text
stringlengths 14
5.77M
| meta
dict | __index_level_0__
int64 0
9.97k
⌀ |
|---|---|---|
Der Waldensteiner Bach ist ein linker Zufluss der Lavant in Kärnten (Österreich). Das Einzugsgebiet liegt in den Gemeinden Preitenegg und Wolfsberg.
Lage
Das Einzugsgebiet des Waldensteiner Bachs liegt im Nordosten von Kärnten an der Grenze zur Steiermark. Es wird im Norden begrenzt von den südlichen Ausläufern der Packalpe und im Süden von der Koralpe.
Der Bach entspringt südöstlich des Bärofen im Gemeindegebiet von Wolfsberg in 1620 Meter Seehöhe. Er fließt nach Norden in die Gemeinde Preitenegg, biegt nach Westen ab und bildet dort die Gemeindegrenze zwischen Preitenegg im Norden und Wolfsberg im Süden. Nach Waldenstein vereinigt er sich mit seinem größten Zufluss dem Auerlingbach. Dieser entwässert das Gebiet östlich des Bergkogels bis zur steirischen Grenze, dem Packsattel und dem Höhenrücken auf dem Preitenegg liegt. Die letzten drei Kilometer fließt der Waldensteiner Bach wieder durch Wolfsberger Gemeindegebiet und mündet nach insgesamt rund 20 Kilometern bei Twimberg in 600 Meter Meereshöhe in die Lavant.
Das Einzugsgebiet umfasst 108,9 Quadratkilometer, davon entfallen 56,8 Quadratkilometer auf den Auerlingbach.
Geschichte
Das Gebiet des Waldensteiner Baches gehört zu einer Region, die in den letzten 2 Millionen Jahren nicht vergletschert war. Hier zeigt sich die Vergänglichkeit der Landschaft mit jungen Tälern, die in eine glatte Reliktlandschaft eingeschnitten sind. Die Erosionsraten betragen durchschnittlich 49 (±8) Millimeter pro 1000 Jahren in der Reliktlandschaft und 137 (±15) Millimeter pro 1000 Jahren für Einzugsgebiete in der eingeschnittenen Landschaft. In einem genauer untersuchten Gebiet des Waldensteiner Baches war die Erosionsrate 241 (±22) bei einer Einschnittstiefe von 375 (±23) Metern.
Die Bereiche des unteren Auerlingbachs und des Waldensteiner Bachs ab Waldenstein sind hochwassergefährdet. Im Jahr 2022 wurde ein Hochwasserschutzprojekt für dieses Gebiet gestartet.
Geologie
Am oberen Ende des Waldensteiner Baches herrschen Zentrale Gneisquarzite vor. In etwa 1250 Meter ist lokal ein Marmor eingequetscht. Nördlich der See Eben zeigen die Gesteine einen Granat-Glimmerschiefer-Charakter.
Orte
Die größten Orte im Einzugsgebiet des Waldensteiner Bachs sind Preitenegg, Waldenstein und Twimberg.
Wanderwege
Der Nord-Süd-Weitwanderweg 05 verläuft entlang der nördlichen und östlichen Grenze des Einzugsgebietes.
Einzelnachweise
Koralpe
Pack- und Stubalpe
Preitenegg
Geographie (Wolfsberg, Kärnten)
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 5,837
|
\section{Introduction}
Pulse shaping in multicarrier transmission is a key ingredient for high rate wireless links.
Furthermore it is the standard tool to mitigate the interference caused by doubly dispersive channels.
Most multicarrier schemes like conventional OFDM exploiting guard regions (a cyclic prefix),
pulse shaped OFDM and OFDM/OQAM can be jointly formulated.
Hence we focus on a transmit baseband signal $s(t)$ given as
\begin{equation}
\begin{aligned}
s(t)
&=\sum_{(mn)\in{\mathcal{I}}}x_{mn}e^{i2\pi mFt}\gamma(t-nT)
=\sum_{(mn)\in{\mathcal{I}}}x_{mn}\gamma_{mn}(t)
\end{aligned}
\label{equ:txsignal}
\end{equation}
where $i$ is the imaginary unit and
$\gamma_{mn}\overset{\text{def}}{=}{\boldsymbol{S}}_{(nT,mF)}\,\gamma$ are time-frequency shifted
versions of the transmit pulse $\gamma$, i.e. shifted according to the lattice
$T\mathbb{Z}\times F\mathbb{Z}$. It is also beneficial to consider different lattice
structures \cite{strohmer:lofdm2} on which our contribution will apply as well.
The time-frequency (or phase space) shift operator ${\boldsymbol{S}}_{(\tau,\nu)}$ is intimately
connected to unitary representations of the Weyl-Heisenberg group as we will elaborate later on.
Therefore (\ref{equ:txsignal}) is also known as Weyl-Heisenberg or Gabor signaling.
The coefficients $x_{mn}$ in (\ref{equ:txsignal}) are the complex
data symbols at time instant $n$ and subcarrier index $m$ with the property
$\EX{\boldsymbol{x}\Bx^*}={\mathbb{I}}$ ($\cdot^*$ means conjugate transpose) where \mbox{$\boldsymbol{x}=(\dots,x_{mn},\dots)^T$}.
The indices $(mn)$ range over some doubly-countable index set ${\mathcal{I}}$,
referring to the data burst to be transmitted.
We will denote the linear time-variant channel by $\boldsymbol{\HH}$ and the
additive white Gaussian noise process (AWGN) by $n(t)$.
The received signal is then
\begin{equation}
\begin{aligned}
r(t)=(\boldsymbol{\HH} s)(t)+n(t)
=\iint \boldsymbol{\Sigma}(\tau,\nu)({\boldsymbol{S}}_{(\tau,\nu)}s)(t)d\tau d\nu + n(t)
\end{aligned}
\label{eq:rxsignal}
\end{equation}
with $\boldsymbol{\Sigma}(\tau,\nu)$ being a realization of the ''channel spreading
function''. In practice $\boldsymbol{\Sigma}(\tau,\nu)$ is causal and has finite support.
We used here the notion of
the WSSUS channel. In the WSSUS assumption the
channel is characterized by the second order statistics of $\boldsymbol{\Sigma}(\tau,\nu)$, i.e.
\begin{equation*}
\EX{\boldsymbol{\Sigma}(\tau,\nu)
\overline{\boldsymbol{\Sigma}(\tau',\nu')}}=\boldsymbol{C}(\tau,\nu)\delta(\tau-\tau')\delta(\nu-\nu')
\end{equation*}
where $\boldsymbol{C}(\tau,\nu)$ is the scattering function.
Without loss of generality we assume $\lVert\boldsymbol{C}\rVert_1=1$.
To obtain the data symbol $\tilde{x}_{kl}$ the receiver does the
projection on $g_{kl}\overset{\text{def}}{=}{\boldsymbol{S}}_{(lT,kF)}g$, i.e.
\begin{equation*}
\tilde{x}_{kl}=\langle g_{kl},r\rangle=\int\overline{g}_{kl}(t)r(t)dt
\end{equation*}
By introducing the elements $H_{kl,mn}\overset{\text{def}}{=}\langle g_{kl},\boldsymbol{\HH}\gamma_{mn}\rangle$ of the
channel matrix $H\in\mathbb{C}^{{\mathcal{I}}\times{\mathcal{I}}}$,
the multicarrier transmission can be formulated as the linear equation
\mbox{$\tilde{\boldsymbol{x}}=H\boldsymbol{x}+\tilde{\boldsymbol{n}}$},
where $\tilde{\boldsymbol{n}}$ is the vector of the projected noise having a
power of ${\sigma^2}$ per component.
We assume that the receiver has perfect channel knowledge
(given by $\boldsymbol{\Sigma}(\tau,\nu)$), i.e.
single carrier based equalization in the absence of noise would be
$\tilde{x}^{\text{eq}}_{kl}=\tilde{x}_{kl}/H_{kl,kl}$, with
\begin{equation*}
\begin{aligned}
H_{kl,kl}
&=\langle g_{kl},\boldsymbol{\HH}\gamma_{kl}\rangle
=\iint \boldsymbol{\Sigma}(\tau,\nu)\langle g_{kl},{\boldsymbol{S}}_{(\tau,\nu)}\gamma_{kl}\rangle d\tau d\nu\\
&\overset{\text{def}}{=}\iint \boldsymbol{\Sigma}(\tau,\nu)e^{-i2\pi(\tau kF-\nu lT)}{\mathbf{A}}_{g\gamma}(\tau,\nu) d\tau d\nu
\end{aligned}
\end{equation*}
where ${\mathbf{A}}_{g\gamma}(\tau,\nu)=\langle g,{\boldsymbol{S}}_{(\tau,\nu)}\gamma\rangle$
is the cross ambiguity function of the pulse pair $\{g,\gamma\}$.
\section{Problem Statement}
Considering only single carrier equalization, it is
natural to require $a\overset{\text{def}}{=}|H_{kl,kl}|^2$ (the channel gain) to be maximal and the interference power
$b\overset{\text{def}}{=}\sum_{(kl)\neq(mn)}|H_{kl,mn}|^2$ to be minimal as possible. This addresses the concept of
{\it pulse shaping}. However to be practicable,
the pulses should be adapted to the second order statistics only, given by $\boldsymbol{C}(\tau,\nu)$
and {\bf not} to a particular channel realization $\boldsymbol{\Sigma}(\tau,\nu)$.
Hence, we aim at maximization of
\begin{equation*}
\text{\small{\rm{SINR}}}\overset{\text{def}}{=}{\frac{\Ex{\boldsymbol{\HH}}{a}}{{\sigma^2}+\Ex{\boldsymbol{\HH}}{b}}}
\end{equation*}
by proper design of $\gamma$ and $g$. Up to very few
special cases the analytical solution of this global optimization problem (jointly non-convex in $(\gamma,g)$)
is unknown. However numerical optimization methods
are presented in \cite{schafhuber:pimrc02,schniter:allerton03,jung:spawc2004}. Following our previous work \cite{jung:spawc2004} we simplify
the problem by proposing a relaxation, which separates the problem into two steps.
Upper bounding \mbox{$\Ex{\boldsymbol{\HH}}{b}\leq B_\gamma - \Ex{\boldsymbol{\HH}}{a}$} gives a lower bound on
$\text{\small{\rm{SINR}}}$ (see \cite{jung:spawc2004}), where $B_\gamma$ is the so called Bessel bound of $\{\gamma_{mn}\}$
\cite{christensen:framesandrieszbases}.
In this paper we focus on the first step only where $\Ex{\boldsymbol{\HH}}{a}$ should be maximized. This gives
the following optimization problem
\begin{equation}
\begin{aligned}
\{\gamma^{\text{\rm (opt)}},g^{\text{\rm (opt)}}\}
&=\arg\max{\Ex{\boldsymbol{\HH}}{a}}\\
&=\arg\max_{\lVert\gamma\rVert_2=\lVert g\rVert_2=1}{\int|{\mathbf{A}}_{g\gamma}(\tau,\nu)|^2d\mu}
\end{aligned}
\label{eq:gainoptimization}
\end{equation}
where $d\mu\overset{\text{def}}{=}\boldsymbol{C}(\tau,\nu)d\tau d\nu$.
In this context it was first introduced in \cite{kozek:nofdm1} respectively
\cite{kozek:thesis}, but similar problems already
occurred in radar literature much earlier.
In particular for the elliptical symmetry of $\boldsymbol{C}(\tau,\nu)$ Hermite functions
establish local extremal points as found in \cite{kozek:thesis}. The
scaling rule for fixed pulses was studied in \cite{kozek:thesis,liu:orthogonalstf}.
Also it is possible to find a
close relation to the channel fidelity and minimum output entropy states
in quantum information theory as we will show later on.
In particular the important class of Gaussian scattering profiles (corresponding to classical
bosonic quantum channels) was already addressed in \cite{arxiv:0404005} and
\cite{arxiv:0409063}.
Out of the scope of this paper is the second step, in which the minimization of $B_\gamma$
(which depends on $\gamma^{\text{\rm (opt)}}$) is achieved. This well known procedure \cite{strohmer:lofdm2}
(in the case of $TF>1$ that is to find the ''nearest'' orthogonal Gabor basis with respect to the ${\mathcal{L}}_2$-norm) is also described in \cite{jung:spawc2004}.
Unfortunately the resulting pulses will be in general again a suboptimal solution of ($\ref{eq:gainoptimization}$).
See \cite{strohmer:dualgaborframes} for a discussion of this problem.
Nevertheless, this separation and therefore (\ref{eq:gainoptimization}) opens up
analytical insights into the pulse design problem.
\section{Contributions}
The original formulation of the pulse design problem in (\ref{eq:gainoptimization})
hides the internal group structure induced by the time-frequency shift operators.
In this paper we derive a lower bound for the optimization
functional (\ref{eq:gainoptimization}) on which we can exploit this structure explicitely.
Moreover we sketch that the results will hold in the direct problem with minor restrictions.
We present an operator--algebraic reformulation by utilizing representation theory
of the Weyl--Heisenberg group. Our approach relates the optimal pulses to approximate
eigenstates of pseudo differential operators. The procedure naturally embeds
the concepts of pulse scaling and optimal time-frequency offsets (or phase space displacement).
Then we extent our framework to
provide exact solutions for the class of Gaussian scattering profiles.
Because the underlying theory is partially
not very common in multicarrier community we will give a short introduction to the few properties
we will need for our investigation. More details can be found in \cite{folland:harmonics:phasespace}.
\subsection{The Weyl-Heisenberg Group and Pseudo differential Operators}
The two families of shift operators ${\boldsymbol{S}}_{(\tau,0)}$ and ${\boldsymbol{S}}_{(0,\nu)}$
are unitary representations of the group corresponding to
the real line $\mathbb{R}$ with addition as group operation. The extension to $\mathbb{R}^2$ in the sense
of
\begin{equation}
{\boldsymbol{S}}_{(\alpha,\beta)}\cdot {\boldsymbol{S}}_{(\gamma,\delta)}=e^{-i2\pi\alpha\delta}{\boldsymbol{S}}_{(\alpha+\gamma,\beta+\delta)}
\label{eq:shift:commutationrule}
\end{equation}
is not closed because of the phase factor.
Closeness is achieved by introducing the torus ($\mathbb{T}$) as the third variable, i.e.
\begin{equation*}
e^{i2\pi\phi}{\boldsymbol{S}}_{(\alpha,\beta)}\cdot e^{i2\pi\psi}{\boldsymbol{S}}_{(\gamma,\delta)}=
e^{i2\pi(\phi+\psi-\alpha\delta)}{\boldsymbol{S}}_{(\alpha+\gamma,\beta+\delta)}
\end{equation*}
The corresponding group $\mathbb{H}=\mathbb{R}\times\mathbb{R}\times\mathbb{T}$ with the group law
$(\alpha,\beta,\phi)(\gamma,\delta,\psi)=(\alpha+\gamma,\beta+\delta,\phi+\psi-\alpha\delta)$ is called
the (reduced\footnote{The addition in third component is taken to be
mod $1$. Otherwise this yields the (full) polarized Heisenberg group with non-compact center.})
polarized {\it Heisenberg group} (HG).
The HG can be represented as a group of upper triangular matrices by the
group homomorphism
\begin{equation*}
(\alpha,\beta,\phi)\rightarrow
H(\alpha,\beta,\phi)=
\left(\begin{array}{ccc}
1 & \alpha & \phi \\
0 & 1 & \beta \\
0 & 0 & 1 \\
\end{array}\right)
\end{equation*}
where the group action is matrix multiplication.
The matrices $h(\alpha,\beta,\phi)=H(\alpha,\beta,\phi)-1$ written with
$d=(1,0,0)$, $x=(0,1,0)$ and $e=(0,0,1)$
as $h(\alpha,\beta,\phi)=\alpha h(d)+\beta h(x)+\phi h(e)$ are clearly isomorphic to
$\mathbb{R}^3$ and with the matrix commutator they turn into a Lie algebra. The Lie bracket
in this case is $\left[(\alpha,\beta,\phi),(\gamma,\delta,\psi)\right]\overset{\text{def}}{=}(0,0,\alpha\delta-\beta\gamma)$.
Due to the bilinearity of the Lie bracket this can be shortly written as the
{\it Heisenberg Commutation Relations}, i.e.
\mbox{$\left[d,x\right]=e\,\,\,\left[x,e\right]=0\,\,\,\left[d,e\right]=0$}.
That this is exactly the Heisenberg algebra connected to the HG follows from
$h(\alpha,\beta,\phi)^2=h(0,0,\alpha\beta)$ and $h(\alpha,\beta,\phi)^n=0$ for $n>2$.
The exponential map of the matrix $h(\alpha,\beta,\phi)$ is then given as
\begin{equation}
\begin{split}
e^{h(\alpha,\beta,\phi)}
&=\sum_{n=0}^\infty\frac{h(\alpha,\beta,\phi)^n}{n!}
=1+h(\alpha,\beta,\phi)+\frac{1}{2}h(0,0,\alpha\beta)\\
&= H(\alpha,\beta,\phi+\frac{1}{2}\alpha\beta)
\raisetag{2em}
\end{split}
\label{eq:heisenberg:nilpotent}
\end{equation}
Thus, it maps the Heisenberg algebra to the unpolarized HG.
The series expansion is finite (the elements $h(\alpha,\beta,\phi)$ are nilpotent
endomorphisms).
Returning to the polarized Heisenberg group we transform finally
\mbox{$H(\alpha,\beta,\phi)=H(0,0,-\frac{1}{2}\alpha\beta)e^{h(\alpha,\beta,\phi)}$}.
To establish the connection to ${\boldsymbol{S}}_{(\alpha,\beta)}$ considered as operators on ${\mathcal{S}}(\mathbb{R})$
(the Schwartz space of rapidly decreasing functions) we have to switch to the so called
{\it Schr\"odinger representation}.
In this picture the hermitian operators ${\boldsymbol{X}}$ and ${\boldsymbol{D}}$ with
\begin{equation*}
\begin{split}
({\boldsymbol{X}} f)(t)&\overset{\text{def}}{=} tf(t) \\
({\boldsymbol{D}} f)(t)&\overset{\text{def}}{=} \frac{1}{2\pi i}f'(t)
\end{split}
\end{equation*}
setup a basis representation for the Heisenberg Lie algebra.
The skew-hermitian operators $2\pi i{\boldsymbol{X}}$
(generates the frequency shifts), $2\pi i{\boldsymbol{D}}$ (generates the time shifts)
and $2\pi i\boldsymbol{E}=2\pi i$ ($E$ is the identity) correspond
to $x,d$ and $e$. They give again the Heisenberg commutation rules,
hence linear combinations of them fulfill the same Lie bracket (the commutator of linear operators) and consequently
\mbox{$(\tau,\nu,s)\rightarrow d\rho(\tau,\nu,s)=2\pi i(s+\nu{\boldsymbol{X}}+\tau{\boldsymbol{D}})$}
is again a Lie algebra isomorphism for the Heisenberg algebra.
As in (\ref{eq:heisenberg:nilpotent}) the HG is then given by exponentiation, i.e. the so called Weyl operator is
given as
\begin{equation*}
\begin{aligned}
\rho(\tau,\nu,s)
&=e^{d\rho(\tau,\nu,s)}=e^{2\pi i(s+\nu{\boldsymbol{X}}+\tau{\boldsymbol{D}})}
=e^{2\pi is}e^{\pi i\tau\nu}{\boldsymbol{S}}_{(-\tau,\nu)}
\end{aligned}
\end{equation*}
With $\rho(\tau,\nu)\overset{\text{def}}{=}\rho(\tau,\nu,0)$ we have
${\boldsymbol{S}}_{(\tau,\nu)}=e^{\pi i\tau\nu}\rho(-\tau,\nu)=e^{\pi i\tau\nu}e^{2\pi i(\nu{\boldsymbol{X}}-\tau{\boldsymbol{D}})}$, i.e.
integrals over shift operators as in (\ref{eq:rxsignal}) are in fact pseudo differential operators \cite{folland:harmonics:phasespace}
of the following spreading representation (the Weyl transform)
\begin{equation*}
\begin{aligned}
\sigma({\boldsymbol{D}},{\boldsymbol{X}})=\iint \fourier{\sigma}(\tau,\nu)e^{2\pi i(\nu{\boldsymbol{X}}+\tau{\boldsymbol{D}})}d\tau d\nu
\end{aligned}
\end{equation*}
$\fourier{\sigma}(\tau,\nu)$ is called the spreading function (or representing function, i.e.
the 2D Fourier transform of the symbol $\sigma(d,x)$ of the operator $\sigma({\boldsymbol{D}},{\boldsymbol{X}})$).
\subsection{The WSSUS Pulse Design Problem}
Straight forward calculation
shows now that the squared magnitude of the cross ambiguity function
$|{\mathbf{A}}_{g\gamma}(\tau,\nu)|^2$ can be written in the following form
\begin{equation}
\begin{aligned}
|{\mathbf{A}}_{g\gamma}(\tau,\nu)|^2
&=\langle g,{\boldsymbol{S}}_{(\tau,\nu)}\gamma\rangle\langle\gamma,{\boldsymbol{S}}^*_{(\tau,\nu)}g\rangle\\
&=\Trace{\,G{\boldsymbol{S}}_{(\tau,\nu)}\Gamma{\boldsymbol{S}}^*_{(\tau,\nu)}}
\end{aligned}
\end{equation}
where $G$ ($\Gamma$) is the (rank-one) orthogonal projector onto $g$ ($\gamma$).
By that transformation we emphasize that $\Gamma$ undergoes a linear transformations
before being projected onto $g$. This special kind of linear transformation is also
called a unitary evolution, which preserve the spectrum of $\Gamma$ (in our
case the rank). This obviously does not hold in generality if taking the sum over
different unitary evolution of the same argument.
Hence, we collect them together by defining affine maps $A$ and $\tilde{A}$ such that
\begin{equation}
\begin{aligned}
\Ex{\boldsymbol{\HH}}{a}
&=\Trace{\,G[\int{\boldsymbol{S}}_{(\tau,\nu)}\Gamma{\boldsymbol{S}}^*_{(\tau,\nu)}}d\mu]
\overset{\text{def}}{=}\Trace{\,G A(\Gamma)}\\
&=\Trace{\,\Gamma[\int{\boldsymbol{S}}^*_{(\tau,\nu)}G{\boldsymbol{S}}_{(\tau,\nu)}}d\mu]
\overset{\text{def}}{=}\Trace{\,\Gamma\tilde{A}(G)}\\
\end{aligned}
\label{eq:cpmap:introduction}
\end{equation}
The main reason for this reformulation is the notion of {\it completely positive
maps} (CP-maps) \cite{stinespring:positivefunctions}
which directly apply on the pulse design problem.
CP-maps like $A(\cdot)$
received much attention due to its application in quantum
information theory.
Before going more in detail, let us define $\mathcal{T}_1$ as the set
of trace class operators. The set
\mbox{$M_1\overset{\text{def}}{=}\{z\,|\,z\in\mathcal{T}_1,z=z^*,z\geq0,\Trace{\,z}=1\}$}
is a convex subset of $\mathcal{T}_1$. With $Z$ we will denote the extremal
boundary of $M_1$, which is the set of all orthogonal rank-one projectors.
With the definition of $\tilde{A}$ in (\ref{eq:cpmap:introduction}) follows that
$\tilde{A}$ is adjoint of $A$ with respect to the inner product $\Trace{X^*Y}$.
Due to $\lVert\boldsymbol{C}\rVert_1=1$ both maps are trace preserving $\Trace{A(X)}=\Trace{X}$.
Moreover they are hermiticity preserving $A(X)^*=A(X^*)$ and
entropy increasing $X\succ A(X)$ ($\succ$ is the partial
order due to eigenvalue majorization).
The complete positivity and the trace-preserving property is ensured by
\begin{equation*}
\begin{aligned}
\int d\mu {\boldsymbol{S}}^*_{(\tau,\nu)}{\boldsymbol{S}}_{(\tau,\nu)}={\mathbb{I}}
\end{aligned}
\end{equation*}
With this framework we can write now the optimization problem as
\begin{equation}
\begin{aligned}
\max_{G,\Gamma\inZ}{\Trace{\,G A(\Gamma)}}
\end{aligned}
\label{eq:optimization:trace}
\end{equation}
where $\Gamma$ represent the transmitter and the CP-map $A(\cdot)$ represent
the ''averaged'' action of the channel and $G$ is the receiver.
This formulation is similar to the channel fidelity in quantum information
processing. In fact - the problems are equivalent if considering so called
pure states. The initial preparation of a pure quantum state (the symbol to transmit)
is represented by a so called rank-one density operator (in our case $\Gamma$). The quantum
channel is represented by a CP-map $A(\cdot)$ having again a density operator as its output.
The measurement (the detection of the transmitted symbol) is performed in our case with $G$.
Obviously either $G$ or $\Gamma$ can be dropped in the optimization, i.e.
\begin{equation}
\begin{aligned}
\max_{G,\Gamma\inZ}{\Trace{\,G A(\Gamma)}}
=\max_{\Gamma\inZ}{\lVert A(\Gamma)\rVert_\infty}
=\max_{G\inZ}{\lVert \tilde{A}(G)\rVert_\infty}
\end{aligned}
\label{eq:reformulation:pulsedesign:rankone}
\end{equation}
where $\lVert\cdot\rVert_\infty$ denotes the operator norm.
This measure represents the maximum achievable
purity of the output of a quantum channel with pure states as input.
Turning back to the language of WSSUS signaling, this represents the maximum achievable
''energy'' which can be collected by a single pulse if communicating with the optimal
pulse $\gamma$ over a large ensemble of WSSUS channels. CP-maps over the Heisenberg group
have some more important properties. One is the covariance property with respect to
group elements which follows from (\ref{eq:shift:commutationrule}), i.e.
\begin{equation}
A({\boldsymbol{S}}_{(\tau,\nu)}\Gamma{\boldsymbol{S}}^*_{\tau,\nu})={\boldsymbol{S}}_{(\tau,\nu)}A(\Gamma){\boldsymbol{S}}^*_{(\tau,\nu)}
\label{eq:covariance}
\end{equation}
The physical meaning is that (\ref{eq:optimization:trace}) is invariant with respect to
common time--frequency shifts of $G$ and $\Gamma$. A trivial but important conclusion is that Weyl-Heisenberg (Gabor)
signaling is a reasonable scheme, which guarantees the same performance on all lattice points.
Alternatively it can be viewed in the quantum picture as symbol alphabet
of pure states achieving all the same fidelity.
From (\ref{eq:covariance}) follows furthermore that different maps $A_1$ and $A_2$ commute, i.e.
$A_1\circ A_2=A_2\circ A_1$.
Coming back to the formulation of pulse design problem in (\ref{eq:reformulation:pulsedesign:rankone})
we can finally relax the constraint set from $Z$ to $M_1$ which gives
\begin{equation}
\begin{aligned}
\max_{\Gamma\inM_1}{\lVert A(\Gamma)\rVert_\infty}
=\max_{G\inM_1}{\lVert \tilde{A}(G)\rVert_\infty}
\end{aligned}
\end{equation}
provided by the convexity of $\lVert\cdot\rVert_\infty$ and linearity of $A(\cdot)$.
To the authors knowledge this reformulation of the pulse design criterion
as a convex maximization problem seems to be new.
Without further investigations of the analytical structure of $A(\cdot)$
such global-type optimization problems are in general difficult to solve.
Therefore we will emphasize in the following more on the Heisenberg group structure contained in
$A(\cdot)$.
\subsection{The Schr\"odinger Representation}
The connection between Weyl operators (the unitary representations of the Weyl-Heisenberg
group in the Schr\"odinger picture) and ${\boldsymbol{S}}_{(\tau,\nu)}$ will reveal the fundamental role
of Gaussians in WSSUS signaling. We will show this first in a simpler lower bound analysis
which mainly admits the same maximizer as the original problem (given in the appendix).
Thus, coming back now to (\ref{eq:gainoptimization}) and let
$(\tau_0,\nu_0)$ be an arbitrary offset between $g$ and $\gamma$ in the time-frequency plane,
hence we define $\tilde{\gamma}={\boldsymbol{S}}_{(\tau_0,\nu_0)}\gamma$.
\begin{equation}
\begin{split}
\Ex{\boldsymbol{\HH}}{a}
&=\int |{\mathbf{A}}_{g\gamma}(\tau,\nu)|^2d\mu
=\int |\langle g,{\boldsymbol{S}}_{(\tau-\tau_0,\nu-\nu_0)}\tilde{\gamma}\rangle|^2d\mu\\
&=\int |\langle g,\rho(-\tau+\tau_0,\nu-\nu_0)\tilde{\gamma}\rangle|^2d\mu\\
&\geq(\int|\langle g,\rho(-\tau+\tau_0,\nu-\nu_0)\tilde{\gamma}\rangle| d\mu)^2\\
&\geq|\int\langle g,\rho(-\tau+\tau_0,\nu-\nu_0)\tilde{\gamma}\rangle d\mu|^2\\
&=|\langle g,[\int\rho(-\tau+\tau_0,\nu-\nu_0)d\mu]\tilde{\gamma}\rangle |^2
\overset{\text{def}}{=}|\langle g,\mathcal{L}\tilde{\gamma}\rangle |^2
\raisetag{6em}
\end{split}
\label{eq:lowerbound}
\end{equation}
In the latter we used Jensen's inequality\footnote{It can be shown that ${\mathbf{A}}_{g\gamma}\inL_1(\mu)$.}
($\int d\mu=\lVert\boldsymbol{C}\rVert_1=1$, see also \cite{jung:spawc2004}).
We will use now (\ref{eq:lowerbound}) for further analytical studies. The bound
becomes sharp iff $\xi{\mathbf{A}}_{g\gamma}(\tau,\nu)\in\mathbb{R}$ is constant on $\text{supp}\,\boldsymbol{C}$
for some $\xi\in\mathbb{T}$, hence is well suited for
underspread channels.
The operator $\mathcal{L}$ is a pseudo differential operator with spreading function
$\fourier{\sigma}(\tau,\nu)=-\boldsymbol{C}(\tau_0-\tau,\nu+\nu_0)$.
\begin{equation*}
\mathcal{L}=\iint-\boldsymbol{C}(\tau_0-\tau,\nu+\nu_0)e^{2\pi i(\nu{\boldsymbol{X}}+\tau{\boldsymbol{D}})}d\tau d\nu
\end{equation*}
\noindent{\bf Local approximation:}
The nilpotent property with respect to the matrix product celebrated in (\ref{eq:heisenberg:nilpotent})
unfortunately does not translate into the Schr\"odinger picture, so that
\begin{equation}
\begin{split}
{\boldsymbol{S}}_{(\tau,\nu)}
&=e^{\pi i\tau\nu}\rho(-\tau,\nu)\\
&=e^{\pi i\tau\nu}[1+d\rho(-\tau,\nu)+\frac{1}{2}d\rho(-\tau,\nu)^2]+o(2)\\
&\approx e^{\pi i\tau\nu}[1+2\pi iK-2\pi^2K^2]
\raisetag{3.5em}
\vspace*{1em}
\end{split}
\label{eq:shift:localapprox}
\end{equation}
with the hermitian operator $K\overset{\text{def}}{=}\nu{\boldsymbol{X}}-\tau{\boldsymbol{D}}$ holds only as an approximation
(for $\tau$ and $\nu$ being small), i.e. gives a local approximation of $\mathcal{L}$ which is
\begin{equation*}
\begin{aligned}
L\overset{\text{def}}{=} C_{00}
&+2\pi i(C_{01}{\boldsymbol{X}}-C_{10}{\boldsymbol{D}})\\
&-2\pi^2(C_{02}{\boldsymbol{X}}^2+C_{20}{\boldsymbol{D}}^2-C_{11}[{\boldsymbol{X}}{\boldsymbol{D}}+{\boldsymbol{D}}{\boldsymbol{X}}])\\
\end{aligned}
\end{equation*}
where $C_{mn}=\iint\boldsymbol{C}(\tau,\nu)(\tau-\tau_0)^m(\nu-\nu_0)^n$
are the moments of the scattering function around $(\tau_0,\nu_0)$.
Because ${\boldsymbol{X}}$ and ${\boldsymbol{D}}$ are hermitian operators, $L$ is hermitian too if
$C_{mn} i^{m+n}\in\mathbb{R}$ for
$m,n=0,1,2$. In this case the optimization problem is an eigenvalue problem.
Moreover then it follows that $g=\alpha L\tilde{\gamma}$ for some $\alpha\in\mathbb{C}$, because only in this case
equality in $|\langle g,L\tilde{\gamma}\rangle|\leq\lVert g\rVert_2\lVert L\tilde{\gamma}\rVert_2$
is achieved. $L$ can be made hermitian if we choose
$\tau_0=\lVert\tau\boldsymbol{C}\rVert_1$ and $\nu_0 =\lVert \nu\boldsymbol{C}\rVert_1$, so
that $C_{10}=C_{01}=0$. Thus we have
\begin{equation*}
L= C_{00}-2\pi^2[C_{02}{\boldsymbol{X}}^2+C_{20}{\boldsymbol{D}}^2-C_{11}({\boldsymbol{X}}{\boldsymbol{D}}+{\boldsymbol{D}}{\boldsymbol{X}})]
\end{equation*}
which is an hermitian differential operator of second order.
With
\begin{equation*}
(d_\alpha f)(t)\overset{\text{def}}{=} \frac{1}{\sqrt{\alpha}}f(t/\alpha)
\end{equation*}
we define now dilated functions
\mbox{$g_\alpha\overset{\text{def}}{=} d_\alpha g$}, \mbox {$\tilde{\gamma}_\alpha\overset{\text{def}}{=} d_\alpha\tilde{\gamma}$}
and the dilated operator $L_\alpha\overset{\text{def}}{=} d_\alpha L d_{1/\alpha}$.
Using furthermore that
\begin{equation}
\begin{split}
d_\alpha{\boldsymbol{X}} d_{1/\alpha}&=\frac{1}{\alpha}{\boldsymbol{X}}\\
d_\alpha{\boldsymbol{D}} d_{1/\alpha}&=\alpha{\boldsymbol{D}}\\
\end{split}
\label{eq:operator:dilation}
\end{equation}
we get
\begin{equation*}
\begin{aligned}
\langle g_\alpha,L_\alpha\tilde{\gamma}_\alpha\rangle
=\langle g_\alpha, C_{00}&-2\pi^2[\frac{C_{02}}{\alpha^2}{\boldsymbol{X}}^2+C_{20}\alpha^2{\boldsymbol{D}}^2\\
&-C_{11}({\boldsymbol{X}}{\boldsymbol{D}}+{\boldsymbol{D}}{\boldsymbol{X}})]
\tilde{\gamma}_\alpha\rangle
\end{aligned}
\end{equation*}
and with $\alpha^4=C_{02}/C_{20}$ the phase space symmetric version
\begin{equation*}
\begin{aligned}
L_\alpha=2\pi^2\sqrt{C_{02}C_{20}}\{\kappa-[{\boldsymbol{X}}^2 + {\boldsymbol{D}}^2-
C_{11}({\boldsymbol{X}}{\boldsymbol{D}}+{\boldsymbol{D}}{\boldsymbol{X}})]\}
\end{aligned}
\end{equation*}
where the constant is $\kappa=\frac{C_{00}}{2\pi^2\sqrt{C_{02}C_{20}}}$ (with
our WSSUS assumptions follows also $C_{00}=1$).
For simplicity let us assume that the shifted scattering
function is separable yielding $C_{11}=0$. In the general case the $C_{11}$-term
can be removed using a proper symplectic transformation
(see for example \cite{taylor:noncommutative:harmonic:analysis}).
The eigenfunction of the so called sub-Laplacian (or the harmonic oscillator
Hamiltonian) ${\boldsymbol{X}}^2 + {\boldsymbol{D}}^2$
are the Hermite functions $h_n$, with $({\boldsymbol{X}}^2 + {\boldsymbol{D}}^2)h_n=\frac{2n+1}{2\pi}h_n$.
Therefore it follows that
\begin{equation*}
L_\alpha h_n=(C_{00}-\pi\sqrt{C_{02}C_{20}}(2n+1))h_n
\end{equation*}
Hence in local approximation the maximization problem is solved by
$h_0$, i.e. $g=d_{1/\alpha}h_0$ and $\gamma={\boldsymbol{S}}^{-1}_{(\tau_0,\nu_0)}d_{1/\alpha}h_0$
which are both scaled and proper separated Gaussians (the ground state of the harmonic oscillator).
This is an important (and expected) result for the pulse design problem in WSSUS channels. It includes
the concepts of pulse scaling (by $d_{1/\alpha}$) and proper phase space displacement
(by ${\boldsymbol{S}}^{-1}_{(\tau_0,\nu_0)}$) as natural operations.
However this approximations is only valid for \mbox{$C_{02}C_{20}\ll 1$} (underspread channel), such that
$(C_{00}-\pi\sqrt{C_{02}C_{20}}(2n+1))>0$. We obtain the same solutions in the original problem
if we apply this approximation (see the appendix). Next we will derive cases where this approximation turns out to be
is exact. \\[1em]
\noindent{\bf Gaussian scattering functions:}
Let us assume that after performing proper pulse scaling and separation the scattering function is given
as the symmetric Gaussian $\boldsymbol{C}(\tau,\nu)=\frac{\alpha}{2}e^{-\frac{\pi}{2}\alpha(\tau^2+\nu^2)}$
where $0<\alpha\in\mathbb{R}$. If $\alpha\gg 1$ the channel is underspread.
It can be shown that then $\mathcal{L}$ essentially self--adjoint, hence the maximum in
(\ref{eq:lowerbound}) is again achieved by eigenfunctions of $\mathcal{L}$. Operators having such spreading functions
are contained in the so called {\it oscillator semigroup} \cite{howe:oscillatorsemigroup} and for $\alpha>1$
they have the representation \cite{folland:harmonics:phasespace}
\begin{equation*}
\mathcal{L}=e^{-2\pi\text{arcoth}\,\alpha({\boldsymbol{X}}^2+{\boldsymbol{D}}^2)}
\end{equation*}
Thus we have that
$\mathcal{L}\cdot h_n=e^{-(2n+1)(\text{arcoth}\,\alpha)}h_n$, hence $h_0$ is the optimum
of (\ref{eq:lowerbound}).
The special case $\alpha=1$ can be included by observing that then
$\boldsymbol{C}(\tau,\nu)\sim{\mathbf{A}}_{h_0h_0}(\tau,\nu)$. Such pseudo differential operators perform simple
projections, in this case onto the span of $h_0$. Note
that for $\fourier{\sigma}(\tau,\nu)=\langle \phi,\rho(\tau,\nu)\psi\rangle$ follows
\begin{equation*}
\begin{split}
\langle g,\sigma({\boldsymbol{D}},{\boldsymbol{X}})\gamma\rangle
&=\langle\fourier{\sigma},\langle g,\rho(\cdot,\cdot)\gamma\rangle\rangle\\
&=\langle\langle \phi,\rho(\cdot,\cdot)\psi\rangle,\langle g,\rho(\cdot,\cdot)\gamma\rangle\rangle\\
&=\langle{\mathbf{A}}_{\phi\psi},{\mathbf{A}}_{g\gamma}\rangle
=\langle g,\phi\rangle\langle\psi,\gamma\rangle
\end{split}
\end{equation*}
Thus $\sigma({\boldsymbol{D}},{\boldsymbol{X}})$ is a rank one projector
(orthogonal in the case $\psi=\phi$) if $\langle\phi,\psi\rangle=1$.\\[1em]
Finally we conclude that for underspread channels the Gaussian pulse shape is an approximate solution of
(\ref{eq:lowerbound}) which becomes more optimal as the support of $\boldsymbol{C}$ decreases.
Furthermore the solution is exact for a Gaussian scattering function.
In the quantum channel context the same arguments hold for coherent states (phase space translated Gaussians).
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 7,846
|
How Did Contemporary Furniture, Become?
Contemporary furniture evolved in the 20th century. Furniture back then was more about intricate patterns and carvings, where pieces were made to dominate a room. Designers of furniture then began to design for functionality rather than splendour. Focusing more on simplified designs that enhanced a room and at the same time could be used in a more functional role.
Once the contemporary furniture movement picked up pace, it changed how designers and consumers thought about furniture and furnishing room interiors. It also changed the way materials are used in the creation of contemporary furniture. You could call early contemporary designers the first eco-friendly manufacturers. Today, the issue still remains, much is made of the materials that are used to help create interior furnishings. Mainstream furnishing giants like Ikea, who mass-produce, flat-packed, contemporary ranges are forever making statements to defend that they are environmentally friendly. Such is the movement towards greener issues. Consumer awareness of how a product was made has placed environmental issues at the very heart of modern contemporary furniture pieces.
Then finally, is the environment that the furniture will be part of. Personal taste is a big part on how we furnish the interiors of our homes and workplaces. Many various styles exist, however, minimalist is probably the most fashionable in today's environment. Remember, contemporary furniture is more about an individuals choice and taste, where customisation rules supreme and the furniture compliments the room within which it is placed.
Simplistic, elegant and organic are three words that I would form in any sentence to help describe what modern furniture design is to a person. Of course, when it comes to writing about modern furniture, words never quite do it justice. There is only one way to truly connect with these design pieces and that is through visual participation. Modern furniture design is forever evolving and can create powerful emotions between a piece and a person. It is this connection, that is the hardest thing to explain in written words.
Much like the way artists move people with their canvas creations, thus modern furniture design also moves. It can be from humorous, novelty chairs to sleek and elegant dining tables. Materials also play a part in the creation of modern furniture design. Recycled plastics, traditional wood, chrome and fabrics bring these creative masterpieces to life.
Our featured modern furniture design products today are the characteristic novelty chairs. UK Contemporary Furniture love these quirky, yet functional, novelty chairs, designed to blend with any personality.
Contemporary furniture is defined as pieces of furniture that are belonging to the same period of time. Meaning that contemporary furniture that was designed in the sixties was furniture that was modern and fitted with the design styles for that era (the sixties). So, when people talk of contemporary furniture or on trend kitchens, it may mean modern furniture of today or modern furniture from a time in the past. Confusing, isn't it. However, a sixties contemporary piece of furniture may still look modern in today's environment, its all about personal judgement and style.
What does your door say about you? Does your front exterior door look inviting and stylish or are you more of a stay well away from my abode type person. Y'see, doors do communicate a message to those who live outside your habitual domain. I'm not meaning the small stickers, either, telling cold-calling salesmen to stay well away. There are very many styles of doors of which to choose to help fit in with your neighbourhood environment.
From slick Italian wooden finished doors to decorative glass doors to sliding double doors that just drip with design elegance. We have featured a few modern styles, to help you see the benifits of fitting a contemporary door to your house. Please use the comments to add any favourites of your own.
With the new school term about to start, many parents will be giving a big sigh of relief as they watch their little ones walk out the door and head towards the school gates for the first time in almost seven weeks. As parents, we always try to encourage our kids to do well academically and this often means study time at home. Well, as parents, we all know trying to get your kids to study at home is easier said than done – social networking sites, X-Boxes, Playstations and Wii games consoles can often take precedence over school study time.
One of the best ways to help encourage your kids to study is to have a little area that is totally dedicated for that purpose, here at UK Contemporary Furniture we have a wide range of modern home office furniture and chairs that will fit into any study/work room making sure that there is no distractions while your children are hard studying.
It used to be, that when we thought of tv stands, we conjured up images of large, solid, stiff clumps of varnished wood that unsightly dominated the corner of your living room. Thankfully, in recent times, designers and manufactures have transformed the look of the humble tv stand creating stunning design pieces made from both modern and traditional materials. In many homes in Britain the television is still the centre piece or vocal point of a room so it is fitting that the furniture which supports it should also have that contemporary, designer feel to it as well.
Over the last few years, the cost of high definition televisions has dropped dramatically, making them more appealing and affordable to everyone. One of the discussions/fallouts that most couples have when selecting their new flat screen TV, apart from the size issue, will be where to place it once they get it home. Some will prefer it hanging on the living room or bedroom wall, while others believe that having it rest on a television stand is the more appealing look.
One of the best furniture materials is oak; this wood has been used for centuries and has lasted the tests of time so that in itself tell's us something. Did you know that it takes an oak tree about 150 years to reach maturity? And with around 400 species it's not surprising that it comes in so many different shades? This wood often has a tight grain giving it a beautiful effect and with the red or white shade it will compliment any home.
If your looking for a piece of wooden furniture that will compliment your taste (no matter, what that is) then look no further than the Large Fama Sideboard this stunning piece will sit perfectly in any living room, dining room or even your kitchen. In today's society, quality is everything, and you get what you pay for, well this is the exception, this is an exceptional piece and an exceptional price.
When it comes to choosing contemporary furniture for your home or office, many residents are starting to realise the true beauty of Solid Oak Furniture. As well as it's natural texture and tone, oak furniture will give it's owners many years of pleasure, which is why solid oak is now one of the most popular materials people choose when it comes to redecorating their homes.
One of the reasons people choose oak furniture is due to the way the grain runs through the wood, giving each piece a unique finish thus making each design special. That alone is why oak furniture has the ability to turn an ordinary room into something warm, inviting and special. People that purchase oak furniture understandably want it to be the focal point in their new room.
If you're looking to turn your living room, dinning room, bedroom or even home office into somewhere special then have a look through our online catalogue, we're bound to have that special piece you desire at a cost you can afford.
With the start of a New Year a lot of us will be starting it with new rules and regulations and one may be to eat as a family around a dining table – no more eating off your lap and watching TV. Sitting around a dinner table gives us a sense of family belonging. But what if you don't have a dining room table? How do you start to look for one that will suit your preferred needs?
Dinner tables come in many sizes, colours and styles and they can be used for a variety of purposes from helping the kids with their homework, to the traditional Sunday roast to hosting dinner parties. Even when they are not in use they can become a centre piece to your room with a nice vase and some freshly, picked flowers.
When looking for a dining table the first thing to do is to find out what space you have available. No point in buying one that seats 6 people when you have limited space and the dining chairs will not fit.
Remember, when choosing a dining room table do not compromise on quality, purchase a solid wood table and insist on one which is not made using veneers?
What is about the sight of the Sun that makes all a lot more friendly, people walking down the street, smiling even stopping and taking the time to chat, "isn't it a lovely day" can be heard all over the high street. As the weather begins to warm up many residents will hopefully be planning the traditional summer BBQ parties in their gardens.
After the cold depressing winter months there is nothing more relaxing (and better) than sitting in the garden enjoying your favourite tipple in the hot evening sun. As the days get longer and hotter, you will see people in their gardens cutting grass, pruning shrubs and privets, and then there is the smell of BBQ food meandering through the air.
Garden parties are great, friends, neighbours and relations all enjoying each others company while swapping stories – mostly about how cold the weather has been. As with all parties there is often a lack of places to sit, and most end up standing in the house (normally your kitchen). However, be different, this year by purchasing the Figure Hugger Bar Stool you will ensure that all your guests will be comfortable whether they are sitting outdoors or indoors.
]]> http://www.uk-contemporary-furniture.co.uk/garden-party-furniture/feed/ 0 Contemporary Furniture | Modern Living Room Furniture http://www.uk-contemporary-furniture.co.uk/contemporary-furniture-modern-living-room-furniture/ http://www.uk-contemporary-furniture.co.uk/contemporary-furniture-modern-living-room-furniture/#respond Wed, 25 Jan 2017 10:54:45 +0000 http://www.uk-contemporary-furniture.co.uk/?p=29 upholstery fabrics, and colours, contemporary furniture can be anything you want it to be.
One of our favourites is Arne Jacobsen, creator of the Egg chair, Ant chair, and Swan chair. We are also partial to the creations of Charles and Ray Eames, but we have many more names to choose from among a product list containing thousands of items. When you shop with us, choice is one word you'll have to get used to. We work with furniture manufacturers from all over the world to bring you the best pieces we can find. Whether you're looking for designer furniture for the bedroom, dining room, or living room, you'll find something you like among our many and varied offerings.
Our selection of television furniture is available for every room in your home or office. We invite you to spend a few minutes and take a look around. We are proud to offer you not only the highest quality contemporary furniture, but also an online environment which lets you shop easily from your own home. That means you shop at a pace determined by your preferences rather than those of the salesman. When you decide you're ready to order our secure online platform will guide you through the entire process. We accept all major credit cards and are ready to ship your order quickly through our vast network of couriers.
Contemporary furniture is all about finding that specific style that speaks of who you are. Unlike more traditional furniture, which is normally confined to a certain set of styles, fabrics, and colours, contemporary furniture can be anything you want it to be. That's why it's so important when shopping for contemporary furniture to find a company offering as many choices as possible. At UKCF, that's exactly what we do. We are one of the UK's leading online contemporary furniture sites featuring some of the best names in the industry.
It has to be said that when people start decorationg their homes, they normally concentrate on doing the living room and kitchen first, followed by the bedrooms then last of all comes the humble bathroom. Tiles, wood panelling, slate or wooden floors and a host of halogen lighting can all add the desired effect to this traditionally small room.
The contents of your bathroom normally consist of a bath/shower, toilet, basin and not a lot of room for anything else, however, clever use of lighting, mirrors and suitable furniture can give the illusion of space – making this usually small room seem much larger.
When it comes to bathroom furniture use light or natural colours, also try using furniture that runs vertically rather than bulky furniture this again will give the illusion of space. One of the best pieces of furniture that ticks all the boxes is the Oak Tall Bathroom Unit, this solid oak unit will fit ideally into any bathroom – no matter the size.
Anyone who has attempted to redesign their home will tell you that decorating the kids' rooms often gives them the hardest time. As not everyone has the same modern furniture design aspirations and ideas as yourself, most notably the kids themselves.
This will undoubtebly lead to squabbles which inturn leads to arguments that eventually lead to tears and me heading out the door and down the pub for some peace and quiet.
Now, we all know kids try to act older than they already are, so choosing the right décor may prove difficult, if you get it right you'll be the hero (for a bit, anyway) but getting it wrong and you'll probably spend more time down the pub than you originally planned.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 3,353
|
I thought I'd share an excerpt of FLYING TO THE LIGHT for folks who were interested in reading a little bit of the novel. I've been getting some really nice press recently and I'm very excited to say the sequel is scheduled to be released early 2013. Please enjoy!
Michael was so terrified he couldn't breathe. His safe, little world had shifted out of its comfortable orbit the moment his parents were kidnapped, propelling him to places he never dreamed imaginable. Suddenly, there was horror everywhere and nothing was what it appeared to be. How could life go so terribly wrong in just a few hours? He had had no time to think. He couldn't plan, but just reacted to what was happening. A storm, more volatile and dangerous than a level five tornado had turned his world inside out and all he could do was attack it head-on, swallowing his fear and living strictly on the adrenaline pumping furiously throughout his body, hoping against hope his luck wouldn't run out.
The only thing he knew was that now his little brother was his responsibility. An innocent six-year old who for some reason the world wanted to claim as its own to exploit.
Michael wouldn't let them win. He would protect his brother with his life.
He glanced at Danny, who had fallen asleep in the passenger seat. He could see his brother's furrowed brow and pale face in the flickering glow of the highway lamplights. Michael felt like they'd been through a war, racing right along with the military guys in the Middle East right now.
Just as the rain began hitting the windshield, Danny started to whimper.
"Don't worry. I won't let anything happen to you." He wondered if he could fulfill that simple promise. He took one hand off the wheel and patted his brother's head. Danny immediately calmed. Still, he wouldn't have responded even if Michael had shouted. Danny was deaf, and that, among other things, was one of the reasons they were in this car fleeing across the country. They were running far away from their home, trying desperately to get to a small town in northern California. Mr. Daley had said there was a man there who could help them. Daley was Michael's high school biology teacher and the only person in Rockland County who wasn't out for the glory, the money, the prize of scientific discovery, or the chance to see the light.
If you're interested, it's available on Amazon for only $3.99 at http://www.amazon.com/Flying-to-the-Light-ebook/dp/B00666JMOM/ref=sr_1_2?ie=UTF8&qid=1347934974&sr=8-2&keywords=flying+to+the+light also on B&N and Amazon UK too!
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 850
|
package io.shardingsphere.core.parsing.parser.clause;
import io.shardingsphere.core.parsing.lexer.LexerEngine;
import io.shardingsphere.core.parsing.lexer.token.DefaultKeyword;
import io.shardingsphere.core.parsing.lexer.token.Keyword;
import io.shardingsphere.core.parsing.lexer.token.Symbol;
import io.shardingsphere.core.parsing.parser.sql.dml.insert.InsertStatement;
import lombok.RequiredArgsConstructor;
/**
* Insert into clause parser.
*
* @author zhangliang
*/
@RequiredArgsConstructor
public abstract class InsertIntoClauseParser implements SQLClauseParser {
private final LexerEngine lexerEngine;
private final TableReferencesClauseParser tableReferencesClauseParser;
/**
* Parse insert into.
*
* @param insertStatement insert statement
*/
public void parse(final InsertStatement insertStatement) {
lexerEngine.unsupportedIfEqual(getUnsupportedKeywordsBeforeInto());
lexerEngine.skipUntil(DefaultKeyword.INTO);
lexerEngine.nextToken();
tableReferencesClauseParser.parse(insertStatement, true);
skipBetweenTableAndValues(insertStatement);
}
protected abstract Keyword[] getUnsupportedKeywordsBeforeInto();
private void skipBetweenTableAndValues(final InsertStatement insertStatement) {
while (lexerEngine.skipIfEqual(getSkippedKeywordsBetweenTableAndValues())) {
lexerEngine.nextToken();
if (lexerEngine.equalAny(Symbol.LEFT_PAREN)) {
lexerEngine.skipParentheses(insertStatement);
}
}
}
protected abstract Keyword[] getSkippedKeywordsBetweenTableAndValues();
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 4,716
|
{"url":"http:\/\/www.ck12.org\/analysis\/Constant-Derivatives-and-the-Power-Rule\/lesson\/Constant-Derivatives-and-the-Power-Rule-MAT-ALY\/","text":"<img src=\"https:\/\/d5nxst8fruw4z.cloudfront.net\/atrk.gif?account=iA1Pi1a8Dy00ym\" style=\"display:none\" height=\"1\" width=\"1\" alt=\"\" \/>\n\n# Constant Derivatives and the Power Rule\n\n## Derivative of a constant is zero and \\frac {d}{dx}[x^n] = nx^{n-1} .\n\nEstimated6 minsto complete\n%\nProgress\nPractice Constant Derivatives and the Power Rule\n\nMEMORY METER\nThis indicates how strong in your memory this concept is\nProgress\nEstimated6 minsto complete\n%\nConstant Derivatives and the Power Rule\n\nThe power rule is a fantastic \"shortcut\" for finding the derivatives of basic polynomials. Between the power rule and the basic definition of the derivative of a constant, a great number of polynomial derivatives can be identified with little effort - often in your head!\n\n### Constant Derivatives and the Power Rule\n\nIn this lesson, we will develop formulas and theorems that will calculate derivatives in more efficient and quick ways. Look for these theorems in boxes throughout the lesson.\n\n#### The Derivative of a Constant\n\nTheorem: If where c is a constant, then .\n\nProof: .\n\nTheorem: If is a constant and is differentiable at all , then . In simpler notation\n\n#### The Power Rule\n\nTheorem: (The Power Rule) If n is a positive integer, then for all real values of x\n.\n\n### Examples\n\n#### Example 1\n\nFind for .\n\nIf for all , then for all .\n\nWe can also write .\n\n#### Example 2\n\nFind the derivative of .\n\n..... Restate the function\n\n..... Apply the commutative law\n\n..... Apply the power Rule\n\n..... Simplify\n\n#### Example 3\n\nFind the derivative of .\n\n..... Restate\n\n..... Rules of exponents\n\n..... By the commutative law\n\n..... Apply the power rule\n\n..... Simplify\n\n..... Simplify again\n\n..... Use rules of exponents\n\n#### Example 4\n\nFind the derivative of\u00a0.\n\nSpecial application of the power rule:\n\n#### Example 5\n\nFind the derivative of\u00a0.\n\nRestate the function:\n\nUsing rules of exponents (from algebra):\n\nApply the power rule:\n\nSimplify:\n\nRules of exponents:\n\nSimplify:\n\n#### Example 6\n\nFind the derivative of\u00a0.\n\nRestate the function:\n\nRules of exponents:\n\nPower rule:\n\nSimplify:\n\nRules of exponents:\n\n### Review\n\n1. State the power rule.\n\nFind the derivative:\n\n1. Given , find the derivative when .\n2. Given , what is ?\n3. when\n4. Given , what is ?\n\n### Notes\/Highlights Having trouble? Report an issue.\n\nColor Highlighted Text Notes\n\n### Vocabulary Language: English\n\nderivative\n\nThe derivative of a function is the slope of the line tangent to the function at a given point on the graph. Notations for derivative include $f'(x)$, $\\frac{dy}{dx}$, $y'$, $\\frac{df}{dx}$ and \\frac{df(x)}{dx}.\n\nproof\n\nA proof is a series of true statements leading to the acceptance of truth of a more complex statement.\n\ntheorem\n\nA theorem is a statement that can be proven true using postulates, definitions, and other theorems that have already been proven.","date":"2017-01-22 04:21:01","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 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\": 4, \"texerror\": 0, \"math_score\": 0.8604927659034729, \"perplexity\": 2459.3748206596815}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2017-04\/segments\/1484560281332.92\/warc\/CC-MAIN-20170116095121-00326-ip-10-171-10-70.ec2.internal.warc.gz\"}"}
| null | null |
using namespace gl;
namespace globjects
{
GLuint FramebufferImplementation_Legacy::create() const
{
GLuint framebuffer;
glGenFramebuffers(1, &framebuffer); // create a handle to a potentially used framebuffer
glBindFramebuffer(s_workingTarget, framebuffer); // trigger actual framebuffer creation
return framebuffer;
}
void FramebufferImplementation_Legacy::destroy(GLuint id) const
{
glDeleteFramebuffers(1, &id);
}
GLenum FramebufferImplementation_Legacy::checkStatus(const Framebuffer * fbo) const
{
fbo->bind(s_workingTarget);
return glCheckFramebufferStatus(s_workingTarget);
}
void FramebufferImplementation_Legacy::setParameter(const Framebuffer * fbo, GLenum pname, GLint param) const
{
fbo->bind(s_workingTarget);
glFramebufferParameteri(s_workingTarget, pname, param);
}
GLint FramebufferImplementation_Legacy::getAttachmentParameter(const Framebuffer * fbo, GLenum attachment, GLenum pname) const
{
fbo->bind(s_workingTarget);
GLint result = 0;
glGetFramebufferAttachmentParameteriv(s_workingTarget, attachment, pname, &result);
return result;
}
void FramebufferImplementation_Legacy::attachTexture(const Framebuffer * fbo, GLenum attachment, Texture * texture, GLint level) const
{
fbo->bind(s_workingTarget);
if (texture == nullptr)
{
glFramebufferTexture(s_workingTarget, attachment, 0, level);
}
else
{
switch (texture->target())
{
case GL_TEXTURE_1D:
glFramebufferTexture1D(s_workingTarget, attachment, texture->target(), texture->id(), level);
break;
case GL_TEXTURE_2D:
case GL_TEXTURE_RECTANGLE:
case GL_TEXTURE_CUBE_MAP_POSITIVE_X:
case GL_TEXTURE_CUBE_MAP_POSITIVE_Y:
case GL_TEXTURE_CUBE_MAP_POSITIVE_Z:
case GL_TEXTURE_CUBE_MAP_NEGATIVE_X:
case GL_TEXTURE_CUBE_MAP_NEGATIVE_Y:
case GL_TEXTURE_CUBE_MAP_NEGATIVE_Z:
case GL_TEXTURE_2D_MULTISAMPLE:
glFramebufferTexture2D(s_workingTarget, attachment, texture->target(), texture->id(), level);
break;
default:
glFramebufferTexture(s_workingTarget, attachment, texture->id(), level);
break;
}
}
}
void FramebufferImplementation_Legacy::attachTextureLayer(const Framebuffer * fbo, GLenum attachment, Texture * texture, GLint level, GLint layer) const
{
fbo->bind(s_workingTarget);
glFramebufferTextureLayer(s_workingTarget, attachment, texture ? texture->id() : 0, level, layer);
}
void FramebufferImplementation_Legacy::attachRenderBuffer(const Framebuffer * fbo, GLenum attachment, Renderbuffer * renderBuffer) const
{
fbo->bind(s_workingTarget);
renderBuffer->bind();
glFramebufferRenderbuffer(s_workingTarget, attachment, GL_RENDERBUFFER, renderBuffer->id());
}
void FramebufferImplementation_Legacy::setReadBuffer(const Framebuffer * fbo, GLenum mode) const
{
fbo->bind(GL_READ_FRAMEBUFFER);
glReadBuffer(mode);
}
void FramebufferImplementation_Legacy::setDrawBuffer(const Framebuffer * fbo, GLenum mode) const
{
fbo->bind(GL_DRAW_FRAMEBUFFER);
glDrawBuffer(mode);
}
void FramebufferImplementation_Legacy::setDrawBuffers(const Framebuffer * fbo, GLsizei n, const GLenum * modes) const
{
fbo->bind(GL_DRAW_FRAMEBUFFER);
glDrawBuffers(n, modes);
}
} // namespace globjects
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 6,237
|
Two new infill wells drilled this year on the Tambar field offshore Norway, have been brought online, Faroe Petroleum, a partner in the Aker BP-operated field said on Friday. Faore said the two new development wells' initial performance exceeded pre-drill expectations.
"During the fourteen days since both new wells have been on stable production, the Tambar field has produced at an average flow rate of c. 18,507 boepd," the company said.
The Tambar development project consists of two new infill wells, both of which have now been completed and put on stream, and the installation of gas lift in three existing wells, which is expected to complete in Q3 2018, to increase overall field life, production and reserves.
The two infill wells targeted undrained areas in the north and south of Tambar, which had been identified through application of a number of different modelling techniques, including 4D seismic technology. The initial flow rates from the two infill wells have exceeded pre-drill expectations.
On April 20, 2018, Faroe said it reached a record net production level of c 19,275 boepd. The 2018 overall production guidance currently remains unchanged at 12-15,000 boepd average for the year.
Faroe said the higher rate of production resulting from the new Tambar wells counters the lower than expected overall production in Q1, notably in relation to the previously announced temporary loss of production from the Trym field, caused by a downstream export fault which has now been rectified, and the scheduled cessation of production from Oselvar, for which the company is being compensated.
Graham Stewart, CEO of Faroe Petroleum, said the new production wells have increased Tambar production "markedly" and, combined with the ongoing installation of gas lift, this will potentially lead to the extension of Tambar field life by up to ten years.
Posted on April 27, 2018 with tags Aker BP, Faroe Petroleum.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 8,390
|
<!--
Copyright (c) 2015 The Polymer Project Authors. All rights reserved.
This code may only be used under the BSD style license found at http://polymer.github.io/LICENSE.txt
The complete set of authors may be found at http://polymer.github.io/AUTHORS.txt
The complete set of contributors may be found at http://polymer.github.io/CONTRIBUTORS.txt
Code distributed by Google as part of the polymer project is also
subject to an additional IP rights grant found at http://polymer.github.io/PATENTS.txt
-->
<link rel="import" href="../packages/polymer_elements/iron_selector_nodart.html">
<dom-module id="test-content-element">
<template>
<iron-selector id="selector" selected="{{selected}}" selectable="[[selectable]]" attr-for-selected="id">
<content></content>
</iron-selector>
</template>
</dom-module>
<script>
Polymer({
is: 'test-content-element',
properties: {
selectable: String,
selected: {
type: String,
notify: true
}
}
});
</script>
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 3,942
|
Without a question Warped is the definitive guide to the lost eighth season of Star Trek: The Next Generation. I say that with 100% conviction.
Never in my life have I read such made up, crackers stories as I read in this book - and that's kind of the point.
Warped is the brainchild of Mike McMahon, creator of the @TNG_s8 Twitter account which has taken the ball and quite literally sped off into the distance with it never to be seen again. Warped follows the hurriedly made and never released final final season of Picard and crew. Made on a fraction of a shoestring budget we get to explore every area of the series in some of the most bizarre episodes never made.
McMahon has created the ultimate spoof companion detailing not just 26 stories as if they were real episodes but also covers bloopers and background material to flesh out the "reality" of the situation.
I did approach the book with some skepticism as to how this would manage to translate from the Twitter limit of 140 characters into a full blown 250+ page book but the end product exceeded my expectations and then some.
McMahon has truly dug into the foundations of The Next Generation and observed the nuances which made the characters - or at least kept cropping up in the show - and has used them just as the writers of The Next Generation did, creating stories around them. This however is a bit different and nowhere near as serious. Picard's terrible luck with turbolifts comes back to haunt him, the buddying of Geordi and Data is in full swing and Beverly just can't find enough desks to bang her fist on. Thing is once you've read them here in their enhanced and more noticeable form you'll see the seven seasons of the show in a slightly different way. Each character has their unique characteristics which are played on with each story but there's even more.
The plots themselves are cleverly thought out, swaying into car-crash B movie territory with over the top sci-fi premises with a severe helping of humour thrown in it really is no holds barred as the series veers into facepalm central - but hey, it was like this to ensure there would be no ninth season(!!!). McMahon hasn't just successfully written an "episode guide" but has taken advantage of a tested book formula to play out other aspects of the season's supposed production.
While stories of combining crew members, Barclay deaths and Q interruptions are great it's actually made all the better for the additional material outside of the A and B plots provided. Dropping in notes regarding "real" Borg with their own herder required for one episode or an overabundance of juvenile crew in another thanks to Bring Your Child to Work Day made this a book I couldn't put down for days. Every episode had something different that made me laugh and made the wife wonder how I could find anything to do with Star Trek funny. Honestly there are some really hilarious sections in here, my favourites being from the highlight episode, Barclay's Day. I won't ruin it but all. I can say is Space Snakes.
Cleverly there are even errors and in-jokes dotted through the book which fans will love. There are a couple at the expense of Voyager which are very on point as well as referencing back to previous episodes from the show, even being able to make fun of some of the real sillier moments such as the Exocomps from The Quality of Life or Geordi's poor luck with the ladies. Deciding to write this from a production perspective detailing the shoddy - purposefully shoddy - work on season eight, the author has created an hilariously believable experience which is fortunately maintained over all 26 synopses.
Some of the lists dropped into the episodes are laugh out loud funny; alternate Tasha's (they turn up all the time y'know), oddball space pirates and many more. I think it's one of those books I couold go back to just to highlight those more eccentric points of the series. McMahon has managed to get every situation just ridiculous enough that it suits the characters we know from The Next Generation perfectly yet doesn't destroy the "real" Star Trek universe if you get where I'm going.
It's a book that absolutely shows the author's love and understanding of the series to a great extent indeed to a point where he has more than competently poked fun at the show and succeeded in producing something extremely readable, entertaining and more than worthy of sitting alongside the other official series companions.
Every story has something different, a new character nuance that's exploited and while it's not a factual book about the series it's absolutely worth getting hold of because it will, I guarantee, entertain you at every page turn. Now just go and buy it.
Warped is available right now from Simon and Schuster priced £9.99 ISBN 9781476779058. Go on, get it.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 9,927
|
Newysports.com: For Passion, For Football
New-FM News
NEWFM Fixtures
MOTM Awards 2019
Zone League
Newysports Schedule
PREVIEW: New-FM Northern League One Round 16
1st Grade:
Singleton Strikers (9th) v New Lambton F.C (8th)
Last Time They Met: New Lambton 4-2 Singleton
It's going to be an interesting encounter up at Howe Park between the in-form Singleton Strikers and a desperate New Lambton side. The Strikers were gallant in a recent win over Wallsend with just nine players due to two red cards. Then they showed quality to overcome rivals Cessnock City on Tuesday night.
That win over Cessnock all but ruled out any chance of Singleton picking up the wooden spoon and now there's a chance for them to jump over New Lambton into 8th spot with a win on Saturday. The involvement of defensive recruit Stuart Thomson has been vital for the Strikers as has another former Cooks Hill player in Bowen Bagnall further forward. Both were instrumental in their win on Tuesday night.
New Lambton haven't played a match since the 27th May and no doubt it's been a very frustrating time for all involved. Unfortunately the Eagles' home base of Alder Park has been an issue for the club and improvements to the pitch will be top of the do to list in the off-season to prevent an abundance of wash-outs next year.
On the park, the Eagles do have some games in hands and it isn't out of the realm of possibility that a winning streak could propel them towards the top four but make no mistake, this game has to be a win and nearly every fixture after that for Andrew Packer's side to make finals. It's a tough ask but the Eagles will take it one game at a time. With only one win in their last five, it's important for them to show quality here.
Prediction: It's hard to predict just what New Lambton might bring to the table after not playing a game in so long but Singleton were good value in their last two home games and it wouldn't surprise to see them step up to the plate once again. Maybe a draw?
My Tip: Singleton Strikers 2-2 New Lambton
Under 23's:
Kickoff: 12:45pm
Singleton found a way to win late against rivals Cessnock City on Tuesday night. The Strikers are still in finals calculations and that three points picked up midweek could kick-start a run all the way to the finals. Last time they met New Lambton, the Strikers managed to spur themselves to a massive 4-3 victory in a topsy-turvy encounter at Alder Park.
New Lambton's finals hopes by their own admission are extremely slim so it's fair to say that Singleton being the home team may just have a little bit more to play for. The Eagles haven't played since May 27th which will no doubt mean that match fitness will be lacking for the visitors. Make no mistake, any slim hope of finals will be far over if they fail to overcome Singleton.
Prediction: Fitness is a big thing in football, in particular match fitness. No games in around four weeks makes it tough for New Lambton and their lack of form makes it tougher. Singleton fought hard for a result on Tuesday and they definitely need to pick up three points at home to keep their finals hopes high.
Kickoff: 11:00am
Singleton 19's picked up a solid 2-0 over Wallsend two weeks or so ago but they'd fall to a disappointing 1-0 loss to rivals Cessnock City on Thursday night. A penalty miss proved costly and the Strikers missed a chance to keep even on points with 6th-placed Kahibah. Like many at this point, their finals hopes are over and it's simply all about playing for pride at this point.
New Lambton return to action for just their second game in the last month. Their previous outing was a 2-2 draw with Kahibah and the Eagles know too well that they've got to up the ante to try and force their way into the top four. They were fairly comfortable winners last time against Singleton with a solid first half setting the foundation for a 3-1 win.
Prediction: Having top goalscorer Lachlan Jones back in 19's makes New Lambton just that much more of a threat going forward, especially when things click between Jones and Luke Homer up top. They could score goals for fun against a Singo side down on confidence following their loss on Thursday.
Singleton Strikers (11th) v New Lambton F.C (5th)
Kickoff: 9:15am
Avoiding the wooden spoon is the aim for the Singleton 17's and they took a big step towards that goal with just their second win of the season on Thursday over rivals Cessnock City. A 3-1 win drew the Strikers level with Thornton and was arguably their best result of the season.
New Lambton have only played once in the last month which makes it tough to keep match fitness ticking over. That result was a 1-1 draw against Kahibah that saw the Eagles miss a chance to move into the top four. They sit in 5th position and games like this against low opposition will be key if they want to play Semi Finals football which is no doubt the minimum aim.
Prediction: Last time these sides met, it finished 6-2 to New Lambton. The Eagles haven't found that sort of goalscoring prowess in recent times so a lower scoring game might be the result up at Howe Park. Singleton did well on Thursday but New Lambton should have the quality to win this one.
Thornton Redbacks (4th) v West Wallsend SFC (6th)
Last Time They Met: West Wallsend 1-6 Thornton Redbacks
If it wasn't for the rivalry of Kahibah v Cooks Hill, this would be the Match of the Round. All eyes will be on Thornton Park on Saturday as the Redbacks look to end a barren spell of results and hammer themselves back into serious minor premiership contention. Winless in their last three, their last result was a late 2-2 draw with Cessnock with two goals conceded in the final moments.
With two games in hand, Darrel McAllister's side are about to enter another very busy period that could see them wind out on top when the dust settles. They seem to produce well on short-turnarounds but after a month without a match, it's hard to predict what they might throw up on Saturday. Needless to say, they need to get things kick-started soon or perhaps find themselves out of the top four.
West Wallsend are the ideal opponents on Saturday for neutral's sake. You never quiet know what Gary Rowe's side will do. Their 4-1 loss at the start of the month to Toronto-Awaba was as poor as any they've had this year (including a 6-1 hammering by Thornton) but their last start 2-1 win over Kahibah showed what the Bluebells are capable of when they find their best form.
Discipline was thrown out the window in a fiery encounter last time this side's met with West Wallsend quiet frankly losing the plot. They've improved that for the most part and they're a good side at leading from the front. The first goal in this game will be paramount to who might win it.
Prediction: This game is make or break for West Wallsend. Their finals hopes could be done and dusted with a loss but a win could really throw them into contention. If they play like last week it'll be a tough ask for Thornton but the Redbacks on home turf simply need to up the ante. Very tough to call but Redbacks to win.
My Tip: Thornton Redbacks 2-1 West Wallsend
Under 23's Match of the Round:
Thornton Redbacks (3rd) v West Wallsend SFC (4th)
The Match of the Round sees two sides that Newysports watched last weekend take to the field. Thornton Redbacks were fast starters in horrendous conditions against lowly Toronto-Awaba and were never in doubt of winning a match that finished in the 58th minute. A much-needed 3-0 win to get back into winning form after some troubling times as of late.
West Wallsend did it a lot harder in their game as they survived a back and forth fist fight with Kahibah. The Bluebells came away with a thrilling 4-3 victory at Johnston Park to move into the top four. Suddenly they find themselves with a chance to jump into third spot. Not a place some expected them to be after results like the one against the Redbacks last time, a 6-1 humbling at home.
Prediction: It's not a must-win game by any stretch but the victor if there happens to be one will take a load of confidence with them out of this game and into the finals race. Home advantage might help the Redbacks who were the more convincing of the two sides last weekend.
Thornton Redbacks (4th) v West Wallsend SFC (2nd)
Thornton picked up a 4-1 win over Toronto last weekend but something about their performance seemed a little off. It was very similar to when Westy put a 4-1 scoreline past Toronto just two weeks prior. Not a convincing display but just doing enough to see off a poor side. The Redbacks need to find that next level to push themselves further into the top four.
West Wallsend are in a battle of their own for the minor premiership with an unbeaten Belswans side. The Bluebells got the job done against Kahibah last week and if that game proved anything they are very dangerous from set-piece opportunities, particularly in the air with some big and tall players. Fortunately Thornton have some defenders who can match it in the air.
Prediction: This could turn out to be a really entertaining match in the Under 19's. There's still doubters out there about Thornton so a win over West Wallsend will surely give their confidence a huge boost. Home field advantage is key but the Bluebells on paper have the attacking quality to edge it.
Thornton Redbacks (10th) v West Wallsend SFC (2nd)
Thornton 17's were fortunate to not have a mud bath of a match against Toronto last week. Lyall Peacock Field really started to become a slog during the 19's but it was still admittedly tough conditions for the Redbacks as they lost 2-0 to the Stags in the Under 17's. Not their worst performance but another week that showed a distinct lack in the attacking third.
West Wallsend were beaten last weekend for the first time this year. Kahibah were lucky in their 1-0 win and it left the Bluebells highly disappointed and they'll no doubt look to get right back to work knowing that the minor premiership is still alive for them. Good teams bounce straight back from disappointment.
Prediction: West Wallsend are coming off their first loss of the season which spells likely doom for Thornton's chances of an upset. The Redbacks just don't have the players to score enough goals and their defence isn't solid enough to record clean sheets.
1st Grade Match of the Round:
Kahibah F.C (5th) v Cooks Hill United (1st)
Last Time They Met: Cooks Hill 3-4 Kahibah
Here we go again. After the madness of the first fixture at the AthField, Kahibah will once again take on Cooks Hill. Times haven't been great for either side. Kahibah's recent form has been patchy and includes a fairly length stint of wash-outs and a bye. It's been frustrating for the likes of Andre Gumprecht who thrives on a routine and keeping his squad match-fit.
The Green and Blacks took on Hamilton Olympic in a decent trial hit-out a week and half ago but last week by their lofty standards simply forgot to show up against West Wallsend. A lazy performance and one or two players who were hooked at half-time for varying reasons was all part of a horror day at the office for Kahibah.
Cooks Hill have amazingly not won a game since May 15th. They are winless in their last four. Even that game, a 5-4 win over New Lambton wasn't convincing. It's hard to say that the wheels have fallen off Graham Law's side's charge towards the minor premiership but the stats are damning.
An 89th minute leveller by Wallsend broke Cooks Hill hearts last weekend, a few weeks after Singleton did the same thing at the AthField. Through all of this, the Cookers have remained on the top of the table but they need to find their best now or face a certain drop from top spot. It's arguably the catalyst for their minor premiership hopes this Sunday.
Prediction: Both sides want to win, both sides need to win, so expect quiet an entertaining tussle at Kahibah Oval. It's highly doubtful it'll be a goalfest like last time with neither side showing much attacking-wise in recent outings. A draw, as much as neither side want it could be the end result.
My Tip: Kahibah F.C 1-1 Cooks Hill United
Kahibah 23's could almost call it a day for their finals chances if not for two vital games in hand. They'll need to upset the ladder-leaders though to really kick-start some self-belief. They fought hard last week with Jackson Gosling securing a hat-trick but their resolve let them down late on. Another tough result to take for the side.
Cooks Hill were beaten 3-0 at home by Wallsend last weekend. The Cookers have been superb all season but it's become very obvious that they don't have Wallsend's number just yet. The Red Devils have beaten them twice but this is a side that will look to get right back to winning ways and make last weekend a distant memory.
Prediction: Kahibah will try to raise the bar on home turf against the ladder-leaders and were gallant in a late 1-0 loss last time against Cooks Hill. The quality has shone through for the Cookers all season and it wouldn't surprise to see it happen again here.
Kahibah F.C (6th) v Cooks Hill United (3rd)
So often this season Kahibah 19's have failed to put their best foot forward in particular games. Last week's loss to West Wallsend was a case of this but Thursday night saw the side overcome Toronto-Awaba 4-1 at Kahibah Oval. Finals are a stretch but wins over Cooks Hill on Sunday and Thornton next week could see them re-enter calculations. It's a big ask though for a side that haven't put together any consistency this season.
Cooks Hill are coming off a rare loss as they fell to a disappointing 1-0 loss in the final minute against lowly Wallsend last week. It was a huge turnaround from Cooks Hill's 8-1 win over the same opposition two months ago. Suddenly Michael Best's side need to produce a result to keep firmly in the race for the minor premiership.
Prediction: Kahibah will have a depleted squad with captain Luke Jones injuring his ankle on Thursday night and Lucas Emmett-Mourlet and Cal Brzozowski both missing this fixture. Cooks Hill will be fired up after that loss last weekend and they'll come to Kahibah Oval with a lot of belief.
Under 17's Match of the Round
Kahibah F.C (6th) v Cooks Hill United (2nd)
Kahibah are a club that aimed for finals in every grade at the beginning of the year but it's fast becoming likely that only 1st Grade will play in August. The 17's took a huge step towards finals as they handed Westy their first loss last Saturday but on Thursday, it was an all too disappointing 3-1 loss to Toronto. A tough night for the Green and Blacks and not good for confidence.
Cooks Hill 17's were the only Cookers side to win at the AthField last weekend as they kept their form going well. They're well in the race for the minor premiership although Toronto overtook them with that win over Kahibah. They produced a thrilling late goal to draw level with Kahibah last time and no doubt Bruce O'Donnell will want his side to produce a more solid effort across the 80 minutes this time round.
Prediction: Kahibah are a side that excel with confidence and belief but that's just not there consistently. They failed to penetrate Toronto's defence on Thursday and this challenge against Cooks Hill will be a tough one. The Cookers have the best defence in the comp and it might help them in this one.
Wallsend F.C (7th) v Cessnock City (11th)
Last Time They Met: Cessnock City 1-4 Wallsend
No-one gave Wallsend much hope against Cooks Hill last week but the Red Devils showed that famous Wallsend F.C belief to turn-around a shocking 9-0 loss the week before to stun Cooks Hill with a late 1-1 draw thanks to a goal from Isaac Reeves. It was a good day out for Chris Gallagher's side.
The Red Devils are most likely out of finals contention but they can certainly earn some respect with a late form patch to try and solidify a higher mid-table finish. Blooding some youngsters for the future is also going to be a focus no doubt, especially against sides like Cessnock who are doing the same thing. Dan Strazzari is one that comes to mind.
The Hornets were in midweek action against local rivals Singleton and it was all-round just a shocking night at the office for Lino Gatti's side. The gaffer has indicated that goalkeeper Mat Zechel might be dropped to 23's after a tough night for him in front of a vocal Singleton home crowd. 23's keeper Dylan Cheetham has certainly impressed as of late.
Cessnock can trouble sides on their best day but results like Tuesday night and the 4-1 loss to Wallsend last time show that it's still a work in progress. The wooden spoon is now almost a certainty unless Gatti's men can string together at least two or three wins in their remaining 6 games.
Prediction: After a confidence-boosting win last weekend, Wallsend will be hopeful of overcoming a winless Cessnock side. If the Hornets show up to fight for it, it'll be a challenge but if they produce similar to their showing at Singleton, it could be a long day at the office.
My Tip: Wallsend F.C 4-1 Cessnock City
Wallsend F.C (2nd) v Cessnock City (10th)
What a result for Wallsend last weekend at the AthField. Their second win over ladder-leaders Cooks Hill this season and amazingly their eighth-straight victory. They are the most in-form side in New-FM football and they are a definite title contender in the Under 23's. Despite being a long way behind Cooks Hill in goals scored, the Red Devils are the most solid defensive unit in the 23's this season.
Cessnock City enjoyed a small form patch, winning a game or two but have slipped up in form and luck in the last couple. A 2-1 loss to Belswans at home ended a two-game winning streak and then the Hornets were beaten by the same score on Tuesday night thanks to the latest of goals from Singleton to see them off at Howe Park.
Prediction: It's good to see so many youngsters in the Cessnock 23's, albeit forced in there due to a distinct lack of players and depth. The future is a decent one but the present is still being worked on. Wallsend are in too good of form to let it slip now. A ninth-straight win could be on the cards. They've enjoyed the 3-0 score this year, three of their last four have ended 3-0.
Both Wallsend and Cessnock took big steps towards avoiding the wooden spoon in their last fixtures. The Red Devils travelled to Cooks Hill last week and turned around an 8-1 loss to the former ladder-leaders with the latest of 1-0 wins thanks to a Jacob Rae goal. No doubt Wallsend's best win of the season and one they celebrated in a big way.
Cessnock City bounced back from a 5-0 hammering at the hands of Belmont-Swansea to knock off local rivals Singleton 1-0 on Thursday night. A quality strike from Jeremy Noone would give the Hornets just their third win of the season and draw them level with Toronto-Awaba. With the bottom four all playing each other this weekend, it could be an important week in the race to avoid the spoon.
Prediction: Cessnock did well to beat Singleton but Wallsend are on their day a tougher challenge than the Strikers. The Red Devils beat Cooks Hill last week and home advantage will play into their favour. It'll be interesting to see who goes in goals after back-up keeper Jackson Bartlett started last week and picked up a clean sheet.
Wallsend F.C (7th) v Cessnock City (9th)
Back-to-back losses for Wallsend 17's has all but ended their run for potential finals football but on their day, the Red Devils possess a very strong side that can trouble good teams and really cause headaches for lower sides. They ended Cessnock's dream start to the year with a 6-1 hammering last time they met the Hornets.
Cessnock City picked up a result against Southy but a loss to Belswans at home saw them return to familiar losing ways and it'd only get worse as they fell to Singleton on Thursday night. The wooden spoon is a distinct possibility for the Hornets who have slid down the table in horrendous fashion during the middle portion of the season.
Prediction: Both sides are coming off two-straight losses which makes this one an interesting fixture but with home advantage, it's safe to say Wallsend enter as the favourites. Cessnock are coming off a loss to Singleton and confidence will be low after that.
South Cardiff (3rd) v Toronto-Awaba (10th)
Last Time They Met: Toronto-Awaba 1-0 South Cardiff
There's been a lot of talk about Cooks Hill's lack of form and the likes of Belswans surging up the table but South Cardiff have gone under the radar and continue to sit fairly comfortably inside the top four. Dino Fajkovic's side are one who take each game as it comes and things are certainly travelling well for the boys from Ulinga Oval.
Southy have won their last four games but more impressively have kept a clean sheet in each of the last three. Only some poor displays and 4 losses has kept them lower than top spot. One of those losses was an unlucky 1-0 loss to Toronto-Awaba. It's quiet possibly that South Cardiff could win the minor premiership if they continue their good form and results go their way.
Toronto-Awaba were washed-out in the final moments before kick-off for the second time this year at Lyall Peacock Field last week as torrential rain left the pitch unplayable. Adding to their woes, the same torrential rain left Kahibah Oval closed for Tuesday's catch-up game.
Their last match was that superb 4-1 victory over West Wallsend and if they play in a similar fashion then they'll feel like a win is a possibility against South Cardiff. With Cessnock losing to Singleton midweek, the Stags dropped to 10th but they sit fairly comfortably clear of cellar-dwellers Cessnock with a seven point advantage.
Prediction: Toronto-Awaba were good value in their 4-1 win over Westy last time out but Southy's form has been superb and their defence has stood firm with three clean sheets in three games. Ulinga Oval can be a tough place to win at and the Stags might find it tricky to overcome the odds against the 3rd-placed side.
My Tip: South Cardiff 2-0 Toronto-Awaba
South Cardiff (7th) v Toronto-Awaba (11th)
South Cardiff led a cluster of teams on the edge of the top four but have fallen to 7th, albeit with a game or two in hand on teams above them. This team will no doubt be aiming for the top four but consistency has let them down. A loss to Cessnock last time out was not a good day for them.
Toronto 23's fell to a 3-0 loss to Thornton last weekend in a game that was abandoned with twenty minutes to go due to unplayable conditions at Lyall Peacock Field. They were disappointed to have their game stopped but at 3-0 down, it wasn't likely that the comeback was on. They're playing for pride and it can be a really dangerous thing as they have simply nothing to lose at this point.
Prediction: With respect to Toronto, they sit bottom and their record speaks for itself. No team that's serious about playing finals will drop points at this stage to Toronto. It's a home game for Southy and it's time for them to remind the 23's competition that they are a finals contender.
South Cardiff's season has been a tough one in the Under 19's but they managed to pick up an important 2-1 win over Cessnock City last time out. Goalscoring has been a major issue for the Gunners but they managed to beat Toronto last time they played. A result for Southy on Sunday will go a big way towards moving them away from the bottom of the table.
Toronto-Awaba have endured a frustrating few weeks and the scoreline of 4-1 is becoming an all too familiar sight. Lindsay Jamieson's side recruited regular 1st Graders Jessie Cowie and Cody Armstrong for their game against Kahibah and whilst it was an improved showing on last week's loss to Thornton, the Stags still went down to a disappointing defeat. Cessnock's win only makes things worse as the wooden spoon looms again for Toronto.
Prediction: South Cardiff deserve to be the favourites on form but Toronto's streak of 4-1 losses will surely end as the Gunners don't have the capacity to bang in four, regardless of opponents. Southy to win but a close one like the last time they met.
South Cardiff (8th) v Toronto-Awaba (1st)
Southy 17's are coming off a two-week break after last week's wash-out and the bye before that. Their last outing was a disappointing 1-1 draw with a Cessnock City side that seemingly every team in the competition have beaten at some stage this year. It hasn't been a good year for Southy and it doesn't get any easier on Sunday.
Toronto-Awaba did enough to see off lowly Thornton last weekend and then they backed that up with a confidence-boosting 3-1 result over Kahibah on Thursday night. Jackson Treiger was quiet impressive on the night and has proved to be a handy sidekick to the free-scoring Thomas Appleby. The Stags moved back to the top of the table with that result over Kahibah.
Prediction: South Cardiff have failed to trouble the top sides this season and Toronto-Awaba's form going into this is fairly impressive. The Stags should have too much quality in this one and it's only a question of what the score might end up. Expect a one or two goal win.
Under 15's Fixtures:
Posted by Ty Stedman at 15:21 No comments:
Labels: New-FM, Newcastle Football, Preview, Sports, Tips
West Wallsend SFC (6th) v Kahibah FC (4th)
Last Time They Met: Kahibah 3-0 West Wallsend
West Wallsend were pushing towards the top four until their last result showed that work still needs to be done. The Bluebells were disappointing in a convincing 4-1 loss at Toronto-Awaba and now sit six points behind Kahibah who do have two games in hand, not including this weekend's fixture.
Gary Rowe's side have boasted a decent record at home this season and will need to use that to turn around their last result against Kahibah, a poor 3-0 loss in a mid-week fixture. In the race for finals, the Bluebells simply need to find that next level and take all three points from their higher-ranked opponents.
Kahibah saw another wash-out midweek against New Lambton and have not played a competitive game since May 21st. Andre Gumprecht and co. were quick to contact Michael Bolch and Hamilton Olympic for a quickly thrown together trial match and went down 3-0 to the NPL powerhouse at the LMRFF on Wednesday.
Kahibah were 5-1 victors last time out in an easy display against Cessnock and will need more to overcome West Wallsend. Having a fully-fit side will be key for the Green and Blacks as well as the quality of David Hodgson who has haunted Westy in recent years, including the earlier fixture this season.
Prediction: Kahibah are yet to start their second half of the season and if things go to plan they'll be hoisting the minor premiership in seven or eight weeks' time. Consistency is something Kahibah strive for and their defensive record is an indication of just how hard it is to beat them.
My Tip: West Wallsend 0-3 Kahibah FC
West Wallsend left it as late as they possibly could to see off lowly Toronto-Awaba last time out and despite being a fairly lucky unconvincing result, the three points was indeed vital as it lifted the Bluebells into the top four, at the top of an abundance of teams in the middle of the table, including Kahibah.
Kahibah's midweek trip to New Lambton was again postponed and the side will have been without a game since May 21st when and if they take the field at Johnston Park this weekend. Their finals hopes rely heavily on results and starting the second half of their season will be vital. Expect a very focused performance in a game against a side they drew 2-2 last time out. Two very even squads on paper.
Prediction: Kahibah sit seven points behind Westy but with two crucial games in hand. Their second half of the season begins on Saturday and really it's a mystery what they might produce after a month without a fixture. Westy weren't convincing last time out so it's anyone's guess who may win.
West Wallsend SFC (3rd) v Kahibah FC (6th)
Kahibah did Westy a favour on Thursday by taking a point away from New Lambton and Westy coach Steve Thomson not that long ago claimed that Kahibah were one of the better opponents his side have faced. There's a real competitive respect between the two sides and both teams know it's a big game.
Westy can keep pace with the top sides in the minor premiership race with a win whilst Kahibah really need all three points to remain in finals calculations. It's a key game for both sides and last time they played it was a tight one with Kahibah making a costly mistake or two.
Prediction: West Wallsend will have extra fitness after Kahibah played on a heavy Alder Park surface on Thursday night. The Bluebells didn't have to get out of second gear in a 4-1 victory at Toronto and will need more to see off Kahibah. Still, the top four side deserve to head into this as the favourites.
West Wallsend SFC (1st) v Kahibah FC (7th)
West Wallsend 17's picked up a dramatic late win over minor premiership rivals Toronto-Awaba two weeks ago to move to the top of the table and remain unbeaten in 2017. The ladder-leaders will no doubt look to rise to the next level as the finals series begins to loom in the distance.
Kahibah were in action on Thursday, their first game since May 21st and were frustrated in a 1-1 draw with New Lambton. The same scoreline last time against Westy too as they conceded a heartbreaking late goal. They were that close to beating the unbeaten Bluebells in that midweek game and the Green and Blacks will see their finals hopes boosted in a big way with a win here.
Prediction: Expect a fired-up Kahibah as frustration is boiling over and the Green and Blacks no doubt want to put a run of luckless results behind them. Westy have stood form all season and remain unbeaten so it wouldn't surprise to see them stand strong again.
Cooks Hill United (1st) v Wallsend FC (8th)
Last Time They Met: Wallsend 1-3 Cooks Hill
This is one of the more interesting fixtures this season for various reasons. Firstly is the fact that Cooks Hill haven't won in their last three games. They have South Cardiff and Kahibah chasing them down and suddenly it looks like a dog-fight for the minor premiership.
With a huge game against Kahibah next week, Graham Law will want his side to simply return to winning ways in one way or another. Luck has let them down but the reality is that two points from three games isn't what they're aiming for. A break last week might have helped with injuries and players away on vacation. Expect a more refined effort this time round.
Cooks Hill can ill-afford to get ahead of themselves against a Wallsend side that suffered one of the more embarrassing losses in recent years of the New-FM competition. Chris Gallagher's side got it all wrong at Blacksmiths last week and it wouldn't surprise to see a raft of changes on Saturday as they look to plug the huge holes that saw them concede nine last week.
Luke Alexander and Kyle Macorig both returned from injury and both were nowhere near the quality they normally show; young defender Darcy Hall was no match for Tega Marcus and will likely be switched out for a more experienced defender this week and goalkeeper James Wand cost his side two or three goals on the day. It was terrible from front to back and side to side. And it doesn't get any easier this weekend.
Prediction: The result rests on Wallsend. If they show up like last week it'll end in tears once again, maybe not to the tune of 9-0 but Cooks Hill can easily bag 5 or 6. The Cookers at home cannot drop any more points. Graham Law wants that minor premiership and it's time for the leaders to return to their best.
My Tip: Cooks Hill 5-1 Wallsend FC
Cooks Hill United (1st) v Wallsend FC (2nd)
If all goes to plan this may be the Grand Final in waiting. There's no doubt these two sides are setting the bar high in the 23's and so far, 14 weeks in, no teams have been able to consistently match them. Cooks Hill have at least four goals each of their last five and an impressive 21 goals in their last three games.
The only side Cooks Hill 23's are yet to slay happens to be Wallsend. The Red Devils were 3-2 winners in a midweek fixture back in April and it remains the only blemish on the Cookers' near-perfect league record. This game is a big one for both and could go a long way towards determining the minor premiers and indeed the title favourites.
Prediction: Cooks Hill are back at home and will be adamant on picking up a win over their top of the table rivals. Cookers' coach Ben Nichols, formerly of Wallsend, will know this is a step up in competition and it wouldn't be a surprise to see the home side win a tight one, rather than a lopsided scoreline.
Cooks Hill United (2nd) v Wallsend FC (10th)
It's not often the Cookers' 19's have had to back up from a loss this season. Their Round 12 loss to Belswans was just their second defeat of the campaign and with last week's glamour match with West Wallsend washed out, they can use this match as a chance to return to winning ways and re-install some confidence.
Wallsend shouldn't be easybeats but it was all too simple when Cooks Hill blasted them 8-1 at The Gardens and it was that same scoreline that saw the Red Devils demolished at Blacksmiths last weekend. Confidence is completely gone and it'll be a challenge for coach Craig Tselembis to get his side believing they can find something in this match.
Prediction: It's not a matter of who will win. Cooks Hill have surely got three points if they manage to find their best football. Whether it'll be two or three or eight is the question. Wallsend's best young defenders are in 23's and 1st Grade and it's a big reason as to why this 19's side have been nowhere near good enough. Cookers by a few.
My Tip: Cooks Hill 6-1 Wallsend
Cooks Hill United (2nd) v Wallsend FC (6th)
Cooks Hill return to action following a solid win over Belswans to remain in contention for the minor premiership. The Cookers are a significant threat to West Wallsend in the race for 1st place and they're a side that have shown throughout the first 14 weeks that they know how to win games and sometimes win games ugly like they did against Wallsend last time.
The Red Devils travelled to Belswans last week in a vital fixture and were fairly underwhelming in a 3-1 loss. That result dented their finals hopes and with so many good teams above them, it'll be a tough ask for them to make it into the top four. A win here would do wonders but a loss could see them lose more vital ground on the four.
Prediction: Cooks Hill found a way to get over the line against Wallsend last time via a late penalty after a topsy-turvy match-up. That three points is one of many that the Cookers have had to fight for. Building confidence in the back end of the year will be the target and a win here would do them well.
Cessnock City (11th) v Belmont-Swansea UFC (5th)
Last Time They Met: Belswans 1-1 Cessnock
Cessnock City should have beat Belmont-Swansea at Blacksmiths earlier this season. They put in an excellent display on the day but unfortunately it's been rarely seen since. Last time out they were beaten 2-0 by South Cardiff and suddenly they sit six points adrift of Singleton in the race to avoid the wooden spoon.
Lino Gatti's side have a horror goal difference compared to their Hunter Valley rivals and even if they beat Singleton next Tuesday, they'll still need to pick up a result or two elsewhere to avoid the spoon. The good news for Cessnock is that they can play with freedom and without fear. You never quiet know if you're going to wind up in a tight battle with the bottom-placed side.
Belmont-Swansea crushed Wallsend into the ground last weekend and if Cessnock show a similar display to the one that saw them dispatched easily by Kahibah not too long ago, you can expect more carnage. But it's hard to judge.
The 5th-placed Belswans simply need to keep winning and hope that one of the top four sides will drop off the pace. This side is too good to not play finals football and that is the absolutely minimum target for Josh Rufo and co. Tega Marcus' four goal haul last week took him to 16 for the season, more than any other 1st Grade player.
Prediction: Like the Cooks Hill v Wallsend game, there should only be one winner but by how much is the question. Cessnock are a tricky side to produce but they'll find it tough against an in-form Belswans. If Chris Gazzard and Sean Matthews click and combine with Tega Marcus and others further forward, it could be a long afternoon at Turner Park.
My Tip: Cessnock City 1-4 Belmont-Swansea
Cessnock's sudden revival in the Under 23's has come as a bit of a shock but the Hornets are producing the goods on the back of some much needed confidence and suddenly any chance of picking up the dreaded wooden spoon is a distant memory. Finals are perhaps out of reach but they can certainly impact a lot of sides trying to battle it out for the likely last two spots in the top four.
Belswans were putting up a good fight against Wallsend last week but similar to their heavy loss to Cooks Hill the week before, a young Belswans side just didn't have the quality to match it with the top two side. It's good to see 17's players like Harry Dyer stepping up into this grade and really making an impact. Finals aren't out of the question and it's games like this one that they need to win.
Prediction: Belswans got to play last weekend which no doubt helped their flow and after back to back games against the two pace-setting sides, they'll be happy to see a less challenging task against Cessnock. Still, the Hornets are in good form in the 23's so it's a tough one to call.
Cessnock City (11th) v Belmont-Swansea UFC (1st)
Six straight losses has the Cessnock 19's sitting firmly on the bottom of the table but with a game in hand on the team above them Wallsend. Their last victory came almost two months ago and the less said about their last match against Belswans the better. Their defence simply did not show up at all in that 8-0 loss.
Belswans are really starting to find their peak form and it's a danger to every team in the competition. The defending champions are still unbeaten since last May and have won their last five games to suddenly sit on top of the table. An 8-1 hiding of Wallsend last week brought back memories of that 8-0 win over Cessnock. They'll head into this one as heavy favourites.
Prediction: Anything is an improvement on the last time this sides played for Cessnock. But still, there's no chance of them winning this if Belswans produce their best football. It could be a very long day if Cessnock don't step up to the challenge.
Cessnock City (9th) v Belmont-Swansea UFC (5th)
Cessnock's hapless run of form in the Under 17's came to an ending of sorts as they picked up a rare point against South Cardiff last time out. It was a little bit of a positive moment after some horror results in recent times. They also defeated Belswans last time they met so that plays into their favour slightly.
Belswans 17's picked up a vital 3-1 win over Wallsend last weekend and with the rest of the round washed out, it helped them gain a little ground on the top four. Some of the players have found some confidence and this side are certainly one to watch in the race for finals football. It's a must-win game for them no doubt.
Prediction: Cessnock's draw with Southy brought some much needed relief to their poor form but Belswans are a different challenge all together. Belswans' attackers have been spoken about but defensively they've been quiet impressive at times. Harrison Rapp has been a constant stand-out for them.
New Lambton FC (7th) v South Cardiff (2nd)
Last Time They Met: South Cardiff 4-0 New Lambton
New Lambton's finals hopes rest on this weekend's result against South Cardiff. A win won't throw them into the top four or anything along those lines but a loss will surely spell an end to their slimming finals hopes, especially if the teams above can all pick up three points.
So often this season the Eagles have risen to the challenge when asked and have revenge on their mind after a shock 4-0 loss to Southy at Ulinga Oval on Good Friday. Head Coach Andrew Packer is a tough to please and that heavy loss to his old side would have hurt like no other. New Lambton will likely aim up to really make Southy work for it.
The Gunners have gone about their business fairly routinely as of late. A 2-0 win over Cessnock lifted them to second spot and if Cooks Hill stumble again then top spot could await Denis Fajkovic's side. Southy will not get ahead of themselves though. They know the job is still nowhere near complete and consistence performance is needed.
With a low goal differential and Belswans within range of them, South Cardiff need to keep up the winning form to continue their run inside the top four. A loss won't hurt them too much with loads of football still to follow but a win could really hammer them into the top four and arguably end the Eagles' finals tilt.
Prediction: As of writing, there was a chance this game could be moved to Ulinga if any rain fell on Alder Park. If the game does go ahead at Alder, expect an unpredictable and tough surface. It could be a very gritty low-scoring affair between the two sides. But then again, Southy did the work easily on a wet and difficult Ulinga Oval on Good Friday.
My Tip: New Lambton 1-2 South Cardiff
New Lambton FC (9th) v South Cardiff (5th)
Confidence has fallen that New Lambton 23's can make the finals and suddenly every fixture going forward is a key one if they are a chance of taking advantage of what is a very tight 23's competition. They were stunned by Southy last time they met with the Gunners scoring late to take all three points at Ulinga Oval.
South Cardiff have been a prime example of the lack of consistency in the Under 23's competition. They remain just one win outside the top four but a whopping 17 points behind ladder-leaders Cooks Hill. Southy's depth has been tested at times with 1st Grade requiring some 23's from time to time and the results just haven't been there some weeks.
Prediction: It's tough to call between two sides keen to build some foundation for a finals run. New Lambton's last outing against West Wallsend was another disappointing result. It's a big character test in belief and that's something that may just favour Southy.
New Lambton welcomed back key attackers Lachlan Jones and Maurice Bechelli on Thursday night but it wasn't enough for the Eagles to see off Kahibah in a 2-2 draw. The word unconvincing comes to mind and it was one of New Lambton's worst displays of the season. The two dropped points sees them drop off the pace in the race for the minor premiership.
South Cardiff have never got going in 2017 in the Under 19's. Their last result was a drought-breaking win over lowly Cessnock but they need to continue that winning form to remain any hope of pulling off a shock and making finals. It's highly doubtful but a respectable back end of the year will be the target for the Gunners.
Prediction: New Lambton were poor on Thursday. Something was off about the display and it wouldn't surprise to see a more refined effort on Sunday. Southy's defence hasn't exactly been solid but attacking-wise they've been the lowest scoring side in the Under 19's. Eagles to win if they're full strength.
New Lambton dropped off the pace in Thursday's 1-1 draw with Kahibah. They led 1-0 but the latter stages of the game saw the Eagles almost lose it. Goalkeeper Jasper Burley was once again a key factor for them as well as Liam Walker who produced another important goal. They've played a lot better games than that on Thursday.
Southy 17's are a long way out of finals contention and no result showed that more than a 1-1 draw with a vastly out of form Cessnock City side last time out. The Gunners did have an injury crisis in the 17's and their form has never got going, similar to the club's 19's side.
Prediction: With Kahibah having a game in hand and Belswans right behind them, New Lambton know that their top four spot is one that's not exactly safe. Winning games is key and it's safe to expect them to raise the bar a little bit against a struggling Southy side. Eagles to win.
Toronto-Awaba (9th) v Thornton Redbacks (3rd)
Last Time They Met: Thornton 5-2 Toronto
Toronto's stunning 4-1 victory over West Wallsend put an end to any belief that the Stags could wind up with a wooden spoon and there's one or two die-hards that believe the side can pull off a miracle and perhaps sneak into finals. Make no doubt, any hope of that rests on if they can overcome Thornton.
That win over Westy was one of the best performances from Toronto-Awaba in recent years. The team played some good football and were surprisingly clinical in front of goal. If they had more of that this season then they may have been higher on the table. They'll take it one game at a time and on home turf, who knows what they may produce with some confidence behind them.
Thornton's minor premiership tilt has halted, their form has gone out the window and now some predict that their spot inside the top four could soon belong to Belswans. The panic light isn't flickering just yet but a loss to Toronto will have the Redbacks looking for answers. They can't keep this poor form going.
A 2-2 draw with Cessnock last time out was a poor result after the Redbacks lead 2-0 with five minutes to go. Toronto will obviously be fully aware of that result and the form that Darrel McAllister's side is currently in. There were seven goals last time they met but expect less on a heavy Lyall Peacock surface with rugby league playing on the ground on Saturday.
Prediction: This is a tough one to call. Toronto were good value last time out but if Thornton can find their best then they'll be a tough side to stop. Nathan McAllister has been vital for his father's side and Jamie Subat will again be a key up front but the Stags have some confidence lately. Tough to call.
My Tip: Toronto-Awaba 1-1 Thornton Redbacks
Toronto-Awaba (11th) v Thornton Redbacks (3rd)
A much more positive performance for Toronto 23's last time out as they pushed West Wallsend to the limits and lost with the final kick of the game. Still, that was another loss and it leaves them seven points behind 10th-placed Cessnock. Avoiding the wooden spoon is a big ask and that's no doubt what the Stags are aiming to do.
Thornton's last game came all the way back on May 28th so no doubt they'll be keen to get back onto the field. Three-straight losses for the previously in-form side has seen them drop off the pace of Cooks Hill and Wallsend but fortunately they have a a few points in hand on the chasing pack and an extra game. Getting back to winning form will do them a world of good.
Prediction: Toronto were smashed last time they met Thornton but the Redbacks will be down on confidence after no wins in their last three. Still, the 3rd-placed side should have the quality to see off the cellar-dwellers but it might not be as simple as that.
Toronto-Awaba (9th) v Thornton Redbacks (5th)
Toronto-Awaba failed to put up much of a challenge against West Wallsend in their last match. The Stags are a long way from finals contention but they do have a chance to pick up a win or two in coming weeks and move away from Cessnock at the bottom of the table. Last time they played Thornton, they went down 2-1 in a hotly contested match-up.
The Redbacks picked up a 1-0 win over Cessnock last time out but it was the sort of victory that doesn't fill onlookers with confidence. Thornton have stumbled in recent times and find themselves outside the top four, albeit with a game in hand on a few higher-placed sides. It's most likely a five-way race for the four finals spots and it's games like this that teams can ill-afford to drop.
Prediction: Toronto will see Connor Morgan return from suspension which gives their attacking potential a big boost but it's further back where things went wrong against West Wallsend. An early goal would do the Stags wonders but another home loss could be on the cards.
Toronto-Awaba (3rd) v Thornton Redbacks (10th)
Toronto-Awaba 17's lost what was a potential finals preview against West Wallsend last time out. The Stags fought hard but were arguably outplayed and lost late on to an unfortunate cross-turned-goal. Still, this team is one that is confident and looks finals bound if they can produce good form and beat teams like Thornton.
The Redbacks picked up a break-through win in their last game as they downed Cessnock City 2-0. It brought a range of emotions to the Thornton squad but mostly that of relief as they finally got to sing the team song. Whether that kick-starts a form run of not is yet to be seen and the Redbacks will be lacking match fitness after three weeks off.
Prediction: Thornton were decent in that win over Cessnock but this game is on another level entirely. The Stags need to bounce back to winning ways with some teams like Belswans and Kahibah potentially chasing down a spot inside the top four. Thornton defenders will need to keep an eye out for 17's top scorer Thomas Appleby.
NewySports.com
Ty Stedman
Ty's Jersey Collection
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 4,245
|
Q: Looking up SessionContext in JBoss 7.2 For various reasons I need to perform a manual lookup of SessionContext. In JBoss5, the solution
InitialContext initialContext = new InitialContext();
SessionContext sessionContext = (SessionContext) initialContext.lookup("java:comp/EJBContext");
has served med well, but from JBoss 7 I instead get a
javax.naming.NameNotFoundException: EJBContext -- service jboss.naming.context.java.global.EJBContext
Has something changed in how context is looked up in JBoss 7.2, or is my deployment lacking anything vital? For reference, standard injection works fine, this is the only lookup that fails. Or am I doing something terribly wrong (besides performing a manual lookup of SessionContext)?
A: According to specification of Java EJB (this one is for EJB 3.2. but nothing changed about EJBContext from previous one, EJB 3.x), you can inject EJBContext into your components either using annotation @Resource or manually via lookup (section 11.15):
The container must make a component's EJBContext interface available either through injection
using the Resource annotation or in JNDI under the name java:comp/EJBContext
Standard way of looking up for EJB resource is via EJBContext.lookup method but there is also JNDI way which is the only possibilities if you don't have already EJBContext:
Context initCtx = new InitialContext();
EJBContext ejbCtx = (EJBContext) initCtx.lookup("java:comp/EJBContext");
This is exactly what you did, so what is wrong? There are two things, which one I'm not sure about. First, with manually lookup it's sometime needed to assign resource to component with annotation at class level:
@Resource(name = "EJBContext", type = javax.ejb.EJBContext)
public class MyComponent {
...
}
but I'm not sure is it needed for EJBContext as well, I guess not. The second thing, more important and critical - according to specification once again:
EJBContext objects accessed through the naming environment are only valid within the bean
instance that performed the lookup.
this one is section 11.15.1, and the next one, section 11.15.2:
The Container Provider is responsible for providing an appropriate EJBContext object to the refer-
encing component. The object returned must be of the appropriate specific type for the bean requesting
injection or performing the lookup—that is, the Container Provider must return an instance of the SessionContext interface to referencing session beans and an instance of the MessageDrivenCon-
text interface to message-driven beans.
Those both mean that injection and lookup for EJBContext are only valid in Enterprise Java Beans, so those which are annotated with @MessageDriven, @Stateful, @Singleton or @Stateless (or described as EJBs in deployment descriptor file, also as EJB 2.x Specification). Maybe your component isn't valid EJB and it's why lookup isn't working? This is only suggestion of course.
There's one more possibilities to get EJBContext (more correctly SessionContext). Your component should implements SessionBean interface which has setSessionContext(SessionContext sessionContext) method. This method should be invoked by EJB container every time when component is used (injected somewhere, invoked by client or timeout, especially when it's created) and inside this method you should assign sessionContext parameter to bean's field.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 9,607
|
100% uptime in US&EU since 2013!
We know that a reliable service is important for you and we take a great pride in keeping a superb uptime. Our engineers are watching our network closely 24/7 so you can rest assured that your visitors and customers will get the data whenever they need it at the best speed possible.
CloudHarmony, a well respected network magazine, has published a report comparing uptimes of various CDN services. And guess what – we are amongst the very best of them.
During the last year, our 365-day uptime reached 99.9995%. In Europe, US and Australia&Oceania, it was full 100%.
The full report can be found here: https://cloudharmony.com/status-1year-of-compute-and-storage-and-dns-and-cdn-group-by-regions-and-provider (filter CDN services).
As we understand that a report is one thing and own experience another, we provide our potential clients with a 14-day absolutely free trial. Sign up and make sure our service meets your needs even before you make your first payment or insert your CC number. Because we believe that's fair.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 6,072
|
Tim Sebastian (born 13 March 1952) is a television journalist and novelist. He is the moderator of Conflict Zone and The New Arab Debates, broadcast on Deutsche Welle. He previously worked for the BBC, where he hosted The Doha Debates and was the first presenter of HARDtalk. He also presented Bloomberg TV's The Outsider, an India-focused debating programme.
He won the BAFTA (British Academy of Film and Television Arts) Richard Dimbleby award in 1981 and Britain's prestigious Royal Television Society Interviewer of the Year award in 2000 and 2001.
Education
Sebastian was born in London, England. He was educated at Westminster School, a fee-paying independent school in Central London. He holds a BA degree in Modern Languages from New College, Oxford and speaks both German and Russian.
He has a Diploma in Journalism Studies from Cardiff University, graduating in 1974.
Journalist
Sebastian began his journalism career at Reuters in 1974, moving to the BBC as foreign correspondent in Warsaw in 1979. He became BBC's Europe correspondent in 1982. At that time in 1983 in Wrocław, during Martial law in Poland, Tim Sebastian interviewed Kornel Morawiecki, the leader of the Polish anti-Soviet and anti-communist underground organization Fighting Solidarity, hiding from the Polish security service.
Between 1984 and 1985 (until his expulsion from the USSR in 1985) he was BBC's correspondent for Moscow in 1984 and then for Washington from 1986 to 1989.
Sebastian has worked for The Mail on Sunday, and has contributed to The Sunday Times.
Interviewer
Memorable Hard Talk interviews with world leaders included US Presidents Bill Clinton Donald Trump and Jimmy Carter, Archbishop Desmond Tutu, Singapore's first Prime Minister Lee Kuan Yew, and the last leader of the Soviet Union Mikhail Gorbachev. He currently hosts Conflict Zone, a one-on-one interview show on Deutsche Welle's international English-language channel.
In March 2016, he interviewed the leader of the German party AfD, Frauke Petry.
Debate moderator
Sebastian is a frequent moderator of major conferences, seminars and forums across the globe.
He was the chairman of The Doha Debates, a Qatar Foundation programme that was broadcast monthly on BBC World News where it was the highest-rated weekend programme. The Debates were founded by Sebastian in 2004 and their fifth series began in September 2008.
Following the political and social unrest in Egypt and Tunisia in early 2011, Tim Sebastian founded The New Arab Debates, which have been held in Egypt, Tunisia and Jordan and were broadcast on Deutsche Welle English as well as regional television channels. The debates are also held in Arabic and hosted by Egyptian TV presenter Mai El Sherbiny.
Awards
In 1982, Sebastian was awarded the British Academy of Film and Television Arts Richard Dimbleby Award and was named Television Journalist of the Year by the Royal Television Society. Additionally he has twice won the Royal Television Society's Interviewer of the Year Award for his HARDtalk interviews.
Personal life
His daughter is CNN journalist Clare Sebastian.
Bibliography
Non-fiction
Nice Promises: Tim Sebastian in Poland (1985)
I Spy in Russia (1986)
Novels
The Spy in Question (1988)
Spy Shadow (1989)
Saviour's Gate (1991)
Exit Berlin (1992)
The Memory Church (1993)
Last Rights (1993)
Special Relations (1994)
War Dance (1995)
Ultra (1997)
References
External links
"The Doha Debates" website
Official biography of Tim Sebastian
Interview in the Gulf Times: 'Sebastian is looking beyond the stereotypes'
Interview in Oryx Magazine: 'Courting Controversy'
Tim Sebastian and the Doha Debates
"The New Arab Debates" website
1952 births
Living people
People educated at Westminster School, London
British reporters and correspondents
BBC newsreaders and journalists
BBC World News
BBC World Service presenters
English male journalists
English television presenters
BAFTA winners (people)
20th-century English novelists
English male novelists
Qatar Foundation people
20th-century English male writers
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 3,205
|
Q: Working programmatically with an HTTPResponse in Django I am working on an app which would enable a preview function for a model. Models marked as preview-able would allow for changes to be made in the Django admin interface and previewed on site using the same view as would an object of that type normally use to render itself, but rendered instead with the new (unsaved) object in it's place.
This is a pretty easy task to do in a bespoke fashion when you know the views or templates ahead of time. But I want this to be reusable and simple.
What I Was Thinking
My idea would be to apply the resolve() urls function to the existing (saved) object's get_absolute_url() value to discover the view used dynamically. Then call that view, get the returned HTTPResponse and alter it in some fashion before returning it myself.
The Problem
It seems that by the time the HTTPResponse is returned by the object's natural view the HTML has already been rendered by the template engine.
So I guess the question is: Is there a way to get at a HTTPResponse before the template is rendered and alter the context variables before that happens.
If not then could the same idea be implemented in another fashion. Would using middleware change anything (your still working with a HTTPResponse object there as well).
Thanks,
Marcus
P.S. If you come up with a radically different methodology to solve this problem, I will be sure to attribute that concept to you in the app documentation (despite it being a small app right now).
A: It is not trivially possible no, the easiest way would actually be to write your own template context processor that checks for example if something like GET['preview'] is set, then sets dictionary values based on some other GET or POST data. Furthermore when other variables are added it should make sure these don't overwrite the existing values set by this method (otherwise the view would override it anyway with some other data).
One remark however: completely unintrusive behaviour will often lead to erroneous behaviour. If the view does not know of this preview functionality and e.g. it expects a valid id or redirects to an error page, your preview won't work (as you don't really have a valid id). Choosing for views that know of this preview functionality is indeed some more work but will certainly be correct. You could try to make it more generic by using decorators or callable view classes (which can be derivable from some generic base) instead of view methods.
A completely different approach that is only slightly 'view' intrusive, I assume you do not want to save the model so it doesn't show up in public listings. An approach could be to add a 'preview' field and use managers to restrict lookups, so something like this:
class NoPreviewManager(models.Manager):
def get_query_set(self):
return super(MyModelManager, self).get_query_set().filter(preview=False)
class MyModel(models.Model):
... #other fields
preview = models.BooleanField()
objects = NoPreviewManager()
allobjects = models.Manager()
In all normal views you can just use MyModel.objects so previews won't be listed. In the object-specific view you use MyModel.allobjects to also enable defailed views of previews. This way you don't have to do weird view hijacking things, but you should take care preview objects get cleaned up if they aren't promoted to real objects. Note that you can also combine many of this logic into a base class.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 4,528
|
Our Products >> Florence Freedom Baseball Fundraiser !
Have Fun and & Support Families with ASD!! $5 from every $10 ticket will go back to Families with ASD!!
Join us Sunday, August 13th. for the Florence Freedom Ball Game! Gates will open up at 5:05pm and first pitch will be at 6:05pm.
For every ticket that your organization purchases the Freedom will be donating $5 back to the organization. Ticket price at $10 for Reserved seating!
Families with ASD has reserved 100 tickets so we can all sit together!
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 1,851
|
\section{TRIAD}
\label{sec:tiara}
The TPL aware detailed router (TRIAD) focuses on accomplishing detailed routing for all given nets and generating a highly decomposable routing outcome with low yield loss.
The routing model of NEMO is adopted here \cite{Routing_Li_TCAD07}\cite{Routing_Chang_ISPD08}.
This work proposes a technique to make TECG work on the routing model of NEMO.
With the aid of TECG, TRIAD can generate stitches which cannot be generated by adopting the conventional DPL stitch generation scheme.
Therefore, TRIAD can update the routing cost based on the number of stitches and TPL conflicts.
Figure \ref{fig:flow} shows the routing flow of TRIAD.
Firstly, all multi-pin nets are decomposed into two-pin nets.
A TPL-aware routing which allows the stitch generation at the cost of increasing routing cost is applied to route all two-pin nets.
If one two-pin net is routed without any conflicts, then the layout and TECG are updated.
Otherwise, TRIAD checks whether conflicts can be generated in current iteration while TRIAD is prohibited to generate TPL conflicts in the first few iterations.
If the TPL stitch generation is not allowed, TRIAD rips up routed nets to release the routing resource and then re-routes the two-pin net.
Otherwise, the layout and the TECG are directly updated.
\begin{figure}[bt!]
\centering
\includegraphics[width=0.45\textwidth]{fig/flow.eps}
\caption{Overall flow of TRIAD.}
\label{fig:flow}
\end{figure}
\subsection{TECG on the Gridless Routing Model}
When constructing contours for routed wire segments, TRIAD also constructs \textit{shadows} for routed wire segments presented by vertices in $\mathcal{G^C}$.
One shadow denotes the TPL effect region of its attached routed wire segment.
TRIAD constructs shadows by extending routed nets by $hw_w+sp_{tp}$.
Figure \ref{fig:rmodel}(a) shows three extracted PMTs with intersected shadows.
The vertices in $\mathcal{G^C}$ attached to the corresponding shadows assist TRIAD in detecting TPL conflicts when the potential routing wire segments pass through a PMT.
The CG vertex representing the potential routing wire segment connects to the CG vertices representing routed wire segments by passing through their corresponding shadows, and the path propagation of TRIAD thus becomes aware of TPL conflicts.
Figures \ref{fig:rmodel}(b)--(d) illustrate the path propagation of TRIAD.
Figure \ref{fig:rmodel}(b) shows five routed wire segments, a PMT with three shadows, and one TECG.
In Fig. \ref{fig:rmodel}(b), one routing wire segment passes through the PMT, sequentially inserting one vertex $F$ in $\mathcal{G^C}$ and one token $T_4$ in $\mathcal{G^T}$.
The routing wire segment represented by $F$ passes through three shadows of $A$, $B$, and $C$.
Therefore, TRIAD iteratively connects $F$ to $A$, $B$, and $C$.
After connecting $F$ to $A$ and $B$, $T_1$ and $T_3$ are merged into $T_5$ as shown in Fig. \ref{fig:rmodel}(c).
In Fig. \ref{fig:rmodel}(d), TRIAD detects one TPL conflict by connecting $F$ and $C$ because $token(F)$ and $token(C)$ equal $T_5$.
\begin{figure}[bt!]
\centering
\subfloat[]{\includegraphics[width=0.24\textwidth]{fig/routing_model0.eps}}
\subfloat[]{\includegraphics[width=0.24\textwidth]{fig/routing_model2.eps}}
\subfloat[]{\includegraphics[width=0.24\textwidth]{fig/routing_model4.eps}}
\subfloat[]{\includegraphics[width=0.24\textwidth]{fig/routing_model5.eps}}
\caption{TECG on routing model of NEMO: (a) TRIAD constructs shadows to represent TPL effect region of routed wire segments; (b) PMT contains three shadows of three routed wire segments with TECG, and TRIAD inserts one vertex $F$ in $\mathcal{G^C}$ and a token $T_4$ in $\mathcal{G^T}$ to represent the routing wire; (c) connecting $F$ to $A$ and $B$ iteratively merges $T_3$ and $T_4$ into $T_5$; (d) TRIAD detects one coloring conflict after connecting $F$ and $C$.}
\label{fig:rmodel}
\end{figure}
\subsection{TPL Stitch Generation Scheme}
\label{sec:tiara_stitch}
After detecting TPL conflicts, TRIAD splits one of the terminal vertices of the conflicting edge to differ the assigned token by generating stitches, if possible.
The DPL stitch generation scheme inserts one stitch in one wire when the wire contains at least one segment that is not passed by shadows of other wire segments.
\textcolor[rgb]{0.00,0.00,0.00}{
Based on the DPL stitch generation scheme, in Fig. \ref{fig:rmodel}(d), no stitch can be inserted in the routing wire segment because the routing wire segment is entirely overlapped by the shadows of routed wire segments.
However, the TPL stitch generation scheme is quite different from the DPL stitch generation scheme.}
With the assistance of TECG, TRIAD can generate stitches at the wire segment even if which is entirely passed by shadows of other wires.
\textcolor[rgb]{0.00,0.00,0.00}{
Before introducing the proposed TPL stitch generation scheme, some definitions are given in the following.}
\textcolor[rgb]{0.00,0.00,0.00}{
\begin{define}[\textbf{Shadowy Interval}]
One shadowy interval, denoted as $\varphi$, is one interval of one wire segment, and one wire segment may contain several shadowy intervals.
Let $S^T_{shd}(\varphi)$ be the set of tokens represented by the shadow set passing through $\varphi$.
For any two adjacent shadowy intervals $\varphi_i$ and $\varphi_j$, $S^T_{shd}(\varphi_i)$ and $S^T_{shd}(\varphi_j)$ cannot be identical.
\end{define}
}
\textcolor[rgb]{0.00,0.00,0.00}{
\begin{define}[\textbf{Splittable Shadowy Interval}]
Given one wire segment \textit{w} represented by one CG vertex $v^c$ and one SCC $scc \in S^{SCC}$ containing $token(v^c)$.
Let $\varphi_{i}$ and $\varphi_j$ be two adjacent shadowy intervals of $w$.
One shadowy interval $\varphi_i$ is called \textit{splittable} when $|S^T_{shd}(\varphi_i) \cap S^T_{shd}(\varphi_j) \cap scc| $ is less than two
\end{define}
}
\textcolor[rgb]{0.00,0.00,0.00}{
\begin{define}[\textbf{Splittable CG Vertex}]
Let $V_{adj}^\mathcal{C}(v^c)$ denote the adjacent vertex set of one vertex $v^c \in V^\mathcal{C}$.
Given one SCC $scc = (token(v^c), token(v_{ad1}), token(v_{ad2})) \in S^{SCC}$ where $v^c_{ad1} \in V_{adj}^\mathcal{C}(v^c)$, $v^c_{ad2} \in V_{adj}^\mathcal{C}(v^c)$, and $v^c_{ad1} \neq v^c_{ad2}$.
One CG Vertex $v^c$ is called \textit{splittable} when $v^c$ contains a set of splittable shadowy intervals that can split $v^c$ into a CG vertex set $V^\mathcal{C}_{SPLIT}$ where $\forall v^c_s \in V^\mathcal{C}_{SPLIT}, token(v^c_s)$ connects to at most two tokens of $scc$
\end{define}
}
\begin{algorithm}[bt]
\caption{TPL Stitch Generation}
\label{alg:st_gen}
\begin{algorithmic}[1]
\REQUIRE One CG vertex $v^c$ to be split, one CG vertex set $V^\mathcal{C}_{c}(v^c)$ adjacent to $v^c$ where $\forall v^c_{adj} \in V^\mathcal{C}_{c}(v^c), token(v^c) = token(v^c_{adj}) = T_c$, one SCC $scc$ containing $T_c
\STATE Compute shadowy intervals $S^{splt}$ in wire segments represented by $v^c$ for $scc$;
\FORALL{Shadowy interval $\varphi \in S^{splt}$}
\IF{$|S^T_{shd}(\varphi)|>2$}
\STATE Increase the routing cost by one $penalty_{unsolvable}$;
\STATE \textbf{break};
\ENDIF
\ENDFOR
\STATE $num_{st} := 0$;
\STATE $\varphi_{st\_cand} := \varphi_{st}$ := $NULL$;
\STATE $S^T_{passed}$ := $\emptyset$;
\STATE Topologically sort $S^{splt}$;
\FORALL{Shadowy interval $\varphi \in S^{splt}$}
\IF{$|S^T_{shd}(\varphi)| = 1$}
\STATE $\varphi_{st\_cand} := \varphi$;
\ENDIF
\STATE $S^T_{passed} := S^T_{passed} \cup S^T_{shd}(\varphi)$;
\IF{$|S^T_{passed}| > 2$}
\STATE Generate one stitch at $\varphi_{st\_cand}$;
\STATE $++num_{st}$;
\STATE $S^T_{passed} := \emptyset$;
\FORALL{Shadowy interval $\varphi_{passed}$ between $\varphi$ and $\varphi_{st}$}
\STATE $S^T_{passed} := S^T_{passed} \cup S^T_{shd}(\varphi_{passed})$;
\ENDFOR
\STATE $\varphi_{st} := \varphi_{st\_cand}$;
\ENDIF
\ENDFOR
\STATE Increase the routing cost by $num_{st} \times penalty_{st}$;
\end{algorithmic}
\end{algorithm}
One wire segment to be split represented by one CG vertex $v^c$ may contain several splittable shadowy intervals for one SCC.
Generating stitches at all splittable shadowy intervals introduces unnecessary stitches, sequentially degrading the yield.
To minimize the number of required stitches, the TPL stitch generation algorithm is proposed in Algorithm \ref{alg:st_gen}.
All shadowy intervals $S^{splt}$ in wire segments represented by $v^c$ are firstly computed (line 1).
One conflicting edge cannot be solved by splitting one wire segment with one shadowy interval passed by more than two shadows because two adjacent CG vertices must be assigned to the same token after splitting.
After detecting one unsolvable conflicting edge, the routing cost is directly increased by one unsolvable penalty $penalty_{unsolvable}$ (lines 2--7).
One token set $S^T_{passed}$ is initially set as empty (line 10).
Before generating stitches based on $S^{splt}$, $S^{splt}$ is firstly topologically sorted (line 11), followed by sequentially checking the shadowy interval $\varphi \in S^{splt}$.
If $|S^T_{shd}(\varphi)|$ equals one, $\varphi$ is recorded as the potential position $\varphi_{st\_cand}$ to generate one stitch (lines 13--15).
Inserting $S^T_{shd}(\varphi)$ into $S^T_{passed}$ (line 16) may cause $|S^T_{passed}|$ to exceed two, requiring generating one stitch at $\varphi_{st\_cand}$ (lines 17--19).
Then $S^T_{passed}$ is set as empty, and the tokens attached to shadows passing through the shadowy intervals between $\varphi_{st}$ and $\varphi$ are inserted into $S^T_{passed}$ (lines 20--23).
Finally, $\varphi_{st}$ is set as $\varphi_{st\_cand}$ to record the latest stitch position (line 24).
After all shadowy intervals in $S^{splt}$ are checked, the routing cost is increased based on the number of generated stitches (line 27).
Notably, the token $T_c$ that causes conflicting edges may belong to more than one SCC.
Therefore, Algorithm \ref{alg:st_gen} is applied to each SCC that contains $T_c$ to generate necessary stitches.
Figures \ref{fig:tpl_st}(a) and \ref{fig:tpl_st}(c) show a small part of one TECG where CG vertices $A, B, C, D$ and $U$ represent four routed wire segments and one routing wire segment, respectively.
Notably, $token(A)$ equals $T_1$; $token(B)$ and $token(D)$ equal $T_2$/$T_1$; and $token(C)$ and $token(U)$ equal $T_3$ in Fig. \ref{fig:tpl_st}(a)/(c).
The routing wire segment in Fig. \ref{fig:tpl_st} contains seven shadowy intervals $\varphi_1, \varphi_2, \cdots,$ and $\varphi_7$, and Algorithm \ref{alg:st_gen} generates stitches by sequentially checking these shadowy intervals.
In Fig. \ref{fig:tpl_st}(a), $|S^T_{passed}|$ equals three when checking $\varphi_4$, resulting in generating one stitch at $\varphi_3$ as shown in Fig. \ref{fig:tpl_st}(b).
However, if $token(D)$ is assigned to $T_1$ as shown in Fig. \ref{fig:tpl_st}(c), the edge $(D, U^2)$ in Fig. \ref{fig:tpl_st}(b) becomes conflicting.
Similarly, for the TECG in Fig. \ref{fig:tpl_st}(c), one stitch is generated in $\varphi_3$ as shown in Fig. \ref{fig:tpl_st}(d) followed by setting $S^T_{passed}$ as $\{T_2, T_3\}$.
When checking $\varphi_7$, $S^T_{passed}$ equals $\{T_1, T_2, T_3\}$, requiring generating another stitch at $\varphi_5$ as shown in Fig. \ref{fig:tpl_st}(d).
To solve the conflicting edge in Fig. \ref{fig:rmodel}(d), one stitch is generating at the routing wire segment by splitting $F$ into $F^1$ and $F^2$ as shown in Fig. \ref{fig:tpl_st}(e).
\begin{figure}[bt!]
\centering
\subfloat[]{\includegraphics[width=0.24\textwidth]{fig/st_1_bf.eps}}
\subfloat[]{\includegraphics[width=0.24\textwidth]{fig/st_1_af.eps}}
\subfloat[]{\includegraphics[width=0.24\textwidth]{fig/st_2_bf.eps}}
\subfloat[]{\includegraphics[width=0.24\textwidth]{fig/st_2_af.eps}}
\subfloat[]{\includegraphics[width=0.24\textwidth]{fig/stitch_generation.eps}}
\caption{\textcolor[rgb]{0.00,0.00,0.00}{Example of TPL stitch generation schemes: (a)(c) DPL stitch generation scheme cannot insert any stitch; (b) generating one stitch can solve the conflict edge in (a); (d) generating two stitch can solve the conflict in (c); (e) TPL stitch generation of Fig. \ref{fig:rmodel}(d).}}
\label{fig:tpl_st}
\end{figure}
\section{Experimental Results}
\label{sec:experiment}
\begin{table}[bt!]
\centering
\caption{Statistics of benchmarks}
\label{tb:stat}
\begin{tabular}{|c||c|c|c|c|c|}
\hline
\multirow{2}{*}{Circuit} & \multirow{2}{*}{Size $(\mu m^2)$} & \# & \# & \# 2Pin & \# \\
& & Layer & Net & Net & Pin \\
\hline
\hline
s5378 & 217.5 $\times$ 119.5 & 3 & 1694 & 3124 & 4818 \\
\hline
s9234 & 202.0 $\times$ 112.5 & 3 & 1486 & 2774 & 4260 \\
\hline
s13207 & 330.0 $\times$ 182.5 & 3 & 3781 & 6995 & 10776 \\
\hline
s15850 & 352.5 $\times$ 194.5 & 3 & 4472 & 8321 & 12793 \\
\hline
s38417 & 572.0 $\times$ 309.5 & 3 & 11309 & 21035 & 32344 \\
\hline
s38584 & 647.5 $\times$ 336.0 & 3 & 14754 & 28177 & 42931 \\
\hline
\end{tabular}
\end{table}
\begin{table*}[t]
\centering
\caption{Comparison between wirelength, stitches, conflicts, and runtime of the greedy approach (GREED) and TRIAD}
\label{tb:result}
\begin{tabular}{|c||c|c||c|c||c|c||c|c|}
\hline
\multirow{2}{*}{Circuit} & \multicolumn{2}{|c||}{ Wirelength ($nm$)} & \multicolumn{2}{|c||}{\# Stitch} & \multicolumn{2}{|c||}{\# Conflict} & \multicolumn{2}{|c|}{Runtime (\textit{s.})} \\
\cline{2-9}
& GREEDY & DPLAG & GREEDY & DPLAG & GREEDY & DPLAG & GREEDY & DPLAG \\
\hline
\hline
s5378 & 382900 & 381170 & 165 & 0 & 0 & 0 & 9.47 & 14.36 \\
\hline
s9234 & 286503 & 284608 & 157 & 0 & 1 & 0 & 8.62 & 9.78 \\
\hline
s13207 & 910055 & 903705 & 405 & 1 & 2 & 0 & 25.38 & 49.11 \\
\hline
s15850 & 1131665& 1124715 & 371 & 0 & 2 & 0 & 40.83 & 95.80 \\
\hline
s38417 & 2457675& 2461940 & 1528 & 0 & 3 & 0 & 122.7 & 443.75 \\
\hline
s38584 & 3211985& 3204160 & 1264 & 2 & 2 & 0 & 168.97 & 660.38 \\
\hline \hline
Ave. & 100\% & 99.46\% & 560.67 & 0.50 & - & - & 1 & 2.41\\
\hline
\end{tabular}
\end{table*}
\textcolor[rgb]{0.00,0.00,0.00}{
The algorithm herein was implemented in C++ language on a workstation with 4-Core 2.4 GHz CPU and 82GB memory.}
A total six benchmarks \cite{Routing_MARS} are adopted in this work.
We scale all benchmarks, including routing area and features size, to approach the target process node.
Table \ref{tb:stat} shows the corresponding statistics.
\textcolor[rgb]{0.00,0.00,0.00}{
The minimum resolution (half-pitch) for pushing the 193\textit{nm} lithography's single exposure limit is around 40\textit{nm}.
Thus, to print 20\textit{nm} half-pitch, we need double patterning, and to print 10\textit{nm} half-pitch, we need quadruple patterning.
The minimum coloring spacing for single exposure lithography is fixed, i.e., around 40\textit{nm}.
The purpose of multiple patterning is to push for smaller resolution (half-pitch).
Therefore, to the first order, the minimum coloring spacing would be \textit{n} times minimum wire spacing (i.e., half-pitch) of the \textit{n} patterning lithography.
The minimum coloring spacing $sp_{tp}$ is set as three times of the minimum wire spacing.
}
As there is no other TPL aware router published, to demonstrate the effectiveness of the proposed algorithm, a greedy approach (GREED) is developed based on TRIAD for comparison.
GREED only contains three colors for each layer and greedily determines the colors of routing wire segments.
In GREED, the colors of routed wire segments are fixed.
GREED adopts the same routing flow of TRIAD without TECG.
Notably, TRIAD and GREED are prohibited to generate conflicts in the first fifteen iterations for fair comparison.
Table \ref{tb:result} shows the wirelength, the number of stitches (\# Stitch), the number of unsolvable conflicts (\# Conflict), and runtime of GREED and TRIAD.
\textcolor[rgb]{0.00,0.00,0.00}{
TRIAD produces no conflicts in all cases and only introduces one and two stitches in $s13207$ and $s38548$, respectively, while GREED only generates conflict-free results in one case with total 3890 stitches.
Moreover, the average wirelength of TRIAD is less than that of GREED by 0.54\% because GREED has to detour the routed colored wire segments to avoid generating TPL conflicts.
Thus, GREED requires more detours than TRIAD does.
Compared to GREED, TRIAD can generate conflict-free results in all cases at the cost of an average 2.41$\times$ of runtime.
For the largest case $s38584$, the runtime of TRIAD is less than four times of that of GREED.}
The most runtime spends of the graph reduction which provides TRIAD high coloring-flexibility to generate TPL-friendly results.
\section{Preliminaries and Problem Formulation}
\label{sec:preliminaries}
\subsection{Conflict Graph}
Kahng \textit{et al.} \cite{DPL_Kahng_TCAD10} first adopted a conflict graph (CG) to maintain the relationship among wire segments for the DPL layout decomposition.
A vertex in one CG represents one wire segment in a layout.
An edge between two vertices, $v_i$ and $v_j$, in one CG is generated when the minimum spacing between the wire segments represented by $v_i$ and $v_j$ is smaller than minimum coloring spacing, denoted as $sp_{dp}$.
One DPL coloring conflict occurs when there is an odd number of connected vertices in a cycle in one CG.
\subsection{Routing Model}
The detailed routing can be classified into grid-based one and gridless one based on the utilized routing models.
While utilizing the routing resources in a dense layout better than the conventional grid-based routers do, the gridless routers construct more complex data structures than grid-based routers owing to the ability to accommodate the routing rules in the routing graph.
Besides, to fit the demand of regular layout designs, gridless routers can also generate on-grid routing wires with an on-grid feature.
Two conventionally adopted gridless routers are tile-based one and implicit connection graph-based one, which possess the advantages of low path propagation complexity and fast routing graph construction, respectively \cite{Routing_TILER} \cite{Routing_DUNE}.
NEMO \cite{Routing_Li_TCAD07}\cite{Routing_Chang_ISPD08} is an implicit connection graph-based router with both the benefits of tile-based and implicit connection graph-based routers.
Before each routing, NEMO expands each obstacle and routed net by half of a wire width $hw_w$ and one wire spacing $sp_w$ to generate contours as shown in Fig. \ref{fig:nemo}(a).
NEMO constructs the implicit connection graph by extracting all borders of contours (the dotted lines in Fig. \ref{fig:nemo}(b)).
In the propagation stage, NEMO performs the path propagation by identifying the adjacent \textit{pseudo-maximum horizontally/vertically stripped tiles} (PMTs) of the last PMT in the minimum-cost path and then expanding the connected PMT list.
The path propagation is repeated until the PMT containing the target is reached.
Accordingly, NEMO generates routing wire segments by retracing the routing result and then places new wire segments on the layout.
In Fig. \ref{fig:nemo}(b), three PMTs are passed from $S$ to $T$, and NEMO traces them to construct the final routing result.
\begin{figure}[bt]
\centering
\subfloat[]{\includegraphics[width=0.23\textwidth]{fig/NEMO1.eps}}
\hspace{0.01\textwidth}
\subfloat[]{\includegraphics[width=0.23\textwidth]{fig/NEMO2.eps}}
\caption{Routing graph construction of NEMO \cite{Routing_Li_TCAD07}\cite{Routing_Chang_ISPD08}: (a) contour generation for each routed wire; (b) routing graph construction and PMT extraction.}
\label{fig:nemo}
\end{figure}
\subsection{Problem Formulation}
\begin{prob}[\textbf{TPL Aware Detailed Routing Problem}]
The minimum coloring spacing of TPL $sp_{tp}$ indicates that two wire segments need to be assigned to different masks when their spacing is smaller than $sp_{tp}$.
Given a netlist and $sp_{tp}$, the detailed routing for all nets is performed to minimize the number of stitches and TPL conflicts.
\end{prob}
\section{Supplemental Material}
\subsection{Details of Detailed Routing}
Lee algorithm \footnote{\small{C. Y. Lee, "An Algorithm for Path Connection and its Application," \textit{IRE Trans. on Electronic Computers}, EC-10, 1961.}} is the most well known routing method to find a connection from a source point to a target point.
Lee algorithm-based routers contain two main stages for each route: \textit{propagation} and \textit{retracing}.
In the propagation stage, the router iteratively explores adjacent routing units to approach the target point.
Each propagation step also records the routing cost.
After reaching the target point, the routing connection is generated by retracing the propagation order on the grid map.
\subsection{Innovative Conflict Graph}
Lin and Li \cite{DPL_Lin_DAC10} proposed an \textit{innovative conflict graph} (ICG) based on \cite{DPL_Kahng_TCAD10} to enable detailed routers to escape the suboptimum of coloring caused by a greedy assignment.
ICG contains \textit{determined} and \textit{undetermined vertices} to represent routed and routing wire segments, respectively.
An edge in ICG indicates the spacing of the wire segments represented by its terminal vertices is smaller than $sp_dp$.
Notably, there are \textit{determined} and \textit{undetermined edges} in ICG to represent the DPL relation between routed/routing and routed/routed wire segments.
Each determined vertex is assigned a pseudo color to represent the potential color.
An ICG comprises several innovative conflict subgraphs (ICSGs), and two ICSGs have no determined and undetermined edges between them.
An odd-length cycle in ICG indicates a coloring conflict.
Figure \ref{fig:icg} shows a routing result and its ICG where solid/dashed border rectangles represent routed/routing wires, solid/dashed border circles represent determined/undetermined vertices, and solid/dashed lines represent determined/undetermined edges.
Each determined vertex is assigned a pseudo color to represent the potentially final color.
This work adopts the concept of ICG to implement the insertion of vertices and edges in TECG caused by routing wire segments.
\begin{figure}[h]
\centering
\subfloat[]{\includegraphics[width=0.18\textwidth]{fig/ICG_layout.eps}}
\hspace{0.04\textwidth}
\subfloat[]{\includegraphics[width=0.18\textwidth]{fig/ICG.eps}}
\caption{Example of ICG: (a) layout contains three routed wires and one wire to be routed; (b) ICG of (a).}
\label{fig:icg}
\end{figure}
\section{Introduction}
As manufacturing process node enters the nano-meter era, the gap between the illumination wavelength of 193\textit{nm} and the target process node becomes increasingly larger.
The semiconductor industry encounters the limitation of manufacturing sub-22\textit{nm} due to the delay of the next generation lithograph (NGL) such as extreme ultraviolet (EUV) and E-beam direct write \cite{Du2012}.
To bridge the gap, double patterning lithography (DPL) is adopted, which decomposes a single layer into two masks (colors) to increase the pitch and enhance the resolution \cite{Kahng_SSDRC_07}.
Deploying DPL involves two challenges.
\textit{Layout decomposition} requires assigning two features to opposite colors (masks) if their spacing is less than a specific spacing, denoted as $sp_{dp}$.
One \textit{coloring conflict} occurs when two features whose spacing is less than $sp_{dp}$ cannot be assigned to different masks.
\textit{Stitch generation}, as the second challenge, is used to solve the coloring conflicts at the cost of yield loss due to the high sensitivity of stitches due to the overlay error.
However, some coloring conflicts cannot be solved even after the stitch generation, e.g., native conflicts.
Figure \ref{fig:DPL} shows that one un-decomposable layout (Fig. \ref{fig:DPL}(a)) becomes decomposable after generating one stitch (Fig. \ref{fig:DPL}(b)).
Figure \ref{fig:DPL}(c) depicts one layout containing native conflicts in which the spacing between arbitrary two features is less then $sp_{dp}$.
To further shrink the process nodes below 22\textit{nm}, the paradigm of DPL can be extended to the triple patterning lithography (TPL) to compensate the delay of NGL.
If single exposure half-pitch is about 40\textit{nm}, the 193\textit{nm} lithography could be used to manufacture the 11\textit{nm} process node \cite{TPL_Yan_SW}.
Compared to DPL, TPL contains one additional mask and can easily solve the native conflicts of DPL.
In the example of Fig. \ref{fig:DPL}(c), TPL assigns the three features to three masks, respectively.
Successfully carrying out the layout decomposition requires layouts containing no native conflicts.
Considering DPL in the layout synthesis, especially in the detailed routing stage, facilitates generating layouts without native conflicts.
Cho \textit{et al.} \cite{DPL_Cho_ICCAD08} developed the first DPL-friendly detailed routing approach which greedily determined the masks of routed wire segments to avoid generating layouts with native conflicts.
Gao and Macchiarulo \cite{DPL_Gao_DATE2010} proposed lazy color decision and last conflict segment recording to enhance the DPL-aware detailed routing based on \cite{DPL_Cho_ICCAD08}.
Lin and Li \cite{DPL_Lin_DAC10} developed a deferred coloring assignment-based gridless detailed routing flow to escape from the suboptimum that may be reached by adopting the greedy coloring strategy.
Yuan and Pan \cite{DPL_WISDOM} spread wires to simultaneously minimize the number of conflicts and stitches, while introducing as less the layout perturbation as possible.
On TPL, previous researches only focus on the layout decomposition.
Cork \textit{et al.} \cite{TPL_Cork} applied a SAT solver to decompose layouts into three colors.
Bei \textit{et al.} \cite{TPL_Bei_ICCAD11} proposed a novel vector programming formulation for TPL decomposition and applied a semidifinite programming (SDP) to solve the problem effectively.
Chen \textit{et al.} \cite{TPL_LD_SATP} and Mebarki \textit{et al.} \cite{TPL_LD_Mebarki} proposed a self-aligned triple patterning (SATP) process to extend the 193\textit{nm} immersion lithography to half-pitch 15\textit{nm} patterning.
\begin{figure}[b!]
\centering
\subfloat[]{\includegraphics[height=0.09\textheight]{fig/DPLLayout.eps}}
\hspace{0.04\textwidth}
\subfloat[]{\includegraphics[height=0.09\textheight]{fig/DPLDecomposedLayout.eps}}
\hspace{0.08\textwidth}
\subfloat[]{\includegraphics[height=0.09\textheight]{fig/DPL3.eps}}
\caption{Challenges of DPL: (a) layout cannot be directly decomposed into two masks; (b) layout becomes decomposable after splitting one feature by generating one stitch; (c) layout contains native conflict.}
\label{fig:DPL}
\end{figure}
Similar to DPL, generating TPL-friendly layouts, especially in the detailed routing stage, becomes urgent as TPL is being considered and adopted in the industry \cite{Lucas2012}.
Generating TPL-friendly layouts is more difficult than the TPL layout decomposition while the TPL decomposition has been shown as a NP-complete problem \cite{TPL_Bei_ICCAD11}.
The DPL coloring conflicts can be easily detected by finding an odd-length cycle in a conflict graph (CG) \cite{DPL_Kahng_TCAD10}\cite{DPL_Lin_DAC10}\cite{DPL_Gao_DATE2010}, which cannot be applied to detect TPL coloring conflicts.
The greedily coloring approach such as \cite{DPL_Cho_ICCAD08} can be directly applied to generate TPL-friendly layouts.
However, greedily determining the colors of routed wire segments significantly sacrifices the flexibility of coloring assignment, which may result in generating native conflicts and introducing unnecessary stitches.
Moreover, the complexity of CG impedes directly detecting TPL coloring conflicts with high flexibility of coloring assignment in one CG.
Figure \ref{fig:motivation}(a) depicts one layout with eight features.
One greedy coloring approach sequentially colors features ($A$, $B$, $C$, $D$, $E$, $F$, $G$, $H$) in Fig. \ref{fig:motivation}(a) with colors ($c_1$, $c_2$, $c_3$, $c_1$, $c_2$, $c_3$, $c_1$, $c_2$).
Figure \ref{fig:motivation}(b) displays the coloring result, in which $G$ and $H$ become un-colorable.
Nevertheless, the layout can be colored without any un-colorable feature as shown in Fig. \ref{fig:motivation}(c).
Therefore, a TPL conflict detection with \textit{high coloring-flexibility} and \textit{low detection-complexity} is desired in a correct-by-construction approach.
\begin{figure}[bt]
\centering
\subfloat[]{\includegraphics[width=0.14\textwidth]{fig/motivation_layout.eps}}
\hspace{0.02\textwidth}
\subfloat[]{\includegraphics[width=0.14\textwidth]{fig/motivation_greedy.eps}}
\hspace{0.02\textwidth}
\subfloat[]{\includegraphics[width=0.14\textwidth]{fig/motivation_opt.eps}}
\caption{Effects of coloring ordering to TPL coloring result: (a) layout contains eight features; (b) sequentially coloring ($A$, $B$, $C$, $D$, $E$, $F$, $G$, $H$) with colors ($c_1$, $c_2$, $c_3$, $c_1$, $c_2$, $c_3$, $c_1$, $c_2$) causes $G$ and $H$ un-colorable; (c) coloring result without conflicts exists.}
\label{fig:motivation}
\end{figure}
In this work, a \textit{token graph-embedded conflict graph} (TECG), comprising a token graph (TG) and a conflict graph (CG), is proposed to enable detailed routers to generate TPL-friendly layouts by a correct-by-construction approach.
One TG is used to maintain the coloring relation among different vertex sets in one CG.
In one TG, one \textit{strictly colored component} (SCC) is constructed to fix the coloring relation among certain vertex sets in one CG.
We apply the proposed TECG to a detailed routing model \cite{Routing_Li_TCAD07}\cite{Routing_Chang_ISPD08} to implement a \textit{triple patterning lithography aware detailed router} (TRIAD).
During the path searching, TRIAD adopts the TECG to detect if any TPL conflict occurs by the current routing wire segment.
After detecting solvable TPL conflicts, TRIAD utilizes the TECG to generate stitches in wire segments.
\textcolor[rgb]{0.00,0.00,0.00}{
With the assistance of TECG, TRIAD can generate stitches which cannot be generated by adopting the conventional DPL stitch generation scheme.
}
Notably, the TPL stitch generation scheme can split one wire segment into several segments even when the wire is entirely intersected by the TPL effect regions of other wire segments.
The main contribution of this paper is to realize a TPL aware detailed router TRIAD with the following two novel techniques:
\begin{itemize}
\item A TECG is proposed to assist detailed routers in detecting the TPL conflicts in a correct-by-construction approach while keeping high coloring-flexibility.
\item A TPL stitch generation scheme is proposed to generate stitches which may not be generated by adopting the conventional DPL stitch generation scheme.
\end{itemize}
The remainder of the paper is organized as follows: Section \ref{sec:preliminaries} presents the basic concepts and the problem formulation.
Sections \ref{sec:TECG} and \ref{sec:tiara} introduce the proposed TECG and TRIAD, respectively.
Next, Section \ref{sec:experiment} summarizes the experimental results.
Brief conclusions are drawn in Section \ref{sec:conclusion}.
\section{TECG}
\label{sec:TECG}
One conflict graph (CG) is used to maintain the \textit{physical} coloring relations among all wire segments.
The higher routed ratio, the higher complexity of CG.
In Fig. \ref{fig:motivation}(c), the decomposable layout is acquired only when $D$, $G$, and $H$ are assigned to the identical color, which indicates that maintaining the consistent coloring relations among disconnected vertices in one CG can assist detailed routers in generating TPL-friendly results.
However, maintaining certain coloring relations among non-adjacent vertices in one CG is quite difficult.
Therefore, one \textit{token graph} (TG) is proposed to maintain the \textit{logical} coloring relation among sets of wire segments.
Before introducing the proposed TG, the terminology of CG is defined as follows.
\begin{define}[\textbf{CG}]
A CG $\mathcal{G^C}$ = ($V^\mathcal{C}, E^\mathcal{C}$) contains a vertex set $V^\mathcal{C}$ representing all wire segments in one layer and an edge set $E^\mathcal{C}$ representing the minimum distance of two vertices, $v^c_i \in V^\mathcal{C}$, $v^c_j \in V^\mathcal{C}$, and $i \neq j$, is smaller than $sp_{tp}$.
\end{define}
\begin{define}[\textbf{Token}]
A token represents a potential color.
Each vertex $v^c_i \in V^\mathcal{C}$ is assigned a token $T$ to represent its potential color, denoted as $token(v^c_i) = T$.
Each token $T$ contains a CG vertex set, denoted as $V_t^\mathcal{C}(T) \subset V^\mathcal{C}$ where $\forall v^c_j \in V_t^\mathcal{C}(T), token(v^c_j) = T$, to indicate all vertices in $V_t^\mathcal{C}(T)$ is assigned to $T$.
\end{define}
\begin{define}[\textbf{TG}]
A TG $\mathcal{G^T}$ = ($V^\mathcal{T}, E^\mathcal{T}$) comprises a vertex set $V^\mathcal{T}$ representing all tokens in one layer and an edge set $E^\mathcal{T}$.
Each edge in $E^\mathcal{T}$ between two tokens, $T_i \in V^\mathcal{T}, T_j \in V^\mathcal{T}$ and $i \neq j$, represents that there exists at least one edge in $\mathcal{G^C}$ between $V_t^\mathcal{C}(T_i)$ and $V_t^\mathcal{C}(T_j)$.
\end{define}
\begin{define}[\textbf{Strictly Colored Component (SCC)}]
One SCC is defined as one three-tuple $(T_i, T_j, T_k)$ where $T_i \in V^\mathcal{T}, T_j \in V^\mathcal{T}$, and $T_k \in V^\mathcal{T}$ comprise one three-clique in $\mathcal{G^T}$.
TG may contain a set of SCCs.
\end{define}
\begin{define}[\textbf{TECG}]
A TECG $\mathcal{G^{TC}}$ comprises one CG $\mathcal{G^C}$ and one TG $\mathcal{G^T}$.
A CG/TG may comprise several connected components, and each component is a subgraph of CG/TG and named as CSG/TSG.
TPL conflict can be detected by finding a conflicting edge $e^c \in E^\mathcal{C}$ between $v^c_i \in V^\mathcal{C}$ and $v^c_j \in V^\mathcal{C}$ where $token(v^c_i) = token(v^c_j)$.
\end{define}
The proposed TG enables detailed routers to maintain high coloring-flexibility.
Instead of assigning physical colors into wire segments, tokens are used to represent the potential colors.
Therefore, one TG may contain more than three tokens even when TPL provides only three colors in each layer.
Figures \ref{fig:tecg} depicts the TECG of the layout in Fig. \ref{fig:motivation}(c).
In Fig. \ref{fig:tecg}(a), $A$ and $C$ are assigned to $T_1$; $B$ and $E$ are assigned to $T_2$; $D$, $G$, and $H$ are assigned to $T_3$; and $F$ is assigned to $T_4$.
The corresponding TG as shown in Fig. \ref{fig:tecg}(b) contains four tokens, $T_1$, $T_2$, $T_3$, and $T_4$, and one SCC $scc=(T_1, T_2, T_3)$.
Therefore, the coloring result in Fig. \ref{fig:motivation}(c) can be obtained by assigning $T_1$, $T_2$, and $T_3$ to $c_1$, $c_2$, and $c_3$, respectively, while $T_4$ can be assigned to $c_1$ or $c_2$.
Notably, the number of vertices and edges of TG is much less than that of CG.
\begin{figure}[bt!]
\centering
\subfloat[]{\includegraphics[width=0.2\textwidth]{fig/cg_tecg1.eps}}
\hspace{0.04\textwidth}
\subfloat[]{\includegraphics[width=0.13\textwidth]{fig/tg_tecg1.eps}}
\caption{TECG of layout in Fig. \ref{fig:motivation}(c): (a) CG; (b) TG.}
\label{fig:tecg}
\end{figure}
\subsection{Token Graph Reduction}
The coloring relation between non-adjacent tokens may become consistent after inserting an edge in one TG.
Merging these tokens can effectively compact TG to facilitate TPL conflict detection.
Two disconnected tokens $T_w \in V^\mathcal{T}$ and $T_x \in V^\mathcal{T}$ are merged when there exists one SCC $scc=(T_x, T_y, T_z)$ in $\mathcal{G^T}$ where $T_y$ and $T_z$ connect to $T_w$.
After merging $T_w$ and $T_x$, the adjacent tokens of $T_w$ and $T_x$ connect to the merged token, which conduces to further graph reduction.
Algorithm \ref{alg:tg_update} depicts the algorithm of TG\_Update with two connected tokens $T_i$ and $T_j$.
TG\_Update finds if there exists one $scc_i = (T_i, T_{i1}, T_{i2}) \in S^{SCC}$ where $T_j$ connects to $T_{i1}$ but not $T_{i2}$.
If $scc_i$ exists, $T_j$ and $T_{i2}$ are merged (lines 1--3).
Otherwise, TG\_Update tries to merge $T_i$ with one token in an existing SCC in a similar scenario (lines 5--7).
When the above two conditions cannot be met, TG\_Update finds if any SCC, such as $scc_{com} = (T_i, T_j, T_k) \in S^{SCC}$, contains $T_i$ and $T_j$.
If $scc_{com}$ exists and $T_i$ and $T_j$ have an other common adjacent token $T_{com}$, $T_k$ and $T_{com}$ are merged (lines 9--11).
If no tokens can be merged and $T_i$ and $T_j$ have one common adjacent token $T_{com}$, TG\_Update generates one SCC, and recursively calls itself until no more tokens/SCCs can be merged/generated (lines 12--16).
\begin{algorithm}[bt!]
\caption{TG\_Update}
\label{alg:tg_update}
\begin{algorithmic}[1]
\REQUIRE Two connected tokens $T_i \in V^\mathcal{T}$ and $T_j \in V^\mathcal{T}$, one SCC set $S^{SCC}$
\STATE Find $scc_i = (T_i, T_{i1}, T_{i2}) \in S^{SCC}$ such that $T_j$ connects to $T_{i1}$ but not $T_{i2}$;
\IF{$scc_i$ exists}
\STATE Token\_Merging($T_j$, $T_{i2}$);
\ELSE
\STATE Find $scc_j$ = $(T_j, T_{j1}, T_{j2}) \in S^{SCC}$ such that $T_i$ connects to $T_{j1}$ but not $T_{j2}$;
\IF{$scc_j$ exists}
\STATE Token\_Merging($T_i$, $T_{j2}$);
\ELSE
\STATE Find $scc_{com} = (T_i, T_j, T_k) \in S^{SCC}$;
\IF{$scc_{com}$ exists \AND $T_i$ and $T_j$ have one common adjacent token $T_{com} \neq T_k$}
\STATE Token\_Merging($T_k$, $T_{com}$);
\ELSIF {$T_i$ and $T_j$ have one common adjacent token $T_{com}$}
\STATE Generate $scc_{new} = (T_i, T_j, T_{com})$;
\STATE $S^{SCC} := S^{SCC} \cup {scc_{new}}$;
\STATE TG\_Update($T_i$, $T_j$);
\ENDIF
\ENDIF
\ENDIF
\end{algorithmic}
\end{algorithm}
Assume that $T_w$ and $T_x$ are merged into $T_{mrg}$.
Let $V_{ad}^\mathcal{T}(T_w)$ and $V_{ad}^\mathcal{T}(T_x)$ be the adjacent vertex sets of $T_w$ and $T_x$ in $\mathcal{G^T}$, respectively.
After merging $T_w$ and $T_x$, the adjacent vertex set of $T_{mrg}$ is $V_{ad}^\mathcal{T}(T_{mrg}) = V_{ad}^\mathcal{T}(T_w) \cup V_{ad}^\mathcal{T}(T_x)$.
Therefore, token merging reduces $|V^\mathcal{T}|$ and $|E^\mathcal{T}|$ by one and $|V_{ad}^\mathcal{T}(T_w)| + |V_{ad}^\mathcal{T}(T_x)|-|V_{ad}^\mathcal{T}(T_{mrg})|$, respectively.
Notably, some redundant TG edges are removed after token merging, and the inserted edges of the merged token can further benefit in simplifying TG.
Algorithm \ref{alg:token_merging} displays the algorithm of merging two tokens $T_w$ and $T_x$.
Merging $T_w$ and $T_x$ into $T_{mrg}$ requires $scc \in S^{SCC}$ being updated by replacing $T_w$ or $T_x$ by $T_{mrg}$ (lines 1--5).
The replacement may cause two SCCs to contain the same tokens, resulting in redundant SCCs.
After updating SCCs, the redundant SCCs are removed, if any (line 7).
The additional edges of the merged token can further assist in graph reduction. (lines 8--10).
\begin{algorithm}[bt!]
\caption{Token\_Merging}
\label{alg:token_merging}
\begin{algorithmic}[1]
\REQUIRE $T_w \in V^\mathcal{T}$, $T_x \in V^\mathcal{T}$, one SCC set $S^{SCC}$
\STATE Merge $T_w$ and $T_x$ into $T_{mrg}$;
\FORALL {$scc \in S^{SCC}$}
\IF{$scc$ contains $T_w$ or $T_x$}
\STATE Update $scc$ by replacing $T_w$ or $T_x$ by $T_{mrg}$;
\ENDIF
\ENDFOR
\STATE Remove redundant SCC from $S^{SCC}$;
\FORALL {token $t \in \{V_{ad}^\mathcal{T}(T_{mrg})-\{V_{ad}^\mathcal{T}(T_w) \cap V_{ad}^\mathcal{T}(T_{x})\}\}$}
\STATE TG\_Update($T_{mrg}$, $t$);
\ENDFOR
\end{algorithmic}
\end{algorithm}
\subsection{TECG Update}
In the routing stage, the vertices representing routing wire segments are inserted into one CG, and new tokens are generated in one TG to represent the potential colors of routing wire segments.
When $v^c_i \in V^\mathcal{C}$ and $v^c_j \in V^\mathcal{C}$ are connected by an edge, an edge in one TG between $token(v^c_i)$ and $token(v^c_j)$ needs to be generated, if necessary.
Algorithm \ref{alg:tecg_update} shows TECG\_Update by connecting $v^c_i \in V^\mathcal{C}$ and $v^c_j \in V^\mathcal{C}$.
Firstly, $v^c_i$ and $v^c_j$ are connected (line 1).
One TPL conflict is detected when $token(v^c_i)$ and $token(v^c_j)$ are identical (lines 2--3).
If $token(v^c_i)$ and $token(v^c_j)$ in the TG are disconnected, $token(v^c_i)$ and $token(v^c_j)$ are connected, and then \textit{TG\_Update} is used to compact the TG, if possible (lines 4--6).
\begin{algorithm}[bt!]
\caption{TECG\_Update}
\label{alg:tecg_update}
\begin{algorithmic}[1]
\REQUIRE $\mathcal{G^{TC}}$, $v^c_i \in V^\mathcal{C}$ and $v^c_j \in V^\mathcal{C}$ to be connected
\STATE Connect $v^c_i$ and $v^c_j$ in $\mathcal{G^C}$;
\IF{$token(v^c_i) = token(v^c_j)$}
\STATE Detect one TPL conflict;
\ELSIF{$token(v^c_i)$ and $token(v^c_j)$ are disconnected in $\mathcal{G^T}$}
\STATE Connect $token(v^c_i)$ and $token(v^c_j)$;
\STATE TG\_Update($token(v^c_i)$, $token(v^c_j)$);
\ENDIF
\end{algorithmic}
\end{algorithm}
Figure \ref{fig:tecg_update}(a) depicts one TECG with one CG containing seven vertices, one TG containing seven vertices and ten edges, and two SCC $scc_1 = (T_1, T_2, T_7)$ and $scc_2=(T_3, T_4, T_5)$.
Connecting $C$ and $G$ (the dashed line) in the CG generates the connection between $T_3$ and $T_7$ (the dashed line) in the TG.
Therefore, TG\_Update$(T_3, T_7)$ merges $T_1$ and $T_3$ into $T_{1'}$, and the two SCCs are updated by replacing $T_1$ and $T_3$ with $T_{1'}$ as shown in Fig. \ref{fig:tecg_update}(b).
Next, because $T_5$ is not a common adjacent token of $T_1$ and $T_3$ in Fig. \ref{fig:tecg_update}(a), TG\_Update$(T_{1'}, T_5)$ merges $T_2$ and $T_5$ into $T_{2'}$ followed by updating SCCs by replacing $T_2$ and $T_5$ with $T_{2'}$ as shown in Fig. \ref{fig:tecg_update}(c).
Similarly, $T_4$ is not a common adjacent token of $T_2$ and $T_5$ in Fig. \ref{fig:tecg_update}(c) so TG\_Update$(T_4, T_{2'})$ updates TG as shown in Fig. \ref{fig:tecg_update}(d).
Notably, after replacing $T_4$ and $T_7$ with $T_{3'}$, the three tokens of two SCCs become identical, requiring removing one redundant SCC.
Therefore, one SCC is removed as shown in Fig. \ref{fig:tecg_update}(d).
Finally, TG\_Update$(T_{2'}, T_6)$ is called because $T_6$ is not a common adjacent token of $T_2$ and $T_5$ in Fig. \ref{fig:tecg_update}(b).
Figures \ref{fig:tecg_update}(e) depicts the updated TECG with one SCC $(T_{1'}, T_{2'}, T_{4'})$ where the assigned token of each vertex in CG is also updated.
Notably, before connecting $C$ and $G$ in the CG, the number of TG vertices, TG edges, and SCCs are seven, ten, and two, respectively.
After connecting $C$ and $G$ in CG, the number of TG vertices, TG edges, and SCCs are reduced by four, seven, and one, respectively, which indicates the proposed graph reduction technique effectively reduces the complexity of the TG.
Therefore, the graph reduction technique of TECG can significantly reduce the complexity of the TPL conflict detection.
\begin{figure}[bt!]
\centering
\subfloat[]{\includegraphics[width=0.285\textwidth]{fig/tecg_update_bf.eps}}
\subfloat[]{\includegraphics[width=0.13\textwidth]{fig/tg_update_1.eps}}
\hspace{0.16\textwidth}
\subfloat[]{\includegraphics[width=0.13\textwidth]{fig/tg_update_2.eps}}
\subfloat[]{\includegraphics[width=0.13\textwidth]{fig/tg_update_3.eps}}
\hspace{0.01\textwidth}
\subfloat[]{\includegraphics[width=0.285\textwidth]{fig/tecg_update_af.eps}}
\caption{\textcolor[rgb]{0.00,0.00,0.00}{
Example of TECG update:
(a) TECG before connecting $C$ and $G$ in CG with two SCCs $(T_1, T_2, T_7)$ and $(T_3, T_4, T_5)$;
(b) updated TG after merging $T_1$ and $T_3$ in (a) into $T_{1'}$;
(c) updated TG after merging $T_2$ and $T_5$ in (b) into $T_{2'}$;
(d) updated TG after merging $T_4$ and $T_7$ in (c) into $T_{3'}$;
(e) updated TECG after merging $T_{3'}$ and $T_6$ in (d) into $T_{4'}$.
}
}
\label{fig:tecg_update}
\end{figure}
\subsection{Implicit Edge in TG}
Two tokens cannot be assigned to the same color when they connect to each other in one TG.
We observe that two non-adjacent vertices in one TG cannot be assigned to one color when certain topology appears in TG.
Notably, there might be other patterns that are not observed.
An \textit{implicit TG edge} between two non-adjacent tokens, such as $T_i$ and $T_j$, is generated when all the following conditions are satisfied:
\begin{enumerate}
\item TG contains two SCCs $(T_x, T_y, T_z)$ and $(T_p, T_q, T_r)$, where
\item $T_x$ and $T_p$ connect to each other;
\item $T_i$ connects to both $T_y$ and $T_q$; and
\item $T_j$ connects to both $T_z$ and $T_r$.
\end{enumerate}
Without loss of generality, there are three colors $(c_1, c_2, c_3)$ can be used to color all tokens in one TG.
Figure \ref{fig:itge}(a) depicts one TG contains the specific topology with two SCCs $scc_1 = (T_3, T_4, T_5)$ and $scc_2 = (T_6, T_7, T_8)$.
Suppose that $T_1$ and $T_2$ are assigned to $c_1$.
Notably, $T_4$/$T_5$ and $T_7$/$T_8$ can only be assigned to $c_2$ and $c_3$ due to the connection to $T_1$/$T_2$, resulting in that $T_3$/$T_6$ must be assigned to $c_1$.
However, there exists one edge between $T_3$ and $T_6$.
Therefore, $T_1$ and $T_2$ must be assigned to different colors, and one implicit edge is generated between $T_1$ and $T_2$ as shown in Fig. \ref{fig:itge}(b).
In TECG\_Update, after TG\_Update (Algorithm \ref{alg:tecg_update}: line 6),a set of implicit TG edges $IE$ is generated by checking if the above conditions are satisfied.
For each implicit edge $ie \in IE$, TG\_Update checks if the TG can be further reduced by inserting $ie$.
\begin{figure}[bt!]
\centering
\subfloat[]{\includegraphics[width=0.18\textwidth]{fig/itge1.eps}}
\hspace{0.04\textwidth}
\subfloat[]{\includegraphics[width=0.18\textwidth]{fig/itge2.eps}}
\caption{Example of implicit TG edge: (a) $T_1$ and $T_2$ are disconnected in TG; (b) implicit TG edge between $T_1$ and $T_2$ is inserted to indicate that colors of $T_1$ and $T_2$ must differ.}
\label{fig:itge}
\end{figure}
\subsection{Rip-up Operation of TECG}
Rip-up and re-route is a necessary means to accomplish detailed routing when some nets cannot be routed successfully.
Due to the page limitation, we skip the rip-up operation of TECG in the main article.
Detailed routers rip up routed wire segments to release routing resource for unroutable nets.
After ripping up one wire segment, the corresponding vertex $v^c$ in CG needs to be removed from CG, which requires disconnecting $v^c$ and $v^c_i \in V^\mathcal{C}_{ad}(v^c)$.
Without loss of generality, two disconnected tokens $T_w \in V^\mathcal{T}$ and $T_x \in V^\mathcal{T}$ are merged into $T_{mrg}$ when the following two conditions are satisfied.
\begin{enumerate}
\item There exists one SCC $scc = (T_x, T_y, T_z)$ in $\mathcal{G^T}$, and
\item $T_w$ connects to $T_y$ and $T_z$.
\end{enumerate}
Then $scc$ is updated as $scc_{mrg} = (T_{mrg}, T_y, T_z)$.
Disconnecting two vertices in CG may break one of the two merging conditions, resulting in \textit{token splitting}.
Figure \ref{fig:s_tecg_ripup}(a) depicts one TECG containing one SCC $scc=(T_2, T_3, T_4)$.
Connecting $A$ and $B$ makes $T_1$ and $T_4$ be merged into $T_5$ as shown in Fig. \ref{fig:s_tecg_ripup}(b).
After disconnecting $B$ and $C$, $token(B) = T_2$ and $token(C) = T_3$ is also disconnected in TG, resulting in the destruction of $scc$.
Notably, $token(A)$ and $token(D)$ are identical due to the appearance of $scc$.
When $scc$ is destructed, $token(A)$ and $token(D)$ differ by splitting $T_5$ into $T_6$ and $T_7$ as shown in Fig. \ref{fig:s_tecg_ripup}(c).
To facilitate token splitting, \textit{merging edges} in CG are generated when two tokens are merged.
The merging edge generation and removal are introduced followed by token splitting.
\begin{figure}[tb!]
\centering
\subfloat[]{\includegraphics[width=0.15\textwidth]{fig/s_tecg_bf_ripup0.eps}}
\subfloat[]{\includegraphics[width=0.15\textwidth]{fig/s_tecg_bf_ripup.eps}}
\subfloat[]{\includegraphics[width=0.15\textwidth]{fig/s_tecg_af_ripup.eps}}
\caption{TECG after rip-up: (a) TECG before connecting $A$ and $B$ in CG; (b) updated TECG; (c) TECG after disconnecting $B$ and $C$ in CG.}
\label{fig:s_tecg_ripup}
\end{figure}
\subsubsection{Merging Edge Generation}
Merging two tokens $T_w \in V^\mathcal{T}$ and $T_x \in V^\mathcal{T}$ into $T_{mrg}$ reassigns $token(v^c)$ as $T_{mrg}$ where $v^c \in V^\mathcal{C}_t(T_w) \cup V^\mathcal{C}_t(T_x)$.
Before merging $T_w$ and $T_x$, \textit{merging edges} are generated between $V^\mathcal{C}_t(T_w)$ and $V^\mathcal{C}_t(T_x)$.
Before introducing merging edge generation, \textit{merging pattern} is firstly defined as follows.
\begin{define}[\textbf{Merging Pattern}]
A merging pattern in CG is defined as a four-tuple $(v^c_{cnt1}$, $v^c_{cnt2}$, $v^c_{brdg1}$, $v^c_{brdg2})$ where $token(v^c_{cnt1})$ = $T_w$ and $token(v^c_{cnt2})$ = $T_x$ are being merged, $token(v^c_{brdge1}) = T_y$, $token(v^c_{brdge2})$ = $T_z$, and there exists one SCC $scc = (token(v^c_{cnt2})$, $token(v^c_{brdg1})$, $token(v^c_{brdg2}))$ in TG.
\end{define}
Algorithm \ref{alg:s_MP_generation} shows the algorithm of merging pattern generation before two tokens $T_w$ and $T_x$ are merged.
Firstly, a merging pattern set $S^{MP}$ is set as empty.
Each vertex $v^c_w \in V^\mathcal{C}_t(T_w)$ tries to generate merging patterns with vertex $v^c_x \in V^\mathcal{C}_t(T_x)$.
Two adjacent vertices of $v^c_w$, such as $v^c_{ad1} \in V^\mathcal{C}_{ad}(v^c_w)$ and $v^c_{ad2} \in V^\mathcal{C}_{ad}(v^c_w)$ where $token(v^c_{ad1}) = T_y$ and $token(v^c_{ad2}) = T_z$, are identified.
Then the adjacent vertex of $v^c_{ad1}$, such as $v^c_{adx} \in V^\mathcal{C}_{ad}(v^c_{ad1})$ where $token(v^c_{adx}) = T_x$, is found to generate one merging pattern with $v^c_w$, $v^c_{ad1}$, and $v^c_{ad2}$ (lines 2--9).
\begin{algorithm}[bt!]
\caption{Merging Pattern Generation}
\label{alg:s_MP_generation}
\begin{algorithmic}[1]
\REQUIRE Two tokens $T_w \in V^\mathcal{T}$ and $T_x \in V^\mathcal{T}$ to be merged, one SCC $scc$ = $(T_x, T_y, T_z)$
\STATE Merging pattern set $S^{MP} := \emptyset$;
\FORALL {$v^c_w \in V^\mathcal{C}_t(T_w)$}
\FORALL {$v^c_{ad1} \in V^\mathcal{C}_{ad}(v^c_w)$, $v^c_{ad2} \in V^\mathcal{C}_{ad}(v^c_w)$ where $token(v^c_{ad1})$ = $T_y$, $token(v^c_{ad2})$ = $T_z$}
\FORALL {$v^c_{adx} \in V^\mathcal{C}_{ad}(v^c_{ad1})$ where $token(v^c_{adx})$ = $T_x$}
\STATE Generate one merging pattern $mp$ = $(v^c_w$, $v^c_{adx}$, $v^c_{ad1}$, $v^c_{ad2})$;
\STATE $S^{MP} := S^{MP} \cup \{ mp \}$;
\ENDFOR
\ENDFOR
\ENDFOR
\RETURN $S^{MP}$;
\end{algorithmic}
\end{algorithm}
One merging pattern $mp$ = $(v^c_{cnt1}$, $v^c_{cnt2}$, $v^c_{brdg1}$, $v^c_{brdg2})$ generates one merging edge, denoted as $e^c_{mrg}(T_mrg)$ between $v^c_{cnt1}$ and $v^c_{cnt2}$ where $T_{mrg}$ is the merging token.
Notably, $v^c_{brdg1}$ and $v^c_{brdg2}$ are called the bridge vertices of $e^c_{mrg}(T_{mrg})$.
Each vertex $v^c \in \mathcal{G^C}$ contains a merging edge set, denoted as $E_{BRDG}(v^c)$, to represent the set of merging edges where $v^c$ is one bridge vertex of $e^c_{mrg} \in E_{BRDG}(v^c)$.
Figure \ref{fig:s_ME}(a) depicts the TECG with merging edges of that in Fig. \ref{fig:s_tecg_ripup}(b) before merging $T_1$ and $T_4$.
After one merging pattern $mp = (A, D, B, C)$ is generated, one merging edge $e_{mrg}$ is generated between $A$ and $D$.
Notably, $B$ and $C$ are the bridge vertices of $e_{mrg}$, resulting in $E_{BRDG}(B) = E_{BRDG}(B) \cup \{e_{mrg}\}$ and $E_{BRDG}(C) = E_{BRDG}(C) \cup \{ e_{mrg}\}$.
Then $T_1$ and $T_4$ are merged into $T_5$ as shown in Fig. \ref{fig:s_ME}(b).
\begin{figure}[bt!]
\centering
\subfloat[]{\includegraphics[width=0.23\textwidth]{fig/s_ME_bf.eps}}
\subfloat[]{\includegraphics[width=0.23\textwidth]{fig/s_ME_af.eps}}
\caption{Merging pattern and merging edge generation: (a) TECG before merging $tokne(A)$ and $token(D)$; (b) TECG after merging $tokne(A)$ and $token(D)$ into $T_5$.}
\label{fig:s_ME}
\end{figure}
\subsubsection{Merging Edge Removal}
Removing one vertex $v^c \in V^\mathcal{C}$ requires disconnecting $v^c$ and $v^c_{ad} \in V^\mathcal{C}_{ad}(v^c)$.
One merging pattern $mp$ = $(v^c_{cnt1}$, $v^c_{cnt2}$, $v^c_{brdg1}$, $v^c_{brdg2})$ is removed when the connection between any vertex pair in $mp$ is removed, if any.
After one merging pattern is removed, the corresponding merging edge is also removed from CG.
Figure \ref{fig:s_MEV}(a)/(b) depicts that the merging edge between $A$ and $D$ in Fig. \ref{fig:s_ME}(b) is removed after disconnecting $A$/$B$ and $C$.
\begin{figure}[bt!]
\centering
\subfloat[]{\includegraphics[width=0.16\textwidth]{fig/s_MERV1.eps}}
\hspace{0.04\textwidth}
\subfloat[]{\includegraphics[width=0.16\textwidth]{fig/s_MERV2.eps}}
\caption{Merging edge removal: (a) disconnecting $A$ and $C$ removes the merging edge between $A$ and $D$ in Fig. \ref{fig:s_ME}(b); (b) disconnecting $B$ and $C$ removes the merging edge between $A$ and $D$ in Fig. \ref{fig:s_ME}(b).}
\label{fig:s_MEV}
\end{figure}
\subsubsection{Token Splitting}
\begin{algorithm}[bt!]
\caption{Token Splitting}
\label{alg:s_TSPLIT}
\begin{algorithmic}[1]
\REQUIRE One TECG $\mathcal{G^{TC}}$ with TG $\mathcal{G^T}$ and CG $\mathcal{G^C}$, token $T_{victim}$ to be split
\FORALL{$T_{ad} \in V^\mathcal{T}_{ad}(T_{victim})$}
\STATE $V^\mathcal{T}_{ad}(T_{ad}) := V^\mathcal{T}_{ad}(T_{ad}) - \{T_{victim}\}$;
\ENDFOR
\STATE Apply BFS to find the connected component set $S^{CONNECT} \in V^\mathcal{C}_t(T_{victim})$ connected by merging edges of $T_{vicrim}$;
\FORALL{Connected component $cc \in S^{CONNECT}$}
\STATE Generate and insert one token $T_{extra}$ in $\mathcal{G^T}$;
\STATE $V^\mathcal{C}_t(T_{extra}) := \emptyset$;
\STATE $V^\mathcal{T}_{ad}(T_{extra}) := \emptyset$;
\FORALL{ CG vertex $v^c \in cc$}
\STATE $token(v^c) := T_{extra}$;
\STATE $V^\mathcal{C}_t(T_{extra}) := V^\mathcal{C}_t(T_{extra}) \cup \{v^c\}$;
\FORALL{Adjacent CG vertex $v^c_{ad}$ of $v^c$}
\STATE $V^\mathcal{T}_{ad}(T_{extra}) := V^\mathcal{T}_{ad}(T_{extra}) \cup \{token(v^c_{ad})\}$;
\STATE $V^\mathcal{T}_{ad}(token(v^c_{ad})) := V^\mathcal{T}_{ad}(token(v^c_{ad})) \cup \{T_{extra}\}$;
\ENDFOR
\ENDFOR
\ENDFOR
\STATE Remove $T_{victim}$ from $\mathcal{G^T}$;
\end{algorithmic}
\end{algorithm}
\begin{figure}[bt!]
\centering
\subfloat[]{\includegraphics[width=0.37\textwidth]{fig/s_TSPLIT1.eps}}
\subfloat[]{\includegraphics[width=0.37\textwidth]{fig/s_TSPLIT2.eps}}
\subfloat[]{\includegraphics[width=0.38\textwidth]{fig/s_TSPLIT3.eps}}
\caption{Token splitting: (a) one TECG where $A$, $D$, and $G$ are connected by merging edges due to $T_1$; (b) disconnecting $I$ and $H$ removes the merging edge between $A$ and $G$; (c) disconnecting $A$ and $C$ removes the merging edge between $A$ and $D$, resulting in $token(A)$ must differ from $token(G)$ and $token(D)$.}
\label{fig:s_TSPLIT}
\end{figure}
For each token $T_i \in V^\mathcal{T}$, any two vertices, such as $v^c_i \in V^\mathcal{C}_t(T_i)$ and $v^c_j \in V^\mathcal{C}_t(T_i)$, connect to each other by merging edges of $T_i$.
When $v^c_i$ and $v^c_j$ cannot connect to each other by merging edges of $T_i$ after removing edges in CG, $T_i$ needs to be split into at least two tokens.
The connectivity of merging edges can be checked by applying depth-first search (BFS).
Algorithm \ref{alg:s_TSPLIT} shows the algorithm of token splitting.
Firstly, the adjacent token $T_{ad} \in V^\mathcal{T}_{ad}(T_{victim})$ and $T_{victim}$ is disconnected (lines 1--3).
BFS is then applied to find the connected component set $S^{CONNECT}$ where all vertices in each component $cc \in S^{CONNECT}$ are connected by merging edges of $T_i$ (line 4).
For each connected component $cc \in S^{CONNECT}$, one extra token $T_{extra}$ is generated and inserted in TG.
For each CG vertex $v^c \in cc$, $token(v^c)$ is assigned to $T_{extra}$ (lines 5--11).
The adjacent tokens of $T_{extra}$ is updated based on the tokens of all adjacent vertices of $v^c$ (lines 12--15).
Finally, $T_{victim}$ is removed from TG (line 18).
Figure \ref{fig:s_TSPLIT}(a) depicts one TECG containing one CG with nine vertices, fifteen edges, and three merging edges of $T_1$ and one TG with seven tokens and nine edges.
$A$, $D$, and $G$ connect to each other by the three merging edges of $T_1$ because they are assigned to $T_1$.
After disconnecting $H$ and $I$, the merging edge of $T_1$ between $A$ and $G$ is removed, and $T_6$ and $T_7$ are also disconnected as shown in Fig. \ref{fig:s_TSPLIT}(b).
$A$, $D$, and $G$ still connect to each other by the remaining two merging edges of $T_1$.
Then, disconnecting $A$ and $C$ removes the merging edge of $T_1$ between $A$ and $D$, sequentially generating two connected component in terms of the connectivity of merging edges of $T_1$.
Therefore, $T_1$ is split into $T_8$ and $T_9$, resulting in that $A$ is reassigned to $T_8$ while $D$ and $G$ are reassigned to $T_9$.
The adjacent tokens of $T_8$ and $T_9$ are updated according to the connection in CG.
Figure \ref{fig:s_TSPLIT}(c) displays the TECG after splitting $T_1$ into $T_8$ and $T_9$.
\section{Conclusion}
\label{sec:conclusion}
The detailed routing is a key optimization stage for TPL.
To effectively detect TPL conflicts with high coloring-flexibility, this work proposes a token graph-embedded conflict graph (TECG) with a graph reduction technique.
\textcolor[rgb]{0.00,0.00,-.00}{
This work develops a TPL aware detailed router (TRIAD) with the TPL stitch generation to solve TPL conflicts..
With the aid of TECG, TRIAD can generate stitches in one wire even when the wire is entirely intersected by the TPL effect regions of other wires.
Experimental results show that the routing results have no TPL conflicts and introduces total three stitches for two cases with 0.54\% decrement in wirelength at the cost of 2.41$\times$ of runtime.}
The future work focuses on the density-driven TPL aware detailed routing.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 4,902
|
The Nation of Ulysses was an American punk rock band from Washington, D.C., formed in spring 1988 with four members. Originally known as simply "Ulysses," the first mark of the group consisted of Ian Svenonius on vocals and trumpet, Steve Kroner on guitar, Steve Gamboa on bass guitar, and James Canty on drums. Tim Green joined the band late in 1989 as a guitarist and the band became "Nation of Ulysses." Nation of Ulysses disbanded in the autumn of 1992, having failed to complete their third album. After the breakup, Svenonius, Canty, and Gamboa went on to form the short-lived Cupid Car Club and The Make-Up. Tim Green went on to help create The Fucking Champs, a mostly-instrumental trio out of San Francisco, and later Concentrick, a solo project with a focus on ambient music.
Nation of Ulysses was known for their far-left politics, their extremely physical live performances, and their unique take on punk culture and fashion. In total, Nation of Ulysses released three full-length albums, and two vinyl EPs released on Dischord Records, and were featured on a number of compilation albums on a variety of record labels.
History
The band formed in spring 1988, initially composed of four members – Ian Svenonius on vocals and trumpet, Steve Kroner on guitar, Steve Gamboa on bass guitar, and James Canty on drums – and known simply as "Ulysses." In late 1989, Tim Green joined the band as a second guitarist and the band was renamed to "Nation of Ulysses."
In 1990, before the band released any official albums, Ian Svenonius was featured as teen-oriented Sassy Magazine'''s first "Sassiest Boy in America." He was interviewed at length in the magazine's October issue, going into some depth about the band's sound and political motivations.
In 1991 they released their first full-length album, 13-Point Program to Destroy America on Dischord Records. This was followed a year later by Plays Pretty for Baby, also on Dischord. During the recording of the band's follow-up to Plays Pretty for Baby, Steve Kroner left the band. The remaining quartet continued to record, but eventually dissolved in 1992. In a later interview, Svenonius explained the reason for the split: "Nation of Ulysses broke up because the epoch changed with the advent of digital music and the Nirvana explosion. We were faced with what's now known as indie rock, a sort of vacuous form. We had to determine our next move and this [the forming of the Make-Up] is it."
After the band's dissolution, Svenonius went on to form the short-lived Cupid Car Club with James Canty and Steve Gamboa, the Make-Up (again with Canty and Gamboa), and Weird War. Green went on to become an engineer and record producer and joined the prog-heavy metal band The Fucking Champs.
Though relatively short-lived, Nation of Ulysses' influence has been substantial: they have been cited as influences for bands such as Glassjaw, The (International) Noise Conspiracy, The Hives, Thursday, Refused, Boysetsfire, Bikini Kill, Rocket From The Crypt, Antioch Arrow, The Locust, Death From Above 1979, At The Drive-in, LCD Soundsystem, Bis and Huggy Bear, among many others.
Recordings
During Nation of Ulysses' four years of activity, they released only two full-length albums: 13-Point Program to Destroy America in 1991 and Plays Pretty for Baby in 1992, both released on Dischord Records. After releasing Plays Pretty for Baby, the band began recording a third full-length album, but Steve Kroner separated from the band before recording was completed. The remaining quartet continued to record, but the group eventually dissolved before the record's completion. In 2000, six songs from those sessions, in addition to four new tracks recorded live, were compiled and released posthumously as The Embassy Tapes.
In addition to their three full-length albums, Nation of Ulysses released two vinyl EPs. The first, a self-titled EP, was the band's first official release, but went out of print when the three tracks from it were included in 13-Point Program to Destroy America. The second EP, released in 1992 under the title The Birth of the Ulysses Aesthetic (the synthesis of speed and transformation), also went out of print when its tracks were later released on Plays Pretty for Baby. The band were also featured in a number of compilation albums on a variety of record labels.
In 1992, NOU was on tour with Bikini Kill, and when they arrived in Memphis, the found that the promoter for the show hadn't actually promoted the show. So, while out on the town flyering the streets and ducking into record stores, NOU ran into Primal Scream, a band which had been in town recording their own album. Primal Scream then decided to attend NOU's show at the Antenna Club, and then proceeded to invite NOU to record with them in the studio the next day. Nation of Ulysses then recorded five songs with Primal Scream in nine hours, though of course only after a quick errand for some tin foil to smoke a bit of crack, according to Tim Green. These recordings are known as Memphis Demos, and include the demos of N-Sub Ulysses, 50,000 Watts, Mockingbird, Yeah, and Shakedown.
Musical ideology and style
Nation of Ulysses' music was noisy and manic, but they also had a strong free-jazz influence. The group embodied a rejection of the 1960s and 1970s music and styling by rejecting drug use and advocating that punk youth dress nicely and sensibly. To this end, the liner notes of 13-Point Program to Destroy America states the band's aim "To dress well, as clothing and fashion, are the only things which we -- the kids -- being utterly disenfranchised, have any control over." Much of the band admitted to not knowing how to play their instruments well, stating "All you need is a concept. There's no reason you have to sound like Led Zeppelin."
Political concepts
Nation of Ulysses described themselves not as a rock 'n' roll group in the traditional sense, but "as a political party" and as "a shout of secession." Explaining their intent, Svenonius said "it's basically a new nation underground for the dispossessed youth colony. It's all about smashing the old edifice, the monolith of rock and roll."
Allmusic's Steve Huey described Nation of Ulysses' philosophy as "a relentlessly provocative (and entertaining) jumble of teenage rock 'n' roll rebellion, leftist radicalism, anarchist punk polemics, and abstract intellectual rambling, [...] [which gives the sense of] an off-kilter, almost tongue-in-cheek approach to a 'perpetual 18-year old's' view of America, and life in general." This tongue-in-cheek political attitude was echoed by a spoken-word introduction to the song "The Sound of Jazz to Come", from their 1992 Plays Pretty for Baby, in which the band describes themselves as "the seriously unserious, reverently irreverent, amoral moralists."
Asked about their use of the medium they claimed to counteract – rock 'n' roll – Svenonius declared
"Well, it's a camouflage, to allow for movement, revolutionary liberation from the constraints of everyday composure, basically allowing anybody to move in anyway that they want and to lift spirit to a plateau to destroy 'parent culture.' "
Nation of Ulysses claimed to make weapons, not records. Discussing their second release, Svenonius asserted: "it's like a blueprint for the destruction of the Parent Culture. It's like a zip gun ... It's an instruction pamphlet for kids on how to destroy their home life, you know, their domestic state."
Although their first album was named 13-Point Program to Destroy America, Nation of Ulysses didn't align themselves with a particular political philosophy: "We don't usually address normal political dictums. We aim toward the everyday fixtures of life, like aesthetics, sound, non-spoken things that are inherently political in nature instead of, like bogus politicians who focus on glossy surface issues which avoid any kind of revolutionary change."
Ulysses Speaks
Nation of Ulysses published a zine called "Ulysses Speaks," which was an extension of their ideology expressed in their music and liner notes. The zines espoused what they referred to as "The Ulysses Aesthetic," which was a mix of 1960s and 1970s radical politics, French Situationist writings, and juvenile delinquency. The zine was distributed at live shows as well as made available by writing the band. A total of 9 issues were published.
Live performances
The band was known for their extremely physical performances, during some of which Svenonius recalls breaking his arm, his leg, and breaking his head open on numerous occasions. Audience members were also hurt during performances. Svenonius described Nation of Ulysses performances as "an extraordinary freak-out kind of thing [...] really masochistic, lots of blood [...] cacophonous, and violent, and aggressive."
Discography
Studio albums13-Point Program to Destroy America (Dischord) (1991)Plays Pretty for Baby (Dischord) (1992)The Embassy Tapes (Dischord) (2000)
Studio EPsNation of Ulysses (Dischord) (1991)The Birth of the Ulysses Aesthetic (the synthesis of speed and transformation)'' (Dischord) (1992)
References
External links
Official site on Southern Records
Official site Dischord page
Archive of Ulysses zine
Musical groups from Washington, D.C.
American post-hardcore musical groups
Dischord Records artists
Musical groups established in 1988
Musical groups disestablished in 1992
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 9,226
|
{"url":"https:\/\/www.gradesaver.com\/textbooks\/math\/other-math\/thinking-mathematically-6th-edition\/chapter-1-problem-solving-and-critical-thinking-1-2-estimation-graphs-and-mathematical-models-exercise-set-1-2-page-25\/1","text":"## Thinking Mathematically (6th Edition)\n\na. $19,465,200$ b. $19,465,000$ c. $19,470,000$ d. $19,500,000$ e. $19,000,000$ f. $20,000,000$\nThe provided population of the state of New York in the exercise is $19,465,197$. To do this exercise, we need to remember how we round numbers: 1. Look at the digit to the right of the digit where rounding is to occur. 2a. If the digit to the right is $5$ or greater, add $1$ to the digit to be rounded. 2b. If the digit to the right is less than $5$, do not change the digit to be rounded. 3. Replace all digits to the right with zeros. a.Rounding to nearest hundred: The hundreds digit is $1$, and the digit behind it is $9$, which is greater than $5$; therefore, we have to increase the hundreds digit by $1$, giving us the following number: $19,465,200$ b.Rounding to nearest thousand: The thousands digit is $5$, and the digit behind it is $1$, which is less than $5$; therefore, we have to keep the thousands digit the same, giving us the following number: $19,465,000$ c.Rounding to nearest ten thousand: The ten thousands digit is $6$, and the digit behind it is $5$, and therefore we have to increase the ten thousands digit by $1$, giving us the following number: $19,470,000$ d.Rounding to nearest hundred thousand: The hundred thousands digit is $4$, and the digit behind it is $6$, which is greater than $5$; therefore, we have to increase the hundred thousands digit by $1$, giving us the following number: $19,500,000$ e.Rounding to nearest million: The millions digit is $9$, and the digit behind it is $4$, which is less than $5$; therefore, we have to keep the millions digit the same, giving us the following number: $19,000,000$ f.Rounding to nearest ten million: The ten millions digit is $1$, and the digit behind it is $9$, and therefore we have to increase the ten millions digit by $1$, giving us the following number: $20,000,000$","date":"2018-07-21 15:51:56","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.22602319717407227, \"perplexity\": 295.0705775098696}, \"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-2018-30\/segments\/1531676592636.68\/warc\/CC-MAIN-20180721145209-20180721165209-00216.warc.gz\"}"}
| null | null |
{"url":"https:\/\/biz.libretexts.org\/Bookshelves\/Finance\/Book%3A_International_Finance__Theory_and_Policy\/09%3A_The_AA-DD_Model\/9.01%3A_Overview_of_the_AA-DD_Model","text":"# 9.1: Overview of the AA-DD Model\n\n\u2022 Contributed by No Attribution by request\n\u2022 Anonymous by request\n\nlearning objective\n\n1. Understand the basic structure and results of the AA-DD model of national output and exchange rate determination.\n\nThis chapter describes the derivation and the mechanics of the AA-DD model. The AA-DD model represents a synthesis of the three previous market models: the foreign exchange (Forex) market, the money market, and the goods and services market. In a sense, there is really very little new information presented here. Instead, the chapter provides a graphical approach to integrate the results from the three models and to show their interconnectedness. However, because so much is going on simultaneously, working with the AA-DD model can be quite challenging.\n\nThe AA-DD model is described with a diagram consisting of two curves (or lines): an AA curve representing asset market equilibriums derived from the money market and foreign exchange markets and a DD curve representing goods market (or demand) equilibriums. The intersection of the two curves identifies a market equilibrium in which each of the three markets is simultaneously in equilibrium. Thus we refer to this equilibrium as a superequilibrium.\n\n## Results\n\nThe main results of this section are descriptive and purely mechanical. The chapter describes the derivation of the AA and DD curves, explains how changes in exogenous variables will cause shifts in the curves, and explains adjustment from one equilibrium to another.\n\n1. The DD curve is the set of exchange rate and GNP combinations that maintain equilibrium in the goods and services market, given fixed values for all other exogenous variables.\n2. The DD curve shifts rightward whenever government demand ($$G$$), investment demand ($$I$$), transfer payments ($$TR$$), or foreign prices ($$P_{\u00a3}$$) increase or when taxes ($$T$$) or domestic prices ($$P_{}$$) decrease. Changes in the opposite direction cause a leftward shift.\n3. The AA curve is the set of exchange rate and GNP combinations that maintain equilibrium in the asset markets, given fixed values for all other exogenous variables.\n4. The AA curve shifts upward whenever money supply ($$M^{S}$$), foreign interest rates ($$i_{\u00a3}$$), or the expected exchange rate ($$E_{\/\u00a3}^{e}$$) increase or when domestic prices ($$P_{}$$) decrease. Changes in the opposite direction cause a downward shift.\n5. The intersection of the AA and DD curves depicts a superequilibrium in an economy since at that point the goods and services market, the domestic money market, and the foreign exchange market are all in equilibrium simultaneously.\n6. Changes in any exogenous variable that is not plotted on the axes (anything but $$Y$$ and $$E_{\/\u00a3}$$) will cause a shift of the AA or DD curves and move the economy out of equilibrium, temporarily. Adjustment to a new equilibrium follows the principle that adjustment in the asset markets occurs much more rapidly than adjustment in the goods and services market. Thus adjustment to the AA curve will always occur before adjustment to the DD curve.\n\n## Connections\n\nThe AA-DD model will allow us to understand how changes in macroeconomic policy\u2014both monetary and fiscal\u2014can affect key aggregate economic variables when a country is open to international trade and financial flows while accounting for the interaction of the variables among themselves. Specifically, the model is used to identify potential effects of fiscal and monetary policy on exchange rates, trade balances, GDP levels, interest rates, and price levels both domestically and abroad. In subsequent chapters, analyses will be done under both floating and fixed exchange rate regimes.\n\nkey takeaways\n\n\u2022 The AA-DD model integrates the workings of the money-Forex market and the G&S model into one supermodel.\n\u2022 The AA curve is derived from the money-Forex model. The DD curve is derived from the G&S model.\n\u2022 The intersection of the AA and DD curves determines the equilibrium values for real GNP and the exchange rate.\n\u2022 Comparative statics exercises using the AA-DD model allow one to identify the effects of changes in exogenous variables on the level of GDP and the exchange rate, while assuring that the Forex, the money market, and the G&S market all achieve simultaneous equilibrium.\n\nexercise\n\n1. Jeopardy Questions. As in the popular television game show, you are given an answer to a question and you must respond with the question. For example, if the answer is \u201ca tax on imports,\u201d then the correct question is \u201cWhat is a tariff?\u201d\n\u2022 At the intersection of the AA and DD curves, the goods and services market, the money market, and this market are simultaneously in equilibrium.\n\u2022 The term used to describe the type of equilibrium at the intersection of the AA curve and the DD curve.","date":"2021-04-21 02:33:19","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 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\": 0.515883207321167, \"perplexity\": 1696.0083038383268}, \"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-2021-17\/segments\/1618039503725.80\/warc\/CC-MAIN-20210421004512-20210421034512-00160.warc.gz\"}"}
| null | null |
\section{Introduction}
\hspace{5mm}It is obvious that our natural physical intuition distinguishes
four dimensions in a natural correspondence with the material reality.
Consequently, the four dimensionality plays special role in almost all
modern physical theories.
On the other hand, it is an well known fact that, in order to create the
Relativity Theory, Einstein was forced to use the Riemannian geometry
instead of the classical Euclidean geometry, the first one representing the
natural ma\-the\-ma\-ti\-cal model for the local \textit{isotropic}
space-time. But, there are recent studies of physicists which suggest a
\textit{non-isotropic} perspective of the space-time (for example, in
Pavlov's opinion \cite{Pavlov}, the concept of inertial body mass emphasizes
the necessity of study of local non-isotropic spaces). Obviously, for the
study of non-isotropic physical phenomena, the Finsler geometry is very
useful as ma\-the\-ma\-ti\-cal framework.
The studies of Russian scholars (Asanov \cite{Asanov[1]}, Mikhailov \cit
{Mikhailov}, Garas'ko and Pavlov \cite{Garasko-Pavlov}) emphasize the
importance of the Finsler geometry which is characterized by the total
equality of all non-isotropic directions. For such a reason, Asanov, Pavlov
and their co-workers underline the important role played by the Berwald-M
\'{o}r metric (whose Finsler geometry is deeply studied by Matsumoto and
Shimada in the paper \cite{Mats-Shimada}
\begin{equation*}
F:TM\rightarrow \mathbb{R},\mathbb{\qquad }F(y)=\left(
y^{1}y^{2}...y^{n}\right) ^{\frac{1}{n}},
\end{equation*
in the theory of space-time structure and gravitation, as well as in unified
gauge field theories. Because any of such directions can be related to the
proper time of an inertial reference frame, Pavlov considers that it is
appropriate as such spaces to be generically called \textit
"multi-dimensional time"} \cite{Pavlov}. In the framework of the $4
-dimensional linear space with Berwald-Mo\'{o}r metric (i.e. the
four-dimensional time), Pavlov and his co-workers \cite{Garasko-Pavlov},
\cite{Pavlov} offer some new physical approaches and geometrical
interpretations such as:
1. physical events = points in the 4-dimensional space;
2. straight lines = shortest curves;
3. intervals = distances between the points along of a straight line;
4. light pyramids $\Leftrightarrow $ light cones in a pseudo-Euclidian space.
For such geometrical and physical reasons, this paper is devoted to the
development on the $1$-jet space $J^{1}(\mathbb{R},M^{4})$ of the
Finsler-like geometry (together with a theoretical-geometric gravitational
field theory) of the \textit{rheonomic Berwald-Mo\'{o}r metric
\begin{equation*}
\mathring{F}:J^{1}(\mathbb{R},M^{4})\rightarrow \mathbb{R},\qquad \mathring{
}(t,y)=\sqrt{h^{11}(t)}\sqrt[4]{y_{1}^{1}y_{1}^{2}y_{1}^{3}y_{1}^{4}},
\end{equation*
where $h_{11}(t)$ is a Riemannian metric on $\mathbb{R}$ and
(t,x^{1},x^{2},x^{3},x^{4},y_{1}^{1},y_{1}^{2},y_{1}^{3},y_{1}^{4})$ are the
coordinates of the $1$-jet space $J^{1}(\mathbb{R},M^{4})$.
The differential geometry (in the sense of distinguished (d-) connections,
d-torsions, d-curvatures, gravitational and electromagnetic geometrical
theories) produced by a jet rheonomic Lagrangian function $L:J^{1}(\mathbb{R
,M^{n})\rightarrow \mathbb{R}$ is now completely done in the author's paper
\cite{Neagu-Rheon}. We point out that the geometrical ideas from \cit
{Neagu-Rheon} are similar, but however distinct ones, with those exposed by
Miron and Anastasiei in the classical Lagrangian geometry \cite{Mir-An}. In
fact, the geometrical ideas from \cite{Neagu-Rheon} (the jet geometrical
theory of the \textit{rheonomic Lagrange spaces}) were initially stated by
Asanov in \cite{Asanov[2]} and developed further by the author of this paper
in the book \cite{Neagu Carte}.
In the sequel, we apply the general geometrical results from \cit
{Neagu-Rheon} to the rheonomic Berwald-Mo\'{o}r metric\textit{\ }$\mathring{
}$.
\section{Preliminary notations and formulas}
\hspace{5mm}Let $(\mathbb{R},h_{11}(t))$ be a Riemannian manifold, where
\mathbb{R}$ is the set of real numbers. The Christoffel symbol of the
Riemannian metric $h_{11}(t)$ i
\begin{equation*}
\varkappa _{11}^{1}=\frac{h^{11}}{2}\frac{dh_{11}}{dt},\qquad h^{11}=\frac{
}{h_{11}}>0.
\end{equation*
Let also $M^{4}$ be a manifold of dimension four, whose local coordinates
are $(x^{1},x^{2},x^{3},x^{4})$. Let us consider the $1$-jet space $J^{1}
\mathbb{R},M^{4})$, whose local coordinates ar
\begin{equation*}
(t,x^{1},x^{2},x^{3},x^{4},y_{1}^{1},y_{1}^{2},y_{1}^{3},y_{1}^{4}).
\end{equation*
These transform by the rules (the Einstein convention of summation is used
throughout this work)
\begin{equation}
\widetilde{t}=\widetilde{t}(t),\quad \widetilde{x}^{p}=\widetilde{x
^{p}(x^{q}),\quad \widetilde{y}_{1}^{p}=\dfrac{\partial \widetilde{x}^{p}}
\partial x^{q}}\dfrac{dt}{d\widetilde{t}}\cdot y_{1}^{q},\qquad p,q
\overline{1,4}, \label{tr-rules}
\end{equation
where $d\widetilde{t}/dt\neq 0$ and rank $(\partial \widetilde{x
^{p}/\partial x^{q})=4$. We consider that the manifold $M^{4}$ is endowed
with a tensor of kind $(0,4)$, given by the local components $G_{pqrs}(x)$,
which is totally symmetric in the indices $p$, $q$, $r$ and $s$. Suppose
that the d-tensor
\begin{equation*}
G_{ij11}=12G_{ijpq}y_{1}^{p}y_{1}^{q},
\end{equation*
is non-degenerate, that is there exists the d-tensor $G^{jk11}$ on $J^{1}
\mathbb{R},M^{4})$ such that $G_{ij11}G^{jk11}=\delta _{i}^{k}.$
In this geometrical context, if we use the notation
G_{1111}=G_{pqrs}y_{1}^{p}y_{1}^{q}y_{1}^{r}y_{1}^{s}$, we can consider the
rheonomic Finsler-like function (it is $1$-positive homogenous in the
variable $y$)
\begin{equation}
F(t,x,y)=\sqrt[4]{G_{pqrs}(x)y_{1}^{p}y_{1}^{q}y_{1}^{r}y_{1}^{s}}\cdot
\sqrt{h^{11}(t)}=\sqrt[4]{G_{1111}(x,y)}\cdot \sqrt{h^{11}(t)}, \label{F}
\end{equation
where the Finsler function $F$ has as domain of definition all values
(t,x,y)$ which verify the condition $G_{1111}(x,y)>0.$ If we denote
G_{i111}=4G_{ipqr}(x)y_{1}^{p}y_{1}^{q}y_{1}^{r}$, then the $4$-positive
homogeneity of the "$y$-function" $G_{1111}$ (this is in fact a d-tensor on
J^{1}(\mathbb{R},M^{4})$) leads to the equalities
\begin{equation*}
G_{i111}=\frac{\partial G_{1111}}{\partial y_{1}^{i}},\quad
G_{i111}y_{1}^{i}=4G_{1111},\quad G_{ij11}y_{1}^{j}=3G_{i111},
\end{equation*
\begin{equation*}
G_{ij11}=\frac{\partial G_{i111}}{\partial y_{1}^{j}}=\frac{\partial
^{2}G_{1111}}{\partial y_{1}^{i}\partial y_{1}^{j}},\quad
G_{ij11}y_{1}^{i}y_{1}^{j}=12G_{1111}.
\end{equation*}
The \textit{fundamental metrical d-tensor} produced by $F$ is given by the
formul
\begin{equation*}
g_{ij}(t,x,y)=\frac{h_{11}(t)}{2}\frac{\partial ^{2}F^{2}}{\partial
y_{1}^{i}\partial y_{1}^{j}}.
\end{equation*
By direct computations, the fundamental metrical d-tensor takes the for
\begin{equation}
g_{ij}(x,y)=\frac{1}{4\sqrt{G_{1111}}}\left[ G_{ij11}-\frac{1}{2G_{1111}
G_{i111}G_{j111}\right] . \label{g-(ij)-general}
\end{equation
Moreover, taking into account that the d-tensor $G_{ij11}$ is
non-degenerate, we deduce that the matrix $g=(g_{ij})$ admits the inverse
g^{-1}=(g^{jk})$. The entries of the inverse matrix $g^{-1}$ ar
\begin{equation}
g^{jk}=4\sqrt{G_{1111}}\left[ G^{jk11}+\frac{G_{1}^{j}G_{1}^{k}}{2\left(
G_{1111}-\mathcal{G}_{1111}\right) }\right] , \label{g+(jk)-general}
\end{equation
where $G_{1}^{j}=G^{jp11}G_{p111}$ and $2\mathcal{G
_{1111}=G^{pq11}G_{p111}G_{q111}.$
\section{The rheonomic Berwald-Mo\'{o}r metric}
\hspace{5mm}Beginning with this Section we will focus only on the \textit
rheonomic Berwald-Mo\'{o}r metric}, which is the Finsler-like metric (\ref{F
) for the particular cas
\begin{equation*}
G_{pqrs}=\left\{
\begin{array}{ll}
\dfrac{1}{4!}, & \{p,q,r,s\}\text{ - distinct indices}\medskip \\
0, & \text{otherwise.
\end{array
\right.
\end{equation*
Consequently, the rheonomic Berwald-Mo\'{o}r metric is given b
\begin{equation}
\mathring{F}(t,y)=\sqrt{h^{11}(t)}\cdot \sqrt[4]
y_{1}^{1}y_{1}^{2}y_{1}^{3}y_{1}^{4}}. \label{rheon-B-M}
\end{equation
Moreover, using preceding notations and formulas, we obtain the following
relations
\begin{equation*}
G_{1111}=y_{1}^{1}y_{1}^{2}y_{1}^{3}y_{1}^{4},\quad G_{i111}=\frac{G_{1111}}
y_{1}^{i}},
\end{equation*
\begin{equation*}
G_{ij11}=\left( 1-\delta _{ij}\right) \frac{G_{1111}}{y_{1}^{i}y_{1}^{j}
\text{ (no sum by }i\text{ or }j\text{),}
\end{equation*
where $\delta _{ij}$ is the Kronecker symbol. Because we hav
\begin{equation*}
\det \left( G_{ij11}\right) _{i,j=\overline{1,4}}=-3\left( G_{1111}\right)
^{2}\neq 0,
\end{equation*
we fin
\begin{equation*}
G^{jk11}=\frac{(1-3\delta ^{jk})}{3G_{1111}}y_{1}^{j}y_{1}^{k}\text{ (no sum
by }j\text{ or }k\text{).}
\end{equation*
It follows that we have $\mathcal{G}_{1111}=(2/3)G_{1111}$ and
G_{1}^{j}=(1/3)y_{1}^{j}$.
