair URL that uniquely identifies a keypair in a hardware wallet.\n\n## Supported Hardware Wallets\u200b\n\nThe Solana CLI supports the following hardware wallets:\n\n## Specify a Keypair URL\u200b\n\nSolana defines a keypair URL format to uniquely locate any Solana keypair on a hardware wallet connected to your computer.\n\nThe keypair URL has the following form, where square brackets denote optional fields:\n\nusb:\/\/<MANUFACTURER>[\/<WALLET_ID>][?key=<DERIVATION_PATH>]\n\nWALLET_ID is a globally unique key used to disambiguate multiple devices.\n\nDERVIATION_PATH is used to navigate to Solana keys within your hardware wallet. The path has the form <ACCOUNT>[\/<CHANGE>], where each ACCOUNT and CHANGE are nonnegative integers.\n\nFor example, a fully qualified URL for a Ledger device might be:\n\nusb:\/\/ledger\/BsNsvfXqQTtJnagwFWdBS7FBXgnsK8VZ5CmuznN85swK?key=0\/0\n\nAll derivation paths implicitly include the prefix 44'\/501', which indicates the path follows the BIP44 specifications and that any derived keys are Solana keys (Coin type 501). The single quote indicates a \"hardened\" derivation. Because Solana uses Ed25519 keypairs, all derivations are hardened and therefore adding the quote is optional and unnecessary.","date":"2023-01-31 16:13:54","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.28959009051322937, \"perplexity\": 8577.124000742526}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2023-06\/segments\/1674764499888.62\/warc\/CC-MAIN-20230131154832-20230131184832-00161.warc.gz\"}"} | null | null |
/**
* Created by clstrfvck on 19/04/2017.
*/
//This is a wrapper for the first sign on view/page.
import React from "react"
import { connect } from 'react-redux'
import { passwordChange, credentialChange, verify } from '../actions/validationActions'
import FirstPassword from './FirstPassword'
import axios from "axios"
import MuiThemeProvider from 'material-ui/styles/MuiThemeProvider';
import Redirect from 'react-router-dom'
class Validate extends React.Component {
constructor(props) {
super(props);
}
componentWillMount() {
//if(!this.props.verified){this.props.verifyHash(this.props.match.params.hash); console.log('Verifyin\'')}
}
render() {
return (
<MuiThemeProvider>
<div className="login__wrapper">
{this.props.navigateToRoot ?
this.props.history.push("/") : <FirstPassword {...this.props}/>}
</div>
</MuiThemeProvider>
)
}
}
const mapDispatchToProps = (dispatch) => {
return {
credentialChange: (key, data) => dispatch(credentialChange(key, data)),
verifyHash: (hash) => {
dispatch({type:"VERIFICATION_ATTEMPT", payload: null});
if(hash.length>0) {
axios.get('/api/user/verify/'+hash)
.then(response => {dispatch(verify(response.data))})
.catch(err => {
console.log(err);
dispatch({type: "VERIFICATION_FAILURE", payload: "Something went wrong. Please try again."})
})
} else {
dispatch({type: "VERIFICATION_FAILURE", payload: "Something went wrong. Please try again."})
}
},
resetPassword: (first, second, hash) => {
//console.log(first,second,hash);
dispatch({type: "VALIDATION_ATTEMPT"});
if(first && first === second) {
let crypto = require('crypto');
let passhash = crypto.createHash('sha512').update(first).digest('hex');
let cpasshash = crypto.createHash('sha512').update(second).digest('hex');
axios.post('/api/user/validate', {
password: passhash,
cpassword: cpasshash,
hash: hash
})
.then(response => {
dispatch(passwordChange(response.data))
})
.catch(err => {
console.log(err);
dispatch({type: "VALIDATION_FAILURE", payload: "Something went wrong. Please try again"})
})
} else {
dispatch({type: "VALIDATION_MISMATCH",
payload: "Palun tee kindlaks et sisestasid kaks identset parooli!"})
}
}
}
};
const mapStateToProps = (state) => {
return {
loading: state.validation.loading,
loggedIn: state.user.loggedIn,
validationError: state.validation.validationError,
password: state.validation.password,
cpassword: state.validation.cpassword,
navigateToRoot: state.validation.navigateToRoot
};
};
export default connect(
mapStateToProps,
mapDispatchToProps
)(Validate); | {
"redpajama_set_name": "RedPajamaGithub"
} | 8,689 |
DSS-162 First Floor
Pocket C, SECTOR-14, HISAR
sales@bigfootbroadband.in
Welcome to Bigfoot Broadband Services,fastest internet provider in HISAR and Fatehabad
raviraiya August 5, 2016 No Comments
Give TRAI's Broadband Plan a Hand
Screams of pain normally accompany the release of a Consultative Paper by the Telecom Regulatory Authority of India, but the one on Broadband released last week has been met with a deafening silence. This worries all of us in the internet industry.
Is the silence symptomatic of Indian telecom players' and policy maker's long standing disinterest in broadband?
TRAI's thought on how to make broadband more affordable and better quality mark a revolutionary departure from India's normal laissez-faire telecom policy stance. It actually proposes an Rs 32,000 crore government initiative to build India's Information Super Highway. , This, when done, will impact our economy more, much more, than even the Golden Quadrangle, that network of highways that is being built to connect our great cities.
This initiative does not come a moment too soon. For a country that has taken bold new initiatives in expanding education and health care and all aspects of our national infrastructure, we have treated the most important infrastructure of the modern knowledge economy, a broadband infrastructure, with benign neglect. As a result, India's Broadband record is dismal. Broadband prices in India are the highest in the world (with the exception of Myanmar). India's broadband connections are a mere 9 million.
Part of the reason for such benign neglect is an under-appreciation by our policy makers and public about the role of broadband in a modern economy. Talk of building better physical roads or bridges and we can easily imagine what this entails and what benefits that brings. Talk of broadband and many go, 'Oh, that's what my teenage son uses to download mp3 music!'
Broadband is that but it is also much more. Broadband is what will drive electronic commerce which in turn will make our big business more efficient, and allow our small businesses to reach out to world export markets. It is also what future-oriented companies like Aravind Eye Hospital use to deliver low cost, high quality medical services. It is the backbone on which high quality school and college education can be delivered cost-effectively to our vast population. And it is the base on which eGovernance initiatives rest.
There is also an ideological misunderstanding behind this benign neglect: many policy makers and the Indian elite may be read the wrong lessons into India's massive private-sector lead mobile phone expansion. Why not leave broadband expansion to the private sector and they will do what they did for mobile phones: raise international capital, compete with each other, bring down prices and expand the industry.
But this, as I said, is a misunderstanding. Broadband infrastructure is like a bridge or an intercity highway: costly to build and on which the financial returns may come only in 15 to 20 years. The mobile-voice businesses get to profitability much sooner and this make private equity capital much more available for mobile voice services and very difficult for broadband data services. If the State does not build it, no one else will.
How, you may ask, have the US and Europe done it? The answer to this is that by the time internet came around in the mid 1990's, the high quality copper or fibre infrastructure was already built out. All they needed to do in those countries was to build internet services over the same infrastructure. In India, there is next to no such infrastructure even today. Somebody has to build it.
In spite of that head start, many advanced economies are doing even more: the United States Federal Government has already put our $110 bn in 2004 and $350 bn in 2005 and continue to spend at similar level to bring broadband to America's rural areas. The national governments of Britain, Australia and Japan have done or are in the middle of similar levels of spending.
Why not leave it to the Mobile Phone companies to offer broadband services through wireless, you may ask. After all, haven't they bid gigantic amounts for broadband wireless spectrum for this very purpose? The answer to this is that no doubt they will, but because of the very nature of wireless broadband technology, such services will cost Rs 1500 to Rs 2000 a month- excellent for the lap-top toting executive but too expensive for middle class India.
For broadband to get to the 100 million households who make up 40% of all households in India, we need a service which is priced no more than Rs 200 per month, not Rs 2000 per month. And we need this service with no ifs and no buts: no conditions that limit the amount of data you can download and no conditions on the time of day when you can use it.
TRAI proposes to get there by 2014, that is, in four years from now.
TRAI's grand vision is to take broadband fibre right up to 374,000 villages at a cost of Rs 32,000 crore. TRAI estimates that Rs 18,000 crore of this is to be spent on the manual labour of digging trenches and laying the cable and the balance Rs 13,000 crore is the cost of fibre optic cable and telecom equipment. They suggest that the manual labour component be done National Rural Employment Guarantee Scheme. The equipment cost of Rs 13,000 crore, they suggest, be met from the Universal Service Obligation Fund.
They also propose that a National Fibre Agency be created to execute this massive project. Once this core network is built, private sector companies like Cable Operators, Cyber Cafes and Internet Service Providers can tap into this and create a vibrant reseller market taking the service to consumer homes, schools and offices.
Rarely, has a government policy making group set out such a carefully thought-out and visionary plan.
Let's give TRAI a hand.
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We provide fastest internet broadband connection in hisar. With over 5 years of experience we'll ensure that you're always getting the best guidance from the most professional company in the industry.
Address: DSS-162 First Floor, Pocket C, SECTOR-14, HISAR 125001
Phone: 9812561898,9518446443
E-mail: sales@bigfootbroadband.in
Website: www.bigfootbroadband.in
© 2016 Bigfoot Broadband. All Right Reserved. Powered By Provinus Solutions | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 9,021 |
Q: Perl created files with future timestamps on FreeBSD 9.3 I just encountered some strange behavior with Perl 5.16.3 on FreeBSD 9.3-RELEASE-p3. We've got a cron job which runs every five minutes and generates some text status files. I just happened to list the contents of the output directory and saw that the timestamps for some of the files were in the future! The files are created like this:
if (open(OUT, "> $status_file_path")) {
print OUT "$status_info\n";
close OUT;
}
Now, the file handle OUT is used in several places, however it is opened and closed within the same block as shown above. And like I said, out of ten files, only a few had future dates when displayed using ls.
For example, files with the current date had timestamps like 04/02/2015 20:29:46, files with future timestamps were out in November, e.g. 11/10/2015 09:38:41.
What might be going on here?
EDIT
I've got two tests running:
1) a perl script running a loop of 1000 iterations, sleeping a random time up to 10 seconds between iterations, using the open/print/close logic to create an output file and abort the script if the file's modification time is in the future.
2) a cron entry to touch a test file every minute, e.g. touch /home/test/test_file_date_with_cron.txt
TEST RESULTS
Neither of the tests generated output files with a timestamp in the future.
This is scary.
EDIT 2
Here is the filesystem info, the files are written in the /usr directory.
# df -h
Filesystem Size Used Avail Capacity Mounted on
/dev/gpt/gprootfs 2G 133M 1.7G 7% /
devfs 1.0k 1.0k 0B 100% /dev
/dev/gpt/gpusrfs 431G 3.8G 392G 1% /usr
procfs 4.0k 4.0k 0B 100% /proc
EDIT 3
Running the script outside of cron for several hundred iterations didn't duplicate the problem. HOWEVER, I just found some other files, which are created by a CGI script which have the future dates:
-rw-r--r-- 1 test test 5783 Nov 10 2015 Config.xml_20150210_104151
-rw-r--r-- 1 test test 34548 Nov 10 2015 Config2.xml_20150210_104151
-rw-r--r-- 1 test test 6105 Nov 10 2015 Config.xml_20151109_232210
-rw-r--r-- 1 test test 34554 Nov 10 2015 Config2.xml_20151109_232210
-rw-rw-r-- 1 root test 2075 Nov 9 2015 Config.xml_20151109_231055
-rw-rw-r-- 1 root test 1232 Nov 9 2015 Config2.xml_20151109_231055
These are archive files, which get moved and renamed with the file's mtime timestamp. Note that BOTH ls and Perl's stat() function report the future date -- stat() is used to generate the file's timestamp portion of the name.
Looking at the first entry, ls reports "Nov 10 2015", whereas when the CGI script processed it, Perl's stat() reported "20150210_104151", i.e. "Feb 02 2015" which is most likely correct.
Further down, we see ls showing "Nov 10 2015" and stat() reported "20151109_232210", i.e. "Nov 09 2015".
A: Finding those additional archived config files helped me track down the cause, which was as others have suggested, that the system date and timezone changed.
From: 1447147328 and America/Adak
To: 1426637771 and America/New_York
What was throwing me off, was that I thought the cron script wrote ALL of the output files each time it executes, but that's not the case. The files have different "refresh intervals".
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 6,708 |
(John Clanton/Tulsa World via AP)
No big Powerball winner; jackpot now up to $750 million
DES MOINES, Iowa (AP) — No one has won the big Powerball prize, so the estimated jackpot now grows to $750 million — the fourth-largest lottery jackpot in U.S. history.
The next drawing will be Saturday.
On Wednesday night, when it was at $620 million, the Powerball jackpot looked sort of puny given all the attention lavished on the $1.537 billion Mega Millions jackpot won in South Carolina on Tuesday. But with two giant prizes in one week, it was hard not to compare.
Million Dollar Mega Millions Ticket Sold In Pennsylvania
South Carolina Convenience Store Sold $1.537 Billion Ticket
Only three lottery jackpots have been larger than the next Powerball prize.
No one has won the Powerball jackpot since Aug. 11, when a man from Staten Island, New York, won $245.6 million.
This story has been corrected to show that only three lottery jackpots have been larger than the next Powerball prize, instead of five. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 88 |
What impact did Spanish colonization have on the indigenous peoples of the Americas?
How did Spanish colonization affect the indigenous peoples?
What was the impact of the Spanish on the indigenous cultures of the Americas?
What impact did the Spanish colonization have on the Americas?
What is the impact of Spanish colonizers to our culture?
What was the biggest impact that Spanish colonization had on indigenous Californians?
How did Spanish colonization impact the new world?
How did colonization affect the culture and tradition of our country?
What is the impact of colonialism?
What did the Spanish contribute to Philippine culture?
In fact, the greatest impact of Spanish contact with the indigenous peoples was the introduction of 'Old World' diseases that decimated their populations. These diseases included smallpox and measles, for which the indigenous population had no immunity.
The Spanish colonization however had major negative impacts on the indigenous people that settled in Trinidad such as the decrease of the population, family separation, starvation and the lost of their culture and tradition. The most prominent amongst them all was genocide and annihilation.
Altered Lifestyles The Spanish altered Indian life in many ways. Their intrusion resulted in changing tribal customs and religious traditions. Tribal alliances were shifted and new rivalries were developed. Indians lost their land, their families, and their lives.
Colonization ruptured many ecosystems, bringing in new organisms while eliminating others. The Europeans brought many diseases with them that decimated Native American populations. Colonists and Native Americans alike looked to new plants as possible medicinal resources.
THIS IS AMAZING: How do you say what do you mean in Spanish in Spanish?
Spanish Colonization (1565-1898)
Because Spain controlled the Philippines so early and for so long, they were a massive influence to the modern Filipino culture. The biggest influence still seen to this day is religion. The majority of religion practiced in the Philippines is still Roman Catholic, at 79.5%.
They established missions to convert Native Americans to Christianity, pueblos, or towns for the Spanish settlers, and ranchos, large land grants for agricultural development. The Spanish impact on California can still be seen in many ways in California.
The arrival of Europeans in the New World in 1492 changed the Americas forever. Over the course of the next 350 years: Spain ruled a vast empire based on the labor and exploitation of the native population. Conquistadors descended on America with hopes of bringing Catholicism to new lands while extracting great riches.
One impact of colonization is "pyschocultural marginality" or the loss of one's cultural identity along with social and personal disorganization. Such impact is produced when people are denied access to their traditional culture, values and norms leading to historical trauma and cultural alienation (Dalal, 2011).
Colonialism's impacts include environmental degradation, the spread of disease, economic instability, ethnic rivalries, and human rights violations—issues that can long outlast one group's colonial rule.
THIS IS AMAZING: Frequent question: Is capital feminine in Spanish?
The Spaniards introduced Christianity (the Roman Catholic faith) and succeeded in converting the overwhelming majority of Filipinos. At least 83% of the total population belongs to the Roman Catholic faith. The American occupation was responsible for teaching the Filipino people the English language. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 1,224 |
{"url":"https:\/\/crypto.stackexchange.com\/questions\/12600\/proof-of-shared-secret-through-key-derivation","text":"# Proof of shared secret through key derivation\n\nAlice gives a random key $K$ (e.g. 32 bytes long) to Bob through a secure channel.\n\nBob want to prove to Alice through an unsecured channel that he knows the key.\n\n\u2022 Is it secure for Bob to send $s||KDF(s||K)$ \u2212with $s$ a random tag, say 8 bytes long\u2212 ? It seems Alice can recompute $KDF(s||K)$ and compare. Nobody can deduce K from the message.\n\u2022 Provided tags are one time use only, and Alice keeps track of which have already been used, how many different proofs could Bob send before an observer could potentially figure out K ?\n\u2022 Is there a way to improve that function ? Any specific key word, name of algorithm or protocol about that topic ?\n\nEdit : I understand HMAC is the function I am looking for. Is it safe to use it for that purpose ? Can an observer learn anything about the key from many HMAC ?\n\n1. Alice sends a 32-byte random number $n$ to Bob.\n2. Bob replies with $H(K || n)$.\nThen an eavesdropper or even a MITM attacker will never learn $K$ faster than bruteforce.","date":"2020-04-09 05:39:44","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.31283998489379883, \"perplexity\": 1241.9044056901973}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-16\/segments\/1585371829677.89\/warc\/CC-MAIN-20200409024535-20200409055035-00069.warc.gz\"}"} | null | null |
# Maritime Networks in the Mycenaean World
In this book, Thomas F. Tartaron presents a new and original reassessment of the maritime world of the Mycenaean Greeks of the Late Bronze Age. By all accounts a seafaring people, they enjoyed maritime connections with peoples as distant as Egypt and Sicily. These long-distance relationships have been celebrated and much studied; by contrast, the vibrant worlds of local maritime interaction and exploitation of the sea have been virtually ignored. Tartaron argues that local maritime networks, in the form of "coastscapes" and "small worlds," are far more representative of the true fabric of Mycenaean life. He offers a complete template of conceptual and methodological tools for recovering small worlds and the communities that inhabited them. Combining archaeological, geoarchaeological, and anthropological approaches with ancient texts and network theory, he demonstrates the application of this scheme in several case studies. This book presents new perspectives and challenges for all archaeologists with interests in maritime connectivity.
Thomas F. Tartaron is Associate Professor of Classical Studies at the University of Pennsylvania, where he is also Chair of the Art and Archaeology of the Mediterranean World Graduate Group and a Consulting Scholar in the Mediterranean Section of the University of Pennsylvania Museum of Archaeology and Anthropology. He has been a Colburn Fellow and Fulbright Fellow at the American School of Classical Studies at Athens. He has participated in numerous excavations and regional surveys in Greece, Iraq, Albania, and the United States. His current field project, the Saronic Harbors Archaeological Research Project, co-directed with Daniel J. Pullen, has exposed a unique Mycenaean harbor settlement that may have been one of Mycenae's main ports on the Aegean Sea. This work is supported by the National Science Foundation (USA) and a number of private foundations. Tartaron has published many articles on Greek prehistory and archaeological method and theory in edited volumes and in journals such as Antiquity, Hesperia, and the Journal of Archaeological Research. His previous book, Bronze Age Landscape and Society in Southern Epirus, Greece (2004), was published in the British Archaeological Reports International Series.
# Maritime Networks in the Mycenaean World
Thomas F. Tartaron
University of Pennsylvania
CAMBRIDGE UNIVERSITY PRESS
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© Thomas F. Tartaron 2013
This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press.
First published 2013
Printed in the United States of America
_A catalog record for this publication is available from the British Library._
_Library of Congress Cataloging in Publication Data_
Tartaron, Thomas F.
Maritime networks in the Mycenaean world / Thomas Tartaron.
p. cm.
Includes bibliographical references and index.
ISBN 978-1-107-00298-2 (hbk.)
1. Navigation -- Greece -- History -- To 1500. 2. Coastal archaeology -- Greece -- Methodology. 3. Coast changes -- Greece -- History. 4. Civilization, Mycenaean. 5. Greece -- Commerce -- History, Ancient. 6. Aegean Sea -- Navigation -- History -- To 1500. I. Title.
VK16.T37 2014
387.50938'09013--dc23 2012042713
ISBN 978-1-107-00298-2 Hardback
Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party Internet Web sites referred to in this publication and does not guarantee that any content on such Web sites is, or will remain, accurate or appropriate.
Dedicated to the memory of my father, Francis X. Tartaron, Jr.
## Contents
Figures
Tables
Preface
Acknowledgments
1 The Problem of Mycenaean Coastal Worlds
2 Mycenaeans and the Sea
3 Ships and Boats of the Aegean Bronze Age
4 The Maritime Environment of the Aegean Sea
5 Coasts and Harbors of the Bronze Age Aegean: Characteristics, Discovery, and Reconstruction
6 Concepts for Mycenaean Coastal Worlds
7 Coastscapes and Small Worlds of the Aegean Bronze Age: Case Studies
8 Conclusions and Prospects
Notes
References
Index
## Figures
1.1 Map of the Mediterranean and Aegean with important Bronze Age sites indicated
2.1 View of the shaft graves of Grave Circle A, as it appeared at Mycenae
2.2 Aerial view of the citadel, Mycenae
2.3 Linear B social pyramid
2.4 Linear B tablet PY Tn 996
2.5 Colin Renfrew's (1975) modes of exchange
2.6 The Flotilla Fresco, West House Room 5, Akrotiri
2.7 "Flagship" from the Flotilla Fresco, Akrotiri
2.8 Clay boat model, Asine LH IIIC
3.1 Mortise-and-tenon joinery
3.2 Painted keel on boat model, Asine LH IIIC
3.3 Bird-head stempost decoration on a straight-sided alabastron
3.4 Steering oar on an Early Cycladic III sherd
3.5 Earliest Mediterranean depiction of a sailing vessel, on a Gerzean jar, Egypt
3.6 Steatite seal with a ship and possible steering oar
3.7 LH II signet ring showing awning structure, Tiryns
3.8 Ikria from Flotilla Fresco ships, Akrotiri
3.9 Ikrion frieze from West House Room 4, Akrotiri
3.10 MH II sherd showing armed figures aboard a ship, Kolonna
3.11 Fragment of the Silver Siege Rhyton, Mycenae
3.12 Kynos A galley with decked hull, LH IIIC Middle
3.13 Ship rowed from "Departure Town," Flotilla Fresco, Akrotiri
3.14 Ship under sail, Flotilla Fresco, Akrotiri
3.15 Incised image of a boat with human and animal, Korphi t'Aroniou, Naxos
3.16 Green steatite seal showing two men in a boat with fish swimming underneath, MM I, Malia (W808)
3.17 Fragmentary boat from an LH IIIC pictorial krater, Kynos
3.18 Sailing ship from an LH IIIC stirrup jar, Skyros
3.19 Corfiot reed boat (papyrella) at sea
4.1 Centers of cyclogenesis in the Mediterranean
4.2 Regional winds of the Mediterranean
4.3 Basic dynamics of land and sea breezes
4.4 Mediterranean currents and water circulation
4.5 General sea-surface circulation flow in the Aegean
4.6 Typical positions of major cyclonic and anticyclonic gyres in the Aegean
4.7 Distribution, size, and intensity plots of eddies in the Aegean
4.8 Satellite image of Kapsali Bay, Kythera
4.9 Visibility of land from the sea in the Mediterranean
4.10 Sun, sea spirals, and fish incised on an Early Cycladic frying pan, Louros Athalassou cemetery, Naxos
4.11 Hypothetical Aegean Bronze Age sea routes
4.12 Hypothetical long-distance sea routes in the eastern Mediterranean
4.13 Example of a star-structure compass, Caroline Islands, Micronesia
5.1 Classification of coasts by relative motion of the shoreline
5.2 Movement of sediments along Mediterranean coasts
5.3 View of the sequence of ridges and swales at the mouth of the Acheron River, Epirus
5.4 View of tombolo, Paximadi Cape, Euboea
5.5 View of an estuary in South Carolina, United States
5.6 Example of a lagoon and barrier system on coastal Elis
5.7 Formation of the Scamander plain, Troy
5.8 Reconstruction of ship sheds at Kommos
5.9 Hypothetical reconstruction of an artificial harbor at Pylos
5.10 Aerial photograph of submerged harbor remains near Naples
5.11 Plan of submerged remains at Caesarea Maritima
5.12 Correlation of biofacies and sedimentary facies in the Ancient Harbour Parasequence
5.13 Topographic typology of Bronze Age anchorages: high-energy coasts
5.14 Topographic typology of Bronze Age anchorages: low-energy coasts
5.15 Flow chart of methods in geoarchaeology and paleogeography
5.16 Location of Liman Tepe in the Bay of Izmir region
5.17 Reconstruction of the Bronze Age coastline at Liman Tepe
6.1 Map of a hypothetical Mycenaean maritime culture region
6.2 Broodbank's PPA versions 1–4, based on different initial and growth conditions
6.3 Maps of cost-weighted path distance for eastbound and westbound journeys in the eastern Mediterranean
7.1 Map of the Saronic Gulf region with important Bronze Age sites indicated
7.2 Comparative ranges of transportation modes in the Saronic Gulf region
7.3 Map showing the locations of corridor houses and fortifications in the EB II Aegean
7.4 Site plan of Bronze Age Kolonna, Aigina
7.5 "Master of Animals" pendant from the Aigina Treasure
7.6 Objects from the Aigina MH II "shaft grave"
7.7 Distribution of Aiginetan "gold mica" pottery exports
7.8 Map of Aigina showing the locations of known MH sites
7.9 Map of early Mycenaean sites in the Saronic region
7.10 Map of late Mycenaean sites in the Saronic region
7.11 General plan of Mycenaean Kanakia, Salamis
7.12 Partial plan of excavated Mycenaean structures, Ayios Konstantinos, Methana
7.13 Digital terrain model of the Korphos region
7.14 Aerial photograph of the Kalamianos site
7.15 GIS plan of architecture and other features at Kalamianos
7.16 Example of large-rubble construction of Mycenaean buildings at Kalamianos
7.17 Reconstructed coastlines and harbor basins at Kalamianos
7.18 Ballast pile identified in inshore waters at Kalamianos
7.19 SHARP survey zones and survey units
7.20 Ancient architectural remains in the SHARP survey area
7.21 Satellite image with locations of stone cairns and enclosures
7.22 View and drawing of a small cairn on the Pharonisi peninsula
7.23 View of an EBA stone enclosure
7.24 View of Stiri and adjacent polje, with location of the Mycenaean site indicated
7.25 Differential GPS plan of Mycenaean architecture at Stiri
7.26 Plan of architectural features at the "saddle site" north of Kalamianos
7.27 Monumental Mycenaean agricultural terrace walls at Stiri
7.28 Map of the southeastern Aegean and southwestern Anatolian coast
7.29 Three-dimensional map of the Latmian Gulf at maximum marine transgression, circa 4000 BP
7.30 Map showing the topography of Bronze Age Miletos and vicinity
7.31 Mycenaean elements in the southeastern Aegean
7.32 Area map of Thessaly, with important Neolithic and Bronze Age sites indicated
7.33 Map of the changing coastline of the Bay of Volos
7.34 Architectural plan of LBA Dimini
## Tables
1.1 Chronological framework for the Aegean Bronze Age
3.1 Variations in the calculation of dimensions of the Flotilla Fresco ships
3.2 Hypothetical LBA small boat types and functions
3.3 Range of fishing practices defined by setting
3.4 Bronze Age seacraft performance characteristics
4.1 Potential navigational aids to Bronze Age seafaring
5.1 Sedimentary facies in the lower Acheron valley
5.2 Mediterranean harbors associated with topographical types
6.1 Framework for Mycenaean maritime cultural landscapes
7.1 Chronological chart for Kolonna
7.2 Classes of EBA and LBA architectural remains in the Korphos region
## Preface
This book is inspired by a keen interest in coastal archaeology, cultivated during twenty years of fieldwork in coastal regions of mainland Greece. Over this time, I have collected empirical data from three regional landscape archaeology projects with extensive coastal components: the Nikopolis Project (1991–95), the Eastern Korinthia Archaeological Survey (1998–2002), and the Saronic Harbors Archaeological Research Project (2007–11), which have allowed me to address Mycenaean coastal exploitation at multiple spatial and temporal scales. As I worked through these data and tried to arrive at a more comprehensive understanding of coastal life, I became increasingly aware of, and frustrated by, the gaps in our knowledge about coastal exploitation in the Mycenaean period and the selective treatment it has received in the scholarly literature. It seemed that local-scale maritime networks were only rarely discussed, and that the coastal communities that participated in them were largely ignored. The topic deserves more comprehensive, systematic treatment than it has received to date. This book constitutes my attempt to suggest a refocused and more holistic research agenda. The elements of this approach are both conceptual and methodological, but perhaps most importantly, they must be transferable to practice in the field, where only by generating robust empirical data can we begin to close this knowledge gap. Accordingly, I offer one detailed case study and two "sketches" to demonstrate the application of this approach and to suggest some directions for future research. I hope to make a helpful contribution to Aegean Bronze Age archaeology, but I also intend this work to be sufficiently general that archaeologists working on maritime and coastal problems in any world area might find it useful in their own investigations.
## Acknowledgments
My love of coastal archaeology has been nurtured over two decades along the shores of Epirus and the Corinthia. I am grateful first to the directors of the Nikopolis Project (James Wiseman and Kostas Zachos) and the Eastern Korinthia Archaeological Survey (Timothy Gregory and Daniel Pullen) for allowing me to indulge my interests. I want especially to recognize Daniel Pullen, with whom I co-direct the Saronic Harbors Archaeological Research Project, who has been an extraordinary colleague and friend. The fundamental ideas about coastscapes and small worlds are ones we crafted together, and he has been unfailingly supportive of me during the gestation of this book and my concurrent progress toward academic tenure. Over the years, Heather Lechtman, Curtis Runnels, and Jeremy Rutter have been mentors whose intellectual influence on my work has been great. Cyprian Broodbank is a colleague whose work has had a tremendous impact on my thinking, as will be evident in the following pages. All of these collaborations have blossomed into long-term associations and friendships that I value deeply.
In the field, I have had the privilege of working together with a remarkable group of geoarchaeologists, including Mark Besonen, Joe Boyce, Rick Dunn, Zhichun Jing, Jay Noller, Rip Rapp, Ed Reinhardt, Richard Rothaus, Tjeerd van Andel, Lisa Wells, and Eberhard Zangger. My understanding of coastal geomorphology and paleocoastal reconstruction has been the direct result of their patient and benevolent teaching, and their broad-minded approach to the interaction of environment and culture. Because I constantly stress the importance of high-quality empirical data, I want also to thank all of the professionals, students, and volunteers – far too many to name here – who walked the fields, mapped the features, and collected the artifacts and other data that form the basis for the kind of study presented here.
No fieldwork can take place in Greece without the support of the regional archaeological authorities. In all of the above-named projects, our teams were fortunate to have the assistance and backing of the relevant ephorates. For the Saronic Gulf case study presented in this book, Daniel Pullen and I enjoyed a fruitful and harmonious collaboration with Konstantinos Kissas, Panayiota Kasimi, and Vasilis Tasinos of the 37th Ephoreia of Prehistoric and Classical Antiquities; and Demetrios Athanasoulis of the 25th Ephorate of Byzantine Antiquities, both in Corinth. I want especially to recognize Panayiota Kasimi, herself an expert in Mycenaean archaeology, for her firm but always collegial oversight of our project. The kindness and support of the people of Korphos have also been unforgettable. We fell in love immediately with this beautiful fishing village and its people, who accepted us, helped us, told us their stories, and supported our efforts to uncover a lost piece of their cultural heritage.
The bulk of this book was written while I was on leave from the University of Pennsylvania in 2009–10, with the generous support of the Loeb Classical Library Foundation and the School of Arts and Sciences of the University of Pennsylvania. I thank my wonderful colleagues in the Department of Classical Studies, who gave advice and other forms of support as I toiled to bring this project to a successful conclusion.
For discussion, advice, offprints, and other information, I am grateful to Eleni Balomenou, Philip Betancourt, Giuliana Bianco, John Bintliff, Emma Blake, Joe Boyce, Hariclia Brecoulaki, Michael Cosmopoulos, Jack Davis, Amy Dill, Eleni Drakaki, Michael Galaty, Walter Gauß, Tim Gregory, Nick Kardulias, Margaretha Kramer-Hajos, Lynne Kvapil, Joseph Maran, Jeremy McInerney, Guy Middleton, Nicoletta Momigliano, Sheila Murnaghan, Bill Parkinson, Jeremy Rutter, Vas f aho lu, Philip Sapirstein, Kim Shelton, Malgosia Siennicka, Carol Stein, Sherry Stocker, Tatiana Theodoropoulou, Lita Tzortzopoulou-Gregory, Aleydis Van de Moortel, Sofia Voutsaki, Malcolm Wiener, and James Wright.
For kind permission to reproduce images or assistance in obtaining permissions, I am grateful to Tim Bekaert, Ira Block, Alexandra Christopoulou, David Davison, Stuart Dawrs, Sharon Day, Madeleine Donachie, Alice Essenpreis, Carol Hershenson, Jenni Hjohlman, Amalia Kakissis, Justin Leidwanger, Kevin McMahon, Dimitri Nakassis, Julie Nemer, Andrew Reinhard, Colin Renfrew, Mimi Ross, Jeremy Rutter, Maria Shaw, Steve Thomas, Theodor Troev, and Michael Wedde. For assistance in drafting several figures, I thank Juliana Di Giustini and Felice Ford.
I owe a great debt to the professional and responsive staff at Cambridge University Press. Beatrice Rehl and her associates, Amanda Smith, Anastasia Graf, and Isabella Vitti, guided me through the complex process with skill and always expressed confidence in the final product. Peggy Rote at Aptara, Inc., was extraordinarily helpful and patient in the final stages of assembling the manuscript. Many thanks also to the anonymous readers: two who recommended the book for publication, and one who read the entire manuscript and gave a final positive assessment. Finally, I have two personal acknowledgments. My wife Juliana Di Giustini has given the love, support, and stability to my life that have allowed me to complete what at times seemed like an impossible task. Finally, this book is dedicated to the memory of my father, Francis X. Tartaron, Jr. Even as he wondered how I would ever make a living in this profession, he and my mother supported me unconditionally. I miss our weekly phone conversations, when he regaled me with all of the latest archaeological news he had culled from the newspapers and television. This book is dedicated with love to you, Dad.
## One The Problem of Mycenaean Coastal Worlds
The archaeological, textual, and iconographic evidence for the Late Bronze Age (LBA) eastern Mediterranean indicates that the Mycenaeans of mainland Greece and the Aegean islands were a seafaring people and key participants in economic and political interactions with Egypt and the Near East, channeled through extensive maritime connections (Fig. 1.1; Table 1.1). The premise of this book is that despite an apparently rich record of engagement with the sea, and the keen interest scholars have shown in elucidating it, we remain surprisingly ignorant about many of its aspects. First, we know little about where Mycenaean anchorages and harbors were, or how they were used. Second, although much attention has been devoted to long-distance "international" connections with the states, empires, and emporia of the eastern Mediterranean, comparatively little consideration has been extended to networks of maritime relations operating at regional and (especially) local scales within the Mycenaean world. Third, we currently lack a systematic body of method and theory to allow us, on the one hand, to identify and reconstruct the coastal nodes and maritime routes that made up small- and medium-scale networks; and, on the other hand, to understand how they functioned within the broader social, political, and economic realities of their day.
1.1 (a) Map of the Mediterranean; (b) detail of the Aegean region, showing the main regions and sites mentioned in the text.
Table 1.1. Chronological framework for the Aegean Bronze Age
* * *
* * *
This work offers a close examination of these lacunae, with three specific aims: (1) to present a more balanced picture of maritime interactions, emphasizing that small- and medium-scale connections are more representative of the activities of most Mycenaean coastal communities than long-distance voyaging; (2) to present a set of concepts and methods for identifying and interpreting evidence for coastal exploitation and maritime interaction; and (3) by means of case studies, to illustrate the practical applications of these ideas and to advocate for new directions in research on Mycenaean "coastal worlds."
### An Archaeological and Historical Problem
At the outset, it will be useful to define some of the terms and concepts fundamental to this study to banish, as much as possible, ambiguity from the arguments and to reveal the assumptions that underlie them. As each section unfolds, further concepts will be defined in a similar way.
To begin with a basic question, what is the overlap, if any, in the terms ship and boat? In practice, for maritime historians and archaeologists the difference resides simply in size and complexity (McGrail : 60), and there is no clear boundary or threshold in these properties that marks the transition from one to the other. Alternative distinctions, such as open-seaworthiness or specific function, are no more than general rules of thumb that cannot be sustained if applied too rigorously. We know from countless ethnographic and historical examples in the South Pacific and elsewhere that small, simple vessels are used regularly for long, open-sea journeys, and similar boats made lengthy open-sea crossings as early as 40,000 years ago when the continent of Australia was first colonized. Therefore, while conceding the general pattern that small boats are used primarily for shorter-distance coast-hopping or navigation of rivers and inland waterways, larger ships may be used for those same purposes and small boats might venture on long journeys. Similarly, function must be demonstrated and not assumed, so associating a narrow set of functions with a particular hull type can be misleading or wrong. Care should be taken when using terms that embed function, such as warship or trading vessel, to allow for multifunctional or hybrid designs. In this study, the generic terms vessel and craft, which carry no implication of size, complexity, or function, will often be used in ambiguous cases or when a comment applies equally to ships and boats.
The coastal nodes of a Mycenaean maritime network might be characterized using such terms as anchorage, harbor, or port, which though sometimes used interchangeably, will have specific definitions for our analysis. An anchorage is any coastal location at which a vessel can be brought to a safe landing position, by any means including being pulled up onto a sandy shore, lying at anchor in shallow offshore waters, or being moored to a natural feature or an artificial construction such as a quay or jetty. There is no necessary implication in this term of the existence of durable, artificial constructions to accommodate vessels, or of a permanent settlement associated with these activities. Many anchorages, past and present, are used episodically, often tied seasonally to environmental conditions and agricultural calendars and providing temporary safe haven in times of danger at sea. The term harbor carries the stronger implication that certain coastal locations are earmarked for the role of accommodating maritime traffic. The morphological attributes of harbors range from entirely natural embayments with few or no artificial constructions to enhance their maritime functions, to fully artificial harbors fashioned by means of breakwaters, quays, and elaborate drainage and maintenance systems. Still, there is no requirement that a permanent settlement accompany a harbor, although the greater the maritime traffic or the number of artificial enhancements, the more likely that this will be the case. The connotation of the term port, finally, is of the existence of a "port town," thus a permanent settlement with a primary function as a major node in a maritime network. The port town typically possesses more than the bare essentials to accommodate maritime traffic: there may be complex facilities for storage, recording, and exchange of commodities; processing of raw materials; transshipment to interior regions or further seaward destinations; and quartering of crews for short- or longer-term residence. These three terms are hierarchical in the sense that ports by definition incorporate the properties of harbors and anchorages, whereas harbors are also anchorages. It is important to retain clear distinctions because this relationship may not work in the other direction: by these definitions, anchorages may not be harbors and harbors may not be ports. These distinctions will be useful for determining the roles and facilities that were or were not present in a given case.
The problem of locating the coastal nodes described by these terms arises from a set of interrelated factors that together engender low archaeological visibility. Unlike later commercial and military harbors of Greek and Roman times, evidence that the Mycenaeans built permanent harbor installations with features such as quays, breakwater structures, lighthouses, or even artificial harbors (Marriner and Morhange ) is decidedly lacking. It remains entirely possible that Mycenaean sailing ships, along with smaller boats powered by oars or paddles, were pulled onto sandy shores or anchored off the coast, as depicted in the somewhat earlier "Flotilla Fresco" in the West House at Akrotiri on the island of Thera (Morgan Brown ; Warren ), largely without the use of built harbor constructions that would leave archaeological traces.
An equally significant obstacle to identifying these locations is geomorphological change since the Bronze Age. Modern Aegean coastlines are poor indicators of their configuration in the Mycenaean era. Global sea-level rise affects the Aegean modestly, on the order of +3 to 5 meters since the Bronze Age (Lambeck , ). Potentially more transformative are catastrophic tectonic events that cause coasts to lift up or subside. Greece lies in a zone of contact between two tectonic plates (the African and Eurasian) whose interaction shapes the Greek land mass and archipelago through deep fault systems, earthquakes, volcanism, and orogenesis. The consequences of tectonically induced uplift and subsidence can be notoriously localized; thus, there is no valid pan-Aegean model for the changes in form and relative sea level on a given segment of shoreline. Another group of anchorages, including river mouths, deltas, and lagoons/estuaries, has disappeared through sedimentation caused naturally and often accelerated by human activities. In view of the complexities of coastal change, any comprehensive study of Aegean Bronze Age coastlines requires integrated programs of coastal geomorphology and archaeology, with both terrestrial and underwater components. Specifically, the archaeological methodology espoused here closely integrates methods of detection (remote sensing, Geographic Information Systems, archaeological and geoarchaeological surface survey) with subsequent investigation (terrestrial and underwater geology, extensive and intensive survey, terrestrial and underwater excavation, ethnography and oral history). As we shall see, the results of such analyses tend not to be broadly valid beyond the immediate settings under study.
There is also a lack of clarity, and a strong bias, concerning the scale and nature of Mycenaean maritime interaction. Scholarly interest tends to focus on long-distance voyaging, involving the exchange of elite goods and raw materials of vital interest to the palaces, such as copper and tin. The image of Mycenaean trading fleets sailing around the eastern Mediterranean, putting in at major ports, is alluring but surely misleading. We are not certain that Mycenaean ships routinely sailed to Egypt, for example, rather than obtaining Egyptian goods at emporia like Ugarit on the Syrian coast, or from ships visiting Greece from the east. More importantly, such long-distance connections were dwarfed in quantity by dense networks of local and regional maritime connections among Mycenaean communities. The latter routes and relationships have received little attention, but they must have dominated the use of anchorages, large and small, on Aegean coasts.
There were many shades of activity in the spectrum between local and international interaction. Local and microregional maritime networks are best expressed by the concept of the "small world" (Broodbank : 175–210), composed of communities bound together by intensive, habitual interactions due to geography, traditional kinship ties, or other factors. There may be a high level of interdependence and communities may come to think of themselves as forming a natural entity, defined by the dense web of connections that supports a combination of political, social, and economic relationships. Small worlds are nested within larger regional and interregional economic and sometimes political networks. Small worlds evolve and change over time, as I seek to demonstrate in a diachronic reconstruction of a Bronze Age small world in the Saronic Gulf (Chapter 7). The inhabitants of Kolonna on the island of Aigina dominated this small world of coastal settlements for nearly a millennium, until the expanding palace at Mycenae broke it apart, incorporating Saronic communities into broader Aegean networks. Within that millennium, however, it is possible to detect waxing and waning of the relationships of small coastal settlements with each other and with Aigina as the attention of Kolonna's inhabitants shifted into and away from the Gulf.
Moving beyond the local context, regional-scale networks within the Mycenaean world are typically measured by the distribution of imported artifacts. Often, it is possible to trace the movements of durable commodities through stylistic analysis (e.g., of painted pottery) or through archaeometric analyses that isolate physico-chemical "fingerprints" by which an artifact's place of origin can be identified. The connections thus recognized between centers and regions may be direct or indirect, and may involve the movement of goods and ideas without the implication of strong political ties or asymmetrical power relationships. For example, during the Mycenaean period the northern Corinthia exhibits clear affinities to the Argolid in material culture classes including architecture and pottery, but virtually all of the painted fineware that emulates Argive types was made locally (Morgan : 349–61). In spite of a long scholarly tradition that Mycenae dominated the entire Corinthia politically, inspired in part by the Homeric catalogue of ships, there is little evidence to support this claim (Pullen and Tartaron ; Tartaron ). Nevertheless, the growth of Mycenaean states did involve expansion into adjacent territories by economic, political, and, most likely, military means. This diachronic process has been fleshed out from excavation, survey, and mortuary data for the palatial centers at Pylos (Bennet ; Bennet and Shelmerdine ) and Mycenae (Cherry and Davis ; Tartaron ; Voutsaki , , ; Wright ).
### Defining Mycenaean Coastal Worlds
A central concept of this book is the "Mycenaean coastal world." The term Mycenaean itself is in need of deconstruction, since it is variously used to describe a chronological period (the Late Bronze Age or Late Helladic [LH] period, c. 1600–1050) occurring in a geographical area (the central and southern Greek mainland and some Aegean islands) with fuzzy and shifting boundaries, a widespread style in material culture (architecture, iconography, pottery forms, etc.) that is said to achieve a kind of koiné in the high palatial period of LH IIIA2–early LH IIIB (roughly, 1370–1250), a complex sociopolitical system based on the palaces and recorded using a syllabic script (Linear B) that represents an archaic form of the Greek language, and sometimes even an ethnicity, a particularly problematic construct for a prehistoric world. Each of these senses of the term has provoked debate. Quite apart from the challenge of fixing chronometric dates based on radiocarbon determinations or synchronisms with Egyptian and other calendars, it is not altogether obvious when the Aegean became Mycenaean or when it ceased to be so. For instance, do we mark the end of the Mycenaean world with the collapse of the palaces circa 1200 BC, or do we recognize that Mycenaean people and their material culture traditions persisted for more than a century afterward, even experiencing a kind of revival in LH IIIC Middle (Thomatos )? In fact, the picture is variable: sharp endings at some centers and in some regions; long twilights elsewhere.
A similar lack of sharp boundaries attends the geographical extent of the Mycenaean world, and here the issue is sometimes framed as a search for the "limits" or "boundaries" of the Mycenaean culture area, as manifest in attenuation of material or material culture traits (Kilian ). A good example is the attempt to define a series of zones and boundaries in Thessaly marking incremental cultural distance from the Mycenaean world (Feuer , ). As coarse tools, these can be useful, though drawing boundaries can be a misleading exercise, for the obvious reason that the spatial pattern of participation in Mycenaean culture is far more complex than a set of map polygons within which uniform cultural engagement is implied (Galaty and Parkinson : 8–9; Tartaron : 165–67). Ongoing discoveries in the Bay of Volos area are pressing the question of whether coastal Thessaly was "Mycenaeanized" or fully Mycenaean (i.e., having no fundamental cultural differences from Mycenaean centers in the Argolid, for example), the latter having been claimed recently by Vasiliki Adrimi-Sismani ().
This discourse begs the question, of course, of whether there can be a list of material and cultural traits by which to include or exclude a settlement or region as Mycenaean, and what measure of "drift" from such a trait list is tolerable for inclusion. This approach assumes the existence of a "core area" comprising the Peloponnese and central Greece, which shared most aspects of material culture and practice: pottery forms and decoration; domestic, military, sacred, and mortuary architecture; and common practices reflected in agropastoral economies, ritual practice, and burial customs. By implication, those living outside the core area were not Mycenaean, though some will have been "Mycenaeanized" by contact or colonization.
The point of view taken in this book is that while it is possible, and at times even useful, to identify a core area in which aspects of Mycenaean material culture and practice were broadly shared, such a zone was neither monolithic in cultural or geographic terms, nor static over time. If the standard is the material culture of the palaces, or that of a specific region such as the Argolid, then places "between and beyond" participated to varying degrees over time subject to local conditions such as accessibility and social organization (Tartaron ). The trouble with normative representations of the Mycenaean world based on trait lists and stylistic attributes is that they give a false impression of uniformity within the core area by suppressing the local and regional variability on which an illuminating and culturally rich narrative of interaction might be based. Those studying frontier areas in Thessaly or distant pockets of Mycenaean coastal presence elsewhere have challenged the sharp Mycenaean/non-Mycenaean cultural dichotomy, but just as important is the realization that a comparable dynamic of accommodation, resistance, and negotiation – to borrow the words of Andrew and Susan Sherratt (: 330), "an organic process of cultural encounter and dialectic" – was occurring in the heartland of the Peloponnese and central Greece.
The notion of "coastal worlds" is rendered in the plural to emphasize two closely related points. First, at any given coastal location, diachronic change is inevitable. With the passage of time, any coastal area may experience geomorphic or topographic change, foundations and destructions, reorientation of relationships and connections, and many other transformations. Therefore, Bronze Age coastal history is a complex narrative, not merely a series of fixed points on a map or a normative characterization that masks changes over centuries or millennia. Thus, for any coastal area that we study, we must deploy diverse perspectives and analytical tools and we must find a way to represent its dynamism.
Second, the term signals my alignment with certain postmodern ideas about landscape, particularly the notion that a multiplicity of culturally constructed landscapes constitutes the experiences of different sets of actors at any given place and time (Anscheutz et al. ; Ashmore and Knapp ). Properly conceived, this perspective does not ignore the realities of the physical world in which people live, or the role of environment in shaping human societies, but in an important way it allows us to address the varied perceptions, ideas, and cultural notions that allowed coastal dwellers to construct a comprehensible world. For instance, Mycenaean sailors had a need to compose a multifaceted maritime world of coastscapes, islandscapes, and seascapes in order to interact with the natural forces of sea and sky, and with the people and places they encountered en route – indeed, to survive. Apart from practical knowledge of ship technology, navigation, and environmental conditions such as currents and winds, ship captains needed to carry a mental map of landmarks, seamarks, and safe anchorages along a series of potential routes. This information was constituted in a symbolic world of named features, places, and meteorological forces; that is, a "habitus" (Bourdieu ) of maritime knowledge passed from one generation of seafarers to the next. In the characters and places of Homer's post–Bronze Age Odyssey, both realistic and fantastical, we may discern traces of a seafarer's phenomenology. At all times, the sea inspired ambivalence, with its paradoxical roles as giver of bounty (fish, exotic objects and ideas) and taker of lives.
These diverse theoretical strands come together in the concept of the "coastscape" (Pullen and Tartaron ), inspired by a postmodern interpretation of landscape and serving as the main analytical and interpretive lens for maritime coastal activity at the local scale. The coastscape takes its place among the constellation of specialized rubrics derived from landscape archaeology and applied to the maritime context, including "islandscapes" (Broodbank : 168–76; Frieman ; Rainbird ) and "seascapes" (Cooney ). From the land-based perspective of a modern, urbanized world of paved roads and mechanized vehicles, coasts are often seen as peripheral, linear and narrow, and liminal or transitional. In a coastscape perspective, however, coasts have a certain centrality as meeting places between the sea and the interior. They are nodes of connectivity and integrative spaces, and as such they were historically privileged locations while at the same time exposed to dangers from both land and sea. This exposure contributed to complex historical sequences. In short, coastal spaces were hotspots in the Bronze Age that witnessed the interactions of everyday life, but also provided the setting for pivotal events and for the exchange of ideas that stimulated profound change. It is possible, therefore, to write an alternative narrative in which coastlines are central settings for economic and social history.
A final, yet crucial, point is that coastal worlds are not merely the linear feature of the coastline or the anchorages and settlements that might be found there. They also encompass offshore waters with their opportunities and dangers, the full visual field (i.e., viewshed) of a coastal location, the arteries connecting the coast to the interior and its resources and people, and the dense network of local maritime connections that constitute the coastal world. In the Mycenaean palatial period, a coastal settlement could access the productive capacity of the interior hinterland, while at the same time functioning as a node oriented primarily to the sea and the Mycenaean maritime economy.
### Organization of the Book
The book consists of eight chapters. In Chapter 2, "Mycenaeans and the Sea," after a brief summary of the cultural and historical background of the Mycenaean period, I examine the geographical and chronological patterns of Mycenaean maritime activity in the Mediterranean, and then outline the relevant categories of evidence, with comments on each. Chapter 3, "Ships and Boats of the Aegean Bronze Age," outlines the salient features that are known or can be inferred about Bronze Age seacraft, using a range of physical, iconographic, textual, and ethnographic evidence. An attempt is made to trace the evolution of different hull types and to assess their performance characteristics, including the smaller boats that should have been the workhorses of local-scale maritime connectivity. In Chapter 4, "The Maritime Environment of the Aegean Sea," I analyze the full range of environmental phenomena, from global to local, that combined to generate the conditions of seafaring and coastal habitats. Further, I discuss the practices of navigation in the Aegean and the formation and perpetuation of maritime communities and their specialized knowledge. Chapter 5, "Coasts and Harbors of the Bronze Age Aegean: Characteristics, Discovery, and Reconstruction," outlines the environmental processes of coastal evolution over the last 6,000 years in the Mediterranean, and their impact on Aegean coastlines. I emphasize the need for programs of paleocoastal reconstruction at the local scale and present the elements of a rational field methodology for recovering Aegean Bronze Age anchorages. The aim of Chapter 6, "Concepts for Mycenaean Coastal Worlds," is to provide a theoretical framework for a multiscalar model of Mycenaean maritime interaction. This model consists of four distinct but nested maritime interaction spheres: the coastscape, the small world, the regional/intracultural maritime interaction sphere, and the interregional/intercultural maritime interaction sphere. Social network analysis is advocated as a means to understand connectivity at these different scales. Chapter 7, "Coastscapes and Small Worlds of the Aegean Bronze Age: Case Studies," puts these concepts and methods to work in one detailed (the Saronic Gulf) and two less detailed (Miletos/Latmian Gulf, Dimini/Bay of Volos) case studies. In each example, paleocoastal reconstruction was a key element in reconstituting the physical setting of coastscapes that were embedded to varying degrees over time in small-world networks. Archaeological evidence is used to track the waxing and waning of these small worlds and to measure their participation in larger-scale connectivity. The concluding Chapter 8 revisits the main themes – theoretical, methodological, and applied – and attempts to summarize my position on how Mycenaean coastal worlds can be both reconstituted and rethought.
The central goal of this book is to advocate for a reorientation of our intellectual energies away from international-scale maritime relations, toward the scale of the coastscape and the small world, not because there is no more to learn from le grand trafic maritime, but because a wealth of information about the preponderance of maritime lives and interactions remains largely untapped. Building on the work of many others before me (e.g., Braudel ; Broodbank ; Horden and Purcell ), I recommend a set of conceptual and methodological tools with which to rationalize and systematize the task of drawing out this information. If we wish to achieve a holistic understanding of Mycenaean maritime activity, and particularly if we hope to offer Aegean Bronze Age data as cross-cultural comparative material (Parkinson ; Parkinson and Galaty 2009b: 11–22), we ought to build up from local-scale coastscapes and networks – a "bottom-up" approach – so that interregional trade is not disembodied from its own social and cultural realities.
## Two Mycenaeans and the Sea
In this chapter, I establish the necessary evidentiary background for the study of Mycenaean maritime activity. I begin with a brief historical sketch of the Mycenaean period, charting the emergence, growth and prosperity, and decline and collapse of complex society on the Greek mainland. Next, I summarize the geographical and temporal variability of Mycenaean maritime activity in the Aegean and beyond, and comment on the nature of those interactions. Finally, I describe and evaluate the main sources of evidence for maritime activity that will be applied in subsequent chapters.
### A Brief Historical Sketch of the Mycenaean Period
#### Emergence of Mycenaean Civilization
The first signs of the emergence of complex society on the Greek mainland during the Bronze Age occurred with the precocious appearance of wealth and social differentiation in the shaft graves at Mycenae (Fig. 2.1), roughly contemporary with the neopalatial Minoan palaces on Crete (see Fig. 1.1, Table 1.1). The richness and quantity of the imported goods buried with the dead are especially striking against the background of a materially poor Middle Bronze Age (MBA; Rutter , pp. 124–47, 151–55). This conspicuous consumption and display of exotic wealth is seen as a deliberate strategy to proclaim status through access to distant lands and their luxury materials (Voutsaki , , , ). Because the exotic objects and pictorial art of the shaft graves show close ties with Minoan Crete, at that time at the apex of its neopalatial prowess, it has often been suggested that small groups of nascent elites on the mainland cultivated a "special relationship" with one or more Minoan palaces to gain access to exotic materials and artisans. The Minoan influence is certainly real and even profound in some areas, including iconography, ceramic forms and styles, metalworking, and to some extent funerary architecture, but the graves and their furnishings betray many other sources and craft traditions, including Anatolian and Egyptian, but more prominently of local or other mainland origin. Still, Minoan expansion may have been the catalyst for this transformation: the influx of Minoan and Cycladic goods disrupted the egalitarian social structures of Middle Helladic (MH) Greece with novel ideas and ways to distinguish oneself through the creation and expression of prestige (Voutsaki , ). These new objects and styles were put to work as political capital through conspicuous ritual deposition in tombs. The execution of certain objects in distinct Minoan technique and style, but depicting mainland-oriented themes such as warfare and hunting, suggests the presence at Mycenae of Cretan artisans.
2.1 Artist's reconstruction of Grave Circle A, as it appeared at Mycenae circa 1210 BC. Drawing by Piet de Jong, Annual of the British School at Athens 25: plate XVIII. Reproduced with the permission of the British School at Athens.
Meanwhile, in Messenia in the southwestern Peloponnese, a competitive process was underway, marked by the occurrence of early tholos tombs at several sites. Regional-scale research has traced the diachronic histories of several small settlements, demonstrating their changing fortunes and functions over time, first within a competitive environment and later as part of a palace state centered at Pylos. Excavations and surface surveys have documented the growth of settlement at Pylos from the beginning of the Shaft Grave Era to the formation and expansion of the palace (Bennet ), along with an apparent nucleation of population around the palace after 1400 as formerly active settlements in the hinterland diminished in importance (Bennet and Shelmerdine ).
2.2 Aerial view of the citadel, Mycenae. Courtesy of Ira Block.
#### The Palatial Period
By 1400 (LH IIIA1 in pottery terms), a handful of palace-based states had emerged on the Greek mainland (Mycenae and Tiryns in the Argolid, Pylos in Messenia, Thebes in Boeotia, and perhaps at Athens, Orchomenos, and elsewhere), apparently because their political elite were able to eliminate rivals in a regional competition for hegemony (Bennet and Davis ; Fig. 2.2). These states controlled relatively extensive territories and developed strongly hierarchical political, social, and economic systems (Fig. 2.3), best understood at Pylos thanks to the discovery of an archive of more than 1,100 clay tablets inscribed in Linear B, a syllabic script that represents an archaic form of the Greek language (Chadwick ; Ventris and Chadwick ; Fig. 2.4). The earliest Linear B archive comes from the palace at Knossos during the period of Mycenaean occupation there. It consists of approximately 4,000 clay tablets from two horizons, the first dating to circa 1400 and the second to the mid-fourteenth century. Most of the remaining tablets are found at the mainland palaces, dating primarily to the destructions at the end of the thirteenth century. In addition to those 1,100 at Pylos, Thebes has yielded around 430, Mycenae 73, Tiryns 24, and Midea 3 (Bennet : 181). Linear B signs were also painted onto "inscribed" stirrup jars manufactured on Crete and distributed mainly to the palatial sites.
2.3 Linear B social pyramid. Courtesy of Dimitri Nakassis.
2.4 Linear B tablet PY Tn 996, recording bathtubs and vessels of gold and bronze. Courtesy of the Department of Classics, University of Cincinnati.
The archive at Pylos contains administrative records of materials entering and exiting the palace; allocations of raw materials to craft workers, of dependent workers and their rations to various projects and industries, and of food and drink for feasts; taxation and land records; conscription of personnel to man fleets; and other transactions (Shelmerdine and Bennet ). There are no signs of literary, legal, historical, or liturgical content. The "top-down" perspective of the tablets has encouraged a tendency to portray the Mycenaean palaces as impersonal structures whose managerial control was "pervasive, monolithic and monopolistic" (Bennet : 25), based partly on comparison with obsolete notions of an "Asiatic" palatial economy in the Near East (Cherry and Davis : 94–95). Recently, scholars have reassessed the evidence and increasingly asserted the existence of palatial and nonpalatial sectors of the economy. In reality, these were not entirely separate, non-intersecting realms of activity, but in certain areas of agriculture and craft production the palaces may have shown little interest or exerted little control (Galaty ; Halstead 1992a, 1992b, , ; Parkinson ). Commodities produced from ubiquitous sources, such as pottery and marine products, may have circulated in independent, local markets (Hruby ; Knappett ; Palaima ; Whitelaw 2001a).
The Mycenaean palaces prospered in the fourteenth and thirteenth centuries, corresponding to ceramic phases LH IIIA and LH IIIB. Settlement numbers rose and reached a second-millennium peak in this period. In the northeastern Peloponnese, settlements effectively doubled around the Saronic Gulf (Siennicka : figs. 1, 2), and the palace state at Mycenae pursued an expansive policy by incorporating the Nemea Valley, the Berbati Valley, and the Saronic Gulf region in succession (Schallin ; Tartaron ; Wright ). Although Mycenae was the greatest palace center on the Greek mainland, its scale was modest when compared with contemporary Near Eastern cities (Hope Simpson ; Whitelaw 2001b), with a continuously settled area of more than 32 hectares and a population around 6,400, assuming a density of 200 people per hectare (Bennet : 187; French : 64). The extent to which Mycenae dominated the particularly complex political environment of the Argolid remains a matter of some debate: in addition to fortified citadels at Mycenae, Tiryns, and Midea, urban centers flourished at Argos, Lerna, Nauplion, and Asine (Kilian ). Although Mycenae was never equaled in wealth and complexity (Voutsaki ), so many substantial settlements may indicate political instability with the likelihood of shifting alliances and threats (Voutsaki : 56, : 102–104). The lack of systematic survey of the Argive Plain hampers finer-resolution information on the impact of the interactions among the major centers, as reflected at local scales. Tiryns is assumed to be the main harbor for the Argolid; paleocoastal reconstruction places the shore about one kilometer from Tiryns' walls in the Late Bronze Age (Zangger 1994a), though as elsewhere no physical traces of harbor facilities remain. In LH IIIB, strong similarities between Mycenae and Tiryns in the architecture of citadel and palace, along with comparable (but not identical) import records (Cline : 89–90), suggest that they worked in concert and not as adversaries (Maran ).
In the fourteenth and thirteenth centuries, communication within the Mycenaean world flourished, giving rise to many commonalities in material culture and practice. This so-called koiné is best seen in fineware pottery styles of the LH IIIA and IIIB1 phases (gradually disintegrating in LH IIIB2), close similarities in Linear B scribal language, and a widespread Mycenaean religious ideology represented by ubiquitous anthropomorphic and zoomorphic figures and figurines. Yet the expansion and consolidation of Mycenaean culture was not a uniform phenomenon at any scale. A survey of Mycenaean influence in the different geographical regions adjacent to the Greek mainland, such as the Cycladic and Dodecanese islands, northern Greece, and the northern Aegean islands, shows a variable penetration of Mycenaean culture that does not correlate with simple determinants such as distance or ease of access (Mee : 365–81). For example, during the palatial period, Mycenaean influence in the Cyclades is seen largely in the transformation of pottery styles, while on Rhodes in the more distant Dodecanese islands, Mycenaean pottery, weapons, and jewelry were placed in Mycenaean-style chamber tombs starting in the later fifteenth century. Regional variability is also apparent in the core area, as attested by persistent regionalism in architecture (Darcque ) and pottery (Mommsen et al. ; Mountjoy , ). Similarly, the organization of political and economic life in the Mycenaean heartland existed on a conceptual continuum from direct palatial control to independence from any center. The temporal and geographical expression of these variations was a patchwork in which rates of consolidation of outlying territories varied, and regional and microregional identities were often paramount in the absence of subordination to a palace state (Tartaron ).
#### Decline and Collapse
At the end of LH IIIB2, circa 1200–1190 BC, the Mycenaean palace states collapsed amidst destructions and abandonments at the palaces themselves as well as broadly across the Mycenaean world. The distinctive palatial buildings and fortifications, along with the organization and administration of the state as recorded in the Linear B archives, disappeared forever. There are two important caveats regarding the seemingly sudden and dramatic collapse, however: first, destructions and disruptions began considerably earlier at many sites, revealing a more protracted process of decline leading up to the final collapse; and second, material culture and ways of life that are recognizably Mycenaean did not vanish, but carried on through a tumultuous time of reconfiguration and change in the twelfth century, before finally fading out by Submycenaean times, circa 1075 BC.
At the end of LH IIIB1, circa 1250, localized destructions by fire occurred at Tiryns' citadel and in the elaborate houses (the Oil Merchant group, the Panayia houses) outside the citadel walls at Mycenae. In Boeotia, an initial destruction affected part of the palace at Thebes, and a more generalized devastation seems to have put the vast fortified site at Gla out of commission. Some of the damage may have been caused by earthquakes, but in many of these cases a violent destruction at human hands cannot be ruled out. There is incontrovertible evidence that in the decades that followed, inhabitants of the palatial centers feared a protracted assault or siege. At Mycenae and Tiryns, the fortification walls were strengthened and expanded to encompass previously undefended portions of the lower slopes, and at Mycenae much of the workshop and storage activity that had been situated outside the walls was relocated inside. New fortification walls were also built at Midea and Athens. Most revealing of the dangerous environment was the construction at Mycenae, Tiryns, and Athens of subterranean passages inside the citadel leading to underground springs outside the walls. Pylos, by contrast, was unfortified in the palatial period, but warnings of impending danger have been read into a set of Linear B tablets recording the mobilization of 800 rowers, with the heading "Thus the watchers are guarding the coasts" (Chadwick : 175). It is not certain whether this recruitment was an emergency measure or a routine annual conscription.
None of these security measures prevented the final destructions of the palaces or the collapse of the palatial system that accompanied them, by circa 1190 BC. Nonpalatial centers, both large and small, were also destroyed or abandoned. The twelfth century was one of considerable upheaval, a complex patchwork of migration, depopulation, and refugee settlement on the mainland (Middleton : 71–92). Entire regions such as Messenia and Laconia were largely depopulated; estimates of the overall decrease in population in the Mycenaean heartland range as high as 75% for the period 1200–1000 (Tandy : 20; but Dickinson : 93–98 doubts this figure). Other settlements and regions became centers for resettlement of refugees. The mountains of Arcadia may have absorbed some of the population fleeing the troubles in Messenia, while at Tiryns the population actually swelled as the Upper and Lower Citadels were reoccupied and the Lower Town expanded, perhaps with refugees from destroyed or abandoned towns of the Argolid (Maran ). Much of the settlement within the walls of Mycenae was rebuilt and occupied at a diminished level throughout the twelfth century. Meanwhile, areas on the edges of the former Mycenaean world with good access to the sea prospered in the postpalatial period: the Ionian Islands, the coastal northwestern Peloponnese, the east coast of Attica, and certain of the Aegean islands flourished, perhaps energized by immigration. Some refugees migrated out of the Aegean entirely, finding their way particularly to Cyprus and the Levantine coast (Yasur-Landau ). Even on the devastated mainland, the chaotic early decades of the twelfth century gave way to a modicum of stability – a final "twilight" of the Aegean Bronze Age – in the second half of the twelfth century, corresponding to the ceramic phase designated LH IIIC Middle (Thomatos ).
For more than a century, scholars have sought explanations for the rapid and permanent collapse of the palace states, giving rise to a range of hypotheses of varying plausibility (helpfully summarized in Deger-Jalkotzy ; Dickinson : 41–57; Drews ; Middleton ; Schofield : 174–82). Each focuses on a single cause or related set of causes deemed to have been a significant trigger setting into motion the process of collapse. Although a detailed examination of these hypotheses is beyond the scope of this historical summary, the main categories of explanation can be mentioned: (1) external invasions by "Dorian" descendants of Herakles in the generations after the Trojan War (Herodotus 1.56–57; Thucydides 1.12), or the marauding "Sea Peoples" mentioned in the Egyptian records of Merneptah and Ramesses III in the late thirteenth and twelfth centuries BC; (2) internal disturbances including internecine warfare among the palace states or social unrest within a palace state; (3) natural disasters such as droughts or earthquakes, whose effects were sufficiently widespread to precipitate a palatial collapse in a short period of time; and (4) inherent economic/political instability that caused the palaces to collapse of their own weight.
No single explanation has been found satisfactory to account for the wide-spread and roughly synchronous demise of the palaces. We will probably never know which factors caused the decline and fall of each of the palace states, but we can be sure that no single cause can account for the process. A multicausal approach recognizes that the Mycenaean kingdoms were not optimized for "sustainability," as we would term it today, suffering from inherent organizational weaknesses, notably overspecialization and overcentralization, which made the state susceptible to internal and external threats. Earthquakes, droughts, diseases, and disruptions in trade may have exacerbated social stresses in a strongly hierarchical society, but the combination of factors will have been different in each case, and the cause and effect relationships among primary and secondary triggers will remain speculative. It is important to bear in mind that the collapse of complex society in the Aegean involved mainly the dismantling of an elite superstructure, leaving behind a considerably less complex agropastoral society that was still culturally Mycenaean.
Long-distance exchange in raw materials and exotic finished goods is central to the story of both prosperity and collapse. Mycenaean palatial elites relied on imports for the purposes of self-definition via conspicuous consumption and display, and to ensure the loyalty of potentially adversarial factions among the aristocratic elements of society (Bennet ; Sherratt ). In the fourteenth and early thirteenth centuries, the Mycenaeans participated in an eastern Mediterranean world increasingly interconnected by maritime networks; as archaeological, archaeometric, and textual data show, goods flowed relatively efficiently, promoting general prosperity. There is some evidence, however, that in the second half of the thirteenth century, maritime trade routes to the palaces were disrupted, possibly a repercussion of the unstable political climate in the eastern Mediterranean. There is a marked reduction in the amount of Mycenaean pottery exported to the east in LH IIIB2, and imported material at Mycenaean sites seems to decrease at the same time, though the quantities of imports are too small for statistical validity or to form definitive patterns. At Pylos, several Linear B tablets detailing the careful rationing of bronze to as many as 400 smiths in the kingdom have been interpreted to reflect a shortage of metal circa 1200 (Chadwick : 140–41). Such a shortage is possibly corroborated by the presence mainly of scrap metal in the Gelidonya shipwreck and absence of metal in the Point Iria wreck, both dated to around 1200, in contrast to the abundance of copper and tin in the Uluburun wreck a century earlier, but other explanations for this difference are possible (see below). Some of the trouble may have come at the hands of pirates and coastal raiders, the sort of seaborne marauders that later coalesced in Egyptian narratives as the Sea Peoples. It could also be the case that the Mycenaeans, always marginal actors in the east, were cut out of long-distance trade routes and rendered irrelevant as Near Eastern polities had more important concerns close to home (Sherratt : 222–24, 237). A strong orientation to the sea survived the collapse, however: in the twelfth century, many prosperous settlements were situated near the coast and maritime trade was apparently just as important as it had been in the palatial period (Dickinson : 69).
### Mycenaean Long-Distance Maritime Activity
As outlined in Chapter 1, research on Mycenaean maritime activity has focused almost exclusively on long-distance trade. The following discussion characterizes the variable patterns of long-distance maritime connections with different areas of the eastern and central Mediterranean over time, before moving to a consideration of the nature of this activity at scales from local to interregional.
#### The Geography and Chronology of Mycenaean Maritime Activity
As mentioned above, complex society on the Greek mainland emerged with much influence from Minoan Crete in the seventeenth to sixteenth centuries, corresponding to ceramic phases MH III–LH IIA. Minoan dominance of sea lanes in the Aegean and long-distance connections with the eastern Mediterranean would endure for some time afterward, until widespread destructions on Crete at the end of LM IB brought the neopalatial period to a close around the middle of the fifteenth century. Those destructions left Knossos the only functioning palace on Crete, apparently controlled by a Mycenaean elite who recorded in Linear B the administration of the palace and a broad swath of western to east-central Crete. The Mycenaean presence at Knossos was accompanied by the introduction across Crete of mainland styles of pottery, fresco iconography, and burial practice. The influx of people and material culture from the mainland may have been the result of an invasion that caused the destructions around the island; alternatively a group of Mycenaeans may have exploited an internal crisis to seize control.
On the mainland, the later fifteenth century was still a time of competition and consolidation of regional political hegemony, in advance of the establishment of territorial palace states in the early fourteenth century (LH IIIA1). The mainland polities were not particularly active in maritime ventures in the Aegean or eastern Mediterranean in LH I–II (Mee : 381), most likely because of Minoan control of sea routes. Starting in LH IIIA, the Mycenaeans broadened their overseas contacts into the Aegean and beyond, superseding Cretan interests and inheriting Minoan maritime trade routes to the east. At Miletos on the coast of Asia Minor, a Minoan colony (Miletos IV) was replaced by a Mycenaean colony (Miletos V) encompassing ceramic phases LH IIIA1–IIIA2, roughly 100 years between the late fifteenth and fourteenth centuries, before suffering a major destruction. Mycenaean Miletos is surely to be identified with Millawanda, the coastal base of the Ahhiyawa in the Hittite texts (Niemeier : 103–105).
In general, Mycenaean objects (mainly painted pottery) begin to appear in quantity in the Aegean, Cyprus, the Levant, and Egypt by LH IIIA1 or LH IIIA2. In LH IIIA and early LH IIIB, most of these Mycenaean vessels were exported from the Greek mainland; a good percentage can be traced to production centers in the Argolid by chemical characterization of their fabrics (Mommsen et al. ; Zuckerman et al. ). Subsequently, there is a shift in LH IIIB2–IIIC toward local production of Mycenaean-style pottery, and where a Greek provenience can be established, a greater diversity of production centers is indicated. Interestingly, imports to the Aegean from the Near East and Egypt are still heavily biased toward Crete, particularly Kommos and Knossos, in LH/LM IIIA (Cline : 92, ). This may be a good illustration of the tenacity of economic relations in spite of political changes and other disruptions (Horden and Purcell : 343–44). A long series of coarse to medium-coarse transport stirrup jars was produced on Crete from the fifteenth to twelfth centuries and used from Sardinia in the west to Egypt and the Levant in the east, including at Mycenaean palace centers in the Argolid and Boeotia (Ben-Shlomo et al. ; Haskell et al. ; Maran ). In LH IIIB, most imports to the Aegean found their way to the mainland, primarily to a few palatial centers (Mycenae, Tiryns, and Thebes).
An entirely different picture emerges for Mycenaean activity in the central Mediterranean (Blake ; Mee : 379–81). Already in LH I Mycenaean pottery appears in southern Italy, Sicily, and the Aeolian islands. In these areas Minoan influence is minimal, and the Mycenaean presence may reflect a freedom to search for alternative sources of raw materials, probably metals. During LH IIIA these contacts expanded; many more sites in Italy have Mycenaean pottery. By LH IIIB, however, contacts diminished sharply with Sicily and the Aeolian islands, while relations with Sardinia and southern Italy strengthened. In the central Mediterranean, Mycenaean pottery was exclusively imported in LH I–IIIA, but in LH IIIB local imitations became increasingly common and by LH IIIC they were predominant. Italian imports in the Aegean are rare, with one exception: at Kommos, which through LH IIIA had been a major importer of goods from the east, objects from the central Mediterranean began to arrive in LH IIIA and by LH IIIB dominated the foreign assemblage (Cline : 90, table 58). Kommos is the closest thing we know of to an international emporion in the Aegean (Rutter ; Shaw ).
In the postpalatial world of the twelfth century, maritime interaction continued and even prospered, but with significant changes in scale and content (Dickinson : 197–205). The well-organized and regulated system of diplomatic and commercial exchanges in the Near East disappeared, and this had consequences for the Aegean as well. The large, standardized cargoes of metal ingots and goods shipped in large transport containers, referred to in Near Eastern and Egyptian texts and demonstrated so spectacularly in the Uluburun shipwreck, no longer made their way to the Aegean. Very few transport stirrup jars or Mycenaean fineware vessels left the Aegean. Yet Near Eastern ships continued to voyage west in search of metals. Along the way, they stopped on Attica's eastern coast at Perati, where the community probably controlled the silver mines at Lavrion. Further north, Lefkandi and Mitrou may have been stops on the route to metal sources on the Chalkidiki peninsula. Elsewhere, good harbors on Crete and the Greek mainland offered stopping points en route to the metal sources of Sardinia and Italy.
On the whole, maritime connections were as important in LH IIIC as in the palatial period, particularly with potentially dangerous conditions in the interior, but the scope of the exchanges was modified. Some exotic goods from distant places still reached the coastal settlements of the Aegean, but these were mostly small items that we might be tempted to characterize as trinkets. In the LH IIIC Middle chamber tomb cemetery at Perati, the grave goods include Egyptian, Syrian, Mesopotamian, and Cypriot seals, scarabs, amulets, and beads (Thomatos ). The scarcity of Aegean-made transport stirrup jars and other fineware vessels outside the Aegean suggests that perfumed oils, wine, and probably textiles were no longer manufactured for export. There is a perceived shift toward regional and local exchange networks in a less regulated and more opportunistic environment. Vigorous trade continued within the Aegean orbit, as attested by the movement of pottery and the mutual influences of local styles. Regions such as the Euboean Gulf preserved their strong maritime orientation and took on distinct identities characteristic of maritime small worlds (Crielaard 2006). But the interregional contacts that did survive were of great importance, particularly for the introduction of new metal styles and technologies. The technology of ironworking seems to have been developed on Cyprus and was learned there by Greek immigrants in the postpalatial period (Iacovou , 2008a, 2008b; Pickles and Peltenburg ), and recently introduced metal artifact types arrived from northern Italy, including long pins, fibulae, Naue Type II swords, daggers, and "flame"-shaped spearheads (Dickinson : 204). Cyprus seems to have been of capital importance in this era, pursuing an independent agenda with active ties in the Levant, in the central Mediterranean with Sardinia, and on the Greek mainland with Euboea, Attica, and Tiryns (Maran ). The notion of a sharp swing from maritime exchange dominated by interregional networks to a narrowed focus on local and intra-Aegean relations is misleading, for two reasons. First, ongoing discoveries have shown that long-distance ties were not cut as completely as once thought. Second, as argued in Chapter 1, local- and regional-scale networks must always have dominated the total picture of maritime interactions. The thriving interregional relations of the fourteenth and thirteenth centuries were more the product of exceptional circumstances; being so extraordinary, they draw attention away from other scales of interaction.
#### The Frequency and Volume of Interregional Trade
There are different ways of interpreting the thousands of Aegean Bronze Age objects, predominantly pottery, which have been recovered around the eastern and central Mediterranean (Cline ; Lambrou-Phillipson ; Leonard ; van Wijngaarden ). Looking at the same evidence, one person can find the wide contacts and varied contexts impressive and view the pottery as the tip of an iceberg of perishable items such as textiles and foodstuffs that must have moved along with the durable ceramics. A second person would emphasize the very small quantities that are attested from the perspective of time and space, perhaps even running a simple quantitative analysis to determine that the corpus amounts to 0.5 objects imported to the Aegean from the eastern Mediterranean per year over six centuries (Cherry : 111–12), or using a more sophisticated approach, fewer than ten contacts per decade (Parkinson : 16–25). As always, the truth must lie somewhere in the middle, but it is difficult to know where in the absence of so much key evidence.
The Linear B tablets are of little help. They contain rare allusions to the palaces' external relations, but none explicitly mentions merchants or long-distance maritime exchange. Exchanges of some kind are implicit in the recording of certain exotic commodities like sesame and ivory, and slave women hailing from the coast of Asia Minor who may have arrived at Pylos as war booty (summarized by Cline : 198–99). We also learn from the Pylian tablets that the palace built and manned ships in the service of the state (Palaima ). A few instances of regional interaction are recorded: a tablet at Mycenae mentions a transaction with Thebes (Cline : 128–31; Killen : 268; Chadwick : 80–81), and another at Thebes documents exchanges with towns on Euboea (Aravantinos et al. ).
From one perspective, the implication that one might infer from the Linear B archives that interregional exchange was an insignificant sector of the Mycenaean economy does not square well with the emerging archaeological record from around the Mediterranean. Durable Mycenaean imports, chiefly fine pottery and bronze weapons and implements, continue to be found in modest amounts in the eastern and central Mediterranean; under these circumstances, it is puzzling that so few "Orientalia" and "Occidentalia" are turning up in Aegean contexts (Cline : 167–68).
The apparent contradiction between the textual and archaeological evidence has engendered a divergence of opinion regarding the scale of Late Bronze Age trade in the Mediterranean, with opposing camps that may be labeled "minimalist" and "maximalist." (For an analysis of the roots of this debate in economic anthropology, see Tartaron 2001a.) Among the minimalists, Anthony Snodgrass is most prominent in denying a significant role for commercial trade in the Bronze Age economy, maintaining that a redistributive center would only send its ships abroad "...when it needs resources from overseas, and this may be very infrequently" (Snodgrass : 18). For the minimalists, the archaeological record is meager evidence (Manning and Hulin ), best explained in terms of infrequent, directional elite gift exchange that has nothing to do with money, markets, or private enterprise. Independent merchants have little place in this tightly regulated maritime economy (Chadwick : 156–58).
The maximalist school tends to view Bronze Age trade as extensive, driven by market forces, and involving a substantial contribution from private merchants. Eric Cline (: 163–64) has recently restated his position on trade between the Aegean and eastern Mediterranean: (1) Trade was mainly directional to the major palace centers of the Aegean, with secondary redistribution from those centers; (2) Trade was predominantly commercial, with some gift exchanges occurring at the diplomatic level; (3) The primary traded goods were wines, perfumes, oils, and metals; (4) Crete was the main recipient of imported goods from the eastern Mediterranean in the seventeenth to fourteenth centuries (LH/LM I–IIIA); (5) The Greek mainland was the main recipient of imported goods from the eastern Mediterranean in the thirteenth to mid-eleventh centuries (LH/LM IIIB–IIIC); (6) Crete interacted mainly with the East in LH/LM I–IIIA, and primarily with the West in LH/LM IIIB–IIIC; (7) The Greek mainland interacted with the West, and to a lesser extent, the East in LH/LM I–IIIA, and with the East in LH/LM IIIB–IIIC; (8) Egypt monopolized trade with the Aegean during LH/LM I–II, and shared the Aegean trade with Syro-Palestine, Cyprus, and Italy during LH/LM IIIA–IIIC. Cline (: 106, : 164) has also reiterated his opinion that the scale of trade encompassing Egypt, the Near East, Italy, and the Aegean in the Late Bronze Age rivals that of today in economic complexity and political motivation. Sherrratt and Sherratt (: 376) concur that the Bronze Age economy was a market economy in the formal sense.
The evidence of shipwrecks points unmistakably to a large, but archaeologically invisible, trade in raw materials and organic substances that do not normally survive in the archaeological record (Bass ). The Bronze Age shipwrecks excavated by George Bass and Cemal Pulak on Turkey's southern coast at Cape Gelidonya and Uluburun (see below for details) provide a glimpse of the breadth and cosmopolitan origin of material being transported around the Mediterranean (Bass , 2005b; Pulak , , ). The main cargo of the Uluburun wreck comprised 10 tons of copper ingots, 1 ton of tin ingots, 175 glass ingots, a ton of terebinth resin stored in 130 of 145 Canaanite jars, and large amounts of varied Cypriot pottery. Also included were rare and exotic materials such as ebony logs, raw hippopotamus and elephant ivory, finished ivory objects, ostrich eggshells, seals, an Egyptian scarab bearing the name of Nefertiti, and vessels of faience, gold, and tin. A few swords and utilitarian objects such as knives, razors, chisels, oil lamps, and fishing equipment belonged to the crew. Numerous sets of pan-balance weights in the Near Eastern standard attest to the presence of a Semitic (not Aegean) merchant or merchants among the crew. Thanks to careful recovery methods and subsequent analyses, specialists were able to identify a range of organic remains including grains, nuts, spices, olives, figs, and pomegranates, as well as various branches and rushes used as dunnage in packing the main cargo (Knapp ; Haldane ). A dozen cultural groups – Canaanite, Egyptian, Cypriot, Mycenaean, Assyrian, Baby-lonian, Kassite, Nubian, Baltic, Balkan, eastern Near East, and possibly Sicilian – are represented in the raw and finished materials (Pulak : 256). With the exception of the pottery, most of the material from the Uluburun wreck would rarely survive in a terrestrial archaeological context: the organic material would have long since disappeared, and most ingots would have been melted down or otherwise rendered unrecognizable in antiquity. The remains preserved on the Uluburun wreck permit us to imagine the range of goods conveyed around the eastern Mediterranean at the time of the Mycenaean palaces.
The excavator suggests that the Uluburun ship originated in a Syro-Canaanite port, and was bound westward toward the Aegean with a probable final destination at one of the Mycenaean palaces, "...an official dispatch of an enormously rich and valuable cargo of raw materials and manufactured goods largely intended for a specific destination" (Pulak : 220). As such, it is perhaps not a "microcosm" of international trade in the fourteenth and thirteenth centuries, as has been claimed (Cline : 100; criticized by Burns : 15), but it could be used to buttress Snodgrass' vision of rare, directional gift exchange. Further support might be drawn from Pulak's (: 218) interpretation of some of the personal effects as belonging to two Mycenaean Greeks on board, whom he construes as emissaries accompanying the rich cargo to its final destination in the Aegean (see also Bachhuber : 134–46).
The Gelidonya shipwreck represents a different, less elite form of maritime traffic, however. It contained mainly metal ingots along with scrap metal and metalworking tools, and is interpreted by the excavator as a private Levantine merchant vessel with a metal tinker on board, engaged in tramping from port to port (cabotage) in search of opportunities to buy, sell, fabricate, and repair metals (Bass : 75). A third LBA shipwreck, which went down at Point Iria in the Argolic Gulf, is highly ambiguous because the small cargo consists only of a mix of Cypriot, Cretan, and Greek mainland pottery (Phelps et al. ). The ship may have originated in any of those places, and it may have been plying local networks rather than long-distance routes (see discussion of shipwrecks below). It is also crucial to consider that textual and archaeological evidence from the Bronze Age eastern Mediterranean demonstrates the conflation of private (entrepreneurial) and state-sponsored (palatial) merchant activity (Bachhuber : 355; Knapp ; Manning and Hulin : 273; Wiener : 264). Even if the primary function of the Uluburun ship was to implement directional trade from one royal court to another, the diversity of the cargo and the presence of many balance weight sets suggest merchants conducting trade for profit at ports of call at Ugarit and on Cyprus.
A compromise between the minimalist and maximalist positions presents itself in the realization of multipurpose voyages and mixed cargoes, and when the volume of trade is distinguished from its significance. Sherratt and Sherratt (: 354) suggest that although the quantities of goods transported over long distances were small relative to total production, their importance should not be underestimated, for they represent the efforts of a minority to acquire goods possessing tremendous social significance. A good case can be made that imported goods arriving in the Aegean from the eastern Mediterranean were imbued with a kind of symbolic power derived from the distances traveled and their mysterious cultural origins (Broodbank ; Helms ; Knapp ). To those who possessed them and controlled their distribution, they imparted esoteric knowledge and prestige, which could be manipulated to legitimize and maintain real social and political power. Bryan Burns () rightly emphasizes that the true significance of imported goods lies not in their source or any meaning they held in their originating culture, but rather in how they were assimilated into indigenous frameworks of meaning in the destination contexts, and hoarded, displayed, or dispatched to enhance status and influence social action. The seemingly narrow distribution of imports in the Aegean may be the result of a deliberate strategy on the part of elites to control quantities as well as content, because exotic objects lose their power when they are widely distributed.
As might be expected, the cumulative evidence favors neither of the extreme ends of the minimalist/maximalist spectrum. With recent research highlighting limits on the control that palaces could exert geographically and over specific sectors of the economy, it is implausible that all long-distance maritime activity was state sponsored or that there was no scope for merchants to pursue private profit. Nevertheless, the concentration of exotic imported goods at palatial centers on Crete and the mainland is a powerful indicator that elites at these centers actively sought to monopolize access to certain valued products. They were successful at doing so because they uniquely possessed sufficient capital and political authority to acquire and transport high-value commodities over long distances (Bennet : 191). Their interest lay mainly in precious metals and other exotic materials that conferred both symbolic prestige (jewelry, vessels in precious metals, ivory plaques and inlays, etc.) and practical advantage (weapons, superior tools). Perhaps nonpalatial merchants dealt mainly in subsistence goods and utilitarian pottery and tools, and circulated in regional and local rather than international, cross-cultural networks. The Gelidonya and Point Iria wrecks might be seen in this light.
#### Direct and Indirect Contact
The notion that Mycenaeans were great seafarers voyaging across the Mediterranean is widely held among scholars who focus on the Greek world, engendered by the broad distribution of Mycenaean artifacts as well as the influence of the Trojan War epics. For an earlier generation of scholars, it was Minoan and then Mycenaean sailors, merchants, and craftsmen who forged the link between the Aegean and the eastern Mediterranean and managed cross-cultural trade between these areas (Kantor ). This was partly (over)compensation for prevailing ex oriente lux frameworks proclaiming the diffusion of knowledge from the civilized Near East to a barbarian Europe (e.g., Childe ). In recent years, the archaeological basis for assigning the Mycenaeans such a prominent role in eastern Mediterranean maritime networks has been deconstructed and reassessed (Bass ), and the Homeric epics have been decoupled from a historical Bronze Age context (Bennet ; Morris ; Raaflaub ). Near Eastern and Egyptian texts, along with the strong Syro-Canaanite character of the Uluburun and Gelidonya shipwrecks, have counteracted the Hellenocentric bias. Yet the pendulum continues to swing back and forth: world-systems theory was exploited initially to characterize the Aegean as a dependent "periphery" to a dominant Near Eastern "core." Subsequently, Nick Kardulias has restored some agency to the Aegean with concepts of "negotiated peripherality" (Kardulias ) and "linked maritime exchange cycles" in which Minoans and Mycenaeans participated in eastern Mediterranean trade on relatively equal footing (Kardulias ). The applicability of world-systems approaches remains a matter of vigorous debate (papers in Parkinson and Galaty 2009a) and many are not convinced (Burns : 18–19; Cherry ; Cline ).
Moving beyond theoretical characterizations, the empirical evidence for Mycenaean long-distance voyaging beyond the Aegean is indirect and mostly circumstantial. No Bronze Age boat or part thereof has survived that is unequivocally a Mycenaean vessel. No indisputable evidence exists that Mycenaean ships visited Egypt or the farthest reaches of the Levantine coast, as opposed to trading through Syro-Canaanite or Cypriot middlemen. The case against visits to Egypt involves a qualitative change: after the Minoans were depicted as Keftiu bearing gifts to Pharaoh on wall paintings in tombs from the reigns of Hatshepsut (reign circa 1479–1457) to Amenhotep II (reign circa 1427–1400), the Mycenaeans were not depicted in the same way, nor were they mentioned in the Amarna archive at a time when large quantities of Mycenaean fineware pottery reached Akhenaten's capital of Akhetaten (Tell el-Amarna). As noted above, there is a lag of about a century between the Mycenaean occupation of Knossos circa 1450 and the widespread appearance of LH IIIA2 pottery at Amarna and many other sites (Judas ). It took even longer – until the end of the fourteenth century – before Egyptian exports to the Aegean found their way to the mainland more abundantly than to Crete.
There is some evidence for direct contact with Egypt, however. The so-called Aegean List inscribed on the back of one of five statue bases at the mortuary temple complex of Amenhotep III (reign circa 1390–1352) is a roster of place names from the Aegean. Following the ethnics Tanaja (thought to refer to mainland Greece) and Keftiu (the Cretans) are the names Amnisos, Phaistos, Kydonia, Mycenae, Thebes, Kato Zakro, Methana, Messana, Nauplion, Kythera, Ilios (Troy), Knossos, Amnisos again, and Lyktos. The list has been interpreted as the itinerary of a royal diplomatic mission, perhaps to reaffirm relations with old Minoan trading partners and to visit the mainland centers during the transition from Minoan to Mycenaean control of sea routes into and out of the Aegean (Cline : 194; Hankey ). Cline (: 194) even links a number of faience plaques, faience scarabs and seals, and a frit vase, found mainly at Mycenae and all inscribed or painted with the cartouche of Amenhotep III or his wife Tiyi, with this same diplomatic mission. These objects, most of which were found in later (LH IIIB) contexts, could equally have arrived at various times and in diverse ways. At the very least, however, the Aegean List does show that the Egyptian court was aware of the most important LBA centers on Crete and the mainland. A final bit of intriguing evidence comes from Amarna. A papyrus fragment dated roughly to the mid-fourteenth century by an associated LH IIIA2 stirrup jar bears a painted scene in which a group of Mycenaean-looking soldiers rush to the aid of a fallen Egyptian (Schofield and Parkinson ). If this scene reflects a real state of affairs, it suggests the presence of Mycenaean mercenaries in Egypt. According to one recent hypothesis, the imported Mycenaean stirrup jars at Amarna, shown to have been manufactured in the Argolid, held olive oil presented to Akhenaten's royal court as part of a diplomatic exchange, the first of many such exchanges that led to the widespread cultivation of the olive in Egypt (Kelder ).
Many favor the Cypriots as the principal intermediaries between Mycenaean Greece and the Near East in the fourteenth and thirteenth centuries (e.g., Bell : 368; Hankey ; Mee : 377; Sherratt : 97–98). There are solid arguments for this position. Cypriot pottery is distributed alongside Mycenaean fineware at Levantine sites like Ugarit, Sarepta, and Tell Abu Hawam, but in greater quantities. The amount of Mycenaean pottery at Cypriot sites like Enkomi is considerably greater than is found at the Levantine sites, but the vessel types are quite similar. Significantly, Cyprus had a history of relations with the Aegean, adopting Cypro-Minoan script and importing pictorial kraters and amphoras specially made in the Argolid for the Cypriot and Syro-Canaanite market. Susan Sherratt (: 97–98) proposes that in the thirteenth and twelfth centuries, it was Cypriot small-scale commercial traders who linked the central and eastern Mediterranean in an unprecedentedly direct way, bringing Sardinia and the head of the Adriatic into contact with the eastern Mediterranean. According to Sherratt, the Cypriots not only distributed Mycenaean pottery in the eastern Mediterranean, they came to the Aegean to get it, visiting other Aegean ports of call such as Kommos regularly (Rutter ). Against Sherratt's notion is the fact that while Mycenaean pottery imports in Cyprus peaked in LH IIIA, Cypriot exports to the Greek mainland did not become substantial before LH IIIB (Cline : 196).
Until a few years ago, the Mycenaeans were absent in Canaanite archives, but now scholars have identified two letters of the late thirteenth or early twelfth century at Ugarit that appear to give the Akkadian version of the Hittite name for the Mycenaeans, Ahhiyawa (Cline : 178; Lackenbacher and Malbran-Labat : 237–38; Singer : 250–52). It is possible that these documents establish some sort of presence for the Mycenaeans at Ugarit.
Another key text implies that the Mycenaeans may have ventured east beyond Cyprus. In a letter to King Sausgamuwa of Amurru, on the northern Syrian coast, Hittite King Tudhaliya IV (reign circa 1237–1209 BC) instructs his vassal to prevent the ships of Ahhiyawa from coming into contact with the Assyrians through the ports of Amurru (Cline ; Güterbock : 136). The obvious implication is that the ships of Ahhiyawa (i.e., Mycenaeans) had reached these ports in the past.
Cline (: 170–73) is steadfast in his opinion that Mycenaean ships were making direct voyages to Egypt and the Levant, although he acknowledges that the evidence is inconclusive. He emphasizes that the Minoans and Mycenaeans did not lack the navigational or organizational skills to make these voyages; thus, the rules of eastern ports or actions like the Hittite embargo may better explain the presence or absence of Mycenaean merchant ships. Often lost in the discussion is the fact that few scholars have expressed doubts that from LH I, Mycenaean ships were sailing to Italy and Sicily. Although maritime routes to the west did not involve a lengthy open-sea voyage similar to that from Crete to Egypt, there were plenty of environmental and navigational challenges along the routes across the Ionian and Adriatic Seas to southern Italy, Sicily, the Aeolian Islands, and eventually Sardinia. An implicit assumption about Mycenaean engagement with the west is that the indigenous inhabitants of the central Mediterranean did not possess the requisite seafaring technology or economic organization to travel east, an impression strengthened by the virtual absence of their finished goods in the Aegean. Whether or not this assumption is correct, the same cannot be said about Minoan or Mycenaean navigational capabilities. It is worth considering instead that the Mycenaeans were able to obtain the commodities they wanted most – Cypriot copper, tin from further east, luxury materials such as ivory – from visiting Eastern ships or by sailing to emporia on Cyprus or the northern Syrian coast, which served as collection points for products from far and wide. In these circumstances, the Mycenaeans may have had little need to make direct voyages to Egypt, which accordingly would have been infrequent.
#### Mechanisms of Trade
From the evidence outlined above, it becomes apparent that Mycenaean extra-Aegean transactions involved a diversity of settings, participants, and mechanisms of transfer. To get a better sense of the range of trade scenarios, it is worth looking back to a classic treatment of ancient trade in which Colin Renfrew (: 41–43) postulated 10 "modes of trade" that specified the participants and the locations where exchanges took place (Fig. 2.5). Renfrew's modes are as follows, with comments added:
(1) Direct access: B has direct access to a resource found at a without the involvement of A, and without concern for territorial boundaries; there is no actual exchange transaction. The clearest example in Aegean prehistory is the assertion that in the Early Bronze Age (EBA), obsidian was obtained directly from Melos without the mediation of the Melians (Renfrew et al. ; Torrence ).
(2) Home-base reciprocity: B visits A at A's home base, exchanging product b for product a. These are direct, face-to-face exchanges that are most characteristic of local interaction networks, but could also apply to longer, regional- or interregional-scale interactions involving, for example, a Cypriot ship sailing to Tiryns to drop off a shipment of copper ingots in exchange for pictorial pottery and textiles.
(3) Boundary reciprocity: A and B meet at a common boundary to exchange goods. It is often difficult to establish the locations of Bronze Age political boundaries, but one could imagine such trade occurring at frontier zones between Mycenaean polities, between Mycenaean and partly Mycenaeanized zones (e.g., the Bay of Volos with interior Thessaly), or at the outskirts of Miletos. The connotation of neutrality inherent in boundary reciprocity gives these spaces the quality of quasi-emporia or quasi-"ports of trade" as envisioned by Karl Polanyi ().
(4) Down-the-line trade: commodities travel across successive territories in a series of home-base or boundary reciprocity exchanges. In this case, goods move step by step through exchanges between agents departing from their home bases. A good example would be a shipment of Mycenaean pottery from the Argolid conveyed by sea to Enkomi on Cyprus; from there, a load is taken to Ugarit or Tell Abu Hawam along with other goods, and some of these vessels are subsequently transported along overland routes to Canaanite and Mesopotamian sites in the interior.
(5) Central-place redistribution: A takes produce to p, rendering it to P and receiving something in return; B takes produce to p and receives from P some of A's produce. This is one of Polanyi's classic "forms of integration" by which "primitive" economies were organized (Polanyi ; Sahlins ). Mycenaean palaces were often cast as centers of redistribution, but the ambiguity of the concept and the implication of a center's altruism in reallocating resources based on others' needs has led many to advocate the use of alternative concepts such as mobilization, in which goods and services are mobilized upward to the palatial center, and there used primarily to support palatial projects and to enhance the wealth and power of the elite (Bennet : 190; Earle ; Halstead ).
(6) Central-place market exchange: A takes produce to p and exchanges it there with B for B's produce. P is not directly involved in the exchange. This mode best describes open market towns, or periodic market days hosted by a particular town. These surely existed in the Bronze Age Aegean, as they have virtually everywhere where there is historical evidence. We might imagine fish, pottery, and other everyday items circulating in such markets outside of palatial oversight in the kingdom of Pylos, for example, but they have not been identified on the ground.
(7) Freelance (middleman) trading: Middleman C exchanges with A at a, and with B at b. C is not controlled by A or B. This form of trade is often referred to as cabotage or tramping, and differs from down-the-line trade in that the merchant or middleman operates independently of the sellers and buyers of the commodities he takes on or discharges en route. Renfrew (: 44) remarks that freelance trading is more efficient than down-the-line trade, since it reduces the number of trips and thus a good deal of effort. The Gelidonya shipwreck is often seen as a tramping vessel carrying a metal tinker and roaming from port to port.
(8) Emissary trading: B sends emissary B', an agent under his jurisdiction, to a to exchange goods with A. The Uluburun shipwreck, possibly with Mycenaean emissaries aboard and seemingly headed toward the Aegean with a return cargo destined for a Mycenaean palace, fits this profile.
(9) Colonial enclave: B sends emissaries B' to establish enclave b' in the vicinity of a in order to exchange with A. The most famous example from the Bronze Age is the Assyrian trading colony or karum installed at Kanesh in southeastern Anatolia from circa 1920–1835 BC (Veenhof ). An enclave of several thousand Assyrian merchants lived in the lower town, and their commercial transactions with their Anatolian hosts are recorded in roughly 20,000 cuneiform tablets from at least 70 distinct archives. Their trading agenda was relatively specific: Afghan tin and Babylonian woolen textiles were imported from Assur; in exchange, huge quantities of silver and lesser amounts of gold were exported to Assur (Veenhof : 338–39).
(10) Port of trade: Both A and B send emissaries A' and B' to a central place (port of trade) that is outside of the jurisdiction of either. Polanyi () envisioned the port of trade as a universal type: a place and a mechanism for the safe transfer of goods in a cross-cultural setting. Polanyi's model differs from Renfrew's definition in that the host society might exercise jurisdiction over the port of trade and participate in the business transacted there. The notion of a neutral space was integral to Polanyi's conception because of the potential threat posed by the indigenous and foreign communities to one another. An emporion such as Ugarit on Syria's Mediterranean coast fulfilled this role for a range of trading partners in the eastern Mediterranean; it may be significant that the settlement of Ugarit/Ras Shamra was separated by a kilometer from the port town (port of trade) at Minet el-Beidha.
2.5 Renfrew's modes of exchange. Renfrew : 42, fig. 10. Courtesy of Colin Renfrew.
### Sources of Evidence and Source Criticism
In the concluding section of this chapter, I briefly summarize the sources of evidence for Mycenaean maritime activity, highlighting the strengths, weaknesses, and potential of each category. The main categories are
(1) Archaeological: The spatial, chronological, and typological distribution of objects of Mycenaean origin found outside the Mycenaean world, and foreign objects found at Mycenaean sites;
(2) Textual: Linear B archives from the Mycenaean palaces bearing on maritime matters, and references to Mycenaeans in Anatolian, Egyptian, or Near Eastern archives;
(3) Iconographic: Artistic representations of Aegean persons or activities found in non-Aegean contexts, and representations of ships and seafaring activities in Aegean contexts;
(4) Shipwrecks: A handful of LBA shipwreck deposits recovered in Greek and eastern Mediterranean waters;
(5) Environmental: Harbor sites and evidence for changing coastal landscapes, as well as data on weather, winds, and currents;
(6) Ethnographic: Analogies based on information drawn from historical, recent, and living societies.
#### Archaeological Evidence
The distribution in time and space of the Mycenaean goods exported from the Aegean and the foreign imports into the Aegean has been well documented in catalogues by Cline () and others (Judas ; Lambrou-Phillipson ; Leonard ; Phillips ; van Wijngaarden ), and the corpus has been subjected to extensive comment and interpretation. As Cline (: 164–65) notes, the corpus is not growing very rapidly, and it has inevitable limitations. Yet there are undeniable patterns in the data that enable us to ask informed questions about the dynamics of cross-cultural networks of interaction. Some of these questions can be illuminated using archaeometric characterization studies, such as petrographic and chemical analyses of ceramic fabrics to address sourcing as well as technological styles and traditions. We can only speculate on the magnitude of the perishable, mainly organic, material that moved through these networks, as well as the metals that were destroyed, recycled, and otherwise removed from the Bronze Age archaeological record. But judging from the textual records recovered in the Aegean, Egypt, and the Near East, the circulation of perishable material must have been immense. We are left with a database that is meaningful and amenable to interpretation, but the small quantities of material and the frequent ambiguities as to the precise dates of objects and their find contexts create challenging problems for quantitative analysis and other measures of significance. Regardless of how the data are analyzed quantitatively, the inevitable conclusion will be that "...long-distance exchange with the Aegean was infrequent, sporadic, and consisted primarily of the importation of small prestige items" (Parkinson : 17–18). Just how to factor in materials that do not survive, and whether these would significantly change the picture, are thorny questions. Furthermore, recovering the exchange events and mechanisms by which the material was transported from its production source to the consumer (and any subsequent transfers in an object's "life history") may be impossible in an environment of webs of interaction where down-the-line trade, cabotage, and other complex modes of exchange overlap. The stability of categories of use and meaning of an object between the originating and consuming cultures is a further complication, though sometimes its contextual associations are revealing. Changes in practice (e.g., burial or cult practices) can indicate the transmission of intangible ideas where archaeological evidence for meaningful contact is otherwise lacking.
Exchange from region to region within the Aegean world can often be identified by the movement of artifacts with distinctive form, style, or manufacturing tradition. The more local the exchanges become, the less likely that there will be detectable differences in material culture, but if a center of production for a distinctive product is known, for example the volcanic rock-tempered storage and cooking wares of Kolonna on Aigina, movements at very close range can be discerned (see the detailed case study on the Saronic Gulf in Chapter 7).
#### Textual Evidence
Contemporary textual references to Mycenaean maritime activity or to the interactions of Mycenaeans with people and places beyond the Aegean are few. They include sporadic references in the Linear B archives, and a small number of mentions of Mycenaeans in Egyptian, Hittite, and Canaanite texts (Bennet 2008: 181; Cline : 175–79).
There are several possible explanations for the near silence of the Linear B archives on exchange within the Mycenaean world and without. One has to do with the narrow temporal scope of the documents, because the tablets were meant as temporary records and so only those referring to current or very recent activities were preserved. Based on the seasonality of the activities and plants mentioned and not mentioned in the Pylian tablets, it is believed that the destruction of the palace occurred in spring and that the activities recorded therein do not extend back further than six months or so (Chadwick : 191–92). If that is the case, we might imagine that most maritime activity involving palatial oversight would have been in hiatus over the winter. Shipbuilding (Pylos tablets Vn 46 and Vn 879), however, would be ongoing in preparation for the return of major maritime endeavors in the spring. A second explanation is that the palaces did not control maritime trade and thus would not be expected to record an activity outside of palatial purview. It is hard to believe, however, given the attention to building and manning ships, and the priority placed on acquiring and controlling access to raw and exotic materials that could only have arrived by sea, that there would be no meaningful palatial involvement in maritime matters. A third possibility, mentioned already, is that there was an interruption of maritime trade toward the end of the thirteenth century, cutting off seaborne imports. The disturbances that led several palaces to enhance their fortifications and excavate access tunnels to external subterranean springs may also have severed overland communications. Despite these hypothetical disruptions, the Linear B archive at Pylos seems to record normal palatial activity, including feasts and sacrifices, maintenance of palatial industries with their dependent workers, and collection of agricultural goods that seem not to be in short supply.
Mycenaean Greece appears in Egyptian texts as Tanaja, and in Hittite and Canaanite texts as Ahhiyawa and related terms (Cline : 178). The Aegean List from Kom el-Hetan ensures that the Egyptians were aware of the Mycenaean palatial centers, and the combination of copious Mycenaean fineware and the painted papyrus depicting Mycenaean soldiers at Amarna helps to counterbalance the absence of mention in the Amarna archive. The Mycenaean Greeks are sometimes counted among the Ekwesh (= Ahhiyawa?) listed in the Merneptah Stele commemorating the victory of Merneptah (reign circa 1213–1203) over a combined force of Libyans and mercenaries from the northern seas, and they may have formed part of the loosely organized marauders that Ramesses III repulsed circa 1176 BC, according to the Medinet Habu inscription. The recently identified references to Mycenaeans in the archive at Ugarit add support to the contention that Mycenaeans did voyage beyond Cyprus to Levantine shores.
More extensive than these, however, are a number of Hittite documents that have real historical significance. Presuming that Ahhiyawa refers to a Mycenaean kingdom to the west of the Anatolian coast – and there is plenty of debate on whether that center was located at Mycenae, Thebes, Rhodes, or elsewhere – it is possible to follow the Mycenaeans causing mischief in the western provinces of the Hittite Empire in the fourteenth and thirteenth centuries from their coastal base at Millawanda (Miletos; Latacz : 73–128; Niemeier , ). The Ahhiyawa are first mentioned (in the earlier form Ahhiya) in the Madduwatta Letter, sent by Arnuwanda (reign circa 1400–1375 BC) to a rebellious vassal chief of the same name to complain of his traitorous activities. The letter recounts that Attarasiya, the "man of Ahhiyawa" and possibly the equivalent of the Helladic name Atreus, joined Madduwatta to raid Alashiya (Cyprus), a Hittite dependency (Bryce : 129–35; Neimeier 2003: 104). Here we have unambiguous evidence of Mycenaeans participating in a naval raid, covering more than 500 kilometers in straight-line distance to reach Cyprus' westernmost shores. For the better part of the next two centuries, the Ahhiyawa enter the documentary record periodically in both peaceful and hostile interactions. For a time, the king of Ahhiyawa was accorded the title "My Brother," in the formulaic language of Near Eastern royal diplomacy, for example when Hattusili III (reign circa 1265–1240) remonstrated with the king of Ahhiyawa over the harboring of a fugitive; later in the thirteenth century, the Mycenaeans lost control of Miletos and the name of the king of Ahhiyawa was erased from diplomatic documents. It is difficult to overestimate the importance of the Hittite texts as a unique source of information about Mycenaean activity in the Aegean, and more generally regarding the political geography of Asia Minor and the eastern Aegean. Their usefulness is fully realized when the archaeological record, particularly at Miletos, is seen to corroborate some of the events and relationships described in the texts. Miletos is the subject of a brief case study in Chapter 7.
Early Greek literary texts, particularly the Odyssey of Homer, are often called upon for insight on Mycenaean seafaring. We must be skeptical about attempts to read Bronze Age realities into a work set in writing half a millennium after the demise of the palaces, particularly as a growing consensus perceives the world that Homer describes as an Iron Age one (Bennet ; Morris ; Raaflaub ). Nevertheless, it is worth asking whether Homer's descriptions of the conditions of seafaring might apply to the Bronze Age. To what extent did ships and navigational technology (e.g., celestial navigation) change between the Late Bronze and Late Geometric periods? Had winds, currents, and other environmental conditions (such as harbor silting or built harbor constructions) changed in significant ways? Equally important is the possibility that persistent traditions of seafaring knowledge and certain maritime mentalités that had survived for centuries might be recoverable from Homer. These topics are explored in Chapters 3 and 4.
#### Iconographic Evidence
A large and constantly expanding literature addresses and debates the technical characteristics of ship form and construction in the Late Bronze Age Mediterranean. (General overviews can be found in Basch ; Mark ; McGrail ; Wachsmann ; Wedde .) During the Bronze Age, images of ships were painted onto pottery and wall frescoes and incised, engraved, embossed, and impressed onto pottery, stone, seals and sealings, and metal jewelry. A surprising fact is that Aegean Bronze Age civilizations, ostensibly linked so closely with the sea, depicted watercraft so infrequently. A recent catalogue of ship representations from the Aegean Bronze Age counts approximately 80 painted ship images from sherds or whole vessels; more than 20 on the painted wall frescoes from the West House at Akrotiri on the island of Thera and a few others from fragmentary frescoes at Pylos, Iklaina, and Mycenae; 138 from seals and sealings; 43 incised, impressed, engraved, or embossed on ceramic, stone, or metal objects; nearly 50 watercraft models; and a further 26 as Cretan hieroglyphic or Linear A signs (Wedde ). In all then, we have fewer than 400 surviving images of ships for a period of 2,000 years.
Inferences about the ships depicted in iconographic representations are often controversial because there is no consistent artistic convention. Bronze Age artists did not necessarily attempt or achieve faithful reproductions of the details of actual ships. This discrepancy may have resulted from deliberate stylization and abstraction (innovation), imitation of contemporary examples of ship iconography, or the use of a pattern book of ship icons. Often, the medium severely limited the amount of accurate detail that could be rendered: the miniscule working surface of seals necessitated executing many details in abbreviated, schematic form, and leaving others out altogether. Two-dimensional representations on any medium make it difficult to estimate a ship's beam or to interpret features such as appendages to the decks, bows, or sterns. Even in the larger formats of frescoes and pottery surfaces, details may be represented in ambiguous or stylized fashion, leaving modern scholars to speculate on the meaning of painted lines and other flourishes. These contingencies can make it difficult to reconstruct a typology of vessel types or to identify regional variability in ship construction.
A particular frustration is that within this limited corpus, virtually nothing is known about small boats of the kind that should have formed the backbone of short-distance maritime connectivity. Such small craft stand little chance of preservation as shipwrecks except in the kinds of contexts – peaty or waterlogged deposits, for example – that are rare in the Aegean. In the absence of hull remains, wrecks of small vessels, with their diminutive cargoes, are unlikely to be recognized as such by even the most fine-grained underwater surveys. Certain boat models and seals are thought to represent small craft, though scale is difficult to infer from them, and in general artists apparently preferred to depict the more impressive galleys and trading ships of the time. For this reason, the "Flotilla Fresco" from Akrotiri on the Cycladic island of Thera takes on a disproportionate significance in understanding Bronze Age Aegean small craft. The fresco's narrative scene shows a full range of vessel types, including several small canoe-like boats and other vessels of modest size, propelled by paddles, oars, and sails.
##### The Flotilla Fresco
The so-called Flotilla Fresco was part of a frieze that adorned the upper walls of Room 5 in the West House at Akrotiri on Thera. It occupied the southeastern corner of the east wall and the south wall of the small, 4 × 4 meter room (Warren : 117, fig. 1). Opposite on the north wall, scenes of a shipwreck or sea battle are depicted, along with marching soldiers. The flotilla scene shows a procession of ships of various kinds, apparently leaving one coastal settlement (the "Departure Town") and arriving at the harbor of a second (the "Arrival Town"; Fig. 2.6). The second town is built above a double harbor formed by a larger and a smaller embayment divided by a narrow promontory. In both towns, spectators gather near the shore and in houses upslope amidst an atmosphere of excitement. The flotilla is composed of seven large sailing ships, highly ornamented with decorative elements, ikria (stern cabins), and awning structures amidships under which well-dressed individuals sit. Only one of the ships is moving under sail, however. The rest are being paddled awkwardly by men straining to reach the water line. Seven other boats are part of the scene: one medium-sized boat is being rowed out of the Departure Town; two small boats with awning structures are moored in the larger of the harbors at the Arrival Town; three small canoes are drawn up on the beach of the smaller harbor of the Arrival Town; and one canoe is being rowed out to meet the arriving flotilla, perhaps to act as a pilot guiding the large ships to safe anchorage.
2.6 The Flotilla Fresco, West House Room 5, Akrotiri. Doumas : 68, fig. 35. Courtesy of the Thera Foundation – Petrikos M. Nomikos.
An interpretation of the scene as representing a ceremonial procession of some type is persuasive for many reasons (Marinatos 1974a; Morgan Brown ; Strasser ; Warren ). Most notable are the unusual circumstances of the ships and their occupants, which suggest that this could not be a routine military or commercial exercise. The large ships are all decorated at the prow and stern, and some have hulls painted with figural motifs or abstract designs. The most elaborate (Spyros Marinatos' "flagship") has a hull decorated with lions and dolphins, and a tall mast from which dress-ship lines festooned with crocus-bloom pendants run to the prow and stern (Fig. 2.7). The fact that six of the seven ships are being paddled is curious, because by LH I this archaic means of propulsion would have long been abandoned in favor of oar and sail, particularly for ships of this size as the paddlers' exertions attest. The other installations aboard the ships are similarly out of the ordinary, if not so quaint. The ikria are small stern cabins consisting of a framework of wooden posts hung with a screen of oxhide or woven fabric (Shaw , ; Wedde : 132–34), where the shipmaster or an honored guest might sit. Ikria are also found as large-scale fresco motifs in Room 4 of the West House, and at Mycenae (Shaw , ). Awning structures cover seated passengers who take no part in paddling the ship; boar's tusk helmets suspended from the roof of some awnings indicate that the passengers are soldiers. Their decorative tunics, stiff-looking and hiding their limbs, resemble the costume later worn by elite chariot passengers depicted on pictorial kraters of the thirteenth century BC. The ikria and awnings would have been a great hindrance to the efficient conduct of military or commercial business; it is believed that they were detachable structures that could be assembled for special occasions and then disassembled easily (Morgan : 139; Shaw : 55). The nature of the ceremony is a matter of speculation, but two of the more plausible suggestions are a nautical festival celebrating the inauguration of a new sailing season in spring (Morgan : 143–45; Morgan Brown ), and a cultic procession commemorating the third-millennium Cycladic longboats (Wachsmann : 108–113), which were sufficiently impressive and symbolically charged for their time to have lived long in memory. Because there is only one ship under sail, the ceremony might even reenact the introduction of the sailing ship to Crete toward 2000 BC.
The Flotilla Fresco is the single most important visual document of the Late Bronze Age for coming to terms with Aegean seafaring. It shows a range of ships and boats with sufficient realism that we can ascertain details about their form and construction, rigging and sails, methods of propulsion, and possible functions, in spite of the ceremonial modifications. The coastal scene also confirms some basic features of Bronze Age seafaring. The Arrival Town's double harbor, formed by a narrow headland with flanking embayments, was a preferred harbor topography in the Bronze Age. These are clearly natural harbors, as no durable harbor constructions are visible; instead three small canoes are pulled up onto the sandy beach of the smaller harbor, while two medium-sized vessels appear to be anchored in the middle of the larger harbor. The larger harbor may have a deeper approach suitable for bigger seacraft. The Minoans did have one kind of dedicated harbor structure, the ship shed, located some distance from the shore and used for storage of ships and nautical equipment during the winter months (see Chapter 5). A portion of the fresco on the north wall of West House Room 5 depicts soldiers marching to the right of a large building partitioned into narrow, open galleries facing the shore, very similar in form to Building P at Kommos, the best known archaeological example of a ship shed. Building P is dated to the later fourteenth century BC, some two centuries or more later than the Flotilla Fresco, but another putative ship shed at Gournia may have been built in MM IIIA (Watrous ), earlier than the Flotilla Fresco and indicating a long tradition into which the example at Akrotiri fits comfortably.
A detailed discussion of the watercraft depicted in the Flotilla Fresco is presented in Chapter 3, and further comment on the harbor setting is made in Chapter 4. The main limitation of the Flotilla Fresco for our purposes is the chronological gap between the massive eruption that destroyed Akrotiri and the earliest verifiable representations of Mycenaean ships and boats. The debate over the date of the eruption is as contentious as ever (see recently the papers in Warburton ), but whether it occurred in the later seventeenth century or sometime in the sixteenth, the connection between the ships and boats of the Flotilla Fresco and the earliest ships represented in Mycenaean art in the later fourteenth century (LH IIIA2) would seem tenuous, but recent revelations of ship frescoes at Pylos and Iklaina may help to clarify the evolution of ship technology during the intervening period (Chapter 3).
2.7 "Flagship" from the Flotilla Fresco, Akrotiri. Wedde : Catalogue 616. Courtesy of Michael Wedde.
Portrayals of Mycenaeans abroad are rare, and they seem only to be recognizable as soldiers wearing boar's tusk helmets. In addition to the Amarna papyrus, there is a warrior, possibly wearing a boar's tusk helmet, painted onto a sherd found at Hattusa and dated to circa 1400 BC (Niemeier : 105, fig. 4). Mycenaeans were not among those painted on tomb walls in Egypt, as were the Cretans before them. Nor do any of the Aegean-style painted frescoes from the East – at Tel Kabri in Israel, Alalakh and Qatna in Syria, and Tell el-Dab'a in Egypt – appear to have Mycenaean connections (Cline et al. ).
##### Pottery and Other Media
Continuing on the theme of the relationship between the Minoan/Cycladic and the Mycenaean seafaring traditions, illustrations of distinctly Mycenaean ships on pottery and other media are curiously late. The Mycenaean oared galley, which may have been an entirely new design (Wedde : 29), is first portrayed on pottery in LH IIIB, but not in great numbers, and other types of craft are generally not depicted at all. Surprisingly, images of the galley on pottery increase after the collapse of the palaces in LH IIIC, a pattern that has social and artistic dimensions that are explored in Chapter 3. Ship images on pottery are subject to all of the ambiguities of artistic representation, with the additional challenge of working on a curved surface.
Boat models in clay are the only three-dimensional representations of Bronze Age watercraft (Fig. 2.8). Models help give a sense of the beam, as well as features such as keels (either painted, or represented three-dimensionally as swellings or protrusions in the clay) and thwarts. Some surely represent small boats rather than the larger ships that predominate in all other media, but their typically crude execution makes determination of dimensions speculative. Of the approximately 50 whole or fragmentary boat models, more than 20 have been found on the mainland, and these range in date from LH IIIA to LH IIIC (Wedde : 307–312). EBA boat models come from Crete and the Cyclades; during the MBA they are limited to Crete; and in the LBA they are found on Crete, in the Cyclades, and on the mainland. Most of the boat models survive as fragments only, limiting the amount of useful information that can be extracted from them.
Seals and sealings with ship imagery come almost exclusively from Crete, where the corpus numbers more than 125, with increasing frequency through the Bronze Age until the destruction of the neopalatial palaces at the end of LM IB (Wedde : 331–49). Fewer than ten have been found on the mainland, in contexts ranging from EH III/MH I to LH IIIB. These undoubtedly represent sporadic arrivals from Crete. As mentioned above, realistic details are difficult to render on such small surfaces, and to make matters more challenging there is a large subclass, the so-called talismanic seals, with particularly abstract designs (Wedde : 134–41).
#### Shipwrecks
Shipwrecks are of immeasurable value in offering single-event horizons and the possibility of full assemblages of associated material being preserved together, though the remains are still susceptible to decay, movement or destruction by currents or sea creatures, and mixing with other wrecks and dumping events. They are direct material evidence for the form and construction of ships and boats; for the cargoes, personal effects, and subsistence items taken on board; and for the location of the vessel when it foundered. The frequent preservation of organic material can open a world of information about the use of substances that rarely survive in the terrestrial archaeological record; the Uluburun project is a shining example of the vast amount of new information that can be gleaned from specialist analyses of the excavated materials. Indirectly, the ship's contents can enlighten about the origin, movements, and final destination of the doomed voyage; about the modes of exchange and purposes of travel represented; and about the social, political, and economic conditions of the time.
2.8 Clay boat model, Asine LH IIIC. After Spathari : 51, fig. 51.
Three LBA shipwrecks in the Mediterranean have been thoroughly excavated and published to date: the Uluburun and Cape Gelidonya wrecks off Turkey's southern coast, and the wreck at Point Iria in the Argolic Gulf on the Greek mainland. A fourth excavated shipwreck, a Minoan ship of MM IIB date off the coast of Crete at Pseira, is nearing publication (Hadjidaki and Betancourt –2006, ). A fifth, near eytan Deresi on the southwestern coast of Turkey, was fully excavated and described in preliminary reports (Bass ; Margariti ), but is not considered here because there is a high probability that it does not actually belong to the Bronze Age (Bass 2005d). Other underwater concentrations of artifacts that have been interpreted as Bronze Age shipwrecks, although without the benefit of timber remains or systematic excavations, include Aegean wrecks off Dokos island (EBA) and at Kyme off the coast of Euboea (LBA, with nineteen "pillow" type copper ingots); and a number of scattered cargo sites on the Israeli coast that are difficult to interpret (Lolos ; Wachsmann : 205–211). In the absence of excavation, we cannot rule out that the remains may represent palimpsests of material from multiple wrecks, or mixed shipboard and land-based debris near harbor sites.
The problem of working back to the Bronze Age ships represented by these sites is exacerbated by the near absence of surviving ship remains. Only the Uluburun and Gelidonya shipwrecks have produced small amounts of hull remains, on which basis the Uluburun ship is estimated to have been 15 meters long and 5 meters wide, and the Gelidonya ship 9–10 meters long but of uncertain width. This material provides crucial information on such aspects of shipbuilding as general hull construction and joinery methods, but is insufficient to furnish independent confirmation of inferences drawn from iconography on a host of unresolved questions. Neither of these wrecks is likely to be a Mycenaean vessel, but as types they must have been familiar in the Aegean.
The most interesting aspect of the four excavated shipwrecks is that they possess contrasting cargo assemblages and exemplify a range of distinctive modes of exchange, as defined above. The Uluburun wreck, with its enormously valuable cargo of precious metals and other luxuries, epitomizes the directional, emissary exchange so vividly described in Near Eastern texts, while at the same time carrying large quantities of non-elite commodities such as poor-quality Cypriot pottery, which merchants on board operating in freelance mode could trade at ports of call (Pulak , ). The Gelidonya ship also carried raw metals: 34 copper oxhide ingots (about one ton), 20 bun and 19 slab ingots of bronze, and a few badly degraded tin ingots; more prominent, however, is an inventory of more than 250 pieces of bronze scrap along with intact tools for coppersmithing, and the absence of luxury finished objects and raw materials such as ivory (Bass , , , ). The Gelidonya ship is seen as a prime example of independent, entrepreneurial cabotage, that is, freelance exchange, but with a specific focus on itinerant metal working. This in itself is fascinating, because itinerant artisans are often postulated to explain the appearance of foreign styles, but rarely are the toolkit and the peripatetic lifestyle of the itinerant craftsman so clearly captured. The Point Iria and Pseira shipwrecks seem to have been small ships plying local networks with cargoes of foodstuffs and other staples transported in utilitarian pots, though there are some points of difference. The cargo of the Point Iria wreck is composed exclusively of pottery (a small anchor found nearby may also belong to the wreck) of three distinct groups: eight Cypriot pithoi, eight Cretan LM IIIB2 transport stirrup jars, and nine assorted Helladic vessels, mainly storage jars and amphoras, but also including a decorated deep bowl that helps to date the entire assemblage to LH IIIB2, probably closer to 1200 BC. With this sort of mixed assemblage, several interpretations of the ship's origin, movements, and mission are possible. It may have left a home base on Cyprus, traveled to Crete, picking up commodities packed in coarse stirrup jars, and continued on to the Argolid – where we know Tiryns had ties to Cyprus and Crete at the time – to conduct business and pick up mainland goods, before finally wrecking on the shallows off Point Iria. Because these transport vessels are so portable, however, the ship could equally have come from Crete, or perhaps most likely of all, it belonged to a local trader carrying produce to neighboring settlements in the Gulf in recycled storage vessels (Dickinson : 35). Cheryl Ward (: 157) cogently interprets the Point Iria wreck as "...the cargo of a small, open boat, not unlike a small modern caique, taking on merchandise at a central location and then traveling within a close-knit network of settlements along the coast." She also reminds us that the detailed and imaginative reconstructions of the ship and its activities that have been published have no basis in material evidence (Ward : 157; see Vichos and especially fig. 16). Finally, the Pseira wreck is dated to MM IIB (roughly the middle to late eighteenth century) by dozens of complete transport amphoras and hole-mouthed jars in local fabrics of the Mirabello region (Hadjidaki and Betancourt –2006: 84–85; P. P. Betancourt, personal communication, 2011). The local provenience and utilitarian function of the pottery have led the excavator to the provisional interpretation of the wreck as a small ship engaged in local-scale trade around the Gulf of Mirabello, which we might imagine as the center of a maritime small world. In the Point Iria and Pseira shipwrecks we have, at last, a glimpse of local-scale maritime networks.
#### Environmental Evidence
As emphasized in the first chapter, the environmental settings of Mycenaean coastal worlds are anything but static; they are constantly undergoing modification by human and natural agents. Some systems are more stable than others. As we shall see in Chapter 4, the weather- and climate-related phenomena that make up the Mediterranean maritime environment – including winds, currents, temperature, and storm patterns – fluctuate on a regular basis but in their broad patterns have not diverged significantly since the Bronze Age. The ability of mariners to cope with environmental challenges through technology and experiential knowledge may have changed radically through the ages, however, so we cannot assume, for example, that Mycenaean captains and navigators possessed the same skill set as Homer's or Hesiod's seafarers.
The changes that have occurred in coastal landscapes and seascapes are generally more extreme, and these are examined in Chapter 5. Coastlines are exceptionally vulnerable to a host of natural alterations resulting from earth processes operating at scales from global to local. Global sea-level change has had a strong effect in some areas. In particular, a maximum marine transgression some 6,000 or 7,000 years ago that flooded Mediterranean land masses and created vast embayments on many Mediterranean coastlines was followed by a stabilization of global (eustatic) sea level and the gradual infilling of these bays with sediments over time. One result is that certain important harbor sites that were open to the sea in the Bronze Age – Troy, Ephesos, and Miletos on Asia Minor's Aegean coast are striking examples – are now stranded literally kilometers inland today. An even more insidious process – because it is more localized and harder to detect on the landscape – is tectonic activity. In the earthquake-prone Aegean region, subsidence and uplift are common coastal processes. There may be regional tendencies – for example, the Corinthia, where the Corinthian Gulf coast is rising while the Saronic Gulf coast is subsiding – but the complexity of the fault systems is such that sites separated by only a few kilometers on the same coastline can have quite distinct tectonic histories (Nixon et al. 2009). Human activities may promote change in coastal embayments, usually by accelerating rates of sedimentation through practices such as extensive agriculture and grazing, which release soil to be eroded and transported to the sea.
Even relatively small changes in a coastline can have transformative effects, rendering a once-inviting harbor unusable by separating it from the sea, making the approach too shallow, altering its protective configuration, or leaving a new set of navigational hazards in the inshore waters. These changes exacerbate the already low visibility of Mycenaean anchorages that can be attributed to the fact that Bronze Age communities did not build permanent harbor facilities such as piers, quays, or breakwaters, so far as we know. Instead, they pulled smaller boats onto the beach, or anchored just offshore, exactly as depicted in the Flotilla Fresco at Akrotiri. Changes in the coastline play a key role in understanding all of the coastscapes examined as case studies in Chapter 7 (Kalamianos, Miletos, and Dimini).
Integrated methods of geomorphological observation, geological coring, and geophysical prospection for paleocoastal reconstruction are well established in the Mediterranean, and the database of case studies of the coasts of Greece and Turkey is constantly growing. The basic principles and many examples of this work are examined in Chapter 5, and the prospects for integrating such programs with archaeological investigations of local and regional maritime worlds are assessed. These studies are essential, but they are expensive and time consuming; they often require many seasons of fieldwork and the results of various analyses may appear only gradually over a period of years. Particular care must be taken that the archaeologist and earth scientist share compatible understandings of the problems being addressed and the expectations of results.
#### Historical and Ethnographic Evidence
One of the benefits of gathering information from historical, recent, and living societies is that, like a well-preserved shipwreck, they offer categories of data that are nearly impossible to recover in a prehistoric archaeological context, and they provide insight on how materials are created, used, and discarded in living societies. And like shipwrecks, they can alert the archaeologist to new ways of thinking. The dangers of simplistic application of ethnographic analogy are well known; the past is indeed a "foreign country" and the gulf that divides the Bronze Age from today must be kept clearly in focus. Yet ethnographic and ethnoarchaeological studies may provide plausible comparative data and even models for Mycenaean coastal worlds. There are numerous ethnographic studies of maritime societies and maritime cultural landscapes around the world, a good many in Oceania, where ancient traditions of seafaring persisted on isolated islands until recently. Several categories of information might be queried for useful comparisons and contrasts; among these are shipbuilding and navigational technology, the organization of maritime coastal communities, and the transmission of maritime knowledge from one generation to the next. In some of these areas, the practices of South Pacific seafarers are exceptionally enlightening, for example the preservation and transmission of maritime knowledge, and the status of the maritime community within the larger society. I incorporate discussion of these points as they relate to Mycenaean seafaring in several chapters.
It may be possible to get this kind of information closer to home. As part of the Saronic Harbors Archaeological Research Project (SHARP), which I co-direct with Daniel J. Pullen (Tartaron et al. ), our colleague Lita Tzortzopoulou-Gregory collected a number of oral histories in interviews with older residents of Korphos, until recently a traditional fishing village. A major focus of these interviews has been to record as much information as possible about life in a maritime coastal community prior to the adoption of motorized boats and before the advent of the modern economy; that is, before the Second World War. The results are, in my opinion, scintillating, because they bring to life aspects of the maritime orientation and economic organization of the community, as well as patterns of interaction by sea that are not merely economic but also social, in the same Saronic setting where the Mycenaeans prospered in the thirteenth century BC. Information from the oral histories is assessed in the Saronic Gulf case study in Chapter 7.
### A Note about Theory
This book is not conceived as a heavily theoretical work, but an essential component of my approach is a theoretical framework that explicitly identifies a structure for maritime interaction at all spatial and temporal scales, and offers a model for the way that these systems are expected to work that is empirically testable. I provide this in Chapter 6, where I begin by focusing on the uniqueness of the coastal zone as a useful empirical category. Then, in order to construct a dynamic framework for interaction in Aegean coastal zones, I adopt the maritime cultural landscape (Westerdahl ) as an overarching concept, and proceed to define four distinct but nested maritime interaction spheres that comprise it, based primarily on geographical scale but also incorporating cultural dimensions. From smallest to largest, these are the coastscape, the small world, the regional/intracultural maritime interaction sphere, and the interregional/intercultural maritime interaction sphere. I draw on social network theory to explain the mechanisms by which these relationships are established and change over time. These concepts are applied to the case studies explored in Chapter 7. More generally, my hope is that this theoretical framework is sufficiently useful and flexible to be adopted in modified form in a range of times and places.
## Three Ships and Boats of the Aegean Bronze Age
An essential baseline for understanding the coastal interface between sea and land in the Aegean Bronze Age is knowledge of the ships and boats active at that time: the range of their forms and functions, their operating limits and geographical ranges. A voluminous and constantly expanding scholarship exists on all aspects of Aegean Bronze Age seacraft (Basch ; McGrail ; Tzalas , 1995a, , , ; Wachsmann ; Wedde ). Much of this discussion remains speculative because, as we have seen, there are few surviving physical remains, and many of the details shown on images and models of Bronze Age vessels are highly ambiguous as to identity and function. Moreover, certain classes of craft that must have existed are represented poorly or not at all. The objective of this chapter is to summarize the salient features that are known or can be inferred about Bronze Age, particularly Mycenaean, seagoing and coast-riding vessels. Here and in the chapters to follow, ship images are cited using the numbering system of Michael Wedde's catalogue in Towards a Hermeneutics of Aegean Bronze Age Ship Imagery (Wedde ). In an image citation, a reference such as W612 simply means Wedde, catalogue number 612 (one of the Flotilla Fresco vessels). The virtues of Wedde's system are that it is rational and easy to use, each item is illustrated and discussed thoroughly, a handy concordance with other catalogues is included, and it is easily accessible in libraries.
### General Characteristics of Mediterranean Bronze Age Ships and Boats
All evidence suggests that Mediterranean Bronze Age vessels were constructed hull first rather than frame first, and there is in fact no certain evidence of frames in the small amount of hull material recovered to date (Pulak : 616). The earliest known frame-first ships date to the mid-first millennium AD, based on recent finds from Tantura Lagoon near Haifa (Kahanov , ) and the Theodosian Harbor at Istanbul (Pulak ). There were two basic methods of joining the planks and other structural members of the ship: mortise-and-tenon joinery, and lacing or sewing (Fig. 3.1). The two Mediterranean Bronze Age shipwrecks from which pieces of the hull have been preserved, those at Uluburun and Cape Gelidonya, were both made with locked mortise-and-tenon joinery (Bass ; Pulak ). Mortise-and-tenon joinery is well known from earlier and contemporary Egyptian vessels (McGrail : 60), but the technique of locking or pegging the tenons into place with treenails was possibly an innovation of Canaanite shipbuilders in the mid-second millennium BC. The locked mortise-and-tenon joint was part of a wider repertoire of woodworking techniques, as illustrated by roughly contemporary tables excavated at MB II Jericho (circa 1750–1650 BC: Wachsmann : 240–41, fig. 10.28) and in Shaft Grave V at Mycenae (Muhly ). Despite the fact that the Uluburun and Gelidonya hulls were joined in this way, ships with sewn planking persisted well into historical times. The hulls of a number of Mediterranean wrecks from the period 600–100 BC are either fully sewn or partly sewn and partly joined by mortise and tenon (Mark : 45–68; McGrail : 61). Most experts accept, however, that the Uluburun and Gelidonya ships were of Levantine or Cypriot, not Mycenaean, origin. There is no consensus on when shipwrights in the Aegean world began to employ mortise-and-tenon joints. Some interpret the boat that Odysseus fashioned to depart from Calypso's island (Odyssey 5.234–57) as a mortise-and-tenon joined boat (e.g., Casson : 217–19), which would establish the Late Geometric period as the minimum age for this technique's appearance in the Aegean. Others, however, read into the same passage a sewn vessel (Mark : 94), and Samuel Mark (: 63–64) in fact places the transition from sewn to mortise-and-tenon joinery in the sixth century BC, as a specific modification to accommodate large, heavy amphora cargoes and burgeoning polis-supported navies. We cannot assume that the Mycenaeans used this technique, but even if they did, many hulls, particularly those of smaller ships and boats, undoubtedly continued to be partially or completely sewn.
3.1 Mortise-and-tenon joinery. Drawing by Felice Ford after Wachsmann : 216, fig. 10.2.
3.2 Painted keel on boat model, Asine LH IIIC. After Vichos and Lolos : 333, fig. 21.
The Uluburun ship had a keel that projected into the interior of the hull, rather than outward as was typical of most ancient Mediterranean hulls (Pulak : 618–19). Boat models from the Mycenaean world of late palatial and postpalatial times show the keel as a painted longitudinal line on the inner bottom surface (e.g., Kynos LH IIIC models W332, W333; Fig. 3.2) or a protrusion on the external bottom surface (Mycenae LH IIIC model W312; Karaminou : 446).
A range of appendages and devices on the bow, stempost, and stern are portrayed on the ships of the Flotilla Fresco (Wedde : 119–30, figs. 9–11; see Fig. 2.7). The larger vessels have bowsprits, or spars, running from the stempost as decorative devices or to fasten the stays (Wedde : 215). Those on the ships participating in the procession are long and decorated with symbols of birds, dolphins, butterflies, and suns. These sprits were apparently detachable (Wedde : 120). Elaborate bowsprits are not characteristic of Mycenaean vessels, but the stempost (the upright continuation of the keel at the bow) often terminates in the head of a bird or other animal, sometimes rendered realistically and in other cases abstractly (Fig. 3.3). These figured stemposts are diagnostic of late Mycenaean ship depictions of LH IIIB and LH IIIC (Wedde's [: 123–24] Skyros and Tragana clusters). The prevalence of bird heads as figureheads accords well chronologically with the depiction of birds (along with other motifs including fish, bulls, octopi, and chariot scenes) often rendered in a similar fashion on painted Aegean and Aegean-style pictorial pottery beginning in LH IIIA in the Argolid at Mycenae and Tiryns (Günter 2000). Subsequently in LH IIIB, an industry centered in the Argolid produced pictorial vessels for export to Cyprus. In LH IIIC, this tradition continued with the "Close Style" at Mycenae and Tiryns, and with other local styles in Greece, the eastern Aegean Islands, coastal Asia Minor, and Cyprus. This tradition then influenced the Philistine Monochrome and Bichrome pictorial pottery as migrants from the Aegean and Cyprus contributed culturally to the formation of the historical Philistines in the twelfth and eleventh centuries (Bunimovitz ; Dothan ; Yasur-Landau , ). Thus the pattern of bird motifs on pottery and ship representations demonstrates continuity bridging the Aegean palatial and postpalatial worlds and involving broad maritime contacts (Meiberg ).
3.3 Bird-head stempost decoration on a straight-sided alabastron, LH IIIC Middle, Tragana. Wedde : Catalogue 643, after Korres : 200. Courtesy of Michael Wedde.
Ships and boats were propelled by one or more of three instruments: paddle, oar, or sail (Wachsmann : 247–54). Anyone who has paddled a canoe and rowed an oared boat will immediately understand the difference in these means of propulsion. Paddling was the earlier form and rowing an innovation in which the pivoting of an oar on a grommet or oarlock increased power and used energy more efficiently. Although paddling continued in use during the Bronze Age in small craft and for cultic use, oared vessels were well established by the later third millennium in Egypt. In the contemporary EBA Aegean, longboats of the Cycladic islands employed up to 25 or more paddlers (Broodbank : 99). It is possible that the shift from paddle to oar as the primary means of propulsion took place as part of an infusion of maritime technology that also brought the sail to the Aegean near the end of the Early Bronze Age. This transformation is evident in the changing depictions on pottery, seals, and models that characterize the developmental sequence from Wedde's "Syros" (EC/EM/EH II) to "Platanos" (EM III–MM III) types (Wedde : 45–52). These changes had other implications for hull design, including broadening the beam to accommodate the mast and rigging as well as the positioning of oars and oarsmen.
Another prominent feature related to propulsion was the steering oar (or quarter rudder, taking the name from its usual position projected over the starboard quarter near the stern), almost certainly attached to the side of the hull by some kind of strap. The steering oar's function is to redirect water past the hull to impart a turning motion to the vessel, and by 3000 BC in Egypt it was found to be a necessary aid to steering. The earliest clear depiction in the Aegean of a steering oar, in this case with a tiller attached, comes from an Early Cycladic III askos from Phylakopi (W416; Wedde : 314; Fig. 3.4). There is a general evolution of the steering oar during the Bronze Age, particularly in the form of the blade (Wedde : 60–62, fig. 7). Earlier depictions from the Cyclades and Crete show a spindle- or leaf-shaped blade, while the Mycenaean blade of LH IIIC was larger and thicker, with a triangular shape. Normally, one steering oar is depicted on the starboard quarter, but rarely there are two (e.g., the ship under sail in the Akrotiri Flotilla Fresco; W617) or even three.
3.4 Early steering oar on an Early Cycladic III sherd. Wedde : Catalogue 416, after Atkinson et al. : pl. V.8c. Courtesy of Michael Wedde.
The earliest certain depiction of a sail in the Mediterranean occurs in Egypt on a Naqada II (Gerzean) pottery vessel dated between 3500 and 3100 BC (Fig. 3.5). The image depicts a ship bearing a single square sail positioned well forward toward the prow, in clear contrast to the conventional positioning of the sail amidships in Bronze Age iconography and models. The position of the sail forward of the center of the hull's profile has been interpreted as part of an early evolutionary stage in ship configuration (Casson : 19), but to modern ship designers, the center of the sail area is properly shifted forward if the vessel is to sail into the wind or with the wind direction forward of the beam (Tilley ).
3.5 Earliest Mediterranean depiction of a sailing vessel, on a Naqada II (Gerzean) jar, Egypt. © Trustees of the British Museum.
3.6 Steatite seal with a ship and possible steering oar, Siteia district, EM III or MM I. Wedde : Catalogue 707, after Xenaki-Sakellariou : pl. 18.79a. Courtesy of Michael Wedde.
The sail was probably introduced to Crete via Egypt just before 2000 BC (Broodbank : 341–47; Yule : 164–66). The earliest certain evidence for the sail in the Aegean comes from a series of Minoan seals from EM III and EM III–MM I contexts (Wedde : 331–33; W701–713), showing ships with a single mast amidships, two or three fore- and backstays, and variable numbers of oars. At least one (W707) may show a steering oar (Fig. 3.6). The high, sweeping stern- and stemposts form the crescent shape characteristic of Cretan vessels through the MBA and into the early phases of the LBA, as is plainly shown in the Akrotiri fresco a half-millennium later.
Until the last phase of the LBA, the square or rectangular sail was stretched between a yard and a fixed boom, and furled by lowering the yard to the boom. This boom-footed rig presented certain limitations (Wachsmann : 248–54). With the fixed boom, the ship had limited ability to sail into the wind; when not traveling before the wind, the crew's options narrowed to lying at anchor or taking up oars. Further, the sail could not be taken in, so to reduce sail the crew was forced to remove the sail and raise a smaller one, not a simple matter with an unwieldy cable system. The results of preliminary sailing experiments with a replicated boom-footed square rig do not contradict these assessments (Raban and Sterlitz ).
A significant innovation of the Bronze Age was the brailed rig with a loose-footed sail, which replaced the boom-footed rig after 1200 BC. In this new configuration, the boom disappeared altogether, replaced by lines (brails) attached to the foot of the sail and threaded up the sail through brailing rings sewn onto the sail. The sail could now be furled by simply pulling on the brails to raise it up to the yard, saving considerable time, effort, and manpower (Wachsmann : 251) and making the ship more responsive to changing conditions at sea. The transition from boom-footed to brailed rig is clearly illustrated in the Aegean. When rigging is identifiable in images of LH/LM IIIB (thirteenth century BC) or earlier, it is almost invariably of the boom-footed type, but most LH IIIC examples of the post-1200 BC period employ the new brailed rig (Wachsmann : 251; Wedde : 80–87). Elsewhere, shortly after 1200, the naval battle scene from the north wall of the mortuary temple of Ramesses III at Medinet Habu shows the ships of both Egyptians and Sea Peoples with brailed rigs (Casson : 36–38, fig. 61; Raban ). The brailed, loose-footed sail seems not to be an Egyptian or Aegean innovation, however. Some interpret the Sea Peoples' ships as Syro-Canaanite (Wachsmann : 163–98). Earlier depictions of Syro-Canaanite ships may or may not carry brailed rigs. An oft-cited example is a craft of Syro-Canaanite type painted in the tomb of Nebamun at Egyptian Thebes, dating to the reign of the Eighteenth Dynasty pharaoh Amenhotep II in the last quarter of the fifteenth century BC. The rigging has been interpreted as supporting a loose-footed sail (Raban : 355), but the part of the painting where the boom would be positioned is not preserved, and several scholars have reconstructed the ship with a standard boom (Wachsmann : 45–47, figs. 3.6–3.8).
