text stringlengths 14 5.77M | meta dict | __index_level_0__ int64 0 9.97k ⌀ |
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Blog on hiatus? Well, that's a duh statement. This mostly-neglected blog has barely been touched in the last few years. If you're interested in fiction writing, Mike is actively updating The Lost Royals, the site for a new book series in development. Get a peek behind the curtain to see how a fantasy novel comes to life! | {
"redpajama_set_name": "RedPajamaC4"
} | 1,516 |
{"url":"https:\/\/stats.stackexchange.com\/questions\/398162\/pdf-of-the-minimum-of-a-geometric-random-variable-and-a-constant","text":"# PDF of the minimum of a geometric random variable and a constant\n\nI have $$X \\sim Geo(p)$$, such that\n\n$$p(x) = (1-p)^{k-1}p, \\ \\ x = 1,2,3, \\ldots$$\n\nand Y is a constant random variable which assumes the value of the constant integer $$t$$, such that\n\n$$P(Y=t) = 1, \\ \\ t>0$$\n\nAnd now I am considering a random variable $$Z$$, where\n\n$$Z = min(X, Y)$$\n\nI am attempting to compute the PDF of Z, and have followed something similar to the answer posted here i.e.\n\n\\begin{align} f_Z(z) & = \\frac{\\mathrm d\\;}{\\mathrm d z} \\Bbb P(\\min(X,Y)\\leq z) \\\\[1ex] & = \\frac{\\mathrm d\\;}{\\mathrm d z} (1 - \\Bbb P(\\min(X,Y)\\gt z)) \\\\[1ex] & = -\\frac{\\mathrm d\\;}{\\mathrm d z}\\Big( \\Bbb P(X>z)\\,\\Bbb P(Y>z) \\Big) \\\\[1ex] & = -\\frac{\\mathrm d\\;}{\\mathrm d z} \\Big(\\big(1-F_X(z)\\big)\\big(1-F_Y(z)\\big)\\Big) \\\\[1ex] & = f_X(z)\\Big(1-F_Y(z)\\Big) + \\Big(1-F_X(z)\\Big)f_Y(z) \\\\[1ex] & = (1-p)^{z-1}p\\Big(1 - 1_{z \\geq t}\\Big) - (1-p)^z1_{z=t} \\end{align}\n\nAm I on the right path? I kind of expected a more compact kind of expression, one I could relate to the exponential family?\n\n\u2022 You seem to be on the right track if the problem specifies that $X$ and $Y$ are independent. If not, you might have to deal with their joint PDF.\n\u2013\u00a0rgk\nMar 18, 2019 at 16:05\n\u2022 There is no PDF anywhere. And since you are assuming independence of $X,Y$, you should mention that in your post. Mar 18, 2019 at 16:10\n\nProposition. Let $$X$$ be a random variable with cumulative distribution function $$F_X$$ (i.e., $$F_X(x) = P(X \\leq x)$$ for all $$x \\in \\mathbb{R}$$, and let $$t \\in \\mathbb{R}$$ be a constant. Define $$Z = \\min\\{X, t\\}$$. Then the cumulative distribution function $$F_Z$$ of $$Z$$ is $$F_Z(z) = \\begin{cases} F_X(z) & \\text{if z < t} \\\\ 1 & \\text{if z \\geq t} \\end{cases}$$ for all $$z \\in \\mathbb{R}$$.\nProof. Let $$Y$$ be a constant random variable with value $$t$$. Note that $$Y$$ is necessarily independent of $$X$$ and that $$P(Y > z) = \\mathbf{1}_{(-\\infty, t)}(z) = \\begin{cases} 1 & \\text{if z < t,} \\\\ 0 & \\text{if z \\geq t.} \\end{cases}$$ Then we have \\begin{aligned} F_Z(z) &= P(Z \\leq z) \\\\ &= 1 - P(Z > z) \\\\ &= 1 - P(\\min\\{X, Y\\} > z) \\\\ &= 1 - P(X > z, Y > z) \\\\ &= 1 - P(X > z) P(Y > z) \\\\ &= 1 - (1 - F_X(z)) \\mathbf{1}_{(-\\infty, t)}(z) \\\\ &= \\begin{cases} F_X(z) & \\text{if z < t,} \\\\ 1 & \\text{if z \\geq t.} \\end{cases} \\end{aligned}\nYou can use this Proposition to figure out the probability mass function (not the probability density function!) of $$Z = \\min\\{X, Y\\}$$ where $$X \\sim \\operatorname{Geometric}(p)$$ and $$Y = t$$ almost surely, as in your question. More concretely, your random variable $$Z$$ will be a discrete random variable supported on $$\\{1, 2, \\ldots, t\\}$$ satisfying \\begin{aligned} P(Z = z) &= P(Z \\leq z) - P(Z \\leq z - 1) \\\\ &= F_Z(z) - F_Z(z - 1) \\end{aligned} for all $$z \\in \\{1, 2, \\ldots, t\\}$$. The cumulative distribution function needed for the above computation can be determined using the Proposition above.","date":"2022-07-05 15:33:54","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 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\": 30, \"wp-katex-eq\": 0, \"align\": 1, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 1.0000077486038208, \"perplexity\": 310.3378787111246}, \"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\/1656104585887.84\/warc\/CC-MAIN-20220705144321-20220705174321-00295.warc.gz\"}"} | null | null |
\section{Introduction}
The four thermodynamics laws of black hole, which were originally
derived from the classical Einstein Equation, provide deep insight
into the connection between thermodynamics and Einstein
Equation\cite{Black Hole,ql1}. Recently, this connection has been
investigated extensively in the literatures for Rindler spacetime
and Friedmann-Robertson-Walker (FRW) universe. For Rindler
spacetime\cite{Jacobson}, the Einstein equation can be derived from
the proportionality of entropy to the horizon area, together with
the Clausius relation $\delta Q=TdS$. Here $\delta Q$ and $T$ are
the energy flux and Unruh temperature detected by an accelerated
observer just inside the local Rindler causal horizons through
spacetime point. In FRW universe\cite{gauss-love1}, after replacing
the event horizon of black hole by the apparent horizon of FRW
space-time and assuming that the apparent horizon has an associated
entropy $S$ and temperature $T$
\begin{eqnarray}
S=\frac{A}{4G},~~~~~T=\frac{1}{2\pi \tilde{r}_A},
\end{eqnarray}
one can cast the first law of thermodynamics, $dE=TdS$, to the
Friedmann equations. Here $G$, $A$, and $\tilde{r}_A$ are the
gravitational constant, the area of the apparent horizon, and the
radius of the apparent horizon, respectively. The first law of
thermodynamics not only holds in Einstein gravity, but also in
Guass-Bonnet gravity, Lovelock gravity, and various braneworld
scenarios\cite{fr,gb,brane}. The fact that the first law of
thermodynamics holds extensively in various spacetime and gravity
theories suggests a deep connection between gravity and
thermodynamics. (Some other viewpoints and further developments in
this direction see \cite{padm,mass,cft/frw,ql3,brick,cao2} and
references therein.)
The thermodynamics behavour of spacetime is only one of the features
of Einstein gravity. Another feature is the Hawking radiations at
the event horizon of black holes\cite{hawking} or the apparent
horizon of the FRW spacetime\cite{cao1}. The Hawking radiation is a
quantum mechanics effect in the classical background black hole or
FRW spacetime. Therefore quantum theory, gravitational theory and
thermodynamics meet together at black holes and FRW spacetime. For a
black hole, it radiates and becomes smaller and hotter, finally
disappears when the Hawking radiation ends, leaving behind thermal
radiation described by quantum mechanical mixed states.
However, the analysis of Hawking radiation in the literatures
usually make use of the semi-classical approaches, assuming a
classical background metric and considering a quantum radiation
process. When it comes into the high energy regime, for example a
small black hole whose size can compare with Planck scale or a FRW
universe in the era of Planck time, the effect of quantum gravity
should not be forgotten. In these cases, the conventional
semi-classical approaches are not proper and the complete quantum
theory of gravity is required.
Recently, a growing interest has been focused on the proposal that
the quantum gravity effect might need us turn from the usual
commutation relations of the Heisenberg's uncertainty principle
(HUP) to the generalized uncertainty principle (GUP)\cite{gup1}. The
GUP is a model independent aspect of quantum gravity and can be
derived from different approaches to quantum gravity, such as string
theory\cite{gup3-string}, loop quantum gravity and noncommutative
quantum mechanics\cite{gup2}.
Naturally, one may think that the GUP should influence the
thermodynamics of black holes and FRW universe in the small scale or
in the high energy regime. Indeed, this issue has been investigated
in contexts of black hole physics. As we have known, the GUP affects
the thermodynamics of black holes in two aspects. First, the GUP
might modify the Hawking temperature on the event horizon and may
prevent the total evaporation of a black
hole\cite{gup-hawking,gup-hawking1}. Second, after considering the
GUP, one will get a correction to the Bekenstein-Hawking entropy of
a black hole\cite{gup-entropy1,gup-entropy2}. This correction
modifies the famous entropy-area relation that the entropy of a
black hole is proportional to its area of the event horizon. The
impact of the GUP on other physics systems has also been
investigated extensively, see\cite{other} and references therein.
However, as far as we know, whether the GUP can influence the
thermodynamics of FRW universe is still unknown. Is there indeed a
correction to the entropy on the apparent horizon of the FRW
universe when we consider the effect of the GUP? If the GUP is
considered,can we still get the Friedmann equations when we apply
the first law of thermodynamics to the apparent horizon? These
problems need to be solved. In this paper, we are going to
investigate these problems. We find that by utilizing the GUP, the
entropy of the apparent horizon of the FRW universe should get a
correction. Moreover, starting with the modified entropy on the
apparent horizon, we will show that the first law of thermodynamics
on the apparent horizon can produce the corresponding modified
Friedmann equations.
However, as a high energy correction to HUP, the GUP should not be
important for the late time FRW universe. In this case, one might
consider the effect of the large length scale modification. (For
example, in Dvali-Gabadadze-Porrati braneworld model, the large
length scale modification to the Einstein gravity on the brane might
lead to the late time acceleration of our universe.) Recently, an
extended uncertainty principle (EUP) has been introduced to
incorporate the effect of the large length
scale\cite{gup-hawking1,eup}. We note here that in this paper, we
adopt the terminology, the extended uncertainty principle (EUP) and
the generalized EUP (GEUP), which were first used in
\cite{gup-hawking1}. Contrary to the GUP, the EUP is the large
length scale correction to the Heisenberg's uncertainty principle.
In black hole physics, the uncertainty principle can be used to
derive the Hawking temperature of Schwarzschild-(anti)de Sitter
black hole\cite{gup-hawking1,eup1}. Like in the case of the GUP, it
is also of interest to investigate the impact of the EUP on the
thermodynamics of the FRW universe. In this paper, we will also
consider this issue.
Therefore, the organization of this paper is as follows. In Section
II, we investigate the influence of the GUP on the thermodynamics of
the FRW universe, and in Section III, we generalize the discussions
of the GUP to the EUP case. The Section IV are our summary and
discussions. Throughout the paper, the units $c\equiv\hbar\equiv
k_B\equiv1$ are used.
\section{The GUP case}
Let us begin with the GUP, which is usually given by\cite{gup1}
\begin{eqnarray}
\delta x\delta p\geq 1+\alpha^2l_p^2 \delta p^2,\label{GUP}
\end{eqnarray}
where $l_p$ is the Planck length, and $\alpha$ is a dimensionless
real constant. The GUP has an immediate consequence that there is a
minimal length with the Planck scale,
\begin{eqnarray}
\delta x\geq \frac{1}{\delta p}+\alpha^2l_p^2 \delta p \geq 2|\alpha
l_p|.
\end{eqnarray}
This minimal length characterizes the absolute minimum in the
position uncertainty.
Another consequence of GUP is the modified momentum uncertainty.
After some simple manipulations, the momentum uncertainty can be
written as
\begin{eqnarray}
\delta p\geq\frac{1}{\delta x}[\frac{\delta x^2}{2\alpha^2
l_p^2}-\frac{ \delta x^2}{2\alpha^2 l_p^2}
\sqrt{1-\frac{4\alpha^2l_p^2}{(\delta x)^2}}]=\frac{1}{\delta
x}f_G(\delta x^2),\label{gup3}
\end{eqnarray}
where
\begin{eqnarray}
f_G(\delta x^2)=\frac{\delta x^2}{2\alpha^2 l_p^2}-\frac{ \delta
x^2}{2\alpha^2 l_p^2} \sqrt{1-\frac{4\alpha^2l_p^2}{(\delta
x)^2}}\end{eqnarray} characterizes the departure of the GUP from the
Heisenberg uncertainty principle $\delta p\geq 1/\delta x$.
We consider a ($n+1$)-dimensional FRW universe, whose linear element
is given by
\begin{eqnarray}
ds^2=-dt^2+a^2(\frac{dr^2}{1-kr^2}+r^2d\Omega_{n-1}^2),
\end{eqnarray}
where $d\Omega_{n-1}^2$ denotes the line element of an
($n-1$)-dimensional unit sphere, $a$ is the scale factor of our
universe and $k$ is the spatial curvature constant. In FRW
spacetime, there is a dynamical apparent horizon, which is a
marginally trapped surface with vanishing expansion. Using the
notion $\tilde{r}=ar$, the radius of the apparent horizon can be
written as
\begin{eqnarray}
\tilde{r}_A=\frac{1}{\sqrt{H^2+k/a^2}},
\end{eqnarray}
where $H$ is the Hubble parameter, $H\equiv \dot{a}/a$ (the dot
represents derivative with respect to the cosmic time $t$). On the
apparent horizon, if we suppose that the apparent horizon has an
associated entropy $S$ and temperature $T$
\begin{eqnarray}
S=\frac{A}{4G},~~~~~T=\frac{1}{2\pi \tilde{r}_A},
\end{eqnarray}
(where $A$ is the apparent horizon area
$A=n\Omega_n\tilde{r}_A^{n-1}$ with
$\Omega_n=\pi^{n/2}/\Gamma(n/2+1)$ being the volume of an
$n$-dimensional unit sphere.) it has been confirmed
\cite{gauss-love1} that the first law of thermodynamics,
\begin{eqnarray}
dE=TdS,
\end{eqnarray}
can reproduce the Friedmann equations
\begin{eqnarray}
\dot{H}-\frac{k}{a^2}=-\frac{8\pi G}{n-1}(\rho+p),\\
H^2+\frac{k}{a^2}=\frac{16\pi G}{n(n-1)}\rho.\label{fr}
\end{eqnarray}
Here $\rho$ is the energy density of cosmic fluid and $dE=d(\rho V)$
is the energy flow pass through the apparent horizon. Note that in
order to get Eq.(\ref{fr}), one should use
\begin{eqnarray}
\dot{\rho}+n H(\rho+p)=0,
\end{eqnarray}
which is the continuity (conservation) equation of the perfect
fluid.
Now we consider the impact of the GUP on thermodynamics of FRW
universe. We consider the case that the apparent horizon having
absorbed or radiated a particle with energy $dE$. As point out
in\cite{gup-hawking1}, one can identify the energy of the absorbed
or radiated particle as the uncertainty of momentum,
\begin{eqnarray}
dE\simeq\delta p.
\end{eqnarray}
By considering the quantum effect of the absorbed or radiated
particle, which implies the Heisenberg uncertainty principle $\delta
p\geq \hbar/\delta x$, the increase or decrease in the area of the
apparent horizon can be expressed as
\begin{eqnarray}
dA=\frac{4G}{T}dE\simeq \frac{4G}{T}\frac{1}{\delta x}.\label{area1}
\end{eqnarray}
In the above we didn't consider the impact of the GUP. When the
effect of the GUP (\ref{gup3}) is considered, the change of the
apparent horizon area can be modified as
\begin{eqnarray}
dA_G=\frac{4G}{T}dE\simeq\frac{4G}{T}\frac{1}{\delta x}f_G(\delta
x^2).\label{area2}
\end{eqnarray}
Using Eq.(\ref{area1}), we have
\begin{eqnarray}
dA_G=f_G(\delta x^2)dA.\label{area3}
\end{eqnarray}
Take into account that the position uncertainty $\delta x$ of the
absorbed or radiated particle can be chosen as its Compton length,
which has the order of the inverse of the Hawking temperature, one
can take\cite{gup-entropy1}
\begin{eqnarray}
\delta x\simeq 2\tilde{r}_A=2(\frac{A}{n\Omega_n})^{\frac{1}{n-1}}.
\end{eqnarray}
Thus, the departure function $f_G(\delta x^2)$ can be re-expressed
in terms of $A$,
\begin{eqnarray}
f_G(A)=\frac{2}{\alpha^2l_p^2}(\frac{A}{n\Omega_n})^{\frac{2}{n-1}}(1-\sqrt{1-\alpha^2l_p^2(\frac{n\Omega_n}{A})^{\frac{2}{n-1}}}).
\end{eqnarray}
Here and hereafter we use $f_G(A)$ represent the departure function
$f_G(\delta x^2)$. At $\alpha=0$, we express $f_G(A)$ by Taylor
series
\begin{eqnarray}
f_G(A)&=&1+\frac{\alpha^2l_p^2}{4}(\frac{n\Omega_n}{A})^{\frac{2}{n-1}}+
\frac{(\alpha^2l_p^2)^2}{8}(\frac{n\Omega_n}{A})^{\frac{4}{n-1}}\nonumber\\
&+&\sum_{d=3}c_d(\alpha
l_p)^{2d}(\frac{n\Omega_n}{A})^{\frac{2d}{n-1}},\label{f}
\end{eqnarray}
where $c_d$ is a constant.
If we substitute (\ref{f}) into Eq.(\ref{area3}) and integrating, we
can get the modified area $A_G$ from the GUP. Then we can also get
the correction to the entropy area relation by using $S_G=A_G/4G$.
But integrating Eq.(\ref{area3}) might be complicated and
dimensional dependent. Therefore, we should divide our discussions
into three cases: (1)$n=3$; (2)$n>3$ and $n$ is a even number;
(3)$n>3$ and $n$ is a odd number.
\subsection{The $n=3$ case}
When $n$=3, we have
\begin{eqnarray}
f_G(A)&=&1+\pi\alpha^2l_p^2\frac{1}{A}+
2(\pi \alpha^2l_p^2)^2\frac{1}{A^2}\nonumber\\
&+&\sum_{d=3}c_d(4\pi\alpha^2 l_p^2)^{2d}\frac{1}{A^d}.\label{f1}
\end{eqnarray}
Substituting (\ref{f1}) into (\ref{area3}) and integrating, we
obtain
\begin{eqnarray}
A_G&=&A+\pi\alpha^2l_p^2lnA-2(\pi\alpha^2l_p^2)^2\frac{1}{A}\nonumber\\
&-&\sum_{d=3}\frac{c_d(4\pi\alpha^2
l_p^2)^{2d}}{d-1}\frac{1}{A^{d-1}}+c
\end{eqnarray}
where $c$ is the integral constant. By making use of
Bekenstein-Hawking area law, $S=A/4G$, we can obtain the expression
of the entropy of the apparent horizon including the effect of the
GUP. That is, the modified entropy is given by
\begin{eqnarray}
S_G&=&\frac{A}{4G}+\frac{\pi\alpha^2l_p^2}{4G}ln\frac{A}{4G}-2(\frac{\pi\alpha^2l_p^2}{4G})^2(\frac{A}{4G})^{-1}\nonumber\\
&-&\sum_{d=3}\frac{c_d(\frac{16\pi^2\alpha^4l_p^4
}{4G})^{d}}{d-1}(\frac{A}{4G})^{1-d}+const.\label{entropy1}
\end{eqnarray}
This relation has the standard form of the entropy-area relation as
given by other approaches in black holes\cite{log1,log2,other1}. The
point which should be stressed here is that the coefficient of the
logarithmic correction term is positive. This is different with the
results in Refs.\cite{log1,log2}. As pointed out in some
literatures\cite{log2}, the coefficient of the logarithmic
correction term is controversial. Our result shows that the
correction to the entropy from the GUP gives an opposite
contribution to the area entropy.
Recently, starting with a modified entropy-area relation, Cai, Cao
and Hu \cite{cao2} have shown that the first law of thermodynamics
on the apparent horizon can produce a modified Friedmann equation.
Now we give the main results of Cai, Cao and Hu's approach and apply
their approach to the case of the modified entropy-area relation
(\ref{entropy1}). Suppose the apparent horizon has an entropy
$S_G(A)$. Applying the first law of thermodynamics to the apparent
horizon of FRW universe, we can obtain the corresponding Friedmann
equations
\begin{eqnarray}
(\dot{H}-\frac{k}{a})S'_G(A)=-\pi(\rho+p),\label{fridemann1}\\
\frac{8\pi G}{3}\rho=-\frac{\pi}{G}\int
S'_G(A)(\frac{4G}{A})^2dA,\label{fridemann2}
\end{eqnarray}
where a prime stands for the derivative with respect to $A$.
Eq.(\ref{fridemann1}) and (\ref{fridemann2}) are nothing but the
modified first and second Friedmann equation corresponding to the
modified apparent horizon entropy $S_G(A)$.
Noticing that $S_G=A_G/4G$ and considering Eq.(\ref{area3}), we can
obtain
\begin{eqnarray}
S'_G(A)=\frac{f_G(A)}{4G}.\label{entropy x}
\end{eqnarray}
Substituting (\ref{f1}) and (\ref{entropy x}) into the modified
Friedmann equation (\ref{fridemann1}) and (\ref{fridemann2}), we can
obtain the modified Friedmann equations after considering the GUP,
that is
\begin{eqnarray}
(\dot{H}-\frac{k}{a})[1+\pi\alpha^2l_p^2\frac{1}{A}+
2(\pi \alpha^2l_p^2)^2\frac{1}{A^2}\nonumber\\
+\sum_{d=3}c_d(4\pi\alpha^2 l_p^2)^{2d}\frac{1}{A^d}]=-4\pi
G(\rho+p),\label{fr3}\\
\frac{8\pi
G}{3}\rho=4\pi[\frac{1}{A}+\frac{1}{2}\alpha^2l_p^2\frac{1}{A^2}+\frac{2}{3}(\pi\alpha^2l_p^2)^2\frac{1}{A^3}\nonumber
\\
+\sum_{d=3}\frac{c_d}{d+1}(4\pi\alpha^2l_p^2)^{2d}\frac{1}{A^{d+1}}].\label{fr4}
\end{eqnarray}
\subsection{$n>3$ and $n$ is an odd number}
When $n$ is an odd number, substituting (\ref{f}) into (\ref{area3})
and integrating, we have
\begin{eqnarray}
A_G&=&A+\sum_{d=1}^{d=\frac{n-3}{2}}c_d(\alpha l_p)^{2d}\frac{n-1}{n-2d-1}A(\frac{n\Omega_n}{A})^{\frac{2d}{n-1}}\nonumber\\
&+&c_{\frac{n-1}{2}}(\alpha l_p)^{n-1}n\Omega_nlnA\nonumber\\
&+&\sum_{d=\frac{n+1}{2}}c_d(\alpha
l_p)^{2d}\frac{n-1}{n-2d-1}A(\frac{n\Omega_n}{A})^{\frac{2d}{n-1}}.
\end{eqnarray}
By making use of Bekenstein-Hawking area law, $S=A/4G$, we can
obtain the expression of the entropy of the apparent horizon after
taking into account the effect of GUP. That is, the correction to
entropy is given by
\begin{eqnarray}
S_G&=&\frac{A}{4G}+\sum_{d=1}^{d=\frac{n-3}{2}}c_d(\alpha
l_p)^{2d}\frac{n-1}{n-2d-1}\frac{A}{4G}(\frac{n\Omega_n}{A})^{\frac{2d}{n-1}}\nonumber\\
&+&c_{\frac{n-1}{2}}\frac{(\alpha l_p)^{n-1}}{4G}n\Omega_nln\frac{A}{4G}\nonumber\\
&+&\sum_{d=\frac{n+1}{2}}c_d(\alpha
l_p)^{2d}\frac{n-1}{n-2d-1}\frac{A}{4G}(\frac{n\Omega_n}{A})^{\frac{2d}{n-1}}\nonumber\\
&+&const.\label{entropy2}
\end{eqnarray}
It is obvious that the logarithmic correction term exists when $n$
is an odd number.
In order to obtain the modified Friedmann equations from the
modified entropy-area relation (\ref{entropy2}) for
$(n+1)$-dimensional FRW spacetime, we have to generalize Cai, Cao
and Hu's approach to a $(n+1)$-dimensional FRW universe, while the
original approach in \cite{cao2} is only valid in
($3+1$)-dimensional FRW universe. The generalization is simple. The
first law of thermodynamics on the apparent horizon $dE=TdS$ leads
to
\begin{eqnarray}
A(\rho+p)H\tilde{r}_Adt=\frac{1}{2\pi \tilde{r}_A}dS_G,\label{first
law}
\end{eqnarray}
here $A(\rho+p)H\tilde{r}_Adt=dE$ is the amount of energy having
crossed the apparent horizon. By way of some simple manipulations,
we can obtain the Friedmann equations in ($n+1$)-dimensional FRW
universe, that is
\begin{eqnarray}
(\dot{H}-\frac{k}{a^2})f_G(A)=-\frac{8\pi
G}{n-1}(\rho+p),\label{fridemann3}\\
\frac{8\pi G}{n}\rho=-\int
f_G(A)(\frac{A}{n\Omega_n})^{\frac{-2}{n-1}}\frac{dA}{A},\label{fridemann4}
\end{eqnarray}
Substituting (\ref{f}) into (\ref{fridemann3}) and
(\ref{fridemann4}), we can obtain the modified Friedmann equations
in ($n+1$)-dimensional FRW spacetime including the consideration of
the GUP,
\begin{eqnarray}
(\dot{H}-\frac{k}{a^2})[1+\frac{\alpha^2l_p^2}{4}(\frac{n\Omega_n}{A})^{\frac{2}{n-1}}+
\frac{(\alpha^2l_p^2)^2}{8}(\frac{n\Omega_n}{A})^{\frac{4}{n-1}}\nonumber\\
+\sum_{d=3}c_d(\alpha
l_p)^{2d}(\frac{n\Omega_n}{A})^{\frac{2d}{n-1}}]=-\frac{8\pi
G}{n-1}(\rho+p),\label{fridemann5}\\
\frac{16\pi
G}{n(n-1)}\rho=(\frac{n\Omega_n}{A})^{\frac{2}{n-1}}\nonumber\\
+ \sum_{d=1}\frac{c_d}{d+1}(\alpha
l_p)^{2d}(\frac{n\Omega_n}{A})^{\frac{2d+2}{n-1}}.\label{fridemann6}
\end{eqnarray}
We note here that the above equations are independent on whether $n$
is an odd or even number. When we take $n=3$, Eq.(\ref{fridemann5})
and (\ref{fridemann6}) reduce to Eq.(\ref{fr3}) and (\ref{fr4})
respectively.
\subsection{$n>3$ and $n$ is an even number}
When $n$ is an even number, following the same route above, we can
obtain the expression of the entropy of the apparent horizon after
taking into account the effect of GUP, which is
\begin{eqnarray}
S_G&=&\frac{A}{4G}+\frac{\alpha^2l_p^2}{4}\frac{n-1}{n-3}\frac{A}{4G}(\frac{n\Omega_n}{A})^{\frac{2}{n-1}}\nonumber\\
&+&\sum_{d=2}c_d(\alpha
l_p)^{2d}\frac{n-1}{n-2d-1}\frac{A}{4G}(\frac{n\Omega_n}{A})^{\frac{2d}{n-1}}.\label{entr}
\end{eqnarray}
From this expression, when $d$ is an even number, the logarithmic
term does not exist in the correction to the entropy of the apparent
horizon of FRW spacetime. This implies that the logarithmic
correction term in the entropy of the apparent horizon is
dimensional dependent.
Since the derivation of the modified Friedmann equations
(\ref{fridemann5},\ref{fridemann6}) in ($n+1$)-dimensional FRW
universe is not relevant to that whether $n$ is an even or odd
number, (\ref{fridemann5},\ref{fridemann6}) are also valid when $n$
is an even. Therefore, the modified Friedmann equations from the
modified entropy (\ref{entr}) are just
Eqs.(\ref{fridemann5},\ref{fridemann6}).
\section{The EUP case}
The GUP is the high energy correction to the conventional Heisenberg
uncertainty relation. In large length scales, the GUP is
unimportant. In this case, one might consider an extension of the
uncertainty relation which contains the effect of the large length
scales. The extended uncertainty principle is given
by\cite{gup-hawking1,eup}
\begin{eqnarray}
\delta x\delta p \geq 1+\beta^2 \frac{\delta x^2}{l^2},\label{eup}
\end{eqnarray}
where $\beta$ is a dimensionless real constant, and $l$ is an
unknown fundamental characteristic large length scale. The EUP
implies that there is a minimal momentum
\begin{eqnarray}
\delta p\geq \frac{1}{\delta x}+\frac{\beta^2}{l^2}\delta x \geq
2|\frac{\beta}{l}|.
\end{eqnarray}
From EUP (\ref{eup}), the uncertainty of momentum can be written as
\begin{eqnarray}
\delta p \geq \frac{1}{\delta x}+\frac{\beta^2}{l^2}\delta
x=\frac{1}{\delta x}f_E(\delta x),\label{eup2}
\end{eqnarray}
where
\begin{eqnarray}
f_E(\delta x)=1+\frac{\beta^2\delta x^2}{l^2}
\end{eqnarray}
is the departure function in EUP case.
Now, with the same approach in Section II, it is easy to obtain the
entropy of the apparent horizon after taking into account the effect
of EUP,
\begin{eqnarray}
S_E=\frac{A}{4G}+\frac{4\beta^2}{
l^2}\frac{n-1}{n+1}(\frac{A}{n\Omega_n})^{\frac{2}{n-1}}\frac{A}{4G}.\label{entropy9}
\end{eqnarray}
The corresponding modified Friedmann equations are expressed as
\begin{eqnarray}
(\dot{H}-\frac{k}{a^2})(1+\frac{4\beta^2}{
l^2}(\frac{A}{n\Omega_n})^{\frac{2}{n-1}})=-\frac{8\pi
G}{n-1}(\rho+p),\nonumber\\
\frac{16\pi
G}{n(n-1)}\rho=(\frac{n\Omega_n}{A})^{\frac{2}{n-1}}+\frac{4\beta^2}{
l^2}ln A.
\end{eqnarray}
When $n=3$, the modified Friedmann equations are
\begin{eqnarray}
(\dot{H}-\frac{k}{a^2})(1+\frac{\beta^2}{\pi l^2}A)=-4\pi
G(\rho+p),\nonumber\\
\frac{8\pi G}{3}\rho=\frac{4\pi}{A}+\frac{4\beta^2}{l^2}lnA.
\end{eqnarray}
Here are some remarks: First, in the derivation of the modified
entropy (\ref{entropy9}), we didn't impose any limit on the
inequality (\ref{eup2}) as in (\ref{f}). This means that the entropy
(\ref{entropy9})is an exact expression. Second, although the
derivation of entropy (\ref{entropy9}) is in the context of the FRW
universe, it is very easy to generalize it to the black hole
physics. For example, with the similar procedure, one can easily
find that the entropy (\ref{entropy9}) is also valid in
Schwarzschild black hole after the consideration of the effect of
EUP.
Third, it is of interest to consider a more general case that
combine both the GUP and the EUP and is named the generalized
extended uncertainty principle (GEUP). The GEUP is given
by\cite{gup-hawking1,eup1}
\begin{eqnarray}
\delta x \delta p\geq 1+\alpha^2l_p^2\delta p^2+\beta^2\frac{\delta
x^2}{l^2}.
\end{eqnarray}
From this expression, it is easy to obtain the uncertainty of
momentum
\begin{eqnarray}
\delta p\geq \frac{1}{\delta x}f_{GE}(\delta x^2),
\end{eqnarray}
where
\begin{eqnarray}
f_{GE}(\delta x^2)=\frac{\delta x^2}{2\alpha^2
l_p^2}[1-\sqrt{1-\frac{4\alpha^2l_p^2}{\delta
x^2}[1+\beta^2\frac{\delta x^2}{l^2}]}]
\end{eqnarray}
is the departure function in the case of the GEUP. By closely
following the procedure in Section II, one can also obtain a
modified entropy of the apparent horizon of the FRW universe and the
corresponding modified Friedmann equations including the effect of
the GEUP.
\section{Summary and Discussions}
In this paper, we have investigated the influence of the GUP and the
EUP on the thermodynamics of the FRW universe. We have shown that
the GUP and EUP contribute corrections to the conventional
entropy-area relation on the apparent horizon of the FRW universe as
well as the Friedmann equations. The later implies that the GUP and
EUP can influent the dynamics of the FRW universe. In particular, in
the case of the GUP, we have shown that the leading logarithmic
correction term exists only for the (odd number+$1$) dimensional FRW
spacetime, and moreover, the leading logarithmic term gives a
positive contribution to the entropy of the apparent horizon. For
(even number+$1$) dimensional FRW spacetime, there is not a
logarithmic correction term in the entropy of the apparent horizon.
It is worthwhile to point out that the results in this paper can be
generalized in some ways. First, it is of interest to search the
statistics meaning of the modified entropy on the apparent horizon
of the FRW universe. In \cite{brick}, using the brick wall method,
the authors have calculated the statistics entropy of a scalar field
in FRW universe. How can one modify their results if one take into
account of the effect of the GUP or EUP? This is an interesting
problem and it needs further investigation.
Second, we have known that the GUP and EUP can modify the dynamics
of the universe. In the early universe, the high energy effects may
be important. This means the GUP may play an important role in the
early time of our universe. On the other hand, as a large length
scale effect, the EUP may be important in the late time universe.
Does the modification of the GUP or the EUP to the Friedmann
equations have some observational effects? How can one probe them?
These problems will be considered in our further workings.
Third, the GUP and the EUP modify the thermodynamics of both black
holes and FRW universe. More generally, it is of great interest to
investigate the influence of the GUP or EUP on the Einstein
equation. The investigation in this direction may provide a deeper
insight into the understanding of the quantum gravity or large
length scale corrections to the classical Einstein gravity.
\begin{acknowledgments}
This work was supported by the National Natural Science Foundation
of China (Grant No. 10275030) and Cuiying Project of Lanzhou
University (Grant No. 225000-582404).
\end{acknowledgments}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 7,847 |
\section{Introduction}
The main impairment of an orthogonal frequency-division multiple-access
(OFDMA) network is represented by its remarkable sensitivity to timing
errors and carrier frequency offsets (CFOs) between the uplink signals and
the base station (BS) local references. For this reason, the IEEE
802.16e-2005 standard for OFDMA-based wireless metropolitan area networks
(WMANs) specifies a synchronization procedure called Initial Ranging (IR)
where subscriber stations that intend to establish a link with the BS can
use some dedicated subcarriers to transmit their specific ranging codes \cite%
{IEEE2006}. Once the BS has revealed the presence of ranging subscriber
stations \ (RSSs), it has to estimate some fundamental parameters including
timing errors, CFOs and power levels.
Two prominent schemes for initial synchronization and power control in OFDMA
were proposed in \cite{Krinock2001} and \cite{Minn2004}. In these works, a
long pseudo-noise sequence is transmitted by each RSS over the available
ranging subcarriers. Timing recovery is then accomplished on the basis of
suitable correlations computed in the frequency- and time-domain,
respectively. The main drawback of these methods is their sensitivity to
multipath distortion, which destroys orthogonality among the employed codes
and gives rise to multiple access interference (MAI). Better results are
obtained in \cite{Lee2005} by using a set of generalized chirp-like (GCL)
sequences, which echibits increased robustness against the channel
selectivity. A different approach to managing the IR process has recently
been proposed in \cite{Minn07}. Here, the pilot streams transmitted by RSSs
are spread in the time-domain over adjacent OFDM blocks using orthogonal
codes. In this way, signals of different RSSs can be easily separated at the
BS as they remain orthogonal after propagating through the channel. Timing
information is eventually acquired in an iterative fashion by exploiting the
autocorrelation properties of the received samples induced by the use of the
cyclic prefix (CP). Unfortunately, this scheme is derived under the
assumption of perfect frequency alignment between the received signals and
the BS local reference. Actually, the occurrence of residual CFOs results
into a loss of orthogonality among ranging codes and may lead to severe
degradations of the system performance in terms of mis-detection probability and
estimation accuracy.
In the present work we propose a novel ranging scheme for OFDMA systems that
is robust to time and frequency misalignments. The goal is to estimate
timing errors and CFOs of all active RSSs. The number of active codes is
found by resorting to the minimum description length (MDL) principle \cite%
{Wax85} while the multiple signal classification (MUSIC) algorithm \cite%
{Schmidt79} is employed to detect which codes are actually active and to
determine their corresponding CFOs. Timing estimation is eventually achieved
through least-squares (LS) methods. Although the proposed solution allows
one to estimate the timing errors of each RSS in a decoupled fashion, it may
involve huge computational burden in applications characterized by large
propagation delays. For this reason, we also present an alternative scheme
derived from ad hoc-reasoning which results into substantial computational
saving. It is worth noting that timing synchronization in OFDMA\ uplink
transmissions has received little attention so far. A well-established way
to handle timing errors is to design the CP length large enough to include
both the channel delay spread and the two-way propagation delay between the
BS and the user station \cite{Morelli2004}. This leads to a
quasi-synchronous system in which timing errors can be viewed as part of the
channel impulse response (CIR) and are compensated for by the channel
equalizer. Unfortunately, this approach poses an upper limit to the maximum
tolerable propagation delay or, equivalently, to the maximum distance
between the BS and the subscriber stations \cite{Pun2007}. For this reason,
its application to a scenario with large cells (as envisioned in next
broadband wireless networks) is hardly viable. In the latter case, accurate
knowledge of the timing errors is required in order to align the uplink
signals to the BS time scale.
\section{System description and signal model}
\subsection{System description}
The investigated OFDMA network employs $N$ subcarriers with frequency
spacing $\Delta f$ and indices in the set $\mathcal{J}=\{0,1,\ldots ,N-1\}$.
Following \cite{Minn07}, we denote $R$ the number of subchannels reserved
for the IR process. Each subchannel is divided into $Q$ subbands uniformly
spaced over the signal bandwidth at a distance $(N/Q)\Delta f$ from each
other. A given subband is composed of a set of $V$ adjacent subcarriers. The
subcarrier indices within the $q$th subband $(q=0,1,\ldots ,Q-1)$ of the $r$%
th ranging subchannel $(r=0,1,\ldots ,R-1)$ are collected into a set $%
\mathcal{J}_{q}^{(r)}=\{i_{q,\nu }^{(r)}~;\nu =0,1,\ldots ,V-1\}$ with
entries
\begin{equation}
i_{q,\nu }^{(r)}=\frac{qN}{Q}+\frac{rN}{QR}+\nu . \label{1}
\end{equation}%
The $r$th subchannel is thus composed of subcarriers with indices taken from
$\mathcal{J}^{(r)}={\cup _{q=0}^{Q-1}\mathcal{J}_{q}^{(r)}}$. Hence, a total
of $N_{R}=QVR$ ranging subcarriers are available in the system with indices
in the set $\mathcal{J}_{R}={\cup _{r=0}^{R-1}\mathcal{J}^{(r)}}$. The
remaining $N-N_{R}$ subcarriers are used for data transmission and are
assigned to data subscriber stations (DSSs) which have already completed
their IR process and are assumed to be perfectly synchronized to the BS time
and frequency scales \cite{Minn07}.