Replacing now the preceding computed entities into the formulas (\re
{g-(ij)-general}) and (\ref{g+(jk)-general}), we ge
\begin{equation}
g_{ij}=\frac{\left( 1-2\delta _{ij}\right) \sqrt{G_{1111}}}{8}\frac{1}
y_{1}^{i}y_{1}^{j}}\text{ (no sum by }i\text{ or }j\text{)}
\label{g-jos-(ij)}
\end{equation
an
\begin{equation}
g^{jk}=\frac{2(1-2\delta ^{jk})}{\sqrt{G_{1111}}}y_{1}^{j}y_{1}^{k}\text{
(no sum by }j\text{ or }k\text{).} \label{g-sus-(jk)}
\end{equation}
Using a general formula from the paper \cite{Neagu-Rheon}, we find the
following geometrical result:
\begin{proposition}
For the rheonomic Berwald-Mo\'{o}r metric (\ref{rheon-B-M}), the \textit
energy action functional
\begin{equation*}
\mathbb{\mathring{E}}(t,x(t))=\int_{a}^{b}\sqrt
y_{1}^{1}y_{1}^{2}y_{1}^{3}y_{1}^{4}}\cdot h^{11}\sqrt{h_{11}}dt
\end{equation*
produces on the $1$-jet space $J^{1}(\mathbb{R},M^{4})$ the \textit
canonical nonlinear connection
\begin{equation}
\Gamma =\left( M_{(1)1}^{(i)}=-\varkappa _{11}^{1}y_{1}^{i},\text{
N_{(1)j}^{(i)}=0\right) . \label{can-nlc=0}
\end{equation}
\end{proposition}
Because the canonical nonlinear connection (\ref{can-nlc=0}) has the spatial
components equal to zero, it follows that our subsequent geometrical theory
becomes trivial, in a way. For such a reason, in order to avoid the
triviality of our theory and in order to have a certain kind of symmetry, we
will use on the $1$-jet space $J^{1}(\mathbb{R},M^{4})$, by an "a priori"
definition, the following nonlinear connection:
\begin{equation}
\mathring{\Gamma}=\left( M_{(1)1}^{(i)}=-\varkappa _{11}^{1}y_{1}^{i},\text{
}N_{(1)j}^{(i)}=-\frac{\varkappa _{11}^{1}}{3}\delta _{j}^{i}\right) .
\label{nlc-B-M}
\end{equation}
\section{Cartan canonical connection. d-Torsions and d-curvatures}
\hspace{5mm}The importance of the nonlinear connection (\ref{nlc-B-M}) is
coming from the possibility of construction of the dual \textit{adapted base
} of distinguished (d-) vector field
\begin{equation}
\left\{ \frac{\delta }{\delta t}=\frac{\partial }{\partial t}+\varkappa
_{11}^{1}y_{1}^{p}\frac{\partial }{\partial y_{1}^{p}},\text{ }\frac{\delta
}{\delta x^{i}}=\frac{\partial }{\partial x^{i}}+\frac{\varkappa _{11}^{1}}{
}\frac{\partial }{\partial y_{1}^{i}},\text{ }\dfrac{\partial }{\partial
y_{1}^{i}}\right\} \subset \mathcal{X}(E) \label{a-b-v}
\end{equation
and distinguished covector field
\begin{equation}
\left\{ dt,\text{ }dx^{i},\text{ }\delta y_{1}^{i}=dy_{1}^{i}-\varkappa
_{11}^{1}y_{1}^{i}dt-\frac{\varkappa _{11}^{1}}{3}dx^{i}\right\} \subset
\mathcal{X}^{\ast }(E), \label{a-b-co}
\end{equation
where $E=J^{1}(\mathbb{R},M^{4})$. Note that, under a change of coordinates
\ref{tr-rules}), the elements of the adapted bases (\ref{a-b-v}) and (\re
{a-b-co}) transform as classical tensors. Consequently, all subsequent
geometrical objects on the $1$-jet space $J^{1}(\mathbb{R},M^{4})$ (as
Cartan canonical connection, torsion, curvature etc.) will be described in
local adapted components.
Using a general result from \cite{Neagu-Rheon}, by direct computations, we
can give the following important geometrical result:
\begin{theorem}
The Cartan canonical $\mathring{\Gamma}$-linear connection, produced by the
rheonomic Berwald-Mo\'{o}r metric (\ref{rheon-B-M}), has the following
adapted local components
\begin{equation*}
C\mathring{\Gamma}=\left( \varkappa _{11}^{1},\text{ }G_{j1}^{k}=0,\text{
L_{jk}^{i}=\frac{\varkappa _{11}^{1}}{3}C_{j(k)}^{i(1)},\text{
C_{j(k)}^{i(1)}\right) ,
\end{equation*
where, if we use the notatio
\begin{equation*}
A_{jk}^{i}=\frac{2\delta _{j}^{i}+2\delta _{k}^{i}+2\delta _{jk}-8\delta
_{j}^{i}\delta _{jk}-1}{8}\text{ (no sum by }i,\text{ }j\text{ or }k\text{),}
\end{equation*
the
\begin{equation*}
C_{j(k)}^{i(1)}=A_{jk}^{i}\cdot \frac{y_{1}^{i}}{y_{1}^{j}y_{1}^{k}}\text{
(no sum by }i,\text{ }j\text{ or }k\text{).}
\end{equation*}
\end{theorem}
\begin{proof}
Via the Berwald-Mo\'{o}r derivative operators (\ref{a-b-v}) and (\ref{a-b-co
), we use the general formulas which give the adapted components of the
Cartan canonical connection, namely \cite{Neagu-Rheon
\begin{equation*}
G_{j1}^{k}=\frac{g^{km}}{2}\frac{\delta g_{mj}}{\delta t},\quad L_{jk}^{i}
\frac{g^{im}}{2}\left( \frac{\delta g_{jm}}{\delta x^{k}}+\frac{\delta g_{km
}{\delta x^{j}}-\frac{\delta g_{jk}}{\delta x^{m}}\right) ,
\end{equation*
\begin{equation*}
C_{j(k)}^{i(1)}=\frac{g^{im}}{2}\left( \frac{\partial g_{jm}}{\partial
y_{1}^{k}}+\frac{\partial g_{km}}{\partial y_{1}^{j}}-\frac{\partial g_{jk}}
\partial y_{1}^{m}}\right) =\frac{g^{im}}{2}\frac{\partial g_{jm}}{\partial
y_{1}^{k}}.
\end{equation*}
\end{proof}
\begin{remark}
The below properties of the d-tensor $C_{j(k)}^{i(1)}$ are true (see also
the papers \cite{At-Bal-Neagu} and \cite{Mats-Shimada})
\begin{equation}
C_{j(k)}^{i(1)}=C_{k(j)}^{i(1)},\quad C_{j(m)}^{i(1)}y_{1}^{m}=0,\quad
C_{j(m)}^{m(1)}=0\text{ (sum by }m\text{)}. \label{equalitie-C}
\end{equation}
\end{remark}
\begin{remark}
The coefficients $A_{ij}^{l}$ have the following values
\begin{equation}
A_{ij}^{l}=\left\{
\begin{array}{ll}
-\dfrac{1}{8}, & i\neq j\neq l\neq i\medskip \\
\dfrac{1}{8}, & i=j\neq l\text{ or }i=l\neq j\text{ or }j=l\neq i\medskip \\
-\dfrac{3}{8}, & i=j=l
\end{array
\right. \label{A-(ijk)}
\end{equation}
\end{remark}
\begin{theorem}
The Cartan canonical connection $C\mathring{\Gamma}$ of the rheonomic
Berwald-Mo\'{o}r metric (\ref{rheon-B-M}) has \textbf{three} effective local
torsion d-tensors
\begin{equation*}
\begin{array}{c}
P_{(1)i(j)}^{(k)\text{ }(1)}=-\dfrac{1}{3}\varkappa
_{11}^{1}C_{i(j)}^{k(1)},\quad P_{i(j)}^{k(1)}=C_{i(j)}^{k(1)},\medskip \\
R_{(1)1j}^{(k)}=\dfrac{1}{3}\left[ \dfrac{d\varkappa _{11}^{1}}{dt
-\varkappa _{11}^{1}\varkappa _{11}^{1}\right] \delta _{j}^{k}
\end{array
\end{equation*}
\end{theorem}
\begin{proof}
A general $h$-normal $\Gamma $-linear connection on the 1-jet space $J^{1}
\mathbb{R},M^{4})$ is characterized by \textit{eight} effective d-tensors of
torsion (for more details, please see \cite{Neagu-Rheon}). For our Cartan
canonical connection $C\mathring{\Gamma}$ these reduce to the following
\textit{three} (the other five cancel)
\begin{equation*}
{P_{(1)i(j)}^{(k)\text{ }(1)}={\dfrac{\partial N_{(1)i}^{(k)}}{\partial
y_{1}^{j}}}-L_{ji}^{k}},\quad R_{(1)1j}^{(k)}={\dfrac{\delta M_{(1)1}^{(k)}}
\delta x^{j}}}-{\dfrac{\delta N_{(1)j}^{(k)}}{\delta t}},\quad
P_{i(j)}^{k(1)}=C_{i(j)}^{k(1)}.
\end{equation*}
\end{proof}
\begin{theorem}
The Cartan canonical connection $C\mathring{\Gamma}$ of the rheonomic
Berwald-Mo\'{o}r metric (\ref{rheon-B-M}) has \textbf{three} effective local
curvature d-tensors
\begin{equation*}
\begin{array}{c}
R_{ijk}^{l}=\dfrac{1}{9}\varkappa _{11}^{1}\varkappa
_{11}^{1}S_{i(j)(k)}^{l(1)(1)},\quad P_{ij(k)}^{l\text{ }(1)}=\dfrac{1}{3
\varkappa _{11}^{1}S_{i(j)(k)}^{l(1)(1)},\medskip \\
S_{i(j)(k)}^{l(1)(1)}={{\dfrac{\partial C_{i(j)}^{l(1)}}{\partial y_{1}^{k}}
-{\dfrac{\partial C_{i(k)}^{l(1)}}{\partial y_{1}^{j}}
+C_{i(j)}^{m(1)}C_{m(k)}^{l(1)}-C_{i(k)}^{m(1)}C_{m(j)}^{l(1)}.
\end{array
\end{equation*}
\end{theorem}
\begin{proof}
A general $h$-normal $\Gamma $-linear connection on the 1-jet space $J^{1}
\mathbb{R},M^{4})$ is characterized by \textit{five} effective d-tensors of
curvature (for more details, please see \cite{Neagu-Rheon}). For our Cartan
canonical connection $C\mathring{\Gamma}$ these reduce to the following
\textit{three} (the other two cancel)
\begin{equation*}
\begin{array}{l}
\medskip {R_{ijk}^{l}={\dfrac{\delta L_{ij}^{l}}{\delta x^{k}}}-{\dfrac
\delta L_{ik}^{l}}{\delta x^{j}}}+L_{ij}^{m}L_{mk}^{l}-L_{ik}^{m}L_{mj}^{l},}
\\
\medskip {P_{ij(k)}^{l\;\;(1)}={\dfrac{\partial L_{ij}^{l}}{\partial
y_{1}^{k}}}-C_{i(k)|j}^{l(1)}+C_{i(m)}^{l(1)}P_{(1)j(k)}^{(m)\;\;(1)},} \\
S_{i(j)(k)}^{l(1)(1)}={{\dfrac{\partial C_{i(j)}^{l(1)}}{\partial y_{1}^{k}}
-{\dfrac{\partial C_{i(k)}^{l(1)}}{\partial y_{1}^{j}}
+C_{i(j)}^{m(1)}C_{m(k)}^{l(1)}-C_{i(k)}^{m(1)}C_{m(j)}^{l(1)},
\end{array
\end{equation*
wher
\begin{equation*}
{C_{i(k)|j}^{l(1)}=}\frac{\delta {C_{i(k)}^{l(1)}}}{\delta x^{j}}+
C_{i(k)}^{m(1)}L_{mj}^{l}}-{C_{m(k)}^{l(1)}L_{ij}^{m}}-
C_{i(m)}^{l(1)}L_{kj}^{m}}.
\end{equation*}
\end{proof}
\begin{remark}
The curvature d-tensor $S_{i(j)(k)}^{l(1)(1)}$ has the propertie
\begin{equation*}
S_{i(j)(k)}^{l(1)(1)}+S_{i(k)(j)}^{l(1)(1)}=0,\quad S_{i(j)(j)}^{l(1)(1)}=
\text{ (no sum by }j\text{).}
\end{equation*}
\end{remark}
\begin{theorem}
The following expressions of the curvature d-tensor $S_{i(j)(k)}^{l(1)(1)}$
hold good:
\begin{enumerate}
\item $S_{i(j)(k)}^{l(1)(1)}=0$ for $\{i,$ $j,$ $k,$ $l\}$ distinct indices;
\item $S_{i(i)(k)}^{l(1)(1)}=-\dfrac{1}{16}\dfrac{y_{1}^{l}}{\left(
y_{1}^{i}\right) ^{2}y_{1}^{k}}$ ($i\neq k\neq l\neq i$ and no sum by $i$);
\item $S_{i(j)(i)}^{l(1)(1)}=\dfrac{1}{16}\dfrac{y_{1}^{l}}{\left(
y_{1}^{i}\right) ^{2}y_{1}^{j}}$ ($i\neq j\neq l\neq i$ and no sum by $i$);
\item $S_{i(j)(k)}^{i(1)(1)}=0$ ($i\neq j\neq k\neq i$ and no sum by $i$);
\item $S_{i(l)(k)}^{l(1)(1)}=\dfrac{1}{16y_{1}^{i}y_{1}^{k}}$ ($i\neq k\neq
l\neq i$ and no sum by $l$);
\item $S_{i(j)(l)}^{l(1)(1)}=-\dfrac{1}{16y_{1}^{i}y_{1}^{j}}$ ($i\neq j\neq
l\neq i$ and no sum by $l$);
\item $S_{i(i)(l)}^{l(1)(1)}=\dfrac{1}{8\left( y_{1}^{i}\right) ^{2}}$ (
i\neq l$ and no sum by $i$ or $l$);
\item $S_{i(l)(i)}^{l(1)(1)}=-\dfrac{1}{8\left( y_{1}^{i}\right) ^{2}}$ (
i\neq l$ and no sum by $i$ or $l$);
\item $S_{l(l)(k)}^{l(1)(1)}=0$ ($k\neq l$ and no sum by $l$);
\item $S_{l(j)(l)}^{l(1)(1)}=0$ ($j\neq l$ and no sum by $l$).
\end{enumerate}
\end{theorem}
\begin{proof}
For $j\neq k$, the expression of the curvature tensor $S_{i(j)(k)}^{l(1)(1)}$
takes the form (no sum by $i$, $j$, $k$ or $l$, but with sum by $m$)
\begin{eqnarray*}
S_{i(j)(k)}^{l(1)(1)} &=&\left[ \frac{A_{ij}^{l}\delta _{k}^{l}}
y_{1}^{i}y_{1}^{j}}-\frac{A_{ik}^{l}\delta _{j}^{l}}{y_{1}^{i}y_{1}^{k}
\right] +\left[ \frac{A_{ik}^{l}\delta _{ij}y_{1}^{l}}{\left(
y_{1}^{i}\right) ^{2}y_{1}^{k}}-\frac{A_{ij}^{l}\delta _{ik}y_{1}^{l}}
\left( y_{1}^{i}\right) ^{2}y_{1}^{j}}\right] + \\
&&+\left[ A_{ij}^{m}A_{mk}^{l}-A_{ik}^{m}A_{mj}^{l}\right] \frac{y_{1}^{l}}
y_{1}^{i}y_{1}^{j}y_{1}^{k}},
\end{eqnarray*
where the coefficients $A_{ij}^{l}$ are given by the relations (\ref{A-(ijk)
).
\end{proof}
\section{Geometrical gravitational theory produced by the rheonomic
Berwald-Mo\'{o}r metric}
\hspace{5mm}From a physical point of view, on the 1-jet space $J^{1}(\mathbb
R},M^{4})$, the rheonomic Berwald-Mo\'{o}r metric (\ref{rheon-B-M}) produces
the adapted metrical d-tenso
\begin{equation}
\mathbb{G}=h_{11}dt\otimes dt+g_{ij}dx^{i}\otimes dx^{j}+h^{11}g_{ij}\delta
y_{1}^{i}\otimes \delta y_{1}^{j}, \label{gravit-pot-B-M}
\end{equation
where $g_{ij}$ is given by (\ref{g-jos-(ij)}). This may be regarded as a
\textit{"non-isotropic gravitational potential"}. In such a physical
context, the nonlinear connection $\mathring{\Gamma}$ (used in the
construction of the distinguished 1-forms $\delta y_{1}^{i}$) prescribes,
probably, a kind of \textit{\textquotedblleft interaction\textquotedblright }
between $(t)$-, $(x)$- and $(y)$-fields.
We postulate that the non-isotropic gravitational potential $\mathbb{G}$ is
governed by the \textit{geometrical Einstein equations
\begin{equation}
\text{Ric }\left( C\mathring{\Gamma}\right) -\frac{\text{Sc }\left(
\mathring{\Gamma}\right) }{2}\mathbb{G=}\mathcal{KT},
\label{Einstein-eq-global}
\end{equation
where Ric $\left( C\mathring{\Gamma}\right) $ is the \textit{Ricci d-tensor}
associated to the Cartan canonical connection $C\mathring{\Gamma}$ (in
Riemannian sense and using adapted bases), Sc $\left( C\mathring{\Gamma
\right) $ is the \textit{scalar curvature}, $\mathcal{K}$ is the \textit
Einstein constant} and $\mathcal{T}$ is the intrinsic \textit{stress-energy}
d-tensor of matter.
In this way, working with the adapted basis of vector fields (\ref{a-b-v}),
we can find the local geometrical Einstein equations for the rheonomic
Berwald-Mo\'{o}r metric (\ref{rheon-B-M}). Firstly, by direct computations,
we find:
\begin{proposition}
The Ricci d-tensor of the Cartan canonical connection $C\mathring{\Gamma}$
of the rheonomic Berwald-Mo\'{o}r metric (\ref{rheon-B-M}) has the following
effective local Ricci d-tensors
\begin{equation}
\begin{array}{l}
\medskip R_{ij}=R_{ijm}^{m}=\dfrac{1}{9}\varkappa _{11}^{1}\varkappa
_{11}^{1}S_{(i)(j)}^{(1)(1)}, \\
P_{i(j)}^{\text{ }(1)}=P_{(i)j}^{(1)}=P_{ij(m)}^{m\text{ }(1)}=\dfrac{1}{3
\varkappa _{11}^{1}S_{(i)(j)}^{(1)(1)},\medskip \\
S_{(i)(j)}^{(1)(1)}=S_{i(j)(m)}^{m(1)(1)}=\dfrac{7\delta _{ij}-1}{8}\dfrac{
}{y_{1}^{i}y_{1}^{j}}\text{ (no sum by }i\text{ or }j\text{){.}
\end{array}
\label{Ricci-local}
\end{equation}
\end{proposition}
\begin{remark}
The local Ricci d-tensor $S_{(i)(j)}^{(1)(1)}$ has the following expression
\begin{equation*}
S_{(i)(j)}^{(1)(1)}=\left\{
\begin{array}{ll}
-\dfrac{1}{8}\dfrac{1}{y_{1}^{i}y_{1}^{j}}, & i\neq j\medskip \\
\dfrac{3}{4}\dfrac{1}{\left( y_{1}^{i}\right) ^{2}}, & i=j
\end{array
\right.
\end{equation*}
\end{remark}
\begin{remark}
Using the third equality of (\ref{Ricci-local}) and the equality (\re
{g-sus-(jk)}), we deduce that the following equality is true (sum by $r$)
\begin{equation}
S_{i}^{m11}\overset{def}{=}g^{mr}S_{(r)(i)}^{(1)(1)}=\frac{5-14\delta
_{i}^{m}}{4}\cdot \frac{1}{\sqrt{G_{1111}}}\cdot \frac{y_{1}^{m}}{y_{1}^{i}
\text{ (no sum by }i\text{ or }m\text{).} \label{S-ridicat}
\end{equation
Moreover, by a direct calculation, we obtain the equalitie
\begin{equation}
\sum_{m,r=1}^{4}S_{r}^{m11}C_{i(m)}^{r(1)}=0,\quad \sum_{m=1}^{4}\frac
\partial S_{i}^{m11}}{\partial y_{1}^{m}}=\frac{3}{\sqrt{G_{1111}}}\dfrac{1}
y_{1}^{i}}. \label{equalities-S-ridicat}
\end{equation}
\end{remark}
\begin{proposition}
The scalar curvature of the Cartan canonical connection $C\mathring{\Gamma}$
of the rheonomic Berwald-Mo\'{o}r metric (\ref{rheon-B-M}) is given b
\begin{equation*}
\text{Sc }\left( C\mathring{\Gamma}\right) =-\frac{9h_{11}+\varkappa
_{11}^{1}\varkappa _{11}^{1}}{\sqrt{G_{1111}}}.
\end{equation*}
\end{proposition}
\begin{proof}
The general formula for the scalar curvature of a Cartan connection is (for
more details, please see \cite{Neagu-Rheon}
\begin{equation*}
\text{Sc }\left( C\mathring{\Gamma}\right)
=g^{pq}R_{pq}+h_{11}g^{pq}S_{(p)(q)}^{(1)(1)}.
\end{equation*}
\end{proof}
Describing the global geometrical Einstein equations (\re
{Einstein-eq-global}) in the adapted basis of vector fields (\ref{a-b-v}),
we find the following important geometrical and physical result (for more
details, please see \cite{Neagu-Rheon}):
\begin{theorem}
The local \textbf{geometrical Einstein equations} that govern the
non-isotropic gravitational potential $\mathbb{G}$ (produced by the
rheonomic Berwald-Mo\'{o}r metric (\ref{rheon-B-M})) are given by
\begin{equation}
\left\{
\begin{array}{l}
\medskip \dfrac{\xi _{11}h_{11}}{\sqrt{G_{1111}}}=\mathcal{T}_{11} \\
\medskip \dfrac{\varkappa _{11}^{1}\varkappa _{11}^{1}}{9\mathcal{K}
S_{(i)(j)}^{(1)(1)}+\dfrac{\xi _{11}}{\sqrt{G_{1111}}}g_{ij}=\mathcal{T}_{ij}
\\
\dfrac{1}{\mathcal{K}}S_{(i)(j)}^{(1)(1)}+\dfrac{\xi _{11}}{\sqrt{G_{1111}}
h^{11}g_{ij}=\mathcal{T}_{(i)(j)}^{(1)(1)
\end{array
\right. \label{E-1}
\end{equation
\medski
\begin{equation}
\left\{
\begin{array}{lll}
0=\mathcal{T}_{1i}, & 0=\mathcal{T}_{i1}, & 0=\mathcal{T}_{(i)1}^{(1)}
\medskip \\
0=\mathcal{T}_{1(i)}^{\text{ }(1)}, & \dfrac{\varkappa _{11}^{1}}{3\mathcal{
}}S_{(i)(j)}^{(1)(1)}=\mathcal{T}_{i(j)}^{\text{ }(1)}, & \dfrac{\varkappa
_{11}^{1}}{3\mathcal{K}}S_{(i)(j)}^{(1)(1)}=\mathcal{T}_{(i)j}^{(1)}
\end{array
\right. \label{E-2}
\end{equation
\medskip where
\begin{equation}
\xi _{11}=\frac{9h_{11}+\varkappa _{11}^{1}\varkappa _{11}^{1}}{2\mathcal{K}
. \label{CSI}
\end{equation}
\end{theorem}
\begin{remark}
The local geometrical Einstein equations (\ref{E-1}) and (\ref{E-2}) impose
as the stress-energy d-tensor of matter $\mathcal{T}$ to be symmetrical. In
other words, the stress-energy d-tensor of matter $\mathcal{T}$ must verify
the local symmetry condition
\begin{equation*}
\mathcal{T}_{AB}=\mathcal{T}_{BA},\quad \forall \text{ }A,B\in \left\{ 1
\text{ }i,\text{ }_{(i)}^{(1)}\right\} .
\end{equation*}
\end{remark}
By direct computations, the local geometrical Einstein equations (\ref{E-1})
and (\ref{E-2}) imply the following identities of the stress-energy d-tensor
(sum by $r$):\bigskip
$\bigskip \mathcal{T}_{1}^{1}\overset{def}{=}h^{11}\mathcal{T}_{11}=\dfrac
\xi _{11}}{\sqrt{G_{1111}}},\quad\mathcal{T}_{1}^{m}\overset{def}{=}g^{mr
\mathcal{T}_{r1}=0,\quad$
$\bigskip \mathcal{T}_{(1)1}^{(m)}\overset{def}{=}h_{11}g^{mr}\mathcal{T
_{(r)1}^{(1)}=0,\quad\mathcal{T}_{i}^{1}\overset{def}{=}h^{11}\mathcal{T
_{1i}=0,$
$\bigskip\mathcal{T}_{i}^{m}\overset{def}{=}g^{mr}\mathcal{T}_{ri}=\dfrac
\varkappa _{11}^{1}\varkappa _{11}^{1}}{9\mathcal{K}}S_{i}^{m11}+\dfrac{\xi
_{11}}{\sqrt{G_{1111}}}\mathcal{\delta }_{i}^{m},$
$\bigskip \mathcal{T}_{(1)i}^{(m)}\overset{def}{=}h_{11}g^{mr}\mathcal{T
_{(r)i}^{(1)}=\dfrac{h_{11}\varkappa _{11}^{1}}{3\mathcal{K}
S_{i}^{m11},\quad\mathcal{T}_{\text{ \ }(i)}^{1(1)}\overset{def}{=}h^{11
\mathcal{T}_{1(i)}^{\text{ }(1)}=0,$
$\bigskip\mathcal{T}_{\text{ \ }(i)}^{m(1)}\overset{def}{=}g^{mr}\mathcal{T
_{r(i)}^{\text{ }(1)}=\dfrac{\varkappa _{11}^{1}}{3\mathcal{K}}S_{i}^{m11},$
$\bigskip\mathcal{T}_{(1)(i)}^{(m)(1)}\overset{def}{=}h_{11}g^{mr}\mathcal{T
_{(r)(i)}^{(1)(1)}=\dfrac{h_{11}}{\mathcal{K}}S_{i}^{m11}+\dfrac{\xi _{11}}
\sqrt{G_{1111}}}\mathcal{\delta }_{i}^{m},$ where the d-tensor $S_{i}^{m11}$
is given by (\ref{S-ridicat}) and $\xi _{11}$ is given by (\ref{CSI}).
\begin{corollary}
The stress-energy d-tensor of matter $\mathcal{T}$ must verify the following
\textbf{geometrical conservation laws} (summation by $m$)
\begin{equation*}
\left\{
\begin{array}{l}
\bigskip \mathcal{T}_{1/1}^{1}+\mathcal{T}_{1|m}^{m}+\mathcal{T
_{(1)1}^{(m)}|_{(m)}^{(1)}=\dfrac{\left( h^{11}\right) ^{2}}{8\mathcal{K}
\dfrac{dh_{11}}{dt}\left[ 2\dfrac{d^{2}h_{11}}{dt^{2}}-\dfrac{3}{h_{11}
\left( \dfrac{dh_{11}}{dt}\right) ^{2}\right] \cdot \dfrac{1}{\sqrt{G_{1111}
} \\
\bigskip \mathcal{T}_{i/1}^{1}+\mathcal{T}_{i|m}^{m}+\mathcal{T
_{(1)i}^{(m)}|_{(m)}^{(1)}=\dfrac{\varkappa _{11}^{1}\xi _{11}}{18}\cdot
\dfrac{1}{\sqrt{G_{1111}}}\cdot \dfrac{1}{y_{1}^{i}} \\
\mathcal{T}_{\text{ \ }(i)/1}^{1(1)}+\mathcal{T}_{\text{ \ }(i)|m}^{m(1)}
\mathcal{T}_{(1)(i)}^{(m)(1)}|_{(m)}^{(1)}=\dfrac{\xi _{11}}{6}\cdot \dfrac{
}{\sqrt{G_{1111}}}\cdot \dfrac{1}{y_{1}^{i}}
\end{array
\right.
\end{equation*
where (summation by $m$ and $r$)\bigskip
$\bigskip \mathcal{T}_{1/1}^{1}\overset{def}{=}\dfrac{\delta \mathcal{T
_{1}^{1}}{\delta t}+\mathcal{T}_{1}^{1}\varkappa _{11}^{1}-\mathcal{T
_{1}^{1}\varkappa _{11}^{1}=\dfrac{\delta \mathcal{T}_{1}^{1}}{\delta t},$
$\bigskip \mathcal{T}_{1|m}^{m}\overset{def}{=}\dfrac{\delta \mathcal{T
_{1}^{m}}{\delta x^{m}}+\mathcal{T}_{1}^{r}L_{rm}^{m}=\dfrac{\delta \mathcal
T}_{1}^{m}}{\delta x^{m}},$
$\bigskip \mathcal{T}_{(1)1}^{(m)}|_{(m)}^{(1)}\overset{def}{=}\dfrac
\partial \mathcal{T}_{(1)1}^{(m)}}{\partial y_{1}^{m}}+\mathcal{T
_{(1)1}^{(r)}C_{r(m)}^{m(1)}=\dfrac{\partial \mathcal{T}_{(1)1}^{(m)}}
\partial y_{1}^{m}},$
$\bigskip \mathcal{T}_{i/1}^{1}\overset{def}{=}\dfrac{\delta \mathcal{T
_{i}^{1}}{\delta t}+\mathcal{T}_{i}^{1}\varkappa _{11}^{1}-\mathcal{T
_{r}^{1}G_{i1}^{r}=\dfrac{\delta \mathcal{T}_{i}^{1}}{\delta t}+\mathcal{T
_{i}^{1}\varkappa _{11}^{1},$
$\bigskip \mathcal{T}_{i|m}^{m}\overset{def}{=}\dfrac{\delta \mathcal{T
_{i}^{m}}{\delta x^{m}}+\mathcal{T}_{i}^{r}L_{rm}^{m}-\mathcal{T
_{r}^{m}L_{im}^{r}=\dfrac{\varkappa _{11}^{1}}{3}\dfrac{\partial \mathcal{T
_{i}^{m}}{\partial y_{1}^{m}},$
$\bigskip \mathcal{T}_{(1)i}^{(m)}|_{(m)}^{(1)}\overset{def}{=}\dfrac
\partial \mathcal{T}_{(1)i}^{(m)}}{\partial y_{1}^{m}}+\mathcal{T
_{(1)i}^{(r)}C_{r(m)}^{m(1)}-\mathcal{T}_{(1)r}^{(m)}C_{i(m)}^{r(1)}=\dfrac
\partial \mathcal{T}_{(1)i}^{(m)}}{\partial y_{1}^{m}},$
$\bigskip \mathcal{T}_{\text{ \ }(i)/1}^{1(1)}\overset{def}{=}\dfrac{\delta
\mathcal{T}_{\text{ \ }(i)}^{1(1)}}{\delta t}+2\mathcal{T}_{\text{ \
(i)}^{1(1)}\varkappa _{11}^{1},$
$\bigskip \mathcal{T}_{\text{ \ }(i)|m}^{m(1)}\overset{def}{=}\dfrac{\delta
\mathcal{T}_{\text{ \ }(i)}^{m(1)}}{\delta x^{m}}+\mathcal{T}_{\text{ \
(i)}^{r(1)}L_{rm}^{m}-\mathcal{T}_{\text{ \ }(r)}^{m(1)}L_{im}^{r}=\dfrac
\varkappa _{11}^{1}}{3}\dfrac{\partial \mathcal{T}_{\text{ \ }(i)}^{m(1)}}
\partial y_{1}^{m}},$
$\mathcal{T}_{(1)(i)}^{(m)(1)}|_{(m)}^{(1)}\overset{def}{=}\dfrac{\partial
\mathcal{T}_{(1)(i)}^{(m)(1)}}{\partial y_{1}^{m}}+\mathcal{T
_{(1)(i)}^{(r)(1)}C_{r(m)}^{m(1)}-\mathcal{T
_{(1)(r)}^{(m)(1)}C_{i(m)}^{r(1)}=\dfrac{\partial \mathcal{T
_{(1)(i)}^{(m)(1)}}{\partial y_{1}^{m}}.$
\end{corollary}
\begin{proof}
The conservation laws are provided by direct computations, using the
relations (\ref{equalitie-C}) and (\ref{equalities-S-ridicat}).
\end{proof}
\section{Some physical remarks and comments}
\subsection{On gravitational theory}
\hspace{5mm}It is known that in the classical Relativity theory of Einstein
(which characterizes the gravity in an isotropic space-time) the tensor of
matter must verify the conservation law
\begin{equation*}
\mathcal{T}_{i;m}^{m}=0,\quad \forall \text{ }i=\overline{1,4},
\end{equation*
where "$;$" means the covariant derivative produced by the Levi-Civita
connection associated to pseudo-Riemannian metric $g_{ij}(x)$ (the
gravitational potentials).
Comparatively, in our non-isotropic gravitational theory (with respect to
the rheonomic Berwald-Mo\'{o}r metric (\ref{rheon-B-M})) the conservation
laws are replaced with ($i=\overline{1,4}$)
\begin{equation*}
\begin{array}{l}
\bigskip \mathcal{T}_{1}=\dfrac{\left( h^{11}\right) ^{2}}{8\mathcal{K}
\dfrac{dh_{11}}{dt}\left[ 2\dfrac{d^{2}h_{11}}{dt^{2}}-\dfrac{3}{h_{11}
\left( \dfrac{dh_{11}}{dt}\right) ^{2}\right] \cdot \dfrac{1}{\sqrt{G_{1111}
} \\
\mathcal{T}_{i}=\dfrac{\varkappa _{11}^{1}\xi _{11}}{18}\cdot \dfrac{1}
\sqrt{G_{1111}}}\cdot \dfrac{1}{y_{1}^{i}},\quad\mathcal{T}_{(i)}^{(1)}
\dfrac{\xi _{11}}{6}\cdot \dfrac{1}{\sqrt{G_{1111}}}\cdot \dfrac{1}{y_{1}^{i
}
\end{array
\end{equation*}
wher
\begin{equation*}
\begin{array}{l}
\medskip \mathcal{T}_{1}\overset{def}{=}\mathcal{T}_{1/1}^{1}+\mathcal{T
_{1|m}^{m}+\mathcal{T}_{(1)1}^{(m)}|_{(m)}^{(1)}, \\
\medskip \mathcal{T}_{i}\overset{def}{=}\mathcal{T}_{i/1}^{1}+\mathcal{T
_{i|m}^{m}+\mathcal{T}_{(1)i}^{(m)}|_{(m)}^{(1)}, \\
\mathcal{T}_{(i)}^{(1)}\overset{def}{=}\mathcal{T}_{\text{ \ }(i)/1}^{1(1)}
\mathcal{T}_{\text{ \ }(i)|m}^{m(1)}+\mathcal{T
_{(1)(i)}^{(m)(1)}|_{(m)}^{(1)}
\end{array
\end{equation*}
By analogy with Einstein's theory, if we impose the conditions ($\forall $
i=\overline{1,4}$
\begin{equation*}
\left\{
\begin{array}{l}
\medskip \mathcal{T}_{1}=0 \\
\medskip \mathcal{T}_{i}=0 \\
\mathcal{T}_{(i)}^{(1)}=0
\end{array
\right.
\end{equation*
then we reach to the system of differential equations
\begin{equation}
\left\{
\begin{array}{l}
\dfrac{dh_{11}}{dt}\left[ 2\dfrac{d^{2}h_{11}}{dt^{2}}-\dfrac{3}{h_{11}
\left( \dfrac{dh_{11}}{dt}\right) ^{2}\right] =0\medskip \\
9h_{11}+\varkappa _{11}^{1}\varkappa _{11}^{1}=0
\end{array
\right. \label{DEs}
\end{equation
Obviously, because we have $h_{11}>0$, we deduce that the DEs system (\re
{DEs}) has not any solution. Consequently, we always hav
\begin{equation*}
\left[ \mathcal{T}_{1}\right] ^{2}+\left[ \mathcal{T}_{i}\right] ^{2}+\left[
\mathcal{T}_{(i)}^{(1)}\right] ^{2}\neq 0,\quad \forall \text{ }i=\overline
1,4}.
\end{equation*}
In our opinion, this fact suggests that our geometrical gravitational theory
(produced by the rheonomic Berwald-Mo\'{o}r gravitational potential (\re
{gravit-pot-B-M})) is not suitable for media whose stress-energy
d-components ar
\begin{equation*}
\mathcal{T}_{AB}=0,\quad \forall \text{ }A,B\in \left\{ 1,\text{ }i,\text{
_{(i)}^{(1)}\right\} .
\end{equation*
However, it is important to note that at "infinity"
\begin{equation*}
\text{(this means that }y_{1}^{i}\rightarrow \infty ,\quad \forall \text{ }i
\overline{1,4}),
\end{equation*
our Berwald-Mo\'{o}r geometrical gravitational theory seems to be
appropriate even for media characterized by a null stress-energy d-tensor of
matter. This is because at "infinity" the stress-energy local d-tensors tend
to become zero.
\subsection{On electromagnetic theory}
\hspace{5mm}In the paper \cite{Neagu-Rheon}, a geometrical theory for
electromagnetism was also created, using only a given Lagrangian function $L$
on the 1-jet space $J^{1}(\mathbb{R},M^{4})$. In the background of the jet
relativistic rheonomic Lagrange geometry from \cite{Neagu-Rheon}, we work
with an \textit{electromagnetic distinguished }$2$\textit{-form
\begin{equation*}
\mathbb{F}=F_{(i)j}^{(1)}\delta y_{1}^{i}\wedge dx^{j},
\end{equation*
wher
\begin{equation*}
F_{(i)j}^{(1)}=\frac{h^{11}}{2}\left[
g_{jm}N_{(1)i}^{(m)}-g_{im}N_{(1)j}^{(m)}+\left(
g_{ir}L_{jm}^{r}-g_{jr}L_{im}^{r}\right) y_{1}^{m}\right] ,
\end{equation*
which is characterized by some natural \textit{geometrical Maxwell equations}
(for more details, please see \cite{Neagu-Rheon})
In our particular case of rheonomic Berwald-Mo\'{o}r metric (\ref{rheon-B-M
) and nonlinear connection (\ref{nlc-B-M}), we find the electromagnetic $2
-form $\mathbb{F}:=\mathbb{\mathring{F}}=0.$ Consequently, our Berwald-M
\'{o}r geometrical electromagnetic theory is trivial. In our opinion, this
fact suggests that the rheonomic Berwald-Mo\'{o}r metric (\ref{rheon-B-M})
has rather strong gravitational connotations than electromagnetic ones. This
is because, in our geometrical approach, the Berwald-Mo\'{o}r
electromagnetism is trivial.
\textbf{Acknowledgements.} The author thanks Professor V. Balan for his
encouragements and useful suggestions.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 5,351
|
\section{Acknowledgements}
We thank CERN for the very successful operation of the LHC, as well as the
support staff from our institutions without whom ATLAS could not be
operated efficiently.
We acknowledge the support of ANPCyT, Argentina; YerPhI, Armenia; ARC,
Australia; BMWF and FWF, Austria; ANAS, Azerbaijan; SSTC, Belarus; CNPq and FAPESP,
Brazil; NSERC, NRC and CFI, Canada; CERN; CONICYT, Chile; CAS, MOST and NSFC,
China; COLCIENCIAS, Colombia; MSMT CR, MPO CR and VSC CR, Czech Republic;
DNRF, DNSRC and Lundbeck Foundation, Denmark; EPLANET, ERC and NSRF, European Union;
IN2P3-CNRS, CEA-DSM/IRFU, France; GNSF, Georgia; BMBF, DFG, HGF, MPG and AvH
Foundation, Germany; GSRT and NSRF, Greece; ISF, MINERVA, GIF, DIP and Benoziyo Center,
Israel; INFN, Italy; MEXT and JSPS, Japan; CNRST, Morocco; FOM and NWO,
Netherlands; BRF and RCN, Norway; MNiSW, Poland; GRICES and FCT, Portugal; MERYS
(MECTS), Romania; MES of Russia and ROSATOM, Russian Federation; JINR; MSTD,
Serbia; MSSR, Slovakia; ARRS and MVZT, Slovenia; DST/NRF, South Africa;
MICINN, Spain; SRC and Wallenberg Foundation, Sweden; SER, SNSF and Cantons of
Bern and Geneva, Switzerland; NSC, Taiwan; TAEK, Turkey; STFC, the Royal
Society and Leverhulme Trust, United Kingdom; DOE and NSF, United States of
America.
The crucial computing support from all WLCG partners is acknowledged
gratefully, in particular from CERN and the ATLAS Tier-1 facilities at
TRIUMF (Canada), NDGF (Denmark, Norway, Sweden), CC-IN2P3 (France),
KIT/GridKA (Germany), INFN-CNAF (Italy), NL-T1 (Netherlands), PIC (Spain),
ASGC (Taiwan), RAL (UK) and BNL (USA) and in the Tier-2 facilities
worldwide.
\section{Introduction}
\label{sec:intro}
The single top-quark signature is sensitive to many models of new
physics~\cite{Tait1:2000sh}. Single top-quark production in the Standard Model (SM) has been measured
at the LHC in the $t$-channel~\cite{Aad:2012ux,CMS:2012ep}
and in association with a $W$ boson ($Wt$-channel)~\cite{ATLAS-WtEvidence,CMS:2012yv}.
Searches for resonant production
of a new particle which decays with a single top-quark have been carried out in the
$s$-channel production of a top~quark together with a
$b$~quark~\cite{Aad:2012ej,CMS:2012sc}. This Letter presents
the first search for a resonance decaying to a single top-quark
and a $W$~boson~\cite{RS:2012}.
Here we consider the production of an excited quark $b^*$
which decays to a single top-quark and a $W$~boson. This is the first search for
excited-quarks coupling to the third generation of fermions.
Previous searches for excited quarks have focused on their strong
interactions~\cite{Aad:2011fq,Chatrchyan:2011ns}, as well as their electromagnetic
interactions~\cite{Aaron:2009iz,ATLAS:2011ai}
with SM~quarks. These searches exploit the coupling between the excited quark and
up or down quarks in the proton.
Here the production of excited-quarks coupling primarily to the third generation
of SM~quarks is investigated. This coupling occurs for example in Randall--Sundrum
models that address the strong interaction
sector~\cite{Cheung:2007bu,Fitzpatrick:2007sa}
or in models with a heavy gluon partner, such as composite Higgs
models~\cite{Bini:2011zb,Vignaroli:2012sf,Vignaroli:2012si}.
The $b^*$~quark is produced singly through its coupling to a
$b$~quark and a gluon, as shown in Fig.~\ref{fig:BpFeynman}.
\begin{figure}[!h!tbp]
\begin{center}
\includegraphics[width=0.9\columnwidth]{bp_feynman}
\caption{Leading-order Feynman diagram for single-$b^*$-quark production
and decay to $Wt$. }
\label{fig:BpFeynman}
\end{center}
\end{figure}
The Lagrangian describing this interaction is given by ~\cite{AdeRujula1984253,Baur:1987ga}
\begin{equation}
{\mathcal L} = \frac{g_s}{2\,\Lambda} G_{\mu\nu}\,
\overline{b}\, \sigma^{\mu\nu}
\biggl(\kappa^b_L P_L + \kappa^b_R P_R\biggr) b^*
+ \mathrm{h.c.}\, , \label{eq:productionL}
\end{equation}
where $g_s$ is the strong coupling, $G_{\mu\nu}$ the gauge field tensor of the
gluon and $\Lambda=m_{b^*}$ the scale of the new physics.
$P_L$ and $P_R$ are the left- and right-handed
projection operators and $\kappa^b_L$ and $\kappa^b_R$ are the respective coupling
strengths. This analysis is thus complementary to excited-quark searches focusing
on the coupling to the first generation~\cite{:2012ak,Aad:2011yn,Aad:2011fq}.
Single $b^*$-quark production
can also reveal the chiral nature of the excited bottom-quark~\cite{RS:2012}.
In addition to the chromomagnetic coupling,
the $b^*$~quark investigated here also
has weak couplings, as in a general class of new physics
models where new heavy particles stabilise the Higgs-boson mass at the electroweak
scale~\cite{Hill:2002ap,Martin:2009bg,Kumar:2010vx,Holdom:2009rf,Alok:2010zj}.
In such models, the heavy quarks can have left-handed or right-handed
couplings to the $W$~boson or can be vector-like with equal strength for both
couplings. The Lagrangian describing the electroweak decay of the $b^*$~quark,
shown in Fig.~\ref{fig:BpFeynman}, is
\begin{equation}
{\mathcal L} = \frac{g_2}{\sqrt{2}} \,W^+_{\mu} \,\overline{t} \gamma^{\mu}\,
\bigl(g_L P_L + g_R P_R \bigr) \,b^* + \mathrm{h.c.} \, ,
\label{eq:decayL}
\end{equation}
where $g_2$ is the $\mathrm{SU(2)}_L$ weak coupling and $g_L$ and $g_R$ are the coupling
strengths for left-handed and right-handed couplings, respectively.
While the search is general and considers any resonance decaying into the $Wt$ signature,
three specific $b^*$-quark coupling scenarios are considered in order to extract
$b^*$-quark coupling and mass limits:
left-handed ($\kappa^b_L$, $g_L$ non-zero and $\kappa^b_R=g_R=0$),
right-handed ($\kappa^b_L=g_L=0$ and $\kappa^b_R$, $g_R$ non-zero) and
vector-like ($\kappa^b_L=\kappa^b_R=\kappa^b_{L/R}$ and $g_L=g_R=g_{L/R}$ non-zero)
production and decay.
Limits are derived as a function of the $b^*$-quark mass as well as the couplings
$\kappa^b_{L,R}$ and $g_{L,R}$. These limits take into account both the change of the
production cross-section and the $b^*\rightarrow Wt$ decay branching ratio,
which depend on the couplings and the $b^*$-quark mass. The branching ratio to
$Wt$ varies between 20\% at $m_{b^*}=300~\GeV$ and 40\% at higher values, with decays to $bg$,
$bZ$ and $bH$ also allowed. Contributions from non-$Wt$ decay modes that may
increase the $b^*$-quark acceptance of this analysis are not considered,
resulting in conservative limits.
Signal event yields presented in the following tables are calculated with $\kappa^b_L=g_L=1$
and $\kappa^b_R=g_R=0$.
For a left-handed $b^*$ at $\sqrt{s}= 7\TeV$ with $\kappa^b_L=g_L=1$ and $\kappa^b_R=g_R=0$, the leading-order cross section times
branching ratio to $Wt$ is 0.80~pb for $m_{b^*}=900\GeV$~\cite{RS:2012}.
The uncertainties due to the choice of factorisation and renormalisation scales are evaluated by varying the scales between $m_{b^*}/2$ and $2 \times m_{b^*}$, and those due to the choice of PDF by comparing results obtained using the CT10~\cite{Lai:2010vv}, MRST~\cite{springerlink:10} and NNPDF~\cite{Ball:2010de} sets. These uncertainties are added in quadrature to yield cross-section uncertainties ranging from 12\% at $m_{b^*}=300\GeV$ to 25\% at $m_{b^*}=1200\GeV$.
This channel proceeds via two $W$~bosons from $b^*$-quark and top-quark decays.
At least one $W$~boson is required to decay to a lepton (electron or muon).
The analysis is performed separately
in the dilepton and lepton+jets final states. The lepton+jets channel has the
advantage that the invariant mass of the $b^*$~quark can be reconstructed,
whereas the dilepton channel benefits from smaller backgrounds. A discriminant
that separates the $b^*$-quark signal from the backgrounds is defined in each final
state. Limits on $b^*$-quark production are obtained from a combined Bayesian
analysis of both discriminant distributions.
\section{The ATLAS detector}
The ATLAS detector~\cite{ATL-2008-001} is a general purpose detector with a precise tracking system, calorimeters
and an outer muon spectrometer. The inner tracking system consists of a silicon pixel
detector, a silicon microstrip tracker, and a straw-tube transition radiation tracker.
This system is immersed in a $2~\mathrm{T}$ axial magnetic field produced by a solenoid and
provides charged particle tracking and identification in the pseudorapidity\footnote{ATLAS
uses a right-handed coordinate system with its origin at
the nominal interaction point (IP) in the centre of the detector and the $z$-axis
along the beam pipe. The $x$-axis points from the IP to the centre
of the LHC ring, and the $y$-axis points upwards. Cylindrical coordinates $(r, \phi)$
are used in the transverse plane, $\phi$ is the azimuthal angle around the beam pipe.
The pseudorapidity $\eta$ is defined in terms of the polar angle $\theta$ as
$\eta = -~\rm{ln}(\tan\theta/2)$.} region $|\eta|<2.5$.
The central calorimeter system consists of a liquid-argon electromagnetic sampling
calorimeter with high granularity and an iron/scintillator tile calorimeter providing
hadronic energy measurements in the central pseudorapidity range ($|\eta|<1.7$).
The endcap and forward regions are instrumented with liquid-argon calorimeters for both
electromagnetic and hadronic energy measurements up to $|\eta|=4.9$.
The muon spectrometer is operated in a toroidal magnetic
field provided by air-core superconducting magnets and includes tracking chambers
for precise muon momentum measurements up to $|\eta|=2.7$ and trigger chambers
covering the range $|\eta|<2.4$.
\section{Data and simulated samples}
\label{sec:DataMC}
This analysis uses data collected with the ATLAS detector in 2011,
corresponding to an integrated luminosity of
$4.7\pm 0.2~\ifb$~\cite{ATL-LUMI,ATL-CONF-2011-116} of $7\TeV$ proton--proton ($pp$)
collisions delivered by the LHC.
The data are selected using single-electron or single-muon
triggers whose efficiencies reach their plateau at $25\GeV$ and $20\GeV$,
respectively~\cite{Aad:2012xs,Hristova:1485638}.
The data must also pass stringent quality requirements~\cite{ATLAS-JETCLEAN}.
Events are selected if they contain at least one primary vertex candidate
with at least five associated tracks.
The signal is modelled using
{\sc MadGraph5}~\cite{Alwall:2011uj} and the {\sc CTEQ6L1} parton distribution
functions (PDFs)~\cite{Nadolsky:2008zw}. Events with single top-quarks in the $t$-channel are
generated with the {\sc AcerMC}~\cite{SAMPLES-ACER} generator,
using the MRST~LO** PDF set~\cite{Sherstnev:2007nd}. {\sc MadGraph5} and
{\sc AcerMC} are interfaced to {\sc Pythia}~\cite{SAMPLES-PYTHIA} for parton showering and modelling
of the underlying event.
Other processes producing single top-quarks and top-quark pairs ($\ttbar$) are modelled
with the next-to-leading-order(NLO) generator {\sc MC@NLO}~\cite{SAMPLES-MCNLO} using the
{\sc CT10} PDF set~\cite{Lai:2010vv}, interfaced to {\sc Herwig}~\cite{SAMPLES-HERWIG} for parton
showering and {\sc Jimmy}~\cite{SAMPLES-JIMMY} for the underlying event.
{\sc Alpgen}~\cite{SAMPLES-ALPGEN} is used to model vector boson ($W$ and $Z$)
production in association with jets as well as diboson processes ($WW$, $WZ$ and $ZZ$)
using the {\sc CTEQ6L1} PDF set. It is interfaced to {\sc Herwig} for parton shower
modelling.
In the lepton+jets analysis the diboson processes are modelled with {\sc Herwig} only.
Decays of $\tau$ leptons are handled by {\sc Tauola}~\cite{Davidson:2010rw}.
A top-quark mass of $172.5\GeV$~\cite{PDG} is assumed.
Approximate next-to-next-to-leading-order(NNLO) cross-section calculations are used to normalise the \ttbar~\cite{Aliev:2010zk} ({\sc Hathor})
and single top-quark samples~\cite{Kidonakis:2011wy,Kidonakis:2010tc,Kidonakis:2010ux},
while the vector boson and diboson samples are normalised using calculations with {\sc MCFM}~\cite{Campbell:2010ff}
at NNLO and NLO, respectively.
A variable number of additional $pp$ interactions (pile-up) are
overlaid on simulated events, which are then weighted to reproduce the distribution
of the number of collisions per bunch crossing observed in data.
All samples are passed through a {\sc GEANT4}-based simulation~\cite{SAMPLES-G4} of
the ATLAS detector~\cite{ATL-2010-005_sgtop} and are then reconstructed using the same
procedure as for collision data.
\section{Physics object selection}
\label{sec:selection}
Electron candidates are reconstructed from
clusters of energy deposits in the calorimeter~\cite{ATL-ELEC-PERF}.
The transverse energy $E_\mathrm{T}$ of electron candidates is required to be larger than
$25\GeV$ and their pseudorapidity is required to be $|\eta|<2.47$.
Electrons in the barrel--endcap transition region of the calorimeter, corresponding
to $1.37<|\eta|<1.52$, are not considered. Selected electrons must pass a set of
``tight'' quality criteria~\cite{ATL-ELEC-PERF} and the electons must be matched to
a track reconstructed in the inner tracking system. Electrons must also be isolated
from close-by tracks in a cone of $\Delta R =\sqrt{(\Delta\eta)^2+(\Delta\phi)^2} <0.3$
and from calorimeter energy deposits not belonging to the electron candidate in a cone of $\Delta R < 0.2$.
The isolation requirements on the sum of transverse momenta of tracks in the cone and on the sum of energy deposits
in the calorimeter in the cone are chosen as a function of $\pt$ and $\eta$ such that an efficiency of 90\%
for electrons in the simulation is achieved.
Muon candidates are reconstructed from matching tracks in the muon spectrometer and inner tracking system.
Muons are required to have transverse momentum $\pt > 25\GeV$
and $|\eta|<2.5$ and fulfil tight quality criteria~\cite{ATL-2010-012-2}. Muons
must be isolated from close-by tracks in a cone of $\Delta R <0.3$
and from energy deposits in the calorimeter in a cone of $\Delta R < 0.2$.
The sum of transverse momenta of tracks in the cone must not exceed $2.5\GeV$ and the sum of energy deposits in the calorimeter in the cone must be below $4\GeV$.
In order to reject events in which a muon emitting a hard photon is also reconstructed
as an electron, events are vetoed if a selected electron--muon pair shares the same
track.
Jets are reconstructed from clusters of energy deposits in the
calorimeter~\cite{ATLAS-TOPO-CLUSTER}
using the anti-$k_{t}$ algorithm~\cite{Salam:2008} with a radius parameter $R=0.4$.
These jets are calibrated to the hadronic energy scale
through $\pT$- and $\eta$-dependent scale factors, which are derived from simulation.
An additional uncertainty due to residual differences between simulation and data
is applied in the analysis~\cite{ATLAS-JES-2011}.
Jets are required to have $\pt > 30~(25) \GeV$ and $|\eta| < 2.5$ in the
dilepton (lepton+jets) channel.
The ratio of the scalar sum of the $\pt$ of tracks associated with the jet and the
primary vertex to the scalar sum of the $\pt$ of all tracks associated with the jet
must be at least 0.75 to reject jets from pile-up interactions.
Muons overlapping with jets within $\Delta R<0.4$ are removed and the jet is kept.
The closest jet overlapping with electrons within $\Delta R <0.2$
is removed and the electron is kept. If electrons subsequently still overlap with
any remaining jet within $\Delta R< 0.4$, they are removed.
Information about jets containing $b$~quarks~\cite{ATLAS-BTAG} is also used
in the lepton+jets channel. A neural network combines lifetime-related
information reconstructed from the tracks associated with each jet.
At the chosen working point, the $b$-tagging algorithm has an efficiency of
70\% (20\%/0.7\%) for jets containing $b$~quarks ($c$~quarks/light quarks or gluons)
in a simulated $t\bar{t}$ sample.
The missing transverse momentum $\met$ is calculated using topological clusters of energy deposits in the
calorimeter and corrected for the presence of muons~\cite{ATLAS-MET}.
\section{Event selection in the dilepton channel}
\label{sec:dilepton}
The event selection and background modelling in the dilepton channel is the same as in the
ATLAS measurement of the single top-quark production in the $Wt$-channel~\cite{ATLAS-WtEvidence}.
Candidate events must contain exactly two leptons ($ee$, $\mu\mu$ or $e\mu$) with
opposite electric charge and exactly one jet. At least one
of the leptons in each event must match the corresponding trigger-level object.
No $b$-tagging requirement is made since the dominant background from $\ttbar$
production also contains $b$~quarks. The $\met$ is required to be
greater than $50 \GeV$. In the $ee$ and $\mu\mu$ channels, the invariant mass of the
lepton pair, $m_{\ell\ell}$, is required to be outside the $Z$~boson mass window:
$m_{\ell\ell}<81 \GeV$ or $m_{\ell\ell}>101 \GeV$.
In all three channels, the \Ztt\ background is reduced by a dedicated veto, which
requires the sum of the azimuthal angle differences between each lepton and the
$\met$ vector to be greater than $2.5~\mathrm{rad}$. After all cuts, the acceptance for
signal events with $m_{b^*}=800\GeV$ in which both $W$~bosons decay leptonically
(to either $e$ or $\mu$) is 26\%.
The main background, accounting for 63\% of the total, comes from $\ttbar$
events in which one of the two jets originating from $b$~quarks is not detected.
The second largest background is
from SM $Wt$ production, which has the same final state as the $b^*$-quark signal, and
accounts for 13\% of the total background. Diboson events produced in association with
jets account for 12\% of the total background.
With the exception of single- and diboson samples, these backgrounds are taken from NLO simulation and
are normalised to their NNLO theoretical predictions.
Drell--Yan (DY) events contribute a small background of 7.3\% to the sum of
$ee$ and $\mu\mu$ channel events. The events are taken from the simulation and normalised
to data using a
two-dimensional sideband region with low $\met$ and/or $m_{\ell\ell}$ outside of the $Z$~boson mass window~\cite{ATLAS-WtEvidence}.
The contribution from $\tau\tau$ final states, where both $\tau$ leptons decay
leptonically, is estimated from simulated samples, with the normalisation checked
in an orthogonal data sample obtained by reversing the $Z\to\tautau$ veto cut
described above. $Z\to\tautau$ events account for 0.7\% of the total background.
The small background from jets that are misidentified as primary leptons and from non-prompt leptons (fake dileptons)
is modelled
and normalised using data~\cite{Aad:2010ey}.
It accounts for 4\% of the background.
The predicted event yields for the backgrounds and signal at a few mass
points are compared to data in Table~\ref{tab:dilepyields}.
The $\pt$~distributions of the two leptons and the jet are shown in
Fig.~\ref{fig:kinematicsLL1}.
\begin{table}[htdp]
\begin{center}
\caption{Observed and predicted event yields in the dilepton channel
with normalisation uncertainties. The signal yields are calculated with
$\kappa^b_L=g_L=1$ and $\kappa^b_R=g_R=0$.}
\label{tab:dilepyields}
\begin{tabular}{l*{1}{r@{\ $\pm$\ }l}*{1}}
Process & \multicolumn{2}{c}{Event yield} \\
\hline \hline
$b^*$ (400 \GeV) & 1250 & 170 \\
$b^*$ (600 \GeV) & 211 & 32 \\
$b^*$ (800 \GeV) & 41 & 8 \\
$b^*$ (1000 \GeV) & 8.9 & 1.9 \\
$b^*$ (1200 \GeV) & 2.1 & 0.5 \\
\hline
$Wt$ & 293 &21 \\
$t\bar{t}$ & 1380 &140 \\
Diboson & 255 &63 \\
$Z\to\ee$ & 41 &4 \\
$Z\to\mumu$ & 118 &12 \\
$Z\to\tautau$& 14 &9 \\
Fake dileptons & 90 &90 \\
\hline
Total expected bkg. & 2190 &180\\
Total observed & \multicolumn{2}{c}{ 2259}\\
\end{tabular}
\end{center}
\end{table}
\begin{figure}
\subfigure[]{
\includegraphics[width=0.9\columnwidth]{paper_ll1j_MC11c_v11_LeadingLeptonPt_flat.eps}
\label{fig:ll_lpt}
}
\subfigure[]{
\includegraphics[width=0.9\columnwidth]{paper_ll1j_MC11c_v11_SubLeadingLeptonPt_flat.eps}
\label{fig:ll_lpt2}
}
\subfigure[]{
\includegraphics[width=0.9\columnwidth]{paper_ll1j_MC11c_v11_Jet1Pt_flat.eps}
\label{fig:ll_jpt}
}
\caption{Kinematic distributions comparing data to predictions in the dilepton channel for
\subref{fig:ll_lpt} the leading lepton $\pt^\mathrm{lep1}$,
\subref{fig:ll_lpt2} the sub-leading lepton $\pt^\mathrm{lep2}$ and
\subref{fig:ll_jpt} the jet $\pt^\mathrm{jet}$.
The hatched band shows the uncertainty due to the background normalisation.
The last bin includes overflows. }
\label{fig:kinematicsLL1}
\end{figure}
A discriminating variable that separates the signal from the backgrounds is
$H_\mathrm{T}$, the scalar sum of the transverse momenta of the leptons, jet and
$\met$. The $H_\mathrm{T}$ distribution is shown in Fig.~\ref{fig:ll_ht}.
~
\begin{figure}[!h!tbp]
\centering
\includegraphics[width=0.9\columnwidth]{paper_ll1j_MC11c_v11_HT_logy.eps}
\caption{$H_\mathrm{T}$ distribution for data and background expectation for the dilepton
channel.
The hatched band shows the uncertainty due to the background normalisation.
The signal for a $b^*$-quark mass of $800\GeV$ is also shown.
}
\label{fig:ll_ht}
\end{figure}
\section{Event selection in the lepton+jets channel}
\label{sec:lepjets}
The analysis in the lepton+jets channel follows the same
background modelling strategy
as the cross-section measurement for single top-quark production in the t-channel~\cite{Aad:2012ux}.
Events are required to have either exactly one muon and $\met >25\GeV$ or exactly one
electron and $\met > 30\GeV$, as well as exactly three jets with $\pt>25\GeV$. Exactly
one of the jets is required to be $b$-tagged to reduce backgrounds.
The lepton must also match the corresponding trigger object.
Additional requirements are made to reject multijet events, which tend to have low
\MET\ and a low transverse mass\footnote{The transverse mass, $m_\mathrm{T}^W$, is calculated from the lepton transverse momentum $\pt^\mathrm{lep}$ and the difference of the azimuthal angle, $\Delta \phi$, between the $\met$ and $\pt^\mathrm{lep}$ vector as $m_\mathrm{T}^W = \sqrt{2 \met p_\mathrm{T}^\mathrm{lep} (1-\cos(\Delta \phi(\met,p_\mathrm{T}^\mathrm{lep})))}$} of the lepton--$\met$~system, $m_\mathrm{T}^W$.
In the muon channel events are required to have $m_\mathrm{T}^W+\MET > 60 \GeV$, while in
the electron channel a requirement of $m_\mathrm{T}^W > 30 \GeV$ is made.
The acceptance for signal events
with $m_{b^*}=800\GeV$ in which one of the $W$~bosons decays leptonically
($e$ or $\mu$) and the other hadronically is 9\%.
In this channel, one of the largest backgrounds is $W$+jets production for which
the normalisation and
flavour composition (the heavy-flavour fraction, HF, includes $b$~quarks and $c$~quarks)
are derived from data~\cite{Aad:2012qf}.
The overall normalisation is determined from the charge asymmetry between $W^+$ and
$W^-$ production in three-jet events without the $b$-tag requirement.
The flavour composition is determined in two-jet events by comparing the predicted
$W$+jets yields to data with and without a $b$-tag requirement. The resulting
normalisation and flavour scale factors are then applied to $b$-tagged $W$+3-jets events.
About 37\% of the total background comes from $W$+jets events, including 28\% from
events with heavy flavour.
Backgrounds from $\ttbar$ yield 41\% of the total background and single top-quark production in the
$t$-, $s$- and $Wt$-channel
9\%.
The multijet background is obtained using a data-based
approach by comparing the numbers of events passing loose and tight
lepton identification criteria~\cite{:2012hg}. It accounts for 9\% of the total
background. Smaller backgrounds from $Z$+jets and diboson processes are normalised
to their theoretical predictions and contribute 4\%.
The predicted event yields are compared to data in Table~\ref{tab:lepjetsyield}.
The distributions of the $\pt$ of the highest-$\pt$ jet and $\met$ are shown in
Fig.~\ref{fig:kinematicsLJ}.
~
\begin{table}[!h!tpb]
\begin{center}
\caption{Observed and expected event yields in the lepton+jets channel
with normalisation uncertainties. The signal yields are calculated with
$\kappa^b_L=g_L=1$ and $\kappa^b_R=g_R=0$.
\label{tab:lepjetsyield}}
\begin{tabular}{l*{1}{r@{\ $\pm$\ }l}}
Process
& \multicolumn{2}{c}{Event yield} \\
\hline \hline
$b^*$ (400 \GeV) & 12100 &1600 \\
$b^*$ (600 \GeV) & 1950 &300 \\
$b^*$ (800 \GeV) & 370 &70 \\
$b^*$ (1000 \GeV) & 79 &17 \\
$b^*$ (1200 \GeV) & 20 &5 \\
\hline
$Wt$ & 1660 & 120 \\
single top $s,t$-channel & 1960 & 140 \\
$t\bar{t}$ & 15700 & 1600 \\
$W$+light jets & 3200 & 400 \\
$W$+jets HF & 10900 & 1400 \\
Diboson & 327 & 16 \\
$Z$+jets & 1300 & 800 \\
Multijet & 3500 & 1700 \\
\hline
Total expected bkg. & 38500 & 2900 \\
Total observed & \multicolumn{2}{c}{38175}\\
\hline\hline
\end{tabular}
\end{center}
\end{table}
\begin{figure}
\subfigure[]{
\includegraphics[width=0.9\columnwidth]{ljplotjpt0.eps}
\label{fig:lj_jpt}
}
\subfigure[]{
\includegraphics[width=0.9\columnwidth]{ljplotmet.eps}
\label{fig:lj_met}
}
\caption{Kinematic distributions comparing data to predictions in the lepton+jets
channel for
\subref{fig:lj_jpt} the $\pt^\mathrm{jet1}$ of the highest-$\pt$ jet and
\subref{fig:lj_met} $\met$.
``Other top'' includes $\ttbar$, $s$- and $t$-channel single top-quark production.
The hatched band shows the uncertainty due to the background
normalisation. The last bin includes overflows.
}
\label{fig:kinematicsLJ}
\end{figure}
\begin{figure}[!h!tbp]
\includegraphics[width=0.9\columnwidth]{ljplotbpm.eps}
\caption{
Reconstructed mass distribution for data and background expectation for the
lepton+jets channel.
``Other top'' includes $\ttbar$, $s$- and $t$-channel single top-quark production.
The hatched band shows the uncertainty due to the background normalisation.
The signal for a mass of $800\GeV$ is also shown.
The last bin includes overflows.}
\label{fig:lj_m}
\end{figure}
In the lepton+jets channel it is possible to reconstruct the candidate
$b^*$-quark mass from the decay products. The only missing information is the
neutrino longitudinal momentum, which is set to zero. The resulting reconstructed
mass provides good discrimination between background and signal, as
shown in Fig.~\ref{fig:lj_m}.
\section{Systematic uncertainties}
\label{sec:SYSTEMATICS}
Systematic uncertainties affecting the signal acceptance and the background normalisation
are considered, together with uncertainties affecting the shape of the discriminant
distributions.
The main experimental source of systematic uncertainty comes from the limited
knowledge of the
jet energy scale~\cite{ATLAS-JES-2011}, which carries an uncertainty of 2--7\% parameterised
as a function of jet $\pt$ and $\eta$. The presence of a $b$~quark in
the jet adds an additional uncertainty of 2--5\% to the jet energy scale uncertainty,
depending on the jet $\pt$.
Other jet-related uncertainty sources are the jet energy resolution,
jet reconstruction efficiency
and $b$-tagging efficiency~\cite{ATLAS-BTAG}. Lepton-related uncertainties come from trigger and
identification efficiencies as well as the lepton energy scale and resolution. Event-related
uncertainties are due to the modelling of multiple proton-proton
interactions and the underlying event as well as $\met$~\cite{ATLAS-MET}. The uncertainty on the
integrated luminosity is $3.9\%$~\cite{ATL-LUMI,ATL-CONF-2011-116}.
Simulation uncertainties include modelling of the hard process,
parton shower and hadronisation, and initial- and final-state radiation.
These have been
assessed for the $\ttbar$ background events by comparing different generators ({\sc Powheg} and {\sc MC@NLO}),
different shower models ({\sc Pythia} and {\sc Herwig}),
and for $\ttbar$ and signal events different settings for the amount of additional radiation~\cite{ATL-ISR-FSR}.
Other sources of theoretical uncertainty include the normalisation for
$\ttbar$ ($^{+7\%}_{-10\%}$)~\citep{Moch:2008ai,langenfeld:2009tc,Beneke,Aliev:2010zk},
single top-quark ($\pm 7\%$)~\cite{Kidonakis:2011wy,Kidonakis:2010ux,Kidonakis:2010tc}
and diboson ($\pm 5\%$ with an additional 24\% per extra jet) production~\cite{Aad:2010ey},
as well as the choice of PDF. The latter was assessed using the
CT10~\cite{Lai:2010vv}, MRST~\cite{springerlink:10} and NNPDF~\cite{Ball:2010de} sets.
Additional uncertainties affect the data-driven background estimation.
The uncertainty on the DY background normalisation in the dilepton channel is 10\%
for $ee$ and $\mu\mu$ final
states and 60\% for $\tau\tau$ final states. The uncertainty on the fake-dileptons
normalisation in the dilepton channel is 100\%.
The uncertainty on the $W$+jets normalisation in the lepton+jets channel
is 13\%. The $W$+jets flavour composition has two additional uncertainties: the
HF contribution has a relative uncertainty of 6\%,
and the $W_{bb}$/$W_\mathrm{HF}$ ratio has an uncertainty of 17\%.
The multijet background normalisation in
the lepton+jets channel has an uncertainty of 50\%.
\section{Statistical analysis}
\label{sec:limit}
Both the $H_\mathrm{T}$~distribution in the dilepton channel and the reconstructed
mass distribution in the lepton+jets channel show good agreement between the
data and the background model. These two discriminants are used to set limits on the
$b^*$-quark signal using a Bayesian analysis technique~\cite{Caldwell:2008fw}. The likelihood function is defined as
\begin{equation}
{\cal L}(\mathrm{data}|\sigma_{b^*}) = \prod_{k}
\frac{\mu_k^{n_k} e^{-\mu_k}}{n_k!}\prod_{i}G_i\, ,
\label{eqn:lhoodtemp}
\end{equation}
where $k$ is the index of the discriminant template bin, running over both analysis
channels; $\mu_k = s_k + b_k$ is the sum of predicted signal and background yields;
$n_k$ is the observed yield and
$G_i$ is a Gaussian prior for the $i$th systematic uncertainty.
A flat prior is assumed for the signal cross-section.
Upper limits on the $b^*$-quark production cross-section times branching ratio to $Wt$
are set at the 95\% credibility level (CL) for a series of $b^*$~masses at
$100\GeV$ intervals.
\begin{figure}[!h!tbp]
\centering
\includegraphics[width=8.0cm]{MC11c_v11HTcomblimit_vsMass.eps}
\caption{Expected and observed limits at the 95\% CL as a function of the $b^*$-quark mass.
Also shown is the theory prediction for $b^*$-quark production with couplings
$\kappa^b_L=g_L=1$ and $\kappa^b_R=g_R=0$, including PDF and scale uncertainties.}
\label{fig:limit}
\end{figure}
The observed and expected cross-section limits as
a function of the $b^*$-quark mass for the left-handed coupling scenario
($\kappa^b_L=g_L=1$ and $\kappa^b_R=g_R=0$) are shown in Fig.~\ref{fig:limit}, where the
expected limit and its uncertainty
are derived from ensembles of background-only pseudo-datasets. The intersection of the
theoretical cross-section and the observed (expected) cross-section limit defines the
observed (expected) $b^*$-quark mass limit.
The observed lower limit on the $b^*$-quark mass for this left-handed coupling
scenario is $870\GeV$ with an expectation of
$910\GeV$. When considering only the dilepton channel, the observed (expected) limit on
the $b^*$-quark mass is $800\GeV$ ($820\GeV$); for the lepton+jets channel, the limits are $800\GeV$ ($830\GeV$).
Limits are also computed for models with right-handed and vector-like couplings of the
$b^{*}$~quark.
Setting $\kappa^b_L=g_L=0$ and $\kappa^b_R=g_R=1$,
the observed lower mass limit is $920\GeV$ with an expected limit of $950\GeV$.
Setting $\kappa^b_L=\kappa^b_R=g_L=g_R=1$,
the observed lower mass limit is $1030\GeV$ with an expected limit of $1030\GeV$.
At each mass point, the corresponding cross section is parameterised as a function of the
couplings $\kappa^b_{L,R}$ and
$g_{L,R}$ in order to extract coupling limits in each of the three $b^*$-quark coupling
scenarios.
The resulting limit contours are shown in Fig.~\ref{fig:limit_2d}. The coupling limits
increase as the theoretical cross-section decreases with $b^*$~mass, except for the region
between $400\GeV$ and $500\GeV$ where the backgrounds decrease rapidly with increasing mass
(see Figs.~\ref{fig:ll_ht} and~\ref{fig:lj_m}).
\begin{figure}[!h!tbp]
\centering
\subfigure[]{
\includegraphics[width=8.0cm]{MC11c_v11HTcombobv_limit_vsMass3D.eps}
\label{fig:limit_2dL}
}
\subfigure[]{
\includegraphics[width=8.0cm]{MC11c_v11HTcombobv_limit_vsMass3DRight.eps}
\label{fig:limit_2dR}
}
\subfigure[]{
\includegraphics[width=8.0cm]{MC11c_v11HTcombobv_limit_vsMass3DVLQ.eps}
\label{fig:limit_2dVLQ}
}
\caption{Limit contours at the 95\% CL as a function of the coupling parameters
for several different $b^*$-quark masses, for
\subref{fig:limit_2dL} left-handed $b^*$~quarks,
\subref{fig:limit_2dR} right-handed $b^*$~quarks and
\subref{fig:limit_2dVLQ} vector-like $b^*$~quarks.
}
\label{fig:limit_2d}
\end{figure}
\section{Summary}
\label{sec:conclusion}
A search for a singly produced excited $b^*$-quark
in $4.7~\ifb$ of data collected with the ATLAS detector in $pp$ collisions at
$\sqrt{s}=7\TeV$ has been presented.
This is the first search for excited-quarks coupling to the third generation.
It considers the
dilepton and lepton+jets final states. Limits are computed as a function of the
$b^* g b$ and $b^* W t$~couplings in three different scenarios.
For purely left-handed couplings and unit strength chromomagnetic coupling,
$b^*$~quarks with mass below 870 GeV are excluded at the 95\% credibility level.
\input{Acknowledgements}
\bibliographystyle{atlasnote}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 8,426
|
{"url":"https:\/\/ctan.org\/ctan-ann\/id\/mailman.33.1316369417.2268.ctan-ann@dante.de","text":"# CTAN Update: bidi\n\nDate: September 18, 2011 8:10:14 PM CEST\nOn Sun, 18 Sep 2011 Vafa Khalighi submitted an update to the bidi package. Summary description: Support for bidirectional typesetting in plain TeX and LaTeX. License type: lppl Announcement text:\n1) Documentation for usage of bidi package in Plain XeTeX is added. 2) New command \\eqnewif is added that makes old and new bidi conditionals identical, e.g. \\if at rl and \\if at RTL. Generally this new command \\eqnewif makes two conditionals identical such that whenever first conditional is true, the second one is also true and whever the first conditional is false, the second contional is also false. For more details see the documentation. 3) Support for lettrine package is added. 4) Support for the new package paracol by Hiroshi Nakashima is added.\nThis package is located at http:\/\/mirror.ctan.org\/macros\/latex\/contrib\/bidi . More information is at http:\/\/www.ctan.org\/pkg\/bidi (if the package is new it may take a day for that information to appear). We are supported by the TeX Users Group http:\/\/www.tug.org . Please join a users group; see http:\/\/www.tug.org\/usergroups.html .\nThanks for the upload. For the CTAN Team Rainer Sch\u00f6pf\n\n## bidi \u2013 Bidirectional typesetting in plain TeX and LaTeX, using XeTeX\n\nA convenient interface for typesetting bidirectional texts with plain and . The package includes adaptations for use with many other commonly-used packages.\n\n Package bidi Version 35.10 2019-10-19 Copyright 2007\u20132019 Vafa Khalighi Maintainer bidi-tex GitHub OrganisationVafa Khalighi\n\nmore","date":"2019-10-20 03:31:01","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\": 0.8894818425178528, \"perplexity\": 11568.690682139757}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"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-2019-43\/segments\/1570986702077.71\/warc\/CC-MAIN-20191020024805-20191020052305-00113.warc.gz\"}"}
| null | null |
Chronic dry eyes cause the upper surface of the eyeball to be dry due to lack of moisture. Often this condition is accompanied by soreness or inflammation with bouts of irritating itchiness. Chronic dry eye relief can only be obtained, if we can diagnose and prevent the causes.
If you want chronic dry eye relief, you should be aware of the below-listed causes for the ailment.
Eye stress: If your eyes get strained due to over stressing, then it might lead to evaporation of the natural lubricant present underneath the eyelids, encircling the eyeball. In some case, the eyes can also become light sensitive and can react or have blurry vision. Looking at the desktop monitor or your mobile phone or any electronic device for an extended period can give you eye strain. In such cases, chronic dry eye relief can be obtained through heat compressors.
Allergens: Allergens like pet, chemicals, pollen particles, foreign objects or dust particles can inflict an itchy feeling when they come in contact with the eye surface. These can act as triggers for dry eyes. Avoiding polluted places is considered to be a safety measure in ensuring chronic dry eye relief.
Hormonal imbalances: Many times it is observed that women taking birth control pills, or those during pregnancy or menopause undergo hormonal changes that affect their body functionality including the eyes, thereby making them dry.
Eye infection: Certain eye infections like blepharitis and conjunctivitis can lead to dry eye syndrome. Eye infection often leads to eye scratching. In such cases of dry, itchy eyes, treatment in the form of artificial drops according to the doctor's prescription can be used.
Contact lenses: Contact lenses are also one among the top causes of dry eyes. In many cases, these lenses affect the cornea and tend to swell the eyes, leading to itchiness and dryness as the water glands of the eyes are affected.
Dry eyes can create a lot of difficulty in vision, but it can be cured and prevented as well. The prevention, however, is only possible by knowing the reasons behind the dryness and taking required measures in the right direction.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 8,652
|
Tanjung Badur är en udde i Indonesien. Den ligger i provinsen Banten, i den västra delen av landet, km väster om huvudstaden Jakarta.
Terrängen inåt land är huvudsakligen platt, men söderut är den kuperad. Havet är nära Tanjung Badur åt nordväst. Den högsta punkten i närheten är Pasir Batubale, meter över havet, km söder om Tanjung Badur. Runt Tanjung Badur är det ganska tätbefolkat, med invånare per kvadratkilometer.
Tropiskt regnskogsklimat råder i trakten. Årsmedeltemperaturen i trakten är °C. Den varmaste månaden är september, då medeltemperaturen är °C, och den kallaste är maj, med °C. Genomsnittlig årsnederbörd är millimeter. Den regnigaste månaden är december, med i genomsnitt mm nederbörd, och den torraste är september, med mm nederbörd.
Kommentarer
Källor
Halvöar i Banten
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 2,673
|
\section{Auxiliary Lemmas}\label{appen:A}
\begin{lemma}\label{aux:lem:5}
Suppose $\{\varphi_i\}_i$ is a positive sequence that satisfies $\lim\limits_{i\rightarrow\infty}i(1 - \varphi_{i-1}/\varphi_i) = \varphi$. Then, for any $p \geq 0$, we have
\begin{equation*}
\lim\limits_{i\rightarrow\infty}i\rbr{1 - \frac{\varphi_{i-1}^p}{\varphi_i^p}} = p\cdot \varphi.
\end{equation*}
\end{lemma}
\begin{proof}
By the condition, we know
\begin{equation*}
\frac{\varphi_{i-1}}{\varphi_i} = 1 - \frac{\varphi}{i} + o\rbr{\frac{1}{i}}.
\end{equation*}
Thus, we have
\begin{equation*}
i\rbr{1 - \frac{\varphi_{i-1}^p}{\varphi_i^p}} = i\rbr{1 - \cbr{1 - \frac{\varphi}{i} + o\rbr{\frac{1}{i}}}^p } = p\varphi + o(1).
\end{equation*}
This completes the proof.
\end{proof}
\begin{lemma}\label{aux:lem:1}
Let $\{\varphi_i\}_i$ be a positive sequence. If $\lim\limits_{i\rightarrow\infty}i(1 - \varphi_{i-1}/\varphi_i) = \varphi<0$, then $\varphi_i \rightarrow 0$ as $i\rightarrow \infty$.
\end{lemma}
\begin{proof}
By Lemma \ref{aux:lem:5}, we know for any positive constant $p$,
\begin{equation*}
\lim\limits_{i\rightarrow \infty} i \rbr{1-\frac{\varphi_{i-1}^p}{\varphi_i^p}} = p\cdot \varphi.
\end{equation*}
Choosing $p$ large enough such that $p\varphi <-1$, the Raabe's test indicates that $\sum_{i=0}^{\infty}\varphi_i^p<\infty$. This implies $\varphi_i \rightarrow 0$ and we complete the proof.
\end{proof}
\begin{lemma}\label{aux:lem:2}
Let $\{\phi_i\}_i$, $\{\varphi_i\}_i$, $\{\sigma_i\}_i$ be three positive sequences. Suppose we have
\begin{equation}\label{pequ:A1}
\lim\limits_{i\rightarrow \infty} i\rbr{1 - \frac{\phi_{i-1}}{\phi_i}} = \phi,\quad \quad\lim\limits_{i\rightarrow\infty}\varphi_i = 0, \quad\quad \lim\limits_{i\rightarrow \infty} i\varphi_i = \tilde{\varphi}
\end{equation}
for a constant $\phi$ and a (possibly infinite) constant $\tilde{\varphi} \in(0, \infty]$. For any $l\geq 1$, if we further have
\begin{equation*}
\sum_{k=1}^{l}\sigma_k + \frac{\phi}{\tilde{\varphi}}>0,
\end{equation*}
then the following results hold as $t\rightarrow \infty$
\begin{equation}\label{pequ:A2}
\begin{aligned}
& \frac{1}{\phi_t}\sum_{i=0}^{t}\prod_{j = i+1}^t\prod_{k=1}^{l}(1-\varphi_j\sigma_k)\varphi_i\phi_i \longrightarrow \frac{1}{\sum_{k=1}^{l}\sigma_k + \phi/\tilde{\varphi}},\\
& \frac{1}{\phi_t}\cbr{\sum_{i=0}^{t}\prod_{j = i+1}^t\prod_{k=1}^{l}(1-\varphi_j\sigma_k)\varphi_i\phi_i a_i + b\cdot \prod_{j = 0}^t\prod_{k=1}^{l}(1-\varphi_j\sigma_k) } \longrightarrow 0,
\end{aligned}
\end{equation}
where the second result holds for any constant $b$ and any sequence $\{a_t\}_t$ such that $a_t\rightarrow 0$.
\end{lemma}
\begin{proof}
For any scalar $A$, we have
\begin{align*}
& \frac{1}{\phi_t}\sum_{i=0}^{t}\prod_{j = i+1}^t\prod_{k=1}^{l}(1-\varphi_j\sigma_k)\varphi_i\phi_i - A \\
& = \frac{1}{\phi_t}\prod_{j = 0}^t\prod_{k=1}^{l}(1-\varphi_j\sigma_k)\cbr{\sum_{i=0}^{t}\prod_{j = 0}^i\prod_{k=1}^{l}(1-\varphi_j\sigma_k)^{-1}\varphi_i\phi_i - A\phi_t\prod_{j = 0}^t\prod_{k=1}^{l}(1-\varphi_j\sigma_k)^{-1} }.
\end{align*}
For the last term, we have
\begin{align*}
& A\phi_t\prod_{j = 0}^t\prod_{k=1}^{l}(1-\varphi_j\sigma_k)^{-1}\\
& = \sum_{i=1}^{t} \rbr{A\phi_i \prod_{j = 0}^i\prod_{k=1}^{l}(1-\varphi_j\sigma_k)^{-1} - A\phi_{i-1}\prod_{j = 0}^{i-1}\prod_{k=1}^{l}(1-\varphi_j\sigma_k)^{-1} } + A\phi_0\prod_{k=1}^{l}(1-\varphi_0\sigma_k)^{-1}\\
& = \sum_{i=1}^{t}A\phi_i \prod_{j = 0}^i\prod_{k=1}^{l}(1-\varphi_j\sigma_k)^{-1}\cbr{1 - \frac{\phi_{i-1}}{\phi_i}\prod_{k=1}^{l}(1-\varphi_i\sigma_k)} + A\phi_0\prod_{k=1}^{l}(1-\varphi_0\sigma_k)^{-1}.
\end{align*}
Combining the above two displays, we obtain
\begin{align}\label{pequ:A3}
& \frac{1}{\phi_t}\sum_{i=0}^{t}\prod_{j = i+1}^t\prod_{k=1}^{l}(1-\varphi_j\sigma_k)\varphi_i\phi_i - A \nonumber\\
& = \frac{1}{\phi_t}\prod_{j = 0}^t\prod_{k=1}^{l}(1-\varphi_j\sigma_k)\bigg\{\sum_{i=1}^{t}\prod_{j = 0}^i\prod_{k=1}^{l}(1-\varphi_j\sigma_k)^{-1}\phi_i\cbr{\varphi_i - A\rbr{1 - \frac{\phi_{i-1}}{\phi_i}\prod_{k=1}^{l}(1-\varphi_i\sigma_k)} } \nonumber \\
& \quad\quad + \phi_0\prod_{k=1}^{l}(1-\varphi_0\sigma_k)^{-1}\rbr{\varphi_0 - A}\bigg\} .
\end{align}
We aim to select $A$ such that the middle term in \eqref{pequ:A3} is small. By \eqref{pequ:A1}, we know
\begin{equation*}
\frac{\phi_{i-1}}{\phi_i} = 1 - \frac{\phi}{i} + o\rbr{\frac{1}{i}} = 1 - \frac{\phi}{\tilde{\varphi}}\cdot \varphi_i + o(\varphi_i),
\end{equation*}
where the second equality is due to $1/(i\varphi_i) = 1/\tilde{\varphi} + o(1)$ (which is true even if $\tilde{\varphi} = \infty$). Furthermore,
\begin{equation*}
\prod_{k=1}^{l}(1-\varphi_i\sigma_k) = 1 - \varphi_i\sum_{k=1}^{l}\sigma_k + o(\varphi_i).
\end{equation*}
Combining the above two displays, we have
\begin{align}\label{pequ:A4}
& \varphi_i - A\rbr{1 - \frac{\phi_{i-1}}{\phi_i}\prod_{k=1}^{l}(1-\varphi_i\sigma_k)} = \varphi_i - A\cbr{1 - \rbr{1 - \frac{\phi}{\tilde{\varphi}}\cdot\varphi_i + o(\varphi_i)}\rbr{1 - \varphi_i\sum_{k=1}^{l}\sigma_k + o(\varphi_i)} } \nonumber\\
& = \varphi_i - A\rbr{\frac{\phi}{\tilde{\varphi}} + \sum_{k=1}^l\sigma_k} \varphi_i +o(\varphi_i).
\end{align}
Thus, we let $A = 1/(\sum_{k=1}^l\sigma_k + \phi/\tilde{\varphi})$ and \eqref{pequ:A3} leads to
\begin{align*}
\frac{1}{\phi_t}\sum_{i=0}^{t}\prod_{j = i+1}^t\prod_{k=1}^{l}&(1-\varphi_j\sigma_k)\varphi_i\phi_i - \frac{1}{\sum_{k=1}^l\sigma_k + \phi/\tilde{\varphi}} \\
& = \frac{1}{\phi_t}\prod_{j = 0}^t\prod_{k=1}^{l}(1-\varphi_j\sigma_k)\bigg\{\sum_{i=1}^{t}\prod_{j = 0}^i\prod_{k=1}^{l}(1-\varphi_j\sigma_k)^{-1}\phi_i\cdot o(\varphi_i) \\
&\quad\quad + \phi_0\prod_{k=1}^{l}(1-\varphi_0\sigma_k)^{-1}\rbr{\varphi_0 - \frac{1}{\sum_{k=1}^l\sigma_k + \phi/\tilde{\varphi}}}\bigg\}.
\end{align*}
Comparing the above display with \eqref{pequ:A2}, we notice that the first result in \eqref{pequ:A2} is implied by the second result. Thus, it suffices to prove the second result. We define
\begin{equation}\label{pequ:A5}
\Psi_t = \frac{1}{\phi_t}\cbr{\sum_{i=0}^{t}\prod_{j = i+1}^t\prod_{k=1}^{l}(1-\varphi_j\sigma_k)\varphi_i\phi_i a_i + b\cdot \prod_{j = 0}^t\prod_{k=1}^{l}(1-\varphi_j\sigma_k) },
\end{equation}
then
\begin{align*}
\Psi_t & = \frac{1}{\phi_t}\cbr{ \varphi_t\phi_ta_t +\prod_{k=1}^{l}(1-\varphi_t\sigma_k)\rbr{\sum_{i=0}^{t-1}\prod_{j = i+1}^{t-1}\prod_{k=1}^{l}(1-\varphi_j\sigma_k)\varphi_i\phi_i a_i + b\cdot \prod_{j = 0}^{t-1}\prod_{k=1}^{l}(1-\varphi_j\sigma_k)} }\\
&\stackrel{\mathclap{\eqref{pequ:A5}}}{=} \;\; \frac{\phi_{t-1}}{\phi_t}\prod_{k=1}^{l}(1-\varphi_t\sigma_k)\Psi_{t-1} + \varphi_t a_t.
\end{align*}
By \eqref{pequ:A4}, we know that
\begin{equation*}
\frac{\phi_{t-1}}{\phi_t}\prod_{k=1}^{l}(1-\varphi_t\sigma_k) = 1 - \rbr{\frac{\phi}{\tilde{\varphi}} + \sum_{k=1}^l\sigma_k}\cdot \varphi_t + o(\varphi_t).
\end{equation*}
Since $\sum_{k=1}^l\sigma_k + \phi/\tilde{\varphi}>0$, we immediately know that for a constant $c>0$ and for all large enough $t$,
\begin{equation*}
|\Psi_t| \leq (1 - c\varphi_t) |\Psi_{t-1}| + \varphi_t |a_t|.
\end{equation*}
Let $t_1$ be a fixed integer. We apply the above inequality recursively and have for any $t\geq t_1+1$,
\begin{equation*}
|\Psi_t| \leq \prod_{i = t_1+1}^{t}(1 - c\varphi_i)|\Psi_{t_1}| + \sum_{i=t_1+1}^{t}\prod_{j = i+1}^t(1 - c\varphi_j)\varphi_i|a_i|.
\end{equation*}
For any $\epsilon>0$, since $a_i\rightarrow 0$, we select $t_1$ such that $|a_i|\leq \epsilon$, for all $i\geq t_1$. Then, the above inequality leads to
\begin{align*}
|\Psi_t| & \leq \prod_{i = t_1+1}^{t}(1 - c\varphi_i)|\Psi_{t_1}| + \epsilon \sum_{i=t_1+1}^{t}\prod_{j = i+1}^t(1 - c\varphi_j)\varphi_i = \prod_{i = t_1+1}^{t}(1 - c\varphi_i)|\Psi_{t_1}| + \frac{\epsilon}{c} \big\{1 - \prod_{j=t_1+1}^{t}(1 - c\varphi_j)\big\} \\
& \leq |\Psi_{t_1}|\exp\rbr{-c\sum_{i=t_1+1}^{t}\varphi_i} + \frac{\epsilon}{c}.
\end{align*}
Since $n\varphi_i \rightarrow \tilde{\varphi}\in(0, \infty]$, $\sum_{t}\varphi_t \rightarrow \infty$. Thus, for the above $\epsilon>0$, there exists $t_2>t_1$ such that $|\Psi_{t_1}|\exp\rbr{-c\sum_{i=t_1+1}^{t}\varphi_i}\leq \epsilon/c$, $\forall t\geq t_2$, which implies $|\Psi_t|\leq 2\epsilon/c$. This means $|\Psi_t|\rightarrow0$ and we complete the proof.
\end{proof}
\begin{lemma}\label{aux:lem:3}
For any scalars $a, b$, we have $P\rbr{a<{\mathcal N}(0,1)\leq b} \leq b-a$. Furthermore, if $0<a\leq b$, then $P\rbr{a<{\mathcal N}(0,1)\leq b} \leq b/a-1$.
\end{lemma}
\begin{proof}
The first part of statement holds naturally due to the fact that the density function of the standard Gaussian $\exp(-t^2/2)/\sqrt{2\pi}\leq 1$ for any $t\in\mathbb{R}$. Moreover, for $0<a\leq b$, we have
\begin{multline*}
P\rbr{a<{\mathcal N}(0,1)\leq b} = \int_{a}^{b} \frac{1}{\sqrt{2\pi}}\exp(-t^2/2)\; dt \leq \frac{b-a}{\sqrt{2\pi}}\exp(-a^2/2) \\
= \rbr{\frac{b}{a}-1}\frac{a}{\sqrt{2\pi}}\exp(-a^2/2) \leq \frac{b}{a}-1,
\end{multline*}
where the last inequality is due to the fact that $a\exp(-a^2/2) \leq 1$ for all $a$; the first inequality uses $0<a\leq b$. This completes the proof.
\end{proof}
\begin{lemma}\label{aux:lem:4}
Let $A_t$, $B_t$, $C_t$ be three variables depending on the index $t$; let $\Phi(z) = P({\mathcal N}(0,1)\leq z)$ be the cumulative distribution function of standard Gaussian variable. Suppose for the index $t$,
\begin{equation}\label{aequ:1}
\sup_{z\in\mathbb{R}}\abr{P\rbr{A_t \leq z} - \Phi(z)} \leq a_t, \quad |B_t|\leq b_t,\quad |C_t|\leq c_t\quad \text{almost surely}
\end{equation}
where $a_t, b_t\geq 0$ and $0\leq c_t< 1$. Then, we have
\begin{equation*}
\sup_{z\in\mathbb{R}}\abr{P\rbr{\frac{A_t+B_t}{\sqrt{1+C_t}}\leq z } - \Phi(z) } \leq a_t + b_t + \frac{c_t}{\sqrt{1-c_t}}.
\end{equation*}
\end{lemma}
\begin{proof}
We only prove the result for any $z> 0$. The result of $z\leq0$ can be shown in the same way. We know from \eqref{aequ:1} that
\begin{equation*}
\frac{A_t-b_t}{\sqrt{1+c_t}} \leq \frac{A_t + B_t}{\sqrt{1+C_t}} \leq \frac{A_t+b_t}{\sqrt{1-c_t}}\quad\quad \text{almost surely}.
\end{equation*}
Therefore, we have
\begin{align*}
& P\rbr{\frac{A_t + B_t}{\sqrt{1+C_t}} \leq z}\geq P\rbr{\frac{A_t + b_t}{\sqrt{1-c_t}} \leq z} = P\rbr{A_t\leq z(1-c_t)^{1/2}-b_t} \stackrel{\eqref{aequ:1}}{\geq} \Phi(z(1-c_t)^{1/2}-b_t) - a_t \\
& = \Phi(z) - P\rbr{z(1-c_t)^{1/2}-b_t<{\mathcal N}(0,1)\leq z(1-c_t)^{1/2}} - P\rbr{z(1-c_t)^{1/2}<{\mathcal N}(0,1)\leq z} -a_t \quad (z\geq0)\\
& \geq \Phi(z) - b_t - \rbr{\frac{1}{\sqrt{1-c_t}} -1} - a_t\quad (\text{by Lemma \ref{aux:lem:3}})\\
& \geq \Phi(z) - b_t - \frac{c_t}{\sqrt{1-c_t}} -a_t.
\end{align*}
On the other hand, we have
\begin{align*}
& P\rbr{\frac{A_t + B_t}{\sqrt{1+C_t}} \leq z}\leq P\rbr{\frac{A_t - b_t}{\sqrt{1+c_t}} \leq z}= P\rbr{A_t\leq z(1+c_t)^{1/2}+b_t}\stackrel{\eqref{aequ:1}}{\leq}\Phi(z(1+c_t)^{1/2}+b_t) + a_t\\
& = \Phi(z) + P\rbr{z<{\mathcal N}(0,1)\leq z(1+c_t)^{1/2}} + P\rbr{z(1+c_t)^{1/2}< {\mathcal N}(0,1)\leq z(1+c_t)^{1/2}+b_t} + a_t\\
& \leq \Phi(z) + \rbr{(1+c_t)^{1/2}-1} + b_t+a_t \quad (\text{by Lemma \ref{aux:lem:3}})\\
& \leq \Phi(z) + c_t + b_t + a_t.
\end{align*}
Combining the above two displays completes the proof.
\end{proof}
\section{Proofs of Section \ref{sec:3}}
\subsection{Proof of Lemma \ref{lem:2}}\label{pf:lem:2}
By \eqref{equ:AL:der} and the definition of $(\Delta{\boldsymbol{x}}_t, \Delta{\boldsymbol{\lambda}}_t)$, we have
\begin{align}\label{pequ:1}
&\begin{pmatrix}
\nabla_{{\boldsymbol{x}}}\mathcal{L}_{\mu, \nu}^t\\
\nabla_{{\boldsymbol{\lambda}}}\mathcal{L}_{\mu, \nu}^t
\end{pmatrix}^T\begin{pmatrix}
\Delta{\boldsymbol{x}}_t\\
\Delta{\boldsymbol{\lambda}}_t
\end{pmatrix} \nonumber\\
& \stackrel{\mathclap{\eqref{equ:AL:der}}}{=} \begin{pmatrix}
\Delta{\boldsymbol{x}}_t\\
\Delta{\boldsymbol{\lambda}}_t
\end{pmatrix}^T\begin{pmatrix}
I + \nu \nabla_{{\boldsymbol{x}}}^2\mathcal{L}_t & \mu G_t^T\\
\nu G_t & I
\end{pmatrix}\begin{pmatrix}
\nabla_{{\boldsymbol{x}}}\mathcal{L}_t\\
c_t
\end{pmatrix} \nonumber\\
& \stackrel{\mathclap{\eqref{equ:Newton}}}{=} - \begin{pmatrix}
\Delta{\boldsymbol{x}}_t\\
\Delta{\boldsymbol{\lambda}}_t
\end{pmatrix}^T\begin{pmatrix}
I + \nu \nabla_{{\boldsymbol{x}}}^2\mathcal{L}_t & \mu G_t^T\\
\nu G_t & I
\end{pmatrix}\begin{pmatrix}
B_t & G_t^T\\
G_t & {\boldsymbol 0}
\end{pmatrix}\begin{pmatrix}
\Delta{\boldsymbol{x}}_t\\
\Delta{\boldsymbol{\lambda}}_t
\end{pmatrix} \nonumber\\
& = - \begin{pmatrix}
\Delta{\boldsymbol{x}}_t\\
\Delta{\boldsymbol{\lambda}}_t
\end{pmatrix}^T\begin{pmatrix}
B_t + \nu \nabla_{{\boldsymbol{x}}}^2\mathcal{L}_tB_t + \mu G_t^TG_t & G_t^T + \nu\nabla_{{\boldsymbol{x}}}^2\mathcal{L}_tG_t^T\\
G_t + \nu G_tB_t & \nu G_tG_t^T
\end{pmatrix}\begin{pmatrix}
\Delta{\boldsymbol{x}}_t\\
\Delta{\boldsymbol{\lambda}}_t
\end{pmatrix} \nonumber\\
& \stackrel{\mathclap{\eqref{equ:upper:bound}}}{\leq} -\Delta{\boldsymbol{x}}_t^TB_t\Delta{\boldsymbol{x}}_t + \nu\Upsilon_B\Upsilon_u\|\Delta{\boldsymbol{x}}_t\|^2 - \mu\|G_t\Delta{\boldsymbol{x}}_t\|^2 - 2\Delta{\boldsymbol{\lambda}}_t^TG_t\Delta{\boldsymbol{x}}_t \nonumber\\
& \quad + \nu(\Upsilon_u + \Upsilon_B)\|\Delta{\boldsymbol{x}}_t\|\|G_t^T\Delta{\boldsymbol{\lambda}}_t\| - \nu\|G_t^T\Delta{\boldsymbol{\lambda}}_t\|^2 \quad \text{(also use Assumption \ref{ass:1})} \nonumber\\
& \stackrel{\eqref{equ:Newton}}{\leq} - \Delta{\boldsymbol{x}}_t^TB_t\Delta{\boldsymbol{x}}_t + \nu\Upsilon_B\Upsilon_u\|\Delta{\boldsymbol{x}}_t\|^2 - \mu\|c_t\|^2 + 2c_t^T\Delta{\boldsymbol{\lambda}}_t +\frac{\nu(\Upsilon_u+\Upsilon_B)^2}{2}\|\Delta{\boldsymbol{x}}_t\|^2 \nonumber\\
&\quad - \frac{\nu}{2}\|G_t^T\Delta{\boldsymbol{\lambda}}_t\|^2 \quad \text{(also use Young's inequality)} \nonumber\\
& \leq - \Delta{\boldsymbol{x}}_t^TB_t\Delta{\boldsymbol{x}}_t + \nu(\Upsilon_B+\Upsilon_u)^2\|\Delta{\boldsymbol{x}}_t\|^2 - \mu\|c_t\|^2 + \frac{8}{\nu\gamma_{G}}\|c_t\|^2 + \frac{\nu\gamma_{G}}{8}\|\Delta{\boldsymbol{\lambda}}_t\|^2 \nonumber\\
& \quad - \frac{\nu\gamma_{G}}{4}\|\Delta{\boldsymbol{\lambda}}_t\|^2 - \frac{\nu}{4}\|G_t^T\Delta{\boldsymbol{\lambda}}_t\|^2\quad \text{(Young's inequality and Assumption \ref{ass:1})} \nonumber\\
& \stackrel{\mathclap{\eqref{equ:Newton}}}{=} - \Delta{\boldsymbol{x}}_t^TB_t\Delta{\boldsymbol{x}}_t + \nu(\Upsilon_B+\Upsilon_u)^2\|\Delta{\boldsymbol{x}}_t\|^2 - \rbr{\mu - \frac{8}{\nu\gamma_{G}}}\|c_t\|^2 - \frac{\nu\gamma_{G}}{8}\|\Delta{\boldsymbol{\lambda}}_t\|^2 - \frac{\nu}{4}\|B_t\Delta{\boldsymbol{x}}_t + \nabla_{{\boldsymbol{x}}}\mathcal{L}_t\|^2 \nonumber\\
& \leq - \Delta{\boldsymbol{x}}_t^TB_t\Delta{\boldsymbol{x}}_t + \nu(\Upsilon_B+\Upsilon_u)^2\|\Delta{\boldsymbol{x}}_t\|^2 - \rbr{\mu - \frac{8}{\nu\gamma_{G}}}\|c_t\|^2 - \frac{\nu\gamma_{G}}{8}\|\Delta{\boldsymbol{\lambda}}_t\|^2 - \frac{\nu}{8}\|\nabla_{{\boldsymbol{x}}}\mathcal{L}_t\|^2 + \frac{\nu\Upsilon_B^2}{4}\|\Delta{\boldsymbol{x}}_t\|^2 \nonumber\\
& \leq - \Delta{\boldsymbol{x}}_t^TB_t\Delta{\boldsymbol{x}}_t +2\nu(\Upsilon_B+\Upsilon_u)^2\|\Delta{\boldsymbol{x}}_t\|^2 - \rbr{\mu - \frac{8}{\nu\gamma_{G}}}\|c_t\|^2 - \frac{\nu\gamma_{G}}{8}\|\Delta{\boldsymbol{\lambda}}_t\|^2 - \frac{\nu}{8}\|\nabla_{{\boldsymbol{x}}}\mathcal{L}_t\|^2,
\end{align}
where the second last inequality uses $\|B_t\Delta{\boldsymbol{x}}_t + \nabla_{{\boldsymbol{x}}}\mathcal{L}_t\|^2 \geq \|\nabla_{{\boldsymbol{x}}}\mathcal{L}_t\|^2/2 - \|B_t\Delta{\boldsymbol{x}}_t\|^2\geq \|\nabla_{{\boldsymbol{x}}}\mathcal{L}_t\|^2/2 - \Upsilon_B^2\|\Delta{\boldsymbol{x}}_t\|^2$.
To further simplify \eqref{pequ:1}, we decompose the step $\Delta{\boldsymbol{x}}_t$ as
\begin{equation*}
\Delta{\boldsymbol{x}}_t = \Delta{\boldsymbol{u}}_t + \Delta{\boldsymbol{v}}_t,\quad \text{ where } \Delta{\boldsymbol{u}}_t \in \text{span}(G_t^T) \text{ and } G_t\Delta{\boldsymbol{v}}_t = {\boldsymbol 0}.
\end{equation*}
Then, we have
\begin{align*}
& - \Delta{\boldsymbol{x}}_t^TB_t\Delta{\boldsymbol{x}}_t +2\nu(\Upsilon_B+\Upsilon_u)^2\|\Delta{\boldsymbol{x}}_t\|^2 \\
& = -\Delta{\boldsymbol{u}}_t^TB_t\Delta{\boldsymbol{u}}_t - 2\Delta{\boldsymbol{u}}_t^TB_t\Delta{\boldsymbol{v}}_t - \Delta{\boldsymbol{v}}_t^TB_t\Delta{\boldsymbol{v}}_t +2\nu(\Upsilon_B+\Upsilon_u)^2\|\Delta{\boldsymbol{x}}_t\|^2 \\
& \leq \Upsilon_B\|\Delta{\boldsymbol{u}}_t\|^2 + 2\Upsilon_B\|\Delta{\boldsymbol{u}}_t\|\|\Delta{\boldsymbol{v}}_t\| - \gamma_{RH}\|\Delta{\boldsymbol{v}}_t\|^2 +2\nu(\Upsilon_B+\Upsilon_u)^2\|\Delta{\boldsymbol{x}}_t\|^2 \quad \text{(Assumption \ref{ass:1})}\\
&\leq \rbr{\Upsilon_B + \frac{2\Upsilon_B^2}{\gamma_{RH}}}\|\Delta{\boldsymbol{u}}_t\|^2 - \frac{\gamma_{RH}}{2}\|\Delta{\boldsymbol{v}}_t\|^2 +2\nu(\Upsilon_B+\Upsilon_u)^2\|\Delta{\boldsymbol{x}}_t\|^2 \quad \text{(Young's inequality)}\\
& = \rbr{\Upsilon_B+\frac{2\Upsilon_B^2}{\gamma_{RH}}+ \frac{\gamma_{RH}}{2} }\|\Delta{\boldsymbol{u}}_t\|^2 - \rbr{\frac{\gamma_{RH}}{2} - 2\nu(\Upsilon_B+\Upsilon_u)^2}\|\Delta{\boldsymbol{x}}_t\|^2\\
& \leq \rbr{\Upsilon_B+\frac{2\Upsilon_B^2}{\gamma_{RH}}+ \frac{\gamma_{RH}}{2} } \frac{1}{\gamma_{G}}\|c_t\|^2 - \rbr{\frac{\gamma_{RH}}{2} - 2\nu(\Upsilon_B+\Upsilon_u)^2}\|\Delta{\boldsymbol{x}}_t\|^2 \\
& \leq \frac{4\Upsilon_B^2}{\gamma_{RH}\gamma_{G}}\|c_t\|^2 - \rbr{\frac{\gamma_{RH}}{2} - 2\nu(\Upsilon_B+\Upsilon_u)^2}\|\Delta{\boldsymbol{x}}_t\|^2 \quad\quad (\text{use } \gamma_{RH} \leq 1 \leq \Upsilon_B),
\end{align*}
where the second last inequality uses the fact that $\Delta{\boldsymbol{u}}_t = G_t^T\Delta\bar{{\boldsymbol{u}}}_t$ for some vector $\Delta\bar{{\boldsymbol{u}}}_t\in\mathbb{R}^m$, and $\|c_t\|^2 \stackrel{\eqref{equ:Newton}}{=}\|G_t\Delta{\boldsymbol{x}}_t\|^2 = \|G_t\Delta{\boldsymbol{u}}_t\|^2 = \|G_tG_t^T\Delta\bar{{\boldsymbol{u}}}_t\|^2\geq \gamma_{G}\|G_t^T\bar{{\boldsymbol{u}}}_t\|^2 = \gamma_{G}\|\Delta{\boldsymbol{u}}_t\|^2$, and the inequality is due to Assumption \ref{ass:1}.
Combining the above display with \eqref{pequ:1}, we have
\begin{align*}
\begin{pmatrix}
\nabla_{{\boldsymbol{x}}}\mathcal{L}_{\mu, \nu}^t\\
\nabla_{{\boldsymbol{\lambda}}}\mathcal{L}_{\mu, \nu}^t
\end{pmatrix}^T\begin{pmatrix}
\Delta{\boldsymbol{x}}_t\\
\Delta{\boldsymbol{\lambda}}_t
\end{pmatrix} & \leq - \frac{\nu\gamma_{G}}{8}\nbr{\begin{pmatrix}
\Delta{\boldsymbol{x}}_t\\
\Delta{\boldsymbol{\lambda}}_t
\end{pmatrix}}^2 - \frac{\nu}{8}\nbr{\begin{pmatrix}
\nabla_{{\boldsymbol{x}}}\mathcal{L}_t\\
c_t
\end{pmatrix}}^2 - \rbr{\frac{\gamma_{RH}}{2}-2\nu(\Upsilon_B+\Upsilon_u)^2 - \frac{\nu\gamma_{G}}{8}}\|\Delta{\boldsymbol{x}}_t\|^2\\
& \quad -\rbr{\mu - \frac{8}{\nu\gamma_{G}} - \frac{4\Upsilon_B^2}{\gamma_{RH}\gamma_{G}} - \frac{\nu}{8}}\|c_t\|^2.
\end{align*}
Thus, the statements hold provided
\begin{equation*}
\frac{\gamma_{RH}}{2} \geq 2\nu(\Upsilon_B+\Upsilon_u)^2 + \frac{\nu\gamma_{G}}{8}\quad \text{ and } \quad\mu \geq \frac{8}{\nu\gamma_{G}} + \frac{4\Upsilon_B^2}{\gamma_{RH}\gamma_{G}}+\frac{\nu}{8},
\end{equation*}
which is implied by \eqref{cond:munu} using $(\gamma_{G}\vee \gamma_{RH})\leq 1\leq (\Upsilon_B\wedge\Upsilon_u)$.
This completes the proof.
\subsection{Proof of Lemma \ref{lem:3}}\label{pf:lem:3}
By \eqref{equ:update} and \eqref{equ:taylor}, we have
\begin{align}\label{pequ:2}
\mathcal{L}_{\mu, \nu}^{t+1} & \leq \mathcal{L}_{\mu, \nu}^t + {\bar{\alpha}}_t\begin{pmatrix}
\nabla_{{\boldsymbol{x}}}\mathcal{L}_{\mu, \nu}^t\\
\nabla_{{\boldsymbol{\lambda}}}\mathcal{L}_{\mu, \nu}^t
\end{pmatrix}^T\begin{pmatrix}
{\bar{\Delta}}{\boldsymbol{x}}_t\\
{\bar{\Delta}}{\boldsymbol{\lambda}}_t
\end{pmatrix} + \frac{\Upsilon_{\mu, \nu}{\bar{\alpha}}_t^2}{2}\nbr{\begin{pmatrix}
{\bar{\Delta}}{\boldsymbol{x}}_t\\
{\bar{\Delta}}{\boldsymbol{\lambda}}_t
\end{pmatrix}}^2 \nonumber\\
& = \mathcal{L}_{\mu, \nu}^t + {\bar{\alpha}}_t\begin{pmatrix}
\nabla_{{\boldsymbol{x}}}\mathcal{L}_{\mu, \nu}^t\\
\nabla_{{\boldsymbol{\lambda}}}\mathcal{L}_{\mu, \nu}^t
\end{pmatrix}^T(I + C_t)\begin{pmatrix}
\Delta{\boldsymbol{x}}_t\\
\Delta{\boldsymbol{\lambda}}_t
\end{pmatrix} + {\bar{\alpha}}_t\begin{pmatrix}
\nabla_{{\boldsymbol{x}}}\mathcal{L}_{\mu, \nu}^t\\
\nabla_{{\boldsymbol{\lambda}}}\mathcal{L}_{\mu, \nu}^t
\end{pmatrix}^T\cbr{\begin{pmatrix}
{\bar{\Delta}}{\boldsymbol{x}}_t\\
{\bar{\Delta}}{\boldsymbol{\lambda}}_t
\end{pmatrix}-(I + C_t)\begin{pmatrix}
\Delta{\boldsymbol{x}}_t\\
\Delta{\boldsymbol{\lambda}}_t
\end{pmatrix} } \nonumber\\
& \quad + \frac{\Upsilon_{\mu, \nu}{\bar{\alpha}}_t^2}{2}\nbr{\begin{pmatrix}
{\bar{\Delta}}{\boldsymbol{x}}_t\\
{\bar{\Delta}}{\boldsymbol{\lambda}}_t
\end{pmatrix}}^2,
\end{align}
where $C_t$ is defined in Lemma \ref{lem:1}(b).
Furthermore, by Lemma \ref{lem:2} and Lemma \ref{lem:1}, we have
\begin{align*}
& \begin{pmatrix}
\nabla_{{\boldsymbol{x}}}\mathcal{L}_{\mu, \nu}^t\\
\nabla_{{\boldsymbol{\lambda}}}\mathcal{L}_{\mu, \nu}^t
\end{pmatrix}^T(I + C_t)\begin{pmatrix}
\Delta{\boldsymbol{x}}_t\\
\Delta{\boldsymbol{\lambda}}_t
\end{pmatrix} \\
& \leq -\frac{\nu\gamma_{G}}{8}\cbr{\nbr{\begin{pmatrix}
\Delta{\boldsymbol{x}}_t\\
\Delta{\boldsymbol{\lambda}}_t
\end{pmatrix}}^2 + \nbr{\begin{pmatrix}
\nabla_{{\boldsymbol{x}}}\mathcal{L}_t\\
c_t
\end{pmatrix}}^2} + \|C_t\|\nbr{\begin{pmatrix}
\nabla_{{\boldsymbol{x}}}\mathcal{L}_{\mu, \nu}^t\\
\nabla_{{\boldsymbol{\lambda}}}\mathcal{L}_{\mu, \nu}^t
\end{pmatrix}}\nbr{\begin{pmatrix}
\Delta{\boldsymbol{x}}_t\\
\Delta{\boldsymbol{\lambda}}_t
\end{pmatrix}} \quad (\text{Lemma \ref{lem:2}})\\
&\stackrel{\mathclap{\eqref{equ:AL:der}, \eqref{equ:upper:bound}}}{\leq} -\frac{\nu\gamma_{G}}{8}\cbr{\nbr{\begin{pmatrix}
\Delta{\boldsymbol{x}}_t\\
\Delta{\boldsymbol{\lambda}}_t
\end{pmatrix}}^2 + \nbr{\begin{pmatrix}
\nabla_{{\boldsymbol{x}}}\mathcal{L}_t\\
c_t
\end{pmatrix}}^2} + \rho^\tau \rbr{1+(2\nu +\mu)\Upsilon_u}\nbr{\begin{pmatrix}
\nabla_{{\boldsymbol{x}}}\mathcal{L}_t\\
c_t
\end{pmatrix}}\nbr{\begin{pmatrix}
\Delta{\boldsymbol{x}}_t\\
\Delta{\boldsymbol{\lambda}}_t
\end{pmatrix}} \quad (\text{Lemma \ref{lem:1}})\\
& \leq -\frac{\nu\gamma_{G}}{8}\cbr{\nbr{\begin{pmatrix}
\Delta{\boldsymbol{x}}_t\\
\Delta{\boldsymbol{\lambda}}_t
\end{pmatrix}}^2 + \nbr{\begin{pmatrix}
\nabla_{{\boldsymbol{x}}}\mathcal{L}_t\\
c_t
\end{pmatrix}}^2} + 2\rho^\tau\mu\Upsilon_u\nbr{\begin{pmatrix}
\nabla_{{\boldsymbol{x}}}\mathcal{L}_t\\
c_t
\end{pmatrix}}\nbr{\begin{pmatrix}
\Delta{\boldsymbol{x}}_t\\
\Delta{\boldsymbol{\lambda}}_t
\end{pmatrix}}\quad (1\leq \Upsilon_u \text{ and } 1+2\nu \leq \mu)\\
& \leq - \rbr{\frac{\nu\gamma_{G}}{8} - \rho^\tau\mu\Upsilon_u}\cbr{\nbr{\begin{pmatrix}
\Delta{\boldsymbol{x}}_t\\
\Delta{\boldsymbol{\lambda}}_t
\end{pmatrix}}^2 + \nbr{\begin{pmatrix}
\nabla_{{\boldsymbol{x}}}\mathcal{L}_t\\
c_t
\end{pmatrix}}^2}.
\end{align*}
Thus, if $\rho^\tau \leq \nu\gamma_{G}/(16\mu\Upsilon_u)$, we have
\begin{equation}\label{pequ:3}
\begin{pmatrix}
\nabla_{{\boldsymbol{x}}}\mathcal{L}_{\mu, \nu}^t\\
\nabla_{{\boldsymbol{\lambda}}}\mathcal{L}_{\mu, \nu}^t
\end{pmatrix}^T(I + C_t)\begin{pmatrix}
\Delta{\boldsymbol{x}}_t\\
\Delta{\boldsymbol{\lambda}}_t
\end{pmatrix} \leq -\frac{\nu\gamma_{G}}{16}\cbr{\nbr{\begin{pmatrix}
\Delta{\boldsymbol{x}}_t\\
\Delta{\boldsymbol{\lambda}}_t
\end{pmatrix}}^2 + \nbr{\begin{pmatrix}
\nabla_{{\boldsymbol{x}}}\mathcal{L}_t\\
c_t
\end{pmatrix}}^2}.
\end{equation}
Moreover, we deal with the last two terms on the right hand side of \eqref{pequ:2}.
We note that
\begin{align}\label{pequ:4}
& \mathbb{E}\sbr{\begin{pmatrix}
{\bar{\Delta}}{\boldsymbol{x}}_t\\
{\bar{\Delta}}{\boldsymbol{\lambda}}_t
\end{pmatrix} \mid \mathcal{F}_{t-1}} = \mathbb{E}\sbr{\mathbb{E}\sbr{\begin{pmatrix}
{\bar{\Delta}}{\boldsymbol{x}}_t\\
{\bar{\Delta}}{\boldsymbol{\lambda}}_t
\end{pmatrix} \mid \mathcal{F}_{t-2/3}}\mid \mathcal{F}_{t-1}} = \mathbb{E}\sbr{(I+C_t)\begin{pmatrix}
{\tilde{\Delta}}{\boldsymbol{x}}_t\\
{\tilde{\Delta}}{\boldsymbol{\lambda}}_t
\end{pmatrix} \mid \mathcal{F}_{t-1}}\quad \text{(Lemma \ref{lem:1}(b))} \nonumber\\
& \stackrel{\mathclap{\eqref{equ:Newton}}}{=} -(I+C_t)K_t^{-1}\mathbb{E}\sbr{\begin{pmatrix}
{\bar{\nabla}}_{{\boldsymbol{x}}}\mathcal{L}_t\\
c_t
\end{pmatrix}\mid \mathcal{F}_{t-1}} = -(I+C_t)K_t^{-1}\begin{pmatrix}
\nabla_{{\boldsymbol{x}}}\mathcal{L}_t\\
c_t
\end{pmatrix}\quad \text{(Assumption \ref{ass:2})} \nonumber\\
& \stackrel{\mathclap{\eqref{equ:Newton}}}{=} (I+C_t)\begin{pmatrix}
\Delta{\boldsymbol{x}}_t\\
\Delta{\boldsymbol{\lambda}}_t
\end{pmatrix},
\end{align}
and
\begin{align}\label{pequ:5}
& \mathbb{E}\sbr{\nbr{\begin{pmatrix}
{\bar{\Delta}}{\boldsymbol{x}}_t\\
{\bar{\Delta}}{\boldsymbol{\lambda}}_t
\end{pmatrix} - (I+C_t)\begin{pmatrix}
\Delta{\boldsymbol{x}}_t\\
\Delta{\boldsymbol{\lambda}}_t
\end{pmatrix}}^2 \mid \mathcal{F}_{t-1}} \nonumber\\
& \leq 3\mathbb{E}\sbr{\nbr{\begin{pmatrix}
{\bar{\Delta}}{\boldsymbol{x}}_t - {\tilde{\Delta}}{\boldsymbol{x}}_t\\
{\bar{\Delta}}{\boldsymbol{\lambda}}_t - {\tilde{\Delta}}{\boldsymbol{\lambda}}_t
\end{pmatrix}}^2\mid \mathcal{F}_{t-1}} + 3\mathbb{E}\sbr{\nbr{\begin{pmatrix}
{\tilde{\Delta}}{\boldsymbol{x}}_t - \Delta{\boldsymbol{x}}_t\\
{\tilde{\Delta}}{\boldsymbol{\lambda}}_t - \Delta{\boldsymbol{\lambda}}_t
\end{pmatrix}}^2\mid \mathcal{F}_{t-1}} + 3\|C_t\|^2\nbr{\begin{pmatrix}
\Delta {\boldsymbol{x}}_t\\
\Delta{\boldsymbol{\lambda}}_t
\end{pmatrix}}^2 \nonumber\\
& \leq 3\rho^\tau\mathbb{E}\sbr{\nbr{\begin{pmatrix}
{\tilde{\Delta}}{\boldsymbol{x}}_t\\
{\tilde{\Delta}}{\boldsymbol{\lambda}}_t
\end{pmatrix}}^2\mid \mathcal{F}_{t-1}} + 3\mathbb{E}\sbr{\nbr{\begin{pmatrix}
{\tilde{\Delta}}{\boldsymbol{x}}_t - \Delta{\boldsymbol{x}}_t\\
{\tilde{\Delta}}{\boldsymbol{\lambda}}_t - \Delta{\boldsymbol{\lambda}}_t
\end{pmatrix}}^2\mid \mathcal{F}_{t-1}} + 3\rho^{2\tau}\nbr{\begin{pmatrix}
\Delta {\boldsymbol{x}}_t\\
\Delta{\boldsymbol{\lambda}}_t
\end{pmatrix}}^2 \quad \text{(Lemma \ref{lem:1}(b,c))} \nonumber\\
& = 3(\rho^\tau + \rho^{2\tau})\nbr{\begin{pmatrix}
\Delta{\boldsymbol{x}}_t\\
\Delta{\boldsymbol{\lambda}}_t
\end{pmatrix}}^2 + 3(1+\rho^\tau)\mathbb{E}\sbr{\nbr{\begin{pmatrix}
{\tilde{\Delta}}{\boldsymbol{x}}_t - \Delta{\boldsymbol{x}}_t\\
{\tilde{\Delta}}{\boldsymbol{\lambda}}_t - \Delta{\boldsymbol{\lambda}}_t
\end{pmatrix}}^2\mid \mathcal{F}_{t-1}} \quad \text{(bias-variance decomposition)} \nonumber\\
& \stackrel{\mathclap{\eqref{equ:Newton}, \eqref{equ:upper:bound}}}{\leq}\;\;3(\rho^\tau + \rho^{2\tau})\Upsilon_K^2\Upsilon_u^2 + 3(1+\rho^\tau)\Upsilon_K^2\mathbb{E}[\|{\bar{g}}_t - \nabla f_t\|^2\mid \mathcal{F}_{t-1}] \quad (\text{also use } \|K_t^{-1}\|\leq \Upsilon_K) \nonumber\\
&\leq 3(1+\rho^\tau)\Upsilon_K^2(\rho^\tau\Upsilon_u^2+\Upsilon_m) \quad \text{(Assumption \ref{ass:2}\eqref{equ:BM:a})}.
\end{align}
Thus, using \eqref{pequ:4} and \eqref{pequ:5}, we know
\begin{align}\label{pequ:6}
&\mathbb{E}\sbr{{\bar{\alpha}}_t\begin{pmatrix}
\nabla_{{\boldsymbol{x}}}\mathcal{L}_{\mu, \nu}^t\\
\nabla_{{\boldsymbol{\lambda}}}\mathcal{L}_{\mu, \nu}^t
\end{pmatrix}^T\cbr{\begin{pmatrix}
{\bar{\Delta}}{\boldsymbol{x}}_t\\
{\bar{\Delta}}{\boldsymbol{\lambda}}_t
\end{pmatrix}-(I + C_t)\begin{pmatrix}
\Delta{\boldsymbol{x}}_t\\
\Delta{\boldsymbol{\lambda}}_t
\end{pmatrix} } \mid \mathcal{F}_{t-1}} \nonumber\\
& \stackrel{\mathclap{\eqref{pequ:4}}}{=} \;\; \mathbb{E}\sbr{\rbr{{\bar{\alpha}}_t - \frac{\beta_t+\eta_t}{2}} \begin{pmatrix}
\nabla_{{\boldsymbol{x}}}\mathcal{L}_{\mu, \nu}^t\\
\nabla_{{\boldsymbol{\lambda}}}\mathcal{L}_{\mu, \nu}^t
\end{pmatrix}^T\cbr{\begin{pmatrix}
{\bar{\Delta}}{\boldsymbol{x}}_t\\
{\bar{\Delta}}{\boldsymbol{\lambda}}_t
\end{pmatrix}-(I + C_t)\begin{pmatrix}
\Delta{\boldsymbol{x}}_t\\
\Delta{\boldsymbol{\lambda}}_t
\end{pmatrix} } \mid \mathcal{F}_{t-1}} \nonumber\\
& \stackrel{\mathclap{\eqref{equ:sandwich}}}{\leq}\frac{\eta_t-\beta_t}{2}\mathbb{E}\sbr{\nbr{\begin{pmatrix}
\nabla_{{\boldsymbol{x}}}\mathcal{L}_{\mu, \nu}^t\\
\nabla_{{\boldsymbol{\lambda}}}\mathcal{L}_{\mu, \nu}^t
\end{pmatrix}}\nbr{\begin{pmatrix}
{\bar{\Delta}}{\boldsymbol{x}}_t\\
{\bar{\Delta}}{\boldsymbol{\lambda}}_t
\end{pmatrix}-(I + C_t)\begin{pmatrix}
\Delta{\boldsymbol{x}}_t\\
\Delta{\boldsymbol{\lambda}}_t
\end{pmatrix}}\mid \mathcal{F}_{t-1} } \nonumber\\
& \stackrel{\mathclap{\eqref{equ:upper:bound}}}{\leq}\frac{\eta_t-\beta_t}{2}(1+(2\nu+\mu)\Upsilon_u)\Upsilon_u\mathbb{E}\sbr{\nbr{\begin{pmatrix}
{\bar{\Delta}}{\boldsymbol{x}}_t\\
{\bar{\Delta}}{\boldsymbol{\lambda}}_t
\end{pmatrix}-(I + C_t)\begin{pmatrix}
\Delta{\boldsymbol{x}}_t\\
\Delta{\boldsymbol{\lambda}}_t
\end{pmatrix}}\mid \mathcal{F}_{t-1}} \nonumber\\
& \leq (\eta_t-\beta_t)\mu\Upsilon_u^2\sqrt{\mathbb{E}\sbr{\nbr{\begin{pmatrix}
{\bar{\Delta}}{\boldsymbol{x}}_t\\
{\bar{\Delta}}{\boldsymbol{\lambda}}_t
\end{pmatrix}-(I + C_t)\begin{pmatrix}
\Delta{\boldsymbol{x}}_t\\
\Delta{\boldsymbol{\lambda}}_t
\end{pmatrix}}^2\mid \mathcal{F}_{t-1}}}\quad (1\leq \Upsilon_u \text{ and } 1+2\nu\leq \mu) \nonumber\\
&\stackrel{\mathclap{\eqref{pequ:5}}}{\leq}2\mu\Upsilon_K\Upsilon_u^2(1+\rho^\tau)(\sqrt{\Upsilon_m}\vee \Upsilon_u)
(\eta_t-\beta_t) \leq 4\mu\Upsilon_K\Upsilon_u^2(\sqrt{\Upsilon_m}\vee\Upsilon_u)(\eta_t-\beta_t),
\end{align}
and
\begin{align}\label{pequ:7}
&\mathbb{E}\sbr{\nbr{\begin{pmatrix}
{\bar{\Delta}}{\boldsymbol{x}}_t\\
{\bar{\Delta}}{\boldsymbol{\lambda}}_t
\end{pmatrix}}^2\mid \mathcal{F}_{t-1}} \nonumber\\
& \stackrel{\mathclap{\eqref{pequ:4}}}{=}\; \nbr{(I+C_t)\begin{pmatrix}
\Delta{\boldsymbol{x}}_t\\
\Delta{\boldsymbol{\lambda}}_t
\end{pmatrix}}^2 + \mathbb{E}\sbr{\nbr{\begin{pmatrix}
{\bar{\Delta}}{\boldsymbol{x}}_t\\
{\bar{\Delta}}{\boldsymbol{\lambda}}_t
\end{pmatrix} - (I+C_t)\begin{pmatrix}
\Delta{\boldsymbol{x}}_t\\
\Delta{\boldsymbol{\lambda}}_t
\end{pmatrix} }^2 \mid \mathcal{F}_{t-1}} \nonumber\\
& \stackrel{\mathclap{\eqref{equ:Newton},\eqref{equ:upper:bound}}}{\leq}\;\;(1+\rho^\tau)^2\Upsilon_K^2\Upsilon_u^2 + \mathbb{E}\sbr{\nbr{\begin{pmatrix}
{\bar{\Delta}}{\boldsymbol{x}}_t\\
{\bar{\Delta}}{\boldsymbol{\lambda}}_t
\end{pmatrix} - (I+C_t)\begin{pmatrix}
\Delta{\boldsymbol{x}}_t\\
\Delta{\boldsymbol{\lambda}}_t
\end{pmatrix} }^2 \mid \mathcal{F}_{t-1}}\quad (\text{also use Lemma \ref{lem:1}(b)}) \nonumber\\
&\stackrel{\mathclap{\eqref{pequ:5}}}{\leq} (1+\rho^\tau)^2\Upsilon_K^2\Upsilon_u^2 + 3(1+\rho^\tau)\Upsilon_K^2(\rho^\tau\Upsilon_u^2+\Upsilon_m)
\leq 16\Upsilon_K^2(\Upsilon_u^2\vee\Upsilon_m).
\end{align}
Combining \eqref{pequ:7} with \eqref{pequ:6} and \eqref{pequ:3}, plugging into \eqref{pequ:2}, and using \eqref{equ:sandwich}, we obtain
\begin{multline*}
\mathbb{E}[\mathcal{L}_{\mu, \nu}^{t+1}\mid \mathcal{F}_{t-1}] \leq \mathcal{L}_{\mu, \nu}^t - \frac{\nu\gamma_{G}\beta_t}{16}\cbr{\nbr{\begin{pmatrix}
\Delta{\boldsymbol{x}}_t\\
\Delta{\boldsymbol{\lambda}}_t
\end{pmatrix}}^2 + \nbr{\begin{pmatrix}
\nabla_{{\boldsymbol{x}}}\mathcal{L}_t\\
c_t
\end{pmatrix}}^2} \\
+ 4\mu\Upsilon_K\Upsilon_u^2(\sqrt{\Upsilon_m}\vee\Upsilon_u)(\eta_t-\beta_t) + 8\Upsilon_{\mu, \nu}\Upsilon_K^2(\Upsilon_u^2\vee\Upsilon_m)\eta_t^2.
\end{multline*}
This completes the proof.
\subsection{Proof of Theorem \ref{thm:1}}\label{pf:thm:1}
Note that the condition of $\tau$ in \eqref{cond:tau} implies that we can select $(\mu, \nu)$ to satisfy \eqref{cond:munu} and have $\rho^\tau\leq \nu\gamma_{G}/(16\mu\Upsilon_u)$ with $\rho = 1-\gamma_{S}$.
Thus, by Lemma \ref{lem:3}, we have
\begin{equation*}
\mathbb{E}[\mathcal{L}_{\mu, \nu}^{t+1} - \min_{\mathcal{X}\times\Lambda}\mathcal{L}_{\mu, \nu}\mid \mathcal{F}_{t-1}] \leq \mathcal{L}_{\mu, \nu}^t - \min_{\mathcal{X}\times\Lambda}\mathcal{L}_{\mu, \nu} - \frac{\nu\gamma_{G}\beta_t}{16}\|\nabla\mathcal{L}_t\|^2 + \tilde{\Upsilon}_{\mu,\nu}(\chi_t+\eta_t^2).
\end{equation*}
By Robbins-Siegmund theorem (see \cite{Robbins1971convergence} or \cite[Theorem 1.3.12]{Duflo1997Random}), we immediately obtain $\sum_t \beta_t\|\nabla\mathcal{L}_t\|^2<\infty$.
Since $\sum_t \beta_t = \infty$, we have $\liminf_{t\rightarrow\infty}\|\nabla \mathcal{L}_t\| = 0$.
Furthermore, we observe that
\begin{multline}\label{pequ:33}
\mathbb{E}\sbr{\nbr{\begin{pmatrix}
{\boldsymbol{x}}_{t+1} - {\boldsymbol{x}}_t\\
{\boldsymbol{\lambda}}_{t+1}-{\boldsymbol{\lambda}}_t
\end{pmatrix}}^2} = \mathbb{E}\sbr{\mathbb{E}\sbr{\nbr{\begin{pmatrix}
{\boldsymbol{x}}_{t+1} - {\boldsymbol{x}}_t\\
{\boldsymbol{\lambda}}_{t+1}-{\boldsymbol{\lambda}}_t
\end{pmatrix}}^2\mid \mathcal{F}_{t-1}}} \\ \stackrel{\eqref{equ:update}}{\leq}\eta_t^2\mathbb{E}\sbr{\mathbb{E}\sbr{\nbr{\begin{pmatrix}
{\bar{\Delta}}{\boldsymbol{x}}_t\\
{\bar{\Delta}}{\boldsymbol{\lambda}}_t
\end{pmatrix}}^2 \mid \mathcal{F}_{t-1}} }\stackrel{\eqref{pequ:7}}{\leq} 16\Upsilon_K^2(\Upsilon_u^2\vee\Upsilon_m)\cdot \eta_t^2.
\end{multline}
Summing over $t=1$ to $\infty$, exchanging the expectation and sum by applying Fubini's theorem \citep[Theorem 1.7.2]{Durrett2019Probability}, and noting that $\sum_t\eta_t^2<\infty$, we obtain
\begin{equation*}
\mathbb{E}\sbr{\sum_{t=1}^{\infty}\nbr{\begin{pmatrix}
{\boldsymbol{x}}_{t+1} - {\boldsymbol{x}}_t\\
{\boldsymbol{\lambda}}_{t+1}-{\boldsymbol{\lambda}}_t
\end{pmatrix}}^2}<\infty.
\end{equation*}
This implies $\sum_{t=1}^{\infty}\|({\boldsymbol{x}}_{t+1} - {\boldsymbol{x}}_t, {\boldsymbol{\lambda}}_{t+1}-{\boldsymbol{\lambda}}_t)\|<\infty$ almost surely and, thus, $\|({\boldsymbol{x}}_{t+1}-{\boldsymbol{x}}_t, {\boldsymbol{\lambda}}_{t+1}-{\boldsymbol{\lambda}}_t)\|\rightarrow0$ as $t\rightarrow\infty$ almost surely.
Suppose $\lim_{t\rightarrow\infty}\|\nabla\mathcal{L}_t\|\neq 0$, then $\limsup_{t\rightarrow\infty}\|\nabla\mathcal{L}_t\| = \epsilon >0$.
Then, there exist two index sequences $\{t_{1,i}\}_i, \{t_{2,i}\}_i$ with $t_{1,i+1}>t_{2,i}> t_{1,i}$, and for all $i = 1,2,\ldots$, we have
\begin{equation}\label{pequ:8}
\|\nabla\mathcal{L}_{t_{1,i}}\| \geq \epsilon/2,\quad\quad \|\nabla\mathcal{L}_j\| \geq \epsilon/3\; \text{ for } j = t_{1,i}+1,\ldots, t_{2,i}-1, \quad\quad \|\nabla\mathcal{L}_{t_{i, 2}}\| < \epsilon/3.
\end{equation}
Since $\sum_t\beta_t\|\nabla\mathcal{L}_t\|^2<\infty$, we know
\begin{equation}\label{equ:20}
\infty>\sum_{i=1}^{\infty}\sum_{j=t_{i,1}}^{t_{i,2}-1}\beta_j\|\nabla\mathcal{L}_j\|^2 \stackrel{\eqref{pequ:8}}{\geq} \frac{\epsilon^2}{9}\sum_{i=1}^{\infty}\sum_{j=t_{i,1}}^{t_{i,2}-1}\beta_j.
\end{equation}
Furthermore, by \eqref{pequ:33}, we have
\begin{equation*}
\mathbb{E}\sbr{\nbr{\begin{pmatrix}
{\boldsymbol{x}}_{t_{2,i}} - {\boldsymbol{x}}_{t_{1,i}}\\
{\boldsymbol{\lambda}}_{t_{2,i}} - {\boldsymbol{\lambda}}_{t_{1,i}}
\end{pmatrix}}} \stackrel{\eqref{pequ:33}}{\leq} 4\Upsilon_K(\Upsilon_u\vee\sqrt{\Upsilon_m})\sum_{j=t_{i,1}}^{t_{i,2-1}}\eta_j \stackrel{\eqref{equ:sandwich}}{=} 4\Upsilon_K(\Upsilon_u\vee\sqrt{\Upsilon_m})\cbr{\sum_{j=t_{i,1}}^{t_{i,2-1}}\beta_j + \sum_{j=t_{i,1}}^{t_{i,2}-1}\chi_j}.
\end{equation*}
Summing over $i=1$ to $\infty$, and noting that $\sum_{i=1}^{\infty}\sum_{j=t_{i,1}}^{t_{i,2-1}}\beta_j<\infty$ by \eqref{equ:20} and $\sum_{i=1}^{\infty}\sum_{j=t_{i,1}}^{t_{i,2-1}}\chi_j \leq \sum_{j=1}^{\infty}\chi_j<\infty$, we exchange the expectation and sum by applying Fubini's theorem again. We know that the sequence $\{({\boldsymbol{x}}_{t_{2,i}} - {\boldsymbol{x}}_{t_{1,i}}, {\boldsymbol{\lambda}}_{t_{2,i}} - {\boldsymbol{\lambda}}_{t_{1,i}})\}_i$ converges to zero as $i\rightarrow\infty$ with probability one.
This contradicts with $\|\nabla\mathcal{L}_{t_{1,i}}\| \geq \epsilon/2$ and $\|\nabla\mathcal{L}_{t_{i, 2}}\| < \epsilon/3$ in \eqref{pequ:8}.
We complete the proof.
\subsection{Proof of Corollary \ref{cor:1}}\label{pf:cor:1}
Applying Lemma \ref{lem:3} and taking full expectation, we know for some constants $h_1, h_2>0$~that
\begin{equation*}
\mathbb{E}[\mathcal{L}_{\mu, \nu}^{t+1}] \leq \mathbb{E}[\mathcal{L}_{\mu, \nu}^t] - h_1\beta_t\mathbb{E}[\|\nabla\mathcal{L}_t\|^2] + h_2(\chi_t + \eta_t^2), \quad \forall t\geq 0.
\end{equation*}
Rearranging the inequality and summing over $t = 0$ to ${\mathcal T}_{\epsilon}-1$, we obtain
\begin{align*}
& h_1\sum_{t=0}^{{\mathcal T}_{\epsilon}-1}\mathbb{E}[\|\nabla\mathcal{L}_t\|^2] \leq \sum_{t=0}^{{\mathcal T}_{\epsilon}-1}\frac{1}{\beta_t}\rbr{(\mathbb{E}[\mathcal{L}_{\mu, \nu}^t]-\min_{\mathcal{X}\times\Lambda}\mathcal{L}_{\mu, \nu}) - (\mathbb{E}[\mathcal{L}_{\mu, \nu}^{t+1}] - \min_{\mathcal{X}\times\Lambda}\mathcal{L}_{\mu, \nu}) } + h_2\sum_{t=0}^{{\mathcal T}_{\epsilon}-1}\frac{\chi_t + \eta_t^2}{\beta_t} \\
& \leq \frac{\mathbb{E}[\mathcal{L}_{\mu, \nu}^0]-\min_{\mathcal{X}\times\Lambda}\mathcal{L}_{\mu, \nu}}{\beta_0} + \sum_{t = 1}^{{\mathcal T}_{\epsilon}-1}\rbr{\frac{1}{\beta_t} - \frac{1}{\beta_{t-1}}}(\mathbb{E}[\mathcal{L}_{\mu, \nu}^t]-\min_{\mathcal{X}\times\Lambda}\mathcal{L}_{\mu, \nu}) + h_2\sum_{t=0}^{{\mathcal T}_{\epsilon}-1}\frac{\chi_t + \eta_t^2}{\beta_t}.
\end{align*}
Using the fact that $\min_{\mathcal{X}\times\Lambda}\mathcal{L}_{\mu, \nu} \leq \mathbb{E}[\mathcal{L}_{\mu, \nu}^t] \leq \max_{\mathcal{X}\times\Lambda}\mathcal{L}_{\mu, \nu}$ and denoting $\Delta\mathcal{L}_{\mu, \nu} = \max_{\mathcal{X}\times\Lambda}\mathcal{L}_{\mu, \nu} - \min_{\mathcal{X}\times\Lambda}\mathcal{L}_{\mu, \nu}$, we further have
\begin{align*}
h_1\sum_{t=0}^{{\mathcal T}_{\epsilon}-1}\mathbb{E}[\|\nabla\mathcal{L}_t\|^2] & \leq \rbr{\Delta\mathcal{L}_{\mu, \nu}\vee h_2}\cbr{\frac{1}{\beta_0} + \sum_{t=1}^{{\mathcal T}_{\epsilon}-1}\rbr{\frac{1}{\beta_t} - \frac{1}{\beta_{t-1}}} + \sum_{t=0}^{{\mathcal T}_{\epsilon}-1}\frac{\chi_t + \eta_t^2}{\beta_t}}\\
& = \rbr{\Delta\mathcal{L}_{\mu, \nu}\vee h_2}\cbr{\frac{1}{\beta_{{\mathcal T}_{\epsilon}-1}} + \sum_{t=0}^{{\mathcal T}_{\epsilon}-1}\frac{\chi_t + \eta_t^2}{\beta_t} } = \rbr{\Delta\mathcal{L}_{\mu, \nu}\vee h_2}\cbr{{\mathcal T}_{\epsilon}^a + \sum_{t=0}^{{\mathcal T}_{\epsilon}-1}\frac{\chi_t + \eta_t^2}{\beta_t}}.
\end{align*}
For the last term on the right hand side, we have
\begin{align*}
\sum_{t=0}^{{\mathcal T}_{\epsilon}-1}\frac{\chi_t + \eta_t^2}{\beta_t} & = \sum_{t=0}^{{\mathcal T}_{\epsilon}-1}\cbr{(t+1)^{a-b} + (t+1)^a\rbr{(t+1)^{-2a} + 2(t+1)^{-(a+b)} + (t+1)^{-2b} }}\\
& \leq \sum_{t=0}^{{\mathcal T}_{\epsilon}-1}(t+1)^{a-b} + 4\sum_{t=0}^{{\mathcal T}_{\epsilon}-1}(t+1)^{-a} = 5 + \sum_{t=1}^{{\mathcal T}_{\epsilon}-1}\cbr{(t+1)^{a-b} + 4(t+1)^{-a}}\\
&\leq 5 + \int_0^{{\mathcal T}_{\epsilon}-1}(t+1)^{a-b} + 4(t+1)^{-a}dt \quad (\text{by the convexity of } x^{p} \text{ with } p<0)\\
& \leq \begin{cases*}
5 + \frac{{\mathcal T}_{\epsilon}^{1+a-b}}{1+a-b} + \frac{4{\mathcal T}_{\epsilon}^{1-a}}{1-a} & \text{ if } $1+a>b$,\\
5 + \log({\mathcal T}_{\epsilon}) + \frac{4{\mathcal T}_{\epsilon}^{1-a}}{1-a} & \text{ if } $1+a=b$,\\
5 + \frac{1}{b-a-1}+ \frac{4{\mathcal T}_{\epsilon}^{1-a}}{1-a} & \text{ if } $1+a<b$.
\end{cases*}
\end{align*}
Combining the above two displays, dividing ${\mathcal T}_{\epsilon}$ on both sides, and using ``$\lesssim$" to neglect constant factors (i.e., not depending on ${\mathcal T}_{\epsilon}$), we obtain
\begin{align*}
\epsilon^2 & \leq \rbr{\frac{1}{{\mathcal T}_{\epsilon}}\sum_{t=0}^{{\mathcal T}_{\epsilon}-1}\mathbb{E}[\|\nabla\mathcal{L}_t\|] }^2 \leq \frac{1}{{\mathcal T}_{\epsilon}}\sum_{t=0}^{{\mathcal T}_{\epsilon}-1}\rbr{\mathbb{E}[\|\nabla\mathcal{L}_t\|]}^2 \leq \frac{1}{{\mathcal T}_{\epsilon}}\sum_{t=0}^{{\mathcal T}_{\epsilon}-1}\mathbb{E}[\|\nabla\mathcal{L}_t\|^2]\\
& \lesssim \begin{cases*}
\frac{1}{{\mathcal T}_{\epsilon}^{1-a}} + \frac{1}{{\mathcal T}_{\epsilon}^{b-a}} + \frac{1}{{\mathcal T}_{\epsilon}^a} & \text{ if } $1+a>b$,\\
\frac{1}{{\mathcal T}_{\epsilon}^{1-a}} + \frac{1}{{\mathcal T}_{\epsilon}^a} & \text{ if } $1+a=b$,\\
\frac{1}{{\mathcal T}_{\epsilon}^{1-a}} + \frac{1}{{\mathcal T}_{\epsilon}^a} & \text{ if } $1+a<b$,\\
\end{cases*} \quad (\text{use } 1/{\mathcal T}_{\epsilon} \leq 1/{\mathcal T}_{\epsilon}^a \text{ and } \log({\mathcal T}_{\epsilon})/{\mathcal T}_{\epsilon} \lesssim 1/{\mathcal T}_{\epsilon}^a),\\
& \lesssim \begin{cases*}
\frac{1}{{\mathcal T}_{\epsilon}^{b-a}} + \frac{1}{{\mathcal T}_{\epsilon}^a} & \text{ if } $1>b$,\\
\frac{1}{{\mathcal T}_{\epsilon}^{1-a}} + \frac{1}{{\mathcal T}_{\epsilon}^a} & \text{ if } $1\leq b$,
\end{cases*} = \frac{1}{{\mathcal T}_{\epsilon}^{(1\wedge b)-a}} + \frac{1}{{\mathcal T}_{\epsilon}^a} \lesssim \frac{1}{{\mathcal T}_{\epsilon}^{a\wedge (1-a)\wedge (b-a)} }.
\end{align*}
This completes the proof.
\subsection{Proof of Theorem \ref{thm:2}}\label{pf:thm:2}
Using $\|\Delta_t\|\leq \omega_t$ in Assumption \ref{ass:4}, we have
\begin{align}\label{pequ:25}
\|K_t - K^\star\| & \stackrel{\mathclap{\eqref{equ:Kt}}}{=} \nbr{\begin{pmatrix}
B_t - \nabla_{{\boldsymbol{x}}}^2\mathcal{L}^\star & (G_t)^T - (G^\star)^T\\
G_t - G^\star & {\boldsymbol 0}
\end{pmatrix}} \stackrel{\eqref{equ:B_t}}{\leq} \nbr{\begin{pmatrix}
\frac{1}{t}\sum_{i=0}^{t-1}{\bar{\nabla}}_{{\boldsymbol{x}}}^2\mathcal{L}_i - \nabla_{{\boldsymbol{x}}}^2\mathcal{L}^\star & (G_t)^T - (G^\star)^T\\
G_t - G^\star & {\boldsymbol 0}
\end{pmatrix}} + w_t \nonumber\\
& \leq \nbr{\frac{1}{t}\sum_{i=0}^{t-1}{\bar{\nabla}}_{{\boldsymbol{x}}}^2\mathcal{L}_i - \nabla_{{\boldsymbol{x}}}^2\mathcal{L}^\star} + \nbr{G_t-G^\star} + \omega_t \nonumber\\
& \leq \nbr{\frac{1}{t}\sum_{i=0}^{t-1} {\bar{H}}_i - \nabla^2f_i} + \frac{1}{t}\sum_{i=0}^{t-1}\nbr{\nabla_{{\boldsymbol{x}}}^2\mathcal{L}_i - \nabla_{{\boldsymbol{x}}}^2\mathcal{L}^\star} + \|G_t-G^\star\| + \omega_t \nonumber\\
& \stackrel{\mathclap{\eqref{Lip:mL}}}{\leq} \nbr{\frac{1}{t}\sum_{i=0}^{t-1} {\bar{H}}_i - \nabla^2f_i} + \frac{\Upsilon_L}{t} \sum_{i=0}^{t-1}\nbr{\begin{pmatrix}
{\boldsymbol{x}}_i - {\boldsymbol{x}}^\star\\
{\boldsymbol{\lambda}}_i - {\boldsymbol{\lambda}}^\star
\end{pmatrix}} + \Upsilon_L\|{\boldsymbol{x}}_t-{\boldsymbol{x}}^\star\| + \omega_t.
\end{align}
Since $({\boldsymbol{x}}_t-{\boldsymbol{x}}^\star, {\boldsymbol{\lambda}}_t-{\boldsymbol{\lambda}}^\star) \rightarrow {\boldsymbol 0}$ as $t\rightarrow\infty$, and by the fact that $a_t\rightarrow a$ implies $\frac{1}{t}\sum_{i=0}^{t-1}a_i\rightarrow a$ for any sequence $a_t$ (known as Stolz–Ces\`aro theorem), it suffices to show that $(\sum_{i=0}^{t-1} {\bar{H}}_i - \nabla^2f_i)/t$ converges.
In fact, noting from Assumption \ref{ass:2}\eqref{equ:BM:d} that $\mathbb{E}[{\bar{H}}_i \mid \mathcal{F}_{i-1}] = \nabla^2 f_i$ and $\mathbb{E}[\|{\bar{H}}_i - \nabla^2f_i\|^2\mid \mathcal{F}_{i-1}]\leq \Upsilon_m$, we know $(\sum_{i=0}^{t-1} {\bar{H}}_i - \nabla^2f_i)/t$ is a square integrable martingale.
Thus, \cite[Theorem 1.3.15]{Duflo1997Random} suggests that for any $\upsilon>0$,
\begin{equation}\label{pequ:26}
\nbr{\frac{1}{t}\sum_{i=0}^{t-1} {\bar{H}}_i - \nabla^2f_i} = o\rbr{\sqrt{\frac{(\log t)^{1+\upsilon}}{t}}}.
\end{equation}
Combining \eqref{pequ:25} and \eqref{pequ:26}, we complete the proof.
\section{Proofs of Section \ref{sec:4}}
\subsection{Proof of Lemma \ref{lem:4}}\label{pf:lem:4}
Using the scheme in Algorithm \ref{alg:1}, we have
\begin{align*}
&\begin{pmatrix}
{\boldsymbol{x}}_{t+1} - {\boldsymbol{x}}^\star\\
{\boldsymbol{\lambda}}_{t+1} - {\boldsymbol{\lambda}}^\star
\end{pmatrix} \stackrel{\mathclap{\eqref{equ:update}}}{=}\begin{pmatrix}
{\boldsymbol{x}}_t-{\boldsymbol{x}}^\star\\
{\boldsymbol{\lambda}}_t - {\boldsymbol{\lambda}}^\star
\end{pmatrix} + {\bar{\alpha}}_t\begin{pmatrix}
{\bar{\Delta}}{\boldsymbol{x}}_t\\
{\bar{\Delta}}{\boldsymbol{\lambda}}_t
\end{pmatrix}\\
& = \begin{pmatrix}
{\boldsymbol{x}}_t-{\boldsymbol{x}}^\star\\
{\boldsymbol{\lambda}}_t - {\boldsymbol{\lambda}}^\star
\end{pmatrix} + \varphi_t\begin{pmatrix}
{\bar{\Delta}}{\boldsymbol{x}}_t\\
{\bar{\Delta}}{\boldsymbol{\lambda}}_t
\end{pmatrix} + \rbr{{\bar{\alpha}}_t -\varphi_t} \begin{pmatrix}
{\bar{\Delta}}{\boldsymbol{x}}_t\\
{\bar{\Delta}}{\boldsymbol{\lambda}}_t
\end{pmatrix}\\
&= \begin{pmatrix}
{\boldsymbol{x}}_t-{\boldsymbol{x}}^\star\\
{\boldsymbol{\lambda}}_t - {\boldsymbol{\lambda}}^\star
\end{pmatrix} +\varphi_t(I + C_t)\begin{pmatrix}
{\tilde{\Delta}}{\boldsymbol{x}}_t\\
{\tilde{\Delta}}{\boldsymbol{\lambda}}_t
\end{pmatrix} + \varphi_t\cbr{\begin{pmatrix}
{\bar{\Delta}}{\boldsymbol{x}}_t\\
{\bar{\Delta}}{\boldsymbol{\lambda}}_t
\end{pmatrix} - (I + C_t)\begin{pmatrix}
{\tilde{\Delta}}{\boldsymbol{x}}_t\\
{\tilde{\Delta}}{\boldsymbol{\lambda}}_t
\end{pmatrix}} + \rbr{{\bar{\alpha}}_t - \varphi_t} \begin{pmatrix}
{\bar{\Delta}}{\boldsymbol{x}}_t\\
{\bar{\Delta}}{\boldsymbol{\lambda}}_t
\end{pmatrix}\\
&\stackrel{\mathclap{\eqref{equ:Newton}}}{=}\begin{pmatrix}
{\boldsymbol{x}}_t-{\boldsymbol{x}}^\star\\
{\boldsymbol{\lambda}}_t - {\boldsymbol{\lambda}}^\star
\end{pmatrix} - \varphi_t(I + C_t)K_t^{-1}\begin{pmatrix}
{\bar{\nabla}}_{{\boldsymbol{x}}}\mathcal{L}_t\\
c_t
\end{pmatrix} + \varphi_t\cbr{\begin{pmatrix}
{\bar{\Delta}}{\boldsymbol{x}}_t\\
{\bar{\Delta}}{\boldsymbol{\lambda}}_t
\end{pmatrix} - (I + C_t)\begin{pmatrix}
{\tilde{\Delta}}{\boldsymbol{x}}_t\\
{\tilde{\Delta}}{\boldsymbol{\lambda}}_t
\end{pmatrix}} + \rbr{{\bar{\alpha}}_t - \varphi_t} \begin{pmatrix}
{\bar{\Delta}}{\boldsymbol{x}}_t\\
{\bar{\Delta}}{\boldsymbol{\lambda}}_t
\end{pmatrix}\\
& = \begin{pmatrix}
{\boldsymbol{x}}_t-{\boldsymbol{x}}^\star\\
{\boldsymbol{\lambda}}_t - {\boldsymbol{\lambda}}^\star
\end{pmatrix} - \varphi_t(I+C_t)K_t^{-1}\begin{pmatrix}
\nabla_{{\boldsymbol{x}}}\mathcal{L}_t\\
c_t
\end{pmatrix} - \varphi_t(I+C_t)K_t^{-1}\begin{pmatrix}
{\bar{g}}_t - \nabla f_t\\
{\boldsymbol 0}
\end{pmatrix} \\
& \quad + \varphi_t\cbr{\begin{pmatrix}
{\bar{\Delta}}{\boldsymbol{x}}_t\\
{\bar{\Delta}}{\boldsymbol{\lambda}}_t
\end{pmatrix} - (I + C_t)\begin{pmatrix}
{\tilde{\Delta}}{\boldsymbol{x}}_t\\
{\tilde{\Delta}}{\boldsymbol{\lambda}}_t
\end{pmatrix}} + \rbr{{\bar{\alpha}}_t - \varphi_t} \begin{pmatrix}
{\bar{\Delta}}{\boldsymbol{x}}_t\\
{\bar{\Delta}}{\boldsymbol{\lambda}}_t
\end{pmatrix}\\
&\stackrel{\mathclap{\eqref{rec:def:b}}}{=} \; \begin{pmatrix}
{\boldsymbol{x}}_t-{\boldsymbol{x}}^\star\\
{\boldsymbol{\lambda}}_t - {\boldsymbol{\lambda}}^\star
\end{pmatrix} - \varphi_t(I+C_t)K_t^{-1}\nabla\mathcal{L}_t + \varphi_t\btheta^t+ \rbr{{\bar{\alpha}}_t - \varphi_t} \begin{pmatrix}
{\bar{\Delta}}{\boldsymbol{x}}_t\\
{\bar{\Delta}}{\boldsymbol{\lambda}}_t
\end{pmatrix}\\
& = \begin{pmatrix}
{\boldsymbol{x}}_t-{\boldsymbol{x}}^\star\\
{\boldsymbol{\lambda}}_t-{\boldsymbol{\lambda}}^\star
\end{pmatrix} - \varphi_t(I+C_t)(K^\star)^{-1}\nabla\mathcal{L}_t - \varphi_t(I+C_t)\cbr{K_t^{-1} - (K^\star)^{-1}}\nabla\mathcal{L}_t + \varphi_t\btheta^t+ \rbr{{\bar{\alpha}}_t - \varphi_t} \begin{pmatrix}
{\bar{\Delta}}{\boldsymbol{x}}_t\\
{\bar{\Delta}}{\boldsymbol{\lambda}}_t
\end{pmatrix}\\
&\stackrel{\mathclap{\eqref{rec:def:d}}}{=}\; \cbr{I - \varphi_t(I+C_t)}\begin{pmatrix}
{\boldsymbol{x}}_t-{\boldsymbol{x}}^\star\\
{\boldsymbol{\lambda}}_t-{\boldsymbol{\lambda}}^\star
\end{pmatrix} - \varphi_t(I+C_t)(K^\star)^{-1}\bpsi^t - \varphi_t(I+C_t)\cbr{K_t^{-1} - (K^\star)^{-1}}\nabla\mathcal{L}_t \\
&\quad + \varphi_t\btheta^t+ \rbr{{\bar{\alpha}}_t - \varphi_t} \begin{pmatrix}
{\bar{\Delta}}{\boldsymbol{x}}_t\\
{\bar{\Delta}}{\boldsymbol{\lambda}}_t
\end{pmatrix}\\
& \stackrel{\mathclap{\eqref{rec:def:c}}}{=}\; \cbr{I - \varphi_t(I+C^\star)}\begin{pmatrix}
{\boldsymbol{x}}_t-{\boldsymbol{x}}^\star\\
{\boldsymbol{\lambda}}_t-{\boldsymbol{\lambda}}^\star
\end{pmatrix} + \varphi_t(\btheta^t + \bdelta^t) + \rbr{{\bar{\alpha}}_t - \varphi_t} \begin{pmatrix}
{\bar{\Delta}}{\boldsymbol{x}}_t\\
{\bar{\Delta}}{\boldsymbol{\lambda}}_t
\end{pmatrix}.
\end{align*}
We then apply the above result recursively and have
\begin{align*}
\begin{pmatrix}
{\boldsymbol{x}}_{t+1} - {\boldsymbol{x}}^\star\\
{\boldsymbol{\lambda}}_{t+1} - {\boldsymbol{\lambda}}^\star
\end{pmatrix} & = \cbr{I - \varphi_t(I+C^\star)}\begin{pmatrix}
{\boldsymbol{x}}_t-{\boldsymbol{x}}^\star\\
{\boldsymbol{\lambda}}_t-{\boldsymbol{\lambda}}^\star
\end{pmatrix} + \varphi_t(\btheta^t + \bdelta^t) + \rbr{{\bar{\alpha}}_t - \varphi_t} \begin{pmatrix}
{\bar{\Delta}}{\boldsymbol{x}}_t\\
{\bar{\Delta}}{\boldsymbol{\lambda}}_t
\end{pmatrix}\\
& = \cbr{I - \varphi_t(I+C^\star)} \bigg\{\cbr{I - \varphi_{t-1}(I+C^\star)}\begin{pmatrix}
{\boldsymbol{x}}_{t-1}-{\boldsymbol{x}}^\star\\
{\boldsymbol{\lambda}}_{t-1}-{\boldsymbol{\lambda}}^\star
\end{pmatrix} + \varphi_{t-1}(\btheta^{t-1}+\bdelta^{t-1}) \\
& \quad + ({\bar{\alpha}}_{t-1} - \varphi_{t-1})\begin{pmatrix}
{\bar{\Delta}}{\boldsymbol{x}}_{t-1}\\
{\bar{\Delta}}{\boldsymbol{\lambda}}_{t-1}
\end{pmatrix} \bigg\} + \varphi_t(\btheta^t + \bdelta^t) + \rbr{{\bar{\alpha}}_t - \varphi_t} \begin{pmatrix}
{\bar{\Delta}}{\boldsymbol{x}}_t\\
{\bar{\Delta}}{\boldsymbol{\lambda}}_t
\end{pmatrix}\\
& = \cdots = \mathcal{I}_{1,t}+\mathcal{I}_{2,t}+\mathcal{I}_{3,t}.
\end{align*}
This shows the first part of the result.
Moreover, if Assumptions \ref{ass:2}, \ref{ass:3} hold, we know $\mathbb{E}[{\bar{g}}_i - \nabla f_i \mid \mathcal{F}_{i-1}] = {\boldsymbol 0}$ by Assumption \ref{ass:2}, and
\begin{equation}\label{equ:tower}
\mathbb{E}\sbr{\begin{pmatrix}
{\bar{\Delta}}{\boldsymbol{x}}_i\\
{\bar{\Delta}}{\boldsymbol{\lambda}}_i
\end{pmatrix} - (I + C_i)\begin{pmatrix}
{\tilde{\Delta}}{\boldsymbol{x}}_i\\
{\tilde{\Delta}}{\boldsymbol{\lambda}}_i
\end{pmatrix} \mid \mathcal{F}_{i-1}} = \mathbb{E}\sbr{\mathbb{E}\sbr{\begin{pmatrix}
{\bar{\Delta}}{\boldsymbol{x}}_i\\
{\bar{\Delta}}{\boldsymbol{\lambda}}_i
\end{pmatrix} - (I + C_i)\begin{pmatrix}
{\tilde{\Delta}}{\boldsymbol{x}}_i\\
{\tilde{\Delta}}{\boldsymbol{\lambda}}_i
\end{pmatrix} \mid \mathcal{F}_{i-2/3} } \mid \mathcal{F}_{i-1}} = {\boldsymbol 0}
\end{equation}
by Lemma \ref{lem:1}(b).
Thus, $\mathbb{E}[\btheta^i\mid \mathcal{F}_{i-1}] = {\boldsymbol 0}$ and $\btheta^i$ is a martingale difference.
\subsection{Proof of Lemma \ref{lem:5}}\label{pf:lem:5}
Let us denote $\text{rank}(S) = r$ in the proof.
Since $K_t$, $K^\star$ have full rank, we know $\text{rank}(K_tS) = \text{rank}(K^\star S) = r$.
Let $K^\star S = ED F^T$ be truncated singular value decomposition of $K^\star S$.
We have
\begin{equation*}
E \in \mathbb{R}^{(d+m)\times r}, \quad F\in \mathbb{R}^{q\times r}, \quad E^TE = F^TF = I, \quad D = \diag(D_1, \ldots, D_r)\;\; \text{ with } D_1\geq \ldots\geq D_r>0.
\end{equation*}
Similarly, we let $K_t S = E'D'(F')^T$.
By direct calculation, we have
\begin{equation}\label{UU:1}
\|K_tS(S^TK_t^2S)^\dagger S^TK_t - K^\star S(S^T(K^\star)^2S)^\dagger S^TK^\star\| = \|EE^T - E'(E')^T\|.
\end{equation}
Define the principle angles $\theta_p$ between $\text{span}(E)$ and $\text{span}(E')$ to be $\theta_p = (\theta_{p,1}, \ldots, \theta_{p,r})$, so that~$E^TE'$ has singular value decomposition $E^TE' = P\cos(\theta_{p})Q^T$, where $P, Q\in \mathbb{R}^{r\times r}$ are orthonormal matrices and $\cos(\theta_{p}) = \diag(\cos(\theta_{p,1}), \ldots, \cos(\theta_{p,r}))$ (similar for $\sin(\theta_p)$).
We further let $E^\perp\in \mathbb{R}^{(d+m)\times (d+m-r)}$ be the complement of $E$, and express $E'$ as
\begin{equation}\label{UU:2}
E' = EA + E^\perp B.
\end{equation}
Then, $E^TE' = A = P\cos(\theta_{p})Q^T$ and $I = (E')^TE' = A^TA+B^TB$.
By the above formulation,
\begin{align}\label{UU:3}
\|EE^T - E'(E')^T\|\; & \stackrel{\mathclap{\eqref{UU:2}}}{=} \; \nbr{(E, E^\perp)\begin{pmatrix}
I - AA^T & -AB^T\\
-BA^T & -BB^T
\end{pmatrix}\begin{pmatrix}
E^T\\
(E^\perp)^T
\end{pmatrix}} = \nbr{\begin{pmatrix}
I - AA^T & -AB^T\\
-BA^T & -BB^T
\end{pmatrix}} \nonumber\\
& \leq \nbr{\begin{pmatrix}
I - AA^T & {\boldsymbol 0}\\
{\boldsymbol 0} & -BB^T
\end{pmatrix}} + \nbr{\begin{pmatrix}
{\boldsymbol 0} & AB^T\\
BA^T&{\boldsymbol 0}
\end{pmatrix}} \nonumber\\
&\leq \max\{\|I - AA^T\|, \|BB^T\|\} + \|AB^T\| \nonumber\\
& = \max\{\|I - AA^T\|, \|I - A^TA\|\} + \|AB^T\| \nonumber\\
& = \|\sin(\theta_{p})\|^2 + \sqrt{\|P\cos(\theta_{p})\sin^2(\theta_{p})\cos(\theta_{p})P^T\|} \nonumber\\
& = \|\sin(\theta_{p})\|^2 + \|\sin(\theta_{p})\cos(\theta_{p})\| \leq 2\|\sin(\theta_{p})\|.
\end{align}
On the other hand, by Wedin's $\sin(\Theta)$ theorem \cite[(3.1)]{Wedin1972Perturbation}, we know that
\begin{equation}\label{UU:4}
\|\sin(\theta_{p})\| \leq \frac{\|(K^\star - K_t)S\|}{D_r}.
\end{equation}
Moreover, we let $F_r$ be the $r$-th column of $F$ and note that
\begin{equation*}
D_r^2 = F_r^TS^T(K^\star)^2S F_r \geq (\sigma_{\min}(K^\star))^2F_r^TS^TSF_r.
\end{equation*}
Since $\text{kernel}(K^\star S) = \text{kernel}(S)$ and $F_r\in \text{kernel}^\perp(K^\star S)$, we know $F_r \in \text{kernel}^\perp(S) = \text{span}(S^T)$.~Thus, $F_r^TS^TSF_r \geq \lambda_{\min}^+(S^TS)$, where $\lambda_{\min}^+(S^TS) = (\sigma_{\min}^{+}(S))^2$ is the least positive~eigenvalue of $S^TS$.
Therefore, we have
\begin{equation}\label{UU:5}
D_r \geq \sigma_{\min}(K^\star)\sigma_{\min}^{+}(S).
\end{equation}
Combining all above derivations, we have
\begin{multline*}
\|K_tS(S^TK_t^2S)^\dagger S^TK_t - K^\star S(S^T(K^\star)^2S)^\dagger S^TK^\star\| \stackrel{\eqref{UU:1}}{=}\|EE^T - E'(E')^T\| \stackrel{\eqref{UU:3}}{\leq} 2\|\sin(\theta_{p})\| \\
\stackrel{\eqref{UU:4}}{\leq} \frac{2\|K_t-K^\star\|\cdot \|S\|}{D_r}\stackrel{\eqref{UU:5}}{\leq} \frac{2\|K_t-K^\star\|}{\sigma_{\min}(K^\star)}\cdot\frac{\|S\|}{\sigma_{\min}^{+}(S)}.
\end{multline*}
This completes the proof.
\subsection{Proof of Corollary \ref{cor:2}}\label{pf:cor:2}
Noting that $\|K_tS(S^TK_t^2S)^\dagger S^TK_t\| \leq 1$, we apply dominated convergence theorem \citep[Theorem~1.6.7]{Durrett2019Probability} and Lemma \ref{lem:5}, and almost surely have
\begin{equation*}
\lim\limits_{t\rightarrow \infty}\mathbb{E}[K_tS(S^TK_t^2S)^\dagger S^TK_t\mid {\boldsymbol{x}}_t, {\boldsymbol{\lambda}}_t] = \mathbb{E}[K^\star S(S^T(K^\star)^2S)^\dagger S^TK^\star],
\end{equation*}
where the expectation is taken over randomness of $S$ only.
This shows $C_t\rightarrow C^\star$.
Further, denoting $A_t = I - \mathbb{E}[K_tS(S^TK_t^2S)^\dagger S^TK_t\mid {\boldsymbol{x}}_t, {\boldsymbol{\lambda}}_t]$ and $A^\star = I - \mathbb{E}[K^\star S(S^T(K^\star)^2S)^\dagger S^TK^\star]$, we have
\begin{align*}
\nbr{C_t-C^\star} & = \nbr{A_t^\tau - (A^\star)^\tau}\leq \nbr{A_t^{\tau-1}(A_t-A^\star)} + \nbr{(A_t^{\tau-1} - (A^\star)^{\tau-1})A^\star}\\
& \leq \|A_t-A^\star\| + \nbr{A_t^{\tau-1} - (A^\star)^{\tau-1}}\quad (\|A_t\| \vee \|A^\star\| \leq 1)\\
& \leq \tau \|A_t-A^\star\| \leq \tau \mathbb{E}\sbr{\nbr{K_tS(S^TK_t^2S)^\dagger S^TK_t - K^\star S(S^T(K^\star)^2S)^\dagger S^TK^\star}\mid {\boldsymbol{x}}_t, {\boldsymbol{\lambda}}_t}\\
&\leq \frac{2\tau\|K_t-K^\star\|}{\sigma_{\min}(K^\star)} \mathbb{E}\sbr{\frac{\|S\|}{\sigma_{\min}^+(S)}} \leq \frac{2\tau\Upsilon_S}{\sigma_{\min}(K^\star)}\nbr{K_t-K^\star}\quad (\text{by Assumption \ref{ass:5}}).
\end{align*}
This completes the proof.
\subsection{Proof of Lemma \ref{lem:6}}\label{pf:lem:6}
We note that
\begin{equation}\label{def:I1t}
\mathcal{I}_{1,t} \stackrel{\eqref{rec:a}}{=} \sum_{i=0}^t\prod_{j=i+1}^{t}\cbr{I - \varphi_j(I+C^\star)}\varphi_i\btheta^i \stackrel{\eqref{equ:Eigen}}{=}U \sum_{i=0}^t\prod_{j=i+1}^{t}\cbr{I - \varphi_j\Sigma}\varphi_i U^T\btheta^i.
\end{equation}
Since $\mathbb{E}[\btheta^i\mid \mathcal{F}_{i-1}] = {\boldsymbol 0}$, $\mathcal{I}_{1,t}$ is a martingale.
We hence aim to apply the strong law of large~number \citep[Theorem 1.3.15]{Duflo1997Random}, the central limit theorem \citep[Corollary 2.1.10]{Duflo1997Random}, and the Berry-Esseen bound \citep[Theorem 2.1]{Fan2019Exact} for martingales to show each result.
We~first~characterize the conditional covariance of $\mathcal{I}_{1,t}$ defined as (cf. \citep[Proposition 1.3.7]{Duflo1997Random})
\begin{equation}\label{I_1t_q}
\langle\mathcal{I}_{1}\rangle_t \coloneqq U\sum_{i=0}^{t}\prod_{j=i+1}^{t}\cbr{I - \varphi_j\Sigma}\varphi_i^2 U^T\mathbb{E}[\btheta^i(\btheta^i)^T\mid \mathcal{F}_{i-1}]U\big(\prod_{j=i+1}^{t}\cbr{I - \varphi_j\Sigma}\big)^TU^T.
\end{equation}
For the term $\mathbb{E}[\btheta^i(\btheta^i)^T\mid \mathcal{F}_{i-1}]$, we have
\begin{align}\label{var:pro}
&\hskip-0.5cm \mathbb{E}[\btheta^i(\btheta^i)^T\mid \mathcal{F}_{i-1}] \stackrel{\eqref{rec:def:b}}{=} \mathbb{E}\bigg[\cbr{(I+C_i)K_i^{-1}\begin{pmatrix}
{\bar{g}}_i - \nabla f_i\\
{\boldsymbol 0}
\end{pmatrix}-\cbr{\begin{pmatrix}
{\bar{\Delta}}{\boldsymbol{x}}_i\\
{\bar{\Delta}}{\boldsymbol{\lambda}}_i
\end{pmatrix}- (I+C_i)\begin{pmatrix}
{\tilde{\Delta}}{\boldsymbol{x}}_i\\
{\tilde{\Delta}}{\boldsymbol{\lambda}}_i
\end{pmatrix} } } \nonumber\\
&\hskip-0.5cm \quad \quad \cbr{(I+C_i)K_i^{-1}\begin{pmatrix}
{\bar{g}}_i - \nabla f_i\\
{\boldsymbol 0}
\end{pmatrix}-\cbr{\begin{pmatrix}
{\bar{\Delta}}{\boldsymbol{x}}_i\\
{\bar{\Delta}}{\boldsymbol{\lambda}}_i
\end{pmatrix}- (I+C_i)\begin{pmatrix}
{\tilde{\Delta}}{\boldsymbol{x}}_i\\
{\tilde{\Delta}}{\boldsymbol{\lambda}}_i
\end{pmatrix} } }^T \mid \mathcal{F}_{i-1}\bigg] \nonumber\\
& \hskip-0.5cm \stackrel{\mathclap{\eqref{equ:tower}}}{=}\; (I+C_i)K_i^{-1}\mathbb{E}\sbr{\begin{pmatrix}
{\bar{g}}_i - \nabla f_i\\
{\boldsymbol 0}
\end{pmatrix}\begin{pmatrix}
{\bar{g}}_i - \nabla f_i\\
{\boldsymbol 0}
\end{pmatrix}^T\mid \mathcal{F}_{i-1}}K_i^{-1}(I+C_i) \nonumber\\
& \hskip-0.5cm + \mathbb{E}\sbr{\cbr{\begin{pmatrix}
{\bar{\Delta}}{\boldsymbol{x}}_i\\
{\bar{\Delta}}{\boldsymbol{\lambda}}_i
\end{pmatrix}- (I+C_i)\begin{pmatrix}
{\tilde{\Delta}}{\boldsymbol{x}}_i\\
{\tilde{\Delta}}{\boldsymbol{\lambda}}_i
\end{pmatrix}}\cbr{\begin{pmatrix}
{\bar{\Delta}}{\boldsymbol{x}}_i\\
{\bar{\Delta}}{\boldsymbol{\lambda}}_i
\end{pmatrix}- (I+C_i)\begin{pmatrix}
{\tilde{\Delta}}{\boldsymbol{x}}_i\\
{\tilde{\Delta}}{\boldsymbol{\lambda}}_i
\end{pmatrix}}^T \mid \mathcal{F}_{i-1}} \eqqcolon \mathcal{J}_{1,i} + \mathcal{J}_{2,i}.
\end{align}
For the term $\mathcal{J}_{1,i}$, we apply Assumption \ref{ass:2} and have
\begin{equation*}
\mathbb{E}[({\bar{g}}_i-\nabla f_i)({\bar{g}}_i-\nabla f_i)^T \mid \mathcal{F}_{i-1}] = \mathbb{E}[{\bar{g}}_i{\bar{g}}_i^T\mid \mathcal{F}_{i-1}] - \nabla f_i\nabla^T f_i.
\end{equation*}
We also have
\begin{align*}
& \nbr{\mathbb{E}[{\bar{g}}_i{\bar{g}}_i^T - \nabla f({\boldsymbol{x}}^\star; \xi)\nabla^T f({\boldsymbol{x}}^\star; \xi)\mid \mathcal{F}_{i-1}]} \\
& \leq 2\mathbb{E}\sbr{\|{\bar{g}}_i - \nabla f({\boldsymbol{x}}^\star;\xi)\|\cdot \|{\bar{g}}_i\| \mid \mathcal{F}_{i-1}} + \mathbb{E}\sbr{\|{\bar{g}}_i - \nabla f({\boldsymbol{x}}^\star;\xi)\|^2\mid \mathcal{F}_{i-1}}\\
& \leq 2\sqrt{\mathbb{E}\sbr{\|{\bar{g}}_i - \nabla f({\boldsymbol{x}}^\star;\xi)\|^2\mid \mathcal{F}_{i-1}}}\sqrt{\mathbb{E}\sbr{\|{\bar{g}}_i\|^2\mid \mathcal{F}_{i-1}}} + \mathbb{E}\sbr{\|{\bar{g}}_i - \nabla f({\boldsymbol{x}}^\star;\xi)\|^2\mid \mathcal{F}_{i-1}},
\end{align*}
and
\begin{equation*}
\mathbb{E}\sbr{\|{\bar{g}}_i - \nabla f({\boldsymbol{x}}^\star;\xi)\|^2\mid \mathcal{F}_{i-1}} \leq \mathbb{E}[\sup_{{\boldsymbol{x}}\in\mathcal{X}}\|\nabla^2f({\boldsymbol{x}};\xi)\|^2]\cdot \|{\boldsymbol{x}}_i - {\boldsymbol{x}}^\star\|^2 \stackrel{\eqref{equ:BM:e}}{\leq} \Upsilon_m\|{\boldsymbol{x}}_i - {\boldsymbol{x}}^\star\|^2,
\end{equation*}
and
\begin{equation*}
\mathbb{E}[\|{\bar{g}}_i\|^2\mid \mathcal{F}_{i-1}] = \|\nabla f_i\|^2 + \mathbb{E}[\|{\bar{g}}_i - \nabla f_i\|^2\mid \mathcal{F}_{i-1}] \stackrel{\eqref{equ:upper:bound}, \eqref{equ:BM:a}}{\leq} \Upsilon_u^2 + \Upsilon_m\leq 2(\Upsilon_u^2\vee\Upsilon_m).
\end{equation*}
Thus, we combine the above three displays, and have
\begin{equation}\label{Lip:map}
\nbr{\mathbb{E}[{\bar{g}}_i{\bar{g}}_i^T - \nabla f({\boldsymbol{x}}^\star; \xi)\nabla^T f({\boldsymbol{x}}^\star; \xi)\mid \mathcal{F}_{i-1}]} \leq 2\sqrt{2}(\Upsilon_u^2\vee\Upsilon_m)(\|{\boldsymbol{x}}_i - {\boldsymbol{x}}^\star\| + \|{\boldsymbol{x}}_i-{\boldsymbol{x}}^\star\|^2)\rightarrow 0.
\end{equation}
Thus, we obtain
\begin{equation}\label{var:lim}
\lim\limits_{i\rightarrow \infty}\mathbb{E}[({\bar{g}}_i-\nabla f_i)({\bar{g}}_i-\nabla f_i)^T \mid \mathcal{F}_{i-1}] = \mathbb{E}[\nabla f({\boldsymbol{x}}^\star; \xi)\nabla^T f({\boldsymbol{x}}^\star; \xi)] - \nabla f({\boldsymbol{x}}^\star)\nabla^T f({\boldsymbol{x}}^\star).
\end{equation}
Further, by Theorem \ref{thm:2}, we know $K_i\rightarrow K^\star$ as $i\rightarrow \infty$; and Corollary \ref{cor:2} shows $C_i\rightarrow C^\star$.
Thus, we use the definition \eqref{equ:Omega} and have
\begin{equation}\label{J_1i}
\mathcal{J}_{1,i} = (I+C^\star){\Omega^\star}(I+C^\star) + O(\mathcal{K}_{1,i})
\end{equation}
with $\mathcal{K}_{1,i} \rightarrow 0$ as $i\rightarrow \infty$ almost surely.
For the term $\mathcal{J}_{2,i}$, we have (recall from Section \ref{sec:2} that ${\boldsymbol{z}}_{i,\tau} = ({\bar{\Delta}}{\boldsymbol{x}}_i, {\bar{\Delta}}{\boldsymbol{\lambda}}_i)$ and ${\tilde{\boldsymbol{z}}}_i = ({\tilde{\Delta}}{\boldsymbol{x}}_i, {\tilde{\Delta}}{\boldsymbol{\lambda}}_i)$)
\begin{align}\label{J_2i}
\mathcal{J}_{2,i}& = \mathbb{E}\sbr{({\boldsymbol{z}}_{i,\tau} - (I+C_i){\tilde{\boldsymbol{z}}}_i)({\boldsymbol{z}}_{i,\tau} - (I+C_i){\tilde{\boldsymbol{z}}}_i)^T\mid \mathcal{F}_{i-1}}\stackrel{\eqref{equ:z:recur}}{=} \mathbb{E}[(\tilde{C}_i - C_i){\tilde{\boldsymbol{z}}}_i{\tilde{\boldsymbol{z}}}_i^T(\tilde{C}_i^T-C_i^T) \mid \mathcal{F}_{i-1}] \nonumber\\
& \stackrel{\mathclap{\eqref{equ:Newton}}}{=} \mathbb{E}\sbr{(\tilde{C}_i - C_i)K_i^{-1}\begin{pmatrix}
{\bar{\nabla}}_{{\boldsymbol{x}}}\mathcal{L}_i\\
c_i
\end{pmatrix}\begin{pmatrix}
{\bar{\nabla}}_{{\boldsymbol{x}}}\mathcal{L}_i\\
c_i
\end{pmatrix}^TK_i^{-1}(\tilde{C}_i^T - C_i^T) \mid \mathcal{F}_{i-1} } \nonumber\\
& = \mathbb{E}\sbr{(\tilde{C}_i - C_i)K_i^{-1}\begin{pmatrix}
{\bar{g}}_i - \nabla f_i\\
{\boldsymbol 0}
\end{pmatrix}\begin{pmatrix}
{\bar{g}}_i - \nabla f_i\\
{\boldsymbol 0}
\end{pmatrix}^TK_i^{-1}(\tilde{C}_i^T - C_i^T) \mid \mathcal{F}_{i-1} } \nonumber\\
& \quad + \mathbb{E}\sbr{(\tilde{C}_i - C_i)K_i^{-1}\nabla\mathcal{L}_i\nabla^T\mathcal{L}_iK_i^{-1}(\tilde{C}_i^T - C_i^T) \mid \mathcal{F}_{i-1} } \nonumber\\
& \quad + \mathbb{E}\sbr{(\tilde{C}_i - C_i)K_i^{-1}\begin{pmatrix}
{\bar{g}}_i - \nabla f_i\\
{\boldsymbol 0}
\end{pmatrix}\begin{pmatrix}
\nabla_{{\boldsymbol{x}}}\mathcal{L}_i\\
c_i
\end{pmatrix}^TK_i^{-1}(\tilde{C}_i^T - C_i^T) \mid \mathcal{F}_{i-1} } \nonumber\\
& \quad + \mathbb{E}\sbr{(\tilde{C}_i - C_i)K_i^{-1}\begin{pmatrix}
\nabla_{{\boldsymbol{x}}}\mathcal{L}_i\\
c_i
\end{pmatrix}\begin{pmatrix}
{\bar{g}}_i - \nabla f_i\\
{\boldsymbol 0}
\end{pmatrix}^TK_i^{-1}(\tilde{C}_i^T - C_i^T) \mid \mathcal{F}_{i-1} }.
\end{align}
For the last two terms, we apply the tower property of conditional expectation by first conditioning~on the randomness of $\zeta_i$ to take expectation over the randomness of $\xi_i$, and then taking expectation~over the randomness of $\zeta_i$.
This is allowed even $\zeta_i$ is generated after $\xi_i$ in the algorithm, because $\xi_i$ and $\zeta_i$ are independent.
In particular, we have (similar for the second last term in \eqref{J_2i})
\begin{align*}
& \mathbb{E}\sbr{(\tilde{C}_i - C_i)K_i^{-1}\begin{pmatrix}
\nabla_{{\boldsymbol{x}}}\mathcal{L}_i\\
c_i
\end{pmatrix}\begin{pmatrix}
{\bar{g}}_i - \nabla f_i\\
{\boldsymbol 0}
\end{pmatrix}^TK_i^{-1}(\tilde{C}_i^T - C_i^T) \mid \mathcal{F}_{i-1} }\\
& = \mathbb{E}\sbr{(\tilde{C}_i - C_i)K_i^{-1}\begin{pmatrix}
\nabla_{{\boldsymbol{x}}}\mathcal{L}_i\\
c_i
\end{pmatrix}\begin{pmatrix}
\mathbb{E}[{\bar{g}}_i - \nabla f_i\mid \mathcal{F}_{i-1}\cup \zeta_i]\\
{\boldsymbol 0}
\end{pmatrix}^TK_i^{-1}(\tilde{C}_i^T - C_i^T) \mid \mathcal{F}_{i-1} }\\
& = \mathbb{E}\sbr{(\tilde{C}_i - C_i)K_i^{-1}\begin{pmatrix}
\nabla_{{\boldsymbol{x}}}\mathcal{L}_i\\
c_i
\end{pmatrix}\begin{pmatrix}
\mathbb{E}[{\bar{g}}_i - \nabla f_i\mid \mathcal{F}_{i-1}]\\
{\boldsymbol 0}
\end{pmatrix}^TK_i^{-1}(\tilde{C}_i^T - C_i^T) \mid \mathcal{F}_{i-1} } = {\boldsymbol 0}.
\end{align*}
For the second term in \eqref{J_2i}, it converges to zero almost surely as $i$ goes to infinity since $\|\tilde{C}_i\|\vee\|C_i\|\leq 1$, $\|K_i^{-1}\|\leq \Upsilon_K$, and $\nabla\mathcal{L}_i\rightarrow 0$.
For the first term in \eqref{J_2i}, we have
\begin{align*}
&\mathbb{E}\sbr{(\tilde{C}_i - C_i)K_i^{-1}\begin{pmatrix}
{\bar{g}}_i - \nabla f_i\\
{\boldsymbol 0}
\end{pmatrix}\begin{pmatrix}
{\bar{g}}_i - \nabla f_i\\
{\boldsymbol 0}
\end{pmatrix}^TK_i^{-1}(\tilde{C}_i^T - C_i^T) \mid \mathcal{F}_{i-1} }\\
& = \mathbb{E}\sbr{(\tilde{C}_i - C_i)K_i^{-1}\begin{pmatrix}
\mathbb{E}\sbr{({\bar{g}}_i - \nabla f_i)({\bar{g}}_i - \nabla f_i)^T\mid \mathcal{F}_{i-1}\cup \zeta_i} & {\boldsymbol 0}\\
{\boldsymbol 0} & {\boldsymbol 0}
\end{pmatrix}K_i^{-1}(\tilde{C}_i^T - C_i^T) \mid \mathcal{F}_{i-1} }\\
& = \mathbb{E}\sbr{(\tilde{C}_i - C_i)K_i^{-1}\begin{pmatrix}
\mathbb{E}\sbr{({\bar{g}}_i - \nabla f_i)({\bar{g}}_i - \nabla f_i)^T\mid \mathcal{F}_{i-1}} & {\boldsymbol 0}\\
{\boldsymbol 0} & {\boldsymbol 0}
\end{pmatrix}K_i^{-1}(\tilde{C}_i^T - C_i^T) \mid \mathcal{F}_{i-1} } \\
& \stackrel{\mathclap{\eqref{equ:tC}}}{\longrightarrow} \mathbb{E}[(\tilde{C}^\star - C^\star){\Omega^\star}((\tilde{C}^\star)^T - C^\star)] = \mathbb{E}[\tilde{C}^\star{\Omega^\star}(\tilde{C}^\star)^T] - C^\star{\Omega^\star} C^\star.
\end{align*}
Again, the convergence here is due to the dominated convergence theorem, \eqref{var:lim}, and $K_i\rightarrow K^\star$; and the expectation is taken over the randomness of $\tau$ sketch matrices $S_1,\ldots, S_\tau$ only.
Thus, combining the above two displays with \eqref{J_2i}, we have
\begin{equation}\label{J_2ii}
\mathcal{J}_{2,i} = \mathbb{E}[\tilde{C}^\star{\Omega^\star}(\tilde{C}^\star)^T] - C^\star{\Omega^\star} C^\star + O(\mathcal{K}_{2,i})
\end{equation}
with $\mathcal{K}_{2,i}\rightarrow 0$ as $i\rightarrow \infty$ almost surely.
Combining \eqref{J_2ii}, \eqref{J_1i}, and \eqref{var:pro}, we obtain
\begin{equation*}
\mathbb{E}[\btheta^i(\btheta^i)\mid \mathcal{F}_{i-1}] = \mathbb{E}[(I+\tilde{C}^\star){\Omega^\star}(I+\tilde{C}^\star)^T] + O(\mathcal{K}_{1,i}+\mathcal{K}_{2,i}).
\end{equation*}
By the definition of $\langle \mathcal{I}_1\rangle_t$ in \eqref{I_1t_q}, let us denote
\begin{equation}\label{Gamma}
\Gamma = U^T\mathbb{E}[(I+\tilde{C}^\star){\Omega^\star}(I+\tilde{C}^\star)^T] U,
\end{equation}
and for any $k, l\in \{1,\ldots,d+m\}$, the $(k,l)$ entry of the matrix $U^T\langle \mathcal{I}_{1}\rangle_t U$ can be written as
\begin{equation*}
[U^T\langle \mathcal{I}_{1}\rangle_t U]_{k,l} = \sum_{i=0}^{t}\prod_{j = i+1}^t(1 - \varphi_j\sigma_k)(1-\varphi_j\sigma_l)\varphi_i^2(\Gamma_{kl} + r_{i,kl}),
\end{equation*}
where $r_{i,kl}\rightarrow 0$ as $i\rightarrow \infty$ almost surely.
Noting from \eqref{equ:sigma_rho} that $\sigma_k+\sigma_l \geq 2(1 - \rho^\tau)$, and using the condition \eqref{cond:varphi}, Lemmas \ref{aux:lem:1}, \ref{aux:lem:2}~lead~to $[U^T\langle \mathcal{I}_{1}\rangle_t U]_{k,l}/\varphi_t\rightarrow \Gamma_{kl}/(\sigma_k+\sigma_l + \varphi/\tilde{\varphi})$ as $t\rightarrow \infty$ almost surely.
Thus, we have
\begin{equation}\label{equ:I_t_var}
\frac{1}{\varphi_t}\cdot \langle \mathcal{I}_{1}\rangle_t \stackrel{a.s.}{\longrightarrow} U(\Theta\circ \Gamma)U^T \stackrel{\eqref{equ:Xi}}{=} \Xi^\star.
\end{equation}
Thus, \cite[Theorem 1.3.15]{Duflo1997Random} shows that \eqref{equ:I_1t:as} holds.
This shows the first part of the results.
For the second part of the results, we assume a higher order moment condition $\eqref{equ:BM:b}$.
We have
\begin{align}\label{pequ:27}
&\mathbb{E}[\|\btheta^i\|^3\mid \mathcal{F}_{i-1}]\stackrel{\mathclap{\eqref{rec:def:b}}}{\leq}4\rbr{\mathbb{E}\sbr{\nbr{(I+C_i)K_i^{-1}\begin{pmatrix}
{\bar{g}}_i - \nabla f_i\\
{\boldsymbol 0}
\end{pmatrix}}^3 \mid \mathcal{F}_{i-1}} + \mathbb{E}[\|{\boldsymbol{z}}_{i,\tau} - (I+C_i){\tilde{\boldsymbol{z}}}_i\|^3\mid \mathcal{F}_{i-1}]} \nonumber\\
& \stackrel{\mathclap{\eqref{equ:z:recur}}}{\leq} 4\rbr{8\Upsilon_K^3\mathbb{E}[\|{\bar{g}}_i - \nabla f_i\|^3\mid \mathcal{F}_{i-1}] + \mathbb{E}[\|(\tilde{C}_i - C_i){\tilde{\boldsymbol{z}}}_i\|^3\mid \mathcal{F}_{i-1}]}\quad (\text{also use } \|C_i\|\leq 1, \|K_i^{-1}\|\leq \Upsilon_K) \nonumber\\
& \stackrel{\mathclap{\eqref{equ:BM:b}}}{\leq} 4\rbr{8\Upsilon_K^3\Upsilon_m+ 8\mathbb{E}[\|{\tilde{\boldsymbol{z}}}_i\|^3\mid \mathcal{F}_{i-1}]} \quad (\text{also use } \|\tilde{C}_i\|\vee\|C_i\|\leq 1) \nonumber\\
& \stackrel{\mathclap{\eqref{equ:Newton}}}{\leq} 4\rbr{8\Upsilon_K^3\Upsilon_m + 8\Upsilon_K^3\mathbb{E}[\|{\bar{\nabla}}\mathcal{L}_i\|^3\mid \mathcal{F}_{i-1}]} \quad (\text{also use } \|K_i^{-1}\|\leq \Upsilon_K) \nonumber\\
&\stackrel{\mathclap{\eqref{equ:Newton}}}{\leq} 4\rbr{8\Upsilon_K^3\Upsilon_m + 8\Upsilon_K^3\cbr{4\|\nabla\mathcal{L}_i\|^3 + 4\mathbb{E}[\|{\bar{g}}_i - \nabla f_i\|^3\mid \mathcal{F}_{i-1}]}} \nonumber\\
& \stackrel{\mathclap{\eqref{equ:BM:b}}}{\leq} 4\rbr{8\Upsilon_K^3\Upsilon_m + 8\Upsilon_K^3\cbr{4\Upsilon_u^3 + 4\Upsilon_m}}\quad (\text{also use } \eqref{equ:upper:bound}).
\end{align}
Thus, $\btheta^i$ has bounded third moment; and \cite[pp. 554]{Wang1995Asymptotic} together with \eqref{equ:I_t_var} give the result (a).
For (b), we verify Lindeberg's condition.
For any $\epsilon>0$, we have
\begin{align*}
&\frac{1}{\varphi_t}\sum_{i=0}^{t}\mathbb{E}\big[\big\|\prod_{j=i+1}^{t}\cbr{I - \varphi_j(I+C^\star)}\varphi_i\btheta^i\big\|^2\cdot {\boldsymbol{1}}_{\nbr{\prod_{j=i+1}^{t}\cbr{I - \varphi_j(I+C^\star)}\varphi_i\btheta^i}\geq \epsilon\sqrt{\varphi_t} } \mid \mathcal{F}_{i-1}\big]\\
& \leq \frac{1}{\epsilon\varphi_t^{3/2}}\sum_{i=0}^{t}\mathbb{E}\big[\big\|\prod_{j=i+1}^{t}\cbr{I - \varphi_j(I+C^\star)}\varphi_i\btheta^i\big\|^3\mid \mathcal{F}_{i-1}\big]\\
& \stackrel{\mathclap{\eqref{equ:Eigen}}}{=} \frac{1}{\epsilon\varphi_t^{3/2}}\sum_{i=0}^{t}\mathbb{E}\big[\big\|\prod_{j=i+1}^{t}\cbr{I - \varphi_j\Sigma}\varphi_iU^T\btheta^i\big\|^3\mid \mathcal{F}_{i-1}\big].
\end{align*}
To show the right hand side converges to zero, it suffices to show that each entry of the vector on the right hand side converges to zero.
In particular, it suffices to show that for any $1\leq k\leq d+m$,
\begin{equation*}
\frac{1}{\epsilon\varphi_t^{3/2}}\sum_{i=0}^{t} \prod_{j=i+1}^{t}\abr{1-\varphi_j\sigma_k}^3\varphi_i^3\mathbb{E}[|[U^T\btheta^i]_k|^3\mid \mathcal{F}_{i-1}] \longrightarrow 0\quad\quad \text{ as } \quad t\rightarrow\infty.
\end{equation*}
By \eqref{pequ:27} and the fact $\mathbb{E}[|[U^T\btheta^i]_k|^3\mid \mathcal{F}_{i-1}] \leq \mathbb{E}[\|\btheta^i\|^3\mid \mathcal{F}_{i-1}]$, we only need to show $\sum_{i=0}^{t} \prod_{j=i+1}^{t}|1-\varphi_j\sigma_k|^3\varphi_i^3 = o(\varphi_t^{3/2})$.
Without loss of generality, we suppose $1 - \varphi_j\sigma_k\geq 0$ for all $j\geq1$, and show
\begin{equation}\label{pequ:28}
\sum_{i=0}^{t} \prod_{j=i+1}^{t}(1-\varphi_j\sigma_k)^3\varphi_i^3 = o(\varphi_t^{3/2}).
\end{equation}
Otherwise, since $\varphi<0$ from \eqref{cond:varphi}, Lemma \ref{aux:lem:1} shows that $\varphi_i\rightarrow 0$.
Thus, there exists $\tilde{t}$ such that $1 - \varphi_j\sigma_k\geq 0$, $\forall j\geq \tilde{t}$.
Therefore,
\begin{align}\label{pequ:29}
\sum_{i=0}^{t} \prod_{j=i+1}^{t}\abr{1-\varphi_j\sigma_k}^3\varphi_i^3 & = \sum_{i=0}^{\tilde{t}-2}\prod_{j=i+1}^{t}\abr{1-\varphi_j\sigma_k}^3\varphi_i^3 + \sum_{i=\tilde{t}-1}^{t}\prod_{j=i+1}^{t}(1-\varphi_j\sigma_k)^3\varphi_i^3 \nonumber\\
& = \prod_{j=\tilde{t}}^{t}(1-\varphi_j\sigma_k)^3 \sum_{i=0}^{\tilde{t}-2}\prod_{j=i+1}^{\tilde{t}-1}\abr{1-\varphi_j\sigma_k}^3\varphi_i^3 + \sum_{i=\tilde{t}-1}^{t}\prod_{j=i+1}^{t}(1-\varphi_j\sigma_k)^3\varphi_i^3 \nonumber\\
& = \sum_{i=\tilde{t}-1}^{t}\prod_{j=i+1}^{t}(1-\varphi_j\sigma_k)^3(\varphi_i')^3,
\end{align}
where
\begin{equation*}
\varphi_{\tilde{t}-1}' = \rbr{\sum_{i=0}^{\tilde{t}-2}\prod_{j=i+1}^{\tilde{t}-1}\abr{1-\varphi_j\sigma_k}^3\varphi_i^3 + \varphi_{\tilde{t}-1}^3}^{1/3}, \quad \text{ and }\quad \varphi_i' = \varphi_i,\quad \forall i\geq \tilde{t}.
\end{equation*}
Note that \eqref{pequ:29} has the same form as \eqref{pequ:28}, and $\varphi_i'$ differs from $\varphi_i$ only at $i = \tilde{t}-1$.
Thus, \eqref{pequ:29} and \eqref{pequ:28} have the same limit.
For \eqref{pequ:28}, we apply Lemma \ref{aux:lem:5}, and note that
\begin{equation*}
\lim\limits_{i\rightarrow\infty}i\rbr{1 - \frac{\varphi_{i-1}^2}{\varphi_i^2}} \stackrel{\eqref{cond:varphi}}{=} 2\varphi\quad\quad \text{ and }\quad\quad 3\sigma_k + 2\varphi/\tilde{\varphi}\stackrel{\eqref{cond:varphi}}{>}0.
\end{equation*}
Thus, Lemma \ref{aux:lem:2} suggests that
\begin{equation*}
\sum_{i=0}^{t} \prod_{j=i+1}^{t}(1-\varphi_j\sigma_k)^3\varphi_i^3 = O(\varphi_t^2).
\end{equation*}
This verifies \eqref{pequ:28} and further verifies Lindeberg's condition.
Thus, the central limit theorem of martingale in \cite[Corollary 2.1.10]{Duflo1997Random} leads to (b).
For (c), we apply \citep[Theorem 2.1]{Fan2019Exact} with $\epsilon = \sqrt{\varphi_t}$, $\delta = 0$, $\rho=1$ (in their notation), as proved for verifying Lindeberg's condition above, and obtain the normalized result immediately.
This completes the proof.
\subsection{Proof of Lemma \ref{lem:7}}\label{pf:lem:7}
We note that
\begin{equation}\label{def:I2t}
\mathcal{I}_{2,t} \stackrel{\eqref{rec:b}}{=} \sum_{i=0}^t\prod_{j=i+1}^{t}\cbr{I - \varphi_j(I+C^\star)}({\bar{\alpha}}_i - \varphi_i){\boldsymbol{z}}_{i,\tau} \stackrel{\eqref{equ:Eigen}}{=} U\sum_{i=0}^t\prod_{j=i+1}^{t}\cbr{I - \varphi_j\Sigma}({\bar{\alpha}}_i - \varphi_i)U^T{\boldsymbol{z}}_{i,\tau}.
\end{equation}
Thus, for any $1\leq k\leq d+m$, we have
\begin{equation*}
[U^T\mathcal{I}_{2,t}]_k = \sum_{i=0}^{t}\prod_{j=i+1}^{t}(1 - \varphi_j\sigma_k)({\bar{\alpha}}_i - \varphi_i)[U^T{\boldsymbol{z}}_{i,\tau}]_k.
\end{equation*}
For the same reason as \eqref{pequ:28} and \eqref{pequ:29}, we suppose for any $j\geq 0$ that $1 - \varphi_j\sigma_k \geq 0$.
Then, we know
\begin{align}\label{equ:UI_2tk}
& \abr{[U^T\mathcal{I}_{2,t}]_k} \leq \frac{1}{2}\sum_{i=0}^{t}\prod_{j=i+1}^{t}|1 - \varphi_j\sigma_k|\chi_i \abr{[U^T{\boldsymbol{z}}_{i,\tau}]_k} = \frac{1}{2}\sum_{i=0}^{t}\prod_{j=i+1}^{t}(1 - \varphi_j\sigma_k)\chi_i \abr{[U^T{\boldsymbol{z}}_{i,\tau}]_k} \nonumber\\
& = \frac{1}{2}\sum_{i=0}^{t}\prod_{j=i+1}^{t}(1 - \varphi_j\sigma_k)\chi_i \mathbb{E}\sbr{\abr{[U^T{\boldsymbol{z}}_{i,\tau}]_k} \mid \mathcal{F}_{i-1}} \nonumber\\
&\quad + \frac{1}{2}\sum_{i=0}^{t}\prod_{j=i+1}^{t}(1 - \varphi_j\sigma_k)\chi_i \cbr{\abr{[U^T{\boldsymbol{z}}_{i,\tau}]_k} - \mathbb{E}\sbr{\abr{[U^T{\boldsymbol{z}}_{i,\tau}]_k} \mid \mathcal{F}_{i-1}} } \eqqcolon \mathcal{J}_{3,t,k} + \mathcal{J}_{4,t,k}.
\end{align}
Intuitively, $\mathcal{J}_{3,t,k}$ dominates $\mathcal{J}_{4,t,k}$ since the latter measures the error to the mean.
We precisely show such result in the following.
We first show that $\abr{[U^T{\boldsymbol{z}}_{i,\tau}]_k}$ has bounded variance.
We have
\begin{multline}\label{pequ:9}
\mathbb{E}\sbr{\cbr{\abr{[U^T{\boldsymbol{z}}_{i,\tau}]_k} - \mathbb{E}\sbr{\abr{[U^T{\boldsymbol{z}}_{i,\tau}]_k} \mid \mathcal{F}_{i-1}} }^2\mid \mathcal{F}_{i-1} } \\
\leq \mathbb{E}\sbr{\abr{[U^T{\boldsymbol{z}}_{i,\tau}]_k}^2\mid \mathcal{F}_{i-1}} \leq \mathbb{E}\sbr{\|{\boldsymbol{z}}_{i,\tau}\|^2\mid \mathcal{F}_{i-1}}\stackrel{\eqref{pequ:7}}{\leq} 16\Upsilon_K^2(\Upsilon_u^2\vee \Upsilon_m).
\end{multline}
Thus, $\mathcal{J}_{4,t,k}$ is a square integrable martingale.
Its variance is bounded by
\begin{align*}
\langle\mathcal{J}_{4,k}\rangle_t & \coloneqq \frac{1}{4}\sum_{i=0}^t\prod_{j=i+1}^{t}(1 - \varphi_j\sigma_k)^2\chi_i^2\mathbb{E}\sbr{\cbr{\abr{[U^T{\boldsymbol{z}}_{i,\tau}]_k} - \mathbb{E}\sbr{\abr{[U^T{\boldsymbol{z}}_{i,\tau}]_k} \mid \mathcal{F}_{i-1}} }^2\mid \mathcal{F}_{i-1} }\\
& \stackrel{\eqref{pequ:9}}{\leq}4\Upsilon_K^2(\Upsilon_u^2\vee \Upsilon_m)\sum_{i=0}^t\prod_{j=i+1}^{t}(1 - \varphi_j\sigma_k)^2\chi_i^2.
\end{align*}
Using \eqref{cond:varphi} and \eqref{cond:chi}, we know
\begin{multline}\label{com:chi:phi}
\lim\limits_{i\rightarrow \infty}i\rbr{1 - \frac{\chi_{i-1}^2/\varphi_{i-1}}{\chi_i^2/\varphi_i}} = \lim\limits_{i\rightarrow \infty}i\rbr{1 - \frac{\chi_{i-1}^2}{\chi_i^2} + \frac{\chi_{i-1}^2}{\chi_i^2}\rbr{1 - \frac{\varphi_i}{\varphi_{i-1}}} }\\
= \lim\limits_{i\rightarrow \infty}i\cbr{\rbr{1 - \frac{\chi_{i-1}}{\chi_i}}\rbr{1+ \frac{\chi_{i-1}}{\chi_i}} - \frac{\chi_{i-1}^2}{\chi_i^2}\frac{\varphi_i}{\varphi_{i-1}}\rbr{1 - \frac{\varphi_{i-1}}{\varphi_i}} } = 2\chi - \varphi.
\end{multline}
Further, \eqref{cond:chi} and \eqref{equ:sigma_rho} imply $2\sigma_k + (2\chi - \varphi)/\tilde{\varphi} >0$.
Thus, Lemma \ref{aux:lem:2} leads to $\langle\mathcal{J}_{4,k}\rangle_t = O(\chi_t^2/\varphi_t)$; and the strong law of large number \citep[Theorem 1.3.15]{Duflo1997Random} suggests that
\begin{equation}\label{pequ:30}
\mathcal{J}_{4,t,k} = o\rbr{\sqrt{\chi_t^2/\varphi_t \cdot\{\log(\varphi_t/\chi_t^2)\}^{1+\upsilon'} }} = o\rbr{\sqrt{\chi_t^2/\varphi_t\cdot \{\log(1/\chi_t)\}^{1+\upsilon'} }} \stackrel{\eqref{cond:chi}}{=} o (\chi_t/\varphi_t).
\end{equation}
For the term $\mathcal{J}_{3,t,k}$, we have
\begin{equation}\label{pequ:10}
\mathcal{J}_{3,t,k} \leq \frac{1}{2}\sum_{i=0}^{t}\prod_{j=i+1}^{t}(1 - \varphi_j\sigma_k)\chi_i \sqrt{\mathbb{E}[\abr{[U^T{\boldsymbol{z}}_{i,\tau}]_k}^2 \mid \mathcal{F}_{i-1}]} \stackrel{\mathclap{\eqref{pequ:9}}}{\leq} 2\Upsilon_K(\Upsilon_u\vee\sqrt{\Upsilon_m}) \sum_{i=0}^{t}\prod_{j=i+1}^{t}(1 - \varphi_j\sigma_k)\chi_i.
\end{equation}
Using \eqref{cond:varphi} and \eqref{cond:chi}, and the fact that
\begin{equation*}
\lim\limits_{i\rightarrow \infty} i\rbr{1 - \frac{\chi_{i-1}/\varphi_{i-1}}{\chi_i/\varphi_i}} \stackrel{\eqref{com:chi:phi}}{=} \chi - \varphi,
\end{equation*}
and $\sigma_k + (\chi-\varphi)/\tilde{\varphi} >0$ (as implied by \eqref{cond:chi}), we apply Lemma \ref{aux:lem:2} and obtain
\begin{equation}\label{pequ:31}
\mathcal{J}_{3,t,k} = O(\chi_t/\varphi_t).
\end{equation}
Plugging \eqref{pequ:31}, \eqref{pequ:30} into \eqref{equ:UI_2tk}, we complete the proof.
\subsection{Proof of Lemma \ref{lem:8}}\label{pf:lem:8}
Based on the definition of $\mathcal{I}_{3,t}$ in \eqref{rec:c}, we have the recursion
\begin{equation}\label{pequ:18}
\mathcal{I}_{3,t+1} = \cbr{I - \varphi_{t+1}(I+C^\star)}\mathcal{I}_{3,t} + \varphi_{t+1}\bdelta^{t+1}.
\end{equation}
By Assumption \ref{ass:1}, we have
\begin{align}\label{pequ:32}
& \nbr{\bdelta^t}\; \stackrel{\mathclap{\eqref{rec:def:c}}}{\leq}2\rbr{\|(K^\star)^{-1}\|\|\bpsi^t\| + \|K_t^{-1} - (K^\star)^{-1}\|\cdot \|\nabla\mathcal{L}_t\|} + \|C_t - C^\star\|\cdot\nbr{\begin{pmatrix}
{\boldsymbol{x}}_t-{\boldsymbol{x}}^\star\\
{\boldsymbol{\lambda}}_t-{\boldsymbol{\lambda}}^\star
\end{pmatrix}} \quad (\|C_t\|\leq 1)\nonumber\\
& \leq 2\Upsilon_K\Upsilon_L\nbr{\begin{pmatrix}
{\boldsymbol{x}}_t-{\boldsymbol{x}}^\star\\
{\boldsymbol{\lambda}}_t-{\boldsymbol{\lambda}}^\star
\end{pmatrix}}^2 + 2\Upsilon_K^2\Upsilon_u\|K_t - K^\star\|\cdot \nbr{\begin{pmatrix}
{\boldsymbol{x}}_t-{\boldsymbol{x}}^\star\\
{\boldsymbol{\lambda}}_t-{\boldsymbol{\lambda}}^\star
\end{pmatrix}} + \|C_t - C^\star\|\cdot\nbr{\begin{pmatrix}
{\boldsymbol{x}}_t-{\boldsymbol{x}}^\star\\
{\boldsymbol{\lambda}}_t-{\boldsymbol{\lambda}}^\star
\end{pmatrix}}.