Various features depicted on the decks and hulls were probably detachable furniture reserved for ceremonial occasions. On both larger and smaller ships in the Flotilla Fresco, a prominent framework composed of vertical stanchions and horizontal roofing beams forms an awning-like structure that occupies almost half the length of the hull (see Fig. 2.7). The framework creates several compartments in which numerous robed figures are seated. Because boar's tusk helmets hang from some of the compartments' roofs, along with what may be weapons stacked on top, these figures have been interpreted as representations of soldiers (Warren : 119), though they could be VIPs of another type (Morgan Brown ). This structure is limited mainly to the Akrotiri fresco of LM IA and the Miniature Wall Painting at Ayia Irini of slightly later date in LM IB/LH II (W672–76; Wedde : 327–28). At that time, Ayia Irini in the northern Cyclades had strong ties to the southern Cycladic and Minoan worlds. A gold signet ring found near Tiryns and dated on stylistic grounds to LH II depicts a similar structure, with two passengers seated face to face under the awning (Fig. 3.7). Several salient points emerge from studies of the awnings. They occur on frescoes in the Cyclades and seals on Crete, in a relatively narrow time horizon in the middle of the second millennium BC, but with the exception of the (surely imported) Tiryns ring, they are not found on the Greek mainland at any time. They appear in pictorial contexts that are strongly ritual or ceremonial in nature – the elaborate ornamentation, fanfare, and deliberately archaizing propulsion at Akrotiri, and intimations of feasting at Ayia Irini. Since such large structures would have proved a hindrance both to sailing and rowing, as well as cargo capacity, it is safe to conclude that these were detachable structures assembled onboard for special events and disassembled afterward. Ships configured with this special furniture may not have been taken very far from shore, if the Akrotiri and Ayia Irini scenes are any indication.
3.7 LH II signet ring showing awning structure, Tiryns. © Hellenic Ministry of Education and Religions, Culture and Athletics/Archaeological Receipts Fund. Courtesy of the National Archaeological Museum, Athens.
Another kind of deck furniture has greater relevance for the Mycenaean world. An elaborately decorated ship cabin, or ikrion (pl. ikria), present on the sterns of all of the larger ships in the Flotilla Fresco procession, was in essence a screen of oxhide or woven fabric on a framework of poles and crossbars, which presumably housed a seat for the ship's captain or some other important official (Shaw , ; Wedde : 132–34; Fig. 3.8). The ikrion was unroofed – as clarified by the heads of occupants protruding over the top of the framework – and open on the side facing the bow. Like the awning structure, it was detachable, but it may have been mounted on a permanent platform built onto the stern deck (Shaw : 56, fig. 5). Maria Shaw (: 55) characterizes the ikrion as a light, flexible tent-like structure that could easily be disassembled and stored.
Apart from the Flotilla Fresco, images of ships with ikria are relatively rare and date mainly to LM I/II. They include a seal (W910) of MM IIIB–LM I type possibly found near Thebes, a number of "talismanic" seals predominantly from Crete or of unknown provenience, and a few examples of Linear A sign *86 that may incorporate ikria. The renderings of ship components in these last two categories, highly stylized in the talismanic seals and schematic in the Linear A script, make readings as ikria a matter of interpretation. A stone vase with relief decoration found near Epidauros contains a depiction of a ship with an ikrion in an anomalously late LH IIIB context (W642; Wedde : 324). Like awning structures, ikria appear in ceremonial scenes. They are not depicted on Late Mycenaean galleys shown engaged in naval battle, for example, or on any of the nonceremonial ships in the Akrotiri fresco.
3.8 Ikria from two of the Flotilla Fresco ships, Akrotiri. Wedde : Catalogue 614 (top), after Marinatos 1974b: 140, fig. 26; and 615 (bottom). Courtesy of Michael Wedde.
Another class of ikrion representations forges a more direct link with the Mycenaean world. The main decoration on the walls of Room 4 of the West House at Akrotiri, adjacent to the room containing the Flotilla Fresco, was a continuous frieze of eight painted ikria (Wachsmann : 94; Warren : 119; Fig. 3.9). M. Shaw (, ) has convincingly reconstructed painted stucco fragments excavated in 1886 in a small room just north of the Megaron of the palace at Mycenae as part of a comparable frieze of at least four ikria. The room is interpreted as belonging to a domestic quarter within the palace complex, similar to the inferred function of West House Room 4. The Mycenae ikria, along with additional fresco fragments from the Mycenaean palace at Thebes that may illustrate an ikrion in association with a female wearing a flounced skirt, and the Epidauros relief vessel mentioned above, combine to make a strong case for the survival of this particular emblem through a half-millennium of changing relationships and ship forms. Found in highly elite contexts at Akrotiri, Mycenae, and Thebes, the ikrion can be understood as a symbol of nautical power transmitted among those elites whose power rested partially in the control of sea routes by which access to raw materials and privileged relationships was secured. The recent recognition that fragments of a wall painting from Hall 64 at the Mycenaean palace at Pylos constitute parts of a ship with a brailed rig (Shaw , ) highlights the underappreciated role of the nautical realm in the visual language of Mycenaean power.
3.9 Ikrion frieze from West House Room 4, Akrotiri. Shaw : 176, ill. 8. Courtesy of Maria C. Shaw and the Archaeological Institute of America.
### Types of Mycenaean Seacraft
It is possible to recognize certain distinct types of vessels plying the Aegean in the LBA, and to hypothesize the existence of other types for which we have little or no direct evidence. Two basic functional types that have been projected onto the Bronze Age data are the merchantman and the galley (Wedde : 609). The merchantman was a trading vessel designed to maximize cargo capacity. To do so, the space available for rowers and other crew was reduced. With diminished capacity for propulsion by rowing, the merchantman was a true sailing ship that could operate with a minimal crew, but relied on wind power and used oars for limited tasks such as maneuvering within harbor areas. The design of the merchantman hull favored a broader beam, i.e., a larger width to length ratio, to achieve greater capacity and enhanced stability when loaded. Merchantmen were not, however, depicted by Aegean Bronze Age artists, and the true merchantman does not appear in the pictorial record until the late sixth century BC. The date of its initial use will have been somewhat earlier, perhaps as a response to a constellation of novel conditions in the Greek world, including the opening of the Western markets as a result of the colonizing movement, and the naval capacities of the burgeoning poleis that were able to suppress piracy sufficiently for a dedicated cargo ship with few defenses to become viable (Wedde : 845).
Egyptian and Near Eastern representations record a range of forms for merchant ships (Wachsmann : 9–60), but the Uluburun ship is probably typical of the merchant vessels plying the Aegean in the LBA. In view of the ship's apparent westward route and the possibility of two Mycenaean individuals aboard (Pulak ), the design of such Syro-Canaanite ships must have been widely known. The Uluburun ship, estimated from hull remains to have been around 15 meters long and 5 meters wide, carried a cargo weighing approximately 20 tons (Pulak : 615). This 1:3 width to length ratio with its substantial storage capacity may be typical of vessels designed primarily for maritime trade in the LBA eastern Mediterranean. The smaller Gelidonya ship, estimated at 9--10 meters in length but uncertain width, has been associated with a "traveling smith, or tinker" because of the tools and scrap metal that formed much of its cargo (Bass 2005b: 51), and must represent another, poorly substantiated, class in the continuum of Bronze Age seacraft.
Neither the ships of Uluburun/Gelidonya type nor the iconographic Mycenaean galleys fit the functional end members of the round-bottomed trading ship or the oared warship, respectively. Wedde (: 610–12) argues for the existence of multifunctional hulls in the Bronze Age, with intermediate or hybrid forms such as the "cargo galley" or "merchant galley" developing near the end of the Bronze Age and in the Iron Age (see also Casson : 65–8, 157–68). There must have been many intermediate designs that constituted distinct solutions to competing desires for increased storage capacity, speed, and ideal propulsion methods (oar or sail). As compromises, these versatile ships could be called upon for speedy transport of warriors and other important persons, time-sensitive messages, or cargo in need of rapid delivery, perhaps including perishable commodities. The Akrotiri fresco illustrates several distinct classes of craft that form a baseline for reconstructing the variety of Mycenaean ships and boats (see below).
The pictorial corpus of Mycenaean seacraft is instead dominated by oared galleys with long, narrow hulls designed to maximize the number of rowers for the purpose of high speed regardless of wind conditions (Wedde : 609). Although the galley carried a mast and sail, it was a less efficient sailing ship than the merchantman with a greatly reduced cargo capacity. Pictorial vases of LH IIIC from Kynos on the coast of East Lokris in central Greece clearly show galleys engaged in naval warfare (Dakoronia , ), and the pedigree of Aegean ships involved in warfare or piracy can be traced back at least to MH II with the painted representation on a pithos sherd from Kolonna on Aigina of armed figures aboard a ship with a curved hull (W511; Fig. 3.10), if not earlier (Höckmann 2001). The Mycenaean galley was not strictly a dedicated warship, however, since it is frequently shown without warriors and in contexts that imply ritual and other nonmilitary activities. The true warships of Archaic and Classical Greece were the result of a gradual evolution that began with the Mycenaean galley, with incremental innovations that moved the design toward the single purpose of naval warfare.
3.10 MH II sherd showing armed figures aboard a ship, Kolonna. Drawing by Felice Ford after Siedentopf : pl. 38.162.
### Development of the Mycenaean Galley
The oared galley is, practically speaking, the only type of distinctly Mycenaean vessel in the pictorial record. Mycenaean merchantmen may have existed, and small working boats certainly did, but we have no indisputable representations of either. Even the Mycenaean oared galley does not appear iconographically until LH IIIB, before being depicted much more frequently in LH IIIC, leaving a period of several hundred years in Early Mycenaean and early palatial times with almost no evidence of seafaring on the Greek mainland. Because Minoan-style vessels continue to be depicted in small numbers until LM/LH IIIB, it is possible to bridge the gap by assuming that Minoan-style vessels were used by the Mycenaeans up to and including a short period of coexistence in LH IIIB (Wedde : 844–45), but the examples of Minoan-style ships and boats are few on the mainland. A terracotta model from Tanagra (W319) of LH IIIA–B date that shares formal characteristics with Theran and Minoan ship images is perhaps representative of a mainland type influenced by Minoan shipbuilding traditions. Certainly, given the heavy influence of Minoan Crete on the emergence of complex society on the mainland in the Shaft Grave Era, also the time of greatest Minoan maritime expansion in the neopalatial period, such a transfer of technology would be unsurprising. On the other hand, because a Mycenaean figural tradition did not emerge in any medium until mature palatial times, the lacuna may be simply part of the general situation with respect to artistic representation, and not an indication that seafaring was of little importance to Mycenaean polities or that Mycenaean shipbuilders lacked their own distinctive practices.
Here it is appropriate to mention the recent discovery of ships in frescoes from two sites in Messenia that help to bridge the chronological and typological gap between the ships of the Flotilla Fresco and the LH IIIC galley images. These important and as yet unpublished finds, from Pylos (Brecoulaki et al. ; Stocker and Davis ) and Iklaina (Cosmopoulos ), may prove to be intermediate types along this evolutionary path, although it is important to remember that artistic representations may not reflect actual shipbuilding traditions. The Pylos "naval fresco" as reconstructed to date shows three ships in fragmentary state (and possibly the steering oar of a fourth?) that are quite varied in their shape and appearance. What is perhaps most striking is that two of the three ships have the strongly crescentic shape of the Minoan/Cycladic tradition, while the profile of the third is flatter and curves only as it rises to the (apparent) sternpost. The hull of the best-preserved ship is brightly painted with a multicolored zigzag pattern, and three oars extend over the side of the ship into the water. At the stern an ikrion can be discerned, as well as a large foliate steering oar, with two secondary steering oars, one to port and the other to starboard. Only a small fragment of the second crescentic boat is preserved, but two oars are in the water and a steering oar is visible. The third ship seems to have an awning structure and an ikrion. The single fresco fragment from Iklaina shows a portion of a boat with two rowers under an awning structure with their oars in the water. Behind them, the head of a figure is visible behind a curved sheet that must represent an ikrion built into the awning structure. The side of the hull is decorated with painted spirals, and two dolphins swim alongside. Like the third ship in the Pylos fresco, the shape of the hull is flatter, with a gentle curve toward the prow, which is not preserved. Interestingly, the rowers appear to be backing the boat, stern-first, into anchorage.
The early date of the Iklaina fresco fragment, recovered in a reportedly secure LH IIB–IIIA1 context, makes it unique in shedding light on the transmission and adaptation of the Aegean tradition on the mainland. Michael Cosmopoulos identifies both mainland and Minoan features in the fragment, and concludes that it shows a Helladic adaptation of Minoan iconographic motifs. A significant difference between the earlier Minoan/Cycladic tradition and the later LH IIIC representations of the Mycenaean galley is that in the former, rowers are often painted realistically as men, as in the Flotilla Fresco and the Iklaina fragment, with realistic details of the hair, face, arms, and clothing. In LH IIIC rowers are schematic shapes if they are depicted at all; often the oars stand in for the men. No rowers are preserved in the Pylos fresco thus far, but the ships also blend mainland and Minoan/Cycladic features at a much later date in LH IIIB, testifying to the continuing influence of the Minoan iconographic tradition. It is not certain whether these two examples represent a peculiarly Messenian tradition, or one that was more widespread on the mainland but rarely preserved. In any case, these frescoes do not contradict Wedde's conjecture that Minoan-style vessels were used on the mainland in the LBA and coexisted for a time with the newly developing galley in LH IIIB.
According to Wedde (: 29), the Mycenaeans invented the oared galley, a radical departure in naval architecture, sometime after 1400 BC. The Mycenaean galley of LH IIIB and IIIC differed in important respects from the Cycladic and Minoan ships that preceded it. Minoan ships, including the seacraft depicted in the Akrotiri Flotilla Fresco, appear to have developed gradually out of the Early Cycladic II longboat tradition illustrated in the ceramic "frying pans," two lead plaques from Naxos, a sherd from Orchomenos, and a model from Palaikastro (Wedde : 610, : 256 [types II–IV]). Wedde (: 30) argues for a single, multifunctional hull type depicted on Minoan seals and the Akrotiri fresco, one equally capable of carrying trade goods, ferrying people on religious processions, or transporting warriors. He considers this a definitively Minoan type of "oared sailing ship" because although they had crews of oarsmen, their design was primarily for sailing. They were versatile in that a small crew could operate them if room was needed for cargo storage, but they also performed efficiently as naval ships.
When representations of Mycenaean galleys appeared rather abruptly in LH IIIB on pictorial pottery, they presented a radically different hull configuration, possibly redesigned from the keel up, rather than the result of a gradual evolution (Wedde : 30–32, : 257–61). The Mycenaean galley loses the crescent shape and sweeping extremities of the Akrotiri ships, replaced instead by vertical stern- and stemposts. The stempost usually rises higher, and is surmounted by a bird or bird head device, in some cases realistic and in others abstract. The fundamental transformation was from Minoan oared sailing vessel to Mycenaean rowed galley. Minoan-style hulls, including the Akrotiri ships stripped of their ceremonial add-ons and with sails restored, sailed efficiently while resorting to oar power mainly when compelled by environmental conditions – dead calm seas, entering or leaving harbors, rounding headlands, or avoiding lee shores. Their general-purpose shape was not optimized for any one task such as rapid movement of troops or other personnel, raiding or defending, or maximum cargo capacity, but they could modify their configuration or propulsion method as the situation demanded (McGrail : 121–22).
The Mycenaean galley moved toward a more purpose-built design. Although equipped with a mast and sail, it featured a long, narrow hull that emphasized oar-driven speed at the expense of wind power and storage capacity. Some basic structural characteristics can be distilled from pictorial representations and models of LH IIIB and IIIC (Wachsmann : 130–53). Along the length of the extended hull, a frequently occurring ladder-like painted design represents an open rowers' gallery with vertical stanchions defining each rower's station (see Fig. 3.3). Oars corresponding to these stations are usually visible against and below the hull, and in rare cases the oarsmen themselves are added. The origin of the penteconter (50-oared galley) in the Mycenaean galley is confirmed by the common occurrence of 25 rowing stations and oars, with slight variations in the number probably the result of artistic license or insufficient space. Thirty-oared galleys (triaconters) are also depicted, and 10- and 20-oared vessels may have been standard types as well.
A logical measure of the contrast in shape between the Minoan all-purpose hull and the later Mycenaean galley ought to be the width to length ratio, but the estimation of this ratio from two-dimensional representations presents many potential errors in both dimensions (Wedde : 101–10). If paddlers, rowers, or oars are depicted, a rough length can be calculated by recourse to the interscalmium, or distance required between rowers or paddlers for them to execute their task. Once the length for this "motor section" is derived, the length of the bow and stern sections can be established as percentages of the depicted length of the motor section. Estimates of width (or beam) are less reliable, since two-dimensional representations provide no information. They are conventionally derived from some combination of the width required to accommodate two rowers or paddlers across, general considerations of hydrodynamic properties and shipbuilding traditions, and Bronze Age boat models.
The difficulties with this approach are easily illustrated by the widely divergent dimensions calculated for the length and beam of one of the large ships in the Akrotiri fresco (Table 3.1). With length and beam varying by as much as a factor of two, these calculations reconstruct vessels of very different size, hull shape, technical performance at sea, and likely function. It is overly optimistic to hope that Bronze Age artists sought to produce drawings at standard scales, and lacking the dimension of width, definitive reconstructions are practically impossible to generate based solely on two-dimensional images (McGrail : 120). The boat models present the third dimension of beam, but are considered unreliable for calculating dimensions of real vessels because of their typically crude execution. Interestingly, however, the beam to length ratio of most models hovers in the range of 1:3. Such a ratio implies a different kind of craft than the large Akrotiri ships, the Early Cycladic longboats (perhaps circa 1:10), or the Mycenaean and later galleys (in the 1:7 or 1:8 range), but roughly matches the ratio of the Uluburun ship. There is reason to believe, therefore, that some boat models represent cargo hulls of the day, though they could equally represent small boats, similar at least in size to those illustrated at Akrotiri. By and large, the models do not seem to depict galleys. TAB
Table 3.1. Calculated dimensions of a large ship in the Akrotiri Flotilla Fresco (after McGrail : table 4.2)
* * *
Author| Beam| Length| Beam/Length ratio
---|---|---|---
Gifford| 2.6| 17.6| 1:6.77
Gillmer| 3.7| 24.0| 1:6.49
Toby| 2.2| 34.0| 1:15.45
Giesecke| 4.0| 35.0| 1:8.75
* * *
The Mycenaean galley offered certain performance advantages over the Minoan oared sailing vessel. It was a speedier ship with its long, narrow profile and emphasis on propulsion from the motor section and greater proportional waterline length. Although lighter overall, it seated more rowers. Being lighter, it was easier to draw out of the water onto anchorages lacking offshore mooring. It was less dependent on favorable winds, but in many situations ship captains would have preferred to wait out favorable winds rather than try to row long distances. The steering mechanism was a significant improvement: the Mycenaean triangular steering oar, evolved from its Minoan spindle-shaped counterpart, was a forerunner of the Early Iron Age to Archaic steering oar, with possibly superior hydrodynamic properties.
The drawbacks to the galley design are that it could not easily be operated with a skeleton crew (Wedde : 32); its cargo capacity was reduced – a long, narrow vessel with most room taken up by oarsmen would not have made an efficient trading vessel; it was less efficient at sailing; and crew fatigue on long journeys must have played a greater role.
If the Mycenaean galley was a purpose-built ship, what were the roles for which it was designed? To answer this question, we must begin with the historical circumstances of the Mycenaean period, and consider what need there would have been for a fast seagoing ship with a large, potentially heavily armed crew and relatively little cargo space. This was an era of intensive trade and diplomatic relations among the greater and lesser powers of the eastern Mediterranean. Although the Mycenaeans may have been largely peripheral to the main sphere of interaction to the east, their presence – direct or indirect – is attested by the large quantities of fineware pottery vessels that arrived in Egypt, the Syro-Canaanite coast, and Cyprus in the fourteenth and thirteenth centuries, corresponding to pottery phases LH IIIA and LH IIIB (see Chapter 2). There is reasonable doubt about whether the pottery was carried by Mycenaean ships to all these destinations, and it seems unlikely in any case that the galley would have been an effective means to transport such commodities, except in modest quantities that make little pure economic sense. Perhaps galleys participated in quasi-diplomatic or gift exchange missions, which might result, for example, in the kind of concentration of Mycenaean pottery seen at Tell el-Amarna during the reign of Akhenaten in the mid-fourteenth century.
There is ample evidence, both direct and indirect, that piracy and naval warfare were concerns that contributed to the design of the Mycenaean galley. Already in the middle to late third millennium BC, heavily fortified coastal settlements had appeared on the Aegean islands and the Aegean coastlines of the Greek mainland and western Asia Minor. Weapons, including daggers of copper and bronze as well as sling stones, have been recovered from some of the fortified sites, and the Cycladic longboat has been linked to the implicit hostilities as a raiding ship (Branigan ; Doumas ). Later in MH II, armed figures are depicted aboard a long-oared vessel on a pithos sherd from Kolonna on Aigina Island (Fig. 3.10). This is fitting, since Kolonna was a sea power surpassed only by Minoan Crete in the MBA Aegean area (see Chapter 7). Mycenae's great period of awakening, the Shaft Grave Era, occurred in MH III–early LH IIA. The frequent martial themes illustrated on artifacts of the Shaft Graves form a striking contrast to previous Aegean imagery, and at least one object, the Silver Siege Rhyton from Shaft Grave IV, depicts a seaborne attack upon a fortified coastal settlement (Fig. 3.11). The narrative scene on the north wall of West House, Room 5 at Akrotiri, contemporary with the Shaft Grave Era in full flower, is interpreted by some as a seaside battle involving both naval warfare and seaborne attacks on a coastal town (Warren : 117–18). Others prefer to see it as a shipwreck scene (Morgan Brown ).
3.11 Fragment of the "Silver Siege Rhyton" showing a seaborne assault on a fortified coastal town. Mycenae, Grave Circle A, Shaft Grave IV. National Archaeological Museum 481.
Although ships of the palatial period, LH IIIA–IIIB, are poorly represented iconographically, contemporary texts and archaeological sites record Mycenaean maritime forays to the east. By the late fifteenth century, Mycenaean emigrants were active in southwestern Asia Minor and the islands of the southeastern Aegean, where their settlements succeeded Minoan colonies at Miletos on the Anatolian mainland and at Ialysos on Rhodes. In the Hittite texts from Hattusa, ships of Ahhiyawa (Mycenaeans) are reported in various actions on the sea in the fourteenth and thirteenth centuries (see Chapter 2). The Mycenaean ships that participated in a naval raid on Cyprus and helped fugitives from Hittite justice escape by sea from Asia Minor must have been galleys (Bryce ; Neimeier 2003). On the other hand, an apparent Hittite embargo designed to prevent Mycenaean ships from reaching the Syrian coast in the late thirteenth century (Cline ; Güterbock : 136) may have targeted either military or commercial traffic, including Mycenaean merchant hulls if they did exist.
A small corpus of Linear B tablets from Pylos, dating to the final days of the palace's existence circa 1200 BC, concern the deployment of rowers to man ships around the coastlines of the kingdom (particularly, tablets An 1, An 610, and An 724: Chadwick : 173–79). The close match between the different crew sizes that can be estimated from iconographic representations (with 20, 30, and 50 as units) and the requisition in Pylos tablet An 610 of approximately 600 rowers, a multiple of any of these apparently standard galley types, makes it reasonably certain that the ships in question were galleys, and that the palace was able to control the fleet and the personnel to operate it. The role or mission of the ships listed in these tablets is not known, but the description "Thus the watchers are guarding the coastal region" (PY tablets An 657, 654, 519, 656, 661) has been interpreted, in view of the impending destruction of the palace, as indicating an anxious effort to defend the kingdom against imminent attack from the sea (Chadwick : 173–75). At the very least these recruitments involve naval or military operations rather than trade. On the other hand, the rowers are called up according to their villages of residence using the same system of proportional ratios employed for taxation, with sailors obliged to contribute service in return for land grants (Killen ). It is thus possible that the recruitment of rowers in Pylos tablets An 1, An 610, and An 724 represents a regular, annual activity and not a desperate measure taken at a time of extreme danger (Palaima : 285–86). Most normal palatial activities continued to the very end, including craft and industrial activities, sacrifices and feasting, and the routine oversight and documentation of many aspects of the agricultural economy (Shelmerdine : 351–62).
Representations of Mycenaean galleys increase dramatically in LH IIIC, ironically perhaps because of the collapse of the palaces. It is at that time that we first see the galley depicted explicitly in scenes of naval warfare, on pictorial pottery from Kynos in East Lokris (Dakoronia , , ; Fig. 3.12). The Kynos ships in particular have been compared with the roughly contemporary depictions of Sea Peoples' ships at Medinet Habu, and the similarities have been cited as evidence of a significant Aegean component among the Sea Peoples (Wachsmann , : 171–72). This close relationship has been disputed, however. Seán McGrail (: 125) finds closer parallels for the Sea Peoples' vessels with Levantine ships, while Vassilis Petrakis (: 3–4) characterizes similarities such as the bird-head device as imprecise, emphasizing instead the differences in bow and stern morphology to suggest that the vessels may be of an entirely different type. Thus, ship iconography lends little weight to the assertion, better left to other forms of evidence, that Aegean refugees formed part of the movements of the Sea Peoples and participated in their raids.
3.12 Kynos A galley with decked hull, LH IIIC Middle. Wedde : Catalogue 6003, after Dakoronia : 122, fig. 2. Courtesy of Michael Wedde.
It is worth asking whether the galleys and warriors engaged in combat on the Kynos vessels are meant to show pitched sea battles between the navies of two polities, or instead the predatory actions of pirates. Piracy, in the form of quick-strike raids on coastal towns, was probably a fixture of Aegean existence at least since EH II, when fortified coastal settlements became widespread. It has been remarked that there is a fine line between trading and raiding, and the same can be said for the distinction between piracy – "informal warfare" – and interpolity warfare on the seas. The Near Eastern and Egyptian documents of the LBA are replete with references to coastal raids by a variety of agents, orchestrated both by recognizable political entities (such as Ahhiyawa) and shadowy, stateless groups such as the Sea Peoples. From the perspective of modern world history, quick-strike guerilla campaigns and terrorist actions of elusive, nonstate groups pose particular problems for large states, because traditional diplomatic and military solutions tend to be ineffective. It is easy to comprehend why piracy, a form of guerilla warfare by sea, was of such concern to eastern Mediterranean states of the Bronze Age that preferred to settle their differences through diplomatic channels or traditional land battles. They were not easily able to respond in a timely or effective manner to piratical attacks, as is abundantly clear in the Hittites' protracted troubles at the hands of the renegade actor Piyamaradu, or Ugaritic texts from the Rap'anu archive containing desperate communications from the king of Ugarit to his counterpart in Alashiya (Cyprus) regarding devastating coastal raids perpetrated by unnamed enemies:
> My father, behold, the enemy's ships came
>
> (here); my cities (?) were burned, and they
>
> did evil things in my country. Does not my
>
> father know that all my troops and chariots (?)
>
> are in the Hittite country, and all my ships
>
> are in the land of Lycia?...Thus, the country
>
> is abandoned to itself. May my father know it:
>
> the seven ships of the enemy that came here inflicted much damage upon us.
>
> (RS 20.238, transl. M. Astour [: 255])
The responses of states to piracy could include organized and aggressive pur-suit – the systematic sweeping of pirates from the seas attributed by Thucydides to the Minoans, or the historically attested Roman efforts to extinguish piracy in the Mediterranean during the pax Romana – and/or enhanced defensive capabilities. Defensive strategies might involve fortified harbors, coastal installations such as watchtowers for monitoring the sea and sending fire signals, placement of major settlements some distance inland, strategic deployment of naval fleets, and improved intelligence operations.
Historical texts of the post-Bronze Age period portray piracy as not only ubiquitous, but in certain contexts a not dishonorable profession. Thucydides (1.4) famously attributed to King Minos of Crete the organization of the first navy in the Hellenic world, with which he extended his rule over the Aegean and eliminated piracy therein. This passage is fundamental to the highly controversial notion of a Minoan thalassocracy in the Middle and early Late Bronze Age, but it also reflects increasingly negative attitudes toward piracy in the Classical period, a time in which unfettered trade and communication by sea were vital to highly developed state societies. At an earlier time reflected in the Homeric epics, however, the piratical life was imbued with much greater ambiguity. The practices of piracy and warfare, as well as those who practiced them, were poorly differentiated; both (as we would distinguish them) were aspects of the violent life of a seagoing warrior (de Souza : 16–19). In the Odyssey, pirates can be viewed with suspicion as "reckless wanderers of the sea...who risk their lives to prey on other men" (3.71–74; also 9.252–55), but the acquisition of booty through plunder is also an honorable means to achieve high status. Odysseus' false tale of a Cretan upbringing (14.191–265) includes episodes of plunder that bring him wealth and respect, and in this passage it is particularly difficult to distinguish warfare from piracy. Ultimately, however, Odysseus' Cretan adventurer is taken prisoner in a botched raid on Egyptian shores. Two further stories in Book 9 find Odysseus' ships raiding coastal settlements on their return from the Trojan War, with similar results. Upon their departure from Troy, the convoy was blown northwest to the Aegean coast of Thrace, where they attacked Ismaros, a town of the Ciconian Thracians (9.39–61). After killing the men and carrying off their wives, livestock, and other wealth, Odysseus' men lingered, feasting in spite of Odysseus' entreaties to return to the ships. The surviving Cicones rallied local forces and routed the attackers, killing many and driving the rest to the sea. Later, it was Odysseus himself who foolishly insisted on staying on to explore the island of the Cyclopes, with the result that many of his men lost their lives in Polyphemus' cave (9.172–402). Almost comically, each of these stories ends in disaster because in their lust for booty, Odysseus and his men ignore the cardinal rule of piracy: strike quickly and get out fast. The piratical way of life is revealed as dangerous, but this level of ineptitude suggests that the Trojan War heroes turned to plundering coastal towns out of need, not by choice, and with little experience of the art.
The galley, whether in Mycenaean or later Homeric form, was the ideal pirate ship: light enough to be beached, optimized for speed with its powerful motor section and capable of rapid strike and retreat regardless of wind conditions, battle-ready with a large number of oarsmen doubling as warriors, and endowed with sufficient storage space for modest quantities of plunder to be distributed among the convoy. The typical pattern of piratical raids, whether described in LBA diplomatic letters or Homeric epics, is that when they are executed in quick-strike fashion, few towns or even empires are able to respond quickly enough to prevent attackers from escaping with their loot (Bradford : 4–5).
Although the Mycenaean galley began a movement away from the all-purpose design, it nevertheless assumed a variety of roles, some of them similar to those inferred for Cycladic longboats of a millennium earlier: raiding, trading of low-bulk cargo, and elite voyaging (Broodbank : 100). To these we might add rapid transport of personnel and messages, defensive deployment against pirates and enemy navies, and ultimately pitched naval battles like the ones depicted on the Kynos sherds. The apparent shift toward more frequent use in warlike situations is not surprising in light of textual and pictorial evidence from the eastern Mediterranean recording naval warfare as a relatively common aspect of LBA interrelations.
### Social and Historical Impact of the Galley
Iconographic representations of the Mycenaean galley are virtually absent until the mature palatial phase of LH IIIB, but since the Mycenaean world had experienced tremendous growth economically and politically by LH IIIA2 – including the emergence of palaces at Mycenae, Tiryns, Thebes, and Pylos – there is reason to believe that the galley was part of this transformation. One motivation for the galley design may have been to extend the range of maritime forays in search of raw materials and trade contacts. The rapid development of the galley could be explained in terms of a feedback loop between a dramatic increase in overseas interaction in LH IIIA2–IIIB1 on the one hand, and innovations in technology on the other (Wedde : 29). As social and economic conditions gave impetus to technological development, the enhanced galley in turn expanded the Mycenaean world. That these circumstances may not have resulted in the development of the merchantman, as would seem logical from an economic perspective, must indicate the high priority for a ship able to defend itself and make headway under widely varying wind and current conditions.
The fact that so much of the pictorial evidence comes from LH IIIC is a striking detail that has received insufficient attention. Why the concentration of galley iconography in the postpalatial period, when it might have been expected to peak instead in LH IIIA2–IIIB1, the heyday of Mycenaean overseas contacts and international trade? Was it merely the sluggish development of a palatial pictorial/figural tradition that delayed the depiction of prominent objects already well established in daily life, or was there something particular about the relationship between the palaces and the seagoing fleets that made such illustration inappropriate?
In pursuit of answers to these questions, it is interesting to speculate on the role of the galleys in the final, turbulent years of the Mycenaean palace system in LH IIIB2. They may have had a dual, paradoxical effect: prolonging the life of declining palace centers by securing lifelines to sources of supply, while at the same time fostering the rise of maritime communities that may have contributed to the downfall of palaces if alternative power centers materialized at coastal nodes in the periphery of the palaces. The latter possibility refers to Wedde's (2005: 33–6) suggestion that a "galley subculture" emerged where galleys were built and beached, and where captains and crews lived, because galley rowing led to the formation of teams commanded by a captain and a helmsman. These became embryonic power centers. So long as the palaces were able to maintain loyalty and order by distributing benefits to key individuals and groups within the kingdom, they could manage maritime activities and the crews that carried them out. But severe disturbances beginning in the mid-thirteenth century seem to have been accompanied by disruptions in long-distance trade. Whether or not these disruptions were at least in part external and systemic to the eastern Mediterranean in general, any breakdown in the flow of goods and services to and from the palaces had the potential to strain or sever the relationships that buttressed palatial power. If nascent maritime communities existed, they may have transformed themselves from agents working in the palatial interest to dangerous male populations capable of using their specialized knowledge and access to distant sources of supply to create alternative centers of power. Closed maritime communities are known throughout history in all areas of the world; at times, they become alternative quasi-societies with their own distinctive ideologies, practices, and social structures (Adams : 304–306; Muckelroy : 221–5, 240–42).
The implication of this argument is that in Messenia, these groups may have helped to bring down the palace. Wedde points to rowers' coastal towns mentioned in the Pylos tablets as possible bases, but if so they were short lived: after 1200 Messenia was nearly depopulated, so no putative coastal power centers survived the palatial collapse, casting doubt on Wedde's scenario for Messenia or any other palatial territory. The palace was clearly involved in building (Vn 46, Vn 879) and staffing (An 610, An 724) ships. The fact that the Pylos tablets belong to a period of no more than one year, and perhaps even less, prior to the final destruction and abandonment of the site implies that the palace was able to summon military personnel up to the very end, unless we propose that the rowers never reported, or that the absent rowers in An 724 were defectors. In the final days, much of the normal business of the palace continued, all recorded routinely and meticulously by palace scribes.
An alternative interpretation of the tablets concerning rowers is that they were requisitioned by the palace as part of an organized exodus of the elite in the face of impending disaster (Wachsmann ). The archaeological record at Pylos is plausibly read as a deliberate abandonment in which people, livestock, and valuables were removed and replaceable items such as pottery were left behind (Wachsmann : 496–98). Refugees from Pylos and other Mycenaean centers fled to the east, where some settled in Cyprus and later migrants founded the Philistine cities of the southern Levant; to remote interior locations such as Arcadia; or to Greece's coasts at places like Lefkandi, Perati, and Kynos, from which maritime connections with the world beyond the Aegean would be continued or resumed. In this scenario, the galley teams become saviors who evacuated the elites instead of rivals intent on bringing the palaces down.
We now know that maritime commerce did not simply wither away after 1200 BC, but was reconfigured in the wake of dramatic social and economic transformations in the eastern Mediterranean. Many prosperous centers of LH IIIC were coastal, contradicting the once commonly held notion that Aegean communities abandoned coastal sites in fear of marauding Sea Peoples, turned their backs on overseas connections, and eked out a pastoral "Dark Age" existence. The unambiguous pictorial representation of galleys engaged in sea battles on the Kynos sherds, and the general spike in illustration of Mycenaean galleys on pottery of LH IIIC, give evidence that fleets of galleys played an important military role in postpalatial times, both to protect communities and their assets, and to serve as offensive weapons to carry out swift seaborne raids. The capability to build fleets of galleys and organize defense and overseas trade implies that some form of hierarchical society with sufficient organizational infrastructure persisted in LH IIIC. It even seems possible that galleys were one element of a revived – or preserved – elite culture, along with feasting, hunting, chariot riding, seafaring, and warfare on land and sea, which found expression on pictorial kraters in the Euboean Gulf area in LH IIIC Middle (Crielaard : 282). Perhaps the galley warriors depicted on the Kynos sherds, and the shipbuilders and ship captains they imply, are examples of Wedde's "galley subculture," elevated to a new level of prominence in the economic and social conditions of LH IIIC Greece, and the ancestors of Odysseus and Homer's other seafaring heroes (Wedde : 36).
Returning to the original observation of concentrated depictions of galleys in LH IIIC, none of the hypotheses just cited addresses why the palaces did not invest more liberally in ship representations as expressions of their power and reach, but in postpalatial times such images may have been fitting expressions for newly empowered groups focused strongly on the sea. Most of the great palace centers – Mycenae, Thebes, Pylos, Knossos – were situated inland and drew the bulk of their resources and power from fertile agropastoral hinterlands. In the postpalatial era, smaller-scale polities lacked the means to control vast territories, and so may have preferred to organize into "small worlds" of settlements occupying good anchorages with modest agricultural hinterlands, relying on one another for protection and more generally for social and economic viability. Galleys and other types of seagoing craft, commanded by sea captains who now perhaps wielded the power of basileis as they do in the Odyssey, forged the essential links that bound these reconfigured "maritime cultural landscapes" (Westerdahl ). It is conceivably for these reasons that galleys became a popular subject for pictorial pottery.
### Small Boats
To put it plainly, we know virtually nothing of the range of small boats the Mycenaeans used. There are no physical remains, no pictorial representations, and no textual references. This poses a problem for a central claim of the present study, that short-distance contacts constituted the main maritime interactive spheres of Mycenaean small worlds, since boats of various forms and functions should have been the workhorses that maintained these close connections. Are these craft, therefore, irrevocably lost to us? The answer is not quite, although we are left to speculate on the basis of mostly indirect evidence. The relevant classes of evidence we do possess are (1) contemporary or near-contemporary pictorial representations, such as Egyptian tomb paintings, seal engravings, and the Flotilla Fresco at Akrotiri; (2) Bronze Age boat models that may or may not represent small boats; (3) worldwide ethnographic data on "traditional" boat building and use at sea; and (4) experiments in building and navigating replicas of ancient ships derived from analysis of sources 1–3 above. One factor that tends to legitimize comparison with contemporary and near-contemporary non-Aegean, as well as worldwide ethnographic, specimens is the long-term conservatism so frequently observed in form and function of small craft. Consider that the large ships of antiquity, from galleys and merchantmen to the great naval and grain ships of the Roman Empire, have all long since disappeared from use as obsolete – it would be absurd to contemplate building such craft today except for historical interest. These ships are technology driven and frequent modifications are part of a competitive process for greater cargo capacity, more speed or superior sailing capabilities, or better defensive or offensive characteristics. By contrast, small boats similar in many respects to those depicted in the Akrotiri fresco and in Egyptian tomb paintings and models are still to be found in scattered parts of the world. This is so because throughout time they have facilitated the fulfillment of local (or "microregional") subsistence and social needs among societies under conditions that necessitate the use of low-cost, readily available materials with minimal technology including simple tools such as adzes, saws, and bowdrills of metal or stone. This has given rise to innumerable local traditions and endless variations of boat form, "conditioned by the geography of the local waters, climate, purposes for which the boat was needed, availability of materials for their construction, tradition of craftsmanship which grew up among the boatbuilders and the general state and nature of the culture of the people building them" (Greenhill and Morrison : 20). Nevertheless, it comes as little surprise that a Middle Kingdom Egyptian papyrus skiff (Jones : fig. 26), an Early Cycladic model canoe of hammered lead (W105), a canoe appearing in the Flotilla Fresco, and a modern dugout canoe of the Caroline (Gladwin ) or Solomon (Feinberg ) islands of the Pacific should have such similar forms as a result of comparable basic materials – including the physical properties and behavior of wood – and uses. In fact, a "papyrus" skiff (papyrella) remained in use on Kerkyra (Corfu) in Greece into the 1980s for local fishing and lobster trapping. (Tzalas 1995b reports that despite the colloquial term papyrus, the plant is actually Ferula communis L., the giant fennel.) Harry Tzalas remarks on the close resemblance of a reconstructed papyrella, built by a local craftsman with no knowledge of ancient boats, to the Bronze Age Egyptian skiff, attributing this to the limitations of the long-stemmed plants themselves, commenting "...it would have been difficult, if not impossible, to obtain a different form" (Tzalas 1995b: 446). Small-scale coastal and island societies with modest access to raw materials, manpower, and technology always require simple water transport for fishing and other subsistence activities, as well as transport within the small worlds that are defined by their social and economic networks. The basic needs of Mycenaean coastal communities were no different; thus it is reasonable to proceed with an examination of the forms of evidence enumerated above.
At the outset, we might propose a series of boat types, and identify the range of functions they may have served in LBA coastal settings (Table 3.2). If a coastal community hosted a harbor that admitted large galleys and trading ships, certain classes of small boats facilitating access in and out of the harbor would be required, especially if the water was shallow or underwater hazards necessitated the approach of larger vessels via specific channels. This kind of boat, which we might refer to as a pilot or guide, has been recognized in the small canoe (W622) being rowed out to meet the ceremonial ships at the Arrival Town in the Flotilla Fresco; three identical canoes are beached on the shore. Another type of harbor vessel would assist in loading and unloading cargoes where ships could not approach closely to shore. The ideal form for these boats would combine storage capacity with shallow draft – most likely a smallish craft propelled by oars and maximizing beam while minimizing draft within the constraints of local conditions. Because of the often windy and turbulent conditions of Aegean coastlines, large, flat barges like those plying the Nile in the Bronze Age would not have been feasible.
Table 3.2. Hypothetical Late Bronze Age small boat types and functions
* * *
Vessel type| Propulsion| Range| Functions
---|---|---|---
Pilot/guide| Rowed or paddled| Local/harbor| Guide incoming ships into harbor channels and away from hazards; to mooring near shore or on offshore islands
Barge| Poled or paddled| Local/harbor| Load and offload cargo
Canoe: dugout; papyrus or reed| Paddled| Local| Fishing, local travel
Rowboat| Rowed| Local/regional| Fishing, local and regional trade and social communication
Coasting vessel| Rowed or sailed| Local/regional| Regional trade and social communication
* * *
Vessels specifically dedicated to subsistence needs would include boats for fishing and trapping in coastal waters and wetlands. The empirical evidence for fishing in the prehistoric Aegean is not overwhelming (Powell ). Judith Powell assembled the evidence from faunal remains, fishing equipment, and iconographic representation available in the mid-1990s. It is safe to say that few of the earlier archaeological investigations surveyed by Powell routinely sieved, let alone floated, archaeological deposits to recover botanical and other tiny to microscopic remains. In recent years, however, sieving and flotation have become standard practices of most Aegean projects, with the result that among the small finds recovered by this process are fish bones, fishhooks, and sinkers. As these data are incorporated into archaeological knowledge of Aegean prehistory, a more balanced assessment of the contribution of fishing to local subsistence will be possible. To cite the example of one site central to our discussion, Kynos has produced substantial numbers of fish bones, fishhooks, lead weights (for nets), and shells from edible and inedible shellfish (Dakoronia : 287).
It is important to consider the range of different methods and settings for fishing, broadly defined. One meaningful way to differentiate fishing activities and their requirements is by setting (Table 3.3). Although it is difficult to avoid a measure of arbitrariness (e.g., the five-kilometer boundary between inshore and offshore fishing is arbitrary and will vary with local conditions) or to imagine that each of these categories correlates with different equipment, practices, or targeted resources, they do prompt us to contemplate fishing more broadly in terms of the comparative difficulty of exploiting different species, the methods of fishing and the required technology including boats, and the risks and rewards as one ventures further from shore.
Table 3.3. Range of fishing practices defined by setting (after Pickard and Bonsall : 274)
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Type of fishing| Setting
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Land-based fishing| All fishing practices that can be conducted from land, without the use of watercraft
Inshore fishing| Fishing activities conducted using watercraft up to 5 km from shore
Offshore fishing| Fishing conducted more than 5 km from shore
Open-sea fishing| Fishing conducted out of sight of land
Deep-sea fishing| Offshore or open-sea fishing in deep waters
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There are many methods of land-based fishing that do not require watercraft, focused in the shallow waters just offshore or in coastal wetlands. Among these are casting a fishing line from shore; fishing with hooks or spears while wading or swimming; diving for shellfish or sponges; setting traps for fish, lobsters, octopi, and other marine life in shallow wading water; and collecting crustaceans on sandy beaches (Powell : 82–138; for ethnographic examples of fishing strategies, see Feinberg : 22–24, 124–32; Gladwin : 27–32). Wetlands provide another set of land-based activities that may involve fishing, fowling, and collecting shelled animals and amphibians.
At a distance from shore, fishing from boats can involve many of the same basic techniques – diving, spearing, fishing with nets or with hook and line – adapted to the prevailing conditions at sea. These techniques are equally practicable from dugout canoes or plank-built rowboats, though modifications such as outriggers attached to dugout canoes may be specialized adaptations for fishing or other tasks (Feinberg : 51–59). Ethnographic research on island societies suggests that few engaged regularly in offshore or open-sea fishing, except under conditions of scarce terrestrial resources (Pickard and Bonsall : 276); nevertheless, open-sea and deep-sea fishing are not unknown (Feinberg : 23). Bones of bluefin tuna are present in Upper Mesolithic levels at Franchthi Cave on the Greek mainland, approximately 9000–8000 BP. The remains of these pelagic fish suggested deep-sea fishing already at that time (Rose ; Runnels : 247–48), but recent assessments are more skeptical, pointing out the small quantities of the bones and the possibility that tuna could be caught by nets as they enter inshore waters to feed (Perlès ; Pickard and Bonsall : 283).
No Linear B texts are known to refer to fish or fishermen (Palaima : 284), probably meaning that fishing was not controlled directly by the palaces and thus was not part of scribal recording systems. Like clay, fish and other marine resources were ubiquitous and it was not feasible for the palace to control access to this basic subsistence resource in coastal areas. This is not to say that the palaces took no interest in products of the sea: the palace may have had several sources of indirect supply from which to choose, as was apparently the case with pottery at Pylos (Tartaron : 105–106). It merely means that the palaces did not attempt to monopolize marine resources or control their movement through the kingdom, as the Pylian palatial elite did with precious metals and the Knossian palace did with sheep. Silence in texts should not be taken to mean that marine resources were little exploited, since in the historical period a comparable paucity of references to fisherman and their products reflects their low social status in spite of ample evidence that fish were eaten and generally appreciated. The frescoes of young fisherman at Akrotiri and the frequent depiction of fish and sea life on pottery (particularly in Minoan Crete) suggest their importance. During postpalatial times, fisherman working communally to trap fish with nets were depicted on painted pottery at Kynos and Aplomata (Yasur-Landau : 91–92, figs. 3.38, 3.39). On the other hand, to date stable isotope analysis of skeletons has failed to demonstrate a significant contribution of marine protein to the diet of Bronze Age individuals, even at coastal sites (Petroutsa and Manolis ; Triantaphyllou et al. ). The extent to which this result is a methodological issue of measurement remains a matter of debate (Hedges ), so it is not yet safe to conclude that Bronze Age coastal dwellers eschewed marine dietary resources. It will be interesting to see whether flotation of archaeological deposits in the future supports the isotopic finding of near absence of marine products in the diet.
Another type of craft plying local and regional waters would fall under the broad rubric of coasting vessel. The coasting vessel, or coaster, has the primary function of "sailing along or near a coast, or running between ports along a coast," typically engaged in coasting trade "carried on by water between neighboring ports of the same country, as distinguished from foreign trade or trade involving long voyages" (Webster ). This functional definition offers little help in imagining what such vessels would have looked like in the Aegean Bronze Age, but we might expect certain characteristics, notably shallow-hulled ships and boats that can negotiate reefs and other hazards where deeper-hulled seagoing ships cannot. Their construction could be less heavy and robust than that of ships built to withstand long voyages on the open sea, the latter perhaps best exemplified by the Uluburun wreck. Perhaps we can recognize a coasting vessel in the modest dimensions of a craft at Akrotiri (W612; Fig. 3.13), rowed out of the harbor of the Departure Town with five rowers under a (temporary?) framework, a helmsman operating a single steering oar, and a seated figure behind him at the stern. Other fragmentary boats appearing in the Flotilla Fresco (W632, 633) may be of the same class, and we might even include the lone boat under sail (W617), once the ceremonial embellishments (ikrion [?], framework, hull decoration) were removed (Fig. 3.14). Coasting boats of this scale, propelled by oar and sail, are central to the notion of maritime small worlds because they played the primary role in creating and maintaining the networks that constituted them.
3.13 Ship rowed from "Departure Town," Flotilla Fresco, Akrotiri. Wedde : Catalogue 612. Courtesy of Michael Wedde.
3.14 Ship under sail, Flotilla Fresco, Akrotiri. Wedde : Catalogue 617. Courtesy of Michael Wedde.
With these hypothetical statements about the form and function of boats operating in Mycenaean small coastal worlds, we may continue with a closer examination of the lines of evidence proposed above.
#### Egyptian Boats
A broad range of depictions of ships and boats has survived from Bronze Age Egypt, the result of a rich visual culture, excellent preservation even of organic material, and the fundamental place of water travel in everyday life as well as in the deeply symbolic and religious structures of the Egyptian worldview. These representations consist of tomb paintings, temple reliefs, boat models mainly from tombs, and remains of actual full-scale boats. Boat models were crafted in wood, ivory, clay, and metals including gold and silver. The vast majority of the surviving examples are of wood, hinting at an entire class of models that may have existed in the Aegean, but that will have disappeared in poor preservation conditions.
The boats found in Egyptian tombs served a range of essential functions for the deceased: utilitarian vessels provided for traveling, carrying loads, hunting and fishing, and pleasure cruising; funerary boats conveyed the mummy across the Nile for burial and on journeys to Abydos and other sacred sites; and magical boats carried pharaohs on eternal journeys to cross the sky by day and the underworld by night (Jones : 27). One of the principal types of utilitarian watercraft, the use of which spans the entire Bronze Age from the Old to the New Kingdom, was a skiff built from bundles of bound papyrus plants. These utilitarian craft were used locally to carry light loads and for fishing and fowling in wetlands (Jones : 36). The papyrella used until recently on Corfu, mentioned above, has a form similar to the Egyptian skiff, a result of the comparable working properties of the papyrus and giant fennel, and was used in a comparable range of settings and for similar tasks. The simple construction of these canoes, made by lashing together bundles of cut stalks with vegetable fibers or leather strips, fulfilled the requirements of low cost and low technology for families with few resources for elaborate and expensive watercraft.
#### Iconographic Images of Small Boats
Apart from the four small skiffs or canoes (W622–25) from the Flotilla Fresco, there are few other Bronze Age images that can be interpreted unambiguously as realistic representations of small boats. There is a large corpus of seals and a few sealings and gold signet rings (W701–981; Wedde : 331–49), almost exclusively from Crete and ranging in date from EM III to LM III with a peak in MM III–LM I, which depict boats or ships of various kinds in "cultic" or "ritual" scenes. Within this corpus several subcorpora can be distilled. One such group (W901–912; Wedde : fig. 18) features anthropomorphic figures of deities or worshippers in boats along with other standard elements of Minoan cult scenes, including trees (W904) and tall structures surmounted by horns of consecration (W908). Another group comprises the so-called talismanic seals, characterized by the use of a tubular drill and broad cutting stone to create heavy geometric elements, resulting in highly abstracted motifs (Onassoglu 1985; Wedde : 134–41, figs. 12–14). When ships or boats appear on talismanic seals, typically only the bow and a portion of the length of the hull are shown, along with a bird device on the bow and a highly stylized ikrion, outlined by net and lunette patterns, rising from the hull (Wedde : figs. 12–14). The trouble with identifying these images of watercraft as small boats is that their dimensions often cannot be estimated on human scale if no figures are depicted, and for many we cannot rule out that the artist sought to illustrate fantastical or magical craft rather than boats faithful to real-life examples. Thus, the cultic group (W901–912) depicts craft carrying an individual or a small group of figures suggesting a small boat, but with the figures themselves often appearing at distinctly different scales, it is difficult to infer how large a vessel the artist imagined, if that was even an important detail at all. Consider another example, an Early Cycladic graffito cut into white marble from Korphi t'Aroniou on Naxos (W413; Fig. 3.15). This pictograph presents a narrative scene of a flat-hulled boat with a prominent rising bow, onto which a quadruped (goat?), more than half as long as the boat itself, has been loaded while a human holding implements in each hand steps onto a spur at the stern. Perhaps the intended narrative is the short-distance transport of an animal or two to grazing lands or for trading with another coastal community, but I find it equally compelling to understand this graffito as a kind of shorthand depiction of a much bigger event, such as the departure scene of an early colonizing expedition in which these figures stand in for many animals and people setting out in larger ships – for example, from the Cyclades to Ayia Photia on Crete in the EBA. In rare instances, we can be more certain that the artist intended to illustrate a small boat. A green steatite seal, found in a MM I context at the palace at Malia (W808), depicts two men occupying most of the space in a flat-hulled boat lacking a mast or other rigging, with five parallel, subvertical appendages to the hull that probably represent oars (Fig. 3.16). Below the boat, six fish swim randomly about. The iconographic elements and the minimal rendering of this scene suggest a simple fisherman's rowboat at work. Other cases are more ambiguous. For example, the lip of a Mycenaean LH IIIC pictorial krater recently excavated at Kynos preserves portions of the mast, forestay, backstay, and the extremity of either the stern or the bow of a sailing vessel (Dakoronia : 286–87; Fig. 3.17). No sail is visible, but a single standing figure operates an oar, indicating the possibility of a harbor scene. Fanouria Dakoronia (: 287) reads the vessel as a fishing boat, pointing to similarities with the ship depicted on the LH IIIC stirrup jar from Skyros (W655), to which she ascribes a similar function (Fig. 3.18). Others have not generally taken the Skyros vessel as a fishing craft, however, and this once again underscores problems of confidence in the recognition of small boats.
3.15 Incised image of a boat with human and animal, Korphi t'Aroniou, Naxos, Early Cycladic. After Wedde : Catalogue 413.
3.16 Green steatite seal showing two men in a boat with fish swimming underneath, Malia MM I. Wedde : Catalogue 808, after Van Effentere 1980: 72, fig. 98. Courtesy of Michael Wedde.
3.17 Fragmentary boat from an LH IIIC pictorial krater, Kynos. After Dakoronia : 290, fig. 11.
3.18 Motif of a sailing ship from an LH IIIC stirrup jar, Skyros. Skyros Archaeological Museum A77.
#### Boat Models
There is reason to believe that small craft are better represented among Bronze Age model boats. Wedde's catalogue lists 50 models, 9 from the EBA, 5 from the MBA, and 36 from the LBA, with the largest group in LH/LM III. Unlike iconographic images, models, as three-dimensional objects, permit measurement of beam and can be used to calculate ratios of length, width, and depth of hull, but only a small number of the models are sufficiently preserved to allow measurement of all these dimensions. The pattern, cited above, that the measurable models cluster around a 1:3 width to length ratio suggests to Wedde (: 108) either a class of Aegean Bronze Age boats that were beamier than two-dimensional images suggest, similar perhaps in ratio and function to the Uluburun merchant ship, or that the models depict mainly small boats. In favor of the latter interpretation is the generally simple (sometimes crude) execution of the models and the lack of elaborate attachments and decorations; often a few tholes or thwarts are the only molded attachments if any are present at all. Rarely a mast step or stump is present to indicate a sailing vessel (W301, 314, 323). Some models, notably in LH/LM III, were painted with banded and other linear decoration, or in some cases even more elaborate motifs, with clear parallels in painted fineware pottery and figurines (see Figs. 2.8, 3.2). The simplicity of many models and the width to length ratios are suggestive, but not conclusive, that these models are more representative of small boats than beamy cargo ships. Because many come from the mainland as well as islands (including Crete) under Mycenaean influence in LH III, the boat models form a significant body of material to illuminate coastal interactions at regional and local scales.
### Ethnographic Analogies
The benefits and dangers of using ethnographic analogy to illuminate poorly understood aspects of the distant past were discussed in Chapter 2. When considering the forms and functions of boats, it is questionable whether analogies from contexts distant in space and time or from recent and historical times in Greece will be of much help, beyond the general assertion, already made, that the technology of small boats used in local and microregional settings is more conservative and enduring than that of larger seagoing ships. A more persuasive application of ethnographic data parses the relationships between the physical and performance characteristics of a boat, the mariners who sail it, and the relationships of maritime societies with the sea and with other maritime societies. Much of this information is important explicitly because we possess no tangible evidence to reconstruct nonmaterial aspects of Mycenaean maritime life, and ethnographic accounts provide new and diverse ways of thinking about them. The insights that can be gained from ethnographic data are taken up mainly in subsequent chapters in discussions of navigation and intersocietal interactions.
As indicated by the earlier quote from Greenhill and Morrison, local ship technology results from a complex combination of environmental and social conditions. The factors that make boats similar or different across space and time are conditioned on the one hand by the building materials, tools, and general level of technology available, and on the other hand by the tasks to be carried out in boats, which are shaped by subsistence patterns and by social needs (relationships of trade, kinship, friendship, intermarriage, etc.) within society and with people in distant locations. Because of the complex interplay of these factors, boats in societies facing similar environmental conditions and having comparable social structures may be dissimilar, while quite similar boats may be produced by societies sharing few social and environmental circumstances. The latter case is possible because certain universal factors come into play in simple boat technology. One of these concerns the widespread use of trees and plants as sources of construction material. Humans learned independently in many places that long tree trunks could be hollowed out to make dugout canoes, and long experience with the hydrodynamics of the form resulted in various modifications that responded to conditions both local and universal. In the Aegean, the dugout canoe was apparently used during the Neolithic and persisted into the EBA, attested by four hammered lead boat models from EC II Naxos (W105–108) whose forms are remarkably similar to canoes in recent Pacific island traditions. Although the Early Cycladic longboat is an apparent descendant of the dugout canoe, dugout construction was replaced by plank-built ships and boats already in the third millennium BC. As we have seen, stalks of different plants (papyrus and giant fennel) had analogous uses in Bronze Age Egypt and modern Corfu, resulting in boats of similar form and function in two vastly different social settings. One way to make sensible use of comparative data on boat form and function is to draw upon performance characteristics derived from ethnographic observation as well as experimental testing of reconstructed "ancient" boats.
#### Performance
The performance of a ship or boat at sea depends on multiple and interacting technical, environmental, and human factors. The technical characteristics of the vessel govern the parameters of its behavior at sea and its tolerances under a range of maritime conditions, modified by alternative configurations and loads that may change from voyage to voyage. Thus, a merchant's sailing ship, like the one that wrecked at Uluburun, will have a sharply different performance profile from an Early Cycladic paddled longboat, quite independent of any environmental or human variables. Environmental factors include macro- or mesoscale forces such as weather and prevailing winds and currents, which in the Mediterranean are often subject to pronounced seasonality. The configuration of coastlines controls the distance between landfalls and the landing places that are suitable for different kinds of vessels; the distribution and contour of land masses in the sea influence visibility and navigational possibilities. The human dimension must take account of experience, knowledge of habitual and alternate routes, navigational knowledge and skill, and crew attributes such as technique, strength, and stamina. In the next chapter I consider in detail the environmental and human contribution to maritime travel; here I focus on what we can reconstruct of the technical performance characteristics of the different types of vessels we have proposed for the Mycenaean world.
Estimation of performance does not depend on a vessel's technical specifications alone, of course. Typically, when attempts are made to evaluate the performance of ancient ships on the basis of the technical parameters of their design, environmental and human factors are neutralized, held "constant," or "averaged." The resulting projections are usually optimized or maximum performance limits. For this reason, ethnographic reporting offers the advantage that the complete interplay of technical, environmental, and human factors can be observed and assessed, and the reasons for success or failure to achieve optimal performance can be identified. To move toward valid comparisons between Aegean Bronze Age mariners and those living in our time in the Pacific, Alaska, or elsewhere in the world, the similarities and differences in all three classes of variables must be identified and assessed. Experiments with reconstructed ancient vessels have the advantage of sailing the same seas as their ancient counterparts, potentially facing many of the same environmental conditions specific to Aegean seafaring. The challenge with these experiments is instead to reach a satisfactory level of confidence that the physical specifications, and to a lesser extent the human performance characteristics, are right.
Let us consider some published data that have been offered as baselines for vessel performance in the Aegean Bronze Age. Cyprian Broodbank (: 101–106, 341–48) uses ethnographic data, calculations based on ancient hull remains, and experimental archaeology with "broadly analogous boats" to carefully outline the performance implications, as well as the profound social and economic transformations that attended the transition from paddled canoes and longboats to the first sailing ships in the late third millennium BC (Table 3.4). The advent of sailing technology, probably transmitted first to Crete via contacts with Egypt, brought in train a series of new performance capabilities. The ability to harness wind power increased the speed of voyaging significantly while also making longer voyages feasible by conserving muscle power. A sailing ship could now voyage from Crete to Egypt in the four days that were previously required for a longboat to reach Crete from the mid-Cyclades (Broodbank : 345). This greater speed and range effectively shrank the Aegean and reconfigured maritime relationships. On Crete, the new maritime technology and the contacts it fostered with Egypt and the Eastern Mediterranean proved to be contributing factors to the rise of complex society and ultimately to the emergence of palaces and their elites, who exerted strong cultural and even political influence over the Cycladic islands in the first half of the second millennium BC.
Table 3.4. Optimized performance characteristics for different types of Aegean Bronze Age seacraft (adapted and expanded from Broodbank : tables 3, 12)
* * *
* * *
The sail made new sea routes possible by allowing the ship to make considerable headway against the wind (Broodbank : 345–46), though it remains controversial just how well they did so, and how often voyages of any length were taken against the wind (Tilley 1999). In this regard, nevertheless, the sailing ship was clearly superior to the galley in offering possibilities for sailing to windward. With their limited sailing ability, galleys had recourse only to short bursts of exhausting rowing to advance into the wind. The implications of this limitation for sea voyaging are worthy of consideration. In a later period, Viking galleys served admirably in roles requiring speed, rapid deployment and escape, and transport of warriors. But as long-distance, open-sea sailing vessels, Alec Tilley (: 424) comments, "Wonderful though their galleys were, they made a long ocean voyage an adventure for heroes, not a profitable venture for merchants." The situation for Mycenaean galleys must have been similar, even in the smaller world of the Aegean. Their movements must have involved coast-hopping with frequent stops, making long open-sea crossings only in the expectation of favorable winds. Still, we are left with a curious lack of evidence for true Mycenaean sailing ships capable of voyaging throughout the Mediterranean as merchant vessels.
Historical records indicate that in antiquity, ship captains often preferred to wait out fair winds rather than risk sailing to windward for extended portions of a journey. Sailors possessed techniques for advancing against headwinds, but progress was often slow and the effort arduous. In the early 1970s, ethnographer Richard Feinberg accompanied a sailing canoe on a 50-kilometer voyage, into the wind, from Anuta to Patutaka in the Solomon Islands (Feinberg : 89–91, 133–47, fig. 19). The navigator's technique was to take a favorable tack as far as possible under sail, then lower the sail and paddle back to the point where the initial tack could be resumed, marking out a zigzag path. Using this slow, strenuous process, the voyage took twenty hours. The trip home, running under a brisk wind, took only six.
The challenge presented by headwinds in the era before the sail appeared in the Aegean would have been that much greater, as the experiments conducted with the Corfiot papyrella illustrate (Tzalas 1995b; Fig. 3.19). Tzalas' point of departure was to evaluate the hypothesis that the obsidian that found its way from Melos to Franchthi Cave already in the Mesolithic period was transported on paddled reed boats, similar to those used in Egypt in the Bronze Age and in use until recently in Corfu, making the long voyage in a series of coastwise and open-sea segments. After resolving problems with construction techniques and recruitment of a five-person paddling crew, the team wished to set out from near Franchthi, only to discover that the circumnavigation of the Saronic Gulf would be so long and arduous as to give no advantage over acquisition through overland transport from a coastal anchorage much closer to Melos. The team subsequently chose to begin the voyage at Lavrion in Attica. This finding in itself deserves comment. In recent decades, it has become commonplace among Mediterranean archaeologists and historians (myself included) to assert that under conditions of ready access to the sea and underdeveloped terrestrial infrastructure, travel by sea would have been a more efficient and less arduous means of maintaining contacts with other coastal and near-coastal communities near and far. As a counterbalance to the land-focused reality in which most modern Western scholars live, this "corrective" has had some validity and utility. Yet the pendulum may have swung too far, such that the difficulties of sea travel are now underestimated, and long histories of overland connectivity disregarded. Still, Tzalas' claim that a coast-hugging voyage from Franchthi to Melos circumnavigating the Saronic Gulf "would have required much more time than the combination of a land voyage from the Argolis to Attica and a sea crossing from Lavrion to Melos" (Tzalas 1995b: 450) is jarring, since we would normally regard the overland journey from Lavrion to Franchthi as particularly long and difficult. Part of the solution may lie in a number of coastal nodes, lost in the rise of global sea level since the Mesolithic, through which obsidian may have been exchanged in down-the-line fashion. It should be remembered that obsidian was recovered in only miniscule quantities from Mesolithic levels at Franchthi (Perlès ), making the notion of direct Franchthi to Melos runs all the more unlikely. If such intermediate settlements did exist, obsidian distribution would have entailed a series of short-distance trade expeditions by land or sea, a very different scenario. In the bigger picture, assertions such as these that run counter to current ways of thinking challenge us to develop more precise knowledge about how exchange worked at diverse scales, including the role of overland traffic.
3.19 Experimental Corfiot papyrella on the Aegean sea. The Papyrella Voyage is a project of the Hellenic Institute for the Preservation of Nautical Traditions. Courtesy of the photo archive of Theodor Troev.
As the papyrella made its way from Lavrion to Melos in early October, with two or three island stops scheduled, it encountered unseasonably (though not extraordinarily) rough weather, including heavy rain, high winds, and waves of 1.2–1.5 meters in height. As a result, in addition to seven days of paddling at sea, another eight days were spent anchored at Seriphos when severe weather and winds of 7 and 8 Beaufort made conditions at sea too dangerous. Thus, a relatively modest sea voyage of 120 kilometers might consume an entire month – or more, since the return voyage would face opposing sea currents and a greater chance of headwinds.
The adverse environmental conditions proved to be a blessing in disguise, because they tested the design and addressed the research question in a way that perfect weather and a trouble-free voyage could not. The general seaworthiness of the Corfiot boat proved that the simplest kinds of vessels constructed with basic, universally available tools and materials were capable of island-hopping voyages in the Aegean. The technical and environmental problems encountered en route illustrate vividly some of the challenges the Aegean presents for small-boat captains.
By harnessing wind power, sailing ships reduced the need for human propulsion, allowing for smaller crews and increased cargo space. Merchant vessels of the LBA, of which the Uluburun ship is perhaps representative, improved sailing capabilities and cargo capacity while sacrificing speed and the ability to operate in shallow anchorages. As we have seen, the design of the Mycenaean galley moved in the opposite direction toward a fast rowing ship that maximized crew at the expense of storage and sailing capability.
Broodbank (: 346–47) indicates some important implications of the arrival of the sailing ship for coastal inhabitants. The increased speed, range, and cargo space triggered the establishment of new maritime networks as any particular place on the Aegean coast could now be reached more often, and from much more distant points of origin. The Aegean became smaller and contacts expanded. With greater range and cargo capacity, the transport of perishable bulk staples became feasible over longer distances, presenting an opportunity for small coastal communities, for whom self-sufficiency and highly localized subsistence networks had been a matter of survival, to expand into larger, nucleated settlements sustained by regional-scale exchange networks.
The transition from canoes and other light boats with minimal draft to broader, more heavily laden ships propelled primarily by wind power also meant that many shallow and/or exposed beachfronts could not accommodate the new ships. Sailing ships, with heavy sails, rigging, and cargo, could not be dragged out of the water, instead requiring anchoring or mooring in sufficiently deep water off the coast; nor could they easily get underway against strong winds. The result of these new requirements was a partial shift from the opportunistic use of a proliferation of small anchorages to the establishment of major dedicated harbors at the more limited number of suitably sheltered, deep-water anchorages that the Aegean offers.
The archaeological record of the Cyclades in EC III–MC I is plausibly interpreted to manifest these transformations. Old island centers lacking sheltered anchorages, such as Chalandriani-Kastri on Syros, declined while new nucleated settlements with excellent harbors (Akrotiri, Phylakopi, Paroikia on Greater Paros) grew and flourished (Broodbank : 347–49). External influences, particularly from the emerging powers on Crete, expanded interaction spheres in the Aegean. Although Broodbank emphasizes the nucleation of settlement and maritime activity around the sheltered harbors once sailing ships were in place, I think it is important to assert that small boats and anchorages would not have disappeared from use. Although the configuration of many networks changed, with old relationships broken and new ones initiated, networks of local, regional, and interregional scope persisted in parallel or in nested arrangements that performed different but often complementary functions (more on this in Chapters 4 and 6). Certain coastal settlements will have participated in networks at all these scales, while others, generally smaller or less advantageously sited, did not. Some communities, especially on small islands lacking sheltered harbors and extensive, agriculturally productive hinterlands, were bound to suffer in the reconfigured environment, and Chalandriani-Kastri may be a good example. Yet there were alternative ways to adapt, and many small coastal communities must have been able to ensure their survival by reaffirming traditional links, establishing new ones, or strengthening ties to inland communities.
Building large sailing ships was not necessarily in the best interests or capabilities of small coastal communities controlling minor anchorages. Sails can equally be installed on small boats, even canoes, which could still use all the old landfalls. Thus, the same benefits of an expanded maritime horizon could be realized by small settlements, helping to compensate for periodic reorientations or interruptions in traditional relationships.
One area in which greater knowledge and attention are needed is in the relationship between coastal settlements and their insular or continental interior territories. Small islands, particularly relatively infertile ones such as most of the Cyclades, have little agriculturally productive land, and this has shaped their maritime histories and reliance on external sources of supply. On continental shores and large fertile islands such as Crete or Corfu, relationships with inland dwellers can be important. The complementarity of resources and commodities can stimulate symbiotic relationships, in spite of common difficulties in communication caused by mountainous, broken topography. A town situated on a typical, small coastal plain with limited arable land may seek foodstuffs, timber, and other products from the interior, and these sources may have been more dependable than trade by sea, particularly during the months outside of the sailing season. The Uluburun wreck demonstrates that preserved foods, such as olives and wine, could be transported long distances by ship, but we do not know how pervasive the practice was, especially among small communities. Inhabitants of the interior would have desired certain kinds of imported raw and finished goods otherwise unavailable to them, such as obsidian from Melos, volcanic stone from Aigina or Methana for ground-stone implements, and metals for fashioning tools and the prestige objects with which they were often buried.
For coastal communities with recourse to relationships by both land and sea, the balance must have fluctuated over time with changing political, economic, and security conditions. The potential of regional-scale archaeological projects to describe and assess the mix of coastal–inland relationships in the Bronze Age is great, but still far from realized. This topic is addressed from a methodological point of view in Chapter 5.
How, then, do these observations apply to the Mycenaeans, who plied the Mediterranean some five millennia after the Mesolithic inhabitants of Franchthi Cave, and the better part of a millennium after the introduction of the sail to Crete? In the LBA, sailing technology was long established in the Aegean; the Mycenaeans may have learned these skills directly from the Minoans, or perhaps through the mediation of the Aiginetans at Kolonna. Their maritime world was expansive within the Aegean, and perhaps they also undertook voyages to Egypt and the Levant – or at least as far as Cyprus and Ugarit – and to the central Mediterranean to visit southern Italy, Sicily, and the Lipari Islands. Yet, these far-flung contacts did not spell an end to local and regional connectivity by land or by sea. In any complex society, no matter how hierarchical, there are multiple nested economies, and the Mycenaeans were no different. Nowhere is this better illustrated than in the territory controlled by the palatial center at Pylos, where Linear B archives and archaeological discoveries reveal a hierarchical society with complex economies, of which the palace controlled only certain key elements. Local and microregional networks of interdependence, which coastal inhabitants shared with inland neighbors or their counterparts on further shores, constituted economies every bit as real, and surely as prevalent, as the "palace economies" or the long-distance exchange of highly visible commodities. One aim of this book, pursued explicitly in the case studies, is to uncover the archaeological signatures of these smaller-scale coastal economies.
### Conclusions: Mycenaean Ships and Boats
This chapter has examined the evidence for Mycenaean ships and boats, and summarized what can be surmised of their forms, functions, and performance characteristics. Only the galley is sufficiently widely attested in a range of media to allow a reasonable understanding of all of these categories. A case has been made for several other kinds of ships and boats, for which we have equivocal evidence or no evidence at all. This case has been an interpretive exercise relying on diverse information: contemporary non-Aegean boats, the Flotilla Fresco at Akrotiri, ethnographic data, shipwrecks, and interpretation of iconography and boat models. From this information, it is possible to suggest that the galley was joined in the Mycenaean repertoire by varied types of vessels that we might label with designations like canoe, fishing boat, rowboat, pilot, coasting vessel, merchant vessel, and so on.
Teasing out a fuller roster of ships and boats, and considering their likely physical and performance attributes, sheds light on the seas, coasts, harbors, and simple anchorages where they would have been active. With their properties and requirements in mind, the next chapter examines the physical characteristics of coastal settings in Greece, and the other parameters – environmental conditions and human skill – of travel to and from them.
## Four The Maritime Environment of the Aegean Sea
> The navigation of the Aegean Sea, though easy, requires constant attention, and a place of shelter should always be kept in view, so that safety may be assured before dark in the event of an approaching gale; the weather may become so thick that among the labyrinth of islands the land may be hardly seen in time to avoid it. (USNOO 1971a: 18)
Having considered the range of ships and boats that the Mycenaeans used, in this chapter I examine the environmental conditions for seafaring and navigation in the LBA Aegean area. The scope of inquiry must now broaden to include everything from global weather systems to minute local variations, since processes at all these scales interact to generate conditions at sea. I reconstruct many of these environmental parameters of navigation based on both modern data and ancient evidence, and give examples of their consequences for seafaring. I then discuss the practice of navigation in the Aegean, and conclude with speculation about the formation of maritime communities in the Bronze Age, and the means by which they and their knowledge were reproduced and perpetuated through time.
### Environmental Conditions for Navigation
The environmental conditions for navigation in the Aegean Sea are produced by complex interactions of atmospheric, hydrospheric, and lithospheric (terrestrial) forces operating at different scales, i.e., global; basin-scale (Mediterranean); sub-basin-scale (the eastern or western basin of the Mediterranean, or one of its constituent bodies of water such as the Aegean Sea); mesoscale (e.g., contained within the Aegean); or microscale (local; Robinson et al. : 1). Global atmospheric systems account for much of overall Mediterranean variation, but they are modified by interactions with smaller-scale processes, from basin-scale to microscale, to produce the localized conditions that navigators encounter in the coastal and offshore waters of the Aegean (Oddo et al. ). The flows of air and water at all scales are driven by gradients in pressure, temperature, and density, as well as by terrestrial and submarine topography. What follows is a summary of the most salient environmental factors that impact navigation of the eastern Mediterranean, taking account of recent meteorological and oceanographic research and addressing the specific conditions of the Bronze Age.
#### Global-Scale Processes
At the global level, interactions between the earth's atmosphere and oceans generate massive high- and low-pressure systems that create climate and weather around the world. These systems have distinct seasonal patterns through the year in response to temperature and pressure gradients, solar and lunar gravitational effects, and other factors. In the late Holocene, they have established "typical" patterns or oscillations of different scales and durations. For example, the North Atlantic Oscillation (NAO) describes a pattern in winter of low pressure centered over Iceland while high pressure prevails over the Azores. The NAO manifests itself through precipitation, sea-level pressure, sea-surface-temperature storm tracks, and temperature, and is most pronounced between December and March as a result of an increased contrast in sea/air temperature (Mayhew ). But these systems are also subject to shifts that may be periodic and predictable, such as the El Niño and La Niña cycles of the Southern (Pacific) Oscillation, or aberrations of unpredictable timing and duration. An extreme shift in the high- and low-pressure centers of the NAO has been implicated in the "Little Ice Age" that brought intense cold to western Europe from the fourteenth to the mid-nineteenth centuries AD, spreading famine and affecting the course of history (Fagan ).
Historians and archaeologists have advanced a relatively simple model that places the Mediterranean at the junction of four global atmospheric systems, the interactions of which create the characteristic Mediterranean climate of mild, wet winters and hot, dry summers, and set the conditions for basin- and sub-basin-scale weather patterns (Agouridis : 3; Grove and Rackham : 25–27; Pryor : 15–17). Each system produces successive waves of pressure cells and fronts that enter the Mediterranean through gaps in blocking mountain ranges (Pryor : 15). Generally, systems tracking eastward from the Atlantic interact with the warm Mediterranean water and cold mountain air, as well as with fronts moving south from the Eurasian continent, to produce complex and variable localized weather (Pryor : 16). In winter, the Mediterranean climate is dominated by the effects of the North Atlantic low over Greenland and Iceland (part of the NAO) and the continental Mongolian high. Storms and winds generated by the North Atlantic low track eastward and affect mainly the western basin of the Mediterranean, while the Mongolian high prevails in the eastern basin. In summer, the North Atlantic low retreats toward the pole and the Atlantic subtropical high (Atlantic Oscillation) over the Azores intensifies, exerting strong effects over the western basin. At the same time, the Mongolian high retreats and the Indo-Persian low prevails over the eastern basin. Recent meteorological and climatological research shows the effects of global-scale systems on the Mediterranean to be considerably more complex and far-reaching (see, e.g., Basharin ; Bengtsson et al. ; Hahmann et al. ; Park ; Rodwell and Hoskins ; Xoplaki ; Xoplaki et al. ), but even simplistic models demonstrate that shifts and disturbances in large-scale atmospheric systems around the world can have substantial effects on climate and weather at human temporal and spatial scales in the Mediterranean.
#### Basin-Scale Atmospheric Processes
In addition to establishing a seasonal climate regime, global-scale weather patterns affect atmospheric conditions in the Mediterranean in a variety of ways that impact the maritime environment. Cyclogenesis, the formation of cyclonic storms, is more intense in winter in the Mediterranean than anywhere else in the world (Fig. 4.1). This activity is largely controlled by global air flows and climatic trends over Europe, including the influence of the NAO. Cyclones form readily because of the sharp temperature gradient between the cold winter air, particularly from major mountain ranges, and the relatively warm sea surface, which releases heat into the air above. Mediterranean cyclones are smaller than large oceanic cyclones and hurricanes (most are less than 650 kilometers in maximum radius, compared to 1,000 to 2,000 kilometers in the Atlantic), and have shorter lives (averaging 28 hours versus three to three and a half days in the Atlantic). Cyprus and the Aegean Sea are locations where cyclones tend to reach full strength, primarily in winter. Not all of these cyclones are intense and dangerous, but those that are intense are closely linked to high-impact weather, including strong winds accompanied by torrential rains and heavy seas with swells and choppy, short-frequency waves, as is characteristic of the Aegean (Homar et al. 2007). These conditions may generate storm surges that occasionally visit extensive damage on coastal areas.
4.1 Centers of cyclogenesis in the Mediterranean. After Brody and Nestor : VI-22, fig. VI-15.
No general wind current dominates the entire Mediterranean basin at any time of year; instead, the variability in topography and global weather patterns prevailing over different parts of the Mediterranean at different seasons gives rise to a number of regional winds (USNOO 1971b: 50–52; Fig. 4.2). Three regional winds, the meltemi (pl. meltemia; the ancient Etesians), the sirocco, and the bora especially affect the Aegean region. The prevailing winds of the Aegean are northerly throughout the year, but are most marked during the summer months. The meltemia are moderately strong and persistent northerly winds generated by a substantial pressure gradient between a deep summer heat low over the Red Sea and an area of relatively high pressure over the Eurasian continent. During July and August the meltemia blow steadily over the Aegean, most strongly in the southeastern Aegean, where they interact with landforms to attain a frequency of 80% or more. Such winds can blow especially strongly when funneled into constricted spaces, such as the Doro Channel between Euboea and Andros, or the straits in the Cretan Arc separating Kythera, Crete, Karpathos, and Rhodes (Soukissian et al. 2002: 186–88).
4.2 Regional winds of the Mediterranean.
The bora is a cold, dry continental northerly wind that enters the Adriatic Sea through the Trieste gap, where it sometimes encounters a passing depression in winter, spawning violent squalls and gale-force winds.
The sirocco (known regionally under various names), by contrast, is a southerly wind originating in the desert regions across the entire expanse of northern Africa. Siroccos are dry winds that are hot in summer and warm in winter. They transport a great amount of dust, forming haze over the Aegean and contributing to the aeolian content of Mediterranean terra rossa clays (Durn ; Yaalon ). Affecting mainly the southern Aegean, the sirocco can be accompanied by rain when it intersects with a low-pressure system moving eastward across the Mediterranean, primarily in winter, or when it encounters high relief after absorbing moisture from the sea. When such conditions are absent, the sirocco produces cloudless, hazy conditions by day with low stratus clouds and heavy dew at night (USNOO 1971b: 51).
Regional winds can be complemented, or even augmented, in coastal areas by more localized wind formational processes, including land and sea breezes, mountain and valley winds, and the effects of winds encountering topographic obstacles. Land and sea breezes are persistent, regular features that result from the differential heating and cooling of adjacent land and sea masses (Morton : 51–53; Fig. 4.3). In a typical diurnal cycle, after sunrise the land heats faster because it absorbs heat from sunlight more efficiently but to superficial depths, whereas the sea reflects more sunlight, loses heat through evaporation, and dissipates heat through a greater depth. As temperatures rise through the morning, air pressure builds over land much faster than at sea as warm air ascends, creating a pressure differential that initiates a flow of air well above ground level from land out to sea, and a compensating flow of cooler air from sea to land at ground level. The latter is the sea breeze felt onshore by mid-morning, increasing in strength through the afternoon. As sunset approaches and temperatures begin to drop, the pressure gradient decreases and a period of calm develops in the evening. At night, the opposite process generates a land breeze. As cooling proceeds, a pressure gradient develops as heat is lost more quickly on land. Higher pressure develops over the sea, initiating a flow of warm, rising air toward land, with the compensating flow of cooler air, the land breeze, out to sea at ground level. Another period of calm prevails in the hours around dawn as temperature and pressure differentials are once again minimized. Because heating induces a larger gradient than cooling, and because sunlight is variable, sea breezes are both stronger and more variable than land breezes. Sea breezes are also more prominent in summer because uninterrupted sunny days and intense heating create particularly strong temperature and pressure gradients. Strong sea breezes can effectively supersede the effects of regional winds in local coastal settings. They can be a help or a hindrance to navigators, pushing boats toward safe anchorage or dangerous shallows.
4.3 Basic dynamics of land and sea breezes.
Mountain (katabatic) and valley (anabatic) winds are generated by the same diurnal cycles that create land and sea breezes (Morton : 53–56). In this case, valley bottoms heat up during the day more rapidly than surrounding peaks and ridges because they are sheltered from winds and their shape promotes radiational heating from enclosing slopes. As air from the valley bottom is heated by the sun (again, more prominently in summer), it flows up the valley walls as an upslope wind, often forming cumulus clouds above the valley rim that may produce rain. Denser, cooler air sinks into the valley to replace the warm, rising air. This mixing creates a more turbulent mass of cooler air that blows along the valley axis in the direction of the prevailing regional winds. At night, the slopes and bottoms cool at a faster rate than they warm during the day, initiating a rush of cold, dense air downslope into the valley and along its axis as a mountain wind (also known as a gravity or drainage wind). Valley and mountain winds are stronger in sunny or clear conditions because of stronger associated temperature and density gradients, and they can be forcefully augmented by regional winds such as the meltemi. Valley and mountain winds affect coastal areas when mountain or stream valleys extend to the coast. A good example is the vardari wind that blows down the Axios River valley to the sea in the Thermaic Gulf, which is irregular and can be quite strong.
A type of wind of great concern to navigators is that produced by the displacement of air currents upon encountering topographic obstacles (Morton : 56–61). This displacement can be vertical or horizontal, and the turbulence caused by the disturbance of the air flow ranges from minimal to violent, depending on a host of factors including wind speed, the stability of the air mass, the angle of the contact, and the shape and height of the topographic feature.
Vertical displacement of wind over an island, promontory, or high coastline produces turbulence as slower moving air at or near sea level is pushed up through faster moving air aloft; the faster the wind speed, the greater the turbulence. If the air mass is stable, that is, not subject to strong differentials in temperature and pressure with altitude, the air passing over the obstacle returns to sea level by flowing down the lee side, but this flow is characterized by strong winds accompanied by atmospheric eddies, wind gusts, and lulls. These conditions produce the well-known phenomenon of violent winds and squalls on the lee side of islands and coastal features. The southern coast of Crete is an excellent example. Psiloriti (2,457 meters) and the other peaks of the central Cretan mountain range place a formidable barrier in the path of northerly winds that blow for most of the year. The extreme vertical displacement of these air masses sends violent winds down the lee side of the mountains to the southern coast, and spawns squalls and gales. This coast is not particularly hospitable in any case, possessing relatively few and widely spaced anchorages, but even anchorages with good water depth and holding ground are generally vulnerable to bad weather from the south in winter and the strong northerly squalls year-round. Modern sailing manuals are full of warnings about sudden, violent squalls that blow down from the mountains, threatening to pull boats from their moorings and anchors, or to dash them against shores and shallows (e.g., USNHO : 111–21). Conditions such as these will have had a significant influence on sea routes, and thus equally on the form that maritime networks of economic and social relations took.
If the air displaced vertically is unstable, usually involving a strong temperature and moisture gradient with altitude, the warm, moist air continues to rise in the atmosphere, causing condensation and forming rain-producing cumulonimbus clouds. These rains tend to fall in concentrated bursts, often as thunderstorms accompanied by cold, powerful downdrafts that spread over the sea, spawning squalls over great distances (Morton : 59–60).
Thus, the effects of topography can intensify winds as well as weather, either episodically, seasonally, or perennially. Yet even if these effects are known and expected, they are irregular and unpredictable because the factors – wind speed and direction, form and orientation of obstacles, season and time of day – are so variable alone and in combination (Morton : 60).
Horizontal displacement occurs when an air stream encounters a low-lying topographic feature, such as a long promontory or a low, broad island, causing a flow of air around one or both sides of the barrier. These flows do not typically produce high winds, except where displaced air is funneled into a narrow gap or strait. The accelerated winds and currents in the narrow strait between Euboea Island and the Greek mainland, or the strait running from the Bosporus to the Dardanelles, are characteristic results of the funneling process of flows of air and water.
#### Basin-Scale Surface Water Circulation and Currents
Atmospheric interactions with the sea also play a key role in determining water circulation in the Mediterranean Sea. This circulation is complex because the Mediterranean is large enough to be subject to the same dynamics that characterize global ocean circulation (Robinson et al. : 285, : 1). The Mediterranean Sea is open only to the Atlantic Ocean at the Straits of Gibraltar and to Black Sea waters through the Dardanelles straits. The Mediterranean basin is divided into roughly equal western and eastern sub-basins, with the effective boundary between them lying at the Sicilian straits. Sea circulation is forced by water exchanges with the Atlantic Ocean and Black Sea, winds, buoyancy effects of different water masses at the surface due to water density (temperature and salinity) contrasts (thermohaline circulation), and topographic features including islands, coasts, narrows, and bathymetry. There are three types of water masses defined by depth: surface waters, intermediate waters, and deep and bottom waters extending to the sea floor. On an annual basis, evaporation, particularly intense in summer, exceeds the total input of rain and river outflows into the Mediterranean, resulting in water fluxes from the Atlantic Ocean and Black Sea. Most of the deficit is rectified by Atlantic water through Gibraltar (71%), with much smaller contributions from rivers (25%) and the Dardanelles (4%; Agouridis : 3). Thus, water exchanges with the atmosphere drive the influx of lower-density ocean waters, which in turn determines the thermohaline circulation of water and many of the characteristics of the marine ecosystem in each basin (Zervakis et al. : 1846).
Water from the Atlantic enters as a coherent surface stream because of its lower salinity, and thus low density, relative to the Mediterranean water. The flux is most intense in summer when evaporation over the Mediterranean is highest, creating a current of six knots or more. This buoyant stream flows eastward, becoming denser and less coherent as it mixes with other water masses and is affected by evaporation and convection. Along the way, it exhibits instabilities as it interacts with powerful gyres (see below), and bifurcates into multiple pathways, yet it reaches the Levantine coast as a distinct stream (Fig. 4.4).
4.4 Mediterranean currents and water circulation. Drawing by Felice Ford after Roussenov et al. : 13,516, fig. 1.
4.5 General sea-surface circulation flow in the Aegean. After Papageorgiou : 209, fig. 3.
By contrast, the volume of Black Sea Water (BSW) that enters the Mediterranean through the Dardanelles is much smaller and produces significant effects only in the Aegean Sea (Kourafalou ; Kourafalou and Tsiaras ; Fig. 4.5). The BSW brings less saline and thus less dense surface water into the Aegean to spread over the warmer and more saline intermediate waters. The general cyclonic (counterclockwise) flow of water around the Aegean is promoted by the north- and westward trajectory of the powerful plume of BSW exiting the narrow Dardanelles Strait. Driven by buoyancy, winds, and the complicated topography of the Northern Aegean, the BSW is deflected to the west along the rim of the northern Aegean coast, and subsequently driven by strong northerly winds down the eastern coast of Greece. Following seasonal patterns, the current is partially deflected to the Cycladic islands and the central Aegean, and partially pushed southward along the eastern coast of the Peloponnese, where it meets the eastward flow of Mediterranean waters from the Ionian Basin and is entrained by these and by the land masses of the Cretan Arc to flow east into the southern Aegean. Finally, the flow joins the Asia Minor Current (AMC), carrying warm, highly saline Levantine waters north along the eastern coast of the Aegean. When the AMC meets the outflow of the BSW, an intense thermohaline front is formed, which is partly responsible for the strength of the Dardanelles current. In this way the general cyclonic circulation around the Aegean rim is completed.
4.6 Typical positions of major cyclonic and anticyclonic gyres in the Aegean. Data from Olson et al. ; Sayin et al. .
At sub-basin scale, Aegean currents are constantly affected by gyres and eddies. Gyres are oceanic surface currents driven by the interaction of the strong Dardanelles outflow with the topography of the Aegean sub-basin, including narrow straits and underwater basins, ridges, and plateaus; the complex shape of the shoreline; and the prevailing northerly winds (Fig. 4.6). Their spiral form is due to the interaction of pressure gradients and the Coriolis effect, which deflects water flow to the right in the Northern Hemisphere and to the left in the Southern Hemisphere. Sub-basin gyres typically have diameters in the range of 200 to 350 kilometers. They can be cyclonic (rotating counterclockwise) or anticyclonic (rotating clockwise) and their duration can be characterized as permanent, recurrent, or transient. They may also exhibit strong seasonal, interannual, or multiannual behaviors. They are distinct from, but analogous to, atmospheric cyclones and anticyclones. The current temperature and sea surface pressure generated by gyres may exert measurable effects on regional temperature and rainfall as they interact with atmospheric forces.
Similar hydrospheric interactions give rise to mesoscale eddies throughout the Aegean, and it is these that most directly affect local – including coastal – currents. Eddies are small-scale currents of water moving against the main current with a circular motion; in the Mediterranean they have diameters in the range of 10 to 14 kilometers, about one-fourth the size of oceanic eddies (Robinson et al. : 5). Like gyres, they can be cyclonic or anticyclonic, and they may be transient or longer lived. They are fairly evenly distributed throughout the Aegean, but anticyclonic eddies occur with greater frequency around the edges of the basin, while cyclonic eddies are more prevalent toward the basin interior.
Eddies are spawned where currents are interrupted by topographic features, or where differences in pressure, temperature, or density exist. Fields of transient eddies form along the border swirl flow of sub-basin-scale gyres; they may then break off to meander in the open sea (Fig. 4.7). In the Aegean, the inflow of BSW at the Dardanelles sets into motion a series of conditions in which eddies emerge, beginning with strong anticyclonic eddies that form in response to the high pressure of the narrow outflow channel and the thermohaline front between the buoyant BSW and the denser Aegean waters. These anticyclones then spread out along the path of the BSW as effects of density gradients. Along the way, the irregular features of the shoreline, for instance headlands or peninsulas and deep inlets, modify the path and motion of currents, spawning eddies that can ride along linear coastlines and fill bays. Donald Olson and colleagues () describe the effects of eddies in the large Thermaic Gulf (with Thessaloniki located at its head). Some of the BSW flow enters the gulf, creating a dense pattern of eddies, dominated by cyclones at the mouth of the bay and anticyclones within it. The anticyclonic rotation forms a rim current that is intensified by the inflow of three major rivers (the Axios, the Aliakmon, and the Loudias) into the gulf. The current is further complicated by the effects of two northerly winds, the meltemi and the local vardari, a strong northwesterly valley wind that blows irregularly from the Axios (ancient Vardar) river valley (Olson et al. : 1904), as well as a sea breeze that forces strong diurnal surface currents within the gulf (Hyder et al. ). Contexts such as the Thermaic Gulf amply demonstrate multiscalar atmospheric, hydrospheric, and terrestrial forces interacting to create local conditions that are highly malleable and distinctly seasonal (Kourafalou and Barbopoulos ).
4.7 Distribution, size, and intensity plots of eddies in the Aegean. Olson et al. : 1914, fig. 15. Courtesy American Meteorological Society.
#### Waves
Wind stress, the drag or tangential force exerted on the earth's surface by adjacent layers of moving air, is the principal engine of oceanic wave formation (Morton : 30–37). These waves travel in the direction of the wind, gaining height and strength so long as the wind continues in the same direction and the waves do not encounter topographic obstructions. Once the wind stops or changes direction, the wave continues some distance under its own momentum, sometimes causing the clashing of waves moving in different directions. The gales, squalls, and other stormy weather typical of the Aegean can produce rough seas with high waves and swells. Because of the irregular pattern of wind flows and gusts, waves in the open Aegean tend to be choppy, short-frequency waves that would have been dangerous for the small craft of the Bronze Age (Broodbank : 101). The concentration of stormy weather and rough seas in winter explains the general avoidance in antiquity of open-sea sailing in the Aegean from October to March. Once waves approach coastal areas, they are affected by several factors that determine their contact with the coast and the effects they might visit upon watercraft and coastal settlements. Among these factors are the form and magnitude of the wave, measured by wave height and wavelength; wind speed and direction; the angle of contact with the coastline; and surface and subsurface topography along the coast. The configuration of the coastline, including its shape and the extent and form of any submarine coastal shelf, may serve to magnify or minimize the force of waves breaking against the land mass. In general, shorelines with long promontories and gradually sloping underwater shelves extending far out to sea induce waves to break well offshore, minimizing potential wave damage. By contrast, shorelines that slope steeply into the sea with little offshore hindrance to water movement can magnify the violence of waves breaking on the shore. Considerations of protection from damaging waves are similar to those of exposure to winds, and figured in the placement of harbors in the Bronze Age.
The combination of intense weather events, northerly winds, waves, topography, and swift current flow with complex patterns of eddies affecting the western coast of the Aegean is implicated in many shipwrecks in recorded history. Perhaps the most famous of these befell the Persian fleet in three separate incidents during invasions of Greece in the early fifth century BC. In 492, Darius's fleet under Mardonius was wrecked by a northerly gale while rounding the Athos peninsula (Herodotus 6.44). Many of these ships were battered against the rocky peninsula. Later, Xerxes' fleet suffered two major shipwrecks in the summer of 480 BC. The first of these occurred on the long and nearly harborless Magnesia coast near Artemision/Cape Sepias (Herodotus 7.188–92). There, a great four-day storm driven by meltemi winds from the northeast roiled the sea in high waves, sinking ships caught in the open sea and dashing others onto the rocks along the rugged coastline. Remarkably, Herodotus records that locals called these storms "Hellespontian" (i.e., originating in the Dardanelles). The third disaster took place while the fleet was sailing off the "Hollows of Euboea," that is, the eastern coast of the southern half of Euboea, before entering the Doro Channel (Herodotus 8.12–13). In that event, a violent storm driven by northerly winds rose at night and dashed ships against the coastal rocks. These calamities present a characteristically Aegean cocktail of hazards: violent squalls and heavy seas that arise at night, forced by high northerly winds and swift, shifting coastal currents, driving ships toward nearly invisible rocks and shoals on a rugged coastline with few opportunities for safe anchorage. The Persian shipwrecks also illustrate the way that phenomena at all scales, from global to local, interact to produce conditions experienced at human scale.
### Aegean Variability
By virtue of its geographical position – a long north to south axis closed in on the north by the land masses of Greece, Anatolia, and the Balkans, and in the south opening into the broad eastern Mediterranean – as well as a complex topography, the Aegean Sea creates a unique maritime environment. The environmental forces that drive atmospheric and hydrospheric circulation converge and cross-cut the Aegean basin in such a way that local conditions, resulting from the interaction of forces operating at all scales, are highly diverse and changeable. The greatest differences are evident as one moves from north to south, through successive influences of continental, Mediterranean, and desert climatic zones, but significant contrasts also occur between the mainland and islands, as well as between contrasting topographic settings regardless of geographical location. Furthermore, seasonal patterns in environmental conditions give a very different character to a given coastal setting through the year. A brief survey of atmospheric conditions in the Aegean, and an example of seasonal variation at a single Aegean port, will highlight some of these contrasts.
#### Atmospheric Conditions in the Aegean Sub-Basin
The Aegean Sea is not excessively windy, cloudy, or stormy when compared with ocean waters such as the North Atlantic. Gales and storms of Beaufort 8 and higher (winds in excess of 62 kilometers per hour) occur in 5% or fewer observations, while light to moderate squalls are somewhat more frequent at 10% in winter, but only 2% in summer (USNHO : 10). Dead calms are also rare, however.
Summer disturbances differ from those in winter. In summer, low-pressure disturbances tend to pass over the Eurasian continent, well north of the Aegean, or remain to the west of Greece. In frequency and severity these are greatly reduced from winter, rarely interrupting the prevailing northerly winds or local sea and land breezes.
Winter gales may arise from several directions, associated with depressions moving east-northeast past southern Greece from Libya, or east-southeast from Italy to cross the Greek mainland. Winter winds are more variable and unstable because of the greater number of high- and low-pressure systems that move through the Mediterranean, which determine the occurrence and severity of winds and storms. Low-pressure systems (depressions) crossing the Mediterranean from the west are often preceded by strong southerly winds. Once a storm passes over the Aegean, the wind shifts through several directions and brings showers and squalls. At other times, cold polar air from high-pressure systems over the Eurasian continent may pass to the south, pushing against the depression and generating strong winds from the north or northeast, accompanied by showers, sleet, or even snow. These storms are particularly dangerous at night on the windward side of coasts and islands because of low visibility and heavy seas; ships encountering these conditions are well advised to make for southern, lee shores. If the continental high enters the Aegean with no depression to the south, skies will be clear with strong northerly winds. The general winter pattern is that the northern Aegean remains chiefly under the influence of the continental high, and thus northerly winds, while the southern Aegean experiences frequent low-pressure systems and a mix of southerly and northerly winds. The frequency of strong winds varies by island or coastal location, and by windward or leeward position. Temporally, the greater frequency of storms and dangerous sailing conditions in winter defined the nonsailing season, as we know from ancient Greek sources beginning with Hesiod (Works and Days 620–95), but these conditions did not preclude maritime communication completely, as we shall see.
The prevailing winds from the north are most marked during the summer meltemia. Although meltemia are fair-weather winds, when blowing strongly they may kick up "white squalls," so named after the appearance of agitated water in sunlight. Locally, however, these and other winds can vary widely due largely to topography, and in summer to land and sea breezes. While land breezes are always light, sea breezes may be strong enough to disrupt or even eliminate the effects of the meltemi. An example of this phenomenon is the coastal promontory at Gytheion in Laconia. There, the meltemi is dissipated by the Taygetos mountains to the north, which shield the Gytheion area. Instead, sea and land breezes dominate during the day, with gusts off the mountains occurring at night. When forceful, sea and land breezes can be used strategically by navigators to make progress against prevailing winds and currents. By contrast, at Kythera Island off the south coast of the Peloponnese facing Gytheion, the meltemi becomes the prevailing wind in late summer as the sea warms and consequently the sea breeze generated by land/sea temperature differentials diminishes (USNHO : 11). The north-facing anchorages are particularly vulnerable to northerly and westerly winds (Heikell : 144–48).
Cloud cover is light overall, with summer skies seldom overcast, but heavier in winter when most precipitation falls. When clouds do occur, a diurnal pattern of minimal cloudiness in the morning, followed by increasing clouds in the afternoon and dissipation in the evening, is common throughout the Aegean. During winter, cloud cover averages 50%, with little variation from November to April. Great banks of clouds shrouding mountain summits often precede bad weather, especially in the northern Aegean.
Consistent with a characteristic Mediterranean climate, the Aegean region experiences distinct wet and dry seasons, with most rain falling between October and March (Xoplaki : 10–12, figs. 1.1, 1.2). Driven by the great weather engines described above, the precipitation of early winter is often heaviest as warm, moist air forming over the open Mediterranean comes into contact with cold, drier polar air lying over the Eurasian continent.
Rainfall patterns across the Aegean basin are variable; the frequency, timing, and amount of rain a particular locality receives depend on several factors. In the northern Aegean, southerly winds contribute most of the rain, though northerly systems sometimes deposit snow. In the southern and central Aegean, southerly, westerly, and northerly winds can all bring rainfall. Because Mediterranean depressions carry abundant moisture, the islands of the southern Aegean, exposed to eastward-tracking depressions from the central Mediterranean, receive more rain than the Greek mainland, but in lower-frequency, higher-volume events. Rains seldom last for more than a few days at a time. On the mainland, rain occurs most frequently in the afternoon, while over the sea and islands, rainfall at night or early morning is more common.
4.8 Satellite image of Kapsali Bay, Kythera. Image © Google Earth, © 2011 European Space Imaging.
Sometimes rain effects are topographic in origin. The northwestern Greek province of Epirus is an area of high rainfall because storm systems crossing the mainland from the west are halted by the Pindos mountain range. There, the clouds release rain ("orographic rainfall": Morton : 64–65), which then drains down the western side of the Pindos to feed large perennial streams and springs. A corresponding "rain shadow" region on the eastern side of the Pindos is much drier. In the mountainous Greek terrain, such rain surplus and deficit relationships are common on both macro- and microregional scales, detectable, for example, even between Athens and its northern suburbs. All areas of the Aegean are also subject to potentially large interannual fluctuations in rainfall, which in drier areas such as the Cycladic Islands can have dire implications for subsistence (Broodbank : 76–78).
#### Seasonality at Kapsali Bay, Kythera
The harbor at Kapsali Bay, on Kythera's southern coast, provides a typical example of the seasonally changing environmental conditions experienced at a southern Aegean anchorage (Naval Research Laboratory ). The southwest-facing harbor exhibits a classic configuration favored in the Bronze Age: a double-lobed harbor formed by a protruding headland with small, indented bays on either side (Fig. 4.8; see Chapter 5 for the configuration of Bronze Age harbors). The headland separates a larger, less sheltered western harbor from a smaller, shallower eastern harbor that is better sheltered by a narrow entrance. Kapsali Bay has relatively few modern built installations that artificially alter the impact of seasonal weather conditions. Boats may find mooring within the bays or lie at anchor just outside them.
Facing southwest, Kapsali Bay is exposed mainly to winds and weather from the west and south. The steep relief of the land to the north shields the harbor from the prevailing northerly winds, but when these are particularly strong, powerful gusts and squally weather can be produced by the vertical displacement of winds to the lee side of this prominent topographic feature. At the same time, coastal sea breezes flowing north are strongest in summer and may offset the prevailing northerlies, resulting in light wind conditions in the afternoon and early evening.
In spring, a long transition occurs between winter and summer conditions. Migratory winter cyclones may continue to arrive from the west well into May. Southerly siroccos peak in frequency and intensity in spring, and if particularly strong may make the harbor unusable for small craft (Heikell 2007: 146). They bring cloudy conditions with light rain and dust (producing "red rain"), with winds of 22 to 33 knots and waves of 1.1 to 2.1 meters. Sirocco events may last for several days, but because they develop slowly over a day or two and are preceded by altocumulus clouds, they can be anticipated and proper precautions taken. When stormy conditions are not present, northerly winds prevail.
Summer brings typical Mediterranean conditions: a nearly cloud-free, precipitation-free sky with warm temperatures and few hazardous weather concerns. Northerly winds prevail, notably the meltemi, which may be felt at varying strength from May to October. Strong meltemi winds may persist for days at a time, but the effects are usually diminished by a counteracting sea breeze. The onset and first day of a meltemi event may be marked by thunderstorms in May to June and September to October, but in July and August only by scattered altocumulus clouds.
In autumn, the onset of the winter pattern is usually rapid, occurring around the third week of October. The first storms of the season are migratory cyclones from the west. Though these are not as intense as later winter weather, they often catch sailors unprepared and cause damage for this reason.
The most difficult navigational conditions occur in winter. The prevailing northerly bora winds may reach 41 to 47 knots and cause early morning temperatures to drop to near freezing, with little warming during the day. The most hazardous conditions are initiated by migratory cyclones approaching from the west, accompanied by low clouds, heavy rain, high winds, and reduced visibility. As they pass over, winds may shift abruptly from northerly, to easterly, southeasterly, and finally southwesterly with ever increasing wind speeds. These storms produce waves of 3.3 to 4.3 meters, in which case watercraft anchored outside the sheltered bays are fully exposed to squalls and battering waves.
Kapsali Bay, like any other Aegean landfall, presents distinctive local characteristics due to its geographical location, exposure, and configuration, while at the same time experiencing a typical range of seasonal conditions generated by the interplay of the broader environmental forces examined in this chapter.
### Implications of the Environment for Aegean Navigation
The foregoing examination of the maritime environment of the Aegean region highlights several implications for Bronze Age seafaring. One of these is that although there are certain broadly predictable patterns of climate, weather, and conditions at sea, these can turn unpredictable and changeable at the scale of human experience. The sudden squalls, storms, and heavy seas reported from antiquity to the modern day often arise with little warning even when the general patterns of hazardous weather are well known. The captains of the Persian fleets, with their imperfect knowledge of local conditions along the western Aegean coast, learned this to their cost. Apart from weather hazards lasting for a few hours to a few days, longer-term anomalies arise because the circulation of air and water in the Aegean is governed by systems operating at multiple, interacting scales. The more extreme case of Europe's "Little Ice Age" shows that centuries-long environmental shifts are possible.
We can assume that similar kinds of anomalies, at least those of short duration, occurred during the Mycenaean period, though we have no empirical evidence for them. A great deal of speculation has surrounded climatic disturbance, usually a decadal-scale pattern of drought, as a possible contributor to the collapse of the Mycenaean palatial system (Carpenter ), but our data – whether from pollen, tree rings, ice cores, terrestrial and deep-sea sediments, or other proxy measures – lack sufficient geographical and chronological resolution to demonstrate this or indeed any specific environmental anomaly during the Mycenaean period. The most we can say is that by the mid-Holocene (roughly 5000 BC), the basic climatic conditions that we experience in the Mediterranean area today were in place (Loy and Wright : 40; McCoy ), and global (eustatic) sea level had stabilized at just a few meters below current levels (Lambeck , ). It is likely, therefore, that the environmental anomalies of recent centuries, for which we have historical accounts and some meteorological data, approximate those encountered in the LBA. This means that Mycenaean sea captains faced weather-related hazards similar to those encountered by their later counterparts prior to motorized watercraft and advanced navigational aids.
The winds and currents described in this chapter also made it possible for boats – whether paddled, rowed, or sailed – to make progress on the sea. These movements normally followed the prevailing winds and sea currents, but knowledge of local winds, coastal currents, and land and sea breezes often made it possible to move against the dominant forces in short segments. This ability to move about locally also broadened the horizon of maritime activity by extending the sailing season, and significantly promoted the kind of short-distance connectivity that led to the formation of the maritime "small worlds" to be described in Chapter 7.
Equally important to Mycenaean navigation were the topographies of coastal landscapes. These were normally stable over the scale of a human lifetime, though as we will see in Chapter 5, modern coastal configurations are often dramatically different from those of the Bronze Age. Working in favor of Mycenaean navigators were the thousands of islands and highly irregular coastlines of the Aegean, indented with an abundance of natural harbors and anchorages. It was possible to traverse most of the Aegean, and in fact much of the eastern Mediterranean, with land always in sight, the main exception being the Libyan Sea between Crete and northern Africa (Fig. 4.9). When bad weather conditions arose, some form of safe anchorage was usually within reach, but it must be emphasized that the hazards of shallow coastal waters, if not known to the crew or in adverse weather, could be just as dangerous as running out to the open sea. Because suitable landfall was rarely more than a day's travel from any harbor within the Aegean, there was little need for night voyaging unless one left the Aegean Sea proper (by choice or when blown off course).
For these reasons, coast- and island-hopping facilitated movement by sea in the Aegean, and travel by sea was bound to play a fundamental role in the development of early Greece. The evolution of navigational skills is apparent in the archaeological record (Papageorgiou : 199–200). Beginning with indirect evidence of sea travel in the presence of Melian obsidian at Franchthi Cave in the late ninth or early eighth millennium BC, subsequent millennia saw traces of inhabitation on several islands and the colonization of Crete in the seventh millennium. Over the following 3,000 years, Aegean islands and mainland coastlines were gradually settled, until during the EBA (third millennium), proto-urban settlements and dense maritime communications were established (Broodbank ; Cherry ). These networks, along with technologies including the sail, were well developed by the time the Mycenaeans emerged as a complex society with ambitions beyond the Greek mainland.
The omnipresence of coasts and islands was also a hindrance to safe navigation, however, and this was a fundamental condition for the formation of navigational techniques in the Aegean. The quote that opens this chapter, taken from a twentieth-century sailing manual, succinctly makes the point that navigating the Aegean was more a problem of pilotage through complex coastal and near-coastal topographies under the influence of varied environmental conditions than a matter of celestial navigation, dead reckoning, or other means of way-finding on the open sea. Thus, despite the obvious advantage of constant visibility of land, knowledge of local coastal conditions was paramount to ensure safe passage and landfall. Ancient texts are replete with stories of ships run afoul of coastal hazards while trying to make safe landing. To cite an example, one of the misadventures of St. Paul's long sea journey from Palestine to Rome was shipwreck on the coast of Malta (Acts of the Apostles 27.6–28.10). After being blown off course from Crete for several days by a raging storm, the ship's captain spied a bay with a sandy beach on the Maltese coast, but on approaching, the ship ran aground on a sandbar, became wedged, and was destroyed by the pounding waves. Such disasters as befell the Persians in 492 and 480 BC or St. Paul in the early years of the Roman Empire were undoubtedly common, and underscore not only a general need for knowledge of local conditions – or better perhaps to highlight the consequences of not having that knowledge – but also support the argument for the prevalence of short-distance connections among those with local experience, leading to the formation of maritime "small worlds." A detailed examination of coastal landforms and navigational conditions in the Aegean in Chapter 5, along with case studies in Chapter 7, will permit these arguments to be developed in full.
4.9 Visibility of land from the sea in the Mediterranean. After Broodbank : 40, fig. 4.
These peculiarities distinguish the Aegean from other seas with which it is often compared. For example, ethnographic accounts of navigation among aboriginal South Pacific islanders are often found appealing as comparative material because they offer detailed studies of navigational techniques that do not rely on modern instruments or advanced seacraft, not to mention insights into maritime communities and their social and economic links with similar communities both near and far. Among these insights is a better understanding of how a pre-literate society can develop and disseminate sophisticated systems of navigation based on detailed observations of celestial bodies and other environmental features.
Yet although many of these insights are useful and even compelling, fundamental differences must be kept in mind. Broodbank (: 38–43) examines the similarities and differences of low-technology voyaging in the Mediterranean, Caribbean, and southwestern Oceania. In spite of comparable challenges inherent to small wind- or human-powered craft, and even some similarities in social and economic organization, Broodbank points to fundamental differences between the Cycladic Islands of the Early Bronze Age and these other areas. One of these is the basic geography of the island chains. In the South Pacific, islands are much more isolated, separated from one another and from any continent by great distances with a seemingly boundless sea intervening. These geographical factors have had a profound effect on the way that social and economic relationships developed over time among the scattered island communities. To a much greater extent, the proximity of Aegean islands to each other and to continental land masses created a profusion of potential maritime routes and situated coastal communities within range of recurrent external influences.
To Broodbank's observations we might add that a very different kind of navigational skill was required for a navigator departing in a small sailing canoe from one tiny Pacific island for another, often hundreds of nautical miles away. The main objective was to avoid missing a tiny speck of land in a vast sea, since the consequence of deviating even a few degrees off course might be to be lost in the open sea. Indeed, from time to time, canoes departed, never to be seen again. Other voyages turned into long adventures of missing the target (due to navigational error or inclement weather), only to turn up at some other island, where the errant crew might remain for a time before returning (Feinberg : 25–31). Most such voyages were successful, however, and to make this possible an elaborate and complex system of navigation using celestial observation of the movements of the sun and stars was developed, partly independently and partly through shared traditions, by many island societies in the South Pacific (Feinberg : 87–118; Gladwin : 145–213; Thomas : 73–85). By contrast, so long as the Mycenaeans were traveling within the Aegean, Ionian, or Adriatic Seas, or along the coasts of the Levant or southern Asia Minor, they could choose to avoid the open sea, instead navigating by coastal and island landforms and relying on their knowledge of safe anchorages en route.
This is not to say that Mycenaeans did not undertake open-sea voyages, or that they were ignorant of celestial navigation. Within the Aegean, ship captains often had a choice between a circuitous coastal route and a more direct open-sea voyage. The decision involved considerations of the ship's structural fitness and equipment (sailing rig, oars, crew, provisions, etc.), the captain's and helmsman's experience, the expected environmental conditions (winds, currents, weather), and the urgency of a speedy arrival (McGrail : 88–89). These dilemmas figured prominently in the return of the Homeric heroes from Troy (McGrail : 88). After arriving on Lesbos, Nestor, Diomedes, and Menelaus debated two island-hopping routes for their return to the Peloponnese, one sailing through the narrow channel between Chios and the Anatolian coast before turning west to the northern Cyclades and thence to Euboea and Attica; and the other keeping west of Chios between that island and Psyra before also turning west to the Cyclades and beyond. Nestor, after inquiring of Zeus, chose instead a direct open-sea voyage to the southern coast of Euboea, keeping Psyra to port (Odyssey 3.169–79). Interestingly, none of these routes is favored by winds or currents, yet Nestor reports that his ships, taking the open-sea route, were able to run before a freshening wind astern, reaching Karystos in southern Euboea by dawn the next day. The route ostensibly favored by environmental conditions was long, possibly involving a crossing toward Skyros in the northern Sporades before turning south to catch the prevailing winds and currents along the eastern coast of Euboea. Matching environmental expectations more closely is a journey found in Odysseus' phony tale of his youthful exploits on Crete (Odyssey 14.252–57), in which he describes a plausible sea voyage from Crete to Egypt, riding a northerly wind down the lee side of Crete and arriving at the Nile in just over four days. The journey from the Aegean to Egypt illustrates the fact that when ancient ships left the familiar confines of the Aegean, they encountered extended periods of days (or even weeks if things went wrong) in the open sea, often out of sight of land. Because these trips involved sailing at night, some means of staying the course was necessary. Homer's sea captains were familiar with stellar navigation: after leaving Calypso's island, Odysseus set a course for seventeen days with reference to the Pleiades, the "late-setting Ploughman" (Arctophylax), Ursa Major (our Big Dipper), and Orion (Odyssey 5.270–77). There is an important difference between the use of prominent stars and constellations to determine orientation and rough geographic position, and the Pacific Islanders' technique of "star path steering," which relies on an intimate knowledge of the rising and setting azimuths of numerous stars, as well as the ability to correct for winds and currents en route (Davis : 298–99). There is no evidence in the Homeric text that Odysseus used this technique in his voyage from Calypso's island (Davis : 299–300).
The fact that night sailing by means of celestial navigation was reasonably well established by the time Homer's epics were set to writing, circa 700 BC, is no proof that Mycenaean sailors were similarly capable more than a half-millennium earlier (McGrail : 89). Indeed, many modern scholars are skeptical of LBA celestial navigation (e.g., Chryssoulaki : 79; Lambrou-Phillipson : 13). Yet there are reasons to believe that such knowledge already existed in the Bronze Age. Christos Agouridis (: 17) develops an argument in favor of celestial navigation in the Aegean Early Bronze Age that, although lacking direct archaeological evidence, is in many respects compelling. The material evidence that is preserved reflects intensive engagement with the sea by EB II, including a proliferation of fortified coastal settlements and the movement of goods by sea throughout the Aegean. Given that the most sophisticated sea-going vessel of the period was the paddled longboat with a maximum daily range of 40 to 50 kilometers (Broodbank : 102), some journeys must have involved overnights on the sea. Meanwhile, a novel suite of iconographic symbols related to sea, sky, and maritime travel emerged in the Cyclades in EB II (Broodbank : 249–53, fig. 81). The ceramic "frying pans" of Syros type carry incised depictions of longboats, fish, sun/star symbols, and abstract "sea spirals" that probably represent billowing waves. The co-occurrence of incised fish, sea spirals, and a sun/star on an early frying pan from the Louros Athalassou cemetery on Naxos (Fig. 4.10) is suggestive of celestial navigation. Adding to the material evidence, Agouridis invokes ethnographic parallels of widespread celestial navigation and open-sea voyaging among Pacific and Caribbean islanders to argue that people engaged intimately with the sea and making long sea crossings will naturally observe the position and movements of heavenly bodies and realize their potential for tracking one's location and destination, likely out of necessity while on the sea at night.
The contrast between seafaring as presented in the Homeric epics and as known from archaeological evidence of the LBA eastern Mediterranean may shed further light on the question of celestial navigation. Jan-Paul Crielaard () observes that long-distance maritime relations in the LBA were far greater in scale, more complex in organization and infrastructure, and served a wider range of exchange relationships than those depicted in the Iliad and Odyssey. The dense networks of trade in raw materials and finished goods, the elaborate harbor facilities of Egypt and the Levant, and the long-distance political relationships among the great states of the eastern Mediterranean had largely evaporated in the Early Iron Age, and do not figure in the epics. Crielaard's principal aim is to show that seafaring provides additional support for the argument, widely accepted today, that the world of the Homeric epics should be situated close to Homer's own time, if to any historical period at all (Bennet ; Morris ). But another implication of his arguments can be expressed as a question: If long-distance seafaring was better organized and more sophisticated in the LBA, would it be surprising to learn that Mycenaeans sailed at night and navigated by the stars, as did Homer's sea captains? The Homeric texts, together with iconographic evidence for a gradual development of the oared galley from late Mycenaean times to the Geometric period (see Chapter 3), do not imply significant technological progress. Although this is not conclusive proof, I am inclined to agree with Agouridis (: 17) that the use of celestial navigation for nighttime voyages on the open sea is virtually certain for the Aegean Bronze Age. Moreover, it may be that there was a division in Mycenaean maritime societies between master navigators, able to steer ships on long open-sea journeys by celestial navigation, and those whose knowledge was limited to local and daytime trips. This would parallel the recent situation in the South Pacific, where most ocean travel occurred within a short distance of a given island and the majority of men became adept at it, but only a handful achieved the level of navigational skill required for long open-sea journeys (Feinberg : 88–91).
4.10 Sun, sea spirals, and fish incised on an Early Cycladic frying pan, Louros Athalassou cemetery, Naxos. © Hellenic Ministry of Education and Religions, Culture and Athletics/Archaeological Receipts Fund. Courtesy of the National Archaeological Museum, Athens.
Even when Mycenaean captains chose to avoid open-sea voyaging when possible, it does not mean that their vessels hugged the coast too closely (Mark : 139–42; Morton : 143–72). Apart from the navigational hazards, known or unknown that lurked in coastal waters (pilotage along a coastline at night was particularly dangerous), the purpose of a voyage affected the desirability of a coastwise route. There might be little reason to stray far from shore for fishing excursions, or for visits to nearby coastal communities for social or trading purposes. Longer trading voyages involving multiple stops en route to buy and sell goods might also necessitate a coastwise, harbor-to-harbor strategy in which the vessel traveled a safe distance from the coast, but could quickly make for a harbor or anchorage to trade or to find shelter. In this case, the route reflected not fear of open-sea voyaging, but economic sense. As mentioned above, ships traveling near-coastal routes might also take advantage of coastal breezes and currents to make headway with or against prevailing winds. On the other hand, when the intention was to travel quickly from one place to another with little interest in intervening locations, a direct route over open sea was often preferable if environmental and human conditions permitted.
### Sailing Routes in the Aegean
Although the proximity of myriad islands and coastlines presented almost unlimited hypothetical routes crisscrossing the Aegean, certain dominant and habitual routes may have emerged over time. There can be little doubt that environmental factors strongly influenced the routes that sea travelers followed around the Aegean in the Bronze Age. Winds, currents, topography, and seasonal weather patterns combined with technologies of shipbuilding, propulsion, and navigation to give shape to a range of possible routes and seasonal schedules. If these factors were sufficiently determinative, certain well-traveled sea lanes could be formalized over long periods of time, with a consequently strong impact on the pattern of social relations. Here a crucial question arises: Did environmental forces play a determining role in coastal settlement patterns, and even in maritime social relations? That is, did coastal communities arise or flourish because they were positioned advantageously with respect to sea paths favored by environmental forces such as winds and currents? Further, were their external relations patterned, or even determined, by movement along these favorable routes? If so, it should be possible to create a simple predictive model for the existence of significant Bronze Age coastal settlements at well-placed nodes along these routes.
Models that emphasize the role of environment on the formation of maritime networks in the Aegean Bronze Age have often been proposed, perhaps most cogently in recent years by Agouridis () and Despoina Papageorgiou (, ). This approach has generally involved creating maps of hypothetical sea lanes based on environmental factors, particularly currents and/or winds, and then plotting the known archaeological sites for a given period to assess the fit between these two sets of data. Agouridis (: 6–15) shows how the currents and winds may have facilitated the dense connections implied by the archaeological evidence. He stresses that proximity can be misleading: islands close to one another but separated by rough seas may have had little contact (Agouridis : 19). Papageorgiou strongly emphasizes the role of sea-surface circulation, determined by wind- and density-driven sea currents: "It is clear that the sea circulation contributed decisively to forming the routes of prehistoric ships in every direction, and to establishing sea lanes between all the mainland coasts and the Aegean islands" (Papageorgiou : 204). On the basis of sea circulation patterns, she proposes six principal Aegean sea routes, with numerous potential subroutes (Fig. 4.11). These include both open-sea and coastal routes, since Papageorgiou believes that open-sea and off-season travel were practiced whenever and wherever environmental conditions were permissive. Having established these routes, Papageorgiou examines the archaeological evidence for site location and activity in the Neolithic and Early Bronze Age. She concludes that prehistoric coastal settlements "...were invariably founded on small promontories and peninsulas, close to protected bays and safe anchorages," and "...located at crucial points on the sea lanes" (Papageorgiou : 216). On a larger stage, winds and currents are generally believed to have dictated a counterclockwise motion to long-distance trade in the eastern Mediterranean in the Bronze Age (Fig. 4.12).
4.11 Hypothetical Aegean Bronze Age sea routes. After Papageorgiou : 210, fig. 4.
Yet cultural factors may have figured more prominently in the establishment and evolution of maritime connections than allowed in these reconstructions. In Chapter 3, we saw how the introduction of the sail in the Cyclades at the end of the Early Bronze Age may have initiated a reconfiguration of prominent coastal nodes and the networks they served (Broodbank : 347–49). Indeed, Broodbank has injected a stronger cultural element into the formation and configuration of maritime cultural networks in the prehistoric Aegean. He maintains that the mutability of environmental conditions and social and economic relations "...makes it unlikely that prevailing currents and winds would determine the locations of maritime centres," and further that "[i]t is more likely that social geography in combination with technological parameters for early seafaring defined such centres and fashioned the sea-paths linking them" (Broodbank : 94). From this perspective, environment, technology, and society converged in various ways to favor or hinder maritime connections in the Bronze Age Aegean. Among the conditions motivating such connections were kinship relations; demographic needs for exogamy and labor sharing; unequal resource distribution, which presented opportunities for exchange and possibly led to "social storage" arrangements among communities to buffer against resource failure; inter-elite relations; and mutual protection against predatory raiding. This view aligns closely with Peregrine Horden and Nicholas Purcell's () examination of historical interaction patterns around the Mediterranean, which reveals countless cases in which human tendencies to connection and mobility trump the ostensible constraints of topography on land and environmental conditions at sea. These connections then come to define regions of different scales and types, which are shaped as much by social as environmental imperatives. Thus, it is plainly inadequate to map maritime interactions based solely on environmental factors. It is the human response, operating within the opportunities and constraints of natural environment and socioeconomic milieu, which is decisive. Seafarers may develop innovative technologies (e.g., the sail) to alter the relationship between human and environment in their favor, or they may forge connections in spite of environmental challenges by finding ways to work around the more prohibitive obstacles. Once set in motion, maritime connections can create a cumulative history that serves to perpetuate relationships regardless of whether or not they reflect economic or environmental logic through changing times. For this reason, unraveling the history of particular connections can be revealing about how relationships among people and communities may persevere in spite of the vagaries of time.
One way to take these ideas further is to think in terms of a feedback or dialectical relationship between environmental and cultural variables, an annales history in which the relationships of humans with the sea, and with other humans across the sea, unfold over different scales of time and space (Braudel ). Building on the data discussed so far, we might propose that the parameters of seafaring involve the interplay of long-term structures of the longue durée (timeless circulation patterns that create the Mediterranean maritime climate, simple technologies of shipbuilding and navigation), medium-term patterns (conjunctures) of the moyenne durée (major climatic oscillations, technological innovations in boat construction and navigation, political and economic relationships on the scale of centuries), and historically contingent événements that may be single events or short-term developments (collapse of the Mycenaean palaces, colonial forays, wars, short-term climatic variations). Johan Rönnby () calls these "maritime durées" and proposes that it may be possible to identify experiences and mentalités unique to coastal inhabitants around the world and through time, an observation that is incorporated into the concept of coastscape, developed in Chapter 6.
4.12 Hypothetical long-distance sea routes in the eastern Mediterranean. After Wachsmann : 296, fig. 13.1.
Scale affects historical trajectories in distinctive ways. The analytical focus advocated in this book on local-scale interactions owes a debt to Horden and Purcell's emphasis on the durability of short-distance interactions among "microregions," with considerably more unstable linkages to the shifting fortunes of supraregional entities. This emphasis results in a very different history from the traditional "Great Men and Battles" version, but one that reflects more faithfully the true rhythm and full scope of Mediterranean life. One of these alternative histories can be written about short-distance maritime connections, which offer a fundamentally different view from the great interregional trading routes on which most scholarship has focused. Although rarely studied, maritime small worlds of the Aegean Bronze Age offer a window on long-lived, though not necessarily unchanging or even stable, interactions. The durability of short-distance relations results from several factors that are typically absent in large-scale, cross-cultural contacts: (1) they are easier to maintain from a practical point of view, since distances and environmental obstacles are less inhibitive; (2) they are often founded on long-established and deeply embedded social ties; and (3) the communities they bind may be less susceptible to changing political fortunes if they lie outside the mainstream of momentous historical events. By contrast, interregional connections are more vulnerable to environmental and political forces, and are the first to be broken in turbulent times.
In effect, the analysis of maritime small worlds compresses space while expanding time. It is therefore possible to build a diachronic narrative of continuity and change in which small worlds are the focus and larger power centers and spheres of influence move into and out of view over time. The intent of this exercise is not a return to historical particularism, but rather on the one hand an interest in these people and places in their own right, and on the other a desire to assemble robust data in the service of regional-scale archaeological reconstructions, as well as comparative models that may embrace scales from regional to global (e.g., Parkinson and Galaty ; Wright ). Too often, these reconstructions and models have a flimsy foundation in data or rely on data sets that are not comparable. By building up from many well-documented local settings, we are better able to clarify their political and economic role in larger webs of interaction, and to analyze the nature and limits of influence wielded by power centers in peripheral areas (Tartaron ). Taking advantage of the benefits of this shift in perspective for the Mycenaean period relies on a comprehensive and interdisciplinary examination of the archaeological and environmental record.
### Bronze Age Navigation in Practice: Navigational Aids
In the previous section, we established the vital nature of coastal pilotage for navigation in the Aegean. Pilotage was aided both by physical features that marked out maritime routes, warned of dangers, or promised safe haven; and by a body of maritime wisdom about them that systematized knowledge into practice. Let us first specify the aids that might have been available to Mycenaean navigators, and then consider how knowledge of their existence and use was transmitted over time.
The terminology describing navigational aids can be confusing. The terms landmark, seamark, and skymark would seem to indicate navigational cues existing on land, sea, and sky, respectively. This is exactly the usage adopted here, disregarding the more expansive definition of seamark that one often encounters, which includes all features that can be seen from the sea (Table 4.1). The specific combination of marks relevant to a given voyage cross-cuts these categories according to the nature of the travel: long distance or short run; day or night; coastwise or open sea; planned or opportunistic journey; familiar or unfamiliar routes; friendly or hostile waters; fair or foul weather. A local trip from one coastal community to another might require knowledge of only a few coastal hazards and landmarks, while a voyage of some length could involve a great many of these navigational marks and a much greater store of knowledge and navigational skill. A sea captain leading a Mycenaean trading mission from the Argolid to Knossos on Crete would have called upon his knowledge of winds, currents, islands, and celestial bodies, as well as a solid knowledge of Cretan coastal topography, wind conditions, and coastal settlements.
Table 4.1. Navigational aids potentially available in the Bronze Age. Artificial constructions are shaded.
* * *
* * *
#### Landmarks
Mountain ranges or peaks with distinctive shapes served as good general markers for staying on course, since they were visible from great distances. For more precise delineation of routes, however, sea cliffs and headlands were particularly useful. In historical times, as recorded by Strabo and others, prominent headlands functioned as nodes in longer journeys conceived in terms of multiple headland-to-headland segments (Morton : 186–88). Thus, one such voyage in the northern Aegean was defined as a series of headlands (Poseidonion, Sepias, Kanastron, Nymphaion, Akrathos) separating the intervening gulfs (Maliac, Thermaic, Toronaean, Singitic, Strymonic; Strabo, Geographica 7.fr. 32). Because of their high visibility and vital role in seafaring, headlands have been favored locations to place various kinds of manmade landmarks meant to be seen from the sea. As early as Homer, conspicuous burial mounds were placed on headlands at the Dardanelles (for Patroclus: Iliad 7.85–91) and at Circe's island (for Elpenor: Odyssey 11.71–78). An archaeological example is the erection of large cairn-like structures, which may have taken the form of tall towers before their collapse over time, on high coastal ridges of the northeastern Peloponnese in EH II (Tartaron, Pullen, and Noller ; Tartaron et al. : 626). Such structures could have been fitted out as lighthouses or signal stations, but this is mere speculation. Apart from their potential as conspicuous landmarks, the monumentality of their construction may have warned off those with hostile intent. Whatever the original motivation for constructing mounds, cairns, or in later times, temples on coastal promontories, they served both to advertise and to provide useful navigational markers to passing vessels. In certain periods of particular orientation to the sea, for example EH II and LH III, settlements might be found on coastal promontories, but it was often not practical to settle on them, sometimes for reasons of defense, but more often due to untenable distances to water and arable land. Nevertheless, promontories are primary locations to search for traces of ancient structures related to maritime activity.
Many coastal promontories or sea cliffs were distinctive for their unusual shapes or striking colors. In historical times, these were defining properties of such topographies, giving rise to place names incorporating colors (e.g., leukos for white, melanos for black, erythros for red) or animal shapes (e.g., Kynosoura for dog's tail, Onougnathos for ass's jaw). A crystalline limestone sea cliff, shining brilliant white in the sun, could be a striking and easily recognizable marker along a sea route; the pervasive limestone geology of the Greek mainland explains the widespread occurrence of forms of leukos for coastal landforms. Naming prominent features along a route transformed space to place; once a place is named and described, it becomes part of the known world and thus less terrifying than a vast, unfamiliar sea. Regrettably, for the Bronze Age we have only the place names from the Linear B tablets and a few other likely Aegean toponyms from Egyptian and Hittite sources. Of these, only a few belong demonstrably to coastal locations and none seems to refer to an evocative coastal landform.
Of crucial navigational importance were the coastal features that pointed toward safe haven, supplies, and trading or raiding opportunities. Long beach strands, river mouths, estuaries, and embayments formed a constellation of potential stopping-places en route, but these landmarks had to be supplemented with knowledge of the characteristics of the shallows to be negotiated to reach them. Human settlements were situated in many of these places, visible in the structures of buildings, walls, roads, and cemeteries; the disposition of their inhabitants toward visitors from the sea had also to be taken into account.
#### Seamarks
Like mountains, islands aided in keeping a ship on course, and literary evidence from Homer forward portrays navigators steering by keeping specific islands to port or starboard. Islands, of course, possess features both of landmarks and seamarks; all the landmarks enumerated above could be found on islands. Yet a long maritime tradition beginning with Homer characterizes islands as objects protruding from the sea that often possess a capacity for motion or floating not attributed to continental land masses.
On the open sea, wave patterns, driven by sea currents and winds, indicated general direction as well as developing weather conditions. While land was still out of sight, observations of fish could distinguish deep-sea from shallow-water species, and sightings of seabirds might permit an estimate of their typical ranges from land. Upon reaching inshore waters, to avoid danger crews had only a sounding lead or pole as an instrument to augment the information they could collect visually and their prior knowledge of local bathymetry and topography. Visual indicators of submarine features included breaking and deflected waves, protruding rocks and shoals, and water color and odor. Waves breaking far from shore indicated a shallow approach and the need for caution to avoid running aground on a shelf or invisible rocks. Changes in water color, typically from deeper to lighter shades of blue, indicated movement into shallower waters, as did the odor of waters permeated with coastal sea plants and animals, and often commingled with fresh water from estuaries, river mouths, and springs.
Human constructions on the sea, including breakwaters, jetties, and mooring posts, were recognizable infrastructural elements of harbors in ancient times. Yet it remains to be demonstrated that Mycenaeans built any of these, in contrast to the built features of contemporary Egyptian or Mesopotamian river harbors, or even the near-shore ship sheds that the Minoans built at Kommos and other sites (Shaw ; Shaw and Shaw ; see Chapter 5). Nor is there any indication that buoys or other channel markers were used in the Bronze Age.
#### Skymarks and Celestial Bodies
In addition to the use of the sun and stars for navigational purposes, as argued above, other celestial signs were available to aid Bronze Age mariners. Wind direction could be matched to known regional or local winds to ascertain general orientation, and wind speed might portend heavy seas or oncoming gales. Darkening skies and clouds were also important indicators of impending conditions at sea. The sudden disengagement of low-lying banks of clouds from the summits of islands and coastal mountains may portend strong winds or gales (USNHO : 12). On Crete, the summer meltemi is often heralded by "a fleecy bank of white clouds" enveloping the summit of Psiloriti and neighboring peaks (USNHO : 116), and further west the appearance of "long black, sausage-shaped" clouds over Cape Drepanon in winter is a sure sign of the onset of a heavy southeastern gale, accompanied by rough seas and a long, heavy swell (USNHO : 139). Sometimes, these massive banks of clouds pointed toward land masses that were not yet visible.
Smoke from fires of diverse origins might also come into view before land was sighted. Forest fires of natural origin occur periodically, as part of a healthy cycle of vegetative death and regeneration. Anthropogenic fires associated with settlement activities included cooking and heating at the domestic hearth, burning in agricultural fields and pastoral complexes, and industrial activities such as pottery firing and metal working. Accidental fires at settlements were also common, set off by lightning or the careless handling of an oil lamp. Signaling by fire or smoke or even the use of rudimentary lighthouses set up on cairns was possible in the Bronze Age but there is no evidence of formal lighthouses in Greece until the late Classical period (Mark : 160; Morton : 212). The use of fire for everyday activities was inevitable and must have made it difficult for any coastal settlement to conceal itself behind a ridge or in a hidden valley.
#### Phenomenology of the Voyage
To our knowledge, Bronze Age navigators did not possess maps and charts, so how then can we describe the mental process of the voyage? Often, the navigator is said to have carried "mental maps" or "cognitive maps" of the desired route and its features (Frake ; Lewis ; Oatley ). This notion has been criticized by Alfred Gell () as flawed and inadequate, in an argument that is to some extent semantic, focusing on what we mean by map, but with interesting substantive implications. Gell distinguishes map from image: the former refers to a model of reality in Cartesian space, showing all of the possible routes and territories that could be encountered, but without the point of view of the sailor and with no particular sequence of movement indicated. Gell asserts that navigators in mapless societies do not call up cognitive versions of such all-inclusive devices (it would anachronistic to claim that they do), but rather they work from a compendium of images that constitutes a perspectival experience of the seascape. The actual mental process combines knowledge of fixed spatial relationships of landscape features (referred to by Gell as "non-token-indexical" statements or beliefs) with a constant monitoring en route of the ship's relative position ("token-indexical" statements or beliefs). For instance, "Chios is south of Lesbos" is a non-token-indexical statement (i.e., it is assumed to always be true and does not depend on where the ship is currently located), while "The ship is currently passing Chios, so it must be south of Lesbos" is a statement that may be true for some part of the journey, but depends on the ship's position relative to these fixed features. If one's final destination is Lesbos, this token-indexical determination permits decision making and the action of setting or maintaining a course to the north. Navigational success is secured by knowledge both of the fixed relationships between known places, and of one's relative position among them, and Gell views these as distinct but complementary cognitive processes.
If these relationships have a form more like a series of mental images than a Cartesian map, then the knowledge and the mental process can be described as phenomenological, experiential, and embodied, but here an important distinction must be made. Gell explicitly critiques Pierre Bourdieu's "practical mastery theory" (Bourdieu : 2), an early phenomenological perspective on landscape that proved a fertile source for the development of the postmodern phenomenologies of Christopher Tilley and others (Bender ; Tilley ). Bourdieu envisions the landscape not as Cartesian map space, but as a "practical" space that constantly morphs in response to the shifting perspective of a subject moving through it. In this kind of subjective, embodied experience, there is no objectively definable landscape, but instead a myriad of landscapes – both tangible and intangible – come into being through the contingent experiences of each individual subject. Practical mastery of maritime routes would involve committing to memory a series of images of landscape features from point of departure to destination, and then linking them en route. Gell rejects this relativistic view of landscape, insisting that landforms have a reality (spatial, substantive) independent of our momentary perceptions of them. Landscape change lies in the realm of geology and other longer-term processes; the change that occurs during a sea voyage is rather our position relative to these fixed features, so we must have external, fixed reference points in order for decisions based on our relative position to make sense. In Gell's phenomenology, the navigator's mental process is to refer to a kind of non-Cartesian mental map consisting of known features in fixed spatial relationships; in order to use this information, he must "match the images produced at particular map-coordinates with perceptual images of the surrounding terrain" (Gell : 282).
To this point, the discussion of phenomenology has concerned one type of voyage only, albeit a common one: the daytime route that is known to a maritime community, but sufficiently long or complex to require aids to way-finding, such as mental maps and landscape images. The mental images that facilitated such journeys in the Bronze Age are likely to have been similar to those of later times in Homer and in the periploi of Archaic to Roman times: islands, headlands, rivers, bays, and other coastal features that functioned as nodes in segmented journeys and potential places of refuge.
Yet there were several other types of sea voyages for which the mental processes and practices were quite different. One of these was the very short journey, along a coastline to a neighboring settlement or across a gulf, for which way-finding was unnecessary. Another type was the night voyage that relied on stellar navigation. Using stars and constellations to maintain general direction is much like using islands and headlands as landmarks, and this is precisely the way Odysseus was instructed by Calypso to navigate to the island of the Phaeacians (Odyssey 5.270–77). By contrast, the star path steering practiced by the mariners of the South Pacific is tantamount to the creation of a Cartesian map of the heavens, which could be read like a modern road map. Finally, some voyages extended beyond the limits of the maritime world known to the navigator, as for example when a ship was blown off course or when pioneering seafarers ventured into unknown waters. In those situations, the crew had no references at all and were forced to fall back on their experience with landmarks, seamarks, and skymarks to travel safely.
#### Types of Maritime Communities in the Bronze Age Aegean
A fundamental assertion, which forms the basis for the archaeological reconstructions to follow, is that in the Bronze Age Aegean there were two distinct realms of maritime activity, entailing different levels of knowledge and experience. A basic, or generalized, realm of activity comprised short-distance travel not requiring advanced way-finding or environmental and social knowledge about distant locations and peoples. A broad range of fishermen, farmers, traders, and other coastal dwellers possessed the resources and skills to undertake simple trips to local fishing grounds, along adjacent coastlines, across straits and gulfs, and out to nearby islands. The risks of this kind of sea travel were comparatively minimal, and consequently harbors large and small must have been busy with small-boat, short-range activity.
A specific, advanced level of knowledge was essential for medium- and long-distance voyaging; medium distance is here defined as nonlocal, intrabasin (Aegean, Ionian) travel, and long distance as venturing beyond these basins. This was the realm of the master shipwright, the expert navigator and helmsman who commanded detailed information about seas and coastlines, and the crews trained in rowing and sailing. It was a subsociety organized in part to safeguard and transmit a body of nautical knowledge to the succeeding generation. Because such mastery was held by few, it often afforded preferential access to exotic goods, raw materials, and esoteric knowledge, and thus it could be a source of considerable social power.
Both generalized and advanced maritime knowledge persisted over long periods of time, the former largely "under the radar" of mechanisms of centralized control. The more specialized knowledge and performance of the professional mariner was of potentially great interest to centralized political entities that depended on communication by sea. Thus at Pylos, personnel and ships were co-opted by the palace, whether to form state fleets or to work as private contractors in the service of the state. Yet the political centralization characteristic of EH II or the Mycenaean palatial period must be seen as a kind of periodic interlude in the more typically loose political structure of the Bronze Age and Early Iron Age. At most times, maritime communities must have operated independently or semi-independently. In the next section, the formation and perpetuation of the maritime knowledge of these specialized communities is examined.
### Transmission of Maritime Knowledge and the Habitus of Maritime Life
Because for all intents and purposes LBA Greek society was pre-literate, the essentials of maritime life were communicated by word and by nonverbal demonstration. Several kinds of expertise were involved: shipbuilding and repair; techniques of paddling, rowing, and sailing; knowledge of the indicators of winds, currents, and weather; and detailed familiarity with a host of principal and alternative sea routes. Mastery of this body of knowledge was neither easily obtained nor commonly held in society; it required physical skills, mental powers of memorization as well as adept thinking in difficult situations, and plenty of experience on the sea. It was a specialized life and throughout recorded history, communities of seafarers have existed in various degrees of segregation from the rest of society. These maritime "closed communities" (Muckelroy : 221–25, 240–42), "ship societies" (Adams : 304), or possibly in the Mycenaean case a "galley subculture" (Wedde : 33–36; see Chapter 3) formed alternative social entities that did not simply reproduce at smaller scale the wider societies of which they were a part. Indeed, a maritime community might look little like its host society: it might be exclusively male, and constituted by family or kin groups specialized in maritime pursuits. Although hierarchical relationships were essential aboard ship due to the hazardous conditions of seafaring, status in the maritime community might be achieved – according to experience and skill – rather than ascribed on the basis of social class, wealth, or kinship ties (Adams : 305–306). Master sailors, navigators, and helmsmen would always have high status relative to less skilled and experienced crew members, as a matter of necessity.
Historical records and ethnographic studies provide many examples of multi-generational seafaring social groups with strong incentives to maintain and perpetuate the maritime closed community. The practical problem that so much information had to be learned and experience gained could best be addressed by beginning to inculcate knowledge at a young age, within the context of the family. Maritime knowledge was worth guarding as a potential source of power and independence. In early times, those with access to distant places with their exotic products and esoteric knowledge possessed a special, perhaps even mystical, status as Broodbank (: 289–90) suggests for the longboat voyagers of the EBA Cyclades (see also Helms ). The independence of maritime communities varied with the strength and interest of the state. In the Linear B tablets An 1, An 610, and An 724 from Pylos, we learn that the palace was able to levy ships and rowers from several villages in the last months of its existence. Similarly, in Viking-era Scandinavia, powerful families ruling proto-towns were able to outfit "levy fleets" with men from surrounding farmsteads and villages (Rönnby : 75; Westerdahl : 10). Yet in the Mycenaean period, and presumably in many other times and places, much of the coastline of the mainland and islands lay beyond the political reach of any palace state, and even within palatial territories, state control could hardly have been all encompassing, whether by force or accommodation. This does not imply that long-distance trade missions involving the acquisition of raw materials and luxury items from distant ports of trade necessarily lay in the hands of independent maritime communities, but short- and medium-distance social and economic sea travel must normally have carried on with little or no palatial interference. These traditions, and the connections they established, time and again outlived centralized states.
Because maritime knowledge was content heavy and complex, some means were needed to transmit a body of maritime knowledge within and across generations. Ethnographic studies of seafaring in the Pacific are a good point of departure, since they offer detailed reconstructions of maritime knowledge and its transmission in a nonliterate society. From a very young age, boys in these communities learn the basic skills of seafaring: they accompany family members on short voyages for fishing or visiting, they assist in the construction of canoes, and build toy models of them. In the Caroline Islands, youngsters begin to learn the complex system of stellar navigation through a series of game-like exercises designed to convey the key concepts and to train the mind for memorization and situational thinking. An instructional "star-structure" compass, representing the great circle of the horizon divided into 32 points where the stars are observed to rise and set, is assembled from simple materials such as stones, shells, and reeds (Fig. 4.13). Sitting together in a boathouse in the evening, experienced navigators guide novices through a set of rigorous exercises including the following (Goodenough and Thomas : 5–7):
Island-Looking: In this memorization and orientation exercise, navigators drill pupils on the locations of islands by starting from an island and, moving around the compass, naming the places that lie in each direction. The exercise is then repeated starting from another island. Advanced students are able to name the locations of reefs, shoals, and other seamarks.
Sea-Knowing: In this exercise, students learn the sea lanes that lie between the islands and reefs by giving them names that refer to the rising and setting of specific stars that occur along them.