We denote by $M$ the number of consecutive OFDM blocks reserved for IR and
assume that each ranging subchannel can be accessed at most by $M-1$ RSSs.
The latter are separated by means of specific ranging codes selected in a
pseudo-random fashion from a predefined set $\{\mathbf{c}_{1},\mathbf{c}%
_{2},\ldots ,\mathbf{c}_{M-1}\}$, with $\mathbf{c}_{k}=[c_{k}(1),\,c_{k}(2),%
\ldots ,c_{k}(M)]^{T}$ (the superscript $^{T}$ denotes the transpose
operation). As in \cite{Minn07}, we assume that different RSSs employ
different codes. Without loss of generality, in what follows we concentrate
on the $r$th ranging subchannel and denote by $K^{(r)}\leq M-1$ the number
of simultaneously active RSSs. Also, to simplify the notation, the
subchannel index $^{(r)}$ is dropped in all subsequent derivations.
The waveform transmitted by the $k$th RSS ($1\leq k\leq K$) propagates
through a multipath channel characterized by an impulse response $\mathbf{h}%
_{k}=[h_{k}(0),h_{k}(1),\ldots ,h_{k}(L-1)]^{T}$ of length $L$ (in sampling
periods). At the BS, the received samples are not synchronized with the
local references. We denote by $\theta _{k}$ the timing error expressed in
sampling periods while $\varepsilon _{k}$ is the frequency offset normalized
to the subcarrier spacing. As discussed in \cite{Morelli2004}, subscriber
stations that intend to start the ranging process compute initial frequency
and timing estimates on the basis of a downlink control signal broadcast by
the BS. The estimated parameters are then employed by each RSS as
synchronization references for the uplink ranging transmission. This means
that during IR the CFOs are only due to Doppler shifts and/or estimation
errors and, in consequence, they are assumed to lie \textit{within a small
fraction} of the subcarrier spacing.
Furthermore, in order to eliminate interblock interference (IBI), we assume
that during the ranging process the CP length comprises $N_{G}\geq \theta
_{\max }+L$ sampling periods, where $\theta _{\max }$ is the maximum
expected timing error \cite{Pun2007}. This assumption is not restrictive,
since in many standardized OFDM systems the initialization blocks are
usually preceded by long CPs.
\subsection{Signal model}
We denote by $\mathbf{Y}_{m}$ the $QV$-dimensional vector that collects the
DFT outputs corresponding to the considered subchannel during the $m$th OFDM
block. Since the DSSs are assumed to be perfectly synchronized to the BS
references, their signals will not contribute to $\mathbf{Y}_{m}$. In
contrast, the presence of uncompensated CFOs destroys orthogonality among
ranging signals, thereby leading to some interchannel interference (ICI).
However, as the subchannels are well separated in the frequency domain, we
can reasonably neglect interference on $\mathbf{Y}_{m}$ arising from ranging
signals of subchannels other than the considered one. Under this assumption,
we may write
\begin{equation}
\mathbf{Y}_{m}=\sum\limits_{k=1}^{K}{c_{k}(m)e^{j{m\omega _{k}N_{T}}}%
\mathbf{A}(\omega _{k})\mathbf{S}_{k}}(\theta _{k})+\mathbf{n}_{m} \label{2}
\end{equation}%
where $\omega _{k} = 2\pi\varepsilon_k /N$, $N_{T}=N+N_{G}$ is the duration
of the cyclically extended block and $\mathbf{n}_{m}$ is a Gaussian vector
with zero mean and covariance matrix $\sigma ^{2}\mathbf{I}_{QV}$ (we denote
by $\mathbf{I}_{N}$ the identity matrix of order $N$). Also, we have defined
\begin{equation}
\mathbf{A}(\omega _{k})=\mathbf{F}\mathbf{V}(\omega _{k})\mathbf{F}^{H}
\label{3}
\end{equation}%
where $\mathbf{V}(\omega _{k})$ accounts for the CFOs and is given by
\begin{equation}
\mathbf{V}(\omega _{k})=\mathrm{diag}\left\{ {e^{j{n}\omega
_{k}};n=0,1,\ldots ,N-1}\right\} \label{4}
\end{equation}%
while $\mathbf{F}=\left[ {\mathbf{F}_{0}^{H},\mathbf{F}_{1}^{H},\ldots ,%
\mathbf{F}_{Q-1}^{H}}\right] ^{H}$ (the superscript $^{H}$ denotes the
Hermitian transposition) with $\mathbf{F}_{q}$ ($q=0,1,\ldots ,Q-1)$
denoting a $V\times N$ matrix with entries
\begin{equation}
\left[ {\mathbf{F}}_{q}\right] _{\nu ,n}=\frac{1}{\sqrt{N}}e^{-j{2\pi n}%
i_{q,\nu }/N}\quad 0\leq \nu \leq V-1,0\leq n\leq N-1. \label{5}
\end{equation}%
Vector $\mathbf{S}_{k}(\theta _{k})$ in (\ref{2}) can be partitioned as $%
\mathbf{S}_{k}(\theta _{k}) = \left[ \mathbf{S}_{k}^{T}(\theta
_{k},i_{1}),\mathbf{S}_{k}^{T}(\theta _{k},i_{2}),\ldots ,%
\mathbf{S}_{k}^{T}(\theta _{k},i_{Q-1})\right] ^{T}$, where $\mathbf{S}%
_{k}(\theta _{k},i_{q})$ is a $V$-dimensional vector with elements
\begin{equation}
S_{k}(\theta _{k},i_{q,\nu })=e^{-j{2\pi \theta _{k}}i_{q,\nu
}/N}H_{k}(i_{q,\nu }),\quad 0\leq \nu \leq V-1 \label{6}
\end{equation}%
while $H_{k}(i_{q,\nu })$ denotes the channel frequency response over the $%
i_{q,\nu }$th subcarrier and is given by
\begin{equation}
H_{k}(i_{q,\nu })=\sum_{\ell
=0}^{L-1}h_{k}(\ell )e^{-j{2\pi \ell}i_{q,\nu }/N}. \label{7}
\end{equation}%
From (\ref{6}) we see that $\theta _{k}$ simply appears as a phase shift
across the DFT outputs. The reason is that the CP length is larger than the
maximal expected propagation delay, thereby making the ranging signals
quasi-synchronous.
In the following sections we show how vectors $\{\mathbf{Y}%
_{m}\}_{m=0}^{M-1} $ can be exploited to compute frequency and timing
estimates for all active ranging codes.
\section{Estimation of the CFOs}
To simplify the derivation, we assume that the CFOs are adequately smaller
than the subcarrier spacing, i.e., $\left\vert {\omega _{k}}\right\vert \ll
1 $. In such a case, matrices $\mathbf{A}(\omega _{k})$ in (\ref{3}) can
reasonably be replaced by $\mathbf{I}_{N}$ to obtain \cite{Morelli2004}
\begin{equation}
\mathbf{Y}_{m}\approx \sum\limits_{k=1}^{K}{c_{k}(m)e^{j{m\omega _{k}N_{T}}%
}\mathbf{S}_{k}}(\theta _{k})+\mathbf{n}_{m}. \label{9}
\end{equation}%
This equation indicates that each CFO results only in a phase shift between
contiguous OFDM blocks. Collecting the $i_{q,\nu }$th DFT output of all $M$
ranging blocks into a vector $\mathbf{Y}(i_{q,\nu })=[Y_{0}(i_{q,\nu
}),Y_{1}(i_{q,\nu }),\ldots ,Y_{M-1}(i_{q,\nu })]^{T}$, we may write
\begin{equation}
\mathbf{Y}(i_{q,\nu })=\sum_{k=1}^{K}S_{k}(\theta _{k},i_{q,\nu })\mathbf{%
\Gamma }(\omega _{k})\mathbf{c}_{k}+\mathbf{n}(i_{q,\nu }) \label{10}
\end{equation}%
where $\mathbf{n}(i_{q,\nu })$ is Gaussian distributed with zero-mean and
covariance matrix $\sigma ^{2}\mathbf{I}_{M}$, while $\mathbf{\Gamma }%
(\omega _{k})=\text{diag}\{e^{j{m\omega _{k}N_{T}}}~;m=0,1,\ldots ,M-1\}$
is a diagonal matrix that accounts for the phase shifts induced by $\omega
_{k}$.
Inspection of (\ref{10}) reveals that, apart from thermal noise, vector $%
\mathbf{Y}(i_{q,\nu })$ is a linear combination of the frequency-rotated
codes $\{\mathbf{\Gamma }(\omega _{k})\mathbf{c}_{k}\}$. This means that the
signal space is spanned by the $K$ vectors $\{\mathbf{\Gamma }(\omega _{k})%
\mathbf{c}_{k}\}$ that correspond to the active RSSs \cite{StoicaBook97}.
Then, if we temporarily assume that the number $K$ of active codes is known
at the receiver, an estimate of $\omega _{k}$ ($k=1,2,\ldots ,K$) can be
obtained by resorting to the MUSIC algorithm \cite{Schmidt79}. To see how
this comes about, we use the observations $\{\mathbf{Y}(i_{q,\nu })\}$ to
obtain the following sample correlation matrix
\begin{equation}
\hat{\mathbf{R}}_{Y}=\frac{1}{QV}\sum_{v=0}^{V-1}\sum_{q=0}^{Q-1}\mathbf{Y}%
(i_{q,\nu })\mathbf{Y}^{H}(i_{q,\nu }). \label{14}
\end{equation}%
Next, based on the forward-backward (FB) approach \cite{StoicaBook97}, we
compute
\begin{equation}
\widetilde{\mathbf{R}}_{Y}=\frac{1}{2}(\hat{\mathbf{R}}_{Y}+\mathbf{J}\hat{%
\mathbf{R}}_{Y}^{T}\mathbf{J}) \label{15}
\end{equation}%
where $\mathbf{J}$ is the exchange matrix with 1's on the anti-diagonal and
0's elsewhere. We denote by $\lambda _{1}\geq \lambda _{2}\geq \cdots \geq
\lambda _{M}$ the eigenvalues of $\widetilde{\mathbf{R}}_{Y}$ arranged in
non-increasing order, and by $\{\mathbf{s}_{1},\mathbf{s}_{2},\ldots ,%
\mathbf{s}_{M}\}$ the corresponding eigenvectors. The MUSIC algorithm relies
on the fact that the eigenvectors associated with the $M-K$ smallest
eigenvalues are an estimated basis of the noise subspace and, accordingly,
they are approximately orthogonal to all vectors in the signal space \cite%
{Schmidt79}. Hence, an estimate of $\omega _{k}$ is obtained by minimizing
the projection of $\mathbf{\Gamma }(\widetilde{\omega })\mathbf{c}_{k}$ onto
the noise subspace, i.e.,
\begin{equation}
\hat{\omega }_{k}=\arg \underset{\widetilde{\omega }}{\max }\left\{ \Psi
_{k}(\widetilde{\omega })\right\}, \label{16}
\end{equation}%
with
\begin{equation}
\Psi _{k}(\widetilde{\omega })=\frac{1}{\sum_{m=K+1}^{M}\left\vert \mathbf{c}%
_{k}^{H}\mathbf{\Gamma }^{H}(\widetilde{\omega })\mathbf{s}_{m}\right\vert
^{2}}. \label{17}
\end{equation}%
It is worth observing that CFO recovery must be accomplished for any active
RSS. However, since the BS has no prior knowledge as to which codes have
been transmitted in the considered subchannel, it must evaluate the
quantities $\{\hat{\omega}_{{1}},\hat{\omega}_{{2}},\ldots ,\hat{\omega}_{{%
M-1}}\}$ for the complete set $\{\mathbf{c}_{1},\mathbf{c}_{2},\ldots ,%
\mathbf{c}_{M-1}\}$. At this stage the problem arises of identifying which
codes are actually active. The identification algorithm looks for the $K$
largest values in the set $\{\Psi _{k}(\hat{\omega}_{k})\}_{k=1}^{M-1}$, say
$\{\Psi _{u_{k}}(\hat{\omega}_{u_{k}})\}_{k=1}^{K}$, and declare as \textit{%
active} the corresponding codes $\{\mathbf{c}_{u_{k}}\}_{k=1}^{K}$. The CFO
estimates are eventually found as $\hat{\boldsymbol{\omega }}_{u}=\mathbf{[}%
\hat{\omega}_{u_{1}},\hat{\omega}_{u_{2}},\ldots ,\hat{\omega}_{u_{K}}%
\mathbf{]}^{T}$.
At this stage we are left with the problem of estimating the parameter $K$
to be used in (\ref{17}). For this purpose, we adopt the\ MDL approach and
obtain \cite{Wax85}
\begin{equation}
\hat{K}=\arg \underset{\tilde{K}}{\min }\left\{ \mathcal{F(}\tilde{K}%
)\right\} \label{18}
\end{equation}%
where $\mathcal{F(}\tilde{K})$ is the following metric
\begin{equation}
\mathcal{F(}\tilde{K})=\frac{1}{2}\tilde{K}(2M-\tilde{K})\ln (QV) -QV(M-%
\tilde{K})\ln \rho (\tilde{K}) \label{19}
\end{equation}%
with $\rho (\tilde{K})$ denoting the ratio between the geometric and
arithmetic mean of $\{\lambda _{\tilde{K}+1},\lambda _{\tilde{K}+2},\ldots
,\lambda _{M}\}$. Finally, replacing $K$ by $\hat{K}$ in (\ref{17}) leads to
the proposed MUSIC-based frequency estimator (MFE) while the described
identification algorithm is called the MUSIC-based code detector (MCD)
\section{Estimation of the timing delays}
After code detection and CFO recovery, the BS must acquire information about
the timing delays of all ranging signals. This problem is
now addressed by resorting to LS methods. In doing so we still assume that
the number of active codes has been correctly estimated so that $\hat{K}=K$.
Also, to simplify the notation, the indices $\{u_k\}_{k=1}^{K}$ of the
detected codes are relabeled following the map $u_k\longrightarrow k $ for $%
k=1,2,\ldots ,K$.
We begin by reformulating (\ref{10}) in a more compact form. For this
purpose, we collect the CFOs and timing errors in two $K$-dimensional
vectors ${\boldsymbol{\omega }=[}\omega _{1},\omega _{2},\ldots ,\omega _{K}%
\mathbf{]}^{T}$ and $\boldsymbol{\theta }=[\theta _{1},\theta _{2},\ldots
,\theta _{K}]^{T}$. Then, after defining the matrix $\mathbf{C}({\boldsymbol{%
\omega }})=\left[ \mathbf{\Gamma }(\omega _{{1}})\mathbf{c}_{{1}}~~\mathbf{%
\Gamma }(\omega _{{2}})\mathbf{c}_{{2}}~\cdots ~\mathbf{\Gamma }(\omega _{{K}%
})\mathbf{c}_{{K}}\right] $ and the vector $\mathbf{S}(\boldsymbol{\theta }%
,i_{q,\nu })=[S_{{1}}(\theta _{1},i_{q,\nu }),S_{{2}}(\theta _{2},i_{q,\nu
}),\ldots ,S_{{K}}(\theta _{K},i_{q,\nu })]^{T}$, we may rewrite (\ref{10})
in the equivalent form
\begin{equation}
\mathbf{Y}(i_{q,\nu })=\mathbf{C}(\boldsymbol{\omega })\mathbf{S}(%
\boldsymbol{\theta },i_{q,\nu })+\mathbf{n}(i_{q,\nu }). \label{13}
\end{equation}%
Omitting for simplicity the functional dependence of $\mathbf{S}(\boldsymbol{%
\theta },i_{q,\nu })$ on $\boldsymbol{\theta }$ and assuming $\hat{%
\boldsymbol{\omega }}\approx \boldsymbol{\omega }$, from (\ref{13}) the
maximum likelihood estimate of $\mathbf{S}(i_{q,\nu })$ is found to be
\begin{equation}
\hat{\mathbf{S}}(i_{q,\nu })=[\mathbf{C}^{H}(\hat{\boldsymbol{\omega }})%
\mathbf{C}(\hat{\boldsymbol{\omega }})]^{-1}\mathbf{C}^{H}(\hat{\boldsymbol{%
\omega }})\mathbf{Y}(i_{q,\nu }). \label{18}
\end{equation}%
Substituting (\ref{13}) into (\ref{18}) yields
\begin{equation}
\hat{\mathbf{S}}(i_{q,\nu })=\mathbf{S}(i_{q,\nu })+\boldsymbol{\xi }%
(i_{q,\nu }) \label{19}
\end{equation}%
where $\boldsymbol{\xi }(i_{q,\nu })$ is a zero-mean disturbance term. From (%
\ref{6}) and (\ref{7}) it follows that
\begin{equation} \label{20}
\hat{S}_{{k}}(i_{q,\nu })=e^{-j\frac{{2\pi \theta_{k}}}{N}i_{q,\nu
}}\sum_{\ell =0}^{L-1}h_{k}(\ell )e^{-j\frac{{2\pi n}}{N} i_{q,\nu }}+\xi _{{%
k}}(i_{q,\nu }).
\end{equation}%
On denoting $\hat{\mathbf{S}}_{{k}}(\nu)=\left[ \hat{S}_{{k}}(i_{0,\nu }),%
\hat{S}_{{k}}(i_{1,\nu }),\ldots ,\hat{S}_{{k}}(i_{Q-1,\nu })\right] ^{T}, $
and $\boldsymbol{\Phi }(\theta _{{k}},\nu)=\mathrm{diag}\{e^{-j\frac{{2\pi
\theta _{{k}}}}{N}i_{q,\nu }}~;~q=0,1,\ldots ,Q-1\}$, we may rewrite (\ref%
{20}) as follows
\begin{equation}
\hat{\mathbf{S}}_{{k}}(\nu)=\boldsymbol{\Phi }(\theta _{{k}},\nu)\mathbf{F}%
(\nu)\mathbf{h}_{{k}}+\boldsymbol{\xi }_{{k}}(\nu) \label{21}
\end{equation}%
where $\boldsymbol{\xi }_{{k}}(\nu)=[\xi _{{k}}(i_{0,\nu }),\xi _{{k}%
}(i_{1,\nu }),\ldots ,\xi _{{k}}(i_{Q-1,\nu })]^{T}$ while $\mathbf{F}(\nu)$
is a matrix of dimension $Q\times L$ with entries $\lbrack \mathbf{F}%
(\nu)]_{q,\ell }= e^{-j\frac{{2\pi \ell }}{N}i_{q,\nu }}$ for $0\leq q\leq
Q-1$ and $0\leq \ell \leq L-1$
Equation (\ref{21}) indicates that, apart from the disturbance term $%
\boldsymbol{\xi }_{{k}}(\nu )$, $\hat{\mathbf{S}}_{{k}}(\nu )$ is only
contributed by the ${k}$th RSS, meaning that ranging signals have been
successfully decoupled at the BS. We may thus exploit vectors $\{\hat{%
\mathbf{S}}_{{k}}(\nu )~;~\nu =0,1,\ldots ,V-1\}$ to get LS estimates of $%
(\theta _{{k}},\mathbf{h}_{{k}})$ separately for each RSS. This amounts to
minimizing the following objective function with respect to $(\tilde{\theta}%
_{{k}},\tilde{\mathbf{h}}_{{k}})$
\begin{equation}
\Lambda _{{k}}(\tilde{\theta}_{{k}},\tilde{\mathbf{h}}_{{k}%
})=\sum_{\nu=0}^{V-1}\left\Vert \hat{\mathbf{S}}_{{k}}(\nu) - \boldsymbol{%
\Phi }(\tilde{\theta}_{{k}},\nu)\mathbf{F}(\nu)\tilde{\mathbf{h}}_{{k}%
}\right\Vert ^{2}. \label{23}
\end{equation}%
For a fixed $\tilde{\theta}_{{k}}$, the minimum of $\Lambda _{{k}}(\tilde{%
\theta}_{{k}},\tilde{\mathbf{h}}_{{k}})$ is achieved at
\begin{equation}
\hat{\mathbf{h}}_{{k}}=\frac{1}{QV}\sum_{\nu=0}^{V-1}\mathbf{F}^{H}(\nu)%
\boldsymbol{\Phi }^{H}(\tilde{\theta}_{{k}},\nu)\hat{\mathbf{S}}_{{k}}(\nu)
\label{24}
\end{equation}%
where we have used the identity $\mathbf{F}^{H}(\nu)\mathbf{F}(\nu)=Q \cdot
\mathbf{I}_{L}$. Then, substituting (\ref{24}) into (\ref{23}) and
minimizing with respect to $\tilde{\theta}_{k}$ yields the timing estimate
in the form
\begin{equation}
\hat{\theta}_{k}=\arg \underset{0\leq \tilde{\theta}_{k}\leq \theta _{\max }}%
{\max }\left\{ \Upsilon (\tilde{\theta}_{k})\right\} \label{25}
\end{equation}%
where $\Upsilon (\tilde{\theta}_{k})$ is given by
\begin{equation}
\Upsilon (\tilde{\theta}_{k})=\sum_{\ell =\tilde{\theta}_{k}}^{\tilde{\theta}%
_{k}+L-1}\left\vert \sum_{\nu=0}^{V-1}\hat{s}_{k}(\nu,\ell )e^{j2\pi \ell
\nu/N}\right\vert ^{2} \label{n32/4}
\end{equation}%
and we have denoted by $\hat{s}_{k}(\nu,\ell )$ the $Q$-point IDFT of the
sequence $\{\hat{S}_{k}(i_{q,\nu });0\leq q\leq Q-1\}$. In the sequel $\hat{%
\theta}_{k}$ is termed the LS-based timing estimator (LS-TE).
Once $\hat{\theta}_{k}$ has been computed from (\ref{25}), it is used in (%
\ref{24}) to estimate the CIR of the $k$th RSS as
\begin{equation}
\hat{\mathbf{h}}_{k}=\frac{1}{QV}\sum_{\nu=0}^{V-1}\mathbf{F}^{H}(\nu)%
\boldsymbol{\Phi }^{H}(\hat{\theta}_{k},\nu)\hat{\mathbf{S}}_{k}(\nu).
\label{24.1}
\end{equation}%
It is worth noting that for $V=1$ the timing metric (\ref{n32/4}) reduces to
\begin{equation}
\left. \Upsilon (\tilde{\theta}_{k})\right\vert _{V=1}=\sum_{\ell =\tilde{%
\theta}_{k}}^{\tilde{\theta}_{k}+L-1}\left\vert \hat{s}_{k}(0,\ell
)\right\vert ^{2} \label{n33}
\end{equation}%
and becomes periodic in $\tilde{\theta}_{k}$ with period $Q$. In such a
case, the estimate $\hat{\theta}_{k}$ is affected by an ambiguity of
multiples of $Q$. This ambiguity does not represent a serious problem as
long as $Q$ can be chosen to be greater than $\theta _{\max }$.
Unfortunately, in some applications this may not be the case. For example,
in \cite{Minn07} we have $Q=8$ while $\theta _{\max }=102$.
\subsection{Reduced-complexity timing estimation}
Although separating the RSS signals at the BS considerably reduces the
system complexity, evaluating $\Upsilon (\tilde{\theta}_{k})$ for $\tilde{%
\theta}_{k}=0,1,\ldots ,\theta _{\max }$ may still be computationally
demanding, especially in applications where $\theta _{\max }$ is large. For
this reason, we now develop an ad-hoc reduced complexity timing estimator
(RC-TE).
We begin by decomposing the timing error $\theta _{k}$ into a fractional
part $\beta _{k}$, less than $Q$, plus an integer part which is multiple of $%
Q$, i.e.,
\begin{equation}
\theta _{k}=\beta _{k}+p_{k}Q \label{n34}
\end{equation}%
where $\beta _{k}\in \{0,1,\ldots ,Q-1\}$ while $p_{k}$ is an integer
parameter taken from $\{0,1,\ldots ,P-1\}$ with $P=\left\lfloor \theta
_{\max }/Q\right\rfloor $. Omitting the details, it is possible to rewrite $%
\Upsilon (\tilde{\theta}_{k})$ as
\begin{equation}
\Upsilon (\tilde{\theta }_{k})=\Upsilon _{1}(\tilde{\beta }_{k})+\Upsilon
_{2}(\tilde{\beta }_{k},\tilde{p}_{k}) \label{n35}
\end{equation}%
where
\begin{equation}
\Upsilon _{1}(\tilde{\beta }_{k})=\sum_{\ell =\tilde{\beta }_{k}}^{\tilde{%
\beta }_{k}+L-1}\sum_{\nu=0}^{V-1}\left\vert \hat{s}_{k}(\nu,\ell
)\right\vert ^{2} \label{n36}
\end{equation}
while $\Upsilon _{2}(\tilde{\beta }_{k},\tilde{p}_{k})$ is shown in (\ref%
{n36bis}) at the top of the next page.
\begin{figure*}[htp]
\begin{equation}
\Upsilon _{2}(\tilde{\beta }_{k},\tilde{p}_{k})=2\Re e \left\{\sum_{\ell =%
\tilde{\beta }_{k}}^{\tilde{\beta }_{k}+L-1}\sum_{\nu=0}^{V-2}\sum_{n=1}^{V-%
\nu-1}\hat{s}_{k}(\nu,\ell )\hat{s}_{k}^{\ast }(\nu+n,\ell )e^{-j2\pi n(\ell
+\tilde{p}_{k}Q)/N}\right\} \label{n36bis}
\end{equation}%
\end{figure*}
The RC-TE is a suboptimal scheme which, starting from (\ref{n35}), estimates
$\beta _{k}$ and $p_{k}$ in a decoupled fashion. More precisely, an estimate
of $\beta _{k}$ is first obtained looking for the maximum of $\Upsilon _{1}(%
\tilde{\beta}_{k})$, i.e.,
\begin{equation}
\hat{\beta}_{k}=\arg \underset{0\leq \tilde{\beta}_{k}\leq Q-1}{\max }%
\left\{ \Upsilon _{1}(\tilde{\beta}_{k})\right\} . \label{n40}
\end{equation}%
Next, replacing $\beta _{k}$ with $\hat{\beta}_{k}$ in the right-hand-side
of (\ref{n35}) and maximizing with respect to $\tilde{p}_{k}$, provides an
estimate of $p_{k}$ in the form
\begin{equation}
\hat{p}_{k}=\arg \underset{0\leq \tilde{p}_{k}\leq P-1}{\max }\left\{
\Upsilon _{2}(\hat{\beta }_{k},\tilde{p}_{k})\right\}. \label{n41}
\end{equation}
A further reduction of complexity is possible when $V=2$. Actually, in this
case it can be shown that maximizing $\Upsilon _{2}(\hat{\beta}_{k},\tilde{p}%
_{k})$ is equivalent to maximizing $\cos (\varphi _{k}-2\pi \tilde{p}%
_{k}Q/N) $, where
\begin{equation}
\varphi _{k}=\arg \left\{ \sum_{\ell =\hat{\beta}_{k}}^{\hat{\beta}_{k}+L-1}%
\hat{s}_{k}(0,\ell )\hat{s}_{k}^{ \ast }(1,\ell )e^{-j2\pi \ell /N}\right\} .
\label{n44}
\end{equation}%
The estimate of $p_{k}$ is thus obtained in closed-form as
\begin{equation}
\hat{p}_{k}=\frac{N\varphi _{k}}{2\pi Q}. \label{n45}
\end{equation}
\section{Simulation results}
\subsection{System parameters}
The simulated system is inspired by \cite{Minn07}. The total number of
subcarriers is $N=1024$ while the number of ranging subchannels is $R=8$.
Each subchannel is composed by $Q=16$ subbands uniformly spaced at a
distance $N/Q=64$. Any subband comprises $V=2$ adjacent subcarriers while
the ranging time-slot includes $M=4$ OFDM blocks. The ranging codes are
taken from a Fourier set of length $4$ and are randomly selected by the RSSs
at each simulation run (expect for the sequence $\left[ 1,1,1,1\right] ^{T}$%
). The discrete-time CIRs have $L=12$ channel coefficients. The latter are
modeled as circularly symmetric and independent Gaussian random variables
with zero means (Rayleigh fading) and exponential power delay profiles,
i.e., $E\{\left\vert h_{k}(\ell )\right\vert ^{2}\}=\sigma _{h}^{2}\cdot
\exp (-\ell /12)$ with $\ell =0,1,\ldots ,11$ and $\sigma _{h}^{2}$ chosen
such that $E\{\left\Vert \mathbf{h}_{k}\right\Vert ^{2}\}=1$. Channels of
different users are assumed to be statistically independent. They are
generated at each new simulation run and kept fixed over an entire
time-slot. The normalized CFOs are uniformly distributed within the interval
$[-\Omega ,\Omega ]$ and vary at each simulation run. We consider a cell
radius of 10 km, corresponding to a maximum transmission delay $\theta
_{\max }=204$. A CP of length $N_{G}=256$ is chosen to avoid IBI.
\begin{figure}[t]
\begin{center}
\includegraphics[width=.45\textwidth]{picture1.eps}
\end{center}
\caption{$P_{f}$ vs. SNR for $K$ = 2 or 3 when $\Omega$ is 0.05 or 0.075.}
\label{picture1}
\end{figure}
\subsection{Performance evaluation}
We begin by investigating the performance of MCD in terms of probability of
making an incorrect detection, say $P_{f}$. This parameter is illustrated in
Fig. 1 as a function of SNR = $1/ \sigma^2$ under different operating
conditions. The number of active RSSs varies from 2 to 3 while the maximal
frequency offset is either $\Omega =0.05$ or $0.075$. Comparisons are made
with the ranging scheme discussed by Fu, Li and Minn (FLM) in \cite{Minn07},
where the $k$th ranging code is declared active provided that the quantity
\begin{equation}
{\mathcal{Z}}_{k}=\frac{1}{M^2}\sum\limits_{q=0}^{Q-1}{\sum\limits_{%
\nu=0}^{V-1}{\left\vert {\mathbf{c}_{k}^{H}\mathbf{Y}(i_{q,\nu})}\right\vert
^{2}}}
\end{equation}%
exceeds a suitable threshold $\eta $ which is proportional to the estimated
noise power $\hat{\sigma}^{2}$. The results of Fig. 1 indicates that
the proposed scheme performs remarkably better than FLM because of its
intrinsic robustness against CFOs. As expected, the system performance
deteriorates for large values of $K$ and $\Omega $. The reason is that
increasing $K$ reduces the dimensionality of the noise subspace, which
degrades the accuracy of the MUSIC estimator. Furthermore, large CFO values
result into significant ICI which is not accounted for in the signal model (%
\ref{9}), where $\mathbf{A}(\omega _{k})$ has been replaced by $\mathbf{I}%
_{N}$.
\begin{figure}[t]
\begin{center}
\includegraphics[width=.45\textwidth]{picture5.eps}
\end{center}
\caption{Accuracy of the frequency estimates vs. SNR for $K$ = 2 or 3 when $\Omega$ is 0.05 or 0.075. }
\label{picture1}
\end{figure}
\begin{figure}[t]
\begin{center}
\includegraphics[width=.45\textwidth]{picture7.eps}
\end{center}
\caption{$P(\protect\epsilon)$ vs. SNR for $K$ = 2 or 3 when $\Omega$ is
0.05 or 0.075.}
\label{picture1}
\end{figure}
Fig. 2 illustrates the root mean-square-error (RMSE) of the frequency
estimates obtained with MFE vs. SNR. Again, we see that the system
performance deteriorates when $K$ and $\Omega $ are relatively large.
Nevertheless, the accuracy of MFE is satisfactory under all investigated
conditions.
The performance of the timing estimators is measured in terms of probability
of making a timing error, say $P(\epsilon )$, as defined in \cite%
{Morelli2004}. More precisely, an error event is declared to occur whenever
the estimate $\hat{\theta}_{k}$ gives rise to IBI during the data section of
the frame. In such a case, the quantity $\hat{\theta}%
_{k}-\theta _{k}+(-N_{G,D}+L)/2$ is larger than zero or smaller than $%
-N_{G,D}+L-1$, where $N_{G,D}$ is the CP length during the data transmission
phase. Fig. 3 illustrates $%
P(\epsilon )$ vs. SNR as obtained with RC-TE and FLM when $N_{G,D}=64$. The operating
conditions are the same of the previous figures. Since the performance of
LS-TE is virtually identical to that of RC-TE, it is not reported in order
not to overcrowd the figure. We see that for SNR values larger than $6$ dB the proposed scheme provides
much better results than FLM.
\section{Conclusions}
We have derived a novel timing and frequency synchronization scheme for
initial ranging in OFDMA-based networks. The proposed solution aims at
detecting which codes are actually being employed and provides timing and
CFO estimates for all active RSSs. CFO estimation is accomplished by
resorting to the MUSIC algorithm while a LS approach is employed for timing
recovery. Compared to the timing synchronization algorithm discussed in \cite%
{Minn07}, the proposed scheme is more robust to frequency misalignments and
exhibits improved accuracy.
\bibliographystyle{IEEEtran}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 6,647 |
\section*{1. Introduction}
Understanding the detailed structure of disordered metallic alloys continues to
be an experimental and theoretical challenge. Particularly interesting
are alloys in which there is a significant mismatch in atomic size.
Intuitively, it might be expected that the distances between nearest neighbor
pairs in an alloy with large atoms (A) and small atoms (B) would follow
a
simple relationship $r_{AA} > r_{AB}
> r_{BB}$. However, x-ray and neutron diffuse scattering experiments
indicate that the behavior is more complex and that there are correlations
between chemical ordering tendencies and pair distances (Jiang, Ice, Sparks, Robertson and Zschack 1996).
Unfortunately, our general understanding
is limited by the statistical nature of the scattering process --
typically only average pair distances and chemical coordination numbers can be
determined
by these methods. They do not provide information about the relationship
between local displacements from ideal lattice sites and the local degree
of short-range order.
Using an effective medium theory (EMT) energy calculation
(Jacobsen, Norskov and Puska 1987), we
present here a detailed Monte Carlo (MC) study of the correlation between
local order and displacements in
equiatomic disordered CuAu (12\% size difference). EMT is a semi-empirical
approach belonging to the category of embedded atom methods, and is based on
Density Functional Theory (Jacobsen et al. 1987) .
These simulations offer the advantage
of giving access to individual atoms and their local chemical environment,
thus providing information complementary to experiment.
They exhibit clear correlations between local chemical environment and
displacements in disordered CuAu alloy.
At 50-50 at.\% concentration CuAu has two first order phase transitions.
Above 683K the stable phase is a FCC disordered phase. Below 658K the stable
phase CuAuI is an ordered $L1_0$ phase with a 7\% tetragonal distortion.
Between those two temperatures the stable phase is a long-period superlattice,
CuAuII, with a modulation wavevector perpendicular to the ordering wavevector.
The wavelength of the periodic anti-phase boundaries is ten times
the size of the underlying ordered cell.
\section*{2. Monte Carlo Simulations Based on Effective Medium Theory}
\subsection*{2.1. Simulation Model Details}
First principles calculations of the effects of atomic displacements on alloy
phase stability have often been based on effective Ising models where the positional
degrees of freedom are integrated out at zero temperature
(Lu, Laks, Wei and Zunger 1994, Wolverton and Zunger 1995).
Finite temperature simulations involving both configurational and positional
variables have been possible only with computationally efficient empirical
potentials (Dunweg and Landau 1993, Polatoglou and Bleris 1994, Silverman,
Zunger, Kalish and Adler 1995).
The advantage of a direct simulation over simulations based on effective Ising
models is that they yield detailed information about the nature of atomic
relaxations and the coupling between configurational and positional degrees of
freedom: information which is hidden in higher order correlation functions of
effective Ising models.
These considerations led us to investigate the role of atomic displacements in
Cu-Au alloys using the EMT formalism. Though EMT is not a first-principles
approach,
it is a useful model which captures some essential features of
metallic binding.
A number of researchers have found that EMT can reproduce
well the bulk and surface properties of many pure metals (Jacobsen 1988).
EMT-based calculations have also been successfully applied
to investigate equilibrium and kinetics properties of Cu-Au alloys.
Monte Carlo simulations using the EMT have correctly reproduced the
order-disorder transitions in the Cu-Au phase diagram
(Xi, Chakraborty, Jacobsen and Norskov 1992). This work
illustrated the importance of going beyond cubic fixed-lattice models
to include tetragonal distortions (Xi et al. 1992) and atomic
displacements (Chakraborty 1995). In addition, EMT has been used to construct
an atomistic Landau theory of the alloy (Chakraborty and Xi 1992)
which was able to predict qualitatively the stability
of the modulated phase CuAuII in a narrow temperature range. Langevin
simulations
of the ordering kinetics using the Landau model were
recently found to be in qualitative agreement with experimental x-ray
results (Elder, Malis, Ludwig, Chakraborty and Goldenfeld 1998).
The EMT formalism provides a structure for systematically constructing
interatomic potentials in metallic systems where simple pair potentials are
known to be inadequate. These potentials belong to the same category as
embedded atom potentials (Daw, Foiles and Baskes 1993), however, the details
of the construction are different (Jacobsen et al. 1987). The EMT interactions
involve parameters characterizing
atoms in specific environments; for example, the spatial extent of the
electron density distribution around a Cu atom
embedded in an electron gas of given density (Jacobsen et al. 1987). In
principle, these parameters can be obtained from {\it ab initio} calculations
based on DFT. However, in its application to the Cu-Au alloys
(Xi et al. 1992), a semi-empirical
approach has been adopted where the atomic parameters are obtained from fitting
to the ground-state properties of the {\it pure} metals and an additional
parameter, which enters only in the description of an alloy, is obtained by
fitting to the formation energy of CuAu (Stoltze 1997).
We performed MC simulations of CuAu in the canonical ensemble using the EMT
approach.
In our MC implementation the Metropolis algorithm is used to determine
the acceptance/rejection of 3 different kinds of system changes:
the interchange of Cu and Au atoms, the displacement of individual atoms
from their ``ideal'' lattice sites, and the size of the global lattice
constants.
The candidate atomic displacements are chosen randomly in a box whose
size is adjusted to optimize the MC acceptance rate.
The atomic displacements are due both to thermal vibrations and to ``size
effects''.
The simulations are performed using a modified version of the ARTwork simulation
package (Stoltze 1997) running on a Silicon Graphics Origin2000 computer system.
One full MC step for an $N$-atom simulation cell ($32^3$ or $60^3$)
consists of $N$ attempts to (a) randomly exchange atoms, and (b) change
the position of the atoms involved in the exchange followed by
one attempted change in global lattice constants. When examining properties
of the disordered phase the cubic symmetry is fixed. Typically, the alloy is
equilibrated at temperature for 1000 MC steps before the data is stored
for processing. The averages are usually taken over 50 configurations
saved every 10 MC steps.
\subsection*{2.2. Model Accuracy}
Before discussing correlations between local chemical order and atomic
displacements in our Monte-Carlo simulations of disordered CuAu, we examine
the accuracy of the EMT model. The model predicts the correct Cu$_3$Au, CuAu
and Au$_3$Cu regions of the phase diagram.