\end{align}
Using the fact that $K_t\rightarrow K^\star$ (cf. Theorem \ref{thm:2}) and $C_t\rightarrow C^\star$ (cf. Corollary \ref{cor:2}), we know
\begin{equation}\label{pequ:19}
\bdelta^t = o(\|({\boldsymbol{x}}_t - {\boldsymbol{x}}^\star, {\boldsymbol{\lambda}}_t - {\boldsymbol{\lambda}}^\star)\|).
\end{equation}
Furthermore, we use $\|C^\star\|\leq \rho^\tau$ and know that for any $a\in(0, 1)$, there exists a threshold $t_1$ such that for any $t \geq t_1$,
\begin{align*}
\|\mathcal{I}_{3,t+1}\| & \leq \cbr{1 - \varphi_{t+1}(1-\rho^\tau)}\|\mathcal{I}_{3,t}\| + \varphi_{t+1}\cdot o\rbr{\nbr{\begin{pmatrix}
{\boldsymbol{x}}_{t+1} - {\boldsymbol{x}}^\star\\
{\boldsymbol{\lambda}}_{t+1} - {\boldsymbol{\lambda}}^\star
\end{pmatrix}}}\\
& \leq \cbr{1 - \varphi_{t+1}(1-\rho^\tau) + o(\varphi_{t+1})}\|\mathcal{I}_{3,t}\| + \varphi_{t+1}\cdot o(\|\mathcal{I}_{1,t}\|+\|\mathcal{I}_{2,t}\|) \quad (\text{Lemma \ref{lem:4}})\\
& \leq\cbr{1 - a(1-\rho^\tau)\varphi_{t+1}}\|\mathcal{I}_{3,t}\| + \varphi_{t+1}\cdot o(\|\mathcal{I}_{1,t}\|+\|\mathcal{I}_{2,t}\|).
\end{align*}
We apply the above inequality recursively, and obtain
\begin{multline}\label{pequ:11}
\|\mathcal{I}_{3,t+1}\| \leq \prod_{j=t_1+1}^{t+1}\cbr{1 - a(1-\rho^\tau)\varphi_j}\|\mathcal{I}_{3,t_1}\| \\
+ \sum_{i=t_1+1}^{t+1}\prod_{j=i+1}^{t+1}\cbr{1 - a(1-\rho^\tau)\varphi_j}\varphi_io(\|\mathcal{I}_{1,i-1}\|+\|\mathcal{I}_{2,i-1}\|).
\end{multline}
We apply Lemmas \ref{lem:6} and \ref{lem:7} for bounding $\|\mathcal{I}_{1,i-1}\|$ and $\|\mathcal{I}_{2,i-1}\|$.
In particular, we note that for any $\upsilon\geq 0$,
\begin{align*}
& \lim\limits_{i\rightarrow\infty}i \rbr{1 - \frac{\sqrt{\varphi_{i-2}\cbr{\log(1/\varphi_{i-2})}^{1+\upsilon} }}{\sqrt{\varphi_{i-1}\cbr{\log(1/\varphi_{i-1})}^{1+\upsilon} }}} = \lim\limits_{i\rightarrow \infty}i\rbr{1 - \frac{\sqrt{\varphi_{i-2}}}{\sqrt{\varphi_{i-1}}} + \frac{\sqrt{\varphi_{i-2}}}{\sqrt{\varphi_{i-1}}}\rbr{1 - \frac{\cbr{\log(1/\varphi_{i-2})}^{\frac{1+\upsilon}{2}}}{\cbr{\log(1/\varphi_{i-1})}^{\frac{1+\upsilon}{2}}}} }\\
&\stackrel{\mathclap{\eqref{cond:varphi}}}{=} \lim\limits_{i\rightarrow\infty} i \rbr{1 - \frac{\sqrt{\varphi_{i-2}}}{\sqrt{\varphi_{i-1}}}} + \lim\limits_{i\rightarrow\infty} i \rbr{1 - \frac{\cbr{\log(1/\varphi_{i-2})}^{\frac{1+\upsilon}{2}}}{\cbr{\log(1/\varphi_{i-1})}^{\frac{1+\upsilon}{2}}}}\\
&\stackrel{\mathclap{\eqref{cond:varphi}}}{=} \frac{\varphi}{2} + \lim\limits_{i\rightarrow\infty} i \rbr{1 - \frac{\cbr{\log(1/\varphi_{i-2})}^{\frac{1+\upsilon}{2}}}{\cbr{\log(1/\varphi_{i-1})}^{\frac{1+\upsilon}{2}}}}. \quad (\text{Lemma \ref{aux:lem:5}})
\end{align*}
Furthermore, we have
\begin{multline*}
\lim\limits_{i\rightarrow\infty} i\rbr{1 - \frac{\log(1/\varphi_{i-2})}{\log(1/\varphi_{i-1})}} = \lim\limits_{i\rightarrow\infty}\frac{i\log(\varphi_{i-2}/\varphi_{i-1})}{\log(1/\varphi_{i-1})} = \lim\limits_{i\rightarrow\infty}\frac{i\log\rbr{1 + (\varphi_{i-2}-\varphi_{i-1})/\varphi_{i-1}}}{\log(1/\varphi_{i-1})}\\
= \lim\limits_{i\rightarrow\infty}\frac{i\cbr{\frac{\varphi_{i-2} - \varphi_{i-1}}{\varphi_{i-1}} + O\rbr{\frac{(\varphi_{i-2} - \varphi_{i-1})^2}{\varphi_{i-1}^2}}} }{\log(1/\varphi_{i-1})} = \lim\limits_{i\rightarrow\infty}\frac{-\varphi}{\log(1/\varphi_{i-1})} = 0,
\end{multline*}
where the last equality uses the fact that $\varphi_i\rightarrow 0$, as implied by Lemma \ref{aux:lem:1}.
Combining the above two displays with Lemma \ref{aux:lem:5}, we have
\begin{equation}\label{pequ:12}
\lim\limits_{i\rightarrow\infty}i \rbr{1 - \frac{\sqrt{\varphi_{i-2}\cbr{\log(1/\varphi_{i-2})}^{1+\upsilon} }}{\sqrt{\varphi_{i-1}\cbr{\log(1/\varphi_{i-1})}^{1+\upsilon} }}} = \frac{\varphi}{2}.
\end{equation}
Moreover, we have
\begin{equation}\label{pequ:13}
\lim\limits_{i\rightarrow \infty}i\rbr{1 - \frac{\chi_{i-2}/\varphi_{i-2}}{\chi_{i-1}/\varphi_{i-1}}} \stackrel{\eqref{com:chi:phi}}{=} \chi - \varphi.
\end{equation}
Letting $a$ be any scalar such that
\begin{equation*}
0<\frac{-\varphi/\tilde{\varphi}}{2(1-\rho^\tau)}\vee \frac{-(\chi-\varphi)/\tilde{\varphi}}{1-\rho^\tau}<a<1,
\end{equation*}
which is guaranteed to exist due to \eqref{cond:varphi} and \eqref{cond:chi}, we can see that
\begin{equation}\label{pequ:14}
a(1-\rho^\tau) + \frac{\varphi}{2\tilde{\varphi}} >0 \quad\quad \text{and}\quad\quad a(1-\rho^\tau) + \frac{\chi-\varphi}{\tilde{\varphi}}>0.
\end{equation}
Thus, combining \eqref{pequ:11}, \eqref{pequ:12}, and \eqref{pequ:13} with Lemma \ref{aux:lem:2}, we obtain the results for both \eqref{equ:BM:a} and \eqref{equ:BM:b}.
This completes the~proof.
\subsection{Proof of Lemma \ref{lem:9}}\label{pf:lem:9}
Combining \eqref{pequ:25} and \eqref{pequ:26}, for any $\upsilon>0$, we have
\begin{align}\label{pequ:15}
& \|K_t-K^\star\| \leq \frac{\Upsilon_L}{t}\sum_{i=0}^{t-1}\nbr{\begin{pmatrix}
{\boldsymbol{x}}_i - {\boldsymbol{x}}^\star\\
{\boldsymbol{\lambda}}_i - {\boldsymbol{\lambda}}^\star
\end{pmatrix}} + \Upsilon_L\|{\boldsymbol{x}}_t-{\boldsymbol{x}}^\star\| + o\rbr{\sqrt{\frac{(\log t)^{1+\upsilon}}{t}}} + \omega_t \nonumber\\
& = \frac{\Upsilon_L}{t}\nbr{\begin{pmatrix}
{\boldsymbol{x}}_0 - {\boldsymbol{x}}^\star\\
{\boldsymbol{\lambda}}_0 - {\boldsymbol{\lambda}}^\star
\end{pmatrix}} + \Upsilon_L\sum_{i=1}^{t-1}\prod_{j=i+1}^{t}\rbr{1 - \frac{1}{j}}\frac{1}{i}\nbr{\begin{pmatrix}
{\boldsymbol{x}}_i - {\boldsymbol{x}}^\star\\
{\boldsymbol{\lambda}}_i - {\boldsymbol{\lambda}}^\star
\end{pmatrix}} + \Upsilon_L\|{\boldsymbol{x}}_t-{\boldsymbol{x}}^\star\| + o\rbr{\sqrt{\frac{(\log t)^{1+\upsilon}}{t}}} + \omega_t \nonumber\\
& = \Upsilon_L\sum_{i=1}^{t-1}\prod_{j=i+1}^{t}\rbr{1 - \frac{1}{j}}\frac{1}{i}\cbr{a_i + O\rbr{\frac{\chi_i}{\varphi_i}}} + \Upsilon_L\rbr{a_t + O\rbr{\frac{\chi_t}{\varphi_t}}} + o\rbr{\sqrt{\frac{(\log t)^{1+\upsilon}}{t}}} + \omega_t,
\end{align}
where, by Theorem \ref{thm:3}, $a_i = o(\sqrt{\varphi_i}\cbr{\log(1/\varphi_i)}^{(1+\upsilon)/2})$ under \eqref{equ:BM:a} and $a_i = O(\sqrt{\varphi_i\log(1/\varphi_i)})$~under \eqref{equ:BM:b}.
We claim that $\varphi > -2$.
Otherwise, $\varphi +1.5 \leq -0.5<0$.
We apply Lemma \ref{aux:lem:5}~and~have
\begin{equation*}
\lim_{t\rightarrow\infty}t\rbr{1 - \frac{\varphi_{t-1}(t-1)^{1.5}}{\varphi_t t^{1.5}}} = \lim_{t\rightarrow\infty} t\rbr{1 - \frac{\varphi_{t-1}}{\varphi_t} + \frac{\varphi_{t-1}}{\varphi_t} \rbr{1 - \frac{(t-1)^{1.5}}{t^{1.5}}}} \stackrel{\eqref{cond:varphi}}{=}\varphi + 1.5<0.
\end{equation*}
Then, Lemma \ref{aux:lem:1} suggests that $\varphi_t t^{1.5}\rightarrow 0$, which cannot hold under \eqref{cond:varphi}.
Thus, $\varphi>-2$.
Using \eqref{pequ:12} and Lemma \ref{aux:lem:2}, and noting that $1 + \varphi/2>0$, we obtain
\begin{equation}\label{pequ:16}
\sum_{i=1}^{t-1}\prod_{j=i+1}^{t}\rbr{1 - \frac{1}{j}}\frac{a_i}{i} = O(a_t).
\end{equation}
Furthermore, we deal with the term that involves $\chi_i/\varphi_i$ in \eqref{pequ:15}.
Without loss of generality, we suppose $\chi\geq 3\varphi/2$.
Otherwise, we know
\begin{equation*}
\lim\limits_{t\rightarrow\infty}t\rbr{1 - \frac{\chi_{t-1}/\varphi_{t-1}^{3/2}}{\chi_t/\varphi_t^{3/2}}} \stackrel{\eqref{com:chi:phi}}{=}\chi - \frac{3\varphi}{2}<0.
\end{equation*}
This implies that $\chi_t/\varphi_t = o(\sqrt{\varphi_t}) = o(a_t)$ and, thus, all terms $O(\chi_i/\varphi_i)$ in \eqref{pequ:15} are negligible and the argument of the lemma holds immediately.
Supposing $\chi\geq 3\varphi/2$ and noting that
\begin{equation*}
\lim\limits_{t\rightarrow\infty}t\rbr{1 - \frac{\chi_{t-1}/\varphi_{t-1}}{\chi_t/\varphi_t}} \stackrel{\eqref{com:chi:phi}}{=} \chi-\varphi\quad\quad \text{ and }\quad\quad 1+\chi-\varphi \geq 1+\frac{\varphi}{2}>0,
\end{equation*}
we obtain from Lemma \ref{aux:lem:2} that
\begin{equation}\label{pequ:17}
\sum_{i=1}^{t-1}\prod_{j=i+1}^{t}\rbr{1 - \frac{1}{j}}\frac{1}{i}\cdot O\rbr{\frac{\chi_i}{\varphi_i}} = O\rbr{\frac{\chi_t}{\varphi_t}}.
\end{equation}
Combining \eqref{pequ:15}, \eqref{pequ:16}, \eqref{pequ:17} together, and noting that $o(\sqrt{(\log t)^{1+\upsilon}/t }) = o(a_t)$ under \eqref{equ:BM:a} (as~implied by the fact that $t\varphi_t\rightarrow \tilde{\varphi}\in(0, \infty]$), we complete the proof.
\subsection{Proof of Lemma \ref{lem:11}}\label{pf:lem:11}
Combining \eqref{pequ:32}, Theorem \ref{thm:3}, Lemma \ref{lem:9}, and Corollary \ref{cor:2}, we have for any~$\upsilon>0$,
\begin{align*}
\|\bdelta^t\| & = O\rbr{\varphi_t\log(1/\varphi_t)} + O\rbr{\frac{\chi_t^2}{\varphi_t^2}} + o\rbr{\sqrt{\varphi_t\log(1/\varphi_t)} \cdot \sqrt{\frac{(\log t)^{1+\upsilon}}{t}}} + o\rbr{\frac{\chi_t}{\varphi_t}\sqrt{\frac{(\log t)^{1+\upsilon}}{t}}}\\
& \quad + O\rbr{\sqrt{\varphi_t\log(1/\varphi_t)}\cdot \omega_t} + O\rbr{\frac{\chi_t}{\varphi_t}\cdot \omega_t}\\
& \stackrel{\mathclap{\eqref{cond:varphi_new}}}{=} O\rbr{\varphi_t\log(1/\varphi_t)} + o\rbr{\sqrt{\varphi_t\log(1/\varphi_t)} \cdot \sqrt{\frac{(\log t)^{1+\upsilon}}{t}}} + O\rbr{\sqrt{\varphi_t\log(1/\varphi_t)}\cdot \omega_t}.
\end{align*}
We plug the above bound into the recursion \eqref{pequ:18} and apply Lemma \ref{aux:lem:2}.
In particular, we note that
\begin{align*}
& \lim_{t\rightarrow\infty} t\rbr{1 - \frac{\varphi_{t-1}\log(1/\varphi_{t-1})}{\varphi_t\log(1/\varphi_t)}} \stackrel{\eqref{pequ:12}}{=} \varphi,\\
& \lim_{t\rightarrow\infty} t\rbr{1 - \frac{\sqrt{\varphi_{t-1}\log(1/\varphi_{t-1})}\sqrt{(\log(t-1))^{1+\upsilon}/(t-1) } }{\sqrt{\varphi_t\log(1/\varphi_t)} \sqrt{(\log t)^{1+\upsilon}/t} } } = \frac{\varphi}{2}-\frac{1}{2},\\
& \lim_{t\rightarrow\infty} t\rbr{1 - \frac{\sqrt{\varphi_{t-1}\log(1/\varphi_{t-1})}\omega_{t-1} }{\sqrt{\varphi_t\log(1/\varphi_t)} \omega_t } } = \frac{\varphi}{2}+\omega.
\end{align*}
Thus, Lemma \ref{aux:lem:2} suggests that $\mathcal{I}_{3,t}$ has the same order as $\bdelta^t$ provided
\begin{equation*}
1-\rho^\tau + \frac{\varphi}{\tilde{\varphi}} >0, \quad 1-\rho^\tau + \frac{\varphi - 1}{2\tilde{\varphi}}> 0,\quad 1-\rho^\tau + \frac{\varphi/2+\omega}{\tilde{\varphi}}>0.
\end{equation*}
The above conditions are implied by \eqref{cond:varphi} and \eqref{cond:varphi_new}.
If $\omega_t=0$ for all sufficiently large $t$, the results hold trivially.
This completes the proof.
\subsection{Proof of Theorem \ref{thm:4}}\label{pf:thm:4}
By Lemmas \ref{lem:7} and \ref{lem:11}, we know that
\begin{align*}
& \sqrt{1/\varphi_t} \cdot \mathcal{I}_{2,t} = O(\chi_t/\varphi_t^{3/2}),\\
& \sqrt{1/\varphi_t} \cdot \mathcal{I}_{3,t} = O(\sqrt{\varphi_t}\log(1/\varphi_t)) + o\rbr{\frac{\sqrt{\log(1/\varphi_t)(\log t)^{1+\upsilon}} }{\sqrt{t}}}+ O\rbr{\sqrt{\log(1/\varphi_t)}\cdot \omega_t}.
\end{align*}
Using Lemma \ref{aux:lem:5} and the fact that
\begin{align*}
& \lim\limits_{t\rightarrow\infty}t\rbr{1 - \frac{\sqrt{\varphi_{t-1}}\log(1/\varphi_{t-1})}{\sqrt{\varphi_t}\log(1/\varphi_t)}} = \frac{\varphi}{2}<0,\\
& \lim\limits_{t\rightarrow\infty}t\rbr{1 - \frac{\sqrt{\log(1/\varphi_{t-1})(\log (t-1))^{1+\upsilon}} /\sqrt{t-1}}{\sqrt{\log(1/\varphi_t)(\log t)^{1+\upsilon}} /\sqrt{t}}} = -\frac{1}{2}<0,\\
& \lim\limits_{t\rightarrow\infty}t\rbr{1 - \frac{\sqrt{\log(1/\varphi_{t-1})}\cdot \omega_{t-1}}{\sqrt{\log(1/\varphi_t)}\cdot \omega_t}} = \omega<0,
\end{align*}
we know $\sqrt{1/\varphi_t} \cdot \mathcal{I}_{2,t} = o(1)$ and $\sqrt{1/\varphi_t} \cdot \mathcal{I}_{3,t}= o(1)$ almost surely.
Thus, the Slutsky's theorem together with Lemma \ref{lem:6} lead to the asymptotic normality.
Furthermore, Lemma \ref{aux:lem:4} with $C_t = 0$, $B_t=\mathcal{I}_{2,t}+\mathcal{I}_{3,t}$ leads to the Berry-Esseen bound.
This completes the proof.
\subsection{Proof of Lemma \ref{lem:13}}\label{pf:lem:13}
We note that
\begin{multline}\label{pequ:20}
\nbr{\Xi^\star - \Xi_t} \leq \nbr{\Xi^\star - \frac{1}{2+\varphi/\tilde{\varphi}}\mathbb{E}[(I+\tilde{C}^\star){\Omega^\star}(I+\tilde{C}^\star)^T]} \\
+ \frac{1}{2+\varphi/\tilde{\varphi}}\nbr{\mathbb{E}[(I+\tilde{C}^\star){\Omega^\star}(I+\tilde{C}^\star)^T] - {\Omega^\star}} + \frac{1}{2+\varphi/\tilde{\varphi}}\nbr{{\Omega^\star} - \Omega_t}.
\end{multline}
For the first term in \eqref{pequ:20}, we have
\begin{align*}
& \nbr{\Xi^\star - \frac{1}{2+\varphi/\tilde{\varphi}}\cdot\mathbb{E}[(I+\tilde{C}^\star){\Omega^\star}(I+\tilde{C}^\star)^T]} \nonumber\\
& \stackrel{\mathclap{\eqref{equ:Xi}}}{=} \nbr{\rbr{\Theta - \frac{1}{2+\varphi/\tilde{\varphi}} {\boldsymbol{1}}\b1^T}\circ U^T\mathbb{E}[(I+\tilde{C}^\star){\Omega^\star}(I+\tilde{C}^\star)^T]U } \nonumber\\
& \leq \nbr{\Theta - \frac{1}{2+\varphi/\tilde{\varphi}}{\boldsymbol{1}}\b1^T} \cdot \|\mathbb{E}[(I+\tilde{C}^\star){\Omega^\star}(I+\tilde{C}^\star)^T]\|\quad (\|A\circ B\|\leq \|A\|\cdot \|B\|) \nonumber\\
& \leq 4\nbr{\Theta - \frac{1}{2+\varphi/\tilde{\varphi}}{\boldsymbol{1}}\b1^T} \|{\Omega^\star}\|,\quad (\|\tilde{C}^\star\|\leq 1)
\end{align*}
and for any $1\leq k, l\leq d+m$,
\begin{align*}
\abr{\Theta_{k,l} - \frac{1}{2+\varphi/\tilde{\varphi}}} &= \abr{\frac{1}{\sigma_k+\sigma_l+\varphi/\tilde{\varphi}} - \frac{1}{2+\varphi/\tilde{\varphi}}} \\
& = \frac{\abr{2 - \sigma_k-\sigma_l}}{(\sigma_k+\sigma_l+\varphi/\tilde{\varphi})(2+\varphi/\tilde{\varphi})}\stackrel{\eqref{equ:sigma_rho}}{\leq} \frac{2\rho^\tau}{(2-2\rho^\tau + \varphi/\tilde{\varphi})(2+\varphi/\tilde{\varphi})}\\
&\stackrel{\mathclap{\eqref{cond:varphi}}}{\leq} \frac{2\rho^\tau}{(2- 2(1+\varphi/\tilde{\varphi}) + \varphi/\tilde{\varphi})(2+\varphi/\tilde{\varphi})} = \frac{2\rho^\tau}{-\varphi/\tilde{\varphi}(2+\varphi/\tilde{\varphi})}.
\end{align*}
Therefore, the above two displays lead to
\begin{equation}\label{pequ:21}
\nbr{\Xi^\star - \frac{1}{2+\varphi/\tilde{\varphi}}\cdot\mathbb{E}[(I+\tilde{C}^\star){\Omega^\star}(I+\tilde{C}^\star)^T]} = O(\rho^\tau).
\end{equation}
For the second term in \eqref{pequ:20}, we have
\begin{align*}
\nbr{\mathbb{E}[(I+\tilde{C}^\star){\Omega^\star}(I+\tilde{C}^\star)^T] - {\Omega^\star}} & \leq \|C^\star{\Omega^\star}\| + \|{\Omega^\star} C^\star\| + \|\mathbb{E}[\tilde{C}^\star{\Omega^\star}(\tilde{C}^\star)^T]\| \quad (\text{since } \mathbb{E}[\tilde{C}^\star] = C^\star)\\
& \leq O(\rho^\tau) + \|\mathbb{E}[\tilde{C}^\star{\Omega^\star}(\tilde{C}^\star)^T]\|. \quad(\|C^\star\|\leq \rho^\tau)
\end{align*}
Furthermore, since ${\Omega^\star}\preceq \Upsilon_K^2\Upsilon_m\cdot I$, we obtain
\begin{align*}
{\boldsymbol 0} & \preceq \mathbb{E}[\tilde{C}^\star{\Omega^\star}(\tilde{C}^\star)^T]\preceq \Upsilon_K^2\Upsilon_m\mathbb{E}[\tilde{C}^\star(\tilde{C}^\star)^T] \\
& = \Upsilon_K^2\Upsilon_m\mathbb{E}\sbr{\cbr{\prod_{j=1}^\tau(I - K^\star S_j(S_j^T(K^\star)S)^\dagger S_j^TK^\star)} \cbr{\prod_{j=1}^\tau(I - K^\star S_j(S_j^T(K^\star)S)^\dagger S_j^TK^\star)}^T }\\
& = \Upsilon_K^2\Upsilon_m\mathbb{E}\bigg[\cbr{\prod_{j=2}^\tau(I - K^\star S_j(S_j^T(K^\star)S_j)^\dagger S_j^TK^\star)}\mathbb{E}[(I - K^\star S_1(S_1^T(K^\star)S_1)^\dagger S_j^TK^\star)\mid S_{2:\tau}]\\
&\quad\quad \cbr{\prod_{j=2}^\tau(I - K^\star S_j(S_j^T(K^\star)S_j)^\dagger S_j^TK^\star)}^T\bigg] \\
& \preceq \Upsilon_K^2\Upsilon_m\rho\mathbb{E}\sbr{\cbr{\prod_{j=2}^\tau(I - K^\star S_j(S_j^T(K^\star)S)^\dagger S_j^TK^\star)} \cbr{\prod_{j=2}^\tau(I - K^\star S_j(S_j^T(K^\star)S)^\dagger S_j^TK^\star)}^T } \\
& \preceq \Upsilon_K^2\Upsilon_m\rho^\tau\cdot I,
\end{align*}
where the second inequality from the end is from Assumption \ref{ass:3} and Corollary \ref{cor:2}; and the last inequality applies the same derivation for sketch matrices $S_{2:\tau}$.
Combining the above two displays,
\begin{equation}\label{pequ:24}
\nbr{\mathbb{E}[(I+\tilde{C}^\star){\Omega^\star}(I+\tilde{C}^\star)^T] - {\Omega^\star}} \leq O(\rho^\tau).
\end{equation}
For the third term in \eqref{pequ:20}, we have
\begin{multline*}
\|\Omega_t - {\Omega^\star}\|
\stackrel{\eqref{equ:Omega}}{=} O(\|K_t - K^\star\|) \\+ O\rbr{\nbr{\frac{1}{t}\sum_{i=0}^{t-1}{\bar{g}}_i{\bar{g}}_i^T - \rbr{\frac{1}{t}\sum_{i=0}^{t-1}{\bar{g}}_i}\rbr{\frac{1}{t}\sum_{i=0}^{t-1}{\bar{g}}_i}^T - \mathbb{E}[\nabla f({\boldsymbol{x}}^\star;\xi)\nabla^Tf({\boldsymbol{x}}^\star; \xi)] - \nabla f({\boldsymbol{x}}^\star)\nabla^T f({\boldsymbol{x}}^\star) }}.
\end{multline*}
Furthermore, we have
\begin{align*}
& \nbr{\frac{1}{t}\sum_{i=0}^{t-1}{\bar{g}}_i{\bar{g}}_i^T - \rbr{\frac{1}{t}\sum_{i=0}^{t-1}{\bar{g}}_i}\rbr{\frac{1}{t}\sum_{i=0}^{t-1}{\bar{g}}_i}^T - \mathbb{E}[\nabla f({\boldsymbol{x}}^\star;\xi)\nabla^Tf({\boldsymbol{x}}^\star; \xi)] - \nabla f({\boldsymbol{x}}^\star)\nabla^T f({\boldsymbol{x}}^\star) }\\
& \leq \nbr{\frac{1}{t}\sum_{i=0}^{t-1}{\bar{g}}_i{\bar{g}}_i^T - \mathbb{E}[\nabla f({\boldsymbol{x}}^\star;\xi)\nabla^Tf({\boldsymbol{x}}^\star; \xi)]} + \nbr{\rbr{\frac{1}{t}\sum_{i=0}^{t-1}{\bar{g}}_i}\rbr{\frac{1}{t}\sum_{i=0}^{t-1}{\bar{g}}_i}^T - \nabla f({\boldsymbol{x}}^\star)\nabla^T f({\boldsymbol{x}}^\star)}.
\end{align*}
We take the first term as an example, while the second term has the same guarantee following the same derivations.
We note that
\begin{multline*}
\nbr{\frac{1}{t}\sum_{i=0}^{t-1}{\bar{g}}_i{\bar{g}}_i^T - \mathbb{E}[\nabla f({\boldsymbol{x}}^\star;\xi)\nabla^Tf({\boldsymbol{x}}^\star; \xi)]} \leq \nbr{\frac{1}{t}\sum_{i=0}^{t-1}({\bar{g}}_i{\bar{g}}_i^T) - \mathbb{E}[{\bar{g}}_i{\bar{g}}_i^T\mid \mathcal{F}_{i-1}]} \\
+\nbr{\frac{1}{t}\sum_{i=0}^{t-1}\mathbb{E}[{\bar{g}}_i{\bar{g}}_i^T\mid \mathcal{F}_{i-1}]- \mathbb{E}[\nabla f({\boldsymbol{x}}^\star;\xi)\nabla^Tf({\boldsymbol{x}}^\star; \xi)]}.
\end{multline*}
By \eqref{equ:BM:c}, we know the first term on the right hand side is a square integrable martingale.
The strong law of large number \cite[Theorem 1.3.15]{Duflo1997Random} suggests that
\begin{equation*}
\nbr{\frac{1}{t}\sum_{i=0}^{t-1}({\bar{g}}_i{\bar{g}}_i^T) - \mathbb{E}[{\bar{g}}_i{\bar{g}}_i^T\mid \mathcal{F}_{i-1}]} = o\rbr{\sqrt{\frac{(\log t)^{1+\upsilon}}{t}}}.
\end{equation*}
By \eqref{Lip:map}, \eqref{pequ:15}, \eqref{pequ:16}, \eqref{pequ:17}, the second term on the right hand side can be bounded by
\begin{equation}\label{pequ:35}
\nbr{\frac{1}{t}\sum_{i=0}^{t-1}\mathbb{E}[{\bar{g}}_i{\bar{g}}_i^T\mid \mathcal{F}_{i-1}]- \mathbb{E}[\nabla f({\boldsymbol{x}}^\star;\xi)\nabla^Tf({\boldsymbol{x}}^\star; \xi)]}
= O\rbr{\sqrt{\varphi_t\log(1/\varphi_t)}} +O\rbr{\frac{\chi_t}{\varphi_t}}.
\end{equation}
Combining the above five displays with Lemma \ref{lem:9}, we have
\begin{equation}\label{pequ:34}
\|\Omega_t - {\Omega^\star}\|= O\rbr{\sqrt{\varphi_t\log(1/\varphi_t)}} +O\rbr{\frac{\chi_t}{\varphi_t}} + o\rbr{\sqrt{\frac{(\log t)^{1+\upsilon}}{t}}} + \omega_t.
\end{equation}
Combining \eqref{pequ:20}, \eqref{pequ:21}, \eqref{pequ:24}, and \eqref{pequ:34}, we complete the proof.
\subsection{Proof of Corollary \ref{cor:3}}\label{pf:cor:3}
We let $\tau$ large enough such that $\bw^T\Xi^\star\bw \neq {\boldsymbol 0}$.
We note that
\begin{equation*}
\frac{\bw^T({\boldsymbol{x}}_t - {\boldsymbol{x}}^\star,{\boldsymbol{\lambda}}_t-{\boldsymbol{\lambda}}^\star)}{\sqrt{\bw^T\Xi_t\bw}} = \frac{\bw^T({\boldsymbol{x}}_t - {\boldsymbol{x}}^\star,{\boldsymbol{\lambda}}_t-{\boldsymbol{\lambda}}^\star)}{\sqrt{\bw^T\Xi^\star\bw} \cdot \sqrt{1 + \frac{\bw^T\Xi_t\bw - \bw^T\Xi^\star\bw}{\bw^T\Xi^\star\bw}}}.
\end{equation*}
Thus, combining Lemmas \ref{lem:13}, \ref{aux:lem:4} with Theorem \ref{thm:4}, we complete the proof.
\subsection{Proof of Theorem \ref{thm:5}}\label{pf:thm:5}
By the Raabe's test, we know that \eqref{cond:1} implies that \eqref{cond:step} holds.
Thus, the convergence of $\|\nabla\mathcal{L}_t\|$ comes from Theorem \ref{thm:1}.
The convergence of $K_t$ comes from Theorem \ref{thm:2}.
Furthermore, we note that \eqref{cond:2} implies \eqref{cond:varphi} and \eqref{cond:chi}.
To show this, we first note from \eqref{cond:1} that $\chi - \beta <0$.
Thus, $\chi_t = o(\beta_t)$ (cf. Lemma \ref{aux:lem:1}).
This implies $\beta_t\leq \varphi_t \leq \beta_t + o(\beta_t)$, and further $\lim\limits_{t\rightarrow\infty}t\varphi_t = \lim\limits_{t\rightarrow\infty}t\beta_t = \tilde{\beta}$.
Moreover, we have
\begin{align*}
& \lim\limits_{t\rightarrow\infty} t\rbr{1 - \frac{\varphi_{t-1}}{\varphi_t}} = \lim\limits_{t\rightarrow\infty} t\rbr{1 - \frac{2\beta_{t-1} + \chi_{t-1}}{2\beta_t+\chi_t}} = \lim\limits_{t\rightarrow\infty} t\rbr{1 - \frac{\beta_{t-1}}{\beta_t} + \frac{\beta_{t-1}}{\beta_t}\cbr{1 - \frac{2+\chi_{t-1}/\beta_{t-1}}{2+\chi_t/\beta_t}}}\\
& = \beta + \lim\limits_{t\rightarrow\infty} t \rbr{ 1 - \frac{2+\chi_{t-1}/\beta_{t-1}}{2+\chi_t/\beta_t} } = \beta + \frac{1}{2}\lim\limits_{t\rightarrow \infty}t\rbr{\frac{\chi_t}{\beta_t} - \frac{\chi_{t-1}}{\beta_{t-1}}} \quad (\text{since } \chi_t = o(\beta_t))\\
& = \beta + \frac{1}{2}\lim\limits_{t\rightarrow\infty} \frac{\chi_t}{\beta_t}\cdot t\rbr{1 - \frac{\chi_{t-1}}{\chi_t}\cdot \frac{\beta_t}{\beta_{t-1}}} = \beta + \frac{\chi - \beta}{2}\lim\limits_{t\rightarrow\infty} \frac{\chi_t}{\beta_t} = \beta.
\end{align*}
Thus, \eqref{cond:2} implies \eqref{cond:varphi} and \eqref{cond:chi}; and the asymptotic convergence rates of $({\boldsymbol{x}}_t, {\boldsymbol{\lambda}}_t)$ and $K_t$ come from Theorem \ref{thm:3} and Lemma \ref{lem:9}.
Finally, it is easy to see \eqref{cond:1}, \eqref{cond:2}, and \eqref{cond:3} imply \eqref{cond:varphi_new}. In fact, it suffices to show $1-\rho^\tau + (\beta-1)/(2\tilde{\beta})>0$. When $\tilde{\beta} = \infty$, it holds naturally. When $\tilde{\beta}\in(0, \infty)$, then we know $\beta = -1$ (otherwise, $\beta\in(-1,-0.5)$ implies $t\beta_t\rightarrow\infty$). Thus, $1-\rho^\tau + (\beta-1)/(2\tilde{\beta})=1-\rho^\tau + \beta/\tilde{\beta} >0$, as implied by \eqref{cond:2}. Thus, the asymptotic normality and Berry-Esseen bound come from Theorem \ref{thm:4} and Corollary \ref{cor:3}.
\section{Introduction}\label{sec:1}
We consider solving constrained stochastic nonlinear optimization problems of the form:
\begin{equation}\label{pro:1}
\begin{aligned}
\min_{{\boldsymbol{x}}\in \mathbb{R}^d}\;\; &f({\boldsymbol{x}} ) = \mathbb{E}[f({\boldsymbol{x}}; \xi)],\\
\text{s.t.} \;\; & c({\boldsymbol{x}}) = {\boldsymbol 0},
\end{aligned}
\end{equation}
where $f:\mathbb{R}^d\rightarrow\mathbb{R}$ is a stochastic objective function that involves a random variable $\xi\sim {\mathcal P}$,~following the distribution ${\mathcal P}$, and $c:\mathbb{R}^d\rightarrow \mathbb{R}^m$ provides deterministic equality constraints.
Problems of this form appear widely in a variety of applications, including optimal control \citep{Birge1997State}, multi-stage optimization \citep{Pflug2014Multistage}, PDE-constrained optimization \citep{Rees2010Optimal}, constrained~maximum likelihood estimation \citep{Onuk2015Constrained}, and constrained deep~neural~networks \citep{Chen2018Constraint}.
In practice, the variable $\xi$ corresponds to a data point; $f({\boldsymbol{x}}; \xi)$ is the loss at data point $\xi$ when using parameter ${\boldsymbol{x}}$ to fit the model; and $f({\boldsymbol{x}})$ is the expected loss.
Deterministic constraints are also common in many real examples. They can characterize the physics of systems, encode prior model knowledge, or reduce searching complexity.
Numerous methods have been proposed for solving constrained optimization problems \citep{Bertsekas1982Constrained, Nocedal2006Numerical}. Compared to most of the existing literature on solving constrained optimization by either penalty methods, augmented Lagrangian methods, or sequential quadratic programming (SQP), we consider here a stochastic objective in Problem \eqref{pro:1}, whose exact function evaluation, gradient, and Hessian are inaccessible due to the expensive calculation of the expectation. However, their stochastic estimates are accessible by drawing samples from ${\mathcal P}$. Recently, several algorithms built on stochastic SQP (StoSQP) have been proposed for solving \eqref{pro:1}. We point the~reader to \cite{Na2021Adaptive, Na2021Inequality, Berahas2021Sequential, Berahas2021Stochastic, Berahas2022Accelerating, Curtis2021Inexact}, and will review these related literature in Section \ref{sec:1.4}. Although these works all showed global convergence for a variety of StoSQP schemes (with or without line search, with or without inequality constraints etc.), a ``finer'' understanding of StoSQP is missing.
By ``finer'' understanding of StoSQP, we mean the (local) convergence rate, the (worst-case)~iteration or sample complexity, and the stationary distribution of the iteration sequence.
These aspects are important since, compared~to global convergence, they characterize the behavior of the iterates more precisely and demonstrate the efficiency of the algorithm. Furthermore, the primal-dual solution $({\boldsymbol{x}}^\star, {\boldsymbol{\lambda}}^\star)$, especially the primal solution ${\boldsymbol{x}}^\star$, is the optimal model parameter that minimizes the expected loss. Performing statistical inference on the model parameter, such as constructing a confidence region for ${\boldsymbol{x}}^\star$ or conducting a hypothesis testing $H_0: \bw^T{\boldsymbol{x}}^\star = 0$ v.s. $H_1: \bw^T{\boldsymbol{x}}^\star \neq 0$ for a direction $\bw\in\mathbb{R}^d$, is necessary for drawing any \textit{statistically significant} conclusions. Such a statistical inference task can be performed once we understand the stationary distribution of stochastic iterates.
In this paper, we make progress towards understanding the aforementioned aspects for StoSQP schemes. We complement the existing literature by establishing the asymptotic~convergence rate as well as the asymptotic normality for the \textit{last} iterate of an Adaptive Inexact StoSQP framework, shortened as~\texttt{AI-StoSQP}. As a result, we can perform statistical inference on $({\boldsymbol{x}}^\star, {\boldsymbol{\lambda}}^\star)$~based~on~the iterates $\{({\boldsymbol{x}}_t, {\boldsymbol{\lambda}}_t)\}_t$ generated by AI-StoSQP. As a byproduct, we also show the worst-case iteration complexity. By the nature of the algorithm, the iteration complexity of AI-StoSQP is consistent with the sample complexity. In the remaining of this section, we first briefly introduce AI-StoSQP and connect it with the existing schemes in Section \ref{sec:1.2}. Then, we summarize main results and contributions in Section \ref{sec:1.3}, followed by reviewing related literature on both constrained optimization and unconstrained optimization in Section \ref{sec:1.4}.
\subsection{Algorithm sketch}\label{sec:1.2}
This paper studies a StoSQP framework called AI-StoSQP. As suggested by the name, we allow the scheme to \textit{adaptively} select random stepsize ${\bar{\alpha}}_t$, although we require ${\bar{\alpha}}_t$ to be controlled~by~$\beta_t\leq {\bar{\alpha}}_t \leq \beta_t +\chi_t \eqqcolon\eta_t$ with $\beta_t, \chi_t$ being prespecified deterministic sequences. We also allow the scheme to solve Newton systems \textit{inexactly} via randomized iterative solvers, e.g., with the \textit{sketch-and-project}~method.
In particular, given the current primal-dual iterate $({\boldsymbol{x}}_t, {\boldsymbol{\lambda}}_t)$, AI-StoSQP performs the following three steps (detailed descriptions are presented in Section \ref{sec:2}):
\noindent $\bullet$ Step 1: We generate a \textit{single} sample $\xi_t\sim {\mathcal P}$ to estimate the gradient ${\bar{\nabla}}_{{\boldsymbol{x}}}\mathcal{L}_t$ and the Hessian ${\bar{\nabla}}_{{\boldsymbol{x}}}^2\mathcal{L}_t$ (with respect to ${\boldsymbol{x}}$) of the Lagrangian $\mathcal{L}({\boldsymbol{x}}, {\boldsymbol{\lambda}}) = f({\boldsymbol{x}})+{\boldsymbol{\lambda}}^Tc({\boldsymbol{x}})$. Then, we construct a modified Hessian $B_t$ based on the \textit{averaged Hessian} $\frac{1}{t}\sum_{i=0}^{t-1}{\bar{\nabla}}_{{\boldsymbol{x}}}^2\mathcal{L}_i$, and use $B_t$ to form the full Hessian $K_t$ of the Lagrangian. See \eqref{equ:B_t} and \eqref{equ:Kt} for the expressions of $B_t$ and $K_t$.
\vskip4pt
\noindent$\bullet$ Step 2: We solve Newton system $K_t{\tilde{\boldsymbol{z}}}_t = -{\bar{\nabla}}\mathcal{L}_t$ \textit{inexactly}, by performing a \textit{fixed}, say $\tau$, number of iterations of the sketch-and-project method. Then, we obtain the inexact Newton direction~${\boldsymbol{z}}_{t,\tau} = ({\bar{\Delta}}{\boldsymbol{x}}_t, {\bar{\Delta}}{\boldsymbol{\lambda}}_t)$. See \eqref{equ:pseduo} for the updating rule of the sketch-and-project method.
\vskip4pt
\noindent$\bullet$ Step 3: We adaptively select a random stepsize ${\bar{\alpha}}_t$ within the interval $[\beta_t, \eta_t]$, where $\beta_t, \eta_t=\beta_t+\chi_t$ are deterministic prespecified sequences. Finally, we update the iterate by $({\boldsymbol{x}}_{t+1}, {\boldsymbol{\lambda}}_{t+1}) = ({\boldsymbol{x}}_t, {\boldsymbol{\lambda}}_t) + {\bar{\alpha}}_t({\bar{\Delta}}{\boldsymbol{x}}_t, {\bar{\Delta}}{\boldsymbol{\lambda}}_t)$ and repeat from Step 1.
\vskip5pt
\noindent\textbf{Connections to the existing schemes}. The above StoSQP framework is related to the existing schemes \citep{Na2021Adaptive, Na2021Inequality, Berahas2021Sequential, Berahas2021Stochastic, Berahas2022Accelerating, Curtis2021Inexact, Curtis2021Worst}, but it has several important enhancements.
First, different from \cite{Na2021Adaptive, Na2021Inequality} but following from \cite{Berahas2021Sequential, Berahas2021Stochastic, Curtis2021Inexact, Curtis2021Worst}, we study a fully stochastic setup where only a single sample is generated in each~iteration for estimating the objective gradient. However, those fully stochastic schemes did not estimate the objective Hessian, and simply let the modified Hessian $B_t$ be identity matrix in the experiments. Differently, we~estimate the objective Hessian (with the same sample). This extra computation is necessary for local analysis even for deterministic problems \citep{Nocedal2006Numerical}. We emphasize that our modified Hessian $B_t$ is constructed based on the averaged Hessian $\frac{1}{t}\sum_{i=0}^{t-1}{\bar{\nabla}}_{{\boldsymbol{x}}}^2\mathcal{L}_i$ excluding the $t$-th iteration, i.e., not based on a single noisy Hessian estimate. This choice of Hessian is more effective in practice (e.g., a recent study on stochastic Newton method has demonstrated the effectiveness of such \textit{Hessian averaging} \citep{Na2022Hessian}), and it is critical for our analysis. In particular, $B_t$ is deterministic conditional on $({\boldsymbol{x}}_t, {\boldsymbol{\lambda}}_t)$, and converges to $\nabla_{{\boldsymbol{x}}}^2\mathcal{L}^\star$ if $({\boldsymbol{x}}_t, {\boldsymbol{\lambda}}_t)$ converges to $({\boldsymbol{x}}^\star, {\boldsymbol{\lambda}}^\star)$.
Second, we solve Newton systems inexactly and randomly via the sketch-and-project method. This method was initially proposed by \cite{Gower2015Randomized} for solving general linear systems. We refer to that work for other equivalent interpretations of the method and for specific matrix-free examples (e.g., randomized Kaczmarz method). \cite{Berahas2021Sequential, Berahas2021Stochastic, Curtis2021Worst} solved Newton systems exactly. \cite{Curtis2021Inexact} solved Newton systems inexactly but deterministically (via conjugate gradient (CG) and minimum residual (MINRES) methods). In addition, we perform a fixed number of iterations of the solver, so that the per-iteration computational cost does not blow-up. In contrast, \cite{Curtis2021Inexact} gradually vanished the approximation error. On the~other hand, we should mention that \cite{Curtis2021Inexact} is more adaptive than AI-StoSQP; the residual of their solver is controlled by either KKT residual or feasibility residual, both of which are computed from the iterates (although converge to zero), while our iteration budget $\tau$ is given and fixed.
Third, we allow for using any random stepsize ${\bar{\alpha}}_t$, as long as it ensures to lie in the interval $[\beta_t, \eta_t]$.~For~example, the designed stepsize selection schemes in \cite{Berahas2021Sequential, Berahas2021Stochastic, Curtis2021Inexact, Curtis2021Worst} all fit in our framework.
In this paper, we call AI-StoSQP a framework for two reasons. First, AI-StoSQP is trimmed from a complete StoSQP scheme. It preserves all the features that are essential for local asymptotic analysis (i.e. $t\rightarrow\infty$). However, for a complete StoSQP scheme, one may insert another step between Step 2 and Step 3 to select suitable penalty parameter for a certain merit function. The selected parameter may kick in $\beta_t$, $\eta_t$ as a multiplier. See \cite{Berahas2021Sequential} for the usage of the $\ell_1$ merit function, $f({\boldsymbol{x}}) + \mu\|c({\boldsymbol{x}})\|_1$. The step of penalty parameter selection is important for global analysis, while is negligible for local analysis. This is because the penalty parameter is both upper and lower bounded by deterministic thresholds, and always stabilizes for large $t$ (under suitable conditions). See, for example, \cite[Section 3.2.2]{Berahas2021Sequential} and \cite[Lemma~4.4]{Na2021Adaptive}. Thus, the local behavior of $({\boldsymbol{x}}_t, {\boldsymbol{\lambda}}_t)$ is fully characterized by the direction $({\bar{\Delta}}{\boldsymbol{x}}_t, {\bar{\Delta}}{\boldsymbol{\lambda}}_t)$ and the controlled sequences $\{\beta_t, \eta_t\}$, which are preserved by AI-StoSQP.~(Note~that the deterministic SQP has the same trim, where a unit stepsize is employed and the local (quadratic/superlinear) behavior is established by purely investigating the nature of (modified) Newton system). Second, AI-StoSQP does not suggest particular designs in Step 2 and Step 3, thus providing much flexibility. One can adopt different sketching matrices in Step 2, driven by the structure of the objective; and more importantly, can adopt different procedures to select ${\bar{\alpha}}_t$ in Step 3. For example, \cite{Berahas2021Sequential, Berahas2021Stochastic, Curtis2021Inexact, Curtis2021Worst} utilized either $\ell_1$ or $\ell_2$ merit functions, and projected a random quantity into the interval $[\beta_t, \eta_t]$ to obtain ${\bar{\alpha}}_t$. One can design a similar procedure for augmented Lagrangian merit function, as used in \cite{Na2021Adaptive, Na2021Inequality}. However, such a design would also fit in the presented framework. Thus, our analysis on AI-StoSQP is broadly applicable.
\subsection{Main results and contributions}\label{sec:1.3}
We study AI-StoSQP with \textit{decaying} $\beta_t$ and $\chi_t$. We establish three main results informally summarized as follows. The rigorous statements are stated in Sections \ref{sec:3} and \ref{sec:4}.
\begin{enumerate}[label=(\alph*),topsep=0pt]
\setlength\itemsep{0.0em}
\item \textbf{Iteration complexity:} AI-StoSQP can take at most $O(\epsilon^{-4})$ iterations to achieve $\epsilon$-stationarity for the expected KKT residual.
\item \textbf{Asymptotic convergence rate:} we have $\|({\boldsymbol{x}}_t-{\boldsymbol{x}}^\star, {\boldsymbol{\lambda}}_t-{\boldsymbol{\lambda}}^\star)\| = O(\sqrt{\beta_t\log(1/\beta_t)}) + O(\chi_t/\beta_t)$ almost surely. Thus, if $\chi_t = O(\beta_t^{3/2})$ and $\beta_t$ decays polynomially in $t$ (as commonly employed in practice), then the error of the last iterate vanishes sublinearly locally.
\item \textbf{Asymptotic normality:} we have $1/\sqrt{\beta_t}\cdot ({\boldsymbol{x}}_t-{\boldsymbol{x}}^\star, {\boldsymbol{\lambda}}_t-{\boldsymbol{\lambda}}^\star)\stackrel{d}{\longrightarrow}{\mathcal N}(0, \Xi^\star)$, where the covariance $\Xi^\star$ depends on the sketching distribution employed in the iterative solver. Furthermore, we establish the Berry-Esseen bound to measure quantitatively how quickly the distribution function of $({\boldsymbol{x}}_t, {\boldsymbol{\lambda}}_t)$ converges, and we provide a practical estimator for $\Xi^\star$.
\end{enumerate}
\noindent Our results contribute to the literature on StoSQP mentioned in Section \ref{sec:1.2}. \cite{Na2021Adaptive, Na2021Inequality, Berahas2021Sequential, Berahas2021Stochastic, Berahas2022Accelerating, Curtis2021Inexact} showed global convergence for various StoSQP schemes; \cite{Curtis2021Worst} showed iteration complexity for an \textit{exact} StoSQP with \textit{constant} $\beta_t$; none of them provided a local view of StoSQP. To be specific, our main results (a)-(c) lead to the following four-fold contributions.
\begin{enumerate}[label=(\arabic*),topsep=0pt]
\setlength\itemsep{0.0em}
\item We show that the KKT residual $\|\nabla\mathcal{L}_t\|$ converges to zero almost surely, which differs~from the convergence in expectation showed in prior works. Thus, our local rate also holds almost~surely.
\item Our result (a) relies on a \textit{non-asymptotic} convergence rate of $\frac{1}{t}\sum_{i=0}^{t-1}\mathbb{E}[\|\nabla\mathcal{L}_i\|]$ for decaying~$\beta_t$, while \cite{Curtis2021Worst} showed a similar non-asymptotic result for constant $\beta_t$. By non-asymptotic we mean the result holds for any $t\geq 0$, which distinguishes from the asymptotic results in (b) and (c) that hold for sufficiently large $t$ (i.e., (b) and (c) are local results).
\hskip 0.3cm (\cite{Curtis2021Worst} also established $O(\epsilon^{-4})$ iteration complexity, however, we would like to mention that two works are not fully comparable. That work involves the step of penalty parameter selection for the $\ell_1$ merit function, which complicates the analysis; requires extra conditions; and is inessential for our analysis in (b) and (c). In this sense, (a) is only our by-product result, and is not as complete as the one in \cite{Curtis2021Worst}. However, \cite{Curtis2021Worst} can only perform finite SQP iterations since $\beta_t$ in their scheme is set as the reciprocal of the iteration length.)
\item Our results (b) and (c) show the local behavior of the last StoSQP iterate, and we provide a statistical view of the scheme. To our knowledge,~such a statistical view is missing in all existing literature on constrained optimization. However, it is important in real parameter estimation problems. When we apply StoSQP for estimating the true model parameter ${\boldsymbol{x}}^\star$, a natural task is to infer ${\boldsymbol{x}}^\star$ given stochastic iterates generated by StoSQP. The results (b) and (c) precisely characterize the uncertainty of the scheme, which in each iteration includes the randomness of sample, the randomness of stepsize, and the randomness of solver. Such characterization enables the inference of ${\boldsymbol{x}}^\star$ (and ${\boldsymbol{\lambda}}^\star$) based on StoSQP iterates, and is novel to the literature.
\item For unconstrained stochastic optimization (we review in Section \ref{sec:1.4}), the asymptotic convergence analysis involving a random stepsize is open even for first-order methods, and involving an inexact randomized Newton direction is open for second-order methods. This paper directly achieves both for constrained stochastic optimization.
\end{enumerate}
\subsection{Literature review}\label{sec:1.4}
This paper relates to prior work both on constrained optimization and on unconstrained optimization.
\vskip5pt
\noindent \textbf{Constrained optimization}. As mentioned, there is a growing body of literature on designing various StoSQP schemes for solving Problem \eqref{pro:1}. Compared to penalty methods and augmented Lagrangian methods, SQP methods preserve the problem structure, are more robust to initialization, and do not suffer ill-conditioning issues.
\cite{Berahas2021Sequential} designed a very first StoSQP scheme. At each iteration, the scheme adaptively selects a penalty parameter of the $\ell_1$ merit function to ensure the Newton direction generates a sufficient decrease on the merit function; and then selects a random stepsize ${\bar{\alpha}}_t$ based on sequences $\beta_t$ and $\chi_t = O(\beta_t^2)$, such that $\beta_t\leq {\bar{\alpha}}_t\leq \eta_t=\beta_t+\chi_t$.
An alternative StoSQP scheme is designed in \cite{Na2021Adaptive}, where the authors embedded stochastic line search into StoSQP to get rid of the prespecified sequences $\beta_t, \chi_t$. That algorithm is more adaptive than the algorithm of \cite{Berahas2021Sequential}, while requiring a more stringent setup for line search---one has to generate batch samples with an increasing batch size in each iteration.
Subsequently, \cite{Berahas2021Stochastic} designed a StoSQP to remove constraint qualification condition; \cite{Curtis2021Inexact} designed an inexact StoSQP to solve Newton systems approximately; \cite{Na2021Inequality} designed an active-set StoSQP to enable inequality constraints; \cite{Berahas2022Accelerating} designed an accelerated StoSQP by applying variance reduction technique on finite-sum objective.
All these works showed global convergence of different StoSQP schemes. In addition to these works, \cite{Oztoprak2021Constrained, Sun2022Trust} considered optimization with noisy functions. Those analyses require known, deterministic, and bounded noise, and thus they are not suited for the considered problems in \eqref{pro:1}.
\vskip 5pt
\noindent\textbf{Unconstrained optimization}. The asymptotic rate of convergence and asymptotic normality have been established for the averaged stochastic gradient descent (ASGD) \citep{Polyak1992Acceleration}. Subsequently, results on the asymptotic normality of other first-order methods and on the construction of the covariance estimator have been reported \citep{Chen2020Statistical, Chen2021First, Zhu2021Online}. The~vast majority of the existing works on the asymptotic analysis of SGD focused on the averaged iterate, and excluded the stepsize~$1/t$ due to some technical challenges. Recently, the asymptotic analysis of stochastic Newton methods has been proposed. \cite{Bercu2020Efficient} designed a Newton scheme for logistic objective,~and \cite{Boyer2020asymptotic} extended to general strongly convex objectives. Compared to the literature on first-order methods, both works showed the normality of the \textit{last} iterate with $1/t$ stepsize. However, those analyses are not applicable for Problem \eqref{pro:1} due to the following reasons.
First, \cite{Bercu2020Efficient} and \cite{Boyer2020asymptotic} studied regression problems, where the Hessian is sum of rank one matrices so that the schemes directly update the Hessian~inverse via Sherman–Morrison formula. This step is not suitable for our general objective. Second, those two papers computed the Hessian inverse to have exact Newton direction. Although the Hessian inverse requires less computation for regression task, this is not the case for Problem \eqref{pro:1}. Differently,~we~allow the scheme to solve Newton systems inexactly and randomly. Third, those two papers only studied $1/t$ deterministic stepsize, while we enable an adaptive \textit{random} stepsize, and the controlled sequence $\beta_t$ can have a general decay rate. Our analysis demonstrates that such extension is nontrivial since the covariance $\Xi^\star$ depends on the decay rate. Fourth, the Berry-Esseen bound is missing in these two papers, which provides a quantitative understanding on the convergence of the distribution function of the iterate. Fifth, our covariance $\Xi^\star$ is more complex than those in the two papers due to the additional sources of randomness (e.g., the randomness in the stepsize and the solver). Therefore, we have to provide a computable estimator of $\Xi^\star$ to make our theory practical.
We would also like to mention a different, but important, line of literature on solving stochastic objectives (with or without constraints) via sample-average approximation (SAA) methods. See, for example, \cite{Shapiro1993Asymptotic, Kleywegt2002Sample, Ahmed2008Solving} for applying SAA on different problems and \cite{Ruszczynski2003Stochastic, Shapiro2014Lectures} for surveys. The SAA methods approximate a stochastic objective with some sampling schemes (such as Monte~Carlo), and apply deterministic solvers for solving the approximated objective. In contrast, we consider stochastic approximation setup, where we apply a stochastic solver (i.e. StoSQP) for solving the original (stochastic) objective. The solver utilizes stochastic gradient and Hessian of the objective that are estimated by sampling.
\vskip 5pt
\noindent\textbf{Notation}.
We use boldface letters to denote column vectors, except that ${\boldsymbol 0}$ may also denote the~zero matrix.
$I$ denotes the identity matrix, whose dimension (and the dimension of ${\boldsymbol 0}$) is clear from~the context.
We use $\|\cdot\|$ to denote the $\ell_2$ norm for vectors and the spectral norm for matrices.
For scalars $a, b$, $a\vee b = \max(a,b)$ and $a\wedge b = \min(a,b)$.
We use $O(\cdot)$ to denote big O notation in the usual sense.
In particular, $f(x) = O(g(x))$ means $|f(x)|$ or $\|f(x)\| \leq Cg(x)$ for a constant~$C$ that is independent~of~$x$.
When $x$ is the iteration index $t$, $f_t=O(g_t)$ if $|f_t|$ or $\|f_t\|\leq Cg_t$ for sufficiently large $t$.
In this case, we use $f_t=o(g_t)$ if $|f_t|/g_t\rightarrow 0$ as $t\rightarrow \infty$.
For a sequence of compatible matrices $\{A_i\}_i$, we let $\prod_{k=i}^{j}A_k = A_jA_{j-1}\cdots A_i$ if $j\geq i$ and $I$ if $j<i$ (similar for scalar sequence).
We use $\bar{(\cdot)}$ to denote a random quantity that depends on a generated sample (except for the iterate).
We reserve the notation $G({\boldsymbol{x}})$ to denote the Jacobian matrix of constraints, that is $G({\boldsymbol{x}}) = \nabla^Tc({\boldsymbol{x}}) = (\nabla c_1({\boldsymbol{x}}),\ldots,\nabla c_m({\boldsymbol{x}}))^T\in \mathbb{R}^{m\times d}$.
\vskip5pt
\noindent\textbf{Structure of the paper}.
We introduce AI-StoSQP in Section \ref{sec:2}; and we establish global~almost sure convergence in Section \ref{sec:3}.
We establish the asymptotic convergence rate and asymptotic normality for AI-StoSQP iterates in Section \ref{sec:4}.
Experiments and conclusions are presented in Sections \ref{sec:5} and \ref{sec:6}, respectively. We defer all proofs to the appendix.
\section{An Adaptive Inexact StoSQP Framework}\label{sec:2}
In this section, we present AI-StoSQP framework. Let $\mathcal{L}({\boldsymbol{x}},{\boldsymbol{\lambda}}) = f({\boldsymbol{x}}) + {\boldsymbol{\lambda}}^Tc({\boldsymbol{x}})$ be the Lagrangian function of Problem \eqref{pro:1} where ${\boldsymbol{\lambda}}\in \mathbb{R}^m$ is the dual vector.
Under certain constraint qualifications, a necessary condition for $({\boldsymbol{x}}^\star, {\boldsymbol{\lambda}}^\star)$ being a local solution is the KKT conditions: $\nabla_{{\boldsymbol{x}}}\mathcal{L}^\star = \nabla f({\boldsymbol{x}}^\star) + G^T({\boldsymbol{x}}^\star){\boldsymbol{\lambda}}^\star = {\boldsymbol 0}$ and $\nabla_{{\boldsymbol{\lambda}}}\mathcal{L}^\star = c({\boldsymbol{x}}^\star) = {\boldsymbol 0}$.
We let $B_0 = I \in \mathbb{R}^{d\times d}$. Given the current iterate $({\boldsymbol{x}}_t, {\boldsymbol{\lambda}}_t)$, we let $c_t = c({\boldsymbol{x}}_t)$ (similarly, $G_t = G({\boldsymbol{x}}_t)$, $\nabla\mathcal{L}_t = \nabla\mathcal{L}({\boldsymbol{x}}_t,{\boldsymbol{\lambda}}_t)$, etc.) for shorthand. AI-StoSQP performs the following three steps.
\vskip 5pt
\noindent\textbf{Step 1: Estimate the gradient and Hessian}.
We generate a single sample $\xi_t\sim {\mathcal P}$ and~compute
\begin{equation*}
{\bar{g}}_t = \nabla f({\boldsymbol{x}}_t; \xi_t)\quad\quad \text{ and }\quad\quad {\bar{H}}_t = \nabla^2f({\boldsymbol{x}}_t; \xi_t).
\end{equation*}
Based on this, we let ${\bar{\nabla}}_{{\boldsymbol{x}}}^2\mathcal{L}_t = {\bar{H}}_t + \sum_{i=1}^m({\boldsymbol{\lambda}}_t)_i\nabla^2c_i({\boldsymbol{x}}_t)$ be the estimated Hessian of the Lagrangian.
We also define the modified Hessian (used in the Newton system \eqref{equ:Newton}) as
\begin{equation}\label{equ:B_t}
B_t = \frac{1}{t}\sum_{i=0}^{t-1}{\bar{\nabla}}_{{\boldsymbol{x}}}^2\mathcal{L}_i + \Delta_t,
\end{equation}
where $\Delta_t = \Delta({\boldsymbol{x}}_t, {\boldsymbol{\lambda}}_t)$ is a regularization term to let $B_t$ be positive definite in the space $\{{\boldsymbol{x}}: G_t{\boldsymbol{x}} = {\boldsymbol 0}\}$, which can simply be Levenberg-Marquardt type modification.
Intuitively, we hope $\Delta_t$ vanishes~when $({\boldsymbol{x}}_t, {\boldsymbol{\lambda}}_t)$ converges to a local solution.
We emphasize that we do not use $\bar{B}_t$ notation since $B_t$ (and $\Delta_t$) is deterministic, conditioned on $({\boldsymbol{x}}_t, {\boldsymbol{\lambda}}_t)$. $B_t$ does not utilize the sample $\xi_t$ via ${\bar{H}}_t$. In other~words, the calculation of ${\bar{H}}_t$ is only for preparation of the next iteration.
\vskip 3pt
\noindent\textbf{Step 2: Solve the Newton system}.
We let ${\bar{\nabla}}_{{\boldsymbol{x}}}\mathcal{L}_t = {\bar{g}}_t + G_t^T{\boldsymbol{\lambda}}_t$ and solve the Newton system
\begin{equation}\label{equ:Newton}
\begin{pmatrix}
B_t & G_t^T\\
G_t & {\boldsymbol 0}
\end{pmatrix}\begin{pmatrix}
{\tilde{\Delta}}{\boldsymbol{x}}_t\\
{\tilde{\Delta}}{\boldsymbol{\lambda}}_t
\end{pmatrix} = -\begin{pmatrix}
{\bar{\nabla}}_{{\boldsymbol{x}}}\mathcal{L}_t\\
c_t
\end{pmatrix}.
\end{equation}
Instead of solving \eqref{equ:Newton} exactly, we apply randomized iterative solvers introduced in \cite{Gower2015Randomized}, which are competitive or better than deterministic solvers in many cases \citep{Strohmer2008Randomized}. The core idea is the \textit{sketch-and-project} step. In particular, we let
\begin{equation}\label{equ:Kt}
K_t = \begin{pmatrix}
B_t & G_t^T\\
G_t & {\boldsymbol 0}
\end{pmatrix}, \quad {\bar{\nabla}}\mathcal{L}_t = \begin{pmatrix}
{\bar{\nabla}}_{{\boldsymbol{x}}}\mathcal{L}_t\\
c_t
\end{pmatrix}, \quad {\tilde{\boldsymbol{z}}}_t =\begin{pmatrix}
{\tilde{\Delta}}{\boldsymbol{x}}_t\\
{\tilde{\Delta}}{\boldsymbol{\lambda}}_t
\end{pmatrix},
\end{equation}
and consider solving $K_t{\boldsymbol{z}}_t = -{\bar{\nabla}}\mathcal{L}_t$ with solution ${\tilde{\boldsymbol{z}}}_t$. The $j$-th recursion has the form (${\boldsymbol{z}}_{t,0} = {\boldsymbol 0}$)
\begin{equation}\label{equ:sap}
{\boldsymbol{z}}_{t,j+1} = \arg\min_{{\boldsymbol{z}}}\|{\boldsymbol{z}} - {\boldsymbol{z}}_{t,j}\|^2\quad \quad \text{subject to} \quad S_{t,j}^TK_t{\boldsymbol{z}} = -S_{t,j}^T{\bar{\nabla}}\mathcal{L}_t,
\end{equation}
where $S_{t,j} \in \mathbb{R}^{(d+m)\times q}$ is a random sketching matrix. Its dimension $q$ can also be random. Let us denote its randomness by variable $\zeta_{t,j}$. We assume $\{\zeta_{t,j}\}_j$ are independent and identically~distributed, and they are also independent of $\xi_t$. That is, how we generate the sketching matrix for solving the Newton system \eqref{equ:Newton} is independent of how we generate samples for estimating the derivatives~of~$f$. (This is reasonable in practice.) By this setup, we have $S_{t,j}\stackrel{iid}{\sim} S$ and $S$ denotes sketching distribution.
By an equivalent formula of \eqref{equ:sap} in \cite[(2.7)]{Gower2015Randomized}, we derive a recursion
\begin{equation}\label{equ:pseduo}
{\boldsymbol{z}}_{t,j+1} = {\boldsymbol{z}}_{t,j} - K_tS_{t,j}(S_{t,j}^TK_t^2S_{t,j})^\dagger S_{t,j}^T(K_t{\boldsymbol{z}}_{t,j} +{\bar{\nabla}}\mathcal{L}_t),
\end{equation}
where $(\cdot)^\dagger$ denotes the Moore–Penrose pseudoinverse.
Although \eqref{equ:pseduo} involves a pseudoinverse, when $q=1$ the quantity $S_{t,j}^TK_t^2S_{t,j}$ reduces to a scalar and we then have a matrix-free update.~See~the randomized Kaczmarz method in \cite{Strohmer2008Randomized}, for example.
We perform~$\tau\geq 1$ iterations of \eqref{equ:pseduo} and let $({\bar{\Delta}}{\boldsymbol{x}}_t, {\bar{\Delta}}{\boldsymbol{\lambda}}_t) \coloneqq {\boldsymbol{z}}_{t,\tau}$ be the inexact Newton direction.
With \eqref{equ:pseduo} and defining $C_{t,j} = I - K_tS_{t,j}(S_{t,j}^TK_t^2S_{t,j})^\dagger S_{t,j}^TK_t$, we have
\begin{equation}\label{equ:z:recur}
{\boldsymbol{z}}_{t,\tau} - {\tilde{\boldsymbol{z}}}_t = C_{t,\tau-1}({\boldsymbol{z}}_{t,\tau-1}-{\tilde{\boldsymbol{z}}}_t) = \cdots = \rbr{\prod_{j = 0}^{\tau-1}C_{t,j}}({\boldsymbol{z}}_{t,0}-{\tilde{\boldsymbol{z}}}_t) = -\rbr{\prod_{j = 0}^{\tau-1}C_{t,j}}{\tilde{\boldsymbol{z}}}_t\coloneqq \tilde{C}_t{\tilde{\boldsymbol{z}}}_t.
\end{equation}
Theoretically, we can show that, if $\mathbb{E}[I - C_{t,j}]$ is invertible, then $\|\mathbb{E}[C_{t,j}]\|<1$.
This implies that $\|\mathbb{E}[\tilde{C}_t\mid {\boldsymbol{x}}_t, {\boldsymbol{\lambda}}_t]\|$ vanishes to zero exponentially in $\tau$, and further implies that $\|\mathbb{E}[{\boldsymbol{z}}_{t,\tau} - {\tilde{\boldsymbol{z}}}_t\mid {\boldsymbol{x}}_t, {\boldsymbol{\lambda}}_t, \xi_t]\|$ vanishes to zero exponentially in $\tau$ due to \eqref{equ:z:recur}.
Moreover, if $\mathbb{E}[I - C_{t,j}]$ is positive definite, a similar guarantee holds for $\mathbb{E}[\|{\boldsymbol{z}}_{t,\tau} - {\tilde{\boldsymbol{z}}}_t\|\mid {\boldsymbol{x}}_t, {\boldsymbol{\lambda}}_t, \xi_t]$.
We refer to \cite[Section 4]{Gower2015Randomized} for more details, and we state necessary results later (cf. Lemma \ref{lem:1}) to make our paper self-contained.
We also mention that we need $\tau$ to be large, but it is independent of $t$. In other words, we do not require a more and more precise Newton direction approximation as the iteration proceeds. This is in contrast to the inexact scheme in \cite{Curtis2021Inexact}, which controls the approximation error by the KKT residual and a stepsize related~sequence~$\beta_t$ (cf. \eqref{equ:sandwich}). In other words, \cite{Curtis2021Inexact} controls the inexactness more adaptively, while the approximation error also has to diminish to zero.
\vskip3pt
\noindent\textbf{Step 3: Update the iterate with a random stepsize}.
With the inexact direction $({\bar{\Delta}}{\boldsymbol{x}}_t, {\bar{\Delta}}{\boldsymbol{\lambda}}_t)$, we update the iterate $({\boldsymbol{x}}_t, {\boldsymbol{\lambda}}_t)$ with a random stepsize ${\bar{\alpha}}_t$
\begin{equation}\label{equ:update}
\begin{pmatrix}
{\boldsymbol{x}}_{t+1}\\
{\boldsymbol{\lambda}}_{t+1}
\end{pmatrix} = \begin{pmatrix}
{\boldsymbol{x}}_t\\
{\boldsymbol{\lambda}}_t
\end{pmatrix} + {\bar{\alpha}}_t\begin{pmatrix}
{\bar{\Delta}}{\boldsymbol{x}}_t\\
{\bar{\Delta}}{\boldsymbol{\lambda}}_t
\end{pmatrix}.
\end{equation}
Certainly, ${\bar{\alpha}}_t$ depends on the randomness of $\xi_t$ and $\{\zeta_{t,j}\}_j$. We allow for any adaptive scheme for selecting ${\bar{\alpha}}_t$, but we require ${\bar{\alpha}}_t$ to satisfy a sandwich condition:
\begin{equation}\label{equ:sandwich}
0< \beta_t\leq {\bar{\alpha}}_t \leq \eta_t\quad \text{ with }\quad \eta_t = \beta_t + \chi_t.
\end{equation}
Here, $\{\beta_t, \eta_t\}$ are deterministic prespecified upper and lower bound sequences, and $\chi_t$ is the gap.~We will impose conditions on these sequences later. The particular adaptive schemes designed in \cite{Berahas2021Sequential, Berahas2021Stochastic, Curtis2021Inexact, Curtis2021Worst} all satisfy the above sandwich condition (e.g., \cite[Lemma 3.6]{Berahas2021Sequential}); and our conditions imposed later on $\{\beta_t, \eta_t, \chi_t\}$ are satisfied by those studies as~well.
It seems that upper and lower bounds in \eqref{equ:sandwich} reduce the difficulty for studying a random stepsize.
However, as revealed by the series of works in \cite{Berahas2021Sequential, Berahas2021Stochastic, Curtis2021Inexact, Curtis2021Worst}, the analysis involving a random stepsize is intrinsically different from the one for non-adaptive schemes; additional terms arise due to the adaptivity; and adaptive schemes have promising practical benefits.
These differences inspire us to enable a random stepsize in the framework (under the restriction~\eqref{equ:sandwich}).
When $\chi_t = 0$, we~arrive at a non-adaptive scheme.
We combine the above three steps and summarize the framework in Algorithm \ref{alg:1}. We provide a remark to further discuss the condition \eqref{equ:sandwich}.
\begin{algorithm}[!t]
\caption{An Adaptive Inexact StoSQP (AI-StoSQP) Framework} \label{alg:1}
\begin{algorithmic}[1]
\State\textbf{Input:} initial iterate $({\boldsymbol{x}}_0, {\boldsymbol{\lambda}}_0)$, positive sequences $\{\beta_t, \eta_t = \beta_t+\chi_t\}$, integer $\tau>0$, $B_0 = I$;
\For{$t = 0,1,2,\ldots$}
\State Generate $\xi_t$ and compute ${\bar{g}}_t = \nabla f({\boldsymbol{x}}_t;\xi_t)$, ${\bar{H}}_t = \nabla^2 f({\boldsymbol{x}}_t;\xi_t)$, ${\bar{\nabla}}_{{\boldsymbol{x}}}^2\mathcal{L}_t = {\bar{H}}_t + \sum_{i=1}^m({\boldsymbol{\lambda}}_t)_i\nabla^2c_i({\boldsymbol{x}}_t)$;
\State Compute the modified Hessian $B_t = \frac{1}{t}\sum_{i=0}^{t-1}{\bar{\nabla}}_{{\boldsymbol{x}}}^2\mathcal{L}_i + \Delta_t$ to make it positive definite in the space $\{{\boldsymbol{x}}: G_t{\boldsymbol{x}} = {\boldsymbol 0}\}$;
\State Generate $\{\zeta_{t,j}\}_{j=0}^{\tau-1}$ from certain distribution to formulate $\{S_{t,j}\}_{j=0}^{\tau-1}$, and apply \eqref{equ:pseduo} for $\tau$~times;
\State Select any random stepsize ${\bar{\alpha}}_t$ with $\beta_t\leq {\bar{\alpha}}_t\leq \eta_t$, and update the iterate as \eqref{equ:update};
\EndFor
\end{algorithmic}
\end{algorithm}
\begin{remark}
We should point out that the condition \eqref{equ:sandwich} excludes the promising stochastic line search method, as studied in \cite{Na2021Adaptive, Na2021Inequality}.
This is because there is no clear decaying trend for random stepsize that is selected by line search.
On the other hand, line search step requires to generate more and more samples to have a precise estimation for the gradient and Hessian, which is inapplicable under our fully~stochastic~setup (i.e., we only generate a single sample per iteration).
\end{remark}
To end this section, we introduce additional notation for Algorithm \ref{alg:1} that we will use later.~As mentioned in Step~2, $\zeta_t = \{\zeta_{t,j}\}_{j=0}^{\tau-1}$ denote random variables for generating sketching matrices $\{S_{t,j}\}_{j=0}^{\tau-1}$ at the $t$-th iteration.
We also allow the stepsize ${\bar{\alpha}}_t$ to depend on another random variable $\psi_t$ in addition to $\xi_t$ and $\zeta_t$ (cf. Line 6 in Algorithm \ref{alg:1}).
For the generated sequence $\{(\xi_t, \zeta_t, \psi_t)\}_t$, $\{\mathcal{F}_t\}_t$~is~its adapted filtration; that is $\mathcal{F}_t = \sigma(\{\xi_i, \zeta_i, \psi_i\}_{i=0}^t)$, $\forall t\geq 0$, is the $\sigma$-algebra generated by the randomness of $\{\xi_i, \zeta_i, \psi_i\}_{i=0}^t$.
We also let
\begin{equation*}
\mathcal{F}_{t-2/3} = \sigma(\{\xi_i, \zeta_i, \psi_i\}_{i=0}^{t-1}\cup \xi_t),\quad\quad \mathcal{F}_{t-1/3} = \sigma(\{\xi_i, \zeta_i, \psi_i\}_{i=0}^{t-1}\cup \xi_t\cup\zeta_t),
\end{equation*}
and have $\mathcal{F}_{t-1}\subseteq \mathcal{F}_{t-2/3}\subseteq \mathcal{F}_{t-1/3} \subseteq\mathcal{F}_t$.
For consistency, $\mathcal{F}_{-1}$ is the trivial $\sigma$-algebra.
Algorithm~\ref{alg:1} has a generating process as follows: given $({\boldsymbol{x}}_t, {\boldsymbol{\lambda}}_t)$, we first generate $\xi_t$ to estimate~the~gradient ${\bar{g}}_t$ and Hessian ${\bar{H}}_t$ and derive $\mathcal{F}_{t-2/3}$; then we generate $\zeta_t$ to obtain inexact Newton direction and derive $\mathcal{F}_{t-1/3}$; then we \textit{may} generate $\psi_t$ to select the stepsize ${\bar{\alpha}}_t$ and derive $\mathcal{F}_t$. For some stepsize selection procedures like \cite{Berahas2021Sequential}, ${\bar{\alpha}}_t$ is fully determined by $\xi_t, \zeta_t$ so that no random variable $\psi_t$ has to be generated. In this case, we have $\mathcal{F}_{t-1/3} = \mathcal{F}_t$. By our setup, it is easy to see that the random quantities in Algorithm \ref{alg:1} have the following recursion
\begin{align*}
\sigma({\boldsymbol{x}}_t, {\boldsymbol{\lambda}}_t)\cup\sigma(B_t) & \subseteq \mathcal{F}_{t-1}, \quad\hskip8pt \sigma({\bar{g}}_t)\cup\sigma({\bar{H}}_t)\cup\sigma({\tilde{\Delta}}{\boldsymbol{x}}_t,{\tilde{\Delta}}{\boldsymbol{\lambda}}_t)\subseteq\mathcal{F}_{t-2/3},\\
\sigma({\bar{\Delta}}{\boldsymbol{x}}_t, {\bar{\Delta}}{\boldsymbol{\lambda}}_t) & \subseteq \mathcal{F}_{t-1/3}, \quad \sigma({\bar{\alpha}}_t)\cup\sigma({\boldsymbol{x}}_{t+1}, {\boldsymbol{\lambda}}_{t+1})\cup\sigma(B_{t+1}) \subseteq \mathcal{F}_t.
\end{align*}
We also let $(\Delta{\boldsymbol{x}}_t, \Delta{\boldsymbol{\lambda}}_t)$ be the solution of \eqref{equ:Newton} but replace ${\bar{\nabla}}_{{\boldsymbol{x}}}\mathcal{L}_t$ with $\nabla_{{\boldsymbol{x}}}\mathcal{L}_t$.
\section{Global Almost Sure Convergence}\label{sec:3}
In this section, we establish an almost sure convergence for the KKT residual~$\nabla\mathcal{L}_t$ of Algorithm \ref{alg:1}, under standard assumptions. This type of convergence guarantee differs from the convergence in expectation, that is $\liminf_{t\rightarrow\infty}\mathbb{E}[\|\nabla\mathcal{L}_t\|] = 0$, established in \cite{Berahas2021Sequential, Berahas2021Stochastic, Berahas2022Accelerating, Curtis2021Inexact}.
On the other hand, those convergence in expectation results may be reformed to almost sure convergence by applying our following analysis on the $\ell_1$ (or $\ell_2$) merit function.
We utilize an exact augmented Lagrangian merit function to show the convergence. This function has the form
\begin{equation}\label{equ:AL}
\mathcal{L}_{\mu, \nu}({\boldsymbol{x}}, {\boldsymbol{\lambda}}) = \mathcal{L}({\boldsymbol{x}}, {\boldsymbol{\lambda}}) + \frac{\mu}{2}\|c({\boldsymbol{x}})\|^2 + \frac{\nu}{2}\|\nabla_{{\boldsymbol{x}}}\mathcal{L}({\boldsymbol{x}}, {\boldsymbol{\lambda}})\|^2\quad\quad \text{for } \mu, \nu>0.
\end{equation}
The augmented Lagrangian \eqref{equ:AL} was initially proposed by \cite{Pillo1979New}, and it has been adopted in SQP schemes for different problems \citep{Na2021fast, Na2021Global}.
The advantage of this augmented Lagrangian is that it is differentiable, and the inner product between Newton direction and the gradient $\nabla\mathcal{L}_{\mu, \nu}$ with properly chosen $\mu, \nu$ is sufficiently~negative to endure inexactness.
By a simple calculation, we have
\begin{equation}\label{equ:AL:der}
\begin{pmatrix}
\nabla_{{\boldsymbol{x}}}\mathcal{L}_{\mu, \nu}\\
\nabla_{{\boldsymbol{\lambda}}}\mathcal{L}_{\mu, \nu}
\end{pmatrix} = \begin{pmatrix}
I + \nu \nabla_{{\boldsymbol{x}}}^2\mathcal{L} & \mu G^T\\
\nu G & I
\end{pmatrix}\begin{pmatrix}
\nabla_{{\boldsymbol{x}}}\mathcal{L}\\
c
\end{pmatrix}
\end{equation}
(the evaluation point has been suppressed). We will show that $({\boldsymbol{x}}_t, {\boldsymbol{\lambda}}_t)$ of Algorithm \ref{alg:1} decreases~$\mathcal{L}_{\mu, \nu}$ in expectation in each step, and finally converges to a KKT point.
\subsection{Assumptions and preliminary results}\label{sec:3.1}
We begin by stating assumptions for showing global convergence.
\begin{assumption}\label{ass:1}
We assume $f, c$ are twice continuously differentiable, and there exists a convex compact set $\mathcal{X}\times \Lambda$ that contains the~iterates $\{({\boldsymbol{x}}_t, {\boldsymbol{\lambda}}_t)\}_t$ generated by Algorithm \ref{alg:1}.
We also assume $\nabla^2\mathcal{L}$ and $\nabla f$ are $\Upsilon_L$-Lipschitz continuous in $\mathcal{X}\times \Lambda$.
That is, for any $({\boldsymbol{x}}, {\boldsymbol{\lambda}}), ({\boldsymbol{x}}', {\boldsymbol{\lambda}}')\in \mathcal{X}\times \Lambda$,
\begin{equation}\label{Lip:mL}
\|\nabla^2\mathcal{L}({\boldsymbol{x}}, {\boldsymbol{\lambda}}) - \nabla^2\mathcal{L}({\boldsymbol{x}}', {\boldsymbol{\lambda}}')\| \leq\Upsilon_L \|({\boldsymbol{x}}-{\boldsymbol{x}}', {\boldsymbol{\lambda}}-{\boldsymbol{\lambda}}')\|,\quad \|\nabla f({\boldsymbol{x}}) - \nabla f({\boldsymbol{x}}')\|\leq \Upsilon_L\|{\boldsymbol{x}}-{\boldsymbol{x}}'\|.
\end{equation}
Furthermore, we assume that $G_t$ has full row rank, and that $\Delta_t$ is chosen such that $B_t$ satisfies~$\|B_t\|\leq \Upsilon_B$, and ${\boldsymbol{x}}^TB_t{\boldsymbol{x}}\geq \gamma_{RH}\|{\boldsymbol{x}}\|^2$, for any ${\boldsymbol{x}}\in\{{\boldsymbol{x}}: G_t{\boldsymbol{x}} = {\boldsymbol 0}\}$, for some constants $0<\gamma_{RH}\leq 1\leq \Upsilon_B$.
\end{assumption}
Assumption \ref{ass:1} is a standard assumption in SQP analysis \citep{Bertsekas1982Constrained, Nocedal2006Numerical}.
The compactness of $\mathcal{X}\times \Lambda$ ensures that $\mathcal{L}_{\mu, \nu}$ is lower bounded, and there exists a constant $\Upsilon_u\geq 1$ such that
\begin{equation}\label{equ:upper:bound}
\|\nabla^2\mathcal{L}({\boldsymbol{x}}, {\boldsymbol{\lambda}})\| \vee \|\nabla\mathcal{L}({\boldsymbol{x}}, {\boldsymbol{\lambda}})\|\vee \|\nabla f({\boldsymbol{x}})\| \leq \Upsilon_u, \quad \text{ for any } ({\boldsymbol{x}}, {\boldsymbol{\lambda}})\in \mathcal{X}\times\Lambda.
\end{equation}
The convexity of $\mathcal{X}\times \Lambda$ ensures the Taylor expansion of $\mathcal{L}_{\mu, \nu}$ at any iterate.
The Lipschitz continuity of $\nabla^2\mathcal{L}$ relaxes the thrice continuous differentiability of $f, c$, as assumed in \cite[Chapter 4.3]{Bertsekas1982Constrained}.
Note that imposing conditions on $\nabla^2\mathcal{L}$ is equivalent to imposing the same conditions on its components $\nabla_{{\boldsymbol{x}}}^2\mathcal{L}$ and $G({\boldsymbol{x}})$, although the Lipschitz constant may be different.
The Lipschitz continuity of $\nabla f$ is implied by the compactness of the set $\mathcal{X}$, while we express it out explicitly with constant $\Upsilon_L$. With the above setup and a simple calculation, we know from \eqref{equ:AL:der} that $\nabla\mathcal{L}_{\mu, \nu}$ is also Lipschitz continuous in $\mathcal{X}\times \Lambda$.
That is, for any $({\boldsymbol{x}}, {\boldsymbol{\lambda}}), ({\boldsymbol{x}}', {\boldsymbol{\lambda}}')\in \mathcal{X}\times \Lambda$,
\begin{multline}\label{equ:taylor}
\|\nabla\mathcal{L}_{\mu, \nu}({\boldsymbol{x}}, {\boldsymbol{\lambda}}) - \nabla\mathcal{L}_{\mu, \nu}({\boldsymbol{x}}', {\boldsymbol{\lambda}}')\| \\ \leq \cbr{\rbr{1+(2\nu+\mu)\Upsilon_u}\Upsilon_u + (2\nu+\mu)\Upsilon_u\Upsilon_L} \|({\boldsymbol{x}}-{\boldsymbol{x}}', {\boldsymbol{\lambda}}-{\boldsymbol{\lambda}}')\| \eqqcolon \Upsilon_{\mu, \nu}\|({\boldsymbol{x}}-{\boldsymbol{x}}', {\boldsymbol{\lambda}}-{\boldsymbol{\lambda}}')\|.
\end{multline}
Assumption \ref{ass:1} also assumes that $G_t$ has full row rank, which is a common constraint~qualification to ensure the uniqueness of the dual solution.
By the compactness of $\mathcal{X}$, we have $G_tG_t^T\succeq \gamma_{G} I$ for some constant $0<\gamma_{G} \leq 1$.
Together with the conditions on $B_t$, we know that the Newton system \eqref{equ:Newton} has a unique solution (cf. \cite[Lemma 16.1]{Nocedal2006Numerical}), and that the KKT matrix inverse $K_t^{-1}$ is uniformly bounded.
We denote by $\|K_t^{-1}\| \leq \Upsilon_K$ for $\Upsilon_K \geq 1$.
We also impose the bounded moment condition on stochastic gradient ${\bar{g}}_t$ and Hessian ${\bar{H}}_t$.
\begin{assumption}\label{ass:2}
We assume $\mathbb{E}[{\bar{g}}_t\mid {\boldsymbol{x}}_t] = \nabla f_t$, $\mathbb{E}[{\bar{H}}_t \mid {\boldsymbol{x}}_t] = \nabla^2f_t$, and assume following moment conditions \textit{when needed}: for a constant $\Upsilon_m\geq 1$,
\begin{subequations}\label{equ:BM}
\begin{align}
\text{gradient}\quad (\text{bounded 2nd moment}):\quad\quad &\mathbb{E}[\|{\bar{g}}_t - \nabla f_t\|^2\mid {\boldsymbol{x}}_t]\leq \Upsilon_m, \label{equ:BM:a}\\
(\text{bounded 3rd moment}):\quad\quad &\mathbb{E}[\|{\bar{g}}_t - \nabla f_t\|^3\mid {\boldsymbol{x}}_t]\leq \Upsilon_m, \label{equ:BM:b}\\
(\text{bounded 4th moment}):\quad\quad &\mathbb{E}[\|{\bar{g}}_t - \nabla f_t\|^4\mid {\boldsymbol{x}}_t]\leq \Upsilon_m, \label{equ:BM:c}
\intertext{and}
\text{Hessian}\quad (\text{bounded 2nd moment}): \quad\quad & \mathbb{E}[\|{\bar{H}}_t - \nabla^2 f_t\|^2\mid {\boldsymbol{x}}_t]\leq \Upsilon_m, \label{equ:BM:d}\\
(\text{bounded 2nd moment}):\quad\quad & \mathbb{E}[\sup_{{\boldsymbol{x}}\in\mathcal{X}}\|\nabla^2f({\boldsymbol{x}};\xi)\|^2] \leq \Upsilon_m. \label{equ:BM:e}
\end{align}
\end{subequations}
\end{assumption}
We write $\mathbb{E}[\cdot\mid {\boldsymbol{x}}_t]$ to express the conditional variable clearly.
It can also be written as $\mathbb{E}[\cdot\mid \mathcal{F}_{t-1}]$, which means that the expectation is taken over randomness of $\xi_t\sim {\mathcal P}$.
For conditions \eqref{equ:BM}, we~do not require all of them at once, but we impose them step by step.
In fact, \eqref{equ:BM:c} implies \eqref{equ:BM:b}, which implies \eqref{equ:BM:a}.
By the compactness of $\mathcal{X}$ (which implies the boundedness of $\|\nabla^2f_t\|$), \eqref{equ:BM:e} implies \eqref{equ:BM:d} although for a different constant.
We mention that \eqref{equ:BM:e} is imposed even for asymptotic analysis of ASGD \citep{Chen2020Statistical}.
It ensures the Lipschitz continuity of the mapping ${\boldsymbol{x}}\rightarrow \mathbb{E}[\nabla f({\boldsymbol{x}}; \xi)\nabla^T f({\boldsymbol{x}}; \xi)]$ (as proved in \eqref{Lip:map}).
See \cite[Assumption 3.2(2) and Lemma 3.1]{Chen2020Statistical} for the discussion.
We note that \cite[Assumption (A1c)]{Boyer2020asymptotic} directly assumed the mapping is continuous.
We prefer to adopt \eqref{equ:BM:e} due to two reasons.
First, \eqref{equ:BM:e} has a clear connection to \eqref{equ:BM:d}.
It is satisfied by a variety of objective functions, such as least squares regression $f({\boldsymbol{x}};\xi) = (\xi_y - {\boldsymbol{x}}^T\xi_{{\boldsymbol{x}}})^2/2$ and logistic regression $f({\boldsymbol{x}}; \xi) = \log\rbr{1+\exp\rbr{{\boldsymbol{x}}^T\xi_{{\boldsymbol{x}}}}} - \xi_y\cdot {\boldsymbol{x}}^T\xi_{{\boldsymbol{x}}}$, where $\xi = (\xi_y, \xi_{{\boldsymbol{x}}})$ is the response-feature pair with $\xi_y\in\mathbb{R}, \xi_{{\boldsymbol{x}}}\in \mathbb{R}^d$ for least squares regression and $\xi_y\in\{0, 1\}, \xi_{{\boldsymbol{x}}}\in \mathbb{R}^d$ for logistic regression, as long as $\xi_{{\boldsymbol{x}}}$ has bounded $4$-th moment.
This condition is more intuitive and checkable than the continuity of the mapping.
Second, although our asymptotic normality and Berry-Esseen bound also hold under the continuity condition on the mapping, \eqref{equ:BM:e} further enables us to design a practical covariance estimator with an explicit convergence rate (cf. Lemma \ref{lem:13}).
The covariance matrix estimation is not considered in \cite{Boyer2020asymptotic}.
In this section, we only require \eqref{equ:BM:a} for studying the convergence of $\nabla\mathcal{L}_t$ and require \eqref{equ:BM:d} for studying the convergence of $K_t$.
In the next section, we require higher order moment condition for conducting asymptotic analysis.
In terms of the distribution of sketching matrices of randomized iterative solvers, we need the following assumption.
\begin{assumption}\label{ass:3}
For any $t\geq 0$, we assume that the sketching matrices $S_{t,j}\stackrel{iid}{\sim} S$ with distribution $S$ satisfies
\begin{equation}
\mathbb{E}[K_tS(S^TK_t^2S)^\dagger S^TK_t \mid {\boldsymbol{x}}_t,{\boldsymbol{\lambda}}_t]\succeq \gamma_{S} I\quad \quad \text{ for } \gamma_{S}>0.
\end{equation}
\end{assumption}
Assumption \ref{ass:3} is required by \cite[Theorem 4.6]{Gower2015Randomized} to ensure that the iterates generated by randomized solvers converge in expectation.
It can be easily verified for some sketching matrices.
For example, in randomized Kaczmarz method, $S = \be_i$ with equal probability where $\be_i$ is the $i$-th canonical basis of $\mathbb{R}^{d+m}$.
Then, we have
\begin{equation*}
\mathbb{E}[K_tS(S^TK_t^2S)^\dagger S^TK_t \mid {\boldsymbol{x}}_t,{\boldsymbol{\lambda}}_t] \succeq \frac{\mathbb{E}[K_tS S^TK_t \mid {\boldsymbol{x}}_t,{\boldsymbol{\lambda}}_t]}{\max_j [K_t^2]_{j,j}} = \frac{K_t^2}{(d+m)\cdot \max_j [K_t^2]_{j,j}} \succeq \frac{I}{(d+m)\kappa (K_t^2)},
\end{equation*}
where $[K_t^2]_{j,j}$ denotes the $(j,j)$-entry of $K_t^2$ and $\kappa(K_t^2)$ denotes the condition number of $K_t^2$ (it is independent of $t$ due to the compactness of $\mathcal{X}\times\Lambda$).
In this case, Assumption \ref{ass:3} is implied by Assumption \ref{ass:1}.
Assumption \ref{ass:3} directly leads to the following result.
\begin{lemma}[Guarantees of randomized solvers]\label{lem:1}
Under Assumption \ref{ass:3}, the following statements hold for all $t\geq 0$.
\noindent(a): $0<\gamma_{S} \leq 1$.
\noindent(b): $\mathbb{E}[{\boldsymbol{z}}_{t,\tau} - {\tilde{\boldsymbol{z}}}_t\mid {\boldsymbol{x}}_t,{\boldsymbol{\lambda}}_t,\xi_t] = -(I - \mathbb{E}[K_tS(S^TK_t^2S)^\dagger S^TK_t \mid {\boldsymbol{x}}_t,{\boldsymbol{\lambda}}_t])^\tau{\tilde{\boldsymbol{z}}}_t \eqqcolon C_t{\tilde{\boldsymbol{z}}}_t$, and $\|C_t\|\leq \rho^\tau$ with $\rho = 1-\gamma_{S}<1$.
\noindent(c): $\mathbb{E}[\|{\boldsymbol{z}}_{t,\tau} - {\tilde{\boldsymbol{z}}}_t\|^2\mid {\boldsymbol{x}}_t,{\boldsymbol{\lambda}}_t,\xi_t ] \leq \rho^\tau \|{\tilde{\boldsymbol{z}}}_t\|^2$.
\end{lemma}
\begin{proof}
We know that
\begin{equation*}
\gamma_{S} \leq \|\mathbb{E}[K_tS(S^TK_t^2S)^\dagger S^TK_t \mid {\boldsymbol{x}}_t, {\boldsymbol{\lambda}}_t]\|\leq \mathbb{E}[\|K_tS(S^TK_t^2S)^\dagger S^TK_t\| \mid {\boldsymbol{x}}_t, {\boldsymbol{\lambda}}_t] \leq 1,
\end{equation*}
where the second inequality is by Jensen's inequality; the third inequality is since $K_tS(S^TK_t^2S)^\dagger S^TK_t$ is a projection matrix.
This shows (a).
(b) follows from \eqref{equ:z:recur} and the independence among $\{\zeta_{t,j}\}_j$.
(c) is proved by \cite[Theorem 4.6]{Gower2015Randomized}.
\end{proof}
\subsection{Almost sure convergence}\label{sec:3.2}
We now set the stage to establish the global almost sure convergence. The first result shows that $(\Delta{\boldsymbol{x}}_t, \Delta{\boldsymbol{\lambda}}_t)$ is a descent direction of $\mathcal{L}_{\mu, \nu}^t = \mathcal{L}_{\mu, \nu}({\boldsymbol{x}}_t, {\boldsymbol{\lambda}}_t)$ if $\mu$ is sufficiently large and $\nu$ is sufficiently small. A similar result has been shown in \cite[(5.15)]{Na2021fast}, while we provide an alternative proof that is more straightforward than there.
\begin{lemma}\label{lem:2}
Under Assumption \ref{ass:1}, we have
\begin{equation*}
\begin{pmatrix}
\nabla_{{\boldsymbol{x}}}\mathcal{L}_{\mu, \nu}^t\\
\nabla_{{\boldsymbol{\lambda}}}\mathcal{L}_{\mu, \nu}^t
\end{pmatrix}^T\begin{pmatrix}
\Delta{\boldsymbol{x}}_t\\
\Delta{\boldsymbol{\lambda}}_t
\end{pmatrix} \leq - \frac{\nu\gamma_{G}}{8}\cbr{\nbr{\begin{pmatrix}
\Delta{\boldsymbol{x}}_t\\
\Delta{\boldsymbol{\lambda}}_t
\end{pmatrix}}^2 + \nbr{\begin{pmatrix}
\nabla_{{\boldsymbol{x}}}\mathcal{L}_t\\
c_t
\end{pmatrix}}^2} ,
\end{equation*}
provided that
\begin{equation}\label{cond:munu}
\nu \leq \frac{\gamma_{RH}}{5(\Upsilon_B +\Upsilon_u)^2} \quad \text{ and }\quad \mu\nu \geq \frac{9}{\gamma_{G}}.
\end{equation}
\end{lemma}
\begin{proof}
See Appendix \ref{pf:lem:2}.
\end{proof}
Combining Lemmas \ref{lem:1} and \ref{lem:2}, we are able to show the following recursion.
\begin{lemma}\label{lem:3}
Under Assumptions \ref{ass:1}, \ref{ass:2}\eqref{equ:BM:a}, \ref{ass:3}, and suppose that $(\mu, \nu)$ satisfies \eqref{cond:munu} and~$\rho^\tau \leq \nu\gamma_{G}/(16\mu\Upsilon_u)$, we have \begin{equation*}
\mathbb{E}[\mathcal{L}_{\mu, \nu}^{t+1}\mid \mathcal{F}_{t-1}] \leq \mathcal{L}_{\mu, \nu}^t - \frac{\nu\gamma_{G}\beta_t}{16}\|\nabla\mathcal{L}_t\|^2 + \tilde{\Upsilon}_{\mu, \nu}(\chi_t +\eta_t^2)
\end{equation*}
where $\tilde{\Upsilon}_{\mu, \nu} = 4\mu\Upsilon_K\Upsilon_u^2(\sqrt{\Upsilon_m}\vee\Upsilon_u)\vee 8\Upsilon_{\mu, \nu}\Upsilon_K^2(\Upsilon_u^2\vee\Upsilon_m)$.
\end{lemma}
\begin{proof}
See Appendix \ref{pf:lem:3}.
\end{proof}
A similar recursion to Lemma \ref{lem:3} for the $\ell_1$ merit function was established in \cite{Berahas2021Sequential}.
By Lemma \ref{lem:3}, we are able to show an almost sure convergence of $\|\nabla\mathcal{L}_t\|$.
\begin{theorem}[Convergence of the KKT residual]\label{thm:1}
Consider Algorithm \ref{alg:1} under Assumptions \ref{ass:1}, \ref{ass:2}\eqref{equ:BM:a}, \ref{ass:3}.
Suppose $\tau$ satisfies
\begin{equation}\label{cond:tau}
\tau \geq \frac{12\log 2 + 5\log(\Upsilon_B+\Upsilon_u) + 2\log(1/(\gamma_{RH}\gamma_{G}))}{\log(1/(1-\gamma_{S}))},
\end{equation}
and $\{\beta_t, \eta_t = \beta_t+\chi_t\}$ satisfies
\begin{equation}\label{cond:step}
\sum_{t=0}^{\infty} \beta_t = \infty, \quad\quad \sum_{t=0}^{\infty} \eta_t^2 <\infty, \quad\quad \sum_{t=0}^{\infty}\chi_t <\infty,
\end{equation}
then we have $\|({\boldsymbol{x}}_{t+1}-{\boldsymbol{x}}_t, {\boldsymbol{\lambda}}_{t+1}-{\boldsymbol{\lambda}}_t)\|\rightarrow0$ and $\|\nabla\mathcal{L}_t\|\rightarrow 0$ as $t\rightarrow\infty$ almost surely.
\end{theorem}
\begin{proof}
See Appendix \ref{pf:thm:1}.
\end{proof}
Based on Theorem \ref{thm:1}, we provide a corollary about the worst-case iteration complexity~of~Algorithm \ref{alg:1}, which recovers the result (a) introduced in Section \ref{sec:1.3}. We do not need the condition \eqref{cond:step} since we provide the convergence rate of averaged expected KKT residual, $\sum_{j=0}^{t-1}\mathbb{E}[\|\nabla\mathcal{L}_j\|^2]/t$, instead of providing the convergence rate of $\|\nabla\mathcal{L}_t\|$.
\begin{corollary}\label{cor:1}
Consider Algorithm \ref{alg:1} under Assumptions \ref{ass:1}, \ref{ass:2}\eqref{equ:BM:a}, \ref{ass:3}.
Suppose $\tau$ satisfies \eqref{cond:tau}, $\beta_t = (t+1)^{-a}$, $\chi_t = (t+1)^{-b}$ where $a\in(0, 1)$ and $a<b$, we define ${\mathcal T}_{\epsilon} = \inf_{t}\cbr{t\geq 1: \mathbb{E}[\|\nabla\mathcal{L}_t\|] \leq \epsilon}$.
Then, we have
\begin{equation*}
{\mathcal T}_{\epsilon} = O\rbr{\epsilon^{-\frac{2}{a\wedge (1-a)\wedge (b-a)}}}.
\end{equation*}
In particular, if $b = 2a$ (as used in \cite{Berahas2021Sequential}), then ${\mathcal T}_{\epsilon} = O(\epsilon^{-2/\cbr{a\wedge (1-a)}})$.
In this case, the best complexity is achieved by $a = 1/2$ and we have ${\mathcal T}_{\epsilon} = O(\epsilon^{-4})$.
\end{corollary}
\begin{proof}
See Appendix \ref{pf:cor:1}.
\end{proof}
Before studying the convergence of the Hessian matrix $K_t$, we provide two remarks for Theorem \ref{thm:1} and Corollary \ref{cor:1}.
\begin{remark}\label{rem:2}
Theorem \ref{thm:1} shows that all limiting points of the iteration sequence are stationary, while it does not suggest that $({\boldsymbol{x}}_t, {\boldsymbol{\lambda}}_t)$ always converges to a stationary point or a local solution.
This is similar to deterministic SQP guarantees \cite[Theorem 18.3]{Nocedal2006Numerical}.
The~convergence of $({\boldsymbol{x}}_t, {\boldsymbol{\lambda}}_t)$ is equivalent to the convergence of $\nabla\mathcal{L}_t$ for problems where the objective is strongly convex and constraints are affine.
For nonlinear problems, if $\mathcal{X}\times\Lambda$ has a single stationary point inside, then $({\boldsymbol{x}}_t, {\boldsymbol{\lambda}}_t)$ also converges to that point.
Furthermore, if that point is a strict local solution (by chance), then $({\boldsymbol{x}}_t, {\boldsymbol{\lambda}}_t)$ converges to that strict local solution.
To~facilitate our analysis on the convergence rate of StoSQP, we assume in Assumption \ref{ass:4} that $({\boldsymbol{x}}_t, {\boldsymbol{\lambda}}_t)\rightarrow ({\boldsymbol{x}}^\star, {\boldsymbol{\lambda}}^\star)$ for a \textit{strict} local solution $({\boldsymbol{x}}^\star, {\boldsymbol{\lambda}}^\star)$, whose Hessian $K^\star = \nabla^2\mathcal{L}^\star$ is invertible (as implied by (i) $\nabla_{{\boldsymbol{x}}}^2\mathcal{L}^\star$ is positive definite in $\{{\boldsymbol{x}}: G^\star{\boldsymbol{x}} = {\boldsymbol 0}\}$; (ii) $G^\star$ has full row rank).
Note that $({\boldsymbol{x}}_t, {\boldsymbol{\lambda}}_t)\rightarrow({\boldsymbol{x}}^\star,{\boldsymbol{\lambda}}^\star)$ is equivalent to assuming that $({\boldsymbol{x}}_t, {\boldsymbol{\lambda}}_t)$ jumps into a small neighborhood of~$({\boldsymbol{x}}^\star,{\boldsymbol{\lambda}}^\star)$ for a sufficiently large $t$.
This is because $\|({\boldsymbol{x}}_{t+1}-{\boldsymbol{x}}_t, {\boldsymbol{\lambda}}_{t+1}-{\boldsymbol{\lambda}}_t)\| \vee \|\nabla\mathcal{L}_t\|\rightarrow0$ implies~that there exists an attraction neighborhood around $({\boldsymbol{x}}^\star, {\boldsymbol{\lambda}}^\star)$---once $({\boldsymbol{x}}_t, {\boldsymbol{\lambda}}_t)$ lies in the neighborhood, then all the following iterates stay in the neighborhood.
We emphasize that targeting on an~iteration~sequence that converges to a \textit{strict} local solution is a routine setup for convergence rate analysis of deterministic SQP (e.g., \cite{Bertsekas1982Constrained, Lucidi1990Recursive, Sun2006Optimization, Liu2011Sequential}); and it is reasonable for StoSQP as well, given the convergence guarantee in Theorem \ref{thm:1} matches the one of deterministic SQP, up to a a zero-probability event.
\end{remark}
\begin{remark}\label{rem:cor}
Corollary \ref{cor:1} complements the existing iteration complexity result in \cite{Curtis2021Worst},~in the sense that it~shows the complexity with decaying sequences $\{\beta_t, \eta_t\}$ while \cite[Theorem 2]{Curtis2021Worst} showed the complexity with constant sequences $\beta_t=\beta$ and $\eta_t = \beta + \beta^2$.
Our analysis allows one to set $\chi_t$ to be different higher orders of $\beta_t$, instead of always setting $\chi_t = O(\beta_t^2)$ as did in that work.
Our result indicates that, to achieve $\epsilon$ stationarity, $O(\epsilon^{-4})$ iterations are required at most with $\beta_t = 1/\sqrt{t+1}$ and $\chi_t = 1/(t+1)$.
The complexity of $O(\epsilon^{-4})$ matches that work as well.
We should mention that \cite{Curtis2021Worst} studied a more practical algorithm that involves the step of merit parameter selection. We remove such a step since we focus on~(local) asymptotic convergence rate.
In addition, both Corollary \ref{cor:1} and \cite{Curtis2021Worst} differ~from our main analyses in Section \ref{sec:4}. Here, we provide a \textit{non-asymptotic} convergence rate for the \textit{averaged expected} KKT residual (i.e., the result holds for any $t$); while in Section \ref{sec:4} we will provide an \textit{asymptotic almost sure} convergence rate for the \textit{last} iterate $({\boldsymbol{x}}_t, {\boldsymbol{\lambda}}_t)$ (i.e., the result holds for sufficiently large $t$).
\end{remark}
Finally, we study the convergence of $K_t$ defined in \eqref{equ:Kt}. We need the following (local) assumption.
\begin{assumption}\label{ass:4}
We assume $({\boldsymbol{x}}_t, {\boldsymbol{\lambda}}_t) \rightarrow ({\boldsymbol{x}}^\star, {\boldsymbol{\lambda}}^\star)$ almost surely to a strict local solution $({\boldsymbol{x}}^\star, {\boldsymbol{\lambda}}^\star)$ that satisfies (i) $G^\star$ has full row rank, (ii) $\nabla_{{\boldsymbol{x}}}^2\mathcal{L}^\star$ is positive definite in the space $\{{\boldsymbol{x}}: G^\star{\boldsymbol{x}} = {\boldsymbol 0}\}$.
Furthermore, we assume $\|\Delta_t\| \leq \omega_t$ with a deterministic sequence $w_t\rightarrow 0$.
\end{assumption}
Assumption \ref{ass:4} assumes that the iteration sequence converges to a local solution.
By Remark~\ref{rem:2}, it enables us to study the local rate of the iterates.
Assumption \ref{ass:4} also assumes that the Hessian modification $\Delta_t$ vanishes, which is standard in deterministic SQP analysis.
In fact, a naive way to choose $\Delta_t$ is based on checking the sign of index of inertia of the KKT matrix $K_t$.
When the inertia has correct sign, i.e., $d$ positive eigenvalues~and $m$ negative eigenvalues, we do not need any modification.
By this way, we have $\omega_t =0$ for all sufficiently large $t$.
We now show that $K_t$ converges to $K^\star = \nabla^2\mathcal{L}^\star$ in the next theorem.
\begin{theorem}[Convergence of the Hessian matrix]\label{thm:2}
Suppose Assumptions \ref{ass:2}\eqref{equ:BM:d} and \ref{ass:4} hold, and $\nabla^2\mathcal{L}$ is $\Upsilon_L$-Lipschitz continuous (cf. \eqref{Lip:mL}).
We have $K_t\rightarrow K^\star$ where $K_t$ is defined in \eqref{equ:Kt}.
\end{theorem}
\begin{proof}
See Appendix \ref{pf:thm:2}.
\end{proof}
\section{Asymptotic Convergence Rate and Normality}\label{sec:4}
In this section, we provide an explicit convergence rate for the iterate $({\boldsymbol{x}}_t, {\boldsymbol{\lambda}}_t)$ when $({\boldsymbol{x}}_t, {\boldsymbol{\lambda}}_t)$ converges to a~strict local solution $({\boldsymbol{x}}^\star, {\boldsymbol{\lambda}}^\star)$. Our main results are presented in Theorems \ref{thm:3} and \ref{thm:4}, which certify the results (b) and (c) introduced in Section \ref{sec:1.3}. These results extend the results of Section \ref{sec:3}, where we showed that the KKT residual $\|\nabla\mathcal{L}_t\|\rightarrow 0$ and the Hessian $K_t\rightarrow K^\star$. We summarize all analyses in Sections \ref{sec:3} and \ref{sec:4}, and present a complete result of AI-StoSQP in Theorem~\ref{thm:5}.
To study the convergence rate, we first establish an iteration recursion in Section \ref{sec:4.1}. The~recursion consists of three terms, which we analyze in Section \ref{sec:4.2}. Then, we establish the asymptotic convergence rate and asymptotic normality in Section \ref{sec:4.3}. To preform statistical inference of $({\boldsymbol{x}}^\star, {\boldsymbol{\lambda}}^\star)$ in practice, we propose a covariance estimator in Section \ref{sec:4.4}. All global and local results are then summarized in Section \ref{sec:4.5} with a discussion on the conditions on the sequences $\{\beta_t, \chi_t\}_t$.
\subsection{Iteration recursion}\label{sec:4.1}
From a high-level view, we can show that Algorithm \ref{alg:1} generates a complicated stochastic process
\begin{equation}\label{equ:Markov}
\begin{pmatrix}
{\boldsymbol{x}}_{t+1} - {\boldsymbol{x}}^\star\\
{\boldsymbol{\lambda}}_{t+1} - {\boldsymbol{\lambda}}^\star
\end{pmatrix} =(1 - {\bar{\alpha}}_t) \begin{pmatrix}
{\boldsymbol{x}}_t - {\boldsymbol{x}}^\star\\
{\boldsymbol{\lambda}}_t - {\boldsymbol{\lambda}}^\star
\end{pmatrix} + {\bar{\alpha}}_t\begin{pmatrix}
\btheta_{{\boldsymbol{x}}}^t\\
\btheta_{{\boldsymbol{\lambda}}}^t
\end{pmatrix} + {\bar{\alpha}}_t\begin{pmatrix}
\bdelta_{{\boldsymbol{x}}}^t\\
\bdelta_{{\boldsymbol{\lambda}}}^t
\end{pmatrix},
\end{equation}
where $(\btheta_{{\boldsymbol{x}}}^t, \btheta_{{\boldsymbol{\lambda}}}^t)$ is a martingale difference with $\mathbb{E}[(\btheta_{{\boldsymbol{x}}}^t, \btheta_{{\boldsymbol{\lambda}}}^t) \mid \mathcal{F}_{t-1}] = {\boldsymbol 0}$, brought by generating a~sample $\xi_t\sim{\mathcal P}$ and solving Newton systems via randomized solvers; and $(\bdelta_{{\boldsymbol{x}}}^t, \bdelta_{{\boldsymbol{\lambda}}}^t)$ is the remaining error~term.
Compared to the studies on unconstrained ASGD and stochastic Newton~\citep{Polyak1992Acceleration, Chen2020Statistical, Bercu2020Efficient, Boyer2020asymptotic}, the~two~main~components of AI-StoSQP---adaptivity and inexactness---kick in all terms in the recursion \eqref{equ:Markov}, and they lead to a much more challenging analysis. In short, we have to deal with a random stepsize ${\bar{\alpha}}_t$, and $\btheta^t = (\btheta_{{\boldsymbol{x}}}^t, \btheta_{{\boldsymbol{\lambda}}}^t)$ and $\bdelta^t = (\bdelta_{{\boldsymbol{x}}}^t, \bdelta_{{\boldsymbol{\lambda}}}^t)$ also contain the approximation error from randomized~solvers.
We formalize the recursion \eqref{equ:Markov} in the following lemma. To ease notation, we let $\varphi_t = (\beta_t+\eta_t)/2$.
\begin{lemma}\label{lem:4}
Algorithm \ref{alg:1} generates a sequence of iterates $({\boldsymbol{x}}_t, {\boldsymbol{\lambda}}_t)$ that can be expressed as
\begin{align*}
\begin{pmatrix}
{\boldsymbol{x}}_{t+1} - {\boldsymbol{x}}^\star\\
{\boldsymbol{\lambda}}_{t+1} - {\boldsymbol{\lambda}}^\star
\end{pmatrix} & = \mathcal{I}_{1,t} + \mathcal{I}_{2,t} + \mathcal{I}_{3,t}
\end{align*}
where
\begin{subequations}\label{rec}
\begin{align}
\mathcal{I}_{1,t} & = \sum_{i=0}^t\prod_{j=i+1}^{t}\cbr{I - \varphi_j(I+C^\star)}\varphi_i\begin{pmatrix}
\btheta_{{\boldsymbol{x}}}^i\\
\btheta_{{\boldsymbol{\lambda}}}^i
\end{pmatrix}, \label{rec:a}\\
\mathcal{I}_{2,t} & = \sum_{i=0}^t\prod_{j=i+1}^{t} \cbr{I - \varphi_j(I+C^\star)}\rbr{{\bar{\alpha}}_i - \varphi_i}\begin{pmatrix}
{\bar{\Delta}}{\boldsymbol{x}}_i\\
{\bar{\Delta}}{\boldsymbol{\lambda}}_i
\end{pmatrix}, \label{rec:b}\\
\mathcal{I}_{3,t} & = \prod_{i=0}^{t}\cbr{I - \varphi_i(I+C^\star)}\begin{pmatrix}
{\boldsymbol{x}}_0 - {\boldsymbol{x}}^\star\\
{\boldsymbol{\lambda}}_0 - {\boldsymbol{\lambda}}^\star
\end{pmatrix} + \sum_{i=0}^t\prod_{j=i+1}^{t}\cbr{I - \varphi_j(I+C^\star)}\varphi_i\begin{pmatrix}
\bdelta_{{\boldsymbol{x}}}^i\\
\bdelta_{{\boldsymbol{\lambda}}}^i
\end{pmatrix}, \label{rec:c}
\end{align}
\end{subequations}
and (see Lemma \ref{lem:1}(b) for the definition of $C_i$)
\begin{subequations}\label{rec:def}
\begin{align}
& \hskip -0.4cm C^\star = -(I - \mathbb{E}[K^\star S(S^T(K^\star)^2S)^\dagger S^TK^\star])^\tau, \label{rec:def:a}\\
& \hskip -1cm \begin{pmatrix}
\btheta_{{\boldsymbol{x}}}^i\\
\btheta_{{\boldsymbol{\lambda}}}^i
\end{pmatrix} = -(I+C_i)K_i^{-1}\begin{pmatrix}
{\bar{g}}_i - \nabla f_i\\
{\boldsymbol 0}
\end{pmatrix} + \cbr{\begin{pmatrix}
{\bar{\Delta}}{\boldsymbol{x}}_i\\
{\bar{\Delta}}{\boldsymbol{\lambda}}_i
\end{pmatrix} - (I + C_i)\begin{pmatrix}
{\tilde{\Delta}}{\boldsymbol{x}}_i\\
{\tilde{\Delta}}{\boldsymbol{\lambda}}_i
\end{pmatrix}}, \label{rec:def:b}\\
& \hskip -1cm \begin{pmatrix}
\bdelta_{{\boldsymbol{x}}}^i\\
\bdelta_{{\boldsymbol{\lambda}}}^i
\end{pmatrix} = -(I+C_i)\cbr{(K^\star)^{-1}\begin{pmatrix}
\bpsi_{{\boldsymbol{x}}}^i\\
\bpsi_{{\boldsymbol{\lambda}}}^i
\end{pmatrix} + \cbr{K_i^{-1} - (K^\star)^{-1}}\begin{pmatrix}
\nabla_{{\boldsymbol{x}}}\mathcal{L}_i\\
c_i
\end{pmatrix} } - (C_i-C^\star)\begin{pmatrix}
{\boldsymbol{x}}_i - {\boldsymbol{x}}^\star\\
{\boldsymbol{\lambda}}_i - {\boldsymbol{\lambda}}^\star
\end{pmatrix},\label{rec:def:c}\\
& \hskip -1cm \begin{pmatrix}
\bpsi_{{\boldsymbol{x}}}^i\\
\bpsi_{{\boldsymbol{\lambda}}}^i
\end{pmatrix} = \begin{pmatrix}
\nabla_{{\boldsymbol{x}}}\mathcal{L}_i\\
c_i
\end{pmatrix} - K^\star\begin{pmatrix}
{\boldsymbol{x}}_i-{\boldsymbol{x}}^\star\\
{\boldsymbol{\lambda}}_i-{\boldsymbol{\lambda}}^\star
\end{pmatrix}. \label{rec:def:d}
\end{align}
\end{subequations}
Further, under Assumptions \ref{ass:2}, \ref{ass:3}, $\btheta^i = (\btheta_{{\boldsymbol{x}}}^i, \btheta_{{\boldsymbol{\lambda}}}^i)$ is a martingale difference~with $\mathbb{E}[\btheta^i\mid \mathcal{F}_{i-1}] = {\boldsymbol 0}$.
\end{lemma}
\begin{proof}
See Appendix \ref{pf:lem:4}.
\end{proof}
From Lemma \ref{lem:4}, we see that the iteration recursion consists of three terms. $\mathcal{I}_{1,t}$ is a martingale, which further consists of the randomness of generating sample $\xi_t$ to estimate $\nabla f_t$ and~the randomness of solving Newton system \eqref{equ:Newton}. $\mathcal{I}_{2,t}$ characterizes the randomness of the stepsize ${\bar{\alpha}}_t$. $\mathcal{I}_{3,t}$ contains all the remaining terms.
We will show that $\mathcal{I}_{1,t}$ term provides the asymptotic convergence rate and asymptotic normality, ensured by the central limit theorem for martingales; and that $\mathcal{I}_{2,t}$ and $\mathcal{I}_{3,t}$ terms only contribute higher order errors as long~as we set $\chi_t = \eta_t-\beta_t$ properly.
To facilitate our analysis, we state here a preliminary continuity property for the projection~matrix $K_tS(S^TK_t^2S)^\dagger S^TK_t$, appearing in quantities $C_{t,j}, C_t$ etc. (cf.~\eqref{equ:z:recur} and Lemma \ref{lem:1}(b)). It is fairly easy to see that if $S^TK_t^2S$ is invertible (i.e., $S$ has full column rank), then the projection matrix is continuous in $K_t$ (because $(\cdot)^{-1}$ is a continuous function). However, to enable more general cases, we do not require $S$ to have full column rank. The continuity property of the~projection matrix still holds, ensured by Wedin's $\sin(\Theta)$ theorem \citep{Wedin1972Perturbation}.
\footnote{In fact, the pseudo-inverse is continuous when rank is not perturbed (which is the case in our paper). In particular, $\|A^\dagger - B^\dagger\|$ is small if $\text{rank}(A) = \text{rank}(B)$ and $\|A-B\|$ is small. See \cite[Theorem 4.1]{Wedin1973Perturbation} for details.}
\begin{lemma}\label{lem:5}
Suppose $K_t, K^\star \in \mathbb{R}^{(d+m)\times(d+m)}$ have full rank, for any $S\in \mathbb{R}^{(d+m)\times q}$, we have
\begin{equation*}
\|K_tS(S^TK_t^2S)^\dagger S^TK_t - K^\star S(S^T(K^\star)^2S)^\dagger S^TK^\star\| \leq \frac{2\|K_t-K^\star\|}{\sigma_{\min}(K^\star)}\cdot\frac{\|S\|}{\sigma_{\min}^{+}(S)},
\end{equation*}
where $\sigma_{\min}(\cdot)$ denotes the least singular value and $\sigma_{\min}^{+}(\cdot)$ denotes the least nonzero singular value.
\end{lemma}
\begin{proof}
See Appendix \ref{pf:lem:5}.
\end{proof}
Lemma \ref{lem:5} not only suggests that the projection matrix $K_tS(S^TK_t^2S)^\dagger S^TK_t$ is continuous in $K_t$ for any $S$ (given the full rankness of $K_t$, cf. Assumption \ref{ass:1}), but also provides an explicit bound on characterizing the difference of two projection matrices. In particular, the difference of two projection matrices is proportional to the difference of two Hessians.
We introduce the following condition on the sketching distribution $S$ and present a corollary.
\begin{assumption}\label{ass:5}
The sketching distribution $S$ satisfies $\mathbb{E}[\|S\|/\sigma_{\min}^+(S)] \leq \Upsilon_S$ for some $\Upsilon_S\geq 1$.
\end{assumption}
Assumption \ref{ass:5} is equivalent to assuming that the condition number of $S$, that is $\|S\|\|S^\dagger\|$, has bounded expectation.
This assumption is mild, and is satisfied with $\Upsilon_S=1$ for any sketching distribution if $S$ is a vector, such as $S=\be_i\in\mathbb{R}^{d+m}$ for $i=1,\ldots,d+m$ with~equal probability as used in randomized Kaczmarz method.
\begin{corollary}\label{cor:2}
Suppose $K_t, K^\star$ have full rank and $K_t\rightarrow K^\star$ almost surely, then $C_t\rightarrow C^\star$ almost surely.
Furthermore, if Assumption \ref{ass:5} holds, then we quantitatively have
\begin{equation*}
\|C_t-C^\star\| \leq \frac{2\tau\Upsilon_S}{\sigma_{\min}(K^\star)}\|K_t - K^\star\|.
\end{equation*}
\end{corollary}
\begin{proof}
See Appendix \ref{pf:cor:2}.
\end{proof}
By Corollary \ref{cor:2}, we know that $\bdelta^t$ in \eqref{rec:def:c} satisfies $\bdelta^t = o(\|({\boldsymbol{x}}_t - {\boldsymbol{x}}^\star, {\boldsymbol{\lambda}}_t-{\boldsymbol{\lambda}}^\star)\|)$.
This is because $\bpsi^t = O(\|({\boldsymbol{x}}_t - {\boldsymbol{x}}^\star, {\boldsymbol{\lambda}}_t-{\boldsymbol{\lambda}}^\star)\|^2)$, $K_t\rightarrow K^\star$, and $C_t\rightarrow C^\star$.
See Lemma \ref{lem:8} for details.
\subsection{Bounding $\mathcal{I}_{1,t}$, $\mathcal{I}_{2,t}$, $\mathcal{I}_{3,t}$ terms}\label{sec:4.2}
We bound $\mathcal{I}_{1,t}$, $\mathcal{I}_{2,t}$, $\mathcal{I}_{3,t}$ in \eqref{rec:a}, \eqref{rec:b}, \eqref{rec:c}, respectively. To simplify the presentation, we~introduce additional notation. We let $I+C^\star$ have an eigenvalue decomposition expressed as
\begin{equation}\label{equ:Eigen}
I+C^\star = U\Sigma U^T\quad \text{ with } \quad\Sigma = \diag(\sigma_1, \ldots, \sigma_{d+m}).
\end{equation}
Under Assumption \ref{ass:3}, full rankness of $K_t$ and $K^\star$, and $K_t\rightarrow K^\star$, Lemma \ref{lem:1}(b) and Corollary \ref{cor:2} imply that $\|C^\star\|\leq \rho^\tau$. Also, we note from \eqref{rec:def:a} that $C^\star\preceq {\boldsymbol 0}$.
Thus, for any $i=1,\ldots, d+m$,
\begin{equation}\label{equ:sigma_rho}
0<1-\rho^\tau\leq \sigma_i \leq 1\quad\quad\text{with } \rho = 1-\gamma_{S}.
\end{equation}
We generate sketch matrices $S_1,\ldots,S_\tau\stackrel{iid}{\sim}S$ and define a random matrix
\begin{equation}\label{equ:tC}
\tilde{C}^\star = -\prod_{j=1}^{\tau}(I - K^\star S_j(S_j^T(K^\star)^2S_j)S_j^TK^\star ).
\end{equation}
Obviously, we have $\mathbb{E}[\tilde{C}^\star] = C^\star$.
We also define
\begin{equation}\label{equ:Omega}
{\Omega^\star} = (K^\star)^{-1}\begin{pmatrix}
\mathbb{E}[\nabla f({\boldsymbol{x}}^\star; \xi)\nabla^T f({\boldsymbol{x}}^\star; \xi)] - \nabla f({\boldsymbol{x}}^\star)\nabla^T f({\boldsymbol{x}}^\star) & {\boldsymbol 0}\\
{\boldsymbol 0} & {\boldsymbol 0}
\end{pmatrix}(K^\star)^{-1}.
\end{equation}
With the above definitions, the next lemma establishes the asymptotic convergence rate, asymptotic normality, and Berry-Esseen bound for the martingale $\mathcal{I}_{1,t}$.
Recall that $\varphi_t = (\beta_t+\eta_t)/2$.
\begin{lemma}\label{lem:6}
Under Assumptions \ref{ass:1}, \ref{ass:2}(\ref{equ:BM:a}, \ref{equ:BM:e}), \ref{ass:3}, \ref{ass:4} and suppose
\begin{equation}\label{cond:varphi}
\lim\limits_{t\rightarrow\infty} t\rbr{1 - \frac{\varphi_{t-1}}{\varphi_t}} = \varphi<0, \quad\quad \lim\limits_{t\rightarrow\infty} t\varphi_t = \tilde{\varphi}\in(0, \infty],\quad\quad 1-\rho^\tau + \frac{\varphi}{\tilde{\varphi}}>0.
\end{equation}
Then, for any $\upsilon>0$, we have
\begin{equation}\label{equ:I_1t:as}
\mathcal{I}_{1,t} = o\rbr{\sqrt{\varphi_t\cbr{\log(1/\varphi_t)}^{1+\upsilon} }} \quad \text{ almost surely}.
\end{equation}
Furthermore, if \eqref{equ:BM:b} holds, then we have
\noindent(a) (asymptotic rate) $\mathcal{I}_{1,t} = O(\sqrt{\varphi_t\log(1/\varphi_t)})$ almost surely.
\noindent(b) (asymptotic normality) $\sqrt{1/\varphi_t}\cdot\mathcal{I}_{1,t} \stackrel{d}{\longrightarrow}{\mathcal N}(0, \Xi^\star)$ where
\begin{equation}\label{equ:Xi}
\Xi^\star = U\rbr{\Theta\circ U^T\mathbb{E}\sbr{(I+\tilde{C}^\star){\Omega^\star}(I+\tilde{C}^\star)^T U} } U^T\quad \text{with}\quad [\Theta]_{k,l} = \frac{1}{\sigma_k+\sigma_l + \frac{\varphi}{\tilde{\varphi}}}.
\end{equation}
Here, $\circ$ denotes the matrix Hadamard product.
\noindent(c) (Berry-Esseen bound) For any vector $\bw = (\bw_{{\boldsymbol{x}}}, \bw_{{\boldsymbol{\lambda}}})\in \mathbb{R}^{d+m}$ such that $\bw^T\Xi^\star\bw \neq 0$, we have
\begin{equation*}
\sup_{z\in\mathbb{R}}\abr{P\rbr{\frac{\sqrt{1/\varphi_t}\cdot \bw^T\mathcal{I}_{1,t}}{\sqrt{\bw^T\Xi^\star\bw}} \leq z} - P\rbr{{\mathcal N}(0,1)\leq z}} = O\rbr{\sqrt{\varphi_t\log(1/\varphi_t)}}.
\end{equation*}
\end{lemma}
\begin{proof}
See Appendix \ref{pf:lem:6}.
\end{proof}
As we mentioned after Assumption \ref{ass:2} in Section \ref{sec:3}, the condition \eqref{equ:BM:e} in Lemma \ref{lem:6} can be weakened to \eqref{equ:BM:d} if we additionally assume the mapping, ${\boldsymbol{x}}\rightarrow \mathbb{E}[\nabla f({\boldsymbol{x}}; \xi)\nabla^T f({\boldsymbol{x}}; \xi)]$, is continuous. (This is because \eqref{var:lim} holds immediately by the continuity of the mapping, and all the remaining proofs still apply.)
Next, we characterize $\mathcal{I}_{2,t}$ defined in \eqref{rec:b}.
Recall that $\chi_t = \eta_t-\beta_t$ is the gap of upper and lower bounds for the stepsize ${\bar{\alpha}}_t$.
\begin{lemma}\label{lem:7}
Under Assumptions \ref{ass:1}, \ref{ass:2}\eqref{equ:BM:a}, \ref{ass:3}, \ref{ass:4} and suppose \eqref{cond:varphi} holds.
Furthermore, we suppose $\chi_t$ satisfies for a constant $\chi$, and a positive constant $\upsilon'>0$ that
\begin{equation}\label{cond:chi}
\lim\limits_{t\rightarrow\infty} t\rbr{1 - \frac{\chi_{t-1}}{\chi_t}} = \chi,\quad 2(1 - \rho^\tau) + \frac{2\chi - \varphi}{\tilde{\varphi}}>0, \quad \cbr{\log(1/\chi_t)}^{1+\upsilon'}=O(1/\varphi_t).
\end{equation}
Then, we have
\begin{equation*}
\mathcal{I}_{2,t}= O\rbr{\frac{\chi_t}{\varphi_t}} \quad \text{almost surely}.
\end{equation*}
\end{lemma}
\begin{proof}
See Appendix \ref{pf:lem:7}.
\end{proof}
Next, we characterize $\mathcal{I}_{3,t}$ defined in \eqref{rec:c}.
\begin{lemma}\label{lem:8}
Under Assumptions \ref{ass:1}, \ref{ass:2}(\ref{equ:BM:a}, \ref{equ:BM:e}), \ref{ass:3}, \ref{ass:4} and suppose \eqref{cond:varphi} and \eqref{cond:chi} hold.~Then, for any $\upsilon>0$,
\begin{equation*}
\mathcal{I}_{3,t} = o\rbr{\sqrt{\varphi_t\cbr{\log(1/\varphi_t)}^{1+\upsilon}}} + o(\chi_t/\varphi_t) = o(\mathcal{I}_{1,t}+\mathcal{I}_{2,t})\quad \text{almost surely}.
\end{equation*}
Furthermore, if \eqref{equ:BM:b} holds, the above result holds with $\upsilon = 0$.
\end{lemma}
\begin{proof}
See Appendix \ref{pf:lem:8}.
\end{proof}
\subsection{Asymptotic rate and normality}\label{sec:4.3}
We combine Lemmas \ref{lem:4}, \ref{lem:6}, \ref{lem:7}, \ref{lem:8}, and derive the following asymptotic convergence rate for the iterates. The proof is omitted.
\begin{theorem}[Asymptotic convergence rate]\label{thm:3}
Under Assumptions \ref{ass:1}, \ref{ass:2}(\ref{equ:BM:a}, \ref{equ:BM:e}), \ref{ass:3}, \ref{ass:4} and suppose \eqref{cond:varphi} and \eqref{cond:chi} hold.~Then, for any $\upsilon>0$,
\begin{equation*}
\nbr{({\boldsymbol{x}}_t-{\boldsymbol{x}}^\star, {\boldsymbol{\lambda}}_t-{\boldsymbol{\lambda}}^\star)} = o\rbr{\sqrt{\varphi_t\cbr{\log(1/\varphi_t)}^{1+\upsilon}}} + O(\chi_t/\varphi_t)\quad \text{almost surely}.
\end{equation*}
Furthermore, if \eqref{equ:BM:b} holds, we have
\begin{equation*}
\nbr{({\boldsymbol{x}}_t-{\boldsymbol{x}}^\star, {\boldsymbol{\lambda}}_t-{\boldsymbol{\lambda}}^\star)} = O\rbr{\sqrt{\varphi_t\log(1/\varphi_t)}} + O(\chi_t/\varphi_t)\quad \text{almost surely}.
\end{equation*}
\end{theorem}
From Theorem \ref{thm:3}, we see that the asymptotic convergence rate consists of two terms.
The first term comes from the strong law of large number for a martingale that characterizes the random sampling in each iteration (cf. $\mathcal{I}_{1,t}$ in \eqref{rec:a}).
The second term comes from the adaptivity of random stepsizes (cf. $\mathcal{I}_{2,t}$ in \eqref{rec:b}).
It disappears when $\chi_t=0$, i.e., setting ${\bar{\alpha}}_t = \beta_t$ deterministically.
From Lemma \ref{lem:8}, we see that all the remaining terms contained in $\mathcal{I}_{3,t}$ only contribute higher~order errors.
We note that \cite{Berahas2021Sequential, Berahas2021Stochastic, Curtis2021Inexact} set $\chi_t = O(\beta_t^2)$.
In this case, $\chi_t = O(\varphi_t^2)$ and, hence, the first term dominates the convergence rate.
In fact, our theorem suggests that $\chi_t = O(\beta_t^{3/2})$ is sufficient to have the first term dominate, which allows a wider interval for selecting the stepsize ${\bar{\alpha}}_t$.
We will transfer conditions \eqref{cond:varphi} and \eqref{cond:chi} to some conditions on the sequence $\{\beta_t, \chi_t\}_t$ in~Theorem \ref{thm:5}, and justify them in Lemma \ref{lem:10}.
Before that, we establish the asymptotic normality and Berry-Esseen bound for the iterate $({\boldsymbol{x}}_t, {\boldsymbol{\lambda}}_t)$.
To show these results, we need an explicit convergence rate for $K_t$, which further allows us to provide a better rate for the term $\mathcal{I}_{3,t}$.
\begin{lemma}\label{lem:9}
Under Assumptions \ref{ass:1}, \ref{ass:2}(\ref{equ:BM:a}, \ref{equ:BM:e}), \ref{ass:3}, \ref{ass:4} and suppose \eqref{cond:varphi} and \eqref{cond:chi} hold.~Then, for any $\upsilon>0$, we have
\begin{equation*}
\|K_t-K^\star\| = o\rbr{\sqrt{\varphi_t\cbr{\log(1/\varphi_t)}^{1+\upsilon}}} + O\rbr{\frac{\chi_t}{\varphi_t}}+ \omega_t \quad \text{almost surely}.
\end{equation*}
Furthermore, if \eqref{equ:BM:b} holds, we have
\begin{equation*}
\|K_t-K^\star\| = O\rbr{\sqrt{\varphi_t\log(1/\varphi_t)}} + O\rbr{\frac{\chi_t}{\varphi_t}} +o\rbr{\sqrt{\frac{(\log t)^{1+\upsilon}}{t}}}+ \omega_t \quad \text{almost surely}.
\end{equation*}
\end{lemma}
\begin{proof}
See Appendix \ref{pf:lem:9}.
\end{proof}
We provide a better rate for $\mathcal{I}_{3,t}$. The following lemma differs from Lemma \ref{lem:8} in the bound of $\bdelta^t$.
Using Lemma \ref{lem:9}, we can have a more precise bound on $\bdelta^t$ than \eqref{pequ:19} in the proof of Lemma \ref{lem:8}. We need to impose Assumption \ref{ass:5} and make use of Corollary \ref{cor:2}.
We only present the result under \eqref{equ:BM:b} and $\chi_t = o(\varphi_t^{3/2})$.
This is because the asymptotic normality and Berry-Esseen bound, where we will apply the result, require both of them.
In particular, \eqref{equ:BM:b} is required for the asymptotic normality of $\mathcal{I}_{1,t}$ in Lemma \ref{lem:6}.
$\chi_t = o(\varphi_t^{3/2})$ is required to ensure that, compared to $\sqrt{1/\varphi_t}\cdot \mathcal{I}_{1,t}$, $\sqrt{1/\varphi_t}\cdot \mathcal{I}_{2,t}= o(1)$ contributes higher order~errors (cf. Lemma \ref{lem:7}) and does not affect the normality.
\begin{lemma}\label{lem:11}
Under Assumptions \ref{ass:1}, \ref{ass:2}(\ref{equ:BM:b}, \ref{equ:BM:e}), \ref{ass:3}, \ref{ass:4}, \ref{ass:5}, and suppose \eqref{cond:varphi} and \eqref{cond:chi} hold.
Suppose also that
\begin{equation}\label{cond:varphi_new}
\lim\limits_{t\rightarrow\infty} t\rbr{1 - \frac{\omega_{t-1}}{\omega_t}} = \omega<0, \quad 1-\rho^\tau + \frac{\varphi + (-1\wedge 2\omega)}{2\tilde{\varphi}}>0, \quad \chi<\frac{3\varphi}{2}.
\end{equation}
Then, for any $\upsilon>0$,
\begin{align*}
\mathcal{I}_{3,t} & = O\rbr{\varphi_t\log(1/\varphi_t)} + o\rbr{\sqrt{\varphi_t\log(1/\varphi_t)} \cdot \sqrt{\frac{(\log t)^{1+\upsilon}}{t}}} + O\rbr{\sqrt{\varphi_t\log(1/\varphi_t)}\cdot \omega_t} \quad \text{almost surely}.
\end{align*}
The above result also holds if $\omega_t=0$ for all sufficiently large $t$.
\end{lemma}
\begin{proof}
See Appendix \ref{pf:lem:11}.
\end{proof}
Combining Lemmas \ref{lem:4}, \ref{lem:6}, \ref{lem:7}, and \ref{lem:11}, we derive the following theorem.
\begin{theorem}[Asymptotic normality and Berry-Esseen bound]\label{thm:4}
Under Assumptions \ref{ass:1}, \ref{ass:2}(\ref{equ:BM:b}, \ref{equ:BM:e}), \ref{ass:3}, \ref{ass:4}, \ref{ass:5}, and suppose \eqref{cond:varphi}, \eqref{cond:chi}, and \eqref{cond:varphi_new} hold.
Then,
\begin{equation*}
\sqrt{1/\varphi_t}\cdot\begin{pmatrix}
{\boldsymbol{x}}_t - {\boldsymbol{x}}^\star\\
{\boldsymbol{\lambda}}_t-{\boldsymbol{\lambda}}^\star
\end{pmatrix}\stackrel{d}{\longrightarrow}{\mathcal N}({\boldsymbol 0}, \Xi^\star),
\end{equation*}
where $\Xi^\star$ is in \eqref{equ:Xi}.
Further, for any vector $\bw = (\bw_{{\boldsymbol{x}}}, \bw_{{\boldsymbol{\lambda}}})\in \mathbb{R}^{d+m}$ such that $\bw^T\Xi^\star\bw \neq 0$ and any $\upsilon>0$, we have
\begin{multline*}
\sup_{z\in\mathbb{R}}\abr{P\rbr{\frac{\sqrt{1/\varphi_t}\cdot \bw^T({\boldsymbol{x}}_t - {\boldsymbol{x}}^\star,{\boldsymbol{\lambda}}_t-{\boldsymbol{\lambda}}^\star)}{\sqrt{\bw^T\Xi^\star\bw}} \leq z} - P\rbr{{\mathcal N}(0,1)\leq z}}\\
= O\rbr{\sqrt{\varphi_t}\log\rbr{1/\varphi_t}} + O\rbr{\frac{\chi_t}{\varphi_t^{3/2}}} + o\rbr{\frac{\sqrt{\log(1/\varphi_t)(\log t)^{1+\upsilon}} }{\sqrt{t}}}+ O\rbr{\sqrt{\log(1/\varphi_t)}\cdot \omega_t}.
\end{multline*}
The above results also hold if $\omega_t=0$ for all sufficiently large $t$.
\end{theorem}
\begin{proof}
See Appendix \ref{pf:thm:4}.
\end{proof}
\subsection{An estimator of the covariance matrix }\label{sec:4.4}
Theorem \ref{thm:4} shows the asymptotic normality of $({\boldsymbol{x}}_t, {\boldsymbol{\lambda}}_t)$. However, the covariance matrix $\Xi^\star$ \eqref{equ:Xi} has a complex form. It depends on ${\Omega^\star}$ in \eqref{equ:Omega} and an expectation of a random matrix $\tilde{C}^\star$ in \eqref{equ:tC}, both of which are inaccessible in practice. To perform statistical inference for $({\boldsymbol{x}}^\star, {\boldsymbol{\lambda}}^\star)$, we propose a simple estimator for $\Xi^\star$. Our estimator is independent of the sketching distribution employed in randomized solvers.
\begin{lemma}\label{lem:13}
We let
\begin{equation}\label{equ:Xi_t}
\Omega_t = K_t^{-1}\begin{pmatrix}
\frac{1}{t}\sum_{i=0}^{t-1}{\bar{g}}_i{\bar{g}}_i^T - \rbr{\frac{1}{t}\sum_{i=0}^{t-1}{\bar{g}}_i}\rbr{\frac{1}{t}\sum_{i=0}^{t-1}{\bar{g}}_i}^T & {\boldsymbol 0}\\
{\boldsymbol 0} & {\boldsymbol 0}
\end{pmatrix}K_t^{-1} \quad \text{ and }\quad \Xi_t = \frac{\Omega_t}{2+\varphi/\tilde{\varphi}}.
\end{equation}
Under Assumptions \ref{ass:1}, \ref{ass:2}(\ref{equ:BM:c}, \ref{equ:BM:e}), \ref{ass:3}, \ref{ass:4} and suppose \eqref{cond:varphi} and \eqref{cond:chi} hold.
Then, for any~$\upsilon>0$, we have
\begin{equation*}
\|\Xi^\star - \Xi_t\| =O(\rho^\tau) + O\rbr{\sqrt{\varphi_t \log(1/\varphi_t)}} +O\rbr{\frac{\chi_t}{\varphi_t}} + o\rbr{\sqrt{\frac{(\log t)^{1+\upsilon}}{t}}} + \omega_t.
\end{equation*}
\end{lemma}
\begin{proof}
See Appendix \ref{pf:lem:13}.
\end{proof}
Lemma \ref{lem:13} leads to the following corollary, which shows Berry-Esseen bound with true covariance being replaced by $\Xi_t$.
\begin{corollary}\label{cor:3}
Under Assumptions \ref{ass:1}, \ref{ass:2}(\ref{equ:BM:c}, \ref{equ:BM:e}), \ref{ass:3}, \ref{ass:4}, \ref{ass:5}, and suppose \eqref{cond:varphi}, \eqref{cond:chi}, and \eqref{cond:varphi_new} hold. Then, for any vector $\bw = (\bw_{{\boldsymbol{x}}}, \bw_{{\boldsymbol{\lambda}}})\in \mathbb{R}^{d+m}$ such that $\bw^T\Xi_t\bw \neq 0$, for any $\upsilon>0$ and all sufficiently large $\tau$, we have
\begin{multline*}
\sup_{z\in\mathbb{R}}\abr{P\rbr{\frac{\sqrt{1/\varphi_t}\cdot \bw^T({\boldsymbol{x}}_t - {\boldsymbol{x}}^\star,{\boldsymbol{\lambda}}_t-{\boldsymbol{\lambda}}^\star)}{\sqrt{\bw^T\Xi_t\bw}} \leq z} - P\rbr{{\mathcal N}(0,1)\leq z}}\\
= O\rbr{\sqrt{\varphi_t}\log\rbr{\frac{1}{\varphi_t}}} + O\rbr{\frac{\chi_t}{\varphi_t^{3/2}}} + o\rbr{\frac{\sqrt{\log(1/\varphi_t)(\log t)^{1+\upsilon}} }{\sqrt{t}}}+ O\rbr{\sqrt{\log(1/\varphi_t)}\cdot \omega_t} + O(\rho^\tau).
\end{multline*}
The above result also holds if $\omega_t=0$ for all sufficiently large $t$.
\end{corollary}
\begin{proof}
See Appendix \ref{pf:cor:3}.
\end{proof}
Combining Corollary \ref{cor:3} with Theorem \ref{thm:4}, we see that Berry-Esseen bound of using the~covariance estimator $\Xi_t$ only leads to an additional $O(\rho^\tau)$ term. This is negligible for a large $\tau$.
\subsection{A complete understanding of AI-StoSQP}\label{sec:4.5}
We summarize all results in Sections \ref{sec:3} and \ref{sec:4}, and rephrase all conditions \eqref{cond:varphi}, \eqref{cond:chi}, and \eqref{cond:varphi_new} using the input sequences $\{\beta_t, \chi_t = \eta_t-\beta_t\}$.
\begin{theorem}[Global and local convergence of AI-StoSQP]\label{thm:5}
The following results of AI-StoSQP in Algorithm \ref{alg:1} hold. The assumptions are gradually strengthened.
\noindent\textbf{(a) Global convergence.}
Suppose Assumptions \ref{ass:1}, \ref{ass:2}(\ref{equ:BM:a}), \ref{ass:3} hold; $\tau$ satisfies \eqref{cond:tau}; and the input sequence $\{\beta_t, \chi_t\}_t$ satisfies
\begin{equation}\label{cond:1}
\lim\limits_{t\rightarrow\infty} t\rbr{1 - \frac{\beta_{t-1}}{\beta_t}} = \beta\in[-1,-\frac{1}{2}),\quad \lim\limits_{t\rightarrow\infty} t\beta_t = \tilde{\beta}\in(0, \infty], \quad \lim\limits_{t\rightarrow\infty}t\rbr{1 - \frac{\chi_{t-1}}{\chi_t}} = \chi<-1,
\end{equation}
then $\|\nabla\mathcal{L}_t\|\rightarrow 0$ as $t\rightarrow\infty$ almost surely. Furthermore, if \eqref{equ:BM:d} and Assumption \ref{ass:4} hold, then $K_t\rightarrow K^\star$ as $t\rightarrow\infty$ almost surely.
\noindent\textbf{(b) Local asymptotic convergence rate.} If \eqref{equ:BM:d} is strengthen to \eqref{equ:BM:e}, and for a constant $\upsilon'>0$,
\begin{equation}\label{cond:2}
1-\rho^\tau + \frac{\beta}{\tilde{\beta}}>0,\quad 2(1-\rho^\tau) + \frac{2\chi - \beta}{\tilde{\beta}}>0, \quad \{\log(1/\chi_t)\}^{1+\upsilon'} = O(1/\beta_t),
\end{equation}
then, for any $\upsilon>0$,
\begin{equation*}
\begin{aligned}
\nbr{\begin{pmatrix}
{\boldsymbol{x}}_t- {\boldsymbol{x}}^\star\\
{\boldsymbol{\lambda}}_t - {\boldsymbol{\lambda}}^\star
\end{pmatrix}} & = o\rbr{\sqrt{\beta_t\cbr{\log(1/\beta_t)}^{1+\upsilon}}} + O(\chi_t/\beta_t)\\
\|K_t-K^\star\| & = o\rbr{\sqrt{\beta_t\cbr{\log(1/\beta_t)}^{1+\upsilon}}} + O(\chi_t/\beta_t) + \omega_t
\end{aligned} \quad\quad \text{almost surely.}
\end{equation*}
Furthermore, if \eqref{equ:BM:a} is also strengthen to \eqref{equ:BM:b}, then
\begin{equation*}
\begin{aligned}
\nbr{\begin{pmatrix}
{\boldsymbol{x}}_t- {\boldsymbol{x}}^\star\\
{\boldsymbol{\lambda}}_t - {\boldsymbol{\lambda}}^\star
\end{pmatrix}} & = O\rbr{\sqrt{\beta_t\log(1/\beta_t)}} + O(\chi_t/\beta_t)\\
\|K_t-K^\star\| & = O\rbr{\sqrt{\beta_t\log(1/\beta_t)}} + O(\chi_t/\beta_t) + o\rbr{\sqrt{(\log t)^{1+\upsilon}/t}} + \omega_t
\end{aligned}\quad \quad \text{almost surely.}
\end{equation*}
\noindent\textbf{(c) Asymptotic normality and Berry-Esseen bound.} If Assumption \ref{ass:5} holds and
\begin{equation}\label{cond:3}
\lim\limits_{t\rightarrow\infty}t\rbr{1 - \frac{\omega_{t-1}}{\omega_t}} = \omega<0,\quad\quad 1-\rho^\tau + \frac{\beta + 2\omega}{2\tilde{\beta}} >0, \quad\quad \chi < \frac{3\beta}{2}
\end{equation}
(all following results also hold if $\omega_t=0$ for large $t$; in this case, conditions that involve $\omega_t$ can be removed), then
\begin{equation*}
\sqrt{1/\beta_t}\cdot\begin{pmatrix}
{\boldsymbol{x}}_t-{\boldsymbol{x}}^\star\\
{\boldsymbol{\lambda}}_t-{\boldsymbol{\lambda}}^\star
\end{pmatrix}\stackrel{d}{\longrightarrow}{\mathcal N}({\boldsymbol 0}, \Xi^\star)
\end{equation*}
where $\Xi^\star$ is defined in \eqref{equ:Xi}, and for any vector $\bw$ such that $\bw^T\Xi^\star\bw\neq 0$,
\begin{multline}\label{BE:bound}
\sup_{z\in\mathbb{R}}\abr{P\rbr{\frac{\sqrt{1/\beta_t}\cdot \bw^T({\boldsymbol{x}}_t - {\boldsymbol{x}}^\star,{\boldsymbol{\lambda}}_t-{\boldsymbol{\lambda}}^\star)}{\sqrt{\bw^T\Xi^\star\bw}} \leq z} - P\rbr{{\mathcal N}(0,1)\leq z}}\\
= O\rbr{\sqrt{\beta_t}\log\rbr{\frac{1}{\beta_t}}} + O\rbr{\frac{\chi_t}{\beta_t^{3/2}}} + o\rbr{\frac{\sqrt{\log(1/\beta_t)(\log t)^{1+\upsilon}} }{\sqrt{t}}}+ O\rbr{\sqrt{\log(1/\beta_t)}\cdot \omega_t}.
\end{multline}
\noindent\textbf{(d) Practical covariance estimator.} If \eqref{equ:BM:b} is further strengthen to \eqref{equ:BM:c}, then we can~employ the estimator $\Xi_t$ in \eqref{equ:Xi_t}, and a similar Berry-Esseen bound holds if we replace $\bw^T\Xi^\star\bw$ by $\bw^T\Xi_t\bw$---only an additional $O(\rho^\tau)$ term appears on the right hand side of \eqref{BE:bound}.
\end{theorem}
\begin{proof}
See Appendix \ref{pf:thm:5}.
\end{proof}
We emphasize that the asymptotic convergence rate requires Assumptions \ref{ass:1}, \ref{ass:2}(\ref{equ:BM:a}, \ref{equ:BM:e}), \ref{ass:3}, \ref{ass:4}, while the asymptotic normality and Berry-Esseen bound require us to impose Assumption \ref{ass:5} and strengthen \eqref{equ:BM:a} to \eqref{equ:BM:b}, although the asymptotic rate is also improved under stronger conditions.
To adopt the covariance estimator $\Xi_t$, \eqref{equ:BM:b} need to be further strengthen to \eqref{equ:BM:c}.~However, even for the most stringent conditions \eqref{equ:BM:c} and \eqref{equ:BM:e}, they are quite common in the literature \citep{Chen2020Statistical}, and they enable wide applications including constrained least squares and~constrained logistic regression.
We let $\beta_t, \chi_t$ be polynomial in $t$ (which is convenient for implementation), and discuss the conditions \eqref{cond:1}, \eqref{cond:2}, \eqref{cond:3} on the input sequence $\{\beta_t,\chi_t\}_t$.
\begin{lemma}\label{lem:10}
Suppose $\beta_t = c_1/t^{c_2}$ and $\chi_t = \beta_t^{c_3}$.
Then,
\noindent (a) \eqref{cond:1} is satisfied if $c_1>0$, $c_2\in (0.5, 1]$, $c_3>1/c_2$.
\noindent (b) \eqref{cond:1}, \eqref{cond:2} are satisfied if
\begin{equation*}
c_2 = 1, c_3>1, c_1 >\frac{1\vee (c_3-0.5)}{1-\rho^\tau} \quad\quad \text{ OR } \quad\quad c_2\in(0.5, 1), c_1>0, c_3>1/c_2.
\end{equation*}
\noindent(c) \eqref{cond:1}, \eqref{cond:2}, \eqref{cond:3} are satisfied if
\begin{equation*}
c_2=1, c_3>1.5, c_1>\frac{(0.5-\omega)\vee (c_3-0.5)}{1-\rho^\tau}\quad\quad \text{ OR }\quad\quad c_2\in(0.5,1), c_1>0, c_3>1.5\vee1/c_2.
\end{equation*}
\end{lemma}
\begin{proof}
The result holds immediately by noting that $\beta = -c_2$, $\tilde{\beta} = c_1$ if $c_2 = 1$ and $\infty$ if $c_2<1$, and $\chi = -c_2c_3$.
\end{proof}
With the setup in Lemma \ref{lem:10}, the result in Theorem \ref{thm:5} can be further simplified.
For instance, we let $c_2\in(0.5, 1)$, $c_1>0$, $c_3=2$.
Then, \eqref{cond:1}, \eqref{cond:2}, \eqref{cond:3} are satisfied.
Thus, under Assumptions \ref{ass:1}, \ref{ass:2}(\ref{equ:BM:b}, \ref{equ:BM:e}), \ref{ass:3}, \ref{ass:4}, \ref{ass:5}, we have
\begin{equation*}
\nbr{\begin{pmatrix}
{\boldsymbol{x}}_t- {\boldsymbol{x}}^\star\\
{\boldsymbol{\lambda}}_t - {\boldsymbol{\lambda}}^\star
\end{pmatrix}} = O\rbr{\sqrt{\frac{\log t}{t^{c_2}}}}
\end{equation*}
and
\begin{equation*}
\sup_{z\in\mathbb{R}}\abr{P\rbr{\frac{\sqrt{t^{c_2}}\cdot \bw^T({\boldsymbol{x}}_t - {\boldsymbol{x}}^\star,{\boldsymbol{\lambda}}_t-{\boldsymbol{\lambda}}^\star)}{\sqrt{c_1\bw^T\Xi^\star\bw}} \leq z} - P\rbr{{\mathcal N}(0,1)\leq z}}
= O\rbr{\frac{\log t}{\sqrt{t^{c_2}}}} + O\rbr{\sqrt{\log t}\cdot \omega_t}.
\end{equation*}
The $95\%$ confidence interval of $\bw^T({\boldsymbol{x}}^\star, {\boldsymbol{\lambda}}^\star)$ is given by
\begin{equation}\label{equ:CI}
\bw^T({\boldsymbol{x}}_t, {\boldsymbol{\lambda}}_t) \pm \frac{1.96}{t^{c_2/2}}\cdot \sqrt{c_1\bw^T\Xi_t\bw}.
\end{equation}
We emphasize that our asymptotic results also hold (and the convergence rate does not change) if we solve Newton systems exactly, and/or adopt deterministic stepsizes.
The covariance matrix for solving Newton systems exactly is $\Xi^\star = {\Omega^\star}/(2+\beta/\tilde{\beta})$.
However, it is not clear to us whether the same results hold when we approximately solve Newton systems via deterministic solvers, such as with CG and MINRES methods as employed in \cite{Curtis2021Inexact}.
In that case, Lemma \ref{lem:1} does not hold and significant adjustments of the analysis are required.
\section{Numerical Experiment}\label{sec:5}
In this section, we provide experimental results of Algorithm \ref{alg:1}. We develop a Julia~implementation \citep{Siqueira2020CUTEst.jl}; and we consider solving problems in CUTEst test set \citep{Gould2014CUTEst}, in particular using problems \texttt{BYRDSPHR, HS7, HS48} as examples to validate our main theoretical results. For each problem, we perform $10^5$ SQP iterations and, in each iteration, we perform~$50$ randomized Kaczmarz iterations for solving the Newton system. That is, for each $t$, we let $S_{t,j}\stackrel{iid}{\sim} \text{Uniform}\{\be_1,\ldots,\be_{m+d}\}$ for $j=1,\ldots,50$, and perform the iteration \eqref{equ:pseduo} with ${\boldsymbol{z}}_{t,0} = {\boldsymbol 0}$. The true solution of each problem is solved by applying IPOPT solver \citep{Waechter2006implementation}. See a Julia implementation of the solver at \url{https://github.com/jump-dev/Ipopt.jl}.
Given an iterate ${\boldsymbol{x}}_t$, we generate ${\bar{g}}_t\sim{\mathcal N}(\nabla f_t + \sigma^2(I+{\boldsymbol{1}}\b1^T))$ where ${\boldsymbol{1}}\in\mathbb{R}^d$ is an all-one vector.
We also generate the $(i,j)$ and $(j,i)$ entries of ${\bar{H}}_t$ from ${\mathcal N}((\nabla^2f_t)_{i,j}, \sigma^2)$.
Here,~$\nabla f_t$~and~$\nabla^2f_t$ are provided by the package of CUTEst.
Among the considered problems, we~vary~$\sigma^2\in\{10^{-8}, 10^{-4}, 10^{-2}, 10^{-1}, 1\}$, and vary $\beta_t \in \{2/t^{0.5}, 2/t^{0.6}\}$.
We also let $\chi_t = \beta_t^2$ and $\eta_t=\beta_t+\chi_t$, and randomly pick the stepsize ${\bar{\alpha}}_t\sim\text{Uniform}([\beta_t, \eta_t])$.
Here, we let $\beta_t$ decay slower to favor larger stepsizes.
We note that, although $\beta_t=2/t^{0.5}$ is not allowed in our asymptotic analysis (see Lemma \ref{lem:10}), we still implement it to investigate the performance of this setup.
In addition, for the Hessian modification, we let
\begin{equation*}
\Delta_t = (-\lambda_{\min}(Z_t^T\sum_{i=0}^{t-1}{\bar{\nabla}}_{{\boldsymbol{x}}}^2\mathcal{L}_i Z_t)/t + 0.1)\cdot I\quad \text{whenever}\quad \lambda_{\min}(Z_t^T\sum_{i=0}^{t-1}{\bar{\nabla}}_{{\boldsymbol{x}}}^2\mathcal{L}_i Z_t)<0.
\end{equation*}
Here, $Z_t\in \mathbb{R}^{d\times (d-m)}$ has orthogonal columns that span the space $\{{\boldsymbol{x}}\in \mathbb{R}^d: G_t{\boldsymbol{x}} = {\boldsymbol 0}\}$, which~is~obtained by QR decomposition in our implementation.
Our code for implementing other problems in CUTEst test set is publicly available at \url{https://github.com/senna1128/Inf-StoSQP}.
\vskip5pt
\noindent\textbf{Convergence behavior.} We first study the convergence behavior of Algorithm \ref{alg:1}.
Figures \ref{fig:1} and \ref{fig:2} show the convergence plots of the KKT residual $\|\nabla\mathcal{L}_t\|$, the iteration error $\|({\boldsymbol{x}}_t-{\boldsymbol{x}}^\star, {\boldsymbol{\lambda}}_t-{\boldsymbol{\lambda}}^\star)\|$, and the Hessian error $\|K_t-K^\star\|$.
We note that, by the Lipschitz continuity of the Hessian, Theorem \ref{thm:5} implies that the KKT residual $\|\nabla\mathcal{L}_t\|$ has (at least) the same asymptotic convergence rate as the iterate, that is $O(\sqrt{\beta_t\log(1/\beta_t)})$ in our experiments.
From Figure \ref{fig:1}, we observe that the setting $\beta_t=2/t^{0.5}$ works well on problems \texttt{HS7, HS48}, while it does not work well on \texttt{BYRDSPHR} in terms of the iteration and the Hessian errors (see Figures \ref{C112} and \ref{C113}). However, the KKT residual still converges to a level below the theoretical prediction (see Figure \ref{C111}). Thus, under this setup, Algorithm \ref{alg:1} converges to a stationary point that is different from the~one returned by IPOPT. The sequence $\beta_t=2/t^{0.6}$ enhances the performance of~Algorithm~\ref{alg:1} on \texttt{BYRDSPHR} for both the iteration and the Hessian errors.
From all figures in Figures \ref{fig:1} and \ref{fig:2} (we should focus on the KKT residual plots in Figures \ref{C111} and \ref{C211} for \texttt{BYRDSPHR}), we see that the theoretical convergence rate precisely catches the asymptotic behavior of Algorithm \ref{alg:1}.
\vskip 5pt
\noindent\textbf{Asymptotic normality.}
We next study the asymptotic normality of the iterate $({\boldsymbol{x}}_t, {\boldsymbol{\lambda}}_t)$. To~visualize the results, we only plot the histograms for $[{\boldsymbol{x}}_t]_1-{\boldsymbol{x}}^\star_1$ and $[{\boldsymbol{\lambda}}_t]_1-{\boldsymbol{\lambda}}^\star_1$; that is we only focus on the first coordinate of the primal and dual solutions. We regard the first half of the iterations as burn-in period; thus we only plot the errors for the second half of the iterations. For each histogram, we fit a normal density by matching the mean and variance. Figures \ref{fig:3}, \ref{fig:4}, \ref{fig:5} show the histograms of three problems \texttt{BYRDSPHR, HS7, HS48}, respectively.
From Figure \ref{fig:5} (i.e. histograms of problem \texttt{HS48}), we see that all iteration errors with both~setups of $\beta_t = 2/t^{0.5}$ and $\beta_t=2/t^{0.6}$ follow a normal distribution with mean around zero. This is consistent with our theorem. From Figure \ref{fig:4}, we see that the histograms of $[{\boldsymbol{x}}_t]_1-{\boldsymbol{x}}^\star_1$ and $[{\boldsymbol{\lambda}}_t]_1-{\boldsymbol{\lambda}}^\star_1$ under the setup of $\beta_t=2/t^{0.6}$ and $\sigma^2=10^{-1}$ do not fit a normal density (see Figures \ref{H234} and \ref{H244}). In fact, from Figures \ref{C221} and \ref{C222}, we know that $({\boldsymbol{x}}_t, {\boldsymbol{\lambda}}_t)\nrightarrow({\boldsymbol{x}}^\star, {\boldsymbol{\lambda}}^\star)$ but $\|\nabla\mathcal{L}_t\|\rightarrow 0$ under this setup. Thus, the scheme converges to a stationary point that differs from the one given by IPOPT. Similar situations happen for few cases in Figure \ref{fig:3}. In particular, Figure \ref{H143} suggests that $[{\boldsymbol{\lambda}}_t]_1 \nrightarrow {\boldsymbol{\lambda}}^\star_1$ with $\beta_t=2/t^{0.6}$ and $\sigma^2=10^{-2}$, which is consistent with Figure \ref{C212}. However, Figure \ref{C211} suggests that $\|\nabla\mathcal{L}_t\|\rightarrow 0$ in this case. Overall, the histograms in Figures \ref{fig:3}, \ref{fig:4}, \ref{fig:5} suggest that the iterate $({\boldsymbol{x}}_t,{\boldsymbol{\lambda}}_t)$ follows a normal distribution with mean $({\boldsymbol{x}}^\star, {\boldsymbol{\lambda}}^\star)$ when it converges to a local solution.
\vskip 5pt
\noindent\textbf{Confidence interval.}
We finally test the effectiveness of the proposed covariance estimator. We construct $95\%$ confidence interval for the solution $({\boldsymbol{x}}^\star, {\boldsymbol{\lambda}}^\star)$. We take ${\boldsymbol{x}}^\star_1 + {\boldsymbol{\lambda}}^\star_1$ as an example. The formula of $95\%$ confidence interval is given by \eqref{equ:CI}, where $\Xi_t$ is estimated by Lemma \ref{lem:13}, and $\bw = (1,0,\ldots,0,1,0\ldots,0)$, i.e., $\bw_1 = \bw_{d+1}=1$ and all other entries are zero.
Figure \ref{fig:6} shows the confidence intervals generated by the last 100 iterations. We observe from Figures \ref{I51}-\ref{I65} that the confidence intervals cover the true value of problem \texttt{HS48} for all $\sigma^2$ varying from $10^{-8}$ to 1, under both setups of $\beta_t=2/t^{0.5}$ and $2/t^{0.6}$. For problem \texttt{HS7}, we observe from Figures \ref{I31}-\ref{I45} that the confidence intervals also cover the true value, except for one setup where $\beta_t=2/t^{0.6}$ and $\sigma^2=10^{-1}$. As explained, the failure of this setup is because $({\boldsymbol{x}}_t, {\boldsymbol{\lambda}}_t)$ does not converge to the solution returned by IPOPT. For problem \texttt{BYRDSPHR}, we observe from Figures \ref{I11}-\ref{I25} that the confidence intervals are improved significantly from $\beta_t=2/t^{0.5}$ to $\beta_t=2/t^{0.6}$. The confidence intervals with the former sequence only cover the true value when $\sigma^2=10^{-2}$, while the confidence intervals with the latter sequence cover the true value except when $\sigma^2=10^{-2}$.
We also observe from Figure \ref{fig:6} that the power of our confidence interval construction is significant.
In particular, the length of intervals varies from $10^{-3}$ to $10^{-1}$ when $\sigma^2$ varies from $10^{-8}$ to $1$; and such a small length implies a high statistical power, which enables a precise inference for the solution in practice (the solution of the optimization problem is often the true model parameter, and the inference of the model parameter is an important statistical problem).
\section{Conclusion}\label{sec:6}
We investigated a stochastic sequential quadratic programming framework called AI-StoSQP.
In particular, we allow the scheme to employ any method to select random stepsizes within an interval $[\beta_t, \eta_t]$; and we allow the scheme to solve Newton systems via randomized iterative solvers.
For~such a framework, we established the asymptotic convergence rate and asymptotic normality for the iterates.
Under suitable conditions and with proper decaying $\beta_t$ and $\eta_t$, we showed that the KKT residual $\|\nabla\mathcal{L}_t\|$ converges to zero \textit{almost surely}.
Moreover, if the sequence converges to a strict local minimum satisfying constraint qualifications, then the~\textit{last} iterate $({\boldsymbol{x}}_t, {\boldsymbol{\lambda}}_t)$ has a local rate $O(\sqrt{\beta_t\log(1/\beta_t)}) + O(\chi_t/\beta_t)$; and $\sqrt{1/\beta_t}\cdot({\boldsymbol{x}}_t-{\boldsymbol{x}}^\star, {\boldsymbol{\lambda}}_t-{\boldsymbol{\lambda}}^\star)$ converges to a Gaussian distribution with a nontrivial covariance matrix.
The covariance matrix depends on the decay rate of $\beta_t$ and the sketching distribution in the randomized solvers. An explicit form of the covariance matrix was provided.
For practical purpose, we also provided a practical estimator for~the covariance matrix.
Our analysis answered an important open problem---what is the (local) asymptotic behavior of StoSQP iterates under the fully stochastic setup. Our analysis precisely characterized the uncertainty of the Markov process generated by StoSQP schemes, which consists of the randomness of sample, randomness of solver, and randomness of stepsize. Since our analysis is independent of the choice of merit functions, we know that designing a StoSQP scheme with differentiable merit functions~under the fully stochastic setup \textit{does not} bring advantages over non-differentiable merit functions (such as the $\ell_1$ merit function), in terms of the local rate. However, our analysis does not allow one to select the stepsize by doing stochastic line search. The question still remains open whether adopting differentiable merit functions in stochastic line search can overcome the Maratos effect, lead to a faster local rate, and bring advantages over non-differentiable merit functions (as they can in deterministic line search). Therefore, the local analysis of StoSQP under the stochastic line search setup deserves studying in the future work.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 789
|
\section{INTRODUCTION}
\label{sec:intro}
The hypothesis that the early universe underwent a period of accelerating expansion, called inflation, has become an essential mechanism for explaining the flatness and homogeneity of the universe and the fluctuations found in the cosmic microwave background (CMB)\cite{Guth, Linde, Albrecht,1982PhRvL..49.1110G,1983PhRvD..28..679B}.
Inflation predicts the existence of primordial gravitational waves that would have produced a unique polarization pattern called the B-modes on the CMB\cite{1997PhRvL..78.2054S,1997PhRvD..55.7368K}. Measurement of the amplitude of these gravitational waves can be used to infer the energy scale of the potential driving the expansion\cite{2009AIPC.1141...10B}. Detection of this signal would be a dramatic confirmation of the inflation paradigm and significantly tighten constraints on inflationary models.
The Cosmology Large Angular Scale Surveyor (CLASS) is a new ground-based instrument designed to search for the large angular scale B-mode signal of the CMB from the Atacama Desert in northern Chile. This instrument will consist of four separate telescopes: one observing at 40 GHz, two observing at 90 GHz and one observing at 150 GHz. Each bandpass is optimized to observe the CMB through atmospheric windows of high transmission. Foreground removal, not sensitivity, is expected to limit the detection of the B-mode signal\cite{CMBforegrounds}. Making observations at multiple frequencies will help to distinguish between the CMB signal and the spectrally distinct synchrotron and polarized dust emission foreground signals.
Each telescope will employ a Variable-delay Polarization Modulator (VPM) as a fast front-end polarization modulator. This technique will allow the instrument to measure the polarization signal of over $65\%$ of the sky from the Atacama, and could be moved to the Northern hemisphere to observe more sky if appropriate. By targeting the B-mode signal on scales larger than $10^\circ$, CLASS will be able to detect the same B-mode amplitude as experiments with many more detectors but focused on the fainter small scale fluctuations.
A full description of the CLASS instrument and its overall science objectives will be given in a future paper. In this paper, we focus on the optical design for the 40 GHz telescope. In Section \ref{sec:overview}, we give a description of the telescope and the procedure followed to generate the design. Section \ref{sec:analysis} analyzes the performance of the design. An overview of the construction plans for the optical components is described in Section \ref{sec:construction}. Stray light and detector loading considerations from spill are discussed in Section \ref{sec:stray_light}. Section \ref{sec:tol} describes the tolerances for the design. Finally, Section \ref{sec:conclusions} presents conclusions of this work.
\section{Design Overview}
\label{sec:overview}
The CLASS 40 GHz telescope will measure the polarization of the CMB from 33 to 43 GHz with 36 transition edge sensor (TES) detector pairs. Each detector pair is coupled through microstrip waveguide (MSWG) to a planar orthogonal mode transducer (OMT) that defines the polarization basis of the detector. The passband definition, required for bolometric detectors, is performed on chip by MSWG filters between the TES and OMT. Each TES pair, the band defining filters, and the associated OMT are lithographically fabricated on a detector chip. Further details of the CLASS detectors are discussed in Rostem et al.\cite{detectors} In the time reversed sense, modes launched from the OMT pass through a smooth-walled horn antenna and through the rest of the telescope system to the sky.
Independent of the detector properties, the science goal of detecting the faint large angular scale B-mode signal translates into the following requirements for the optical design:
\begin{enumerate}
\item The telescope must have a fast polarization modulator to distinguish the cosmic signal from telescope and atmospheric drifts, i.e. $1/f$-noise.
\item The modulator must be the first optical component in the telescope to prevent instrument polarization systematic errors.
\item The resolution should be near $1.5^\circ$ to resolve polarization fluctuations greater than $10^\circ$ and to characterize foregrounds on scales smaller than the targeted B-mode signal.
\end{enumerate}
We use horn antennas as the primary beam forming apparatus. The telescope then redirects and magnifies the symmetric beams formed by the horns. The detector spacing is 38 mm; this is a practical limit to allow enough room for detector mounting and readout circuitry. The telescope design, therefore, is constrained by its two ends - the first optical component will be a polarization modulator and the focal plane will use horn antennas separated by 38 mm as the primary beam forming components.
The immediate consequence of the resolution requirement is that the entrance pupil must be $\sim37$ cm in diameter. Placing a VPM near this entrance pupil satisfies the fast front-end modulator requirements - further details of the modulator are included in section \ref{sec:VPM}.
With the detector spacing and the entrance pupil specified, the only remaining telescope parameter is the focal length, or equivalently the focal ratio $f/$. Faster $f/$ choices would increase the field of view (FOV) making it difficult to avoid blockage. Furthermore, the differential light path length through the VPM depends upon the angle of incidence on the device. The VPM efficiency can be optimized more successfully for a smaller spread of angles. Slower systems, on the other hand, make the cryogenic system exceptionally long. While there was no strict optimization, an $f/2$ system is a practical compromise between cryogenic system size and workable FOV.
Since the target scale ($10^\circ$ and larger) is larger than the beam, the mapping speed of the telescope increases as the beam separation on the sky increases and as the spill efficiency increases. For this design, the beam separation is set by the 2.4 $f/ \cdot \lambda$ horn spacing, and the beam truncation is dominated by the cold aperture stop with 10 dB edge taper.
Satisfying the above design criteria required the development of an entirely new type of telescope - the design of which is discussed in the following sections. A ray trace of the final design is shown in Figure \ref{fig:raytrace}. The optical design was generated and optimized using the ray tracing software ZEMAX\footnote{www.radiantzemax.com}. The first element, illustrated as a plane mirror in the Figure, is the VPM. Following the VPM, the primary and secondary mirrors route the signal through a Zotefoam\footnote{The window will consist of laminated sheets of HD30 manufactured by Zotefoams PLC, www.zotefoams.com} window into a cryostat where high density polyethylene (HDPE) lenses focus the signal onto the horn array. Infrared (IR) blocking filters are situated between the window and the first lens to allow the lenses to cool. The first lens will be cooled to 4 K and the second will be cooled to 1 K. Parameters summarizing the telescope are in Table \ref{tab:telescope_summary}.
\begin{figure}[t]
\begin{center}
\includegraphics[width=5in]{RayTrace}
\end{center}
\caption[example]
{ \label{fig:raytrace} A ray trace of the 40 GHz CLASS telescope is shown for a selection of fields. The first optical element is the VPM. For the purpose of optimizing the telescope design, the VPM is modeled as a flat mirror. After the polarization of the sky signal has been encoded by the VPM, the light is routed by two mirrors into a cryogenic receiver. Two cold high-density polyethylene lenses form the final image on the focal plane.}
\end{figure}
\begin{table}[b]
\caption{\label{tab:telescope_summary}Summary parameters for the CLASS 40 GHz telescope}
\begin{center}
\begin{tabular}{p{2.5in}c}
\toprule
Beam FWHM & $1.5^\circ$ \\
Edge taper of cold aperture stop & $\sim10$ dB \\
$f/$ at 10 dB & 2 \\
Number of detector pairs & 36 \\
Entrance Pupil diameter & $\sim 37$ cm \\
Horn spacing in the focal plane & 2.4 $f/ \cdot \lambda$ at 38 GHz \\
FOV & $14^\circ \times 19^\circ$ \\
\bottomrule
\end{tabular}
\end{center}
\end{table
\subsection{The Variable-delay Polarization Modulator}
\label{sec:VPM}
CMB polarization instruments must modulate the incoming light to distinguish the cosmic signal from detector, instrument, and atmospheric drifts, e.g. $1/f$ noise. The modulation strategies employed by most other CMB telescopes involve fast azimuth scanning or rotating a wave plate. Azimuth scanning, unfortunately, modulates both the CMB polarization and the CMB temperature anisotropy synchronously. An experiment adopting this strategy must employ additional techniques to guard against systematic errors mixing these signals. Rapidly rotating a half-wave plate overcomes this limitation but adds the complication of needing to understand the detailed spectral transmission of the wave plate including any anti-reflection (AR) coatings. Modulated thermal emission must also be accounted for in warm wave plates, and fast rotating cryogenic wave plates have proven extremely difficult to deploy. Furthermore there are practical limits to the diameter of wave plates that can be manufactured. This size constraint limits how close to the sky the wave plate can be placed in the optical train, especially for low frequencies.
CLASS will utilize the new VPM technology as a front-end modulator to distinguish the cosmic signal from various noise sources while circumventing the challenges faced by other techniques. This modulator consists of a polarizing wire array backed by a parallel movable mirror. The operating principle of a VPM has been discussed previously\cite{vpm_theory}, and an example of a VPM fielded in the Hertz instrument is discussed in Krejny et al.\cite{vpm_example}. A few aspects of VPMs that guide the design of this optical system are highlighted here.
\begin{description}
\item[The VPM is the first optical component.] The scalable mechanical construction of a VPM enables the device to be placed at the entrance pupil of a modest-to-small size telescope. Since the resolution required to target large angular scales is low, a VPM with an aperture diameter of 60 cm is sufficient.
\item[The VPM is the polarization modulator.] Each modulation cycle of the VPM is an \emph{absolute} measurement of a Stokes parameter in a particular direction. No scan modulation is required.
Since data can be collected while drift scanning or slow azimuth scanning, the tolerance analysis need not include telescope bending modes that would otherwise be present during scan turnarounds.
\item[The VPM transmission spectrum is simple.] The VPM is a purely reflective device. The lack of dielectric substances make the spectral characteristics of the VPM simple to understand. Chuss et al. \cite{2012ApOpt..51..197C} have identified resonant conditions that exist for particular VPM arrangements, but these situations are well understood and can be included in the transfer function template of the VPM.
\end{description}
Figure \ref{fig:VPMcartoon} shows a schematic of the VPM layout. An incoming signal is separated into its two orthogonal linear polarizations by the wire array. The component polarized parallel to the wire array is reflected, while the component with the polarization orthogonal to the wires passes unimpeded. The transmitted component then reflects off the movable mirror and passes back through the wire array. The relative phase between the two reflected components is specified by controlling the separation between the wires and the mirror.
\begin{figure}[t]
\begin{center}
\subfigure[]{\label{fig:VPMcartoon}
\includegraphics[width=0.35\textwidth]{VPMcartoon}}
\subfigure[]{\label{fig:Poincare}
\includegraphics[width=0.35\textwidth]{Poincaresphere}}
\end{center}
\caption[example]
{ \label{fig:VPM} (a) A schematic of the VPM operation is shown. The input sky signal is separated by the polarizing grid into two orthogonal components. One component is reflected from the wires, and the other component passes through the wires and is reflected by the movable mirror. The two reflected signals combine to form the modulated output signal. (b) The dashed path on the Poincar\'{e} sphere illustrates the change in sensitivity to the sky basis defined Stokes parameters. As the mirror moves, the measured parameter modulates between Stokes Q and V.}
\end{figure}
The signal for a given direction on the sky can be represented by Stokes parameters $I_{sky}$, $Q_{sky}$, $U_{sky}$ and $V_{sky}$. The orthogonal antenna pair of the detector projected through the optics onto the sky naturally defines $Q_{det}$ through detector differencing. When the detector basis projected onto the VPM is rotated $45^\circ$ with respect to the VPM wires, the polarization sensitivity of the detector to the sky signal as a function of the relative phased delay, $\phi$, is
\[Q_{det} = Q_{sky} \cos \phi + V_{sky} \sin \phi. \]
Figure \ref{fig:Poincare} shows a portion of this modulation cycle on the Poincar\'{e} sphere.
Periodically the telescope will execute $45^\circ$ rotations about the telescope boresight. Such a rotation changes the detector sensitivity relative to the sky-fixed Stokes basis to
\[Q_{det} = U_{sky} \cos \phi + V_{sky} \sin \phi .\]
In this way CLASS will have complete coverage of the Stokes parameters for every point within the accessible sky region. The $V_{sky}$ astrophysical signal is expected to be zero and will serve as critical null observations for systematic error checking.
\subsection{Fore-optics design and description}
\label{sec:fore_optics}
The telescope fore-optics consist of the VPM and the two curved mirrors shown in Figure \ref{fig:raytrace}. The two curved mirrors create an image of the cold 4 K aperture stop on the central region of the VPM. In this location, the diameter of the VPM is minimized for a fixed entrance pupil size. There is no magnification requirement of this relay system for this low resolution telescope. Note the entrance pupil is purposefully tilted with respect to the VPM to appear circular when viewed from the center of the FOV.
\begin{figure}[t]
\begin{center}
\includegraphics[width=6.5in]{MirrorParents2}
\end{center}
\caption[example]
{ \label{fig:foreoptics} The relative geometry of the fore-optics mirrors is shown. Each ellipse is rotated about its major axis to generate the surface of the mirrors. The portion of the ellipsoids used for the mirrors are indicated by thicker lines on the ellipses. The darker ellipse generates the primary. Crosses are placed at the focal points of the ellipses. The large `X' mark indicates the location of the cold stop and the entrance pupil. The smaller `x' indicates the intermediate virtual pupil. Both mirrors have circular perimeters when projected along the central ray light path. The dotted lines indicate this projection direction for each mirror. The projected diameters of the primary and secondary mirrors are 114 cm and 116 cm respectively. }
\end{figure}
The fore-optics were optimized as an afocal system. It was important during this step to temporarily place the stop surface in ZEMAX at the VPM. With the stop as the first optical surface, ZEMAX efficiently launches all necessary rays for the optimization. If the stop surface was kept in its physical location, the aperture stop vignetting function would depend upon the exact size, shape, and location of all prior optical elements. Optimization for this off-axis tilted system would not be computationally feasible. In the temporary `false stop' arrangement, the goal of the fore-optics system is to map plane waves, apodized by the stop, to aberration-free spherical waves at the physical location of the stop. The location and shape of the two mirrors were allowed to vary in order to optimize the following criteria:
\begin{itemize}
\item Minimize the wave front error (WFE) of the outgoing spherical waves for each incoming plane wave direction.
\item Ensure marginal rays were sufficiently far from mirror edges and the cryostat edges to prevent aperture blockage.
\item Minimize the distance between the intersection of the chief rays for all sampled directions at the cold stop.
\item Minimize the distance between the top marginal rays for all fields and a point 15 cm from the chief ray center in the desired pupil plane.
\item Ensure the distance between the two powered mirrors was not larger than 2.2 meters (to limit the size of the telescope).
\item Ensure the intended aperture stop plane is always orthogonal to the chief ray of the central field.
\end{itemize}
A damped least squares (DLS) optimization resulted in an initial design, but the ZEMAX `global search' optimization was used to generate the final design. Once the optimization converged to an acceptable solution, the `false stop' surface in ZEMAX was removed and a stop with 30 cm diameter was placed in the actual aperture stop location.
Single off-axis mirror designs for the fore-optics were ruled out because of unavoidable polarization distortion\cite{1987IJIMW...8.1165M}. In addition to correcting this distortion, the degrees of freedom added by using a second mirror enable an improved entrance pupil. Figure \ref{fig:foreoptics} shows the relationship between the two fore-optics mirror shapes resulting from the optimization. Both mirrors are off-axis sections of ellipsoids generated by rotating the parent ellipse shape about it major axis. The mirrors are conveniently defined by the focal points of the parent ellipse and a point at the center of each mirror. These coordinates are given in Table \ref{tab:mirror}. This virtual pupil is then imaged onto the VPM by the primary mirror. The location of these points are indicated by `X' marks in the Figure.
\begin{table}[t]
\begin{center}
\caption{\label{tab:mirror}Parameters defining the shape and relative geometry of the fore-optics. The coordinate system origin is the vertex of the primary mirror ellipsoid.}
\begin{tabular}{lll}
Point & x (cm) & y (cm)\\
\toprule
$f_{p1}$ & 208.543348 & 0\\
$f_{p2}$ & 446.483014 & 0 \\
$ f_{s1}$ & 40.759849 & 245.895577 \\
$f_{s2}$ & -106.455190 & 76.696615 \\
$m_p$ & 73.410690 & 192.517211\\
$ m_s$ & 11.566104 & 198.577111\\
Stop Center & 83.430267 & 37.653512 \\
\bottomrule
\end{tabular}
\end{center}
\end{table}
\subsection{Re-imaging optics}
\label{sec:reimaging_optics}
Light from a specified field passing through the cold stop is nearly a spherical wave. The re-imaging optical system is therefore more concerned with re-imaging these nearly-spherical waves to a flat focal surface than correcting aberrations.
The lenses for the re-imaging optics are made of HDPE. A survey of the literature suggests the room temperature index of refraction for HPDE is near $n=1.526$ at 40 GHz\cite{2006ApOpt..45.5168M,afsar85}. Since the linear dimensions of HDPE contract by $2\%$ when cooled from 300 K to 4 K\cite{HDPE_contraction}, we used the Lorentz-Lorenz relation\cite{Jackson}
\[ \rho \propto \frac{n^2 -1}{n^2+2}\]
to scale the room temperature index to $n=1.564$ at 4 K and 1 K. This cold index value is assumed for the re-imaging optics design.
Designs using a single HDPE lens were found to have diffraction limited performance across the required FOV, but the lens for such a design was too large to be practically cooled. A design using two smaller and thinner lenses was adopted instead. The symmetric signal through the stop enables both lenses and the focal plane to be co-axial with the aperture stop.
Several criteria were used to optimize the re-imaging optics
\begin{itemize}
\item The WFE for each field at the focal plane was minimized.
\item The focal ratio ($f/$) was optimized to be near 2.
\item Moderate clearances between the pupil lens and the aperture stop, between the two lenses, and between the final lens and the focal plane were enforced.
\item All field chief rays should intersect the focal plane as close as possible to $90^\circ$
\item Each lens must be thinner that 5.4 cm with a minimum edge thickness of 1.1 cm.
\end{itemize}
Similar to the fore-optics, a DLS optimization produced an initial design, and a global search produced the final design. Both lens for the final design are convex-convex with on-axis ellipsoidal surfaces. Each surface is conveniently described through the sag equation
\[ z = \frac{ r^2/R}{1+ \sqrt{1-(1+\kappa)r^2/R^2}}\]
where $z$ is the distance from the vertex tangent plane to the lens surface, $r$ is the radial coordinate off the lens axis, $R$ is the radius of curvature at the vertex of the lens, and $\kappa$ is the conic constant. Table \ref{tab:lens} gives the shape parameters and location of each lens. Note the AR coating thicknesses are not included in the separation distances provided.
\begin{table}[t]
\begin{center}
\caption{\label{tab:lens} Geometric parameters for the re-imaging optics. AR coating thicknesses are not included in the distance and thickness parameters.}
\begin{tabular}{lccc}
\toprule
\multicolumn{4}{l}{Distance between the stop and the first lens = 2.999992}\\
\midrule
\multicolumn{4}{c}{\textbf{First Lens parameters at 4 K}}\\
\midrule
Side & R (cm) & $\kappa$ & central thickness (cm)\\
\midrule
1 & 152.420594 & --3.790630 & 4.659434 \\
2 & 53.948154 & -1.705192 & \\
\midrule
\multicolumn{4}{l}{Distance between lenses = 56.151877}\\
\midrule
\multicolumn{4}{c}{ \textbf{Second Lens parameters at 1 K}}\\
\midrule
Side & R (cm) & $\kappa$ & central thickness (cm)\\
\midrule
1 & 122.103054 & -22.013666 & 5.4 \\
2 & 49.471025 & -2.654538 & \\
\midrule
\multicolumn{4}{l}{Distance between the second lens and the focal plane = 21.998085}\\
\bottomrule
\end{tabular}
\end{center}
\end{table}
\subsection{Focal plane}
\label{sec:focal_plane}
The telescope ultimately couples the signal from the sky to a flat focal plane. The focal plane uses 36 smooth-walled horn antennas to guide the incoming radiation to the CLASS detectors. Each horn will be smooth-walled with a monotonic profile\cite{LZthesis}. While this architecture is more difficult to design than corrugated horns, the smooth interior surface of the horn is easier and cheaper to machine and competitive performance is achievable. The horn design procedure for this telescope is described by Zeng et al.\cite{2010ITAP...58.1383Z}. To be compatible with the telescope, the horns have been designed to produce a 10 dB edge taper at $f/2$. This constraint, combined with minimizing return-loss and cross-polar\footnote{We exclusively use Ludwig's third definition\cite{Xpol} of cross-pol in this paper.} response, form the basic constraints for the horn optimization. The horn/telescope system design is further described in section \ref{sec:analysis}. The horn arrangement in the focal plane and the horn shape are shown in Figures \ref{fig:FPlayout} and \ref{fig:profile} respectively.
\begin{figure}[t]
\begin{center}
\subfigure[]{\label{fig:FPlayout}
\includegraphics[width=2in]{FPlayout}}
\subfigure[]{\label{fig:profile}
\includegraphics[width=1.4in]{FeedProfile2}}
\subfigure[]{\label{fig:hornbeam}
\includegraphics[width=3.1in]{FeedBeams}}
\end{center}
\caption[example]
{ \label{fig:fp} (a) The arrangement of the horns in the focal plane is shown. The horns are arranged in two hex-close-packed halves. The nearest-neighbor horns are separated by 38 mm. (b) The horn profile of the 40 GHz smooth walled horn is shown. The length of the contoured section of the horn is 100 mm. The horn aperture inner diameter is $\sim 36$ mm. A short section of circular waveguide is include in the Figure. (c) The predicted far field beam pattern of the horn is shown along with the best fit Gaussian beam to the region between the vertical lines. The vertical lines are at $\pm 14^\circ$ and show the angle represented by marginal rays in the ray trace.}
\end{figure}
\section{Performance analysis}
\label{sec:analysis}
Due to the speed rays can be traced by a computer through the optical system, the ray-calculated WFE metric was used for optimizing the location and shape for all optical surfaces. The maximum rms WFE across the FOV is $8.74 \times 10^{-3}$ waves for the final design. The rms WFE is frequently used by ray tracing software to approximate the Strehl ratio, S, of the telescope via the equation
\[S = e^{-(2 \pi \sigma)^2} \]
where $\sigma$ is the rms WFE referenced to the geometric ray centroid at the focal plane.
This approximation is accurate when the peak of the point spread function (PSF) is well defined and the Strehl ratio is not far \mbox{from 1}. By convention Strehl ratios above 0.8 are considered diffraction limited. From Figure \ref{fig:StrehlRatio}, we see the Strehl ratios for this telescope are $>0.996$ for all points within the FOV.
\begin{figure}
\begin{center}
\includegraphics[width=4in]{StrehlRatioPlot}
\end{center}
\caption[example]
{ \label{fig:StrehlRatio} The Strehl ratio for all points within the FOV are displayed. The Strehl Ratio is above 0.998 for all field points within the FOV.}
\end{figure}
While the Strehl ratio is a useful metric to understand the image quality of a telescope, the fact that the aperture stop is only 30 cm (38 waves) in diameter indicates diffraction effects must be included to describe the telescope performance. To calculate the beams produced by the telescope on the sky and to predict the loading on the detectors from diffraction around the optical elements, the beam pattern of the horns in the focal plane must be included in the analysis. The field distribution at the aperture of the horn is known, but it is convenient to simplify the horn beam pattern to a Gaussian beam fit to the main beam of the far-field pattern of the horn. This Gaussian beam accurately represents the portion of the beam passing through the cold stop. Figure \ref{fig:hornbeam} shows the far field beam and the best fit Gaussian to the main beam. The 11.5511 mm diameter waist producing the best fit beam was launched from the focal plane in the location of the pixel being studied.
The physical optics propagation (POP) feature in ZEMAX uses scalar diffraction theory to progressively calculate the field distribution on each optical surface. Scalar diffraction theory can not predict the field distribution on aperture edges, but all edges are under-illuminated since the primary beam forming components are the horns themselves.
Beyond the scalar field approximation, ZEMAX also assumes the field component along the direction of travel of the wave is small compared to the transverse field components. This assumption breaks down for very fast systems (typically $\sim f/1$). Since the fastest beams in this telescope are $f/2$, the ZEMAX POP feature is acceptable for understanding the propagation of the fields through the telescope itself. Figure \ref{fig:illum} shows the illumination for a central pixel on the VPM, primary, and secondary. The size of each element was selected to maximize the telescope gain and limit warm spill over the edges. The secondary mirror was enlarged the most because proximity to the intermediate focal surface between the mirrors causes a slow Airy-like radial intensity drop-off and because all spill beyond this element views the \mbox{300 K }environment. Spill over the primary will largely be directed toward the cold sky and therefore does not require as aggressive oversizing.
\begin{figure}
\begin{center}
\includegraphics[width=6.8in]{EllementIllumination}
\end{center}
\caption[example]
{ \label{fig:illum} The illumination of the VPM, primary, and secondary are shown for horn 22. The VPM edges are under illuminated to prevent modulated side lobes. The clear Airy pattern on the secondary is due to the proximity to the intermediate focal surface between the primary and secondary mirrors. }
\end{figure}
The final propagation step from the entrance pupil of the telescope to the sky was calculated using GRASP10\footnote{www.ticra.com}. In GRASP10, the field configuration from ZEMAX is converted to a fictional current distribution that would have produced the input field distribution. The fictional current distribution is then used to calculate the far-field beam pattern. Figure \ref{fig:beams} shows the predicted co-polar and cross-polar beams of a typical pixel. The FWHM of the beams on the sky is $\sim1.5^\circ$. The cross-pol peaks are seen to be more than 40 dB below the main beam peaks. The main beam is circular with the first side lobe showing some asymmetry, but is roughly 23 dB lower than the main beam peak.
\begin{figure}
\begin{center}
\includegraphics[width=6.5in]{SkyBeams}
\end{center}
\caption[example]
{ \label{fig:beams} The GRASP10 calculated co-polar (left) and cross-polar (right) beams for the telescope are shown for a central pixel. The peaks of the cross-polar pattern are more than 40 dB lower than the co-polar peak. The beams for other pixels are similar. The $45^\circ$ rotation of the detector with respect to the plane of symmetry of the telescope is evident in the cross-polar beam pattern.}
\end{figure}
\section{construction plan}
\label{sec:construction}
The VPM for the 40 GHz telescope will consist of a planar array of regularly spaced wires backed by a parallel and movable aluminum faced honeycomb mirror. The lightweight, high stiffness properties of the honeycomb material enable us to rapidly translate the mirror while staying parallel to the wire array. A prototype VPM wire array has been constructed with a 50 cm clear aperture\cite{grid}. The prototype array uses 64 $\mu$m diameter gold plated tungsten wire regularly spaced at 200 $\mu$m pitch. The minimum resonant frequency of the wires was measured to be above 150 Hz, well above the modulation frequency of the VPM. To maintain good phase definition, the wires are planar within 1\% of a wave. The techniques used for the prototype are expected to scale to the required 60 cm clear aperture without significant modification. The polarization transfer function of the prototype has been measured, and the device performs as expected\cite{2011RScI...82h6101E}.
The primary and secondary mirrors will both be open-back light-weighted monolithic aluminum structures. Figure \ref{fig:secondary} shows a sectioned view of the secondary mirror. This mirror has a minimum 3 mm face sheet with 5 mm thick webbing arranged in a triangular pattern. A single 14 mm thick web encircles the back of the mirror for mounting. Standard CNC machining techniques are more than sufficient for achieving the generous tolerances required for this low frequency channel. The mirrors, however, will be constructed such that tolerances limits even for the 90 GHz channel are satisfied. This will enable the same mechanical design to be iterated for the higher frequency channels and will not significantly increase the difficulty of manufacturing. The surfaces will be hand-finished to a $\sim 2$ $\mu$m rms surface roughness to remove possibly polarizing tool marks. This roughness will degrade the gain by the factor
\[\alpha = e^{-\left( 4 \pi \epsilon/\lambda \right)^2}\]
where $\epsilon$ is the rms surface roughness \cite{Ruze}. For an $\epsilon=2$ $\mu$m surface finish, the gain degradation will be negligible for the 40 and 90 GHz bands. At the same time, the surface is still rough enough to avoid specularly reflecting high frequency IR and visible light. Avoiding specular reflection of higher frequency signals will help prevent unintentionally focusing energy on the cryostat or other portions of the telescope.
\begin{figure}
\begin{center}
\includegraphics[width=5in]{secondaryLeafSpring}
\end{center}
\caption[example]
{ \label{fig:secondary} A sectioned view of the secondary mirror is shown. The face sheet of the mirror is 3 mm thick. 5 mm wide webs are arranged in a triangular pattern to stiffen the mirror. A single 14 mm wide web encircles the mirror and ensures the mirror stiffness is greater than the mounting hardware stiffness. Three mounting plates are arranged in a regular triangular pattern of the the thick web. Tapped holes are placed at webbing intersections to mount thermal monitoring and control hardware. The primary mirror will have an analogous construction.}
\end{figure}
HDPE was selected as the lens material for its low dielectric loss, ease of machining, ease of AR coating, and the availability of raw material in sufficiently large pieces to construct the lenses. Each lens will be AR coated by a simulated dielectric layer. This layer will consist of small holes drilled in a rectangular array across the lens surface. The designed geometric lens parameters will be enlarged to accommodate the $2\%$ integrated linear expansion when warmed from 4 K to 300 K \cite{HDPE_contraction}. Thermal radiation blocking filters at 60 K and at 4 K will be used to ensure the lenses get cold. These filters will consist of metal-mesh frequency selective surfaces and neutral density dielectric slabs.
The aperture stop will be constructed from a formed ring of steel loaded epoxy called steelcast\cite{steelcast}. The ideal cold stop would be an infinitesimally thin aperture allowing all radiation within a defined radius to pass unimpeded and all radiation beyond that radius to be blocked. Unfortunately such a surface does not exist at any wavelength. More accurately, a stop should be expressed as a surface with a particular absorption as a function of incident angle and polarization. To increase the loss, and approach the perfectly absorbing ideal, the thickness of the absorber must be a significant fraction of a wavelength (7 mm). Even in this case, the stop is not perfectly `black.' To decrease the edge illumination on the stop, a series of axillary rings with slightly larger apertures will be placed on both sides of the geometric-optics defined stop. Each aperture will gradually clip the beam with the central geometrically located ring dominating the clipping. Sharp et al. have constructed a similar baffle design for the GISMO instrument\cite{2008SPIE.7020E..66S,sharp2}.
\section{Stray Light and Spill}
\label{sec:stray_light}
The frustum of rays defined by the FOV edges and the cleanly imaged entrance pupil on the VPM represents, to first order, the only path of light to the focal plane. Neglecting diffraction effects, the only way for sources beyond the FOV to illuminate the focal plane would be diffusively scattering radiation into the light path. This behavior was approximated by imagining the optical components and support structure scatter geometric rays with a Lambertian probability distribution peaking in the direction of the local surface normal. Sources scattering significant power into the beam were eliminated, baffled or blackened. The most troubling surfaces was found to be the 4 K baffling between the lenses and the 1 K baffling between the focal plane and final lens. As a first effort to eliminate this effect, the walls are blackened with steelcast. Glint baffles, in the form of ribs, will surround the optical path to further eliminate large angular scattering.
Diffraction of light around the optical components is also a source of extra loading on the detectors. To protect against warm spill from Earth-shine, Moon-shine, Sun-shine or other strong sources of in-band radiation, the primary and secondary mirror must be oversized and a co-moving ground shield will surround the telescope. To predict the loading from diffraction beyond these elements, a series of POP calculations was performed in ZEMAX. As described in section \ref{sec:analysis}, a Gaussian beam representing the main beam of the horn is transformed through the system to predict the illumination of each surface. Keeping track of the fraction of the beam lost at each surface and normalizing via $A\Omega = \lambda^2$ at the sky, the loading from spill beyond each surface can be calculated\cite{1966raas.book.....K}. Combining the POP results with an atmosphere model generated by ATM\footnote{ATM is authored and maintained by Juan Pardo. The software is available at \mbox{/www.mrao.cam.ac.uk/~bn204/alma/atmomodel.html}} (assuming 1 mm precipitable water vapor and $45^\circ$ elevation angle), we find the expected detector loading is $\sim 2.2$ pW. Table \ref{tab:loading} breaks the sources of loading down into categories. The total loading is dominated by the atmosphere.
\begin{table}[t]
\begin{center}
\caption{\label{tab:loading}Predicted total thermal load on the detector assuming a generic IR filtering scheme.}
\begin{tabular}{ll}
Source of loading & Power (pW)\\
\toprule
CMB & 0.225\\
Atmosphere & 1.34\\
Warm spill & 0.136\\
Cold spill & 0.19\\
Telescope emission & 0.274 \\
\midrule
Total & 2.17\\
\end{tabular}
\end{center}
\end{table}
\section{Tolerances}
\label{sec:tol}
Monte Carlo (MC) simulations were performed to determine the tolerance for 144 degrees of freedom (DOF) of the optical design. These DOF included the shape, position, tip, tilt, thickness, material properties, and surface irregularity for all optical components. For each MC realization, all DOF were randomly perturbed around a nominal value according to a normal distribution with a specified width. The secondary is then refocused via three steps of the DLS algorithm to minimize the WFE at the focal plane for each MC realization. The WFE of all sampled fields was then recorded.
Based upon 21,000 MC simulations, the tolerances for the optical design are extremely forgiving. The generous tolerances for the telescope are to be expected for this long wavelength band. The most challenging tolerance is setting the distance between the focal plane and nearest lens within 2 mm - owing to the relatively fast $f/2$ focal ratio at the focal plane. All tolerances should be achievable by standard machining and assembly techniques. For the MC simulations, 98\% of the simulated telescope instances still had Strehl ratios greater than 0.993 across the entire FOV.
\section{Conclusions}
\label{sec:conclusions}
To study inflation, CLASS will use four telescopes at three frequencies to search for the B-mode signal in the CMB and thus infer the energy scale of the potential driving inflation. The 40 GHz channel of CLASS will be an essential component in the full multi-frequency CLASS instrument. To this end, we have developed a new telescope design for ground based observation of CMB polarization at 40 GHz with a front-end VPM modulator. This new design separates the key tasks of imaging the cold stop onto the VPM and imaging the sky onto the focal plane by delegating the former task to a two mirror fore-optics section and the later to a two lens re-imaging cold optics section. The telescope will cleanly map the horn antenna beams from the focal plane to $1.5^\circ$ FWHM beams on the sky. The peaks of the cross-polar response of the telescope are expected to be at least 40 dB lower than the main beam peak. A tolerance analysis indicates the presented design delivers robust performance for a wide range of perturbations. Design and construction of the VPM and optical components has begun, with full telescope integration to follow.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 5,837
|
{"url":"https:\/\/math.stackexchange.com\/questions\/939766\/probability-between-two-dice-games","text":"# Probability between two dice games\n\nTwo games, both use un-biased 6 sided dice.\n\ngame A, Sam throws one die 4 times. He wins if he rolls at least a 6 game B, he has 24 turns, and each time he rolls two dice simultaneously. He wins if he rolls at least one \"double six\"\n\nWhich game is Sam most likely to win?\n\nI have a good idea of how to work this out and i have done majority of this i just wanted to double check that i am correct.\n\nWhat i have done:\n\ngame A sample space = $24$ $(1,2,3,4,5,6 * 4)$\n\nE = all outcomes of rolling a 5\n\n|E| = 4 $((6),(6),(6),(6))$\n\nprobability (6) = $1\/6$\n\nGame B Sample space = |S|=$36*4 = 864$\n\nE = all outcomes with (6,6) |E| = $26$\n\nprobability (6,6) = $24\/864$ = $1\/36$\n\nIs my working out correct?\n\nThe sample space is to some degree at our disposal. However, it is convenient, if we can manage it, to have a sample space in which all outcomes are equally likely.\n\nFor Game A, a convenient space is the set of all ordered quadruples $(a,b,c,d)$, where each of the symbols ranges over the numbers $1$ to $6$. The numbers $a,b,c,d$ record, in order, the result of the first toss, the second, and so on.\n\nThe sample space has $6^4$ elements. We count the favourables, the sequences which have at least one $6$. It is easier to count the sequences that have no $6$. There are $5^4$ of them.\n\nSo the probability of at least one $6$ is $\\frac{6^4-5^4}{6^4}$. This is approximately $0.517746913$. That is a good approximation of the probability of winning Game A.\n\nNow we find the probability of at least one double $6$ in $24$ tosses. Here the sample space is the set of all sequences of length $24$, where the elements of the sequence are all ordered pairs $(x,y)$, where $x$ records the number on the gree die, and $y$ records the number on the blue die. There are $36^{24}$ such sequences.\n\nThere are $35$ equally likely ways to throw two dice and not get a double $6$. So there are $35^{24}$ \"bad\" sequences of length $24$ consisting of bad double-throws only. Thus the probability that in $24$ double throws there will be at least one double $6$ is $\\frac{36^{24}-35^{24}}{26^{24}}$. Thus the probability of winning Game B is about $0.491403876$.\n\nGame A is somewhat more favourable for Sam than Game B.\n\n\u2022 I have a question for game B as to why the total number of possible outcomes is $26^24$ ? Shouldnt it be $36^24$ \u2013\u00a0I hate math Sep 24 '14 at 23:26\n\u2022 It depends how you count. I chose equally likely outcomes. When you toss a red die and a green die, there are $36$ equally likely outcomes. For $24$ tosses, that gives $36^{24}$. There are other ways of solving. Like probability of a double-six is $1\/36$. The probability of not double-six in one toss is $35\/36$. The probability this happens $24$ times in a row is $(35\/36)^{24}$. The probability it doesn't happen (so we get at least one double-six) is $1-(35\/36)^{24}$. Same as what I wrote in the answer. \u2013\u00a0Andr\u00e9 Nicolas Sep 25 '14 at 0:09\n\u2022 Another question i have is how would i prove the following argument? A double six in a single turn in game B is 1\/6 as likely as rolling a six in one turn in game A. But there are 6 times as many turns in game B as game A. Thus the two games are equally good bets \u2013\u00a0I hate math Sep 25 '14 at 5:19\n\u2022 Well, the detailed calculations show that this intuitively appealing argument is not correct. What is correct is that the expected (mean) number of $6$'s in the first game is equal to the expected number of double-$6$ in the second game. The question you ask is very interesting. It is closely related to a question the Chevalier de M\u00e9r\u00e9 asked Pascal, in the early days of probability theory. Back then, there was a degree of confusion between expectation and probability. \u2013\u00a0Andr\u00e9 Nicolas Sep 25 '14 at 5:25\n\u2022 what ive done is $(36^1-35^1)\/(35^1)$ = 0.028% and $(6^1-5^1)\/6 = 1\/6 = 0.16$ not quite sure what to do with this information \u2013\u00a0I hate math Sep 25 '14 at 6:04\n\nYour sample spaces are not correct. For the first one, you have six possible rolls each time, so the size of the sample space is $6^4$, but that is not needed. The chance of getting at least one six in four rolls is one minus the chance of getting no sixes. What is the chance of not getting a six on one roll? Now you need to succeed at that four times in a row. Similarly for the second problem, what is the chance of not getting double sixes on one roll (of two dice)? To not get a double six in 24 rolls, you need to succeed at this 24 times in a row.","date":"2019-05-21 12:59:37","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.7944502234458923, \"perplexity\": 142.5299983475056}, \"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-2019-22\/segments\/1558232256381.7\/warc\/CC-MAIN-20190521122503-20190521144503-00231.warc.gz\"}"}
| null | null |
package com.taisho6339.man.crawler.service;
import com.taisho6339.man.crawler.model.EmpTagRelation;
import com.taisho6339.man.crawler.model.Employee;
import com.taisho6339.man.crawler.repository.EmpTagRelationRepository;
import org.springframework.beans.factory.annotation.Autowired;
import org.springframework.stereotype.Service;
import java.util.List;
/**
* Created by sakamohiroki on 2016/05/31.
*/
@Service
public class EmpTagRelationService {
@Autowired
private EmpTagRelationRepository empTagRelationRepository;
public EmpTagRelation save(EmpTagRelation relation) {
return empTagRelationRepository.save(relation);
}
public List<EmpTagRelation> findByTagId(Long tagId) {
return empTagRelationRepository.findByTagId(tagId);
}
public List<EmpTagRelation> findByEmpId(Long empId) {
return empTagRelationRepository.findByEmpId(empId);
}
public EmpTagRelation findByTagIdAndEmpId(Long tagId, Long empId) {
return empTagRelationRepository.findByTagIdAndEmpId(tagId, empId);
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 3,125
|
The Assyrian Democratic Party (, ; short: ADP) is an Assyrian political party active in Syria, that traditionally represents the interests of the Eastern Assyrians of the Khabur valley. Although aiming for the peaceful implementation of democracy in Syria, the party has generally sided and cooperated with the Ba'athist government since the 1990s. In course of the Syrian Civil War, the Assyrian Democratic Party has come to be closely affiliated with the Khabour Guards and Nattoreh. It is part of the Syrian Democratic Council of Rojava.
Ideology
Since its foundation, the Assyrian Democratic Party has been considered to be "overtly sectarian", regarding the Eastern Assyrians as the only "true" Assyrians while distrusting the Western Assyrians. Since the outbreak of the Syrian Civil War, the party has however gradually broadened its aims and become more inclusive. In 2014 it began to work closely with other Assyrian and Christian denominations in the "General Authority of the Chaldean Syriac Assyrians", a committee whose stated aims also include ensuring social equality and more rights for all Christians of al-Hasakah Governorate. In 2017, the party declared that it wanted a new constitution for Syria that recognized all Assyrians as native people of Syria and granted them "cultural, administrative and other rights".
The ADP also officially rejects partisanship and supports the peaceful implementation of democracy in Syria, and is social democratic according to Al-Araby Al-Jadeed. Nevertheless, the party has generally sided and cooperated with the Ba'athist government since the 1990s, and supported pro-government militias in the civil war. The ADP advocates the unity of Syria, though has begun to support plans for the country's federalization since 2017. By July 2018, an ADP member stated that the "federal model that was set up [in northern Syria] is satisfying as we feel sufficiently represented".
History
Foundation and early years
The Assyrian Democratic Party was founded in 1978, when a splinter faction under Adam Homeh seceded from the Assyrian Democratic Organization (ADO). From its outset, the ADP understood itself as proponent of rights for the Eastern Assyrians as opposed to the Western Assyrians, who dominated the ADO. Over time, a strong rivalry developed between the pro-opposition ADO and the ADP, which shifted increasingly closer to the Ba'athist government in the 1990s. As result, the Assyrian Democratic Party came to present itself as the pro-government alternative to the ADO, and when taking part in various Syrian parliamentary elections, supported Assyrian candidates that were not strongly opposed to the rule of the al-Assad family.