Sea Brothers: The student learns all the named sea lanes that lie along the same star compass coordinates. If a navigator forgets the star path between two islands, he may remember that the path is "brother" to another sea lane, and thus recover the forgotten coordinates.
Coral Hole Stirring: To reinforce the mastery of the star courses, this game imagines a chase sequence between a fisherman and a parrot fish. As the fisherman probes a hole in the reef at a given island, the fish darts off to a neighboring island, then another, and so on until eventually returning to the starting point. Each hole has a name that substitutes for the name of the island.
4.13 Example of a star-structure compass, Caroline Islands, Micronesia. Thomas : 81. Courtesy of Stephen D. Thomas.
With these and other exercises, aspiring navigators are initiated, in a systematic way, into the knowledge they will need to voyage successfully at sea. As they sail with master navigators, they recall this information and put the principles into practice. Some, but not all, will become master navigators themselves, responsible for transmitting knowledge to the next generation.
The community of sailors in the Caroline Islands takes on the characteristics of a closed maritime society, with selective membership and an oral tradition that is both durable and conservative. The apprenticeship of navigators and sailors (exercises, training at sea) is rigorous, with constant practice and layers of redundancy that facilitate mastery of crucial knowledge. In this way the society and its fundamental store of knowledge are faithfully reproduced and protected. The cosmology of the navigational star structure, and the alternative names for islands and sea lanes, are part of an esoteric seafaring lore known only within the group. This lore also incorporates stories with practical and moral lessons, which are embedded in chants that are structured, metrically and tonally, to aid in memory recall. The meaning of the chants is often cryptic, and only learned with the interpretive commentary of a teacher. The settings for this instruction are special places (the boathouse, at sea) exclusive to the maritime community. Because sea travel is at once both essential to island life and yet formidable and often perilous, master navigators are also ritual specialists who communicate with spirits of navigation, observe taboos, and perform rituals to ward off dangers at sea.
To place these ethnographic observations in a more general and time-transgressive model, Bourdieu's concept of habitus (Bourdieu , ) is a useful way to frame the process by which maritime social groups and their knowledge are perpetuated. This is so because habitus explains how and why traditional lifeways are reproduced and preserved over long periods of time, while at the same time mediating between the extremes of determinism – that is, that human perceptions and practices are determined by pre-existing structures over which people have no control – and voluntarism, which accords humans unfettered freedom to make choices that shape the conditions of their lives.
According to Bourdieu, the habitus is a system of dispositions in an individual or social group toward conceptions, activities, or behaviors that are structured and perpetuated by a "present past" of similarly structured practices and worldviews (Bourdieu : 53–55). In other words, each generation inherits a durable structure that defines the parameters of thought and action in which it can operate, and within which practice is ultimately generated. The habitus is not defined by explicit laws, but instead by the unconscious adherence to these inherited structures. Bourdieu did not view the habitus as a purely determinative mechanism in which individuals and groups are without choices, however. Instead, there exist "...an infinite number of practices that are relatively unpredictable"; nevertheless, freedom to choose is not actually unlimited because "...the habitus tends to produce all the 'reasonable,' 'common-sense' behaviors (and only those) which are possible within the limits of these regularities" (Bourdieu : 55).
To many observers, Bourdieu's actors possess too little agency, leaving insufficient scope to account for innovation ("creation of novelty": Bourdieu : 55) outside the bounds of inherited knowledge, particularly internally generated change as distinct from that which is externally stimulated or enforced. An alternative viewpoint is expressed in the "structuration theory" of Anthony Giddens (Giddens , ; Nakassis : 19–23), which posits a different relationship between structure and agency. Giddens envisions structure and agency as forming an inseparable duality. Although individuals and groups are undeniably shaped by the durable structures they inherit through no choice of their own, actors are knowledgeable, able to make conscious choices that are not only unconstrained by the limits of the habitus but that also may reshape it. Conscious choices need not have intended or predictable outcomes; they often lead down contingent paths to changes that could not have been predicted and for which the chains of agency are known only in hindsight. In this manner the elements of structure – traditions, worldviews, practices – interact with agency in a dynamic process of continuity and change, of mutual shaping (Giddens : 14). A simple example is linguistic change: language is a dynamic structure in which everyday speech reproduces inherited syntax and semantics, but also introduces changes that one day may be additions to, or modifications of, the structure of language (Nakassis : 21).
The habitus, in Bourdieu's sense, aptly describes the power of tradition and the conservative maintenance and transmission of maritime knowledge over long periods of time. This is particularly the case because maritime communities depended on the effective reproduction of large bodies of specialized knowledge to protect their lifeways and in a literal sense their very lives. Their habitus cannot have been overly rigid, however, because adaptation and innovation were equally essential to their survival. To continue the ethnographic example, elements of innovation are evident among the Caroline island navigators. Often, a master navigator died before imparting to his pupils his full commentary on seafaring lore, leaving them to develop their own interpretations based on the knowledge they had managed to acquire as well as their personal experiences. The new versions were frequently "quite different from the original and yet still workably consistent with reality" (Goodenough and Thomas : 13). Further, there is a tendency for navigators to elaborate on the lore in displays of virtuosity. Some individuals voyage to more distant places along star paths to discover islands previously unknown, which they may equate with previously mythical places. It is also the case that Caroline islanders have adapted to the availability of new materials and sailing (including navigational) equipment from the West. Some new knowledge can be accommodated within the existing habitus, for example, by fitting newly discovered places and new commentaries into the dynamic, living lore. Technological change, particularly that initiated by contact with the West, may precipitate deeper structural changes, and the fate of the traditions of the maritime communities of the Pacific – their habitus – is far from clear in the twenty-first century. Yet the recent history of the Caroline Island navigators amply demonstrates a closed maritime community striving, both consciously and unconsciously, to perpetuate and protect their habitus, while also creating and making choices about information that pushes beyond the bounds of their inherited structures.
### Mycenaean Maritime Communities
We ought now to ask whether we can detect maritime communities in the Mycenaean world possessing a specialized body of maritime knowledge. There is little direct evidence for them, though it is possible to make certain inferences from indirect evidence, and from what we know of maritime communities of other times and places. There are certain general statements we can advance, for example that the nautical life involves specialized skills and occupations. Shipbuilding and maintenance, navigation, and even rowing and other crew duties all require experience and advanced skills, which must have been transmitted through a system of master–apprentice relationships. A consistent conclusion drawn from recent trials of replica ships – from "Neolithic" reed boats (Tzalas 1995b), to Bronze Age galleys (Severin , ), to the reconstructed trireme Olympias (Rankov ) – is that an inexperienced crew requires a long apprenticeship before it can properly propel and control an unfamiliar vessel. The triremes of the Classical Athenian navy were manned by professionalized crews of free citizens, metics, and foreign mercenaries who practiced during peacetime (Hale ); a similar program of training may have been followed by the captain, helmsman, and rowers of the Mycenaean galley.
It may be possible through archaeological means to identify the coastal habitations and workplaces of maritime communities, but this remains a challenge because so few harbors, at least of the Mycenaean period, have been recognized and studied. This topic will occupy much of the remainder of the book. For now, let us explore how the early Greeks represented their maritime pursuits in word and image, through the Linear B tablets, iconographic images, and the later Homeric epics, particularly the Odyssey. In these sources, we might recognize the bare outlines of a maritime durée, of persons in action or a persistent mentalité expressed in words.
#### Linear B Evidence
It is worth recalling that the Mycenaean galley was the largest and most technologically advanced machine of its time, and the resources and organizational capacity of the palaces may have been required to build and maintain a galley fleet. Although it is uncertain how extensively Mycenaean ships traveled around the Mediterranean in pursuit of trade and diplomatic missions (see the discussion in Chapter 2), certain materials, particularly metals, were imported from overseas and controlled carefully by the palaces. The Linear B tablets from Pylos provide compelling textual evidence for palatial oversight of building, finding crews for, and sailing ships, at least in Messenia (Palaima : 284–86). The personnel drafted for these tasks are listed or named. Shipbuilders (na-u-do-mo) are mentioned on two tablets at Pylos. On Vn 865, na-u-do-mo is the heading of a list of 12 names, each followed by the numeral 1, probably indicating 12 shipbuilders assigned to construct one ship each. On Na 568, a group of men with this title is recorded with a single place name and an exemption of 50 units of flax while they are supervising construction. The title e-re-e-u, found on several tablets at Pylos, seems to refer to an official in charge of rowers. The rowers themselves were recruited from a number of coastal towns, apparently as part of an annual levy organized using the same principles as Mycenaean taxation; they performed their services in exchange for land use rights (Killen ). The association of at least one of the absent rowers on tablet An 724 with the lawagetas, a military leader, suggests that these records refer to a military, rather than commercial, fleet.
At least seven names occurring in the Linear B archives at Knossos and Pylos derive from roots associated with maritime activities, translatable as "Ship-Famous," "Fine-Ship," "Swift-Ship," "Ship-Starter," "Shipman," "Fine-Harborer," and "Fine-Sailing" (Palaima : 284). Intriguingly, however, the men with these names appear mainly as herders, as well as two bronze smiths at Pylos; none is recorded pursuing a maritime activity. One way to interpret the lack of correlation between name and occupation is that as the palace states reorganized and consolidated their economies, workers from across the economic spectrum were co-opted to the large, palatially controlled industries, including metallurgy and other craft manufacturing, and the production of wool on Crete and linen at Pylos. One of the political consequences of economic restructuring by Minoan and Mycenaean states tended to be that traditional ties between communities and their landscapes and subsistence practices were severed (Haggis ). Even so, skilled seamen would have been prime candidates for the annual levy of rowers. We might imagine that many such names existed among members of specialized maritime communities throughout the Mycenaean world; on the other hand, these could be very old names that had long since lost any exclusive association with the original occupation, much as names like Smith, Miller, and Carter have done in English-speaking countries today.
The Linear B scribes did not concern themselves with localized coastal activities such as fishing, presumably because the palaces had neither the interest nor the ability to exert direct control over a ubiquitous resource with little potential for profitable specialization. Products of the sea must have been widely available in local markets, and the palace may have used agents or other means of acquiring them as needed. The practices of fishing and short-distance voyaging for social and economic purposes must have continued, both within and beyond the palatial territories, without hiatus during the Mycenaean period. In other words, these traditional activities tend to persist regardless of the vagaries of political organization.
The matter, discussed in Chapters 2 and 3, of whether maritime commercial activity was carried out in state-controlled ships or commissioned to private merchant vessels that may have been partially or fully independent of the palace, is unlikely to be resolved soon, barring the discovery of a series of tablets that explicitly addresses overseas trade. The same may be said for Wedde's () argument from the Pylos tablets that a "galley subculture," composed of captains, helmsmen, and rowers, arose on the Messenian coast to challenge the authority of the palace in LH IIIB (Chapter 3). This is one plausible interpretation of the apparently ominous circumstances under which rowers were recruited in one document (made up of tablets An 657, 654, 519, 656, and 661) bearing the heading, "Thus the watchers are guarding the coastal regions" (Chadwick : 173–79). Whether the galley crews helped to topple the palace, or instead came to the aid of the palatial elites in a time of grave danger (Wachsmann ), we may infer minimally the existence of a seafaring community with specialized, highly valued knowledge and skills – helmsmen and navigators being most obviously in demand. To what extent that community operated independent of the palace, we cannot say with certainty, but we can propose with some confidence that the state controlled some aspects of seafaring – such as military operations and long-distance trade – while leaving the rest in private hands. Thus, the Linear B archives do not offer an unambiguous portrait of a maritime community, but the pattern of palatial political and economic interests revealed in the tablets points to an aggressive but limited oversight of nautical affairs, leaving much room for traditional coastal activities.
#### Iconography
The Flotilla Fresco at Akrotiri, depicting a range of ships and boats, gives some sense of the complement of vessels one might see in a busy Bronze Age harbor. The small rowed skiff W622 seems to play the role of pilot, but must have functioned just as well as a fishing boat or short-distance coaster. The fragmentary but contemporary fresco from Ayia Irini, Kea (W672–676), shows either crews or harbor personnel walking on shore and mixing something in large, tripod cauldrons. Behind them is an ashlar building that suggests parallels with the possible ship shed on the north wall of Room 5 of the West House, as well as the foundations of ship sheds known archaeologically on Crete. In the frescoes, a visual distinction made between the crews, dressed only in loincloths, and the honored guests, who wear elaborate robes, emphasizes the difference in status between elites and non-elites. A recent interpretation of the Flotilla Fresco views all these men, seamen and foot soldiers alike, as members of a victorious warrior "coalition": "soldiers, priests, sea captains, and lowly paddlers – whose membership in hierarchical but cooperative coalitions ensures island power" (Chapin : 142).
These glimpses into maritime life in the cultural sphere of the Minoan palaces of the neopalatial period may serve as a bridge to Wedde's Mycenaean galley subculture. Wedde traces these groups, and with them a continuity in the traditions of seafaring, into the EIA through the iconography of the galley, which peaks in LH IIIC and persists with only gradual modification in the EIA. This continuity may be seen as a strong indication that seafaring communities and their traditions survived intact despite the collapse of political centralization.
#### Homeric Epics
The commission of the Homeric epics to writing postdates the fall of the Mycenaean palaces by more than 400 years, but as the oldest examples of Greek literature they are closest in time to the events of a distant past about which the poet sings. Furthermore, the Odyssey is, at least superficially, a seafarer's tale. Although it is well established that Bronze Age survivals in Homer are few (Bennet ; Finley ; Raaflaub 1996), it might be possible to recognize a mentalité or elements of a maritime habitus that persisted from the LBA into historical times.
Seafaring is not glorified in Homer. Rather, heroes like Menelaus, Nestor, and Odysseus emphasize the ardor, the danger, the terror, and the loss of life at the hands of malignant forces of nature and implacable gods and monsters. The Odyssey abounds in stories of ships blown off course, stranded in windless seas, or wrecked altogether. A recurring theme is that these disasters are visited upon crews that have offended a powerful god in some way. The audience listening to a bard singing these tales could not fail to grasp the implication that seafaring was not for the faint of heart or for the uninitiated. Many would have had personal knowledge of a ship or a person lost at sea.
In the Odyssey, there are three main groups of sea travelers: warrior-heroes and their followers, pirates, and merchants. They are distinguished on the basis of who they are – their status and social identity – rather than on their actions at sea, because their realms of activity – war, trade, and raiding – overlap to a surprising extent. As we have seen, the Trojan War heroes often acted as pirates do, plundering coastal regions for needed supplies or treasure. They also were the beneficiaries, if not the actual agents, of long-distance trade for raw materials and other exotic commodities (Tandy : 75). Pirates pursued organized, "informal" warfare, and their ships and crews must have been fitted out to wage war. Merchants, too, might raid when the opportunity arose.
Odysseus and the rest of the Trojan War heroes formed an elite warrior class that ruled small, dispersed chiefdoms, more analogous politically with an emerging aristocracy of Homer's time – the basileus and his followers of the eighth century – than with the rulers of the Mycenaean world. In Homer, the social and economic structure was based on the oikos, consisting of the extended family (usually three generations living together) and its land, animals, slaves, and all other assets. Ships were owned by prosperous families, whose patriarchs could assemble a crew from among the dependants of the oikos, or hire them from elsewhere. This world is more similar to Viking-era Scandinavia (Rönnby ) than to the Mycenaean palatial era.
Eighth-century basileis, like their epic counterparts, constructed and maintained their status by differentiating themselves from the rest of society by means of what David Tandy (: 141–65) terms "tools of exclusion." The most prominent of these devices were inter-elite, reciprocal hospitality and gift-giving; councils and feasts; hero cult and warrior burial; and indeed the epics themselves, which can be seen as part of the program of elite self-definition and exclusion (Tandy : 141, 152). A detailed treatment of this program is beyond the scope of the present work (for different perspectives, see Antonaccio ; Morris , ; Osborne : 70–136; Tandy ; Whitley ), but a few examples from Homer and from the archaeological record give a sense of it. Telemachus receives a lavish reception at Pylos and Sparta once he is recognized as the son of Odysseus, and thus as a member of the elite warrior class. His arrival at Pylos interrupts a great feast at the seashore for which 81 bulls were sacrificed to Poseidon (Odyssey 3.1–10), obviously well beyond the resources of all but the wealthiest of men. The audience halls of Alcinous, Odysseus, Nestor, and Menelaus find parallels in several large buildings on the Greek mainland and islands (Mazarakis Ainian : 363–67; Tandy : 144–49). Beginning in the later eighth century, cult places dedicated to epic heroes, including the Menelaion at Sparta and possibly shrines to Agamemnon at Mycenae and Odysseus on Ithaca, "...paralleled social display and epic poetry as means of legitimating social and economic inequality...[t]he cults of epic heroes can be seen as part of that ideology: their worship confirmed social realities for those in power, who claimed them as ancestors" (Antonaccio : 64). For the Homeric leaders, long-distance maritime travel was an essential contribution to the identity of the warrior-chief: to prove one's mettle by warring and raiding in distant lands, to procure exotic material and objects whose possession conferred status, and to establish alliances with other elites.
Those who could be identified as merchants did not belong to the elite class. Homeric heroes display ambivalence toward trade for profit, which was generally left to professional merchants and, especially, Phoenicians. At one point a Phaeacian youth insults Odysseus' fitness for (elite) athletic competition by saying, "You look more like the captain of a merchant ship, plying the seas with a crew of hired hands and keeping a sharp eye on his cargo, greedy for profit" (Odyssey 8.161–64). In Hesiod's near-contemporary Works and Days, a common farmer could take up sailing for part of the year in order to sell his produce for profit (Works and Days 619–94). There is no reason to believe, however, that elites of Homer's time did not participate in trade voyages, particularly those involving gift exchanges or luxury materials and products. The apparent disdain of trade among the Homeric heroes may be a bit of aristocratic dissimulation aimed at creating social distance from professional merchants.
The sea tales in the Odyssey, the gods, monsters, and natural hazards that Odysseus encounters in his wanderings, are a potentially rich source of information about ancient attitudes toward the sea, but they are notoriously difficult to interpret. Is it possible to find in them traces of a more ancient maritime lore? One way to interpret these stories is as mariners' tall tales, plucked from hoary oral tradition and rendered increasingly fantastical with retelling over time. Exaggeration and dissociation from reality may reflect a combination of conditions inherent to both early seafaring and oral tradition. The terror of encountering storms and hostile natives in uncharted waters would provide a source of vivid recollections and dire cautionary tales. Exaggeration might serve several purposes: the enhancement of a good story; the use of hyperbole to impress upon the listener a visceral sense of danger, whether in a didactic or performative context; or as a way to maintain the secrecy of routes and places by distorting their locations and inflating the hazards associated with them. All manner of anecdotes could be believable given a hazy knowledge of some part of the world. In the Odyssey, once the action moves from the Aegean and eastern Mediterranean westward into the central Mediterranean, places and characters shift from the real and identifiable (e.g., Pylos, Sparta, the Nile River) to an unreal world of monsters, goddesses, and the realm of the dead. The commission of the Homeric epics to writing occurred just as the colonizing movement west to Sicily and Italy was underway; the origins of some of the sea tales in the Odyssey might be traced to the imperfect knowledge about these new worlds.
It is also plausible that these stories contain elements of phenomenological itineraries full of periplus-like sailing directions, mnemonic devices for recalling details of long voyages, and cautionary tales told in metaphorical and possibly encoded terms. If we imagine the Greeks trying to come to grips with new lands and essentially uncharted seas in the eighth and seventh centuries BC, several fragments of sailing instructions can be recognized in the Odyssey: (1) segmented journeys from headland to headland and island to island; (2) storms and gales with realistic effects and reactions by Odysseus' crews; (3) coastal hazards including narrow passages and submerged rocks and coastal shelves; (4) rough seas with currents, waves, whirlpools, and eddies; and (5) potentially hostile natives. Naming and animating these people, places, and forces of nature in the form of narrative stories served practical purposes. They were incorporated into a vital body of maritime knowledge as mnemonic devices to aid in the recognition of a route and the hazards that lay along it. Naming and describing transforms unknown to known, and begins the process of domestication of an unfamiliar part of the world.
It is not particularly useful in the present context to try to chart Odysseus' route or to identify the places where he encountered gods and monsters, as so many "Homeric geographers" have done (e.g., Bérard –9; Bradford ; Severin ; Stanford ). To a bard's audience, any explicit association between the fantastical places and characters inhabiting locations beyond the familiar geographical realm and real-world routes and inhabitants would be meaningless detail. The epics are not geographical treatises, after all, and although the audience would expect some adherence to what they knew of the world and the Trojan War story, much of the entertainment value of the performance was in being lifted out of the here and now to journey to times and places literally out of this (i.e., the audience's) world.
Thus, instead of trying to match epic elements to the real world, it makes sense to think of Odysseus' encounters as representative of universal experiences at sea. The proximal impetus for some of the stories might have been the colonizing effort in the central Mediterranean and northern Africa, but the pattern of their construction may be much more ancient. To cite a few examples, the "Clashing Rocks" that appear in the Argonautica (the myth of the voyage of Jason and the Argonauts) are often identified with the mouth of the Bosporus, the entrance into the Black Sea (Apollonius Rhodius, Argonautica 2.555–606). These two floating masses of rock crashed together whenever a ship tried to pass between them, demolishing the ship and its crew. The origin of this story may lie in the difficulty of passing from the Dardanelles through the Bosporus into the Black Sea, because of the combination of prevailing northerly winds and the strong current of the Black Sea outflow. Ships could only make headway by waiting for a strong southerly/westerly wind to propel them against the current. The danger of being driven against the rocks if the wind failed may have prompted the story of the clashing rocks, but it is easy to see how such an apparition could be inspired by any turbulent narrows. A distinct but similar obstacle, the "Wandering Rocks," was encountered both by Odysseus (Odyssey 12.61–72) and Jason (Argonautica 4.785–88). Similarly, Charybdis, the personification of a treacherous whirlpool, was encountered by these two heroes (Odyssey 12.235–44; Argonautica 4.920–60). It may be significant that once the heroes passed safely, these dangers were effectively neutralized: once Jason had passed between the Clashing Rocks, they became fixed in place (Argonautica 2.606–607); and the Sirens were said to have committed suicide after they failed to entice Odysseus to shore. These resolutions may be metaphors for successfully incorporating these hazards, and the means to overcome them, into the knowledge system of the maritime community.
Into the monstrous appearance and practices of the creatures with whom Odysseus and his men came into contact can be read assorted fears about hostile indigenous people. Again and again, they met inhabitants living in incomprehensibly un-Greek ways, from the Cyclopes who lived in isolation from one another and failed to take any account of the rich agropastoral and maritime potential of their offshore island (Odyssey 9.116–39), to the horrific cannibalism of Scylla, the Cyclopes, and the Lystraegonians. Cannibalism is a virtually universal taboo, which throughout history has been falsely projected by colonizers onto indigenous people, out of ignorance, fear, or a need to regard them as less than human (Biber ; Clemmer ; Obeyesekere ). False or exaggerated representations of the "other" in seafaring lore served to underscore the caution necessary when making incipient contacts with people in the absence of a common language or customs. These concerns were often justified.
Could some of these elements be much older than the period of westward colonization? Richard Martin argues that the Odyssey tells of the end of a heroic tradition; the poem and its characters speak to a vanishing past from a diminished present, and for Iron Age bards performing the tale this heroic age was that much more remote (Martin : 240; Murnaghan : 140–41). In the Odyssey, the Trojan War heroes are now dead or returned home to relatively uneventful lives with no comparably heroic successors in sight. Among the lost traditions was adventure on the boundless sea, and hard men like Odysseus who embraced both its peril and its magic. The death of the old maritime tradition is signaled when the Phaeacian ship returning home from conveying Odysseus to Ithaca is turned into stone by an angry Poseidon (Odyssey 13.125–83). With this act, the maritime life of these consummate seafarers is "fossilized" and passes into the past.
We may bring this discussion back to the Mycenaeans by noting the strong evidence that Mycenaean ships had already arrived in the southern Italy and Sicily at the beginning of the Mycenaean period, by LH I if not a little earlier. They, not the Minoans or the colonizers of Homer's day, were the Aegean pioneers in the region. How did they assemble the information they needed to make the voyage safely, and how did they structure knowledge to instruct and transmit it across generations? I submit that the process must not have changed much over the centuries, and that fragments of maritime lore embedded in the Odyssey were probably drawn from much older traditions of building phenomenological itineraries and structuring narratives of exploration. Obviously, in any long-lived oral tradition there is a process of selection and updating of material that prevents us from retrojecting stories directly and literally back to Mycenaean times. Nevertheless, the Odyssey and other early Greek literature, when examined alongside archaeological and ethnographic evidence, can contribute to a better understanding of the cognitive and practical process of building and maintaining maritime knowledge over periods of centuries.
### Conclusions
The environmental conditions of seafaring in the Aegean area are complex outcomes of earth processes in nested scales from global to local. These interactions create patterns of climate, weather, currents, and winds, which are in turn influenced by the Aegean's unique geographical position and topographic configuration, with its profusion of islands and deeply indented coastlines. These parameters undoubtedly shaped the development of maritime life in the Bronze Age, but social factors were also crucial in the formation and maintenance of maritime connections, sometimes in the face of environmental obstacles.
Because of the relatively small scale of the Aegean Sea and the dense packing of islands within it, navigation involved coastal pilotage to a much greater extent than open-sea navigational techniques. Two distinct levels of seafaring expertise existed: a general level sufficient for local fishing and short-range travel to nearby coasts and islands; and a specialized, sophisticated knowledge of environmental conditions, navigational techniques, routes, and distant places for medium- and long-range travel. A landlocked farmer like Hesiod with a basic knowledge of seamanship could make short sea crossings to trade his wares. The professional seaman engaged in longer voyages, on the other hand, drew upon a complex store of maritime knowledge. Such knowledge was systematized within a discrete maritime community as a body of lore, the transmission of which involved hands-on training, formal sailing instructions, and vivid, metaphorical narratives designed as mnemonic devices and in-group esoterica. Because this information was communicated orally, it would not be expected to survive in the material record, but the early Homeric literature may preserve traces of this lore, and ethnographies of modern low-technology seafarers offer enlightening insights into the structure of maritime communities, the content of their knowledge, and the mechanisms of its transmission. The combined evidence of texts, iconography, and archaeological materials shows that maritime communities and their specialized expertise survived relatively intact through stable and unstable periods.
It is now time to address a problem outlined in Chapter 1: Where are the Aegean harbors and landfalls of the LBA, and why do we know so little about them?
## Five Coasts and Harbors of the Bronze Age Aegean
Characteristics, Discovery, and Reconstruction
For all the indications that sea travel was fundamental to the Mycenaean political economy, we have little evidence for their coastal anchorages, let alone developed harbors or port towns, in the Aegean. We must ask why this is; what, if anything, we can do to recover and investigate them; and why we should want to do so in the first place. These questions form the focus of this chapter.
To tackle the last question first, the importance of identifying specific landing sites instead of merely characterizing the types that would have existed on Bronze Age Aegean coasts (e.g., Morton : 6–7) must be established. Many anchorages will have been small, susceptible to alteration over time, and not necessarily accompanied by a settlement or a conspicuous artifact scatter. Is it worth the time and effort to search for them? Surely it is if we are serious about understanding the networks of interactions by land and sea that amounted to the connectivity of daily life, particularly if, following Horden and Purcell (2000: 123), our aim is to reveal "...the various ways in which microregions cohere, both internally and also one with another – in aggregates that may range in size from small clusters to something approaching the entire Mediterranean." Only by knowing the locations of the coastal nodes in these networks can we fully make use (and sense) of archaeological data bearing on interaction at all scales.
Dedicated harbors as well as anchorages used only intermittently or opportunistically must have been abundant in many regions during Mycenaean times, both because of the morphology of the Greek coastline and because they were needed. As communication by sea expanded, a multitude of anchorages, large and small, was required to ensure that voyaging was as safe as possible, and to facilitate economic and social relationships. Ship captains needed access to safe anchorages to shelter from winds and storms, to procure provisions, and to enter into various kinds of transactions, including trading and raiding. Running out to sea to escape a storm was a maneuver of last resort; to seek a coastal haven was much preferred. At times, ships and their crews were forced to wait for favorable winds, and this could occupy days or weeks during which their needs for sustenance would have to be met. The potential hostility of the local population was a complicating factor that must often have forced crews to seek alternative landing sites. Another function of small, scattered anchorages might have been as convenient pickup points for agropastoral products to be transported by sea in local and regional trade networks (Rothaus et al. : 40).
Too often, however, the existence of safe, suitable anchorage is taken to be self-evident or based on guesswork that is quite possibly wrong. Often, it seems, the problem of long-term coastal change is simply left unexamined. A common mistake is to assume, implicitly or explicitly, that Bronze Age and modern coastal morphology are essentially the same. Even when some attempt is made to infer changes based on observations of modern coastal landscapes, in the absence of geomorphological analysis the conclusions reached can be misleading or wrong, and are at best limited in their ability to lead to a genuine understanding of ancient coastal environments. The actual changes to Aegean coastlines over time vary widely, but they tend to have localized causes that can only be charted with geoarchaeological techniques. In this chapter, I describe the process of working back to Bronze Age coastlines, and stress the importance of doing so in a systematic way.
### Conditions of Discovery
There are few places on the modern coastline to search for Bronze Age anchorages and harbors that can truly be called "obvious." Locations that boast fine harbor basins today, or that preserve harbor works from historical periods of the past, are no sure indicators of high potential for harbors of the Bronze Age. One reason for this is changes in maritime practice – mainly ship technology – over time. It is undoubtedly true that small, shallow anchorages perfectly suitable for Mycenaean shipping could not accommodate the large, heavily laden military and commercial fleets of later Greek and Roman antiquity. Thus, to limit the search to prominent historical and modern harbors would be to miss most of the Mycenaean maritime landscape.
A more important reason is pervasive change over time in the physical topography of the coastline, caused by short- and long-term geomorphological processes. The most prominent of these are sea-level change, sedimentation, marine erosion, and tectonics. Eustatic sea-level change, caused by the cycling of ocean waters into and out of the polar ice caps, has had a profound effect on the world's coastlines since the end of the last Ice Age. Following the last glacial maximum circa 18,000 years ago, at which time eustatic sea level lay at 90 to 150 meters below present sea level, global warming induced rapid sea-level rise, interrupted only by the cool Younger Dryas event of 10,000 to 11,000 years ago, until circa 6000 BP (van Andel ; Wells : 151). In the Mediterranean, maximum sea transgression into coastal areas was reached circa 6000 BP, followed by a stabilization of eustatic sea level. Since that time, global eustatic sea level has slowly risen by no more than five meters or so. Since the LBA, eustatic sea-level rise is only a few meters, a figure that can account neither for observed vertical changes in relative sea level caused by tectonic uplift and subsidence, nor the lateral and vertical changes effected by coastal sedimentation and erosion.
Moreover, the discovery potential for Mycenaean harbors is not the same for all parts of Greece and the Aegean, because regional-scale geomorphological histories may make detection of ancient harbors relatively easier or more difficult. As a result of regional tectonics, some coastlines abound in natural coves and inlets, while others can be characterized as virtually harborless coasts. Faulting of the Greek land mass into mountains and intervening valleys in a general northwest–southeast alignment, followed by inundation of the coasts by the sea, has produced the characteristic pattern on Greece's Aegean (east-facing) shores of promontories separated by deep gulfs, with numerous smaller inlets on a generally rocky coast (Morton : 15–16). The Aegean coast of Asia Minor developed deep inlets reaching tens of kilometers inland at places like Miletos, Ephesus, and Troy. These great estuary and delta systems hosted superb Bronze Age harbors, but have silted in completely in historical times, leaving once-great harbors stranded many kilometers inland.
There are other distinctive patterns. Some coastlines are formed by major faults that separate a mountain ridge from an adjacent collapsed and submerged block; these coasts are steep and linear, with few inlets or offshore islands, and thus little safe anchorage (Morton : 137). Examples of such "harborless" coasts occur in Epirus and Thessaly (the two flanks of the northern Greek mainland), and on the long northeastern coast of Euboea. They tend to be cliffbound and very deep – difficult places to find holding ground for an anchor or to come ashore for provisions. In a few places, perennial rivers have broken through to the coast, forming broad estuaries and deltas. In Epirus north of the Ambracian Gulf, marine embayments or estuaries at the mouths of the Acheron and Kalamas (Thyamis) Rivers provided safe haven and access to fertile hinterlands. If it can be established that the separation of the two blocks forming such coastlines occurred before the Bronze Age, the candidates for harbor locations are few and the search is simplified.
By contrast, the long, west-facing coastline of Elis in the western Peloponnese has witnessed several sequences of accretional barrier and lagoon formation since the end of the Mesolithic period, with the result that most evidence of Bronze Age coastal landforms and archaeological sites is now buried under meters of sand and wetland deposits. There, investigation of Bronze Age coastal environments requires long-term programs of geological coring, and results remain tentative even after years of study (Kraft et al. ).
The lack of knowledge about Aegean Bronze Age coastlines is also partly a symptom of the general historical trajectory of maritime archaeology in the Mediterranean, which in the twentieth century was dominated by studies of ships and shipwrecks, at the expense of harbors and coastal landscapes (Marriner and Morhange : 137–44). The emphasis on ships continues today in both the Old and New Worlds (e.g., Bass 2005a; Blue et al. ; Gould ). The shipwrecks at Uluburun, Gelidonya, and Point Iria have generated extraordinary information about LBA seafaring and maritime trade, but as restricted spatial and temporal events, they constitute a small part of a much larger picture (Marriner and Morhange : 180). When harbors were the object of study, they tended to be the largest and best-known artificial harbors of the Greco-Roman world. Prehistoric and small, natural anchorages and harbors were rarely investigated, nor were there frequent efforts to reconstruct entire "maritime cultural landscapes" formed by networks of landing sites, coastal settlements, and their ties with other communities by land and sea (Westerdahl ). In this respect, Mediterranean maritime archaeology is less advanced than in northwestern Europe or the Baltic area (e.g., Chapman and Chapman ; Cunliffe ; Ilves ). Moreover, until recently, maritime archaeology in Greek waters lagged behind other areas in the Mediterranean, such as the Levantine and Turkish Aegean coasts. This can be attributed in part to highly restrictive controls placed on underwater archaeology by the Greek underwater archaeological authority (known as Enalion), particularly with respect to foreign teams. As a result, most of the major projects and innovations in underwater technique were developed on the coasts of countries like Israel and Turkey, where permitting was more liberal (Bass 2005c; Raban ). In a policy shift that occurred after 2000, Enalion has invited collaborations with foreign teams, and many are now underway alongside an expanded agenda for Enalion's own projects.
It must be pointed out that a long tradition of coastal geomorphology has existed in the Aegean, targeting coastal change over time in both local and regional settings, for example Franchthi Cave (van Andel and Sutton ), Asine (Zangger 1994b), Tiryns and the Argive Plain (Niemi and Finke ; Zangger , 1994a), Pylos (Kraft et al. ), Dimini (Zangger ), Troy (Kraft, Kayan et al. ; Kraft, Rapp et al. ; Rapp and Gifford ), Ephesus (Kraft et al. ), Miletos (Brückner ), and the deltas of the Alpheios (Kraft et al. ) and Acheron (Besonen et al. ) Rivers. In Greece, this work could generally be accomplished under more easily acquired permits granted by the geological service (IGME). Yet the specific areas in need of development in the Aegean are two: first, there is a lack of systematic programs of research that identify and investigate local and regional maritime cultural landscapes in a holistic way; and second, programs of coastal reconstruction have often not been closely coordinated with terrestrial surveys and excavations, in terms of research design and planning, shared fieldwork and resources, and joint publication. For example, although all the programs of coastal paleogeography cited above sought to address specific archaeological questions, most were not integrated closely with a particular excavation or survey on land. Meanwhile, excavations and regional archaeological surveys on islands and in coastal areas have rarely extended their focus beyond the shoreline, with the result that the potential of integrative concepts, such as the maritime cultural landscape, which situate coastal regions within interactive networks that unite inland, coast, and sea, has scarcely been tapped (Berg ). Some vestiges of the former restrictions remain as mandated by national legislation covering all archaeological activity, including a limited number of available permits for foreign researchers and a clear distinction between terrestrial and underwater permits (these come under separate authorities, and a project can rarely expect to hold both land and underwater permits in a single year). Yet there are many ways to make collaborative research work within the present regulations: much paleocoastal fieldwork can still be accomplished on land under geological (IGME) permits, and it is possible to bring together mixed Greek and foreign nationals to work in a given locality under separate projects and permits. Nevertheless, moving toward a truly holistic coastal archaeology of maritime cultural landscapes will require fully integrated programs of archaeological and geomorphological investigation of sea and land in coastal regions.
### Geomorphology of Coastal Zones
Coastlines are among the Earth's most dynamic geomorphological settings. George Rapp and John Kraft (1994: 71–72) list the characteristic and interrelated processes: "local tectonism; eustatic sea level change; climatic change; ocean currents and wave regimes; the nature and frequency of catastrophic events; sources, types, and quantities of sediments available and the resultant aggradation and progradation of deltaic floodplains into erosional and tectonically derived embayments; and the nature and intensity of human activity." Over time, even the most stable coastal environments evolve as sediments accumulate or shorelines erode. Local relative sea level moves up or down in response to changes in the absolute level of the land (tectonics, isostasy), sea (eustasy), or both. Climate and topography control the flow of energy (tides, waves, currents, sediment flux) in a coastal system. The tectonic environment (faults, tectonic events) impacts topography and relative sea level, and sometimes introduces changes to coastal topography that are catastrophic from a human perspective. Finally, human impacts on coastal environments, notably in the form of increased sedimentation, have been detected since the Neolithic period in the Mediterranean. The following discussion summarizes briefly the most important processes and materials involved in long-term coastal evolution, following the thorough treatments of the topic by Lisa Wells () and Michael Summerfield (: 313–42).
Broadly speaking, a first-order distinction can be made between open coasts and protected coastlines (Wells : 150). Many coastlines cannot be characterized as completely open or completely sheltered, but for a coastal location to have been viable as a harbor, it must have been sheltered to at least a considerable degree from winds and waves. Often, because of shifts in wind direction and intensity during the course of the year, the viability of a harbor may respond to seasonal or even day-to-day conditions.
Open coasts are exposed to the full impact of winds and wind-driven waves, creating high-energy environments and landforms dominated by beaches or rocky shorelines. Coastal erosion is common on open coasts, and sediments tend to be relatively coarse with abundant organic material including woody debris or carbonate particles. By contrast, on protected coastlines, wave energy is attenuated by refraction, creating low-energy environments exemplified by estuaries and tidal marshes in which sedimentation dominates coastal evolution and landform development. Sediments are relatively fine grained and may contain highly organic deposits such as peat. Protected coastlines are ideal locations for sheltered harbors.
### Long-Term Coastal Evolution
Coastal evolution can be characterized in terms of the relative motion of the shoreline over time in response to the processes outlined by Rapp and Kraft above, and by the resulting landforms, which can appear and disappear with successive periods of coastal change. Joseph Curray (; see also Wells : 154–55) proposed a classification of coasts based on their relative motion over a discrete period of time: (1) progradational coasts grow seaward (prograde) as a result of sedimentation; (2) transgressive coasts are submerged as a result of relative sea-level rise; (3) recessive coasts erode landward (marine erosion); (4) regressive coasts emerge as relative sea level falls; and (5) aggradational coasts grow vertically (aggrade) when the rates of sea-level rise and sedimentation are roughly equal (Fig. 5.1). The early Holocene record of the Mediterranean, outlined above, provides a good example of rapid and widespread transgression as global sea levels rose. The maximum transgression is often marked by wave-cut notches in a former sea cliff, or other signs of a paleoshoreline stranded far inland from the modern shoreline. After eustatic levels stabilized circa 6000 BP, these transgressive coasts shifted to progradational, recessive, or aggradational, depending on the rate and dynamics of sedimentation and erosion in a given location. The Aegean coasts of Turkey provide dramatic examples of deep embayments created by the late Pleistocene to early Holocene marine transgression, followed after circa 6000 BP by a gradual but inexorable progradation of tens of kilometers in the major river deltas. The relationship of sea and land created by the interplay of erosion and sedimentation can be altered locally by tectonic movements.
5.1 Classification of coasts by relative motion of the shoreline. After Wells : 154, fig. 6.2. Courtesy of University of Utah Press.
#### Sediment Supply to Coastal Zones
There are three main sources of sediment supply to the coastal zone: (1) the coastal landforms themselves, notably sea cliffs; (2) the land inland from the coast; and (3) material transported from offshore (Summerfield : 324–25). The volume and nature of the sediments are controlled by geology, tectonics, climate, oceanography, and the topography of the sediment source area (Wells : 155). On open coastlines, high wave energy can erode coastal landforms, causing recession and liberating sediment to be transported and reworked. The amount of sediment thus produced depends on wave energy and the degree of consolidation of the cliffs or other landforms suffering erosion.
By far the greatest sediment load, however, is brought to the coast from inland by rivers. On average, mid-latitude coastlines receive more than 90% of their sediment from fluvial sources. Where drainage systems are long and broad (e.g., the Nile, Tigris, Euphrates, Maeander), the material transported to the shore will be predominantly fine grained as coarser clasts are deposited well upstream. By contrast, short, steep drainages carry coarser materials to the coast. In Greece, there are relatively few large, perennial rivers with broad drainage basins (to name two, the Peneios and the Spercheios). Most rivers have a seasonal flow that is heaviest in the winter months as a result of rains that trigger high-energy, swift-flowing torrents capable of transporting much coarse material. As a result, a typical stony beach will consist of a mix of coarser and finer material.
Free sediment is transported in the interface of coast and inshore waters in several ways (Summerfield : 325; Wells : 155–57). Tides and wave action move sediment continuously and more or less perpendicularly into and away from shore. Wind-driven waves can be significant agents of erosion and sediment transport on open coastlines, but because the Mediterranean is a virtually tideless sea, the effect of tides on coastal processes is minimal. Where offshore winds are strong and persistent, substantial amounts of airborne sand-sized and smaller grains can be deposited onshore (aeolian transport). Yet the dominant movement of sediments along Mediterranean coasts is due to oceanic currents (Fig. 5.2). Generally, the fine clay- to silt-sized sediments escape offshore in suspension and are carried away by oceanic currents. The coarser (sand-sized and greater) material ordinarily remains in the littoral zone, to be entrained in longshore currents, which transport material parallel to shore through a process called longshore drift (also known as littoral drift). The most important contributor to this material is alluvial sediment, but sediments from the erosion of coastal landforms as well as material previously entrained in ocean currents can also be present. The rate and volume of longshore transport along a coastline are controlled by current velocity and wave energy, as well as the angle at which waves strike the coast. The impact of longshore drift on coastal configuration depends on the presence or absence of features of coastal topography and inshore bathymetry that act to impede or facilitate sediment movement. Sediment remains in transit until a sediment sink drains it offshore or an impediment causes deposits to form against it and build up over time. Thus, on linear coasts, longshore drift may proceed with little interruption or effect on coastal configuration. Where coasts have irregular configurations, however, as is so often the case in Greece, longshore sediments become trapped against headlands, offshore islands, delta formations, and other prominent features (Fig. 5.2). These deposits contribute to the development of landforms such as barrier islands, spits, and sandbars, behind which lagoons form. The elongated form of these deposits reflects the prevailing direction and strength of the longshore currents. The long and complex history of longshore-derived landforms of coastal Elis shows these interactions quite clearly (Kraft et al. ). Coastal locations downcurrent from significant sediment traps are characteristically starved of sediment, and these features will be absent. Littoral sediments can also be drained offshore by deep-sea canyons.
The overall contribution of sedimentation to coastline configuration can be measured in terms of a sediment budget: if the rate of sediment delivery exceeds the capacity for sediment transport away from the coastal zone, accretion will occur; if not, sediment will remain in transit until it reaches a sediment trap that captures it or conducts it out to sea. It is important to note that even at the local scale, erosion and sedimentation can co-occur. In the region of the lower Acheron River valley of southwestern Epirus, the tectonic structure of the linear Ionian coastline promotes coastal erosion and efficient longshore transport, while within the once-broad embayment at the mouth of the Acheron River, a sharply progradational sequence since the Neolithic period has been documented (Besonen et al. ).
5.2 Movement of sediments along Mediterranean coasts. After Wells : 156, fig. 6.3. Courtesy of University of Utah Press.
### Coastal Landforms
The interaction of land, sea, and sky generates a wide array of coastal landforms, which in general terms can be categorized as destructional or constructional (Summerfield : 325–41). The destructional group is smaller, comprising mainly wave-cut cliffs and shore platforms. The constructional landforms are many and varied, including beaches, barrier islands and lagoons, estuaries and tidal features, deltas, coastal dunes, and reefs.
#### Coastal Cliffs and Shore Platforms
Coastal cliffs are steep, often vertical, slopes that rise above the shoreline. Fault-derived cliffs may plunge precipitously into deep water, suffering little erosion since wave energy against them is minimized by reflection of swell waves. In shallower waters, cliffs are susceptible to basal erosion by the action of breaking waves. Particularly on open coasts, wave action undercuts sea cliffs, cutting a notch around mean sea level. Over time, an overhang forms that becomes increasing unstable and eventually collapses into the sea. This material, along with other gravity-entrained colluvium from the slope, is available for reworking and transport in the littoral zone. This erosional process forms recessive shorelines: as the coastal cliffs retreat, horizontal shore platforms are left behind at the basal level where wave cutting occurred. These wave-cut or intertidal platforms are common in Greece where high wave energies combine with easily eroded limestone strata. Once formed, they are subject to weathering and abrasion. The rate of cliff erosion varies over time: in particularly stormy years, wave energy and weather can accelerate coastal erosion. If the process continues, however, eventually the cliff will be protected because wave energy is expended crossing the platform.
The sequence of shore platforms and sea cliffs may leave physical traces on land or underwater, particularly where coasts are uplifted or submerged, respectively. The Corinthia in southern Greece preserves examples of both (Hayward : 16–17; Pirazzoli et al. ; Wells 2001: 173–75, fig. 6.7). The northern Corinthian plain, bordering the Gulf of Corinth, comprises a stair-stepping sequence of risers and treads that represent Pleistocene coastal cliffs and shore platforms, subsequently uplifted and subjected to faulting and erosion. Beginning in the Neolithic period, these uplifted features were attractive locations for repeated human habitation because they afforded expansive views and defensive possibilities (Tartaron et al. : 496). To the east, the Corinthian port at Kenchreai on the Saronic Gulf is partially submerged as a result of approximately 2.3 meters of tectonic subsidence over the last two millennia. As many as five submerged notches, cut by wave action, dissolution, and bioerosion, have been documented in and around Kenchreai, each representing a paleoshoreline formed during a period of relative sea-level stability (Noller et al. ).
#### Beaches
A beach is a shore built of unconsolidated sediment (Hamblin and Christiansen : 398). Beach sediments form at the dynamic interface of land and sea, where the shoreline is constantly exposed to wave action and littoral currents. Most beaches consist of sand- or pebble-sized clasts (thus, sand or pebble beaches) because wave energy is usually sufficient to remove silt and clay from the littoral zone. Waves and currents sort beach sediments both vertically and laterally, and round and polish the clasts. These well-known processes produce a predictable set of sedimentary structures that allow the identification of fossil beach deposits (Wells : 158, table 6.1).
Beaches are composed of both organic and inorganic material, but in Greece, carbonate pebble beaches are especially common because of the pervasive carbonate geology (limestone and other biogenic carbonates). Beachrock, intertidal sediment indurated by calcite or aragonite cement, forms readily because of the abundance of carbonate in solution in near-shore waters (Wells : 158). Submerged or uplifted beachrock can indicate the approximate lateral positions of former shorelines, particularly in the nearly tideless Mediterranean.
The progradation of sediment seaward is sometimes manifest as sets of sandy, shore-parallel beach ridges and intervening swales. William Tanner () identifies four types, based on their origin. The most common are swash-built ridges, which occur where there is a constant source of sandy sediments, but longshore currents are insufficient to remove them. The sediment is instead reworked by modest storm waves (swash) into narrow mounds with low relief along the shoreline. The lower-lying surface between the new ridge and the adjacent landward ridge floods seasonally, filling with sandy near-shore sediment (e.g., backswamp or marshy deposits), thus forming the ridge-swale set. This is a process of gradual accretion, with each ridge representing an old beach and the ridge system marking the advance of the shoreline seaward over time. Formation can be relatively rapid, however: Tanner (: 159–60) suggests an average interval of around 50 years for sandy swash-built ridges on ocean coasts. A striking example can be found in Greece at the mouth of the Acheron River in Epirus (Besonen et al. : 216), where concentric accretionary beach ridges and swales have formed in historical times in an embayment well sheltered from longshore currents (Fig. 5.3).
Beach deposits can also grow outward to join with an offshore island by a narrow neck known as a tombolo. This typically occurs because the offshore island behaves as a natural breakwater, creating a wave shadow along the coast behind it, where sediment begins to build outward from the shore. Subsequently, longshore drift moves sediments onto and around the tombolo (Hamblin and Christiansen : 399; Fig. 5.4). Tombolos can enhance the natural qualities of an anchorage by creating a double-harbor configuration, and in historical times they were emulated by built causeways. The island of Mochlos off Crete's northern coast is believed to have been connected to the mainland by a tombolo in the Bronze Age (Branigan 1991: 97), and the same may have been true of the southern Cretan harbor at Kommos (Shaw : 55).
5.3 View of the sequence of ridges and swales at the mouth of the Acheron River, Epirus. Courtesy of Nikopolis Project archives.
#### Estuaries and Tidal Landforms
An estuary is a partially enclosed coastal body of water that communicates with the open sea, but is protected from the action of open ocean waves by its topography or some form of barrier that absorbs and refracts wave energy (e.g., sand bars or barrier islands). Usually the term is applied to inlets into which rivers or streams flow, causing fresh and marine waters to intermingle (Fig. 5.5). These transitional zones between river and ocean environments experience both marine processes (tides, waves, saline water influx) and fluvial processes (freshwater influx, sedimentation), contributing diverse nutrients that make estuaries among the world's most productive natural habitats.
5.4 View of tombolo, Paximadi Cape, Euboea. Courtesy of Tim Bekaert.
5.5 View of an estuary in South Carolina, USA. Courtesy of the National Oceanic and Atmospheric Administration.
Estuaries trap fine-grained sediments, leading to the formation of a variety of landforms such as tidal mudflats, tidal marshes and swamps, and tidal inlets. In the Aegean, where tidal currents are weak, mud and clay are deposited on shallow mudflats in the lower intertidal zone. In the upper (landward) intertidal zone, continued deposition of mud can cause vertical accretion and eventually the formation of a brackish marsh above the normal high tide level. These different landforms exhibit characteristic sedimentary structures and contents (Wells : 158–59, 162). Mudflat sediments typically consist of a mix of silt and clay with a high iron content and in situ molluscan fauna. Tidal marshes are rich in peat, comprising very fine-grained clay and silt with a high organic content. In tectonically stable contexts, marshes build up and out, gradually infilling the estuarial basin.
Estuaries and their associated landforms are highly significant to the study of ancient coastlines. Because of their micro- and macrofossil content, as well as datable organic material, they provide key evidence for coastal evolution. Equally important is the fact that estuaries were a far more prominent feature of coastal landscapes in the Bronze Age than they are today, offering quiet environments ideal for anchorage; these would have been desirable harbor locations throughout the Mediterranean (Raban ). Great estuaries at major river mouths are obvious targets for Bronze Age harbors, but because of the tendency of estuaries to fill in over time, smaller estuarial harbors might only be revealed by systematic geomorphological prospection.
#### Barrier Islands and Lagoons
Lagoons are tidal inlets of shallow marine or brackish water that are separated from the sea by a barrier. Mediterranean lagoons are typically sheltered by barrier islands, which are elongated offshore sand ridges extending parallel to the shoreline and separated from it by the lagoon (Summerfield : 330). Barrier islands range from a few meters in width and a few hundred meters in length to long islands a few kilometers in width and hundreds of kilometers in length. Lagoon waters are replenished by tidal inlets that perforate the barrier at irregular and migrating intervals. In response to changes in sediment supply, relative sea level, and climate, barriers may form, retreat landward, or disappear.
Lagoons fill with coarser sediment predominantly supplied by longshore drift. Because lagoons are shallow, they are susceptible to infilling and are strongly influenced by precipitation and evaporation. When evaporation rates are high, salinity in a lagoon can exceed that of the sea, and stranded lagoons can form salt pans or salt lakes. The Akrotiri Salt Lake on the southern tip of Cyprus was probably part of a lagoonal system in the Bronze Age that was later isolated from the sea (Blue 1997: 35–38). Salt deposits were surely exploited in the Bronze Age for subsistence and trade.
The western coast of the Peloponnese is a prominent example of lagoon and barrier formation. On the coast of Elis, coastal landform change has been dominated for the last 8,000 years by longshore redistribution of sediments from the Alpheios and Peneus rivers into an extensive Holocene barrier accretion and lagoonal system (Kraft et al. : 4; Fig. 5.6). In these settings, both sedimentation from fluvial inputs and erosion and littoral transport from high wave energy have shaped Elean and Messenian shorelines into long, sandy beach strands (hence, Homer's "sandy Pylos"). Hans-Jeorg Streif (1964, cited in Kraft et al. : 5) identifies three major alluvial terraces in the Alpheios River system and correlates them with three episodes of coastal barrier accretion at the delta. These are roughly dated to EBA, LBA, and Classical to modern (in several subphases).
#### Deltas
Deltas form where rivers deposit sediment directly into a standing body of water, such as a lake or the ocean, more rapidly than it can be redistributed by coastal processes. The sediment load deposited by deltas into seas and estuaries creates progradational complexes of river sediments reworked by littoral or estuarine processes. There are two essential components of a river delta: (1) the delta front, comprising the shoreline and the gently sloping, submerged offshore zone; and (2) the delta plain, an extensive lowland area landward of the delta front, made up of active and abandoned distributary channels fanning out over the plain (Summerfield : 333–36). The terrain between the channels is occupied by floodplains, levees, tidal flats, marshes, and lakes. All of these features can be recovered in a geological core, and organic material is usually available to establish a chronometric framework for the sequence.
5.6 Example of a lagoon and barrier system on coastal Elis. Kraft et al. : 14, fig. 5. Courtesy of the Trustees of the American School of Classical Studies at Athens.
The structure and associated landforms of a delta depend on the interaction of the sediment-carrying stream with ocean currents and waves. Because in Mediterranean embayments, fetch – the distance wind can travel unimpeded by landforms – is limited and tidal effects are minimal, deltas are subject to limited modification by coastal processes (Summerfield : 333). Thus, on a continuum of delta types dominated by tides, waves, and rivers, most Mediterranean deltas are fluvially dominated (Summerfield : 334–35, table 13.4, fig. 13.22). Tectonic and climatic factors also play an important role in delta morphology (Summerfield : table 13.5). If a delta region is tectonically stable, the delta plain will aggrade as it progrades; if subsiding, it will form overlapping sedimentary lobes as it progrades; and if rising, river distributaries will cut down into and rework previously deposited sediments. Precipitation and temperature control the type and amount of the vegetation cover. Once rooted, plants trap sediments and contribute to the formation of peat.
The evolution of Mediterranean river deltas since the mid-Holocene deceleration of marine transgression has been the work of both natural and human agents. As sea-level rise slowed, river sediments began to aggrade (build vertically) and prograde (build laterally seaward) in sheltered and shallow marine embayments. Wave energy was low, and thus erosion and littoral transport were not prominent mechanisms. Instead, the deltaic material prograded gradually toward the open sea in a landscape of multiple active and abandoned river channels, distributary levees and swamps, brackish to saline lagoons, and isolated ponds or lakes ranging from saline through freshwater (Fig. 5.7). At Troy, it was only as the delta coast approached the Dardanelles that littoral currents and increased wave action deposited sands in nearshore shoals and beaches (Kraft, Rapp et al. : 164), and the same is true of the Maeander River, across the mouth of which a long, arcuate barrier ridge has developed in recent times (Brückner ). Kraft and colleagues (Kraft, Kayan et al. : 367, fig. 4) cite the delta of the Spercheios River, emptying into the Gulf of Malia on Greece's eastern coast, as a modern analogue where similar landforms produced by the progradation of the river delta can be observed. There, a complex landscape of meandering levees and backswamps, distributional channels, and pervasive coastal marshes has formed as the delta gradually progrades into the deep, sheltered Gulf of Malia. Because of the absence of high wave energy and littoral drift at the current delta shoreline, no coastal barriers have formed.
#### Coastal Sand Dunes
Coastal sand dunes can form in a range of settings, including deltas, coastal plains, and embayments. They are aeolian landforms that develop where certain conditions exist: coastlines where waves and currents interact with abundant loose, sand-sized sediments available for transport; persistent onshore or alongshore winds blowing for at least part of the year at 16 kilometers per hour or more; and flat or low-relief terrain immediately inland of the coastline (Bauer and Davidson-Arnott ; Martínez et al. ; Pye ). Dune formation occurs when winds blow dry sand particles landward from the beach; the main sources of the sand are exposed offshore sandbars and river-mouth and other backshore sediments. Objects inland from the coast, such as plants, logs, or human constructions, interrupt the wind flow, causing sand to be deposited in drifts around them. These drifts act as barriers to sand movement, and grow over time to landforms ranging from small hillocks to vast dune systems tens of meters in elevation.
Sediment supply is the key limiting factor to dune formation. Fluvial systems are noteworthy for the large amount of sediment that they can contribute to dune formation: the Nile River has supplied sufficient sands to result in the formation of vast belts of dune ridges (El Banna and Frihy ). In Greece, coastal dunes are common, but because of the prevalence of rocky (as opposed to sandy) shorelines and mountainous coastal topography, long, continuous costal dune systems such as are found on French or Dutch coastlines are rare. Instead, coastal dunes tend to occur in isolated embayments between promontories and cliffs, or as more extensive but discontinuous coastal dune systems, for example in the western Peloponnese or on Euboea's eastern coast (Heslenfeld et al. : 338; Spanou et al. : 235–37, fig. 1). They tend to exist where barrier islands or wave-dominated depositional landforms occur, often as integral elements of barrier and lagoon systems (Kraft et al. ).
5.7 Formation of the Scamander plain, Troy. Kraft, Kayan et al. : 365, fig. 2. Courtesy of Springer Science+Business Media.
Coastal dunes perform several functions beneficial to the stability of coastlines and potentially to humans as well. They shield low-lying coastlines from violent storm winds and waves, and inhibit coastal erosion. They protect against saltwater intrusion into wetlands and dry lands behind them, and they support a diverse flora and fauna (Spanou et al. ; Sýkora et al. ). Thus, they form an integral part of coastal ecology and the resources to be found there.
#### The Anthropogenic Contribution
The human role in coastline formation has been conspicuous in two processes: the acceleration of sedimentation to prograding river deltas, and the creation of artificial harbors. It has long been noted that the shift to settlement in sedentary agropastoral communities in the Mediterranean coincided with mid-Holocene delta formation and shoreline progradation in the estuaries that had resulted from the late Pleistocene–early Holocene marine transgression. Human agency in soil loss – caused by stripping vegetation cover through forest clearing, agriculture, and grazing – has been implicated in increased sediment load to streams and the resulting expansion of deltas and related coastal landforms. But it is not easy to demonstrate a primary human role in this process, particularly for periods as remote as the Neolithic and Bronze Ages. There are three related problems: contemporaneity, causality, and degree of impact (Halstead ). The palynological, geoarchaeological, and archaeological data used to assess human impact on sediment load to coastal areas can often be dated only approximately, with the result that the contemporaneity of the impacts seen in these data sets is highly uncertain. Even when the chronological correlation is fairly secure, human causality, as opposed to a variety of climatic and other environmental changes, can be difficult to establish. Finally, the magnitude of the human impact, given the level of population and the specific activities in which communities were engaged, must be evaluated. Can it be demonstrated that Bronze Age population levels in and around coastal drainage systems were sufficiently large and their subsistence practices sufficiently destructive to account for significantly accelerated delta formation and coastline progradation?
In Greece, the most detailed documentation of putative human agency in landscape destabilization resulting in erosion and catastrophic soil loss comes from the southern Argolid. The regional surveys of the Argolid Exploration Project in the 1970s to the 1990s amassed a large body of geological and archaeological data that seemed to indicate human agency in certain episodes of massive Holocene soil erosion (Runnels , 2000; van Andel et al. , 1990; Zangger 1994a). One of these (the Pikrodafni alluvium) was dated broadly by pottery sherds to the end of EH II, and was concentrated in valleys where FN–EH II settlement was extensive (van Andel et al. : 113). The Pikrodafni alluvium is dominated by debris flows: "...chaotic beds of ill-sorted, largely angular boulders, cobbles, and pebbles, surrounded and supported by a matrix of finer material" (van Andel et al. : 111), likely the result of sheet erosion of slopes made vulnerable to soil loss when vegetation cover was devastated by drought, fire, or clearing. Although a change of climate to drier conditions that reduce vegetation may instigate sheet erosion and result in debris flows, the investigators attributed the Pikrodafni alluvium to careless slope clearance and the "...eventual failure of EH agriculture to contain the loss of soil" (van Andel et al. : 117). This conclusion has been questioned on the grounds of chronological and causal ambiguity as outlined above (e.g., Bintliff ; Butzer ; Endfield ; Moody , 2000; Whitelaw ). This debate highlights the problems and underscores the need to assess each event on its own merits before wider inferences about regional land–human relationships can be made.
In general, it is unlikely that the effect of human subsistence activities was such that large natural harbors were strongly affected in the Bronze Age, and particularly it is doubtful that humans made a greater contribution to sedimentation than the combination of climatic oscillations and natural sediment transport after the mid-Holocene stabilization of eustatic sea level. This conclusion is supported by much geomorphological research that shows that rapid infilling of major natural embayments and estuaries in the eastern Mediterranean has been a phenomenon of much more recent times (Brückner : 122–25; Kraft, Kayan et al. ; Raban ). On the other hand, small natural inlets and harbors based on barrier and lagoon systems must always have been more susceptible to both anthropogenically and naturally induced sedimentation (Kraft et al. ), migrating and going in and out of practical use with much greater frequency.
The creation of artificial harbors in the Bronze Age Aegean is unlikely, but cannot be ruled out entirely. This kind of harbor is an artificial estuary or lagoon, where breakwaters have been constructed to reduce wave energy, creating a quiet and sheltered environment in which vessels may operate. At the same time, by minimizing the normal forces of waves and littoral currents, artificial harbors promote the net accumulation of sediment. A universal feature of artificial harbors is that they must be maintained by dredging or by some means of flushing sediment from the harbor basin. All of the processes that occur in natural estuaries also occur in artificial harbors, but they may be accelerated due to intensive human presence. Because the maintenance of harbors responds to the ebb and flow of political and economic conditions, the eventual infilling and abandonment of artificial harbors are all but inevitable (Wells : 171).
Artificial harbors leave distinctive signatures in the geoarchaeological record, which allow them to be distinguished from natural harbors. Nick Marriner and Christophe Morhange (2007: 175–77) have identified a fairly consistent geomorphological sequence repeated throughout the eastern Mediterranean, which they term the "Ancient Harbour Parasequence" (AHP). The AHP comprises the depositional history of the harbor basin, from the natural pre-harbor state to postabandonment, with the following surfaces (boundaries) and deposits (facies):
(1) The Maximum Flooding Surface (MFS) marks the maximum marine transgression circa 6000 BP. It forms the lower boundary of the sediment archive, and laterally the farthest landward position of the coast. The deposit is characterized by coarse sand and pebbles.
(2) After 6000 BP, beach sands began to aggrade naturally, overlaying the MFS with little or no human contribution. Net sediment supply increased as the coastline prograded and the basin began to fill.
(3) The Harbour Foundation Surface (HFS) marks the incipient human modification of the basin, in the form of built harborworks, to create a sheltered harbor basin. A sharp transition from coarse beach sands to fine-grained silts and clays characterizes the sedimentology of the harbor basin. In most of the Mediterranean, this surface postdates the Bronze Age. Human exploitation of natural low-energy basins in the Bronze Age is rarely measurable on the basis of granulometry, but can sometimes be detected in subtle patterns of molluscan micro- and macrofossil assemblages (Marriner and Morhange : 176).
(4) The Ancient Harbour Facies (AHF) refers to the stratigraphic sequence of deposits during active use of an artificial harbor. Enhanced harbor engineering through time is evident in increasingly fine deposits (silts and plastic clays) through the Roman period, and remnants of harbor architecture (moles, breakwaters, quays, etc.), artifacts, and other anthropogenic debris are often present. The AHF may generate a diagnostic assemblage of macro- and microfauna, as well as a strong geochemical signature from human pollutants.
(5) The Harbour Abandonment Surface (HAS) records the initial (semi-)abandonment of the harbor basin, often after the Late Roman period. It corresponds to the deterioration or abandonment of maintenance of harbor infrastructure, and is marked by a transition from fine-grained harbor silts and clays to coarse sands and gravels.
(6) The Harbour Abandonment Facies (HAF) registers a return to "natural" conditions after the harborworks have deteriorated to the point that the basin is exposed to higher-energy wave action and the formation of coarse-grained sand and gravel beaches.
The Ancient Harbour Parasequence framework has been applied by this investigative group to ancient harbors including Beirut (Marriner et al. ), Sidon and Tyre (Marriner and Morhange ), and Marseille (Morhange et al. ), and they have also fitted the geostratigraphy of other harbors, such as Caesarea Maritima (Reinhardt and Raban ), into this scheme (Marriner and Morhange : 172–74, fig. 29).
### Natural versus Artificial Harbors in the Mycenaean World
As observed above, it is widely assumed that neither artificial harbor basins nor durable built harbor infrastructure existed in the Aegean Bronze Age. Centuries later, Homer and Hesiod were barely aware of artificial harbors (Morton : 106), and the image of Odysseus' crews dragging their ships onto sandy beaches has been held up as representing standard practice in the Bronze Age. But is this really the case? Blind acceptance of the notion that all Bronze Age harbors were natural, and that ships were simply pulled up onto sandy beaches in the Homeric style (e.g., Iliad 1.485–86), has been justly criticized by Avner Raban (: 136) on the grounds that it has prevented investigators from looking for artificial harbor works or from accepting evidence for them. Nevertheless, to date there remains little evidence for built harbor infrastructure in the Mycenaean world. The Minoans, perhaps because of their earlier contact with Egyptian and Near Eastern seafaring societies, seem to have been more advanced in this respect. At a small number of Cretan coastal sites, large buildings divided into long, narrow galleries have been interpreted as ship sheds, used for storage of ships and nautical equipment during the winter months. The clearest case is the excavated Building P at Kommos (J. Shaw ; M. Shaw ; Shaw and Shaw ; Fig. 5.8); at more than 37 meters long and 5.6 meters wide, the galleries of Building P could accommodate the largest of the ships depicted in the Akrotiri Flotilla Fresco (see Table 3.1). Other candidates for ship sheds include the "Shore Building" or "Shore House" at Gournia (Fotou ; Shaw and Shaw : 852, n. 16; Watrous ), as well as unexcavated foundations and cuttings at Malia and Nirou Chani (Marinatos : 146; Raban : 139–40; Shaw : 425–28). Elsewhere, cuttings and ruins of built features have been proposed as channels, moles, and tombolos associated with Minoan harbors (Chryssoulaki 2005; Hadjidaki 2004: 54–56). These features cannot at present be dated directly; typically, they are considered to be Minoan based on close spatial, but not stratigraphic, association with remains of Minoan buildings or artifacts. Nothing comparable has been identified on the Greek mainland.
5.8 Reconstruction of ship sheds at Kommos. Shaw : 425, fig. 9, drawing by Giuliana Bianco. Courtesy of the Trustees of the American School of Classical Studies at Athens.
Even the well-dated Building P is beset by interpretive issues that are crucial for the present discussion. Building P was constructed over the ruins of LM I Building J/T during pottery phase LM IIIA2, corresponding to the later fourteenth century BC. This was a time of new beginnings in the Mesara region; at nearby Ayia Triada, a megaron-style building was constructed, similar to contemporary Mycenaean megara on the mainland. A spirited debate continues on whether the Mycenaeans exercised political control over large parts of Crete during this time. If so, Building P could be a Mycenaean construction, as it has sometimes been called (Haggis ; Yasur-Landau : 50). The excavators do not share this view, however (Shaw and Shaw ). They find no close parallel for Building P on the Mycenaean mainland, and they continue to believe that the Mycenaeans exercised neither direct political control nor extensive cultural influence over the Mesara in the Late Bronze Age. Instead, they point to iconographic evidence for the non-Mycenaean origin of the Cretan ship shed (J. Shaw : 429–32; M. Shaw ). At Akrotiri, a portion of the Flotilla Fresco on the north wall of Room 5 depicts soldiers marching to the right of a large building partitioned into narrow, open galleries facing the shore, very similar in form to Building P. The fresco at Ayia Irini on Kea depicting a seaside scene may also preserve the corner of such a building (Shaw : 430–31). The idea of an earlier Minoan tradition finds support at Gournia: on the putative ship shed there, Vance Watrous (: 13) comments, "Similar in material, masonry, and monumental scale to the palace at Gournia, the ship-shed galleries seem to have been built at roughly the same time as the palace, probably in MM IIIA." Thus, the Cretan ship sheds bring us no closer to definitive proof of Mycenaean built harbor infrastructure.
If the Mycenaeans did not erect built structures to enhance coastal topography, in one case at least they seem to have created an artificial harbor basin at a remove from the coastline. At Romanou near Pylos, according to Eberhard Zangger and associates, the Mycenaeans created an artificial harbor basin by means of a sophisticated hydraulic engineering project (Zangger et al. : 619–23; Fig. 5.9). A natural depression comprising soft fossil dunes some 500 meters from the Bronze Age shoreline was widened and deepened to serve as the harbor basin. A channel approximately 40 to 50 meters wide was dug to connect this basin to a small natural cove at the coast that had probably been the original anchorage. To prevent longshore sediments from silting up the harbor basin, the perennial Selas River was diverted upstream to provide a steady flow of clean water to flush the basin. The Selas was first diverted into a lake, and from there an outlet controlled the outflow of clean overspill water, which was conducted by an artificial canal to the harbor basin. The diversion of the Selas River has been dated to the LBA by establishing the radiocarbon chronology of a sharp drop in terrestrial sediments in cores from the Osmanaga Lagoon, the natural outlet for the river prior to the diversion (Zangger et al. : 622). The result was a sheltered and readily defensible inner harbor.
The components of this harbor virtually disappeared from the landscape once the authority that maintained it was gone; it was only recognized through a careful and expert geomorphological analysis, and to date remains the only known Mycenaean artificial harbor. Nevertheless, this approach to harbor construction is entirely in keeping with the ambitious hydraulic engineering projects at which the Mycenaeans excelled, the most prominent being the drainage and water management of the Kopais Basin in Boeotia (Knauss ), and the Kofini Dam near Tiryns (Zangger 1994a). The work of geomorphologists reminds us that Bronze Age coastal features, whether natural or artificial, are difficult to detect, yet it is possible to read them from the modern landscape if one is attuned to the traces they leave behind and adept in the techniques for recovering them. Although it is unlikely that many harbors as elaborate as the one at Romanou existed in Mycenaean times, this discovery does provide an example of the kind of harbor engineering of which the Mycenaeans were capable, and offers impetus to continued search for other engineered harbors.
5.9 Hypothetical reconstruction of an artificial harbor at Pylos. Drawing by Felice Ford after Zangger et al. : 619, fig. 46.
Apart from natural harbors that were used during the Bronze Age, a variety of coastal wetlands could be exploited for diverse uses and resources. Deltas, estuaries, and lagoons are merely part of a broader series of coastal landscapes characterized by high biomass and biodiversity, which may also include lakes, bogs, river floodplains, spring-fed wetlands, and seasonally inundated dolines and poljes (Van de Noort and O'Sullivan : 36–64). From such coastal settings in the Mediterranean, resources were readily available such as reeds and rushes for architectural construction; clay for pottery, mudbrick, and other architectural applications; and fish, waterfowl, amphibians, and mollusks. Wetlands are underappreciated as a resource for Bronze Age communities partly because in the modern world their settings are peripheral and they are rapidly disappearing through reclamation and other forms of human interference.
### Geoarchaeological Methods for Reconstructing Coastal Landscapes
The investigation of ancient coastal landscapes by geoarchaeological means is a well-established tradition with a rich literature, but one that continues to evolve with technological advances and new perspectives influenced in part by recent trends in landscape archaeology, giving rise to a "theoretically informed landscape approach" (Breen and Lane : 469–70; Marriner and Morhange : 180). The methodology of coastal geoarchaeology involves extensive work in both the field and laboratory, and a wide range of materials as proxies for past conditions and processes. The basic principles and techniques are widely published and demonstrated by case studies (e.g., Marriner and Morhange ; Rapp and Hill : 75–81; Summerfield : 313–42, 433–55; Wells ). To be truly effective, these techniques must be performed in combination, and to realize their potential to illuminate the human past, they must be integrated with archaeological investigation.
The purpose of this section is to summarize the elements of a robust geoarchaeological investigation of a coastal landscape, following the framework outlined by Marriner and Morhange (2007; see also Brückner : 125–26; Goiran and Morhange ) that broadly divides the work between field and laboratory techniques.
#### Field Techniques
Geomorphological survey involves the initial mapping of coastal landforms, typically starting from maps (geological, topographic, soils), images (satellite, aerial photographs), and archaeological information (chronological, distributional) on sites and features in the coastal zone. These documents may reveal a wealth of information, and form a baseline for the research design. High-resolution satellite images and low-altitude aerial photographs often yield visual evidence for coastal landforms, such as barrier and lagoon systems, or evidence for ancient harbors, such as uplifted harbor basins or submerged harbor installations (Fig. 5.10).
Subsequent ground truthing, both on land and under water, allows natural and anthropogenic features to be studied firsthand in order to form provisional hypotheses about the morphology, genesis and developmental sequences, and chronology of coastal landforms. In many cases, landforms may be visible, for example, dune ridges, infilled lagoons, estuaries, river and stream features, beach ridges, and fossil sea cliffs, but they must be constrained chronologically and present morphologies should not be extrapolated to the past without detailed analysis. Indications of sea-level change may be apparent in submerged buildings, stranded harbor basins, wave-cut notches on fossil sea cliffs, erosion benches (platforms), and other terrestrial and underwater features. Road cuts, irrigation ditches, foundation trenches, and eroded coastal cliffs are common targets of opportunity providing windows onto past depositional and erosional sequences. There are two essential principles that must guide observations about sea-level change, however. First, the critical consideration is relative sea level; that is, the relationship between land and sea at any location is affected by erosional, depositional, and tectonic processes, and not merely by the absolute changes suggested by global or regional eustatic sea-level curves. Consequently, the second principle is that each coastal location has its own relative sea-level curve, determined by local tectonic controls as well as distinct erosional and depositional characteristics. Rapp and Kraft (1994: 73) emphasize that composite regional sea-level curves, or those borrowed from "nearby" localities, usually lead to errors in interpretation.
5.10 Aerial photograph of submerged harbor remains near Naples. Reprinted from Marriner and Morhange : 151, fig. 13, with permission of Elsevier.
Observations of visible coastal landforms and archaeological features often form the basis for speculation regarding the layout and date of ancient harbors. This tradition is particularly strong with respect to purported Minoan harbors on Crete (Chryssoulaki 2005; Hadjidaki 2004; Marinatos ; Raban ; Shaw ). Typically, the evidence might include artifact concentrations, foundations of Minoan-style walls extending into the sea, anchors found in the nearby sea bed, or cuttings in bedrock that cannot be dated. The arguments for the existence of these harbors often cite the prominence of Minoan seafaring in the Bronze Age (Chryssoulaki 2005; Hadjidaki 2004), an important consideration but not a directly relevant form of evidence. Thus, although reasonable hypotheses may be derived from archaeological observations and general historical arguments, little can be said with real conviction, and little detailed information can be obtained, in the absence of a full geoarchaeological assessment. Marriner and Morhange (2007: 162) cite the cautionary tale of the harbor at Kition-Bamboula on Cyprus, which two researchers depicted as a cothon based on modern engravings and field observations, before a geoarchaeological investigation revealed this to be erroneous (Morhange et al. ).
Geophysical survey employs nondestructive techniques to detect remotely subsurface features on land and under water. In terrestrial archaeological contexts, extensive use has been made of ground-penetrating radar, magnetometry, electromagnetics, and resistivity (Kvamme ; Sarris and Jones ). Each instrument and technique has its optimal use conditions and detection characteristics. Because harbors and other coastal features are often buried in sediment, geophysical surveys can frequently reveal the outlines of buried features, provided that their properties (magnetism, conductivity, density, etc.) contrast sufficiently with the surrounding matrix, and that they are located vertically within the detection limits of the geophysical instrument. An excellent example is the buried harbor and urban area of Portus, where Simon Keay and colleagues integrated the results of magnetometry, resistance survey, ground-penetrating radar, and electrical resistivity tomography to reveal much of the plan of Rome's principal imperial port (Keay et al. ). These results were confirmed and expanded with programs of coring and excavation.
Comparable techniques can be used to detect features on the sea floor and buried in marine sediments beneath it, typically by towing geophysical equipment on a rig attached to a small boat. Side-scan sonar and sub-bottom profiler surveys are used to create bathymetric maps that measure depths of sediment and bedrock, giving indications of the shape and orientation of ancient harbor basins over time (Lafferty et al. ; Papatheodorou et al. ). Marine magnetic surveys are well established and have often yielded spectacular results where underwater features with strong magnetic signals are present. A magnetic survey off the coast of Caesarea Maritima in Israel – Herod's Roman harbor – has clarified a number of questions concerning its construction and use. The magnetic data revealed the hydraulic concrete foundations for the ruined harbor moles by detecting the volcanic ash (pozzolana) in the concrete as a strong magnetic anomaly (Boyce et al. ; Fig. 5.11). These results allowed new interpretations of the shape and construction of the now-submerged outer harbor, defined by the moles and additional breakwater structures. Detected in the same survey were numerous low-relief mounds with elevated magnetic signals, on the sea bed beyond the outer harbor. These features, subsequently investigated by jet probing and excavation, turned out to be ships' ballast piles, identified by a mixture of igneous and metamorphic boulders and fired pottery exhibiting strong magnetic properties (Boyce et al. ). It was even possible, through careful mapping and examination, to hypothesize the formation process of the ballast piles by ships anchored around a designated anchorage point (Boyce et al. : 1524). Ballast piles associated with Bronze Age pottery have recently been discovered off the Saronic Gulf coast at Kalamianos (Tartaron et al. 2011; see Chapter 7).
5.11 Plan of submerged remains at Caesarea Maritima. From Boyce et al. : 132, fig. 8. Reproduced with permission of Blackwell Publishing Ltd.
Geophysical techniques are nondestructive and non-intrusive, yet they can often supply rapid and reliable information about the location, depth, and nature of buried or submerged features without excavation. They may play the key role of guiding subsequent coring and excavation strategies, and more generally helping to form hypotheses and research questions. A geophysical project will more than pay for itself if it saves the investigator from a misguided research design. There are also disadvantages to geophysical surveys. There is no chronological control on the stratigraphy and archaeological features they detect, and the interpretation of anomalies often involves a high degree of subjectivity if their forms and signals are not transparently characteristic of known features. Ground truthing, in the form of excavation, coring, or other means of direct examination, is usually required to verify the identity of potential archaeological features. The entire enterprise of archaeological geophysics relies on a high level of expertise and expensive equipment that may not be easily available or affordable. Nevertheless, geophysical surveys are indispensable for the investigation of buried harbors and submarine features.
Coastal stratigraphy is the central concept of the geoarchaeological method, which entails direct observation of the geological record to locate, characterize, and reconstruct physical evidence of natural and anthropogenic processes that contribute to the long-term evolution of coastal environments. As we have seen, observations of the modern surface can be misleading or ambiguous in their relationship to the past, but all major processes that generate change (cited above) leave a record in local sediments that can be decoded by careful analysis. Ideally, one would excavate a number of sections of the vertical and horizontal dimensions of coastal and underwater sediments, but this approach is rarely feasible in terms of resources (including time and money) and permissions. Thus, geological core drilling is the method of choice to obtain sediment samples for analysis. The decision about where to sample and how many cores to take is a critical aspect of the research design. Although it is obvious that enhancing the resolution of the study, that is, by increasing the number of cores and decreasing the spatial interval at which they are taken, should lead to higher confidence in the results, given the usual constraints a well-rationalized sampling design is of the utmost importance. Moreover, no coring program, no matter how large or well designed, will provide all the answers hoped for, but will generate new questions that can only be addressed with further research. A perfect illustration is the decades of coring work dedicated to reconstructing the marine embayment and coastal paleogeographies of ancient Troy (Kraft, Kayan et al. ; Kayan et al. ). Hundreds of cores have now been taken and analyzed from the plain around Hisarlik, yet although the general outlines of the evolution of this coastal landscape are reasonably well understood, the details of interpretation continue to be debated and new questions remain to be addressed.
Many macroscopic attributes of the core sample are described in the field, using one of a number of recording systems (e.g., Folk ). In the lower Acheron River valley of southern Epirus, Mark Besonen and colleagues recorded lithology, grain-size distribution, color when wet using the Munsell soil color chart, sediment consistency, plant and animal macrofossils, pedogenic characteristics (structure, sesquisoxide/reduction mottling, and calcium carbonate filaments or nodules), and chance finds such as pottery fragments (Besonen et al. : 209). The depth of the core varies with the equipment used and the nature of subsurface layers encountered. Hand-augering is the conventional technique, because the instrument is portable and widely owned by earth science departments and private geological firms. With hand-operated equipment, it is not uncommon to reach impenetrable stony layers well before the maximum length of the auger. More elaborate power corers generally do not have this limitation (Kayan et al. : 382–84), but their use is far more expensive and they may not be readily available in country. Once the attributes of the core are described, all or part of each sediment column is packed in aluminum foil or some other protective material for transport to the laboratory.
#### Laboratory Techniques
The core that reaches the laboratory preserves a stratified, and thus diachronic, record of the sedimentary sequence at a specific location. It is a storehouse of information, but decisions must be made about the tests that will be performed and the materials that will be tested, as well as the resolution at which the examinations are to be made. The tighter the sampling interval, the higher the resolution of the data and the better the reconstruction of coastal stratigraphy and coastal history (including chronology); Marriner and Morhange (2007: 165) recommend five centimeters or less. Time, money, and expertise for analysis are costs that should be resolved before coring begins.
After initial description of sedimentary structures, the cores are separated into subsamples to be used for analyses of different proxy materials. These analyses broadly involve sedimentology, biostratigraphy, and geochemistry.
Sedimentological analysis allows the investigator to characterize the sedimentary structures of a core sample and to identify the environmental facies that are represented in the stratified deposit.
The structural characteristics of the sediment provide important clues to the depositional environment and the source of the material that formed the deposit. Typically, subsamples are analyzed for color, grain size, microfossils, organic matter, and calcium carbonate content; organic material is extracted for radiocarbon (usually AMS) dating and sherds or other cultural material are noted (Jing and Rapp : 159, fn. 4).
Granulometry – the analysis of grain size, shape, sphericity, roundness, and sorting of clastic particles in a sediment – can indicate the mechanisms of transport and deposition and distinguish between high- and low-energy formational environments in a coastal area. Sediment texture refers in part to the range of particle sizes present; these are determined by a process of wet and dry sieving to separate the gravel-, sand-, silt-, and clay-sized fractions (Rapp and Hill : 22–23, figs. 2.2, 2.3). These fractions are then weighed and their relative proportions are plotted graphically and statistically. The resulting ratios can indicate different kinds of coastal environments; for example, predominance of gravel and sand may indicate a fossil beach; predominant sand may indicate littoral deposits such as sandbars, barrier reefs, and islands; and predominant silts and clays may indicate an artificial harbor basin. The association between grain size distribution and paleolandform is not always straightforward, however. Rapp and Hill (1998: 38–39) identify five factors influencing grain size distribution: (1) the type of source rock and the original size of grains; (2) the type of transporting medium; (3) abrasion and solution during transportation; (4) sorting of size fractions before deposition; and (5) the depositional environment. Thus, the sources of sediment (e.g., fluvial, aeolian, longshore), the distance of transport, and the reworking of sediments in the coastal environment are some of the variables in working back to coastal paleoenvironments. Artifacts that may be present as clasts in the sand- and gravel-sized fractions are of great interest when recovered from cores.
Cores are also sampled for biogenic material, which can provide conclusive evidence for the depositional environments in which particular species lived. Biostratigraphy entails the identification of fossil organisms and the study of their temporal and spatial distribution in order to record the changing abundance and species composition, and thereby to reconstruct the depositional environments (biofacies) that these organisms characterize. The main fossil types are marine mollusks, ostracods, and foraminifera, each being both abundant in coastal waters and specific in its environmental tolerances to depth, salinity, and temperature. For this reason, they are excellent indicators of depositional environment and accordingly are used widely in paleoenvironmental reconstruction of coastal regions.
There are different advantages and disadvantages to these fossil types. Marine mollusks tend to be abundant, with known environmental tolerances by species. In situ shells are useful for radiocarbon dating. Mollusk shells are often the primary component of coastal middens that may be detected during archaeological or geomorphological survey. Ostracods are microcrustaceans that leave a calcite bivalve carapace, the morphological characteristics of which can provide taxonomic and phylogenetic information (Marriner and Morhange : 170). They tend to be ubiquitous in both fresh and marine waters, their carapaces preserve well, and because of their small size a great number can often be extracted from a core subsample. They are excellent indicators of paleoenvironment because their composition, population density, and population diversity vary as a function of water temperature, water salinity, water depth, and anthropogenic impacts (Marriner and Morhange : 170). As a result, ostracod species are strongly correlated with very specific depositional environments.
Foraminifera are single-celled organisms with tests divided into chambers that accumulate during growth. They are among the most ubiquitous shelled organisms, often yielding more than one million living specimens per cubic meter, and they live in all marine habitats from intertidal to deepest ocean, from tropics to poles. Their mineralized tests preserve well, so a small sediment sample is likely to provide a large and statistically valid assemblage. They do not, however, live in freshwater environments, and only a few species live in brackish contexts. Further, individual species are highly specific to environment and are quick to respond to environmental change. Thus, foraminifera are highly useful for determining a range of paleoenvironmental characteristics, including sea-level change, paleoclimate, temperature, salinity, carbonate chemistry, diet, and nutrient conditions (Marriner and Morhange : 170).
In the laboratory, fossil organisms are extracted and assigned to general ecological assemblages or biofacies, such as freshwater, brackish lagoon, marine lagoon, coastal, and marine. These categories are determined, as explained above, on the basis of the environmental tolerances of the species present, but also by geochemical analysis to determine the stable isotope ratios of oxygen and carbon in the shells of ostracods and foraminifera. The ratio of the two stable isotopes of oxygen, 16O and 18O, in ocean waters is a function of global evaporation rates, which in turn are proxy measures for long-term climate fluctuations as well as water temperature and salinity (for principles, see Rapp and Hill : 104–105). This ratio is also recorded in the shells of fossil marine and freshwater organisms because the shell is formed by the precipitation of carbonates from the surrounding water. The 18O/16O ratio, expressed as variance from present mean sea water (δ18O in parts per thousand [‰]), is determined mainly by the temperature and salinity of water; negative values record progressively depleted 18O and consequently warmer and less saline water, while positive values reflect enriched 18O and colder, more saline water (Deith ). Similarly, the ratio of the two stable isotopes of carbon, 13C and 12C, imparts information about temperature and salinity as well as the presence of carbon from decomposed plants. When stable isotope ratios are combined with the species' ecological preferences and tolerances, the depositional environment can often be determined conclusively (e.g., Goodman et al. ).
The ultimate aim of geoarchaeological analysis of paleocoastal environments is to identify the changing coastal environments of deposition, or facies, over time. Facies are defined as the characteristics of a rock unit (in this case, a sedimentary deposit) that reflect the condition of its origins and differentiate it from adjacent units. Each sedimentary facies identified in the core has an essentially uniform character that reflects its origin and distinguishes it from adjacent units laterally and vertically (Rapp and Kraft : 72). Biofacies derived from microfaunal analysis are correlated with sedimentary facies resulting from the granulometric analysis (Fig. 5.12). The chronometric framework for the sedimentary sequence is usually provided by radiocarbon determinations on organic material present in the core, including shells, charcoal, wood fragments, peat, charred roots, and other plant remains. Datable sherds can also provide chronological information. When a number of sediment cores are taken across a coastal landscape, the sedimentary units can be correlated to construct a broad record of changing coastal landforms over time.
The work of Besonen and colleagues in the lower Acheron River valley provides an example of facies designations (Besonen et al. ). On the basis of geoarchaeological fieldwork using the techniques described above and others (Besonen et al. : 209), they identified 14 distinct sedimentary facies on the modern landscape, which correlate to paleoenvironments that existed at one time or another in the Holocene (Table 5.1). They divide the facies into two broad depositional systems: the fluvial depositional system, consisting of the sedimentary environments landward of the shoreline, and the deltaic nearshore system, comprising those seaward of the shoreline but within the marine embayment (Besonen et al. : 212–16). These data, combined with an array of radiocarbon dates from core material, allowed for the reconstruction of coastline configuration and coastal environments over time (Besonen et al. : figs. 6.13–6.15). Because the terminology used to name these facies is not standard across the Mediterranean, it can take some effort to work out the correlations from one site to another.
5.12 Correlation of biofacies and sedimentary facies in the Ancient Harbour Parasequence. Reprinted from Marriner and Morhange : 177, fig. 31, with permission of Elsevier.
Table 5.1. Sedimentary facies in the lower Acheron valley (Besonen et al. : 213–16)
* * *
Facies| Main component(s)
---|---
(1) River channel| Coarse lag deposits and bars
(2) Subaerial natural levees| Ridges of coarse and muddy sand formed by deposition of coarse overbank deposits
(3) Crevasse splays| Sand- to mud-sized flood deposits
(4) Floodplain| Flat ground adjacent to river channel, mainly silt- and clay-sized flood deposits
(5) Backswamp| Transitional zone between floodplain and shallow lake; perpetually saturated, organic-rich muds and clays; freshwater ostracods
(6) Shallow freshwater lakes and pools| Clay-sized particles; sparse freshwater ostracods and gastropods
(7) Delta-top marsh (fresh to brackish)| Organic-rich peat and peaty mud; brackish and freshwater microfauna
(8–10) Active delta front: distributary channel; distributary mouth bar; subaqueous levee| Continuity of subaerial natural levees underwater; particle size ranges from sand and sandy gravels (delta distributary channel) to sand and silt (subaqueous levee); brackish to marine microfauna
(11–12) Lower delta front; prodelta| Basinward of active delta front; low organic; coarser deposits than active delta front; brackish to marine microfauna
(13) Interdistributary bay| Shallow open body of water; silts and clays; abundant brackish to marine microfauna
(14) Concentric accretionary beach ridge and swale sets (see Fig. 5.3)| Facies (7)–(9) provide constant source of sandy sediment reworked by wave action, then piled over regular wave deposits by spring and winter storm waves and kept in place because Phanari Bay too well sheltered for longshore currents to entrain them
* * *
### Classification of Bronze Age Harbors
Several typological schemes have been developed to classify ancient anchorages and harbors. Generally, archaeologists have offered typologies based on idealized coastal topography projected to the time of use in antiquity (e.g., Blue 1997; Chryssoulaki 2005; Raban ; Shaw ), while geologists use geomorphological formation and history as the main criteria (e.g., Marriner and Morhange ).
Lucy Blue (1997: 31–34: figs. 1, 2), basing her typology on previous work by N. C. Flemming, distinguishes two categories of Bronze Age anchorages: those on high-energy, cliff-lined coasts, and those on low-energy, low-lying coasts. Anchorages on the former occur in (1) natural bays; (2) almost enclosed bays; (3) bays on either side of an anvil-shaped headland; (4) lee of a promontory; (5) sheltered valleys; and (6) lee of offshore islands or reefs (Fig. 5.13). On the latter, anchorages are found (1) on the banks of navigable rivers that empty into the sea; (2) in inland lakes upriver; (3) in natural embayments; (4) in river deltas; and (5) in lagoons or estuaries (Fig. 5.14). Blue's typology takes into consideration not only coastal configuration and landforms but also systemic energy; her distinction between high- and low-energy environments correlates fairly well with Wells' (2001: 150) first-order division of open versus protected coastlines. Other published descriptions of the range of Aegean or eastern Mediterranean anchorages are less detailed or are not typologies per se. Stella Chryssoulaki's (2005: 82–83, fig. V) typology is much simplified relative to Blue's and focused more narrowly on landforms rather than processes, but she does explicitly address certain features, such as sunken peninsulas and tombolos, which are particularly relevant to Minoan Crete. Using Blue's topographical typology, it is possible to assign a number of Aegean anchorages to one of her types (Table 5.2).
5.13 Topographic typology of Bronze Age anchorages: high-energy coasts. Drawing by Felice Ford after Blue 1997: 33, fig. 1.
5.14 Topographic typology of Bronze Age anchorages: low-energy coasts. Drawing by Felice Ford after Blue 1997: 34, fig. 2.
Table 5.2. Mediterranean anchorages associated with topographical types
* * *
| Topographic type| Mediterranean examples
---|---|---
High-energy| Natural bay| Zakros, Souda (Crete)
| Almost enclosed bay| Mezapos, Kiparissi (Laconia)
| Bays either side of anvil-shaped headland| Pseira (Crete); Ayia Irini (Kea); Vayia (Corinthia)
| In lee of a promontory| Many, depending on wind direction
| Sheltered valley| Ayiofarango (Crete), Vathi (Matala, Crete)
| Anchorage in lee of islet or offshore reef| Amnisos, Kommos (Crete)
Low-energy| Riverine| Israeli coast
| Inland lake up river| Acherousian Lake (Epirus, post-Bronze Age); Enkomi Ayios Iakovos (Cyprus)
| Natural embayment| Many bays and gulfs, e.g., Glykys Limin (Epirus); Latmian Gulf (western Anatolia)
| Delta| Nile Delta; Cilician Delta (southern Anatolia)
| Lagoonal| Western Peloponnesian coast (Elis, Messenia); Hala Sultan Tekke (Cyprus)
* * *
Geomorphologists Marriner and Morhange (2007: 146–62, figs. 7–9) propose a complex harbor classification based on four variables: (1) proximity to the modern coastline; (2) position relative to sea level; (3) sedimentary environments; and (4) taphonomy (how a harbor came to be fossilized in the sedimentary record). According to this typology, there are buried urban, buried land-locked, buried lagoonal, buried fluvial, submerged, uplifted, and eroded harbors (Marriner and Morhange : fig. 7). These can further be organized as lying on unstable coasts (those exhibiting tectonic subsidence or uplift) or stable coasts (all others, not affected by tectonic activity; Marriner and Morhange : fig. 8). Another geomorphological perspective is preservation potential: buried harbors have good preservation potential by virtue of being buried; uplifted and submerged harbors have medium preservation potential; and eroded harbors have poor preservation potential (Marriner and Morhange : fig. 9).
The advantage of geomorphological typologies is that they are ostensibly based on fieldwork that has established the diachronic history of deposition, erosion, and movement of the shoreline. This is not to say that these geomorphological reconstructions cannot be misleading or wrong, as the case of Kition-Bamboula (Morhange et al. ) demonstrates. As mentioned above, archaeological discussions of ancient coastlines that retroject modern topography into the distant past without careful consideration of geomorphological history, or that fail to present explicit evidence for proposed configurations, must be treated with caution. Blue's topographic typology is based on an understanding and application of geomorphological principles, but in the wrong hands the typology can make it seductively easy to simply match modern satellite images of coastal areas, which are so easily accessible online, with these idealized types. The topographic and geomorphological typologies are complementary data sets that should be used in combination.
### A Systematic Approach to Detecting Bronze Age Harbors
The investigation of ancient harbors is by definition both a geoarchaeological problem of coastal landforms and processes, and one that requires more traditional means of terrestrial and underwater archaeology. In view of the rich geomorphological scholarship on the Mediterranean and the availability of interested coastal geomorphologists, there is no justifiable philosophical impediment to systematic, joint archaeological–geomorphological programs to detect ancient anchorages in a narrow sense, and more broadly to investigate Bronze Age coastal worlds. These investigations are interdisciplinary as is all good archaeology, and they involve terrestrial and maritime fieldwork proceeding hand in hand. There are abundant models for the implementation of pedestrian survey (e.g., Banning ; Cherry et al. ; Tartaron et al. ), terrestrial excavation (e.g., Drewett ; Hodder ), and underwater survey and excavation focused particularly on shipwrecks (e.g., Gould : 21–64; Muckelroy : 24–58), but the real need going forward is for the design of programs of close collaboration between archaeologists and geoarchaeologists from inception to interpretation.
Models of integrative research, relying on constant collaboration across disciplines in the field and "intensive exchange of information, ideas, and procedures from the planning stage through final publication" (van Andel : 28), are at hand in the eastern Mediterranean. The renewed excavations at Çatalhöyük by Ian Hodder and colleagues practiced a "reflexive archaeology" that assembled specialists from diverse disciplines to observe the recovery of material "at the trowel's edge" (Farid ; Hodder , 2000). The Eastern Korinthia Archaeological Survey (EKAS) applied a comparable philosophy to pedestrian surface survey by integrating experts from all participating disciplines in each stage and in every aspect of the project (Tartaron et al. : 463–64). The fieldwork application of this philosophy entailed the "embedding" of geomorphologists in archaeological survey teams, and archaeologists with teams engaged in geomorphological mapping and soil description. The atmosphere of mutual enlightenment engendered by shared research allows collaborators to gain a realistic understanding of the opportunities, limitations, and reasonable outcomes of each others' fields. The goal as applied to coastal research is to pool the acquired knowledge in order to learn about the human and natural interface of land and sea, in the service of a broader, more holistic study of maritime cultural landscapes; in the words of Ina Berg (: 21), "...an inclusive archaeology of islands, sea and coasts." But, as Berg cogently emphasizes, this is not merely a matter of cross-disciplinary communication, but instead requires a commitment to breaking down the conceptual divide between land and sea that has characterized Mediterranean maritime archaeology (see Chapter 6).
Like any archaeological investigation, a joint archaeological–geomorpho-logical program of coastal research should proceed by a sequence of coherent, logical steps (Fig. 5.15). One might begin, as the present study did, with a vexing problem (where are the Mycenaean-era anchorages?), or any set of compelling, carefully defined archaeological or anthropological questions that can be applied to a geographical and chronological horizon of interest. The precise study area can be refined based on the research questions and tempered by constraints of time, money, and permit restrictions. The research questions should be formulated in an explicit theoretical framework (as we know, all archaeological research has a theoretical orientation, whether acknowledged or not), from which a series of testable hypotheses can be generated. From these preliminary deliberations, appropriate search strategies and field methods can be developed. For coastal projects, archaeological and geomorphological fieldwork may occur both on land and in/on the sea.
Strategies for the initial discovery of Bronze Age anchorages in a survey universe (a long coastline, for example) of relatively unknown site potential can make use of a host of distinctive approaches, such as the following:
(1) Investigation of known historical harbors; many have earlier histories as natural anchorages of the Bronze Age or Early Iron Age (e.g., Sidon, Tyre, Liman Tepe).
(2) Examination of modern natural anchorages to test, through geoarchaeological means, whether they existed as suitable anchorages in the Bronze Age.
(3) Focus on deltas and river mouths. Because these were favored as natural harbors in the Bronze Age, programs of geological coring may recover evidence of buried deltaic and estuarine systems of that age.
(4) Collection of information from local inhabitants, who often know of coastal and underwater archaeological sites. In addition, oral histories and archival data giving evidence of human activities in historical and modern times can provide analogies and insights into emic perceptions of maritime cultural landscapes (see Chapter 7).
(5) Intensive geomorphological and archaeological surface survey along the entire littoral zone of the study area to assess evidence for coastal landforms and to recover traces, however abundant or sparse, of human activity (with selective excavation, where possible, occurring at a later phase).
(6) Systematic search based on models of coastal exploitation. The Coasts and Harbors Survey, a subproject of EKAS, developed a GIS-driven probabilistic model for prehistoric harbor locations based on environmental and cultural variables (Rothaus et al. ; Tartaron et al. ). Ground-truthing the model resulted in the discovery of two major Bronze Age anchorages at Vayia and Kalamianos on the Corinthia's Saronic coast, and subsequent geoarchaeological investigation and archaeological prospection over the years has generally validated the principles and results of the model. In a different way, geoarchaeological analysis of the coast of Elis led to predictive statements about the locations of buried Bronze Age sites and coastal landforms (Kraft et al. ). In both cases, a Bronze Age coastline has been partially reconstructed from a modern littoral virtually devoid of recognizable anchorages, a first step to reconstituting humanized coastal worlds.
Other discovery strategies, not generally practiced in the search for Mediterranean Bronze Age anchorages, are prominent elsewhere. In Scandinavia, the preferred methodological tools are phosphate analysis and place-name studies (Ilves ; Westerdahl : 9–11). The method of identifying coastal settlements by detecting dark soils with high phosphate content, followed by test pitting, was established already in the early decades of the twentieth century by Olof Arrhenius (Ilves : 154), but has been little attempted in the Mediterranean (the so-called stables at Gla are an exception: Iakovidis , 2001). Similarly, the use of the many place names with transparent or plausible associations with maritime activity, as well as old maps and charters, to find harbors, is a significant element in Scandinavian maritime archaeology because of the primary focus on historical (Viking Age and later) periods. As we have seen, the names of relatively few Aegean Bronze Age settlements are known, and none has a transparent linguistic association with the coast or sea.
5.15 Flow chart of methods in geoarchaeology and paleogeography. Brückner : 126, fig. 2. Courtesy of Springer Science+Business Media.
### Liman Tepe, Aegean Turkey: A Brief Example
Recent research in the Bay of Izmir region of Aegean Turkey provides a compelling example of integrative research and a striking demonstration of the need for comprehensive geoarchaeological investigation to properly reconstruct ancient coastlines and harbors. The coastal settlement at Liman Tepe was a significant Bronze Age center, and in later times the site of the Ionian city of Klazomenai. Liman Tepe was particularly important in the EBA, when a fortified citadel was built. Finds from the limited excavations to date within the settlement have revealed a vigorous local culture with strong ties to other areas in the Aegean Islands and the Greek mainland, as shown by imported pottery as well as an architectural complex that may have functional affinities with the "corridor houses" of the EH II Aegean ( aho lou 2005).
From the archaeological and historical information, it was assumed that a major harbor was located at Liman Tepe, but until the late 1990s, paleocoastal reconstructions were limited to interpretations of modern topography and the distribution of archaeological materials from surface survey. Based on information from these two sources, it was believed that in the EBA, a large part of the enclosure wall, now submerged, extended to the north and in the northwest formed a large pier with an attached built breakwater, preserved as a ruined underwater structure ( aho lou 2005: 98; Fig. 5.16). It was also thought that the later Archaic-period harbor occupied a large embayment delineated by the distribution of surface and near-surface architecture and artifacts (Ersoy ).
Because these reconstructions relied on tenuous data, a new archaeological and geoarchaeological program was begun in 1999 and continued in years following, featuring underwater excavations and a multiproxy geoarchaeological study (Goodman et al. , 2009). The geoarchaeological data set was derived from terrestrial cores, sea-bed cores, and grab samples from the walls of the underwater excavation trenches. These materials were subjected to microfaunal analysis of shells, foraminifera, and other microfossils to determine presence/absence, ubiquity, and species and their environmental tolerances; grain size analysis; determination of δ18O and δ13C values relative to local sea water; presence/absence of archaeological materials; radiocarbon dating; and chronological and stratigraphic association of deposits (Goodman et al. : 1270–72, 2009: 97–98). Using these proxy measures, the researchers were able to identify the following environmental facies: terrestrial, supratidal, wetland, foreshore, lagoon, upper shoreface, and artificial harbor basin (Goodman et al. : 1272–77, 2009: 98–100).
5.16 Location of Liman Tepe in the Bay of Izmir region. Satellite image © 2011 Google Earth, © 2011 Digital Globe.
The historical interpretations engendered by these results conform in important ways to current understandings of long-term regional processes in the eastern Mediterranean, but explicitly contradict the previous reconstructions of the ancient coastline at Liman Tepe (Goodman et al. : 100–102). The data are in basic agreement with an early Holocene marine transgression, followed by eustatic sea-level-rise deceleration circa 6000 BP. The deceleration resulted in a positive sediment budget, and excess sediments were transported by longshore currents to form sandbars, and ultimately beach barriers with extensive lagoons behind them. It was during the EBA that an ideal combination of nearshore lagoons and a tombolo joining the mainland to an offshore island created several natural, sheltered contexts to anchor or beach seafaring vessels. By the end of the LBA, however, these longshore sediments isolated the lagoons from the sea, as indicated by a change from brackish-marine to fresh water, ending the viability of Liman Tepe's harbors (Fig. 5.17). There is no evidence for a harbor of any kind for centuries thereafter, until artificial harbor constructions were first built circa 800 BC. During this Early Iron Age interim of at least 200 years, archaeological evidence indicates a diminished community with few maritime connections.
5.17 Reconstruction of the Bronze Age coastline at Liman Tepe. Goodman et al. : 102, fig. 3B. Courtesy of John Wiley & Sons, Inc.
It is of particular importance that the geoarchaeological study allowed the investigators to reject the previous coastal constructions (Goodman et al. : 101–103). The submerged built feature was associated not with the EBA, but with the much later built harbor of the Archaic and Classical period. This finding is significant because, along with the revelation that a number of natural anchorages existed along the EBA coastline, it helps to sustain the notion that built harbor constructions were rare or absent in the Bronze Age Aegean. Earlier reconstructions also showed a broad bay east of Liman Tepe in classical times (Ersoy ; Goodman et al. : fig. 1), but the geomorphological data demonstrate that such a bay existed only in the EBA.
The implications of these results are clear for understanding the history of the site and its maritime connectivity over time. Once again, we see that however meticulous or well intentioned, reliance on modern surface topography or archaeological patterns often leads to flawed reconstructions. Thus, the geoarchaeological study of Liman Tepe's paleocoastline offers a strong argument for the type of interdisciplinary work advocated in this chapter. Additionally, as Beverly Goodman and colleagues (2009: 103) point out, the Liman Tepe study presents a contrast to previous studies in western Anatolia focused on sites at the mouths of large rivers, such as Ephesus or Troy. It illustrates that major coastal change occurs also in less dynamic environments, even in the absence of plentiful sediment supply from an alluvial source. This point of view is especially germane as the scale of analysis moves toward finer-grained, local coastal contexts.
### Conclusion
In this chapter, I have attempted to address critically the widely held impression that the land masses of the Greek mainland and Aegean Islands have been essentially stable over the period from the Bronze Age to the present. By necessity, I have emphasized the physical and environmental characteristics of coastal settings. In the next chapter, I present theoretical perspectives that emphasize the social aspects of living in coastal and maritime settings. Thereafter, the physical and social-symbolic background will be firmly in place, and the two distinct approaches can be joined in the case studies of Chapter 7.
## Six Concepts for Mycenaean Coastal Worlds
> The ensemble of Mediterranean lands...has an inside-out geography in which the world of the sea is "normal" (the interior), and the land is the fringe, its marginality increasing with its distance from the water. (Horden and Purcell : 133)
Having presented arguments for the recovery of the physical spaces of ancient coastlines, I seek in this chapter to integrate the spatial, temporal, and social dimensions of coastal dwelling in a broader conceptual framework that emphasizes networks of interaction. Such a framework must be dynamic: it must accommodate expansion and contraction of networks, reconfiguration in the number and location of nodes in a network, and multiple and overlapping networks operating at different geographic and temporal scales and serving different purposes. These changes and adjustments can be influenced by environmental and technological factors, but to a much greater extent they are determined by political, economic, and social relationships (e.g., Horden and Purcell : 53–88). On the most fundamental level, however, the conceptual framework should address whether the coastal mode of existence is sufficiently distinctive to merit theoretical consideration as an entity or category in its own right. I begin, therefore, with a discussion of how coastal people are positioned relative to the worlds of land and sea around them, and how consequently they played a special role in the connectivity of the Mycenaean world. Subsequently, I present a framework for Mycenaean maritime connectivity based on several conceptual categories related to scales of interaction. Finally, I explore how this framework can accommodate current ideas about connectivity and social networks.
### The Unique Status of the Coastline?
Life in the twenty-first century can color one's view of movement about the landscape, particularly when one's point of reference is the hyper-developed "first world." The elaboration of an infrastructure of roads, railways, and airports has marginalized sea travel for most people, and this change has affected Greece as much as any other European country, despite its long association with the sea. The ability to travel safely around Greece on good roads has minimized sea travel and has located coastal areas in a conceptual periphery: coastlines are now destinations to which one drives on occasion to escape the rigors of continental life. Most sea travel consists of short jumps from mainland to island or island to island while on holiday. But in the LBA this relationship is likely to have worked in the opposite way. It is commonly observed that in antiquity it was easier to travel around the Greek mainland by sea than by traversing the rugged interior. This principle calls attention to the virtual lack of built roads in the Bronze Age (those around Mycenae being the primary exception), the limited hauling capacity of humans and pack animals over rough tracks relative to the considerable cargo capacity of ships such as those recovered at Uluburun and Gelidonya, and the fact that few locations on the mainland are far from the sea. Though true enough, this principle predictably threatens to become an overcorrection that underestimates overland traffic, which must have been continuous and perhaps even predominant in months when sailing was difficult.
An even more obvious point is that nodes in a maritime network are coastal, and for this reason alone coastal places and people are essential to any network analysis that involves connections by sea. But can we say more? Can we mark out a distinctive role for coastal people that transcends mere geography to convey something meaningful about the implications of inhabiting the coast during Mycenaean times? I think it is possible by re-centering the coastline, and to do this it is necessary to embrace the notion that webs of interaction articulate coastal dwellers to both land and sea.
The first step is to recognize the simultaneous liminality and centrality of the coastline. The coast occupies a physically liminal space at the interface of land and water. Coastal people occupy an ecotone – a transition between contrasting ecological zones and their productive resources. This liminality is both economic and social; that is, the shore is the space where people and products from the interior meet those arriving by sea. Coastal communities play a mediating role in these encounters, and the coast becomes a central – not merely liminal and certainly not peripheral – place. As implied in the quote opening this chapter, it is inland people who have the greater challenge establishing and maintaining connectivity in a Mediterranean setting such as the Aegean. In Chapter 7, oral histories from inhabitants of the modern village of Korphos will be used to illustrate the ways that coastal people may manipulate their centrality in economic transactions.
There are several implications of the centrality of coastlines. People inhabiting them need not be oriented to the sea, but it may be impossible to escape this orientation when the sea offers resources and brings other people, wanted or unwanted. Coastal dwellers are in a sense amphibious since they become adept at negotiating both land and sea, and mediate between terrestrial and maritime worlds. They possess distinctive and specialized knowledge about how to negotiate local winds, weather, and other navigational hazards. In a more figurative sense, coastal areas have been central to history and tradition as the settings for armed conflict, whether we invoke the coastal raiding of the Odyssey or the great battles at Troy, Marathon or Thermopylae, or Navarino. The recurrent raids and invasions suffered by coastal settlements mark them as exposed and vulnerable. Fortified coastal settlements such as those of the EBA Aegean reflect both a maritime orientation and the need to protect communities from the dangers of piratical raids.
Although I stop short of advocating subdisciplinary status for coastal archaeology, I do suggest in this book that coastal areas are sufficiently unique in their geomorphology, geographic position, and interactive potential that they merit special study. It may be useful to draw a brief comparison with "island archaeology" in the Mediterranean. Broodbank () explicitly called for an island archaeology approach that focuses on Aegean-scale connectivity and interaction, while rejecting then-prevalent notions of insularity and island biogeography that treat individual islands as isolated "laboratories" for cultural evolution (inspired of course by the studies of Darwin and his successors on the effects of isolation in biological evolution on islands; see Cherry : 241–45; Patten : 1–5, 19–34). In a remarkable bit of irony, Paul Rainbird in his The Archaeology of Islands (Rainbird ) argues against the very concept of island archaeology. While supportive of Broodbank's general approach, he rejects the commitment to islands as units of analysis as a "fatal flaw" (2007: 43). He instead favors a theoretical shift to an archaeology of the sea that encompasses islands and the littorals of mainlands as a complement to archaeology on land, as well as a shift away from emphasis on the material and environmental record to the postprocessual project of trying to recover the lives and perspectives of mariners and coastal dwellers. The coastal archaeology that I advocate draws something from both; specifically, Broodbank's stronger commitment to empirical evidence (both environmental and cultural), and Rainbird's interest in teasing out the individual and (especially) the maritime community. I make no a priori distinction between coastal communities situated on mainlands and those on islands, because the coastline always unites land and sea, and because the way that a coastline functions does not strictly depend on its mainland or island location. Small islands may comprise little more than coastline without a significant terrestrial hinterland, but that situation is simulated on Greek continental coastlines where a narrow coastal strip is backed by a rugged, mountainous interior. Similarly, the experience of coastal dwelling on larger islands may not be fundamentally different from that on continental coastlines. I follow both of these scholars in advocating an approach that encompasses maritime, terrestrial, and coastal, while placing particular emphasis on the distinctiveness, analytical potential, and historical significance of the coastal zone.
### A Framework for Maritime Cultural Landscapes
The concept of the maritime cultural landscape, coined and subsequently developed by Christer Westerdahl, is widely cited and used as a framework for discussion of maritime interaction, but its application is somewhat problematic in that it remains highly generalized and ambiguous. The maritime cultural landscape, according to an early translation into English, "...comprises the whole network of sailing routes, old as well as new, with ports and harbours along the coast, and its related constructions and remains of human activity, underwater as well as terrestrial" (Westerdahl : 6). Although Westerdahl recognized the "immaterial, cognitive or indicatory" aspects of maritime life, for him these were preserved mainly in place names (Westerdahl : 6). Such a definition encompasses virtually everything with little consideration of variability or scale – it could equally describe a local or an international network, a Bronze Age or Medieval world – and gives insufficient scope to the immaterial cultural dimensions of living and moving within maritime landscapes. In later writing, Westerdahl increasingly incorporated experiential and cognitive elements, but this did not fundamentally change the broad-brush quality of his concept. The tendency of landscape studies to ambiguity about the nature of what landscape is, or can be, and the proliferation of often conflicting definitions of landscape, have been noted as both a strength – as a capacious space for theoretical exploration and interpretation – and a weakness since a lack of clarity inevitably attenuates the force of the concept (Duncan : 10–11).
Rather than bending the concept of maritime cultural landscape to my particular purposes, I propose to allow it to stand as an all-embracing foundation onto which I will construct an explicit framework for Mycenaean maritime connectivity that is based on nested scales of interaction from local to international. This multiscalar framework, summarized in Table 6.1, requires some general explication before each category is examined in turn. The hierarchy inherent in this framework is purely geographical and does not necessarily imply hierarchical relations of power. For example, a maritime small world composed of a number of coastscapes may be persistently controlled politically or economically by a powerful entity, or it may be heterarchical – that is, lacking a hierarchy of power or susceptible to changing hierarchical relationships over time. Further, leaving aside political power, interaction at any of these scales need not imply long-term stability as a social or economic network. The physical and social boundaries between these categories are fuzzy and prone to change as networks are modified by internal and external forces; they may expand or contract, solidify or fragment, be impermeable or porous. Normally, cohesion should decrease with the increasing geographical scope of an interaction sphere; a central assertion of Chapter 4 was that local networks are more stable and permanent than long-distance ones, and thus interregional and intercultural networks would be expected to be the most vulnerable to change. Yet social network theory (see below) predicts that "shortcuts" can expand networks and make longer-distance connections more efficient, and at any time two or more large, powerful sites with abundant resources may bypass local and regional networks to forge direct, long-distance connections.
Table 6.1. A framework for Mycenaean maritime cultural landscapes
* * *
* * *
As envisioned here, the spheres of maritime activity are nested geographically in increasingly larger webs of interaction, which together form the maritime cultural landscape of a particular cultural entity at a particular moment in time. As noted in Table 6.1, however, I also inject elements of temporality and maritime specialization to express the idea that as connections grow more distant, they will normally involve fewer trips and personnel from the local area, though connections made with distant places, whether indirect or direct, active or passive, may possess tremendous social significance (Broodbank ; Helms ; Sherratt and Sherratt ).
It may be convenient to characterize the maritime cultural landscape by reference to the largest sphere of interaction detectable at a given moment, since it can be assumed that societies engaged in interregional networks will possess intracultural networks and small worlds as well. Not all levels of interaction need exist at a given time, however: obviously, connectivity is more far-reaching in some eras than in others, and for certain expanses of time a coastal community may lack the ships and master sailors to actively participate in long-range voyaging. Furthermore, these coastal communities always respond in some way to the ebb and flow of the wider webs of communications around them. In periods when long-distance connectivity is limited by technology or interrupted by political conditions, maritime relations may occur predominantly or solely in the realm of small worlds and intracultural regions. Even under such conditions, goods and people may travel long distances, cutting across major geographic and cultural boundaries, but doing so in smaller segments. One thinks of the Aegean in EB II, when despite the limitations of human-powered seacraft, exotic and quotidian items circulated throughout the coasts and islands, most likely by coast- and island-hopping. The maritime cultural landscapes of these coastal inhabitants were situated entirely within the Aegean archipelago. The collapse of this vigorous connectivity in EB III reduced most interaction to a more local scale. Fluctuations in the scale of maritime connectivity, which might be read in the archaeological record, allow for diachronic reconstructions that lend flexibility and historicity to the multiscalar framework. We would expect connectivity within the Mycenaean world to vary through time, moving through the many phases from the Shaft Grave Era to the postpalatial period; and by location in the diverse environmental and cultural settings of Greece. Although my approach privileges coastscapes and small worlds, these are clearly embedded in larger webs of connectivity, and influenced by events unfolding in distant places. In epistemological terms, I regard the finer-scale networks as building blocks in the archaeological process of recovering a more accurate understanding of how entire "world systems" work.
I turn now to defining the spheres of interaction that make up the Mycenaean maritime cultural landscape (Table 6.1). Although they are inspired by the conceptual categories of Horden and Purcell (), Broodbank (), Sherratt and Sherratt (), and others, they do not correlate exactly to any of these. My aim has been to make these spheres flexible, yet specific to the conditions of the LBA Aegean. They may not work particularly well when applied to other places and times, but similar principles might be used to tailor these concepts to any study of maritime connectivity.
#### Coastscapes
Coastscape refers to the coastal zone characterized by habitation and interaction, and by practice and perception. In the coastscape I include the following components: (1) the linear or convoluted shoreline and the adjacent coastal lowland that may be inhabited and exploited by maritime communities; (2) the connective routes and openings into the interior, which are often dendritic and follow natural paths connecting coast and hinterland (e.g., streams, mountain passes). The landward limit of the coastscape is often defined by ridges or mountains that block views to the interior and impede easy passage; (3) the inshore waters that are utilized on a daily basis for economic and social purposes; and (4) the visual seascape, the everyday field of view that defines the cognitive horizon in the seaward direction, in recognition of a continuous cognitive landscape for which the land–sea interface is no boundary. Hypothetically, therefore, the limits of the coastscape extend from the coast to the connective passes inland and the visible seaward horizon, but topography has much to do with the size and scope of a coastscape. The coastline may be short or long, and the coastal lowland ranges from narrow and relatively isolated from the interior, to broad and relatively open to the interior. The visual seascape may be a broad horizon, or it may be obscured by coastal ridges, offshore islands, or neighboring mainland. The visual seascape of one coastscape may overlap with that of others.
Coastscapes are instantiated by practice. Coastal zones are distinguished by the inhabitation of coastal communities, and by their maritime and terrestrial activities. Everyday engagement with the sea involves fishing, local travel along the coast or to offshore islands, and pilotage and other interactions with ships and boats attempting to make landfall. The fishing boat and the coasting vessel, along with the pilot to assist larger ships to anchorage, would be characteristic of activity in the coastscape (Table 6.1). A coastscape may possess one or several anchorages, varying in size, depth, and exposure to winds and waves. It is a common practice in conditions of severe weather or strong winds to move boats from one anchorage to another nearby that is better protected, if one is available (Malkin et al. : 1). Small anchorages may be used only seasonally or opportunistically to collect produce to be picked up by passing ships (Rothaus et al. : 40), or as shelters from violent weather. The quality of a coastscape's harbors and anchorages for specific purposes plays a role, though not necessarily a determinative one, in its connectivity. Maritime activities occurring in the coastscape are understood to be nonspecialized in the sense that they do not require sophisticated navigational skills or knowledge of complex environmental phenomena. Within the confines of the coastscape and the small world, the part-time seafarer would possess adequate knowledge of nearby anchorages and their hazards, and it would not be overly risky to project good weather conditions for a trip lasting from a few hours to a day or two.
Coastscapes also witness a range of productive activities on land. Coastal settlements are rarely organized for maritime pursuits alone; land in coastal lowlands and accessible uplands is exploited, often heavily, for agricultural, pastoral, and wetland resources. These activities may not produce all of the goods required or desired by the community, which must as a consequence establish exchange relationships with maritime and inland partners. The movement of people and products between the coast and the interior has been documented as a pervasive mode of connectivity throughout antiquity, sometimes attaining a form of symbiosis. As at sea, routes and connections to the interior often overcome ostensibly formidable topographies; and like the visible features that mark a maritime network, routes may be indicated by villages, temples, and other monuments. Local guides may be needed to perform duties comparable to coastal pilots. New tracks to and through the interior are carved out in response to political conditions, for example to avoid taxes and hide from oppressors during the Ottoman period in Greece (Horden and Purcell : 131).
Coastscapes share many properties of Horden and Purcell's "microecologies" and "microregions," but there are crucial differences. For Horden and Purcell, microregions appear to be aggregates of microecologies that may embrace coasts, lowlands, uplands, and interiors depending on existing environmental zones and, particularly, human efforts to integrate them. Their examples of the Biqa valley of the Lebanon and the central Cyrenaica demonstrate as much (Horden and Purcell : 54–59; 65–74). The Biqa also shows that their microregions can be completely inland, with little or no regular contact with the sea. The coastscape, by contrast and by definition, centers on the shoreline and assumes a maritime orientation. The high mountainous areas and the interior zones beyond, forming an integral part of some of Horden and Purcell's microregions, lie beyond the realm of the coastscape. In marking this difference, I do not deny the existence or the significance of these broader interactive zones, and as mentioned above the interactions between interior and coast may be essential to sustaining maritime coastal life. Rather, the framework offered here is meant as an analytical and interpretive tool to address the particular problem of defining the nodes and networks of maritime connectivity. Coastscapes serve as these nodes, and through descriptions of them and of their variability across space and time, they become more than dots on a map or points on a graph of connectivity.
#### Maritime Small Worlds
The use of the term small world to describe a kind of social network is now widespread in sociology, anthropology, ancient history, and archaeology (e.g., Broodbank : 175–210; Pullen and Tartaron ; Sherratt and Sherratt ; Watts ; Watts and Strogatz ), but as a concept it is defined and applied in various, sometimes incompatible, ways. Explicit definition is therefore essential, and my formulation of the small world follows from the logic of the maritime cultural landscape framework as the most local-level aggregation of nodes, that is, coastscapes. Maritime small worlds are interaction spheres that form as aggregates of many neighboring coastscapes; they might also be called local worlds. Their cohesion results from, and they are in fact constituted by, habitual face-to-face interaction based on proximity and various kinds of social and economic ties. The communities that make up a small world commonly share cultural traditions, language, social networks such as kinship ties and intermarriage, mutual protection arrangements, and dense economic relations. Often they are united by economic interdependence if resources are unevenly distributed or if subsistence is precarious.
As noted above, the political and economic organization of small worlds may be characterized by the presence or absence of hierarchy; often the horizontal ties of coastscape communities are more prominent than ranking or vertical hierarchies. In Chapter 7 we shall see how a Bronze Age small world in the Saronic Gulf oscillated between cohesion and fragmentation, and it will also be possible to consider economic hegemony as distinct from political control.
Intervisibility can be an important component of the cohesion of small worlds. Lines of sight are perhaps the most powerful integrative factor in the phenomenological world of a coastal dweller or mariner (Horden and Purcell : 124–26). Chains of mutual visibility among coastscapes give an experiential impetus to the coalescence of small worlds, but universal intervisibility rarely extends to an entire Aegean small world, owing to the mountainous and complex topography. Some landscapes lend themselves well to the formation of small worlds by their geography and topography, such as semi-enclosed gulfs like the Argolic and Saronic, or long facing coastlines separated by narrow straits, for example the eastern Greek mainland and the western coast of Euboea. Even under the ideal conditions of the Saronic Gulf, whereas Aigina is plainly visible from most coastal locations, parts of the gulf are always invisible from any coastal vantage point. In this case, visibility combines with the peculiar conditions of maritime travel within the confines of the gulf – in contrast to those encountered once outside – to demarcate, at least hypothetically, the outlines of a maritime small world.
Distance and travel time are also crucial variables in the definition of small worlds. Actual linear distance is relevant only in its general relationship to travel time; alone it is an insufficient measure of the effort required to maintain contact between two nodes. A more useful measure of distance is a travel time index that provides a travel path textured by the resistance of sea or land to progress, taking into account, at minimum, the effects of winds and currents. Maritime travel must take into account the common voyage that requires one day out, but four or five days return because of winds and currents. Carl Knappett and colleagues (Knappett et al. : 1021) stress that actual travel times should replace physical distance as quantitative input in network models, but they were not able to quantify this variable in their first attempt (see below). David Conlin (: 179–84) presented a mathematical model to simulate the effects of wind and currents on travel time, and Justin Leidwanger () has made a start at texturing the surface of the sea with wind and current data using Geographic Information Systems (GIS) software. Soon these various trajectories will surely converge so that we can systematically address observations such as that of Agouridis (: 19), who points out that environmental conditions can make islands in close geographic proximity "distant" in terms of potential for interaction.
Although travel time parameters are to an extent contingent on specific journeys, habitual face-to-face interaction places broad limits on scale. The geographic extent of a small world depends on the environmental configuration of the seascape (exposure to currents, waves, and weather patterns), seafaring technology, and strength of the relationships among members. These factors tend to be mutually reinforcing at the local level: the greater predictability of weather conditions for short-range journeys promotes access by sea, virtually year-round in many cases. Frequent contact facilitates the establishment of strong economic and social ties, potentially including kinship and social storage relations. As these relations develop, interaction becomes more frequent. Thus begins a history that, because of the density and strength of personal ties, may endure through the boom and bust cycles of larger-scale political and economic entities. Durable is not the same as immutable, however. Small worlds are not immune to the effects of environmental shifts or catastrophes, internal conflict and power shifts, or external developments.
The maximum daily range of a Bronze Age sailing ship of between 100 and 150 kilometers (Broodbank : 345, table 12; Knappett et al. : 1014, opt for the lower figure of 100, which I follow here) should set the outer limits of what we might call "local," and thus the greatest expanse of a LBA maritime small world from end to end. Beyond this, we move into the realm of open-sea or night voyaging, with their requirements of advanced skills and knowledge and the higher probability of unfamiliar seas and coasts, as well as unforeseen weather emergencies. In practice, even if we imagine short hops on long itineraries, the norm for a small world should be smaller in scale. The fishing boat, the rowboat, and the coasting vessel would be more the rule than the large sailing ship, which might operate from a major harbor (Table 6.1). By contrast, smaller boats would require only informal anchorages offering a sandy strand, or even something like the innumerable tiny docks bearing the name skala (from the Italian scala) for the few rock-cut steps used to board a vessel (Constantakopoulou : 222–23). Maritime travel within the small world would thus remain primarily in the realm of the nonspecialist.
Specific to the small world are three specialized applications of social and economic connectivity that are worthy of mention as distinctive of the Mediterranean: the phenomena of the "goat island"; the porthmeutike, or short-distance ferrying of goods, people, and animals; and the peraia, here defined as a coastal region controlled by an island lying opposite (Constantakopoulou : 200–26). Goat islands refer somewhat more generally than the name suggests to small, offshore islands and islets (inhabited or uninhabited) that are used to expand agricultural and pastoral production. These have been a prominent feature of Aegean seascapes since antiquity: Christy Constantakopoulou (: 200–14) documents many Aegean examples from ancient literary and epigraphic sources, beginning with Thrinacia, where Helios' cattle and sheep were pastured in the Odyssey (12.127–30), and mentions several modern instances. Others have substantiated this practice with archaeological and ethnographic evidence, as for example, in the work of Nick Kardulias, Timothy Gregory, and colleagues in the area of the Saronic Gulf. Their teams performed architectural and surface artifact surveys on a number of offshore islands and islets, and if those were occupied, they interviewed residents. On the small, waterless island of Evraionisos in the western Saronic Gulf (dimensions 1000 × 400 meters), they found evidence of many periods of use, most prominently LBA and Late Roman (Kardulias et al. ). The discovery of LH IIIB and IIIC artifacts, along with the foundations of fortification walls possibly of Mycenaean date, is consistent with widespread evidence of use of near-shore islands in the LBA (Hope Simpson ). The presence of cisterns, fortifications, and other durable structures suggested to the researchers the exploitation of marginal niches during times of economic and demographic expansion (Kardulias et al. : 17). The island, lacking water and arable land, could be used as a lookout and for grazing sheep and goats, but it could not long sustain occupation without a lifeline to the mainland. The success of productive activity on what amount to tiny rocks protruding from the sea depended on certain conditions of connectivity; specifically, the economic expansion of the Mycenaean palatial period made the exploitation of Evraionisos viable by enhancing connectivity and incorporating new nodes into an ever denser web – in this case at the level of the small world.
Ethnographic and ethnoarchaeological work directed by Kardulias on the island of Dokos off the coast of the southern Argolid examined the adaptations of modern herders (Kardulias ). These observations help to fill in the many gaps in our knowledge of the use of such islands in antiquity with plausible adaptation scenarios. Dokos, also waterless, is more strategically located than Evraionisos on a sealane between Cape Malea and Athens. Yet in a variety of ways, the inhabitants are equally dependent upon external connections for survival. In order to make optimal use of their scarce resources, they have adopted a mixed subsistence and settlement strategy. Kardulias learned that the 22 families living on Dokos in 1945 herded sheep and goats, grew wheat and olives, maintained domestic gardens, foraged for wild plants, and collected water in cisterns for human and animal use. Yet in spite of their risk-buffering behaviors and attempts at self-sufficiency, this economy could only function if articulated to the markets and resources of the larger island of Hydra and the mainland around Hermione. On Hydra they sold animals, cheese, and milk. In addition, many of the men worked away from Dokos for much of the year. Some were sponge divers in far-flung Aegean locations, others did wage work on Hydra, and still others rented pasture on the mainland for about half the year because grazing on Dokos was inadequate. The rent for these grazing rights was usually paid in kind with milk, cheese, and wool. Thus, even a "goat island" witnessed multifaceted economies (reminiscent of the "traditional" Mediterranean economy: Butzer ; Halstead ) closely linked to larger economic nodes and structures (Kardulias : 38–39). Evraionisos and Dokos provide an intriguing glimpse at how the small parts of a maritime small world might have fit into the local-scale Bronze Age economy.
The practice of ferrying people across short expanses of sea (porthmeutike) is well attested in Greek literature and inscriptions; the routes between Attica and the Saronic islands and between the Greek mainland and Euboea seem to have been especially heavily traveled in classical times (Constantakopoulou : 222–26). This sort of short-distance traffic must have been a ubiquitous feature and a fundamental mechanism of connectivity in a small world: a lifeline for inhabitants of offshore islands, and the means to maintain social ties among coastal communities. Interestingly, Kardulias' informant on Dokos used one of his own boats to transport tourists to and from the various coastal towns and islands in the vicinity (Kardulias : 42).
The island–peraia relationship that Constantakopoulou describes from literary and epigraphic sources for the Classical and Hellenistic Aegean involves a powerful island state possessing coastal lands on the mainland opposite; often these territories were quite large, but susceptible to expansion or contraction with shifting political fortunes. Such relationships could have existed in the Bronze Age, but it is important to note that Constantakopoulou's peraiai are explicitly political possessions, even if a primary motivation for holding one must often have been economic (e.g., control of mines or agricultural land). Certainly there were powerful island centers in the Bronze Age Aegean, but control of any kind extending to the mainland, economic let alone political, is something to be demonstrated and not assumed. This question will be addressed in Chapter 7 with regard to Aigina's relationship with Saronic coastal settlements.
How and why, then, do small worlds cohere? My definition is intended to address that question. Connectivity and interaction explain how, but the question why is more complex and less amenable to resolution by archaeological means. Small worlds are not determined solely by environment or geography, although proximity is tautologically essential to their configuration. Despite the fundamental influence of environmental factors, small worlds are "culturally defined unities" (Broodbank : 175) that result from conscious decisions to forge connections with nearby communities. In constructing his network analysis of EBA Cycladic interaction spheres, Broodbank (: 176–77) considered population growth, maritime travel ranges, and climatic variability with respect to resources as potential engines for the formation of local-scale interaction spheres, before deciding to emphasize the first of these. In Broodbank's model, population increase (simulating the rise in population from the Neolithic to the peak of complexity in EB II) altered the number, size, density, and location of small-world clusters in the Aegean; a similar outcome might be expected for an analysis tracking population rise on mainland Greece from the EH III–MH II demographic crash through the Mycenaean palatial period.
Renfrew (: 10–11) enumerated eleven social and economic motivations for travel to engage in various types of material- and nonmaterial-oriented interactions: to trade one's goods, to obtain others' goods, to participate in social gatherings, to seek knowledge or wisdom that will impart prestige, to visit a distant holy place as a pilgrim, to train or learn a skill, to find work or a better living situation, to serve as a mercenary, to find a spouse, to visit relatives or friends, and to serve as a sanctioned emissary. From this list we might extrapolate some more specific strategies that center on social ties that helped to bind communities into small worlds. The dispersal of kin or descent groups may originate in periods of low population density and limited mobility, when communities must look outward for intermarriage as a way to widen the gene pool and ensure reproductive viability, and for economic accommodations to counteract resource variability or unforeseen shortfalls. Along with these relationships come various social obligations that might include periodic gatherings for feasts or other kinds of rituals promoting solidarity and group identity. The maintenance of kinship ties by means of inter-island voyaging has been the topic of detailed ethnographic studies, particularly in Oceania (e.g., Hage and Harary ; Hage and Marck ). We might also imagine economic interests born not of necessity but of desire: both within and beyond small worlds, we observe in the archaeological record a widespread preference for Melian obsidian over locally available chert for stone tool manufacture, or for Aiginetan cooking and storage pottery over locally manufactured functional equivalents.
We should also consider certain proximity effects. The phenomenology of lines of sight among island and mainland coasts created an everyday visual world that invited interaction and inhibited isolation. The rough geographic limits that I have suggested for small worlds distinguish them from larger spheres of interaction because they represent a range of routine travel without the need for specialized nautical and navigational technology (given the technology of the time). To emphasize the point that small worlds should be more stable over time than larger spheres of interaction, we can observe in the historical record that favorable economic and political conditions for cross-cultural, long-distance trade wax and wane: complex networks collapse, as the great eastern Mediterranean state system did circa 1200 BC; the seas may be infested with pirates, etc. Under any conditions, small worlds may lack the capacity to participate in larger interaction spheres. Thus, small worlds would be characterized by certain kinds of economic transactions involving face-to-face trading between producers and consumers, or small-scale redistribution from key coastal nodes. In Renfrew's framework, home-base reciprocity, boundary reciprocity, central-place redistribution, central-place market exchange, and localized down-the-line trade could all exist in a small world (see Fig. 2.5). Yet any small world could contain a major harbor – even a colonial enclave or port of trade – that articulated the local area to regional or interregional networks, bringing in other forms of interaction, including cabotage or directed emissary trade. In such cases, the different spheres of interaction blend, and it may not be possible to separate them, either conceptually or in the archaeological record. For example, the Pseira (Hadjidaki and Betancourt –2006, ) and Point Iria (Phelps et al. ) shipwrecks are compatible with small-world connectivity. But we cannot be sure whether the Point Iria ship was Cypriot, having traveled a very long distance to Crete and then to the Argolid; a Cretan ship moving Cypriot goods with their own; or a local boat plying the Argolic Gulf with nonlocal transport vessels that had been recycled numerous times – and still other scenarios are possible. We can speak more confidently about short-haul trade regarding the Pseira ship. The spread of material at the wreck site, along with the utilitarian function and local provenience of the MM IIB transport amphoras and hole-mouthed jars in fabrics of the Mirabello region (Hadjidaki and Betancourt –2006: 84–85; P. P. Betancourt, personal communication, 2011), indicate a small ship operating in a small maritime world centered on the Gulf of Mirabello.
There are several models of local- or small-scale maritime networks in the ancient Aegean, but their logic and geographic scale do not necessarily match those presented here. As is the case for coastscapes, Horden and Purcell's microregions and "definite places" may be entirely terrestrial, and when they include coasts, they may also encompass several interior zones. The "mini island networks" of Constantakopoulou (: 176–227) are more in line with the scale and maritime focus of my small worlds; however, they are explicitly political clusters of islands and it is not clear how she would map economic networks for the Classical and Hellenistic Aegean. Sherratt and Sherratt () include the term small world in the title of an oft-cited article, but the text has little to do with local-scale connectivity, and in fact the term itself is not repeated in the body of the article. If anything, the networks they describe are long-distance and cross-cultural. Similarly, Irad Malkin envisions Greek colonization of the eighth to sixth centuries "...turning the vast Mediterranean and the Black Sea into a 'small world'" (Malkin : 5). This radically more expansive definition follows closely the "small world networks" as imagined by social network analysts (Watts and Strogatz ). In social network theory, small-world networks emerge when the addition of a few key links joins smaller, neighboring clusters, creating paths and shortcuts to more distant clusters. The network proximity of these models need not equate to physical proximity (Leidwanger : 89). To be sure, shortcuts and direct long-distance connections link small worlds (or coastscapes within small worlds) to regional and interregional networks, even by bypassing the members of the local small world altogether. When they do so, however, they are not operating in small-world spheres of interaction (as I understand them), either geographically or conceptually.
My small worlds find closer parallels with the "clusters" of social network analysis (Scott : 129–33). Clusters are aggregates of points (or nodes) that form high-density areas in a network graph, and that separate from other clusters. Since not all points within such a cluster need be adjacent (e.g., the cluster can be elongated with many points intervening between end members; Scott : 131, fig. 7.2:ii), this pattern of network proximity could be comparable to the geographic configuration of an elongated maritime small world with many coastscapes arrayed along its length. An entity that exists below the level of the cluster is the "neighborhood," which encompasses all the points to which a given point is adjacent, or connected directly by a single line or step. Thus, conceptually at least, clusters could correlate with my definition of small worlds and neighborhoods with fragments thereof consisting of neighboring coastscapes, but caution is warranted regarding the concept of distance. The distance between two points in a network analysis is defined as the length of the shortest path connecting them, where the length of a path equals the number of lines and intervening points it takes to get from one point to the other. This may or may not approximate a realistic travel itinerary, but in any case it does not represent geographic distance and it does not address social or environmental friction to interaction.
Broodbank's small worlds seem to provide the closest fit. He repeatedly refers to small worlds as "local worlds" or "local interaction networks" (Broodbank : 175–76), and by locating a number of small worlds within the Cyclades, adopts a geographical scale that is commensurate in magnitude with those I advocate, for example within the Saronic or Argolic Gulf. Many of his criteria for linking nodes into networks, mentioned above, are adopted in the chapter to follow when practicable. The most important criterion for his network analysis, population change, will be difficult to quantify for many periods in the absence of good cemetery data. Along similar lines, James Wright () has proposed that social groups in the Bronze Age Aegean formed at scales he calls local, locality, and regional. For Wright, the local consists of a community and its territory, and the locality embraces several interlinked communities "among which interaction is so common as to regard them as extensions of each community" (Wright : 806). In principle, local and locality correlate closely with my notions of coastscape and small world, respectively, although many of Wright's examples are drawn from internal valleys with primarily or exclusively overland interactions. In a more maritime context, Wright (: 808) wonders whether the island of Aigina was a single locality or rather consisted of multiple localities in the MBA. This is a question that searches beyond the coastscape to explore the internal relations of an island, analogous with examining the connections of coastal settlements with interior hinterlands. Although this is not the primary focus of the present study, fleshing out these relationships will indeed be a crucial part of the analysis of coastscapes and small worlds.
Lastly, there is the question of whether the earliest Greek literature – Homer or Hesiod – might illuminate the maritime small world. For instance, do the contingents of towns mustering ships in the Catalogue of Ships in Iliad Book 2 make up something like small worlds? Many contingents are composed primarily of men from inland locations, but if we consider the maritime regions of the northeastern Peloponnese, the organization of the realms is curious from the point of view of both network logic and archaeological evidence. Agamemnon's fleet draws men from a vast hinterland stretching north, encompassing the northern Corinthia, and west to the truly distant Achaian towns of Helike and Aigion. Such a realm makes little economic or political sense, given the difficulty of connectivity and the sheer distances over this rugged terrain. Moreover, there is little archaeological evidence to support a direct presence from Mycenae in the northern Corinthia and Achaia, but instead a great interest in the Argolic and Saronic Gulf regions (Pullen and Tartaron ; Tartaron ), which in the Iliad are controlled by other men. The Saronic Gulf itself is carved into three separate contingents: one from Athens, one from Salamis, and Diomedes' contingent of nine towns on the Saronic and Argolic Gulfs. A political division along these lines could conceivably exist, though nothing like it is known from the Bronze or Early Iron Age. It certainly does not map well onto the logic of the maritime small world in social or economic terms.
Hesiod's adventures in sea trade and his crossing from the Greek mainland to Euboea (Works and Days 619–94) are more credible, reflecting the kinds of interactions one would expect within a small world. The part-time maritime pursuits of the farmer in early Greece, operating as a local-scale merchant of his produce, exemplify the nonspecialist nature of much seafaring in a small-world context. Hesiod's crossing to Euboea to perform at funeral games was likely made on one of many ferries engaged in the kind of porthmeutike routes mentioned above. Despite the omnipresence of the gods in the Works and Days, Hesiod paints a realistic and unflinching portrait of agricultural life and the pragmatics of local politics, one which inspires more confidence in its utility for illuminating small-world activities.
#### Regional/Intracultural Maritime Interaction Sphere
New problems are encountered when trying to discern the transition from the local context to something larger, a medium scale that we might call regional, and then again from the regional to the interregional or international. It has always been tricky to define the medium-scale region, since one could use many criteria: geographical or environmental regions delimited by transitions from one climatic regime to another, or from one gulf, basin, or other oceanographic feature to another; economic regions based on the extent and intensity of trade relations; social regions based on the distribution of language or cultural traits such as architecture or religious practices; political regions based on territory controlled by a state; or some combination of these. For prehistoric periods, the absence of texts (or near absence, in the Mycenaean case) forces reliance on reading from the surviving archaeological remains the unifying factors that make up a region.
A crucial distinction must be made at this juncture. By region, I am referring to maritime regions and the scale at which connectivity at sea – measured by technology, travel times, social relations, and other environmental and human factors – permits extralocal interaction spheres to cohere at any given moment. This is very different from terrestrial regions, where a different set of topographic and cultural constraints causes territories to coalesce or fragment into what we call "regions." We should not expect them to map onto one another very closely. For the Mycenaean world, I prefer to define medium-scale regional maritime interaction as (1) occurring in a geographical space beyond the small world, and thus outside the realm of habitual, face-to-face interaction; but (2) still characterized by participation in a recognizably Mycenaean material culture and, to the extent that we can know, common customs and beliefs; and (3) occurring less frequently than interactions in the small world, but considerably more often than those involving long-distance, cross-cultural relations. This definition necessitates speaking of a Mycenaean "culture area." Such an area was neither static nor delimited by hard boundaries; nor was it continuous in space. Attempts have been made to draw lines around a map of the Mycenaean world, sometimes distinguishing between a "core area" and a "periphery" (and more recently, "semi-periphery" and "margin"; Feuer , ; Kardulias , ; Kilian ; Parkinson and Galaty 2009a; Sherratt and Sherratt ). Questions arise about the outer limits of a Mycenaean culture area: Was a maritime voyage from Mycenae to Knossos on Crete in LH/LM III an intracultural or cross-cultural journey? The answer depends on how pervasive one believes the Mycenaean presence was on Crete at the time, and on that point there is vigorous disagreement (Burke ; various articles in Driessen and Farnoux ; Preston , ). In a similar vein, would the Mycenaean (or is it Mycenaeanized?) settlement at coastal Dimini in Thessaly (Adrimi-Sismani ; Pantou ) be considered part of the Mycenaean cultural sphere? Probably so, but what about sites like Assiros or Toumba Thessalonikis in Macedonia (Andreou and Kotsakis ; Buxeda i Garrigós et al. ; Wardle ), where contact with the Mycenaean world is apparent in pottery styles and decorations, which exist however in thriving local, non-Mycenaean, settlements? Probably not, but what factors determine inclusion or exclusion?
For the purposes of the present analysis, and from the perspective of a sea traveler, I suggest that intracultural interaction should include those places where a ship's captain could expect to find a substantial population of people speaking Greek and observing recognizable customs, such as ritual, funerary, and domestic practices. Other conformities with material culture, such as town planning and architectural design or pottery forms and decorations, were more variable and might not have been so central to Mycenaean identity. Roughly, this "Mycenaean maritime culture region" would encompass the Aegean Islands and the Aegean coasts of the Greek mainland as far north as the Bay of Volos; Crete from LM IIIA if not earlier; and the coasts and islands of the Ionian Sea as far north as the Ambracian Gulf, but would exclude much of the Aegean coast of Asia Minor north of Cape Mykale (Fig. 6.1). A regional, intracultural sphere of maritime interaction defined in this way is of course open to numerous objections. There are many areas within it, both coastal and inland, that by any definition were only tenuously Mycenaean; conversely, there are a few plausible Mycenaean coastal colonies and points of frequent contact beyond these areas, such as at Glykys Limin on the southern coast of Epirus (Tartaron : 145–77), and in southern Italy and Sicily (but see Blake for a strong challenge to the notion of Mycenaean colonies there). Nonetheless, keeping in mind the fuzzy, shifting, and discontinuous nature of the Mycenaean cultural sphere over the course of the LBA (Tartaron , ), it should be possible to carve out hypothetical regional interaction spheres within the confines of this map.