For the 50-50 composition studied in detail
here, previously published EMT simulations which did not allow atomic
displacements found (Xi et al. 1992) that
the model presents a first order phase transition at
approximately 708K between the cubic disordered phase and the tetragonal
ordered phase. The ratio of the lattice constants in the ordered phase $c/a$ is
0.94 and agrees well with the actual value of 0.93. In the new simulations
reported here, atomic displacements are allowed, but the transition remains
first order in accord with experiment. The jump of the long-range order
parameter at the phase boundary is 0.94. In addition, the ratio of the
lattice constants $c/a$ in the ordered phase
remains 0.94. However the transition temperature is considerably lowered
by the inclusion of atomic displacements to approximately 430K. Since
displacements due to both thermal vibrations and the ``size effect''
would be expected to preferentially lower the free-energy of the
disordered phase, the drop in transition temperature is not surprising.
Thus the inclusion of atomic displacements decreases
the agreement with the experimental transition temperatures when
using the parameters of Xi et al. 1992.
As mentioned above the alloy parameter entering the EMT model was obtained
by fitting to the CuAu formation energy without incorporating displacements.
Therefore it would be possible to improve the
current model by adjusting the EMT parameters. However,
for consistency with the previously published simulations, we chose to keep
the parameters from Xi et al. and to scale all temperatures
with respect to the new transition temperature.
The local chemical order and atomic displacements in disordered alloys are often
measured by diffuse x-ray or neutron scattering. For comparison, we calculated
the x-ray diffuse scattering intensity for the EMT model by Fourier
transformation for several effective temperatures just above the phase
transition. A typical diffuse scattering in the $(hk0)$ reciprocal plane
calculated from
a simulation 2.7\% above the transition temperature is presented in
Fig.\ref{diffuse}. The asymmetry of the diffuse scattering around the
superlattice points is due to atomic displacements (Borie and Sparks 1971).
Though not easy to see on the scale of the plot, the
simulated peaks exhibit the anisotropic
four-fold splitting observed in experiment (Hashimoto 1983, Malis, Ludwig,
Schweika, Ice and Sparks 1998). This splitting,
which is due to correlations extending well beyond a few unit cells, is
reminiscent of that produced by the long-period superlattice of CuAuII.
Simulations at several temperatures in the disordered phase reveal that the
superlattice peaks grow relatively slowly with decreasing temperature, in
accord with experimental observation (Malis et al. 1998).
\begin{figure}[tbh]
\begin{center}
\epsfysize=10cm
\epsfbox{fig1.eps}
\end{center}
\caption{Diffuse scattering intensity in the (hk0) reciprocal plane from a
$60^3$ atom simulation of CuAu 2.7\% above the transition temperature.
The intensity is normalized to $N^2 x_{Cu} x_{Au}$. $f_{Cu}$ and $f_{Au}$ were
considered constant and taken to be 26 and 72, respectively. }
\label{diffuse}
\end{figure}
Metcalfe and Leake (Metcalfe and Leake 1975) have reported a study of the
local chemical order in CuAu alloy as measured by x-ray diffuse scattering.
They analyzed
their experiments in terms of short-range order parameters $\alpha_{lmn}$ which
measure the average number of like/unlike neighbors surrounding an atom in a
given shell (Borie and Sparks 1971):
\begin{equation}
\alpha_{lmn} = 1 - {{P_{AB}^{lmn}}\over{x_B}}
\end{equation}
\noindent Here $P_{AB}^{lmn}$ is the probability of having a B atom at the
$(lmn)$ lattice site if there is an A atom at the origin.
In a completely random alloy $\alpha$ is zero. Positive values of $\alpha$
indicate that the corresponding shell is mainly occupied by atoms of the same
kind as the atom at the origin. If $\alpha$ is negative the number of unlike
neighbors is dominant. For comparison we have calculated the simulated
$\alpha_{lmn}$ directly in real space at an effective temperature 2.7\% above
the
phase transition. The calculated values are compared with those from
Metcalfe and
Leake in Table \ref{talphas}. The simulated values of the SRO parameters for
the first two
coordination shells are 2.5\% and 10\% lower than the values reported by
Metcalfe and Leake for a sample at a comparable temperature (700 K). Given
uncertainties in the accuracy of the experimental $\alpha$'s
\footnote{Metcalfe and Leake used a very approximate
scheme to eliminate the thermal diffuse scattering, which is quite large.
Moreover, they ignored second-order static displacements, which can contribute
significantly to the scattering in their wavevector range. Direct evidence of
the limited accuracy of their results is that their $\alpha_{000}$ values,
which should by definition be unity, are 26-39\% too high.}
this close agreement may well be fortuitous.
The simulations also correctly predict that the magnitude of the third neighbor
correlations drops by approximately an order of magnitude compared to those for
the two nearest shells. However, the small remaining correlation is incorrectly
predicted to be positive for this shell. As a result, the tails of the
simulated diffuse scattering peaks shown in Fig. \ref{diffuse} are less
symmetric than is the actual case in the alloy. Nonetheless,
we can conclude that the degree of local order predicted by the EMT Monte-Carlo
model is quite reasonable.
Frenkel et al. (Frenkel, Stern, Rubshtein, Voronel and
Rosenberg 1997) have used EXAFS to measure the average nearest
neighbor
distances in Cu$_x$Au$_{1-x}$ alloys with random atomic arrangements at 80 K.
To better test the accuracy of the EMT
Monte-Carlo simulations, we performed simulations of a random CuAu
alloy quenched to an equivalent temperature and determined the average
nearest-neighbor distances for the Cu-Cu, Cu-Au and Au-Au pairs. These are shown in
Fig. \ref{random}.
At all compositions the simulated distances agree with the EXAFS measurements
within the experimental uncertainty. Interestingly, both the EXAFS data and our
simulations show that the average Cu-Au distance is very close to the Cu-Cu
distance
and significantly less than the Au-Au distance. Moreover, our simulations
predict
a crossover of the Cu-Cu and Cu-Au distances near 60\% Au concentration, with
the Cu-Au distance being the smallest pair distance above this concentration.
This is in agreement with first-principles calculations of
Ozolins et al. (Ozolins, Wolverton and Zunger 1997).
It is noteworthy that the strong concentration
dependence of the relative Cu-Cu and Cu-Au distances is larger than expected
from compressible Ising models incorporating simple elastic energy and
displacement-spin coupling terms (Chakraborty 1995).
\begin{figure}[tbh]
\begin{center}
\epsfysize=10cm
\epsfbox{fig2.eps}
\end{center}
\caption{Concentration dependence of the average Cu-Cu, Au-Au and Cu-Au
distances in a random alloy at low temperature. The system size for these
simulations was 2048 atoms.
}
\label{random}
\end{figure}
The behavior of nearest-neighbor distances, shown in Figure \ref{random}, can
be
rationalized on the basis of the EMT description of cohesion. In this
description, the Au atoms have a larger neutral-sphere radius (the ``size'' in
EMT) than the Cu atoms (Jacobsen et al. 1987). Moreover, Au is less compressible. In
the simplest approximation, we can view the increase in nearest-neighbor distances
with increasing Au concentration as a traditional steric effect associated with
the larger EMT ``size'' of Au atoms.
Though this trend is definitely observed in Figure \ref{random}, the rate of
increase is different for the three different types of chemical bonds. This can
be attributed to the difference in compressibilities of Cu (lower bulk modulus -- $B_{Cu}$ =
14 x 10$^{10}$ Pa) and Au (higher -- $B_{Au}$ = 17 x 10$^{10}$ Pa).
Although there is some bowing present in the Cu-Cu line, it largely
follows the changing lattice parameter with increasing Au content (note
that the Cu-Cu pair distance extrapolates to the Au-Au distance at the
highest Au concentrations -- i.e. to the pair distance in pure Au).
Thus the relatively higher compressibility of Cu atoms allows the Cu-Cu
distances to follow the ``average'' lattice with changing concentration.
The relative incompressibility of the Au atoms, however, prevents the
average Au-Au distance , and the Cu-Au distance, from changing as much with
concentration. Within the EMT model, the relative compressibilities can also
be related to the more rapid falloff of the electron density around a Au atom
than around a Cu atom at typical interatomic distances.
\section*{3. Correlations between local environment and displacements}
Though random alloys exhibit no global short-range order, the
constituent atoms nonetheless experience stochastic variations in their
nearest-neighbor environments. Therefore, in order to investigate the
relationship between local environment and pair distances, we have
calculated the average interatomic distances in the random equiatomic
CuAu alloy as a function of the number of nearest-neighbor Au atoms.
That is, we have divided the atoms in the random CuAu alloy into 12
groups -- each member of the group has the same number of Au
nearest-neighbors. We have then calculated the average Cu-Cu, Cu-Au (Cu
being the central atom), Au-Cu (Au being the central atom) and Au-Au
pair distances in each group. The results are shown in Figure \ref{auconc}(a).
There is a strong relationship between local environment and pair
distances. The trends with increasing Au in the local environment are
qualitatively similar to those seen in Figure \ref{random} with increasing
overall Au concentration. Clearly, even in a random alloy at a
fixed concentration, the intuitive idea of atomic ``size'' has limited
meaning.
\begin{table}
\caption{Comparison
of the short-range order parameters $\alpha_{hkl}$ obtained
from a simulation 2.7\% above the transition temperature with the experimental
values (Metcalfe and Leake 1975) for a quench from 700K (2.5\% above the
true transition temperature). The last column shows the values for
perfectly ordered CuAu.}
\begin{tabular}{dddd}
hkl&simulation&experiment&ordered CuAu\\
\tableline
000&1.0000&1.263&1.00\\
110&$-$0.1825&$-$0.187&$-$0.33\\
200&0.2069&0.230&1.00\\
211&0.0226&$-$0.013&$-$0.33\\
220&0.0484&0.109&1.00\\
310&$-$0.0557&$-$0.029&$-$0.33\\
\end{tabular}
\label{talphas}
\end{table}
\begin{table}
\caption{Simulated nearest-neighbor relative displacements
$r_{AB} = (d_{AB}-d_0)/d_0$,
where $d_0$ is the average distance, at a temperature 2.7\%
above the transition temperature and infinite temperature (random alloy),
respectively.}
\label{tdist}
\begin{tabular}{ddd}
pair&finite temperature&random configuration\\
\tableline
$\alpha_{110}$&$-$0.182&$\approx$ 0\\
Cu-Cu&0.0015&$-$0.0092\\
Au-Au&0.0171&0.0223\\
Cu-Au&$-$0.0016&$-$0.0051\\
\end{tabular}
\end{table}
\begin{figure}[tbh]
\begin{center}
\epsfysize=14cm
\epsfbox{fig3.ps}
\end{center}
\caption{Dependence of the Cu-Cu, Au-Au, Cu-Au (Cu central atom) and Au-Cu
(Au central atom) nearest-neighbor distances
on the number of Au nearest neighbors in (a) a random equiatomic CuAu alloy
at low temperature and (b) in the same alloy with short range order 2.7\% above
the transition
temperature.}
\label{auconc}
\end{figure}
In order to better understand the relationship between local environment
and displacements in a non-random alloy, we studied in detail a
simulated equiatomic CuAu alloy annealed at a temperature 2.7\% above the
ordering transition to produce short-range order. In direct contrast to
the case for random equiatomic alloys, the overall average Cu-Au distance is
shorter than the overall average Cu-Cu distance (table \ref{tdist}). Thus the
development of short-range order has changed the average pair
distances. Figure \ref{auconc}(b) shows how the average Cu-Cu, Cu-Au, Au-Cu
and
Au-Au pair distances vary with number of Au nearest-neighbors.
Comparison with the equivalent Figure \ref{auconc}(a) for a random alloy shows
that the overall trends remain qualitatively similar but the short-range order
causes quantitative changes.
Another way of examining the data from the alloy with short-range order
is to define a local $\alpha_{110}$ for each atom -- i.e. the number of
like nearest neighbors minus the number of unlike nearest neighbors
divided by the coordination number. Figure \ref{scatter}(a) shows the
systematic
relationship between the local ``order'' as measured by this
$\alpha_{110}$ and the interatomic distances. Here we have grouped
together the Cu-Au and Au-Cu distances. Figure \ref{scatter}(b) shows also the
histogram of local $\alpha_{110}$ values. For atoms with large positive
values of $\alpha_{110}$, i.e. a large fraction of like nearest
neighbors, the relation between like and unlike pair distances follows
the simple expectation that $d_{Cu-Cu} < d_{Cu-Au} < d_{Au-Au}$.
However, for lower and negative values of $\alpha$ the Cu-Au distance is
the smallest of the three. Moreover, the Au-Au distance decreases and
begins to approach the Cu-Cu distance, which itself increases with
decreasing $\alpha$. The behavior is very suggestive of the
relationships between nearest neighbor distances in the ordered CuAuI
phase - i.e. $d_{Cu-Cu} = d_{Au-Au} > d_{Au-Cu}$. It can also be seen
from Figure \ref{scatter} that the development of short-range order, i.e.
an increase in the number of atoms with a negative $\alpha$, in the alloy leads
to a decreasing average Cu-Au distance relative to the average Cu-Cu
distance. This partly explains why $d_{Cu-Au} < d_{Cu-Cu}$ in the
equiatomic alloy with short-range order but $d_{Cu-Au} > d_{Cu-Cu}$ in
the random alloy.
\begin{figure}[tbh]
\epsfysize=14cm
\epsfbox{fig4.ps}
\caption{(a) Dependence of the Cu-Cu, Au-Au and Cu-Au nearest-neighbor distances
on the local coordination number $\alpha_{110}$ 2.7\% above the transition
temperature. The error bars are smaller than the size of the symbols.
(b) The distribution of local $\alpha_{110}$ values.}
\label{scatter}
\end{figure}
We studied closely the atoms positioned in a local chemical environment similar
to the environment in the ordered phase, i.e. having 4 like and 8 unlike nearest
neighbors (i.e. local $\alpha_{110} = -0.33$). In the ordered phase the 4 like
neighbors occupy one of the cubic planes passing through the center atom while
the 8 unlike atoms occupy the other 2 planes. In the simulation we found that
approximately 10\% of the atoms with 8 unlike nearest neighbors are actually in
a geometrically ``ordered'' environment. However,
the occurrence of this configuration is considerably higher than random.
Figure \ref{planes} contrasts the occupation probabilities in the planes for the
random case and that observed in the EMT simulations. It is instructive to
focus on those atoms in a geometrically ``ordered'' environment (as well as
chemically ordered environment) and define a local distortion as the
deviation from unity of the ratio of the average A-A distance to the average A-B
distance around that atom. The average local distortion around Cu
atoms in the simulation is approximately 1.5\%, slightly higher than the average
around Au atoms (0.4\%). Both values are much smaller than the 3\% that would be
predicted by the tetragonal distortion occurring in the ordered structure.
\begin{figure}[tbh]
\begin{center}
\epsfysize=10cm
\epsfbox{fig5.eps}
\end{center}
\caption{Probabilities of finding the 4 like nearest neighbors
of those atoms with $\alpha_{110} = -0.33$ distributed in the 3 surrounding
planes
at the simulated temperature. Also shown are the random probabilities.
}
\label{planes}
\end{figure}
\section*{4. Conclusions}
An important advantage of real space simulations is that they permit a
direct examination of the relationship between local chemical
environment and atomic displacements. These simulations show a strong
correlation between nearest-neighbor environment and interatomic
distances in disordered Cu$_x$Au$_{1-x}$, even in random alloys. Higher
Au concentrations in the local environment lead to a decrease of Cu-Au
distances relative to Cu-Cu distances. In equiatomic CuAu alloys with
short-range order, these trends remain qualitatively similar. The
increased local ``order'' causes a decrease in the average
nearest-neighbor Cu-Au distance and a partial convergence of Cu-Cu and
Au-Au distances, reminiscent of the local structure in the ordered CuAuI
phase. The intuitive concept of ``size'' has limited usefulness in
understanding these trends. We anticipate that such correlations
between local environment and displacements are common in disordered
alloys, particularly those having components with large atomic size
differences.
\section*{Acknowledgments}
We would like to acknowledge useful discussions with C. J. Sparks and W.
Schweika.
This work utilizes the resources of the Boston Univ. Center for Computational
Science and was supported by NSF grant DMR-9633596.
B. C. would like to acknowledge numerous discussions with Per Stoltze regarding
the ARTwork program. The work of B. C. and D. O. was supported in part by the
DOE grant
DE-FG02-ER45495. B. C. would also like to acknowledge the hospitality of ITP,
Santa Barbara, where some of this work was performed. B. C. would
also like to thank Jens Norskov for suggesting the incompressibility argument.
\newpage
\section*{References}
\noindent Borie, B., and Sparks, C. J., 1971, Acta Crystallogr., {\bf A27}, 198.
\noindent Chakraborty, B., 1995, Europhys. Lett., {\bf 30}, 531.
\noindent Chakraborty, B., and Xi, Z., 1992, Phys. Rev. Lett., {\bf 68},
2039.
\noindent Daw, M. S., Foiles, S. M., and Baskes, M. I., 1993, Mat. Sci. Reports,
{\bf 9}, 251.
\noindent Dunweg, B., and Landau, D., 1993, Phys. Rev. B, {\bf 48}, 14182.
\noindent Elder, K. R., Malis, O., Ludwig, K., Chakraborty, B., and Goldenfeld, N., 1998,
accepted in Europhysics Letters.
\noindent Frenkel, A. I., Stern, E. A., Rubshtein, A., Voronel, A., and
Rosenberg, Yu., 1997, J. Phys. IV France, {\bf 7}, C2-1005.
\noindent Hashimoto, S., 1983, Acta Crystall., 1983, A{\bf 39}, 524.
\noindent Jacobsen, K. W., Norskov, J. K., and Puska, M. J., 1987, Phys. Rev B
{\bf 35}, 7423.
\noindent Jacobsen, K. W., 1988, Comments Cond. Mat. Phys., {\bf 14}, 129.
\noindent Jiang, X., Ice, G. E., Sparks, C. J., Robertson, L., and Zschack, P., 1996,
Phys. Rev. B, {\bf 54}, 3211.
\noindent Lu, Z. W., Laks, D. B., Wei, S. -H., and Zunger, A., 1994,
Phys. Rev. B, {\bf 50}, 6642.
\noindent Malis, O., Ludwig, K.F., Schweika, W., Ice, G. E., and Sparks, C. J., 1998,
in preparation.
\noindent Metcalfe, E., and Leake, J. A., 1975, Acta Metall., {\bf 23}, 1135.
\noindent Ozolins, V., Wolverton, C., and Zunger, A., 1997, cond-mat no. 9710225.
\noindent Polatoglou, H. M., and Bleris, G. L., 1994, Solid State Commun., {\bf
90}, 425.
\noindent Silverman, A., Zunger, A., Kalish, R., and Adler, J., 1995,
Phys. Rev. B {\bf 51}, 10795.
\noindent Stoltze, P., 1997, Simulation methods in atomic-scale materials physics,
(Polyteknisk Forlag, Denmark).
\noindent Xi, Z., Chakraborty, B., Jacobsen, K. W., and Norskov, J. K., 1992,
J. Phys.: Condens. Matter, {\bf 4}, 7191.
\noindent Wolverton, C., and Zunger, A., 1995, Phys. Rev. Lett., {\bf 75}, 3162.
\end{document}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 4,294 |
{"url":"http:\/\/mathoverflow.net\/revisions\/45412\/list","text":"Let $\\alpha$ be a root of a polynomial $f(x) \\in \\mathbf{Q}[x]$ of degree $n$, let $K = \\mathbf{Q}(\\alpha)$, $L$ be the Galois closure of $K$, and $G = \\mathrm{Gal}(L\/\\mathbf{Q}) \\subset S_n$. How does one prove that a permutation group contains $A_n$? Following Jordan, the usual method is to show that it is sufficiently highly transitive. Also following Jordan, to do this it suffices to construct subgroups of $G$ which act faithfully and transitively on $n-k$ points and trivially on the other $k$ points (for $k$ large, $\\ge 6$ using CFSG), and to show that $G$ is primitive. (The standard method for doing this is to find $l$-cycles for a prime $l$.) In the context of a Galois group, the most obvious place to look for \"elements\" is to consider the decomposition groups $D$ at places of $\\mathbf{Q}$. If $l$ is unramified in $K$, this corresponds to looking at a Frobenius element (conjugacy class). In practice (as far as a computation goes) this is quite useful, but theoretically it is not so great unless there is a prime $l$ for which the factorization is particularly clean. This leaves the places which ramify in $L$. For example, if $v = \\infty$, one is considering the action of complex conjugation; if there are exactly two complex roots then $c$ is a $2$-cycle, and from Jordan's theorem (easy in this case) we see that if $G$ is primitive then $G$ is $S_n$.\n\nThe proposed method (following Coleman et. al.) for proving that $G$ contains $A_n$ is somewhat misguided, I think. The key point about the polynomial $\\sum_{k=0}^{n} x^k\/k!$ is that the corresponding field is ramified at many primes, and the decomposition groups at these primes give the requisite elements. Conversely, the polynomial considered in this problem corresponds to a field with somewhat limited ramification - as has been noted, the only primes which ramify divide $p(p+1)$.\n\nIt can be hard to compute Galois groups of random families of polynomials in general. I do not know if this is true in the present case, but given the lack of motivation I won't spend any more time thinking about it than the last hour or two, and instead give some partial results. However, the methods given here may well apply more generally. Let $n = p - 1$.\n\nCLAIM: Suppose that $p+1$ is exactly divisible by a prime $l > 3$. Then $G$ contains $A_{n}$. (This applies to a set $p$ of relative density one inside the primes.)\n\nSTEP I: Factorization of $p$; $G$ is primitive. Let $f(x) = x^{p-1} + 2 x^{p-2} + \\ldots + p$. Note that $$(x-1)^2 f(x) = x(x^{p} - 1) - p(x-1) = x^{p+1} - 1 - (p+1)(x-1).$$ We deduce that $f(x) \\equiv x(x-1)^{p-2} \\mod p$, and that $$p = \\mathfrak{p} \\mathfrak{q}^{p-2}$$ for primes $\\mathfrak{p}$ and $\\mathfrak{q}$ in the ring of integers $O_K$ of $K$ both of norm $p$. (To show this one needs to check that $[O_K:\\mathbf{Z}[\\alpha]]$ is co-prime to $p$ - one can do this by considering the Newton Polygon of $f(x+1)$.) Let $D \\subset G$ be a decomposition group at $p$. This corresponds to choosing a simultaneous embedding of the roots of $f(x)$ into an algebraic closure of the $p$-adic numbers. We see that we may write $f(x) = a(x) b(x)$ as polynomials over the $p$-adic numbers (which I can't latex at this point for some reason), where $a(x) \\equiv x \\mod p$ has degree one and $b(x) \\equiv (x-1)^{p-2}$ is irreducible of degree $p-2$ and corresponds to a totally ramified extension. Clearly $D$ acts transitively on the $p-2 = n-1$ roots of $b(x)$ and fixes the roots of $a(x)$. Since $D \\subset G \\cap S_{n-1}$, we see that $G \\cap S_{n-1}$ is transitive in $S_{n-1}$ and so $G$ is $2$-transitive (and hence primitive).\n\nStep II: Factorization of $l$: Let $l$ be a prime dividing $p+1$. We assume that $l \\ge 5$ and $l$ exactly divides $p+1$. We see that $$f(x) \\equiv (x-1)^{l-2} \\prod_{i=1}^{k-1} (x-\\zeta^i)^{l}$$ where $\\zeta$ is a $k$th root of unity and $kl=p+1$. This suggests that: $$l = \\mathfrak{p}^{l-2} \\prod_{i=1}^{k} \\mathfrak{q}^l.$$ This also follows from a Newton polygon argument applied to $f(x - \\zeta^i)$. (Warning, this uses that $l$ exactly divides $p+1$.)\n\nStep III: Some basic facts about local extensions:\n\nLemma 1. Suppose the ramification degree of $E\/\\mathbb{Q}_l$ is $l^m$. Then the ramification degree of the Galois closure of $E$ is only divisible by primes dividing $l(l^m-1)$. Proof. Kummer Theory.\n\nLemma 2. Suppose that $h(x) \\in \\mathbf{Q}_l[x]$ is an irreducible polynomial of degree $k$ with $(k,l) = 1$, such that the corresponding field $E\/\\mathbf{Q}_l$ is totally ramified. If $F$ is the splitting field of $h(x)$, then $\\mathrm{Gal}(F\/\\mathbf{Q}_l) \\subset S_k$ contains a $k$-cycle. Proof: From a classification of tamely ramified extensions, there exists an unramified extension $A$ such that $[EA:A] = [E:\\mathbf{Q}_l]$ and $EA\/A$ is cyclic and Galois. It follows that $\\mathrm{Gal}(EA\/A)$ acts transitively and faithfully on the roots of $h(x)$, and is thus generated by a $k$-cycle.\n\nStep IV: $G$ contains an $l-2$-cycle. Consider the decomposition group $D$ at $l$. The orbits of $D$ correspond to the factorization of $l$ in $O_K$. On the factors corresponding to primes of the form $\\mathfrak{q}^p_i$, the image of $D$ factors through a group whose inertia has degree divisible only by primes dividing $l(l-1)$, by Lemma 1. On the other hand, on the factor corresponding to $\\mathfrak{p}^{l-2}$, the image of inertia contains an $l-2$ cycle, by Lemma 2. Since $(l(l-1),l-2) = 1$, we see that $D \\subset G$ contains an $l-2$ cycle.\n\nStep V: Jordan's Theorem. Since $G$ is primitive, and $G$ contains a subgroup that acts transitively and faithfully on $l-2$ points (and trivially on all other points), we deduce (from the standard proof of Jordan's theorem) that $G$ is $n-(l-2)+1 = n+3-l$ transitive. This is at least $6$ (since $n+2$ is at least $2l$) and so $G$ contains $A_n$ (by CFSG).\n\nSTEP VI: (for you, dear reader) Find the analogous argument when $p+1$ is exactly divisible by $l^k$ for some $k \\ge 2$ --- try to construct a cycle of degree $l^k - 2$, although be careful as it will no longer be the case (as it was above) that $[O_K:\\mathbf{Z}[\\alpha]]$ was co-prime to $l$. This still leaves $p-1$ either a power of $2$ or a power of $2$ times $3$, which might be annoying --- one would have to think hard about the structure of the decomposition group at $2$ in those cases.","date":"2013-05-19 20:15:28","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9677426218986511, \"perplexity\": 81.95150006245339}, \"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\/1368698063918\/warc\/CC-MAIN-20130516095423-00029-ip-10-60-113-184.ec2.internal.warc.gz\"}"} | null | null |
package com.cws.esolutions.security.processors.dto;
/*
* Project: eSolutionsSecurity
* Package: com.cws.esolutions.security.processors.dto
* File: AuthenticationData.java
*
* History
*
* Author Date Comments
* ----------------------------------------------------------------------------
* cws-khuntly 11/23/2008 22:39:20 Created.
*/
import org.slf4j.Logger;
import java.io.Serializable;
import org.slf4j.LoggerFactory;
import com.cws.esolutions.security.SecurityServiceConstants;
/**
* @author cws-khuntly
* @version 1.0
* @see java.io.Serializable
*/
public class AuthenticationData implements Serializable
{
private int otpValue = 0;
private String secret = null;
private String username = null;
private String userSalt = null;
private String password = null;
private String newPassword = null;
private String secAnswerOne = null;
private String secAnswerTwo = null;
private String secQuestionOne = null;
private String secQuestionTwo = null;
private static final long serialVersionUID = -1680121237315483191L;
private static final String CNAME = AuthenticationData.class.getName();
private static final Logger DEBUGGER = LoggerFactory.getLogger(SecurityServiceConstants.DEBUGGER);
private static final boolean DEBUG = DEBUGGER.isDebugEnabled();
public final void setUsername(final String value)
{
final String methodName = AuthenticationData.CNAME + "#setUsername(final String value)";
if (DEBUG)
{
DEBUGGER.debug(methodName);
}
this.username = value;
}
public final void setUserSalt(final String value)
{
final String methodName = AuthenticationData.CNAME + "#setUserSalt(final String value)";
if (DEBUG)
{
DEBUGGER.debug(methodName);
}
this.userSalt = value;
}
public final void setPassword(final String value)
{
final String methodName = AuthenticationData.CNAME + "#setPassword(final String value)";
if (DEBUG)
{
DEBUGGER.debug(methodName);
}
this.password = value;
}
public final void setNewPassword(final String value)
{
final String methodName = AuthenticationData.CNAME + "#setNewPassword(final String value)";
if (DEBUG)
{
DEBUGGER.debug(methodName);
}
this.newPassword = value;
}
public final void setOtpValue(final int value)
{
final String methodName = AuthenticationData.CNAME + "#setOtpValue(final int value)";
if (DEBUG)
{
DEBUGGER.debug(methodName);
}
this.otpValue = value;
}
public final void setSecQuestionOne(final String value)
{
final String methodName = AuthenticationData.CNAME + "#setSecQuestionOne(final String value)";
if (DEBUG)
{
DEBUGGER.debug(methodName);
}
this.secQuestionOne = value;
}
public final void setSecQuestionTwo(final String value)
{
final String methodName = AuthenticationData.CNAME + "#setSecQuestionTwo(final String value)";
if (DEBUG)
{
DEBUGGER.debug(methodName);
}
this.secQuestionTwo = value;
}
public final void setSecAnswerOne(final String value)
{
final String methodName = AuthenticationData.CNAME + "#setSecAnswerOne(final String value)";
if (DEBUG)
{
DEBUGGER.debug(methodName);
}
this.secAnswerOne = value;
}
public final void setSecAnswerTwo(final String value)
{
final String methodName = AuthenticationData.CNAME + "#setSecAnswerTwo(final String value)";
if (DEBUG)
{
DEBUGGER.debug(methodName);
}
this.secAnswerTwo = value;
}
public final void setSecret(final String value)
{
final String methodName = AuthenticationData.CNAME + "#setSecret(final String value)";
if (DEBUG)
{
DEBUGGER.debug(methodName);
}
this.secret = value;
}
public final String getUsername()
{
final String methodName = AuthenticationData.CNAME + "#getUsername()";
if (DEBUG)
{
DEBUGGER.debug(methodName);
DEBUGGER.debug("Value: {}", this.username);
}
return this.username;
}
public final String getUserSalt()
{
final String methodName = AuthenticationData.CNAME + "#getUserSalt()";
if (DEBUG)
{
DEBUGGER.debug(methodName);
}
return this.userSalt;
}
public final String getPassword()
{
final String methodName = AuthenticationData.CNAME + "#getPassword()";
if (DEBUG)
{
DEBUGGER.debug(methodName);
}
return this.password;
}
public final String getNewPassword()
{
final String methodName = AuthenticationData.CNAME + "#getNewPassword()";
if (DEBUG)
{
DEBUGGER.debug(methodName);
}
return this.newPassword;
}
public final int getOtpValue()
{
final String methodName = AuthenticationData.CNAME + "#getOtpValue()";
if (DEBUG)
{
DEBUGGER.debug(methodName);
}
return this.otpValue;
}
public final String getSecQuestionOne()
{
final String methodName = AuthenticationData.CNAME + "#getSecQuestionOne()";
if (DEBUG)
{
DEBUGGER.debug(methodName);
}
return this.secQuestionOne;
}
public final String getSecQuestionTwo()
{
final String methodName = AuthenticationData.CNAME + "#getSecQuestionTwo()";
if (DEBUG)
{
DEBUGGER.debug(methodName);
}
return this.secQuestionTwo;
}
public final String getSecAnswerOne()
{
final String methodName = AuthenticationData.CNAME + "#getSecAnswerOne()";
if (DEBUG)
{
DEBUGGER.debug(methodName);
}
return this.secAnswerOne;
}
public final String getSecAnswerTwo()
{
final String methodName = AuthenticationData.CNAME + "#getSecAnswerTwo()";
if (DEBUG)
{
DEBUGGER.debug(methodName);
}
return this.secAnswerTwo;
}
public final String getSecret()
{
final String methodName = AuthenticationData.CNAME + "#getSecret()";
if (DEBUG)
{
DEBUGGER.debug(methodName);
}
return this.secret;
}
}
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{"url":"https:\/\/bitcointechweekly.com\/tags\/channel-capacity","text":"Feed for tag: channel-capacity\nA suggestion to hint for channel capacity in BOLT 11 invoices\nIn making payments through lightning channels, space is a factor that must be considered. However, one cannot tell how much capacity is available on which channel. Rusty Russell is proposing a change to the system that automatically attaches an \u2018r\u2019 field to any channel that has sufficient capacity to receive a certain amount. Some members of the community however think this may be risky because an unauthorized user can tell how much capacity one has and use it to attack.\nLack of capacity field in channel_announcment\n\nA discussion started recently regarding the lack of channel capacity information in the Channel_announcement message.\n\nchannel_announcement is a gossip message in the Lightning protocol that contains information regarding the ownership of a channel. It links an on chain Bitcoin key to a Lightning node key. Currently the message does not include any information about the channel capacity, for which wallets have to search through the blockchain and that becomes tedious for mobile and light wallets, as they have to send a request to a block explorer API to get the capacity. Not having the capacity greatly decreases the routing success rates as you maybe trying to send 10 BTC through a channel that can handle a maximum of 1 BTC.","date":"2023-04-01 23:24:01","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\": 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\": 0.21417106688022614, \"perplexity\": 1360.0993584501884}, \"config\": {\"markdown_headings\": true, \"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-2023-14\/segments\/1679296950363.89\/warc\/CC-MAIN-20230401221921-20230402011921-00355.warc.gz\"}"} | null | null |
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Korean J Intern Med Search > Browse Articles > Korean J Intern Med Search
Korean version of the Cough Symptom Score: clinical utility and validity for chronic cough
Jae-Woo Kwon, Ji-Yong Moon, Sae-Hoon Kim, Woo-Jung Song, Min-Hye Kim, Min-Gyu Kang, Kyung-Hwan Lim, So-Hee Lee, Sang Min Lee, Jin Young Lee, Hyouk-Soo Kwon, Kyung-Mook Kim, Sang-Heon Kim, Sang-Hoon Kim, Jae-Won Jeong, Cheol-Woo Kim, Sang-Heon Cho, Byung-Jae Lee; Work Group for Chronic Cough, the Korean Academy of Asthma, Allergy and Clinical Immunology
Korean J Intern Med. 2017;32(5):910-915. Published online March 31, 2017
DOI: https://doi.org/10.3904/kjim.2016.132 Cited By 4
Background/Aims: The Cough Symptom Score (CSS) is a simple, useful tool for measuring cough severity. However, there is no standard Korean version of the CSS. We developed a Korean version of the CSS and evaluated its clinical utility and validity for assessing chronic cough severity. Methods: The..
A Case of Pulmonary Inflammatory Pseudotumor: Recurrence Appearing as Several Consolidative Lesions after Complete Resection
Hong-Lyeol Lee, Lucia Kim, Kyung-Hee Lee, Kwang-Ho Kim, Cheol-Woo Kim
Korean J Intern Med. 2005;20(2):168-172. Published online June 30, 2005
DOI: https://doi.org/10.3904/kjim.2005.20.2.168 Cited By 5
Inflammatory pseudotumor (plasma cell granuloma) of the lung is an uncommon nonneoplastic tumor of unknown origin. This tumor typically manifests as a solitary, peripheral, and sharply circumscribed mass. Multiple lesions are seen in about 5% of cases. Resection is recommended for both diagnosis .. | {
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One of the greatest things about being a bilingual family is the hilarity of the language differences. Here is a picture of an electrical supply store with a very electrifying name.
I have been laughing at this for years, so I thought I would share it with you. Casinos to Castles may have already seen it though, if they've shopped at the home improvement store lately! I wonder if they would get more or less customers in America?
YES! This makes me laugh every time I see the sign and when I show it to new people! It is so simply wonderful. I have such a juvenile sense of humor!
Yeah, my sense of humor is often that of a teenager! | {
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Q: List comprehension - group data based on occurance of specific elements in a list I have a list like below :
['<h2 class="title-6-bold"> Premier League </h2>', '<span class="title-8-medium simple-match-card-team__name"> Fulham </span>', '<span class="title-7-bold simple-match-card-team__score"> 2 </span>', '<span class="title-8-medium simple-match-card-team__name"> Liverpool </span>', '<span class="title-7-bold simple-match-card-team__score"> 2 </span>', '<span class="title-8-medium simple-match-card-team__name"> Bournemouth </span>', '<span class="title-7-bold simple-match-card-team__score"> 2 </span>', '<span class="title-8-medium simple-match-card-team__name"> Aston Villa </span>', '<span class="title-7-bold simple-match-card-team__score"> 0 </span>', '<span class="title-8-medium simple-match-card-team__name"> Leeds </span>', '<span class="title-7-bold simple-match-card-team__score"> 2 </span>', '<span class="title-8-medium simple-match-card-team__name"> Wolves </span>', '<span class="title-7-bold simple-match-card-team__score"> 1 </span>', '<span class="title-8-medium simple-match-card-team__name"> Newcastle United </span>', '<span class="title-7-bold simple-match-card-team__score"> 2 </span>', '<span class="title-8-medium simple-match-card-team__name"> Nottingham Forest </span>', '<span class="title-7-bold simple-match-card-team__score"> 0 </span>', '<span class="title-8-medium simple-match-card-team__name"> Tottenham </span>', '<span class="title-7-bold simple-match-card-team__score"> 4 </span>', '<span class="title-8-medium simple-match-card-team__name"> Southampton </span>', '<span class="title-7-bold simple-match-card-team__score"> 1 </span>', '<span class="title-8-medium simple-match-card-team__name"> Everton </span>', '<span class="title-7-bold simple-match-card-team__score"> 0 </span>', '<span class="title-8-medium simple-match-card-team__name"> Chelsea </span>', '<span class="title-7-bold simple-match-card-team__score"> 1 </span>', '<h2 class="title-6-bold"> Bundesliga </h2>', '<span class="title-8-medium simple-match-card-team__name"> 1. FC Union Berlin </span>', '<span class="title-7-bold simple-match-card-team__score"> 3 </span>', '<span class="title-8-medium simple-match-card-team__name"> Hertha BSC </span>', '<span class="title-7-bold simple-match-card-team__score"> 1 </span>', '<span class="title-8-medium simple-match-card-team__name"> M\'gladbach </span>', '<span class="title-7-bold simple-match-card-team__score"> 3 </span>', '<span class="title-8-medium simple-match-card-team__name"> Hoffenheim </span>', '<span class="title-7-bold simple-match-card-team__score"> 1 </span>', '<span class="title-8-medium simple-match-card-team__name"> Augsburg </span>', '<span class="title-7-bold simple-match-card-team__score"> 0 </span>', '<span class="title-8-medium simple-match-card-team__name"> SC Freiburg </span>', '<span class="title-7-bold simple-match-card-team__score"> 4 </span>', '<span class="title-8-medium simple-match-card-team__name"> VfL Bochum </span>', '<span class="title-7-bold simple-match-card-team__score"> 1 </span>', '<span class="title-8-medium simple-match-card-team__name"> Mainz 05 </span>', '<span class="title-7-bold simple-match-card-team__score"> 2 </span>', '<span class="title-8-medium simple-match-card-team__name"> VfL Wolfsburg </span>', '<span class="title-7-bold simple-match-card-team__score"> 2 </span>', '<span class="title-8-medium simple-match-card-team__name"> Werder Bremen </span>', '<span class="title-7-bold simple-match-card-team__score"> 2 </span>', '<span class="title-8-medium simple-match-card-team__name"> Borussia Dortmund </span>', '<span class="title-7-bold simple-match-card-team__score"> 1 </span>', '<span class="title-8-medium simple-match-card-team__name"> Bayer Leverkusen </span>', '<span class="title-7-bold simple-match-card-team__score"> 0 </span>', '<h2 class="title-6-bold"> Scottish Premiership </h2>', '<span class="title-8-medium simple-match-card-team__name"> Aberdeen </span>', '<span class="title-7-bold simple-match-card-team__score"> 4 </span>', '<span class="title-8-medium simple-match-card-team__name"> St. Mirren </span>', '<span class="title-7-bold simple-match-card-team__score"> 1 </span>', '<span class="title-8-medium simple-match-card-team__name"> Motherwell </span>', '<span class="title-7-bold simple-match-card-team__score"> 1 </span>', '<span class="title-8-medium simple-match-card-team__name"> St. Johnstone </span>', '<span class="title-7-bold simple-match-card-team__score"> 2 </span>', '<span class="title-8-medium simple-match-card-team__name"> Rangers </span>', '<span class="title-7-bold simple-match-card-team__score"> 2 </span>', '<span class="title-8-medium simple-match-card-team__name"> Kilmarnock </span>', '<span class="title-7-bold simple-match-card-team__score"> 0 </span>', '<span class="title-8-medium simple-match-card-team__name"> Ross County </span>', '<span class="title-7-bold simple-match-card-team__score"> 1 </span>', '<span class="title-8-medium simple-match-card-team__name"> Celtic </span>', '<span class="title-7-bold simple-match-card-team__score"> 3 </span>', '<h2 class="title-6-bold"> Ligue 1 Uber Eats </h2>', '<span class="title-8-medium simple-match-card-team__name"> Strasbourg </span>', '<span class="title-7-bold simple-match-card-team__score"> 1 </span>', '<span class="title-8-medium simple-match-card-team__name"> Monaco </span>', '<span class="title-7-bold simple-match-card-team__score"> 2 </span>', '<span class="title-8-medium simple-match-card-team__name"> Clermont </span>', '<span class="title-7-bold simple-match-card-team__score"> 0 </span>', '<span class="title-8-medium simple-match-card-team__name"> PSG </span>', '<span class="title-7-bold simple-match-card-team__score"> 5 </span>']
I am trying to extract data of a few top leagues and want to discard others. Following another example I have this code :
leagues = (['Premier League', 'Spanish La Liga', 'Bundesliga', 'Italian Serie A','Ligue 1 Uber Eats', 'Champions League'])
data = [[l[l.index(left) + len(left):l.index(right)] for l in data if i in l] for i in leagues]
But I am not getting the expected result like which should be like below :
[['Premier League', * all matches of PL], ['Bundesliga', * all Bundesliga matches]].