Activities during the civil war
When the Syrian Civil War broke out in 2012, the ADP openly opposed the Syrian opposition, with some of its members even joining the government militias (Popular Committees) in Qamishli. The Assyrian Democratic Party also expressed support for the pro-government Sootoro.
In late 2013, the ADP lamented that the Free Syrian Army had occupied the Assyrian villages in the Khabur valley, saying that even though the FSA fighters did not harass the locals, their presence caused the area to become a target for government attacks. This in turn forced the Assyrians to flee their homes. The ADP also protested against the declaration of the autonomous region "Rojava" by the Kurdish-led administration in northern Syria around that time. Party president Ninos Isho stated in early 2014 that it is "not logical to have 30 percent of the population [the Kurds] in this area rule over the other 70 percent [the Arabs and Assyrians]", and that "the Kurdish political groups must accept real power-sharing". In September 2014, the Assyrian Democratic Party put up a shelter in Qamishli for Yazidis who had fled the Sinjar massacre, and distributed food, clothing and medicine to them. The party also participated in the formation of the "General Authority of the Chaldean Syriac Assyrians" in the next month. This committee aimed at strengthening the cooperation of the different Christian churches, parties and organizations in al-Hasakah Governorate, and ensuring the rights of all Christians in the region. Among the participants was also the pro-Kurdish Syriac Union Party.
After the assassination of one of its commanders by the YPG, the Khabour Guards (a small Assyrian self-defense militia) broke their ties with the Syriac Union Party and aligned themselves with the ADP around mid-2015. In November 2015, the ADP issued a statement condemning purported human rights violations in Rojava by the governing Democratic Union Party (PYD). Despite this, the ADP joined the PYD-led Syrian Democratic Council one month later, with Wail Mîrza serving as the party's representative in the assembly. In January 2017, the Assyrian Democratic Party reached a deal with the PYD, according to which the Khabour Guards and Nattoreh would become the sole security force for the Assyrian villages in the Khabur valley. In return, the Khabour Guards and Nattoreh joined the Syrian Democratic Forces, while the ADP agreed to support the PYD's federalism project for Syria. Nevertheless, the party continues to advocate "the unity of Syria, as a country and people". On 13 April, PYD forces officially handed over the valley's villages to the Khabour Guards and Nattoreh, though the YPG kept a military base near Tell Tamer. The Assyrian Democratic Party said that this was a first step towards establishing Assyrian self-administration in the Khabur valley.
Party President Ninos Isho also stated during a meeting with Russian deputy foreign minister Mikhail Bogdanov in April 2017 that the party wants a new Syrian constitution that grants self-determination, more rights and recognition as native people to the Syrian Assyrians. The Assyrian Democratic Party also took part in the Northern Syria regional elections in December 2017 as part of the PYD-led "Democratic Nation List". Party official Wail Mîrza called the election "historical", and further said that such elections are what Assyrians "are striving for all over Syria". The ADP took part in the Sochi peace talks of January 2018.
On 20 September 2018, the Assyrian Democratic Party announced the creation of a united military leadership for Nattoreh and the Khabour Guards, known as "General Command of the Assyrian Forces". The Khabour Guards already left the "Assyrian Forces" in the following December.
References
Works cited
1978 establishments in Syria
Assyrian political parties
Assyrians in Syria
Organizations of the Syrian civil war
Political parties established in 1978
Political parties in Syria
Political parties in the Autonomous Administration of North and East Syria
Political parties of minorities in Syria
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 9,813
|
Q: For any finite simple graph $G$, can we always find an intersection graph of some set such that it is isomorphic to $G$?
Let $G=(V,E)$ be a finite simple graph. Does there exist a finite set
$S$ and $S_G\subseteq P(S)$ such that the intersection graph $H$ of
$S_G$ is isomorphic to $G$, i.e. $H\cong G$.
Here's my idea:
If I can find such an intersection graph $H$ that $H\cong G$, then the proposition is proved.
Let $V=\{v_1,\dots,v_n\}$ and $E=\{e_1,\dots,e_m\}$, where $n,m$ are both positive integers since $G$ is finite. Now, let $S=V$, and $S_G=E\cup \{\{v_i\}\}$, where $v_i$ satisfies the condition that $v_i\neq e_j\cap e_k$ for all $j,k\le m$. That is, $v_i$ is a vertex with degree 1. We define the set of edges $F$ by
$$f=\begin{cases}
\{v_i,e_i\}&\text{ if }v_i\in S_G\ \text{and}\ v_i\text{ is incident with }e_i\\
\{e_j,e_k\}&\text{ for all }e_j,e_k\in E\text{ with }j\neq k
\end{cases}$$
Then $H=(S_G,F)\cong G$.
I find my idea works well for closed graph, but not so well for others. There must be some problems in my idea. Anyone has any idea or can modify my solution?
A: I don't understand your solution. Here's how I'd do it. Let $G=(V,E)$ be a simple graph, finite or infinite. Let $S=V\cup E.$ (Of course, if $G$ is finite, then $S$ is finite.) For each vertex $v\in V$ let $f(v)=\{v\}\cup\{e\in E:e\text{ is incident with }v\}\in P(S)$ and let $S_G=\{f(v):v\in V\}\subseteq P(S).$ Then $f:V\to S_G$ is bijective and, for distinct vertices $v_1,v_2\in V,$ we have $v_1$ adjacent to $v_2$ if and only $f(v_1)\cap f(v_2)\ne\emptyset.$ That is, $f$ is an isomorphism from the graph $G$ to the intersection graph of $S_G.$
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 4,472
|
\section{Introduction}
High-speed solar wind streams are emanated from coronal holes (CHs), where the magnetic field opens into interplanetary space enabling plasma to escape from the Sun. The speed of the enhanced solar wind is positively correlated to the coronal hole area \citep{nolte76}. CHs are rather slowly evolving features in the solar corona (several hours to days), occurring over all latitudes and appearing, due to low plasma density, as dark regions in EUV image data \citep{cranmer09}. We can, therefore, assume that the solar wind speed from the same solar source is persistent in time which was shown with a statistics over 50 years by \cite{owens13}. Based on this, typically single viewpoint in-situ measurements are used to perform persistence models for solar wind speed forecasting, giving very reasonable results \citep[see e.g.][]{owens13}. A similar performance as a persistence model built on ACE (Advanced Composition Explorer) measurements, has the empirical solar wind forecasting (ESWF\footnote{ESWF is a service provided by ESA's Space Situational Awareness Program (SSA) within the Heliospheric Expert Service Center and is operational at the University of Graz since October 2016 (\href{http://swe.uni-graz.at}{swe.uni-graz.at}).}) model which is built on the linear \textit{CH area-solar wind speed} relation using an empirical formula that relates the CH area observed remotely in the EUV wavelength range with the speed at 1~AU \citep[see][]{vrsnak07,rotter12,rotter15}. With this, ESWF is able to give a forecast of solar wind bulk speed at 1~AU distance, having a lead time of about 4~days and an accuracy of $\pm$1~day in the arrival time and $\pm$150~km/s in the impact speed \citep{reiss16}. \cite{jian15} compared, among others, different numerical models running at the Community Coordinated Modeling Center (CCMC), and found uncertainties in the arrival times of 0.5--3 days \citep[see also][]{lee09,gressl14}.
For short time series and during minimum solar activity (2007--2009), very high correlations between solar wind data from the twin STEREO (Solar Terrestrial Relations Observatory) satellites were found for separation angles up to 30$^{\circ}$ \citep{opitz09}. \cite{turner11} concluded that models based on data from spacecraft located behind Earth in their orbit around the Sun (like STEREO-B and since mid of 2015 STEREO-A) perform even better than ACE based persistence models \citep[see also][]{li04}. Hence, by using in-situ measurements from spacecraft located East of Earth, we are able to improve solar wind forecasting at Earth. However, we must not forget that variations in the structure of a CH do have effects on the emanated solar wind. \cite{gomez11} found from using STEREO data, a clear increase in the in-situ measured solar wind speed for a CH that was expanding. Much stronger effects on the persistence of the solar wind flow are caused by coronal mass ejections (CMEs), but are temporally and spatially restricted \citep[see e.g.][]{temmer17}.
Here, we present a new concept of persistence modeling by adding two important aspects, i) multi-viewpoint observations and ii) taking into account the history and evolution of the solar wind due to changes of its solar sources. With this we increase the lead time of solar wind forecast and improve its performance. We make use of STEREO in-situ measurements, combined with remote sensing EUV observations from two viewpoints (STEREO and Earth view) to assess changes in the areas of CH. Determined changes in the CHs are used to adapt the STEREO based forecast accordingly. To derive the likelihood of persistence of solar wind speed, we first present for the time range 2008--2012 a thorough statistical analysis between STEREO-B persistence results and ACE. We compare the results in a continuous and an event-based approach, with and without CMEs, and relate the derived changes in the CH areas with changes in the measured solar wind speed. Finally, we show the application of the new concept in an operational mode in 2017 using beacon STEREO-A in-situ and EUV, and SDO (Solar Dynamics Observatory) EUV data for real-time solar wind forecasting.
\section{Data and Methods}
Our statistical analysis spans the time range 2008--2012 (to cover with STEREO-B a similar spacecraft position as we currently have with STEREO-A, and to explore a possible L5 mission which corresponds to the time range around autumn 2009 and beginning of 2010) covering the minimum phase of solar cycle 23 and the increasing to maximum phase of solar cycle 24. The statistics will, therefore, comprise time ranges of low and high solar activity that are compared with each other. During 2008--2012, the STEREO-B position covers the angular range 23--131$^{\circ}$ and in 2017, STEREO-A covers 144--129$^{\circ}$, both East of Earth. The year 2008 is very suitable for statistical calibration, due to i) STEREO-B was located in close angular distance to Earth, hence, evolutionary effects of CHs should be small and ii) transient events are sparse due to very low solar activity. We show the application of the new concept to real-time solar wind data from STEREO-A covering the time range January--August in 2017. Figure~\ref{fig:where} illustrates the angles between the spacecraft STEREO-B and STEREO-A with Earth for the time ranges under study (separation angles are given in HEE longitude, neglecting latitude). The real-time forecast of the solar wind speed at Earth has currently (2017) a lead time of about 10 days and will decrease as STEREO-A moves further West (e.g., 4.5 days for the L5 location reached end of 2020).
\begin{figure}
\centering
\includegraphics[width=0.9\columnwidth]{fig1.pdf}
\caption{\small Positions of STEREO spacecraft for the time under study. Left: STEREO-B position spans the angular range 23--131$^{\circ}$ East of Earth and is used for the statistical evaluation. Right: Results from the evaluation are used to run the persistence model based on STEREO-A data for the angular range 144--129$^{\circ}$ East and will be monitored further on, with special focus on the L5 position which will be reached end of 2020.}
\label{fig:where}
\end{figure}
\subsection{Definition of metrics and persistence models based on in-situ data}\label{data}
We make use of in-situ measurements with 1~min time cadence, from STEREO-B \citep[PLASTIC;][]{galvin08} and ACE \citep[SWEPAM;][]{mccomas98}, to identify streams of enhanced solar wind speed. Before performing the model algorithms and analysis, all in-situ measurements are linearly interpolated onto a 6~h time grid. For the detection of ``events'' we use the same algorithm as \cite{reiss16}: a requirement for a peak to be detected is a minimum speed of 400~km/s before and after the increase, having a peak prominence of at least 60~km/s (which does not need to be monotonically increasing).
A ``hit'' (true positive; TP) is defined if a predicted event overlaps within a time window of $\pm$2~days with a detected event in the reference data set (i.e., ACE). A ``false alarm'' (false positive; FP) is given when an event is predicted but not detected, whereas a ``miss'' (false negative; FN) is a detected event which has no predicted event associated. From the number of hits, false alarms, and misses we calculate well-established skill measures such as the probability of detection (POD), the false alarm ratio (FAR), and the threat score (TS). The TS is a measure of the overall model performance ranging from 0 to 1, where the best TS is 1 and the worst TS is 0. Additionally, the bias (BS) indicates whether the number of observations is underestimated ($BS < 1$) or overestimated ($BS > 1$). This event-based validation approach is supplemented by an error analysis of the predicted solar wind speed timelines for which we compute the arithmetic mean, mean absolute error, and root mean square error, standard deviation and correlation coefficient. For each hit (TP) we compare the predicted and the measured speeds and check whether the solar wind speed was over- or underestimated. In order to make the results comparable, we define the fraction of the underestimated speed (RUS = ratio underestimated speed, with 1$-$RUS = overestimated speed). For a more detailed description on the metrics used we refer the interested reader to the forecast validation tool developed in \cite{reiss16}. As CMEs cause transient peaks in the in-situ measured solar wind speed, we make a separate analysis excluding CMEs from the analysis, for which we use ready catalogs maintained by Richardson \& Cane for ACE \citep[194 CMEs for 2008--2012; see][for a description of the catalog]{richardson10} and for STEREO-B from L.~Jian \citep[189 CMEs for 2008--2012; see][for a description of the catalog]{jian06}. We note that for the real-time application in 2017, STEREO-A data are used and that CMEs were not excluded.
The realization of a forecast from a persistence model depends on the values measured at earlier time steps given by the separation angle between measured and forecast position (solar angular rotation speed is about 13.2$^\circ$/day). We present three different persistence models that are evaluated against actual ACE measurements: \textit{ACE+27}, based on ACE data forward shifted in time by 27.2753~days \citep[cf.][]{owens13}; \textit{STEREO} persistence, based on STEREO data dynamically forward shifted, according to the angle between the STEREO-B (STEREO-A) spacecraft and Earth; the new concept model \textit{STEREO+CH}, based on the STEREO persistence model taking into account the evolution of CH.
\subsection{EUV data reduction and coronal hole evolution}
Variations in the sources of enhanced solar wind speed, i.e., CHs, cause changes in the measured bulk velocity at 1~AU. To calculate these variations, we extract fractional CH areas from EUV data in STEREO \citep[EUVI;][]{wuelser04}, SoHO \citep[EIT;][]{delaboudiniere95}, and SDO \citep[AIA;][]{lemen12}. The fractional areas, $A(t)$, are derived within a meridional slice of $\pm 7.5^\circ$ around the central meridian, for which a linear relation with the measured solar wind speed after 4~days at 1~AU exists \cite[for more details see][]{vrsnak07,rotter12}. For the extraction we utilize 1024$\times$1024 image data provided by the SoHO/EIT instrument in the 195\AA~band (2008-2010) the SDO/AIA instrument in the 193\AA~band (2010-2012, 2017) with a time cadence of 6~hours, and the STEREO/SECCHI/EUVI instrument in the 195\AA~band (STEREO-B: 2008--2012, STEREO-A: 2017 - beacon/quick-look data) with a time cadence of 1~hour. To match the in-situ data, all EUV data are linearly interpolated onto a 6~h time grid.
Using the same extraction technique as for ESWF (a histogram-based segmentation method from the intensity distribution in the full-disk EUV images), we apply a threshold value of $f \times$median on-disk intensity, where $f=0.35$ for AIA, $f=0.47$ for EIT (AIA and EIT areas were cross-checked for the overlapping time range in 2010), and $f=0.32$ for STEREO. For CHs spanning their areas beyond $\pm 60^\circ$ heliographic latitude and longitude, we use an additional multiplication factor of 1.6 (to take into account projection effects as CHs appear less dark close to the limb). For further details see \cite{reiss16}.
Based on this we define the ``CH evolution ratio'' r$_{\rm CH}$ by,
\begin{equation}
r_{\rm CH} = A_{\rm ST} / (A_{\rm Earth}+A_{\rm ST}) \,,
\end{equation}
where A$_{\rm ST}$ (A$_{\rm Earth}$) refers to the fractional CH areas as derived from STEREO (Earth view, SDO or SoHO). To get a more smooth variation in the CH evolution ratio we use a cutoff for A$_{\rm ST, Earth}<$0.02. From this we compute the median $r_{\rm CH_{med}}$. Deriving $r_{\rm CH} \sim r_{\rm CH_{med}}$ we can assume that the CH fractional areas stay rather constant, hence, reveal a low degree of evolution. For $r_{\rm CH} \lessgtr r_{\rm CH_{med}}$ the solar wind sources underwent some changes and due to expanding/decaying CH areas the persistence model may under-/over-estimate the solar wind speed in its forecast for Earth. We note that $r_{\rm CH}$ cannot be calculated for 21\% of the time range due to gaps in EUV image data (data gaps from EIT: 15\%; AIA: 8\%; 2\% overlapping time). All derived values of $r_{\rm CH}$ are shifted in time by $+$4~days in order to match the corresponding in-situ measurements.
\section{Investigating coronal hole evolution and response in solar wind speed}\label{CH:dist}
Figure~\ref{fig:histo} shows the distribution of $r_{\rm CH}$ over the entire time range under study and separately for 2008. For the calibration year 2008, we derive a rather symmetric distribution with a median of 0.53, giving evidence that most of CH areas evolve slowly over time spans of $\sim$2 days and during low solar activity phases (as expected). The distribution of $r_{\rm CH}$ over the entire time range is rather asymmetric with a median of 0.59, referring to an evolutionary effect in CH areas. As the spacecraft separation (and solar activity) increases, we find a higher rate of decaying than expanding fractional CH areas (cf.\,results in Section~\ref{CH:speed}). Either the decay phase of CHs is of longer duration compared to the expansion phase, or as the solar activity picked up more CHs decayed. Clearly, persistence models based on future L5 missions, need to be updated/corrected as the solar wind sources may evolve over time spans of $\sim$4.5 days.
\begin{figure}[h]
\centering
\includegraphics[width=1\columnwidth]{fig2.pdf}
\caption{\small Histogram for the CH evolution ratio covering 2008--2012 (left panel) and separately for the year 2008 (right panel). The derived median is given by a red vertical line. }
\label{fig:histo}
\end{figure}
To assess different levels of persistence (i.e.\,how strongly CHs evolve), we calculate for each year separately the median $r_{\rm CH_{med}}$ and its median absolute deviation (MAD) $r_{\rm CH_{MAD}}$. In order to relate the derived distribution of $r_{\rm CH}$ to differences in the measured in-situ solar wind speed between ACE and STEREO, we calculate the differences between the measured solar wind bulk speed $v=v_{\rm ACE}-v_{\rm STEREO}$, the median $v_{\rm med}$ and its $v_{\rm MAD}$. All values are given separately for each year in Table~\ref{tab:CHrat}, together with the arithmetic mean and standard deviation as well as the standard error of the mean and median. On average, the differences in the solar wind speed between the two spacecraft show no clear trend of increased values for larger separation angles. Compared to that, the CH evolution ratio increases constantly from 2008 to 2012 (median from 0.53 to 0.63; cf.\,results in Section~\ref{CH:dist}).
\begin{table}
\caption[]{Statistical results (median, MAD, standard error of the median ($\mathit{SE}_{\rm med}$), arithmetic mean, standard error of the mean ($\mathit{SE}_{\rm mean}$), standard deviation) of solar wind speed differences, $v$ in km/s derived from ACE$-$STEREO (CMEs excluded), and CH evolution ratio values, $r_{\rm CH}$, separately listed for each year covering 2008--2012 as well 2017. }
\label{tab:CHrat}
$$
\begin{array}{l|cccc|cccc}
\hline
\noalign{\smallskip}
Year & v_{\rm med} \pm v_{\rm MAD} & {\mathit{SE}_{\rm med}} & \tilde{v} \pm 1\sigma & {\mathit{SE}_{\rm mean}} & r_{\rm CH_{med}} \pm r_{\rm CH_{MAD}} & {\rm \mathit{SE}_{med}} & \tilde{r_{\rm CH}} \pm 1\sigma & {\rm \mathit{SE}_{mean}} \\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
2008 & 1.52 \pm 37.57 & 2.18 & 8.13 \pm 66.30 & 1.74 & 0.534 \pm 0.083 & 0.004 & 0.536 \pm 0.114 & 0.003 \\
2009 & 12.08 \pm 36.49 & 1.99 & 13.16 \pm 60.88 & 1.59 & 0.559 \pm 0.088 & 0.004 & 0.563 \pm 0.122 & 0.003 \\
2010 & 7.68 \pm 49.46 & 2.93 & 8.78 \pm 89.24 & 2.34 & 0.605 \pm 0.114 & 0.005 & 0.601 \pm 0.156 & 0.004 \\
2011 & 5.70 \pm 48.80 & 3.17 & 7.00 \pm 96.68 & 2.53 & 0.609 \pm 0.109 & 0.005 & 0.604 \pm 0.141 & 0.004 \\
2012 & 9.31 \pm 51.12 & 2.98 & 11.51 \pm 90.82 & 2.38 & 0.632 \pm 0.124 & 0.005 & 0.625 \pm 0.158 & 0.004 \\
2017 & 7.74 \pm 63.01 & 4.30 & 11.80 \pm 113.87 & 3.43 & 0.508 \pm 0.101 & 0.005 & 0.524 \pm 0.145 & 0.005 \\
\noalign{\smallskip}
\hline
\end{array}
$$
\end{table}
\subsection{Relating CH evolution ratio to solar wind speed}\label{CH:speed}
Figure~\ref{fig:ratio} shows for the years 2008--2012 and 2017, the CH evolution ratio for decaying ($r_{\rm CH}>r_{\rm CH_{med}}$) and expanding CHs ($r_{\rm CH}<r_{\rm CH_{med}}$) versus $v$. A linear fit, separately calculated for expanding CHs and decaying CHs, shows that, on average expanding CHs tend to be related to a positive speed difference, i.e.\,an increase in the solar wind speed (as would be expected). However, we note that this trend is not very distinct. The years 2008 and 2017 reveal the most obvious relation between expanding CH areas and positive speed difference. In comparison, decaying CHs seem to be related to negative as well positive speed differences. There is no consistent trend found for decaying CHs to be related to solar wind streams of decreasing speed. We also note that results from 2009 show the least scattering maybe due to the deep solar minimum and the flat heliospheric current sheet.
This (weak) trend enables to give for expanding CHs an uncertainty estimate in the forecast speed of the persistence model, that should improve hit/miss rates. We define different persistence levels for $r_{\rm CH}$, based on the derived MAD values (as robust measure and less sensitive to outliers). \textit{high persistence}: $r_{\rm CH_{med}}-0.7 r_{\rm CH_{MAD}} \leq r_{\rm CH} < r_{\rm CH_{MAD}}$; \textit{medium persistence}: $r_{\rm CH_{med}}-1.4 r_{\rm CH_{MAD}} \leq r_{\rm CH} < r_{\rm CH_{med}}-0.7 r_{\rm CH_{MAD}}$; \textit{low persistence}: $r_{\rm CH_{med}}-1.4 r_{\rm CH_{MAD}} < r_{\rm CH}$. As shown in Figure~\ref{fig:ratio}, within each level we visually display the derived median values for the speed and the speed distribution as a box-and-whisker plot (for completion also for decaying CHs). On average, the interquartile range of the three different persistence levels for expanding CHs spans speed differences up to about $+$1.5$v_{\rm MAD}$. Each persistence level, high/medium/low, is then related to speed uncertainties of $+$0.5$v_{\rm MAD}$/$+$1.0$v_{\rm MAD}$/$+$1.5$v_{\rm MAD}$. With this we assess the speed uncertainty that we expect in the solar wind speed forecast when the CH fractional area undergo changes.
\begin{figure}
\centering
\includegraphics[width=1.\columnwidth]{fig3.pdf}
\caption{\small CH evolution ratio versus solar wind speed differences between ACE and STEREO in-situ measurements (CME events excluded). To show the overall trend, we overplot a linear fit (black line) to the data of the category expanding CHs and separately for decaying CHs. We mark the median of the CH evolution ratio of each year (vertical solid line) and give for each performance class the calculated MAD ranges (vertical dashed lines), for high performance $\pm$0.7MAD, for medium performance $\pm$1.4MAD, and for low performance outside $\pm$1.4MAD. Within each category we give a box-and-whisker plot showing the median (blue asterisk), the difference between the lower and upper quartile (blue box; interquartile range or IQR), and the upper and lower whisker given by 1.5$\times$IQR. Horizontal lines refer to the speed median (gray) and MAD ranges with $\pm$0.5MAD (blue solid), $\pm$1MAD (blue dashed). Note that CMEs are not excluded for 2017.}
\label{fig:ratio}
\end{figure}
\subsection{STEREO+CH persistence model}
To get a solar wind speed forecast that is updated with the information about a possible evolution of the source of the solar wind stream, we use the STEREO persistence model and modify its results. According to the obtained trend between expanding CHs and positive speed difference (cf.\,Figure~\ref{fig:ratio}), we simply increase the derived solar wind forecast from STEREO during those time ranges which are related to expanding CHs ($r_{\rm CH}<r_{\rm CH_{med}}$). As described in the previous section, for high persistence levels we increase the derived STEREO forecast speed by $+0.5 v_{\rm MAD}$, for medium persistence we increase by $+1 v_{\rm MAD}$, for low persistence we increase the speed by $+1.5 v_{\rm MAD}$ (cf.\,Table~\ref{tab:CHrat}). Considering that decaying CHs affect the solar wind speed in a less clear trend than expanding CHs, no change is applied for time ranges revealing $r_{\rm CH}>r_{\rm CH_{med}}$. As these uncertainties represent an upper limit of the forecast speed, we calculate the arithmetic mean between the maximum envelope of the modified speed profile and the original STEREO persistence forecast. This completes the STEREO+CH persistence model from which we derive a new forecast of solar wind speed having intensified and slightly broader speed profiles for those streams that are related to expanding CHs. Data gaps due to the lack of EUV data are filled by the original results from the STEREO persistence model. Figures~\ref{fig:test} and~\ref{fig:test2012} show for 2008 and 2012 the STEREO solar wind speed forecasting curve (red line) together with the estimated speed uncertainties (gray line) from the CH evolution and the resulting forecast curve, the STEREO+CH persistence model (blue line). The evolution of the CH ratio area and estimated speed uncertainties are given in the middle and bottom panels, respectively. We note that the STEREO persistence model has a variable lead time depending on the spacecraft location - as studied here, for up to 10.9~days. In comparison, the derived uncertainties for the STEREO+CH persistence model are based on the information of CH evolution (EUV data from Earth view are compared to STEREO), and therefore restricted to a lead time of 4~days.
\begin{figure}
\centering
\includegraphics[width=1.\columnwidth]{fig4.pdf}
\caption{\small For the year 2008 we show in the top panel the results from the STEREO persistence model (red profile), the derived speed uncertainties using CH evolution information (gray profile), and the new STEREO+CH forecast (blue profile) - for time ranges with EUV data gaps no speed uncertainty can be given and the STEREO profile is used as forecast (thin red profile); in the middle panel we give the evolution of the CH ratio as derived from EUV data together with its median and $+$0.7/$+$1.4MAD levels (vertical solid, dotted and dashed lines); in the bottom panel the speed uncertainties are shown.}
\label{fig:test}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=1.\columnwidth]{fig5.pdf}
\caption{\small Same as Figure~\ref{fig:test} but for the year 2012.}
\label{fig:test2012}
\end{figure}
\section{Performance comparison between different models}
We present the performance of ACE+27, STEREO, and STEREO+CH persistence in comparison to ACE measurements using an event-based and continuous forecast validation approach (see Section~\ref{data}). Figures~\ref{fig:p27}, \ref{fig:p4}, and \ref{fig:ST-CH} show for the time range 2008--2012 the evolution of the solar wind bulk speed measured at ACE in comparison to the forecasts from the ACE+27, STEREO, and STEREO+CH persistence model, respectively. The number of identified peaks in the speed profiles, together with the number of hits, false alarms, and misses are given in the legend of each plot. Time ranges during which CMEs occurred are marked. With black dots that are put below or above a hit (blue crosses), we show that the peak speed was under- or overestimated by the forecast. For comparison, we present in Figure~\ref{fig:2017} the real-time application (using beacon data) for the three models during 2017 (January-August)\footnote{These results use the $r_{\rm CH_{med}}$ as derived over the year 2017. For the real-time application, this $r_{\rm CH_{med}}$ is adapted every month.}. The results for the time range under study, and separately for the years 2008, 2012, and 2017 are summarized in Tables~\ref{tab:events} and~\ref{tab:cont}.
Inspecting Table~\ref{tab:events} we obtain, compared to ACE+27, in general, a better performance of the models STEREO and STEREO+CH with the highest values for POD and lowest for FAR. For comparison, we give the results also for CMEs included in the statistics. For 2008 almost no changes are revealed in the performance of the models by excluding the few (3) CME events which occurred. Interestingly, for 2012 a slightly better performance is revealed when CMEs are included, that might be related to CMEs occurring close to CHs \cite[see e.g.,][]{dhuys14}. As we focus in our study on solar wind streams, we describe the performance of the models derived by excluding CMEs from the time series. In total (2008--2012) the number of hits can be increased from 111 (ACE+27) to 119 (STEREO) and 124 (STEREO+CH), while the number of misses can be decreased from 81 (ACE+27) to 67 (STEREO) and 62 (STEREO+CH). Also, the false alarms are fewer for STEREO (61) and STEREO+CH (64) compared to ACE+27 (79). Inspecting TS values, we obtain an improvement from ACE+27 (TS=0.41) to STEREO (TS=0.49) and STEREO+CH (TS=0.50). ACE+27 looses 51\% performance comparing low and high solar activity (TS=0.57 in 2008 and 0.28 in 2012), while for STEREO it is 42\% (0.67/0.39 in 2008/2012) and STEREO+CH 30\% (0.65/0.46 in 2008/2012). Results for the continuous values of the measured and predicted solar wind speed, as their mean and median values, errors, standard deviations and correlation coefficients, are given in Table~\ref{tab:cont} and show that STEREO and STEREO+CH are able to forecast the measured speed profile better than ACE+27, with RMSE of the order of 80~km/s for STEREO and STEREO+CH, and 88~km/s for ACE+27. On average, for ACE+27 around 46\% of the detected hits underestimate the peak speed, 45\% for STEREO and 42\% for STEREO+CH. Using STEREO and STEREO+CH we obtain a correlation coefficient of 0.72 and 0.71, respectively, between measured and predicted solar wind speed while lower, 0.56, for ACE+27. When investigating the entire time span (2008--2012), by excluding CMEs from the time series the correlation coefficient increases for all models. The accuracy of the arrival time for each hit lies for all models in the range of $\pm$1~day.
\begin{table}
\caption[]{Event based statistics for ACE+27, STEREO and STEREO+CH persistence models covering the time range 2008--2012 and 2017 (January--August). ALL = 2008--2012; YES = CMEs occurrences are included in the statistical analysis; NO = CMEs are excluded from the statistical analysis; Total = number of all predicted peaks; TP = hit; FP = false alarm; FN = miss; FAR = False Alarm Ratio; TS = Threat Score (worst: TS = 0, best: TS = 1); BS = Bias (BS $<$ 1 underestimation; BS $>$ 1 overestimation); POD = Probability of Detection;
}
\label{tab:events}
$$
\begin{array}{lllllllll}
\hline
Data & Total & \mathit{TP} & \mathit{FP} & \mathit{FN} & \mathit{FAR} & \mathit{TS} & \mathit{BS} & \mathit{POD} \\ \hline
\multicolumn{9}{c}{\mathit{ACE}+27} \\ \hline
2008/YES & 41 & 30 & 11 & 12 & 0.29 & 0.57 & 0.98 & 0.73 \\ \hline
2008/NO & 41 & 30 & 11 & 12 & 0.29 & 0.57 & 0.98 & 0.73 \\ \hline
2012/YES & 44 & 22 & 22 & 22 & 0.50 & 0.33 & 1.00 & 0.50 \\ \hline
2012/NO & 33 & 15 & 18 & 21 & 0.58 & 0.28 & 0.92 & 0.45 \\ \hline
ALL/YES & 215 & 126 & 89 & 90 & 0.42 & 0.41 & 1.00 & 0.59 \\ \hline
ALL/NO & 190 & 111 & 79 & 81 & 0.42 & 0.41 & 0.99 & 0.58 \\ \hline
2017/YES & 30 & 18 & 12 & 13 & 0.42 & 0.42 & 0.97 & 0.60 \\ \hline
\multicolumn{9}{c}{\mathit{STEREO}~persistence~model} \\ \hline
2008/YES & 42 & 34 & 8 & 8 & 0.19 & 0.68 & 1.00 & 0.81 \\ \hline
2008/NO & 41 & 33 & 8 & 8 & 0.20 & 0.67 & 1.00 & 0.80 \\ \hline
2012/YES & 43 & 25 & 18 & 19 & 0.43 & 0.40 & 0.98 & 0.58 \\ \hline
2012/NO & 30 & 18 & 12 & 16 & 0.47 & 0.39 & 0.88 & 0.60 \\ \hline
ALL/YES & 208 & 137 & 71 & 79 & 0.37 & 0.48 & 0.96 & 0.66 \\ \hline
ALL/NO & 180 & 119 & 61 & 67 & 0.36 & 0.49 & 0.97 & 0.66 \\ \hline
2017/YES & 33 & 20 & 13 & 11 & 0.35 & 0.45 & 1.06 & 0.61 \\ \hline
\multicolumn{9}{c}{\mathit{STEREO+CH}~persistence~model} \\ \hline
2008/YES & 41 & 33 & 8 & 9 & 0.21 & 0.66 & 0.98 & 0.80 \\ \hline
2008/NO & 40 & 32 & 8 & 9 & 0.22 & 0.65 & 0.98 & 0.80 \\ \hline
2012/YES & 47 & 29 & 18 & 15 & 0.34 & 0.47 & 1.07 & 0.62 \\ \hline
2012/NO & 36 & 22 & 14 & 12 & 0.35 & 0.46 & 1.06 & 0.61 \\ \hline
ALL/YES & 214 & 143 & 71 & 73 & 0.34 & 0.50 & 1.00 & 0.67 \\ \hline
ALL/NO & 188 & 124 & 64 & 62 & 0.33 & 0.50 & 1.01 & 0.66 \\ \hline
2017/YES & 33 & 19 & 14 & 12 & 0.39 & 0.42 & 1.06 & 0.58 \\ \hline
\end{array}
$$
\end{table}
\begin{table}
\caption[]{Statistics on continuous values for solar wind speed [km/s] covering the time range 2008--2012 and 2017 (January--August). ALL = 2008--2012; YES = CMEs occurrences are included in the statistical analysis; NO = CMEs are excluded from the statistical analysis; ME = mean error; MAE = mean average error; RMSE = root mean square error; meanP = mean speed predicted; stddevP = standard deviation predicted; meanM = mean speed measured; stddevM = standard deviation measured; CC = correlation coefficient; RUS = ratio of underestimated speed (calculated for hits only);
}
\label{tab:cont}
$$
\begin{array}{lrlllrrrlr}
\hline
Data & \mathit{ME} & \mathit{MAE} & \mathit{RMSE} & meanP & stddevP & meanM & stddevM & \mathit{CC} & \mathit{RUS} \\ \hline
\multicolumn{9}{c}{ACE+27} \\ \hline
2008/YES & -6.62 & 57.96 & 77.13 & 458.96 & 116.48 & 452.34 & 114.89 & 0.67 & 0.33 \\ \hline
2008/NO & -6.95 & 58.07 & 77.31 & 460.10 & 116.75 & 453.15 & 115.44 & 0.67 & 0.33 \\ \hline
2012/YES & -0.29 & 73.44 & 98.92 & 407.90 & 81.91 & 407.61 & 82.38 & 0.31 & 0.45 \\ \hline
2012/NO & -0.68 & 71.13 & 95.39 & 408.81 & 82.78 & 408.13 & 81.49 & 0.49 & 0.40 \\ \hline
ALL/YES & -1.84 & 67.32 & 90.23 & 412.92 & 94.53 & 411.08 & 93.67 & 0.53 & 0.45 \\ \hline
ALL/NO & -2.36 & 66.03 & 88.31 & 414.33 & 96.22 & 411.97 & 94.83 & 0.56 & 0.46 \\ \hline
2017/YES & -1.43 & 81.86 & 108.41 & 462.42 & 113.20 & 460.99 & 110.95 & 0.62 & 0.61 \\ \hline
\multicolumn{9}{c}{STEREO~persistence~model} \\ \hline
2008/YES & 8.26 & 46.39 & 63.79 & 444.08 & 116.20 & 452.34 & 114.89 & 0.80 & 0.59 \\ \hline
2008/NO & 8.13 & 46.16 & 63.73 & 445.62 & 117.02 & 453.75 & 115.96 & 0.78 & 0.58 \\ \hline
2012/YES & 6.54 & 69.90 & 94.44 & 401.06 & 93.64 & 407.61 & 82.38 & 0.54 & 0.44 \\ \hline
2012/NO & 11.62 & 66.00 & 88.27 & 396.62 & 91.37 & 408.24 & 82.64 & 0.71 & 0.44 \\ \hline
ALL/YES & 8.69 & 59.46 & 82.71 & 402.39 & 99.02 & 411.08 & 93.67 & 0.61 & 0.47 \\ \hline
ALL/NO & 9.79 & 57.15 & 79.28 & 402.15 & 99.71 & 411.94 & 95.08 & 0.72 & 0.45 \\ \hline
2017/YES & 14.22 & 87.26 & 116.60 & 446.77 & 101.59 & 460.99 & 110.95 & 0.61 & 0.65 \\ \hline
\multicolumn{9}{c}{STEREO+CH~persistence~model} \\ \hline
2008/YES & -1.70 & 46.78 & 63.16 & 454.05 & 121.25 & 452.34 & 114.89 & 0.80 & 0.39 \\ \hline
2008/NO & -1.94 & 46.53 & 63.01 & 455.69 & 122.08 & 453.75 & 115.96 & 0.77 & 0.38 \\ \hline
2012/YES & -12.17 & 72.84 & 96.85 & 419.77 & 97.20 & 407.61 & 82.38 & 0.53 & 0.45 \\ \hline
2012/NO & -7.00 & 68.58 & 90.24 & 415.24 & 95.30 & 408.24 & 82.64 & 0.62 & 0.45 \\ \hline
ALL/YES & -5.12 & 60.45 & 83.25 & 416.20 & 102.31 & 411.08 & 93.67 & 0.62 & 0.44 \\ \hline
ALL/NO & -3.86 & 57.99 & 79.62 & 415.80 & 103.05 & 411.94 & 95.08 & 0.71 & 0.42 \\ \hline
2017/YES & -6.60 & 84.88 & 113.60 & 467.59 & 104.07 & 460.99 & 110.95 & 0.64 & 0.42 \\ \hline
\end{array}
$$
\end{table}
\section{Summary and discussion}
We present the concept of a new persistence model (STEREO+CH) to forecast solar wind speed using the advantage of multi-viewpoint satellite data. The model is based on a STEREO persistence method using STEREO-B (and -A in 2017) in-situ measurements shifted forward by a variable time span ($\sim$2--10 days) according to the angle of the STEREO spacecraft with respect to Earth. To this model we apply the information on CH evolution by comparing CH areas extracted in EUV data from STEREO and Earth perspective. By analyzing the evolution of CHs from different satellites we tracked the history of a CH and how changes in the CH area affect the solar wind speed arriving at 1~AU.
Compared to ACE+27 (ACE in-situ measurements shifted forward by a full solar rotation), STEREO and STEREO+CH use the most recent available information to forecast the solar wind speed. This is well reflected when comparing the ``hit and miss'' performance of the different models, revealing a clear improvement between ACE+27 and STEREO, and further on between STEREO and STEREO+CH. By taking into account the evolution of CHs, compared to ACE+27, the STEREO+CH model is able to reduce the misses by about 23\%, the false alarms by about 19\%, and increases the hit rate by about 12\%. In comparison to STEREO, STEREO+CH can reduce the misses by about 8\%, increase the hits by about 4\%, but produces a higher false alarm rate by 5\%. Especially during times of enhanced activity (as in 2012), the performance is better for STEREO+CH compared to ACE+27 and STEREO. The additional information of CH evolution is however not able to deliver more accurate solar wind speed forecasts than using the STEREO persistence model without CH information. In general, the correlation between forecast and measured speed increases when excluding CME events, hence, for operational persistence models an automatic detection of CMEs from in-situ data would be an advantage \citep[see e.g.][]{vennerstroem15}.
The STEREO+CH model produces an intensified profile for those solar wind streams that are related to expanding CHs, while suppressing those which are related to decaying CHs. This is in favor for the prediction as it increases the number of hits and reduces the number of misses. However, we find that the derived CH evolution ratio cannot be related in a simple linear way to the observed speed variations. On average, we obtain that expanding CHs more likely cause an increase in the related solar wind speed stream, while decaying CHs show less clear or no trends at all related to a decrease in the speed. Only for the year 2017 we derive distinct results showing that decaying CHs cause a decrease in the solar wind speed. On a statistical basis, for larger separation angles (and increased solar activity) we observe more decaying than expanding CHs. From this we may speculate that either the decay phase of a CH is of longer duration compared to its expansion phase or, as the solar activity picked up in 2012, more CHs decayed. An increased solar activity contributes to a more frequent and eruptive opening of magnetic field (CME events), hence, maybe a faster evolutionary phase of CHs. The behavior of CHs and solar wind evolution has a solar cycle dependence and varies with changes in the underlying as well global magnetic field \citep[e.g.][]{wang90,luhmann09,petrie13}. In that respect, also polar coronal holes influence the solar wind speed in the ecliptic plane, however, the interplay between polar and low lying coronal holes is not fully understood yet \citep[see also a recent review by][]{cranmer17}. \cite{miyake12} concluded from investigating STEREO in-situ data during 2007--2009 for predicting the Kp-index, that a simple correlation method of solar wind measurement at separated solar longitude is not enough even though the correlation is generally high. Also our study revealed that solar wind high-speed streams related to decaying CHs undergo various speed changes that cannot be covered by a simple statistical approach. As we observe a large fraction of decaying CHs, STEREO+CH is not able to deliver more accurate forecasts. In order to refine future models, the relation between solar surface structures as magnetic field and CHs, and in-situ solar wind plasma and magnetic field needs to be investigated in more detail to better understand the physics behind.
The study shows the importance for an L5 mission covering at least EUV and magnetic imager, and in-situ plasma and magnetic field instruments, to explore in more detail the mid-term evolution of solar surface structures and their effects in the solar wind. The real-time application of the STEREO+CH persistence model and comparison to in-situ measurements can be found at $\href{http://swe.uni-graz.at}{swe.uni-graz.at}$ under \textit{Services}.
\begin{acknowledgements}
M.T. acknowledges the support by the FFG/ASAP Programme under grant no. 859729 (SWAMI). J.H.\, is supported by the CCSOM project funded by BELSPO (BRAIN-be). We thank the ACE SWEPAM instrument team and the ACE Science Center for providing the ACE data. SDO data are courtesy of the NASA/SDO and the AIA science team, STEREO data are courtesy of the NASA/STEREO and the SECCHI and PLASTIC science teams. M.A.R.\,acknowledges the Austrian Science Fund (FWF): J4160-N27. We also would like to thank the anonymous reviewers for their careful reading of our manuscript and their helpful comments.
\end{acknowledgements}
\begin{figure}
\centering
\includegraphics[angle=90,width=0.9\columnwidth]{fig6.pdf}
\caption{\small Solar wind speed measured by ACE (red curve) covering the time range 2008--2012 with the forecast from the ACE+27 persistence model overlaid (blue curve). Extracting the R\&C list, we mark detected CME events at ACE (magenta; ICME at ACE) and during the time span covered by the persistence model ACE+27, i.e.\,during the subsequent rotation (green; ICME at ACE+27). Triangles denote the detected peaks in each profile, which are labeled as a hit (blue cross), false alarm (red circle), or miss (red cross). Black dots below or above a hit symbol marks that the actually measured speed was under- or overestimated by the forecast. }
\label{fig:p27}
\end{figure}
\begin{figure}
\centering
\includegraphics[angle=90,width=.9\columnwidth]{fig7.pdf}
\caption{\small Same as Figure~\ref{fig:p27}, but overlaid with STEREO-B persistence model forecast results. Detected CME events at ACE (R\&C list) and STEREO-B (Jian list) are marked in magenta and green. }
\label{fig:p4}
\end{figure}
\begin{figure}
\centering
\includegraphics[angle=90,width=.9\columnwidth]{fig8.pdf}
\caption{\small Same as Figure~\ref{fig:p4}, but overlaid with STEREO+CH model results. }
\label{fig:ST-CH}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=1.\columnwidth]{fig9.pdf}
\caption{\small Results for the year 2017 using ACE+27 (top), STEREO (middle), and STEREO+CH (bottom) persistence with the same symbols and description as given for Figures~\ref{fig:p4} and~\ref{fig:ST-CH}. }
\label{fig:2017}
\end{figure}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 6,914
|
{"url":"https:\/\/www.gradesaver.com\/textbooks\/math\/precalculus\/precalculus-6th-edition-blitzer\/chapter-6-section-6-7-the-dot-product-exercise-set-page-792\/21","text":"Precalculus (6th Edition) Blitzer\n\n${{38.70}^{{}^\\circ }}$\nLet the angle between $\\mathbf{v}$ and $\\mathbf{w}$ be $\\theta$ such that the angle between the vectors $\\mathbf{v}$ and $\\mathbf{w}$ can be obtained using the formula $\\theta ={{\\cos }^{-1}}\\left( \\frac{\\mathbf{v}\\cdot \\mathbf{w}}{\\left| \\mathbf{v} \\right|\\left| \\mathbf{w} \\right|} \\right)$ as, \\begin{align} & \\theta ={{\\cos }^{-1}}\\left( \\frac{\\mathbf{v}\\cdot \\mathbf{w}}{\\left| \\mathbf{v} \\right|\\left| \\mathbf{w} \\right|} \\right) \\\\ & ={{\\cos }^{-1}}\\left( \\frac{\\left( 6\\mathbf{i}+0\\mathbf{j} \\right)\\cdot \\left( 5\\mathbf{i}+4\\mathbf{j} \\right)}{\\left( \\sqrt{{{\\left( 6 \\right)}^{2}}} \\right)\\left( \\sqrt{{{5}^{2}}\\mathbf{+}{{4}^{2}}} \\right)} \\right) \\\\ & ={{\\cos }^{-1}}\\left( \\frac{6\\cdot 5+0\\cdot 4}{\\left( 6 \\right)\\left( \\sqrt{41} \\right)} \\right) \\\\ & ={{\\cos }^{-1}}\\left( \\frac{30}{\\sqrt{1476}} \\right) \\end{align} Solve ahead to get the result as, \\begin{align} & \\theta ={{\\cos }^{-1}}\\left( \\frac{30}{\\sqrt{1476}} \\right) \\\\ & ={{38.70}^{{}^\\circ }} \\end{align} Hence, the angle between $\\mathbf{v}$ and $\\mathbf{w}$ is $\\theta ={{38.70}^{{}^\\circ }}$.","date":"2021-04-21 08:54:02","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\": 2, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 1.0000100135803223, \"perplexity\": 3734.223187199013}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-17\/segments\/1618039526421.82\/warc\/CC-MAIN-20210421065303-20210421095303-00469.warc.gz\"}"}
| null | null |
Теосо́фське товариство (самоназва — Теософічне товариство; від теософія) — міжнародна громадська організація. Девіз Товариства: Satyat Nasti Paro Dharmah (санскрит); прийнятий (але не точний) переклад на англійську:There is no religion higher than truth («Немає релігії понад істину»).
Історія створення
Товариство було засновано 17 листопада 1875 в Нью-Йорку Оленою Петрівною Блаватською, полковником Генрі Олкоттом і .
Відомий показовий факт. При реєстрації Теософського товариства в Сент-Луїсі в 1880 році в американському суді суддя А. Александер засвідчив:
«Прохач не являє собою релігійну громаду <…> Слід відзначити, що в ст. 2 Статуту цього Товариства слово "релігія" вживається у множині. Викладання релігійних навчань є освітньо-просвітницька діяльність, а не релігійна. "Сприяти вивченню релігій" у деякому сенсі означає сприяти вивченню історії людства. Попутно зауважу, що у Товариства немає релігійного символу віри або особливого культу».
З 1882 штаб-квартира Товариства знаходиться в Адьярі, Індія.
Три мети Теософського Товариства, закріплені в його статуті
Заснувати ядро всесвітнього братерства незалежно від раси, віри, статі, касти тощо.
Заохочувати порівняльне вивчення релігій, філософії і наук.
Дослідити непояснені закони природи і приховані сили людини.
Видатні люди — члени Теософського Товариства
Олена Блаватська, російська філософ, вчений, просвітитель, письменниця, засновниця Теософського Товариства
Томас Едісон, американський винахідник
Вільям Крукс, провідний фізик і хімік XIX століття, Президент лондонського королівського товариства (1913—1915)
Каміль Фламмаріон, знаменитий французький астроном
Вільям Джемс, американський філософ і психолог
Мотілал Неру, батько першого прем'єр-міністра незалежної Індії Джавахарлала Неру.
Вільям Батлер Єйтс, ірландський англомовний поет, драматург. Лауреат Нобелівської премії з літератури 1923.
Анна Кінгсфорд, одна з перших жінок в Англії з вченим ступенем у галузі медицини.
Доля Товариства після смерті О. П. Блаватської
Після смерті Блаватської її продовжувачами стали Олкотт і Джадж, а також англійська письменниця Анні Безант. У 1895 Джадж, внаслідок конфлікту з Олкоттом і Безант, відокремився й самостійно очолив «Американську секцію» Товариства.
Організація, очолювана Олкоттом і Безант продовжує базуватися в Індії і відома як The Theosophical Society — Adyar; секція ж, очолювана Джаджом, нині відома просто як The Theosophical Society, але для ясності часто додають international headquarters, Pasadena, California (адреса штаб-квартири в Пасадені, Каліфорнія).
Третя структура, відома як The United Lodge of Theosophists (ULT), відокремилася від другої в 1909.
Президенти Теософського Товариства
Полковник Г. С. Олкотт став першим президентом Теософського Товариства і очолював його з 1875 по 1895 роки (з 1895 по 1907 рр. — його частини в Адьярі).
Анні Безант керувала з 1907 по 1933 рр..
Джордж С. Арундейл — з 1934 по 1945 рр..
— з 1946 по 1953 рр..
Нілаканта Шрі-Рам — з 1953 по 1973 рр..
Джон Коутс — з 1973 по 1979 рр..
Радга Берньє — з 1980 по 2013 рр..
Тім Бойд є президентом Товариства з 2014 р. і до цього дня.
Пам'ять
У 1975 урядом Індії була випущена пам'ятна марка, присвячена 100-річчю заснування Теософського Товариства. На марці зображено печатку Товариства і його девіз: «Немає релігії вище істини».
Див. також
Ісус Христос у вченні Теософського товариства
Примітки
Джерела
Посилання
Theosophical society — Adyar . Міжнародний офіційний сайт Теософського товариства (Адьяр).
САЙТ ТЕОСОФСЬКОГО ТОВАРИСТВА В УКРАЇНІ
Blavatsky Net — Theosophy . Сайт про О. П. Блаватську й теософію.
Міжнародні громадські організації
Теософія
Філософські напрямки та школи
Організації Нью-Йорка
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 1,253
|
\section{Introduction}
\label{intro}
The propagation of high frequency surface acoustic waves (SAWs) on a crystalline piezoelectric substrate are exquisitely sensitive to their physical environment. For this reason they have proven to be a versatile tool for studying a wide variety of physical systems placed in close proximity to the surface wave. These range from molecular adsorbates\cite{Bal89,Roc09} to two-dimensional electron systems\cite{wix86,wil932,esk96,pol16,fri17,Lan18} to superconducting qubits\cite{Gus14,Man17}, just to name a few. SAWs have also found utility in studying quantum fluids where they have been used to explore the bulk acoustic properties of liquid $^{3}$He and $^{4}$He\cite{Arz67,Tok79,Che82,Aok03,Aok032} and to investigate superfluidity of extremely thin films (2-10 monolayers) of $^{4}$He\cite{Mor80,Che822,Kos90}. However, to our knowledge, no surface acoustic wave experiments have been performed on helium films having a thickness on the order of $\sim 10-100$ nm, where one would expect strong coupling of the SAW to excitations in the liquid helium surface. Moreover, since films of this thickness are routinely used as a substrate in experiments with electrons on helium\cite{And97,Mon04}, understanding the coupling of a piezoelectric surface wave to the helium film is a prerequisite for future SAW measurements of the non-degenerate two-dimensional electron system that can be created on the surface of the liquid.
In light of this, we have performed pulsed measurements of the attenuation, $\alpha$, of high frequency SAWs on the surface of lithium niobate caused by thin films of liquid helium. We find that for superfluid films with a thickness of $\approx 60 - 70$ nm the SAW attenuation anamolously exceeds that of bulk helium at the same temperature. We speculate that this increased attenuation is due to the electrostrictive coupling between the piezoacoustic wave and excitations in the helium film surface.
\section{Experiment}
\label{expt}
The SAW attenuation experiments reported here were performed using a YZ cut lithium niobate (LiNbO\textsubscript{3}) single crystalline wafer as a substrate for SAW propagation. The lithium niobate substrate was diced into a rectangle with a length of 20 mm and a width of 10 mm. To excite and detect SAWs, two identical interdigitated transducers (IDT) having 40 pairs of $3 \mu$m wide fingers were patterned on the surface a LiNbO$_{3}$ chip in the form of a delay line, having a length of 16.5 mm, along the crystallographic x-axis using conventional photo-lithography (see Fig.~\ref{fig1}a). The transducers were fabricated from aluminum and had a thickness of approximately 70 nm and a width of 4 mm defining the width of the SAW beam within the delay line. We note that in this geometry the IDTs are positioned 1.75 mm from the diced edge of the chip.
\begin{figure*}
\includegraphics{Fig1-eps-converted-to.pdf}
\caption{Pulsed surface acoustic wave experimental setup. (a) Schematic of the lithium niobate chip used for exciting and detecting SAW via aluminium interdigitated transducers (IDT), which were directly fabricated onto the surface of the substrate. (b) Block diagram of the circuit used for measuring the attenuation of pulsed SAWs. As described in the text, by measuring the time-of-flight signal we are able to disentangle direct capacitive crosstalk between the two interdigitated transducers as well as multiply reflected SAW pulses.}
\label{fig1}
\end{figure*}
Resonant SAWs are launched by applying a high frequency signal between the transducer fingers of the exciter to create an elastic distortion of the piezoelectric substrate beneath the IDT. In this configuration, Rayleigh mode surface acoustic waves with components parallel (longitudinal) and perpendicular (transverse) to the wave propagation are launched along the LiNbO$_{3}$ surface toward the detector IDT. The fundamental resonant frequency of our Rayleigh wave SAW device is $\nu = v_{s}/\lambda = 291$ MHz, dictated by the IDT finger periodicity, $\lambda = 12~\mu$m, and the speed of sound in YZ-cut LiNbO$_{3}$, $v_{s} = 3488$ m/s. The frequency response of the SAW delay used in our attenuation measurements was characterized using an Agilent N5230A vector network analyzer. Fig.~\ref{fig2} shows the measured transmission coefficient, $S_{12}$, of the delay line as a function of frequency at $T = 1.55$ K in vacuum.
\begin{figure}
\includegraphics{Fig2-eps-converted-to.pdf}
\caption{Frequency dependence of the transmission coefficient ($S_{12}$) of the SAW delay line at $T = 1.55$ K in vacuum. The resonant peak at $\nu = 296$ MHz is associated with the generation of surface acoustic waves in the IDT delay line.}
\label{fig2}
\end{figure}
The resonance in the transmitted power at $\nu \simeq$ 296 MHz is associated with the generation of SAW in the substrate. The measured resonance is a few percent larger than the expected value, likely due to an increase in the elastic moduli of LiNbO$_{3}$ at cryogenic temperature.
Fig.~\ref{fig1}b shows a diagram of the circuit used to measure the attenuation of the piezoelectric waves. Pulsed SAWs are created by gating a continuous wave signal using a fast solid-state switch. For the measurements reported here the pulses had a width of $3~\mu$s and were repeated at a frequency of 1 kHz. The received SAW signal at the detector IDT was amplified by 30 dB and measured using a calibrated crystal diode detector (Krytar Model 203AK S/N 00277). A boxcar integrator was used for time-of-flight measurements. This mode of operation allows us to disentangle the piezoacoustic signal from direct capacitive crosstalk between the two IDTs as well as multiply reflected surface acoustic wave pulses within the delay line.
To study the surface acoustic wave attenuation in liquid $^{4}$He, the SAW device was mounted inside of a hermetically sealed copper cell attached to a closed cycle 1K cryostat. Helium gas was supplied into the cell at T $\simeq$ 1.55 K through a capillary fill line. A resistive thermometer located outside of the cell and calibrated relative to the known superfluid transition temperature of $^{4}$He was used to measured the temperature of the liquid helium. The liquid helium volume admitted into the cell was determined by varying the pressure in a calibrated standard volume of 260 cc at room temperature. The thickness of the helium film can be estimated from\cite{Pob92},
\begin{equation}
d = \Big(\frac{\gamma}{\rho g H}\Big)^{1/4},
\end{equation}
where $\gamma$ is the van der Waals constant, $\rho$ is the mass density of $^{4}$He, $g$ is the acceleration due to gravity, and $H$ is the distance from the LiNbO$_{3}$ surface down to the liquid helium level in the reservoir volume in the cell. Three-dimensional modeling of the experimental cell open volume was used to calculate $H$ from the volume of helium admitted into the cell from the calibrated volume at room temperature.
\section{Results and Discussion}
\label{RandD}
Before presenting our results on thin liquid helium films, we first demonstrate that we are able to use our measurement setup to reproduce the known temperature dependence of Rayleigh wave attenuation by bulk liquid helium. Rayleigh mode surface acoustic waves are attenuated by contact with a bulk liquid due to the surface-normal component of the particle motion and that of the co-propagating piezoelectric field. These components generate longitudinal compressional waves upward into the liquid, dissipating most of the acoustic energy\cite{Arz67,Dran70}. This method of energy loss is closely related to the problem of the Kaptiza resistance based on the acoustic mismatch between liquid helium and a solid substrate\cite{Pob92,Pol09}, whereby longitudinal waves having a velocity $v$ are emitted into the liquid from the substrate surface at an angle $\phi = \arcsin(v/v_{s})$. In addition to this mechanism, the in-plane shear component of the SAW can also radiate energy into a fluid over the viscous penetration depth,
\begin{equation}
L = \sqrt{\frac{\eta}{\pi \nu \rho}}
\end{equation}
where $\eta$ and $\rho$ are the viscosity and mass density of the fluid. However, SAW energy lost to shear is two to three orders of magnitude smaller than longitudinal absorption in our temperature range and is negligible even for our measurements in thin helium films.
Our results for the temperature dependence of the SAW attenuation produced by bulk liquid helium are shown in Fig. \ref{fig4} as the solid red curve. For comparison we also show the prediction for the attenuation (dashed red curve) based on the loss of SAW energy in the form of longitudinal compressional waves\cite{Dran70}. As expected, our measurements are in good correspondence with theoretical prediction. Moreover, our results for bulk helium are also in good agreement with previous measurements of the attenuation of piezoelectric Rayleigh waves by liquid helium\cite{Arz67,Tok79,Che82,Aok03}.
\begin{figure}
\includegraphics{Fig3-eps-converted-to.pdf}
\caption{Attenuation, $\alpha$, of Rayleigh waves on LiNbO$_{3}$ in contact with bulk liquid helium. Our measurements (solid red curves) at $\nu=296$ MHz are in good agreement with the theory (dashed red curve) of Dransfeld and Salzmann\cite{Dran70} for energy loss due to radiation of longitudinal compressional waves into the liquid. (Inset) The local minimum of the SAW attenuation at $T_{\lambda}= 2.17$ K is associated with the transition from normal to superfluid $^{4}$He.}
\label{fig3}
\end{figure}
Unexpectedly, however, we find that when the thickness of the liquid helium layer is sufficiently reduced an anomalously large attenuation can be induced. In Fig.~\ref{fig4} we show measurements of the SAW attenuation (solid blue data) at $T = 1.55$ K while increasing the volume of helium in the cell by small increments.
\begin{figure}
\includegraphics{Fig4_Rev4-eps-converted-to.pdf}
\caption{Attenuation, $\alpha$, of Rayleigh waves on LiNbO$_{3}$ in contact with thin films of liquid helium at $T = 1.55$ K and $\nu = 296$ MHz. The measurements are made by incrementally admitting small amounts of helium gas into the cell from a standard volume at room temperature (lower horizontal axis). The vertical dashed line indicates the estimated thickness of the helium film on the LiNbO$_{3}$ surface based on Eq.~ 1 and the height $H$ from the substrate to the bulk helium reservoir level in the cell (top horizontal axis). For comparison, the horizontal solid red line indicates the SAW attenuation due to bulk liquid helium at the same temperature. The calculated thickness of the helium film for the final two data points above the bulk value are 73 nm and 76 nm. Finally, we note that the IDT fingers are expected to fill completely with helium via capillary action for very small amounts of helium introduced into the cell, \emph{i.e.} already at H = 0.46 mm}
\label{fig4}
\end{figure}
With an increasing amount of helium we observe a rapid increase in the SAW attenuation once the thickness of the superfluid film reaches $\approx$ 60 - 70 nm ($H \approx 0.28$ mm) and continues to increase and quickly exceeds the attenuation produced by bulk helium.
For reference we plot the SAW attenuation measured in bulk helium at the same temperature as the red horizontal line in Fig.~\ref{fig4}. We emphasize that this anomalously large attenuation is reproducible on multiple fillings of the experimental cell and additionally is insensitive to the tilt of the cryostat with respect to vertical\footnote{With further increasing film thickness the SAW attenuation will eventually recover the value measured for bulk helium. However in this crossover regime ($d \sim 1-10 \mu$m) the film thickness on the substrate is exceedingly sensitive to fluctuations in $H$ and also to the tilt of the cryostat leading to non-reproducibility. Future experiments, using microchannel geometries could allow for controlled measurement in this regime of film thickness.}. Moreover, we find that this phenomenon is not restricted to superfluid helium but rather persists into the normal state.
To our knowledge, no prior measurements of the attenuation of high frequency SAW have been made in this regime of helium thickness nor is there a theory describing the coupling of SAWs to such films. We speculate that the occurrence of the attenuation anomaly reported here for thin helium films can qualitatively be understood in terms of a coupling between the piezoelectric surface wave on the LiNbO$_{3}$ substrate and excitations in the helium surface. While the physical displacement of atoms associated with the SAW is at the angstrom scale, the SAW piezoelectric field has a spatial extent on the order of the SAW wavelength, which in our case is $\simeq 12~\mu$m. In fact, it is well-known that an electric field gradient can exert a force to move liquid helium via electrostriction, an effect which has even been utilized in developing a number of recent superfluid optomechanical systems\cite{Har16,Lor17,Sou17,Chi17}. It is possible that an electrostrictive coupling between the LiNbO$_{3}$ surface wave and the liquid helium surface conspire to produce a new mode of energy loss not present in bulk helium. Finally, we note that our observation of enhanced attenuation in thin \emph{normal} helium films is able to rule out the coupling of SAWs to third sound as the sole mechanism for increased SAW energy loss.
\section{Conclusion}
\label{conclude}
In conclusion, we have performed high frequency SAW attenuation measurements in thin films of superfluid and normal $^{4}$He where we find an anomalously large loss of energy from the piezoelectric surface wave into the liquid. We suspect that this increased attenuation is associated with electromechanical excitation of the helium film surface via a coupling of the SAW electric field to the dielectric constant of liquid helium.
\begin{acknowledgements}
We are grateful to M.I. Dykman and W.P. Halperin for helpful discussions. This work was supported by the NSF (Grant no. DMR-1708331).
\end{acknowledgements}
\bibliographystyle{spphys}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 5,640
|
She did a great job. We made paper bag turkeys last year, and I think we will have to do it again this year!
You always have the cutest crafts to go with the books y'all read! This turkey is super!
Very cute craft. That book looks great. I think it would be wonderful to share with my 2 year old and baby!
I'm eagerly waiting for my Thanksgiving books I ordered to come in. Hopefully we'll get some cute Thanksgiving crafts in this year.
Oh that turkey is adorable. Thanks for doing this linky each and every week it's great.
This is a pretty complicated project, and Emily did great. I can totally see her reasoning about 3 feathers.
I love her paper bag turkey! She did a great job on it!
Her turkey looks great. I really want to try this now.
We just checked that book out from the library and our kids love it! This would be a fun craft to go along.
I just found your blog via Little Page Turners. I linked a few projects of my own and am a new follower. I love doing these kinds of things with my kids!
The book looks really cute. The paper bag turkey is adorable. I love crafts with different steps that the girls are able to do themselves.
Cute Turkey! There are a bunch of turkeys this week it looks like...good for me to add to my list hehehe!
What a cute looking turkey. Emily did such a great job.
Super cute turkey! One of these days, I would love it if you would share if you keep crafts and if so, how you store them! I almost dread making lots of crafty things (unless they lie flat) because I have a hard time throwing anything away.
I LOVE your site and your creative ideas! I wanted you to know that I listed you as a resource for teaching little ones in my printable Thanksgiving Unit.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 1,482
|
\section{Introduction}
The well-developed techniques for controlling and manipulating
Bose-Einstein condensates (BECs) of spinor atoms provide an
excellent opportunity to explore novel magnetism and quantum phases.
In a magnetic trap, atoms with different magnetic moments are
subjected to different forces, so it is very difficult to confine a
spinor BEC involving all possible spin states. However, since the
laser-atom interaction is determined by the induced electric dipole
moment, an optical trap may confine every spin state to preserve the
``vector'' property of spinor atoms. This allows one to trap a true
spinor BEC, which involves an ensemble of Bose atoms condensed in a
coherent superposition of all possible hyperfine states. In this
way, several experimental groups have successfully demonstrated
spinor BECs of $^{23}$Na \cite{Stamper-Kurn1998,Miesner1999} and
$^{87}$Rb \cite{Matthews1999, Barrett2001} atoms.
The ground states and some low-energy excitations of a spinor BEC
were theoretically analyzed by Ohmi and Machida \cite{Ohmi1998} and
Ho \cite{Ho1998}.
It has been shown experimentally that all three spin components of a
spin-1 condensate can be either miscible or immiscible with one
another where the immiscibility will lead to formation of spin
domains \cite{Stenger1998}. This has been confirmed numerically
using the Gross-Pitaevskii equation and Thomas-Fermi approximation
\cite{Isoshima1999}. Using the single-mode approximation (SMA), the
ground state population dynamics of a spin-1 BEC have been studied
by Law \emph{et al.} \cite{Law1998} and Pu \emph{et al.}
\cite{Pu1999}.
They have also found that the ground state is a superposition of
collective spin states (Fock states) and cannot be expressed as a
product of individual spin states. This shows the collective
behavior of all three spin components. Ho and Yip \cite{Ho2000},
found the antiferromagnetic ground state to be a fragmented
condensate with large particle number fluctuations as stated in the
references \cite{Law1998,Pu1999}. This fragmented condensate
gradually deforms into a more stable coherent state as the strength
of the external field gradient increases. Recently, Rizzi \emph{et
al.} \cite{Rizzi2008} applied the DMRG method to determine the phase
diagram for one-dimensional spin-1 bosons. In accordance with
Imambekov \emph{et al.} \cite{Imambekov2003} and Yip \cite{Yip2003},
they showed that the dimerized state is among the ground states. The
quantum phases such as the polar phase, nematic phase and spin
singlet phase were discussed by Demler and Zhou \cite{Demler2002}.
In one dimension (1D), spinor Bose gases have a ferromagnetic ground
state in the absence of spin-dependent forces
\cite{Lieb,Yang-Li,Guan2007}. However, the spinor Bose gas can have
either a ferromagnetic or an antiferromagnetic ground state in the
presence of spin-exchange interaction \cite{Ho1998,Ho2000,Ueda}.
Very recently, Cao \emph{et al.} \cite{Cao2007} proved that there
exists an integrable point in scattering parameter space for 1D
spin-1 bosons with both delta-function contact interaction and
spin-exchange interaction. This model provides an important
benchmark to understand spinor BECs and spin liquids in low
dimensions. From the exact Bethe ansatz (BA) solution, Cao \emph{et
al.} found that the ground state is a spin singlet in the absence of
an external field. Essler \emph{et al.} \cite{Essler2009} then
proved that the low energy physics in the weak repulsive coupling
regime can be described by a spin-charge separated theory of an
effective Tomonaga-Luttinger Hamiltonian and an $O(3)$ nonlinear
sigma model. In this weak coupling limit, both the collective
pairing fluctuations and spin fluctuations dominate the low-lying
excitations. Using the BA equations and the effective field theory,
Essler \emph{et al.} calculated the scaling dimensions and the
large-distance asymptotics of correlation functions of the model.
Such a spin liquid phase was also previously investigated by Zhou
\cite{Zhou2001} through the introduction of the Weyl representation
of $SU(2)$.
In this paper, we investigate quantum liquid phases in the 1D
integrable system of spin-1 bosons \cite{Cao2007} with strongly
repulsive and antiferromagnetic spin-exchange interactions. We
derive the thermodynamic Bethe ansatz (TBA) equations on the basis
of particle-hole excitations \cite{Yang1969} and spin-strings
\cite{Takahashi1971}. From these equations we find that for the
strong coupling regime there is a large energy gap in the lowest
spin excitation. We also show that spin fluctuations are frozen out
under a strong external field at zero temperature. When the external
magnetic field exceeds the lower critical field $H_{c1}$, the energy
gap vanishes and the charge excitations evolve into two gapless
modes of singlet pairs and the branch of magnetic quantum number
$m_F=1$ atoms. The external field may break a singlet pair into two
unpaired atoms of $m_{F}=1$ under a sufficiently strong magnetic
field. A ferromagnetic Tonks-Girardeau gas of $m_F=1$ atoms appears
if the external field exceeds the upper critical field $H_{c2}$. The
singlet pairs and unpaired $m_{F}=1$ bosons coexist in the
intermediate region $H_{c1}<H<H_{c2}$. We show that for strong
coupling the low energy physics of the gapless phase is described by
the universality class a two-component Luttinger liquid as long as
the spin dynamics are frozen out. Moreover, from the TBA equations,
we obtain exact results for the ground state energy and magnetism
for the system with an external magnetic field, which provide an
exact phase diagram and the universality class of quantum phase
transitions for the integrable spinor $F=1$ Bose gas with strongly
repulsive and antiferromagnetic spin-exchange interactions.
\section{The model}
We consider $N$ particles confined in 1D to a length $L$ with
delta-function type density-density interactions and spin-spin
interactions between two atoms. In first quantized form, the
Hamiltonian of this model is given by \cite{Ho1998}
\begin{equation}
H = -\frac{\hbar^{2}}{2m}\sum_{i=1}^{N}\frac{\partial^{2}}{\partial
x_{i}^{2}}+\sum_{i<j}[c_{0}+c_{2}\mathbf{F}_{i}\cdot\mathbf{F}_{j}]\delta(x_{i}-x_{j})
+E_{z}\label{Ham}
\end{equation}
where $\mathbf{F}_{i}$ are spin-1 operators,
$c_{0}=(g_{0}+2g_{2})/3$ and $c_{2}=(g_{2}-g_{0})/3$ are the
interaction parameters which are related to the $s$-wave scattering
lengths $a_{F}$ in the spin-0 and spin-2 channels, and
$g_{F}=4\pi\hbar^{2}a_{F}/m$ with $m$ being the mass of the atoms.
The term $E_{z}$ accounts for the Zeeman energy which will later be
given explicitly. Throughout this paper we use the dimensionless
units of $\hbar=2m=1$ for convenience. We are interested in the
antiferromagnetic case $c_{2}>0$ and $c_{0}=c_{2}=c$ where this
model is exactly solvable by the BA \cite{Cao2007}. The model that
we examine also has repulsive density-density interactions since the
interaction parameter $c_{0}=c_{2}>0$. Repulsive interactions
result in an effective attraction in the two-body scattering matrix
for the spin-0 channel and an effective repulsion in the scattering
matrix for the spin-2 channel. This naturally leads to the formation
of singlet bound pairs in the spin-0 channel \cite{Cao2007}. Due to
the existence of the spin exchange interaction in the Hamiltonian,
the number of particles in a particular spin state ($m_{F}=1,0,-1$)
is no longer conserved because spin transmutation is allowed to
occur. The scattering between two particles of spin $m_{F}=1$ and
$m_{F}=-1$ can produce two particles of spin $m_{F}=0$ and
vice-versa. The only conserved quantities are the total particle
number $N$ and the total spin in the $z$-component $S^{z}$. This
model possesses $U(1)$ symmetry for charge conservation and $SU(2)$
symmetry which corresponds to spin conservation. For weak
interaction and in the absence of an external field, the spin
dynamics is described by the $O(3)$ non-linear sigma model which can
be separated from the BA equations \cite{Essler2009}. The charge
sector on the other hand is described by collective pairing density
fluctuations of free boson fields.
The BA equations for this Hamiltonian acting on a totally symmetric
Bose wavefunction are \cite{Cao2007}
\begin{eqnarray}
\exp(\mathrm{i}k_{j}L) &=& -\prod_{l=1}^{N}e_{4}(k_{j}-k_{l})
\prod_{\alpha=1}^{M}e_{-2}(k_{j}-\Lambda_{\alpha}),\nonumber\\
\prod_{l=1}^{N}e_{2}(\Lambda_{\alpha}-k_{l}) &=&
-\prod_{\beta=1}^{M}e_{2}(\Lambda_{\alpha}-\Lambda_{\beta}),
\label{BA}
\end{eqnarray}
where $j=1,\ldots,N$ and $\alpha=1,\ldots,M$. $\{k_{j}\}$ is the set
of quasimomenta for the particles and $\{\Lambda_{\alpha}\}$ is the
set of spin rapidities that characterize the internal spin degrees
of freedom. The quantum number $M$ is a conserved quantity
satisfying the relation $M=N-S^{z}$ and the function
\begin{equation}
e_{n}(x)=\frac{x+\mathrm{i}nc'}{x-\mathrm{i}nc'}
\end{equation}
where $c'=c/4$. The energy $E=\sum_{j}k_{j}^{2}$ and total momentum
$p=\sum_{j}k_{j}$ of the system can be obtained by solving the
coupled BA equations for the sets $\{k_{j}\}$ and
$\{\Lambda_{\alpha}\}$.
\section{The TBA equations}
In the thermodynamic limit $N,L\rightarrow\infty$ with the ratio
$N/L$ finite, the sets of solutions $\{k_{j}\}$ and
$\{\Lambda_{\alpha}\}$ of the BA equations take certain forms. As
mentioned in ref. \cite{Cao2007}, the $k_{j}$'s and
$\Lambda_{\alpha}$'s can form complex pairs
$k_{j}=\lambda_{j}\pm\mathrm{i}c'$ and
$\Lambda_{j}=\lambda_{j}\pm\mathrm{i}c'$ where $\lambda_{j}$ is
real. In Fig. 1, we show a schematic configuration of the
quasimomenta and spin rapidities for the ground state. Notice that
each pair of $k_{j}$'s share the same real part as a corresponding
pair of $\Lambda_{\alpha}$'s. The bound states are associated with a
pair of $m_F=\pm1$ bosons or two $m_F=0$ bosons. In the absence of
an external field, this bound state is created by the operator
$A^{\dagger}\equiv
[(a_0^{\dagger})^2-2a^{\dagger}_1a^{\dagger}_{-1}]/\sqrt{3}$
\cite{Ueda}. In the presence of a sufficiently strong magnetic
field, the singlet bound state of two $m_{F}=0$ bosons is not
energetically favored \cite{Ho2000}. In addition to that, we also
have real $k_{j}$'s and $\Lambda$-strings of the form
$\Lambda_{\alpha}^{n,j}=\Lambda_{\alpha}^{n}+\mathrm{i}(n+1-2j)c'$,
$j=1,\ldots,n$ as solutions. The spin-strings characterize the spin
wave fluctuations. In the thermodynamic limit, the grand partition
function is $Z=tr(\mathrm{e}^{-H/T})=\mathrm{e}^{-G/T}$
\cite{Yang1969,Takahashi1999} where the Gibbs free energy $G=E
+E_{\rm Z}-\mu N-TS$, chemical potential $\mu$, Zeeman energy
$E_{\rm Z}=-HS^z$ and entropy $S$ are given in terms of the
densities of charge bound states and spin-strings which are subject
to the BA equations (\ref{BA}).
The equilibrium states are determined by minimizing the Gibbs free
energy, which gives rise to a set of coupled nonlinear integral
equations -- the TBA equations, i.e.
\begin{eqnarray}
\varepsilon_{1}(k) &=&
k^{2}-\mu-H-Ta_{4}\ast\ln\left(1+e^{-\varepsilon_{1}(k)/T}\right)\nonumber\\
&&+T[a_{1}-a_{5}]\ast\ln\left(1+e^{-\varepsilon_{2}(k)/T}\right)
\nonumber\\
&&-T\sum_{n=1}^{\infty}[a_{n-1}+a_{n+1}]\ast\ln\left(1+e^{-\phi_{n}(k)/T}\right),\nonumber\\
\varepsilon_{2}(k) &=&
2(k^{2}-c'^{2}-\mu)+T[a_{1}-a_{5}]\ast\ln\left(1+e^{-\varepsilon_{1}(k)/T}\right)\nonumber
\\
&&
+T[a_{2}-a_{4}-a_{6}]\ast\ln\left(1+e^{-\varepsilon_{2}(k)/T}\right),\nonumber
\\
\phi_{n}(k)&=&nH+T[a_{n-1}+a_{n+1}]\ast\ln\left(1+e^{-\varepsilon_{1}(k)/T}\right)\nonumber\\
&&+T\sum_{m=1}^{\infty}T_{mn}\ast\ln\left(1+e^{-\phi_{n}(k)/T}\right).\label{TBA}
\end{eqnarray}
We present a detailed derivation of the TBA equations in the
Appendix. In the above equations, the convolution $f\ast g(x)$ and
the functions $a_{n}(x)$ are defined in equations
(\ref{Convolution}) and (\ref{a}). The function $T_{mn}(x)$ is also
given in the Appendix. The TBA equations are expressed in terms of
the dressed energies $\varepsilon_{1}(k)$, $\varepsilon_{2}(k)$ and
$\phi_{n}(k)$ for unpaired states, paired states and spin-strings,
respectively. The dressed energy $\varepsilon_{1}(k)$ depends not
only on the chemical potential $\mu$ and the external field $H$ but
also on the interactions between unpaired bosons and singlet pairs
as well as the spin fluctuations characterized by the spin-strings
$\phi_{n}(k)$. Physically, the dressed energies measure the
energies over the ``Fermi surfaces''. We clearly see that spin
fluctuations are coupled to the dressed energy of unpaired $m_{F}=1$
bosons $\varepsilon_{1}(k)$ through the last term in the first
equation in (\ref{TBA}). The spin flippings caused by thermal
fluctuations are described by the last equation where the magnon
excitations in $m_{F}=1$ bosons are described by an effective
ferromagnetic spin-spin interaction. There is no such spin
fluctuation coupled to the dressed energy of bound pairs due to its
spin neutral effect.
In the strong coupling limit, the dressed energies
$\varepsilon_{1}(k)$ and $\varepsilon_{2}(k)$ marginally depend on
the pressures of each other, see the dressed energy-dependent terms
in the first and second equations. This is similar to the
configuration for the attractive Fermi gas
\cite{GBLB,Iida2007,ZGLBO}. The only difference is that here the
unpaired bosons may scatter between themselves whereas in the
attractive Fermi gas unpaired fermions do not scatter among
themselves. The pressure per unit length of the system is derived
from the expression $p=-\partial G/\partial L$ as
\begin{eqnarray}
p&=&\frac{T}{2\pi}\int_{-\infty}^{\infty}\ln\left(1+e^{-\varepsilon_{1}(k)/T}\right)dk\nonumber\\
&&+\frac{T}{\pi}\int_{-\infty}^{\infty}\ln\left(1+e^{-\varepsilon_{2}(k)/T}\right)dk,
\end{eqnarray}
where the first term corresponds to the pressure for unpaired
bosons and the second term to the pressure for singlet pairs.
\section{Quantum phase transitions and magnetism}
Solving the TBA equations (\ref{TBA}) imposes a formidable challenge
due to the involvement of infinitely many spin-strings. Here we
shall focus on the ground state properties and quantum phase
transitions driven by an external magnetic field. Following the
method developed in \cite{GBLB,HFGB}, we state two conditions to
proceed on, namely we consider: (I) the scenario where we are in the
ground state with $T\rightarrow 0$, and (II) the strong coupling
limit $c\gg 1$. With these two conditions, we can obtain a series
expansion in terms of the coupling strength $1/c$ for various
thermodynamic quantities, as we shall see later. The strong
interaction condition should be easily reached because generally the
interaction energy is much larger than the kinetic energy for a
dilute gas in 1D with finite interaction strength $c$.
When $T\rightarrow 0$, the TBA equations (\ref{TBA}) simplify to
\begin{eqnarray}
\varepsilon_{1}(k)&=&k^{2}-\mu-H+a_{4}\ast\varepsilon_{1}^{-}(k)
+[a_{5}-a_{1}]\ast\varepsilon_{2}^{-}(k)\nonumber \\
\varepsilon_{2}(k)&=&2(k^{2}-c'^{2}-\mu)+[a_{5}-a_{1}]\ast\varepsilon_{1}^{-}(k)\nonumber \\
&&+[a_{6}+a_{4}-a_{2}]
\ast\varepsilon_{2}^{-}(k),\label{TBA-d}
\end{eqnarray}
where the dressed energies $\varepsilon^{-}_a(k)$ with $a=1,\,2$
imply that we only consider the domain where the function
$\varepsilon_{a}(k)<0$. The negative part of the dressed energies
$\varepsilon_{a}(k)$ for $k\le Q_{a}$ corresponds to occupied states
in the dressed energies while the positive part of $\varepsilon_{a}$
corresponds to unoccupied states. The integration boundaries $Q_{a}$
characterize the ``Fermi surfaces'' at $\varepsilon_a(Q_{a})=0$.
There are no $\Lambda$-strings involved in the ground state (all
$\phi_n(k)$ are not occupied), thus the dressed energy equations
evolve into two coupled dressed energies. This characterizes the
scattering among singlet bound pairs and unpaired bosons. They
provide complete phase diagrams and information of quantum phase
transitions with respect to the Zeeman splitting parameter $H$ and
the chemical potential $\mu$. The pressure of the system can be
represented in a neater way if we introduce the following notation
when $T\rightarrow 0$:
\begin{eqnarray}
\nonumber p &=&
-\frac{1}{2\pi}\int_{-\infty}^{\infty}\varepsilon_{1}^{-}(k)dk-\frac{1}{\pi}\int_{-\infty}^{\infty}\varepsilon_{2}^{-}(k)dk
\\ &\equiv& p_{1}+p_{2}.
\end{eqnarray}
Every thermodynamic quantity with a subscript 1 (or 2) corresponds
to unpaired states (or paired states).
When $c\gg 1$, we can take a Taylor expansion of the functions
$a_{n}$. Throughout this paper, we only keep track of terms up to
order $1/c^{2}$. Higher order corrections can be calculated in a
straightforward manner. In this limit, equation (\ref{TBA-d})
becomes (up to order $1/c^2$)
\begin{eqnarray}
\nonumber \varepsilon_{1}(k)&\approx&
k^{2}-\mu-H-\frac{p_{1}}{2c'}+\frac{4p_{2}}{5c'},\\
\varepsilon_{2}(k)&\approx &2(k^{2}-c'^{2}-\mu)+\frac{8p_{1}}{5c'}+\frac{p_{2}}{12c'}.\label{T-E}
\end{eqnarray}
We then integrate equations (\ref{T-E}) between the ``Fermi points''
$\pm Q_{1}$ and $\pm Q_{2}$ so that we can re-write the equations in
terms of $p_{1}$ and $p_{2}$. This gives
\begin{eqnarray}
-2\pi p_{1} &\approx & \frac{2}{3}Q_{1}^{3}-2\mu Q_{1}-2H
Q_{1}-\frac{p_{1}Q_{1}}{c'}+\frac{8p_{2}Q_{1}}{5c'},\nonumber\\
-\pi p_{2}&\approx &\frac{4}{3}Q_{2}^{3}-4c'^{2}Q_{2}-4\mu
Q_{2}+\frac{16p_{1}Q_{2}}{5c'}+\frac{p_{2}Q_{2}}{6c'}.\label{T-P}
\end{eqnarray}
We also make use of the fact that the dressed energies
$\varepsilon_{1}(k)$ and $\varepsilon_{2}(k)$ vanish at the ``Fermi
points'' i.e., $\varepsilon_{1}(\pm Q_{1})=0$ and
$\varepsilon_{2}(\pm Q_{2})=0$,
\begin{eqnarray}
Q_{1}^{2}&\approx &\mu+H+\frac{p_{1}}{2c'}-\frac{4p_{2}}{5c'},\nonumber\\
Q_{2}^{2}&\approx &\mu+c'^{2}-\frac{4p_{1}}{5c'}-\frac{p_{2}}{24c'}.
\end{eqnarray}
Substituting the ``Fermi points'' into equations (\ref{T-P}) and
then re-arranging and iterating the terms yield
\begin{eqnarray}
p_{1}&\approx &\frac{2\mu_{1}^{3/2}}{3\pi}+\frac{\mu_{1}^{3/2}}{4\pi}\left(\frac{p_{1}}{2\mu_{1}c'}
-\frac{4p_{2}}{5\mu_{1}c'}\right)^{2}\nonumber\\
&&
+\frac{\mu_{1}^{3/2}}{\pi}\left(\frac{p_{1}}{2\mu_{1}c'}
-\frac{4p_{2}}{5\mu_{1}c'}\right),\nonumber\\
p_{2}&\approx &\frac{8\mu_{2}^{3/2}}{3\pi}+\frac{\mu_{2}^{3/2}}{\pi}\left(\frac{4p_{1}}{5\mu_{2}c'}
+\frac{p_{2}}{24\mu_{2}c'}\right)^{2}\nonumber\\
&& -\frac{4\mu_{2}^{3/2}}{\pi}\left(\frac{4p_{1}}{5\mu_{2}c'}
+\frac{p_{2}}{24\mu_{2}c'}\right),
\end{eqnarray}
where we denote the effective chemical potentials
$\mu_{1}\equiv\mu+H$ and $\mu_{2}\equiv\mu+c'^{2}$ for the unpaired
and paired bosons. From the relations $n=\partial p/\partial\mu$ and
$nm^{z}=\partial p/\partial H$ and after some lengthy iterations, we
arrive at the expressions for the chemical potentials of unpaired
and paired bosons,
\begin{eqnarray}
\mu_{1}&\approx &
\pi^{2}n^{2}\left[(m_{z})^{2}\left(1-\frac{16m^{z}}{3\gamma}+\frac{32(1-m^{z})}{5\gamma}\right)\right.\nonumber\\
&&\left.+\frac{2(1-m^{z})^{3}}{15\gamma}\right],\nonumber
\\
\mu_{2}&\approx &\frac{\pi^{2}n^{2}}{16}\left[(1-m^{z})^{2}\left(1+
\frac{4(1-m^{z})}{9\gamma}+\frac{32m^{z}}{5\gamma}\right)\right.\nonumber
\\
&&\left. +\frac{512(m^{z})^{3}}{15\gamma} \right],\label{mu}
\end{eqnarray}
where $\gamma=c/n$. Substituting these two equations back into
$p_{1}$ and $p_{2}$ gives the pressures
\begin{eqnarray}
p_{1}&\approx &\frac{2}{3}\pi^{2}n^{3}(m^{z})^{3}\left(1-\frac{6m^{z}}{\gamma}+\frac{48(1-m^{z})}{5\gamma}\right),\nonumber\\
p_{2}&\approx &\frac{1}{24}\pi^{2}n^{3}(1-m^{z})^{3}\left(1+\frac{(1-m^{z})}{2\gamma}+\frac{48m^{z}}{5\gamma}\right).
\end{eqnarray}
Further, the free energy can be obtained as
\begin{eqnarray}
\nonumber F &\approx & \nonumber\frac{1}{3}\pi^{2}n^{3}(m^{z})^{3}\left(1-\frac{4m^{z}}{3\gamma}+\frac{32(1-m^{z})}{5\gamma}\right)\nonumber\\
&&+\frac{1}{48}\pi^{2}n^{3}(1-m^{z})^{3}\left(1+\frac{(1-m^{z})}{3\gamma}+\frac{32m^{z}}{5\gamma}\right)\nonumber\\
&&
-\frac{c^{2}}{16}n(1-m^{z})-Hnm^{z}+O\left(\frac{1}{\gamma^{2}}\right).
\end{eqnarray}
To find the ground state energy, we can use the relation
$E=F+Hnm^{z}$. There is also an alternative way to derive the ground
state energy based on the definition $E=\sum_{j}k_{j}^{2}$ and the
distribution of $\{k_{j}\}$ in quasimomenta space. Indeed, we show
that the energy per unit length derived from the discrete BA
equations (\ref{BA}) for arbitrary magnetization,
\begin{eqnarray}
\nonumber \frac{E}{L} &=&
\frac{1}{3}\pi^{2}n_{1}^{3}\left(1+\frac{2(32n_{2}-10n_{1})}{5c}+\frac{3(32n_{2}-10n_{1})^{2}}{25c^{2}}\right)\nonumber
\\ && +\frac{1}{6}\pi^{2}n_{2}^{3}\left(1+\frac{2(48n_{1}+5n_{2})}{15c}+\frac{3(48n_{1}+5n_{2})^{2}}{225c^{2}}\right)\nonumber\\
&&-\frac{n_{2}c^{2}}{8}+O\left(\frac{1}{c^{3}}\right)
\end{eqnarray}
coincides with the TBA result $E=F+Hnm^{z}$ up to the order of $1/c$
through the relations $n_{1}=nm^{z}$ and
$n_{2}=\frac{n}{2}(1-m^{z})$ where $n_{1}$ and $n_{2}$ are the
density of unpaired and paired bosons, respectively. However, the
dressed energy formalism provides a more elegant way to study
quantum phase transitions \cite{BGOT}.
For strong coupling, a pair of two bosons becomes stable because the
binding energy $\epsilon_b=\frac{\hbar^2}{2m}\frac{c^2}{8}$ can
exceed the kinetic energy. Therefore, the ground state in the
absence of an external field is characterized by an empty ``Fermi
sea'' for unpaired bosons and a fully filled ``Fermi sea'' for bound
pairs. From the dressed energy equations (\ref{TBA-d}), we find that
quantum phase transitions driven by an external field can be
determined by the energy transfer relation
$H=\mu_{1}-\mu_{2}+c^{2}/16$, i.e.,
\begin{eqnarray}
&&H \approx
n^{2}\left[\frac{\gamma^{2}}{16}+\pi^{2}(m^{z})^{2}\left(1-\frac{112m^{z}}{15\gamma}
+\frac{32(1-m^{z})}{5\gamma}\right.\right.
\nonumber \\
&& \left.+\frac{164(m^{z})^{2}}{5\gamma^{2}}-\frac{1792m^{z}(1-m^{z})}{25\gamma^{2}}
+\frac{768(1-m^{z})^{2}}{25\gamma^{2}}\right) \nonumber \\
&&
-\frac{\pi^{2}(1-m^{z})^{2}}{16}\left(1-\frac{76(1-m^{z})}{45\gamma}+\frac{32m^{z}}{5\gamma}+\frac{768(m^{z})^{2}}{25\gamma^{2}}\right. \nonumber \\ &&
\left.\left.-\frac{167(1-m^{z})^{2}}
{180\gamma^{2}}-\frac{1216m^{z}(1-m^{z})}{75\gamma^{2}}\right)\right]\label{H-E}
\end{eqnarray}
where we have used the relations
$\mu_{\kappa}=\frac{\partial}{\partial
n_{\kappa}}\left(E/L+n_{2}\epsilon_{b}\right)$ for $\kappa=1,2$ to
obtain expressions for the chemical potentials up to order
$1/\gamma^{2}$.
The lower critical field $H_{c1}$ diminishes the gap, thus a phase
transition from a singlet ground state into a gapless phase, where
two dressed energies of the paired and unpaired bosons couple to
each other, occurs when $H>H_{c1}$. When the external field exceeds
the upper critical field $H_{c2}$, all singlet bound pairs are
broken which leads to a ferromagnetic Tonks-Girardeau Bose gas. The
lower and upper critical fields are found by letting $m^{z}=0$ and
$m^{z}=1$ in (\ref{H-E}), with result
\begin{eqnarray}
H_{c1}&\approx &\frac{n^{2}}{16}\left[\gamma^{2}-\pi^{2}\left(1-\frac{76}{45\gamma}-\frac{167}{180\gamma^{2}}\right)
\right],\nonumber\\
H_{c2}&\approx &\frac{n^{2}}{16}\left[\gamma^{2}+16\pi^{2}\left(1-\frac{112}{15\gamma}+\frac{164}{5\gamma^{2}}
\right)\right].
\end{eqnarray}
In Fig. 2, we show the magnetization vs external field for different
values of the interaction strength $c$. We see clearly that for
$H<H_{c1}$ there is no breaking of bound pairs. The magnetization
gradually increases from zero to $n$ as $H$ gradually approaches
$H_{c2}$. The phase transitions across $H_{c1}$ and $H_{c2}$ are of
second order. In the vicinities of $H_{c1}$ and $H_{c2}$, the
leading order of the respective normalized magnetizations are given
by
\begin{eqnarray}
m_{1}^{z}&\approx &\frac{8(H-H_{c1})}{\pi^{2}n^{2}}\left(1+\frac{86}{15\gamma}-\frac{2813}{450\gamma^{2}}\right),\nonumber\\
m_{2}^{z}&\approx &1-\frac{(H_{c2}-H)}{2\pi^{2}n^{2}}\left(1+\frac{72}{5\gamma}-\frac{2536}{25\gamma^{2}}\right).
\end{eqnarray}
which show a linear dependence on the external field near the
critical points. For an external field $H_{c1}<H<H_{c2}$, the
singlet paired state and unpaired state coexist. They form a
two-component Luttinger liquid in this gapless phase.
In Fig. 3, we show the ground-state phase diagram in the $n-H$
plane. As $n \to 0$, both critical fields approach the same value
$H_c=\epsilon_{\rm b}/2$. The solid (dashed) lines correspond to the
two critical fields for the case $c=20$ ($c=40$). The ferromagnetic
phase of all atoms in state $\left|F=1,m_{F}=1\right\rangle$ appears
above the critical field $H_{c2}$, the singlet phase of singlet
pairs appears below the critical field $H_{c1}$ and the mixed phase
of atoms in state $\left|F=1,m_{F}=1\right\rangle$ and singlet pairs
appears between the two critical fields.
\section{The spin and charge velocities}
In 1D systems, spin-charge separation is the hallmark of many-body
physics \cite{Giamarchi2003}. The collective charge excitations are
described by sound modes with a linear dispersion.
The spin excitations are gapped with a dispersion
$\epsilon_{\nu}(p)=\sqrt{\Delta_{\nu}^2+v_{\nu}^2p^2}$ where
$\Delta_{\nu}$ is the excitation gap and $v_{\nu}$ is the spin
velocity in spin branch $\nu$.
This leads to the phenomenon of spin-charge separation. A method has
been proposed to probe this phenomenon experimentally in a 1D system
of interacting electrons at low energies \cite{Jompol2009}.
To calculate the charge velocity, we need to find the energy of the
lowest excited state that does not involve breaking any pairs. In
the absolute ground state where $H=0$, the system is only made up of
fully paired states below the ``Fermi level'' and the total momentum
of the system is zero. This is achieved when there is no magnetic
field present. To excite the system, we allow the pair with the
largest momentum to leave the ``Fermi sea'' and let the excited
state have a total momentum of $p$, i.e., $\sum_{j}k_{j}=p$. We then
calculate its total energy $E=\sum_{j}k_{j}^{2}$. The difference
between the excitation energy and the ground state energy is equal
to the charge velocity times $p$. The energy difference in the
thermodynamic limit is thus calculated to be
\begin{equation}
E(p)-E_{0}=\pi
n_{2}p\left(1+\frac{2n_{2}}{3c}\right)+O\left(\frac{1}{c^{2}}\right).
\end{equation}
Therefore in terms of the total particle number $n=N/L$ and
interaction strength $c$, the charge velocity is
\begin{equation}
v_{c}=\frac{\pi
n}{2}\left(1+\frac{1}{3\gamma}\right)+O\left(\frac{1}{c^{2}}\right).
\end{equation}
An alternative way to calculate the charge velocity for the singlet
ground state is through the relation
\begin{equation}
v_{c}=\sqrt{\frac{2L}{n}\left(\frac{\partial^{2}E_{0}}{\partial
L^{2}}\right)}.
\end{equation}
Both methods yield the same result.
The spin velocity on the other hand is calculated by considering the
lowest excited state where one pair is broken into two unpaired
states. Both unpaired states will occupy opposite ends of the
momentum distribution so that the excitation energy is minimized.
The total momentum of the excited state can be parameterized by $p$
in the same manner as before. We can equate the energy difference
between the excited state and the fully paired ground state to the
energy dispersion $\epsilon(p)$. In the thermodynamic limit,
\begin{equation}
E(p)-E_{0}=\frac{c^{2}}{8}+\frac{p^{2}}{2}\left(1+\frac{64n_{2}}{5c}\right)+O\left(\frac{1}{c^{2}}\right)
\equiv\epsilon(p).
\end{equation}
From the original relation
$\epsilon(p)=\sqrt{\Delta^{2}+v_{s}^{2}p^{2}}$, we obtain the
relation $\epsilon(p)=\Delta+\frac{v_{s}^{2}p^{2}}{2\Delta}$ in the
limit $\Delta\gg 1$ where the gap is very large. Comparing both
expressions for the dispersion energy, we can easily verify that
$\Delta=c^{2}/8$ and
\begin{equation}
v_{s}^{2}=\frac{c^{2}}{8}\left(1+\frac{64n_{2}}{5c}\right).
\end{equation}
Hence the spin velocity is
\begin{equation}
v_{s}=\frac{c}{2\sqrt{2}}\left(1+\frac{16n}{5c}\right)+O\left(\frac{1}{c^{2}}\right).
\end{equation}
The spin velocity is divergent due to a large energy gap as $c\to
\infty$. This demonstrates that there is spin-charge separation over
the singlet ground state. We also note that this phenomenon depends
on the state of the system within an external field. Essler \emph{et
al.} \cite{Essler2009} showed that spin and charge velocities are
equal in the weak coupling limit when there is no external field
involved.
In the gapless phase when $H_{c1}<H<H_{c2}$, the ground state
($T\rightarrow 0$) of this system is conformally invariant
\cite{Blote1986,Affleck1986}. The excitations close to the ``Fermi
surfaces'' in unpaired and pair branches have linear dispersions.
The finite-size corrections to the ground state energy are given by
\begin{equation}
E_{0}=Le_{0}^{\infty}-\frac{\pi c}{6L}(v_{c}^{(1)}+v_{c}^{(2)})
\end{equation}
where the central charge $c=1$ for this system, $E_{0}$ is the
ground state energy for the finite system and $e_{0}^{\infty}$ is
the ground state energy density for the infinite system. The charge
velocities for unpaired and paired bosons are given explicitly by
the expressions
\begin{eqnarray}
v_{c}^{(1)}&\approx &2\pi
n_{1}\left(1+\frac{2(32n_{2}-10n_{1})}{5c}+\frac{3(32n_{2}-10n_{1})^{2}}{25c^{2}}\right),\nonumber\\
v_{c}^{(2)}&\approx &\pi
n_{2}\left(1+\frac{2(48n_{1}+5n_{2})}{15c}+\frac{3(48n_{1}+5n_{2})^{2}}{225c^{2}}\right).
\end{eqnarray}
In this phase, spin fluctuations are frozen out and thus the charge
density fluctuations dominate the ground state and is effectively
described by the universality class of a two component Luttinger
liquid.
\section{Conclusion}
In conclusion we derived the TBA equations for a system of 1D spin-1
bosons with repulsive density-density and antiferromagnetic spin
exchange interactions and solved the TBA equations for the zero
temperature case in the strong coupling limit. We obtained the
ground state energy, chemical potentials, critical fields and
magnetization in terms of interaction strength and the external
magnetic field. We also presented an exact phase diagram of strongly
interacting spin-1 bosons which facilitates experimental analysis
of phase segments. For the weak coupling limit, the collective
excitations in the charge sector is described by a
Tomonaga-Luttinger liquid, whereas the spin dynamics is described by
the $O(3)$ non-linear sigma model \cite{Essler2009}. However, for
the strong coupling limit, spin fluctuations can be suppressed by a
strong external magnetic field. The density fluctuations thus evolve
into a two-component Luttinger liquid. At zero temperature, the
model exhibits three quantum phases: singlet pairs of two bosons for
external field $ H<H_{c1}$; a fully-polarized Tonks-Girardeau gas
phase of $m_F=1$ bosons for $H> H_{c2}$; and a mixed phase of
singlet pairs and unpaired $m_{F}=1$ atoms for an intermediate field
$H_{c1} < H < H_{c2}$. The phase transitions in the vicinities of
$H_{c1}$ and $H_{c2}$ are of second order with a
linear-field-dependent magnetization. Our results provide a new
aspect of this model, namely spin liquid v.s. Luttinger liquid
behavior.
\vspace{1cm}
This work has been supported by the Australian Research Council.
We thank Profs J.-P. Cao, S. Chen and Y.-P. Wang for helpful
discussions. C.L. thanks Yu.S. Kivshar for support.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 500
|
{"url":"https:\/\/matheducators.stackexchange.com\/questions\/5685\/at-what-stage-during-an-undergrad-degree-should-a-student-move-beyond-just-doin","text":"# At what stage during an undergrad degree should a student move beyond just 'doing' the equations?\n\nI ask \"At what stage during an undergrad degree should a student move beyond just 'doing' the equations?\" because I am taking some Year 1 maths papers in 2015. I am worried I won't be able to actually see the deeper meaning and reasoning behind the equations. I may be able to plug numbers into the formula and remember the procedures to follow, but I feel this is different to really knowing what is going on and why.\n\nI asked a 3rd year student and he said during years 1 and 2 that I shouldn't get too worried about the deeper meaning, rather the teachers\/lecturers are wanting to see that I can do the work and arrive at the correct answer.\n\nIs anyone able to shed some light on this and how they see it, or share their experiences?\n\nI am not fresh out of high school, rather an adult trying to get an education later in life.\n\nthanks\n\n\u2022 This is a very tricky subject and depends very much on what you mean by \"just 'doing' the equations\", e.g. some mathematicians never go beyond manipulating symbols. If you are struggling with geometrical interpretation of a derivative then I would be concerned; on the other hand, in logic or universal algebra one might not have any intuitions for a long time. \u2013\u00a0dtldarek Oct 30 '14 at 10:53\n\u2022 @dtldarek A geometric interpretation of a derivative is something I'd never thought of. My experience with derivatives so far has been along the lines of \"here is x^2, in this case apply the power rule, so it is now 2x\". So while I may have come up with the answer, I start to wonder what does it really mean? how do you prove that ? and what does it look like to look at ? I guess now it should include what is its geometric representation. My problem is as I have just come across derivatives I don't understand the material that proves why it works (it seems like a whole new topic) \u2013\u00a0user3754366 Oct 31 '14 at 21:40\n\n## 1 Answer\n\nMy experience has been that this can happen as early as you like.\n\nI teach College Algebra, Calculus, and Differential Equations at a community college. In all of these courses, it is possible to see connections and ask \"why\" questions. I truly believe that if you start asking why and finding out what is going on, it is significantly easier to learn and retain the material.\n\nEven as early as the identity $x^0 = 1$, it is very productive to ask why!\n\nYour professors will almost certainly have office hours available. Appearing in their office to ask these questions is a very good idea.","date":"2021-04-17 17:52:06","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\": 0.5664673447608948, \"perplexity\": 275.9154072102579}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"absolute_threshold\": 20, \"end_threshold\": 15, \"enable\": false}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-17\/segments\/1618038461619.53\/warc\/CC-MAIN-20210417162353-20210417192353-00119.warc.gz\"}"}
| null | null |
\section{Introduction}
Computed tomography (CT) has been widely applied in clinical, industrial and other areas ~\cite{ref1}. However, when metallic implants exist in scanning regions, the effects of beam hardening, photon starvation, scattering and nonlinear partial volume effect become prominent, giving rise to severe radial streak artifacts in reconstructed CT images, which may seriously affect the subsequent analysis or diagnosis tasks ~\cite{ref2}. As a result, it is of great importance to develop an efficient method to reduce metal artifacts while preserving tissue details.
\begin{figure}
\begin{center}
\centering\includegraphics[width=0.45\textwidth]{11111.png}
\end{center}
\caption{A representative slice processed using different MAR methods. (A) Metal-corrupted image, (B) CNNMAR, (C) DuDoNet, (D) IDOL-Net}
\label{fig0}
\end{figure}
\section{Method}
\subsection{Overview}
During the past decades, many metal artifact reduction (MAR) methods have been proposed ~\cite{ref3}. These methods can be roughly divided into three groups: sinogram completion, iterative reconstruction and image post-processing. 1) Sinogram completion methods regard projection data in the metal trace region as missing, which are estimated using image interpolation or inpainting methods ~\cite{ref4}~\cite{ref5}~\cite{ref6}~\cite{ref7}~\cite{ref8}~\cite{ref9}~\cite{ref10}~\cite{ref11}. This kind of method is difficult to maintain continuity at the boundaries of metal trace in sinogram domain and secondary artifacts will be introduced in the reconstructed CT images ~\cite{ref12}. 2) Iterative reconstruction methods usually formulate the objective function aided by prior information, which can be solved with an iterative manner ~\cite{ref13}~\cite{ref14}~\cite{ref15}~\cite{ref16}. Nevertheless, these approaches are usually time-consuming [3] and some parameters need to be set manually. 3) Image post-processing methods directly remove artifacts in reconstructed image ~\cite{ref17}~\cite{ref18}, but they have to entail risks of distorting the anatomic structure ~\cite{ref19}.
Recently, increasing attention has been paid to deep learning (DL). As a result, many DL-based MAR approaches were proposed, which also can be grouped into three different categories according to their operation domain. For sinogram-domain methods, projection data in metal trace region is replaced with the values generated by neural networks. Image-domain methods dedicate to suppressing metal artifacts directly without accessing projection data. In the last few years, dual-domain methods achieve impressive performance, which simultaneously leverage the information from both image and sinogram domains. However, current dual-domain methods usually process dual-domain data in a specific order, which implicitly imposes a certain priority prior into MAR and may ignore the latent information interaction between both domains. On the other side, as shown in recent studies \cite{ref19}~\cite{ref20}~\cite{ref21}~\cite{ref32}, it is preferred to produce a prior image as the auxiliary input to improve the performance of networks. As a result, how to obtain a high-quality initial estimation becomes another critical point for DL-based MAR methods.
To tackle these problems mentioned above, in this paper, we propose a novel Interactive Dual-dOmain paraLlel network for CT MAR, dubbed as IDOL-Net. The proposed IDOL-Net is composed of two modules as depicted in Fig. 1. The disentanglement module introduces the idea of disentanglement representation to generate high-quality prior sinogram and image as the complementary inputs. The follow-up refinement module consists of two parallel and interactive branches that simultaneously operate on image and sinogram domain, attempting to fully exploit the latent information interaction between both domains. The intermediate results exported from both branches are fused to exchange the useful features from each other. Finally, the image quality is further enhanced by the refinement module. A representative slice processed using different MAR methods is given in Fig. \ref{fig0}.
The main contributions of this work are summarized as follows:
(1) A novel dual-domain network, IDOL-Net, which can simultaneously leverage the information from both sinogram and image domains, is proposed to estimate a complex end-to-end mapping from joint metal-affected sinogram and image to artifact-free image.
(2) Disentanglement representation is introduced to generate a high-quality hybrid prior estimation from the initial metal-affected sinogram and reconstructed image.
(3) A refinement module containing two parallel and interactive branches is designed to simultaneously extract the intermediate features from both complementary domains, which are delivered and fused during the reconstruction procedure.
(4) Extensively experimental results on the simulated and clinical datasets demonstrate the effectiveness compared to other state-of-the-art MAR methods in both qualitative and quantitative aspects.
\section{Related works}
\subsection{Single-domain DL-based MAR}
Early MAR methods directly applied DL on single-domain data, i.e., sinogram or image reconstructed by filter backprojection (FBP). For sinogram domain, neural networks are utilized to synthesize the missing data in the metal trace region. Park et al. used U-Net ~\cite{ref23} to repair inconsistent sinograms by removing the primary metal-induced beam hardening factors along the metal trace boundaries ~\cite{ref22}. In ~\cite{ref24}~\cite{ref25}~\cite{ref26}, generative adversarial net (GAN) and its variant were introduced to improve the inpainting performance. Partial convolution ~\cite{ref27}, which is more suitable to sinogram completion, was employed in ~\cite{ref28} and ~\cite{ref29}. However, it is hard to guarantee the sinogram continuity at the boundaries of metal trace and secondary artifacts will appear consequentially. On the other hand, Gjesteby et al. first applied a simple plain CNN to suppress the artifacts in image domain ~\cite{ref30}. Zhang et al. proposed to fuse the results obtained using different methods as the prior image and replace the metal-affected region in the sinogram with the corresponding part from the prior image ~\cite{ref34}. In ~\cite{ref31}, the authors presented to combine the result of NMAR ~\cite{ref8} and the detail image extracted by guided filter as hybrid inputs to a dual-stream network. Liao et al. introduced the idea of unsupervised disentanglement representation to recover the clean image with unpaired training samples ~\cite{ref36}. The main limitation for this method lies in that when the metallic implants get larger, the artifacts cannot be completely removed and the anatomic structures are prone to distortion.
\subsection{Dual-domain DL-based MAR}
Recently, dual-domain methods become mainstream. For instance, in ~\cite{ref20}, two sub-networks are jointed via a Radon inversion layer to process the linear-interpolated sinogram and its reconstructed image in sequence. To further improve the performance, the authors in ~\cite{ref21} proposed to insert metal mask projection into the sinogram network. Yu et al. pointed out that there would be some tiny anatomical structure changes if image domain enhancement is adopted at the final stage ~\cite{ref19}. Thus, an image domain network is trained at first to produce a prior image with good quality to guide the subsequent sinogram domain learning. Peng et al. utilized partial convolution operator to recover the contaminated values in sinogram and trained an auxiliary sub-network in image domain to further refine the details ~\cite{ref32}. However, since these methods usually process the sinogram and image in a specific order, which ignores the latent information interaction, there is still room to improve the performance by searching for a better network architecture for dual-domain information fusion.
\begin{figure*}
\begin{center}
\centering\includegraphics[width=1\textwidth]{IDOL-pic318.png}
\end{center}
\caption{Overview of the proposed MAR method.}
\label{fig1}
\end{figure*}
\section{Method}
\subsection{Overview}
As depicted in Fig. \ref{fig1}, the proposed IDOL-Net is composed of two modules, the disentanglement module and refinement module. In the disentanglement module, we try to obtain accurate prior estimations, including both initial sinogram and image. In the refinement module, the preliminarily corrected sinogram and image from disentanglement module are further refined aided by interactive feature fusion. The details of both modules are elaborated as follows.
\subsection{Disentanglement module}
Current DL-based methods ~\cite{ref19}~\cite{ref20}~\cite{ref21}~\cite{ref34} have demonstrated the effectiveness in introducing the prior image or sinogram separately as the auxiliary input for MAR. However, although it is natural to combine both image and sinogram as the prior inputs, none of these methods have explored this possibility. To take full advantage of the prior information from both domains, unlike other DL-based methods, which use linear interpolation or U-Net to obtain the coarse prior sinogram or image, the proposed disentanglement module introduces the idea of disentanglement representation to produce a more accurate hybrid prior estimation.
As illustrated in the left part of Fig. \ref{fig1}, the original metal-affected sinogram \(S_{ma}\) is fed into two networks, \(G_{S\_metal}\) and \(G_{S\_clean}\), which respectively extract the metal-only sinogram \(S_{metal}\) and the content sinogram \(S_{clean}\) from \(S_{ma}\):
\begin{equation}
\label{eq1}
S_{metal}= G_{S\_metal} \ (S_{ma} )
\end{equation}\begin{equation}
\label{eq2}
S_{clean}= G_{S\_clean}\ (S_{ma} )
\end{equation}
Meanwhile, the artifact-corrupted image \(X_{ma}\) reconstructed from \(S_{ma}\) using FBP is also processed by two networks, \(G_{X\_art}\) and \(G_{X\_clean}\) , which respectively estimate the artifact image \(X_{art}\) and the artifact-free image \(X_{clean}\) :
\begin{equation}
\label{eq3}
X_{art}= G_{X\_art}\ (X_{ma} )
\end{equation}
\begin{equation}
\label{eq4}
X_{clean} = G_{X\_clean}\ (X_{ma} )
\end{equation}Unlike the traditional MAR networks, which usually utilize one network to generate clean sinogram or image, the proposed disentanglement module imposes additional constraints on the extracted residuals, i.e., the metal-only sinogram and the artifact image, producing a purer decomposition on the original sinogram and image. The loss function of the disentanglement module is defined as:
\begin{equation}
\label{eq5}
\mathcal L_{dis}=\mathcal L_{clean}+\alpha_{1}\mathcal L_{art}+\alpha_2\mathcal L_{ma}
\end{equation}
Three components are formulated respectively as
\begin{equation}
\label{eq6}
\mathcal L_{clean}=||X_{clean}-X_{clean}^{gt}||_{1}+\theta_{1}||S_{clean}-S_{clean}^{gt}||_{1}
\end{equation}
\begin{equation}
\label{eq7}
\mathcal L_{art}=||X_{art}-X_{art}^{gt}||_{1}+\theta_{2}||S_{metal}-S_{metal}^{gt}||_{1}
\end{equation}
\begin{equation}
\begin{aligned}
\label{eq8}
\mathcal L_{ma}&=||(X_{clean}+X_{art})-X_{ma}||_{1}\\&+\theta_{3}||(S_{clean}+S_{metal})-S_{ma}||_{1}
\end{aligned}
\end{equation}
where \(X_{clean}^{gt}\) , \(S_{clean}^{gt}\) , \(X_{art}^{gt} \) and \(S_{metal}^{gt}\) are the corresponding labels. \(\mathcal{L}_{clean}\) denotes the reconstruction loss to facilitate \(X_{clean}\approx X_{clean}^{gt}\) and \(S_{clean}\approx S_{clean}^{gt}\); \(\mathcal{L}_{art}\) indicates the reconstruction loss to impose \(X_{art}\approx X_{art}^{gt}\) and \(S_{metal}\approx S_{metal}^{gt}\); \(\mathcal{L}_{ma}\) is adopted to ensure that the decomposition is invertible and without loss of information.
For simplicity, in this paper, \(G_{S\_metal}\) , \(G_{S\_clean}\) , \(G_{X\_art}\) and \(G_{X\_clean}\) share the same structure, which is presented in Fig. \ref{fig2}. A simple plain CNN with four residual blocks is adopted as the unified network architecture.
\begin{figure*}
\begin{center}
\centering\includegraphics[width=1\textwidth]{f2coarse.png}
\end{center}
\caption{Fig. 2: The details of \(G_{S\_metal}\) , \(G_{S\_clean}\) , \(G_{X\_art}\) and \(G_{X\_clean}\) . C: channel numbers, K: kernel, S: stride, and P: padding sizes.}
\label{fig2}
\end{figure*}
\subsection{Refinement module}
\begin{figure}[t]
\begin{center}
\includegraphics[width=1\linewidth]{f4im.png}
\end{center}
\caption{Illustrations of the \(G_{im}\). K: kernel, S: stride, P: padding sizes and D: dilation.}
\label{fig3}
\end{figure}
\begin{figure}[t]
\begin{center}
\includegraphics[width=1\linewidth]{f3sino.png}
\end{center}
\caption{Illustrations of the \(F_{sino}\). K: kernel, S: stride, P: padding sizes and D: dilation.}
\label{fig4}
\end{figure}
Existing studies demonstrate that simply processing the corrupted sinogram or image with one single network has limited performance ~\cite{ref19}~\cite{ref20}. Based on this consideration, a refinement module is proposed to further improve the image quality. Its structure is shown in the right part of Fig. \ref{fig1}. There are two branches to separately handle sinogram and image data. Both branches have four same blocks, i.e., \(G_{im}^{i}\) for image branch and \(G_{sino}^{i}\) for sinogram branch. \(X_{clean}\) and \(S_{clean}\) generated by the disentanglement module are employed as the inputs of the proposed refinement module. FBP is applied on \( S_{clean}\) to obtain \(X_{clean}^{fbp}\), which is concatenated with \(X_{clean}\) as the initial input of the first \(G_{im}^{1}\). Similarly, forward projection (FP) is applied on \(X_{clean}\) to obtain \(S_{clean}^{fp}\), which is concatenated with \(S_{clean}\) as the initial input of the first \(G_{sino}^{i+1}\) . Similar operations are performed after each \(G_{im}^{i}\) and \(G_{sino}^{i}\). FBP and FP are applied to the intermediate sinogram and image results. Then the outputs of both FBP and its corresponding \(G_{im}^{i}\) are concatenated and fed into \(G_{im}^{i+1}\). The outputs of both FP and its corresponding \(G_{sino}^{i}\) are concatenated and fed into \( G_{sino}^{i+1}\). For simplicity, U-Net is adopted as the backbone of both \(G_{sino}^{i}\) and \(G_{im}^{i}\) and we halve the channel numbers to reduce the computational cost. To suppress the nonlocal artifacts in image domain, nonlocal network modules ~\cite{ref37} are integrated into \(G_{im}^{i}\). The architecture of \(G_{im}^{i}\) is illustrated in Fig. \ref{fig3}. For sinogram branch, a sinogram completion network, denoted as \(F_{sino}^{i}\), is adopted to restore projection data within the metal trace mask and the structure of \(F_{sino}^{i}\) is depicted in Fig. \ref{fig4}. Inspired by the idea of mask pyramid network (MPN) ~\cite{ref33}, the metal mask projection is leveraged to alleviate discontinuity at boundaries of metal trace.
Specifically, in a refinement block, given the metal mask \(M \) and metal trace mask \(M_t\), there are four outputs, outputs of \(G_{sino}^{i+1}\) and \(G_{im}^{i+1}\) recognized as \(S_{sino}^{i+1}\) and \(X_{im}^{i+1}\), forward projection of \(X_{im}^{i+1}\) denoted as \(S_{im}^{i+1}\) and filtered back-projection of \(S_{sino}^{i+1}\) referred to as \(X_{sino}^{i+1}\) .
\begin{equation}
\label{eq9}
S_{res}^{i+1} = F_{sino}^{i+1}(S_{im}^{i}\ ,S_{sino}^{i}\ ,M_{t})
\end{equation}
\begin{equation}
\begin{aligned}
\label{eq10}
S_{sino}^{i+1} &= G_{sino}^{i+1}(S_{res}^{i+1} ,S_{ma} ,M_{t})\\&=S_{res}^{i+1}\odot M_{t}+S_{ma}\odot(1-M_{t})
\end{aligned}
\end{equation}
\begin{equation}
\label{eq11}
X_{im}^{i+1} = G_{im}^{i+1}(X_{im}^{i}\ ,X_{sino}^{i})+X_{im}^{i}
\end{equation}
\begin{equation}
\label{eq12}
S_{im}^{i+1} = \mathcal{FP} (X_{im}^{i+1})
\end{equation}
\begin{equation}
\label{eq13}
X_{sino}^{i+1} = \mathcal{FBP} (S_{sino}^{i+1})
\end{equation}
where i=0,1,2,3. When i=0, \(S_{sino}^{1}=S_{clean}\), \(S_{im}^{1}=S_{clean}^{fp}\), \(X_{im}^{1}=X_{clean}\) and \(X_{sino}^{1}=X_{clean}^{fbp}\).
The first three refinement blocks share the same structure. The last refine block has similar architecture, but it has no FP operation on \(X_{im}^{4}\). Then, we concatenate \(X_{im}^{4}\) and \(X_{sino}^{4}\) of the last block as the inputs of the fusion layer \(\mathcal{F}\), which is a \(1\times1\) convolution layer, to obtain the final result \(X_{corr}\) as
\begin{equation}
\label{eq14}
X_{corr} = \mathcal{F} (X_{im}^{4}\ ,X_{sino}^{4})
\end{equation}
To optimize the refinement module, \(\mathcal{L}_1 \) loss is adopted to minimize the differences between \(X_{corr} \) and the ground truth:
\begin{equation}
\label{eq15}
\mathcal{L}_{corr} = ||(X_{corr}-X_{gt}^{clean})\odot (1-M_{t})||_1
\end{equation}Thus, the total loss function can be written as:
\begin{equation}
\label{eq16}
\mathcal{L} = \mathcal{L}_{dis}+\gamma\mathcal{L}_{corr}
\end{equation}
where \(\alpha_1\), \(\alpha_2\), \(\theta_1\),\( \theta_2\), \(\theta_3\) and \(\gamma\) are hyper-parameters to balance the weights of different components. We manually set \(\alpha_1=\alpha_2=1.0\) , \(\theta_1=\theta_2=\theta_3=0.06\) and \(\gamma=10.0\) in this manuscript.
\section{Experiments}
\subsection{Data set}
In our experiments, the manually segmented metal masks dataset ~\cite{ref34}, which contains simulated dental fillings, spine crews and hip prostheses, is utilized to produce the metal masks. We randomly selected 1000 CT images from DeepLesion dataset ~\cite{ref35} and 90 metal masks to build our training set, which contains totally 90000 images. Another 200 CT images and ten masks were used to form the testing set with totally 2000 images. The same procedure used in ~\cite{ref34}~\cite{ref36} is adopted to synthesize the metal-affected CT images. Images were resized to 256×256 to reduce the computational cost. Without loss of generality, 2D parallel-beam geometry was employed. There were 361 views evenly distributed from 0° to 180° and 367 bins received photons. In addition, the pixel values were clipped to [0, 4095], which coincided with the real scenario.
\subsection{Implementation details}
The \textit{ODL} library was adopted to implement FBP and FP. The whole network was trained in an end-to-end manner with the \textit{Pytorch} deep learning framework and was optimized using the Adam optimizer with the parameters \((\beta_{1},\beta_{2})=(0.5,0.999)\). The learning rate was set to 0.0002 initially and halved for every 10 epochs. We trained our network for 100 epochs on a workstation with four Nvidia 1080Ti GPUs.
\begin{table*}[]
\centering
\label{table1}
\begin{tabular}{@{}cccccccc@{}}
\toprule
\hline
PSNR/SSIM & 1&2 &3 &4 &5 &Average\\
\midrule
\hline
Uncorrected & 15.93/0.7101 & 15.14/0.6617 & 14.86/0.6417 & 15.14/0.6572 & 15.32/0.6564 & 15.33/0.6673 \\
LI & 32.86/0.9545 & 32.60/0.9442 & 30.76/0.9330 & 27.26/0.8934 & 28.94/0.8853 & 30.74/0.9224 \\
NMAR & 32.54/0.9564 & 31.87/0.9394 & 31.20/0.9359 & 27.91/0.9035 & 29.21/0.9000 & 30.83/0.9270 \\
CNNMAR & 32.78/0.9665 & 32.80/0.9641 & 33.24/0.9612 & 32.61/0.9517 & 32.46/0.9455 & 32.72/0.9574 \\
ADN & 30.92/0.9335 & 30.03/0.9162 & 32.08/0.9393 & 26.45/0.9136 & 31.78/0.9407 & 30.58/0.9290 \\
DuDoNet & 38.77/0.9673 & 38.57/0.9648 & 36.44/0.9642 & 33.03/95.33 &34.26/0.9531 & 36.82/0.9777 \\
IDOL-Net & \textbf{42.01/0.9870} &\textbf{ 42.37/0.9878} & \textbf{40.67/0.9900} & \textbf{35.26/0.9782 }& \textbf{38.30/0.9842} & \textbf{40.20/0.9858}\\
\bottomrule
\hline
\end{tabular}
\caption{Quantitative comparison of different methods on the simulated dataset. 1-5 represents the sizes of metallic implants in ascending order.}
\end{table*}
\begin{figure*}
\begin{center}
\centering\includegraphics[width=0.8\textwidth]{r1.png}
\centering\includegraphics[width=0.8\textwidth]{r2.png}
\centering\includegraphics[width=0.8\textwidth]{r3.png}
\end{center}
\caption{Comparisons with different MAR methods on simulation data with small, middle and big metallic implant. (A1-A3) reference images; (B1-B3) metal corrupted images; (C1-C3) corresponding results of LI; (D1-D3) NMAR; (E1-E3) CNNMAR; (F1-F3) ADN; (G1-G3) DuDoNet and (H1-H3) IDOL-Net.}
\label{fig5}
\label{fig:short}
\end{figure*}
\begin{table*}[]
\centering
\label{table2}
\begin{tabular}{@{}cccccccc@{}}
\toprule
\hline
Methods & Refine-Net & Clean-IDOL & Art-IDOL &Prior-Im &Prior-Sino &IDO-Net\\
\midrule
\hline
PSNR & 36.35 & 36.65 &37.52 &31.42 &27.30 & \textbf{40.20} \\
SSIM &0. 9660 & 0.9716 & 0.9723 &0.9418 &0.6005 & \textbf{0.9858} \\
\bottomrule
\hline
\end{tabular}
\caption{Quantitative comparison of different variants of our method on the simulated dataset.}
\end{table*}
\subsection{Experimental results on the simulated data set}
To validate the effectiveness of the proposed IDOL-Net, several methods, including LI ~\cite{ref6}, NMAR ~\cite{ref8}, CNNMAR ~\cite{ref34}, ADN ~\cite{ref36}, DuDoNet ~\cite{ref20} were involved for comparison. LI and NMAR are widely used interpolation-based methods. CNNMAR takes advantage of the outputs of different MAR methods to generate a prior image. ADN is an advanced unsupervised image-domain method. DuDoNet is a state-of-the-art dual-domain approach. All the methods were implemented with the publicly released code or strictly implemented according to the original paper. Structural similarity (SSIM) and peak signal to noise ratio (PSNR) were employed as the quantitative metrics.
\textbf{1) Quantitative comparisons:} Table \ref{table1} shows the average quantitative results on the simulated dataset. It is observed that all the methods can significantly improve the PSNR and SSIM values and DL-based methods achieve better scores than traditional MAR methods. DuDoNet and IDOL-Net gain obvious improvements in terms of both metrics, which quantitatively demonstrated the merits of dual-domain methods. In addition, our method produces a noticeable advantage over DuDoNet in terms of both metrics for all the metal sizes, with an average improvement of 3.3dB in PSNR and 0.81 in SSIM.
\textbf{2) Qualitative comparisons:} Fig. \ref{fig5} shows the representative results of different methods on simulated data with different metal sizes. For better visualization, the metal masks are painted in red. Due to the fact that LI and NMAR are interpolation-based methods directly discarding the projection data in metal trace, the metal information is lost and the corrected sinogram cannot keep the continuity at the boundaries of metal trace, leading to blurred tissues around the metals and remarkable secondary artifacts. Although CNNMAR fuses the results of different MAR methods, the improvement is quite limited. ADN achieves better performance on structure recovery around the metal implants than LI, NMAR and CNNMAR, and even DuDoNet in some cases, but it also can be noticed that in the second case (Fig. \ref{fig5} (F2)), the shadow artifacts are not well suppressed and the details covered by the artifacts are hard to identify. This phenomenon probably lies in that ADN is essentially a post-processing method, which does not leverage the sinogram information. Compared to other methods, DuDoNet and the proposed IDOL-Net have the best performance on artifact reduction, but DuDoNet cannot recover the structures and tissues around the metal well, which can be clearly observed in the magnified yellow boxes in Fig. \ref{fig5} (G2) and (G3). Our proposed IDOL-Net not only effectively removes most artifacts but also recovers the details better than all the other methods in all cases.
\subsection{Experimental results on clinical data}
To evaluate the robustness of our IDOL-Net for clinical practice, clinical CT images were tested and Fig. \ref{fig6} shows the visual comparisons of one representative slice processed using different methods. The metal was segmented by the threshold of 2000 HU and the metal mask is painted in red for better visualization. The test image was normalized to the same range as the training data and the results were obtained using the model trained with simulated dataset used in the previous subsection. In Fig. \ref{fig6} (A), the metal leads to severe artifacts, which covers some tissues, especially near the metal. It is observed that LI, NMAR and DuDoNet cannot retrieve the tissues around the metal, but due to the utilization of dual-domain information, DuDoNet outperforms most methods in artifact reduction. Since the shadow artifacts around the metal is very heavy, ADN cannot do much about this, which is similar to the situation in simulated dataset. It is observed that IDOL-Net suppresses most of the metal artifacts and effectively preserves the anatomical structures around the metals, which demonstrates its potential for real clinical application.
\subsection{Analytical studies}
In this subsection, we investigate the effectiveness of different network modules and our ablation experiments use the following configurations.
\begin{figure}[]
\begin{center}
\centering\includegraphics[width=0.5\textwidth]{clinical.png}
\end{center}
\caption{Visual comparison with different MAR methods on a clinical CT image. (A)-(G) respectively represent uncorrected CT images and corrected results using LI, NMAR, CNNMAR, ADN, DuDoNet and IDOL-Net.}
\label{fig6}
\end{figure}
(a): Refine-Net: network trained without both prior image and sinogram;
(b): Clean-IDOL: network that only learns clean parts of CT images and sinograms followed by refinement module;
(c): Art-IDOL: network that only learns residual parts of CT images and sinograms followed by refinement module;
(d): Prior-Im: network with disentanglement module followed by only image refinement branch;
(e): Prior-Sino: network with disentanglement module followed by only sinogram refinement branch; and
(f): Our full model with both disentanglement and refinement modules.
\textbf{1) Effectiveness of prior estimation.}
Most dual-domain MAR methods usually use original or LI-interpolated sinogram and corresponding reconstructed CT images as inputs, or adopted LI's corrected results as inputs. However, the first case results in incomplete artifact reduction and maintains some obvious artifacts in final corrected images. In addition, since LI is just a rough estimation, which gives rise to details lost around the metal. A possible solution is to generate both good quality prior sinogram and image. To investigate the effectiveness of the prior image and sinogram generated by our proposed disentanglement module, a network without this module, termed Refine-Net, is trained and we take metal-affected CT image and sinogram as inputs. Table 2 shows the quantitative results. It is clear to see that aided by prior image and sinogram, our model achieves much better score in terms of both metrics. As shown in In Fig. \ref{fig7} (B), there are obvious artifacts indicated by a blue arrow.
\textbf{2) Effectiveness of \(\mathcal{L}_{ma}\).}
There are two ways to obtain high quality prior image and sinogram. One is just predicting the clean data directly from contaminated data. The other is learning residual part and get the clean part by subtracting the learned residual data from original corrupted data. However, these methods may fail to separate the clean data or residuals from original data completely. In other words, there could be some useful information remaining in the restored data or residuals. To tackle this problem, inspired by disentangling representation, we learn both clean part and residual part simultaneously. In addition, we adopt \(\mathcal{L}_{ma}\) to produce a lossless decomposition. To investigate the effectiveness of \(\mathcal{L}_{ma}\), we trained two networks, Clean-IDOL and Art-IDOL, without \(\mathcal{L}_{ma}\). It can be seen in Table 2 that both Clean-IDOL and Art-IDOL achieve lower SSIM and PSNR values than IDOL-Net. In Fig. \ref{fig7}, Clean-IDOL introduced secondary artifacts indicated by green arrow. Art-IDOL shows slight anatomic structure distortion marked by yellow dotted box. It is noticeable that the result of IDOL-Net is most consistent with the referenced image in all the methods.
\textbf{3) Effectiveness of parallel branches.}
To show the effectiveness of our proposed interactive dual-domain parallel branches, we trained two networks, dubbed Prior-Im and Prior-Sino, which are only with image domain branch or sinogram domain branch, respectively. It can be seen in Table 2 that Prior-Sino gains lower SSIM and PSNR scores than other methods. The possible reason lies in that Prior-Sino restored projection data within metal trace region using the original contaminated data and the distortion introduced by inaccurate result will expand to the whole reconstructed image, which heavily lower both quantitative scores calculated in image domain. Prior-Im significantly improves the qualitative and quantitative results compared to Prior-Sino, but as shown in Table 2 and Fig. \ref{fig7}, our IDOL-Net have superior performance to Prior-Im, which provides impressive support for our claim that the proposed interactive dual-domain parallel branches are efficient.
\textbf{4) Discussion of refinement blocks.}
To sense the impact of the number of network blocks in refinement module, different block numbers (from 1 to 5) are tested. TABLE 3 lists the quantitative results and it can be observed that the model with four blocks attains the best quantitative score.
\begin{table}[]
\centering
\label{table2}
\scalebox{0.9}{
\begin{tabular}{@{}cccccccc@{}}
\toprule
\hline
Methods & 1 Block & 2 Block & 3 Block &4 Block & 5 Block \\
\midrule
\hline
PSNR & 37.70 & 39.40 &39.34 & \textbf{40.20} &39.37\\
SSIM &0. 9794 & 0.9827 & 0.9831 &\textbf{0.9858} &0.9808 \\
\bottomrule
\hline
\end{tabular}
}
\caption{The proposed method with different numbers of refinement blocks.}
\end{table}
\begin{figure}[!htb]
\begin{center}
\centering\includegraphics[width=0.5\textwidth]{ab1.png}
\end{center}
\caption{Visual comparison with different MAR methods on a clinical CT image. (A)-(G) respectively represent ground truth and corrected results using Refine-Net, Clean-IDOL, Art-IDOL, Prior-Im, Prior-Sino and IDOL-Net.}
\label{fig7}
\end{figure}
\section{Conclusion}
In this work, we present a novel interactive dual-domain parallel network for CT MAR, which contains two main modules. The disentanglement module aims to generate high quality prior sinogram and image as the initial inputs for the following refinement module, which is composed of two interactive parallel branches to exchange the useful information extracted from the intermediate results of both sinogram and image domains. The whole framework is trained in a end-to-end manner so that the dual-domain architecture can not only benefit from each other but also alleviate information loss. Experiment results on simulated and clinical data demonstrate that our proposed method can effectively suppress most artifacts and preserve tissues details. In the future work, we will investigate how will the segmentation result influence the final MAR result. Integrating the segmentation task into our MAR framework is also an interesting topic.
{\small
\bibliographystyle{ieee_fullname}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 3,963
|
Q: How should I properly add constraints to an AVCaptureVideoPreviewLayer? I have a Navigation Bar at the top of my view. I'm adding an AVCaptureVideoPreviewLayer but the way it gets positioned, there is a gap between the bottom of the Nav Bar and the top of the Preview Layer.
self.view.backgroundColor = UIColor.redColor()
var navBarFrame = CGRectMake(0, 0, self.view.frame.width, 64.0)
var navBar = UINavigationBar(frame: navBarFrame)
var navItem = UINavigationItem()
navItem.title = "zzzz"
navBar.pushNavigationItem(navItem, animated: false)
self.view.addSubview(navBar)
// init device input
var error: NSErrorPointer!
var deviceInput: AVCaptureInput = AVCaptureDeviceInput.deviceInputWithDevice(captureDevice, error: error) as AVCaptureInput
self.stillImageOutput = AVCaptureStillImageOutput()
// init session
self.session = AVCaptureSession()
self.session.sessionPreset = AVCaptureSessionPresetPhoto
self.session.addInput(deviceInput as AVCaptureInput)
self.session.addOutput(self.stillImageOutput)
// layer for preview
var previewLayer: AVCaptureVideoPreviewLayer = AVCaptureVideoPreviewLayer.layerWithSession(self.session) as AVCaptureVideoPreviewLayer
previewLayer.frame = self.view.bounds
self.view.layer.addSublayer(previewLayer)
How can I ensure that the Preview Layer is constrained to the bottom of the Nav Bar?
A: Design a view in IB with the constraints you want. Add the preview layer to this view instead, you also need to update your layer size when the view size changes, use the layoutSubviews method from your view controller to get updates when the size changes.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 33
|
{"url":"http:\/\/projecteuclid.org\/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.aop\/1176989274","text":"### Almost Sure Bootstrap of the Mean Under Random Normalization\n\nSteven J. Sepanski\nSource: Ann. Probab. Volume 21, Number 2 (1993), 917-925.\n\n#### Abstract\n\nWe consider the problem of when the bootstrap sample mean, appropriately normalized and centered, converges in distribution along almost every sample path. We allow the normalizing sequence to be an arbitrary sequence of positive random variables. It is proved that the only possible normalizing sequence is essentially $(\\sum^n_{i = 1}X^2_i)^{1\/2}$. Furthermore, if the bootstrap sample mean converges along almost every sample path, then either the variance is finite or else the distribution of $X$ is extremely heavy tailed. In the latter case, the distribution of the bootstrap sample mean is completely determined by how many times the maximum order statistic from the original random sample is repeated in the bootstrap sample. The necessary condition on how heavy the tails must be is $(\\sum^n_{i = 1}|X_i|^p)^{1\/p}\/(\\sum^n_{i = 1}X^2_i)^{1\/2} \\rightarrow 1$ almost surely for all $p \\in (0, \\infty\\rbrack$. Furthermore, we show that in this case the limit of the bootstrap sample mean normalized by $(\\sum^n_{i = 1}X^2_i)^{1\/2}$ is Poisson with mean 1.\n\nFirst Page:\nPrimary Subjects: 62E20\nSecondary Subjects: 62F12, 60F05\nFull-text: Open access","date":"2013-06-19 03:00:08","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9306671023368835, \"perplexity\": 266.4265557778344}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2013-20\/segments\/1368707440258\/warc\/CC-MAIN-20130516123040-00028-ip-10-60-113-184.ec2.internal.warc.gz\"}"}
| null | null |
Self-Concept, Depression, Anxiety and Stress among women: The effects of Behavioral Exercise Therapy, Exercise Therapy and Harmony Stimulation
Aram Hamldar 1
Alireza Salemi Khameneh 2
Zohreh Sadeghi 3
1 Department of Clinical Psychology, Varamin-Pishva Branch, Islamic Azad University, Varamin, Iran
2 1Department of Clinical Psychology, Varamin-Pishva Branch, Islamic Azad University, Varamin, Iran<br/>2PhD, Department of Clinical Psychology, Varamin-Pishva Branch, Islamic Azad University, Varamin,
3 PhD, Department of Clinical Psychology, Varamin-Pishva Branch, Islamic Azad University, Varamin, Iran
10.30483/rijm.2019.249718.0
Background: The mental health of women with overweight is at a higher risk comparing with women without overweight. Objectives: The present study aims to examine the effect of behavioral exercise therapy, exercise therapy and harmony stimulation on self-concept, depression, anxiety and stress among obese women in Tehran, 2017. Materials and Methods: The method was semi-experimental, pre-test and post-test with control group design. The population of the study consisted of women aged 20 years and older (not more than 40 years) in Tehran's tenth province. Convenience sampling method was used and the statistical sample included 40 persons. The therapy consisted of behavioral exercise therapy, exercise therapy and harmony stimulation in 10 sessions around 1 hour (only for the experimental group). Data collection was done based on Beck's self-concept questionnaire and DASS (Depression, Anxiety and Stress) scale. The data analysis (descriptive and inferential) was conducted by SPSS20 software. Results: The results showed the therapies were effective on depression, anxiety and stress in obese women. Moreover, the self- concept of women with obesity increased after the therapies (p < 0.05). Conclusion: It can be settled that the mental health of women with obesity can be changed by behavioral exercise therapy and exercise therapy.
Volume 7, Issue 3.4
Article View: 2
PDF Download: 4
Hamldar, A., Salemi Khameneh, A., Sadeghi, Z. (2019). Self-Concept, Depression, Anxiety and Stress among women: The effects of Behavioral Exercise Therapy, Exercise Therapy and Harmony Stimulation. Razavi International Journal of Medicine, 7(3.4), 15-18. doi: 10.30483/rijm.2019.249718.0
Aram Hamldar; Alireza Salemi Khameneh; Zohreh Sadeghi. "Self-Concept, Depression, Anxiety and Stress among women: The effects of Behavioral Exercise Therapy, Exercise Therapy and Harmony Stimulation". Razavi International Journal of Medicine, 7, 3.4, 2019, 15-18. doi: 10.30483/rijm.2019.249718.0
Hamldar, A., Salemi Khameneh, A., Sadeghi, Z. (2019). 'Self-Concept, Depression, Anxiety and Stress among women: The effects of Behavioral Exercise Therapy, Exercise Therapy and Harmony Stimulation', Razavi International Journal of Medicine, 7(3.4), pp. 15-18. doi: 10.30483/rijm.2019.249718.0
Hamldar, A., Salemi Khameneh, A., Sadeghi, Z. Self-Concept, Depression, Anxiety and Stress among women: The effects of Behavioral Exercise Therapy, Exercise Therapy and Harmony Stimulation. Razavi International Journal of Medicine, 2019; 7(3.4): 15-18. doi: 10.30483/rijm.2019.249718.0
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 4,081
|
{"url":"https:\/\/www.shaalaa.com\/question-bank-solutions\/nature-roots-formula-method_1198","text":"# Solution - Nature of Roots\n\nAccount\nRegister\n\nShare\n\nBooks Shortlist\nConceptNature of Roots\n\n#### Question\n\nSolve the equation by using the formula method.\n3y2\u00a0+7y + 4 = 0\n\n#### Solution\n\nYou need to to view the solution\nIs there an error in this question or solution?\n\n#### Similar questions VIEW ALL\n\nSolve for x:\n\n1\/(x+1)+2\/(x+2)=4\/(x+4),\u00a0x\u00a0\u2260 -1, -2, -3\n\nview solution\n\nIf\u00a0x=2\/3\u00a0and\u00a0x\u00a0=3 are roots of the quadratic equation ax2 + 7x + b = 0, find the values of a and b.\n\nview solution\n\nFind the values of k for which the quadratic equation (k + 4) x2 + (k + 1) x + 1 = 0 has equal roots. Also find these roots.\n\nview solution\n\nFind the nature of the roots of the following quadratic equations.\u00a0If the real roots exist, find them;\n\n3x^2\u00a0-\u00a04sqrt3x\u00a0+ 4 = 0\n\nview solution\n\nIs it possible to design a rectangular park of perimeter 80 and area 400 m2? If so find its length and breadth\n\nview solution\nSolution for concept: Nature of Roots. For the course 9th - 10th SSC (English Medium)\nS","date":"2017-10-21 19:41:59","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\": 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\": 0.42429864406585693, \"perplexity\": 896.4828847224661}, \"config\": {\"markdown_headings\": true, \"markdown_code\": false, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2017-43\/segments\/1508187824894.98\/warc\/CC-MAIN-20171021190701-20171021210701-00707.warc.gz\"}"}
| null | null |
STARTTLS Considered Harmful
There are two ways that otherwise plain text protocols can provide encryption with TLS. The first way is to listen on two ports: one port that is always plain text, and a second port that is always encrypted with TLS. The other way is to use a single port on which communication starts out unencrypted, but can be "upgraded" to a TLS encrypted connection using an application-level command specific to the protocol. HTTP/HTTPS uses exclusively the first approach, with ports 80 and 443. The second approach, called STARTTLS, is used by SMTP, XMPP, IMAP, and POP3, though several of those protocols also support the first approach.
There's a clear bias for STARTTLS in the IETF's email standards. The use of alternative TLS-only ports for IMAP, POP3, and SMTP was never formally standardized: people just started doing it that way, and although port numbers were registered for the purpose, the registration of the encrypted SMTP (SMTPS) port (465) was later rescinded. When the IETF finally standardized the use of TLS with IMAP and POP3 in 1999, they prescribed the use of STARTTLS and gave several reasons why STARTTLS should be used instead of an alternative TLS-only port. Briefly, the reasons are:
Separate ports lead to a separate URL scheme, which means the user has to choose between them. The software is often more capable of making this choice than the user.
Separate ports imply a model of either "secure" or "not secure," which can be misleading. For example, the "secure" port might be insecure because it's using export-crippled ciphers, or the normal port might be using a SASL mechanism which includes a security layer.
Separate ports has caused clients to implement only two security policies: use TLS or don't use TLS. The desirable security policy "use TLS when available" would be cumbersome with the separate port model, but is simple with STARTTLS.
Port numbers are a limited resource.
Except for reason four, these reasons are pretty terrible. Reason one is not very true: unless the software keeps a database of hosts which should use TLS, the software is incapable of making the choice between TLS and non-TLS on behalf of the user without being susceptible to active attacks. (Interestingly, web browsers have recently started keeping a database of HTTPS-only websites with HSTS preload lists, but this doesn't scale.)
Reason three is similarly dubious because "use TLS when available" is also susceptible to active attacks. (If the software detects that TLS is not available, it doesn't know if that's because the server doesn't support it or if it's because an active attacker is blocking it.)
Reason two may have made some sense in 1999, but it certainly doesn't today. The export cipher concern was mooted when export controls were lifted in 2000, leading to the demise of export-crippled ciphers. I have no idea how viable SASL security layers were in 1999, but in the last ten years TLS has clearly won.
So STARTTLS is really no better than using an alternative TLS-only port. But that's not all. There are several reasons why STARTTLS is actually worse for security.
The first reason is that STARTTLS makes it impossible to terminate TLS in a protocol-agnostic way. It's trivial to terminate a separate-port protocol like HTTPS in a software proxy like titus or in a hardware load balancer: you simply accept a TLS connection and proxy the plain text stream to the backend's non-TLS port. Terminating a STARTTLS protocol, on the other hand, requires the TLS terminator to understand the protocol being proxied, so it can watch for the STARTTLS command and only upgrade to TLS once the command is sent. Supporting IMAP/POP3/SMTP isn't too difficult since they are simple line-based text protocols. (Though you have to be careful - you don't want the TLS terminator to misfire if it sees the string "STARTTLS" inside the body of an email!) XMPP, on the other hand, is an XML-based protocol, and do you really want your TLS terminator to contain an XML parser?
I care about this because I'd like to terminate TLS for my SMTP, IMAP, and XMPP servers in the highly-sandboxed environment provided by titus, so that a vulnerability in the TLS implementation can't compromise the state of my SMTP, IMAP, and XMPP servers. STARTTLS makes it needlessly difficult to do this.
Another way that STARTTLS harms security is by adding complexity. Complexity is a fertile source of security vulnerabilities. Consider CVE-2011-0411, a vulnerability caused by SMTP implementations failing to discard SMTP commands pipelined with the STARTTLS command. This vulnerability allowed attackers to inject SMTP commands that would be executed by the server during the phase of the connection that was supposed to be protected with TLS. Such a vulnerability is impossible when the connection uses TLS from the beginning.
STARTTLS also adds another potential avenue for a protocol downgrade attack. An active attacker can strip out the server's advertisement of STARTTLS support, and a poorly-programmed client would fall back to using the protocol without TLS. Although it's trivial for a properly-programmed client to protect against this downgrade attack, there are already enough ways for programmers to mess up TLS client code and it's a bad idea to add yet another way. It's better to avoid this pitfall entirely by connecting to a port that talks only TLS.
Fortunately, despite the IETF's recommendation to use STARTTLS and the rescinding of the SMTPS port assignment, IMAP, POP3, and SMTP on dedicated TLS ports are still widely supported by both server and client email implementations, so you can easily avoid STARTTLS with these protocols. Unfortunately, the SMTPS port is only used for the submission of authenticated mail by mail clients. Opportunistic encryption between SMTP servers, which is extremely important for preventing passive eavesdropping of email, requires STARTTLS on port 25. And modern XMPP implementations support only STARTTLS.
Moving forward, this shouldn't even be a question for new protocols. To mitigate pervasive monitoring, new protocols should have only secure versions. They can be all TLS all the time. No need to choose between using STARTTLS and burning an extra port number. I just wish something could be done about the existing STARTTLS-only protocols.
Hi, I'm Andrew. I run SSLMate, which makes SSL certificates easy through automation, great software, and friendly support.
I blog about security, PKI, Linux, and more. If you liked this post, check out my other posts or subscribe to my RSS feed.
My email address is andrew@agwa.name. I'm AGWA at GitHub and @__agwa on Twitter.
Older (View Archive)
LibreSSL's PRNG is Unsafe on Linux [Update: LibreSSL fork fix]
SHA-1 Certificate Deprecation: No Easy Answers
Reader George on 2014-08-13 at 07:54:
With the recent backtracking of IETF from making HTTP 2.0 strictly HTTPS, I'm not too hopeful that new protocols would only have secure versionns.
Andrew Ayer on 2014-08-13 at 14:24:
You may be right, but HTTP 2.0 isn't a great example because it's not an all new protocol but rather an update to an existing protocol that has been widely deployed without TLS.
Reader Charles on 2014-08-13 at 11:41:
thanks. Very interesting. One question: how can a client protect itself against a TLS downgrade attack? How can it tell the difference between an SMTP server that has TLS but with a man in the middle and an SMTP server with no TLS?
And an additional question: do you know if smtp clients usually do have protections against downgrade attacks?
If by "SMTP client" you mean a mail user agent (MUA) like Thunderbird, Mail.app, Outlook, etc., then I think the popular ones probably do it correctly. I'd be concerned about less popular ones that have received less scrutiny.
If by "SMTP client" you mean a mail transfer agent (MTA) that's sending mail server-to-server, then they intentionally allow downgrades, because not all MTAs accept mail over TLS. Since this is intentional and currently unavoidable, I'm not really talking about this case. I'm more concerned with the client to server scenario which is virtually always expected to be secure.
I guess most of the MUA these days are mobile phones. I would be surprised if clients like Thunderbird are still used massively (and when they are, they are usually connected inside a protected corporate environment).
I was referring to both actually. Would the gmail MTA be exposed to MITM attacks when delivering an email to an smtp server? This is a pretty big weakness as we know private communications is the focus of sovereign spying. Is there any real danger these days to not allowing an unencrypted connection? Surely no reputable smtp MTA wouldn't support TLS, only spammer bots.
Would the gmail MTA be exposed to MITM attacks when delivering an email to an smtp server?
Yes, gmail's MTA intentionally lets itself downgrade, which is vulnerable to MITM attacks. Unfortunately many legitimate email servers still don't support TLS. Fortunately, Google is leading the way in trying to get other providers to support TLS <https://www.google.com/transparencyreport/saferemail/>.
The google link is a bit of an eye opener. These are pretty major ISP who don't bother crypting any of their customers correspondence. And would be curious to know how many of the ISP who crypt their traffic actually check the validity of the certificates.
Some see the NSA as a predator, but it is really a fox helping itself in an open henhouse!
You raise yet another issue, which is checking certificates. I would guess that virtually no email provider checks certificates. The problem is that there's no way to know what the name on the certificate should be. Ideally it would match the domain of the email address you're sending to, but this is rarely the case because domains usually delegate their mail to servers under other domains using MX records. And without DNSSEC, the MX delegation is completely unauthenticated, making it pointless to check that the certificate name matches the name in the MX record.
For example, gmail.com's smallest-priority MX server is `gmail-smtp-in.l.google.com`. The names on `gmail-smtp-in.l.google.com`'s certificate are:
DNS:aspmx.l.google.com, DNS:alt1.aspmx.l.google.com, DNS:alt2.aspmx.l.google.com, DNS:alt3.aspmx.l.google.com, DNS:alt4.aspmx.l.google.com, DNS:gmail-smtp-in.l.google.com, DNS:alt1.gmail-smtp-in.l.google.com, DNS:alt2.gmail-smtp-in.l.google.com, DNS:alt3.gmail-smtp-in.l.google.com, DNS:alt4.gmail-smtp-in.l.google.com, DNS:gmr-smtp-in.l.google.com, DNS:alt1.gmr-smtp-in.l.google.com, DNS:alt2.gmr-smtp-in.l.google.com, DNS:alt3.gmr-smtp-in.l.google.com, DNS:alt4.gmr-smtp-in.l.google.com, DNS:mx.google.com, DNS:aspmx2.googlemail.com, DNS:aspmx3.googlemail.com, DNS:aspmx4.googlemail.com, DNS:aspmx5.googlemail.com.
gmail.com doesn't appear anywhere in that list. So even though the name of the MX server does appear in that list, since the MX record lookup was unauthenticated, a program has no way of knowing that `gmail-smtp-in.l.google.com` is `gmail.com`'s true MX server, and not a bogus server set up by an active attacker who can manipulate DNS.
Reader Jarek on 2014-08-13 at 13:30:
One question: how can a client protect itself against a TLS downgrade attack? How can it tell the difference between an SMTP server that has TLS but with a man in the middle and an SMTP server with no TLS?
Using DANE records in DNSSEC. It's quite recent thing, and DNSSEC still isn't such widely deployed, but it's supported by both Postfix and Exim SMTP servers (I think it's disabled by default, though).
So am I right to understand that the DANE record would say "this SMTP server supports TLS" and therefore a client would not accept an unencrypted connection with this server even if the server responds it does not support TLS.
But the article says that "it's trivial for a properly-programmed client to protect against this downgrade attack". I presume the author wasn't referring to using DANE records (which as you point is not widely supported)?
Yes, DANE can do that. Also, it can specify CA signing the server certificate (because even if TLS is enforced, attacker can MITM the connection, as certificates aren't validated. Without DANE they can't, because there is no common database of trusted CAs and each server trusting other CAs would lead to chaos and unreliability). For instance, for the Postfix implementation see http://www.postfix.org/TLS_README.html#client_tls_dane.
It's not only not trivial, but also impossible for a client to protect against downgrade attacks without additional tricks. Those tricks can include DANE, manual configuration (not scalable) and some magic caching (but I wouldn't say it'd be a properly-programmed client, what would happen if remote server simply turns off TLS?). And even with the insane-magic-caching, we still have the common trust anchor problem. So well... the author is wrong here (at least in SMTP case, interactive clients or freshly designed protocols might be in better position).
As for wide support... It's supported by two very (most?) popular SMTP servers (Postfix and Exim), so it's not that bad. But it appeared only recently (is available only in relatively fresh versions) and requires additional configuration (especially validating DNSSEC resolver). But bigger problem is on the other side (target) -- it's not widely deployed.
But it has to ba said, that even the opportunistic encryption with possible downgrade attacks is beneficial. Often the monitoring is passive (attacker just sniffs received traffic without tampering it) and downgrade attacks require active MITMing. So, while not perfect, it's still better than nothing.
Reader Lie on 2016-04-26 at 15:33:
Downgrade attack is trivial to protect, a client will simply require STARTTLS and simply drops the connection when the server does not support ESMTP and encryption.
DNSSEC and DANE protects against different vulnerabilities, which is that MUAs and MTAs can't reliably verify the server certificates. In the DNSSEC and DANE scenario, the attacker MITM the encrypted connection and it receives encrypted connection from the client and then makes encrypted connection to the mail server. In other words, even if both client and server refuses unencrypted connection, that is all futile if the client cannot verify the server's certificate. DNSSEC signs DNS record to protect DNS record from being tampered by malicious or compromised recursive DNS resolvers, and DANE embeds the TLS certificate inside DNSSEC-signed records to avoid reliance on CA, whose security model is totally broken for emails. Additionally, there's also DNSCrypt/DNSCurve that encrypts the connection to the DNS server to protect privacy against eavesdropping.
The client has to know that the server supports TLS somehow. Someone else mentioned DNSSEC+DANE, but that's not widely deployed and it's questionable whether it ever will be.
In practice the way it works with mail clients is you tell it when configuring your account. For example Thunderbird has a drop-down box called "Connection Security" with the options "None", "STARTTLS", and "SSL/TLS". (Many mail clients have something similar.) The concern is that the "STARTTLS" would be vulnerable to the downgrade attack, while "SSL/TLS" wouldn't be. (Note: I'm extremely confident Thunderbird does this correctly. The concern is with other, less widely used software.)
Reader Pino on 2017-12-09 at 17:49:
I would expect "STARTTLS" to mean "Connect unencrypted, and disconnect if STARTTLS fails." Is that what happens in current Thunderbird?
Anonymous on 2014-08-13 at 12:32:
"prescribed", not "proscribed". They're opposites.
Thanks for the correction!
Reader Bron Gondwana on 2014-08-13 at 21:31:
Nice to see more awareness of this issue. We have blogged on this at FastMail as well, and it's the reason that we ONLY support the SSL ports for IMAP and POP3.
https://www.fastmail.fm/help/technical/ssltlsstarttls.html
Reader Dave Cridland on 2014-10-18 at 20:04:
OK, so I think you're wrong, but where to start? Mostly, I think you've decided that you want to write Titus in a certain way, and since STARTTLS doesn't allow you to do this, therefore STARTTLS is bad.
So first off: Most protocols using STARTTLS involve relatively few client/server pairings. It's not clear whether STARTTLS would have been useful on the web - I suspect it would have been, but history didn't go that way - but the web does have the issue that there are a vast number of services to track. But an XMPP client (or an MUA) can, trivially, know which services offer TLS. STARTTLS allows clients to discover this trivially, without user interaction, and additionally allows this information to be cached. It's best to think of STARTTLS advertising as - like SASL mechanism advertising - a once-only, account-configuration thing. After that mental adjustment, you're really down the the counter-argument that it's an additional round-trip, and I can argue that one with you as well if you really want.
A key factor in the "different URI" debate is that I have no way of knowing whether https://www.google.co.uk/ is the same service as http://www.google.co.uk/ - the differing scheme actually imposes that a client must assume otherwise. On the other hand, STARTTLS protocols are effectively mandated to offer the same service (though many refuse access to non-protected sessions).
Finally, while SNI has helped, there are protocols which need application data to select the correct certificate to use.
I do agree that TLS is now the only interesting security layer, however - while GSSAPI still gives you good security, most people will want TLS.
It might help, though, to understand that there's lots of other things Titus can't do because of its nature. It can't handle outgoing sessions, it can't handle certificate selection outside SNI, it can't handle client authentication, it cannot let a server note an improper TLS shutdown, it prevents channel binding, and so on. Every one of these things is important.
Many of these things are impossible to implement if Titus is the dumb pass-through it currently is. I shall leave it as a trivial exercise to the reader to figure out an architecture that supports them - and incidentally support STARTTLS - without including a protocol parser.
titus was the impetus for this blog post, but I had disliked STARTTLS long before I wrote titus, and I present other, unrelated arguments for why STARTTLS is a bad idea.
But an XMPP client (or an MUA) can, trivially, know which services offer TLS. STARTTLS allows clients to discover this trivially, without user interaction, and additionally allows this information to be cached. It's best to think of STARTTLS advertising as - like SASL mechanism advertising - a once-only, account-configuration thing.
This isn't secure, because a MitM can strip the STARTTLS advertisement, so a user has to know, and specify, whether a service supports TLS or not. It's OK for opportunistic encryption, and caching does help, but you don't need STARTTLS for this. The insecure service could advertise the presence of a secure service on a different port, and the client could then reconnect with TLS on that port. This is essentially how HTTP and HTTPS work, when you redirect a HTTP connection to a HTTPS URI.
A key factor in the "different URI" debate is that I have no way of knowing whether https://www.google.co.uk/ is the same service as http://www.google.co.uk/ - the differing scheme actually imposes that a client must assume otherwise
I really don't think that's a practical concern, and it's already a widely-followed convention that these different URIs are the same service. Besides, moving forward we should be phasing out our insecure services, so there will only be one URI anyways.
SNI should definitely handle that, especially if proposals to encrypt the SNI name come to fruition.
It's true that titus can't handle these but fortunately they aren't that important. Can you point to a widely-used protocol on the Internet where any of these things are important? (I'll concede outgoing SMTP connections over TLS.)
Any other solution would require modifications to the application. I am actually working on the design of a "libtitus" that would let applications easily initiate titus-secured outgoing connections, or accept incoming connections with client certificates. But modifying applications isn't always possible, and we've got many widely-used Internet protocols which don't typically use client certificates, channel binding, etc. where a dumb pass-through would be completely sufficient if not for the fact that they are needlessly using STARTTLS to immediately upgrade a connection which could have been TLS in the first place.
Yes, downgrade attacks are already done by at least one ISP: http://arstechnica.com/tech-policy/2014/11/condemnation-mounts-against-isp-that-sabotaged-users-e-mail-encryption/
This line caught my eye from your post Andrew: "Opportunistic encryption between SMTP servers, which is extremely important for preventing passive eavesdropping of email, requires STARTTLS on port 25"
Now, I have been using 465 port with SSL for ages when sending mail.
So does this mean that unless I use port 25 with STARTTLS there is never going to be opportunistic encryption between SMTP servers if I continue using 465 ? I mean, even tought my mail will be encrypted between my client and server, the continuing server-to-server delivery is not going to be encrypted if not using port 25?
Generally mail servers don't accept mail from other MTAs over port 465, just mail from authenticated MUAs, so if you try to use it for server-to-server delivery your mail might be encrypted but it won't go anywhere. I doubt this will change, especially since port 465 is unofficial and has actually been reassigned by IANA.
That makes port 25 the only viable option for server-to-server delivery. Some mail servers don't support STARTTLS on port 25 at all, in which case your mail to them will be unencrypted. Others servers support STARTTLS, but if your ISP or another active attacker MitMs the connection, your mail to that server will be compromised. This is regardless of what port your MUA uses to submit mail.
The status quo is that there is approximately zero authenticated encryption of server-to-server email delivery because of backwards compatibility with mail servers that don't support any encryption, as well as a lack of a standard for how certificates names for SMTP servers should work. Possible solutions are DNSSEC+DANE (which Postfix already supports) and/or the EFF's STARTTLS Everywhere project: https://github.com/EFForg/starttls-everywhere
Reader Gary Mort on 2015-05-05 at 16:06:
There is an invalid premise, you have assumed that using TLS means the communication is secure. Due to the root certificate structure, in the real world TLS security is situational. Since a certificate issued by ANY root CA is considered just as valid as any other root CA, it is simple to compromise[implementation may or may not be difficult, but the technical process itself is simple]. The long running CNNIC controversy shows that the root process is flawed and it took 5 years to reach the point where they got sloppy and were exposed. [Since they control much of the TCP/IP infrastructure in their country, it is trivial for them to provide redirect traffic to their own proxies and provide fake certificates only for targeted individuals - something they have already been caught doing in the past]
As such, using the word "secure" when referring to any central authority registration scheme is demonstrably false. What so called security experts mean is "it is secure, assuming that...." and their long standing avoidance of mentioning the assumptions they based it on makes most security professionals opinions worthless.
Starttls is an interesting protocol in that it gives the end user simple access to make a choice of how "secure" they want their communications to be, Always use, Prefer to use, or Do not use.
Most of the other "options" built into TLS and other protocols are not provided in any meaningful way for the end user[who is the one who should be making the decisions about what they want].
Security nihilism at its finest.
In reality, there is a large class of attackers who are thwarted by mandatory, authenticated encryption, but have no problem at all attacking opportunistic encryption. To suggest that we aim for the lesser level of security because there may be some attackers who can break the higher level of security is ridiculous.
Reader Mike Spooner on 2017-08-27 at 11:23:
Re "dedicated TLS ports are still widely supported by ... client email implementations".
On the desktop and perhaps tablets too, that assertion is mostly accurate, but on mobile ('droid phones, iOS) it really isn't, even as late as 2017... most such apps give encryption config choices of "TLS", "TLS (dont check certificate)" or "None" - however, this almost always actually means "STARTTLS" or "None", and not straight TLS.
Frustrating!
Your comment will be public. If you would like to contact me privately, please email me. Please keep your comment on-topic, polite, and comprehensible.
Your Name: (Optional; will be published)
Your Email Address: (Optional; will not be published)
Your Website: (Optional; will be published)
Blank lines separate paragraphs.
Lines starting with ">" are indented as block quotes.
Lines starting with two spaces are reproduced verbatim.
Text surrounded by *asterisks* is italicized.
Text surrounded by `back ticks` is monospaced.
URLs are turned into links.
Use the Preview button to check your formatting.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 4,001
|
\section{Introduction}
The mechanical force exerted by light on dielectric objects enables capturing, manipulation,
and analysis of particles at the micro- and nano-meter scales \cite{Ashkin1986,Dienerowitz2008}.
Often, especially in biological applications, the experimental set-up of an optical trap
involves a liquid host medium \cite{Molloy2002}. The dynamics of particles in such
traps is influenced not only by the optical and Brownian forces but by the drag force of
the liquid as well. If operating in the neighborhood of an absorption band,
a significant thermal flow may be induced by the light itself, and needs to be taken
into account \cite{Peterman2003, Weinert2008, Schermer2011}. The analysis of the host-medium
hydrodynamics is important for understanding the observed non-conservative motion of
trapped particles \cite{Wu2009}, which at the moment is mostly
attributed to the presence of the so-called scattering force, Brownian `motors' \cite{Khan2011},
and thermal/thermophoretic flows. The emergent field of optofluidics also requires a better understanding
of the mechanical interaction of light with liquids \cite{Psaltis2006,Louchev2008,Yang2009,Ryu2010}.
A question arises whether absorption/extinction
is the only mechanism by which a light-driven hydrodynamic flow may be induced.
Can a flow be created in a lossless liquid?
Since the host liquid is typically optically homogeneous the question can be narrowed down
to the mechanical forces exerted by the electromagnetic field inside a homogeneous dielectric.
Yet, no simple answer is available at the moment.
The two dominant approaches, associated with the names of Abraham and Minkowski,
although differing on the form of the mechanical momentum of the electromagnetic field,
seem to agree that the time-averaged force density (the Helmholtz force density)
in a homogeneous medium should be exactly zero \cite{Frias2012}. On the contrary, the third approach,
that treats the induced (polarization) currents on equal footing with the externally imposed ones, leads to
a non-zero Lorentz force density. Thus, one faces a fundamental question: does the electromagnetic
field exert a mechanical force on the polarization currents induced by that same field?
It has been shown that all three approaches give the same
total time-averaged mechanical force on a finite homogeneous solid body, even though the Helmholtz force
density contributes only at the boundary surface while the Lorentz force density acts across the whole volume of the
object \cite{Dienerowitz2008}. However, recently it has been suggested that a deformable homogeneous body would respond
differently to these forces \cite{Frias2012}. Although, the internal stresses induced by light in thin films
may appear to be difficult to detect experimentally,
if such internal stresses do exist, they might induce hydrodynamic flows that are much easier
to observe and should, in fact, be taken into account during the optical trapping.
The quantitative analysis of light-driven flows for a typical single-beam optical trap presented here
is aimed at working out the consequences of the induced-current Lorentz force density model.
We consider both the transparent and absorptive spectral regions.
Our simulations demonstrate that according to this model the light-driven flows
should be significant (in the order of tens of micrometers per second) even with small
laser powers ($\sim 1$~mW). Such flows have very distinctive patterns that could be detected
in an experiment.
The paper is organized as follows. First, we briefly describe the three forms of the
electromagnetic momentum conservation law and the associated time-averaged force densities.
We devote a separate section to the question of modeling of the electromagnetic field in a
single-beam optical trap, where we show that a standard Gaussian beam model is not applicable
for the purposes of force computations as it fails to satisfy the momentum conservation law.
We resort to an alternative method that mimics the Gaussian beam by a linear superposition
of exact fundamental solutions of the Maxwell equations.
Then, we introduce the mechanical coupling between the electromagnetic field and the host liquid
via the body force term of the incompressible Navier-Stokes equations and the Boussinesq approximation
for the thermally-driven flows. Finally, we present the results of
numerical experiments and our conclusions.
\section{Models of the Electromagnetic Force Density}
There exist two mathematically equivalent views on the electromagnetic field in
media. In vacuum the Maxwell equations have the form:
\begin{align}
\label{eq:Maxwell}
\begin{split}
-\nabla\times\mathbf H +\varepsilon_{0}\partial_{t}\mathbf E&=-\mathbf J^{\rm ext},
\\
\nabla\times\mathbf E +\mu_{0}\partial_{t}\mathbf H&=-\mathbf 0,
\end{split}
\end{align}
with $\mathbf E(\mathbf x,t)$, $\mathbf H(\mathbf x,t)$ denoting the
electric and magnetic field strengths, and $\mathbf J^{\rm ext}(\mathbf x,t)$ --
the external (i.e., independent of the field) electric current density.
A medium different from vacuum can be introduced either via the concept
of electric and magnetic fluxes $\mathbf D$ and $\mathbf B$ entering the Maxwell equations as
\begin{align}
\label{eq:MaxwellDB}
\begin{split}
-\nabla\times\mathbf H +\partial_{t}\mathbf D&=-\mathbf J^{\rm ext},
\\
\nabla\times\mathbf E +\partial_{t}\mathbf B&=-\mathbf 0,
\end{split}
\end{align}
or via the induced electric and magnetic current densities
$\mathbf J^{\rm ind}$ and $\mathbf K^{\rm ind}$ as
\begin{align}
\label{eq:MaxwellInduced}
\begin{split}
-\nabla\times\mathbf H +\varepsilon_{0}\partial_{t}\mathbf E&=-\mathbf J^{\rm ext}-\mathbf J^{\rm ind},
\\
\nabla\times\mathbf E +\mu_{0}\partial_{t}\mathbf H&=-\mathbf K^{\rm ind},
\end{split}
\end{align}
Supplied with appropriate constitutive relations, for example,
\begin{align}
\label{eq:ConstRelLossless}
\begin{split}
\mathbf D&=\varepsilon\mathbf E,\;\;\;\mathbf B=\mu\mathbf H,
\\
\mathbf J^{\rm ind}&=(\varepsilon-\varepsilon_{0})\partial_{t}\mathbf E,\;\;\;\mathbf K^{\rm ind}=(\mu-\mu_{0})\partial_{t}\mathbf H,
\end{split}
\end{align}
both formulations lead to exactly the same Maxwell's equations for the fields $\mathbf E$ and $\mathbf H$.
In general, the momentum conservation law has the form:
\begin{align}
\label{eq:MomentumConservationGeneral}
\nabla\cdot{\mathbb T}-\partial_{t}\mathbf P = \mathbf f,
\end{align}
where ${\mathbb T}$ is the stress tensor density, $\mathbf P$ is the momentum density,
and $\mathbf f$ is the force density, which is our main concern. The actual form of this law depends on
the definition and interpretation of the terms. For example, in the lossless case described
by the constitutive relations (\ref{eq:ConstRelLossless}) the Abraham and Minkowski expressions for
the force density in the part of the domain without external currents/charges are
\begin{align}
\label{eq:ForceAbraham}
\mathbf f^{\rm A}&=\mathbf f^{\rm H}+\frac{\varepsilon\mu-1}{c^{2}}\partial_{t}(\mathbf E\times\mathbf H),
\\
\label{eq:ForceMinkowski}
\mathbf f^{\rm M}&=\mathbf f^{\rm H},
\end{align}
where
\begin{align}
\label{eq:ForceHelmholtzLossless}
\mathbf f^{\rm H}=-\frac{1}{2}(\mathbf E\cdot\mathbf E)\nabla\varepsilon-\frac{1}{2}(\mathbf H\cdot\mathbf H)\nabla\mu
\end{align}
is the Helmholtz force density. In a general medium this force density is defined as:
\begin{align}
\label{eq:ForceHelmholtz}
\mathbf f^{\rm H}=\frac{1}{2}\left(\mathbf D\cdot\nabla\mathbf E-\mathbf E\cdot\nabla\mathbf D\right)
+\frac{1}{2}\left(\mathbf B\cdot\nabla\mathbf H-\mathbf H\cdot\nabla\mathbf B\right).
\end{align}
Obviously, after time averaging over a period of harmonic
oscillations the Helmholtz force will be the only non-zero contribution to both
the Abraham and Minkowski force densities and it will be zero
in a homogeneous lossless medium, i.e.,
\begin{align}
\label{eq:ForceAMaveraged}
\langle\mathbf f^{\rm A}\rangle=\langle\mathbf f^{\rm M}\rangle=\langle\mathbf f^{\rm H}\rangle=0,
\;\;\;\varepsilon,\mu=\text{const.}
\end{align}
The Abraham and Minkowski expressions follow from the fluxes-based approach to the
medium (\ref{eq:MaxwellDB}).
If, on the other hand, one starts with the induced-currents approach (\ref{eq:MaxwellInduced}),
then the force density turns out to be:
\begin{align}
\label{eq:ForceLorentz}
\begin{split}
\mathbf f^{\rm L}&=\rho_{\rm e}^{\rm ind}\mathbf E+
\rho_{\rm m}^{\rm ind}\mathbf H
+
\mu_{0}\mathbf J^{\rm ind}\times\mathbf H-\varepsilon_{0}\mathbf K^{\rm ind}\times\mathbf E,
\\
\rho_{\rm e}^{\rm ind}&=\int_{0}^{t}\nabla\cdot\mathbf J^{\rm ind}\,dt',
\;\;\;
\rho_{\rm m}^{\rm ind}=\int_{0}^{t}\nabla\cdot\mathbf K^{\rm ind}\,dt'.
\end{split}
\end{align}
In a lossless homogeneous medium this generalized Lorentz force density reduces to:
\begin{align}
\label{eq:ForceLorentzLossless}
\mathbf f^{\rm L}&=\mu_{0}(\varepsilon-\varepsilon_{0})(\partial_{t}\mathbf E)\times\mathbf H-
\varepsilon_{0}(\mu-\mu_{0})(\partial_{t}\mathbf H)\times\mathbf E.
\end{align}
Assuming time-harmonic fields with the angular frequency $\omega$ and
a linear non-magnetic homogeneous dispersive medium we perform the averaging of the general expression
(\ref{eq:ForceLorentz}) over a period of oscillations and arrive at the following result:
\begin{align}
\label{eq:ForceFrequencyDomain}
\begin{split}
\langle\mathbf f^{\rm L}\rangle&=
\frac{1}{2}\omega\mu_{0}(\varepsilon'-\varepsilon_{0})\text{Im}\{\hat{\mathbf S}\}
+\frac{1}{2}\omega\mu_{0}\varepsilon''\text{Re}\{\hat{\mathbf S}\},
\end{split}
\end{align}
where the complex Poynting vector is defined as
\begin{align}
\label{eq:PoyntingComplex}
\hat{\mathbf S}=\hat{\mathbf E}\times\hat{\mathbf H}^{*},
\end{align}
Here, the complex field amplitudes $\hat{\mathbf E}$, $\hat{\mathbf H}$ satisfy the frequency-domain
Maxwell's equations, $(\dots)^{*}$ denotes the complex conjugation, and
$\varepsilon(\omega)=\varepsilon'(\omega)+i\varepsilon''(\omega)$
is the complex permittivity of the host liquid.
The heat produced by the electromagnetic field is given by the following standard
time-averaged expression derived from the energy conservation law:
\begin{align}
\label{eq:HeatEM}
q=\frac{1}{2}\omega\varepsilon''\hat{\mathbf E}\cdot\hat{\mathbf E}^{*}.
\end{align}
\section{Single-Beam Optical Trap}
\begin{figure*}[t]
\includegraphics[width=17cm]{residuals}
\caption{Normalized residuals of the Maxwell equations (left) and the momentum conservation
law (right) as a function of the beam parameter $\vert s\vert$, see eq.~(\ref{eq:GaussianBeamParameter}),
for different discretization levels ($n=32, 64, 96, 128$ uniformly spaced grid-points along each of the $10\,\mu\text{m}$
sides of a cube).
Solid lines -- dipole-based model, dashed lines -- Gaussian beam approximation. As the beam waist gets wider
(smaller $\vert s\vert$ values) the residual of the Maxwell equations for the Gaussian
approximation becomes smaller and is limited by the discretization error. The
residual of the momentum conservation law stays relatively high irrespectively of the discretization level
for the Gaussian approximation, whereas it is consistently smaller and improves with finer discretization levels
for the dipole model of the beam.}
\label{fig:Residuals}
\end{figure*}
In optics it is common to model the spatial distribution of the field of a single-beam optical trap
as a tightly focused Gaussian beam. Unfortunately, this model is not really suitable for the calculation of the
force density since it violates the conservation of momentum to an extent that may
invalidate the results of numerical experiments. Indeed, the expression
(\ref{eq:ForceFrequencyDomain}) represents the electromagnetic
force density only if the field quantities appearing in it are the exact solutions of the frequency-domain
versions of the Maxwell equations (\ref{eq:MaxwellDB}) or (\ref{eq:MaxwellInduced}),
whereas the Gaussian beam is not an exact solution of the Maxwell equations.
Figure~\ref{fig:Residuals} (left) gives the norm of the residual
of the frequency-domain Maxwell's equations computed numerically via the finite-volume
approximation of the equations on different progressively refined grids. The horizontal axis is the Gaussian
beam parameter
\begin{align}
\label{eq:GaussianBeamParameter}
\vert s\vert = \frac{\lambda_{0}}{2\pi w_{0} \vert n\vert},
\end{align}
where $\lambda_{0}$ is the vacuum wavelength of light, $n$ is the complex refractive index of the medium,
and $w_{0}$ is the beam waist. Thus, the larger is the relative beam waist $w_{0}/\lambda_{0}$, the smaller is
$\vert s\vert$. The residual of the Gaussian approximation (dashed lines) reduces for larger waists, generally following the
${\mathcal O}(\vert s\vert^{2})$ trend -- the order of the approximation. It also improves with the grid refinement.
This, however, is merely a consequence of the finite-volume approximation of the spatial derivatives in the Maxwell
equations.
Figure~\ref{fig:Residuals} (right) shows the residual of the frequency-domain
version of the momentum conservation law (\ref{eq:MomentumConservationGeneral}), where the
left-hand side was estimated using the numerical surface integration over the elementary cells.
Here too the Gaussian approximation (dashed lines) improves with the increase in the beam waist. However, refining the discretization
does not help any more. Hence, we conclude that the error is mainly due to the analytical mismatch in the spatial
structure of the fields.
A way to improve the single-beam optical trap model is to represent the electromagnetic field as a superposition
of exact analytical solutions of the Maxwell equations due to elementary dipole sources.
With the help of these fundamental solutions we can model a focused laser beam by distributing an array of dipoles
over a plane just outside the computational domain and tuning their amplitudes, phases, and polarizations (dipole moments) so that they reproduce
the spatial pattern of the electric field of an ideal focused Gaussian beam passing through that plane.
Basically, this can be viewed as a numerical implementation of the Huygens principle.
Solid lines in Figure~\ref{fig:Residuals} demonstrate that the residuals
for the dipole-based model of the beam stay roughly constant for various beam waists
in the Maxwell equations and become smaller for larger beam waists in the momentum conservation law.
Moreover, refining the discretization helps to reduce both residuals making the dipole-based
beam model numerically superior with respect to the Gaussian approximation when calculating the
force densities, especially for tightly focused beams.
Our numerical experiments
showed that dipole-based beams very closely resemble ideal Gaussian beams for large waists and contain
expected diffraction artifacts (side-lobes) with smaller waists. For example, Figure~\ref{fig:Beams} shows
the result for two linearly polarized beams with different wavelengths and the same desired waist.
The computational domain is a cube with $10\,\mu\text{m}$ sides.
The dipole array is situated $2\,\mu\text{m}$ outside the domain and represents a
$15\lambda\times15\lambda$ square aperture uniformly filled with $32$ tuned point dipoles in each direction.
It is important to realize that the dipole model is not always able to achieve the waist size
of the original Gaussian beam it mimics (see the contour lines in Figure~\ref{fig:Beams}). This is the price one
pays for a better conservation of momentum.
\begin{figure}[t]
\hspace*{-0.3cm}
\includegraphics[width=8.5cm]{beams}
\caption{Intensity profiles of two focused beams with the same desired waists but different wavelengths
modeled via the superposition of dipoles tuned to mimic ideal Gaussian beams
(top: $\lambda_{0}=1.2\,\mu\text{m}$, bottom: $\lambda_{0}=3.2\,\mu\text{m}$).
Actual waists are shown as contour lines.
The larger wavelength beam (bottom row) corresponds to a smaller relative desired waste (i.e. larger
beam parameter $\vert s\vert$) and deviates from the Gaussian beam while conserving the momentum.
Both beams have the same power, and the drop in the intensity at $\lambda_{0}=3.2\,\mu\text{m}$
is due to the increased absorption in water.}
\label{fig:Beams}
\end{figure}
The power emitted by the dipole array into the liquid is determined and adjusted by computing the
integral of the normal component of the Poynting vector over the (virtual) interface between the computational
domain and the array. Also, in this paper we neglect the reflection of light at the interface between the
vessel and the surrounding medium, considering the whole medium to be optically homogeneous, although the
liquid is confined to a finite spatial domain. The embarrassingly parallel nature of the superposition principle
allowed us to compute the electromagnetic field and the corresponding force density very efficiently by exploiting
the computational power of the Graphics Processing Unit (GPU).
\section{Light-Driven Incompressible Fluid}
The Navier-Stokes equations for an incompressible Newtonian fluid are (see e.g. \cite{Batchelor2000})
\begin{align}
\label{eq:DivLaw}
&\nabla\cdot\mathbf v=0,
\\
\label{eq:NavierStokes}
&\rho_{0}\partial_{t}\mathbf v +\rho_{0}(\mathbf v \cdot\nabla) \mathbf v-\mu\nabla^{2} \mathbf v+\nabla p = \mathbf f,
\end{align}
where $\mathbf v(\mathbf x,t)$ is the local velocity of fluid, $p(\mathbf x,t)$ is the pressure, $\rho_{0}$ is the mass density,
$\mu$ is the viscosity and $\mathbf f(\mathbf x,t)$ is the applied volumetric body force density.
Since light is partially absorbed by the liquid, we need to consider the following heat equation:
\begin{align}
\label{eq:HeatLaw}
&\rho_{0} c_{\rm p}\partial_{t}T+\rho_{0} c_{\rm p}(\mathbf v\cdot\nabla)T
-k\nabla^{2}T=q,
\end{align}
where $T(\mathbf x,t)$ is the temperature, $c_{\rm p}$ is the specific heat capacity at constant pressure,
and $q(\mathbf x,t)$ is the heat source density.
To model the motion of fluid due to the heat-driven expansion we
use the Boussinesq approximation, which amounts to splitting the body force into
two parts:
\begin{align}
\label{eq:Boussinesq}
\mathbf f=\langle \mathbf f^{\rm L}\rangle+\rho \mathbf g,
\end{align}
where $\mathbf g$ is the acceleration due to gravity, and $\langle \mathbf f^{\rm L}\rangle$ is the
time-averaged Lorentz force density (\ref{eq:ForceFrequencyDomain}).
The modified mass density $\rho(\mathbf x,t)$ is described by
\begin{align}
\label{eq:DensityLaw}
\rho=\rho_{0}\left[1-\beta(T-T_{0})\right],
\end{align}
where $T_{0}$ is the reference temperature (before the heat source $q$ is applied), and $\beta$
is the thermal expansion coefficient of the fluid.
In the present paper we consider the stationary flows only, so that all quantities above are
considered to be time-independent. This also simplifies the equations (\ref{eq:NavierStokes})
and (\ref{eq:HeatLaw}). We impose the no-slip boundary condition $\mathbf v=0$ at the
walls of the vessel.
The numerical solution of the above coupled system of non-linear equations was implemented
in the open-source software environment OpenFOAM \cite{OpenFOAM}, which is based on the finite-volume
discretization scheme and features a rich set of robust algorithms.
In particular, we employed the iterative SIMPLE (Semi-Implicit Method for Pressure-Linked Equations) algorithm
\cite{Jasak1996,Patankar1980} that converged reasonably fast to the expected tolerance in all our numerical experiments.
We have also tested the convergence of the numerical solution (to a stable result) as a function of the
discretization step and observed the expected second-order behavior.
\section{Analysis of Flows}
\begin{figure}[t]
\includegraphics[width=8cm]{dispersion}
\caption{Real and imaginary parts of the complex relative permittivity of water in the neighborhood of
the absorption peak at $3\,\mu\text{m}$. Circles indicate different wavelengths ($1.2\,\mu\text{m}$,
$2.4\,\mu\text{m}$, $2.8\,\mu\text{m}$, and $3.2\,\mu\text{m}$)
considered in our numerical experiments.}
\label{fig:Dispersion}
\end{figure}
\begin{figure}[t]
\hspace*{-0.5cm}
\includegraphics[width=4cm]{streams1_2mum}
\includegraphics[width=8cm]{quiver1_2mum}
\caption{Induced flow at $1.2\,\mu\text{m}$ (low optical losses). Top: typical
stream lines (beam propagates vertically upwards through the middle of the domain).
Bottom: quiver plot in the plane orthogonal to the beam axis showing the four characteristic quarters.}
\label{fig:Streams1_2mum}
\end{figure}
\begin{figure*}[t]
\hspace*{-0.5cm}
\includegraphics[width=4.5cm]{stream02_3_2mum}
\includegraphics[width=4.5cm]{stream06_3_2mum}
\includegraphics[width=4.5cm]{stream07_3_2mum}
\includegraphics[width=4.5cm]{stream03_3_2mum}
\hspace*{-0.5cm}
\includegraphics[width=4.5cm]{stream04_3_2mum}
\includegraphics[width=4.5cm]{stream01_3_2mum}
\includegraphics[width=4.5cm]{stream05_3_2mum}
\includegraphics[width=4.5cm]{stream08_3_2mum}
\caption{Stream lines of the induced flow at $3.2\,\mu\text{m}$ (high optical losses). Each plot shows a single stream line seeded at a particular location and propagated for the same large number of steps with the Runge-Kutta algorithm. In each quarter only two stream lines are simple closed loops (both shown). Other stream lines have a more complex shape (the typical ones are shown). The flow patterns in the other three quarters of the domain are similar.
}
\label{fig:Streams3_2mum}
\end{figure*}
In this section we describe the flows that would be induced in pure water at the reference temperature $T_{0}=300\,\text{K}$
if the Lorentz force model was indeed true. We have computed the induced flows at several wavelengths
in the neighborhood of the pronounced absorption peak at $3\, \mu\text{m}$ shown in Figure~\ref{fig:Dispersion},
\cite{ref:Hale1973}. The wavelengths $1.2\,\mu\text{m}$, $2.4\,\mu\text{m}$, $2.8\,\mu\text{m}$, and $3.2\,\mu\text{m}$
indicated in Figure~\ref{fig:Dispersion} cover the regions of transparency, as well as normal and anomalous
dispersion allowing to distinguish between the effects of the different terms in the
Lorentz force density (\ref{eq:ForceFrequencyDomain}) and the heat source (\ref{eq:HeatEM}).
The most interesting results were obtained at $1.2\,\mu\text{m}$ where
the water is almost lossless and the body force (\ref{eq:ForceFrequencyDomain}) is
dominated by the first term while the heat source (\ref{eq:HeatEM}) is almost zero, and
at $3.2\,\mu\text{m}$ where, although the absorption is significant, it does not extinct the beam too soon and
lets it propagate some length.
The container and the computational domain of our simulations is a
cube with $10\,\mu\text{m}$ sides. The Gaussian beam (re-modeled by the dipole array) propagates along one of the Cartesian axes coinciding with the edge of the domain
and is linearly polarized along one of the other edges. We have simulated a whole range of incident powers with the
results presented below corresponding to $1\,\text{mW}$.
In all our experiments the spatial pattern of the flow appears to be divided
into four quarters by the two planes intersecting at the beam axis.
One of the planes is the polarization plane of the beam and the other plane is orthogonal to it.
Despite the presence of gravity, which in our case acts along the
polarization direction, i.e., orthogonal to the beam axis, the flow patterns are largely symmetric about
the mentioned two planes.
Due to the assumed incompressibility of water the resulting stationary flows are divergence-free.
Since the water container is closed and there are no sources or sinks the stream lines should be closed.
This is exactly what we observe in Figure~\ref{fig:Streams1_2mum} where the stream lines and the
quiver plot of the induced flow at $1.2\,\mu\text{m}$ are shown. The axis of the beam is vertical and
its direction of propagation is upwards in this and subsequent 3D plots.
In each of the aforementioned quarters the stream lines represent simple closed
concentric loops centered around a line parallel to the beam axis. The direction of flow along each such loop is
always away from the beam axis along the polarization direction and coming back towards the beam along the
orthogonal direction, see Figure~\ref{fig:Streams1_2mum} (bottom).
The velocity along the loops is highest at the point closest to the beam axis reaching $\sim 87\,\mu\text{m/s}$.
The flow pattern changes when we consider the wavelengths where the optical absorption in water
becomes significant introducing the second term in the body force (\ref{eq:ForceFrequencyDomain}).
For example, in Figure~\ref{fig:Streams3_2mum} we show some of the stream lines of the flow induced
at $3.2\,\mu\text{m}$. There are just two simple closed loops in each quarter in this case
(both are shown). The rest of the stream lines have a more complicated shape featuring a large number
of windings. It is also apparent that the stream lines are no longer orthogonal to the beam axis.
The direction of flow is determined by both the polarization and the propagation direction of the beam
in this case. Although the heat term (\ref{eq:HeatEM}) is no longer zero and the direction of gravity breaks
the symmetry of the problem, the overall flow pattern remains almost symmetric about the
same two orthogonal planes as in the lossless case.
Our simulations show that the flows induced by the beams with higher powers have the same spatial pattern
as low-power flows and the computed velocity scales linearly with the body force up to $50\,\text{mW}$.
This is a clear indication of the low Reynolds number regime, meaning that in the future studies a linear Stokes
approximation can be used to simplify the computational model.
\section{Conclusions}
We have demonstrated that the induced Lorentz force model
(\ref{eq:ForceLorentz})--(\ref{eq:ForceFrequencyDomain}) results in significant hydrodynamic flows
in the neighborhood of a single-beam optical trap in a liquid environment.
The existence of such flows would suggest the form of the electromagnetic momentum
conservation law (\ref{eq:MomentumConservationGeneral}) different from the ones due to
Abraham and Minkowski, which after time averaging both feature the Helmholtz force density
that equals zero in a homogeneous liquid.
Of course, the fact that such flows have not been directly observed so far (except, perhaps, in
\cite{Ryu2010,Schermer2011,Khan2011}) could simply indicate their absence.
These induced flows, however, are almost invisible by their nature and may be difficult to detect
with other means.
Normally a flow pattern can be visualized using small tracer particles, such as ink. This is not straightforward
in the present case, since both the flow and the tracer particle will be influenced by the laser beam.
The equation of motion for a small spherical particle with mass $m$ and position $\mathbf x(t)$ is
\begin{align}
\label{eq:NewtonLaw}
m\frac{d^{2}\mathbf x}{dt^{2}}=\mathbf F_{\rm em} + \mathbf F_{\rm drag},
\end{align}
where $\mathbf F_{\rm em}$ and $\mathbf F_{\rm drag}$ are the electromagnetic force and
the fluid drag force acting on the particle.
Considering a sufficiently small tracer particle without optical losses the electromagnetic force can be
approximated by the gradient force:
\begin{align}
\label{eq:GradientForce}
\mathbf F_{\rm em}\approx\mathbf F_{\rm grad}=\pi r^{3} {\rm Re}(\varepsilon_{\rm w})
\frac{\varepsilon_{\rm p}/\varepsilon_{0}-1}{\varepsilon_{\rm p}/\varepsilon_{0}+2}\nabla \vert \mathbf E\vert^{2},
\end{align}
where $\varepsilon_{\rm w}$ and $\varepsilon_{\rm p}$ are the permittivities of the water and particle respectively.
The Stokes drag is given by $\mathbf F_{\rm drag}=6\pi\mu_{\rm w} r \mathbf u$,
where $\mu_{\rm w}$ is the dynamic viscosity of water, $r$ is the particle radius, and $\mathbf u$ is the particle velocity
with respect to the stationary fluid. In the present case, however, the fluid is already in motion due to the
induced flow. Let the velocity of this flow with respect to the stationary reference frame be $\mathbf v$. Then,
the drag force is $\mathbf F_{\rm drag}=6\pi\mu_{\rm w} r (\mathbf v- d\mathbf x/dt)$, and the equation of motion becomes
\begin{align}
\label{eq:EquationOfMotion}
\begin{split}
m\frac{d^{2}\mathbf x}{dt^{2}}&+6\pi\mu_{\rm w} r\frac{d\mathbf x}{dt}=
\\
&\pi r^{3} {\rm Re}(\varepsilon_{\rm w})
\frac{\varepsilon_{\rm p}/\varepsilon_{0}-1}{\varepsilon_{\rm p}/\varepsilon_{0}+2}\nabla \vert \mathbf E\vert^{2}
+ 6\pi\mu_{\rm w} r\mathbf v.
\end{split}
\end{align}
Paradoxically, if the tracer velocity $d\mathbf x/dt$ coincides with the velocity $\mathbf v$ of the fluid flow, i.e., the tracer serves its purpose
and moves with the flow, then the drag force disappears, the tracer will be pulled by the optical gradient force and
thus no longer follow the flow. On the other hand, if the gradient force pulls the
particle in the direction opposing the flow, the friction term will slow it down and the drag force will
reduce the trapping efficiency.
Fortunately, the difference in the dependence on the particle radius ($r^{3}$ versus $r$)
means that sufficiently small tracers will mostly follow the induced flow, since the gradient optical force
acting on them will be much smaller than the drag force. However, such tracers
will have to be much smaller than the wavelength making it difficult to actually {\it see} their motion.
Finally, the equation of motion (\ref{eq:EquationOfMotion}) does not include the stochastic Brownian term which
will make tracers jump from one streamline to another, further complicating the detection of the simulated
flow patterns. Hence, a detailed analysis of the particle dynamics and a robust experimental technique
are needed before the induced flows predicted in this paper can be confirmed or truly ruled out.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 9,471
|
Kórsun-Shevtchênkivski (; ) é uma pequena cidade localizada no Oblast de Tcherkássi, no centro da Ucrânia. A cidade fica às margens do rio Ros, e é o centro administrativo do Raion Kórsun-Shevtchênkivski.
História
O nome da cidade vem da cidade grega de Quersoneso (; ; traduzida como Kórsun) na península da Criméia. Uma fortaleza foi fundada em Kórsun em 1032, pelo príncipe Yaroslav, o Sábio, do Rus' de Kiev, e serviu de proteção a Kiev de nômades das estepes do sul. Em 1240, Kórsun foi destruída por Batu Khan, e em 1584 uma base militar foi estabelecida na cidade.
Nos primeiros tempos modernos, o lugar pertencia à Comunidade Polaco-Lituana, durante a qual outra fortaleza foi construída e a cidade recebeu os direitos de Magdeburgo (seu reconhecimento como cidade autônoma). Em 1630, rebeldes cossacos liderados por Taras Fedorovitch atacaram a cidade e destruíram sua guarnição polonesa. A cidade foi arrasada pelas forças polonesas durante a rebelião cossaca de 1637, liderada por Pavlo Pavliuk. Em 1648, ela foi palco da Batalha de Korsuń, durante a Revolta Khmelnytsky. Em 1768, durante a rebelião Koliivschina, a guarnição polonesa foi destruída pelas forças de Maksim Zalizniak.
Em 1793, Kórsun foi anexada ao Império Russo. Em 1903, uma das maiores fábricas de tintas de todo o Império Russo foi construída em Kórsun. No período da Segunda Guerra Mundial (1941-1945), o Exército Vermelho soviético derrotou as forças alemãs na área em torno de Kórsun, durante o episódio conhecido como Bolsão de Kórsun. Em 14 de fevereiro de 1944, Kórsun foi liberada das forças alemãs.
A agricultura e a economia agrícola de Kórsun foram logo reconstruídas nos anos do pós-guerra. Até 1944 a cidade era conhecida simplesmente como Kórsun, mas mais tarde ao seu nome original foi adicionada uma homenagem a Taras Shevchenko, um famoso poeta e artista ucraniano.
Economia
Kórsun-Shevtchênkivski possui a estação ferroviária Kórsun, na linha férrea Kiev-Zvitkovo. ALém da atividade agrícola, Kórsun-Shevtchênkivski também possui muitas industrias, incluindo fábricas mecânicas, uma fábrica de material de construção, uma fábrica de asfalto, uma indústria vinícola, e fábricas de costura.
Cultura
Um complexo de parques pertencente ao antigo palácio da família nobre Lopukhinikh-Demidovikh, considerado um dos melhores complexos de parques naturais no estilo do romantismo na Ucrânia. O parque foi construído em 1782 a pedido do nobre Stanisław Poniatowski, mais tarde rei da Polónia e do Grão-Duque da Lituânia. Em meados do século XIX, o parque foi decorado com muitas esculturas. Além disso, pequenas pontes foram adicionadas a ele. Área total do parque é de 97 hectares.
O palácio da nobre família Lopukhinikh-Demidovikh.
A reserva histórica-cultural de Kórsun-Shevtchênkivski.
O museu dedicado à história da Batalha de Kórsun-Shevtchênkivski.
Cidades gêmeas e irmãs
Chojnice, Polônia
Gifhorn, Alemanha
Ligações externas
korsun.ic.ck.ua - Site da reserva histórico-cultural da cidade
heraldry.com.ua - Brasão de armas de Kórsun-Shevtchênkivski (ucraniano)
sunsite.berkeley.edu - Mapa topográfico soviético em escala 1:100.000
Cidades do oblast de Tcherkássi
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 291
|
Mia Negovetić (Rijeka, 5. listopada 2002.) mlada je hrvatska pjevačica iz Rijeke poznata po pobjedi u dječjem pjevačkom talent showu Zvjezdice koji se prikazivao na RTL.
Popularnost diljem svijeta stekla je objavljivanjem videa na kojem pjeva pjesmu "Listen" koju originalno izvodi Beyoncé, ali i sudjelovanjem u američkom televizijskom showu Little Big Shots čiji su autori Steve Harvey i Ellen DeGeneres. Nakon nastupa u poznatom američkom showu, u medijima su se pojavili brojni komentari u kojima se od Beyoncé tražilo da sklopi suradnju s Miom. Na službenoj Facebook stranici američkog pjevačkog showa The Voice pojavila se snimka uz komentar "Beyoncé, čuvaj se!"
Jedan od najznačajnijih Mijinih nastupa bio je u kolovozu 2015.godine povodom obilježavanja 20. godišnjice Oluje kada je izvela hrvatsku himnu "Lijepa naša domovino". Mia je na pjevanju himne na mimohodu nosila majicu s vukovarskom golubicom, a na nogama startasice koje je vukovarsko Borovo dizajniralo posebno za ovu prigodu pa startasice na mašni imaju simbole pletera. Živi u Rijeci sa svojim roditeljima, Igorom i Kristinom te sestrom Lanom. Osim navedenih obradila je pjesme "All by Myself" (Celine Dion), "Kao rijeka" (duet s Vannom), "Caruso" (duet s Tonijem Cetinskim), "Suus" (Rona Nishliu).
Nakon uspjeha s obradama, došao je red i za autorske stvari.
Krajem 2019. godine pojavila se informacija kako je Mia dogovorila suradnju s poznatim švedskim autorima Linneom Deb, Anderzom Wrethovom i Dennizom Jamom što je potvrđeno u jednom Instagram postu koji je pjevačica sama objavila.
Početkom 2020. godine Mia je nastupila na Zagrebačkom festivalu s pjesmom Na pola puta do svemira koju je sama napisala, dok je glazbu i aranžman radio poznati riječki producent Olja Dešić.
Također je potvrđena kao jedna od natjecatelja na Dori 2020. s pjesmom When it comes to you te se potvrđuje da su švedski autori pomogli kod pisanja ove pjesme.
Na samom natjecanju, Mia potvrđuje status favorita te na kraju osvaja sjajno 2. mjesto, s tim da joj je pobjeda umakla za svega 300 glasova publike.
Tijekom 2020. godine, Mia je izdala novi singl Pusti koji je naišao na uspjeh.
15. prosinca objavljeno je da će nastupiti na festivalu Dora 2021. godine s pjesmom She's like a dream. Pjesmu su ponovno napisali švedski autori Linnea Deb, Denniz Jamm i Denise Kertes koji su zaslužni i za prošlogodišnji veliki uspjeh na istom festivalu.
Diskografija
Singlovi
Nagrade i priznanja
2021. Porin – Nominirana za novog izvođača godine
Izvori
Vanjske poveznice
Novi list
Životopisi, Rijeka
Hrvatski pjevači
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 5,272
|
Q: VueJS Tailwind Contain div within itself without impacting others So, I am going ABSOLUTELY crazy here. I have attached a pic showing the issue. It is as follows:
*
*The layout on the right is a user list
*The middle content for chat should be y-scroll
*The bottom div should be locked in place and not move.
For some reason, the way it works now the scroll is all over the place and I feel like I should be doing this a lot better. It just feels "off" to me. I am normally a backend guy and I am struggling with this badly and I don't know how best to resolve this.
I have attached my code for assistance please and thank you!!