6.1 Map of a hypothetical Mycenaean maritime culture region.
The entire area in question is not large, and a sailing vessel should have been able to traverse most of it within several days or less. But once a ship and its crew broke free of the confines of the small world, the nature of seafaring changed, with different actors and different agendas. We can now assume the presence of specialist seafarers, members of the kind of maritime community described in Chapter 4, who possessed specialized knowledge of navigation and of the natural and social conditions existing at distant places. Their ships would now have to be open-seaworthy, and perhaps more purpose designed to optimize for hauling cargo or transporting personnel over longer distances. Because of the topography of the Aegean archipelago, extended trips could be made by coast- and island-hopping, minimizing the need for frequent open-sea or night sailing.
At regional scales, different modes of trade become prominent, including down-the-line and freelance (cabotage) trade as sailors move beyond the realm of habitual contacts to a world inhabited by comprehensible, yet increasingly unfamiliar, people and places. This aspect of the regional is similarly evoked by Wright (: 806): "...contact and travel among these localities is regarded as a departure from the safe and familiar."
Regional spheres of interaction are best measured by material culture studies that define the distribution of chronologically and culturally sensitive artifacts, features, and practices. Traditional methods of stylistic and formal analysis of pottery, stone tools, weapons, jewelry, seals and sealings, wall paintings, and other objects can be combined with archaeometric analyses to trace the movements of goods and illuminate the social dimensions of production, distribution, and consumption. This kind of synergy is illustrated, for instance, by the way that Penelope Mountjoy's stylistic study of regional variation in Mycenaean painted pottery (Mountjoy ) is informed by the results of chemical and petrographic analyses undertaken over several decades, most recently by the Bonn group (Hein et al. ; Mommsen et al. ).
The archaeological record points to a material culture koiné gradually enveloping most of the Mycenaean core area in the palatial period. This trend culminates in the mature stages of LH IIIA, continues into LH IIIB, but diminishes in the second half of the thirteenth century in LH IIIB2. Goods and information flowed relatively freely within the Mycenaean world. Imports were common, but style and technology were also transmitted, as attested by local imitations of widely disseminated artifact types. Petrographic analysis has demonstrated that pottery shapes and decorative styles were often imitated, using not only local materials, but also fabric recipes and manufacturing techniques specific to local potting communities. For example, during MH the spread of fine gray burnished ware across central Greece, the Corinthian Gulf, and the western Peloponnese, or the distribution of lustrous decorated ware that links Kythera, Laconia, and the Argolid, can be said to mark out intracultural regions of economic interaction (Wright : 809–10).
Only a few mentions of intra-Aegean trade exist in the Linear B tablets: most notably, Mycenae Tablet X508 records the transfer of a type of cloth to Thebes (Bennet : 201–202), and exchanges between Thebes and the apparently subordinate towns of Karystos and Amarynthos on the island of Euboea involve the movement of wool to Euboea and pigs, goats, sheep, and cattle to Thebes (Chadwick : 99; McInerney : 64; Palaima : 106). These latter interactions parallel Hesiod's short journeys from Boeotia to Euboea, and seemingly reflect the geographical extent of the Theban state and possibly a microregion of the scope described by Horden and Purcell. Various explanations have been advanced for the scarcity of references to intra-Aegean exchange, which patently contradicts the archaeological record. In view of the thousands of extant tablets that document other aspects of the economy in considerable detail, we can rule out accidents of preservation; and given the archaeological evidence it is difficult to argue that the interactions were simply infrequent. The most plausible explanations center on the palaces' administrative practices: if these exchanges were seasonal or irregular, they may have occurred outside the limited time frame of the administrative cycle preserved by the destructions, or they may have belonged to a different, perhaps higher, administrative level not recorded on clay tablets (Bennet : 202). It is also possible to suggest that most such connections had been interrupted by social disturbances in the months and years leading up the collapse of the palaces. Nevertheless, the homogeneity of the Linear B script, the language it represents, and the administrative system it served at the palaces indicate a significant level of intracultural interaction, at least at the elite level, which complements the archaeological evidence for the interconnectedness of the Mycenaean polities.
#### Interregional/Intercultural Maritime Interaction Sphere
Continuing the logic of the expanding geographical and cultural scope, the interregional/intercultural maritime interaction sphere involves interactions and networks that extend beyond the Mycenaean maritime culture area. As outlined in Chapter 2, Mycenaean extra-Aegean connections are often deduced by plotting the distribution of Mycenaean objects found beyond the Aegean and the non-Aegean objects found in Mycenaean contexts (Burns : 36–40; Cline ; Lambrou-Phillipson ; van Wijngaarden ). The historical sketch and the discussion of the problems of interpreting the very partial evidence presented there need not be repeated, but a few key points might be emphasized (see Table 6.1).
We do not know how often, how far, or from where, Mycenaean ships might have ventured beyond familiar waters. Sporadic visits of Mycenaeans to far-flung lands beyond the Aegean seem assured for Cyprus and the northern Levantine coast in the East, as well as the shores of southern Italy and Sicily in the West. The catalogue of imported objects, which we might have expected to travel home with Mycenaean merchants, is neither impressive quantitatively nor widely distributed geographically, however. A dominant role for ships operated by intermediaries based at Cypriot or Syro-Canaanite entrepôts (represented by the Uluburun and Gelidonya wrecks) in maintaining long-distance networks cannot be discounted. It is likely that a series of intermediate nodes was interposed between the East and the Mycenaean heartland, with down-the-line and freelance trading the rule and direct voyages or diplomatic missions the exception. Around the rim of the Aegean, Crete, with its long history of engagement with the East, and Miletos or Rhodes, enjoying proximity and close contact with the Hittites, were well positioned as ports of call and transshipment points. Whether ships were controlled by palaces or independent merchants, there was ample opportunity to mix private enterprise with official business. Long-distance voyaging was the realm of seagoing ships manned by master navigators and seasoned sailors with knowledge of open-sea sailing and experience with sea lanes and hazards en route to distant places. Traversing intercultural space, these ships entered ultimately into the realms of intracultural regions and small worlds, instantiating the intersection and overlap of the nested scales of interaction described in this chapter. Once inside a small world, the xenoi relied on their accumulated knowledge and the accommodation offered by locals for safe landfall at one or more anchorages. This would also have been the case for ships traveling long distances within the Aegean world, as when a ship from the Argolic Gulf made landfall in a Cretan small world at the Gulf of Mirabello. The case for Mycenaean permanent presence in enclaves or colonies beyond Aegean and Ionian shorelines is equivocal at best and dubious at worst.
In closing this section, it needs to be stressed that geographically defined spheres of interaction are always contingent to time and place. I have chosen to define four maritime interaction spheres based on geographical scale, frequency of interaction, and cultural identity. They are meant to fit what we know about the political organization and maritime technology of the LBA, as well as the moderate distances and dense distribution of islands that characterize the Aegean region. It would make little sense to apply these spheres without modification to the widely dispersed islands of Oceania, or to the enormously different conditions of political organization and maritime technology of Greece during the Roman period. Even in the Bronze Age, we can observe coastscapes emerge, thrive, decline, and disappear; small worlds alternate between cohesion and fragmentation; and long-distance connectivity of Aegean polities with the eastern and central Mediterranean fluctuate over time with the political and economic conditions of the day. For this reason, our frameworks must be flexible enough to accommodate a diachronic history of adjustment and reconfiguration. In the following chapter, I will attempt to show how such histories can be written within the maritime cultural landscape framework.
### Connectivity and Social Network Theory
Following Fernand Braudel (), a central theme of Horden and Purcell's The Corrupting Sea is that the extreme fragmentation of Mediterranean coastlands and islands encourages intensive local interactions by sea, while relatively easy maritime communication allows these to be expanded to form larger networks as environmental, economic, and political conditions permit. The emphasis on microregions as the fundamental building block for networks of all sizes is no accident: "The short hops and unpredictable experiences of cabotage are...the basic modality for all movements of goods and peoples in the Mediterranean before the age of steam" (Horden and Purcell : 365). Malkin et al. (: 1) describe the symbiosis of two tiny, neighboring Aegean islands, Herakleia and Schinoussa. Herakleia's harbor is protected from the southeast, Schinoussa's from the northwest. As winds and weather change, fishermen and yachtsmen rush to move their boats from one island to the other – a relatively common occurrence that illustrates another element of Horden and Purcell's paradigm, the instability and unpredictability of the Mediterranean climate. The coastal world (or "mental horizon") of these islanders consists first of their village; second of their linked harbors; and third of the Aegean and Mediterranean beyond. This conceptualization of the maritime world conforms well to the coastscapes, small worlds, and larger spheres of interaction proposed in this chapter. Malkin and colleagues (: 1) call the relationship between Herakleia and Schinoussa a "fractal" of Mediterranean networks: operating at a small, local scale, but exhibiting dynamics that can extend to the whole of the Mediterranean. This analogy from the natural world is an intriguing one, but debatable – is the architecture of a local maritime network repeatable and remarkably similar at any scale? Because of the quantum leap in knowledge and professionalism, as well as the different kinds of exchanges that characterize the transition from small worlds to larger spheres of interaction, I am skeptical that this should be so.
Yet these scholars rightly question the notion that autarchy or true isolation existed in the Aegean, insisting instead that microregions and microecologies, while distinctive, cannot be separated from the wider networks of which they form a part. Balanced against the certainty that in aggregate, small-scale interaction moved far more goods and people than did long-haul traffic, as well as the likelihood that rather few Bronze Age coastal dwellers ever ventured far from home, must be the realization that the ways that microregions interact and form clusters (i.e., connectivity) are as important as their internal features. As Horden and Purcell (: 465) put it, "The wider historical context is as potent a factor in the workings of the microecology as is the local physical environment or the human responses to it." Still, I would defend the separation (however fuzzy) of the different scales of interaction for analytical purposes, in agreement with Michael Galaty and colleagues (Galaty, Parkinson et al. 2009: 43), who caution that they possess contrasting dynamics that cannot be revealed when conflated.
The ways that connectivity works highlight both the corporate responses of (to us, faceless) societies to opportunities and constraints of the physical and social environment, and the decisions and actions of intrepid individuals and small groups. Horden and Purcell's () "four definite places" and the hundreds of other examples they offer to illustrate historical connectivity draw attention to the insights that emerge from written documents and the limits of our knowledge in the absence of them. They offer a kind of template for the establishment of maritime relations that accords agency to individual coastal settlements and their inhabitants: two ports, located a short voyage apart, have goods to exchange and people whose mutual interests and understanding of each other promote friendly relations. They make a pact to encourage exchange, and set about improving their port facilities. Local agency is a common feature of social networks, which are often understood as self-organizing systems "...due to the local decisions made by the individual vertices" (Barabási and Albert : 512). Once established, such relations can become durable, carrying on in spite of the vagaries of environment, politics, and economics in the wider world, and even if the rationality and expediency of the connection itself is lost (Horden and Purcell : 128). A reason for this may be the deep social ties that have developed over time. In periods of growth, the information, resources, services, and people flowing along these paths stimulate the creation of new vertices and paths, resulting in clusters that coalesce at larger and larger scales; thus, the formation of world systems from microregions.
### Network Models and the Aegean Bronze Age
Two important attempts have been made at spatial modeling of maritime networks in the Aegean Bronze Age: Broodbank's () for the Cycladic Islands in the EBA, and that of Knappett and colleagues () more broadly for the Aegean, with a focus on the MBA. Network analysis has not been explicitly applied to the Mycenaean period.
Broodbank's network model consists of a Proximal Point Analysis (PPA) that simulates interaction networks given certain assumptions about the number and location of interacting nodes (in this case, settlements). PPA predicts patterns of connections between points distributed in space, conventionally by connecting each point with the three closest to it. The webs formed by these connections generate network clusters, as some points accumulate more links by virtue of their proximity to a larger number of other points. The denser clusters hypothetically mark out interaction "centers" where communication ought to flow most easily. The actual placement of points in the model is a problem given the fragmentary nature of the archaeological record. Broodbank addresses this by placing known sites on the map and then proceeding to add points to simulate the growth of population over time. The number and location of these points is determined by varying the amount of land area required to place a point on an island. He creates four different models (PPA 1–4) by adding a point for every 150, 100, 75, and 50 square kilometers, respectively (Fig. 6.2), and then compares the results to the apparent settlement patterns of the Neolithic to EBA Cyclades. PPA 1, with only 19 points, is taken to resemble the Neolithic–EB I pattern of low-density networks in which larger settlements and longer-distance connections are vital for survival. By contrast, the more heavily populated and highly connected EB II is simulated best by PPA 4, in which 54 points yield more localized, high-density networks and maximal small-island participation.
6.2 Broodbank's PPA versions 1–4, based on different initial and growth conditions. Broodbank : 184, fig. 53. Courtesy of Cambridge University Press.
Broodbank's PPA is, like any other model, laden with assumptions, simplifications, and choices that affect its utility in representing reality (Knappett et al. : 1010). In this case, it is assumed that communities interact most intensely with their nearest neighbors, since longer journeys are riskier and more time consuming. Sea travel is taken to be uniform in all directions. Sites are deemed to be of roughly equal size and distributed evenly in space among the islands. The links between them are similarly undifferentiated: one node can connect with any other directly or through a series of short hops as constrained by the available propulsion technologies of rowing and paddling. The only variables that can be adjusted to induce change are the number of sites in the system or the number of links each site can form. While this set of rules and assumptions obviously oversimplifies and distorts the reality of these networks, it is important to recognize that Broodbank's PPA was designed for the limited geographical world of the Cyclades and for a time in which boats were propelled by human power. Settlements were smaller and less hierarchically organized than in the later Minoan and Mycenaean palatial worlds. Within these bounded circumstances, the model can claim some success in explaining the presence or absence of certain centers around the Cyclades. For example, no such center is known on the large island of Andros, which in spite of possessing fresh water and arable land, is shown to be out of the mainstream of near-neighbor connectivity; conversely the settlement of Daskaleio-Kavos flourished on tiny, resource-poor Keros in the Erimonisia group by virtue of its position in a dense web of crisscrossing links. The fit, in terms of prominence or insignificance of sites, between the model and the archaeological record is not perfect, but Broodbank's analysis does demonstrate that network centrality was an important factor in the role that specific settlements played in maritime connectivity and in the way that clusters of small worlds were constructed. Yet PPA is limited in its application – it is not likely to simulate eras with large travel ranges well, or translate easily to greater geographical scales.
Knappett et al. () sought to build on Broodbank's beginnings to devise a more sophisticated network model with wider applicability. Their model, which they characterize as "imperfect optimisation," uses a detailed mathematical equation to express the notion that participants in a network tend to strike a reasonable, though never perfectly optimal, balance between the costs and benefits of maintaining maritime connections. The model is far more flexible because it incorporates more of the variables that influence connectivity and allows the weight of each variable to be modified, either experimentally or to reflect current understandings of the archaeological record.
To assess the likelihood of connection between two sites, or the connective potential of any single site in a network, each site is coded with several variables, including a kind of estimate of importance based on site size, population, and available resources. Where the connectivity between any two sites is concerned, the energy required to maintain contacts is a combination of the physical distance between them and the fraction of effort each devotes to the interaction. These values lead to an equation:
where H yields a quantitative representation of the energy balance between the costs of supporting the local population (P) versus the costs of maintaining links (T), and the benefits of exploiting local resources (R) versus the benefits of maintaining links (E). The characters κ, λ, j, and μ are constants that record the relative importance or weight accorded to each factor. The flexibility of this model resides in accommodating a broader number of variables that can influence connectivity, as well as the ability to alter variables as population, settlement patterns, or technologies (e.g., seafaring or mining) evolve. At the same time, the ratios of the constants can be modified to reflect different cost–benefit relationships between local affairs and nonlocal interaction.
Even with this more sophisticated and encompassing approach, Knappett's model carries its own assumptions and simplifications. A central assumption is the network centrality of large sites, like Knossos, which are more connected than smaller sites and attract new connections preferentially because of their greater ability to acquire and control the resources needed to sustain and benefit from overseas contacts. Further, large communities tend to target each other, creating longer-distance connections and network hierarchies. This "gravitational pull" can aid in linking distant settlements and holding large-scale networks together. The disproportionate role of well-positioned nodes finds strong support in network theory. Albert-László Barabási and Réka Albert () describe two common properties of networks: continuous growth by the addition of new vertices (i.e., nodes or points), and preferential attachment by which new vertices attach disproportionately to sites that are already well connected. These patterns can be observed in many kinds of social networks, such as the linking of documents in the World Wide Web or the preferential citation of certain articles in scientific literature (Barabási and Albert : 510). These dynamics have several implications for Aegean Bronze Age maritime networks. One is that a node that acquires more connections than another one will accumulate them at an increasing rate, causing the difference in connectivity between the two to multiply as the network grows (Barabási and Albert : 511). Such conditions may help explain the emergence of Mycenae during the Shaft Grave Era, with its wide access to exotic goods, or the rise of Knossos to an unparalleled position on Crete. Furthermore, the rapid growth in disparity between well- and less well-connected sites may result in the kind of explosive growth to prominence that seems to describe Mycenae in the Shaft Grave Era.
With the advance of seafaring technology and the emergence of large centers in protopalatial Crete, conditions were set for Aegean-scale networks to grow, requiring a model of greater scope and variability than Broodbank's PPA. Knappett and colleagues published a few variations on the imperfect optimisation model, including one in which the benefits of trade were incrementally increased (λ = 1.0, 2.0, 3.0, and 4.0: Knappett et al. : 1015–1016, fig. 4). At each increment, the links between geographically distant areas of the Aegean – the mainland, Cycladic Islands, Crete, the Dodecanese and Asia Minor – strengthened and particularly well-positioned sites such as Akrotiri on Thera became crucial "intermediate" nodes in holding the larger network together, in spite of their modest size. Removing these nodes, as when Thera was destroyed in a volcanic cataclysm in the middle second millennium BC, can (and did) cause major disruptions in the broader network (Knappett et al. ).
When a PPA analysis was performed on the same data set according to Broodbank's rules, however, Crete failed to link to the Cyclades, and the Dodecanese Islands were entirely isolated from all other Aegean networks (Knappett et al. : 1019, fig. 7). This is hardly surprising as it is difficult to generate large-scale networks when a node can only connect to its three nearest neighbors. Social network theory predicts the constant addition of new vertices and the creation of shortcuts between well-connected nodes, linking local clusters into "small worlds" (Watts and Strogatz ) and further to large-scale networks that may feature direct connections between powerful centers, or emporia that attract connections from the entire sailing universe.
Knappett's model can be manipulated to simulate admirably enough the kinds of networks that plausibly existed in the MBA Aegean, but it too has limitations, mainly the ambiguity of the quantitative value of the variables and constants that the mathematical equations use. How exactly, we might ask, are numerical values calculated for abstract concepts or data fields for which there is only fragmentary information? A few examples will illustrate this problem. The means of quantifying the "fraction of effort" that one site puts into its relationship with another is not explained; it seems perhaps to be equated with the fraction of trade, but how this is derived is unknown (Knappett et al. : 1014, fig. 2). Part of the effort of maintaining the connection involved the ease or difficulty of intersite travel, but in this model distance and travel remain simplistic. While Knappett and colleagues recognize the need to arrive at travel times ("daily transport distance") rather than simple linear distances, the model does not incorporate any calibrations and we are left with a uniform, essentially friction-free sea. In a dissertation on Roman maritime trade in the eastern Mediterranean, Leidwanger (: 90–121) takes initial steps toward a textured seascape by constructing a GIS model using wind speed and directional data, as well as historical sources and sea-trial data from the replica ship Kyrenia II, to arrive at a friction factor, which is finally converted into a buffered map of sailing days from a given location. These maps of "cost weighted path distance" can be used to estimate the differences in travel for outbound and return voyages. For instance, applying the friction factor, the sailing time from Ialysos on Rhodes in the southeastern Aegean to Salamis on eastern Cyprus should be around four days, whereas the voyage in the other direction could take more than eleven (Leidwanger : figs. 2.7, 2.10; Fig. 6.3). This kind of information would be valuable as an input for Knappett's distance and effort variables. Even these advances leave room for development, since one could envision friction factors incorporating currents as well as a stochastic element for storms and other hazards.
6.3 Maps of cost-weighted path distance for eastbound and westbound journeys in the eastern Mediterranean. (a) Sailing days from Ialysos, Rhodes; (b) Sailing days from Salamis, Cyprus. Courtesy of Justin Leidwanger.
Other ambiguities arise as an inescapable result of the fragmentary nature of the archaeological record. Population (let alone population density) and carrying capacity figures are simply not available for most LBA settlements, at least not without wide margins of error. Lacking reliable quantitative data for these and other categories, calculations of site importance or cost–benefit for local and long-distance interaction are necessarily open to challenge. It is the double quandary of acquiring robust numerical values for structural features and then translating them through mathematical equations into social behavior that has led many historians and archaeologists to adopt a cautious attitude (Malkin et al. : 6). One could continue the discussion along these lines, but it is sufficient to conclude at this point that social network analyses hold promise, as yet not realized. With further development and robust data, they have the potential to reconstruct maritime social and economic networks for prehistory, and to identify the dominant engines driving their formation and change over time. The purpose of the present critique is not to advance a skeptical position, or to deliver a new, improved network model. Instead, I wish to expose the potential pitfalls of working with quantitative models that, if not invested with robust empirical data, give results that may seem authoritative but are in fact illusory. These are concerns of which Broodbank and Knappett were quite aware; hence their cautionary statements. I also want to alert readers that in the case studies to follow in the next chapter, data are often not amenable to quantification. In such cases, I follow Malkin and colleagues in adopting more qualitative characterizations of the maritime networks that I seek to analyze.
## Seven Coastscapes and Small Worlds of the Aegean Bronze Age
Case Studies
The purpose of this chapter is to bring together the various conceptual and empirical approaches outlined in previous chapters in order to apply them to real-world times and places of the Aegean Bronze Age. In three case studies, this chapter suggests how we might write diachronic histories of maritime connectivity at local to regional scales of interaction. One lengthy case study is drawn from the "heartland" of the Mycenaean world in the Saronic Gulf, followed by two brief portrayals of potential coastscapes and small worlds, one focused on Miletos on the coast of southwestern Asia Minor, and the other on Dimini and neighboring sites on the Bay of Volos, which are meant to suggest opportunities for further research along the lines advocated in this book.
This exercise aims to reveal the trajectories over time of coastscapes that may range from isolated to highly connected, and of small worlds that oscillate between cohesion and fragmentation, which often means alternating between hierarchical and heterarchical or nonhierarchical organizational structures. It focuses both on internal dynamics and on the ways that external stimuli – opportunities, threats, and greater historical currents – impinge to play often profound roles in local and small-regional histories. Placing a primary emphasis on coastscapes and small worlds means eliciting rich local contexts from which to build out to broader spheres of interaction (Galaty, Parkinson et al. 2009; Tartaron ; Wright : 808, 815). These case studies construct histories in the Mediterranean, because only when these are robust can they offer comparative material to the grand project of history of the Mediterranean (Horden and Purcell ).
### Making and Breaking a Small World: The Saronic Gulf, 3000–1200 BC
The essential aim of this case study is a diachronic reconstruction of a Bronze Age maritime small world in the Saronic Gulf. The inhabitants of Kolonna on the island of Aigina dominated this small world of many coastscapes – coastal settlements dotting the islands and mainland – from the middle of the EBA until the early phases of the LBA, when the expanding palace at Mycenae broke it apart, incorporating Saronic communities into broader Aegean networks. Over its life, this small world alternated between cohesion and fragmentation, as Kolonna responded to conditions within the Gulf and without, often initiated by events taking place beyond the Saronic and affecting large parts of the Aegean. I will attempt to write this history primarily from two vantage points: from the center at Kolonna; and from the small Bronze Age settlement at Kalamianos, built upon a gently inclined shoreline near Korphos on the Saronic's western coast. Kalamianos was a rather minor player for most of the period under consideration, only achieving prominence in EH II and LH IIIB. Other settlements in Kolonna's orbit will be called upon to fill in aspects of the story.
#### The Physical Environment of the Saronic Gulf
The Saronic Gulf occupies a central place in Greek maritime history, in part because of favorable sailing conditions and a strategic geographical situation (Fig. 7.1). It is partially enclosed by the land masses of the Argolid, the Corinthia, and Attica; as a consequence, winds, waves, and currents are moderate compared with more open areas of the Aegean Sea (Heikell : 17, 29; Soukissian et al. 2002). The winds are reasonably consistent, especially in the summer months. The meltemi blows from the north to northeast, beginning fitfully in July and increasing to full strength in August to early September before diminishing thereafter. It generally blows in the Beaufort 3–5 range (gentle breeze to fresh breeze), though in peak season it reaches 5–6 (fresh breeze to strong breeze) and occasionally higher. Winter winds are less consistent and winds up to gale force are more frequent, though hardly common. Waves are rarely significant enough to be damaging to coastal areas or dangerous to maritime traffic, and there is a minimal effect from the currents of the Black Sea Waters, mainly in the form of some strong anticyclonic eddies at the mouth of the Gulf (Olson et al. ). These currents, combined with meltemi winds, can make for a bumpy ride departing the Saronic for the Cyclades (Heikell : 29, 52). Isolated storms with associated squalls may arise at any time of year, though they are not common in the warmer months and they seldom last for more than an hour or two. The Saronic has relatively few dangerous reefs and rocks, excepting those quite close to shore and those in the narrows between small islands or between islands and offshore islets. While these mean characteristics establish the Saronic as an inviting maritime environment, there are local variations and exceptions to each. To give two examples: hazardous shoal waters extend southwest from the harbor at Aigina town (ancient Kolonna) through the islet of Metopi to Angistri; and strong westerly to northwesterly winds can blow from the Gulf of Corinth to produce severe gusts along the western side of the Saronic south to Epidauros (Heikell : 61, 74–75). Further, each anchorage has unique characteristics that vary during the course of the year; the reader is referred to the discussion of Kapsali Bay, Kythera in Chapter 4 for an account of typical Aegean variability.
7.1 Map of the Saronic Gulf region with important Bronze Age sites indicated. Pullen and Tartaron : 147, fig. 14.1. Courtesy of the Cotsen Institute Press.
Sea travel in the Saronic is enhanced by large and small islands and moderate distances throughout. No trip within the Gulf approaches the 100-kilometer daily range proposed in the previous chapter for Bronze Age sailing under normal conditions. Even paddled longboats of the EBA could complete virtually any one-way trip in a single day given the 40-kilometer range proposed by Broodbank (: 287–289), and many round trips were possible in a day or less (Fig. 7.2). There are many islands of all sizes in the Saronic, even if we discount the tiny rocks that could not accommodate even a small boat. There are two particularly large islands, Aigina and Salamis, and in this category we might also count the presqu'isle of Methana, attached by the narrowest of necks to the Peloponnesian mainland but behaving in most respects as an island. Just a bit smaller is the island of Poros, in this case separated from the mainland only by a narrow channel several kilometers south of Methana. Angistri to the west of Aigina is somewhat smaller than Poros, but after this there is a drop-off to very small and tiny islands with far fewer usable anchorages. Nevertheless, for the reasonably experienced sailor there is shelter and good anchorage within reach throughout the Gulf.
7.2 Comparative ranges of transportation modes in the Saronic Gulf region. Pullen and Tartaron : 154, fig. 14.4. Courtesy of the Cotsen Institute Press.
The Saronic Gulf is a crossroads by sea and land. By sea, it is the entrance from the open Aegean to the land masses of western Attica and the northeastern Peloponnese. From the Cyclades, the Saronic is the sea passage to the Isthmus of Corinth, and by crossing that narrow neck of land, to the Corinthian Gulf and the West. The presence of Aiginetan pottery at coastal sites on the Corinthian Gulf in the MBA suggests that the Isthmus was already used for that purpose. The Isthmus was also the land connection from southern to central Greece. At various times in history, states centered at Aigina, Corinth, Athens, and Nauplion laid claim to control of the Saronic as a fundamental basis of their economic and political power.
#### The Social Environment of the Saronic Gulf
An argument for the existence of a Saronic small world can continue with a phenomenological perspective. Aigina is situated in the geographical center of the Saronic Gulf, with the land masses of Attica, the Corinthia, and the Argolid nearly encircling it. Intervisibility to and from Aigina is exceptionally high: Aigina is a large island with a distinctive shape – the pointed peak of Mt. Oros is unmistakable – that looms on the near horizon from most coastal vantage points. With some exceptions, the Saronic coastline is rugged, with an abundance of small anchorages attached to diminutive coastal plains or to sheer coastal cliffs. Small coastal settlements tend to be perched on headlands or limited coastal lowlands backed by high mountains that block views, and easy access, to the interior. Thus the everyday field of view is directed toward the sea, to other coastal settlements, and especially to Aigina. Looking upon the Saronic, one perceives not boundless sea, but islands and coasts occupying much of the horizon at distances manageable for small craft. The phenomenological experience of inhabiting one of these communities reverses the common expression of looking out to sea, by giving the sense instead of settlements orbiting around and looking into the center at Kolonna. The visual element of connectivity so keenly highlighted by Horden and Purcell finds a perfect expression in the Saronic. We may hypothetically suggest that intervisibility, combined with moderate distances and relatively easy sea voyaging, promoted the perception of being part of an organically constituted, coherent world. Opportunities for forging ties with other coastal settlements must certainly have flowed from these advantages, but at times there must also have been social imperatives, including mutual arrangements to buffer the risk of resource failure, and the practice of exogamy to maintain the genetic viability of small communities and to cement the social ties needed to perpetuate vital relationships (Bintliff ).
#### Kolonna and the Bronze Age Saronic Small World
The promontory of Kolonna on the northwestern coast of Aigina was occupied during the Neolithic period at least as early as the fourth millennium BC. The natural advantages of the site are evident: it is elevated 12 meters above sea level and protected by cliffs on three sides, with a double embayment to the south and north and abundant arable land to the east (Felten : 12). Although the shallow harbor at Kolonna – later Aigina town – was considered in Antiquity to be among the most hazardous in the Aegean to approach, it repeatedly served as the main port of powerful Aiginetan states from the Bronze Age to the Archaic period. This disadvantage did not outweigh the location's other natural benefits, or the social and economic will of the city's inhabitants to succeed in spite of environmental shortcomings.
An incipient maritime small world may have come into being in the Saronic as early as the Late or Final Neolithic. It has been demonstrated that Aigina was the main source of andesite for millstones in Attica and the Peloponnese by the later Neolithic period (Runnels 1985a), and a "Saronic" fabric that appears macroscopically to be tempered with volcanic-related inclusions characteristic of Aigina is common among the FN to EBA pottery sherds recovered during a recent surface survey in the Korphos region (D. Pullen, personal communication 2001). The center at Kolonna comes into clearer focus in a mature phase of EB II in the Aegean. This was a time of increasing social complexity that witnessed the emergence of chiefdoms, the erection of fortifications at many sites, and vigorous exchange of exotic items with presumably high social value, including bronze daggers and tools, metal jewelry, fine drinking and pouring vessels of metal, and ceramic and marble vessels and figurines; in short, an era of "international spirit" (Renfrew : 451–55). The relatively undifferentiated pattern of small farmsteads and hamlets in the preceding EB I period was transformed by a striking expansion of settlement and the appearance of large settlements, particularly at coastal locations oriented to maritime activity (Broodbank : 279–87; Konsola ; Pullen ). This was also the time of the monumental "corridor houses" with long passages flanking the internal rooms found on the Greek mainland and at Kolonna itself (Fig. 7.3). These structures have been variously interpreted as palaces, administrative centers, residences of prominent families or lineages, or even hotels or meeting halls for traders (Felten ; Nilsson ; Pullen ; Shaw ; Weingarten ; Wiencke ).
7.3 Map showing the locations of corridor houses and fortifications in the EB II Aegean. After Tartaron, Pullen, and Noller : 147, fig. 3.
Bronze Age Kolonna is a highly complex archaeological site, with nine separate urban phases or "cities," including massive fortification walls that were modified and strengthened over a period of 500 years (Table 7.1). From the EBA to the beginning of the LBA, roughly 2500 to 1400 BC, Kolonna was a site without peer in the Aegean outside of the brilliant Minoan civilization on Crete to the south (Rutter : 125–30). Some believe that Kolonna achieved the first Aegean state-level society after the Minoans and before the Mycenaeans (Niemeier ).
During EH II (Kolonna phases II–III; circa 2700–2200 BC), Kolonna was a modest settlement of mudbrick houses, but had already begun to distinguish itself from other coastal and island sites in the Saronic and beyond. There is evidence of economic specialization in the production of textiles in the "Färberhaus" (phase III) and storage of agricultural surplus in the "House of the Pithoi" (phase III; Felten ). The monumental corridor house known as the "Weißes Haus" of phase III (along with its predecessor the "Haus am Felsrand" of phase II) may have played a central administrative role in the community. In its layout and construction, the Weißes Haus exhibits particularly close parallels to the House of the Tiles at Lerna, indicating early and meaningful relations (Shaw ; Wiencke : 298–303). Ongoing excavations at Kolonna are revealing a number of large buildings in phase III, however, so the former impression of the Weißes Haus as singular in its size and complexity may be giving way to the picture of "...an accumulation of more or less homogeneous self-sufficient unities" (Felten : 13).
By the latter centuries of the second millennium in EH III (Kolonna phases IV–VI early; circa 2200–2000 BC), Kolonna had been transformed into one of the most significant urban centers of the Aegean: a densely populated, heavily fortified town with monumental stone buildings and sophisticated town planning with buildings arranged in insulae separated by gravel roads. Beginning in EH III, pottery was imported, and stylistic influences on local pottery were adopted, from the Peloponnese, central Greece, and the Cycladic islands (Gauß and Smetana : 329, : 167); and by the beginning of the MBA, these same areas had begun to import Aiginetan tableware, storage vessels, and cooking pots (Lindblom : 40–42, 131–32). There is some evidence in phase IV for a copper-smelting operation.
By EH III, all around Aigina the "international spirit" had broken down, ushering in a period of diminished activity and even abandonment over much of the Saronic and northeastern Peloponnese, which endured until the last phases of the MBA. Although there are variations across the area, the trend is a strong one that is clearly documented by both survey and excavation data (Wright : 119–28). The inland Nemea Valley in the southern Corinthia is a particularly well-studied example, having been targeted by an intensive surface survey and a long-term excavation at Tsoungiza, its most prominent prehistoric settlement. These investigations indicate that the valley suffered virtually complete abandonment from sometime in EH III to MH III/LH I, a phenomenon sometimes known as the "Middle Helladic hiatus" (Cherry and Davis : 151–55; Wright : 119–28; Wright et al. : 628–29). The reasons for this nadir in human activity are not well known: in the case of the Nemea Valley, flooding of the valley floor has been postulated (Cherry and Davis : 155–56); elsewhere, finds of daggers, spear points, and sling stones at fortified coastal sites in the Aegean suggest violent destructions (Branigan ; Doumas ).
Table 7.1. Chronological chart for Kolonna (after Wild et al. 2010: 1020, table 3)
* * *
* * *
Notes:
(a) long absolute chronology for the Aegean Bronze Age based on the few presently published 14C dates;
(b) historical chronology based on the Egyptian Chronology and its relations to the Aegean;
(c) with subphases (A1, A2, etc.);
(d) time range for the respective confidence level (1σ and 2σ);
(e) no time range is given for boundaries at the beginning or the end of the sequence or hiatus.
By contrast, Kolonna, almost uniquely in the southern mainland region, grew in prosperity and complexity through MH (circa 2000–1600), establishing relations beyond the Saronic with central and northern Greece (Maran ; Sarri ), the Cycladic Islands (Crego ; Gauß and Smetana ; Nikolakopoulou ; Overbeck ), the Argolid (Nordquist : 44, 50–51; Philippa-Touchais ; Touchais ; Zerner : 156–58, : 48–50), and Minoan Crete (Gauß ; Gauß and Smetana : 61–65; Hiller ). The prosperity of Kolonna's MBA inhabitants is evident in the material remains. By MH I, the community had expanded beyond the fortification wall to an "inner extension" or "inner suburb" that was then enclosed with a less imposing wall; still later, in early Mycenaean times, a further "outer extension" enlarged the urban area to almost the entire promontory (Fig. 7.4). Notable is the so-called Large Building Complex, founded early in MH just inside the massive fortification wall, and persisting until early Mycenaean times spanning several major architectural phases (Gauß and Smetana ). The footprint of the complex may have reached 680 square meters in the MBA, making it one of the largest known structures on the mainland; it has been interpreted as a mansion with a possible administrative function suggested by a clay stamp and a clay seal (Gauß and Smetana : 172). The finds from the Large Building Complex include enormous amounts of pottery and faunal remains. The pottery of the complex's second architectural phase (Kolonna IX) comprises imports from the Cyclades and Minoan Crete, locally manufactured vessels of Minoan type, Aiginetan matt-painted (Siedentopf ), and solid painted. The imported and imitation Minoan pottery demonstrates not merely close exchange relations with Crete, but also the possibility that Minoan craftsmen (potters, at least) were resident on Aigina (Gauß ; Hiller ). The local vessels of Minoan type exhibit significant departures from Aiginetan potting traditions: they are wheelmade, they lack the omnipresent potters' marks found on contemporary Aiginetan vessels, and their forms are dominated by small, open shapes and cooking ware (Gauß and Smetana : 63, 66). Other objects that testify to Minoan influence, if not presence, are an ashlar block with a Minoan-style double-axe mason's mark reused in a Late Roman context (Niemeier : 78), a Minoan-type loomweight, fragments of three Minoan stone vases, a ceremonial stone hammer, Minoan jewelry, a stone kernos, and fragments of a potter's wheel (Hiller : 199).
7.4 Site plan of Bronze Age Kolonna, Aigina. After Gauß and Smetana : 58, fig. A.
Analysis of botanical, faunal, and human skeletal remains from the recent excavations at the Large Building Complex has revealed important information about how some inhabitants of Kolonna lived and died in the MBA (Forstenpointner et al. ; Galik et al. ; Kanz et al. ). The plant remains are dominated by the domesticated grain crops emmer wheat, bread wheat, and barley, with lentils as the main identifiable pulses. Grape, fig, and olive were also cultivated. The faunal assemblage consists of 3,178 terrestrial and 1,772 aquatic specimens. The terrestrial animals are overwhelmingly domesticated livestock, predominantly sheep/goat (66%), with lesser amounts of pig (20%) and cattle (14%). Only miniscule numbers of wild animal bones are present. This is a fairly standard faunal assemblage for the MBA and LBA, although the mix of domesticates varies and Gerhard Forstenpointner and colleagues note that the high percentage of sheep and goat is more characteristic of the Aegean Islands and Crete than the mainland, where cattle are more prominent. The remains suggest a mixed livestock economy in which both primary products (meat, hides) and secondary products (milk, hair, wool) were used, but a large percentage of animals were not slaughtered before four to five years of age. The inhabitants of the Large Building Complex also consumed fish, shellfish, and snails. Mollusks (bivalves and gastropods) make up 67% of the marine assemblage. Fish are perhaps underrepresented because of poor preservation of small bones, yet several species including dentex, pandora, sea bream, grouper, barracuda, and mullet indicate a mix of near-shore and open-sea fishing. Remains of fins, ribs, and scales imply processing on site. Alfred Galik and colleagues also find closer parallels for the marine assemblage in Middle and Late Minoan Crete (e.g., Kommos) than in contemporary mainland sites. Taken together, these studies portray a varied and robust diet, but it must be remembered that the material comes only from the limited context of the Large Building Complex, an apparently elite setting where residents might be expected to have access to a better diet than others at Kolonna or in other settlements on the island.
The study of 48 subadult human skeletons recovered from intramural burials – chosen because Kolonna's adult cemeteries have not been located – produced results that support the impression of a generally prosperous community. The burials come from excavations of the last 20 years and range chronologically from EH I to LH (subphase not specified). Although these individuals died in utero (stillbirth), immediately or shortly after birth, or within the first year of life, there are few signs of malnutrition of the mother during pregnancy, or stress response in respiration, nutrition, or blood circulation after birth. Instead, death is more often attributed to perinatal failure: prematurity, congenital defects, acute diseases, and birth complications occurring at or immediately after birth (Kanz et al. : 483–84). Stillbirth and death shortly after birth were surely common, unavoidable occurrences in the Bronze Age. A lingering question is whether meaningful trends can be extracted from a small sample spread over almost 2,000 years, but if it is accepted that the data fairly represent general trends in the health of Kolonna's population, a comparison with children at Lerna and Asine shows a much lower occurrence of malnutrition at Kolonna as measured by rates of dental hypoplasia and other indicators of metabolic problems.
The wealth and wide connections of Kolonna's inhabitants are suggested by the so-called Aigina Treasure. The mysterious history of this hoard, if that indeed is the right term for it, is well known (Higgins ), but recently new information has emerged, leading to a conference in which the historiography of the treasure was updated (Williams ) and the objects were reanalyzed stylistically and technically (Fitton ). The hoard is a spectacular collection of gold jewelry, comprising earrings, pendants, diadems, bracelets, necklaces, rings, and plaques, with lapis lazuli, amethyst, jasper, and rock crystal beads as secondary decorations (Fitton et al. ; Fig. 7.5). There is a basic consensus among scholars that the treasure probably did originate on Aigina in the MBA and should be viewed as a group that may have been looted from a MH tomb. Most accept that the pieces could have been made in an Aiginetan workshop, but not necessarily all in the same generation. The widest divergence of opinion concerns the identity of the craftspeople and the techniques and stylistic influences intrinsic to the individual pieces. Stefan Hiller () supposes that Minoan artisans, part of a small but affluent colony residing on Aigina, created such jewelry mainly for their own community, at the same time as their fellow expatriates manufactured Minoan-style vessels. While Hiller's scenario assumes that most of the objects find their closest parallels in Minoan typology and iconography, other scholars favor comparanda from the Near East, Anatolia, Egypt, or the Greek mainland as inspirations for individual objects (various contributions to Fitton ; Koehl ). Perhaps the most useful statement one can make is that the Aigina Treasure underscores the unusual wealth and wide foreign connections that the community at Kolonna enjoyed in the MBA. The treasure seems to represent a synthesis of influences, perhaps filtered through Cretan connections and individuals.
7.5 "Master of Animals" pendant from the Aigina Treasure. © Trustees of the British Museum.
The significance of the Aigina Treasure is highlighted by the more recent discoveries at Kolonna of an EH III hoard and a warrior's grave of MH II. The hoard, excavated in 2000 in House 19 of the "inner town," bears some similarities with the later Aigina Treasure in its content and wide geographical affinities. It consists of a number of gold pins with loop terminals, gold and silver bracelets, several gold and silver pendants with embossed and wire decoration, and one or more necklaces with beads of gold, silver, carnelian, faience, and rock crystal (Felten : 15, : 34–35). The traditions from which these pieces come include northeastern Aegean, Anatolian, Levantine, Mesopotamian, and Cretan. This hoard has several important implications. It implies that in EH III an elite group already existed that could assemble such a rich collection of precious jewelry, and thus the Aigina Treasure may be part of a much longer local tradition. Furthermore, since those who hid the jewelry lived in a period before the earliest Minoan objects appeared in MH I, they were apparently able to forge such far-flung connections without Cretan intermediaries.
The warrior's grave is conventionally known as the Middle Bronze Age Shaft Grave of Aigina, and it is explicitly offered as a forerunner of, and possible model for, the somewhat later shaft graves at Mycenae (Kilian-Dirlmeier ). Opinion is divided on whether it is a true shaft grave, however; according to Oliver Dickinson's (: 56) widely recognized definition, a shaft grave comprises a rectangular shaft cut into soft rock and earth, with built or rock-cut ledges some way down the shaft on which a roof of wooden beams would rest, creating a cavity for the burial chamber below it. The roof was covered with clay and the shaft above it was then filled with earth, stone, and sometimes offerings from a funerary meal. A tumulus might finally be raised above the grave. The Aigina grave does not entirely match this definition, in that the cut shaft is extremely shallow, with most of the grave built up of limestone rubble. There is no indisputable ledge, though Imma Kilian-Dirlmeier has plausibly detected a horizontal row of flat stones that could have served to hold in place a roof that does not survive (Kilian-Dirlmeier : 17, fig. 4). Others have classified the burial as a "built tomb" or a "built cist" (Cavanagh and Mee : 27; Hiller : 138–39). The consequence of this debate is that it may not be possible to hold up the Aigina tomb as the model for the form of the later shaft graves at Mycenae, Lerna, and Ayios Stephanos; it must be pointed out, however, that the earliest shaft graves in Grave Circle B (MH IIIA–IIIB) at Mycenae do not display the fully developed, canonical form of the later (end of MH to LH IIA) examples (Graziadio ).
On the other hand, the prominent location and contents of the shaft graves at Aigina and Mycenae betray certain shared conceptions of the status and treatment of the deceased. Both were built in extraordinarily conspicuous locations just outside of the contemporary settlement's walls – in the case of the Aigina shaft grave, against the outer face of the enclosure wall of the inner extension during Kolonna IX. This may have been a unique honor; unlike those at Mycenae, the grave seems not to have been part of a cemetery, unless the latter was destroyed by construction during later periods. Kilian-Dirlmeier (: fig. 35) restores a 2-meter-thick tumulus over the shaft grave at Kolonna; the grave circles at Mycenae may have been covered by a low mound, separate mounds over individual graves, or no mound at all, but at both sites these reconstructions remain hypothetical (Mylonas : 89–90).
The grave offerings at Kolonna are often thought of as a sampling, on a more modest scale, of the riches to come in Grave Circle A at Mycenae, but a better comparison is Grave Circle B, closer in date to the Kolonna shaft grave and less opulent in grave goods. The contents of the Kolonna burial include a bronze sword with a gold hilt and ivory pommel; several bronze daggers, including one with a decorated gold sheet molded around the handle; a bronze spear point; a gold diadem decorated with repoussé crosses; a gold knife with gold animal-head fittings; boar's tusk plaques from several helmets; six obsidian arrowheads; Minoan pottery of mature Kamares style dating to MM II; Middle Cycladic pottery from Melos and perhaps elsewhere in the Cyclades; and local matt-painted and plain vessels for drinking, eating, pouring, and storage (Kilian-Dirlmeier ; Fig. 7.6). It has been noted that the artisanship and decoration of the metal objects reflect mainland rather than Cretan traditions (Hiller : 37), a claim supported by the similarity of motifs on the molded gold sheet to those on locally manufactured matt-painted pottery (Kilian-Dirlmeier : 57). The pottery fits well with the assemblage of local and imported wares in Kolonna settlement IX, ceramic phase I, chronologically equivalent to MH II (Gauß and Smetana : 63, 66) and roughly contemporary with the Aigina Treasure.
7.6 Examples of imported and high-status objects from the Aigina MH II "shaft grave." (a) Kamares Ware vessel, imported from Crete, MM II; (b) gold diadem; (c) bronze sword with ivory pommel and gold fittings. After Kilian-Dirlmeier : p. 28, fig. 27:16 (vessel); p. 19, fig. 6:9 (diadem); p. 18, fig. 5:1 (sword).
All of these artifact types are present in abundance in the shaft graves at Mycenae. Like many of the interments in Grave Circles A and B, the Kolonna shaft grave contains a warrior burial of a type that persists through the Mycenaean period and survives the collapse of the palaces (Deger-Jalkotzy ). That the elite individuals and families marked out by these shaft graves enjoyed preeminent status within the community is demonstrated by their setting and rich offerings, but it is specifically the warrior status of the individual buried at Kolonna that prefigures the striking (and decidedly un-Minoan) Mycenaean preoccupation with martial equipment and iconography, as well as the possibly decisive role of violence in the emergence of the Mycenaean palace states (Acheson ; Bennet and Davis ; but cf. Wolpert ). The twin concerns with maritime and warlike pursuits (and perhaps even with naval warfare) are highlighted in a small number of MH Aiginetan matt-painted barrel jars decorated with ships and in one case a scene of armed warriors aboard a rowed ship (see Fig. 3.10; Rutter : 128–30; Siedentopf : fig. 4, pl. 38.162). There are very few Aiginetan pottery vessels deposited with the dead in Grave Circles A and B, undermining notions that Aigina had direct involvement in Mycenae's emergence to complexity. At the close of the MH period, however, Kolonna's long-standing relationship with Crete may have provided a conduit for Mycenae's initial contacts with the Minoan world. More likely than this is that Kolonna's massive fortification walls, paralleled in the contemporary Aegean only at Troy and Kea (Niemeier : 75), and the precocious warrior burial, exerted a strong influence on an aspiring elite familiar with the prowess and the products of the island polity.
During the MH demographic free fall in Attica and the northeastern Peloponnese, the Aiginetans leapfrogged these areas to establish longer-distance trade relations with central Greece, the Cycladic Islands, and Crete. The impressive distribution of Aiginetan pottery plots the maritime routes over which the cargoes were moving, as well as overland routes by which fewer pots made their way to inland settlements (Fig. 7.7). Goods from Aigina may have been transferred across the Isthmus of Corinth to sites in central Greece along the Corinthian Gulf (e.g., Kirrha, Eutresis) through intermediaries living in the northern Corinthian plain. A number of sites in this intermediate zone, including Korakou, Gonia, Peridkaria, Aetopetra, Arapiza, and Ayios Gerasimos, seem to have been occupied from EH III through the Mycenaean period (Lambropoulou : 144). They seem to have coexisted in a stable, heterarchical settlement pattern over much of the Bronze Age (Pullen and Tartaron : 148, 150–52). During MH, their only detectable external contacts were with Aigina, indicated by the presence of matt-painted, red-slipped and burnished, and coarse plain and cooking vessels in Aiginetan gold-mica fabric. At Gonia, these types constitute 19% of the total ceramic assemblage; at Korakou the figure is 9% (Lambropoulou : 145).
7.7 Distribution of Aiginetan "gold mica" pottery exports. After Rutter : 127, fig. 12, with additions.
Because there has never been a systematic site survey on Aigina, the handful of known MH sites have been discovered as the result of informal explorations or as chance finds. In the mid-1990s, the MH catalogue consisted of eight confirmed sites and eleven uncertain sites (Fig. 7.8). These sites are mainly sherd scatters or occasionally graves, but beyond Kolonna architecture is lacking. It is at present impossible to know if this pattern is a fair representation of reality, and we are not in a position to answer Wright's (: 808) query concerning whether there were centers on the island apart from Kolonna serving as magnets for small villages and hamlets. Very small amounts of MBA material have been recovered in excavations at Lazarides, an elevated site in east-central Aigina with views over most of the Saronic (Sgouritsa ), and at the location of the later temple of Aphaia in the northeastern corner of the island (Pilafides-Williams 1998: 82–83, 156). These and other sporadic finds are not suggestive of alternative centers, or of a complex hierarchy of sites below Kolonna. There is a comparable dearth of MH I–II sites around the Saronic, but an important exception is the recent discovery at Megali Magoula near Galatas, across from Poros, of a small but impressive settlement enclosed by an elliptical fortification wall (Konsolaki-Yiannopoulou 2003a, ). The MH pottery is a mixture of Peloponnesian and Aiginetan types with a chronological concentration in MH II. Alongside mainland gray Minyan and Argive Minyan, much of the fine to semi-fine matt-painted pottery is Aiginetan, including large and small basins and a few examples of cylindrical pyxides and barrel jars. Megali Magoula prospered along with Kolonna IX and X, perhaps in part by serving as intermediary for Aiginetan products with trade partners in places like Lerna and Asine (Konsolaki-Yiannopoulou : 73).
7.8 Map of Aigina showing the locations of known MH sites. After Kilian-Dirlmeier : 109, fig. 62.
At the end of the MBA, Kolonna X (MH III–early LH) witnessed a further expansion of the town to the east, enclosed by yet another wall in early LH, this time of large rubble construction reminiscent of cyclopean masonry. The ceramic evidence suggests that the outward focus that the Aiginetans had maintained on more distant trading partners during the Middle Helladic hiatus shifted back to the regions surrounding the Saronic Gulf, where two related transformations were taking place starting in MH III/LH I: the "colonization" of the interior of the northeastern Peloponnese, which saw resurgent populations establishing new sites or reoccupying old ones that had been effectively abandoned since the late third millennium (Rutter : 42–43); and the social, political, and economic developments of the Shaft Grave Era, most prominently the emergence of complexity at Mycenae. The Aiginetan ceramic industry responded to the increased demand for household pottery closer to home by expanding production in a range of standardized and specialized forms: larger closed and open vessels including water jars, barrel jars, and kraters; smaller drinking and eating vessels such as goblets, kantharoi, and handleless bowls; and four types of cooking pots (Rutter : 36). A pottery kiln dating to the early years of LH that was recently excavated in the southwestern part of the Large Building Complex may have played a role in the increased production. The Saronic small world centered on Aigina was thus revived, starting in MH III and peaking in LH I–II. This was the era of the greatest cohesion of the Kolonna-centered Saronic world, and for most sites in the Saronic and northeastern Peloponnese, the time of greatest abundance of Aiginetan imports (Lindblom : 41–42).
Mycenae was not yet connected in any meaningful way to this network, but soon would be. Before we turn to the expansion of Mycenae, it is worth reflecting on why Kolonna had become such a monumental settlement with such broad contacts, and why the pottery produced on the island was one of few Aegean products to be so widely disseminated. It was partly a matter of Aigina's fortunate geographical position, and the opportunities for efficient transport by sea. It had also to do with the excellent sources of clay and temper to which potters at Kolonna had access. Moreover, Kolonna filled a power vacuum, surviving and flourishing while communities all around disintegrated, by forging new ties with more distant partners. A distinct distribution pattern had developed by the late MH for two main ceramic production and export industries: Aiginetan; and lustrous decorated wares centered in the southern Peloponnese or Kythera (Zerner ). In the southern Peloponnese, there is much lustrous decorated and little Aiginetan; in central Greece and Attica, the situation is reversed; and in the northeastern Peloponnese, there is much of both (Rutter : 36).
Many scholars have focused on the intrinsic properties of the Aiginetan pottery itself relative to local and imported alternatives (e.g., Zerner ). The Aiginetan product was more standardized in its form, the result of consistent forming and firing practices, including levigation, uniform clay composition, and controlled firing conditions (Philippa-Touchais : 110), lending the impression of greater reliability. Its well-executed and attractive matt-painted decoration was appreciated for its aesthetic properties, inspiring local imitation. There is also strong evidence of superior performance for the pots' intended uses (Rutter : 42). The cooking ware was lighter in weight but better made and more durable than the norm; the porosity of the fabric inhibited cracking during expansion and contraction cycles, while the volcanic rock temper apparently possessed favorable thermal expansion characteristics. The result was higher thermal shock resistance and fewer failures under thermal stress. The several forms of water jug (stamnoi, hydrias, amphoras, and large jugs) were larger, lighter, with thinner walls, thus more practical for transporting water, and their porosity promoted evaporation of moisture through the body wall and into the atmosphere, keeping the liquids they contained cooler.
While the performance characteristics of Aiginetan pottery have long been acknowledged, in recent years scholars have attributed to the trade in Aiginetan pottery far more profound influences. Anna Philippa-Touchais (: 110–12) asserts that the aesthetic of Aiginetan MBA pottery not only inspired imitations at Argos and elsewhere, but actually created a network of "common references," a kind of koiné of instantly recognizable shapes, fabrics, and technical excellence that attained an ideological value for local elites wishing to display their connections with an external world in the context of communal feasting.
This sentiment is echoed in studies of Aiginetan ceramics in Thessaly and Boeotia, and at Lerna. Despite the fact that imported Aiginetan vessels are quite rare in Thessaly, Joseph Maran () believes that "Magnesia polychrome," manufactured in or around Pefkakia beginning in MH II, emulates the shapes and decoration of Aiginetan matt-painted pottery. According to Maran, the adoption of these novel table and cooking vessels actually transformed methods of food preparation and consumption. These new practices became strategies in communal eating and drinking ceremonies to emphasize the connection of those who possessed them to elite practices in distant southern Greece. As at Argos, aspiring elites sought to differentiate themselves in society through the use of such exotic objects. Maran sees the spread of this influence, which began with exposure to a limited number of genuine Aiginetan specimens, to the northern Aegean and the Izmir region (Maran : 174). In Boeotia, the aesthetics of Aiginetan pottery had a strong effect by MH II, as potters began to combine Minyan and matt-painted styles. This interaction can be traced through a succession of changes from yellow and red Minyan matt-painted, to polychrome mainland in MH III, and ultimately to Mycenaean style (Sarri : 163). At Lerna, a massive collection of broken pottery and animal bones in the fill of two shaft graves of LH I, representing funerary meals that must have involved hundreds or even thousands of participants, contains Aiginetan pottery in the amount of more than 50% of between 15,000 and 18,000 sherds (Lindblom ). In such an obviously communal and symbolically charged event, vessels manufactured at Kolonna, an impressively fortified place possessing a maritime fleet and advanced technological knowledge, could serve as a powerful demonstration that the followers of the deceased had access to a network of social relations beyond the reach of most members of the communities on the Argive Plain (Lindblom : 126). It may have been especially important to display wealth and esoteric knowledge if one purpose of the ceremony was to transfer rights and privileges to an heir of the deceased under potentially contentious circumstances. We might imagine that the Lerna shaft grave deposit represents the kind of competitively charged communal event that Philippa-Touchais and Maran have in mind for Argos and Pefkakia. The social ramifications implicit in the acquisition and use of Aiginetan wares thus extend well beyond the economic value of the pots or the exchange networks that moved them.
An even more direct influence may have been at work in Aigina's relationship with the settlement at Ayia Irini on the island of Kea (Crego , ; Overbeck ; Overbeck and Crego ), just outside the Saronic Gulf. Ayia Irini IVa was founded in a developed phase of Middle Cycladic after a hiatus spanning the end of Early Cycladic (Ayia Irini III) and the earliest part of the Middle Cycladic. The settlement was apparently colonized from outside, with an intrusive ceramic repertoire including a system of potters' marks; immediate engagement in vigorous trade with the mainland, the Cyclades, and Crete; and an impressive fortification wall. Donna May Crego (: 843) points out that there is little evidence for traditional women's crafts, and burials of the period are not yet known, suggesting to her the initial settlement of Ayia Irini IVa by a male, commercially oriented installation rather than a typical village. As for the origin of the settlers, in an earlier article John Overbeck and Crego (: 305) pointed to central Greece, perhaps Boeotia, on the strength of the abundance of mainland pottery types such as gray Minyan. More recently, in something of a reassessment, Crego () relocates the settlers to Aigina, highlighting shared elements that add up to a special relationship between the two islands. She sees links to Kolonna in the fortification wall and the system of potters' marks. More salient still are indications of close relations in the ceramic assemblages (Crego : 842–45). Although true Aiginetan matt-painted pottery makes up only around 3% of the pottery corpus of phase IVa at Kea, locally produced yellow-slipped (12%) appears to be an emulation of Aiginetan matt-painted adapted to local clays. Further, the two settlements exchanged vessel types rarely found outside their local contexts: at Kolonna the old and new excavations, as well as the shaft grave, have yielded a range of Keian vessels, including the rare white-on-gray, found in numbers matching those known on Kea itself. In parallel, potters at Ayia Irini manufactured barrel jars and bulbous jars in yellow-slipped fabrics, imitating the shape and appearance of Aiginetan matt-painted prototypes. The latter shape is rare outside Aigina. Crego concludes that Ayia Irini IVa was founded from Kolonna as a trade station to distribute Aiginetan products and to provide access to the metal deposits at nearby Lavrion on the Attic mainland. The wide contacts of the new settlement can be explained by Kolonna's existing maritime network of ties to the mainland, Cyclades, and Crete. In the subsequent phase IVb, commercial interests continued, but the far greater occurrence of burials and women's equipment suggests a fully formed village and an incipient Kean identity separate from Aigina. The dominant influence of Aigina had declined by the late MBA (phase V), when Minoan pottery was imported and imitated, Minoan architectural styles were adopted, and Linear A script was used (Davis : 195). By the following phase VI, corresponding to the beginning of the LBA, Minoan influence was pervasive in every aspect of material culture. If Crego's interpretation of Ayia Irini IVa is accepted (and there are certainly alternative explanations of the evidence; e.g., Davis : 194–96), it shows Kolonna in an expansive mode, extending its small world beyond the confines of the Saronic Gulf.
Aigina's unusual success in production and export, amounting to the better part of a millennium of competitive advantage, might be further illuminated if we think in terms of connectivity. Recalling the discussion of social network theory in the previous chapter, we can suggest that the principle of preferential attachment (Barabási and Albert ), by which new vertices attach disproportionately to sites that are already well connected, applies forcefully to Kolonna's situation in the MBA and early LBA. Kolonna was a peer of several highly developed EH II communities in the northeastern Peloponnese and Cycladic islands, but unlike most others survived the EH III decline as a prosperous community, filling a yawning power vacuum. Although the growth of the Aiginetan potting industry was perhaps stimulated by contact with protopalatial Crete, this cannot explain the initiation of exchange relations with the Cyclades, the Peloponnese, and central Greece, for which the role of intrepid and enterprising individuals must have been decisive. By means of this precocious outreach, Kolonna became more "connected" than any other settlement in the region. As demographic recovery proceeded and new settlements were established in MH III–LH I, a period of continuous growth began with the addition of new vertices and new paths between them, but the huge competitive advantage held by Aiginetan producers in terms of experience, efficiency, and established connections meant that these new nodes connected to Aigina preferentially, in agreement with the ceramic evidence from the Saronic and surrounding areas. Under conditions of continuous growth and preferential attachment, a node that acquires more connections than others will accumulate them at an increasing rate, causing the difference in connectivity to multiply as the network grows (Barabási and Albert : 511). I suggested in Chapter 6 that this dynamic might illuminate the emergence of Mycenae during the Shaft Grave Era or the dominant position of Knossos in the neopalatial period, but we can now apply the same idea to Kolonna's long-term prominence from EH III to LH II. This process, the impetus for which may have originally been economic, was a key factor leading to a situation where the emergence of rival centers of political power is suppressed, as argued by Pullen and Tartaron () for Kolonna's relationship with the Saronic region and beyond. A consideration of connectivity within the framework of network theory augments the interpretations of the ceramic evidence, outlined above, to begin to answer Wright's (: 808) question: "How do we assess the regional influence or connectedness of Aegina beyond [the Saronic Gulf] area?"
#### Kolonna and Mycenae in the Late Bronze Age
The expansion of Mycenae's economic and political interests was destined to transform the Saronic Gulf entirely, but this was more a gradual process than the execution of a strategic plan at any one point in time. A brief survey of the evidence of pottery in regions to the north and east of Mycenae is enlightening on this point. The areas of the southeastern Corinthia north of Mycenae, such as the Nemea and Longopotamos Valleys, have been considered natural targets for Mycenae to expand into virtually empty landscapes in the early years of the Shaft Grave Era in MH III–LH I (Cherry and Davis ). But the ceramic evidence suggests otherwise, indicating a strong measure of independence in the early Mycenaean period (Morgan : 358–61; Mountjoy : 197; Rutter , , ; Wright : 124–26). Jeremy Rutter (: 452–55) has observed that the pottery used by the first group to resettle Tsoungiza finds close parallels not in the Argolid but in late MH graves in the North Cemetery at Corinth. The MH III assemblage is parochial, with a few imports from Aigina, but only general stylistic links with the Argolid and the Corinthia (Morgan : 360). In LH I, Mycenaean-style fineware is rare while imported Aiginetan gold-mica storage, cooking, and mixing vessels comprise between 7% and 10% of the total pottery assemblage (Rutter : 12; Lindblom : 41), with smaller numbers of Cycladic and Cretan pots possibly obtained through Aiginetan intermediaries. Tsoungiza may have looked not south to Mycenae, but west toward the thriving center at Aidonia at this time (Wright : 125). It is not until LH IIA that a significant connection can be demonstrated with Mycenae. Although imports of Aiginetan utilitarian vessels held steady at approximately the same levels as in LH I (Rutter : 82–85, table 1), trench EU 10 produced high-quality Mycenaean fineware, including a Vapheio cup and four piriform jars so similar to examples from Mycenae that they may have come from the same workshop (Mountjoy : 199; Rutter : 74–75, 79). By this time, then, Tsoungiza was being drawn into Mycenae's orbit, although we cannot say with certainty that Tsoungiza had been incorporated politically as opposed to simply participating in economic transactions with an emerging center of pottery production and trade at Mycenae (Rutter 1993: 91). Indeed, in LH IIB both Mycenaean and Aiginetan imports actually declined and the LH IIIA1 subphase is not well known (Mountjoy : 200).
The more distant northern Corinthia was slow to adopt the Mycenaean style. At LH I Korakou, there are a few sherds only of LH I style, and a small number in the palatial and pseudo-Minoan styles of LH IIA (Davis ). Instead the main connection in the early Mycenaean period was with Aiginetan trade networks. As mentioned above, this relationship began in the MBA, but by LH I the inhabitants of Korakou were importing a range of Aiginetan cookware, kraters, and large storage and pouring vessels (Davis : 241, 258–59; Lindblom : 41; Morgan : 351; Mountjoy : 199–200). MH traditions persisted longer in the northern Corinthia than in the Argolid: in the East Alley, gray Minyan, matt-painted, and yellow Minyan wares were found together with sherds of Mycenaean LH I and LH II styles (Davis : 256–57).
Mycenaean LH I style is also rare at Kolonna and at the circum-Saronic settlements that imported pottery primarily from Aigina throughout the MBA and early Mycenaean period (Lindblom : 43, table 9; Siennicka : 181–84). Relatively few sites with good early Mycenaean deposits have been published, and these have produced few examples of Mycenaean LH I. In Attica, it is exceedingly rare; Kiapha Thiti has few sherds if any at all (Maran : 205; Mountjoy : 491–92). Megali Magoula (Galatas) has produced some sherds of Mycenaean painted LH I style from the mounds of earth covering two early tholos tombs; this material seems earlier than the tombs themselves, reflecting settlement pottery rather than grave goods (Konsolaki-Yiannopoulou : 73). If Megali Magoula flourished in MH because of access to the Aiginetan economy, the tholos tombs appear to indicate a later prosperity tied to relations with the Argolid and beyond.
Commenting on exchange systems in LH I, Mountjoy (: 20, 492) finds it surprising that lustrous decorated and other early Mycenaean styles should be so rare in the Saronic and the Corinthia, despite the easy voyage from the Gulf of Argos, where they are found in abundance. She notes that the shapes in which Aiginetan workshops specialized, including hydrias, amphoras, and kraters, do not duplicate the fine tableware of LH I style, so redundancy is not an explanation. She speculates that Aiginetan activity might account for the lack of pottery decorated in the LH I style, and that Lerna and Kolonna may have had separate interaction spheres. This seems correct, but I would go further to suggest exclusionary practices – a deliberate strategy of protectionism reflecting not only economic hegemony but also a final phase of Aiginetan political muscle.
LH II marks a transition when Mycenaean pottery of palatial and pseudo-Minoan type found its place at Aigina, Kiapha Thiti, and Athens by LH IIA. Both of these classes were produced locally at Kolonna and Athens (Mountjoy : 492). Among the pseudo-Minoan types, the marine style is found at Kolonna, Athens, Thorikos, and Eleusis. But in that same period Aiginetan imports still made up 7–10% of the corpus at Tsoungiza and 20% at Kiapha Thiti (Maran : 204–211). Mycenaean LH IIB pottery is still relatively little known in Attica, except for some graves at the Athenian Agora, until masses of later LH IIB pottery were dumped into wells on the south slope of the Athenian acropolis (Mountjoy : 492–93). Also included in these deposits is late matt-painted ware, possibly an Aiginetan product.
The appearance of Mycenaean pottery for the first time in substantial quantities marks the initiation of a shift, played out over a period of maybe 50 to 100 years and essentially accomplished in LH IIIA – the early Mycenaean palatial period in the fourteenth century – by which Mycenae swallowed the Saronic Gulf into its economic and political orbit. It is no coincidence that Kolonna's export industry seems to have gone into decline sometime during LH IIIA1, around the time of the establishment of the first verifiable palace at Mycenae (Lindblom : 129–30). The chronological period represented by LH IIIA1 is barely detectable at Kolonna, and few LH IIIA2 deposits in the Aegean have produced Aiginetan imports (Lindblom : 129). By that time, Mycenaean fineware and utilitarian vessels had superseded most Aiginetan shapes throughout Kolonna's former sphere of influence. (Nevertheless, exports of Aiginetan storage and cooking vessels continued in LH IIIB and IIIC, owing to their superior working qualities as well as the momentum of long-term relationships by which they were exchanged [Lindblom : 41; Zerner : 55].) It is reasonable to assume that this shift in production and consumption patterns reflects the appropriation of the export market by Mycenaeans from the Argolid, but there are also clear signs of political expansion of Mycenae during the palatial period into the southwestern Corinthia and the Saronic Gulf, though probably not the northern Corinthia.
At Tsoungiza in the southwestern Corinthia, a ceremonial feasting deposit of LH IIIA2 (trench EU 9) consisting of cattle bones; drinking, serving, and cooking vessels; and a fragmentary ceramic female figure has been interpreted as the remains of a regional feast intended to cement alliances between elites at Mycenae and Tsoungiza (Dabney et al. ). The analysis of a pit with contents dating to LH IIIB1 shows that residents of tiny Tsoungiza had access to the same range and quality of pottery as Mycenae, indicating a close link but not necessarily strict control (Thomas ; this may already have been true in LH IIA: Rutter : 90). Patrick Thomas also reinterpreted the so-called potters' shop in House B at Zygouries as a workshop for the manufacture of perfumed olive oil, implying a close link with Mycenae's interests in LH IIIB (Thomas ). In the broader sweep of the Mycenaean era, the southwestern Corinthia was only gradually incorporated into the political economy of the Argolid. Wright (: 127) has associated the Nemea Valley with a "periphery model," in which such regions exhibit considerable autonomy, participating in alternative social and economic networks before being incorporated into palatial economies to varying degrees in LH III.
A different pattern prevails in the northern Corinthia. There, the numbers of Aiginetan as well as other imported vessels declined in LH IIIA2. During the palatial period, Corinthian fineware shows strong stylistic connections with the Argolid in both forms and decorative motifs, but virtually all pottery vessels and terracotta figurines are believed to have been made locally (Morgan : 353). The absence of true imports from the Argolid makes it highly unlikely that Mycenae dominated the northern Corinthia politically or established a permanent presence there (Pullen and Tartaron ; Tartaron ).
In the Saronic Gulf, the process of Mycenaean expansion into the region is not easily appreciated because few contexts spanning early to later Mycenaean are available, and in general the early Mycenaean remains are inferior in quantity and quality to those of the later Mycenaean phases (Siennicka ). Ongoing investigations at the MH–LH settlement of Megali Magoula offer a window onto the process by which Mycenaean influences insinuated themselves into the Saronic Gulf region (Konsolaki-Yiannopoulou 2003a, ). Located in the southwestern corner of the Gulf, with manageable overland and maritime access to the Argive Plain and the Argolic Gulf, Megali Magoula was well positioned to be an intermediate link between the two bodies of water. As we have seen, a prosperous community of the MBA had strong ties to Aigina, and MH III–LH I sherds found in the fill of the somewhat later tholos tombs show continuity into the LBA. Of the three tholoi, Tomb 3 seems to be earliest, dating perhaps to LH I based on pottery and weapons tenuously associated with the burial(s). The form of the tomb, built entirely above ground with a circular chamber and no dromos, recalls the EM–MM tholoi of southern Crete; Eleni Konsolaki-Yiannopoulou (: 72–73) proposes that it may represent, along with the Vagenas tomb in Messenia, a link between Cretan tombs and Helladic tholoi – though of course there is nothing approaching a consensus about the origin of the Helladic tholos (Rutter : 139; Voutsaki : 42–43). If such a connection existed, it might have been part of the cultural expansion of Minoan Crete that affected Ayia Irini at the dawn of the LBA.
Tombs 1 and 2 are more recognizably Mycenaean tholoi, the architectural features and pottery of which indicate a date in LH IIB for their construction and earliest burials. They are quite different in form. Tomb 1 is a very large tholos (D = 11.8 meters) of Pelon's Class C built mainly above ground with an artificial tumulus heaped over it (Konsolaki-Yiannopoulou 2003a: 165–75). Elements of the tomb's construction find parallels in early tholos tombs in Attica, Messenia, and the northeastern Peloponnese. The Mycenaean pottery, while not found in undisturbed burial contexts, indicates that the tholos was in use from LH IIB to LH IIIB. Tomb 2 is a very small example (D = 3.8 meters) of Pelon's Class A, rare in the northeastern Peloponnese but common in Messenia, where Minoan influences were strongly felt (Nelson ; Pelon ). A construction date in LH IIB is also favored, with continuing use in LH III and a concentration of Mycenaean pottery in LH IIIA2–IIIB1 (Konsolaki-Yiannopoulou 2003a: 177–78). Initial use of these tombs in LH II coincides with the first wave of Mycenaean pottery in the Saronic, and we might imagine elites at Megali Magoula now taking their cues from the families burying their dead in early tholoi in Messenia and the Argolid, keeping in mind that the fertilization of Mycenaean culture from Crete was still ongoing. As Kolonna lost its preeminent position in the Saronic in LH IIIA, the wider area of Mycenaean Troizen around Megali Magoula flourished, indicated for example by the rich chamber tomb cemetery at nearby Apatheia, where evidence for libations as part of elaborate funerary rituals parallels similar traces in the Megali Magoula tholoi (Konsolaki-Yiannopoulou ). Following Konsolaki-Yiannopoulou's (: 73) suggestion that "[t]he fall of Aegina and the rise of Mycenaean Troezen are two parallel phenomena, which may not be disconnected...," it is reasonable to perceive in these changing fortunes the moment at which Mycenaean presence in the Saronic began to have political, not just economic or cultural, ramifications.
The archaeological record shows unambiguously what a momentous shift this was (Figs. 7.9, 7.10). The number of known sites around the Saronic increases almost twofold in late Mycenaean times when corrected for phase durations, and numerous new settlements indicate a dynamic expansion (Siennicka : 184–89). Some sites that had long been occupied continued to flourish; for example, in Attica, Eleusis and Ayios Kosmas experienced prosperity and expansion, and the long-established settlements of the northern Corinthian plain carried on as before. But many more of the settlements were new foundations of the palatial period, as Figure 7.10 clearly shows. With some variations, they adopted the typical repertoire of Mycenaean material culture, including pottery forms and styles, architectural techniques, burial customs, and cult practices; in short, they participated in the Mycenaean cultural koiné that formed rapidly in LH IIIA and remained in place until it began to fragment in later LH IIIB. To give a sense of the range of palatial-period communities in the Saronic Gulf region, I will next describe briefly two settlements, Kanakia on Salamis Island and Ayios Konstantinos on the Methana peninsula, before taking up a third, Korphos-Kalamianos, at much greater length. (For a more inclusive survey of LH IIIA–IIIB Saronic settlements, see Siennicka : 184–89.)
7.9 Map of early Mycenaean sites in the Saronic region. After Siennicka : 180, fig. 1.
Kanakia was an acropolis-type settlement of LH IIIA–IIIC date in the southwestern corner of Salamis, built on a series of terraces with retaining walls on and around a pair of neighboring peaks (Lolos ). The site overlooks two harbors, with a broad viewshed encompassing much of the Saronic Gulf. The built area covers approximately 4.5 hectares, with structures varying in size and plan separated by roads and courtyards (Fig. 7.11). Free-standing structures with one, two, and three rooms have been identified, along with true megara, trapezoidal buildings, and corridor-type buildings such as are known in LH IIIB contexts at Mycenae, Tiryns, and elsewhere. There are also at least two complexes of multiple, attached buildings on the upper areas of the acropolis. The site is unfortified, but the approaches are steep and a system of watch towers seems to have been in place.
7.10 Map of late Mycenaean sites in the Saronic region. After Siennicka : 180, fig. 2.
7.11 General plan of Mycenaean Kanakia, Salamis. After Lolos : 238, fig. 4.
Excavations since 2000 have focused on structures within the building complexes of LH IIIB–IIIC date. The structures often rested on multiple levels conforming to the terraced topography; an example is building IA, a LH IIIB corridor house built on two levels with an upper level devoted to working areas where stone tools, pottery, and traces of mineral pigments were found, and a lower-level cellar where pottery vessels were stored. Building IA forms part of a larger industrial complex with buildings IB and IΔ; this compound comprises more than forty rooms and spaces for workshops, storerooms, auxiliary rooms, corridors, courtyards, and paths. The finds of querns, grinders, whetstones, spindle whorls, beads, a hoard of bronze tools in IΔ, and everyday pottery of LH IIIB2–LH IIIC Early are consistent with this interpretation. Some evidence of cult has been found in a couple of buildings, in the form of a number of clay anthropomorphic and animal figurines, the former mainly of phi and psi type, but these attest to ritual practice in household or workshop contexts only. Overall, the settlement as revealed to date reflects a working community; as yet no building of truly palatial character has been uncovered. Yet the size of the settlement, the quality of the architecture, and the presence of imported goods suggest that this was an important settlement. Architectural details such as columned entrances (propylaia), a large "double megaron" (building Γ, considered by the excavator to be a ruler's residence: Lolos : 235), and a unique, massive tower-like structure attached to a twin gate that controls access to a triangular space all point to a community of some wealth and power. Pottery was imported from the Argolid, Attica, and Aigina – in the last case the cooking pots, some with potters' marks, which were still circulating in palatial times. In the industrial area of IB, a large fragment of a Cypriot copper oxhide ingot was found, and also of Cypriot origin or inspiration, a piece of a ceramic wall bracket from building IΔ of a type known from Tiryns, and from the same context a coarseware stirrup jar marked in a Cypriot fashion.
Kanakia is best interpreted as the seat of a local ruler well connected to Mycenaean political and economic networks; with probably fine harbors, it must have been a destination for maritime traffic in the Saronic Gulf. Salamis was a busy place in LH IIIA–IIIB, with a large number of settlements and cemeteries that have not been adequately investigated (Anastasiou-Alexopoulou ). In the early twelfth century, Salamis was apparently a destination for refugees of the palatial collapse and Kanakia may have been one of several sites on the island to receive them until circa 1150, when it was finally abandoned.
Ayios Konstantinos is a small village of the Mycenaean palatial period, situated on a high ridge overlooking the southeastern coast of the Methana peninsula. Unlike Kanakia, the settlement had no easy access to the sea, and so probably supported an agropastoral community exploiting terrestrial resources and routes. Yet among its humble buildings it housed a remarkable sanctuary, important for numerous reasons: its inconspicuous position within a simple village; the in situ condition of the remains, which permits chronology and ritual performance to be reconstituted; and the distinctiveness of the cult objects, which show local variability that cannot be characterized as a chronological effect (Hamilakis ; Hamilakis and Konsolaki ; Konsolaki-Yiannopoulou , , , 2003b). The cult centered on the small Room A (4.3 × 2.6 meters), whose furnishings consisted of a floor of mixed earth and pebbles, a stepped bench in the northwest corner opposite the entrance, a low platform along the south wall, a podium in the center of the room, and a hearth in the southeast corner (Fig. 7.12). The finds date the use of the room to LH IIIA–LH IIIB. On and around the bench, excavators found more than 150 terracotta figurines, tripod altar tables, pottery, and a triton shell similar to those found in Minoan shrines. The corpus of figurines is unusual in that it consists mainly of bovids (cattle and oxen) and horses, with several rare groups including horses with helmeted riders, horses with chariot groups, and ridden and yoked oxen. The standard Mycenaean female figurines that are so abundant elsewhere are virtually absent. Other aspects of the sanctuary are well attested elsewhere, however. Like most Mycenaean cult places outside the palaces, this sanctuary lacks monumental construction or decorative elaboration. The pottery includes kylikes, bowls, alabastra, and rhyta, all common ritual shapes. Certain structural features, a stepped bench on which figurines were displayed, and platforms on the wall opposite the bench and in the center of the room, probably served as attention-focusing devices in the rituals and connect this sanctuary with others such as the Temple in the Cult Centre at Mycenae. Of utmost significance is the hearth, which was filled with ash and animal bones as well as scattered sherds from tripod cooking pots. Analysis of the faunal remains revealed a predominance of burnt juvenile pig bones, with lesser representation of sheep and goat (Hamilakis ; Hamilakis and Konsolaki ). The presence of all body parts suggests that these animals were burnt offerings (holocausts) to the deity rather than meals roasted for human consumption. The destruction of the entire animal body is perhaps to be understood in terms of the symbolic consumption of the offering by the deity (Hamilakis and Konsolaki : 145). This is the first evidence found in a primary use context for burnt animal offerings in Mycenaean Greece, although the practice of animal sacrifice followed by human consumption was certainly widespread (Hamilakis and Konsolaki : 144). In such close quarters, the performance of ritual at Ayios Konstantinos may have created an embodied sensory experience of food, drink, music (the triton shell used as a horn), and symbolic communication with deities and ancestors through the sights and smells of burnt offerings (Hamilakis and Konsolaki : 146–47).
7.12 Partial plan of excavated Mycenaean structures, Ayios Konstantinos, Methana, with Room A indicated. Konsolaki-Yiannopoulou : 26, fig. 1. Courtesy of the Swedish Institute at Athens.
The anomalous features at Ayios Konstantinos are difficult to assess, since we possess few Mycenaean sanctuaries and thus do not know the true range of variation. We do not know whether the sanctuary was autonomous, serving the needs of a small rural community, or tethered to a regional center, such as Megali Magoula (Konsolaki-Yiannopoulou , 2003b). Ayios Konstantinos may have been like one of the outlying communities to which the palaces sent animals for sacrifices and feasting, as attested in the Linear B archives at Pylos and interpreted from a large deposit of animal bones and tableware at Tsoungiza (Bennet : 33; Dabney et al. ).
Kolonna itself was occupied throughout the Mycenaean palatial period, as we know from pottery and burials, but there is little architecture that can be definitively attributed to LH IIIA–IIIB, and the surviving material is sufficiently meager that the continuing status of Kolonna as a center of major political and economic importance is in doubt. There are mitigating circumstances, however. The necropolis on nearby Windmill Hill indicates a sizable population, and extensive leveling in the Archaic and Hellenistic periods has obliterated at least some of the earlier architectural complexes. Remains of buildings and terraces underneath later structures, exposed in recent excavations in the West Complex and the south slope, may be part of the "missing" fourteenth to thirteenth century center (Felten : 18–19; Felten et al. ). The ceramic material and the tombs demonstrate that Kolonna had been incorporated into the Mycenaean koiné, while imports from Cyprus and the southeastern Aegean show that Kolonna remained connected to regional and interregional maritime trade.
Elsewhere on Aigina, there are ample signs that influences from the Argolid were pervasive in the palatial period. The later sanctuary of Aphaia in the northeastern corner of the island was possibly an open-air hill sanctuary already in the LBA (Pilafidis-Williams ). The presence of standard terracotta human and animal figurines implies the adoption of Mycenaean cult practices. Neutron activation analyses carried out on sherds and figurine fragments from the site identified an origin in the Argolid for a high percentage of both groups (Pilafidis-Williams : 166–81). If we combine this evidence with the limited but growing material from Kolonna, a picture emerges of an island thoroughly invested by Mycenaean influences from the Argolid no later than LH IIIA2, and possibly earlier.
The critical juncture at which hegemony in the Saronic passed from Kolonna to Mycenae seems therefore to fall sometime early in LH IIIA, i.e., the first half of the fourteenth century. This has been seen as some form of conflict or competition (Pullen and Tartaron 2007), but the nature of the interaction and resulting transformation is unclear. Was it a violent takeover of territory and trade routes, or was it an evolutionary process in which Mycenae's superior resources and broader networks of relations around the Aegean and beyond gradually rendered Kolonna irrelevant? There is no obvious evidence of destruction at Kolonna in this period, or necessarily of retrenchment; indeed, recent excavations indicate that "...the whole enlarged settlement was in use at least until LH IIIB" (Felten : 19). Nor is there much clarity about Mycenae's specific endeavors abroad since the early Mycenaean period there is known mainly from burials, and even LH IIIA settlement in and around the citadel is poorly known because of the extensive rebuilding programs in LH IIIB (French et al. ; Shelton ).
On balance, the second scenario seems more likely and has been offered as a partial explanation for the emergence of Mycenae to prominence in the Argolid (Voutsaki , , , ). Sofia Voutsaki (: 113–14) makes a compelling case that Mycenae outmaneuvered Argive rivals such as Lerna and Asine to forge strong ties with partners on Aigina, the Cyclades, Kythera, and Crete. This network of alliances, giving access to exotic goods and raw material wealth – displayed or fashioned into high-status items deposited ultimately in monumental tombs – allowed elites at Mycenae to differentiate themselves from their counterparts in the Argolid and to position themselves, in social network terms, to accumulate ties preferentially and thus to suppress competition. As mentioned above, a similar scenario has been proposed with Kolonna as the dominant node in the Saronic Gulf, and Kolonna may even have played a role in suppressing the emergence of a palace state in the Corinthia (Pullen and Tartaron : 157). Nevertheless, groups in the Argolid at Asine, Argos, Midea/Dendra, Tiryns, and elsewhere continued to bury exotic items and other forms of wealth with their dead at least through LH IIIA, before the concentration of wealth in burials was increasingly restricted to Mycenae in LH IIIB (Burns : 168–90).
Thus, we can establish the likelihood, but not the certainty, that it was Mycenae that carved out maritime networks in the Saronic Gulf before LH IIIB. Given this ambiguity, it is the smaller settlements located in between Kolonna and Mycenae, such as Megali Magoula, with material spanning LH IIB–LH IIIA, and the later foundations at Kanakia and Ayios Konstantinos, through which we witness the gradual transfer of the Saronic region from the Aiginetan to the Mycenaean sphere of influence. The last location considered in this case study, the coastal site of Korphos-Kalamianos, presents another perspective on the Bronze Age Saronic maritime small world as a settlement that alternated over time between prominence and insignificance, between high and low connectedness. A consideration of this settlement from the dawn of the Bronze Age to the end of the Mycenaean palatial period will help to round out our diachronic narrative.
#### Korphos-Kalamianos and the Saronic Small World
In 2001, members of the Eastern Korinthia Archaeological Survey (EKAS) discovered a large Mycenaean architectural complex at the location Kalamianos near the village of Korphos, on the rugged Saronic coast of the southeastern Corinthia (Fig. 7.13; Rothaus et al. ; Tartaron et al. ). The importance of the site was instantly clear: walls and foundations of buildings of Mycenaean type, some of them monumental, are exposed on the surface of the gentle seaside slope above the cape known as Akrotirio Trelli, covering almost eight hectares on land and an unknown further extent now submerged underwater (Fig. 7.14). In 2006, the Saronic Harbors Archaeological Research Project (SHARP), which I co-direct with Daniel J. Pullen, was constituted for the purpose of initiating investigations on the site and in its surroundings. From 2007 to 2009, a first phase of surface investigations was carried out, comprising detailed mapping and architectural study, a surface survey on the site and in a zone of seven square kilometers around it, geomorphological and environmental research, initial underwater investigations, the recording of oral histories, and various specialist studies of the artifacts collected by the survey. (For a detailed preliminary report, see Tartaron et al. .) SHARP hopes to undertake excavations at Kalamianos at a future date.
7.13 Digital terrain model of the Korphos region.
7.14 Aerial photograph of the Kalamianos site. Balloon photograph by Kostas Xenikakis and Symeon Gesafides.
The site consists of an urban settlement preserved as stone architectural foundations and walls occupying approximately 4.5 hectares set within a town wall enclosing around eight hectares (Fig. 7.15). The "empty" 3.5 hectares seem to have been used for agricultural terraces and to quarry the settlement's building stone. Because of a unique convergence of tectonic activity, erosion, and human history, these features are exposed on the surface, giving us a rare opportunity to study a virtually complete Mycenaean settlement. The buildings employ a characteristic Mycenaean large-stone and -rubble construction, with foundations and walls preserved in situ, surrounded by massive stone collapse that indicates the considerable height of the original walls (Fig. 7.16). To date we have recorded over 1,200 walls and more than 50 buildings.
7.15 GIS plan of architecture and other features at Kalamianos. Photo by author.
7.16 Example of large-rubble construction of Mycenaean buildings at Kalamianos. Photo by author.
Although Kalamianos witnessed human activity at detectable levels during much of the Bronze Age, the urban settlement was a new foundation, laid out with a strong measure of central planning in a short period of time beginning around 1300 BC or a little earlier. Most buildings are oriented roughly to the cardinal directions, with long axes either north–south or east–west. Yet neither the layout nor the buildings themselves are uniform across the site. In certain areas, multiroom buildings cluster to form complexes, whereas elsewhere buildings are free-standing and often set at a distance from one another. Moreover, some of the buildings can be described as monumental while others are more modest in size and architectural elaboration. These contrasts suggest some form of differentiation that may be social, functional, chronological, or some combination thereof.
The chronology of the Kalamianos site was firmly established by a gridded intensive surface survey. Artifacts and features were recorded in regular 25 × 25 meter grids, and special collections were made from the interior spaces and rubble cores of intact buildings. The canonical masonry of the walls provides a rough chronology in the palatial period (circa 1400–1200 BC), but the retrieval of LH IIIB pottery built into the cores of the walls of many buildings provides a terminus post quem that indicates a construction date in the thirteenth century. A preliminary analysis of the pottery collected at Kalamianos shows how dominant Mycenaean material is relative to all other periods. If we remove the unidentifiable sherds, LBA makes up 86% with Late Roman coming in a distant second at 5.5%. Also significant is the fact that we have not yet recognized LH IIIC material, meaning that Kalamianos was likely abandoned by around 1200, and so may be closely tied to the palaces and their fate. Postabandonment phases from LH IIIC through Hellenistic are virtually absent.
##### Geomorphology of an Unlikely Harbor
We have strong evidence that Kalamianos was a harbor settlement in the Mycenaean palatial period, and we have come to believe that it served as Mycenae's principal Saronic harbor in the thirteenth century. Yet we could never have imagined making such bold statements upon first encountering the site. Kalamianos is by no means an obvious location for an ancient harbor: a shallowly submerged peninsula off the coast makes it impossible for even small boats to avoid the shoals and approach the shore today. We approached the Korphos region as most observers would (e.g., Conlin : 77), assuming that if an ancient harbor were to be found, it would be located in the sheltered, inviting Korphos Bay (Fig. 7.13), but Kalamianos provides a perfect illustration of the point, emphasized in Chapter 5, that we cannot assume that ancient Aegean coastlines possessed the same configurations as their modern counterparts.
The modern coastline in the Korphos region is rugged, dominated by a rocky shoreline that plunges to water depths of three or more meters, with the exception of Korphos Bay. Despite its rugged structure, the Saronic coast offers an abundance of small, sheltered anchorages. This was surely true in ancient times as well, but the configuration of the shoreline has changed dramatically since the Bronze Age due to tectonic displacements. In the Corinthia, tectonic movements have occurred along several major regional extensional fault systems with a complex history of differential fault motions. In low-lying, shallow water contexts like Kalamianos, these forces can bring about significant changes in coastal configuration with even small changes in relative sea level. The narrow land shelf at Kalamianos slopes gently into the shallow offshore waters, with depths of only several meters within 125 meters of the shoreline, after which the sea floor drops abruptly to 50 meters, and within 500 meters from shore reaches more than 100 meters depth. This feature is known to local fisherman as the "chasm," and is exploited as a particularly fertile fishing ground that has sustained the fishing trade for generations.
We have followed multiple lines of geomorphological evidence to reconstruct the coastline and harbor basins of the Bronze Age. Recently, a Canadian-American team collaborating with EKAS determined that the coastline of Korphos Bay, about three kilometers west of Kalamianos and just southwest of Korphos village, has undergone net subsidence during the Holocene as a result of co-seismic fault motion (Nixon et al. ). From a series of cores taken in a salt marsh, they identified up to five phases of local coastal subsidence since the mid-Holocene, associated with seismic events resulting in rapid relative sea-level rise. The transgressive events were recognized by shifts in the abundance of microfossils (foraminifera, thecamoebians) in marsh sediments and correlated with tidal notches in the inshore area. They estimate a relative sea-level rise of about four meters in the last 5,500 years. Members of the same team recognized several beachrock platforms at depths up to 5.9 meters in the inshore areas adjacent to Kalamianos (Rothaus et al. ; Nixon et al. ). These cemented beach deposits were formed in the supratidal zone close to sea level and provide a useful indicator of former sea level (Kelletat ; Vousdoukas et al. ). In spite of the proximity of these two locations, their tectonic histories are not identical; Nixon and colleagues report that Korphos Bay and Kalamianos have distinct and independent sequences controlled by different fault blocks (Nixon et al. : 51–52). This result illustrates how localized tectonic effects can be, with serious implications for coastline reconstruction, while the shared indications of multiple subsidence events support the archaeological evidence of submerged Bronze Age structures and artifacts off the coast at Kalamianos.
The next step toward identifying the configuration of the Bronze Age coastline and harbor basin was taken in 2009, when a collaborative project was initiated between the Canadian Institute in Greece and the Greek Ephorate of Underwater Antiquities (Enalion). More than 400 line kilometers of bathymetry, side-scan sonar, sub-bottom seismic, and magnetic survey data were acquired within a ten-square-kilometer expanse of sea in the Korphos region using a seven-meter Zodiac inflatable survey boat. The bathymetric survey generated a detailed map of the sea-bed relief around the site, and determined the location and configuration of beachrock ridges identified by previous work, which were then mapped using Differential Global Positioning System (DGPS) equipment. The sub-bottom seismic and magnetic survey data provided information on sediment thickness, bedrock structure, and location of buried ballast and pottery materials within the harbor basin. Underwater diver surveys were conducted using scuba equipment to investigate the submerged beachrock platforms and other targets identified by the geophysical survey. These were documented with underwater video and samples were obtained at several locations for ongoing laboratory analysis (grain size, micropalaeontology, pottery studies) and AMS radiocarbon dating of shell materials.
##### Results
Based on the results of these studies, Joseph Boyce has constructed a preliminary model of the evolving Bronze Age paleoshoreline configuration (Fig. 7.17). The bathymetry clearly identifies a submerged bedrock promontory extending east from Akrotirio Trelli and a drowned isthmus that formerly connected the small islet with the mainland coast. The submerged isthmus divides the inshore area into two separate lagoonal basins (the "western" and "eastern" basins in Fig. 7.17a). Two distinct beachrock platforms (BR-1, BR-2) appear in the bathymetry mapping and were confirmed by diver survey. BR-1 consists of two mound-like beachrock outcrops located on the submerged isthmus, about 100 meters from shore. The mounds are up to 1.2 meters in height, 30 to 40 meters in length, and about 20 meters in width. Both outcrops are elongated roughly parallel with the modern shore and have a basal water depth of 3.2 to 3.6 meters. Cemented into the calcarenite of BR-1 are thousands of Mycenaean sherds, constituting around 30% to 50% of the beachrock volume and showing little sign of post-depositional reworking or biological alteration. This condition is consistent with rapid burial, as with a tectonic event, in a supratidal low-energy beach environment. The lowermost beachrock platform (BR-2) occurs at a depth of 5.4 to 5.8 meters on the western margin of the submerged promontory. The beachrock is about 0.4 to 0.6 meters in height and incorporates well-preserved sherds of EH pottery making up 10% to 20% of the beachrock volume. This pottery also preserves surface decorative features and lacks significant post-depositional reworking or biological alteration, consistent with rapid burial. Because beachrock forms at the interface of shore and sea, and because the Aegean is nearly tideless, we know that at one time BR-1 and BR-2 were shoreline positions. The pottery cemented into the platforms gives terminus post quem dates for the formation of the beachrock; that is, BR-1 could not have formed prior to the Mycenaean period, and BR-2 must have formed in the EBA or later. Yet because the condition of the pottery suggests rapid burial and not gradual transport or wearing away of surfaces, and because our examination of the potsherds to date indicates segregation of the pottery phases with little mixing of earlier or later material, it is highly likely that the broken sherds were incorporated into the deposits roughly during the time of their use, whether as the refuse of normal harbor activities or the result of a catastrophic tectonic event.
7.17 Reconstructed coastlines and harbor basins at Kalamianos. Courtesy of Joseph I. Boyce, Despina Koutsoumba, and the Trustees of the American School of Classical Studies at Athens.
The provisional chronology derived from the associated pottery allows us to assign the BR-1 shoreline to LH III (circa 1400–1200 BC) and the BR-2 shoreline to an EH phase (circa 2700–2200 BC). As reconstructed, during the LH (Mycenaean) phase the islet was much more extensive than at present (approximately 500 square meters) but separate from the mainland. The bedrock promontory on the east side of Akrotirio Trelli would have provided a sheltered anchorage site (western basin) with a deep-water approach, the extent of which is approximate because the thickness of the post-Mycenaean sediment fill has yet to be established in seismic and core data. During the Mycenaean phase, small boats could have been pulled up onto shore, and larger ships may have anchored in the western basin or moored at the offshore island. The process of onloading and offloading may have generated much of the broken pottery preserved in BR-1. The western basin would have provided a sheltered anchorage during periods when the dominant winds were blowing from the north or west to southwest, accounting for most wind patterns throughout the year. During periods when winds were blowing from the east and southeast, the offshore island offered some protection from winds and along with the submerged promontory diminished wave energy, but ships might also anchor off the western side of Akrotirio Trelli.
During the EH phase, the local relative sea level was about 5.4 meters below present and the island was connected to the mainland via an isthmus that stood 1.0 to 1.5 meters above sea level. Together, the island and isthmus formed a natural recurved breakwater about 250 meters long and 40 to 50 meters wide, creating a well-protected double harbor configuration with many options for moving watercraft as required by weather conditions and a sufficiently deep approach to permit even the largest seagoing vessel of the day – the Cycladic longboat – to anchor close to shore.
Other important clues to the location of anchorage sites were obtained from the distribution of ships' ballast, which can be detected by a magnetic gradiometer survey even when buried at some depth (Boyce et al. ). Magnetic surveys in the eastern and western harbor basins at Kalamianos identified a number of magnetic "hotspots" found by subsequent examination to be associated with accumulations of volcanic ballast stones and pottery, which have a significant induced and remnant magnetization compared to the local limestone bedrock and seafloor sediments. Diver reconnaissance surveys of the western basin identified a number of small ballast stone piles and a large, partially exposed ballast mound consisting mainly of andesitic boulders and limestone cobbles (Fig. 7.18). The exposed portion of the ballast mound is four to five meters in diameter and includes scattered Mycenaean pottery fragments. Mapping the distribution of magnetic anomalies and recording their sources is helping to pinpoint the locations of anchorage sites. An intriguing and possibly telling pattern in the magnetic data shows numerous anomalies all around Kalamianos, but few in Korphos Bay. This pattern seems to confirm that Kalamianos was the area's primary anchorage, and there is some evidence that the modern Korphos Bay may have been primarily a wetland in the Bronze Age.
7.18 Ballast pile identified in inshore waters at Kalamianos. Courtesy of Joseph I. Boyce, Despina Koutsoumba, and Trustees of the American School of Classical Studies at Athens.
#### Beyond the Site: Korphos as a Bronze Age Regional Center
SHARP undertook a systematic surface survey of seven square kilometers outside the walls of Kalamianos from 2007 to 2009, using both intensive and extensive methods (Fig. 7.19; Tartaron et al. ). The survey aimed to contextualize Kalamianos in its wider microregional setting, in order to better understand how the harbor town was sustained by and connected to its interior hinterland. We were astonished to discover that the survey area was nearly as rich as Kalamianos itself in ancient architectural remains, similarly exposed on the modern surface (Fig. 7.20). The bulk of this architecture dates either to the EBA or LBA, and we quickly realized that these were the periods in which the Korphos region came to some kind of prominence in the past. Each of these periods is characterized by three kinds of architectural remains (Table 7.2). Taking each in turn, we shall see that there are similarities, but also interesting differences in the locations where people chose to build, and in their overall use of the landscape.
7.19 SHARP survey zones and survey units.
7.20 Ancient architectural remains in the SHARP survey area, marked in black.
Table 7.2. Classes of EBA and LBA architectural remains in the Korphos region
* * *
Early Bronze Age| Quantity| Late Bronze Age| Quantity
---|---|---|---
Habitation sites (Kalamianos and Stiri)| 2| Habitation sites (Kalamianos and Stiri)| 2
Stone cairns| Approximately 25| Agricultural terrace walls| Dozens to hundreds of preserved segments
Elliptical stone enclosures| Approximately 20| Fortified stone enclosures| 2
* * *
##### Early Bronze Age
The earliest material recovered in survey is pottery belonging to the Final Neolithic/Early Helladic; although it is difficult to be more specific chronologically for unstratified material using only formal criteria, there are reasons to believe that most of these sherds date no earlier than the end of the long FN period or the beginning of EH. Remarkably, the whole range of vessel forms, from tableware to cooking and storage pots, contain inclusions (almost certainly temper) characteristic of the volcanic suite of minerals found on Aigina. The reddish, iron-oxide-rich fabric may derive from local terra rossa clays, tempered with crushed volcanic rock retrieved from andesite imported from Aigina as raw material or as finished ground stone implements. The appearance of Aiginetan volcanic rock in the Korphos region at this early date is consistent with Curtis Runnels' (1985a) finding that the island exported volcanic millstones by Late Neolithic, and it implies that Aigina was already becoming a center in the Saronic region. Close connections with Aigina already in the EBA are not surprising, since Kolonna and Kalamianos are intervisible sites; indeed, on a clear afternoon it is possible to make out the archaeological site of Kolonna from Kalamianos.
EBA material (architecture, pottery, lithics) is spread throughout the survey area. During the EBA, there were two substantial settlements in the Korphos region: a seaside settlement at Kalamianos – now mostly submerged and without obvious architectural remains preserved on land – and a quite large settlement at Stiri, perched high above Kalamianos on a coastal bluff (Fig. 7.19). The most striking feature of the poorly preserved settlement at Kalamianos is an obsidian workshop now eroding from gravels near the modern water's edge. All stages of the reduction process, from raw nodules with preserved cortex to finished blades, are present. Inland from Kalamianos, obsidian is ubiquitous, but occurs mainly as finished blades and flakes, although cores and other pieces show that tool making took place on a small scale in many locations.
The settlement at Stiri is marked by discrete fields of stone on the sea-facing southern slope, replete with EBA pottery and stone artifacts, which represent the locations of collapsed structures. Several thousand pottery sherds and obsidian flakes and blades were concentrated over an area of approximately four hectares on the peak and surrounding slopes, making Stiri a very substantial settlement for its main period of occupation in EH II. The pottery encompasses a wide range of types and decorative styles, indicating a thriving domestic settlement with local and long-distance trade contacts. Two likely motivations for occupying this location are the vast viewshed over the Saronic Gulf, and ready access to a number of small but well-watered upland basins ideal for small-scale agriculture.
A second architectural manifestation of the EBA comprises about 25 enigmatic cairns – amorphous stone piles distributed both on the ridge of the Pharonisi peninsula north and west of Kalamianos, and in the upland basins to the north (Fig. 7.21). These cairns can be distinguished from modern field clearance piles by their form, erosional features, and artifactual associations. Pottery retrieved from their interiors is of FN–EH II type, with no certain later material. They occur in a larger and a smaller size that seem to relate to different functions. The larger cairns are similar to those we have investigated in the last decade at Vayia and Vassa in the northeastern Peloponnese, where collapsed but partly preserved wall faces suggest an originally squarish, perhaps tower-like form that we interpreted as collapsed bastions in enclosure walls around EH II settlements (Tartaron, Pullen, and Noller ). Those cairn groups tend to snake through the landscape, with large cairn mounds connected by wide linear stone piles that we interpreted as walls. There are only two certain cairns of this scale in the survey area, and unlike at Vayia and Vassa, their relationship to known settlements is unclear.
7.21 Satellite image with locations of stone cairns (diamonds) and enclosures (ovals). Satellite image © 2009 Google Earth, © 2009, Digital Globe.
The smaller cairns cannot be interpreted in the same way. They are far more numerous (23 of 25), and their form is more limited to a round or elliptical mound of stone, without radiating linear features. In one of the smaller cairns, illicit digging revealed a cavity or chamber built up in a corbelling technique, which suggests the strong possibility of a burial chamber, though no finds were discovered inside (Fig. 7.22). Despite the differences between the small and large cairns, they share one intriguing feature: most have one or more depressions in their upper surfaces, suggesting a collapsed cavity such as the one revealed illicitly. It is thus possible that all of the cairns are funerary monuments, with simpler and grander versions.
7.22 View and drawing of a small cairn on the Pharonisi peninsula. Drawn by Giuliana Bianco.
The collapsed cairns of the Korphos region would have been taller than they appear today, and more visible on the landscape locally and from a distance. They were placed in prominent ridge-top locations with expansive viewsheds and high intervisibility with other EBA cairns and enclosures. Typically, they overlooked both the sea and adjacent arable land, suggesting that they were meant to be seen from the sea, and possibly also served as territorial markers manifesting the claims of a living community to land and resources through explicit links of descent from ancestors who occupied them in the past (Murphy ; Saxe ). Whatever the range of functions, the cairns in the Korphos area can now be associated with at least a regional, and not simply a local, tradition in the northeastern Peloponnese.
The final architectural type comprises approximately 20 walled stone enclosures, found in virtually every part of the survey area, which can now be confidently dated to the EBA on the basis of associated artifacts – we have not recovered later material from the features themselves – as well as geomorphological observations. Though predominantly elliptical in form, they range from round to elliptical to squarish and vary in size from 15 × 12 meters to 25 × 30 meters, translating roughly to between 125 and 700 square meters of internal space (Fig. 7.23). The locations and viewsheds of the enclosures provide the best clues to their functions (Fig. 7.21). Most have excellent views both to the sea and to nearby arable land below them. Almost every enclosure is potentially intervisible with at least one other, and many with several others, although we lack information on vegetation cover and we cannot definitively establish that the enclosures are all contemporary. Yet it appears that they were placed systematically and strategically on the landscape with a carefully rationalized social purpose. Some potential functions include monitoring stations with views to the sea, to agricultural territories, and to an upland basin-to-basin route running west to give access to the interior Corinthia and the Argolid; collection structures for agricultural produce; animal enclosures; territorial markers; or forts or similar defensive complexes. Perhaps they combined all of those functions as the strongholds of extended family or kin groups arrayed across a contested agropastoral landscape. Historical and ethnographic examples of contested landscapes resulting in functionally comparable structures can be found in Greece and the Balkans (e.g., Galaty in press; Galaty, Lee et al. ; Karakatsianis ; Mangalakova ; Wagstaff ).
7.23 View of an EBA stone enclosure.
There also seems to be a chronological and conceptual association between the cairns and enclosures. The stone enclosures are closely associated spatially with cairns at several locations, but elsewhere they seem to be isolated or semi-isolated from other architectural complexes. Yet both are embedded in the same webs of intervisibility: cairns are visible from enclosures and vice versa. Most decisive, however, is the fact that at two or three places, cairns and enclosures are combined into a single feature. Together, these features present a highly humanized, and perhaps competitive, rural landscape in EH II.
The implication of this busy countryside is that the Korphos region was an important EH center, a coastscape interacting with other coastal centers in a period of high connectivity. The importation of Aiginetan andesite and obsidian from Melos, 170 kilometers distant, shows Kalamianos already operating as a major harbor. Whether the obsidian was obtained directly on Melos by Korphiote voyagers, purchased from visiting Cycladic traders, or acquired through Aiginetan intermediaries, the fact remains that Kalamianos is one of relatively few import and primary reduction centers known in the Aegean, joining the list of such sites as Lerna in the Argolid (Hartenberger and Runnels ; Runnels 1985b), the Fournoi cluster in the southern Argolid (Kardulias : 64–71; Kardulias and Runnels : 104–108; Van Andel and Runnels : 89–91), and Romanou in Messenia (Parkinson ). In the reconstructed EBA coastline, the obsidian workshop lies close to the shore of the western basin.
In EH II, the sequence of corridor houses and other signs of emerging complexity at Kolonna indicate that it continued to be the center of a Saronic small world, with cultural ties, though not yet regular trade relations, with other coastal settlements with fortifications and corridor houses. Kalamianos/Stiri played a significant role in the Saronic small world, though it is difficult to know if labels such as "peer of Kolonna" or "secondary center" are appropriate. The Saronic Gulf was apparently a vibrant place in EH I–II before a precipitous decline sometime in EH III. EH material has been found in almost every area subjected to excavation or regional survey; perhaps the most compelling demonstration of general vitality emerges from systematic, intensive surveys. The Methana Survey team, though by their own account working in a "rough and rocky place," nevertheless recorded fifty-one sites with EH pottery in the limited confines of the Methana peninsula, roughly evenly divided between EH I and EH II (Mee and Taylor : 42–51). SHARP has located dozens of scatters of EH sherds in its own small survey area, many of these not directly associated with the EH sites or architectural features described above.
The data from SHARP and Methana are again informative about the period of abandonment, or at least retrenchment, between EH III and the end of the MH period. SHARP has produced no certain EH III or MH pottery, with the possible exception of a few sherds with standard Aiginetan potters' marks that may fall sometime in MH I–LH I. On the Methana peninsula, both EH III and MH sherds are rare, though present (Mee and Taylor : 51–52). This is precisely the period in which the vibrant Saronic small world of the EBA collapsed, compelling Kolonna to refocus its energy beyond the Saronic Gulf. That the gulf was not an entirely empty seascape, however, is underscored by the recent discovery on Salamis of a large MH II–III acropolis-type settlement at Sklavos, on the island's southern coast facing Aigina (Lolos ). It seems, therefore, that scattered pockets of the Saronic still supported substantial communities, while most places were reduced to tiny hamlets or abandoned altogether.
##### Late Bronze Age
During the LBA, the Mycenaean harbor settlement at Kalamianos was the main, anchoring center of the Korphos region. Pottery recovered at the site indicates that a settlement of modest size had taken root in the fourteenth century – just less than 10% of the LH assemblage at Kalamianos belongs securely to LH IIIA. From this inconspicuous beginning, in the early thirteenth century, i.e., the LH IIIB1 pottery phase, the urban harbor complex was built and became one of the more important sites in the Saronic region. It was also in the thirteenth century that the Mycenaeans developed the hinterland to harness the agricultural and pastoral potential of the lowland and upland zones, in support of Kalamianos' maritime (and perhaps overland) connection to the Mycenaean economy. The physical traces of this expansion include a second substantial settlement built at Stiri over a part of the old EH site, a large fortified enclosure in the territory between Kalamianos and Stiri, and dozens of agricultural terrace walls of Mycenaean date.
The Mycenaean settlement at Stiri sits on a ridge overlooking a double-lobed basin to the west, and the sea to the east and south (Fig. 7.24). As at Kalamianos, the foundations and lower walls of several distinct complexes of well-constructed buildings are exposed on the surface, preserving the plan of the settlement in its apparent entirety (Fig. 7.25). At around 1.4 hectares in extent, Mycenaean Stiri is less than one-fifth the size of Kalamianos. The masonry technique is essentially the same as at Kalamianos, although Stiri lacks the monumentality of some buildings at the harbor site, and the varied building plans do not match those at Kalamianos particularly closely. Yet the buildings are remarkable in their own right, such as the sprawling central structure 13-III that consists of between 35 and 40 rooms. The Mycenaean artifacts recovered within rooms and wall cores belong exclusively to LH IIIB, showing that Stiri was a later foundation than Kalamianos, but also that the two settlements overlap chronologically in that phase.
7.24 View of Stiri and adjacent polje, with location of the Mycenaean site indicated.
7.25 Differential GPS plan of Mycenaean architecture at Stiri.
Location must have been an important factor in the role Stiri played in the Mycenaean coastal world of the thirteenth century BC. Perched on a high sea cliff with an unobstructed view of Kalamianos, Stiri was undoubtedly in constant communication with the harbor town below. A sweeping viewshed extending from Athens and Salamis in the northeast to Aigina and Methana in the east and southeast allowed the inhabitants to monitor seaborne traffic on the Saronic. A second important function is suggested by the basin west of the site, which is well watered by springs and winter rains, making agriculture and pastoralism possible on a relatively large scale. Intensive cultivation of wheat and olives has been practiced here in recent times, along with grazing of sheep and goats on wheat stubble and in the wooded hills all around. This productive landscape may have been systematically developed to provide staple crops, animals and their secondary products, and trade goods to the harbor community at Kalamianos.
In the sloping terrain between Kalamianos and Stiri, a large Mycenaean walled enclosure was situated in a saddle between two low peaks immediately north of Kalamianos, consisting of a large space enclosed by partially preserved fortification-grade walls that can be traced for about 180 meters (Fig. 7.26). Within the presumed interior, there are many terrace walls of Mycenaean type on the north slope facing a large basin that may have been another locus of agricultural activity. Also preserved inside the fortified area is one of the elliptical stone enclosures of EBA date. To the south, the sea view is limited, but the harbor at Kalamianos is plainly visible. This site may have been the agricultural estate of a high-status family or individual, connected closely with the settlement at Kalamianos and perhaps with elite families there.
7.26 Plan of architectural features at the "saddle site" north of Kalamianos.
In the territories adjacent to the two main settlement sites, the Mycenaeans invested heavily in terrace wall construction, apparently to maximize agricultural potential in this stony, semi-arid landscape. The terrace walls are the subject of a recent dissertation by Lynne Kvapil, whose important contribution has been to systematize the documentation and dating of terraces throughout the study area (Kvapil ). Systems of terracing dating to the Mycenaean period have been found at Kalamianos itself, on the western slopes of the hill north of Kalamianos, on the south-facing slope of the Pharonisi peninsula, in the saddle area described above, and on the steep south slope below Stiri. At Stiri in particular, the slope facing the sea was terraced with massive walls in Mycenaean masonry technique, sections of which survive on contours from top to bottom (Fig. 7.27). One aspect of their construction that ties them closely to Mycenaean architectural practice is the use of stones with flat outer faces and long, triangular trailing edges that help to bond the wall with the terrace behind it. This technique is also clearly seen in wall building at the two settlement sites. The southern slope at Stiri is the steepest on which the Mycenaeans built terraces, partially explaining their monumentality, but just as important was their prominent visibility from the sea. As at Kalamianos, monumentality and high visibility seem to have been as integral to the design of the built environment as their utilitarian function. Although there were several basins suitable for agricultural exploitation in both the lowland and upland zones, it seems that the Mycenaeans felt the need to maximize agricultural yields to support the population of this microregion and perhaps to generate a surplus for shipment from the harbor at Kalamianos.
7.27 Monumental Mycenaean agricultural terrace walls at Stiri.
#### Comparing Early and Late Bronze Age Exploitation
There are similarities, but also important differences, in EH and LH patterns of activity in the Korphos region. The Mycenaean inhabitants occupied Kalamianos and Stiri as their primary settlements, as had their counterparts in EH. These locations make sense as the lowland and upland anchors of the region, giving access to the sea at Kalamianos and agropastoral resources as well as panoramic viewsheds at Stiri. In LH IIIB, Kalamianos was a much larger and more important settlement than Stiri, while in the EBA the relationship was reversed. Mycenaean habitation at Stiri was confined primarily to the ridge top; the steep south-facing slope was apparently used only for agricultural purposes as many segments of strongly built terrace walls survive but counts of LBA artifacts are quite low.
Beyond the two main settlement sites, the differences in the distribution of remains and use of the landscape are striking. The most distinctive difference is that EBA activity, measured both by architecture and portable artifacts, was much more extensive throughout the survey area, while the Mycenaean activity pattern was more spatially limited, focused on the habitation sites and their immediate surroundings. A likely explanation for this difference is that EBA activity was the result of a long development, begun already in FN, and thus a "settling into" the landscape. Depending on how far back into FN the activity began (and this we do not know at present), a period of a millennium or more is indicated to the end of EH II. The EBA signature developed over a relatively long period of gestation leading to a flourishing and complex society in EH II.
The Mycenaean distribution, on the other hand, reflects a deliberate but short-lived transformation of physical and social landscapes in which emigrants, most likely from Mycenae, arrived in the late fourteenth century to a sparsely populated area, built a harbor at Kalamianos, and developed the hinterland to support it. The identification of Mycenae as colonizer of Kalamianos rests on circumstantial evidence, which taken together presents an argument that we have found persuasive, if not yet conclusive. It is perhaps most accurate to say that the evidence draws us to the Argolid, with Mycenae consistently the most plausible option. The Mycenaean fineware collected from Kalamianos and Stiri exhibits general affinities with the Argolid, while the architecture offers compelling parallels in construction techniques and monumentality (Tartaron et al. ). The large-rubble construction of buildings at Kalamianos can be classified as Type III cyclopean masonry in Claire Loader's (1999: 27–31) typology, characteristic of the Argolid and other Mycenaean core areas. The Mycenaean agricultural terrace walls, particularly at Stiri, show strong similarities to those in the vicinity of Mycenae itself. It is also possible to make a case that Kalamianos was the most conveniently located anchorage offering Mycenae access to the Saronic Gulf, Attica, and the Isthmus of Corinth, particularly since the evidence of Mycenae's presence in the northern Corinthia is slim (Pullen and Tartaron ; Tartaron ). There is a modern land route beginning at Korphos or Stiri, which follows a series of interconnected east- to west-trending basins and passes through Angelokastro and Limnes, before joining the Mycenaean road at Berbati to finally reach Mycenae after a journey of approximately 50 kilo-meters on foot. This route is attested by villagers in the Korphos/Sophiko area, and members of SHARP have made the walk on several occasions in a single day, requiring between nine and thirteen hours depending on fitness. It is by no means an easy walk, but even making allowances for ancient tracks rather than modern roads, a two-day journey with a donkey would not have been difficult.
Mycenae's interest beyond the connectivity offered by the maritime station was agricultural intensification in small pockets of fertility, while the upland zone also monitored the sea and connected the region to routes leading to the interior of the Corinthia and the Argolid. The timeframe of their arrival in the late fourteenth or beginning of the thirteenth century, as suggested by the ceramic evidence, coincides with the explosion of sites with Mycenaean characteristics on the islands and shores of the Saronic (see Fig. 7.10). By the late fourteenth century, the penetration of Mycenaean material culture was profound, encompassing not only styles of architecture and pottery, but also burial and cult practices, including the objects that accompanied them – such as the ubiquitous anthropomorphic and zoomorphic figurines that might betoken the propagation of a state religion.
Mycenae's presence in the Korphos region was intense but brief, lasting perhaps only 100 or 125 years, before the abandonment of the region circa 1200 BC, roughly synchronous with the collapse of the palace state at Mycenae. The brevity of Mycenae's presence precluded expansion into all niches in the landscape as a normal consequence of development and growth. However, the substance and monumentality of the Mycenaean constructions suggests that they were built for permanence, and surely long-term growth was expected before it was truncated by the collapse of the palaces.
#### Korphos and the Saronic World through Time
The results of SHARP's field studies permit the outlines of a diachronic narrative for the Korphos region to be interwoven with that of Kolonna and other communities to develop a larger history of the Saronic maritime small world.
Sometime before the beginning of the third millennium BC, potters in the Korphos area were importing Aiginetan volcanic stone, which they crushed and used as temper in the full range of functional pottery classes. Incipient settlement in the SHARP survey zone in FN/EH I grew steadily, culminating in a highly developed economic and social exploitation of the landscape in EH II. During that phase, raw obsidian was imported from Melos and worked into tools at a workshop overlooking the western basin of the harbor at Kalamianos. Stiri was a large settlement well situated for agropastoral subsistence and for expansive views over the Saronic Gulf. The pottery assemblage at Stiri represents a full domestic suite, and shows that the inhabitants were connected to sources of contemporary shapes and decorative styles. The Korphos region can thus be counted among the nucleated and socially complex coastal centers of the EH II Aegean. At that time, Kolonna, with its large settlement and two phases of a grand corridor house, was the most important settlement in the Saronic and well along its trajectory toward regional domination. This was a period of cohesion in the Saronic small world as settlements in places like Kalamianos, Kiapha Thiti, and the Methana peninsula interacted with Aigina, and, although it is nearly impossible to prove, surely with neighboring small settlements as well.
From EH III to the beginning of the Mycenaean palatial period, Kalamianos is almost invisible archaeologically, like so many other small settlements of the northeastern Peloponnese. This hiatus lasted even longer than for the many communities that were founded or revitalized in MH III or LH I. The scant evidence of human presence, consisting of a few sherds at Kalamianos with standard Aiginetan potmarks, is insufficient to project more than a sparse, inward-focused population engaged in agropastoral pursuits, with limited external contacts between 2200 and 1400 BC. This dramatic depopulation prevailed throughout the Saronic, with the principal exception of Kolonna, which exploded into complexity with continuous expansion of the settlement, characterized by massive building programs of fortifications and dwellings. Kolonnans now developed long-distance contacts with Minoan Crete, the Cycladic islands, central Greece, and the interior Peloponnese, in part to compensate for the deep reduction in connectivity within the Saronic Gulf. They imported pottery and may have hosted a small enclave of Minoan potters, but soon Aiginetan potters developed their own highly successful export industry that persisted well into the Mycenaean palatial period, and for specific shapes even to its very end.
The recolonization of the northeastern Peloponnese and the lands bordering the Saronic in MH III–LH I, and the events of the Shaft Grave Era in the Argolid, seem to have drawn Kolonna's attention back to the Saronic Gulf. The earliest phases of the LBA marked a time of prosperity and high connectivity between Aigina and the settlements on the islands and coasts of the Saronic, along with more distant partners in Attica, the northeastern Peloponnese, central Greece, and the Aegean Islands. Mycenaeans from the Argolid expanded their interests and exports only gradually into the Saronic region. Mycenaean-style painted pottery of LH I–IIA is rare in the circum-Saronic area. The Saronic small world, although nested geographically within the Helladic realm, may have been culturally distinct from the emerging Mycenaean palatial state in the Argolid, and seems to have resisted its expansion into the Saronic Gulf. By LH IIB, when Mycenaean fineware pottery had begun to appear around the Saronic, the inhabitants of Megali Magoula in the Troizenia were building tholos tombs, perhaps signaling the establishment of a Mycenaean foothold on the western shores of the Saronic. During the crucial transition to early LH IIIA, Kalamianos was part of a contested periphery – the setting for a competitive process in which Mycenae extended its sphere of influence into the Saronic Gulf at the expense of Aigina. The foundation of a number of centers large and small in LH IIIA, such as Kanakia and Ayios Konstantinos, coincided with the decline of the Aiginetan pottery export industry and the adoption of Mycenaean cult practices at Aphaia.
The founding of a port town at Kalamianos, probably by Mycenae circa 1300 BC or slightly earlier, served two objectives: first it was a foothold and safe haven for maritime economic and military activity in the Saronic, and second it was a definitive statement of Mycenae's ascendancy. This statement is encoded in the monumentality of the architecture at Kalamianos and the terrace walls at Stiri, quite remarkable in contrast to other Saronic settlements of the period, marking Kalamianos as a second-order center and probably Mycenae's principal Saronic harbor. This display of power was probably not specifically aimed at Aigina, since Kolonna by that time was no longer a legitimate threat. Instead, it was a characteristic habit of the Mycenaeans of the Argolid to build monumental structures as an advertisement of power, from the shaft graves and tholos tombs to the fortification walls and elaborate buildings on their citadels. The imposing architecture at Kalamianos and Stiri was meant to be seen from the sea.
Kalamianos was a coastscape and the anchor of a maritime microregion characterized by highly developed internal organization, which was at the same time the creation of the wider Mycenaean world, to which it was closely connected by both sea and land. The Korphos region was developed to support the role of Kalamianos as a working harbor town, giving rise to a second substantial settlement at Stiri and a system of agricultural terracing. Kalamianos was not a long-lived settlement, however. The rapid and intensive development of this microregion ceased abruptly circa 1200 BC, when Kalamianos and the other sites were abandoned, suggesting a strong association with the fate of the palaces and many other settlements that were destroyed or abandoned at that time.
#### Oral History and Kalamianos
In the absence of written records in prehistory, different forms of ethnographic and ethnoarchaeological research can contribute to enlightening hypotheses about the conditions of seafaring and the social and economic networks that prevailed in ancient small worlds. Members of SHARP were fortunate to be able to interview elder residents of Korphos, who described details of life in the village in the years during and before World War II, when there were no paved roads to Korphos and no motorized seacraft, yet the Saronic Gulf was teeming with social and economic activity. Lita Tzortzopoulou-Gregory conducted a program of interviews between May 2007 and June 2009, which I have mined for the observations that follow. Of particular relevance to the topic of this chapter are the relationships that the inhabitants of Korphos maintained with the inland village of Sophiko on the one hand, and the coasts and islands of the Saronic on the other.
Prior to the Second World War, Korphos was a fishing and seafaring village, with perhaps 90% of the male population engaged in fishing or merchant activities on the Saronic Gulf. Young boys learned by doing, taking to the boats at a young age to accompany their fathers and grandfathers on their rounds. The more ambitious or better connected aspired to be sea traders because there was good money in it. The fishermen were generally poor, as fish were plentiful and cheap throughout the Gulf. Their work provided subsistence and fish to exchange with farmers, shepherds, and forest workers for needed commodities. There were approximately 30 families living in Korphos, each owning at least one fishing boat or caique. As many as 60 rowboats, fishing boats, caiques, and small sailing boats were anchored at Korphos. Most of the boats were built at Perama on Salamis island.
The consensus among the seagoing Korphiotes is that the Saronic is a relatively trouble-free body of water to navigate. They use the word limni (lake) to describe it, asserting that the winds and currents are not especially dangerous, and the shallows and other hazards are few. This is not to say that environmental conditions had little effect on voyages. One experienced seaman reported that the trip from Korphos to Aigina in a small sailing boat could take anywhere from three to seven hours, depending on the winds. On longer trips, the merchants would overnight at ports of call in their boats before setting off for home the next morning; they generally did not travel on the Saronic at night.
The fishermen worked in local waters and preferred the fishing ground between Kalamianos and the small island of Ayios Petros offshore. It was there that the shallow waters off Kalamianos gave way to the steep drop-off of the sea bed, known locally as the "chasm," where the catch was plentiful. The fishermen rarely ventured more than a few kilometers from Korphos. In winter, fishing continued but kept close to shore. In addition to subsistence use, fish and seafood were transported by donkey to Sophiko, a trip of approximately one and a half hours by an old path that followed a stream bed west of town to the upland basins that open west to the interior Corinthia. One older woman remembers bartering for goods with Sophiko residents who did not have cash to pay for the fish.
Korphos was, in the early twentieth century, a proti skala, a major port in the Saronic trade, and this afforded the sea traders a more varied life, intimately connected with both inland producers and the merchants at ports and anchorages around the Saronic. Farmers and herders from Sophiko village owned most of the land in the hinterland of Korphos, and they engaged in several traditional pursuits. Farmers grew cereals, chiefly wheat but also barley, and tended olives, mainly for their own subsistence needs with the surplus traded in Korphos and elsewhere. Wheat and barley were also grown in the limited lowland basins, including the one directly above the Kalamianos site. Sheep and goat were herded in the upland areas and their primary and secondary products were offered in trade for maritime products and services. The most prevalent occupation in the upland zones around Korphos, however, was forest work. Wood, charcoal (mainly from bushes and bush roots), and pine resin were harvested in this heavily forested region and brought on donkeys to Korphos for shipment abroad. The sea traders purchased these varied products and exported them to Saronic markets, either in their own boats or in larger ships they contracted for the purpose. It was not only at Korphos that these products were collected for shipment. Often, when a farmer's fields or trees were closer to one of the many tiny anchorages in the area, the produce would be brought down and picked up there. One resident reported that the sea traders often took advantage of the inland producers who were dependent on sea transport by bargaining for unfairly low prices.
There was not a single dominant port in the Saronic, but instead a handful of large, bustling nodes of maritime connectivity. Several interviewees recalled bringing wood, charcoal, resin, and manure to markets at Piraeus, Eleusis, Salamis, Aigina, Poros, Nea Epidauros, and elsewhere. Frequently, a port town specialized in processing certain material or had high demand for specific products. At Eleusis there were factories processing resin, while charcoal and wood were in demand at all of the above-named ports. In exchange, the Korphiotes sought food and staples. From Aigina they imported flour and water jugs (even in modern times tempered with the volcanic inclusions that enhanced their performance), fruits and vegetables from Nea Epidauros, and foodstuffs and water from Piraeus, among many other items. On returning to Korphos, the merchants brought their wares to Sophiko and sometimes beyond, where local buyers acquired them and distributed them further on. The forest industries have long since become economically unprofitable. There are few uses for charcoal, and pine resin, once used in turpentine and other chemical products, has been superseded by synthetic substitutes, while the popularity of resinated wine has declined in recent years. A few farmers continue to harvest resin on a small scale.
Fresh water was and remains scarce in the village, and this was perhaps a strong incentive for Bronze Age people to settle at Kalamianos instead. Water was retrieved from coastal sites such as Nea Epidauros, Kyra island, and occasionally Kenchreai. One informant describes four men regularly taking a four-meter-long rowboat to Nea Epidauros to fill 150-kilogram barrels with water, taking turns rowing one and a half hours each way. Tiny Kyra island, several kilometers off Kalamianos, had a fine though not copious spring where fishermen would often fill up. In the years after World War II, small boats brought water daily from Piraeus or Salamis as part of government programs. Fetching water by boat was a summer activity, since cisterns in the village filled amply with winter rains. Women and girls traveled by boat or donkey to Kalamianos to wash clothing in two brackish wells there.
Some of the more intrepid seafarers ventured outside the Saronic, one mentioning that he had sailed out to islands such as Siros, and along the eastern Peloponnesian coast. We might think of these as the modern counterparts of the "expert" sailors discussed in previous chapters. Many Korphiotes spent some part of their adult life in the merchant marine, aboard big ships engaged in international commerce. They all returned to the village and their families after several years at sea.
Kinship relations with Sophiko were close, and there was much intermarriage. People also found spouses in Aigina and Salamis; many Korphiotes emigrated to Salamis and Aigina after marriage. This is one demonstration that social imperatives such as maintaining genetic and demographic viability bound together coastal communities in a small world. Another example is that children from Korphos, Nea Epidauros, and other coastal villages were sent to Aigina for high school because these small communities could support nothing more than a one-room elementary school. The notion presented in Chapter 5 that the landward limits of the coastscape were generally the passes and the first-encountered inland nodes finds support in the movements of the Korphiote merchants, as well as the fact that there was little interaction with Corinth before the modern road was built to join the Corinth–Epidauros coastal highway in the 1960s.
When prompted concerning the general orientation of the community, the informants were unanimous that the Korphiotes have always thought of themselves as an island people: they looked to the sea for their livelihood, wore island dress, listened to island music and danced island dances, and created networks of interaction with coastal and island people in the Saronic. They contrasted their outlook with that of the Sophikites, whom they considered inland, "mountain" people. That they nevertheless maintained close social and economic ties with Sophiko indicates the dual orientation of a maritime coastal community, and exemplifies the inland--coastal symbiosis that is an important feature of the dynamism of coastal life. Perhaps the coastal–inland symbiosis between Korphos and Sophiko in modern times is analogous to the relationship between Kalamianos and Stiri in the LBA. Several interviewees spoke of a pre-modern switchback walking path from the lowland north of Kalamianos up the steep slope to Stiri, used to access the eleventh-century church of Panayia Stiris; thus, although the two sites seem mutually inaccessible, people on foot with their donkeys have managed to overcome an environmental obstacle to preserve connectivity in this microregion.
The Korphos–Sophiko system in the early modern period bears the stamp of a microregion in Horden and Purcell's terms, and Korphos emerges as a coastscape and a maritime coastal community. The people of Korphos forged the link between the terrestrial and maritime worlds and facilitated the exchange of desired commodities. The sea merchants truly occupied a position of centrality with respect to connectivity around the Saronic. Young boys were inculcated in the seafaring life and the essential knowledge was passed down within families, much as we have seen among South Pacific societies. In the first half of the twentieth century, the Saronic Gulf was a vibrant modern small world, with a proliferation of nodes on coasts and islands and innumerable crisscrossing paths connecting them.
What use are these oral histories to us as we contemplate life in the coastscapes of the Mycenaean period? With the customary caution against equating modern times with eras of the remote past, it is possible to suggest that the challenges and opportunities encountered by these two peoples inhabiting a Saronic small world bear many similarities. The traditional lifeways of early twentieth-century people and their Mycenaean counterparts in the Korphos region were not qualitatively dissimilar; they possessed comparable technologies of subsistence and seafaring. They lived at times of modest prosperity and vigorous interaction, when both were highly connected to spheres of interaction on land and sea. Much will have been different, of course; to name just one example, the structures of political power are not comparable. Nevertheless, the information we obtained from local residents tends to support the picture I have constructed from archaeological and ethnographic data, and therefore it seems appropriate to add it to the diverse strands of evidence bearing on the reconstruction of Mycenaean coastal worlds. The theoretical underpinnings of this position rest in a structure–contingency framework (Bintliff ; Tartaron : 158–59): essentially, there are long-term structures, corresponding in annales terms to the forces of the longue durée that influence the configuration of societies and their interactions with the world around them. Among the most important are the environment (including physical geology and geography, climate, and resources) and the human subsistence technologies (agropastoral, maritime) and other adaptive mechanisms (culture) that allow populations to survive and sometimes thrive over time. By establishing structural similarities between two societies or periods, it is acceptable to take the comparisons further, but this may not be done by ignoring the differences, which may reside already in the structural realm but are most salient in medium-term political and economic patterns (conjonctures) and in decisive events (événements). It is in the interplay of long-term forces with shorter temporal and smaller spatial contexts that historical contingency arises, giving each locality and community a unique history. My contention is not that we can simply equate the Korphiotes of the early twentieth century with their counterparts at Kalamianos in the LBA. Rather, given key structural similarities of environment, technology, and location in the Saronic Gulf, their respective engagements and worldviews on facing the sea – their connectivity and interaction patterns – may also share important parallels, at least hypothetically as we await future phases of investigation.
#### Discussion
When considered in terms of a maritime cultural landscape framework, we observe the fluctuation of the Bronze Age Saronic maritime small world between cohesion and fragmentation, as demographic patterns and external opportunities drew Aigina's attention into and away from the Gulf. The hegemony of Aigina in this small world, at least economically, seems to have begun already in the later Neolithic. From there, the Saronic maritime small world developed steadily to a peak in EH II, collapsed from EH III to MH III, revived in the Shaft Grave Era to reach a second peak in LH I–II, until finally (though gradually) Mycenae usurped Kolonna's traditional role. For the coastal communities dotting the coasts and islands of the Saronic Gulf, this transformation entailed not only a new master, but new cultural material and practices, and a reorientation of maritime relations and connections. In effecting this transformation, Mycenae broke apart the old Saronic world and incorporated the region into a larger world of land and sea connections.
I hope to have made a few central points in this extended case study by interweaving the stories of Kolonna and Kalamianos, ones that can be applied usefully to other cases. The Saronic was susceptible to the emergence of maritime small worlds because visual contact, relative ease of movement by sea, and moderate distances facilitated connectivity and the experiential sense of a coherent world. Taking a bottom-up perspective, we can propose that this is important because most Mycenaeans lived and died mostly or wholly within these small-scale settings. For more than a millennium, Kolonna, with a fortunate location and important natural resources, established itself as a center interacting with small peripheral settlements in the Saronic as well as more distant trading partners. But precisely because small worlds are nested in larger-scale spheres of influence and respond to the consequences of external developments, they are prone to change over time. The Middle Helladic hiatus shows that, as Horden and Purcell emphasized, social forces often trump environmental imperatives; we cannot simply map maritime relations according to currents, winds, and distances.
By following Kolonna and Kalamianos, we see the Saronic small world responding both to internal dynamics and to shifting centers of power and demographic trends played out beyond the Saronic. Kalamianos became prominent only in periods of strong supra-local connectivity: EH II with its nucleation of population and strong maritime orientation, and LH III with the incorporation of large territories by the Mycenaean palaces. In each case, the harbor at Kalamianos and its hinterland were developed to articulate with economic and political systems of greater scope than the Saronic. If we break down these broad patterns, we could write a different history for each coastscape, reflecting varied effects of, and responses to, dynamics both internal and external to the Saronic. The story of Kalamianos is different from those of Megali Magoula, Kiapha Thiti, or the Salaminian settlements at Kanakia and Sklavos, nuancing but not diminishing the validity of the broad diachronic and spatial patterns. The same dynamism pertains to the shape and extent of the regional/intracultural sphere over time. The changing distribution of Aiginetan pottery (excepting rare distant outliers) is a useful measure of Kolonna's regional sphere of interaction in a given phase (Fig. 7.7).
Tracking the long-term history of the Saronic leads to the realization that Kolonna and Mycenae exercised very different styles of center–periphery leadership. The evidence from Kalamianos and other sites suggests that when the Mycenaeans infiltrated the western shores of the Saronic, they colonized, built massively, developed local economies, and in some cases extended a measure of political control. By comparison, the Saronic small world of the Aiginetans seems decidedly underdeveloped. Certainly, Kolonna exercised economic hegemony, benefiting from control over trade in the Saronic and extending its export networks to the nearby mainland and islands. Yet one looks in vain for sites with monumental Aiginetan-style architecture, or other signs of intensive political or economic development of the Saronic. As such, the coastscapes of the Saronic were not exactly like the peraia of later times (Constantakopoulou ; Horden and Purcell : 133), because the elements of political control and direct economic exploitation from the island state that seem to have been essential in the Classical period were lacking.
In attempting to understand the coastscape at Kalamianos and its role in the Saronic small world, the ability to reconstruct the Bronze Age coastline was decisive, and this will be true also in the two brief case studies to which I now turn.
### Potential Coastscapes and Small Worlds: Miletos and Dimini
In this concluding section, I offer brief outlines of two additional places where there is high potential for identification of coastscapes and small worlds. These observations on Miletos and the Latmian Gulf, and Dimini and the Bay of Volos, are not detailed analyses, but rather explorations of ways that a maritime cultural landscape perspective might be illuminating in understanding the Mycenaean-period activity in these coastal settings. The main principle guiding the selection was that reasonable amounts of both archaeological and paleocoastal information should exist.
#### Miletos and the Latmian Gulf
The former Latmian Gulf (now virtually closed) is a striking example of a deep marine embayment created by flooding of a low-lying coastal shelf during the pan-Mediterranean Holocene marine transgression (Figs. 7.28, 7.29). At the peak of the transgression circa 6000 to 5000 BP, the gulf may have extended 40 to 50 kilometers inland, but there is some evidence that relative sea level was actually highest circa 2500 BC (Bay and Schröder n.d.; Brückner ; Herda et al. ; Knipping et al. ; Müllenhoff et al. ). At the termination of the marine transgression, the process of infilling of the gulf by delta progradation of the Maeander (Menderes) River began, assisted by the instability of the natural Mediterranean environment and augmented by variable long-term human impacts. A German geoarchaeological team placed more than 100 sediment cores in the Maeander floodplain in order to reconstruct the advancing coastline in the context of human activity (e.g., Brückner : 121–27). Using methods similar to those described in Chapter 5, they relied mainly on macro- and microfaunal analysis to determine diverse environments of deposition (marine, littoral, lacustrine, terrestrial). Radiocarbon dating of organic material furnished a chronological framework, which was supplemented by archaeological and historical information.
7.28 Map of the southeastern Aegean and southwestern Anatolian coast. Drawing by Felice Ford.
7.29 Three-dimensional map of the Latmian Gulf at maximum marine transgression, circa 4000 BP. After Bay and Schröder n.d., fig 3.
The progradation of the shoreline toward the Aegean was gradual through the Bronze Age, though a modest increase in sediment load can be attributed to the erosional effects of expanded goat herding in the second millennium BC (Knipping et al. : 368, table 1). A rapid and massive increase in the rate of sedimentation occurred only in the first millennium BC (Bay and Schröder n.d., figs. 4, 5). During the Mycenaean period, the gulf still penetrated some 30 kilometers inland, and the promontory of Miletos consisted of two main islands, one formed by Home Tepe and Kale Tepe and the other the area of the later temple of Athena, which may or may not have been connected to the mainland by a tombolo (Brückner : 129–30); in short, Miletos was part of an archipelago-like coastal landscape facing onto a still-vast Latmian Gulf (Fig. 7.30). All around the islands and coastal areas there will have been natural anchorages and small coastal plains suitable for habitation. The climate was favorable, with moderate temperatures and adequate rainfall to support agriculture. Other natural resources such as timber and building stone were plentiful (Greaves : 8–16). The Maeander valley was also a communication corridor connecting the sea with east–west land routes to the interior. Along those routes metals and other products from the interior of Anatolia may have been passed along to the Aegean (Greaves : 32–37).
7.30 Map showing the topography of Bronze Age Miletos and vicinity. After Brückner : 128, fig. 3.
Sporadic German excavations since the beginning of the twentieth century have demonstrated that the LBA at Miletos witnessed first intensive Minoan, then Mycenaean, influence (Niemeier , ). Early excavations established three LBA "building periods," essentially confirmed by more recent campaigns. The first building period corresponds to Minoan presence in Miletos phase IV, succeeded by Mycenaeans in the second (Miletos V) and third (Miletos VI) building phases.
Miletos V encompasses pottery phases LH IIIA1–2, from the late fifteenth to the end of the fourteenth century. Wolf-Dietrich Niemeier (, , ) makes a strong case that in the second building period, there already was a Mycenaean colony at Miletos. The architectural remains are meager, and two rectilinear buildings in the Athena temple area may or may not show Mycenaean influence (Niemeier : 30–31). But in other ways, the settlement is overwhelmingly Mycenaean. The pottery – including painted fineware, unpainted, and domestic coarseware – is predominantly Mycenaean with virtually no indigenous Anatolian types. Seven kilns from this period are known, including mainland Greek and Cretan types, establishing Miletos as an important center of pottery production (Niemeier ). Slight evidence exists for cult activity in the form of a terracotta phi-type figurine (Niemeier : 33). No cemetery associated with the settlement is known. The second building period ended in a destruction dated by pottery to the LH IIIA2/IIIB1 transition, which has been linked to the Hittite conquest of Millawanda circa 1315 BC (Niemeier : 38). As we have seen, scholarly opinion increasingly endorses the equations Ahhiyawa = Mycenaean Greeks and Millawanda = Miletos. Millawanda was a foothold for the kingdom of Ahhiyawa on the western coast of Asia Minor, and Miletos is far and away the most likely candidate for Millawanda, linguistically and archaeologically.
After the destruction of Miletos V and possible control by the Hittites for some period of time, the settlement regained its Mycenaean character in the thirteenth century. The third building period, Miletos VI, has yielded LH IIIB–LH IIIC pottery in large quantities, much of it locally made. Although the architectural remains have been mostly obliterated by later construction, one corridor-type building similar to thirteenth-century examples at Mycenaean mainland centers is partially preserved. A cemetery at De irmentepe, 1.5 kilometers southwest of the Athena temple, can now be associated with Miletos VI. It includes 11 chamber tombs of canonical Mycenaean type, containing LH IIIB–IIIC pottery and Mycenaean weapons and jewelry. The evidence of cult and administration is again slight: a psi-type figurine and two pithos sherds of local manufacture that may have Linear B signs incised on them (Niemeier : 36–37). The date of the final destruction of Miletos VI has been ambiguous, but the last Mycenaean pottery has recently been placed in transitional LH IIIB/LH IIIC Early or LH IIIC Early, which by comparison with material at Ugarit suggests a date in the neighborhood of 1185 BC, at the time of general unrest in the eastern Mediterranean (Mountjoy ).
Miletos was unquestionably the most important Mycenaean settlement on the coast of Asia Minor, and there are similarities in its position within the Latmian Gulf to Kolonna's status in the Saronic Gulf at an earlier time. The scale of the two bodies of water is comparable, and the role of intervisibility among the coastal settlements must have been equally important in creating a Latmian maritime small world with numerous coastscapes engaged in dense webs of interactions. Like the Saronic Gulf, the Latmian Gulf is an ideally circumscribed body of water with which to pursue a study of interaction networks at small to medium scale. A similar sentiment is expressed by Nicoletta Momigliano, based on her study of material from Iasos. She characterizes Iasos in the early LBA as a community open to maritime traffic and exchange, but acting only within a regional sphere of interaction in the Aegean; most of the pottery is of Anatolian type while actual imports from Crete, the Cyclades, the Dodecanese, and further afield are rare (Momigliano ). She stresses that we should pay more attention to smaller-scale exchange networks and cabotage as the chief mechanism of moving material. (Of course, this is a fundamental theme for Horden and Purcell, and for the present work.) If sites like Miletos, Trianda, and Seraglio were the emporia of the LBA, Iasos is more representative of the kind of settlement we would expect to find at good anchorages on the shores of the Latmian Gulf.
It is possible to also think about larger-scale interaction spheres into which Miletos was incorporated, thanks to a protracted dialogue among archaeologists, philologists, and historians about the nature and intensity of interactions between the Mycenaeans and the inhabitants of Anatolia's western coast. Long ago, it was noticed that, roughly speaking, the regions south of the Mykale peninsula (i.e., the northern promontory of the Maeander valley) possess a much richer record of contact with the Mycenaean world than those to the north, not only in the quantity of items but also in the presence of material categories that are deemed to reflect actual settlement or some form of engagement well beyond simple trade or episodic visits (e.g., cult objects, burial practices, domestic pottery; Fig. 7.31). The patterns are relatively uncontroversial, but some see colonies or other forms of permanent presence, while others see selective adoption or acculturation. (Compare Mountjoy and Niemeier for a sampling of the debate.)