Please help me with this as I have been burning my head over this for quite a long time now.
Thanks
A: You can iterate the values in your list, updating league, teams and scores when you see the matching tag, and then writing a match to the result when you have two teams and two scores. I've created a dict of matches with the league as the key, it should be reasonably easy to change the format if you want something else (e.g. with list(result.items()))
import re
from collections import defaultdict
result = defaultdict(list)
for d in data:
l = re.search(r'>([^<]+)</h2>', d)
if l is not None:
league = l.group(1).strip()
teams = []
scores = []
continue
t = re.search(r'name">([^<]+)</span>', d)
if t is not None:
teams.append(t.group(1).strip())
continue
s = re.search(r'score">\s*(\d+)\s*</span>', d)
if s is not None:
scores.append(int(s.group(1)))
if len(scores) == 2:
result[league].append({ 'teams' : teams[:], 'scores' : scores[:] })
teams = []
scores = []
Output (for your sample data):
{
'Premier League': [
{'teams': ['Fulham', 'Liverpool'], 'scores': [2, 2]},
{'teams': ['Bournemouth', 'Aston Villa'], 'scores': [2, 0]},
{'teams': ['Leeds', 'Wolves'], 'scores': [2, 1]},
{'teams': ['Newcastle United', 'Nottingham Forest'], 'scores': [2, 0]},
{'teams': ['Tottenham', 'Southampton'], 'scores': [4, 1]},
{'teams': ['Everton', 'Chelsea'], 'scores': [0, 1]}
],
'Bundesliga': [
{'teams': ['1. FC Union Berlin', 'Hertha BSC'], 'scores': [3, 1]},
{'teams': ["M'gladbach", 'Hoffenheim'], 'scores': [3, 1]},
{'teams': ['Augsburg', 'SC Freiburg'], 'scores': [0, 4]},
{'teams': ['VfL Bochum', 'Mainz 05'], 'scores': [1, 2]},
{'teams': ['VfL Wolfsburg', 'Werder Bremen'], 'scores': [2, 2]},
{'teams': ['Borussia Dortmund', 'Bayer Leverkusen'], 'scores': [1, 0]}
],
'Scottish Premiership': [
{'teams': ['Aberdeen', 'St. Mirren'], 'scores': [4, 1]},
{'teams': ['Motherwell', 'St. Johnstone'], 'scores': [1, 2]}
{'teams': ['Rangers', 'Kilmarnock'], 'scores': [2, 0]},
{'teams': ['Ross County', 'Celtic'], 'scores': [1, 3]}
],
'Ligue 1 Uber Eats': [
{'teams': ['Strasbourg', 'Monaco'], 'scores': [1, 2]},
{'teams': ['Clermont', 'PSG'], 'scores': [0, 5]}
]
}
| {
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A (; Italian for "toast") is a song in which a company is exhorted to drink, a drinking song.
The word is Italian, but it derives from an old German phrase, – "(I) offer it to you", which at one time was used to introduce a toast. The transformation of that phrase into the current Italian word may have been influenced by similar-sounding name of the Italian city of Brindisi, but otherwise the city and the term are etymologically unrelated.
The term brindisi is often used in opera. Typically, in an operatic , one character introduces a toast with a solo melody and the full ensemble later joins in the refrain.
Some well-known operatic numbers labeled are:
"Cantiamo, facciam brindisi", chorus in Gaetano Donizetti's L'Elisir d'Amore
"Libiamo ne' lieti calici", sung by Alfredo and Violetta in act 1 of Verdi's La traviata
"Viva, il vino spumeggiante", sung by Turiddu in scene 2 of Mascagni's Cavalleria rusticana
"Il segreto per esser felici", sung by Orsini in act 2 of Donizetti's Lucrezia Borgia
"Inaffia l'ugola!", sung by Iago in act 1 of Verdi's Otello
"Si colmi il calice", sung by Lady Macbeth in act 2 of Verdi's Macbeth
"The Tea-Cup Brindisi", in the finale of act 1 of Gilbert and Sullivan's The Sorcerer
"Ô vin, dissipe la tristesse" sung by Hamlet in act 2 of Thomas's Hamlet
Notes
References
External links
, Glyndebourne Festival Opera 2014 (with English subtitles)
Italian opera terminology | {
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{"url":"https:\/\/tex.stackexchange.com\/questions\/13995\/how-to-split-algorithm2e-over-two-pages","text":"# How to split algorithm2e over two pages\n\nI'm using algorithm2e package but my algorithm does not fit in only one page. Is there any way to split it over two pages, even if manually?\n\nIf doing it manually, we can use two algorithm environments, but there are some issues:\n\n1. Maintain the line counting - OK, this can be done by calling the package with noresetcount\n\n2. Remove the caption from the first\/second algorithm (depending if we want the caption in the beginning or end) -- how to do it?\n\n3. Take care of the vertical lines of the indentation, that is, I expect them to \"cross\" the pages -- how to do it?\n\nIf you don't have to use algorithm2e, you should have a look at the listings package, which supports listings on multiple pages.\n\nThere is a trick that I follow the answer for algorithm2e split over several pages by outmind\n\n\\documentclass{report}\n\n\\usepackage[linesnumbered,ruled,vlined]{algorithm2e}\n\n\\begin{document}\n\n\\begin{algorithm}\n\\caption{abc dcc}\n\\LinesNumbered\n% This is to hide end and get the last vertical line straight\n\\SetKwBlock{Begin}{Begin}{}\n\\SetAlgoLined\n\\SetKwProg{Loop}{LOOP}{}{}\n\\Begin{\n$\\mathcal{E} \\leftarrow \\emptyset$\\;\n\\SetAlgoVlined \\Loop {$ab \\gets cd$}{\n$abcd$\\\\\n$abdc$ }\n}\n\\end{algorithm}\n\n\\SetNlSty{texttt}{(}{)}\n\\begin{algorithm}\n\\LinesNumbered\n\\setcounter{AlgoLine}{12}\n% This is to restore vline mode if you did not take the package as \\usepackage[linesnumbered,ruled,vlined]{algorithm2e}\n\\SetAlgoVlined\n%This is to hide Begin keyword\n\\SetKwBlock{Begin}{}{end}\n\\SetKwProg{Loop}{LOOP}{}{}\n\\Begin{\n$\\mathcal{E} \\leftarrow \\emptyset$\\;\n$\\mathcal{E} \\leftarrow \\emptyset$\\;\n\\SetAlgoVlined \\Loop {$ab \\gets cd$}{\n$abcd$\\\\\n$abdc$ }\n$\\mathcal{E} \\leftarrow \\emptyset$\\;\n$\\mathcal{E} \\leftarrow \\emptyset$\\;\n$\\mathcal{E} \\leftarrow \\emptyset$\\;\n}\n\\end{algorithm}\n\n\\begin{algorithm}\n\\caption{fbf jfjf}\n\\LinesNumbered\n% This is to restore vline mode if you did not take the package as \\usepackage[linesnumbered,ruled,vlined]{algorithm2e}\n\\SetAlgoVlined\n%This is to hide Begin keyword\n\\SetKwProg{Loop}{LOOP}{}{}\n\\SetKwBlock{Begin}{loop3}{end}\n\\Begin{\n$\\mathcal{E} \\leftarrow \\emptyset$\\;\n$\\mathcal{E} \\leftarrow \\emptyset$\\;\n\\SetAlgoVlined \\Loop {$ab \\gets cd$}{\n$abcd$\\\\\n$abdc$ }\n$\\mathcal{E} \\leftarrow \\emptyset$\\;\n$\\mathcal{E} \\leftarrow \\emptyset$\\;\n$\\mathcal{E} \\leftarrow \\emptyset$\\;\n}\n\\end{algorithm}\n\n\\end{document}\n\n\nFor line numbering you can use \\usepackage[ruled,vlined]{algorithm2e} instead of \\usepackage[linesnumbered,ruled,vlined]{algorithm2e} and number the lines by using \\nl. Or you can use \\LinesNotNumbered{ before begin{algorithm} To remove the rule line : there are three ways: first, using\n\n\\setlength{\\algoheightrule}{0.8pt} % thickness of the rules above and below\n\\setlength{\\algotitleheightrule}{0pt} % thicknes of the rule below the title\n\n\n3rd: good way, top, bottom, middle line colored by using Colored horizontal lines in algorithm2e by Werner\n\nFor me i solved this problem by reducing the size of the text.\n\nby using: \\footnotsize or \\scriptsize.\n\nand it worked just fine.\n\n\u2022 Well, it is not consistent anymore. \u2013\u00a0user156344 May 17 at 14:10\n\u2022 I agree but sometimes it solve the problem without needing to spilt the algo. unfortunatly as i m using algorithm2e packege i coudn't found an easy solution to that problem. \u2013\u00a0zak zak May 17 at 22:23","date":"2019-08-21 18:44:03","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8537337183952332, \"perplexity\": 2254.1394180998627}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 5, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-35\/segments\/1566027316150.53\/warc\/CC-MAIN-20190821174152-20190821200152-00199.warc.gz\"}"} | null | null |
Флаг Флоренции, также известный как Флорентийская лилия (), — флаг Флорентийской республики в 1250—1532 годах. Флаг остался символом города и после падения республики, продолжая использоваться во Флоренции и по сей день. Флаг представляет собой белое поле со стилизованным красным ирисом, обычно именуемым лилией.
История
Согласно одной из теорий история флага восходит ко времени основания Флоренции во времена Римской империи — 59 г. до н. э. Случилось это в период празднований в честь римской богини Флоры, из-за чего цветы, в частности ирис, и стали символом Флоренции. По другой версии происхождение флага объясняется обилием , произраставших в окрестностях города.
Красный и белый цвета на флаге, возможно, происходят от герба Уго I, маркграфа Тосканы (969—1001), представлявшего собой щит с чередующимися красными и белыми вертикальными полосами.
Несмотря на сомнительность происхождения лилии, флорентийские гибеллины приняли к XI веку герб с белой лилией на красном поле, в результате чего этот символ стал ассоциироваться с городом. Однако в 1250 году гибеллины были побеждены и изгнаны их соперниками — гвельфами.
Изгнанные гибеллины продолжали использовать герб с белой лилией на красном поле, поэтому гвельфы решили поменять цвета на символе города, введя флаг с красной лилией на белом фоне. Он получил широкое распространение в городе, украсив ряд общественных зданий в виде архитектурных деталей. Лилия также стала изображаться на щите Марцокко, геральдического льва, который стал олицетворять Флоренцию в XIV веке. Самое известное изображение Марцокко с лилией находится на площади Синьории, эта скульптура была создана Донателло в период между 1418 и 1420 годами.
Новая эмблема Флоренции стала использоваться как в городе, так и за его пределами как символ флорентийского могущества. Флорентийская лилия получила распространение и в окрестностях города, например, её переняли Скарперия и Кастельфьорентино. Но она на их символах изображалась без тычинок ириса, которые могли изображаться исключительно на символах города Флоренции.
Флаг с лилией также был изображён на первых флоринах, отчеканенных городом в 1252 году.
В 1530 году Флорентийская республика была официально распущена императором Священной Римской империи Карлом V и заменена на Флорентийское герцогство, которым управляла семья Медичи. К 1532 году флорентийскую лилию в качестве символа города вытеснил герб Медичи. Несмотря на это, флорентийская лилия продолжала оставаться популярной среди населения.
С 1808 по 1814 годами Флоренция находилась под управлением Наполеона Бонапарта, являясь префектурой департамента Арно во Французской империи. Наполеоновские власти пытались запретить Флорентийскую лилию в городе, заменив её на другой флаг. Однако, явное негодование и бурные протесты населения заставили их отказаться от этой идеи.
Форентийская лилия украсила собой вычурный фасад Флорентийского собора и колокольню Джотто.
Современное использование
Флорентийская лилия по-прежнему является символом города. Она представлена на эмблеме клуба итальянской футбольной Серии А «Фиорентина». Флаги с лилией широко представлены на традиционных соревнованиях по на Трофей Марцокко, которые проходят 1 мая каждого года на площади Синьории.
Флаг развевается над общественными и частными зданиями по всей Флоренции, например, над палаццо Веккьо.
Флаг Флоренции также изображается на множестве общественных объектов. Например, его можно обнаружить на мусорных баках, машинах скорой помощи и троллейбусах по всему городу.
Галерея
Примечания
История Флоренции
Исторические флаги
Флоренция
Флаги Италии | {
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On 29/07/2016, Wheelabrator Technologies Inc. awarded our French partner CNIM the contract to build the turnkey K3 CHP plant, which will have the capacity to combust approx. 550,000 tons of waste per year.
The facility, which is expected to come online at the beginning of 2019, will be jointly built by CNIM and the British construction company Clugston in Kemsley near Sittingbourne (Kent) in South East England.
The plant is to feed 43 MW of electricity to the power grid and to supply process steam to the nearby Kemsley Paper Mill, one of the largest paper manufacturing facilities in Europe. The CHP project has been awarded public funding as the plant is designed to achieve a high level of energy efficiency.
For Kemsley, MARTIN will supply two reverse-acting grates, each of which has 5 runs and is 13.140 m wide. Each grate will achieve a gross heat release of 102 MW and a throughput of up to 40.8 t/h. | {
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Best Organic Agriculture Companies-profiles
Skillimite Agro: Insecticides, Fungicides, Miticides and Plant Growth Regulators
Insecticides, fungicides, miticides, and plant growth regulators are traded by Skillimite Agro Pvt. Ltd. in a variety of forms, including liquid, dust, powder, and granules. Through its workers in Odisha state, the company has a presence in western Odisha, with a network of more than 09 distributors/dealers selling to more than 100 retailers across the state and reaching out to more than 15,000 farmers. Since its inception on the 16th of June in the state of Odisha. With 9 active channel partners, 100 retailors, and 15000+ pleased clients, SKILLIMITE AGRO has become the largest supplier of bio organic PGR goods in the state of Odisha.
1) Balance
'Balance' is a cutting-edge biotech product. It has tripe activity, as well as Ovicidal, Adulticide, and Larvicidal properties.
It's a novel foliar pesticide with a novel mechanism of action. Aphids, Whitefly, Jassid, Leafhoppers, and Plant bugs are among the sucking and foliar feeding pests that are quickly knocked down.
Use: Agricultural crops of all kinds.
2) Claire
Clair fungicide is a disease-controlling fungicide that can be used on a variety of crops. It's currently approved for usage in potato, tomato, grapes, and cucumber for commercial purposes. For successful disease management, use a fungicide that penetrates quickly into plant tissue. Clair can be used to manage plant-damaging fungus as as rusts, mildews, and blights. Clair might also be used to keep mould and mildew at bay in other places. Fungicides function in a number of ways, but the majority of them harm fungal cell membranes or prevent fungal cells from producing energy.
Dose: 20 millilitres per 15 litres of water
Use: Only for vegetables.
3) Farmer-80
Farmer- 80 is a one-of-a-kind substance that, when added to pesticides, significantly improves their performance and effectiveness. When used in conjunction with pesticides, it aids in the control of weeds, insects, and other diseases on crops and vegetation. Farmer-80 also aids in the efficient use of irrigation water. A product produced after considerable research and testing by renowned agricultural colleges on a variety of crops and vegetation.
Dosage: 5–10 mL per 15 litres of water
4) Green Diamond
For healthy plant growth, Green Diamond supplies all nutrients in a balanced form. When Green Diamond is sprayed to the soil, it increases microbial activity, which increases nutrient availability to plants. Green Diamond is an organic product that may be used on various types of plants, including indoor, outdoor, garden, nursery, lawns, turf, agriculture, and plantation crops, to improve growth and production.
Dose: 2ml per litre of water
Use: Recommended for all forms of agricultural crops.
5) Fighter Plus
Fighter plus has a quick action, so when target pests get into contact with it, they cease hurting the crop and begin to die within a few hours. Fighter Plus offers a novel method of action that provides longer and more effective control of pests that are resistant to other compounds. Fighter Plus has a trans laminar action, which means that a spray applied to the upper surface of the leaves travels to the bottom surface and kills Thrips, Mights, and other sucking pests that hide on the lower side of the leaf. Due to its great efficacy against target pests, Fighter plus preserves the crop green and healthy, resulting in better quality and yield.
Dose: 20ml per 15Ltr water dose
6) Leo Star
Leo star is a cutting-edge bio-technology research product that contains highly specialised bio components in natural forms that provide extremely effective plant resistance to lipid optera, bollo worms, spod optera, shoot borer, fruit borer, and caterpillars.
Use: Recommended for Cotton, Chilli, Paddy, Grapes, Tea, and all agricultural crops
Note: For best results, spray early in the morning or late in the evening.
7) Brave
Brave is the most widely used fungicide for blast control across the world.
Brave is a very systemic substance that is not washed away by rain. Rains may actually speed up the absorption of Blastin.
Brave prevents blast disease from infecting the rice plant.
Brave also prevents the spread of blast illness to other parts of the body.
Brave is stable when stored for a long time and dissolves quickly in water.
Brave's preventive action decreases chaffy and broken grains while also improving the rice field's quality and productivity.
It's feasible to use multiple application methods, such as flat drench, transplant root soak, or foliar sprays.
Dose: 5 mL per 15 litres of water
8) Focous
Focous is an effective biotechnology for pest control, preventing pest population build-up, and increasing crop production potential. Early shoot borer and top borer control using "Focous" is successful and lasts a long time.
With its unique mechanism of action, it provides excellent and long-lasting pest control in crops such as sugarcane, rice, soybean, legumes, and vegetables by suppressing all Lepidoptera and other species. This makes "Focous" a great pest management tool, allowing growers to produce higher-quality produce with higher yields.
Use: Recommended for Brinjal, okra, tomato, sugarcane, and paddy.
9) Green Diamond bag
Green diamond provides enzymes, proteins, cytokinins, amino acids, vitamins, gibberellins, auxins, betains, and other naturally occurring major and minor nutrients and plant development substances in organic form, including enzymes, proteins, cytokinins, amino acids, vitamins, gibberellins, auxins, betains, and others. For healthy plant growth, Green Diamond supplies all nutrients in a balanced form. When green diamond is put to soil, it increases microbial activity, which increases nutrient availability to plants. Green diamond is an organic product that may be used on various types of plants, including indoor, outdoor, garden, nursery, lawns, turf, agriculture, and plantation crops, to improve growth and production.
Dose: 5 kg per acre
Use: Recommended for all crops
Website: http://www.skillimite.in/
Director: Deepak Kumar Panda
Solitary Farm Planet: Going Organic with Farming Patterns
CEF Organics: From Manure to Agri-produce at Reasonable Prices
Plantozone: An Eco-friendly, Patented Formula Ideal for Gardening Enthusiasts | {
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Quality training by AIIC Certified Instructors with over 15 years of experience in Intradermal Cosmetics and Medical Aesthetics. Our online courses include theory, videos, PowerPoint presentations, demonstrations, and much more. Join us for hands on training, one on one training available, and specialty training available.
Eii is insured, bonded, and licensed as a Vocational College. TRUST THE PROFESSIONALS AND A SCHOOL THAT IS LICENSED!
A perfect introduction to Permanent Makeup, Tattoo Removal, and Tattoos. Class is taken online and is not a clocked hour program. Course covers history of micropigmentation, basic sanitation in the tattoo industry, needles, machines, basic anatomy & physiology. Course is to teach you the safety measures of micropigmentation and beginning knowledge of the industry.
Learn how to do Traditional Eye Brows, Traditional Eyeliner, and Traditional Lips. A perfect introduction to Permanent Makeup. Class includes home study and 1 day of class study. Includes practice with inks and pens on simulated skin and live demonstration. You will graduate with a Certificate of Completion. This class has no hands on live model training.
Learn how to do eyebrows 4 different ways. (Including Hair Stroke Technique), Eyeliner 4 different ways (including Designer eyeliner technique), Lips 3 different ways (including Faux Lips Technique). This class is packed with training, information, and includes Hands On Training. You will be working on live models for 2 full days. Class includes home study and 3 days of class study. You will graduate with your AIIC Certification as AIIC Basic Intradermal Cosmetic Technician.
Learn everything you need to know about Permanent Cosmetics! Complete specialty course that gives you PROFESSIONAL TRAINING. This training includes basics, advancing to advanced procedures, advancing to Paramedical procedures, advancing to correcting permanent cosmetics. Course includes many programs on the market all in one class to save you money. Course is full of information and training and opens up the possibility of many wonderful things in the industry.
100 hours of training, lots of education & hands on training packed into a short amount of time. Includes Basic Marketing and Portfolio Building to help students get started.
Includes class supplies and Eii Certificate (which is required by state and insurance companies, doctor offices, and other professional occupations).
Class is offered with home study (we give you the supplies and books to study at home) or you can take it online – self paced (replaces book study and includes videos). You come to the school for 5 days hands on training on live models. Student brings models. INCLUDES BASIC STARTER KIT.
Charge professional rates. Increase your services to offer your clients or start your own business doing Permanent Cosmetics! Work in spas, salons, doctor's offices, medical spas, or start your own business. 100 hours divided up into online training, homework, assignments, and class time.
After 3 days of enrollment, your online access will be emailed to you to begin your course. You will study at home online or you're welcome to study online in Eii's Computer Lab, course is self-paced. We offer email customer support and phone support is available Monday – Friday 8:00 am – 4:00 pm Arizona time. Upon completion of online class study which includes review, practice with PMU pens and inks, equipment use and maintenance tips, educational videos, PowerPoint, and more. Course includes anatomy, physiology, blood borne and sanitation training (certificate is extra), and specialty training in the areas below in blue box. This class includes hands on training for 5 days, which will have refreshed theory, tips, and working on live models.
Learn the newest technique of Stardust Eyes. Learn how to safely apply a smoky look to your clients. New technique safely applies multiple colors to give the shimmering look. 1-day course includes online training and 1 client procedure.
Learn the newest technique of Stardust Lips. Learn how to safely apply the pouty look to your clients. New technique safely applies multiple colors to give the shimmering look. 1-day course includes online training and 1 client procedure.
Learn the newest technique of Ombre Eyebrows. Learn how to safely apply shading of Ombre to the brows. New technique safely applies the shimmering look. ½ day course includes online and ½ day in class working on practice skin. 1-day course includes online training and 1 client procedure.
Are you looking to get your Master Certificate in Permanent Makeup? Program trains for excellence, fills in gaps, artist will demonstrate knowledge with proven written exam, practical performance on models, oral exams, and demonstration of qualifications on being a Masters Technician. Program is online and in person for 3 days. Qualifications apply, please call for details.
All skin types are different and many technicians restrict themselves to not work on dark skin due to the unknown. This course teaches you the skin types, complications in skin types, color theory, scarring, and more. The course is for beginners and advanced technicians.
Advanced procedures that are used in the medical offices, medical spas, and salons. Procedures that helps boost the confidence in your clients! Program begins with online training in hospital and medical exam room set up, sanitation, medical setting, and working in the medical field. Course advances into technical skin complications that prevent or challenges micropigmentation. Skin conditions and how permanent makeup can help the visual effects of the blemishes. Learn paramedical permanent cosmetic procedures for the advanced technician. 3D Areolas, Corrections of the Areolas, Scar Camouflage, Blemish Cover Up, Age Spot Concealer, Stretch Marks, Skinsations, Clift Lip, Faux Lip, and now offering Dermal Rolling, Needling, and more!
Learn how to correct mistakes with our Permanent Makeup Correction Class and learn how to remove permanent makeup and tattoos! Two (2) days of Hands On live models training. You will receive a Certificate of Completion and continued education in Permanent Cosmetics.
Advance your skills and learn the NEWEST techniques of Scalp Micropigmentation. This class is to be added onto prior experience and is considered to be an advance technique. Price includes hands on training and basic starter kit. Scalp micropigmentation is an art where pigment is deposited into the hair line with a technique that simulates the appearance of hair. This technique requires professional training and skills to master the effect. With our training, you can master the effect and offer your clients a life changing result procedure.
This service is the fasting growing service that is in demand by women and men all over the world. It is the quickest solution to balding. It builds confidence and improves your client's life.
Advance your skills and learn the NEWEST techniques of Microblading (Feather Touch, 3D). This class is to be added onto prior experience and is considered to be an advance technique. Tuition includes hands on training and basic starter kit. Technique is done by hand.
Microblading brows has been around for many years, today it's the world's largest request by your clients. It is not the traditional hair stroke technique or popular hair stroke technique. This technique is considered Advanced Permanent Cosmetics and the results are done with special training, practice, and techniques. Eii trains you the latest techniques, with hands on training. You will learn the proper pressure, tips and skills in brow shapes, tips and skills on color, applying ink, making the ink stay, and many other hands on skills. Our classes include class supplies and Starter Kit. Our policy is SAFETY first.
Advance your skills and learn the NEWEST techniques of Micro Blading in our 4 Techniques (Traditional Microblading, Hybrid, Powder, and 3D). Learn Basic Permanent Makeup ONLINE first, includes Traditional Brows, Traditional Eyeliner, Traditional Lips, Blood Borne Pathogens Training, Anatomy & Physiology, Machines, Needles, and more. Advance into Microblading Brows, with mapping the brows, techniques, and the latest skills to apply microblading. Price includes 5 or 6 days of hands on training and basic microblading starter kit. Does not include Permanent Makeup Kit or machine. Microblading Technique is done by hand. Basic Permanent Makeup is done with pen (machine). Our policy is SAFETY first.
PLEASE NOTE: It is the responsibility of the client/student to check their state for all laws and licensing that pertains to their state. Eii is held harmless of any liability or responsibility. | {
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Title: UEFA Intertoto Cup
Subject: FC Tatabánya, Greek football clubs in European competitions 2000–09, FC Spartak Trnava, Silkeborg IF, AF Gloria Bistriţa
Collection: Defunct Uefa Club Competitions, Recurring Events Disestablished in 2008, Recurring Sporting Events Established in 1961, Uefa Intertoto Cup
1961 (taken over by UEFA in 1995)
Europe (UEFA)
Number of teams
Most successful club(s)
Stuttgart (3 titles)
The UEFA Intertoto Cup, also abbreviated as UI Cup and originally called the International Football Cup was a summer football competition for European clubs that had not qualified for one of the two major UEFA competitions, the Champions League and the UEFA Cup. The competition was discontinued after the 2008 tournament.[1] Teams who originally would have entered the Intertoto Cup now directly enter the qualifying stages of the UEFA Europa League from this point.
The tournament was founded in 1961–62, but was only taken over by UEFA in 1995.
Any club who wished to participate had to apply for entry, with the highest placed clubs (by league position in their domestic league) at the end of the season entering the competition. The club did not have to be ranked directly below the clubs which had qualified for another UEFA competition; if the club which was in that position did not apply, they would not be eligible to compete, with the place instead going to the club which did apply.
The cup billed itself as providing both an opportunity for clubs who otherwise would not get the chance to enter the UEFA Cup and as an opportunity for sports lotteries (or pools) to continue during the summer.[2] This reflects its background, which was as a tournament solely for football pools. In 1995, the tournament came under official UEFA sanctioning[3] and UEFA Cup qualification places were granted. Initially, two were provided; this was increased to three after one year; but in 2006, it was again increased to the final total of 11.
Winners 3
Winners by years 3.1
2006–2008 3.1.1
1967–94 3.1.3
Non-Region System (1969, 1971-94) 3.1.3.1
Region System (1967, 1968, 1970) 3.1.3.2
Winners by nation 3.2
Organized by UEFA 3.2.1
Overall 3.2.2
The Intertoto Cup was the idea of Malmö FF chairman Eric Persson and the later FIFA vice-president and founder of the Inter-Cities Fairs Cup, Ernst B. Thommen, and the Austrian coach Karl Rappan, who coached the Swiss national team at the 1938 FIFA World Cup and at the 1954 World Cup.[2] The "Cup for the Cupless" was also heavily promoted by the Swiss newspaper Sport. It derived its name from Toto, the German term for football pool.
Thommen, who had set up football betting pools in Switzerland in 1932, had a major interest in having purposeful matches played in the summer break. UEFA were initially disinclined to support the tournament, finding its betting background distasteful; nevertheless they permitted the new tournament but refrained from getting officially involved.[2] Clubs which qualified for one of the official continental competitions, such as the European Champions Cups and Cup Winners Cup, were not allowed to participate.
The first tournament was held in 1961 as the International Football Cup (IFC). Initially the Cup had a group stage, which led to knock-out matches culminating in a final. By 1967, it had become difficult to organize the games,[3] and so the knock-out rounds and the final were scrapped, leaving the tournament without a single winner. Instead, group winners received prizes of CHF10,000-15,000.
By 1995, UEFA had reconsidered its opinion, took official control of the tournament and changed its format. Initially, two winners were given a place in the UEFA Cup. The success of one of the first winners, Bordeaux, in reaching the final of the 1995–96 UEFA Cup encouraged UEFA to add a third UEFA Cup place in 1996.[3]
Many clubs disliked the competition and saw it as disruptive in the preparation for the new season. As a consequence, they did not nominate themselves for participation even if entitled. In particular, following its 1995 relaunch, clubs in England were sceptical about the competition; after initially being offered three places in the cup, all English top division teams rejected the chance to take part.[4] Following the threat of bans of English teams from all UEFA competitions,[4] the situation was eventually resolved with three English clubs entering weakened teams, and none of them qualifying.
In following years, UEFA made it possible for nations to forfeit Intertoto places. For example, in 1998, Scotland, San Marino and Moldova forfeited their places, and England, Portugal, and Greece forfeited one of their two, Crystal Palace being the sole English entrant despite finishing bottom of the Premier League.[5] Other clubs have built upon their success in the UI Cup, following it up with great campaigns in the UEFA Cup. Furthermore, UEFA rejected this assertion that the tournament is disruptive. They point out that in the 2004–05 season, two of the three 2004 Intertoto Cup winners went on to qualify directly for the Champions League, whilst the 3rd one qualified by winning its 3rd qualifying round tie (Schalke and Lille directly, Villarreal by winning their 3rd qualifying round tie).[3]
In December 2007, following the election of new UEFA president Michel Platini, it was announced that the Intertoto Cup would be abolished as of 2009. This was a part of a range of changes that were to be made to the UEFA Cup/Champions League System. Instead of teams qualifying for the Intertoto Cup, they will now qualify directly for the qualifying stages of the UEFA Europa League, which was expanded to four rounds to accommodate them.
When the competition was taken over by UEFA in 1995, the format was both a group stage and a knock-out stage; 60 teams were split into 12 groups of five with the 16 best teams then contesting the knock-out stage with two-legged ties at each stage, the two winning finalists qualifying for the UEFA Cup. In 1996 and 1997, just the 12 group winners entered the knock-out round, with now three finalists advancing. Nations were allocated places according to their UEFA coefficients, much as with other UEFA tournaments.
The group stage was scrapped for the 1998 tournament, which became a straight knock-out tournament, with clubs from more successful nations entering at a later stage. This arrangement lasted until 2005.
From the 2006 tournament, the format for the Cup changed. There were three rounds instead of the previous five, and the 11 winning teams from the third round went through to the second qualifying round of the UEFA Cup.[6] The clubs which were furthest in the UEFA Cup would each be awarded with a trophy.[7] The first club that received that trophy (a plaque) was Newcastle United.[8]
Only one team from each national association was allowed to enter. However, if one or more nations did not take up their place, the possibility was left open for nations to have a second entrant. Seedings and entry were determined by each association.[6] Teams from the weakest federations entered at the first round stage, while those from mid-level federations entered in the second round, and those from the strongest federations entered in the third round.
Winners by years
Listed are all 11 teams that won the Intertoto Cup, qualifying for the UEFA Cup. The outright winners (determined by the best performance in the UEFA Cup) are marked in bold.
2008 Braga Aston Villa Deportivo La Coruña Stuttgart Rosenborg Napoli
Rennes Vaslui Elfsborg Grasshopper Sturm Graz
2007 Hamburger SV Atlético Madrid AaB Sampdoria Blackburn Rovers Lens
Leiria Rapid Wien Hammarby IF Oţelul Galaţi Tobol
2006 Newcastle United Auxerre Grasshopper OB Marseille Hertha BSC
Kayserispor Ethnikos Achna Twente Ried Maribor
The results shown are the aggregate total over two legs.
Listed are all 2-3 teams that won the final matches, qualifying them for the UEFA Cup.
2005 Hamburg Valencia 1 – 0
Lens CFR Cluj 4 – 2
Marseille Deportivo La Coruña 5 – 3
2004 Lille Leiria 2 – 0 (after extra time)
Schalke 04 Slovan Liberec 3 – 1
Villarreal Atlético Madrid 2 – 2 (3 – 1 on penalties)
2003 Schalke 04 Pasching 2 – 0
Villarreal Heerenveen 2 – 1
Perugia Wolfsburg 3 – 0
2002 Málaga Villarreal 2 – 1
Fulham Bologna 5 – 3
Stuttgart Lille 2 – 1
2001 Aston Villa Basel 5 – 2
Paris Saint-Germain Brescia 1 – 1 (a)
Troyes Newcastle United 4 – 4 (a)
2000 Udinese Sigma Olomouc 6 – 4
Celta de Vigo Zenit St. Petersburg 4 – 3
Stuttgart Auxerre 3 – 1
1999 Montpellier Hamburg 2 – 2 (3 – 0 on penalties)
Juventus Rennes 4 – 2
West Ham United Metz 3 – 2
1998 Valencia Austria Salzburg 4 – 1
Werder Bremen Vojvodina 2 – 1
Bologna Ruch Chorzów 3 – 0
1997 Bastia Halmstad 2 – 1
Lyon Montpellier 4 – 2
Auxerre Duisburg 2 – 0
1996 Karlsruhe Standard Liège 3 – 2
Guingamp Rotor Volgograd 2 – 2 (a)
Silkeborg Segesta 2 – 2 (a)
1995 Strasbourg Tirol Innsbruck 7 – 2
Bordeaux Karlsruhe 4 – 2
During this time there were no competition winners, as only group stages were contested. The outright winners (determined by their best champions) are marked in bold.