<div class="w-full">
<div class="sm:hidden">
<label for="tabs" class="sr-only">Select a tab</label>
<select id="tabs" name="tabs" class="block w-full focus:ring-indigo-500 focus:border-indigo-500 border-gray-300 dark:border-gray-600">
<option>Room 1</option>
<option>Room 2</option>
<option>Room 3</option>
<option>Room 4</option>
</select>
</div>
<div class="hidden sm:block sticky top-0 z-10">
<div class="border-b border-gray-200 dark:border-gray-600">
<nav class="-mb-px flex bg-white dark:bg-gray-900" aria-label="Tabs"><a href="#" class="bg-indigo-800 text-gray-50 w-1/4 py-4 px-1 text-center border-b-2 dark:text-gray-50 dark:border-gray-600 font-medium text-sm" aria-current="page">Room 1</a><a href="#" class="text-gray-500 hover:text-gray-700 hover:border-indigo-300 w-1/4 py-4 px-1 text-center border-b-2 dark:text-gray-50 dark:border-gray-600 font-medium text-sm">Room 2</a><a href="#" class="text-gray-500 hover:text-gray-700 hover:border-indigo-300 w-1/4 py-4 px-1 text-center border-b-2 dark:text-gray-50 dark:border-gray-600 font-medium text-sm">Room 3</a><a href="#" class="text-gray-500 hover:text-gray-700 hover:border-indigo-300 w-1/4 py-4 px-1 text-center border-b-2 dark:text-gray-50 dark:border-gray-600 font-medium text-sm">Room 4</a></nav>
</div>
</div>
<section class="h-full w-full flex-1 flex scrollbar-thin hover:scrollbar-thumb-indigo-700 dark:hover:scrollbar-thumb-indigo-700 dark:scrollbar-thumb-gray-600 dark:scrollbar-track-gray-800 scrollbar-thumb-gray-400 scrollbar-track-gray-50 overflow-y-scroll">
<ul class="divide-y dark:divide-gray-700 divide-dotted divide-gray-200 w-full">
<li class="bg-gray-300 py-2 px-3">
<div class="flex space-x-3">
<img class="h-6 w-6 rounded-full" src="https://images.unsplash.com/photo-1480455624313-e29b44bbfde1?ixid=MnwxMjA3fDB8MHxzZWFyY2h8Nnx8bWFsZXxlbnwwfHwwfHw%3D&ixlib=rb-1.2.1&auto=format&fit=crop&w=900&q=60" alt="">
<div class="flex-1 space-y-1">
<div class="flex items-center justify-between">
<h3 class="text-sm font-medium">Username</h3>
<p class="text-sm text-gray-500 dark:text-gray-200">1h</p>
</div>
<div class="flex items-center justify-between">
<p class="text-sm text-gray-500">test</p>
<p class="text-sm bg-blue-500 p-1 rounded text-gray-50">public</p>
</div>
</div>
</div>
</li>
<li class="bg-gray-300 py-2 px-3">
<div class="flex space-x-3">
<img class="h-6 w-6 rounded-full" src="https://images.unsplash.com/photo-1480455624313-e29b44bbfde1?ixid=MnwxMjA3fDB8MHxzZWFyY2h8Nnx8bWFsZXxlbnwwfHwwfHw%3D&ixlib=rb-1.2.1&auto=format&fit=crop&w=900&q=60" alt="">
<div class="flex-1 space-y-1">
<div class="flex items-center justify-between">
<h3 class="text-sm font-medium">Username</h3>
<p class="text-sm text-gray-500 dark:text-gray-200">1h</p>
</div>
<div class="flex items-center justify-between">
<p class="text-sm text-gray-500">test</p>
<p class="text-sm bg-blue-500 p-1 rounded text-gray-50">public</p>
</div>
</div>
</div>
</li>
<li class="bg-gray-300 py-2 px-3">
<div class="flex space-x-3">
<img class="h-6 w-6 rounded-full" src="https://images.unsplash.com/photo-1480455624313-e29b44bbfde1?ixid=MnwxMjA3fDB8MHxzZWFyY2h8Nnx8bWFsZXxlbnwwfHwwfHw%3D&ixlib=rb-1.2.1&auto=format&fit=crop&w=900&q=60" alt="">
<div class="flex-1 space-y-1">
<div class="flex items-center justify-between">
<h3 class="text-sm font-medium">Username</h3>
<p class="text-sm text-gray-500 dark:text-gray-200">1h</p>
</div>
<div class="flex items-center justify-between">
<p class="text-sm text-gray-500">test</p>
<p class="text-sm bg-blue-500 p-1 rounded text-gray-50">public</p>
</div>
</div>
</div>
</li>
<li class="bg-gray-300 py-2 px-3">
<div class="flex space-x-3">
<img class="h-6 w-6 rounded-full" src="https://images.unsplash.com/photo-1480455624313-e29b44bbfde1?ixid=MnwxMjA3fDB8MHxzZWFyY2h8Nnx8bWFsZXxlbnwwfHwwfHw%3D&ixlib=rb-1.2.1&auto=format&fit=crop&w=900&q=60" alt="">
<div class="flex-1 space-y-1">
<div class="flex items-center justify-between">
<h3 class="text-sm font-medium">Username</h3>
<p class="text-sm text-gray-500 dark:text-gray-200">1h</p>
</div>
<div class="flex items-center justify-between">
<p class="text-sm text-gray-500">test</p>
<p class="text-sm bg-blue-500 p-1 rounded text-gray-50">public</p>
</div>
</div>
</div>
</li>
<li class="bg-gray-300 py-2 px-3">
<div class="flex space-x-3">
<img class="h-6 w-6 rounded-full" src="https://images.unsplash.com/photo-1480455624313-e29b44bbfde1?ixid=MnwxMjA3fDB8MHxzZWFyY2h8Nnx8bWFsZXxlbnwwfHwwfHw%3D&ixlib=rb-1.2.1&auto=format&fit=crop&w=900&q=60" alt="">
<div class="flex-1 space-y-1">
<div class="flex items-center justify-between">
<h3 class="text-sm font-medium">Username</h3>
<p class="text-sm text-gray-500 dark:text-gray-200">1h</p>
</div>
<div class="flex items-center justify-between">
<p class="text-sm text-gray-500">test</p>
<p class="text-sm bg-blue-500 p-1 rounded text-gray-50">public</p>
</div>
</div>
</div>
</li>
<li class="bg-gray-300 py-2 px-3">
<div class="flex space-x-3">
<img class="h-6 w-6 rounded-full" src="https://images.unsplash.com/photo-1480455624313-e29b44bbfde1?ixid=MnwxMjA3fDB8MHxzZWFyY2h8Nnx8bWFsZXxlbnwwfHwwfHw%3D&ixlib=rb-1.2.1&auto=format&fit=crop&w=900&q=60" alt="">
<div class="flex-1 space-y-1">
<div class="flex items-center justify-between">
<h3 class="text-sm font-medium">Username</h3>
<p class="text-sm text-gray-500 dark:text-gray-200">1h</p>
</div>
<div class="flex items-center justify-between">
<p class="text-sm text-gray-500">test</p>
<p class="text-sm bg-blue-500 p-1 rounded text-gray-50">public</p>
</div>
</div>
</div>
</li>
<li class="bg-gray-300 py-2 px-3">
<div class="flex space-x-3">
<img class="h-6 w-6 rounded-full" src="https://images.unsplash.com/photo-1480455624313-e29b44bbfde1?ixid=MnwxMjA3fDB8MHxzZWFyY2h8Nnx8bWFsZXxlbnwwfHwwfHw%3D&ixlib=rb-1.2.1&auto=format&fit=crop&w=900&q=60" alt="">
<div class="flex-1 space-y-1">
<div class="flex items-center justify-between">
<h3 class="text-sm font-medium">Username</h3>
<p class="text-sm text-gray-500 dark:text-gray-200">1h</p>
</div>
<div class="flex items-center justify-between">
<p class="text-sm text-gray-500">test</p>
<p class="text-sm bg-blue-500 p-1 rounded text-gray-50">public</p>
</div>
</div>
</div>
</li>
<li class="bg-gray-300 py-2 px-3">
<div class="flex space-x-3">
<img class="h-6 w-6 rounded-full" src="https://images.unsplash.com/photo-1480455624313-e29b44bbfde1?ixid=MnwxMjA3fDB8MHxzZWFyY2h8Nnx8bWFsZXxlbnwwfHwwfHw%3D&ixlib=rb-1.2.1&auto=format&fit=crop&w=900&q=60" alt="">
<div class="flex-1 space-y-1">
<div class="flex items-center justify-between">
<h3 class="text-sm font-medium">Username</h3>
<p class="text-sm text-gray-500 dark:text-gray-200">1h</p>
</div>
<div class="flex items-center justify-between">
<p class="text-sm text-gray-500">test</p>
<p class="text-sm bg-blue-500 p-1 rounded text-gray-50">public</p>
</div>
</div>
</div>
</li>
<li class="bg-gray-300 py-2 px-3">
<div class="flex space-x-3">
<img class="h-6 w-6 rounded-full" src="https://images.unsplash.com/photo-1480455624313-e29b44bbfde1?ixid=MnwxMjA3fDB8MHxzZWFyY2h8Nnx8bWFsZXxlbnwwfHwwfHw%3D&ixlib=rb-1.2.1&auto=format&fit=crop&w=900&q=60" alt="">
<div class="flex-1 space-y-1">
<div class="flex items-center justify-between">
<h3 class="text-sm font-medium">Username</h3>
<p class="text-sm text-gray-500 dark:text-gray-200">1h</p>
</div>
<div class="flex items-center justify-between">
<p class="text-sm text-gray-500">test</p>
<p class="text-sm bg-blue-500 p-1 rounded text-gray-50">public</p>
</div>
</div>
</div>
</li>
<li class="bg-gray-300 py-2 px-3">
<div class="flex space-x-3">
<img class="h-6 w-6 rounded-full" src="https://images.unsplash.com/photo-1480455624313-e29b44bbfde1?ixid=MnwxMjA3fDB8MHxzZWFyY2h8Nnx8bWFsZXxlbnwwfHwwfHw%3D&ixlib=rb-1.2.1&auto=format&fit=crop&w=900&q=60" alt="">
<div class="flex-1 space-y-1">
<div class="flex items-center justify-between">
<h3 class="text-sm font-medium">Username</h3>
<p class="text-sm text-gray-500 dark:text-gray-200">1h</p>
</div>
<div class="flex items-center justify-between">
<p class="text-sm text-gray-500">test</p>
<p class="text-sm bg-blue-500 p-1 rounded text-gray-50">public</p>
</div>
</div>
</div>
</li>
<li class="bg-gray-300 py-2 px-3">
<div class="flex space-x-3">
<img class="h-6 w-6 rounded-full" src="https://images.unsplash.com/photo-1480455624313-e29b44bbfde1?ixid=MnwxMjA3fDB8MHxzZWFyY2h8Nnx8bWFsZXxlbnwwfHwwfHw%3D&ixlib=rb-1.2.1&auto=format&fit=crop&w=900&q=60" alt="">
<div class="flex-1 space-y-1">
<div class="flex items-center justify-between">
<h3 class="text-sm font-medium">Username</h3>
<p class="text-sm text-gray-500 dark:text-gray-200">1h</p>
</div>
<div class="flex items-center justify-between">
<p class="text-sm text-gray-500">test</p>
<p class="text-sm bg-blue-500 p-1 rounded text-gray-50">public</p>
</div>
</div>
</div>
</li>
<li class="bg-gray-300 py-2 px-3">
<div class="flex space-x-3">
<img class="h-6 w-6 rounded-full" src="https://images.unsplash.com/photo-1480455624313-e29b44bbfde1?ixid=MnwxMjA3fDB8MHxzZWFyY2h8Nnx8bWFsZXxlbnwwfHwwfHw%3D&ixlib=rb-1.2.1&auto=format&fit=crop&w=900&q=60" alt="">
<div class="flex-1 space-y-1">
<div class="flex items-center justify-between">
<h3 class="text-sm font-medium">Username</h3>
<p class="text-sm text-gray-500 dark:text-gray-200">1h</p>
</div>
<div class="flex items-center justify-between">
<p class="text-sm text-gray-500">test</p>
<p class="text-sm bg-blue-500 p-1 rounded text-gray-50">public</p>
</div>
</div>
</div>
</li>
<li class="bg-gray-300 py-2 px-3">
<div class="flex space-x-3">
<img class="h-6 w-6 rounded-full" src="https://images.unsplash.com/photo-1480455624313-e29b44bbfde1?ixid=MnwxMjA3fDB8MHxzZWFyY2h8Nnx8bWFsZXxlbnwwfHwwfHw%3D&ixlib=rb-1.2.1&auto=format&fit=crop&w=900&q=60" alt="">
<div class="flex-1 space-y-1">
<div class="flex items-center justify-between">
<h3 class="text-sm font-medium">Username</h3>
<p class="text-sm text-gray-500 dark:text-gray-200">1h</p>
</div>
<div class="flex items-center justify-between">
<p class="text-sm text-gray-500">test</p>
<p class="text-sm bg-blue-500 p-1 rounded text-gray-50">public</p>
</div>
</div>
</div>
</li>
<li class="bg-gray-300 py-2 px-3">
<div class="flex space-x-3">
<img class="h-6 w-6 rounded-full" src="https://images.unsplash.com/photo-1480455624313-e29b44bbfde1?ixid=MnwxMjA3fDB8MHxzZWFyY2h8Nnx8bWFsZXxlbnwwfHwwfHw%3D&ixlib=rb-1.2.1&auto=format&fit=crop&w=900&q=60" alt="">
<div class="flex-1 space-y-1">
<div class="flex items-center justify-between">
<h3 class="text-sm font-medium">Username</h3>
<p class="text-sm text-gray-500 dark:text-gray-200">1h</p>
</div>
<div class="flex items-center justify-between">
<p class="text-sm text-gray-500">test</p>
<p class="text-sm bg-blue-500 p-1 rounded text-gray-50">public</p>
</div>
</div>
</div>
</li>
<li class="bg-gray-300 py-2 px-3">
<div class="flex space-x-3">
<img class="h-6 w-6 rounded-full" src="https://images.unsplash.com/photo-1480455624313-e29b44bbfde1?ixid=MnwxMjA3fDB8MHxzZWFyY2h8Nnx8bWFsZXxlbnwwfHwwfHw%3D&ixlib=rb-1.2.1&auto=format&fit=crop&w=900&q=60" alt="">
<div class="flex-1 space-y-1">
<div class="flex items-center justify-between">
<h3 class="text-sm font-medium">Username</h3>
<p class="text-sm text-gray-500 dark:text-gray-200">1h</p>
</div>
<div class="flex items-center justify-between">
<p class="text-sm text-gray-500">test</p>
<p class="text-sm bg-blue-500 p-1 rounded text-gray-50">public</p>
</div>
</div>
</div>
</li>
<li class="bg-gray-300 py-2 px-3">
<div class="flex space-x-3">
<img class="h-6 w-6 rounded-full" src="https://images.unsplash.com/photo-1480455624313-e29b44bbfde1?ixid=MnwxMjA3fDB8MHxzZWFyY2h8Nnx8bWFsZXxlbnwwfHwwfHw%3D&ixlib=rb-1.2.1&auto=format&fit=crop&w=900&q=60" alt="">
<div class="flex-1 space-y-1">
<div class="flex items-center justify-between">
<h3 class="text-sm font-medium">Username</h3>
<p class="text-sm text-gray-500 dark:text-gray-200">1h</p>
</div>
<div class="flex items-center justify-between">
<p class="text-sm text-gray-500">test</p>
<p class="text-sm bg-blue-500 p-1 rounded text-gray-50">public</p>
</div>
</div>
</div>
</li>
<li class="bg-gray-300 py-2 px-3">
<div class="flex space-x-3">
<img class="h-6 w-6 rounded-full" src="https://images.unsplash.com/photo-1480455624313-e29b44bbfde1?ixid=MnwxMjA3fDB8MHxzZWFyY2h8Nnx8bWFsZXxlbnwwfHwwfHw%3D&ixlib=rb-1.2.1&auto=format&fit=crop&w=900&q=60" alt="">
<div class="flex-1 space-y-1">
<div class="flex items-center justify-between">
<h3 class="text-sm font-medium">Username</h3>
<p class="text-sm text-gray-500 dark:text-gray-200">1h</p>
</div>
<div class="flex items-center justify-between">
<p class="text-sm text-gray-500">test</p>
<p class="text-sm bg-blue-500 p-1 rounded text-gray-50">public</p>
</div>
</div>
</div>
</li>
<li class="bg-gray-300 py-2 px-3">
<div class="flex space-x-3">
<img class="h-6 w-6 rounded-full" src="https://images.unsplash.com/photo-1480455624313-e29b44bbfde1?ixid=MnwxMjA3fDB8MHxzZWFyY2h8Nnx8bWFsZXxlbnwwfHwwfHw%3D&ixlib=rb-1.2.1&auto=format&fit=crop&w=900&q=60" alt="">
<div class="flex-1 space-y-1">
<div class="flex items-center justify-between">
<h3 class="text-sm font-medium">Username</h3>
<p class="text-sm text-gray-500 dark:text-gray-200">1h</p>
</div>
<div class="flex items-center justify-between">
<p class="text-sm text-gray-500">test</p>
<p class="text-sm bg-blue-500 p-1 rounded text-gray-50">public</p>
</div>
</div>
</div>
</li>
<li class="bg-gray-300 py-2 px-3">
<div class="flex space-x-3">
<img class="h-6 w-6 rounded-full" src="https://images.unsplash.com/photo-1480455624313-e29b44bbfde1?ixid=MnwxMjA3fDB8MHxzZWFyY2h8Nnx8bWFsZXxlbnwwfHwwfHw%3D&ixlib=rb-1.2.1&auto=format&fit=crop&w=900&q=60" alt="">
<div class="flex-1 space-y-1">
<div class="flex items-center justify-between">
<h3 class="text-sm font-medium">Username</h3>
<p class="text-sm text-gray-500 dark:text-gray-200">1h</p>
</div>
<div class="flex items-center justify-between">
<p class="text-sm text-gray-500">test</p>
<p class="text-sm bg-blue-500 p-1 rounded text-gray-50">public</p>
</div>
</div>
</div>
</li>
<li class="bg-gray-300 py-2 px-3">
<div class="flex space-x-3">
<img class="h-6 w-6 rounded-full" src="https://images.unsplash.com/photo-1480455624313-e29b44bbfde1?ixid=MnwxMjA3fDB8MHxzZWFyY2h8Nnx8bWFsZXxlbnwwfHwwfHw%3D&ixlib=rb-1.2.1&auto=format&fit=crop&w=900&q=60" alt="">
<div class="flex-1 space-y-1">
<div class="flex items-center justify-between">
<h3 class="text-sm font-medium">Username</h3>
<p class="text-sm text-gray-500 dark:text-gray-200">1h</p>
</div>
<div class="flex items-center justify-between">
<p class="text-sm text-gray-500">test</p>
<p class="text-sm bg-blue-500 p-1 rounded text-gray-50">public</p>
</div>
</div>
</div>
</li>
<li class="bg-gray-300 py-2 px-3">
<div class="flex space-x-3">
<img class="h-6 w-6 rounded-full" src="https://images.unsplash.com/photo-1480455624313-e29b44bbfde1?ixid=MnwxMjA3fDB8MHxzZWFyY2h8Nnx8bWFsZXxlbnwwfHwwfHw%3D&ixlib=rb-1.2.1&auto=format&fit=crop&w=900&q=60" alt="">
<div class="flex-1 space-y-1">
<div class="flex items-center justify-between">
<h3 class="text-sm font-medium">Username</h3>
<p class="text-sm text-gray-500 dark:text-gray-200">1h</p>
</div>
<div class="flex items-center justify-between">
<p class="text-sm text-gray-500">test</p>
<p class="text-sm bg-blue-500 p-1 rounded text-gray-50">public</p>
</div>
</div>
</div>
</li>
</ul>
</section>
<section aria-labelledby="chat-footer" class="h-auto w-full sticky shadow-2xl bg-gradient-to-br from-gray-100 to-gray-300 dark:from-gray-900 dark:to-gray-900 border-l dark:border-gray-800 border-gray-200 bottom-0 min-w-full flex-1 flex flex-col lg:order-last">
<div>
<span class="pl-10 pt-2 relative z-0 inline-flex shadow-sm rounded-md">
<button type="button" class="relative inline-flex items-center px-3 py-1 rounded-l-md border border-gray-300 bg-white dark:border-gray-700 dark:bg-gray-900 text-sm font-medium text-gray-700 hover:bg-gray-50 focus:z-10 focus:outline-none focus:ring-1 focus:ring-indigo-500 focus:border-indigo-500">
<svg xmlns="http://www.w3.org/2000/svg" fill="none" viewBox="0 0 24 24" stroke="currentColor" class="w-4 h-4 text-gray-600 dark:text-gray-50">
<path stroke-linecap="round" stroke-linejoin="round" stroke-width="2" d="M10 20l4-16m4 4l4 4-4 4M6 16l-4-4 4-4"></path>
</svg>
</button>
<button type="button" class="-ml-px relative inline-flex items-center px-3 py-1 border border-gray-300 bg-white dark:border-gray-700 dark:bg-gray-900 text-sm font-medium text-gray-700 hover:bg-gray-50 focus:z-10 focus:outline-none focus:ring-1 focus:ring-indigo-500 focus:border-indigo-500">
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 20 20" fill="currentColor" class="w-4 h-4 text-gray-600 dark:text-gray-50">
<path fill-rule="evenodd" d="M8 4a4 4 0 100 8 4 4 0 000-8zM2 8a6 6 0 1110.89 3.476l4.817 4.817a1 1 0 01-1.414 1.414l-4.816-4.816A6 6 0 012 8z" clip-rule="evenodd"></path>
</svg>
</button>
<button type="button" class="-ml-px relative inline-flex items-center px-3 py-1 border border-gray-300 bg-white dark:border-gray-700 dark:bg-gray-900 text-sm font-medium text-gray-700 hover:bg-gray-50 focus:z-10 focus:outline-none focus:ring-1 focus:ring-indigo-500 focus:border-indigo-500">
<svg xmlns="http://www.w3.org/2000/svg" fill="none" viewBox="0 0 24 24" stroke="currentColor" class="w-4 h-4 text-gray-600 dark:text-gray-50">
<path stroke-linecap="round" stroke-linejoin="round" stroke-width="2" d="M4 16v1a3 3 0 003 3h10a3 3 0 003-3v-1m-4-8l-4-4m0 0L8 8m4-4v12"></path>
</svg>
</button>
<button type="button" class="-ml-px relative inline-flex items-center px-3 py-1 border border-gray-300 bg-white dark:border-gray-700 dark:bg-gray-900 text-sm font-medium text-gray-700 hover:bg-gray-50 focus:z-10 focus:outline-none focus:ring-1 focus:ring-indigo-500 focus:border-indigo-500">
<svg xmlns="http://www.w3.org/2000/svg" fill="none" viewBox="0 0 24 24" stroke="currentColor" class="w-4 h-4 text-gray-600 dark:text-gray-50">
<path stroke-linecap="round" stroke-linejoin="round" stroke-width="2" d="M8 11V7a4 4 0 118 0m-4 8v2m-6 4h12a2 2 0 002-2v-6a2 2 0 00-2-2H6a2 2 0 00-2 2v6a2 2 0 002 2z"></path>
</svg>
</button>
<div class="relative inline-block text-left">
<div>
<button id="headlessui-menu-button-1" type="button" aria-haspopup="true" class="-ml-px relative inline-flex items-center px-3 py-2 rounded-r-md border border-gray-300 bg-white dark:bg-gray-900 dark:border-gray-700 dark:hover:bg-gray-600 text-sm font-medium text-gray-700 dark:text-gray-50 hover:bg-gray-50 focus:z-10 focus:outline-none focus:ring-1 focus:ring-indigo-500 focus:border-indigo-500">
Send Option
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 20 20" fill="currentColor" class="-mr-1 ml-2 h-5 w-5">
<path fill-rule="evenodd" d="M5.293 7.293a1 1 0 011.414 0L10 10.586l3.293-3.293a1 1 0 111.414 1.414l-4 4a1 1 0 01-1.414 0l-4-4a1 1 0 010-1.414z" clip-rule="evenodd"></path>
</svg>
</button>
</div>
<!---->
</div>
</span>
<div class="pr-10 pl-10 pb-5 h-full mt-1 flex rounded-md shadow-sm">
<div class="relative flex items-stretch flex-grow focus-within:z-10">
<div class="absolute inset-y-0 left-0 pl-3 flex items-center pointer-events-none">
<svg xmlns="http://www.w3.org/2000/svg" fill="none" viewBox="0 0 24 24" stroke="currentColor" class="h-5 w-5 text-gray-400">
<path stroke-linecap="round" stroke-linejoin="round" stroke-width="2" d="M12 4.354a4 4 0 110 5.292M15 21H3v-1a6 6 0 0112 0v1zm0 0h6v-1a6 6 0 00-9-5.197M13 7a4 4 0 11-8 0 4 4 0 018 0z"></path>
</svg>
</div>
<input type="text" name="email" id="email" class="focus:ring-indigo-500 focus:border-indigo-500 block w-full rounded-none rounded-l-md pl-10 sm:text-sm border-gray-300" placeholder="Type your message...">
</div>
<button id="switchTheme" class="-ml-px bg-indigo-700 relative inline-flex items-center space-x-2 px-4 py-1 border border-gray-300 dark:border-transparent text-sm font-medium rounded-r-md text-gray-700 bg-transparent hover:bg-gray-100 focus:outline-none focus:ring-1 focus:ring-indigo-500 focus:border-indigo-500">
<svg xmlns="http://www.w3.org/2000/svg" fill="none" viewBox="0 0 24 24" stroke="currentColor" class="h-5 w-5 text-gray-50">
<path stroke-linecap="round" stroke-linejoin="round" stroke-width="2" d="M10.325 4.317c.426-1.756 2.924-1.756 3.35 0a1.724 1.724 0 002.573 1.066c1.543-.94 3.31.826 2.37 2.37a1.724 1.724 0 001.065 2.572c1.756.426 1.756 2.924 0 3.35a1.724 1.724 0 00-1.066 2.573c.94 1.543-.826 3.31-2.37 2.37a1.724 1.724 0 00-2.572 1.065c-.426 1.756-2.924 1.756-3.35 0a1.724 1.724 0 00-2.573-1.066c-1.543.94-3.31-.826-2.37-2.37a1.724 1.724 0 00-1.065-2.572c-1.756-.426-1.756-2.924 0-3.35a1.724 1.724 0 001.066-2.573c-.94-1.543.826-3.31 2.37-2.37.996.608 2.296.07 2.572-1.065z"></path>
<path stroke-linecap="round" stroke-linejoin="round" stroke-width="2" d="M15 12a3 3 0 11-6 0 3 3 0 016 0z"></path>
</svg>
<span class="text-gray-50">Send</span>
</button>
</div>
</div>
</section>
</div>
A: I think you just want to make the outer div flex column (flex-col), and then set max height (max-h-screen)..
<div class="w-full flex flex-col max-h-screen">
<div class="sm:hidden">
</div>
<section class="w-full flex scrollbar-thin hover:scrollbar-thumb-indigo-700 dark:hover:scrollbar-thumb-indigo-700 dark:scrollbar-thumb-gray-600 dark:scrollbar-track-gray-800 scrollbar-thumb-gray-400 scrollbar-track-gray-50 overflow-y-scroll">
</section>
<section aria-labelledby="chat-footer" class="h-auto w-full sticky shadow-2xl bg-gradient-to-br from-gray-100 to-gray-300 dark:from-gray-900 dark:to-gray-900 border-l dark:border-gray-800 border-gray-200 bottom-0 min-w-full flex-1 flex flex-col lg:order-last">
</section>
</div>
Demo
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 8,231
|
#
# Dedication
To my dad, for encouraging my love of the fantastic.
I miss you.
# Contents
Dedication
1. Memory
2. Shadows
3. Scorch
4. Watchers
5. Words
6. Gratitude
7. Rebirth
8. Newsoul
9. Lake
10. Questions
11. Blue
12. Spiraling
13. Jungle
14. Crevice
15. Weeper
16. Truth
17. Key
18. Crash
19. Transform
20. Explode
21. Smoke
22. Absence
23. Freeze
24. Fade
25. Snowfall
26. Demonstration
27. Skeletons
28. Trapped
29. Immortality
30. Sacrifice
31. Heartbeat
Back Ads
Acknowledgments
About the Author
Credits
Copyright
About the Publisher
#
# MEMORY
MY LIFE WAS a mistake.
As long as I'd been alive, I'd wanted to know why I'd been born. Why, after five thousand years of the same souls being reincarnated, my soul had slipped through the cracks of existence and burdened the people of Heart with such newness.
No one could tell me how I happened, not until the night I'd found my way into the temple with no door, trapping myself with the entity called Janan.
"Mistake," he'd said. "You are a mistake of no consequence."
I knew, as I'd always known, that I was a soul asunder.
Outside the temple, the night had spiraled into chaos. Sylph burned, and dragons rained acid from the thunder-torn sky. The numinous light of the temple had vanished. The father I'd never known appeared and told me the same as Janan: I was an experiment gone wrong.
My life might have begun as a mistake, but I wouldn't let it end as one.
Spring slipped across Range, a verdant blanket stitched with new life. Trees blossomed and young animals peeked from the forest, and the people of Heart cleared a stretch of land north of the city, just beyond the geysers and mud pits that steamed and bubbled as winter eased its grip on the world.
Instead of crops, they planted dozens of black obelisks, each carved with loving words, achievements, and the name of a darksoul: a soul who wouldn't be reincarnated; a soul lost during the battle of Templedark.
Every citizen of Heart took on a task. They gathered physical reminders to place by the obelisks, combed through records to find videos of darksoul friends, or assisted in the construction of the Templedark Memorial.
Sam and Councilor Sine combined their efforts, composing music and writing laments. They created different melodies and lyrics for every darksoul. I wanted to help, though I didn't know most of the darksouls well enough to contribute.
When spring bowed to summer and the memorial was finished, everyone in Heart met on North Avenue and formed two lines.
Two by two, we passed beneath the Northern Arch.
Two by two, we filed out of the white city.
Two by two, we entered the Templedark Memorial.
Our lines split there, and we followed the iron bars of the fence. Wind gusted through, making the whole place smell of roses and tinges of sulfur from a nearby geyser. Steam drifted through the cerulean sky.
The procession took ages. By the time we all arrived, people stood three deep around the field of high monuments. Everything was silent, save rustling leaves and the gasp-heave of weeping. Next to me, my best friend, Sarit, squeezed my hand tight and blinked tears off her dark lashes. Our dresses tugged in the wind while we waited.
A bell tolled in the center of the memorial, one peal for each soul lost.
What happened after death? Where did you go? What did you do? The scariest possibility was that we might. just. stop.
After another moment of aching hush, Sine pulled away from the perimeter and took a microphone. "Today, we gather to remember those who fell during Templedark. We come to honor their lives and deaths, and begin the long process of healing not only our bodies and city, but also our souls...."
Most people kept their heads down, the weight of grief so evident in their slumped postures I feared they might collapse. Others stood stoic, blank, as though their minds were somewhere very far away.
But here and there I caught eyes seeking mine; I exchanged sad smiles with almost-friends. Most were people I'd warned about dying during Templedark. There wasn't much to say about it, but they were nice to me, and our encounters were always cautiously hopeful.
Sine finished her speech.
One at a time, someone stood for each darksoul to recount lifetimes and memories. Sam and Sine performed the music they'd written. Small screens went into the base of each obelisk, set to play a video of the darksoul, or play a recorded copy of the music written for them.
Then we turned our attention to the next darksoul.
At the end of the day, we filed out of the memorial, same as we'd come in. Friends stayed at Sam's house with us, but everyone was so raw with sadness there was no joy in the companionship, and the next morning, we walked back to Templedark Memorial.
It took four days to remember the lives of almost eighty souls, and as we left the field of black obelisks for the final time, people kept glancing at the empty places in the back: room for more darksouls, because we couldn't be sure about when a few people had died. Some might still come back.
Over the next weeks, some people went on like it never happened, but there were rumors of people sleeping in the market field or destroying everything in their homes. Others supposedly didn't leave their houses for weeks at a time.
I went back to my lessons—what lessons were still being offered—and tried to find happiness with my friends and music, but the strangeness of the community's behavior smothered me. No one seemed to heal.
As summer hurtled toward autumn, the mood sagged from melancholy to disconsolate, and the pulse in the walls grew unbearable. The city wall. The Councilhouse walls. Even the exterior walls of everyone's homes. The slow throb of life inside stone made my skin try to crawl off.
I couldn't take it anymore.
"I have to get out," I told Sam. "I need to get away. Will you go with me?"
"Anywhere," he said, and kissed me.
We left Heart just before summer faded into memory.
"You've been quiet," Sam said as we left behind the geysers and mud pits, the fumaroles and rime-whitened trees.
"Nothing's wrong." Oops. We hadn't gotten to that part of his questioning yet.
He snorted. "Okay. What's on your mind?"
I lengthened my strides to keep up with Sam and Not as Shaggy as His Father, the pony that bore most of our bags. We called him Shaggy for short. My backpack straps dug into my shoulders, but I carried only a few essentials—in case we somehow got separated—plus the temple door device, and my notebook. Sam had taken to calling it my diary, but I didn't keep track of my days in there.
"Nothing in particular, I guess." I glanced back at Heart, from here just a seemingly endless expanse of white ripples and curves over the plateau. The immense central tower stood partially obscured by late-summer foliage. The city looked peaceful from far away. "I feel better getting out of there."
"The walls?" He said it like he understood, but the walls didn't feel bad to anyone but me.
"Yeah." I slipped my thumbs beneath my backpack straps, relieving some of the pressure on my shoulders. "Did you see Corin when we went through the guard station?"
"Corin?" Sam raised an eyebrow. "He didn't do anything."
"No, he didn't." I kicked a fallen branch off the road. Pine needles scraped the cobblestone. "He just sat there at his desk. He didn't say anything. He didn't acknowledge us. He barely moved."
"He's grieving," Sam said gently. "He lost souls very close to him."
"Then why does he go to the guard station every day?"
"What else should he be doing?"
"I don't know. Staying home? Staying with a friend?"
Sam's eyes were dark as dusk, and his voice deep with a hundred lifetimes. "It doesn't always make sense, the way others grieve. I can't imagine what I'd be like if I'd lost you, but it would probably seem very strange to others."
Because I was the newsoul, and why would anyone grieve that much over me?
Then again, I knew how I'd behaved during Templedark. Fearing for Sam's life, I'd hurtled through fields of dragon acid, dodging sylph and laser fire. I'd felt like someone other than myself, like I might do something crazy if I didn't find Sam, because how could my world be right without him?
"I don't like the way grief feels," I said at last. "And I don't like the way it feels when other people grieve." Which sounded like I thought they should avoid the emotion because it made me uncomfortable. No, what did I really mean? "After the dragons attacked the market, I wanted you to feel better. I wanted to do anything I could to help, to make you stop hurting, but I didn't know how. I tried and..."
Sam nodded. "It makes you feel helpless."
"I don't like that."
"Me neither." He pushed a strand of black hair from his eyes. "I've felt like that about you, helpless to make you feel better."
"Really?"
He flashed a strained smile. "When we first met. You trapped the sylph in an egg, letting your hands get burned so you could rescue me."
Sylph. Just the word made me shudder and check the woods for unnatural shadows. Too easily, I could remember the inferno racing through my hands, up my forearms, and the red-and-black skin all bubbling with blisters.
"You tried to be so strong after that," Sam said. "And you were strong, but I knew how much it must have hurt. I wanted to take the pain from you, but I couldn't. I felt helpless."
"Even though we'd just met?"
Sam only smiled and touched my hand, and we shifted to the safer topics of music he wanted me to learn, and debating whether or not Sarit would actually make good on her threat to come after us if we didn't return to Heart before winter.
Late summer bathed Range in shades of green. Clouds drifted across the sky, catching and tangling on mountains like gauze. A hawk careened in from above, calling his territory, and a family of weasels startled at the sound. They tumbled to hide in the brush, even though the hawk was far away.
When night fell, we set up a tent and sleeping bags and discussed music over dinner, then went outside to take turns on the flute he'd packed. I liked waking up across the tent from him; seeing his messy hair and sleepy smile first thing chased away my lingering fears and sadness.
We made good progress across Range, and finally we reached our destination: Purple Rose Cottage.
The last time I'd seen Purple Rose Cottage, the roof bore daggers of ice, and the path uphill had been slippery with snow. Li had stood in the doorway, tall and beautiful and fierce, and she'd given me a broken compass so I'd lose my way and fall prey to sylph.
Now Sam and I stepped out of the forest shade and trudged up the hill. Sunlight warmed my face and arms and made the cottage glow brown and almost unfamiliar with how welcoming it looked. Rosebushes huddled around the wall, indigo blooms just fading as summer came to a close. Vegetables lay half-eaten and rotted in the garden; no one had been here to harvest and put them away for winter.
We spent a couple of days getting the cottage cleaned up, arranging our things in the bedrooms and kitchen, and not discussing anything more difficult than who was in charge of coffee each morning. It was nice living with Sam without the heartbeat-filled walls boxing us in.
Our third evening in Purple Rose Cottage, Sam asked me to wait for him outside.
The cool air gave me goose bumps, but I waited on the grass by a bush of indigo roses. Low sunlight shot around the cottage, casting the forest in shadow and gold-green and hints of russet. The door shut, and Sam walked over carrying a large basket.
"Help me with this?" he asked. Together, we spread a blanket on the grass to sit, and his eyes shone in the dimness. "I want to give you something." From the basket, he removed a long wooden box. Faint light from the window made the polish glisten. When had he packed that? "This is for you."
"You didn't have to get anything for me. I have everything I need."
He smiled and regarded the box, his hands covering the gilt latches. "It's a gift, like friends gave Tera and Ash for their rededication ceremony."
That had been a special occasion, celebrating their eternal love. Today was nothing, as far as I could remember. Still, the idea of a gift delighted me, and I tried to squeeze my fingers between his to look. There were patterns carved into the wood, but I couldn't see them. "What is it?"
His hands trembled as he pulled up the latches, and the box was soundless as he turned up the lid.
Light glimmered across two lengths of silver, catching on a row of keys and delicate swirls engraved into the metal.
It was a flute, one I'd never seen before.
A rush of wind stirred the trees and stole my quiet "Oh" as Sam pulled the flute from its case and pieced it together. His eyes were dark, wide with anticipation and something else as he offered the instrument with both hands. "It's beautiful," I whispered.
"I hoped you would like it." The flute nearly vanished in his hands, though it seemed normal-sized when I rested my fingertips on the cool metal. "Take it," he urged. "It's yours."
"Why?" My question didn't stop my fingers from wrapping around the flute, from pulling it to my lips. My breath hissed over the mouthpiece as my fingers found their places on the keys.
The heat of his body warmed me as he leaned closer. "Here." He nudged my right thumb farther down the tube. "And your chin." He tilted my face up slightly, his fingers lingering over my skin.
Our eyes met, both of us suddenly aware of his other hand flat on my ribs, unconsciously adjusting my posture. "Better?" I breathed.
He watched my mouth and nodded. "Play for me."
Play what? He hadn't brought out music. But as sunlight began to fade, making the indigo roses turn ink-dark and early snow glow on the mountaintops, I played a long, low note that filled the cottage clearing with a haunting reverberation.
The note created a bubble of warmth around us. It tangled around vines, caught in rosebushes, and pushed out toward the mountains that rose like distant walls. I found a breath, and my fingers climbed a half step up.
The flute stretched its sound. It fit me as precisely as though it had once been part of my body and now we were reunited. My hands and mouth and lungs knew this flute, and I knew this flute would do anything I could ask of it, and more.
I climbed notes until a pattern emerged, as sweet and haunting as the flute's sound. The melody took shape and flew on sure, steady wings. Music filled me until it seemed I might burst.
When I lowered the flute, Sam leaned toward me, a satisfied smile on his lips. "It suits you."
"It's perfect." I caressed the silver, engravings sharp and new beneath my fingertips. They looked like ivy, or something delicate and twisty. "Did you make it?"
"Some. I had a friend do a lot of the work. How was I going to hide it from you otherwise?"
The metal was warm from my playing, and I couldn't stop staring at the way it looked in my hands. It was perfect. "I want to play it all the time."
"Good." Sam grinned widely. "Because you will." His tone turned conspiring. "I wrote some duets for us."
My heart stumbled over itself. "Really?"
"I want to keep this moment forever, the way you're smiling right now."
"You may." I placed the flute in my lap and brushed my hands over my mouth, pretending to grab my smile as though it were bits of wool or clouds. "Here." I pressed my imaginary smile into his hands. "This is for you."
He held his fists against his heart and laughed. "It's just what I always wanted."
"I have more whenever you want them."
"All I have to do is give you new instruments?"
I shrugged. "We might be able to find other things worthy of smiles."
He cupped my cheek and kissed me. "Ana, I..." The way his voice had softened, deepening with emotion, made me shiver. He pulled back. "I'll get you a jacket."
Whatever he'd been ready to say before faded into the cool night. "No, you know what would help me warm up? If you got the other flute and music."
"You're ready to start now?" He lifted an eyebrow.
"You can't give me a pretty new flute and expect me just to put it away." I clutched the instrument to my chest.
"Then I'll be right back." He kissed me again, then got up and vanished into the cottage, turning on the front light as the door shut behind him. Good idea, if we were to read music.
Alone but for the trees and roses and a few birds settling in, I lifted my flute and found a simple melody. Somewhere in the woods, a bird repeated a few notes. I smiled and played again, and the bird sang back.
Strange, but I couldn't identify the bird. It didn't sound like a shrike or mockingbird. A thrush? No, the voice was too otherworldly.
Peering into the darkness, I played a few measures of my minuet—the one I'd written not long before Templedark—and the bird...something...sang it back. It wasn't a bird.
"What are you doing?" Sam came outside again, his arms filled with a stand, a book of music, and his flute.
"There's something out there." I couldn't see. The front light stretched and vanished only halfway down the path, and the trees huddled beyond its reach. Rosebushes shivered in a cool breeze, and in the woods, someone moaned long and mournful.
My stomach dropped. I knew that sound.
"Sylph." The light made harsh shadows across Sam's face. "Is that a sylph? Here?"
"It didn't sound like a sylph before. I thought it was a bird. It was mimicking my playing."
Shock flickered in Sam's expression as he squinted into the dark. "Surely they wouldn't be this far into Range. Or—mimic you."
I licked my lips and played four notes, and the repeat came from closer. Just beyond the light, a shadow writhed. Then another, to the left, and a third still in the forest. There were so many, maybe as many as there'd been the night they chased me off a cliff, into Rangedge Lake.
Sylph burned, reeked of ash and fire, and they were without substance. The lore was complicated and contradictory. Some said they were shadows brought to a terrible half-life, thanks to fumes and heat from the caldera beneath Range. Skeptics maintained sylph were simply another of the planet's dominant species, like dragons or centaurs or trolls; people should be cautious, but not assign them any special history or powers.
Whatever they were, I'd had more than enough experience with them for one lifetime.
"Sam." I hardly recognized my voice, so opposite the storm of fear building inside me. "Get all the traps you can find."
Several more sylph picked up the notes, singing as though it were a short round of music. The sound grew, pressing closer, and abruptly stopped.
A sense of waiting grew heavy in the air. A heartbeat later, a sylph whistled a scale.
Sam touched my elbow. "You need to get inside. The walls are protected."
"Protected. Not sylph-proof." I lifted my flute. "I think—" My breath hissed across the mouthpiece and made all the sylph tense, push closer. I retreated until my skirt caught in a rosebush; thorns pricked through the cloth. "I think my playing keeps them distracted. Get the eggs. Set the traps. If the sylph attack, I'll go inside."
And hope I was fast enough to reach the door before they burned me alive.
"I'll hurry." Sam vanished into the cottage.
Heat billowed from all sides as the sylph swarmed closer. Heart pounding, I began to play.
#
# SHADOWS
DARK TENDRILS FLICKERED in and out of the light. The moaning grew softer as I played a major scale—and they sang it back.
Every scale I played, every arpeggio and trill, the sylph echoed it and hummed closer. Heat brushed against my skin like breath as the shadows drew ever nearer, but did not attack. The scent of ozone filled the clearing, though, and the front light seemed to grow dimmer.
"Good Janan!" A boy's voice came from the bottom of the path.
Every sylph went rigid and shrieked, and a wave of heat rolled toward the cottage. I gagged on the taste of ash, and sweat prickled over my skin.
"Stop!" The word was out before I could consider the wisdom of shouting, but the sylph froze. Adrenaline surged through me, making my head buzz with terror and my voice too high and pinched. "Stay where you are," I called to the newcomer. "Stay out of their way."
Silence. Either he had run, or he was doing as I said.
I couldn't breathe through the heat. Too easily, I could recall the sensation of a sylph burning my hands. The blaze, the lightning pain, and then nothing.
These hadn't burned me—yet—and if music would keep them from trying, I'd give them music. Sam would be out soon with the sylph eggs. I hoped.
Sweat pooled between my chin and the flute as the heat intensified, but I could feel their attention shifting back to me as I drew a breath, struggled to focus, and blew a stream of air across the mouthpiece. Haltingly, I played one of the first sonatas I'd learned. It was a sweet, unassuming thing called "Honey," which Sam had named for Sarit and her apiary five or six lifetimes ago.
My hands and jaw shook, but after a few moments, the sylph heat faded. One or two tried to sing along, and more caught on as I kept playing.
The sylph danced, black knotting with black. Ropes of darkness reached toward the stars, twisting with one another until they melted into one writhing shape.
They seemed to...enjoy the music. A little more confident, I stepped closer and they moved back—as though I were a light they couldn't stand to be near. But they kept singing, kept twisting. They kept dancing, even as we moved away from the cottage.
Sylph had always been terrifying shadow predators, but these were behaving unlike any sylph I'd ever met. Not like the ones that had chased me on my eighteenth birthday, or the one that had burned my hands the day after. They weren't even like the ones that had been at Templedark, though those had behaved strangely as well, fleeing from my father.
But this. Dancing. This was not sylphlike behavior at all.
The sonata came to an end. I smothered a moment of panic—would they be angry?—but the sylph hmmed and murmured the melody here and there, like echoes or making sure they hit the right notes.
One at a time, sylph drifted down the path, humming as they went.
Brush rustled, and a flashlight beam bounced across the yard as the newcomer hurried out of their path. When they were gone, the boy climbed the hill, sagging under the weight of his enormous backpack. "What did you do?" he asked.
I clutched my flute to my chest, waiting for my heartbeat to slow to a normal speed. I had no idea what I'd done. They heard the music, sang along, and went away. It was very odd behavior.
The boy didn't wait for an answer. He pulled his backpack off and dropped it to the ground beside him, glancing over his shoulder like he thought the sylph might change their minds. Did they have minds? They were incorporeal shadows, affecting the world only with their heat. My hands prickled with memory of sylph burns and my phoenix feeling from months ago. The pain had been excrutiating, but when it was over, my scars had been burned away.
"Were they after you?" I asked.
He shook his head. "I don't think so. I was walking here and heard your playing. I thought you might be—" He shrugged the words off. "Then I saw the sylph as I approached the path. That's it."
"Hmm." I looked beyond him into the forest, but nighttime hid everything, especially sylph.
"I'm sorry," he said, offering his hand. "I've been rude. I don't think we've met in this life. Cris."
"Cris." I glanced at the cottage as Sam's rushed footfalls came toward the front door. "Purple rose Cris."
He made a smile that might have been a grimace. "Yes."
"Sorry, I meant blue." According to everyone, Cris had bet he could grow the perfect blue rose, supposedly a genetic impossibility. Four lifetimes of rose breeding later, everyone said the results were purple, and Cris left his cottage. This cottage, which people called Purple Rose Cottage to mock his attempt.
"Don't worry about it." Another smile-grimace. Cris was tall and narrow, with sharp points at his cheekbones and chin, accented by short hair. Physically, he was maybe only a couple of years older than Sam and me. In reality...
They were all much, much older.
The front door flew open, and Sam stood there with an armful of sylph eggs. He scanned the clearing, breath heaving. "Where are they?"
"They flew away." The bar of keys on my flute dug into my ribs where I held it too tightly. "We got Cris in trade."
"Cris." Sam's voice slipped, and there was something while the boys looked at each other—something I couldn't understand.
"Dossam. I heard you were..." Cris shifted his gaze to me. "Then you must be Ana."
"Yes."
Awkwardness pulled in all directions: the awkwardness of being me, the newsoul; the sylph that had seemed happy to go away after singing; whatever history Sam and Cris had. Friendship? Hate? Some sort of falling-out? Sam hadn't talked about Cris much, and everything I'd ever read about or by Cris—mostly gardening notes—made him seem like someone who kept to himself.
"Sorry," Sam said, coming back to himself. "The sylph are gone?"
I nodded.
"Then we should get inside before they come back. Cris, are you staying?" Sam backed into the cottage and dropped the sylph eggs in a basket, making a metallic clatter. Then he hurried to help me with the blanket and music.
I glanced at Cris, inclining my head toward the door: another invitation. It was his cottage anyway. I didn't know if he built it specifically for the roses, or if he'd built it long before, but it carried their name.
He grabbed his backpack and followed me up, eyeing the roses as he walked past. "Someone's been taking care of these." He lifted an eyebrow at me. "You?"
"They didn't deserve to be abandoned just because they weren't what you expected." The words cut out sharper than I intended, and both Cris and Sam winced as we filed inside. "Sorry," I muttered.
"I'll make tea." Sam shut the door. "You still prefer coffee, Cris?"
"Please." Cris smiled—sort of—and left his backpack by the basket of sylph eggs. "I wasn't expecting to find anyone here."
"You'll stay, of course. We'll work out sleeping arrangements." Sam took Cris's jacket and hung it on a peg, while Cris looked between us as though he were reevaluating something. Was he surprised that Sam and I didn't share a room? A bed?
A few minutes later, Cris had washed up and Sam was in the kitchen, boiling water and preparing mugs. Cris and I sat in the front room, me on the threadbare sofa and him on the chair across the low table. Neither of us said anything, and my thoughts flashed back to the sylph and their strange actions. What had they been doing?
"I thought you'd be bigger," Cris said.
"What?"
He had the decency to blush. "Sorry. I just meant that you're the newsoul. Even being away for four years, I've still managed to hear the fuss everyone makes. I thought you'd be giant or have tentacles, but you're not. You're kind of pretty."
"Oh. Um." I wished I had something to do with my hands. Anything. Besides Sam and Sarit, no one had ever said I was pretty. Sam's friend Stef had called me cute, but that hardly seemed the same thing. "Thanks. I guess."
"So you're studying music with Dossam?"
A thrill raced through me, and I couldn't stop myself from grinning at the flutes and music resting on the table. It had always been my dream to study with Dossam. Sam. I'd wanted music from the first moment I heard it, and Sam gave it to me every day. But Cris didn't need to know that much about me. I just nodded.
"What about the roses? You took care of them, even though you thought no one wanted them."
"People don't want a lot of things, but they get them anyway." Such as newsouls, or roses of indeterminate color. "I liked the roses for what they were."
Cris offered a dazzling smile, like I'd just said something amazing or profound. "I'm glad someone appreciated them."
"Hmph." I wished Sam would hurry with the tea. Then I could pretend to focus on not spilling. "We had things in common, the roses and me. That's all." I wanted to kick myself for being rude, but Sam came into the room with a tray of mugs and rescued me from more humiliation. The way he looked at me said he knew it, too.
"Where have you been traveling, Cris?" Sam sat beside me and offered a mug of tea. I wrapped both my hands around it, grateful for the distraction.
"Lots of places. I went across the continent, cataloguing different species of plants, their rate of growth, looking for more edible plants that we might be able to grow in Heart...."
"You walked the whole way?" I asked. "For four years?"
He nodded. "That's the best way to see plants you might like to eat."
No wonder he was as thin as a wire. But he looked strong and sharp, like he could walk across the world. I didn't know much about lands outside Range, but I knew this continent was huge, with mountains, plains, deserts, and marshes. You could walk a thousand leagues from the east to the west and still miss so much. That is, as long as nothing killed you the moment you stepped foot outside of Range.
"Didn't you get lonely?"
"Sometimes, but I had my SED." He patted his breast pocket. "Which is how I heard about something called Templedark. What happened?"
I shuddered, and Sam pressed a strong hand on my spine. "My father made Templedark," I said. Though maybe I shouldn't claim Menehem. I hadn't known him—only through his diaries and the way just his name seemed to make everyone roll their eyes. I'd met him for only a short while the night of Templedark, before he died. "Menehem did something to the temple to stop Janan from being able to reincarnate anyone who died that night. He captured dozens of sylph from outside of Range, then released them in Heart. Dragons came that night, too."
Cris jerked his gaze toward Sam, who'd gone still and pale at the mention of dragons. "And you—" Cris smoothed his perplexed expression. "You made it through. That's good."
"Ana saved me." Sam's hand settled on my hip, pulling us close. "She saved me from dragons twice."
Questions stretched in the air between Cris and us, a piano wire pulled so taut it might snap. "So, Ana," Cris said, "you know about Dossam and dragons?"
I nodded.
Sam was still ashen. "I told her about the way they come after me. She knows."
The thinking line had carved itself between Sam's eyebrows; sometimes it was a worry line, or a stress line. I rested my hand on his knee and drew his gaze, and when our eyes met, the line melted away. "It's okay," I murmured. "I'll protect you from the dragons." It was a joke, mostly, just to make him smile. Because what could I do against dragons? They'd killed him thirty times.
Thirty.
But Sam wove his fingers with mine and smiled. "I know you will." It didn't sound like teasing at all.
"Fascinating." Cris wrapped his hands around his mug, his tone light and amused, but tinged with something sad. He sipped his coffee, as though to hide the emotion. "One newsoul, and Sam's problem with dragons is fixed."
"I wouldn't say that." I glanced toward the window, like sylph or dragons might be peeking in right now. "There've been two dragon attacks since I came."
"They always come in twos." Cris rested his mug on his knee. "You were just unlucky enough to be here for their first visit in quite some time."
"And we were all unlucky enough they chose to come during Templedark." Sam lowered his eyes, the memory of Templedark still fresh and heavy. "The sylph and the dragons proved too much. Everyone panicked. We lost more than we should have before anyone realized what Menehem had done: he made the temple go dark."
When I closed my eyes, I could still see the strange darkness where the iridescent light of the temple should be. Except it shouldn't illuminate. What kind of building glowed in the dark?
One with Janan in it.
"Stopping reincarnation. What a thing to do." Cris shook his head, then leveled his gaze on me. "Did Menehem—before? You?"
Oh, Cris was quick. "By accident. That was why he left eighteen years ago—to find out what he'd done." I shrugged, feigning nonchalance. "Only Menehem knows why he wanted to end so many others. Maybe he'll tell us when he's reborn."
That wasn't quite true, I knew. But not knowing how Cris felt about Janan—some people really cared, while others hadn't believed for millennia—I didn't say any more. Menehem had given me two explanations. The first made it sound as though he were doing me a favor: attempting to let more newsouls be born.
The other reason Menehem had given me seemed most genuine: he'd wanted to prove Janan's existence either true or false. It had been scientific curiosity, nothing more.
Cris glared into his coffee. "I'm sure the Council will be very curious to find out exactly how he created Templedark so they can prevent anyone from ever doing it again."
"I'm sure they will be." Did my voice shake? It seemed like Menehem's research notes were a bright beacon shining from my room. He'd left them for me after he died, and I hadn't wanted to leave them in Heart. The folders and diaries, the door device, the mysterious books I'd stolen from the temple—it was a wonder everyone didn't know about them just from the guilt on my face.
But I wasn't ready to tell anyone about my visit into the temple or Menehem's research, and Sam had agreed. I didn't know exactly how the Council would react, but it definitely wouldn't be good.
Sam looked at Cris, a strange and awkward hopefulness in his tone. "You're on your way back to Heart now?"
"I think I'd better be," Cris said. "Sine's message indicated they'd need assistance reorganizing the genealogies now that so many won't return."
"I'm sure they'll appreciate your help," Sam said, not explaining to me how a gardener would be useful for genealogies.
They talked until everyone's mug was empty, keeping the conversation to simple things, like the best road to take into Heart, and warnings about bears and wolves in certain parts of the forest. They concluded with a polite argument about who would take the other bedroom, and Sam won, which meant he slept on the sofa.
As the calming herbs in my tea took effect, I wished the boys good night and went into my room, trying desperately not to think of the sylph.
Moaning wind roused me from fiery dreams.
My bedroom looked the same as it always had, dusty wood floors and walls all bathed in darkness, but something was different. Not the shadows, but the sounds. The wind had never made this particular wailing in the eighteen years I'd lived in Purple Rose Cottage.
I went to the window and pushed open the shutters.
Stars blazed far away, trees hugged the earth and sky, and the rosebushes breathed perfume that didn't quite mask the lingering reek of ashes. The night was perfectly still, but the moan persisted.
A shadow moved.
They twisted all along the path up to the cottage, whistling, humming, singing. A melody I'd played earlier lifted and faded in the strange song. A moment later, another familiar tune piped up, and the others built on it with harmony and countermelody. Unearthly music filled the night, subtle enough that it might have been wind on the corner of the cottage. Strange enough that it had pulled me from sleep.
There had to be a dozen sylph outside my bedroom, and though they were eyeless, I could feel them looking at me.
A whimper escaped my throat.
A gasp sounded in the front room, and blankets crumpled to the floor. Soft thumps made their way to my bedroom door. Sam. I knew the cadence of his footsteps.
I raced to the door and dragged it open.
In the dimness, Sam glanced me over, as though to make sure I wasn't bleeding—why would I be bleeding?—and then swept me up in a tight hug. "Are you all right? I heard you—"
He stilled as the sylph sang outside, echoes of music he'd composed.
"Oh." His breath rustled my hair as he released me, and together we made our way back to the window. Warm air pushed inward, smelling faintly of ash and ozone.
One by one, the sylph finished their music.
One by one, the sylph drifted down the cottage path, leaving nothing more threatening than a song.
"What does it mean?" Sam whispered. He cocked his head, as though listening for sounds of Cris stirring in the other bedroom, but relaxed. Cris must have been a deep sleeper, or tired from walking everywhere.
"It means I need to stop avoiding Menehem's research. The sylph were terrified of him during Templedark, and it was his research on the sylph that affected Janan's temple. I need to understand why. And why they'd sing outside my window." Though it was unlikely Menehem would be able to answer that question. As far as I could tell, he'd never been concerned with thoughts or feelings or motivations of others; he couldn't grasp them.
Sam dropped back his head in resignation. Our peace was too short-lived. "What do you want to do?"
I stared into the darkness, but nothing moved, and the sylph odor abated. "I wish we could stay outside of Heart, just playing music all the time. But in houses. I don't want to walk around for four years like Cris."
"Pianos are too heavy to carry in a backpack, anyway." He kissed my forehead, stubble scratching my cheek. "You know people there like you."
"Sarit, Stef, Sine—other people whose names begin with S."
He chuckled. "Armande, Lidea, Wend, Rin, Orrin, Whit. Lots of others. Templedark was horrible, but it did show people you cared. How many did you save that night?"
I didn't answer because I didn't know. The night had been so frantic, and mostly I'd been looking for Sam.
Warm fingers curved over my cheek, and he drew my gaze upward. "Are you worried they'll change their minds about you?"
How did he always know my real fears? "No one calls me nosoul anymore, but how long will that last when they find out sylph don't chase me anymore? Cris saw them reacting to my playing."
"He won't tell anyone. You can trust Cris."
I wished I had Sam's confidence that people would remember I wasn't out to destroy their existence. Maybe that was why I was reluctant to look into Menehem's research, but I couldn't let fear of others' reactions stop me anymore.
"All right. We go east of Range, where Menehem did his experiments." I closed the shutters, locked them. "I don't want anyone to know we're going there. The Council won't like it."
"No," Sam whispered darkly. "They won't."
"We leave as soon as Cris does." And then, I hoped, I would find out what Menehem had done to the sylph, and their connection to Janan. But mostly, I needed to find out what they wanted from me.
#
# SCORCH
CRIS LEFT THE cottage as the sun rose. I lay in my lumpy bed, listening to a stranger move through the next room, through the washroom, through the front room. I was still getting used to other sounds in this house belonging to Sam, not Li, and Cris's sounds were different yet. His steps were longer than Sam's, and not...heavier, but more solid somehow.
Just as I realized I should see him off, low voices came from the front room. "Will you tell Ana I said good-bye?"
"She's awake, if you want to tell her yourself." Sam's voice was groggy, but he'd probably awakened as soon as Cris's feet hit the floor.
"I'm sure she'd rather go back to sleep. I'll see you both in Heart." Cris hesitated. "You mentioned she's taking lessons from people. Perhaps she'll want to try gardening."
"Perhaps." Sofa springs made their someone-sitting-up groan, but it wasn't Li's motion. It would never be Li again. Sam's voice came almost wistfully. "It was good seeing you again, Cris."
"And you." A moment later, the front door squeaked and shut.
It was hard to find time to practice while traveling, but Sam insisted there was more to music than playing an instrument. Theory was just as important, and we listened to as much music as we talked about, our SEDs synced to play the same things.
Sonatas, minuets, arias, symphonies: these things accompanied us through the forest, all golden-green woven with the fire of oncoming autumn as we walked northeast through Range.
"Are you worried sylph will hear the music and come after us?" I asked Sam once.
"No." He paused. "Not too worried while we're in Range, at least. And we have sylph eggs. Purple Rose Cottage is at the edge of Range, so there weren't as many traps between them and us. It's unlikely they'd follow us."
Unlikely, but not impossible. Sylph had been doing all sorts of unlikely things lately. A few heat-sensing traps might not make much of a difference. "What about when we get to Menehem's laboratory?" I asked. The map he'd included in his diaries indicated the building was just beyond the border of Range, drifting into troll territory.
"We'll have to be more careful there, but I'm sure the building itself is well protected."
"Hmph."
I shouldn't have been worried, though. When we arrived at the coordinates Menehem had left, we found an ugly iron building the size of a barn. Solar panels covered the roof, while cisterns hugged the sides.
Trees stumps dotted the area, some as big as dinner tables. Here and there, the grass had been scorched black. Not by lightning. Sylph? But how?
I knelt and dragged my fingers through fine, midnight powder. Ash. It trickled away in a gust of wind, leaving my fingers stained with dusk.
Sam stopped beside me. "What do you think happened?"
As if I had any clue.
"Not sure." I pulled out my SED and made a quick video of the entire area. "Eerie," I muttered as I saved it in a private, protected folder Stef had taught me how to make. I doubted she realized exactly what I'd be doing with my privacy, though; she'd probably assumed it was simply because I didn't want to risk any of them finding my secrets, since I hadn't yet started a diary for eventual sharing like everyone else.
A message flashed in the corner of my screen. Sarit had sent a photo of a jar of honey with a teal ribbon tied around it, and "For Ana" written in her flowing script.
"What are you smiling about?" Sam nudged me with his elbow.
"Sarit." I showed him the photo. "I think it must be a form of bribery."
"She misses you. I would, too." Sam gazed up at the monstrosity of a building while I sent a message back to Sarit, letting her know her bribe would have more effect if I hadn't remembered to pack a small jar. She would try again when I ran out. "Ready to go in?" Sam asked.
"Ugh. I can just imagine how comfortably we'll be living until we go back to Heart."
He chuckled and motioned toward the cloudy sky. "At least we won't get rained on. Do you want to take the bags in and I'll put Shaggy in the stable?"
I looped bags over my shoulder and blew Sam a kiss as I headed for the front door. Menehem had left me a key and a code, though it could have been easily broken by anyone who cared enough to try. A soul-scanner would have been more secure, but maybe he'd been planning on leading me here; I wasn't in the main database and wouldn't have been able to get in. He couldn't have predicted I'd have Sam with me.
Inside smelled like something had died months ago. Certainly the building was protected from sylph, thanks to all the iron, but it wasn't protected from dust, small animals, or general grossness.
Lights flickered on as I dropped the bags and stepped into the front section, filled with cabinets and rickety furniture for a parlor, bedroom, and kitchen. Another room—a washroom, I hoped—was blocked off in the back.
Beyond the front area, I found a lab with tons of equipment I couldn't identify, huge glass and steel containers, and stuff. It looked like Menehem had been collecting lab-type junk for a lifetime.
A stair to the upper story revealed a dark data console and a small library's worth of research. It seemed he'd also stored off-season clothes and supplies here, because I discovered crates of jackets, skis, and other things. The scent of cedar—to ward off bugs—flooded the area.
"Ana?" Sam's voice came from below, and I clomped back down the stairs. "Anything exciting up there?" He was gazing around the lab when I found him, probably looking for a mop or button that would miraculously clean the layers of dust and grime. Menehem hadn't even been gone a year, but it didn't take long for nature to start reclaiming things.
He probably hadn't been the cleanest person to start with.
"Just lots and lots of research and junk." I sighed. "This is going to be like reopening the cottage, but even worse, isn't it?"
"Do you want to sleep in here with everything like this?" He lifted an eyebrow.
"We could sleep outside. I'd risk the sylph."
"How about cleaning the living area today, then we'll worry about the rest?"
"Fine." I dragged out the word, but mostly I was complaining to complain. I didn't mind cleaning up if Sam was nearby. "But my cooperation comes with a price."
"What's that?" His posture relaxed, voice warming like he knew already. And when I smiled and tilted my face upward, he kissed me so sweetly my entire body hummed with adoration and desire. Could anyone else ever make me feel this complete?
No. Only Sam.
It had always been Sam.
Almost a week later, we'd tossed out a decaying raccoon and scrubbed the living quarters and lab until they didn't make us want to run back to Heart and shower. Sam fished twigs, dead bugs, and a snake out of the cisterns—I double-checked that the water purifier had new filters and solution—and finally we were able to get started on the research we'd come to do.
We sat at the splintering kitchen table, diaries and papers spread out around us. I pointed at a notebook. "This journal matches up with what we already know: he'd been trying to find ways to stop sylph. He started with iron and was looking for ways to make sylph power the eggs with their own life force; that way they'd keep sylph trapped long after the batteries ran out. But that didn't work, so he went back to chemicals."
"He was always best with chemicals," Sam agreed.
"During the first Templedark—the night Ciana died—he was doing his experiments in the market field." Then Ciana hadn't been reborn. I'd replaced her.
"Because, of course, that's a logical place to do experiments."
"You know Menehem." My heart pinched for only a moment—if Menehem hadn't been so irresponsible, I wouldn't be here—but Sam touched my hand. His knee bumped mine. His comment had been only about Menehem, and involved zero regret about the way things had turned out, even though the world had lost Ciana. I tried to smile.
Strong fingers tightened over mine, and he lifted his eyebrow, waiting for something. Acceptance. I was getting better at reading Sam, if not many others.
I smiled again, squeezed his hand, and we both relaxed. "So whatever he was doing in the market field," I went on, "there was some kind of minor explosion, and a vapor went up. That's when the temple went dark."
"From the gas," Sam said. "Then he came here to figure out how to reproduce the mistake, because he didn't know what he'd done to get that reaction."
"Right." I flipped a few pages and pointed at a list. "These are the chemicals he used." It was a long list.
"I don't know what those are."
"Hormones, some of them. I recognize a few from Micah's biology lessons." I glanced toward the lab in the back. "There are stores of the chemicals in there. Most of them are labeled, even. And he wrote down the final recipe, though I'd like to study his experiments a bit more first."
"First? Before trying it yourself?" Sam frowned. "I don't know if that's a good idea."
I flinched. "You don't think I'd try to make another Templedark, do you?"
"No, I know you'd be doing it for research, but what if the Council finds out? We both know what they'd assume."
I slumped and planted my chin on my fist. "You're right."
"Besides, you've told me that sylph were fleeing Menehem during Templedark. That makes me think he was hurting them."
"Are you worried about hurting sylph, Dossam?" I flashed a dry smile.
He spoke gently. "I just don't think you'd want to hurt anything, even sylph."
I lowered my eyes. "No, not even sylph." After the weeks it had taken for my hands to recover from sylph burns, I might not have minded. But the night of Templedark, when Meuric had led me into the temple and tried to trap me, I'd stabbed him in the eye with a knife and shoved him beneath an upside-down pit. He'd fallen upward, body still flailing. That had been self-defense, but the guilt still writhed inside me. I should have come up with a better solution to my problem, but it was too late now.
Sam put his arms around me.
"I don't want to hurt them," I said, "but the more I understand about this, the more I understand about Janan. Whatever Menehem did, it stopped Janan for a little while. The rest of you don't feel it, but the white walls feel horrible to me. And the temple makes me feel—" I blinked away tears. "He's not good, Sam. Whatever Janan is, it's bad. It's evil."
"All right." Sam pressed himself against me, as though he could shield me from something like Janan. As though he could even comprehend my fear of Janan when he didn't fear Janan at all. I probably sounded crazy to him, thinking the heat and pulse of the walls were wrong. My seemingly irrational dislike of sleeping close to the exterior walls of buildings was unique, but I couldn't even lean against the wall. It made my stomach twist with unease.
I was right, though. There was something off about Janan. Inside the temple, he'd called me a mistake, which implied that he had a plan. He'd also said I was "of no consequence," which implied that he didn't view me as a threat.
I aimed to be a threat.
Sam combed his fingers through my hair, down the back of my neck. "I wish I understood what it feels like for you. I wish I could make it right."
He didn't want to make me right. He wanted to make things with Janan right.
I liked that he didn't think I was wrong. I liked that he believed me. That he trusted me, in spite of how I must have looked.
The building creaked in the wind as night settled, and my hair muffled Sam's words. "I'm just worried that if we go too far into Menehem's research, regardless of our intentions, someone will think we're creating another Templedark."
"Even our possessing his research will be too much for some people," I whispered. "Maybe I have more friends now, but Meuric wasn't alone in his feelings about newsouls. Not nearly." Right off, I could think of five people who'd made their dislike clear, and lots more who just didn't bother acknowledging me.
Sam nodded, his expression etched with frustration.
"I don't want anyone to think I want another Templedark, but Menehem's poison is the only thing I know that affects Janan. I just—I want a weapon, Sam. You gave me a knife when I told you someone followed me home one night. A knife won't work against Janan. We only know one thing that affects him, and this is it. I want to understand how. I want to discover if maybe there's another way I can protect myself." I wanted to feel safe, but that would never happen in Heart, and I wouldn't ask Sam to spend this lifetime in a dusty cabin just for me.
"Let's go through the rest of Menehem's research," Sam said. "I'm sure he recorded videos and every possible variation in his results. Will that help?"
"It's a start."
#
# WATCHERS
SAM WAS ALREADY sleeping on the sofa when the noise came, a soft shriek of wind that sent splinters of fear through my chest. I scrambled for the window.
Dusk had fallen, and the view from the window nearest my bed revealed only twin mountains against starlight, and lots of trees in between. Brittle leaves rushed in the wind, and I relaxed. Real wind. Real wind in a strange place. I didn't know the sounds of this building like I knew those of Purple Rose Cottage. I wasn't familiar with the particular way wind cut across the iron corner in the northeast, or which trees groaned. I didn't know their voices.
The sound remained, but the branches, half-dressed with autumn, became motionless.
A square of light fell from my window onto the grass when Sam clicked on the nearest lamp. "What is it?" He stopped at the foot of the bed, yawning.
"They're watching." I grabbed my flashlight from the nightstand, gave the tube a few sharp twists, and shone the light toward the woods.
Shadows skittered away, yelping and whining, but they didn't come closer. When I pulled the beam toward the lab again, the shadows relaxed and resumed their places at the tree line.
"Watching?" Sam touched my shoulder and peered out from behind me. "How many are there?"
"A lot." I closed the window and pulled the shade. We were probably safe inside the iron building. Probably. "Do you think any of these are the same sylph that attacked me on my birthday?"
"I don't know." Sam clicked off the light. "If they are, why behave differently now?"
Mysteries and more mysteries.
The sylph didn't leave that night, or the next, or the next. They never moved closer, never threatened or attacked, but they were always there. Watching.
Over the next few weeks, I learned why it had taken Menehem eighteen years to re-create and perfect the results of the first Templedark.
The process of creating and dispersing the poison was a complicated one. Sam and I watched video after video of Menehem explaining different theories and tests to the camera. The hundreds of combinations ran together until one finally gave the response Menehem had been looking for.
Sam and I sat curled on the sofa together, his arm around my shoulders. I had a notebook balanced on my knees so I could write down stray thoughts. The screen, which Menehem had hidden in a wall, showed a summer day with my father bustling about the yard with cans of aerosol poison, which he'd created using a machine in the back of the lab.
"Aerosol," he explained to the camera for the hundredth time, "has proven to be the most effective delivery system. It allows the hormones to be both solid and suspended midair. For sylph, both corporeal and incorporeal, almost a paradox themselves, fighting them with a substance that behaves the same way seems the most logical.
"The problem has been finding just the right amount of each hormone and timing the exposure, but I believe that I've finally found a combination that will work. I started with..."
He droned on for a while, repeating many of the same things he'd said before. Then he ambled toward a large speaker by the building and flipped a switch. Music crackled and settled, and a haunting piano sonata flowed across the small field and toward the nearby creek. Music streamed toward the mountains, filling the entire area with melody.
As they'd done in nearly every other video, sylph appeared in the distance.
Shadows glided toward the speaker, writhing like flames. Tendrils of darkness shot out of them, like hands reaching for the sky. Under the familiar sounds of Sam's piano playing, the sylph voices rose up to sing along.
I glanced at Sam. "Is that weird? That they like music so much?" As I had, Menehem seemed to have discovered their response to music by accident. Then he'd begun using it as a lure.
"Maybe. Who can tell with sylph?"
Perhaps they thought Menehem had captured one of their own. Would they have cared if Menehem had trapped another sylph?
On the screen, sylph drifted around the yard, ignoring the small canisters placed about. When there were nearly a dozen sylph singing along to the sonata, Menehem pressed another button.
The canisters spewed aerosol, hissing loudly. The sylph ignored it; if these were the same sylph as had been in previous videos, they were used to this part, too. The gas had never done anything to them.
This time, the sylph dropped.
Two or three at first. They twitched and seemed to glance around—how could they glance if they had no eyes?—and then sank into puddles of darkness.
Another sylph shimmered and fell. And another.
Soon, Menehem flicked off the speaker and the field was silent.
"I did it," he said. "Finally, I did it!" Menehem jumped and whooped, giving me an odd sense of embarrassment for him.
Sam shifted uncomfortably, and I doodled roses in the margins of my notebook while we waited for Menehem to compose himself on the screen.
"It looks like they just fall asleep," Sam said.
"The music draws them in, and the gas puts them to sleep." I nodded and leaned forward as Menehem approached one of the sylph puddles. I almost felt bad for them, being experimented on. Sam was right. I didn't want to hurt them—though these didn't look like they were hurt, exactly.
Menehem knelt by the nearest sylph and pulled a device from his pocket. "Temperature is abnormally low for a sylph." He stuck his hand into the sleeping sylph. "It's warm, but not uncomfortable."
My heart jumped and sped. "Sam." Why would anyone put their hand inside a sylph?
"I see it." He touched my hands, squeezed them. "They're fine now. Healed completely, remember?"
I nodded, but the sensation of burning wasn't something I'd ever be able to forget.
As we watched, a couple of sylph twitched and shifted in their sleep. Could sylph dream?
Suddenly, the one Menehem had touched shot into the air, towering over the chemist. Grass sizzled, and the sylph shrieked so loud and high that Sam and I both covered our ears.
Other sylph awoke, equally enraged. Smoke billowed where they burned the grass. They converged on Menehem and—
And they seemed to think about it. Something passed among the sylph. Communication? I couldn't tell. It was so fast, and they were still keening....
At once, all dozen sylph fled the area, leaving patches of scorched earth behind. Menehem slumped to the ground, an unused sylph egg rolling from his hand. He'd almost been killed.
He'd almost been burned alive, but the sylph decided not to do it.
"What..." I stared at the screen until Menehem soberly got up to declare this portion of the experiment finished. The video stopped. "Did he even realize the sylph chose not to kill him?"
"Hard to say with Menehem." Sam switched to the next video but didn't play it yet. "It didn't last very long, what he did to the sylph. A few minutes at best."
I checked one of Menehem's diaries. "This says the initial dose was small. He eventually increased the doses, but they grew resistant."
Sam nodded. "And the dose he gave to Janan?"
Oh. If it affected both sylph and Janan the same way, then it was logical to assume Janan would develop a tolerance, too. I checked Menehem's notes, flipping several pages toward the end. "It was a massive dose. At least a hundred times larger than the biggest dose ever given to the sylph."
"So what he did during Templedark won't work again."
I shook my head. "No, looking at the logs, the tolerance grew quickly and exponentially in sylph. If anything were to affect Janan again, it would have to be—I can't even comprehend. There'd have to be a lot of it, and it would take months to make, even with that machine in the back doing all the work. The delivery would have to be unbelievably enormous."
"Yeah." Sam considered for a moment, then touched my wrist. "At least there's something good in this."
Was it bad? I wasn't sure how I felt about it, let alone whether it was good or bad.
"Should anyone accuse you of wanting to attempt another Templedark, we have proof it isn't possible."
Newsouls were supposed to be impossible, yet I'd been born.
Getting into the temple should have been impossible, yet I'd been inside.
Poisoning Janan again wasn't impossible. With a bigger dose and a bigger delivery system, it could be done. I just didn't know how. Or whether I should.
Inside the temple, Meuric had hinted about something horrible happening on Soul Night: the spring equinox of the Year of Souls. That threat nipped at my thoughts.
"We still can't tell anyone about the poison," I said. "I don't want the Council—or anyone—to know we came here and looked. They'll assume the wrong thing. They'll assume I'm like Menehem, and I'm not."
"I know." Sam lurched up from the couch and paced the room, shoulders and back stiff.
After a while of watching him walk ditches in the floor, I asked, "Are you okay?"
"Yes." He stopped and sighed. "No. Sorry."
Sorry for saying yes when he didn't mean it? Or sorry that he wasn't okay? I waited for him to go on.
"I've never gotten involved in disputes. I don't like them. Even in the beginning, I stayed away from conflict." Emotions shifted over his face, and he looked at me. "I'm on your side, Ana. Every time. Before, it was easy to stay out because I didn't care. I just made music, and no one expected anything more from me. But with you, I do care."
And by being with me, the controversial newsoul, his life had changed. What he'd been before—notable for only his music—was gone. Now he was notable for living with, and frequently kissing, the newsoul, and that forced him to take a side. Mine.
"I'm on your side," he said again. "But I have to admit the idea of being in something is frightening."
I pushed my notebooks to the sofa and crossed the floor to Sam. His cheeks were warm beneath my palms, and stubble scraped my skin. I wanted to—something. Thank him. Reassure him. Make him know how much I appreciated him and cared about him. Express everything I felt, but nothing that found its way to my tongue felt big enough. So I brushed a kiss over his mouth and stayed silent. His hands tightened on my hips.
Moments spiraled between us, ripe with words unsaid, until finally I pulled away and gathered up my notebooks to work at the table. He'd relaxed a little; that was what I'd wanted.
"What are the sylph?" A book slid from my pile and hit the floor with a loud slap.
"Shadows?" Sam bent and retrieved the book smoothly, and sat across the table from me. "Fire? I'm not sure what you're asking. They're just sylph."
"But they're—" I dropped to the chair. "Are they like people? Do they think? Have emotions? Societies?" They seemed like creatures with reason in the videos we'd just watched. They'd made choices.
Choices I didn't understand.
"I don't know." Sam eyed me askance. "What are you thinking?"
"I'm not sure. I mean, we know centaurs live in communities, right? They have language, traditions, and hierarchy. They go on hunts together."
He nodded.
"And trolls? They're the same?"
"Different, but yes. They live in communities, too."
"What about dragons?" I didn't want to ask him about dragons, all things considered, but I was chasing a point. An idea. A question.
"From what we've been able to tell, yes. And rocs nest with a mate and care for their chicks until they're old enough, like eagles do. Unicorns live in herds. They're not human, none of them, but they do seem to be more." He studied me with those dark eyes. "You're trying to figure out sylph."
"Aren't you?" I glanced out the window, where ranks of sylph still guarded the building. "We know so much about everything else surrounding Range, but not sylph. We see them in swarms sometimes, or off by themselves, but we don't know if they stay with the same group or just join up with any other sylph they meet. We don't know if they eat, how they think, whether they reproduce. Are there a limited number? We haven't been able to kill them, but what about other creatures? Centaurs are intelligent. Can they kill sylph?"
Sam just stared at me like I'd grown a second head. "I don't know. Sylph are all over the world, rather than generally keeping to one region, like we do, or dragons or centaurs or trolls."
"There are bears and flies all over the world, too, but sylph aren't like bears or flies."
"No," he agreed. "They're much different."
"But how? Please tell me you at least understand the questions I'm asking, even if you don't know the answers."
"Of course." Sam frowned, eyebrows drawing together to carve the thinking line. "Why wouldn't I understand your questions?"
Lightning struck inside my chest. He couldn't even remember that sometimes he didn't. It was okay, though. Most people couldn't remember things about the temple, or Janan, or understand why I asked so many questions. I said gently, "What's inside the temple, Sam?"
"Nothing. It's empty."
"How do you know that? Have you been inside?"
He shook his head, looking confused. "There's no door."
My fingers skimmed the silver box I'd taken from Meuric during my trip into the temple. The door device was too dangerous to leave behind. Doors could be created, and the temple wasn't empty—not completely. I'd stabbed Meuric and left him there, and I'd taken a stack of books from inside the temple; there were still more there.
Sam followed the motion of my hand. "What is that?"
"A key, I think." I waved his next question away. We'd had this conversation a few times already. He just couldn't remember. "And in your life before this one, you went north?"
"Yes. There were dragons." A shudder passed through him.
I wished I hadn't brought it up.
"But before the dragons, you said you came upon a huge white wall."
Slowly, he nodded. "There was snow everywhere. The wall was pitted with weather. It seemed familiar, but strange, too." The haze vanished from his eyes. "We were talking about sylph."
No, we were talking about why he might not understand my questions.
And knowing what I did about reincarnation, what entity was responsible for it, I could make a pretty good guess as to why Sam—and everyone else—struggled with certain subjects.
Janan didn't want them to know.
Janan didn't want them to question.
Janan kept a huge secret in that temple, in those books, and somehow the sylph were connected to it.
I just had to find out what it was—and use it against him.
#
# WORDS
AFTER MORE RESEARCH and note consolidation than I could stand, I slumped on the rickety bed to check my SED. It had been chirping with messages all morning; they were from Sarit, asking me to call.
I shifted to the communicator function.
"Ana!" Sarit's voice squeaked with joy. "What have you been doing all day? I've been waiting and waiting."
I giggled. Sarit was amazing. No other oldsoul would complain about waiting. Everyone else, Sam included, was ridiculously patient. "You know Sam. There's always more work with him."
Sam glanced up from the sofa where he was writing in his diary. He looked adorably baffled about why he was getting blamed for something, and I winked.
"Oooh." Sarit drew out the sound. "Work. I'm sure that's all you've been doing."
"Well." I giggled again, remembering earlier when we'd taken a break to cook lunch, and later had to eat burned rice and vegetables because we'd been too busy kissing to remember to stir.
"That's what I thought." Sarit laughed. "I can't wait for you to come home. I want to see the flute and hear the duets. Just the idea is making me wilt."
"That's why you're one of my favorite people." I leaned back on my pillow. "But tell me about the fifty thousand messages you left. What's so important?"
"Okay, there are two things. First, did you feel the earthquake?"
"Earthquake? No." I glanced at Sam, my eyebrow raised. "Did you feel an earthquake?"
He shook his head.
"Neither of us did," I said to Sarit. "Was it big? Is everyone okay?"
"Oh, yeah." She sounded breezy. "Everyone's fine. It was small, really. There are always earthquakes in Range, though most are too tiny to feel. But you know how everyone gets. They're all wishing Rahel were still here; she looked after the geological and geothermal aspects of Range. It just makes people feel safer to have someone like her saying there's no danger."
"Ah." I shifted, hating when anyone mentioned a darksoul. It wasn't that I wanted people to pretend they'd never been here, but friends' pain was too sharp. "What was the second thing?"
Her tone lightened. "Well, you remember how you were asking about what happens during a rebirth?"
"Yeah..."
"But no one invited you and you said you felt awkward just showing up?"
"Yes?" After Templedark, several pairings had been approved by the Council; they needed to start getting souls back, and not everyone who'd died that night became a darksoul. Lots would return, like Menehem.
"Lidea was asking about you last night. She wanted to know when you and Sam would be back, because she was hoping you'd be there when she gives birth."
"Really?" I bounced. "You're not tricking me so I'll come back to Heart early, are you?"
"No!" Sarit lowered her voice, as though telling me a secret. "She said she wouldn't be alive now if it weren't for you. And she was asking about you, really. Not even Sam, but he's probably expected if you're going."
"Wow. Okay. When is it?"
"They're thinking it will be just a couple of weeks"—her tone turned sly—"so you should probably come home now."
I snorted. "I knew there was a catch."
"And a bribe. Come home and I'll give you another jar of honey. A bigger jar. I'm sure you've run out of the little one by now."
It was true. "You do know the way to my heart. I'll tell Sam I can't pass up an offer like that."
"Good! Okay, go tell him. See you soon, my little moth!"
"Ew. Really?"
"I have a whole list of bugs for you. Bye!" She hung up.
"What did she call you this time?" Sam put down his diary and stretched.
"A moth." I checked for other messages, but Sarit's were the only ones. "I think she's trying to wear me down about the butterfly thing."
"Is it working?" He stood and glanced out the window. His expression didn't change, though, which meant the sylph must still be out there.
"Nope. She can call me whatever bug she wants. I'm not a butterfly."
"No." Sam gave a quiet half smile. "You aren't."
Shortly after we'd met, Sam had compared my life to a butterfly's, saying that to others I was fleeting, inconsequential. I'd long ago forgiven the insult, but I'd made the mistake of wearing a butterfly costume to a rededication masquerade earlier this year. The nickname had stuck, mostly as an endearment, though knowing my distaste for it, Sarit searched tirelessly for alternatives.
"It sounds like we're going back to Heart sooner than scheduled?" Sam hesitated, then sat on the corner of the bed. He'd been sleeping on the sofa only a few strides away, and he claimed he wasn't uncomfortable there, but I kept wondering if we might both be more comfortable if he were here. With me. I didn't say anything, though.
"Yep." I stretched to put my SED on the small nightstand, next to my private notebook. "We've been invited to a rebirth, and I really want to go. I think we're done here, anyway."
Sam tracked my motions, something deep and undefinable in his eyes when I settled against the pillow again. "You know I'd go anywhere with you, Ana."
I smiled. "Keep saying that and I might start believing you."
"It's true." He scooted closer, now by my hips. "Where do you want to go?"
"To the moon?"
He grinned. "I like that you think big."
"What about to the bottom of the ocean?" I'd never even seen the top of the ocean, but why stop there? "We could go to the very bottom and explore. Can you imagine what kind of creatures must live under all that water?"
"I think you can, and that's what I—" He dragged in a breath. "I want to tell you something."
"What is it?" I pushed myself up straight, and suddenly we were very close together and the mattress sagged awkwardly under our combined weight. He hooked his arm around my waist to keep me from falling over as I let my hands slide downward so my fingers curved over his arms.
"Ana—" He kissed me, gentle and sweet, but filled with an intoxicating urgency. His arm tightened around me, drawing me nearer. He kissed a trail down my neck, as far as he could go before my shirt collar stopped him, and then he stayed there. Breathing hard.
I almost asked again what he wanted to tell me, but maybe I didn't want to know. Maybe it was something bad and that was why he'd kissed me like that. Maybe he thought it would be so horrible I'd never speak to him again, but surely he knew he meant everything to me.
"Sam?" I combed my fingers through his hair, soft and thick and dark. I liked the way he kept it, a barely contained disaster. "What is it?" I whispered.
He drew himself up, kissed me again, and spoke the words against my lips. "I love you, Ana."
My breath caught in my chest.
The words. They made my heart beat faster. I wanted to be able to tell him how I felt, and what he must have been waiting to hear, but even thinking the words made me sweat. Nosouls couldn't love. That was what my mother had told me for eighteen years, and she'd slapped me if I even said the word.
But I wasn't a nosoul. Newsoul, yes. Still, was I truly capable and deserving of love?
"It's okay," he whispered, and his worry turned into understanding. Of course he understood; he always did. "It's okay if you can't tell me. Or if you don't feel the same way about me. I just wanted to make sure that you knew I do love you."
The words sent chills across my skin. "Thank you." I tried to smile to reassure him, but nothing would come.
"I love you." He said it as though repeating himself would convince me.
The tightness spread up to my throat, making tears blur my vision.
"Ana." He turned my face up, brushing his thumbs across my damp cheeks. "Why are you crying?"
"I don't know." A sob erupted with an ugly heave. I couldn't breathe anymore.
Sam wrapped his arms around me, holding me tight against his chest. My tears soaked his shirt, and when my nose ran, he pressed a clean handkerchief into my hand. I clutched the square of white cloth. Clutched Sam. Part of me wanted him to go away so I could cry in peace, but I didn't want him to leave.
He rocked me until my sobs waned to sniffles, not asking why again. And he didn't take back the words. That word. I didn't want him to take it back. I wanted him to feel that way about me.
I wanted it. I couldn't bear if he took it away.
"Lie down," he whispered.
I did, wiping a dry corner of the handkerchief over my face as he pulled a blanket over me.
"Do you want a cup of tea?"
"No." Weeping had shredded my voice. "I want you."
"Okay." He bent to untie his shoes, then kicked them off and stretched out on the narrow bed beside me. Achingly close. Too close, because our knees and elbows got in each other's way, even through the blanket. Not close enough.
I shut my eyes so I didn't have to see his concerned expression, his confusion, or the hurt. If I could have explained somehow, I would have, but there was nothing where an explanation should be.
He brushed hair off my face, strands stuck to damp skin, and finally I drifted in and out of restless sleep. But he was there every time I opened my eyes, stirring awake when I moved.
Darkness blanketed the world as I crept up to wash my face and throw the handkerchief in the laundry pile. Night had fallen. Outside, sylph crooned half-familiar songs, making me pause.
"Ana." Sam rolled over to face me, and I darted back into bed before he decided to go to the sofa.
The blanket was still warm from his body, even though he lay on top of the covers. "Thank you," I whispered.
"For what?" He leaned on his elbow, face just above mine in the darkness. Kisses breezed over my forehead, my cheeks.
For not running away when I started crying. For not taking the words back. For saying the words in the first place.
"I don't know. For everything."
Dawn slanted through the east-facing windows in gold banners. Sam was in the kitchen area, preparing coffee, and all our bags were waiting by the front door.
He scraped the sides of the empty honey jar before stirring my coffee, then smiled over. "Hey."
"Are the sylph still out there?" I rubbed sleep from my eyes and scooted off the bed. The rest of the covers were cold; he must have been up a long time, getting ready for us to leave. "Do you think they'll let us by without trouble?"
"They haven't bothered us yet." He took our mugs and came to sit beside me, handing me my coffee.
It was true. We'd gone outside to clean, to get fresh air, to take care of Shaggy, and the sylph hadn't done anything more sinister than study us.
Ceramic warmed my hands as I breathed in bitter and sweet steam. "Sam, about last night..."
He inclined his head toward me, black hair falling across his eyes.
In all the months I'd known him, he'd never said my feelings were stupid. He never made me feel wrong or dumb. He'd always taken me seriously. I could trust him.
"I don't know why I reacted like that, after you said—" I stared into my coffee. "I didn't want to start crying. It's embarrassing. I'm sorry."
He caressed my cheek, my neck. "There's no reason to be embarrassed or sorry. It's fine. I think...I think I understand."
"Can you explain it to me?" I choked a laugh. "Because I don't understand at all."
"No, because if I'm wrong, I'll be really embarrassed."
"Thank you." I leaned on his shoulder.
He kissed the top of my head, and we stayed quiet as sunlight moved across the floor. "After someone followed you home, I gave you a knife."
"Yes." I shivered closer to him.
"Did it make you feel safer?"
I considered. At first, I'd tried to squirm away from the weapon, but it was beautiful and had later saved me from Meuric. I carried it everywhere now, though I tended to use it for holding down paper more than stabbing.
"Yes," I said at last.
"All right." His voice grew distant, disturbed.
"What is it?"
"Nothing you need to worry about." Sam was a rule follower. He didn't do things that might get him in trouble. He plain didn't think about trouble at all.
But whatever he had in mind now, it was as un-Sam-like as the sylph outside.
#
# GRATITUDE
A COUPLE OF hours later, the sylph watched us go. They stayed by their trees, moaning pitifully and huddling together when a chill wind snapped through, but they made no motion to follow us. I had pockets full of sylph eggs, just in case.
As Sam and I headed west into the woods, the pony on a lead behind us, the sylph wailed and sang part of a symphony we'd been listening to the other night.
I shivered deeper into my coat. What did they want? Nothing in the lab had offered an explanation. Aside from the poison, I was just as confused as before.
"Come on," Sam said, gentle as ever when pulling me from fearful contemplation. "We need to decide something very important."
"What's that?" I tugged my hat over my ears and adjusted my fingerless mittens, trapping in as much heat as possible.
"Which duet we'll play for Sarit first. Do you have a preference?"
I grinned and let him distract me with talk of music for the next several hours, though both of us kept checking over our shoulders for shadows that didn't belong.
During our hike from Purple Rose Cottage to Menehem's lab, autumn had only been creeping into the leaves, weaving red and gold and russet with the green. Now, as we pushed toward Heart, an autumn carpet crunched beneath our boots.
A deep roar sounded, long and rumbling. I stiffened and reached for my knife—as if it would do any good if we were about to encounter a bear—but Sam just took my forearm and drew me off the road.
"Stand back here." As the roar grew louder and higher, Sam slipped one hand around mine and held tight to Shaggy's harness with the other.
It wasn't a bear growl; the sound was too long and even and mechanical. A low-flying air drone approached in a torrent of leaves. Metal glinted in dappled sunlight, the only thing I could see through the leaf storm, and the noise grew so shrill I covered my ears.
Then the drone was gone, its sound falling lower as it vanished down the road. Leaves rained down on the sides of the road, showers of gold and red and russet, leaving the cobblestones mostly clear.
"It's safe now." Sam drew Shaggy and me back onto the road.
"A drone to clear the roads?" I gazed after the thing, but it was long gone. Only flurries of autumn leaves gave evidence to its passing. "How does it know where to go? And why is it so loud?" Labor drones were typically quiet.
"There are sensors under the roads, which tell if there's anything covering it for long periods of time. Rain doesn't matter, and moving traffic doesn't set it off, but snow and lots of leaves do. Even dead animals. It can tell what kind of material is covering the stone, and appropriate drones get sent out."
"Stef's idea?" It sounded like something she would insist on, keeping the roads clear even when traffic outside of Heart was uncommon.
"And the noise they make." Sam tugged Shaggy's lead and the pony snorted, ears twisting to listen to the retreating drone. "We found out quickly that with the quieter models, animals didn't know what to do. With the noise, they tend to run."
"Instead of waiting to get hit on the head?"
"Exactly. Now," he said, "we need to talk about your posture when you play your flute. You keep letting the end drop. Is it too heavy?" His tone was teasing.
"No," I mumbled, because he was right. It was just laziness.
"You'll get a better sound if you hold your flute straight."
"I know, I know. Do you keep yourself up at night coming up with new things to correct me on?"
He chuckled. "Do you keep yourself up at night coming up with new ways to see if I'm paying attention?"
"That's exactly what I do." Sighing happily, I lifted my face to the perfume of autumn. The scents of turning leaves and decaying grass tickled through me, and as we left the sylph and lab behind, a knot inside my chest began to loosen.
A new tangle formed as we drew closer to Heart, and two days later we woke to clouds falling across the sky in great splashes. The air, so recently crisp and exciting, now felt heavy and close with waiting. As we finished packing our belongings, I hunched deep into my raincoat, wishing it would just rain already.
The sky rumbled and the ground shook. Water poured from the clouds, soaking the remainder of the journey to Heart with misery.
The rain pounded on us at all hours, dripping through the autumn foliage and revealing thin spots in our tent that night. The temperature dipped, and by the time we approached Heart the next day, my wool clothes were sodden and smelly, chafing my skin. I entertained vivid fantasies of a hot shower.
At last, the city wall shone white atop the plateau, and beside me, Sam muttered something in relief. Beneath his hood, his expression melted into what mine must have looked like when we'd left Heart.
But seeing the pristine white tower that soared into the clouds, muscles in my neck and shoulders crawled with tension, and all I could think about were Janan's words to me: mistake. You are a mistake of no consequence.
I jerked my gaze downward, pulled in a breath, and twisted my hands around in my mitts to distract myself. Even with the sylph, our weeks away had erased the stress I'd barely realized I'd been living with in Heart. And it took only one look to bring it back.
"Are you all right?" Sam's voice came just over the pounding rain. "Ana?"
I nodded. "Let's get it over with."
Geysers steamed fiercely in the cold, making the whole plateau misty and difficult to navigate.
"Stay on the road," Sam reminded me, though it wasn't necessary. Some of the ground here was very thin; beneath us, an immense chamber of magma pulsed and boiled, releasing its energy in bursts of steaming water and bubbling mud. Nevertheless, I let him guide me to the Eastern Arch, and waited while he pressed his hand against the soul-scanner. A moment later, it allowed our entrance.
Inside the guard station, we dried ourselves and Shaggy, and sent the pony with the guard on duty so he could be fed. The guard smiled at me. I sort of recognized him from Templedark. Had I warned him against dying? The whole night had been too chaotic for clear memories, and he took Shaggy and left before I could ask.
"Do you want to wait here until it stops raining?" Sam asked when we were alone.
"No, we might as well walk to the other side of the city now. Who knows how long it will be before the storm passes?" I pulled out my SED and sent a message to Sarit to meet us at Sam's. "But we are going to have a big jar of honey waiting for us. Sarit likes rain, right?"
Sam grinned and hefted four of our bags onto his shoulders, leaving two for me. Just those were more than heavy enough.
We headed outside, into Heart, trudging under the weight of our belongings. East Avenue was dark and quiet, except for the driving rain, so we hurried down the road without interruption. Mills and warehouses of the industrial quarter watched us from the south; evergreen trees blocked the northeastern residential quarter, leaving only the occasional street as proof that people lived there.
As we entered the market field—the wide expanse of cobblestone surrounding the temple and Councilhouse—Sam moved to walk between the temple and me. He didn't say anything about it, and he knew I didn't like the temple, but I wasn't sure it was an entirely conscious move, either.
We turned onto South Avenue. A side road here and there, and finally we came to his walkway, covered with wet leaves and broken twigs. The fruit trees were bare, and at one side of the house, chickens and cavies' buildings were nearly invisible in the rain.
"Ready to get out of the weather?" Sam asked, hitching his load of bags again.
Yes, definitely, but I wasn't eager to box myself into one of the identical houses of white stone. Walls shouldn't have heartbeats. They shouldn't. And as welcoming as Sam's house was otherwise—pine-green shutters and doors, rosebushes below the windows, and a generous garden—it was still made from that stone. It had windows and doors in the same places as every other house in Heart. It wasn't natural.
Still, I didn't want to stand out in the rain staring. I followed Sam indoors and dropped my bags on the mud rug. Water soaked the wool threads immediately, turning the gray a shade darker.
Sam stripped off his outer clothes and boots, leaving them behind as he moved between all the instruments in the parlor. The sheets covering the piano, harpsichord, cello—all the large instruments on the floor—had been moved away already, probably courtesy of Stef or Sarit.
I unloaded all my wet belongings and clothes, muscles creaking with relief, then hurried up the spiral staircase and into my washroom for a shower.
When I was warmed through, dry, and clothed in a dove-gray sweater and thick black pants, I skipped downstairs to find Sarit and Stef making tea in the kitchen.
"Ana!" Sarit abandoned the kettle and wrapped me in a hug. "You're back! And just in time. I got a message earlier saying that Lidea went to the rebirthing center this afternoon, and Wend will send a message when we should come. They'd have been so sad if you couldn't make it."
"Ugh, the rain, though." I pulled my damp hair into a quick bun. "Our tent had a leak. I've had quite enough of the rain."
"But you're going, right?" Sarit narrowed her dark eyes at me. "Because I will put you in one of my tiny bags and carry you if I have to."
"I'll go! Anything but the tiny bag." With Sarit appeased, I shifted and hugged Stef before accepting the hot mug of tea she offered. "Missed you."
She shook back a length of blond hair and kissed my forehead. "You too."
We headed into the parlor, where Stef and I sat on the sofa, and Sarit perched on the piano bench. "Think he'll notice?" She glanced toward the stairs; Sam was somewhere up there, finishing washing or unpacking. I didn't even want to think about unpacking, but my bag stared at me from the door, waiting.
"Oh, he'll notice," Stef said. "But he won't mind."
Sarit grinned and caressed the row of ebony and ivory piano keys. "I call this one 'Bumble, Bumblebee.' It's for you, Ana."
I laughed and leaned back to listen while she played a silly tune that seemed to involve more picking notes at random than anything. Eventually Sam came down and sat on the arm of the sofa next to me, and everyone caught up with one another.
In all my years of living in Purple Rose Cottage with Li, I'd never imagined this: sitting in Dossam's elegant parlor, surrounded by glorious instruments I'd only dreamt of seeing, and listening to my friends discuss their days.
I had friends.
It was more than I could have hoped.
Stef was fierce and intimidating, possessing this grace so practiced it was unconscious after all these generations. Sitting next to her always made me feel skinny and awkward. And while Stef looked like sunshine, Sarit looked like nighttime, with dramatic dark hair and eyes. They were both so beautiful it hurt.
But they were my friends. My friends. They liked me for some reason. And Sam—Sam had said he loved me.
I leaned back, scribbling happiness into my notebook and listening to the melody of my friends' voices.
Sam glanced over and lifted an eyebrow. "Diary?"
With a shrug and a smile, I closed my notebook. I'd show him what was inside when I was ready.
After a half hour of talking and drinking tea, the patter of rain let up, and Sarit checked her SED. "Looks like Lidea is ready for us. We should go while it's safe to walk outside without gills."
We put on coats and grabbed umbrellas, and the four of us went out, Sam and Stef walking together, and Sarit with me. The world smelled of damp grass and leaves, fresh, in spite of the way everything was dying at winter's approach.
"Long trip away," she muttered. "Just you and Sam. A romantic gift of a flute. So have you"—she lifted an eyebrow suggestively—"you know?"
Had I what? Had Sam and I done something together? Something that warranted a suggestive eyebrow? She must have thought I'd be embarrassed to talk about it—which meant whatever it was, I hadn't done it. "No." I bit my lip.
"Really? The way you two were at the rededication, I'd have thought months ago."
Heat rose to my throat and cheeks when I thought about the masquerade, Sam as the shrike and the way he danced with me. My face really burned when I remembered what happened after the masquerade, the way he'd touched me and made me long for something I couldn't name.
But then Meuric and Li had taken Sam prisoner, and forced me to live with Li until Templedark, when I'd escaped. After that... "We slowed down," I said. "Really slowed. Nothing has been the same since the rededication. That night was unique."
In both wonderful and horrible ways.
She nodded. "But you're happy with him? Slow, but good?"
"Yes, very." Nervousness fluttered inside of me. "He said he lo—" The word stuck on my tongue, and Sarit waited for me to finish, her dark gaze patient. I gathered the syllables in my mouth again. "He said he loves me."
A dozen reactions flickered across Sarit's face—I caught shock and joy and confusion—before her expression softened into understanding. "And you think...what?"
I shrugged.
"I want to know." She bumped my shoulder with hers and lowered her voice, though Sam and Stef were far ahead of us now. "Did you say it back?"
She wouldn't think badly of me. I could be honest. "I didn't."
"Did you want to?"
I pulled my flashlight from my coat and swept the beam across the wet cobblestones. It wasn't quite dark enough to need it, but they'd all keep going long after it was too dark to see. They could navigate Heart with their eyes closed. Sam had done it once when I'd bet him he couldn't.
Sarit touched my wrist as we turned the corner onto South Avenue. "It's okay if you didn't want to say it. Or you just couldn't." She was an echo, just as sweet and understanding as he'd been. Both of them made my heart feel like it might burst with wonder.
"I don't know yet," I said at last. I couldn't even explain to myself why I'd started crying when he told me, and I didn't want to bother Sarit with it. Not now. This would be a happy night. "But he told me again the next day. And the days after."
"Good." She was quiet as we headed toward the Councilhouse, where the hospital waited. "Don't worry about that other thing."
Ah, the thing that might be embarrassing to talk about. I bit my lip, half wishing she'd been more clear about it, half relieved she hadn't. "Okay."
"It'll happen when you're ready. Just...He loves you, Ana. If he said it, he means it. And I love you too. I'm still really glad you're here."
"Why?" I whispered, hardly able to believe she'd said it, too. How easy she made it sound, just freely giving love.
Sarit stopped and regarded me with a wry smile. "Just accept it, Ana. You can't stop your friends from loving you. You can't stop Sam from feeling the way he does. You know I admire that you question things, but this—this doesn't have to be one of them."
Gratitude kindled inside of me, almost choking my words. "Thank you," I said, and we hurried after Sam and Stef.
#
# REBIRTH
THE COUNCILHOUSE WAS an immense building with a wide half-moon staircase spreading out from the front. A huge landing waited at the top, just ahead of a series of double doors; sometimes the landing was used as a stage for outdoor concerts and dances, or just announcements. Though since Templedark, there hadn't been much call for celebration.
We climbed the stairs—two at a time for those with longer legs—and stepped around columns and crumbling statues. The human-made parts of the Councilhouse were old and falling apart, worse after Templedark. Nothing had been the same since Templedark.
Sam held the door open for us, and we headed toward the hospital wing. Now that we were almost there, it took all my effort not to skip. "I'm excited to see a rebirth. Do you think she'll let me touch the baby?"
"Probably." Sam fell into step with Stef, behind Sarit and me.
"Good," I said. "That was your last chance to tell me if asking would be rude."
Stef lowered her voice, intentionally just loud enough for me to hear. "Next she'll be wanting one of her own."
One of my own?
A baby?
"Right." I glanced over my shoulder to find Sam looking fascinated by the wall, and Stef wearing a smirk. "Because what I really need is to be responsible for someone. Because I'd be really good at giving a baby everything it needs, thanks to the great example I got from Li." I choked out the last words. I had no idea if I ever wanted to have a baby, but it certainly wasn't next on my list of things to do.
Darkness flashed through Sam's eyes, but he didn't say anything.
"The Council is approving a lot of couples right now. I bet you'd be approved, too." Stef acted oblivious to my response, or Sam's discomfort. "We can check the genealogies later to make sure they'd say yes. It's embarrassing when they say no."
The Council had to be careful of accidental inbreeding and unfavorable genes being passed along, and no one wanted to be responsible for future generations suffering poor eyesight or genetic disorders. The practice made me uncomfortable, but to everyone else, it was a way of taking care of their bodies.
She went on. "I think—"
Sam interrupted her, voice deep and dark. "Let it go, Stef."
"Fine. I was just being interested in your lives."
Sam gave a long sigh; that was what the end of his patience sounded like, which I knew from experience. "Passive aggression doesn't suit you," he said. "If you want to talk about this, then we should. But later."
"Later," Stef whispered, tone all pity, and I could almost feel her glare on the back of my head. "I guess I do have that in my favor."
I probably hadn't been meant to hear the last part. My face burned with shame and grief at my own inevitable demise. We didn't know for sure, of course, whether I'd be reincarnated after this life, but it didn't seem likely.
Next to me, Sarit's expression was twisted with discomfort.
"Here we are." I spoke mostly to pretend like I hadn't heard Stef's comment, though everyone probably knew better.
The birthing center was a warm, open section of the hospital wing, with silk walls pinned in place by metal shelves. We hurried past the lit Soul Tellers' office, toward the rebirth room with its cheery decorations and array of medical equipment—just in case. They'd stopped using most of it a century ago.
As we entered the crowded room, buzzing conversation paused and people glanced up to see who'd arrived. Lidea was propped up on the bed with her eyes closed, surrounded by a team of birthing assistants.
"Shouldn't be long now," Stef said, folding her coat. "You can put your belongings on a shelf, Ana. Looks like we'll have to stand, though. All the chairs are taken."
"Why are there so many people here?" I placed my coat and umbrella next to her things. "There's got to be at least four dozen. Are they all going to watch her have a baby?"
"Yep." Stef flashed a smile, almost like an apology for her insensitivity earlier.
I'd have to remember this kind of thing attracted crowds, because in the unlikely event I did ever have a baby, someone would be in charge of shutting the door.
Sam took my hand and guided Sarit, Stef, and me through the crowd of people chatting, speculating on who'd come back.
"Look," someone muttered, "the nosoul is here."
Shock spiraled though me, shame not far behind. I wasn't a nosoul. I wasn't.
Few people used the word "nosoul" anymore, so what had changed? Perhaps it was this birth: Lidea had gotten pregnant after Templedark, and everyone was nervous.
Still, I kept my face down as I walked, as though I could hide from the words.
"She'll curse Lidea," and "She's already cursed everyone. Her and Menehem. They planned Templedark," and "Dossam with her. He's no better."
Sam's hand tightened painfully around mine, but neither of us acknowledged the speakers. As much as I wanted to defend myself, this wasn't the time. Maybe I shouldn't have come. The last thing Lidea needed was for my presence to start a fight.
"Often," Sam said, as though we hadn't heard a bunch of people talking about me, "we can predict who'll be born, since there aren't a lot of possibilities. Maybe two or three. Their best friends usually attend the birthing to welcome them back."
We found a spot by the back wall, and I said, "A lot of people lost their friends."
Sam kept his voice soft as he turned his attention to the bed and birthing assistants gathered around. "Yeah."
Wend, Lidea's partner, stood beside her, petting her hair and whispering encouragement. Nearby, someone said she was pushing now, so it wouldn't be much longer.
I stood on my toes, but from our corner, I couldn't see more than Wend's head. There were too many people in front of us, half of them standing. I tugged Sam's sleeve. "I can't see."
Sam eyed the rows of people, and my nice view of their shoulders. "Go to the front." He nudged me. "I'll wait here."
I hesitated—some of these people hated me—but I refused to let them stop me from seeing my friend. I squeezed Sam's hand, then maneuvered through the crowd before I missed anything else. Right in time to see Micah, one of the birthing assistants, adjust the sheet over Lidea's legs and—Ew. It was really going to come out of her.
Sarit sidled up next to me. "Thought you could use some company." Protection, she meant, but I wasn't going to complain.
"Wow." I tried not to gape as Lidea groaned at another contraction. "That can't feel good."
Someone glared at me, and Sarit giggled.
Lidea grunted and—around a white-smocked birth assistant—I saw her face, lined with concentration. Her eyes were closed as if there was nothing else in the world. Just her and the baby.
True, most people weren't watching, but she made a lot of weird noises I'd have been embarrassed about. No one seemed to care, though.
It wasn't long before a last push brought the baby and its cry. Everyone cheered and called out, "Welcome back!" while Micah gave the baby to Lidea, who was flushed and sweaty, but grinned happily. Wend unbundled a small blanket and laid it across both of them.
"He's healthy!" Micah's shout made everyone cheer again. She put a dark green cap, embroidered with tiny ospreys and elk, on his head.
Sarit leaned toward me and muttered, "It's a running joke that there are only five or six newborn caps in existence. Everyone just passes them back and forth."
I giggled. "It does look suspiciously familiar."
After a few minutes, the cheering quieted and a pair of Soul Tellers stepped forward. Sarit and I ducked away, back to Sam and Stef.
"That was amazing," I whispered, pressing my spine against Sam's chest, relaxing when his arms circled my waist. "And kind of gross. It must have hurt."
"I'm sure she'd be happy to share, if you asked her."
I couldn't decide whether he was making a joke or not. Why would anyone want to talk about childbirth?
Maybe I'd see if the library had a book on it, instead.
Aside from Lidea's cooing to the baby until he calmed, the room grew silent as Emil, one of the Soul Tellers, approached the bed with a small device. It was a soul-scanner, like those used around the city to restrict access to armories and other secret places.
"Baby soul-scanners?" I asked.
Stef nodded. "They're new for the Soul Tellers, only fifty or so years old. Before that, Soul Tellers did blood tests, which were less reliable. They measured chemicals they believed the soul produced."
I hmmed. Sam had once mentioned that certain tests hadn't been reliable, and people would be called the wrong name until they were old enough to complain about it.
"Soul-scanners have been around much longer, of course," Stef went on, "but they work by measuring vibrations of the soul inside the body. Newborns tend to have erratic and excited souls. It took a lot of work to get around that."
"Huh." Maybe they'd thought the scanner was broken when I was born, if the technology was that new. Maybe they'd tried three or four times, and with different scanners, just to make sure.
"Hold his hand still," said Emil. "We should know in just a few minutes." They pressed the baby's palm against the scanner face and then tucked the blanket tighter. Being born must have been terribly shocking, and cold, but he stayed quiet, tucked against Lidea's chest.
Everyone in the room stared at Emil, all anticipation and hope that this baby was their best friend who'd been lost the night of Templedark. The number of possibilities was staggering, but worse, underneath rode a current of fear: glances at me, muttered prayers to Janan, and objects clutched to their chest.
The last must have been things belonging to whomever they hoped would return. A box, a key, a silk fan.
Emil lowered the device and gazed around, eyes settling briefly on me. I tensed as another wave of anxiety passed through the silent room. "Is something wrong with him?" The words barely formed in my mouth, and Sam squeezed me, as if to caution.
"Who is he?"
Lidea's expression twisted with worry. "Please just tell me."
Emil faced her, his tone sober. "He's a newsoul."
#
# NEWSOUL
I WASN'T ALONE.
I wasn't the only one.
I wanted to be sick.
All eyes fell on me, and the first ones I saw were angry and accusing. Sam's arms grew tight around me, ready to protect me from the inevitable storm. "Ana..."
Sam followed my gaze to the large man on the other side of the room, slowly standing, his glare locked on me. The man was enormous, with shoulders so wide he made Sam look small. Close-cropped brown hair made him look bald, and a few days' worth of stubble darkened his face. His name was Merton; I'd seen him leading anti-newsoul speeches and complaints to the Council.
Anti-Ana speeches, because there was only me.
Until now.
"This is your fault." He seemed bigger for all the rage building up beneath his words. As though anger were contagious, the room began to boil with it. "Meuric was right. Li was right. This one was only the beginning of replacements. Now Lidea has borne another."
On the bed, Lidea stared at the infant in her arms, like she wasn't sure what to do now. Tears trickled down her sweat-streaked face.
"Nosouls will replace everyone," someone in the back shouted. Panic pitched his voice high, and then it was lost under the wave of suspicion-filled mutters.
"We're being invaded!" Merton shouted.
A small cry of agreement went up, hesitant at first, gaining voices swiftly.
"When sylph infest the city," Merton roared, "what do we do? Capture them and send them beyond Range."
People nodded emphatically. A few cheered.
"When centaurs hunt in our forests," Merton went on, "we drive them out with gas that erodes the bonds holding together their two aspects."
My stomach dropped, but Merton held everyone's rapt attention. He looked just as eager to say what they were all waiting to hear.
"Now we need to learn to defend ourselves against this new threat."
He thought of us as monsters. This baby who'd barely drawn breath, and me. Several people thundered agreement. With Merton as the conductor, the shouts and rage crescendoed.
The baby wailed, and Lidea held him close, but she wept too. My friends yelled in my defense, and the birthing assistants ordered people to leave the room. No one obeyed. People kept shouting and pointing, pressing closer to me as the scowls and glares deepened. They practically burned.
Their heat filled me, leaving no room for disbelief or shock. How could I be shocked when some of these people had treated me with nothing but hatred?
But as the shouting grew and the baby screamed, my own anger replaced my fear. Like a geyser, pressure built inside of me, boiling with the heat of the cacophony all around—like the power of the Range caldera. I was ready to erupt.
"Stop!" I wrested myself from Sam's grip and climbed onto a chair. "Enough!"
They all stared—birthing assistants, observers, and Soul Tellers—and I imagined geyser steam wafting through the room, stunning them into silence. Only the baby cried, and then Lidea put him on her breast.
Silence.
Oops. Everyone was looking at me.
On the bed, Lidea cradled the baby to her. Sweat dripped down her temples, and her skin flushed bronze. The room smelled of salt and copper and other things I couldn't identify.
I focused on the geyser feeling, how furious I'd been about everyone scaring the baby, threatening to kill him as if he were some kind of monster.
They would not hurt him. I wouldn't let them.
"I was led to believe that you were all rational people who knew how to behave around an infant." My voice shook. So much for being strong like a geyser. "If you want to yell, do it outside. This isn't the place."
No one moved; I wasn't sure this was better than the yelling.
"If not for the baby, please show a little consideration for Lidea. Or don't you care about her anymore?"
That shamed a few people into slinking out of the room. I stayed on my chair as they passed.
"Anyone else?" I mimicked an angry expression Li had always used to force me to confess when I'd been listening to music. It seemed to work, though I felt more like a chipmunk addressing a room full of wolves. "We're here to celebrate a birth. If you can't do that simply because he's a newsoul, you're welcome to leave."
More people left. More than before. A few had the decency to look ashamed. I didn't bother hiding my disgust for any of them.
Across the room, Merton stood there with his arms crossed, his face crimson and contorted with rage. He stalked toward me.
Everyone watched, and Sam eased toward my chair, but when Merton reached me, he just glowered and walked around me—to the door.
I tried not to let my relief show. If he'd attacked me, there'd have been little my friends could have done. Merton was huge. And strong.
But he was gone for now. I focused on breathing, and trying not to crumple under the stares of birthing assistants, observers, and friends. Most of the hostile people had gone ahead of Merton, so why did my heart speed up now? Surely I should have been able to say something coherent in front of people who didn't completely hate me.
"I believe the tradition is to welcome newborns." Welcome them back, anyway. But this one hadn't been here before. He was like me. Newsoul. "I'll go first." I ached for him, this unnamed child facing an existence like mine. At least he wouldn't be the only one.
Sam offered a hand to help me off the chair, and I accepted. The last thing I needed was to fall on my face.
As I approached Lidea's bed, I imagined what the scene must have been like when Li gave birth to me, and the Soul Tellers announced I wasn't anyone. There'd probably been fewer people in the room. And all of Ciana's friends would have been there.
Ciana, whom I'd replaced.
I doubted anyone had welcomed me to the world.
I stopped by Lidea's bed. Someone had pulled sheets all around her and wiped sweat off her face, though her skin remained flushed with heat and anger and the labor of birth. Black hair hung in tendrils over her shoulders; the baby's hand reached upward at nothing, losing his fingers in the tangles.
Sam stood next to me, and everyone else queued behind him. Except Wend, Lidea's partner; he didn't leave her side.
I searched for the right words, but what did you say to someone who'd had a newsoul? Apologizing seemed wrong, because this wasn't bad, and I'd had nothing to do with it. The only thing that made me sorry was knowing how much everyone already hated him.
"Thank you." Lidea's smile was strained. "For making them stop. For making them leave."
"I couldn't let them continue." What if they'd hurt him? He was tiny, all splotchy red and brown skin, and his face scrunched up with the stress of being born.
She lowered her eyes. "The idea of having a newsoul—it was terrifying. And"—her voice caught with the confession—"humiliating. But holding him now, I'm glad he's here. I love him completely."
My throat tightened from choking back tears. "He's lucky to have you."
"I'm going to do everything in my power to make sure he's happy. I'll keep him safe."
So would I. "I'd like to welcome him. Will you name him?"
"I thought about naming him after you, in honor of your standing up for him." Her eyes were only for the baby. She didn't see the way my mouth fell open. "But that would be confusing, and I don't want to start a trend of all newsouls being Ana."
I hoped not. Li had told me they'd chosen my name because it was part of Ciana's name, symbolizing the life I'd taken from her. The name also meant "alone" and "empty."
"It was a generous thought, but not necessary." And I didn't deserve that kind of honor.
Lidea caressed his round cheeks, small nose. "It did give me another idea." Sweeter anticipation filled the room. "Anid is close."
My heart felt swollen as I reached, glanced at Lidea for permission, and then touched Anid's tiny hand. He didn't seem to notice. "Hi, Anid. Welcome to the world." My voice trembled as I whispered, "I'm really glad you're here."
We were in this together now. Neither of us were alone. Asunder.
He looked toward me with wide, dark blue eyes. He was beautiful, and I wasn't ready to move on, but people waited behind me, so I touched Lidea's hand, then Wend's, and gave Sam a turn. As the line moved, I watched how everyone else was with the baby, trying to memorize the faces of those who'd stayed. Were they friendly, or just polite?
After everyone but the birthing assistants had left, I offered Anid my finger again. His fist closed around it immediately.
"Don't let anyone call you a nosoul again," I whispered to him. "If they do, tell me and I'll take care of it for you."
Lidea looked amused. "Are you corrupting him already?"
"Just a little." I smiled so she'd know I wasn't serious.
"I'm worried," Lidea confessed. "After earlier, all that yelling." She squeezed her eyes shut, and tears shimmered across her lashes. "What if they really try to hurt him?"
Wend appeared by her side, hand on her shoulder. "Nothing will happen to him." When Lidea twisted toward him, he leaned over to hug her.
Sam touched my elbow and murmured, "Ready to go?" I nodded, and we said our good-byes, fetched our belongings, and headed for the exit.
It was raining again when we went outside, and fully dark now. Only the temple glowed, shedding watery light across the market field. Without conversation, we headed back to the southwest quarter of the city where all our homes were located. Sarit and Stef broke off onto their streets, close to ours.
Inside and dried off, I said, "Sam," before realizing I'd spoken.
He paused on his way to the piano, one hand drifting over my hip as he faced me. With his face in shadow, Sam's eyes were even darker, more mysterious, and heavier with the weight of centuries. Millennia.
"Once, you called me a butterfly, because my existence seems so fleeting to everyone else in Heart."
A line formed between his eyes. "Ana—"
"I know you didn't mean it to hurt me, and I know you've apologized a thousand times." I swallowed nerves caught in my throat. "That doesn't make my existence less potentially ephemeral. I could die and never be reincarnated."
"Please don't say that," he whispered.
"You, Stef, Sarit, others—you've made the Year of Hunger bearable. I didn't think I could have friends until you proved me wrong." I reached up for his shoulders, let my hands slide along the backs of his arms. "But the beginning of my life was terrible, and half the people still treat me like I'm responsible for Templedark and every other horrible thing that's ever happened."
He looked downward, like I blamed him for others' actions. "I'm sorry."
"Don't be. None of it's your fault. I just meant to say, I don't want Anid to grow up like I did."
"Lidea and Wend will care for him. So will we."
I nodded. "But it's not enough. You saw what happened in there. People were anxious to welcome back a friend, and then it was terrible. Within minutes, people were talking about killing him. If that's any indication of the rest of the city's reaction to his birth, when other newsouls start coming, there won't be anywhere safe. Not in the city. I need to make it safe. Somehow."
"Ana." Sam stepped so close I had to drop my head back to meet his eyes, and the way he said my name—it was same reverence people used in their prayers to Janan. My insides knotted up as he touched my jaw and kissed me. Softly, gently, aching with restraint. "Anything you need from me, just ask. I promise, we'll give these newsouls the chance you never had."
Hearing it in those words made everything so clear. Sam understood me better than I understood myself, and he'd known what I needed all along.
#
# LAKE
I'D BEEN RIGHT about the shift in Heart over the next couple of weeks.
Twice, when I went out on my own, someone threw rocks at me. People jeered and called me names. At the market, people refused to sell things to me without one of my friends there. Strange calls came on my SED, just loud breathing. Stef traced them for me and blocked them from calling again. Then Sam started getting calls.
I tried to ignore it. The rock throwing and SED calling were new, but all in all it wasn't much different from when I'd first arrived in Heart. The fear and anger were the same.
Every morning, Sam and I had music lessons and practice. I took Council-required lessons in the afternoon—they'd kick me out of the city if I didn't—and my monthly progress report was coming up. After my long trip to Purple Rose with Sam, I should have been trying to squeeze in more study to make up for time lost, but Lidea called and asked if Sam and I wanted to go to the lake with some friends.
Absolutely.
"What about your paper on the history of geothermal energy?" Sam asked, not quite hiding his smirk as we walked to Lidea's house.
"I'm sure you can see how devastated I am about going to the lake on the last warm day of the year. Spending time with you, with friends—Ugh. I don't know how I'll make it through the afternoon." I grinned and slipped my hand into his.
With Lidea, Wend, Anid, and a handful of other friends in tow, we headed out the Southern Arch, toward Midrange Lake. It was the biggest lake in Range, and mostly used for the city's fish and water supply, but there were a few beaches set aside for enjoyment. Sam and I had gone a couple of times over the summer.
Geyser steam wafted across the barren land between the city wall and the forest, reeking of sulfur. I wrinkled my nose until the wind shifted to blow the stink away from the path.
I held on to Sam's hand, listening while Stef and Orrin inquired about the baby's health, and Whit and Armande discussed the effort to rebuild sections of Heart that had been destroyed during Templedark.
"The Council isn't even trying," Armande complained. "Have you noticed the statues by the Councilhouse? And the relief over the front? Not to mention the stairs."
"Those things are hardly as important as rebuilding the mills and agricultural areas." Whit shook his head. "Lots of private gardens and livestock were destroyed, if not by sylph or dragon acid, then by drone fire and neutralizing chemicals. Even with sharing and appropriating supplies from"—his voice caught—"the darksouls, it's going to be a hard winter."
"Because the Council stored food in buildings that won't stand up to dragon acid."
"Armande," Whit said gently, "even if they'd put everything in the Councilhouse, it could have been destroyed just as easily. Templedark, remember? The walls were useless."
The white stone had repaired itself when Janan awakened, so some people preferred to believe the cracked temple had been only a nightmare.
"Anyway," Whit continued, "you've completely changed the subject. You're upset about the statues and stairs, but don't you think that's a little shallow, considering all the things that need to be repaired?"
Armande snorted. "Maybe so, but I'm the one who has to look at them every day."
"You don't have to set up your stall. Let people make their own pastries if it's so difficult to look at pockmarked statues."
Armande pressed his palm to his chest. "You're condemning even more people to starvation. Or, at the very least, bad breakfast. Besides, our art is a testament to our society. It's a symbol of our achievements, like your library and Sam's music. It's something to be proud of, and we should take care of it."
I thought about that as we stepped into the shade of fir trees and headed down a smooth path that thumped solidly beneath my boots. We were off the thinnest part of the caldera.
Sam held aside a low-hanging branch for me, then ducked under after.
"Thanks." I glanced back; the branch was as big as my arm. "If you'd left it, we could have had matching bruises on our foreheads."
He laughed. "That's not as romantic as matching hats or belts."
"And that's not romantic at all. Did anyone really do that?"
"Some did. About a thousand years ago." He rolled his eyes, but his grin widened. "I don't think I was ever so happy to see a fashion pass. The hats got worse every year. Taller, bigger feathers, ridiculous shapes. It was terrible."
"Did you ever wear matching"—I couldn't believe it was a real thing—"hats or belts?"
He shot a look that said I'd wasted my breath asking. Of course he hadn't. He didn't even like attending the rededication of souls.
My tone slipped toward mocking. "I should have known not to question your sense of fashion."
Sam squeezed my hand, his smile full of mischief. "If you asked, I'd probably find us matching hats."
"You're such a tease."
Another ten minutes and we arrived at the beach, all sand and frothy gray water, veiled by evergreen trees on three sides. Immense snow-blanketed mountains stood on the horizon like walls, shaded blue and gray in this weather. These walls, unlike the one around the city, made me feel safe. Protected.
"The beach looks bigger today," Sam said, as we came off the narrow path, the only access to the beach.
Orrin glanced southward and scowled. "It is. The water line is lower."
"What does that mean?" I asked.
"Nothing, probably." Orrin and Whit exchanged looks, and Orrin shrugged. "We've had a lot of small earthquakes. Nothing you would have felt, and earthquake swarms don't necessarily mean anything. They're just part of living on the caldera."
I knew that. "But would a tiny earthquake change the level of the lake?"
"Maybe." He gazed toward the water, probably wishing Rahel—the soul who'd been responsible for monitoring these things—were still alive. "A crack might have opened in the bottom of the lake. We're on such a thin crust of land here. But I'm sure it's nothing to worry about."
"If you say so." People used the lake for water and fish, so if the level dropped, surely that would have an impact on life in Heart. But I didn't want to get into an argument on such a nice day. When the group stopped in the middle of the beach, I helped Sam spread out a blanket and then squatted by the basket of snacks. Surely Armande had packed honey-glazed buns.
"Don't worry about the caldera," Orrin said, crouching next to me. "Whit and I are taking up some of Rahel's work. If you're interested, you're welcome to join us when you have time."
"Thanks. I might."
He smiled and peered into the basket. "Have you seen muffins?"
Soon everyone was chatting, laughing, listening to waves brush sand. A few cranes and herons braved the day, but most waterfowl had already migrated south. The baby cried, but Lidea held him close, wrapped up in soft wool blankets, threaded with cinnamon-colored buffalo yarn for extra warmth.
Wend flirted with Lidea, while the others talked about their projects or music they were hoping to play together. After admiring Anid's tiny fingers and nose and ears—all pointed out by Lidea, as if I couldn't figure it out myself—I mostly lay on the blanket for hours, writing in my secret notebook and taking in the afternoon's thin near-winter sunlight and the happy sound of friends' voices.
The voices stopped.
Suddenly conscious of the change, I looked over my shoulder to follow everyone's stares.
By the forest, shadows twisted toward sunlight. Five sylph. Ten. They emerged from the forest, silent across the sand.
Dread rushed through me, chased by fear. How had they gotten here? What did they want?
"Do we have sylph eggs?" Stef whispered, reaching for the nearest bag.
"No." I didn't have to look. Why would we have sylph eggs when we were so close to Heart? Midrange Lake should have been safe. There had to be a hundred sylph traps all through the forest between here and Menehem's laboratory.
No sylph eggs. What did we have?
"Protect Anid," I said, standing. "When you can get him out of here, do it." The sylph on the edges of Range hadn't done anything more threatening than sing at us, but here, with more people? With Anid?
I couldn't take the chance of them hurting him.
"What are you doing?" Armande asked, even as everyone stood and made a protective circle around Lidea and Anid. As if that would stop sylph.
No sylph eggs. My jacket pocket held my knife, the temple key, and my SED.
Unearthly cries shivered across the beach as I drew the SED from my pocket and sent a quick message to Councilor Sine, asking for guards and sylph eggs at the lake. Then I shifted to the music player.
"Get behind me." My voice quivered, and my heart beat too fast, but toward the forest the sylph had stopped, and they were looking at me. "When they're distracted, head toward the path. Don't run or they'll chase. They're predators. They can't help but chase." No one knew whether sylph somehow ate what they burned, but they would chase.
"Don't be stupid," Stef said. "No one's leaving you."
"Please." I shot her a desperate look. "Please just trust me." Maybe these were the same sylph. Maybe they weren't. I had to try.
"I'm staying with you." Sam touched my shoulder, looking uncertain about my plan but determined to remain at my side.
Grateful for his presence, I set my SED volume on high, and strains of a nocturne floated on the air.
The sylph, which had all been curious before, snapped to alertness. All ten focused on me as I stepped to the right, away from the path. Away from my friends.
The breeze picked at the melody, sweeping it across the beach and toward the sylph. They followed the sound bit by bit, edging closer to me like they were afraid I'd take the music away.
The SED had good speakers for its size—Stef had designed everything, consulting Sam for sound quality—so the music was loud and clear as I lured the sylph away from the group. The line of shadows followed me, entranced by the long chords and arpeggios.
"Go." I tried to keep my voice level, hoping Stef, Lidea, and the others would hear me. "While they're distracted."
Orrin and Armande motioned for Lidea and the baby to go first, toward the path off the beach. Pine boughs rustled, but the sylph didn't turn. They watched me, slipping closer as I bent and placed the SED on the ground. I backed away, and they writhed toward the device, seeming to stare down at it.
Their cries were like wind over canyons: hollow, melancholy, eerie. Heat rolled off them in waves, reeking of ash and death. Any sane creature made of flesh and bone knew to stay away from sylph.
That was my plan as well. Soon my friends would be on the path to the city, and then Sam and I would follow. I'd have to leave the SED to keep the sylph distracted, but surely the Council would replace it. But how would I explain this? Call it an accident? Stef and everyone else had seen me pull out my SED with a purpose.
While my friends escaped, sylph swirled around the device as though dancing. Their moans carried across the graying beach, and then one of their cries hit a high note at the same time as the nocturne. The cacophony snapped, a sensation like walking into a crowded room—and suddenly understanding individual voices and words. One mournful wail became the melody, while another sang countermelody. They all chose parts, like they were members of one of Sam's orchestras.
Near the path, Stef—the last to escape—turned around, her mouth hanging open. I motioned for her to hurry, and she turned back to the path.
The nocturne swirled around the beach, above the swish of waves and alongside the rustling of pine boughs. It was beautiful, all of them singing. Part of me wanted to lose myself in the haunting sound, but I knew better.
The nocturne ended.
Sylph fluttered around the SED, waiting. A tendril of shadow hovered over the small device, but nothing happened.
I glanced over my shoulder. Stef and the others were still visible between trees, and if anyone from the guard station was coming, I couldn't hear them.
Sam and I weren't nearly close enough to the path off the beach. Not that reaching the path would magically make us safe. Sylph could fly between trees and catch us in no time.
I shot Sam a pleading look, silently urging him to stay put while I started toward the sylph and my SED again. He nodded once, watching me with an intense protectiveness. But he'd let me do what I needed to do. He always did. And all I had to do was program the SED to play enough music to give us time to get back to Heart.
Sylph whistled and moaned, watching my approach like I might attack them. My heartbeat thundered in my ears as the sun dipped behind mountain peaks. Golden shafts of light spilled across the beach and the dusk-gray water, and the sylph grew darker. Taller. They hummed at the crunch of my boots on sand.
"Back away," I whispered, and put my hands palm-out as though I could push them. "Back away."
A line of inky shadows shuddered away from the SED, keeping an even distance from me. Sam gasped.
Shivering in spite of the heat billowing off the creatures, I bent and retrieved the SED. "Stay back," I murmured, tapping the SED's face and selecting all of Sam's music. Typically, selling recorded music was part of how he earned enough credit to feed himself, but as his student—and other things—I was able to access all his music. I'd been grateful before. Now his music would save lives.
The opening chords of Sam's Phoenix Symphony flowed from the speakers. I put the device back on the sand. The sylph stuttered forward in unison, and stopped when I held up my hands.
A weak, panicked laugh escaped me. I was holding up my hands? The same hands a sylph had burned less than a year ago? Sometimes my idiocy astounded even me.
As before, the sylph began to dance, writhing like dark flames. They flowed in and out of one another, moving closer to the SED as I took measured steps away.
Soon, one picked up the melody, sang the notes just behind the piano, violins, and flutes. Close. So close.
Sam reached for me when I glanced backward. I moved as quickly as I dared. My heart raced and my hands shook, but I didn't want to give the sylph any more indication of my fear.
I was halfway back to Sam when one of the sylph trilled and turned its eyeless gaze on me. The weight of its attention made me stagger as dusk deepened across the beach.
"What?" My words came as a whisper.
The sylph trilled again, twisting closer to me as it took up its part in the music once more.
It wanted me to sing? I stayed where I was, boots planted in the sand, conscious of Sam behind me. But when the sylph trilled again, I hummed the next few measures of the melody.
As though electricity surged through the sylph, they all shivered straighter, taller, and closer to me. They seemed eager and—welcoming?
That sounded ridiculous, even in my head, but the last thing I wanted to do was anger them. I held up my hands as though pushing, stepped forward, and kept humming.
They breezed backward, eerily intent while they sang.
I could feel Sam's attention on my back, and hear the edges of hushed orders in the forest. A rescue, I hoped, because it was obvious the sylph wouldn't let me go. At least one had recognized my plan to escape. If I tried again, they might attack.
The sylph eased back as I moved forward. Music played between us, warm and joyful, with a flute duet and bass thudding like a heartbeat. We all—the sylph and me—stepped on the beats.
"What in the name of Janan?" A boy's voice carried across the twilight beach, stunning the sylph, stunning me.
Someone swore. "What is she doing?"
"She controls the sylph."
I spun around, too shocked to respond, and suddenly the sylph fanned around me like an escort or army or dark wings, all heat and ashy reek. They keened, voices so high my ears ached, and two of them shot forward to attack the intruders.
"Stop!"
At my cry, the sylph halted and their wailing silenced. Breath rasped across the gloom, the only sound in a pause between movements of the Phoenix Symphony.
"How did she do that?" The growling voice was familiar. Merton? That guy was everywhere.
"Janan has indeed forsaken us," someone else muttered. "Nosouls will be our ruin."
"Shut up," I hissed. "I don't control sylph any more than you control the weather or your own reincarnation. They like music. That's all." Was it?
Councilor Deborl stepped forward, holding a brass egg the size of two fists. "We'll take care of them now, Ana." Physically, he was younger than me, but he held himself with all the importance of his rank. Even his tone was a reminder of his true age.
Darkness shuddered on either side of me: sylph. Heat bloomed against my bare face and hands; I could even feel it through my coat. But the sylph never moved too close—close enough to boil me alive, anyway. Any sylph was always too close.
And yet, they'd responded to music. Now, with the second movement beginning on the SED behind us, one sylph hummed quietly along with the bass line.
I glanced at Sam; the way he stared at the sylph told me he heard it, too.
"Do the eggs hurt the sylph?" I bit my lip, regretting the question immediately. Now Deborl and the guards would think I sympathized with sylph.
"Does it matter?" Deborl moved forward, and a dozen others followed. They all carried eggs. "Sylph will burn you alive. They almost did before, remember? And they were just about to attack Lidea and the other newsoul. I thought you cared about them."
"I do!" Too shrill, too desperate, but the sylph waited beside me as Deborl, Merton, and the others approached. "I do care, but look, they're not hurting anyone. What if they just left on their own?"
Sam shook his head, a warning.
"Ana," Deborl said as he neared the first pair of quivering shadows. The sylph didn't move. Why didn't they flee? They would be trapped. "Ana, clearly the sylph listen to you. You're a special soul."
Hah.
But I couldn't forget that night at Purple Rose Cottage, or the way they'd stood guard outside Menehem's lab. Why were they following me everywhere?
He continued. "Word of this will spread. If you use your gift to help us, perhaps the popular view of newsouls will change."
Oh. He wanted me to tell the sylph to get inside the eggs, though why would they do that just because I said? Anyway, they weren't supposed to be able to understand words.
They weren't supposed to be able to come inside Range, either. These had been very clever, and very determined to get here. Why? To sing at me again? To attack Anid?
They hadn't attacked him, though. They'd moved toward him, yes, but a group of adults posed no barrier if the sylph had wanted to kill him. They could have killed all of us.
But they hadn't.
They'd chosen not to kill Menehem during his experiments.
"Ana?" Sam moved closer, though sylph stood between us and he looked torn. Risk the sylph to stand by me, or stay put.
Near the first two sylph, Deborl and Merton twisted the eggs to activate them, and within seconds both shadows would be sucked in—
"Run!" I shouted. "Fly away. Go!"
Obsidian-black shadows shrieked and surged into the woods, moving around people and the eggs meant to trap them.
People yelled, Sam rushed to my side, and soon Deborl and his guards surrounded me. Blue targeting lights flashed against my coat: the guards aimed lasers at my chest.
"What are you doing?" Sam stepped in front of me, reaching behind himself to touch me, make sure I wasn't dead. "You can't shoot her."
Deborl motioned, and everyone lowered their weapons. Targeting lights flickered off. "No one is shooting anyone."
Yet.
"Ana," the Councilor said, "why did you tell the sylph to flee?" There was no fear in his voice, only calculating curiosity.
I stepped out from behind Sam. I didn't need a human shield. What would I have done if he'd gotten shot? "It was the right thing to do." My voice shook. I swallowed and tried again. "They hadn't hurt anyone, and they were listening to me. I don't know why."
"So you took their side?" Deborl cocked his head.
"I didn't take their side. I accomplished the same thing you were trying to do, but without trapping them inside eggs, and without anyone accidentally getting burned. They'll go away now." I hoped.
"Hmm. Perhaps." Deborl reminded me of Meuric, the Council's former Speaker, and the boy I'd killed inside the temple. They were both short and skinny, physically younger than me, and devoted to Janan—though Deborl's devotion seemed to depend on the season, the phase of the moon, and whoever happened to be standing within earshot.
I hadn't trusted Meuric; I didn't trust Deborl, either.
I stood as tall as I could make myself, trying not to shiver in the evening breeze, and with the adrenaline fading from my system. "We're going to leave now." My voice trembled.
"Very well." Deborl twisted his sylph egg to deactivate it, then pressed the cold object into my hands. "Try not to get into trouble between here and the Southern Arch. And"—his gaze flickered to Sam—"I expect to see both of you in the Council chamber in the morning. Tenth hour."
"But we have—" Music practice, but Sam touched my hand and shook his head. "Fine." I turned away to reclaim my SED, still determinedly playing the second movement of the Phoenix Symphony. Sand swished as Deborl, Merton, and the guards headed up the path.
"Are you all right?" Sam touched my shoulder, my cheek. "I can't believe they threatened to shoot you."
He wanted to know if I was all right because of Deborl and the guards. Not because of the sylph. The sylph, as crazy as it seemed, had been ready to protect me. From people.
Oh, how our lives had changed. "I'm all right." I hugged Sam close, my cheek pressed against his chest so I could hear his racing heartbeat. "We're both all right." Because they'd pointed lasers at him, too.
Then, in silence, we packed what was left of our afternoon with friends and trudged toward Heart.
The huge outer wall blocked the sky as we drew near. Solar panels and antennae glimmered like needles in the moonlight. From the center of the city, the temple rose into the clouds, a shining beacon.
I kept my eyes on the Southern Arch, nearly big enough for a dragon to fly through, but the temple seemed to watch my approach no matter how I avoided looking at it.
Janan's presence hung over the city as thick as ash. I imagined I could feel the heat of molten rock and boiling mud churning just beneath my feet. If Janan cared about his people at all, why had he built Heart over the most powerful volcano on the planet? Surely not even the temple would survive if Range erupted.
"What will we tell the Council?" Sam pressed his palm to the soul-scanner, and the gate swung open.
"I'm not sure." I bit my lip, confused and frustrated and ready to collapse into bed. "They'll think I like sylph now. Or that I'm like Menehem."
One thing was for sure: I'd just made life for newsouls a lot worse.
#
# QUESTIONS
IN THE MORNING, Sam and I headed to the Councilhouse, a firm plan in mind: deny. They would get nothing about the research Menehem had left to me, and even less about the lab east of Range.
I fidgeted with my notebook as we headed up South Avenue, wondering if I'd done a good enough job hiding the folders and diaries Menehem had left, and the books I'd stolen from the temple. I still hadn't figured out how to read the temple books, though not for lack of trying.
"Let's walk around the right side," Sam said as we approached the market field.
I craned my neck to see why we were heading for a different doorway into the Councilhouse, but all I could see ahead were people walking, talking, sipping from cups of Armande's coffee. "What's going on?" I was too short to see over the crowd.
"Nothing," Sam said too quickly, and winced when I eyed him askance. "One of Merton's public rants. He's on the steps getting people worked up."
"Ugh." Fortunately, there were more doors into the Councilhouse. After what happened yesterday, I really didn't want to go anywhere near Merton.
"Just last night, the newsoul set sylph free by Midrange Lake," Merton cried. "She controlled them. They did as she said. I was there. Janan forbid, but what if all newsouls have this power?"
Shouts rose up, sounding afraid, defiant, angry.
Merton roared louder. "Newsouls will rip Heart asunder! We've spent five thousand years perfecting our lifestyle and honing our talents, and now this."
I sighed and stared at the cobblestones. "I'm sorry, Sam."
"Why?" He walked between Merton and me, guiding me through a thin spot in the crowd.
A few people sneered at me. One shouted, "Sylph-lover!" but most just frowned and turned away. Maybe they didn't completely believe what Merton was saying. It did seem too fantastic.
"For dragging you even deeper into this mess." I ducked through the door when Sam hauled it open for me. "After our trip"—I didn't specify where, in case anyone overheard us—"and what we learned there, you must be pretty nervous."
Darkness flashed in his eyes, something he wasn't telling me, but it vanished quickly. "I want you to feel safe. I'd never regret your feeling safe." He followed me inside. "If I can't give you that, I at least want you to have answers. I'll help you find them however I can."
"I know people have started calling your SED to yell, now that Stef blocked them on mine." I hated that they were trying to make him miserable, too.
He shrugged. "It's okay. I can deal with them."
Why? Why would he endure all this for me? Was this what it meant to be loved? If you loved someone, could love make you strong like that?
I hoped I could become that strong.
Sam rested his palm on the small of my back as we walked through the ornate halls of the Councilhouse. Paintings lined the walls, most depicting faraway places with cliffs or endless stretches of sand. Closer to the library, there was a painting of tropical fish in a coral reef; that was one of my favorites, though I'd never been to such a place. One day, I would. I hoped.
When we reached the Council chamber, we were told to wait. I filled the time by writing in my notebook. Sam spent the time frowning at a wall. "The piano needs a little work, don't you think?"
I glanced up. "Maybe?"
"It sounds off. I'm going to look at it when we get home."
The piano sounded spectacular to me, but I didn't have his ear, so I just smiled and leaned on his shoulder.
When we were called, I followed him into the Council chamber and dropped my notebook onto the table, which ran the length of the room. It was an ancient piece made from a dozen species of wood, inlaid with beautiful swirls of metal. Once a month, the Council called me in for a progress report; while they droned on about the importance of mathematics, which I already knew, I had ample time to search for patterns across the smooth face.
Ten Councilors sat across from Sam and me, some familiar faces, some new since Templedark. Four Councilors had been confirmed dead that night, and the fifth, Meuric...they'd never find him. The replacements were mostly young, one barely past his first quindec, the age when people were allowed to start working again.
"Hello, Dossam. Hello, Ana." Councilor Sine brushed aside a wisp of gray hair that had escaped her bun. "This session is closed for now, but later the recording will be archived and available, all right?"
It wasn't really a question, so I didn't respond.
She went on. "The Council has been informed of yesterday's incident. Please, tell us about it."
"Okay." My heartbeat fumbled as I sat and tried not to pay attention to everyone looking at me. "A bunch of us went to the beach yesterday. Sylph came. I brought out my SED to message you."
"I remember," said Sine. "Go on."
"Then I turned on the music."
"Why?" asked Deborl.
I was a terrible liar. "Um." A really terrible liar. "I think Menehem mentioned something to me during Templedark. He said music calmed sylph."
"And you never told us that before?" Councilor Frase lifted an eyebrow. "That would have been very useful information to have."
"I forgot. I only remembered at the lake." How much deeper could I dig this lie? It made me feel dirty, even though they'd throw me out of Heart if they knew the truth.
"Then the sylph followed your orders," Councilor Antha said. "Yesterday, at the lake. Reports say you shouted at them to flee, and they did."
"Do you have any idea why they did that?" Sine laced her fingers, not at all the friendly Councilor she'd been when we first met. Now she was the Speaker, always looking to see where people had cracks in their armor, and whether they might be lying. Her attention made me want to shatter.
"I—" The lines I'd prepared seemed like someone else's words now. Everyone would know I was lying, and I couldn't look to Sam for help, because these questions weren't for him. "I don't think sylph are stupid," I blurted.
"Oh?" Sine waited.
"Well, people saw it yesterday: sylph sang along with music. They knew enough to recognize it as music and sing along with it without ever—I assume—having heard it before."
"That's fascinating," Councilor Finn said, and his tone turned mocking. "But did you tell them to run away by singing it?"
I cringed. "No. What I'm saying is that if they're smart enough to recognize music, maybe they recognize the similarities between humans. They fled Menehem during Templedark. Isn't it possible they realized I look like him? Or perhaps they saw me with him that night and remembered?"
The Councilors exchanged glances, frowns.
"We don't know what Menehem did to the sylph." My lie grew a little more confident. "And we won't until he's reborn."
They muttered at one another, and Sam gave me an encouraging look. This was his part. "Tell them the other thing you remembered about Templedark," he said.
I bit my lip—real nervousness, not feigned—and the room quieted again. "I forgot about this too. I'm sorry, the night was just so—"
"It's all right." Sine almost looked like her old self again, like she cared about me. "People do tend to forget traumatic things. It's your mind's way of protecting itself."
It seemed unlikely I could feel worse about all these lies, but if I didn't tell them something, they'd keep pressing me. As long as I didn't tell them how I knew things, I could give them some peace, and a reason to stop being so suspicious of me.
"The other thing Menehem told me was that whatever he did to Janan, it wouldn't work again. No one would be able to make another Templedark."
Several people exhaled and sat back in their chairs. No more newsouls. No more oldsouls would be lost. I hated not knowing how I felt about that, like I should be relieved too, but pieces of me felt disappointed and guilty. Why did I get to live? Why not all the others, too?
Would there only be seventy-three of us, and then we'd die, forgotten after our generation?
"So that's your theory?" Deborl asked. "Menehem experimented on the sylph for eighteen years and they decided you were in on it too. That's why they listened to you out there?"
It sounded stupid when he said it. And it was stupid. But it was better than claiming I had no clue—or admitting I'd been to Menehem's lab and knew all about his research.
"Let it go, Deborl." Sine didn't look at him. Wrinkles spiderwebbed across her face. They were deeper than when I'd first met her. Visible stress of being the Speaker. "Ana's brought us valuable information, and whether or not she should have told the sylph to go into the eggs doesn't matter anymore. They might not have done it, anyway."
"The problem," Deborl said, "is that Ana made a choice. She chose them."
Sine eyed me, disappointment flashing through her expression. "That is true."
"Can you blame her?" Sam asked. "If she did choose sylph, can you blame her at all, considering how people have treated her? Deborl, surely you recall that your friend Merton suggested newsouls should be killed like centaurs."
Deborl and the other Councilors had the decency to look ashamed.
"Let's not forget that ever since Anid's birth, people have been throwing rocks at Ana, and the Council's response has been to tell her not to fight back. Not to defend herself." Sam stood hunched over the table, leaning on his fists. "Even if you don't think she's worthy of being treated like a human, what happened to the law you passed about not letting people attempt to kill her? It's very poor leaders who won't enforce their own laws."
Frase and Deborl lurched to their feet. "I think that's enough," said the latter. "Dossam, we understand you're frustrated—"
"Frustrated?" Sam pulled himself straight. "We went to Purple Rose just to get away from people looking at Ana like Templedark was her fault. Lidea won't even take Anid through the city without several friends along. She doesn't trust people not to try to kill him."
"We can't control everyone's actions—" started Finn.
Sam raised his voice. "You say Ana made a choice. So has the Council. I'd have chosen the sylph, too."
"All right." Sine rose, balancing herself with long, wrinkled fingers on the table. "That is enough."
I edged closer to Sam, humiliated he'd had to stand up for me, but grateful he was brave enough to do it.
"The Council has been busy with city repairs and seismic studies. I'm afraid that we've been unable to focus on as many things as we'd like." Sine glanced at everyone in turn. "However, we should make this issue higher priority. The arrival of more newsouls is no longer a possibility; it is a promise."
Sam, still with an edge, said, "So you'll discuss it now? Ana and I can wait while you all decide to uphold the laws you put in place. It shouldn't take long."
"No, not right now." She slumped into her chair. "Now, Ana, I'm sorry, but I must ask you to leave the room." Exhaustion filled Sine's voice. "We need to discuss some things that wouldn't be appropriate for you."
Because it was so much worse than what I'd already been through? I scowled. "I can handle it."
She sighed and glanced at Deborl and the other Councilors. "It's not that we don't think you can. It's—Do you remember the law that was passed a few years ago?"
Ugh. The law that didn't allow anyone to be a citizen unless they'd owned a home in Heart for a hundred years. They wouldn't even have let me into the city if it hadn't been for Sam offering to become my guardian and ensure that I was properly educated. Sam and I had done everything the Council asked, including lessons with every type of work someone would teach me, monthly progress reports, and a curfew.
"The next part of our meeting is for citizens only," Finn said.
Sam's knuckles were white as he gripped the edge of the table. "So you care about this law, but not—"
I touched his elbow. "It's not worth it." If we provoked them too much, they might threaten to revoke his guardianship.
Anger-clouded eyes met mine, and he'd drawn his mouth into a thin line. I pressed my hand against his elbow until his expression grew easier. "As you wish."
"I'll meet you outside." I gathered my coat from the back of my chair and my notebook from the table, and left without acknowledging anyone else.
#
# BLUE
WHEN I CAME out the side door, the market field was still busy with people walking around, chatting, and listening to music on their SEDs, but not as crowded as before. Merton's group was gone, though the effects of his speech lingered. People eyed me with distaste, and some had gathered into small circles of gossip.
I slumped on a bench and fumbled for my mitts.
"Hey, Ana." Armande sat next to me and offered a paper cup of coffee.
"Thanks." I balanced it on my knee and watched a group of children chase one another through the market field. They weren't really children, though. They were five-thousand-year-old children, burning off the excess energy of their age. Would I know what that felt like, being a kid again but remembering everything I did now? I wanted the chance—ached for it—and acceptance.
"Don't worry, Ana." Armande gave me an awkward sideways hug, somehow knowing what I was thinking about. If I was that easy to read, surely my lies in the Council chamber had been, too.
"Did Lidea and Anid get home safely yesterday?" I asked.
He nodded. "Thanks to you. Wend is with her, of course, and Stef waited a few hours to make sure everything was all right. I think she's rather taken with Anid." Armande grinned. He was Sam's father in this life, so the physical similarities between them were striking: dark hair they both wore perfectly shaggy, wide-set eyes, and strong builds. But that was where their likeness ended. Sam was quiet and graceful; Armande was outgoing and...less graceful.
I liked trying to figure out which traits were inherited each generation, and which traits had become habits.
"How'd it go in there?" He jerked his chin toward the Councilhouse.
I sipped my coffee, letting the heat flood through me. "The Council is angry with me."
"The Council is always angry."
"Deborl thinks I can control sylph."
Armande snorted. "That's like saying you control dragons. Ridiculous."
I tried to smile, but I couldn't forget the way the sylph had responded to my voice, to physical gestures, and my words when I shouted for them to flee. Maybe they'd have fled simply because I was shouting.
"It's curious how there were so many, though. Aside from Templedark, we haven't had an attack that size in centuries."
It hadn't even been an attack. Maybe. No one had been hurt—besides my reputation—so did it still count? In hindsight, it seemed like the sylph had just wanted to look at us.
We sat in silence while I waited on Sam, and Armande...made sure no one threw rocks. His stall was close enough to keep an eye on it while he kept an eye on me, too. I hated that, but I really didn't want the girl across the way to yell at me, or the guy on the Councilhouse steps to call me names, so I said nothing.
"I'm worried about Anid." I placed my coffee on the bench beside me. "About how he and other newsouls will grow up. The Council isn't going to do anything."
I couldn't help but remember my first conversation with Councilors. Sam and I had just reached Heart, and I wasn't allowed into the city. They'd insisted the no-Ana law was because they hadn't been sure the city could support newsouls. Who would feed us and teach us? But there'd only been me.
Now there were two.
Soon there could be more.
"I expect there will be fierce debates from both sides, not just from those afraid of change. Lots of people like you and are looking forward to meeting more newsouls." Armande patted my shoulder fondly. "If nothing else, the next few months will give you an idea of whom to avoid."
I hated knowing this was something I had to do, not only for myself but for other newsouls. "At least when policies are finally made, we'll know what kind of things to watch out for. Like if they say it's legal to throw rocks at us. I think I still have a bruise from the last one."
Armande didn't laugh.
"Ana!" Cris towered above the crowd, features sharp in the near-winter sunlight as he moved toward us.
I waved.
"I didn't realize you knew Cris," Armande said, voice low and tinged with something I couldn't identify. Memories? The past? Definitely something he didn't want to tell me.
"We met him at Purple Rose Cottage when he was on his way back here." I took another sip of my coffee as Cris approached and sat on the other side of me, placing a rose across my knees. Velvety indigo petals shivered in the breeze, and settled as I brushed my fingers up to the tips. It was the same kind of rose I'd tended in Purple Rose Cottage, though the thorns had been clipped off this one. "Where did you get this?"
He cocked his head, shadowing his expression. "I didn't abandon them all."
Oh, right. Like I'd accused him of doing. "I'm glad to hear that. I didn't realize you'd kept growing them."
"It's not something someone stops doing just because other people don't agree about color."
"Technology didn't agree either," said Armande. "They tested whether the color registered more red—like purple—or blue."
Cris smiled. "What do you think, Ana? Blue or purple?"
I held up my hands, torn between being stunned and pleased someone had asked my opinion. "I'm not getting into this." My chuckle came out high and shaky. "This is clearly an inflammatory topic, and I think it's safer not to have an opinion."
Cris laughed. "Very well. I was more curious whether you'd like to continue gardening. You've been taking lessons from everyone, right? Are you still interested in roses?"
I nodded toward the southwestern residential quarter. "I've been tending the roses at Sam's house. It's not nearly as involved as what you're used to, I'm sure, but I enjoy it."
"That's good to hear." He motioned toward the rose still on my lap. "Were you interested in learning more about the genetics and how to begin projects like these roses? We've actually learned a lot about human genetics by breeding plants to see what traits pass on."
That was something I didn't want to hear about—how carefully the Council and geneticists decided who could and couldn't have children. Maybe I was only sensitive to it because I was new, or maybe they'd become desensitized after living with the awkwardness for millennia.
But since I was interested in the first part—making new kinds of roses and things that required more gardening knowledge—I said, "Sure. I need to check my schedule to see what days are free. Last week I had to learn about automated sewer maintenance. Soon I'll be accompanying Stef and a few others into a mine to rescue a broken drone. I'm supposed to help fix it." I made a face. More than likely, I'd be holding a flashlight.
"Gardening won't be quite as physically exhausting as that."
"You can't trick me. I've fought weeds before." I fondled the rose petals, soft against my fingertips peeking out from my mitts. It was just like the roses from the cottage, even the sweet scent. "We usually go to lessons in the afternoon, unless another time is better for you."
"We? Sam goes too?" He raised an eyebrow.
I frowned. "Is that not okay? The Council makes him report everything." Plus, it was nice having him around in case we ran across someone like Merton—not that I would admit that out loud.
"It's fine." His expression had darkened, though. "I just didn't realize Sam accompanied you. But please call when you're ready to schedule."
"Thanks. I look forward to it." I offered back the indigo rose, but he shook his head.
"That's for you." With a quick smile, he headed off, almost lost in the crowd again, except he was so much taller than everyone. Did he have to duck to get through doorways? How did someone even get that tall? I stared enviously as he vanished behind a crumbling statue of someone riding a horse.
Armande shifted his weight and hmmed. "That was kind of odd."
"I agree. Why would he just give me a rose when people are supposed to pay for them?" Maybe he'd request payment during the lesson, or hint at Sam later since I, of course, had no credit.
A dark figure appeared around the Councilhouse, hair tousled as he scanned the thinning crowd for me.
I placed the rose and my coffee on the bench and met Sam halfway. He hugged me so tightly my feet lifted off the ground, and then he pressed his mouth against my neck.
"Everything okay?" His coat collar muffled my voice.
"Better now that I have you again." But he didn't sound happy, and across the market field, someone made rude comments to their companion. Something about how boring it must be, being with a newsoul. Sam cringed and dug his fingers into my coat. "Don't listen to them. There's nothing boring about you."
Face red, I stood on my toes to kiss his cheek. "Thank you for standing up for me in the Council chamber."
"I don't know if it will make any difference, but it's better than silence." His voice sobered further. "Stay away from Merton if you can."
That guy again. Had he gone into the Council chamber after I left? Perhaps that was why Sine wanted me to leave. Perhaps she yelled at him for the way he'd behaved in the hospital, and the way he kept holding public rants. "I hadn't planned on asking him to be my new best friend."
Sam didn't smile. "I think he's the person who attacked us after the masquerade. With Li." He held me tighter when I winced. "I don't have proof, but I asked Sine if someone would keep watch on him. For now, the best we can do is avoid him."
"I understand." I glanced at the temple, so high it made me dizzy to watch the clouds drift around it. "Can we go home now?" When I was aware of it, the building seemed more solid, taller and wider and hungrier. It seemed like it hated me, and if a building could hate, it would be the one Janan inhabited.
Sam kissed my forehead. "Home sounds good." He slipped his hand around mine as we headed over to wish Armande a good day.
"Hi, Sam." Armande stood, and his gaze flickered toward the bench where I'd been sitting.
Sam's followed. "Someone gave you a rose?"
It took me a second to realize that was a question for me, not Armande. "Cris did. I'd accused him of abandoning the ones at Purple Rose Cottage. I guess he wanted to prove he hadn't." I plucked it off the bench. "He offered gardening lessons. I said we'd call him."
"Okay." Sam and Armande exchanged more silent communication—most of which came from Armande—but it was the kind that came from knowing each other forever. I couldn't read it.
When we left Armande with his pastry stall, I tossed my coffee cup in a recycle bin and asked, "What was that look about?"
Sam didn't answer.
Okay. Question for later, then. "What happened in the Council chamber after I left?"
He just shook his head and didn't speak until we turned onto his street, like he'd been putting the words together the entire walk home. "They wanted me to remember the truth about newsouls."
#
# SPIRALING
"THE TRUTH ABOUT newsouls?" I couldn't breathe.
"No one's sure how to respond to Templedark," he admitted at last. "At first, the community was in shock. We reacted how we always react to battles: tend to the wounded; rebuild the city. We could do that in our sleep. But eventually, we woke up and realized." Sam's voice broke, and he stopped walking. "So many souls are gone forever. We'll never see them again. No one knows what happens after you die like that."
Almost a year ago, he'd said the scariest thing he could think of was no longer existing. True death.
Living in Heart and witnessing Templedark gave me new appreciation for how frightening that thought was. I still didn't know what would happen to me when I died.
I didn't want to stop existing either.
"People are born in patterns. For me, it's just usually being male and being born in the Year of Songs. Nothing special. But others have the same mother or father so often it's eerie. Most keep their close friends through generations."
I knew all that. Sam and Stef had been friends since the beginning—five thousand years—and Whit and Orrin had practically built the library together in the first Year of Binding.
Sam went on. Fire-colored leaves floated to the ground behind him. "Some of those best friends and perpetual parents are gone. I keep thinking, what if Stef had been one of them? Or Sarit or Armande or Sine? They've been my friends for thousands of years."
I couldn't imagine. Didn't want to imagine. I just wanted him to stop hurting.
He began walking again, fast, clipped steps like he could outrun the pain. "People want revenge." His words almost didn't carry over the breeze, the rustle of conifers, and the tapping of naked deciduous branches. "But Menehem is gone, at least for now. There's no one to punish."
Waiting for his return had to be unsatisfying. I was the next logical choice.
"The Council wants to search your room for anything Menehem might have left you."
"Why?" I hugged my notebook to my chest as we turned onto his walkway. A chill breeze tugged at the rose in my fist, and leaves skittered across the cobblestones.
"They're afraid Menehem might have left clues for you, and they're afraid of what would happen if you knew how to put Janan to sleep."
"Oh. Even though I just told them it isn't possible?" Maybe they'd seen through my lie after all. The thought made me sick and dizzy. "Anyway, how could they think I'd risk sacrificing my friends? Or you?"
For a moment, I hoped he might joke about being upset that he wasn't my friend, but he just turned his face to the sky and sighed.
"You know I'd never risk you." The wind nearly stole my words away. I stepped closer, heart aching. "You know I'm not like Menehem. I don't want anyone to get hurt. I'd never do what he did. You know that, right?"
"I know." He stared far away, cracks showing in his normally calm demeanor. They'd planted something nasty inside of him, and it was growing, bursting out. "I think they're imagining what it might mean, you not being the only newsoul anymore. Having more has never been a possibility before, but if you knew how to do it—"
"I'd never risk you. You know how I feel." Didn't he? Maybe he didn't, if I couldn't say it. "And it seems like everyone else knows how I feel, too." Given how often they gossiped about our relationship.
I shifted my belongings to one hand and touched his shoulder. We stood there in the middle of the walkway, underneath one of the skeletal fruit trees and a sky full of clouds. Chickens and cavies rustled in their pens, softly clucking and wheeking as they waited to be fed.
The world moved around us while I waited for him to look at me. While I waited for him to believe me.
"You know how I feel," I repeated, heart twisting into knots. "But maybe the newsouls being born, like Lidea's baby, won't have the same problems I did." I stopped myself before adding, "Still do," but only just. He knew.
"Are you"—his words came like dread—"happy that newsouls are being born? That you're not the only one?" His face revealed no hints of his true question.
"Yes? No?" I dropped my hands to my sides, notebook and rose still clutched in my fist. "It's not safe for newsouls, and I'm terrified we'll never be accepted. So no, I'm not happy they're being born into this life. And I'm not happy that darksouls are gone. Friends, families. I did everything I could to avoid losing anyone."
"I remember." The words became white mist, and he didn't look at me.
"Some souls aren't coming back. There's nothing we can do for them now. So in that, I am happy newsouls are being born. It's better than no one being born." Gooseflesh prickled over my skin as I stared at the sky, searching for answers in cloud formations. "Ever since Anid was born—since I realized I hadn't just gotten stuck or left behind five thousand years ago—I've been thinking there must be a place full of souls waiting for a turn at life. Waiting and waiting, never having a chance because Janan makes someone else reincarnate instead."
His voice turned low and careful. "And now almost eighty will have a chance. Do you think that's a fair trade?"
"Nothing is fair. Not even souls being reincarnated for a hundred lives while newsouls never get one."
"Well, now they'll live, and Devon won't. Neither will Larkin or Minn. Neither will Enna, my current mother, or four Councilors." His voice shook with barely restrained grief. "They were here five thousand years. They were part of our lives. Julid, one of the greatest inventors, is lost forever. Rahel kept watch on Range, making sure we never overhunted, making sure the caldera wasn't going to erupt. People who were necessary to our lives are gone. Thanks to Menehem's meddling, the entire world has changed. You've tried to understand that, I know, but you can't. Not this life. Maybe not your next, either."
My heartbeat raced in my ears. My notebook and rose dropped, purple-blue petals vibrant against the gray stone, like paint on canvas. Shouts itched to get out, and I almost succumbed. I didn't. He was already hurting enough.
Instead, I turned up my chin, keeping my gaze and voice steady. "If not for Menehem's meddling, I wouldn't be here."
His mouth dropped and his eyes went wide. "Ana..."
I scooped up my belongings, swallowing anger. We were both right, and he knew it. There was no good answer. There was no fair answer. "Let's just go in." My voice rasped with tears.
Sam watched me a moment longer, then nodded and went for the door. I trailed after him, and when he sat at the piano—to work on it or practice, I wasn't sure—I headed up the spiral staircase, through the hallway, and to my bedroom. Not even watching Sam play the piano could lift my mood right now.
Like every room upstairs, mine had interior walls made of sheets of silk, and pinned together by delicately carved wooden shelves. So when Sam started playing downstairs, I could hear every note perfectly. He began with scales and warm-ups, playing with such force that his discontentment and confusion cascaded through the house.
Jaw clenched to cage frustration, I gathered up the books I'd stolen from the temple. To keep anyone from noticing them, I'd hidden them separately, in drawers or behind other books. With the Council's promise to search my room, I would need to come up with better spots.
But for now, I sat at my desk and placed one of the books in front of me.
More than ever, I needed to understand Janan, and what was happening with the newsouls. I hadn't magically been able to decipher the symbols in the books yet, but I'd definitely never be able to read them if I didn't try.
The binding creaked when I opened the first book. Dashes of ink stood dark on pale paper, grainy and thick, as if it had been made hundreds of years ago. I let my thoughts drift as I searched the page for anything familiar, and Sam's practicing seeped into my consciousness like water. His practice sounded better than my playing, even when he stopped to work through a section. His music was beautiful even when he was angry and exasperated, emotions spiraling out of control.
Spiraling.
Spirals.
Snail shells. Rose petals. Hurricane clouds. Faraway galaxies.
The nonsense markings jerked into place.
When I blinked, they were random again. Nevertheless, I'd found the pattern, like when I'd first taught myself to read, or when Sam had played music and I'd been able to follow the dots and bars—but never for more than a few seconds. At first.
I pushed the book aside and opened another and another, making a rainbow of ancient texts across my desk.
I couldn't read anything, and it took practice to see it again, but every page in every book had the same structure: a spiral.
Seeing the spiral was difficult at first. After straining my eyes for an hour, I realized my problem: I'd assumed the lines, for lack of a better term, were all the same size, like bars of music were all the same height.
But like looking into a pit with stairs spiraling down, they appeared smaller toward the center. A two-dimensional representation of something three-dimensional. I'd seen it in my mathematics studies, but it wasn't part of my curriculum, so I hadn't had time to pursue it.
Once I realized that, I could see the spiral as clearly as any other line of text, though the characters themselves still made no sense. Not to mention why they'd go in a spiral, forcing the reader to turn the book around and around.
I copied symbols into a notebook to view them flat, but they still looked like random scratches.
Downstairs, Sam's playing stopped, and he played the same note several times, as though testing it; he'd said earlier he wanted to work on the piano.
I put in my SED earpieces and tapped the screen for a random recording of his music. There was so much, I hadn't managed even a quarter of it in my months here, and I still had my favorites and pieces I had to study for lessons. A random piece would be good for me.
A flute sang, low and breathy, reminding me of earth. I'd listened to Sam's playing enough to recognize his vibrato, and the power that lurked behind the gentle sound. A lute joined in a moment later with a light, delicate voice, and soon both played together in an unfamiliar minor key.
The rhythm unfurled oddly, unpredictable almost, though there was a pattern I could almost hear—
Then I lost it.
The peculiar beauty swept me along in the sweetness and warmth, and just as it ended, I glanced at the title on the screen. Blue Rose Serenade.
Shivers marched up my spine.
The second player...
I pressed my hands over my mouth as though I could smother the stab of hurt. Why couldn't Sam really be a boy my age, with no more experience than I had? No past lives, past loves.
Why couldn't he be only for me?
I hated feeling jealous. It was petty, and I knew he loved me. He'd told me. And still my inability to believe he'd choose me over anyone—it squirmed in my gut and made me sick.
I turned the music down as the next piece came on, letting nocturnes and minuets seep into my thoughts while I focused on the temple books.
"This looks like a crescendo symbol."
I jumped as Sam's forefinger touched the paper. I hadn't heard him come into my room, but there he was, leaning on the corner of my desk.
Blushing, I removed the SED earpieces and shrugged. "Maybe. Or grow, expand, increase, swell. Or none of those things. Chances are just as high it means something else." Still, I wrote "Crescendo?" next to the lines.
"How are you getting these markings?" He didn't sound skeptical that I saw them, just curious.
"Here." I slid one of the books toward him and grabbed a pencil. "Watch." Lightly, so I could erase later, I traced a spiral under the text, starting from the center.
"Oh." Sam glanced at the other books and flipped a few pages, just as I had done. "That's incredible. I don't suppose you've translated everything but the crescendo symbol, hmm?"
"No, unfortunately." I leaned back in my chair, stretching cramped muscles. "But I've looked at these things how many times? I'm glad for any progress."
"I've no doubt." He picked up the rose, which I'd left on the edge of my desk. It was tiny in his hands, delicate, and the way he gazed at it was more mysterious than the books. "What else are you looking at here? I see the size changes from the center to the outside."
"It does, and I couldn't tell you if you read it outside to inside, or inside to outside. Or why anyone would write in a spiral, making you have to turn the book around."
"It does seem like a lot of trouble."
"I've tried to write down when I see symbols in patterns, but it's hard to tell when I'm not even sure of the direction of the text." I spun my notebook to face him. "Does anything else look familiar?" Maybe if more were music symbols, that would offer a place to start. But he shook his head.
"Not yet."
I let my thoughts wander through all the information I'd learned about Heart, its history, and where people had come from. He'd told me about tribes, people discovering Heart already built.
"Once, you told me you'd found bones in the agricultural quarter?" I watched him from the corner of my eye. "They might have been from a civilization before you."
He wore caution like a mask. "That was a long time ago."
I refused to be discouraged. "If people lived in Heart before you, perhaps these were their books."
"Perhaps."
How unhelpful. I tried again. "Do you remember anything? Any writing on rocks or trees? Anything like this?" Knowing who wrote it might give clues to what it said.
"Ana, that was a long time ago." His gaze dropped toward the rose bloom, cupped in his hand like a puddle of twilight. "And it wasn't my specialty. I avoided the agricultural quarter whenever I could. The only thing I wanted to do then was carve whistles that sounded like my favorite birds."
"Whose specialty was it? We can look at their early diaries. Or just ask." People expected me to be interested in strange things, and as long as I wasn't rescuing sylph, I doubted anyone would mind.
Of course, after the sylph incident, they probably minded when I breathed.
Sam avoided my eyes. "We'd have to talk to Cris."
"I thought he grew roses." I nodded toward the one Sam held.
"He does. They've always been his love, like music is for me, but his talent was more practical in the earlier generations."
I supposed no one cared which animal hide made a better drum skin if they really wanted to use it as clothes. I managed a smile and nod, because I knew how it felt to be useless.
Sam gazed through me, though. He had that familiar somewhen-else expression. "Cris had a way of making things grow, and finding the right spot to plant crops, which can be difficult over the caldera. The ground isn't always thick enough to support anything with roots deeper than grass."
That fit with what I knew of all attempts to dig beneath Heart. The sewer had been especially tricky.
"Cris was the first to find skeletons in the ground. It's possible he saw something else while clearing farmland. An object with one of these symbols on it." Sam came back to himself, back to the present. "Something you could use for reference."
Something I could use for reference?
I didn't want to be the one who figured things out. Everyone else was so old and experienced. Why couldn't they do it? Why couldn't I just focus on music and making the city safe for newsouls?
"Ana?" His voice was soft.
Without even realizing, I had hunched over the notebook, buried my face in my arms.
He touched the base of my neck, caressed all the way down my spine. He was solid and warm, and I wished things were the same as before we'd come back to Heart. Life hadn't been perfect then, but I hadn't felt this rift.
Chasm. Fissure. Canyon. Even with his palm on the small of my back, I felt like the entire Range caldera stretched between us.
I pulled away. "Let's call him for a gardening lesson. Tomorrow afternoon, if he can fit us in." I copied several symbols onto a fresh sheet of paper. "I'll ask if he's seen any of these and say"—I bit my lip—"I caught you doodling, but you couldn't remember where you'd seen them before."
"Okay." His features twisted into a mask of uncertainty.
I started closing the books, but paused when I remembered the look between Armande and Sam when he'd discovered the rose. And the awkwardness between Sam and Cris in Purple Rose Cottage. I hadn't thought much about it then, but...then there was the Blue Rose Serenade. "Did you want to ask?"
He cocked his head and searched me, as though I wore the correct answer on my face. "I'd rather not," he said after a moment.
Because he thought that was what I wanted to hear?
No. As I studied him, his expression shifted like shadows on darkness. Memory. "What happened? Did he do something to you?"
"No." Sam laid the rose back on the desk, voice deepening. "He's never done anything awful to me, or to anyone else. He's one of the best souls in Heart."
"So what is it?" Maybe I didn't want to know, but the question was out.
Sam strode toward the window, where he did not answer me, just gazed outside like he'd rather be anywhere else.
Tough. Surely I deserved some answers. I followed him, but paused when I noticed him leaning his forehead on the exterior wall. Paintings and furniture covered most of it, but here by the window was a clear spot. And he'd touched it. For comfort? Revulsion shuddered through me, and his worn expression made me bite back my questions about his relationship with Cris. For now.
"If Cris can't help me with some of these symbols," I said, "I have to go back into the temple and look for clues. Maybe Janan will answer me."
"No." Sam gripped my arm.
I looked up so sharply my neck stung.
"Ana." His jaw clenched and his voice pulled taut. "Don't you understand that I love you?"
I recoiled. Why would he ask that? "Apparently I'm too stupid to understand."
"You've told me how terrible it was in there and—" He paused, looking frantic while he searched for memories. He had enough difficulty remembering I'd been in there; anything more was almost impossible. "You can't even bear this wall, let alone standing next to the temple. How would you manage inside?"
Confusion flashed in his eyes—perhaps the question of how I would get in, because he couldn't remember the key I carried—and his grip tightened painfully around my arm. I wrenched myself away.
He must have realized he'd hurt me, because he held his hands before him in surrender. "Sorry. I'm sorry." He said it as a lament, breathing hard and staring at his hands like he didn't know whose they were. "If you want to go, I can't stop you. I won't try. But I will go with you."
"Thank you," I whispered. I had never imagined anyone could feel that strongly about me. "Because I'd rather not go alone."
He lifted one hand, hesitated, and caught my chin to tilt up my face.
Our eyes met, and everything inside of me twisted.
His thumb slid along my jaw while his forefinger held me up. If I spoke, I'd nudge his hand off me. I closed my eyes and let my head drop back as he slid his palms across my cheeks and into my hair.
His mouth was warm and soft. We kissed like a bow and violin strings. I wasn't sure who was which, but we made a melody that lasted only a breath.
He pulled away a fraction. "I didn't mean to start fights."
"I know." I kissed him again, my fingertips grazing the smooth skin of his jaw. His cheeks, his throat, his ears. Barely-there touches that made him shiver and sigh.
"I lived ten lifetimes in that kiss, and it still wasn't enough." He tucked a strand of my hair behind my ear. "I was weak in the Council chamber, after you left. They knew just how to exploit all my insecurities."
"Is that an excuse?"
"No." He retreated to sit on the corner of my bed. "Yes, it is an excuse, but it shouldn't be. I'm sorry, Ana."
Sorry because something terrible happened? Sorry because the Council had pressured and he'd slipped, telling them about Menehem's lab? Something worse? I could imagine a thousand horrible things he might apologize for.
"Why?" I couldn't stop the shaking in my voice.
"For letting their talk get to me and"—he slumped, elbows braced on his knees—"I don't know. I'm angry about Templedark. It hurts thinking about the darksouls." He buried his face in his hands. "When I see Menehem again, I can't say what I'll do."
He wasn't the only one to feel like that, either, but at least he didn't want to punish me for what Menehem had done.
Sam met my eyes, apology in his expression. "But I wouldn't want to undo anything that allowed you to be with us. Lidea feels the same about Anid." He looked so torn. "No matter how horrible Templedark was, it allowed for newsouls and you're right. That's better than no one being born at all."
I flashed a tight smile. He'd been right, too: I couldn't feel the same pain he did. That didn't make my caring any less, though.
"Sometimes good things come from unexpected places. Life out of death. No scars after a sylph burn." I showed him my pale, pencil-smudged hands. "And roses that taught me how to care for things, even though no one else thought the roses' color was good enough."
Sam glanced past me, toward the bloom on the desk. "How did you get so wise, Ana?"
"Someone strong and patient showed me." I sat next to him, looping my arm with his. "Will you say it again? The thing you said that night at Menehem's lab." It probably wasn't fair to ask him to say it when I couldn't say it back, but that didn't stop me from wanting to hear it more.
He must have caught the tension in my voice, because he twisted to face me, expression anxious. "You don't think I'd stop loving you, do you? Or change my mind?"
"No." Maybe a little.
"We might fight or disagree sometimes, but that doesn't change that I love you."
What a powerful feeling, love, able to withstand time and distance and disagreements. No wonder I wanted it so badly. "I haven't forgotten what Li told you," he said, "that nosouls can't love." He lifted our hands to his chest, fingers knotted with mine. "I haven't forgotten the way you tried to run away when you accidentally said the word 'love' that day in the cabin."
I couldn't forget it either, when he'd asked what made me happy and I'd answered, Music. I'd slipped, used a word I knew I shouldn't.
Love. I'd said I loved Dossam, his music.
I hadn't known Sam was Dossam then.
He kissed my fingers. "You may think you aren't capable of love, but I feel you are. I know you are." His breath came warm against my skin. "But don't feel rushed or pressured. I can wait if you need time."
How could he be so confident when I could hardly accept his emotions toward me? "It helps. Knowing someone can"—I gathered my courage—"love me, it helps."
His smile grew relieved. "I'll tell you as many times as you need to hear it, so you'll never doubt." He touched my cheek. "A hundred times? A thousand?"
"Start now and I'll tell you when." Part of me wanted to cry again, not from fear or disbelief, but from joy. As incredible as it was, Sam—Dossam—loved me, and he wanted me to understand. To believe.
I was Ana who Had Love.
Sam swept his fingers through my hair, down my arm. "All right." His voice was light and deep and open. "I love you because you're clever. I love you because you're talented." He touched my chin. "I love you because you have a perfect smile. I love you because you bite your lip when you're nervous and I think it's adorable."
I ducked my face. "Go on."
"I love you because you're good and honest. I love you because you're brave." His tone shifted, filled with melody that made me shiver inside. "I love you because you're strong. I love you because you don't let anything get in the way of doing what's right."
He went on, touching my hands and hair as he spoke. His words kindled a fire inside of me. I grew familiar with each sound, each letter. I memorized the softness in his voice, and the way he made "love" sound different and the same every time.
Maybe he was right: I didn't have to decide whether I could love. Not right now. All I had to do was accept and enjoy the idea that someone else could love me.
#
# JUNGLE
CRIS SAID HE'D be happy to fit us in, so the next afternoon, Sam and I headed through the city, toward the northeast quarter.
The walk through the market field involved no fewer than three rude gestures, two rocks—one that Sam caught before it hit me—and at least a dozen not-quite-hushed conversations discussing my relationship with Sam or sylph.
I kept my head down while he navigated the crowd, not relaxing until we reached North Avenue. "How does someone make a living gardening?" I asked, because I didn't want to talk about what people were saying about me.
Sam eyed me askance, but let me avoid the subject. "Same as with music. He grows things people want. His passion is roses, but he also works in the agricultural quarter. He's the most knowledgeable person when it comes to growing seasons, which crops to plant where, and when to send the harvesting drones out."
"Sounds like the city would starve without him."
"Probably." A note of pride and respect filled his voice. "But he gives lessons as well, or assists when someone does something seemingly irreparable to their private gardens."
And hadn't Cris said he helped geneticists' research by breeding different plants to see what traits were passed on? "I don't understand how anyone can get so much done and still have time for hobbies and friends."
Sam's grip slackened. "It's best to keep busy. A lot of tasks no one wants to do are automated now, like mining or recycling waste, but other things"—his gaze shifted into the distance—"it's better to do ourselves, even when we could have machinery do it for us. Five thousand years is a long time, and there can be joy in mundane tasks."
"That's why you always write music by hand, even though Stef could create a program to make it easier?"
He nodded. "I enjoy the process, even when I make mistakes and have to go back a hundred measures."
"You haven't had much time for that lately." Aside from the music he'd written for darksouls and the memorial, anyway. He was too busy walking around Range with me, escorting me to lessons, doing all the things the Council required of him if he wanted me to stay in Heart. He'd put so many things aside for me.
Sam shook his head. "I've had a lot of time to do a lot of things, and I'll always find time for what I enjoy. Don't forget, I do enjoy you."
His words warmed me as we continued to Cris's house.
I didn't have to ask to know when we arrived: the entire yard was a garden. Vines climbed over an iron archway wrought into silhouettes of hawks and storks and grouse. Hedges lined the path toward the house, hidden behind immense trees.
From the main walkway, more paths broke off like cracks in glass. One grew into a tiny wooden bridge—posts capped with flowerpots—that went over a stream so small it wouldn't get your ankle wet to step in it. Benches, birdbaths, and huge stone flowerpots with leaves spilling over the sides stood in a tiny clearing. Statues of the Range megafauna lurked in corners or at a fountain, as though lapping water.
Leaves hissed in the wind, and ancient maple trees rattled. Mourning doves cooed, jays and wrens and shrikes sang, and a woodpecker tapped rhythm. The scents of green and water and flowers replaced the fumarole stench, and I drew a long breath, smiling.
"What is it?" Sam touched my elbow.
I looked up at him, a dark figure against the bright sky and foliage. "I can hear music."
"Don't let it go. Keep it in your head until we get home and you can write it down." His voice lowered as he leaned toward me. "I want to find out if it's the same music I hear."
His was probably better than mine, but I smiled. "This way?" I motioned in the direction we'd been going.
To either side of the white stone house, which was covered with climbing roses, a pair of long glass buildings reached just as tall. Their windows were fogged, but it was impossible to ignore the green inside, and my heart jumped when I caught sight of familiar indigo roses near the door of one.
I squeezed Sam's hand. "Where do you think he is?"
His tone was easy. Happy, almost. "Somewhere in the garden, I assume."
Really helpful. The entire place was a maze, shades of green plants, gray cobbles and stonework, and the occasional squirrel or chipmunk peering from hidden houses someone had built as nests. That seemed like something Cris would do, sheltering animals others treated like pests.
He emerged from a greenhouse and waved us closer. "I was just cleaning up for your visit," he said as we approached. "Come inside. I think you'll enjoy this."
Though I smiled and thanked him, I felt clumsy trying to watch my step, to make sure I trampled nothing. Sam, of course, glided through easily, and the plants barely seemed affected by his passage. I watched enviously, trying to find the same footing through a patch of tall—I didn't even know what they were without blooms—plants, but my foot slipped on a rock, and I had to grab his shoulder for balance.
"Step this way," Cris said, offering a hand. "I just watered that area, so it's still damp. Sorry."
I nodded, keeping one hand on Sam, and used the other to take Cris's. We made it safely over a cluster of slick stones without incident, then onto a path that led to the greenhouse door.
The air glowed verdant with the many-tiered shelves running the length of the building. It was hot and humid, a weird shift from the coolness outside. No breeze, either.
But the colors were amazing. Shades of green certainly dominated, leaves and stems and buds, but splashes of orange and yellow and pink made dizzying patterns on shadows and glass.
I slipped away from Sam and Cris, letting my bag drop as I tried to slow my frantic heartbeat. There were so many roses, all shapes and colors, and the sweet scent was overwhelming. I felt like I could open my mouth and breathe it all in, capture the perfume in my chest, next to my heart.
He didn't have just white roses, but ivory and cream and old lace; and not just red roses, but ruby and scarlet and burgundy. I leaned to smell individual flowers, fiery petals tickling my nose and chin.
My face must have burned as bright as the roses when I glanced up to find both boys watching me. Sam had picked up my bag and hung back while Cris approached.
"These are Phoenix roses," he said, indicating the ones I'd just been sniffing. "Do you like them?"
I gazed at the perfect red, the spiral of petals, and the spicy scent so thick I could almost taste it. "Very much."
Cris chuckled. "That's not a surprise. They're Sam's favorite, too."
My face grew hotter as I stared at the rose.
"It still surprises me to see roses in colors other than blue," I said, before the awkward silence could fester. "I only saw the ones at Purple Rose Cottage for eighteen years." Because Li had never bothered to teach me colors, it had taken me years to figure out the difference between purple and blue, what with the name of the cottage. I'd thought they were two names for the same color.
"Blue, huh?" Cris raised an eyebrow. "I thought you weren't getting into that debate."
"I've had some time to think about it."
Cris grinned like I was his new favorite person.
For the next hour, we followed him around the greenhouse, Sam with his hands shoved in his pockets, and me with a notebook, scribbling to keep up with his lecture. Later, I'd copy everything again into more readable handwriting.
"The pruning shears are here," Cris said, motioning to a shelf with empty pots and jugs of liquid. "Especially in the greenhouse, you'll want to be careful to disinfect the shears between every plant. Otherwise you can spread a disease."
My pencil stopped over the paper. "Disease? I didn't know plants could—" No, that was wrong. I'd seen trees in the forest with strange fungus growing on them. "Never mind. But in the greenhouse? In the wild makes sense, but everything is safe here, isn't it?"
"Humidity." Like that explained everything. "I want to go over the main ways of reproducing roses and the results you can expect from them. Growing seasons, when to put fertilizer on them, when to prune. That kind of thing."
"It sounds like a lot for one afternoon." Not to mention the symbols I wanted to ask him about, if only I could find an opening.
"We can schedule lessons. Every week or every month." His gaze flickered toward Sam so, for a moment, I wasn't sure who the next words were for. "Whatever works for you."
I answered before Sam had a chance to look around awkwardly. "Every week would be great."
Cris beamed and drew me toward the workbench, explaining the difference between cuttings and budding.
We spent the next three hours in the greenhouse, me filling up pages of my notebook, before Cris declared that was all for the first lesson. We headed outside. Wind snaked between trees and bushes, stealing perspiration off my forehead and the back of my neck.
"So you'll call when you figure out a time to come every week?" Cris asked as Sam wandered off to look at something growing in a stone basket held by a stone rabbit.
I nodded. "Before we go, I was curious about something. Sam said you were the best person to ask."
Cris glanced at Sam, expression blank, and returned his attention to me. "Okay."
I pulled the folded paper from my pocket. "I caught Sam doodling and asked what they were. He said maybe something he saw a long time ago, but he couldn't remember exactly."
Cris raised his eyebrows. "And he thought I might know?"
I gave a one-shouldered shrug. "They look old, and I heard there were remnants of things before everyone came to Heart. And that you discovered most of those things because you put the agricultural quarter together."
"Hmm." Cris studied the paper, turning it on its side and upside down. "Some of these look familiar, but even if I'd seen them before, I couldn't tell you if they meant anything."
"I was hoping you might remember something like a label." I shifted my weight to one hip. "I know it's unlikely."
"Sorry. It was a long time ago"—he sounded just like Sam when he said that—"and we didn't keep records like we do now."
"Oh." I couldn't stop the disappointment in my tone as I reached for the page. It wasn't like I'd thought he'd have all the answers, but even a hint would have been useful.
"If you don't mind, I'll hold on to this. Maybe I'll think of something later and need to double-check." He glanced again at the paper as I nodded. "If you're looking for evidence of a civilization here before us, don't forget there were centaurs and trolls all through Range before we settled. They're not all bright, but they're not without their own means of written communication."
"Like scratchy drawings?"
"Sure. Or any other number of things. But I'm not the one to ask about that. There are books in the library you could start with. If you still have questions, I can give you a few names of people who—"
"Who don't mind newsouls?" I hazarded.
"Yeah." He smiled, looking relieved. "It's awkward to say that without saying that."
"You'll have to get used to it. With me, just be blunt. It's not like I don't already know what half the people think. The newsouls coming now will learn soon, too."
"Thank you for the advice." His gaze slipped to the paper again, and his smile faltered.
"What is it?" It was the same dazed look Sam wore when he suddenly remembered something he shouldn't.
"Several lifetimes ago I traveled to the jungles along the equator." His tone drifted, almost singsong. "The air was as thick and hot as a greenhouse, and the plants were incredible. They were immense, and everywhere, taking up every bit of earth. The air buzzed with the noise of bugs and animals calling territory."
I could feel it. Hear it. What an alien cacophony it must have been.
"You couldn't even drink the water. It wasn't safe." He traced one of the symbols on the page, paper fluttering in the fading autumn breeze. "And then it seemed like out of nowhere I came across these piles of enormous stone, so old and weathered some of them were breaking apart, but I could see where they'd once made a wall."
"What kind of wall?" I whispered.
The remembrance began to fade. "I'm not sure...."
"Was it white?"
"What?" He blinked, and the memories vanished. "I'm sorry. I must have been thinking about something else."
"You were telling me about stones you found in the jungle. You said they'd once been a wall. Was it white?"
Cris shook his head. "I...don't remember that. Sorry." He slipped the paper into his pocket. "But thank you for coming this afternoon. It was nice seeing you again."
When we finished polite small talk and farewells, Sam and I headed from the garden maze. His voice came low and soft. "Nothing?"
If Cris couldn't remember the conversation about the wall in the jungle, neither would Sam if I told him. And the fact that Cris had difficulty remembering it made me think Janan was involved somehow. Ugh. If only he'd said a little more. Described a heartbeat in the stone, maybe. Except the wall had crumbled, which meant what?
Sam had found a wall in the north, in dragon territory. Cris had found one in the jungle. Neither could recall the encounters clearly.
"Ana?" Sam touched my shoulder, looking worried. "Are you all right?"
"Yeah, sorry." I shook away my thoughts of other cities for another time. "Cris said he'll look at the symbols more."
"Then he will." He said it with utter certainty.
It killed me not knowing what had happened between them. They thought so highly of each other, and yet.
"So what now?" he asked.
I pressed my palm over the pocket where the temple key rested. "What I said I was doing before: going into the temple to look for more clues." I sounded as enthusiastic as I would have at the idea of cutting off my hand with a rusty knife, but I was still glad he'd demanded to go with me.
"Oh, right." He didn't sound upset or disappointed. More as though I'd just reminded him of something. "You can get in."
"Yes. I have a key, remember? It makes doors."
"I remember. It's silver."
I stared at him. He'd never remembered before. Cracks were loosening the magic that kept his memories locked. It had never been challenged, and if everyone had the same selective amnesia, then it didn't have to be good magic. But he'd spent so much time with me, with my questions—
I shivered with hope. Maybe I could break the magic.
#
# CREVICE
THROUGH THE WINDOW glass, the sky turned velvet indigo as the sun hovered below the city wall and horizon.
My backpack grew heavy as I filled it with dried fruits and crackers, bottles of water, and painkillers. When I'd met Meuric in the temple before, and he'd tried to trap me, he said I'd never get hungry or thirsty. Maybe that was true, but I didn't want to take chances.
"Got enough stuff?" Sam said as he came into my bedroom and watched me shove a small blanket into the bag. "Sure you don't want to add the piano? I bet you can make it fit."
I made a show of looking back and forth between him and the bag. "I'm not convinced you can carry all that."
He pressed his hand over his heart in mock indignation. "I could. And I'd carry all your books. Your flute. Your rose, too."
"Oh! My rose!" I grabbed it off my desk and threaded the stem through my braid. "Even if I can't take the piano, I should be able to take something good. Besides you, I mean. I'm glad you're coming." The rose was starting to dry, though. Petals rasped under my fingertips. "How's that?"
"Beautiful." Sam snapped my backpack closed and pressed his mouth into a line.
"What? You don't like my hair like this?" Too bad for him; I did. I'd get a hundred roses and put them in my hair.
"Oh, I do." He put on his coat, then pulled the backpack on over it. "It just startled me for a moment. Cris used to wear roses in her hair, too."
Her hair.
Blue Rose Serenade.
"Ah. Clearly it's the best way to show off flowers." I tugged my coat sleeves over my arms, then checked my pockets for the important things. SED, knife, water bottle, temple key, and notebook. Not that I would have time to write in my notebook in the temple—or I'd have too much time—but I didn't like going out without it.
The awkward moment passed, and Sam kissed my cheek before we headed downstairs and out the door.
My SED chirped with a message from Sarit, and I stifled a laugh as we took to the dark streets of Heart. "Sarit just said to have fun and to make sure you massage my shoulders. I wish we really were just sneaking out for a few days of romance. It sounds like a lot more fun."
"I think so, too." Sam walked close to me, cutting his steps short so I didn't have to run to keep up.
With a sigh, I put the SED back in my pocket and took my flashlight out instead. The moon shone brightly, but it wasn't quite enough for someone who hadn't been walking around Heart for five thousand years.
As we came to the road, I caught sight of the temple rising above the city, and the white glow of shifting patterns. It was almost hypnotic.
"What's it like in there?" Sam asked. I'd warned him several times already, about the everywhere-light and the unsound, but that was knowledge he kept losing. The forgetting magic had cracked, not shattered.
I told him again as we walked to the market field, and his face grew pale and drawn, lined with fear. "You don't have to go." I spoke gently, and he really didn't need to go, but I wanted him to. I didn't want to go by myself. The time I'd been in there had been terrifying. Having Sam with me would make it easier.
"I'm going," he said, and in the temple light, I caught his determination, and that strength he got from loving me. It made him brave.
Answers beckoned from across the market field. I couldn't help but imagine everyone all across Range looking up one night to see a strange, beautiful light, five thousand years ago. Of course they'd been drawn to the city. Sam had said they'd lived in tribes for a while, fighting over Heart before they realized it could easily house everyone. Maybe they'd been fighting over the light, too, if it brought them comfort.
My stomach turned. I couldn't believe I was going in again. Willingly.
For the newsouls, for answers, I would do anything.
I stashed my flashlight away and took a quick drink of water before heading across the market field. There was no one out this late, so the way was clear as we approached.
There was a crevicelike place where the Councilhouse and temple huddled together; Sam had told me earlier that in a few of the back rooms, there were spots on the walls that glowed at night, though none of them were big enough to use the key to create a door. It would have to be done outside.
"Ready?" I pulled out the key and squeezed into the hidden place. It was just big enough for elbow room—for me. Sam stood a little outside.
"Yeah." He took a deep breath, as though preparing himself, but tensed instead and looked over his shoulder. He swore quietly. "It's Stef."
"Sam!" Stef's voice carried across the market field. "What are you doing?"
Sam swore again. "What will happen if we just go in with her looking? Will she forget?"
"I don't know." I really didn't, but being pinned between these two buildings made me itch. "Go see what she wants."
He nodded. "I'll hurry." Then he trotted toward Stef, who was halfway to the temple, and halfway to spotting me clutching the key and ready to make a door.
I held still while they greeted each other.
"Going somewhere?" Stef motioned to the backpack.
"Ana and I are taking a short trip out of Heart. Didn't you get my message?"
"Yes, but you're here in the middle of Heart. In the middle of the night." She put her hands on her hips.
"So are you."
I swallowed a groan. This wasn't going to end well.
"I," Stef said, "am going home after working on Orrin's data console, since he insists he needs one at his house, too. I have been working on it for the last seven hours, because he decided he wants to track seismic activity in Range." Her pause was sharp, daring. "Where's Ana?"
"She's waiting on me. So we can go." Sam shifted his weight and didn't glance back at me, but his shoulders twisted like he wanted to.
"At your house? At a guard station? We can walk together." She hooked her arm with his. "Come on."
"No, it's fine." Sam pulled away, and it seemed unlikely anyone had ever looked more suspicious.
The buildings pulsed around me, making my skin prickle. Being this close to the temple made the faint taste of acid crawl up the back of my throat.
Stef's false cheer faded. Her posture straightened and her voice deepened, showing real hurt. "What's going on, Dossam? You're always off with Ana, caught up in your own private quests no one understands. You left Heart because you said Ana wanted time away, and that's great, Sam. She's cute, and I'm glad you're having a nice time with her. You both deserve happiness.
"But ever since you came back to Heart, you just look more and more stressed. Whatever you did in Purple Rose must not have been very relaxing or fun. We've been friends for thousands of years. You don't have to tell me everything that happened, but don't pretend I don't know you've been hiding something."
I wanted to shrink until I vanished between the cobblestones. She meant Menehem's lab. It weighed on him, what we'd learned, but it seemed like there was something more. Something he hadn't told me, either.
"Stef—"
She cut him off. "Your friends are worried. The Council—well, you know the Council. They're looking for a reason to toss Ana—and the other newsoul—out of Heart."
"They wouldn't." Sam shook his head. "They wouldn't, because we've done everything they've demanded."
"They're waiting for you to make a mistake." Her voice lost some of its bite. "I just wish you'd let me help. How can we be best friends when you don't let me into your life?"
Sam bowed his head. "We are best friends. But we've had five thousand years."
"And she's still working on her nineteenth. I know. So you'll spend the next seventy years shutting me out. And if she's reborn, what then? Do I cease to matter?"
"You know that isn't true—"
"What about the rest of your friends? You hardly visit like you used to."
"What are you talking about?" Sam raised his voice. "I see people as often as I always have. More, perhaps. But I've always needed time alone. You know that."
"You're never alone anymore. She's always with you. And when you go out to see people, it's for her. Introductions, lessons. Everything you do is about her." Her anger made the last words fall like punches.
There wasn't much Sam could say to that, and he seemed to know it. He had devoted a lot of his time to me. The moments he took to think about his response gave Stef another opening.
"You know what they're saying," she said, "about Ana and the sylph. About newsouls and the sylph."
"It's not true." He didn't sound even slightly convincing.
"I was there, Sam. I saw Ana go right for her SED. I saw her when she immediately knew how to distract the sylph long enough for the others to get away. And I saw what happened with the sylph when Deborl and everyone came with the eggs."
"Surely you don't believe—"
"What am I supposed to believe? You don't talk to me about things anymore. People keep asking me questions, because they think I must know what's going on, but the only things I ever hear are rumors." Her voice cracked. "I miss you. I miss how things used to be."
Sam's shoulders slumped.
This fight would last forever, and I couldn't stay hidden in the crevice any longer. Every moment made me feel worse, and listening to them...
I couldn't go out there. Stef had shown all her anguish, and she would be furious if she knew I'd overheard. She'd never hurt me, not like Li would have if I'd witnessed that kind of vulnerability, but I didn't want her to be angry with me, nonetheless.
Sam couldn't end this—Stef wouldn't let him—and I couldn't stay trapped here between walls that made me itch. Sam would know where I went.
Silver shone in temple light as I lifted the key and pressed the shapes engraved into the metal and squeezed. A gray door swirled into existence.
With one last look at Sam and Stef arguing in the market field, I stepped into the temple.
#
# WEEPER
NO SOUND EXISTED inside the temple, not even ringing in my ears, like silence after a loud noise. Temple silence was thicker than regular silence, like stone was thicker than air.
I clutched the door device to my chest, waiting for my eyes to adjust to the everywhere-light that left no shadows. The glow that emanated from the white walls wasn't actually bright, but the reflections and lack of darkness made my eyes water.
Mysteries surrounded the temple like a cocoon. Everyone knew it was empty, and yet no door existed—not without the key I held. As far as I knew, the only other person who'd been inside the temple was Meuric.
The air pulsed with the temple's heartbeat, making my skin prickle. Janan was here. "Hello?"
No answer. Just the flattening of my voice in dead air.
Wishing I had the backpack, I tucked the door device into my pocket and tried to decide which way to go. The room was immense, though I didn't think it was the chamber from the last time I'd found myself in the temple. Neither was it the hall with books, or the room with an upside-down pit where I'd killed Meuric.
Carefully, I strode across the chamber toward an archway, nearly invisible in the strange light. My footfalls made no sound, and not because I was trying for stealth. Sound simply did not carry.
Moans rippled through the walls.
I halted and waited, but they didn't return, so I continued along my original path. I couldn't let Janan intimidate me just because he was a powerful, incorporeal being older than everyone in Range. Just because—by all accounts—he held dominion over life and death and reincarnation.
Right. None of that was intimidating.
There were no stairs at the archway like last time. It just opened into another room, and when I crossed the threshold, the archway vanished, cutting me off from the original chamber.
The new room was smaller, with archways scattered across the walls that made gentle ripples like curtains. They did nothing to create shadows, but successfully conjured a headache behind my eyes. I pulled out my flashlight, gave it a few twists, and shone it across the room.
It wasn't perfect, but at least I could tell how far away things were, judging by the size of the beam.
I couldn't trust my perception completely. The last time I'd been here, I'd found stairs that looked as though they went down, but actually went up. Nothing in this place was what it seemed.
The key's weight in my pocket suggested I could make things easier for myself while in the temple, but I had no idea how to do that. Too bad Meuric hadn't left instructions.
Determined to stop wishing for things I didn't have, I slipped through another archway and lurched into a sideways room.
I yelped and dropped my flashlight. It flew left and shattered against the wall—or another floor.
My feet stayed planted on the floor where I walked, but my weight pulled to my left, as though I stood on a wall. The other floor was shiny and lumpy, bubbling around the shards of my flashlight like an unfortunate batch of cheese soup I'd once made. All the cheese had coagulated and the milk scorched; the house had smelled terrible for hours.
In the temple, there were no scents, save for what outsiders brought in.
Awkwardly, I sidled through the nearest archway and staggered as gravity righted itself underneath me. My stomach flipped, and I swallowed repeatedly until I was sure I wouldn't throw up.
The room was small, only the size of my washroom. An empty white box with no archways, not even the one I'd come through. Only the occasional groan and gurgle shivered through the tiny room.
Suddenly, the air grew sharp and crushing. The heartbeat pulsed louder until it rattled in my ears, and my chest ached with the fight to breathe. It seemed all the air was being sucked away.
"Now what, Janan?" I could barely speak.
No answer.
I withdrew the door device and jabbed at random symbols. The silver box swirled in my fading sight until I wasn't sure I was actually pressing buttons, just hitting and jamming my fingers. I felt right side up and upside down, and on both of my sides. All at once. Acid crept into my throat.
My body ached as though I were being ripped apart, and my lungs burned with all the air pushing and sucking and swirling around. Vision grayed, and the only thing I could hear was the incessant weeping and moaning.
Janan's hollow whisper silenced everything. "That is not for you." It came from everywhere and nowhere. A place on the nearest wall rippled as though something moved beneath the stone, or inside it. I tried not to look because it made my vision worse, but it was impossible to ignore.
"Let me go." I gasped at the thinning air. "I'll keep pushing buttons."
Pressure gathered around the lump inside the wall. For a moment, it looked human-shaped, though its proportions were wrong. Limbs too long, waist too narrow, head too wide.
Then the shape scattered in all directions, ripples smoothing into the glowing stone. A black archway shimmered where the shape had been, and noise returned in waves.
Whispering.
Moaning.
Weeping.
The air remained stifling, but I could breathe. My vision returned to normal as I replaced the key in my pocket and staggered toward the opening. Losing the key would surely end with my being trapped forever.
I'd gone through a black archway before. It had been as quick as stepping into another room, like any other archway, though they looked frightening.
This time, I stepped into ink and starless night. The blackness coated my skin like oil and made breath...what I imagined it would be like to breathe liquid and not die. It sloshed through my nose and windpipe, and I felt ever nearer to drowning.
Three more steps and I still wasn't through. I stretched out my arms to feel the walls, but there weren't any. The archway either led into an empty black room, or I hadn't made it through before the portal vanished.
That meant I was trapped in the walls. With Janan.
Groans and whines pursued me like sylph. There was no telltale heat or strange singing, only the heartbeat and pressure, and what might have been my hair—or someone's fingernails—brushing my arms.
I ran.
The wailing grew all around, tangible, and Janan whispered right by my ear, "You wanted somewhere to go. Now you have everywhere."
I pushed my legs harder, away from his voice, but the fingernails scraping my skin never ceased. If I stopped, he'd hurt me worse. He didn't have to say it.
When I slowed enough to wrestle out my SED, hoping for some kind of illumination, the onyx air only swallowed the light. If anything, the darkness closed in further, though I couldn't fathom how complete blackness could become even more perfect.
Hours passed. Or longer. It was impossible to measure time, if time even mattered in here, but my hips and legs ached and I had the vague sensation that I should be hungry or thirsty.
And then I was, because I knew I should be. I slowed to grip my stomach. I was starving, though Meuric had said before that I wouldn't need to eat or drink in here.
"I am hungry, too," Janan murmured, "and I am sure you are delicious."
My hiccups fell flat on the liquid air. I wished Sam were here. I wished we didn't know about Janan. I wished we were sitting at the piano playing a duet, our legs pressed together because neither of us were thinking about music, not really. I wished it all so hard that for a moment I thought I was there, but then a scream cut through the blackness, and I remembered the temple and running and Janan.
"No tears." Not Janan. Not a real voice, either, but a thought that wasn't mine. "The Devourer is incorporeal. He has never been able to touch the other one."
My feet caught on themselves and I stumbled, dropped, and hit the floor. Stabbing pains raced up my palms and knees as I searched the darkness for the non-voice. If it wasn't me and it wasn't Janan, perhaps it was one of the weepers.
I struggled to catch my breath, then fumbled through my coat for the bottle of water and drank half of it. The sensation of claws on skin never faded, but the non-voice was right. The feeling was just in my mind, and stopped when I rubbed my palms over my face and neck and hands.
Janan's words, and the weeper's—they meant something, but my head was too fuzzy to let me think clearly. The darkness remained overpowering.
Maybe I was blind. No matter how I forced my eyes open wider, I never caught light. I tried my SED again. A white glow pierced the dark, but illuminated only blackness when I held the screen to the floor. And blackness all around.
Trembling, I tried to send a message to Sam, but the SED beeped in error. I put it away and pushed myself to my feet. I couldn't let the screaming get to me, or the crying, or the fingernails raking across my skin. They weren't real.
They weren't.
Determined not to let Janan stop me, I stepped forward, and the whole world changed.
#
# TRUTH
BRIGHT WHITE SURROUNDED ME.
I crumpled to the floor, clutching my face and stinging eyes as pressure drained and the weeping no longer followed. Now just the hiss and scrape of cloth, ragged breathing that wasn't mine, and a reek like copper and ammonia so strong it made my head spin.
I wasn't alone.
"What are you doing here?"
The voice was broken, garbled and raspy at the same time, and came from across what appeared to be the bottom of a large hole, though a stairway spiraled up.
I wiped tears from my eyes and focused on the dark lump of bones and rags. Blood stained his face and hands, and a rotted wound hunched like a spider where I'd stabbed his eye out. But the other seemed to work, and it watched me.
"Meuric."
"Nosoul."
He couldn't be alive. It wasn't possible. I'd shoved him under the upside-down pit. The fall must have shattered every bone in his body. It had been months. And still.
I felt only a little better knowing I hadn't actually killed him. And then I felt much worse, imagining the pain he must have been in all this time, trapped at the bottom of a pit with stairs offering a way out—except his bones were splintered and he couldn't move from this spot.
Blood and other fluids seeped around his filthy clothes, but the rest of the floor was clean. No, he definitely hadn't moved.
"You tried to kill me," he gasped.
"After you tried to trap me in here so you could tell everyone I was dead."
Dried blood cracked and flaked when he smiled. Black rot filled the creases between his teeth. "And now I'm trapped. Does that make you feel better?"
"No." His stench made my head spin. I squatted on the floor and leaned against the wall for balance. It didn't help the dizziness, but my back and hips creaked with relief.
The pit was ten paces across. A fair size. When I looked up, the opening was invisible with the everywhere-light. It must have been deep enough to shatter all his bones, and shallow enough so he wouldn't die. How cruel of Janan to arrange that.
"Why aren't you dead?"
He laughed, like bubbles rising from the mud pits around Heart. Then wheezing and coughing, then groaning and silence.
I almost wanted to help him, but couldn't bring myself to go near him while he remained slumped, breath whistling as though there were holes in his lungs or throat. I couldn't get over the creeping feeling that, if I did go over, his body would miraculously mend and he'd grab me.
That thought coiling in my gut, I pressed my spine to the wall and sat properly, waiting for him to regain the strength to speak. How long had it been for him? As long as it had been on the outside?
"Janan won't let me die." His good eye was trained on me. "Do you have the key?"
I pressed my hands to my knees. I didn't want to slip and reveal the key's location.
"I need it," he whispered, managing to lift one arm toward me. "I need it to live after Soul Night. You have to give it back."
"What happens on Soul Night?" I'd come here for answers, after all, though I hadn't expected Meuric to provide them.
He wheezed laughter. "You won't stop it."
I stood, trying to make myself formidable. "What happens?"
"Give me the key." His glare followed me as I marched toward him. "Give, and I'll tell you."
Not a chance. He'd said he needed it to live after Soul Night, so what happened to everyone without a key?
I hovered just out of arm's reach, ready to run for the stairs if he so much as shifted his weight. "You've been down here for months," I muttered. "You must be very hungry. And thirsty. When was the last time you had anything to drink?"
His eye widened, and he groaned.
I felt sick taunting him like this, but I knelt so I was level with him. "Tell me what you know, and I'll give you the rest of my water."
His thirst must be horrible, even if he hadn't been thinking about it before. Janan couldn't fix everything...as evidenced by Meuric's broken body.
"So thirsty." The eye closed. The other remained a rotted hole, impossible not to look at; its reek rode the steady heartbeat of the temple. There were no screams currently, just muffled whimpering, as though they were waiting to find out what I'd do.
I checked to make sure the stairs were still an option. "If you tell me what's going to happen, I'll give you water."
"Soul Night."
The spring equinox of the Year of Souls. "Yes, I know that's when it happens."
He nodded. It was frightening how ancient he looked now, though this body was only fifteen years old. Months of dehydration and starvation, incredible physical damage...If he'd succeeded in trapping me in here before Templedark, this could have been me.
"I didn't think it would work." His once-high voice sounded like gravel now. "His plan seemed too fantastic, but if anyone could succeed, it would be Janan, so I convinced everyone to let him try. And then he did it. He really did it."
"What did he do?" I wanted to shake him and force him to speak clearly. Instead, I stayed on one knee, ready to bolt.
"He made himself greater. He made people like phoenixes." Meuric held out his hand again. "Water."
"That's not an answer." Phoenixes were another dominant species, like centaurs or trolls, but they appeared to reincarnate as people did.
They were rare—reports said there were perhaps a dozen in the entire world—but once someone had observed a phoenix in the jungles on a southern continent. It built a nest of dry brush, then settled down as though to lay an egg. Instead, it exploded into a rain of sparks and died.
The explorer had stayed at the pyre for hours, trying to figure out why the creature had done that. And then sunlight broke through the jungle canopy and shone on the ashes, dazzling him. When his vision cleared, a tiny phoenix chirped. It looked at him with the same ancient expression the other had worn, and then it flew off, trailing sparks and ash.
"It is an answer." Meuric's garbled voice grew panicked. "Water."
"No. What is Janan trying to do?"
"What has he already done, you mean." His good eye squeezed shut. "You're so stupid. It's already done. Soul Night is inevitable now. He will rise."
"Like a phoenix?"
"No. No, nothing like that. Come Soul Night, you won't care about phoenixes. No one will. Birth is so painful."
Okay. Something terrible would happen. We'd gone over that. Maybe he didn't know exactly what would happen. Or maybe he was too crazy to express how awful it would be.
I forced myself to meet his good eye, though he seemed to have trouble focusing. "When I came here before, I found books. But I don't know who wrote them, and I can't read the symbols."
"No one wrote them. They were simply written." He groaned and dropped his hand. "Give me water. You promised."
"Tell me how to read the books."
"Same way you'd read anything. Learn the language." Oil-dark fluid seeped from his ruined eye, down the crevices of his face, and into cracked lips. He swallowed it.
"What's the connection between sylph and Janan?"
"Janan is nothing like sylph!"
"Don't lie to me. I know there's a connection." The poison wouldn't have worked on both of them otherwise.
"He is greater than them. He has always been greater, and they deserve to be cursed."
Cursed? "What are sylph?"
"They are betrayers!"
"Did they betray Janan? Did he curse them?" Maybe all their attacks on Heart were about revenge. But why did they seem to like me?
"Oh, they betrayed Janan," Meuric said. "But he didn't have to curse them. I don't know who did, but if I had to guess, I'd guess a phoenix did it."
A phoenix. No, that seemed too incredible.
"Give me water!" Meuric's body tipped toward me.
I stood and stepped backward in one motion. "You're not getting anything until you give me answers. Real answers."
"There are no real answers."
"Look, Meuric." Ugh, wrong thing to say, because he grinned widely.
I fought hard not to gag. Meuric's odor of ammonia and bile made my headache increase. Soon my body would stop breathing out of self-defense.
I tried again. "Here." I pulled the bottle of water from my coat. "Half-full." I sloshed it. "I'll give you this water, but you have to answer questions for me."
"What questions?"
I put the water away and found my notebook, wishing I had the list I'd given to Cris. Still, I remembered lots of the symbols, and I flipped to a blank page and began drawing. "See this mark? What does it mean?" I showed him the symbol that looked like a crescendo.
"Less."
"What?"
"It means less than. Math. Or it could mean 'speak louder.' I don't know. Context. You must tell me more for me to tell you anything. Honestly, I can't believe how stupid you are. Do you think I'm a data console, able to call up information when you press the correct buttons? Or a vision pool? Oh, I remember those. We used to think the hot springs would give us visions if we stood there and inhaled the fumes long enough. And they did give us visions! But not of the future or past or anything useful. Headaches. Like you're giving me now."
I blinked and glanced at the page, desperately hoping it wasn't a math symbol and that all the books weren't written entirely in mathematical equations.
"Okay, let's try another. Maybe it will be less ambiguous." I offered a symbol that looked like an up arrow, but with four points along the shaft rather than one at the top.
"Hmm. Another."
The next was a circle with a dot in its center.
"Still wanting answers from those books." Meuric shook his head, as though disappointed but not surprised.
"Do you know what these mean?"
"Of course."
"You must tell me everything. No leaving out details. If I think you're lying, I won't give you this water."
"Very well." Meuric coughed, flecks of blood and mucus spattering across the floor. "The second symbol means rising or higher. Ascending. You may sometimes read it as Janan, though it isn't his name, simply a reference to him. The third symbol means city, or Heart—but only Heart in the way the other means Janan."
"How do you tell which it means?"
For someone in his condition, he did an admirable job of looking at me like I was an idiot. "Context. Of course."
"Oh, of course," I muttered, scribbling notes to myself. "What about the first symbol? The 'less than' mark."
"It is but a modifier, changing the meanings of the words around it." He gave examples of how the symbol might affect others.
I showed him several more symbols and he answered readily, the whole time grinning as if he believed I would regret all this questioning. But I continued on, and he told me how and why different meanings might be assigned to different marks. Then, too soon, I couldn't remember any others well enough to ask about them. If only I'd found the stack of books again when I came in.
"Okay, you can have the water now." I put my notebook away and retrieved the bottle.
"Yes! Give it." Meuric lifted his arm, which drooped in unnatural places. When I handed him the bottle, it fell from his grasp and rolled across the floor. As it bounced against the far wall and settled, he just stared, desolate and unable to go after it.
Pity gnawed at me, and I fetched it for him. "Do anything I even think might be an attack, and I'll shove this in your other eye. Got it?"
Meuric nodded, as I removed the top and held the bottle in front of him. All he had to do was lean forward, but I didn't think he could. He should have been dead. Bone shards should have pierced all his organs and he shouldn't be breathing, let alone talking.
Whatever Janan had done to Meuric, it wasn't a favor.
I tilted the bottle over him until water trickled into his mouth. He drank, sputtered, coughed, and I backed far away. I didn't trust all those sudden movements.
"Answer a few more questions and I'll give you the rest." Unless he started coughing on me again. Maybe I could leave the bottle next to him and call it the end of our agreement. But he couldn't drink it on his own. I hated that I felt obligated to make sure he got what he'd bargained for.
"You want to know how to stop Janan. There is no way to stop him, least of all for you. You are nothing to Janan. Insignificant." He kept staring at the bottle, even as water dribbled down his chin.
"I'm not insignificant to you. I have the water." I shook the bottle again. All this protest. All this insistence on my insignificance. Meuric was afraid of me, of what I might do, because I was the only one against Janan who could remember everything others were supposed to forget. Because I was new. Different. Asunder.
Maybe special.
I steeled my voice. "Now tell me how to stop him."
"Nothing can stop him. Already the world quivers with anticipation." He glared up with his good eye, and the bad one gaped wider. "Why are you even here? You should have been like these screams, these crying souls never born."
Terror flurried inside of me, and I whispered, "What do you mean?"
"You weren't supposed to be born. You keep interfering and because of you, more oldsouls have been taken from Janan forever. More newsouls escape." Meuric cackled, rough and bubbling. "But it doesn't matter. You came too late to have any effect on him. He won't notice the loss of your tiny spark."
"But the others?" My tongue might have been paper as I asked, "Will he notice the darksouls, and the newsouls born in their place?"
Meuric settled into the position I'd first seen him in, obscured by ratty cloth and blood. "He may notice, but it's too late to stop him. Your trials are for naught. You've secured a few short years for yourself, and a few short breaths for others. But the death you'll soon experience will surely be a hundred times worse than your original fate."
My boots hissed on stone as I backed toward the stairs. "And what was my original fate?" I asked, thinking of the weeper and what it had called Janan. The Devourer.
When he grinned, a cracked and bloodied tooth dropped from its socket. "The same fate of all newsouls, caught to allow an oldsoul to be reborn. The same fate of all the newsouls you hear right now with their little screams and lives never lived.
"They're being eaten."
#
# KEY
THE WATER BOTTLE dropped to the floor, spilling open, and Meuric howled with laughter.
I threw myself up the spiral stairs, around in circles higher and higher. My thighs burned and my head throbbed, but I ignored my own pain. It was nothing. Janan was replacing souls, letting the old live and keeping the new for himself. The weeper, the non-voice that had comforted me in the blackness, was being consumed.
As I climbed, the sobbing and wailing grew louder, and I imagined the souls were calling me back, though whether to save them or die with them, I wasn't sure.
Up the stairs, I emerged into a spherical room. I didn't stop running, and the entire chamber rolled under my feet as though I were trapped inside a giant ball.
Remembering how the upside-down pit had sucked Meuric upward, I stopped while the hole was still on the side of the room, then fumbled for the door device with my pulse thundering in my ears. I pressed the combination that had opened a path to freedom before. Gray misted on the white stone, and I ran into scorching daylight.
Even as the door vanished, Meuric's words pursued me: they're being eaten.
All the weeping, all the whispered cries for help. Newsouls.
Light rained around me and the temple pressed against my back, echoing my pounding heartbeat. All I could see were the cobblestones under my boots and my shaking hands as I thrust the key into my pocket. I blinked to clear my vision, but it didn't help.