7.31 Mycenaean elements in the southeastern Aegean. Drawing by Felice Ford, after Niemeier : 103, fig. 1.
We need not get bogged down in these issues to make the simple suggestion that the zone south of Mykale, termed by Mountjoy (: 33, fig. 1) the "Lower East Aegean–West Anatolian Interface," should correspond to the regional/intracultural maritime interaction sphere (see Table 6.1) in which Miletos operated. Mountjoy (: 47–51) proposes that this Lower Interface is the kingdom of Ahhiyawa itself. This is another, much more complex, debate beyond the scope of the present discussion (see Niemeier , for the view that the kingdom of Ahhiyawa must be on the Greek mainland), but certainly the Lower Interface roughly demarcates the network in which familiar cultural materials and information moved with relative ease by sea. In LH IIIC, this zone became the core of the "East Aegean Koine" (excepting Rhodes: Mountjoy : 52–63). For a Mycenaean crew departing Miletos, voyaging beyond the Lower Interface into the Central and Upper Interfaces might have been tantamount to a cross-cultural adventure, though perhaps not particularly daunting to an experienced sailor.
It is difficult to say, in my ignorance of the area, whether a targeted archaeological prospection of the former Latmian Gulf, taking as its starting point the excellent geoarchaeological work, might succeed in populating the LBA small world. Some survey work has been done, but mostly in the vicinity of Miletos itself (Lohmann , , ) and mainly with an interest in the historical periods (but see Marchese ). Colluvial and alluvial deposits will have buried many early sites (Greaves : 40), but it is also true that Mycenaean artifacts are found on hills and in the hinterland away from Miletos, not restricted to the coast as in the period of Minoan presence (Greaves : 56). It may be interesting to attempt an investigation of some part of the lower Maeander valley from a Maritime Cultural Landscape perspective.
#### Dimini and the Bay of Volos
The Bay of Volos, on the Aegean coast of Thessaly, presents another attractive setting for Mycenaean coastal activity (Fig. 7.32). Well sheltered by its position deep within the Pagasitic Gulf, the bay was the gateway from the sea to the rich Thessalian plain, already the destination of the earliest agropastoralists of the Greek Neolithic. Paleocoastal reconstruction of the bay shows that following the maximum marine transgression circa 6000 BP, at which time the sea penetrated three kilometers inland of its modern position, a series of human impacts and natural sedimentation processes caused the shore to prograde rapidly, so that by the EBA, the coastline had moved 1.5 to 2.0 kilometers seaward (Fig. 7.33; Zangger ). The location of the shoreline in the LBA is not known precisely, but it likely averaged two kilometers from the maximum marine transgression, or a little more than one kilometer inland from the modern coast. In addition to abundant arable and pasture land nearby, coastal dwellers could exploit marine resources and trade for timber and other forest products from the Pindos mountains.
7.32 Area map of Thessaly, with important Neolithic and Bronze Age sites indicated. After Andreou et al. : 261, fig. 1.
7.33 Map of the changing coastline of the Bay of Volos. Drawing by Felice Ford, after Zangger : 3, fig. 1.
Ringing the LBA Bay of Volos were a small number of large, nucleated settlements, most prominently Dimini, Kastro (Volos), and Pefkakia. By that time, Dimini was a little more than 2.5 kilometers from the bay, but Kastro and Pefkakia had always been and remained coastal sites. Each of these sites was inhabited through much of the Bronze Age, rarely with a hiatus or a shift in settlement location. Intrasite complexity was well established at the beginning of the Mycenaean palatial period, expressed in the construction of LH II–IIIA tholos tombs near Dimini and Kastro, and built chamber tombs at Pefkakia. Thus, by LH IIIA, one group in society built monumental structures and buried their dead in monumental tombs, while others lived and died more simply. All three sites suffered major destructions at the end of LH IIIB2; Dimini was reoccupied on a small scale in the beginning of LH IIIC, but by the end of LH IIIC Early was abandoned. Only Kastro persisted through LH IIIC and into Protogeometric and Geometric times (Batziou-Efstathiou ).
Thessaly is usually considered a periphery of the Mycenaean world, in spite of a large number of sites, both on the coast and in the interior, that were heavily Mycenaeanized. Bryan Feuer (, , , ) has modeled Thessaly as a periphery exhibiting decreasing integration with the Mycenaean world as one moves from the coast to the interior, in three zones that he terms the "inner border" (i.e., the coastal zone), the "outer border," and the "frontier" (e.g., Feuer : fig. 5). Based on this pattern, Nikolas Papadimitriou () characterizes Thessaly as both center and periphery. Adrimi-Sismani () argues, however, that the entire region should be considered a fully integrated part of the Mycenaean world, having in common with it settlement patterns, intrasite settlement structure, tomb types, cult practices, pottery and other material culture, and a similar historical trajectory. For the coastal area, at least, this claim has merit and continuing discoveries tend to support it.
Much of Adrimi-Sismani's case rests on her excavations at the remarkable site of Dimini. She has touted Dimini as a Mycenaean palace center, probably the Iolkos of Homer and the saga of Jason and the Argonauts (Adrimi-Sismani , ). Excavations from 1977 to 1997 revealed a Mycenaean settlement of around 10 hectares founded east of the Neolithic mound at the end of the fifteenth century (Adrimi-Sismani , , , ). The site has two main architectural phases, in LH IIIA and LH IIIB. The later (thirteenth-century) settlement was divided into an eastern and a western zone by a wide road running north–south (Fig. 7.34). The western zone was an elite, or at least public, sector segregated from humbler domestic dwellings east of the road. The western sector centered on Megaron A and Megaron B, two large megaron-style corridor buildings, defined by Panagiota Pantou (: 39) as structures that comprise "...a megaron-type unit flanked on one side by a long corridor and a series of smaller rooms (secondary wing)." These buildings were constructed of rubble stone foundations and mudbrick superstructures. A monumental gateway with three axial columns gave access to a forecourt and then to a peristyle courtyard before the megaron unit could be reached in Megaron A. In the series of small rooms to the south, separated from the megaron unit by a long corridor, evidence was found of food storage and preparation, as well as tools for potting and jewelry manufacture. Here too was found a fragment of a stone weight with a Linear B inscription (Adrimi-Sismani : fig. 15.4). Megaron B was even more interesting, with plentiful evidence for cult activity and feasting (Adrim-Sismani : 165). In the middle of the vestibule at the eastern end of the megaron unit lay an H-shaped altar of clay attached to an elliptical platform and two perforated, triangular mudbricks. A painted mug found in situ in front of the altar suggests the pouring of libations. In three small attached rooms to the south, cups holding the remains of animal bones were recovered. Outside the southern entrance to the large western room of the megaron unit, 16 small Mycenaean clay figurines were found next to a large limestone slab with cavities, suggesting a function as a kernos for the placement of cult offerings. The northern auxiliary wing contained many storage, cooking, and serving vessels, and just outside the building middens of animal and fish bones, seashells, and broken pottery may be the refuse of feasts. The two large rooms of the megaron unit were found nearly empty, but considering their size and the finds from adjacent areas, they may have been locations for communal eating and drinking, cult ceremonies, and other kinds of public gatherings (Pantou : 386–87). The evidence from Dimini indicates the existence of an intrasite social hierarchy with two tiers: an elite ruling and priestly caste living in the western sector and burying their dead in two tholos tombs at the site, and a larger group of commoners engaged in agropastoral and craft occupations and burying their dead in modest cist graves (Pantou : 389). Adrimi-Sismani (: 167) labels Dimini a palace center and the controlling hub of a regional hierarchy in which Dimini "...combines all the features of an administrative, financial, and religious center, and consequently it is the only settlement in Thessaly that clearly displays organization and social elements...of a true center."
7.34 Architectural plan of LBA Dimini. Drawing by Felice Ford, after Pantou : 388, fig. 5.
Leaving aside Dimini's possible mythical connections, not all accept the designation of the Megaron A/Megaron B complex as palatial, or of Dimini as a regionally dominant center. In a thorough and methodical reassessment of the archaeological evidence in the Volos region, Pantou (, ) has challenged many of Adrimi-Sismani's interpretations. Her disagreements fall in two main areas. First, she asserts that the "palace" at Dimini is not palatial. Although the plans (corridor buildings with megaron units, storage, industrial, and cult areas) and some of the activities (e.g., feasting, cult) carried out in Megara A and B emulate those of the Mycenaean palaces, the materials used (stone socle and mudbrick superstructure), the modest elaboration (e.g., simple plastered floor and walls with some painted colors but no frescoes, no ashlar blocks), and the size (falling into Pascal Darcque's [] intermediate, not palatial, category) fall far short of their counterparts at Mycenae, Tiryns, Pylos, and elsewhere. Further, the discovery of part of a stone weight with a Linear B inscription does not constitute evidence for "...the presence of an accounting system that monitored the movement of products manufactured in the complex" (Adrimi-Sismani : 168).
Second, Dimini was perhaps not the "administrative, financial, and religious center" of the Volos region. Pantou (: 383) finds striking similarities in architecture, burial types, and material culture among the settlements at Dimini, Kastro, and Pefkakia. For example, only minor differences in elaboration and grave furnishings exist when one compares tholoi with tholoi and cist graves with cist graves across the region. A two-tiered social hierarchy of ruling elites and commoners existed at each site, manifest in contrasts of architectural elaboration and burial monuments, but in Pantou's view this did not extend to a regional hierarchy (an opinion already expressed by Andreou et al. : 272–73). Instead, she envisions a stable socioeconomic environment with a heterarchical rather than hierarchical relationship among the sites. Dimini, Kastro, and Pefkakia were independent communities with their own internal hierarchies, but with regard to one another display overlapping, redundant features and functions. The settlements are three to five kilometers apart, intervisible, and unfortified. They lack smaller satellite settlements. In this the Volos region differs from the inland settlement pattern (the Lake Karla region, Almyros Plain, and Pharsala region), where large settlements are surrounded by satellites, probably small agricultural or pastoral settlements (Adrimi-Sismani : 171–74). The contrast must partly reflect a stronger orientation toward maritime and industrial pursuits at the coast, but Pantou (: 386) does caution that systematic, intensive surveys are needed to be sure that small sites have not been missed.
If Pantou's reconstruction of the Volos area without a central-place hierarchy is correct, it may be similar to the situation in the northern Corinthian plain, where Pullen and I have argued for long-term social and economic stability in a heterarchical arrangement of settlements (e.g., Gonia, Perdikaria, Korakou) spaced at regular intervals and exploiting similar resources in a generous environment (Pullen and Tartaron ; Pantou [: 394] notes the similarity herself). Such a stable milieu may in fact inhibit the emergence of an overarching palace center (Haggis ; Pullen and Tartaron : 148). This is in contrast to the Saronic Gulf: although Aiginetan dominance was politically underdeveloped, Kolonna was nevertheless the undisputed central place settlement and economic power driving the maritime small world for a millennium. The Mycenaean features in the Volos region might be explained primarily by acculturation, since there is strong evidence of connections with southern Greece already in the MBA. The reader will recall Maran's argument that by MH II, potters in coastal Thessaly were emulating the shapes and decorative schemes of matt-painted Aiginetan pottery (his "Magnesia polychrome"), and from there the influences traveled along with Thessalian products to the northeastern Aegean islands in MH and early LH (Cultraro ; Maran ). By the time the Mycenaean palaces emerged in the Peloponnese and Boeotia, an elite familiar with southern materials and practices was in place and eager for practical and symbolic markers of power (Adrimi-Sismani ).
These observations help us to better define coastscapes and small worlds in the Volos region and beyond. The Bay of Volos may comprise a series of coastscapes within a small world defined by the Pagasitic Gulf. To the south, the Almyros plain and the western and southern coasts of the Pagasitic Gulf have produced several LH sites and five small tholos tombs in the Pteleos area, despite patchy investigation (Adrimi-Sismani : 173). Even less information is available about the Gulf's eastern promontory. It remains likely, however, that the Bay of Volos, with three major, independent settlements, was the main port area for the Pagasitic Gulf, and Pefkakia may have served as the principal harbor. Heterarchy does not mean simply the absence of hierarchy, however, but the possibility of shifting hierarchies and nonhierarchical configurations over time. Thus, in the Volos region we see that the main settlement at Dimini suffered a hiatus between EH III and MH II; Pefkakia was particularly prosperous and outward looking in the EBA and MBA; tholos tomb use continued in LH IIIB only at Dimini; and only Kastro survived beyond LH IIIC Early. With these and many more observations on individual site histories we can tease out the subtleties of their interrelationships over time.
Casting an eye beyond the Gulf to the regional scale, the early interactions with the nearby Sporades and the northern Aegean islands as far as Lemnos trace out one part of the regional interaction sphere. Another obvious and important maritime route ran south into the narrow Euboean Gulf, the safer side of Euboea for navigation, giving access to Attica, the Cyclades, and farther on the Saronic Gulf and the eastern Peloponnese. The North Euboean Gulf, with many coastal Mycenaean sites, was surely another small world that would reward investigation (Crielaard ; Kramer-Hajos ; Nikolopoulos ; Van de Moortel and Zahou ).
Placing an area like the Bay of Volos in a maritime cultural landscape framework may be simply a matter of posing the question from that point of view. One could systematically gather information on the exploitation of marine resources (e.g., fish and shellfish at Dimini), the physical traces of harbor activities at Pefkakia, the evidence of extralocal contacts in the material culture assemblages (e.g., Aiginetan influence on the MBA pottery repertoire; a Canaanite amphora at Dimini), and compare these across the sites. Were the intervisible communities at Dimini, Kastro, and Pefkakia acting in concert in connecting to networks within and beyond the Pagasitic Gulf, or were they acting independently? Was Pefkakia the main harbor for all three? Returning to the question of surface survey coverage in the region, Pantou (: 386) doubts that we understand the nature and degree of integration of the coastal area with the interior because there have not been systematic, intensive surveys. How much would such a survey change the picture we now have of large, solitary coastal settlements articulating with very differently organized habitation and production in the interior? How much could systematic survey add to the more "empty" eastern and western land masses enclosing the Pagasitic Gulf, and how would that change our reconstruction of coastscapes and maritime small worlds in the area? The Pagasitic Gulf is a fascinating case study in the extension of Mycenaean influence along maritime routes, and despite a spate of new discoveries and the extraordinary work at sites like Dimini, there is much more that could be learned with problem-oriented research on maritime networks at the local and small-regional scale.
### Conclusion
The aim in presenting one detailed and two brief case studies of Mycenaean maritime worlds has been not only to demonstrate a particular approach, but also to try to convince the reader that this approach offers the possibility of alternative histories that are truly meaningful because they reveal details about the fabric of Mycenaean life as experienced by most coastal and near-coastal dwellers. The scale of analysis appears to be justified because to a surprising extent, each region in the Mycenaean world was unique, due to the varied environmental and historical conditions that are expressed in the structure and contingency of the long-, medium-, and short-term processes of annales history. Just how striking these contrasts can be is shown in a brief comparative analysis of seven Bronze Age "settlement regions" on or near the North Euboean Gulf by Margaretha Kramer-Hajos (: 114–17). Despite being contiguous and occupying a relatively small part of Greece, they exhibit sharp differences in political organization, site types and locations, burial practices, monumental works, and other social and cultural characteristics. Surely this result validates the microregional framework of Horden and Purcell, and the focus of this book on the local and microregonal scale. Nevertheless, we must not lose sight of the bigger picture: the results of the analysis of coastscapes and small worlds form the robust data sets that can make big-picture and cross-cultural studies more than "cherry picking" from trait lists for superficial similarities and differences (Tartaron : 134, ).
In the concluding chapter, I shall restate the main points of the study, and discuss prospects for future research along the same lines.
## Eight Conclusions and Prospects
In this brief concluding chapter I revisit a few central topics and offer some thoughts on where the approach advocated in this book might take us in the future.
### A Conceptual Reorientation
During my twenty years of archaeological research in coastal areas of Greece, I have repeatedly confronted a disconnect between the usual narratives of Mycenaean maritime connectivity, focused on long-distance exchange and elite cargoes, and the lives I imagined for people living in the small coastal sites I often encountered in survey. Others had identified the same problem, albeit using different approaches. Horden and Purcell's () attempt to write a history of the Mediterranean from the point of view of microregions and short-distance connectivity was an eye-opening inspiration, as was Broodbank's () network analysis of small maritime worlds in the Cycladic islands. A group of archaeologists, among them Broodbank, Rainbird (), and Berg (), identified another problem in the segregation of land from sea, both conceptually and in fieldwork. As Berg (: 19) points out, the interest of continental and island surveys has generally stopped at the water's edge. To some extent, this problem arises because the sea–land divide is formalized in Greece by an administrative structure that places sea and land in the purview of two entirely separate authorities, with the result that permits to work on land do not extend to the sea and vice versa. Yet there is no intellectual justification for upholding this division. Islands are not in fact isolated "laboratories" of cultural evolution, and the sea is not a flat and featureless "liquid plain" serving only as connective space, but rather it is a textured, richly humanized place permeated by opportunity and danger, and animated by daily activity and maritime lore. Land and sea are not experienced separately in coastal regions, so our studies should not compartmentalize them either.
To the extent that I have any evangelical agenda in this study, it is to advocate for a shift in scholarly attention to the local scale of coastscapes and small worlds. Many of the arguments for shifting focus to the local scale have already been made convincingly. Horden and Purcell (: 123) regard microregions as the basic units of connectivity that may coalesce in larger and larger aggregates that effectively cross-cut environmental zones and geographical scales, while Broodbank showed that even in the modest expanse of the Cycladic islands, a non-uniform and fluctuating pattern of local-scale maritime interactions over time distinctly affected the overall configuration of connectivity in the island chain in the EBA. A fundamental argument in the present work is that the interactions of daily life and travel occurred overwhelmingly at close range, yet the excessive attention to long-distance maritime networks has created gaps in knowledge of the local scale. As an antidote, the coastscape is offered not as a periphery or as a boundary between land and sea, but as a uniquely central and integrative place articulating terrestrial and maritime worlds. If we wish to characterize life for the vast majority of coastal dwellers in the Mycenaean period, coastscapes and small worlds are appropriate units of analysis.
Coastscapes and small worlds are no more isolates than are islands. Like the microregions of Horden and Purcell, they coalesce and fragment, form larger aggregrates by joining other small worlds, and then devolve once again into local entities. They are routinely affected or even transformed by external influences and events: the Saronic Gulf small world was profoundly affected first by the mysterious Middle Helladic hiatus, and later by the successive influences of the Minoans and the Mycenaeans. This multiscalar dynamism is a strong indication that a renewed interest in local-scale entities should not be misunderstood as a return to historical particularism. Moreover, robust data from local and regional scales safeguards against superficial characterizations when we attempt to write larger narratives of the Mycenaean world or the eastern Mediterranean, or mine the data for cross-cultural comparisons.
### A Conceptual and Methodological Toolkit
It is not a simple matter to "find" a coastscape or a small world. We generally do not know where Bronze Age anchorages were: sites and anchorages may be lost to coastal processes, and there is no evidence as yet that Mycenaeans built durable harbor structures that might aid in identification. The Linear B tablets tell us very little about coastal and maritime activity, and the LBA is too remote from Homer to hope that we can learn much from the epic poems. There is no surviving hull material from a Mycenaean boat, and the artistic representations depict a narrow range of seagoing vessels or are difficult to interpret.
These problems present theoretical and methodological challenges. It is first essential to recognize that coastscapes and small worlds are theoretical constructs devised by archaeologists to bring order to a world they know only dimly from fragmentary evidence. They have no empirical reality independent of our typological frameworks; thus, we designate coastscapes and small worlds, we do not discover or recognize them. The framework presented in Table 6.1 is a set of models that simplify and order spatial data in an attempt to illuminate the operation of maritime networks at contrasting scales. The difficulty I encountered in trying to define the geographical scales and their attributes reflects the complexity of human spheres of interaction: boundaries are fluid and porous, networks of different size and shape overlap in space, and change is constant. The best outcome for such a framework is that the individual models – each of the spheres of interaction a testable hypothesis – present a scenario that plausibly fits the distribution of artifacts and other material evidence, and that together credibly portray the complexity of a multiscalar system. As I pointed out, this particular framework is designed for the geographical and cultural milieu of the LBA Aegean; it would not likely be valid for other times and places without extensive modification. Yet it is a tool; I offer it here with the hope that it will be tested and refined as needed. Much of the benefit of the framework resides in the way it can facilitate systematic thought about a particular problem. It may be useful in thinking through research designs, and in interpreting results.
There is nothing shockingly new in the methods I propose, since there are long traditions in both archaeology and geoarchaeology of recovering the kinds of data that could lead to the reconstitution of coastscapes and small worlds. The collaboration of archaeologists and geoarchaeologists at Kalamianos is hardly unique, but it does illustrate how interdisciplinary research can restore the essential elements of a Bronze Age coastal world. In other settings, it may not be as easy to recover ancient shores or the settlements and activity areas associated with them. A striking outcome of a long-term program of paleogeographic reconstruction of western Peloponnesian coastlines is that Bronze Age sites along much of the coast, which existed in a lagoon and barrier environment, are now buried under colluvium, alluvium, and lagoonal mud (Kraft et al. : 33–35). In such cases, geophysical remote sensing may assist in the recovery of lost sites. Most of these techniques are expensive and require lengthy analyses – even the rapid results of modern geophysical methods normally must be verified by excavation – but as I have emphasized throughout, it is difficult to reconstitute maritime networks at any scale without baseline information on the Bronze Age configuration of the coastline in the area of interest.
### Maritime Coastal Communities
The maritime activities of coastal inhabitants are elusive. We learn little about them from the Linear B tablets, apart from a few cases of shipbuilding and rowing in state-controlled ships. The archaeological record contains limited faunal and artifactual evidence for the use of marine resources (Powell ), and the results of stable isotope analyses have not yet demonstrated a significant contribution of marine protein to the diet of individuals at coastal sites (Petroutsa and Manolis ; Triantaphyllou et al. ). This appears to fly in the face of reason – depictions of fish and fishing on pottery and the frescoes of fishermen at Akrotiri tend to inspire greater confidence. Perhaps further development of the methods and standards used for the stable isotope studies (Hedges ), and more careful excavation recovery methods, will resolve this contradiction.
I chose to focus on three related aspects of maritime activity: navigational skill, the transmission of maritime knowledge across generations, and the organization of maritime activity within the larger social setting of the coastal community. I proposed that two types of individuals were active on the sea: non-specialized fishermen and farmers operating in inshore waters and traveling to nearby destinations; and master navigators who were capable of directing long sea voyages beyond the confines of the small world. Hesiod, with his short crossings to Euboea, is representative of the first group, while Homer's hard sea captains belong to the second. This division too is fluid, but it agrees with ethnographic information from the South Pacific, where most sea travel is local and only a small percentage of men attain the status of master navigator capable of leading long voyages.
Ethnographic data are informative on the other two aspects. Among South Pacific islanders, navigational knowledge is sophisticated and is protected within the maritime community – a subset of the larger village community – by physical segregation in boathouses and at sea, and by an esoteric and secret body of maritime knowledge and lore. Part of this lore involves vivid stories about markers and hazards en route to specific destinations. I suggested that the stories of hideous monsters and conspicuous burial tumuli in the Odyssey might be echoes of the kinds of maritime lore shared about distant and unfamiliar places in the Bronze Age. Specialized expertise in the mechanics of navigation and the features of specific itineraries constituted a habitus of maritime knowledge that was vital to the success of the maritime community, and it is not surprising that we should find traces of it embedded in metaphorical tales. Although there are no equivalent structures in the Mycenaean world, the Minoan ship sheds may have served a secondary function as meeting places for the seafaring community.
### Building Networks
The network analyses of Broodbank (), with updates and modifications suggested by Knappett and colleagues (, ) and Leidwanger (), are promising advances toward an understanding of how networks actually form, expand, and contract. The more recent versions address specific shortcomings by adding a range of cultural and environmental variables, the values of which can be altered individually or covaried to simulate different conditions (Knappett), and by adding texture to the sea to produce more realistic travel times (Leidwanger). The social network studies of Duncan Watts and Steven Strogatz (Watts ; Watts and Strogatz ) and Albert-László Barabási (Barabási ; Barabási and Albert ) examine the behavior of networks, showing how shortcuts extend networks over long distances, and explaining why some sites are perennially better connected than most others, and expand in connectivity more rapidly than others. I suggested that these properties of social networks may help to explain the prominence of Knossos, Mycenae, and Kolonna at various moments in prehistory. Human behavior can be difficult to model, however, and social network models as they currently exist for the Aegean can be described as a work in progress.
### A Universe of Coastscapes and Small Worlds
Malkin and colleagues (: 7) record a pithy quote from Barabási: "Networks are present everywhere. All we need is an eye for them" (Barabási : 7). The same can be said for coastscapes and small worlds; if there is a central message to this book, it might be expressed in the exhortation: "They are out there; go out and find them!" Bearing in mind that assigning a coastscape or small world is an act of interpretation and not observation (one does not really "find" them), the number of Mycenaean maritime small worlds one could investigate is practically unlimited. It primarily requires a shift in thinking about the archaeological record in terms of maritime cultural landscapes. Several archaeologists have already put a comparable approach to work, as, for example, Momigliano () for the vicinity of Iasos in the southeastern Aegean and Crielaard () and Kramer-Hajos () for the Euboean Gulf. Other possibilities come immediately to mind. The Argolic Gulf would be a challenging case study, because of the complex and ambiguous nature of the relationships of the major sites around the Gulf and further inland (Mycenae, Midea, Tiryns, Argos, Asine, Lerna, Nafplion, etc.; see Sjöberg ; Voutsaki , ). Although important paleocoastal work has been accomplished (Zangger , , 1994a, 1994b), the lack of systematic surface survey on much of the territory bordering the Gulf hinders discussion of the human landscape. Another large study might target the western Messenian coast of the Peloponnese, drawing upon the rich combination of archaeological survey and excavation, geoarchaeological studies, and the Pylian Linear B archives (Bennet ; Bennet and Shelmerdine ; Davis ; Zangger et al. ). On Crete, the Gulf of Mirabello and the Isthmus of Ierapetra, with the sites of Mochlos, Pseira, Kavousi, Gournia, and Vrokastro, would make an intriguing study. As much as anything, limitations of space prevent me from pursuing these case studies in the present work.
This book presents a set of methodological and conceptual approaches to support a particular vision of coastal archaeology, and strives to demonstrate what that approach would look like when applied to archaeological case studies. Despite the broad awareness of a comprehensive agenda for island archaeologies (Broodbank ; Rainbird ) and maritime cultural landscapes (Westerdahl ), the translation of these ideas into practice has been slow, as Berg () stresses. I have tried to be equally explicit in defining an archaeological problem – a lack of balance in our knowledge of the Mycenaean maritime world – and offering a complete set of tools to attack it. I hope to stimulate discussion, but even more to encourage new field studies and analyses inspired by the maritime cultural landscape concept.
## Notes
Chapter 2: Mycenaeans and the Sea
There are other useful frameworks that address the same issue. For example, Bernard Knapp and John Cherry (: 123–55) simplified Renfrew's list with four overlapping mechanisms of Late Bronze Age trade that emphasize the locus of control: centralized control, localized control, freelance trade, and gift exchange.
Chapter 3: Ships and Boats of the Aegean Bronze Age
McGrail (: 133) notes, however, that Egyptian shipwrights already knew of locked mortise-and-tenon joinery in the mid-third millennium BC, as superstructural elements of the Khufu funerary boat used the technique. It is uncertain why they chose not to join the hull planks in this way.
According to McGrail (: 138), in the Mediterranean even sewn ships relied to some extent on treenails or mortise-and-tenon fastenings within the seams with lacings across the seams. Thus, the status of an ancient Mediterranean tradition of solely sewn hulls comparable to that of northwestern Europe is uncertain.
The use of a pole for propulsion, known from Bronze Age Egypt but more suitable in riverine contexts, cannot be confirmed for the Aegean. Uniquely, the Stathatos seal (W910), of MM III–LM I date, may show two females poling a vessel that also has five oars below the hull.
There is enormous confusion about the relationship between the terms steering oar and quarter rudder, starting from their very definitions. There appears to be no universal agreement; depending on the source, they can be synonyms, hierarchically related (one is a kind of the other), or completely distinct (the rudder developed from and superseded the steering oar). For different perspectives, see Block : 8–9; Mark : 121–22; Mott : 2; and Runyan . Because most studies of Aegean Bronze Age ships use the term steering oar, with some justification, I follow that convention here.
A painted ceramic disk, recently discovered at As-Sabiyah in Kuwait in a context contemporaneous with Ubaid 3, the second half of the fifth millennium BC, depicts a boat with a bipod mast but no visible sail (Carter ). If we assume a sailing vessel, this would now be the earliest known use of mast and sail.
The gradual migration of the sail toward amidships in Egypt can be traced in representations from the Old to the New Kingdom: Jones : 36–51, Plates V, VI, and VIII; Raban and Sterlitz : 655.
The painted larnax from Gazi, Crete (W608) is a potential exception if the LM IIIB date is correct.
Note should also be taken of the so-called talismanic glyptic representations of ships, bearing stylized, abstract elements on and above hulls that could be sails, awning structures, ikria (see below), or other structures. These enigmatic images, with a chronological range between MM III and LM II, have long been associated with ritual and magic (Wedde : 134–41).
These "types" are of course the creation of modern observers. Apart from possibly failing to approximate the way Mycenaean people formed distinctions about watercraft, they very likely suppress real variability (see below).
I am grateful to Hariclia Brecoulaki and Sharon Stocker for allowing me to mention the naval fresco, and for sending me an image of it in advance of publication.
I am grateful to Michael Cosmopoulos for his kind correspondence concerning the Iklaina fresco fragment, and for sending me a copy of the paper cited here.
The assumption that the highly abstracted form of the stempost device of the Iron Age continues to represent a bird, in a unbroken chain of continuity from Mycenaean times (Wachsmann : 177–97), is challenged by Wedde (: 837–43), who interprets abstract curving devices of the Iron Age as horns.
de Souza (: 16) advances a more extreme view that does not recognize coastal plundering as piracy: "People using ships to plunder coastal settlements are not called pirates, so they cannot really be said to be practising piracy." His subsequent discussion of Homeric and other sources appears to contradict this position.
This question forms part of a greater inquiry into strategies of self-representation among ruling groups. The Minoan and Mycenaean civilizations did not share the Egyptian and Near Eastern propensity for iconographic and narrative emphasis on individual rulers, for example. Several archaeologists have modeled these contrasting tendencies in terms of "network" (centralized, exclusionary) vs. "corporate" (decentralized, inclusive) strategies of leadership and self-representation (Blanton et al. ; Feinman ). The Mycenaeans, with their sharply hierarchical social structure revealed in the Linear B archives, are usually placed farther toward the "network" end of this spectrum than the Minoans: Borgna ; Parkinson and Galaty .
This orientation may be partially a consequence of the inward-focused Middle Bronze Age societies from which they emerged.
Mention should also be made of the remains of a small wooden boat of the EBA–MBA transition recently discovered at Mitrou in central Greece (Van de Moortel and Zahou ). Conservation of this find is ongoing and little information is available at present.
This discussion does not take into account fishing in rivers or lakes. Because few rivers in Greece are perennial, especially in the southern and central regions and the Aegean islands, they are not a significant source of fish. Lakes of substantial size are also relatively few in Greece, but these presumably would have been well stocked with fish. The eels from Lake Kopais were an especial delicacy in classical times (Aristophanes, Acharnians l. 940–950; Pausanias 9.24.2; see Vika et al. ). Kopais was partially drained in the Mycenaean palatial period to reclaim land for agriculture (Knauss ).
Wedde's assessment is, of course, built upon the assumption that the artists tried to approximate the dimensions of real vessels of which they had some knowledge.
The sailing estimate from Crete to Egypt is based in part on a reference in the Odyssey (14.255–57) in which a sailing ship from Crete riding a fresh north wind made the Nile delta on the fifth day. The underlying assumption is that the performance traits of Aegean Bronze Age sailing ships were similar to those described by Homer. In view of the gradual development of ships like the Mycenaean galley in the Early Iron Age, this assumption has some validity.
Tilley (: 423) defines a true sailing ship as "...one that can make some headway under sail up wind" (italics in original), and continues, "I would like to distinguish it as sharply as possible from a galley." Modern sailing boats are not considered capable of sailing directly into the wind, and so must undertake tacking maneuvers at various angles to windward.
Tzalas (1995b: 453–54) also suggests technical improvements that would increase speed and minimize the rather excessive drift that the canoe experienced. These steps might reduce travel time somewhat.
Chapter 4: The Maritime Environment of the Aegean Sea
A thorough and highly readable account by Jamie Morton, The Role of the Physical Environment in Ancient Greek Seafaring (Morton ), covers most of this ground in far more detail than is possible here. His work, however, focuses on the historical period and to a large extent provides a commentary on a wide range of textual references to Aegean seafaring found in literature from Homer to the Roman period. Some of the conditions of seafaring that he describes – for example with respect to ships, navigational knowledge, and the organization of maritime trade – were quite distinct from those that prevailed earlier in the Mycenaean period. Nevertheless, this is important source material and most of his observations on the seafaring environment remain valid for the Bronze Age.
During a recent junior world championship sailing event in the Thermaic Gulf at Thessaloniki in July 2009, the competition was forced off the water by a strong vardari wind: <http://470.org/content.asp?id=1700>, accessed March 24, 2010.
The recent discovery of Lower Paleolithic habitation in the Plakias region of Crete, an island for at least the last five million years, implies open-sea navigation at least 130,000 years ago: Strasser et al. . If confirmed, these results will have a pro-found impact on our knowledge of early seafaring and the dispersal of early humans.
Several periploi are extant in fragmentary or near-complete form. Among the more complete are The Periplus of Hanno the Navigator (sixth century BC), The Massaliote Periplus (sixth century BC?), The Periplus of Pseudo-Scylax (fourth or third century BC), The Periplus of Scymnus of Chios (late second century BC), The Periplus of the Erythraean Sea (first century BC), and the Periplus Ponti Euxini (early second century AD). The following passage from the Periplus Ponti Euxini, written by Arrian (best known for his history of Alexander the Great), exemplifies the use of these features as segments in a longer journey:
> From the Phasis, we passed the navigable river
>
> Charies; there are 90 stades between the two.
>
> From the river Charies, we sailed on another 90
>
> stades to the river Chobos, and there we
>
> anchored....From the Chobos, we passed the
>
> navigable river Sigame; it is approximately 210
>
> stades from the Chobos. After the Sigame is the
>
> river Tarsouras; there are 120 stades between the
>
> two. The river Hippos is 150 stades beyond the
>
> Tarsouras, and the Astelephos 30 beyond the Hippos.
>
> (10.i–iii; transl. A. Liddle)
In this regard, it is interesting that one of the contingents of rowers listed on An 610 came from the island of Zakynthos and may have held mercenary status.
Hesiod himself had sailed only as far as Euboea, a short crossing from the mainland, and as a passenger at that. The advice he dispenses about the sea is common wisdom; he can hardly be said to have any specific knowledge about seafaring. The implication that land-locked farmers regularly owned ships and took to the sea is intriguing, but we cannot tell from this passage how far they ventured. According to the model presented in this book, these must have been relatively short-distance voyages, for reasons of knowledge but also because perishable commodities would not keep for long.
Tim Severin (: 132–44) gives a vivid description of the extraordinary difficulties of passing through the Bosporus from the Dardanelles in the Argo, a reconstructed "Bronze Age" twenty-oared galley.
Chapter 5: Coasts and Harbors of the Bronze Age Aegean: Characteristics, Discovery, and Reconstruction
The main area of disagreement I have with Morton () and Papageorgiou () concerns the pervasiveness of coastal change, especially of tectonic origin, and the importance of identifying coastal configuration at a specific, local scale. They are right that in general the processes of change are the same as in the past (by simple uniformitarian principles) and that the same range of coastal features existed in the Bronze Age as do now. But I believe that coastal change is more dynamic than they allow, resulting in shifting anchorages over time, and I hope to argue persuasively in this chapter for acquiring detailed knowledge of local coastal settings as a necessary first step toward understanding Mycenaean maritime networks at all scales.
Isostasy refers to the rebound of land masses formerly under ice sheets, causing a drop in relative sea level (isostatic compensation) in glacial margins. Because this is not a process that affects the Mediterranean, I do not consider it further.
Tanner () outlines an alternate mechanism in which the ridge and swale sets are formed by a sea-level rise-and-fall couplet with amplitude from 5 to 30 centimeters.
Although similar in plan, the Minoan structures differ from later Classical ship sheds in their location away from the shore; thus, they were not used for launching ships as were the Classical ship sheds (Shaw and Shaw 1999: 369).
It may be significant that Ellen Davis () considers the Ayia Irini frescoes to have been painted by traveling Cretan artists.
Van de Noort and O'Sullivan (2006: 36–37) emphasize that certain types of biogenic wetlands, notably peatlands, actually rank among the poorest biomass producers in the world because of saturation, which deprives plants and animals of nutrients, and high acidity. Mediterranean wetlands generally have a riverine or estuarine origin, with high biomass and biodiversity.
Here I use this term in a generic sense to indicate the recovery of a continuous core; see Rapp and Hill : 192–94 for distinctions among the terms coring, drilling, and augering.
Other biogenic clasts, including diatoms and pollen, are not included here as they have not played a significant role in paleocoastal reconstruction to date. For details, see Marriner and Morhange : 170–71.
Chapter 6: Concepts for Mycenaean Coastal Worlds
Brad Duncan (: 10) points out that although Westerdahl coined a term that has come into common use, a long tradition of ethnographic, archaeological, and anthropological research in the Pacific had already addressed many of the same issues using oral traditions, toponymy, and specialized local knowledge to illuminate identity and belonging to place among maritime communities.
W. V. Harris (: 6, n. 15) observes that while the microregion is a central concept in The Corrupting Sea, it is nowhere defined explicitly enough to prevent ambiguity in attempting to apply it.
Constantakopoulou (: 231) acknowledges that the term peraia has a broader application to any region that is controlled or possessed by a state. The narrower sense employed here is, however, the most commonly attested.
Here we must treat Crete as a mainland or mini-continent, and not an island. It has not been suggested that the small islands off the Cretan coast possessed peraiai; rather, islands such as Mochlos and Pseira acted as "gateways" to harbors on Crete's northern coast (Betancourt ; Betancourt and Banou ; Branigan ).
Malkin's important book, which incorporates his prior work on colonization, cross-cultural interaction, identity, and ethnicity into a network theory framework, arrived too late for me to consider in detail in the present work.
Leone Porciani () observes that Horden and Purcell tend to emphasize peaceful relations, while underplaying the role of aggressive and hostile interactions.
This summary of Broodbank's PPA model is partly excerpted from Tartaron 2001b: 228.
For an overview of the intellectual history of network analysis, see Scott : 7–38.
Chapter 7: Coastscapes and Small Worlds of the Aegean Bronze Age: Case Studies
Because changes in the coastline at a local scale since the Bronze Age can be considerable, this statement should be qualified simply by keeping in mind that today's good anchorages need not be the same as those in the Bronze Age, but on average anchorages of comparable kind and quality were available.
If the story that the hoard was found in the nineteenth century in a LH IIIB chamber tomb on Windmill Hill is correct, this would mean that the hoard was originally looted in the Bronze Age and redeposited in the chamber tomb, or that it was preserved as an heirloom collection for several hundred years before its ultimate deposition in the LH IIIB burial.
The contributors to Fitton do not come to a consensus about the date of the Aigina Treasure, though a date between later MH I and MH II seems to be favored. The treasure and the shaft grave are not so readily comparable because the artifact composition between a jewelry hoard and a warrior burial are functionally distinct. Hiller () believes that the gold objects from the two contexts represent different traditions, one Minoan (the jewelry of the Aigina Treasure) and one from the mainland (the shaft grave).
Much of the following discussion closely follows excerpts from Tartaron : 171–72 and Tartaron et al. , with some modifications and additions. These recent writings continue to reflect my current thinking on the matter.
The Saronic Harbors Archaeological Research Project is carried out under the auspices of the American School of Classical Studies at Athens, with the approval of the 37th Ephoreia of Prehistoric and Classical Antiquities and the 25th Ephoreia of Byzantine Antiquities, and a permit issued by the Greek Ministry of Culture. For their kind support, we wish to thank Konstantinos Kissas and Panayiota Kasimi from the 37th Ephoreia and Demetrios Athanasoulis from the 25th Ephoreia. We gratefully acknowledge financial support from the Institute for Aegean Prehistory, the U.S. National Science Foundation (Grant BCS-0810096), the Stavros S. Niarchos Foundation, the Loeb Classical Library Foundation, the Arete Foundation, the Florida State University, the University of Pennsylvania, and Norwich University.
Investigations of the underwater areas of Kalamianos and the Korphos Bay region are undertaken as a joint Greek-Canadian project, under the direction of Despina Koutsoumba of the Ephoreia of Enalion Antiquities and Joseph Boyce of McMaster University, representing the Canadian Institute in Greece. This project is independent of, but in close cooperation with, SHARP.
The reconstructed shoreline positions are approximations based on the modern bathymetric contours and do not take into account the effects of sediment accumulation and compaction following the submergence of the beachrock platforms. These parameters will be clarified in future studies.
Amy Dill and D. J. Pullen, personal communication. A number of samples have been selected for a future program of petrographic and chemical analysis aimed at clarifying the source of the potting materials as well as aspects of production and consumption.
Recent discussions with colleagues have raised the likelihood that similar cairns may exist in many places in southern Greece that have not previously been recognized as EBA features.
The interviews were much more extensive than those I consulted here. The oral histories will be the subject of future publication by Tzortzopoulou-Gregory.
For a brief but vivid description of a traditional charcoal-burning operation in the Souli region of southern Epirus, see Newby : 168–170.
In the post-Bronze Age era, the progradation was asymmetrical, with the bifurcation of the Maeander into northern and southern branches, of which the northern branch caused a far more rapid advance of the coastline in the northern and central Gulf than in the south (Knipping et al. : fig. 1; Müllenhoff et al. ).
Another tholos tomb was discovered at Kazanaki, but it is sufficiently distant from the settlements under discussion that it may have been attached to an undiscovered settlement.
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## Index
Abydos
accretional barrier see coastal landforms
acculturation ,
Acheron River (Epirus) 141–142, 147–150, , 169–170
Adriatic Sea 29–30, ,
Aegean islands , , 17–18, , , , , , , , , , 282–283
Aegean List 28–29,
Aeolian islands ,
aeolian landforms , ,
aerial photographs 162–163,
Aetopetra (Corinthia)
Agamemnon ,
agency, human , 128–129, 204–205
in soil loss
Agouridis, Christos 112–114,
agriculture see agro-pastoral adaptations
agro-pastoral adaptations , , 70–71, , , , , , , , , 260–265, , , , ,
Ahhiyawa , 29–30, , , ,
Aidonia (Corinthia)
Aigina , , , , , , , , 213–243, , 258–259, 264–270
see also Kolonna
Aigina Shaft Grave 223–225,
Aigina treasure 222–225
Aigion (Achaia)
Akhenaten 28–29,
Akrotiri (Thera) , 37–41, , 52–58, 61–64, 71–72, 75–77, , , , 159–160, ,
see also West House; Flotilla Fresco
Akrotiri Salt Lake (Cyprus)
Akrotirio Trelli (Corinthia) , ,
Alalakh (Turkey)
Alashiya ,
Alcinous
Almyros Plain (Thessaly)
Alpheios River (Elis) ,
Amarynthos (Euboea)
Ambracian Gulf ,
Amenhotep II ,
Amenhotep III
Amnisos (Crete) ,
Amurru 29–30
anabatic wind
Anatolia (Turkey) , , 33–34, , , , , , , , , , 178–180, , 272–276
anchorages , 4–6, 9–10, , , , 86–89, , , , , , , , 139–142, 149–151, , , 171–176, , , , 213–216, , 250–251, , , , ,
defined
discovery of 139–143
typologies of 171–174
Ancient Harbour Facies ,
Ancient Harbour Parasequence 157–158,
Andros (island) ,
Angelokastro (Corinthia)
Angistri (island) 214–215
annales history , ,
Apatheia (Argolid)
Aphaia sanctuary (Aigina) , ,
Aplomata (Naxos)
Arapiza (Corinthia)
Arcadia ,
archipelago, Aegean , , , n:273
Argive Plain , , ,
Argolic Gulf , , , 197–198, , ,
Argolid 6–8, , , , , , , , , , , , , , 233–236, , 242–243, 256–257, 262–265
Argolid Exploration Project
Argonautica of Apollonius of Rhodes
Argos (Argolid) , , , ,
Arrhenius, Olof
artisans , , ,
Asia Minor , , , , , , , , , , , 274–275
Asia Minor Current
Asine (Argolid) , , , , , , ,
Assiros (Macedonia)
Assyrians , ,
Athens (Attica) , , , , , ,
Acropolis south slope
Agora
Athos peninsula
Attica , , , , , , , , 215–216, , , , , , , 263–264,
Ayia Irini (Kea) 54–55, , , , 230–231,
Ayia Photia (Crete)
Ayia Triada (Crete)
Ayios Gerasimos (Corinthia)
Ayios Konstantinos (Methana) , 240–243,
Ayios Kosmas (Attica)
Ayios Petros (island)
Babylonians ,
ballast 164–165, ,
Barabási, Albert-László ,
barley ,
barrier islands see coastal landforms
basileus
bathymetry , , ,
beaches see coastal landforms
Beaufort wind scale , ,
Beirut
Berbati Valley (Argolid) ,
Berg, Ina , ,
Besonen, Mark ,
Biqa valley (Lebanon)
Black Sea , ,
Black Sea Water (current) ,
Blue, Lucy 171–172
boar's tusk helmets , , ,
boats
canoes , 39–40, , 72–74, , 81–84, , , ,
defined
fishing boats , , , , , , , , ,
models , , 42–43, , 50–52, , 61–63, , , , 80–81, ,
reed papyrella , 76–77, , 84–86
rowboats , , , , ,
Boeotia , , , , , , ,
bogs see wetlands
Bonn group
bora wind , ,
Bosporus ,
Bourdieu, Pierre , 128–129
Boyce, Joseph
Braudel, Fernand
breakwaters , , , , , , , , , ,
Broodbank, Cyprian , 86–87, , , , , , , 205–206, , , 285–286,
Building P, Kommos , 159–160
burial mounds
Burns, Bryan
cabotage , , , , , , , ,
Caesarea Maritima (Israel) , ,
cairns, stone 119–120, , , 254–257
Canaanite jars ,
Canaanites see Syro-Canaanite
Canaanite texts 35–36
cannibalism 136–137
Cape Drepanon (Crete)
Cape Gelidonya wreck see shipwrecks
Cape Malea (Laconia)
Cape Mykale (Turkey) ,
Caroline Islands , 126–127,
carrying capacity
Çatalhöyük (Turkey)
Catalogue of Ships, Homeric ,
cattle , , , ,
ceramic petrography ,
Chalandriani-Kastri (Syros) ,
Chalkidiki peninsula (Macedonia)
chamber tombs , , , ,
Chios (island) ,
Chryssoulaki, Stella
cisterns , ,
"Clashing Rocks,"
cliffs, coastal see coastal landforms
climate see environment, Mediterranean
Cline, Eric 24–26, , ,
coastal configuration , ,
aggradational coasts 143–144, 157–158,
anthropogenic impacts 156–158,
high- and low-energy coasts , , , , 171–174,
open coasts 144–148
progradational coasts 143–147, , 152–154, , , , 272–273,
protected coasts , , , , 171–172, , , ,
recessive coasts ,
regressive coasts
stable and unstable coasts
submerged coasts , , 148–149, 162–165, , , , 244–250
transgressive coasts , ,
uplifted coasts 148–149, ,
coastal geomorphology , 142–161
geomorphological survey 162–164
marine erosion 140–148, 162–163
sedimentation , , 140–158, 164–171, 179–180, , 273–277
coastal landforms
accretional barriers and barrier islands , , 150–157, , ,
beach ridges and swales ,
beachrock , 247–248,
beaches , , , , 119–120, , 146–152, , 158–159, , , 170–171, , , 248–250
biostratigraphy 168–171
constructional and destructional 147–148
deltas , 141–143, , , 152–154, , , , , , ,
dunes 154–155, ,
estuaries , , , , , 150–152, , , , , ,
facies , , 167–171,
levees 153–154,
lagoons , , , , 152–157, 160–162, , 172–174, 178–179, ,
reefs , , , , ,
salt pans, marshes, and lakes ,
sea cliffs , , , , , , , , 162–163, , ,
sedimentology , 167–168,
stratigraphy , 166–168,
swamps , , ,
tidal marshes and mudflats , , , 153–154, ,
tidal notches , , ,
wetlands 73–74, , , , , ,
coast- and island-hopping , , , , , ,
coasting vessel , 75–76, , ,
coastscape
see also maritime interaction spheres
defined 9–11, 188–190
coastwise voyaging ,
cognitive landscapes , ,
colonization
of Aegean islands , , , n:231
internal, MH III/LH I ,
of Mediterranean 57–58, 135–137, ,
Mycenaean , , , , , , ,
complexity, social-political , , , , 19–20, , , , , , , 194–195, 217–218, , 226–228, , , , , ,
Conlin, David
Constantakopoulou, Christy , ,
contested landscapes ,
copper
artifacts
ingots , , ,
smithing ,
trade of , ,
core and periphery , , , , , , ,
core area, Mycenaean 7–8, , 200–201, ,
cores, geological , , 166–170, , , ,
Corfu see Kerkyra
Corinthia , , , , , , , , , , 232–237, , , , , ,
Corinthian Gulf , , ,
Coriolis effect
corridor houses
Early Bronze Age , 217–218, , ,
Mycenaean , , 279–280
Cosmopoulos, Michael
cost weighted path distance , n:210
cothon
Crego, Donna May
Cretan Arc ,
Cretan hieroglyphic
Crielaard, Jan-Paul
cult see sanctuaries and cult centers
culture area, Mycenaean , , see maritime culture region, Mycenaean
Curray, Joseph
cyclogenesis 92–93
cyclonic and anticyclonic currents 97–101
cyclonic and anticyclonic winds 92–93,
cyclopean masonry see masonry, Mycenaean
Cyclopes , 136–137
Cypro-Minoan script
Cyprus , , , 25–26, , , , , 50–51, 63–64, , , , , , , , , 209–210, n:242
Dakoronia, Fanouria
Darcque, Pascal
Dardanelles 97–100, , , ,
Darwin, Charles
Daskaleio-Kavos (Keros)
"definite places," ,
De irmentepe (Turkey)
deltas see coastal landforms
Dickinson, Oliver
diet see stable isotope analysis
Dimini (Thessaly) 277–283
Diomedes ,
diver survey ,
Dokos (island) , 193–194
dolines
Dorian invasion
Doro Channel ,
dunes see coastal landforms
East Aegean Koine
East Alley, Korakou (Corinthia)
Eastern Korinthia Archaeological Survey (EKAS) , , ,
ebony
ecotones
eddies 99–102, ,
Egypt
Aegean imports in , ,
chronology
exports to Aegean 21–22,
harbors in ,
in Homer's Odyssey , 111–112
influence in Aegean ,
maritime contacts with , , 21. 28–30, 82–83, ,
Mycenaean mercenaries in
objects on Uluburun shipwreck n:25
and the "Sea Peoples," 19–20
ships and boats , 51–54, , , 76–77
texts and archives , , 28–29, 34–36, ,
tomb paintings ,
El Niño and La Niña
electromagnetics see geophysical survey
Eleusis (Attica) , 237. 267
Elis , , 152–153, ,
emporion ,
Enkomi (Cyprus) , ,
environment, Mediterranean
climate, general , 90–92, , , , 137–138, , , , , , , ,
currents , , , , 97–102, , , , , 114–115, , , , 135–136, , , , , 153–154, , , , , , ,
drought , ,
geographic scales 90–91
gradients in temperature, density, pressure 91–96, ,
precipitation 91–97, , 103–106, 145–146, 152–153, ,
salinity 97–99, , , , 168–169
seasonality , , , , , 91–93, , 99–100, 103–108, , 144–145, , , ,
storms 91–93, , 101–109, , 139–140, , , , , n:213
tides 143–144, , 148–153, , , 247–248,
water masses , 97–99
waves , , , 101–102, , , , , , 143–150, 152–155, 157–158, , , ,
weather systems 90–92
winds , , , , , 68–69, , 83–86, 92–97, 100–104, , , , , 114–115, , , , , 139–140, , , 153–155, , , , , , 213–214, , ,
see also bora; meltemi ; sirocco ; vardari
Ephesos (Turkey)
Ephoreia of Underwater Antiquities (Enalion), Greece ,
Epidauros (Argolid) , , , 267–268
estuaries see coastal landforms
ethnography and ethnoarchaeology , , , 46–47, , , 80–84, 109–110, , 125–129, 192–195, , 265–269,
Euboea , , , , , , , , , , , , , ,
Euboean Gulf , , 283–284,
eustasy see sea level
Eutresis (Boeotia)
événement.
See also annales history
Evraionisos (island) 192–193
ex oriente lux
exogamy ,
experimental voyages , , 84–86
facies see coastal landforms
faience , ,
faunal remains , , , 220–221, , ,
Feinberg, Richard
Ferula communis (giant fennel) , ,
fetch
Feuer, Bryan
figurines , , , , , , , , ,
figs ,
fish and shellfish , , , , , , , , , , , , ,
fishing , 72–76, 78–80, , , , , , , , , , , , ,
flaked stone see lithics
Flemming, N. C.
floodplains , , , , n:273
Flotilla Fresco, Akotiri (Thera) , 38–41, , , , 54–56, 60–62, 71–73, 75–77, , , ,
foraminifera see macro- and microfauna
forest resources 266–267,
formal and stylistic analysis , , , , , , , , ,
Forstenpointner, Gerhard
fortifications, coastal 64–67, , , , , 217–218, , , , 230–231, , , 258–260, ,
Fournoi cluster
fractals
frame-first construction see ship construction
frescoes see wall painting
"frying pan," Cyladic , ,
Franchthi Cave (Argolid) , 84–85, , ,
Galik, Alfred
galley , , 57–71, 83–84, , , 132–133,
see also ships, Mycenaean
"galley subculture," 69–71, , 132–133
Gelidonya see shipwrecks
Gell, Alfred 122–123
genetic viability ,
geochemical analysis ,
Geographic Information Systems (GIS) , ,
geomorphology see coastal geomorphology
geophysical survey ,
Gibraltar, Straits of
Giddens, Anthony 128–129
Gla (Boeotia) ,
glacial maximum 140–141
Glykys Limin (Epirus) ,
goats see pastoralism
"goat island," 192–193
gold , , , , 222–225
Gonia (Corinthia) ,
Goodman, Beverly
Gournia (Crete) , 159–160, n:290
Grave Circles A and B see shaft graves, Mycenae
Gregory, Timothy
ground stone see lithics
ground truthing , ,
gyres 99–100
Gytheion (Laconia)
habitus of maritime life , 128–129, 133–138,
Hall 64, Pylos
harborless coasts ,
Hattusa (Turkey) ,
Hattusili III
harbors
defined
sheltered (protected) 86–87, 106–107, , , , 157–158, , , , , , , 250–251,
typologies 171–174
headlands see promontories
Helike (Achaia)
Herakleia (island) 203–204
Hermione (Argolid)
Hesiod, Works and Days , , , , , ,
heterarchy, social-political , , , 281–282
hierarchy, social-political , , , , , , , , , , , 280–282
Hiller, Stefan 222–223
Hisarlik see Troy
Hittites ,
Hittite texts , 29–30, 36–37, , , ,
see also texts
Hodder, Ian
Holocene paleoenvironments , , 144–145, 152–153, 156–157, , , ,
Homer, Iliad , , 159. 197–198
Homer, Odyssey , , , 67–68, , 111–112, , , , 133–137, , ,
Homeric Catalogue of Ships ,
Horden, Peregrine, and Nicholas Purcell 116–117, , 188–190, , , 203–204, , 269–270, 284–286
House of the Tiles, Lerna
hull-first construction see ship construction
Hydra (island)
Ialysos (Rhodes) , ,
Iasos (Turkey) 275–276,
ice cores
Iklaina (Messenia) , , ,
ikrion (pl. ikria) , 39–40, 55–57, , ,
Iliad of Homer see Homer, Iliad
"imperfect optimisation" model 206–210
Indo-Persian low pressure system n:92
ingots
copper , , , ,
glass
tin ,
Institute for Geology and Mineral Exploration (IGME) ,
interior, coastal relations with , , , , , , , 188–190, 196–197, , , , , , , , , ,
"international spirit," EH II 217–218
interregional/intercultural maritime interaction sphere see maritime interaction spheres
interscalmium
intertidal zone , ,
intervisibility
in coastscapes and small worlds , 215–216,
of enclosures and cairns at Korphos
Iolkos (Thessaly)
Ionian Islands
Ionian Sea , , , , , ,
Iron Age , , , , , , ,
iron technology
island archaeology 184–185
island biogeography
islandscapes
isostasy see sea level
Isthmus of Corinth , ,
Isthmus of Ierapetra (Crete)
Italy , , , , , , , , ,
Israel , , , ,
Ithaca (island)
ivory , , , , , , ,
Izmir, Bay of 178–179,
Jericho
Kalamas River (Epirus)
Kalamianos (Corinthia) , , , 243–271
as coastscape in Saronic small world 263–265
as Early Bronze Age regional center 253–258
as Late Bronze Age regional center 258–263
oral history 265–269
paleocoastal reconstruction 246–251
terrace walls , , , 260–263,
Kanakia (Salamis) 237–240, , ,
Kanesh karum (Turkey)
Kapsali Bay (Kythera) 105–107
Kardulias, P. Nick , 192–193
Karpathos (island)
Karystos (Euboea) ,
Kastro (Thessaly) , ,
katabatic wind
Kato Zakro (Crete)
Kavousi (Crete)
Keay, Simon
keel see ship construction
Keftiu
Kenchreai (Corinthia) ,
Kerkyra (Corfu island) , , , ,
Keros (island)
Kiapha Thiti (Attica) , ,
Kilian-Dirlmeier, Imma
kinship , , , , , , ,
Kirrha (Phocis)
kilns see pottery
Kition-Bamboula (Cyprus) ,
Klazomenai (Liman Tepe, Turkey)
Knappett, Carl 205–211
Knossos (Crete) , , , , , , , , ,
Kofini Dam (Argolid)
koiné, cultural , , , , , ,
Kolonna (Aigina ) , , , , , , , , , 218–234, 242–243, , , 270–271
connectivity of 231–232
EH III hoard
Färberhaus
faunal remains
Haus am Felsrand
House of the Pithoi
human skeletal remains
and Kalamianos 263–265, 270–271
Large Building Complex 220–222,
marine resources 221–222
Minoan residents at ,
plant remains
potting industry 226–230
relations with Mycenae 232–237, 242–243,
"Shaft Grave," 223–226
sphere of influence 226–231,
urban phases 218–220,
Weißes Haus
Windmill Hill
Kom el-Hetan (Egypt)
Kommos (Crete) , , , , , , , ,
Konsolaki-Yiannopoulou, Eleni
Korakou (Corinthia) , ,
Korphi t'Aroniou (Naxos)
Korphos (Corinthia) , , 243–248
as Bronze Age regional center 251–265
coastal geomorphology of 246–248
as modern proti skala
Kraft, John 143–144
Kramer-Hajos, Margaretha
Kvapil, Lynne
Kydonia (Crete)
Kyme (Euboea)
Kynos (East Lokris) , , 65–68, , , , ,
Kyra (island)
Kyrenia II
Kythera (island) , , 104–107, , ,
Laconia , , ,
lagoons see coastal landforms
Lake Karla (Thessaly)
Lake Kopais (Boeotia)
lakes 152–154, 160–161, , ,
land breeze 94–95, ,
landmarks see navigation
landscape archaeology , , , , ,
Latmian Gulf (Turkey) , 272–273, 275–276
Lavrion (Attica) , 84–85,
lawagetas ,
Lazarides (Aigina)
lee shores , , , , , n:171
Lefkandi (Euboea) ,
Leidwanger, Justin ,
Lerna (Argolid) , , , , , , , , ,
House of the Tiles
shaft graves ,
Levant , , , , 28–30, , , , , , , , , , ,
levy fleets ,
Lesbos (island)
Libyan Sea
lighthouses , ,
Liman Tepe (Turkey) , 178–180
liminality, of coasts ,
Limnes (Argolid)
Linear A , ,
see also texts
Linear B , 14–18, 20–21, , , 35–36, 64–65, 74–75, , , , 130–132, 201–202, , , , , 286–288,
see also texts
lithics , , , 253–254
flaked stone
ground stone , , , , 263–264
"Little Ice Age," ,
Loader, N. Claire
longboats, Cycladic , , , , , , 81–83, , , , n:251
Longopotamos Valley (Corinthia)
longshore currents and deposits 146–147, , , , , ,
longue durée
see also annales history
Louros Athalassou (Naxos) 112–113
"Lower East Aegean–West Anatolian Interface,"
Lystraegonians
macro- and microfauna , , , ,
foraminifera 168–169, , ,
mollusks , ,
ostracods 168–171
Maeander (Menderes) River (Turkey) , , 272–274, 276–277
magnetometry see geophysical survey
Malia (Crete) 78–79,
Malia, Gulf of (Thessaly) ,
Malkin, Irad , ,
Malta
maps
Cartesian 122–124,
cognitive/mental , 122–123
cost-weighted path distance 209–210
Maran, Joseph ,
Marathon (Attica)
Marinatos, Spyros
masonry, Mycenaean , , , ,
maritime coastal communities 46–47, 86–88, 123–126, , 183–184, , 288–289
at modern Korphos 266–270
Mycenaean 69–70, , , 124–125, 130–133,
Pacific Ocean 109–110, 126–129
maritime culture region, Mycenaean , 199–202
see also culture area, Mycenaean
maritime cultural landscapes , , 142–143, 175–176, 185–190, 270–272, , 289–290
maritime durées ,
maritime interaction spheres , , 185–203,
maritime knowledge 26–27, , , , 107–124, 130–136, , n:184, , , , 203–204, n:288
transmission of 46–47, 125–129, , ,
maritime lore 127–129, 135–138, ,
maritime small worlds , 10–11, , , , 71–72, , , , 117–118, 185–187, 190–198, 203–204, , , 286–289
see also maritime interaction spheres
in the Bay of Volos 277–283
in the Latmian Gulf 271–277
in the Saronic Gulf 213–271
maritime trade
disruptions in 19–20, ,
in obsidian , 84–87, , , , ,
interregional (long-distance) , 19–37, , , 117–118, , 132–134, , 195–196, 202–203, , , 285–286
local-scale , , 26–27, 31–32, , 44–45, , 75–76, 86–88, 117–118, , , , , , 195–196, , , , , 266–268
mechanisms of 30–33
in metals , , , , , , , , ,
regional-scale , , , , , , , 86–88, , 195–196, 198–202, , , , ,
Mark, Samuel
Marriner, Nick ,
marshes see coastal landforms
Martin, Richard
maximum marine transgression see sea level
McGrail, Séan
Medinet Habu (Egypt) , ,
Megali Magoula (Argolid) 227–228, 234–236, 242–243, ,
megaron (pl. megara)
Megara A and B, Dimini 279–281
megara, Kanakia 238–239
megaron, Ayia Triada
Megaron, Mycenae
Melos (island) , 84–85, , , , ,
meltemi wind (ancient Etesians) 93–95, , , , , ,
Menelaion (Laconia)
Menelaus , ,
mentalité , , ,
mercenaries , , ,
merchants 24–27, , , , 133–135, , , 266–269
Merneptah Stele
Mesara Plain (Crete) 159–160
Mesopotamia , , ,
Messenia 13–14, 60–61, 69–70, 130–132, , , , ,
metals trade see maritime trade
metal vessels , , ,
Methana (Argolid) , , , , 240–241, , ,
Methana Survey Project
Metopi (island)
microecologies and microregions , , , , , , , , 189–190, , , 203–205, , , , 284–286
Minet el-Beidha (Syria)
"Middle Helladic hiatus," , , , ,
Midea (Argolid) , , , ,
Miletos (Turkey) , , 36–37, , , , , , , 272–277
Millawanda (Mycenaean Miletos) , 36–37, 274–275
millstones, Aiginetan ,
"mini island networks,"
Minoan Crete
boat models
chronology
colonies , ,
environmental conditions , , ,
frescoes in Near East
harbors , , , , 163–164, ,
under Mycenaean influence , ,
maritime trade networks 20–21, 24–25, 27–29, , ,
pottery , , , , ,
relations with Ayia Irini ,
relations with Kolonna , 220–226, , ,
relations with Mycenaean world 12–13, , , , , , 233–234, , ,
seals and sealings , 53–55, 77–78
ships , 59–63,
ship sheds , , , 159–160,
sites in "Aegean List," 28–29
"thalassocracy," 66–67
Mirabello, Gulf of (Crete) , , ,
Mitrou (East Lokris) ,
Mochlos (Crete) ,
mollusks see macro- and microfauna
Momigliano, Nicoletta
Mongolian high pressure system
Morhange, Christophe ,
mortise-and-tenon joinery 49–50
see also ship construction
Mountjoy, Penelope ,
moyenne durée
see also annales history
Munsell soil color chart
Mycenae (Argolid) , 12–14, 16–18, , , , 36–37, , , , , , , , , , , , , , , , 223–228, 232–235, , , 242–243, , 262–265, 270–271, ,
Naqada II (Gerzean)
na-u-do-mo (shipbuilder)
Nauplion (Argolid) , ,
"naval fresco," Pylos
navigation
celestial , 109–113, ,
coastal hazards , , , , , , , , , , ,
coastwise voyaging ,
dead reckoning
landmarks , 118–120, ,
master navigators , 124–129, , ,
night voyaging , 111–113, , , ,
open-sea voyaging , , , 83–84, 110–114, , , , ,
sailing ranges, daily , , , , , , , , , , 214–215
seamarks , , 120–121, , n:126
segmented journeys , , 119–120, , ,
skymarks , 121–122,
"star path steering," ,
winter voyaging , , , 103–104, ,
windward sailing 83–84
Naxos (island) , , , 112–113
Nea Epidauros (Argolid) 267–268
Nebamun, tomb of (Egypt)
Nemea Valley (Corinthia) , , , ,
Nestor , ,
network analysis see social networks
neutron activation analysis
Niemeier, Wolf-Dietrich
Nile River , , , , , ,
Nirou Chani (Crete)
North Atlantic Oscillation 91–93
obsidian see maritime trade
Oceania , , ,
"Occidentalia,"
Odyssey of Homer see Homer, Odyssey
offshore islands , , , , , , , , , 188–189, 192–193, , ,
oikos 133–134
Oil Merchant group, Mycenae
olives and olive oil , , , , , , ,
Olson, Donald
Olympias (replica trireme)
opportunistic use of anchorages , ,
oral histories , , , , , 265–270
Orchomenos (Boeotia) ,
organic remains , , , , , , , ,
"Orientalia,"
orographic rainfall
Osmanaga Lagoon (Messenia)
ostracods see macro- and microfauna
Ottoman period
overland routes and connections , , , , 84–85, , , 188–189, , , , , , 266–268
paddling
see also ships, propulsion
Pagasitic Gulf (Thessaly) , 282–283
palaces, Mycenaean
collapse , 17–20, 69–70, ,
control of trade 64–65, , 124–126, 130–132, 202–203
emergence 12–14, , , ,
Linear B texts and trade 35–37, 130–132, ,
long-distance trade 20–30
political and economic system 7–8, 14–17, 74–75,
Palaikastro (Crete)
paleocoastal reconstruction , , , 142–143, 162–171, 174–180, ,
palynology
Panayia Houses, Mycenae
Pantou, Panagiota 279–280
Papadimitriou, Nikolas
Papageorgiou, Despoina
papyrella boat, Corfu , 76–77, , 84–86
papyrus boats, Egypt , 76–77,
Paroikia (Paros)
pastoralism (sheep and goat) , , , , , 192–193, , , , , , , ,
Pefkakia (Thessaly) , , 281–283
Peloponnese 7–8, , , , , , , , , , 197–198, , 215–216, , , 228–229, , , 254–256, , ,
Peneios Riveer (Thessaly)
Peneus River (Elis)
penteconter ,
peraia , ,
Perama (Salamis)
Perati (Attica) ,
Perdikaria (Corinthia)
"periphery model,"
periplus, pl. periploi, ,
Petrakis, Vassilis
Phaeacia , ,
Phaistos (Crete)
Pharonisi peninsula (Corinthia) , ,
Pharsala (Thessaly)
phenomenology , 122–124, 135–137, 190–191, , 215–216
Philippa-Touchais, Anna 229–230
Philistines 50–51,
Phoenicians
Phylakopi (Melos) ,
phosphate analysis 176–177
pigs , ,
pilotage , 72–73, , 108–109, , , , , ,
Pindos Mountains ,
piracy , , , 66–68, , , ,
Piraeus (Attica) 267–268
place-name studies , , , 176–177,
Point Iria (Argolid) wreck see shipwrecks
Polanyi, Karl 32–33
poljes ,
Poros (island) 214–215, ,
ports , , , , , , , , , , , , , , ,
defined
port of trade , , ,
porthmeutike 192–194,
Portus (Italy)
pottery
Aiginetan gold mica fabric 226–227,
Aiginetan matt painted , 225–227, 229–231, ,
Argive Minyan
characterization studies , , , ,
Fine Grey Burnished
Gray Minyan , ,
Kamares Ware 224–225
Keian White-on-Grey
Keian Yellow Slipped
kilns ,
Lustrous Decorated , , ,
Magnesia polychrome ,
Marine Style
as marker of interaction 6–8, , 20–25, 28–29, 44–45, 50–51, , , , 199–201, 215–218, , , 226–237, , , 263–265, , 274–276, , 282–283
Philistine 50–51
Polychrome Mainland
potters' marks , , , ,
pseudo-Minoan ,
Red Minyan
thermal properties
Vapheio cup
Yellow Minyan ,
Powell, Judith
practical mastery theory
private enterprise , , , , 131–132, 202–203
promontories and headlands , , , , 100–101, 104–106, , 119–120, , , , , , , , , , , , , ,
Proximal Point Analysis 205–208
Pseira (Crete) 43–45, , , ,
Psiloriti, Mt. (Crete) ,
Psyra (island)
Pteleos (Thessaly)
Pullen, Daniel ,
Purcell, Nicholas see Horden, Peregrine, and Nicholas Purcell
Pylos (Messenia) , 13–14, , , n:20, , 35–36, 56–57, , 64–65, , 70–71, , , 124–125, 130–132, , , , 160–161, ,
quarter rudder
see also ships, propulsion
Qatna (Syria)
Raban, Avner
radiocarbon dating , , 167–170, , ,
rain shadow
Rainbird, Paul ,
Ramesses III ,
Rap'anu archive, Ugarit
Rapp, George 143–144
Ras Shamra see Ugarit
regional/intracultural interaction sphere see maritime interaction spheres
reflexive archaeology
Renfrew, Colin , , 194–195
replica ships , , ,
see also experimental voyages
research design , , , , ,
resistivity see geophysical survey
Rhodes (island) , , , , , 209–210,
rigging see ship construction
risk buffering ,
roads and paths , , 188–189, , , , , ,
Romanou (Messenia) 160–161,
Rönnby, Johan
rowers , 57–58, 60–65, 69–70, , , 130–132
rowing
see also ships, propulsion
Runnels, Curtis
Rutter, Jeremy
sail
boom-footed 53–54
brailed 53–54
impact of 83–84, 86–87, 115–116
introduction of , ,
positioning of 52–53
sailing see navigation
sailing season , , 103–104, ,
Saint Paul
Salamis (Cyprus) 209–210
Salamis (island) , , 237–240, , , 266–268
sanctuaries and cult centers
Aigina, sanctuary of Aphaia ,
Ayios Konstantinos, Room A 240–242
Dimini, Megaron B 279–280
Ithaca, shrine to Odysseus (?)
Mycenae, Cult Centre
Mycenae, shrine to Agamemnon (?)
Sparta, Menelaion
Sardinia , , , ,
Sarepta (Lebanon)
Saronic Gulf
as small world 213–271
maritime environment in 213–215
trade, pre-World War II 265–269
Saronic Harbors Archaeological Research Project (SHARP) , 243–269
satellite imagery ,
Scandinavia , , 176–177
scarabs , ,
Schinoussa (island)
Scylla and Charybdis 136–137
sea breeze 94–95, , , ,
sea lanes see sea routes
sea level
eustatic , , 140–141, 143–144, , ,
isostatic
maximum marine transgression , , , , , ,
relative , , , , , , , , , ,
seals and sealings see Minoan Crete
seamarks see navigation
Sea Peoples , , , 65–66,
sea routes , , , , , , , , , 82–83, , 110–119, 122–123, , , , , , ,
seascapes , , , , , , , ,
sediment budget ,
sedimentology , 167–168,
Selas River (Messenia)
self-organizing systems
Seraglio (Kos)
Seriphos (island)
sewn joinery see ship construction
eytan Deresi (Turkey)
Shaft Grave era , , , , , , , 232–233, ,
"shaft grave," Kolonna see Kolonna
shaft graves, Lerna ,
shaft graves, Mycenae , , , , 223–225,
Shaw, Maria 55–56
sheep see pastoralism
sheltered anchorages
see also harbors, sheltered
Sherratt, Andrew ,
Sherratt, Susan ,
shipbuilding , , 59–60, , ,
ship captains , , , 69–71, , , , 109–113, , , , , , ,
ship construction
frame-first 48–49
hull-first 48–49
mortise-and-tenon joinery 49–50
sewn joinery 49–50
ship sheds , , , 159–160
ship societies 125–134, 288–289
ships
see also boats
defined ,
iconographic representations 37–41, , , 59–61, 64–66, 68–71, 77–80, 132–133
merchantman 57–59, , , , , ,
merchant ships , , ,
Mycenaean galleys , , 58–65, 68–71, 83–84, , 132–133, n:186
performance 62–64, 81–87
propulsion 51–54
shipwrecks
Cape Gelidonya , 25–28, , 43–44, , , , , ,
Point Iria , 26–27, 43–45, ,
Pseira 43–45, , 195–196
Uluburun , 25–26, , , 42–44, 49–50, , , , , , , , , ,
side-scan sonar see geophysical survey
Sidon (Lebanon) ,
Sirens
Sicily , , , , , ,
silver and silver mining , , ,
Silver Siege Rhyton 64–65
sirocco wind 93–94,
skala ,
Sklavos (Salamis) ,
skymarks see navigation
Skyros (island) , 78–80,
small worlds see maritime small worlds
Snodgrass, Anthony ,
social networks
growth of networks , , ,
network models 205–211
preferential attachment of nodes , 231–232,
social network theory and analysis , , , 196–197, 203–205
social storage ,
Solomon Islands (Oceania) ,
Sophiko (Corinthia) , 266–269
southern Argolid , ,
Southern Oscillation
Spercheios River ,
Sporades Islands ,
springs , , , , ,
stable isotope analysis , ,
star-structure compass 126–127
steering oar 51–53, , ,
Stiri (Corinthia) , , , 258–265, 268–269
stirrup jars , , , , , 78–79,
straits , 96–97, , ,
Streif, Hans-Jeorg
Strogatz, Steven
structuration theory 128–129
structure-contingency framework
Summerfield, Michael
surface survey
archaeological , , 13–14, , , , , , , , , , , 243–246, 251–263, , , , ,
geoarchaeological 162–163, , , 247–248
swamps see coastal landforms
Syro-Canaanite (Syro-Palestinian) , , , , , , , , , ,
Syros (island) , ,
talismanic seals , ,
Tanagra (Boeotia)
Tanaja ,
Tandy, David
Tanner, William
Taygetos Mountains
tectonic events
coastal subsidence and uplift , , , , 148–149, , , ,
tectonic structure , , , , , , , , ,
Telemachus
Tel Kabri (Israel)
Tell Abu Hawam (Israel) ,
Tell el Dab'a (Egypt)
Tantura Lagoon (Israel)
terra rossa soils ,
terebinth resin
terrace walls, Mycenaean , , , 260–263,
see also Kalamianos
textiles , ,
texts
Canaanite/Ugaritic , , ,
Cypro-Minoan
Egyptian 19–20, , 28–29, ,
Hittite , 29–30, , 36–37, , , ,
Linear A , ,
Linear B , 14–18, 20–21, , , 35–36, 64–65, 74–75, , , , 130–132, 201–202, , , , , 286–288,
Thebes (Boeotia) , , , , , , , , , ,
Thebes (Egypt)
Theodosian Harbor (Turkey)
Thera (Santorini)
see also Akrotiri; flotilla fresco
eruption
Thermaic Gulf , ,
thermohaline circulation
thermohaline front ,
Thermopylae (Phthiotis)
Thessaly , , , 277–283
Thomas, Patrick
tidal marshes and mudflats see coastal landforms
Tilley, Alec
Tilley, Christopher
tholos tombs , , , , 280–281, 282–283
Thorikos (Attica)
Thucydides , 66–67
Tiryns (Argolid) , 16–17, , , , , , 54–55, , , 238–239, , ,
tombolo 149–150, , , ,
"tools of exclusion,"
Toumba Thessalonikis (Macedonia)
trade and exchange see maritime trade
tramping see cabotage
travel times 191–192
see also cost weighted path distance
tree-ring dating (dendrochronology)
Trianda (Rhodes)
triaconter
Trieste gap
Troizen (Argolid) ,
Trojan War , , 67–68, , , 136–137,
Troy, coastal reconstruction , , , 154–155, ,
Tsoungiza (Argolid) , 233–235,
Tudhaliya IV
tuna fishing
Turkey see Anatolia
Tyre (Lebanon) ,
Tzalas, Harry ,
Tzortzopoulou-Gregory, Lita ,
Ugarit (Syria) , , , 32–33, , , ,
Uluburun see shipwrecks
Vagenas tholos tomb (Messenia)
vardari wind ,
Vassa (Argolid)
Vayia (Corinthia) , ,
viewsheds , , , , ,
Vikings , , ,
volcanic rock, Aiginetan , , , , , , 263–264,
Volos, Bay of , , , , 277–283
Voutsaki, Sofia
Vrokastro (Crete)
wall painting
Akrotiri, West House Room , 39–40, 56–57
Akrotiri, West House Room 5, "Flotilla Fresco."
see also Flotilla Fresco
Akrotiri, West House Room 5, young fisherman fresco , n:288
Ayia Irini, miniature fresco , ,
Egyptian tomb paintings , ,
Iklaina, fresco fragment 60–61
Minoan frescoes in Near East
Mycenae, Megaron, fresco fragments , ,
Pylos, Hall 64 fresco fragments 56–57,
Thebes, fresco fragments
"Wandering Rocks,"
Ward, Cheryl
warfare, naval 58–59, 64–68, , ,
warriors , 58–59, , 66–68, , 132–134, 223–226
Watts, Duncan
wave-cut cliffs and notches see coastal landforms
way-finding , 123–124
weapons
sling stones , , 224–225
spear points , ,
swords and daggers , , ,
Wedde, Michael
Weißes Haus see Kolonna
Wells, Lisa
Westerdahl, Christer
West House, Akrotiri , , 38–40, 56–57, ,
see also Flotilla Fresco
wetlands see coastal landforms
wheat , , ,
winds see environment, Mediterranean
winter voyaging see navigation
world-systems , ,
World Wide Web
Wright, James , ,
xenoi
Zangger, Eberhard
Zygouries (Corinthia)
| {
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} | 1,026 |
Sci Rep. 2018 Jun 11;8(1):8805. doi: 10.1038/s41598-018-27181-y.
Metabolomic and microarray analyses of adipose tissue of dapagliflozin-treated mice, and effects of 3-hydroxybutyrate on induction of adiponectin in adipocytes.
Nishitani S1, Fukuhara A2,3, Shin J1,4, Okuno Y1, Otsuki M1, Shimomura I1.
Departments of Metabolic Medicine, Osaka University Graduate School of Medicine, Suita, Osaka, Japan.
Departments of Metabolic Medicine, Osaka University Graduate School of Medicine, Suita, Osaka, Japan. fukuhara@endmet.med.osaka-u.ac.jp.
Departments of Adipose Management, Osaka University Graduate School of Medicine, Suita, Osaka, Japan. fukuhara@endmet.med.osaka-u.ac.jp.
Departments of Diabetes Care Medicine, Osaka University Graduate School of Medicine, Suita, Osaka, Japan.
Sodium/glucose cotransporter 2 (SGLT2) inhibitor improves systemic glucose metabolism. To clarify the effect of dapagliflozin, we performed gene expression microarray and metabolomic analyses of murine adipose tissue. Three groups of mice were used; non-diabetic control KK mice (KK), diabetic KKAy mice (KKAy), and KKAy mice treated with dapagliflozin (KKAy + Dapa). Plasma glucose levels were significantly reduced in KKAy + Dapa compared with KKAy. Food consumption was larger in KKAy + Dapa than KKAy, and there were no significant differences in body and adipose tissue weight among the groups. Metabolomic analysis showed higher levels of many intermediate metabolites of the glycolytic pathway and TCA cycle in KKAy than KK, albeit insignificantly. Dapagliflozin partially improved accumulation of glycolytic intermediate metabolites, but not intermediate metabolites of the TCA cycle, compared with KKAy. Interestingly, dapagliflozin increased plasma and adipose 3-hydroxybutyric acid (3-HBA) levels. Microarray analysis showed that adipocytokines were downregulated in KKAy compared with KK mice, and upregulated by dapagliflozin. In vitro, 3-HBA induced β-hydroxybutyrylation of histone H3 at lysine 9 and upregulation of adiponectin in 3T3-L1 adipocytes independent of their acetylation or methylation. Our results suggest that 3-HBA seems to provide protection through epigenetic modifications of adiponectin gene in adipocytes.
Changes in body weight, food consumption, water intake, blood glucose level, and organ weight of the three mice groups. Three groups of mice are used; non-diabetic control female KK mice (KK), diabetic female KKAy mice (KKAy), and female KKAy mice treated with dapagliflozin (KKAy + Dapa). (a) Body weight, (b) food consumption, and (c) water intake were measured weekly for 5 weeks. (d) Fasting blood glucose levels were measured every two weeks. Data are mean ± SEM (n = 6). **p < 0.01, ***p < 0.001, KK versus KKAy. #p < 0.05, ##p < 0.01, ###p < 0.001, KKAy versus KKAy + Dapa, by one-way ANOVA followed by post hoc analysis (Tukey-Kramer test). (e) Weight of subcutaneous WAT (subWAT), periovarian WAT (ovaWAT), mesenteric WAT (mesWAT), BAT, liver, kidney, skeletal muscle after 5 weeks of dapagliflozin treatment. Data are mean ± SEM (n = 6). *p < 0.05, **p < 0.01, ***p < 0.001, by one-way ANOVA followed by post hoc analysis (Tukey-Kramer test).
Plasma levels of glycated HbA1c, insulin, NEFA, TG, adiponectin, and 3-HBA. (a) Glycated HbA1c (HbA1c), (b) plasma insulin, (c) plasma NEFA, (d) plasma TG, (e) plasma adiponectin, and (f) plasma 3-HBA levels were measured after 5 weeks of treatment with dapagliflozin. Data are mean ± SEM (n = 6). *p < 0.05, **p < 0.01, ***p < 0.001, by one-way ANOVA followed by post hoc analysis (Tukey-Kramer test).
Results of metabolomic and microarray analyses of periovarian WAT (ovaWAT). All analysis were performed after 5 weeks of dapagliflozin treatment. (a) Relative levels of metabolites and gene expressions associated with glycolytic and tricarboxylic cycle (TCA cycle) pathways. Relative levels of metabolites and gene expressions were surrounded by solid line and bottled line, respectively. (b,c) Relative expression levels of genes associated with adipocytokines (b) and the metabolism of ketone body (c). Data are normalized to the values of metabolites or gene expression levels of KK mice, and expressed as mean ± SEM (n = 4). *p < 0.05, **p < 0.01, ***p < 0.001, by one-way ANOVA followed by post hoc analysis (Tukey-Kramer test). G6P, glucose 6-phosphate; F6P, fructose 6-phosphate; F1,6P, fructose 1,6-diphosphate; GAP, glyceraldehyde 3-phosphate; 1,3-DPG, 1,3-di phosphoglycerate; 3-PG, 3-phosphoglycerate; 2-PG, 2-phosphoglycerate; PEP, phosphoenolpyruvate; AcCoA, acetyl CoA; OA, oxaloacetate; cis-Aco, cis-aconitate; 2-OG, 2-oxoglutarate; SucCoA, succinyl-CoA; 3-HBA, 3-hydroxybutyrate; Hk2, hexokinase 2; Gpi1, glucose phosphate isomerase 1; Pfkp, phosphofructokinase, platelet; Aldoc, aldolase C; Gapdh, glyceraldehyde-3-phosphate dehydrogenase; Pgk1, phosphoglycerate kinase-1; Pgam1, phosphoglycerate mutase 1; Eno1, enolase 1; Pkm, pyruvate kinase, muscle; Pdha1, pyruvate dehydrogenase E1 alpha 1; Pdhb, pyruvate dehydrogenase (lipoamide) beta; PPAR-γ, peroxisome proliferator activated receptor gamma; MCP-1, Monocyte chemoattractant protein-1; PAI-1, plasminogen activator inhibitor-1; IL-6, interleukin 6; TNF-α, tumor necrosis factor alpha; MCT1, monocarboxylic acid transporters 1; MCT2, monocarboxylic acid transporters 2; MCT4, monocarboxylic acid transporters 4; SCOT, succinyl-CoA-3-oxaloacid CoA transferase.
Effects of 3-HBA on mRNA expression levels and intracellular protein levels in 3T3-L1 adipocytes. On day 7 after differentiation, the medium of 3T3-L1 cells were replaced with KRBB supplemented with 0 or 10 mM 3-HBA and incubated for 24 hour. (a,f) The relative mRNA expression levels of adiponectin (a), PPAR-γ (b), MCP-1 (c), PAI-1 (d), IL-6 (e), and TNF-α (f) in 3T3-L1 adipocytes were measured by quantitative real-time PCR. Data are normalized to the level of 36B4 mRNA, and expressed as mean ± SEM (n = 6). (g) Intracellular protein levels of adiponectin and β-Actin in 3T3-L1 adipocytes using western blot analysis. Left panel; Representative western blot analysis. Right panel; Quantitative analysis of adiponectin contents in the left panel. Data are mean ± SEM (n = 3). *p < 0.05, **p < 0.01, ***p < 0.001.
Effects of 3-HBA on adiponectin gene in 3T3-L1 adipocytes. (a) Promoter analysis of adiponectin gene in 3T3-L1 adipocytes. A series of fragments of the 5′-flanking region of the human adiponectin gene were subcloned upstream of the luciferase reporter gene as described in Materials and Methods. On day 7 after differentiation, each promoter/reporter construct was transfected into 3T3-L1 adipocytes, and next day, the media of 3T3-L1 cells were replaced with KRBB supplemented with or without 10 mM 3-HBAs or 1 μM pioglitazone. Luciferase activity was measured after 24-hr incubation. Luciferase values were normalized by an internal CMV-Renilla control and expressed as relative luciferase activity. Data are mean ± SEM (n = 3). (b) Adiponectin promoter region bisulfite sequencing analysis of 3T3-L1 adipocytes. On day 7 after differentiation, the media of 3T3-L1 cells were replaced with KRBB supplemented with 0 or 10 mM 3-HBA and incubated for 24 hr. Top panel; Each circle represents sequencing results of independent clones. Open circles: unmethylated CpGs, solid circles: methylated CpG. The CpG position relative to upstream transcription start site of mouse adiponectin gene is shown below each column. Bottom panel; Percentage of 5-methylcytosine. Data are mean ± SEM of three independent samples (n = 3). (c) ChIP-qPCR analysis of histone H3 tail at lysine 9 modifications on the adiponectin gene in 3T3-L1 adipocytes. On day 7 after differentiation, the media of 3T3-L1 cells were replaced with KRBB supplemented with 0 or 3 mM 3-HBA and incubated for 24 hr. The genomic DNA was precipitated by antibodies against β-hydroxybutyrylated histone H3 at lysine 9 (H3K9bhb), acetylated histone H3 at lysine 9 (H3K9ac), di-methylated histone H3 at lysine 9 (H3K9me2). ChIP signals of each region of adiponectin gene were detected by quantitative real-time PCR and normalized to input signal as relative to input (%). Data are mean ± SEM (n = 3). *p < 0.05, **p < 0.01, ***p < 0.001. | {
"redpajama_set_name": "RedPajamaC4"
} | 3,202 |
GIO Stadium
Junior Representatives
NRL Nines
NSW Cup Preview: Mounties v Warriors
Matt Buxton
Fri 28 Aug 2015, 01:24 PM
Mounties v New Zealand Warriors: Competition heavyweights clash
St Marys Leagues Stadium, Saturday 29 August, 3pm
In a thrilling 2015 VB NSW Cup finals preview, Mounties host the Warriors in what certainly is the most anticipated match of the round.
The Wyong v Wentworthville match will finish just minutes before kickoff, and the score line of that game will determine how much Mounties will have to win by if they are any chance of taking out the minor premiership.
However, with a top-four spot secured, Mounties coach Steve Antonelli insists that the number-one position on the ladder is not their major focus.
"We haven't spoken about [the minor premiership] too much," says Antonelli.
"We're in the top four, so we haven't spoken much on where we will end up on the table."
Mounties have an incredibly impressive attacking record, topping the league with a whopping average of 29 points per game.
Defence is where they let themselves down in the middle of the year, but it has improved in recent times for the team that has only lost one match since round 17.
Antonelli made it clear that defence will be the primary focus heading into the game against the Warriors.
"The last three weeks we've set ourselves goals defensively, our focus [this week] is on defence," says Antonelli.
"If you can defend you're halfway there to winning the game. In the last six weeks, our individual defence has improved 50 per cent, we're doing a lot of work on it."
The Warriors are still mathematically a chance of winning the minor premiership, but it is highly unlikely, as they would have to win by a significant margin and hope other results go their way.
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\section{Introduction and preliminary results}\label{sectI}
The problem of finding exactly solvable quantum systems has always raised a lot of interest in the community of physicists and mathematicians (see for instance \cite{carinena14} and references therein). The reason is obvious: from a physical side, a quantum system is meant to describe some (relevant) phenomenon, mainly at microscopic scale. Solving the Schr\"odinger equation with a potential not considered so far can represent a serious mathematical challenge.
With this in mind, along the years several general ideas have been proposed to produce new solvable models out of old ones. For instance, if one knows the eigenvalues and eigenvectors of a given Hamiltonian $H$, with $H=H^\dagger$, one can construct new related Hamiltonians related to $H$ by some similarity operator, or by intertwining operators, or, if $H$ can be Darboux factorized, by considering its super-symmetric partner. In each of these ways these new Hamiltonians have spectra and eigenvectors which can be deduced from the ones of $H$. It should also be stressed that, in recent years, and in particular since the publication of the famous paper by Bender and Boettcher in 1998, \cite{ben1}, similar strategies have been extended also to the case in which the original Hamiltonian $H$ is not necessarily self-adjoint, $H\neq H^\dagger$.
In recent years an alternative strategy to construct a solvable model working with self-adjoint Hamiltonians has been proposed by one of us (E.C.) and his coworkers in a series of papers on what has been called {\em generalized Heisenberg algebras} (GHA), \cite{curado,curado1,curado2}. This strategy is mainly based on the existence of suitable intertwining and commutation relations, and on the existence of a certain function related to them. On a different side, deformations of the canonical (anti-)commutation relations have proved to be quite useful for deducing eigenvalues and eigenvectors of certain Hamiltonians appearing in the literature devoted to non-Hermitian quantum mechanics. We refer to \cite{baginbagbook} for a recent review. Here we somehow merge the two approaches showing how a GHA can be deformed to include in the framework Hamiltonians which are manifestly non self-adjoint, and how eigenvalues and biorthogonal eigenvectors can be constructed explicitly.
The paper is organized as follows: in the rest of this section we briefly discuss some essential aspects of the GHA. In Section \ref{sectII} we propose our deformed version of this algebra. We present in Section \ref{sectIII} several examples. Those devoted to the P\"oschl-Teller potential and its limit case which is the infinite square-well potential are examined in detail. In Section \ref{sectIV} we discuss the relation of the deformed GHA with the so-called non linear ${\mc D}$-pseudo bosons, \cite{fbnlpbs}, while Section \ref{sectconcl} contains our conclusions. To keep the paper self-contained, the Appendix contains some useful formulas on the Gegenbauer polynomials and definitions on biorthogonal sets used in the paper.
\subsection{An introduction to GHA}\label{sectAITGHA}
Let $c$ and $H=H^\dagger$ be two operators satisfying the following conditions:
\begin{equation}
cH=f(H)c,\qquad\mbox{and}\qquad [c,c^\dagger]=f(H)-H,
\label{11}\end{equation}
where $f(x)$ is a given function of $x$, known as {\em the characteristic function} of the GHA. It has to be stressed that both these equalities should be properly defined, at least if $c$ or $H$, or both, are unbounded operators. This is the case, for instance, in the simple case when $H=c^\dagger c$, with $[c,c^\dagger]=1 \!\! 1$. When this is so, formulas (\ref{11}) should be understood in the sense of unbounded operators. For instance, following \cite{baginbagbook}, we can assume that a set ${\mc D}$ exists, dense in the Hilbert space $\mc H$, stable under the action of $c$, $c^\dagger$, $H$ and $f(H)$ such that
$$
cH\varphi=f(H)c\,\varphi, \qquad \mbox{and} \qquad (c\,c^\dagger-c^\dagger\,c)\varphi=(f(H)-H)\varphi,
$$
for all $\varphi\in{\mc D}$. Of course, $f(H)$ can be naturally defined using functional calculus, since $H=H^\dagger$, \cite{rs}. From now on, when no confusion can arise, we will use the simpler notation in (\ref{11}).
From (\ref{11}) we easily deduce that $[c,H]=(f(H)-H)c$, and $[c^\dagger,H]=-c^\dagger(f(H)-H)$. Let us suppose $H$ positive,
and having $\hat e_0 \in \mathcal{H}$ as a normalized ground-state of $H$, $H \hat e_0=\epsilon_0 \, \hat e_0$.
For concreteness we assume $\epsilon_0>0$. If $\hat e_0$ belongs to ${\mc D}$, we can introduce the vectors $\hat e_n=(c^\dagger)^n\hat e_0$, $n\geq0$, and they all belong to ${\mc D}$ as well, and they are all eigenstates of $H$ with eigenvalues $\epsilon_n$, defined recursively as $\epsilon_n=f(\epsilon_{n-1})$, $n\geq1$. In other words, we have
\begin{equation}
H\hat e_n=\epsilon_n\hat e_n,
\label{12}\end{equation}
$n\geq0$, with $\epsilon_n$ as above. Of course, self-adjointness of $H$ implies that, if the multiplicity of each $\epsilon_n$ is one, different $\hat e_n$ are mutually orthogonal: $\left<\hat e_n,\hat e_k\right>=0$ if $n\neq k$. However they are, in general, not normalized. To fix the normalization it is useful to check first if $c$ is a lowering operator, i.e. if $c\hat e_n$ is proportional to $\hat e_{n-1}$. For that it is necessary to check first that $c\hat e_0=0$. This may follow from an explicit computation, but it may also be deduced using the following general result:
\begin{prop}\label{prop0}
Let ${\cal F}_{\hat e}=\{\hat e_n,\,n\geq0\}$ be complete in $\mc H$. Then $c\, \hat e_0=0$.
\end{prop}
\begin{proof}
Using the orthogonality of the vectors $\hat e_n$ and their definition we have
$$
\left<\hat e_n,(c\hat e_0)\right>=\left<c^\dagger\hat e_n,\hat e_0\right>=\left<\hat e_{n+1},\hat e_0\right>=0
$$
for all $n\geq0$. Then the result follows from our assumption on ${\cal F}_{\hat e}$.
\end{proof}
Using now induction on $n$ we can prove that, if $c\, \hat e_0=0$, then, for $n\geq1$,
\begin{equation}
c\, \hat e_n=\sqrt{\epsilon_n-\epsilon_0} \,\hat e_{n-1}, \qquad \mbox{ and }\qquad c^\dagger \,c\, \hat e_n=(\epsilon_n-\epsilon_0)\hat e_{n}.
\label{13}\end{equation}
The proof is easy and is based on (\ref{11}), and will not be given here. Rather than this we comment that, in order for the framework to make sense, the function $f(x)$ must be such that
\begin{equation}
\epsilon_n>\epsilon_0,
\label{14}\end{equation}
for all $n>0$. This is because, from the second equality in (\ref{13}), we have
$$
\left<\hat e_n,c^\dagger \,c\hat e_n\right>=\|c\hat e_n\|^2=(\epsilon_n-\epsilon_0)\|\hat e_{n}\|^2,
$$
for all $n\geq1$, which is only compatible, if $\hat e_n\neq0$, with (\ref{14}). This, of course, imposes some constraint on the characteristic function $f(x)$, for instance that $f(x)$ is strictly increasing, as we will assume from now on. This also guarantees that each $\epsilon_n$ is not degenerate. Now, it is not hard to compute the right normalization to produce an orthonormal (o.n) basis of eigenvectors of $H$. They are
\begin{equation}
e_n=\frac{1}{\sqrt{(\epsilon_n-\epsilon_0)!}}\,\hat e_n=\frac{1}{\sqrt{(\epsilon_n-\epsilon_0)!}}\,(c^\dagger)^n\hat e_0,
\label{15}\end{equation}
where $(\epsilon_n-\epsilon_0)!=(\epsilon_n-\epsilon_0)(\epsilon_{n-1}-\epsilon_0)\cdots(\epsilon_1-\epsilon_0)$, $n\geq1$ and $0! =1$. These vectors satisfy the following relations:
\begin{equation}
\left\{
\begin{array}{ll}
c^\dagger e_n=\sqrt{\epsilon_{n+1}-\epsilon_0}\,\,e_{n+1},\\
c e_n=\sqrt{\epsilon_{n}-\epsilon_0}\,\,e_{n-1},
\end{array}
\right.
\label{16}\end{equation}
which together imply that
\begin{equation}
cc^\dagger e_n=\left(\epsilon_{n+1}-\epsilon_0\right)\,e_{n}, \qquad c^\dagger c \,e_n=\left(\epsilon_{n}-\epsilon_0\right)\,e_{n},
\label{17}\end{equation}
for all $n\geq0$. In particular these imply that the original Hamiltonian $H$ can be written as $H=c^\dagger c+\epsilon_01 \!\! 1$, so that $H-\epsilon_01 \!\! 1$ is factorizable and coincides with $c^\dagger c$. This suggests to introduce a second operator constructed using $c$ and its adjoint, the so-called SUSY-Hamiltonian $H_{Susy}=cc^\dagger+\epsilon_01 \!\! 1$. The eigenvectors of $H_{Susy}$ are the same as those of $H$, while its eigenvalues are simply shifted:
\begin{equation}
H_{Susy}e_n=\epsilon_{n+1}e_n,
\label{18}\end{equation}
$n\geq0$. $H_{Susy}$ plays also a role in the determination of the characteristic function of the GHA. In fact, since $H_{Susy}=[c,c^\dagger]+c^\dagger c+\epsilon_01 \!\! 1=f(H)-H+H=f(H)$, they are really the same object.
\section{Deformed GHA}\label{sectII}
\subsection{General considerations}\label{sectII_1}
In this section we will show how the general structure considered in Section \ref{sectAITGHA} can be extended in order to include Hamiltonian operators which are not necessarily self-adjoint.
The starting point of our analysis is the following definition:
\begin{defn}
\label{def1}
Let $a$, $b$ and $h$ be three operators defined on some dense domain ${\mc D}$ of the Hilbert space $\mc H$. Let us assume that ${\mc D}$ is stable under their action and under the action of their adjoints. We say that $(a,b,h)$ are {\em compatible} if, for all $\varphi\in{\mc D}$, the following equalities hold:
\begin{equation}
hb\,\varphi=bf(h)\,\varphi, \qquad ah\,\varphi=f(h)a\,\varphi,
\label{21}
\end{equation}
for some fixed, strictly increasing, function $f(x)$.
\end{defn}
\vspace{2mm}
{\bf Remarks:--} (1) The first remark is that, if $h$ is bounded and $f(x)$ admits an expansion in power series $f(x)=\sum_{n=0}^\infty c_n x^n$ convergent inside an interval $|x|<M$, then $f(h)$ can be defined as $f(h)=\sum_{n=0}^\infty c_n h^n$, at least if $\|h\|<M$. On the other hand, if $h$ is unbounded but self-adjoint, $h=h^\dagger$, $f(h)$ can be defined by means of the spectral theorem. If $h$ is unbounded and not self-adjoint, defining $f(h)$ can be more complicated, but still it can be done in some cases, for instance if $h$ is similar to another operator $H$, $H=H^\dagger$, at least if the similarity map is given by a bounded operator with bounded inverse. In the most general case we can define $f(h)$ via its action on a basis of $\mc H$. Of course, the natural choice of basis would be, when possible, the set of eigenstates of $h$.
(2) In principle there is no reason a priori for taking a single function $f(x)$ in both equalities in (\ref{21}). For instance, we could assume that $hb\varphi=bf_b(h)\varphi$ and $ah\varphi=f_a(h)a\varphi$ for two different functions $f_a(x)$ and $f_b(x)$. However, in view of what we will do next, this would not be a useful choice. Moreover, the reason why we have assumed here that $f(x)$ is strictly increasing is because of what was discussed in the previous section on $\epsilon_n-\epsilon_0$, and because we want to avoid some of the eigenvalues of $h$ to be degenerate.
(3) If $h=h^\dagger$ and $a=b^\dagger$ the two equalities in (\ref{21}) collapse into a single one, which is the one considered in \cite{curado1,curado2} and reviewed in Section \ref{sectAITGHA}. Otherwise, they are different. Interestingly, they can be seen as two different intertwining relations between $h$ and $f(h)$, one due to $a$ and the other to $b$. In principle, these two equations are really independent, since $a$ and $b$ are unrelated, so far. However, in the following, see (\ref{25}), they are assumed to satisfy a suitable commutation rule, so that they are, in fact, connected. Of course, with this in mind, the whole machinery of intertwining relations, see \cite{intop,bagint1,bagint2}, could be considered in connection with our operators. However, this analysis is not particularly relevant for us now and it is postponed to a future paper.
\vspace{3mm}
\begin{prop}\label{prop1}
Let $(a,b,h)$ be compatible operators and let $\epsilon_0$ be a fixed (non-negative) real number. Let us call $\epsilon_{n+1}=f(\epsilon_n)$, $n\geq0$. Suppose now that two non-zero vectors, $\varphi_0$ and $\psi_0$, in ${\mc D}$ do exist such that $h\varphi_0=\epsilon_0\varphi_0$ and $h^\dagger \psi_0=\epsilon_0\psi_0$. Let us call
\begin{equation}
\varphi_n=\frac{1}{\sqrt{(\epsilon_n-\epsilon_0)!}}\,b^n\varphi_0\qquad \psi_n=\frac{1}{\sqrt{(\epsilon_n-\epsilon_0)!}}\,(a^\dagger)^n\psi_0,
\label{22}\end{equation}
for all $n\geq0$, where $(\epsilon_n-\epsilon_0)!$ is defined below Eq. (\ref{15}).
Then:
\begin{equation}
h\varphi_n=\epsilon_n\varphi_n,\qquad h^\dagger \psi_n=\epsilon_n\psi_n,
\label{23}\end{equation}
for $n\geq0$. Moreover,
\begin{equation}
\left<\psi_n,\varphi_m\right>=\delta_{n,m}\left<\psi_n,\varphi_n\right>.
\label{24}\end{equation}
\end{prop}
\begin{proof}
Formulas in (\ref{23}) can be proved by induction using (\ref{21}). For instance, for $n=1$ we have $$
h\varphi_1=\frac{1}{\sqrt{(\epsilon_1-\epsilon_0)!}}\,h\,b\varphi_0=\frac{1}{\sqrt{(\epsilon_1-\epsilon_0)!}}\,b\,f(h)\varphi_0=f(\epsilon_0)
\frac{1}{\sqrt{(\epsilon_1-\epsilon_0)!}}\,b\,\varphi_0=\epsilon_1\varphi_1.
$$
Let us now assume that $h\varphi_n=\epsilon_n\varphi_n$ for a fixed $n$. To prove that $h\varphi_{n+1}=\epsilon_{n+1}\varphi_{n+1}$ we observe that
$$
h\varphi_{n+1}=\frac{1}{\sqrt{(\epsilon_{n+1}-\epsilon_0)}}\,h\,b\varphi_n=\frac{1}{\sqrt{(\epsilon_{n+1}-\epsilon_0)}}\,b\,f(h)\varphi_n=
f(\epsilon_n)
\frac{1}{\sqrt{(\epsilon_{n+1}-\epsilon_0)!}}\,b\,\varphi_n=\epsilon_{n+1}\varphi_{n+1}.
$$
A similar proof holds for the vectors $\psi_n$, by using the adjoint of the equality $ah=f(h)a$. Formula (\ref{24}) is a simple consequence of the eigenvalues equations in (\ref{23}), and of the fact that the various $\epsilon_n$ are all different.
\end{proof}
\vspace{2mm}
{\bf Remarks:--} (1) Due to the stability of ${\mc D}$ it is clear that all the vectors $\varphi_n$ and $\psi_n$ belong to ${\mc D}$.
(2) For all $n\geq1$ the quantities $\epsilon_n-\epsilon_0$ are strictly positive, and $\{\epsilon_n\}$ is a strictly increasing sequence.
(3) The above result could be generalized to the situation in which the eigenvalues of $h$ and of $h^\dagger$ are different. However, doing so, we would lose the isospectrality of these two operators, which on the other hand we prefer to keep, since it has very useful consequences. {This kind of generalization has been discussed, for instance, in \cite{bagcomplex}, in finite
dimensional Hilbert spaces.}
\vspace{3mm}
Now, formulas (\ref{22}) show that $b$ and $a^\dagger$ are raising operators for the vectors in ${\cal F}_\varphi=\{\varphi_n, \,n\geq0\}$ and ${\cal F}_\psi=\{\psi_n, \,n\geq0\}$ respectively. We expect that $a$ and $b^\dagger$ are lowering operators. However, this is not true in general, but it is true if the following commutation rule between $a$ and $b$ is satisfied:
\begin{equation}
[a,b]=f(h)-h,
\label{25}\end{equation}
at least on ${\mc D}$. Of course, this also implies that $[b^\dagger,a^\dagger]=f(h^\dagger)-h^\dagger$ on ${\mc D}$. The following lowering equations can now easily be proved, if $a\varphi_0=b^\dagger\psi_0=0$\footnote{As in Proposition \ref{prop0} these equalities are surely true if the sets ${\cal F}_\varphi$ and ${\cal F}_\Psi$ are complete.}:
\begin{equation}
a\varphi_n=\sqrt{\epsilon_n-\epsilon_0}\,\varphi_{n-1},\qquad b^\dagger\psi_n=\sqrt{\epsilon_n-\epsilon_0}\,\psi_{n-1},
\label{26}\end{equation}
for all $n\geq1$. Also, we get
\begin{equation}
\left\{
\begin{array}{ll}
ba\varphi_n=(\epsilon_n-\epsilon_0)\varphi_n,\qquad\quad ab\varphi_n=(\epsilon_{n+1}-\epsilon_0)\varphi_n\\
a^\dagger b^\dagger\psi_n=(\epsilon_n-\epsilon_0)\psi_n,\qquad\, b^\dagger a^\dagger\psi_n=(\epsilon_{n+1}-\epsilon_0)\psi_n,
\end{array}
\right.
\label{27}\end{equation}
which show that $\varphi_n$ is an eigenstate of both $ba$ and $ab$, while each $\psi_n$ is eigenstate of both $a^\dagger b^\dagger$ and $b^\dagger a^\dagger$.
Now, under the assumptions of Proposition \ref{prop1}, if $\left<\psi_0,\varphi_0\right>=1$, it is a standard computation to check that $\left<\psi_n,\varphi_m\right>=\delta_{n,m}$.
\begin{defn}\label{defDGHA}
The set of commutation rules in (\ref{21}) and (\ref{25}) satisfied by the operators $a$, $b$ and $h$ define a {\em deformed generalized Heisenberg algebra} (DGHA).
\end{defn}
\vspace{2mm}
{\bf Remark:--} In view of what we have seen in Section \ref{sectAITGHA}, and of the eigenvalue equations in (\ref{27}), two SUSY Hamiltonians can be introduced for $h$ and $h^\dagger$, and their eigenvalues and eigenvectors can be easily be deduced out of those above.
\subsection{Constructing a deformed GHA from a GHA }
\label{DGHA}
Let $c$ and $H$ be operators satisfying the GHA as discussed in Section \ref{sectI}. As we have seen, they satisfy the following:
$$
cH=f(H)c,\qquad [c,c^\dagger]=f(H)-H,
$$
where $H=H^\dagger$ and $f(x)$ is a strictly increasing function. If we call $e_0$ the ground state of $H$, we require that $ce_0=0$ and that $e_0\in{\mc D}$, a suitable dense subset of $\mc H$. Let now $S$ be an invertible operator which leaves ${\mc D}$ stable, together with $S^{-1}$. Then, introducing $a=ScS^{-1}$, $b=Sc^\dagger S^{-1}$, $\varphi_0=Se_0$, $\psi_0=(S^{-1})^\dagger e_0$ and $h=SHS^{-1}$, it is clear that these new operators and vectors satisfy all properties required in Section \ref{sectII_1}. In particular, (\ref{21}) and (\ref{25}) are satisfied. Hence two biorthogonal sets can be constructed as in (\ref{22}), and these are eigenstates of $h$ and $h^\dagger$. Notice that, as it is discussed extensively in \cite{baginbagbook}, these two sets in general are not bases for the Hilbert spaces, even if they turn out, quite often in concrete examples, to be complete. However, if both $S$ and $S^{-1}$ are bounded, then ${\cal F}_\varphi$ and ${\cal F}_\psi$ are biorthogonal Riesz bases. Otherwise they are ${\mc D}$-quasi bases, see Appendix and reference \cite{baginbagbook}.
\vspace{2mm}
These simple steps show how a GHA can be modified in order to get a DGHA. In a certain sense, this result can be inverted: under natural assumptions, any DGHA gives rise to a GHA. Let us consider the operators $(a,b,h)$ satisfying Definition \ref{defDGHA}. To simplify the situation, we assume here that the sets ${\cal F}_\varphi$ and ${\cal F}_\psi$ constructed as discussed in Proposition \ref{prop1} are biorthogonal Riesz bases. Then the operators $S_\varphi f:=\sum_{n}\left<\varphi_n,f\right>\varphi_n$ and $S_\psi g:=\sum_{n}\left<\psi_n,g\right>\psi_n$ can be defined in all of $\mc H$. Moreover, because of our assumption, a bounded operator $R$ exists, with bounded inverse $R^{-1}$, and an o.n. basis ${\cal F}_v=\{v_n\}$ such that $\varphi_n=Rv_n$ and $\psi_n=(R^{-1})^\dagger v_n$, for all $n$. Then we easily deduce that $S_\varphi=RR^\dagger$, while $S_\psi=S_\varphi^{-1}$. Hence $S_\varphi$ and $S_\psi$ are both bounded and positive. Hence their (unique) positive square roots exist. If, for simplicity, we assume that also $S_\varphi^{1/2}$, $S_\psi^{1/2}$ leave ${\mc D}$ invariant, then the operator $c=S_\psi^{1/2}aS_\varphi^{1/2}$ also maps ${\mc D}$ in ${\mc D}$. This surely happens if ${\mc D}={\cal L}_\varphi\cap {\cal L}_\psi$, where ${\cal L}_\varphi$ is the linear span of the $\varphi_n$'s and ${\cal L}_\psi$ is the linear span of the $\psi_n$'s. Notice that both ${\cal L}_\varphi$ and ${\cal L}_\psi$ are dense in $\mc H$,since ${\cal F}_\varphi$ and ${\cal F}_\psi$ are Riesz bases for $\mc H$, and we are here assuming that their intersection is dense as well. The adjoint of $c$, $c^\dagger$, turns out to be $c^\dagger=S_\psi^{1/2}bS_\varphi^{1/2}$ on ${\mc D}$. Now, if we introduce a new operator $H$ on ${\mc D}$ as $Hg=S_\psi^{1/2}hS_\varphi^{1/2}g$, $g\in{\mc D}$, and new vectors $e_n=S_\varphi^{1/2}\psi_n$, we go back to what discussed in Section \ref{sectAITGHA}.
\section{Some classical examples}\label{sectIII}
We now discuss some classical examples which fit our assumptions,
and we will deform them according with what discussed in Section \ref{DGHA}.
\subsection{P\"oschl-Teller potentials}
As a concrete example of the general scheme presented above, we first consider the quantum model of a one-dimensional particle subjected to the symmetric P\"oschl-Teller potential
$$
V_{\lambda}(x)=\frac{\lambda(\lambda-1)}{\sin^2x}\, ,
$$
where $\lambda\geq1$ and $x\in (0,\pi)$. For $\lambda > 1$, this potential is a regularization of the infinite square well ($\lambda =1$) and extrapolates both the latter and the harmonic oscillator (for small $\vert x-\pi/2\vert$). The Hamiltonian of the particle is, fixing
$\hbar=2m=1$, $$H_{\lambda}=-\frac{\mathrm{d}^2}{\mathrm{d} x^2}+V_{\lambda}(x)\, , $$
and the eigenvalue equation for $H_{\lambda}$ can be explicitly solved, (see for instance \cite{antoine_etal_01} and references therein):
\begin{equation}
H_{\lambda}\,e^{\lambda}_n(x)=\epsilon^{\lambda}_n e^{\lambda}_n(x)\, ,
\label{28}\end{equation}
where $\epsilon^{\lambda}_n=(n+\lambda)^2$ and
\begin{equation}
e^{\lambda}_n(x)=K^{\lambda}_n\,\sin^{\lambda} x\,\mathrm{C}_n^{\lambda}\left(\cos x\right)\,.
\label{29}\end{equation}
Here
$$
K^{\lambda}_n=\Gamma(\lambda)\frac{2^{\lambda-1/2}}{\sqrt{\pi}}\sqrt{\frac{n!(n+\lambda)}{\Gamma(n+2\lambda)}}
$$
is a normalization constant and $\mathrm{C}_n^{\lambda}$ is the Gegenbauer polynomial of degree $n$ \cite{magnus66}. The set $\{e^{\lambda}_n(x)\}$ is an orthonormal basis.
We can check that $\epsilon^{\lambda}_{n+1}=(\sqrt{\epsilon^{\lambda}_n}+1)^2$, so that $f(x)=(\sqrt{x}+1)^2$ is the characteristic function for the system.
\vspace{2mm}
{\bf Remark:--}
For the Hamiltonian $H_{\lambda}$ a SUSY approach has been discussed in \cite{bergasiyou10,bersiyou12}, with corresponding lowering and raising operators, $A_{\lambda}= \dfrac{\mathrm{d}}{\mathrm{d} x} -\lambda \cot x$ and $A^{\dag}_{\lambda}= -\dfrac{\mathrm{d}}{\mathrm{d} x} -\lambda \cot x$. In this case one finds the Darboux factorization $H_{\lambda}= A^{\dag}_{\lambda} A_{\lambda} +\epsilon^{\lambda}_0$, and one gets $H_{\lambda+1}=A_{\lambda}\, A^{\dag}_{\lambda} + \epsilon^{\lambda}_0$ for its partner. However, this representation of $H_\lambda$ does not satisfy the assumptions of a GHA since $A_{\lambda}$ and $A^{\dag}_{\lambda}$ are not ladder operators. In fact, they shift both the polynomial degree and the parameter $\lambda$ as
$$
A_{\lambda}\, e^{\lambda}_n(x) = \sqrt{\epsilon^{\lambda}_n-\epsilon^{\lambda}_0 }\, e^{\lambda+1}_{n-1}(x)\, , \quad A^{\dag}_{\lambda}\, e^{\lambda+1}_n(x) = \sqrt{\epsilon^{\lambda}_{n+1}-\epsilon^{\lambda}_0 }\, e^{\lambda}_{n+1}(x)\,.
$$
as expected from the SUSY quantum mechanics formalism, which is not what should be satisfied. In fact, to be relevant for our purposes, we should find a factorization which leaves $\lambda$ unchanged, while changing $n$. Hence, let us introduce the following ladder operators,
\begin{equation}
\label{Bnl}
B_{\lambda}= -\sin x \frac{\mathrm{d}}{\mathrm{d} x}+\cos x\,(\hat{N}_{\lambda}+\lambda)\, , \quad B^{\dag}_{\lambda}= \sin x \frac{\mathrm{d}}{\mathrm{d} x}+(\hat{N}_{\lambda}+\lambda +1)\,\cos x\, ,
\end{equation}
where the diagonal ``number'' operator $\hat{N}_{\lambda}$ is defined by its action on the basis \eqref{29} as
\begin{equation}
\label{Nl}
\hat{N}_{\lambda}\, e^{\lambda}_n(x)= n e^{\lambda}_n(x)\,.
\end{equation}
The actions of the operators \eqref{Bnl} on the basis are easily derived from \eqref{uEn} and \eqref{u2dEn} by putting $u = \cos x$:
\begin{align}
\label{BEn}
B_{\lambda} \, \mathcal{E}^{\lambda}_{n}(u)&=\sqrt{\frac{n(n+\lambda)(n+2\lambda-1)}{n-1+\lambda}} \mathcal{E}^{\lambda}_{n-1}(u)\,,\\\label{BdEn} B^{\dag}_{\lambda} \, \mathcal{E}^{\lambda}_{n}(u)&= \sqrt{\frac{(n+1)(n+1+\lambda)(n+2\lambda)}{n+\lambda}} \mathcal{E}^{\lambda}_{n+1}(u)\, .
\end{align}
The next step is to build the operator $c$ corresponding to the Hamiltonian $H_{\lambda}$. We first derive from \eqref{BEn} the diagonal operator
\begin{equation}
\label{BdB}
B^{\dag}_{\lambda}\,B_{\lambda} = \frac{\hat{N}_{\lambda}(\hat{N}_{\lambda}+ \lambda)(\hat{N}_{\lambda}+\lambda-1)}{\hat{N}_{\lambda} -1 +\lambda}= \left(G_{\lambda}\left(\hat{N}_{\lambda}\right)\right)^{-2}\, \left(H_{\lambda} -\lambda^2\right)\, ,
\end{equation}
where, for $\lambda >1$, the strictly increasing positive bounded function $G_{\lambda}(t)$ is defined by
\begin{equation}
\label{Gx}
G_{\lambda}(t)= \sqrt{\frac{(t+2\lambda)(t-1+\lambda)}{(t+2\lambda -1)(t+\lambda)}}\, , \quad 0 <\sqrt{\frac{2(\lambda-1)}{2\lambda-1}} \leq G_{\lambda}(t) < 1
\end{equation}
The limit case of the infinite square well, for which $\lambda =1$, deserves a particular treatment and will be examined in the sequel.
We can rewrite $G_{\lambda} (t)$ as:
\begin{equation}
\label{GT}
G_{\lambda}(t)= T_\lambda^{1/2}(t-1) T_\lambda^{-1/2}(t) \, ,
\end{equation}
where
\begin{equation}
\label{Tlambda}
T_{\lambda}(t)= \frac{t+ \lambda }{t+2 \lambda } \, ,
\end{equation}
which is also positive bounded with bounded inverse and it is a
monotonically increasing function, from $1/2$ ($t=0$) to $1$ (t $\to \infty$).
We can now introduce our operators corresponding to $c$ and $c^{\dag}$,
\begin{align}
\label{cnl}
C_{\lambda}=& B_{\lambda} \,G_{\lambda}(\hat{N}_{\lambda}) = T_{\lambda}^{1/2}(t) B_{\lambda} T_{\lambda}^{-1/2}(t) \\
C^\dagger_{\lambda}=&G_{\lambda}(\hat{N}_{\lambda})\, B^{\dag}_{\lambda} =T_{\lambda}^{-1/2}(t) B_{\lambda}^\dagger T_{\lambda}^{1/2}(t) \, ,
\end{align}
where we have used the commutation relations $\mathcal{G}(\hat{N}+1) B_\lambda = B_\lambda \mathcal{G}(\hat{N})$, $\mathcal{G}(\hat{N}-1) B_\lambda^\dagger = B_\lambda^\dagger \mathcal{G}(\hat{N})$,
valid for any smooth function $\mathcal{G}(\hat{N})$.
It is easy to check that, as simplified ladder operators, they obey all expected GHA properties for this example of potential. In particular:
\begin{align}
\label{Clad}
C_{\lambda}\, \mathcal{E}^{\lambda}_{n}(u)&= \sqrt{n(n+2\lambda)}\, \mathcal{E}^{\lambda}_{n-1}(u)= \sqrt{\epsilon_n^{\lambda} - \epsilon_0^{\lambda}}\,\, \mathcal{E}^{\lambda}_{n-1}(u)\, , \\
\label{Clad}
C^{\dag}_{\lambda}\, \mathcal{E}^{\lambda}_{n}(u)&= \sqrt{(n+1)(n+1+2\lambda)}\, \mathcal{E}^{\lambda}_{n+1}(u)= \sqrt{\epsilon_{n+1}^{\lambda} - \epsilon_0^{\lambda}}\,\, \mathcal{E}^{\lambda}_{n+1}(u)\, .
\end{align}
\begin{equation}
\label{CLH}
C_{\lambda}\, H_{\lambda}= f\left(H_{\lambda}\right)\,C_{\lambda} \, , \ \mbox{with} \ f\left(H_{\lambda}\right)= \left(\sqrt{H_{\lambda}} + 1\right)^2\,.
\end{equation}
\begin{equation}
\label{ }
C^{\dag}_{\lambda}\, C_{\lambda}= H_{\lambda}- \epsilon_0^{\lambda}\,I\, , \quad C_{\lambda}\,C^{\dag}_{\lambda}= f\left(H_{\lambda}\right) - \epsilon_0^{\lambda}\,I\,,\quad \left[C_{\lambda}\,, \,C^{\dag}_{\lambda}\right]= f\left(H_{\lambda}\right) - H_{\lambda}\,.
\end{equation}
A similar physical realization of the P\"oschl-Teller potential was also recently realized, see \cite{RMECLRpra2017}. There, the P\"oschl-Teller creation and annihilation operators are written in a different way, but they are completely equivalent to our $C_\lambda$ and $C_\lambda^\dagger $ operators.
\subsubsection{Deforming P\"oschl-Teller}
If we now want to produce a DGHA, the easiest procedure consists in fixing a bounded operator with bounded inverse, and working as proposed at the beginning of Section \ref{DGHA}. For that let us illustrate the procedure by picking the following function of $x$,
$ S(x) = (1+2x)/(1+x)$.
It is clear that this can be considered as a bounded multiplication operator, with bounded inverse, for all $x\in [0,\pi]$. Hence we can use it and $S^{-1}(x)$ to deform the system, similarly to what we have done before: $a_\lambda=S(x)C_\lambda S^{-1}(x)$, $b_\lambda=S(x)C_\lambda^\dagger S^{-1}(x)$, $\varphi_0^\lambda(x)=S(x)e_0^\lambda(x)$ and $\psi_0^\lambda(x)=S^{-1}(x)e_0^\lambda(x)$. In particular, the Hamiltonian takes the following explicit expression:
\begin{equation}
h_\lambda=S(x)H_\lambda S^{-1}(x)=-\frac{\mathrm{d}^2}{\mathrm{d} x^2}+\frac{2}{(1+x)(1+2x)}\,\frac{\mathrm{d}}{\mathrm{d} x}+V_\lambda(x)-\frac{4}{(1+x)(1+2x)^2},
\label{ptham}\end{equation}
which is manifestly non self-adjoint. This describe an Hamiltonian with a new potential $V_\lambda(x)-\frac{4}{(1+x)(1+2x)^2}$, plus a term which is proportional to a given function of $x$ multiplying the momentum operator. For this Hamiltonian, and its adjoint $h_\lambda^\dagger$, the eigenvectors can be deduced as discussed in Section \ref{sectII_1}.
\subsection{The infinite square well}
As already stated, the infinite square well can be viewed as a particular case of the P\"oschl-Teller potentials, with $\lambda=1$. So, in principle, we just adapt our previous results to this particular situation simply by fixing this value for $\lambda$. However, we follow here a slightly different choice for the operator $S$ used to deform the GHA.
We also simplify our notations by dropping the subscript ``1", which is proper to the
infinite square well. Hence, we now consider the Hamiltonian of an infinite well in the interval $[0,\pi]$:
\begin{equation}
H=-\,\frac{\mathrm{d}^2}{\mathrm{d} x^2} + V(x)\, ,
\label{iw1}
\end{equation}
where
$$
V(x)=\left\{
\begin{array}{ll}
0\qquad\qquad\,\, x\in(0,\pi)\, , \\
\infty\qquad\qquad \mbox{elsewhere}
\end{array}
\right.\, .
$$
Eigenvalues and corresponding eigenfunctions read $\epsilon_n=(n+1)^2$ and $e_n(x)=\sqrt{\frac{2}{\pi}}\,\sin(n+1)x$ respectively. Hence $He_n=\epsilon_ne_n$, $n\geq0$, and the characteristic function $f(x)$ for the GHA is as before: $f(x)=(\sqrt{x}+1)^2$. The function $G(t)$ becomes
\begin{equation*}
G(t)= \frac{\sqrt{t\,(t+2)}}{t+1}\, ,
\end{equation*}
i.e., the ratio of the geometric mean of $t$ and $t+2$ to the arithmetic one.
The ladder operators \eqref{cnl} assume their simplest expression:
\begin{align}
\label{iw2}
C &= B \, G(\hat N) = T^{1/2}(\hat N) \, B \, T^{-1/2}(\hat N) \\
C^\dagger &= G(\hat N) \, B^\dagger = T^{-1/2}(\hat N) \, B^\dagger \, T^{1/2}(\hat N)\, ,
\end{align}
with $T(t)= (t+1)/(t+2)$ in agreement with Eq. (\ref{Tlambda}) for $\lambda =1$.
$\hat N$ is still the usual number operator $\hat N e_n(x)=ne_n(x)$, i.e., $\hat N = \sqrt{H}-1$.
\subsubsection{Deforming the infinite square-well potential}
What we want to do here is to deform this example in the way we have proposed in Section \ref{DGHA}: for that, let $\sigma(x)$ be a real function, which is supposed to satisfy the following bounds: $0<\sigma_m\leq\sigma(x)\leq\sigma_M<\infty$, for almost all $x\in[0,\pi]$. This means that the inverse of $\sigma(x)$ exists. Moreover, using the spectral theorem, we can define $\sigma\left(\hat N +1\right)$ and this operator commutes with $H$. Therefore, if we take $S:=\sigma\left(\hat N +1\right)$, it follows that $h=H$. Moreover, $\varphi_0=Se_0=\sigma(1)e_0$, while $\Psi_0=(S^{-1})^\dagger e_0=(\sigma(1))^{-1}e_0$. Hence, $\varphi_0$ and $\Psi_0$ differ from $e_0$ only for a normalization, and they satisfy $\left<\varphi_0,\Psi_0\right>=1$. The operators $a$ and $b$ act on $e_n(x)$ in a (formally) easy way. This follows from the fact that $Se_n=\sigma\left(\hat N+1\right)e_n=\sigma(n+1)e_n$. Then, for instance,
$$
b\,e_n=SC^\dagger S^{-1}e_n=\sigma\left(\hat N +1\right) C^\dagger \sigma\left(\hat N+1\right)^{-1}e_n=(\sigma(n+1))^{-1}\sigma\left(\hat N+1\right) C^\dagger e_n=$$ \begin{equation}=(\sigma(n+1))^{-1}\sigma\left(\hat N +1\right)\sqrt{\epsilon_{n+1}-\epsilon_0}\,e_{n+1}=
\frac{\sigma(n+2)}{\sigma(n+1)}\,\sqrt{\epsilon_{n+1}-\epsilon_0}\,e_{n+1},
\label{iw3}\end{equation}
with a similar result for $a$. We see that $b$ is still a raising operator, also with respect to the original o.n. basis ${\cal F}_e$, but with a slightly different coefficient, which involves $\sigma$.
\vspace{2mm}
A different conclusion is deduced if we consider a different choice of the operator $S$. In particular, if we take now $S$ to be the following multiplication operator: $S=(\sigma(x))^{-1}$, where $\sigma(x)$ is as before. Then $h$ turns out to be different from $H$, in general. In fact:
$$
h=-\frac{\sigma^{\prime\prime}(x)}{\sigma(x)}-\frac{2\sigma'(x)}{\sigma(x)}\,\frac{\mathrm{d}}{\mathrm{d} x}-\frac{\mathrm{d}^2}{\mathrm{d} x^2},
$$
which is manifestly non self-adjoint. Let us see what happens with the particular choice of $\sigma(x)=\alpha+\cos(k_0x)$, where $\alpha>1$ and $k_0\geq1$ is a fixed natural number. Of course we have $\sigma_m=\alpha-1>0$ and $\sigma_M=\alpha+1<\infty$. A simple computation shows that
$$
\varphi_0(x)=\sqrt{\frac{2}{\pi}}\,\frac{\sin(x)}{\alpha+\cos(k_0x)}, \qquad \Psi_0(x)=\alpha e_0(x)+\frac{1}{2}\left(e_{k_0+1}(x)-e_{k_0-1}(x)\right).
$$
We see that, while $\Psi_0(x)$ is just a linear combination of three elements of ${\cal F}_e$, $\varphi_0(x)$ is an infinite series of such elements.
The analogous of formula (\ref{iw3}) can now be deduced using the equality
$$
S^{-1}e_n(x)=\alpha e_n+\frac{1}{2}\left(e_{n+k_0}(x)+e_{n-k_0}(x)\right),
$$
which follows from some well known trigonometric identities. We restrict here to the case $n\geq k_0$. The opposite case can be easily deduced by simply using the parity properties of $e_n(x)$. We get
$$
b\,e_n=\frac{1}{\alpha+\cos(k_0x)}\left(\alpha\sqrt{\epsilon_{n+1}-\epsilon_0}\,e_{n+1}+\frac{\sqrt{\epsilon_{n+1+k_0}-\epsilon_0}}{2}e_{n+k_0+1}+
\frac{\sqrt{\epsilon_{n+1-k_0}-\epsilon_0}}{2}e_{n-k_0+1}
\right),
$$
which clearly shows how $b$ is no longer a raising operator, in this case, for the family $\{e_n\}$. The action of $a$ on $e_n$ can be deduced in a similar way. As for $h$, we get
$$
h=\frac{1}{\alpha+\cos(k_0x)}\left(k_0\cos(k_0x)+2k_0\sin(k_0x)\,\frac{\mathrm{d}}{\mathrm{d} x}-(\alpha+\cos(k_0x))\frac{\mathrm{d}^2}{\mathrm{d} x^2}\right).
$$
This operator looks extremely different from the one in (\ref{ptham}), even when $\lambda$ in $h_\lambda$ is fixed to be one. This is a consequence of the two different choices of the similarity operator $S$ in this case, and in (\ref{ptham}).
\subsection{The harmonic oscillator}
\subsubsection{A familiar preliminary}
Let $c$ be the standard bosonic lowering operator, satisfying the canonical commutation rule $[c,c^\dagger]=1 \!\! 1$, and let $H_0=c^\dagger c$. Of course, $H_0=H_0^\dagger$. Working in the coordinate representation, $e_0(x)=\frac{1}{\pi^{1/4}}e^{-x^2/2}$, $x \in (-\infty,\infty)$, is a function annihilated by $c$: $ce_0=0$. Now, taking $f(x)=x+1$ and identifying ${\mc D}$ with the set of test functions ${\cal S}(\Bbb R)$, a DGHA is trivially recovered if $a=c$, $b=c^\dagger$ and $h=H_0$. In this case, clearly, $\varphi_0(x)=\psi_0(x)=e_0(x)$, and ${\mc D}$ is stable under the action of $c$ and $c^\dagger$. Also, $f(x)$ is strictly increasing.
The physical realization of the operators $c = (1/\sqrt{2}) (x+ i d/dx)$ and its adjoint $c^\dagger$ is well-known, of course, and they satisfy the Weyl-Heisenberg algebra.
\subsubsection{Deforming the harmonic oscillator}
In order to implement a non self-adjoint deformation of the harmonic oscillator, according to
Section \ref{DGHA}, we have to find a positive, bounded, with a bounded inverse, multiplication operator $S(x)$. As the variable $x \in (-\infty,\infty)$, a function like $S(x) =2+ \tanh(x)$ satisfy all necessary requirements of Section \ref{DGHA}. This function increases, monotonically, from $1$ ($ t \to -\infty$) up to $3$ ($t \to \infty$).
Using $S(x)$ and its inverse, $S^{-1}(x)$, the harmonic oscillator can be deformed in the following way:
$a = S(x) \, c \, S^{-1}(x)$, $b = S(x) \, c^\dagger \, S^{-1}(x)$, $\varphi_0(x) = S(x) e_0(x)$ and $\psi_0(x) =
S^{-1}(x) e_0(x)$. The new Hamiltonian takes the form:
\begin{equation}
\label{hhod}
h = S(x) \, H_0 \, S^{-1}(x) = -\frac{\mathrm{d}^2}{\mathrm{d} x^2}+ 2 (1-\tanh(x)) \,\frac{\mathrm{d}}{\mathrm{d} x} - 2 (1-\tanh(x)) + \frac{x^2}{2} \, ,
\end{equation}
which is clearly non self-adjoint. This system has an effective potential $V_{eff}(x) =\frac{x^2}{2} - 2 (1-\tanh(x)) $, shown in Fig \ref{potentialho}, that is non symmetric and is slightly displaced from the origin. There is also a term that is a function of $x$ multiplied by the derivative $\mathrm{d} / \mathrm{d} x$, which seems to appear any time we use a function of $x$ to deform the GHA.
\begin{figure}
\begin{center}
\includegraphics[width=3in]{potential_HO.pdf}
\caption{Effective potential of a deformed non self-adjoint harmonic oscillator, with $S(x) = 2 + \tanh(x)$. }
\label{potentialho}
\end{center}
\end{figure}
\section{Relations with nonlinear pseudo-bosons}\label{sectIV}
\subsection{${\mc D}$-pseudo bosons}
Let $A$ and $B$ be pseudo-bosonic operators in the sense of \cite{baginbagbook}, i.e., $[A,B]= 1 \!\! 1$ and $B$ is supposed to be not equal to $A^{\dag}$, and let ${\mc D}$ be the dense domain left stable by these operators and by their adjoints. In this situation the assumptions in Section \ref{sectII_1} are satisfied if we take $h=BA$ and $f(x)=x+1$, as for the harmonic oscillator. The sets of eigenvectors ${\cal F}_\varphi$ and ${\cal F}_\psi$ can be biorthogonal (Riesz) bases, or ${\mc D}$-quasi bases, see Appendix, depending on the explicit details of the original pair of operators $(A,B)$, as widely discussed in \cite{baginbagbook} and in references therein.
\subsection{Examples related to ${\mc D}$-pseudo bosons}
The pseudo-bosonic operators $A$ and $B$ used in the previous example can be used to construct a new class of examples. For that we define new operators
$$
a=A, \quad b=N_0^kB,
$$
where $N_0=BA$ and $k$ is a fixed positive integer, $k=1,2,3,\ldots$. We further define $N=ba=N_0^{b+1}$ and $h=N=N_0^{k+1}$. It is possible to check that, putting $f(h)=[a,b]+h$, we have $f(h)=(N_0+1 \!\! 1)^{k+1}$, for all fixed $k$. Of course, since the eigenvalues of $N_0$ are just the natural numbers, including zero, $f(h)$ is positive and increasing. The equalities in (\ref{21}) can be further checked explicitly, as a consequence of the following equalities:
$$
AN_0^k=(N_0+1 \!\! 1)^kA,\qquad BN_0^k=(N_0-1 \!\! 1)^kA, \qquad N_0^kB=B(N_0+1 \!\! 1)^k,
$$
for all $k=0,1,2,\ldots$.
Few years ago, \cite{fbnlpbs}, the concept of PBs was generalized to consider quantum systems in which the eigenvalues of the Hamiltonian (and its adjoint) do not depend linearly on the quantum number labeling the eigenstates. Few examples were discussed in \cite{fbnlpbs}, and in other and more recent papers. What we will show now is that there is a strong connection between these nonlinear pseudo-bosons (NLPBs) and the DGHA discussed in Section \ref{sectII}. To show that, we briefly recall how NLPBs are constructed.
Let us consider a strictly increasing sequence $\{\epsilon_n\}$: $0=\epsilon_0<\epsilon_1<\cdots<\epsilon_n<\cdots$. Further, let us consider two
operators $A$ and $B$ on $\mc H$, and let us suppose that there exists a set ${\mc D}\subset\mc H$ which is dense in $\mc H$, and which is stable under the action of $A, B$ and their adjoints.
\begin{defn}\label{def2}
We will say that the triple $(A,B,\{\epsilon_n\})$ is a family of ${\mc D}$-non linear pseudo-bosons (${\mc D}$-NLPBs) if the following properties hold:
\begin{itemize}
\item {\bf p1.} a non zero vector $\Phi_0$ exists in ${\mc D}$ such that $A\,\Phi_0=0$;
\item {\bf p2.} a non zero vector $\eta_0$ exists in ${\mc D}$ such that $B^\dagger\,\eta_0=0$;
\item {\bf { p3}.} Calling
\begin{equation} \Phi_n:=\frac{1}{\sqrt{\epsilon_n!}}\,B^n\,\Phi_0,\qquad \eta_n:=\frac{1}{\sqrt{\epsilon_n!}}\,{A^\dagger}^n\,\eta_0, \label{55} \end{equation} we
have, for all $n\geq0$, \begin{equation} A\,\Phi_n=\sqrt{\epsilon_n}\,\Phi_{n-1},\qquad B^\dagger\eta_n=\sqrt{\epsilon_n}\,\eta_{n-1}. \label{56}\end{equation}
\item {\bf { p4}.} The set ${\cal F}_\Phi=\{\Phi_n,\,n\geq0\}$ is a basis for $\mc H$.
\end{itemize}
\end{defn}
Of course, since ${\mc D}$ is stable under the action of $B$ and $A^\dagger$, it follows that $\Phi_n, \eta_n\in {\mc D}$, for all $n\geq0$. Notice that ${\mc D}$-PBs are recovered by fixing $\epsilon_n=n$. Notice also that the set ${\cal F}_\eta=\{\eta_n,\,n\geq0\}$ is automatically a basis for $\mc H$ as well. This follows from the fact that, calling $M=BA$, we have
$M\Phi_n=\epsilon_n\Phi_n$ and $M^\dagger\eta_n=\epsilon_n\eta_n$. Therefore, choosing the normalization of $\eta_0$ and $\Phi_0$ in such a way
$\left<\eta_0,\Phi_0\right>=1$, ${\cal F}_\eta$ is biorthogonal to the basis ${\cal F}_\Phi$. Then, it is possible to check that ${\cal F}_\eta$ is the unique
basis which is biorthogonal to ${\cal F}_\Phi$. We refer to \cite{fbnlpbs} for more details.
\vspace{2mm}
To connect NLPBs to DGHA we first observe that, if we consider a DGHA with $\epsilon_0=0$, this automatically gives rise to a family of ${\mc D}$-NLPBs. For that it is sufficient to identify $(A,B,\Phi_0,\eta_0)$ in Definition \ref{def2} with the quantities $(a,b,\varphi_0,\psi_0)$ introduced in Section \ref{sectII}, respectively. In fact, with this identification, conditions {\bf p1}, {\bf p2} and {\bf p3} are surely satisfied. Moreover, ${\cal F}_\Phi$ is a basis if ${\cal F}_\varphi$ is a basis.
\vspace{2mm}
It is also possible to check that the opposite holds, at least under some further minor assumption: for that we start with a family of ${\mc D}$-NLPBs, and we identify $(a,b,h,\varphi_0,\psi_0)$ with $(A,B,BA,\Phi_0,\eta_0)$. Moreover, we identify also $f(h)$ with $AB$. Notice that $f(h)=AB=[A,B]+h$ so that (\ref{25}) is automatically satisfied, at least if $[A,B]$ can be written in terms of $h$ and the resulting $f(x)$ is increasing. This becomes, in the case of ${\mc D}$-PBs, $f(h)=h+1 \!\! 1$, as we have already found before. In the general case, it is easy to see that $\left<\varphi_n,[A,B]\varphi_n\right>=(\epsilon_{n+1}-\epsilon_n)\|\varphi_n\|^2$, for all $n$. This, however, does not imply that $\left<f,[A,B]f\right>$ is automatically positive, since ${\cal F}_\varphi$ is not an o.n. basis. However it is yet a strong indication that $[A,B]$ is positive. This can be explicitly checked at least on those $f\in{\mc D}$ for which each $\left<f,\varphi_n\right>\left<\psi_n,f\right>$ is non negative, at least if $\inf_n(\epsilon_{n+1}-\epsilon_n)>0$. In fact, in this case, we have
$$
\left<f,[A,B]f\right>=\sum_n(\epsilon_{n+1}-\epsilon_n)\left<f,\varphi_n\right>\left<\psi_n,f\right>\geq \inf_n(\epsilon_{n+1}-\epsilon_n)\sum_n\left<f,\varphi_n\right>\left<\psi_n,f\right>=$$
$$=\inf_n(\epsilon_{n+1}-\epsilon_n)\|f\|^2>0,
$$
using the fact that ${\cal F}_\varphi$ and ${\cal F}_\psi$ are biorthogonal (or ${\mc D}$-quasi) bases. Finally notice that the commutation rules for DGHA in (\ref{21}) are trivially satisfied with our choices: $hb=(ba)b$, and $bf(h)=b(ab)$. Also, $ah=a(ba)$, while $f(h)a=(ab)a$, and we see that, in fact, (\ref{21}) are satisfied.
\vspace{2mm}
\subsection{Quons}
In \cite{fbnlpbs} it is discussed how quons are connected with ${\mc D}$-NLPBs. Therefore it is not a surprise that quons are connected to DGHA. Let us first consider {\em ordinary quons}, i.e. operators $c$ and $c^\dagger$ obeying the following commutation rule: $c\,c^\dagger-q\,c^\dagger\,c=1 \!\! 1$, where $q\in[-1,1]$. Of course, $q=-1$ gives back CAR, while if $q=1$ we recover CCR. In general, if we introduce $a=c$, $b=c^\dagger$ and $h=c^\dagger c$, it is easy to check that they give rise to a DGHA with $f(x)=qx+1$, which is increasing if $q\in]0,1]$. Therefore, under this limitation, we recover the algebraic structure discussed in Section \ref{sectI}.
The same conclusion can be found if we consider a deformed version of quons, see \cite{bagquons2}. In this case we have two operators, $a$ and $b$, with $b\neq a$, satisfying $a\,b-q\,b\,a=1 \!\! 1$, where $q\in[-1,1]$, and we define $h=ba$. Of course these commutation rules should be defined on a dense set, possibly stable under the action of the operators involved in the game. Once again, also in this case, it is possible to show that these operators give rise to a DGHA with the same $f(x)$ as for the ordinary quons.
\section{Conclusions}\label{sectconcl}
We have shown how GHA can be deformed using ideas borrowed from the theory of pseudo-bosons, and that, in this way, biorthogonal sets of eigenvectors of the related, non self-adjoint operators, can be explicitly constructed. This strategy has been applied to several examples, and relations with NLPBs and quons have also been described. We plan to consider more applications and construct new quantum solvable models adopting our ideas. { In particular, it will be interesting to see what our strategy can give when taking the systems in \cite{quons} as starting points.}
\section*{Acknowledgements}
FB acknowledges support from the GNFM of Indam and from the University of Palermo. JPG acknowledges the CBPF for financial support, and EMFC acknowledges the Brazilian scientific agencies CNPq and FAPERJ for financial support.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 3,128 |
Christian Alsted (født 18. november 1961 i Horsens er biskop i Metodistkirken for Norden og Baltikum.
Alsted er født i Horsens i 1961 og blev student fra Horsens Gymnasium i 1980. Han er uddannet teolog på metodistkirkens nordiske teologiske seminarium i Gøteborg (Sverige) i 1984 og har en Doctor of Ministry-grad fra Asbury Theological Seminary, Kentucky (USA) fra 2002.
Han var præst i metodistkirken i Esbjerg og Varde fra 1984 til 1989, og fra 1989 til 2009 var han præst i Jerusalemskirken i København. Alsted blev valgt til biskop for metodistkirken i Danmark, Sverige, Norge, Finland, Estland, Letland og Litauen i februar 2009 og startede som biskop 1. maj samme år. Han blev valgt for en periode på otte år og genvalgt i oktober 2016.
Bispekontoret er i København ved siden af Jerusalemskirken. Alsted har som biskop for syv lande omkring 180 rejsedage om året. Der er omkring 24.000 metodister i stiftet, heraf omkring halvdelen i Norge.
Alsted har haft en række tillidsposter i metodistkirken, blandt andet formand for Metodistkirkens Historiske Selskab og Landsarkiv, for Copenhagen Gospel Festival og for Metodistkirkens Salmebogskommission. Internationalt har han været formand for verdensrådet for koordinering af United Methodist Churchs forskellige virksomheder og for European Methodist Council, som er en paraplyorganisation for tolv metodistkirker i Europa.
Christian Alsted har skrevet Worship Change to Reach Non-Christians in the Traditional Danish Evangelical Free Church. Han var ansvarshavende redaktør for Salmer & Sange for Metodistkirken i Danmark (Salmebog 2006) og er medforfatter til flere bøger om kirkelige emner.
Christian Alsted er gift med Elisabeth Flinck, og de har tre voksne børn og svigerbørn og børnebørn.