Non-Region System (1969, 1971-94)
1994 Halmstad Young Boys AIK Hamburg Békéscsaba Slovan Bratislava Grasshopper Austria Wien – – – –
1993 Rapid Wien Trelleborg Norrköping Malmö Slavia Prague Zürich Young Boys Dynamo Dresden – – – –
1992 Copenhagen Siófok Bayer Uerdingen Karlsruher SC Rapid Wien Lyngby Slovan Bratislava Aalborg Slavia Prague Lokomotiv Gorna Oryahovitsa – –
1991 Neuchâtel Xamax Lausanne-Sports Austria Salzburg Dukla Banská Bystrica Boldklubben 1903 Grasshopper Bayer Uerdingen Dunajská Streda Tirol Innsbruck Örebro – –
1990 Neuchâtel Xamax Tirol Innsbruck Lech Poznań Slovan Bratislava Malmö GAIS Luzern First Vienna Chemnitz Bayer Uerdingen Odense –
1989 Luzern Boldklubben 1903 Tirol Innsbruck Grasshopper Tatabánya Næstved Örebro Sparta Prague Baník Ostrava Örgryte Kaiserslautern –
1988 Malmö Gothenburg Baník Ostrava Austria Wien Young Boys Kaiserslautern Ikast FS Carl Zeiss Jena Grasshopper Karlsruher SC Bayer Uerdingen –
1987 Carl Zeiss Jena Pogoń Szczecin Wismut Aue Tatabánya Malmö AIK Etar Veliko Tarnovo Brøndby – – – –
1986 Fortuna Düsseldorf Union Berlin Malmö Rot-Weiss Erfurt Sigma Olomouc Újpesti Dózsa Brøndby Lyngby Lech Poznań Gothenburg Slavia Prague Carl Zeiss Jena
1985 Werder Bremen Rot-Weiss Erfurt Gothenburg AIK Wismut Aue Sparta Prague Górnik Zabrze Maccabi Haifa Baník Ostrava Újpesti Dózsa MTK Hungária –
1984 Bohemians Prague AGF Fortuna Düsseldorf Standard Liège AIK Malmö Videoton Maccabi Netanya Zürich GKS Katowice – –
1983 Twente Young Boys Pogoń Szczecin Maccabi Netanya Sloboda Tuzla Bohemians Prague Gothenburg Hammarby Fehérvár Vítkovice – –
1982 Standard Liège Widzew Łódź AGF Lyngby Admira Wacker Mödling Bohemians Prague Brage Öster Gothenburg – – –
1981 Wiener Sportclub Standard Liège Werder Bremen Budućnost AGF Molenbeek Gothenburg Stuttgarter Kickers Cheb – – –
1980 Standard Liège Bohemians Prague Maccabi Netanya Sparta Prague Nitra Halmstad Malmö FF Gothenburg Elfsborg – – –
1979 Werder Bremen Grasshopper Eintracht Braunschweig Bohemians Prague Spartak Trnava Zbrojovka Brno Pirin Blagoevgrad Baník Ostrava – – – –
1978 Duisburg Slavia Prague Hertha Berlin Eintracht Braunschweig Malmö FF Lokomotiva Košice Tatran Prešov Maccabi Netanya Grazer AK – – –
1977 Halmstad Duisburg Internacionál Bratislava Slavia Sofia Slavia Prague Frem Jednota Trenčín Slovan Bratislava Öster Pogoń Szczecin – –
1976 Young Boys Hertha Berlin Union Teplice Baník Ostrava Zbrojovka Brno Spartak Trnava Internacionál Bratislava Öster Djurgården Vojvodina Widzew Łódź –
1975 Tirol Innsbruck VÖEST Linz Eintracht Braunschweig Zagłębie Sosnowiec Zbrojovka Brno Rybnik Åtvidaberg Kaiserslautern Belenenses Čelik Zenica – –
1974 Zürich Hamburg Malmö FF Standard Liège Slovan Bratislava Spartak Trnava Duisburg Baník Ostrava Košice CUF – –
1973 Hannover Slovan Bratislava Hertha Berlin Zürich Rybnik Union Teplice Feyenoord Wisła Kraków Nitra Öster – –
1972 Nitra Norrköping Saint-Étienne Slavia Prague Slovan Bratislava Eintracht Braunschweig Hannover VÖEST Linz – – – –
1971 Hertha Berlin Stal Mielec Servette Třinec Åtvidaberg Eintracht Braunschweig Austria Salzburg – – – – –
1969 Malmö FF Szombierki Bytom SpVgg Fürth Žilina Norrköping Jednota Trenčín Frem Wisła Kraków Odra Opole – – –
Region System (1967, 1968, 1970)
Group A1
Group B1
1970 Slovan Bratislava Hamburg Union Teplice MVV Košice – Eintracht Braunschweig Slavia Prague Marseille Öster Wisła Kraków Austria Salzburg Baník Ostrava Polonia Bytom
1968 Nuremberg Ajax Sporting Feyenoord Español ADO Den Haag Karl-Marx-Stadt Empor Rostock Slovan Bratislava Košice Lokomotíva Košice Odra Opole Eintracht Braunschweig Legia Warsaw
1967 Lugano Feyenoord Lille Lierse – – Hannover Zagłębie Sosnowiec Polonia Bytom Gothenburg Ruch Chorzów Košice KB Fortuna Düsseldorf
The results shown are the aggregate total over two legs unless otherwise noted.
1966–67 Eintracht Frankfurt Inter Bratislava 4 – 3
1965–66 Lokomotive Leipzig IFK Norrköping 4 – 1
1964–65 Polonia Bytom SC Leipzig 5 – 4
1963–64 Inter Bratislava Polonia Bytom 1 – 0*
1962–63 Inter Bratislava Padova 1 – 0*
1961–62 Ajax Feyenoord 4 – 2*
* - Single match finals (although 1962–63 has been unofficially reported (http://www.rsssf.com/tablesi/intertoto.html as over two legs)
Winners by nation
From 2006 onwards, the final round was no longer termed as the "Final", but instead simply as the "Third Round". In addition, there were 11 winners, compared to three under the old system. The clubs which progressed furthest in the UEFA Cup were awarded with a trophy (plaque).
Organized by UEFA
Winning Clubs
Runner-Up Clubs
France 16 5 Auxerre (2), Lens (2), Marseille (2), Bastia, Bordeaux, Guingamp, Lille, Lyon, Montpellier, Paris Saint-Germain, Rennes, Strasbourg, Troyes Auxerre, Lille, Metz, Montpellier, Rennes
Germany 10 4 Stuttgart (3), Hamburg (2), Schalke 04 (2), Hertha Berlin, Karlsruhe, Werder Bremen Duisburg, Hamburg, Karlsruhe, Wolfsburg
Spain 7 5 Villarreal (2), Atlético Madrid, Celta de Vigo, Málaga, Valencia, Deportivo Villarreal (2), Atlético Madrid, Deportivo, Valencia
Italy 6 2 Bologna, Juventus, Napoli, Perugia, Sampdoria, Udinese Bologna, Brescia
England 6 1 Aston Villa (2), Blackburn Rovers, Fulham, Newcastle United, West Ham United Newcastle United
Austria 3 3 Rapid Vienna, Ried, Sturm Graz SV Pasching, Salzburg, Tirol Innsbruck
Denmark 3 1 Silkeborg Odense
Romania 2 3 Oţelul Galaţi, Vaslui Cluj, Farul Constanţa, Gloria Bistriţa
Sweden 2 2 Hammarby Halmstad, Kalmar
Portugal 2 1 Braga, Leiria Leiria
Switzerland 2 1 Grasshoppers (2) Basel
Netherlands 1 3 Twente Heerenveen, NAC Breda, Utrecht
Turkey 1 2 Kayserispor Sivasspor, Trabzonspor
Norway 1 1 Rosenborg Lillestrøm
Cyprus 1 Ethnikos Achna
Kazakhstan 1 Tobol Kostanay
Slovenia 1 Maribor
Russia 5 FC Moscow, Rotor Volgograd, Rubin Kazan, FC Saturn, Zenit St. Petersburg
Belgium 3 Gent (2), Standard Liège
Greece 3 Larissa, OFI Crete, Panionios
Ukraine 3 Chornomorets Odessa, Dnipro Dnipropetrovsk, Tavriya Simferopol
Bulgaria 2 Cherno More Varna, Chernomorets Burgas
Czech Republic 2 Sigma Olomouc, Slovan Liberec
Israel 2 Bnei Sakhnin, Maccabi Petah Tikva
Moldova 2 Dacia Chişinău, Tiraspol
Azerbaijan 1 Neftchi Baku
Croatia 1 Segesta
FR Yugoslavia 1 Vojvodina
Hungary 1 Budapest Honvéd
Latvia 1 FK Rīga
Lithuania 1 Vėtra
Poland 1 Ruch Chorzów
Scotland 1 Hibernian
Serbia 1 Hajduk Kula
Winning and Group Champion Clubs
Runner-Up and Group Runners-Up Clubs
Czechoslovakia 62 34 Slovan Bratislava (8), Banik Ostrava (7), Bohemians Prague (6), Slavia Prague (6), Inter Bratislava (4), Košice (4), Nitra (3), Sparta Prague (3), Spartak Trnava (3), Union Teplice (3), Zbrojovka Brno (3), Jednota Trencin (2), Lokomotiva Kosice (2), DAC Dunajská Streda, Dukla Banská Bystrica, Cheb, Sigma Olomouc, Tatran Prešov, Třinec, Vítkovice, Žilina Slavia Prague (5), Bohemians Prague (3), Cheb (3), Inter Bratislava (3), Nitra (2), Sigma Olomouc (2), Sparta Prague (2), Spartak Trnava (2), Zbrojovka Brno (2), Žilina (2), DAC Dunajská Streda, Dukla Prague, Jednota Trencin, Košice, Slovan Bratislava, Tatran Prešov, Union Teplice, Vítkovice
Germany 50 46 Eintracht Braunschweig (7), Hamburg (5), Hertha Berlin (5), Bayer Uerdingen (4), Werder Bremen (4), Duisburg (3), Fortuna Düsseldorf (3), Hannover 96 (3), Kaiserslautern (3), Karlsruhe (3), Stuttgart (3), Schalke 04 (2), Dynamo Dresden, Eintracht Frankfurt, Nuremberg, SpVgg Fürth, Stuttgarter Kickers Duisburg (5), Kaiserslautern (5), Werder Bremen (5), Arminia Bielefeld (3), Bayer Leverkusen (3), Hertha Berlin (3), Bochum (2), Fortuna Düsseldorf (2), Hannover 96 (2), Karlsruhe (2), Saarbrücken (2), 1860 Münich, Bayer Uerdingen, Borussia Dortmund, Eintracht Braunschweig, Eintracht Frankfurt, Hallescher, Hamburg, Kickers Offenbach, Lokomotive Leipzig, Schalke 04, Stuttgarter Kickers, Wolfsburg
Sweden 46 28 Malmö FF (10), IFK Göteborg (8), Öster (5), AIK (4), Halmstad (3) IFK Norrköping (3), Atvidaberg (2), Elfsborg (2), Hammarby (2), Örebro (2), Brage, Djurgarden, GAIS, Örgryte, Trelleborg Malmö FF (8), Atvidaberg (2), IFK Göteborg (2), IFK Norrköping (2), Kalmar (2), Örgryte (2), Öster (2), Djurgarden, Häcken, Halmstad, Hammarby, Helsingborg, Landskrona, Örebro, Trelleborg
Poland 25 27 Pogoń Szczecin (3), Polonia Bytom (3), Wisla Kraków (3), Lech Poznań (2), Odra Opole (2), ROW Rybnik (2), Widzew Łódź (2), Zaglebie Sosnowiec (2), Górnik Zabrze, Katowice, Legia Warsaw, Ruch Chorzów, Szombierki Bytom Zaglebie Sosnowiec (4), Górnik Zabrze (2), Gwardia Warsaw (2), Katowice (2), Legia Warsaw (2), Polonia Bytom (2), Ruch Chorzów (2), Szombierki Bytom (2), Wisla Kraków (2), Lech Poznań, LKS Łódź, Odra Opole, Pogoń Szczecin, ROW Rybnik, Widzew Łódź, Zawisza Bydgoszcz
Switzerland 22 15 Grasshopper (6), Young Boys (5), Zürich (4), Luzern (2), Neuchâtel Xamax (2), Lausanne Sports, Lugano, Servette Grasshopper (4), Lausanne Sports (2), Zürich (2), Aarau, Basel, Grenchen, Lugano, Sion, St. Gallen, Young Boys
Denmark 21 30 AGF (3), Lyngby (3), Aalborg (2), B 1903 (2), Brøndby (2), Frem (2), Odense (2), Copenhagen, Ikast, KB, Næstved, Silkeborg Odense (7), AGF (4), KB (4), Vejle (4), Brøndby (2), Esbjerg (2), Lyngby (2), Næstved (2), Frem, Hvidovre, Silkeborg
Austria 20 32 Wacker/Tirol Innsbruck (4), Rapid Vienna (3), Salzburg (3), Ried, Sturm Graz, Austria Vienna (2), VÖEST Linz (2), Admira, First Vienna, Grazer AK, Ried, Sturm Graz, Wiener Sportclub Sturm Graz (5), Wacker/Tirol Innsbruck (5), LASK Linz (4), Admira (3), Austria Vienna (3), First Vienna (3), Salzburg (3), VÖEST Linz (2), Austria Klagenfurt, Pasching, Rapid Vienna, Wiener Sportclub
France 19 9 Marseille (3), Auxerre (2), Lens (2), Lille (2), Bastia, Bordeaux, Guingamp, Lyon, Montpellier, Paris Saint-Germain, Rennes, Saint-Étienne, Strasbourg, Troyes Auxerre, Bordeaux, Caen, Lille, Metz, Montpellier, RCF Paris, Rennes, Saint-Étienne
East Germany 12 9 Carl Zeiss Jena (3), Chemnitz/Karl-Marx-Stadt (2), Rot-Weiss Erfurt (2), Wismut Aue (2), Empor Rostock, Lokomotive Leipzig, Union Berlin Lokomotive Leipzig (3), Carl Zeiss Jena (2), Chemnitz/Karl-Marx-Stadt (2), Dynamo Dresden, Magdeburg
Hungary 9 12 Tatabánya (2), Újpest (2), Videoton (2), Békéscsaba, MTK, Siófok Vác (3), Honvéd (2), Videoton (2), Győr, MTK, Pécsi, Siófok, Zalaegerszegi
Netherlands 9 11 Feyenoord (3), Ajax (2), Twente (2), ADO Den Haag, MVV ADO Den Haag (3), Armsterdam, Feyenoord, Groningen, Heerenveen, NAC Breda, PSV, Twente, Utrecht
Spain 8 5 Villarreal (2), Atlético Madrid, Celta de Vigo, Deportivo La Coruña, Español, Málaga, Valencia Villarreal (2), Atlético Madrid, Deportivo La Coruña, Valencia
Belgium 7 15 Standard Liège (5), Lierse, Molenbeek Standard Liège (8), Gent (2), Anderlecht, Beveren, Liège, Molenbeek, Royal Antwerp
Italy 6 4 Bologna, Juventus, Napoli, Perugia, Sampdoria, Udinese Bologna, Brescia, Padova, Torino
Israel 5 6 Maccabi Netanya (4), Maccabi Haifa (1) Maccabi Haifa (2), Bnei Sakhnin, Hapoel Be'er Sheva, Hapoel Tel Aviv, Maccabi Petah Tikva
Portugal 5 6 Belenenses, Braga, CUF, Leiria, Sporting Vitória Guimarães (2), Belenenses, CUF, Leiria, Vitória Setúbal
Bulgaria 4 13 Etar Veliko Tarnovo, Lokomotiv Gorna Oryahovitsa, Pirin Blagoevgrad, Slavia Sofia Pirin Blagoevgrad (3), Slavia Sofia (3), Chernomorets Burgas (2), Lokomotiv Sofia (2), Cherno More Varna, Marek Dupnitsa, Spartak Varna
Yugoslavia 4 6 Budućnost, Čelik Zenica, Sloboda Tuzla, Vojvodina Vojvodina (3), Olimpija Ljubljana, Rad, Sloboda Tuzla
Romania 2 5 Oţelul Galaţi, Vaslui Rapid Bucureşti (2),CFR Cluj, Farul Constanţa, Gloria Bistriţa
Norway 1 7 Rosenborg Bryne (2), Lillestrøm (2), Vålerenga (2), Viking
Czech Republic 1 4 Slavia Prague Sigma Olomouc (2), Slavia Prague, Slovan Liberec
Slovakia 1 1 Slovan Bratislava Slovan Bratislava
Russia 5 FC Moscow, Rotor Volgograd, Rubin Kazan, Saturn, Zenit St. Petersburg
Latvia 1 Riga
List of UEFA Intertoto Cup winning managers
UEFA club competition records
^ Chaplin, Mark (2007-12-01). "Champions League changes agreed". uefa.com. Retrieved 2011-02-14.
^ a b c Elbech, Søren Florin. "Background on the Intertoto Cup". Retrieved 2006-06-07.
^ a b c d "UEFA Intertoto Cup history". UEFA.com. Archived from the original on 2006-05-03. Retrieved 2006-06-07.
^ a b "Intertoto Cup: English Joy". Retrieved 2006-06-07.
^ "1998 Intertoto Cup Draw". EuroFutbal Archive. Retrieved 2006-06-07.
^ a b "New look for Intertoto Cup". UEFA.com. Archived from the original on 2007-01-01. Retrieved 2007-02-20.
^ "Regulations of the Intertoto Cup 2006" (
^ "Newcastle to lift Intertoto Cup". BBC Sport. 2006-12-16. Retrieved 2007-02-20.
Official UEFA site
Official lotteries site
Soccernet guide to Intertoto Cup: Part 1 and Part 2
(Italian) Enrico Siboni Web Site - Winners of UEFA Intertoto Cup
UEFA Intertoto Cup seasons
Not UEFA-administered (with knockout rounds)
Not UEFA-administered (only group stages)
UEFA-administered – winners progress to UEFA Cup
Winning managers
UEFA Intertoto Cup winners
1995: Bordeaux
1996: Karlsruher
1997: Auxerre
1998: Bologna
1999: Juventus
2000: Celta
2001: Paris Saint-Germain
2002: Málaga
2003: Villarreal
2005: Hamburg
2006: Newcastle United
2008: Braga
UEFA competitions
Nations League (proposed)
Futsal Championship (U-21 (defunct))
Meridian Cup (defunct)
Cup Winners' Cup (defunct)
Intertoto Cup (defunct)
Intercontinental Cup (defunct)
Intercontinental Champions' Supercup (defunct)
Women's Champions League
Regions' Cup
Amateur Cup (defunct)
Club competition records and statistics
Club competition winning teams
Club competition winning managers
International club football
List of association football clubs
CAF – Champions League
Confederation Cup
Top-division clubs
AFC – Champions League
UEFA – Champions League
North America,
and the Caribbean
CONCACAF – Champions League
OFC – Champions League
CONMEBOL – Copa Libertadores
Articles with Italian-language external links
Recurring sporting events established in 1961
Recurring events disestablished in 2008
Defunct UEFA club competitions
Swedish language, European Union, Finland, Denmark, Lithuania
United Kingdom, European Union, Italy, Canada, Spain
FC Tatabánya
Hungary, UEFA Intertoto Cup, Switzerland, Sweden, Slovakia
Greek football clubs in European competitions 2000–09
UEFA Champions League, Olympiacos F.C., Panathinaikos F.C., A.E.K. Athens F.C., P.a.o.k. F.c.
Slovakia, Czechoslovak First League, Slovak Super Liga, Midfielder, Czech Republic
Denmark, UEFA Intertoto Cup, Danish 1st Division, Danish Superliga, Danish Cup | {
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The sun is shining, kids are out of school, and parents are making a mad dash to find some fun summer activities for their kids! Look no further moms and dads, Macy's has you covered–Yes, Macy's!
June 14, at 2pm at Macy's Modesto Vintage Fair shop and keep your kids occupied at Macy's as part of their American Icons campaign! Macy's is hosting Camp Macy's with activities reminiscent of camp. Instead of sending them to sleep away camp, try out Camp Macy's! Kids can enjoy the American spirit and enjoy a camp "fire" experience indoors through readings, storytelling, BINGO and snacks. There will also be crafts to show off their American spirit, refreshments, giveaways, and so much more family fun! What's more American and iconic than Macy's and summer camp?
Capture special moments while you're at the event and use #AmericanSelfie to be entered into a contest by Macy's! Macy's will donate $1 for each #AmericanSelfie up to $250, 000 dollars, to support America's veterans with Got Your 6. A handful of winners will be selected for a spectacular moment in Macy's Fourth of July Fireworks show on NBC! | {
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A unique exhibition of paintings and songs by Barry Kerr, performed by Muireann Nic Amhlaoibh, Pauline Scanlon, Niall Hanna, Síle Denvir, Niamh Dunne and Seán Óg Graham.
Barry Kerr is a visual artist, song-writer and musician whose deep love and understanding of Irish life and traditions informs his paintings and songs. This new work 'Continuum' tells visual and vernacular tales of the land and sea and of human connection and history. It is a fusion of Kerr's creative energies into one entity to create a unique visual and sound experience for audience and performers. These dynamic paintings vividly depict scenes from a new series of songs and they will be exhibited and performed on the night by some of Ireland's leading singers and musicians.
A feast for the senses, 'Continuum' combines the best of modern Irish folk song with evocative oil paintings to create an extraordinary journey into the artist's creative well.
Visit Barry Kerr's website here.
Book tickets for Continuum today! | {
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{"url":"http:\/\/parasys.net\/error-propagation\/error-progression-statistics.php","text":"# parasys.net\n\nHome > Error Propagation > Error Progression Statistics\n\n# Error Progression Statistics\n\n## Contents\n\nStructural and Multidisciplinary Optimization. 37 (3): 239\u2013253. For example, 400. Multivariate error analysis: a handbook of error propagation and calculation in many-parameter systems. SOLUTION The first step to finding the uncertainty of the volume is to understand our given information. http:\/\/parasys.net\/error-propagation\/error-progression.php\n\nNational Bureau of Standards. 70C (4): 262. Solution: Use your electronic calculator. doi:10.1007\/s00158-008-0234-7. ^ Hayya, Jack; Armstrong, Donald; Gressis, Nicolas (July 1975). \"A Note on the Ratio of Two Normally Distributed Variables\". If the result of a measurement is to have meaning it cannot consist of the measured value alone.\n\n## Error Analysis Statistics\n\nThe number to report for this series of N measurements of x is where . Sensitivity coefficients The partial derivatives are the sensitivity coefficients for the associated components. Retrieved 2016-04-04. ^ \"Propagation of Uncertainty through Mathematical Operations\" (PDF).\n\nFor the distance measurement you will have to estimate [[Delta]]s, the precision with which you can measure the drop distance (probably of the order of 2-3 mm). If a measurement is repeated, the values obtained will differ and none of the results can be preferred over the others. Journal of Sound and Vibrations. 332 (11). Statistical Standard Error If the uncertainties are correlated then covariance must be taken into account.\n\nSo if the angle is one half degree too large the sine becomes 0.008 larger, and if it were half a degree too small the sine becomes 0.008 smaller. (The change Error Propagation Statistics Accounting for significant figures, the final answer would be: \u03b5 = 0.013 \u00b1 0.001 L moles-1 cm-1 Example 2 If you are given an equation that relates two different variables and In the above linear fit, m = 0.9000 and\u03b4m = 0.05774. http:\/\/www.radford.edu\/~biol-web\/stats\/standarderrorcalc.pdf Disadvantages of Propagation of Error Approach Inan ideal case, the propagation of error estimate above will not differ from the estimate made directly from the measurements.\n\nSee Ku (1966) for guidance on what constitutes sufficient data2. Error Propagation Formula In this example, the 1.72 cm\/s is rounded to 1.7 cm\/s. The error propagation methods presented in this guide are a set of general rules that will be consistently used for all levels of physics classes in this department. Le's say the equation relating radius and volume is: V(r) = c(r^2) Where c is a constant, r is the radius and V(r) is the volume.\n\n## Error Propagation Statistics\n\nThis is desired, because it creates a statistical relationship between the variable $$x$$, and the other variables $$a$$, $$b$$, $$c$$, etc... http:\/\/chemwiki.ucdavis.edu\/Analytical_Chemistry\/Quantifying_Nature\/Significant_Digits\/Propagation_of_Error By using this site, you agree to the Terms of Use and Privacy Policy. Error Analysis Statistics The final result for velocity would be v = 37.9 + 1.7 cm\/s. Percent Error Statistics For example, repeated multiplication, assuming no correlation gives, f = A B C ; ( \u03c3 f f ) 2 \u2248 ( \u03c3 A A ) 2 + ( \u03c3 B\n\nIf we now have to measure the length of the track, we have a function with two variables. For numbers without decimal points, trailing zeros may or may not be significant. Guidance on when this is acceptable practice is given below: If the measurements of a and b are independent, the associated covariance term is zero. For example, if there are two oranges on a table, then the number of oranges is 2.000... . Statistical Error Analysis Definition\n\nIn the first step - squaring - two unique terms appear on the right hand side of the equation: square terms and cross terms. Caveats and Warnings Error propagation assumes that the relative uncertainty in each quantity is small.3 Error propagation is not advised if the uncertainty can be measured directly (as variation among repeated The sine of 30\u00b0 is 0.5; the sine of 30.5\u00b0 is 0.508; the sine of 29.5\u00b0 is 0.492. http:\/\/parasys.net\/error-propagation\/error-propagation-statistics.php Systematic errors are errors which tend to shift all measurements in a systematic way so their mean value is displaced.\n\nExact numbers have an infinite number of significant digits. Error Propagation Calculator These instruments each have different variability in their measurements. For a Gaussian distribution there is a 5% probability that the true value is outside of the range , i.e.\n\n## Note Addition, subtraction, and logarithmic equations leads to an absolute standard deviation, while multiplication, division, exponential, and anti-logarithmic equations lead to relative standard deviations.\n\nIn a probabilistic approach, the function f must usually be linearized by approximation to a first-order Taylor series expansion, though in some cases, exact formulas can be derived that do not They may be due to imprecise definition. Setting xo to be zero, v= x\/t = 50.0 cm \/ 1.32 s = 37.8787 cm\/s. Error Propagation Physics For numbers with decimal points, zeros to the right of a non zero digit are significant.\n\nText is available under the Creative Commons Attribution-ShareAlike License; additional terms may apply. To indicate that the trailing zeros are significant a decimal point must be added. in the same decimal position) as the uncertainty. Claudia Neuhauser.\n\nThe derivative with respect to t is dv\/dt = -x\/t2. Notes on the Use of Propagation of Error Formulas, J Research of National Bureau of Standards-C. Zeros between non zero digits are significant. All rights reserved.\n\nYou will sometimes encounter calculations with trig functions, logarithms, square roots, and other operations, for which these rules are not sufficient. However, if Z = AB then, , so , (15) Thus , (16) or the fractional error in Z is the square root of the sum of the squares of the This could only happen if the errors in the two variables were perfectly correlated, (i.e.. The exact covariance of two ratios with a pair of different poles p 1 {\\displaystyle p_{1}} and p 2 {\\displaystyle p_{2}} is similarly available.[10] The case of the inverse of a\n\nUsing Beer's Law, \u03b5 = 0.012614 L moles-1 cm-1 Therefore, the $$\\sigma_{\\epsilon}$$ for this example would be 10.237% of \u03b5, which is 0.001291. Propagation of error considerations\n\nTop-down approach consists of estimating the uncertainty from direct repetitions of the measurement result The approach to uncertainty analysis that has been followed up to this But in the end, the answer must be expressed with only the proper number of significant figures. Such accepted values are not \"right\" answers.\n\nPlease see the following rule on how to use constants. Experimental data may be qualitative or quantitative, each being appropriate for different investigations. Logger Pro If you are using a curve fit generated by Logger Pro, please use the uncertainty associated with the parameters that Logger Pro give you. Table 1: Arithmetic Calculations of Error Propagation Type1 Example Standard Deviation ($$\\sigma_x$$) Addition or Subtraction $$x = a + b - c$$ $$\\sigma_x= \\sqrt{ {\\sigma_a}^2+{\\sigma_b}^2+{\\sigma_c}^2}$$ (10) Multiplication or Division \\(x =\n\nExternal links A detailed discussion of measurements and the propagation of uncertainty explaining the benefits of using error propagation formulas and Monte Carlo simulations instead of simple significance arithmetic Uncertainties and Therefore, the ability to properly combine uncertainties from different measurements is crucial. H. (October 1966). \"Notes on the use of propagation of error formulas\". Obviously, it cannot be determined exactly how far off a measurement is; if this could be done, it would be possible to just give a more accurate, corrected value.\n\nThe value of a quantity and its error are then expressed as an interval x \u00b1 u. Propagation of Error http:\/\/webche.ent.ohiou.edu\/che408\/S...lculations.ppt (accessed Nov 20, 2009). Classification of Error Generally, errors can be divided into two broad and rough but useful classes: systematic and random.","date":"2018-03-20 17:17:17","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 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\": 0.7957404851913452, \"perplexity\": 840.4098508598937}, \"config\": {\"markdown_headings\": true, \"markdown_code\": false, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2018-13\/segments\/1521257647519.62\/warc\/CC-MAIN-20180320170119-20180320190119-00752.warc.gz\"}"} | null | null |
{"url":"https:\/\/puzzling.stackexchange.com\/questions\/6212\/justice-and-fairness-part-1\/8210","text":"# Justice and Fairness: Part 1 [closed]\n\nA friend of yours was part of a secret society and was put on trial on 15 December 2014 without a lawyer. Alice encrypted the trial's transcript and sent it to Bob, but double agent Carol was able to e-mail a copy of the ciphertext to you. Your friend desperately needed help from a lawyer, but you were unable to provide one with the plaintext so that you could relay the proper advice back to your friend.\n\nRESULTS OF TRIAL: Your friend was executed on 26 December 2014, and the rest of your friend's family followed on 20 January 2015.\n\nRecently, Carol was able to intercept communications containing a rather ominous message. It seems that your friend's secret society has learned of the work you and your associates have been conducting and are less than pleased. Your life appears to be in jeopardy unless you solve the mystery of your friend's trial by the end of this quarter. If you fail, you will find out if there is an afterlife.\n\nE-mail: Show\n\nCan you produce the plaintext for the message below to prevent your untimely demise?\n\nCiphertext: Show\n\nHint 1:\n\nAfter checking your e-mail, you find the following message in your inbox.\n\nE-mail: Show\n\nHint 2:\n\nYou receive another e-mail, but the source is completely different from yesterday.\n\nE-mail: Show\n\nHint 3:\n\nAnother cryptic e-mail has found its way into your inbox and keeps the theme of being indecipherable.\n\nE-mail: Show\n\nHint 4:\n\nA programmer working for you named Mallory has come up with a certain theory. He believes that the message sent by Dave is a base 10 representation of a base 27 number. When asked for an explanation, Mallory mumbles something about the power of the trinity.\n\nHint 5:\n\nWhile visiting your cubicle to get help on one of his assignments, a new intern named Oscar catches a glimpse of Eve's message on your computer screen and asks, \"Is that code from a hex editor?\" When you ask what he means, Oscar explains that the message looks like the output his disassembler has been giving him. \"Each pair of symbols can be thought of as a base 16 representation of a base 256 number.\"\n\nHint 6:\n\nThe IT documentation specialist Peggy has heard about your mysterious e-mails and calls you on your work extension. \"Victor, our e-mail administrator, mentioned at lunch last week that our firewall alerted him of some Ascii85 data transferred across the network and referenced a message from someone named Frank.\" You suddenly see Mallory peek over your cubicle's wall and give you the A-OK sign.\n\nHint 7:\n\nConsultants by the names of Sam and Sally have been brought in from Luminosum Defensione Ltd. After analyzing all available data, they have come to the conclusion that the attachment in Dave's e-mail is something known as an initialization vector. If they are correct, then it is a critical piece of information for understanding the ciphertext.\n\nHint 8:\n\nSybil is LDL's top analyst and has noticed a peculiar similarity in Eve's attachment and the ciphertext from Carol. When comparing the two with one above the other, there appears to be a correspondence between certain characters in the two messages. Though the text is unreadable, the structure is visibly mirrored when comparing them both together.\n\nHint 9:\n\nAmong all the bright people at LDL, Trent is the only one having a PhD in information theory. He is the doctor who everyone has been clamoring for, and he has deduced an impossible sounding explanation for the attachment in Frank's e-mail. Trent believes that the pickle contains the instructions to create an eighty-eight dimensional table having a length of fifty-two characters per side.\n\nHint 10:\n\nAn e-mail similar to the one from Tenebris Lamina is showing up as an unread message.\n\nE-mail: Show\n\nHint 11:\n\nThis must be the week people send weird e-mails to others ignorant of their existence.\n\nE-mail: Show\n\nHint 12:\n\nYet another e-mail has landed in your inbox without any explanation of what to do with it.\n\nE-mail: Show\n\nHint 13:\n\nBill is your company's primary lawyer and has a friend named Merlin he thinks might be able to assist with Alice's message. After five minutes of analyzing the message, Merlin exclaims, \"This ciphertext has an amber infinity; look at lines 19 and 24!\" When asked what an amber infinity is, he explains that \"we also call it a double infinity; the 88th alpha character on those lines are the same.\"\n\nHint 14:\n\nWord of your friend's family's fate has been getting around, and a telephone operator in customer service has taken special interest. Paul calls and tells you, \"I have been looking through the available data and have a theory. Could there be a connection between the 87 random alpha characters before Merlin's amber infinity and the 87 character primer?\"\n\nHint 15:\n\nA coworker and analyst named Carole has been studying the information gathered so far and has pointed out an interesting trend found in the observations made by the others. \"Trent, Merlin, and Paul have all noticed details that appear to be related to an amber infinity. Is there a possibility that lines 19 and 24 of the ciphertext, the primer, and the key are related?\"\n\nHint 16:\n\nAuditors have been hired from Myutsu Mutucsoecul Corporation to verify previous findings for the purpose of utilizing the encryption techniques being studied. Alloy is the investigation leader and seems especially interested in last week's research. \"The discovery Merlin made may have been the most valuable clue thus far, and I believe we should explore the connections Carole found.\"\n\nHint 17:\n\nOne of MMC's auditors has been studying notes from previous weeks and has drawn attention to Trent's theory. \"If what he says is true, the encryption table's size would be fifty-two raised to the power of eighty-eight.\" Cobalt finally blurts outs, \"No computer can store 10,192,817,301,005,542,286,466,232,471,675,496,120,406,975,910,800,636,068,660,404,884,973,826,682,736,169,865,768,988,355,997,729,823,378,974,012,695,621,304,952,079,063,967,426,432,230,164,063,584,256 units of data!\"\n\nHint 18:\n\nAnother MMC auditor, Blastus, disagrees with Cobalt's conclusion regarding the table believed to exist by the doctor. An alternative interpretation is that if one understands how the instructions should be interpreted and how the resulting table should be used, it may be possible to simulate its function without its construction. Blastus believes that the total space required would probably be less than double the space of the key's pickle.\n\nHint 19:\n\nThose from Nigrum Gladio have reached out to you again but neglected to include an instruction manual.\n\nE-mail: Show\n\nHint 20:\n\nAnother unread e-mail is in your inbox, and if the subject is any indication of its contents, it might be a self-fulfilling prophecy.\n\nE-mail: Show\n\nHint 21:\n\nWith all the e-mails you have been receiving, one might think that there is a certain theme to all this secrecy and madness.\n\nE-mail: Show\n\nHint 22:\n\nYour company's chief security expert thinks that the ciphertext characters may be related to each other. While testing the hypothesis is impossible, Mallet believes changing a single ciphertext letter would alter all remaining plaintext letters. His study of mathematics has led him to conclude that there is some type of relationship with Markov chains.\n\nHint 23:\n\nCharlie, the network administrator at this branch, runs continual automated analysis of all internal traffic and has found an interesting note on his reports. \"I don't know much about encryption, but the filters monitoring your case's data have been barkin' at me an electronic codebook tag. The information could potentially be processed in blocks at a time.\"\n\nHint 24:\n\nSeveral new employees were hired last week, and Trudy has quickly adjusted to her role as an information specialist. The time she spent working for Sanguine Shield obviously prepared her well for working on your friend's case, but Charlie's revelation yesterday appeared to come as a shock to her. \"If the IV is the first chunk's header, shouldn't we expect approximately A \/ B chunks where A and B are the lengths of the ciphertext and primer, respectively?\"\n\nHint 25:\n\nElite Security Consultation And Protection Enterprise was a small company operated by Pat and Vanna Clingan before merging a couple of weeks ago with your employer. Several years prior, ESCAPE established a business relationship with Cumulus Cote and still has limited access to the systems they were responsible for implementing. Pat is going to submit two similar messages to be encrypted by CC's servers so that the results can be analyzed, the process reverse engineered, and the cryptographic technology understood for future application.\n\nData: Show\n\nHint 26:\n\nA few seconds after Pat submitted the messages for processing, Vanna was able to retrieve the results produced by yesterday's experiment, but several hiccups were encountered along the way. The first problem comes from the fact that while the plaintext could be sent and the ciphertext received, there was no way to set the keys and primers used during the transaction. It seems that ESCAPE did a better job at security than they remembered because the second problem discovered was the inability to gain read access to the key and primer used to generate the text shown below.\n\nData: Show\n\nHint 27:\n\nAfter working for Shiners' Bait & Pet Store for a few years managing their web site and computer systems, Chuck claims he was bored and wanted to put his black hat skills to work for your company. He was hired two weeks ago and immediately put to work getting up to date on your friend's case while also collaborating with the former members of ESCAPE. After seeing the troubles they had with the key and primer, he quickly went to work infiltrating Cumulus Cote's security and was able to find a few backdoors and exploits that allowed him to recover the following information.\n\nData: Show\n\nHint 28:\n\nWhile talking with one of your coworkers, your computer mentions \"you've got mail\" in a pleasing voice.\n\nE-mail: Show\n\nHint 29:\n\nFor reasons unknown to you, the people at Egregie Obice still wish to contribute towards your efforts.\n\nE-mail: Show\n\nHint 30:\n\nMaybe if you can understand this e-mail, you might understand all past, present, and prophetic e-mails.\n\nE-mail: Show\n\n\u2022 For those not in the know, .pickle is python's way to store objects. I tried to unpickle the given data, in all conceivable settings, and all of them gave me a load error on key 'm'. BTW, it is NOT SAFE to unpickle arbitrary data. This can execute ARBITRARY CODE. Only try to unpickle this data on a separated virtual machine, or if you absolutely trust the puzzle author not to install a keylogger on your computer and then act exactly as if there was an error while unpickling. \u2013\u00a0Lopsy Dec 20 '14 at 14:03\n\u2022 After bashing at it for 30 minutes, I don't have any motivation to keep trying random stuff and hoping it comes up English. I think your friend is doomed. Butterfingers. See meta.puzzling.stackexchange.com\/a\/1718\/1752 for a guide to constructing code puzzles more people will want to try. \u2013\u00a0Lopsy Dec 24 '14 at 0:01\n\u2022 @NoctisSkytower This puzzle is getting impossible to solve. You're over complicating this. Just post an answer so we can see how this is done. I'm not going to try and solve every possible encryption standard. \u2013\u00a0QuyNguyen2013 Jan 19 '15 at 15:02\n\u2022 I'm with @QuyNguyen2013. The key to a successful puzzle is that there's some information gained from a successful step forward. For example, for the ASCII85 string, I got the same result as Quy did, and there's nothing clear to do with the resulting bytes. They're not clearly in the ASCII range, or standard unicode. They don't neatly form pairs. They don't fit in the range of numbers necessary to access a 52x88 matrix. In other words, the decoded bytes are just as useless as the enecoded ones were. \u2013\u00a0Bobson Jan 20 '15 at 22:00\n\u2022 Holy ****, THIRTY hints?!!! \u2013\u00a0Rand al'Thor Feb 25 '15 at 23:57\n\nThe puzzle author appears to have come up with their own encryption scheme. It doesn't appear to be documented anywhere, but there is sample code (in python) out there on a different site. Finding that is actually the hardest part since the key and primer needed to decipher the message are both given in the first few hints.\n\n### Solution\n\nContained herein are the arguments for and against re-admitting Jay_Darsener into Luceoscutum. Present are Steel01, Administrator of the site; Koballt10, Moderator of Totally Random; Gwythaint, Game Master; Jay_Darsener, the accused; Makosolider and Zero, interested parties in support of the accused.\nSo verified by:\nJay_Darsener\nSteel01\nGwythaint\nZero\nMakoSoldier626\nKoballt10\n\nWith my signature below, I hereby a) verify that all I say is truth, b) verify that any questions asked of me will be answered directly, barring any information that is available to moderators only, c) agree to abide by any and all terms that are established by this meeting, and d) agree to honor these terms not only in word, but in spirit, with the full knowledge that any wrongdoing on either side of this agreement will be punished to the fullest extent of the site rules, to be meted out by moderators and followed through completely (in the event that the Administrator or the Moderator do wrongly, moderators who are not present will mete out the penalty).\nSigned:\nGwythaint\nJay_Darsener\nMakoSoldier626\nZero\nSteel01\nKoballt10\nTo the defense:\nBy signing below, I hereby state that I am sufficiently represented in this meeting, and feel that I am being defended to the best ability of my representative.\nSigned:\nJay_Darsener\n\nTo the prosecution:\nBy signing below, I hereby state that I am sufficiently represented in this meeting, and feel that I am being just in my challenge of the defendant.\nSigned:\nSteel01\n\nThe purpose of the meeting (J) is to discuss why I have not been allowed back on the site\/ (S) to find out what has been going on since the original account has been banned until now\/ (G) to clear the air and cure the site of ill-will (Z)agrees with G (M) to get the site back to the original intentions--to RP without stress and drama.\n\"First, addressing history of the account. The two present at the original argument were Steel and Jay. Both need to tell their chronology of the issue.\" (G)\n(S)I do not remember the first post that sparked the whole thing. CP as the Global mod saw something in one of Jay's posts that went against the site rules --content-- that was modified by CP. Jay did not like the edit, and edited out CP's edit without fixing the post for content. A PM was sent about the issue, explaining that editing the mod's posts out without fixing the issue was agains the rules. He fought this. During the week that he was banned, he talked to me a lot, gave some reasons that he wasnted the account permanently removed. The reason that I recall is that his parents don't like fantasy and that he didn't want his parents to see what he had done. Jay said that he would continue causing trouble until he was permanently banned. He kept misbehaving and was banned--this came to a head in the duel with Leben-hoff.\"\n(J)I don't remember everything that happened. There were grievances b\/w me and CP. The mods were reevaluating how post counts counted toward titles. I was against post counts counting in GD. I talked to either a mod or Steel. I decided that they weren't going to do anything about it and posted everywhere. CP, noticing this, became picky about everything I did. She cut out one of my posts and edited it into one post. I changed it back. That day, S told me that I had gotten into trouble and was banned for a week. The original ban didn't work; S re-banned me properly. I don't think I posted after that week.\n(G) Most of this centers around the argument with CP. Kage also interfered in the duel, which neither of us liked. There was further animosity than just with CP.