I gasped at air, gulping the scent of sweat and burned coffee and sulfur from an erupting geyser beyond the wall. Steam wafted across the agricultural quarter, through the orchards and fields. Two more geysers erupted in the north and east, their loud gush and whoosh audible even over the market field din. Water sprayed high, reaching over the immense city wall.
Hands closed over my shoulders and yanked me close, and I screamed.
A man I'd never seen before shoved me and slammed me against the temple. Lightning snapped through my vision and thoughts, and I cried out as the stranger pinned me to the wall. I couldn't get away. The temple thrummed against my spine, and the back of my head ached where it had hit. The stranger dug into my pocket and seized the temple key.
"This," he growled, "doesn't belong to you." He grabbed the front of my coat, jerked me around so I hit the wall again, and then he was gone.
My head pounded as I struggled to find my feet, to go after him, but I staggered a few steps and hit the ground. Stone scraped my palms and fingertips, all gritty and cold. I stared up at the real world, such a shock after an eternity of solitude.
At least two dozen people milled around the market field. Some gaped at me. I hadn't seen them before, hadn't thought to be mindful when I emerged from the temple. There wasn't supposed to be a door. Had they seen the man attack me? Had they noticed my appearance?
Had anyone heard the souls crying? The temple loomed behind me, immense and infinitely horrible. Maybe it wasn't a heart, but a stomach.
I tried to track the man who'd attacked me, but my eyes were bleary with pain and grief. His large form stopped by a smaller one—Deborl?—and moved on. I lost him.
I'd lost the key. I'd lost my biggest advantage.
I collapsed over my knees and sobbed.
"Ana!" Sam fell beside me, wrapped his arms around me. "Where have you been? What happened?"
"Someone took the key."
"Your key? Who?"
"I don't know." I buried my face in Sam's shirt and let tears fall. My eyes were heavy with the weight of them, like I could cry seas.
"Ana," he murmured. "Oh, Ana. You're safe now."
I didn't have the breath to tell him I wasn't worried about myself. It was the others. It should have been me, too, except Menehem's experiment had gone wrong. His meddling.
Trying to swallow my sobs so we wouldn't draw a crowd, I burrowed deeper into Sam's embrace. I inhaled the scent of sunshine on his skin, shampoo in his hair, and coffee on his breath as he squeezed me tighter.
"I was so afraid for you, but you're here now. You're safe. You're safe." He whispered comforting nonsense while he peeled my hair off my wet cheeks and neck. I smelled salty, sweaty, and perhaps I'd carried Meuric's odors of blood and pee, because Sam dragged his hands over me as though searching for injuries.
My worst injuries were on the inside.
A narrow shadow dropped over us. Sam's weight shifted when he looked up, and his voice rumbled in his chest against my ear. "What?"
"Just checking to make sure everything was okay." Councilor Deborl's voice was strained, as though he were trying to make it deeper than it really was.
"Thanks, but we're fine." Sam stood, drawing me with him. I had just enough time to dry my cheeks, not that it mattered. Dark stains on Sam's shirt revealed the oceans of my crying.
"When people scream, it's rude to leave them in the middle of the market field." Deborl leveled his glare on me. "Especially when her guardian is the one to frighten her so badly."
I edged closer to Sam. "There was someone else. He pushed me and took—"
Deborl cocked his head. "And took what?"
Took the key, but I wasn't supposed to know about the key. No one was supposed to be able to remember it, and what if the stranger hadn't just paused by Deborl, but given him the key, too? If I accused Deborl of having the key, there'd be questions of how I came to possess it. Questions like what happened to Meuric, and why had I been hiding such an important object?
I slumped against Sam. "The man shoved me. He was big...." Everyone was big compared to me. "He had brown hair. He walked right by you."
"I'll look for him," Deborl said, but he didn't leave.
"Everything is okay, Deborl." Sam kept his voice even, and only the way his arm tightened around me belied his tone. "Thank you for checking."
Deborl glanced between us, scratching his chin where red lines marked cuts from shaving. "I hope you haven't been letting her get hurt a lot. After all, the Council trusts you to care for her." His eyes narrowed when he smiled. "You know, half the population thinks she's responsible for Templedark, and the other half isn't convinced that she's not. And now they're talking about the incident with the sylph."
Sam's hands curled into fists, and his shoulders pulled back as though he was ready to hit Deborl. "Ana did more to mitigate a slaughter during Templedark than anyone. And where were you that night? Did you turn over and go back to sleep?"
Their argument had begun drawing curious looks. Cris strode toward us as though on a mission. Most others just stared.
"Stop," I said. "Both of you." I couldn't imagine how my voice didn't shake. I locked my knees to stay upright, but it just made me light-headed.
Deborl smirked.
"Hello, Cris," I said as he approached behind Deborl. I was so sore and tired. Maybe someone else could keep Sam and Deborl from coming to blows. Then I could curl up on a nice rock and go to sleep for a year.
He nodded in greeting, exchanging a questioning look with Sam. Something heavy passed between them, though I couldn't decipher the flickers of expression.
"Is this your plan, Sam? Get people to feel sorry for newsouls by parading around a tearful Ana? It won't work. It's pathetic." Deborl sneered. "People will never accept newsouls. Everyone knows you're blinded by"—he eyed me—"whatever you two do together. Disgusting."
Sam's arm tightened around me. "Don't you have something important to do?" He glared at Deborl. "Maybe you could find whoever assaulted Ana?"
The Councilor showed teeth when he smiled. "Little Ana missed her progress report the other day, and you haven't rescheduled. Some Councilors are wondering if she doesn't really want to be a member of the community."
The other day?
"I told you, she was sick—"
Sam had needed to make an excuse for me?
"You have until the end of this week to report to the Council." Deborl's glare didn't shift away from me. "That's in two days. Be there no later than tenth hour, or your status as Dossam's ward will be revoked and you will be exiled from Range." With that, he marched into the dispersing crowd. A couple people patted him on the back, pleased with the idea that I might be kicked out.
Armande strode up, coffee in hand. He offered the paper cup to me, and I clutched it to my chest, trying to absorb its warmth.
"So." Cris turned to Sam. "I see you found her."
"You were looking for me?" He'd known exactly where I was. He'd been ready to go, too. How did I go from being sick to missing? What happened to the original plan of letting everyone assume we were off kissing somewhere?
"You were missing." Sam's fingers curled over the small of my back, as though to draw me close again. "We all went out to look for you that night, and the next. Cris and Armande stayed out late with me every night, but we couldn't find you."
That night? The next night? Every night? How many had there been? I reached back and touched the rose I'd braided into my hair, but it felt the same as it had when I'd put it there: a little brittle, but certainly not that old.
"We were all worried," Cris said. "Sarit is a wreck. Someone should call her."
My head throbbed so hard I could barely think. I just wanted to sleep, but the temple loomed at my back, a thousand times more frightening than it had ever been. Meuric's words still haunted me. The souls still haunted me.
I licked my lips. "How long was I"—not in there, not with Armande and Cris present—"missing?"
"A week." Sam's expression was sober, lines around his mouth and between his eyes. His skin was pallid, his eyes bloodshot and circled with hollowed darkness. "You've been missing for a week."
My cup slipped from my hands and slammed onto the cobblestones. The lid popped off and coffee splashed over shoes and pant hems, but I couldn't muster the energy to apologize, let alone back away from the liquid flying everywhere.
Coffee seeped through the cracks in stones, like rot dribbling from Meuric's eye—
Sam caught me when my knees buckled. "It's all right now. I'll take you home."
#
# CRASH
I MADE IT as far as South Avenue before my legs refused to work anymore, so Sam carried me. Safe in his arms, I closed my eyes and listened to the melody of voices.
"Where was she?" Cris asked. "I'd thought you must have found her this morning and you both came to the market field...."
"I don't know," Sam said. I couldn't tell whether he remembered where I'd been. "I wish Deborl had minded his own business."
Armande snorted. "You know he can't. Just as I bake, you play music, and Cris gardens, Deborl must interfere in others' business. It's the only thing he's got going for him."
With my face pressed into Sam's coat, I managed a smile.
Sam tightened his hold on me. "Someone told Lidea that Ana was missing. She's been calling every hour, worried Ana had been kidnapped, and they might come after Anid next. She refuses to leave her house, and she had Stef set up all manner of monitoring systems in the baby's room. Not that it matters, because Lidea sleeps next to his crib to guard him."
Guilt burrowed in my stomach. A week. It hadn't felt like a week. My rose...
I drifted in and out, and it seemed like forever before they carried me up the front steps and through the parlor.
A cup was pressed against my mouth, and water trickled in. I swallowed hesitantly at first, but as my throat grew used to the motion, I gulped the water down until my stomach hurt.
Bundled in blankets on the sofa, surely I was safe.
Sam showed the other two to the door, thanking them. It might have been my state or blurry vision, but while Sam seemed easy with Armande, his posture changed when he faced Cris. Slumped shoulders, weight shifted toward the other boy. Cris stood like his mirror.
"You didn't have to do so much," Sam said. "But I'm grateful. Thank you."
"She seems nice." Cris hesitated. "Well, a bit testy, but I suspect she's nice underneath all those thorns."
"When we first met, she had scars all over her hands. It took me a while to figure out how she'd gotten them." Sam hooked his thumbs in his pockets. "Or why they looked familiar."
Cris held up his hands; I couldn't see clearly from my place on the sofa, or with the current foggy state of my vision, but I imagined they were both looking at the scars he wore, too. You'd think someone who had been tending roses for hundreds of years might figure out about gloves.
"I saw the roses at the cottage." Cris lowered his hands. "She did a good job with them. Maybe I'll bring a few more by to cheer her up."
"She'd like that." They spoke a moment more, offers of further assistance, and Cris turned to leave. "Hey." Sam shifted his weight and his tone lightened. "I always thought your roses were blue."
Cool fingers touched my cheek. "Ana?"
"Mmm?" I tilted my head toward the window, where light could burn beyond my eyelids; I didn't want to wake up in the dark.
"Where did you go?" He sounded broken. Shattered. He sat on the edge of the sofa. "I looked everywhere for you."
My arms were too heavy to lift all the way to his face, so I settled for his elbows and dragged him downward. "You really don't remember?"
"You didn't tell me. I thought we were going somewhere together, but I can't remember. I had a backpack. I tried to call you."
The memory magic had closed over the cracks in my absence. I groaned.
"It's okay," Sam murmured. "We can talk about it later, if you want. I've called Lidea and Sarit. They want to come see you."
Opening my eyes was painful. No way could I smile for guests. "Not now."
"Not now," he agreed. "Can I get anything for you?"
I spoke without thinking. There was one thing I always needed. "Music. Play for me."
Sam kissed my forehead and retreated to the piano in the center of the parlor. Long, low notes filled the room, bouncing off the polished wood and stone figurines. This room was meant for music, and I sank into the sound as though it were a pile of feathers.
I dreamt of black rooms and black tears, and my fate narrowly avoided.
I awoke trapped in the tangled embrace of blankets. I thrashed and tumbled off the sofa, ran for the nearest washroom, and lost everything in my stomach.
Outside the washroom, I heard Sam growling into his SED. "Tell them to postpone the deadline. She's in no condition to leave right now....
"She's very ill....
"No, she was getting better, and then someone attacked her in the market field. Deborl cornered her right after....
"You're the Speaker, Sine. Overrule them....
"Stand up for her. Stand up for all the newsouls and do something to help."
More than he knew, someone had to stand up for them. Someone had to stop Janan from hurting newsouls. Someone had to.
I had to.
I sobbed until I crashed into dreams again.
When I finally opened my eyes without panicking, Sam brought tea and a plate of buttered toast. The lines and dark smudges were gone from his face, so I must have slept for quite a while.
I'd lost a week in the temple, lost more time sleeping after my escape. If I kept this up, I wouldn't have any memories at all. I might as well be one of the newsouls trapped in the everywhere-light and darkness.
I lowered my teacup mid-sip, and Sam brushed a tear off my cheek. That was all I had left: a few tears. No energy left for a big cry.
"I wish I hadn't gone in." I gulped my tea and set the cup aside, scrubbed my palms against my face. I really wanted a shower. A week of real sleep. No nightmares. "Where are my things? My notebook?" I needed to work on translating the temple books.
"In your room. Do you want to go up?"
"After I finish this."
Sam frowned, but waited while I ate my toast and found my feet. I felt like a memory of myself, after no food, after crying. It made me heavy and light at the same time, and I swayed on aching legs. Were they thinner than before? If I took off my clothes and looked in the mirror, could I count my ribs? I felt so hollow.
I managed to get upstairs without crumbling, without forgetting I wasn't still climbing out of Meuric's pit. Sam followed me into my room, staying close while I found clean clothes. He didn't speak when I went to shower.
Hot water burned off layers of memory. The reek of sideways and spherical rooms, the rancid odor of Meuric and his eye, and the stench of my own sweat. I watched it spiral into the drain.
Dressed again, I sat next to Sam on my bed. "Did you sleep in the parlor today? Last night?" My window showed a deep purple sky, a pale dusting of stars. Evening.
"I'm afraid of what will happen if I look away from you."
"If you were afraid I'd been kidnapped, why did you tell everyone I was sick?"
A line of thought formed between his eyes. "We checked everyone who's ever publicly acted against you, like Merton, but I was afraid that—no matter what actually had happened—people would find a way to twist the truth. You were kidnapped because everyone hates you, or you ran away to live with the sylph. I don't know. Scared people are creative people. They would have come up with something, so if I only said you were sick and no one knew the truth—that you were missing—I could control what people said."
"Sam." I tried not to imagine how frightened I'd have been if our positions were reversed. I couldn't blame him for the way he watched me now. "Sam," I whispered again, because the only thing I could say was his name.
He pressed his hand over mine, resting on my lap. "I don't think I've ever been as afraid as when I couldn't find you that night." His breath was long and shaky. "I've been inside every darksoul home, every warehouse and building in both the agricultural and industrial quarters, and every closet in the Councilhouse. I don't think I slept for more than five minutes at a time.
"When we first met, you asked about the scariest thing I could think of."
The day had been clear and cold, filled with questions. I hadn't even known who he was then, just that he pulled strangers from frozen lakes. I wished he could pull me from the frozen shock now. "I remember," I whispered. "You said not knowing what would happen if you died and didn't come back. Where would you go? What would you do?" My gut twisted.
"When I couldn't find you that night, I realized that wasn't my answer anymore." He pulled my hand up, placed it over his heart. The beat raced under my fingertips. "If you asked me now, I'd say the scariest thing I can imagine is losing you."
I didn't know how to respond.
"I wish I could tell you all the things you make me feel. I tried putting them into music, but even that wasn't strong enough."
I wanted to ask how he knew, how he could tell the difference between love and infatuation. But I couldn't force my mouth to form the words, because then he kissed my fingers one at a time and my focus sharpened, narrowed to all the places we touched. Our knees, his hands over my wrist, his lips on my knuckles.
When each finger had a kiss, he turned my hand palm up and cupped it over his cheek. "You're part of me, part of my existence." Muscles in his jaw shifted under my fingers. "Everything was dimmer without you."
If he'd been the one missing, I'd have crawled onto him to keep him from leaving ever again. Even in my imagination, I could feel him beneath me, bones and muscles and the solid presence of him. In my imagination, he lay there beneath me and never left.
I was both relieved and disappointed that he didn't have the same impulse. Or he had better restraint.
Sighing, he released my hand. "I'm still not sure you won't vanish if I'm not holding you." He glanced at my fingers, now curled on my knee. He started to reach again, but hesitated. Maybe he did want to crawl on top of me after all. "But you just got back, and there are so many things we need to do, which means anything I want will wait. And whatever happened to you, it must have been terrible."
The odor of Meuric's nest, the blackness with weepers, and Janan's voice by my ear. My breath came like a stutter.
Sam tucked my damp hair behind my ear. "Can you tell me?"
"You don't want to know," I whispered, hating myself for all the terrible things I was about to make him feel. "But it's important that you do, anyway."
He waited.
"First, you have to know that for a little while, you knew exactly where I was. You were going to go inside the temple with me."
"That's not possible."
"It is. I had a key." But it was gone now. Had the stranger given it to Deborl? What would they do with it? "We were going to go in together. You insisted, and I didn't want to go by myself. But Stef spotted you and I had to go in alone, before I lost the chance. It's just, Janan plays with your memories. You aren't allowed to know certain things, so you forget them, and you don't question inconsistencies because none of you notice them."
"That sounds crazy," he whispered. "We remember everything, from the very first lifetime."
"You don't." I touched his hand. "You don't remember everything. And that's not the only thing." I told him what Janan did to souls like mine.
#
# TRANSFORM
AFTER TELLING SAM everything that happened in the temple, I didn't have the energy to attempt translating the books, though I'd hoped to try.
Instead, I started crying again, and Sam grew somber and distant as he led me downstairs. Dusk had fallen long ago, leaving only lamps and reflections off polished wood to illuminate the parlor. I wrapped myself in blankets on the sofa, listening to his footfalls in the kitchen. Cabinet doors opened and closed, boiling water hissed, and a spoon clanked on ceramic as he stirred honey into my tea.
He left the mug on an end table for me, then went to work on the piano, adjusting strings beneath the gleaming maple lid, then testing pitches. Every so often, he'd stop working to play, always making sure to ask if I had any requests, but most of the time I was content to watch and listen.
Nocturnes and preludes lulled me into dozing, and I awakened to find morning had arrived, covered in a film of snow. Sam and I dressed warmly and headed to the Councilhouse for my very late monthly progress report.
Predictably, the Council quizzed me mercilessly on my supposed sickness and symptoms, expressing false sympathy. Well, Sine's concern might have been real. She worked hard to steer the conversation back to my progress report, but the general suspicion was clear: the Council thought I was up to something.
And wasn't I? I'd discovered Menehem's poison-making machine, Janan's terrible hunger, and their fellow Councilor alive inside the temple. I possessed the only unaltered memory, books from the temple, and—until recently—the key to the temple. Sylph sang for me.
It wouldn't matter that Janan had even more sinister plans for Soul Night. The Council couldn't trust someone like me.
Fortunately, Sam had foreseen the Council's questions about my illness and prepared me, so I described sleeping through a fever that involved lots of snot and throwing up.
"I died from that once," Sam added as we descended the Councilhouse stairs. An icy breeze scoured the market field, though it didn't deter devoted gossips and workers.
"Um." I hunched beneath my coat hood, conscious of glares in my direction. Merton was out again, reminding people about the sylph incident at the lake, and how disgusting it was that Sam was in a romantic relationship with me. The Council's advice on this was the same as it had been: ignore it. "If you died from the illness," I asked, "is it a miracle I'm alive?"
He slipped his hand around mine and squeezed. "Well, yes. But that was several lifetimes ago. Medicine has come a long way since then. Don't worry. The medic who supposedly treated you is a good friend. She won't say anything if they ask."
"Oh, good."
We stopped at Armande's pastry stall, sipping coffee and eating muffins until he was satisfied I wasn't starving to death. Sam kept checking his SED, but otherwise held a long conversation with Armande about what they each planned to have for lunch. It seemed suspicious to me, but we sat a good distance away from the temple and Merton's gathering, and Armande continued giving me snacks. I didn't complain, but I couldn't ignore the voices from the Councilhouse steps.
"Newsouls are a plague," a woman shouted. "Punishment for our lack of devotion to Janan."
Her theory and the truth were as far apart as the sea and the stars, but it was a popular sentiment.
"They have no skills," said a man. "Why should we feel obligated to care for anyone who can't offer anything to the community? We don't have resources to shelter and feed them. What happens if there are more and more? There are—were—a million of us. And only a million. We used to think we were the only souls in existence, but that's been"—the man's voice thinned, like he didn't believe what he was about to say—"proven false. Now whatever limit was set has been broken. What happens when they outnumber us?"
I glanced at Sam and Armande just in time to see them cringe.
It was a good question. I didn't know, either. Of course, this man was leaping to conclusions. For all anyone knew, newsouls might be limited, too. Eventually, by counting how many newsouls were born, they'd be able to tell how many oldsouls had truly been lost during Templedark. At least seventy-two. Probably more. But it seemed to me, once we reached that number, that would be it.
Then we'd either be reincarnated or we wouldn't.
At noon, Sam wished Armande a good day, and we headed back to the southwestern residential quarter. Snow flurries pushed through the streets, and the day was just cold enough to allow a layer of white on the ground.
When we got home, tracks in the snow led to the front door and away, scuffed enough that I couldn't tell anything about them except the intruder had been through a lot. Light seeped from the parlor windows. Perhaps the Council had finally made good on their threat to have my room searched. If they took my books and research, and Deborl had the key—
Fear splintered through me. "Sam?"
"It's all right." He took my hand and drew me to the door, where I caught a sweet scent. And when I stepped inside, roses transformed the parlor into another world entirely.
Shades of red and blue clustered in vases on tables and shelves. They rested alone on the piano's music stand and on the edge of each step of the staircase. They peeked from stands, from instrument cases, from behind the decorations that served beauty and acoustics.
The perfume was intoxicating, so rich I could taste it. A subtler, spicy aroma filled me up, warming me as the front door shut and Sam stopped beside me, wearing a smile. "I like it."
"Did Cris run out of room in his greenhouse?"
Sam chuckled. "Not as far as I know."
I drifted through the room, touching petals. "I like the way they're all mixed together, the red and blue. Are these"—I bent to sniff one—"Phoenix roses?" They had more petals than the blue roses I was used to, like ruffles of wisp-thin paper.
"They are. As many blue and Phoenix roses as he could stand to lose." Sam tugged off his boots and leaned against the piano, tracking my progress through the room. "I haven't seen you look this happy in a long time."
"It's like a greenhouse exploded in our parlor and left—" I swept my hands around. They were everywhere, changing the way light and color caught my attention, drawing my eyes to places I hadn't looked since I'd first come to Heart. They were by the cello, resting on the harpsichord, and threaded through my music stand.
And by my new flute, resting on its stand and polished to a shine, lay the most perfect blue rose I'd ever seen, with smooth petals so flawless they didn't look real. The bloom bent under my fingertips, as soft as air.
I turned. "Why would you do this?"
He smiled as I stepped into his embrace, and his arms wrapped strong and solid around my waist. "Why not?" He pulled me tight, and when I lifted my eyes to his, he kissed me.
I lost myself in the brush of his lips, the thrill of his fingers against my cheek and neck and shoulder, and thump of his heartbeat under my palm. So engaged in the way his mouth fit with mine, I almost missed the purr of my coat being unzipped. When he paused his kissing, I stepped back, and he slipped my coat off my shoulders; I dropped my arms and the cloth fell with a soft whump.
"I love you. Have I told you that since you got back?" He curled his hands over my hips and didn't wait for me to answer. "I want to tell you every hour. Every minute. Ever since you returned, all I can think about is how close I came to not having you at all. And how close you came to being—" He looked away, expression grim.
"You remember that?" I would be so grateful if I didn't have to keep explaining it, or reminding him that I hadn't actually been missing. "You remember everything I told you about being inside the temple?"
He nodded, looking wrecked. "I keep remembering it."
"And the white wall in the north? Right before the dragons?" I bit my lip.
Recognition flickered through him, but he shook his head. "No. A little, but no." He grew quiet, seeming distracted by my hair. It tugged and tingled across my scalp where he pulled his fingers through the waves. "There are things I should remember, but I don't."
"Yeah." My heart thudded.
"You remember them."
I offered a pale smile, relief that my newness was good for something. "I wasn't reborn."
"And there are things I'm remembering because of you."
"Yes." At least, it seemed to be my doing. It was unlikely that after five thousand years, the magic would suddenly begin breaking down in the middle of this lifetime. I was the only thing that had changed.
A tiny sense of importance surged through me.
"I'm glad you're here," he said.
I looped my arms around his shoulders and pulled close. "Because I make you remember things?" I didn't want to think about Janan right now. I wanted Sam to kiss me.
"Because of a lot of reasons." He read my mind, or read the way our bodies pressed together, only bunched clothes between us.
Our kiss stopped time, stopped thoughts. All I knew was the feel of his mouth, the gasp and shuddered breath, and the calluses of his hands on my back. Cool air fluttered where he'd lifted my shirt, sharp contrast to the way he made me burn with desire. I didn't have words for what I wanted with him, but if I could push closer and closer—
"No one would believe Ana was terribly ill for a week if they saw this." Amusement filled Sarit's voice, and I spun to find her—and Cris and Stef—crowding the kitchen doorway. "Oh, I'm sorry." Sarit grinned, not sounding sorry at all. "I didn't mean to interrupt, but I thought you should know we brought lunch."
My face ached with embarrassment, but I didn't miss the way Cris looked sort of blank and Stef looked...upset? Angry? I couldn't tell.
"Lunch?" My voice sounded pinched, and I wasn't sure if I could actually eat after Armande fed me half his pastry stall, but to make the awkward moment go away, I'd eat anything.
The five of us spent the next hour over plates filled with roast cavy and vegetables, catching up, and admiring the roses.
"I recruited Stef and Sarit's help. I didn't think you'd mind them wandering around the parlor." Cris's plate was empty, but he eyed mine, which was still half-full. He couldn't still be growing, as tall as he was. Surely he couldn't. But when I surrendered my leftovers, he seized them as though he hadn't eaten in days.
"I don't mind at all." Sam grinned and found my hand under the table. "Stef lives here part-time anyway, and lately we have Sarit more often than not, too."
"To be completely honest," Sarit said, "I must admit that my increased presence since another musician moved in is not a coincidence." She winked at me. "In fact, didn't you miss your practice this morning? You should probably play for us now. Call it payment for all the work we did arranging these roses."
Before I could come up with a response, her SED chirped and she excused herself, vanishing around the corner to the other end of the parlor. Cheerfulness drained from her tone as she spoke, and when she returned, she almost looked her age.
"That was Lidea. Someone smashed a window in Anid's room. Lots of his things were taken. He wasn't there, but the threat was clear. Lidea is a wreck, and Wend doesn't know what to do." She pressed her mouth in a line. "I'm not sure Wend is handling the stress well. Everyone he lost during Templedark, and now this? It isn't the first time they've received threats, but it's certainly the worst."
I couldn't think around the rushing in my head. Someone had tried to hurt Anid.
As much as I wanted to be shocked that anyone would do this...I knew how I'd grown up, how Li had always treated me, and how people still leered at me. They would keep trying to hurt Anid.
"This will only escalate," I whispered, and everyone faced me.
"Ana, dear." Stef's tone turned comforting. "Lidea is strong. She'll make sure Anid is safe. You shouldn't worry about it."
"No." My voice broke as I lurched to my feet. "I must worry about it. Newsouls will keep coming, and they'll all face this kind of hatred. If I don't stand up for them, who will?"
"We all will," Cris said. "We're your friends. We want to help."
Sam gazed at me, waiting. He looked proud, which made my heart flutter.
"I know what to do," I said after a moment, and counted days in my head. Less than a week, but maybe... "I have an idea, but I need to speak to people. Tonight."
Part of me was ready for them to try talking me out of it. A smaller part expected laughter and placation. But Stef's expression grew serious, focused, and she pulled out her SED. "All right. Who do you want?"
Relief poured through me. "Trustworthy people. You guys. Lidea and Wend. Orrin and Whit. Armande."
"What about Sine?" Cris asked.
I shook my head. "I think this would conflict with her office too much." She'd been different toward me lately, anyway. Probably because she was the Speaker now and the Council pressured her more than ever, but her being the Council Speaker made the decision easier.
I listed off a few more people, and everyone was on their SEDs, sending messages. Warmth replaced the horror of Sarit's announcement. I could do something. I might not be able to do anything for the souls inside the temple, but I would convince the Council that newsouls deserved to be treated like real people.
Even though I'd invited them, everyone's arrival still surprised me.
Some, like Moriah and Lorin, were Sam's friends who'd given me lessons in various subjects. But Whit and Orrin were my friends, and liked to tease me about how much time I spent in the library. More than a few times, they'd tried to convince me to become an archivist with them.
Lidea, Wend, and Anid arrived last, the baby bundled in a hundred blankets. Wend hauled a small nursery in a bag, shooting me a strange look as he followed Lidea inside.
Armande appropriated the kitchen to make coffee and tea, and after everyone had a turn cooing at Anid and admiring the roses, they settled on chairs, music benches, and the sofa, waiting to find out why I'd asked them here.
Well, there was no way I could see everyone from the floor, and Sam wouldn't appreciate it if I stood on top of the piano. I climbed up the first few stairs, leaning my elbows on the rail so I could look at everyone.
From his place beside Stef, Sam gave me an encouraging smile. He made me feel strong.
I gathered my thoughts and cleared my throat, and everyone looked up. "I want to start by reminding you what happened the night Anid was born.
"It was, from what I understand, a normal rebirthing. Lots of people were present, hoping a friend would be reincarnated. But when the Soul Tellers announced Anid was new, everything changed. Some of you were there. You remember how people yelled, threatened him, even though he hadn't done anything except be born."
People nodded, and Lidea held Anid to her chest as though she relived those minutes, not knowing whether the crowd would hurt her child. Her eyes shone with tears, and Wend sat stiffly next to her, his expression hard.
"The fact is, more newsouls are going to be born, and there shouldn't be a need to guard the birthing room. I know people are afraid of what this means, or angry that some souls aren't coming back. Those both are perfectly reasonable reactions, but—"
I stopped myself before getting into the same discussion Sam and I had after the Council pulled him aside. I thought it was better that newsouls were being born—rather than no one being born—but for others, newsouls would be a constant reminder of Templedark and the souls who'd been lost.
"My point is—" I smoothed the shaking from my voice, needing to sound stronger. "Unless we do something, people will continue acting out against newsouls. I'm sure you've all heard Merton and his friends in the market field, yelling about me."
"Anid, too," Lorin added.
At least Merton had a reason to yell about me. The way sylph behaved around me was suspicious. But Anid hadn't done anything.
"I want to tell you what it was like growing up. Not just because of Li"—people hissed at her name—"but being different, and understanding how different and hated I was before I could even speak. You need to understand what it means to be a newsoul: knowing everyone wishes you were the darksoul you replaced."
Haltingly, I spoke about the previous Soul Night, now nearly a quindec ago. I tried not to pay attention to the winces and mutters as I recounted how the revelers had stared at me from across the campfire. I told them how I'd needed to teach myself to read and do chores. How I'd always known nothing I did was new or innovative; someone else had already accomplished it, or figured out a better way.
"It's humiliating to be new. To be the only one." My voice dipped low as I found Anid cradled in Lidea's arms. "And now there are all these new people coming. They could be anything. Scientists, explorers, musicians, warriors. But they're going to feel out of place and confused, always knowing what happened to allow them to have a life. They might feel guilty for something they had no control over. They might feel like a mistake."
Sam tensed, his unease a silent reminder of all the times he urged me to know I wasn't responsible for Ciana's absence. But knowing didn't mean it was easy to believe; the people who threw rocks at me knew I hadn't done anything to Ciana. So did Merton, but he still ranted about me at every opportunity.
"I want to talk to people who are pregnant," I said. "Any of them could give birth to a newsoul, and don't you think most of them will want basic rights and protection for their children?" Surely they weren't all like Li. Lidea wasn't; she gave me hope. "I wasn't even allowed into the city without a lot of bargaining with the Council and many of you agreeing to help. I don't want anyone else to have to go through that kind of fight, just to be allowed to live with the rest of civilization.
"We need to make people understand that the newsoul they give birth to will—" My voice caught like I didn't know how to say the word. Maybe I hadn't until now. "Their child will love them no matter what. And they'll need to be loved, too."
Sam sat up straighter, this time at the word. It felt strange in my mouth.
He probably wondered if I'd loved Li in spite of everything. Her death had upset me, but I'd never loved her.
"If more people knew, it might help." My voice faltered. I tried to look anywhere but others' eyes. The harp or honeycomb shelf. Maybe they'd all think I was making eye contact with everyone, just hadn't reached them yet. "What I mean to say is, it's worth discussing newsoul rights. The break-in at Lidea's is inexcusable. What were they going to do to him? Kill him?"
Across the room, Lidea shuddered and held Anid close. Next to her, Wend shifted and stared at me, as though surprised I could consider such awful things happening.
"Anid—and the others who will be here soon—are worthy of a champion. They'll bring new ideas and insights into the world, but right now there are no laws to protect them. How can they ever feel like part of the community if no one will stand up for them?"
"I agree." Cris flashed a wide smile from the back of the crowd. "We've been so consumed with the loss after Templedark, we haven't thought of what we're about to gain. Nearly a hundred new people."
"They've talked about it in Council meetings," Stef said, "but of course they don't come up with solutions or anything concrete. They keep circling the issue like there's all the time in the world."
I nodded. "I guess it's easy to forget that time is different for us. You do have time. Newsouls...we don't know yet." And probably wouldn't until I died.
"And like Ana said," Armande added, "the newsouls will have their own talents and ideas. We should be ready to embrace that, to encourage it."
Lidea glanced at her baby. "We weren't ready for Ana, and in spite of knowing he was a possibility, we weren't ready for Anid. But they won't be the last."
"There's still time for him," I said. "To him, every second will count. Days will seem like years, and years will seem like centuries."
And for everyone else, those days and years went by as fast as heartbeats.
Sam dropped his gaze, and Stef watched him from the corner of her eye. For a moment, she looked softer.
"I'll talk to anyone who wants to know what it's like to be a newsoul. I'll tell you anything you want to know." My mouth had a mind of its own. I hadn't meant to make such a huge offer—tell strangers about Li laughing the first time I menstruated?—but as soon as the words came out, I decided to stick with them. This was for Anid, and those not yet born. For those who would never be born.
"That's very generous of you," said Lidea. "I actually had a few questions, but I was hesitant to ask."
"I'll help however I can." I forced myself to move on to the next step, the reason I'd actually brought them all here. "The first thing I want you to do is meet up with friends and figure out whether they would be open to supporting newsouls. I expect most won't be, but we have to try."
Orrin lifted his eyebrows. "I don't think you'll have as much trouble as you're imagining."
"And that's where the rest of us become necessary?" Moriah guessed.
"Exactly." I relaxed as everyone said they'd help. My idea wasn't stupid after all. Orrin thought people would be receptive. "I made a list of pregnant women I know"—minus a couple I knew about through Sarit's gossip but wasn't supposed to—"and thought we could start with them."
"Seems reasonable." Sarit's smile was all innocence, as if she hadn't given me most of my information. "We can all speak to a couple of people, give them the basic proposal, and if necessary we can set up a meeting with you."
"It sounds so much easier when you say it." I grinned. "But we'll have to do this quickly, because the next part comes on market day, when everyone is in the market field."
Cris took a sip of coffee. "That's less than a week away."
"Yes, which means we have a lot of work to do, if everyone is willing to help."
"It will be easier with everyone helping," Sarit said, and all my friends nodded.
I couldn't believe it. I'd asked them here because I hoped they would help, but the confirmation made my heart swell with gratitude. "First," I said, "we need to call Sine and make sure we can use the Councilhouse stairs. Sam is going to play his piano."
Sam looked surprised but delighted, and a few people cheered.
"While we have everyone's attention, I will speak up for newsouls. I want the Council to hear people supporting newsoul rights, for them to know that people are discussing the arrival of newsouls, too. Not just people who hate us." I urged strength into my voice. "And if anyone else wants to say something so I'm not alone up there, I'd really appreciate it." Sam, Sarit, and Orrin raised their hands to volunteer. Others weren't far behind.
I stepped down from the stairs, meeting Sam's eyes, and smiled when he mouthed, "I'm proud of you."
Warmth filled me as I took a seat on the sofa arm, my list in hand so we could decide who would do what. This might actually work.
#
# EXPLODE
AFTER EVERYONE FINISHED deciding who would do what for our market day demonstration, we broke into smaller groups. I talked with Lidea, giving her more details about what I wished I'd been taught growing up. Cris, Moriah, and Orrin discussed the possibility of lessons for parents of newsouls, and newsouls themselves.
The melody of voices livened the parlor, bright as noon, and the scent of roses warmed everyone. The baby cried, and Lidea carried him to the other side of the room for a change. When she came back, she offered to let me hold him.
He wasn't heavy, but he startled in my arms. Then he went still. "He's beautiful." Much more than the first time I'd seen him, all wet and red. Lidea and Wend probably didn't want to hear that, though.
"I know." She bounced on her toes. "He's the most amazing thing. I love him more than anyone. I mean"—she eyed Wend, who fake-pouted—"he's tied with someone else for my greatest love."
They flirted back and forth until Cris and Sarit walked over. "I just had the best idea," Sarit announced.
"We." Cris rolled his eyes. "We had an idea."
"Sure. Cris and I had an idea." Sarit leaned against the back of the sofa. "It had to do with roses. We think they should go all over the Councilhouse stairs when we speak, like you have here tonight. Not only would it be pretty, but Cris was telling me how special the blue ones are to you."
"We have things in common." I smiled, imagining the Councilhouse columns wreathed in red and blue. Phoenix roses for oldsouls. Blue roses for newsouls. "Are there enough roses?"
"We might have to steal some of these," Cris said, "but I think we'll make it. Sarit volunteered to do all the arranging, and I can run to the cottage and get a few more blue ones if necessary."
"Thank you." I hugged both Cris and Sarit, gratitude filling me up. Maybe—hopefully—others would notice the significance of the roses, too, and see how beautiful they all looked together. Heart could be like that.
"The stage will be gorgeous! Don't you think so?" Lidea nudged Wend, who nodded.
"Now it's my turn to hold Anid." Sarit held out her hands. "Give him, or no flowers for you, ladybug."
I laughed and handed him over, and when Sarit, Lidea, and Wend moved toward the piano, Cris sat on the arm of the sofa and lowered his voice. "I've been thinking about those symbols of yours. I meant to bring the list."
"Oh." I shuddered, too easily remembering Meuric trapped in the tower, the grating of his voice, the fluid seeping from his eye. His delight when he told me Janan was consuming newsouls. I clutched my stomach and tried to swallow the acid taste in the back of my throat.
"Are you all right?" Cris touched my shoulder. "What's wrong?"
"I think I need some water."
Cris slipped off the sofa and led me to the stairs. "Sit here out of the way. I'll get you a glass."
When I was settled on one of the upper steps, looking down at all my guests chatting and admiring the roses, I tried to relax. I was out of the temple. Safe. I would help the newsouls coming to Heart. I would learn to read the books from the temple. I would discover the connection between Janan and the sylph. I would...what?
I still had no idea what Janan had planned for Soul Night.
"Focus," I whispered to myself, wrapping my fingers around the stem of a blue rose. First the newsouls.
Stef's voice came from just below my stair. "Did you see her holding Anid earlier? Babies holding babies."
I clenched my jaw.
"Ana is an adult," Sam said. "Almost four years past her first quindec. If newsouls had full rights, she could have had a job years ago."
I appreciated him standing up for me, but it wasn't like I'd always known what I was going to do. I liked learning about everything.
"Physically," Stef said, "nearly four years past first quindec describes you too. But that's physical. She's cute, and anyone can see why you like her, but stop pretending that five thousand years don't matter."
"She's accomplished more in these last months than many of us did in entire lifetimes. Even before we met her, she'd taught herself how to do things it took us ages to learn. She hasn't been a child in a very long time."
I certainly didn't feel like a child.
Sam spoke so quietly I almost didn't hear. "There are a million things she can teach us, simply by virtue of being new and seeing things differently."
"Like Templedark?"
His voice was a razor. "Ana rescued both of us that night. And hundreds more. Everything else was Menehem. You know that. Ana is no more responsible for his actions than you."
"You're really hopeless about this, aren't you?" Stef gave a long sigh, and her tone turned to steel. "Listen, Dossam. People are talking about your relationship with her. Whatever you've done with her? Inappropriate. Whatever you want to do with her? Inappropriate. She's five thousand years younger than you, and even if she doesn't know better, you should."
I squeezed my eyes shut, glad I was on the stairs where no one noticed me. More than anything I wanted to march over and tell her to mind her own business, but there were still so many people around, all chatting and having a nice time.
"People don't know anything about it. One day," he growled, "you're going to have to accept it. I don't care if she's eighteen or eighteen hundred. I love her more than—"
"What?" Stef's voice was low and dangerous. "More than music? More than me? More than everyone you've known for the last five millennia?" She paused, and the silence was heavy like the moments between a lightning strike and a roll of thunder. "More than all the darksouls?"
I gripped the rose stem so tightly it cracked. Did Sam love me like that? Was that kind of love even possible?
"Yes." His word was barely a breath. "More than all that."
Relief and horror poured through me. Stef had listed herself in there.
"It's unfair to ask me to rank feelings," Sam muttered.
Then Cris was back with a glass of water, still much taller even when he sat a step lower. "Were you just listening to them down there?" He kept his voice soft, and when I shrugged, he leaned on his elbow toward me. "Don't let it get to you. She's probably hearing a lot of cruel gossip—probably more than you or Sam, since she's been his friend so long and people know..."
"She loves him," I whispered into my water.
Cris lowered his eyes and nodded. "It makes people do strange things sometimes."
"It's fine." I put down the rose I'd been holding and took a long drink of water, trying to think of a way out of this. Cris was nice, and I'd have to remember to talk to him about the symbols from the books, but right now, I just wanted to make it through the night. "Thanks for the water. I'm going to see if anyone wants to play some music."
Cris unfolded himself to rise, then offered a hand to help me up. I headed downstairs with him and took my flute off its stand.
That was enough to draw looks. Sam and Stef first, both dark-faced from their fight. Then the hum of conversations drained, and others began to claim instruments or find stands. Most, as far as I knew, played at least a few instruments. They were Sam's friends, after all.
When everyone had chosen, Whit and Armande ran upstairs to the music library and returned with appropriate music for each instrument. Sam adjusted the lights so everyone could read.
We started with scales to warm up, then moved on to a few pieces we all knew. At first I thought they'd chosen simple songs because I was new, but after a few squeaks from Lorin on the oboe, I realized they weren't going easy on me.
As unlikely as it was, I was a better player than a few of these people; I practiced several hours a day, while they practiced when they felt like it and when Sam scheduled a group performance.
Of course, as soon as Moriah, Orrin, and Whit played a dizzying reel on a cello, violin, and clarinet, my pride vanished. They practiced every day. Someday, I would be as good as them. Better.
Sarit sang a ballad to Stef's piano. Others moved in with duets, trios, their favorite pieces. They made lots of trips to the music library upstairs, and I had a brief surge of worry that they'd stumble into my room and find the temple books hidden all over, but no one was gone long enough for that. Anyway, Sam would have heard someone walking into the wrong room.
My heart swelled as we played more group pieces, broke off into more small ensembles. How did I get this lucky? Friends—surely they all counted as friends now—who were willing to help with the newsouls, and play music with me? It was too incredible. Too wonderful.
Music swirled around the parlor, bringing the room to life. I struggled not to grin around my embouchure as we came to the coda of a waltz.
"What about"—Stef hmmed—"Blue Rose Serenade? Did anyone see the lute?"
"Um." Sam glanced at Cris as awkward silence fluttered through the room. People glanced at me, at Sam, at Cris, at the roses all around. "One of the strings snapped. I haven't replaced it yet."
"Maybe whatever you and Ana are working on?" Cris lowered the clarinet he'd been playing.
"Oh, I don't know," I said. We'd been practicing flute duets, and Sarit and Stef had listened a few times, but more people?
"Come on." Sarit batted her eyes. "Let everyone hear."
Sam gave me a look like it was up to me, so I nodded, mostly to stop the awkwardness. He had never mentioned Blue Rose Serenade before. He'd probably forgotten about it when he'd given me the code to add all his music to my SED.
Later, I vowed, I'd ask what had happened between them.
Sam warmed up on the other flute while I found our music, and my heart thudded at the weight of everyone watching. Listening.
But as soon as he met my eyes and silently counted off, my fear evaporated. I stood up straighter, rolled the flute in where my high notes tended to go sharp, and played like I never had before. Every time I glanced at Sam for a fermata or tempo change, he looked as though he wanted to smile.
Before I was ready, we came to the last note and held it until Sam nodded, and our duet ended.
When everyone left, I shut the door and locked it, and Sam and I cleaned up, talking about who needed to practice their favorite instruments more. I wanted to ask him about his fight with Stef, but how did one even open a conversation like that?
"What's this?" Sam bent to retrieve something from the floor. He pinched a tiny wire between his fingers. "From a flute?"
"Ugh." I checked mine. Sure enough, the wire was a spring that had popped out. A set of keys flopped pitifully. "Lorin was messing with it earlier. Guess she got a little enthusiastic when she was pretending she knew how to play."
Sam smirked and handed me the spring. "She's banned from holding flutes from now on. Will you take this upstairs? I'll show you how to fix it tomorrow."
I nodded and carried the flute and spring to the workroom, which held a dozen instruments in various stages of repair, as well as tools to fix them. Building and repairing instruments wasn't Sam's job, but he insisted it was important for every musician to know the basics. All this looked like more than basics to me, though.
I left my flute on the workbench and headed back downstairs. Just as I reached the last step, thunder tore in the north, and the ground rippled. "Was that an earthquake?" Range was constantly shifting, but most earthquakes were too tiny to feel.
"No." Sam was pale, staring northward with wide eyes. "I think it was an explosion."
I raced for my coat and then into the cold. Nearby, an orange glow raged against the dark sky.
Sam followed me out, a water bottle in his pocket and his SED pressed to his ear. "Alert all the guards and medics. Hurry."
We ran down the walkway as secondary explosions shattered the night. Crashes and bangs, pops and screeching. I ran as fast as my legs would carry me. Sam raced across the street ahead, not waiting.
From the fiery glow beyond a line of trees, smoke poured into the black sky, obscuring the moon and stars. The acrid reek burned through my nose and throat, caught in my lungs. The sting made my eyes water, and we hadn't yet come in sight of the fire, except where its light shot between trees.
The roar of flame covered the sound of my footfalls beating the ground—I could only feel the heavy thud, thud travel up my legs—and the shaking in my breath from icy air.
Hot brilliance blinded me as we broke through the trees, reaching the house. Dark and light. I couldn't see because my eyes didn't know how to adjust.
I crashed into Sam. He held his sleeve over his face, using his free hand to press a wet cloth into my hands. I wondered where he'd gotten the water, but then remembered the bottle stuffed into his pocket.
"Put this over your nose and mouth." The fire tried to consume his words. The rush and groan were louder now that we stood in the yard. Heat billowed toward us, bringing smoke and sparks. In spite of the inferno, I was glad for my coat to protect my skin. I'd already experienced enough burns to last several lifetimes.
"We have to help whoever's inside the house." I squinted at the trees silhouetted between bands of firelight. "Who lives here?"
"You're not going anywhere until you put that over your face." He pressed the handkerchief over my nose. Breathing turned wet and heavy, more difficult than inhaling smoke. He wasn't wearing a wet handkerchief.
"But you—" I bit off my words when I saw his expression, like pieces of him were being ripped out and hurled into the flames.
"I can't stop you from going in," he said, probably at a normal volume, but the fire made his words soft. "I can only try to help you make it out alive."
I pinned the handkerchief against my face and gave a curt nod. We dashed toward the house.
Outbuildings had collapsed in the initial wave, but the white stone remained solid. Janan's doing. At least the entire structure wouldn't fall in on us, though there was still furniture to dodge inside.
And the fire.
We went in. All the doors and windows had blown out—glass crunched under my boots—and tables and chairs were nigh unrecognizable. Everything was black and red, blazing hot. I didn't even have time to sweat beneath my wool coat; the dry air sucked the moisture out of me, and out of my handkerchief as I searched through the terrible heat and burning remains of someone's home.
The fire roared and rushed. It seemed impossible I'd be able to hear anything else, but I caught the ragged sound of coughing. Metals clattered against stone.
Smoke and fire. Debris piled up. I couldn't find the source of the noise.
There, between a fallen bookcase and the remains of a large stringed instrument, lay a woman on her side, facing away from me.
At my approach, she rolled over. Her stomach bulged like Lidea's had while she'd been pregnant. Geral. I'd taken lessons from her, about building roads and constructing outbuildings.
I rushed for her, screaming her name, and as I jumped over the wreckage I lost track of Sam.
"Geral!" Smoke suffocated my voice as I reached her, but her face twisted with confusion. "I'm here to help."
Her eyes focused as I pressed the mostly dry handkerchief over her nose and mouth. It took some maneuvering and shifting her weight, but finally I got her arm around my shoulders and used every muscle in my legs and back to haul her up. By my ear, her breath came shallow and weak from inhaling smoke.
We turned toward the direction I thought I'd come, but the room had changed. Beams had fallen, burning brightly. Blackening rubble blocked our path. And where was Sam?
I coughed at the smoke singeing my lungs, and shielded my face with my free arm like Sam had. It didn't help.
"This way." Maybe it wasn't a good idea to talk, but it helped me focus through the dizzying heat, and she relaxed—only slightly.
Windows and doors were in the same place on every house, and there was some kind of opening on every wall. Any direction was better than standing still. I guided Geral, both of us coughing. Only the fact that the upper level had collapsed on the other side of the house saved us from suffering worse; the smoke had somewhere to go.
I hoped Sam wasn't over there.
Our journey to the wall was unbearably hot. My eyes watered, and Geral was too heavy for me to carry, but that wouldn't stop me. We stumbled again and again.
I reminded myself—maybe out loud—that I'd endured worse, with my hands inside a sylph. But this was everywhere, and I wasn't alone. Geral counted on me to get her to safety.
The world swam with blackness. I staggered, Geral heavy on my shoulders, but as my knees hit the ground, a cool mist bathed my face.
Someone lifted Geral away from me.
I tried to watch where they took her, but I was blind now that I'd left the too-bright house. No matter how much I blinked, my vision wouldn't work right after peering through smoke and heat. Maybe my eyes had boiled out.
Cold pressed against my face, then air. Fresh air. I inhaled as deeply as my lungs would allow, like I'd never get another clean breath again.
Strong arms encircled me, picked me up, and I was carried away from the heat and roaring fire. My skin cooled when I sat on the ground, and at last my vision fizzled toward normal. A youthful face floated before me.
"Sam?" Was that my voice sounding so wispy? I sucked on air from the mask again. Coughed. Breathed.
Sandy hair and sharp features. Cris shook his head and smiled. "Wrong admirer. Sam is over there with Stef. He got Orrin out."
Orrin had been here? My head pounded, and I tried to focus. Sam was okay. Cris had given me air. I was sitting on the hard, cold ground.
"I thought you'd be across the city by now." My voice sounded like a toad. That wasn't much of an improvement.
"I stayed to visit with Orrin and Geral. A little after I left, I heard the explosion." He gazed around the ruins. "Good thing you and Sam got here so quickly."
"Will she be okay?" I couldn't find her in the mass of people around the house. They aimed hoses at the building, spraying the same mist I'd stumbled into.
His tone was gentle, and so was the way he wiped a cloth over my face; it came away soot-black. "I don't know."
I appreciated his honesty.
The fire died, leaving only electric emergency lamps to light the ruins. Smoke still rose like giant sylph as people shouted orders, darted around. Their silhouettes were strange and long in the illumination, but I saw Councilors Deborl and Sine speaking. Arguing? I couldn't tell from where I stood.
When I lowered the mask—I'd forgotten I still held it to my face—I caught a familiar shape across the yard, sitting near a tangle of fallen and blackened pine trees. Sam.
Stef crouched over him, hands on his shoulders. I couldn't hear their conversation, but Stef glanced my way, darkness obscuring her expression.
She was in love with him. Cris probably was too. I could only think of maybe six people who wouldn't be.
Gravity dragged at me, but Cris caught my elbows and kept me from slumping over. The mask wasn't so lucky. It bounced when it hit the dirt.
The reek of smoke permeated the air, but everything seemed so quiet now that the fire was out. The roaring, blaring, consuming fire. All around, people were still gathered in groups, talking and pointing at various places on the house.
Strange that the white stone remained as if nothing had happened. I hadn't expected anything less from Janan, though. I'd seen the temple mend itself after Templedark, and other structures of white stone withstand onslaughts they shouldn't. It was wrong. Creepy.
"Can you stand up?" Cris held my shoulders.
"I don't know." But I gave it a try, climbing to my feet, using Cris's shoulder for balance. Across the yard, Sam got up, too, and started toward me, leaving Stef to trail after him. "Thank you for helping me." I was always too slow with politeness, but at least I'd remembered this time.
"Of course." Cris smiled. "We've got a lot of work to do, you especially. I can't let you go around with smoky lungs when market day is so close."
Lights shone in a mobile medical vehicle. Geral was probably in there. "I hope she and the baby are okay."
"They have a chance because of you."
I wasn't sure how that was supposed to make me feel. Good? Proud? Mostly I felt overwhelmed and exhausted.
"Ana." Sam's deep voice filled me, sweeter than smoke, sweeter than the burst of fresh air from the mask. Soot and ash stained his face and clothes.
I stepped forward into his arms, relieved just to touch him. Warm. Solid. Real. Neither of us had gotten burned up.
He swayed, but stayed upright as my weight settled against him. "I'm so glad you're okay," he murmured into my hair. "I lost track of you in the smoke. I was worried." Hands pressed hard on the small of my back.
"Any idea what caused the explosion?" Cris asked behind me. I'd forgotten he was here.
Sam shook his head. "Let's not bother Geral about it, but Orrin is over there. We can ask if they were doing anything unusual."
"Should I walk Ana back to your house, Sam?" That was Stef. I'd forgotten she was here, too. "No need to burden her with this, and she looks like she could use some rest."
I peeled myself off Sam. "I'm fine. Besides, Stef, I'm sure your scientific mind will be more useful here than taking me home."
She looked ready to argue—probably that I was so young and shouldn't be exposed to such horrors—but just then light bloomed on the far side of the city. The ground trembled.
"Was that—" she started, but seemed incapable of completing the thought. Like it was too terrible to comprehend.
The words were ash in my mouth. "Another one."
This was not an accident.
#
# SMOKE
THERE WERE THREE more explosions, each an hour after the one before. Stef tried to send me home every time, but I refused. Sam and Cris never backed her up, and her annoyance devolved into a glower.
"She shouldn't be here," she told Whit. "She's too young. This will traumatize her."
I turned to watch flames die under the fire-suppressant mist. Floodlights burned across the city, and smoke billowed into the sky, so completely veiling the stars they might not exist anymore except in memory. I'd grown used to geyser steam rising at all hours, but this was nothing like that. Smoke plumed dark and angry, evidence of destruction and hatred.
We waited for the sixth explosion. Everyone wore tight faces and worry, but we stood there by the smoldering ruins of the fifth house and nothing happened.
I stared at the white shell of the house—now streaked with cinders and dust, but whole—and hated Janan. I hated him for what he did to newsouls, how he'd deceived everyone for so long, and that he'd never let anything happen to his precious white city as long as he was awake.
My hand found my knife inside my coat pocket, the cool rosewood handle smooth under my soot-darkened fingers. As I had in Menehem's laboratory, I wished for a weapon against Janan. Something that would hurt him.
But even if it were possible, Janan reincarnated souls who meant everything to me; I wouldn't be able to do it.
That made me hate him more.
Sam drew me homeward, into his own white shell of a house. I could sometimes forget about the exterior walls with all the parlor instruments, the honeycomb shelves, and the perfume of roses.
At some point I must have showered, because when I realized I was sitting on the sofa, tense and waiting for another explosion, my clothes were clean and my hair wet. I no longer stank of smoke and ash. A glass of juice sat on a table beside me, half drunk.
Unnerving.
Sam came downstairs, wearing nightclothes the colors of winter forests. Hollows darkened under his eyes, and he carried weariness like chains. "It will be dawn in a couple hours."
Would the sun even shine through all that smoke? "I can't sleep. Maybe ever again."
Birdsong skittered outside, hesitant. Sam sat at the piano like he'd play accompaniment, but his hands rested on his knees, unmoving. "I know. I keep thinking what if our house is next?"
Our house. I liked that he said that, though I wished our house were his cabin in the woods, or Purple Rose Cottage. "It won't be."
He raised an eyebrow.
"I'm not pregnant."
"Oh." His posture evaporated as he realized what I had: every one of the attacked homes had belonged to someone carrying.
Some had been no more than a month or two, so it wasn't obvious like with Geral. Two women had died in the explosions, and a third had miscarried. The three babies might have been newsouls. If they were oldsouls, they'd be reborn none the wiser. But newsouls...
Escaped Janan's hunger only to die before they were born.
Sam swore so softly I almost couldn't hear the words, but they hissed through me and left smears of despair in their wake. "Someone did this on purpose."
Of course someone had.
"No one has resorted to this kind of terror in three thousand years." He faced the nearest window, looking somewhen-else, like he always did when talking about the distant past. "Violence like this only infuriates people. They'll be reborn and take revenge again and again until they feel they've had justice."
And every death and rebirth meant another newsoul went to Janan.
"Honestly," he muttered, "it's easier to live with one another and stop fighting, no matter how much you hate someone. They're not going away. Ever." He glanced at me, eyes shadowed. "Until recently."
"I'm afraid that was the point." I crossed the floor to the piano bench. He stayed on the edge, facing away from me now, so I put my legs on either side of the bench and pressed myself behind him, my cheek on his shoulder blade, my thighs against his. "Someone wanted to send us a message."
One of his hands fell to my knee. "They attacked Geral's house first because she lives close to us. We wouldn't be able to miss it."
"I didn't even know Orrin and Geral were together."
Sam gave more of a breath than a chuckle. "He keeps quiet about his relationships. Don't feel bad that you didn't notice."
I squeezed my arms around his ribs. Comforting my stupid worries even when...this...was happening.
"Can I tell you something?" He sounded like I felt. Kind of desperate. Depressed.
"You can tell me anything." Because I wanted to know everything about him.
He threaded his fingers with mine, our hands wrapped together in delicate knots. "I used to worship Janan. We all did, long ago. And now I find out that he'd have consumed your soul if not for an experiment gone wrong."
Didn't he know I thought about that every day? Every hour since my return from the temple?
"I keep wondering how this happened. Why we get to live and newsouls get..." His whole body trembled against mine. I could imagine the dark, angry look in his eyes. "When we first arrived in Heart, you asked if I felt betrayed that Janan never aided us in a crisis, never protected us."
"I remember. You said you did, a little, and that you wanted to believe you were here for a purpose."
"That's right." His voice grew quiet. Distant. "And now I discover that all along our purpose has been to replace newsouls. Our purpose has been to help feed a monster."
The words made me shudder.
"I do feel betrayed," he said. The rest seemed to tumble out. "We worshipped him, and he used us. I resent the time I spent caring, even though it was forever ago, and I resent everyone who tried to shame me when I stopped. It hurts, this betrayal."
His confession and anguish filled me, made me want to shatter. I wanted to draw it out of him, encourage and comfort him like he always did for me. "Why did you worship him?"
"For most of us, it was Meuric's suggestion. We listened to what he said. The rest was us simply not wanting to feel alone." He sounded melancholy. "The inscription on the temple said he created us, gave us souls. It said he was responsible for our reincarnation. It said he would protect us."
"We know the reincarnation part is true," I whispered. "But it's not as if no one's ever changed written history to suit their own purposes. You told me Deborl changed and omitted things from the history books I read in Purple Rose."
Sam nodded.
"So let's go by what we know. In five thousand years, Janan never protected you?"
"He kept the walls impenetrable, but since he is the temple and his heartbeat runs throughout the walls, that might be him protecting himself more than anything."
I squeezed his hands. "Did he help you when you needed him?"
Sam shook his head.
"Does he love you?" I asked.
Silence.
Roses filled the parlor, evidence of love: Sam's love for me, Cris's love for his garden, and my friends' love for one another. I'd seen over and over how they demonstrated their love, and it made them giving, compassionate, and kind. Sam's love made him brave, selfless, and willing to protect me when I needed it.
Maybe Sam was thinking about the same things, because he said, "It seems love is the most important thing. Someone who doesn't love us and uses us to hurt others was never worth our devotion." He let out a long sigh, deflating. "I know now he's real, but before Templedark, I hadn't believed in Janan for thousands of years. But then, when I did, it was comforting thinking there might be some kind of power protecting us."
"There might be. It's just not Janan."
"I love your hope. You make me want to have hope, too." Sam kissed my knuckles, and exhaustion filled his tone. "Are you really all right after earlier?"
"I am. Really." Though I would never forget the heat and horror. "What will happen now? Will there be an investigation into the explosions? Will the survivors be cared for?"
Sam nodded. His damp hair brushed my skin when I lifted my face. He smelled like soap and shampoo, no traces of tonight's events.
"There will be an investigation, but it will take a long time to question so many people. It was already night when this happened, and most live alone. There will be few alibis, few reasons to believe anyone. People who've actively spoken out against newsouls will be the first suspects."
"That's a good start." It wasn't enough, though. "I doubt everyone who hates newsouls is stupid enough to blab about it before planting explosives."
Sam snorted. Had that been a dumb thing to say? I started to lean back, but he tightened his fingers around mine, keeping me pressed against his spine.
"You're right," he said. "There are people who hate newsouls but will have kept quiet until they were ready to act."
I shuddered, wondering who those people might be. "It's easier when you know who your enemies are."
"Really is."
I bit my lip, like that would keep in what I didn't want to say. Maybe I held my breath, too, and my heart slowed down, or maybe I tensed. At any rate, Sam knew I was trying not to just blurt out my every thought.
"What is it?" he asked, and I could almost hear the resigned smile.
I pressed my face against his back, though his shirt was tight against his skin so there wasn't anywhere to bury myself. "Promise you won't be upset."
He just kissed my forefinger. Not a promise.
"It"—at least I didn't have to explain what it was—"happened so soon after the meeting, whoever did it must have known our plan."
He was very still. A statue of a musician.
"Maybe I'm wrong. Maybe it's just a coincidence that it happened so soon after I asked for help. People have been throwing rocks and breaking into nurseries. Maybe this was their next step and it had nothing to do with tonight." I peeled off Sam, off the piano bench. He hadn't moved, but I couldn't stay still. In spite of the wretched day that wouldn't end, everything in me itched to run. To go somewhere. The grief inside me needed to escape.
I stalked around the parlor as though my anxiety might leak out from the force of my feet hitting the floor. I made the perimeter twice before Sam came to life again and began tracking my progress.
"I hope you're right," he said at last. "I hope it is just a coincidence, because the idea of one of our friends being responsible is too horrible."
My insides twisted into knots as I stopped before him. We were both exhausted and heartsore. Maybe right now wasn't the best time for this talk.
When the light shifted in the window behind him, casting him as a silhouette, he looked my age. I trailed my fingertips over the soft curves of his cheekbones, down his freshly shaven jaw, and across the thick lines of his eyebrows. At my touch, he swallowed hard, and little by little the tension drained from his shoulders and neck. He dropped back his head. His lips parted and his breath shallowed.
"I miss being outside of Heart," I said. "I want those days in Purple Rose before the sylph came." I caressed the frown line between his eyes, how I always knew he was thinking too hard about something. Then I found the line by his mouth, a long curve from his smile.
"Me too." His hands breezed over my hips. I didn't fight the urge to lean toward him, press my body against his.
Just us, the parlor instruments, and the early morning quiet, it was easy to imagine we were the only people. No explosions, no sylph, no Janan.
My fingers came to the end of their wanderings on his forehead. I smoothed back his hair and kissed him. "Sam." A shiver ran through his body when I said his name. "Will you sleep in my room with me?"
A wicked smile flashed, and his hands dipped from my hips to my bottom to my thighs. "I'd like that."
My skin burned with his touch, even through my clothes, and I ached to discover what he might feel like without the layers of cloth between us.
Before I could suggest it, he drew away. His hands fell to his knees and the longing faded from his eyes, shifting to regret. "But I should sleep in my own bed." He spoke softly, but that didn't dampen the pain of his words.
I stood there like a moron, feeling trapped in another rejection, trapped in the memory of when I'd first arrived in Heart. We'd faced each other in the kitchen so close, so tense I'd thought he might kiss me. But he hadn't.
He stared at his hands, as though remembering the same event. "Ana." Just a breath. "I do want to, but maybe we shouldn't."
"Why?" I knew why. I'd heard why earlier.
"I..." Resolve steeled his voice. "It wouldn't be proper."
Proper.
"I heard your fight with Stef." My voice trembled with effort to speak softly. "The night of the masquerade, after you were arrested, Li kept saying things like that. She insisted the way we danced was inappropriate. Then Deborl said it that day outside the temple. People have said it in the market field. Have you been hearing it from friends, too?"
"Everyone has opinions."
"And why should we mind? Have I given you a reason to believe I care if other people think our relationship is inappropriate?"
"I'm glad you're not worried what they think." He closed his eyes, expression drawn as though he'd rather be having any conversation but this one. "But have you considered that despite how you feel, this might truly be inappropriate?"
My mouth fell open.
"Stef had a point. I'm old, Ana." He shoved himself to his feet, all fire and passion. "It doesn't matter what I look like. The truth is that I have done so many things in other lifetimes. I don't mean composing symphonies or exploring the world beyond Range. I mean intimate two-people-alone things."
Pieces of me were unraveling. Was he trying to hurt me?
"I hate that." My heart thundered. I'd just wanted to be near him while we slept, and suddenly everything spun out of control, all my unspoken fears and insecurities so bright and blinding. "Every time you remind me how much older and more experienced you are—I hate that. You think I don't know?"
"I think you don't care."
"Well, I don't." I was a liar. I did care, but not nearly as much as I did other times. "I want things—whatever kind of things—to go normally. Whenever they're supposed to happen, that's when I want them to happen."
His face was stone. "That's the problem. Normally both parties know all the details. They have the experience, even if it's not with each other. This relationship is different. There's nothing normal about it. How am I supposed to know how far to go with you? How am I supposed to know when you're ready, and for what? I want to be honorable and do the right thing, but I don't know what that is."
"You could let me decide." I crossed my arms. "Aren't both people supposed to have a vote in a relationship?"
He shifted his weight, myriad expressions crossing his face before he settled back to the same stone as before. "Do you know what you'd be deciding?"
Caught. My face ached with scowling. "I'm an adult, Sam. Nearly four years past my first quindec. You said that just last night."
He towered over me, body tense and voice sharp. "Really, Ana."
I resisted the urge to back away. "Like many things I had to figure out on my own, the books I had access to didn't specify how to do certain activities."
"So you don't know. You can't make an informed decision like that."
"You could tell me."
He massaged his forehead. "I can't even imagine how strange it would be for you to hear about it. Even thinking about how I'd explain it makes the whole thing seem a lot less fun. It might even sound scary."
"No, that isn't what I meant." I shook my head. "I meant showing me with—with you. Like you promised the night of the masquerade." He'd said he had a thousand things to show me, places he'd kiss me or touch me. My whole body ached with anticipation under his hands, and I'd thought he felt the same way. I said more softly, "Don't you want to?"
"Yes." He sounded raw. "Yes, but I don't want to take advantage of you."
"Your stupid honor is going to make me crazy. As far as I can tell, Sam, we're going to spend the rest of our potentially short lives not doing anything more than kissing."
He looked uncertain. A crack in the stone. "You could ask Sarit?"
How could he be so clueless? "You're missing the point."
He waited.
"I should be able to count on you, but you're telling me I can't."
"Ana—"
"No. I understand this whole thing is weird. You don't know how to reconcile what has always been acceptable and what you feel is honorable in this case. I've always admired your need to do the right thing, so I appreciate it. Really."
He didn't look convinced, and it was hard to believe that less than a day ago, we'd been standing here by the piano, surrounded by roses, kissing, his hands up the back of my shirt....
"We may not be able to decide whether our relationship is or isn't appropriate. We have emotions invested." I struggled to steady my voice. "But we can decide if we care about appropriateness. If you don't care, then we'll decide together what we do."
His voice was rough. "And if I do care?"
"Then I suppose nothing will ever change." Or everything would. "I don't want to be sixty and still unenlightened about these matters."
"I'm sure by then—"
"It will be appropriate?" My head buzzed with exhaustion and sadness. "When does that happen? When do I magically become old enough for you? There will always be five thousand years between us."
"I don't know." He dropped his gaze. "I just don't. I'm sorry."
Ugh. I saw his dilemma, but that didn't change the fact that we weren't going anywhere until he made a choice. It was our relationship, so what other people thought shouldn't matter. "I'm going to bed."
He nodded.
Why couldn't he just be whatever I wanted, whenever I wanted? Why did things Stef said have to matter so much? Why couldn't Sam truly be eighteen—almost nineteen—like me so we didn't have to deal with any of his issues from being so old and my being so new? I didn't care. Usually. He shouldn't care either.
I almost asked him to reconsider my offer. Instead, I just said, "Good night," and turned away. My courage was as thin as silk, but I held it around me like armor and urged myself up the stairs, dragging the remains of my dignity.
#
# ABSENCE
WHEN I GOT up a few hours later, I started coffee and took care of all the chores. I hadn't slept well—or at all—and even during a crisis, chickens and cavies needed to be fed.
Then, at the kitchen table with a cup of coffee, I closed my eyes and inhaled steam, absorbing the silence of no explosions and no fighting with Sam.
The scrape of ceramic on stone yanked me out of my peace. Sam poured coffee at the counter, his face lined with exhaustion. Just seeing him from the corner of my eye, he might have been a stranger. Even his clothes were rumpled.
I settled into a comfortable glower when he faced me.
"Do you want to see if we can talk to the survivors in the hospital?" His voice was hoarse with no sleep. "See if they saw anyone?"
"I was already going to do that." I gulped down the rest of my coffee and stood. "Are you ready to go?"
"I guess." He combed his fingers through his hair—it didn't make much of a difference—and finished his coffee.
When we were dressed for the chilly weather, we headed toward the Councilhouse. He didn't try to make excuses for earlier this morning, which was good. He didn't even talk to me. Just as well. It left me time to focus on not paying attention to the ashy reek, or the rubble strewn around Geral's property.
Charred bits of something littered the road. Sam picked them up. To carry to a recycling bin, I supposed. I couldn't let him feel morally superior, so I grabbed some, too.
We dropped everything in the appropriate bins when we reached South Avenue, then turned north, and I couldn't help but see the temple. White on gray sky, though it wasn't just smoke up there now. Clouds thickened, threatening snow or sleet.
I shivered and eased my strides closer to Sam. He was nice enough to pretend not to notice.
"Tonight," I said, so he'd think my walking closer to him was about secrecy rather than comfort, "I'm going to work on translating the books. Cris said he meant to bring over the paper I gave him before, so I want to get that, too." I had the notes I'd gotten from Meuric safe in my pocket.
"Okay." Sam kept walking.
We wandered through the hospital wing of the Councilhouse until one of the medics told us where Geral and the other two survivors were being treated. I wrinkled my nose at the scent of rubbing alcohol and burned flesh—a reek too familiar to me. My hands were folded up and tucked beneath my chin before I realized.
Sam touched my back. "This way."
I flinched, but followed through double doors that led into a reception area the size of Sam's parlor, with walls of white synthetic silk sheets, pinned in place by steel shelves; the walls seemed to glow in all the light. People at the desk glanced up at our entrance, then back to their work.
"Sam. Ana." Sine approached, her gray hair pulled into a tight bun. She wore a medic's smock and gloves, and a deep frown. "Is something wrong?"
"We came to see Geral and the others," Sam said. "Do you know anything about who caused the explosions?"
"I think you mean what caused them." She glanced around the room; a lanky teenage girl watched us, while another man—Merton?—muttered into his SED as he vanished behind a partition. Sine spoke at a normal volume. "It was only gas leaks and corroded wires. Walk with me over here."
Sam's face was stone as he nodded, and we headed into a hallway off the main chamber. Several curtained rooms waited on one side. Recovery rooms.
We went all the way to the end of the hall and took the last room. It was unoccupied, as were the five before it. Sine must have wanted a lot of privacy.
She motioned at the chairs around the bed. "Sit close so I don't have to yell."
Sam and I scooted our chairs toward hers.
"For now, the Council is giving the gas story." Her voice was so quiet I strained to hear. "But I assume you two have already figured out what really happened."
"Someone hates newsouls." I wanted to be sick.
"Yes." She leveled her gaze on me. "I can stop you from investigating this, but I won't. I know this is something you're passionate about, Ana. I want to caution you, though, before you do anything reckless."
Because someone had told her about the meeting last night? Or she just knew?
She went on. "Whoever planted those explosives is already willing to risk Council repercussions, not to mention several lifetimes of people exacting revenge. Hurting—or killing—either of you isn't going to be a problem."
"But the law about killing me—"
She shook her head. "They don't care, Ana. Any of those unborn could have been newsouls. The law protects them, too, but..."
"There should be better laws." I crossed my arms, and neither Sam nor Sine disagreed. "What about Lidea and Anid?"
"Wend took everything they needed to my house. Whoever is doing this won't suspect Lidea and Anid are with me, at least for a few more days. Hopefully we'll have answers by then."
"I suppose you've already questioned the survivors?" Sam asked.
She nodded. "As much as we could. Some were badly burned, and their medication is making them, ah, interesting to talk to. But Geral was asking about you, Ana. And you'll be relieved to know that, while the shock did send her into labor, she gave birth this morning. They're both fine."
"She did? They are?" I twisted in my chair like I'd be able to see through the layers of silk walls. "When can I see her?"
"Now, if you'd like. She's in the first room in this hall."
I was up and at the curtain before I realized they weren't with me. "You aren't coming?"
"We have a couple of things left to discuss," said Sam. "I'll join you shortly." He sat straight with his hands on his knees. How had they both known to stay? How had one communicated to the other there was something they wanted to talk about without me? There must have been some signal I'd missed.
I hated being new. I hated being excluded. "Okay." I shoved the curtain closed behind me, though it wasn't terribly accommodating with slamming. It swished and floated back into creepy templelike perfection.
I found Geral's room. There was nowhere to knock, so I swished the curtain around until she laughed and told me to come in.
She reclined on the bed, a swaddled baby in her arms. The medics and birthing assistants had cleaned her up, but there was a bandage on her forearm and stitches along a cut on her jaw. I should have found her sooner, before she'd gotten injured.
"Would you like to come the rest of the way in? If someone else stops by, they're going to run over you." Her smile would have been serene if not for the flinch at her cut. "I was hoping you'd visit," she said as I crossed the room.
"Oh." I tucked my hands into pockets. "I actually came to ask if you saw who blew up your house, but—"
"It was a gas leak and corroded wires."
I tipped my face downward and raised my eyebrows. Stef had used that look on Sam several times, and it always made him tell the truth.
She gave a breathy laugh and hugged the baby tighter. It was sleeping, and I couldn't tell if it was a boy or girl. "Well, no, I didn't see anyone. Neither did Orrin. Cris was there a while, but he wouldn't have hurt us."
"No, of course he wouldn't have." I stared at the wall, wishing I knew why the arsonist had chosen last night. Coincidence? My meeting?
"Would you like to see my baby?" Geral whispered, tired and hopeful and sad, and I managed to feel even worse than I already did. She'd just wanted to share her newborn, and I'd been caught up in other problems.
"Sure." I stood. "I'm sorry. Everything is just overwhelming. Who did you have?"
"Ariana." She tugged the baby's knit hat downward, though it had been just fine a second ago. "I hope you don't mind."
"Why would I?" And then: "Oh." There'd never been an Ariana. It was a new name for a newsoul. "Oh." The last came out perhaps more startled than I'd intended.
"I'm going to be a good mother." Defensiveness edged Geral's tone. "And you of all people should be accepting—"
"I am!" I pressed my hand to my mouth. "Sorry. I keep getting surprised. I'm as used to people being reborn as everyone else. I may not have had five thousand years for it, but it's still been my entire life." And that felt like a long time.
Her gaze flickered toward the curtain as someone entered. "It's all right. After last night, I'm nervous about how people will react to her. I already love her so much."
Why couldn't she have been my mother? Or Lidea? Aside from the fact that they were currently too young.
"I understand." I gazed at Ariana, her dark skin and silky hair. I wanted to tell her how she'd escaped Janan, how others hadn't been so lucky. Even this life where people threw rocks at newsouls—it was better than never having the chance.
And I wanted to tell her I'd do anything to protect her, because we newsouls had to stick together.
I didn't say any of that, though. Not in front of Geral and whoever else had come into the room. Sam. I recognized the sound of clothes rustling as he shifted his weight. "I was just alarmed by her name," I said at last. "But I'm honored."
"You saved her life." Geral blinked away tears.
My face burned. "Next time a newsoul is in danger, I'll send Sam. So far I've got two named after me, and he doesn't have any."
"I am a little jealous," he said from the foot of the bed. "She's beautiful, Geral. Congratulations."
We chatted a little more, and then it was time to go. "Keep her safe," I whispered as I hugged Geral. "Don't trust anyone."
Sam and I finished speaking with the other two survivors, but they knew even less than Geral. We left the Councilhouse and sat on a market field bench as the sun dipped toward the west. Not that it made much of a difference. The clouds from this morning had thickened, and delicate snowflakes spiraled toward the ground. I caught one on my mitten, but it melted.
"How do you feel about it?"
I couldn't tell if Sam was asking about the snowflake or the direction the planet rotated, so I just raised my eyebrows and waited for him to figure out why I wasn't responding.
"Ariana. Anid. People naming newsouls after you."
The temple loomed behind me, already lit with shifting patterns: signs of the evil entity inside. He'd wanted to consume us, the newsouls. So we had that in common already. Now our names. "It doesn't matter how I feel about it."
"Of course it does."
If how I felt about anything mattered, why didn't he tell me what he and Sine had been so secretive about? He could have asked how I felt about not understanding the sylph, or not having had a chance to study the temple books.
Instead, he wanted to know how I felt about newsouls being named after me?
"I think a lot of people whose names sound like Ana are going to consider changing them."
"So as not to be associated with newsouls?" He sipped from a bottle of water like he didn't really care about my answer. I wanted to rip it from him and hurl it across the market field.
There were people out, though. Councilors loitered on the steps—Deborl kept glancing at me—and a couple holding hands walked by. One muttered, "Sylph lover," and probably wished she had a rock to hurl.
A trio of children caught snowflakes on their tongues; I'd done the same thing when I was that age, but it was surprising to see anyone else do it. These children were five thousand years old. A characteristic of their physical age, perhaps, like people currently teenagers were attracted to other teenagers.
I didn't look at Sam as I spoke. "We should speak to everyone who was at the meeting last night. Everyone who knew our plans. Not to accuse them of anything. Just—Just to see if they have any ideas."
Sam capped his water and checked the time on his SED. "Where first?"
I hadn't expected it to be that easy. Maybe he knew better than to argue with me. "Let's start on the far side of the city and work our way home."
"That could take a while." Ah, there was the disapproval I'd been waiting for.
"Did you have plans this evening?" I wanted to work on the temple books, but I could do that after he went to bed.
He glanced at the sky, snow dropping harder now, and shrugged. "Best start with Cris. He'll be out covering the plants, if he hasn't done it already."
Sam set a brisk pace, leaving no breath for talk. No problem. I lengthened my strides, but still had to take two for his every one. Also no problem. It helped stave off the cold.
Darkness loomed by the time we reached Cris's house, though in the temple light, and light from our SEDs, it was easy to see nothing had been covered or moved into greenhouses like Sam had predicted. Nor could we hear Cris's movements.
Most flowering plants had closed themselves for winter, but they were still vulnerable to the cold; I'd lost several roses at Purple Rose Cottage before I realized they needed protection, just like people.
"Where is he?" Sam marched down the path and headed toward one of the greenhouses. "Check the other one. If he's not there, we'll look inside."
I almost snapped at him for telling me what to do, but the last thing we needed was another argument. Fuming silently, I did as ordered.
Inside the greenhouse, warmth and humidity washed over me, a sudden and unpleasant shift from the crispness outside. "Cris?"
There was no response as I walked between the rows of orchids and other flowers I didn't recognize. This was the second greenhouse, the one I hadn't been inside yet; I wished Cris were here to tell me what all these flowers were.
I turned off the light and shut the door behind me, and met Sam on the front steps, picking the lock. Why did anyone bother to lock things?
He ushered me in first, out of the falling snow. "I called his SED. He didn't answer."
"He might have gone back to Purple Rose Cottage to get those roses. He said he might." I stepped farther into the cluttered house. "Cris?" I yelled again. Only the eerie quiet answered, thickened by the sheet of white forming outside. If he'd gone to Purple Rose, surely he would have covered his garden here first.
Plants and journals filled the parlor and all connecting rooms I could see. Shelves held pots and trays of seeds. Heat lamps stood in two corners, though I couldn't tell what they warmed. It was practically another greenhouse, though some of these plants looked edible. The whole place smelled green and loamy and floral.
I followed Sam into the kitchen. "What's that?"
He was in the process of lifting a tray of seedlings and picking out a folded sheet of paper from beneath it. "This is yours."
How could he tell? "Yeah, he said he had a few thoughts."
The paper was damp and smudged with soil, but Sam carefully unfolded it on the tabletop to reveal the list I'd given Cris after our gardening lesson. "Look." He brushed away dirt.
I pressed my shoulder against his and peered at the new lines on the page. "'Gate or portal? Arch?'" The symbol next to Cris's guesses did look like an archway, but only if I tilted my head.
"That seems reasonable enough."
Hmming, I swept more dirt aside. Damp grains stuck to my fingers. "I remember this one." I tapped a symbol that was a pair of vertical wavy lines, thick slashes between them like shading. "'Shadow. Darkness. Nighttime.' I was looking at it the wrong way."
"How do you mean?"
Thoughts snapped, clicked together like the first time I'd understood a waltz had three beats, not four. Suddenly it made sense.
I bounced on my toes. "I get it!"
Sam put on his most expectant look. "The writing?"
"No, why paper cuts hurt worse than knife wounds." I rolled my eyes. "Of course I meant the writing."
"All right. I don't get it."
I made my fingers like a spider on the paper and turned it around and around. "This is what I was doing when I was trying to read the spiral. Turning the book upside down when I reached the top of the spiral. That's also how I copied the symbols, like this one." I pointed at the one Cris had marked "gate."
"But?"
"Why would anyone write like that in something as unwieldy as a book? They'd spend all their reading time turning the book around and getting dizzy. This symbol"—again I pointed at the gate symbol—"was on the side of the spiral when I copied it. That's why it's sideways now."
Understanding bloomed on Sam's face. "So you read in a spiral, but all the symbols are oriented the same, no matter their location."
"Exactly." I bounced again, and Sam twitched a smile. "I get it! I love that feeling. I want to go read all the books right now."
He stared at me like I'd grown a second set of eyes. "You said lo—" His mouth made a line as he looked away. "Well, Cris isn't here. Shall we try the next person?"
As soon as he spoke, I halted mid-bounce. I'd said love. Out loud. Did I mean it? Did he expect me to say it to him now? There was a huge difference between loving a feeling or event—and loving a person.
I felt like a whirlwind, with all my thoughts and emotions. Or maybe they were whirlwinds, and I was just a butterfly or blue rose.
"Sure." Trying—and failing—to pretend like nothing happened, I shook the rest of the soil off the paper and put it in my pocket. Cris had left only a few guesses, and they might be wrong, but they'd be good places to start.
"Whit is next." He led me through the maze of potted plants and out the door. Snow fell thicker, a solid white coat. "I don't think the weather will let up any time soon. We may have to quit early, before it gets too difficult to walk. Home is on the other side of the city."
As we emerged onto the road again, I looked southwest toward our house, but there was only dark and snow. And the temple light making a million flakes shimmer as they fell.
The dark streets remained empty, our passage the only sound. I wished we were at home having music lessons, because playing in a group last night had given me ideas. And music was far less hurtful than thinking about the explosions, or our argument.
Cold swirled and made me shiver as we passed by a white shell, which had once been someone's house. Now the occupant was gone, lost to Templedark. Someone had cleared away the debris from outbuildings. I wondered if there was anything left inside, or if the darksoul's belongings would stay there until they rotted—a memory of someone loved and lost.
We kept walking. The silence and weight of history drowned me.
"What happened between you and Cris?" My words turned into mist, barely visible in the temple light.
"It's nothing." Roughness edged Sam's voice.
I knew better than to push, but—"I don't think it was nothing to him. I see the way you are together, and the way he looks at you."
I didn't think he was going to say anything at first, but then: "It was two lifetimes ago." He had that somewhen-else tone again. Good memories or bad? Suddenly I wished I hadn't asked. "Cris was working on the roses, and I was composing a nocturne about them, so I asked to stay with her a while and study how they grew, how she cared for them."
Sam had lived in Purple Rose Cottage? With Cris? I tried to imagine I'd always felt his presence there, even before I became aware of music and what it meant to me. "What happened?"
"It was fine. I went between there and my cabin, learned more about roses than I thought possible, and after a while, we grew to appreciate each other's company—more than I want to talk to you about."
"More than I want to hear about, I'm sure." I wanted to pretend he was really only eighteen and everything he was telling me had actually happened to someone else. I wanted to pretend he'd only ever loved me. "The song you composed—"
"Songs have words. You can't use 'song' for everything."
I smiled. "Your song ended up being a serenade? For Cris?"
He nodded, his movement barely discernible in the darkness. "We played it as a duet. I'd mostly forgotten about it."
Would he forget about the waltz he'd written for me? Most nights, I fell asleep listening to it on my SED. It wasn't as good as hearing Sam play it on the piano for the first time, but it always made me happy, made me remember the evening I'd discovered he wasn't just Sam, he was Dossam, the musician.
Oblivious to the way my heart tied itself into knots, he continued. "After that, it was my fault. We wanted different things, we argued, and she told me not to come back to the cottage until I was less selfish. So I left. I could have stayed and tried to work things out, or find a compromise, but I didn't. By the time I was reincarnated, I realized I regretted my decision."
"What did you fight over?"
He glanced at me and shook his head. "I don't—I don't want to talk about that."
It must have been huge. Dedication-of-souls huge? What else could drive them apart if they still looked at each other awkwardly, hopefully? I couldn't forget my first morning in Sam's house, when Stef had whispered, Don't let him break your heart, sweetie. He never settles.
Now I knew part of that was because she loved him. Cris loved him. He hadn't stayed with either of them.
And did I love him? The word still made me choke. Even more frightening was the sudden understanding that my feelings for him—whatever they were—might be bigger than his feelings for me. I didn't want to end up like Stef and Cris, pining lifetimes later.
Cold sapped moisture from my skin. I licked my lips and ducked into my scarf. "So after a while," I said, "you regretted the decision not to find a way to work it out?"
He nodded and guided me around a corner. Snow built up in yards and on trees, reflecting temple light enough to illuminate our path. "I've lived long enough to know there are things worth regretting, but there's nothing you can do to change the past. And yet, sometimes it works out anyway, in ways you don't expect."
Did he mean me? I couldn't bring myself to ask. The things I wanted to say and do but didn't know how—they felt like a wall between us. "Do you still regret it? Whatever it was you couldn't agree on?"
"I regret that I hurt her so badly. And that we didn't speak for a hundred years because of it. By the time he presented the roses and no one thought they were blue—that was both of our last generations—I felt like saying anything would just make it worse for both of us."
My face did something between a smile and a grimace. "I hate admitting when I'm wrong, too."
Sam pulled out his SED. The glow shone on his frown, and the line between his eyes. After a moment's hesitation, he tapped the screen a few times and pressed the device to his ear.
I blinked away light and let my eyes adjust to the darkness once more. "I'm worried we haven't heard from him. Even more worried he didn't care for his garden."
"Me too." Sam replaced the SED in his pocket. "After the blue rose challenge was first issued, Cris packed up everything and built his cottage so he could work without everyone watching and criticizing his progress.
"One spring, he came back to Heart for supplies. It had been an especially unfriendly winter, but it was warm when he set out. Of course, as soon as he got home, a blizzard came through. He'd left his plants ready for spring, so they were still delicate. As soon as he realized how bad the weather was going to get, he turned his horse around. He made the entire trip in a day and a half and saved all his roses at the last minute. Didn't lose even a leaf."
That sounded like the Cris I sort of knew, and solidified my worry. Had something happened to him?
"There." Sam pointed to a glow ahead. "Whit's home, at least."
It was almost a relief to go back to thinking about explosions. Thinking about Sam's long history of one-sided relationships—My heart couldn't take it.
#
# FREEZE
INSIDE WHIT'S HOUSE, warmth prickled across my face as I unwrapped my scarf and pulled off my coat. I'd just have to put them on again, but I didn't want to risk sweating inside and then freezing outside later.
"We were curious if you'd heard anything about the explosions last night." Sam pocketed his mittens. His cheeks were flushed dark with chill.
"Just what everyone was told. I saw Jac's house go up last night." He glanced at me, his expression somber. "She was on our list of people to speak with. So were most of the other victims."
"All of them were," I said, "but I didn't bring up a couple last night because only a few people were supposed to know." Sarit had told me privately she'd talk to them.
"How did you know, then?" He cocked his head.
I shrugged and fiddled with my mittens. "Sometimes people just tell me things. I don't know why." Mostly a lie. People told Sarit, and Sarit told me because she didn't think it was fair if I didn't know just as much gossip as everyone else.
"I see." Whit sat on the arm of his sofa, a monstrosity of faded gray and orange fabric that dominated the room. The rest was all bookcases and what looked like old board games on a long table. "I wish I had answers for you, but I came right here after the meeting. I walked part of the way with Lorin and Armande, but eventually we did have to go our separate paths. Orrin stayed behind to visit Geral. Cris, too."
I nodded. "Have you seen Cris today?"
Whit stared through a bookcase. "No, but that doesn't mean anything. People often go days or weeks without seeing even close friends."
That sounded crazy and lonely to me. I wanted to see my friends all the time. But maybe friendship was different when you'd been at it five thousand years.
"He isn't home, and all his plants are uncovered." Sam looked worried again. "We were just there."
"Well, that's a bad sign." Whit scowled.
"To put it mildly." Sam didn't smile. "I was just telling Ana about the time he raced back to Purple Rose Cottage to beat the frost."
"Cris would do anything for those plants." Whit shook his head, a fond smile tugging at his mouth. Then it dropped, like he remembered Cris was missing. "I'll call a few of his friends. Maybe they know something."
"It's strange," I said, "that the explosions would happen just after the meeting. It could be coincidence, but..."
Whit shook his head. "I can't imagine anyone in that crowd doing something like that. They're all good people. You chose well."
The compliment drifted by. I'd chosen well, but somehow, people had still been hurt. I should have done something different. Something better. "The Council is telling everyone it was gas leaks and corroded wires. They should be putting all the pregnant women somewhere safe."
"Keeping them together makes them an easy target," Whit said.
"Then not together. There are lots of places in Heart that aren't being used right now."
"Don't take this the wrong way, Ana, but it's unlikely anyone on the Council would tell you what they have planned. They might very well be doing exactly what you've suggested, but the fewer people who know the details, the safer everyone will be." Whit leaned on a table, near a board game with tricolored tiles and pieces shaped like horses in various stages of rearing or running. "I wish I could give you answers."
"What about Deborl?" I asked.
Whit lowered his voice. "He's a Councilor."
"Who hates newsouls." Maybe I didn't know Deborl well, but I knew enough about him and his choice of friends. Merton had attacked me, spoken out against me, said those horrible things after Anid was born. And Deborl hadn't seemed to care when someone attacked me in the market field. "Do you think anyone might have let it slip to Deborl—"
"That fast?" Whit shook his head. "Everyone was at Sam's for a long time after the discussion. No one left early, right? No one had time to speak to anyone, accidentally reveal our plans, and then the second person go out and set explosives. There just wasn't time."
How long did it take to set up an explosive and get away? Or not get away, if it was Deborl? He'd been at Geral's. "SED messages."
Neither Sam nor Whit argued with that possibility.
"What are you trying to prove?" Red veined Whit's eyes; I was upsetting him. "Do you want someone to have betrayed us? Why are you pushing so hard?"
"Someone has to." My throat tightened, making my voice pinched and desperate. "I hate the idea of someone betraying us, but I swore I'd protect newsouls to the best of my ability. I have to."
Both men stayed silent, just watching me like I might burst.
At last, Whit spoke softly. "Would it be easier if one of our friends were somehow responsible for this?"
"Easier than watching more newsouls die." I swallowed hard. "Easier than not being able to do anything at all."
Whit glanced at Sam, something passing between them, and then Sam touched my elbow. "We'd better go."
I wanted to apologize to Whit, but I wasn't sure what it'd be an apology for. Instead, I thanked him for his time as I pulled on all my warm clothes again. Sam and I headed out.
"I can't protect newsouls from Janan." My eyes stung with tears and cold. "I can't pull them out of the temple and bring them to life, no matter how much I wish I could. But I should at least be able to protect the ones who escaped. I should be able to protect them from people."
Who was I kidding? I could barely protect myself.
My hand fell on my tiny knife, and I squeezed it until my knuckles burned. Not much protection.
"Let's go." Sam sounded like he didn't know how to respond to my confession. I didn't blame him. I wouldn't have known, either.
Before, snow had left a white sheet on the ground; now it coated cobblestones like a blanket.
"I think we should go home," Sam said, linking his arm with mine. I wasn't ready for this kind of closeness, but he knew his way around the city in the dark. I tightened my arm with his.
"But we need to speak to everyone."
"Not tonight."
"And if there are more explosions? I won't be able to live with myself if another newsoul dies because we stopped just short of catching this person." There was no wind and the snow fell in silence, but my voice still rose as if we stood in the middle of a blizzard. Icy air snaked inside my clothes, making me tremble.
"Ana, you're shivering already, and we haven't been out but two minutes. How many times do you expect me to keep you from frostbite or hypothermia?" He brought his face so close to mine I could feel the heat of his words. His skin. "You enjoy making me worry, don't you?"
"No, I hate it." There wasn't much vehemence, though. "I want to do the right thing."
"Sometimes"—he tugged me closer to him—"that means not freezing your fingers off. We still have tomorrow. Anything that happens between now and then is not your fault. Let's go home."
"Fine." I hated when he was right. Snow was piling up; if we waited too long, getting home would be more of a challenge than either of us could handle, especially on empty stomachs. "But first thing tomorrow, we're either going to see people, or be making a lot of calls."
He glanced toward the sky, though it was just dark with swirls of snow. "Calls, unless this lets up. Which I doubt."
I almost asked how he knew, but right. He was five thousand years old. He could probably tell by the smell or the size of snowflakes.
Our trek back to the southwestern residential quarter was long and cold and slow. We passed the temple—Sam had somehow maneuvered so he walked between the tower and me—and still had a long way to go when the wind kicked up. What had been a beautiful, if annoyingly timed, snowfall became rough and stinging.
Snow flew horizontally down South Avenue. It howled like a sylph as it cut through narrow places in the industrial quarter. Trees whipped in a frenzy. Sharp wind scoured the cobblestones clear, and if not for Sam, it might have carried me off, too. I was a rose petal in a snowstorm.
Drifts stood knee-high against buildings, though Sam managed to find walkable paths. I held tight to him, wishing we were already home. My legs ached with cold and fighting the wind. My muscles burned with exertion, and it felt like I should be sweating, but frigid air stole the ability. It was hard to breathe.
Once we reached our street, thick conifers buffered us from the wind. The night was black and snow. My eyes burned. Every bit of me was freezing, even inside my wool coat and mittens.
"Just a little farther." Sam drew me to our walkway, where more evergreens sheltered us from the screaming wind. He breathed hard, too.
Finally, we reached the house, and Sam's mitten slipped on the doorknob as he spoke. "I wanted to ask you something. You've been talking about making your own decisions, wanting to do things for yourself." He tried the knob again, but snow and wool slid across each other.
"And?" I scrubbed my mittens on my coat and grasped the knob, a dim shape in the glow from the window. It turned.
"Do you want your own house? Li's or Ciana's?" His words tumbled over one another as the door swung open. "I'm sure Sine could convince the Council if you did."
I felt like I had a mouthful of snow as I stared up at him. Both houses were across the city, in the northeastern residential quarter. Had he changed his mind? Decided he loved Stef or Cris more?
Maybe remembering why he and Cris had separated made him realize the same thing would happen with me.
Or—I'd probably gone too far, ruining things for him all the time. The Council, the talk with Whit, the way I'd dragged him into my research of sylph and Menehem's machine. Nothing had gone right for him since he'd found me in Rangedge Lake.
How was I supposed to respond? Say yes, I wanted to leave? I didn't. I wanted to stay, because even when I was mad at him, I still liked being with him. But if I said I didn't want to leave, he'd say okay whether or not he really meant it. And I'd keep ruining things for him. There was no right answer.
Sam wasn't even looking at me as I stood there in the doorway like an idiot. He'd dropped my arm and taken one step into the parlor, and he didn't move.
I shuffled the rest of the way in and edged around him. If I was going to cry, I'd at least do it where my tears wouldn't freeze on my eyes. The door slammed behind me, leaving us in silence. "I don't understand," I whispered at last.
"Me neither," he murmured. He wore a stricken expression. But he'd been the one to suggest it.
No, that wasn't why he was upset. I blinked through the tears blurring my vision. The parlor was different.
Destroyed.
Every instrument had been completely demolished.
#
# FADE
"NO." SAM DROPPED to the floor with a thump, staring confusedly at the wreckage. He gathered up bits of something now unidentifiable and turned it over in his hands, looking lost. His agony might kill me.
Low light illuminated the parlor, golden on the hardwood floor, braided rugs, honeycomb shelves between here and the kitchen.
Every time I thought I was free from the horror, though, my eyes were drawn back to the splintered maple wood and shattered ivory keys strewn about the parlor. Ebony keys dashed into slivers. Snapped wires curled on themselves as though trying to hide. Hammers and levers and tuning pins, often overlooked pieces that made the music happen. He'd wanted me to understand them while learning to play the piano, understand their true importance.
Here they were. On the floor.
Then, my eyes acknowledged a twisted length of silver that wasn't part of the piano. Sam's flute, its keys stripped off, leaving gaping black holes across its body.
Cracked pronghorn bones, shredded osprey feathers. Heavy curves of carved wood lay scattered across the floor with harp strings hanging on like cut ligaments. The corpse of a violin rested at my feet, bow broken in half.
I stepped over it, careful not to damage it further. As if that mattered. Glass crunched under my shoes, and I winced, but Sam didn't notice. He stayed by the door, staring as though dead.
Centuries' worth of instruments lay destroyed on the floor.
Blue rose petals were scattered like drops of paint. Their stems and stamens dripped from vases and off shelves. Only the Phoenix roses had been left unharmed.
I scanned the parlor like there might be something left, but even the careful stacks of cases had been demolished, and the walls raided. Some of the carved shelves hung at awkward angles.
Dreading what I'd find, I crossed the battlefield and checked the kitchen, but everything was eerily normal. I could almost hear echoes of mocking laughter.
Sam didn't look at me as I stepped around a crushed sheet of maple, remnants of the piano lid. I tried not to look at him, either, but it was hard to ignore the way he shook his head, muttering to himself. Then, a wild darkness in his eyes, he hurled a length of metal at the wall. It clattered against the wooden shelves, bringing a cascade of rose petals.
Heart breaking for him, I climbed the stairs to check for more damage or anything missing, but it was hard to see what was missing when it wasn't there.
The rooms between our bedrooms held the oldest surviving instruments, sealed in airtight containers to slow decay. They appeared untouched, and so did the workroom and library of sheet music, recordings, and notes on how all his instruments had been constructed.
The harp in his bedroom stood whole. It wasn't much, but it might help, if only I could get him up here to see it. My bedroom looked the same as it had earlier, but I checked all my hiding places anyway.
The books I'd stolen from the temple were missing. So was Menehem's sylph research.
First the temple key. Now the books and research. They had everything.
Almost. They didn't have the translations I'd gotten from Meuric and Cris; those were still in my coat.
My fingers felt like ice as I dialed Sine and told her about the break-in. My voice was too calm, as though my body did all these things on its own now.
"I'm sorry, Ana," Sine said. "Do you want me to send someone over to help clean?"
Outside, the wind howled. Snow pattered on the window. "No." I stared at the empty hiding places and touched the pocket where I used to keep the key. "You aren't going to like this, but can you have someone watch Deborl and Merton?" I wished I knew the name of the guy who'd stolen the key, but I couldn't even remember what he looked like, besides big and scary.
"Deborl and Merton? You don't think they'd—"
"I think they both hate me. I can't prove they've done anything, but—" My voice broke. "Please, Sine."
"All right." Resigned, she hung up.
I put my SED back in my pocket, feeling defeated. They'd taken everything.
Downstairs, the front door stood ajar, and snow dusted the floor. Sam was nowhere in sight.
I leapt off the last few steps and hurtled outside. Snow and darkness veiled the night, but a black shape marched down the walkway.
"Sam!"
He didn't stop.
I raced after him, steps heavy with cold and snow, and caught him just as he turned onto the road. "Sam!" Without thinking, I grabbed his arm.
He spun, and his palm landed on my chest—
There was no force behind the almost-blow. His muscles tensed under my hands as he must have realized who'd run after him. "Ana." Wind captured my name and carried it far away.
"Where are you going?" Only faint light came from the house; I couldn't see his face, and the cold made me shiver so hard I might fall apart.
"I'm going to find who did this. I'm going to hurt them." That wasn't his voice at all. In all the time I'd known him, he'd never sounded so broken. "They—My instruments. Everything I've worked for."
"I know." Even in the dark, my hands could find his face, just as they could find piano keys without looking. "Do you know who did this?"
He shook his head; his skin was icy under my palms, and all the rage was burning out. "I have to go. I'll find someone."
"Come inside."
"I have to find—"
"No, Sam. Not right now." We'd both freeze if we didn't get in soon; already, shivers racked through me, and I could barely speak through the cold. "Let's go inside."
Head dropped, he gathered me into a tight, uncomfortable hug. He shivered, too. Or wept. I couldn't tell, except he spoke the same muffled words over and over. "They're gone. I can't believe they're gone."
I had no words of comfort. There was no way to fix this, so I held still and let his grief wash over me in torrents.
Not soon enough, we went back inside and shut the door.
"Let's get your coat off." My words hissed harsh and loud in the too-silent room. I peeled off his gloves and hat and dropped them into a basket, then helped with the buttons and zippers on his coat. Our snow-crusted shoelaces were almost impossible with the burn of ice, but we managed.
His focus drifted to the piano as we reached the stairs, and he was silent as I guided him to his bedroom. There, he collapsed by his pillows, face streaked with sorrow.
I sat beside him and held his hands, warming them, wishing for anything but this. His instruments hadn't been just one lifetime's work, but many. I wondered if that made him feel like none of those lives had happened now.
After a minute, he leaned his head on mine. "Who would do this?" His tone was hollow, hopeless.
I didn't voice my suspicions. It wouldn't help. "What do you need right now?" I grimaced. He probably needed his instruments, and for me not to ask stupid questions.
He sighed and looked at the ceiling, misery making lines around his eyes and mouth. Cold still stained his skin red, and we both needed hot showers to warm our insides, but I couldn't see Sam caring right now.
"I don't know." He closed his eyes when I stroked his face. His skin was cold, but he didn't respond to my touch. "I don't think there's anything."
"Okay." I'd find warmer blankets, at least. I wanted to hold him, share heat, but I couldn't forget what he'd asked me on the doorstep. Did I want to leave? "Everything in the library and workroom is fine, including construction notes. I'll start cleaning, but is there anything you need me to save to help you rebuild?"
"Build new instruments?" He made it sound like the most horrible thing.
"I assumed you'd want to."
"Yeah. I guess. I hadn't thought that far ahead." His breath came raspy, and I couldn't imagine rebuilding lifetimes of instruments, either. But I didn't want to just leave everything where it was, in case he came downstairs. "The ivory," he said at last. "It's from far away, and it's hard to get more. But only if the pieces look like they're worth gluing together."
He told me a few more things, then let me help him lie down. I stacked blankets on top of him, wool and silk and bison fur, and went downstairs to heat soup and tea. When I brought them up on a tray, I forced several good sips into him before leaving the room. If I had just lost a thousand years of work, I would want to be alone, not awkwardly trying to accept someone's comfort when there was no way they could understand the chasm inside.
In the parlor, I picked up a few pieces of ivory, but most looked useless. Little was salvageable. Either the intruder knew exactly what to destroy, or had just decided to smash anything that looked important. Even the steel frame had been heated and melted so it would never be useful again.
Sam's flute was a wreck of silver. I hugged the remains to my chest, and blue petals floated out of the tube. Whoever had destroyed it had thought it was my new one. They couldn't tell the difference.
It was likely the instruments had been distractions, which was even more upsetting. But the books and diaries were gone. How long until they discovered Menehem's lab? How long until they discovered I'd been there?
The Council had suspected I'd been given Menehem's research, but no one should have known about the temple books.
No one should have known, but someone did.
I worked until my muscles clenched and sleep threatened every time I blinked. Since I couldn't move things outside right now, I set them by the door, a blanket beneath to keep from further damaging the floor.
Too worn to go upstairs, I dropped myself onto the sofa and woke when dawn speared my eyes from a crack in the shutters.
Outside, the snow was piled as high as my knees, and though the sun shone, more clouds huddled over the horizon, barely visible around trees and the immense city wall.
My lungs ached as I lugged broken instruments outside; when the snow thawed, maybe Armande or Orrin would help me separate materials for recycling. But now, I just needed them out of the house. If the sight of them hurt my heart, Sam's must be shattered.
To keep him busy for a while, I brought up more tea and soup. The other mug and bowl were only half-empty, but that was better than nothing.
"You should shower." I sat next to him on the bed. "You stink." As if I didn't smell like sweat, too.
"Doesn't matter." That wasn't Sam's voice. At least not the Sam I knew. Too rough, shredded into black ribbons. "It's all gone."
I wanted to touch him, hug him close, but my muscles wouldn't budge when I tried. "Finish your food and shower. I'll come back up in a little while."
Though Stef's house was usually only a five-minute walk, it took longer in the snow, and I was shivering when I arrived. Her place had the same outbuildings and snow-frosted fruit trees as Sam's, but was sparser, as she didn't garden or keep animals herself but helped tend Sam's in exchange for a share.
I took the steps two at a time and banged on the door.
Wind rattled evergreens, making a loose board on a shed bang in a staccato tempo. Otherwise, the place was silent, waiting for more snow.
Either she wasn't home, or she was avoiding me after the fight she'd had with Sam. I bit my lip and tried the doorknob. It turned.
I'd only been in her house a few times. When it was her turn for giving me lessons, she hadn't wanted to lug over the equipment for teaching basic machine repair; we'd started with water pumps and ended with solar panels. Mostly she went to Sam's if they wanted to visit.
Before I lost my nerve, I pulled open the door and stepped inside and stomped snow from my boots. Sunlight streamed through the parlor windows, glowing across the hardwood floor, landing on the small piano pressed against one wall. While Sam's walls were delicate shelves, most of Stef's were made of bookcases stuffed with notes and diagrams on fascinating subjects like automatic recycling machines.
"Stef?" I slipped around the chairs and sofa, with faded, patched upholstery and blankets thrown across the backs. She had more rooms on the first floor than Sam, most of them filled with inventions in various stages of completion. The stairs were hidden away in a corner, leading to the equally packed second story.
Floorboards creaked under my weight. I listened for any noises other than my own—nothing—and crept around the house, finding a library, a washroom, and a bedroom. Like Sam, she was usually male, but she didn't keep separate bedrooms for male and female incarnations. She just tossed her extra things in trunks for a lifetime, so now her bedroom was filled with dresses.
I started to leave, but a familiar photograph caught my attention. Hating myself for the intrusion, I looked closer. The photo I'd recognized was of two men, their arms slung around each other's shoulders, both smiling. That was Sam and Stef in their previous lifetimes. Other photos on the shelf were new to me, but I recognized some of Sam's previous incarnations. Sometimes he was alone, but most of the time he was with another person. Stef, I assumed.
Next to the photos rested a stack of papers: letters in Sam's handwriting, written while he was on trips and saved up until he returned to Heart to deliver them. I skimmed only a couple of them, loathing myself as I did because they were private, but they only talked about places he was going and things he saw that she might like.
There were a lot of them.
The last photograph was of the Sam I knew, sly smile and dark, messy hair. I recognized the shirt, too; I'd helped him choose it during a summer market day. For a moment, I thought she must have taken it while I was trapped inside the temple. Surprising that he let her, because he hated being photographed. But his head was turned and one arm was outstretched. He held a smaller hand in his. Mine. My hand was the only part of me in the picture.
I stepped away.
Half of me expected Stef to appear in the hallway and demand to know what I was doing, but the house remained quiet. Feeling confused and betrayed and jealous, I left the room.
I'd known they had history. I'd even seen photos from previous lifetimes where he was kissing someone. It bothered me, but sometimes I could imagine those Sams weren't my Sam. Those had been older, occasionally female, sometimes overweight or too skinny. I could find pieces of my Sam in all of them, but I could trick myself when it hurt.
She loved him. I couldn't imagine why anyone wouldn't. It was the intensity of her feelings I hadn't anticipated.
"How hurt does someone need to be to do something desperate?" I whispered, then felt sick. Stef would never hurt Sam like that. She might antagonize him, try to convince him that our relationship was improper. But she would never destroy what Sam loved most. Never.
"I'm sorry," I said, even though she wasn't here to hear it. It had been a petty, jealous thought, and I scrubbed my hands over my face as though I could wipe it away.
Time to go home. I went outside, finding sunlight had dimmed as gray clouds covered the sky, ready to drop more snow.
I shivered with winter chill by the time I opened the door to Sam's house again. The parlor was still a wreck, and the upstairs was quiet. Hopefully he was sleeping.
Fending off tears, I found a large bin and continued throwing away unsalvageable pieces of Sam's instruments. Any time the bin got heavy, I carried it outside and dumped it out with the rest.
When I couldn't stand any more, I climbed upstairs to shower and change into something not covered in sweat and dirt and splintered memories of a hundred broken instruments. Outside, snow fell heavy and white and wet.
It was almost night by the time I called Stef's SED. No answer. Nothing from Cris, either. Where could they be? Worry gnawed deeper; I tried Sarit.
"Hey, Ana."
"Thank goodness." I slumped to the sofa, relief like a waterfall through me. "You're there."
"Yeah, freezing my tail off. Cris didn't answer his door yesterday morning, and there weren't enough blue roses in the greenhouse. I'm on my way to Purple Rose to see if I can salvage any from there. You owe me. A hundred concerts, at least. Write a song for me while you're at it, cricket."
I shook my head, even though she couldn't see me. "With this snow, they're probably already gone. Just come home." She might have been my best friend, but she was also crazy.
"No way. I'm getting those roses for you. I'll keep them alive with my sunny personality."
"You're insane." I stared around the wreck of a parlor and tried to breathe right. "I'm glad I can get hold of you, though. Stef and Cris aren't answering. They're not at home."
"Cris still isn't there?" Worry crept into her words.
"His garden is collecting snow. And when Sam and I came back—" My voice caught. I tried again. "Sarit, someone destroyed the instruments. All of them."
"Oh." Her voice softened, deepened. "Oh, Ana. Your flute too?"
"No." I took a shaky breath. "It was in the workroom. Lorin accidentally popped a wire out, and Sam was going to show me how to fix it."
"But everything in the parlor..."
I gazed at a length of steel I hadn't been able to pick up. "Even the piano. Especially the piano." The words choked me, and my throat tightened with tears.
She didn't speak.
"And you know about the explosions, right?" When I closed my eyes, I could still see the fire, the smoke. I could still feel Geral's weight in my arms. "They're telling me to stop."
"How do they know what you're doing?" she asked.
"I don't know." I squeezed the SED, wishing I could tell her about the books, the key, the research—everything. I could tell her about the fight Sam and I had, and that he'd asked if I wanted to leave, but not right now. Not when she was so far away. "I wish you were here," I whispered into the SED.
"Me too." She hesitated. "You aren't going to give up, are you?"
"No." I clenched my jaw. "No, they can tell me to stop, but I won't. I'm not giving up."
"Good." She sighed, and a minute shivered past. "I've been riding hard to Purple Rose. The road has been snowy, but fine. A drone will come through if it gets bad."
"So you'll be home soon?"
"Yeah, a few more days. This horse is going to hate me, though." Something clanked in the background. "I've been calling my people. I checked in with Lidea and Moriah, and they've been in touch with their groups. Everyone is doing their part. You just get ready for yours and don't worry about the rest of us."
"That's hard to do with Cris and Stef missing." With explosions, people destroying parlors, and nursery break-ins, anything could have happened.
"I'll call their lists. It's fine, Ana. I'm sure they'll turn up soon." She didn't sound convinced, though. "I bet Sam could use your company right now. Go be with him, and I'll talk to you soon. Love you."
The SED clicked, and she was gone. Just in case, I tried Stef and Cris again. They didn't answer, so I left messages. Then I readied another tray of food for Sam, hoping he'd finished the last, hoping he'd gotten up to bathe.
He hadn't. He didn't break his intense study of the floor. His scowl never eased as I replaced his food tray.
Heavy with dread and worry, I did the only thing I could think of that might rouse him from his misery. I sat at the tall harp and positioned my hands like he'd shown me a few months ago—right hand close, left hand far—and plucked at the first string my fingers found, then the next.
On the bed, facing the other wall, Sam sat taller. He tilted his head.
I played another string, and another. Long, low ringing filled the bedroom like gentle snow. It was slightly out of tune, but I didn't know how to fix that. I'd only played the harp a few times before, though the strings on my fingertips, the curve of wood against my shoulder—they felt natural.
My fingers wandered into familiar patterns from Sam's brief lessons. I played a simple tune, belatedly recalling how to work the pedals to change key. My playing wasn't what anyone would call good, but as I continued, I heard silverware clank on ceramic, a mug thunk on the nightstand. A few minutes later, the shower started.
He came back into the room—water still running in the background—while I fumbled across a series of notes I couldn't remember; I was used to having music in front of me.
"Here." He took my hand and placed it on the correct string. "The arpeggio begins here." His fingers fell off mine, skin grazing skin.
I nodded, continued playing, and watched while he took clothes from his wardrobe and drawers, then went into the washroom. Steam wafted from the door he'd left ajar.
My music soared through the house, even when my fingertips started to hurt and I lost track of which strings were which. I needed the music, too.
Shower water silenced, and a few minutes later Sam appeared in clean clothes, his hair chafed damp against a towel. He sat on the bed near me while I kept playing the harp.
"I remember building it," he murmured, almost a countermelody against the delicate harp. "The piano. I remember covering it with coats of clear finish to let the natural wood shine through, fitting the cloth into corners and creases to ward away bubbles and drips. It felt like it warmed under my hands, like it was alive. I could already hear all the music I'd make. Preludes and nocturnes, sonatas and waltzes."
My fingers found a darker melody to match his mood.
"I never imagined choosing a favorite instrument, but even before I played the first note, I thought the piano could be it.
"Each piece of ivory and ebony came from faraway lands. I carved and polished every one myself. I cut the maple from forests near Range, and mined the ore—to be smelted and purified for wires and such—with my own hands."
Which hands were those? Ten generations ago?
"It took half a lifetime to plan and gather the materials, learn the necessary skills for constructing what I envisioned. I couldn't do it all by myself—some things just need more hands—but I worked so hard on it. When it was complete, I was an old man and my fingers ached from all that I'd done to create this thing, but when I touched the keys and played the first notes, it was so beautiful. So wonderful. Even now, I can almost hear the echoes of music from centuries ago."
I leaned my cheek against the smooth wood of the harp and let my hands rest on my knees. The music faded.
He watched me with dark, haunted eyes, his damp hair pressed against his skin. Anguish shone raw on his face: the strained set of his mouth, the way he made breathing look like the hardest thing in the world.
"I didn't make other people's pianos. I gave the construction plans to people who could do a better job. I'm a musician, nothing else. But I was proud of that piano."
"Nothing I say will help." I lowered my eyes. "I'm sorry."
"Your music helped." He reached as though to touch my arm, but I couldn't stop remembering what he'd suggested before we came in and found the parlor. He wanted me to leave. He wouldn't have suggested it unless he meant it.
I pulled away from him; I had to protect what was left of my heart, too. "The temple books are gone," I said, standing up. "And Menehem's research."
Sam said nothing.
"Stef isn't answering her SED. I went over there to see her, but she's gone."
He dropped his gaze. "She probably decided to wait out the snowstorm with another friend. I doubt she felt welcome here."
"Because you two were fighting." About me. Did that make it my fault? "She should have answered her SED, anyway. I called a million times and left a million messages."
He clenched his jaw. "She's angry with me. Maybe she's ignoring you by association."
I doubted that was it, but I wished he were right. Stef avoiding me was better than Stef being missing.
"We've been at such odds lately." He dragged in a deep breath. "I thought she would be happy I was happy. I don't understand why she's been acting like this."
Really? He didn't understand? How could someone with so much history and experience be so oblivious?
I'd reached the end of what I could take. Every piece of me felt like it was vibrating so fast it might fall off. A piano wire. A harp string. I'd spent the last day dragging off pieces of instruments I loved, too, to be sorted into scraps later. I'd frozen, seen friends killed, and Sam had asked if I wanted to leave. So what did it matter if I told him?
"She's in love with you, Sam. Really, really in love." My throat ached, and my heart felt dashed into a thousand pieces. "She's jealous that you've spent so much time with me. She just wants you back."
He was shaking his head. "No. We've had relationships in the past, but nothing like you mean. She can't."
"Because you said?" I raised an eyebrow. "You don't get to say how other people feel or don't feel. You can choose to be blind, but that doesn't change what everyone else sees. She loves you."
He seemed lost, like he didn't know where we were or who I was, let alone the language I spoke.
But I'd told him. Now he had to choose what to do with the information; I'd already decided what I'd do with everything he hadn't said. "I've been thinking about what you asked. I'll go." Speaking the words aloud made them true.
"Why? Where?"
"Li's or Ciana's, like you said. Maybe Sarit's until I get my own things." I bit my lip, wondering at what point my heart would crumble under the weight of my decision. Any second now. "I hope you don't mind if I stay here until the snowstorm is over."
His mouth dropped open, and he just stared for what seemed like hours. Like after the instruments, this was going to break him. I couldn't feel bad, though. Wouldn't. He'd suggested it. I'd have stayed forever if I thought he wanted me.
But in the hours that were really minutes, he didn't beg me not to leave. He didn't say he hadn't meant it. When I stood, his gaze just followed me up. Then I was a shattered blown-glass blue rose, and every step away from him made my shards clatter and chime.
#
# SNOWFALL
AS I LEFT his room, I wanted him to stop me. I wanted it so badly I could almost hear the perfect words he'd say to convince me to stay, but when I breathed, those words were lost. They'd never existed. I braced myself on the nearest shelf as my vision tunneled and faded, and up and down became the same direction. Another step. If I could just make it to my bedroom—
Arms wrapped around me and my knees buckled. "No." Sam's cheek grazed mine, fresh-shaved stubble. "Don't go. I need you."
I jerked out of his arms. "You asked if I wanted to move out. You can't take back a question like that. Words don't just go away."
His voice came from behind me, soft and stricken. "I didn't say you had to go." But his tone sounded like he was figuring it out, how there had been no right answer to his question. Did I want to leave? Live somewhere else?
No, I wanted to be here. I wanted him, the music. "I don't want you to worry about what's appropriate or not, or feel like you need to make those decisions without me." The words barely fit in my mouth. "I know I must seem very young to you, and why would anyone trust me to make choices about anything important? But I've been deciding things on my own my whole life, because no one else ever cared enough to help. Not until you."
Behind me, there was only silence.
How could my heart hurt this much? It shouldn't be possible that it ached more than my sylph-burned hands. "I don't feel young," I whispered, "and I don't feel like anything we had was inappropriate. I still don't care what others think. I still don't think it's inappropriate for us to touch or kiss. Maybe strange, but strange and inappropriate are different things."
And maybe I was talking to empty air. Should I turn around?
"I am an idiot." He said it like tumbling, like if he didn't get it out quickly enough, we'd both fall apart. But weren't we doing that already? "I asked if you wanted to leave because I wanted you to know you could. I don't want you to feel trapped here."
I stared at my socked feet and focused on breathing, suddenly aware of the entire house around us. Rooms filled with books and instruments, bedrooms with personal things, the parlor that used to be a haven, and the white shell around everything. Snow and wind beating on that shell.
He held his hand near mine, not touching. "I hate what people say about you. Everyone knows we live together, and everyone knows how I feel." His words rustled hair across the back of my neck, making me shiver. "The assumptions about us aren't kind."
I knew.
"I don't need that kind of protection, Sam. I've lived with gossip my entire life. I can deal with what other people think or assume. Whatever is appropriate for them—they made those rules for them. Not for me.
"While I am"—I snorted—"lucky to have the benefit of everyone's experience and wisdom, the truth is it's been so long since any of you were truly my age that you can't fathom what it's like. Even if you do remember, the world is different now. You've made the world different. That leaves me with the responsibility of deciding what is or isn't appropriate. If they want, other newsouls might be able to use my experience to decide when they're ready, but who knows how the world will have changed by then?" According to Meuric, nothing would matter after Soul Night, anyway.
"So does that mean you're staying with me?"
"Is that what you want?" Hope blossomed in my heart, but what happened the next time someone suggested a five-thousand-year-old teenager and a real teenager shouldn't be together?
"More than anything, I want you."
What happened the next time he saw Stef?
But he'd followed me out here to apologize. He'd danced with me at the masquerade, maybe even attended because of me. He'd been ready to go into the temple so I wouldn't have to be alone.
I slid my heel back and let my weight follow until I pressed against his chest. His arms closed around me. Warmth filled me everywhere he touched.
"Ana," he whispered. "I only wanted to do right by you, but I should have talked to you about it, too. Better than I did the other morning."
"You and your stupid sense of honor." My words held no bite. I was too drained, and he'd already apologized. Asking him to do it again would diminish the words.
"I agree." He kissed the tip of my ear, sending prickles of heat all down my right side. His arms stayed around me, and when I tilted my head and he kissed my neck, it was as though we'd never left the masquerade. Only the music of our heartbeats and wind outside, surrounded only by silk and wood and cool air.
"Try not to be so dumb again." I faced him, took his hand, and tried not to think about what I was admitting. "I'm not that strong, Sam. I can't forget the past as easily as you. For me, it's all right here, smushed together. Not stretched over thousands of years."
He cupped my cheek and nodded, his jaw clenched tight.
"I've never been able to trust anyone before." And the things I didn't say out loud, but hoped he understood: please don't hurt me again; be the person I need you to be; show me what it means to be in love so I can decide whether that's what I feel.
Fingertips traced lines over my cheek, down my jaw. "I'll do my best to deserve your trust."
I lifted my face and kissed him, tasting the salt of my own tears, inhaling the scent of his soap.
He lifted me off the floor, held me tight against him. My skin slipped against my sweater until my toes touched hardwood and air touched my bare spine. He gave a breathy, nervous laugh as he hitched me up again and this time supported me with a hand on the back of my thigh. "Is this okay?" he whispered.
I had lost all power to breathe, but managed to hook my legs around his waist. It was strange, like we were too close and not close enough. His hips moved when he walked, and he kept one hand on my back, and one under my leg so I wouldn't fall.
He placed me at the foot of his bed, and I recaptured my breath as he knelt before me. "You are beautiful." His hands rested on my knees. "And wiser than anyone has given you credit for. The world does need you, Ana. You challenge us, make people think and open their eyes to the truths that we've been ignoring for too long. Sometimes I'm so aware of how close the world came to not having you at all, and it terrifies me. Our immortality is not without a price."
"Neither is my life. There was Ciana, and other darksouls."
He shook his head, black hair falling across his eyebrows. "I'm sorry that I disappoint you sometimes, Ana. I know I'm not perfect. No one is."
I tried not to think about how many times I would inevitably disappoint him. I'd want his forgiveness when I did. I could forgive him now.
"There is something I am good at." He ducked his head as though to hide a blush, and his hands on my legs forced my insides into taut coils of yearning. "At least I hope. I imagine you would tell me if I've been doing it wrong this whole time."
"Music?" I bit my lip. I'd never heard him so much as play an out-of-tune note.
He raised himself, leaned so close his words touched my mouth. "Kissing you."
I couldn't move. "Prove it."
His sly smile flashed as he tilted his head and tipped his chin toward mine. Our lips brushed, but instead of kissing me, he rested his teeth against my skin and gave a gentle squeeze. His voice was so low it rumbled in my stomach, too. "I just wanted to find out if it tasted as good as I imagined."
"And?" He hadn't hurt me, but I could still feel the slight pressure where his teeth had been.
Maybe he'd do it again.
He leaned close and whispered by my ear. "Better."
Wind and snow pattered on the shuttered window while we kissed. He touched my face, throat, collarbone, making me feel like a piano must under strong, skilled fingers. But his movements dragged, and even the cadence of his breath sounded off, as though he was trying not to yawn.
"When was the last time you slept?" I cupped my hand over his cheek, feeling the way his jaw moved when he answered.
"I don't remember."
Not since we'd found the parlor, I was certain. Even before that had merely been a couple of hours in the early morning. He must be exhausted.
"Lie down. I'll turn off the light."
He kissed me again, as if to prove he wasn't that tired, and then stretched across the bed. "Stay with me," he said, as I made the room fall into twilight.
I paused, wanting him to mean it.
"Please," he whispered.
"Okay." I emptied my pockets and laid my belongings on his nightstand. Then I crawled into the bed, facing him. Everything was so dark, I could barely see the shape of his body, and for a moment, my frantic heartbeat seemed the loudest thing.
"Blanket?" He reached around behind him to find the end.
"I am cold," I whispered. And if he heard the shaking in my voice, maybe he'd think it was from chill.
He swept the sheet and down-filled comforter over us. "Closer?"
Yes. Definitely. I reached for him, relieved to find him reaching for me, too. His hands found my waist and pulled me tight against him. "Sam, I don't know—"
His tone sounded like a half smile. "It's okay. We'll figure it out another time. I just want to hold you right now."
That was good. I wanted—something. But I didn't want to do it wrong and embarrass myself. I probably would, anyway, if we ever got that far. But for now, I turned over—awkward in my day clothes—and pressed my back to his chest. Our legs intertwined, and I knotted my hands with his at my chest.
I slept.
And later woke to perfect snow silence outside, no wind or rattle of trees or clucking of chickens. Light seeped in around the shutters. I found which legs were mine and reclaimed them, then turned in Sam's grasp. His hands were slack and heavy with the carelessness of sleep.
He rolled onto his back as I finished turning, and blankets pulled away. The susurrus of silk and our breathing were the only sounds.
Pale light shone around him, making highlights and deep shadows around the ridges of his face and neck, down his torso and arms. Hesitating—what if he woke up?—I combed dark strands of hair off his face, then traced the lines of his cheekbones and smile.
He didn't react; he must have been exhausted.
Brave when he wasn't watching, I pushed onto my elbow to get a better angle, then kissed the same path my fingers had taken. He smelled like laundered sheets and hints of sweat.
My fingers had wandered down his chest while I wasn't paying attention. Through the thin shirt, I explored hills and valleys of muscle, relaxed while he slept. I discovered the plains of his stomach and lifted his shirt to the bottom of his ribs, finding smooth skin, warm with sleep. He moaned.
I froze. "Are you awake?" Barely worthy of being called a whisper.
Muscles tensed beneath my questing fingers. "I am now."
My face might have been on fire as I withdrew, but it was dim enough—I hoped—that he couldn't tell. "Sorry."
He dragged in a shuddering breath and gazed at me for a long moment. "I wasn't expecting that kind of wake-up."
"You didn't think I'd still be here?" I could have gone back to my room, but he'd been so warm and—
"No, I'm glad you were here." He pushed himself up, covers swishing around his legs. His shirt slipped back down, settling askew on his shoulders, and his smile was warm and shy. Boyish. "I like seeing you first thing."
"Oh, good." I doubted it was possible for my face to burn any hotter.
"Just the way you—" He dragged his fingertips from my shoulder to my wrist, making me shiver. "I didn't realize we were doing that now."
What? Touching? We touched all the time. Or maybe I'd ventured into one of those places I didn't know about, just wanted to. Well, this time had been different: he'd been sleeping, which might have been a little creepy of me, but I doubted that was it. My hands on his stomach, though...
My own stomach muscles tightened when I remembered the way he'd caressed me during the masquerade. Tickling. Tingling. Deeper. "Oh." The word came as a breath. "I think we should. Be doing that now, I mean." Maybe right now.
His smile grew slowly, as if he knew my thoughts. I sort of hoped he did. "Did you sleep well?" he asked instead.
"Yes." I scooted to the edge and let my legs dangle off. My toes brushed the floor as I gazed around at the bookshelves and old instruments crowding his bedroom. As long as I kept my back to the exterior wall, it was a safe room, all dimness and comforting things. Music. Sam. "Your bed is softer than mine."
Sam chuckled and sat beside me. "They're exactly the same."
"They are not. Yours is better." I didn't really want to argue, but little bickering neither of us would take seriously—I knew how to deal with that. It was easier than asking him to show me what else we were doing now. I could barely think those words, let alone say them.
"Very well. It is better." His mouth grazed my cheek. "When you're with me."
Eventually, my skin would stain red. Permanently. "Do you think it's still snowing?"
"Sounds like it. Can't you hear?"
I held still, listening as hard as my ears could manage. "It sounds like settling. Breaths drifting and sighing. The quiet groan of trees and roofs as they bear more weight."
"Yes." Covers hissed as he scooted closer and wrapped his arm around my waist. "I love that you hear it, too. That it sounds the same to both of us."
I did, too. "I want to learn everything, Sam. All about music, every instrument. I want to compose things I hear in my head at night—things that aren't yours or anyone else's—and I want to find a way to mimic the sound of snowfall."
His fingers twisted in my sweater, drawing my gaze to meet his wide, dark eyes.
"Maybe you want to do it alone," I whispered, "and I understand if you do. But if you'll accept, I want to help you rebuild everything that was in the parlor."
He kissed me, warm and hard enough to make me dizzy, but his arm around my waist stayed; he didn't let me spin away. "I love you." It was his voice, but his lips rested against mine so my mouth made the shape of the words.
"I wish I could tell you that, too." My heart thudded too quickly. "Whenever you say it, I feel so good and happy. But guilty for keeping the goodness to myself."
"That's not how it works." He kissed me again, as if the act would force me to accept his way of thinking. "Besides, I can wait."
Another benefit of being ancient: immeasurable patience.
My feelings were deep and overwhelming and confusing, but at the same time the emotion filled me with a sense of belonging. This boy. This soul. We were tied together with something stronger than anything physical. With him, I was not a soul asunder.
A quiet rumble came from the front of the house, drawing me to my feet. "What's that?" I grabbed my things from the nightstand and wandered into the hall, to a front-facing window.
"A plow." Sam followed. "It's like the drones we saw on the way back to Heart. There it is." He held a curtain aside, revealing a vehicle with a large scoop on the front. It heaved up to the steps—shoving a pile of snow to block the door—and turned to clear the other half of the walkway.
"Okay, so it works here, but what about people like Cris who have about three places you're allowed to step?"
"The price of filling your walkway is the plows don't clear it for you. And they're not very good about the doors. It's going to be tough to escape. I might need your help."
Because I was so strong. Right. But I caught the way he tried to stop his smile, and I rolled my eyes. "I'm worried about him and Stef." I could see slivers of her house from this window. Or maybe that was just more snow.
Sam released the curtain and leaned on the wall, something I still couldn't make myself do. "Me too."
I checked my SED, but she hadn't replied to my messages. I sent another, and one to Cris, asking again if they were okay. I hated that neither were home during a storm. "Where could they be?"
"Wish I knew." The thinking line deepened between his eyes. "After the explosions and what happened downstairs, their absence is especially worrisome."
"I think it was Deborl. Merton. Their other friends."
Sam frowned. "He's a Councilor."
"So was Meuric, and he tried to lock me in the temple. He got Li and Merton to attack us after the masquerade. Being a Councilor didn't stop him, and it wouldn't stop Deborl."
Sam gazed at nothing down the hallway. "You think he'd set explosives to kill people who might be pregnant with newsouls? Or break into our house and destroy"—his voice hitched—"my instruments?"
"I have no doubt."
Sam reached for my hand, squeezed my fingers. "All right, so what do we do? If he's attacking newsouls, we need proof."
"Sine is having someone watch them."
Sam nodded. "That's a start. Who knows? Maybe he'll get himself caught."
I rather doubted that, but since I'd definitely get caught and thrown in prison—or worse—if I tried to sneak into Deborl's house and see if he had my things, Sine's people would have to do. "You know what still bothers me?"
"I can't even count that high."
I stood on my toes and messed up his hair, then started down the hall. Just being close to the exterior wall made me squirmy. "If the explosions were coincidence—not a response to the meeting—all right. But how did they know about the books and Menehem's research?"
Sam shook his head. "Did you talk to anyone else about it?"
"No." I leaned on the balcony rail. "Well, Cris told me he had some ideas about my symbols, but no one else was with us. Sarit, Lidea, and Wend had just walked away."
"Cris wouldn't have done any of these things."
No, he wouldn't have. "So now they have the key, the books, and the research. They have everything and we don't have anything." I slouched, despair building inside me. How could I protect newsouls if I couldn't even protect a few inanimate objects?
Sam put his arm around my shoulders. "They don't have everything."
I shivered deeper into his embrace. I wanted to say something nice to him, anything to let him know how much I appreciated him and how glad I was we weren't fighting anymore. But I didn't want to sound stupid. There was one way to show him.
I pressed my palms on the balcony railing, overlooking the ruined parlor. "I'm ready to share something with you."
He waited.
I refused to hesitate. "My notebook isn't a diary." I pulled it out and flipped it open to the first page to reveal hand-drawn bars of music, scribbled words in the margins, and doodles everywhere. "Maybe it sort of is, I guess. Just not like the ones everyone else keeps." I gave Sam the notebook. "I don't think I'm very good at being like everyone else."
"I wouldn't want you to be." He sat on the top stair and turned pages, reading the words and music; they were both his language.
I sat next to him, elbows braced on my knees while I fidgeted and felt naked. Paper fluttered as he turned another page, and another. When he hummed a couple of measures, I cringed, but he kept reading without comment. Then he closed the notebook.
"It's not finished," he said, giving it back.
"Not yet." Maybe not ever, but I hadn't been writing it to finish something. I'd been writing emotions, because I didn't always have words for what I wanted. But there was always music, and sometimes it seemed like the most powerful thing in the world.
"Have you played any of it?"
I held the notebook to my chest, pressing the music against my heart so hard it might leave permanent impressions. "I've been too afraid of what it might actually sound like outside my head."
Sam stood and offered his hand. "It may be time to find out."
Maybe he was right.
#
# DEMONSTRATION
DAYS LATER, WE walked to the street and South Avenue, past walls of snow rising as high as my shoulders. Sunlight glittered across the ripples and made the whole city bright. So much light hurt my eyes, but not in the way the temple did. There were still drifts and shadows, dark evergreens against the brilliant snow. White veins shimmered between the cobblestones, and the sky was pale blue, a color almost too impossible to be real.
It was the perfect day for the monthly market, and everything I had planned.
The entire market field had been plowed, along with the wide half-moon stairs leading up to the Councilhouse. It was early, so a few sellers were still assembling their tents and tables, spreading their wares for viewing.
In spite of my coat and mitts and scarf, I shivered as we approached the field, the Councilhouse, the temple pushing into the sky. Cris and Stef were still missing—no one had heard from them—but everyone else had contacted their lists and were prepared to make their speeches this morning. Anticipation and defiance surged through me. Today, my friends and I would show everyone that newsouls were worthwhile. We'd show the Council that some people welcomed newsouls and wanted them to be safe.
I touched my flute case, a velvet-lined tube with a strap that went across my chest; it was easier to carry than the wooden box the flute had come in.
"You'll do fine," Sam said. The market's joyful din clattered across the field as we came in sight of the Councilhouse stairs and wide landing that would double as the stage. Sarit, Lorin, and Moriah were already there, winding evergreen boughs around the columns. "I have to help move the piano from the warehouse. Will you be okay up there?"
"Yep." I stood on my toes to kiss him, then trotted up the stairs, holding my flute case to my chest to keep it from bouncing.
Sarit, Lorin, and Moriah all hugged me, and I began adding the blue roses to the evergreens.
"Sam's getting the piano?" Sarit asked.
I nodded and slipped a rose into the strap on my flute case; I wanted one for my hair later. "The piano they keep over there." I waved my hands toward the industrial quarter with its warehouses and mills. "He already went twice to tune it, but he said he wanted to do one more pass because it's been so long since anyone has played it. And he's, you know, Sam. It has to be perfect or it's not worth playing."
"How's he doing with"—Lorin gave an awkward shrug—"the parlor?"
I bit my lip and glanced at the market, which grew more crowded by the minute. The only space not filled with colorful tents and stalls was an aisle to the steps, where there was a ramp for the piano. Several people watched our work, and rumors about an impromptu concert trickled through the tents. I tried to find anyone looking especially surprised or upset that I hadn't given up on my plan, but most people seemed to be looking forward to hearing Sam play. They didn't know what had happened in his parlor.
"Sam's angry, of course," I said. "Someone destroyed his work. But he could be worse."
"But they didn't get your flute," Lorin said.
"Because someone popped out a spring when she was playing with it, and I had to take it upstairs for repair. It wasn't in the parlor, or they would have." I tried not to imagine my flute twisted up, keys ripped off and holes gaping like empty eye sockets.
Lorin gave me a sideways hug. "Sorry about the spring."
"Thanks for breaking it." I turned to Sarit. "And thank you for getting the roses. I don't know what we would have done without you."
"You would not have roses." Sarit's tone was light, but she glanced northeast, toward Cris's house, and her expression tensed. "I hope he and Stef are okay. I wish they'd call or send a message."
If Stef had been the only one missing, I could have blamed it on her being angry with Sam. Cris, though, wasn't angry with anyone. As far as I knew.
Just as we finished decorating the stage and setting up microphones, Sam and some of his friends appeared with the piano. A few people from the market cheered, while others wore expressions somewhere between curiosity and suspicion.
When Sam had the piano where he wanted it and sat to warm up, I went inside the Councilhouse with Sarit.
"Are you ready?" she asked as we moved away from the glass doors.
"No. Yes." I handed her my flute case so I could take off my coat. No one would take me seriously when all my layers made me look like a bundled-up child. I could shiver for a little while if it meant people paid attention.
"Oh, pretty!" Sarit laid my coat on the back of a chair and started braiding my hair. "When did you get this dress?"
I smoothed the gray ripples of wool and synthetic silk that hung to my ankles—concealing a pair of thick tights so my legs wouldn't freeze. The sleeves hugged my wrists, delicate fingerless mitts covered my hands, and I kept a synthetic silk scarf around my neck. The blue matched the rose Sarit threaded into my braid.
"It's one of Sam's dresses. From before. We had to do a lot of work to make it fit." A few generations ago, he—she?—had been taller and curvier, and wore a lot of dresses. Maybe when you were a boy most lifetimes, you wore dresses when you got the chance. "But I thought it suited today perfectly."
"It looks perfect on you." Sarit stepped back and admired her work with my braid. "Beautiful. Now warm up, or Sam will frown at both of us. I'll get your music."
I pulled my flute from its case and played through warm-up exercises and scales. Outside, Sam played similar exercises on the piano; the powerful sound rattled the series of double doors.
By the time I was warmed up, Sarit had finished organizing my music, which was now written on real music paper and given a temporary ending.
She grabbed the music stand she'd stashed here earlier and nodded toward the doors. "Let's go, dragonfly."
I laughed at the attempted endearment, but just as we reached the door, Councilor Sine burst inside.
"Ana, finally. I've been looking everywhere for you." She took a deep breath, eyeing my dress and flute with uncertainty. "I haven't had any luck locating Cris or Stef. I'm sorry, but I'm sure they're fine."
I scowled, far less sure. "Okay. What about Deborl and Merton? And the guy who shoved me?"
She shifted her weight and shook her head. "I had a few people watch Deborl and Merton, but it sounds like they didn't do anything more suspicious than shovel snow."
I snorted. "I find it suspicious they get up and pee in the morning."
Sine cringed. "I'm sorry I couldn't be of any more help."
Maybe she really was. Mostly, I hoped she was ready to listen to what I had to say, and what my friends had to say.
Sarit went first, taking my stand and music onto the wide landing. She placed it just enough away from the microphone that it wouldn't screech—I hoped.
I clutched my flute and went outside, greeted by cold air, the piano's rich sound, and the fade of conversation around the market as people crowded to look.
"You can do this," Sam murmured from the piano bench. This instrument was dark, as though stained with midnight; it was ink against the white stone and evergreens and blue roses.
My smile felt tight, fake, but as I stood behind the music stand, positioned so I could see both Sam and the crowd gathering below, I reminded myself why I had to do this: for Anid and Ariana, held in their mothers' arms as they paused by a tent with mittens and scarves; for the others who'd be born soon and needed care and protection; for those who would stay trapped in the temple, consumed.
I lifted my flute.
There was a soft click as Sarit turned on the microphones.
Sam nodded. I breathed. A long, low chord rang from the piano. The sound vibrated through stone and into my legs, and the world grew silent as we began to play.
My flute whispered at first, evidence of my fear, but I'd played this before, and I could do it again. At home, I'd practiced with Sam, him humming the chords he'd play on the dark piano, because he'd listened to my music and gazed at me with such wonder that I might have flown.
I'd played it a hundred times with Sam correcting my posture and reminding me that cold air would make me sharp. Now on the stage, I pulled myself straight and let my flute sing.
Melancholy melody drifted across the stage, the deep piano chasing after it. I played loneliness and fear, yearning for things unnameable and shining. The sound caught around people, pushed through tents, and heated the air as I gained confidence. My flute stretched, warm and full and silver, and I played as I never had before.
Music grew, shifted into the richer sounds of courage and hope and desire. The piano provided foundation, encouraging my playing, lifting it and somehow revealing new layers of the flute's voice.
I played of sunsets and snow, the way leaves shifted and fell, and the anticipation of a kiss.
Music moved around the market field, raining from speakers to make people look up, look around. Friends and teachers smiled. Councilors tilted their heads, expressions unreadable. Strangers wore a range of emotions, some I didn't want to see, so I turned back to my music, back to Sam, and he smiled.
The music gasped with a kiss, surged with fear, and loomed long and low and lonesome where I'd written my experiences in the temple. Heavy chords were billowing smoke across the stage, and I ended with the four notes that began the waltz Sam had composed for me when we met, a haunting echo of blossoming love.
I lowered my flute, and no one in the market field moved.
They were waiting, which was exactly what I'd hoped, but it was much scarier when it was actually happening, all their eyes trained on me.
I'd played. I could do this, too.
Heart thumping, I stepped around my music stand and up to the microphone. I lifted my chin and found the words I'd practiced; it wasn't much, because others would do most of the talking. I only needed to make an impression.
"I am Ana, a newsoul. The music you just heard is mine, and this"—I held out my flute, which gleamed in sunlight—"survived in spite of someone's attempt to destroy it and stop me from playing for you today."
A few people in the crowd shifted. Some went back to shopping.
"I've been attacked," I said, lifting my voice. "People have thrown rocks at me. Beaten me. Spread rumors about me. All in response to one transgression: I was born. The same is going to happen to Lidea's baby, and Geral's, and maybe some of yours.
"The reactions to our new knowledge—that more newsouls will be born—have been varied and complicated. Some people have been welcoming. Others have not. I can't ask that everyone accept us. I know that won't happen. But this is my plea to you, the people of Heart, and the Council: protect newsouls. Before dismissing us as inconsequential, give us a chance to prove that we are worthwhile."
I smiled—sort of—and walked toward Sarit, who waited by a column, wearing a wide grin. Sam got up to speak, and I tried to relax. My part was over. Everyone else would do the rest.
"You were great, firefly," Sarit whispered. She took my flute and headed inside to put it away while I listened to Sam.
His words came like a song. "I met Ana when she escaped a swarm of sylph by leaping into Rangedge Lake. That was the first thing I knew about her: she would rather choose her own destiny.
"The next day, we encountered another sylph. In order to rescue me, she burned her hands, even after having been told that any significant sylph burn would grow and kill the victim. A lie, as we all know. But that didn't stop her. That was the second thing I learned about Ana: she is selfless.
"Ana taught herself how to read, memorize music, and survive. Many of you have had the privilege of teaching her and have seen how quickly she acquires new skills. Her very first night in Heart, I left her in my parlor while I cleaned up. When I returned, she was sitting at my piano"—his voice cracked—"and she'd already figured out how to read music. Not long after, she composed her own minuet. The beautiful piece you heard today is only her second composition."
My face ached with heat, from the people staring at me. He wasn't supposed to brag about me, just encourage discussion. This was embarrassing.
"Yet when she arrived in Heart, she was not welcomed. In her absence, a law had been made to keep her from living as an adult, though she was already three years past her first quindec. She wasn't allowed farther than the guard station until she agreed to lessons and curfews and progress reports, as though she were less than human. Less than everyone else simply because she is new."
I wanted to find a cozy rock to hide under. If it were possible for a face to glow with so much blood rushing upward, mine did. People kept looking at me and hmming.
"During Templedark," Sam said, his voice deeper, "when Menehem told her his intentions, Ana did everything in her power to save souls. She warned everyone of the price of dying during those hours. She sought me out when I'd gone to fight—and she rescued me again, this time from a dragon.
"Have any of you ever seen me not die when a dragon was trying to kill me?"
A few people chuckled nervously.
"That is what I want you to understand when I tell you we need newsouls. We need them to have privileges and rights, just like the rest of us. We need to encourage their talents and growth. No one will deny education is necessary, but Ana has proven ten times over that she can be trusted, and she will do anything in order to protect our community. It's her community as well."
Just as Sam finished speaking, screams flashed throughout the crowd. A commotion pushed its way between tents, coming toward the stage. Men in black coats dragged something behind them.
I walked back to Sam and the microphone to get a better look. "What's going on?" My voice carried from speakers everywhere as screams grew louder and people hurried to get out of the black-coated men's way.
One was Merton; his huge frame was impossible to mistake as he crashed up the half-moon stairs. Deborl hurried after him, and between them...
Meuric.
The reek of his putrid wounds heralded his appearance, all broken and seeping like he'd been before.
I staggered back. Sam caught me, arms tight around my waist.
"Is that Meuric?" His tone was incredulous, and the microphone dropped it all across the market field. People rushed like colliding waves, many away from Meuric's decaying body, and even more toward because they couldn't see the horror; they'd only heard Sam's words.
Meuric did not move by his own power. Merton carried him, while Deborl made a show of assistance. Other Councilors rushed in, though I couldn't guess their intentions. They wanted to help him? Keep people away?
"Where's a medic?" Sam leaned toward the microphone. "Rin, we need you on the stairs."
"Don't bother." Deborl shoved his way to the microphone. "Meuric isn't going to live. His bones have been shattered. His eye was carved out. He's been starving for months."
"How is he still alive?" Councilor Frase scrambled up the steps, gaping at the mess on Meuric's clothes and the way his body drooped. "Oh, Janan. Give mercy!" At the top of the steps, Frase bent over and threw up.
I gagged on the miasma of decay and vomit, backing toward the columns and piano like they could save me. Sam turned gray, trying vainly to hide my eyes, as though I hadn't seen this before in the smothering quiet of the temple.
Screams crescendoed as Merton positioned Meuric's fading body where everyone could see it. The crowd pushed around to the front of the steps, leaving tents and stalls untended. Shouts of disgust rang out.
Deborl spoke into the microphone and motioned at Meuric. "This is what the newsoul has done. She obtained a key to the temple, to our temple, and took Meuric inside, where she all but killed him. To mock Janan, she left him there, broken. I know such beliefs have fallen out of favor, but Meuric was once called Janan's Hallow. And to leave him there in this state is one of the highest insults."
The screams became cacophony, deafening. Sam grabbed my hand and tried to pull me toward the Councilhouse, but I felt like stone. I couldn't look away from Meuric, and from Deborl, because he was right. I had left Meuric there. I'd stabbed him, kicked him into the pit, and then abandoned him. And even when I found him in the temple again, I did nothing.
"Ana!" Sam yanked me, and I stumbled into his chest. "Come on. We have to go."
Go where? But I followed, glancing back to see Meuric crumple to the stage. His head lolled, and as Deborl raged and the crowd surged, I caught one last look at Meuric: the black rot between his teeth when he grinned at me, and the awareness fading from his good eye as he finally died.
I ran with Sam, not sure there was anywhere safe to go, but it was better than watching this.
Sam reached for one of the glass doors, and just as it swung open, I saw the reflection of dozens of people pressing close behind me.
Someone grabbed my shoulders and ripped me away from Sam. I shouted and jerked my elbow behind me. Bone hit soft tissue—a stomach?—and I started back to Sam, but more people appeared.
Hands grabbed from every direction, taking my arms and shoulders and hair. They found Sam, too, and immediately I lost sight of him.
I struggled, but so many bodies created walls around me. I couldn't get away as they push-dragged me somewhere I couldn't see. Above everything, Deborl's voice thundered.
"This is what newsouls do! This is what they will keep doing to us: killing us, destroying us, replacing us."
The bodies blocking me moved aside, revealing me to the mob below. Tents had been thrown on their sides, tables knocked over. People pushed up the stairs, reaching.
I screamed for Sam, for Sarit or any of my friends. Where were Lidea and Geral with their newsouls? What would happen to them?
Someone kicked the backs of my legs and I dropped. Bone slammed on stone, and it felt like my knees shattered, but I could still move my toes. I blinked and breathed through dizziness.
"My friends," Deborl cried, "we cannot accept newsouls. They will rip us apart. For her crimes, the newsoul will be punished."
Cheering rose up. Someone shouted against it, but that voice was quickly silenced.
Fingers gouged into my skin, keeping me on the ground as Deborl approached. He leaned close, whispering into my ear. "You might have thought you could stop Janan. You can't. Nothing can stop him. Meuric failed, but Janan has chosen a new Hallow. I will be the one who welcomes him when he ascends on Soul Night." Deborl gripped my chin and yanked my face around. His eyes narrowed. "And you will be where you belong, trapped where you should have stayed before you were born."
I tried to wrestle away from the people who held me, but they were too strong. Bruises formed under their fingers. I wanted to scream, to make some kind of response, but the noise and heat and rage overwhelmed me.
Deborl shoved me away as he stood. "Take her to the temple wall. I'll put her with the others."
My captors hauled me up, carrying me awkwardly so I couldn't fight or flail. Every time I struggled, their grips got tighter and my existence grew fuzzier.
People banged against me as Deborl's friends carried me through the crowd. No matter how I fought, they kept hold, and nothing I did led to freedom. We left the worst of the crowd soon, and moved between tents. I saw cobblestones, shoes, and trash on the ground. Never my captors' faces.
Until they slammed me against the temple wall, and then I looked up to see Wend. Lidea's partner. Anid's father.
I choked. "You?"
"I do love Lidea," he said, "but the newsoul is not right. He's not natural." Wend backed away, but before I could think about running, I found the blue targeting lights on my chest. The others had laser pistols aimed at me.
"Why not?" I asked. "Other animals live and die and are never reborn."
"We have souls," Wend said.
One of the others chuckled. "Some of us, anyway."
I wanted to be horrified at how Wend felt no attachment to Anid, that he didn't care at all that Anid's existence was partly his doing. But I remembered Li, and how she hated me, how she resented me because I represented everything that terrified her most: the unknown.
"We have Janan." Deborl came around after us, drawing the silver temple key from his pocket. "Janan gives us every life."
"What about phoenixes?" I couldn't stop staring at the key as he pressed the symbols I'd only guessed at.
"Janan is only for humans. For souls." Deborl sneered and nodded at Wend. "Get her."
Wend grabbed my arm as a door misted into existence on the temple. Did they all know about the temple? Was that how Wend knew what symbols Cris and I had been talking about? And how they knew what to take from Sam's house?
Deborl dragged open the door, and reality hit me. They were going to throw me in.
I struggled, squirming away just long enough for someone to shoot the cobblestone in front of me. Stone sizzled as Wend grabbed me back.
"I'd like to break your bones and gouge your eye before putting you in there." Deborl shoved me into the doorway; I stood half in and half out of the temple. "That way you can feel the pain you put Meuric through. Unfortunately, I only have time for this, but it will do."
He reached back, and Merton slapped a laser pistol into his hand. To shoot me? To burn me just enough so I suffered forever inside the temple? I didn't have a key this time. There'd be no way out.
I searched for a path between the men. Deborl, Wend, Merton, and strangers were too thick. There was nowhere to go.
The targeting light flashed on my shoulder.
Wend lurched forward and shoved me.
Just as gray veiled the outside, I saw Deborl turn and shoot Wend. For saving me the pain of being injured inside the temple?
Wend's body crumpled.
I fell backward into the temple.
#
# SKELETONS
I TUMBLED INTO the white chamber, all painful glow of everywhere-light and the deafening throb of Janan's heartbeat. I skidded to a stop in the middle of the floor, clutching my head and groaning.
"Ana?" The heavy air smothered the deep voice. The human voice.
I looked up to find Cris and Stef sitting together on the far side of the chamber. Their clothes were ripped, and scrapes crisscrossed their hands and faces.
"Oh. I've been trying to find you two." I struggled to stay upright. "For days."
"Days?" Cris climbed to his feet and started toward me. "What do you mean?"
"You've been missing." I took a deep breath and tried not to think about where I was, but souls began to whisper and cry. The truth was impossible to forget. It was all around me: the incredible nothing that should have swallowed me, too.
"Not days, though. Deborl and some of his friends grabbed me," Stef said, following Cris, "but that was just this morning."
I shook my head, but decided not to burden her with the truth just yet.
"Do you have your SED?" Without my permission, she dug into my pocket.
"It doesn't work here," I warned, and checked to see where we were. Not that it made a difference. Most places in the temple looked alike, all big white chambers and archways. Whispers and murmurs rippled, souls cried. There were no words for how much I didn't want to be here.
"How do you know?" Stef tapped the SED screen like it'd do magic.
Cris offered me a hand up. "I could have sworn they shoved us into the temple, but there's no door."
"This is the temple. Sorry. I've been here before." I bit my lip. "This is my third time."
They both stared at me, confusion bright. "How is that possible?" Cris asked.
The weeping and unsilence surrounded me, heavier and thicker for no reason except that we were trapped without the key. It would be impossible to tell how long we'd been in here, or what was going on outside. The everywhere-light glowed with ever-unwavering determination.
"Meuric had a device. Right before Templedark, he tricked me into coming here, then followed with the intention of leaving me locked in so I wouldn't cause trouble. I took the key from him." And then trapped him in here, caught between life and death. Now he was out, finally dead on the steps of the Councilhouse.
Stef raised her eyebrows. "And you've been coming and going since? Why?"
"Not because I like it here. I need to learn what Janan's trying to hide. I came here before because I thought I could find answers." I almost wished for ignorance again; it had hurt less than the truth. "Now I have even more questions."
"Oh." Stef shifted and handed back my SED. "Well, feel free to start explaining things to me any minute. Even the questions."
"Okay." I stuffed my SED into my pocket, wishing I'd brought my knife instead. It was at home, since my dress had only one small pocket, but if I'd known I was going to get shoved into the temple again... "Have you been exploring?" As much as I hated moving around the temple without the key, especially when I wasn't sure if they'd throw Sam in after us, it would give me the illusion of doing something.
"A little," Cris said. "But it's empty."
They clearly hadn't reached the spherical room, or the sideways-gravity room. Lucky them. "Stay close, then." We headed toward the nearest archway, and I began telling them the truth about Templedark, my disappearance since then, and the books I was trying to translate.
I told them what Janan was doing to newsouls.
"No," Stef whispered. "Surely no."
Cris's eyes widened with horror. "Why? How? How could that possibly be?"
"Meuric told me," I said. "He might have lied, but I don't think so." Even as I said it, cries grew louder, thicker on the smothering air until they were like black smoke clinging to our clothes and skin. Cris and Stef said nothing, just looked like they wanted to be sick.
It was painful, watching them react to the truth about newsouls. I changed the subject. "I found the guesses you left in your house, Cris. For the symbols."
Cris looked up. "You were in my house?"
"We couldn't find you outside and it was snowing. None of your plants were covered, so we were worried."
"Ah." He glanced nowhere, as though he could see his frost-coated roses. "I was studying the symbols again when someone knocked. I tucked the paper under the tray, and then Deborl, Merton, and a bunch of others took me."
"Why would he take you?" I stepped off a narrow stairway, onto a floor that looked like white water. It held my weight. For now.
"I don't know." He eyed the floor like it might change its mind about being solid. "Well, Deborl did ask about books and symbols. He said he wanted to know what I knew, which wasn't anything, since you didn't give me details." The last part sounded a little accusing, but I forgave him because it was my fault he'd been kidnapped.
"He asked me the same questions," said Stef. "But I really didn't know anything, because you didn't give me even a hint." That definitely sounded accusing, but I forgave her because she was right and it was my fault she'd been kidnapped, too.
Deborl must have assumed Sam and I had told her because they were best friends. If Sarit hadn't left for Purple Rose Cottage, would they have taken her, too? And I couldn't stop wondering what they were doing with Sam right now.
I found an archway out of the water-floor room quickly, before I lost control of my stomach. Stef looked green, too.
The souls around us continued weeping.
"I didn't remember the symbols from anything in Range like you thought I might." Cris's voice was low as we entered a long hall, white everywhere; I dragged my fingers along one wall to make sure I didn't walk into it. "The symbols came from writing I'd seen in the far south, in jungles. I was collecting samples of plants for medicine and experiments, and found giant ruins. A white wall..." His voice grew soft and faraway.
I'd been right the day of my gardening lesson: the wall had been white, like the wall Sam found in the north.
"I climbed to the top of one of the tallest pieces to get a good view. It was hard to make out with trees and vines and creatures everywhere, but it looked like the wall had once been a ring, like the one around Heart, but there was no evidence of a city inside. Only a razed building in the center, with enough rubble it might have been as big as our temple."
"The ruins looked like a circle with a dot in the middle?"
Cris nodded.
That was the symbol Meuric had said meant Heart or city, but there'd been no city in the jungle. There'd have been some kind of evidence otherwise, even if the jungle had mostly overgrown it. "And the symbols?"
"They were etched into the stone, though erosion made them difficult to see. When I left, it was so hard to remember."
Meuric had said no one wrote the books, that they were simply written. But the language seemed to be from the jungle, where phoenixes lived and burned and died and lived again. So what were the books doing here?
Cris focused again, confusion magic evaporating. "Did my guesses help? It was a long time ago."
"Yes, definitely." I wished I'd actually been able to study them, now that I'd acquired a few translations. "You helped a lot. I hadn't realized I'd been looking at some of the symbols sideways."
He offered a warm smile. "I'm glad you trusted me enough to ask."
Stef shot me a dark look, a vivid contrast to the white all around. I wanted to say something comforting to her, but I didn't know what. We were stuck here together, me and two people who loved Sam, and the object of our affections on the outside. Maybe hurt or imprisoned. Who knew what else Deborl had told everyone?
The truth was bad enough.
The hall ended in a black archway. I hesitated, uncertain about this one, though I couldn't tell why. It was the same as all the other black archways, midnight on white.
"That's easier to look at, at least." Cris rubbed his eyes.
"The crying stopped." Stef glanced at me. "Are we going through?"
She was asking me? Perhaps I'd inadvertently given them the idea I knew my way around. "Yeah, I suppose. Keep watch for anything that might help us escape."
There wouldn't be anything. The key was gone. Nothing would help us escape, but they needed the comfort.
We walked through the archway.
The circular chamber beyond was not like the rest of the temple. Here, the walls glowed red, and inky shadows lurked beneath skeletons chained in tarnished silver shackles. Thousands of skeletons. Maybe a million.
A wide pit waited in the center, large enough for a piano to fall through. Like a spider straddling the hole, a white table stood above it. One body, perfectly preserved, rested on the table with a knife thrust into his chest. His own hands held it in.
Stef's voice dropped low and heavy. "What is this?"
"I've never seen it before." I couldn't move. Everywhere there were skeletons, yellow bones clean of flesh and fabric. They sat on tiers around the room, heads lolled to the sides, bound hands on their laps or the stone beside them.
I'd never seen so much death before, not even in graveyards Sam had shown me. Those had been peaceful, all iron and stonework, flowers and vines. They were bodies kept in mausoleums and caskets where they belonged.
"This one is different," Stef called from across the pit and the man on the stone table.
I stared at the table man as I rounded the pit, not too close. He was short and thick, with bushy brown hair on his head and face. His jaw jutted forward as though he'd died focusing on something important. Mostly, he looked strong, like he could wrestle a troll and win.
"Ana." Cris touched my shoulder. Where Stef crouched, another skeleton slumped in its shackles, but away from the rest. It lay prostrate in the middle of the floor, arms outstretched as though bowing to the man on the table.
"That's not the weirdest part." Stef stepped away from the shackled one to reveal a second, which appeared to have been cast aside. Limbs flailed, bones barely held together by worn ligaments. It looked like if anyone touched it, the skeleton would collapse into a pile of dust.
I gazed along the walls, along the ranks of gaping eye sockets and lower jaws hanging precariously. "There." I pointed to an empty spot. Silver shackles sat unlocked on the white stone. "Someone put that one over here."
What Deborl had said about replacing Meuric—
"What's a Hallow?" The question was out before I realized I'd spoken. Deborl actually had replaced Meuric. Physically.
"That's a word I haven't heard in a long time." Stef cocked her head. "Meuric claimed the title in the beginning, saying he had a special connection to Janan, but he didn't seem to do anything, really. He eventually stopped talking about it."
I fiddled with my scarf, the cool length of silk only a pale comfort. "Meuric was the first Hallow," I said, gazing at the skeletons on the floor. "Whatever he was supposed to do, he failed when I trapped him in here. Deborl replaced him."
Cris stood next to me, towering. "But why? What does it matter?"
"Meuric and Deborl both said something about Janan rising. Ascending."
"That sounds familiar," Stef muttered. "Ascending."
I waited, but she didn't elaborate. "Meuric was convinced that if he had the key, he would survive Soul Night."
"That's in three months." Cris shook his head. "But we have a Soul Night every fifteen years. We all survive it. What makes this one different?"
Time? Whatever Janan was working toward, was five thousand years long enough? Meuric had been convinced it was happening soon, even before the temple turned him crazy. "If surviving Soul Night requires the key, and the Hallow gets the key, that would certainly be motivation to do whatever Janan wants."
"And what does Janan want?" Cris asked. "Rising? Ascending?"
Not rising like a phoenix, Meuric had said. Something else. Something sinister.
I pointed at the two on the floor. "Those two are Meuric and Deborl." I swept my arms around the room. "And the rest of these are you. All of you. Sam, Sarit, Orrin, Whit, Armande, Sine—everyone."
Cris and Stef gasped.
"What happened here?" It was probably mean of me to ask, since they couldn't remember. Janan didn't want them to remember, or know about the other white walls and towers around the world, or consider certain paradoxes enough to know they were ridiculous.
He did something to them every time they were reincarnated, but maybe now that they were inside the temple, memories would return.
Stef focused inward, a line carved between her eyes. "Janan was our leader. He used to be a man. A human."
I glanced at the body on the table. "Him?"
"Him," she repeated. "He wasn't even anything special. He was our leader, but he was just a human."
How incredible, all this because of one man.
Stef's jaw muscles clenched, and her knuckles turned white with strain. "Every time I think I have it, it slips away."
"It's all right." I laid my hand on her shoulder. "Just tell me whenever you know something. I won't forget."
Sometimes, being new had its advantages.
"You said he was your leader. Just a man." I spoke as much for their benefit as my own. Maybe it would spark more memories. "Had you discovered Heart yet?"
"No." Cris frowned. "And we weren't in tribes across Range like I thought. We were all together. All of us except for Janan. We were going to him."
"The story I was told was this: no one agrees how you got here, but you lived in different tribes. Then you all discovered Heart and fought over it until you realized it was big enough for everyone. That was the first time you came together, all million of you."
"Yes, that's right." Cris shook his head. "But that's not right. That's not what happened. Janan was our leader, but he'd been wrongly imprisoned. Everyone came to free him. The city appeared later. After...after we did something."
I motioned to the table. "Somehow he ended up there. And somehow you all ended up sitting around the room with chains connecting you. How?"
"I don't remember," Stef whispered. "I know Meuric bound us in the chains, then bound himself beside the altar and told Janan we were ready. I remember white and wind everywhere—and the very next thing is standing just outside the city wall. We all thought we'd just arrived, but no one knew why we'd come." She gestured around the room. "Whatever happened in here, it tied us to him forever. It changed him, made him both less and more at the same time. It made us reincarnate."
Unsilence thickened in the moments between her words, and all of us realized the answer to my biggest question.
I wasn't going to be reincarnated.
Definitely not.
I hadn't been here five thousand years ago. I didn't have a skeleton chained to the walls.
When I died, I'd be gone. Gone, and no one would remember me but through pieces of music and the few notebooks I kept.
I wanted to sit, or speak, or breathe, but it seemed ice radiated from the blue rose in my hair, freezing first my thoughts, then every other piece of me. No matter what I did now—whether or not I escaped here, saved newsouls, and stopped Janan—when I died, that was it. No lifetimes with Sam. No helping to rebuild his instruments, no learning how to play them all, no writing music that sounded like snowfall.
My heart shattered, glass on stone.
Then Janan spoke.
#
# TRAPPED
"MISTAKE. IT HAS returned."
Janan's voice hit me from all directions, huge and deep and overwhelming. I blinked away threatening tears and glanced at the man on the table, but he remained dead.
Despair splintered through me. I was a mistake. Asunder. And after this life I wouldn't return. I would never be like everyone else.
"You must leave. This place is not for you." Janan's words ripped around the room, and red light gathered on the domed ceiling. It brightened, sucking all the crimson from the walls until they glowed hot white.
The presence faded, leaving the red to bleed back into the walls. Everything became how it had been a few minutes ago. Except my new knowledge of my...temporariness.
"Was that Janan?" Stef's face was pale and drawn as she gazed around. She lowered her voice. "He knows we're here?"
"He does now." I hugged myself. "Usually he doesn't pay attention to me unless I'm fiddling with the key. He can't touch you—he's incorporeal—but he can change the walls. Once he locked me in a small room."
"How did you escape?" Cris asked.
"I threatened to keep pushing buttons on the key." Deep breaths. In and out. I focused on anything but the idea of dying and never coming back. "I don't know how it works, but it must make Janan uncomfortable if he didn't want me poking at it."
Cris nodded. "I suppose if the walls are his body, it'd be like me making your arms move around for you. A trust you'd only give your Hallow."
Janan's voice boomed again, violent as thunder shaking the ground. "You do not belong here!"
I jumped, my bones feeling like they might vibrate out, and tried not to stare at the body on the table. Or the skeletons along the walls. There wasn't really a safe place to look.
"Should we go?" Stef asked, once the rumbling died. Her voice trembled. I hated seeing Stef frightened; she was always so confident.
"No. If we leave this room, he'll trap us somewhere. I don't have the key. There's no way I'd be able to get us out."
"So we stay here?" Cris looked dubious.
"Doesn't that trap us here instead?" Stef eyed the door like she might run for it.
"I've never been here." I didn't want to be here now. "Maybe Deborl made a mistake when he left. Maybe going somewhere in the temple makes it more likely to be available again. All I can tell you is that I've been in the temple twice before, and I've never seen this room."
Stef shook her head. "We're still trapped."
"Important things happened here. This place holds answers. We won't find anything in the rest of the temple."
"I just want to go." Stef edged toward the archway. "There's no way out from here, but the rest—"
"There are no exits." I balled my fists. "I know it's hard to understand, but there is no way out. I don't have the key."
"Sam will free us." Cris looked hopeful.
"No, he won't." The temple had no temperature—not too hot or cold—but chills raced across my body, and I shivered. "He won't save us, because he won't be able to. It was a mob out there. I don't even know if Sam's okay." The last words choked me.
Both of them wore pinched expressions, and Cris touched my elbow. "I'm sure he's fine. He's probably working out a plan to take the key right now."
I shook my head and described what I'd seen. No need to go easy on them. I didn't see how our situation could get any worse.
"Ana!" Stef jabbed a finger toward the archway we'd come from. It was gone. "He locked us in. Now we really can't get out."
"We wouldn't have gotten out anyway!" The constant pulse of the temple made my head throb. "What about that is so confusing? No matter where we are, we're not leaving. Sam can't rescue us. Deborl wouldn't free us if his life depended on it. No one else knows how to use the key. We're stuck."
"All right!" Cris rubbed his forehead as though to press a headache into a smaller, more manageable size. "Both of you, please. We need a plan."
"Like what? Escape?" I scowled and gestured around the room. "The only thing I see is the hole under the altar, and I don't recommend it."
"We're just as bad off as if we'd left," Stef muttered, just loud enough that it was meant to be heard.
I shook my head. "If we'd left, we would be stuck in a tiny box."
"You don't know that. You're guessing on all of this." Stef loomed over me. "You've been leading us nowhere the whole time, with no idea where you were going or what you wanted to accomplish."
"At least I was doing something." My fingernails dug into my palms, carving crescents into my skin. "You'd have just stayed there all confused. Did you even try to escape Deborl? Or were you too busy being mad because Sam and I didn't tell you what had been going on?"
"Don't pretend like you know anything about me or the way I feel." Her face was pink in the red glow, and I had pushed too hard.
I didn't care. "I know enough." She kept antagonizing Sam about his closeness to me, but she was the person keeping a stash of his photos and letters. I wanted to hurt her. "I know how you disguise your feelings for Sam. You fooled him, but he's used to your flirting and never took it seriously. But I know you meant it."
She stared at me like I'd said she had chicken feet or hands growing from her head, but that was Stef. Pretending even now she didn't really care.
I knew I shouldn't, but I said it anyway, my voice low and too calm. "I told him that you love him."
Her face went blank.
A wiser person would have stopped there, but I went on anyway. "If you were as brave as you claim, you'd have told him lifetimes ago."
"And I suppose you have? No, that would ruin your tortured newsoul existence." Her voice grew stronger, angrier. "You can't let yourself be happy, can you? Well maybe this will fix it: you're not coming back. There's no skeleton in here with your name on it, so when you die, that's it. Gone. I'll still love Sam, and thanks for telling him, by the way. Now he'll have time to figure out a response in our next lives when you're not here. Are you happy now? You really are as tragic as you think you are, butterfly."
She might as well have stabbed me; it hurt the same.
There was a whole list of things I shouldn't do, including asking if she could find his skeleton among all these—I could—and telling her about his reaction when he found out how she felt. But I didn't do any of them, because it would be cruel and petty. Not that I'd been much above cruel and petty so far, but I didn't want her to hate me forever. And, romantic feelings aside, she was still Sam's best friend.
"There's no point in arguing about it, Stef." My voice was more level than it had ever been, but surely she could hear the strain. "Because we're trapped. We're never leaving this room."
#
# IMMORTALITY
AFTER STARTLING STEF and Cris into silence, I marched over to Meuric's skeleton and kicked the skull.
Whatever magic had been holding it together must have failed when the shackles came off. The skull skittered across the floor and dropped into the pit beneath the table.
I kicked an arm, and several bones cracked against one another, the floor, and a table leg. Pieces of Meuric dropped out of sight, making no sound on their way down. If they ever reached an end, I didn't hear the clatter.
Still angry, I kicked his ribs, hips. Smaller bones turned to powder as my boot hit them. "I hate you," I hissed, as the last of Meuric vanished.
Stepping around the dust, I almost felt bad for kicking Meuric down a pit again.
"Okay," I whispered to myself, and knelt by the skeleton now caught in shackles. Deborl. I hated him, too. More than I hated Meuric. The hate twisted inside me like a snake, uncomfortable, but clean and sharp and determined.
I reached for the tarnished silver cuff to search for a lock. If Janan didn't have a Hallow, maybe he couldn't ascend on Soul Night. It would make being trapped in here worth it, if I could save everyone else from the fate hurtling toward them.
As I touched the shackle, electricity zapped through me. I screamed, and lost feeling in my right side. My arm hung uselessly.
"Ana?" Cris hurried toward me, looking around for whatever had attacked me.
I shook away the buzzing in my head. "Don't touch the chains."
He sat with me until feeling in my fingers returned, and then, more carefully, I climbed onto the table and tried to kick Janan off.
He might have been a human-shaped lump of silver himself. He didn't move. Cris even joined me, but no matter what we tried, we couldn't budge him.
The knife, however, did come away when we worked together. Cris pried Janan's grip off the handle just enough so I could slip it out. The blade was silver at the base, but the end looked as though it had been dipped in liquid gold. Janan's hands returned to their original position, but now they held only a memory of the knife.
I didn't have anything to do with it, though, so I left it on the table.
"You will die!" shouted the walls, incorporeal Janan.
"Why don't you lick my shoe?" I propped my boot on dead-Janan's face. "You won't do anything to us. Not here."
Red light swirled around the chamber, and Janan's screams resounded through the room as he called me names I'd never imagined could be put together like that.
But he was without substance, and we were already trapped.
"You're just a human, like us!" Not quite true—he was powerful, incorporeal, and consumed and reincarnated souls—but he'd started out human. Reminding him of that was satisfying. "Just a short human!"
"Is that your plan?" Stef asked when the screams faded. "Annoy him until he kicks us out?"
"No. I'm working on a better one." I flashed a tight, fake smile. "This is just the beginning." I kicked dead-Janan's head, but numbness rushed through my toes as though I'd kicked a block of ice.
I hopped off the table and marched around the perimeter of the room.
A few minutes or an hour later, Cris fell into step beside me and I said, "If you're here to chastise me for being mean to Stef, I don't want to hear it." I twisted my scarf in my hands, hating my obvious fidget, but I couldn't stay still.
"No, I thought you'd probably done enough of that yourself."
"Mmm." Noncommittal. I'd picked it up from Sam, and it seemed to work for whatever the other person wanted to hear.
"I've been trying to figure out how to get out of here," I said. The archway was still missing, and there weren't any signs. No words or pictures to indicate what we should do next. "No key means we can't control the walls. We can't make doors or do anything useful. The good news is we won't get hungry or thirsty, as long as we don't think too hard about it."
"Great, thanks. Deborl didn't feed us before he trapped us. Do you have any idea how long it's been?"
"A day? A week? Five minutes?" I shrugged. "Time passes differently here, and not even at a consistent rate."
Moriah had told me time mostly mattered to the person measuring it, which had made me laugh because she built clocks. SEDs and clocks didn't work in here, but now I was extra aware of every second and how they carried me closer to my end.
"So what we need is someone who can make doors."
I raised an eyebrow. "Well, yes. Pretty much."
"Stef?" He waved in her direction. "You don't happen to have anything on you that would make a door in the wall, do you?"
Her glower was dragon acid. "Go roll around in rosebushes, Cris."
"I don't think she appreciates your humor," I muttered. As if I could blame her. At this rate, we'd be out of Janan's way in a few days because we'd have killed one another. Well, Stef would kill me, then Cris, and then she'd be here all by herself. And I wouldn't feel bad for her.
"Few really do." He kept pace with me easily. "Why are you walking?"
"It feels like if I stop, then I give up. But I don't know what to do." My throat tightened with the confession. He was going to think I was weak, just like Stef did.
"Hey." He tugged my arm. I stumbled and he caught me, one hand on my back. "Sorry. Hey." He faced me, expression serious. "We're going to find a way out, okay? And then you'll rescue Sam from the angry mob, reclaim your books, and find a way to stop Janan from ascending."
"So while I do all these miracles, you'll be where?" My whole body ached, and I really wanted to lose myself in the piano, but it was gone. Smashed. And my flute? Sarit had put it in the Councilhouse, but they might have found it.
Cris said, "I've been remembering, too."
I waited.
"Being here has made me remember a lot of things we're not supposed to know. The memories are so old they feel like dreams or someone else's life, but I know they're real." He looked more serious than I'd ever seen him. No hint of a smile, no friendly stance. He looked sad. "I remember what Janan said he was going to do."
"What is that?" I whispered.
"He wants to be immortal."
"But—"
"True immortality. Not like we are, trapped in an endless cycle of birth and death and rebirth. And not like what he is now, trapped in these walls. Before, when he was still human, there was nothing in this tower. No rooms or light or shifting walls. It was meant to be a prison."
Even before he started switching old and newsouls, he'd been imprisoned? "Why was he here? Who put him here?" Whatever he'd done, it must have been terrible, and as far as I could see, he was only getting worse.
"Before all this"—Cris gestured around—"Janan took his best warriors on a quest for immortality. People were so afraid of everything, like dragons and centaurs and trolls—"
"And sylph?"
Cris cocked his head. "No, we hadn't seen sylph yet. Only after."
"Okay." That was odd, though. "Go on."
"Well, he said he discovered the secret to immortality, but that phoenixes were jealous: they didn't want anyone else to know their secret. They made this prison—and prisons all over the world—and locked Janan and his warriors away, one in each tower so they'd never band together again."
"Phoenixes." I'd known they were real, but I'd never heard of them making prisons or really doing much besides flying around, burning up, and rising from their own ashes. Well, Meuric had said a phoenix cursed the sylph, but Meuric had been crazy. Maybe. "The other prisons were towers, like this one? With a wall around?"
"I never saw them, but I think so. I think when we came to rescue Janan from his prison, it was just a tower and a wall."
"Like the one you saw in the jungle."
He nodded.
And like the one Sam had seen in the north, I guessed. But none of those towers had anything like Janan. If they did, they wouldn't have been affected by weather and life. So what had happened to those prisons and prisoners?
Cris seemed somewhen else, heavy with his memories. "We all went to rescue Janan, but instead, he said the secret to immortality meant he had to stay in the prison—for a while. He said phoenixes had made this tower, so it was already infused with their magic. And the rest of us were to wait for his success and return." Cris gazed around the red-lit chamber. "Can you imagine five thousand years existing only in stone, just waiting?"
"He's eating newsouls." I clenched my jaw. "I'm having trouble sympathizing with him."
"I didn't mean—" Cris lowered his eyes. "Sorry. I didn't mean it like that. It's just, five thousand years. That's a long time."
So long I could hardly imagine it. "I shouldn't have snapped. I'm just exhausted."
"I understand." Cris flashed a pale smile. "Janan shed his mortality, but souls still need something to contain them."
What did that mean for sylph, then? It seemed hard to believe that anything without a soul could love music as much as they seemed to.
"All this time, he's been waiting, growing, gaining power. If he ascends on Soul Night and becomes truly immortal, no need to consume newsouls to survive, then he won't need to reincarnate us."
"What about the Hallow? Meuric said if he had the key, he would live."
Cris smiled grimly, voice low and filled with hurt. "Why should Janan bother? We'll be unnecessary, even Meuric and Deborl. With Janan free of the temple, there will be no need for someone to guard the key."
The key. Another thousand questions revolved around that little box. Where had it come from? "The night of Templedark, Meuric said that birth isn't pretty. It's painful."
"Add the Range caldera to that," Cris said, "and you have—nothing. When it erupts, there will be nothing left but Janan."
I wanted to be sick. I hadn't even considered the caldera, but the earthquake swarms, the lake level...
The caldera beneath Range wasn't just moving through one of its natural cycles. No, it was getting ready to erupt. There should have been lots of warning. There should have been years of evidence beforehand. But nothing about Janan was natural; the unrest in the caldera must have been his doing.
When Range erupted, the devastation would be complete. The ground would be ripped apart. Lava would pour across the forest, killing everything in its path. Ash would fill the air, blocking the sun. The world's temperature would drop dramatically.
Not that anyone would be around to see that happen.
Heart—even Range—would be a hole in the ground.
Cris shoved his hands in his pockets, frowning at nothing. "Soul Night is still months away. There's still time to stop him if you can escape."
"By 'you' I assume you mean 'we.'"
"No, I mean you. And Stef if she'd like to escape as well."
From the other side of the room, Stef called, "What?" and stood. "You thought of a way out?"
Cris nodded as she rounded the stone table. "Ana, I have to confess something first." His tone made me shiver.
"What?"
"Please understand the last thing I want to do is hurt you, but"—he glanced at Stef, who didn't react—"I think you need to know."
I waited.
"Janan is using us, yes, switching oldsouls and newsouls to feed himself. But he didn't deceive us or trap us, in spite of these chains. We were told he'd gain knowledge and power to protect us when he returned, truly immortal. All we had to do was bind ourselves to him and he'd do the rest. We were afraid of the world, and of him, so we said yes." Cris gestured around the room. "We all made the agreement to be bound. We chose to be reincarnated."
It must have seemed like such an easy decision; after all, who wanted to die when you could live forever? "You didn't know about the newsouls?" Surely they hadn't known. Sam had been horrified when he learned the truth, and Stef and Cris were the same. The people I knew would never make that trade.
"Understand that we were young," he whispered, his face ashen. "We were young and in a dangerous land that spat boiling water and mud. There were dragons and centaurs, trolls and rocs, plus the regular animals that live in Range. Half our number had already been killed on the journey here. We were—still are—terrified of death."
Stef dropped her gaze. "It was selfish and desperate, but those were wilder times."
"No." I spoke as if denying it would change anything.
My heart beat itself into knots. I wanted to say I'd never make that decision, but how I felt now—knowing that no matter what I did, my life would be short—I might accept such a bargain. One more life with Sam, with music, with everything I ever wanted. All it would cost was someone who'd never know what they missed.
This would have been so much easier if I could have hated everyone for what they'd done.
Cris closed his eyes. "I don't want to think about how many souls that is, especially considering how frequently some people die."
"Hundreds of millions of newsouls." Stef's voice turned raspy. "I'm so sorry, Ana."
I was sick and aching. Sam had made the deal, too. Sam who loved me.
It stabbed like betrayal no matter how I reminded myself it was so long ago. My Sam. My friend Sarit. Lidea, who loved Anid so much. Geral, who thought Ariana was the most precious creature. All my friends. Everyone I'd ever trusted.
They'd all made the deal.
The people of Heart were so terrified of newsouls replacing them, but in truth, they'd been replacing newsouls for five thousand years.
A sob choked out, but I wiped my cheeks and tried to put the grief and anger aside. I was too worn to deal with it now. "Okay. So what's your plan? How does remembering how Janan started all this help?"
Cris was quiet for so long I thought he didn't really have a plan. "Someone needs to be able to open a door. I'll do it."
"Without the key?"
He closed his eyes. "A key. Not the key."
It took me a minute to follow. "No. You can't."
"I'm the only one who can."
"No." I scrambled to my feet, heart collapsing in on itself. "I won't let you sacrifice yourself."
"I'm sorry, Ana." He stood, too, with ten times more grace. "It has to be me. The world still needs Stef."
"The world still needs you." I was yelling at a rock, because he just shook his head. "Society would have never understood farming without you. Greenhouses. Fields. Orchards. That's because of you."
"That was thousands of years ago." He touched my arm, but I batted him away. "Now I grow roses. A noble endeavor, but not necessary for survival."
"What?" Stef peered between us. "What are you talking about? Why don't you want him to open a door?"
"Because without the key, there's only one way to make a door," I said.
She shook her head, looking weary. "Please remember I've been kidnapped and starved."
"Cris"—I pointed and growled his name—"thinks he's going to do whatever Janan did: get rid of his body; become part of the temple."
"What?" Stef was on her feet in an instant, shrieking at Cris.
"If you do it, you'll be as bad as him. You'll have to consume souls to survive, and someone will have to be the Hallow, and how will both you and Janan fit in the walls? I'm sure he won't be happy about sharing his space with you."
Stef stood inches from Cris, yelling as loud as she could while he stayed silent, waiting. "Why do you think this is going to work? For all you know, you'll just stick a knife in your chest and die."
"Even if it does work," I said, "in five thousand years everyone will have to stop you and they'll feel bad because you're otherwise nice."
Stef and I both stopped to breathe at the same time, and Cris cut in.
"First of all, I don't have followers like Janan did." He motioned around the room at our skeletal audience. "If I'm not reincarnating anyone, I won't get souls. These skeletons are bound in chains. They're bound to him."
"What if it changes?" My throat hurt from yelling, and my head throbbed with anger and betrayal. "What if suddenly you're supposed to switch souls?"
"I wouldn't do it." He sounded so calm and certain, like he didn't think it would be a temptation. "Ana, I promise. Knowing what I do, knowing you, I understand what we sacrificed so long ago." He touched my hand, softly enough that I could barely feel his fingers tremble. "I'm so sorry, Ana. We don't deserve your forgiveness, but I can try to put things right."
"How is that?" I wanted to hate him and his stupid plan, but now that I wasn't yelling, my body felt limp and heavy.
"I will become part of the walls, like Janan, then open a door."
"No." I crossed my arms. "This is a crazy plan. You don't even know if it will work."
"Wouldn't you need a Hallow?" Stef asked. "I'm not chaining myself up like those two." She pointed at Deborl, and one of Meuric's toe bones I'd missed.
"There's no need for a Hallow." He smiled at her, all grim determination. "Janan needed one to help bind his followers and guard the key, but I won't. No souls. No sacrifices."
"You're talking about sacrificing yourself." My words squeaked out. This wasn't happening. It couldn't be.
"For you." He took my hand, his five thousand years evaporating. He looked young and scared, just like I felt, and his hand sweated over mine. "You haven't had a hundred lifetimes, and even this one has just begun. There's so much you still have to experience. No matter what happens with all this"—he gestured around the temple—"I need to give you a chance."
A million things happened inside me at once, most prominently my heart squeezing up to my throat, and my stomach flip-flopping. Grateful and sick and filled up with misery.
"Cris, no." I didn't want to die, though, or be trapped forever. I wanted to live, to have experiences. I wanted to see the world with my single short life. But Cris...
"Think of it as a gift, if it helps. One you can't turn down."
Stef stood nearby, eyes round as if she'd begun to accept what he was going to do.
"Janan is too strong. You can't beat him," I whispered, half saying the words because I knew I should. "He's had five thousand years to gain power. You will be new and weak. He won't let you stay in the walls." He needed to see how futile his plan was.
"I only need a few moments to open a door for you." He cupped my cheek with his free hand.
"What happens if he kills you? Will you be reincarnated?"
"For a newsoul's sake," Cris said, "I hope not."
But I didn't want him to be gone forever. Where would he go? What would he do?
"Ana, you have to live. You have to get out of here, stop Janan from destroying Heart, and live this life. Do everything you can. Don't waste it. Promise me."
"We'll find another way." Why couldn't he see?
"When? How? There's nothing here but skeletons." His eyes were glassy, and he blinked several times as though trying not to cry.
"Please don't." I looked to Stef for help, but she just watched us with a hard expression, like ice.
Just as I turned to him again, Cris leaned forward and kissed me. Not long, and not desperate. I barely had a chance to register the way his lips tasted like tears before he drew back, looking as surprised as I felt.
"I thought you were in love with Sam." That wasn't what I wanted to say, but it saved me from having to think too hard about the simultaneous thrill and fear and stress of what had just happened. I still didn't understand why Sam wanted to kiss me, let alone anyone else.
"I will always be in love with Dossam." He focused inward, somewhen-else. He didn't mean my Sam, but a Sam from lifetimes ago. "And I love you," he whispered, coming back to the present. "Not like Sam does, not nearly. But that's why you have to live. I couldn't bear to let anything happen to you when you've just begun, and I couldn't bear Sam's pain if he lost you."
My breath was too heavy, crushing me from the inside. I couldn't let him do this, but I wanted to escape. I wanted to live and be loved and not die. Pieces of me were becoming resigned to it, even welcoming his fate because it meant I might be free.
Stef was still ice. No hope of strength from her.
Cris squeezed my hand. I'd forgotten he hadn't let go. "You're going to live," he said. "You're going to make it out of the temple, and then you're going to use everything you've learned to stop Janan. Save the newsouls."
I hated myself as I nodded, and warmth trickled down my cheeks. He was crying, too, but I didn't know what to say to other people who cried. Instead I just hugged him. His wiry body tensed before his arms went around me, too.
If I spoke, I would be undone. Everything in me would spill out. So I squeezed him until he pried himself loose and said, "I shouldn't have kissed you. I hope you can forgive me."
Because I still couldn't speak, I pressed my fingers to my lips and nodded, and hoped he knew that I understood. He was afraid.
"Be ready to run," he said, "because I have no idea how long it will take, or how long it will last. If I have time, once you're free, I'll try—I don't know. Maybe I can save the souls he's trapped here."
Was that even possible? Maybe it was to the boy who'd ride across Range to save his roses from frost.
"You don't have to," Stef whispered. "I could."
"The world has more need for a scientist and engineer than a gardener, especially right now." He hugged her as well, and kissed her cheeks. "Please don't kill each other after I'm gone."
Gone.
He was going to do it now? Shouldn't he wait?
My legs were numb, my arms useless. My voice had long since abandoned me. I wanted to tell him to stop, to reconsider, but it would only delay the inevitable. He'd already decided, and I selfishly wanted to go home.
Without regard for my silent urging him to wait, Cris climbed onto the table next to Janan, found the knife, and lay down.
I wished I had something strong or brave to say, something that might give him a breath of reassurance. But I had nothing to offer. I was useless.
Stef stood next to me, put her arm around my waist. Crying, I leaned my head on her shoulder and watched Cris settle on the stone and position the knife above his heart. He was really going to do it. There had to be another way, and I was crying instead of figuring it out.
"Please, Cris." The temple smothered my words. Please don't. Please wait. Please come back.
He turned his head to look at us, managed a grim smile, and closed his eyes. Silver and gold flashed in red light as the knife pierced.
He died.
#
# SACRIFICE
I SCREAMED.
Fingers dug into my arms, through my sleeves, and Stef yelled my name over and over. I strained against her, reaching for Cris on the table. His eyes were dull and glassy; his knuckles were white around the knife hilt.
No matter how I struggled, Stef was stronger. I rushed toward Cris, but Stef yanked me back and shoved me to the floor, pinning me. "Stop it!" she yelled.
But I wasn't flailing anymore. I was too busy watching a white light bleed into the table.
The light expanded, flooding around the table legs that stretched over the pit. It was so bright I had to squint as the glow encompassed Cris's body.
Tears leaked down my face, from despair and shock and light. All the air swept inward, wind rattling bones and snatching at our clothes; I caught my scarf as it tried to flee my neck. Deborl's skeleton skidded on the floor toward the pit, as though all the air were being sucked down. It strained against the shackles.
The glow flared so bright I had to close my eyes. I wanted to close my ears as the wind howled around table legs.
Beneath me, the floor moved, slick against my clothes.
No, I was moving on the floor, both Stef and me. Shrieking wind pulled us, even as Stef scrambled to help me off my back. Wind-deaf and light-blind, we had to feel our way as the pull grew stronger, like gravity was shifting.
My heart hammered with a surge of adrenaline.
"We have to find something to hold on to!" I couldn't tell if she heard me over the rush and keen, but I reached—her arm reached with mine—and felt along the floor, trying to dig my toes in.
"No!" Janan's voice filled the room, thunder and waterfall-crashing.
I fought the wind's pulling, the way air thinned, and I lost track of Stef. Twice, I felt her bump against me, but I focused on not sliding as red light pulsed beyond my eyelids, and white light burned and moved. Even with my eyes closed, I saw silhouettes of my hands splayed on the floor, desperate for traction.
And then Cris's voice: "Ana. Stef. Go."
I couldn't help but sob. He'd done it. Done something. "Cris!" My voice was lost under Janan's rage and the wind still sucking toward the pit. Bones clacked, and silver chains rattled and clanked.
Janan roared words I didn't know, had never heard. His voice was pressure on my skin, hot as a sylph turned solid.
"Ana, now!" Cris again, like sparks catching and burning. "Please."
It was his desperation that made me open my eyes. A gray archway waited ahead of me, just a few paces away, and mostly in the floor so I wouldn't even have to stand. He'd done it. Freedom. His plan had worked.
Jaw clenched, gasping at thin air, I clawed toward the misty portal and hooked my fingers on the bottom lip. I just had to pull myself up and tumble out. Quickly, too, because the outline wavered, shot with streaks of black and white. Changing its destination.
If I didn't hurry, Janan would seize control.
"Go, Ana!" Cris again, choked and smothered. Lights and air pulsed all around the chamber as the two battled within the temple walls.
Stef. I couldn't find her.
Digging my fingers into the stone—what would happen if the archway vanished altogether?—I adjusted myself to get a better look around the room. I shouted her name, but she wouldn't hear me over the stampede of Janan's rage.
The table. If I squinted right, I could make out arms looped around the near table leg, and Stef straining to keep herself from being sucked the rest of the way in.
She had bumped against me before. Nudging me away from the pit?
Her attempt at heroism had almost gotten her killed, too.
I had a scarf, but even if I had been strong enough to hold on to the archway with one hand and pull her up with the other, it wasn't long enough.
There was no asking Cris for help. The shrieking and wind grew worse, and Cris cried out in pain. I had no idea what Janan could do when they were mostly without substance, but the wailing sounded like stars dying.
I pulled myself far enough to the arch and braced my elbow inside it, then lifted my leg as high as I could. My heel caught the edge. Terrified every motion would make me slip, I tied one end of the scarf around my ankle, making sure the knot was secure.
Leg down again, the scarf whipped in the wind, close to Stef but not close enough. I couldn't see her face in the searing light, but her arms didn't move from around the table leg.
Chest muscles aching with the strain of holding on, I switched to my hands again, so now instead of my upper body at the archway, only my head peeked in.
Stef—I hoped Stef—tugged on the scarf, but the weight wasn't enough to make me believe she'd taken a good hold. It wasn't constant pulling.
Sam would never forgive me if I got this far and didn't save her. I took three breaths as deep as I could, wind stinging my throat and eyes, and lowered myself farther so my arms stretched before me. Only my fingers stayed in the archway as the sucking wind grew stronger.
Red flashed like bloody lightning, and the cacophony grew worse. But then there was steady tugging on the scarf as Stef grabbed hold and began climbing.
"Please let the knot hold," I whispered.
The scarf yanked on my foot, and Stef was more weight for me to keep up. My hands were numb as I struggled to hold on, struggled to keep my foot flexed so the scarf wouldn't slip off. My muscles shook.
A hand closed around my ankle, and another on my calf. My own scream was lost in the din as I begged my arms to pull us up again. If I could just get my elbows over the edge, I would be able to fall through the hole.
Stef used me like a rope, climbing as I worked to bend my arms. The wind pulled and pushed, and lights flared. I focused on breathing, focused on the archway stretched above me. Freedom. If only Stef's arms weren't wrapped around my waist.
She must have been pushing with her feet, because a nudge gave me the weightlessness and strength to move my left shoulder over the lip and hang on with my elbow. Now I pushed instead of pulled, but fire still ripped through my arms and chest as I gained enough strength to move my upper half over the archway.
Stef reached for the edge with one arm. Her other around me slipped.
"Just a little farther," I urged. The wind stole my voice.
Chasms of concentration lined her face. She clenched her jaw tighter, reached again, and caught hold enough to pull herself up next to me.
The archway had been gray when Cris opened it, but now it was midnight dark. Relief for my eyes, but I was pretty sure that meant Cris wasn't in control anymore, and no matter how much I shouted his name, the archway didn't change.
Stef leaned toward me, shouted by my ear. "Why aren't we leaving?"
My tortured voice wasn't even as loud as hers, but I tried. "Gray means outside. Black or white means inside."
She looked ready to cry, but nodded and hauled herself higher on the archway. One foot on the left edge, one on the right. She positioned herself over it like a spider waiting to pounce.
I understood. The second the portal turned to gray, we were going through. I hastened to follow her example, screaming to Cris as loud as I could that we were ready.
But when it did flicker and the black became smooth gray, I wasn't prepared. My foot had slipped and I was trying to push myself up with just one leg. All my muscles felt shredded, though, too worn to move.
Stef grabbed my wrist and dragged me through the gray archway just as it began to change.
Silence.
Real silence, not the temple unsilence where not even my ears would ring.
And air, windy and cold, but it didn't try to pull me places. It was thick enough to breathe.
Frigid skin pressed against mine, and I opened my eyes to see Sine above me. Her mouth moved as though she spoke, but I couldn't hear, so I just blinked and breathed and waited for my muscles to melt. For now, at least, they were too cold to hurt. I reveled in the ability to lie flat on my back and not be moving.
"Ana." Sine sounded far away. "You have to get up."
I turned my head to find Stef staring up at Councilor Frase. She looked the way I felt. Dull. Not really here.
The market field cobblestones had never been so beautiful.
"Ana!" Sine's shout brought me back to myself. "Get up before I find someone to carry you."
That didn't sound like a bad idea at the moment, but as I regained control over my body, I remembered market day, Deborl's speech, Meuric dying in front of everyone, and the resulting mob.
I sat up so quickly Sine almost didn't dodge fast enough. "Where is Sam?" I tried to make my eyes focus on her again, but I'd moved too fast, and dizziness swarmed inside my head.
"Hospital." She stood and offered a hand. I climbed up by myself when I saw Stef finding her way to a more vertical position, too. "With everything that happened the other day, he received a few serious injuries, but he'll live. He just woke up an hour ago."
I wanted to feel numb, not vainly try to patch the cracked dam of emotions. Sam. Cris. Janan. Soon I was going to break.
Just not in front of anyone. Please.
"What day is it?"
"You've been missing for two days."
It felt like a month. Maybe Cris had managed one last favor, letting us out as close to the time we went in as possible.
The dam inside me strained. I should have stopped Cris. I'd as good as killed him.
"Where's Deborl? I'm going to electrocute him and then set him on fire—" Stef gasped as she leaned on Frase's shoulder, hiding her face.
"Deborl and his friends are in prison."
"Prison?" I could hardly imagine good news anymore. "What about Wend? He was there, too." Though Deborl had shot him....
Sine combed her fingers through my tangles. "Wend is dead." Lines creased her face as she frowned, and a tear dropped from crevice to crevice. "None of them will trouble newsouls again, though it's only fair to tell you that they were not ignored."
"I need Sam." I needed to tell him everything that had happened.
"Of course. Corin, please fetch Dossam." She signaled to someone behind me—Corin, presumably—and footsteps retreated. "Where is Cris? They said he was with you."
I gazed at the temple, cold and white and not quite as evil if Cris was still in there. Sam had said Cris had never done anything terrible to anyone. Even after learning they'd all sacrificed newsouls for reincarnation, I still believed that. He'd sacrificed himself for us now.
But I couldn't answer Sine's question.
I was going to break.
I wasn't sure how long I stood there, holding myself together with nothing but threads, but eventually a familiar shadow fell next to mine.
My muscles felt like liquid as I lifted my hand just enough that Sam's closed around it, and then his arm closed around the rest of me.
The dam broke and everything spilled out. Sam hugged me so tightly I couldn't breathe, or maybe the sobs choked me. He touched my hair and face, kissed me. His affection was featherlight, as though he was afraid of crushing me.
I cried into his shirt even though there were other people here. Stef, Sine, Frase. People I didn't know. I wanted to hide, but I was afraid I wouldn't be able to walk. Even now, Sam mostly held me up.
Sam, who, five thousand years ago, had taken immortality knowing the price. How could I ever look at him the same way?
But I couldn't bear to pull away from him. Maybe I wouldn't tell him; it would be hard enough for both of us to deal with the fleetingness of my existence.
I would just die.
Where would I go? What would I do?
So lost in myself, and in Sam's arms, I almost didn't notice the commotion around the curve of the temple.
"What's going on?" I swallowed more tears.
"Sylph. Don't worry. They'll capture it and set it free outside Range." He started to adjust his hold on me, but I straightened and pulled away. "What is it?" Concern lined his face.
"I just had a horrible thought." I wanted to be wrong, but my mind worked no matter how I tried to ignore it. "Help me get there before they put it in an egg."
He looked uncertain, but kept me upright as I limped toward the crowd gathered around a panicked sylph. The tall shadow hummed and sang, caught in the circle of people with brass eggs. It could have burned any of them, but it stayed in the center and shifted as though trying to decide what to do.
Then it saw me.
I gathered my strength and gave Sam's hand a squeeze. "Let me through." My voice cracked, and I had to say it again, but the team with sylph eggs backed off. Maybe they remembered Deborl's claims that I could control sylph.
I stepped through the line of people, Sam close behind, and Stef after him. The column of smoke and shadow grew still and its songs silent. It looked at all of us and slumped, somewhere between relief and exhaustion.
It was too human.
"We shouldn't have let him do it, Stef." I lifted my hand toward the black smoke. People hissed, but when my fingers passed through, there was only uncomfortable warmth. The sylph hummed, calmer.
I raised my other palm toward the midnight curls, but it shivered away from me as heat grew, like it had lost control.
"Oh." Stef sounded like she wanted to be sick. "Cris?"
The sylph twitched—acknowledgment—and a tendril of shadow blossomed like a black rose, then fell to my feet.
I clutched my chest, my heart caged inside. We'd let him sacrifice himself for us, and now he was cursed—
Cursed.
Sylph were cursed.
Cris had said there'd been no sylph in the beginning. I still didn't know how they'd been cursed, but I knew what Cris had done.
"Oh, Cris."
The shadow rose vanished, and the sylph floated between a pair of guards—who stepped aside to let him pass. He flowed like ink down East Avenue, and Sine muttered into her SED. "There's a sylph going through the Eastern Arch. Open the gates wide and let him be."
#
# HEARTBEAT
AFTER SPENDING A few days in the hospital, I was taken to the Council chamber. The remaining Councilors were there—nine now, since Deborl was in prison—but none of them looked happy to see me. Most just stared at the items on the table: a stack of leather-bound books, a handful of diaries, and a small silver box.
This wasn't quite everything Deborl had stolen from me, but these things were the most incriminating. The music...
I slumped in my chair, grateful when Sam sat next to me; they hadn't let anyone visit while I was in the hospital.
"Today's session is closed." Sine focused on me, her gaze hard and holding back all emotion. "And it will probably stay closed. Typically we are in favor of sharing our decisions with everyone in Heart, but this—Ana." She said my name like heartbreak.
Everyone stared at me, but I didn't look away from Sine. I just waited.
"Deborl's methods were reprehensible, but he did uncover several unfortunate truths.
"First, you were in possession of Menehem's research." She pressed her palm on the diaries, as though she could crush them into dust. "The same research that describes how he created Templedark. You lied to us. You hid information regarding our existence and our history. Regardless of whether Menehem left the research in your care, it was never yours to keep."
I clenched my jaw and said nothing, because nothing I might say would help. I had kept the research. I had lied. Those things were true.
"Second, there's Meuric."
The name alone conjured memories of his stench, his grating voice, and his manic laughter when he told me that Janan ate souls. I shuddered and swallowed the taste of acid in my throat. Below the table, Sam took my hand and squeezed.
"Is what Deborl said true, Ana?" Emotion cracked through Sine's voice when she asked, "Did you kill Meuric?"
Had I? I'd thought so before, but then he was alive in the temple. He would have stayed alive, but Deborl brought him out. Both Deborl and I were responsible, but if I hadn't stabbed and kicked Meuric to begin with... "It was self-defense. He tricked me into the temple. He was going to trap me there. We fought. I won."
"And you decided not to tell anyone." Sine glanced at Sam, probably knowing I'd told him, but if she was going to punish him for not coming forward, she wasn't going to do it now. "That brings me to the third complaint." She touched the temple key and books, and confusion flickered across her face. The other Councilors, too, seemed unsure what they were looking at.
"Those were Meuric's," I offered. Technically the books weren't, but he could have shared them with the community. He'd decided not to.
"And yet," Sine said, "when you came into possession of them, you hid them."
It wasn't like anyone else would have remembered them. Deborl would have taken everything, and I would have no answers.
Maybe I had too many answers.
"Do you remember what happened to Cris?" My voice caught on his name.
The Councilors glanced at one another, muttering, until Sine shook her head. "He was killed during the mob on market day."
My fists balled up and my jaw ached from clenching it, but there was no point in arguing. They wouldn't remember that Cris had become a sylph, or what Janan did to newsouls, or that they'd all agreed to bind themselves to him in the first place. The forgetting magic was too strong.
They'd only remember that they didn't trust me. That I'd lied. That I'd kept things from them.
"Ana." Sine leaned on the table. "I know the newsouls are important to you."
She had no idea.
"The Council had several emergency meetings after market day. We did listen to what you had to say, and we've already put laws in place to make sure newsouls are protected. Anid and Ariana are safe. So are any others born."
And me? I'd never feel safe again. Neither would those inside the temple. Still, it was more than I'd expected. "Thank you."
"But," she said, "I'm afraid given what we've discussed today, the Council has decided to revoke your status as a guest in Heart."
Everything inside of me spun, dropped, slammed. She couldn't do that.
"Be reasonable—," Sam started, but Sine held up a hand to stop him.
"This was not an easy decision to come to." Sine lifted her voice, glancing between Sam and me. "We agonized over it, trust me, but the fact of the matter is that Ana has not held up her end. By lying and withholding important information and items, she betrayed our trust."
"No." Sam's voice was low. Dangerous. "You betrayed her trust. She hasn't been safe in Heart since she arrived. And would you have believed her, even if she'd come forward? People have tried to kill her; Deborl—a Councilor—and his friends set explosives to try and kill more newsouls. The moment she arrived in Heart, the Council betrayed her by making up laws to prevent her from joining society."
Sine closed her eyes, and her tone was much too calm. "As a result of your actions, Ana, I'm sorry to say you are no longer welcome in the city. Dossam will no longer be your guardian. While we will not force you beyond Range—that would surely be a death sentence—you are hereby exiled from Heart."
"No!" Sam lunged across the table, and within heartbeats, every Councilor was up and shouting.
Exiled.
Screaming and fighting built around me. I stared at my hands on my lap.
Exiled.
Asunder after all.
"I'll go." I stood, and the cacophony stopped; Frase and Finn had Sam pinned against the wall already, fists drawn back as though they were going to punch him. It hadn't been much of a fight, one against many. "I'll go," I said again, "as long as you keep your promise about the other newsouls."
"Of course we will." Sine nodded at the men holding Sam. "Please, all of you. Stop this. Sam, I'm disappointed in you."
Sam muttered a few unfamiliar curses as he jerked away from the Councilors. "You deserve everything coming to you."
I frowned—no one deserved what Janan would do—but I just turned for the Council chamber doors. "I need to pack a few things."
"That's fine." Now that it was over, Sine was gentle. "We can give you two days."
That didn't give me much time to pack and say good-bye to friends, but it was more time than I'd expected.
Afternoon fell in pale splashes across Heart. Sam and I walked home, not talking about the Council's decision. Instead, he called and vented to Stef, who immediately went to demand an appeal. Other friends joined in, once they heard the news, but when they came over later, everyone said the Council had refused to hear them.
When Stef and Sarit left late that night, Sam and I changed into nightclothes and settled in the remains of our parlor.
"I can't believe this is really happening." He sounded far away.
"They'll never accept me." I closed my flute into its travel case, grateful it had survived the mob. "They don't trust me, and they won't believe the truth. But I believe Sine when she says she'll take care of the other newsouls. They have a chance. As far as Janan goes—what can I do against him when the Council has everything?"
"You won't give up against him, though."
"No. But what I want to do, I can't do here."
"What's that?" Hope colored the edges of his voice.
"I'm going to find the sylph." I was closer to finding answers to the questions I'd asked in Menehem's lab, but I needed to know more. And Cris was out there. Somewhere. Maybe—No, he'd already given everything.
Sam smiled grimly. "I'm going with you."
Joy sparked inside of me. "Are you sure? I'd never ask you to leave—"
"I'd go anywhere with you." He touched my cheek. "It doesn't matter where or how far, or even why. I want to be with you no matter what."
"Thank you." Heartbeat thudding in my ears, I met his eyes and let my emotions bubble into words. "I love you, Sam."
It was easy to say. I could love. And I did.
Sam swept me into his arms and hugged me so tight I couldn't breathe, whispering his love again and again. His promises sat warm on my throat, trapped in my hair and shirt collar, and I imagined they wrapped around me like armor.
"I always have," I said into his hair. "I've loved you since I first heard your music, and saw how you wrote about it." I kissed his throat, twisted my fingers in his shirt. "I loved you when you saved me from the lake and put your breath inside me." His face, his hair, his shoulders—everywhere my hands could find, they did. "I loved you that day in the library when you showed me your past lives, and at the masquerade before I was totally sure you were the shrike."
"All those times?" He pressed his cheek against mine.
"And more. When you took care of my hands, when you found me outside the temple. I even loved you when I was angry with you. Maybe especially then." While I spoke, I'd settled myself on his lap, facing him. His heart thudded against mine. "I think no matter what happens, I'll always love you."
Even though five thousand years ago, he'd made the choice to sacrifice newsouls, I still loved him. I couldn't help it.
He'd changed so much since then. The entire world had changed.
We fell asleep on the sofa, tangled in blankets. My cheek rested on his chest, and my arms looped around him. I loved the way he felt underneath me, and the way his hand rested on my ribs. I loved the occasional soft snore.
There were so many times I should have said it since meeting him. I should have told a lot of other people, too.
It was the last night before the new year. The Year of Hunger was passing into memory as I breathed.
I slipped from Sam's arms and let my nightgown fall straight, then took careful steps around the last of the piano wreckage. Dried and delicate rose petals still speckled the floor, like flecks of cracked blue paint.
"Ana?" Sam watched me from the sofa, my hand extended halfway to the exterior wall. "What are you doing?"
I shook my head, dropped my arm. "Just...seeing." I'd feel better about leaving if everyone was safe.
He sat up, blankets tangled around him, his shirt askew and half-unbuttoned. "There's something I need to tell you. Something I did at Menehem's lab."
I picked up a rose petal; it rasped and crunched in my fingers.
"I asked you about the knife. You'd said it made you feel better. You'd said you wanted something like that against Janan."
The petal crumbled and bits fluttered to the floor, and I couldn't forget his guilty expression anytime we'd spoken of Menehem's lab. I'd felt bad about keeping the truth from people, but he'd seemed to take the guilt even harder. "What did you do?"
He met my eyes and drew a shaky breath. "I turned on the machine. It's been making the poison since we left."
"Oh." A hundred emotions flooded through me, shock and dread and gratitude.
"You said it would take an incredible amount to affect Janan even for a moment. I don't understand much about it. I just turned on the machine and input the ratios from the dose that worked on sylph. It may not do anything. It may be for nothing. But I wanted to give you a knife."
"Thank you." I wasn't ready to believe it would work, but my heart swelled with what he'd done for me. Turning on the machine went against his nature, but he loved me and wanted me to feel safe. "So that's where we go first, before the Council looks into it. Unless..."
I touched my fingertips to the exterior wall.
The white stone was warm, as usual, but it didn't do anything untoward. I dared a moment of relief.
Shouldn't have.
The heartbeat pulsed faster than ever. I snatched my hand back, echoes of Janan's roaring in my head, visions of Cris a sylph behind my eyes. He'd freed us, but nothing more.
Midnight struck.
The Year of Souls had begun.
# Back Ads
# ACKNOWLEDGMENTS
ETERNAL THANKS TO:
Lauren MacLeod, my agent, and Sarah Shumway, my editor. Thank you for always challenging me to do better and inspiring me to try harder. I can't imagine this journey without you ladies.
The entire team at Katherine Tegen Books, including Amy Ryan, Brenna Franzitta, Casey McIntyre, Esilda Kerr, Joel Tippie, Katherine Tegen, Laurel Symonds, Lauren Flower, and Megan Sugrue. Hearts and flowers to you all.