Referencer
Metodistiske præster
Biskopper fra Danmark | {
"redpajama_set_name": "RedPajamaWikipedia"
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{"url":"http:\/\/gmatclub.com\/forum\/i-think-570-is-inlove-with-me-153013.html?sort_by_oldest=true","text":"Find all School-related info fast with the new School-Specific MBA Forum\n\n It is currently 27 Oct 2016, 18:38\n\nGMAT Club Daily Prep\n\nThank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.\n\nCustomized\nfor You\n\nwe will pick new questions that match your level based on your Timer History\n\nTrack\n\nevery week, we\u2019ll send you an estimated GMAT score based on your performance\n\nPractice\nPays\n\nwe will pick new questions that match your level based on your Timer History\n\nEvents & Promotions\n\nEvents & Promotions in June\nOpen Detailed Calendar\n\nI think 570 is inlove with me.\n\nAuthor Message\nTAGS:\n\nHide Tags\n\nIntern\nJoined: 24 Feb 2012\nPosts: 36\nGMAT 1: 680 Q43 V40\nWE: Law (Consulting)\nFollowers: 2\n\nKudos [?]: 14 [4] , given: 116\n\nI think 570 is inlove with me.\u00a0[#permalink]\n\nShow Tags\n\n18 May 2013, 12:57\n4\nKUDOS\nI took the GMAT Prep Exam 1 today, and for the 3rd time running, I scored a 570.\n\nSituation: My exam date is on the 24th of this month i.e. this coming Friday. Wonderful. Time to faint.\n\nTo be honest though, I'm not feeling tense oddly enough (with a 570 score, I guess it's best not to feel anything for a while), and after a 10 minute introspection (half of which was spent devouring a red velvet cupcake), I decided to write this post as an antithesis to the usual 700+ score debriefs that drive this forum and though my awe-inspiring mistakes, ensure that people who are starting out with prep steer clear of them. You've probably heard all them all and most of the points are intuitive but we just don't learn do we!!!!\n\nPlease step away from the door:\n\n1. Dragging your prep on for way to long - I've been prepping on-off for about a year now. Ok that's a lie, I've been on-off for about a year and a half. And every, single time I think about it, I want to go back in time and kick my silly little ass for not taking the exam seriously from Day 1. Once you decide to head down this path, that's it. Give it your all for the next 2-3 months. If you procrastinate like I did, you'll end up giving up on your life, your free time, and your piece of mind until you take the blasted thing. I could have worked on extracurricular activities (something that's still a blank on my sheet), slept more, read more, or just do anything but try to pretend to study!\n\n2. Take a prep exam as soon as you're done reviewing the basics - I'm so used to exams that test you on rote learning. Learn your numbers and dates and then you vomit them out on exam day. But not the GMAT. Noooo. While I knew what to expect, I did not expect the test to be so ...harsh! So you should know that the impending 24th is actually the second time I've booked the exam. My first date was actually way back in August of last year but I had to skip it because of work commitments. No loss there though because back then, I took my first GMAT prep test a whole 2 days before the exam. Yep. 2 days. And how you know why I call myself 'iissosmarts'. And that's the first time I received my first 570. I was\/am an avid time-logger though. Every time I practiced any problems, I always kept track of my time. But I learnt that that tracking time for a set of 10 questions is no way to prepare for a 75 minute session.\n\n3. Don't listen to them. Listen to me - book the exam only when you're prepared. I'm warning you! - Almost every post I've come across says that you should go ahead and just book a date - it's your self-imposed time bomb and will push you towards making sure that you stay the course on your prep, and while that makes total sense, it obviously hasn't worked for me and I'm now $500 down the hole. The past 2 months have been crazy for me in terms of work and I thought that by pushing myself to set a date, I would in turn push myself to stick to my schedule. Can you say EPIC FAIL?! I haven't even taken a IR mock test yet!!!!! 4. Be a little wary of the posts that are on here - Especially when you're starting out, don't start hitting up the 700+ type-questions-list thread like I did. It intimidated me and added a lot of unnecessary stress. Without even solidifying my concepts, I tried to solve some of the tough ones and I just got more confused, especially when a few were written in something resembling hieroglyphs. These are currently the only 4 points I can think of (it is 1.11am here). Hopefully I don't have to add anymore. More than anything, this is a post for me - to motivate myself because I am going to continue. Given how badly I've behaved with my prep, I know I can do better, much better. But I'm not going to cancel my exam date. If nothing else, I'll at least familiarize myself with the test environment. So I'm going to head to the exam center, sign in, and do the best I can. I might see Mr. 570 again for the fourth time, but I don't care because I'll be seeing his sorry ass for the last time!! MBA Blogger Joined: 13 Apr 2013 Posts: 104 Concentration: General Management, Nonprofit Schools: Cambridge'16 (S) GMAT 1: 510 Q25 V36 GMAT 2: 510 Q29 V31 GMAT 3: 590 Q32 V38 WE: Research (Non-Profit and Government) Followers: 3 Kudos [?]: 30 [1] , given: 9 Re: I think 570 is inlove with me. [#permalink] Show Tags 19 May 2013, 07:05 1 This post received KUDOS Good luck with the exam. Thanks very much for the post. Found it encouraging as I also got myself psyched out with the exam by focusing too much on 700 stuff and not focusing enough on my own game. Posted from my mobile device Intern Joined: 24 Feb 2012 Posts: 36 GMAT 1: 680 Q43 V40 WE: Law (Consulting) Followers: 2 Kudos [?]: 14 [0], given: 116 Re: I think 570 is inlove with me. [#permalink] Show Tags 19 May 2013, 12:27 RESET Day 1: May 19, 2013. Verbal is killing me at the moment. All this while my prep has been too quant-focused and I've probably spent about less than 1\/5th of my study time on verbal. Needless to say, the lack of verbal prep is now coming back to haunt me. The funny thing is I was great in CR! In the beginning, I was hitting an accuracy rate of 87% but then hubris set in and I let my CR prep slide. Most of the verbal questions I answered incorrectly at yesterday's Prep Test were CR. Motherf%#^&@. Lesson learnt: Don't go on too long without practicing a particular set of questions, no matter how good you think you already are in that area. What I did today 1. Quant - Because quant is still something I need to pick up speed on, I attempted the first 30 PS questions and the first 20 DS from OG13. I've already solved all the quant problems from OG12 and while OG13 is mostly repetition, doing the sums all over again has helped seer some of the basic math principles into my head. One thing I did differently this time, I made sure I went through all the 50 questions I just worked on on the GMAT club forum. Irrespective of whether I got them right or not (happy to report I got only a total of 5 wrong; yeah I know they're the 'easy' ones but at this point I'm going to savour every little progress!). I've downloaded the GMAT Toolkit App on my ipad (a fantastic app btw!) and one of the great things about it is that the 'OG Tracker' contains forum links to each of the OG quant questions (all added by bunuel). So it was easy for me to navigate directly to the forum discussion for each OG question. If you don't have the app, just Google the question up and the first links are usually the Gmatclub's links. Anyway this time consuming exercise has already opened my eyes to a few new ways of solving specific problems. Even if I answered a question correctly and it was probably one of the easiest questions in the book, I made it a point to read up on the solutions listed on the forum. I had set aside a time of 2 hours to solve all 50 questions and review the answers, but I ended up spending 3.5 hours instead. Well spent. Another site which I refer to for OG problems is GMAT quantum. I don't think I've ever heard anyone mention this site before. While some of the videos may seem lengthy, Mahendra Dabral does more than an excellent job of explaining OG problems. 2. CR - I've been using the MGMAT books for prep and I love them. Their quant books are excellent for math novices like me. I think their CR book is fine too but I have decided to give Power Score CR a try. I know it's not advisable to keep switching between books and methods, but I've just heard so many good things about this book that I have to at least glance at it. I've only finished reading till the 2nd chapter and I can already see that this book does appear to be slightly better than MGMAT's. I wish I had picked this up in the beginning but I was put off by how thick the book was . That being said, because I've been practicing the MGMAT technique, I'm going to try and stick to it, while maybe interspersing it with any Power Score tips that might compliment it. After completing chapter 2 and the practice questions after it, I attempted 10 questions from OG (medium level). 3 out of 10 wrong But to be fair, I worked on the 10 questions after coming back from a heavy buffet dinner with my sister where I did drink a couple of glasses of wine That's another thing I've learnt - not to put my life on hold because of this. All this while, I haven't been hanging out with my friends, going out, or even allowing myself to read my plethora of books because I've been trying so hard to not let anything make me lose my focus. But I think depriving myself all this while only made it worse because I kept spending half my study time thinking about how nice the weather was outside and how I'd much rather be doing a, b, c, or d instead of this!So if you want to do a, b, c, or d, go and do it. Just as long as you have the discipline to come back and continue where you left off. It's only been a day for me, but living this new 'philosophy' has already made me a less snappy person and it has helped me plan my time more. Intern Joined: 24 Feb 2012 Posts: 36 GMAT 1: 680 Q43 V40 WE: Law (Consulting) Followers: 2 Kudos [?]: 14 [0], given: 116 Re: I think 570 is inlove with me. [#permalink] Show Tags 21 May 2013, 11:12 Day 3: May 21, 2013. Now work is killing me. I keep hoping I can sneak in some time at work to study but to no avail. Trying to put in a minimum of 2-3 hours of study every weekday but it's tough. Lesson learnt: Not rocket science, but when possible, take time off from work to completely focus. I know people who haven't done so, and still manage to get a good score, but that means their prep has been extended over 6-7 months. One guy who posted a 700+ debrief (can't remember the exact post) said that while he was on holiday in a foreign land (Bali?), he studied for hours every day in a local coffee shop. Sounds crazy...or is it really? A 2-week solid vs. 6 months of prep. Going to see if I can get at least a week off before D-Day. What I did today\/yesterday 1. Quant - 25 PS - 10 'easy' range, 10 'medium' and 5 'hard'. The classification is based on what's written in the GMAT Toolkit App. Realized that not all of the quant questions have forum links built into the App. Bleh. 2. CR - Done with the 4th chapter of Power Score CR and I can now say that this is much better than MGMAT's CR book. I especially love the 'summaries' that are provided at the end of each chapter - really helps you drill down to the important points among all that verbose. Not that I've seen any improvement in my CR score (still at a ratio of 3:7) - but I blame that on my mixing of methodologies. Going to try and stick to Power Score. 3. SC - Re-reading MGMAT's SC book. Great book. One must read it at least twice. Too tired to do any practice questions. Intern Joined: 24 Feb 2012 Posts: 36 GMAT 1: 680 Q43 V40 WE: Law (Consulting) Followers: 2 Kudos [?]: 14 [0], given: 116 Re: I think 570 is inlove with me. [#permalink] Show Tags 24 May 2013, 03:44 TEST DAY!! Day 6: May 24, 2013. As you probably know, I went ahead and took the exam because I figured I had nothing to lose. Since B schools only look at your best score anyway, I thought I might as well at least acquaint myself with the whole test-day' experience. Well what do y'know - I took the exam today, and my score was, guess what, 570!!! Q36,V32 (Does it even matter if I add the break up? A sh***y score, is a sh***y score). Not that it was a complete shocker, but I thought that some improvement might be had, given the 3-4 days of study I put in between today and my mock test. New found respect for the GMAT prep software though. It does not lie! Test experience: 1. Test Center I was feeling fairly confident, that I would be sitting for the exam a second time, because of which my anxiety levels were at a minimum (some of amount of stress was there, but mostly because I was afraid I wouldn't reach the test center on time given Mumbai's notorious traffic). There are now two test centers in Mumbai. The one I went to, the Churchgate branch, is only a month old. The office is surprisingly small, and seats only 4 people at a time. The administrators were very helpful though (one of the fellas said that 570 was a \"good score\" given how I prepped for it, yeah right!) . I asked them when I should book my next appointment so as to stay clear of the crowds and they told me to avoid weekends. There was a guy in my batch who would randomly start muttering loudly to himself and normally I would turn around and tell him to pip down, but he sounded like he was in pain! Almost every second question would elicit a 'tut, tut' or a sigh from his mouth. Lol. Another guy behind me let out a \"Yes!\" after the end of his session. He must have surpassed the 700 mark. If you're Indian, you know you're not going to say a \"Yesss!\" unless you've crossed the 700 mark! 2. The Test. AWA topic was very straight forward, but me being me, I ended up adding to many points to my essay. I guess I got a little too excited but in the end, I couldn't completely review it within the allotted time. IR was surprisingly not as tough as I expected. There were a few that stumped be but my main problem was time - I actually enjoyed answering some of the questions until I looked up and realized that had 59 seconds left and I was still on question number 9! 'Enjoyment' flew out of the window and I just selected random answers for the rest of the segment without even bothering to read the question. Lesson learnt Do not allow myself to be completely submerged in a question! No matter how interesting! I thought quant started off well (yeah, thought wrong!!) until I hit a plateau in the middle. The thing about a few of the questions was that I hadn't seen these type of questions before - I have gone through the OG12 and OG Quant, including a few questions available online, but I just didn't know how to go about solving them. I was fairly confident about quant basics before but it looks like I'm going to have to dig deeper. Verbal. Damn you verbal. With quant questions, you sort of know when you're right or just guessing, but with verbal, you can never tell, which makes analyzing how you did on verbal so much harder. I know that if I'm going to have to pull my score up considerably, I have got to start concentrating on verbal. Going to aim around mid\/late June as my next exam date. I have to think of a way to excuse myself from work for at least a week. I'll invent an ailing grandmother or some impending marriage in the family, I don't care but I will get the time off for it!! Going to unwind for the rest of the day. Will watch a movie and then get cracking on verbal. While I'm not terribly annoyed at my score, I am terribly annoyed by the fact that I can't immediately start working on my applications. The more I put this off, the less time I have for everything else. Sucks. To-do list for the weekend: 1. Finish off Power Score CS. At this point, I really wouldn't be spending too much time on theory, but I can't relax when I know I haven't finished a book. 2. Continue re-reading MGMAT SC; 3. Go through the GMAT Club Math Book v3 (I think that's the latest version). I've never gone though before, but after glancing through after I got back, it appears to cover a few shortcuts\/formula not mentioned in the MGMAT books. Here's to gearing up for round 2!! VP Status: Final Lap Up!!! Affiliations: NYK Line Joined: 21 Sep 2012 Posts: 1096 Location: India GMAT 1: 410 Q35 V11 GMAT 2: 530 Q44 V20 GMAT 3: 630 Q45 V31 GPA: 3.84 WE: Engineering (Transportation) Followers: 38 Kudos [?]: 510 [0], given: 70 Re: I think 570 is inlove with me. [#permalink] Show Tags 25 May 2013, 21:21 I can understand your state of mind, but you should refrain from taking exam, If you are not scoring close to what you intend to score on actual test day. Moreover, GMAT prep is an actual score predictor, no matter how many times you take the exam, unless youmemorize the option choice. Consider kudos if my post helps!!! Archit Moderator Joined: 10 May 2010 Posts: 825 Followers: 25 Kudos [?]: 394 [1] , given: 192 Re: I think 570 is inlove with me. [#permalink] Show Tags 26 May 2013, 03:19 1 This post received KUDOS 570 is in love with you because you apparently don't love the GMAT . Just Kidding. Don't read too many debriefs and try to incorporate every tip out there. Just read the debrief that makes sense to you and stick with the approach. If you haven't read , go ahead and read these epic debriefs: 1.) The debrief on how to structure your GMAT Prep if you want a time bound schedule http:\/\/www.urch.com\/forums\/just-finishe ... post152529 2.) If you are looking for a more flexible schedule this might be a good approach http:\/\/www.beatthegmat.com\/770-50q-46v- ... 95399.html 3.) The best debrief for miscellaneous GMAT preparation tips. gmatclubbing-administered-770-50q-45v-86239.html _________________ The question is not can you rise up to iconic! The real question is will you ? Manager Status: Never ever give up on yourself.Period. Joined: 23 Aug 2012 Posts: 152 Location: India Concentration: Finance, Human Resources GMAT 1: 570 Q47 V21 GMAT 2: 690 Q50 V33 GPA: 3.5 WE: Information Technology (Investment Banking) Followers: 10 Kudos [?]: 274 [1] , given: 35 Re: I think 570 is inlove with me. [#permalink] Show Tags 26 May 2013, 05:26 1 This post received KUDOS dude, don't get discouraged, fight back. I know what it likes to be on 570, I have been through it. You're not alone. We'll help you make a come back, just that you've to trust yourself. The feeling you'll get after scoring 650+ will be out of this world. don't give up. Posted from my mobile device _________________ Don't give up on yourself ever. Period. Beat it, no one wants to be defeated (My journey from 570 to 690) : http:\/\/gmatclub.com\/forum\/beat-it-no-one-wants-to-be-defeated-journey-570-to-149968.html Intern Joined: 24 Feb 2012 Posts: 36 GMAT 1: 680 Q43 V40 WE: Law (Consulting) Followers: 2 Kudos [?]: 14 [0], given: 116 Re: I think 570 is inlove with me. [#permalink] Show Tags 26 May 2013, 19:28 AbhiJ wrote: 570 is in love with you because you apparently don't love the GMAT . Just Kidding. Don't read too many debriefs and try to incorporate every tip out there. Just read the debrief that makes sense to you and stick with the approach. If you haven't read , go ahead and read these epic debriefs: 1.) The debrief on how to structure your GMAT Prep if you want a time bound schedule http:\/\/www.urch.com\/forums\/just-finishe ... post152529 2.) If you are looking for a more flexible schedule this might be a good approach http:\/\/www.beatthegmat.com\/770-50q-46v- ... 95399.html 3.) The best debrief for miscellaneous GMAT preparation tips. gmatclubbing-administered-770-50q-45v-86239.html Hey thanks for links! I've come across the first link before but not the others, and they're equally helpful! Thank you! Intern Joined: 24 Feb 2012 Posts: 36 GMAT 1: 680 Q43 V40 WE: Law (Consulting) Followers: 2 Kudos [?]: 14 [0], given: 116 Re: I think 570 is inlove with me. [#permalink] Show Tags 04 Jun 2013, 06:27 Day 17: June 4, 2013. Unfortunately, I'm not done with MGMAT SC. Was hoping to complete it yesterday but with work being the way it is, I haven't been able to put in that many hours during the week. But let's not dwell on that! The GREAT news: 1. CR has improved considerably. My accuracy rate is now getting close to 80% (and I'm talking about the harder sets on OG); 2. Starting Wednesday, I'll be off for about 3 weeks!! Albeit I will have to head to the office about once a week but it's better than nothing. Going to try and stretch it to a month. Btw, I received an e-mail from walker (the guys behind GMAT ToolKit). Starting today at 7am CST, the new GMAT Toolkit 2 will be available at a reduced price of only$0.99 (\\$24.99 value) up until noon CST. Get it if you can. I cannot rave enough about the old iteration and I'm sure the new version will be amazing as well!\nManager\nStatus: SLOGGING : My son says,This time Papa u will have to make it : Innocence is BLISS\nJoined: 16 Jan 2012\nPosts: 210\nLocation: India\nWE: Sales (Energy and Utilities)\nFollowers: 3\n\nKudos [?]: 44 [0], given: 30\n\nRe: I think 570 is inlove with me.\u00a0[#permalink]\n\nShow Tags\n\n04 Jun 2013, 10:03\nhi buddy......u r a rocker...n thats not an adulation but i mean it....the deep introspection , internalisation of the rights n wrongs , coupled with such a happy go attitude......the smooth flow of ur text makes it such a good read....the spirit to take the bit bxn the teeth and that too with such ease.......its really envious.\n\nmark my words man...u wll surely get your life back n that too the way u want\n\nplz dont change urself....this sheer attitude of urs will take u places rather than a poor 3 digit score that stares at u at the end of 4 hrs.\nIntern\nJoined: 07 Oct 2012\nPosts: 23\nLocation: India\nConcentration: General Management, Finance\nGMAT 1: 640 Q45 V33\nWE: Engineering (Other)\nFollowers: 0\n\nKudos [?]: 11 [1] , given: 7\n\nRe: I think 570 is inlove with me.\u00a0[#permalink]\n\nShow Tags\n\n04 Jun 2013, 10:42\n1\nKUDOS\nHi Dude,\nI felt some of the same things when i too got 570, but slowly & surely, you will get a hang of what gmat is about. (currently am scoring above 660 in gmat prep). what i found is -\nIt is mostly about clearing your concepts. Any source you find to be good-math websites,mgmat books, high school books, whatever.\nYou should not jump into gmat problems. You need to get the quant concepts in order for eg, & then practice at the 500 level, then at 600 level, there you might also get stuck. There are some skills that may take time.\nYou will then need extra practice & other strategies to manage time\/panic\/estimation etc & lots of practice from official sources (read OGs) before you break that 650 barrier.\nWhen you reach there, you will find the window for that 700 figure, but not before.\nIts hard, but frankly what makes it that much harder is our full time jobs & full time lives.Getting four weeks off to study for the gmat seems like a wow situation, but it seldom happens.\nAnd for sure, for a 700+ score posted here, there might be hundreds of 500s never posted.\nAll the best !\nManager\nStatus: SLOGGING : My son says,This time Papa u will have to make it : Innocence is BLISS\nJoined: 16 Jan 2012\nPosts: 210\nLocation: India\nWE: Sales (Energy and Utilities)\nFollowers: 3\n\nKudos [?]: 44 [0], given: 30\n\nRe: I think 570 is inlove with me.\u00a0[#permalink]\n\nShow Tags\n\n04 Jun 2013, 11:36\n@iwantmylifeback :just a thought......if u r planning to give quant a shot in the arm....u can get in touch with ian stewart, the most revered quant guru ianstewart@gmail.com.......number theory n inequalities, the 2 most dreaded ones, will seem to be\nIntern\nJoined: 24 Feb 2012\nPosts: 36\nGMAT 1: 680 Q43 V40\nWE: Law (Consulting)\nFollowers: 2\n\nKudos [?]: 14 [0], given: 116\n\nRe: I think 570 is inlove with me.\u00a0[#permalink]\n\nShow Tags\n\n06 Jun 2013, 09:03\nskmskm wrote:\nhi buddy......u r a rocker...n thats not an adulation but i mean it....the deep introspection , internalisation of the rights n wrongs , coupled with such a happy go attitude......the smooth flow of ur text makes it such a good read....the spirit to take the bit bxn the teeth and that too with such ease.......its really envious.\n\nmark my words man...u wll surely get your life back n that too the way u want\n\nplz dont change urself....this sheer attitude of urs will take u places rather than a poor 3 digit score that stares at u at the end of 4 hrs.\n\nLol well thank you for that and you're right, I really shouldn't let a 3 digit score run my life!! That sentence really helped drive home the fact that in the end, it really is just a number But...I am going to see this through!\nRe: I think 570 is inlove with me. \u00a0 [#permalink] 06 Jun 2013, 09:03\nSimilar topics Replies Last post\nSimilar\nTopics:\n17 How the Gmat ripped me a new one. 710 the night before to 570 16 20 Sep 2014, 16:51\n2 570 (prep) to 690 (actual) in 6 weeks - I'll take it! 2 15 Oct 2011, 14:19\n1 I think i will call it a day now........ 11 30 Nov 2010, 11:15\nAfter 3 Months of Preparation, i've got 570!? PLZZ Advice! 15 24 Sep 2008, 05:14\ni think i'm done with the gmat... 4 07 Aug 2008, 21:47\nDisplay posts from previous: Sort by","date":"2016-10-28 01:38:17","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.3452955186367035, \"perplexity\": 4177.655234728988}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2016-44\/segments\/1476988721415.7\/warc\/CC-MAIN-20161020183841-00394-ip-10-171-6-4.ec2.internal.warc.gz\"}"} | null | null |
Tag Archives: Teacher
Worst School Shooting In U.S. History Kill's 22 Children And 7 Teachers
Posted by letsfindthem on December 14, 2012
The shooter – believed to be 24-years-old from New Jersey – is confirmed dead and body still inside the school
A parent of the gunman was found dead at his home in New Jersey
Second gunman now in custody
One entire classroom is unaccounted for, sources said
One teacher said masked gunman started firing out shots from kindergarten classroom shortly after 9.30am
One student said there were bullets whizzing by him in the hallway
At least 100 rounds are believed to have been fired
Female principal and school psychologist believed to have been targeted
12/14/12 NEWTOWN, CT — At least 29 people are believed to be dead – including 22 children – after a gunman believed to be a student's father opened fire today at an elementary school in Connecticut, making it the worst school shooting in U.S. history.
Police were dispatched to Sandy Hook Elementary School in Newtown after they received reports of shots being fired by a gunman in one of the kindergarten classrooms at 9.35am.
There are preliminary reports a student's 24-year-old father was the shooter and that he went from classroom-to-classroom firing at students and teachers with a .223-caliber rifle.
One of his parents was found dead at his New Jersey home around 2pm.
An entire classroom of students is unaccounted for and frantic parents are rushing to the school desperately seeking information on the safety of their children.
The gunman is dead and his body still at the scene. It is unclear if he shot himself or was brought down by an officer.
A second man who was seen running from the school right after the shooting was taken into custody around 1pm after he was found in nearby woods.
Traumatized students were seen being led out of the school crying and holding hands
A woman waits to hear about her sister, a teacher, following the tragic shooting this morning which has shocked the quiet suburban community
Chaotic scenes at the school as police work to secure the area and bodies are carried out of the school
A black Honda believed to belong to the shooter has been cordoned off
CLICK DAILYMAIL.CO.UK FOR FULL ARTICLE & VIDEO
You may also like to read: 'There were screams coming over the intercom… and the children were told to close their eyes': Terrified students recall horrifying scene after gunman opened fire at Connecticut elementary school
Leave a comment Posted in Uncategorized Tagged 2012, Associated Press, Connecticut, CT, Dailymail.co.uk, Elementary school, FBI, Mass Shooting, Murdered Children, New England, New Jersey, Newtown, NJ, Northeast, Sandy Hook Elementary School, School shooting, Shot, Teacher, United States | {
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\section{Introduction}
One of the most interesting manifestations of the principle of
antisymmetrization of two-fermion systems is the existence of two
types of configurations for the Helium atom. They correspond to
singlet and triplet states and are usually denoted as para- and
ortho-Helium states, showing different distributions of energy
levels.
The study of the Helium atom is not a closed subject \cite{Tan}. For
instance, calculations of some energy levels of the confined Helium
atom \cite{Bar} and refinements in the para- and ortho-Helium evaluations
\cite{Dua} have been presented in the literature. In has also been
shown that the ionization properties of the Helium atom are strongly
dependent on the type of configuration \cite{Pla}. The antiprotonic
Helium \cite{Ead}, the system produced by the capture of an
antiproton by a $He^+$ ion, has also extensively been studied.
In this Letter we suggest a new line to study the interesting
properties of para- and ortho-Helium. We shall show that it is
possible to prepare the Helium atom in a metastable superposition of
para and ortho states. The scheme of preparation is in principle
very simple. It is based on the capture by an $He^+$ ion of an
electron, which must be prepared in a superposition of spin states.
As we shall see later, in order to obtain a superposition one must
only demand to the Hamiltonian describing the capture process to be
symmetric. We should stress that this scheme appears quite naturally
during the double ionization of He with intense laser fields. It has
been shown that, for laser intensities below $5 \times 10^{15}$
W/cm$^2$, the relevant path for ionization is a non-sequential
process, in which the two electrons entangled are emitted with the
same direction \cite{walker}. Since the probability for single
ionization is always higher than the double, there is also a
fraction of $He^+$ so the electron capture is also feasible. Note
however that this process involves three particles, therefore the
final superposition of He states is entangled with the surviving
electron. Up to our knowledge, superpositions of ortho and para
states have only been considered in the context of Helium collisions
\cite{JPB} (see section 3).
The proposed states would be interesting in several aspects. From a
fundamental point of view they would provide us with a situation
where the linearity of quantum mechanics has not been explored
before, that in which the antisymmetrization postulate and exchange
effects must be taken into account. Superpositions of different
energy states of an atom have previously been considered in the
literature. An atom can be prepared in a superposition of excited
energy states using an impulsive excitation such that its Fourier
spectrum contains frequency components corresponding to the energy
intervals between ground and excited states. The rate of spontaneous
emission of atoms prepared in this manner can oscillate in time.
These quantum beats differ from the usual exponential decay expected
for atoms in well-defined energy states \cite{Sil}. Quantum beats,
manifested as modulations in the absorption rates, are also present
in the absorption of light by atoms in superposition states
\cite{Wol}. However, in these (energy-type) superpositions the
exclusion principle does not play any role.
From a more practical point of view, it has been discussed the
existence of collisional velocity changes associated with atoms in
superposition states \cite{Mos}. It has also been signaled that
atomic superposition states are sensitive to phase dependent
properties of radiation fields and, consequently, could be employed
as detectors \cite{Kni}. Moreover, we can expect that many
properties of the atom such as absorption rates, ionization energy ,
etc. will differ in normal and superposition states and could be
useful to understand the mechanisms involved in these processes.
Before considering these possibilities we must analyze the stability
of the states of the superposition. In general, as we shall see in
Sect. 3, there are only stationary states in some particular cases,
when the ortho and para states are degenerate or almost degenerate.
In the absence of stationary states we must consider the existence
of metastable states, which could be studied with high resolution
laser spectroscopy techniques. We shall study the superposition of
ortho- and para-1s2s states, showing that it can be prepared as a
long lifetime metastable superposition state. Moreover, in
principle, this state is accessible to interesting experimental
verifications such as the modification of the amplitude of the
quantum beats and the variation of the mean lifetime and of the
fluctuations of the decaying rate. In this context it must be
signaled that the metastable $1s2s$ ortho-Helium state has been Bose
condensed \cite{Sci}.
\section{Preparation of the superposition}
First of all we show how to prepare the superposition. On the one
hand, we must have an $He^+$ ion with the electron in a well-defined
state of the spin component along a given axis, for instance
$|\uparrow >_z$. We take as axis of reference the $z$ one, denoting
by $|\uparrow >_z$ and $|\downarrow >_z$ the two possible states. We
can determine that the electron is in the correct state by a direct
measurement or by preparation. In the last case we should have an
$He^{++}$ ion, which captures an electron in the state $|\uparrow
>_z$.
On the other hand, once prepared the $He^+$ ion with the electron in
the $|\uparrow >_z$ state, we must have an electron in a
superposition state $|\phi >= \alpha |\uparrow >_z + \beta
|\downarrow >_z$, with $|\alpha |^2 +|\beta |^2=1$. This step can be
easily done by preparing the electron in a well-defined state along
other spin axis. For instance, the up and down components of the
spin along the two axes orthogonal to the $z$ one are $(|\uparrow
>_z + |\downarrow >_z)/\sqrt{2}$ and $(|\uparrow >_z -i |\downarrow
>_z)/\sqrt{2}$. An alternative procedure, as discussed above, is to
consider the non-sequential double ionization of He.
Now we can make interact the electron in state $|\phi >$ with the
$He^+$ ion, whose state is described by the ket $|He^+ >$.
If $He^+$ would only interact with an electron in state $|\uparrow
>_z$, capturing it, the map describing the interaction
would be
\begin{equation}
|He^+> |\uparrow >_z \rightarrow |He_{or}>
\end{equation}
We would obtain an ortho-Helium atom because we have assumed that
the other electron was also in the state $|\uparrow >_z$.
On the other hand, if the ion $He^+$ would interact with an electron
in the state $|\downarrow >_z$, the evolution would be
\begin{equation}
|He^+> |\downarrow >_z \rightarrow |He_{pa}>
\end{equation}
In this case we would obtain a para-Helium state.
Finally we move to the most interesting situation, that with the
incident electron in a superposition of spin states. If the capture
of the electron by the ion is a linear process (we discuss this
point later) we have
\begin{equation}
|He^+> |\phi > \rightarrow \alpha |He_{or}> + \beta |He_{pa}>
\end{equation}
which is a superposition of the Helium atom in para and ortho states.
We note that there is no superselection rule preventing the
superpositions considered here. From all the
superselection rules presented so far in the literature the only one
that is related to our proposal is that preventing the existence of
superpositions with different values of $(-1)^{2J}$, representing
$J$ the modulus of the total angular momentum, ${\bf J}={\bf L}+{\bf S}$, with
${\bf L}$ the orbital angular momentum and ${\bf S}$ the spin.
The two states of the superposition must have the same value of
$(-1)^{2J}$. In other words, $2J$ must be in both cases an even or
an odd integer. This is so in our case because the values of $l$ are
$0,1,2...$ and $s=1/2$. Then $2J=2l+1$ that is always odd.
We discuss now the linearity of the capture process. The Hamiltonian
describing the capture must take into account two different types of
interactions: (i)The electromagnetic, spin-orbit, spin-spin, etc.
usual interactions in the atom. (ii)The exchange effects associated
with the antisymmetrization of the wavefunction of two identical
electrons. With respect to (i) it is well-known the linearity of the
associated Hamiltonian. We consider now (ii). The wavefunction of
the complete system (nucleus plus the two electrons) must be
antisymmetrized with respect to the variables of the two electrons,
$\Psi ({\bf x},s_x;{\bf y},s_y;{\bf Z};t)= \psi ({\bf x},s_x;{\bf
y},s_y;{\bf Z};t) - \psi ({\bf y},s_y;{\bf x},s_x;{\bf Z};t)$ where
${\bf x}$ and ${\bf y}$ are the spatial coordinates of the two
identical particles, $s_x$ and $s_y$ refer to the spin components
and ${\bf Z}$ includes all the variables related to the nucleus. It
is simple to show adding the Schr\"{o}dinger equations ruling the
evolution of each $\psi $ that $\Psi$ only obeys a linear
Schr\"odinger's equation when $\hat{H} ({\bf x},s_x;{\bf y},s_y;{\bf
Z};t)=\hat{H} ({\bf y},s_y;{\bf x},s_x;{\bf Z};t)$ ($\hat{H}$ is the
Hamiltonian of the system), that is, when the Hamiltonian is
symmetric. Since all the Hamiltonians used in atomic physics fulfill
this condition we must expect the capture process to be linear.
Finally, we briefly discuss the possibility of actually implementing
the scheme suggested here. We start with a sample of $He$ atoms,
which is illuminated by a laser tuned in the adequate frequency to
induce double ionization. Using electric fields (which do not modify
the spins) we can separate the $He^{++}$ ions from $He$ atoms and
$He^+$ ions. Then a beam of electrons in the $|\uparrow >_z$ state
interacts with the sample of $He^{++}$ ions. The $He^+(|\uparrow
>_z)$ ions produced by the capture of one electron are separated,
using again an electric field, from the $He^{++}$ ions and
$He(|\uparrow >_z, |\uparrow >_z )$ atoms. Finally, a beam of
electrons, for instance in the $|\uparrow >_x$ state, interacts with
the sample of $He^+(|\uparrow >_z)$ ions. The ions that capture
electrons become in a superposition state. Using once more an
electric field we can separate them from the $He^+(|\uparrow >_z)$
ions. The beams of electrons can be obtained from sources producing
them in arbitrary random spin states using Stern-Gerlach devices
with adequate orientations. An interesting alternative is provided
by the process of non-sequential double ionization of He in strong
electromagnetic process. In this case, the initial state is an
entangled electron pair ionized from some neighbor atom. The capture
of one of the electrons by a $He^+$ ion, leads to a entangled state
of the Helium atom with the surviving electron. The properties of
such three particle system will be the subject of a future
investigation.
\section{Stationary states of the superposition}
A fundamental question to be answered about the superposition is its
stability. In the quantum realm an atom can be stable because of the
existence of stationary states. We must look for the stationary
states of the superposition, which would be given by the solutions
of the equation
\begin{equation}
\hat{H} (\alpha |He_{or}[n]> + \beta |He _{pa}[m]>)=E_{[n,m]}
(\alpha |He_{or}[n]> + \beta |He _{pa}[m]>)
\end{equation}
where $[n]$ and $[m]$ represent the two sets of indexes
characterizing the stationary states of both types of
configurations. As $\hat{H} |He_{or}[n]> =E_{[n]} |He_{or}[n]>$
and $\hat{H} |He_{pa}[n]> =E_{[m]} |He_{pa}[m]>$ the stationary states of the superposition must obey the relation
\begin{equation}
E_{[n,m]}=E_{[n]}=E_{[m]}
\label{eq:ene}
\end{equation}
In general, the energies of ortho and para states are different. But
still it can happen that although with different energies, the
states be so close to be considered as almost degenerate. As we
shall discuss later in this section these states will decay by
spontaneous emission. Then their energies will have an uncertainty
determined by their mean lifetimes. In this context the natural
criterion to consider two states as almost degenerate is that the
difference between their energies be smaller than the broadening of
the lines. As the mean lifetimes are of the order $10^{-9}s$, the
uncertainty of the energies are $\delta E \approx 10^{-6}eV$ (note
that this value is of the same order of the actual precision on the
measurement of the energy). To see this point in detail let us
analyze the experimental data. By the matter of concreteness we use
the data of the NIST \cite{nis}. It is simple to see that there are
some states for which the condition in Eq. (\ref{eq:ene}) is
fulfilled up to the broadening of the lines (and the experimental
error). For instance, in the configuration $1s4f$ with terms
$^3F^0$, $J=2$ for the ortho and $^1F^0$, $J=3$ for the para we have
an energy difference between both levels $\Delta E=9 \, 10^{-7} eV$,
which is below $\delta E$ (and the experimental error). Similarly,
we have the configuration $1s5f$ with terms $^3F^0$,$J=2$ and
$^1F^0$,$J=3$, or other terms obeying the relation $\Delta E \leq
\delta E$.
Other degeneracy has already been signaled in the literature for
this system. For $d3$ configurations the terms $2P$ and $2H$ are
degenerate due to the symmetries existent in the problem.
We conclude that there are some energy levels of the ortho- and
para-Helium which can be considered as almost degenerate. However,
these states associated with almost degenerate levels would be of no
interest since they would be unstable because of spontaneous
emission. For instance, in the case of the state $1s4f$ the electron
in the state $4f$ would emit photons decaying consecutively to
states $3d$, $2p$ (final state in the ortho case) or $1s$ (final
state for the para configuration). In the absence of stationary
states of interest the most relevant states of the system would be
the metastable ones. We consider then in next section.
We must signal here that $F$ levels are involved in the only (up to
our knowledge) previous consideration of mixed multiplicity states
(see \cite{JPB} and references therein).
They are states that cannot
be considered as pure singlets or triplets ones but rather as
mixtures of them.
They are related to the F-cascade model, in
which it is assumed that in a collision process the excitation
energy is transferred from resonance ($n^1P$)-levels to mixed
$F$-levels: $He(1^1S)+He(n^1P) \rightarrow He(n^{mix}F)+ He(1^1S)$,
where $n^{mix}F$ represents a superposition of $n^1F$ and $n^3F$
states.
\section{Metastable states}
In the Helium atom the $1s2s$ state is
metastable for both para- (lifetime of $19.7 ms$) and ortho-type
(lifetime around $10^8 s$) configurations. In the first case the
decaying occurs through a two-photon electric dipole transition and
in the second via relativistic and spin-orbit interactions. Consequently, we expect their
superposition also to be a metastable state. We
shall explore its properties.
First of all, we note that the energies of the two configurations are
different, $E(1s2s, or)=E_{or} \neq E_{pa}=E(1s2s, pa)$. Then
in addition to the superposition of the spins we must have a small
indetermination in the energy of the state (initially carried by
the incident electron), giving rise also to a superposition of the
energy states. Both superpositions are compatible because energy and
spin are compatible variables, not affected by complementarity
relations.
Next we consider the lifetime of the state. As usual, the lifetime
of a state is evaluated as the inverse of its decay rate, $\tau = 1/
\Gamma $. If we denote by $|\phi _{or}>=|1s2s, or>$, $|\phi
_{pa}>=|1s2s, pa>$ and $|\psi >=|1s1s, pa>$ (the ground state, at
which both decay) the decay rate is given by
\begin{eqnarray}
\Gamma = |<\psi | \hat{H}_*| \alpha \phi _{or}+ \beta \phi _{pa}>|^2 =|\alpha M_{or} e^{-iE_{or}t/\hbar }+ \alpha M_{pa} e^{-iE_{pa}t/\hbar } |^2 = \nonumber \\
|\alpha |^2 \Gamma _{or} + |\beta |^2 \Gamma _{pa} + 2 Re(\alpha ^* \beta M_{or}^* M_{pa} e^{-i(E_{pa}-E_{or})t/\hbar })
\end{eqnarray}
where $M_i= <\psi | \hat{H}_*|\phi _i>$, $i=or,pa$, are the matrix
elements giving the transition rates, $\Gamma _i =|M_i|^2$. Note
that we use $\hat{H}_*$ instead of $\hat{H}$ to remark that
now relativistic and spin-orbit interactions are included.
$\Gamma $ is the instantaneous value of the decay rate. However, it
is unobservable. We must consider the averaged value
$\overline{\Gamma }$ over the time scale $T$ characteristic of the
observations, $ \overline{\Gamma } =1/T \int _0^T \Gamma dt $.
Assuming by simplicity the coefficients $\alpha $ and
$\beta $ and the matrix elements to be real we have for the interference term
$ \Gamma _{\alpha \beta }/T \int _0^T cos (\omega t) dt = \Gamma _{\alpha
\beta } sin(\omega T)/\omega T$ where $\Gamma _{\alpha \beta }=2\alpha \beta M_{or} M_{pa}$ and
$\omega = (E_{pa}-E_{or})/\hbar $. As $\omega \approx 10^{15}
s^{-1}$ and $|sin(\omega T)| \leq 1$ we have that $sin(\omega
T)/\omega T \approx 0$ for any $T$ much larger than $10^{-15}s$.
Consequently, for any realistic $T$ the interference term can be
neglected in the averaged expressions.
Finally, we can write
\begin{equation}
\tau =\frac{1}{|\alpha |^2 \Gamma _{or} + |\beta |^2 \Gamma _{pa} }
\end{equation}
The lifetime of the superposition state ranges between $\tau
_{or}=1/\Gamma _{or}$ and $\tau _{pa}=1/\Gamma _{pa}$. Varying the
coefficients $\alpha $ and $\beta $ we can obtain all the values of
lifetimes in that range. For any value of the coefficients we have a
metastable state.
Although the interference term has not influence on the (mean)
lifetimes its effects
lead to high frequency oscillations
of the decay rate, in a similar way to the quantum beats present in
energy-type superposition states. To emphasize this point we follow
the approach of Ref. \cite{Sil} expressing the coefficients $\alpha
M_{or}$ as $A_{or}e^{-\Gamma _{or}t}$,..., i. e., taking into
account explicitly the dependence of the coefficients on the decay
rate. Hence, the probability for the transition at time $t$ (when
all the amplitudes are assumed to be real valued) is $A_{or} ^2
e^{-2\Gamma _{or} t} + A_{pa} ^2 e^{-2 \Gamma _{pa}t} + 2 A_{or}
A_{pa} e^{-(\Gamma _{or} + \Gamma _{pa})t} cos (\omega t)$, showing
clearly the existence of oscillations around the mean values. These
oscillations, being their amplitude and frequency of the same order
of magnitude than those associated with energy-type superpositions,
are in principle experimentally observable. Moreover, varying the
parameters $\alpha $ and $\beta $ we could modulate the amplitude of
the oscillations.
Another experimental way to test the existence of the superpositions
would be the measurement of the variations of the decaying rate. We
proceed in the standard way, i. e., by counting the number of decays
in a given time interval of observation. From an experimental point
of view the way of measuring the decays is to count the photons
emitted during that interval. As different atoms emit their
radiation independently this photon source is of chaotic type. As it
is well known \cite{Lou} the distribution of photocounts is of
Poisson type, $P_n (T)=<n>^n e^{-<n>}/n!$, provided that the time of
observation $T$ is much longer than the coherence time of the light
(if not the distribution would be super-Poisson). In the above
relation $<n>$ denotes the mean number of photocounts, which in the
semiclassical approximation (for chaotic light the semiclassical and
fully quantum approaches give the same result \cite{Lou}) is given
by the expression $<n>=\xi \overline{I}T$, with $\xi$ the efficiency
of the detectors and $\overline{I}=\overline{I}(t)$ the
cycle-averaged intensity of the light. The intensity is given by the
number of photons emitted at $t$. By definition, this number is
proportional to the number of atoms that decay at $t$, which is
$\Gamma $, and $I(t) \sim \Gamma (t)$. Finally, we must average over
the time of observation, which must be much longer than the
coherence time of the light. Denoting by $\overline{\Gamma}$ this
average over $T$ (which according to our previous results is
time-independent for any realistic choice of $T$) the mean number of
photocounts is $<n>=\overline{\Gamma}T$, where the efficiency factor
has been absorbed in $\overline{\Gamma}$ by simplicity in the
notation. The Poisson distribution can be expressed as
\begin{equation}
P_n(T)= \frac{(\overline{\Gamma } T)^n}{n!} exp(-\overline{\Gamma
}T)
\label{eq:Poi}
\end{equation}
Using the relation $\overline{\Gamma }=|\alpha |^2 \Gamma _{or} +
|\beta |^2 \Gamma _{pa}$ and the well-known expression $(x+y)^n =
\sum x^{n_x} y^{n_y}n!/n_x ! n_y !$, where the summation extends to
all the non-negative integers obeying the relation $n_x + n_y = n$,
we have,
\begin{equation}
P_n (T)= \sum _{n_{or}+ n_{pa}=n} {\cal P}_{n_{or}} (T) {\cal P}_{n_{pa}} (T)
\label{eq:nin}
\end{equation}
where
\begin{equation}
{\cal P} _n(T)= \frac{(|\alpha |^2 \Gamma _{or} T)^{n_{or}}}{n_{or}!} exp(-|\alpha |^2 \Gamma _{or} T)
\end{equation}
that is, the same (\ref{eq:Poi}) distribution, but with the decay
rate replaced by $|\alpha |^2 \Gamma _{or}$, a weighted decay rate.
Therefore, the detection distributions show a characteristic
dependence on $\alpha $ and $\beta $ that could be tested
experimentally. Moreover, the distribution (\ref{eq:nin}) differs
from that expected for a mixture of Helium atoms prepared in
($1s2s$) ortho and para states with weights $|\alpha |^2$ and
$|\beta |^2$, which as it is simple to see is given by the
expression $ P_n^{mix} (T)= \sum _{n_{or}+ n_{pa}=n} |\alpha |^2
P_{n_{or}}^{or} (T) |\beta |^2 P_{n_{pa}}^{pa} (T)$, with $
P_{n_{or}}^{or}$ and $ P_{n_{pa}}^{pa}$ the usual Poisson
distributions with $\Gamma _{or}$ and $\Gamma _{pa}$.
We conclude that, in principle, some peculiar characteristics of the
metastable $1s2s$ superposition state can be observed
experimentally.
\section{Acknowledgments}
We acknowledge partial support from MEC (FIS2006-04151, Consolider
Program SAUUL, CSD2007-00013)
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 5,379 |
Q: How to fetch from an object that is inside arary of array I want to access: order_items > items > order_item_name
{
"order_id": 14858,
"parent_id": 0,
"date_created": "2020-05-17 07:31:42",
"date_created_gmt": "2020-05-17 07:31:42",
"status": "wc-completed",
"customer_id": 9,
"customer": {
"customer_id": 9,
"user_id": 3,
"username": "husam",
"first_name": "Hosam",
"last_name": "amazon",
"email": "ecommerce@amazonfoods.ae",
},
"order_items": [
{
"order_item_id": 152,
"order_id": 14858,
"product_id": 9095,
"shipping_tax_amount": 0,
"item": {
"order_item_id": 152,
"order_item_name": "Amazon Cheddar Cheese, 50g",
"order_item_type": "line_item",
"order_id": 14858
}
},
{
"order_item_id": 153,
"order_id": 14858,
"product_id": 9063,
"shipping_tax_amount": 0,
"item": {
"order_item_id": 153,
"order_item_name": "Amazon 1121 Golden Sella Rice, 5kg",
"order_item_type": "line_item",
"order_id": 14858
}
},
]
},
Since it is in the loop and can be seen here:
const ordersTest = [];
for (let i = 0; i < props.allOrdersState.length &&
props.allOrdersState.length; i++) {
ordersTest.push({
state:props.allOrdersState.length ? props.allOrdersState[i]['customer']['state'] :'',
first_name:props.allOrdersState.length ? props.allOrdersState[i]['customer']['first_name'] :'',
orderItems: [{
itemName:props.allOrdersState.length ? props.allOrdersState[i]['order_items'][i]['item']['order_item_name'] :'',
}],
});
}
I want to access the order_item_name but it gives an error of undefined. however, the first name and other details can be accessed.
A: It looks like each order has multiple orderItems, so I assume you are trying to map them? The length conditionals weren't needed since the for loop would not have run if length equaled 0. Also, there is no "state" property so I changed it to "status".
for (let i = 0; i < props.allOrdersState.length; i++) {
ordersTest.push({
status: props.allOrdersState[i]['status'],
first_name: props.allOrdersState[i]['customer']['first_name'],
orderItems: props.allOrdersState[i].order_items.map((oi) => ({
itemName: oi['item']['order_item_name']
}))
})
}
const props = {
allOrdersState: [
{
"order_id": 14858,
"parent_id": 0,
"date_created": "2020-05-17 07:31:42",
"date_created_gmt": "2020-05-17 07:31:42",
"status": "wc-completed",
"customer_id": 9,
"customer": {
"customer_id": 9,
"user_id": 3,
"username": "husam",
"first_name": "Hosam",
"last_name": "amazon",
"email": "ecommerce@amazonfoods.ae",
},
"order_items": [
{
"order_item_id": 152,
"order_id": 14858,
"product_id": 9095,
"shipping_tax_amount": 0,
"item": {
"order_item_id": 152,
"order_item_name": "Amazon Cheddar Cheese, 50g",
"order_item_type": "line_item",
"order_id": 14858
}
},
{
"order_item_id": 153,
"order_id": 14858,
"product_id": 9063,
"shipping_tax_amount": 0,
"item": {
"order_item_id": 153,
"order_item_name": "Amazon 1121 Golden Sella Rice, 5kg",
"order_item_type": "line_item",
"order_id": 14858
}
},
]
}
]
}
const ordersTest = [];
for (let i = 0; i < props.allOrdersState.length; i++) {
ordersTest.push({
status: props.allOrdersState[i]['status'],
first_name: props.allOrdersState[i]['customer']['first_name'],
orderItems: props.allOrdersState[i].order_items.map((oi) => ({
itemName: oi['item']['order_item_name']
}))
})
}
console.log(ordersTest)
A: You can use the recursive method to find the fields you want.
let answer = [];
function indexOf(obj, to) {
if (obj.hasOwnProperty(to)) {
answer.push(obj[to]);
} else if(Object.prototype.toString.call(obj) === '[object Object]') {
Object.values(obj).filter(item => typeof (item) === 'object').map(item => {
let newObj = item;
indexOf(newObj, to);
})
} else if(Object.prototype.toString.call(obj) === '[object Array]') {
obj.map(item=>{
let newObj = item;
indexOf(newObj, to);
})
}
return answer;
}
My method is a little lax, but it's OK to meet your requirements.
const obj = {
"order_id": 14858,
"parent_id": 0,
"date_created": "2020-05-17 07:31:42",
"date_created_gmt": "2020-05-17 07:31:42",
"status": "wc-completed",
"customer_id": 9,
"customer": {
"customer_id": 9,
"user_id": 3,
"username": "husam",
"first_name": "Hosam",
"last_name": "amazon",
"email": "ecommerce@amazonfoods.ae",
},
"order_items": [
{
"order_item_id": 152,
"order_id": 14858,
"product_id": 9095,
"shipping_tax_amount": 0,
"item": {
"order_item_id": 152,
"order_item_name": "Amazon Cheddar Cheese, 50g",
"order_item_type": "line_item",
"order_id": 14858
}
},
{
"order_item_id": 153,
"order_id": 14858,
"product_id": 9063,
"shipping_tax_amount": 0,
"item": {
"order_item_id": 153,
"order_item_name": "Amazon 1121 Golden Sella Rice, 5kg",
"order_item_type": "line_item",
"order_id": 14858
}
},
]
};
let answer = [];
function indexOf(obj, to) {
if (obj.hasOwnProperty(to)) {
answer.push(obj[to]);
} else if(Object.prototype.toString.call(obj) === '[object Object]') {
Object.values(obj).filter(item => typeof (item) === 'object').map(item => {
let newObj = item;
indexOf(newObj, to);
})
} else if(Object.prototype.toString.call(obj) === '[object Array]') {
obj.map(item=>{
let newObj = item;
indexOf(newObj, to);
})
}
return answer;
}
console.log(indexOf(obj, 'order_item_name'));
A: I just solved the issue. I pushed the orderItems along with firstName and other details to the empty array(ordersTest = [];) and all the items moved to the array and then I accessed each item details and order details from the array.
for (let i = 0; i < props.allOrdersState.length &&
props.allOrdersState.length; i++) {
ordersTest.push({
state:props.allOrdersState.length ? props.allOrdersState[i]['customer']['state'] :'',
first_name:props.allOrdersState.length ? props.allOrdersState[i]['customer']['first_name'] :'',
orderItems: [{
itemName:props.allOrdersState.length ? props.allOrdersState[i]['order_items']:'',
}],
});
}````
Then I accessed each element like:
{row.itemName.map((historyRow) => (
<div style={{border:'1px solid #041e41',padding:'10px'}}>
<Typography>{historyRow.item.order_item_name}</Typography>
<Typography gutterBottom component="div">
<strong> Shipment Amount:</strong> {historyRow.shipping_amount}
</Typography>
<Typography>
<strong> product Quantity:</strong> {historyRow.product_qty}
</Typography>
<Typography>
<strong>Tax Amount:</strong> {historyRow.tax_amount}
</Typography>
</div>
))}
I have mapped over row because I am mapping rows of the table and inside each row i have take collapsed of material UI, so each row contains single record:
<TableBody>
{ordersTest.map((row) => (
<Row key={row.order_id} row={row} />
))}
</TableBody>
And in each single record I am fetching detials and there is again loop over items.
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Trichotosia collina är en orkidéart som först beskrevs av Rudolf Schlechter, och fick sitt nu gällande namn av Peter Francis Hunt. Trichotosia collina ingår i släktet Trichotosia och familjen orkidéer.
Underarter
Arten delas in i följande underarter:
T. c. collina
T. c. govidjoae
Källor
Orkidéer
collina | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 7,818 |
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Welcome to Lafayette Elementary School, located in the heart of Macon County, Tennessee. We invite you to join us in celebrating education and the youth of Lafayette.
MAKE ARRANGEMENTS TO BE HERE ALL 5 DAYS BY 8:00 TO START TESTING!!!
2018-19 Kids Choice Award for TN click HERE to vote!!
LES Library Website link click HERE!
To see the Accelerated Reader(AR) booklist by author click on the image above!!!
To see the Accelerated Reader(AR) booklist by book title click the image above!!!
The content of some hyperlinks is not controlled by Macon County Public Schools. While school staff reviews all links before they are inserted, the content may change. While we strive to maintain appropriate, student friendly resources, these pages may link to other pages that have not been reviewed by staff. The existence of a link should not be an assumed endorsement by Macon County Public Schools. As always, please monitor your child's Internet use. If you find content which you consider inappropriate, please email the our technology coordinator immediately. | {
"redpajama_set_name": "RedPajamaC4"
} | 3,647 |
Tasmanian Young Achiever Awards
2020 Winners of the Tasmanian Young Achiever Awards
2020 Premier's Young Achiever of the Year – Toby Thorpe
St.LukesHealth Healthier Communities Award
Harriet Beattie, 28 of Scottsdale is extremely committed to the health and wellbeing of the Dorset community in a physical, social and emotional capacity, which she has demonstrated through various roles. Harriet is currently a Rural Physical Health worker for the Royal Flying Doctor Service where she delivers group and individual cardiac & pulmonary rehabilitation programs along with other exercise-based care plans for individuals suffering from chronic conditions. Harriet has successfully increased the physical activity of people aged 50-85 and the number of people completing cardiopulmonary rehabilitation living in rural and remote areas. She provides educational health programs and runs a free weekly group exercise program.
Heather & Christopher Chong Community Service & Volunteering Award
Toby Thorpe, 18 of Huonville led the Energy Futures Team at Huonville High School to become the sole Australian finalist in the International Zayed Sustainability Prize and the winner of $100,000 (US). Toby has represented his community twice at the Abu Dhabi Sustainability Week and spoke at multiple e-meetings across the world. He organised the first ever state-wide Climate Leaders Conference in Hobart, attracting over 350 students and professionals and is an active member of the Australian Youth Climate Coalition. Toby volunteers for Switched on Schools and has empowered young people to take action against climate change through the role of Youth Facilitator at Education for Sustainability Tasmania.
First National Real Estate Leadership Award
Gabrielle Dewsbury, 17 of Newnham pioneered the School Strike 4 Climate movement which is a student-led movement fighting for climate justice. She has led multiple Climate Strikes including the first Northern Strike in Tasmania and set up a Student Wellbeing Committee at her school. Gabrielle leads the "Health is Wealth" initiative, which runs workshops focused on mental health and wellbeing and has partnered with Beyond Blue to expand Nationally. Gabrielle is on multiple committees including the Australian Youth Climate Coalition and speaks at conferences and forums around the state. She has won multiple awards and represented Tasmania at the UN Youth Aotearoa Leadership Tour.
Dental South Aboriginal and Torres Strait Islander Achievement Award
Jordy Gregg, 22 of Glenorchy is a passionate teacher, actor writer and film maker. In 2018, Jordy played the part of Laertes in Hamlet which went on to win Tasmania's Best Professional Production. He has been the lead tour guide for the Bay of Fires Walk and worked closely with educational facilities such as the Lambert School where he gives music lessons. Jordy has given talks to students at Clarence High School regarding jobs and potential career paths and taught children at the Aboriginal Children's Center about acting, drama and social dynamic skills. He is currently writing a theatre show which has attracted national interest.
Motors Tasmania Sports Award
Kate Eckhart, 22 of West Hobart is a Canoe Slalom athlete and has represented Australia for almost seven years. During that time she has won countless awards, some of which include; second and third place at the Junior world championships, third place at the senior world championships, third place at the Australian open K1, 6th place at the U/23 World Canoe Slalom and third place at the World Championships U/23 World Canoe Slalom Championships. Kate has held a scholarship for the Tasmanian Institute of Sport and Paddle Australia since 2013 and is an amazing role model for other athletes at the Tasmanian Institute of Sport.
Spirit of Tasmania Tourism and Hospitality Award
Benjamin Coe, 23 of Flowery Gully is a tour guide and supervisor at Platypus House. Ben is responsible for running the tours, assisting staff, and liaising with tour directors. His passion lies with animals and touring, which is evident from the hundreds of customer service reviews highlighting his friendly, enthusiastic persona when hosting the tours. Ben's efforts have led to the opening of a Platypus Exhibit at San Diego Zoo and took him to the Finalist stage in Chilli 90.1fm's Champions of Tourism Awards. Ben's shining personality and positive attitude allows him to lead his team by example and improve the efficiency of the group.
Colony 47 Transition to Work Award
Jan Richards, 18 of Latrobe is a determined, energetic young woman. In 2019 she completed her Certificate II in Community Services focusing on Children's Services and a Certificate III in Hospitality. Jan is currently completing a traineeship within the Child Care industry as well as a Certificate III in Early Childhood Education and Care. She completed these whilst also working part time to support herself. Jan participated in the planning and running of Play Group sessions that were provided to the parents and children within their community.
TADPAC Print Service to the Disability Sector Award
Heidi La Paglia, 26 of Hobart is an advocate for young women with disabilities, like herself. Heidi undertook leadership roles for the Tasmania University Union and National Union of Students where she assisted women with disabilities with their welfare and participation. She worked to assist individuals in accessing the National Disability Insurance Scheme in her roles working for Federal Members of the Australian Labor Party, Senator Carol Brown and the Hon Julie Collins MP. As the Content Officer for Women With Disabilities Australia's virtual centre project, Heidi worked extensively with individuals with disabilities across Australia to develop content for their website 'Our Place' which was launched in March.
Travel Associates Career Achievement Award
Shannon Umgeher, 25 of Murdunna is a young and successful entrepreneur. In 2014 he completed his study of residential drafting at TasTAFE, after which he became Store Manager at the Brighton IGA where he managed 23 staff members. During this time he also established and successfully operated his own garden maintenance business. In 2017 Shannon took the next step in his career and became the owner of the Murdunna Roadhouse, a small service station and general store. Since Shannon purchased the business he has been able to expand the building, product line, and services offered, and has single-handedly doubled the business's stock, customers and sales.
The 2019 Winners with The Honourable Will Hodgman MP, Premier of Tasmania and Jo Palmer, News Presenter, 7 Tasmania
Overall Winner - Premier's Young Achiever of the Year
Matthew Etherington, 23 of SANDY BAY is committed to helping people experiencing disability and disadvantage in his community. Matthew helps promote healthier communities through the Big Issue Community Street Soccer Program, using sports to empower and improve the lives of the disadvantaged. He organised a Mental Health First Aid Initiative at UTAS to prepare over 330 students to exercise self-care, promote mental health and encourage peer support. He is a national Youth Health Forum member and former Head of Welfare at UN Youth and the Tasmanian Youth Local Government. Matthew also led the organising committee of the Red Cross Social Connectedness Summit during Mental Health Week 2018.
The 2019 Premier's Young Achiever of the Year, Matthew Etherington with The Honourable Will Hodgman MP, Premier of Tasmania
Alex Morris Baguley, 28 of LESLIE VALE is proof that a positive attitude can overcome challenges. Despite being diagnosed with several conditions, including Down Syndrome and ADHD, Alex's family fought hard to ensure she had access to the same education and training opportunities as other people her age. After finishing Year 12, she worked at Australian Disability Enterprise. Alex then joined the Hamlet, Inc. training program where she learned about customer service. To date, Alex has become an integral member of the Hamlet team, completing more than 500 hours of work. She is now undertaking the advanced training model, learning Barista basics and further catering skills.
Danny Sutton, Chief Executive Officer, Colony 47 with Alex Morris Baguley
Campbell Remess, 14 of ACTON PARK has handmade and gifted over 1,700 bears to children worldwide through his initiative, "Project 365 by Campbell". He gives presents to children in hospital during Christmas and has helped raised $200,000 for charities. Campbell created Kindness Cruises, a charity which raises funds for surprise cruises for families impacted by cancer and other life-changing difficulties. He has released two books encouraging children to follow their dreams. He is also a regular speaker at schools where he discusses the impact of kindness as opposed to bullying. Campbell received the CNN Heroes, Young Wonder Award in 2016, the first Australia to do so.
Heather Chong, CEO, QEW Orchards with Campbell Remess
Elias Solis, 29 of NEW TOWN wants to bring Latin culture to Hobart. A son of Chilean emigrant parents, he established the Yambu café, a vibrant home for Latin American food, culture and music. Over the last decade, Elias has initiated events celebrating Latin American culture where he was either a producer, leader, musical director or key player in Latin American bands. He organised Pop-Up Day of the Dead Festivals. Elias organised workshops in percussion and hosted Spanish language classes and organised fundraising events for study tours to Cuba for young Tasmanians. He is also a key member of the Latin American Cultural Association of Tasmania.
Nick Harriman, General Manager, Retail & Hospitality, Spirit of Tasmania with Elias Solis
Karita Casimaty, 29 of MOONAH was the first Tasmanian to develop and offer the internationally recognised Duke of Edinburgh Award to young people diagnosed with a disability. She has successful delivered the program since 2015, showcasing her leadership skills, and ensuring that clear expectations and guidelines were outlined from the beginning. In 2018, Karita was asked to deliver a speech to His Royal Highness Prince Edward at Government House during the royal visit, explaining the benefits of the award to young people with disability. Karita is currently the Oakdale Training Service Supervisor, planning and delivering transition to work training to young adults diagnosed with a disability.
Jeremy Pettet, Executive Officer, Uniting VIC/TAS with Karita Casimaty
Zac Romagnoli-Townsend, 25 of NUBEENA advocates for climate justice. A proud Mutwintje man, he is a coordinator for the Seed Indigenous Youth Climate Justice Network. As part of the Network, he coordinates ten young Aboriginal and Torres Strait Islander volunteers and collaborates with the local Australian Youth Climate Coalition with non-Indigenous volunteers. Zac helps facilitate national trainings, gatherings and campaigns as a member of the national core Seed team. One campaign successfully prevented bank funding of an Adani coal mine. Zac facilitates and speaks in workshops and grassroots community organisations to build a social movement to keep all new fossil fuels in the ground.
Dr Jessica Manuela, Director, Dental South with Zac Romagnoli-Townsend
Matthew Etherington, 23 of SANDY BAY is committed to helping the disabled and disadvantaged in his community. Matthew helps promote healthier communities through the Big Issue Community Street Soccer Program, using sports to empower and improve the lives of the disadvantaged. He organised a Mental Health First Aid Initiative at UTAS to prepare over 330 students to exercise self-care, promote mental health and encourage peer support. He is a national Youth Health Forum member and Head of Welfare at UN Youth and the Tasmanian Youth Local Government. Matthew also led the organising committee of the Red Cross Social Connectedness Summit during Mental Health Week 2018.
Peter Murfett, Director, St.LukesHealth with Matthew Etherington
University of Tasmania, Faculty of Education Teaching Excellence Award
Grace Birchall, 24 of SHEARWATER is a compassionate teacher. She teaches English, Geography and Dance at Ulverstone High School. During lunch hours, Grace teaches students who wish to dance outside normal class schedules, and coordinates song and dance performances in a community theatre. During weekends, Grace teaches persons with Down Syndrome at the Bright Stars Troupe, helping to release their inhibitions and perform at community functions. She attends professional development opportunities, continuously seeks feedback about her teaching capabilities and is always the first to implement different classroom strategies. Grace provides an inviting environment, empathises with her individual students and implements mindfulness in her room.
Professor Karen Swabey, Dean & Head of School - Faculty of Education, UTAS with Grace Burchall
Samuel Watson, 18 of ULVERSTONE is an advocate for LGBTIQ rights. He was Head Boy and Student Representative at The Friends' School providing a strong focus on diversity. A senior Navy Cadet, Samuel mentored ten refugees through a ten-day Rotary Windeward Bound Youth Leadership Challenge. He is an Advisory Group member for the Commission for Children and Young People and served on the Hobart Youth Advisory Squad. As 2018 Youth Deputy Premier, his group passed a Gender Equality Bill that was handed to Government. He has also patrolled Tasmanian beaches for over 250 hours and has been his club's Surf Life Saver of the Year.
John McGregor, Director, First National Real Estate McGregor with Samuel Watson
Daniel Watkins, 23 of GROVE and is a competitive athlete who has had an outstanding year in both Kayak and Canoe Slalom. He was selected to represent Australia in the U23 World Championships where he placed a creditable 6th in the C1 and 14th in the K1. He won gold in the U23 Aus Open K1 and the Nationals K1 and C1 divisions. Daniel's biggest achievement to date is winning in the U23 K1 and C1 divisions at 2018 Oceania Championships in Auckland. He makes sure to give back to the community by volunteering his time to help coach and mentor junior paddlers in Hobart.
Campbell York, General Manager, Motors Hobart with Daniel Watkins
The 2018 Winners with The Hon Jacquie Petrusma and Jo Palmer, Southern Cross Television Presenter
Kirby Medcraft, 29 of Lutana is Assistant Principal at Windermere Primary School where she has brought about significant change, particularly in regards to family engagement. Her K-2 'Counting Bags' take home Maths activities had a 100% involvement from families. And the 'Bedtime Stories' afternoon saw over 200 parents attend school to read with their child. In her role, Kirby works closely with early childhood teachers, mentoring them and modeling best practice.
The 2018 Premier's Young Achiever of the Year, Angela Crane who represented Kirby Medcraft with The Hon. Jacquie Petrusma MP
The Coffee Club Arts and Fashion Award
Stephanie Eslake, 27 of Sandy Bay has both media and music degrees. She founded 'CutCommon', Australia's online publication for young classical musicians and was shortlisted in the 2017 NEXT Innovation Award. Stephanie has written program notes for Symphony Orchestras and hosted radio programs. She is the sub-editor for Warp Magazine and Undertow Magazine and co-editor and publications mentor for Platform Magazine. Stephanie was 2017 Young Citizen of the Year for her artistic contribution
Jason Travis, Regional Development Manager TAS, The Coffee Club with Stephanie Eslake
Associate Professor Karen Swabey, Dean and head of School, University of Tasmania Faculty of Education with Angela Crane who represented Kirby Medcraft
Grant Milbourne, 28 of Lenah Valley founded I'ME or Insight Mindfulness Education. The not-for-profit organisation supports youth wellbeing and mental health. They provide meditation retreats and professional development programs. Their school based programs develop stress management skills and strategies for teachers and students. Grant has volunteered over 2,000 hours and organised a concert that raised $14,000 for I'ME. The website and Facebook provide resources, tips and strategies for meditation, managing stress and psychological wellbeing.
Ray Ellis, CEO, First National Real Estate with Grant Milbourne
Olivia Fleming, 21 of Rosny Park founded The Little Help Project during her first year studying medicine. The Project tackles mental health issue and empowers young people. The not-for-profit has 25 volunteers and has helped 8,000 Tasmanians build resilience and self-esteem. Olivia oversees self-defence and development classes and an outreach program educating women about confidence, consent, and boundaries. She has volunteered in over 30, week long programs, facilitated camps and many community programs.
Alderman Heather Chong, CEO, QEW Orchards with Olivia Fleming
Jack Dyson, 20 of Rokeby has made a significant contribution to Cystic Fibrosis Tasmania raising awareness and funds. In his many media appearances he talks openly about his personal challenges with CF. Jack's first You Tube video, "Iron Lungs" has nearly 3,500 views, was filmed from his hospital bed. Using Social Media, he has become an inspirational role model, motivating others with his body building to improve his CF and talking honestly about mental health.
Jeremy Pettet, CEO, TADPAC Print with Jack Dyson
Nadine Ozols, 29 of Bicheno has worked in the health and aged care sector for 10 years and is currently the Public Relations and Development Manager working across May Shaw Health Centre and Aged Care Deloraine. She has facilitated health and wellbeing programs such as Guided Relaxation Sessions, Women's Health Days and Youth Health Days. Nadine is a graduate of the Tasmanian Leaders Program for high potential leaders and has volunteered an estimated 800 hours.
Peter Murfett, Director, St.LukesHealth with Nadine Ozols
Tara Howell, 27 of Launceston founded Blue Derby Pods Ride a three-day, soft-adventure mountain biking experience in Derby. The guests indulge in Tasmanian food and wine, and stay in unique, architecturally designed accommodation pods. Tara created a tourism development plan and received a $500,000 grant. She has a 50-year lease on land in Derby Regional Reserve where the infrastructure has been built. Blue Derby is in its third year of operation, with four employees.
Nick Harriman, General Manager, Retail & Hospitality, Spirit of Tasmania with Tara Howell
Madeleine Fasnacht, 18 of Blackmans Bay was the 2017 Commonwealth Youth Games Closing Ceremony Flag Bearer and named the 2017 Australian Junior Cyclist. She was placed first at the Commonwealth Youth Games in the Individual Time Trial and 3rd at the World Junior Road Championship. She won the 2017 Junior Oceania Road Championships and was 3rd at the Commonwealth Youth Games. Madeleine recently achieved bronze for the under-19s women's time trial at the World Championships.
Campbell York, General Manager, Motors Hobart with Madeleine Fasnacht
Colony 47 Young Indigenous Achievement Award
Madelena Andersen-Ward, 26 of Margate has a Bachelor of Music and is a singer and songwriter. She is actively engaged in the indigenous community through community performances and festivals. She is promotes education to Indigenous youth and has spoken on radio and at schools. Madelena mentored young musicians in the Indigenous Tutorial Assistance Scheme. She is currently of one of the seven contemporary voice roles in a new musical and visual project called A Tasmanian Requiem.
Elizabeth Daly OAM, Northern Manager, for Colony 47 with Madelena Andersen-Ward
The Hon Will Hodgman MP and The Hon Elise Archer MP with the 2017 winners at the Gala Presentation Dinner
Dr Jessica Manuela, 28 of Blackmans Bay is a qualified oral care professional, dedicated to raising awareness of oral health. Opening her own practice two years ago, she already has 3,000 registered patients. Jessica also volunteers 20 to 30 hours each week, speaking with school students about oral hygiene and running community information evenings. She is the chairperson for the Oral Health Promotion Committee of the Tasmanian Dental Association and was awarded the 2012 New Zealand and Australian Society of paediatrics award
Jacob Prehn, 29 of Kingston has dedicated six years to improving Tasmanian Aboriginal health outcomes. He is the volunteer Deputy chairman of the National Aboriginal and Torres Strait Islander Health Workers Association. Jacob works part-time in Aboriginal Research and Leadership at the University of Tasmania and his PHD will focus on men's mental health to improve the health outcomes of Aboriginal people. Jacob is working with the CEO of Karadi to build the state's first Aboriginal men's shed.
Tyler Richardson, 29 of Sandy Bay is the self-taught lead singer and songwriter for punk rock band, Luca Brasi. Tyler has 40 songs credited to him over five record releases; the last three charting on the ARIA chart. Luca Brasi have won multiple awards including Best Punk Band in 2016. The band has performed to 40,000 people across Australia, UK and Europe and their upcoming Australian headline tour is sold out. In addition to passionately promoting Tasmania, Tyler fundraises for LGBTIQ youth, refugees and Indigenous causes.
Mohammad Nourouzi, 23 of South Launceston arrived as a refugee in 2013 with no English. He is a volunteer interpreter and at The Door of Hope and fixes bicycles and cars for those in need. Mohammad is active in his Afghan Hazara community, promoting cultural activities and inclusion. He is a volunteer leader for Tasmania Parks and Wildlife and a Green Army conservation volunteer. Mohammad is also a member of the Migrant Resource Centre's Youth Advisory Network and worked for Red Cross.
Dr Jessica Manuela, 28 of Blackmans Bay is a qualified oral care professional, dedicated to raising awareness of oral health. Opening her own practice two years ago, she already has 3,000 registered patients. Jessica also volunteers 20 to 30 hours each week, speaking with school students about oral hygiene and running community information evenings. She is the chairperson for the Oral Health Promotion Committee of the Tasmanian Dental Association and was awarded the 2012 New Zealand and Australian Society of paediatrics award.
Shai Denny, 22 of Ridgely was born with Downs Syndrome and is a tireless ambassador for people with a disability. Employed with Family Based Care Association, she worked as Co-Director of "PULSE", a project for people with a disability to become involved in arts based performances. Shai created and taught most of the choreography and assisted in all production decisions including the costuming and presentation of acts. Shai volunteers at countless organisations and is currently studying child care.
Motors Group Tasmania Sports Award
Ariarne Titmus, 16 of Launceston is an Australian swimming star. In 2016, she was selected as the youngest team member of the Australian Dolphins World Short Course Swimming team to compete in Canada. She made the finals in three divisions of the World Open Short Course Championships, finishing 4th in two races and 6th in another. She is also ranked 1st in Australia for the short course open 400m freestyle and 8th internationally for the short course 800m freestyle.
University of Tasmania, Faculty of Education, Teaching Excellence Award
Caitlin Cashion, 27 of Huntingfield is an Advanced Skills Teacher at Fairview Primary School and is passionate about helping 'at-risk' and 'dis-engaged' students. She is currently studying a Masters of Education and uses her research to implement programs to engage students. In collaboration with Beacon Foundation, Caitlin led an Alternative Learning Day for year 7 to 10 students providing hands-on, real world relevance and problem solving sessions. Caitlin has been Acting Principal and mentors colleagues through modelling, research and leading professional development sessions.
The Hon Elise Archer MP with the 2016 winners at the Gala Presentation Dinner
Mitch McPherson, 28 of Tranmere established "SPEAK UP! Stay ChatTY" after his younger brother Ty took his own life in 2013. The charity aims to prevent suicide by spreading the message that nothing is so bad that you can't talk about it. Mitch has presented to more than 150 schools and sporting clubs and will roll out a school program this year. The Tasmanian State Football League dedicated one round to raise funds and awareness. Mitch has raised over $120,000 in funds and Speak Up Stay ChatTY has almost 11,000 Facebook followers.
CEO for St.LukesHealth, Chris Williams presented Mitch McPherson with the Healthier Communities Award
AustralianSuper Career Kick Start Award
Charlotte Hunn, 28 of Kingston founded COMET, or Community Engagement Tasmania. COMET trains law students to provide education for disadvantaged and homeless youth on key aspects of criminal law and the justice system. Charlotte was awarded the Sandy Duncanson Social Justice Bursary for The COMET Project. She received the LexisNexis Prize for achieving the highest mark in Criminology and has accepted a position as research assistant on the National Jury Project with the Tasmanian Law Reform Institution. Charlotte has also volunteered for Colony 47 and the Tasmanian Asylum Seekers Support Network.
Business Partnership Manager for AustralianSuperm Randolph Nevis presented Charlotte Hunn with the Career Kick Start Award
Heather & Christopher Chong Community Service Award
Eva MacKinley, 25 of Sandy Bay founded The Last Straw, a campaign to eliminate the use of plastic straws across Australia, encouraging a plastic free and zero waste life. She is Director of Global Partners for Change, a start-up program supporting young Kenyan people to create social change programs for their communities. Eva was the first Tasmanian Ambassador for the 2014 One Young World Summit in Dublin and Youth delegate to the 2013 Rotary International Peace Conference. She is a member of the Tasmanian Youth Consultative Committee and Youth Council.
Alderman Heather Chong presented Eva McKinley with the Community Service Award
Halina Kaufman, 28 of Lenah Valley has had a successful career in jewellery manufacturing with two regional and a National gold medal for WorldSkills. She received an international team leadership challenge scholarship and an International Specialised Skills Institute fellowship. Halina has been a finalist five times in Australian Design Awards. She trained with master engravers in Florence and Louisiana and a specialist diamond setter in The Netherlands. Halina is a judge and project designer for the WorldSkills jewellery competitions and aims in the future to be the Jewellery Expert.
Director of The Coffee Club, John Lazarou presented Halina Kaufman with the Arts and Fasion Award
Samuel Morey, 28 of Berriedale began teaching at St Therese's Catholic Primary school as the PE teacher and in 2014 was appointed Assistant Principal of St Paul's Catholic Primary School in Bridgewater. Sam implemented the extra-curricular 'Kickstart' and 'Accelerator' sessions, for students who have anxiety towards learning and providing acceleration opportunities for capable students. He was awarded a "Growing in Leadership: Emerging Leaders Award" by the Australian Council of Educational Leaders. Sam volunteers his time coaching and facilitating St Therese's soccer, basketball, athletics and Cross Country Teams, and has been appointed mentor and talent development coach with Athletics Tasmania.
Acting Dean of the Faculty of Education, University of Tasmania, Professor Karen Swabey presnented Samuel Morey with the Teaching Excellence Award
Jacob Birtwhistle, 20 of Launceston won the 2015 under-23 Triathlon World Championship. One of the top men on each leg of the triathlon, he separated himself from a five-man front pack to ultimately win the race and take home the World Champion honour. He was the 2014 world junior duathlon and junior world championship silver medallist. Jacob took 1st and 2nd places in four out of seven triathlons last year. Jacob will now challenge for a spot in the Olympic team for Rio.
State Sales Manager for Motors Group Tasmania, Angela Travers presented James' representative with the Sports Award
Carrie Leppard, 26 of Glenorchy has raised over $17,000 for Cystic Fibrosis Tasmania. When her daughter Charlotte was born with CF, Carrie wanted to create community awareness. She has spoken publicly about the importance of undertaking testing for carrying the CF gene, and about the challenges of parenting a child with Cystic Fibrosis. Carrie set up the 'Cystic Fibrosis Support Group and Advice Australia', Facebook page which now has 400 active members. The group allows parents and carers to discuss issues around their child's CF in a supportive environment.
Chairman of TADPAC Print, Ross Copping presented Carrie Leppard with the Service to the Disability Sector Award
Colony 47 Young Aboriginal Achievement Award
Teangi Brown, 21 of Bellerive works at the Tasmanian Museum and Art Gallery, running educational programs and guided tours on Tasmanian Aboriginal culture and history. He is often requested to speak at functions on the history of Tasmanian Aboriginal people. Teangi also successfully operates his own business as an Aboriginal Interpretation guide for schools on cultural camps. He has participated in the Tasmanian Youth Forum and the National Indigenous Youth Parliament. He volunteers for Tasmanian Aboriginal Centre's Land care with Parks and Wildlife and regularly performs Welcome to Country at community events.
CEO of Colony 47, Therese Taylor presnted Teangi Brown with the Aboriginal Achievement Award
Overall Winner - The Premier's Young Achiever of the Year
Dr Lila Landowski, 28 of North Hobart is a neuroscientist investigating nerve regeneration specifically in peripheral neuropathy. She is currently conducting a trial focusing on a treatment for chemotherapy-induced neuropathy, a debilitating condition that affects the nerve fibres carrying information between the brain and body. Her work has been presented at 11 high profile national and international conferences, and she has trained with the Mayo Clinic cementing her status as an emerging leader in the field of peripheral nerve injury. An enthusiastic volunteer, Dr Landowski gives her time to cancer support services and science outreach activities, and also teaches medical students and dementia care professionals about health science.
First National Real Estate Leadership & Innovation Award
Waqas Durrani, 26 of Hobart has been awarded the Vice Chancellor's and Rotary Leadership Awards, and the International Student of the Year for his initiatives in promoting inclusion whilst celebrating student diversity. Speaking five languages, Waqas interprets for refugees and migrants at the Royal Hobart Hospital, Law Courts and Schools. He organised the "Lollywood Gala" at Mona Museum and other multicultural events such as Islam Awareness Week and "Fast With Us" to promote tolerance and harmony. Waqas was also instrumental in initiating changes within the Universities cafeteria to cater for varying cultural dietary requirements.
Mitch McPherson, 27 of Tranmere has over the last two years spent an estimated 1,200 volunteer hours creating his charity "Speak up stay chatTY" after losing his younger brother Ty, to suicide in 2013. He is changing the way in which people look at mental health. Mitch plans community and charity events for "Speak up stay chatTY", including the sale of over 7,000 car bumper stickers, the Stay ChatTY Cup - a local football match in the Tasmanian League between Clarence and Lauderdale and an annual quiz night which has raised over more than $30,000 in the past two years.
Andrew Millhouse, 24 of Kingston completed his Bachelors Degree in Arts/Law with honours in 2014. He has provided countless voluntary hours of support to young people from all walks of life with free facilitation services to a number of organisations that work with young people including AYCC, Students of Sustainability and the Oaktree Foundation. In 2013, Andrew created Slingshot Training - a social enterprise to help young people become the best they can be - support others, overcoming challenges and discovering success. Andrew is currently looking at how Slingshot can support young people experiencing anxiety and depression.
Lyndon Riggall, 25 of Riverside won the Hot Key Books Young Writers' Prize in 2013 for his first children's novel "Charlie in the Dark," and is now working on a second book called "The Drift." Lyndon has a deep commitment to assisting the next generation of writers acting as a judge for the Children's Book Council of Australia, a marker for the Tasmanian Qualifications Authority assessing creative writing folios, and a workshop facilitator for Tasmanian students and teachers. Lyndon is currently compiling a portfolio of short stories which he hopes to submit to the Clarion Writers' Workshop in San Diego.
Zane Littlejohn, 28 of Summerhill is organised, prepared and professional, and values the time necessary for him to support and train practicum teachers through University of Tasmania's student internship program as he has done for the last five years whilst working at City Campus. Working with the Principal and School Community, Zane was instrumental in creating the Culture implementation called the "VOICE" which was designed to give students a voice and the World of Work Program Health and Wellbeing program. He was also the 2014 Tasmanian State League Coach of the Year Award Winner.
Zane Littlejohn was represented by his mother Lisa Hardy
Colony 47 Aboriginal Achievement Award
Kartanya Maynard, 21 of Glenorchy is a student at the UTAS Conservatorium of Music. She won the 2011 Tasmanian Aboriginal Artist of the year, the 2013 Tasmanian Young Aboriginal Person of the Year, was the youngest person ever elected to the Tasmanian Aboriginal State Committee and performs at the Putalina Festival each year. Kartanya is dedicated to cultural revival and education providing vocal coaching to a women's Aboriginal singing group and teaches children about Tasmanian Aboriginal history. She is an interpreter in Aboriginal Culture at the Tasmanian Museum and Art Gallery and plans to write more songs in her language for future generations to follow
Tasmanian Institute of Sport Sports Award
Macey Stewart, 19 of Devonport had an amazing 2014, winning three Junior World Cycling Championships in the Junior Women's Team and Individual Pursuits in Korea. She has recently also been crowned the 2014 junior national track and road cyclist and the overall outstanding junior cyclist of the year.After winning the Junior Women Individual Time Trial at the Road World Championships in Italy last September, Macey is now representing Australia at Senior International level. Awarded the 2012 Devonport Young Citizen of the year, Macey is a great role model for cycling and women's sport and an advocate for safe riding in Tasmania.
Macey Stewart was represented by her brother Andrea Stewart
Meg Cooper, 23 of Sandy Bay gives endless volunteer hours planning and running classes for BrightStars – the Southern Dance Group of Down Syndrome Tasmania and is a strong advocate for people with disabilities. Meg is an excellent role model and friend to her brother who has Down Syndrome, assisting her family with his care. Working within the Arts Program at Cosmos, a non-profit organisation offering learning and leisure opportunities for people with intellectual disabilities, Meg is described as the "gold star standard'" for support workers where it's not just about work, but it's a true vocation
Tasmanian Early Years Foundation Excellence Award
Stacey Hall, 27 of Orielton developed her love for helping children in year 11, choosing to enrol in "working with children" at Rosny College. She has since successfully completed her Diploma of Children's Services and is currently employed at Lady Gowrie Tasmania. Stacey is the Team leader and heads a team of six educators and mentors new educators and students. Stacey leads by example and is always inclusive in her practices ensuring educators are worthy contributors. Working 72 hours each week in the baby's room which is a highly demanding position, she goes above and beyond for the children in her care
St.LukesHealth Health and Wellbeing Award
The 2014 Winners at the Gala Presentation Dinner
Bokart Print Arts and Cultural Development Award and The Premier's Young Achiever of the Year
Joshua Lowe, 27 of Rose Bay has a Bachelor of Dance. In 2007 he founded DRILL Performance Company to provide young people in Launceston with work. Joshua is Artistic Director of DRILL Performance Company in Hobart, Festival Director for Short+Sweet Dance and Program Producer for Yellow Wheel in Melbourne. He has been mentored by eminent Australian choreographer Becky Hilton, Artistic Director Emma Porteus, Adam Wheeler and also Brooke Stamp during a residency in France. Whilst in Europe he saw over 30 performances and participated in 14 workshops. In 2013 he worked on Tasdance's first DanceNET program and in 2014 will undertake a highly competitive Asialink residency and finish choreographing the project with Tasdance for national release.
First National Real Estate Leadership and Innovation Award
Adam Mostogl, 26 of Queenstown is a leader with the Door of Hope Christian Church in Launceston and also runs his own business, illuminate SDF. He started illuminate SDF in 2009 which focuses on delivering meaningful and inspiring experiences for students in Grade 9 and 10 and other young people. The aim is to activate passion and get young people excited and more confident about the business world. The programs are run across Tasmania in partnership with the Tasmanian School of Business and Economics at the University of Tasmania and the Department of Maritime and Logistics Management at the Australia Maritime College. More than 600 students have undertaken the challenge, finding a new understanding and inspiration in their approach and attitude towards learning.
Heather and Christopher Chong Community Service Award
Christopher Ballard, 26 of West Hobart has served the scouting movement for almost ten years. Whether as leader of the Mt Stuart Scout Group increasing membership from 20 to 70, mentoring new leaders in the Lenah Valley Scout Group to re-start their Cubs section, or in a variety of capacities supporting major scouting events, Chris undertakes all of his roles with dedication, commitment and kindness that exemplifies all that Scouting is about. Chris contributes to the long term sustainability of Scouting and has provided input into the National Rover Council to improve national policy. He has received a number of Scouting awards and juggles his service to the scout movement with full time study in Pharmacy, and now works as a Pharmacist.
Lila Landowski, 27 of North Hobart is a PhD student in Neuroscience at the Menzies Research Institute devoting up to 80 hours a week in the laboratory. Lila has attended 13 national and international conferences, presenting her work at eight. Whilst teaching 1st and 2nd year students at the University of Tasmania, Lila is also developing a treatment for nerve damage, for which there currently is no cure. Part of her research received one of only two 2013 National Health and Medical Research Council grants by UTAS, valued at $360,000. Her therapeutic investigation also received a $35,000 grant from the Brain Foundation. The discoveries in her research show compelling evidence that a particular natural molecule can direct nerve regeneration, and may benefit those with neuropathy which may revolutionise treatment.
Caitlin Rice, 25 of North Hobart believes that every child has the right to a quality education and aims to make all learning experiences meaningful and real for her students. A teacher at Levendale Primary, a small school of 17 children, Caitlin taught children in a mixed class of Kindergarten to Grade 2 and Science and History to the Grade 3 and 4's. She also took on additional roles such as Teacher in Charge when the Principal was absent. Caitlin became involved in a number of events supporting the Levendale Community. When the final decision was made to close the school, Caitlin worked tirelessly with both Principal and the Community to ensure that the transition to new schools during 2014 was as smooth as possible for the children.
Mat Goggin Foundation Entrepreneurship Award
Ella Watkins, 18 of West Hobart has been writing and illustrating children's books and also filmmaking for a number of years. She produced her first short film at just 15. Ella published her first book, 'Henry's Holiday' at just 12, and her second book, 'Henry the Goat' in 2011. Ella's books were presented by the State Government to Crown Prince Frederik and Princess Mary. Her book artwork was selected as a part of the Art Rage exhibition. Ella completed year 11 and 12 in one year graduating at 17 with an ATAR score of 99.5. She has also volunteered at an African orphanage and a school in Kenya and completed art courses in Paris, keeping her away from home for five months. Ella is planning a move to Los Angeles to further explore her passions for writing, painting, and filmmaking.
Jessica Norton, 22 of Hobart is living with a number of food allergies. She has taken steps to help condition by founding her own company, Eat Safe. Through her Eat Safe company website, she brings awareness to the community of Tasmania about allergies and dietary choices. The information allows individuals to visit and eat with confidence at various local shops, restaurants, cafes and events. Jessica has created symbols that are simple and eye catching that will become visual icons and will be recognised around Hobart and surrounds. Each symbol represents a different food allergy/sensitivity or personal food choice. The community is now benefitting from the fast increasing amount of knowledge, and the available choices that is generated by Eat Safe.
Hydro Tasmania Environment Award
Patrick Kirkby, 27 of South Hobart has delivered hundreds of workshops, presentations and seminars on topics ranging from climate change, to renewable energy, sustainability, social justice and youth leadership. Patrick's volunteer efforts in the climate sector has found him working across the globe, from Bangladesh to Canada. He took 25 local youth on a 12-day Journey along the mighty Kelani River in Sri Lanka. He nurtured, trained and inspired them as future environmental leaders through life-changing environmental educational experiences. The project was hailed to be Sri Lanka's most successful and innovative environmental education project. Patrick has recently commenced a PhD working on climate change solutions across Asia and building on his life-changing experiences.
Temco Science and Technology Award winner and Southern Cross / Premier's Young Achiever of the Year
Dr Clare Smith, 27 of Drinedary received her PhD in Medical Research in 2012. She works with the Menzies Research Institute investigating a novel therapy against Malaria. A major outcome of her research was discovering an antimalarial compound that could make an impact worldwide. The finding is now protected by a patent, and a clinical trial is about to begin. Her work has won numerous Awards and been presented at conferences and seminars across Australia and Europe. Clare was one of only six Australians to attend the 61st Nobel Laureate meeting for medicine/physiology in Germany. She also received a grant from the Australian European Malaria cooperation to work in the Institute of Molecular Medicine in Lisbon, Portugal.
Emma Flukes, 24 of Lindisfarne is a marine scientist with the Institute for Marine and Antarctic Studies and is completing her PhD. She is investigating the formation of 'barren habitat' caused by long-spined sea urchins and has won numerous awards. This species poses the single largest environmental threat to Tasmanian reefs and the fisheries they support. Emma is an avid underwater photographer and editor for a highly successful local marine magazine. She is involved with rubbish clean ups around waterways and volunteer efforts to conserve marine species both locally and internationally. She is also part of the Young Tasmanian Scientists program.
Emily Pickett, 17 of Moriarty is involved in music, dance and surf lifesaving. She is the youngest ever playing member of the Devonport Brass Band. She has been playing with the band for the past 10 years. Emily makes numerous public appearances including playing the "Last Post" at the Devonport RSL commemorative services. Emily completed her Bronze Medallion and volunteers patrolling beaches as a Surf Life Saver. She is an outstanding student, musician and a volunteer committee member of many school programs and community organisations and most recently became an "Austswim Instructor".
Meg Good, 25 of Hobart is a PhD Candidate at the UTAS Faculty of Law, specialising in environmental and human rights law. Meg has created a number of initiatives including in 2011 two inaugural law school competitions (the UTAS Law Moot Competition and the ISSP Soccer Tournament) which have been run annually by TULS since. In January 2013, she co-ordinated Tasmania's first Animal Law Conference which was funded by a $10,000 Voiceless Grant Meg won in 2012. The Conference was a great success, featuring 18 speakers and participants from across Australia. Meg is a former PASS Leader/ISSP Tutor, and currently works as a lecturer, tutor and research assistant at the law school.
Kate Longey, 28 of Action Park works closely with the refugee community teaching English. She worked as a Youth Pathways Advisor for Colony 47 before coming to Dominic College in 2010 to construct a pilot program for disengaged students called The Magone Program. In 2012, Kate was asked by the Principal to step into the role of Lead Teacher for the program. Kate is also responsible for implementing the "Rock and Water" Program, designed to teach young people safety, integrity, solidarity, self control, self confidence and self respect. Last year, full time student numbers increased by 29% and retention of students within the program has increased by 17%.
Jack Beardsley, 23 of Fern Tree is a Team Manager in the State Emergency Service, Southern Tasmanian Search and Rescue Team and an active member of the Operations Team. Jack took on the Managers role in 2011 and is on call 24/7. He is involved in the coordination, training and management of 30 volunteers. Jack is a qualified first-aid instructor, remote area navigation, land search techniques, four wheeled driving, helicopter operations, wilderness survival and tracking. Last year the Search and Rescue team responded to 11 incidents, with Jack coordinating all of them. He is also actively involved with Camp Quality and the Tasmanian Fire Service.
O Group Trade and Enterprise Achievement Award
Maja Veit, 25 of South Hobart started Silver Hill Fisch in June 2010, producing and selling boutique style seafood sausages. Silver Hill Fisch has won a Silver Medal at the Tasmanian Royal Fine Food Awards. Maja applied her graphic design skills to develop the business logo, signage and stickers and a custom-fitted mobile food van for "easy trade" at small events and markets. Her product is stocked in delis around Hobart and she holds food stalls selling salmon sausages at five Summer Festivals and has a permanent position at the local Farm Gate Market. Maja also has a retail outlet in Queensland. She has completed a Cert IV in Small Business Management and Marketing.
Bokprint Arts Award
Fernando do Campo, 26 of Newstead is a contemporary artist. He recently undertook a three month residency at the Rosamond McCullough Studio at the Cite International des Arts in Paris. In 2012, he was awarded five funded applications including a 2012 Artsbridge Grant by Arts Tasmania. Fernando has a leadership role in supervising the Australian Pavilion at the 2013 Venice Biennale and is part of a mentorship program to oversee and curate "An Awfully Beautiful Place: The Antarctic Art of Stephen Eastaugh" at the Carnegie Gallery, Hobart. He held three solo exhibitions last year and has been selected for a solo exhibition in Launceston.
Premier's Young Achiever of the Year
Cait Clarke, 25 of Railton
Damien Almond, 21 of Rokeby is a well respected community volunteer. For over 13 years, he has been an active member of St John Ambulance and was part of a contingent that assisted the Queensland flood victims. Damien has also been a volunteer with Ambulance Tasmania since 2009. He completed a Cert IV in Basic Emergency Care and was promoted to Volunteer Co-ordinator of St Helens Station in 2011, where he gives over 200 hours each month. Damien also volunteers for St Helens Marine Rescue and is a volunteer Announcer and Production Technician for StarFM Community Radio as well as the Wireless Institute Civil Emergency Network.
Laura Sykes,19 of Sandy Bay is Founder and President of Go Fair Inc, a non profit organisation educating and empowering the community about fair trade and ethical consumerism. Laura works closely with Councils towards fair trade accreditation and organises seminars and campaigns for local businesses and schools. In May 2011, Laura was Event Manager and Coordinator of the Fairtrade Fiesta which attracted over 4,000 people. She is also involved with numerous other community organisations and is State Director of World Vision's youth movement Vision Generation. Laura advocates for fair trade at a local, national and global level.
University of Tasmanian, Faculty of Education Teaching Excellence Award
Holly Barnewall, 26 of Whitemark has embraced the challenges faced by Flinders Island youth. She organises most of the Schools extra curricular activities including camps, movie nights, picnics and BBQ's. With the assistance of the Island Youth Officer, she ensures there are always afterschool and weekend activities for students and is always there for those in need. She introduced a 'transition' subject at school which prepares students for life off the Island including a visit to Launceston or Melbourne. She has rewritten English, SOSE & Related Arts programs to be more relevant and exciting. Holly also organises students in the UTAS Student Voices program.
Print Applied Technology Sports Award
Matt Goss,25 of Berriedale moved through the Tasmanian cycling pathway before heading to Europe to chase his cycling dream. It was an extraordinary year in 2011 for Matt, finishing 2nd in the World Road Cycling Championships, winning the Milan San Remo one day classic and becoming the number 1 ranked cyclist in the world. Matt also competed in the Tour De France, finishing in 2nd place in one stage of the event. He was also awarded the Southern Cross TIS Tasmanian Athlete of the Year Award. Matt is a proud Tasmanian and takes every opportunity to assist with the development of young Tasmanian cyclists.
Emma Flukes,23 of Lindisfarne is a marine scientist with the Institute for Marine and Antarctic Studies and is completing a PhD. She is investigating the formation of 'barren habitat' caused by the long-spined sea urchin and has won numerous awards for her research. This species poses the single largest environmental threat to Tasmanian reefs and the fisheries they support. Emma is also an avid underwater photographer and editor for a highly successful local marine magazine. She is involved with rubbish clean ups around waterways and volunteer groups to conserve marine species. She also volunteered in the Philippines collecting shark research.
Danielle Black, 25 of Montrose owns her own business - maXreaction Graphic Design Studio. Launched in 2005, maXreaction is a very successful and highly sought after design studio that offers complete design services from initial concept through to final production. Danielle's clients include some of Tasmania's best known businesses and events. She also has a Cert IV in Small Business Management. Danielle was asked to join Plants Management Australia in 2009 as Senior Graphic Designer. Her remarkable impact on the business is said to be a major contributor to the company being named 2011 Telstra Tasmanian Business of the Year.
TEMCO Science & Technology Award
Catherine Blizzard, 28 of Hobart graduated with her PhD in Philosophy in 2011.She has commenced a prestigious three year postdoctoral fellowship with the Motor Neuron Disease Research Institute Australia. Catherine was awarded the Menzies Research Institute Tasmanian Student of the Year and a travel fellowship from the International Brain Research Organisation to present her work at the 8th World Congress on Brain Research in Italy. She has won many awards, published research papers in several high ranking international journals and was invited to become the sole author of the International MND Association Quarterly Research Report.
Cait Clarke, 25 of Railton is Chairperson of the Kentish Youth Council with input into Council's strategic direction. She also contributes through the Guiding Coalition and the Social Inclusion Action Group. Cait re-established the Sheffield Girl Guides and was instrumental in the development of the Dreamcasters program nurturing young women into future leaders. She is a driver mentor with the Road Education Volunteers Program. Cait is trained in suicide intervention and youth leadership. She is a single mum and developed the 'Walkers and Talkers' program for young mothers and became the youngest ever Kentish Councillor.
Stanislav Shabala, 27 of Lindisfarne
Tasmania Together Community Service Award
Sarah Perry, 23 of Risdon Vale is a volunteer with the Warrane Mornington Neighbourhood Centre. She was integral in introducing a youth program to the Centre with twenty young people attending each week. The Centre then became involved with the Youth, Community, Ownership, Prevention and Education Project, in conjunction with other local Centres. Sarah worked with the Project Officer to develop a trial holiday program for young people which proved a great success. She also had a governing role in the award winning TOOL Project. Sarah has been President of the Centre since 2008, the youngest President of a Neighbourhood Centre in Tasmania.
Academy of the Arts: UTAS School of Visual and Performing Arts & Tasmanian Polytechnic Arts Award
Michael Lampard, 25 of Battery Point graduated with a Masters of Music Degree and has studied in Düsseldorf and New York City. He has worked with many of Australia's leading organisations and several international orchestras and opera companies in Europe and Asia. Michael has won many awards, recorded a CD and launched HIP-POCKET OPERA with a Gala concert in Hobart. The concert featured performances by some of Hobart's most renowned classical singers including Ben Davidson and Sharon Prero and saw Michael perform with emerging Hobart soprano Melinda Briton.
Spirit of Tasmania Award
Simon French, 27 of Cremorne Simon is a former Junior Australian National Champion in Downhill Mountain Biking. He established 'Dirt Art', a company that constructs mountain bike facilities. He volunteers when not working as a nurse, advocating for ongoing development of mountain bike parks in Glenorchy and Clarence and volunteers and coordinates volunteer days to build tracks and trails. He is President of Hobart Wheelers Dirt Devils Cycling Club and also the Clarence Mountain Bike Park Association. Simon is Vice President of Glenorchy Mountain Bike Park Association and a Downhill Rider Representative for Mountain Bike Australia.
Eddie Ockenden, 24 of Moonah has been a permanent member of the Kookaburras since 2007. He won the 2007 and 2008 Champions Trophy Most Promising Player Award, 2008 World Young Player of the Year and was chosen in the 2008 and 2009 World All Stars Teams. He was part of the bronze medal team at the 2008 Beijing Olympics. 2010 saw Eddie win 3 gold medals in each of the prestigious Champions Trophy, World Cup and Commonwealth Games. While he is now based at the AIS for Hockey in Perth and also plays in the Dutch National League, Eddie still runs coaching clinics and visits schools when he comes home.
Gregory Irons, 28 of Brighton became Director of Bonorong Wildlife Sanctuary at 25, and has since made it his mission to educate and empower the local, national and global community to help save the Tasmanian Wildlife. His FOC emergency wildlife rescue program has attracted more than 200 volunteers. Greg had trained volunteers in rescue, transport and care. He is focused on grass roots education and community empowerment. He networks with other organisations such as the RSCPA and Parks and Wildlife. When Greg isn't rescuing or educating, he is visiting the children's ward of the Royal Hobart Hospital with baby wombats and blue tongue lizards.
Lisa Tedeschi, 27 of Launceston became unwell with a life threatening auto immune disease know as Systemic Lupus Erythematosus whilst studying. Once in remission, she completed certificate studies and established "Ahead in Time Hairdressing Salon" at age 20. After growing the business to 11 staff, she sold the Salon to an employee and started Sebachi Ladies and Men's Fashion clothing store which now has a team of 5. Lisa later developed Mint Bath and Body Beauty before expanding the retail store to include a beauty salon. Lisa has won many business awards and was President of Hair and Beauty Tasmania in 2009, and is currently Vice President of City Prom.
Stanislav Shabala, 27 of Lindisfarneis an ARC Super Science Fellow in the Astrophysics and Environmental Geodesy groups at the University of Tasmania. He was awarded the Bok Prize for the top Astrophysics Honours thesis in Australia in 2003. Stas completed a PhD at the University of Cambridge and held a research fellowship at Oxford. His research is on the physics of black holes and galaxies. He has published proficiently in journals and is an invited speaker at international conferences. His work on black holes has practical applications in climate science and geophysics, helping to answer vital questions about the deformation of our continent and sea level rise.
Camerons Leadership and Innovation Award
Abyilene McGuire, 28 of Blackmans Bay is the Senior Environmental Health Officer at Kingborough Council. She was a 2010 national ambassador for the Year of Women in Local Government and served on the State Board of Environmental Health Australia for eight years. Abyilene has worked in community leadership roles for many years through Girl Guides Australia at local, state, national and international levels. She participated in numerous leadership forums and conferences including the world-first UN Environment Program 'Global Town Hall' meeting on climate change and attended the UN Climate Change Conference in Copenhagen.
Fonterra Agricultural Award
Joe Bennett, 28 of Great Bay established Get Shucked in 2004. The business has become a highly efficient oyster production system, supplying 140,000 dozen high quality oysters each year to the Australian market. Joe employs 3 fulltime staff plus additional shop staff in peak seasons. Get Shucked also splits oysters at their onsite farm shop for local restaurants and for sales direct to the public. The Get Shucked Oyster Shop on Bruny Island had a record year in 2010. A new shop and processing facility are in the planning stage. Joe enjoyed a very successful joint stall with 2010 Premier's Young Achiever of the Year Will Bignell, at the Taste Festival.
Will Bignell, 27 of Sandy Bay
Eva Mackinley, 20 of West Hobarthas not had an easy or comfortable life. She works four jobs to provide for herself, and solely care for her 15 year old brother. Eva has also saved relentlessly to enrol in a Bachelor of Arts with a focus on International Relations. In spite of her hardships, she gives selflessly of her time to improve the lives of others and has influenced the lives of many through the Festival of Dreams program, Tasmanian Youth Forum, World Vision, TasKids Carers, Invisible Children and St Vincent de Paul. In 2009 she was appointed to the Australian Youth Forum Steering Committee. Eva is also involved with Celebrate Tasmania and has a passionate interest in the arts through theatre company Plot.
MyState Financial Arts Award
Josie Hurst, 28 of West Hobarthas a multimedia degree Bachelor of Fine Arts/Bachelor of Information Systems. As the Devonport Regional Gallery Exhibitions and Public Programs Officer, she developed and managed over 50 programs ranging from life drawing classes, contemporary jewellery workshops, graffiti workshops, puppetry, African music, dancing, talks and forums. In 2009 Josie commenced a Masters of Cultural Heritage. She runs education programs at Moonah Arts Centre and is co-curator of youth program Soft Centre, part of Glenorchy Council's Works Festival. She has key involvement and board membership of numerous significant arts organisations in Tasmania.
Danni Murfet, 21 of Launceston is driven to make a difference in the lives of others and to work with the disabled, disadvantaged and those in need. She is working toward her Community Services Diploma and her Disability Services Certificate IV. Danni is currently working for the Northern Residential Support Group as a Disability Residential Support Assistant. With two friends, she started a youth support group "allsorts" for young gay people in Launceston. There are now also groups running in the South and North West. She is also involved with the Youth Network of Tasmania, Tasmanian Youth Forum and volunteers with St John Ambulance and Speak Out Tasmania.
Eddie Ockenden, 22 of Moonah North was the 2008 FIH World Junior Hockey Player of the Year. He is also a member of the FIH 2008 World Team. He was a member of the National Team that qualified for the 2010 World Cup by beating New Zealand in Invercargill in August 2008 and won a Bronze medal at the 2008 Beijing Olympics. He has been playing in the Dutch National League and was a member of the Australian team in the 2009 Men's Champions Trophy Tournament. Eddie was part of the National Team that won the 2010 World Cup recently and also the Hamburg Cup in 2009.
Kirsty Albion, 22 of Lauderdale is a ranger at Mt Field and Maria Island National Parks, Bonorong Wildlife Park and with The Spirit of Tasmania where she educates people in how to protect our natural heritage. She also supports the development of young environmental leaders, believing them to be the solution to the climate crisis. Kirsty lead a delegation of 15 young Australians and 11 Pacific Island youth to the international climate negotiations in Copenhagen. She is building youth climate networks throughout Australia in her role as National Volunteer Co-ordinator for the Australian Youth Climate Coalition, Australia's largest youth organisation. Kirsty is a Founding Member of Project Survival Pacific.
Shanna Sweeney, 28 of Geilston Bay has a Bachelor of Business Administration at UTAS. She worked in a number of Property Manager role's with Quest Serviced Apartments and is into property investing with numerous personal investment properties. More recently Shanna opened Tasmania's first cupcake store. "Cutie Cups" is located in Hobart's Elizabeth St Mall. All ideas, recipes and concepts are wholly and solely Shanna's. She aims to franchise the business and is in discussion with numerous interstate and overseas interests. Shanna sponsors the RSPCA Cupcake Day, Breast Cancer Pink Ribbon afternoon teas, a youth battle of the bands and the "baby-teresa" initiative.
Jessica Andrewartha, 26 of Lenah Valley is a Civil Engineer and the API Research Fellow in the Centre for Renewable Energy and Power Systems at the University of Tasmania. She submitted her PhD thesis in August 2009 and is undertaking research in the areas of renewable energy and experimental fluid dynamics. Jessica is investigating methods to make the delivery of water from dams to hydropower stations more efficient. She has presented her research at international conferences and has had numerous papers published. She lectures at the School of Engineering and volunteers her time to speak at schools. Jessica is Chair of Young Engineers Australia representing over 40,000 engineers nationally.
Forestry Tasmania Regional Initiative Award
Christopher Cusick, 29 of Nugent is an apprentice at Wursthaus. He is President of the Nugent Community & Sports Association and was until recently Secretary of South East Field and Game. Christopher is also on the steering committee for Bendigo Bank in the hope of opening up a community branch in Sorell. He organises "Clean up Australia" day, and events for the Bream Creek Show. The Bream Creek vs Nugent Football Match received a certificate of recognition for Community Event of the Year at the Australia Day awards. A ride on lawn mower race between Nugent & Bream Creek is the main event.
Will Bignell, 27 of Sandy Bay is nearing the completion of his doctorate with UTAS and CSIRO. He is investigating enhancing long chain omega-3 content in Australian lamb and improving sheep production. Will also co-manages the family farm, "Thorpe", at Bothwell, as the 7th generation on the farm. He has a strong passion for sustainable agriculture and produces award winning, high quality produce for his customers. Will has been an active member in the community, donating time on the executives of University clubs and societies, agricultural advisory boards and helping to inspire students to undertake tertiary education.
Scott Brennan, 25 of Lindisfarne
Rio Tinto Alcan Community Service Award
Jessica Jacobson, 20 of Ulverstone spent twelve months overseas as an ambassador for Tasmania after finishing school. She returned with a passion to eradicate poverty. She is involved in Make Poverty History events and workshops and is also involved in Coat Day which has become an international event. Over 50,000 coats have been donated for those in need. Jessica was the longest serving President of youth group Enormity receiving Life Membership. She mentors at schools and speaks to community groups to raise awareness and funds for international human rights issues. Jessica spent time volunteering at an orphanage in India last year. She is also working hard to turn around poverty in Braddon. She was awarded the Central Coast Young Citizen Award for her community service.
Finegan Kruckemeyer, 27 of West Hobarthas had 27 plays commissioned and performed by professional theatre companies in Australia and America. Additionally Finegan has 17 new works set for professional seasons over the next 3 years in Australia, England, China, Spain, Singapore, Scotland and America. He has received key theatre awards and has taught drama and playwriting around Australia for the past 10 years. Finegan created and ran an integrated company in Hobart to include the disabled, has written works and performed in numerous national and international festivals. He is the Editor of national arts magazine Lowdown and sits on various arts boards and panels.
Josh sutton, 23 of Devonport started his apprenticeship as a chef at Essence Food and Wine in October 2002. After a change of business ownership in 2005 he took on the role as Head Chef, managing the complete kitchen operations. He is also the trainer of 3 apprentice chefs and 2 trainee chefs. Josh's highlights include 1st place in the state-wide Whirlpool cooking competition. He was selected as Tasmania's representative Chef in the "Produce of Heaven Campaign" which incorporated a fine dining dinner and tradeshow stand in Taipei, Taiwan. Josh volunteers with numerous organisations including the "Grans Van" charity organisation by preparing food and services to the homeless of Devonport and surrounds.
Scott Brennan, 25 of Lindisfarne was devastated with his 7th place at the 2004 Olympics and was determined to do better. He finished 1st in the national double and single scull in 2006. He suffered injury in 2007 but came back to compete in the Australian men's double scull for the World Championships in Munich setting an Australian record in the regatta and placing 8th. In 2008 he came 1st in the National senior double scull and quad scull and 3rd in the Lucerne and Munich World Cups. Scott went through the Beijing Olympic regatta undefeated in the men's double scull beating world and Olympic champions on the way to winning the gold medal. He has also completed his medical degree.
Jan Zika, 26 of West Hobart is a Climate Scientist at CSIRO Marine and Atmospheric in Hobart and is currently completing his PhD. He was chosen as one of ten students from around the world to receive training in climate dynamics in the US. Jan has developed an innovative method for understanding the movement of heat and CO2 in the Ocean and improves models of the climate and global warming. His new "inverse method" is less sensitive to error than previous methods. His studies of the Southern Ocean and Atlantic are being published in international scientific journals with plans for collaborators to implement his techniques to the Global Ocean. He gives considerable time promoting science at schools and community events. Jan is also a volunteer with Amnesty International.
O Group Trade and Career Achievement Award
Brad Smith, 21 of Legana trades most nights on the US stock market and spends every day growing his businesses. Brad owns and operates 2 mini motocross superstores, runs an importing business and also a wholesale business. He designs his own brand of mini bike and has his own range of clothing, parts and accessories all under the brand name "Braaap". He also owns several on-line businesses and is working towards franchising his specialised retail stores. He spends a great deal of time in China where his bikes are manufactured and attends shows and motocross events. Brad has been responsible for building 4 mini motocross tracks in Tasmania and has a vision to have mini tracks built near all local townships to keep young people off the streets.
Peter While, 27 of Hobartis a post-doctoral researcher in the field of magnetic resonance imaging. He was awarded a University Medal in 2003 and completed his PhD in mathematical physics in less than three years, at age 26. He has a formidable publication history in MRI coil design through numerous international publications and is a highly commended communicator, presenting his work as far afield as Berlin and Honolulu. His most recent work represents an entirely new design methodology for MRI scanners and has been described by experts in the field as a landmark step in MRI research. These designs promise to improve the performance of scanners and patient comfort during scanning and are to undergo testing in Queensland for commercial application.
Melissa Krushka, 26 of Scottsdale has been volunteering for the Australian Breastfeeding Association in Northern Tasmania for over 8 years. She has given over 2,000 voluntary hours to supporting and educating breastfeeding mothers and their families. Melissa provides regular get-togethers for social and educational purposes for mothers in the Dorset area. She is the Locality Mother, Meeting Hostess, Library Officer, Community Educator and is a qualified Trainee Breastfeeding Counsellor. She also provides homed cooked meals for those who need extra help. Melissa annually organises a Parenting Facilities Tent at the Scottsdale Show and organised an ABA accredited Baby Care Room in Scottsdale. She is also President of the Scottsdale P & F Association.
Aaron Mackril, 27 of Lenah Valley
Aaron Mackril, 27 of Lenah Valley was born with Cystic Fibrosis and was not expected to live into his teens. Falling very ill as a teenager, he was given a second chance at life when he received a double lung transplant. Since then, his health has improved and he now works as a Registered Nurse and also dedicates more than 20 hours each week to raising awareness of various causes including Cystic Fibrosis, diabetes and organ donation. Aaron has recently become President of Cystic Fibrosis, is an executive member of Transplant Australia and an integral member of Diabetes Australia working with teen diabetics.
Annika Koops, 24 of Beaumaris was awarded an Artist in Residence position at "Foundation B.a.d" in Rotterdam in early 2007 where she is working and researching. Annika was also awarded the prestigious Keith and Elizabeth Murdock Travelling Fellowship and has travelled extensively around Europe attending cultural events and visiting institutions where she will potentially finish her MFA. She has been a part of several shows and events in 2007 and held a solo exhibition of her paintings and photographic works at BUS Gallery. Recently she completed a show at "Art Rotterdam International Art Fair" and has been awarded a grant by Arts Tasmania for the production of a solo show scheduled for early 2009.
Katy Pakinga, 27 of Burnie believes in making opportunities happen, not waiting for them to happen. In 2003 Katy created the contemporary Burnie Youth Choir and just 4 years later, the Choir won the best show/pop choir at the Performing Arts Challenge in Sydney. Katy has also opened her own successful performing arts studio "Encore". She is an inspirational role model with significant success participating in, and coordinating numerous musical and other events. She has raised the profile of Burnie and Tasmania through many media stories across the country about her successes. Katy was rewarded with the Burnie Young Citizen Award in 2005 for her community contribution .
Scott Brennan, 25 of Lindisfarne achieved exceptional results as a junior rower, and in his first senior event won Gold in the quad scull in Lucerne at the World Cup. Scott was devastated with his seventh place at the 2004 Olympics and was determined to improve. He finished 1st in the interstate single scull in 2005 and 2006 and 1st in national double and single scull in 2006. After suffering a significant injury in 2007, he came back to compete in the Australian men's double scull for the World Championships in Munich setting an Australian record in the regatta and placing 6th in the semi finals. Scott is now focused on the 2008 Olympics and completing his medical degree.
Abyilene Dobson, 25 of Blackmans Bay is an environmental Health Officer with the Kingborough Council and is a Branch Councillor with the Australian Institute of Environmental Health. Her role is to ensure a healthy and safe environment through the implementation of public and environmental health programs. Abyilene conducts numerous education programs and coordinates projects including the Bruny Island Water Supply for which she received an Industry Award for her work. She has also recently undertaken volunteer work in Peru, South America to help improve education and hygiene standards about safety of the water and wastewater management to minimise contamination in the remote Angian mountain villages.
TAFE Tasmania Trade and Career Achievement Award
Leah Brown, 23 of Lindisfarne is a Barrister and Solicitor. She is the first Indigenous female lawyer in Tasmania and was the Aboriginal student representative at UTAS and also represented her community as President of the Aboriginal Centre. She attended the 1999 Youth Reconciliation Convention and the World Indigenous Peoples Convention on Education in Canada in 2002. She is the cofounder of Aboriginal Dance Groups and was awarded the 2003 Aborigine of the Year, later receiving the Centenary Medal of Australia. Leah is now actively involved in assisting with the creation of new government legislation to protect Aboriginal Heritage and assists traditional owners in the area of native title.
James Hamilton, 25 of West Launceston graduated with first class honours with a Bachelor in Engineering, and became the 2005 engineering graduate in profile. In 2007 he was named the Engineers Australia Young Mechanical Engineer of the Year. James has devoted the past two years to the design, commissioning and development of a commercial "vibrocompacted drained aluminium cell" for Rio Tinto Alcan, which reduces the environmental footprint with substantially lower power requirements. James recently initiated the "Engineering Advisory on Industrial Development" organisation and plans to further develop his "Total Environment and Conservation House" electronic education tool which is all about education for the home environment.
Esther Rubenach, 24 of Gray dedicates over 50 hours in voluntary service each week. She is a volunteer ambulance officer and is also trained in road crash rescue in the State Emergency Service. During the devastating bush fires, Esther was called to an accident where a young fire fighter was killed. She volunteered to stay with the body to assist with execration even though burning trees were falling all around. Esther is also a driver for the Community Transport Service and is actively involved as a member and past secretary of the Tasmanian Lymphoedema Centre at St Marys. She is enrolled in Nursing in 2008, but travels to Launceston rather than relocating, so that she can continue her community activities.
Phillip Pullinger, 26 of Hobart
Rio Tinto Alminium Community Service Award
Jessica Brown, 23 of Howrah is passionately committed and dedicated to community service work which has inspired many young people to make volunteering an important part of their lives to. Overcoming significant personal challenge, she has achieved a great deal including an Advanced Diploma and University Degrees. Jessica is proud of her Aboriginality and is a Mentor for the Aboriginal Personal Pathways Program and she is the Primary Carer for her grandmother as well as a Targa Tasmania volunteer. Jessica for many years has contributed significant voluntary hours to the Tasmania Fire Service, and is also a Project Team member and a volunteer Training Instructor for the Fire Service.
islandstate Arts Award
Pete Cornelius, 23 of St Marysreleased his first CD "stepping out in blue" at just 13. In 1999 Pete Cornelius and the DeVilles were formed and on their second tour, along with many Festivals they performed at the Australian Blues Festival in Goulburn and the Frankston International Guitar Festival in Victoria. Pete was awarded the Best Blues/Jazz Guitarist Award at both Festivals. The band has since toured Australia on numerous occasions. The band has produced many CD's and has successfully toured the US twice and also the UK. Pete signed an endorsement deal with Fender, produced a CD with Chicago Blues star Steve Avery and is now also coordinator of the Falmouth Festival.
Robyn McKinnon, 23 of Longford Age is a survivor of sexual abuse. Moving to Tasmania at 18, she now uses her personal life experience to assist hurting and broken young people. Robyn is dedicated to raising awareness of the prevalence and effects of Child Sexual Abuse and developing programs to support young people in their transition to adulthood. She is also coordinator of the Rural Co-Pilots program in the Northern Midlands aimed to help and encourage young people to reach their full potential and has trained over 40 people to become mentors of programs that she organises for young people. Robyn is also a member of the Tasmanian Youth Consultative Committee and other community organisations.
Printing Authority of Tasmania Sports Award
Johanna Allston, 21 of Fern Tree became the Senior and Junior World Champion in Orienteering at only 20. She is the first non European to win an orienteering medal and the first orienteer ever to win a Gold Medal at both Junior and Senior levels. Hanny is also the first junior to win at senior level in the 40 year history of senior World Championships. She is an amazing athlete with an amazing training regime and holds a TIS scholarship. Locally she runs training camps for Tasmanian orienteers and provides coaching and support in running club events. At school Hanny achieved a National TER score of 98.75 and is now studying medicine in and around her sport.
Phillip Pullinger, 26 of Hobart is a medical doctor with a passion for environmental protection and a commitment to public health issues. He has made a significant contribution to a number of key environmental groups including the Cam River Action Group, The Wilderness Society, Doctors For Forests and Environment Tasmania. Phillip was President of the Tarkine National Coalition and was a founding member and is the current convenor of Environment Tasmania a peak environment group focused on forest, marine and coastal conservation with 22 environment group members representing over 5500 Tasmanians.
TAFE Tasmania Career Achievement Award
Adrian Bold, 25 of Hobart established an innovative media business at only 19 years of age while still studying full time at University. 5 years on, his company has grown into a dynamic, sustainable and profitable business, achieving annual sales growth of 200%. Clients range from publicly listed development firms to Government Agencies. Adrian's business produces 3D Digital Content that leads the way internationally in 3D technology, visually communicating new projects before they exist. Adrian has become a sought after speaker, was the 2004 Australasian Student Entrepreneur of the Year and was selected as one of 12 leading global young entrepreneurs.
Luke Bereznicki, 26 of Kingston graduated from the School of Pharmacy with first class honours and was an inaugural recipient of a National Institute of Clinical Studies PhD Scholarship. His research has been focused on strategies to improve the quality of anticoagulants particularly with the drug Warfarin in the hospital and community settings. Luke has published numerous international papers and has won many awards and was the 2006 National Prescribing Service Quality Use of Medicine Student of the Year for his pioneering achievements in the field. Luke is also a lecturer in Pharmacy Practice at the Tasmanian School of Pharmacy.
Karena Brown, 23 of Burnie left school at a young age, working hard to save money towards her vision and dream of owning her own child care business. At only 19 she started "Maddington Child Services". Working long hours from home providing children's parties and "in home" carer services, the business expanded rapidly. Karena rented a derelict property and after totally renovating it, achieved her dream of being able to service her community through her own child care centre. In the past 12 months the business has flourished and Karena has opened another 3 fully licensed centres with plans for another 2. KLB Enterprises is now one of the largest child care facilities in Tasmania.
Jyoti Chuckowree, 26 of South Hobart
Comalco Community Service Award
April Chivers, 18 of Lenah Valley has overcome significant personal adversity, always putting others first. April was extremely active at Claremont College and is an active member of the Glenorchy Youth Task Force participating in many projects such as the 2005 long lunch, big breakfast bash and "Butt Ugly" a very successful anti smoking education program. April prepares and presents workshops, conferences and consults with committees on youth issues and is a member of the Tasmanian Youth Consultative Committee. She volunteers for the Brain Injury Association of Tasmania, MS Society and World Vision. Suffering from depression, April is now working on a project to help other young people with depression.
William Lane, 23 of Sandy Bay is a violinist of local, national and international acclaim. In 2004 William toured Europe with the Australian Youth Orchestra and upon winning the Claudio Alcorso Scholarship fulfilled residencies in Europe and the US. William has performed solo in Italy, Austria, Iceland, the USA and Mexico, with highlights including performances at the International Viola Congress in Iceland and an all Mozart performance at the Ponchielli Theatre in Italy. Recently William won 2 prizes at an international performance competition in Rome.
Robert Gane, 22 of Devonporthas overcome many personal obstacles to achieve. His battles which begun when he was an infant included multiple cancer tumours requiring weekly trips to Melbourne for intensive treatment, tuberculosis and double pneumonia. Despite these setbacks and in remission he has gone on to become a double scholarship winner at University, vice president at his College and a graduate with honours in Engineering being awarded a full blue for representing Tasmania in Rugby Union. Robert plays many sports, is active in Scouts and having been a Camp Quality Camper for many years is now giving back as a Camp Quality Companion encouraging and inspiring young people with cancer.
Mark Jamieson, 21 of Acacia Hills is an AIS Track Endurance Scholarship holder who has won many national and international titles. In 2005 Mark won the Individual Pursuit and the teams' pursuit at the National Track Championships and was named the Australian Track Champs Rider of the Year. Mark came third at the Los Angeles World Track Championships in the Teams Pursuit, second in the Individual Pursuit at the Madrid Track World Cup and won in the Teams Pursuit at the Moscow Track World Cup.
Matthew Goss, 19 of Prospect is an AIS Junior Track/Road Cycling Scholarship holder based in Italy. Matthew has won many national and international titles and was the 2004 Junior World Champion. In 2005 in seniors he won the teams' pursuit and came second in the points' race at the National Track Championships. Matthew came third at the Los Angeles World Track Championships in the Teams Pursuit, first in the Teams Pursuit at the Moscow Track World Cup and first in a stage of the Tour of Japan and second in the Jayco Herald Sun Tour.
Lucy Harlow, 26 of Lenah Valley is currently undertaking her PhD within the School of Plant Sciences at the University of Tasmania. Lucy has volunteered and assisted with numerous Science programs and is researching Toxic algal blooms which are a worldwide threat to aquaculture, fisheries, tourism and public health. Lucy has successfully developed methods for genetic identification of toxic algae using molecular tools. She has designed and developed a gene probe to detect toxic micro-algae in environmental samples, including ship ballast water which will be essential to prevent the introduction of algae to ports. Lucy's research has been published in several prestigious journals.
Chris Johnson, 26 of Hobart is the Station Manager of vibrant Hobart community youth radio station, Edge Radio 99.3FM. Through great vision and energy Chris has taken Edge from a fledgling station in 2002 to become an intrinsic part of Hobart's youth scene. Chris is well positioned to manage the station with qualifications in audio engineering and a Bachelor of Arts in journalism and human geography together with years of technical experience and a solid grounding in local music. Chris has been heavily involved and totally dedicated in all areas of the station from its very beginnings, including sales, promotions, music, technical and management.
Jyoti Chuckowree, 26 of South Hobartcompleted her Bachelor of Science degree with first class honours as a member of the NeuroRepair Group at the University of Tasmania, recently submitting her PhD. Jyoti's research focuses on how the adult brain responds to trauma following injury and that precursor cells contribute to brain healing. These discoveries have highlighted aspects of the brain's response to injury that may be targeted to promote recovery from brain trauma. Jyoti has presented her work at numerous conferences and her research has been published in several prestigious international journals. Jyoti has recently been awarded a CJ Martin Fellowship, which will see her travel to Switzerland to research for 2 years.
Damian Atkins, 26 of Delorainegrew up on a dairy, sheep and beef farm. After his studies, an apprenticeship on his parents operation and a position with a Victorian artificial breeding firm, Damian purchased North West Breeding Services. Damian regularly travels overseas to inspect donor animals, learn new genetic advancements and source genetic material from 40 countries for importation whilst bringing back international best practice. The business has fast become the biggest independent genetics provider in Tasmania and one of the largest in Australia with two of the worlds' largest genetics suppliers currently negotiating to buy Damian's business.
Southern Cross Young Achiever of the Year
Amy Cutler, 25 of West Launceston
Thuy Shaw, 24 of West Hobart arrived in Australia as a "boat person" in the 1980's and knows too well the hardships and trauma that face refugees. The Seven Seas, Seven Sands, 7-day Refugee Week festival is testament to her passion for helping refugees. She advocates for refugees and has educated professionals and community members alike through cross cultural awareness training. Thuy has been a keynote speaker at many conferences talking about barriers and hardships. She also featured in the documentaries "Finding the Tasmanian voice" and "Bonza girls" as well as being amongst the Mercury's Top ten Tasmanians to watch in 2005.
Hobart Ports Corporation Arts Award
Amy Cutler, 25 of West Launceston grew up studying violin, piano and voice. In 1997 she won a scholarship to the Queensland Conservatorium completing her Bachelor of Music in Voice with Professor Janet Delpratt. Amy has performed professionally in Opera, Cabaret and with the Orchestra and Ballet, releasing her first classical CD in 2003. As well as singing, Amy has developed a career writing music, winning the Noise/Triple J remix competition as well as co-writing the theme song for the international children's television show "The Sleepover Club." Amy won 2 National Musicoz awards for original music in 2004 and is currently working on an original orchestral recording.
Matthew Wells, 26 of Glenorchy is Co-Captain of the Australian Mens Hockey Team, first representing Australia in 1995. Matthew was a member of the gold medal winning Junior World Cup team in 1997 before being selected in the national senior squad in 1998. He has represented Australia in many major tournaments, winning gold in the 2004 Olympic Games; bronze in the 2000 Olympic Games; silver in the 2002 World Cup and gold in the 2002 Commonwealth Games. Matthew also plays in the National Hockey League for the Tassie Tigers and some of his awards include the 2001 and 2003 League's Best and Fairest and the 2003 Australian International Player of the Year.
Rachel Anderson, 26 of Glenlusk completed her Science Degree at the University of Tasmania in 2001 and graduated with Honours, majoring in Environmental Science. Rachel has continued her research into saving the habitat and the endangered Ptunarra Brown Butterfly whilst undertaking her PhD. Rachel is involved with many community groups working on various environmental projects. She also works with children in schools helping them undertake environmental research and establish Butterfly Gardens. Rachel is also an Environmental Consultant for Guides Tasmania, encouraging all Units to include environmental activities in their program.
The Apprenticeships Specialist Career Achievement Award
Anthony Scotney, 23 of Sandy Bay graduated from the University of Tasmania in 2002 with a Bachelor of Science majoring in Computer Science and Information Systems. He then established JadeLiquid Software Pty Ltd to develop software tools that would enhance the Java programming language. In 2003 Anthony obtained investment from In-tellinc Pty Ltd to commercialise these innovations. Anthony's first product WebRenderer is widely accepted within the international Java community and has achieved sales into 13 countries spanning North America, Europe and Asia. In two years Anthony has established JadeLiquid as one of Tasmania's most successful emerging technology companies.
Fiona Poke, 25 of South Hobart completed a Bachelor of Science degree with first class honours at the University of Tasmania in 2001 and received several awards including a University Medal. Fiona is completing a PhD focusing on finding and developing ways of improving the wood quality of Eucalyptus globules, the main hardwood plantation species grown in temperate Australia and around the world. This work will help optimize productivity from plantations and reduce environmental impacts through the improvement of wood quality. Fiona has had two papers published in scientific journals and was invited to speak at an International Eucalyptus conference in Portugal.
Matthew Hill, 21 of Norfolk is committed to the importance of valuing young people with his involvement in government youth advisory groups and peak government and non-government youth organisations. He has held management and leadership roles within Derwent Valley Online Access Centres, Derwent Valley Community House, Derwent Valley Economic Renewal Group, New Norfolk Historic Information Centre, New Norfolk Township Revitalisation Committee and the New Norfolk Scout Troop. Matthew has helped organise the New Norfolk Christmas Parade and Derwent Valley Council Australia Day Celebrations and is currently involved in the New Norfolk Lions Club and Bushy Park Show Society.
Premier's Young Achiever of the Year
Dr Roger Chung, 26 achieved first class honours in Biochemistry and in 2003 was awarded his PhD. Roger is currently a Research Fellow within the NeuroRepair Group at the School of Medicine, University of Tasmania. His research focuses on understanding how the brain responds to injury or disease. Roger has discovered the ability of a brain protein named metallothionein to directly promote brain recovery following brain injury. This discovery may have clinical applications leading to the development of a treatment for brain trauma patients. Roger has won many awards, published numerous papers, including the prestigious Journal of Neuroscience, and has presented his work at many conferences and forums internationally
Natalia Rodriguez, 23 is currently studying for a Bachelor of Natural Environment and Wilderness at the University of Tasmania. Natalia volunteers a significant number of hours as the Community Liaison Officer for the Lilydale Landcare Group. She is also very active in Green Corps, Conservation Volunteers, Clean Up Australia, National Tree Planting, the Tamar Region Natural Resource Management Strategy and other Weed Strategy Initiatives.
Emma Butler, 24 was born with cerebral palsy. All of her limbs are affected and she requires daily support. Emma is a driving force for Arts Roar Accessible Arts Project, particularly in her role as secretary, averaging 20 voluntary hours each week. Emma also does a great deal for Independent Services. These services cater for the creative and recreational needs of people with disabilities. Emma also has a passion for directing videos that highlight social issues faced by people with disabilities. Emma is a well-known spokesperson for people with a disability, and regularly speaks at forums, conferences and training sessions.
Ben van Tienen, 20 is an extremely talented and versatile artist. His first award, the "Brenda Hean Memorial Award" came in 1997 when he was only 14. The accolades have just kept coming for his singing and piano performances and recitals. Ben has won the Lady Cross Trophy at the Hobart Eisteddfod for the past three years. He conducted his first musical at 16 and has been the Musical Director for the Tasmanian Song Company for the past three years, as well as directing for Annie, La Cage aux Folles, Jesus Christ Superstar and Chicago to name a few. Ben is now furthering his studies at the Sydney Conservatorium of Music.
Dr Roger Chung, 26 achieved first class honours in Biochemistry and in 2003 was awarded his PhD. Roger is currently a Research Fellow within the NeuroRepair Group at the School of Medicine, University of Tasmania. His research focuses on understanding how the brain responds to injury or disease. Roger has discovered the ability of a brain protein named metallothionein to directly promote brain recovery following brain injury. This discovery may have clinical applications leading to the development of a treatment for brain trauma patients. Roger has won many awards, published numerous papers, including the prestigious Journal of Neuroscience, and has presented his work at many conferences and forums internationally.
Kerry Hore, 22 has been an outstanding and consistent achiever for many years winning numerous awards including the 2003 Tasmanian Institute of Sport Female and Overall Athlete of the Year. Kerry is nearing completion of her Bachelor of Pharmacy at the University of Tasmania, currently studying by correspondence so that she can focus on the 2004 Athens Olympics. Kerry has won state and national titles including Gold in the Womens Double and Quad sculls. In July 2003 Kerry won Gold at the World Cup in Luzern followed by Gold in the Quad sculls at the World Rowing Championships in Milan.
Matthew Young, 26 is a young farmer making a significant contribution on the family vegetable, crop, cattle and sheep farm. Matthew is heavily involved in Rural Youth and has held committee positions for the Devonport Club since joining in 1994. He has also been involved in Agfest since 1995. Matthew has won many awards including Tasmanian Rural Youth Young Farmer of the Year and National Young Farmer of the Year and is on many committees and advisory teams including the National Farmers Young Farmers Forum and Young Rural Leaders.
The Apprenticeship Specialist Career Achievement Award
Fiona Bakes, 25 is the owner operator of Reflections Dance Studio, a high energy modern studio catering for all ages developing and inspiring a wide range of dance and theatrical techniques. Since Fiona has taken ownership of the business, student numbers have doubled and profit has increased by 250%. In addition to the three fully equipped studios' in the Burnie facility, additional classes are held in Wynyard and Ulverstone. Graduation for the 600 students each year is one of the biggest annual events held in Burnie with over 2000 guests in attendance. Fiona's next aim is to introduce drama to the studio. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 8,596 |
\section{Introduction}
Let $\Gamma$ be a countable discrete group, and let $\mathcal{G}$ be a family of groups $G$ equipped with bi-invariant metrics $d_G$. The question of \textit{stability of $\Gamma$ with respect to $\mathcal{G}$} asks whether a map $\varphi : \Gamma \to G \in \mathcal{G}$ that is a homomorphism up to a small error, is close to an actual homomorphism. This can be made rigorous in two different ways, that depend on whether one wants the errors and closeness to be pointwise or uniform.
The notion of ``error" is defined as follows:
\begin{definition}
\label{def:def}
Let $\varphi : \Gamma \to G \in \mathcal{G}$ be a map. We define the \textit{defect of $\varphi$ at $(g, h) \in \Gamma^2$} to be
$$\operatorname{def}_{g, h}(\varphi) := d_G(\varphi(gh), \varphi(g) \varphi(h)).$$ The \textit{defect of $\varphi$} is
$$\operatorname{def}(\varphi) := \sup\limits_{g, h \in \Gamma} \operatorname{def}_{g, h}(\varphi).$$
A sequence $(\varphi_n : \Gamma \to G_n \in \mathcal{G})_{n \geq 1}$ is called a \textit{pointwise asymptotic homomorphism} if $\operatorname{def}_{g, h}(\varphi_n) \xrightarrow{n \to \infty} 0$ for all $(g, h) \in \Gamma^2$; and a \textit{uniform asymptotic homomorphism} if $\operatorname{def}(\varphi_n) \xrightarrow{n \to \infty} 0$.
\end{definition}
Other commonly used terms are \textit{almost-representation} \cite{defstab} for the pointwise notion and \textit{quasi-representation} \cite{Shtern} or \textit{$\varepsilon$-representation} \cite{amenst} for the uniform notion. The notion of ``closeness" is defined as follows:
\begin{definition}
\label{def:dist}
Given two maps $\varphi, \psi : \Gamma \to G \in \mathcal{G}$, we define their \textit{distance at $g \in \Gamma$} to be
$$\operatorname{dist}_g(\varphi, \psi) := d_G(\varphi(g), \psi(g));$$
and their \textit{distance} to be
$$\operatorname{dist}(\varphi, \psi) := \sup\limits_{g \in \Gamma} \operatorname{dist}_g(\varphi, \psi).$$
Two asymptotic homomorphisms $(\varphi_n, \psi_n : \Gamma \to G_n \in \mathcal{G})_{n \geq 1}$ are \textit{pointwise asymptotically close} if $\operatorname{dist}_g(\varphi_n, \psi_n) \xrightarrow{n \to \infty} 0$ for all $g \in \Gamma$; and \textit{uniformly asymptotically close} if $\operatorname{dist}(\varphi_n, \psi_n) \xrightarrow{n \to \infty} 0$.
\end{definition}
This leads to the definition of two notions of stability, that we attribute to Ulam, after \cite{Ulam}:
\begin{definition}[Ulam]
\label{def:stab}
The group $\Gamma$ is \textit{pointwise $\mathcal{G}$-stable} if any pointwise asymptotic homomorphism is pointwise asymptotically close to a sequence of homomorphisms. It is \textit{uniformly $\mathcal{G}$-stable} if any uniform asymptotic homomorphism is uniformly asymptotically close to a sequence of homomorphisms.
\end{definition}
Early mentions of these problems can be found in works of von Neumann \cite{vN} and Turing \cite{Turing}. The problem of pointwise stability of $\mathbb{Z}^2$ with respect to certain families of matrices, for instance self-adjoint and with the operator norm, received a lot of attention during the second half of the twentieth century (see \cite{Lin} and the references therein). In \cite[Chapter 6]{Ulam}, Ulam discusses more generally the question of stability of certain functional equations: because of this, the term \textit{Ulam stability} was introduced in \cite{BOT} to refer to uniform stability with respect to unitary groups equipped with the distance induced by the operator norm (see below). Some of the most common families $\mathcal{G}$ of approximating groups are the unitary groups $\operatorname{U}(n)$ or the symmetric groups $S_n$.
$\operatorname{U}(n)$ is typically considered with a metric induced by a norm defined on $\operatorname{M}_n(\mathbb{C})$. The first example is that of the \textit{operator norm}, where pointwise stability has striking topological and $K$-theoretic interpretations \cite{bundles1, bundles2}, all amenable groups are known to be uniformly stable \cite{amenst}, and groups with non-trivial quasimorphisms are known not to be uniformly stable \cite{BOT}. Another example is that of the \textit{Frobenius norm} $\| A \|_{Frob} := \sqrt{|A_{ij}|^2}$, that is, the norm induced by the embedding of $\operatorname{U}(n)$ into $\mathbb{C}^{n \times n}$: this has the advantage of allowing a cohomological criterion for pointwise stability \cite{GLT}. The third main example is given by the \textit{Hilbert--Schmidt norm} $\| A \|_{HS} := \frac{1}{\sqrt{n}} \| A \|_{Frob}$, which is the normalization of the Frobenius norm: this has the advantage of allowing a $C^\ast$-algebraic characterization of pointwise stability \cite{HSstab}, as well as a simple algebraic characterization of uniformly stable groups among finitely generated residually finite ones \cite{uHS}.
On the other hand the groups $S_n$ are studied with the normalized Hamming distance $d_H(\sigma, \tau) := 1 - \frac{1}{n}|Fix(\sigma^{-1} \tau)|$. Pointwise stability of equations in permutation was initially considered by Glebsky and Rivera \cite{GR}, then by Arzhantseva and P\u{a}unescu \cite{a:comm} who proved that this can be translated to a group property, as in Definition \ref{def:stab}. Since then this pointwise stability problem has been under intense investigation, as well as some variants thereof: flexible \cite{T, surf}, quantitative \cite{quant}, uniform, probabilistic \cite{BChap}, and connections to computer science \cite{test}. \\
The pointwise and uniform problems typically exhibit a very different behaviour. For example, consider the family $\mathcal{G} = \{ (\operatorname{U}(n), \| \cdot \|_{op}) : n \geq 1 \}$ of unitary groups equipped with the metric induced by the operator norm, and the two stability problems with respect to $\mathcal{G}$. On the one hand, $\mathbb{Z}^2$ is not pointwise stable \cite{Voie}, but it is uniformly stable, as are all amenable groups \cite{amenst}. On the other hand, a non-abelian free group of finite rank is not uniformly stable \cite{Rolli}, but it is pointwise stable: if $(\varphi_n)_{n \geq 1}$ is a pointwise asymptotic homomorphism, then letting $\psi_n$ be the unique homomorphism that coincides with $\varphi_n$ on a given free basis, $(\psi_n)_{n \geq 1}$ is pointwise asymptotically close to $(\varphi_n)_{n \geq 1}$. \\
In this paper, we study \textit{ultrametric} versions of these problems, that is, we look at approximating families $\mathcal{G}$ whose groups are ultrametric. The main example throughout the paper will be a $p$-adic analogue of Ulam stability: we choose $\operatorname{GL}_n(\mathbb{Z}_p)$ -- which is maximal compact in $\operatorname{GL}_n(\mathbb{Q}_p)$ -- as an analogue of $\operatorname{U}(n)$ -- which is maximal compact in $\operatorname{GL}_n(\mathbb{C})$. The natural norm on $\mathbb{Q}_p$-vector spaces, that is, the one that preserves the non-Archimedean nature, is the $\ell^\infty$-norm relative to the $p$-adic norm $|\cdot|_p$ on $\mathbb{Q}_p$. Keeping this and the case of $\operatorname{U}(n)$ in mind, there are three norms that one could choose to induce a distance on $\operatorname{GL}_n(\mathbb{Z}_p)$: the operator norm with respect to the $\ell^\infty$-norm on $\mathbb{Q}_p^n$, the norm induced by the embedding into $\mathbb{Q}_p^{n \times n}$ with the $\ell^\infty$-norm, and a normalized version of the latter. It turns out that all of these coincide (Lemma \ref{lem:GL}), and so
$$\| \cdot \| : \operatorname{M}_n(\mathbb{Q}_p) \to \mathbb{R}_{\geq 0} : A = (A_{ij})_{1 \leq i, j \leq n} \mapsto \max\limits_{1 \leq i, j \leq n} |A_{ij}|_p$$
is, in some sense, the canonical norm to consider. It induces a bi-invariant ultrametric $d$, and moreover it reflects the profinite structure of $\operatorname{GL}_n(\mathbb{Z}_p)$: in fact $\|A - I \| \leq p^{-k}$ if and only if $A \equiv I \mod p^k$. We denote by $\operatorname{GL}(\mathbb{Z}_p)$ this family of metric groups, and will focus on this example of approximating family for the statements of the results, mentioning which properties we are using. Each result can be generalized to families of groups satisfying such properties, and the statements will be given in full generality in the paper. \\
To the author's knowledge, the only previous mention of $p$-adic versions of stability is in Kazhdan's work \cite[Proposition 1]{amenst}, where it is shown that for every $n \geq 1$ the standard representative map $\varphi : \mathbb{Z}/p^n\mathbb{Z} \to \mathbb{Z}_p$ satisfies $\operatorname{def}(\varphi) = p^{-n}$ and $\operatorname{dist}(\varphi, \psi) = 1$ for every homomorphism $\psi$. This result does not however show that these groups are unstable with respect to the family $\{ \mathbb{Z}_p \}$: this is indeed never the case as we will see in Proposition \ref{prop:unfin}. However, the fact that these bad estimates for stability arise when looking at maps from a finite $p$-group to a pro-$p$ group is not a coincidence, as appears from the results in Sections \ref{s:vpropi} and \ref{s:char0}. \\
Using only the ultrametric inequality, we prove a relation between the pointwise and uniform stability problems, that as we have seen above does not hold in the Archimedean setting (Theorem \ref{thm:pw_un}):
\begin{theorem}
\label{intro:thm:pw_un}
Let $\Gamma$ be finitely generated and pointwise $\operatorname{GL}(\mathbb{Z}_p)$-stable. Then $\Gamma$ is uniformly $\operatorname{GL}(\mathbb{Z}_p)$-stable. If moreover $\Gamma$ is finitely presented, then the converse holds.