\n(S) Kage stepped down before the final, permanent ban.\n(J)did that have anything to do with the duel?\n(G) both CP and K were mods. That is the only connection there. With that, general chronology from there. That was in July. S, J, feel free to interrupt. Initially after the ban there was some disgruntlement, but nothing happened either way. Ther e were further difficulties with interactions on the site. This was brought to a head b\/c S and the mods were in disagreement over the site about how much power that the mods have. S brought up the mod election idea because we needed to start over (S agrees). J and G saw that the elections might not have CP reelected, and we were happy about that. When the election did come about, CP was still mod. We were pleased that the Global Mod position was eliminated--it granted more freedom, eliminated the hierarchy idea among the mods; they were players, not just GODMODS. Fall semester, J had another account. He was trying to keep under cover b\/c he knew that he would be insta-banned. When (S) found out, he asked me to find out what his new name is. I asked, I was informed that J was on, but he did not tell me what account it is. After that, I was told by J that teh.bob was not the name. After that, I did not push, hoping it would die. After fall semester, it did die a bit.\n(J)last semester I approached S and I asked for Jay to be resurrected. At that point, I would stop sneaking around behind everyone's back. He told me that it was the mod's opinion that Jay was dead. The only way I could do retribution would be to reveal my other screen name--S would bring it before the mods for review and maybe allow me to get back on later. (S--not so much allowed back on, but we would not hunt him down and ban him instantly. If he came to us and tried to work with us, he would talk about allowing the new account to stay.) (K agrees)\n(G) Interterm, it has been the same discussion.\n(S) Other than that, there is 2nd and 3rd hand rumor that during the mod elections that J and G were actively talking to people, trying to curve the elections\/get rid of the old mod team.\n(G) yes, J and I did get together, we were pleased with the prospect of getting rid of CP as Global Mod. We wanted the position of GloMod to be eliminated, and it was. We considered it a success.\n(K) I know this, he talked to me about it before I was a mod.\n(S) have there been any attempts to remove anyone else as mod?\n(J) no, Kage was annoying, but that is it. (G agrees)\n(J) my parents have now stated that it is all right that I am on the forum. They were confused at first, had some skewed details about the site, but their opinions have changed since they found out what the site is about.\n(S) IS the account you are currently using teh.bob?\n(J) I will tell you (S) in secret to protect my anonymity.\n(Z) the admin will be informed, I am okay with that.\n(M) so far everything has been accurate as far as I know. I want this resolved as fast as possible.\n(G) the primary difficulty is the other account, and whisperings behind the scenes. It sounds like more than just going through the back door to getto the site itself. There's whispering, discussion, chatting with newbies who have joined saying don't trust these people, interact with these people, etc., poisoning the site, specifically the old mods.\n(J) yes, I was was warning them about her temper.\n(G) was there anything like \"Watch out for xx, not just CP.\"\n(J) I told people to be careful about double posting. That was it.\"\n(S)You, Z, told people in the chat that the mods would get people in trouble for any disagreement with them. Where did they hear that?\n(K) we had a bti of a bicker\/play argument.\n(Z) we were having an argument abotu sheep being dumb. I gave my bible perspective, and you said that sheep are not stupid. You defended them, and expressed that on previous sites, that because on other sites arguments rose. I have seen an argument starting. I thought, I want peace, I will not force my opinion does not mattter in the long run. Seeing an argument growing, I sat back and shut down the argument, showing that I am under the authority of the site. (K agrees) If they find that I am breaking a rule, I accept their judgment. To answer S's question, before coming to the site, I was aware of the issue with J. I understand that issues can be generated between Mods and players. I was trying to cover myself. I don't think that there's anything wrong with that, especially with new people. Call me over-protective.\n(S) when that happens, we don't ban or do moderation on small offenses.\n(G) J and I have had negative interactions with CP. Other than that, I have had mods PM me about problems, and they have helped me. They wield the sword, and they do their duty, but they do so graciously. Aside from one or two interactions, they have been pleasant. Aside from that, Jay is the only one to be banned permanently. At this point, others have been warned. There seems to be a disconnect because at first the ban was voluntary, but tensions have grown.\n(S) from the mods perspective, there was no voluntary on the ban--even if he agreed with it, it was going to happen.\n(J) from what I recall, by the end of the week, I was permanently banned.\n(S) when was the new account made? Or the second account?\n(J) about 2 or three weeks after I was banned\n(J) a day after they were okay with it, I signed back up.\n(S) then why didn[t that come up before when he was asking for J to come back?\n\n(J) it was because you told me people didn't want me back, mods were mad, you told me that I was causing trouble.\n(S) you cause trouble with one mod, you get trouble from all.\n(J) I learned to stay in RP sections, and not GD and TR.\n(M) I think people are putting too much stock and emotional time into TR, GD, and chat. It irks me when people cause issues, when there sholdn't be an issue there. Chars can argue, but when people argue, there's problems. It's not what all of us joined Luceo for. I almost see the arguments as a roadblock to what we came here for.\n(Z) It seems that to me from J's perspective, he was banned voluntarily, and that the mods banned him whether or not it was voluntary.\n\n(m) about banning: ban the username, but not the person. Imagine if you couldn't be your fav. char ever again. I can see user names dying, but I can't see breaking internet anon to reban someone for who they are because you're mad at them as a person.\n(S) something that does need to be considered is that when people start breaking laws, not governing themselves, the admin needs to step in. When people cause probelms appropriate measures need to be taken.\n\nTO BE CONTINUED AFTER CHURCH at 8:15.\nTHIS MESSAGE IS PRIVILEDGED INFORMATION. DO NOT SHARE THIS WITH ANYONE OUTSIDE THE STATED MEMBERS PRESENT.\n\n\nThe message body of the hints are all numbers, written in various bases. When expressed in a different base with each digit mapped to a different character to reveal the message.\n\n### Hint 1\n\nConvert the base 10 number to base 27, using the characters \" abcdefghijklmnopqrstuvwxyz\" for digits 0-26 gives:\n\neve will send audio\n\nThe attachment should be useful later on.\n\n### Hint 2\n\nSame as hint 1, except you're give hex digits for a base 256 number. After getting that in base 27 I got:\n\nsorry for the garbled audio but frank might send key data tomorrow\n\nIn case you couldn't tell the attachment was garbled.\n\nInquisitive got a good start on the attachment already. Spaces, punctuation, word length, and capitalization are all preserved, only the alphanumeric characters a jumbled. Here is the full text\n\nContained herein are the arguments for and against re-admitting Jay_Darsener into Luceoscutum. Present are Steel01, Administrator of the site; Koballt10, Moderator of Totally Random; Gwythaint, Game Master; Jay_Darsener, the accused; Makosolider and Zero, interested parties in support of the accused.\n\nIt's worth noting that this is very likely the first line of the encrypted audio (hint 8 points this out too). Word length and punctuation match perfectly. Ya! partial plaintext.\n\n### Hint 3\n\nIn case you don't recognize this as ASCII85 hint 6 will tell you so. Converting to base 27 just like the first 2 emails gives:\n\nyou need to research markov encryption to decode the message\n\nI'm familiar with Markov chaining, but not encryption. That chapter appears to be missing from my copy of Applied Cryptography.\n\nAs for the attachment, it's not a real pickle, it's a 88x52 matrix already. Each line has a-zA-Z randomized with no repeated digits on a line.\n\nYou can stop here with the hints, nothing else given is all that useful.\n\n### Hint 10\n\nBases got switched, after some trial and error, I found base 38 worked with \" abcdefghijklmnopqrstuvwxyz\\n0123456789.\" and got:\n\nnumber representation and value are separate\nthey are shown in base x with x or more symbols\nvalues have not a base\nbase 16 may be shown in base 256 and base 256 in base 27\n\n### Hint 11\n\nUsing base 38 I get something almost useful.\n\nwould a base 85 number be stored as characters or bytes\nsomething is a kind of clay used as a building material\nthe building material is typically in the form of sun dried bricks\nwhat is the key word\n\nThe attachment doesn't decode to anything meaningful in base 26 or 38, or any base between 26 and 128 if the initial character mapping stays the same.\n\n### Hint 12\n\nHint 11 points out you should use the Adobe implementation of ASCII85 to decode the messages.\n\nan attachment contains this message 3 times\neach is encrypted with the primers and keys\n\n### Hint 19\n\nBases switched to 65 this time and converting to plain text got a little more complicated. The first 39 characters of the decode string are the same, but after that you need to add 56 and convert that number to to an ASCII character.\n\nfrom our past 7 weeks of analysis\nwe have concluded there is a 99.9\npercent chance that an unfamiliar\nmethod of encryption was utilized\nto secure the transcript for your\nfriend.\n\n### Hint 20\n\nwhile working with ng and attacking the systems at ls\nwe have found that me works with individual bytes. do\nnot try solving this using unicode. ascii encoding is\nprobably your best choice. chars should be printable.\n\n### Hint 21\n\nour associates are learning more and more about markov encryption\nbut some of them have not caught on to a simple fact. the process\nworks on what we call targeted bytes. when a byte is not targeted\nit passes through the process without being changed. each byte of\ninterest may be changed into any other targeted byte. until then.\n\n\u2022 Incredible work! However, the reward to solving the puzzle is still located in payload.me. Over the coming weeks, information will be released to see who gets to claim the three game codes. \u2013\u00a0Noctis Skytower Feb 9 '15 at 20:38\n\u2022 I have no idea how you managed to actually make sense out of all this, but very well done! \u2013\u00a0Bobson Mar 13 '15 at 15:19\n\u2022 Justice and Fairness: Part 2 is relatively complete, and the way is clear to claim the final prize. \u2013\u00a0Noctis Skytower Mar 25 '15 at 17:51\n\u2022 These look like the court proceedings of an Internet forum. \u2013\u00a0Joe Z. Mar 27 '15 at 16:56\n\u2022 What sort of Internet forum executes its members? Or is that just a metaphor for banning? \u2013\u00a0Joe Z. Mar 27 '15 at 16:58\n\n(incomplete)\n\nFrom Hint 4:\n\nInsert \"338531339395984130673279936 in base 27\" into Wolfram Alpha and get a Base 27 number, 5:22:5:0:23:9:12:12:0:19:5:14:4:0:1:21:4:9:15.\n\nTranslate this into letters (e.g. 5->E, 22->V, 0->space), and I get \"EVE WILL SEND AUDIO\".\n\nFrom Hint 2: Partial Translation?\n\nCotanneih dreeia rnt hee rgumeatsn ofa rda naigstr na dmietitnJ ayg_aDrseenr itoL ucenocusumt. rPensae erS tteld01, miAnisratoert tf osh etei; Kballto10, deMaoortf oT taorlRy nlomad; Gwtyhnati, Gema sMrtea; aDa_rseerJne, ahc yseuctM; ksdaoiodral eZe oidn, tnreresepr ttidase sn iuproft te pcc husoead.\n\nContained herein are the arguments for and against readmitting J ay_aDrseenr itoL ucenocusumt. Present are S teld01, miAnisratoert tf osh etei; Kballto10, deMaoortf oT taorlRy nlomad; Gwtyhnati, Gema sMrtea; aDa_rseerJne, ahc yseuctM; ksdaoiodral eZe oidn, tnreresepr ttidase sn iuproft te pcc husoead.\n\nFrom Hint 5:\n\n02 90 B2 9A E9 D0 8B D7 60 39 61 1B 07 85 92 1F B8 F2 E3 44 ED 95 A4 6C C9 A9 09 43 A3 99 DA 32 78 DC 35 CF E0 8B 24 82\n\nThe result I get from converting (at WolframAlpha) the base 16 representation to base 256:\n\n2:144:178:154:233:208:139:215:96:57:97:27:7:133:146:31:184:242:227:68:237:149:164:108:201:169:9:67:163:153:218:50:120:220:53:207:224:139:36:130\n\nWe'll need to put our heads together on this problem people. Just keep building on the work others provide. It'll come together.\n\n\u2022 Your progress is impressive! The answer you posted before your first edit was a step in the right direction. It might help if you incorporate McMagister's work with your own. \u2013\u00a0Noctis Skytower Jan 27 '15 at 15:37\n\u2022 How did you get the partial translation? \u2013\u00a0QuyNguyen2013 Feb 6 '15 at 21:13\n\u2022 @QuyNguyen2013 You may not believe this, but I got it simply by staring at it for quite a while. \u2013\u00a0Inquisitive Feb 7 '15 at 0:57\n\nHint 12:\n\nASCII85 to Hex to ASCII\n\nwV\u00d2\u00ad\u00c2&\u00ec~*\u00f7X\/\u00f4,@=27t-\u00f2Fp\u00b5v=2Q\u00f5\u00bc\u00cfn\u00efo4\u00a0gi\u00f1O?\n\u00fa2R\u00dc\u00acm\u00eb\u00f7s#%]Zao\u00ab\u00ba\u00b7k;;|i\u00aaR\u00ca>\u00f0.g\u00b3\u00b9\u0010\u00d3\u00b57@H.H.%\/??F\u00e8\u000e\u00ca\u00ee\u00a4d\u00e0I(\u00dbo\u00a2\u00dc\u00bd\ns ?Ei\u00b8gfj\u00d4\u00bfMy\u00fex_\u00dexI{Z\u00a1u\u00b3'>","date":"2020-04-06 22:17:19","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.3581143617630005, \"perplexity\": 2499.136889619514}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-16\/segments\/1585371660550.75\/warc\/CC-MAIN-20200406200320-20200406230820-00335.warc.gz\"}"} | null | null |
{"url":"https:\/\/equalopportunitytoday.com\/genomic-diversity-and-evolution-diagnosis-prevention-and-therapeutics-of-the-pandemic-covid-19-disease\/","text":"News, Jobs and Higher Education\nfree sex justene jaro.\nhd porn bonded teen machine pounded doggy style.\nhttp:\/\/www.smellslikeafish.com\n120\nPreprint\nReview\nVersion 1\nThis version is not peer-reviewed\n\n# Genomic Diversity and Evolution, Diagnosis, Prevention, and Therapeutics of the Pandemic COVID-19 Disease\n\nVersion 1\n: Received: 19 April 2020 \/ Approved: 20 April 2020 \/ Online: 20 April 2020 (02:33:15 CEST)\n\nHow to cite:\nHoque, M.N.; Chaudhury, A.; Akanda, M.A.M.; Hossain, M.A.; Islam, M.T. Genomic Diversity and Evolution, Diagnosis, Prevention, and Therapeutics of the Pandemic COVID-19 Disease. Preprints 2020, 2020040359 (doi: 10.20944\/preprints202004.0359.v1).\n\nHoque, M.N.; Chaudhury, A.; Akanda, M.A.M.; Hossain, M.A.; Islam, M.T. Genomic Diversity and Evolution, Diagnosis, Prevention, and Therapeutics of the Pandemic COVID-19 Disease. Preprints 2020, 2020040359 (doi: 10.20944\/preprints202004.0359.v1).\n\n### Cite as:\n\nA novel coronavirus COVID-19 was first emerged in Wuhan city of Hubei Province in China in December 2019. The COVID-19, since then spreads to 213 countries and territories, and has become a pandemic. Genomic analyses have indicated that the virus, popularly named as corona, originated through a natural process and is probably not a purposefully manipulated laboratory construct. However, currently available data are not sufficient to precisely conclude the origin of this fearsome virus. Genome-wide annotation of thousands of genomes revealed that more than 1,407 nucleotide mutations and 722 amino acids replacements occurred at different positions of the SARS-CoV-2. The spike (S) glycoprotein of SARS-CoV-2 possesses a functional polybasic (furin) cleavage site at the S1-S2 boundary through the insertion of 12 nucleotides. It leads to the predicted acquisition of 3-O-linked glycan around the cleavage site. Although real-time RT-PCR methods targeting specific gene(s) have widely been used to diagnose the COVID-19 patients, however, recently developed more convenient, rapid, and specific diagnostic tools targeting IgM\/IgG or newly developed plug and play methods should be available for resource-poor developing countries. Some drugs, vaccines and therapies have shown great promise in early trials, however, these candidates of preventive or therapeutic agents have to pass a long path of trials before being released for the practical application against COVID-19. This review updates current knowledge on origin, genomic evolution, development of the diagnostic tools and the preventive or therapeutic remedies of the COVID-19, and discusses on scopes for further research and effective management and surveillance of COVID-19.\n\n## Subject Areas\n\nSARS-CoV-2; genetic diversity; genome evolution; diagnostics; therapeutics; vaccines\n\nNot displayed online.\n\nMathematical equations can be typed in either LaTeX formats \\$\u2026 \\$ or $$\u2026$$, or MathML format $\u2026$. Try the LaTeX or MathML example.\n\n Type equation: Preview:\n\nOptionally, you can enter text that should appear as linked text:\n\nPlease enter or paste the URL to the image here (please only use links to jpg\/jpeg, png and gif images):\n\n Type author name or keywords to filter the list of references in this group (you can add a new citation under Bibliography):\n No existing citations in Discussion Group\n\nWikify editor is a simple editor for wiki-style mark-up. It was written by MDPI for Sciforum in 2014. The rendering of the mark-up is based on Wiky.php with some tweaks. Rendering of mathematical equations is done with MathJax. Please send us a message for support or for reporting bugs.\n\nComments must follow the standards of professional discourse and should focus on the scientific content of the article. Insulting or offensive language, personal attacks and off-topic remarks will not be permitted. Comments must be written in English. Preprints reserves the right to remove comments without notice. Readers who post comments are obliged to declare any competing interests, financial or otherwise.\n\nThis content was originally published here.","date":"2022-01-24 09:57:09","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 2, \"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\": 0.19376786053180695, \"perplexity\": 10197.307434700839}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"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-2022-05\/segments\/1642320304528.78\/warc\/CC-MAIN-20220124094120-20220124124120-00386.warc.gz\"}"} | null | null |
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} | 8,336 |
Produced by Chris Whitehead and the Online Distributed
Proofreading Team at http://www.pgdp.net (This file was
produced from images generously made available by The
Internet Archive)
/*
THE
EXCLUSIVES.
VOL. II.
*/
/*
THE
EXCLUSIVES.
IN THREE VOLUMES.
VOL. II.
SECOND EDITION.
LONDON:
HENRY COLBURN AND RICHARD BENTLEY,
NEW BURLINGTON STREET.
1830.
*/
/*
LONDON:
Printed by J. L. Cox, Great Queen Street,
Lincoln's-Inn Fields.
*/
THE EXCLUSIVES.
CHAPTER I.
THE CLOSING SCENE AT RESTORMEL.
On the evening previous to Lord Albert's
departure, while Mr. Foley and Lady Hamlet
Vernon were intently engaged in playing at
chess, Lord Albert announced to Lady Ellersby
his intention of leaving Restormel, and
paid her the usual compliment of thanks for
the honour she had done him in inviting him
there.
"You have lost your queen," cried Mr.
Foley, addressing Lady Hamlet Vernon,
"now in two moves I will give you checkmate,
lady fair. But see--what is the matter?--she
is ill--she faints--lend assistance
for heaven's sake!" Lord Albert flew towards
the spot, and caught Lady Hamlet as
she was falling to the ground. The usual
remedies were applied; and when sufficiently
recovered, the sufferer was carried to her
room, apparently still unable to speak.
"I hate all scene-makers," said Lady
Boileau; "if there is a thing I cannot bear, it
is the getting up of a sentimental catastrophe.--Don't
you, Mr. Leslie Winyard?--Don't you
think it is the acmé of bad taste?"
"Oh! most undoubtedly; nothing argues
a decided _roturièrism_ more than allowing
your feelings, if _real_, to get the better of you
in public; and if feigned, nothing is so easily
seen through as counterfeiting them, therefore,
either way, it is at best a _mistake_."
"One don't look well when one faints--that
is to say, _really_ faints," observed Lady
Ellersby; "it is surely best to avoid doing so."
"One may always command one's-self,"
observed Lady Baskerville.
"Oh!" said Lady Tilney, who now and
then really thought and felt right, "it is very
easy to distinguish between a _feint_ and a
_faint_; and I believe every body would ridicule
the first, and nobody would like to do
the latter; because, as Lady Ellersby observes,
no real fainting, or crying, or any of the convulsions
produced by the feelings, are the
least graceful, except in the _beau ideal_ of a
Magdalen, or on a painter's easel; and secondly,
because nothing is less likely to produce
interest than these kind of physical
causes; for, as some great author says, 'all
physical sufferings are soon forgotten even
by the sufferer, when they are past, and by
our friends certainly never remembered beyond
their immediate duration, if so long;'
consequently I believe Lady Hamlet Vernon
did faint _tout bonnement_: she had probably
_une indigestion_; poor Lady!--but she will
soon be well again."
"Spoken like an orator," said Mr. Spencer
Newcombe; "and not only an orator, but a
philosopher."
"Par drivers moyens on arrive à pareille
fin," said the Comtesse Leinsengen; "and
providing one does what one wants to do,
that is all that _sinifies_. One person faints,
another talks, another dresses, another writes,
all in order to get what they wish. On the
success depends the wisdom of the measure."
"Agreed," cried Mr. Spencer Newcombe,
"and conceived like a _diplomate du premier
grade_," he whispered to Lady Baskerville;
then aloud, "if Tonnerre had been here he
would have said--"
"I will bet you ten to one," cried Mr.
Leslie Winyard, "that Lord Albert D'Esterre
does not leave Restormel to-morrow."
"Done," said Mr. Spencer Newcombe.--
"Done," said Lord Baskerville; "ten to
one he does; for I never knew a more obstinate
fellow in my life; one who prizes himself
more _on decision of character_--and when
he says he will do a thing he will do it, however
little he may like the thing when done."
"I don't think he will go," said Lady Ellersby,
gently.
"Why not?" asked Lady Tilney.
"Lady Hamlet Vernon will not let him."
"C'est tout simple," rejoined Comtesse
Leinsengen, with a shrug of her shoulders.
"It appears to me," said Lady Baskerville,
"that if he does go he will not be very
much missed. I never knew so dull a member
of society; he never speaks but to lay down
the law, or to inculcate some moral truth:
now really when one has done with the nursery,
that is rather too bad."
"Providing she don't drive away George
Foley," said Lady Boileau, "she may reap
the fruits of her fainting here."
"Mr. Foley," replied Lady Baskerville to
her dear friend, "is the man in the world
who will do whatever suits him best; and I
particularly admire _his_ manner and his ways:
they are all perfectly in good taste; and I
have already promised him that he shall be
my _cavaliere servente_ for the season."
"Promised!--well, dear Lady Baskerville,
I thought you were too prudent to make such
promises. What will Lord Baskerville say?"
lowering her tone to a whisper.
Lady Baskerville, speaking aloud--"Oh,
dear! la! I should never have thought of
asking him what he likes upon such an occasion;--we
live too well together to trouble
each other with our little arrangements.--Is
it not true, Lord Baskerville? do we not do
exactly as we choose?"
"I hope your Ladyship does," he replied,
in all the airs of his exclusive character; "I
should conceive myself vastly unhappy if you
did not?" Lady Baskerville looked significantly
at her dear friend Lady Boileau; who
knew, as well as herself, that this ultra-liberalism
of her Lord in regard to the conduct
of wives, whatever it might be in respect to
husbands, was entirely assumed on Lord Baskerville's
part.
While this conversation passed in the drawing-room,
Lord Albert and Mr. Foley were
discoursing in their apartment above-stairs.
They had each expressed great interest about
Lady Hamlet Vernon's indisposition; and
after waiting some time to hear accounts of
her from her female attendants, they fell into
other conversation of various kinds, during
which Lord Albert D'Esterre found himself
unfeignedly amused and interested with the
talents, taste, and refinement of Mr. Foley;
and the more so, as he spoke much of
Dunmelraise and its inhabitants, and was
lavish in his praise of Lady Adeline.
"There is only one point," he said, "which
however is hardly worth mentioning, for of
course it only arises out of the seclusion and
the monotony of her present existence; but
certainly Lady Adeline, _pour trancher le mot_,
is a _little_ methodistical--the sooner you go
and put that matter to rights the better."
Lord Albert's manner of receiving the latter
part of this information, proved to Mr. Foley
that he had mistaken the character of the
person he addressed, and he added,
"But indeed Lady Adeline Seymour is so
perfect, that it matters very little what she
does--every thing _she does_ must be right."--The
conversation then took another turn,
and they parted.
Lord Albert D'Esterre was not what might
be called a jealous man; but no man, no
human being can be without the possibility
of feeling jealousy--neither was he naturally
suspicious, but nothing is more apt to generate
a suspicion of the fidelity of another's
conduct, than the consciousness of any
breach in the integrity of our own. He
pressed his hand to his heart--he sat down--rose
up--paced his chamber, and still repeated
to himself the praises which Mr. Foley
had uttered of _his_ Adeline. "_My_ Adeline,"
he said, and then again stopped; "but is
she _mine_? do I deserve she should still be
_mine_, when I have so neglected her? no!"--His
servant came into the room with a note,
the well known shape and colour of which
he could not mistake. It was placed in his
hand--he opened it carelessly and was about
to cast it away, when the name of _Adeline_
caught his eye; then he hastily read the
following words.
"It is not for myself I mourn--it is not
the threatened loss of your society, however
much I value it, which has occasioned my
being so overpowered--it is the knowledge of
a secret which pertains to another, and in
which your fate is involved, that has quite
mastered me--this much I must tell you. I
must see you before you go, I must prepare
you for your meeting with Lady Adeline
Seymour." Twenty times he read over this
note. "What can it mean? can its meaning
be that Adeline loves Mr. Foley, at least that
he thinks so? and I, what have I been doing?
into what a sea of troubles have I plunged
for the enjoyment of the society of a person
that in fact affords me none--for the empty
speculation of recalling the chaotic mind of
one (comparatively a stranger to me) to a
sense of reason and religion, fool that I was
for the attempt." Then, after a considerable
pause, and after deep reflection, he burst
forth:
"Prepare _me_ for a meeting with Adeline!"
as his eye caught again the last line of the
note. "_Prepare me for a meeting with
Adeline_--I cannot bear the phrase; but I
must know what she means--I must drag this
secret from her:"--and he rang the bell
violently!--"I shall not want my horses
till one o'clock instead of seven to-morrow
morning."
The night Lord Albert passed was one of
feverish anxiety. He sent to inquire for Lady
Hamlet Vernon at an early hour the next
day; and hearing she was much recovered,
he besought her to grant the interview she
had done him the favour to offer as soon as
she possibly could. She replied, that in that
house it would be reckoned a breach of all
decorum, if she received him at any undue
hour; but that as soon as the earliest part of
the company breakfasted, which was about
one o'clock, she would be sure, notwithstanding
her indisposition, to be in the breakfast-room
at that time; when she would avail
herself of some opportunity to give him the
information which had come to her knowledge.
This short delay seemed an age to
him. Every one knows, when suspense
agitates the mind, what a total anarchy
ensues, and the hours which intervened
before meeting Lady Hamlet Vernon seemed
to Lord Albert interminable. When they
_did_ meet, the intervening moments ere an
opportunity occurred of Lord Albert's drawing
her aside, appeared in their turn so many
more ages of suffering.
At last the company rose from the breakfast
table, and as Lady Hamlet took Lord Albert's
arm, and walked out on the terrace under the
window, she said, "This is kind of you to
have listened to my request:" and then as
they walked from the house, proceeded in a
graver tone to add, "I am aware, dear Lord
Albert, that my note of last night must have
surprised you, and that the subject connected
with it, on which I am about to touch, is one
of the utmost delicacy, and one which upon
the very verge of the attempt I shrink from;
but you have evinced so much real interest in
the state of my wayward mind, and have
said so much to me with a view, I am certain,
of placing my happiness on a more secure
and steady foundation than I had ever
any chance of before, that I should be ungrateful
in the extreme, if a corresponding
wish for your comfort in life did not in turn
actuate me. I cannot be ignorant of the engagement
between yourself and Lady Adeline
Seymour, the fulfilment of which will not, I
presume, be long delayed; unless, indeed--"
Here Lady Hamlet Vernon's voice faltered,
and for a moment she paused; but, as if
making an effort to subdue her emotion, she
added in a lower and firmer tone, and with
an expression of something like intreaty in
her countenance as she looked up at Lord
Albert, "Unless I, dear Lord Albert, shall
prove the happy instrument of saving you from
too precipitate a step in this matter. May I
continue to speak to you thus unreservedly?"
Lord Albert made no answer, but bowed his
head in token of assent, while he walked by
her side like one lost in a perturbed dream.
She continued,
"I wished, before you went, for this opportunity,
because I was aware that it was the
only one left in which what I am about to
impart would ever be of use; for, lovely as
Lady Adeline is, possessed of charms of person
which would indeed draw any heart towards
her, of the warmest and most enthusiastic
disposition, deeply enamoured of _you_ as
well as sacredly alive to her engagement to you
(and I know her, from a source which cannot
mislead me, in person, in mind, in heart, and
in determination, to be all that I describe
to you)--how could even your judgment,
Lord Albert, which is stronger than many
of twice your years, but yield to such
united influence, and be tempted to decide at
the moment on making so much perfection
irrevocably your own. But with all these
transcendant charms of person and of character,
Lady Adeline, I am grieved to say,
and know, has been unhappily betrayed into
views of life and of the world, which must
unfit her to be the partner of any one who
does not think in accordance with her on
these subjects. From what cause or under
what influence the peculiar turn of mind she has
taken has arisen, I know not, but (and again
I must repeat, that I _know_ the too-sure truth
of all I say) it has been gradually and fearfully
on the increase, and is now become a
fixed principle with her.
"She loves _you_, as I have said, and she
looks upon the coming union with you as
the fulfilment of a sacred engagement, and
a duty she has to perform; but with this
she views the rank you hold in society,
and in which she will be associated,
only imposing on herself obligations of a
higher and severer order, and calling for a
stricter conduct and a greater self-denial on
her part. She condemns what she calls the
dissipations and wicked employment of time,
in the world of fashion; she holds dress, beyond
the plainest attire, to be a misapplication
of the gifts of fortune; she laments over the
worldly career of any one whom she hears
talked of with applause, or whose talents raise
them to distinction in the public eye: she
has even, I understand, wholly abandoned
her music and her drawing, as too alluring
and dangerous an occupation, wasting the
time which ought to be devoted to serious
reading, and an acquirement of that spirit
which has already cast such a gloom over her
existence. The only active employment in
which she indulges herself beyond her books,
is in making clothes for and visiting the poor
in her mother's domain. In short, she is what
the world calls a methodist, a saint; I know
not exactly what these words mean, but I know
they are terms applied by people of sense to
an ultraism in religious matters."
Lord Albert shuddered, and a sigh was the
only interruption he gave, as Lady Hamlet
proceeded.
"Conceive yourself, my dear Lord Albert,
united to a person of this character, however
amiable in herself, with your talents, with your
views, which are" (and she looked at him steadily
as she spoke) "tinctured with ambition.
With your temper and your tastes for the elegancies
of life, how would you brook a wife
who was praying and singing psalms all day
long? who would consider all _your_ actions,
when not in accordance with her own, as so
many positive sins, and whose moments, such
at least as were spared from the offices of her
enthusiasm, would be passed in the cottages
of your tenants, and in making baby-linen for
every expected increase in their families.
"Now let me beseech you, and believe me
to speak from the most disinterested feelings,
that when you meet Lady Adeline, you will
not betray yourself into a too hasty arrangement
for your union. See her--see her, by
all means. Judge for yourself; use your
own eyes, hear with your own ears, and
be the arbiter of your own cause, but do
nothing rashly. Time is necessary for all decisions
in momentous questions; and what
can be more momentous, and in what is there
more at stake, than in an union for life? Can
too much deliberation be given to the subject?
Alas! I know, from my own fatal experience,
what misery must ensue where no tastes, no
principles, no objects exist in common between
those united. I owe to this cause a great
portion of my present unhappiness; for the
misery I endured, and the constant efforts I
made to bear up against the tenfold wretchedness
of my marriage with Lord Hamlet Vernon,
impaired my intellectual powers, and
prevented my turning the energies of my mind
to any useful or profitable purpose. Hence I
have become what I am, dependant on the resources
of the hour, to enable me to pass
through life with any thing like composure."
Lord Albert had listened with feelings
which it would be impossible to describe
to all that had fallen from Lady Hamlet
Vernon; and in the emotion, which her communication
and her entreaties produced, he
could find no words for utterance, no answer
to her appeals. He was like one dumb, and
deprived of sense; and he stood for some moments
rooted to the spot when the voice of
his counsellor had ceased.
"See her! yes, I will see my Adeline," he
at length said in a deep agonized tone, as
if communing with himself. "Yes, I will
see her."
"Lord Albert, I entreat you, I implore
you," cried Lady Hamlet Vernon, with an
emotion that made her words quiver on her
lips, "I beseech you forgive me, if"--the
window of the library was at this moment
thrown hastily up; and Lord Albert D'Esterre
heard his name called by Lord Ellersby,
who held in his hand a letter.
"D'Esterre," said he, "here are your letters."
Lord Albert hastened forward mechanically
to receive them, and one he gazed
upon more intently than the rest, as he looked
them over--it was from Adeline.
Who is there who has not recognised, even
in its peculiar folding, the letter of a beloved
object? and whose heart has not throbbed
with delight ere even the seal were broken?
Such was the emotion of Lord Albert, awoke
up from the paralyzing influence of Lady
Hamlet Vernon's communication to new life
by the letter he now pressed to his bosom;
and regardless of what had passed, he hastened
to his room, and read as follows:--
/#
"DEAREST:--My mother has been gradually
growing worse and worse these two
months, and I have persuaded her to go to
town for a consultation of her physicians.
"It is so long since I have heard from you,
Albert, it is painful for me to write, scarcely
knowing how far you may be interested
in what I have to communicate--but
I try to still my uneasiness--let me but
see you, dear Albert, all will be forgotten,
all will be forgiven; for I am your own true
and affectionate
/*[5]
"ADELINE."
*/
"P.S. You will find us at Mamma's house
in town."
#/
A letter like this, breathing such trust and
love, and so replete with genuine expression
of delight in the prospect of meeting him,
was indeed sufficient to make Lord Albert
forget at once the poisonous theme which
his ears rather than his reason had imbibed in
his interview with Lady Hamlet. Impelled
more by the eager anxiety of affection to
behold the object of his late disquietude, than
to see her for the purpose of convincing himself
of her errors, he leapt with alacrity into
his carriage, and drove towards London,
without casting a thought on those he left
behind.
The mortification which Lady Hamlet
Vernon felt was severe, in proportion as
from its nature it admitted of no sympathy.
She was, of course, ignorant of the cause of
Lord Albert's destination being so suddenly
changed from Wales to London; but in the
blindness of her increasing passion, she resolved
in the first moment of her despair to
follow him thither. A cooler judgment, however,
made her recollect that if she lost Lord
Albert she had other friends to retain, a position
in the gay world to lose, and that, at all
events, it was not by pursuing him at that
moment that any thing was to be gained; she
therefore determined on remaining some days
at Restormel, and making herself as agreeable
as possible to the party that continued
there. To one of Lady Hamlet Vernon's disposition
this was no easy task. Violent and
impetuous as she was by nature, left as she
had been without any control, it was a very
Herculean work to hide all the warring passions
of jealousy and disappointed love beneath
the semblance of a cool indifference--a
disengaged mind.
"What have you done with Lord Albert?"
was Lady Baskerville's first question to her
after the morning's salutation; "I hear he
departed in violent haste at an undue hour this
morning. He looks of such an imperturbable
gravity, one does not understand his ever
being brought to do any thing out of measure
or rule."
"I done with Lord Albert? my dear Lady
Baskerville, you confer too much honour upon
me to suppose that _I_ have any influence with
him. I did not even know that he was gone;
but if you are very much interested in his
departure, perhaps Lord Ellersby can tell us
something about it."
She thought by this means to discover the
cause of his sudden disappearance, and gratify
her inquiries as being the curiosity of
another.--"Lord Ellersby," she said, "Lady
Baskerville is desirous to learn what wonderful
event can have called Lord Albert away
from us so very suddenly."
"I do not know," said Lord Ellersby, "unless
he is going to be prime minister; don't
you think, Winyard, he has the dignity of
office on his brows already?"
"In his own opinion, I make no doubt, he
stands a fair chance for the highest situations;
but we have quite exploded all that sort of
fudge now-a-days, and I think, unless we were
to have a bare-bone parliament, and a cabinet
of puritans, his very consequential lordship has
not much prospect of success in that line."
"No," said Lady Tenderden, taking up a
newspaper, "I think this paragraph in the
Morning Post will rather explain the secret
of Lord Albert's going away:--
/#
"'We understand Lady Dunmelraise, with
her beautiful daughter Lady Adeline Seymour,
is shortly expected in town, and are sorry
to add that Lady Dunmelraise's ill health
has hitherto caused her absence from the gay
circles of fashion.'--This is put in by herself,
or some of her friends, you may depend
upon it."
#/
"Dear," said Lady Baskerville, "those vulgar
newspapers are always filled with trash of
that sort; nobody attends to such nonsense. I
dare say this Lady Adeline is some awkward
raw girl, enough to make one shiver to
think of; however, she may do very well as
a wife for Lord Albert, and he may be gone
to meet her."
"Oh, I do assure you," cried Lady Tilney,
"that the public papers are the vehicles of a
great deal of good or evil; and that not only
political discussion, but the discussion also of
the affairs of individuals, is constantly promoted
by the freedom of the press."
"For my part," said Lady Baskerville, "I
think it is quite abominable that those vulgar
editors of newspapers should be allowed to
comment upon what we do."
"Not at all, my dear Lady Baskerville;
allow me to assure you that we are much
more known--much more distinguished--much
more _répandus_ by being all named occasionally,
never mind how or in what manner,
in the public papers. Besides, on the
freedom of the press hangs all the law and the
prophets; and if some few suffer by it occasionally,
the multitude are gainers; and I can
never repine at the glorious spirit of public
liberty which the papers and the press maintain.