The amazing people who read early versions (or last-minute changes!) of this book: Adam Heine, Beth Revis, Bria Quinlan, Christine Nguyen, C. J. Redwine, Corinne Duyvis, Gabrielle Harvey, Jamie Harrington, Jaime Lee Moyer, Jillian Boeme, Joy Hensley George, Kathleen Peacock, Lisa Iriarte, Myra McEntire, and Wendy Beer. Your comments, encouragement, and kicks in the butt were invaluable for getting this book from pixels to paper. I'm so grateful that you're all in my life.
Warm thanks and adoration to Jeri Smith-Ready, Rachel Hawkins, and Robin McKinley, who said such nice things about Incarnate. Your praise means so much to me, as I'm an incredible fan of your books as well.
Special thanks to the book bloggers, writers, and knit bloggers who helped me launch the Incarnate Theater Treasure Hunt the week Incarnate was released. Your support, enthusiasm, and hard work amaze me.
Adam Heine, Amanda Miller, Amanda at Loves Books Reviews, Amber Mitchell, Amy Fournier, Angel Cruz, Anna Billings, Asheley Tart, Becky Herrick, Best Tanakasempipat, Bonnie Lynn Wagner, Brenna at Ever After Esther, Charlee Vale, Dot Hutchison, Emily Wright, Enna at Squeaky Books, Gabi Becker, Gabrielle Carolina, Hannah Courtney, Jaime from Two Chicks on Books, James at Book Chic Club, Jessica Reigle, Jodie at Uniquely Moi Books, Julie, Katie, Kaye M., Lauren at 365 Days of Reading, Linda Dao, Mary at the Book Swarm, Mei Jiao Ashley Chen, MG Buehrlen, Michelle and Amethyst at Libri Ago, Michelle Villarmia, Rachel, Sana Reddy, Sarah Nicolas, Shanyn Day, Shellie from Creative Reads, Shelley Watters, Stephanie at Poetry to Prose, Stephanie Huber, Susan Adrian, Tammy Moore, and Traci Inzitari.
Jill Roberts, for being the best, most encouraging mother a girl could ask for. Thank you for always believing in me.
Jeff Meadows, for unending patience and understanding, and willingness to listen to book-crazed ramblings at all hours of the day.
Thanks to God, who I will never be able to thank enough.
And thank you, reader, for picking up this book. I hope you enjoy reading it as much as I enjoyed writing it.
# About the Author
JODI MEADOWS lives and writes in the Shenandoah Valley, Virginina, with her husband, a cat, and an alarming number of ferrets. She is a confessed book addict and has wanted to be a writer ever since she decided against becoming an astronaut. You can visit her online at www.jodimeadows.com.
Visit www.AuthorTracker.com for exclusive information on your favorite HarperCollins authors.
# Credits
Jacket photo © 2013 by
Gustavo Marx/MergeLeft Reps, Inc.
Jacket art and design by Joel Tippie
# Copyright
Katherine Tegen Books is an imprint of HarperCollins Publishers.
ASUNDER
Copyright © 2013 by Jodi Meadows
All rights reserved under International and Pan-American Copyright Conventions. By payment of the required fees, you have been granted the non-exclusive, non-transferable right to access and read the text of this ebook on-screen. No part of this text may be reproduced, transmitted, down-loaded, decompiled, reverse engineered, or stored in or introduced into any information storage and retrieval system, in any form or by any means, whether electronic or mechanical, now known or hereinafter invented, without the express written permission of HarperCollins ebooks.
www.epicreads.com
* * *
Library of Congress Cataloging-in-Publication Data
Meadows, Jodi.
Asunder / Jodi Meadows. — 1st ed.
p. cm.
Sequel to: Incarnate.
Summary: After the devastation of Templedark, eighteen-year-old Ana must stand up for the additional newsouls and figure out the mystery of their—and her—existence.
ISBN 978-0-06-206078-5 (trade bdg.)
[1. Fantasy. 2. Reincarnation—Fiction. 3. Identity—Fiction.] I. Title.
PZ7.M5073As 2013
2012009696
[Fic]—dc23
CIP
AC
* * *
12 13 14 15 16 LP/RRDH 10 9 8 7 6 5 4 3 2 1
FIRST EDITION
EPub Edition © January 2013 ISBN: 9780062060808
# About the Publisher
Australia
HarperCollins Publishers (Australia) Pty. Ltd.
Level 13, 201 Elizabeth Street
Sydney, NSW 2000, Australia
<http://www.harpercollins.com.au>
Canada
HarperCollins Canada
2 Bloor Street East – 20th Floor
Toronto, ON, M4W, 1A8, Canada
<http://www.harpercollins.ca>
New Zealand
HarperCollins Publishers (New Zealand) Limited
P.O. Box 1
Auckland, New Zealand
<http://www.harpercollins.co.nz>
United Kingdom
HarperCollins Publishers Ltd.
77–85 Fulham Palace Road
London, W6 8JB, UK
<http://www.harpercollins.co.uk>
United States
HarperCollins Publishers Inc.
10 East 53rd Street
New York, NY 10022
<http://www.harpercollins.com>
|
{
"redpajama_set_name": "RedPajamaBook"
}
| 4,029
|
{"url":"https:\/\/socratic.org\/questions\/a-triangle-has-corners-at-3-5-4-7-and-1-2-how-far-is-the-triangle-s-centroid-fro","text":"# A triangle has corners at (3 ,5 ), (4 ,7 ), and (1 ,2 ). How far is the triangle's centroid from the origin?\n\nMay 17, 2016\n\nThe distance of triangle's centroid from the origin is $5.375$ units\n\n#### Explanation:\n\nThe centroid of a triangle whose vertices are $\\left({x}_{1} , {y}_{1}\\right)$, $\\left({x}_{2} , {y}_{2}\\right)$ and (x_3,y_3)# is given by\n\n$\\left(\\frac{{x}_{1} + {x}_{2} + {x}_{3}}{3} , \\frac{{y}_{1} + {y}_{2} + {y}_{3}}{3}\\right)$\n\nHence centrid of given triangle is $\\left(\\frac{3 + 4 + 1}{3} , \\frac{5 + 7 + 2}{3}\\right)$ or $\\left(\\frac{8}{3} , \\frac{14}{3}\\right)$\n\nand its distance from origin is\n\n$\\sqrt{{\\left(\\frac{8}{3}\\right)}^{2} + {\\left(\\frac{14}{3}\\right)}^{2}} = \\sqrt{\\frac{64}{9} + \\frac{196}{9}} = \\sqrt{\\frac{260}{9}} = \\frac{1}{3} \\times 16.125 = 5.375$","date":"2022-06-30 07:17:28","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 7, \"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\": 0.9862610101699829, \"perplexity\": 998.5815666554724}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-27\/segments\/1656103669266.42\/warc\/CC-MAIN-20220630062154-20220630092154-00489.warc.gz\"}"}
| null | null |
Q: how to get a part of file path? for example I have string:
test = "/var/tmp/test.log"
I want to get a path of this file.
A: Use os.path.dirname():
>>> import os
>>> os.path.dirname('/var/tmp/test.log')
'/var/tmp'
A: >>> import os
>>> os.path.dirname("/var/tmp/test.log")
'/var/tmp'
A: Which part are you interested in? The os.path module has a lot of useful functions. If that fails, a .split("/") would do the trick.
A: if you want to get absolute path, try this
C:>python
Python 2.5.4 (r254:67916, Dec 23 2008, 15:10:54) [MSC v.1310 32 bit (Intel)] on
win32
Type "help", "copyright", "credits" or "license" for more information.
>>> import os.path
>>> test = '.'
>>> os.path.dirname(os.path.abspath(test))
'C:\\'
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 6,623
|
Q: Uniqueness quantification of planes in vector space
*
*Say we have two parallel lines $l_1\parallel l_2$. Note: We know the equations for these
*We know one point on each line, $P\in l_1\land Q\in l_2$.
To find the plane $\pi$, such that $l_1,l_2\parallel\pi$ (both lines also lie on $\pi$), must know at least one coordinate on the plane, as well as two directional vectors.
*
*We know two coordinates, $P$ and $Q$. Of course we could figure out more with our lines
*We know one directional vector already.
$$\vec{v} = (a,b,c)$$
The other vector, according to the task, can be determined by looking at an arbitrary point on each line and figure out the directional vector by subtracting $A$ from $B$.
Of course, given the information, it's comfortable to just calculate $\vec{PQ}$, as we already know these points.
Given two directional vectors and a point, it's easy to figure out the affine form of our plane $\pi:a'x+b'y+c'z+d' = 0$.
Simply look up the normal vector for the plane by calculating the vector product:
$$\vec{n}:(a',b',c') = \vec{v} \times \vec{PQ}$$
$d'$ is solved trivially.
Now, what if I took two other arbitrary points, other than $\vec{PQ}$, such as $\vec{AB}$ where $B$ is a point on $l_1$ and $A$ is a point on $l_2$. After trying this out, it led to the same plane $\pi$. Meaning a plane is uniquely quantified by two parallel lines going through a plane.
The question
When I think about this geometrically, I can't help but to think something is wrong. What stops you from simply rotating the plane around the normal vector? The plane would still go through two lines, yet would, I'm assuming, yield a different equation for the plane.
The two drawn planes below look different to me.
A: A plane extends infinitely in all directions it encompasses. Rotating the plane about a normal vector therefore does not change the plane.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 4,657
|
{"url":"https:\/\/lavelle.chem.ucla.edu\/forum\/viewtopic.php?f=145&t=28533","text":"## Unique Average Rates\n\n$aR \\to bP, Rate = -\\frac{1}{a} \\frac{d[R]}{dt} = \\frac{1}{b}\\frac{d[P]}{dt}$\n\nEmily Warda 2L\nPosts: 29\nJoined: Fri Sep 29, 2017 7:06 am\nBeen\u00a0upvoted: 1 time\n\n### Unique Average Rates\n\nWhen is the unique average rate most useful? And will it always be specifically asked for or should we know instances when to use it and when not to use it?\n\nRebecca Doan 2L\nPosts: 51\nJoined: Thu Jul 27, 2017 3:01 am\n\n### Re: Unique Average Rates\n\nI think the unique average rate is just specific to the question\/ reaction you are given. So when they ask for the unique average rate of the reaction, they are talking specifically about that certain reaction with those stoichiometric coefficients. When you calculate the average rate, the coefficient is not taken into the account, so the average rate can be different from the unique rate.","date":"2020-12-02 20:04:34","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 1, \"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\": 0.5533657073974609, \"perplexity\": 1765.1164875898046}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-50\/segments\/1606141715252.96\/warc\/CC-MAIN-20201202175113-20201202205113-00130.warc.gz\"}"}
| null | null |
Nour El-Refai, née le au Liban, est une actrice et présentatrice de télévision suédoise.
Biographie
Elle est née au Liban de parents d'origine syrienne et libanaise, sa famille déménage pour la Suède pour habiter dans l'Eslöv quelques mois après sa naissance.
En 2014, elle présente les Melodifestivalen 2014.
Filmographie
2004 - Generation Y
2005 - Vem är du?—as Åsa
2006 - Beck – Det tysta skriket (téléfilm)
2006 - Elvismackan (court-métrage)
2006 - Tale of Vampires (Frostbiten) : Cornelia
2006 - När mörkret faller
2009-2010: Ballar av stål as Bitterfittan
2010 - Välkommen åter (séries)
2010 - Wallander – Dödsängeln
2011 : The Crown Jewels (Kronjuvelerna) d'Ella Lemhagen : la sage-femme
2011 - Stockholm - Båstad (TV mini-séries, as Sharon)
2012 - Johan Falk: Spelets regler as Lovisa
2012 - Labyrint (TV series)
2014 - Sköterskan: The Nurse as The Nurse (court-métrage)
2014 - Melodifestivalen 2014
Références
Voir aussi
.
.
Liens externes
Naissance en novembre 1987
Naissance au Liban
Animatrice suédoise de télévision
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 2,638
|
BB-8, occasionally spelled Beebee-Ate, had been a BB-series astromech droid who functioned around thirty years following the Battle of Endor. It had a gun mind, very similar to that of R2 string astromech droids, together with the majority of its body composed of a chunk where the droid's head wrapped. BB-8 was mainly white, with a few orange and silver on its entire body, in addition to a shameful photoreceptor. The droid belonged into Resistance pilot Poe Dameron, whom he accompanied through his flights onto his own T-70 X-wing fighter.
Turn 1 wheel in the other side to rotate the comprehensive mind, and then another to start the entrance hatch and expand the'welding torch'. This faithfully replicated LEPIN BB-8 Robot model also has a display rack, decorative reality plaque and little BB-8 figure, which makes it a fantastic display piece for any office or bedroom.
Includes characteristics genuine detailing, also wheel-activated rotating head and opening with welding torch that is welding.
Also contains a display rack, cosmetic fact plaque along with a little BB-8 figure.
This is a 100% LEGO harmonious Star Wars KO set.
Warning: Choking hazard – small parts not for children under 6 years old.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 9,471
|
Has been working since 1990.
IGK Group helps to create a business environment that enables lenders to minimize financial losses due to unpaid debts.
The success of IGK is due to our strong team of specialists and managers who fully understand the credit management field both internationally and locally. The company brings together experienced and responsible professionals. The result is that our clients receive the highest quality service in the field of credit reporting and debt collection and credit insurance.
IGK Holding was registered in Germany.
IGK info and rating helps you defining your company's credit politics towards your partners. The clients of IGK are the leading credit insurance companies Atradius, Euler Hermes, Credendo Group, Sinosure, K-Sure and many others.
Credit insurance — contribution to the security of your business.
IGK debt collection is used by all World leading credit insurers.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 1,402
|
{"url":"https:\/\/socratic.org\/questions\/how-much-work-would-it-take-to-push-a-6-kg-weight-up-a-9-m-plane-that-is-at-an-i","text":"# How much work would it take to push a 6 kg weight up a 9 m plane that is at an incline of pi \/ 4 ?\n\nI found: $374 J$","date":"2020-05-27 17:05:03","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 1, \"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\": 0.6655834913253784, \"perplexity\": 485.6518788549131}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-24\/segments\/1590347394756.31\/warc\/CC-MAIN-20200527141855-20200527171855-00475.warc.gz\"}"}
| null | null |
Escape The Fate's Robert Ortiz talks about Avenged Sevenfold
Escape The Fate's Robert Ortiz spoke of Avenged Sevenfold in a recent interview with Upstatelive.com, you can read an excerpt below:
I was raised as a metal head. My parents were rockers. They raised me with the classics like Ozzy and Black Sabbath, Led Zeppelin to Aerosmith, Guns n' Roses and Metallica and all that awesome '80s metal and then I fell in love with the big four of thrash metal that was all my shit. And that's what influenced me to play metal, and then led to some of the newer bands like Thrice and Avenged Sevenfold. All that stuff that was new when I was in high school. And, now it's almost everything. If it touches me in some way and I like it and it finds its way into everything I do. I always go back to those classic fuckin' rock and metal bands that will always be in my heart.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 9,471
|
#include "includes.h"
#include "common.h"
#include "common/ieee802_11_defs.h"
#include "wps/wps.h"
#include "../config.h"
#include "../wpa_supplicant_i.h"
#include "../bss.h"
#include "dbus_new_helpers.h"
#include "dbus_dict_helpers.h"
#include "dbus_new.h"
#include "dbus_new_handlers.h"
#include "dbus_common.h"
#include "dbus_common_i.h"
#include "dbus_new_handlers_p2p.h"
#include "p2p/p2p.h"
/**
* wpas_dbus_signal_interface - Send a interface related event signal
* @wpa_s: %wpa_supplicant network interface data
* @sig_name: signal name - InterfaceAdded or InterfaceRemoved
* @properties: Whether to add second argument with object properties
*
* Notify listeners about event related with interface
*/
static void wpas_dbus_signal_interface(struct wpa_supplicant *wpa_s,
const char *sig_name, int properties)
{
struct wpas_dbus_priv *iface;
DBusMessage *msg;
DBusMessageIter iter, iter_dict;
iface = wpa_s->global->dbus;
/* Do nothing if the control interface is not turned on */
if (iface == NULL)
return;
msg = dbus_message_new_signal(WPAS_DBUS_NEW_PATH,
WPAS_DBUS_NEW_INTERFACE, sig_name);
if (msg == NULL)
return;
dbus_message_iter_init_append(msg, &iter);
if (!dbus_message_iter_append_basic(&iter, DBUS_TYPE_OBJECT_PATH,
&wpa_s->dbus_new_path))
goto err;
if (properties) {
if (!wpa_dbus_dict_open_write(&iter, &iter_dict))
goto err;
wpa_dbus_get_object_properties(iface, wpa_s->dbus_new_path,
WPAS_DBUS_NEW_IFACE_INTERFACE,
&iter_dict);
if (!wpa_dbus_dict_close_write(&iter, &iter_dict))
goto err;
}
dbus_connection_send(iface->con, msg, NULL);
dbus_message_unref(msg);
return;
err:
wpa_printf(MSG_ERROR, "dbus: Failed to construct signal");
dbus_message_unref(msg);
}
/**
* wpas_dbus_signal_interface_added - Send a interface created signal
* @wpa_s: %wpa_supplicant network interface data
*
* Notify listeners about creating new interface
*/
static void wpas_dbus_signal_interface_added(struct wpa_supplicant *wpa_s)
{
wpas_dbus_signal_interface(wpa_s, "InterfaceAdded", TRUE);
}
/**
* wpas_dbus_signal_interface_removed - Send a interface removed signal
* @wpa_s: %wpa_supplicant network interface data
*
* Notify listeners about removing interface
*/
static void wpas_dbus_signal_interface_removed(struct wpa_supplicant *wpa_s)
{
wpas_dbus_signal_interface(wpa_s, "InterfaceRemoved", FALSE);
}
/**
* wpas_dbus_signal_scan_done - send scan done signal
* @wpa_s: %wpa_supplicant network interface data
* @success: indicates if scanning succeed or failed
*
* Notify listeners about finishing a scan
*/
void wpas_dbus_signal_scan_done(struct wpa_supplicant *wpa_s, int success)
{
struct wpas_dbus_priv *iface;
DBusMessage *msg;
dbus_bool_t succ;
iface = wpa_s->global->dbus;
/* Do nothing if the control interface is not turned on */
if (iface == NULL)
return;
msg = dbus_message_new_signal(wpa_s->dbus_new_path,
WPAS_DBUS_NEW_IFACE_INTERFACE,
"ScanDone");
if (msg == NULL)
return;
succ = success ? TRUE : FALSE;
if (dbus_message_append_args(msg, DBUS_TYPE_BOOLEAN, &succ,
DBUS_TYPE_INVALID))
dbus_connection_send(iface->con, msg, NULL);
else
wpa_printf(MSG_ERROR, "dbus: Failed to construct signal");
dbus_message_unref(msg);
}
/**
* wpas_dbus_signal_blob - Send a BSS related event signal
* @wpa_s: %wpa_supplicant network interface data
* @bss_obj_path: BSS object path
* @sig_name: signal name - BSSAdded or BSSRemoved
* @properties: Whether to add second argument with object properties
*
* Notify listeners about event related with BSS
*/
static void wpas_dbus_signal_bss(struct wpa_supplicant *wpa_s,
const char *bss_obj_path,
const char *sig_name, int properties)
{
struct wpas_dbus_priv *iface;
DBusMessage *msg;
DBusMessageIter iter, iter_dict;
iface = wpa_s->global->dbus;
/* Do nothing if the control interface is not turned on */
if (iface == NULL)
return;
msg = dbus_message_new_signal(wpa_s->dbus_new_path,
WPAS_DBUS_NEW_IFACE_INTERFACE,
sig_name);
if (msg == NULL)
return;
dbus_message_iter_init_append(msg, &iter);
if (!dbus_message_iter_append_basic(&iter, DBUS_TYPE_OBJECT_PATH,
&bss_obj_path))
goto err;
if (properties) {
if (!wpa_dbus_dict_open_write(&iter, &iter_dict))
goto err;
wpa_dbus_get_object_properties(iface, bss_obj_path,
WPAS_DBUS_NEW_IFACE_BSS,
&iter_dict);
if (!wpa_dbus_dict_close_write(&iter, &iter_dict))
goto err;
}
dbus_connection_send(iface->con, msg, NULL);
dbus_message_unref(msg);
return;
err:
wpa_printf(MSG_ERROR, "dbus: Failed to construct signal");
dbus_message_unref(msg);
}
/**
* wpas_dbus_signal_bss_added - Send a BSS added signal
* @wpa_s: %wpa_supplicant network interface data
* @bss_obj_path: new BSS object path
*
* Notify listeners about adding new BSS
*/
static void wpas_dbus_signal_bss_added(struct wpa_supplicant *wpa_s,
const char *bss_obj_path)
{
wpas_dbus_signal_bss(wpa_s, bss_obj_path, "BSSAdded", TRUE);
}
/**
* wpas_dbus_signal_bss_removed - Send a BSS removed signal
* @wpa_s: %wpa_supplicant network interface data
* @bss_obj_path: BSS object path
*
* Notify listeners about removing BSS
*/
static void wpas_dbus_signal_bss_removed(struct wpa_supplicant *wpa_s,
const char *bss_obj_path)
{
wpas_dbus_signal_bss(wpa_s, bss_obj_path, "BSSRemoved", FALSE);
}
/**
* wpas_dbus_signal_blob - Send a blob related event signal
* @wpa_s: %wpa_supplicant network interface data
* @name: blob name
* @sig_name: signal name - BlobAdded or BlobRemoved
*
* Notify listeners about event related with blob
*/
static void wpas_dbus_signal_blob(struct wpa_supplicant *wpa_s,
const char *name, const char *sig_name)
{
struct wpas_dbus_priv *iface;
DBusMessage *msg;
iface = wpa_s->global->dbus;
/* Do nothing if the control interface is not turned on */
if (iface == NULL)
return;
msg = dbus_message_new_signal(wpa_s->dbus_new_path,
WPAS_DBUS_NEW_IFACE_INTERFACE,
sig_name);
if (msg == NULL)
return;
if (dbus_message_append_args(msg, DBUS_TYPE_STRING, &name,
DBUS_TYPE_INVALID))
dbus_connection_send(iface->con, msg, NULL);
else
wpa_printf(MSG_ERROR, "dbus: Failed to construct signal");
dbus_message_unref(msg);
}
/**
* wpas_dbus_signal_blob_added - Send a blob added signal
* @wpa_s: %wpa_supplicant network interface data
* @name: blob name
*
* Notify listeners about adding a new blob
*/
void wpas_dbus_signal_blob_added(struct wpa_supplicant *wpa_s,
const char *name)
{
wpas_dbus_signal_blob(wpa_s, name, "BlobAdded");
}
/**
* wpas_dbus_signal_blob_removed - Send a blob removed signal
* @wpa_s: %wpa_supplicant network interface data
* @name: blob name
*
* Notify listeners about removing blob
*/
void wpas_dbus_signal_blob_removed(struct wpa_supplicant *wpa_s,
const char *name)
{
wpas_dbus_signal_blob(wpa_s, name, "BlobRemoved");
}
/**
* wpas_dbus_signal_network - Send a network related event signal
* @wpa_s: %wpa_supplicant network interface data
* @id: new network id
* @sig_name: signal name - NetworkAdded, NetworkRemoved or NetworkSelected
* @properties: determines if add second argument with object properties
*
* Notify listeners about event related with configured network
*/
static void wpas_dbus_signal_network(struct wpa_supplicant *wpa_s,
int id, const char *sig_name,
int properties)
{
struct wpas_dbus_priv *iface;
DBusMessage *msg;
DBusMessageIter iter, iter_dict;
char net_obj_path[WPAS_DBUS_OBJECT_PATH_MAX], *path;
iface = wpa_s->global->dbus;
/* Do nothing if the control interface is not turned on */
if (iface == NULL)
return;
os_snprintf(net_obj_path, WPAS_DBUS_OBJECT_PATH_MAX,
"%s/" WPAS_DBUS_NEW_NETWORKS_PART "/%u",
wpa_s->dbus_new_path, id);
msg = dbus_message_new_signal(wpa_s->dbus_new_path,
WPAS_DBUS_NEW_IFACE_INTERFACE,
sig_name);
if (msg == NULL)
return;
dbus_message_iter_init_append(msg, &iter);
path = net_obj_path;
if (!dbus_message_iter_append_basic(&iter, DBUS_TYPE_OBJECT_PATH,
&path))
goto err;
if (properties) {
if (!wpa_dbus_dict_open_write(&iter, &iter_dict))
goto err;
wpa_dbus_get_object_properties(iface, net_obj_path,
WPAS_DBUS_NEW_IFACE_NETWORK,
&iter_dict);
if (!wpa_dbus_dict_close_write(&iter, &iter_dict))
goto err;
}
dbus_connection_send(iface->con, msg, NULL);
dbus_message_unref(msg);
return;
err:
wpa_printf(MSG_ERROR, "dbus: Failed to construct signal");
dbus_message_unref(msg);
}
/**
* wpas_dbus_signal_network_added - Send a network added signal
* @wpa_s: %wpa_supplicant network interface data
* @id: new network id
*
* Notify listeners about adding new network
*/
static void wpas_dbus_signal_network_added(struct wpa_supplicant *wpa_s,
int id)
{
wpas_dbus_signal_network(wpa_s, id, "NetworkAdded", TRUE);
}
/**
* wpas_dbus_signal_network_removed - Send a network removed signal
* @wpa_s: %wpa_supplicant network interface data
* @id: network id
*
* Notify listeners about removing a network
*/
static void wpas_dbus_signal_network_removed(struct wpa_supplicant *wpa_s,
int id)
{
wpas_dbus_signal_network(wpa_s, id, "NetworkRemoved", FALSE);
}
/**
* wpas_dbus_signal_network_selected - Send a network selected signal
* @wpa_s: %wpa_supplicant network interface data
* @id: network id
*
* Notify listeners about selecting a network
*/
void wpas_dbus_signal_network_selected(struct wpa_supplicant *wpa_s, int id)
{
wpas_dbus_signal_network(wpa_s, id, "NetworkSelected", FALSE);
}
/**
* wpas_dbus_signal_network_enabled_changed - Signals Enabled property changes
* @wpa_s: %wpa_supplicant network interface data
* @ssid: configured network which Enabled property has changed
*
* Sends PropertyChanged signals containing new value of Enabled property
* for specified network
*/
void wpas_dbus_signal_network_enabled_changed(struct wpa_supplicant *wpa_s,
struct wpa_ssid *ssid)
{
char path[WPAS_DBUS_OBJECT_PATH_MAX];
os_snprintf(path, WPAS_DBUS_OBJECT_PATH_MAX,
"%s/" WPAS_DBUS_NEW_NETWORKS_PART "/%d",
wpa_s->dbus_new_path, ssid->id);
wpa_dbus_mark_property_changed(wpa_s->global->dbus, path,
WPAS_DBUS_NEW_IFACE_NETWORK, "Enabled");
}
#ifdef CONFIG_WPS
/**
* wpas_dbus_signal_wps_event_success - Signals Success WPS event
* @wpa_s: %wpa_supplicant network interface data
*
* Sends Event dbus signal with name "success" and empty dict as arguments
*/
void wpas_dbus_signal_wps_event_success(struct wpa_supplicant *wpa_s)
{
DBusMessage *msg;
DBusMessageIter iter, dict_iter;
struct wpas_dbus_priv *iface;
char *key = "success";
iface = wpa_s->global->dbus;
/* Do nothing if the control interface is not turned on */
if (iface == NULL)
return;
msg = dbus_message_new_signal(wpa_s->dbus_new_path,
WPAS_DBUS_NEW_IFACE_WPS, "Event");
if (msg == NULL)
return;
dbus_message_iter_init_append(msg, &iter);
if (!dbus_message_iter_append_basic(&iter, DBUS_TYPE_STRING, &key) ||
!wpa_dbus_dict_open_write(&iter, &dict_iter) ||
!wpa_dbus_dict_close_write(&iter, &dict_iter))
wpa_printf(MSG_ERROR, "dbus: Failed to construct signal");
else
dbus_connection_send(iface->con, msg, NULL);
dbus_message_unref(msg);
}
/**
* wpas_dbus_signal_wps_event_fail - Signals Fail WPS event
* @wpa_s: %wpa_supplicant network interface data
*
* Sends Event dbus signal with name "fail" and dictionary containing
* "msg field with fail message number (int32) as arguments
*/
void wpas_dbus_signal_wps_event_fail(struct wpa_supplicant *wpa_s,
struct wps_event_fail *fail)
{
DBusMessage *msg;
DBusMessageIter iter, dict_iter;
struct wpas_dbus_priv *iface;
char *key = "fail";
iface = wpa_s->global->dbus;
/* Do nothing if the control interface is not turned on */
if (iface == NULL)
return;
msg = dbus_message_new_signal(wpa_s->dbus_new_path,
WPAS_DBUS_NEW_IFACE_WPS, "Event");
if (msg == NULL)
return;
dbus_message_iter_init_append(msg, &iter);
if (!dbus_message_iter_append_basic(&iter, DBUS_TYPE_STRING, &key) ||
!wpa_dbus_dict_open_write(&iter, &dict_iter) ||
!wpa_dbus_dict_append_int32(&dict_iter, "msg", fail->msg) ||
!wpa_dbus_dict_close_write(&iter, &dict_iter))
wpa_printf(MSG_ERROR, "dbus: Failed to construct signal");
else
dbus_connection_send(iface->con, msg, NULL);
dbus_message_unref(msg);
}
/**
* wpas_dbus_signal_wps_event_m2d - Signals M2D WPS event
* @wpa_s: %wpa_supplicant network interface data
*
* Sends Event dbus signal with name "m2d" and dictionary containing
* fields of wps_event_m2d structure.
*/
void wpas_dbus_signal_wps_event_m2d(struct wpa_supplicant *wpa_s,
struct wps_event_m2d *m2d)
{
DBusMessage *msg;
DBusMessageIter iter, dict_iter;
struct wpas_dbus_priv *iface;
char *key = "m2d";
iface = wpa_s->global->dbus;
/* Do nothing if the control interface is not turned on */
if (iface == NULL)
return;
msg = dbus_message_new_signal(wpa_s->dbus_new_path,
WPAS_DBUS_NEW_IFACE_WPS, "Event");
if (msg == NULL)
return;
dbus_message_iter_init_append(msg, &iter);
if (!dbus_message_iter_append_basic(&iter, DBUS_TYPE_STRING, &key) ||
!wpa_dbus_dict_open_write(&iter, &dict_iter) ||
!wpa_dbus_dict_append_uint16(&dict_iter, "config_methods",
m2d->config_methods) ||
!wpa_dbus_dict_append_byte_array(&dict_iter, "manufacturer",
(const char *) m2d->manufacturer,
m2d->manufacturer_len) ||
!wpa_dbus_dict_append_byte_array(&dict_iter, "model_name",
(const char *) m2d->model_name,
m2d->model_name_len) ||
!wpa_dbus_dict_append_byte_array(&dict_iter, "model_number",
(const char *) m2d->model_number,
m2d->model_number_len) ||
!wpa_dbus_dict_append_byte_array(&dict_iter, "serial_number",
(const char *)
m2d->serial_number,
m2d->serial_number_len) ||
!wpa_dbus_dict_append_byte_array(&dict_iter, "dev_name",
(const char *) m2d->dev_name,
m2d->dev_name_len) ||
!wpa_dbus_dict_append_byte_array(&dict_iter, "primary_dev_type",
(const char *)
m2d->primary_dev_type, 8) ||
!wpa_dbus_dict_append_uint16(&dict_iter, "config_error",
m2d->config_error) ||
!wpa_dbus_dict_append_uint16(&dict_iter, "dev_password_id",
m2d->dev_password_id) ||
!wpa_dbus_dict_close_write(&iter, &dict_iter))
wpa_printf(MSG_ERROR, "dbus: Failed to construct signal");
else
dbus_connection_send(iface->con, msg, NULL);
dbus_message_unref(msg);
}
/**
* wpas_dbus_signal_wps_cred - Signals new credentials
* @wpa_s: %wpa_supplicant network interface data
*
* Sends signal with credentials in directory argument
*/
void wpas_dbus_signal_wps_cred(struct wpa_supplicant *wpa_s,
const struct wps_credential *cred)
{
DBusMessage *msg;
DBusMessageIter iter, dict_iter;
struct wpas_dbus_priv *iface;
char *auth_type[6]; /* we have six possible authorization types */
int at_num = 0;
char *encr_type[4]; /* we have four possible encryption types */
int et_num = 0;
iface = wpa_s->global->dbus;
/* Do nothing if the control interface is not turned on */
if (iface == NULL)
return;
msg = dbus_message_new_signal(wpa_s->dbus_new_path,
WPAS_DBUS_NEW_IFACE_WPS,
"Credentials");
if (msg == NULL)
return;
dbus_message_iter_init_append(msg, &iter);
if (!wpa_dbus_dict_open_write(&iter, &dict_iter))
goto nomem;
if (cred->auth_type & WPS_AUTH_OPEN)
auth_type[at_num++] = "open";
if (cred->auth_type & WPS_AUTH_WPAPSK)
auth_type[at_num++] = "wpa-psk";
if (cred->auth_type & WPS_AUTH_SHARED)
auth_type[at_num++] = "shared";
if (cred->auth_type & WPS_AUTH_WPA)
auth_type[at_num++] = "wpa-eap";
if (cred->auth_type & WPS_AUTH_WPA2)
auth_type[at_num++] = "wpa2-eap";
if (cred->auth_type & WPS_AUTH_WPA2PSK)
auth_type[at_num++] =
"wpa2-psk";
if (cred->encr_type & WPS_ENCR_NONE)
encr_type[et_num++] = "none";
if (cred->encr_type & WPS_ENCR_WEP)
encr_type[et_num++] = "wep";
if (cred->encr_type & WPS_ENCR_TKIP)
encr_type[et_num++] = "tkip";
if (cred->encr_type & WPS_ENCR_AES)
encr_type[et_num++] = "aes";
if (wpa_s->current_ssid) {
if (!wpa_dbus_dict_append_byte_array(
&dict_iter, "BSSID",
(const char *) wpa_s->current_ssid->bssid,
ETH_ALEN))
goto nomem;
}
if (!wpa_dbus_dict_append_byte_array(&dict_iter, "SSID",
(const char *) cred->ssid,
cred->ssid_len) ||
!wpa_dbus_dict_append_string_array(&dict_iter, "AuthType",
(const char **) auth_type,
at_num) ||
!wpa_dbus_dict_append_string_array(&dict_iter, "EncrType",
(const char **) encr_type,
et_num) ||
!wpa_dbus_dict_append_byte_array(&dict_iter, "Key",
(const char *) cred->key,
cred->key_len) ||
!wpa_dbus_dict_append_uint32(&dict_iter, "KeyIndex",
cred->key_idx) ||
!wpa_dbus_dict_close_write(&iter, &dict_iter))
goto nomem;
dbus_connection_send(iface->con, msg, NULL);
nomem:
dbus_message_unref(msg);
}
#endif /* CONFIG_WPS */
void wpas_dbus_signal_certification(struct wpa_supplicant *wpa_s,
int depth, const char *subject,
const char *cert_hash,
const struct wpabuf *cert)
{
struct wpas_dbus_priv *iface;
DBusMessage *msg;
DBusMessageIter iter, dict_iter;
iface = wpa_s->global->dbus;
/* Do nothing if the control interface is not turned on */
if (iface == NULL)
return;
msg = dbus_message_new_signal(wpa_s->dbus_new_path,
WPAS_DBUS_NEW_IFACE_INTERFACE,
"Certification");
if (msg == NULL)
return;
dbus_message_iter_init_append(msg, &iter);
if (!wpa_dbus_dict_open_write(&iter, &dict_iter))
goto nomem;
if (!wpa_dbus_dict_append_uint32(&dict_iter, "depth", depth) ||
!wpa_dbus_dict_append_string(&dict_iter, "subject", subject))
goto nomem;
if (cert_hash &&
!wpa_dbus_dict_append_string(&dict_iter, "cert_hash", cert_hash))
goto nomem;
if (cert &&
!wpa_dbus_dict_append_byte_array(&dict_iter, "cert",
wpabuf_head(cert),
wpabuf_len(cert)))
goto nomem;
if (!wpa_dbus_dict_close_write(&iter, &dict_iter))
goto nomem;
dbus_connection_send(iface->con, msg, NULL);
nomem:
dbus_message_unref(msg);
}
#ifdef CONFIG_P2P
/**
* wpas_dbus_signal_p2p_group_removed - Signals P2P group was removed
* @wpa_s: %wpa_supplicant network interface data
* @role: role of this device (client or GO)
* Sends signal with i/f name and role as string arguments
*/
void wpas_dbus_signal_p2p_group_removed(struct wpa_supplicant *wpa_s,
const char *role)
{
DBusMessage *msg;
DBusMessageIter iter;
struct wpas_dbus_priv *iface = wpa_s->global->dbus;
char *ifname = wpa_s->ifname;
/* Do nothing if the control interface is not turned on */
if (iface == NULL)
return;
msg = dbus_message_new_signal(wpa_s->dbus_new_path,
WPAS_DBUS_NEW_IFACE_P2PDEVICE,
"GroupFinished");
if (msg == NULL)
return;
dbus_message_iter_init_append(msg, &iter);
if (!dbus_message_iter_append_basic(&iter, DBUS_TYPE_STRING, &ifname)) {
wpa_printf(MSG_ERROR, "dbus: Failed to construct GroupFinished"
"signal -not enough memory for ifname ");
goto err;
}
if (!dbus_message_iter_append_basic(&iter, DBUS_TYPE_STRING, &role))
wpa_printf(MSG_ERROR, "dbus: Failed to construct GroupFinished"
"signal -not enough memory for role ");
else
dbus_connection_send(iface->con, msg, NULL);
err:
dbus_message_unref(msg);
}
/**
* wpas_dbus_signal_p2p_provision_discovery - Signals various PD events
*
* @dev_addr - who sent the request or responded to our request.
* @request - Will be 1 if request, 0 for response.
* @status - valid only in case of response
* @config_methods - wps config methods
* @generated_pin - pin to be displayed in case of WPS_CONFIG_DISPLAY method
*
* Sends following provision discovery related events:
* ProvisionDiscoveryRequestDisplayPin
* ProvisionDiscoveryResponseDisplayPin
* ProvisionDiscoveryRequestEnterPin
* ProvisionDiscoveryResponseEnterPin
* ProvisionDiscoveryPBCRequest
* ProvisionDiscoveryPBCResponse
*
* TODO::
* ProvisionDiscoveryFailure (timeout case)
*/
void wpas_dbus_signal_p2p_provision_discovery(struct wpa_supplicant *wpa_s,
const u8 *dev_addr, int request,
enum p2p_prov_disc_status status,
u16 config_methods,
unsigned int generated_pin)
{
DBusMessage *msg;
DBusMessageIter iter;
struct wpas_dbus_priv *iface;
char *_signal;
int add_pin = 0;
char peer_obj_path[WPAS_DBUS_OBJECT_PATH_MAX], *path;
int error_ret = 1;
char pin[9], *p_pin = NULL;
iface = wpa_s->global->dbus;
/* Do nothing if the control interface is not turned on */
if (iface == NULL)
return;
if (request || !status) {
if (config_methods & WPS_CONFIG_DISPLAY)
_signal = request ?
"ProvisionDiscoveryRequestDisplayPin" :
"ProvisionDiscoveryResponseEnterPin";
else if (config_methods & WPS_CONFIG_KEYPAD)
_signal = request ?
"ProvisionDiscoveryRequestEnterPin" :
"ProvisionDiscoveryResponseDisplayPin";
else if (config_methods & WPS_CONFIG_PUSHBUTTON)
_signal = request ? "ProvisionDiscoveryPBCRequest" :
"ProvisionDiscoveryPBCResponse";
else
return; /* Unknown or un-supported method */
} else if (!request && status)
/* Explicit check for failure response */
_signal = "ProvisionDiscoveryFailure";
add_pin = ((request && (config_methods & WPS_CONFIG_DISPLAY)) ||
(!request && !status &&
(config_methods & WPS_CONFIG_KEYPAD)));
if (add_pin) {
os_snprintf(pin, sizeof(pin), "%08d", generated_pin);
p_pin = pin;
}
msg = dbus_message_new_signal(wpa_s->dbus_new_path,
WPAS_DBUS_NEW_IFACE_P2PDEVICE, _signal);
if (msg == NULL)
return;
/* Check if this is a known peer */
if (p2p_get_peer_info(wpa_s->global->p2p, dev_addr, 0, NULL, 0) < 0)
goto error;
os_snprintf(peer_obj_path, WPAS_DBUS_OBJECT_PATH_MAX,
"%s/" WPAS_DBUS_NEW_P2P_PEERS_PART "/"
COMPACT_MACSTR,
wpa_s->dbus_new_path, MAC2STR(dev_addr));
path = peer_obj_path;
dbus_message_iter_init_append(msg, &iter);
if (!dbus_message_iter_append_basic(&iter,
DBUS_TYPE_OBJECT_PATH,
&path))
goto error;
if (!request && status)
/* Attach status to ProvisionDiscoveryFailure */
error_ret = !dbus_message_iter_append_basic(&iter,
DBUS_TYPE_INT32,
&status);
else
error_ret = (add_pin &&
!dbus_message_iter_append_basic(&iter,
DBUS_TYPE_STRING,
&p_pin));
error:
if (!error_ret)
dbus_connection_send(iface->con, msg, NULL);
else
wpa_printf(MSG_ERROR, "dbus: Failed to construct signal");
dbus_message_unref(msg);
}
void wpas_dbus_signal_p2p_go_neg_req(struct wpa_supplicant *wpa_s,
const u8 *src, u16 dev_passwd_id)
{
DBusMessage *msg;
DBusMessageIter iter;
struct wpas_dbus_priv *iface;
char peer_obj_path[WPAS_DBUS_OBJECT_PATH_MAX], *path;
iface = wpa_s->global->dbus;
/* Do nothing if the control interface is not turned on */
if (iface == NULL)
return;
os_snprintf(peer_obj_path, WPAS_DBUS_OBJECT_PATH_MAX,
"%s/" WPAS_DBUS_NEW_P2P_PEERS_PART "/" COMPACT_MACSTR,
wpa_s->dbus_new_path, MAC2STR(src));
path = peer_obj_path;
msg = dbus_message_new_signal(wpa_s->dbus_new_path,
WPAS_DBUS_NEW_IFACE_P2PDEVICE,
"GONegotiationRequest");
if (msg == NULL)
return;
dbus_message_iter_init_append(msg, &iter);
if (!dbus_message_iter_append_basic(&iter, DBUS_TYPE_OBJECT_PATH,
&path) ||
!dbus_message_iter_append_basic(&iter, DBUS_TYPE_UINT16,
&dev_passwd_id))
wpa_printf(MSG_ERROR, "dbus: Failed to construct signal");
else
dbus_connection_send(iface->con, msg, NULL);
dbus_message_unref(msg);
}
static int wpas_dbus_get_group_obj_path(struct wpa_supplicant *wpa_s,
const struct wpa_ssid *ssid,
char *group_obj_path)
{
char group_name[3];
if (os_memcmp(ssid->ssid, P2P_WILDCARD_SSID, P2P_WILDCARD_SSID_LEN))
return -1;
memcpy(group_name, ssid->ssid + P2P_WILDCARD_SSID_LEN, 2);
group_name[2] = '\0';
os_snprintf(group_obj_path, WPAS_DBUS_OBJECT_PATH_MAX,
"%s/" WPAS_DBUS_NEW_P2P_GROUPS_PART "/%s",
wpa_s->dbus_new_path, group_name);
return 0;
}
/**
* wpas_dbus_signal_p2p_group_started - Signals P2P group has
* started.Emitted when a group is succesfully started
* irrespective of the role (client/GO) of the current device
*
* @wpa_s: %wpa_supplicant network interface data
* @ssid: SSID object
* @client: this device is P2P client
* @network_id: network id of the group started, use instead of ssid->id
* to account for persistent groups
*/
void wpas_dbus_signal_p2p_group_started(struct wpa_supplicant *wpa_s,
const struct wpa_ssid *ssid,
int client, int network_id)
{
DBusMessage *msg;
DBusMessageIter iter, dict_iter;
struct wpas_dbus_priv *iface;
char net_obj_path[WPAS_DBUS_OBJECT_PATH_MAX];
char group_obj_path[WPAS_DBUS_OBJECT_PATH_MAX];
iface = wpa_s->parent->global->dbus;
/* Do nothing if the control interface is not turned on */
if (iface == NULL)
return;
if (wpas_dbus_get_group_obj_path(wpa_s, ssid, group_obj_path) < 0)
return;
/* New interface has been created for this group */
msg = dbus_message_new_signal(wpa_s->parent->dbus_new_path,
WPAS_DBUS_NEW_IFACE_P2PDEVICE,
"GroupStarted");
if (msg == NULL)
return;
dbus_message_iter_init_append(msg, &iter);
if (!wpa_dbus_dict_open_write(&iter, &dict_iter))
goto nomem;
/*
* In case the device supports creating a separate interface the
* DBus client will need to know the object path for the interface
* object this group was created on, so include it here.
*/
if (!wpa_dbus_dict_append_object_path(&dict_iter,
"interface_object",
wpa_s->dbus_new_path))
goto nomem;
if (!wpa_dbus_dict_append_string(&dict_iter, "role",
client ? "client" : "GO"))
goto nomem;
os_snprintf(net_obj_path, WPAS_DBUS_OBJECT_PATH_MAX,
"%s/" WPAS_DBUS_NEW_NETWORKS_PART "/%u",
wpa_s->parent->dbus_new_path, network_id);
if (!wpa_dbus_dict_append_object_path(&dict_iter, "group_object",
group_obj_path) ||
!wpa_dbus_dict_append_object_path(&dict_iter, "network_object",
net_obj_path) ||
!wpa_dbus_dict_close_write(&iter, &dict_iter))
goto nomem;
dbus_connection_send(iface->con, msg, NULL);
nomem:
dbus_message_unref(msg);
}
/**
*
* Method to emit GONeogtiation Success or Failure signals based
* on status.
* @status: Status of the GO neg request. 0 for success, other for errors.
*/
void wpas_dbus_signal_p2p_go_neg_resp(struct wpa_supplicant *wpa_s, int status)
{
DBusMessage *msg;
DBusMessageIter iter;
struct wpas_dbus_priv *iface;
iface = wpa_s->global->dbus;
/* Do nothing if the control interface is not turned on */
if (iface == NULL)
return;
msg = dbus_message_new_signal(wpa_s->dbus_new_path,
WPAS_DBUS_NEW_IFACE_P2PDEVICE,
status ? "GONegotiationFailure" :
"GONegotiationSuccess");
if (msg == NULL)
return;
if (status) {
dbus_message_iter_init_append(msg, &iter);
if (!dbus_message_iter_append_basic(&iter, DBUS_TYPE_INT32,
&status)) {
wpa_printf(MSG_ERROR,
"dbus: Failed to construct signal");
goto err;
}
}
dbus_connection_send(iface->con, msg, NULL);
err:
dbus_message_unref(msg);
}
/**
*
* Method to emit Invitation Result signal based on status and
* bssid
* @status: Status of the Invite request. 0 for success, other
* for errors
* @bssid : Basic Service Set Identifier
*/
void wpas_dbus_signal_p2p_invitation_result(struct wpa_supplicant *wpa_s,
int status, const u8 *bssid)
{
DBusMessage *msg;
DBusMessageIter iter, dict_iter;
struct wpas_dbus_priv *iface;
wpa_printf(MSG_INFO, "%s\n", __func__);
iface = wpa_s->global->dbus;
/* Do nothing if the control interface is not turned on */
if (iface == NULL)
return;
msg = dbus_message_new_signal(wpa_s->dbus_new_path,
WPAS_DBUS_NEW_IFACE_P2PDEVICE,
"InvitationResult");
if (msg == NULL)
return;
dbus_message_iter_init_append(msg, &iter);
if (!wpa_dbus_dict_open_write(&iter, &dict_iter))
goto nomem;
if (!wpa_dbus_dict_append_int32(&dict_iter, "status", status))
goto nomem;
if (bssid) {
if (!wpa_dbus_dict_append_byte_array(&dict_iter, "BSSID",
(const char *) bssid,
ETH_ALEN))
goto nomem;
}
if (!wpa_dbus_dict_close_write(&iter, &dict_iter))
goto nomem;
dbus_connection_send(iface->con, msg, NULL);
nomem:
dbus_message_unref(msg);
}
/**
*
* Method to emit a signal for a peer joining the group.
* The signal will carry path to the group member object
* constructed using p2p i/f addr used for connecting.
*
* @wpa_s: %wpa_supplicant network interface data
* @member_addr: addr (p2p i/f) of the peer joining the group
*/
void wpas_dbus_signal_p2p_peer_joined(struct wpa_supplicant *wpa_s,
const u8 *member)
{
struct wpas_dbus_priv *iface;
DBusMessage *msg;
DBusMessageIter iter;
char groupmember_obj_path[WPAS_DBUS_OBJECT_PATH_MAX], *path;
iface = wpa_s->global->dbus;
/* Do nothing if the control interface is not turned on */
if (iface == NULL)
return;
if (!wpa_s->dbus_groupobj_path)
return;
os_snprintf(groupmember_obj_path, WPAS_DBUS_OBJECT_PATH_MAX,
"%s/" WPAS_DBUS_NEW_P2P_GROUPMEMBERS_PART "/"
COMPACT_MACSTR,
wpa_s->dbus_groupobj_path, MAC2STR(member));
msg = dbus_message_new_signal(wpa_s->dbus_groupobj_path,
WPAS_DBUS_NEW_IFACE_P2P_GROUP,
"PeerJoined");
if (msg == NULL)
return;
dbus_message_iter_init_append(msg, &iter);
path = groupmember_obj_path;
if (!dbus_message_iter_append_basic(&iter, DBUS_TYPE_OBJECT_PATH,
&path))
goto err;
dbus_connection_send(iface->con, msg, NULL);
dbus_message_unref(msg);
return;
err:
wpa_printf(MSG_ERROR, "dbus: Failed to construct signal");
dbus_message_unref(msg);
}
/**
*
* Method to emit a signal for a peer disconnecting the group.
* The signal will carry path to the group member object
* constructed using p2p i/f addr used for connecting.
*
* @wpa_s: %wpa_supplicant network interface data
* @member_addr: addr (p2p i/f) of the peer joining the group
*/
void wpas_dbus_signal_p2p_peer_disconnected(struct wpa_supplicant *wpa_s,
const u8 *member)
{
struct wpas_dbus_priv *iface;
DBusMessage *msg;
DBusMessageIter iter;
char groupmember_obj_path[WPAS_DBUS_OBJECT_PATH_MAX], *path;
iface = wpa_s->global->dbus;
/* Do nothing if the control interface is not turned on */
if (iface == NULL)
return;
if (!wpa_s->dbus_groupobj_path)
return;
os_snprintf(groupmember_obj_path, WPAS_DBUS_OBJECT_PATH_MAX,
"%s/" WPAS_DBUS_NEW_P2P_GROUPMEMBERS_PART "/"
COMPACT_MACSTR,
wpa_s->dbus_groupobj_path, MAC2STR(member));
msg = dbus_message_new_signal(wpa_s->dbus_groupobj_path,
WPAS_DBUS_NEW_IFACE_P2P_GROUP,
"PeerDisconnected");
if (msg == NULL)
return;
dbus_message_iter_init_append(msg, &iter);
path = groupmember_obj_path;
if (!dbus_message_iter_append_basic(&iter, DBUS_TYPE_OBJECT_PATH,
&path))
goto err;
dbus_connection_send(iface->con, msg, NULL);
dbus_message_unref(msg);
return;
err:
wpa_printf(MSG_ERROR, "dbus: Failed to construct PeerDisconnected "
"signal");
dbus_message_unref(msg);
}
/**
*
* Method to emit a signal for a service discovery request.
* The signal will carry station address, frequency, dialog token,
* update indicator and it tlvs
*
* @wpa_s: %wpa_supplicant network interface data
* @sa: station addr (p2p i/f) of the peer
* @dialog_token: service discovery request dialog token
* @update_indic: service discovery request update indicator
* @tlvs: service discovery request genrated byte array of tlvs
* @tlvs_len: service discovery request tlvs length
*/
void wpas_dbus_signal_p2p_sd_request(struct wpa_supplicant *wpa_s,
int freq, const u8 *sa, u8 dialog_token,
u16 update_indic, const u8 *tlvs,
size_t tlvs_len)
{
DBusMessage *msg;
DBusMessageIter iter, dict_iter;
struct wpas_dbus_priv *iface;
char peer_obj_path[WPAS_DBUS_OBJECT_PATH_MAX], *path;
iface = wpa_s->global->dbus;
/* Do nothing if the control interface is not turned on */
if (iface == NULL)
return;
msg = dbus_message_new_signal(wpa_s->dbus_new_path,
WPAS_DBUS_NEW_IFACE_P2PDEVICE,
"ServiceDiscoveryRequest");
if (msg == NULL)
return;
/* Check if this is a known peer */
if (p2p_get_peer_info(wpa_s->global->p2p, sa, 0, NULL, 0) < 0)
goto error;
os_snprintf(peer_obj_path, WPAS_DBUS_OBJECT_PATH_MAX,
"%s/" WPAS_DBUS_NEW_P2P_PEERS_PART "/"
COMPACT_MACSTR, wpa_s->dbus_new_path, MAC2STR(sa));
path = peer_obj_path;
dbus_message_iter_init_append(msg, &iter);
if (!wpa_dbus_dict_open_write(&iter, &dict_iter))
goto error;
if (!wpa_dbus_dict_append_object_path(&dict_iter, "peer_object",
path) ||
!wpa_dbus_dict_append_int32(&dict_iter, "frequency", freq) ||
!wpa_dbus_dict_append_int32(&dict_iter, "dialog_token",
dialog_token) ||
!wpa_dbus_dict_append_uint16(&dict_iter, "update_indicator",
update_indic) ||
!wpa_dbus_dict_append_byte_array(&dict_iter, "tlvs",
(const char *) tlvs,
tlvs_len) ||
!wpa_dbus_dict_close_write(&iter, &dict_iter))
goto error;
dbus_connection_send(iface->con, msg, NULL);
dbus_message_unref(msg);
return;
error:
wpa_printf(MSG_ERROR, "dbus: Failed to construct signal");
dbus_message_unref(msg);
}
/**
*
* Method to emit a signal for a service discovery response.
* The signal will carry station address, update indicator and it
* tlvs
*
* @wpa_s: %wpa_supplicant network interface data
* @sa: station addr (p2p i/f) of the peer
* @update_indic: service discovery request update indicator
* @tlvs: service discovery request genrated byte array of tlvs
* @tlvs_len: service discovery request tlvs length
*/
void wpas_dbus_signal_p2p_sd_response(struct wpa_supplicant *wpa_s,
const u8 *sa, u16 update_indic,
const u8 *tlvs, size_t tlvs_len)
{
DBusMessage *msg;
DBusMessageIter iter, dict_iter;
struct wpas_dbus_priv *iface;
char peer_obj_path[WPAS_DBUS_OBJECT_PATH_MAX], *path;
iface = wpa_s->global->dbus;
/* Do nothing if the control interface is not turned on */
if (iface == NULL)
return;
msg = dbus_message_new_signal(wpa_s->dbus_new_path,
WPAS_DBUS_NEW_IFACE_P2PDEVICE,
"ServiceDiscoveryResponse");
if (msg == NULL)
return;
/* Check if this is a known peer */
if (p2p_get_peer_info(wpa_s->global->p2p, sa, 0, NULL, 0) < 0)
goto error;
os_snprintf(peer_obj_path, WPAS_DBUS_OBJECT_PATH_MAX,
"%s/" WPAS_DBUS_NEW_P2P_PEERS_PART "/"
COMPACT_MACSTR, wpa_s->dbus_new_path, MAC2STR(sa));
path = peer_obj_path;
dbus_message_iter_init_append(msg, &iter);
if (!wpa_dbus_dict_open_write(&iter, &dict_iter))
goto error;
if (!wpa_dbus_dict_append_object_path(&dict_iter, "peer_object",
path) ||
!wpa_dbus_dict_append_uint16(&dict_iter, "update_indicator",
update_indic) ||
!wpa_dbus_dict_append_byte_array(&dict_iter, "tlvs",
(const char *) tlvs,
tlvs_len) ||
!wpa_dbus_dict_close_write(&iter, &dict_iter))
goto error;
dbus_connection_send(iface->con, msg, NULL);
dbus_message_unref(msg);
return;
error:
wpa_printf(MSG_ERROR, "dbus: Failed to construct signal");
dbus_message_unref(msg);
}
/**
* wpas_dbus_signal_persistent_group - Send a persistent group related
* event signal
* @wpa_s: %wpa_supplicant network interface data
* @id: new persistent group id
* @sig_name: signal name - PersistentGroupAdded, PersistentGroupRemoved
* @properties: determines if add second argument with object properties
*
* Notify listeners about an event related to persistent groups.
*/
static void wpas_dbus_signal_persistent_group(struct wpa_supplicant *wpa_s,
int id, const char *sig_name,
int properties)
{
struct wpas_dbus_priv *iface;
DBusMessage *msg;
DBusMessageIter iter, iter_dict;
char pgrp_obj_path[WPAS_DBUS_OBJECT_PATH_MAX], *path;
iface = wpa_s->global->dbus;
/* Do nothing if the control interface is not turned on */
if (iface == NULL)
return;
os_snprintf(pgrp_obj_path, WPAS_DBUS_OBJECT_PATH_MAX,
"%s/" WPAS_DBUS_NEW_PERSISTENT_GROUPS_PART "/%u",
wpa_s->dbus_new_path, id);
msg = dbus_message_new_signal(wpa_s->dbus_new_path,
WPAS_DBUS_NEW_IFACE_P2PDEVICE,
sig_name);
if (msg == NULL)
return;
dbus_message_iter_init_append(msg, &iter);
path = pgrp_obj_path;
if (!dbus_message_iter_append_basic(&iter, DBUS_TYPE_OBJECT_PATH,
&path))
goto err;
if (properties) {
if (!wpa_dbus_dict_open_write(&iter, &iter_dict))
goto err;
wpa_dbus_get_object_properties(
iface, pgrp_obj_path,
WPAS_DBUS_NEW_IFACE_PERSISTENT_GROUP,
&iter_dict);
if (!wpa_dbus_dict_close_write(&iter, &iter_dict))
goto err;
}
dbus_connection_send(iface->con, msg, NULL);
dbus_message_unref(msg);
return;
err:
wpa_printf(MSG_ERROR, "dbus: Failed to construct signal");
dbus_message_unref(msg);
}
/**
* wpas_dbus_signal_persistent_group_added - Send a persistent_group
* added signal
* @wpa_s: %wpa_supplicant network interface data
* @id: new persistent group id
*
* Notify listeners about addition of a new persistent group.
*/
static void wpas_dbus_signal_persistent_group_added(
struct wpa_supplicant *wpa_s, int id)
{
wpas_dbus_signal_persistent_group(wpa_s, id, "PersistentGroupAdded",
TRUE);
}
/**
* wpas_dbus_signal_persistent_group_removed - Send a persistent_group
* removed signal
* @wpa_s: %wpa_supplicant network interface data
* @id: persistent group id
*
* Notify listeners about removal of a persistent group.
*/
static void wpas_dbus_signal_persistent_group_removed(
struct wpa_supplicant *wpa_s, int id)
{
wpas_dbus_signal_persistent_group(wpa_s, id, "PersistentGroupRemoved",
FALSE);
}
/**
* wpas_dbus_signal_p2p_wps_failed - Signals WpsFailed event
* @wpa_s: %wpa_supplicant network interface data
*
* Sends Event dbus signal with name "fail" and dictionary containing
* "msg" field with fail message number (int32) as arguments
*/
void wpas_dbus_signal_p2p_wps_failed(struct wpa_supplicant *wpa_s,
struct wps_event_fail *fail)
{
DBusMessage *msg;
DBusMessageIter iter, dict_iter;
struct wpas_dbus_priv *iface;
char *key = "fail";
iface = wpa_s->global->dbus;
/* Do nothing if the control interface is not turned on */
if (iface == NULL)
return;
msg = dbus_message_new_signal(wpa_s->dbus_new_path,
WPAS_DBUS_NEW_IFACE_P2PDEVICE,
"WpsFailed");
if (msg == NULL)
return;
dbus_message_iter_init_append(msg, &iter);
if (!dbus_message_iter_append_basic(&iter, DBUS_TYPE_STRING, &key) ||
!wpa_dbus_dict_open_write(&iter, &dict_iter) ||
!wpa_dbus_dict_append_int32(&dict_iter, "msg", fail->msg) ||
!wpa_dbus_dict_append_int16(&dict_iter, "config_error",
fail->config_error) ||
!wpa_dbus_dict_close_write(&iter, &dict_iter))
wpa_printf(MSG_ERROR, "dbus: Failed to construct signal");
else
dbus_connection_send(iface->con, msg, NULL);
dbus_message_unref(msg);
}
#endif /*CONFIG_P2P*/
/**
* wpas_dbus_signal_prop_changed - Signals change of property
* @wpa_s: %wpa_supplicant network interface data
* @property: indicates which property has changed
*
* Sends PropertyChanged signals with path, interface and arguments
* depending on which property has changed.
*/
void wpas_dbus_signal_prop_changed(struct wpa_supplicant *wpa_s,
enum wpas_dbus_prop property)
{
WPADBusPropertyAccessor getter;
char *prop;
if (wpa_s->dbus_new_path == NULL)
return; /* Skip signal since D-Bus setup is not yet ready */
switch (property) {
case WPAS_DBUS_PROP_AP_SCAN:
getter = (WPADBusPropertyAccessor) wpas_dbus_getter_ap_scan;
prop = "ApScan";
break;
case WPAS_DBUS_PROP_SCANNING:
getter = (WPADBusPropertyAccessor) wpas_dbus_getter_scanning;
prop = "Scanning";
break;
case WPAS_DBUS_PROP_STATE:
getter = (WPADBusPropertyAccessor) wpas_dbus_getter_state;
prop = "State";
break;
case WPAS_DBUS_PROP_CURRENT_BSS:
getter = (WPADBusPropertyAccessor)
wpas_dbus_getter_current_bss;
prop = "CurrentBSS";
break;
case WPAS_DBUS_PROP_CURRENT_NETWORK:
getter = (WPADBusPropertyAccessor)
wpas_dbus_getter_current_network;
prop = "CurrentNetwork";
break;
case WPAS_DBUS_PROP_BSSS:
getter = (WPADBusPropertyAccessor) wpas_dbus_getter_bsss;
prop = "BSSs";
break;
case WPAS_DBUS_PROP_CURRENT_AUTH_MODE:
getter = (WPADBusPropertyAccessor)
wpas_dbus_getter_current_auth_mode;
prop = "CurrentAuthMode";
break;
default:
wpa_printf(MSG_ERROR, "dbus: %s: Unknown Property value %d",
__func__, property);
return;
}
wpa_dbus_mark_property_changed(wpa_s->global->dbus,
wpa_s->dbus_new_path,
WPAS_DBUS_NEW_IFACE_INTERFACE, prop);
}
/**
* wpas_dbus_bss_signal_prop_changed - Signals change of BSS property
* @wpa_s: %wpa_supplicant network interface data
* @property: indicates which property has changed
* @id: unique BSS identifier
*
* Sends PropertyChanged signals with path, interface, and arguments depending
* on which property has changed.
*/
void wpas_dbus_bss_signal_prop_changed(struct wpa_supplicant *wpa_s,
enum wpas_dbus_bss_prop property,
unsigned int id)
{
char path[WPAS_DBUS_OBJECT_PATH_MAX];
char *prop;
switch (property) {
case WPAS_DBUS_BSS_PROP_SIGNAL:
prop = "Signal";
break;
case WPAS_DBUS_BSS_PROP_FREQ:
prop = "Frequency";
break;
case WPAS_DBUS_BSS_PROP_MODE:
prop = "Mode";
break;
case WPAS_DBUS_BSS_PROP_PRIVACY:
prop = "Privacy";
break;
case WPAS_DBUS_BSS_PROP_RATES:
prop = "Rates";
break;
case WPAS_DBUS_BSS_PROP_WPA:
prop = "WPA";
break;
case WPAS_DBUS_BSS_PROP_RSN:
prop = "RSN";
break;
case WPAS_DBUS_BSS_PROP_IES:
prop = "IEs";
break;
default:
wpa_printf(MSG_ERROR, "dbus: %s: Unknown Property value %d",
__func__, property);
return;
}
os_snprintf(path, WPAS_DBUS_OBJECT_PATH_MAX,
"%s/" WPAS_DBUS_NEW_BSSIDS_PART "/%u",
wpa_s->dbus_new_path, id);
wpa_dbus_mark_property_changed(wpa_s->global->dbus, path,
WPAS_DBUS_NEW_IFACE_BSS, prop);
}
/**
* wpas_dbus_signal_debug_level_changed - Signals change of debug param
* @global: wpa_global structure
*
* Sends PropertyChanged signals informing that debug level has changed.
*/
void wpas_dbus_signal_debug_level_changed(struct wpa_global *global)
{
wpa_dbus_mark_property_changed(global->dbus, WPAS_DBUS_NEW_PATH,
WPAS_DBUS_NEW_INTERFACE,
"DebugLevel");
}
/**
* wpas_dbus_signal_debug_timestamp_changed - Signals change of debug param
* @global: wpa_global structure
*
* Sends PropertyChanged signals informing that debug timestamp has changed.
*/
void wpas_dbus_signal_debug_timestamp_changed(struct wpa_global *global)
{
wpa_dbus_mark_property_changed(global->dbus, WPAS_DBUS_NEW_PATH,
WPAS_DBUS_NEW_INTERFACE,
"DebugTimestamp");
}
/**
* wpas_dbus_signal_debug_show_keys_changed - Signals change of debug param
* @global: wpa_global structure
*
* Sends PropertyChanged signals informing that debug show_keys has changed.
*/
void wpas_dbus_signal_debug_show_keys_changed(struct wpa_global *global)
{
wpa_dbus_mark_property_changed(global->dbus, WPAS_DBUS_NEW_PATH,
WPAS_DBUS_NEW_INTERFACE,
"DebugShowKeys");
}
static void wpas_dbus_register(struct wpa_dbus_object_desc *obj_desc,
void *priv,
WPADBusArgumentFreeFunction priv_free,
const struct wpa_dbus_method_desc *methods,
const struct wpa_dbus_property_desc *properties,
const struct wpa_dbus_signal_desc *signals)
{
int n;
obj_desc->user_data = priv;
obj_desc->user_data_free_func = priv_free;
obj_desc->methods = methods;
obj_desc->properties = properties;
obj_desc->signals = signals;
for (n = 0; properties && properties->dbus_property; properties++)
n++;
obj_desc->prop_changed_flags = os_zalloc(n);
if (!obj_desc->prop_changed_flags)
wpa_printf(MSG_DEBUG, "dbus: %s: can't register handlers",
__func__);
}
static const struct wpa_dbus_method_desc wpas_dbus_global_methods[] = {
{ "CreateInterface", WPAS_DBUS_NEW_INTERFACE,
(WPADBusMethodHandler) &wpas_dbus_handler_create_interface,
{
{ "args", "a{sv}", ARG_IN },
{ "path", "o", ARG_OUT },
END_ARGS
}
},
{ "RemoveInterface", WPAS_DBUS_NEW_INTERFACE,
(WPADBusMethodHandler) &wpas_dbus_handler_remove_interface,
{
{ "path", "o", ARG_IN },
END_ARGS
}
},
{ "GetInterface", WPAS_DBUS_NEW_INTERFACE,
(WPADBusMethodHandler) &wpas_dbus_handler_get_interface,
{
{ "ifname", "s", ARG_IN },
{ "path", "o", ARG_OUT },
END_ARGS
}
},
{ NULL, NULL, NULL, { END_ARGS } }
};
static const struct wpa_dbus_property_desc wpas_dbus_global_properties[] = {
{ "DebugLevel", WPAS_DBUS_NEW_INTERFACE, "s",
(WPADBusPropertyAccessor) wpas_dbus_getter_debug_level,
(WPADBusPropertyAccessor) wpas_dbus_setter_debug_level,
RW
},
{ "DebugTimestamp", WPAS_DBUS_NEW_INTERFACE, "b",
(WPADBusPropertyAccessor) wpas_dbus_getter_debug_timestamp,
(WPADBusPropertyAccessor) wpas_dbus_setter_debug_timestamp,
RW
},
{ "DebugShowKeys", WPAS_DBUS_NEW_INTERFACE, "b",
(WPADBusPropertyAccessor) wpas_dbus_getter_debug_show_keys,
(WPADBusPropertyAccessor) wpas_dbus_setter_debug_show_keys,
RW
},
{ "Interfaces", WPAS_DBUS_NEW_INTERFACE, "ao",
(WPADBusPropertyAccessor) &wpas_dbus_getter_interfaces,
NULL,
R
},
{ "EapMethods", WPAS_DBUS_NEW_INTERFACE, "as",
(WPADBusPropertyAccessor) wpas_dbus_getter_eap_methods,
NULL,
R
},
{ NULL, NULL, NULL, NULL, NULL, 0 }
};
static const struct wpa_dbus_signal_desc wpas_dbus_global_signals[] = {
{ "InterfaceAdded", WPAS_DBUS_NEW_INTERFACE,
{
{ "path", "o", ARG_OUT },
{ "properties", "a{sv}", ARG_OUT },
END_ARGS
}
},
{ "InterfaceRemoved", WPAS_DBUS_NEW_INTERFACE,
{
{ "path", "o", ARG_OUT },
END_ARGS
}
},
{ "PropertiesChanged", WPAS_DBUS_NEW_INTERFACE,
{
{ "properties", "a{sv}", ARG_OUT },
END_ARGS
}
},
{ NULL, NULL, { END_ARGS } }
};
/**
* wpas_dbus_ctrl_iface_init - Initialize dbus control interface
* @global: Pointer to global data from wpa_supplicant_init()
* Returns: 0 on success or -1 on failure
*
* Initialize the dbus control interface for wpa_supplicantand and start
* receiving commands from external programs over the bus.
*/
int wpas_dbus_ctrl_iface_init(struct wpas_dbus_priv *priv)
{
struct wpa_dbus_object_desc *obj_desc;
int ret;
obj_desc = os_zalloc(sizeof(struct wpa_dbus_object_desc));
if (!obj_desc) {
wpa_printf(MSG_ERROR, "Not enough memory "
"to create object description");
return -1;
}
wpas_dbus_register(obj_desc, priv->global, NULL,
wpas_dbus_global_methods,
wpas_dbus_global_properties,
wpas_dbus_global_signals);
wpa_printf(MSG_DEBUG, "dbus: Register D-Bus object '%s'",
WPAS_DBUS_NEW_PATH);
ret = wpa_dbus_ctrl_iface_init(priv, WPAS_DBUS_NEW_PATH,
WPAS_DBUS_NEW_SERVICE,
obj_desc);
if (ret < 0)
free_dbus_object_desc(obj_desc);
else
priv->dbus_new_initialized = 1;
return ret;
}
/**
* wpas_dbus_ctrl_iface_deinit - Deinitialize dbus ctrl interface for
* wpa_supplicant
* @iface: Pointer to dbus private data from wpas_dbus_init()
*
* Deinitialize the dbus control interface that was initialized with
* wpas_dbus_ctrl_iface_init().
*/
void wpas_dbus_ctrl_iface_deinit(struct wpas_dbus_priv *iface)
{
if (!iface->dbus_new_initialized)
return;
wpa_printf(MSG_DEBUG, "dbus: Unregister D-Bus object '%s'",
WPAS_DBUS_NEW_PATH);
dbus_connection_unregister_object_path(iface->con,
WPAS_DBUS_NEW_PATH);
}
static void wpa_dbus_free(void *ptr)
{
os_free(ptr);
}
static const struct wpa_dbus_property_desc wpas_dbus_network_properties[] = {
{ "Properties", WPAS_DBUS_NEW_IFACE_NETWORK, "a{sv}",
(WPADBusPropertyAccessor) wpas_dbus_getter_network_properties,
(WPADBusPropertyAccessor) wpas_dbus_setter_network_properties,
RW
},
{ "Enabled", WPAS_DBUS_NEW_IFACE_NETWORK, "b",
(WPADBusPropertyAccessor) wpas_dbus_getter_enabled,
(WPADBusPropertyAccessor) wpas_dbus_setter_enabled,
RW
},
{ NULL, NULL, NULL, NULL, NULL, 0 }
};
static const struct wpa_dbus_signal_desc wpas_dbus_network_signals[] = {
{ "PropertiesChanged", WPAS_DBUS_NEW_IFACE_NETWORK,
{
{ "properties", "a{sv}", ARG_OUT },
END_ARGS
}
},
{ NULL, NULL, { END_ARGS } }
};
/**
* wpas_dbus_register_network - Register a configured network with dbus
* @wpa_s: wpa_supplicant interface structure
* @ssid: network configuration data
* Returns: 0 on success, -1 on failure
*
* Registers network representing object with dbus
*/
int wpas_dbus_register_network(struct wpa_supplicant *wpa_s,
struct wpa_ssid *ssid)
{
struct wpas_dbus_priv *ctrl_iface;
struct wpa_dbus_object_desc *obj_desc;
struct network_handler_args *arg;
char net_obj_path[WPAS_DBUS_OBJECT_PATH_MAX];
#ifdef CONFIG_P2P
/*
* If it is a persistent group register it as such.
* This is to handle cases where an interface is being initialized
* with a list of networks read from config.
*/
if (network_is_persistent_group(ssid))
return wpas_dbus_register_persistent_group(wpa_s, ssid);
#endif /* CONFIG_P2P */
/* Do nothing if the control interface is not turned on */
if (wpa_s == NULL || wpa_s->global == NULL)
return 0;
ctrl_iface = wpa_s->global->dbus;
if (ctrl_iface == NULL)
return 0;
os_snprintf(net_obj_path, WPAS_DBUS_OBJECT_PATH_MAX,
"%s/" WPAS_DBUS_NEW_NETWORKS_PART "/%u",
wpa_s->dbus_new_path, ssid->id);
wpa_printf(MSG_DEBUG, "dbus: Register network object '%s'",
net_obj_path);
obj_desc = os_zalloc(sizeof(struct wpa_dbus_object_desc));
if (!obj_desc) {
wpa_printf(MSG_ERROR, "Not enough memory "
"to create object description");
goto err;
}
/* allocate memory for handlers arguments */
arg = os_zalloc(sizeof(struct network_handler_args));
if (!arg) {
wpa_printf(MSG_ERROR, "Not enough memory "
"to create arguments for method");
goto err;
}
arg->wpa_s = wpa_s;
arg->ssid = ssid;
wpas_dbus_register(obj_desc, arg, wpa_dbus_free, NULL,
wpas_dbus_network_properties,
wpas_dbus_network_signals);
if (wpa_dbus_register_object_per_iface(ctrl_iface, net_obj_path,
wpa_s->ifname, obj_desc))
goto err;
wpas_dbus_signal_network_added(wpa_s, ssid->id);
return 0;
err:
free_dbus_object_desc(obj_desc);
return -1;
}
/**
* wpas_dbus_unregister_network - Unregister a configured network from dbus
* @wpa_s: wpa_supplicant interface structure
* @nid: network id
* Returns: 0 on success, -1 on failure
*
* Unregisters network representing object from dbus
*/
int wpas_dbus_unregister_network(struct wpa_supplicant *wpa_s, int nid)
{
struct wpas_dbus_priv *ctrl_iface;
char net_obj_path[WPAS_DBUS_OBJECT_PATH_MAX];
int ret;
struct wpa_ssid *ssid;
ssid = wpa_config_get_network(wpa_s->conf, nid);
#ifdef CONFIG_P2P
/* If it is a persistent group unregister it as such */
if (ssid && network_is_persistent_group(ssid))
return wpas_dbus_unregister_persistent_group(wpa_s, nid);
#endif /* CONFIG_P2P */
/* Do nothing if the control interface is not turned on */
if (wpa_s == NULL || wpa_s->global == NULL ||
wpa_s->dbus_new_path == NULL)
return 0;
ctrl_iface = wpa_s->global->dbus;
if (ctrl_iface == NULL)
return 0;
os_snprintf(net_obj_path, WPAS_DBUS_OBJECT_PATH_MAX,
"%s/" WPAS_DBUS_NEW_NETWORKS_PART "/%u",
wpa_s->dbus_new_path, nid);
wpa_printf(MSG_DEBUG, "dbus: Unregister network object '%s'",
net_obj_path);
ret = wpa_dbus_unregister_object_per_iface(ctrl_iface, net_obj_path);
if (!ret)
wpas_dbus_signal_network_removed(wpa_s, nid);
return ret;
}
static const struct wpa_dbus_property_desc wpas_dbus_bss_properties[] = {
{ "SSID", WPAS_DBUS_NEW_IFACE_BSS, "ay",
(WPADBusPropertyAccessor) wpas_dbus_getter_bss_ssid,
NULL,
R
},
{ "BSSID", WPAS_DBUS_NEW_IFACE_BSS, "ay",
(WPADBusPropertyAccessor) wpas_dbus_getter_bss_bssid,
NULL,
R
},
{ "Privacy", WPAS_DBUS_NEW_IFACE_BSS, "b",
(WPADBusPropertyAccessor) wpas_dbus_getter_bss_privacy,
NULL,
R
},
{ "Mode", WPAS_DBUS_NEW_IFACE_BSS, "s",
(WPADBusPropertyAccessor) wpas_dbus_getter_bss_mode,
NULL,
R
},
{ "Signal", WPAS_DBUS_NEW_IFACE_BSS, "n",
(WPADBusPropertyAccessor) wpas_dbus_getter_bss_signal,
NULL,
R
},
{ "Frequency", WPAS_DBUS_NEW_IFACE_BSS, "q",
(WPADBusPropertyAccessor) wpas_dbus_getter_bss_frequency,
NULL,
R
},
{ "Rates", WPAS_DBUS_NEW_IFACE_BSS, "au",
(WPADBusPropertyAccessor) wpas_dbus_getter_bss_rates,
NULL,
R
},
{ "WPA", WPAS_DBUS_NEW_IFACE_BSS, "a{sv}",
(WPADBusPropertyAccessor) wpas_dbus_getter_bss_wpa,
NULL,
R
},
{ "RSN", WPAS_DBUS_NEW_IFACE_BSS, "a{sv}",
(WPADBusPropertyAccessor) wpas_dbus_getter_bss_rsn,
NULL,
R
},
{ "IEs", WPAS_DBUS_NEW_IFACE_BSS, "ay",
(WPADBusPropertyAccessor) wpas_dbus_getter_bss_ies,
NULL,
R
},
{ NULL, NULL, NULL, NULL, NULL, 0 }
};
static const struct wpa_dbus_signal_desc wpas_dbus_bss_signals[] = {
{ "PropertiesChanged", WPAS_DBUS_NEW_IFACE_BSS,
{
{ "properties", "a{sv}", ARG_OUT },
END_ARGS
}
},
{ NULL, NULL, { END_ARGS } }
};
/**
* wpas_dbus_unregister_bss - Unregister a scanned BSS from dbus
* @wpa_s: wpa_supplicant interface structure
* @bssid: scanned network bssid
* @id: unique BSS identifier
* Returns: 0 on success, -1 on failure
*
* Unregisters BSS representing object from dbus
*/
int wpas_dbus_unregister_bss(struct wpa_supplicant *wpa_s,
u8 bssid[ETH_ALEN], unsigned int id)
{
struct wpas_dbus_priv *ctrl_iface;
char bss_obj_path[WPAS_DBUS_OBJECT_PATH_MAX];
/* Do nothing if the control interface is not turned on */
if (wpa_s == NULL || wpa_s->global == NULL)
return 0;
ctrl_iface = wpa_s->global->dbus;
if (ctrl_iface == NULL)
return 0;
os_snprintf(bss_obj_path, WPAS_DBUS_OBJECT_PATH_MAX,
"%s/" WPAS_DBUS_NEW_BSSIDS_PART "/%u",
wpa_s->dbus_new_path, id);
wpa_printf(MSG_DEBUG, "dbus: Unregister BSS object '%s'",
bss_obj_path);
if (wpa_dbus_unregister_object_per_iface(ctrl_iface, bss_obj_path)) {
wpa_printf(MSG_ERROR, "dbus: Cannot unregister BSS object %s",
bss_obj_path);
return -1;
}
wpas_dbus_signal_bss_removed(wpa_s, bss_obj_path);
wpas_dbus_signal_prop_changed(wpa_s, WPAS_DBUS_PROP_BSSS);
return 0;
}
/**
* wpas_dbus_register_bss - Register a scanned BSS with dbus
* @wpa_s: wpa_supplicant interface structure
* @bssid: scanned network bssid
* @id: unique BSS identifier
* Returns: 0 on success, -1 on failure
*
* Registers BSS representing object with dbus
*/
int wpas_dbus_register_bss(struct wpa_supplicant *wpa_s,
u8 bssid[ETH_ALEN], unsigned int id)
{
struct wpas_dbus_priv *ctrl_iface;
struct wpa_dbus_object_desc *obj_desc;
char bss_obj_path[WPAS_DBUS_OBJECT_PATH_MAX];
struct bss_handler_args *arg;
/* Do nothing if the control interface is not turned on */
if (wpa_s == NULL || wpa_s->global == NULL)
return 0;
ctrl_iface = wpa_s->global->dbus;
if (ctrl_iface == NULL)
return 0;
os_snprintf(bss_obj_path, WPAS_DBUS_OBJECT_PATH_MAX,
"%s/" WPAS_DBUS_NEW_BSSIDS_PART "/%u",
wpa_s->dbus_new_path, id);
obj_desc = os_zalloc(sizeof(struct wpa_dbus_object_desc));
if (!obj_desc) {
wpa_printf(MSG_ERROR, "Not enough memory "
"to create object description");
goto err;
}
arg = os_zalloc(sizeof(struct bss_handler_args));
if (!arg) {
wpa_printf(MSG_ERROR, "Not enough memory "
"to create arguments for handler");
goto err;
}
arg->wpa_s = wpa_s;
arg->id = id;
wpas_dbus_register(obj_desc, arg, wpa_dbus_free, NULL,
wpas_dbus_bss_properties,
wpas_dbus_bss_signals);
wpa_printf(MSG_DEBUG, "dbus: Register BSS object '%s'",
bss_obj_path);
if (wpa_dbus_register_object_per_iface(ctrl_iface, bss_obj_path,
wpa_s->ifname, obj_desc)) {
wpa_printf(MSG_ERROR,
"Cannot register BSSID dbus object %s.",
bss_obj_path);
goto err;
}
wpas_dbus_signal_bss_added(wpa_s, bss_obj_path);
wpas_dbus_signal_prop_changed(wpa_s, WPAS_DBUS_PROP_BSSS);
return 0;
err:
free_dbus_object_desc(obj_desc);
return -1;
}
static const struct wpa_dbus_method_desc wpas_dbus_interface_methods[] = {
{ "Scan", WPAS_DBUS_NEW_IFACE_INTERFACE,
(WPADBusMethodHandler) &wpas_dbus_handler_scan,
{
{ "args", "a{sv}", ARG_IN },
END_ARGS
}
},
{ "Disconnect", WPAS_DBUS_NEW_IFACE_INTERFACE,
(WPADBusMethodHandler) &wpas_dbus_handler_disconnect,
{
END_ARGS
}
},
{ "AddNetwork", WPAS_DBUS_NEW_IFACE_INTERFACE,
(WPADBusMethodHandler) &wpas_dbus_handler_add_network,
{
{ "args", "a{sv}", ARG_IN },
{ "path", "o", ARG_OUT },
END_ARGS
}
},
{ "RemoveNetwork", WPAS_DBUS_NEW_IFACE_INTERFACE,
(WPADBusMethodHandler) &wpas_dbus_handler_remove_network,
{
{ "path", "o", ARG_IN },
END_ARGS
}
},
{ "RemoveAllNetworks", WPAS_DBUS_NEW_IFACE_INTERFACE,
(WPADBusMethodHandler) &wpas_dbus_handler_remove_all_networks,
{
END_ARGS
}
},
{ "SelectNetwork", WPAS_DBUS_NEW_IFACE_INTERFACE,
(WPADBusMethodHandler) &wpas_dbus_handler_select_network,
{
{ "path", "o", ARG_IN },
END_ARGS
}
},
{ "AddBlob", WPAS_DBUS_NEW_IFACE_INTERFACE,
(WPADBusMethodHandler) &wpas_dbus_handler_add_blob,
{
{ "name", "s", ARG_IN },
{ "data", "ay", ARG_IN },
END_ARGS
}
},
{ "GetBlob", WPAS_DBUS_NEW_IFACE_INTERFACE,
(WPADBusMethodHandler) &wpas_dbus_handler_get_blob,
{
{ "name", "s", ARG_IN },
{ "data", "ay", ARG_OUT },
END_ARGS
}
},
{ "RemoveBlob", WPAS_DBUS_NEW_IFACE_INTERFACE,
(WPADBusMethodHandler) &wpas_dbus_handler_remove_blob,
{
{ "name", "s", ARG_IN },
END_ARGS
}
},
#ifdef CONFIG_WPS
{ "Start", WPAS_DBUS_NEW_IFACE_WPS,
(WPADBusMethodHandler) &wpas_dbus_handler_wps_start,
{
{ "args", "a{sv}", ARG_IN },
{ "output", "a{sv}", ARG_OUT },
END_ARGS
}
},
#endif /* CONFIG_WPS */
#ifdef CONFIG_P2P
{ "Find", WPAS_DBUS_NEW_IFACE_P2PDEVICE,
(WPADBusMethodHandler)wpas_dbus_handler_p2p_find,
{
{ "args", "a{sv}", ARG_IN },
END_ARGS
}
},
{ "StopFind", WPAS_DBUS_NEW_IFACE_P2PDEVICE,
(WPADBusMethodHandler)wpas_dbus_handler_p2p_stop_find,
{
END_ARGS
}
},
{ "Listen", WPAS_DBUS_NEW_IFACE_P2PDEVICE,
(WPADBusMethodHandler)wpas_dbus_handler_p2p_listen,
{
{ "timeout", "i", ARG_IN },
END_ARGS
}
},
{ "ExtendedListen", WPAS_DBUS_NEW_IFACE_P2PDEVICE,
(WPADBusMethodHandler)wpas_dbus_handler_p2p_extendedlisten,
{
{ "args", "a{sv}", ARG_IN },
END_ARGS
}
},
{ "PresenceRequest", WPAS_DBUS_NEW_IFACE_P2PDEVICE,
(WPADBusMethodHandler)wpas_dbus_handler_p2p_presence_request,
{
{ "args", "a{sv}", ARG_IN },
END_ARGS
}
},
{ "ProvisionDiscoveryRequest", WPAS_DBUS_NEW_IFACE_P2PDEVICE,
(WPADBusMethodHandler)wpas_dbus_handler_p2p_prov_disc_req,
{
{ "peer", "o", ARG_IN },
{ "config_method", "s", ARG_IN },
END_ARGS
}
},
{ "Connect", WPAS_DBUS_NEW_IFACE_P2PDEVICE,
(WPADBusMethodHandler)wpas_dbus_handler_p2p_connect,
{
{ "args", "a{sv}", ARG_IN },
{ "generated_pin", "i", ARG_OUT },
END_ARGS
}
},
{ "GroupAdd", WPAS_DBUS_NEW_IFACE_P2PDEVICE,
(WPADBusMethodHandler)wpas_dbus_handler_p2p_group_add,
{
{ "args", "a{sv}", ARG_IN },
END_ARGS
}
},
{ "Invite", WPAS_DBUS_NEW_IFACE_P2PDEVICE,
(WPADBusMethodHandler)wpas_dbus_handler_p2p_invite,
{
{ "args", "a{sv}", ARG_IN },
END_ARGS
}
},
{ "Disconnect", WPAS_DBUS_NEW_IFACE_P2PDEVICE,
(WPADBusMethodHandler)wpas_dbus_handler_p2p_disconnect,
{
END_ARGS
}
},
{ "RejectPeer", WPAS_DBUS_NEW_IFACE_P2PDEVICE,
(WPADBusMethodHandler)wpas_dbus_handler_p2p_rejectpeer,
{
{ "peer", "o", ARG_IN },
END_ARGS
}
},
{ "Flush", WPAS_DBUS_NEW_IFACE_P2PDEVICE,
(WPADBusMethodHandler)wpas_dbus_handler_p2p_flush,
{
END_ARGS
}
},
{ "AddService", WPAS_DBUS_NEW_IFACE_P2PDEVICE,
(WPADBusMethodHandler)wpas_dbus_handler_p2p_add_service,
{
{ "args", "a{sv}", ARG_IN },
END_ARGS
}
},
{ "DeleteService", WPAS_DBUS_NEW_IFACE_P2PDEVICE,
(WPADBusMethodHandler)wpas_dbus_handler_p2p_delete_service,
{
{ "args", "a{sv}", ARG_IN },
END_ARGS
}
},
{ "FlushService", WPAS_DBUS_NEW_IFACE_P2PDEVICE,
(WPADBusMethodHandler)wpas_dbus_handler_p2p_flush_service,
{
END_ARGS
}
},
{ "ServiceDiscoveryRequest", WPAS_DBUS_NEW_IFACE_P2PDEVICE,
(WPADBusMethodHandler)wpas_dbus_handler_p2p_service_sd_req,
{
{ "args", "a{sv}", ARG_IN },
END_ARGS
}
},
{ "ServiceDiscoveryResponse", WPAS_DBUS_NEW_IFACE_P2PDEVICE,
(WPADBusMethodHandler)wpas_dbus_handler_p2p_service_sd_res,
{
{ "args", "a{sv}", ARG_IN },
END_ARGS
}
},
{ "ServiceDiscoveryCancelRequest", WPAS_DBUS_NEW_IFACE_P2PDEVICE,
(WPADBusMethodHandler)wpas_dbus_handler_p2p_service_sd_cancel_req,
{
{ "args", "t", ARG_IN },
END_ARGS
}
},
{ "ServiceUpdate", WPAS_DBUS_NEW_IFACE_P2PDEVICE,
(WPADBusMethodHandler)wpas_dbus_handler_p2p_service_update,
{
END_ARGS
}
},
{ "ServiceDiscoveryExternal", WPAS_DBUS_NEW_IFACE_P2PDEVICE,
(WPADBusMethodHandler)wpas_dbus_handler_p2p_serv_disc_external,
{
{ "arg", "i", ARG_IN },
END_ARGS
}
},
{ "ServiceDiscoveryExternal", WPAS_DBUS_NEW_IFACE_P2PDEVICE,
(WPADBusMethodHandler)wpas_dbus_handler_p2p_serv_disc_external,
{
{ "arg", "i", ARG_IN },
END_ARGS
}
},
{ "AddPersistentGroup", WPAS_DBUS_NEW_IFACE_P2PDEVICE,
(WPADBusMethodHandler) wpas_dbus_handler_add_persistent_group,
{
{ "args", "a{sv}", ARG_IN },
{ "path", "o", ARG_OUT },
END_ARGS
}
},
{ "RemovePersistentGroup", WPAS_DBUS_NEW_IFACE_P2PDEVICE,
(WPADBusMethodHandler) wpas_dbus_handler_remove_persistent_group,
{
{ "path", "o", ARG_IN },
END_ARGS
}
},
{ "RemoveAllPersistentGroups", WPAS_DBUS_NEW_IFACE_P2PDEVICE,
(WPADBusMethodHandler)
wpas_dbus_handler_remove_all_persistent_groups,
{
END_ARGS
}
},
#endif /* CONFIG_P2P */
{ "FlushBSS", WPAS_DBUS_NEW_IFACE_INTERFACE,
(WPADBusMethodHandler) &wpas_dbus_handler_flush_bss,
{
{ "age", "u", ARG_IN },
END_ARGS
}
},
{ NULL, NULL, NULL, { END_ARGS } }
};
static const struct wpa_dbus_property_desc wpas_dbus_interface_properties[] = {
{ "Capabilities", WPAS_DBUS_NEW_IFACE_INTERFACE, "a{sv}",
(WPADBusPropertyAccessor) wpas_dbus_getter_capabilities,
NULL, R
},
{ "State", WPAS_DBUS_NEW_IFACE_INTERFACE, "s",
(WPADBusPropertyAccessor) wpas_dbus_getter_state,
NULL, R
},
{ "Scanning", WPAS_DBUS_NEW_IFACE_INTERFACE, "b",
(WPADBusPropertyAccessor) wpas_dbus_getter_scanning,
NULL, R
},
{ "ApScan", WPAS_DBUS_NEW_IFACE_INTERFACE, "u",
(WPADBusPropertyAccessor) wpas_dbus_getter_ap_scan,
(WPADBusPropertyAccessor) wpas_dbus_setter_ap_scan,
RW
},
{ "BSSExpireAge", WPAS_DBUS_NEW_IFACE_INTERFACE, "u",
(WPADBusPropertyAccessor) wpas_dbus_getter_bss_expire_age,
(WPADBusPropertyAccessor) wpas_dbus_setter_bss_expire_age,
RW
},
{ "BSSExpireCount", WPAS_DBUS_NEW_IFACE_INTERFACE, "u",
(WPADBusPropertyAccessor) wpas_dbus_getter_bss_expire_count,
(WPADBusPropertyAccessor) wpas_dbus_setter_bss_expire_count,
RW
},
{ "Country", WPAS_DBUS_NEW_IFACE_INTERFACE, "s",
(WPADBusPropertyAccessor) wpas_dbus_getter_country,
(WPADBusPropertyAccessor) wpas_dbus_setter_country,
RW
},
{ "Ifname", WPAS_DBUS_NEW_IFACE_INTERFACE, "s",
(WPADBusPropertyAccessor) wpas_dbus_getter_ifname,
NULL, R
},
{ "Driver", WPAS_DBUS_NEW_IFACE_INTERFACE, "s",
(WPADBusPropertyAccessor) wpas_dbus_getter_driver,
NULL, R
},
{ "BridgeIfname", WPAS_DBUS_NEW_IFACE_INTERFACE, "s",
(WPADBusPropertyAccessor) wpas_dbus_getter_bridge_ifname,
NULL, R
},
{ "CurrentBSS", WPAS_DBUS_NEW_IFACE_INTERFACE, "o",
(WPADBusPropertyAccessor) wpas_dbus_getter_current_bss,
NULL, R
},
{ "CurrentNetwork", WPAS_DBUS_NEW_IFACE_INTERFACE, "o",
(WPADBusPropertyAccessor) wpas_dbus_getter_current_network,
NULL, R
},
{ "CurrentAuthMode", WPAS_DBUS_NEW_IFACE_INTERFACE, "s",
(WPADBusPropertyAccessor) wpas_dbus_getter_current_auth_mode,
NULL, R
},
{ "Blobs", WPAS_DBUS_NEW_IFACE_INTERFACE, "a{say}",
(WPADBusPropertyAccessor) wpas_dbus_getter_blobs,
NULL, R
},
{ "BSSs", WPAS_DBUS_NEW_IFACE_INTERFACE, "ao",
(WPADBusPropertyAccessor) wpas_dbus_getter_bsss,
NULL, R
},
{ "Networks", WPAS_DBUS_NEW_IFACE_INTERFACE, "ao",
(WPADBusPropertyAccessor) wpas_dbus_getter_networks,
NULL, R
},
#ifdef CONFIG_WPS
{ "ProcessCredentials", WPAS_DBUS_NEW_IFACE_WPS, "b",
(WPADBusPropertyAccessor) wpas_dbus_getter_process_credentials,
(WPADBusPropertyAccessor) wpas_dbus_setter_process_credentials,
RW
},
#endif /* CONFIG_WPS */
#ifdef CONFIG_P2P
{ "P2PDeviceProperties", WPAS_DBUS_NEW_IFACE_P2PDEVICE, "a{sv}",
(WPADBusPropertyAccessor) wpas_dbus_getter_p2p_device_properties,
(WPADBusPropertyAccessor) wpas_dbus_setter_p2p_device_properties,
RW
},
{ "Peers", WPAS_DBUS_NEW_IFACE_P2PDEVICE, "ao",
(WPADBusPropertyAccessor) wpas_dbus_getter_p2p_peers,
NULL, R
},
{ "Role", WPAS_DBUS_NEW_IFACE_P2PDEVICE, "s",
(WPADBusPropertyAccessor) wpas_dbus_getter_p2p_role,
NULL, R
},
{ "Group", WPAS_DBUS_NEW_IFACE_P2PDEVICE, "o",
(WPADBusPropertyAccessor) wpas_dbus_getter_p2p_group,
NULL, R
},
{ "PeerGO", WPAS_DBUS_NEW_IFACE_P2PDEVICE, "o",
(WPADBusPropertyAccessor) wpas_dbus_getter_p2p_peergo,
NULL, R
},
{ "PersistentGroups", WPAS_DBUS_NEW_IFACE_P2PDEVICE, "ao",
(WPADBusPropertyAccessor) wpas_dbus_getter_persistent_groups,
NULL, R
},
#endif /* CONFIG_P2P */
{ NULL, NULL, NULL, NULL, NULL, 0 }
};
static const struct wpa_dbus_signal_desc wpas_dbus_interface_signals[] = {
{ "ScanDone", WPAS_DBUS_NEW_IFACE_INTERFACE,
{
{ "success", "b", ARG_OUT },
END_ARGS
}
},
{ "BSSAdded", WPAS_DBUS_NEW_IFACE_INTERFACE,
{
{ "path", "o", ARG_OUT },
{ "properties", "a{sv}", ARG_OUT },
END_ARGS
}
},
{ "BSSRemoved", WPAS_DBUS_NEW_IFACE_INTERFACE,
{
{ "path", "o", ARG_OUT },
END_ARGS
}
},
{ "BlobAdded", WPAS_DBUS_NEW_IFACE_INTERFACE,
{
{ "name", "s", ARG_OUT },
END_ARGS
}
},
{ "BlobRemoved", WPAS_DBUS_NEW_IFACE_INTERFACE,
{
{ "name", "s", ARG_OUT },
END_ARGS
}
},
{ "NetworkAdded", WPAS_DBUS_NEW_IFACE_INTERFACE,
{
{ "path", "o", ARG_OUT },
{ "properties", "a{sv}", ARG_OUT },
END_ARGS
}
},
{ "NetworkRemoved", WPAS_DBUS_NEW_IFACE_INTERFACE,
{
{ "path", "o", ARG_OUT },
END_ARGS
}
},
{ "NetworkSelected", WPAS_DBUS_NEW_IFACE_INTERFACE,
{
{ "path", "o", ARG_OUT },
END_ARGS
}
},
{ "PropertiesChanged", WPAS_DBUS_NEW_IFACE_INTERFACE,
{
{ "properties", "a{sv}", ARG_OUT },
END_ARGS
}
},
#ifdef CONFIG_WPS
{ "Event", WPAS_DBUS_NEW_IFACE_WPS,
{
{ "name", "s", ARG_OUT },
{ "args", "a{sv}", ARG_OUT },
END_ARGS
}
},
{ "Credentials", WPAS_DBUS_NEW_IFACE_WPS,
{
{ "credentials", "a{sv}", ARG_OUT },
END_ARGS
}
},
{ "PropertiesChanged", WPAS_DBUS_NEW_IFACE_WPS,
{
{ "properties", "a{sv}", ARG_OUT },
END_ARGS
}
},
#endif /* CONFIG_WPS */
#ifdef CONFIG_P2P
{ "P2PStateChanged", WPAS_DBUS_NEW_IFACE_P2PDEVICE,
{
{ "states", "a{ss}", ARG_OUT },
END_ARGS
}
},
{ "DeviceFound", WPAS_DBUS_NEW_IFACE_P2PDEVICE,
{
{ "path", "o", ARG_OUT },
{ "properties", "a{sv}", ARG_OUT },
END_ARGS
}
},
{ "DeviceLost", WPAS_DBUS_NEW_IFACE_P2PDEVICE,
{
{ "path", "o", ARG_OUT },
END_ARGS
}
},
{ "ProvisionDiscoveryRequestDisplayPin", WPAS_DBUS_NEW_IFACE_P2PDEVICE,
{
{ "peer_object", "o", ARG_OUT },
{ "pin", "s", ARG_OUT },
END_ARGS
}
},
{ "ProvisionDiscoveryResponseDisplayPin", WPAS_DBUS_NEW_IFACE_P2PDEVICE,
{
{ "peer_object", "o", ARG_OUT },
{ "pin", "s", ARG_OUT },
END_ARGS
}
},
{ "ProvisionDiscoveryRequestEnterPin", WPAS_DBUS_NEW_IFACE_P2PDEVICE,
{
{ "peer_object", "o", ARG_OUT },
END_ARGS
}
},
{ "ProvisionDiscoveryResponseEnterPin", WPAS_DBUS_NEW_IFACE_P2PDEVICE,
{
{ "peer_object", "o", ARG_OUT },
END_ARGS
}
},
{ "ProvisionDiscoveryPBCRequest", WPAS_DBUS_NEW_IFACE_P2PDEVICE,
{
{ "peer_object", "o", ARG_OUT },
END_ARGS
}
},
{ "ProvisionDiscoveryPBCResponse", WPAS_DBUS_NEW_IFACE_P2PDEVICE,
{
{ "peer_object", "o", ARG_OUT },
END_ARGS
}
},
{ "ProvisionDiscoveryFailure", WPAS_DBUS_NEW_IFACE_P2PDEVICE,
{
{ "peer_object", "o", ARG_OUT },
{ "status", "i", ARG_OUT },
END_ARGS
}
},
{ "GroupStarted", WPAS_DBUS_NEW_IFACE_P2PDEVICE,
{
{ "properties", "a{sv}", ARG_OUT },
END_ARGS
}
},
{ "GONegotiationSuccess", WPAS_DBUS_NEW_IFACE_P2PDEVICE,
{
END_ARGS
}
},
{ "GONegotiationFailure", WPAS_DBUS_NEW_IFACE_P2PDEVICE,
{
{ "status", "i", ARG_OUT },
END_ARGS
}
},
{ "GONegotiationRequest", WPAS_DBUS_NEW_IFACE_P2PDEVICE,
{
{ "path", "o", ARG_OUT },
{ "dev_passwd_id", "i", ARG_OUT },
END_ARGS
}
},
{ "InvitationResult", WPAS_DBUS_NEW_IFACE_P2PDEVICE,
{
{ "invite_result", "a{sv}", ARG_OUT },
END_ARGS
}
},
{ "GroupFinished", WPAS_DBUS_NEW_IFACE_P2PDEVICE,
{
{ "ifname", "s", ARG_OUT },
{ "role", "s", ARG_OUT },
END_ARGS
}
},
{ "ServiceDiscoveryRequest", WPAS_DBUS_NEW_IFACE_P2PDEVICE,
{
{ "sd_request", "a{sv}", ARG_OUT },
END_ARGS
}
},
{ "ServiceDiscoveryResponse", WPAS_DBUS_NEW_IFACE_P2PDEVICE,
{
{ "sd_response", "a{sv}", ARG_OUT },
END_ARGS
}
},
{ "PersistentGroupAdded", WPAS_DBUS_NEW_IFACE_P2PDEVICE,
{
{ "path", "o", ARG_OUT },
{ "properties", "a{sv}", ARG_OUT },
END_ARGS
}
},
{ "PersistentGroupRemoved", WPAS_DBUS_NEW_IFACE_P2PDEVICE,
{
{ "path", "o", ARG_OUT },
END_ARGS
}
},
{ "WpsFailed", WPAS_DBUS_NEW_IFACE_P2PDEVICE,
{
{ "name", "s", ARG_OUT },
{ "args", "a{sv}", ARG_OUT },
END_ARGS
}
},
#endif /* CONFIG_P2P */
{ "Certification", WPAS_DBUS_NEW_IFACE_INTERFACE,
{
{ "certification", "a{sv}", ARG_OUT },
END_ARGS
}
},
{ NULL, NULL, { END_ARGS } }
};
int wpas_dbus_register_interface(struct wpa_supplicant *wpa_s)
{
struct wpa_dbus_object_desc *obj_desc = NULL;
struct wpas_dbus_priv *ctrl_iface = wpa_s->global->dbus;
int next;
/* Do nothing if the control interface is not turned on */
if (ctrl_iface == NULL)
return 0;
/* Create and set the interface's object path */
wpa_s->dbus_new_path = os_zalloc(WPAS_DBUS_OBJECT_PATH_MAX);
if (wpa_s->dbus_new_path == NULL)
return -1;
next = ctrl_iface->next_objid++;
os_snprintf(wpa_s->dbus_new_path, WPAS_DBUS_OBJECT_PATH_MAX,
WPAS_DBUS_NEW_PATH_INTERFACES "/%u",
next);
obj_desc = os_zalloc(sizeof(struct wpa_dbus_object_desc));
if (!obj_desc) {
wpa_printf(MSG_ERROR, "Not enough memory "
"to create object description");
goto err;
}
wpas_dbus_register(obj_desc, wpa_s, NULL, wpas_dbus_interface_methods,
wpas_dbus_interface_properties,
wpas_dbus_interface_signals);
wpa_printf(MSG_DEBUG, "dbus: Register interface object '%s'",
wpa_s->dbus_new_path);
if (wpa_dbus_register_object_per_iface(ctrl_iface,
wpa_s->dbus_new_path,
wpa_s->ifname, obj_desc))
goto err;
wpas_dbus_signal_interface_added(wpa_s);
return 0;
err:
os_free(wpa_s->dbus_new_path);
wpa_s->dbus_new_path = NULL;
free_dbus_object_desc(obj_desc);
return -1;
}
int wpas_dbus_unregister_interface(struct wpa_supplicant *wpa_s)
{
struct wpas_dbus_priv *ctrl_iface;
/* Do nothing if the control interface is not turned on */
if (wpa_s == NULL || wpa_s->global == NULL)
return 0;
ctrl_iface = wpa_s->global->dbus;
if (ctrl_iface == NULL)
return 0;
wpa_printf(MSG_DEBUG, "dbus: Unregister interface object '%s'",
wpa_s->dbus_new_path);
if (wpa_dbus_unregister_object_per_iface(ctrl_iface,
wpa_s->dbus_new_path))
return -1;
wpas_dbus_signal_interface_removed(wpa_s);
os_free(wpa_s->dbus_new_path);
wpa_s->dbus_new_path = NULL;
return 0;
}
#ifdef CONFIG_P2P
static const struct wpa_dbus_property_desc wpas_dbus_p2p_peer_properties[] = {
{ "Properties", WPAS_DBUS_NEW_IFACE_P2P_PEER, "a{sv}",
(WPADBusPropertyAccessor) wpas_dbus_getter_p2p_peer_properties,
NULL, R
},
{ "IEs", WPAS_DBUS_NEW_IFACE_P2P_PEER, "ay",
(WPADBusPropertyAccessor) wpas_dbus_getter_p2p_peer_ies,
NULL, R
},
{ NULL, NULL, NULL, NULL, NULL, 0 }
};
static const struct wpa_dbus_signal_desc wpas_dbus_p2p_peer_signals[] = {
{ NULL, NULL, { END_ARGS } }
};
/**
* wpas_dbus_signal_peer - Send a peer related event signal
* @wpa_s: %wpa_supplicant network interface data
* @dev: peer device object
* @interface: name of the interface emitting this signal.
* In case of peer objects, it would be emitted by either
* the "interface object" or by "peer objects"
* @sig_name: signal name - DeviceFound
*
* Notify listeners about event related with newly found p2p peer device
*/
static void wpas_dbus_signal_peer(struct wpa_supplicant *wpa_s,
const u8 *dev_addr, const char *interface,
const char *sig_name)
{
struct wpas_dbus_priv *iface;
DBusMessage *msg;
DBusMessageIter iter;
char peer_obj_path[WPAS_DBUS_OBJECT_PATH_MAX], *path;
iface = wpa_s->global->dbus;
/* Do nothing if the control interface is not turned on */
if (iface == NULL)
return;
os_snprintf(peer_obj_path, WPAS_DBUS_OBJECT_PATH_MAX,
"%s/" WPAS_DBUS_NEW_P2P_PEERS_PART "/" COMPACT_MACSTR,
wpa_s->dbus_new_path, MAC2STR(dev_addr));
msg = dbus_message_new_signal(wpa_s->dbus_new_path, interface,
sig_name);
if (msg == NULL)
return;
dbus_message_iter_init_append(msg, &iter);
path = peer_obj_path;
if (!dbus_message_iter_append_basic(&iter, DBUS_TYPE_OBJECT_PATH,
&path))
goto err;
dbus_connection_send(iface->con, msg, NULL);
dbus_message_unref(msg);
return;
err:
wpa_printf(MSG_ERROR, "dbus: Failed to construct signal");
dbus_message_unref(msg);
}
/**
* wpas_dbus_signal_peer_found - Send a peer found signal
* @wpa_s: %wpa_supplicant network interface data
* @dev: peer device object
*
* Notify listeners about find a p2p peer device found
*/
void wpas_dbus_signal_peer_device_found(struct wpa_supplicant *wpa_s,
const u8 *dev_addr)
{
wpas_dbus_signal_peer(wpa_s, dev_addr,
WPAS_DBUS_NEW_IFACE_P2PDEVICE,
"DeviceFound");
}
/**
* wpas_dbus_signal_peer_lost - Send a peer lost signal
* @wpa_s: %wpa_supplicant network interface data
* @dev: peer device object
*
* Notify listeners about lost a p2p peer device
*/
void wpas_dbus_signal_peer_device_lost(struct wpa_supplicant *wpa_s,
const u8 *dev_addr)
{
wpas_dbus_signal_peer(wpa_s, dev_addr,
WPAS_DBUS_NEW_IFACE_P2PDEVICE,
"DeviceLost");
}
/**
* wpas_dbus_register_peer - Register a discovered peer object with dbus
* @wpa_s: wpa_supplicant interface structure
* @ssid: network configuration data
* Returns: 0 on success, -1 on failure
*
* Registers network representing object with dbus
*/
int wpas_dbus_register_peer(struct wpa_supplicant *wpa_s, const u8 *dev_addr)
{
struct wpas_dbus_priv *ctrl_iface;
struct wpa_dbus_object_desc *obj_desc;
struct peer_handler_args *arg;
char peer_obj_path[WPAS_DBUS_OBJECT_PATH_MAX];
/* Do nothing if the control interface is not turned on */
if (wpa_s == NULL || wpa_s->global == NULL)
return 0;
ctrl_iface = wpa_s->global->dbus;
if (ctrl_iface == NULL)
return 0;
os_snprintf(peer_obj_path, WPAS_DBUS_OBJECT_PATH_MAX,
"%s/" WPAS_DBUS_NEW_P2P_PEERS_PART "/" COMPACT_MACSTR,
wpa_s->dbus_new_path, MAC2STR(dev_addr));
wpa_printf(MSG_INFO, "dbus: Register peer object '%s'",
peer_obj_path);
obj_desc = os_zalloc(sizeof(struct wpa_dbus_object_desc));
if (!obj_desc) {
wpa_printf(MSG_ERROR, "Not enough memory "
"to create object description");
goto err;
}
/* allocate memory for handlers arguments */
arg = os_zalloc(sizeof(struct peer_handler_args));
if (!arg) {
wpa_printf(MSG_ERROR, "Not enough memory "
"to create arguments for method");
goto err;
}
arg->wpa_s = wpa_s;
os_memcpy(arg->p2p_device_addr, dev_addr, ETH_ALEN);
wpas_dbus_register(obj_desc, arg, wpa_dbus_free,
NULL,
wpas_dbus_p2p_peer_properties,
wpas_dbus_p2p_peer_signals);
if (wpa_dbus_register_object_per_iface(ctrl_iface, peer_obj_path,
wpa_s->ifname, obj_desc))
goto err;
return 0;
err:
free_dbus_object_desc(obj_desc);
return -1;
}
/**
* wpas_dbus_unregister_peer - Unregister a peer object with dbus
* @wpa_s: wpa_supplicant interface structure
* @dev_addr: p2p device addr
* Returns: 0 on success, -1 on failure
*
* Registers network representing object with dbus
*/
int wpas_dbus_unregister_peer(struct wpa_supplicant *wpa_s,
const u8 *dev_addr)
{
struct wpas_dbus_priv *ctrl_iface;
char peer_obj_path[WPAS_DBUS_OBJECT_PATH_MAX];
int ret;
/* Do nothing if the control interface is not turned on */
if (wpa_s == NULL || wpa_s->global == NULL ||
wpa_s->dbus_new_path == NULL)
return 0;
ctrl_iface = wpa_s->global->dbus;
if (ctrl_iface == NULL)
return 0;
os_snprintf(peer_obj_path, WPAS_DBUS_OBJECT_PATH_MAX,
"%s/" WPAS_DBUS_NEW_P2P_PEERS_PART "/" COMPACT_MACSTR,
wpa_s->dbus_new_path, MAC2STR(dev_addr));
wpa_printf(MSG_INFO, "dbus: Unregister peer object '%s'",
peer_obj_path);
ret = wpa_dbus_unregister_object_per_iface(ctrl_iface, peer_obj_path);
return ret;
}
static const struct wpa_dbus_property_desc wpas_dbus_p2p_group_properties[] = {
{ "Members", WPAS_DBUS_NEW_IFACE_P2P_GROUP, "ao",
(WPADBusPropertyAccessor) wpas_dbus_getter_p2p_group_members,
NULL, R
},
{ "Properties",
WPAS_DBUS_NEW_IFACE_P2P_GROUP, "a{sv}",
(WPADBusPropertyAccessor) wpas_dbus_getter_p2p_group_properties,
(WPADBusPropertyAccessor) wpas_dbus_setter_p2p_group_properties,
RW
},
{ NULL, NULL, NULL, NULL, NULL, 0 }
};
static const struct wpa_dbus_signal_desc wpas_dbus_p2p_group_signals[] = {
{ "PeerJoined", WPAS_DBUS_NEW_IFACE_P2P_GROUP,
{
{ "peer", "o", ARG_OUT },
END_ARGS
}
},
{ "PeerDisconnected", WPAS_DBUS_NEW_IFACE_P2P_GROUP,
{
{ "peer", "o", ARG_OUT },
END_ARGS
}
},
{ NULL, NULL, { END_ARGS } }
};
/**
* wpas_dbus_register_p2p_group - Register a p2p group object with dbus
* @wpa_s: wpa_supplicant interface structure
* @ssid: SSID struct
* Returns: 0 on success, -1 on failure
*
* Registers p2p group representing object with dbus
*/
void wpas_dbus_register_p2p_group(struct wpa_supplicant *wpa_s,
struct wpa_ssid *ssid)
{
struct wpas_dbus_priv *ctrl_iface;
struct wpa_dbus_object_desc *obj_desc;
char group_obj_path[WPAS_DBUS_OBJECT_PATH_MAX];
/* Do nothing if the control interface is not turned on */
if (wpa_s == NULL || wpa_s->global == NULL)
return;
ctrl_iface = wpa_s->global->dbus;
if (ctrl_iface == NULL)
return;
if (wpa_s->dbus_groupobj_path) {
wpa_printf(MSG_INFO, "%s: Group object '%s' already exists",
__func__, wpa_s->dbus_groupobj_path);
return;
}
if (wpas_dbus_get_group_obj_path(wpa_s, ssid, group_obj_path) < 0)
return;
wpa_s->dbus_groupobj_path = os_strdup(group_obj_path);
if (wpa_s->dbus_groupobj_path == NULL)
return;
wpa_printf(MSG_INFO, "dbus: Register group object '%s'",
group_obj_path);
obj_desc = os_zalloc(sizeof(struct wpa_dbus_object_desc));
if (!obj_desc) {
wpa_printf(MSG_ERROR, "Not enough memory "
"to create object description");
goto err;
}
wpas_dbus_register(obj_desc, wpa_s, NULL, NULL,
wpas_dbus_p2p_group_properties,
wpas_dbus_p2p_group_signals);
if (wpa_dbus_register_object_per_iface(ctrl_iface, group_obj_path,
wpa_s->ifname, obj_desc))
goto err;
return;
err:
if (wpa_s->dbus_groupobj_path) {
os_free(wpa_s->dbus_groupobj_path);
wpa_s->dbus_groupobj_path = NULL;
}
free_dbus_object_desc(obj_desc);
}
/**
* wpas_dbus_unregister_p2p_group - Unregister a p2p group object from dbus
* @wpa_s: wpa_supplicant interface structure
* @ssid: network name of the p2p group started
*/
void wpas_dbus_unregister_p2p_group(struct wpa_supplicant *wpa_s,
const struct wpa_ssid *ssid)
{
struct wpas_dbus_priv *ctrl_iface;
/* Do nothing if the control interface is not turned on */
if (wpa_s == NULL || wpa_s->global == NULL)
return;
ctrl_iface = wpa_s->global->dbus;
if (ctrl_iface == NULL)
return;
if (!wpa_s->dbus_groupobj_path) {
wpa_printf(MSG_DEBUG,
"%s: Group object '%s' already unregistered",
__func__, wpa_s->dbus_groupobj_path);
return;
}
wpa_printf(MSG_DEBUG, "dbus: Unregister group object '%s'",
wpa_s->dbus_groupobj_path);
wpa_dbus_unregister_object_per_iface(ctrl_iface,
wpa_s->dbus_groupobj_path);
os_free(wpa_s->dbus_groupobj_path);
wpa_s->dbus_groupobj_path = NULL;
}
static const struct wpa_dbus_property_desc
wpas_dbus_p2p_groupmember_properties[] = {
{ "Properties", WPAS_DBUS_NEW_IFACE_P2P_GROUPMEMBER, "a{sv}",
(WPADBusPropertyAccessor) wpas_dbus_getter_p2p_group_properties,
NULL, R
},
{ NULL, NULL, NULL, NULL, NULL, 0 }
};
/**
* wpas_dbus_register_p2p_groupmember - Register a p2p groupmember
* object with dbus
* @wpa_s: wpa_supplicant interface structure
* @p2p_if_addr: i/f addr of the device joining this group
*
* Registers p2p groupmember representing object with dbus
*/
void wpas_dbus_register_p2p_groupmember(struct wpa_supplicant *wpa_s,
const u8 *p2p_if_addr)
{
struct wpas_dbus_priv *ctrl_iface;
struct wpa_dbus_object_desc *obj_desc = NULL;
struct groupmember_handler_args *arg;
char groupmember_obj_path[WPAS_DBUS_OBJECT_PATH_MAX];
/* Do nothing if the control interface is not turned on */
if (wpa_s == NULL || wpa_s->global == NULL)
return;
ctrl_iface = wpa_s->global->dbus;
if (ctrl_iface == NULL)
return;
if (!wpa_s->dbus_groupobj_path)
return;
os_snprintf(groupmember_obj_path, WPAS_DBUS_OBJECT_PATH_MAX,
"%s/" WPAS_DBUS_NEW_P2P_GROUPMEMBERS_PART "/" COMPACT_MACSTR,
wpa_s->dbus_groupobj_path, MAC2STR(p2p_if_addr));
obj_desc = os_zalloc(sizeof(struct wpa_dbus_object_desc));
if (!obj_desc) {
wpa_printf(MSG_ERROR, "Not enough memory "
"to create object description");
goto err;
}
/* allocate memory for handlers arguments */
arg = os_zalloc(sizeof(struct groupmember_handler_args));
if (!arg) {
wpa_printf(MSG_ERROR, "Not enough memory "
"to create arguments for method");
goto err;
}
arg->wpa_s = wpa_s;
os_memcpy(arg->member_addr, p2p_if_addr, ETH_ALEN);
wpas_dbus_register(obj_desc, arg, wpa_dbus_free, NULL,
wpas_dbus_p2p_groupmember_properties, NULL);
if (wpa_dbus_register_object_per_iface(ctrl_iface, groupmember_obj_path,
wpa_s->ifname, obj_desc))
goto err;
wpa_printf(MSG_INFO,
"dbus: Registered group member object '%s' successfully",
groupmember_obj_path);
return;
err:
free_dbus_object_desc(obj_desc);
}
/**
* wpas_dbus_unregister_p2p_groupmember - Unregister a p2p groupmember
* object with dbus
* @wpa_s: wpa_supplicant interface structure
* @p2p_if_addr: i/f addr of the device joining this group
*
* Unregisters p2p groupmember representing object with dbus
*/
void wpas_dbus_unregister_p2p_groupmember(struct wpa_supplicant *wpa_s,
const u8 *p2p_if_addr)
{
struct wpas_dbus_priv *ctrl_iface;
char groupmember_obj_path[WPAS_DBUS_OBJECT_PATH_MAX];
/* Do nothing if the control interface is not turned on */
if (wpa_s == NULL || wpa_s->global == NULL)
return;
ctrl_iface = wpa_s->global->dbus;
if (ctrl_iface == NULL)
return;
if (!wpa_s->dbus_groupobj_path)
return;
os_snprintf(groupmember_obj_path, WPAS_DBUS_OBJECT_PATH_MAX,
"%s/" WPAS_DBUS_NEW_P2P_GROUPMEMBERS_PART "/" COMPACT_MACSTR,
wpa_s->dbus_groupobj_path, MAC2STR(p2p_if_addr));
wpa_dbus_unregister_object_per_iface(ctrl_iface, groupmember_obj_path);
}
static const struct wpa_dbus_property_desc
wpas_dbus_persistent_group_properties[] = {
{ "Properties", WPAS_DBUS_NEW_IFACE_PERSISTENT_GROUP, "a{sv}",
(WPADBusPropertyAccessor)
wpas_dbus_getter_persistent_group_properties,
(WPADBusPropertyAccessor)
wpas_dbus_setter_persistent_group_properties,
RW
},
{ NULL, NULL, NULL, NULL, NULL, 0 }
};
/* No signals intended for persistent group objects */
/**
* wpas_dbus_register_persistent_group - Register a configured(saved)
* persistent group with dbus
* @wpa_s: wpa_supplicant interface structure
* @ssid: persistent group (still represented as a network within wpa)
* configuration data
* Returns: 0 on success, -1 on failure
*
* Registers a persistent group representing object with dbus.
*/
int wpas_dbus_register_persistent_group(struct wpa_supplicant *wpa_s,
struct wpa_ssid *ssid)
{
struct wpas_dbus_priv *ctrl_iface;
struct wpa_dbus_object_desc *obj_desc;
struct network_handler_args *arg;
char pgrp_obj_path[WPAS_DBUS_OBJECT_PATH_MAX];
/* Do nothing if the control interface is not turned on */
if (wpa_s == NULL || wpa_s->global == NULL)
return 0;
/* Make sure ssid is a persistent group */
if (ssid->disabled != 2 && !ssid->p2p_persistent_group)
return -1; /* should we return w/o complaining? */
ctrl_iface = wpa_s->global->dbus;
if (ctrl_iface == NULL)
return 0;
/*
* Intentionally not coming up with different numbering scheme
* for persistent groups.
*/
os_snprintf(pgrp_obj_path, WPAS_DBUS_OBJECT_PATH_MAX,
"%s/" WPAS_DBUS_NEW_PERSISTENT_GROUPS_PART "/%u",
wpa_s->dbus_new_path, ssid->id);
wpa_printf(MSG_DEBUG, "dbus: Register persistent group object '%s'",
pgrp_obj_path);
obj_desc = os_zalloc(sizeof(struct wpa_dbus_object_desc));
if (!obj_desc) {
wpa_printf(MSG_ERROR, "dbus: Not enough memory to create "
"object description");
goto err;
}
/*
* Reusing the same context structure as that for networks
* since these are represented using same data structure.
*/
/* allocate memory for handlers arguments */
arg = os_zalloc(sizeof(struct network_handler_args));
if (!arg) {
wpa_printf(MSG_ERROR, "dbus: Not enough memory to create "
"arguments for method");
goto err;
}
arg->wpa_s = wpa_s;
arg->ssid = ssid;
wpas_dbus_register(obj_desc, arg, wpa_dbus_free, NULL,
wpas_dbus_persistent_group_properties,
NULL);
if (wpa_dbus_register_object_per_iface(ctrl_iface, pgrp_obj_path,
wpa_s->ifname, obj_desc))
goto err;
wpas_dbus_signal_persistent_group_added(wpa_s, ssid->id);
return 0;
err:
free_dbus_object_desc(obj_desc);
return -1;
}
/**
* wpas_dbus_unregister_persistent_group - Unregister a persistent_group
* from dbus
* @wpa_s: wpa_supplicant interface structure
* @nid: network id
* Returns: 0 on success, -1 on failure
*
* Unregisters persistent group representing object from dbus
*
* NOTE: There is a slight issue with the semantics here. While the
* implementation simply means the persistent group is unloaded from memory,
* it should not get interpreted as the group is actually being erased/removed
* from persistent storage as well.
*/
int wpas_dbus_unregister_persistent_group(struct wpa_supplicant *wpa_s,
int nid)
{
struct wpas_dbus_priv *ctrl_iface;
char pgrp_obj_path[WPAS_DBUS_OBJECT_PATH_MAX];
int ret;
/* Do nothing if the control interface is not turned on */
if (wpa_s == NULL || wpa_s->global == NULL ||
wpa_s->dbus_new_path == NULL)
return 0;
ctrl_iface = wpa_s->global->dbus;
if (ctrl_iface == NULL)
return 0;
os_snprintf(pgrp_obj_path, WPAS_DBUS_OBJECT_PATH_MAX,
"%s/" WPAS_DBUS_NEW_PERSISTENT_GROUPS_PART "/%u",
wpa_s->dbus_new_path, nid);
wpa_printf(MSG_DEBUG, "dbus: Unregister persistent group object '%s'",
pgrp_obj_path);
ret = wpa_dbus_unregister_object_per_iface(ctrl_iface, pgrp_obj_path);
if (!ret)
wpas_dbus_signal_persistent_group_removed(wpa_s, nid);
return ret;
}
#endif /* CONFIG_P2P */
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 9,334
|
{"url":"https:\/\/jira.lsstcorp.org\/browse\/DM-26526?focusedCommentId=255677&page=com.atlassian.jira.plugin.system.issuetabpanels:comment-tabpanel","text":"# Can't run RawIngestTask with processes != 1\n\nXMLWordPrintable\n\n#### Details\n\n\u2022 Type: Bug\n\u2022 Status: Done\n\u2022 Resolution: Done\n\u2022 Fix Version\/s: None\n\u2022 Component\/s:\n\u2022 Labels:\n\u2022 Story Points:\n6\n\u2022 Sprint:\nAP F20-4 (September)\n\u2022 Team:\n\u2022 Urgent?:\nNo\n\n#### Description\n\nAttempting to run RawIngestTask with multiple processes on DM-25806, after fixing a pickling problem with Gen3DatasetIngestTask, produces the following stack trace:\n\n Process ForkPoolWorker-1: Traceback (most recent call last): File \"\/software\/lsstsw\/stack_20200515\/python\/miniconda3-4.7.12\/envs\/lsst-scipipe\/lib\/python3.7\/multiprocessing\/process.py\", line 297, in _bootstrap self.run() File \"\/software\/lsstsw\/stack_20200515\/python\/miniconda3-4.7.12\/envs\/lsst-scipipe\/lib\/python3.7\/multiprocessing\/process.py\", line 99, in run self._target(*self._args, **self._kwargs) File \"\/software\/lsstsw\/stack_20200515\/python\/miniconda3-4.7.12\/envs\/lsst-scipipe\/lib\/python3.7\/multiprocessing\/pool.py\", line 110, in worker task = get() File \"\/software\/lsstsw\/stack_20200515\/python\/miniconda3-4.7.12\/envs\/lsst-scipipe\/lib\/python3.7\/multiprocessing\/queues.py\", line 354, in get return _ForkingPickler.loads(res) TypeError: __init__() takes from 1 to 2 positional arguments but 5 were given Process ForkPoolWorker-2: Traceback (most recent call last): File \"\/software\/lsstsw\/stack_20200515\/python\/miniconda3-4.7.12\/envs\/lsst-scipipe\/lib\/python3.7\/multiprocessing\/process.py\", line 297, in _bootstrap self.run() File \"\/software\/lsstsw\/stack_20200515\/python\/miniconda3-4.7.12\/envs\/lsst-scipipe\/lib\/python3.7\/multiprocessing\/process.py\", line 99, in run self._target(*self._args, **self._kwargs) File \"\/software\/lsstsw\/stack_20200515\/python\/miniconda3-4.7.12\/envs\/lsst-scipipe\/lib\/python3.7\/multiprocessing\/pool.py\", line 110, in worker task = get() File \"\/software\/lsstsw\/stack_20200515\/python\/miniconda3-4.7.12\/envs\/lsst-scipipe\/lib\/python3.7\/multiprocessing\/queues.py\", line 354, in get return _ForkingPickler.loads(res) TypeError: __init__() takes from 1 to 2 positional arguments but 5 were given Process ForkPoolWorker-3: Traceback (most recent call last): File \"\/software\/lsstsw\/stack_20200515\/python\/miniconda3-4.7.12\/envs\/lsst-scipipe\/lib\/python3.7\/multiprocessing\/process.py\", line 297, in _bootstrap self.run() File \"\/software\/lsstsw\/stack_20200515\/python\/miniconda3-4.7.12\/envs\/lsst-scipipe\/lib\/python3.7\/multiprocessing\/process.py\", line 99, in run self._target(*self._args, **self._kwargs) File \"\/software\/lsstsw\/stack_20200515\/python\/miniconda3-4.7.12\/envs\/lsst-scipipe\/lib\/python3.7\/multiprocessing\/pool.py\", line 110, in worker task = get() File \"\/software\/lsstsw\/stack_20200515\/python\/miniconda3-4.7.12\/envs\/lsst-scipipe\/lib\/python3.7\/multiprocessing\/queues.py\", line 354, in get return _ForkingPickler.loads(res) TypeError: __init__() takes from 1 to 2 positional arguments but 5 were given ^CProcess ForkPoolWorker-4: Process ForkPoolWorker-5: Traceback (most recent call last): File \"\/software\/lsstsw\/stack_20200515\/python\/miniconda3-4.7.12\/envs\/lsst-scipipe\/lib\/python3.7\/multiprocessing\/pool.py\", line 733, in next item = self._items.popleft() IndexError: pop from an empty deque \u00a0 During handling of the above exception, another exception occurred: \u00a0 Traceback (most recent call last): File \"\/scratch\/krzys001\/ap_verify\/bin\/ingest_dataset.py\", line 29, in result = runIngestion() File \"\/scratch\/krzys001\/ap_verify\/python\/lsst\/ap\/verify\/ap_verify.py\", line 238, in runIngestion ingestDatasetGen3(args.dataset, workspace, processes=args.processes) File \"\/scratch\/krzys001\/ap_verify\/python\/lsst\/ap\/verify\/ingestion.py\", line 628, in ingestDatasetGen3 ingester.run(processes=processes) File \"\/scratch\/krzys001\/ap_verify\/python\/lsst\/ap\/verify\/ingestion.py\", line 479, in run self._ensureRaws(processes=processes) File \"\/scratch\/krzys001\/ap_verify\/python\/lsst\/ap\/verify\/ingestion.py\", line 511, in _ensureRaws Traceback (most recent call last): File \"\/software\/lsstsw\/stack_20200515\/python\/miniconda3-4.7.12\/envs\/lsst-scipipe\/lib\/python3.7\/multiprocessing\/process.py\", line 297, in _bootstrap self.run() File \"\/software\/lsstsw\/stack_20200515\/python\/miniconda3-4.7.12\/envs\/lsst-scipipe\/lib\/python3.7\/multiprocessing\/process.py\", line 99, in run self._target(*self._args, **self._kwargs) File \"\/software\/lsstsw\/stack_20200515\/python\/miniconda3-4.7.12\/envs\/lsst-scipipe\/lib\/python3.7\/multiprocessing\/pool.py\", line 110, in worker task = get() File \"\/software\/lsstsw\/stack_20200515\/python\/miniconda3-4.7.12\/envs\/lsst-scipipe\/lib\/python3.7\/multiprocessing\/queues.py\", line 351, in get with self._rlock: File \"\/software\/lsstsw\/stack_20200515\/python\/miniconda3-4.7.12\/envs\/lsst-scipipe\/lib\/python3.7\/multiprocessing\/synchronize.py\", line 95, in __enter__ return self._semlock.__enter__() self._ingestRaws(dataFiles, processes=processes) KeyboardInterrupt File \"\/scratch\/krzys001\/ap_verify\/python\/lsst\/ap\/verify\/ingestion.py\", line 538, in _ingestRaws Traceback (most recent call last): File \"\/software\/lsstsw\/stack_20200515\/python\/miniconda3-4.7.12\/envs\/lsst-scipipe\/lib\/python3.7\/multiprocessing\/process.py\", line 297, in _bootstrap self.run() File \"\/software\/lsstsw\/stack_20200515\/python\/miniconda3-4.7.12\/envs\/lsst-scipipe\/lib\/python3.7\/multiprocessing\/process.py\", line 99, in run self._target(*self._args, **self._kwargs) File \"\/software\/lsstsw\/stack_20200515\/python\/miniconda3-4.7.12\/envs\/lsst-scipipe\/lib\/python3.7\/multiprocessing\/pool.py\", line 110, in worker task = get() File \"\/software\/lsstsw\/stack_20200515\/python\/miniconda3-4.7.12\/envs\/lsst-scipipe\/lib\/python3.7\/multiprocessing\/queues.py\", line 352, in get res = self._reader.recv_bytes() File \"\/software\/lsstsw\/stack_20200515\/python\/miniconda3-4.7.12\/envs\/lsst-scipipe\/lib\/python3.7\/multiprocessing\/connection.py\", line 216, in recv_bytes buf = self._recv_bytes(maxlength) File \"\/software\/lsstsw\/stack_20200515\/python\/miniconda3-4.7.12\/envs\/lsst-scipipe\/lib\/python3.7\/multiprocessing\/connection.py\", line 407, in _recv_bytes buf = self._recv(4) File \"\/software\/lsstsw\/stack_20200515\/python\/miniconda3-4.7.12\/envs\/lsst-scipipe\/lib\/python3.7\/multiprocessing\/connection.py\", line 379, in _recv chunk = read(handle, remaining) KeyboardInterrupt self.ingester.run(dataFiles, run=None, processes=processes) File \"\/software\/lsstsw\/stack_20200515\/stack\/miniconda3-4.7.12-46b24e8\/Linux64\/obs_base\/20.0.0-36-g2de6156+58b4951e8a\/python\/lsst\/obs\/base\/ingest.py\", line 443, in run exposureData = self.prep(files, pool=pool, processes=processes) File \"\/software\/lsstsw\/stack_20200515\/stack\/miniconda3-4.7.12-46b24e8\/Linux64\/obs_base\/20.0.0-36-g2de6156+58b4951e8a\/python\/lsst\/obs\/base\/ingest.py\", line 361, in prep exposureData: List[RawExposureData] = self.groupByExposure(fileData) File \"\/software\/lsstsw\/stack_20200515\/stack\/miniconda3-4.7.12-46b24e8\/Linux64\/obs_base\/20.0.0-36-g2de6156+58b4951e8a\/python\/lsst\/obs\/base\/ingest.py\", line 280, in groupByExposure for f in files: File \"\/software\/lsstsw\/stack_20200515\/python\/miniconda3-4.7.12\/envs\/lsst-scipipe\/lib\/python3.7\/multiprocessing\/pool.py\", line 737, in next self._cond.wait(timeout) File \"\/software\/lsstsw\/stack_20200515\/python\/miniconda3-4.7.12\/envs\/lsst-scipipe\/lib\/python3.7\/threading.py\", line 296, in wait waiter.acquire() \n\nDiscussion on #dm-middleware revealed that this functionality has never been tested. Investigate this bug in a pure-obs_base context and fix the problem. Unit tests for pickling support of RawIngestTask and related classes are a good starting point.\n\n#### Activity\n\nHide\nKrzysztof Findeisen added a comment -\n\nJim Bosch, can you be more specific? How does the proposed factory convert positional arguments into keyword arguments without knowing what they are? Or are you saying that it's impossible to decouple this code from Task?\n\nShow\nKrzysztof Findeisen added a comment - Jim Bosch , can you be more specific? How does the proposed factory convert positional arguments into keyword arguments without knowing what they are? Or are you saying that it's impossible to decouple this code from Task ?\nHide\nTim Jenness added a comment -\n\nHe's suggesting that from reduce you return a callable trampoline function, rather than the class, that takes positional arguments and then calls the real constructor with keyword arguments. This is a special hand-crafted trampoline function specifically for unpickling of this particular class.\n\nShow\nTim Jenness added a comment - He's suggesting that from reduce you return a callable trampoline function, rather than the class, that takes positional arguments and then calls the real constructor with keyword arguments. This is a special hand-crafted trampoline function specifically for unpickling of this particular class.\nHide\nKrzysztof Findeisen added a comment - - edited\n\nThis ticket ended up scope-creeping a bit to cover DefineVisitsTask as well, which had a similar structure and issues to RawIngestTask.\n\nUnfortunately, most parallelism support cannot be checked through unit tests, because calling either RawIngestTask.run or DefineVisitsTask.run requires a Camera object, which the hypothetical \"DummyCam\" doesn't have. For now, I've used ap_verify to check that everything appears to work with HSC; parallel ingestion should become part of our integration tests once DM-25806 merges.\n\nShow\nKrzysztof Findeisen added a comment - - edited This ticket ended up scope-creeping a bit to cover DefineVisitsTask as well, which had a similar structure and issues to RawIngestTask . Unfortunately, most parallelism support cannot be checked through unit tests, because calling either RawIngestTask.run or DefineVisitsTask.run requires a Camera object, which the hypothetical \"DummyCam\" doesn't have. For now, I've used ap_verify to check that everything appears to work with HSC; parallel ingestion should become part of our integration tests once DM-25806 merges.\nHide\nJim Bosch added a comment -\n\nLooks good!\n\nAny sense for whether the parallelization yields meaningful speedups?\n\nShow\nJim Bosch added a comment - Looks good! Any sense for whether the parallelization yields meaningful speedups?\nHide\nKrzysztof Findeisen added a comment - - edited\n\nSorry, it's hard to tell. I tried scaling up to the full HiTS-2015 dataset, but that revealed that the bottleneck (in ap_verify) was importing all the non-ingested datasets. I'll have to look into that separately (DM-26662).\n\nShow\nKrzysztof Findeisen added a comment - - edited Sorry, it's hard to tell. I tried scaling up to the full HiTS-2015 dataset, but that revealed that the bottleneck (in ap_verify ) was importing all the non-ingested datasets. I'll have to look into that separately ( DM-26662 ).\n\n#### People\n\nAssignee:\nKrzysztof Findeisen\nReporter:\nKrzysztof Findeisen\nReviewers:\nJim Bosch\nWatchers:\nJim Bosch, Krzysztof Findeisen, Tim Jenness","date":"2023-01-29 02:49:12","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 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\": 0.22090764343738556, \"perplexity\": 11752.66475049093}, \"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\/1674764499697.75\/warc\/CC-MAIN-20230129012420-20230129042420-00123.warc.gz\"}"}
| null | null |
"""
Module for parsing the results.
####
# Cluster parsing
####
<base_directory>
<param_iteration>
config.json # SP parameters
cv.pkl # CV splits
<param_iteration>-<cv_iteration>
config.json # SP parameters
stats.csv # Time and accuracy using features
p.pkl # SP permanences
syn_map.pkl # SP synaptic map
sp.pkl # SP instance
te_x.pkl # SP output for each training sample
tr_x.pkl # SP output for each testing sample
####
# Local parsing
####
<base_directory>
<param_iteration>-<cv_iteration>
config.json # SP parameters
stats.csv # Time and accuracy using features
p.pkl # SP permanences
syn_map.pkl # SP synaptic map
sp.pkl # SP instance
te_x.pkl # SP output for each training sample
tr_x.pkl # SP output for each testing sample
cv_results.pkl # Parameter names, parameter values, accuracies
cv_clf.pkl # Grid scores, best score, best params
####
# Storage
####
<param_iteration>, <cv_iteration>
G{packagetree mHTM}
"""
__docformat__ = 'epytext'
# Native imports
import json, os, pkgutil, csv, cPickle
# Third party imports
import numpy as np
# Program imports
from mHTM.parallel import create_runner, execute_runner
def save_results(results, p):
"""
Save the results.
@param results: The array containing the data to save.
@param p: The full path to where the data should be saved.
"""
with open(p, 'wb') as f:
cPickle.dump(results, f, cPickle.HIGHEST_PROTOCOL)
def load_results(p):
"""
Load the results.
@param p: The full path to where the data should be loaded from.
"""
with open(p, 'rb') as f:
return np.array(cPickle.load(f))
def get_missing_results(bp):
"""
Find any missing results.
@param bp: The full path to the directory containing the runs.
@return: A list of the missing experiments.
"""
missing = []
for p in sorted(os.listdir(bp)):
# Only work with valid runs
try:
param_iteration, cv_iteration = [int(x) for x in p.split('-')]
except ValueError:
continue
# Check to see if path exists
p2 = os.path.abspath(os.path.join(bp, p))
if not os.path.exists(os.path.join(p2, 'stats.csv')):
missing.append(p2)
return missing
def get_results(bp, score_method='column'):
"""
Get the results for the experiment.
@param bp: The full path to the directory containing the runs.
@param score_method: The method to used for computing the accuracy. This
must be one the following: "column" - Uses the set of active columns to
compute the accuracy, "prob" - Use the probabilistic version of the input,
or "reduction" - Uses the dimensionality reduced version of the input.
@return: The results and the base paths
"""
# Store the results
results = []
prev_param_iteration = None
method = 'Accuracy: ' + score_method
paths = []
# Get the data
for p in sorted(os.listdir(bp)):
# Only work with valid runs
try:
param_iteration, cv_iteration = [int(x) for x in p.split('-')]
except ValueError:
continue
# Read in the accuracy
with open(os.path.join(bp, p, 'stats.csv')) as f:
reader = csv.reader(f)
for row in reader:
key, value = row
if method in key:
accuracy = float(value)
# Add to data structure
if prev_param_iteration == param_iteration:
results[-1].append(accuracy)
else:
prev_param_iteration = param_iteration
paths.append(os.path.join(bp, p.split('-')[0]))
results.append([accuracy])
return np.array(results), paths
def get_top_paths(bp, k=10):
"""
Get the top paths for each method. This method uses the mean across the
CV runs.
@param bp: The full path to the directory containing the runs.
@param k: Select this many maximum from the results.
@return: A list of the paths with the top results.
"""
# Get top results for each type
top_paths = set()
methods = ('column', 'prob', 'reduction')
for method in methods:
accuracy, paths = get_results(bp, method)
for ix in accuracy.mean(1).argsort()[::-1][:k]:
top_paths.add(paths[ix])
return sorted(top_paths)
def get_top_path(bp):
"""
Get the top path out of all methods. This method uses the mean across the
CV runs.
@param bp: The full path to the directory containing the runs.
@param k: Select this many maximum from the results.
@return: A string containing the top path. This is the best path out of the
methods utilized.
"""
# Get top result
best, best_path = 0., None
methods = ('column', 'prob', 'reduction')
for method in methods:
accuracy, paths = get_results(bp, method)
avg_accuracy = accuracy.mean(1)
ix = avg_accuracy.argsort()[::-1][0]
result, path = avg_accuracy[ix], paths[ix]
if result > best:
best, best_path = result, path
return best_path
def launch_top_runs(top_paths, bp, command, auto_pupdate=False,
partition_name='debug', time_limit='04-00:00:00', memory_limit=2048):
"""
Launch the top runs.
@param top_paths: The full path to the base directory containing the top
results.
@param bp: The new base directory.
@param command: The base command to execute in the runner. Two additional
arguments will be passed - the base directory and the fold index.
@param auto_pupdate: If True the permanence increment and decrement amounts
will automatically be computed by the runner. If False, the ones specified
in the config file will be used.
@param partition_name: The partition name to use.
@param time_limit: The maximum time limit.
@param memory_limit: The maximum memory requirements in MB.
"""
for p in top_paths:
# Path where the run should occur
job_name = os.path.basename(p)
p2 = os.path.join(bp, job_name)
try:
os.makedirs(p2)
except OSError:
pass # Overwrite the files
# Create the runner
runner_path = os.path.join(p2, 'runner.sh')
command_new = '{0} "{1}" "{2}" {3}'.format(command, p, p2,
int(auto_pupdate))
stdio_path = os.path.join(p2, 'stdio.txt')
stderr_path = os.path.join(p2, 'stderr.txt')
create_runner(command=command_new, runner_path=runner_path,
job_name=job_name, partition_name=partition_name,
stdio_path=stdio_path, stderr_path=stderr_path,
time_limit=time_limit, memory_limit=memory_limit)
# Execute the runner
execute_runner(runner_path)
def launch_missing(missing, command, partition_name='debug',
time_limit='00-04:00:00', memory_limit=512):
"""
Launch the missing results on the cluster. It assumes that the convention
<run_instance>-<fold_instance> for the directories was utilized.
@param missing: The missing items.
@param command: The base command to execute in the runner. Two additional
arguments will be passed - the base directory and the fold index.
@param partition_name: The partition name to use.
@param time_limit: The maximum time limit.
@param memory_limit: The maximum memory requirements in MB.
"""
# Execute each missing item
for p in missing:
# Build the SP kargs for the proper path
bn, ix = os.path.basename(p).split('-')
bp = os.path.join(os.path.dirname(p), bn)
with open(os.path.join(bp, 'config.json'), 'rb') as f:
kargs = json.load(f)
kargs['log_dir'] = p
# Dump the arguments to a new file
s = json.dumps(kargs, sort_keys=True, indent=4,
separators=(',', ': ')).replace('},', '},\n')
with open(os.path.join(bp, 'config-{0}.json'.format(ix)), 'wb') as f:
f.write(s)
# Create the runner
runner_path = os.path.join(bp, 'runner-{0}.sh'.format(ix))
job_name = os.path.basename(p)
command_new = '{0} "{1}" {2}'.format(command, bp, ix)
stdio_path = os.path.join(bp, 'stdio-{0}.txt'.format(ix))
stderr_path = os.path.join(bp, 'stderr-{0}.txt'.format(ix))
create_runner(command=command_new, runner_path=runner_path,
job_name=job_name, partition_name=partition_name,
stdio_path=stdio_path, stderr_path=stderr_path,
time_limit=time_limit, memory_limit=memory_limit)
# Execute the runner
execute_runner(runner_path)
if __name__ == '__main__':
# Parameters to launch jobs
mnist_runner_path = os.path.join(pkgutil.get_loader('mHTM.examples').
filename, 'mnist_runner.py')
command = 'python "{0}"'.format(mnist_runner_path)
# Execute each type
types = ('global', 'local')
for type in types:
# Find and launch any missing jobs
bp = 'results/full_mnist-{0}'.format(type)
# missing = get_missing_results(bp)
# if len(missing) > 0:
# print missing
# launch_missing(missing, command, 'work')
# Launch the top jobs
# bp2 = 'results/full_mnist_auto-{0}'.format(type)
# top_paths = get_top_paths(bp)
# launch_top_runs(top_paths, bp2, command, False, 'work')
# Find the top job
print get_top_path(bp)
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 5,130
|
Q: How to combine background images? I have some pictures I am using on my website. I am trying to reduce the http requests. In my header I have my logo floated left and then another picture with special offers etc floated to the right in the same div. My question how do I combine these together and still work out the amount of space I need between them as I am using a fluid layout.
Would you also advise combining the main background image with these logos etc and if yes how do i do this so it still looks correct in a a fluid layout
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 6,209
|
"""
runscope_adapter.py
~~~~~~~~~~
Contains an implementation of an HTTP adapter for Requests that automatically
creates Runscope URLs for requests.
"""
import urlparse
from requests.adapters import HTTPAdapter
class RunscopeAdapter(HTTPAdapter):
"""
A Runscope-aware Transport Adapter for Python Requests. The central
portion of the API.
:param bucket_key: The Runscope bucket key to use for this request.
"""
bucket_key = ""
auth_token = None
def __init__(self, bucket_key, auth_token=None, **kwargs):
self.bucket_key = bucket_key
self.auth_token = auth_token
super(RunscopeAdapter, self).__init__(**kwargs)
def send(self, request, **kwargs):
"""
Sends a PreparedRequest object.
:param request: The Requests :class:`PreparedRequest <PreparedRequest>` object to send.
"""
request.url, port = self.proxify(request.url, self.bucket_key)
if port:
request.headers["Runscope-Request-Port"] = port
if self.auth_token:
request.headers["Runscope-Bucket-Auth"] = self.auth_token
return super(RunscopeAdapter, self).send(request, **kwargs)
def build_response(self, request, response):
"""
Builds a Response object from a urllib3 response.
:param request: The Requests :class:`PreparedRequest <PreparedRequest>` object sent.
:param response: The urllib3 response.
"""
resp = super(RunscopeAdapter, self).build_response(request, response)
return resp
def proxify(self, original_url, bucket_key, gateway_host="runscope.net"):
"""
Take a raw url string and turn it into a valid Runscope URL.
Before:
http://foo.example.com/path
After:
http://foo-example-com-bucket_key.<gateway_host>/path
"""
parts = urlparse.urlsplit(original_url)
# prevent double proxifying (e.g. when handling rewritten redirect locations)
if gateway_host in original_url:
return original_url, port if parts.port else None
clean_host = parts.hostname.replace("-", "~").replace(".", "-")
new_host = "{0}-{1}.{2}".format(clean_host, bucket_key, gateway_host).replace("~", "--")
if parts.username or parts.password:
new_host = "{0}:{1}@{2}".format(parts.username, parts.password, new_host)
port = None
if parts.port:
port = parts.port
return urlparse.urlunsplit((parts.scheme, new_host, parts.path, parts.query, parts.fragment)), port
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 8,168
|
Le district de Hartberg était une subdivision territoriale du land de Styrie en Autriche.
Géographie
Relief
Lieux administratifs voisins
Économie et infrastructures
Communes
Le district de Hartberg était subdivisé en 50 communes :
Bad Waltersdorf
Blaindorf
Buch-Geiseldorf
Dechantskirchen
Dienersdorf
Ebersdorf
Eichberg
Friedberg
Grafendorf bei Hartberg
Greinbach
Grosshart
Hartberg
Hartberg Umgebung
Hartl
Hofkirchen bei Hartberg
Kaibing
Kaindorf
Lafnitz
Limbach bei Neudau
Mönichwald
Neudau
Pinggau
Pöllau
Pöllauberg
Puchegg
Rabenwald
Riegersberg
Rohr bei Hartberg
Rohrbach an der Lafnitz
Saifen-Boden
Sankt Jakob im Walde
Sankt Johann bei Herberstein
Sankt Johann in der Haide
Sankt Lorenzen am Wechsel
Sankt Magdalena am Lemberg
Schachen bei Vorau
Schäffern
Schlag bei Thalberg
Schönegg bei Pöllau
Sebersdorf
Siegersdorf bei Herberstein
Sonnhofen
Stambach
Stubenberg am See
Tiefenbach bei Kaindorf
Vorau
Vornholz
Waldbach
Wenigzell
Wörth an der Lafnitz
Hartberg
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 735
|
\section{Introduction}
Synthetic aperture radar (SAR) is a form of radar that is used for creating 2D or 3D reconstructions of objects. It is generally used in remote sensing. It relies on electromagnetic waves in the microwave spectrum to generate images and hence can see through obstructions. SAR sensor is mounted on a moving platform that travels in the time taken by the emitted pulse to return to the camera, thereby creating a larger perceived aperture. The resolution of the image relies on the aperture of the sensor, regardless of the nature of the aperture, so it can generate relatively higher resolution images with a smaller aperture. SAR's ability to produce high-quality images at night and in adverse weather conditions with relatively smaller physical aperture provides an advantage over traditional optical and infrared imaging systems.
SAR images inherently suffer from a multiplicative noise, called speckle. This type of noise is the result of constructive and destructive interference of coherent reflections scattered by small reflectors inside each resolution cell. The effect of speckle tends to weaken for very high-resolution systems since the number of elemental scatterers within a resolution cell decreases \cite{sar-tutorial-1}. The presence of speckle noise makes image processing and computer vision tasks relatively difficult. Hence, it is important to remove speckle from SAR images to improve the performance and efficiency of various computer vision tasks such as segmentation, object detection, classification, and recognition.
In further sections, we discuss our study and provide our conclusions. In Section 2 we elaborate our study of SAR images, nature of speckle noise and existing algorithms used for speckle reduction. We also share our motivation behind curating a new dataset for this task. In Section 3 we give the reasoning for our approach towards designing the dataset. In Section 4, we train one of the state-of-the-art deep neural network based systems using our dataset and provide the visual and qualitative results for the same.
\section{Literature Study}
The most commonly used model to understand distributed scatterers causing the speckle noise in SAR images is \cite{goodman1976some}:
\begin{equation}
Y = N X,
\end{equation}
where $Y \in \mathbb{R}^{W \times H}$ is the observed image intensity, $X \in \mathbb{R}^{W \times H}$ is the noise-free image, and $N \in \mathbb{R}^{W \times H}$ is the normalized fading speckle noise random variable. Assuming that the SAR image is an average of $L$ looks, one common assumption on $N$ is that it follows a Gamma distribution with unit mean and variance $\frac{1}{L}$. It has the following probability distribution function \cite{ulaby2019handbook}
\begin{equation}
p(N) = \left(\frac{N L}{e}\right)^{L} \frac{e^{-N}}{N} \frac{1}{\Gamma(L)}, N \geq 0, L \geq 1,
\end{equation}
where $\Gamma(\cdot)$ denotes the Gamma function.
Researchers have used statistical techniques like multilooking \cite{oliver2004understanding} \cite{Jakowatz1996}, pixel based filtering \cite{lee-filter} \cite{kuan-filter} \cite{frost-filter} \cite{ppb}, wavelet based filtering \cite{sarbm3d}, etc. for removing speckle. Some of these methods \cite{sar-cnn} transfer the image into the logarithmic domain to transform multiplicative noise into additive noise, but it has its limitations which have been discussed in the upcoming section. Due to the non-local nature of the processing being carried out by some of these methods \cite{lee-filter} \cite{kuan-filter} \cite{frost-filter} \cite{ppb}, they are not able to preserve sharp features and edges, making the task of the further processing even more difficult.
In recent times there has been a meteoric rise in the usage of deep learning in image restoration and enhancement tasks like denoising and super-resolution. Efforts on despeckling have also been able to beat the state-of-the-art in this domain \cite{sar-cnn} \cite{id-cnn}, yet the reproducibility and reliability are very low due to the lack of a publicly available standard dataset for this task.\footnote {Unlike image classification, image segmentation, video classification, object detection, etc. no standard dataset is available on which researchers can develop neural network based SAR image despeckling techniques.}
Moving and Stationary Target Acquisition and Recognition (MSTAR) \cite{mstar} system has been used by many researchers to improve the performance of Automatic Target Recognition (ATR) systems operating over SAR images \cite{mstar-use1} \cite{mstar-use2}, but the images from this model have not been used for training denoising models due to the lack to diversity in images. As described in \cite{BALZ2015102}, a small number of commercial and open-source real-time simulators for SAR images are also available. However, they require a lot of time and computational power for generating the diverse and huge dataset, required by deep learning based image denoising techniques. An attempt to create a systematic framework for the evaluation of any general speckle reduction algorithm is available in G. Di Martino et al. \cite{6351163}, but it still does not resolve the problem of the lack of a training set for deep learning based despeckling techniques.
Authors of Image Despeckling Convolutional Neural Network (ID-CNN) \cite{id-cnn} and Fractional ID-CNN (FID-CNN) \cite{fid-cnn} have used a combination of Uncompressed Colour Image Dataset (UCID) \cite{ucid}, Berkeley Segmentation Data Set and Benchmarks 500 (BSDS500) \cite{bsd-500} and Northwestern Polytechnical University REmote Sensing Image Scene Classification (NWPU-RESISC45) \cite{nwpu-resisc45}, after adding a static amount of speckle noise, for training their models. In F. Lattari et al. \cite{sar-unet}, they have used images from UC Merced Land Use Dataset \cite{uc-merced} and PatternNet \cite{patternnet} to create training and testing datasets respectively. Hence, it is evident that there is a lot of variation in the datasets used by each research study for training and testing their algorithms, which makes it very difficult for anyone to objectively compare the results and determine which is the best algorithm.
It is relatively easy to find actual SAR images from the National Aeronautics and Space Administration's (NASA) Alaska Satellite Facility (Vertex portal) and the European Space Agency's (ESA) Copernicus portal. They have published a huge amount of data from various missions like Sentinel - 1 (C band) and ALOS PALASAR (L band) under open access. They also provide higher resolution data from the missions for research and study requests.
Even though actual SAR images are so freely available in the public domain, it is practically impossible to use them directly for training purposes. This is because all the captured SAR images inherently contain speckle noise and it is not possible to extract a completely noise free patches from actual data and create pairs of noisy and clean image patches required for training a neural network. Converting SAR images to a format usable by the deep learning frameworks is a slow process and requires significant computational power.
Aerial images are closely related to SAR images in terms of structure and type of features that they contain. This makes them the next best candidate for generating simulated SAR images. REmote Sensing Image Scene Classification (RESISC), created by Northwestern Polytechnical University (NWPU) \cite{nwpu-resisc45} is one of the largest collections of aerial images, containing images of 45 different classes with 700 images belonging to each class.
After a very comprehensive study, we are of opinion that research on deep neural networks in this domain is in a nascent state. We feel the need to establish an initial standard dataset which can be used by all the researchers to advance the field, because:
\begin{itemize}
\item SAR images are difficult to handle due to their size.
\item SAR images cannot be directly used for training deep neural networks due to a lack of pairs of clean and noisy images.
\item Currently, there is no standard reference dataset available for this task.
\end{itemize}
We try to address all these issues by developing the very first reference dataset, for this image processing task, as described in the next section.
\section{Proposed Dataset}
We have created a dataset using all the 31500 images from NWPU-RESISC45 \cite{nwpu-resisc45}. A. Moreira et al. \cite{sar-tutorial-1} have established that the variance of the speckle noise model depends upon the actual value of the signal, i.e. the value of noise and the amplitude of the original signal are correlated. F. Argenti et al. \cite{sar-tutorial-2} discuss the following alternative model used for representing speckle noise:
\begin{equation}
J = \eta I
\end{equation}
\begin{equation}
J = I + (\eta - 1) I
\end{equation}
\begin{equation}
J = I + K
\end{equation}
where $J$ is the observed image intensity, $I$ is the noise free image and $K = (\eta - 1)I$ accounts for the multiplicative speckle component of the SAR image and $\eta$ is randomly sampled from a uniform distribution with mean $\mu$ and variance $\sigma$. Furthermore, S. Abramov et al. \cite{Abramov14} suggests that $\sigma \in [0.55, 0.9]$.
We propose to divide images from each category of NWPU-RESISC45 dataset \cite{nwpu-resisc45} in a way that images of each category have noise belonging to the entire possible spectrum of values of $\sigma$ in the training set to make sure that each level of noise has enough diversity. We include the cases of lower noise as compared to the actual SAR images to induce a moderate amount of regularization while training the neural network. It helps the network to become robust while avoiding over-fitting the noisy images.
We are of the opinion that 37 images from publicly available USC-SIPI Image Database \cite{usc-sipi}, Volume 2: Aerials can be borrowed as the cross-validation set. The variance $\sigma$ can be set to any value, but we choose to use the default value provided by Matlab's implementation of $\texttt{imnoise()}$ function \cite{matlab:2018}.
\section{Experimental Results}
In this section, we provide details of the experiments we carried out using our synthetically generated dataset. We decided to baseline our dataset using a CNN architecture inspired by the current state-of-the-art, ID-CNN \cite{id-cnn}. A possible solution to the multiplicative nature of the noise would be to transfer the image to the logarithmic domain. However, that would introduce the problem of negative infinity values where-ever the value of a pixel is exactly $0$. Therefore, we decide to skip this step and try to learn a direct mapping from the original image to the residual noise and thereby to the clean image. P. Wang et al. \cite{id-cnn} and K. Zhang et al. \cite{7839189} provide more detailed reasoning for the design of the architecture. The discussion of the same is beyond the scope of this paper. We dropped the regularizing total variation (TV) term from the loss function as the clean images in the dataset already regularizing the training process. Fig.~\ref{fig:idcnn} shows the CNN architecture we borrowed from ID-CNN. The CNN was trained using ADAM optimizer with the default learning rate $0.001$ and mean squared error as the loss function. The model was trained on a private Kaggle kernel, which provides a CPU with 2 cores and an Nvidia Tesla P100 with 16 GB of RAM.
\begin{figure}[ht]
\captionsetup{justification=centering}
\centering
\includegraphics[width=0.5\textwidth]{images/architecture.pdf}
\caption{Architecture of ID-CNN}
\label{fig:idcnn}
\end{figure}
We compare the performance of the CNN with that of the following 5 despeckling algorithms: Lee filter \cite{lee-filter}, Kuan filter \cite{kuan-filter}, Frost filter \cite{frost-filter}, SAR-BM3D \cite{sarbm3d} and PPB \cite{ppb}. Note that all the parameters are set as suggested in their corresponding papers.
Fig.~\ref{fig:simar_outputs} and Table.~\ref{tab:metrics} contain the qualitative and quantitative results of our experiment respectively. The functions $\texttt{peak\char`_signal\char`_noise\char`_ratio()}$ and $\texttt{structural\char`_similarity()}$ from \textbf{scikit-image} package are used for measuring Peak Signal to Noise Ratio (PSNR) and Structural Similarity Index (SSIM) respectively. Equivalent Number of Looks (ENL) is a metric to compare the relative quality of the image after denoising in the absence of a ground-truth image. It is a parameter of multilook SAR images, which describes the degree of averaging applied to the SAR measurements during data formation and sometimes also post-processing. The following formula has been used for calculating ENL of the images \cite{oliver2004understanding}:
\begin{equation}
ENL = \frac{(mean)^{2}}{variance}
\end{equation}
The biggest motivation for using deep neural networks is to reduce the pre-processing time required for SAR image based pipelines. The time taken by all the methods to denoise a $1024 \times 1024$ image is given in Table~\ref{tab:metrics}. The experiments to measure the time taken for denoising were carried out on a high-performance mobile workstation running Windows 10 operating system. It is equipped with 32 GB of random access memory and Intel\textsuperscript{\textregistered} Core\textsuperscript{TM} i7-8650U CPU @ 2.11 GHz with 4 physical cores and 8 logical cores. We used Python 3.7.6 compiled for MSC v.1916 for a 64-bit processor and Matlab 2019b with Image Processing Toolbox.
\begin{figure*}[ht]
\captionsetup{justification=centering}
\centering
\subfloat[b][Groundtruth]{\includegraphics[width=0.24\textwidth]{images/03.png}}
\hspace{0pt}
\subfloat[b][Noisy Image]{\includegraphics[width=0.24\textwidth]{images/03_noisy.png}}
\hspace{0pt}
\subfloat[b][ID-CNN]{\includegraphics[width=0.24\textwidth]{images/03_idcnn.png}}
\hspace{0pt}
\subfloat[b][SAR-BM3D]{\includegraphics[width=0.24\textwidth]{images/03_sarbm3d.png}}
\hspace{0pt}
\subfloat[b][Lee Filter]{\includegraphics[width=0.24\textwidth]{images/03_lee.png}}
\hspace{0pt}
\subfloat[b][Kuan Filter]{\includegraphics[width=0.24\textwidth]{images/03_kuan.png}}
\hspace{0pt}
\subfloat[b][Frost Filter]{\includegraphics[width=0.24\textwidth]{images/03_frost.png}}
\hspace{0pt}
\subfloat[b][PPB Filter]{\includegraphics[width=0.24\textwidth]{images/03_non-iterative_ppb.png}}
\caption{Visual results of despeckling techniques on an image from USC SIPI Image Database \cite{usc-sipi}}
\label{fig:simar_outputs}
\end{figure*}
\begin{table*}[ht]
\captionsetup{justification=centering}
\caption{Quantitative results of despeckling techniques on an image from USC SIPI Image Database \cite{usc-sipi}}\label{tab:metrics}
\begin{tabular*}{\textwidth}{M{0.1\textwidth} | M{0.1\textwidth} | M{0.1\textwidth} | M{0.1\textwidth} | M{0.1\textwidth} | M{0.1\textwidth} | M{0.1\textwidth} | M{0.1\textwidth}}
\specialrule{.2em}{.1em}{.1em}
Metric & Noisy & Lee filter & Kuan filter & Frost filter & PPB & SAR-BM3D & CNN \\ \specialrule{.1em}{.05em}{.05em}
ENL & 11.237 & 20.852 & 20.950 & 21.080 & 30.351 & \textbf{34.990} & 31.362 \\
PSNR & 18.182 & 22.965 & 22.989 & 22.945 & 25.830 & 26.025 & \textbf{27.119} \\
SSIM & 0.294 & 0.457 & 0.458 & 0.453 & 0.594 & 0.617 & \textbf{0.661} \\
Time taken & - & 38.37 s & 57.89 s & 79.65 s & 137.28 s & 159.74 s & \textbf{3.22 s} \\ \specialrule{.2em}{.1em}{.1em}
\end{tabular*}
\end{table*}
\section{Conclusion}
Synthetic Aperture Radar (SAR) images have innumerable applications but cannot be used to their fullest till date due to a lack of better denoising techniques. We provide a novel synthetic dataset to accelerate the development of better deep neural networks for denoising of SAR images. We also demonstrate the usage of the synthetic data and improvement in the performance of the state-of-the-art convolutional neural network based speckle reduction system. It can potentially change the scope of work and research in this domain.
\section*{Conflict of Interest}
The authors declare no conflict of interest.
\section*{Acknowledgments}
We want to thank Prof. Anitha Modi from Insitute of Technology, Nirma University, India for her valuable comments, contributions and continued support to the project. We are would like to thank the National Aeronautics and Space Administration (NASA) and the European Space Agency (ESA) for providing the data from the Sentinel-1 and the ALOS PALSAR missions and the SNAP 7.0 software. We are grateful to all experts for providing us with their valuable insights and informed opinions ensuring completeness of our study.
\bibliographystyle{IEEEtran}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 8,013
|
using TransactionalNodeService.Proxy.Universal.Collections;
using TransactionalNodeService.Proxy.Universal.Collections.NodeContainers;
namespace TransactionalNodeService.Proxy.Universal
{
internal interface INodeManager
{
void Load(ConnectionSet connectionSet);
void Clear();
void UnionWith(NodeSet nodes);
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 2,638
|
Home News The first digital pill with ingestible sensor Approved by US FDA
The first digital pill with ingestible sensor Approved by US FDA
Digital Intent November 16, 2017 News,
The product, Abilify MyCite (aripiprazole tablets with sensor), was approved on Monday for the treatment of schizophrenia, bipolar disorder and for use as an add-on treatment for depression in adults.
The system works by sending a message from the pill's sensor to a wearable patch. The patch transmits the information to a mobile application so that patients can track the ingestion of the medication on their smartphone, the FDA said in a statement.
Patients can also permit their caregivers and physician to access the information through a web-based portal.
The approval of the so-called digital pill is expected to increase adherence to medication.
"The FDA supports the development and use of new technology in prescription drugs and is committed to working with companies to understand how technology might benefit patients and prescribers," Mathis said.
By Digital Intent at November 16, 2017
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 2,149
|
Q: Prove the set of sequences $c_0$ which converge to zero in $l_{\infty}$ is closed. Prove the set of sequences which converge to zero in $l_{\infty}$ is closed.
Let $x_n(k)\rightarrow x(k)$ as $n\rightarrow\infty$. With $x_n(k)\in c_0$ and $x(k)\in l_{\infty}$.
Let $\varepsilon>0$. Then there exists an $N>0$ such that
$$\parallel x_N-x\parallel_{\infty}:=\sup_{k\in\mathbb{N}}|x_N(k)-x(k)|\leq\varepsilon.$$
Then we have,
\begin{align}
|x(k)| &= |x(k) - x_N(k) + x_N(k)| \\\\
&\leq |x_N(k) - x(k)| + |x_N(k)| \\\\
&\leq \varepsilon + |x_N(k)|\rightarrow \varepsilon\;\; \text{as}\;\; k\rightarrow\infty.
\end{align}
Therefore since $\varepsilon$ was chosen arbitrarily we can conclude that $x(k)\rightarrow0$ and thus that $x(k)\in c_0$
Can someone check my work on this? It seems too slick and painless to be correct.
A: The solution is correct. Just to beef up this post, I'll sketch a slightly different proof: the complement of $l_0$ is open.
If $x\notin l_0$, let $r=\frac12\limsup_{k\to\infty} |x(k)|$. If $\|x-y\|\le r$, then $$\limsup|y(k)| \ge \limsup_{k\to\infty} |x(k)|-r =r$$ hence $y\notin l_0$.
By the way, this is the first time I see notation $l_0$ used for this subspace; all sources I know use $c_0$. I think $l_0$ is prone to confusion.
A: One more proof: $f:\ell_\infty \to \mathbb R$, $x\mapsto \lim\sup |x_n|$ is continuous so that $c_0= f^{-1}(\lbrace 0\rbrace)$ is closed.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 33
|
Q: Java Cucumber - Run Junit Maven project in Command line Trying to do some automated test. I have used below commands to execute the junit file (RunnerTest.java) in command line, getting an error as "could not find class"
Command line commands:
C:\Users\username\workspace\MavenCucumberPrototype\src\test\java\com\cucumber\MavenCucumberPrototype>javac -cp "C:/cjars/*" MavenCucumberPrototype/*.java
C:\Users\username\workspace\MavenCucumberPrototype>java -classpath C:/cjars/junit-4.12.jar org.junit.runner.JUnitCore src.test.java.com.cucumber.MavenCucumberPro
totype.RunnerTest
Maven Project structure
MavenCucumberPrototype
-/src/main/java
-com.cucumber.MavenCucumberPrototype
-/src/test/java
-com.cucumber.MavenCucumberPrototype
-postconn.java
-RunnerTest.java
-Steps.java
-/src/test/resource
-myfeature.feature
A: When I run Maven projects from a command line, I usually do
mvn clean install
or whatever lifecycle phase I want to execute. In your case, it might be sufficient to do
mvn test
The only imprtant thing to notice is that you must execute the command in the same directory as the pom.xml resides.
The error you recieve is nost likely due to a classpath that doesn't contain what you expected.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 5,763
|
Recovery Begins Here
We're open everyday 24/7
Free & confidential
Dual-Diagnosis
Treatment in Florida
Veterans Treatment
Is Focalin Addictive? What Are the Signs and Symptoms?
Home » Is Focalin Addictive? What Are the Signs and Symptoms?
What Is Focalin?
Signs of Stimulant Addiction
Focalin Addiction Treatment
How Dangerous Is Focalin?
Focalin Statistics
Focalin Vs. Adderall: What Are The Differences?
How to Recognize a Focalin Overdose (& Next Steps)
Snorting or Smoking Focalin – Is It Safe?
What Are the Short-Term and Long-Term Effects of Focalin Use?
Focalin is addictive. In fact, the nation's leading drug enforcement authority likens it to cocaine, opium, and morphine in terms of addiction potential.
Focalin, Ritalin, and Adderall — prescription stimulants that treat attention deficit hyperactivity disorder (ADHD) — are designated as Schedule II controlled substances, meaning they carry a high potential for abuse that may lead to psychological or physical dependence.
Put another way, Focalin and those other ADHD medications are among the most addictive substances that have a medical use, along with prescription opioids like OxyContin, Percocet, and Dilaudid.
A Focalin addiction does not carry the same stigma as say a prescription opioid or cocaine dependency. That's because people who abuse ADHD medications use them as study aids, or as a means of attaining high academic or professional achievement. In Western society, productivity and achievement are virtues.
Young people between the ages of 18-25 represent the largest contingent of recreational ADHD drug users. Use among high school-age students is also considerable.
High-schoolers who abused these medications, one student told The New York Times, are "the A students, sometimes the B students, who are trying to get good grades…They're the quote-unquote good kids, basically".
"Sky high" prescription revenue sales of ADHD drugs is further confirmation of their social acceptability. In 2016 alone, the combined revenue of Adderall XR, Focalin, and Ritalin was nearly $646 million, according to Statista, a statistics and market research database. Many of those prescriptions are diverted for sale.
For people with ADHD, Focalin, Ritalin, and Adderall produce a calming effect. For those that do not have the disorder, the medications provide them the 'kick' needed to power through all-night study sessions and homework assignments. The medication affords them enough energy to remain awake in order to take their exams on the following day.
While the long-term outcome of ADHD drug abuse for young people is not known, the short-term effects can range from uncomfortable to life-threatening. Read on to find out more about the signs and symptoms of Focalin addiction and treatment options.
Focalin is the brand name for the manmade central nervous system (CNS) stimulant dexmethylphenidate. The U.S. Food and Drug Administration (FDA) approved Focalin in 2001; an extended-release version of the medication was approved in 2005. The original Focalin comes in doses of 2.5 mg (milligrams), 5 mg, and 10 mg.
The immediate release version of Focalin lasts between four to six hours while the extended release form can sustain its effect for up to 12 hours, according to Drugs.com.
How It Differs from Ritalin And Adderall
Dexmethylphenidate is used to treat ADHD and is derived from methylphenidate, the medication that is marketed under the brand names of Ritalin and Concerta. Because dexmethylphenidate is the more active aspect of methylphenidate, Focalin is twice as strong as Ritalin and Concerta.
Like Adderall, Focalin treats ADHD. However, what differentiates it from Adderall is that it is not an amphetamine, and it has a separate mechanism of action. Adderall also treats narcolepsy. Still, both medications work by boosting the dopamine and norepinephrine neurotransmitters in the brain. Both also have a similar potency, and when abused, they produce a 'high' in users.
How a Focalin Addiction Starts
In fact, like Adderall, prolonged use of Focalin can spur someone to become psychologically dependent. A user could develop a tolerance to Focalin where they will require more of it in order to achieve a previous effect. The abrupt stoppage of the medication can lead to the development of painful and uncomfortable withdrawal symptoms, prompting someone to resume use.
Signs and Symptoms of Stimulant Addiction
Whenever a substance is officially sanctioned for having the same addiction potential as cocaine, the consequences of its use are serious. Like other stimulants, when Focalin is taken in excess or ingested in unintended ways (via injection or by snorting), it can produce incredibly dangerous effects.
The short-term signs of stimulant abuse include:
Increased alertness
Dangerously high body temperatures
Seizures (although these are more rare)
Depression (as part of the post-high "crash")
The sign of a growing addiction starts with an established tolerance followed by dependency. That's when a user's body adapts to the presence of Focalin. Without it, they will begin to experience withdrawal symptoms like nausea, vomiting, stomach cramps, trembling, unusual tiredness, and depression.
When addiction takes hold, they will exhibit visible behavioral signs, which include:
Prioritizing Focalin over hobbies, responsibilities, and relationships
Inability to perform daily tasks
A significant decline in work or school performance
Missing money or valuables to pay for Focalin
Trying to rationalize or make excuses for using Focalin
Lying or being secretive about Focalin use
Noticeable lack of concern with personal hygiene and appearance
Legal problems resulting from abusing Focalin
If you suspect that you or a loved one has a Focalin addiction, it is imperative that you undergo a medically supervised detox along with therapy and counseling to reverse the course. These services are available in professional addiction treatment.
Many people opt to stop using addictive substances on their own instead of seeking professional treatment. However, this is a path that often leads to relapse. The withdrawal symptoms, while not life-threatening, can produce enough discomfort to cause them to return to the drug. This is why professional addiction treatment is vital. It's a medically supervised, clinically sound approach to treating substance addictions.
The first step of professional treatment is a medically supervised detox, where the specific harmful substance and toxins are removed from the body. The stimulant detox and withdrawal process usually aren't severe. Thus, the procedure can be handled in an outpatient setting. However, you will be subject to medical supervision to ensure a safe and comfortable process.
You may also be administered medications to ease your withdrawal symptoms and ameliorate cravings. Common medications used in treatment include:
Antidepressants to minimize the feelings of depression and suicidal thoughts
Anticonvulsants in case of seizures that may occur during the initial withdrawal phase
Mild sedatives to combat the symptoms of insomnia, restlessness, and exhaustion
Anti-anxiety medication to keep stress levels low and lower the risk of panic attacks
After detox, the next step is recovery treatment. A typical recovery program will offer you the tools and skills needed to manage your Focalin addiction over the long haul while providing resources and support networks to help you achieve sustained recovery.
You will work with a therapist or counselor to tailor a treatment plan that will chart the most successful path toward your recovery. You will also be able to choose from a range of treatment options that include individual counseling, support groups, educational workshops, relapse prevention planning, and various individual therapies.
Do not let the veneer of Focalin as a doctor-prescribed medication fool you. Misuse can be dangerous. While Focalin isn't fatal in overdose, it can provoke uncomfortable effects.
The overdose symptoms from Focalin include:
Inappropriate happiness
Widening of pupils
Uncontrollable shaking of a part of the body
Hallucinations (seeing things or hearing voices that do not exist)
According to the 2011 National Survey on Drug Use and Health (NSDUH), 4.9 million people (1.9% of the population), 12 years or older, used methylphenidate or Focalin for non-medical purposes in their lifetime.
In 2016, Focalin and Ritalin generated $282 million in sales revenue, according to Statista.
Between 2005 and 2010, emergency department visits involving the non-medical use of ADHD stimulants like Focalin increased from 5,212 to 15,585 visits; those involving adverse reactions increased from 5,085 to 9,181 visits.
Get Help for Your Focalin Addiction Now
Focalin addiction does not have to control your life. Professional treatment can help you break its hold over you.
Call 855-534-3574 anytime, day or evening, for a free consultation with one of our knowledgeable addiction recovery specialists at The Palm Beach Institute. They can help you locate the right treatment option. Contact us online for more information.
(n.d.). Retrieved from https://www.deadiversion.usdoj.gov/schedules/
Dexmethylphenidate (Oral Route) Side Effects. (2018, October 01). Retrieved from https://www.mayoclinic.org/drugs-supplements/dexmethylphenidate-oral-route/side-effects/drg-20067099?p=1
Dexmethylphenidate: MedlinePlus Drug Information. (n.d.). Retrieved from https://medlineplus.gov/druginfo/meds/a603014.html
Focalin: Uses, Dosage, Side Effects & Warnings. (n.d.). Retrieved from https://www.drugs.com/focalin.html
Rodden, J., & Rodden, J. (2018, March 23). Focalin. Retrieved from https://www.additudemag.com/medication/focalin/
Delphi Health Group
contact@delphihealthgroup.com
We've Helped Thousands Overcome Addiction
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 8,014
|
Event management company, Spyder Lee Entertainment, on Tuesday 5th May 2015 presented winners as well as the runners up of the maiden edition of the GN Bank Awards with their prize packages.
The presentation was done at a colourful event held at the Coconut Grove Regency Hotel in Accra and was attended by the winners, representatives of GN Bank, title sponsors of the awards, executives of Group Nduom and key players within the Ghanaian Arts and entertainment Industry.
The runners up in each category received a certificate of recognition and an investment package of GHS1,000 with the Gold Coast Fund Management. Winners of each category had a prize package worth GHS 5,000 (GHS3,500 cash, a treat for two at the Coconut Grove Regency Hotel, Elmina and a certificate of recognition).
The biggest prize on the night was received by the Favourite Celebrity (Bola Ray of Starr FM) who received a cash amount of GHS 8,500, a treat for two at the Coconut Grove Regency Hotel, Elmina and a certificate of recognition.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 4,716
|
The Harvest Expression shows complex spice notes that balance the Juniper nicely. Cinnamon, lemon verbena, Christmas cake and so much more. This has a distinctly bold mouthfeel with notes of Kaffir lime, fresh cut violets and the essence of cloves and burnt orange peel. This is a gin one can actually sip on its own, and would taste great with just a simple dash of bianco vermouth. A straight gin for the "barrel-finished" gin lover, this one is complex enough, yet still refreshingly vivacious!
Brothers Don and Scott Avellino took an interesting approach to that idea and have begun producing seasonally inspired gins to showcase different botanical mixes that each make for a spirit with gusto! The bottlings are appropriately named "The Revivalist." Given gin's resurgence in the spirits world, it is a logical place for a small, craft distillery to start. Housed in a converted, restored barn, and nestled around 45 miles west of Philadelphia's City Center, Brandywine Branch Distillers began making artisan spirits from grain to bottle in 2016. They use a custom built alembic still and have installed fermentation tanks made of local Douglas Fir, definitely modeled on an "old school" operation. Don, Scott, and their team are dedicated to preserving the traditions of craft and producing spirits that discerning drinkers will want to return to time and again.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 2,128
|
About Semenggoh Orang Utan Centre : One of the World's best places to see semi-wild orangutans in their natural rainforest habitat. Semenggoh is nestled within a 6.8sq km Nature Reserve, home to 25 orangutans. 11 of whom were rescued from captivity or orphaned, 14 Semenggoh-born offsprings.
The adventure begins with the fuzzy "man of the forest" at the Semenggoh Orang Utan Centre. The centre is a 740 hectare stretch of virgin forest, home to the humble man of the forest, the "Orang Utan". Spot Orang Utans emerging from the rainforest during feeding time and learn how the facility functions as a Rehabilitation Centre, where most Orang Utans are orphans or rescued from injuries and illegal animal poaching.
Other than that, visitors will be able to witness other endangered species at Semenggoh. With a wildlife population that varies, it is difficult to say exactly what animals you may encounter. However, the center is a sanctuary to a wide range of wildlife, including rescued gibbons, porcupines, crocodiles and river terrapins. In the surrounding forest, you can even listen in on the cries of rehabilitated gibbons as well as the songs of wild birds.
Depart park and return to hotel.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 8,892
|
\section{Introduction}
Over the era of big data, large companies have the desire of building automated alerting system to continuously monitor the health of their sites and applications. Currently, many solutions are proposed to serve their needs, including Alibaba \citep{alibaba}, Amazon \citep{amazon}, Anodot \citep{anodot}, Baidu \citep{baidu}, AT\&T \citep{att2012}, Facebook \citep{facebook}, Google \citep{google1}, LinkedIn, Microsoft \citep{ren2019time}, Twitter \citep{hochenbaum2017automatic}, and Yahoo \citep{yahoo}.
At eBay, product owners monitor various metrics every day. Each metric potentially contains thousands of time series with different dimension aspects. To effectively monitor these millions of time series and alert automatically, we developed a two-phase approach with anomaly detection and alert retrieval to build eBay automated alerting system.
In our practice, there are three challenges for building the automated alerting system:
\begin{itemize}
\item Lack of true alert labels: it is demanding for domain analysts to label past anomalies in each time series for machine to learn,
\item Scalability and efficiency: we need to detect millions of time series every day and alert in time,
\item Avoid alert spamming: flooded alerts do not help users since not all finding outliers are valid alerts to all users.
\end{itemize}
To tackle the first two challenges, a fast and robust time series anomaly detection algorithm (MMD) is developed. It firstly decomposes a metric time series to "normal" patterns (e.g. trend, seasonality) and a noise/residual part. To gauge the noise level as well as filter out potential anomalies without assuming particular distribution of the noise part, we use Chebyshev's Inequality to determine the anomaly criteria that controls the overall false positive detection rate (type I error outside "normal" range). At the end of the anomaly detection phase, we obtained a list of time series with anomalies that are potentially used to alert users.
Secondly, to increase precision of the potential alerts and reduce alert spamming, we designed an alert retrieval framework that takes input from the anomaly detection phase, combined with additional rules and policies, to generate the final ranked or prioritized alert list to notify users.
In recent years, many anomaly detectors are proposed to tackle above issues, such as Isolation Forest \citep{liu2008isolation}, Argus \citep{att2012}, OC-SVM \citep{amer2013enhancing}, EGAD \citep{yahoo}, Opprentice \citep{liu2015opprentice}, VAE \citep{an2015variational}, RRCF \citep{amazon}, Donut \citep{alibaba}, SR-CNN \citep{ren2019time} and ATAD \citep{zhang2019cross}. Our time-series anomaly detection focuses more on statistical model as \citet{makridakis2018statistical} pointed out that the performance of ML models is lower to that of statistical methods in time-series area. Besides, many existing anomaly detection methods may cause alert spam since they only focus on anomalies without considering the second phase to retrieve valid alerts.
In this paper, we first explain our two-phase approach, especially the efficient and robust moving metric detector; then we discuss the evaluation of alert result and production performance; and conclude with some discussion.
\section{Approach}
Let's assume a business metric is measured daily and represented as a time series. Anomaly detection is to determine if the last observation of the metric is significantly different from what it is supposed to be given the past measurements of the time series. In Figure \ref{fig1}, we illustrate the overall alert detection and alert notification pipeline. The anomaly detection phase comprises of time series signal decomposition and statistical test for anomaly. The alert retrieval phase is to filter out valid alert from recently detected anomalies found in the previous phase.
\begin{figure}[h]
\centering
\includegraphics[width=0.9\columnwidth]{fig-two-phase}
\caption{The Two-Phase Alerting System.}
\label{fig1}
\end{figure}
\subsection{Anomaly Detection Phase}
Our basic assumption is that metric time series data should contain mostly "normal" observations than "abnormal" observations, and we do not require the abnormal data being labeled. In order to identify anomalies, the general scheme is to learn the "normal" patterns in the time series and use it as prediction to test if the last observation is within the "normal" metric value range.
\subsubsection{Time Series Decomposer}
As key part of anomaly detection phase, we decompose each time series to trend, seasonality and residual parts. There are existing time series decomposition implementations, however, they are either sensitive to outliers because using moving average, e.g. classical decomposition \citep{kendall1983advanced}, or with high computation cost of iterations, e.g. STL \citep{cleveland1990stl}. To extract "normal" patterns from time series containing sparse anomalies in the past but without labels, we developed our moving metric decomposer using median to extract robust trend and seasonality in the time series.
For a given time series $X$, we first do a rough estimate of trend ($L$) using symmetric moving average with a sliding time window ($w$), which is the number of observations per cycle in the given time-series $X$. To estimate the $w$ for each metric, we applied a one-time signal frequency estimation using ESPRIT method \citep{roy1989esprit} since the frequency of each metric rarely changes for almost all cases in eBay. Using the trend $L$, we can discover seasonality and reduce part of the variance in $X$.
Let's define trend-removed time series as $L^{\prime}=X-L$. To extract a robust seasonality ($S$), we use the median of historical periodic values of trend-removed time series $L^{\prime}$, instead of simple averaging which is sensitive to outliers.
\begin{equation*}
S_{t} = median(L_{t \pm iw}^{\prime} | i \leq \tau, i \in \mathbb{N})
\end{equation*}
\noindent where $w$ is the number of observations per cycle, $i$ is the number of cycles away from observation at $t$ and $\tau$ is the number of cycles near the observation at $t$ that is used to estimate seasonality $S_t$. In our case, we set $\tau$ as length of the input time series $X$ that contains all the observations.
With extracted seasonality $S$, we then estimate trend once again from the seasonality-removed series as $S^{\prime}=X-S$. We ignore the first estimate of trend $L$ and recompute trend from $S^{\prime}$ as $T$ in the following. Using right alignment to calculate rolling median in a sliding time window ($w$),
\begin{equation*}
T_t^f=median(S_{t-j}^{\prime} | j \leq w, j \in \mathbb{N})
\end{equation*}
\noindent where $j$ is the number of observations previous to observation at $t$. The $f$ indicates it is right alignment.
The right alignment rolling method naturally lags behind current observations, so we estimated a potential bias term using median in the following formula to model the lagging effect. Hence, the extracted trend contains two terms,
\begin{equation*}
T=T^f+median(S^{\prime}-T^f).
\end{equation*}
Compared with STL \citep{cleveland1990stl}, which iteratively estimates trend and seasonality with inner loop and outer loop to moderate outlier impact on trend and seasonality, our method first computes an approximated trend $L$ and extract seasonality $S$, and then extract the final trend $T$ with median as a robust replacement of average. Our method requires less computation with better and robust performance.
\subsubsection{Determine "Normal" Range of a Metric}
With time series decomposer, given time series $X$ is decomposed into trend ($T$), seasonality ($S$) and residual ($R$), which is defined as $R=X-T-S$. We detect anomaly as the most recent observation in a time series whose value differs significantly from the "normal" prediction. To detect anomaly, we have to calculate the "normal" range from the input time series $X$.
In many cases, the noise residual $R$ is assumed to be normally distributed. However, in our approach, we do not assume "normal" residuals distribution, i.e., whether it is Gaussian or not. By using Chebyshev's Inequality, for any real number $k>0$,
\begin{equation*}
P(|R- \hat{\mu} | \geq k\hat{\sigma}^2 ) \leq \frac{1}{k^2}
\end{equation*}
\noindent where the $\hat{\mu}$ is the expected value and $\hat{\sigma}$ is the standard deviation of the residual $R$. Using the above inequality formula, we can define false positive criteria to differentiate anomaly values outside of "normal" region.
The distribution-free "normal" range of $X$ can be calculated with a given expected probability $p$ of seeing an anomaly \citep{amidan2005data},
\begin{equation*}
T+S+\hat{\mu} \pm k\hat{\sigma}
\end{equation*}
\noindent where $k=\frac{1}{\sqrt{p}}$.
To estimate the $\hat{\mu}$ and $\hat{\sigma}$ for "normal" range, \citet{hampel1974influence} pointed out that mean and standard deviation are sensitive to outliers and \citet{leys2013detecting} suggested using median and median absolute deviation (MAD) as robust replacements, as follow:
\begin{align*}
& \hat{\mu} = median(R) \\
& \hat{\sigma} = b \times median(|R_t-median(R)|)
\end{align*}
\noindent where the robust standard deviation $\hat{\sigma}$ is estimated by scaling MAD with a constant $b$ (commonly set as $1.4826$).
Especially with our decomposition method, we have the $\hat{\mu} = median(R)=0$ due to the extra added estimated bias by $median(S^{'}-T^f)$. In our model, the k of the "normal" range can be interpreted as the number of standard deviations away from the prediction ($T+S$). With Chebyshev's Inequality, the "normal" range with $p=0.01$ would be interpreted as at most $1\%$ probability of exceed the range with $k=10$. In this case, we identify anomaly if the value is beyond $10$ standard deviation away from prediction.
\subsection{Alert Retrieval Phase}
Not all anomalies are valid actionable alerts for all users. In eBay practice, we can only send 10 alerts for each user every day, otherwise it will cause alert spamming. In the alert retrieval phase, we developed a ranking model and filtering rules and policies to retrieve valid alerts.
\subsubsection{Ranking of Anomalies}
To retrieve valid alerts and increase precision, we created a point-wise anomaly ranking algorithm with generalized linear ranking model to sort valid alerts from identified anomalies. The ranking score has several components, i.e.,
\begin{itemize}
\item Anomaly deviation severity feature computed using outputs from the previous phase as $f_d=|R-\hat{\mu} |/ \hat{\sigma}$ .
\item User defined 4-level priority (e.g. P1, P2, P3, P4) for each metric as $f_p$ (one-hot encoding).
\item Granularity of time series dimension aspect values as $f_g$, which is the number of dimensions that not rolled up in metric dimension hierarchy. For example, "Country: US" has granularity $f_g=1$ and "Country: US, Device: PC" has granularity $f_g=2$.
\end{itemize}
With above features, our ranking model is
\begin{equation*}
g(p)=w_d f_d+w_p f_p+w_g f_g
\end{equation*}
where $w_p$, $w_g$ and $w_d$ are the weight vector that can be learned from users feedbacks. The $g(p)$ is the ranking score function considering both anomaly severity and importance.
\subsubsection{Retrieval Logic and Rules}
Some anomalies that we detected are true and valid alerts to users, however, they are related and pointing to the same issue. From user perspective, it is one kind of alert spam if flooding very similar alerts to user. As a business requirement to consider the alerts diversity, our retrieval mechanism combined the diversity of the alerts and ranking score with following two steps:
\begin{itemize}
\item For each metric, detected anomalies are ranked by the ranking score g(p). We take the top anomaly from the list and compare the next highest ranked anomaly with the top one by computing the absolute Pearson correlation coefficient, if the coefficient is less than 0.9, which implies different time series pattern, we include it in the final alerts, if not we compare the next one on the list and so on. Overall, we select two alerts for each metric.
\item Retrieve at most 10 alerts across metrics and rank them with score g(p) from selected anomalies in each metric.
\end{itemize}
To reduce spam, many business rules are proposed from our user, such as stop alerting duplicated/similar anomalies in recent k days and alerting after k days continuous exceed interval, which achieved good result in eBay.
\section{Validation and Production Performance}
To evaluate our approach, we crowdsourced alert labels from product owners and domain analysts in eBay for a selected set of metrics to collect alert feedbacks (valid alert or not). Also, on our production environment, every alert that we sent out, are triaged and labeled as a valid alert if it is a truly useful and actionable issue with further investigation by the domain analysts.
For the crowdsourced labels, we showed the time series to some colleagues that labeled them independently. Here are the details of our crowdsourced labeling setup:
\begin{itemize}
\item Each person had the same amount ($15$) of time series to look at out of a pool of $164$ different time-series.
\item Each person was told to label all points that shall trigger alert for each $7$-month (from Jan to July) daily time series.
\item While random, the allocation of time series was designed in such a way that each metric was seen by at least two people.
\end{itemize}
At the end of above alert crowdsourcing collection, we received total $321$ alert feedbacks from $38$ different product owners and domain analysts in eBay.
\subsection{Crowdsourcing Labels Analysis}
It turned out that people, when looking at data independently have very differing views about what constituted a valid alert, despite the fact that they were all given the exact same, clear definition of an alert. In Figure \ref{fig2}, in approximately over half of the crowdsourced alert labels there was no common agreement (agreement ratio $\leq 0.50$) as to whether a given data point was an alert or not. Here, the agreement ratio of one alert is defined as the percentage of people who labeled the observation as an alert over the people who saw the time-series.
\begin{figure}[h]
\centering
\includegraphics[width=0.9\columnwidth]{fig-lack-of-agree}
\caption{Histogram of Agreement Ratio.}
\label{fig2}
\end{figure}
To understand the lack of agreement situation, we talk with domain analysts to build business rules to reduce spamming. For example, some domain's analysts do not expect alerts on just the first day exceed interval since there service exist auto-recovery mechanism, however, they are interested in the $3$ days continuous exceed interval, so we employed the "alerting after k days continuous exceed interval" rule.
\subsection{Validation of Anomaly Detection}
To keep only good quality data, we introduced majority vote rule (agreement ratio $> 0.5$) over the crowdsourced labels, therefore excluding a fair share of data points that were ties. Then, we compared our time series decomposition method with classical method and STL on anomaly detection task using our distribution-free "normal" range.
In order to weigh in recall more than precision during anomaly detection phase, we choose F2-Score as measurement since it focuses on recalling hundreds of potential alert candidates out of millions. We tuned model parameters with grid search that maximize the F2-Score for each anomaly detection method using different time series decomposer with data during Jan-May 2018, and ran the tuned model over an unseen testing set (Jun-Jul 2018), which we had held aside.
\begin{table}[h]
\centering
\scalebox{0.9}{
\begin{tabular}{lllllllllll}
\toprule
Method & Recall & Precision & F2-Score & Processing Time \\
\midrule
Classical & 64.86\% & 28.92\% & 0.519 & 252.1ms \\
STL & 59.46\% & 37.93\% & 0.534 & 156.8ms \\
Ours & 72.97\% & 38.03\% & 0.616 & 90.3ms \\
\bottomrule
\end{tabular}
}
\caption{Performance comparison on crowdsourcing dataset.}\label{table1}
\end{table}
Besides, we compared speed by using average processing time for every 100 time-series (no parallel computing enabled). The implementation of classical method and STL are the $decompose$ and $stl$ functions in R Package $stats$. In Table \ref{table1}, our decomposer shows better F2-Score performance over eBay Metrics with much faster processing speed.
\subsection{Production Performance}
Currently, our alert system monitors several million time-series and process hundreds of gigabytes of data every day. we implemented the two-phase solution on our private cloud as several micro-services, such as the data-driven anomaly detector, alerting rules tagger and ranking scorer. In figure \ref{fig3}, Function as a Service (FaaS) framework is adopted to scale up these micro-services as $\lambda$ functions in the anomaly detection phase. Then, the alert retrieval phase leveraged the tags and scores generated in the previous phase to filter and rank anomalies according to domain analysts' settings.
\begin{figure}[h]
\centering
\includegraphics[width=0.9\columnwidth]{fig-mmd-faas}
\caption{Architecture with Function as a Service Cluster.}
\label{fig3}
\end{figure}
With the 15 servers (each has 8 VCPUs and 16GB RAM) FaaS cluster, Our anomaly detector can handle at least 6.4 thousand time-series per minutes, and we can easily scale up the system by adding additional hardware if needed.
\begin{table}[h]
\centering
\scalebox{0.9}{
\begin{tabular}{lllllllllll}
\toprule
Approach & Weeks & \#Alert & \#Valid Alert & Precision \\
\midrule
First Phase Only & 10 & 164 & 67 & 40.85\% \\
Two Phases & 9 & 118 & 106 & 89.83\% \\
\bottomrule
\end{tabular}
}
\caption{Production performance on valid alert precision.}\label{table2}
\end{table}
To evaluate our alert result, we built an evaluation system to collect alert feedbacks that we sent out. Here, we compared the anomaly detection phase only with the two-phase approach. As in Table \ref{table2}, the extra alert retrieval phase dramatically improved the precision of valid alert.
\section{Conclusion}
To automate product health moving metric alert process at eBay, we developed a two-phase approach to identify anomalies and retrieve valid alerts for different domain users. At anomaly detection phase, we developed the Moving Metric Detector (MMD), which contains a fast and robust time-series decomposition algorithm with better F2-Score performance compared with classical method and STL on anomaly detection task. We leverage Chebyshev's Inequality to determine a distribution-free "normal" range. To avoid alert spamming and improve alert diversity, we designed the alert retrieval phase with a point-wise ranking model and business rules. Our alerting system is adopted by users across eBay and proved to be an effective way to early alarm issues for business needs.
\section{Acknowledgments}
We thank Rong Song, Christine Wu, Nathan Ni, Zhixuan Jia for their insights and expertise that greatly assisted the research, and continued support by eBay infrastructure organization. We would also like to show our gratitude to Giorgio Ballardin, Woody Zhou, Liren Sun, Jianpeng Xu and Jiahui Ruan for their stimulating discussions.
\section{Copyright}
All papers submitted for publication by AAAI Press must be accompanied by a valid signed copyright form. There are no exceptions to this requirement. You must send us the original version of this form. However, to meet the deadline, you may fax (1-650-321-4457) or scan and e-mail the form (pubforms20@aaai.org) to AAAI by the submission deadline, and then mail the original via postal mail to the AAAI office. If you fail to send in a signed copyright or permission form, we will be unable to publish your paper. There are \textbf{no exceptions} to this policy.You will find PDF versions of the AAAI copyright and permission to distribute forms in the AAAI AuthorKit.
\section{Formatting Requirements in Brief}
We need source and PDF files that can be used in a variety of ways and can be output on a variety of devices. The design and appearance of the paper is strictly governed by the aaai style file (aaai20.sty).
\textbf{You must not make any changes to the aaai style file, nor use any commands, packages, style files, or macros within your own paper that alter that design, including, but not limited to spacing, floats, margins, fonts, font size, and appearance.} AAAI imposes requirements on your source and PDF files that must be followed. Most of these requirements are based on our efforts to standardize conference manuscript properties and layout. All papers submitted to AAAI for publication will be recompiled for standardization purposes. Consequently, every paper submission must comply with the following requirements:
\begin{quote}
\begin{itemize}
\item Your .tex file must compile in PDF\LaTeX{} --- ( you may not include .ps or .eps figure files.)
\item All fonts must be embedded in the PDF file --- including includes your figures.
\item Modifications to the style file, whether directly or via commands in your document may not ever be made, most especially when made in an effort to avoid extra page charges or make your paper fit in a specific number of pages.
\item No type 3 fonts may be used (even in illustrations).
\item You may not alter the spacing above and below captions, figures, headings, and subheadings.
\item You may not alter the font sizes of text elements, footnotes, heading elements, captions, or title information (for references and tables and mathematics, please see the the limited exceptions provided herein).
\item You may not alter the line spacing of text.
\item Your title must follow Title Case capitalization rules (not sentence case).
\item Your .tex file must include completed metadata to pass-through to the PDF (see PDFINFO below)
\item \LaTeX{} documents must use the Times or Nimbus font package (you may not use Computer Modern for the text of your paper).
\item No \LaTeX{} 209 documents may be used or submitted.
\item Your source must not require use of fonts for non-Roman alphabets within the text itself. If your paper includes symbols in other languages (such as, but not limited to, Arabic, Chinese, Hebrew, Japanese, Thai, Russian and other Cyrillic languages), you must restrict their use to bit-mapped figures. Fonts that require non-English language support (CID and Identity-H) must be converted to outlines or 300 dpi bitmap or removed from the document (even if they are in a graphics file embedded in the document).
\item Two-column format in AAAI style is required for all papers.
\item The paper size for final submission must be US letter without exception.
\item The source file must exactly match the PDF.
\item The document margins may not be exceeded (no overfull boxes).
\item The number of pages and the file size must be as specified for your event.
\item No document may be password protected.
\item Neither the PDFs nor the source may contain any embedded links or bookmarks (no hyperref or navigator packages).
\item Your source and PDF must not have any page numbers, footers, or headers (no pagestyle commands).
\item Your PDF must be compatible with Acrobat 5 or higher.
\item Your \LaTeX{} source file (excluding references) must consist of a \textbf{single} file (use of the ``input" command is not allowed.
\item Your graphics must be sized appropriately outside of \LaTeX{} (do not use the ``clip" or ``trim'' command) .
\end{itemize}
\end{quote}
If you do not follow these requirements, you will be required to correct the deficiencies and resubmit the paper. A resubmission fee will apply.
\section{What Files to Submit}
You must submit the following items to ensure that your paper is published:
\begin{itemize}
\item A fully-compliant PDF file that includes PDF metadata.
\item Your \LaTeX{} source file submitted as a \textbf{single} .tex file (do not use the ``input" command to include sections of your paper --- every section must be in the single source file). (The only allowable exception is .bib file, which should be included separately).
\item The bibliography (.bib) file(s).
\item Your source must compile on our system, which includes only standard \LaTeX{} 2018-2019 TeXLive support files.
\item Only the graphics files used in compiling paper.
\item The \LaTeX{}-generated files (e.g. .aux, .bbl file, PDF, etc.).
\end{itemize}
Your \LaTeX{} source will be reviewed and recompiled on our system (if it does not compile, you will be required to resubmit, which will incur fees). \textbf{Do not submit your source in multiple text files.} Your single \LaTeX{} source file must include all your text, your bibliography (formatted using aaai.bst), and any custom macros.
Your files should work without any supporting files (other than the program itself) on any computer with a standard \LaTeX{} distribution.
\textbf{Do not send files that are not actually used in the paper.} We don't want you to send us any files not needed for compiling your paper, including, for example, this instructions file, unused graphics files, style files, additional material sent for the purpose of the paper review, and so forth.
\textbf{Do not send supporting files that are not actually used in the paper.} We don't want you to send us any files not needed for compiling your paper, including, for example, this instructions file, unused graphics files, style files, additional material sent for the purpose of the paper review, and so forth.
\textbf{Obsolete style files.} The commands for some common packages (such as some used for algorithms), may have changed. Please be certain that you are not compiling your paper using old or obsolete style files.
\textbf{Final Archive.} Place your PDF and source files in a single archive which should be compressed using .zip. The final file size may not exceed 10 MB.
Name your source file with the last (family) name of the first author, even if that is not you.
\section{Using \LaTeX{} to Format Your Paper}
The latest version of the AAAI style file is available on AAAI's website. Download this file and place it in the \TeX\ search path. Placing it in the same directory as the paper should also work. You must download the latest version of the complete AAAI Author Kit so that you will have the latest instruction set and style file.
\subsection{Document Preamble}
In the \LaTeX{} source for your paper, you \textbf{must} place the following lines as shown in the example in this subsection. This command set-up is for three authors. Add or subtract author and address lines as necessary, and uncomment the portions that apply to you. In most instances, this is all you need to do to format your paper in the Times font. The helvet package will cause Helvetica to be used for sans serif. These files are part of the PSNFSS2e package, which is freely available from many Internet sites (and is often part of a standard installation).
Leave the setcounter for section number depth commented out and set at 0 unless you want to add section numbers to your paper. If you do add section numbers, you must uncomment this line and change the number to 1 (for section numbers), or 2 (for section and subsection numbers). The style file will not work properly with numbering of subsubsections, so do not use a number higher than 2.
If (and only if) your author title information will not fit within the specified height allowed, put \textbackslash setlength \textbackslash titlebox{2.5in} in your preamble. Increase the height until the height error disappears from your log. You may not use the \textbackslash setlength command elsewhere in your paper, and it may not be used to reduce the height of the author-title box.
\subsubsection{The Following Must Appear in Your Preamble}
\begin{quote}
\begin{scriptsize}\begin{verbatim}
\documentclass[letterpaper]{article}
\usepackage{aaai20}
\usepackage{times}
\usepackage{helvet}
\usepackage{courier}
\usepackage[hyphens]{url}
\usepackage{graphicx}
\urlstyle{rm}
\def\rm{\rm}
\usepackage{graphicx}
\frenchspacing
\setlength{\pdfpagewidth}{8.5in}
\setlength{\pdfpageheight}{11in}
\pdfinfo{
/Title (Type Your Paper Title Here in Mixed Case)
/Author (John Doe, Jane Doe)
/Keywords (Input your keywords in this optional area)
}
\title{Title}\\
\author\{Author 1 \ and Author 2\\
Address line\\
Address line\\
\ And\\
Author 3\\
Address line\\
Address line
}\\
\end{verbatim}\end{scriptsize}
\end{quote}
\subsection{Preparing Your Paper}
After the preamble above, you should prepare your paper as follows:
\begin{quote}
\begin{scriptsize}\begin{verbatim}
\begin{document}
\maketitle
\begin{abstract}
\end{abstract}\end{verbatim}\end{scriptsize}
\end{quote}
\subsubsection{The Following Must Conclude Your Document}
\begin{quote}
\begin{scriptsize}\begin{verbatim}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 4,294
|
Q: My website on checking showing 403 error on HTTP Status My website is running fine but when I check on the https://httpstatus.io/, it is showing the 403 error and also on the Google page insights, it is not showing the report.
My website DNS is currently pointing to the Cloudflare and previously I have installed the Cloudflare plugin in my Wordpress and now I have uninstalled it.
If I removed my website from Cloudflare, the problem will be solved or not, Or it will affect my website.
Any help is much appreciated. Thanks In Advance.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 6,708
|
15 Warning Signs That Your Kid May Be Using Drugs
Written by: American Addiction Centers Editorial Staff
Kids are a source of worry for every parent. You look after them for years, and you hope that they end up turning our alright. While there are hundreds of books and articles on how to raise your kids, few really work that well, and it's all down to trial and error. Of course, if your kids end up taking drugs and getting caught, they could end up facing a trail for their errors.
Bad puns aside, it's clear that drug education does a lot for some and little for others. Indeed, government-sponsored drug education programs tend to be somewhat weak. It's therefore vital that you watch for the common signs of drug use in your kids.
Possession of the drug itself is a dead giveaway. While marijuana is fairly distinctive, how do you tell whether a pill has been prescribed or not? The Internet is usually a good resource. Look for the symbol on the pill. Something marked OP will likely be OxyContin, for example. Identify the pill and see what comes up. Alternatively, ask your teen.
Odd smells are another sign. It could be a new interest in deodorant or a heady smell of marijuana-laced smoke. If you don't know what marijuana smells like, it's time to educate yourself. We don't suggest smoking it yourself, but you may be able to ask a friendly cop to show you a sample.
Paraphernalia for drug taking include roll-ups and tin boxes. For other drugs, it could be syringes and burnt teaspoons. If you see a tin box, open it and take a sniff. If it smells like tobacco, it probably is. If it smells of something else, ask your kid about it.
Rapidly changing grades are one of the common consequences of drug addiction or use. If you kid goes from being a straight-A student to getting F's or D's, something's changed. Of course, it could be linked to a number of factors, so tread carefully here.
Glazed expressions may be a sign of addiction, but with some teens, it's hard to tell. Teenagers and twenty-somethings tend not to be the most communicative of creatures, but if your kid starts looking stoned all the time and are accompanied by any of the other factors listed, it's entirely possible he or she is stoned.
Abandoning friends is quite common throughout the teenage years, but it could have a more sinister implication. If your kid starts hanging out with a different crowd who smoke and so on, it could be a phase, but it could be linked to drugs.
Abandoning social activities is another potential sign of drug abuse. Again, interests change throughout your kid's formative years, so tread lightly. It might just be related to a change of tastes.
Evasive answers to questions of where your kid has been can sometimes be linked to drugs. As a parent, you'll never know all the aspects of your kid's life, and sometimes it could be related to your kid's interest in dating.
Behavioral changes are quite common with kids who take drugs. While the moody teen is a stereotype, it's one that holds true. If your kid is jittery in the morning and calmer in the evening, he or she could be taking drugs.
Memory problems sometimes herald drug use. While everyone forgets stuff, if your kid has problems remembering basic things, you might need to question further. Of course, it could also be a sign of medical issues, such as ADHD.
Unexplained injuries can be related to drug or alcohol use. Just as above, however, they could also be related to medical issues or even bullying.
Items or money going missing around your house might mean that your kid is stealing to fund a habit. Keep an open mind, however, as it's just as likely to be a partner or someone else stealing them.
Weight changes are a normal part of teen life, but rapid fluctuations could indicate an addiction. Some teens neglect to eat due to drugs or spend lunch money on an addiction rather than eat.
Your child is more likely to get ill if he or she takes drugs, as the side effects of some drugs partially suppress the immune system. Inhaled drugs can also lead to respiratory problems.
Staying out late is a typical teenage habit, but in combination with things listed above, it's possible that this could be an indication of drug abuse. Of course, it's most likely that the drug of choice is alcohol in this case.
American Addiction Centers Editorial Staff
The editorial staff of DrugAbuse.com is comprised of addiction content experts from American Addiction Centers . Our editors and medical reviewers have over a decade of cumulative experience in medical content editing and have reviewed thousands of pages for accuracy and relevance. Our reviewers consistently monitor the latest research from SAMHSA, NIDA, and other reputable sources to provide our readers the most accurate content on the web.
•residential drug treatment center
•detox for drugs
•drug rehab without insurance
•solutions drug treatment
•substance abuse treatment using ibogaine
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 8,532
|
El municipio de Silver Lake (en inglés: Silver Lake Township) es un municipio ubicado en el condado de Martin en el estado estadounidense de Minnesota. En el año 2010 tenía una población de 519 habitantes y una densidad poblacional de 5,35 personas por km².
Geografía
El municipio de Silver Lake se encuentra ubicado en las coordenadas . Según la Oficina del Censo de los Estados Unidos, el municipio tiene una superficie total de 97.06 km², de la cual 91,24 km² corresponden a tierra firme y (5,99 %) 5,81 km² es agua.