\end{theorem}
The techniques developed for the proof of Theorem \ref{thm:pw_un} also apply further: in Proposition \ref{prop:unfin} we show that if $\mathcal{G}$ is a finite family of compact ultrametric groups, then any finitely generated group is uniformly $\mathcal{G}$-stable. This also does not hold in the Archimedean setting: for instance if $\mathcal{G} = \{ ( \operatorname{U}(1), \| \cdot \|_{op} ) \}$, then a non-abelian free group is not uniformly $\mathcal{G}$-stable \cite{Rolli}. \\
Using the fact that the metric reflects the profinite structure, we are able to reduce the uniform stability problem to residually finite groups (Theorem \ref{thm:rf}):
\begin{theorem}
\label{intro:thm:rf}
Let $\Gamma$ be a group, and $R$ its largest residually finite quotient. Then $\Gamma$ is uniformly $\operatorname{GL}(\mathbb{Z}_p)$-stable if and only if $R$ is. If $\Gamma$ is pointwise $\operatorname{GL}(\mathbb{Z}_p)$-stable, then so is $R$.
\end{theorem}
This theorem implies in particular that a group without finite quotients is uniformly $\operatorname{GL}(\mathbb{Z}_p)$-stable. An analogous result holds for pointwise stability, as we shall shortly see. \\
The techniques developed for the proof of Theorem \ref{intro:thm:rf} also apply further: in Propositions \ref{prop:aut_stab} and \ref{prop:gal_stab} we provide the complete solution to three other stability problems. The first one is with respect to the family $T(R)$ of invertible upper-triangular matrices over a commutative ring $R$; the second one is with respect to the family $\operatorname{Aut}(X^*_\bullet)$ of automorphism groups of regular rooted trees of increasing degrees; the third one is with respect to the family $\operatorname{Gal}(K)$ of Galois groups associated to all Galois extensions of a field $K$ (which needs to admit only countably many finite Galois extensions for the groups to be metrizable: see Lemma \ref{lem:gal_firstcount}). Under natural ultrametrics that reflect a projective structure, we prove that all finitely generated groups are uniformly stable in the last case, and all groups are uniformly stable in the first two cases. \\
Related to the stability problem is the corresponding approximation problem. We attribute the following definition to Gromov, after \cite{Gromov}:
\begin{definition}[Gromov]
\label{def:approx}
A sequence $(\varphi_n : \Gamma \to G_n \in \mathcal{G})_{n \geq 1}$ is \textit{asymptotically injective} if
$$\liminf_{n \to \infty} d_{G_n}(\varphi_n(g), 1) > 0$$
for every $1 \neq g \in \Gamma$. A pointwise asymptotic homomorphism that is also asymptotically injective is called a \textit{$\mathcal{G}$-approximation}: if one exists, $\Gamma$ is said to be \textit{$\mathcal{G}$-approximable}.
\end{definition}
This leads to the important notions of \textit{sofic groups}, when $\mathcal{G} = \{ (S_n, d_H) : n \geq 1\}$, introduced by Gromov \cite{Gromov} and named by Weiss \cite{Weiss}; and \textit{hyperlinear groups}, when $\mathcal{G} = \{ (\operatorname{U}(n), \| \cdot \|_{HS}) : n \geq 1\}$, introduced by Radulescu \cite{Radulescu} in the context of the Connes embedding conjecture \cite{Connes}. These classes of groups are very large, so large that no non-example is known to date. In contrast, the profinite nature of $\operatorname{GL}_n(\mathbb{Z}_p)$ allows to characterize approximation in terms of other well-studied properties (Propositions \ref{prop:approx_1} and \ref{prop:approx_2}):
\begin{theorem}
\label{intro:thm:approx}
A countable group is $\operatorname{GL}(\mathbb{Z}_p)$-approximable if and only if it is LEF (locally embeddable in the class of finite groups). In particular, a finitely presented group is $\operatorname{GL}(\mathbb{Z}_p)$-approximable if and only if it is residually finite.
\end{theorem}
The class of LEF groups was formally introduced by Gordon and Vershik in \cite{LEF}, although it is already present in Malcev's work \cite{Malcev}: we refer the reader to Subsection \ref{ss:rf} for the precise definitions. We are also able to characterize \textit{strong} approximability, where the approximation is required to be a uniform asymptotic homomorphism, as being equivalent to residual finiteness, for arbitrary countable groups. \\
By an argument due to Arzhantseva and P\u{a}unescu \cite{a:comm} (see Lemma \ref{lem:GR}), a group that is both $\mathcal{G}$-approximable and pointwise $\mathcal{G}$-stable is fully residually-$\mathcal{G}$. Using this fact and Theorem \ref{intro:thm:approx}, any group that is LEF but not residually finite is not pointwise $\operatorname{GL}(\mathbb{Z}_p)$-stable (Corollary \ref{cor:cex_stab}). This gives several examples of non-pointwise stable finitely generated groups (Examples \ref{ex:cex} and \ref{ex:cex2}), proving that both Theorem \ref{intro:thm:pw_un} and Theorem \ref{intro:thm:rf} are sharp. Indeed, there exist finitely generated groups that are uniformly stable but not pointwise stable; and there exist finitely generated groups that are not pointwise stable but whose largest residually finite quotient is. Moreover, the techniques developed for the proof of Theorem \ref{intro:thm:approx} allow to characterize approximability for a few other families (Corollaries \ref{cor:approxTR} and \ref{cor:galfin_approx_1}), and to prove a pointwise version of Theorem \ref{intro:thm:rf} (Proposition \ref{prop:lef_q}), where the largest residually finite quotient is replaced by the largest LEF quotient. This last result is analogous to the fact that a group is pointwise stable in permutation if and only if its largest sofic quotient is. \\
Going back to stability, the strongest results that are proven in this paper concern ultrametric families with some restriction on the order of their finite quotients. Namely, using that the groups $\operatorname{GL}_n(\mathbb{Z}_p)$ are virtually pro-$p$, we can prove stability results for fundamental groups of graphs of groups with some restriction on the orders of finite quotients. These include the following classes of examples (see Section \ref{s:vpropi}):
\begin{theorem}
\label{intro:thm:pifree}
The following groups are uniformly $\operatorname{GL}(\mathbb{Z}_p)$-stable:
\begin{enumerate}
\item Groups without finite virtual $p$-quotients.
\item Finitely generated virtually free groups without elements of order $p$.
\item Baumslag--Solitar groups $\operatorname{BS}(m, n)$, whenever $p$ divides exactly one of $m$ and $n$.
\item $\mathbb{Z} \left[ \frac{1}{mn} \right] \rtimes_{\frac{m}{n}} \mathbb{Z}$, for $m, n$ as above, if moreover $(m, n) = 1$ and $1 \neq |m| \neq |n| \neq 1$.
\item Wreath products $G \wr \mathbb{Z}$, whenever $G$ does not surject onto $\mathbb{F}_p$.
\end{enumerate}
\end{theorem}
Groups as in $1.$ include all periodic groups without elements of order $p$ (Example \ref{ex:period}), as well as groups of automorphisms of regular rooted trees of degree smaller than $p$ with the congruence subgroup property (Example \ref{ex:CSP}). Item $3.$, with the appropriate $p$, applies to every non-Hopfian Baumslag--Solitar group and every residually finite Baumslag--Solitar group, with the exception of $\mathbb{Z}^2$ and the Klein bottle group \cite{BS:Hopf}. The group from $4.$ is the largest residually finite quotient of $\operatorname{BS}(m, n)$ \cite{moldavanskii}, so it also provides an example of an infinitely presented pointwise stable group, by Item $3.$ and Theorem \ref{intro:thm:rf}. \\
The proof of Theorem \ref{intro:thm:pifree} relies on the Schur--Zassenhaus Theorem (Theorem \ref{thm:SZ}), which states that any extension of finite groups with coprime orders splits, and that any two splittings are conjugate. The first part is used to prove Item $1.$, the second one is used to treat graphs of groups. All these results are quantitatively precise, in particular, the quantitative estimates involved with stability are optimal. Moreover, the statement about graphs of groups falls both in the framework of \textit{constraint stability} \cite{a:const} and of \textit{stability of epimorphisms} \cite{stepi}, providing new examples of these notions. \\
These results only use that the groups $\operatorname{GL}(\mathbb{Z}_p)$ are virtually pro-$p$, so in particular they also apply to the characteristic $p$ setting, where $\mathbb{Z}_p$ is replaced by $\mathbb{F}_q[[X]]$, for $q$ a power of $p$. But for the case of $\mathbb{Z}_p$ we can make these criteria more flexible: a cohomological argument implies an analogue of the Schur--Zassenhaus Theorem suitable to this setting (Lemma \ref{lem:cohopk}), that yields the following strengthening of Theorem \ref{intro:thm:pifree} (see Section \ref{s:char0}):
\begin{theorem}
\label{intro:thm:vpfree}
The following groups are uniformly $\operatorname{GL}(\mathbb{Z}_p)$-stable:
\begin{enumerate}
\item Groups with a bound on the order of their finite virtual $p$-quotients.
\item Finitely generated virtually free groups.
\item Baumslag--Solitar groups $\operatorname{BS}(m, n)$, whenever $\nu_p(m) \neq \nu_p(n)$.
\end{enumerate}
\end{theorem}
Item $1.$ includes all groups with finite exponent, by Zelmanov's solution of the restricted Burnside problem \cite{RBP_odd, RBP_2}. In Item $3.$ above, $\nu_p$ denotes the $p$-adic valuation: with the appropriate $p$ it applies to every non-residually finite Baumslag--Solitar group \cite{BS:Hopf}, extending the remark following Theorem \ref{intro:thm:pifree}. Also these results are quantitatively precise, and the estimates involved are linear. Both stability results on Baumslag--Solitar groups are part of more general statements on \textit{Generalized Baumslag--Solitar groups} (Corollaries \ref{cor:GBS_p} and \ref{cor:GBS}), which give combinatorial and arithmetic conditions on the underlying weighted graphs that imply stability. \\
The methods used for the proof of Theorem \ref{intro:thm:vpfree} rely strongly on the fact that $\mathbb{Q}_p$ has characteristic $0$, and in particular they cannot be used to determine whether finite $p$-groups are stable with respect to $\operatorname{GL}(\mathbb{F}_q[[X]])$, where $q$ is a power of $p$. Still, we are able to show that stability does hold for $\mathbb{Z}/2\mathbb{Z}$ in characteristic $2$, with a quadratic estimate (Proposition \ref{prop:z2z}). However our method relies on the solution of the similarity problem for representations of $\mathbb{Z}/2\mathbb{Z}$ over finite commutative local rings \cite{Braw2}. The analogous problem for all other $p$-groups in characteristic a power of $p$ is computationally wild \cite{wild}. \\
The stability of virtually free groups can also be proven by another method, which is conceptually very different from the rest of the paper, and is reserved to Section \ref{s:BC}. In \cite{GLT}, the authors consider the family $\mathcal{G} := \{ (\operatorname{U}(n), \| \cdot \|_{Frob}) : n \geq 1 \}$ and prove a cohomological criterion that ensures pointwise $\mathcal{G}$-stability of finitely presented groups. The key feature of the Frobenius norm that is exploited is its \textit{submultiplicativity}, and indeed the same approach works for other submultiplicative norms on $\operatorname{U}(n)$ \cite{pnorm}. The $\ell^\infty$-norm on $\operatorname{GL}_n(\mathbb{Z}_p)$ also has this property (Lemma \ref{lem:GL}), which allows to carry over the arguments. But the non-Archimedean setting has a peculiarity of its own: the cocycles appearing in the proof are moreover bounded, so it is natural to state the result in terms of \textit{bounded cohomology} instead. This is a rich theory over the reals (see e.g. \cite{monod, Frig}), whose study over non-Archimedean fields was recently initiated by the author \cite{mio}. The criterion is the following (Theorem \ref{thm:BC}, Corollary \ref{cor:BC}):
\begin{theorem}
\label{intro:thm:BC}
Let $\Gamma$ be a finitely presented group such that $\operatorname{H}^2_b(\Gamma, E) = 0$ for every Banach $\mathbb{Q}_p[\Gamma]$-module $E$ with a solid norm. Then $\Gamma$ is $\operatorname{GL}(\mathbb{Z}_p)$-stable. In particular, this holds if $\operatorname{H}^2(\Gamma, E) = 0$ for every such $E$.
\end{theorem}
A Banach norm $\| \cdot \|$ on $E$ is said to be \textit{solid} if $\| E \| \subset |\mathbb{Q}_p|_p$: such spaces are isometrically classified \cite[Theorem 2.5.4]{NFA}. The last statement implies that virtually free groups are $\operatorname{GL}(\mathbb{Z}_p)$-stable. Similarly, the analogous statement in characteristic $p$ implies that virtually free groups without elements of order $p$ are $\operatorname{GL}(\mathbb{F}_q[[X]])$-stable. We conjecture that these are the only examples that can be obtained via this theorem (Conjecture \ref{conj}). \\
\textbf{Outline.} In Section \ref{s:preli} we review a few general facts about stability, approximation, residual properties and local embeddings. Moreover, we recall the interplay between lifting, splitting and cohomology, proving a useful technical lemma. In Section \ref{s:fam} we introduce the general framework of ultrametric families which will be the subject of our stability results, focusing on examples. In Section \ref{s:ultra} we treat stability with respect to general ultrametric families, proving Theorems \ref{intro:thm:pw_un} and \ref{intro:thm:rf}. In Section \ref{s:approx} we treat approximation and prove Theorem \ref{intro:thm:approx}. In Section \ref{s:vpropi} we focus on families that are virtually pro-$\pi$ for some set $\pi$ of primes, and prove generalizations of Theorem \ref{intro:thm:pifree}. In Section \ref{s:char0} we focus on the case of $\operatorname{GL}(\mathbb{Z}_p)$ (or more generally $\operatorname{GL}(\mathfrak{o})$ where $\mathfrak{o}$ is the ring of integers of a finite extension of $\mathbb{Q}_p$), prove Theorem \ref{intro:thm:vpfree}, and end by discussing the differences in the case of positive characteristic. Finally, in Section \ref{s:BC} we take a bounded-cohomological approach to stability, proving Theorem \ref{intro:thm:BC}. Section \ref{s:q} is dedicated to open questions and suggestions for further research. \\
\textbf{Remark.} The results of this paper are part of the author's PhD project. \\
\textbf{Acknowledgements.}
I would like to start by thanking my advisor Alessandra Iozzi, as well as Marc Burger and Konstantin Golubev, for the constant support and guidance. Secondly, thanks to Bharatram Rangarajan for introducing me to the subject of stability, as well as the organizers of the Geometry Graduate Colloquium at ETH Zurich for inviting him; and to Alexander Lubotzky and Konstantin Golubev, who first suggested to me the study of $p$-adic stability and bounded cohomology, respectively.
I had the great chance to talk with many people during the development of this project, and for that I would like to thank Dante Bonolis, Michael Chapman, Yves de Cornulier (via MathOverflow), Alexandra Edletzberger, Victor Jaeck, Jan Kohlhaase,
Maxim Mornev, Amritanshu Prasad, Peter Schneider, Zoran \v{S}uni\'c and Michele Zordan for lending me their time, and for the interesting and useful discussions, suggestions and comments. Most of all, thanks to Goulnara Arzhantseva for the invaluable comments, insight, and help with navigating the literature.
This project was started during the first wave of the COVID-19 pandemic, and I am writing these acknowledgements while vaccination is ongoing in Switzerland. Therefore I need to extend a special thank you to my flatmates Etienne Batelier, Victor Jaeck and Lauro Silini, for making working for home a pleasant experience.
\pagebreak
\section{Preliminaries}
\label{s:preli}
\textbf{Notations and conventions.} In the sequel, $p$ always denotes a prime. If $\pi$ is a set of primes, we denote by $\pi'$ its complement, in particular $p'$ is the set of primes other than $p$. For an integer $n$, the $p$-adic valuation of $n$ is denoted by $\nu_p(n)$. That is, $p^{\nu_p(n)}$ is the largest power of $p$ that divides $n$. The set of natural numbers $\mathbb{N}$ starts at $1$. For simplicity, we use $x_n \to x$ instead of $x_n \xrightarrow{n \to \infty} x$ to denote convergence of sequences, whenever this does not lead to confusion. \\
$\Gamma$ denotes a countable discrete group. Given a set $S$ of letters, $F_S$ denotes the corresponding free group. We will always assume that $S \cap S^{-1} = \emptyset$. The trivial homomorphism onto any group will be denoted by $\mathbbm{1}$. If $R \subset F_S$ we denote by $\langle \langle R \rangle \rangle$ its normal closure, and $\langle S \mid R \rangle := F_S/\langle \langle R \rangle \rangle$. Once the presentation is fixed, we denote the projection map by $F_S \to \langle S \mid R \rangle : w \mapsto \overline{w}$. An extension $1 \to N \to \Gamma \to Q \to 1$ will be referred to as an extension of $N$ by $Q$.
\subsection{Stability and approximation}
Let $\mathcal{G}$ be a family of groups equipped with arbitrary bi-invariant metrics. By bi-invariant we mean that if $(G, d_G) \in \mathcal{G}$, and $g, h, k \in G$, we require
$$d_G(g, h) = d_G(kg, kh) = d_G(gk, hk).$$
We denote by $G(\varepsilon)$ the closed ball of radius $\varepsilon > 0$ around the identity. The groups $G$ can thus be seen as topological groups. \\
Most of the paper is concerned with uniform stability. What follows is an equivalent characterization which allows to make the statements more quantitative:
\begin{lemma}
\label{lem:quant}
The following are equivalent:
\begin{enumerate}
\item $\Gamma$ is uniformly $\mathcal{G}$-stable.
\item For all $\varepsilon > 0$ there exists $\delta > 0$ such that whenever $\varphi : \Gamma \to G \in \mathcal{G}$ satisfies $\operatorname{def}(\varphi) \leq \delta$, there exists a homomorphism $\psi : \Gamma \to G$ such that $\operatorname{dist}(\varphi, \psi) \leq \varepsilon$.
\end{enumerate}
\end{lemma}
If the function assigning to $\varepsilon$ the optimal $\delta$ is linear, polynomial, exponential... then we can attach the same adjective to the stability of $\Gamma$.
\begin{proof}
$2. \Rightarrow 1.$ Suppose that $2.$ holds, and let $(\varphi_n : \Gamma \to G_n \in \mathcal{G})_{n \geq 1}$ be a uniform asymptotic homomorphism. For all $\varphi_n$ let $\psi_n : \Gamma \to G_n$ be a homomorphism that minimizes $\operatorname{dist}(\varphi_n, \psi_n)$ up to $1/n$ (we cannot ask that a minimizing one exists in such a general situation). We need to show that $\operatorname{dist}(\varphi_n, \psi_n) \to 0$, so let $\varepsilon > 0$ and let $\delta > 0$ be as in $2.$ for $\varepsilon/2$. Let $N$ be large enough so that $N \geq 2/\varepsilon$ and $\operatorname{def}(\varphi_n) \leq \delta$ for all $n \geq N$. Then $\operatorname{dist}(\varphi_n, \psi_n) \leq \varepsilon/2 + 1/n \leq \varepsilon$. \\
$1. \Rightarrow 2.$ Suppose that $2.$ does not hold. Then there exists $\varepsilon > 0$ with the following property: for all $n \geq 1$ there exists $\varphi_n : \Gamma \to G_n \in \mathcal{G}$ such that $\operatorname{def}(\varphi_n) \leq 1/n$ but for any homomorphism $\psi_n : \Gamma \to G_n$ we have $\operatorname{dist}(\varphi_n, \psi_n) > \varepsilon$. The sequence $\varphi_n$ provides a counterexample to the uniform stability of $\Gamma$.
\end{proof}
Rephrasing stability in terms of presentations (see Proposition \ref{prop:stab_equiv}) allows to give a quantitative characterization of pointwise stability of finitely presented groups, as in \cite{a:comm}. However it does not make sense to go into it here, since in the ultrametric setting uniform and pointwise stability coincide for finitely presented groups, as we will prove in Theorem \ref{thm:pw_un}. \\
The following simple observation, first made in \cite[Theorem 4.3]{a:comm} (see also \cite[Proposition 3]{GR}), is the key to connecting the notions of stability and approximation. Recall that given a group $\Gamma$ and a family of groups $\mathcal{G}$, we say that $\Gamma$ is \textit{residually-$\mathcal{G}$} if for all $1 \neq g \in \Gamma$ there exists a homomorphism $\varphi : \Gamma \to G \in \mathcal{G}$ such that $g \notin \ker(\varphi)$. It is \textit{fully residually-$\mathcal{G}$} if for any finite subset $K \subset \Gamma$ there exists a homomorphism $\varphi : \Gamma \to G \in \mathcal{G}$ such that $f|_K$ is injective. If the groups in $\mathcal{G}$ are residually finite, and $\Gamma$ is residually-$\mathcal{G}$, then $\Gamma$ is residually finite.
\begin{lemma}[Arzhantseva--P\u{a}unescu]
\label{lem:GR}
Let $\Gamma$ be a group that is both $\mathcal{G}$-approximable and pointwise $\mathcal{G}$-stable. Then $\Gamma$ is fully residually-$\mathcal{G}$.
\end{lemma}
This is mostly useful for counterexamples. For instance let $\mathcal{G}$ be a family of residually finite groups. Then if $\Gamma$ is not residually finite, it cannot be simultaneously approximable and pointwise stable: such classes include all families of finite groups, as well as the \textit{profinite families} that will be defined in Section \ref{s:fam}. Similarly, if $\mathcal{G}$ is a family of locally residually finite groups, then the same holds under the additional hypothesis that $\Gamma$ is finitely generated: such classes include all linear groups by a theorem of Malcev \cite{Malcev}. This is the way the authors in \cite{GR} provide the first non-examples of pointwise $(S_n, d_H)$-stable equations. It is also the approach suggested in \cite{a:comm} and successfully realized in \cite{GLT}, by which the authors provide the first example of non-approximable group, with respect to $(\operatorname{U}(n), \| \cdot \|_{Frob})$. \\
Another useful equivalent characterization of pointwise stability, due to Arzhantseva and P\u{a}unescu \cite[Theorem 4.2]{a:comm}, is in terms of ultralimits and ultraproducts. We will only use it in Section \ref{s:BC} and apply it to finitely presented groups, so we state it in this setting which is the one from \cite{a:comm}. In the statement, $\prod\limits_{n \to \omega} G_n$ denotes the \textit{metric ultraproduct} of the $G_n$ with respect to the free ultrafilter $\omega$. That is, $\prod\limits_{n \to \omega} G_n$ is the quotient of the direct product by the normal subgroup $\{ (g_n)_{n \geq 1} : d_{G_n}(g_n, 1_{G_n}) \xrightarrow{n \to \omega} 0 \}$.
\begin{lemma}[Arzhantseva--P\u{a}unescu]
\label{lem:stab_up}
Let $\Gamma = \langle S \mid R \rangle$ be finitely presented, and let $\mathcal{G}$ be a family of groups equipped with bi-invariant metrics. The following are equivalent:
\begin{enumerate}
\item $\Gamma$ is pointwise $\mathcal{G}$-stable.
\item For every free ultrafilter $\omega$ on $\mathbb{N}$ and any sequence $(G_n)_{n \geq 1} \subset \mathcal{G}$, every homomorphism $\Gamma \to \prod\limits_{n \to \omega} G_n$ lifts to a homomorphism $\Gamma \to \prod\limits_{n \geq 1} G_n$.
\end{enumerate}
\end{lemma}
\subsection{Residual properties and local embeddings}
\label{ss:rf}
See \cite[Chapters 2, 3, 7]{Cell} for more detail. \\
Let $\mathcal{C}$ be a class of groups, that for simplicity we assume to be closed under taking subgroups. If the class $\mathcal{C}$ is closed under taking direct products, then a residually-$\mathcal{C}$ group is automatically fully residually-$\mathcal{C}$.
The following result is standard, so we include it here for reference and omit the proof:
\begin{lemma}
\label{lem:largest_rf}
Let $\mathcal{C}$ be a class of groups closed under taking subgroups, $\Gamma$ a group, $K$ the intersection of all normal subgroups of $\Gamma$ such that the quotient is in $\mathcal{C}$, and $R := \Gamma / K$. Then $R$ is the largest residually-$\mathcal{C}$ quotient of $\Gamma$; that is, $R$ is residually-$\mathcal{C}$, and any homomorphism from $\Gamma$ to a residually-$\mathcal{C}$ group factors through $R$.
\end{lemma}
The following will be common examples throughout this paper.
\begin{example}
\label{ex:sym0}
Let $\operatorname{Sym}(\mathbb{Z})$ be the group of permutations of the integers, let $\operatorname{Sym}_0(\mathbb{Z})$ be the subgroup of permutations with finite support, and $\operatorname{Alt}_0(\mathbb{Z})$ the subgroup of even permutations with finite support.
Let $T : \mathbb{Z} \to \mathbb{Z} : n \to (n + 1)$ be the translation. We denote by $G$ the group generated by $\operatorname{Sym}_0(\mathbb{Z})$ and $T$, and by $G^+$ the group generated by $\operatorname{Alt}_0(\mathbb{Z})$ and $T$, which has index $2$ in $G$. Then $G$ splits as a semidirect product $\operatorname{Sym}_0(\mathbb{Z}) \rtimes \langle T \rangle$, and similarly $G^+$ splits as a semidirect product $\operatorname{Alt}_0(\mathbb{Z}) \rtimes \langle T \rangle$. Since $\operatorname{Alt}_0(\mathbb{Z})$ has no non-trivial finite quotients, and $\mathbb{Z}$ is residually finite, the largest residually finite quotient of $G^+$ is $\mathbb{Z}$. Similarly the largest residually finite quotient of $G$ is $\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$.
\end{example}
\begin{remark}
The group $G$ above was first considered by Malcev in \cite{Malcev}. It is commonly referred to as \textit{Houghton's second group}, in referece to \cite{Houghton}, and denoted by $H_2$.
\end{remark}
\begin{example}
\label{ex:wr}
Let $G, H$ be groups, denote $\Sigma_H G = \oplus_{h \in H} G_h$, where each $G_h$ is an indexed copy of $G$, and consider the wreath product $G \wr H = \Sigma_H G \rtimes H$ where $H$ acts on the direct sum by shifting the coordinates. Common examples in combinatorial group theory are the \textit{lamplighter groups}, where $H = \mathbb{Z}$ and $G$ is finite (sometimes the name is used to refer to the specific case of $G = \mathbb{Z}/2\mathbb{Z}$).
Suppose that $H$ is infinite. Then the projection of $G \wr H$ onto its largest residually finite quotient factors through $\operatorname{Ab}(G) \wr H$, where $\operatorname{Ab}(G)$ is the abelianization of $G$. This latter is residually finite if both $\operatorname{Ab}(G)$ and $H$ are residually finite \cite[Theorem 3.2]{wr_rf} (we will mostly be interested in the first statement).
\end{example}
In Section \ref{s:approx} we are concerned with local embeddings, which sit between approximability in the sense of Definition \ref{def:approx} and the corresponding residual property. The notion of local embedding was formally introduced in \cite{LEF} although the idea goes back to Malcev \cite{Malcev}. We refer the reader to \cite[Chapter 7]{Cell} for details and proofs.
\begin{definition}[Gordon--Vershik]
Let $\Gamma, C$ be groups, $K \subset \Gamma$ a finite subset. A map $f : \Gamma \to C$ is a \textit{$K$-local embedding} if $f|_K$ is injective and $f(g)f(h) = f(gh)$ whenever $g, h \in K$.
Let $\mathcal{C}$ be a class of groups (closed under taking subgroups). The group $\Gamma$ is \textit{locally embeddable into $\mathcal{C}$} if for any finite subset $K \subset \Gamma$ there exists a $K$-local embedding $f : \Gamma \to C \in \mathcal{C}$. When $\mathcal{C}$ is the class of finite groups, $\Gamma$ is said to be \textit{LEF}.
\end{definition}
Here is an equivalent characterization of local embeddability (see e.g. \cite[Theorem 7.2.5]{Cell}). In the statement, $\prod\limits_{n \to \omega} C_n$ denotes the \textit{set-theoretic ultraproduct} of the $C_n$ with respect to the free ultrafilter $\omega$. That ism $\prod\limits_{n \to \omega} C_n$ is the quotient of the direct product by the normal subgroup $\{ (g_n)_{n \geq 1} : \{n : g_n = 1_{C_n} \} \in \omega \}$ (equivalently, the metric ultraproduct where the $C_n$ are endowed with the discrete metric).
\begin{proposition}[Gordon--Vershik]
\label{prop:lef_up}
Let $\Gamma$ be a countable group, $\mathcal{C}$ a class of groups. Then $\Gamma$ is locally embeddable into $\mathcal{C}$ if and only if it embeds into $\prod\limits_{n \to \omega} C_n$ for some sequence $(C_n)_{n \geq 1} \subset \mathcal{C}$.
\end{proposition}
The properties of being residually-$\mathcal{C}$ and locally embeddable into $\mathcal{C}$ are related by the following result (see e.g. \cite[Corollary 7.1.14, 7.1.21]{Cell}):
\begin{proposition}[Gordon--Vershik]
\label{prop:lec_rc}
Let $\mathcal{C}$ be a class of groups closed under taking subgroups.
\begin{enumerate}
\item Any fully residually-$\mathcal{C}$ group is locally embeddable into $\mathcal{C}$.
\item Any finitely presented group that is locally embeddable into $\mathcal{C}$ is fully residually-$\mathcal{C}$.
\end{enumerate}
\end{proposition}
Importantly, Item $2.$ does not hold for general finitely generated groups.
\begin{example}
\label{ex:sym0_lef}
The group $G$ from Example \ref{ex:sym0} is finitely generated: by $(12)$ and $T$. It is not residually finite: we computed its largest residually finite quotient in Example \ref{ex:sym0}. However $G$ is LEF \cite{LEF}. Similarly, $G^+$ is finitely generated (by $(123)$ and $T$, or because it has finite index in $G$), not residually finite (again by Example \ref{ex:sym0}), but it is LEF (since a subgroup of a LEF group is clearly LEF).
\end{example}
\begin{example}
\label{ex:wr_lef}
Let $G, H$ be finitely generated LEF groups. Then their wreath product $G \wr H$ is finitely generated and LEF \cite[Theorem 2.4 (ii)]{LEF}. However, if $G$ is non-abelian, then $G \wr H$ is not residually finite by Example \ref{ex:wr}.
\end{example}
A natural framework in which to see this properties is that of \textit{marked groups}, introduced by Grigorchuk in \cite{grigor_mg}. We use the point of view of normal subgroups, as in \cite{Cell} (see also \cite{limit}). Given a countable group $\Gamma$, the set of \textit{$\Gamma$-marked groups} is the set of isomorphism classes of quotients of $\Gamma$, identified with the set of normal subgroups $\mathcal{N}(\Gamma)$ of $\Gamma$. The space of marked groups is this set endowed with the subspace topology $\mathcal{N}(\Gamma) \subset \mathcal{P}(\Gamma) \cong \{ 0, 1 \}^\Gamma$. This topology is totally disconnected, compact and, since $\Gamma$ is suppose to be countable, metrizable. \\
Given a class $\mathcal{C}$ closed under taking subgroups, both the residual property and local embeddability admit characterizations in terms of the space of marked groups (see e.g. \cite[Proposition 3.4.3, Corollary 7.1.20]{Cell}):
\begin{theorem}[Gordon--Vershik]
\label{thm:mg}
Let $\mathcal{C}$ be a class of groups closed under taking subgroups. Let $\Gamma = \langle S \mid R \rangle$ be a countable group, $N := \langle \langle R \rangle \rangle \leq F_S$. Then:
\begin{enumerate}
\item $\Gamma$ is fully residually-$\mathcal{C}$ if and only if there exists a sequence $N_k \to \{ 1 \} \in \mathcal{N}(\Gamma)$ such that $\Gamma/N_k \in \mathcal{C}$ for all $k$.
\item $\Gamma$ is locally embeddable into $\mathcal{C}$ if and only if there exists a sequence $N_k \to N \in \mathcal{N}(F_S)$ such that $\Gamma/N_k \in \mathcal{C}$ for all $k$.
\end{enumerate}
\end{theorem}
This point of view gives many more examples of finitely generated LEF groups that are not residually finite.
\begin{example}
\label{ex:small_canc}
Let $\Gamma = \langle S \mid R \rangle$, with $S$ finite and $R$ possibly infinite, be a presentation satisfying the $C'(1/6)$ small cancellation condition: we call $\Gamma$ a \textit{classical small cancellation group}. Letting as usual $N = \langle \langle R \rangle \rangle \leq F_S$, there exists a sequence $N_k \to N \in \mathcal{N}(F_S)$, where the $N_k$ are normally generated by $k$ elements of $R$. Since the groups $F_S/N_k$ are defined by \textit{finite} $C'(1/6)$ presentation, they are hyperbolic, and moreover they are residually finite. The last statement follows by combining the following three deep results: finitely presented $C'(1/6)$ groups are cubulable \cite{Wise}, hyperbolic cubulable groups are virtually special-cubulable \cite{Agol}, and special-cubulable groups embed into RAAGs \cite{scc}. By Item $2.$ of Theorem \ref{thm:mg}, it follows that $\Gamma$ is LEF. Let us point out that this is a special property of the $C'(1/6)$ small cancellation condition: there exist finitely presented groups satisfying more relaxed small cancellation conditions that are not residually finite \cite{Wiserf}.
On the other hand there are many examples of (infinitely presented) classical small cancellation groups that are not residually finite. A classical example is a group constructed by Pride in \cite{Pride}:
$$\Gamma = \langle a, b \mid a u_1, b v_1, a u_2, b v_2, \ldots \rangle,$$
where $u_n, v_n$ are well-chosen words in $a^n, b^n$ so that the presentation is $C'(1/6)$. For any such choice, this group is infinite and has no proper finite-index subgroup, so it is in particular not residually finite. For more examples see \cite[Section 2]{a:small} and the references therein; the authors also explain how to construct continuum-many isomorphism classes of non-residually finite classical small cancellation groups.
\end{example}
\subsection{Non-Archimedean fields}
\label{ss:nona}
See \cite{NFA} for more detail. \\
Let $(\mathbb{K}, | \cdot |)$ be a normed field. If the group $|\mathbb{K}^\times| \leq \mathbb{R}^\times_{> 0}$ is discrete, it is either trivial, or of the form $r^\mathbb{Z}$, for some $0 < r < 1$, in the latter case we say that $\mathbb{K}$ is \textit{discretely valued}. The norm is \textit{non-Archimedean} if it satisfies the strong triangle inequality, namely $|x + y| \leq \max\{ |x|, |y| \}$. Then the closed ball of radius $1$ is a local ring, called the \textit{ring of integers} and denoted by $\mathfrak{o}$, whose maximal ideal is the open ball of radius $1$, denoted by $\mathfrak{p}$. The quotient $\mathfrak{k} := \mathfrak{o} / \mathfrak{p}$ is called the \textit{residue field} of $\mathbb{K}$, and its characteristic is called the \textit{residual characteristic} of $\mathbb{K}$. \\
If the induced topology is locally compact and non-discrete then $\mathbb{K}$ is called a \textit{local field}. Non-Archimedean local fields are precisely those that are discretely valued and have a finite residue field. It follows that $\mathfrak{o}$ is compact, and that the maximal ideal $\mathfrak{p}$ is principal, generated by an element $\overline{\omega}$ such that $|\overline{\omega}| = r$, using the notation above. Such an element is called a \textit{uniformizer}. By Ostrowski's Theorem a non-Archimedean local field $\mathbb{K}$ is either a finite extension of $\mathbb{Q}_p$ (if it has characteristic $0$) or of $\mathbb{F}_q((X))$ (if it has characteristic $p$, where $q$ is a power of $p$).
\begin{example}
We write elements of $\mathbb{Q}_p$ in the usual series form $x = \sum_{i \geq i_0} a_i p^i$, where $a_i \in \{0, \ldots, p-1\}$ and $a_{i_0} \neq 0$. Then $|x|_p := p^{-i_0}$, so the norm takes values in $p^\mathbb{Z}$: in the previous notation we have $r = p^{-1}$. The ring of integers is $\mathfrak{o} = \mathbb{Z}_p$ with uniformizer $\overline{\omega} = p$ and maximal ideal $\mathfrak{p} = p \mathbb{Z}_p$. The residue field is $\mathfrak{k} \cong \mathbb{F}_p$.
\end{example}
\begin{example}
$\mathbb{F}_q((X))$ is the field of formal Laurent series, so it consists of elements of the form $x = \sum_{i \geq i_0} a_i X^i$, where $a_i \in \mathbb{F}_q$ and $a_{i_0} \neq 0$. Then $|x|_q := q^{-i_0}$, so the norm takes values in $q^\mathbb{Z}$: in the previous notation we have $r = q^{-1}$. The ring of integers is $\mathfrak{o} = \mathbb{F}_q[[X]]$ with uniformizer $\overline{\omega} = X$ and maximal ideal $\mathfrak{p} = X \mathbb{F}_q[[X]]$. The residue field is $\mathfrak{k} \cong \mathbb{F}_q$.
\end{example}
We next review the basics of functional analysis over the local field $\mathbb{K}$, needed in Section \ref{s:BC}. A \textit{normed $\mathbb{K}$-vector space} is a $\mathbb{K}$-vector space $E$ endowed with a norm $\| \cdot \| : E \to \mathbb{R}_{\geq 0}$ that is positive-definite, $\mathbb{K}$-multiplicative, and satisfies the strong triangle inequality. If the induced metric is complete, we say that $E$ is a \textit{$\mathbb{K}$-Banach space}. The norm of $E$ need not take the same set of values as that of $\mathbb{K}$. If this is the case, we say that the norm on $E$ is \textit{solid}. In our case $\mathbb{K}$ is a local field, so it is in particular complete and discretely valued: such Banach spaces are isometrically classified \cite[Theorem 2.5.4]{NFA}. \\
Given two normed $\mathbb{K}$-vector space $E, F$, a linear map $T : E \to F$ is continuous if and only if it is bounded, that is if and only if $\| T \|_{op} = \inf \{ C \geq 0 : \|Tx\|_F \leq C \| x \| \text{ for all } x \in E \} < \infty$. This applies in particular to the case when $E = \mathbb{K}$; then the space of bounded linear maps is called the continuous dual $E^*$, and is a normed $\mathbb{K}$-vector space endowed with the operator norm $\| T \|_{op}$. This is in general not equal to $\sup \{ |Tx|_{\mathbb{K}} : \|x\| \leq 1 \}$, but it is when the norm on $E$ is solid. Since $\mathbb{K}$ is complete, a dual space is always Banach.
\begin{definition}
A \textit{normed $\mathbb{K}[\Gamma]$-module} is a normed $\mathbb{K}$-vector space $E$ with a linear isometric action of the group $\Gamma$. If $E$ is Banach, we say that $E$ is a \textit{Banach $\mathbb{K}[\Gamma]$-module}. If $E = F^*$ is the dual of a $\mathbb{K}[\Gamma]$-module endowed with the operator norm and the dual action, we say that $E$ is a \textit{dual $\mathbb{K}[\Gamma]$-module}.
\end{definition}
The following basic theorem of functional analysis holds also in the non-Archimedean context \cite[Theorem 2.1.17]{NFA}:
\begin{theorem}[Open Mapping Theorem]
\label{thm:open}
Let $T : E \to F$ be a bounded surjective linear map between Banach spaces. Then $T$ is open.
\end{theorem}
\subsection{Splitting and lifting}
\label{ss:preli:split}
See \cite[Chapter IV]{Brown} and \cite[Chapter 7]{Rotman} for more detail. These results will be relevant starting from Section \ref{s:vpropi}. All statements -- apart from Lemma \ref{lem:cohopk} -- are standard, but we remind them here since we will need some specific constructions in the sequel. \\
Let $1 \to N \to E \to Q \to 1$ be a group extension. A \textit{splitting} is a homomorphic section $ \sigma : Q \to E$: if one exists we say that the extension \textit{splits}, which is equivalent to $E \cong N \rtimes Q$, and the image of this section is a \textit{complement of $N$}. Consider splittings $\sigma_1, \sigma_2 : Q \to E$ with complements $Q_1, Q_2$. The complements are \textit{conjugate} if there exists $g \in E$ such that $gQ_1g^{-1} = Q_2$, which implies that there exists $g \in N$ such that $g \sigma_1 g^{-1} = \sigma_2$.
Because of this we may refer to the splittings themselves being conjugate, and the conjugating element may be assumed to lie in $N$. \\
Splitting problems are a special case of the more general \textit{lifting problems}: given a group $\Gamma$, and a homomorphism $\Gamma \xrightarrow{\varphi} G/N$, can this be lifted to a homomorphism $\Gamma \xrightarrow{\psi} G$? Are all lifts conjugate?
\[\begin{tikzcd}
& G \\
\Gamma \\
& {G/N}
\arrow["\varphi"', from=2-1, to=3-2]
\arrow[from=1-2, to=3-2]
\arrow["\psi", dashed, from=2-1, to=1-2]
\end{tikzcd}\]
A splitting problem is the special case in which $\Gamma = G/N$ and $\varphi$ is the identity. But any lifting problem can be reduced to a splitting problem. Consider the pullback $G \times_{\varphi} \Gamma := \{ (x, g) \in G \times \Gamma : xN = \varphi(g) \in G/N \}$, denote by $pr_{1, 2}$ the natural projections, and notice that there is a natural embedding $j : N \to G \times_{\varphi} \Gamma : n \mapsto (n, 1)$. Then we have the following commutative diagram with exact rows:
\[\begin{tikzcd}
1 & N & {G \times_{\varphi} \Gamma} & \Gamma & 1 \\
1 & N & G & {G/N} & 1
\arrow[from=1-1, to=1-2]
\arrow["{pr_2}", from=1-3, to=1-4]
\arrow[from=1-4, to=1-5]
\arrow[from=2-1, to=2-2]
\arrow[hook, from=2-2, to=2-3]
\arrow[from=2-3, to=2-4]
\arrow[from=2-4, to=2-5]
\arrow["{=}"', from=1-2, to=2-2]
\arrow["{pr_1}", from=1-3, to=2-3]
\arrow["\varphi", from=1-4, to=2-4]
\arrow["j", from=1-2, to=1-3]
\end{tikzcd};\]
and the lift $\psi$ exists if and only if the exact sequence on the top row splits.
Moreover, two lifts are conjugate if and only if the corresponding two splittings are conjugate, and by the discussion above the conjugating element may be chosen to lie in $N$. Lifting problems will appear more naturally in this paper, so we will state our results in those terms. \\
The following will be the fundamental tool in Section \ref{s:vpropi} \cite[Theorem 7.41]{Rotman}:
\begin{theorem}[Schur--Zassenhaus]
\label{thm:SZ}
Let $\Gamma, N$ be finite groups such that the orders of $\Gamma$ and $N$ are coprime. Then, whenever $N \unlhd G$, any homomorphism $\Gamma \to G/N$ lifts to a homomorphism $\Gamma \to G$, and any two lifts are $N$-conjugate.
\end{theorem}
The conjugacy statement admits a simple proof in the case in which either $N$ or $\Gamma$ is solvable. This will be the setting for $\operatorname{GL}(\mathfrak{o})$: in fact $N$ will even be a $p$-group. The general case follows from the Odd Order Theorem: if $N$ and $\Gamma$ have coprime order then at least one of them has odd order, and so is solvable. To the author's knowledge, no simpler proof is known. \\
The Schur--Zassenhaus Theorem alone will be enough for Section \ref{s:vpropi}. In Section \ref{s:char0} we will need a stronger lifting criterion, under additional hypotheses on $N$. Recall that given an extension $1 \to N \to E \to Q \to 1$ with $N$ abelian, the action of $E$ on $N$ by conjugacy induces an action of $Q$ on $N$, and so the corresponding cohomology groups $\operatorname{H}^\bullet(Q, N)$. To this extension one associates a second cohomology class $[E] \in \operatorname{H}^2(Q, N)$. A cocycle representing it can be defined as follows: take a (set-theoretic) section $\tilde{\sigma} : Q \to E$, and define $c : Q \times Q \to N : (g, h) \mapsto c(g, h) = \tilde{\sigma}(gh)^{-1} \tilde{\sigma}(g) \tilde{\sigma}(h)$. If the extension splits, let us identify $E$ with $N \rtimes Q$ (which as a set is just the cartesian product $N \times Q$). Then each splitting $\sigma : Q \to N \rtimes Q$ defines a first cohomology class $[\sigma] \in \operatorname{H}^1(Q, N)$. A cocycle representing it can be defined by $c : Q \to N : g \mapsto c(g)$ where $\sigma(g) = (c(g), g) \in N \rtimes Q$.
\begin{theorem}[Schreier]
The extension splits if and only if the associated cohomology class $[E] \in \operatorname{H}^2(Q, N)$ vanishes. Two splittins $\sigma_1, \sigma_2$ are conjugate if and only if the associate cohomology classes $[\sigma_1], [\sigma_2] \in \operatorname{H}^1(Q, N)$ coincide.
\end{theorem}
So cohomology vanishing gives rise to splitting and conjugacy results. The Schur--Zassenhaus Theorem is proven this way: one first reduces to the case where $N$ is an elementary abelian $p$-group, and then uses the fact that if $Q$ has order coprime to $p$, then $\operatorname{H}^n(Q, N) = 0$ for $n \geq 1$. The following lemma strengthens this:
\begin{lemma}
\label{lem:cohopk}
Let $\Gamma, N$ be finite groups, with $N$ a $\mathbb{Z}/p^k \mathbb{Z}$-module, and $\nu_p(|\Gamma|) \leq l \leq k$. Let $H := p^l N$, which is characteristic in $N$. Then, whenever $N \unlhd G$, for any homomorphism $\Gamma \xrightarrow{\varphi} G/H$, the projection $\Gamma \xrightarrow{\overline{\varphi}} G/N$ lifts to a homomorphism $\psi : \Gamma \to G$.
\[\begin{tikzcd}
& G \\
\Gamma & {G/H} \\
& {G/N}
\arrow["\psi", dashed, from=2-1, to=1-2]
\arrow["{\overline{\varphi}}"', from=2-1, to=3-2]
\arrow["\varphi"', from=2-1, to=2-2]
\arrow[from=2-2, to=3-2]
\arrow[from=1-2, to=3-2, bend left = 50]
\end{tikzcd}\]
Moreover, if they exist, any two lifts of $\varphi$ are $N$-conjugate.
\end{lemma}
\begin{remark}
Note that a lift of $\varphi$ is also a lift of $\overline{\varphi}$, but the converse need not hold. The first part of the lemma only guarantees the existence of the latter, and the second one is a statement about the former.
\end{remark}
\begin{proof}
The lift exists if and only if the extension $1 \to N \to G \times_{\overline{\varphi}} \Gamma \to \Gamma \to 1$ splits, which in turn is equivalent to the vanishing of the corresponding cohomology class in $\operatorname{H}^2(\Gamma, N)$. A cocycle representing it is given by $(g, h) \mapsto \tilde{\sigma}(gh)^{-1} \tilde{\sigma}(g) \tilde{\sigma}(h)$, where $\tilde{\sigma} : \Gamma \to G \times_{\overline{\varphi}} \Gamma$ is any (set-theoretic) section. We can choose $\tilde{\sigma} : \Gamma \to G \times_{\varphi} \Gamma \leq G \times_{\overline{\varphi}} \Gamma$, and then the corresponding cocycle will take values in $H$. Similarly, a lift of $\varphi$ defines a class in $\operatorname{H}^1(\Gamma, N)$ such that the cocycle representing it takes values in $H$. Since $H = p^l N$, to show that these classes vanish it suffices to show that $p^l \cdot \operatorname{H}^n(\Gamma, N) = 0$ for $n = 1, 2$. By \cite[Corollary III.10.2]{Brown}, we have $|\Gamma| \cdot \operatorname{H}^n(\Gamma, N) = 0$ for all $n \geq 1$. Write $|\Gamma| = p^a m$ where $a \leq l$ and $(m, p) = 1$. The latter condition ensures that $m$ is a unit in $\mathbb{Z}/p^k \mathbb{Z}$, and so $m \cdot \operatorname{H}^n(\Gamma, N) = \operatorname{H}^n(\Gamma, N)$. Therefore $p^a \cdot \operatorname{H}^n(\Gamma, N) = 0$ and so $p^l \cdot \operatorname{H}^n(\Gamma, N) = 0$, which concludes the proof.
\end{proof}
\pagebreak
\section{Ultrametric families}
\label{s:fam}
The subject of this paper is stability with respect to a family $\mathcal{G}$ all of whose groups are equipped with bi-invariant ultrametrics. Before moving to stability in the Section \ref{s:ultra}, here we prove some basic facts about such families, and present several examples.
\subsection{Basic facts and terminology}
\begin{definition}
We say that the metric group $(G, d)$ is \textit{ultrametric} if the ultrametric inequality holds:
$$d(g, k) \leq \max\{ d(g, h), d(h, k) \} \text{ for all } g, h, k \in G.$$
We say that the family $\mathcal{G}$ is \textit{ultrametric} if every $G \in \mathcal{G}$ is ultrametric. If moreover the groups in $\mathcal{G}$ are compact, the family $\mathcal{G}$ is called \textit{profinite}.
\end{definition}
The most important general property of ultrametric groups is contained in the following lemma. Recall that, given a metric group $(G, d)$, the closed ball of radius $\varepsilon > 0$ around the identity is denoted by $G(\varepsilon)$.
\begin{lemma}
\label{lem:ultra_gp}
Let $(G, d)$ be a group equipped with a bi-invariant ultrametric. Then $G(\varepsilon)$ is a clopen normal subgroup of $G$.
\end{lemma}
\begin{proof}
Closed balls in ultrametric spaces are automatically open. If $g, h \in G(\varepsilon)$, then
$$d(gh^{-1}, 1) = d(g, h) \leq \max\{ d(g, 1), d(1, h) \} \leq \varepsilon.$$
This shows that $G(\varepsilon)$ is a subgroup, and it is normal because $d$ is conjugacy-invariant.
\end{proof}
This allows to quotient out balls, leading to the following definition:
\begin{definition}
Let $(G, d)$ be an ultrametric group. The quotients $G/G(\varepsilon)$ are called \textit{metric quotients} of $G$. Given an ultrametric family $\mathcal{G}$, we denote by $MQ(\mathcal{G})$ the family of metric quotients of groups in $\mathcal{G}$ and all subgroups thereof.
\end{definition}
A metric quotient of $G$ is a discrete, since each $G(\varepsilon)$ is open, and comes equipped with a quotient metric. If $G$ is moreover compact, then all metric quotients are finite. \\
The possibility of quotienting out balls has very strong consequences for stability and approximation. These are based on the following lemma:
\begin{lemma}
\label{lem:mq}
Let $\Gamma$ be a group, $(G, d)$ an ultrametric group and $\varepsilon > 0$. Let $\varphi, \psi : \Gamma \to G$. Then:
\begin{enumerate}
\item $\operatorname{def}(\varphi) \leq \varepsilon$ if and only if the map $\varphi(\varepsilon) : \Gamma \xrightarrow{\varphi} G \to G/G(\varepsilon)$ is a homomorphism.
\item If both $\operatorname{def}(\varphi), \operatorname{def}(\psi) \leq \varepsilon$, then $\operatorname{dist}(\varphi, \psi) \leq \varepsilon$ if and only if the corresponding homomorphisms $\varphi(\varepsilon), \psi(\varepsilon)$ coincide.
\end{enumerate}
\end{lemma}
\begin{proof}
$1.$ The map $\varphi(\varepsilon)$ is a homomorphism if and only if $\varphi(gh)G(\varepsilon) = \varphi(g)\varphi(h) G(\varepsilon)$ for all $g, h \in \Gamma$, which is equivalent to $\operatorname{def}_{g, h}(\varphi) \leq \varepsilon$. \\
$2.$ The homomorphisms coincide if and only if $\varphi(g) G(\varepsilon) = \psi(g) G(\varepsilon)$ for all $g \in \Gamma$, which is equivalent to $\operatorname{dist}_g(\varphi, \psi) \leq \varepsilon$.
\end{proof}
\begin{definition}
With the notation of the previous lemma, we refer to the homomorphisms $\{ \varphi(\varepsilon) : \varepsilon \geq \operatorname{def}(\varphi) \}$ as the \textit{homomorphisms induced by} $\varphi$.
\end{definition}
\subsection{Examples}
A trivial example of bi-invariant metrics falls in the ultrametric framework.
\begin{example}
\label{ex:discr}
Given a discrete group $G$ it is always possible to define a \textit{discrete metric} on it by setting $d(g, h) = 0$ if and only if $g = h$. This is a bi-invariant ultrametric. A family $\mathcal{G}$ of discrete groups equipped with the discrete metric will be called a \textit{discrete family}.
\end{example}
Probabilistic stability problems with respect to this metric are mostly used in property testing (see e.g. \cite{PT} and \cite{BChap}). In our deterministic setting, we will see that stability with respect to such families is less interesting (see Example \ref{ex:discr:stab}). \\
Next, we present two constructions that allow to put natural ultrametrics onto groups, and we apply them to give examples of ultrametric families. Then we move on to the main example that will be treated in this paper, namely integral matrices over non-Archimedean fields. For the rest of this subsection, fix a strictly decreasing sequence $\overline{\varepsilon} = (\varepsilon_k)_{k \geq 0} \subset (0, 1]$ with $\varepsilon_0 = 1$ and $\varepsilon_k \to 0$.
\subsubsection{Groups acting on filtered sets}
Let $\Omega$ be a set with a (possibly finite) filtration $(\Omega_k)_{k \geq 1}$; that is $\Omega_k \subset \Omega_{k+1}$, and $\Omega = \cup_{k \geq 1} \Omega_k$. Let $G$ be a group acting on $\Omega$ preserving each $\Omega_k$. Define $d(g, h) := \varepsilon_k$, where $k$ is the maximal integer such that $g|_{\Omega_k} = h|_{\Omega_k}$, and $k = 0$ if no such integer exists. This is a bi-invariant ultrametric: that it is a left-invariant ultrametric is clear, and right-invariance follows from the fact that $G$ preserves each $\Omega_k$.
\begin{example}
\label{ex:UT}
Let $G = T_n(R)$ be the group of invertible upper-triangular $(n \times n)$ matrices over a commutative ring $R$. Then we can set $\Omega = R^n$ and $\Omega_k = \operatorname{span}\{e_1, \ldots, e_k\}$, so $d_\Omega(g, h) \leq \varepsilon_k$ if and only if the first $k$ columns of $g$ and $h$ are identical. We denote by $T(R)$ the family $\{ (T_n(R), d) : n \geq 1 \}$, or $T(R)(\overline{\varepsilon})$ if we want to emphasize the choice of the sequence $\overline{\varepsilon}$. Similarly we can look at the subgroup $UT_n(R)$ of upper-triangular matrices with ones on the diagonal, and obtain the family $UT(R)$.
Given $\varepsilon > 0$, let $k \geq 1$ be the maximal integer such that $\varepsilon_k \geq \varepsilon$. Then $G(\varepsilon) = G(\varepsilon_k)$ is the subgroup consisting of upper-triangular matrices with a copy of $I_k$ in the upper-left corner. It follows that the metric quotient $G/G(\varepsilon)$ is isomorphic to $T_k(R)$, or $T_n(R)$ if $k > n$. In particular all metric quotients are solvable, and even nilpotent-by-abelian, so $MQ(T(R))$ is contained in the class of nilpotent-by-abelian groups. Similarly metric quotients of $UT_n(R)$ are isomorphic to $UT_k(R)$ for some $k$, in particular they are all nilpotent.
\end{example}
\begin{example}
\label{ex:aut:filt}
See \cite{selfsim} for more detail. Let $X$ be a finite alphabet and $\Omega := X^*$ the regular rooted tree of finite words on $X$. We denote by $\Omega_k$ the set of words of length at most $k$. Then $G = \operatorname{Aut}(\Omega)$, the group of rooted tree automorphisms, can be equipped with this metric, so $d_\Omega(g, h) \leq \varepsilon_k$ if and only if $g$ and $h$ act the same way on $\Omega_k$, or equivalently they act the same way on the $k$-th level of the rooted tree. We denote by $\operatorname{Aut}(X_{\bullet}^*)$ the family $\{ (\operatorname{Aut}(X_n^*), d) : n \geq 1 \}$, where $X_n = \{ 1, \ldots, n \}$, or $\operatorname{Aut}(X_{\bullet}^*)(\overline{\varepsilon})$ if we want to emphasize the choice of the sequence $\overline{\varepsilon}$.
Given $\varepsilon > 0$, let $k \geq 1$ be the maximal integer such that $\varepsilon_k \geq \varepsilon$. Then $G(\varepsilon) = G(\varepsilon_k)$ is the stabilizer of $\Omega_k$, equivalently the stabilizer of the $k$-th level of the rooted tree. It follows that the metric quotient $G/G(\varepsilon)$ is a $k$-fold iterated wreath product of the symmetric group $\operatorname{Sym}(X)$. In particular all metric quotients are finite $\pi$-groups, where $\pi$ is the set of primes $p \leq |X|$. So $MQ(\operatorname{Aut}(X_\bullet^*))$ is the class of all finite groups (recall that we are also including subgroups of metric quotients in our definition of $MQ$).
\end{example}
\subsubsection{Projective limits}
Let $(A_k)_{k \geq 1}$ be an inverse system of discrete groups, indexed by the directed set $\mathbb{N}$, and let $G$ be the corresponding projective limit. Then we can define $d(g, h) = \varepsilon_k$, where $k \geq 1$ is the maximal integer such that $g$ and $h$ have the same image in $A_k$. This is a bi-invariant ultrametric.
Let $G_k$ be the kernel of the quotient map $G \to A_k$. Then $G_k = G(\varepsilon_k)$ and $d(g, h) \leq \varepsilon_k$ if and only if $g G_k = h G_k$. Given $\varepsilon > 0$, let $k \geq 1$ be the maximal integer such that $\varepsilon_k \geq \varepsilon$. Then $G(\varepsilon) = G(\varepsilon_k) = G_k$ and it follows that the metric quotient $G/G(\varepsilon)$ is isomorphic to $A_k$. \\
This construction applies to more general projective limits, where the defining system is countable but not necessarily indexed by $\mathbb{N}$. More precisely, if $G$ is the projective limit of $(A_i)_{i \in I}$, where $I$ is a countable directed set, then we may choose a sequence $(i_k)_{k \geq 1}$ that is order-isomorphic to $\mathbb{N}$ such that the corresponding system $(A_{i_k})_{k \geq 1}$ gives back $G$.
Note that given an inverse system $(A_i)_{i \in I}$, the set $I$ is countable if and only if $G$ admits a countable neighbourhood basis of the identity, which for topological groups is equivalent to being first-countable. This shows that countability is a necessary and sufficient condition to put a metric on $G$, since a metric space is always first-countable.
\begin{example}
\label{ex:prof}
Let $G$ be a first-countable profinite group. Then there exists a strictly nested sequence $(G_k)_{k \geq 1}$ of finite-index normal subgroups that intersect trivially. The metric defined by $d(g, h) = \varepsilon_k$, where $k$ is the maximal integer such that $gG_k = hG_k$, is a bi-invariant ultrametric. The corresponding metric quotients are the finite groups $G/G_k$. We denote this metric by $d = d((G_k)_{k \geq 1}, \overline{\varepsilon})$.
\end{example}
Here are two specific examples of first-countable profinite groups metrized this way. The first we have already seen from another point of view.
\begin{example}
\label{ex:aut:prof}
Using the same notation as in Example \ref{ex:aut:filt}, let $G = \operatorname{Aut}(X^*)$ and $G_k$ be the stabilizer of the $k$-th level. Then $(G_k)_{k \geq 1}$ is a strictly nested sequence of finite-index normal subgroups that intersect trivially. Letting $d = d((G_k)_{k \geq 1}, \overline{\varepsilon})$ be the corresponding metric from Example \ref{ex:prof}, we obtain the same metric as in Example \ref{ex:aut:filt}.
\end{example}
The next example falls into the more general construction, where the set $I$ is countable but not naturally order-isomorphic to $\mathbb{N}$. See \cite{Neuk} for more information.
\begin{example}
\label{ex:gal:prof}
Let $K$ be a field, $L/K$ a Galois extension and $\operatorname{Gal}(L/K)$ the corresponding Galois group: this is the projective limit of the Galois groups of all intermediate finite Galois extensions. Let $G$ be the absolute Galois group, that is, the Galois group of the separable closure $K^{sep}/K$ of $K$: for any other Galois extension $L/K$ the group $\operatorname{Gal}(L/K)$ is a quotient of $G$ by a closed subgroup. Suppose that $G$ is first-countable. Then we can choose a strictly nested sequence $G_k$ of open finite-index normal subgroups, and project $G_k$ onto $\operatorname{Gal}(L/K)$ for every other Galois extension: the resulting groups will have the same property. This defines a metric as in Example \ref{ex:prof} on each $\operatorname{Gal}(L/K)$, with respect to the image of the sequence $(G_k)_{k \geq 1}$ and the sequence $\overline{\varepsilon}$.
We denote by $\operatorname{Gal}(K)$ the profinite family obtained this way, or $\operatorname{Gal}(K)((G_k)_{k \geq 1}, \overline{\varepsilon})$ if we want to emphasize the choices of the sequences $(G_k)_{k \geq 1}$ and $\overline{\varepsilon}$.
\end{example}
Here is a characterization of fields whose absolute Galois group is first-countable:
\begin{lemma}
\label{lem:gal_firstcount}
Let $L/K$ be a Galois extension of a field $K$. Then the following are equivalent:
\begin{enumerate}
\item $\operatorname{Gal}(L/K)$ is first-countable;
\item $L$ is a countable (increasing) union of intermediate finite Galois extensions;
\item There exist only countably many intermediate finite Galois extensions.
\end{enumerate}
\end{lemma}
\begin{proof}
$1. \Leftrightarrow 2.$ A countable nested sequence of finite-index open normal subgroups corresponds to a countable increasing sequence of finite Galois extensions. If we choose the subgroups to form a basis, so to intersect trivially, the corresponding increasing union gives all of $L$, and viceversa. Moreover the existence of a countable union implies the existence of an increasing one, since the compositum of finitely many finite Galois extensions is a finite Galois extension. \\
$2. \Leftrightarrow 3.$ Write $L = \bigcup_{i \geq 1} K_i$, where each $K_i$ is a finite intermediate Galois extension. By the Primitive Element Theorem, each intermediate finite Galois extension is contained in some $K_i$. By Galois Correspondence, there are only finitely many Galois subextensions of each $K_i$. The other implication is clear.
\end{proof}
The first easy example is the following:
\begin{example}
Let $K$ be a countable field. Then $K$ admits only countably many finite Galois extensions, since $K^{sep}$ itself is countable. For instance the absolute Galois group of $\mathbb{Q}$, or a finite field, is first-countable.
\end{example}
The second one is more involved. It relies on Krasner's Lemma \cite[8.1.6]{Neuk}: let $K$ be a non-Archimedean complete normed field, $\alpha \in K^{sep}$ with conjugates $\alpha = \alpha_1, \ldots, \alpha_d$. If $\beta \in K^{alg}$ satisfies $|\beta - \alpha| < |\alpha - \alpha_i|$ for all $i = 2, \ldots, d$, then $K(\alpha) \subset K(\beta)$. This lemma is crucial in the proof that the field $\mathbb{C}_p$ of $p$-adic complex numbers is algebraically closed \cite[Theorem 10.3.2]{Neuk}, and in fact the following proof is very similar to that.
\begin{example}
Let $K$ be a complete non-Archimedean field whose topology is separable. Then $K$ admits only countably many finite Galois extensions. This applies to all non-Archimedean local fields.
\begin{proof}
By hypothesis there exists a countable dense subset $D \subset K$, which we may assume is a field. We claim that $K^{sep} = \bigcup_{\beta \in D^{sep}} K(\beta)$. Since $D^{sep}$ is countable, this allows to conclude thanks to Lemma \ref{lem:gal_firstcount}. So let $\alpha \in K^{sep}$ and let $f(X) \in K[X]$ be its minimal polynomial, whose roots $\alpha = \alpha_1, \ldots, \alpha_d$ are the Galois conjugates of $\alpha$, which are all distinct since $\alpha$ is separable. We can choose a $g(X) \in D[X]$ arbitrarily close to $f(X)$, in fact we can choose $g$ so that $|g(\alpha_j)| = |g(\alpha_j) - f(\alpha_j)|$ is arbitrarily small, for all $j$. Now write $g(X) = \prod (X - \beta_i)$, where $\beta_i \in K^{alg}$: this implies that for all $j$ there exists $i$ such that $|\alpha_j - \beta_i|$ is small, say smaller than $|\alpha_j - \alpha_k|$ for all $j \neq k$. Since $g$ has the same degree as $f$, and the $\alpha_j$ are all different, this association is a bijection. This implies that all $\beta_i$ are distinct, so $g$ is separable. Moreover, if $\beta$ is the root close to $\alpha$, we have $K(\alpha) \subset K(\beta)$ by Krasner's Lemma, and $\beta \in D^{sep}$.
\end{proof}
\end{example}
\subsubsection{Integral matrices}
Let $(\mathbb{K}, | \cdot |)$ be a non-Archimedean field with ring of integers $\mathfrak{o}$, maximal ideal $\mathfrak{p}$ and residue field $\mathfrak{k}$ of characteristic $p \geq 0$. Then the matrix group $\operatorname{GL}(\mathfrak{o})$ comes equipped with the $\ell^\infty$-norm induced by the inclusion into $\operatorname{M}_n(\mathbb{K})$:
$$\| M \| := \max \{ |M_{ij}| : 1 \leq i, j \leq n \}.$$
This induces a distance by $d(A, B) = \| A - B \|$, which we already mentioned in the introduction in the special case $\mathbb{K} = \mathbb{Q}_p$.
\begin{lemma}
\label{lem:GL}
Let $\| \cdot \|$ and $d$ be as above. Then
\begin{enumerate}
\item $d$ is an ultrametric.
\item $\| \cdot \|$ is submultiplicative.
\item $\| \cdot \|$ is equal to the operator norm, where $\mathbb{K}^n$ is also equipped with the $\ell^\infty$-norm.
\item If $A \in \operatorname{GL}_n(\mathfrak{o})$, then $\| A \| = 1$.
\item If $A \in \operatorname{GL}_n(\mathfrak{o})$ and $\| A - B \| < 1$, then $B \in \operatorname{GL}_n(\mathfrak{o})$.
\item $\| \cdot \|$ is invariant under left or right multiplication by elements of $\operatorname{GL}_n(\mathfrak{o})$.
\end{enumerate}
\end{lemma}
\begin{proof}
1. This follows directly from the fact that the norm on $\mathbb{K}$ induces an ultrametric. \\
2. Let $A, B \in \operatorname{M}_n(\mathbb{K})$. Then
$$\| AB \| = \max\limits_{1 \leq i, j \leq n} |(AB)_{ij}| = \max\limits_{1 \leq i, j \leq n} \left| \sum\limits_{k = 1}^n A_{ik} B_{kj} \right| \leq $$
$$ \leq \max\limits_{1 \leq i, j, k \leq n} |A_{ik} B_{kj}| \leq \max\limits_{1 \leq i, k \leq n} |A_{ik}| \cdot \max\limits_{1 \leq j, k \leq n} |B_{kj}| = \|A\| \cdot \|B\|.$$
3. The submultiplicativity also holds for matrix-vector multiplication, with the same proof: $\|Ax\| \leq \| A \| \cdot \|x \|$ for all $x \in \mathbb{K}^n$, and so $\|A\|_{op} \leq \|A\|$. For the converse, Suppose that $|A_{ij}|$ attains its maximum in the $i$-th column $A_i$. Then $\|A\|_{op} \geq \|A e_i\| = \|A_i\| = \|A\|$. \\
4. Suppose that $A \in \operatorname{GL}_n(\mathfrak{o})$; this immediately implies $\|A\| \leq 1$. It also implies that $\det(A) \in \mathfrak{o}^\times$ so $|\det(A)| = 1$. Since $\det(A)$ is a polynomial on $A_{ij}$, this is not possible if $\|A\| < 1$. Note that the converse of $4.$ is not true: if $\| A \| = 1$, then $A$ is integral, but its determinant may lie in $\mathfrak{o} \, \backslash \, \mathfrak{o}^\times$, in which case $A^{-1}$ is not integral. \\
5. Clearly $B \in \operatorname{M}_n(\mathfrak{o})$ so it suffices to show that $\det(B) \in \mathfrak{o}^\times$. This is because $A \equiv B \mod \mathfrak{p}$, so $\det(B \mod \mathfrak{p}) \neq 0 \in \mathfrak{k}$. \\
6. Let $A \in \operatorname{GL}_n(\mathfrak{o})$ and $M \in \operatorname{M}_n(\mathbb{K})$. Then, using 2. and 4.:
$$\|M\| = \|A^{-1} A M \| \leq \|A^{-1}\| \cdot \|AM\| = \|AM\| \leq \|A\| \cdot \|M\| = \|M\|.$$
Therefore $\|AM\| = \|M\|$. Similarly $\|MA\| = \|M\|$.
\end{proof}
The family of groups $\operatorname{GL}_n(\mathfrak{o})$ with the distance $d$ is thus an ultrametric family. We denote this family by $\operatorname{GL}(\mathfrak{o})$. A special case is the family $\operatorname{GL}(\mathbb{Z}_p)$ from the introduction. \\
In case $\mathbb{K}$ is a local field, $\operatorname{GL}(\mathfrak{o})$ is a profinite family, and this norm can also be seen as a special case of Example \ref{ex:prof}. Since $\mathbb{K}$ is discretely valued we can choose as a sequence $\varepsilon_k := |\overline{\omega}|^k$ (where $\overline{\omega}$ is a uniformizer), and $\operatorname{GL}_n(\mathfrak{o})_k$ will be the ball of radius $\varepsilon_k$ around the identity. This can be explicitly identified as the \textit{congruence subgroup}:
$$\operatorname{GL}_n(\mathfrak{o})_k := \{ I + \overline{\omega}^k M : M \in \operatorname{M}_n(\mathfrak{o}) \},$$
where $\overline{\omega}$ is a uniformizer. The metric quotients are the finite matrix groups $\operatorname{GL}_n(\mathfrak{o} / \mathfrak{p}^k)$. There is no restriction on the order of these groups: indeed any finite group embeds into $\operatorname{GL}_n(\mathfrak{o}/\mathfrak{p}) = \operatorname{GL}_n(\mathfrak{k})$ for $n$ large enough. However there is a restriction on the order of the metric quotients of the principal congruence subgroup $\operatorname{GL}_n(\mathfrak{o})_1$, which will be crucial in Sections \ref{s:vpropi} and \ref{s:char0}.
\begin{lemma}
\label{lem:vprop}
With the notation above, the principal congruence subgroup $\operatorname{GL}_n(\mathfrak{o})_1$ is pro-$p$. More precisely, for all $k \geq 1$ there is an isomorphism:
$$\operatorname{GL}_n(\mathfrak{o})_k / \operatorname{GL}_n(\mathfrak{o})_{2k} \to \left( \operatorname{M}_n( \mathfrak{o}/ \mathfrak{p}^k ), + \right).$$
\end{lemma}
\begin{proof}
Define the map $\operatorname{GL}_n(\mathfrak{o})_k \to \operatorname{M}_n(\mathfrak{o}/ \mathfrak{p}^k) : (I + \overline{\omega}^k M) \mapsto M \mod \overline{\omega}^k$. This is a homomorphism:
$$(I + \overline{\omega}^k M)(I + \overline{\omega}^k N) = I + \overline{\omega}^k(M + N + \overline{\omega}^k MN),$$
and the kernel is precisely $\operatorname{GL}_n(\mathfrak{o})_{2k}$.
\end{proof}
\pagebreak
\section{Ultrametric stability}
\label{s:ultra}
Let $\mathcal{G}$ be an ultrametric family. This section is concerned with stability with respect to $\mathcal{G}$, without additional assumptions, the main goals being Theorems \ref{intro:thm:pw_un} and \ref{intro:thm:rf}. The main tool for the proof of Theorem \ref{intro:thm:pw_un} will be to rephrase stability in terms of presentations, following \cite{a:comm}: this is well-known in the general setting, but the ultrametric framework gives better quantitative estimates that we will use in the rest of the paper, so we go through the arguments in detail. The proof of Theorem \ref{intro:thm:rf} will be quite direct thanks to Lemma \ref{lem:mq}. We will end the section by giving complete solutions to uniform stability problems with respect to some of the families introduced in Section \ref{s:fam}.
\subsection{Lifting and inducing asymptotic homomorphisms}
\label{ss:free}
Let $\Gamma = \langle S \mid R \rangle$ be a presentation of $\Gamma$: for the moment we do not impose any finiteness condition. We define defects and distances for homomorphisms that are close to satisfying the relations: this is analogous to how one can look at homomorphisms on $\Gamma$ as homomorphisms that satisfy the relations. The following definitions are due to Arzhantseva--P\u{a}unescu in the finitely presented case \cite{a:comm}, and to Becker--Lubotzky--Thom in the finitely generated case \cite{IRS}.
\begin{definition}
Given a map $\hat{\varphi} : F_S \to G \in \mathcal{G}$ we define the \textit{defect of $\hat{\varphi}$ at $r \in \langle \langle R \rangle \rangle$} and the \textit{defect of $\hat{\varphi}$} to be
$$\operatorname{def}_r(\hat{\varphi}) := d_G(\hat{\varphi}(r), 1_G); \,\,\,\,\,\,\, \operatorname{def}(\hat{\varphi}) := \sup\limits_{r \in R} \operatorname{def}_r(\hat{\varphi}).$$
Given two maps $\hat{\varphi}, \hat{\psi} : F_S \to G \in \mathcal{G}$ we define their \textit{distance at $w \in F_S$} and their \textit{distance} to be
$$\operatorname{dist}_w(\hat{\varphi}, \hat{\psi}) := d_G(\hat{\varphi}(s), \hat{\psi}(s)); \,\,\,\,\,\,\, \operatorname{dist}(\hat{\varphi}, \hat{\psi}) := \sup\limits_{s \in S} \operatorname{dist}_s(\hat{\varphi}, \hat{\psi}).$$
\end{definition}
We will show in Lemma \ref{lem:ultra_lift} the correspondence between these notions ``at the level of $F_S$" and those ``at the level of $\Gamma$" that we defined in the introduction. A good feature of these definitions is that they allow to give a unique quantity for the notions of defect (for finitely presented groups) and of distance (for finitely generated groups), even when dealing with pointwise asymptotic homomorphisms. For dealing with uniform almost-homomorphisms, one would instead have to define the defect with a supremum over \textit{all} relations $r \in \langle \langle R \rangle \rangle$, and the distance with a supremum over \textit{all} words $w \in F_S$. It turns out that this is not necessary under the ultrametric assumption:
\begin{lemma}
\label{lem:ultra_free}
Let $\Gamma = \langle S \mid R \rangle$ and $\hat{\varphi}, \hat{\psi} : F_S \to G \in \mathcal{G}$ a homomorphism. Then:
\begin{enumerate}
\item For every $r \in \langle \langle R \rangle \rangle$ there exists a finite set $\{r_1, \ldots, r_k\} \subset R$ (independent of $\hat{\varphi}$) such that
$$\operatorname{def}_r(\hat{\varphi}) \leq \max\limits_{i} \operatorname{def}_{r_i}(\hat{\varphi}).$$
In particular $\sup\limits_{r \in \langle \langle R \rangle \rangle} \operatorname{def}_r(\hat{\varphi}) = \sup\limits_{r \in R} \operatorname{def}_r(\hat{\varphi}) = \operatorname{def}(\hat{\varphi})$.
\item For every $w \in F_S$ there exists a finite set $\{ s_1, \ldots, s_k \} \subset S$ (independent of $\hat{\varphi}$ and $\hat{\psi}$) such that
$$\operatorname{dist}_w(\hat{\varphi}, \hat{\psi}) \leq \max\limits_i \operatorname{dist}_{s_i}(\hat{\varphi}, \hat{\psi}).$$
In particular $\sup\limits_{w \in F_S} \operatorname{dist}_w(\hat{\varphi}, \hat{\psi}) = \sup\limits_{s \in S} \operatorname{dist}_s(\hat{\varphi}, \hat{\psi}) = \operatorname{dist}(\hat{\varphi}, \hat{\psi})$.
\end{enumerate}
\end{lemma}
\begin{proof}
$1.$ Let $r \in \langle \langle R \rangle \rangle$, and write $r = (w_1 r_1 w_1^{-1}) \cdots (w_k r_k w_k^{-1})$ for $w_i \in F_S$ and $r_i \in R$. Then:
$$\operatorname{def}_r(\hat{\varphi}) = d_G(\hat{\varphi}(r), 1_G) = d_G(\hat{\varphi}(w_1 r_1 w_1^{-1}) \cdots \hat{\varphi}(w_k r_k w_k^{-1}), 1_G) \leq $$
$$\leq \max \{ d_G(\hat{\varphi}(w_2 r_2 w_2^{-1}) \cdots \hat{\varphi}(w_k r_k w_k^{-1}), 1_G), d_G(\hat{\varphi}(w_1 r_1 w_1^{-1}), 1_G) \} \leq \cdots \leq $$
$$\leq \max\limits_{i} d_G(\hat{\varphi}(w_i r_i w_i^{-1}), 1_G) = \max\limits_{i} d_G(\hat{\varphi}(r_i), 1_G) = \max\limits_i \operatorname{def}_{r_i}(\hat{\varphi}).$$
$2.$ Let $w = s_1 \cdots s_k \in F_S$. Then, similarly:
$$\operatorname{dist}_w(\hat{\varphi}, \hat{\psi}) = d_G(\hat{\varphi}(w), \hat{\psi}(w)) = d_G(\hat{\varphi}(s_1) \cdots \hat{\varphi}(s_k) \hat{\psi}(s_k)^{-1} \cdots \hat{\psi}(s_1)^{-1}, 1_G) \leq$$
$$\leq \max\limits_{i} d_G(\hat{\varphi}(s_i) \hat{\psi}(s_i)^{-1}, 1_G) = \max\limits_{i} d_G(\hat{\varphi}(s_i), \hat{\psi}(s_i)) = \max_i \operatorname{dist}_{s_i}(\hat{\varphi}, \hat{\psi}).$$
\end{proof}
Let us now make the connection between such maps and those defined at the level of $\Gamma$:
\begin{lemma}
\label{lem:ultra_lift}
Let $\Gamma = \langle S \mid R \rangle$ and denote by $F_S \to \Gamma : w \mapsto \overline{w}$ the projection.
\begin{enumerate}
\item Let $\varphi : \Gamma \to G \in \mathcal{G}$ be a map. Let $\hat{\varphi} : F_S \to G$ be the unique homomorphism coinciding with $\varphi$ on $S$. Then for any word $w \in F_S$ there exists a finite set $\{(g_1, h_1), \ldots, (g_k, h_k) \} \subset \Gamma^2$ (independent of $\varphi$) such that
$$d_G(\varphi(\overline{w}), \hat{\varphi}(w)) \leq \max\limits_i \operatorname{def}_{g_i, h_i}(\varphi).$$
In particular, if $r \in \langle \langle R \rangle \rangle$ is a relation, then $\operatorname{def}_r(\hat{\varphi}) \leq \max\limits_i \operatorname{def}_{g_i, h_i}(\varphi)$ and $\operatorname{def}(\hat{\varphi}) \leq \operatorname{def}(\varphi)$.
\item Let $\hat{\varphi} : F_S \to G \in \mathcal{G}$ be a homomorphism. Choose a (set-theoretic) section $\sigma : \Gamma \to F_S$ and define $\varphi := \hat{\varphi} \circ \sigma$. Then for every $(g, h) \in \Gamma^2$ there exists a finite set $\{ r_1, \ldots, r_k \} \subset R$ (independent of $\hat{\varphi}$ but depending on $\sigma$) such that
$$\operatorname{def}_{g, h}(\varphi) \leq \max\limits_i \operatorname{def}_{r_i}(\hat{\varphi}).$$
In particular $\operatorname{def}(\varphi) \leq \operatorname{def}(\hat{\varphi})$.
\end{enumerate}
\end{lemma}
The proof that follows is unfortunately heavy in notation, mainly because we are making a rigorous distinction between elements of $S$ and elements of $S^{-1}$. All indices $\varepsilon_i$ could be taken out if we could reduce to the case in which $\varphi(s^{-1}) = \varphi(s)^{-1}$. However this assumption is equivalent to stability of the group $\mathbb{Z}/2 \mathbb{Z}$, which cannot be established in such great generality, as the next example shows.
\begin{example}
\label{ex:2unst}
For each $n \geq 1$, let $d_n$ be the metric on $\mathbb{Z}_p$ defined by the non-Archimedean norm $|\cdot|_p^n$, let $G_n$ denote $\mathbb{Z}_p$ equipped with this metric and $\mathcal{G}$ the corresponding family, which is ultrametric. We claim that $\mathbb{Z}/p\mathbb{Z}$ is not uniformly $\mathcal{G}$-stable (note that the pointwise and uniform notions coincide for finite groups). Indeed, let $\varphi_n : \mathbb{Z}/p\mathbb{Z} \to \mathbb{Z}_p$ be the map sending $k \mod p$ to $k$, for $0 \leq k < p$. Since $\mathbb{Z}_p$ is torsion-free, the only homomorphism $\mathbb{Z}/p\mathbb{Z} \to \mathbb{Z}_p$ is the trivial one, which is at a distance $1$ from $\varphi_n$. On the other hand $\operatorname{def}(\varphi_n) = d_n(p, 0) = p^{-n} \to 0$. Thus $(\varphi_n : \mathbb{Z}/p\mathbb{Z} \to G_n)_{n \geq 1}$ is a uniform asymptotic homomorphism that is not uniformly asymptotically close to any homomorphism.
\end{example}
We proceed with the proof.
\begin{proof}[Proof of Lemma \ref{lem:ultra_lift}]
$1.$ Fix a word $w = s_1^{\varepsilon_1} \cdots s_k^{\varepsilon_k}$ written as a reduced product of elements of $S \sqcup S^{-1}$: here $s_i \in S$ and $\varepsilon_i = \pm 1$. Let
$$\delta := \max\limits_i d_G(\varphi(s_1^{\varepsilon_1} \cdots s_i^{\varepsilon_i}), \varphi(s_1^{\varepsilon_1} \cdots s_{i-1}^{\varepsilon_{i-1}}) \varphi(s_i)^{\varepsilon_i}).$$
Notice that this is bounded by finitely many terms of the form $\operatorname{def}_{g, h}(\varphi)$ for some $(g, h) \in \Gamma^2$: here we are using that $d_G(\varphi(s^{-1}), \varphi(s)^{-1}) \leq \max\{ \operatorname{def}_{s, s^{-1}}(\varphi), \operatorname{def}_{1, 1}(\varphi) \}$. Now $d_G(\varphi(s_1^{\varepsilon_1}), \varphi(s_1)^{\varepsilon_1}) \leq \delta$, and by induction
$$d_G(\varphi(s_1^{\varepsilon_1} \cdots s_i^{\varepsilon_i}), \varphi(s_1)^{\varepsilon_1} \cdots \varphi(s_i)^{\varepsilon_i}) \leq \max \{ d_G(\varphi(s_1^{\varepsilon_1} \cdots s_i^{\varepsilon_i}), \varphi(s_1^{\varepsilon_1} \cdots s_{i-1}^{\varepsilon_{i-1}})\varphi(s_i)^{\varepsilon_i}), $$
$$d_G(\varphi(s_1^{\varepsilon_1} \cdots s_{i-1}^{\varepsilon_{i-1}}), \varphi(s_1)^{\varepsilon_1} \cdots \varphi(s_{i-1})^{\varepsilon_{i-1}}) \} \leq \delta.$$
Therefore
$$d_G(\varphi(\overline{w}), \hat{\varphi}(w)) = d_G(\varphi(s_1^{\varepsilon_1} \cdots s_k^{\varepsilon_k}), \varphi(s_1)^{\varepsilon_1} \cdots \varphi(s_k)^{\varepsilon_k}) \leq \delta.$$
The last statement follows by taking $w$ to be a relation, so $\varphi(\overline{w}) = \varphi(1_\Gamma)$ is close to $1_G$; more precisely $d_G(\varphi(1), 1) = \operatorname{def}_{1, 1}(\varphi)$. \\
$2.$ Fix $g, h \in \Gamma$; and let $r := \sigma(gh) (\sigma(g) \sigma(h))^{-1} \in \langle \langle R \rangle \rangle$. Then
$$\operatorname{def}_{g, h}(\varphi) = d_G(\varphi(gh), \varphi(g)\varphi(h)) = d_G(\hat{\varphi}(\sigma(gh)), \hat{\varphi}(\sigma(g))\hat{\varphi}(\sigma(h)) ) = d_G(\hat{\varphi}(r), 1_G) = \operatorname{def}_r(\hat{\varphi}).$$
The result then follows from Item $1.$ of Lemma \ref{lem:ultra_free}.
\end{proof}
This implies the desired equivalent characterization of stability:
\begin{proposition}
\label{prop:stab_equiv}
Let $\Gamma = \langle S \mid R \rangle$.
\begin{enumerate}
\item $\Gamma$ is pointwise $\mathcal{G}$-stable if and only if for any sequence $(\hat{\varphi}_n : F_S \to G_n \in \mathcal{G})_{n \geq 1}$ such that $\operatorname{def}_r(\hat{\varphi}_n) \to 0$ for all $r \in R$, there exists a sequence $(\hat{\psi}_n : F_S \to G_n)_{n \geq 1}$ such that $\operatorname{def}(\hat{\psi}_n) = 0$ and $\operatorname{dist}_s(\hat{\varphi}_n, \hat{\psi}_n) \to 0$ for all $s \in S$.
\item $\Gamma$ is uniformly $\mathcal{G}$-stable if and only if for any sequence $(\hat{\varphi}_n : F_S \to G_n \in \mathcal{G})_{n \geq 1}$ such that $\operatorname{def}(\hat{\varphi}_n) \to 0$, there exists a sequence $(\hat{\psi}_n : F_S \to G_n)_{n \geq 1}$ such that $\operatorname{def}(\hat{\psi}_n) = 0$ and $\operatorname{dist}(\hat{\varphi}_n, \hat{\psi}_n) \to 0$.
\end{enumerate}
\end{proposition}
\begin{remark}
The condition $\operatorname{def}(\hat{\psi}) = 0$ is equivalent to: $\hat{\psi}$ descends to a homomorphism of $\Gamma$.
\end{remark}
\begin{proof}
Suppose that $\Gamma$ is pointwise $\mathcal{G}$-stable and let $\hat{\varphi}_n$ be a sequence as in the statement of $1.$ Item $2.$ of Lemma \ref{lem:ultra_free} gives a sequence $(\varphi_n : \Gamma \to G_n)_{n \geq 1}$ such that $\operatorname{def}_{g, h}(\varphi_n) \to 0$ for all $(g, h) \in \Gamma^2$. By pointwise stability the asymptotic homomorphism $(\varphi_n)_{n \geq 1}$ is pointwise asymptotically close to a sequence of homomorphisms $(\psi_n : \Gamma \to G_n)_{n \geq 1}$. This lifts to a sequence $(\hat{\psi}_n : F_S \to G_n)_{n \geq 1}$ such that $\operatorname{def}(\hat{\psi}_n) = 0$ and $\operatorname{dist}_s(\hat{\varphi}_n, \hat{\psi}_n) \to 0$ for all $s \in S$.
The converse is similar, using Item $1.$ of Lemma \ref{lem:ultra_free} instead, and the proof for the uniform case is the same.
\end{proof}
So one can take the notions of stability to be defined by maps at the level of the free group that almost descend to $\Gamma$. Then this proposition becomes less obvious than it looks, since it says that this notion does not depend on the choice of the presentation. This fact can also be proven directly by noticing, as in \cite[Section 3]{a:comm}, that this notion of stability of a presentation is not affected by Tietze transformations \\
We can provide a further equivalent characterization of uniform stability by rephrasing the quantitative characterization from Lemma \ref{lem:quant} in these terms:
\begin{corollary}
\label{cor:quant}
Let $\Gamma = \langle S \mid R \rangle$. The following are equivalent:
\begin{enumerate}
\item $\Gamma$ is uniformly $\mathcal{G}$-stable.
\item For all $\varepsilon > 0$ there exists $\delta > 0$ such that whenever $\hat{\varphi} : F_S \to G \in \mathcal{G}$ satisfies $\operatorname{def}(\hat{\varphi}) \leq \delta$, there exists $\hat{\psi} : F_S \to G$ that descends to $\Gamma$ and satisfies $\operatorname{dist}(\varphi, \psi) \leq \varepsilon$.
\end{enumerate}
\end{corollary}
\begin{proof}
The proof is the same as that of Lemma \ref{lem:quant}, using the characterization of uniform stability from Proposition \ref{prop:stab_equiv}.
\end{proof}
One last thing we need to understand is when closeness on the generators implies closeness elsewhere. The following lemma showcases how useful moving up and down from $\Gamma$ to the corresponding free group can be.
\begin{lemma}
\label{lem:ultra_close}
Let $\Gamma$ be generated by a set $S$, and consider two maps $\varphi, \psi : \Gamma \to G \in \mathcal{G}$. Then for all $g \in \Gamma$ there exist finite sets $\{ (x_1, y_1), \ldots, (x_k, y_k) \} \subset \Gamma^2$ and $\{ s_1, \ldots, s_k \} \subset S$ (independent of $\varphi, \psi$) such that
$$\operatorname{dist}_g(\varphi, \psi) \leq \max\limits_i \{ \operatorname{def}_{x_i, y_i}(\varphi), \operatorname{def}_{x_i, y_i}(\psi), \operatorname{dist}_{s_i}(\varphi, \psi) \}.$$
In particular
$$\operatorname{dist}(\varphi, \psi) \leq \max \{ \operatorname{def}(\varphi), \operatorname{def}(\psi), \sup\limits_{s \in S} \operatorname{dist}_s(\varphi, \psi) \}.$$
\end{lemma}
\begin{proof}
Let $\hat{\varphi}, \hat{\psi} : F_S \to G$ be the homomorphisms obtained via Item $1.$ of Lemma \ref{lem:ultra_lift}. That statement allows us to work with $\hat{\varphi}, \hat{\psi}$ instead, up to finitely many defects of $\varphi$ and $\psi$. Then Item $2.$ of Lemma \ref{lem:ultra_free} implies that the distance at $g$ is bounded in terms of the distance at finitely many generators.
\end{proof}
\subsection{Finiteness conditions}
Now we add finiteness conditions on the presentation. The following proposition gives general properties of asymptotic homomorphisms under such hypotheses:
\begin{proposition}
\label{prop:ultra_asy}
\begin{enumerate}
\item Suppose that $\Gamma$ is generated by the finite set $S$, and that $(\varphi_n, \psi_n : \Gamma \to G_n \in \mathcal{G})_{n \geq 1}$ satisfy $\operatorname{dist}_s(\varphi_n, \psi_n) \to 0$ for all $s \in S$. If $\varphi_n, \psi_n$ are pointwise (respectively, uniform) asymptotic homomorphisms, then they are pointwise (respectively, uniformly) asymptotically close.
\item Suppose that $\Gamma$ is finitely presented. Then any pointwise asymptotic homomorphism is pointwise asymptotically close to a uniform asymptotic homomorphism.
\end{enumerate}
\end{proposition}
\begin{proof}
$1.$ This follows directly from Lemma \ref{lem:ultra_close}. \\
$2.$ Let $(\varphi_n)_{n \geq 1}$ be an asymptotic homomorphism. Using Item $1.$ of Lemma \ref{lem:ultra_lift}, we can lift $\varphi_n$ to $(\hat{\varphi}_n : F_S \to G_n)_{n \geq 1}$ such that $\operatorname{def}_r(\hat{\varphi}_n)$ is bounded in terms of finitely many defects of $\varphi_n$ for any relator $r \in R$. Since $R$ is finite, the same holds for $\operatorname{def}(\hat{\varphi}_n)$. Now $\hat{\varphi}_n$ induces a map on $\Gamma$ using Item $2.$ of Lemma \ref{lem:ultra_lift}, thus we obtain maps $\psi_n : \Gamma \to G_n$ such that $\operatorname{def}(\psi_n) \leq \operatorname{def}(\hat{\varphi}_n)$. It follows that $\psi_n$ is a uniform asymptotic homomorphism. It is pointwise asymptotically close to $\varphi_n$ by the previous item.
\end{proof}
Item $2.$ of Proposition \ref{prop:ultra_asy} does not say that pointwise asymptotic homomorphisms are automatically uniform: this is false in general, as the next example shows.
\begin{example}
Consider the map
$$\varphi_n : \mathbb{Z} \to \operatorname{GL}_2(\mathbb{Z}_p) : k \mapsto
\begin{cases}
\begin{pmatrix}
1 & k \\
0 & 1
\end{pmatrix} & \text{if } |k| \leq n \\
I_2 & \text{otherwise.}
\end{cases}$$
Then $\operatorname{def}(\varphi_n) = 1$ for all $n$, while $\operatorname{def}_{g, h}(\varphi_n) \to 0$ for all $(g, h) \in \mathbb{Z}^2$. However note that, by construction, $\varphi_n$ is pointwise (not uniformly) close to a homomorphism, in line with Item $2.$ of Proposition \ref{prop:ultra_asy}.
\end{example}
We can now prove Theorem \ref{intro:thm:pw_un}:
\begin{theorem}
\label{thm:pw_un}
Let $\Gamma$ be finitely generated and pointwise $\mathcal{G}$-stable. Then $\Gamma$ is uniformly $\mathcal{G}$-stable. If moreover $\Gamma$ is finitely presented, then the converse holds.
\end{theorem}
\begin{proof}
Let $(\varphi_n : \Gamma \to G_n \in \mathcal{G})_{n \geq 1}$ be a uniform asymptotic homomorphism. Since $\Gamma$ is pointwise $\mathcal{G}$-stable, this is pointwise asymptotically close to a sequence of homomorphisms $(\psi_n : \Gamma \to G_n)_{n \geq 1}$. By Item $1.$ of Proposition \ref{prop:ultra_asy}, since $\varphi_n$ and $\psi_n$ are both uniform, they are uniformly asymptotically close. \\
Now suppose that $\Gamma$ is finitely presented and uniformly $\mathcal{G}$-stable. Let $(\varphi_n : \Gamma \to G_n \in \mathcal{G})_{n \geq 1}$ be a pointwise asymptotic homomorphism. By Item $2.$ of Proposition \ref{prop:ultra_asy}, this is pointwise asymptotically close to a uniform asymptotic homomorphism, which in turn is uniformly (thus pointwise) asymptotically close to a sequence of homomorphism, by uniform $\mathcal{G}$-stability.
\end{proof}
We will see in Example \ref{ex:cex} that there exist finitely generated groups that are uniformly but not pointwise $\operatorname{GL}(\mathfrak{o})$-stable. This theorem allows to unambiguously talk about \textit{$\mathcal{G}$-stability} for finitely presented groups, since pointwise and uniform stability are equivalent.
\begin{example}
\label{ex:free}
A free group of finite rank is $\mathcal{G}$-stable. This follows immediately by using the characterization of pointwise stability in Proposition \ref{prop:stab_equiv}. It also applies to free groups of countably infinite rank, proving both pointwise and uniform stability.
\end{example}
Although pointwise stability of free groups holds for any family $\mathcal{G}$, as we remarked in the introduction, uniform stability is really special to the ultrametric setting. Indeed, free groups are not uniformly $\mathcal{G}$-stable, for $\mathcal{G} = (\operatorname{U}(n), \| \cdot \|)$, where $\| \cdot \|$ is any $\operatorname{U}(n)$-invariant norm on $\operatorname{M}_n(\mathbb{C})$ \cite{Rolli}, or for $\mathcal{G} = (S_n, d_H)$ \cite{BChap}.
\begin{example}
\label{ex:discr:stab}
Let $\mathcal{G}$ be a discrete family (Example \ref{ex:discr}). Then every group is uniformly $\mathcal{G}$-stable: if $\varphi : \Gamma \to G \in \mathcal{G}$ satisfies $\operatorname{def}(\varphi) < 1$, then $\varphi$ is already a homomorphism. Theorem \ref{thm:pw_un} implies that all finitely presented groups are pointwise $\mathcal{G}$-stable.
\end{example}
Pointwise $\mathcal{G}$-stability need not hold for arbitrary finitely generated groups: if $\mathcal{G}$ is the discrete family of all finite groups and $\Gamma$ is LEF but not residually finite, then $\Gamma$ is not pointwise stable. This will be explained in more detail and generality in Section \ref{s:approx}.
\subsection{Homomorphisms onto metric quotients}
We now move to the proof of Theorem \ref{intro:thm:rf}. The main tool is given by Lemma \ref{lem:mq}, which relates asymptotic homomorphisms to $\mathcal{G}$ with true homomorphisms to $MQ(\mathcal{G})$, the family of metric quotients of $\mathcal{G}$ and subgroups thereof. \\
We start by proving a consequence of Lemma \ref{lem:mq} which essentially gives one direction of Theorem \ref{intro:thm:rf}. Let $G$ be an ultrametric group, and let $\mathcal{C}$ be a class of groups that is closed under taking subgroups, and such that all metric quotients of $G$ are contained in $\mathcal{C}$. By Lemma \ref{lem:largest_rf} there exists a largest residually-$\mathcal{C}$ quotient of $\Gamma$, that we denote by $R$. Lemma \ref{lem:mq} implies that maps onto $G$ with small defect almost factor through $R$.
\begin{lemma}
\label{lem:factor_R}
Let $\varphi : \Gamma \to G$ be such that $\operatorname{def}(\varphi) \leq \varepsilon$. Then there exists a map $\overline{\varphi} : R \to G$ such that $\operatorname{def}(\overline{\varphi}) \leq \varepsilon$ and $\operatorname{dist}(\varphi, \overline{\varphi} \circ \pi_R) \leq \varepsilon$.
\end{lemma}
\begin{proof}
By Item $1.$ of Lemma \ref{lem:mq} we can consider the homomorphisms $\varphi(\varepsilon) : \Gamma \to G/G(\varepsilon) \in \mathcal{C}$ induced by $\varphi$. By the universal property of $R$ (Lemma \ref{lem:largest_rf}) this map factors through a homomorphism $\phi : R \to G/G(\varepsilon)$. Take $\overline{\varphi}$ to be any lift of this homomorphism to a map $R \to G$. Then by construction $\phi : R \xrightarrow{\overline{\varphi}} G \to G/G(\varepsilon)$ is a homomorphism, so by Item $1.$ of Lemma \ref{lem:mq} again $\operatorname{def}(\overline{\varphi}) \leq \varepsilon$. Moreover, the induced homomorphisms $\varphi(\varepsilon), (\overline{\varphi}(\varepsilon) \circ \pi_R) : \Gamma \to G/G(\varepsilon)$ both coincide with $\phi \circ \pi_R$, so by Item $2.$ of Lemma \ref{lem:mq} we have $\operatorname{dist}(\varphi, \overline{\varphi} \circ \pi_R) \leq \varepsilon$.
\end{proof}
For the rest of this section, we let $R$ be the largest residually-$\mathcal{C}$ quotient of $\Gamma$, where $\mathcal{C}$ is a class of groups closed under taking subgroups and containing $MQ(\mathcal{G})$. That is, $R := \Gamma/K$ where $K$ is the intersection of all normal subgroups of $\Gamma$ with quotient in $\mathcal{C}$ (see Lemma \ref{lem:largest_rf}). For instance, if $\mathcal{G} = \operatorname{GL}(\mathfrak{o})$, then $MQ(\mathcal{G})$ is the class of all finite groups, and so $R$ can be taken to be the largest residually finite quotient of $\Gamma$.
\begin{theorem}
\label{thm:rf}
Let $\mathcal{G}, \Gamma, R$ be as above. Then $\Gamma$ is uniformly $\mathcal{G}$-stable if and only if $R$ is. If $\Gamma$ is pointwise $\mathcal{G}$-stable, then so is $R$.
\end{theorem}
\begin{proof}
Suppose that $R$ is uniformly $\mathcal{G}$-stable. We use the characterization from Lemma \ref{lem:quant}: for all $\varepsilon > 0$ there exists $\delta > 0$ such that whenever $\operatorname{def}(\varphi) \leq \delta$ there exists a homomorphism $\psi$ such that $\operatorname{dist}(\varphi, \psi) \leq \varepsilon$. Fix $\varepsilon > 0$ and let $\delta > 0$ be given by uniform stability of $R$; we may assume that $\delta \leq \varepsilon$. Let $\varphi : \Gamma \to G \in \mathcal{G}$ be a map such that $\operatorname{def}(\varphi) \leq \delta$. By Lemma \ref{lem:factor_R} there exists a map $\overline{\varphi} : R \to G$ such that $\operatorname{def}(\overline{\varphi}) \leq \delta$ and $\operatorname{dist}(\varphi, \overline{\varphi} \circ \pi_R) \leq \delta$. By uniform stability of $R$ there exists a homomorphism $\overline{\psi} : R \to G$ such that $\operatorname{dist}(\overline{\varphi}, \overline{\psi}) \leq \varepsilon$. Then $\psi := \overline{\psi} \circ \pi_R : \Gamma \to G$ is a homomorphism and
$$\operatorname{dist}(\varphi, \psi) \leq \max\{ \operatorname{dist}(\varphi, \overline{\varphi} \circ \pi_R), \operatorname{dist}(\overline{\varphi} \circ \pi_R, \overline{\psi} \circ \pi_R) \} \leq \max\{ \delta, \varepsilon \} \leq \varepsilon.$$
Suppose that $\Gamma$ is pointwise $\mathcal{G}$-stable, and let $(\varphi_n : R \to G_n \in \mathcal{G})_{n \geq 1}$ be a pointwise asymptotic homomorphism. Then $(\varphi_n \circ \pi_R : \Gamma \to G_n)_{n \geq 1}$ is also a pointwise asymptotic homomorphism: indeed $\operatorname{def}_{g, h}(\varphi_n \circ \pi_R) = \operatorname{def}_{\pi_R(g), \pi_R(h)}(\varphi_n)$ for all $(g, h) \in \Gamma^2$. By pointwise stability of $\Gamma$ there exists a sequence of homomorphisms $(\tilde{\psi}_n : \Gamma \to G_n)_{n \geq 1}$ that is pointwise asymptotically close to $(\varphi_n \circ \pi_R)_{n \geq 1}$. Since $G$ is residually-$MQ(\mathcal{G})$, we have that $\tilde{\psi}_n$ factors through $R$, and so there exists a homomorphism $\psi_n : R \to G$ such that $\tilde{\psi}_n = \psi_n \circ \pi_R$. Moreover, $\varphi_n$ and $\psi_n$ are pointwise asymptotically close: indeed $\operatorname{dist}_{\pi_R(g)}(\varphi_n, \psi_n) = \operatorname{dist}_g(\varphi_n \circ \pi_R, \tilde{\psi}_n)$.
Similarly, if $\Gamma$ is uniformly $\mathcal{G}$-stable, then so is $R$: the proof is the same.
\end{proof}
\begin{example}
\label{ex:fq_free}
Let $\Gamma$ be a group without non-trivial quotients in $MQ(\mathcal{G})$. Then $\Gamma$ is uniformly $\mathcal{G}$-stable. More precisely, if $\varphi : \Gamma \to G \in \mathcal{G}$ and $\operatorname{def}(\varphi) \leq \varepsilon_k$, then $\operatorname{dist}(\varphi, \mathbbm{1}) \leq \varepsilon_k$. For instance, if $\Gamma$ is a simple group, then it can only be unstable if it belongs to $MQ(\mathcal{G})$.
If $\mathcal{G}$ is profinite, then $MQ(\mathcal{G})$ consists of finite groups, and so any infinite group without non-trivial finite quotients is uniformly $\mathcal{G}$-stable. Examples include Pride's group, which was already discussed in Example \ref{ex:small_canc}, as well as certain finitely presented groups such as Higman's group \cite{Higman} or Burger--Mozes groups \cite{BM}. Recently, more examples have been found in \cite{isomh3} among discrete subgroups of $\mathrm{Isom}(\mathbb{H}^3)$.
If $MQ(\mathcal{G})$ consists of finite $\pi$-groups, for $\pi$ a set of primes, then this statement is a weaker version of Proposition \ref{prop:pifree}.
\end{example}
Let us specialize to the case in which $\mathcal{G}$ is profinite, and so $MQ(\mathcal{G})$ consists of finite groups. The previous example implies that an infinite group without non-trivial finite quotients, such as Pride's example (Example \ref{ex:small_canc}), is uniformly $\mathcal{G}$-stable. However we can exploit Theorem \ref{thm:rf} further than just the case in which $R = \{ 1 \}$.
\begin{example}
\label{ex:sym0_ust}
The group $G^+$ from Example \ref{ex:sym0} has $\mathbb{Z}$ as largest residually finite quotient, which is uniformly $\mathcal{G}$-stable by Example \ref{ex:free}. Therefore $G^+$ is uniformly $\mathcal{G}$-stable. Similarly the largest residually finite quotient of $G$ is $\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$, which as we will see in Corollaries \ref{cor:vfree_p} and \ref{cor:vfree} is uniformly $\operatorname{GL}(\mathfrak{o})$-stable, whenever $\mathbb{K}$ has characteristic other than $2$. Thus $G$ is uniformly $\operatorname{GL}(\mathfrak{o})$-stable.
\end{example}
\begin{example}
\label{ex:wr_ust}
Let $G, H$ be finitely generated, with $H$ infinite and residually finite. Then by Example \ref{ex:wr} the largest residually finite quotient of $G \wr H$ is $\operatorname{Ab}(G) \wr H$. In particular, if $G$ is perfect (that is, $\operatorname{Ab}(G) = \{ 1 \}$) and $H$ is uniformly $\mathcal{G}$-stable, then $G \wr H$ is uniformly $\mathcal{G}$-stable. For example the lamplighter group $G \wr \mathbb{Z}$ is uniformly $\mathcal{G}$-stable for every non-abelian finite simple group $G$, by Example \ref{ex:free}. This will be strengthened for virtually pro-$\pi$ families in Corollary \ref{cor:wr}.
\end{example}
We will see in Example \ref{ex:cex} that the groups from these examples are not pointwise $\operatorname{GL}(\mathfrak{o})$-stable, where $\mathfrak{o}$ is the ring of integers of a non-Archimedean local field. This will prove that pointwise $\mathcal{G}$-stability of $R$ does not imply pointwise $\operatorname{GL}(\mathfrak{o})$-stability of $\Gamma$, and also that a finitely generated uniformly $\mathcal{G}$-stable group need not be pointwise $\mathcal{G}$-stable. \\
Note that in the proof of Theorem \ref{thm:rf}, as well as that of Lemma \ref{lem:factor_R}, we only used that any homomorphism from $\Gamma$ to a residually-$MQ(\mathcal{G})$ group factors through $R$. The same holds for any intermediate quotient, and so we obtain the following generalization of Theorem \ref{thm:rf}:
\begin{corollary}
\label{cor:rf}
Let $K \leq \Gamma$ be a group that is contained in the kernel of the quotient $\Gamma \to R$. Then $\Gamma$ is uniformly stable if and only if $\Gamma/K$ is. If $\Gamma$ is pointwise $\mathcal{G}$-stable, then so is $\Gamma/K$.
\end{corollary}
Another interesting consequence of Theorem \ref{thm:rf} is that the equivalence of Theorem \ref{thm:pw_un} can also be extended to some infinitely presented residually-$MQ(\mathcal{G})$ groups.
\begin{corollary}
\label{cor:fgrf}
Let $\mathcal{G}$ be an ultrametric family, $\mathcal{C}$ a class of groups closed under taking subgroups and containing $MQ(\mathcal{G})$. Let $\Gamma$ be a (finitely generated, residually-$\mathcal{C}$) group, that can be expressed as the largest residually-$\mathcal{C}$ quotient of some finitely presented group. Then $\Gamma$ is pointwise stable if and only if it is uniformly stable.
\end{corollary}
\begin{proof}
By Theorem \ref{thm:pw_un} we only need to show that uniform stability implies pointwise stability. Let $\hat{\Gamma}$ be the finitely presented group from the statement. If $\Gamma$ is uniformly stable, then $\hat{\Gamma}$ is uniformly stable by Theorem \ref{thm:rf}; being finitely presented it is also pointwise stable by Theorem \ref{thm:pw_un}, and so $\Gamma$ is pointwise stable again by Theorem \ref{thm:rf}.
\end{proof}
In case $\mathcal{G}$ is profinite and $\mathcal{C}$ is the class of all finite groups, is tempting to conjecture that all finitely generated residually finite groups have this property, and so the equivalence of Theorem \ref{thm:pw_un} applies to all of them. This is not the case. Indeed, let $\Gamma$ be a finitely generated residually finite group. Suppose that we know:
\begin{enumerate}
\item Whenever $C$ is finitely presented and $C \to \Gamma$ is a surjective homomorphism, $C$ is \textit{large}, that is, it virtually surjects onto $F_2$.
\item There exist finite groups onto which $\Gamma$ does \textit{not} virtually surject.
\end{enumerate}
Then $\Gamma$ cannot be the largest residually finite quotient of a finitely presented group. The paper \cite{covers} contains many examples of such groups.
\begin{example}
Any finitely generated group that surjects onto $\mathbb{Z}$ with locally finite kernel has property $1.$ \cite[Post-Scriptum]{covers}. Thus the Lamplighter group $\mathbb{Z}/2\mathbb{Z} \wr \mathbb{Z}$ (which has property $2.$, since it is metabelian) cannot be the largest residually finite quotient of a finitely presented group.
\end{example}
\begin{example}
Here is a torsion-free example. The Basilica group, introduced in \cite{basilica}, is finitely generated, residually finite, torsion-free, and every proper quotient is solvable, so it has property $2.$ It also has property $1.$ \cite[Section 2]{covers}, so it cannot be the largest residually finite quotient of a finitely presented group.
\end{example}
Compare this with \cite[Corollary 6.9]{limit}: every finitely generated residually free group is the largest residually free quotient of a finitely presented group.
\subsection{Solution to some stability problems}
\label{ss:sol}
We present here the complete solution to three uniform stability problems, with respect to families introduced in Section \ref{s:fam}. The first two are with respect to the families $T(R)$ (Example \ref{ex:UT}) and $\operatorname{Aut}(X^*_\bullet)$ (Example \ref{ex:aut:filt}) and admit a short and direct proof.
\begin{proposition}
\label{prop:aut_stab}
Let $\mathcal{G}$ be an ultrametric family with the following property: for every $G \in \mathcal{G}$ and every $\varepsilon > 0$, the extension $1 \to G(\varepsilon) \to G \to G/G(\varepsilon) \to 1$ splits. Then all groups are uniformly $\mathcal{G}$-stable. In particular, all groups are uniformly $T(R)$-stable and $\operatorname{Aut}(X^*_\bullet)$-stable.
\end{proposition}
\begin{proof}
We use the characterization from Lemma \ref{lem:quant}. By Lemma \ref{lem:mq} it suffices to show that any homomorphism $\varphi : \Gamma \to G/G(\varepsilon)$ lifts to a homomorphism $\psi : \Gamma \to G$. Composing $\varphi$ with a section $G/G(\varepsilon) \to G$, we conclude.
Both $T(R)$ and $\operatorname{Aut}(X^*_\bullet)$ satisfy the hypothesis. The metric quotients of $T_n(R)$ are isomorphic to $T_{n-k}(R)$ for some $1 \leq k \leq n$, and the section is given by the inclusion in the upper-left corner. The metric quotients of $\operatorname{Aut}(X^*_n)$ are isomorphic to the group of automorphisms of words of a given finite length, and the section is given by letting these elements act on the prefix of the appropriate length, and trivially on the other letters.
\end{proof}
\begin{remark}
Note that the proof gives more that uniform stability: it implies moreover that the stability estimate is optimal. That is, if $\operatorname{def}(\varphi) \leq \varepsilon$, then there exists a homomorphism $\psi$ with $\operatorname{dist}(\varphi, \psi) \leq \varepsilon$.
\end{remark}
In Example \ref{ex:cex} we show that there exist (finitely generated) groups that are not pointwise $\operatorname{Aut}(X^*_\bullet)$-stable. \\
Our next goal is to give the complete solution for uniform stability of finitely generated groups with respect to the family $\operatorname{Gal}(K)$ (Example \ref{ex:gal:prof}). We start by proving a stability result for finite families $\mathcal{G}$. When studying uniform stability, even looking at families of a single group is interesting. For instance, when $\mathcal{G} = \{ (\operatorname{U}(1), \| \cdot \|_{op}) \}$, then non-abelian free groups are not uniformly $\mathcal{G}$-stable \cite{Rolli}. However, in the ultrametric setting, such a situation cannot occur:
\begin{proposition}
\label{prop:unfin}
Let $\mathcal{G}$ be a finite profinite family. Then any finitely generated group is uniformly $\mathcal{G}$-stable.
\end{proposition}
This is a uniform version of \cite[Proposition 6]{a:quant}, in the ultrametric setting. For the proof we need the following equivalent characterization of uniform stability in terms of ultralimits, which is a uniform version of \cite[Theorem 4.2]{a:comm} (see Lemma \ref{lem:stab_up}):
\begin{lemma}
\label{lem:stab_uf}
Let $\mathcal{G}$ be any family of groups equipped with bi-invariant ultrametrics and let $\Gamma = \langle S \mid R \rangle$ be a countable group. The following are equivalent:
\begin{enumerate}
\item $\Gamma$ is uniformly $\mathcal{G}$-stable.
\item For every free ultrafilter $\omega \subset \mathcal{P}(\mathbb{N})$ the following holds: for any sequence $(\hat{\varphi}_n : F_S \to G_n \in \mathcal{G})_{n \geq 1}$ such that $\operatorname{def}(\hat{\varphi}_n) \xrightarrow{n \to \omega} 0$, there exists a sequence $(\hat{\psi}_n : F_S \to G_n)_{n \geq 1}$ of homomorphisms that descend to $\Gamma$ such that $\operatorname{dist}(\hat{\varphi}_n, \hat{\psi}_n) \xrightarrow{n \to \omega} 0$.
\end{enumerate}
\end{lemma}
\begin{proof}
We use the characterization of uniform $\mathcal{G}$-stability from Corollary \ref{cor:quant}. \\
$1. \Rightarrow 2.$ Fix an ultrafilter $\omega$ and let $(\hat{\varphi}_n : F_S \to G_n \in \mathcal{G})_{n \geq 1}$ be such that $\operatorname{def}(\hat{\varphi}_n) \xrightarrow{n \to \omega} 0$. For all $\hat{\varphi}_n$, let $\hat{\psi}_n : F_S \to G_n$ be a homomorphism that descends to $\Gamma$ and minimizes $\operatorname{dist}(\varphi_n, \psi_n)$ up to $1/n$. We need to show that $\operatorname{dist}(\hat{\varphi}_n, \hat{\psi}_n) \xrightarrow{n \to \omega} 0$, so let $\varepsilon > 0$ and let $\delta > 0$ be as in Corollary \ref{cor:quant} for $\varepsilon/2 > 0$: this means that if $\operatorname{def}(\hat{\varphi}_n) \leq \delta$ then $\operatorname{dist}(\hat{\varphi}_n, \hat{\psi}_n) \leq \varepsilon/2 + 1/n$. Let $N \geq 2/\varepsilon$. Then if $\operatorname{def}(\hat{\varphi}_n) \leq \delta$ and $n \geq N$ we have $\operatorname{dist}(\hat{\varphi}_n, \hat{\psi}_n) \leq \varepsilon$. Therefore $\{ n \geq 1 : \operatorname{dist}(\hat{\varphi}_n, \hat{\psi}_n) \leq \varepsilon \} \supset \{ n \geq N \} \cap \{ n \geq 1 : \operatorname{def}(\hat{\varphi}_n) \leq \delta \}$. The smaller set belongs to $\omega$ because $\operatorname{def}(\hat{\varphi}_n) \xrightarrow{n \to \omega} 0$ and $\omega$ is free. Thus the larger set is also in $\omega$, and we conclude. \\
$2. \Rightarrow 1.$ Suppose that $\Gamma$ is not uniformly $\mathcal{G}$-stable. By Corollary \ref{cor:quant} there exists $\varepsilon > 0$ and a sequence $(\hat{\varphi}_n : F_S \to G_n \in \mathcal{G})_{n \geq 1}$ such that $\operatorname{def}(\hat{\varphi}_n) \leq 1/n$ but for any sequence of $\hat{\psi}_n : F_S \to G_n$ descending to $\Gamma$ we have that $\operatorname{dist}(\hat{\varphi}_n, \hat{\psi}_n) \geq \varepsilon$. This implies that for every free ultrafilter $\omega$ we have $\operatorname{def}(\hat{\varphi}_n) \xrightarrow{n \to \omega} 0$ while $\lim\limits_{n \to \omega} \operatorname{dist}(\hat{\varphi}_n, \hat{\psi}_n) \geq \varepsilon > 0$ for any sequence $\hat{\psi}_n$ of homomorphisms that descend to $\Gamma$. So $3.$ does not hold.
\end{proof}
\begin{proof}[Proof of Proposition \ref{prop:unfin}]
Let $\Gamma = \langle S \mid R \rangle$ be a finitely generated group. Fix a free ultrafilter $\omega \subset \mathcal{P}(\mathbb{N})$ and let $(\hat{\varphi}_n : F_S \to G_n \in \mathcal{G})_{n \geq 1}$ be a sequence such that $\operatorname{def}(\hat{\varphi}_n) \xrightarrow{n \to \omega} 0$. Since $\mathcal{G}$ is finite, up to restricting to a subset in the ultrafilter we may assume that $G_n = G$ is a fixed group for all $n \geq 1$. Since $G$ is a compact metric space, for all $s \in S$ the sequence $\hat{\varphi}_n(s)$ admits an ultralimit, which we denote by $\hat{\psi}(s) \in G$. Let $\hat{\psi} : F_S \to G$ be the corresponding homomorphism. Then $\hat{\psi}$ descends to $\Gamma$: indeed, for all $r \in R$ we have
$$d_G(\hat{\psi}(r), 1_G) = \lim\limits_{n \to \omega} d_G(\hat{\varphi}_n(r), 1_G) = \lim\limits_{n \to \omega} \operatorname{def}_r(\hat{\varphi}_n) = 0.$$
Moreover, by definition $\operatorname{dist}_s(\hat{\varphi}_n, \hat{\psi}) \xrightarrow{n \to \omega} 0$, and so, since $S$ is finite, $\operatorname{dist}(\hat{\varphi}_n, \hat{\psi}) \xrightarrow{n \to \omega} 0$. We conclude by Lemma \ref{lem:stab_uf}.
\end{proof}
We are now ready to solve uniform stability of finitely generated groups with respect to the family $\operatorname{Gal}(K)$.
\begin{proposition}
\label{prop:gal_stab}
A group is uniformly $\operatorname{Gal}(K)$-stable if and only if it is uniformly $\{ \operatorname{Gal}(K^{sep}/K) \}$-stable. In particular, all finitely generated groups are uniformly $\operatorname{Gal}(K)$-stable.
\end{proposition}
\begin{proof}
We use the notation from Example \ref{ex:gal:prof}: the absolute Galois group of $K$ is denoted by $G$, and the metric is constructed via the sequences $(G_k)_{k \geq 1}$ and $\overline{\varepsilon}$. We denote by $G_k^L$ the image of $G_k$ in $\operatorname{Gal}(L/K)$. \\
We use the characterization of uniform stability from Lemma \ref{lem:quant}. Clearly if $\Gamma$ is uniformly $\operatorname{Gal}(K)$-stable, then it is uniformly $\{ G \}$-stable. Now suppose that $\Gamma$ is uniformly $\{ G \}$-stable; fix $\varepsilon > 0$ and let $\delta > 0$ be as in Lemma \ref{lem:quant}. Since the defect only takes values in $\{ \varepsilon_k : k \geq 0 \}$, we may assume that $\delta = \varepsilon_k$ for some $k$. Let $\varphi : \Gamma \to \operatorname{Gal}(L/K)$ be a map with $\operatorname{def}(\varphi) \leq \varepsilon_k$ and consider the induced homomorphism $\varphi(\varepsilon_k) : \Gamma \to \operatorname{Gal}(L/K) / G_k^L$. Now $\operatorname{Gal}(L/K) / G_k^L \cong G / G_k$, so we may lift this homomorphism to a map $\hat{\varphi} : \Gamma \to G$ such that $\operatorname{def}(\hat{\varphi}) \leq \varepsilon_k$, by Lemma \ref{lem:mq}. By the choice of $\delta = \varepsilon_k$, there exists a homomorphism $\hat{\psi} : \Gamma \to G$ such that $\operatorname{dist}(\hat{\varphi}, \hat{\psi}) \leq \varepsilon$. Then $\psi : \Gamma \xrightarrow{\hat{\psi}} G \to \operatorname{Gal}(L/K)$ is a homomorphism, and $\operatorname{dist}(\varphi, \psi) \leq \varepsilon$. \\
We conclude that $\Gamma$ is $\operatorname{Gal}(K)$-stable. Since $\{ G \}$ is a finite profinite family, the last statement follows directly from Proposition \ref{prop:unfin}.
\end{proof}
\pagebreak
\section{Ultrametric approximation and pointwise stability}
\label{s:approx}
This section constitutes an interlude, in that we leave the question of stability to focus on the related approximation problem. The main goal is to prove Theorem \ref{intro:thm:approx} for the family $\mathcal{G}$; combining this with Lemma \ref{lem:GR} will produce several counterexamples to pointwise stability. \\
Recall from Definition \ref{def:approx} that $\Gamma$ is \textit{$\mathcal{G}$-approximable} if there exists a \textit{$\mathcal{G}$-approximation}, namely an asymptotically injective pointwise asymptotic homomorphism $(\varphi_n : \Gamma \to G_n \in \mathcal{G})_{n \geq 1}$. The following similar notion appears quite naturally in this context:
\begin{definition}
We say that a $\mathcal{G}$-approximation is \textit{strong} if it is moreover a uniform asymptotic homomorphism. If $\Gamma$ admits a strong $\mathcal{G}$-approximation, it is said to be \textit{strongly $\mathcal{G}$-approximable}.
\end{definition}
The results from Section \ref{s:ultra} imply that this is not stronger for finitely presented groups:
\begin{lemma}
\label{lem:approx_fp}
Let $\mathcal{G}$ be an ultrametric family, $\Gamma$ a finitely presented group. If $\Gamma$ is $\mathcal{G}$-approximable, then it is strongly $\mathcal{G}$-approximable.
\end{lemma}
\begin{proof}
By Item $2.$ of Proposition \ref{prop:ultra_asy}, any pointwise asymptotic homomorphism of $\Gamma$ is pointwise asymptotically close to a uniform one. Applying this to a $\mathcal{G}$-approximation gives a uniform asymptotic homomorphism that is still asymptotically injective: a strong $\mathcal{G}$-approximation.
\end{proof}
\subsection{From approximations to local embeddings}
Well-studied approximation properties such as soficity or hyperlinearity are much weaker than residual finiteness, or local embeddability into finite groups. In this subsection we prove that $\mathcal{G}$-approximability, when $\mathcal{G}$ is a profinite family, is stronger. This is essentially a reinterpretation of the interplay between local embeddability and convergence in the space of marked groups \cite{LEF} (see Theorem \ref{thm:mg}).
\begin{proposition}
\label{prop:approx_1}
Let $\Gamma = \langle S \mid R \rangle$ be a countable group. If $\Gamma$ is $\mathcal{G}$-approximable, then $\Gamma$ is locally embeddable into $MQ(\mathcal{G})$. If $\Gamma$ is strongly $\mathcal{G}$-approximable, then $\Gamma$ is fully residually-$MQ(\mathcal{G})$. In particular, if $\Gamma$ is $\mathcal{G}$-approximable and finitely presented, then $\Gamma$ is fully residually $MQ(\mathcal{G})$.
\end{proposition}
\begin{remark}
The last statement follows form the general fact that finitely presented groups that are locally embeddable into $MQ(\mathcal{G})$ are also fully residually-$MQ(\mathcal{G})$ \cite{LEF} (see Proposition \ref{prop:lec_rc}). However it also follows by combining the rest of the proposition with Lemma \ref{lem:approx_fp}.
\end{remark}
\begin{proof}
Let $(\varphi_n : \Gamma \to G_n)_{n \geq 1}$ be a pointwise asymptotic homomorphism, which we lift to $(\hat{\varphi}_n : F_S \to G_n)_{n \geq 1}$ using Lemma \ref{lem:ultra_lift}. Fix an enumeration of $N = \langle \langle R \rangle \rangle$, denote by $N(k)$ the first $k$ elements, and fix a strictly decreasing sequence $\varepsilon_k \to 0$. Up to subsequence, we may assume that $\operatorname{def}_r(\hat{\varphi}_n) \leq \varepsilon_n$ for all $r \in N(n)$, and we look at the induced homomorphism $f_n : F_S \to G/G(\varepsilon_n)$. We get a sequence of $F_S$-marked groups with kernel $N_n = \{ w \in F_S : d(\hat{\varphi}_n(w), 1) \leq \varepsilon_n \}$. Up to subsequence, this converges in $\mathcal{N}(F_S)$ to:
$$\{ w \in F_S : d(\hat{\varphi}_n(w), 1) \leq \varepsilon_n \text{ infinitely often} \} = \{ w \in \Gamma : d(\hat{\varphi}_n(w), 1) \leq \varepsilon_n \text{ almost always} \}.$$
Now $N$ is contained in the left-hand side by choice of the subsequence. If moreover $\varphi_n$ is asymptotically injective, then $N$ contains the right-hand side, and so $N_n \to N \in \mathcal{N}(F_S)$, which implies that $\Gamma$ is locally embeddable into $MQ(\mathcal{G})$ by Item $2.$ of Theorem \ref{thm:mg}. \\
We can do the same for uniform asymptotic homomorphisms, by working over $\Gamma$-marked groups and taking $\varepsilon_n = \operatorname{def}(\varphi_n)$. The corresponding sequence $N_n \in \mathcal{N}(\Gamma)$ converges to $\{ 1 \}$, and we apply Item $1.$ of Theorem \ref{thm:mg}.
\end{proof}
\begin{example}
A group is called \textit{weakly hyperlinear} if it is approximable with respect to some family of compact metric groups \cite{weakly2, a:quant}. Similarly, a group is called \textit{weakly sofic} if it is approximable with respect to some family of finite metric groups \cite{weakly}. Proposition \ref{prop:approx_1} shows that, if we add the hypothesis that the approximating families are ultrametric, then all such groups are LEF.
\end{example}
According to the properties of the class $MQ(\mathcal{G})$, the conclusion of Proposition \ref{prop:approx_1} can be strengthened.
We look at two examples: the family $T(R)$ (Example \ref{ex:UT}) where $R$ is a finite ring, and the family $\operatorname{Gal}(\mathbb{F})$ (Example \ref{ex:gal:prof}), where $\mathbb{F}$ is a finite field.
\begin{corollary}
\label{cor:approxTR}
Let $R$ be a commutative ring, and $\Gamma$ a countable $T(R)$-approximable group. Then there exists a normal subgroup $\Gamma_0 \leq \Gamma$ such that $\Gamma/\Gamma_0$ embeds into $(R^\times)^{\mathbb{N}}$ and $\Gamma_0$ is locally embeddable into $UT(R)$, in particular it is locally embeddable into the class of nilpotent groups.
If moreover $R$ is finite and $\Gamma$ is finitely generated, then $\Gamma$ and $\Gamma_0$ have non-trivial abelian quotients, and $\Gamma/\Gamma_0$ embeds into $(R^\times)^n$ for some $n \geq 1$.
\end{corollary}
\begin{proof}
We will use the equivalent definition of local embeddability in terms of ultraproducts (Proposition \ref{prop:lef_up}) repeatedly throughout the proof.
By Proposition \ref{prop:approx_1}, and since all metric quotients of $T_n(R)$ are of the form $T_k(R)$ (Example \ref{ex:UT}), we know that $\Gamma$ is locally embeddable into $T(R)$. Then $\Gamma$ embeds into an ultraproduct $\prod\limits_{n \to \omega} T_n(R)$. This gives a homomorphism
$$\Gamma \to \prod\limits_{n \to \omega} T_n(R) \to \operatorname{Ab}\left( \prod\limits_{n \to \omega} T_n(R) \right) \cong \prod\limits_{n \to \omega} (R^\times)^n,$$
let $\Gamma_0$ be its kernel. Then $\Gamma_0$ embeds into $\prod\limits_{n \to \omega} UT_n(R)$, and so it is locally embeddable into $UT(R)$.
Now $\Gamma/\Gamma_0$ embeds into $\prod\limits_{n \to \omega} (R^\times)^n$, so it is locally embeddable into $\mathcal{C} := \{ (R^\times)^n : n \geq 1 \}$. Since $\Gamma/\Gamma_0$ is abelian, every finitely generated subgroup is finitely presented and locally embeddable into $\mathcal{C}$, so residually-$\mathcal{C}$ by Item $2.$ of Proposition \ref{prop:lec_rc}, and so it embeds into $(R^\times)^{\mathbb{N}}$. Since $\Gamma$ is countable, $\Gamma/\Gamma_0$ embeds into $(R^\times)^{\mathbb{N}}$, too. If now $R^\times$ is finite, then $\mathbb{Z}$ cannot be residually-$\mathcal{C}$, since there is a bound on the order of cyclic subgroups of $(R^\times)^n$. So $\Gamma/\Gamma_0$ is a torsion group; if moreover $\operatorname{Ab}(\Gamma)$ is finitely generated, then $\Gamma/\Gamma_0$ is finite, and being residually-$\mathcal{C}$ it embeds into $(R^\times)^n$ for some $n \geq 1$. The statement about $\Gamma$ and $\Gamma_0$ having non-trivial abelian quotients is a consequence of \cite{Sol}, where it is proven that this holds for all finitely generated groups that are approximable in the class of finite solvable groups.
\end{proof}
For the family $\operatorname{Gal}(\mathbb{F})$, we can give a full characterization.
\begin{corollary}
\label{cor:galfin_approx_1}
Let $\mathbb{F}$ be a finite field, and $\Gamma$ a group. Then $\Gamma$ is $\operatorname{Gal}(\mathbb{F})$-approximable if and only if it is abelian and all of its finite subgroups are cyclic. In particular, if $\Gamma$ is finitely generated, then it is $\operatorname{Gal}(\mathbb{F})$-approximable if and only if it is of the form $\mathbb{Z}^r \times C$ for some finite cyclic group $C$.
\end{corollary}
\begin{proof}
The absolute Galois group $\operatorname{Gal}(\mathbb{F}^{sep}/\mathbb{F})$ is isomorphic to the profinite completion $\hat{\mathbb{Z}}$ of $\mathbb{Z}$, which in turn is isomorphic to the direct product of $\mathbb{Z}_p$, where $p$ goes through all primes. It follows that any Galois extension of $\mathbb{F}$ has pro-cyclic Galois group, so by Proposition \ref{prop:approx_1}, if $\Gamma$ is $\operatorname{Gal}(\mathbb{F})$-approximable, then it locally embeddable in the class of finite cyclic groups. This implies two things: first, all finite subgroups of $\Gamma$ are cyclic, since a finite group that is locally embeddable in a class automatically blongs to that class. Secondly, $\Gamma$ is abelian; more generally, a group that is locally embeddable in the class of abelian groups is abelian: this follows directly from the ultraproduct characterization (Proposition \ref{prop:lef_up}). \\
We next show that all such groups are approximable. Since approximability is a local property it suffices to show that $\Gamma = \mathbb{Z}^r \times C$ is approximable. We construct the approximations inside the absolute Galois group of $\mathbb{F}$, which we identify with $\hat{\mathbb{Z}} \cong \prod \mathbb{Z}_p$. This is metrized using a nested sequence $G_k$ of finite-index open normal subgroups and a sequence of positive reals $\varepsilon_k \to 0$. Let us make another reduction: if we are able to approximate $C = \mathbb{Z}/p^n\mathbb{Z}$ via maps that take values in $\mathbb{Z}_p \leq \hat{\mathbb{Z}}$, then we are done. Indeed, we can write any finite cyclic group as a direct product of such groups, for a finite set $\{p_1, \ldots, p_i\}$ of distinct primes, and take the direct product of these approximations into $\prod_i \mathbb{Z}_{p_i} \leq \hat{\mathbb{Z}}$. Finally, we can embed $\mathbb{Z}^r$ into a direct product of $\mathbb{Z}_p$ for $r$ distinct primes that we did not use yet.
So we are left to show that we can approximate $C = \mathbb{Z}/p^n\mathbb{Z}$ for some $n \geq 1$ with an approximation taking values in $\mathbb{Z}_p$. Given $k \geq 1$ let $m \geq 1$ be such that $\mathbb{Z}_p \cap G_k = p^m \mathbb{Z}_p$; since the $G_k$ get smaller, $m \to \infty$ as $k \to \infty$. Let $k_n$ be the smallest integer such that the corresponding $m$ is larger than $n$, and let $k \geq k_n$. Then we can embed $C$ into $\mathbb{Z}/p^m \mathbb{Z}$, compose with a section into $\mathbb{Z}_p$, and finally include the latter in $\hat{\mathbb{Z}}$. This gives a map $\varphi : C \to \hat{\mathbb{Z}}$ that projects to an injective homomorphism into $\hat{\mathbb{Z}}/G_k$. It follows that $\operatorname{def}(\varphi) \leq \varepsilon_k$, and moreover $d(\varphi(x), 1) \geq \varepsilon_{k_n}$ for all $x \neq 1$.
\end{proof}
Note that the proof actually shows that these groups are $\{ \operatorname{Gal}(\mathbb{F}^{sep}/\mathbb{F}) \}$-approximable.
\subsection{From local embeddings to approximations}
The first easy examples of sofic and hyperlinear groups are residually finite, and more generally LEF groups. In this subsection we prove that these are also approximable with respect to certain ultrametric families. The precise statement will involve the following concept:
\begin{definition}
\label{def:capprox}
Let $\mathcal{C}$ be a class of groups. We say that $\mathcal{C}$ is \textit{$\mathcal{G}$-approximable} if there exists $\varepsilon > 0$ such that for all $C \in \mathcal{C}$ and for all $\delta > 0$ there exists a map $\eta : C \to G \in \mathcal{G}$ such that $\operatorname{def}(\eta) < \delta$ and $d_G(\eta(x), 1_G) > \varepsilon$ for all $1 \neq x \in C$.
\end{definition}
So a class $\mathcal{C}$ is $\mathcal{G}$-approximable if every group in $C$ is strongly $\mathcal{G}$-approximable, and moreover the injectivity gap can be taken uniformly for the entire class. For several profinite families $\mathcal{G}$, the class of all finite groups is $\mathcal{G}$-approximable:
\begin{example}
\label{ex:approx_gl}
The class of all finite groups is $\operatorname{GL}(\mathfrak{o})$-approximable, where $\mathfrak{o}$ is the ring of integers of a non-Archimedean local field. Indeed any finite group $C$ can be embedded into $\operatorname{GL}_n(\mathfrak{o})$, for $n$ large enough, using permutation matrices. If $\eta$ denotes this embedding, then $\operatorname{def}(\eta) = 0$ and $d(\eta(x), I_n) = 1$ for all $1 \neq x \in C$.
\end{example}
\begin{example}
\label{ex:approx_aut}
The class of all finite groups is $\operatorname{Aut}(X_\bullet^*)$-approximable. Indeed, any finite group $C$ can be embedded into $S_n$, for $n$ large enough, which in turn can be embedded into $\operatorname{Aut}(X_n^*)$ by acting on the first letter of a word. If $\eta$ denotes this embedding, then $\operatorname{def}(\eta) = 0$ and $d(\eta(x), id_{X_n^*}) = 1$ for all $1 \neq x \in C$.
\end{example}
For other families, this definition is quite restrictive:
\begin{example}
\label{ex:galfin_approx_2}
Let $K$ be a field and consider the family $\operatorname{Gal}(K) = \operatorname{Gal}(K)((G_k)_{k \geq 1}, \overline{\varepsilon})$ of Galois groups of Galois extensions of $K$: see Example \ref{ex:gal:prof} for the notation. We claim that an infinite class of finite groups cannot be $\operatorname{Gal}(K)$-approximable.
Suppose that $\mathcal{C}$ is a $\operatorname{Gal}(K)$-approximable family of finite groups, let $\varepsilon > 0$ be the uniform injectivity gap as in the definition and let $k \geq 1$ be such that $\varepsilon \geq \varepsilon_k$. Then for all $C \in \mathcal{C}$ there exists $G/N \in \operatorname{Gal}(K)$ and a map $\eta : C \to G/N$ such that $\operatorname{def}(\eta) \leq \varepsilon_k$ and $d(\eta(x), 1) > \varepsilon \geq \varepsilon_k$ for all $1 \neq x \in C$. The first inequality implies that $\eta$ induces a homomorphism $C \to (G/N) / (G_k N / N) \cong G / G_k N$, and the second one implies that this homomorphism is injective. Therefore $|C| \leq [G : G_k N] \leq [G : G_k]$. Since the inequality holds for any $C$, this implies that $\mathcal{C}$ is finite.
\end{example}
By Proposition \ref{prop:approx_1} a restriction of this kind is necessary. For instance if a finite group does not belong to $MQ(\mathcal{G})$, then it cannot be $\mathcal{G}$-approximable. But when the local embeddings take place in a $\mathcal{G}$-approximable class, then we can prove a converse to Proposition \ref{prop:approx_1}:
\begin{proposition}
\label{prop:approx_2}
Let $\mathcal{C}$ be a $\mathcal{G}$-approximable class of groups and $\Gamma$ a countable group. If $\Gamma$ is locally embeddable into $\mathcal{C}$, then $\Gamma$ is $\mathcal{G}$-approximable. If $\Gamma$ is fully residually-$\mathcal{C}$, then $\Gamma$ is strongly $\mathcal{G}$-approximable.
\end{proposition}
\begin{proof}
Since $\Gamma$ is countable, we can write it as an increasing union of finite sets $(K_n)_{n \geq 1}$. If $\Gamma$ is locally embeddable into $\mathcal{C}$, then for all $n$ we can choose a $K_n$-local embedding $f_n : \Gamma \to C_n \in \mathcal{C}$. Since $\mathcal{C}$ is $\mathcal{G}$-approximable, there exists $\varepsilon > 0$ such that: for all $n$ there exists a map $\eta_n : C_n \to G_n \in \mathcal{G}$ with $\operatorname{def}(\eta_n) \leq 1/n$ and $d_n(\eta_n(x), 1_{G_n}) > \varepsilon$ for all $1 \neq x \in C_n$. Then $\varphi_n : \Gamma \xrightarrow{f_n} C_n \xrightarrow{\eta_n} G_n$ satisfies: $\operatorname{def}_{g, h}(\varphi_n) \leq 1/n$ for all $(g, h)^2 \in K_n^2$, and $d_n(\varphi_n(g), 1_{G_n}) \geq \varepsilon$ for all $1 \neq g \in K_n$. It follows that $\varphi_n$ is a $\mathcal{G}$-approximation.
If $\Gamma$ is fully residually-$\mathcal{C}$, then the $K_n$-local embeddings $f_n : \Gamma \to C_n$ may be chosen to be homomorphisms that restrict to injective maps on $K_n$. Then the resulting $\mathcal{G}$-approximation is strong.
\end{proof}
The requirement that the class $\mathcal{C}$ be $\mathcal{G}$-approximable allows to deduce approximability from the existence of local embeddings of $\Gamma$ into $\mathcal{C}$ without knowing what they look like. But this is not the only way to produce approximations. For instance, by Example \ref{ex:galfin_approx_2}, when $\mathcal{G} = \operatorname{Gal}(K)$ we can only apply Proposition \ref{prop:approx_2} to groups that are locally embeddable into some finite class $\mathcal{C}$ of finite groups, which are necessarily finite. But in Corollary \ref{cor:galfin_approx_1} we saw that a $\operatorname{Gal}(\mathbb{F})$-approximable group can very well be infinite. \\
The proof of Proposition \ref{prop:approx_2} implies something (a priori) slightly stronger than approximation: namely, such groups are $\mathcal{G}$-approximable with a uniform injectivity gap:
$$\inf\limits_{g \in \Gamma} (\liminf\limits_{n \to \infty} d_n(\varphi_n(g), 1_{G_n})) > 0.$$
This is taken to be the definition of $\mathcal{G}$-approximation in some of the literature (see e.g. \cite{a:quant}). This ambiguity is due to the fact that in several approximation problems the two notions coincide: for instance, for sofic groups, this follows from a well-known \textit{amplification trick} due to Elek and Szab\'o \cite{ampl}. \\
Putting together Proposition \ref{prop:approx_1}, Examples \ref{ex:approx_gl} and \ref{ex:approx_aut}, and Proposition \ref{prop:approx_2}, we obtain that a group is LEF if and only if it is $\operatorname{GL}(\mathfrak{o})$-approximable, or $\operatorname{Aut}(X^*_\bullet)$-approximable. In other words, the notions of approximability with respect to these two families coincides with that of approximability with respect to the family of all finite groups equipped with the discrete metric (see Example \ref{ex:discr}). However the \textit{quantitative} versions of approximability are potentially distinct: for instance to embed a finite group into $\operatorname{GL}_n(\mathfrak{o})$ one does not always need the degree $n$ to be equal to the order. The quantitative study of approximation properties was initiated in \cite{a:quant}, and the case of local embeddings into finite groups was recently studied in more detail in \cite{Bradford}.
\subsection{Counterexamples to pointwise stability}
We now apply the results in this section to give counterexamples to pointwise stability. These all stem from the following corollary, which applies to both $\mathcal{G} = \operatorname{GL}(\mathfrak{o})$ and $\mathcal{G} = \operatorname{Aut}(X^*_\bullet)$.
\begin{corollary}
\label{cor:cex_stab}
Let $\mathcal{G}$ be a profinite family, such that the class finite groups is $\mathcal{G}$-approximable. Then if $\Gamma$ is LEF but not residually finite, it is not pointwise $\mathcal{G}$-stable.
\end{corollary}
\begin{proof}
This is just a combination of Lemma \ref{lem:GR} and Propositions \ref{prop:approx_1} and \ref{prop:approx_2}.
\end{proof}
It is worth noticing that there is no hypothesis of finite generation in this statement, in contrast to the analogous statement for families of unitary groups. This is because the groups $\operatorname{GL}_n(\mathfrak{o})$ are not only locally residually finite, as guaranteed for all linear groups by a theorem of Malcev \cite{Malcev}, but they are themselves residually finite, being profinite. \\
In the examples below, $\operatorname{GL}(\mathfrak{o})$ may be replaced with any class satisfying the hypotheses of the corollary, for instance $\operatorname{Aut}(X^*_\bullet)$, or the discrete family of all finite groups (Example \ref{ex:discr}).
\begin{example}
\label{ex:cex}
Let $\Gamma$ be a classical small cancellation group that is not residually finite, for instance Pride's group (Example \ref{ex:small_canc}). Then $\Gamma$ is not pointwise $\operatorname{GL}(\mathfrak{o})$-stable. So even the simplest example of a finitely generated uniformly stable group -- a group without non-trivial finite quotients (see Example \ref{ex:fq_free}) -- need not be pointwise $\operatorname{GL}(\mathfrak{o})$-stable.
\end{example}
\begin{example}
\label{ex:cex2}
Let $G = \operatorname{Sym}_0(\mathbb{Z}) \rtimes \mathbb{Z}$ and $G^+ = \operatorname{Alt}_0(\mathbb{Z}) \rtimes \mathbb{Z}$ be the groups from Example \ref{ex:sym0}. These groups are finitely generated, and their largest residually finite quotient is $\mathbb{Z} \times \mathbb{Z}/2 \mathbb{Z}$, respectively $\mathbb{Z}$, so they are not residually finite. But by Example \ref{ex:sym0_lef} these groups are LEF, therefore they are not pointwise $\operatorname{GL}(\mathfrak{o})$-stable by Corollary \ref{cor:cex_stab}. On the other hand, by Example \ref{ex:sym0_ust} the group $G^+$ is uniformly $\operatorname{GL}(\mathfrak{o})$-stable (and when $\mathbb{K}$ does not have characteristic $2$, so is $G$, by Example \ref{ex:pfree} and Proposition \ref{prop:vpfree}). The same result holds for lamplighter groups by Examples \ref{ex:wr}, \ref{ex:wr_lef} and \ref{ex:wr_ust}, for instance if $G$ is a non-abelian finite simple group, then $G \wr \mathbb{Z}$ is finitely generated, LEF, but not residually finite, so it is not pointwise $\operatorname{GL}(\mathfrak{o})$-stable, even though it is uniformly $\operatorname{GL}(\mathfrak{o})$-stable.
\end{example}
These examples show that two previous results are sharp. First, the last statement of Theorem \ref{thm:pw_un} does not hold for general finitely generated groups: there exists a finitely generated group that is uniformly but not pointwise $\operatorname{GL}(\mathfrak{o})$-stable. Secondly, the converse of the last statement of Theorem \ref{thm:rf} fails in general: there exists a (finitely generated) group that is not pointwise $\operatorname{GL}(\mathfrak{o})$-stable, and whose largest residually finite quotient is pointwise $\operatorname{GL}(\mathfrak{o})$-stable.
\subsection{A pointwise version of Theorem \ref{thm:rf}}
Let $\mathcal{C}$ be a class of groups closed under taking subgroups such that $MQ(\mathcal{G}) \subset \mathcal{C}$, and let $R$ be the largest residually-$\mathcal{C}$ quotient of $\Gamma$, that we assume to be countable as usual. We further assume that $\mathcal{C}$ is closed under taking directed products, from which it follows that residually-$\mathcal{C}$ groups are fully residually-$\mathcal{C}$, and so locally embeddable into $\mathcal{C}$ by Proposition \ref{prop:lec_rc}. Recall from Theorem \ref{thm:rf} that $\Gamma$ is uniformly stable if and only if $R$ is. The methods from this section allow to prove a pointwise version of this theorem, where again the residual property is replaced by local embeddability. To this end let $L$ be the largest quotient of $\Gamma$ that is locally embeddable into $\mathcal{C}$. As in Lemma \ref{lem:largest_rf}, this is the quotient $\Gamma / K$, where $K$ is the intersection of all kernels of morphisms of $\Gamma$ into a group that is locally embeddable into $\mathcal{C}$. By construction $L$ has the factoring property, and the fact that $\mathcal{C}$ is closed under taking direct products implies that $L$ is locally embeddable into $\mathcal{C}$, since this condition can be verified on finite subsets. \\
The following proposition is analogous to the fact that a group is pointwise stable in permutation if and only if its largest sofic quotient is.
\begin{proposition}
\label{prop:lef_q}
Let $\mathcal{G}, \Gamma, L$ be as above. Then $\Gamma$ is pointwise $\mathcal{G}$-stable if and only if $L$ is. If $\Gamma$ is uniformly $\mathcal{G}$-stable, then so is $L$.