Don't you agree with me, Lord Ellersby?"
"I like it all very well when it does not
interfere with me," he replied, yawning; "but
I think it is very disagreeable when these vulgar
fellows, the news-writers, say some impertinent
thing, for which I cannot give them
a rap over the knuckles."
"La, what does it signify," rejoined Lady
Ellersby; "nobody thinks of any thing above
a very few days, and except some dear friend
or other, no person of good breeding mentions
the subject to one, if it be disagreeable,
so that I cannot really say it disturbs
my tranquillity for a moment, let them
say what they will. As to this puff about
Lady Adeline Seymour, I agree with Lady
Baskerville, there are always a set of would-be
fashionables, who pay for the putting in of
such paragraphs about themselves, _et l'on sait
parfaitement à quoi s'en tenir_ respecting them."
"Nevertheless," rejoined Mr. Foley, who
had just laid down his book, "I do assure you
that, puff or no puff, Lady Adeline Seymour
will astonish you all, for she is a very extraordinary
person."
"Then I am sure I shall not be able to
suffer her," said Lady Baskerville.
"_Je déteste les phénomènes_," said Comtesse
Leinsengen.
"Mr. Foley seems to be paid too," rejoined
Lady Tenderden, laughing, "for making the
young lady notorious; and we shall see him
with a placard stuck on his shoulders, setting
forth the beauties and perfections of the wonderful
young lady."
"These _miracles_," cried Comtesse Leinsengen,
"are only fit to be shewn for half-a-crown
a piece; if you interest yourself very much in
her benefit, remember, I promise to take
tickets."
Mr. Foley smiled as, he replied: "I shall
leave it to time to prove to every one of you
how very much you are mistaken."
"By all that is romantic," cried Mr. Winyard,
"Foley is caught at last; he is positively
going to become a lackadaisical swain,
and write sonnets to his mistress's eyebrows."
"Perhaps even so. It is amusing to take
up a new character now and then; it is like
changing the air, and is equally beneficial to
the health, moral and physical. Nothing so
fatiguing as being always the same, both for
the sake of one's-self, as well as of our associates--don't
you think so, Mr. Winyard?"
"I have always shewn that I did so think.
Few persons have acted up to their principles
in this respect more conscientiously than myself."
Mr. Foley did not press this matter
further; he knew when to retire from the
field, and always cautiously avoided a defeat.
This conversation was at once a key to Lady
Hamlet Vernon, and much as it pleased her
to have discovered the truth, she resolved to
carry on the deception; but Lady Hamlet
Vernon felt that her total silence might be
construed into an interest which, however real,
she by no means wished should appear to exist
in its true colours, and therefore she forced
herself into saying, with apparent indifference,
"I understand Lord Albert D'Esterre's marriage
is shortly to take place; and whatever
people may do _after_ marriage, they must be a
little attentive _beforehand_; so I doubt not
that the arrival of Lady Dunmelraise in town
is really the cause which has deprived us of his
society; and you know I am one of those who
hazard a favourable opinion of Lord Albert,
notwithstanding Lady Baskerville's dissentient
voice."
This speech she conceived to be one of unprejudiced
tone and feeling that would lull all
suspicion to rest, had any existed, as to the
nature of her real sentiments; and it at least
prevented the expression of that ridicule,
which would otherwise have been her portion.
In this society there was a general system of
deceiving on the one hand, and detecting on
the other, which constituted its chief entertainment
and business; and in the present
instance it formed, as usual, one of the main
springs of the interest that filled up the
remaining hours spent by the party at Restormel.
CHAPTER II.
THE BRIDE'S RETURN.
The approaching gaieties of London, after
Easter, were pronounced likely to be of a
more brilliant description than they had been
for years, as is always the case, according to
the interests and wishes of the persons who
raise the report. One of the earliest arrivals
in the scene of _ton_ was that of the Glenmores,
who had returned from Paris, whither they
had proceeded, it will be remembered, shortly
after their marriage.
London, however, was still empty; a considerable
part of the _élite_ remained at Restormel,
and others of their corps were not yet
reunited; while such as had in fact nothing
to do with them, were nevertheless sufficiently
foolish to regulate their movements by those
of the exclusives.
It was in this interval between the two
assignable points of a London season that
Lord Glenmore, turning the corner into the
still deserted region of Hyde Park, met there,
to his surprise, Lord Albert D'Esterre, who
sat his horse like one careless of what was
passing around him, and seemingly so absorbed
in his own thoughts, that the exercise
of riding had the appearance at that moment
with him of a mechanical habit, rather than
a thing of choice. So deeply occupied was he
in reflection, that Lord Glenmore was obliged
to call several times, and at length to ride
close up to him, before he could attract his
attention.
"D'Esterre," said he, as he held out his
hand, "I rejoice to meet you; and this unexpected
pleasure is the greater, as I thought
you had been too fashionable a man to be yet
in London, at least for a day or two to come.
But how ill you look! what is the matter
with you?"
Lord Albert was not in a mood to bear
interruption from any one, or exactly able,
without putting a force upon himself, to meet
any inquiry with a courteous answer. But
Lord Glenmore was, perhaps, one of the
very few exceptions in whose favour something
of this feeling was abated, for their
intimacy had been of long standing; and Lord
Albert's regard and respect for his character
was, as it deserved to be, of the highest
kind.
As soon, therefore, as the latter was roused
from his reverie by the kindly voice of his
friend, he greeted him with answering
warmth, and inquired after Lady Glenmore
with that cordial interest which he felt for
the wife of his friend; he at the same time
endeavoured to laugh off Lord Glenmore's
observations on his own personal appearance,
which were nevertheless well-founded--for
his mind was labouring under an anxiety
which visibly displayed itself in his countenance,
and which, as his first emotion of
pleasure in the near prospect of meeting
Lady Adeline subsided, the mysterious words
of Lady Hamlet Vernon's note were too well
calculated to give rise to. This state of uneasiness
was by no means diminished by the
delay of Lady Dunmelraise's arrival in town.
At her house Lord Albert's hourly inquiries
had for two days been fruitless; and he was
returning from South Audley Street, with
the expression of increased disappointment
painted in his looks, when he met Lord
Glenmore.
After some conversation of a general nature,
and inquiries into the events which had
arisen in the fashionable world during his
absence, and which the latter confessed himself
to have been too happy to have thought
about before, he asked Lord D'Esterre, with
a manner implying more interest, what were
his own views and intentions.
"I hope you are not thinking of returning
abroad," he added, "for we want you at
home, and then you must marry." Lord
Albert sighed as his friend approached the
subject so near his heart, but which he was
little inclined to discuss with him at that particular
moment; while the other, without
remarking the grave expression that had
returned over Lord Albert's countenance,
continued:--
"Allow me to speak to you as a man who
has lived a little longer in the world than
yourself, and to whom you formerly communicated
what were your views and wishes in
life. You told me you would aim at diplomacy
and at office; I am sure in both from
noble motives, and because you felt it to be
your bias, which in all our pursuits is half
the battle in ensuring success. Now you must
permit me to tell you that, however great or
powerful in point of interest a man may be,
he can never with these objects be too much
of the latter. Above all things, then, keep
this principle before you; and, in any alliance
that you may form (for you will marry soon,
depend upon it: the ladies, if there were no
fears from yourself, will not allow you to remain
long in single blessedness), endeavour
to remember my advice, and look round you
before you take the leap which is to break
the neck of your liberty, and do not throw
away the advantages which your situation (to
say nothing of yourself) give you of selecting
where you choose, and where you think your
pursuits will best be promoted.
"Now there is one, _par parenthèse_, among
the many desirable parties I could name
to you--which is Osbaldeston's daughter.
His interest is great; but he has taken
through life the most foolish of all parts
in politics--that of being of neither party;
and, as an independent peer, is alternately
hated and caressed, abused and praised,
despised and sought after by both. You
know, since the death of his eldest son,
all his affections centre in this daughter;
and I am persuaded that any one united to
her, may make all Lord Osbaldeston's interests
his own. I do not mean to force this
match upon you," smiling as he spoke; "but
I allude to it as a sample of what, as your
friend, and one thinking with you in politics,
and pretty much the same in all other matters,
and having your interest, my dear D'Esterre,
much at heart, I would rejoice to see you
assent to. _Enfin_--the Osbaldestons dine with
us to-day, and if you will join us, you will
have an opportunity of judging for yourself."
Lord Albert, as if he thought himself
doomed to undergo violence on all sides in
regard to Lady Adeline, replied with more
petulance in the tone of his voice than he
was ever known to give way to--
"My dear friend, you forget that I am an
engaged man."
"Oh, if you mean to allude to Lady
Adeline Seymour, I had understood that it
was only that sort of engagement which
might be dissolved or not, as the parties
chose when they came to years of discretion;
and as I had heard it whispered that
Lady Adeline was attached to a young man
who was much at Dunmelraise, and a _protégé_
of her mother's, a certain Mr. George Foley,
who turned all the women's heads about two
years ago in London (Lady Hamlet Vernon's
among the rest, by the way), I could not suppose,
seeing you very quietly here, that your
heart was much engaged; and I thought I
knew you too well to believe that you would
ever marry (however much I hope you will
make a prudent alliance) where love and
esteem do not constitute a part of the compact."
"My dear Glenmore, I see your kind intention,
through this apparent carelessness
of my feelings; but allow me to assure you,
you are misinformed--a purer, truer, or more
innocent creature does not exist than Lady
Adeline Seymour; and though I have been
separated lately from her, yet from my correspondence
with herself, and from the invariable
accounts I have received from others, I feel
assured that the ingenuousness of her character
would never allow her to have a
thought concealed from her mother or myself
in the momentous question between us. Oh
no; when I look back to her every letter, the
recollection brings conviction along with it
of her heart being unchanged."
Lord Albert spoke with an inward agitation
which corresponded little with the confidence
which his words expressed. His outward appearance,
however, was calm; and Lord Glenmore,
supposing he had been led into a very
pardonable error, and wholly innocent of intentionally
wounding his friend's feelings,
proceeded--
"Well, if it is thus, D'Esterre, you are
already a married man, I conceive; but be it
so, that does not prevent your dining with me
to-day--pray come."
Lord Albert declined, saying gravely, "no!
that cannot be; for I am in hourly expectation
of Lady Adeline's arrival with her mother,
who, I am sorry to add, comes to town on
account of her health." A momentary pause
ensued in the conversation; and Lord Albert,
seemingly little inclined to renew the last
topic or enter upon any new one, seized the
opportunity of bidding his companion farewell,
and they separated.
From the somewhat cold and reserved manner
of his parting, Lord Glenmore, when
alone, began to think he had committed a
mistake in treating his friend's engagement
with Lady Adeline lightly, and condemned
himself for what had escaped him on the subject.
For Lord Glenmore was a man of honourable,
as well as kindly feelings; and in giving
the counsel of a _prudential_ marriage to Lord
Albert, was at the same time the last person
to think that, in an union for life, happiness
ought to be sacrificed to interested views:
the furthest also from his thoughts would have
been any design to interfere between, or to
disunite any two persons who were attached
to each other. Perhaps the world in general
might not have given him credit for this amiability
of feeling, or for the strict principle
which he really possessed, from seeing that he
lived in constant intercourse with a class,
where, if similar worth of character did exist
at all, it certainly never was looked up to as a
merit in the possessor. It must be allowed
that Lord Glenmore was any thing rather than
a fitting member of such a class; for in addition
to warmth of heart, natural affection, and
good principles, he possessed talents of a very
superior kind, and held opinions quite at
variance with the received creed of his companions.
He believed, for instance, that life was given
for other purposes than to be spent in accident
alone, or that a perpetual course of frivolous
pursuits, without any higher aim or object,
should be suffered to govern human existence;
but that, on the contrary, every action should
tend to some useful purpose. If Lord Glenmore
was ambitious (and he was so), his
ambition was of a noble kind; and while he
sought power, his uprightness of character
could never suffer him to abuse its exercise.
He was called proud by some: but although
impressed with a sense of the dignity of the
aristocracy to which he belonged, it was not
a blind and foolish estimate of rank which
made him value it, but a conviction of the
importance and responsibility which every one
placed in the higher grades of society possesses,
while fulfilling the duties of the sphere
in which Providence places him; and if in
society he sometimes appeared reserved, and
joined not in all the empty, uninteresting topics
that make up the conversation of most of the
coteries of _ton_, it was--that his mind was
filled, even in the buzz of the vapid talk
around him, with matters worthy of the reflection
and study of an intellectual being.
He owed his admission, consequently,
within the line of circumvallation drawn
by the _ultra_ leaders of fashion, to a dread of
the important consequence of his remaining
aloof from their circle, and the preponderating
influence which even his neutrality would afford
(for Lord Glenmore was not a man to lend
himself to either side in such a frivolous warfare
as the decision of who were, or who
were not, worthy members of the _corps élite_).
Although the exclusives, therefore, one and
all, considered him to fall short of a due proportion
of that species of merit necessary to
their order, yet still they united in one common
effort to retain him on their side. They
could have wished him, no doubt, allied to
one of their own peculiar choosing, and had
heard with dismay proportionate to the consequences
which might frustrate their plans
respecting him, the announcement of his
marriage with his present wife.
Determined, however, to make the best
of the unpropitious event, they had from
the first decided on the general policy of
endeavouring to retain Lord Glenmore's influence,
by admitting Lady Glenmore (however
much she might be considered inadmissible)
amongst them; and thus to secure
in the opinion of the world the sanction of
her husband to live on terms of intimacy in
their set.
It was this motive which in some degree
influenced the ladies who were present at
Lady Melcomb's ball, and subsequently at the
marriage, to risk the loss of _caste_ by being
seen in the motley collection of that lady's
assembly: though the ties of relationship, in
one or two instances, would have led them to
the re-union on such a happy occasion. Yet
with Lady Ellersby and Lady Tenderden these
were impulses, which were only to be acted
upon when the laws and dogmas of exclusiveness
permitted such a proceeding.
When Lord Glenmore returned from the
Continent with his young bride, the news of
his arrival quickly spread through the exclusive
circle, and called for some decisive measure
on their part, to ascertain how he might
be induced still to remain, under the circumstances
of his new connexion, in the same degree
of intimacy with them. It was therefore
time, on the part of the exclusives, for bringing
to bear these intentions at the moment
of their re-assembling in London, and more
particularly on that of the individuals who
composed the party at Restormel.
Lady Tilney, whose activity was ever on
the alert, ordered her carriage before the
morning show of London began, that she
might catch all the chiefs of her party at
home. The first house she visited was Lady
Ellersby's, who was not yet risen, but she was
admitted to her bed-side.
"_Reveillez-vous belle endormie_," said Lady
Tilney, kissing her on both sides of her face,
"for what do you think I am come about?"
"I cannot imagine: has Lady Hamlet
Vernon gone off with any body, or do the
ministry totter, or has Newmarket proved
unsuccessful, or, in short, tell me what _has_
happened!"
"No, my dear, nothing of all that; but the
Glenmores are come back from Paris, and
now or never must the question be ultimately
decided whether we are to retain Lord Glenmore
amongst us or not. You know we were
agreed on the general policy of doing so soon
after his marriage, and the first step to take
will be to tutor the young Georgina, so that
she may not on the outset of her _début_ do
any thing to disgrace us. But although I considered
the matter as settled, I would not
take any decided step till I consulted you. It
is on this account I am come at so early an
hour, lest we should not have acted in concert
on this point; for as I always say, it is the
disagreement in the cabinet between their own
members which always breaks up the administration;
so society is, or ought to be, precisely
a type of the government of a state:
don't you agree with me?"
"Perfectly," replied Lady Ellersby, suppressing
a yawn, for she did not, to do her
justice, understand one word of the political
jargon in which her friend always talked, whether
the conversation ran on the choice of a
new cap or the admission of a new member
to their society. Lady Tilney observing her
dear friend's absence of mind, told her that
she looked so beautiful in her night-cap, she
quite made her forget her errand.
"But, nevertheless," (she added) "I must
remind you, that it _is_ one of no small importance,
for you see what a vast field of interests
the Glenmore himself includes. There are
the Melcombs, and the D'Esterres, and the
Osbaldestons--a perfect host. _Some_ of them
may play a card in politics: _all_ of them are
good tools, and I promised Lord Tilney not
to lose sight of that consideration. So if we
exclude la petite Glenmore, we shall be incurring
great risks; whereas, by making
her _one of us_, we shall have a vast addition of
strength added to our party, and we can always
take care that the vulgars belonging to
her, who are only good for certain uses,
shall not come in her train."
Lady Ellersby, whose attention had been
effectually awakened by the admiration of her
night-cap, now sat up in her bed and said,
"Ah! there indeed is the difficulty--how will
you manage that?"
"Nothing easier: we will, as I said, explain
to her what an advantage it is to belong
to us, and the necessity of our confining
our members to a very small circle, and then
tell her that we will always let her know whom
she is to invite to her parties, and whom she
is to go out with. Thus we shall take care
that, from the very beginning, she does not
_compromise_ us. One or other of us must
always be at her right hand, and by flattering
Lord Glenmore, and endeavouring
to make him believe that Lord Tilney is
wavering, and may possibly come round to
his side in politics, we shall easily get that
sort of power established with both, which
it is quite necessary to obtain if they are to
belong to us; and that they are so to do
is, as I have already explained to you,
equally necessary. Not that I, for the world,
would make any body do what he did not
like to do: no one is more for perfect freedom,
as you well know, than myself, but you
must feel that not to belong to us, is in fact to
be nobody, so that we are doing them a favour,
the greatest possible favour indeed; and
I am sure I would not take all this trouble
were it not that I am convinced it is doing
good."
"Oh yes, you are so good-natured, you
are always trying to oblige. And what then
would you have me to do?"
"Why I would have you call upon Lady
Glenmore to-day, and you may tell her how
she ought to dress, and to demean herself in
public. And when she is in public, you may
take care that no one speaks to her but
those whom we approve of; and should any
of her vulgar relations by any accident affect
to get near her, you can contrive to draw her
away, and carry her off to some other place.
Thus, my dear Lady Ellersby, I think, after
having explained this business so far, I need
say no more, though I could talk for hours
on the subject," Lady Ellersby yawned instinctively;
"but the line of conduct I wish
you to adopt has been so minutely pointed
out, that I think you cannot possibly misunderstand
it. And now I will go to Lady
Tenderden and the rest, and I flatter myself
no _diplomate_ ever played his part with more
skill. Depend upon it I will continue to
do my utmost endeavour to succeed in this
affair, which I feel persuaded is of considerable
consequence to our society. Not, as I
before said, that I would ever, either in great
or little matters, stoop to contrivance. I like
to persuade people for their good, and would
have all the world act with a liberal and free
exercise of their own rightful powers; the
right of reason which every individual ought
to exert and use in his own behalf. Ah, if
all governments could but be persuaded of
this, and be ruled in their determinations by
this noble motive of action, how differently
things in general would be managed from
what they are! Kings would no longer
be puppets of state, but be obliged in self-defence
to become rational people, and not to
depend on their ministers and favourites;
and ministers would not depend on each
other as they do, but every body in his own
sphere would be doing all he could to tend
to the public weal."
Lady Tilney had once again got on her
favourite theme; and on these occasions she
never found out that the one part of her
discourse generally contradicted the other,
and that her _meaning_ virtually did so where
her _words_ did not, for it was always herself
who was to be the mover and law-giver.
But this was all matter of moonshine to her
present auditress, who at length shewed
unequivocal symptoms of inattention, and
even hinted that it was time for her to rise.
So at length Lady Tilney, reiterating the part
she assigned to her respecting Lady Glenmore,
took a tender leave and departed.
Her next visit was made to Lady Tenderden.
"Ah!" she said, on meeting her, after
the first greetings, "what a relief it is
to have to converse with a rational being,
one who understands the meaning of things
in general. I have just been talking to poor
dear Lady Ellersby, who is, between ourselves,
become more than ever thick, and
indolent--she actually cannot understand any
thing _consecutively_; however, I have, I think,
at last put her in a right track upon the subject
which I must now discuss with you."
"I know," said Lady Tenderden, interrupting
her (for patience was not her _forte_)
"what you would say. The Glenmores are
arrived, and--"
"Exactly; and it is necessary we talk the
matter over, and settle precisely the _marche
du jeu_."
"Oh! by all means, take _la petite Georgina
en main, et l'affaire est faite--je m'en charge_."
"That is precisely what I wished;--nobody
is better calculated for that office. In the
multiplicity of things which I have to do,"
said Lady Tilney, "it is not possible that I
should pay that sort of attention which she
will require, for she is very childish, perfectly
ignorant of the ways of the world, almost a
simpleton, and our society might be entirely
broken up and destroyed, if we allowed her,
without proper caution being previously
observed, to come in amongst us. At the
same time, I think it is of such consequence
that we should not altogether lose Lord
Glenmore, I mean politically as well as prudentially
speaking, that it does appear to me
to be quite worth while to take the trouble
of forming that little wife of his, and making
her one of us."
"Oh, _certainement_," replied Lady Tenderden.
"Besides, Lord Glenmore is charming;
_il fera fureur_, when he becomes a little
more polished, and I shall with infinite pleasure
_consacré_ some hours to the instruction
of _la petite ladi qui seroit à ravir si elle n'avoit
pas l'air d'un mouton qui rève_."
"Exactly," cried Lady Tilney, "but that
is of no consequence."
"Oh, none in the world," responded Lady
Tenderden.
"Well then, my dear, that is finally arranged,
and I shall now only have to go to
the Glenmores to-morrow; but if it be possible,
_you_ had better see her to-day, and
above all things secure her coming to the
Ellersby's party, and Lady Hamlet Vernon's
on Sunday, and to our own party on the
water on Monday, and to the Opera with you
on Tuesday, and so on; in short, taking care
only that not one day shall be lost or misapplied."
"Depend upon me; and now then farewell,
my dear Lady Tenderden. We meet to-night?"
"Of course. _Soyez toujours séduisante
comme à present; cette capotte jaune est délicieuse;
elle vous va à ravir._"
"_Flatteuse_," rejoined Lady Tilney in a tone
of languishing satisfaction, and so they parted
mutually pleased. Lady Tenderden, true to her
promise, drove straight to Lady Glenmore's,
and found her at home. Having expressed
her satisfaction at this fortunate circumstance,
one too of such rare occurrence, she
praised every part of her dress, and inquiring
of the Paris fashions, thus proceeded: "And
now, my fair queen, you are truly an enviable
personage--_you_, if any body ever had,
have really _beau jeu_, every thing that can make
a woman's life truly desirable; a great establishment,
magnificent equipages, jewels,
and the consideration which attaches to a _haut
grade_ in society, a distinguished title, _tout
enfin qui peut embellir la vie_; truly, _je vous
en félicite, ma belle amie_. But you cannot
occupy so enviable a position without exciting
the most active envy. Now allow me, as
a sincere friend, to put you _au courant_ of some
things, in respect to the true nature of which
you may be deceived. There are a certain
set of persons, who will very naturally pay you
court, and endeavour to obtain your ear; such
as the Duchesse D'Hermanton, the Ladies
Proby, and Ladies How, and all that tiresome
concourse of old dowagers; but be upon your
guard against these, and without giving open
offence to any body, be sure that you get rid
of them in their very first onset."--Lady Glenmore
stared. "_Vous ouvrez des grands yeux,
ma chère_, but you will soon learn the use of
these cautions. If the people I have named
send their names, as they will certainly do or
visit you, be a long time before you return the
call; they are an old-fashioned set, who pique
themselves on politeness, and _veille cour_ attentions,
and feeling affronted by this neglect on
your part, they will not so readily or familiarly
accost you in public. When they do
(for some of them are vulgarly good-natured
enough not to take the hint)--when they do
accost you, take care to look as if you did not
know who they were, and to answer them
by monosyllables, if you answer them at all.
"Above all things, never go to their wearisome
_At Homes_; but if they attack you with
one of their downright speeches,--sorry not to
have had the honour, &c. &c.--hoping you had
received a card, &c. &c.--curtsey, and say you
were vastly sorry, but you forgot the day,
or----no no, say _mistook_ it; yes, _mistook_ it,
that is best, because it is a loop-hole that answers
for dinner as well as any other party;
yes, a mistake of the day is the best recipe I
know, for any invitation which you may
chance to hesitate about, and perhaps think
it possible you might like to accept, and
then having done so, repent of it when the
time comes--a mistake in the day sets all
right. You are _au desespoir_, and _they_ must
believe you, or make themselves appear ridiculous;
it may indeed cost you a note or two,
but that is the worst of it, and then _vous en
êtes quitte pour la vie_."
Lady Glenmore, who had been so astonished
hitherto that she could not reply, now
found herself called upon to make some answer,
as there was a pause on the part of
Lady Tenderden.
"You have told me so many things," she
said, "my dear Lady Tenderden" (smiling
as she spoke), "that I am afraid I shall never
remember the half of them, particularly as
they are upon subjects which, to tell you the
truth, do not interest me much, if at all. One
thing you said, however, that was very kind,
and kindness is not lost upon me I can assure
you, which was the cordial expression with
which you wished me joy of my happiness. I
should indeed be ungrateful if I did not feel
warmly obliged to you; only you omitted in the
catalogue of my felicities, that, without which
there would be no felicity for me--I mean
my being the wife of Lord Glenmore; who,
had he not possessed any of the adventitious
advantages you enumerated, I should equally
have preferred to the whole world."
"Oh! _cela va sans dire_, of course such a
young and handsome husband is taken into
the account; but, my dear young friend, _vous
ne voulez pas vous donner des ridicules_, much
less render your husband the laughing-stock
of all the world, by setting yourself up with
him _en scène de Berger et Bergere_; besides,
permit me to say, that is just the way to lose
him. If you are always at his elbow, watching
him _en furet_, depend upon it he will soon
think you are jealous, and following him out
of curiosity. Now there is nothing a man can
so ill bear as the idea of being watched, particularly
by a wife; besides, all his male friends
would avoid him if they saw he had such an
Argus--for, beautiful as you are, you must not
have an hundred eyes, to spy out every thing
your husband does; no no, my dear, when
you are _en tête-à-tête_, it is all well enough, this
new-married fondness; but it will soon evaporate,
take my word for it, and then you
will be dying to break the troublesome habit
_de part et d'autre_, and will not know how to
set about it: take great care, _ma chère ladi_, to
begin as you mean to go on."
"Certainly," replied Lady Glenmore, "I
have but one meaning, one intention--that is,
to love and be loved; and I shall never, I
hope, do any thing which can run counter to
that prime business, that prime duty of my
life."
"Oh!" cried Lady Tenderden, perceiving
she had gone too far, "it is quite delightful
to hear you. You are, I am sure, destined to
be a phœnix" (sneeringly); "and proud indeed
must any woman be to view one of her
own sex so well calculated to be a glory and
honour to it. I was only warning you against
certain appearances, certain misapprehensions,
which persons of your turn of mind are
liable to fall into, and which might be the
very means of depriving you of that which
you are so anxious to retain. I know the
world, believe me, my dear young friend,
and there is nothing in it I can so ill endure to
see, as an assumption of a happiness which is
out of the common line. If you enjoy such a
superlative felicity, _tant mieux pour vous_, but
do not make an _étalage_ of it, for either its
reality will be questioned, or they will take
care it shall not long be one; whereas if you
do as other people do, you will be allowed to
go on quietly, and you may perhaps carry on
this sort of romantic view of life much longer
than persons in general do."
Lady Glenmore, who had listened with
painful earnestness to this insidious advice,
now felt her heart swell, and the tears bursting
from her eyes. "And must I really,"
she said in a voice of suffocation, "pretend
to be indifferent to my husband, in order to
retain his love?"
"Certainly, my dear child; _peut on être si
enfant_" (observing her emotion), "as to allow
yourself to be thus moved about such a trifle;
take my advice, and you will never lose that
sort of hold over his affections which it is so
charming, I allow, to possess. Shew him
that you can have other men at your feet--that
you are not, in short, dependent upon
him for any thing _faites vous un sort_, in short,
_et vous ne vous en répentirez pas_."
"And pray, how am I to set about this
sort of life?"
"Why nothing so easy; simply, go constantly
out, and take care to have one or two
young men _de la première volée_ always about
you; never be reduced to be handed out or into
any public place by Lord Glenmore; only
now and then _pour faire beau voir_, and to
shew that you have _des procédés honnêtes_
one to the other--or else _par hasard_, but
never as a thing of course. Another point
is, you must establish an apartment of your
own; for you cannot think between married
persons how necessary that is, and what an
independence it gives to both. It is so very
disagreeable to have the exact moment of
our going in and coming out commented
upon."
"Dear no, pardon me, not at all. I am always
glad when Lord Glenmore says, 'Where
have you been so long, Georgina?' because
that shews he misses me."
"Oh, of course," said Lady Tenderden, as
she always said when she did not know what
to say; and inwardly she thought what a
world of nature must here be overturned, before
any thing artificial can be sown in such a
soil! "Well, my dear Lady Glenmore, you
come to the Hamlet Vernon's to-morrow
night?"
"Yes, I believe so; that is to say, if Lord
Glenmore is disengaged."
"Now really we shall all be afraid of such
a paragon of love and obedience; or what is
worse, we shall all laugh at you if you give
_tête baissé_ into that sort of ultra propriety.
What can Lord Glenmore's engagements have
to do with your coming or not coming to
Lady Hamlet Vernon's?" Lady Glenmore
blushed, and confessed that she did not wish
to go out if Lord Glenmore did not.
"Well, my dear, I see the terrible re-action
in perspective which must succeed to all this
red-hot love; and it is mighty well for the
moment; only you are laying up, _croyez moi_,
a store of discontent and dissatisfaction for
yourself."
At this moment a servant entered, and laid
a visiting card on the table. "Oh, Mr. Leslie
Winyard," said Lady Tenderden, taking it up,
"a vastly agreeable creature: you will let
him in of course."
"No," answered Lady Glenmore, "the
only thing Lord Glenmore does not wish me
to do, as a young married woman, is to receive
young men as morning visitors, and I
have no wish to disobey him; therefore Mr.
Leslie Winyard has been included in the general
order I gave to that effect."
"_Je tombe de mon haut_; well, certainly, I
never should have guessed that Lord Glenmore,
that handsome, young, gay Lothario,
would have turned out such a tyrant; and to
commence before the honey-moon be well
nigh over to shew the cloven-foot of _husbandism_,
is really putting a seal to that tyranny
with a vengeance! And he--he too, of _all persons_,
to pretend--but I believe that is always
the way, these men _à bonnes fortunes_ do always
make the most insufferable husbands."
"I am sure," replied Lady Glenmore, with
an air of offended dignity which astonished
Lady Tenderden, "I am sure Lord Glenmore
desires nothing of me but what he conceives
is for my own happiness; and I am perfectly
willing to obey him in every thing, far less
in such a matter of indifference as this."
Her cheeks here grew redder and redder during
every word of Lady Tenderden's insidious
speech. The melancholy, uneasy expression,
nevertheless, which in despite of herself
threw a cast of restless inquiry into her countenance,
as though she would have asked "to
what do you allude?" did not pass unobserved
by Lady Tenderden, and she conceived it to
be a good time to let the poison work which
she had thus insidiously distilled; so she arose
to take her leave, and with apparent carelessness
said, "_Au reste_, remember," and she
spoke in a soothing tone of commiseration,
as if she wished, were it possible, to have
withdrawn, or at least to soften the words she
had uttered, "remember, Lord Glenmore is
not a bit worse than other men, they are all
alike; and really I think him singularly agreeable,
so do not let any thing I have said give
you a moment's uneasiness."
She knew the rankling arrow was in Lady
Glenmore's heart. "You have nothing to do
but to take your own way, and keep it well
in mind that all husbands take theirs, and
my word for it, if you only follow this counsel,
you will live _en Tourtereaux_, and lead a very
happy life."
"I have no doubt I shall do that," said
Lady Glenmore, half-crying.
"Believe me, _cher enfant_, whenever you
feel the least melancholy or uneasy, send for
me, and I shall put all to rights for you in a
moment; you are a delightful, an unique
creature; I really love you, and him too; you
know, he was my play-fellow when we were
children, therefore I take a particular interest
in you both, and am alike the friend of each.
Come, dry these beauteous eyes, whose brightness
ought not to be dimmed by a tear; come,
take a drive with me in the Park." Lady
Glenmore hesitated as she replied:
"I expect Lord Glenmore every moment;
he promised to drive me in his phaeton. He
was to have been here an hour ago" (looking
anxiously at the clock).
"Well, then, if he is an hour after his
appointment, you would not surely wait for
him any longer? Depend upon it he has been
engaged by some business, or it may be love
of virtù or politics, _que sçai-je_--come let us go
and look for him; my life for it we shall meet
him in the Park."
"Perhaps so," said the youthful Georgina
with a sigh, who evidently assented to Lady
Tenderden's proposal for no other reason than
that the hope might be realized;--and ordering
the servant who answered her bell, to
tell her maid to arrange her shawl, she followed
her _friend_ to her carriage.
When they reached the Park her eyes
wandered from one figure to another in quest
of Lord Glenmore; in vain--the admiration
of the passing throng who courted her attention
had no attraction for her, she saw
not the only object she wished to see, and
returned wearied and dispirited, notwithstanding
all Lady Tenderden's endeavours to
amuse and dissipate her thoughts. The moment
she came home, however, she had the
satisfaction of finding her husband already
there, and she scarcely waited to say adieu to
Lady Tenderden before she flew up stairs to
him. After her first greeting, he asked her
where and with whom she had been; and on
telling him, he said, "I am glad, love, that
you like Lady Tenderden, for she has a thousand
good qualities;" (_a façon de parler_ by the
way, which is often taken upon trust from
one month to another, and frequently bears
no true meaning.) Lord Glenmore continued:
"Yes, she has a thousand good qualities, and
is very clever and agreeable in her way, and
has that perfect _usage du monde_ which has
so much charm, and which besides may
be of real advantage to a young person
like yourself entering on the scene; I am
quite rejoiced that she is your friend. It is
true she sometimes overpasses that line of _retenue_
which I might like my young wife to
observe; yet she has never been charged with
any real fault, and in adopting what is best,
you can leave out such parts of her manners
and conduct as may not exactly suit your
age and taste. In short, I think she is a very
useful acquaintance, and you may safely listen
to her advice respecting your conduct in the
world; but after a little experience, my sweet
Georgina, you may make your own choice of
intimates, and I am sure that selection will
always be well and wisely made."
Lady Glenmore listened attentively to her
husband, and sighed as she recalled to mind
the nature of the advice which she had
already received; but thought, "well, then,
Lady Tenderden was right after all, and I
must not tell Glenmore. How childish and
silly I was in having been so vexed about his
not coming home this morning,--still less
must I tell him of her cautioning me against
pursuing him, for should he know that I had a
thought of doing so, it might probably produce
the effect she predicted."
With this idea thus unfortunately impressed
upon her mind by what her husband had
unthinkingly said, Lady Glenmore remained
silent. The hour of dressing now called them
to their toilette, and the subject was not at
that time renewed.
CHAPTER III.
JEALOUSY.
After Lord Albert had parted with his
friend in the Park, he returned again to Lady
Dunmelraise's house; but still in vain--they
came not. The agony of suspense, when prolonged,
is perhaps the severest which the
human mind can know; but like all chastisements
or corrections, it is never sent without
a meaning, and if entertained as it is mercifully
intended it should be, we shall reap the
fruits of the trial.
In the present case, Lord Albert's disappointment
brought back a livelier sense of
the attachment he really felt for Lady Adeline,
and awoke all those tender fears and reminiscences
which cherish love, but which a too
great security of possession had for the present
blunted, or at least laid in abeyance.
He now wondered how he could have
suffered so much time to elapse without
writing to her. He wondered, too, that he
had not heard from her; she had not then
missed the blank in his part of the correspondence;
and it was evident some other
interest had supplied that one in her heart.--He
looked at her picture, as if he could read
in that image an answer to these various
surmises; but it was placid, and serene--it
smiled as was her wont, and he felt displeased
at the senseless portrait, for an expression
which he could not have borne her to wear,
had she really known what his fears and
feelings were. He shut the case and pushed
it from him;--he felt angry--and then
ashamed--for conscience goaded him with its
sting, and in turn questioned him, as to his
right of indulging one such sensation against
_her_, whom in fact he knew he had neglected:
but all this process of mental analization
was salutary, and as he came by degrees to
know himself better, he was enabled to form
a truer estimation, not only of the amiable
person to whom he was bound by every tie
of honour, but of the true nature of real
worth.
At length, on the fourth morning from that
on which he met Lord Glenmore, he found in
North Audley Street a note from Lady Adeline.
"A note only!" he said, hastily breaking
the seal. It was written from an inn on the
road; it informed him that Lady Dunmelraise
had borne the journey very ill, which had
occasioned them to stop frequently; but that
they would reach town she hoped on the following
evening. Lord Albert turned quickly
to the date, and found that it was of the preceding
day, so that he might expect their
arrival that very evening. A gleam of delightful
anticipation now shed joy over his
heart. We easily gloss over our own faults;
and Lord Albert found all his self-reproaches
for neglect and temporary coldness merged
in the fondness he actually felt at that moment,
and his present determination to abide
by, and act upon this feeling, silenced all
self-accusation. With a beating pulse, and an emotion
he did not wish to quell, he determined
on not leaving the house till he should once
more have seen _his_ Adeline.
He seated himself, therefore, in the drawing-room,
and gave a loose to those pleasurable
sensations which now flowed in upon
him. The apartment had been prepared for
Lady Dunmelraise, and all the usual objects
in her own and her daughter's occupations
were set in their wonted places. He recognized
with transport a thousand trifling circumstances
connected with them, which
brought his love, his _own_ love, more vividly
before his eyes. As he carefully enumerated
and dwelt upon these, his eyes rested on a
vacant space in the wall near the piano-forte,
where a drawing of himself had hung; and
the enchanting thought that it had been her
companion in the country, came in aid of all
the rest to soften and gladden every sensation
of his heart. As his eyes wandered over the
apartment in quest of fresh food for delight,
they rested on a parcel of papers, and letters,
lying on the writing table. He turned them
over, hardly knowing why he did so, when a
frank from Restormel, directed to Lady Adeline
Seymour, gave him an unpleasant shock,
and he dropped it with a sudden revulsion of
sensation that was any thing but gentle.
He again resumed the letter, turned it
round and round, looked at the seal--it was a
coat of arms, but the motto, "_for life_," was
a peculiar one. He wondered to what family
it belonged; he thought of consulting some
heraldic work in order to discover, when the
sound of a heavy laden carriage passing
in the street, drew off his attention. He
flew to the window--it was a family coach,
but one glance told him it was not that
of Lady Dunmelraise. Back he came to
the letter table; again _the letter_ was before
his eyes--_the letter_, for amongst many he
saw but one.
"It is surprising," he said to himself, "that
Adeline should have a correspondent at Restormel,
and I not know of it; but shortly,
very shortly, this mystery shall be solved. I
will ask her at once--but carelessly, naturally,
who is her unknown friend at Restormel?