Demografía
Según el censo de 2010, había 519 personas residiendo en el municipio de Silver Lake. La densidad de población era de 5,35 hab./km². De los 519 habitantes, el municipio de Silver Lake estaba compuesto por el 97,88 % blancos, el 0,19 % eran afroamericanos, el 0,19 % eran asiáticos, el 1,16 % eran de otras razas y el 0,58 % eran de una mezcla de razas. Del total de la población el 1,54 % eran hispanos o latinos de cualquier raza.
Referencias
Enlaces externos
Municipios de Minnesota
Localidades del condado de Martin (Minnesota)
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 9,127
|
Q: Highcharts data showing wrong I am trying to build a chart for tracking my project status. I have given input as timestamp(in 2018) but it is showing wrong year(which is from 1969).
The code is as follows
<script>
Highcharts.chart('container', {
chart: {
type: 'bar'
},
title: {
text: 'Project Status'
},
xAxis: {
categories: ['BOT POC', 'Analytics POC', 'CRM POC']
},
yAxis: {
type: 'datetime',
labels: {
rotation: -65,
style: {
fontSize: '9px',
fontFamily: 'Verdana, sans-serif'
}
},
crosshair: true,
dateTimeLabelFormats: {
week:"Week from %A, %b %e, %Y",
}
},
tooltip: {
formatter: function() {
return '<b>'+ this.series.name +'</b><br/>'+
this.x +': '+ this.y;
}
},
legend: {
reversed: true
},
plotOptions: {
series: {
stacking: 'normal'
}
},
series: [{
name: 'Scoping In Progress',
data: [1528635194, 1528203194, 1527252794]
}, {
name: 'Scoping Complete',
data: [1528807994, 1528375994, 1527511994]
}, {
name: 'Development in Progress',
data: [1528894394, 1528462394, 1527943994]
}]
});
</script>
The result is as follows
How can the date brought to current date/year?
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 1,793
|
Q: Adding headers after RESTAdapter initialization I am trying to add an Authorization header to my adapter's request after the adapter has been initialized and used. I can add headers in a static way at the time I create my ApplicationAdapter, but I can't seem to get it use the headers in subsequent REST calls. I am trying this:
var auth= "Basic " + hash;
App.ApplicationAdapter.reopen({
headers: {
Authorization: auth
}
});
I have debugged RESTAdapter in the ajax method, and the test for adapter.headers is always undefined.
A: The accepted answer doesn't address the fact that the recommended approach is not working in ember-data. I say recommended since:
https://github.com/emberjs/data/blob/master/packages/ember-data/lib/adapters/rest_adapter.js#L88
https://github.com/emberjs/data/blob/master/packages/ember-data/lib/adapters/rest_adapter.js#L162
and other places in that file.
Further, the issue the OP brings up with of undefined specifically happens here:
https://github.com/emberjs/data/blob/master/packages/ember-data/lib/adapters/rest_adapter.js#L619
So, the following simply does not work:
App.ApplicationAdapter.reopen({
headers: {token: 'reopen_token (NO WORK)' }
});
I've tried to point to this out as an issue but it got closed within an hour:
https://github.com/emberjs/data/issues/1820
Hopefully core will decide to either fix this or remove the comments. But, yes, for now it seems you have to hijack jQuery ajax setup, Ember.$.ajaxPrefilter, or override the ajax on the adapter yourself.
EDIT: So after getting some more feedback from Ember devs, it looks like the core of this issue is trying to reopen an instance already created. So using a computered property when it's defined (so it will update as desired) seems to be the advised approach. Hope that helps (there's a recently merged pull request that makes this more obvious in the comments of referenced file:https://github.com/emberjs/data/pull/1818/files#diff-1d7f5a5b77898df15de501c3c38d4829R108 )
EDIT 2: Got this working in my app so here's the code in case someone else gets stuck:
//app.js
App.ApplicationAdapter = DS.ActiveModelAdapter.extend({
namespace: 'api/v1',
headers: function() {
return {
token: this.get('App.authToken') || localStorage.getItem('token')
};
}.property("App.authToken")
});
//login-controller.js (only action shown..assume `data` has user/pass)
actions: {
login: function() {
$.post('/token/', data).done(function(user) {
App.set('authToken', user.token);
//Above will trigger adapters's header computed property to update
// Transition to previous attempted route
var attemptedTransition = self.get('attemptedTransition');
if(attemptedTransition) {
attemptedTransition.retry();
}
else {
self.transitionToRoute('yourapproute');
}
})
.fail(function(response) {
//fail handling omitted
});
A: The answers are already introduced in official API document.
http://emberjs.com/api/data/classes/DS.RESTAdapter.html#toc_headers-customization
*
*Use computed property with session injection
*or just use volatile computed property
A: You should be able to use $.ajaxPrefilter to add custom headers (or params).
See: http://api.jquery.com/jQuery.ajaxPrefilter/
Ember.$.ajaxPrefilter(function( options, oriOptions, jqXHR ) {
var auth= "Basic " + hash;
jqXHR.setRequestHeader("Authorization", auth);
});
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 4,084
|
{"url":"https:\/\/scienceprog.com\/modeling-of-analog-part-for-dds3-signal-generator\/","text":"# Modeling of analog part for DDS3 signal generator\n\nWhen building AVR DDS2 signal generator there were lots of discussions about signal conditioning in analog part of device. First argument was that LM358 wasn\u2019t the best choice for this purpose. Another one pointed to sine wave that weren\u2019t smooth enough.\n\nAs you can see there are some dents on it. Other waveforms also are distorted especially when higher voltages are selected. This definitely asks for better analog part. Some people suggested to replace LM358 with OPA2134, but it seems to be quite expensive choice. In my opinion low noise general purpose op-amp can be great too. I\u2019m gonna give a try to Texas Instruments TL074 low noise op-amp. It is low power, high slew rate (13V\/us) IC \u2013 almost five times faster than LM358 and for same reasonable price.\n\nIf we look at DDS2 signal generator offset and gain control circuit we can see that there is a problem in it. We regulate offset before amplitude control \u2013 normally offset should be regulated after amplifying signal. And third problem is that offset can be regulated in small interval \u2013 in 5V range while we could do this in -12V \u2026 +12V range. Lets redesign analog part so it would give better result.\n\n### Zeroing offset voltage\n\nFirst of all offset control we can slightly modify first stage so that offset voltage of signal would be set to 0. For instance if we are generating sine-wave on DAC we get maximum 2.5V amplitude signal with 2.5V DC offset. Instead of using POT1 we can calculate resistor divider so that on output we get 0V offset. We assume that all signals coming out of microcontroller have offset 2.5V (when VCC=5V), then we can model following circuit:\n\nCalculating is easy:\n\nWe know that source voltage is Vs=2.5V. Output voltage is Vo=0V. We need gain 1 (-1 for inverting amp). So at inverting op \u2013 amp input we get voltage:\n\n$V_2 = (V_S - V_0)\\frac{R1}{R1+R2} = 1.25V$\n\nFrom from operational amp analysis we know that V- = V+ and currents to both inputs are equal to zero (I- = I+ = 0A). Then We must get 1.25V at V3. so we need to select voltage divider to scale voltage from 5V to 1.25. To do so we chose one fixed resistor value R3 to be 100kOhms and then we calculate R4:\n\n$V_3 = \\frac{R4}{R3+R4}VCC$\n\nevaluate R4:\n\n$R4 = V_3\\frac{R3}{VCC-V_3} = 33333\\Omega$\n\nWe can get 33.33k\u03a9 by connecting two resistors in series: 33k\u03a9 + 330\u03a9 that will give resulting voltage close enough to 0V.\n\n### Optional filter stage\n\nIn the next stage of analog part we are going to add a filter. Since DDS signal generator is capable of outputting various types of signals we don\u2019t want it to be hardwired. For instance if we generate sine wave the we could use filter to make it more smooth while on square wave it would have negative effect by rounding the shape. For this we can add a bypass switch that would allow to insert or bypass filter on demand. What filter are we going to use for this purpose. Since we have plenty of op-amps on TL074 we can make an active filter.\n\nI find Butterworth (Sallen \u2013 Key) low pass filter quite handy since it gives flattest response in band. Lets select parameters for this filter. DDS generator in our case isn\u2019t going to exceed 100kHz frequency. Also we want attenuation in this up to roll \u2013 off frequency close 0dB. So lets increase roll-off frequency up to 200kHz. This value we are going to use while calculating.\n\nResistor and capacitor values of filter can be determined with these simple rules:\n\nR1 = R2, C1 = 2\u0387C2\n\n$f= \\frac{\\sqrt{2}}{{4}\\cdot{\\pi}\\cdot{R2}\\cdot{C2}}$\n\nIf we select R2 = R1 = 33k (have plenty of these)\n\nthen we can calculate C2 as follows:\n\n$C2=\\frac{\\sqrt{2}}{{4}\\cdot{\\pi}\\cdot{R2}\\cdot{f}} = 17pF\\approx{18pF}$\n\nso standard value of C1 = 33pF.\n\nYou can check calculations on line here.\n\n### Signal amplitude gain control\n\nAfter we have adjusted signal offset voltage filtered it (or not) the next part is to adjust amplitude. We need to adjust signal amplitude from 0 to 12V. For this purpose we are gonna use inverting amplifier with potentiometer adjustable gain. Wee need to calculate proper resistor values to get nice gain control over all potentiometer turn. Say we are going to use 47k potentiometer. Lets calculate input resistor value.\n\nAnother known condition is that input voltage from previous stage is 2.5V. Say we want to get 12V amplitude on output we need gain: 12\/2.5 = 4.8. If we turn potentiometer to the max left we get:\n\nR1=47k\/4.8=9.79k~10k. In order to get 0V amplitude we just need to turn potentiometer to the right so gain ration gets close to 0.\n\n### Signal offset regulation\n\nAnd last stage of analog part is signal offset regulation. We want to regulate offset in range of -12V to 12V. The easiest way to do so is to add offset voltage to signal voltage.\n\nSince we already have two inverting cascades we don\u2019t want the last one to be inverted that would lead to inverted signal on output. So we are going to implement non inverting summing:\n\nLets see how to calculate resistor values. R6 and R7 we select to be 100k resistors as they aren\u2019t critical while in recommended range 1k to 1M. More interesting part is gain. Lets see how non inverting summing amplifier output voltage look like:\n\n$Vout=(V_1\\frac{R7}{R6+R7}+V_2\\frac{R6}{R6+R7})\\cdot(1+\\frac{R9}{R8})$\n\nWhat we see here is a voltages added and multiplied by gain. Since our R7 and R8 are equal to 100k we get that only half of these voltages are added. So we need to adjust amplifier gain to 2 in order to operate with full values. So we need:\n\n$2=1+\\frac{R9}{R8}$\n\nafter solving we get that both resistors have to be equal. In order to keep resistor values less scattered we also chose those to be 100k.\n\n### Putting it all together\n\nNow we have all blocks ready and can connect them in to single circuit:\n\nHere we have connected all four modeled parts: offset adjust, low pass filter, amplitude control, ofset control. TL074 chip comes with four operational amplifiers built in so we will get away with single chip and this functionality. If you are interested in simulating here is LTspiece simulation file. In simulator TL074 were replaced with similar TL1359 op amp, so actual results shouldn\u2019t differ much.\n\nAnd corresponding signals on various nodes:\n\nAs you can see on first stage we have a sine signal with small distortion added and its offset is 2.5V. After offset adjust signal still is distorted but it crosses 0V now. After filtering we see no more distortion \u2013 only smooth sine. Then next stage adjusts signal voltage gain and on output we get signal with selected offset somewhere at -5V. Results looks promising so next step will be to put everything in to working project. Comments and suggestions are welcome.\n\n1. LEECH666\n\nAwesome!\n\nI considered building the v2 version of this project, but now I\u2019ll wait for the v3.\n\n2. Joe Tester\n\nHow about digital control for analog stage? We could add another cheap micro. This additional microcontroller could control lcd\/keypad and reset DDS micro to change what it generates. It could also provide interface to a computer for automated tests.\n\n3. I had some thoughts about controlling offset and voltage by using digital potentiometer. Using standard 256 step digital pots don\u2019t allow fine tuning due to significantly large voltage steps. Another problem is additional board space. Probably intuitive potentiometer is good choice for now.\nThe idea of using additional microcontroller sounds great \u2013 especially when performance and flexibility is needed. But again \u2013 simplicity will be lost.\n\n4. Dirk\n\nhi, i won`t build V2.0 but is there some .brd .sch for\nV3.0?\n\nGreetings\n\nDirk\n\n5. Alex Henrique\n\nWhat about to create a function that we can design inside the chip the wave that we desire?\nPerhaps read a potentiometer when we press a button?\n\n6. Practically there is no need for potentiometer. All can be done with same menu buttons. Wave table consist of 256 8-bit values so every sample could be managed one by one selecting its value and storing it. No need for additional pot. Good suggestion.\n\n7. Tom\n\nI\u2019ll second Joe\u2019s opinion. We already have two great simple designs from you (I\u2019m using v2 with great success). For v3 you should pursue performance and flexibility.\n\n8. Gatis\n\nFor those who need signal generator now, just build DDS2 generator schematic with planed DDS3 generators analog part, and it already will be much better. And for those, who wants some arbitrary waveform generator capability, just buy Rigol DG1022, it could be find for 400\\$ shipped. I bought one and I am realy happy with it, but I still use DDS2 for tasks where I need to generate more than two different signals, but now I plan to upgrade just analog part of it.\n\n9. With DDS2 this analog part should work fine.\n\n10. Ffa\n\nCan anyone comment on:\n\n1. variable gain (pot in feedback loop). I guess it varyies freequency response of opamp and some opamps may start to oscillate on gains 20khz). Does anybody build analog part of Ver3 and can verify this?\n\n4. Do we need to set output impedance to some value (say 600om) putting isolating resistor in series on output and terminate with same value close to oscilloscope input?\n\n5. Also datasheet figures measured on 2k load. Do we need to provide some load after each opamp section?\n\n11. Ffa\n\n<<<>\n\n2. I think schematic also misses decoupling 0.1uF caps close to each opamp supply pin to GND, without this it should oscillate.\n\n3. Also not sure regarding 100k resistor values. It should add noise and bouncing\/ringing\/overshooting problem on fast signal edges (e.g. rise\/fall of sqare wave >20khz). Does anybody build analog part of Ver3 and can verify this?\n\n12. Hi Ffa, there are definitely few problems with analog part. You can see few initial tests here: https:\/\/www.scienceprog.com\/avr-dds3-first-tests\/\nIt overshoots and rings on sharp edges. I am not that good at analog electronics so suggestions for improvements are welcome.","date":"2019-01-16 01:57:10","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\": 7, \"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\": 0.41276365518569946, \"perplexity\": 2805.756465114708}, \"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-2019-04\/segments\/1547583656577.40\/warc\/CC-MAIN-20190116011131-20190116033131-00165.warc.gz\"}"}
| null | null |
Q: Does float type coercion always yield the same result as static_cast? Dear fellow stackoverflowers,
Is there some kind of guarantee that coercing a float type to a "wider" one, always yields the same result as performing the according static_cast?
Here an example:
float f = <any float>;
double a = f;
double b = static_cast<double>(f);
// does this always hold?
EXPECT_TRUE(a==b);
Thank you for the information.
A:
Does float type coercion always yield the same result as static_cast?
It seems that by type coercion, you refer to implicit conversion. The answer is yes: If there is an implicit conversion from one type to another, then static cast performs that same conversion.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 7,630
|
Carl Fredrik Billman, född 24 juli 1814 i Sölvesborg, död 14 oktober 1871, var en svensk tenorbasunist vid Kungliga hovkapellet.
Biografi
Carl Fredrik Billman föddes 24 juli 1814 i Sölvesborg. Han var son till en snickarmästare. Billman gick i lära som hattmakare 1825–1827 och blev 1829 inskriven som dragon vid Skånska dragonregementet. Han blev extra trumpetare 1830, stabstrumpetare med sergeants karaktär 1834, kvartermästare 1836 och slutligen fanjunkare 1834. Från 1839 till 1842 vistades han i Leipzig, där han undervisades i basun av Carl Traugott Queisser. Billman uppträde på Gewandhaus konserterna som leddes av Felix Mendelssohn. 1844 gjorde Billman en konsertresa till Norge och anställdes som musikdirektör vid Första Agershuniska brigaden, vilken tjänst han innehade till 1848. Under 1845 uppträdde han i Köpenhamn på Jenny Linds konsert i Ridehuset, han gav även själv en konsert där. På den danske kungen Kristian VIII:s begäran spelade Billman ett solostycke vid en musikfest på Christiansborgs slott. Han avslutade sin tjänst vid Skånska dragonregementet 1853. Billman var från den 1 juli 1854 anställd som tenorbasunist vid Kungliga hovkapellet. Han slutade vid hovkapellet 1871 och avled 14 oktober 1871.
Referenser
Noter
Medlemmar av Kungliga Hovkapellet
Födda 1814
Avlidna 1871
Män
Svenska musiker under 1800-talet
Svenska trombonister
Personer från Sölvesborg
Svenska musikdirektörer
Svenska trumpetare
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 1,897
|
\section{Data analysis}
Each dual-species interferometer image contains four output ports (two for each isotope) separated vertically by $\sim 2.5\,\text{mm}$ on the CCD array. To analyze the images, we bin each port in the vertical direction and construct the asymmetry
\begin{equation}
A(x) = \frac{P_1(x) - P_2(x)}{P_1(x) + P_2(x)}
\end{equation}
of each interferometer, where $P_i(x)$ is the number of counts in the $i^\text{th}$ port as a function of horizontal pixel number. Ideally, the asymmetry is given by
\begin{equation} \label{Eq:simpleInt}
A_1(x) = c \cos \left[k_x (x - x_0) + \phi\right]
\end{equation}
where $c$ is the interferometer contrast, $k_x$ is the detection fringe frequency, $x_0$ is a reference point chosen to be in the center of the port, and $\phi$ is the interferometer phase. Our asymmetry model incorporates two additional effects. First, the interferometer contrast varies slowly as a function of horizontal position due to Rabi frequency inhomogeneity. Second, the photon collection efficiency of the imaging system varies as a function of vertical position. We therefore use the model
\begin{equation}
A_2(x) = \frac{(1 - r) + (1 + r) \exp \left[-(x - x_0)^2 / (2 \sigma^2) \right] A_1(x)}{(1 + r) + (1 - r) \exp \left[-(x - x_0)^2 / (2 \sigma^2) \right] A_1(x)}
\end{equation}
where $\sigma$ characterizes the length scale of the contrast variation and $r$ is the collection efficiency ratio between the two output ports. As $r \longrightarrow 1$ and $\sigma/(x - x_0) \longrightarrow \infty$, this expression reduces to Eq.~\ref{Eq:simpleInt}. A typical fit of this model to the data is shown in Fig.~\ref{fig:asym}. We verified that the fitted interferometer phase $\phi$ is insensitive to the precise form of the model.
\begin{figure}[h!]
\centering
\includegraphics[width=\columnwidth]{8587asymv2.pdf}
\caption{Asymmetries and fits of a single shot. The $x$ axis represents the horizontal pixel number on the CCD array (1 pixel = $170\, \mu\text{m}$).}
\label{fig:asym}
\end{figure}
Each interferometer is fit independently, and the relevant information from each image is the differential phase $\phi_\text{85} - \phi_\text{87}$ between the two interferometers. The EP signal is the linear dependence of the differential phase on the number of photon recoils $n$, where $n \in \{4, 8, 12\}$. This signal was blinded throughout the data-taking and analysis process by adding an unknown offset $n/4 \cdot \phi_\text{blind}$ to each differential phase, where $\phi_\text{blind}$ was randomly chosen from a Gaussian distribution with mean zero and standard deviation $2 \, \text{mrad}$ (corresponding to a $1\sigma$ fractional acceleration difference of $7 \times 10^{-12}$). After removing the blind, the result was published without taking additional data or altering the data analysis in any way.
\section{Detection fringe}
The angles of the interferometry beams are adjusted to create a horizontal phase shift that is proportional to the position at the time of detection. To fix notation, suppose that the initial beamsplitter, mirror, and final beamsplitter pulses occur at times $t = 0$, $t = T$, and $t = 2T$, respectively, and that there is a time interval $\tau$ between the final beamsplitter pulse and the fluorescence detection. Then the horizontal phase shift $\phi_h$ is given by
\begin{equation}
\phi_h = n k \theta_1 x + n k \theta_3 (x + v_x \cdot 2T).
\end{equation}
Here $x$ is the initial horizontal position, $v_x$ is the initial horizontal velocity, $\theta_1$ and $\theta_3$ are the angles of the initial and final beamsplitter pulse, respectively, and we have chosen coordinates so that the mirror pulse angle is zero. To create a detection fringe, we choose $\theta_1$ and $\theta_3$ so that
\begin{equation}
\phi_h = \alpha k\, \big(x + v_x \cdot (2T + \tau)\big)
\end{equation}
where $\alpha$ is a dimensionless number that sets the detection fringe frequency. Comparing the coefficients of $x$ and $v_x$, we have
\begin{equation}
\theta_1 = -\frac{\alpha}{n}\cdot \frac{\tau}{2T}
\end{equation}
and
\begin{equation}
\theta_3 = \frac{\alpha}{n}\cdot \frac{2T + \tau}{2T}.
\end{equation}
We use this technique to implement a detection fringe in large momentum transfer interferometers, taking into account the time interval between sequential pulses in each pulse zone.
\section{Systematic errors}
\subsection{I. Kinematic effects}
We suppress the linear phase gradients associated with all six kinematic DoFs by adjusting the frequency and angles of each pulse zone, as described in the main text. To calibrate the compensation for a particular DoF, the two clouds are differentially displaced in that DoF, and the resulting differental phase shift is observed. The initial position difference in the vertical direction is adjusted by varying the time between the Raman velocity selection pulses, during which the two isotopes have a differential velocity of $2\hbar k/m$. The initial velocity difference in the vertical direction is adjusted by varying the frequencies of the Raman velocity selection pulses. To change the initial position and velocity differences in the horizontal directions, we adjust the position of the optical lattice that is used to launch the atoms. During the EP measurement, the uncertainties in the differential kinematic DoFs are $(65, 20, 20)\, \mu\text{m}$ for $(\Delta z, \Delta x, \Delta y)$ and $(60, 50, 40)\; \mu\text{m/s}$ for $(\Delta v_z, \Delta v_x, \Delta v_y)$.
There is a nonzero shift associated with the vertical velocity difference because the frequency of the Raman velocity selection pulse for each species was chosen to optimize the transfer efficiency of the first beamsplitter pulse, which selects distinct velocity classes of the two isotopes due to the recoil velocity difference. This velocity difference multiplies with a residual velocity-dependent phase gradient that was observed after the EP data-taking was completed.
In order to suppress phase shifts proportional to horizontal velocity by a factor of 1000, the beam angles are set with a precision of $50\,\text{nrad}$. The angle of the retroreflection mirror is controlled by a feedback loop. A position-sensitive detector measures the position of a laser that is reflected off of the back of the mirror, and the mirror angle is adjusted by changing the voltage across the three piezoelectric actuators on which the mirror is mounted.
Since the two species are fit with the same weighting function in the $x$ direction, the differential phase is insensitive to quadratic wavefront perturbations on that axis. However, a quadratic wavefront perturbation in the $y$ direction induces a differential phase due to the cloud width difference. We bound the magnitude of this effect by correlating the width difference in the $y$ direction with the differential phase. In addition, we estimate the phase shift in the $y$ direction from the observed quadratic dependence of the phase in the $x$ direction. This estimate provides the uncertainty quoted in Table I of the main text.
\subsection{II. AC-Stark shift}
In a Bragg interferometer, each atom remains in a single internal state throughout the interferometry sequence, so the interferometer is not sensitive to relative energy shifts between internal states (as in \cite{Zhou2015}). However, the AC-Stark effect induces a differential phase shift if the laser intensity on the interferometer arms differs during the middle pulse zone. As described in the main text, the optical spectrum of the interferometry lasers is designed to reduce the magnitude of the AC-Stark shift.
Fig.~\ref{fig:universe} contains a schematic of the optical spectrum. About $25\%$ of the power of each beam is transferred into the second-order sidebands.
\begin{figure}[h!]
\centering
\includegraphics[width=\columnwidth]{optspectrumv3.pdf}
\caption{Optical spectrum. The carriers, located between the $^{85}$Rb and $^{87}$Rb transitions, are separated by $158\,\text{MHz}$. The frequency difference between the blue sidebands (Bragg pair) is adjusted to be resonant with the two-photon Bragg transition for each pulse. The red sidebands (Compensation pair) are separated by an additional $316\, \text{MHz}$ and do not drive Bragg transitions. AC-Stark compensation for both isotopes is achieved by controlling the intensity ratio between carriers and sidebands. Figure is not to scale.}
\label{fig:universe}
\end{figure}
We compute the Stark shift of the measured spectrum and control the carrier frequencies, carrier amplitudes, and sideband asymmetries to maintain Stark shift compensation. The differential Stark shift depends primarily on the amplitudes of the carriers, which are ideally suppressed to a fractional intensity of $7.5 \times 10^{-4}$ compared to the sidebands.
To bound the systematic error associated with the AC-Stark effect, we operate a dual-species $8\hbar k$ interferometer with twice as many pulses in each pulse zone. The additional pulses (``Stark shift zones'') are detuned from the two-photon resonance and do not cause Bragg transitions, but they double the magnitude of the residual Stark shift. The difference in the differential phase of the $8\hbar k$ interferometer with and without Stark shift zones is below $0.96\, \text{mrad}$, which limits the systematic uncertainty in the EP measurement to $2.7 \times 10^{-12}$.
\subsection{III. Magnetic gradient}
As described in the main text, a magnetic field gradient induces a differential acceleration between the two isotopes through the second-order Zeeman effect. We characterize the magnetic gradient with a $4\hbar k$ $^{87}$Rb gradiometer in which one interferometer is in the magnetically sensitive state $\ket{F=2, m_F=1}$. The other interferometer is in the magnetically insensitive state $\ket{F=2, m_F=0}$ and acts as a phase reference. The two interferometers are spatially overlapped to within $1\, \text{cm}$.
\begin{figure}[h!]
\centering
\includegraphics[width=\columnwidth]{magscanv6.pdf}
\caption{Magnetically sensitive interferometry. Main plot: phase difference between magnetically sensitive and insensitive interferometer as a function of interferometer time. The extrapolations used to estimate the phase shift at $T = 955\, \text{ms}$ are also shown. Inset: measurements with small timing offsets near $T = 655\, \text{ms}$.}
\label{fig:magsan}
\end{figure}
Fig.~\ref{fig:magsan} shows the differential phase of the $^{87}$Rb gradiometer as a function of interferometer time $T$. The gradiometer phase varies by much more than $2 \pi$ as the interferometer time is increased. To determine the scale of the phase variation, we took data with small timing offsets near $T = 655\, \text{ms}$.
The sub-quadratic scaling of the gradiometer phase with $T$ indicates that the magnetic gradient varies over the interferometry region, and the gradiometer phase is proportional to an average of the magnetic gradient on the interferometer trajectory (see supplement of \cite{Overstreet2018} for an analogous calculation). To estimate the average magnetic gradient for the EP measurement, we fit several low-order polynomials to the measured gradiometer phase and extrapolate to $T = 955\, \text{ms}$. Specifically, we use the model $y = a x^3 + b x^2 + c x + d$, with various combinations of $\{a, b, c, d\}$ as free parameters. The quoted uncertainty is derived from the range of values predicted by these extrapolations.
\subsection{IV. Pulse timing}
The phase of an atom interferometer generally depends on the time between each interferometer pulse. We operate the interferometer with a $\sim 3\,\text{ms}$ delay between pulses, which improves contrast and suppresses spurious interferometers associated with imperfect transfer efficiency. The pulses have a Gaussian temporal profile and a maximum Rabi frequency of $20\,\text{kHz}$. The pulse area is controlled with a precision of $0.15\%$ by a photodiode that monitors the intensity of a sampled beam. A timing error $\delta T$ in a single pulse produces a differential phase shift of $\sim 2k(\Delta v_z + \Delta r)\delta T$, where $\Delta r = \hbar k/m_{85} - \hbar k/m_{87}$ is the recoil velocity difference between isotopes. The term proportional to $\Delta v_z$ is already included in the kinematic error, but the term proportional to $\Delta r$ is an additional source of uncertainty if the timing error is not constant when the beamsplitter direction is reversed.
We bound pulse timing errors by using an oscilloscope to record the intensity of the interferometry lasers as a function of time during the interferometry sequence. The center of each pulse is identified by fitting the temporal profile of the pulse to a Gaussian. There is a $40\, \text{ns}$ offset between the expected and observed location of each $\pi/2$ pulse, with the $\pi/2$ pulses occurring earlier than expected. This offset may be caused by the finite bandwidth of the acousto-optic modulator that is used to control the pulse intensity. The associated phase shift reverses with beamsplitter direction and does not scale with interferometer order, so it does not contribute to the EP uncertainty. The remaining pulses are correctly timed to within $5\, \text{ns}$, which corresponds to an additional uncertainty of $4 \times 10^{-14}$ in the EP result.
\subsection{V. Blackbody radiation}
A thermal gradient in the interferometry region creates a blackbody radiation gradient that exerts forces on $^{85}$Rb and $^{87}$Rb \cite{Sonnleitner2013,Haslinger2018}. Although the two isotopes have the same ground-state polarizability, the resulting differential acceleration is nonzero because of the mass difference. We monitor the temperature in the interferometry region with a series of thermocouples placed at $\sim 1\, \text{m}$ intervals along the length of the $10\, \text{m}$ vacuum chamber. The thermal gradient in the vertical direction is below $0.1 \degree \text{C/m}$, and a numerical simulation indicates that the resulting differential acceleration is below $10^{-14}\, g$.
\subsection{VI. Mean field interaction}
Cold intra- and inter-species collisions of $^{85}$Rb and $^{87}$Rb in the freely falling atom clouds lead to a density-dependent energy shift \cite{Aguilera2014,Debs2011}. If the initial beamsplitter transfer is unbalanced, the two interferometer arms will experience different mean field energies. The Bragg lasers are $3\%$ less detuned from the $^{85}$Rb resonance than from the $^{87}$Rb resonance, and the resulting Rabi frequency difference creates unbalanced $^{85}$Rb densities along the interferometer arms. At the first pulse zone, the atom clouds have a horizontal width of $1.5\,\text{mm}$ and a vertical width of $0.7\,\text{mm}$ (FWHM). The associated phase shift is estimated to be $0.2\,\text{mrad}$, nearly independent of interferometer order. As a function of interferometer order, the density decreases due to spontaneous emission by 1.3\% per $4\hbar k$, which leads to a relative error of less than $8 \times 10^{-15}$ in the EP result.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 5,259
|
\section{Introduction}\label{s:intro}
Detailed mapping of nearby star-forming clouds at submm wavelengths within legacy surveys such as the SCUBA2 Gould-belt survey, the \emph{Spitzer}, and more recent Herschel survey of these clouds has revealed their richness in structure. Star-forming clouds occurring at different stages of evolution have various sizes, and shapes. Inter-stellar shocks profoundly affect gas dynamics on a large scale, and are likely to be crucial in producing the observed dense filamentary clouds. I propose to examine this hypothesis numerically.
\section{Description of the problem}
I shall consider a simple test case where a molecular cloud is shocked by an incident shell moving with a supersonic velocity.
The cartoon in Figure 1 demonstrates the relevant physical details of the problem. The computational domain may be conveniently divided into three regions viz. the intercloud medium (ICM), the standing point at the cloud surface, and the interiors of the shocked cloud marked 1, 2 and 3, respectively. The slab, as shown in this cartoon, moves through the ICM from the left to the right with a precollision velocity, $V_{s}$. It forms a standing shock, labelled 2 in Fig. 1,
after colliding with the cloud surface. The impact of this collision generates reflected waves in region 1, while some straddle the cloud surface and others propagate within the post-collision cloud, the so called transmitted-shock, with a velocity, $V_{c}$, in region 3. The reflected shock-waves have been marked with curly arrows in Figure 1. Pressure in each of the three regions will be denoted by $p$ with a subscript 1, 2 or 3, commensurate with the notation introduced in Fig. 1.
\begin{figure}
\begin{minipage}[b]{0.5\linewidth}
\centering
\includegraphics[angle=0,width=6cm]{slabcld.eps}
\caption{A cartoon showing physical details of the slab-cloud system. See text for description.}
\end{minipage}
\hspace{1.cm}
\begin{minipage}[b]{0.5\linewidth}
\centering
\includegraphics[angle=0,width=6cm]{shlexpnsch.eps}
\caption{Schematic representation of the computational domain. Shown here is a periodic box enclosing the cloud confined by an intercloud medium, and a slab approaching it with a precollision velocity, $V_{s}$.}
\end{minipage}
\end{figure}
Any discussion of this problem would be incomplete without appropriate reference to the shock-dynamics, however, for want of space, I shall restrict myself to only quoting some important results which can be derived using the Rankine-Hugoniot jump conditions (e.g. Courant \& Friedrichs 1956). Following the notation introduced above, the excess pressure due to a highly supersonic shock is
\begin{equation}
\frac{p_{2}-p_{0}}{p_{1}-p_{0}}\sim 2 + \frac{1}{\mu^{2}}, \textrm{where}\ \mu^{2} = \frac{\gamma-1}{\gamma+1};
\end{equation}
which for an adiabatic gas constant, $\gamma = 5/3$, produces a pressure-excess of $\sim6$. Pressure within the cloud due to the transmitted shock, $p_{3}$, can be shown to be related to its preshock value, $p_{0}$, as
\begin{equation}
p_{3} = p_{0}\Big[\frac{(1-2M^{2})-1/\gamma}{(1-2M_{+}^{2}) - 1/\gamma}\Big],
\end{equation}
where $M_{+}$ is the Mach number for the incident shock. Observe that, relative to the incident Mach number, there is a fall in pressure at the surface of incidence, the standing point. Finally, the density within the shocked cloud compares with that in the external medium as,
\begin{equation}
\frac{\rho_{3}}{\rho_{0}} = \frac{(\gamma-1)-p_{3}/p_{0}(3\gamma-1)}{p_{3}/p_{0}(\gamma-1) - (3\gamma-1)}.
\end{equation}
For a finite value of this ratio, the velocity of transmitted shock, $V_{c}$, within the cloud can be calculated using the equation of continuity. The resulting effect will be discussed in Section 5 below.
\section{Numerical scheme}
We have used the Lagrangian, particle-based scheme, Smoothed particle hydrodynamics (Monaghan 1992), hereafter referred to as SPH. An SPH particle, in the strictest sense, is not a point particle but has a finite spatial extent defined by a quantity called the smoothing length, $h$. Each SPH particle, characterised by state properties of density, velocity and temperature, interacts with other particles through numerical viscosity. The density of a particle, $\rho_{i}$, is calculated by summing over contributions from nearest neighbours, $N_{neibs}$, of that particle within a radius 2$h$,
\begin{equation}
\rho_{i}\equiv\rho(\mathbf{r}_{i}) = \sum_{j=1}^{Neibs} m_{j}W(\mathbf{r}_{i}-\mathbf{r}_{j},h),
\end{equation}
where $\textbf{r}$ is the position vector of the particle with identifier, $i$. The density of the fluid, $\rho_{s}$, is then simply
\begin{equation}
\rho_{s} = \int W(\mathbf{r} - \mathbf{r'},h)\rho(\mathbf{r'})d\mathbf{r'}
\end{equation}
The search for the nearest neighbours, and the calculation of net force on an individual particle is done using a tree-algorithm (Barnes \& Hut 1986). Local inhomogeneities are smoothed out using a kernel that has been normalised to unity. Gravity for vanishingly small inter-particle separations is smoothed out using a modified kernel that introduces an infinitesimally small repulsive force between closely spaced particles. The Thomas-Couchman kernel, a special type of the M-4 cubic spline, ensures a finite repulsive force between closely interacting particle-pairs (Thomas \& Couchman 1992). The numerical algorithm employed here, Seren, is a well tested code (Hubber \emph{et al.} 2011), and includes all the features described above.
The thermodynamic details of the problem are modelled using a simple barotropic equation of state (EOS), defined by Equation (6) below, that mimics the post-collision temperature-jump, $T_{ps}$, and cools off to the precollision temperature, $T_{iso}$, at relatively higher densities.
\begin{equation}
\frac{P}{\rho} = (k_B/\bar{m})\times
\left\{ \begin{array}{ll}
\Big(\frac{T_{ps}}{\textrm{K}}\Big)\ ; \rho\le10^{-23} \textrm{g cm}^{-3}\\
\Big(\frac{T_{cld}}{\textrm{K}}\Big)\ ; 10^{-23}\textrm{g cm}^{-3} <\rho
\le 10^{-22}\textrm{g cm}^{-3}\\
\Big(\frac{\gamma T_{cld}}{\textrm{K}}\Big)\Big(\frac{\rho}{10^{-22}\textrm{g cm}^{-3}}\Big)^{\gamma-1}; 10^{-22}\textrm{g cm}^{-3}\\
< \rho\le 5\times 10^{-22}\textrm{g cm}^{-3}\\
\Big(\frac{T_{cld}}{\textrm{K}}\Big)\ ; 5\times 10^{-22}\textrm{g cm}^{-3}<\rho \\
\le 10^{-18}\textrm{g cm}^{-3}\\
\Big(\frac{T_{iso}}{\textrm{K}}\Big)\Big[1 + \gamma\Big(\frac{\rho}{10^{-15}\textrm{g cm}^{-3}}\Big)^{\gamma-1}\Big]\ ; \\
\rho > 10^{-18}\textrm{g cm}^{-3};
\end{array} \right.
\end{equation}
where the adiabatic gas constant, $\gamma=5/3$, $k_{B}$ and $\bar{m}$ are respectively the Boltzmann constant, and mean molecular mass.
\section{Initial conditions}
We propose to test the above theory using the scheme outlined in Figure 2. Ordinary SPH particles, representing gas within the cloud and the slab, interact with each other via gravitational and hydrodynamic forces. The ICM confining the cloud, however, is represented by special particles that exert only thermal pressure on other particles. The entire assembly, including the slab, is enclosed in a self-wrapping periodic box, where the periodicity is limited to merely ghosting particles; in other words, a particle leaving through one face of the box enters from the opposite face.
Elsewhere in the literature, we have discussed an ensemble of cases for different choices of resolution (Anathpindika \& Bhatt 2011). In this paper though, the emphasis being on the formation of dense structure within the shocked cloud, I will only discuss the simulation with highest resolution. The precollision cloud was modelled as a sphere having uniform density, and the probability distribution function (PDF) of particles in the virgin cloud peaking at the predicted density, $\rho_{true}$, is shown in Figure 3, which not only demonstrates the stability of the precollison cloud, but also the absence of any spurious dense pockets.\\
\begin{figure}
\begin{minipage}[b]{0.5\linewidth}
\centering
\includegraphics[angle=270,width=5cm]{stability.eps}
\caption{A histogram of the gas density within the precollision cloud. The relatively small spread around the expected density, $\rho_{true}$, indicates the stability of this cloud.}
\end{minipage}
\hspace{1.cm}
\begin{minipage}[b]{0.5\linewidth}
\centering
\includegraphics[angle=0,width=5cm]{HIIshckpress.eps}
\caption{A plot showing the radial variation of pressure within the shocked cloud at different epochs.}
\end{minipage}
\end{figure}
\textbf{Important parameters } Mass of the cloud, $M_{cld}$ = 500 M$_{\odot}$; radius, $R_{cld}$= 0.5 pc; uniform temperature, $T_{iso}$ = 15 K.
The cloud, ICM, and the slab were represented by SPH particles having three different choices of mass, $m$. If $i$, $j$, and $k$ are the respective identifiers of particles in each of the three regions then, $(m_{i}, m_{j}, m_{k})/[\textrm{M}_{\odot}]\equiv(2.2\times 10^{-3},2.53\times 10^{-6}, 1.26\times 10^{-5})$, and the number of particles in each of these regions is respectively (2.4, 2.3, 7.3)$\times 10^{5}$, so that total number of particles, $N_{tot} = 1.2\times 10^{6}$. \\
\textbf{Resolution} The smallest resolvable spatial scale in an SPH calculation, as noted above, is the average smoothing length, $h_{avg}$, so that the volume of a typical SPH particle, $V_{SPH}=\frac{32\pi h_{avg}^{3}}{3}$; and $N_{cld}(32\pi h_{avg}^{3})/3 = 4\pi R_{cld}^{3}N_{neibs}/3$. Thus,
\begin{equation}
h_{avg}\sim \frac{1}{2}\Big(\frac{N_{neibs}}{N_{cld}}\Big)^{1/3}R_{cld},
\end{equation}
is the typical smoothing length of a particle in the test cloud. In the present case, $h_{avg}\sim 1.5\times 10^{-2}$ pc, and $2\mathcal{R}\equiv\lambda_{J}/h_{avg}\sim 13$, where $\lambda_{J} = (\pi a_{0}^{2}/G\rho)^{1/2}$, is the length of the fastest growing unstable mode in a gas body, the Jeans length. The quantity $\mathcal{R}$ defines the number of SPH particles available to resolve the unstable mode, which in this case is $\sim 6$, and therefore satisfies the Truelove criterion of spatially resolving the instability (Truelove \emph{et al.} 1998).
\begin{figure}
\centering
\includegraphics[angle=270,width=12cm]{testHRmontgLR.eps}
\caption{A rendered density plot showing a time sequence of the mid-plane of the shocked cloud, as seen along a direction orthogonal to the plane of the figure. Turbulence induced by the transmitted shock generates clumps and filaments visible in these plots.}
\end{figure}
\section{Result}
The post-collision reflected shock-wave as well as the transmitted shock within the cloud is evident from the plot in Figure 4, that shows the radial variation of pressure at different epochs. The incidence of the front surface of the slab on the cloud causes a jump in pressure at its surface, defined by Equation (1), followed shortly by another, relatively smaller, jump due to the rear surface of the slab shocking the cloud (red-curve). The green curve shows the pressure distribution at a slightly later epoch when the pressure-jump moved leftward relative to its position in the red curve, suggesting an outward propagation of a wave, i.e. the reflected shock moving in the ICM. It can be identified as the flared region around the shocked cloud shown in the rendered density plot of Figure 5.
The weaker transmitted shock that is most important in the evolution of the post-collision cloud, initially causes slight contraction of the cloud, evident from the collection of denser gas in a ring on its periphery. Relative to the incident shock, gas within the cloud is at a lower pressure manifested by a trough in the pressure distribution at the surface of incidence, defined by Equation (2). Inward propagation of this shock generates a turbulent velocity field within the cloud that soon produces fractal structure, in other words the appearance of relatively dense pockets of gas interspersed with rarefied regions, the so called holes. Structure within the shocked cloud grows on a timescale comparable to the crossing time, $t_{c}\sim 2R_{cld}/V_{s}$, of the precollision slab which is much shorter than the growth time, $t_{g}$, of the fastest growing unstable mode of length, $\lambda\sim (\pi v_{eff}^{2}/G\rho_{cld})^{1/2}$, where $v_{eff} = a_{0}+v_{c}$; $v_{c}\sim V_{s}/\mathcal{M_{+}}\sim 0.4$ km/s, and $a_{0}(T=20 K)\sim$ 0.27 km/s, so that $\lambda\sim$ 0.1 pc, which implies, $t_{g}\sim$ 0.14 Myr.
The simulation was terminated when the slab, having traversed the width of the cloud, reached the opposite face of the periodic box. Consequently, further evolution of clumps and other contiguous filaments could not be investigated in this work. We have observed that the incident slab, after colliding with the cloud, suffers severe ablation by the time it reaches the other end of the cloud. The transmitted shock, relative to the incident slab, propagates at a much lower velocity within the post-collision cloud, evident from the rendered density plots in Figure (5). However, as argued in Section 2 above, Equations (8) below show that the transmitted shock in a cold, dense gas could still be supersonic, and therefore, will likely have a significant effect on the internal structure of the cloud. The pressure behind the slab changes on a very short timescale, the dynamical timescale, $t_{d}$, of the slab which can be shown to be
\begin{equation}
t_{d}\sim\frac{R_{cld}}{V_{s2}}\frac{(\gamma + 1)^{2}}{\mathcal{M}(\gamma - 1)},
\end{equation}
where
\begin{displaymath}
V_{s2}\sim \Big(\frac{\rho_{0}}{\rho_{3}}\Big)V_{s1}\sim V_{c},
\end{displaymath}
and $\mathcal{M}\equiv \mathcal{V}_{c}/a_{0}$, is the Mach number for the transmitted shock; see paper I for the derivation. In the present case, $t_{d}\sim 10^{-4}$ Myrs $\ll t_{c}$, implying, the dynamical properties behind the shock must indeed change rapidly. In this light the problem under consideration here, in effect, reduces to one of a weak-shock impinging on a cloud or conversely, a shock interacting with a large cloud. \\
\begin{figure}
\begin{minipage}[b]{0.5\linewidth}
\centering
\includegraphics[angle=270,width=6cm]{HireshPDF.eps}
\caption{The density PDF for gas in the shocked cloud after the slab has traversed the cloud. Bimodal nature of the density distribution is evident from this plot.}
\end{minipage}
\hspace{1.cm}
\begin{minipage}[b]{0.5\linewidth}
\centering
\includegraphics[angle=270,width=6cm]{spectra.eps}
\caption{This the corresponding energy spectrum of gas within the shocked cloud. Like the PDF, this spectrum too shows a distinct segregation. The dotted lines show respective power-law fits; see text for description.}
\end{minipage}
\end{figure}
\textbf{The probability distribution function (PDF)}
The CMF has often been suggested as the likely progenitor of the stellar IMF, and the likely relation between the two distributions has been examined by several authors (e.g. Anathpindika 2011, Hatchell \& Fuller (2008), Nutter \& Ward-Thompson 2007, Motte \emph{et al.} 1998). This proposition makes an investigation in to the origin of the CMF even more pertinent. Numerical simulations like those discussed by V{\' a}zquez-Semadeni (1994), and Padoan \& Nordlund (2002), among several other authors, and the PDF plotted in Figure 6 demonstrate the crucial role of interstellar shocks in generating a lognormal density PDF; although we note that the plot shown here is only semi-lognormal. Interestingly, a lognormal fit has also been attempted for the distribution of masses of dense ($\gtrsim 10^{4}$ cm$^{-3}$) cores in nearby star-forming clouds (e.g. Enoch \emph{et al.} 2008). While the apparent similarity of the dense-phase PDF derived here, with the CMF may perhaps be only a fortuitous coincidence, the likelihood of a causal relationship must be further investigated. Also, in the dense phase, as can be seen in Figure 6, this PDF peaks at $\sim 10^{5}$ cm$^{-3}$ which at 15 K, the precollision temperature within the cloud, corresponds to a characteristic mass, $M_{s}\sim 3$ M$_{\odot}$; incidentally, using a gas-to-star conversion efficiency of 10\% this characteristic mass corresponds to 0.3 M$_{\odot}$, the mass at which the stellar IMF develops a knee before turning over in to the brown-dwarf regime. This test simulation therefore seeks to re-emphasise the importance of interstellar shocks in forming putative star-forming pockets within molecular clouds, and may hold the key to explain the CMF. \\
\textbf{Power-spectrum}
The energy density, $E(k)$, of a turbulent velocity field in the wavevector ($k$) space, the Fourier domain, is related to its Cartesian counterpart through a simple integral,
\begin{equation}
\int_{0}^{\infty}E(k)dk = \frac{1}{2}\sum_{i} v_{i}^{2}.
\end{equation}
The integral on the left-hand side extends over all possible wave-vectors, while the summation on the left-hand side runs over all SPH particles. The resulting power-spectrum for gas within the shocked cloud has been shown in Figure (7), which like the PDF, is also segregated. Although the spectrum for both, the dense as well as the rarefied phase of the gas, is a power-law it is, however, considerably steeper ($\propto k^{-4.5}$) for the former, and Kolmogorov-like ($\propto k^{-1.7}$) for the latter. The Kolmogorov-spectrum, as is well known, applies to an inviscid, incompressible fluid, though the gas here is viscous, but the rarefied phase appears to obey this approximation; it obviously breaks down for the dense phase.
\section{Conclusions}
\begin{enumerate}
\item I have argued in favour of my hypothesis that interstellar shocks could possibly lead to formation of dense clumps, and contiguous filaments in molecular clouds.
\item It has been demonstrated that propagation of a shock renders the density field unstable, and generates structure in it on a rather short timescale. Stars likely to form in these pockets, via energetic feedback, may further inject energy within the gas and quench star-formation in one region, only to trigger it elsewhere.
\end{enumerate}
\textbf{Ancillary remarks }
This test case only examined the balance between self-gravity and thermal pressure as the magnetic field was not included, however, magnetohydrodynamic simulations by for e.g. Padoan \& Nordlund (2011) suggest a considerable modulation of the rate at which protostellar objects form in turbulent gas. Other issues deserving a brief explanation include the effect of numerical resolution, and hydrodynamic instabilities on the shocked cloud. The latter is essentially related to the numerical resolution, and more critically, to the influence of SPH viscosity on dynamically unstable fluid layers. First, a relatively poor resolution tends to suppress dynamical fragmentation. Second, the shearing interaction between the slab and the surface of the cloud is also likely to be unstable to hydrodynamic instabilities such as the Kelvin-Helmholtz instability. The demand on resolving this thin layer, however, is conservatively large and not fulfilled here. It does not though, compromise the arguments presented in favour of the hypothesis examined here.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 3,411
|
Matebeleland's possible oldest woman celebrated her 103rd birthday on the 5th of September 2015 at Mzilikazi Township in Bulawayo.
Eve Mkandla nee Gumede was born on the 5th of September 1912, 19 years after the 1st Matebele Rebellion War of 1893 and just 20 years after the disappearance of King Lobengula in 1894. It was also roughly 10 years after Cecil John Rhodes had died in March 1902.
Going by her given date of birth, it also means that Eve Mkandla nee Gumede was 10 years old when a white only referendum on the status of Southern Rhodesia was held on the 27th of October 1922 to either form a responsible government or join the union of South Africa.
Eve Mkandla nee Gumede was therefore born at the time when colonialists were putting up, entrenching and consolidating their colonial governing structures in both Matebeleland and Mashonaland respectively and in then Northen Rhodesia (Zambia).
1912 is also a year when the largest passenger-liner The Titanic sank in April after hitting an iceberg in the North Atlantic ocean killing more than 1 500 people.
The ship's voyage was from Southampton to New York. Eve Mnkandla nee Gumede was born 5 months after that significant world event.
In that year of 1912 Woodrow Wilson was elected the President of the USA and New Mexico and Arizona were admitted as America's 47th and 48th states respectively. In January of that year the Republic of China had just been established.
Eve Mkandla nee Gumede was married to Jortham Mkandla. She had 6 children with Mkandla. The oldest is 82 years. She has lost 2 of her male children and 1 female. 3 of her children are still around, one male and 2 daughters.
MaMkandla comes from Lupane in a place called Gomoza. She is currently living in Mzilikazi suburb with her relatives having recently moved from Lupane. She walks with the aid of a stick and has a full set of teeth. Her favourite food is ocra, meat, umvushwa ledobi and ibhobola.
Eve's memory is reportedly still very sharp. She is apparently the first in her family to be blessed with such a long life.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 6,421
|
echo -n "creating framework to support automated puppet/r10k workflows"
echo -n "- installing git"
apt-get update -y && apt-get install -y git
echo -n "- changing to /etc"
cd /etc/
rm -rf /etc/puppet
echo -n "- Retrieving Base /etc/puppet"
git clone https://github.com/ppouliot/puppet-etc_puppet puppet
#echo -n "installing base puppetfile for /etc/puppet/modules"
#wget -cv https://raw.githubusercontent.com/ppouliot/puppet-Puppetfile_Env/master/Puppetfile.base -O /etc/puppet/Puppetfile
echo -n "- Installing PuppetLabs Repositories"
cd /tmp
# this is a debian/ubuntu specific command
release=`lsb_release -c | awk '{print $2}'`
echo '### INSTALLING PUPPETLABS APT REPO ###'
wget http://apt.puppetlabs.com/puppetlabs-release-$release.deb; dpkg -i puppetlabs-release-$release.deb
if [ $? $test -eq 1 ]; then
echo "Could not find puppetlabs release for $release. Trying alternative."
wget http://apt.puppetlabs.com/puppetlabs-release-precise.deb; dpkg -i puppetlabs-release-precise.deb
fi
echo -n "- Installing Puppet and other necessary packages"
apt-get update -y && apt-get install -y --force-yes openssh-server puppet ruby ruby-dev
echo -n "- Installing R10K"
gem install r10k
gem install hiera-eyaml
cd /etc/puppet/secure
eyaml createkeys
chown -R puppet:puppet /etc/puppet/secure/keys
chmod -R 0500 /etc/puppet/secure/keys
chmod 0400 /etc/puppet/secure/keys/*.pem
echo -n "- Installing Puppetlabs Supported and Endorsed Modules to BaseModulePath"
cd /etc/puppet && r10k --verbose DEBUG puppetfile install
echo -n "- deploying environment modules"
cd /etc/puppet && r10k --verbose DEBUG deploy environment -vp
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 2,886
|
{"url":"https:\/\/socratic.org\/questions\/how-do-i-find-the-surface-area-of-a-solid-of-revolution-using-polar-coordinates","text":"# How do I find the surface area of a solid of revolution using polar coordinates?\n\nIf a surface is obtained by rotating about the x-axis the polar curve $r = r \\left(\\theta\\right)$ from $\\theta = {\\theta}_{1}$ to ${\\theta}_{2}$, then its surface area A can by found by\n$A = 2 \\pi {\\int}_{{\\theta}_{1}}^{{\\theta}_{2}} r \\left(\\theta\\right) \\sin \\theta \\sqrt{{r}^{2} + {\\left[r ' \\left(\\theta\\right)\\right]}^{2}} d \\theta$.","date":"2022-05-23 08:01:40","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 4, \"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\": 0.9702640771865845, \"perplexity\": 71.95866433280015}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-21\/segments\/1652662556725.76\/warc\/CC-MAIN-20220523071517-20220523101517-00727.warc.gz\"}"}
| null | null |
class Record < ActiveRecord::Base
belongs_to :assignment
end
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 4,067
|
Felix Colgrave (born 29 November 1992) is an Australian director, animator, cartoonist, filmmaker, artist and musician. Distribution of Colgrave's work has, to date, been focused on YouTube where his channel has 1.67 million subscribers. Colgrave mainly uses After Effects for his animations.
As of now, his most popular animation is "Double King" with over 73 million views.
Work
Colgrave has been involved in several commercial projects, having worked for clients such as Vice and Comedy Central, as well as for Trip Tank and Off the Air series. He provided some in-game animations for Bethesda Softworks' Fallout 4.
Colgrave's YouTube channel has twenty six of his 2D surreal, psychedelic, animations, half of which have over a million views each. "DOUBLE KING", has over 50 million views as of February 2021. Colgrave also created the music for the DOUBLE KING animation, which he released as an album known as Royal Noises from Dead Kingdoms.
His animation "Man Spaghetti", has been featured on the animation website known as Cartoon Brew Colgrave has directed a number of music videos, including videos for DJ Mustard, Nicki Minaj & Jeremih's "Don't Hurt Me" Fever The Ghost's "SOURCE", and Shoe's "Egg".
He was credited for doing the storyboards for Childish Gambino's music video for Feels Like Summer.
Colgrave was interviewed at the Sydney Opera House with hosts from Comedy Central and Super Deluxe.
Awards and recognition
The animation "CU" has won the best of the month award on the video-sharing website Vimeo. His follow-up work "Flying Bamboo" was also featured as the opening for the Melbourne International Film Festival.
"The Elephant's Garden" won "Best Australian Film" at the Melbourne International Animation Festival in 2014. It has also gained exposure to the public from various web sources. Both "DOUBLE KING" and "The Elephant's Garden" are featured in the book titled Australian Animation: An International History.
Personal life
Felix Colgrave has been animating since he was a child. One of his first animations, "Last Resort", was created early in 2008 when he was 15 years old.
Colgrave's wife, Zoë is a costume and makeup-artist, known as Trugglet on Instagram They became engaged on 31 October 2016; On 23 February 2019, Colgrave revealed that he has one child with his wife. Currently he lives in Melbourne, Australia.
References
External links
Australian YouTubers
Australian animators
Living people
Australian surrealist artists
1992 births
Music YouTubers
YouTube animators
Artists from Tasmania
Musicians from Tasmania
Australian directors
21st-century Australian artists
Australian filmmakers
Australian cartoonists
YouTube channels launched in 2008
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 8,946
|
Nunatak Ovcyna är en nunatak i Antarktis. Den ligger i Östantarktis. Australien gör anspråk på området. Toppen på Nunatak Ovcyna är meter över havet.
Terrängen runt Nunatak Ovcyna är platt åt nordväst, men åt sydost är den kuperad. Den högsta punkten i närheten är Nunatak Chirikova, meter över havet, kilometer nordväst om Nunatak Ovcyna. Trakten är obefolkad. Det finns inga samhällen i närheten.
Kommentarer
Källor
Berg i Östantarktis
Australiens anspråk i Antarktis
Berg i Antarktis 1000 meter över havet eller högre
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 4,337
|
Les South Sydney Rabbitohs (souvent abrégé en Souths, signifiant Les Petits lapins de Sydney Sud ) est une équipe australienne de rugby à XIII, basée à Redfern, un quartier de la ville de Sydney. Elle participe au championnat australien de rugby à XIII, la National Rugby League et est une des neuf équipes professionnelles de la capitale de la Nouvelle-Galles du Sud. Le club a été fondé en 1908 et fait partie des membres fondateurs de la New South Wales Rugby Football League. South Sydney est avec les Sydney Roosters, la seule équipe fondatrice (foundation club en anglais) à être encore présente dans le championnat de l'élite australienne.
La base des supporters des Rabbitohs se trouvent essentiellement dans le centre-sud et le sud-est de Sydney, cependant l'équipe est également supportée à travers toute la Nouvelle-Galles du Sud. Ils disputent leurs matchs à domicile à l'ANZ Stadium de Sydney.
Au niveau de l'élite du rugby à XIII australien, South Sydney est l'équipe la plus titrée d'Australie avec 21 championnats remportés. Après 43 ans d'attente, le club de Redfern remporte en 2014 le championnat en s'imposant en finale face aux Canterbury Bulldogs. Un dicton dit que Quand Souths va bien, le rugby à XIII va bien (When Souths are going well, rugby league is going well, en anglais).
Palmarès
Championnat d'Australie (21) :
Champion: 1908, 1909, 1914, 1918, 1925, 1926, 1927, 1928, 1929, 1931, 1932, 1950, 1951, 1953, 1954, 1955, 1967, 1968, 1970, 1971, 2014.
Finaliste : 1910, 1916, 1917, 1920, 1923, 1924, 1935, 1937, 1939, 1949, 1952, 1965, 1969 et 2021.
Minor Premierships (17) : 1908, 1909, 1914, 1918, 1925, 1926, 1927, 1929, 1932, 1949, 1950, 1951, 1953, 1968, 1969, 1970, 1989.
Charity Shield (12) :1984, 1988, 1989, 1990, 1991, 1992, 1999, 2002, 2005, 2006, 2008, 2009.
Histoire
Premières années
South Sydney est le troisième club de rugby à XIII créé en Australie après Glebe District Rugby League Football Club et Newtown. Il est ainsi le vétéran de ceux participant au championnat d'élite australien. En 1908, une compétition de rugby à XIII se forme avec les clubs de la classe ouvrière de Sydney, dont parmi eux, South Sydney Rugby Union Club qui quitte le rugby à XV pour jouer aux nouvelles règles adoptées par la New South Wales Rugby League. Le South Sydney District Rugby League Football Club est fondé le au Redfern Town Hall lorsque JJ Giltinan est rejoint sur le podium par le joueur de cricket Victor Trumper et Henry Clement Hoyle, élu travailliste au Parlement de Nouvelle-Galles-du-Sud, devant une importante foule de supporters.
Lors de la saison inaugurale de la NSWRFL en 1908, South Sydney débute le par une victoire 11 à 7 face à North Sydney. South termine en tête à l'issue de la phase régulière et affronte Glebe en demi-finale. Le , le club de Redfern bat ces derniers sur le score de 16 à 3 et rejoint en finale son voisin d'Eastern Suburbs. La finale inaugurale a lieu au Agricultural Society's Ground. Les deux équipes sont handicapées par l'absence des joueurs sélectionnés pour la première tournée des Kangourous en Angleterre. Ce , South Sydney remporte la première saison de ce championnat, 14 à 12. La saison suivante, South conserve son titre, à la suite du forfait de Balmain en finale. Ces derniers voulant protester son mécontentement contre l'organisation du match opposant les Kangourous aux Wallabies en ouverture de la finale, décident de ne pas y prendre part. Les demandes de Balmain de reporter la finale à une date ultérieure sont refusées par la fédération de Nouvelle-Galles du Sud. En 1910, les deux premières équipes de la saison régulière s'affrontent en finale. Le au Sydney Showground, South et Newtown se quittent sur un match nul 4 partout mais ce sont ces derniers qui remportant le championnat en raison de leur première place lors de la saison régulière.
Durant ces premières années, Arthur Hennessy est considéré comme l'homme-clé de South Sydney. En effet, il est à la fois le premier capitaine et entraîneur de l'histoire du club. Il devient aussi le premier capitaine de l'équipe de Nouvelle-Galles du Sud et de l'Australie. De 1908 à 1911, il effectue 26 rencontres sous les couleurs de South.
Après leur troisième et quatrième titre obtenue en 1914 et 1918, South Sydney entame sa première de ses trois périodes dorées, en gagnant 7 des 8 championnats se déroulant de 1925 à 1932. Seul celui de 1930, leur échappe car Western Suburbs les éliminent au stade des demi-finales. En 1925, les Rabbitohs remportent le championnat tout en restant invaincus. Les joueurs symbolisant cette période sont : Alf Blair, Eddie Root, Benny Wearing (144 essais), Alf O'Connor et le seconde ligne, George Treweek.
La domination de Souths durant ces premières années de championnat leur a valu d'être appelé The Pride of the League (la fierté de la Ligue). De 1933 à 1949, le titre échappe aux Rabbitohs qui se hissent à trois reprises en finale (1935, 1939 et 1949). En 1948, South Sydney déménage pour le Redfern Oval et pour sa première rencontre dans son nouveau stade fait un nul 19 partout avec Easts.
Les années 1950
Après avoir raté les demi-finales les quatre années précédentes, les Rabbitohs finissent en tête de la saison régulière de 1949 devant Wests, St. George et Balmain. Cependant, ils sont battus deux fois, en demie 16 à 12 et 19 à 12 en finale, par St. George. Il faut attendre l'année 1950 pour que Souths remportent leur douzième titre
contre Wests 21 à 15. 1951 est une année remarquable pour Souths qui finit la saison régulière 11 points devant Manly. Les deux équipes se retrouvent en finale et les Rabbitohs écrasent les Sea Eagles 42 à 14, ce qui constitue le score le plus élevé pour une finale de championnat.
En 1952, Souths est battu lors d'une finale controversée par Western Suburbs 22 à 12. En effet l'arbitre refuse un essai en début de match pour les Rabbitohs qui semblent valable selon ces derniers. Il s'agit du quatrième et dernier titre pour les Magpies. L'année suivante, les Rabbitohs finissent une nouvelle fois premier de la saison régulière, battent North Sydney en demie et s'imposent en finale contre St. George 31-12, avec un Clive Churchill dominateur. En 1954, le club de Redfern conserve son titre face à Newtown (23-15). En 1955, South Sydney termine difficilement quatrième de la saison régulière et bat Manly dans la douleur 14 à 12 en phases finales. En finale préliminaire, mené 14 à 11 par St. George à quelques minutes de la fin de la rencontre, Souths parvient à s'imposer sur le fil 18 à 14 grâce à une essai de Ian Moir et un but de Bernie Purcell. Durant le finale, alors qu'il ne reste que 6 minutes à jouer, Newtown vire en tête par 11 points à 7. Mais un essai de Col Donohoe converti par Purcell permet à Souths de renverser une nouvelle fois une situation délicate et de gagner le match 12 à 11.
En 1956, Souths échoue en finale préliminaire face à Balmain 36 à 33. Cette année est marquée par le début de règne de 11 ans de St. George sur le championnat (1956-1966).
Clive Churchill est la figure centrale de cette période dorée de Souths. Ce petit arrière, surnommé "The Little Master", est arrivé en provenance de Newcastle en 1947. Après 157 rencontres pour les Rabbitohs et 5 titres de champion, il quitte le club en 1959 pour un club de Brisbane.
Les années 1960 et 70
Après la fin de leur seconde période dorée en 1955, le club de Redfern attend une décennie pour rejouer une finale de championnat (qui détient le record du nombre de spectateurs du Sydney Cricket Ground) perdue 12 à 8 face à St George. En 1967, Souths met fin au règne de 11 années sans partage des Dragons en s'imposant contre Canterbury-Bankstown 12 à 10 grâce à une spectaculaire interception de Bob Mc Carthy. Ce titre correspond au retour de Clive Churchill comme entraîneur qui construit une équipe compétitive autour d'un pack rugueux incluant Ron Coote, Bob McCarthy, Gary Stevens, Bob Moses, John O'Neill, Jim Morgan, Elwyn Walters et le capitaine John Sattler. En 1968, Souths défait Manly en finale 13 à 9 avec notamment un essai de l'ailier Mike Cleary. L'année suivante, les Rabbitohs ne conservent pas leur titre battu 11 à 2 par Balmain. En 1970, South Sydney l'emporte 23 à 12 en finale face à Manly avec deux essais du demi de mêlée Bob Grant. 1971 est l'année de la dernière victoire de Souths en championnat 16 à 10 grâce à trois essais de Branighan, Coote et McCarthy.
Durant cette période faste (quatre titres sur cinq possibles entre 1967 et 1971), les lignes arrières de Souths sont performantes avec l'ailier-buteur Eric Simms, Michael Cleary, Jimmy Lisle, Paul Sait, Bob Honan, Brian James, les frères Ray et Arthur Branighan, Dennis Pittard et Bob Grant. Eric Simms est le meilleur réalisateur de l'histoire du club avec un total de 1 841 points (23 essais, 803 buts et 86 drops). En 1969, ce dernier bat le record de points marqués lors d'une saison (265), détenu auparavant par Dave Brown d'Eastern Suburbs. Simms a réalisé tellement de drops que cela a influencé les autorités à réduire en 1971 sa valeur de 2 à 1 point.
Des difficultés financières commencent à apparaitre pour Souths, ses meilleurs joueurs quittent le club pour rejoindre Manly (John O'Neill, Ray Branighan, Bob Moses) ou Easts (Ron Coote, Michael Cleary, Elwyn Walters) et les performances sportives de Souths s'en ressentent. En 1975, le club termine lanterne rouge de la saison régulière. En 1978, sous l'impulsion de leur nouvel entraîneur Jack Gibson, Souths remporte la compétition de pré-saison (10 à 3 face à Canterbury).
Les années 1980
Les seules compétitions remportées par Souths dans les années 1980 sont la Tooth Cup en 1981 et le World Sevens en 1988. En championnat, le club de Redfern atteint les demi-finale en 1980, 1984, 1986, 1987 et 1989, une année où South Sydney termine premier de la saison régulière. En 1980, les Rabittohs terminent à la cinquième place du championnat et sont éliminés en phases finales par St. George 16 à 5. En 1983, Ron Willey devient le nouvel entraîneur de Souths et l'année suivante les qualifie pour les play-off où après avoir battu Canberra (23 à 4) et Manly (22 à 18), St. George les élimine sur le score de 24 à 6. En 1986, les Rabbitohs accrochent une seconde place en championnat mais ratent complètement leur play-off en perdant 2 matchs contre Canterbury et Balmain. En 1989, South Sydney finit à la première place de la phase régulière pour la dix-septième fois de leur existence mais est éliminé par le futur champion, Canberra, lors de la finale préliminaire. Les stars de Souths lors de cette décennie sont le talonneur Mario Fenech, les avants Les Davidson et Ian Roberts, le demi de mêlée Craig Coleman et l'ouverture Phil Blake.
Les années 1990
Cette décennie commence mal pour South Sydney en finissant dernier du championnat en 1990. Les années suivantes, Souths occupe toujours le bas du tableau. En 1996, les Rabbitohs jouent leur dernier match de championnat à Redfern, face aux South Queensland Crushers. En 1997, à la suite de la guerre de la Super League, deux championnats de rugby à XIII coexistent en Australie, Souths Sydney choisit de rester fidèle à celui organisé par l'Australian Rugby League. L'exode de leurs meilleurs joueurs (Mario Fenech, Les Davidson, Mark Carroll et Craig Field) et même de leurs meilleurs jeunes issus du centre de formation (Jim Dymock, Ian Roberts, Jim Serdaris, Terry Hill, Craig Wing et Braith Anasta) vers d'autres clubs est la principale explication de ces mauvais résultats. La réunification de deux championnats concurrents de l'ARL et de la Super League, pour former la National Rugby League, a lieu en 1998. Dans un souci de rationalisation, ce nouveau championnat doit passer de 20 à 14 clubs à partir de 2000. De nombreuses équipes sont amenées à fusionner ou à disparaître ce qui est le cas pour Souths exclu du championnat à partir de 2000. Durant deux ans, South Sydney va se lancer dans une bataille juridique contre la National Rugby League et News Limited pour réintégrer le championnat. Cette cause obtient un grand soutien populaire avec notamment des marches allant jusqu'à 80 000 personnes pour manifester leur mécontentement du sort subi par Souths. Le , la Cour Fédérale décide de la réadmission des Rabittohs en championnat dès la saison 2002.
Le renouveau des Rabbitohs
En 2002, Nick Pappas, avocat d'affaires et un des principaux acteurs de cette réadmission, succède à l'ancien joueur George Piggins comme président du club. La première rencontre de Souths, après son exclusion, a lieu le au Sydney Football Stadium devant 36 804 spectateurs, dans le cadre du Charity Shield, face à St. George Illawarra. Sans surprise, ce retour à la compétition est particulièrement difficile et le club de Redfern termine à l'avant-dernière place du championnat. En 2003 et 2004, les Rabbitohs finissent lanterne rouge de la saison régulière.
Le , une vaste majorité des membres du South Sydney club (75,8 %) permet à l'acteur hollywoodien Russell Crowe et à l'homme d'affaires Peter Holmes à Court d'obtenir 75 % des parts du club. Russell Crowe, un fervent supporter de South Sydney depuis son enfance, a assisté à de nombreuses rencontres et a soutenu le club lorsque celui-ci était en difficulté financière. Après la guerre de la Super League, Crowe utilise son réseau hollywoodien, pour tenter de convaincre les dirigeants de la National Rugby League de ne pas exclure le club. Cette nouvelle manne financière apportée par les deux principaux actionnaires (3 millions de dollars australien), permet à Souths de recruter des joueurs de niveau international comme les néo-zélandais Roy Asotasi, David Kidwell et Nigel Vagana. En 2007, les Rabbitohs se qualifient, la première fois depuis 1989, pour les phases finales en finissant à la septième place.
Le , Crowe organise un match amical à Jacksonville en Floride et voit son club perdre 24 à 26 face aux anglais des Rhinos de Leeds, devant 12 000 spectateurs. Ces trois dernières saisons, les Rabbitohs ne se qualifient pas pour les phases finales ratant la marche de peu en 2009 et 2010. Pour la saison 2011, le club de Redfern, signe l'international australien, Greg Inglis, élu meilleur joueur du monde en 2009. En 2012, les Rabbitohs terminent à la troisième place de la saison régulière et échouent aux portes de la finale en s'inclinant face aux Bulldogs.
Effectif actuel
Saison
Couleurs et symboles
Emblème et couleurs du club
L'emblème de South Sydney est un lapin blanc qui figure sur leur maillot depuis 1959. L'origine de ce surnom viendrait du fait qu'au début du , les joueurs du club capturaient et vendaient des lapins en criant Rabbitoh dans le quartier de Redfern pour arrondir leur fin de mois. Une autre version est que le terrain de South Sydney était jonché de terriers de lapin, rabbit holes en anglais.
Les couleurs traditionnelles des Rabbitohs sont le rouge cardinal et le vert myrte. Avant son passage au rugby à XIII en 1908, le club local de rugby à XV portaient déjà ses couleurs. Durant la majeure partie de son existence, le maillot de South Sydney est à rayures vertes et rouges. en 1945 et 1946, la tradition est rompue puisque la tunique du club est verte avec un "V" rouge. De 1980 à 1984, le maillot est vert avec une bande centrale rouge avec aux extrémités de celle-ci deux bandes blanches. Avec l'introduction du maillot extérieur à la fin du , le club de Redfern choisit la couleur blanche.
Mascotte
La mascotte de South Sydney est un lapin qui se nomme . Il s'agit de la plus vieille et de la plus célèbre mascotte du Rugby à XIII. En 1968, un supporter ramène des États-Unis un costume de lapin et est porté pour la première fois lors de la finale du championnat 1968 contre Manly. La première personne à endosser ce costume est un certain Albert Clift. Ce dernier a fait don en 1998 de la cloche utilisée lors du premier match du championnat en 1908 à une vente aux enchères destinée à renflouer les caisses du club. La cloche est acquise par le futur propriétaire du club, l'acteur Russell Crowe, pour un montant de 48 000 dollars australiens. Dans la NRL, la mascotte de Souths est la seule à ne pas être rémunérée.
Infrastructures
Stades
Bien que Souths soit un club fondateur de la ligue australienne, il a évolué durant son histoire dans plusieurs stades. Le club de South Sydney, n'a joué véritablement dans son fief, le South Sydney District, qu'entre 1948 à 1987, lorsqu'il joue au Redfern Oval. De 1908 à 1947, et de 1988 à 2005 (excepté les 2 années de sommeil), les rabbitohs disputent leurs matchs à domicile dans le quartier Est de Sydney, au Moore Park, qui se trouve être le fief des Sydney Roosters (anciennement Eastern Suburbs). Depuis 2006, le club joue ses matchs à domicile, dans le quartier Ouest de Sydney, à l'ANZ Stadium.
Durant les premières années de championnat de la New South Wales Rugby League, South Sydney joue le plus souvent ses matchs à domicile au Royal Agricultural Society Ground (1908-1920) ou au Sydney Sports Ground (à partir de 1911). En 1947, le club évolue pour sa dernière saison au Sports Ground, avant de déménager pour le Redfern Oval en 1948. En 1988, les Rabbitohs commencent à jouer dans le nouveau Sydney Football Stadium, construit sur l'ancien Sydney Sports Ground. En 2005, le contrat qui lie les Rabbitohs avec le Sydney Football Stadium, arrive à échéance et le club de Redfern lorgne vers Gosford, North Sydney et le Telstra Stadium (aujourd'hui ANZ Stadium). Finalement, la décision se porte sur ce dernier et en , les Rabbitohs renouvelle jusqu'à fin 2017 le partenariat avec l'ANZ Stadium. En 2008, la ville a investi 19.5 millions pour rénover le Redfern Oval qui à partir de 2009 peut accueillir des rencontres de pré-saison. Certains matchs de Souths sont délocalisés, notamment chaque année au Bluetongue Central Coast Stadium de Gosford et depuis 2009 au Members Equity Stadium de Perth.
Le record de spectateurs dans un match de saison régulière est de lors de la réception de St. George en 1969 au Sydney Cricket Ground. Le record de spectateurs au Sydney Football Stadium, pour un match de saison régulière est de (face aux Sydney Roosters en 2002) et à l'ANZ Stadium de (face aux Bulldogs en 2007). La finale du championnat de 1965, St. George opposant à Souths, détient le record du Sydney Cricket Ground, avec spectateurs.
Installations
Situation financière
Le budget de South Sydney s'élève en 2013 à 23.8 millions de dollars Australiens. Les Rabbitohs sont le second plus important budget de NRL, derrière les Brisbane Broncos (35 millions de dollars Australiens) et devant les Canterbury Bulldogs (22.2 millions de dollars australiens). La majeure partie de leurs recettes proviennent du sponsoring (6.7 m), de la billetterie (5.6 m) et du merchandising (3.7 m). Les principaux sponsors du club sont Crown Resorts (Groupe de loisirs Australien) et DeLonghi. Depuis 2013 et pour trois ans, le groupe Crown Resorts débourse 1 million de dollars Australiens pour être le principal sponsor maillot des Rabbitohs.
Statistiques et records
South Sydney avec 21 championnats remportés, est l'équipe la plus titrée de la National Rugby League dont le dernier titre intervient en 2013. Enfin, il a également remporté la défunte Coupe Panasonic en 1981 (compétition organisée de 1974 à 1989). Le club a la distinction d'avoir marqué le plus grand nombre de points (42), d'essais (8) et de buts (9) lors d'une finale de championnat (tous accomplis en 1951 contre Manly).
Eric Simms détient le record de points inscrits sous le maillot des Rabbitohs (1841 points). Il détient également le record de points inscrits en une saison pour South Sydney avec 265 points en 1969. Nathan Merritt détient quant à lui le record d'essais marqués pour Souths avec 146 essais en 237 rencontres. Le record d'essais en une saison est l'attribut de Les Brennan avec 29 essais lors de la saison 1954. Johnny Graves détient lui le record de points inscrits lors d'un seul match avec 29 points, dont 3 essais, contre Easts en 1952. John Sutton est le joueur détenant le record de rencontres joués avec les Bunnies (259 matchs).
Le plus grand écart de points dans un match à l'actif des Rabbitohs est de 67 points contre Wests en 1910 avec une victoire 67-0. En revanche, leur plus grand écart de points dans un match au passif de Souths est contre les New Zealnd Warriors par 66-0 en 2006.
Bilan du club
Figures historiques
Entraîneurs
Joueurs emblématiques
La South Sydney Dream team est révélée le à Sydney. Cette équipe est composée de 17 joueurs et d'un entraîneur ayant marqué le club de 1908 à 2004.
En 2002, un panel de journalistes, d'anciens joueurs de Souths et d'entraîneurs sélectionne une équipe, appelée The Magnificent XIII, composée de 17 joueurs et d'un entraîneur qui ont marqué le club de 1908 à 2002.
La médaille Georges Piggins récompense chaque année depuis 2003 le meilleur joueur de la saison. Le palmarès est le suivant :
Culture populaire
Rivalités
À travers son histoire centenaire, les Rabbitohs et leurs fans ont construit de nombreuses rivalités dont la plus importante est celle avec les Sydney Roosters, l'autre club-fondateur de la NSWRL encore existant. La rivalité entre les Rabbitohs et les Roosters qui se partagent le territoire du centre-ville de Sydney a commencé dès 1908 quand Souths bat 14 à 12 Eastern Suburbs lors de la première finale. De ce fait, les rencontres entre ces deux voisins forment le plus vieux derby local de la National Rugby League. En 1950, cette rivalité croit en raison d'un conflit sur le territoire des juniors et s'accélère à partir de 1970 avec les transferts de nombreux joueurs-clé entre les deux clubs. Lors de la première journée de la saison 2010 est célébré le entre ces deux équipes avec un bilan de 105 victoires pour South Sydney, 90 pour les Sydney Roosters et 5 matchs nuls. Chaque année, la Coupe Ron Coote est mise en jeu entre ces deux clubs historiques.
Les autres rivaux traditionnels sont Manly (qui a acheté depuis 1970 de nombreux joueurs de Souths) mais aussi les anciens clubs de St George (maintenant avec St. George Illawarra pour l'obtention du Charity Shield) et de Balmain. La rivalité avec Balmain remonte en 1909 lorsque ces derniers refusent d'affronter Souths lors de la finale pour protester contre l'organisation du match entre les Kangaroos et les Wallabies.
Supporters
Les South Sydney Rabbitohs continue d'avoir une importante base de supporters dans leur zone traditionnelle du sud-est de Sydney, malgré le fait d'avoir déménagé du Redfern Oval depuis deux décennies. Outre la ville de Sydney, le club compte de nombreux fans à travers toute l'Australie. Le principal groupe de supporters est The Burrow. En , les Rabbitohs comptent 17 500 membres, soit le nombre le plus important de la National Rugby League. Dans ces 17 500 membres sont inclus 11 000 abonnés, soit le nombre le plus important des clubs de Sydney. Les membres du club proviennent de tous les États australiens et de 22 pays à travers le monde. En 2002, à la suite de la réintégration de Souths en NRL, le club a compté jusqu'à 22 000 membres. Pour la saison 2013, les Rabiitohs rassemble plus de 25 000 abonnés.
Le Groupe 14 est une association de personnes influentes créée avant l'exclusion des Rabbitohs en 1999 et a contribué à la réadmission de Souths en 2002. À la suite de l'annonce de cette exclusion, un grand soutien populaire, sans précédent dans l'histoire du sport australien, s'est formé autour des Rabbitohs avec un rassemblement de 40 000 personnes dans le centre des affaires de Sydney. En 2000 et 2001, plusieurs manifestations de soutien ont lieu dont une rassemblant près de 80 000 personnes. Un fort lien existe entre la communauté aborigène du quartier de Redfern et Souths qui remonte à la formation du Redfern All Blacks Football Club en 1930, le plus vieux club aborigène de rugby à XIII du pays. Actuellement de nombreux joueurs de souths sont d'origine indigène comme Nathan Merritt ou Greg Inglis.
En 2013, le club a connu sa meilleure affluence annuelle avec 22 261 spectateurs de moyenne par matchs.
Notes et références
Annexes
Bibliographie
Liens externes
Site officiel du club
Club de rugby à XIII à Sydney
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 7,262
|
Q: Pictures not centered in carousel I made a carousel using twitter-bootstrap, and it shows a couple pictures. Problem is, all of the pictures are different sizes and, right now, they aren't centered. Is there a way that the carousel can re-size to center each picture horizontally and vertically whenever the picture in question changes?
Here is code so far:
<div class="modal-body">
<div id="picture-carousel" class="carousel slide" data-ride="carousel">
<!-- Indicators -->
<ol class="carousel-indicators">
<li data-target="#picture-carousel" data-slide-to="0" class="active"></li>
<li data-target="#picture-carousel" data-slide-to="1"></li>
<li data-target="#picture-carousel" data-slide-to="2"></li>
</ol>
<!-- Wrapper for slides -->
<div class="carousel-inner">
<div class="item active">
<img src="/images/picture-1.jpg" alt="" />
<div class="carousel-caption">
<p></p>
</div>
</div>
<div class="item">
<img src="/images/picture-2.jpg" alt="" />
<div class="carousel-caption">
<p></p>
</div>
</div>
<div class="item">
<img src="/images/picture-3.jpg" alt="" />
<div class="carousel-caption">
<p></p>
</div>
</div>
</div>
<!-- Controls -->
<a class="left carousel-control" href="#picture-carousel" data-slide="prev">
<span class="glyphicon glyphicon-chevron-left"></span>
</a>
<a class="right carousel-control" href="#picture-carousel" data-slide="next">
<span class="glyphicon glyphicon-chevron-right"></span>
</a>
</div> <!--Carousel-->
</div> <!--modal-body-->
Thanks for any and all tips!
A: Use this Style to center images :-
div.carousel-inner img{
margin: 0 auto;
}
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 1,501
|
>> Study Guides
TAGS: Suriy-i-Haykal (Surih of the Temple)
Prepared as part of Wilmette Institute notes and commentary on the Tablets of Bahá'u'lláh.
Add or read comments or links pertaining to this work here.
See also http://bahai-library.com/bahaullah_summons_lord_hosts.
Tablet of the Temple (Súratu'l-Haykal):
Wilmette Institute faculty notes
by Iraj Ayman
The word "Haykal" (Temple) in Arabic and Persian comes from a Sumerian root. In Sumerian language it referred to temple (Place of worship) or castle. In Arabic and Persian it has many meanings and connotations in addition to the body or temple of a human being, such as, the face of a human, a statute, tall building or tree, corpulent animal or human being. It is also used as the name of the place of religious sacrifice in the temples or synagogue. In Persian it has also been used for a charm or a refuge.
The word has a rich background in literature and religious texts of all Semitic religions. In the Bábí period the Báb has revealed a number of Súrah-ye Herziyyeh composed of names of God, numbers and cryptographic symbols written in pentacle shape. Bábís were carrying them on their body for protection purposes. The Báb had written these charm-like protection prayers in the pentacle shape for men and in circular shape for women. Therefore in the Bábí period men were referred to as Awlel-Hayakil (Those having or carrying the temples) and women were referred to as Zavatu'l-Dava'ir (Those who are the essence of circles). These Súrahs were referred to as Hayakil (pl.haykal). The Báb has also revealed a Tablet, a book, for Dayyan, by the name of Hayakil which is a commentary on the science of numerology. Bahá'u'lláh has referred to this book in some of his Tablets and the mysteries contained in it.
In the Bahá'í era: Bahá'u'lláh has revealed a charm-like prayer, in pentacle form, for protection. It is called Du'a-ye Haykal (Prayer of Haykal). Bahá'u'lláh in a special Tablet has given the exact and rather elaborate instructions on how this prayer should be copied, in what color and with what kind of ink, and how it should be carried. The instruction even includes the amount of the contribution that should be donated by the person who wants to carry this prayer on himself.
Súrih-ye Haykal is one of the Tablets that was revealed in Akká in 1869. Its original text in Arabic is 88 printed page and contains the texts of a number of Tablets addresses to the Kings and Rulers. It should not be mistaken by Lawh-i-Haykal which contains a selection of extracts from the Tablets addressed to the Kings and Rulers of the world and is written in pentacle form at the instruction of Bahá'u'lláh . Someone had asked Bahá'u'lláh about the addressee of Súrih-ye Haykal. In response, He has stated that both the addresser and the addressee is His own self. This point is also evident from certain statements in the Súrih-ye Haykal. Another indication is in the Kitáb-i-Aqdas para 86: "O King of Berlin! Give ear unto the Voice calling from this manifest Temple." Yet an extra indication that Haykal refers to Bahá'u'lláh is in Hidden Words (Arabic): "The temple of being is My Throne."
It should be also added that Bahá'ís in the East particularly in Persia from the very early days were and still are using "Hakal-i-Mobarak" (The Blessed Temple) when they refer to the central figures of the Faith (the Báb, Bahá'u'lláh, 'Abu'l-Bahá and Shoghi Effendi).
"Haykal" in the Writings have been used to allude to other things as well. At the beginning of the Will and Testament of 'Abdu'l-Bahá, it refers to the Cause of God: " All praise to Him Who, by the Shield of His Covenant, hath guarded the Temple of His Cause." Bahá'u'lláh In the Tablet of Nasir refers to Mírzá Yahyá as the Haykal-i-Nar (the Temple of Fire). Also in the Kitáb-i-Aqdas, para 14, seating cross-legged in original Arabic is seating in the form of Haykal-i-Tawhíd (Tabernacle of Unity).
Back to: Study Guides
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 6,443
|
{"url":"http:\/\/mathhelpforum.com\/discrete-math\/127399-induction-inequality.html","text":"# Math Help - Induction for an inequality\n\n1. ## Induction for an inequality\n\nlet a1=2\n\nan+1=3+an^2\/2an\n\nprove n>=1 an>root3\n\n2. hmmm... this is an interesting question, but I'm not sure if I've understood it correctly. Do you mean:\n$\na_{n + 1} = (\\frac{3 + a_n^2}{2a_n})$\n; $n > 1, a_n > \\sqrt{3}\n$\n?\n\nOops, took me a while to figure out what was meant by the question...\n\n$\na_{n + 1} = \\frac{3 + (\\sqrt{3})^2}{2(\\sqrt{3})} = \\frac{3}{\\sqrt{3}} =\\sqrt{3}$\n; substitue $a_n = \\sqrt{3}$\n\nmind you that's not an answer!","date":"2015-05-22 11:58:01","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 4, \"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\": 0.8530371785163879, \"perplexity\": 1403.2389309231323}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2015-22\/segments\/1432207924991.22\/warc\/CC-MAIN-20150521113204-00192-ip-10-180-206-219.ec2.internal.warc.gz\"}"}
| null | null |
JLL changes
By Staff Reporters
JONES Lang LaSalle executive director Peter Agostino has resigned from the position of executive director in Perth.An announcement was made to staff earlier this week, however Mr Agostino was unwilling to discuss his plans for the future.
JONES Lang LaSalle executive director Peter Agostino has resigned from the position of executive director in Perth.
An announcement was made to staff earlier this week, however Mr Agostino was unwilling to discuss his plans for the future.
"It's my intention to continue to work in the property industry but I haven't made any firm decision as to in what capacity that might be," Mr Agostino said.
The Jones Lang LaSalle WA chief was only appointed to the position of executive director in July of 2001, replacing John Wilshaw.
In March this year Mr Agostino was further promoted from national director to regional director for Asia-Pacific.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 4,794
|
\section{Introduction}
Generalization from limited data is the basic cognitive ability of intelligence~\cite{sims2018efficient}. To achieve class-level generalization abilities, researchers develop zero-shot learning (ZSL)~\cite{xian2018zero}, which aims to identify unseen classes without any available images during training. ZSL can also be extended to a more general setting called generalized ZSL (GZSL) which tries to identify both seen and unseen classes at test time~\cite{chao2016empirical}.
\begin{figure}[htbp]
\centering
\includegraphics[width=0.95\linewidth]{figures_banner.pdf}
\caption{Comparison between traditional embedding ZSL methods and our Re-balanced MSE (ReMSE). Traditional embedding ZSL methods generally produce imbalanced error losses on semantic prediction, which have a high co-relation with label value. Our ReMSE could decease the co-relation and obtain a re-balanced error distribution.}
\label{fig:banner}
\end{figure}
Generally, existing ZSL methods utilize various semantic labels, e.g. word2vec~\cite{xian2018zero} and attributes~\cite{lampert2013attribute} as auxiliary information, to obtain knowledge that transfers from seen to unseen classes.
Benefiting from these semantic labels, researchers can build a learnable mapping between visual space and semantic space and classify unseen classes in the mapping space, which is called embedding ZSL.
Recently, ZSL methods have mainly focused on exploiting various complex modules and architectures, such as GCN~\cite{xie2020region} and Transformer~\cite{chen2022transzero} to extract powerful visual features and improve visual-semantic interaction. However, high computational demands caused by such a complex structure pose challenges in practical applications.
In this work, we address the zero-shot problem from a brand new perspective, namely how semantic labels would affect the performance of ZSL methods.
Specifically, we find that almost all existing ZSL models suffer from imbalanced semantic prediction, i.e. the model can accurately predict some semantics but may not for others.
We argue that such problem could significantly limit the performance of ZSL.
In fact, treating every semantic fairly is a prominent topic in current visual recognition systems~\cite{ramaswamy2021fair, park2022fair}.
Essentially, an ideal ZSL model should treat all semantic labels with equal importance.
However, borrowing the imbalanced regression defined in~\cite{yang2021delving}, we find that such imbalanced semantic prediction in ZSL unfortunately suffers from two unique challenges. On one hand, as shown in Fig.~\ref{fig:banner} (a), the imbalanced predictions in ZSL are highly correlated with the value of semantic labels, rather than the number of samples as seen in traditional imbalanced learning~\cite{yang2021delving,ren2021balanced,hu2020learning}; on the other hand, ZSL is multi-label imbalanced, i.e. different semantics follow quite different error distributions between classes. As such, existing general imbalanced regression methods are not suitable for the ZSL scenario. For instance, the recent Balanced MSE~\cite{ren2021balanced}, one important general imbalanced learning method, is rather limited in dealing with ZSL, which is also empirically demonstrated in our experiments.
To this end, we first statistically examine the effect of semantic labels on imbalanced semantic prediction.
Next, leveraging a novel notion of the class-averaged semantic error matrix, we develop a simple yet effective re-balanced strategy, \textbf{Re-balanced Mean-squared Error (ReMSE)} loss, which mitigates the inherited imbalance drawback observed in ZSL.
To penalize under-fitting semantics potentially with more errors, ReMSE relies on the statistics of semantic errors to generate re-weighting factors.
Besides, to adjust different error distributions, we also design two-level re-weighting factors, i.e. 1) Class-level re-weighting that compares errors of the same class across different semantics, and 2) Semantic-level re-weighting that compares errors of the same semantic across classes.
Analytically, we show that minimizing the ReMSE loss tends to minimize the mean error loss as well as the standard deviation of error losses across different semantics, thus certifying the effectiveness of ReMSE in learning a re-balanced visual-semantic mapping.
In summary, our main contributions are three-fold:
\begin{enumerate}
\item[(1)]
It is a first attempt to theoretically and empirically analyze that ZSL is an imbalanced regression problem affected by semantic label values, thus offering a new insight into ZSL.
\item[(2)] A novel loss function ReMSE is designed for ZSL which dynamically perceives multiple error distributions, focusing on under-fitting semantics without increasing inference cost. Furthermore, We show that minimizing the ReMSE loss tends to minimize the mean and variance of the error distributions, leading to a rebalanced ZSL.
\item[(3)]
Extensive experiments on three ZSL benchmarks show that our ReMSE effectively alleviates the imbalanced regression problem. Without bells and whistles, our approach outperforms many state-of-the-arts in ZSL (e.g. Transformers), as well as the imbalanced regression: Balanced MSE.
\end{enumerate}
\section{Related Work}
\label{sec:rel}
\subsection{Zero-shot Learning}
Existing ZSL can be mainly divided into generative methods and embedding methods.
Generative methods utilize generative models, e.g. Generative Adversarial Network~(GAN)~\cite{li2019alleviating,ye2019sr}, Variational AutoEncoder~(VAE)~\cite{li2021generalized}, and Flow models~\cite{shen2020invertible} to synthesize unseen visual features.
In this paper, we focus on the embedding methods, which typically learn the visual-semantic mapping and classify unseen classes in the mapping space.
Most recent embedding ZSL proposals study ZSL from the model perspective, i.e. by introducing more complex modules (e.g. GNN)~\cite{fu2017zero} or frameworks (e.g. Transformer)~\cite{chen2022transzero} to extract highly powerful visual features. However, there are just a few investigations of ZSL from the data perspective.
For example, \cite{akata2015label} shows that $l_{2}$ normalization can compress the noise of semantic labels.
Most of these works do not explicitly analyze the impact of semantics.
In contrast, we focus on how semantic annotations would impact learning difficulty. Identifying the semantic prediction imbalance of ZSL, we propose the novel ReMSE algorithm. Supported by
the rebalancing strategy, simple CNN-based models can lead to superior performance, even on par with other complex models.
\subsection{Imbalanced Learning}
Imbalanced learning has been widely studied but received more and more attention recently due to its extensive application in practical scenarios.
While most imbalanced learning focuses on classification~\cite{cao2019learning,hu2020learning}, imbalanced regression involving continuous and infinite target values was first defined in~\cite{yang2021delving}.
Along this research, Balanced MSE~\cite{ren2021balanced} assumes a balanced distribution to represent the true training and testing distributions.
Albeit its good performance, existing imbalanced regression attributes imbalanced optimization to the imbalanced sample distribution. Differently, we find that imbalanced values of semantic labels also play a major role for ZSL.
As such, we propose the ReMSE method that could obtain re-balanced prediction error distributions across classes.
\section{Preliminary}
The main goal of ZSL is to obtain a classifier that can distinguish visual samples (i.e. images) $\mathcal{X}^{u}$ of unseen classes $\mathcal{C}^{u}$ from the images $\mathcal{X}^{s}$ of seen classes $\mathcal{C}^{s}$, which only appear in the training set, i.e. $\mathcal{C}^{s} \cap \mathcal{C}^{u} = \emptyset$.
Since existing methods utilize the class-level semantic labels $\mathcal{S}$ (e.g. attributes or word2vec) to bridge the gap between seen and unseen classes, we define the training set as $\mathcal{D}^{tr}=\{(\mathbf{x}_i, \mathbf{s}_i, y_i)| \mathbf{x}_i \in \mathcal{X}^s, \mathbf{s}_i \in \mathcal{S}, y_i \in \mathcal{C}^{s}\}$, where $\mathbf{x}_i$ and $\mathbf{s}_i$ represent image of the $i$-th sample and its semantic vector, respectively.
The number of samples in the training set is denoted by $n_{tr}$.
Similarly, we can denote the test set by $\mathcal{D}^{te}=\{(\mathbf{x}_i,\mathbf{s}_i,y_i)| \mathbf{x}_i \in \mathcal{X}^u, \mathbf{s}_i \in \mathcal{S}, y_i \in \mathcal{C}^{u} \}$, where the testing samples are from unseen classes in ZSL setting.
In GZSL setting, the testing samples may be taken from seen classes, in which $\mathcal{X}^u$ and $\mathcal{C}^{u}$ will be replaced by $\mathcal{X}^u \cup \mathcal{X}^s$ and $\mathcal{C}^u \cup \mathcal{C}^s$, respectively. We denote $d_{s}$ and $d_{v}$ as the dimension of semantic label and visual feature.
It is worth mentioning that the values of semantic labels vary in different datasets.
For example, on the dataset CUB~\cite{wah2011caltech}, 312 semantic values range from $0$ to $100$.
We focus on using Embedding ZSL~\cite{xie2019attentive,liu2021goal} in our work. Such methods first use a pre-trained model, such as ResNet, as the backbone for extracting visual feature $\tilde{\mathbf{v}}_{i}$ of the image $\mathbf{x}_i$.
Then a fully connected network (semantic predictor) with a parameter of $\mathbf{W}\in \mathbb{R}^{d_{s} \times d_{v}}$ solves the ZSL task by semantic prediction:
\begin{equation}
f(\tilde{\mathbf{v}}_i) = \mathbf{W} \tilde{\mathbf{v}}_{i}= \tilde{\mathbf{s}}_{i}.
\end{equation}
Typically, some recent work~\cite{xu2020attribute,liu2021goal,du2022boosting} trains embedding ZSL models by leveraging the so-called Semantic Cross-Entropy loss (SCE):
\begin{equation}
\label{eq:SCE}
\mathcal{L}_{SCE} = -\log p_{y_i}(\mathbf{x}_{i}),
\end{equation}
\begin{equation}
\label{eq:CP}
p_{y_i}(\mathbf{x}_i) = \frac{
e^{
\tau\cos \theta (\tilde{\mathbf{s}}_i, \mathbf{s}_i)
}
} {
\sum_{c \in \mathcal{C}_s } e^{
\tau\cos \theta (\tilde{\mathbf{s}}_i, \mathbf{s}_c)
}
} ,
\end{equation}
in which $p_{y_i}(\mathbf{x}_i)$ is the predicted probability of class $y_i$ for the sample $\mathbf{x}_i$, $\tau$ is a scale hyper-parameter, $\theta (\tilde{\mathbf{s}}_i, \mathbf{s}_c)$ is defined as the angle between semantic predictions $\tilde{\mathbf{s}}_i$ of sample $i$ and semantic labels $\mathbf{s}_c$ of class $c$, i.e. $\cos \theta (\tilde{\mathbf{s}}_i, \mathbf{s}_c) = \tilde{\mathbf{t}}_i^{T} \mathbf{t}_c$, where $\tilde{\mathbf{t}}_i$ and $\mathbf{t}_c$ denote the $l_2$-normalized semantic prediction and class semantic label, i.e.
\begin{align}
\tilde{\mathbf{t}}_i = \tilde{\mathbf{s}}_i/\|\tilde{\mathbf{s}}_i\|_2 ,
\mathbf{t}_c = \mathbf{s}_c/\|\mathbf{s}_c\|_2.
\end{align}
Specifically, for ZSL, the test sample $\mathbf{x}_{i} \in \mathcal{X}^u$ can be assigned to the best matching class $c^{\prime}$ from the unseen classes $\mathcal{C}^u$:
\begin{equation}
c^{\prime} = \underset{c \in \mathcal{C}^u}{\arg \max} \,\cos\theta (\tilde{\mathbf{s}}_i, \mathbf{s}_c).
\end{equation}
In the GZSL setting, the testing samples may be taken from seen classes, in which $\mathcal{X}^u$ and $\mathcal{C}^{u}$ will be replaced by $\mathcal{X}^u \cup \mathcal{X}^s$ and $\mathcal{C}^u \cup \mathcal{C}^s$, respectively.
\begin{figure*}[htbp]
\centering
\begin{minipage}[t]{0.48\linewidth}
\centering
\includegraphics[width=9cm]{figures_ErrorDistribution.pdf}
\caption{Class-averaged semantic error matrix on test set of AWA2 and CUB under GZSL setting (rescaled by logarithmic function). It shows (1) error distributions are not balanced; (2) the same semantic has different prediction errors on different classes;
(3) and the same class also has different prediction errors on different semantics.}
\label{fig:ErrorDistribution}
\end{minipage}
\quad
\begin{minipage}[t]{0.47\linewidth}
\centering
\includegraphics[width=8cm]{figures_correspondence.pdf}
\caption{Correlation between semantic label value $t$ and averaged semantic prediction error $m$, which exhibit a strong positive linear relationship regrading Pearson Correlation Coefficient (PCC).}
\label{fig:correspondence}
\end{minipage}
\end{figure*}
\section{Main Methodology}
In this section, we first treat the ZSL as a regression problem. Next, we verify that the imbalanced semantic prediction remains in the existing methods. We then study imbalanced semantic prediction statistically by examining how semantic label values could affect averaged semantic prediction.
Finally, we propose our Rebalanced MSE which could well mitigate the imbalanced semantic prediction issue.
Our Rebalanced MSE ensures that our model not only minimizes these prediction errors, but also tends to equal or balance these prediction errors whilst converging.
Furthermore, to enhance the model's representation ability on different semantics, we further design a novel attention-based baseline, named AttentionNet, which can generate semantic-specific attention maps during training and testing stage.
\subsection{Tackling ZSL via Regression}
\label{sec:tackling}
Compared to training ZSL models with SCE, treating ZSL as a regression task has an unique advantage:
SCE entangles the gradient of one semantic with other semantics, whereas regression losses (i.e. MSE, MAE and so on) do not.
Specifically, by representing the $k^{th}$ row vector of $\mathbf{W}$ as $\mathbf{w}_{k}$ responsible for the $k^{th}$ semantic prediction, we could derive the gradient as
\begin{align}
\label{eq:SCEgrad}
\frac{\partial \mathcal{L}_{SCE}}{\partial \mathbf{w}_{k} } & = \sum_{c\in \mathcal{C}^{s}} \frac{\partial \mathcal{L}_{SCE}}{\partial \tilde{\mathbf{t}}_i^{\top} \mathbf{t}_{c} } \frac{\partial \tilde{\mathbf{t}}_i^{\top} \mathbf{t}_{c} } {\partial \mathbf{w}_{k} } \nonumber \\
& = \sum_{c \in \mathcal{C}^{s}} \underbrace{ \left( p_{y_i}(\mathbf{x}_i) - \mathds{1}_{[c=y_i]} \right) }_{\text{class probability term}} \frac{\partial \tilde{\mathbf{t}}^{\top} \mathbf{t}_{j} } {\partial \mathbf{w}_{k}}.
\end{align}
Obviously, the class probability is determined by the whole semantic prediction. Thus, even if some semantic predictions are not good, the predictions of these semantics can still be further optimized when the class probability is close to 1.
In contrast, regression losses could explicitly perceive the performance of every semantic.
For example, denoting the formulation of MSE:
\begin{equation}
\mathcal{L}_{MSE} = \frac{1}{N} \sum_{i = 1}^N \|\tilde{\mathbf{s}}_i - \mathbf{s}_i\|_2^2,
\end{equation}
we can obtain the gradient for $\mathbf{w}_{k}$:
\begin{equation}
\label{eq:MSE}
\frac{\partial \mathcal{L}_{MSE}}{\partial \mathbf{w}_{k} } = 2(\mathbf{w}_{k}^\top \tilde{\mathbf{v}}_i - s_{ik}) \tilde{\mathbf{v}}_i.
\end{equation}
It is evident that the optimization of $\mathbf{w}_{k}$ would not be interfered by other semantics.
It is worth mentioning that many works~\cite{romera2015embarrassingly, qiao2016less, xu2020attribute, liu2021goal} tried to utilize the regression loss $\mathcal{L}_{MSE}$ as a compensatory loss in ZSL, i.e.,
\begin{equation}
\mathcal{L} = \mathcal{L}_{SCE} + \lambda \mathcal{L}_{MSE},
\end{equation}
where $\lambda$ denote a hyper-parameter.
Although $\mathcal{L}_{MSE}$ penalizes the discrepancy between semantic labels and semantic predictions with the Euclidean distance, it is incompatible with the cosine distance optimized by the SCE loss in ZSL. We have the following proposition:
\begin{shaded}
\begin{proposition}
\label{prop:norm}
For any regressor with the MSE loss and parameter $\mathbf{W}$, given any example $(\mathbf{x}_i,\mathbf{s}_i,y_i)$, when the model tends to be optimal, we have $\|\tilde{\mathbf{s}}_i\|_{2} \rightarrow \|\mathbf{s}_i\|_2 \cos \theta$, where $\theta$ is the angle between the semantic value and its predicted value.
\end{proposition}
\end{shaded}
\begin{proof}
Recalling the formulation in Eqn.~(\ref{eq:MSE}), the MSE loss can be expanded as follows:
\begin{align}
\mathcal{L}_{MSE} = \frac{1}{N} \sum_{i = 1}^N (\|\tilde{\mathbf{s}}_i\|_2^2 - 2\|\tilde{\mathbf{s}}_i\|_2 \|\mathbf{s}_i\|_2 \cos \theta + \|\mathbf{s}_i\|_2^2). \nonumber
\end{align}
By directly calculating the gradient of the MSE loss with respect to the parameter $\mathbf{W}$, we obtain
\begin{align}
\label{eq:MSEgrad}
\frac{ \partial \mathcal{L}_{MSE} }{ \partial \mathbf{W} } & = \frac{1}{N} \sum_{i = 1}^N \frac{ \partial \mathcal{L}_{MSE}} { \partial \|\tilde{\mathbf{s}}_i \|_2 } \frac{ \partial \|\tilde{\mathbf{s}}_i \|_2}{\partial \mathbf{W} } \nonumber \\
& = \frac{1}{N} \sum_{i = 1}^N 2(\|\tilde{\mathbf{s}}_i\|_2 - \|\mathbf{s}_i\|_2 \cos \theta) \frac{ \partial \|\tilde{\mathbf{s}}_i \|_2 }{\partial \mathbf{W} }. \nonumber
\end{align}
Hence, when the gradient tends to be $0$, the magnitude of $\frac{ \partial \|\tilde{\mathbf{s}}_i \|_2 }{\partial \mathbf{W} }$ becomes very small, then we have $\|\tilde{\mathbf{s}}_i\|_2 \rightarrow \|\mathbf{s}_i\|_2 \cos \theta$.
\end{proof}
This proposition shows that the original MSE may not precisely measure how well the semantics fit.
We can see that even if $\|\tilde{\mathbf{s}}_i\|_2 \rightarrow \|\mathbf{s}_i\|_2 \cos \theta$, $\tilde{\mathbf{s}}_i$ and $\mathbf{s}_i$ may still generate a big angle. Therefore, the predicted and ground-truth semantics can still be very inconsistent.
Thus, we normalize both semantic labels and predictions in MSE to obtain the Normalized MSE $\mathcal{L}_{NMSE}(\tilde{\mathbf{s}}_i,\mathbf{s}_i) = \mathcal{L}_{MSE}(\tilde{\mathbf{t}}_i,\mathbf{t}_i)$, which proves fairly compatible with $\mathcal{L}_{SCE}$, since the $\mathcal{L}_{NMSE}= 2- \cos\theta(\tilde{\mathbf{s}}_i, \mathbf{s}_i)$.
It is noted that Balanced MSE~\cite{ren2021balanced} handles imbalanced error distributions by assuming that the distribution of sample is balanced. Its batch-based formulation is
\begin{equation}
\label{eq:BalancedMSE}
\mathcal{L}_{balMSE} = -\log \frac{e^{-\|\tilde{\mathbf{t}}_{i}-\mathbf{t}_{i}\|^2_2/\sigma } }{\sum_{\mathbf{t}_{j} \in B_{\mathbf{t}}} e^{-\|\tilde{\mathbf{t}}_{i}-\mathbf{t}_{j}\|^2_2/\sigma} },
\end{equation}
where $B_{\mathbf{t}}$ is a batch of normalized semantic labels and $\sigma$ is a learnable parameter. It can be viewed as $\mathcal{L}_{NMSE}$ with a regularization to balance sample distribution.
However, as discussed in the next section, we find that the imbalanced error distribution in ZSL is caused by the imbalanced semantic values, rather than the sample size. Thus, the Balanced MSE does not perform well in the ZSL setting.
\subsection{Imbalanced Semantic Prediction}
\label{sec:ISP}
Now, we consider the imbalanced semantic prediction problem in ZSL.
During the training process, we find that previous methods often generate imbalanced semantic error distribution.
To qualitatively illustrate this, in the GZSL setting\footnote{Note that the test samples of unseen classes of ZSL is the same as in GZSL setting. Thus we only need visualize error matrices under GZSL setting.}, we exploit GEMZSL (an advanced ZSL method) to visualize the class-averaged semantic error matrix $M \in \mathcal{R}^{|\mathcal{C}^u \cup \mathcal{C}^s| \times d_{s}}$ on AWA2 and CUB.
Specifically, we collect test samples from every class $l \in \mathcal{C}^u \cup \mathcal{C}^s$ and compute the mean prediction error for each class and each semantic
\begin{equation} \label{eqn:prediction_loss}
m_{lj} = \frac{1}{|\mathcal{D}_{l}|}
\sum_{(\mathbf{x}_i,\mathbf{s}_i,y_i) \in \mathcal{D}_{l}} (\tilde{t}_{ij}-t_{ij})^2,
\end{equation}
where $|\mathcal{D}_l|$ is the number of samples in $\mathcal{D}_l$ and $t_{ij}$ is the $j^{th}$ element of the vector $\mathbf{t}_i$. Notice that $m_{lj}$ indicates the averaged error loss on the $l^{th}$ class for the $j^{th}$ semantic.
The visualization of GZSL on CUB and AWA2 is shown in Fig.~\ref{fig:ErrorDistribution}.
The results in Fig.~\ref{fig:ErrorDistribution} reveal three interesting observations: (1) ZSL models perform unbalanced on semantic prediction problems: for a certain semantic of a certain class, the model may fit well, it may however do poorly for other semantics and classes; (2) the same semantic has different prediction errors on different classes; and (3) the same class also has different prediction errors on different semantics.
To investigate further, in Fig.~\ref{fig:correspondence}, we employ the Pearson Correlation Coefficient (PCC) to quantitatively measure the correlation between the semantic prediction error $m_{lj}$ and semantic label values $t_{ij}$ on three benchmarks.
It is found that this error loss tends to have a high positive correlation with semantic label value. In other words, the larger the semantic label value, the larger the semantic prediction error.
\subsection{ReMSE}
To this end, we try to utilize imbalanced regression methods to balance the semantic errors.
However, in our experiment, previous imbalanced methods~\cite{ren2021balanced,yang2021delving} (i.e. $\mathcal{L}_{balMSE}$) is not suitable for the imbalance semantic prediction problem.
They all rely on the hypothesis: imbalanced test error is caused by imbalanced sample distribution, while in ZSL the imbalanced problem is related to the imbalanced semantic label values, as verified in our work.
Thus, unlike these methods, we directly adjust the loss, which explicitly measures the semantic prediction performance.
\begin{figure*}[ht]
\centering
\includegraphics[width=0.9\linewidth]{figures_ReMSE.pdf}
\caption{Overview of our ReMSE method: The prediction error is computed on the basis that both the semantic and its predicted value are normalized. Our goal is to minimize the weighted prediction error, where the weight factor of prediction error is represented by the average of prediction errors for the classes, including the category-level factors and the semantic-level factors. It is worth noting that once a new prediction error is obtained, its weight need be recalculated. }
\label{fig:ReMSE}
\end{figure*}
First, we design another class-averaged semantic error matrix $M^{\prime} \in \mathcal{R}^{|\mathcal{C}^s| \times d_{s}}$ within every training batch to establish a common scale for error magnitudes and re-balance every error per class and per semantic.
As shown in Fig.~\ref{fig:ErrorDistribution}, for different datasets, underfitting of semantic predictions could be distinct: the same semantic has different prediction errors on different classes, and the same class also has different prediction errors on different semantics.
Thus, given a label $l$ and a semantic $j$, on one hand, we adopt a semantic-level re-weight factor $p_{lj}$ to balance its weight among different semantics of the same class; on the other hand, we adopt a class-level re-weight factor $q_{lj}$ among different classes of the same semantic.
The class-level balancing factor $p_{lj}$ is designed by
\begin{equation}
p_{lj} = \left(\log \frac{
m^{\prime}_{lj}
}{
\min_{c \in \mathcal{C}^s} m^{\prime}_{cj}
} + 1 \right)^{\alpha}, \quad \alpha \ge 0.
\end{equation}
A logarithmic function is used here to avoid potential ratio explosions, and a parameter $\alpha$ is taken to control the scale of re-weighting.
Similarly, the semantic-level balancing factor $q_{lj}$ is calculated as follows:
\begin{equation}
q_{lj} = \left(\log \frac{
m^{\prime}_{lj}
}{
\min_{1 \le k \le d_s} m^{\prime}_{l k}
} + 1 \right)^{\beta}, \quad \beta \ge 0.
\end{equation}
Finally, we obtain the Rebalanced MSE loss function:
\begin{equation}
\label{eq:ReMSE}
\mathcal{L}_{ReMSE} = \frac{1}{N} \sum^{N}_{i=1} \sum^{d_{s}}_{j=1} p_{y_i j} q_{y_i j} e_{ij},
\end{equation}
where $ e_{ij}= (\tilde{t}_{ij}-t_{ij})^2$.
In order to better understand the property of our ReMSE, we reduce our problem to the simplest case, proving that the prediction error for each class will be equal when the algorithm converges.
\begin{shaded}
\begin{theorem}
\label{theorem:rmes} Given two classes of data $\{(x,c_1),(y,c_2)\}$, which are located in a 1-dimensional space, and $\tilde{x}$ and $\tilde{y}$ are the prediction values of the semantic values $x$ and $y$, respectively. The loss function become
\begin{equation}
\mathcal{L}_{ReMSE} = w(x - \tilde{x})^2 + v(y - \tilde{y})^2,
\end{equation}
where $w = p_{1 1}q_{1 1}$ and $v = p_{2 1}q_{2 1}$.
Minimizing $\mathcal{L}_{ReMSE}$ by gradient descent will minimize prediction errors for each class as well as the variance of prediction errors.
\end{theorem}
\end{shaded}
\begin{proof}
For the $t$-th iteration, we have $w_t \ge 1,v_t = 1$ or $w_t = 1, v_t \ge 1$. We know that the prediction errors for the two samples in the $t$-th iteration are $(\tilde{x}_t - x)^2$ and $(\tilde{y}_t - y)^2$.
Without loss of generality, let us assume that $(\tilde{x}_t - x)^2 < (\tilde{y}_t - y)^2$, then we have $w_t = 1$ and $v_t > 1$. We update $\tilde{x}$ and $\tilde{y}$ by gradient descent with a stepsize of $r$ (a small positive constant), and get
\begin{align}
\tilde{x}_{t + 1} &= \tilde{x}_t - 2 r w_t (\tilde{x}_t - x), \\
\tilde{y}_{t + 1} &= \tilde{y}_t - 2 r v_t (\tilde{y}_t - y).
\end{align}
Therefore, we get a new prediction error as follows
\begin{align}
(\tilde{x}_{t + 1} - x)^2 &= (1 - 2 r w_t)^2 (\tilde{x}_t - x)^2, \\
(\tilde{y}_{t + 1} - y)^2 &= (1 - 2 r v_t)^2 (\tilde{y}_t - y)^2.
\end{align}
At this point, since $(1 - 2 r w_t)^2 > (1 - 2 r v_t)^2$, we have
\begin{align}
& (\tilde{y}_{t + 1} - y)^2 - (\tilde{x}_{t + 1} - x)^2 \nonumber \\
& = (1 - 2 r v_t)^2 (\tilde{y}_t - y)^2 - (1 - 2 r w_t)^2 (\tilde{x}_t - x)^2 \nonumber \\
& < (1 - 2 r w_t)^2 \left((\tilde{y}_t - y)^2 - (\tilde{x}_t - x)^2 \right).
\end{align}
Since $(1 - 2 r w_t)^2 < 1$, after a gradient descent, the difference in prediction error between the two classes becomes smaller, that is, these losses become more balanced. The weight $w_t$ and $v_t$ are continuously adjusted until $(\tilde{x}_t - x)^2 = (\tilde{y}_t - y)^2$, at which point we have $w_t = 1$,$v_t = 1$ and $(\tilde{x}_{t + 1} - x)^2 = (\tilde{y}_{t + 1} - y)^2$.
Finally, it is worth noting that after the $t$-th iteration, we have $(\tilde{x}_{t + 1} - x)^2 \le (\tilde{x}_t - x)^2$ and $(\tilde{y}_{t + 1} - y)^2 \le (\tilde{y}_t - y)^2$. Namely, the prediction error for each class decreases as the iteration progresses. Our ReMSE algorithm will adjust the weights so that they end up being balanced. Ideally, when our loss is minimized, the average prediction error for each class is roughly equal.
\end{proof}
It is worth noting that, considering the separability of the loss function along the class dimension and the semantic dimension, if the optimization task of the loss function $\mathcal{L}_{ReMSE}$ is regarded as a $|\mathcal{C}^u| \times d_{s}$ independent 1-dimensional regression problem, then the above conclusion can be easily extended to multi-class and multi-semantic situations.
\subsection{AttentionNet}
\begin{figure}[ht]
\centering
\includegraphics[width=0.95\linewidth]{figures_Innovation.pdf}
\caption{Comparison between previous attention-based ZSL methods and our AttentionNet. The symbols I, F, A and S denote the images, feature maps, attentive features and predicted semantics, respectively. (a) GEMZSL~\cite{liu2021goal} \& APN~\cite{xu2020attribute} use only attention branch to train their backbone, and do not use attention in testing stage. (b) SGAM~\cite{zhu2019semantic} uses attention branch and global branch to predict semantics either locally or globally. (c) In contrast, our AttentionNet utilizes the attention branch in both training and testing stages to produce semantic-specific attentional visual features. Furthermore, we add a semantic-specific feature fusion to avoid the degradation of attentional features.}
\label{fig:innovation}
\end{figure}
\begin{figure*}[ht]
\centering
\includegraphics[width=0.95\linewidth]{figures_AttentionNet.pdf}
\caption{Architecture of our AttentionNet. It contains two innovations: (1) a cross-modal semantic-specific attention module and (2) the semantic-specific fusion. The cross-modal semantic-specific attention module allows the model to pay attention to different local regions according to different semantic attributes to fully exploit both visual and textual semantics. During training and testing, our model can better train end-to-end global and local image features.}
\label{fig:attentionNet}
\end{figure*}
To further improve the accuracy of semantic prediction, we try to utilize additionally the attention mechanism to extract semantic-specific visual features.
However, we observe that many existing attention-based ZSL methods do not fully exploit the capacity of the attention mechanism.
Thus, we design a novel ZSL embedding model called AttentionNet.
The comparison between existing attention-based ZSL and our AttentionNet is illustrated in Fig.~\ref{fig:innovation}.
Several remarks are highlighted as follows.
(1) The traditional methods APN~\cite{xu2020attribute} and GEMZSL~\cite{liu2021goal} only use the attention branch to train the backbone, while in testing stage they abandon it.
Obviously, the attention branch cannot help the model to localize discriminative region at testing stage.
(2) The method SGMA~\cite{zhu2019semantic} utilizes a global branch and an attention branch to extract global and local features and predict semantics individually.
However, a vast amount of research demonstrates that features fusing local and global information is more powerful~\cite{guo2020augfpn,gao2019res2net}.
In addition, the predefined ratio also increases the cost of hyper-
parameter search.
(3) In contrast, our AttentionNet utilizes the attention branch in both training and testing stages so as to fully exploit the capacity of attention.
Besides, our attention branch could generate semantic-specific attentive visual features for different semantics. Moreover, to avoid the degradation of attentional features, we add a feature fusion operator between the attentive features and global features, which shows high effectiveness as empirically demonstrated in our Experiment.
The structure details of our AttentionNet is shown in Fig.~\ref{fig:attentionNet}.
Specifically, AttentionNet is divided into two branches: the upper branch will synthesize semantic features for image attention, while the lower branch will generate visual (image) global features. In the upper branch, we first adopt the word representation model Glove~\cite{pennington2014glove} to obtain word vectors for all $d_{s}$ semantics ($d_{w2v}$ represents its dimension). For visual feature extraction, we adopt ResNet101~\cite{he2016deep} as our backbone. It will extract $W\times H$ regional visual features using $d_{v}$ channels, which is then followed by our cross-modal attention module. We use three fully connected (FC) layers including query layer, key layer, and value layer to project semantic word2vec and visual features into the latent space where visual and semantic will be aligned. Matrix multiplication between $Q$ and $K$ represents the strength of attention (normalized by softmax). Once the attention map is obtained, the attention network will drive the semantic items to automatically focus on specific regions. Next, our network augments the vector $d_v$ to obtain $d_s \times d_v$ features, which are then merged with the upper branch as the input of the semantic prediction network.
Our cross-modal semantic-specific attention module between semantics and images allows our model to implicitly establish relationships between semantic attributes and image features, which could take full advantages of visual and textual modalities.
With the help of the attention mechanism, our model is able to obtain better local image features, which are related to the predicted semantics, and fused with global features in the testing phase to obtain better semantic predictions.
\section{Experiments}
\begin{table*}[htbp]
\centering
\caption{Statistics of datasets.}
\label{table:dataset}
\begin{tabular}{c|c|c|c|c|c|c|c}
\hline
Dataset&\tabincell{c}{Semantic\\dimension} &\tabincell{c}{Semantic\\range}&\tabincell{c}{$\#$ Seen\\classes}&\tabincell{c}{$\#$ Unseen\\classes}&\tabincell{c}{$\#$ Images\\(total)}&\tabincell{c}{$\#$ Images\\(train+val)}&\tabincell{c}{$\#$ Images\\(test unseen/seen)}\\
\hline
AWA2~\cite{lampert2013attribute}&85 & $[0,100] \cup \{-1\}$ &40&10&30475&19832&4958/5685\\
\hline
CUB~\cite{wah2011caltech}&312& $[0,100]$&150&50&11788&7057&2679/1764\\
\hline
SUN~\cite{patterson2012sun}&102& $[0,1]$ &645&72&14340&10320&1440/2580\\
\hline
\end{tabular}
\end{table*}
\begin{table*}[htb]
\centering
\caption{Overall comparison with SOTAs in the setting of ZSL and GZSL.
In ZSL, T1 represents the top-1 accuracy (\%) for unseen classes.
In GZSL, $U$, $S$ and $H$ represent the top-1 accuracy (\%) of unseen classes, seen classes, and their harmonic mean, respectively.
The symbol $^\star$ denotes the results of our implemented version.
The best and next best results among embedding methods are marked with \textcolor{red}{\textbf{red}} and \textcolor{blue}{\textbf{blue}}. }
\begin{tabular}{p{2.5cm}|p{1.0cm}|p{0.05cm}p{0.05cm}p{0.05cm}|p{0.05cm}p{0.05cm}p{0.05cm}p{0.05cm}p{0.05cm}p{0.05cm}p{0.05cm}p{0.05cm}p{0.05cm}}
\hline
& &\multicolumn{3}{|c}{Zero-shot Learning} & \multicolumn{9}{|c}{Generalized Zero-shot Learning}\\
\hline
& & \multicolumn{1}{|c}{AWA2} & \multicolumn{1}{|c}{CUB} & \multicolumn{1}{|c|}{SUN} & \multicolumn{3}{|c}{AWA2} & \multicolumn{3}{|c}{CUB} &\multicolumn{3}{|c}{SUN} \\
\hline
Approach & Refer & \multicolumn{1}{|c}{T1} & \multicolumn{1}{|c}{T1} & \multicolumn{1}{|c|}{T1} & \multicolumn{1}{|c}{U} & \multicolumn{1}{c}{S} & \multicolumn{1}{c|}{H} & \multicolumn{1}{|c}{U} & \multicolumn{1}{c}{S} & \multicolumn{1}{c|}{H} & \multicolumn{1}{|c}{U} & \multicolumn{1}{c}{S} & \multicolumn{1}{c}{H} \\
\hline
\multicolumn{14}{c}{Embedding approaches}\\
\hline
AGEN~\cite{xie2019attentive} & CVPR19 & \multicolumn{1}{|c|}{66.9} & \multicolumn{1}{|c|}{72.5} & \multicolumn{1}{|c|}{60.6} & \multicolumn{1}{|c}{54.7} & \multicolumn{1}{c}{79.1} & \multicolumn{1}{c|}{64.7} & \multicolumn{1}{|c}{63.2} & \multicolumn{1}{c}{69.0} & \multicolumn{1}{c|}{66.0} & \multicolumn{1}{|c}{40.3} & \multicolumn{1}{c}{32.3} & \multicolumn{1}{c}{35.9}\\
DUET~\cite{jia2019deep} & TIP19 & \multicolumn{1}{|c|}{\textcolor{red}{\textbf{72.6}}} & \multicolumn{1}{|c|}{72.4} & \multicolumn{1}{|c|}{-} & \multicolumn{1}{|c}{48.2} & \multicolumn{1}{c}{\textcolor{red}{\textbf{90.2}}} & \multicolumn{1}{c|}{63.4} & \multicolumn{1}{|c}{39.7} & \multicolumn{1}{c}{80.1} & \multicolumn{1}{c|}{53.1} & \multicolumn{1}{|c}{-} & \multicolumn{1}{c}{-} & \multicolumn{1}{c}{-} \\
SGAM~\cite{zhu2019semantic}& NeurIPS19 & \multicolumn{1}{|c|}{68.8} & \multicolumn{1}{|c|}{71.0} & \multicolumn{1}{|c|}{-} & \multicolumn{1}{|c}{37.6} & \multicolumn{1}{c}{\textcolor{blue}{\textbf{87.1}}} & \multicolumn{1}{c|}{52.5} & \multicolumn{1}{|c}{36.7} & \multicolumn{1}{c}{71.3} & \multicolumn{1}{c|}{48.5} & \multicolumn{1}{|c}{-} & \multicolumn{1}{c}{-} & \multicolumn{1}{c}{-} \\
APN & NeurIPS20 & \multicolumn{1}{|c|}{68.4} & \multicolumn{1}{|c|}{72.0} & \multicolumn{1}{|c|}{61.6} & \multicolumn{1}{|c}{56.5} & \multicolumn{1}{c}{78.0} & \multicolumn{1}{c|}{65.5} & \multicolumn{1}{|c}{65.3} & \multicolumn{1}{c}{69.3} & \multicolumn{1}{c|}{67.2} & \multicolumn{1}{|c}{41.9} & \multicolumn{1}{c}{34.0} & \multicolumn{1}{c}{37.6}\\
GEMZSL & CVPR21 & \multicolumn{1}{|c|}{67.3} & \multicolumn{1}{|c|}{77.8} & \multicolumn{1}{|c|}{62.8} & \multicolumn{1}{|c}{64.8} & \multicolumn{1}{c}{77.5} & \multicolumn{1}{c|}{70.6} & \multicolumn{1}{|c}{64.8} & \multicolumn{1}{c}{\textcolor{red}{\textbf{77.1}}} & \multicolumn{1}{c|}{70.4} & \multicolumn{1}{|c}{38.1} & \multicolumn{1}{c}{\textcolor{blue}{\textbf{35.7}}} & \multicolumn{1}{c}{36.9} \\
LSG~\cite{xu2021semi}& TIP21 & \multicolumn{1}{|c|}{61.1} & \multicolumn{1}{|c|}{52.9} & \multicolumn{1}{|c|}{53.4} & \multicolumn{1}{|c}{60.4} & \multicolumn{1}{c}{84.9} & \multicolumn{1}{c|}{70.6} & \multicolumn{1}{|c}{49.6} & \multicolumn{1}{c}{50.4} & \multicolumn{1}{c|}{50.0} & \multicolumn{1}{|c}{\textcolor{red}{\textbf{52.8}}} & \multicolumn{1}{c}{23.1} & \multicolumn{1}{c}{32.2} \\
TransZero~\cite{chen2022transzero} & AAAI22 & \multicolumn{1}{|c|}{70.1} & \multicolumn{1}{|c|}{76.8} & \multicolumn{1}{|c|}{\textcolor{red}{\textbf{65.6}}} & \multicolumn{1}{|c}{61.3} & \multicolumn{1}{c}{82.3} & \multicolumn{1}{c|}{70.2} & \multicolumn{1}{|c}{69.3} & \multicolumn{1}{c}{68.3} & \multicolumn{1}{c|}{68.8} & \multicolumn{1}{|c}{\textcolor{blue}{\textbf{52.6}}} & \multicolumn{1}{c}{33.4} & \multicolumn{1}{c}{\textcolor{red}{\textbf{40.8}}} \\
\hline
APN$^\star$~\cite{liu2021goal}~\cite{xu2020attribute}& NeurIPS20 & \multicolumn{1}{|c|}{68.2} & \multicolumn{1}{|c|}{71.9} & \multicolumn{1}{|c|}{61.0} & \multicolumn{1}{|c}{59.8} & \multicolumn{1}{c}{75.1} & \multicolumn{1}{c|}{66.6} & \multicolumn{1}{|c}{64.4} & \multicolumn{1}{c}{67.8} & \multicolumn{1}{c|}{66.0} & \multicolumn{1}{|c}{41.1} & \multicolumn{1}{c}{34.0} & \multicolumn{1}{c}{37.2} \\
~~~~+Balanced MSE& CVPR22 &\multicolumn{1}{|c|}{68.1} & \multicolumn{1}{|c|}{68.5} & \multicolumn{1}{|c|}{60.6} & \multicolumn{1}{|c}{58.3} & \multicolumn{1}{c}{78.9} & \multicolumn{1}{c|}{67.1} & \multicolumn{1}{|c}{57.0} & \multicolumn{1}{c}{65.5} & \multicolumn{1}{c|}{60.9} & \multicolumn{1}{|c}{41.3} & \multicolumn{1}{c}{34.3} & \multicolumn{1}{c}{37.4} \\
~~~~\textbf{+ReMSE}& \textbf{Ours} &\multicolumn{1}{|c|}{68.3} & \multicolumn{1}{|c|}{72.1} & \multicolumn{1}{|c|}{61.5} & \multicolumn{1}{|c}{63.2} & \multicolumn{1}{c}{74.9} & \multicolumn{1}{c|}{68.5} & \multicolumn{1}{|c}{67.8} & \multicolumn{1}{c}{64.7} & \multicolumn{1}{c|}{66.2} & \multicolumn{1}{|c}{42.9} & \multicolumn{1}{c}{33.7} & \multicolumn{1}{c}{37.7} \\
\hline
GEMZSL$^\star$~\cite{liu2021goal} & CVPR21 &\multicolumn{1}{|c|}{65.7} & \multicolumn{1}{|c|}{75.8} & \multicolumn{1}{|c|}{62.2} & \multicolumn{1}{|c}{\textcolor{blue}{\textbf{62.0}}} & \multicolumn{1}{c}{79.9} & \multicolumn{1}{c|}{69.8} & \multicolumn{1}{|c}{69.9} & \multicolumn{1}{c}{73.2} & \multicolumn{1}{c|}{71.5} & \multicolumn{1}{|c}{37.3} & \multicolumn{1}{c}{\textcolor{red}{\textbf{37.9}}} & \multicolumn{1}{c}{37.6} \\
~~~~+Balanced MSE& CVPR22 &\multicolumn{1}{|c|}{65.3} & \multicolumn{1}{|c|}{75.3} & \multicolumn{1}{|c|}{61.7} & \multicolumn{1}{|c}{60.7} & \multicolumn{1}{c}{81.2} & \multicolumn{1}{c|}{69.5} & \multicolumn{1}{|c}{67.1} & \multicolumn{1}{c}{\textcolor{blue}{\textbf{75.5}}} & \multicolumn{1}{c|}{71.1} & \multicolumn{1}{|c}{46.3} & \multicolumn{1}{c}{30.9} & \multicolumn{1}{c}{37.1} \\
~~~~\textbf{+ReMSE} & \textbf{Ours} & \multicolumn{1}{|c|}{66.1} & \multicolumn{1}{|c|}{76.6} & \multicolumn{1}{|c|}{63.1} & \multicolumn{1}{|c}{61.4} & \multicolumn{1}{c}{81.9} & \multicolumn{1}{c|}{70.2} & \multicolumn{1}{|c}{69.0} & \multicolumn{1}{c}{75.2} & \multicolumn{1}{c|}{72.0} & \multicolumn{1}{|c}{48.8} & \multicolumn{1}{c}{33.6} & \multicolumn{1}{c}{39.8} \\
\hline
\textbf{AttentionNet} & \textbf{Ours} & \multicolumn{1}{|c|}{69.3} & \multicolumn{1}{|c|}{\textcolor{blue}{\textbf{80.2}}} & \multicolumn{1}{|c|}{62.8} & \multicolumn{1}{|c}{\textcolor{red}{\textbf{63.8}}} & \multicolumn{1}{c}{84.6} & \multicolumn{1}{c|}{\textcolor{blue}{\textbf{72.8}}} & \multicolumn{1}{c}{\textcolor{blue}{\textbf{71.9}}} & \multicolumn{1}{c}{74.6} & \multicolumn{1}{c|}{\textcolor{blue}{\textbf{72.9}}} & \multicolumn{1}{|c}{47.1} & \multicolumn{1}{c}{32.8} & \multicolumn{1}{c}{38.6} \\
~~~~+Balanced MSE& CVPR22 & \multicolumn{1}{|c|}{66.4} & \multicolumn{1}{|c|}{79.6} & \multicolumn{1}{|c|}{62.4} & \multicolumn{1}{|c}{59.9} & \multicolumn{1}{c}{83.4} & \multicolumn{1}{c|}{69.7} & \multicolumn{1}{c}{70.7} & \multicolumn{1}{c}{74.9} & \multicolumn{1}{c|}{72.7} & \multicolumn{1}{|c}{47.9} & \multicolumn{1}{c}{33.7} & \multicolumn{1}{c}{39.6} \\
~~~~\textbf{+ReMSE}& \textbf{Ours} & \multicolumn{1}{|c|}{\textcolor{blue}{\textbf{70.9}}} & \multicolumn{1}{|c|}{\textcolor{red}{\textbf{80.9}}} & \multicolumn{1}{|c|}{\textcolor{blue}{\textbf{63.2}}} & \multicolumn{1}{|c}{\textcolor{red}{\textbf{63.8}}} & \multicolumn{1}{c}{85.6} & \multicolumn{1}{c|}{\textcolor{red}{\textbf{73.1}}} & \multicolumn{1}{c}{\textcolor{red}{\textbf{72.8}}} & \multicolumn{1}{c}{74.8} & \multicolumn{1}{c|}{\textcolor{red}{\textbf{73.8}}} & \multicolumn{1}{|c}{47.4} & \multicolumn{1}{c}{34.8} & \multicolumn{1}{c}{\textcolor{blue}{\textbf{40.1}}} \\
\hline
\rowcolor{gray!14}
\multicolumn{14}{c}{Generative approaches}\\
\hline
\rowcolor{gray!}
fCLSWGAN~\cite{xian2018feature}& CVPR18 & \multicolumn{1}{|c|}{-} & \multicolumn{1}{|c|}{57.3} & \multicolumn{1}{|c|}{60.8} & \multicolumn{1}{|c}{56.1} & \multicolumn{1}{c}{65.5} & \multicolumn{1}{c|}{60.4} & \multicolumn{1}{|c}{43.7} & \multicolumn{1}{c}{57.7} & \multicolumn{1}{c|}{49.7} & \multicolumn{1}{|c}{42.6} & \multicolumn{1}{c}{36.6} & \multicolumn{1}{c}{39.4}\\
\rowcolor{gray!10}
DCRGAN~\cite{ye2021disentangling} & TMM21 & \multicolumn{1}{|c|}{-} & \multicolumn{1}{|c|}{61.0} & \multicolumn{1}{|c|}{63.7} & \multicolumn{1}{|c}{-} & \multicolumn{1}{c}{-} & \multicolumn{1}{c|}{-} & \multicolumn{1}{|c}{55.8} & \multicolumn{1}{c}{66.8} & \multicolumn{1}{c|}{60.8} & \multicolumn{1}{|c}{47.1} & \multicolumn{1}{c}{38.5} & \multicolumn{1}{c}{42.4} \\
\rowcolor{gray!10}
HSVA~\cite{chen2021hsva} & NeurIPS21 & \multicolumn{1}{|c|}{-} & \multicolumn{1}{|c|}{62.8} & \multicolumn{1}{|c|}{63.8} & \multicolumn{1}{|c}{56.7} & \multicolumn{1}{c}{79.8} & \multicolumn{1}{c|}{66.3} & \multicolumn{1}{|c}{52.7} & \multicolumn{1}{c}{58.3} & \multicolumn{1}{c|}{55.3} & \multicolumn{1}{|c}{48.6} & \multicolumn{1}{c}{39.0} & \multicolumn{1}{c}{43.3} \\
\rowcolor{gray!10}
CE-GZSL~\cite{han2021contrastive} & CVPR21& \multicolumn{1}{|c|}{70.4} & \multicolumn{1}{|c|}{77.5} & \multicolumn{1}{|c|}{63.3} & \multicolumn{1}{|c}{63.1} & \multicolumn{1}{c}{78.6} & \multicolumn{1}{c|}{70.0} & \multicolumn{1}{|c}{63.9} & \multicolumn{1}{c}{66.8} & \multicolumn{1}{c|}{65.3} & \multicolumn{1}{|c}{48.8} & \multicolumn{1}{c}{38.6} & \multicolumn{1}{c}{43.1} \\
\hline
\end{tabular}
\label{table:OverallComparison}
\end{table*}
\begin{table}[htbp]
\centering
\caption{AUSUC for GZSL. A larger value means a better trade-off between seen and unseen accuracy.}
\begin{tabular}{p{0.1cm}|p{0.1cm}|p{0.1cm}|p{0.1cm}|}
\hline
& \multicolumn{1}{c|}{AWA2} & \multicolumn{1}{|c|}{CUB} & \multicolumn{1}{|c}{SUN}\\
\hline
\multicolumn{1}{c|}{APN} & \multicolumn{1}{|c|}{0.5784} & \multicolumn{1}{|c|}{0.5545} & \multicolumn{1}{|c}{0.2056}\\
\multicolumn{1}{c|}{+Balanced MSE} & \multicolumn{1}{|c|}{0.5825} & \multicolumn{1}{|c|}{0.4973} & \multicolumn{1}{|c}{0.2064}\\
\multicolumn{1}{c|}{\textbf{+ReMSE (Ours)}} & \multicolumn{1}{|c|}{\textbf{0.5840}} & \multicolumn{1}{|c|}{\textbf{0.5552}} & \multicolumn{1}{|c}{\textbf{0.2114}}\\
\hline
\multicolumn{1}{c|}{GEMZSL} & \multicolumn{1}{|c|}{0.5823} & \multicolumn{1}{|c|}{0.6178} & \multicolumn{1}{|c}{0.2211}\\
\multicolumn{1}{c|}{+Balanced MSE} & \multicolumn{1}{|c|}{0.6017} & \multicolumn{1}{|c|}{0.6137} & \multicolumn{1}{|c}{0.1991}\\
\multicolumn{1}{c|}{\textbf{+ReMSE (Ours)}} & \multicolumn{1}{|c|}{\textbf{0.6067}} & \multicolumn{1}{|c|}{\textbf{0.6230}} & \multicolumn{1}{|c}{\textbf{0.2275}}\\
\hline
\multicolumn{1}{c|}{{AttentionNet}} & \multicolumn{1}{|c|}{0.6275} & \multicolumn{1}{|c|}{0.6390} & \multicolumn{1}{|c}{0.2145}\\
\multicolumn{1}{c|}{+Balanced MSE} & \multicolumn{1}{|c|}{0.6007} & \multicolumn{1}{|c|}{0.6373} & \multicolumn{1}{|c}{0.2192}\\
\multicolumn{1}{c|}{\textbf{+ReMSE (Ours)}} & \multicolumn{1}{|c|}{\textbf{0.6476}} & \multicolumn{1}{|c|}{\textbf{0.6516}} & \multicolumn{1}{|c}{\textbf{0.2338}}\\
\hline
\end{tabular}
\label{table:AUSUC}
\end{table}
\begin{table*}[htbp]
\centering
\caption{Results of ablation analysis in the setting of ZSL and GZSL.
AB and GB denote the attention branch and global branch, respectively.
In ZSL, T1 represents the top-1 accuracy (\%) for unseen classes.
In GZSL, $U$, $S$ and $H$ represent the top-1 accuracy (\%) of unseen classes, seen classes, and their harmonic mean, respectively. }
\begin{tabular}{p{2.5cm}|p{1.7cm}|p{0.04cm}p{0.04cm}p{0.04cm}|p{0.04cm}p{0.04cm}p{0.04cm}p{0.04cm}p{0.04cm}p{0.04cm}p{0.04cm}p{0.04cm}p{0.04cm}}
\hline
& &\multicolumn{3}{|c}{Zero-shot Learning} & \multicolumn{9}{|c}{Generalized Zero-shot Learning}\\
\hline
& & \multicolumn{1}{|c}{AWA2} & \multicolumn{1}{|c}{CUB} & \multicolumn{1}{|c|}{SUN} & \multicolumn{3}{|c}{AWA2} & \multicolumn{3}{|c}{CUB} &\multicolumn{3}{|c}{SUN} \\
\hline
Structure & Loss & \multicolumn{1}{|c}{T1} & \multicolumn{1}{|c}{T1} & \multicolumn{1}{|c|}{T1} & \multicolumn{1}{|c}{U} & \multicolumn{1}{c}{S} & \multicolumn{1}{c|}{H} & \multicolumn{1}{|c}{U} & \multicolumn{1}{c}{S} & \multicolumn{1}{c|}{H} & \multicolumn{1}{|c}{U} & \multicolumn{1}{c}{S} & \multicolumn{1}{c}{H} \\
\hline
AttentionNet w/o AB & $\mathcal{L}_{SCE}$ & \multicolumn{1}{|c|}{67.0} & \multicolumn{1}{|c|}{74.8} & \multicolumn{1}{|c|}{62.7} & \multicolumn{1}{|c}{62.1} & \multicolumn{1}{c}{83.1} & \multicolumn{1}{c|}{71.1} & \multicolumn{1}{|c}{66.9} & \multicolumn{1}{c}{72.0} & \multicolumn{1}{c|}{69.3} & \multicolumn{1}{|c}{47.5} & \multicolumn{1}{c}{31.2} & \multicolumn{1}{c}{37.6}\\
AttentionNet w/o GB & $\mathcal{L}_{SCE}$ & \multicolumn{1}{|c|}{64.2} & \multicolumn{1}{|c|}{79.7} & \multicolumn{1}{|c|}{60.9} & \multicolumn{1}{|c}{58.3} & \multicolumn{1}{c}{68.5} & \multicolumn{1}{c|}{68.4} & \multicolumn{1}{|c}{71.5} & \multicolumn{1}{c}{73.9} & \multicolumn{1}{c|}{72.7} & \multicolumn{1}{|c}{46.7} & \multicolumn{1}{c}{30.6} & \multicolumn{1}{c}{37.0}\\
AttentionNet & $\mathcal{L}_{SCE}$ & \multicolumn{1}{|c|}{69.3} & \multicolumn{1}{|c|}{80.2} & \multicolumn{1}{|c|}{62.8} & \multicolumn{1}{|c}{63.8} & \multicolumn{1}{c}{84.6} & \multicolumn{1}{c|}{72.8} & \multicolumn{1}{|c}{71.2} & \multicolumn{1}{c}{74.6} & \multicolumn{1}{c|}{72.9} & \multicolumn{1}{|c}{47.1} & \multicolumn{1}{c}{32.8} & \multicolumn{1}{c}{38.6}\\
AttentionNet & $\mathcal{L}_{SCE}$+$\mathcal{L}_{ReMSE}$ & \multicolumn{1}{|c|}{70.9} & \multicolumn{1}{|c|}{80.9} & \multicolumn{1}{|c|}{63.2} & \multicolumn{1}{|c}{63.8} & \multicolumn{1}{c}{85.6} & \multicolumn{1}{c|}{73.1} & \multicolumn{1}{|c}{72.8} & \multicolumn{1}{c}{74.8} & \multicolumn{1}{c|}{73.8} & \multicolumn{1}{|c}{47.4} & \multicolumn{1}{c}{34.8} & \multicolumn{1}{c}{40.1}\\
\hline
\end{tabular}
\label{table:Ablation}
\end{table*}
\begin{figure*}[htbp]
\centering
\includegraphics[width=0.95\linewidth]{figures_AUSUC.pdf}
\caption{Visualization of Area Under Unseen-Seen Accuracy (AUSUC). Our ReMSE's AUSUC is mostly higher than the baseline model (GEMZSL).}
\label{fig:AUSUC}
\end{figure*}
\begin{figure*}[htbp]
\centering
\includegraphics[width=0.95\linewidth]{figures_AverageErrorMatrix.pdf}
\caption{Class-averaged semantic error matrix on the test set of the datasets CUB and AWA2 (rescaled by logarithmic function).}
\label{fig:OverallErrorDistribution}
\end{figure*}
To demonstrate the effectiveness of our ReMSE, we implement various SOTA ZSL methods and evaluate our ReMSE on multiple metrics both on ZSL and GZSL settings over three popular benchmark datasets. With extensive studies, we show our ReMSE could improve various SOTA models by a significant gap as seen in Sec.~\ref{sec:SOTA}.
We also present the imbalanced semantic regression performance in Sec.~\ref{sec:Regression}.
Finally, we demonstrate the effectiveness of intra-class re-weighting and intra-semantic re-weighting on Sec.~\ref{sec:Ablation}.
\textbf{Datasets.}
We conduct extensive experiments to evaluate the proposed method on three ZSL benchmarks, namely (1) the coarse-grained dataset AWA2~\cite{lampert2013attribute}, one extensive animal dataset composed of 37,322 images from 50 classes (40 seen and 10 unseen) with 85-dim attributes ranged from $0$ to $100$ ($-1$ denotes missing data); (2) the fine-grained bird dataset CUB~\cite{wah2011caltech}, containing 11,788 images in 200 (150 seen and 50 unseen) classes with 312 semantics ranging from 0 to 100; (3) the fine-grained dataset SUN~\cite{patterson2012sun}, a large-scale dataset including 14,340 images from 717 classes (645 seen and 72 unseen) with 102 attributes ranging from 0 to 1.
We divide these data into training and testing sets following~\cite{xian2018zero}, which is widely used in present methods.
\textbf{Baselines \& Implementation Details.}
We examine our ReMSE strategy on three ZSL baselines, i.e., APN, GEMZSL\footnote{It is worth mentioning that the model GEMZSL only utilizes the gaze embedding to build the attention maps, its ability to recognize unseen classes only relies on the semantics provided by the benchmark~\cite{xian2018zero}.}, and AttentionNet.
For fair comparison, we implement APN and GEMZSL following the original training configuration, including their batch size, learning rate, sampling strategy and so on.
For AttentionNet, we only use SCE and our proposed ReMSE loss.
We adopt ResNet101~\cite{he2016deep} pretrained on ImageNet1K~\cite{deng2009imagenet} as the backbone. AttentionNet is optimized with a stochastic gradient descent optimizer with a learning rate of 0.0005, momentum of 0.9, and weight decay of 0.0001.
The batch size for all datasets is set to 32.
All the experiments were run on an NVIDIA Quadro RTX 8000 graphics card with 48GB of memory.
\textbf{Evaluation Protocols.}
We adopt a variety of metrics for comparison.
Specifically, for ZSL, we calculate the top-1 classification accuracy (T1) for unseen classes.
For GZSL, we calculate three kinds of top-1 accuracies, namely the accuracy for unseen classes (denoted as $U$), the accuracy for seen classes ($S$), and their harmonic mean:
\begin{equation}
H = \frac{2 \times U \times S}{U + S}.
\end{equation}
Besides, for GZSL, we report the performance based on the Area Under Seen-Unseen accuracy Curve (AUSUC)~\cite{chao2016empirical}, which evaluates the degree of trade-off between $U$ and $S$ for ZSL.
Finally, we exploit two new metrics, the mean and standard deviation of the class-averaged semantic error matrix, to evaluate the imbalanced performance on semantic regression, i.e. how well the ZSL models can fit the semantic labels and how well the error distribution is balanced, respectively.
\subsection{Comparison with SOTAs}
\label{sec:SOTA}
For ZSL and GZSL tasks, we focus on embedding methods and compare our approach with classical AGEN (CVPR19), DUET (TIP19), SGAM (NeurIPS19), and more recent APN (NeurIPS20), GEMZSL (CVPR21), LSG (TIP21) and even state-of-the-art TransZero (AAAI22).
We also report the performance of various generative methods, including f-CLSWGAN (CVPR18), DCRGAN (TMM21), HSVA (NeurIPS21), CE-GZSL (CVPR21) for comprehensive reference.
For imbalanced regression, we apply a Balanced MSE (CVPR22) on three embedding methods (i.e. APN, GEMZSL, and AttentionNet), which may be the first method in multi-label imbalanced regression.
The results are reported in Table~\ref{table:OverallComparison}. We highlight three main observations: 1) ReMSE can improve the baselines consistently. For example, on the CUB dataset for ZSL task, our ReMSE endows vanilla APN, GEMZSL and AttentionNet with 0.2\%, 0.8\%, and 0.7\% performance gain, respectively, confirming that ReMSE can effectively learn a better visual-semantic mapping. 2) On the CUB dataset, ReMSE achieves the highest score compared with AttentionNet by a considerable gap, i.e. at least 4.1\% higher than all the rest SOTAs for ZSL (Ours $80.9\%$ v.s. TransZero $76.8\%$), and at least $2.3\%$ (w.r.t. H) higher for GZSL (Ours $73.8\%$ v.s. GEMZSL $71.5\%$). 3) Balanced MSE may perform unstable. In some case, it may bring improvements (e.g. when it is integrated into APN for GZSL on SUN and AWA2), but in other cases, it may degrade the performance. Likewise, comparisons on the AUSUC metric (as seen in Table~\ref{table:AUSUC}) also validate that our ReMSE improves all the models with a large gap, again demonstrating the advantage of the rebalancing strategy.
A visualization of the Area Under Unseen-Seen Accuracy (AUSUC) is shown in Fig.~\ref{fig:AUSUC}. We can see that our ReMSE's AUSUC is mostly higher than the baseline model (GEMZSL), which evaluates the trade-off ability of ZSL between unseen accuracy and seen accuracy.
\begin{figure*}[htbp]
\centering
\includegraphics[width=0.95\linewidth]{figures_ErrorCurve.pdf}
\caption{Mean and standard deviation of error distributions on test set. ReMSE could lead to significant drops in terms of both Mean and standard deviation of error distributions.}
\label{fig:ErrorCurve}
\end{figure*}
\begin{figure*}[htbp]
\centering
\includegraphics[width=0.95\linewidth]{figures_PCCCurve}
\caption{Visualization of the Pearson Correlation Coefficient (PCC) between semantic prediction errors and semantic label values during training. Compared with the baseline model, ReMSE makes the Pearson Correlation Coefficient drop significantly. This shows that our ReMSE method can greatly reduce the correlation between semantic prediction errors and semantic label values.}
\label{fig:PCCCurve}
\end{figure*}
\subsection{Validation of Rebalancing Property}
\label{sec:Regression}
To validate that our approach can indeed rebalance errors across different classes and different semantics, we conduct several additional experiments.
First, we visualize the variations of the error distribution on the testing set of CUB in Fig.~\ref{fig:OverallErrorDistribution}.
Darker red means more errors, while darker blue means fewer errors. We can see that with ReMSE, the distribution of prediction errors changes from darker red to blue overall. This clearly shows that our re-weighting could effectively suppress prediction errors without negatively affecting other well-fitting semantic regions.
Second, conducting experiments on GEMZSL, we perform two quantitative comparisons of the mean and standard deviation of the errors distributions, as shown in Fig.~\ref{fig:ErrorCurve}.
It is evident that once ReMSE is applied, both seen or unseen classes, the means and standard deviations drop significantly, implying that the errors are indeed balanced.
We also verify that the semantic predictions of most existing models are unbalanced. Furthermore, imbalanced prediction errors are often associated with semantic labels. To demonstrate that our ReMSE can reduce undesired correlations, we visualize the Pearson Correlation coefficient (PCC) between semantic prediction errors and semantic label values, as shown in Fig.~\ref{fig:PCCCurve}. We can observe that our ReMSE leads that the PCC drops significantly compared to the baseline model, which indicates that our ReMSE can indeed greatly reduce the linear relationship between semantic prediction error and semantic label value, thereby alleviating the imbalanced semantic prediction issue.
\begin{figure*}[htbp]
\centering
\includegraphics[width=0.95\linewidth]{figures_ablation.pdf}
\caption{Effects of re-weighting hyper-parameters $\alpha$ (class-level) and $\beta$ (semantic-level).}
\label{fig:ablation}
\end{figure*}
\subsection{Ablation Study}
\label{sec:Ablation}
\subsubsection{Component analysis}
\label{sec:ComAna}
We conduct ablation studies to verify the effectiveness of our approach.
Table~\ref{table:Ablation} shows the impact of each component.
We first use SCE loss to train a model that only contains global branch (GB) or attention branch (AB).
Next, we fuse these two branches as the full AttentionNet.
After that, our ReMSE are added to our AttentionNet.
From the table, we could get three conclusions: (1) The attention branch might cause some degradation. For example, the model without AB could perform better than the model without GB on AWA2 and SUN. (2) The first three column indicates combining global and attentive features could improve the expressiveness of features, and allow models predict semantics more accurate.
(3) Remarkly, Our proposed ReMSE improves the T1 of ZSL over the model trained by SCE by 1.6\% (AWA2), 0.7\% (CUB) and 0.4\% (SUN), respectively, and the harmonic mean accuracy (H) of GZSL by 0.3\% (AWA2), 0.9\% (CUB) and 1.5\% (SUN), respectively.
This influence verifies the effectiveness of our ReMSE that does not only decrease the mean of errors but also reduce the variance of errors.
\subsubsection{Sensitivity Analysis}
\label{sec:Sensitivity}
We take the SUN and CUB dataset to analyze the Sensitivity of the hyperparameters $\alpha$ and $\beta$ used in the rebalance method. As shown in Fig.~\ref{fig:ablation}, a proper $\alpha$ or $\beta$ can bring improvement of T1, which appears to be higher than the vanilla method.
\begin{figure*}[htbp]
\centering
\includegraphics[width=0.9\linewidth]{figures_attention_vis1.pdf}
\caption{Visualization of attention maps produced by our AttentionNet according for different semantics of unseen images on the dataset CUB. The attention map has a resolution of $7 \times 7$, and is reshaped into $224 \times 224$ to match the image size.}
\label{fig:attention_vis1}
\end{figure*}
\begin{figure*}[htbp]
\centering
\includegraphics[width=0.95\linewidth]{figures_attention_vis2.pdf}
\caption{
The effect of our ReMSE algorithm on the attention map. The settings are the same as in Fig.~\ref{fig:attention_vis1}, but the model results in a more accurate attention map for difficult semantics.}
\label{fig:attention_vis2}
\end{figure*}
\subsection{Visualization of Attention}
We also visualize the attention map of our AttentionNet to qualitatively verify its effectiveness as shown in Fig.~\ref{fig:attention_vis1}.
The figure shows the results of different attention maps according to various semantics.
Obviously, our AttentionNet adaptively detects the semantic regions that are beneficial for prediction.
For example, when the semantics are related to crown, eye and bill, the attention is distributed to heads of birds.
When the semantics are related to wing or upper-parts, the attentive regions become the bodies.
Moreover, we also verify the effectiveness of our ReMSE for the attention, as shown in Fig.~\ref{fig:attention_vis2}.
We can see that with the help of our ReMSE the model corrects its out-of-focus regions.
For instance, for the semantics related to bills, the model incorrectly focuses on (a) the legs, (b) the throat, and (c) the chest.
But with the help of ReMSE, it exactly focuses on bills.
Generally, these figures illustrate that our ReMSE plays a key role in predicting accurately hard semantics.
\section{Conclusion}
In this work, we address the zero-shot learning problem from a brand new perspective of imbalanced learning. We propose the ReMSE strategy and focus on re-balancing the imbalanced error distribution across different classes and different semantics. We set out a series of analyses both theoretically and empirically to validate the rationale of ReMSE in ZSL. Extensive experiments on three benchmark datasets show that ReMSE can consistently improve the three baselines, achieving competitive performance even compared to those sophisticated ZSL methods.
\section*{Acknowledgments}
This work was partially supported by ``Qing Lan Project" in Jiangsu universities, National Natural Science Foundation of China under nos. 61876155 and 62106081, and Jiangsu Science and Technology Programme under no. BE2020006-4.
\newpage
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 4,127
|
Q: Json decoding not decoding into complex object So I have a set of classes:
public class CallQueueRequest
{
public string ACK { get; set; }
public string ERROR { get; set; }
public Agent AGENT { get; set; }
public Skill SKILL { get; set; }
public string TIME { get; set; }
}
public class Agent
{
public string has_result { get; set; }
public List<Agents> agents { get; set; }
}
public class Agents
{
public string display_name { get; set; }
public string time_in_state { get; set; }
public string state { get; set; }
public string callstakentoday { get; set; }
public string avaya_skill_num { get; set; }
}
public class Skill
{
public string has_result { get; set; }
public string num_skills { get; set; }
public List<Skills> skills { get; set; }
}
public class Skills
{
public string display_name { get; set; }
public string avaya_skill_num { get; set; }
public string callsinqueue { get; set; }
public string callstoday { get; set; }
public string abantoday { get; set; }
public string lwt { get; set; }
public string ewt { get; set; }
public string servicelvl { get; set; }
public string avgspeedans { get; set; }
public string talktime { get; set; }
}
And I have this Json:
{
"ACK":"SUCCESS",
"ERROR":null,
"AGENT":{
"has_results":1,
"agents":[
{
"display_name":"John Doe",
"time_in_state":"378",
"state":"Acd",
"callstakentoday":null,
"avaya_skill_num":"81"
},
{
"display_name":"Jane Joe",
"time_in_state":"220",
"state":"Acd",
"callstakentoday":null,
"avaya_skill_num":"81"
}
]
},
"SKILL":{
"has_results":1,
"num_skills":1,
"skills":[
{
"display_name":"QueueName",
"avaya_skill_num":"81",
"callsinqueue":"1",
"callstoday":"29",
"abandtoday":"1",
"lwt":"74",
"ewt":"223",
"servicelvl":"86",
"avgspeedans":"35",
"talktime":"873"
}
]
},
"TIME":1355864270
}
I am using this code:
object qr = JsonConvert.DeserializeObject(jsonString);
This does not seem to be converting from Json to the complex class properly. Can someone assist me with this? I think its just a small mistake.
A: I was able to find some info on what I'm trying to do. If anyone finds this question, here is the answer
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 5,259
|
import testtools
from tempest.api.compute.floating_ips import base
from tempest import config
from tempest.lib.common.utils import data_utils
from tempest.lib import decorators
from tempest.lib import exceptions as lib_exc
CONF = config.CONF
class FloatingIPsNegativeTestJSON(base.BaseFloatingIPsTest):
max_microversion = '2.35'
@classmethod
def resource_setup(cls):
super(FloatingIPsNegativeTestJSON, cls).resource_setup()
# Generating a nonexistent floatingIP id
body = cls.client.list_floating_ips()['floating_ips']
floating_ip_ids = [floating_ip['id'] for floating_ip in body]
while True:
if CONF.service_available.neutron:
cls.non_exist_id = data_utils.rand_uuid()
else:
cls.non_exist_id = data_utils.rand_int_id(start=999)
if cls.non_exist_id not in floating_ip_ids:
break
@decorators.attr(type=['negative'])
@decorators.idempotent_id('6e0f059b-e4dd-48fb-8207-06e3bba5b074')
def test_allocate_floating_ip_from_nonexistent_pool(self):
# Negative test:Allocation of a new floating IP from a nonexistent_pool
# to a project should fail
self.assertRaises(lib_exc.NotFound,
self.client.create_floating_ip,
pool="non_exist_pool")
@decorators.attr(type=['negative'])
@decorators.idempotent_id('ae1c55a8-552b-44d4-bfb6-2a115a15d0ba')
def test_delete_nonexistent_floating_ip(self):
# Negative test:Deletion of a nonexistent floating IP
# from project should fail
# Deleting the non existent floating IP
self.assertRaises(lib_exc.NotFound, self.client.delete_floating_ip,
self.non_exist_id)
class FloatingIPsAssociationNegativeTestJSON(base.BaseFloatingIPsTest):
max_microversion = '2.43'
@classmethod
def resource_setup(cls):
super(FloatingIPsAssociationNegativeTestJSON, cls).resource_setup()
cls.server = cls.create_test_server(wait_until='ACTIVE')
cls.server_id = cls.server['id']
@decorators.attr(type=['negative'])
@decorators.idempotent_id('595fa616-1a71-4670-9614-46564ac49a4c')
def test_associate_nonexistent_floating_ip(self):
# Negative test:Association of a non existent floating IP
# to specific server should fail
# Associating non existent floating IP
self.assertRaises(lib_exc.NotFound,
self.client.associate_floating_ip_to_server,
"0.0.0.0", self.server_id)
@decorators.attr(type=['negative'])
@decorators.idempotent_id('0a081a66-e568-4e6b-aa62-9587a876dca8')
def test_dissociate_nonexistent_floating_ip(self):
# Negative test:Dissociation of a non existent floating IP should fail
# Dissociating non existent floating IP
self.assertRaises(lib_exc.NotFound,
self.client.disassociate_floating_ip_from_server,
"0.0.0.0", self.server_id)
@decorators.attr(type=['negative'])
@decorators.idempotent_id('804b4fcb-bbf5-412f-925d-896672b61eb3')
def test_associate_ip_to_server_without_passing_floating_ip(self):
# Negative test:Association of empty floating IP to specific server
# should raise NotFound or BadRequest(In case of Nova V2.1) exception.
self.assertRaises((lib_exc.NotFound, lib_exc.BadRequest),
self.client.associate_floating_ip_to_server,
'', self.server_id)
@decorators.attr(type=['negative'])
@decorators.idempotent_id('58a80596-ffb2-11e6-9393-fa163e4fa634')
@testtools.skipUnless(CONF.network.public_network_id,
'The public_network_id option must be specified.')
def test_associate_ip_to_server_with_floating_ip(self):
# The VM have one port
# Associate floating IP A to the VM
# Associate floating IP B which is from same pool with floating IP A
# to the VM, should raise BadRequest exception
body = self.client.create_floating_ip(
pool=CONF.network.public_network_id)['floating_ip']
self.addCleanup(self.client.delete_floating_ip, body['id'])
self.client.associate_floating_ip_to_server(body['ip'], self.server_id)
self.addCleanup(self.client.disassociate_floating_ip_from_server,
body['ip'], self.server_id)
body = self.client.create_floating_ip(
pool=CONF.network.public_network_id)['floating_ip']
self.addCleanup(self.client.delete_floating_ip, body['id'])
self.assertRaises(lib_exc.BadRequest,
self.client.associate_floating_ip_to_server,
body['ip'], self.server_id)
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 5,170
|
\section{Introduction}
Consider a smooth probability density defined on some open interval $I \subset{\mathbb{R}}$ and such that all its derivatives vanish at both ends of $I$.
Rolle's theorem shows that the $n-$th derivative vanishes at least $n$ times on $I.$ The density is said to be bell-shaped if the $n-$th derivative vanishes exactly $n$ times on $I$ for every $n\ge 1.$ For $n =1$ this amounts to strict unimodality. For $n=2$ this means, as for the familiar bell curve, that there is one inflection point on each side of the mode and that the second derivative is successively positive, negative, and positive. Rolle's theorem also entails that the zeroes of the successive derivatives of a bell-shaped density strictly interlace, and hence that each derivative has an alternating sign sequence, starting positive. The sign sequence of the third derivative can be observed on a curve in comparing the aberrancy and the slope, as shown by Transon's formula \cite{Sc}, but contrary to the second derivative it requires an educated eye to guess what this sequence must be at first glance. The fourth derivative of a density has also a kinematic interpretation in terms of penosculating conics - see \cite{Sc} and the references therein for a complete account, which however cannot be understood through the sole sequence of its signs.
For a given density whose derivatives vanish at both ends of its definition interval, the usual way to show the bell-shape is to factorize its $n$-th derivative by a positive function and a polynomial function of degree $n$. This entails that this $n$-th derivative vanishes at most, and hence exactly, $n$ times. Basic examples - whose details can be checked by the reader and serve as an exercise in calculus for sophomores - are the Gaussian, Gumbel and centered Student densities on ${\mathbb{R}}$, and the inverse Gamma densities on $(0, +\infty).$ When the density is not explicit, such factorizations are not appropriate anymore. Another criterion for the bell-shape is the ETP character of the associated additive convolution kernel - see Chapter 6.11.C in \cite{K} - but the latter is rather difficult to check for non-explicit densities, and also quite stringent since it is e.g. fulfilled neither by the Student nor by the inverse Gamma densities. In this paper we are interested in the positive $\alpha-$stable random variables $\Xa$ and their densities $\fa, \, 0<\alpha<1.$ The latter are characterized by their Laplace transform and we choose the normalisation
$$\EE[e^{-\lambda \Xa}]\; =\;\int_0^\infty e^{-\lambda x} \fa(x)\, dx\; =\; e^{-\lambda^\alpha}, \quad \lambda \ge 0.$$
It is known - see e.g. the expansions (14.31) and (14.35) in \cite{S} - that $\fa$ is real-analytic on $(0,+\infty)$ and that all its derivatives vanish at zero and at infinity. By Proposition 7.1.3 in \cite{K} and (14.31) in \cite{S}, the kernel $\fa(x-y)$ is not ${\rm TP}_2$ and hence not ETP. The function $f_{1/2}$ is explicit thanks to the identity in law
$$X_{1/2}\;\stackrel{d}{=}\; \frac{1}{4\Gamma_{1/2}}$$
where here and throughout $\Gamma_a$ denotes the standard Gamma random variable of parameter $a > 0,$ and is hence bell-shaped. The functions $f_{1/3}$ and $f_{2/3}$ can be written down in terms of a modified Bessel resp. a confluent hypergeometric function - see e.g. (2.8.31) and (2.8.33) in \cite{Z} - but it does not seem to the author that the associated second order ODE's can provide any substantial information on their bell-shape.
The strict unimodality of $\fa$ was first proved by Ibragimov and Chernin - see the proof of the Theorem in \cite{IC} - and then extended to all real stable densities by Sato and Yamazato - see Theorem 1.4 in \cite{SY}. Since the latter are real-analytic, this actually follows from Yamazato's original theorem \cite{Y1} on self-decomposable distributions. See also \cite{S2} for a short proof. The graphic simulations made by Holt and Crow \cite{HC}, partly reproduced in \cite{Z}, display bell-shaped curves for all real stable densities, at least at the level of the second derivative. A consequence of Theorem 5.1.(ii) in \cite{SY} is that $\fa$ has only one inflection point on the left side of its mode, and more generally that each $n$-th derivative of $\fa$ vanishes only once before the first zero of the $(n-1)$-th derivative. The bell-shape property was claimed in \cite{G1} for all stable densities but a serious mistake on the TP character of the kernel $\fa(xy^{-1})$ - see Remark (f) in \cite{S2} for an explanation - invalidates this result, also at the unimodal level, except for a particular class of stable densities which will be discussed in Section 3.3 and does not include the positive case. Apart from \cite{HC, S2, G1} the bell-shape of stable densities seems to have escaped investigation, even in the most visual case $n=2.$ In this note, we show the full property in the positive case.
\begin{Theo} The densities $\fa$ are bell-shaped.
\end{Theo}
Our main argument comes from Schoenberg's variation-diminishing property and the total positivity of certain infinitely convoluted exponential kernels. The little known fact that $\Xa$ admits infinite exponential sums as additive factors had been noticed in \cite{Y2}. It is used here in conjunction with a precise analysis of the other factor, which contains a certain exponential mixture with no atom at zero and can be chosen in order to have a weak bell-shape property of arbitrary order. In the case when $1/\alpha$ is an integer, another proof of the bell-shape is obtained quickly by a multiplicative factorization with inverse Gamma laws - see Section 3.1 below, but the argument does not extend to the other cases. With this method, the bell-shape of the density of $\log X_{1/n}$ on ${\mathbb{R}}$ can also be established. In Section 3.2, we raise two natural conjectures on the bell-shape of positive self-decomposable densities.
\section{Proof of the theorem}
We first establish an additive factorization of $\Xa$ via standard exponential variables. Recall that a positive random variable $X\sim\mu$ is said to be an exponential mixture (ME) if its law has the form
$$\mu(dx)\; =\; c\delta_0(dx)\; +\; f(x) dx$$
with $c\in [0,1]$ and $f$ a completely monotone function on $(0,+\infty).$ When $c=0$ we will use the notation $X\in{\rm ME}^*,$ and in this case Bernstein's theorem shows that the density $f$ writes
\begin{equation}
\label{ME}
f(x) \; =\; \int_0^\infty \theta e^{-\theta x} \mu(d\theta)
\end{equation}
over $(0, +\infty),$ where $\mu$ is some probability measure on $(0,+\infty).$ In other terms $X\stackrel{d}{=} {\rm Exp}\, (1)\times X_\mu^{-1}$
where here and throughout Exp $(\lambda)$ stands for the exponential law with parameter $\lambda,$ and $X_\mu \sim\mu.$ The following lemma is mostly due to Yamazato - see \cite{Y2} pp. 601-602 - but we rephrase it and give a proof for the reader's convenience.
\begin{Lemm}[Yamazato] One has the independent factorization
\begin{eqnarray}
\label{Dixmude}
\Xa\;\stackrel{d}{=}\;\Ya\; +\; \sum_{n\ge 1} \,\Yan,
\end{eqnarray}
where $\Yan\sim{\rm Exp} ((n\pi/\sin(\pi\alpha))^{1/\alpha})$ for every $n\ge 1$ and $\Ya\in{\rm ME}^*.$
\end{Lemm}
\proof We first write
$$\lambda^\alpha\; =\; \frac{\alpha}{\Gamma(1-\alpha)}\int_0^\infty (1-e^{-\lambda x})\frac{dx}{x^{\alpha+1}}\; =\; \int_0^\infty (1-e^{-\lambda x})\left(\int_0^\infty (c_\alpha u)^\alpha e^{-xu} du\right) dx$$
with the notation $c_\alpha = \sin(\pi\alpha)/\pi.$ This yields the further decomposition
$$\lambda^\alpha\; =\;\int_0^\infty (1-e^{-\lambda x})l_\alpha(x)dx\; +\; \int_0^\infty (1-e^{-\lambda x})\left(\int_0^\infty \left[ (c_\alpha u)^\alpha\right] e^{-xu} du\right) dx$$
where $[.]$ stands for the integer part and
$$l_\alpha(x)\; =\; \int_0^\infty ( (c_\alpha u)^\alpha -\left[ (c_\alpha u)^\alpha\right]) e^{-xu} du$$
is a completely monotone function. By Theorem 51.10 in \cite{S} there exists a random variable $\Ya$ belonging to the Bondesson class such that
$$-\log\EE[e^{-\lambda \Ya}]\; =\; \int_0^\infty (1-e^{-\lambda x})l_\alpha(x)dx, \qquad \lambda \ge 0.$$
Because $0\le (c_\alpha u)^\alpha -\left[ (c_\alpha u)^\alpha\right]\le 1$ for all $u\ge 0$ and $u^{-1}((c_\alpha u)^\alpha -\left[ (c_\alpha u)^\alpha\right])$ is integrable at zero, Steutel's criterion - see Theorem 51.12 in \cite{S} - shows that $\Ya$ is actually ME. We next prove that $\Ya$ has no atom at zero, which will entail $\Ya\in{\rm ME}^*.$ A change of variable and an integration by parts yield
$$l_\alpha(x)\; =\; \frac{x}{c_\alpha}\int_0^\infty e^{-xc_\alpha^{-1}u} \left(\int_0^u (t^\alpha -\left[ t^\alpha\right])\, dt\right) du$$
for every $x>0,$ and it is a bit tedious but elementary to see that
$$\int_0^u (t^\alpha -\left[ t^\alpha\right])\, dt\; \sim\; \frac{[u^\alpha]^{1/\alpha}}{2}\; \sim\; \frac{u}{2}$$
as $u\to+\infty.$ Theorem XIII.5.4 in \cite{F} entails
\begin{equation}
\label{Tauber}
l_\alpha(x)\; \sim\;\frac{c_\alpha}{2x}
\end{equation}
as $x\to 0,$ so that $\Ya$ has infinite L\'evy measure and, by Theorem 27.4 in \cite{S}, no atom at zero. On the other hand, setting $\kappa_{\alpha,n} = (n\pi/\sin(\pi\alpha))^{1/\alpha}$ for every $n\ge 1,$ one has
$$\int_0^\infty \left[ (c_\alpha u)^\alpha\right] e^{-xu} du\; =\; \sum_{n\ge 1} \int_{\kappa_{\alpha,n}}^\infty e^{-xu} du\; =\; \sum_{n\ge 1} \frac{e^{-x\kappa_{\alpha,n}}}{x}\cdot$$
Hence,
\begin{eqnarray*}\int_0^\infty (1-e^{-\lambda x})\left(\int_0^\infty \left[ (c_\alpha u)^\alpha\right] e^{-xu} du\right) dx & = & \sum_{n\ge 1}\int_0^\infty (1-e^{-\lambda x}) \frac{e^{-x\kappa_{\alpha,n}}}{x}dx\\
& = & \sum_{n\ge 1} (\log(\lambda+ \kappa_{\alpha,n}) - \log(\kappa_{\alpha,n}))\\
& = & -\log \EE[ \exp [ -\lambda\sum_{n\ge 1} \Yan]],
\end{eqnarray*}
where the first equality comes from Fubini's theorem and the second from Frullani's. Putting everything together completes the proof.
\endproof
\begin{Rqs} {\em (a) Because $\alpha\in(0,1),$ the sum on the right-hand side of (\ref{Dixmude}) has finite expectation. This shows that $\Xa$ and $\Ya$ have the same $\L_p-$integrability index which is $\alpha,$ excluded. The latter can also be seen from Theorem 25.3 in \cite{S} and the fact that $l_\alpha(x)\sim (\alpha/\Gamma(1-\alpha)) x^{-(\alpha +1)}$ as $x\to +\infty.$
(b) Since the infinite exponential sum has a ${\rm PF}_2$ viz. log-concave density - see the proof of the theorem below for a stronger property - and since the density of $\fa$ is not ${\rm PF}_2,$ one can interpret $\Ya$ as the factor which breaks down the ${\rm PF}_2$ property for $\fa$. Recall indeed that $\fa$ is log-concave until its second inflection point - see Theorem 1.3 (vii) in \cite{SY}. On the other hand $\Ya$ has a log-convex density by H\"older's inequality. It is plausible that $\log\fa$ has only {\em one} inflection point, which can be readily checked for $\alpha =1/2$.}
\end{Rqs}
We now introduce our notations concerning the sequence of signs of a real function defined on $(0,+\infty).$ We set
$${\rm sign}(x)\; =\; \left\{ \begin{array}{cl}+ & \mbox{if $x>0$}\\
0 & \mbox{if $x=0$}\\- & \mbox{if $x<0$}\end{array}\right.\qquad\mbox{and}\qquad \pm^n\; =\; \left\{ \begin{array}{cl}+ & \mbox{if $n$ is even}\\
- & \mbox{if $n$ is odd}\end{array}\right.$$
for every $x\in[-\infty,+\infty], n\in\NN.$ For $\{u_n\} = \{\varepsilon_1, \ldots, \varepsilon_n\}$ some finite sequence in $\{-,0,+\},$ we say that a continuous function $f :\, (0,+\infty) \to{\mathbb{R}}$ is of type $\varepsilon_1\ldots\varepsilon_n$ if it has limits (finite or infinite) at zero and at infinity, vanishes on a finite set, and if the ordered sequence of its signs on $[0,+\infty]$ is given by $\{\varepsilon_1, \ldots, \varepsilon_n\}.$ For brevity, we will write $f\sim\{ u_n\}$ or $f\sim\varepsilon_1\ldots\varepsilon_n$ to express this property. Observe that by the intermediate value theorem, the zero set of $f\sim\varepsilon_1\ldots\varepsilon_n$ corresponds to a subsequence of period 2 in $\{\varepsilon_1, \ldots, \varepsilon_n\}.$ In particular, a function of type $\varepsilon_1 \cdots \varepsilon_n$ vanishes exactly $(n/2 -1)$ times on $(0,+\infty)$ if $n$ is even and either $(n-1)/2$ or $(n-3)/2$ times on $(0,+\infty)$ if $n$ is odd. For example, the $n$-th derivative of a completely monotone function is of type $\pm^n0$ for every $n\ge 0.$ The density function $\fa$ is of type $0\!+\!0$ and, by the strict unimodality of $\Xa,$ its derivative $\fa'$ is of type $0\!+\!0\!-\!0.$ Introducing the sequences
$$\{a_n\}\;=\; 0\!\pm^0\!0\!\pm^1\!0\cdots\pm^n\!0\qquad \mbox{and} \qquad\{b_n\} = \pm^00\!\pm^1\!0\cdots\pm^{n}\!0$$
for every $n\ge 1,$ we can now state the central definition of this paper.
\begin{Def} For every $n\ge 0$, a smooth function $f:(0,+\infty)\to{\mathbb{R}}$ is said to be weakly bell-shaped of order $n$ {\em (${\rm WBS}_n$)} if $f^{(i)}\sim \{a_i\}$ $\forall\,i= 0\ldots n$ and $(-1)^{n+1+i}f^{(i)}\sim \{b_{n+1}\}$ $\forall\, i\ge n+1.$
\end{Def}
The bell-shape property of a function $f$ means that $f^{(i)}\sim \{a_i\}$ $\forall\,i\ge 0,$ so that weakly bell-shaped functions are never bell-shaped. Notice also that ${\rm WBS}_p\,\cap\,{\rm WBS}_q=\emptyset$ if $p\neq q.$ The following result provides our key-argument.
\begin{Prop} Let $X\in {\rm ME}^*$ and $0<\lambda_1\le\lambda_2\le\ldots\le\lambda_n\le\ldots$ be some sequence. For every $n\ge 1$, the independent sum
$X\,+\, {\rm Exp}(\lambda_1) \,+\, \cdots \,+\, {\rm Exp}(\lambda_n)$
has a ${\rm WBS}_{n-1}$ density.
\end{Prop}
\proof Set $f_0$ for the density of $X$ and recall from (\ref{ME}) that it is completely monotone. In particular $f_0$ is real-analytic on $(0,+\infty).$ We denote by $f_n$ the density of $X+ {\rm Exp}(\lambda_1) + \cdots + {\rm Exp}(\lambda_n)$ for every $n\ge 1.$ The latter are connected to one another by the formula
\begin{equation}
\label{Convol}
f_n(x)\; =\; \lambda_ne^{-\lambda_n x}\int_0^x e^{\lambda_n y} f_{n-1}(y) dy,
\end{equation}
and hence solutions on $(0,+\infty)$ to the linear ODE's
\begin{equation}
\label{ODE}
f_n'\; +\; \lambda_n f_n\; =\; \lambda_n f_{n-1}.
\end{equation}
An induction hinging on (\ref{Convol}) and (\ref{ODE}) show that all densities $f_n$ are real-analytic on $(0, +\infty),$ that all derivatives $f_n^{(i)}$ vanish at infinity, and that $f_n^{(i)}(0+) = 0$ for all $n\ge i+1\ge 1.$ In particular one has $f_n \sim \{a_0\}$ for every $n\ge 1.$ Using the complete monotonicity of $f_0$ for the initial step, another induction shows $(-1)^{n+i}f_n^{(i)}(0+) > 0$ for all $i\ge n\ge 0,$ possibly with infinite values. This entails $f_n^{(n)}(0+) > 0$ for all $n\ge 0,$ so that $0+$ is an isolated zero of $f_n^{(i)}$ for all $n\ge i+1\ge 1$ (notice that this cannot be proved directly by analyticity because $0+$ is also a singular zero). The weak bell-shape property will now be established through a more elaborate induction on $n$.\\
We first show that $f_1$ is ${\rm WBS}_0.$ Since $f_1 \sim \{a_0\},$ we know that $f_1'$ vanishes at least once on $(0,+\infty).$ Set $x_1 =\inf\{x > 0, \, f_1'(x) =0\} > 0,$ the strict inequality coming from $f_1'(0+) > 0.$ Differentiating (\ref{ODE}) yields
$$f_1''(x) \; =\; \lambda_1f_0'(x)\; <\; 0$$
for every $x >0$ such that $f_1'(x) = 0.$ This shows that $f_1'$ vanishes only at $x_1,$ whence $f_1'\sim\{b_1\}.$ Again, this entails that $f_1''$ vanishes at least once on $(0,+\infty).$ Set $x_2 =\inf\{x > 0, \, f_1''(x) =0\} > 0,$ the strict inequality coming from $f_1''(0+) < 0.$ Differentiating (\ref{ODE}) further entails
$$f_1'''(x) \; =\; \lambda_1f_0''(x)\; >\; 0$$
for every $x >0$ such that $f_1''(x) = 0.$ This shows that $f_1''$ vanishes only at $x_2,$ whence $-f_1''\sim\{b_1\}.$ Using the complete monotonicity of $f_0,$ the same argument yields $(-1)^{i+1}f_1^{(i)}\sim\{b_1\}$ for every $i\ge 1,$ as required.\\
We next show the induction step and suppose that $f_n$ is ${\rm WBS}_{n-1}$ for some $n\ge 1.$ We have already seen that $f_{n+1}\sim\{a_0\}$ and we will first prove by induction on $i$ that $f_{n+1}^{(i)}\sim \{a_i\}$ for every $1\le i\le n$. We begin with $f_{n+1}'$ for the sake of clarity. Since $f_{n+1}\sim\{a_0\},$ we know that $f_{n+1}'$ vanishes at least once on $(0,+\infty)$. Set $x_1 =\inf\{ x> 0, \; f_{n+1}'(x) = 0\}> 0,$ where the strict inequality comes from the fact that $0+$ is an isolated zero of $f_{n+1}'.$ Differentiating (\ref{ODE}) as above entails
$$ \lambda_n f_n'(x_1) \; =\; f_{n+1}''(x_1)\; \le\; 0,$$
where the inequality comes from the initial profile $0\!+\!0$ of $f_{n+1}'.$ Suppose that $f_{n+1}''(x_1)= 0.$ Then by analyticity $x_1$ is an isolated zero of $f_{n+1}'',$ so that
$$\lim_{x\uparrow x_1}\frac{f_{n+1}''(x)}{f_{n+1}'(x)}\; =\; -\infty.$$
By (\ref{ODE}), this entails $f_n'(x) < 0$ as $x\uparrow x_1.$ But the induction hypothesis yields $f_n'\sim \{b_1\}$ if $n=1$ and $f_n'\sim \{a_1\}$ if $n >1,$ so that one cannot have $f_n'(x_1) = 0,$ a contradiction. Hence $f_{n+1}''(x_1)< 0$ and if there exists $x_2 > x_1$ such that $f_{n+1}'(x_2) = 0,$ then again the induction hypothesis entails $f_n'(x_2) < 0$ so that $f_{n+1}''(x_2)< 0,$ a contradiction. This shows that $f_{n+1}'$ vanishes only once on $(0,+\infty)$ and, all in all, that $f_{n+1}'\sim\{a_1\}.$
We now suppose $f_{n+1}^{(i)}\sim \{a_i\}$ for some $i< n$ and show $f_{n+1}^{(i+1)}\sim \{a_{i+1}\}.$ We know that $f_{n+1}^{(i+1)}$ vanishes at least $i+1$ times on $(0,+\infty),$ and also from the above that $f_{n+1}^{(i+1)}$ has initial profile $0\!+\!0$. Set $\{x_j, \; j\ge 1\}$ for the ordered sequence of its zeroes on $(0,+\infty).$ Again, one gets from (\ref{ODE})
\begin{eqnarray}
\label{Induc}
f_{n+1}^{(i+2)}(x_j)\, =\, \lambda_n f_n^{(i+1)}(x_j)\quad\mbox{and}\quad \lim_{x\uparrow x_j}\frac{f_{n+1}^{(i+2)}(x)}{f_{n+1}^{(i+1)}(x)}\, =\,\lim_{x\uparrow x_j}\frac{f_n^{(i+1)}(x)}{f_{n+1}^{(i+1)}(x)}\, =\, -\infty
\end{eqnarray}
for every $j\ge 1.$ For clarity we consider two separate cases.
(i) If $i+1=n,$ the induction hypothesis entails $f_n^{(i+1)}\sim\{b_n\} =\pm^0 0\cdots\pm^n\!0.$ Set $I^k_n\leftrightarrow\{\pm^k0\}$ for the $k$-th half-closed interval corresponding to each part $\pm^k0$ of the graph of $f_n^{(i+1)}, k =0\ldots n.$ The last equality in (\ref{Induc}) with $j=1$ and the initial profile $0\!+\!0$ of $f_{n+1}^{(i+1)}$ entail that $x_1\in I^{k_1}_n$ with $k_1$ odd. If $f_n^{(i+1)}(x_1) < 0,$ then (\ref{Induc}) with $j=1,2$ and the induction hypothesis show that $x_2\in I^{k_2}_n$ with $k_2$ even. If $f_n^{(i+1)}(x_1) = 0,$ then $x_2\in I^{k_2}_n$ with $k_2 > k_1.$ Iterating the procedure shows that there are at most, and hence exactly, $n=i+1$ zeroes and that $x_k\in I^k_n$ for every $k=1\ldots n.$ All in all this shows that $f_{n+1}^{(i+1)}\sim\{a_{i+1}\},$ as required.
(ii) If $i+1<n,$ one has $f_n^{(i+1)}\sim\{a_{i+1}\} =0\!\pm^0\!0\cdots\pm^{i+1}\!0$ by the induction hypothesis. But adding a $0$ at $0+$ in the profile does not change anything in the above analysis, and the fact that $f_{n+1}^{(i+1)}\sim\{a_{i+1}\}$ follows exactly along the same lines as in (i).\\
We finally show that $(-1)^{n+1+i}f_{n+1}^{(i)}\sim \{b_{n+1}\}$ for every $i\ge n+1.$ We only sketch the proof since the arguments are the same. Consider first the case $i=n+1.$ Since $f_{n+1}^{(n)}\sim \{a_n\},$ we know that $f_{n+1}^{(n+1)}$ vanishes at least $n+1$ times on $(0,+\infty)$ and set $\{x_k, \, k\ge 1\}$ for the ordered sequence of its zeroes. Recall $f_{n+1}^{(n+1)}(0+) > 0$ and, from the induction hypothesis, that $f_{n}^{(n+1)}\sim\pm^10\cdots\pm^{n+1}\!0.$ Set again $I^k_n\leftrightarrow\{\pm^k0\}$ for the $k$-th half-closed interval corresponding to each part $\pm^k0$ of the graph of $f_n^{(n+1)}, k =1\ldots n+1.$ The same reasoning as above shows that necessarily $x_k\in I^k_n$ for all $k$ and hence that $f_{n+1}^{(n+1)}$ vanishes exactly $n+1$ times on $(0,+\infty),$ in other terms $f_{n+1}^{(n+1)}\sim \{b_{n+1}\}.$ It is then easy to show by induction on $i$ that $(-1)^{n+1+i}f_{n+1}^{(i)}\sim \{b_{n+1}\}$ for every $i\ge n+1,$ in using $(-1)^{n+1+i}f_{n+1}^{(i)}(0+) >0$ and the induction hypothesis $(-1)^{n+1+i}f_{n}^{(i)}\sim \pm^10\cdots\pm^{n+1}\!0.$
\endproof
\begin{Rqs}{\em (a) The complete monotonicity of $f_0$ is crucial in the above argument, initializing all induction steps. It seems difficult to extend the proposition to broader classes than ME.
(b) When $X$ itself has an exponential law and all parameters are different, the density of $X\,+\, {\rm Exp}(\lambda_1) \,+\, \cdots \,+\, {\rm Exp}(\lambda_n)$ is a linear combination of $n+1$ distinct exponentials functions. Since the latter define a Chebyshev system - see \cite{K} p. 24, the property ${\rm WBS}_n$ is much easier to prove. We leave the details to the reader.
(c) The method of examining the sign of the $(n+1)$-th derivative at the zero of the $n$-th derivative is used in \cite{IC} for $n=1$ in order to show the strict unimodality of $\fa,$ after some explicit contour integration. The involved computations to extend this argument directly on the further derivatives of $\fa$ seem however extremely heavy, and the author does not believe that they could yield the full bell-shape. See \cite{G2} for an application of the contour integration of \cite{IC} to the asymptotic behaviour of these further derivatives.}
\end{Rqs}
\noindent
{\bf End of the proof.} Fix $n\ge 1$ and set $f_{\alpha,n}$ resp. $g_{\alpha,n}$ for the density of
$$\Xa\, +\, \sum_{k=1}^{n+2} \,Y_{\alpha,k}\qquad\mbox{resp.}\qquad \sum_{k=n+3}^{\infty} \,Y_{\alpha,k}$$
with the notations of the Lemma, so that one has
$$\fa(x)\; =\; \int_0^\infty f_{\alpha,n}(x-y) g_{\alpha,n}(y) dy.$$
By the Proposition, the function $f_{\alpha,n}$ has a $\C^{n+1}$ extension on ${\mathbb{R}}$ and one obtains
$$\fa^{(n+1)}(x)\; =\; \int_0^\infty f_{\alpha,n}^{(n+1)}(x-y) g_{\alpha,n}(y) dy\; =\; \int_0^\infty f_{\alpha,n}^{(n+1)}(y) g_{\alpha,n}(x-y) dy.$$
The Laplace transform of $g_{\alpha,n}$ reads
$$\int_0^\infty e^{-\lambda x} g_{\alpha,n}(x) \, dx\; =\; \prod_{k=n+3}^{\infty}\left( \frac{1}{1+(\sin(\pi\alpha)/k\pi)^{1/\alpha}\lambda}\right)$$
and its reciprocal is of the P\'olya-Laguerre class $\E_1^*$ defined in \cite{K} p. 336. By Theorem 7.3.2. (b) in \cite{K} p. 345, this shows that the kernel $g_{\alpha,n}(x-y)$ is ${\rm TP}_\infty$ on ${\mathbb{R}}\times{\mathbb{R}}.$ Notice that $g_{\alpha,n}(x-y)$ is however not ${\rm STP}_\infty,$ because of the indicator function. Using the notation (3.1) p. 20 in \cite{K} with $I=(0,+\infty),$ Theorem 3.1. (a) p. 21 in \cite{K} and the Proposition entail
$$S^-(\fa^{(n+1)})\; \le\; S^-(f_{\alpha,n}^{(n+1)})\; =\; n+1.$$
Now since $\fa^{(n)}$ has isolated zeroes on $(0,+\infty)$ and vanishes at zero and infinity, Rolle's theorem yields
$$S^+(\fa^{(n)})\; \le\; S^-(\fa^{(n+1)})-1\; \le\; n$$
where we have used the notation (3.2) p. 21 in \cite{K} with $I=(0,+\infty).$ This shows that $\fa^{(n)}$ vanishes at most $n$ times, and hence exactly $n$ times, on $(0,+\infty).$ The proof is complete.
\vspace{-0.4cm
\section{Remarks and open questions}
\subsection{The case when $1/\alpha$ is an integer} In this situation the bell-shape follows easily from the independent factorization
\begin{equation}
\label{Will}
X_{1/n}\; =\; n^{-n} \Gamma_{1/n}^{-1}\,\times\,\cdots\,\times\, \Gamma_{(n-1)/n}^{-1},
\end{equation}
which was pointed out in \cite{Wi} and also basically shown in \cite{K} pp. 121-122. We discard the explicit case $n=2$ and set $g_n$ resp. $h_n$ for the density of $n^{-n}\Gamma_{1/n}^{-1}$ resp. $\Gamma_{2/n}^{-1}\times\ldots\times\Gamma_{(n-1)/n}^{-1}.$ The multiplicative convolution yields
$$f_{1/n}(x)\; =\; \int_0^\infty g_n(xy^{-1})h_n(y)\frac{dy}{y}$$
and one can clearly differentiate under the integral. This entails
$$f_{1/n}^{(i)}(x)\; =\; \int_0^\infty g_n^{(i)}(xy^{-1})h_n(y)\frac{dy}{y^{i+1}}\; =\; \frac{1}{x^i}\int_0^\infty h_n(xy^{-1}) g_n^{(i)}(y) y^{i-1} dy$$
for every $x > 0, i\ge 1.$ The kernel $h_n(xy^{-1})$ is the composition of $(n-2)$ kernels that are ${\rm STP}_\infty$ on $(0, +\infty)\times(0,+\infty)$ by Theorem 2.1 p. 18 in \cite{K}, and is hence itself ${\rm STP}_\infty$ on $(0, +\infty)\times(0,+\infty)$ by the Binet-Cauchy formula. With the above notation for $S^-$ and $S^+$, Theorem 3.1 (b) p. 21 in \cite{K} and the aforementioned bell-shape property of $g_n$ yield
$$S^+(f_{1/n}^{(i)})\; \le\; S^-(g_n^{(i)})\; =\; i$$
for every $i\ge 1$ so that $f_{1/n}$ is bell-shaped, too. The above factorization (\ref{Will}) shows also easily that $\log X_{1/n}$ has a bell-shaped density on ${\mathbb{R}},$ a property which we believe to be true for all $\alpha\in (0,1).$ When $\alpha$ is rational, another multiplicative factorization of $\Xa$ in terms of inverse Beta and inverse Gamma random variables was obtained in Lemma 2 of \cite{S1}. But the latter seems useless for the bell-shape of $\Xa$ or $\log\Xa,$ because inverse or log Beta laws have neither bell-shaped densities nor ${\rm TP}_\infty$ convolution kernels.
\subsection{Positive self-decomposable densities} A positive random variable $X$ is self-decomposable when its Laplace transform reads
$$\EE[e^{-\lambda X}]\; =\; \exp -\left[\gamma_0\lambda \,+ \, \int_0^\infty (1-e^{-\lambda u}) \frac{k(u)}{u} du\right], \qquad\lambda \ge 0$$
for some $\gamma_0\ge 0$ and $k : (0, +\infty)\to{\mathbb{R}}^+$ non-increasing. For example, the positive stable law is self-decomposable with $\gamma_0 = 0$ and $k(x) = \alpha/\Gamma(1-\alpha) x^\alpha.$
It is known and easy to see - see Theorem 27.7 in \cite{S} - that $X$ has a density $f$ when $k\neq 0.$ The unimodality of $f$ was shown by Wolfe \cite{Wo}. In the following we will suppose that $\gamma_0 = 0$ w.l.o.g. and set $f_k$ for the positive self-decomposable density associated with the non-degenerate spectral function $k.$
When $k(0+) =+\infty,$ Theorem 28.4 (ii) in \cite{S} and Lemma 2.5 in \cite{SY} show that $f_k$ is smooth and that all its derivatives vanish at zero and at infinity, and Theorem 1.4 in \cite{SY} entails that $f_k$ is strictly unimodal. When $k(0+) < +\infty,$ it is also known - see Remark 28.6 in \cite{S} - that $f_k$ is not smooth at zero and hence not bell-shaped. In view of the main result of the present paper, one might raise the
\begin{Conju} The density $f_k$ is bell-shaped if and only if $k(0+) =+\infty.$
\end{Conju}
Apart from the positive stable, examples of densities verifying the above conjecture are the inverse Gamma, or more generally the densities of $\Gamma_t^{-a}$ with $t > 0, a \ge 1.$ Indeed, the latter are self-decomposable by the HCM criterion \cite{B}, and bell-shaped by a direct computation. Other self-decomposable examples are infinite exponential sums, whose bell-shape property can be proved exactly in the same manner as for $\fa.$ As pointed out in the introduction, when $k(0+) =+\infty$ Theorem 5.1.(ii) in \cite{SY} already shows that $f_k^{(n)}$ vanishes only once before the first zero of $f_k^{(n-1)},$ for every $n\ge 2.$ The proof of this latter property relies on a repeated use of Steutel's equation - see e.g. Theorem 51.1 of \cite{S}. Analyzing the further zeroes of $f_k^{(n)}$ with the same method seems however not quite obvious, because of the memory involved in this integro-differential equation.
Let us now consider the case where $k(0+)$ is finite. When $k(0+)<1,$ Theorem 1.3 in \cite{SY} shows that $f_k$ is decreasing. When $k(0+) =1,$ the situation is complicated because of the three different types ${\rm I}_2,{\rm I}_3$ and ${\rm I}_4$ of \cite{SY}. In particular $f_k$ may have a non-trivial modal interval - see Lemma 53.2 in \cite{S}. When $k(0+) >n+1$ for some $n\ge 0,$ then Theorem 28.4 (i) in \cite{S} and Lemma 2.5 in \cite{SY} show that $f_k$ is $\C^n$ with derivatives vanishing at zero and at infinity, and again Theorem 1.4 in \cite{SY} entails that $f_k$ is strictly unimodal. The following is hence quite natural.
\begin{Conjd} If $k(0+) >n+1$ for some $n\ge 0,$ then $f_k^{(i)}\sim\{a_i\}$ for all $i = 0\ldots n.$
\end{Conjd}
An example supporting this conjecture is the density $f_{a,b}$ of $-\log \beta_{a,b},$ where $\beta_{a,b}$ is the Beta random variable with parameters $a>0, b >1.$ Indeed, one can show by a direct computation that $f_{a,b}$ is self-decomposable with spectral function
$$k_{a,b}(x)\; =\; \frac{e^{-ax}(1-e^{-bx})}{(1-e^{-x})}$$
so that $k_{a,b}(0+) = b,$ and the explicit formula
$$f_{a,b} (x) \; =\; \frac{\Gamma(a+b)}{\Gamma(a)\Gamma(b)}e^{-ax}(1-e^{-x})^{b-1}$$
entails that $f_{a,b}$ satisfies the required properties.
\subsection{Two-sided stable densities} As mentioned in the introduction, stable densities are all visually bell-shaped on Holt and Crow's graphics, and it is an interesting problem to show the property rigorously. It is known that two-sided stable densities are real-analytic on ${\mathbb{R}}$, never vanish, and that all their derivatives tend to zero at infinity - see Remarks 14.8 and 28.8 in \cite{S}. Hence, their $n$-th derivative vanishes at least $n$ times on ${\mathbb{R}}$ by Rolle's theorem (a more complicated proof of this fact is also given in \cite{G1} pp. 234-235). The strict unimodality follows from Yamazato's theorem \cite{Y1} by analyticity. The article \cite{G1} had claimed the full bell-shape property for all stable densities with an argument relying on a certain integral representation through a Student-type kernel, and a multiplicative TP property. The integral representation is correct and basically amounts to Bochner's subordination and Zolotarev's duality - see Lemma 2 in \cite{G1}. However the multiplicative TP property claimed in Lemma 1 (iv) of \cite{G1} and crucial to bound the number of zeroes - see pp. 236-237 in \cite{G1}, is false in general. Indeed, Schoenberg's theorem shows - see pp. 121-122 and p. 390 in \cite{K} - that the kernel $K_\alpha(x,y)=\fa(e^{x-y})$ is ${\rm STP}_\infty$ if and only if $1/\alpha$ is the reciprocal of an integer. Actually, $K_\alpha$ is not even ${\rm TP}_2$ for $\alpha > 1/2$ - see the main result of \cite{S1}. In a forthcoming paper, we also show that $K_\alpha$ is not ${\rm TP}_n$ as soon as $\alpha > 1/n$ and $1/\alpha$ is not the reciprocal of an integer. In \cite{G1} the skewed Cauchy case is also treated separately via Zolotarev's representation - see (2.8) therein. All in all the main result of \cite{G1} must hence be reduced to the
\begin{Theo}[Gawronski] Two-sided $\alpha-$stable densities are bell-shaped when $\alpha =1,2$ or when $1/\alpha$ is an integer.
\end{Theo}
Notice that this result does not extend directly to the one-sided case in spite of the affirmation made in (iv) p. 239 of \cite{G1}, because the number of zeroes of a function sequence might clearly increase at the continuous limit. In order to cover all stable densities, the multiplicative method of \cite{G1} seems not appropriate because of the poor variation-diminishing properties of $K_\alpha$ - see however \cite{S2} for a somehow related multiplicative point of view, which works at the unimodal level. On the other hand, Yamazato's additive factorization extends to {\em all} two-sided stable densities and delivers yet another proof of their unimodality - see again \cite{Y2} pp. 600-601. Unfortunately the complete monotonicity of the factor $\Ya$ which is crucial in our argument to get the full bell-shape - see the above Remark 2 (a), is clearly lost in the two-sided case. \\
\noindent
{\bf Acknowledgements.} Ce travail a b\'en\'efici\'e d'une aide de l'Agence Nationale de la Recherche portant la r\'ef\'erence ANR-09-BLAN-0084-01.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 1,768
|
Comparison And Contrast Drinking Articles Essay
Comparison and Contrast:
"Too Many Colleges are in Denial about Alcohol Abuse" and "The Battle of the Binge"
"Too Many Colleges are in Denial about Alcohol Abuse" and "The Battle of the Binge" are both two unique pieces of writing that dig deep into the aspects of consuming alcohol, and drinking too much of it, referred to as "binge drinking." They acknowledge the fact that drinking alcohol itself in proper moderation is not bad, but the abuse of alcohol is. "Too Many Colleges are in Denial about Alcohol Abuse" is similar to "The Battle of the Binge" by pinpointing drinking in college specifically, and stressing the problems related to binge drinking and how the change is going to be ...view middle of the document...
The two writings make it clear that the number of binge drinkers whom are attending a university or college is becoming extremely high, creating a dilemma in the United States.
The two articles are also similar in a way that they both place emphasis on the complications and problems that can be acquired from binge drinking, and how it is going to be very difficult to change this situation. In "Too Many Colleges are in Denial about Alcohol Abuse," it is declared that "alcohol abuse is a common, not a marginal, activity at most colleges and we only fool ourselves if we expect marginal efforts to reduce it" (Wechsler 336). The essay further explained the problems related to binge drinking, that those who binge drink are "seven to ten times more likely as non-binge drinkers to fail to use protection when having sex, to engage in unplanned sexual activity, to get into trouble with campus police, to damage property, or to suffer an injury" (Wechsler 338). "The Battle of the Binge," states it is "going to require some larger cultural changes that will make us see the irony, even cruelty, of infantilizing certain young adults…"(Hitt 342). The author believes that moderate drinking has disappeared among students and because alcohol is so restricted, it only worsens the outcomes. He introduces his nephew in his article, explaining, "he knew two students who had died – falling off a cliff, blood poisoning – and five others who had been paralyzed or seriously injured in car accidents because of binging" (Hitt 341). The little details between these works are distinctive, but "Too Many Colleges are in Denial about Alcohol Abuse" and "The Battle of the Binge" both agree that the problems occurring from students are serious and drastic measures must be made to assure that they will not happen again.
"Too Many Colleges are in Denial about Alcohol Abuse" is different than "The Battle of the Binge" because it uses the persuasive technique of logos more throughout the entirety of the essay than "The Battle of the Binge." "Too Many Colleges are in Denial about Alcohol Abuse" uses statistics from a various amount of sources, including the Carnegie Foundation for the Advancement of Teaching, Harvard University, New York Times, Journal of the American Medical Association, Southern Illinois institute, and American Council on Education. The authors take these statistics and relate them to students in order to gain attention to the material being presented, for example: "a study done in 1993 by the Institute for Social Research of the University of Michigan found that 40 percent of the college students...
Other Papers Like Comparison and Contrast Drinking Articles
Comparison and Contrast of the Lottery and the Destructors
1076 words - 5 pages The Lottery vs. The Destructors Comparison and Contrast of The Lottery and The Destructors Elyse Carbajal Liberty University The Lottery vs. the Destructors Thesis In The Destructors by Graham Greene and The Lottery, by Shirley Jackson's, the authors produce stories filled with irony, grim reality, which is filled with symbolism. Moreover, they mask evil, which ultimately showcases how individuals sightlessly follow tradition
Comparison and Contrast Cna Essay
862 words - 4 pages Long term vs. Short term When one thinks of pursing a nursing career, he/she may consider trying to obtain a certified nursing technician license first. A certified nursing technician is a person who assists the patients and nurses to complete tasks that need to be done, such as bathing, feeding, walking, dressing changes, and doing the vital signs that need to be recorded. The nursing technicians do just about everything a nurse does except
Comparison and Contrast of the Steps That Led Russia to Leave Ww1 and the United States to Enter Ww1
1814 words - 8 pages Comparison and Contrast of the Steps that Led Russia to Leave WW1 and The United States to enter WW1 World War 1 was one of the darkest and bloodiest chapters of World History. The heavy number of casualties and the economic loses suffered for all the European countries that took part in the 'Great War' , as it was known later on, are enough proves of the bloodshed and violence of it. Nevertheless, two of these countries had an
A Comparison of the Articles "The Performance Implications of Dynamic Capabilities: the Case of Product Innovation" and "Absorptive Capacity, Environmental Turbulence, and the Complementarity of...
3084 words - 13 pages WHU – Otto Beisheim School of Management A Comparison of the Articles "The Performance Implications of Dynamic Capabilities: The Case of Product Innovation" and "Absorptive Capacity, Environmental Turbulence, and the Complementarity of Organizational Learning Processes" Ulrich Lichtenthaler Seminar on Research and Academic Writing Institute for Management Accounting and Control Submission Date: October 15th, 2012 Submitted to: Jun.-Prof
Discuss the Role of Women as Represented in Art, Photography or Advertising Through Comparison and Contrast of the Work and Intentions of at Least Three Artists from Different Time Periods. What Role...
1767 words - 8 pages Question no. 5 'Men act, woman appear', Berger wrote. Discuss the role of women as represented in art, photography or advertising through comparison and contrast of the work and intentions of at least three artists from different time periods. What role did men play during the period being discussed, and how were they represented? At least one of the artists discussed should be contemporary; discuss changes in representation of women, and why
Comparison and Contrast on Romeo and Juliet and West Side Story
557 words - 3 pages Essay In plays, as in real life, individuals' actions can change the course of events in theirs, as well as, other individuals' lives. In both Romeo and Juliet and West Side Story, the lovers were forced to take their own lives or be killed because of fate and the actions of others. In Romeo and Juliet, one character who is partly responsible for the deaths of Romeo and Juliet is Tybalt. Tybalt enrages Remeo and
Comparison and contrast of the ways of characterization used by the two playwrights in Arthur Miller's "The Crucible" and Tennessee Williams' "Streetcar Named Desire"
535 words - 3 pages : "holding back a full condemnation of her" or " he goes directly to her and grabs her by her cloak, furious"Blanche DuBois is also a not satisfied person , who had been just thrown away by the person she admired and felt safe next to. The description of the scene shows her weak personality that she couldn't stop drinking alcohol, when she wanted to forget what had happened to her. All of these made through the stage directions describing her actions
Binge Drinking Notes
5031 words - 21 pages amounts of alcohol by binge drinking. Binge drinking, or the partying lifestyle of young people may be related to an environment that appears to support heavy drinking. Teenagers report that alcohol is more easily available to them today than it was 5 years ago, and there is a high correlation between availability and use. In addition, alcoholic beverages remain inexpensive in comparison with other beverages, especially beer when purchased in kegs
Nba Lockout
1347 words - 6 pages Adam Rubenacker Mrs. Ellis English 1101 24 October 2011 The Comparison of Two NBA Lockout Articles The NBA lockout is when the current Collective Bargaining Agreement (CBA) between the NBA and the NBA Player's Association (NBAPA) expires. It began on July 1, 2011 and has resulted in the cancellation of regular season games. The NBA wants to implement a "hard" cap and decrease the salary cap, below $50 million. In the meantime
443 words - 2 pages 181 Adjectives, Adverbs, Comparison 184 Common ways of forming adjectives; order of adjectives; adverbs; adjectives and adverbs with similar forms; gradable and ungradable adjectives; comparison 24 VOCABULARY Education and Assessment GRAMMAR 191 197 REVIEW 12 Countable and Uncountable Nouns, Articles, Quantifiers 200 Problems with plurals; uncountable nouns; articles: a, an, the, zero article; quantifiers
1099 words - 5 pages , "make up for lost time." However, a study done by Alexander Wagenaar and PM O'Malley found that when the minimum drinking age was twenty-one, there was a lower use of alcohol after they turn twenty-one. One of the largest arguments in favor of lowering the drinking age is the use of Europe as a comparison. Where as in Europe, where there isn't a prescribed legal age for drinking, the age for obtaining a drivers license is eighteen. Sixteen is
Comparison And Contrast Paper
686 words - 3 pages Contrast and Comparison Paper As an adult have you ever gotten on the bicycle that you had since you were a teenager and decided to go for a spin? How did that work out? It probable did not feel the same as it did way back in the days. You probable could not ride as hard or as long as you could have back then. Then you take a step back and realize that things just aren't the way they used to be. Well the men in "Glory Days" and
Comparison And Contrast Essay
1181 words - 5 pages Thaddeus Walker Comparison and Contrast Essay Ms. LaCroix 24 February, 2012 No Heart No Liver For this paper, I reviewed two fairy tales Grimm's" Little Snow White" and "Gold Tree Silver Tree "two different versions of the same fairy tale. I found both stories enjoyable and having a lot of moral values. They both contain key characters that we can relate to in our lives to a certain extent and, just like most fairy tales good triumphs
Contrast And Comparison Of Operating Systems
3990 words - 16 pages Contrast and Comparison of Operating Systems Team B will take what we have learned from our course POS/355 Operating systems and research to compare and contrast the UNIX/Linux, Microsoft Windows, and MacOS systems. In particular we will contrast and compare the different operating systems in regards to memory management, processing, file management and security. Operating systems at the core is all about the kernel. The kernel in all of the
Photo Editing Software Comparison And Contrast Essay
922 words - 4 pages Photo Editing Software Comparison and Contrast Essay Editing photos can be done with a click of a mouse. There are numerous options for photo editing software. Two great and professional programs are Adobe Photoshop and Adobe Lightroom. In this essay, there will be a comparison of Photoshop and Lightroom, the advantages, disadvantages and similarities. Photoshop and Lightroom both come from Adobe; they do have similar features and are
Jones V Star Credit Corp Essay
Chapter 2 Financial Essay
Urbanization in the Third World Essay
Marketing Essay
Inventory Management and Its Effect on Customer Satisfaction
Dairy Business in Uae Essay
Influences on Learning Essay
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 4,771
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.