\end{proposition}
\begin{proof}
If $\Gamma$ is (pointwise or uniformly) $\mathcal{G}$-stable, then so is $L$, by the same proof as Theorem \ref{thm:rf}.
Suppose that $L$ is pointwise $\mathcal{G}$-stable. By Lemma \ref{lem:stab_up}, this means that any homomorphism of $L$ to a metric ultraproduct of $G_n \in \mathcal{G}$ lifts to the direct product. To prove the same for $\Gamma$ it suffices to show that any homomorphism of $\Gamma$ to a metric ultraproduct of $G_n$ factors through $L$. Now the image of such a homomorphism is $\mathcal{G}$-approximable, so locally embeddable into $MQ(\mathcal{G})$ by Proposition \ref{prop:approx_1}, in particular locally embeddable into $\mathcal{C}$.
\end{proof}
\begin{example}
\label{ex:simplelef}
A finitely presented non-residually-$\mathcal{C}$ group is not locally embeddable into $\mathcal{C}$, by Proposition \ref{prop:lec_rc}. Take such a group, and embed it into a finitely generated simple group $\Gamma$ \cite{simple1, simple2}. Then $\Gamma$ contains a subgroup that is not locally embeddable into $\mathcal{C}$, so it has the same property. Being simple, the corresponding group $L$ from Proposition \ref{prop:lef_q} is trivial, and so $\Gamma$ is pointwise $\mathcal{G}$-stable. For instance, if the family $\mathcal{G}$ is profinite, we can start with any finitely presented non-residually finite group, and obtain a finitely generated non-LEF simple group that is pointwise $\mathcal{G}$-stable. Note that Hall's construction \cite{simple1} always outputs an infinitely presented group. So these are our first examples of finitely generated infinitely presented pointwise stable groups. We will give other explicit examples in Corollary \ref{cor:rfBS_p}.
\end{example}
\pagebreak
\section{Virtually pro-$\pi$ stability}
\label{s:vpropi}
We now specialize our study of stability to profinite families $\mathcal{G}$ whose metric quotients have restricted orders. This will allow us to provide various examples of uniformly stable groups with respect to such families, some of which are listed in Theorem \ref{intro:thm:pifree}. \\
The basic idea of this approach can be traced back to \cite{simGLp}, where the author uses the conjugacy part of the Hall Theorem on solvable groups to prove a kind of stability result for the conjugacy relation, under some coprimality assumption on the order of the elements. Here we will go much further, and this is made possible by the use of the more general Schur--Zassenhaus Theorem (see Theorem \ref{thm:SZ}).
\begin{definition}
Let $\mathcal{G}$ be a profinite family. Given a class $\mathcal{C}$ of finite groups, we say that $\mathcal{G}$ is \textit{virtually pro-$\mathcal{C}$} if there exists some $\varepsilon > 0$ such that $G(\varepsilon)$ is pro-$\mathcal{C}$ for all $G \in \mathcal{G}$. This section we focus on the class $\mathcal{C}$ of $\pi$-groups, where $\pi$ is a fixed set of primes: we say that $\mathcal{G}$ is \textit{virtually pro-$\pi$}.
\end{definition}
The condition that $G(\varepsilon)$ be pro-$\mathcal{C}$ is equivalent to all metric quotients of $G(\varepsilon)$ being in $\mathcal{C}$. Notice that we are asking that the $\varepsilon > 0$ be uniform for the whole family.
\begin{example}
Let $\mathfrak{o}$ be the ring of integers of a non-Archimedean local field whose residue field has characteristic $p$. Then $\operatorname{GL}(\mathfrak{o})$ is virtually pro-$p$: indeed by Lemma \ref{lem:vprop} the principal congruence subgroups $\operatorname{GL}_n(\mathfrak{o})_1$ are pro-$p$, and we can take $\varepsilon = p^{-1}$.
\end{example}
The key to this approach is the interpretation of stability in terms of the following lifting property. A map $\varphi : \Gamma \to G$ with $\operatorname{def}(\varphi) \leq \delta$ induces a homomorphism $\varphi(\delta) : \Gamma \to G/G(\delta)$. Then a homomorphism $\psi : \Gamma \to G$ satisfies $\operatorname{dist}(\varphi, \psi) \leq \varepsilon$, where $\delta \leq \varepsilon$, if and only if it is a lift of the induced homomorphism $\varphi(\varepsilon) : \Gamma \to G/G(\varepsilon)$. This is just a rephrasing of Lemma \ref{lem:mq}.
\subsection{$\pi$-free groups}
\label{ss:pifree}
In this subsection we use the lifting part of the Schur--Zassenhaus Theorem to prove stability with respect to virtually pro-$\pi$ families of groups whose finite quotients have restricted orders.
\begin{definition}
\label{def:pifree}
A group $\Gamma$ is \textit{$\pi$-free} if all of its finite quotients are $\pi'$-groups; that is, if all of its finite quotients have order divisible only by primes not in $\pi$. Equivalently, a group is $\pi$-free if it has no finite virtual $p$-quotients, for any $p \in \pi$.
\end{definition}
Clearly a group is $\pi$-free if and only if it is $p$-free for all $p \in \pi$. This terminology is inspired from the terminology in \cite{Schik, mio} introduced by Schikhof: indeed a group is $p$-free according to Definition \ref{def:pifree} if and only if its profinite completion is $p$-free according to Schikhof's definition. This class of groups clearly contains groups without finite quotients, whose uniform stability was already established in Example \ref{ex:fq_free}. But there are more examples, including residually finite ones.
\begin{example}
Finite $\pi'$-groups are $\pi$-free.
\end{example}
\begin{example}
\label{ex:period}
More generally, let $\Gamma$ be a periodic group without elements of order $p$ for all $p \in \pi$. Then $\Gamma$ is $\pi$-free. Indeed, the order of any element in a finite quotient of $\Gamma$ must divide the order of any preimage thereof. For example Grigorchuk's first group is a finitely generated periodic residually finite $2$-group, so it is $2'$-free.
\end{example}
\begin{example}
\label{ex:CSP}
Let $X$ be a finite alphabet, and let $\operatorname{Aut}(X^*)$ be the group of rooted tree automorphisms (see Example \ref{ex:aut:filt}). Following \cite{branch}, we say that a subgroup $\Gamma \leq \operatorname{Aut}(X^*)$ -- which is necessarily residually finite -- has the \textit{congruence subgroup property} if every finite-index subgroup of $\Gamma$ contains some level stabilizer $\operatorname{Aut}(X^*)_k$.
Let $\pi$ be the set of primes $p \leq |X|$. Then $\operatorname{Aut}(X^*)$ is pro-$\pi$, and so any subgroup $\Gamma \leq \operatorname{Aut}(X^*)$ with the congruence subgroup property is $\pi'$-free. This gives many examples of $\pi$-free groups, among which are many branch groups \cite{branch}, and in particular all Grigorchuk--Guptda--Sidki groups with non-constant defining vector \cite{GGS1, GGS2}. See \cite{GGS3, EGS} for more examples.
\end{example}
\begin{example}
Building on the previous example, there exist finitely generated residually finite torsion-free groups that are $p'$-free \cite{GGS2}.
\end{example}
We now prove stability of such groups. We first prove a quantitative lemma, and deduce the stability statement.
\begin{lemma}
\label{lem:pifree}
Let $\mathcal{G}$ be a virtually pro-$\pi$ family, and let $\varepsilon_0$ be such that $G(\varepsilon_0)$ is pro-$\pi$ for all $G \in \mathcal{G}$. Let $\varphi : \Gamma \to G \in \mathcal{G}$ be such that $\operatorname{def}(\varphi) \leq \varepsilon \leq \varepsilon_0$, and suppose that the image $C$ of $\Gamma$ in $G/G(\varepsilon)$ is a $\pi'$-group. Then there exists a homomorphism $\psi : \Gamma \to G$ such that $\operatorname{dist}(\varphi, \psi) \leq \varepsilon$. Moreover, $\psi(\Gamma) \leq G$ is a finite group isomorphic to $C$, in particular it is a $\pi'$-group.
\end{lemma}
\begin{proof}
Consider the induced homomorphism $\varphi(\varepsilon) : \Gamma \to G/G(\varepsilon)$ given by Lemma \ref{lem:mq}. By hypothesis $C = \varphi(\varepsilon)(\Gamma)$ is a $\pi'$-group. Given $\delta < \varepsilon$, we have the following lifting problem
\[\begin{tikzcd}
&& {G/G(\delta)} \\
\Gamma & C \\
&& {G/G(\varepsilon) \cong (G/G(\delta))/(G(\varepsilon)/G(\delta))}
\arrow["{\varphi(\varepsilon)}", from=2-1, to=2-2]
\arrow[dashed, from=2-2, to=1-3]
\arrow[from=2-2, to=3-3]
\arrow[from=1-3, to=3-3]
\end{tikzcd}\]
Since $G(\varepsilon)/G(\delta)$ is a finite $\pi$-group, by the Schur--Zassenhaus Theorem there exists a lift: a homomorphism $\varphi(\delta) : \Gamma \to G/G(\delta)$ such that the projection onto $G/G(\varepsilon)$ gives back $\varphi(\varepsilon)$. Moreover $\varphi(\delta)(\Gamma) \cong C$, since it is a quotient of $C$ and it surjects onto it. Repeating this process by induction on a sequence $\varepsilon \geq \delta_i \to 0$ gives a sequence of homomorphisms $\varphi(\delta_i) : \Gamma \to G/G(\delta_i)$ that are all compatible with the projections, and such that all images are groups isomorphic to $C$. Since $G$ is the projective limit of the groups $G/G(\delta_i)$, this induces a homomorphism $\psi : \Gamma \to G$ such that $\psi(\varepsilon) : \Gamma \to G/G(\varepsilon)$ coincides with $\varphi(\varepsilon)$ and $\psi(\Gamma)$ is a finite group isomorphic to $C$. Therefore $\operatorname{dist}(\varphi, \psi) \leq \varepsilon$ by Lemma \ref{lem:mq}.
\end{proof}
\begin{proposition}
\label{prop:pifree}
Let $\mathcal{G}$ be a virtually pro-$\pi$ family and $\Gamma$ a $\pi$-free group. Then $\Gamma$ is uniformly $\mathcal{G}$-stable.
\end{proposition}
\begin{proof}
Since $\Gamma$ is $\pi$-free, all finite quotients of it are $\pi'$-groups. Therefore given a map $\varphi : \Gamma \to G$ with small enough defect, the previous lemma applies and $\varphi$ is close to a homomorphism. We conclude by Lemma \ref{lem:quant}.
\end{proof}
Lemma \ref{lem:pifree} shows that the estimate for stability is the optimal one: there exists $\varepsilon_0 > 0$ such that if $\varphi : \Gamma \to G$ satisfies $\operatorname{def}(\varphi) \leq \varepsilon \leq \varepsilon_0$, then there exists a homomorphism $\psi : \Gamma \to G$ such that $\operatorname{dist}(\varphi, \psi) \leq \varepsilon$.
\begin{example}
\label{ex:pfree}
Let $\Gamma$ be a $p$-free group, $\mathfrak{o}$ the ring of integers of a non-Archimedean local field of residual characteristic $p$. Then $\Gamma$ is uniformly $\operatorname{GL}(\mathfrak{o})$-stable. Quantitatively, for any map $\varphi: \Gamma \to \operatorname{GL}_n(\mathfrak{o})$ such that $\operatorname{def}(\varphi) \leq \varepsilon \leq |\overline{\omega}|$ (where $\overline{\omega}$ is a uniformizer), there exists a homomorphism $\psi : \Gamma \to \operatorname{GL}_n(\mathfrak{o})$ such that $\operatorname{dist}(\varphi, \psi) \leq \varepsilon$.
\end{example}
Note that, even for finite groups, the hypothesis of $\pi$-freeness is necessary. Indeed, we saw in Example \ref{ex:2unst} that there exists a family of pro-$2$ groups with respect to which $\mathbb{Z}/2\mathbb{Z}$ is unstable. On the other hand, we will see that $\mathbb{Z}/2\mathbb{Z}$ is $\operatorname{GL}(\mathfrak{o})$-stable also when $\mathbb{K}$ has residual characteristic $2$ (Example \ref{ex:finst} and Proposition \ref{prop:z2z}). \\
The following example is a hint at the relation with bounded cohomology that will be explored in Section \ref{s:BC}.
\begin{example}
\label{ex:npa}
Let $\mathbb{K}$ have characteristic $p$, and let $\Gamma$ be a normed $\mathbb{K}$-amenable group \cite[Definition 1.1]{mio}. Then $\Gamma$ is $\operatorname{GL}(\mathfrak{o})$-stable: indeed such groups are characterized as being locally finite (thus periodic) and without elements of order $p$ \cite[Theorem 6.2]{mio}.
\end{example}
\subsection{Graphs of groups}
\label{ss:gog}
In this subsection we exploit the conjugacy part of the Schur--Zassenhaus Theorem to prove stability with respect to virtually pro-$\pi$ families of several fundamental groups of graphs of groups. This part of the Schur--Zassenhaus Theorem depends on the Odd Order Theorem, but this can be avoided if either the kernel or the quotient of the extension to which the theorem is being applied is solvable. As in the proof of Proposition \ref{prop:pifree}, the extensions to which we apply the Schur--Zassenhaus Theorem are with a $\pi$-kernel and a $\pi'$-quotient, so if $\pi = \{ p \}$ then the kernel is solvable. Similarly if $\pi = \{ p, q \}$ then the kernel is solvable by Burnside's Theorem. The same kind of statements can be given for the quotient. It may also be possible that we know for other reasons that $\mathcal{G}$ is virtually prosolvable: for instance this is the case for $\operatorname{Gal}(\mathbb{F})$ when $\mathbb{F}$ is a finite field (see Corollary \ref{cor:galfin_approx_1}). For the general case, however, we need the full power of the Schur--Zassenhaus Theorem, and so the general statements in this subsection depend on the Odd Order Theorem. \\
Let us fix the definitions and notation (see \cite{Serre} for more detail). Let $X = (V, E)$ be a connected graph with vertex set $V$ and edge set $E$, maps $\pm : E \to V : e^{\pm}$ giving the source and target of an edge, and a fix-point free involution $E \to E : e \mapsto \overline{e}$ reversing the orientation of each edge. A \textit{graph of groups} is composed by the following data: a graph $X = (V, E)$, groups $\Gamma_v$ for all $v \in V$ and $\Gamma_e$ for all $e \in E$ such that $\Gamma_e = \Gamma_{\overline{e}}$, and injective morphisms $\iota_e^{\pm} : \Gamma_e \to \Gamma_{e^\pm}$. By abuse of notation we use $X$ to denote both the graph of groups and the underlying abstract graph.
Let $T$ be a spanning tree of $X$. The \textit{fundamental group} of this graph of groups is the group generated by all vertex groups, together with an element $t_e$ for each $e \in E$, with the additional relations: $t_{\overline{e}} = t_e^{-1}, t_e \iota_e^-(x) t_e^{-1} = \iota_e^+(x)$ for all $x \in \Gamma_e$ (a generating set of $\Gamma_e$ suffices), and $t_e = 1$ if $e \in T$. The isomorphism type of the fundamental group is independent of the choice of $T$. A presentation of the fundamental group is thus given by
$$\langle \{ S_v : v \in V \} \cup \{ t_e : e \in E \} \mid \{ R_v : v \in V \} \cup \{ R_e : e \in E \} \rangle,$$
where $\langle S_v \mid R_v \rangle$ is a presentation of $\Gamma_v$, and $R_e$ are the relations describing the identification $t_{\overline{e}} = t_e^{-1}$, the effect of conjugacy by $t_e$, and the relation $t_e = 1$ if $e \in T$. \\
As in the previous subsection, we first prove a quantitative lemma, and then deduce two stability statements. Since the fundamental group is defined in terms of a presentation, the most natural approach is by working in terms of it, which is possible by Proposition \ref{prop:stab_equiv} and Corollary \ref{cor:quant}.
\begin{lemma}
\label{lem:pifree_gog}
Let $\mathcal{G}$ be a virtually pro-$\pi$ family, and let $\varepsilon_0$ be such that $G(\varepsilon_0)$ is pro-$\pi$ for all $G \in \mathcal{G}$. Let $X$ be a connected graph of groups with vertex groups $\Gamma_v$, edge groups $\Gamma_e$ and edge inclusions $\iota_e^{\pm} : \Gamma_e \to \Gamma_{e^{\pm}}$. Let $\Gamma$ be the fundamental group of $X$, with the standard presentation $\langle S \mid R \rangle$ as above.
Let $\hat{\varphi} : F_S \to G \in \mathcal{G}$ be a map with $\operatorname{def}(\hat{\varphi}) \leq \varepsilon \leq \varepsilon_0$. Suppose further that for all $v \in V$ the restriction of $\hat{\varphi}$ to $F_{S_v}$ descends to a homomorphism $\varphi_v : \Gamma_v \to G$ such that, if $e^{\pm} = v$, the image of $\varphi_v(\iota_e^{\pm}(\Gamma_e))$ in $G/G(\delta)$ is a $\pi'$-group, for all $\delta \leq \varepsilon$.
Then there exists a homomorphism $\hat{\psi} : F_S \to G$ such that $\operatorname{dist}(\hat{\varphi}, \hat{\psi}) \leq \varepsilon$ and $\hat{\psi}$ descends to a homomorphism of $\Gamma$.
\end{lemma}
Before proceeding with the proof, let us comment on how this lemma is of interest independently of our applications (namely Propositions \ref{prop:pifree_gog1} and \ref{prop:pifree_gog2} below and their corollaries in the next subsections). Indeed, it shows that such fundamentals groups of graphs of groups are examples of two notions related to stability, introduced recently in the literature and of which few examples are known so far. \\
First, Lemma \ref{lem:pifree_gog} is a statement about \textit{constraint stability}, a notion introduced by Arzhantseva and P\u{a}unescu in \cite{a:const}. Given a group $\Gamma$ and a subgroup $\Lambda \leq \Gamma$, let us say that $\Gamma$ is \textit{constraint stable} with respect to $\Lambda$, if for any asymptotic homomorphism $(\varphi_n : \Gamma \to G_n)_{n \geq 1}$ such that its restriction to $\Lambda$ is close to a sequence of homomorphisms $(\psi_n : \Lambda \to G_n)_{n \geq 1}$, we can extend $(\psi_n)_{n \geq 1}$ to a homomorphism of $\Gamma$ that is close to $(\varphi_n)_{n \geq 1}$. As usual, this can be formalized to a pointwise notion and a uniform one. Similarly we can talk of $\Gamma$ being stable with respect to a set of subgroups. Then Lemma \ref{lem:pifree_gog} is a statement about constrant stability of $\Gamma$ with respect to the set of vertex subgroups. \\
Secondly, Lemma \ref{lem:pifree_gog} is a statement about \textit{stability of an epimorphism}, a notion introduced by Lazarovich and Levit in \cite{stepi}. We say that an epimorphism $\overline{\Gamma} \to \Gamma$ is \textit{stable} if any asymptotic homomorphism of $\overline{\Gamma}$ that almost descends to $\Gamma$ (where ``almost descends'' is meant as in Proposition \ref{prop:stab_equiv}) is close to a sequence of homomorphisms of $\overline{\Gamma}$ that descend to $\Gamma$. Again, this leads to a pointwise and a uniform notion. Then Lemma \ref{lem:pifree_gog} is a statement about stability of the epimorphism of $((*_v \Gamma_v) * (*_e \langle t_e \rangle))$ onto $\Gamma$. Interestingly, this is precisely the setting in \cite{stepi}, where the authors prove stability of the same epimorphism in the case of virtually free groups, to deduce that all virtually free groups are stable in permutation. \\
We proceed with the proof.
\begin{proof}[Proof of Lemma \ref{lem:pifree_gog}]
By hypothesis $\hat{\varphi}$ already satisfies all relations $R_v$. Our goal is to modify $\hat{\varphi}$ step by step so that it keeps this property, changes by at most $\varepsilon$, and it also satisfies all relations $R_e$. For the rest of this proof, we denote the reduction map $G \to G/G(\delta)$ by $(\cdot \mod \delta)$. So if $A \leq G$, its image in $G/G(\delta)$ is denoted by $A \mod \delta$. \\
First of all, we can choose a set $E_+$ of positively oriented edges, and replace $\hat{\varphi}(t_e)$ by $\hat{\varphi}(t_{\overline{e}})^{-1}$ for all $e \notin E_+$. Since $\operatorname{def}(\hat{\varphi}) \leq \varepsilon$, these two elements are at a distance at most $\varepsilon$, and so this substitution does not affect the other relations being satisfied in $G/G(\varepsilon)$. Similarly we can replace $\hat{\varphi}(t_e)$ by $1$ for all $e \in T$. This leaves us with the conjugacy relations. \\
We start with the edges in $T$: we will modify $\hat{\varphi}$ at the vertex groups so that it still restricts to a homomorphism and moreover it satisfies the amalgamations. Fix a vertex $v_0$ in $X$, with neighbours $v_1, \ldots, v_r$ and edges $e_1, \ldots, e_r \in T$, where $v_0 = e_i^-$ and $v_i = e_i^+$ for $i = 1, \ldots, r$. Let $A_i^- := \varphi_{v_0}(\iota_{e_i}^-(\Gamma_{e_i})) \leq \varphi_{v_0}(\Gamma_{v_0})$ and $A_i^+ := \varphi_{v_i}(\iota_{e_i}^+(\Gamma_{e_i})) \leq \varphi_{v_i}(\Gamma_{v_i})$. By hypothesis, for all $\delta \leq \varepsilon$ the reduction modulo $\delta$ of both $A_i^-$ and $A_i^+$ is a $\pi'$-group. This implies in particular that for all $\delta \leq \varepsilon$ the projection map $A_i^{\pm} \mod \delta \to A_i^{\pm} \mod \varepsilon$ is an isomorphism: indeed the kernel is contained in $G(\varepsilon)/G(\delta)$ which is a finite $\pi$-group since $G(\varepsilon) \leq G(\varepsilon_0)$ is pro-$\pi$.
Now $\operatorname{def}(\hat{\varphi}) \leq \varepsilon$, and the copies of $\Gamma_{e_i}$ in $\Gamma_{v_0}$ and $\Gamma_{v_i}$ are amalgamated in any quotient of $\Gamma$. Thus by Lemma \ref{lem:mq} the reduction modulo $\varepsilon$ of the homomorphisms $\varphi_{v_0} \circ \iota_{e_i}^-, \varphi_{v_i} \circ \iota_{e_i}^+ : \Gamma_{e_i} \to A_i^{\pm}$ is the same, denoted $f(\varepsilon)$. Therefore the reductions modulo $\delta$ of the same maps are two (a priori distinct) solutions to the following lifting problem:
\[\begin{tikzcd}
&& {G/G(\delta)} \\
{\Gamma_{e_i}} & {f(\varepsilon)(\Gamma_{e_i})} \\
&& {G/G(\varepsilon) \cong (G/G(\delta))/(G(\varepsilon)/G(\delta))}
\arrow[dashed, from=2-2, to=1-3]
\arrow[from=2-2, to=3-3]
\arrow[from=1-3, to=3-3]
\arrow[from=2-1, to=2-2]
\end{tikzcd}\]
Since $G(\varepsilon)/G(\delta)$ is a finite $\pi$-group, the Schur--Zassenhaus Theorem implies that these two lifts are $G(\varepsilon)/G(\delta)$-conjugate. Let $t \in G(\varepsilon)$ be a lift of this conjugating element. We can then replace $\varphi_{v_i}$ by $x \mapsto t \varphi_{v_i}(x) t^{-1}$. This is still a homomorphism of $\Gamma_{v_i}$, and since $t \in G(\varepsilon)$ it remains $\varepsilon$-close to $\varphi_{v_i}$, but now the groups $A_i^{\pm}$ coincide modulo $\delta$. We can repeat this process inductively on a sequence $\varepsilon \geq \delta_k \to 0$: at each step we modify $\varphi_{v_i}$ by conjugating it by an element of $G(\delta_k)$, so that the groups $A_i^{\pm}$ are amalgamated modulo $\delta_{k+1}$. This sequence of conjugating elements converges to an element $t \in G(\varepsilon)$, and conjugating by it we have modified $\varphi_{v_i}$ so that it is still $\varepsilon$-close to $\varphi$, but it moreover satisfies the amalgamation $A_i^- = A_i^+$.
We can do this for all $i$, and so we get the desired relation for each edge $e_i$. These modifications are compatible, since they only affect $\varphi_{v_i}$ for $i \geq 1$ and not for $i = 0$. We can now apply the same procedure to all neighbours of $v_i$ connected by edges of $T$ other than $e_0$, and take care of those edges without affecting the behaviour of $\varphi_{v_i}: i = 0, \ldots, r$. Since $T$ is a tree we can keep on doing this until we have covered all edges of $T$. We have thus obtained $\hat{\varphi} : F_S \to G$ with the same properties as before, but now it also satisfies all relations $\{ R_e : e \in T \}$. \\
We next move to edges not in $T$. For such an edge $e$, let $A^{\pm} := \varphi(\iota_e^{\pm}(\Gamma_e)) \leq G$ as before, so that $A^{\pm} \mod \varepsilon$ is a $\pi'$-group. Since $(\hat{\varphi} \mod \varepsilon)$ descends to a homomorphism of $\Gamma$, we know that $A^+$ and $A^-$ are conjugate modulo $\varepsilon$ by $\hat{\varphi}(t_e)$. So $A^+$ and $\hat{\varphi}(t_e)A^-\hat{\varphi}(t_e)^{-1}$ satisfy the same hypotheses as in the previous step. By the same argument, there exists $t \in G(\varepsilon)$ such that $t \hat{\varphi}(t_e) A^- \hat{\varphi}(t_e)^{-1} t^{-1} = A^+$. We can thus replace $\hat{\varphi}(t_e)$ by $t \hat{\varphi}(t_e)$, which is congruent to $\hat{\varphi}(t_e)$ modulo $\varepsilon$. This takes care of all such relations. Note that we have only modified the images of the edge elements, so this does not affect the definition of $\hat{\varphi}$ at the vertex groups, or at the other edge groups. \\
We are left with a homomorphism $\hat{\psi} : F_S \to G$ such that $\operatorname{dist}(\hat{\varphi}|_{F_{S_v}}, \hat{\psi}|_{F_{S_v}}) \leq \varepsilon$ for every vertex $v$, and $d(\hat{\varphi}(t_e), \hat{\psi}(t_e)) \leq \varepsilon$ for every edge $e$, and moreover $\hat{\psi}$ satisfies all of the defining relations of $\Gamma$. Thus $\hat{\psi}$ is $\varepsilon$-close to $\hat{\varphi}$ and it descends to a homomorphism $\Gamma \to G$.
\end{proof}
Here is our first stability result for graphs of groups:
\begin{proposition}
\label{prop:pifree_gog1}
Let $\mathcal{G}$ be a virtually pro-$\pi$ family and $\Gamma$ the fundamental group of a graph of groups such that all vertex groups are uniformly $\mathcal{G}$-stable with a uniform estimate, and such that for every edge $e$ adjacent to a vertex $v$, the image of $\Gamma_e$ in any finite quotient of $\Gamma_v$ is a $\pi'$-group. Then $\Gamma$ is uniformly $\mathcal{G}$-stable.
\end{proposition}
\begin{remark}
By ``uniformly $\mathcal{G}$-stable with a uniform estimate" we mean that the $\delta = \delta(\varepsilon)$ from Lemma \ref{lem:quant} may be chosen uniformly for all $\Gamma_v$. This is automatically satisfied if the graph is finite.
\end{remark}
\begin{proof}
We use the characterization of uniform stability from Corollary \ref{cor:quant}. Let $\langle S \mid R \rangle$ be the standard presentation of $\Gamma$. Fix $0 < \varepsilon \leq \varepsilon_0$. Let $\hat{\varphi} : F_S \to G \in \mathcal{G}$ be a homomorphism with $\operatorname{def}(\hat{\varphi}) \leq \delta$, where $\delta = \delta(\varepsilon)$ is given by uniform $\mathcal{G}$-stability of the $\Gamma_v$. This allows to modify $\hat{\varphi}$ by at most $\varepsilon$ on the vertex generators $S_v$ so that $\hat{\varphi}|_{F_{S_v}}$ descends to a homomorphism of $\Gamma_v$. Now $\hat{\varphi}$ satisfies the hypotheses of Lemma \ref{lem:pifree_gog}, and so there exists $\hat{\psi} : F_S \to G$ such that $\operatorname{dist}(\hat{\varphi}, \hat{\psi}) \leq \varepsilon$ and $\hat{\psi}$ descends to a homomorphism of $\Gamma$.
\end{proof}
Lemma \ref{lem:pifree_gog} shows that the estimate for stability is at least as good as the uniform estimate for the vertex groups. For instance if the graph is finite, this proposition gives as an estimate of stability the one of the vertex group with the least efficient estimate. This slightly different result combines Lemma \ref{lem:pifree_gog} with the last part of Lemma \ref{lem:pifree}:
\begin{proposition}
\label{prop:pifree_gog2}
Let $\mathcal{G}$ be a virtually pro-$\pi$ family and $\Gamma$ the fundamental group of a graph of groups such that for every vertex $v$ the image of $\Gamma_v$ in any finite quotient of $\Gamma$ is a $\pi'$-group. Then $\Gamma$ is uniformly $\mathcal{G}$-stable.
\end{proposition}
\begin{proof}
Let $\langle S \mid R \rangle$ be the standard presentation of $\Gamma$. Fix $0 < \varepsilon \leq \varepsilon_0$ and let $\hat{\varphi} : F_S \to G \in \mathcal{G}$ be a homomorphism with $\operatorname{def}(\hat{\varphi}) \leq \varepsilon$. By Lemma \ref{lem:mq}, this induces a homomorphism $\Gamma \to G/G(\varepsilon)$, whose restriction to $\Gamma_v$ is a $\pi'$-group. Now Lemma \ref{lem:pifree} allows to modify $\hat{\varphi}$ by at most $\varepsilon$ on the vertex generators $S_v$ so that $\hat{\varphi}|_{F_{S_v}}$ descends to a homomorphism $\Gamma_v \to G$ whose image is a finite $\pi'$-group. Then we apply Lemma \ref{lem:pifree_gog} and conclude as in Proposition \ref{prop:pifree_gog1}.
\end{proof}
Here the proof shows that the estimate for stability is optimal.
\subsection{First corollaries}
\label{ss:vpropi:cor}
We now apply Propositions \ref{prop:pifree_gog1} and \ref{prop:pifree_gog2} to obtain examples of uniformly $\mathcal{G}$-stable groups. \\
The following is a direct consequence of Proposition \ref{prop:pifree_gog1}:
\begin{corollary}
\label{cor:pifree_gog}
Let $\mathcal{G}$ be a virtually pro-$\pi$ family. The following groups are uniformly $\mathcal{G}$-stable:
\begin{enumerate}
\item Fundamental groups of connected graphs of groups, with $\pi$-free vertex groups.
\item Fundamental groups of finite, connected graphs of groups, with uniformly $\mathcal{G}$-stable vertex groups and $\pi$-free edge groups.
\end{enumerate}
\end{corollary}
The next corollary relies on Dunwoody's characterization of groups of cohomological dimension (denoted $cd$) at most $1$ \cite{cd1}:
\begin{corollary}
\label{cor:vfree_p}
Let $\mathcal{G}$ be a virtually pro-$\pi$ family. If $cd_{\mathbb{F}_p}(\Gamma) \leq 1$ for all $p \in \pi$, then $\Gamma$ is uniformly $\mathcal{G}$-stable. In particular finitely generated virtually free groups without elements of order $p$, for all $p \in \pi$, are uniformly $\mathcal{G}$-stable.
\end{corollary}
\begin{proof}
By \cite{cd1}, a group has $\mathbb{F}_p$-cohomological dimension at most $1$ if and only if it is the fundamental group of a connected graph of groups whose vertex groups are finite and $p$-free. Even if the underlying graph is infinite, the estimate for uniform $\mathcal{G}$-stability of the vertex groups is uniform (in fact, optimal) by Lemma \ref{lem:pifree}, and so we can apply Proposition \ref{prop:pifree_gog1}. The statement about virtually free groups is the finitely generated case of \cite{cd1}, but it also follows more directly from Stallings's Theorem on groups with infinitely many ends, without going through cohomological dimension (see \cite{Stallings1} for the torsion-free case and \cite[5.A.9]{Stallings2} for the general case).
\end{proof}
Recall from Example \ref{ex:wr_ust} that if $G$ is perfect, then $G \wr \mathbb{Z}$ is uniformly $\mathcal{G}$-stable for any profinite family $\mathcal{G}$. The following corollary of Proposition \ref{prop:pifree_gog2} strengthens this:
\begin{corollary}
\label{cor:wr}
Let $\mathcal{G}$ be a virtually pro-$\pi$ family. If $G$ does not surject onto $\mathbb{F}_p$, for any $p \in \pi$, then $G \wr \mathbb{Z}$ is uniformly $\mathcal{G}$-stable.
\end{corollary}
\begin{proof}
We use the notation from Example \ref{ex:wr}. Note that $G \wr \mathbb{Z}$ is the fundamental group of a loop with vertex group and edge group $G$, where the edge inclusions are $G \xrightarrow{\cong} G_0$ and $G \xrightarrow{\cong} G_1$. So to apply Proposition \ref{prop:pifree_gog2} it suffices to show that in any finite quotient of $G \wr \mathbb{Z}$ the image of $\Sigma_{\mathbb{Z}} G$ has order coprime to $p$. By Example \ref{ex:wr}, this image must be abelian, and a finite abelian group of order divisible by $p$ surjects onto $\mathbb{F}_p$. This is ruled out by the hypothesis.
\end{proof}
Proposition \ref{prop:pifree_gog2} can also be applied to Generalized Baumslag--Solitar groups, which will be the subject of the next subsection. \\
Note that -- except for Item $2.$ of Corollary \ref{cor:pifree_gog} which depends on the stability estimates of the vertex groups -- all other examples have an optimal estimate for stability: see the discussions after the proofs of Propositions \ref{prop:pifree}, \ref{prop:pifree_gog1} and \ref{prop:pifree_gog2}.
\subsection{GBS groups}
\label{ss:GBS}
We now apply Proposition \ref{prop:pifree_gog2} to many Generalized Baumslag--Solitar (from now on: GBS) groups. We refer the reader to \cite{GBS} for more details on GBS groups. \\
Recall that the \textit{Baumslag--Solitar group} $\operatorname{BS}(m, n)$ is defined by the presentation $\langle s, t \mid ts^nt^{-1} = s^m \rangle$, so it is the fundamental group of a loop with vertex group and edge group $\mathbb{Z}$, where the edge inclusions are $\mathbb{Z} \to \mathbb{Z} : 1 \mapsto n, m$. More generally, a \textit{GBS} group is the fundamental group of a finite connected graph of groups $X = (V, E)$ all of whose vertex and edge groups are infinite cyclic. The information on the edge inclusions can be summarized in to \textit{weight functions} $w_{\pm} : E \to \mathbb{Z} \, \backslash \, \{ 0 \}$, that is $\iota_e^{\pm} : \Gamma_e \cong \mathbb{Z} \to \Gamma_{e^{\pm}} \cong \mathbb{Z} : 1 \mapsto w_{\pm}(e)$. Note that it suffices to know $w_+$, or to know $w_\pm$ on a set of positively oriented edges, in order to recover all the information; indeed $\iota_e^- = \iota_{\overline{e}}^+$. We denote the graph of groups associated to a GBS group by $(X, w)$, where $X$ is the underlying graph and $w = (w_-, w_+)$ are the weight functions.
It will be convenient to extend the weight functions from oriented edges to oriented paths. So given an oriented path $P : v_1 \xrightarrow{e_1} v_2 \to \cdots \to v_k \xrightarrow{e_k} v_{k+1}$, we denote by $w_{\pm}(P) := \prod_i w_{\pm}(e_i)$. If $p$ is a prime, we have $\nu_p(w_{\pm}(P)) = \sum_i \nu_p(w_{\pm}(e_i))$. In particular $\nu_p(w_{\pm}(P)) = 0$ if and only if $\nu_p(w_{\pm}(e_i)) = 0$ for all $i = 1, \ldots, k$.
The following corollary to Proposition \ref{prop:pifree_gog2} gives a combinatorial criterion that ensures that a GBS group is $\mathcal{G}$-stable (since such groups are finitely presented, by Theorem \ref{thm:pw_un} we need not specify whether the stability is pointwise or uniform).
\begin{corollary}
\label{cor:GBS_p}
Let $\mathcal{G}$ be a virtually pro-$\pi$ family, and let $\Gamma$ be a GBS group corresponding to the weighted graph $(X, w)$. Suppose that for all $p \in \pi$ there exists a set of oriented cycles $C$ satisfying $\nu_p(w_-(C)) = 0 < \nu_p(w_+(C))$, and that for every vertex $y$ there exists a vertex $x$ belonging to one of these cycles, and a path $x \xrightarrow{P} y$ with $\nu_p(w_+(P)) = 0$. Then $\Gamma$ is $\mathcal{G}$-stable.
\end{corollary}
Note that the condition only requires that such cycles and paths exist for any given $p \in \pi$: we are allowed to choose different ones for each prime in $\pi$. We will prove that such groups satisfy the conditions of Proposition \ref{prop:pifree_gog2}. So we deduce not only stability, but also that the estimate is optimal. The simplest example is that of Baumslag--Solitar groups, which features in Theorem \ref{intro:thm:pifree}. Here there is only one vertex so the condition on the existence of special paths is not needed, and the special cycle is given by the loop.
\begin{corollary}
\label{cor:BS_p}
Let $\mathcal{G}$ be a virtually pro-$\pi$ family and suppose that each $p \in \pi$ divides exactly one of $m, n$. Then $\operatorname{BS}(m, n)$ is $\mathcal{G}$-stable.
\end{corollary}
Here is a more complex example of graph $(X, w)$ that satisfies the conditions of Corollary \ref{cor:GBS_p} where $\pi = \{ p \}$ is a single prime. We draw a set of positively oriented edges $e$, labeled by the weights $(w_-(e), w_+(e))$. Each weight labeled $0$ may be replaced by any integer coprime to $p$, each weight labeled $1$ by any non-zero multiple of $p$, and $*$ by any non-zero integer.
\begin{center}
\begin{tikzpicture}[->,>=stealth,
shorten >=5pt ,
node distance =2.5cm,
semithick]
\node[state] (U) {$u$};
\node[state] (V) [right of=U] {$v$};
\node[state] (X) [right of=V] {$x$};
\node[state] (Y) [right of=X] {$y$};
\node[state] (Z) [right of=Y] {$z$};
\path (U) edge [below] node {$(0, *)$} (V)
(V) edge [below] node {$(0, 1)$} (X)
(X) edge [below] node {$(*, *)$} (Y)
edge [bend right, above] node {$(0, *)$} (U)
(Z) edge [loop, above] node {$(1, 0)$} (Z)
edge [below] node {$(*, 0)$} (Y);
\end{tikzpicture}
\end{center}
If $C : u \to v \to x \to u$, then $\nu_p(w_-(C)) = \nu_p(0) + \nu_p(0) + \nu_p(0) = 0$, while $\nu_p(w_+(C)) \geq \nu_p(1) > 0$, so it satisfies the hypothesis. Similarly, the loop at $z$ satisfies the hypothesis: $\nu_p(0) = 0 < \nu_p(0)$. The only vertex left to check is $y$, and for this we use the path $P : z \to y$, that satisfies $\nu_p(w_+(P)) = \nu_p(0) = 0$. \\
This example also clarifies that although the condition is stated in notation-heavy terms, it is quite easy to check, and there is no need to precisely compute $\nu_p(w_{\pm}(C, P))$. For instance $\nu_p(w_-(C)) = 0 < \nu_p(w_+(C))$ simply means that the negative weights along $C$ are all coprime to $p$, and that there is at least one positive weight that is divisible by $p$. Similarly $\nu_p(w_+(P)) = 0$ simply means that the positive weights along $P$ are all coprime to $p$.
\begin{proof}[Proof of Corollary \ref{cor:GBS_p}]
The proof will be split in a sequence of technical lemmas. Fix a GBS group $\Gamma$ with graph $(X, w)$ and $p \in \pi$. Given a vertex $x$ we denote by $s_x$ the corresponding generator. By Proposition \ref{prop:pifree_gog2} we need to show that for every vertex $x$ the image of $s_x$ in any finite quotient of $\Gamma$ has order coprime to $p$. For the sake of brevity, let us say that such a vertex $x$ is \textit{$p$-free}. The first lemma shows that the condition on the existence of special paths reduces the question to the vertices belonging to special cycles:
\begin{lemma}
\label{lem:GBS_pfree}
Suppose that $x$ is $p$-free, and let $x \xrightarrow{P} y$ be an oriented path with $\nu_p(w_+(P)) = 0$. Then $y$ is also $p$-free.
\end{lemma}
\begin{proof}
By induction on the length of the path, it suffices to show this for paths of length $1$. So suppose that $x \xrightarrow{e} y$ and let $(w_-(e), w_+(e)) = (m, n)$. By definition of the fundamental group, $s_x^m$ is conjugate to $s_y^n$ in $\Gamma$, and by hypothesis $\nu_p(n) = 0$. Let $f : \Gamma \to K$ be a finite quotient of $\Gamma$, let $o_x$ be the order of $f(s_x)$ and $o_y$ the order of $s_y$. Conjugacy implies that $f(s_x)^m$ and $f(s_y)^n$ have the same order, that is $o_x/(o_x, m) = o_y/(o_y, n)$. Since $x$ is $p$-free, $\nu_p(o_x) = 0$, so $\nu_p(o_y) = \nu_p(o_y, n) \leq \nu_p(n) = 0$. Since $K$ was arbitrary, we conclude that $y$ is $p$-free.
\end{proof}
So we only need to show that if $C$ is a cycle such that $\nu_p(w_-(C)) = 0 < \nu_p(w_+(C))$, then every vertex of $C$ is $p$-free. Here is a sufficient condition for a vertex to be $p$-free:
\begin{lemma}
\label{lem:GBS_omn}
Let $x$ be a vertex such that $s_x^m$ is conjugate to $s_x^n$. If $\nu_p(m) = 0 < \nu_p(n)$, then $x$ is $p$-free.
\end{lemma}
\begin{proof}
Let $f : \Gamma \to K$ be a finite quotient of $\Gamma$, and let $o$ be the order of $s_x$. As in the previous lemma, the conjugacy implies that $o/(o, m) = o/(o, n)$ and so $(o, m) = (o, n)$. Since $p$ divides $n$ but not $m$, this is only possible if $p$ does not divide $o$.
\end{proof}
Note that this lemma alone is enough to conclude the proof in the case in which all cycles are loops, in particular it concludes the proof in the case of the Baumslag--Solitar group (which does not even need Lemma \ref{lem:GBS_pfree}). For more general cycles, we use instead the next lemma:
\begin{lemma}
\label{lem:GBS_conjP}
Let $x \xrightarrow{P} y$ be an oriented path. Then $s_x^{w_-(P)}$ is conjugate to $s_y^{w_+(P)}$.
\end{lemma}
\begin{proof}
We prove the statement by induction on the length of $P$. It is clear if $P$ has length $1$. Now suppose that the statement is true for $x \xrightarrow{P} y$ and let us prove it for $x \xrightarrow{P} y \xrightarrow{e} z$ for some edge $e$. By induction hypothesis $s_x^{w_-(P)}$ is conjugate to $s_y^{w_+(P)}$, and so $(s_x^{w_-(P)})^{w_-(e)}$ is conjugate to $(s_y^{w_+(P)})^{w_-(e)}$. Since by definition $s_y^{w_-(e)}$ is conjugate to $s_z^{w_+(e)}$, this implies that $(s_y^{w_-(e)})^{w_+(P)}$ is conjugate to $(s_z^{w_+(e)})^{w_+(P)}$. Thus $s_x^{w_-(P) \cdot w_-(e)}$ is conjugate to $s_z^{w_+(P) \cdot w_+(e)}$.
\end{proof}
In the case in which $P = C$ is a cycle such that $\nu_p(w_-(C)) = 0 < \nu_p(w_+(C))$, Lemma \ref{lem:GBS_conjP} implies that every vertex in $C$ satisfies a relation as in Lemma \ref{lem:GBS_omn}, and so is $p$-free. Lemma \ref{lem:GBS_pfree} allowed to reduce to looking at the vertices in cycles, hence this concludes the proof of Corollary \ref{cor:GBS_p}.
\end{proof}
Corollary \ref{cor:BS_p} has a further consequence. We know from Theorem \ref{thm:pw_un} that for finitely presented groups the notions of pointwise and uniform stability coincide. For finitely generated infinitely presented groups, we have mostly seen examples of \textit{uniform} stability, and non-examples of pointwise stability (Example \ref{ex:cex}), with one family of examples of pointwise stability (Example \ref{ex:simplelef}). But we know from Theorem \ref{thm:rf} that the largest residually finite quotient of a pointwise stable group is pointwise stable. We will apply this to the largest residually finite quotient of the non-residually finite Baumslag--Solitar groups, which were identified by Moldavanskii in \cite{moldavanskii}. Thus we obtain:
\begin{corollary}
\label{cor:rfBS_p}
Let $\mathcal{G}$ be a virtually pro-$\pi$ family and suppose that each $p \in \pi$ divides exactly one of $m, n$. Let $d := (m, n)$ be the greatest common divisor of $m, n$, and suppose that $|m|, |n|$ are distinct from each other and from $1$. Then the group
$$\Gamma = \langle a, b_i : i \in \mathbb{Z} \mid [b_i^d, b_j] = 1, b_i^m = b_{i+1}^n, ab_ia^{-1} = b_{i+1} : i \in \mathbb{Z} \rangle$$
is finitely generated, infinitely presented, and pointwise $\mathcal{G}$-stable.
Writing $m = du, n = dv$, this group fits into an extension
$$1 \to \mathbb{Z} \left[ \frac{1}{uv} \right] \to \Gamma \to (\mathbb{Z}/d\mathbb{Z} * \mathbb{Z}) \to 1,$$
and in case $d = 1$ it is isomorphic to $\mathbb{Z}\left[\frac{1}{mn}\right] \rtimes_{\frac{m}{n}} \mathbb{Z}$.
\end{corollary}
\begin{proof}
By Corollary \ref{cor:BS_p} the group $\operatorname{BS}(m, n)$ is $\mathcal{G}$-stable. The condition on $|m|, |n|$ is equivalent to $\operatorname{BS}(m, n)$ being not residually finite \cite{BS:Hopf}, and the group above is its largest residually finite quotient \cite[Equation (1)]{moldavanskii}, which is pointwise stable by Theorem \ref{thm:rf}. It is infinitely presented by \cite[Theorem 2]{moldavanskii}.
The description of the group is in \cite[Proposition 3]{moldavanskii}, except the author uses the presentation $C = \langle e_k : k > 0 \mid e_k = e_{k+1}^{uv} \rangle$ for the kernel of the extension \cite[Proposition 4]{moldavanskii}. This is isomorphic to $\mathbb{Z}[1/uv]$ under the isomorphism $\varphi : C \to \mathbb{Z}[1/uv] : e_k \mapsto (uv)^{-k}$, with inverse $\varphi^{-1} : \mathbb{Z}[1/uv] \to C : a(uv)^{-k} \mapsto e_k^a$.
\end{proof}
\begin{example}
The group $\mathbb{Z}\left[ \frac{1}{6} \right] \rtimes_{\frac{2}{3}} \mathbb{Z}$ is finitely generated, infinitely presented, residually finite and pointwise $\operatorname{GL}(\mathfrak{o})$-stable, whenever $\mathbb{K}$ has residual characteristic $2$ or $3$.
\end{example}
\pagebreak
\section{$\operatorname{GL}(\mathfrak{o})$-stability}
\label{s:char0}
In this section we focus on the family $\operatorname{GL}(\mathfrak{o})$, where $\mathfrak{o}$ is the ring of integers of a non-Archimedean local field of characteristic $0$. By Ostrowski's Theorem, $\mathbb{K}$ is a finite extension of $\mathbb{Q}_p$, where $p$ is the residual characteristic of $\mathbb{K}$. The stability results will be similar to the ones in Section \ref{s:vpropi}, but more flexible: this will be achieved by applying Lemma \ref{lem:cohopk} (instead of the Schur--Zassenhaus Theorem) to ensure existence or conjugacy in the lifting problems that occur. This will allow to expand the class of examples that we presented in Section \ref{s:vpropi}, some of which are included in Theorem \ref{intro:thm:vpfree}. \\
We fix the following notation for the rest of the section (see Subsection \ref{ss:nona}): $\mathbb{K}$ is a finite extension of $\mathbb{Q}_p$, with a norm $| \cdot |$ that we may assume restricts to the $p$-adic norm of $\mathbb{Q}_p$. Let $\mathfrak{o}$ be the ring of integers, $\overline{\omega}$ a uniformizer, so the maximal ideal of $\mathfrak{o}$ is $\mathfrak{p} = \overline{\omega} \mathfrak{o}$, and $|\overline{\omega}| = r < 1$ so that $|\mathbb{K}^\times| = r^{\mathbb{Z}}$. Since $|p| = |p|_p = p^{-1}$, there exists $a \geq 1$ such that $r^a = p^{-1} = |p|$, that is $p \in \mathfrak{p}^a$. Whenever $n$ does not vary, we denote $G := \operatorname{GL}_n(\mathfrak{o})$ and the congruence subgroups by $G_k := \operatorname{GL}_n(\mathfrak{o})_k$. \\
The last Subsection \ref{ss:poschar} switches to the case in which $\mathbb{K}$ has characteristic $p$: we will prove stability of $\mathbb{Z}/2\mathbb{Z}$ in this case (Proposition \ref{prop:z2z}) and discuss why the method does not work for other finite $p$-groups.
\subsection{Virtually $p$-free groups}
In this subsection we use the lifting part of Lemma \ref{lem:cohopk} to prove stability of groups that are only required to be \textit{virtually} $p$-free. This covers in particular all finite groups. Here is a characterization:
\begin{lemma}
\label{lem:vpfree_char}
Let $\Gamma$ be a group and $p$ be a prime. Then $\Gamma$ is virtually $p$-free if and only if
$$\sup \{ \nu_p(|C|) : C \text{ is a finite quotient of } \Gamma \} < \infty.$$
\end{lemma}
\begin{proof}
Suppose that $\Gamma$ is virtually $p$-free, and let $H$ be a $p$-free finite-index subgroup. A finite-index subgroup of a $p$-free group is $p$-free, so we may assume that $H$ is normal. We claim that the supremum is achieved at $\Gamma/H$. Indeed, let $K$ be any other finite-index normal subgroup of $\Gamma$. Then
$$\nu_p(|\Gamma/K|) \leq \nu_p(|\Gamma/(K \cap H)|) = \nu_p(|\Gamma/H|) \cdot \nu_p(|H/(K \cap H)|) = \nu_p(|\Gamma/H|),$$
where the last equality uses that $H$ is $p$-free.
Conversely, suppose that the supremum is achieved at $\Gamma/H$, where $H$ is a finite-index normal subgroup of $\Gamma$. Then $H$ is $p$-free. Indeed, if $K$ is a finite-index normal subgroup of $H$, let $N \leq K$ be a finite-index normal subgroup of $\Gamma$; then
$$\nu_p(|H/K|) \leq \nu_p(|H/N|) = \nu_p(|\Gamma/N|)/\nu_p(|\Gamma/H|) = 1,$$
where the last equality uses that $\nu_p(|\Gamma/N|) \geq \nu_p(|\Gamma/H|)$, and the latter is maximal.
\end{proof}
Therefore a group is virtually $p$-free if and only if there is a bound on the order of its finite virtual $p$-quotients.
\begin{example}
Let $\Gamma$ be a locally finite group with a bound on the order of its finite $p$-subgroups. Say $\Gamma$ has no subgroup of order $p^k$ (and so no subgroup of order $p^l$ for $l \geq k$). Then it cannot admit a group of order $p^k$ as a virtual quotient. Lemma \ref{lem:vpfree_char} implies that $\Gamma$ is virtually $p$-free.
\end{example}
We now prove the analogues of Lemma \ref{lem:pifree} and Proposition \ref{prop:pifree}. The proofs are essentially the same, but they use Lemma \ref{lem:cohopk} instead of the Schur--Zassenhaus Theorem.
\begin{lemma}
\label{lem:vpfree}
Let $\varphi : \Gamma \to G \in \operatorname{GL}(\mathfrak{o})$ be such that $\operatorname{def}(\varphi) \leq r^{ak} = p^{-k}$ for some $k \geq 1$, and suppose that the image of $\Gamma$ in $G/G_{ak}$ is a group $C$ with $\nu_p(|C|) \leq l < k/2$. Then there exists a homomorphism $\psi : \Gamma \to G$ such that $\operatorname{dist}(\varphi, \psi) \leq r^{a(k - l)} = p^l \cdot p^{-k}$. Moreover, $\psi(\Gamma) \leq G$ is a finite group of isomorphic to a quotient of $C$.
\end{lemma}
\begin{proof}
Let $\varphi_k = \varphi(r^{ak}) : \Gamma \to G/G_{ak}$ denote the induced homomorphism; we also get a homomorphism $\varphi_{k - l} : \Gamma \to G/G_{a(k-l)}$. So we have the following lifting problem:
\[\begin{tikzcd}
& {G/G_{2a(k-l)}} \\
C & {G/G_{ak}} \\
& {G/G_{a(k-l)}}
\arrow[dashed, from=2-1, to=1-2]
\arrow["\varphi_k"', from=2-1, to=3-2]
\arrow["\varphi_{k-l}"', from=2-1, to=2-2]
\arrow[from=2-2, to=3-2]
\arrow[from=1-2, to=3-2, bend left = 50]
\end{tikzcd}\]
Now $G/G_{ak} \cong (G/G_{2a(k-l)})/(G_{ak}/G_{2a(k - l)})$ and $G/G_{a(k-l)} \cong (G/G_{2a(k-l)})/(G_{a(k-l)}/G_{2a(k - l)})$. Moreover by Lemma \ref{lem:vprop} we have an isomorphism
$$G_{a(k-l)}/G_{2a(k-l)} = \operatorname{GL}_n(\mathfrak{o})_{a(k-l)} / \operatorname{GL}_n(\mathfrak{o})_{2a(k-l)} \to \operatorname{M}_n(\mathfrak{o}/\mathfrak{p}^{a(k-l)}).$$
So $G_{a(k-l)}/G_{2a(k-l)}$ is a $\mathbb{Z}/p^{(k - l)} \mathbb{Z}$-module (because $\mathfrak{o}/\mathfrak{p}^{a(k-l)}$ is) and the image under multiplication by $p^l$ is $G_{ak}/G_{2a(k-l)}$. Since $\nu_p(|C|) \leq l$ we are in the situation of Lemma \ref{lem:cohopk}, which means that the lift exists. Lifting this in turn to a map $\psi : \Gamma \to G$, we have $\operatorname{def}(\psi) \leq r^{2a(k - l)}$ and $\operatorname{dist}(\varphi, \psi) \leq r^{a(k-l)}$.
The hypothesis $k > 2l$ implies that the defect of $\psi$ is strictly smaller than that of $\varphi$, so we can apply the above procedure to $\psi$. This leads to a sequence $\psi_i : \Gamma \to G$ such that $\operatorname{def}(\psi_i) \to 0$ and $r^{a(k-l)} \geq \operatorname{dist}(\psi_i, \psi_{i-1}) \to 0$. The latter condition implies that $\psi_i$ is Cauchy with respect to the uniform norm, so it converges to a homomorphism $\psi$. The inequality $\operatorname{dist}(\varphi, \psi) \leq r^{a(k-l)}$ holds because it does for all $\psi_i$, by the ultrametric inequality.
\end{proof}
\begin{proposition}
\label{prop:vpfree}
Let $\Gamma$ be virtually $p$-free. Then $\Gamma$ is uniformly $\operatorname{GL}(\mathfrak{o})$-stable.
\end{proposition}
\begin{proof}
Let $l$ be the supremum from Lemma \ref{lem:vpfree_char}: for any finite quotient $C$ of $\Gamma$, it holds $\nu_p(|C|) \leq l$. Therefore given a map $\varphi : \Gamma \to \operatorname{GL}_n(\mathfrak{o})$ with small enough defect, the previous lemma applies and $\varphi$ is close to a homomorphism. We conclude by Lemma \ref{lem:quant}.
\end{proof}
Lemma \ref{lem:vpfree} shows that the estimate for stability is the linear: if $\varphi : \Gamma \to G$ satisfies $\operatorname{def}(\varphi) < r^{a2l} = p^{-2l}$ (so the $k$ in the lemma is indeed larger than $2l$), then there exists a homomorphism $\psi : \Gamma \to G$ such that $\operatorname{dist}(\varphi, \psi) \leq p^l \operatorname{def}(\varphi)$. Note how this is reminiscent of a well-known generalization of Hensel's Lemma: if $f \in \mathbb{Z}_p[X]$ and $a \in \mathbb{Z}_p$ satisfy $p^r = |f(a)|_p, p^h = |f'(a)|_p$ and $r > 2h$ (that is, $|f(a)|_p < |f'(a)|_p^2$), then there exists a unique root $a'$ of $a$ such that $a' \equiv a \mod p^{r - h}$.
\begin{example}
\label{ex:finst}
All finite groups are $\operatorname{GL}(\mathfrak{o})$-stable. More precisely, let $\Gamma$ be a finite group, and $\nu_p(\Gamma) = l$. Then for every $\varphi : \Gamma \to \operatorname{GL}_n(\mathfrak{o})$ such that $\operatorname{def}(\varphi) \leq p^{-2l}$, there exists a homomorphism $\psi : \Gamma \to \operatorname{GL}_n(\mathfrak{o})$ such that $\operatorname{dist}(\varphi, \psi) \leq p^l \operatorname{def}(\varphi)$. Compare this with Example \ref{ex:2unst} and \cite[Proposition 1]{amenst}.
\end{example}
\begin{example}
A finitely generated group of finite exponent is uniformly $\operatorname{GL}(\mathfrak{o})$-stable. Indeed, by Zelmanov's solution of the restricted Burnside problem \cite{RBP_odd, RBP_2}, any such group has only finitely many finite quotients. In particular, free Burnside groups of finite rank are uniformly $\operatorname{GL}(\mathfrak{o})$-stable.
\end{example}
We saw in Example \ref{ex:npa} that normed $\mathbb{K}$-amenable groups are $\operatorname{GL}(\mathfrak{o})$-stable when $\mathbb{K}$ has characteristic $p$. The following example completes the picture:
\begin{example}
\label{ex:npa2}
Let $\mathbb{K}$ have characteristic $0$, and let $\Gamma$ be a normed $\mathbb{K}$-amenable group \cite[Definition 1.1]{mio}. Then $\Gamma$ is $\operatorname{GL}(\mathfrak{o})$-stable: indeed such groups are characterized as being locally finite and with a bound on the order of their finite $p$-subgroups \cite[Theorem 6.2]{mio}.
\end{example}
\subsection{Graphs of groups}
We now use the conjugacy part of Lemma \ref{lem:cohopk} to strengthen the results on stability of graphs of groups from Subsection \ref{ss:gog} from which we borrow the notation. As before, we start with the analogue of Lemma \ref{lem:pifree_gog} and then prove the analogues of Propositions \ref{prop:pifree_gog1} and \ref{prop:pifree_gog2}. Also here, the lemma gives examples of constraint stability \cite{a:const} and stability of an epimorphism \cite{stepi}: see the discussion after the statement of Lemma \ref{lem:pifree_gog}.
\begin{lemma}
\label{lem:vpfree_gog}
Let $X$ be a connected graph of groups with vertex groups $\Gamma_v$, edge groups $\Gamma_e$ and edge inclusions $\iota_e^{\pm} : \Gamma_e \to \Gamma_{e^{\pm}}$. Let $\Gamma$ be the fundamental group of $X$, with the standard presentation $\langle S \mid R \rangle = \langle S_v, t_e \mid R_v, R_e \rangle$.
Let $\hat{\varphi} : F_S \to G \in \operatorname{GL}(\mathfrak{o})$ be a map with $\operatorname{def}(\hat{\varphi}) \leq r^{ak} = p^{-k}$ for some $k \geq 1$. Suppose further that for all $v \in V$ the restriction of $\hat{\varphi}$ to $F_{S_v}$ descends to a homomorphism $\varphi_v : \Gamma_v \to G$ such that, for all $m \geq k$, if $e^{\pm} = v$, the image $C$ of $\varphi_v(\iota_e^{\pm}(\Gamma_e))$ in $G/G_{am}$ satisfies $\nu_p(|C|) \leq l < k/2$.
Then there exists a homomorphism $\hat{\psi} : F_S \to G$ such that $\operatorname{dist}(\hat{\varphi}, \hat{\psi}) \leq r^{a(k-l)} = p^l \cdot p^{-k}$ and $\hat{\psi}$ descends to a homomorphism of $\Gamma$.
\end{lemma}
\begin{proof}
The proof is essentially the same as that of Lemma \ref{lem:pifree_gog}, using Lemma \ref{lem:cohopk} as we did in the proof of Lemma \ref{lem:vpfree}.
We start by setting $\hat{\varphi}(t_e) = 1$ for all $e \in T$. Next we modify $\hat{\varphi}$ at the vertex groups so that it satisfies the conjugacy relations given by edges in $T$. Using the same induction argument it suffices to treat the case $(v_0 \xrightarrow{e_i} v_i) \in T$: we need to find $t \in G_{a(k-l)}$ that conjugates the image of $\Gamma_{e_i}$ in $\varphi(\Gamma_{v_0})$ to that of $\varphi(\Gamma_{v_i})$. Considering the following lifting problem:
\[\begin{tikzcd}
&&& {G/G_{2a(k-l)}} \\
{\Gamma_{e_i}} & {f_k(\Gamma_{e_i})} && {G/G_{ak}} \\
&&& {G/G_{a(k-l)}}
\arrow[from=2-4, to=3-4]
\arrow[from=1-4, to=2-4]
\arrow[from=2-1, to=2-2]
\arrow[from=2-2, to=1-4, dashed]
\arrow[from=2-2, to=2-4]
\arrow[from=2-2, to=3-4]
\end{tikzcd}\]
and using Lemma \ref{lem:cohopk} to prove that any two lifts of the horizontal arrow are $G_{a(k-l)}$-conjugate, we obtain an element $t \in G_{a(k-l)}$ that conjugates the two images modulo $r^{2a(k-l)}$. Reiterating this process yields a sequence that converges to the desired conjugating element.
Finally we modify $\hat{\varphi}$ at the generators of edges not in $T$, using the same argument.
\end{proof}
\begin{proposition}
\label{prop:vpfree_gog1}
Let $\Gamma$ the fundamental group of a graph of groups such that all vertex groups are uniformly $\operatorname{GL}(\mathfrak{o})$-stable with a uniform estimate, and such that there exists $l \geq 1$ such that for every edge $e$ adjacent to a vertex $v$, the image of $\Gamma_e$ in any finite quotient of $\Gamma_v$ has no subgroup of order $p^l$. Then $\Gamma$ is uniformly $\operatorname{GL}(\mathfrak{o})$-stable.
\end{proposition}
\begin{remark}
By the proof of Lemma \ref{lem:vpfree_char}, the second condition is really asking for the image of $\Gamma_e$ inside $\Gamma_v$ to be ``virtually $p$-free relative to $\Gamma_v$": that is, there exists a finite-index subgroup $\Gamma_e' \leq \Gamma_e$ such that the image of $\Gamma_e'$ in any finite quotient of $\Gamma_v$ is a $p'$-group. We state it by first fixing $l$ because we need this condition to be uniform on the vertices. If however the graph is finite, then the uniformity is automatically satisfied.
\end{remark}
\begin{proof}
As in the proof of Proposition \ref{prop:pifree_gog1}, we start with $\hat{\varphi} : F_S \to \operatorname{GL}_n(\mathfrak{o})$ of small defect, use stability of the vertex groups to modify it at the vertex generators, then apply Lemma \ref{lem:vpfree_gog} to obtain a homomorphism $\hat{\psi} : F_S \to \operatorname{GL}_n(\mathfrak{o})$ close to $\hat{\varphi}$ that descends to a homomorphism of $\Gamma$.
\end{proof}
Lemma \ref{lem:vpfree_gog} shows that the estimate for stability is linear in terms of the uniform estimate for the vertex groups.
\begin{proposition}
\label{prop:vpfree_gog2}
Let $\Gamma$ be the fundamental group of a graph of groups such that there exists $l \geq 1$ such that for every vertex $v$ the image of $\Gamma_v$ in any finite quotient of $\Gamma$ has no subgroup of order $p^l$. Then $\Gamma$ is uniformly $\mathcal{G}$-stable.
\end{proposition}
\begin{proof}
As in the proof of Proposition \ref{prop:pifree_gog2}, we start with $\hat{\varphi} : F_S \to \operatorname{GL}_n(\mathfrak{o})$ of small defect, apply Lemma \ref{lem:vpfree} to modify it at the vertex groups so that it descends to homomorphisms with finite image on each vertex group, and finally apply Lemma \ref{lem:vpfree_gog} to obtain a homomorphism $\psi : F_S \to \operatorname{GL}_n(\mathfrak{o})$ close to $\hat{\varphi}$ that descends to $\Gamma$.
\end{proof}
Here the proof shows that the estimate for stability is linear. More precisely, if $l$ is as in the statement of the proposition, and $\varphi : \Gamma \to \operatorname{GL}_n(\mathfrak{o})$ satisfies $\operatorname{def}(\varphi) \leq r^{2al} = p^{-2l}$, then there exists a homomorphism $\psi : \Gamma \to \operatorname{GL}_n(\mathfrak{o})$ such that $\operatorname{dist}(\varphi, \psi) \leq p^l \operatorname{def}(\varphi)$.
\subsection{Corollaries}
\label{ss:char0:cor}
We now apply Propositions \ref{prop:vpfree_gog1} and \ref{prop:vpfree_gog2} to obtain some examples of uniformly $\operatorname{GL}(\mathfrak{o})$-stable groups, which strengthen those in Subsections \ref{ss:vpropi:cor} and \ref{ss:GBS}.
\begin{corollary}
\label{cor:vpfree_gog}
The following groups are uniformly $\operatorname{GL}(\mathfrak{o})$-stable:
\begin{enumerate}
\item Fundamental groups of finite, connected graphs of groups, with virtually $p$-free vertex groups.
\item Fundamental groups of finite, connected graphs of groups, with uniformly $\operatorname{GL}(\mathfrak{o})$-stable vertex groups and virtually $p$-free edge groups.
\end{enumerate}
\end{corollary}
\begin{proof}
This is a direct consequence of Propositions \ref{prop:vpfree_gog1} and \ref{prop:vpfree_gog2}. We restrict to finite graphs of groups in order to have the integer $l$ in the statements be uniform.
\end{proof}
A special case of Item $2.$ is when edge groups are finite. Such fundamental groups are precisely the finitely presented groups with infinitely many ends, by Stalling's Theorem \cite{Stallings1, Stallings2}.
\begin{corollary}
\label{cor:vfree}
Finitely generated virtually free groups are $\operatorname{GL}(\mathfrak{o})$-stable.
\end{corollary}
\begin{proof}
This follows again from Proposition \ref{prop:vpfree_gog1} and \cite{cd1} or \cite{Stallings1, Stallings2}: such groups are fundamental groups of finite connected graphs of groups, with finite vertex groups. The statement does not specify the type of stability because such groups are finitely presented.
\end{proof}
As for GBS groups, we have a criterion for stability much simpler than that of Corollary \ref{cor:GBS_p}:
\begin{corollary}
\label{cor:GBS}
Let $\Gamma$ be a GBS group corresponding to the graph $(X, w)$. Suppose that there exist a cycle $C$ in $X$ satisfying $\nu_p(w_-(C)) \neq \nu_p(w_+(C))$. Then $\Gamma$ is uniformly $\operatorname{GL}(\mathfrak{o})$-stable.
\end{corollary}
\begin{proof}
The proof is similar to Corollary \ref{cor:GBS_p}. Given a vertex $x$ denote by $s_x$ the corresponding generator. By Proposition \ref{prop:vpfree_gog2} we need to show that for every vertex $x$ the image of $s_x$ in any finite quotient of $\Gamma$ has order divisible by at most a uniformly bounded power of $p$. For the sake of brevity, let us say that such a vertex is \textit{virtually $p$-free}. \\
We only need to show this for a single vertex: we claim that if some $x$ is virtually $p$-free, then all vertices are. Since the graph $X$ is finite and connected, it suffices to show that if $x \xrightarrow{e} y$ is an edge in $e$ and $x$ is virtually $p$-free, then so is $y$. Let $(w_-(e), w_+(e)) = (m, n)$ be the weights of $e$. Then $s_x^m$ is conjugate to $s_y^n$, so as in Lemma \ref{lem:GBS_pfree}, if $o_x, o_y$ are the orders of $s_x, s_y$ in a finite quotient of $\Gamma$, we have $o_x/(o_x, m) = o_y/(o_y, n)$. Thus
$$\nu_p(o_y) = \nu_p(o_x) - \nu_p(o_x, m) + \nu_p(o_y, n) \leq \nu_p(o_x) + \nu_p(n)$$
is uniformly bounded, and $y$ is virtually $p$-free. \\
We are left to show that a vertex $x$ lying on a cycle $C$ satisfying $\nu_p(w_-(C)) \neq \nu_p(w_+(C))$ is virtually $p$-free. Let $m := w_-(C), n := w_+(C)$. By Lemma \ref{lem:GBS_conjP} we know that $s_x^m$ is conjugate to $s_x^n$. Now let $o$ be the order of $s_x$ in a finite quotient of $\Gamma$. As in Lemma \ref{lem:GBS_omn} we have $(o, m) = (o, n)$, and so
$$\min \{ \nu_p(o), \nu_p(n) \} = \nu_p(o, n) = \nu_p(o, m) = \min \{\nu_p(o), \nu_p(m)\}.$$
Since $\nu_p(m) \neq \nu_p(n)$, this is only possible if $\nu_p(o) \leq \min\{\nu_p(m), \nu_p(n)\}$, which gives a uniform bound on $\nu_p(o)$ and concludes the proof.
\end{proof}
We similarly obtain corollaries about the special case of Baumslag--Solitar groups and their largest residually finite quotient:
\begin{corollary}
\label{cor:BS}
Suppose that $\nu_p(m) \neq \nu_p(n)$. Then $\operatorname{BS}(m, n)$ is uniformly $\operatorname{GL}(\mathfrak{o})$-stable.
\end{corollary}
\begin{corollary}
\label{cor:rfBS}
Suppose moreover that $|m|, |n|$ are distinct from each other and from $1$ and let $d := (m, n)$ be the greatest common divisor of $m, n$. Then the group
$$\Gamma = \langle a, b_i : i \in \mathbb{Z} \mid [b_i^d, b_j] = 1, b_i^m = b_{i+1}^n, ab_ia^{-1} = b_{i+1} : i \in \mathbb{Z} \rangle$$
is finitely generated, infinitely presented, and pointwise $\operatorname{GL}(\mathfrak{o})$-stable.
\end{corollary}
Note that the special case $d = 1$, which admits the nicer description $\mathbb{Z} \left[ \frac{1}{mn} \right] \rtimes_{\frac{m}{n}} \mathbb{Z}$, is not more general than Corollary \ref{cor:rfBS_p}: if $\nu_p(m) \neq \nu_p(n)$ and $(m, n) = 1$, then $p$ must divide exactly one of $m, n$. \\
Also in these examples -- except for Item $2.$ of Corollary \ref{cor:vpfree_gog} which depends on the stability estimates of the vertex groups -- we obtain a linear estimate for stability.
\subsection{Positive characteristic}
\label{ss:poschar}
Let $\mathbb{K}$ have characteristic $p > 0$ for the rest of this subsection. The proofs in this section until now all relied on the cohomological Lemma \ref{lem:cohopk}. This cannot have as strong of an analogue in characteristic $p$. For instance if $M$ is an $\mathbb{F}_p$-module, seen as a trivial $\mathbb{F}_p[\mathbb{Z}/p\mathbb{Z}]$-module, then $\operatorname{H}^n(\mathbb{Z}/p\mathbb{Z}, M) \cong M$ for all $n \geq 1$. Therefore, analogues of the stability results we proved so far need a different approach. \\
In Sections \ref{s:vpropi} and \ref{s:char0} many results reduced to statements about finite groups. So we restrict our attention to those, and ask:
\begin{question}
\label{q:fin}
Are all finite groups $\operatorname{GL}(\mathfrak{o})$-stable?
\end{question}
We already know that finite groups without elements of order $p$ are $\operatorname{GL}(\mathfrak{o})$-stable, by Proposition \ref{prop:pifree}. So the next natural question is whether finite $p$-groups are $\operatorname{GL}(\mathfrak{o})$-stable. Using a Theorem of Gasch\"utz \cite[Theorem 7.43]{Rotman} and an argument similar to the one of Lemma \ref{lem:vpfree}, one can show that if $\Gamma$ is a finite group such that all $p$-Sylow subgroups are $\operatorname{GL}(\mathfrak{o})$-stable with a subquadratic estimate, then $\Gamma$ is stable. The hypothesis of the subquadratic estimate is needed in order to have the kernel $G_k/G_{2k}$ of the corresponding lifting problem be abelian: an example due to Zassenhaus implies that this hypothesis is necessary in Gasch\"utz's Theorem \cite[Postscriptum]{Gaschutz}.
However, stability with a subquadratic estimate for finite $p$-groups is too strong of a requirement to have useful applications. Indeed, the next example shows that the estimate for $\mathbb{Z}/p^k\mathbb{Z}$ is, at best, polynomial of degree $p^k$.
\begin{example}
\label{ex:zp:est}
Let $A \in \operatorname{M}_n(\mathfrak{o})$ be such that $\| A \| \leq \varepsilon$. Then, using that $\mathbb{K}$ has characteristic $p$:
$$\| (I + A)^{p^k} - I \| = \| A^{p^k} \| \leq \| A \|^{p^k} \leq \varepsilon^{p^k}.$$
On the other hand, if we chose $A$ so that $\| A^{p^k} \| = \varepsilon^{p^k}$, then $(I + A)$ is $\varepsilon$-far from any matrix $(I + A')$ satisfying $(I + A')^{p^k} = I$.
\end{example}
Notice that for all groups whose stability was proven so far, the estimates were always linear. Thus Example \ref{ex:zp:est} shows that Question \ref{q:fin} is quite different from the stability problems we encountered until now. \\
Here is a special case that admits a positive answer:
\begin{proposition}
\label{prop:z2z}
Let $\mathbb{K}$ have characteristic $2$. Then $\mathbb{Z}/2\mathbb{Z}$ is $\operatorname{GL}(\mathfrak{o})$-stable.
\end{proposition}
The proof will show that the estimate is quadratic: if $\| A^2 - I \| \leq \varepsilon^2$, then there exists $A'$ such that $(A')^2 = I$ and $\| A - A' \| \leq \varepsilon$. This estimate is sharp by Example \ref{ex:zp:est}. \\
Combining Proposition \ref{prop:z2z} with Proposition \ref{prop:pifree_gog1}, we obtain more examples of $\operatorname{GL}(\mathfrak{o})$-stable groups that are not covered by the results in Section \ref{s:vpropi}:
\begin{example}
The infinite dihedral group $D_\infty \cong \mathbb{Z}/2\mathbb{Z} * \mathbb{Z}/2\mathbb{Z}$ is $\operatorname{GL}(\mathfrak{o})$-stable; more generally, free Coxeter groups are $\operatorname{GL}(\mathfrak{o})$-stable. The modular group $\operatorname{PSL}_2(\mathbb{Z}) \cong \mathbb{Z}/2\mathbb{Z} * \mathbb{Z}/3\mathbb{Z}$ is $\operatorname{GL}(\mathfrak{o})$-stable.
\end{example}
The proof of Proposition \ref{prop:z2z} relies on the following theorem \cite[Theorem 3.1]{Braw2}, which classifies similarity classes of involutory matrices over the quotient rings $\mathfrak{o} / \mathfrak{p}^k$:
\begin{theorem}[Brawley--Gamble]
\label{thm:BG}
Let $R$ be a finite commutative local ring of characteristic a power of $2$ and maximal ideal $m$. Let $A \in \operatorname{M}_n(R)$ be such that $A^2 = I$. Then there exists $P \in \operatorname{GL}_n(R), 0 \leq l \leq n/2$ and $B \in I_{n-2l} + \operatorname{M}_{n-2l}(m)$ such that
$$PAP^{-1} = \begin{pmatrix}
0 & I_l & \\
I_l & 0 & \\
& & B
\end{pmatrix}.$$
\end{theorem}
This has the following implication in our setting:
\begin{corollary}
\label{cor:BG}
Let $A \in \operatorname{GL}_n(\mathfrak{o})$ be such that $\| A^2 - I \| \leq r^k < 1$. Then there exists $P \in \operatorname{GL}_n(\mathfrak{o})$ and $B \in I_{n-2l} + \operatorname{M}_{n-2l}(\mathfrak{p})$ such that
$$PAP^{-1} \equiv \begin{pmatrix}
0 & I_l & \\
I_l & 0 & \\
& & B
\end{pmatrix} \mod \mathfrak{p}^k.$$
\end{corollary}
\begin{proof}
Apply Theorem \ref{thm:BG} to the reduction of $A$ modulo $\mathfrak{p}^k$, which is involutory over $\mathfrak{o}/\mathfrak{p}^k$; then lift the matrices $P$ and $B$ to elements of $\operatorname{M}_n(\mathfrak{o})$. Since $k \geq 1$, the lift to $\operatorname{GL}_n(\mathfrak{o})$ of any matrix in $\operatorname{GL}_n(\mathfrak{o}/\mathfrak{p}^k)$ is invertible.
\end{proof}
We proceed with the proof. This will use repeatedly the identity $(I + M)^2 = I + M^2$, which holds since $\mathbb{K}$ has characteristic $2$.
\begin{proof}[Proof of Proposition \ref{prop:z2z}]
Let $\varepsilon > 0$. We will show that if $A \in \operatorname{M}_n(\mathfrak{o})$ is such that $\| A^2 - I \| \leq \varepsilon^2$, then there exists $A' \in \operatorname{M}_n(\mathfrak{o})$ such that $\| A - A' \| \leq \varepsilon$. Since $\| \cdot \|$ takes values in $r^\mathbb{Z}$, we may assume that $\varepsilon =: r^a$, and since the statement is trivial when $\varepsilon \geq 1$, we may assume $a \geq 1$. The proof is by induction on $n$. For $n = 1$, we compute $\varepsilon^2 \geq |a^2 - 1| = |(a - 1)^2| = |a - 1|^2$, and so $|a - 1| \leq \varepsilon$. So let $n > 1$ and suppose by induction that the statement holds up to $(n - 1)$ for all $\varepsilon > 0$. \\
Since $\| A^2 - I \| \leq r^{2a}$, by Corollary \ref{cor:BG} there exist $P \in \operatorname{GL}_n(\mathfrak{o}), 0 \leq l \leq n/2$ and $B \in I_{n-2l} + \operatorname{M}_{n-2l}(\mathfrak{p})$ such that
$$
PAP^{-1} \equiv \begin{pmatrix}
0 & I_l & 0 \\
I_l & 0 & 0 \\
0 & 0 & B
\end{pmatrix} \mod \mathfrak{p}^{2a}.
$$
Now $\| B^2 - I_{n-2l} \| \leq \max \{ r^{2a}, \| P A^2 P^{-1} - I_n \| \} \leq \varepsilon^2$. If $l \geq 1$, it follows by induction that there exists $B' \in \operatorname{M}_{n-2l}(\mathfrak{o})$ such that $(B')^2 = I_{n - 2l}$ and $\| B - B' \| \leq \varepsilon$. Then
$$
A' := P^{-1} \begin{pmatrix}
0 & I_l & 0 \\
I_l & 0 & 0 \\
0 & 0 & B'
\end{pmatrix} P
$$
is the desired matrix. So in case $l \geq 1$, the statement is true for all $\varepsilon > 0$ as well. \\
Assume finally that $l = 0$, and so $A = I + \overline{\omega}^k M$ for some $M \in \operatorname{M}_n(\mathfrak{o})$, where we take $k \geq 1$ to be maximal. If $r^k \leq \varepsilon$ we may take $A' = I$ and conclude. Otherwise:
$$\varepsilon^2 \geq \| A^2 - I \| = \| I + \overline{\omega}^{2k} M^2 - I \| = r^{2k} \| M^2 \|;$$
so $\| (I + M)^2 - I \| = \| M^2 \| \leq (r^{-k} \varepsilon)^2 < 1$. The choice of $k$ implies that $(I + M)$ is not congruent to the identity modulo $\mathfrak{p}$, so $(I + M)$ falls in the previous case. Therefore there exists $M' \in \operatorname{M}_n(\mathfrak{o})$ such that $(I + M')^2 = I$ and $\| (I + M) - (I + M') \| \leq r^{-k} \varepsilon$. Then $A' := I + \overline{\omega}^k M'$ satisfies $(A')^2 = I$ and
$$\| A - A' \| = \| \overline{\omega}^k (M - M') \| = r^k \| (I + M) - (I + M') \| \leq r^k \cdot r^{-k} \varepsilon = \varepsilon.$$
\end{proof}
The fundamental tool for the proof of Proposition \ref{prop:z2z} was Theorem \ref{thm:BG}, which provides simple representatives for each conjugacy class of representations $\mathbb{Z}/2\mathbb{Z} \to \operatorname{GL}_n(\mathfrak{o}/\mathfrak{p}^k)$. A similar result for other finite $p$-groups could similarly be used to prove stability. However the group $\mathbb{Z}/2\mathbb{Z}$ is really special from this point of view. Indeed, it is shown in \cite{wild} that for any other finite $p$-group $\Gamma$ and any commutative local ring $R$ of characteristic a power of $p$, the analogous problem for representations $\Gamma \to \operatorname{GL}_n(R)$ is \textit{computationally wild}. More precisely, this problem contains the problem of simultaneous similarity classes of pairs of matrices over the residue field $R/m$, which is a long-standing open problem in its general form (see e.g. \cite{similarity}). Therefore, although the few known special cases could be used to prove further stability results, the general solution to Question \ref{q:fin} needs a different approach.
\pagebreak
\section{Stability via bounded cohomology}
\label{s:BC}
The goal of this section is to prove the following bounded cohomological criterion for uniform $\operatorname{GL}(\mathfrak{o})$-stability, where $\mathfrak{o}$ is the ring of integers of a non-Archimedean local field $\mathbb{K}$:
\begin{theorem}
\label{thm:BC}
Let $\Gamma$ be finitely presented and suppose that $\operatorname{H}^2_b(\Gamma, E) = 0$ for every Banach $\mathbb{K}[\Gamma]$-module $E$ with a solid norm. Then $\Gamma$ is $\operatorname{GL}(\mathfrak{o})$-stable.
\end{theorem}
Note that by Theorem \ref{thm:pw_un} we do not need to specify whether the stability is pointwise or uniform. The approach follows that in \cite{GLT}, and the reason we restrict to finitely presented groups is the same: the ultraproduct techniques work best for pointwise stability, but the quantitative approach needs a quantity that controls all local defects, and this is only possible for finitely presented groups. The ultrametric inequality gives \textit{bounded} cocycles, and so the bounded cohomological approach is the more natural one to take in this case. By \cite[Corollary 8.7]{mio} (see Proposition \ref{prop:comp}), cohomology vanishing is, a priori, stronger than bounded cohomology vanishing, and this will imply the cohomological analogue of Theorem \ref{thm:BC} (Corollary \ref{cor:BC}). \\
The boundedness of the cocycles is a consequence of the fact that pointwise asymptotic homomorphisms are asymptotically close to uniform ones for finitely presented groups, by Item $2.$ of Proposition \ref{prop:ultra_asy}. So the reader may suspect that a bounded cohomological criterion for uniform stability should also hold in the Archimedean setting. However, in that case the situation is more delicate and requires the introduction of a different cohomology theory, called \textit{asymptotic cohomology} \cite{Bharat}. The advantage of our setting is that, while cocycles are bounded thanks to the uniform nature of the problem, we can still use the same ultraproduct techniques that apply to the pointwise setting of \cite{GLT}. \\
We will use Theorem \ref{thm:BC} to deduce the stability results for virtually free groups we obtained in Sections \ref{s:vpropi} and \ref{s:char0}. However we are not able to produce examples other than these, and we conjecture that in fact this method cannot produce other examples (Conjecture \ref{conj}). We end by discussing how a stronger criterion could potentially produce more examples, and justify why this seems to be a hard problem. \\
For the rest of this section, we fix a non-Archimedean local field $\mathbb{K}$ with ring of integers $\mathfrak{o}$, uniformizer $\overline{\omega}$, maximal ideal $\mathfrak{p} = \overline{\omega} \mathfrak{o}$, residue field $\mathfrak{k}$, and value group $r^{\mathbb{Z}} = |\mathbb{K}^\times|$, where $r = |\overline{\omega}| \in (0, 1)$.
\subsection{Bounded cohomology}
We review here the basics of bounded cohomology of discrete groups that are needed for the rest of the section. For more information, see \cite{monod, Frig} for bounded cohomology over the reals, and \cite{mio} for bounded cohomology over non-Archimedean fields. All of the material presented here is also contained in \cite{mio}. \\
We will work with the bar resolution throughout, since it is the easiest one to treat lifting problems with. Let $\Gamma$ be a group, $E$ a $\mathbb{K}[\Gamma]$-module, without any specified norm. Let
$$\operatorname{C}^n(\Gamma, E) := \{ f : \Gamma^n \to E \},$$
which is a $\mathbb{K}$-vector space with pointwise addition and scalar multiplication. Define the coboundary map $\delta^n : \operatorname{C}^n(\Gamma, E) \to \operatorname{C}^{n+1}(\Gamma, E)$ by the formula:
$$\delta^n(f)(g_1, \ldots, g_{n+1}) := g_1 \cdot f(g_2, \ldots, g_{n+1}) + $$
$$ + \left( \sum\limits_{i = 1}^n (-1)^i f(g_1, \ldots, g_i g_{i+1}, \ldots, g_{n+1}) \right) + (-1)^{n+1} f(g_1, \ldots, g_n).$$
This defines a cochain complex of $\mathbb{K}$-vector spaces $(\operatorname{C}^\bullet(\Gamma, E), \delta^\bullet)$: we denote the cocycles by $\operatorname{Z}^\bullet(\Gamma, E)$, the coboundaries by $\operatorname{B}^\bullet(\Gamma, E)$, and the \textit{cohomology} by $\operatorname{H}^\bullet(\Gamma, E)$. \\
Now suppose that $E$ is a normed $\mathbb{K}[\Gamma]$-module. Let
$$\operatorname{C}^n_b(\Gamma, E) := \{ f : \Gamma \to E : \| f \|_\infty < \infty \} \subset \operatorname{C}^n(\Gamma, E),$$
which is a normed $\mathbb{K}$-vector space with the supremum norm $\| \cdot \|_\infty$, and even a Banach space if $E$ is also Banach. With the same coboundary map, we obtain the cochain complex $(\operatorname{C}^\bullet_b(\Gamma, E), \delta^\bullet)$, the bounded cocycles $\operatorname{Z}^\bullet_b(\Gamma, E)$, the bounded coboundaries $\operatorname{B}^\bullet_b(\Gamma, E)$, and the \textit{bounded cohomology} $\operatorname{H}^\bullet_b(\Gamma, E)$. \\
The inclusion $\operatorname{C}^n_b(\Gamma, E) \to \operatorname{C}^n(\Gamma, E)$ is a chain map, that induces the \textit{comparison map}
$$c^n : \operatorname{H}^n_b(\Gamma, E) \to \operatorname{H}^n(\Gamma, E).$$
The kernel of this map, called \textit{exact bounded cohomology}, is denoted by $\operatorname{EH}^n_b(\Gamma, E)$. In the real case it is very rich and interesting, even in the simple case where $n = 2$ and $E$ is the trivial $\Gamma$-module $\mathbb{R}$, leading to the theory of \textit{quasimorphisms}. However, in the non-Archimedean case, the exact bounded cohomology in degree $2$ is trivial for finitely generated groups \cite[Corollary 8.7]{mio}:
\begin{proposition}
\label{prop:comp}
Let $\Gamma$ be a finitely generated group, $E$ a normed $\mathbb{K}[\Gamma]$-module. Then the comparison map $c^2 : \operatorname{H}^2_b(\Gamma, E) \to \operatorname{H}^2(\Gamma, E)$ is injective.
\end{proposition}
Theorem \ref{thm:BC} applies to groups such that $\operatorname{H}^2_b(\Gamma, E) = 0$ for every Banach $\mathbb{K}[\Gamma]$-module $E$ with a solid norm. The next lemma shows that this vanishing is in some sense uniform.
\begin{lemma}
\label{lem:UBC}
Let $n \geq 1$ and suppose that $\operatorname{H}^n_b(\Gamma, E) = 0$ for any Banach $\mathbb{K}[\Gamma]$-module $E$ with a solid norm. Then there exists $C \geq 1$ such that for any such $E$ and any $z \in \operatorname{Z}^n_b(\Gamma, E)$ there exists a primitive $b \in \operatorname{C}^{n-1}_b(\Gamma, E)$ such that $\|b\|_\infty \leq C \| z \|_\infty$.
\end{lemma}
\begin{proof}
We start by showing that for every Banach $\mathbb{K}[\Gamma]$-module $E$ with a solid norm we can choose a constant $C = C(E)$ that works for $\operatorname{Z}^n_b(\Gamma, E)$ (this is the same proof as in the real case \cite{MM}). The map $\delta^{n-1} : \operatorname{C}^{n-1}_b(\Gamma, E) \to\operatorname{C}^n_b(\Gamma, E)$ is a bounded linear map between Banach spaces. Since $\operatorname{H}^n_b(\Gamma, E) = 0$, it is surjective onto $\operatorname{Z}^n_b(\Gamma, E)$, which is closed in $\operatorname{C}^n_b(\Gamma, E)$ and thus Banach. It follows from the Open Mapping Theorem (Theorem \ref{thm:open}) that $\delta^{n-1}$ is open and so the induced isomorphism $\delta^{n-1} : \operatorname{C}^{n-1}_b(\Gamma, E)/\operatorname{Z}^{n-1}_b(\Gamma, E) \to \operatorname{Z}^n_b(\Gamma, E)$ is bi-Lipschitz, where the left-hand side is endowed with the quotient norm. The Lipschitz constants give the desired result. \\
Next, we need to show that the constant $C(E)$ can be chosen independently of $E$. First, note that it is only necessary to uniformly bound $C(E_i)$, where $\{ E_i \}_{i \in I}$ is the set of Banach $\mathbb{K}[\Gamma]$-modules with a solid norm of cardinality at most $2^{\aleph_0}$. Indeed, any $z \in \operatorname{Z}^n_b(\Gamma, E)$ takes values in some $E_i$, more precisely the smallest $\Gamma$-invariant closed subspace of $E$ containing the countable set $z(\Gamma^n)$. Now take their Banach direct sum, which is the completion of the direct sum $\tilde{E} = \bigoplus_{i \in I} E_i$ with respect to the norm $\| (x_i)_{i \in I} \| := \max_{i \in I} \| x_i \|_{E_i}$ (this is allowed because we have taken a \textit{set} of modules, and not the proper class of all modules). This is a Banach $\mathbb{K}[\Gamma]$-module, the norm is still solid, and we have $C(\tilde{E}) \geq C(E_i)$ for all $i \in I$, which gives the desired uniform bound.
\end{proof}
\subsection{Reducing the defect}
Lemma \ref{lem:stab_up} rephrases stability in terms of a lifting property, which is what allows to use cohomology to prove stability. However the kernel in this lifting problem is not tractable with cohomology: it is not even abelian. The goal of this subsection is to show that a weaker quantitative statement (intuitively: every asymptotic homomorphism is close to one with smaller defect) can be related to a simpler lifting problem, where the kernel is more approachable and will be analyzed in the next subsection. \\
Let us fix some terminology and notation concerning ultrafilters. Fix a free ultrafilter $\omega$. We say that an event $E_n$ holds for \textit{most} $n$ if it holds for a set of $n$ inside $\omega$. Accordingly we denote $\varepsilon_n \neq_\omega 0, \delta_n \leq_\omega \varepsilon_n$, and so on. Given a sequence $\varepsilon_n \neq_\omega 0$, we write $\delta_n = O_\omega(\varepsilon_n)$ if there exists $C \geq 1$ such that $\delta_n \leq_\omega C \varepsilon_n$. The minimal such $C$ can be characterized as $\lim\limits_{n \to \omega} \delta_n/\varepsilon_n$: this limit makes sense since $\varepsilon_n \neq_\omega 0$. If this limit is $0$, we write $\delta_n = o_\omega(\varepsilon_n)$. \\
Let $G_n := \operatorname{GL}_{k_n}(\mathfrak{o})$, and fix a sequence $(\hat{\varphi}_n : F_S \to G_n)_{n \geq 1}$ with $\varepsilon_n := \operatorname{def}(\hat{\varphi}_n) \xrightarrow{n \to \omega} 0$. Since the metric on $G_n$ takes values on $r^{\mathbb{Z}}$, the same holds for the sequence $\varepsilon_n$. Moreover we may assume that $\varepsilon_n \neq_\omega 0$, since otherwise $\operatorname{def}(\varphi_n)$ is already close to a homomorphism. The asymptotic homomorphism $(\hat{\varphi}_n)_{n \geq 1}$ induces a homomorphism onto the ultraproduct (see Lemma \ref{lem:stab_up}). Here we will use a modified ultraproduct that takes into account the sequence $\varepsilon_n$ as well. In analogy with notation which will shortly be introduced, we denote $G(0) := \prod\limits_{n \geq 1} G_n$.
For a sequence $\delta_n \xrightarrow{n \to \omega} 0$, denote
$$N(\delta_n) := \{ (A_n)_{n \geq 1} \in G(0) : \| A_n - I_{k_n} \| \leq_\omega \delta_n \},$$
and similarly $N(O_\omega(\delta_n))$ and $N(o_\omega(\delta_n))$.
\begin{lemma}
$N(\delta_n), N(O_\omega(\delta_n))$ and $N(o_\omega(\delta_n))$ are normal subgroups of $G(0)$.
\end{lemma}
The fact that $N(\delta_n)$ is a subgroup relies strongly on the ultrametric inequality, and will allow to streamline a few arguments, and to make them more quantitatively precise, which allows to use bounded cohomology instead of cohomology. By contrast, in the Archimedean case \cite{GLT} one needs to work with $N(O_\omega(\varepsilon_n))$.
\begin{proof}
We prove the statement for $N(\delta_n)$ (which is the only one specific to the ultrametric case), the rest is similar. Suppose that $(A_n)_{n \geq 1}, (B_n)_{n \geq 1} \in N(\delta_n)$. Then $\| A_n B_n - I_{k_n} \| \leq \max \{ \| A_n B_n - B_n \|, \| B_n - I_{k_n} \| \} \leq_\omega \delta_n$, where we have used that $\| \cdot \|$ is right-invariant. If now $(C_n)_{n \geq 1} \in G(0)$, then $\| C_n A_n C_n^{-1} - I_{k_n} \| = \| A_n - I_{k_n} \| \leq_\omega \delta_n$.
\end{proof}
We will denote $G(\delta_n) := G(0)/N(\delta_n)$. Given a constant $C \geq 1$, we have $\varepsilon_n \leq_\omega C \varepsilon_n$, so $N(o_\omega(\varepsilon_n)) \leq N(\varepsilon_n) \leq N(C \varepsilon_n)$, and so there are quotient maps $G(o_\omega(\varepsilon_n)) \to G(\varepsilon_n) \to G(C \varepsilon_n)$. The asymptotic homomorphism $(\hat{\varphi}_n)_{n \geq 1}$ induces homomorphisms $\varphi(\varepsilon_n) : \Gamma \to G(\varepsilon_n)$, as well as homomorphisms $\varphi(C \varepsilon_n) : \Gamma \to G(C \varepsilon_n)$. This yields to the following lifting problem similar to the one from Lemma \ref{lem:cohopk}:
\[\begin{tikzcd}
& {G(o_\omega(\varepsilon_n))} \\
\Gamma & {G(\varepsilon_n)} \\
& {G(C\varepsilon_n)}
\arrow["\psi", dashed, from=2-1, to=1-2]
\arrow["{\varphi(C \varepsilon_n)}"', from=2-1, to=3-2]
\arrow["{\varphi(\varepsilon_n)}", from=2-1, to=2-2]
\arrow[from=2-2, to=3-2]
\arrow[from=1-2, to=3-2, bend left = 50]
\end{tikzcd}\]
\begin{lemma}
\label{lem:red_lift}
The existence of a solution $\psi$ to the above lifting problem is equivalent to the existence of a sequence $(\hat{\psi}_n : F_S \to G_n)_{n \geq 1}$ such that $\operatorname{def}(\hat{\psi}_n) = o_\omega(\varepsilon_n)$ and $\operatorname{dist}(\hat{\varphi}_n, \hat{\psi}_n) \leq_\omega C \varepsilon_n$.
\end{lemma}
\begin{proof}
The proof is the same as that of \cite[Theorem 4.2]{a:comm} (see Lemma \ref{lem:stab_up}).
\end{proof}
We denote by $E_C := N(C \varepsilon_n) / N(o_\omega(\varepsilon_n)) = \ker \{ G(o_\omega(\varepsilon_n)) \to G(C \varepsilon_n) \}$. As in Subsection \ref{ss:preli:split}, this lifting problem reduces to a splitting problem
$$1 \to E_C \to \left( G(o_\omega(\varepsilon_n)) \times_{\varphi(C \varepsilon_n)} \Gamma \right) \to \Gamma \to 1$$
and in turn, if $E_C$ is abelian, to a cohomology vanishing problem with coefficients in $E_C$. Moreover, as in Lemma \ref{lem:cohopk}, the particular form of this lifting problem implies that the relevant cocycle takes values in $N_1 = N(\varepsilon_n) / N(o_\omega(\varepsilon_n)) = \ker \{ G(o_\omega(\varepsilon_n)) \to G(\varepsilon_n) \}$.
\subsection{Additional structures on the kernel}
In this subsection we show not only that $E_C$ is abelian, which allows to apply cohomology to the above lifting problem, but moreover that it is the closed $C$-ball of a Banach $\mathbb{K}[\Gamma]$-module. This Banach $\mathbb{K}[\Gamma]$-module will be $E := N(O_\omega(\varepsilon_n)) / N(o_\omega(\varepsilon_n))$. We can characterize $N(O_\omega(\varepsilon_n))$ as
$$N(O_\omega(\varepsilon_n)) := \bigcup\limits_{C \geq 1} N(C \varepsilon_n) = \{ (A_n)_{n \geq 1} \in G(0) : \lim\limits_{n \to \omega} \frac{\| A_n - I_{k_n} \|}{\varepsilon_n} < \infty \}.$$
Here we are using that $\varepsilon_n \neq_\omega 0$, and that a bounded sequence always admits an ultralimit. We will denote by $A = (A_n)_{n \geq 1}$ elements of $G(0)$, and if $A \in N(O_\omega(\varepsilon_n))$ we will denote by $[A]$ its image in $E$. We also denote $I := (I_{k_n})_{n \geq 1}$. \\
We start by showing that $E$ is a $\mathbb{K}$-vector space. It already has a well-defined product structure, but to underline that it is abelian we will denote it by $[A] + [B] := [AB]$. The scalar multiplication will be given by a stretch fixing the identity, namely $\lambda [A] = [\lambda A + (1 - \lambda) I] = [I + \lambda(A - I)]$.
\begin{lemma}
With these operations, $E$ is a $\mathbb{K}$-vector space.
\end{lemma}
\begin{proof}
We already know that $(E, +)$ is a group. It is moreover abelian: for this we need to show that given $A, B \in N(O_\omega(\varepsilon_n))$ we have $ABA^{-1}B^{-1} \in N(o_\omega(\varepsilon_n))$. This follows from the submultiplicativity of the norm $\| \cdot \|$ (see Lemma \ref{lem:GL}):
$$\|A_n B_n A_n^{-1} B_n^{-1} - I_{k_n}\| = \|A_n B_n - B_n A_n\| = \|(A_n - I_{k_n})(B_n - I_{k_n}) - (B_n - I_{k_n})(A_n - I_{k_n})\| \leq $$
$$ \leq \|A_n - I_{k_n}\| \cdot \|B_n - I_{k_n}\| = O_\omega(\varepsilon_n^2) = o_\omega(\varepsilon_n).$$
The scalar multiplication is well-defined: indeed
$$\lambda A_n + (1 - \lambda) I_{k_n} = I_{k_n} + \lambda(A_n - I_{k_n}),$$
is close to the identity, thus invertible, when $\|A_n - I_{k_n}\|$ is small enough. It is also easy to see that it is still in $N(O_\omega(\varepsilon_n))$, and that $(\lambda \mu) [A] = \lambda (\mu [A])$. Finally we prove bilinearity of the scalar multiplication. First, given $\lambda, \mu \in \mathbb{K}, [A] \in N(O_\omega(\varepsilon_n))$, we have:
$$\lambda \cdot [A] + \mu \cdot [A] = [I + \lambda(A - I)] + [I + \mu(A - I)] = $$
$$ = [I + (\lambda + \mu)(A - I) + \lambda \mu (A - I)^2] = (\lambda + \mu)[A],$$
where we used that $\| (A_n - I_n)^2 \| = o_\omega(\varepsilon_n)$. Similarly, given $\lambda \in \mathbb{K}, [A], [B] \in N(O_\omega(\varepsilon_n))$, we have:
$$\lambda \cdot [A] + \lambda \cdot [B] = [\lambda(A - I) + I] + [\lambda(B - I) + I] = $$
$$ = [\lambda^2 (A - I)(B - I) + \lambda (A - I) + \lambda(B - I) + I] = [\lambda(A + B - 2I) + I] = $$
$$ = [-\lambda(A - I)(B - I) + \lambda(AB - I) + I] = [\lambda(AB - I) + I] = \lambda \cdot [AB]$$
\end{proof}
Now since $E$ is abelian, the splitting problem after Lemma \ref{lem:red_lift} gives an action of $\Gamma$ on $E$ by conjugacy. Concretely, this action is defined on the free group by $w * [A] = [(\hat{\varphi}_n(w) A_n \hat{\varphi}_n(w)^{-1})_{n \geq 1}]$, and it descends to $\Gamma$ since $\operatorname{def}(\hat{\varphi}_n) \leq_\omega \varepsilon_n$. \\
Next we introduce a norm on $E$, namely $\| [A] \|_E := \lim\limits_{n \to \omega} \frac{\| A_n - I_{k_n} \|}{\varepsilon_n}$.
\begin{lemma}
$\| \cdot \|_E$ is a solid Banach norm on $E$, and the action of $\Gamma$ is isometric.
\end{lemma}
\begin{proof}
$\| \cdot \|_E$ is well-defined on $N(O_\omega(\varepsilon_n))$ and it is zero precisely on $N(o_\omega(\varepsilon_n))$, by very definition of these two spaces. Now
$$\frac{\| A_n B_n - I_{k_n} \|}{\varepsilon_n} \leq \frac{\max \{ \|A_n B_n - B_n\|, \|B_n - I_{k_n}\| \} }{\varepsilon_n} = \max \left\{ \frac{\|A_n - I_{k_n}\|}{\varepsilon_n}, \frac{\|B_n - I_{k_n}\|}{\varepsilon_n} \right\},$$
and taking the ultralimit shows that $\| AB \|_E \leq \max\{ \| A \|_E, \| B \|_E \}$. This implies at once that $\| \cdot \|_E$ is well-defined on $E$, and that it satisfies the strong triangle inequality. That it is $\mathbb{K}$-linear is clear. It takes values in $r^\mathbb{Z}$ because the maximum norm does, so $\| \cdot \|_E$ is a solid norm on $E$. Since $\Gamma$ acts by conjugation by elements of $\operatorname{GL}_{k_n}(\mathfrak{o})$, and the maximum norm is bi-invariant, the action is isometric. \\
We are left to that $E$ is Banach. So let $\{ [A^k] = [(A^k_n)_{n \geq 1}] \}_{k \geq 1}$ be a Cauchy sequence. Explicitly, this means that
$$\| [A^k] - [A^l] \|_E = \lim\limits_{n \to \omega} \frac{\|A^k_n (A^l_n)^{-1} - I_{k_n} \|}{\varepsilon_n} = \lim\limits_{n \to \omega} \frac{\|A^k_n - A^l_n\|}{\varepsilon_n} \xrightarrow{k, l \to \infty} 0.$$
By Tychonoff's Theorem the product space $G(0)$ is compact, so up to subsequence we may assume that a sequence of representatives $A^k$ converges in this topology to some $A \in G(0)$. This means pointwise convergence, that is: $\|A^k_n - A_n\| \xrightarrow{k \to \infty} 0$ for all $n$, although the convergence is not necessarily uniform in $n$. We need to show that $A \in N(O_\omega(\varepsilon_n))$ and that $\|[A^k] - [A]\|_E \xrightarrow{k \to \infty} 0$.
Let $\delta > 0$ be fixed. We will show that there exists some $K = K(\delta) \geq 1$ such that for all $k \geq K$ and for most $n$ we have: $\frac{\|A^k_n - A_n\|}{\varepsilon_n} < \delta$. An application of the triangle inequality then shows that $A \in N(O_\omega(\varepsilon_n))$, and letting $\delta \to 0$ we also have $\omega$-convergence. We choose $K$ to be such that $\|[A^k] - [A^l]\|_E < \delta$ for all $k, l \geq K$. By definition of the ultralimit it follows that
$$X := \left\{ n \geq 1 : \varepsilon_n \neq 0, \, \frac{\|A^k_n - A^l_n\|}{\varepsilon_n} < \delta \, \, \, \forall \, k, l \geq K \right\} \in \omega.$$
For all $n \in X$, let $K_n$ be such that $\|A^l_n - A_n\| < \delta \varepsilon_n$ for all $l \geq K_n$ (we can do this because $n \in X$ and so $\varepsilon_n \neq 0$). Then for all $k \geq K$, given $n \in X$ and choosing $l \geq \max \{ K_n, K \}$ we have:
$$\frac{\|A^k_n - A_n \|}{\varepsilon_n} \leq \max \left\{ \frac{\|A^k_n - A^l_n \|}{\varepsilon_n} , \frac{\|A^l_n - A_n \|}{\varepsilon_n} \right\} < \max \left\{ \delta, \frac{\delta \varepsilon_n}{\varepsilon_n} \right\} = \delta.$$
This concludes the proof.
\end{proof}
By very definition:
\begin{lemma}
\label{lem:ker_ball}
$E_C$ is the closed $C$-ball in $E$ with respect to the norm $\| \cdot \|_E$.
\end{lemma}
We will not go into detail here, but this Banach $\mathbb{K}[\Gamma]$-module $E$ is isometrically $\Gamma$-isomorphic to one which admits a nicer description, namely the matrix ultraproduct $\prod\limits_{n \to \omega} \operatorname{M}_{k_n}(\mathbb{K})$ with the maximum norm. This is the quotient of the subspace of bounded sequences in the direct product by the subspace of sequences $(M_n)$ such that $\| M_n \| \xrightarrow{n \to \omega} 0$, and can be endowed with a natural norm $\| [M] \| = \lim\limits_{n \to \omega} \| M_n \|$ and a $\Gamma$-action by conjugacy via $(\hat{\varphi}_n)_{n \geq 1}$. Then the isometric $\Gamma$-isomorphism is given by
$$\prod\limits_{n \to \omega} M_{k_n}(\mathbb{K}) \to E : [M = (M_n)_{n \geq 1}] \mapsto [(I_n + \varepsilon_n^{-1} M_n)].$$
\subsection{Proof of Theorem \ref{thm:BC}}
We are ready to prove Theorem \ref{thm:BC}. For the rest of this subsection, assume that $\Gamma$ satisfies the cohomology vanishing criterion. Lemma \ref{lem:UBC} has the following consequence for our problem:
\begin{lemma}
\label{lem:UBC_cor}
There exists $C \geq 1$ such that the following holds. For every sequence $(\hat{\varphi}_n : F_S \to G_n)_{n \geq 1}$ such that $\operatorname{def}(\hat{\varphi}_n) \xrightarrow{n \to \omega} 0$ there exists a sequence $(\hat{\psi}_n : F_S \to G_n)_{n \geq 1}$ such that $\operatorname{def}(\hat{\psi}_n) = o_\omega(\operatorname{def}(\hat{\varphi}_n))$ and $\operatorname{dist}(\hat{\varphi}_n, \hat{\psi}_n) \leq_\omega C \varepsilon_n$.
\end{lemma}
\begin{proof}
We let $C$ be as in Lemma \ref{lem:UBC}. By Lemma \ref{lem:red_lift} and the discussion thereafter, the existence of such a sequence is equivalent to the vanishing of a cohomology class in $\operatorname{H}^2_b(\Gamma, E_C)$, which admits a representative cocycle taking values in $E_1$. By Lemma \ref{lem:ker_ball}, cocycles taking values in $E_1$ are cocycles taking values in $E$ whose norm is at most $1$, and so the existence of a primitive taking values in $E_C$ is guaranteed by Lemma \ref{lem:UBC}.
\end{proof}
Now fix a sequence $\varepsilon_n \xrightarrow{n \to \omega} 0$ and define $\operatorname{Hom}_{\varepsilon_n}(\Gamma, \operatorname{GL}(\mathfrak{o})) := \{ (\hat{\varphi}_n : F_S \to G_n)_{n \geq 1} : \operatorname{def}(\hat{\varphi}_n) \leq_\omega \varepsilon_n \}$. We need to show that for any $(\hat{\varphi}_n)_{n \geq 1} \in \operatorname{Hom}_{\varepsilon_n}(\Gamma, \operatorname{GL}(\mathfrak{o}))$ there exists a sequence $(\pi_n : F_S \to G_n)_{n \geq 1}$ that descends to $\Gamma$ and such that $\operatorname{dist}(\hat{\varphi}_n, \pi_n) \xrightarrow{n \to \omega} 0$. Define
$$\operatorname{Hdist}(\hat{\varphi}) := \inf \{ \operatorname{dist}(\hat{\varphi}, \pi) : \pi \text{ descends to } \Gamma \}.$$
Therefore we need to show that for any $(\hat{\varphi}_n)_{n \geq 1} \in \operatorname{Hom}_{\varepsilon_n}(\Gamma, \operatorname{GL}(\mathfrak{o}))$ we have $\operatorname{Hdist}(\hat{\varphi}_n) \xrightarrow{n \to \omega} 0$. \\
We equip the space of functions $\{ f : S \to G_n \}$ with the product topology, making it into a compact space such that the functions $\operatorname{def}$ and $\operatorname{Hdist}$ are continuous. Let $X_n$ be the subspace of maps such that $\operatorname{def}(\hat{\varphi}_n) \leq \varepsilon_n$, which is closed, thus compact. The map
$$ \theta : X_n \to \mathbb{R}_{\geq 0} : \hat{\varphi}_n \mapsto \operatorname{Hdist}(\hat{\varphi}_n) - 2 C \operatorname{def}(\hat{\varphi}_n)$$
is continuous, and so there exists an asymptotic homomorphism $(\hat{\varphi}^M_n)_{n \geq 1} \in \operatorname{Hom}_{\varepsilon_n}(\Gamma, \operatorname{GL}(\mathfrak{o}))$ maximizing $\theta$ for each $n$. We claim that $\theta(\varphi^M_n) =_\omega 0$. \\
Let $(\hat{\psi}_n)_{n \geq 1}$ be given by Lemma \ref{lem:UBC_cor}, so that $\operatorname{def}(\hat{\psi}_n) = o_\omega(\operatorname{def}(\hat{\varphi}_n^M))$ and $\operatorname{dist}(\hat{\varphi}^M_n, \hat{\psi}_n) \leq_\omega C \operatorname{def}(\hat{\varphi}_n^M)$. The first condition implies that $\operatorname{def}(\hat{\psi}_n) \leq_{\omega} \frac{1}{4} \operatorname{def}(\hat{\varphi}_n^M) \leq_\omega \varepsilon_n$. Then we have
$$\operatorname{Hdist}(\hat{\varphi}^M_n) \leq_\omega \operatorname{Hdist}(\hat{\psi}_n) + \operatorname{dist}(\hat{\varphi}^M_n, \hat{\psi}_n) \leq_\omega \operatorname{Hdist}(\hat{\psi}_n) + C \operatorname{def}(\hat{\varphi}^M_n).$$
By maximality:
$$\operatorname{Hdist}(\hat{\psi}_n) - 2C \operatorname{def}(\hat{\psi}_n) \leq_\omega \operatorname{Hdist}(\hat{\varphi}^M_n) - 2C \operatorname{def}(\hat{\varphi}^M_n) \leq_\omega \operatorname{Hdist}(\hat{\psi}_n) - C \operatorname{def}(\hat{\varphi}^M_n),$$
whence
$$\operatorname{def}(\hat{\varphi}^M_n) \leq_\omega 2 \operatorname{def}(\hat{\psi}_n) \leq_\omega \frac{1}{2} \operatorname{def}(\hat{\varphi}^M_n).$$
It follows that $\operatorname{def}(\hat{\varphi}^M_n) =_\omega 0$, so $\hat{\varphi}_n^M$ is a homomorphism for most $n$, and $\operatorname{Hdist}(\hat{\varphi}_n^M) =_\omega 0$ too. In particular $\theta(\hat{\varphi}_n^M) =_\omega 0$, which proves the claim. \\
Finally, since $\hat{\varphi}^M_n$ maximizes $\theta$ for all $n$, for all $(\hat{\varphi}_n)_{n \geq 1} \in \operatorname{Hom}_{\varepsilon_n}(\Gamma, \operatorname{GL}(\mathfrak{o}))$ we have
$$\lim\limits_{n \to \omega} \operatorname{Hdist}(\hat{\varphi}_n) = \lim\limits_{n \to \omega} \theta(\hat{\varphi}_n) \leq \lim\limits_{n \to \omega} \theta(\hat{\varphi}^M_n) = 0,$$
which concludes the proof of Theorem \ref{thm:BC}.
\subsection{Applying the criterion}
\label{ss:conj}
By analogy with the notion of cohomological dimension, let us say that $\Gamma$ has \textit{$\mathbb{K}$-bounded cohomological dimension at most $1$}, denoted $bcd_{\mathbb{K}}(\Gamma) \leq 1$ if it satisfies the condition on Theorem \ref{thm:BC}, that is $\operatorname{H}^2_b(\Gamma, E) = 0$ for every Banach $\mathbb{K}[\Gamma]$-module $E$ with a solid norm. Proposition \ref{prop:comp} and Theorem \ref{thm:BC} together imply:
\begin{corollary}
\label{cor:BC}
Let $\Gamma$ be finitely presented and suppose that $\operatorname{H}^2(\Gamma, E) = 0$ for every Banach $\mathbb{K}[\Gamma]$-module $E$ with a solid norm. Then $\Gamma$ is $\operatorname{GL}(\mathfrak{o})$-stable.
\end{corollary}
In particular, if $\Gamma$ is finitely presented and has $\mathbb{K}$-cohomological dimension at most $1$, then it is stable. Dunwoody characterized such groups in \cite{cd1}: if $\mathbb{K}$ has characteristic $0$, then these are precisely the finitely generated virtually free groups, and if $\mathbb{K}$ has characteristic $p$, these are precisely the finitely generated virtually free groups without elements of order $p$. These examples are already contained in Corollary \ref{cor:vfree} and \ref{cor:vfree_p}, respectively. \\
Corollary \ref{cor:BC} is an analogue of the main theorem in \cite{GLT}: a finitely presented group is $(\operatorname{U}(n), \| \cdot \|_{Frob})$-stable if $\operatorname{H}^2(\Gamma, E) = 0$ for any unitary representation $E$. The so-called Garland method, initially introduced in \cite{Garland} and since then vastly generalized to include even general Banach coefficients \cite{Garland2}, allows to give many examples of groups satisfying this condition. Our criterion asks for vanishing over Banach spaces, which at first glance may seem too restrictive compared to the Archimedean setting. However, on the one hand there is no analogue of Hilbert spaces over non-Archimedean fields \cite[2.4]{NFA}, and on the other hand the hypothesis of the norm being solid has strong implications: such spaces are isometrically classified \cite[Theorem 2.5.4]{NFA}. But even then it seems hard to prove a cohomology vanishing criterion by adapting Garland's method, since the distinction between positive and negative real eigenvalues plays a fundamental role. \\
Theorem \ref{thm:BC} is a priori stronger than Corollary \ref{cor:BC}: it asks for vanishing of $\operatorname{H}^2_b(\Gamma, E)$ which is a subspace of $\operatorname{H}^2(\Gamma, E)$. The hypothesis of $\Gamma$ being finitely presented may play an important role here. Indeed, the comparison map $c^2 : \operatorname{H}^2_b(\Gamma, \mathbb{K}) \to \operatorname{H}^2(\Gamma, \mathbb{K})$ is an isomorphism when $\Gamma$ is finitely presented, and $\mathbb{K}$ is seen as the trivial $\mathbb{K}[\Gamma]$-module \cite[Corollary 8.13]{mio}, although we do not know whether this holds with non-trivial coefficients. \\
We conjecture that the only finitely presented groups satisfying $bcd_{\mathbb{K}}(\Gamma) \leq 1$ are virtually free. More generally:
\begin{conjecture}
\label{conj}
Let $\Gamma$ be an arbitrary group such that $bcd_{\mathbb{K}}(\Gamma) \leq 1$, and let $p$ be the characteristic of the residue field of $\mathbb{K}$. Then the following holds:
\begin{enumerate}
\item If $\mathbb{K}$ has characteristic $p$, then $cd_{\mathbb{K}}(\Gamma) \leq 1$, and so $\Gamma$ is the fundamental group of a graph of groups whose vertex groups are finite without elements of order $p$.
\item If $\mathbb{K}$ has characteristic $0$, then there exists $k \geq 1$ such that $\Gamma$ is the fundamental group of a graph of groups whose vertex groups are finite with $p$-subgroups of order at most $p^k$.
\end{enumerate}
\end{conjecture}
Let us give some motivation behind this conjecture. Let $E$ be a Banach $\mathbb{K}[\Gamma]$-module. Since $\Gamma$ acts isometrically on it, it also acts on the \textit{reduction} $\overline{E}_k$ for all $k \geq 1$, that is, the quotient of the closed ball $B_1$ of radius $1$ by the closed ball $B_{r^k}$ of radius $r^k$. Note that $\overline{E}_k$ is an $(\mathfrak{o} / \mathfrak{p}^k)[\Gamma]$-module, in particular $\overline{E}_1$ is a $\mathfrak{k}[\Gamma]$-module. The reduction plays an important role in the classification of Banach spaces with a solid norm \cite[Theorem 2.5.4]{NFA}: if $X$ is a $\mathfrak{k}$-basis of $\overline{E}_1$, then $E$ is isometrically isomorphic to
$$c_0(X) = \{ f : X \to \mathbb{K} : \# \{ x \in X : |f(x)| < \varepsilon \} < \infty \text{ for all } \varepsilon > 0 \},$$
with the supremum norm.
Suppose that $bcd_{\mathbb{K}}(\Gamma) \leq 1$. By using a dimension-shifting argument, one would prove that $\operatorname{H}^3_b(\Gamma, E) = 0$ for all Banach $\mathbb{K}[\Gamma]$-modules with a solid norm, too. This would need a functorial approach to bounded cohomology over non-Archimedean fields, as in the real case \cite{monod}, but this has yet to be developed (see \cite[Section 7]{mio} for a discussion). Assuming this step, by Lemma \ref{lem:UBC} there exists a constant $C \geq 1$ such that for $n = 2, 3$ and any $z \in \operatorname{Z}^n_b(\Gamma, E)$ there exists $b \in \operatorname{C}^{n-1}_b(\Gamma, E)$ such that $\| b \|_\infty \leq C \| z \|_\infty$.
\begin{claim}
Suppose that, in the above setting, we have $C = 1$. Then $\operatorname{H}^2(\Gamma, \overline{E}_1) = 0$.
\end{claim}
\begin{proof}
Let $z \in \operatorname{C}^2(\Gamma, \overline{E}_1)$. We lift this to an element $t \in \operatorname{C}^2(\Gamma, B_1) \leq \operatorname{C}^2_b(\Gamma, E)$, whose coboundary takes values in $B_r$; in other words $\| t \|_\infty \leq 1$ and $\| \delta^2 t \|_\infty \leq r$. Now $\delta^2 t \in \operatorname{Z}^3_b(\Gamma, E)$, so there exists $c \in \operatorname{C}^2_b(\Gamma, E)$ such that $\delta^2 c = \delta^2 t$ and $\| c \|_\infty \leq \| \delta^2 t \|_\infty \leq r$. We can thus consider $\hat{z} := (t - c) \in \operatorname{Z}^2_b(\Gamma, B_1)$ whose image in $\operatorname{Z}^2(\Gamma, \overline{E}_1)$ is $z$. Then there exists $\hat{b} \in \operatorname{C}^1_b(\Gamma, E)$ such that $\delta^1 \hat{b} = \hat{z}$ and $\| \hat{b} \|_\infty \leq \| z \|_\infty \leq 1$. Denoting by $b$ the image of $\hat{b}$ in $\operatorname{C}^1(\Gamma, \overline{E}_1)$, we conclude that $z = \delta^1 b$ is a coboundary.
\end{proof}
This conclusion is quite strong: for instance by taking $E = c_0(\Gamma)$ we have $\overline{E}_1 \cong \mathfrak{k}[\Gamma]$, and so $\operatorname{H}^2(\Gamma, \mathfrak{k}[\Gamma]) = 0$. This implies that either $\Gamma$ has $\mathfrak{k}$-cohomological dimension at most $1$ (and so it satisfies the conjecture by Dunwoody's characterization), or it has infinite $\mathfrak{k}$-cohomological dimension. Conversely:
\begin{claim}
Suppose that $\operatorname{H}^2(\Gamma, \overline{E}_1) = 0$. Then $\operatorname{H}^2_b(\Gamma, E) = 0$; more precisely for all $z \in \operatorname{Z}^2_b(\Gamma, E)$ there exists a primitive $b \in \operatorname{C}^1_b(\Gamma, E)$ such that $\| b \|_\infty \leq \| z \|_\infty$.
\end{claim}
\begin{proof}
Let $z \in \operatorname{Z}^2_b(\Gamma, E)$. Then the projection $\overline{z} \in \operatorname{Z}^2(\Gamma, \overline{E}_1)$ of the normalization $z / \| z \|_\infty$ admits a primitive $\overline{b} \in \operatorname{C}^1(\Gamma, \overline{E}_1)$. Lifting $\overline{b}$ and rescaling it to an element $b_1 \in \operatorname{C}^1_b(\Gamma, E)$ with $\| b_1 \|_\infty \leq \| z \|_\infty$ we have $\| z - \delta^1 b_1 \|_\infty \leq r \cdot \| z \|_\infty$. We then apply the same procedure to $(z - \delta^1 b_1)$, and so on inductively. This yields a sequence $b_i$ such that $\| b_i \|_\infty \to 0$ and $\| z - (\delta^1 \sum_{j < i} b_j) \|_\infty \to 0$. Since $\operatorname{C}^1_b(\Gamma, E)$ is Banach, $b := \sum b_i$ exists, it satisfies $\| b \|_\infty \leq \| z \|_\infty$ and $\delta^1 b = z$.
\end{proof}
The formulation of the conjecture for the more general case in which $C \geq 1$ is done by analogy with the way Schikhof's notion of $\mathbb{K}$-amenability \cite{Schik} was generalized to the author's notion of normed $\mathbb{K}$-amenability \cite[Definition 1.1]{mio}. Intuitively, Schikhof's notion of $\mathbb{K}$-amenability (where $\mathbb{K}$ is local, or more generally spherically complete) is a ``norm $1$" notion, similar to the condition $C = 1$ above, and the notion only depends on $p$, not on $\mathbb{K}$ or its characteristic. When making this notion more flexible by allowing ``bounded norms", similar to allowing $C \geq 1$ above, we obtained the notion of normed $\mathbb{K}$-amenability, which stays the same for characteristic $p$, and in characteristic $0$ it replaces the absence of elements of order $p$ by a bound on the order of finite $p$-subgroups. \\
Looking at the real setting, one may hope that Theorem \ref{thm:BC} could be strengthened by only asking for vanishing with \textit{dual} $\mathbb{K}[\Gamma]$-modules. For instance, in the real setting all amenable groups have vanishing bounded cohomology with dual coefficients \cite[Chapter 3]{Frig}, and in degree $2$ this even applies to high-rank lattices \cite{burmon}. In our setting, there is a significant obstacle, namely that no infinite-dimensional $\mathbb{K}$-Banach space is reflexive \cite[Corollary 7.4.20]{NFA}, so proving that a Banach $\mathbb{K}[\Gamma]$-module is dual would require an explicit construction of a pre-dual. This seems hard considering that the spaces appearing in our setting are quite complicated: they are matrix ultraproducts with the $\Gamma$-action induced by an asymptotic homomorphism. Using the classification of Banach spaces, and assuming that the degree $k_n \to \infty$ (else Proposition \ref{prop:unfin} applies), we are able to show that all spaces appearing in the proof are isometric to $\ell^\infty(\mathbb{N})$, which is the dual of $c_0(\mathbb{N})$. So there is a chance that these spaces are dual $\mathbb{K}[\Gamma]$-modules, but to show that the action is dual one would probably have to construct an explicit pre-dual. We formulate this as an open question:
\begin{question}
Does stability still hold if in Theorem \ref{thm:BC} bounded cohomology vanishing is assumed only for dual modules?
\end{question}
Even if such a strengthening were possible, all vanishing results over local fields with dual modules that are known so far \cite[Theorem 7.4, Corollary 7.13]{mio} apply to groups whose stability has already been proved in Sections \ref{s:vpropi} and \ref{s:char0}. So such a strengthening would be of interest only if one were able to prove more general vanishing results in degree $2$, possibly by adapting the work of Burger and Monod \cite{burmon} to lattices in $\mathbb{K}$-analytic groups.
\pagebreak
\section{Further remarks and open questions}
\label{s:q}
In this section we survey some open questions on ultrametric stability, give a few partial answers, and propose directions for further research. The first two subsections contain open questions about $\operatorname{GL}(\mathfrak{o})$-stability of certain groups, with special attention to $\mathbb{Z}^2$. Lastly Subsection \ref{ss:q:fam} proposes other ultrametric families whose study may be of interest. We refer the reader to Subsections \ref{ss:poschar} and \ref{ss:conj} for further open questions, about stability of finite groups in positive characteristic and bounded cohomology vanishing, respectively. \\
Throughout this section, $\mathfrak{o}$ is the ring of integers of a non-Archimedean local field $\mathbb{K}$ of residual characteristic $p$, uniformizer $\overline{\omega}$ and value group $| \mathbb{K}^\times | = r^\mathbb{Z}$.
\subsection{Finding non-examples}
Most of this paper has been concerned with giving positive results on stability. The negative results have been few and far apart: in Example \ref{ex:2unst} we constructed a pro-$p$ family $\mathcal{G}_p$ such that $\mathbb{Z}/p\mathbb{Z}$ is not $\mathcal{G}_p$-stable, and in Corollary \ref{cor:cex_stab} we showed that a non-residually finite LEF group is not pointwise $\operatorname{GL}(\mathfrak{o})$-stable. In particular, the following questions remain open:
\begin{question}
\label{q:fg:un}
Does there exist a finitely generated group that is not uniformly $\operatorname{GL}(\mathfrak{o})$-stable?
\end{question}
\begin{question}
\label{q:fgrf:pw}
Does there exist a finitely generated residually finite group that is not pointwise $\operatorname{GL}(\mathfrak{o})$-stable?
\end{question}
\begin{question}
\label{q:fp:st}
Does there exist a finitely presented group that is not $\operatorname{GL}(\mathfrak{o})$-stable?
\end{question}
It would be very surprising if some of these questions had a negative answer. Good candidates seem to be free abelian groups, surface groups, free nilpotent groups and free solvable groups. The case of $\mathbb{Z}^2$ is discussed in detail in Subsection \ref{ss:q:z2}, the other ones are briefly mentioned after Example \ref{ex:z2:est}. Other potential non-examples are given by graphs of groups as in Subsection \ref{ss:gog}, where the coprimality conditions on finite quotients are not satisfied. For instance a good candidate for the uniform part of Question \ref{q:fgrf:pw} is the lamplighter group $\mathbb{Z}/2\mathbb{Z} \wr \mathbb{Z}$: we believe this group not to be uniformly $\operatorname{GL}(\mathfrak{o})$-stable when $p = 2$. This would also give a positive solution to the following question:
\begin{question}
Does there exist a group that is (pointwise, uniformly) $\operatorname{GL}(\mathfrak{o})$-stable in some, but not all, (residual) characteristics?
\end{question}
Analogously, an interesting family of examples to look at is that of \textit{GBS-tree products}, that is, GBS groups whose underlying graph is a tree, to which the criteria from Corollaries \ref{cor:GBS_p} and \ref{cor:GBS} cannot apply. In particular:
\begin{question}
Which torus knot groups $K_{m, n}$ are $\operatorname{GL}(\mathfrak{o})$-stable? Does some condition on $\nu_p(m)$ and $\nu_p(n)$ imply stability, or instability?
\end{question}
All our examples of finitely generated groups that are uniformly but not pointwise stable use Corollary \ref{cor:cex_stab}, so they cannot be residually finite. Therefore we ask a more precise version of Question \ref{q:fgrf:pw}:
\begin{question}
\label{q:fgrf:unnonpw}
Does there exist a finitely generated residually finite group that is uniformly but not pointwise $\operatorname{GL}(\mathfrak{o})$-stable?
\end{question}
Recall from Corollary \ref{cor:fgrf} that if $\Gamma$ is a finitely generated residually finite group that can be expressed as the largest residually finite quotient of a finitely presented group, then the uniform and pointwise $\operatorname{GL}(\mathfrak{o})$-stability of $\Gamma$ are equivalent. We saw in the discussion after Corollary \ref{cor:fgrf} that not all finitely generated groups satisfy this. The examples given there, all coming from \cite{covers}, could provide a positive answer to Question \ref{q:fgrf:unnonpw}. \\
Proposition \ref{prop:lef_q} seems to suggest that by working on the space of marked groups one could adapt the results from Sections \ref{s:vpropi} and \ref{s:char0} to the pointwise setting.
\begin{question}
Do the results from Sections \ref{s:vpropi} and \ref{s:char0} have a pointwise counterpart?
\end{question}
However, this presents more technical subtleties, since one would have to prove lifting results for \textit{local} homomorphisms to the metric quotients: we are not aware of such results, or of a connection to cohomology analogous to the classical one (see Subsection \ref{ss:preli:split}). \\
Before moving on to specific examples in the next subsections, let us comment on why a general method for producing non-examples of stability does not work for $\operatorname{GL}(\mathfrak{o})$. This is commonly known as the \textit{projection trick}. For instance when $\mathcal{G}$ is a family of unitary groups, one starts with $(n+1)$-dimensional irreducible unitary representations of the finitely generated group $\Gamma$, and restricts to the top $(n \times n)$ corner to obtain an asymptotic homomorphism. Assuming this is close to a homomorphism, one arrives at a contradiction with the irreducibility of the initial representation. The same idea works for permutations, by starting with a transitive action of $\Gamma$ on $\{ 1, \ldots, n\}$. To our knowledge this method was first used in \cite{T} to prove that if $\Gamma$ is infinite, sofic and has property (T), then it is not pointwise stable in permutations. It also appears in \cite{BChap} and \cite{uHS}, where it is shown that uniform stability, in permutations and with respect to unitary groups with the Hilbert--Schmidt norm respectively, are very restrictive properties.
It is key in the arguments that by looking at $n$ out of $(n+1)$ entries (or $n^2$ out of $(n+1)^2$) one does not lose much in terms of normalized metrics. This cannot be the case for the $\ell^\infty$-norm on $\operatorname{GL}(\mathfrak{o})$, where a big difference in a single entry is detected as a big difference overall. \\
This projection trick is precisely the motivation behind introducing notions of \textit{flexible stability} \cite{T}, that have proven fruitful in some contexts \cite{surf}. The discussion above shows that it seems hard to define an analogous notion of flexible $\operatorname{GL}(\mathfrak{o})$-stability. Given the rigidity of this context, it is even possible that na\"ive definitions of flexible $\operatorname{GL}(\mathfrak{o})$-stability are equivalent to ordinary $\operatorname{GL}(\mathfrak{o})$-stability.
\begin{question}
Is there a meaningful notion of flexible $\operatorname{GL}(\mathfrak{o})$-stability? Is it different from ordinary $\operatorname{GL}(\mathfrak{o})$-stability?
\end{question}
\subsection{(In)stability of $\mathbb{Z}^2$}
\label{ss:q:z2}
The following is the main open question we would like to draw attention to:
\begin{question}
Is $\mathbb{Z}^2$ $\operatorname{GL}(\mathfrak{o})$-stable?
\end{question}
An answer for free abelian groups of arbitrary finite rank would be ideal, but we stick to $\mathbb{Z}^2$ for this discussion. An algebraic-geometric approach seems to be the most appropriate: indeed in \cite[Theorem D, Example 10.1]{Zordan} the author proves a result -- including an explicit example -- that suggests that $\mathbb{Z}^2$ is not \textit{constraint $\operatorname{GL}(\mathfrak{o})$-stable} with respect to a direct factor (see \cite{a:const} for the definition of constraint stability). The notions of stability and constraint stability can be quite distinct, for instance $\mathbb{Z}^2$ is stable in permutation \cite{a:comm} but not constraint stable with respect to a direct factor \cite{a:const}. Still, Zordan's result is the closest instance to a result on $\operatorname{GL}(\mathfrak{o})$-instability of $\mathbb{Z}^2$ that we were able to find in the literature. \\
Zordan's result is in terms of Lie algebras, not of Lie groups. This is equivalent to our setting: more precisely, instead of working with $\operatorname{GL}_n(\mathfrak{o})$, one could work with $\operatorname{M}_n(\mathfrak{o})$ and prove stability of the ring-theoretic commutator.
\begin{lemma}
\label{lem:z2:lin}
The following are equivalent:
\begin{enumerate}
\item $\mathbb{Z}^2$ is $\operatorname{GL}(\mathfrak{o})$-stable.
\item For all $\varepsilon > 0$ there exists $\delta > 0$ such that if $A, B \in \operatorname{M}_n(\mathfrak{o})$ satisfy $\| AB - BA \| < \delta$, then there exist $A', B' \in \operatorname{M}_n(\mathfrak{o})$ such that $A'B' = B'A'$ and $\| A - A' \|, \|B - B'\| < \varepsilon$.
\end{enumerate}
\end{lemma}
\begin{proof}
We use the characterization from Corollary \ref{cor:quant}. Since $\| ABA^{-1}B^{-1} - I \| = \| AB - BA \|$ by $\operatorname{GL}(\mathfrak{o})$-invariance of $\| \cdot \|$ (Lemma \ref{lem:GL}), instability of $\mathbb{Z}^2$ implies that $2.$ does not hold. Conversely, given matrices $A, B$ contradicting $2.$ for a given $\varepsilon > 0$, the matrices $(I + \overline{\omega} A), (I + \overline{\omega} B)$ contradict the characterization of stability from Corollary \ref{cor:quant}, for a rescaled $\varepsilon$. Indeed, they are invertible by Lemma \ref{lem:GL}, and
$$\| (I + \overline{\omega} A)(I + \overline{\omega} B) - (I + \overline{\omega} B)(I + \overline{\omega} A) \| = r^2 \| AB - BA \|.$$
\end{proof}
Since the norm $\| \cdot \|$ on $\operatorname{M}_n(\mathbb{K})$ coincides with the operator norm (Lemma \ref{lem:GL}), it is tempting to try and adapt Voiculescu's counterexample \cite{Voie} to prove instability of $\mathbb{Z}^2$. Voiculescu's matrices proving that $\mathbb{Z}^2$ is not pointwise stable with respect to $\{ (\operatorname{U}(n), \| \cdot \|_{op}) : n \geq 1 \}$ are the permutation matrix $P$ corresponding to the cycle $(1 \cdots n)$ and the diagonal matrix $D := diag( 1, \omega_n, \omega_n^2, \ldots, \omega_n^{n-1})$, where $\omega_n = e^{\frac{2 \pi i}{n}}$. It is easy to adapt this example to produce asymptotic homomorphisms from $\mathbb{Z}^2$ to $\operatorname{GL}(\mathfrak{o})$, but they are all close to homomorphisms. This is true even for an arbitrary matrix $D$, as long as $P$ is monomial (that is, it has a unique non-zero entry in each row and column):
\begin{lemma}
Let $P \in \operatorname{GL}_n(\mathfrak{o})$ be a monomial matrix, and let $D \in \operatorname{M}_n(\mathbb{K})$. Then there exists $D' \in \operatorname{M}_n(\mathbb{K})$ such that $PD' = D'P$ and $\| D - D' \| \leq \| PD - DP \|$.
\end{lemma}
\begin{proof}
Let $\varepsilon := \| PD - DP \| = \|PDP^{-1} - D \|$. Let $\sigma \in S_n$ be the permutation such that $P_{ij} \neq 0$ precisely when $j = \sigma(i)$, and set $\lambda_i := P_{i \sigma(i)}$, which is in $\mathfrak{o}^\times$ since $P \in \operatorname{GL}_n(\mathfrak{o})$. Then $(PDP^{-1})_{ij} = \lambda_i \lambda_j^{-1} D_{\sigma(i)\sigma(j)}$, so $|D_{\sigma(i)\sigma(j)} - \lambda_i^{-1} \lambda_j D_{ij}| \leq \varepsilon$ for each $(i, j) \in \{ 1, \ldots, n\}^2$. By induction it follows that
$$\left|D_{\sigma^k(i)\sigma^k(j)} - \left( \prod\limits_{l = 0}^{k-1} \lambda_{\sigma^l(i)}^{-1} \lambda_{\sigma^l(j)} \right) \cdot D_{ij}\right| \leq \varepsilon$$
for every $k \geq 0$.
Now choose representatives for each orbit of the diagonal action of $\sigma$ on $\{1, \ldots, n\}^2$, and for each representative $(i, j)$ and each $k \geq 0$ smaller than the size of the corresponding orbit, set
$$D'_{\sigma^k(i) \sigma^k(j)} := \left( \prod\limits_{l = 0}^{k-1} \lambda_{\sigma^l(i)}^{-1} \lambda_{\sigma^l(j)} \right) \cdot D_{ij}.$$
Then $\| D - D' \| \leq \varepsilon$ and $PD' = D'P$.
\end{proof}
In particular if $D \in \operatorname{GL}_n(\mathfrak{o})$ and $\| PD - DP \| < 1$, then $\| D - D' \| < 1$ and so $D' \in \operatorname{GL}_n(\mathfrak{o})$ as well, by Lemma \ref{lem:GL}. This shows that such matrices produce asymptotic homomorphisms that are close to homomorphisms, even with an optimal estimate. Moreover there is no need to modify $P$, so this also does not even work as a counterexample to constraint $\operatorname{GL}(\mathfrak{o})$-stability of $\mathbb{Z}^2$. \\
Let us end by noticing that the stability estimate of $\mathbb{Z}^2$ is, at best, quadratic. So this example really is different from the ones treated in this paper, where the stability estimates were always linear with the exception of Subsection \ref{ss:poschar}.
\begin{example}
\label{ex:z2:est}
Let $A, B \in \operatorname{M}_n(\mathfrak{o})$ be such that $\| A \|, \| B \| \leq \varepsilon < 1$, and consider the map $\varphi : F_2 \to \operatorname{GL}_n(\mathfrak{o})$ sending the generators to $(I + A)$ and $(I + B)$, which are invertible by Lemma \ref{lem:GL}. This homomorphism almost descends to $\mathbb{Z}^2$ with a defect of $\varepsilon^2$:
$$\| (I + A)(I + B) - (I + B)(I + A) \| = \| AB - BA \| \leq \| A \| \cdot \| B \| \leq \varepsilon^2.$$
On the other hand, if $A$ and $B$ are chosen so that $\| AB - BA \| = \varepsilon^2$, then this homomorphism is $\varepsilon$-far from any homomorphism that descends to $\mathbb{Z}^2$.
\end{example}
With the same idea one can show that free abelian groups and surface groups have at best quadratic estimates. For free nilpotent groups, applying the above argument inductively on the length of the lower central series, one can show that the estimate is at best polynomial, with the degree increasing together with the length. Similarly for free solvable groups with the length of the derived series.
\subsection{Other ultrametric families}
\label{ss:q:fam}
Most of this paper was concerned with $\operatorname{GL}(\mathfrak{o})$-stability, where $\mathfrak{o}$ is the ring of integers of a non-Archimedean local field. The groups $\operatorname{GL}_n(\mathfrak{o})$ are compact because $\mathfrak{o}$ is, and compactness played an important role in our arguments, especially to have finiteness of the metric quotients. The general picture could be more complicated:
\begin{question}
Study $\operatorname{GL}(\mathfrak{o})$-stability, where $\mathfrak{o}$ is the ring of integers of a (not necessarily local) non-Archimedean field with residual characteristic $p > 0$. How does it compare to the case of local fields? Does completeness play a role? Does spherical completeness?
\end{question}
It is likely that some results from Section \ref{s:vpropi} and \ref{s:char0} carry over, assuming at least completeness. In this case the residue field $\mathfrak{k}$ is not finite, but at least it has characteristic $p$, which makes it possible to recover some arguments. In case $\mathbb{K}$ has residual characteristic $0$, the analogy with local fields breaks down, so this is likely to need a separate study:
\begin{question}
Study $\operatorname{GL}(\mathfrak{o})$-stability, where $\mathfrak{o}$ is the ring of integers of a non-Archimedean field with residual characteristic $0$. Does completeness play a role? Does spherical completeness?
\end{question}
Another direction in which to generalize $\operatorname{GL}(\mathfrak{o})$ while retaining compactness is to look at other compact $\mathbb{K}$-analytic groups equipped with suitable bi-invariant ultrametrics. For instance, using a result of Segal \cite{Segal}, some of the stability results on graphs of groups from Section \ref{s:char0} could be generalized to any family of compact $p$-adic analytic groups equipped with a suitable metric.
\begin{question}
Study $\mathcal{G}$-stability, for other families $\mathcal{G}$ of compact $\mathbb{K}$-analytic groups equipped with suitable bi-invariant ultrametrics.
\end{question}
In the introduction we mentioned that the $\ell^\infty$-norm on $\operatorname{M}_n(\mathbb{K})$ has the special feature of being at once an ultrametric analogue of the operator norm, of the Frobenius norm, and of the Hilbert--Schmidt norm on $\operatorname{U}(n)$. There is a fourth norm on matrix groups that one could consider, namely the normalized rank, leading to the \textit{rank metric}: the corresponding approximable groups are called \textit{linear sofic} and are studied in \cite{a:lin}. The rank metric can also be defined on non-Archimedean fields, however it is not an ultrametric, so it does not fall in the framework of this paper. Therefore we ask:
\begin{question}
Does the rank metric admit an ultrametric analogue?
\end{question}
For the family $\operatorname{Gal}(K)$ we proved in Proposition \ref{prop:gal_stab} that every finitely generated group is uniformly stable. This essentially followed from the fact that every group in the family is a quotient of the absolute Galois group. Another interesting family of Galois groups that does not have this feature is $\mathcal{G} := \{ (\operatorname{Gal}(\mathbb{Q}_p^{sep}/\mathbb{Q}_p, d_p) : p \text{ prime} \}$, where $d_p$ is a bi-invariant ultrametric obtained as in Example \ref{ex:prof} with respect to a fixed sequence $\overline{\varepsilon}$.
\begin{question}
Study stability with respect to the family $\mathcal{G}$ above.
\end{question}
Our first trivial example of ultrametric family was a family of discrete groups $\mathcal{G}$ equipped with discrete metrics. We saw that stability is less interesting in this setting (Example \ref{ex:discr:stab}) but as we mentioned in the discussion after Example \ref{ex:discr}, probabilistic versions of stability with respect to such families appear often in property testing \cite{PT}. Therefore it would be interesting to develop a general framework of probabilistic ultrametric stability, analogously to what is done for the family $\{ (S_n, d_H) : n \geq 1 \}$ in \cite{BChap}, which could produce new results in property testing.
\begin{question}
Study probabilistic analogues of ultrametric stability.
\end{question}
Finally, it would be interesting to produce and study more example of ultrametric families of finite groups. Beyond discrete families, the only case we treated is the family $T(R)$, where $R$ is a finite commutative ring: we proved in Corollary \ref{cor:approxTR} a result concerning approximation with respect to such families, and stability was treated in Proposition \ref{prop:aut_stab} without the finiteness hypothesis. While we studied in detail an ultrametric analogue of $\operatorname{U}(n)$, it would be interesting to find an ultrametric analogue of $(S_n, d_H)$ and compare the corresponding stability problems.
\begin{question}
Produce new examples of ultrametric families of finite groups, and study the corresponding stability problems.
\end{question}
\pagebreak
\bibliographystyle{alpha}
| {
"redpajama_set_name": "RedPajamaArXiv"
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