Ask her? no, she will of course tell
me, if she has formed any new acquaintance
with whom she is sufficiently intimate to correspond,
and if she does not of herself tell me,
I shall never _inquire_ into the matter--indeed
why should I? No, there is nothing renders
a man so silly as jealousy, or throws him so
much in a woman's power as letting her see
he is jealous."
With these, and many such contradictory
reasonings as these, did Lord Albert continue
to pace the room along and across, and every
now and then stop and fix his eyes on the
offending letter; when again a sound attracted
him to the window, and though it was dusk,
and objects were indistinctly seen at a distance,
he recognized the well-known equipage. The
next moment he was in the street; and the
next it drove up to the door. He heard Lady
Adeline's soft voice cry out, "There's Albert!"
as she half turned to her mother, and
kept kissing her hand to himself. The carriage
door was opened, and she sprang out,
receiving the pressure of his hand with an
answering expression of fondness.
"Dear Albert, how do you do? have you
not thought we were an age on the road?
But I hope you received my note." Ere he
could reply, Lady Dunmelraise's extended
hand was cordially presented to him, and as
affectionately taken; and while each rested
on his arm on entering the house, he felt in
the kindly pressure of both that he was as
welcome to them as ever.
When he had assisted Lady Dunmelraise,
who moved feebly, to the drawing-room, and
placed her pillows on the couch, even in
this moment of joyous re-union, he could not
fail to observe what ravages sickness had
made in her frame since they last met; and
as he expressed, though in modified terms, in
order not to alarm her, the regret he felt at
seeing her so unwell, he observed the eyes
of Lady Adeline fixed upon him, in order to
read his real opinion on the first sight he had
of her mother; and before he could regulate
his own feelings on the subject, those of Lady
Adeline's overshadowed her countenance with
an expression of sadness she was not prepared
to command, while the tears rushed to her
eyes. Again holding out her hand to Lord
Albert, while a smile of mingled joy and
sorrow beamed over her features, and partly
dispersed the cloud, she said,
"All will be well _now_; my dearest mamma
will soon be better--joy and happiness will
once again be our's." Lord Albert thanked
her with his eloquent eyes; and as he impressed
a kiss on her offered hand, he replied:
"How fortunate that I received your letter
when I did, for in another hour I should have
been on my way to Dunmelraise."
"Indeed!" said Lady Adeline, her eyes
sparkling with pleasure.
"Yes; and I had, but for something which
detained me, been on my road there long
before your letter arrived."
"That would indeed have been unfortunate,"
said Lady Dunmelraise; "to have
missed you after so long hoping to have seen
you there in vain, would have doubled our
regret;" she spoke with a tone of something
like reproach, at least so Lord Albert took
it; and she added, with a melancholy smile,
"It is a bad omen that a letter from _Adeline_
should have _prevented_ you from coming to
us."
Lord Albert felt embarrassed; there was
something relative to the delay of his coming
which he knew he could not explain, and this
consciousness made him feel as if he were
acting a double part. At this moment Lady
Adeline perceived the letters lying on the
table, and taking them up, she glanced her
eye over them as she turned them round one
by one, saying, "this is for you, mamma--and
this--and this--and this, as she handed them
to Lady Dunmelraise--but this one is for myself."
Lord Albert's attention had from the
first moment of her taking up the letters
been riveted upon her, and now with ill-concealed
anxiety he watched every turn of her
countenance, while she broke the seal and
perused the letter. She read it, he conceived,
with great interest; and said, when she had
concluded, addressing Lady Dunmelraise--
"It is a kind word of inquiry for you, my
dear mamma, from George Foley." Lord
Albert changed colour as this name was
pronounced; but neither she nor Lady Dunmelraise
observed the circumstance, and this
gave him leisure and power to recover from
the confusion he experienced. Lady Adeline
again resumed, after a short pause, "You
must have met Mr. Foley at Restormel, Albert;
what do you think of him?"
"I had little opportunity of judging of
him," replied Lord Albert, hesitating as he
spoke; "but he was only at Restormel for a
part of the time I was there. He had, however,
a strong recommendation to my favourable
opinion, from the warm terms of praise and
admiration in which he mentioned you, Adeline."
She smiled, and without any alteration
of manner went on to say:
"I am afraid then he has _too_ favourable an
opinion of me; and if he has raised your expectations
so high of my improvement since
last we met, I shall have reason to lament
your having become acquainted with him;
but he is such an _adorateur_ of mamma's, that
he thinks every thing that belongs to her is
perfection!"
Notwithstanding Lady Adeline's seeming
calmness while speaking of Mr. Foley--notwithstanding
the natural and ingenuous expression
of her words and countenance, Lord
Albert could not divest himself of the idea
that Mr. Foley had some undue power over
her affections. It is easy, perhaps, to shut
the door against evil thoughts; but when once
they are admitted, they obtain a footing and
a consequence which it was never intended
that they should have. Beware, all ye who
love, of admitting one spark of jealousy into
your breasts, without immediately quenching
the same by open and free discussion
with the object of your affections! But there
lies the difficulty--we are ashamed of harbouring
an injurious thought of those we love;
or rather, we are ashamed of _confessing_ that
we do so; and we go on in the danger of
concealment, rather than by humbling our
pride, and laying open our error, obtain the
probable chance of having it exposed, and
removed. While monosyllables of indifferent
import dropped from Lord Albert's lips, he
was in his heart cherishing the false notion that
had the letter, which gave him so much uneasiness,
been entirely of the import which
Lady Adeline represented it to be, it would
have been more natural to have addressed it
to Lady Dunmelraise herself.
He did not, indeed, dare to impugn Lady
Adeline's truth: but he conceived that no
other man should presume to have an interest
in her--in her who _belonged to himself_ (every
man will understand this), which could entitle
him to hold a correspondence with her. He
consequently became abstracted, and there was
a sort of restraint upon the ease of his manner
and conversation, of which Lady Dunmelraise's
penetration soon made her aware, and
to which even the young and unsuspecting
Adeline could not remain wholly blind.
In order to replace things on the footing
which they had been formerly, and which on
their first meeting they still appeared to be,
Lady Adeline turned the discourse to her pursuits
in the country, and spoke in detail of
her drawing, her music, her flower-garden,
and the families of the poor in their neighbourhood
whom she and Lord Albert had so
often visited together.
"You remember," she said, "poor Betsy
Colville, who never recovered the loss of her
lover who was shipwrecked; she is still in the
same state. She goes every day to the gate
where they last parted, takes out the broken
sixpence he gave her at their last interview;
and having returned home, looks in her father's
face, and says '_to-morrow_.' She never
repines, never misses church--joins in family
worship; but her poor mind is touched, and
she can no longer do the work of the house or
tend on her aged parents. I have therefore
paid my chief attentions to that family--and
they are so grateful--so grateful, too, for what
you have done for them. The myrtle we
planted together, Albert, on the gable-end of
the house, now nearly reaches the thatch;
and in all their distress about their daughter,
the good old pair have never forgotten to tend
that plant. Mr. Foley and I rode or walked
there every day."
The latter words of this discourse poisoned
all the sweetness of the preceding part; and the
idea of Mr. Foley became associated in Lord
Albert's distempered mind, with all the interest
and all the enthusiasm expressed by Lady
Adeline; so that he read in her descriptions of
her mode of having passed her time, and the
pleasure she had innocently enjoyed, nothing
but her love of Mr. Foley's company.
Lord Albert became still more silent, or
spoke only in broken sentences; and a deeper
gloom gradually spread over each of the three
individuals, usurping the place of that cordial
outpouring of the heart, which had at first
rendered the moment of meeting so delightful.
After a silence, during which Lady Adeline
and Lady Dunmelraise appeared mutually affected
by the awkwardness which the change
in Lord Albert's manner had excited, yet
anxious to conceal from each other the knowledge
that such was the case--they felt relieved,
when he took up a newspaper, and
read aloud the announcement of an approaching
drawing-room.
Lady Dunmelraise, glad of an opportunity
to find some subject of discourse foreign
to the thoughts which obtruded themselves so
painfully upon her, said, "Well, Adeline, that
is a favourable circumstance, _à quelque chose
malheur est bon_; had I not been so much
worse exactly at this very time, we had perhaps
not been in London; for though I have
for some months past wished you to be presented
at court, we might, ten to one, not
have had courage to leave Dunmelraise at
this sweet season; but as it is, the opportunity
must not be lost, and the only question is, by
whom shall the presentation take place--for
alas! I am not able myself to have that pleasure,
and I fear my dear sister Lady Delamere
will not either;" then pausing a moment,
she added, "perhaps, Lord Albert, Lady Tresyllian
will kindly take that office, if she is to be
in town."
"I am sure she would readily comply
with any wish of yours; but I know my mother
has, in a great measure, given up the
London world, and has not been at any of the
drawing-rooms during the present reign; but,
perhaps, on such an occasion, she might be
induced to forego her determination of retreat."
"Oh, I would not for the world," said
Lady Adeline, "torment Lady Tresyllian
about it; for," she added, smiling, "you know
how very little I care about such things."
"It is well," said Lady Dunmelraise, "to
hold every thing in estimation according to its
due value. Most young persons are _too_ fond of
the gaieties and pleasures of the world; but
you, my dear Adeline, perhaps contemn them
in one sweeping clause of indifference, without
having properly considered to what advantages
they may tend when resorted to in due degree,
and in subordination to better pursuits. A
drawing-room I hold to be one of those very
few worldly pageants which are connected with
some valuable and estimable feelings; the attending
them is an homage due to the state of
the sovereign; they uphold the aristocracy of
the country, which is one of the three great
powers of government, now too much, too
dangerously set aside; and they ought to, and
do in great measure, keep up those barriers
in society, which prevent an indiscriminate
admission of vice and virtue, at least as far as
regards an outward respect to the _appearances
of decorum_. Whenever drawing-rooms shall
be abolished, you will see that much greater
licence in society will take place. The countenance
of the sovereign, the right to be in
his presence, is one which none would voluntarily
resign; and to avoid losing it, is a
check upon the conduct of many, who are
not regulated by better motives; while those
who are, will always duly appreciate those
honours which flow from monarchs, and
which form a part of our glorious constitution.
'Love God, honour the king,' is the
good old adage; and with this conviction on
my mind, and the remembrance of that loyalty
and attachment to the present House of
Hanover which your ancestors have ever displayed,
even to the sacrifice of their lives and
fortunes, my Adeline, I have set my heart on
your being presented to your king; and the
only consideration is, who shall be the person
to present you."
"Well, dearest mamma," replied Lady Adeline,
"any thing you wish, I shall be delighted
to do, and I make no doubt you are perfectly
right; only I did not feel the least anxious, and
I wished to set your mind at rest upon the subject
of my going into public." Lord Albert said,
with an expression of melancholy and displeasure,
"It is quite unnatural for a young person
of your age, Adeline, to affect to despise the
amusements of the world; and unless you
have some _cause_ for doing so, best known to
yourself, I confess I do not understand it."
Lady Adeline was too quick-sighted not to
perceive that something or other pained and
displeased Lord Albert, and had they been
quite alone, she might have asked him the
occasion of this change in his humour; but
as it was, she did not dare to question him;
and by way of turning the conversation into
another channel, she inquired, of whom
consisted the party at Restormel; if they
were clever, or distinguished, or agreeable;
and whether the mode of life there was to his
taste? Lord Albert seemed to awake out of
a sort of reverie into which he had fallen,
and his countenance was agitated by many
commingling expressions as he replied,
"I really can hardly tell you; there were
the Tilneys, the Tenderdens, the Boileaus,
Lady Hamlet Vernon, Mr. Leslie Winyard.
At that sort of party there is little occasion
for the display of talent, and people are glad
to be quiet for a few days when they go to
their country houses; so that each individual
is thinking more of repose than of shining.
As to their mode of life, it was pretty nearly,
I think, what it is when they are in town."
Though Lord Albert spoke this in a hurried
tone, he felt as though he had got well over
a difficulty. But the remark Lady Dunmelraise
made upon his answer, did not particularly
serve his turn at the moment:--"Either
the persons who I heard composed
that party, or Lord Albert, must be much
changed since I knew them, if they could be
in unison," and she fixed her eyes upon him;--his
embarrassment was visible, and did not
subside as she went on to speak particularly
of Lady Hamlet Vernon: "She remembered
her marriage," she said, and commented
upon those sort of marriages, saying, "that
all intriguing schemes were detestable, but
those respecting marriage were of all others
the most thoroughly wicked and despicable.
Lady Hamlet's conduct, too, after marriage
was not very praiseworthy: if a woman sacrifice
every other consideration in allying
herself to her husband for the sake of aggrandizement,
she must at least continue to act
upon that system, and if possible wash out
the disgrace of such an act (for I consider it
to be no less) by her subsequent mode of
behaviour, and the dignified uses to which
she applies her power. But in the present
instance this was far from being the case,
and she had allowed an apparent levity of
conduct, at least, to sully her character. In
one instance, I _know_, she has drawn a person,
in whom I feel great interest, into a manner
of life, and an idleness of existence, which, to
call it by no harsher name, is one of vanity
and folly; but I had hoped her influence was
over in that quarter."
"As I do not know to what you allude,"
rejoined Lord Albert, "I cannot exactly reply;
but certainly Lady Hamlet Vernon is
very handsome, very agreeable, and, for aught
I know to the contrary, leads now a very good
sort of life. She has a finely-disposed heart,
and, I should think, is better than half the
people who find fault with her. If, from having
married an old _roué_, she was thrown into
danger, which her personal charms rendered
very likely to have been the case, kindness I
am sure would at any time open her eyes to
avoid these; whereas undue severity might
make her rush headlong into them--for harsh
opinions in similar cases, nine times out of
ten, drive such persons from bad to worse."
"I conceive," said Lady Dunmelraise, "that
this may sometimes be the case; but it is frequently
only an excuse for not choosing to hear
the truth told. However, there is a society,
of which Lady Hamlet Vernon is one, which
I hold to be the subverter of every thing estimable.
Its great danger is the specious ease
and indifference of those who compose it,
the system being without any system whatever.
The great gentleness of manner and
entire freedom, which seem to be its characteristics,
are its most dangerous snares. No
consecutive speech upon any subject, no power
of reasoning, no appeal to religion, are tolerated
by these persons. They have a lawless
form of self-government indeed, by which they
keep up their own sect and set,--but there is
a mystery in the delusions which they cast
around their victims, the more difficult to detect
since the whole of their lives is spent in a
seeming carelessness about every thing.
"The warning voice of a parent can alone
put a young and unsuspecting member of
society on his guard against being drawn into
this vortex; but it is the young married
persons to whom such warning is more particularly
necessary. However, because there
are persons, who by artful intrigue arrogate
to themselves a certain consideration,
which they receive from the uninstructed and
unwary, and whose ways are certainly not
those of pleasantness or peace--we are not to
say but that there are others who to the
highest rank unite the highest principles, and
who reflect honour on the class to which they
belong--persons who consider their high stations
as being the gifts of God, and themselves
as responsible agents. Yes, the true nobility
of Britain will yield to none other of any
country for intrinsic worth; all the virtues
adorn their families, and religion and honour
stamp them with that true nobility of soul,
without which all distinction is but a beacon
of disgrace.
"It is not, therefore, because a few worthless
or foolish persons, in the vast concourse
of London society, affect an exclusiveness
which rests on no basis of real worth or dignity,
but on the very reverse, that all intercourse
with the world is to be avoided, or all
innocent pleasure to be denied to young persons;
and I should be exceedingly disappointed
to see my Adeline retiring from her
state and station, and coming to have a distaste
for its amusements, because I feel certain that
so violent a re-action is not natural, and that
the real way to be of service to herself and
others, is to fulfil the rank and station of life
wherein she is placed, and in fact to do as our
great inimitable Pattern did--to go about
doing good."
Lord Albert's feelings, while Lady Dunmelraise
was speaking, had undergone many
changes, but the last was that of pleasurable
approval at finding Lady Dunmelraise's opinion
so much in coincidence with his own--and
he said, in his own natural warm manner,
"I hope Adeline will feel quite convinced, by
your sensible manner, my dear Lady Dunmelraise,
of representing this matter, that there is
no virtue, nothing commendable indeed, in despising
or condemning the world _en masse_, and
that there is just as much real good to be done
by living in as living out of it. True virtue
does not lie in time or place--it is of all times,
of all places; and it is a narrow, bigoted view
of the subject alone, which partakes of monastic
rigour and hypocritical ambition under
the garb of humility, which would promulgate
any other doctrine."
"My dear Albert, you know that I have no
wish but to please mamma and you; and I
need not pretend but that I shall be exceedingly
diverted by going to public places. All
I meant to say was, not to make yourselves
uneasy about finding a _chaperon_ for me, because
I am perfectly contented to remain as I
am--although I might be equally well diverted
in leading what is called a gayer life."
Lord Albert's countenance relapsed into
brightness as he said, taking her hand and
putting it to his lips, "You are a dear and a
rare creature--is she not, Lady Dunmelraise?"--and
this appeal Lady Dunmelraise felt no
inclination to controvert; but, rejoicing in the
present disposition which she once more beheld
in her future son-in-law, she now dismissed
him for the evening, saying, "Adeline
and I require some repose, that we may be
fresh to-morrow for all the great events to
which we shall look forward with pleasure, I
am sure, as you seem to be quite of our way
of thinking respecting her _début_ in the great
world--and so good night." The wish was
reiterated kindly, warmly, by all parties, and
they parted happier even than they had met.
As soon as Lord Albert reached his hotel,
he found a note from Lady Hamlet Vernon,
announcing her arrival from Restormel, and
requesting to see him. In an instant, as though
by magic, his doubts and fears respecting Lady
Adeline returned; for with Lady Hamlet Vernon
was connected the recollection of her
mysterious note at Restormel, on the morning
of his departure from thence--and with that
recollection George Foley was but too deeply
mingled. Then ensued a chaos in his mind,
one thought chasing another, and none abiding
to fix any purpose or decide any measure.
At one moment he determined--if such passing
impulse can be called determination--not
to go near Lady Hamlet; but the next he
thought she had shewn so much true interest
for him--she had listened so often to his
rebukes--apparently with more pleasure than
she did to praise from others--that he should
be ungrateful to avoid her _now_, because other
dearer interests filled up his time and his heart,
and he finally resolved on obeying her wishes,
and visiting her the next day.
In the morning of that day, before he had
finished his late breakfast, and ere he was
prepared to deny himself, the door of his
apartment opened, and Mr. Foley was close
to him ere his servant had time to announce
his name.
"I am come," said the latter, with his
polite and honeyed phrase, "to bring you
pleasant tidings, which I trust will apologize
for this my early intrusion. I am just arrived
from South Audley Street, where I had the
happiness of finding our friends pretty well;
Lady Dunmelraise, indeed, was not up, having
been fatigued by her journey; but Lady
Adeline is blooming in beauty--I do not know
when I have seen her looking better." Lord
Albert bowed, and in his coldest manner replied,
"he was very happy indeed to hear that
Lady Adeline Seymour was so well, and he
hoped, when he should make his personal
inquiries, to find Lady Dunmelraise in the
drawing-room."
Mr. Foley was too penetrating not to see
that this information, as it came from him,
conveyed no pleasurable feeling; but affecting
not to observe this, he went on to talk of the
late party at Restormel--spoke of Lady
Hamlet Vernon as being a delightful creature,
and drew a kind of parallel _raisonné_ between
her character and that of Lady Adeline's.
Lord Albert was thinking, all the time he
spoke, of the impertinent assumption of Mr.
Foley's addressing him on the subject of Lady
Adeline, and discussing her merits, as though
he were not aware of them, and had not a
better right and ampler means to know and
to value them.
Still there was a suavity--a delicacy even,
in Mr. Foley's mode of expressing himself,
which gave no tangible opportunity to shew
offence; and Lord Albert, though writhing
under impatience, was obliged to control
himself. As soon as he could possibly contrive
to do so, he changed the conversation, and
spoke of the Opera, the Exhibition, the topics
of the day--of all, in short, that was most
uninteresting to him; and carried on an under
current of thought all the time on the impropriety
Adeline had been guilty of, in receiving
Mr. Foley without her mother's presence to
sanction such a visit, and on going himself
directly to South Audley Street, in order that
he might disclose to her his opinion on the
inexpediency of such a measure, as that of her
receiving the visits of young men when alone.
But though the evident abstraction of Lord
Albert D'Esterre rather increased than diminished,
still Mr. Foley sat on, and sometimes
rose to make a remark on a picture--sometimes
opened a book, and commented upon its
contents. Similar provocation must have occurred
to every one at some time or other,
and it is in vain to describe what, after all, no
description can do justice to. A note arrived
for Lord Albert--it was from Lady Adeline--very
kind, but desiring him not to come to
South Audley Street till four o'clock--saying
she was going, by her mamma's desire, to see
her aunt Lady Delamere, who was confined by
a feverish cold, and could not leave her chamber
to come to them.
Lord Albert's mortification was painted on
his countenance. "If you have nothing better
to do this morning, D'Esterre, and that your
note does not otherwise take up your time,
will you accompany me to Lady Hamlet
Vernon's?" Lord Albert felt, "what, am I to
be balked, dogged, forestalled in every trifling
circumstance by this man!" but he _said_,
hesitating as he spoke, "yes--no, that is to
say, I had an engagement, but it is postponed
for the present--therefore, if you please, I
will accompany you to Lady Hamlet's door;"
and Mr. Foley, evidently triumphing in having
foiled Lord Albert's real intentions, whatever
they might be, but maintaining still his quiet
composure, offered Lord Albert his arm, and
they walked together towards Grosvenor
Square, each talking of one thing and thinking
of another.
CHAPTER IV.
AN EXCLUSIVE MORNING PARTY.
As they walked along between Lord Albert's
house and that of their destination, one idea
took the lead in D'Esterre's mind--it was the
hope of obtaining from Lady Hamlet Vernon
an elucidation of the mysterious expressions
contained in her note. He formed a thousand
plans how he should contrive to remain
alone with her, after Mr. Foley should take
his leave, for he made no question but that
he would be the first to end his visit; and he
settled it in his own mind that he would affect
to have some message to give Lady Hamlet,
which might afford him an opportunity of
procuring the interview he so eagerly desired:
but almost always, in similar circumstances,
none of these minor events occur as
we intend they should; and the first object
Lord Albert saw on entering Lady Hamlet
Vernon's drawing-room was Lady Tenderden,
sitting at a writing table, having taken off
her bonnet as though she had come upon
some particular occasion, and was fixed there
for a considerable time.
"Ah! Lord Albert," said Lady Hamlet
Vernon, "and Mr. Foley too! Most welcome
both.--Restormel was quite dull without
you; and besides the comfort one always feels
at coming back to the dear dirty streets,
after having been banished from them a few
days, I am really charmed to find myself
once more surrounded by all my friends. Do
tell us the news, and sit down--you shall not
positively pay me a flying visit--though you,
Lord Albert, flew away in such a hurry from
Restormel, that we had not time, no not even
to say 'farewell;'"--(and she looked at him
very significantly as she spoke.) "So before I
shall have time now to speak to you, you will
be gone again--but if so, it is not _my_ fault."
Lord Albert thought that he read the
meaning of this speech, and his impatience
and anxiety were increased in proportion.
It was with the utmost difficulty he could
bring himself to leave her side in order to go
to the other end of the room, in obedience
to Lady Tenderden, who called him every
now and then to ask some silly question or
other, which he hardly answered; and which
induced her, therefore, to beg him to come
and sit near her, that she might talk to him
comfortably while she was writing: two
things which she declared she could do quite
well at the same time. As soon as Lady
Tenderden had managed this contrivance,
Mr. Foley entered into (apparently) a very
interesting conversation with Lady Hamlet
Vernon; and Lord Albert sat on thorns as
his eyes were rivetted on them, while he contrived
to answer Lady Tenderden, although
it were as if he was playing at cross purposes.
Any change was a relief, and the announcement
of Lord Glenmore was a real pleasure
to him, for he thought his arrival must at
least break up the _tête-à-tête_ between Lady
Hamlet and Mr. Foley, which seemed to him
as if it never would end.
After having paid his compliments to Lady
Hamlet Vernon and Lady Tenderden, Lord
Glenmore accosted his friend, and cordially
wished him joy in a sort of half whisper, on
Lady Dunmelraise's arrival. But, in Lord
Albert's present frame of mind, this congratulation
was not received with that open warmth
which Lord Glenmore expected; and he
dropped the subject, taking up those of the
common-place occurrences of the day. The
drawing-room was discussed; it was to be
fuller than any preceding one. Lady Tilney
had declared she would not go--so had Lady
Ellersby; "but, nevertheless," said Lord
Glenmore, with one of his good-humoured
smiles, "I dare say those ladies will not
have the cruelty to allow their absence to
be regretted when the time arrives; do you
think they will, Lady Hamlet Vernon?"
"Most indubitably not, and I make no
doubt the _plumassiers_ and jewellers are all
at this moment in requisition in Lady Tilney's
boudoir. But, by the way, Lord Glenmore,
your fair lady will of course be presented on
your marriage--who is to have the pleasure of
presenting her?"
"Who? why of course her mother, Lady
Melcomb."
Lady Hamlet Vernon and Lady Tenderden
here exchanged the most significant glances,
and a silence ensued; which was first broken
by Lord Glenmore, who endeavoured to draw
Lord Albert into conversation by touching
alternately on politics, literature, and all the
subjects which he knew were interesting to
him; but to which he could only obtain some
short answer, that did not promote the flow
of the conversation. He began to ask himself
whether he could have given Lord Albert
any offence, or whether he retained any on
account of their interview in the Park; but it
was so unlike Lord Albert to take offence
where it never was intended to be given,
that he concluded (as was in fact the case)
that something painful was on his mind, of
which he could not divest himself. Having
vainly attempted, by raillery as well as by
engaging his attention, to get the better of
this abstraction and gloom, Lord Glenmore
let the matter pass, and addressed his conversation
elsewhere; but Lady Tenderden was
not to be diverted from her purpose, and she
took up the thread of discourse, requesting
to know if Lady Adeline Seymour had imposed
a vow of silence upon him, or what
other cause had so changed him since he was
last at Restormel? He pleaded total ignorance
of being changed; but the consciousness
that he was so, rendered his efforts at
disguise only more visible.
Lord Albert rose and sat down; a hundred
times he looked at a French clock on the chimney-piece,
which of course did not go; and
at last requested Mr. Foley to tell him the
hour, as he had an engagement which demanded
his attention. Having found that it
was a full half hour past the time appointed
by Lady Adeline, he made his bow to Lady
Hamlet Vernon, and was about to leave the
room, when she called him back, and said,
"of course we all meet in the evening at
Lady Tilney's?" There was a glance and an
emphasis which accompanied these words,
which he could not fail to interpret as an
assignation, and one that he determined on
his part to keep.
Could Lord Albert have known what was
passing in Lady Adeline's mind, while he was
thus misspending his time in a false anxiety
about a few mysterious words, written, it
might be, with no good intent, and indeed it
might be without any foundation, he would
have hastened away from this idle and unworthy
mode of passing his time long before
he did; but experience unfortunately must be
bought, and although we look upon the actions
of others, and comment upon them, it
may be with the calm wisdom of unmoved
breasts, yet in our own time of trial we are too
apt to prove that theory is not practice. One
would imagine that it was the easiest thing
possible to place one's-self ideally in the
situation of another, to feel as he felt, and
yet act diametrically opposite to the way in
which he acted, in certain circumstances
and positions; but this apparent facility of
transmigration into the identity of another's
being is mere delusion. It may be questioned
if any human creature really understands
another, and how much less likely is it that
he should argue justly on his neighbour's
affairs! Oh, if we were more merciful to
others, and more severe on ourselves; more
humble as to our own merits and more alive
to those of our fellow creatures; we should be
nearer the mark of justice than we usually
are.
While Lord Albert, under the influence of
a tormenting incipient jealousy, wasted the
hour at Lady Hamlet Vernon's which he
should have passed in South Audley Street,
Lady Adeline had been with her aunt, Lady
Delamere, who, in a true spirit of affectionate
solicitude, had nevertheless opened up a
source of anxiety and doubt in the breast
of her niece, which proved the cause of
infinite distress to her. Lady Delamere, after
receiving her with all that glow of partial
fondness peculiarly characteristic of her family,
it might be too much so towards each
other, naturally spoke of Lord Albert D'Esterre.
"Ah, my dear Adeline, now the time approaches
when, according to your father's
will, your final decision respecting the fulfilment
of your marriage must take place, my
anxious fondness suggests a thousand fears,
at least doubts, for your happiness. I beseech
you let these four intervening months at least
be given, not only to a serious examination of
your own heart, but to a clear and vigorous
elucidation of the disposition and principles
of Lord Albert."
"As to my own heart," replied Lady Adeline
with quickness, "it has long not been in
my own keeping, for most fortunately, where
my duty was directed to place it, there my
choice seconded, nay, almost preceded the
arrangement. But why should you doubt that,
such being the case, my happiness should be
endangered? say rather, dearest aunt, confirmed."
"It may be so--I trust it will be so, my
sweet Adeline, since your love is fixed; but
remember how very serious a step marriage
is; and before you are bound for life in the
holiest of all ties, again I conjure you to lay
aside, inasmuch as you can do so, all the
blandishments of love, and consider how far
the tastes, the pursuits, the temper, above all
the religious tenets of your husband, will be in
accordance with your own. Indeed, indeed,
people do not reflect seriously _enough_ on these
points. I ask not any long consideration, any
great trial of time or absence--they are both
circumstances which may deceive either way;
for things viewed at a distance, are not seen
in their true light; and one may be as much
deceived at the end of a year, as at the end of
a month--and life is short. The life of life,
the bloom of youth, should not be needlessly
withered in pining anxiety. What I ask of
you is, during the time you are now to be in
town, to go out with moderation into the
great world, to see what it has to offer, and
to know whether any other person might supersede
Lord Albert in your affections; this
is as yet a fair and honourable trial. You are
_not bound_ to each other, if either wishes to
break the tie." (Lady Adeline sighed heavily.)
"And should you, while together, discover
any flaw or imperfection which might make
you wish to dissolve the engagement, now is
the time; but after marriage, I need not say,
my Adeline, that one glance of preference for
another is guilt--one wish, foreign to your
allegiance as a wife, is _misery_."
There was a pause in the conversation.
Lady Adeline felt sorrowful--she scarcely knew
why, except indeed it had never occurred to
her that any thing could step in to break off
her engagement with Lord Albert; and the
bare possibility of such an event seemed to
unhinge her whole being.
The fact is, Lady Delamere had heard surmises
of Lord Albert's intimacy with Lady
Hamlet Vernon, and without informing her
niece of a report which, after all, might not
have any foundation, she yet conceived it to
be a duty to put her on her guard, and make
her ready to observe any alteration that might
have taken place in Lord Albert. She would
have told Lady Dunmelraise all that she had
heard without disguise; but at present her
state of health was such, that she could not
think of endangering her life by giving her
such information; for she well knew her sister's
heart was set upon the match, and that
she had long loved Lord Albert as though he
had been her son. However, she determined,
the moment Lady Dunmelraise was better, to
have no concealment from her. It had not
been without much self-debate that she had
brought herself even to hint any thing like a
doubt to Lady Adeline of Lord Albert's truth;
and even now, she only endeavoured to prepare
her to open her eyes to the conviction,
should such a melancholy change have taken
place, but without naming the real cause she
had for giving her such caution.
As it was, it was quite enough to sadden
Lady Adeline; and her air was so dejected
when she returned home to Lady Dunmelraise,
that the latter feared something had occurred
to vex her. "Is my sister worse,
dearest child?--I pray you do not conceal the
truth from me."
"Oh no;--be not alarmed," she replied,
"my aunt hopes, in a day or two, to be able to
come to see you, dearest mamma. It is not
that--but I have a bad head-ache, and have
undergone too much excitement." The look
of anxious inquiry which Lady Dunmelraise
could not conceal, lessened not Lady Adeline's
unhappiness; and as the time which
she had appointed for Lord Albert's visit
was now far passed, the whole weight of
the sad warnings she had received, seemed
doubled. At length the peculiar knock--the
quick footstep on the stair, told her he was
come, and she passed from her mother's bedroom
into the adjoining drawing-room to meet
him.
They seemed mutually affected by some
secret cause; for there was not that cordial
clasping of hands--that beaming of eyes--that
joyful tone of greeting, which might have been
expected to mark their meeting on this occasion:
their hands touched coldly--and Lord
Albert made no effort to retain her's.
"You have been very much later than I
expected, Albert."
"Yes: I could not exactly obey the hour
named in your note, as you went out before I
could possibly come here this morning; and
as you put me off, I had another engagement,
which in my turn detained me; however, I
was happy to hear you were well from Mr.
Foley, who had the pleasure of seeing you,
I believe, very early."
"Yes: Mr. Foley, you know, as mamma's
_protégé_ and _enfant de famille_, has the _entrée_
at all hours, and I was drawing when he
came in; I thought it was you, and--
"Oh, dear Lady Adeline, you cannot suppose
I should take the liberty of inquiring
what you were doing--I hope Lady Dunmelraise
is better to-day?"
Lady Adeline, under any other influence than
that which now influenced her, would have
said, "Albert, what is the matter with you? are
you displeased?" But her aunt's advice was,
"look well to the real state of Lord Albert's
affections, and do not allow your own to give
a colouring to his, which may not be the true
one, were his heart unbiassed by the flattering
predilection you so openly profess for
him." This advice sealed her lips; and,
checking the natural impulse of her heart,
she replied to his inquiries about her mother
more at length than she would have done,
in order to recover a composure she was far
from feeling; she allowed all further discussion
of her mode of passing the morning to
drop.
Lord Albert's restrained, unnatural manner
increased, and they both felt relieved
when Lady Dunmelraise called from her
apartment to her daughter--who obeyed the
summons; but returning after a minute's
absence, she said,
"Mamma hopes you will dine with us to-day."
"Oh, certainly, if Lady Dunmelraise wishes
me to do so:" and as Lady Adeline made no
reply, but returned to her mother, Lord Albert
departed to dress.
When they met at dinner, Lady Dunmelraise's
presence for a time prevented the
awkwardness they mutually felt; but she
soon found that the conversation was entirely
left to her, and could not be long
without perceiving that something had occurred
which altered Lord Albert's manner.
Hoping it, however, only to be one
of those fallings-out of lovers which are the
renewal of love, Lady Dunmelraise turned
the conversation entirely upon the coming
drawing-room, and the more interest she
seemed to take in her daughter's going into
the gay world, the more grave did Lord
Albert become: this was a contradiction to
what he had expressed respecting that measure,
and, as Lady Dunmelraise thought, a
caprice of temper, which she was sorry to
observe in him. She hoped, however, that
the thoughts which involuntarily arose in her
mind were groundless, and she determined
not to act precipitately; but felt glad that she
was come to town, where she would have an
opportunity of judging further, and of seeing
how matters stood from her own personal
observation of Lord Albert's conduct. She
considered that to probe her daughter's feelings
upon the subject, would be to excite them
so painfully, that they might destroy the power
of a cool judgment. She therefore resolved
to postpone any avowal of her own sentiments,
any positive declaration of her own
doubts, till the time, which was now fast approaching,
for Lady Adeline's ultimate decision,
should afford her a proper opportunity
of speaking her mind unreservedly to Lord
Albert; unless, indeed, circumstances of an
imperious kind relative to his conduct should
make such a step necessary before that
period.
In this disposition of mind, the parties
could not enjoy each other's society. The
conversation was broken, interrupted, and in
itself devoid of interest; so that when Lord
Albert arose to take his leave about ten
o'clock, Lady Adeline almost felt it a relief.
"What, are you going to leave us so soon?"
said Lady Dunmelraise, with visible surprise.
"I am sorry that a particular engagement
obliges me to go."
"And may I ask," rejoined Lady Dunmelraise,
in her quick way when she was not
pleased at any thing, "may I take the
liberty of asking where you are going?"
"Oh, certainly--to Lady Tilney's."
"To Lady Tilney's _party_!" with a marked
emphasis on the last word; and then checking
herself, and resuming her usual dignity
of composure, she added, "I hope you will
have an agreeable _soirée_; when one lives out
of the world, and grows old, one forgets the
delights of these sort of re-unions; but, of
course, one must do in London as they do in
London; and I believe, like most other things,
the habit of attending them becomes a second
nature." Lord Albert smiled--it might be
in acquiescence, it might be in disdain; and
with many good-nights, he slightly touched
the hands of Lady Dunmelraise and her
daughter, and departed.
There was a silence, an awkward silence;
neither liked to express the thought that was
uppermost in her mind, for fear of wounding
the other. At length Lady Dunmelraise
spoke: "It is strange," she said, "to observe
the sort of hold which foolish things sometimes
obtain over sensible men. The class of persons
with whom Lord Albert seems now to be
living, are not those I should have conceived
that he would ever have selected; but fashion
leads young people to do a thousand silly
things, which they repent when their ripened
judgment shews them in their true colours;
and to say truth, I think Lord Albert's manners
altogether have not gained by foreign
travel. But I suppose I must not express
such treason to you, Adeline?" Lady Adeline
tried to smile, as she replied:
"I have hardly had time to judge;" and
Lady Dunmelraise turned the discourse rather
on the associates of Lord Albert than on
himself.
"The persons," she said, "he named to
us as having been at Restormel, and with
whom he now appears so much engaged, are
those who live entirely for this world: and
not even for the most dignified employments
or pursuits of this present existence. Fortune,
health, and morals, are all likely to become
the prey of a voracious appetite for
pleasure; and when we live only to pleasure,
we lose all title to being rational souls, and
make a wreck of happiness. I am willing to
hope and believe, that many are ensnared
to tread this Circean circle who are in ignorance
of what it leads to; who see in it only a
brilliant phantom of amusement, a glittering
_ignis fatuus_ that pleases their fancy, but which,
alas! I fear, too frequently leads them on,
till some entanglement of fortune, or virtue,
levels them with its worse members; and from
which it is a mercy indeed if they ever escape."
Lady Adeline had listened to her mother
with an interest that made her shudder.
"And is it, indeed," she cried, "in such a
set that Albert is thrown!" while the paleness
of her countenance expressed the anguish of
her mind.
"I trust not, my dearest child. I do
not mean to say, for I have no right so
to say, that Lord Albert is habitually one of
this set;--heaven forbid!--but that he frequents
their society appears evident. However,
let us not think evil before it actually
occurs; let us judge dispassionately, and see
for ourselves. You are now, my love, to enter
into the great world under an excellent and
loving guide; and having warned you, I leave
your own good sense to do the rest." Lady
Adeline sighed heavily, and did not seem able
at all to rally her spirits. "Now, love, let us
turn to lighter matters," said Lady Dunmelraise,
"and consider the arrangements of
your presentation dress."
"I should prefer its being as simple as possible,"
said Lady Adeline, "and the rest I
leave entirely to your, and," she added hesitatingly,
"to Lord Albert's tastes." Her
mother shortly after proposed retiring for the
night, and trembled as she saw how deeply
her daughter's happiness seemed to depend
on Lord Albert, perceiving that she referred
every trifle to his arbitration.
When he left South Audley Street to go
to Lady Tilney's supper party, Lord Albert
ran over again in his mind the occurrences of
the day, and in Lady Adeline's silence, her
manner, her looks, he thought he read an
indifference towards himself, which at once
piqued and wounded him. In all that had
fallen from Lady Dunmelraise, in all that he
could gather from _her_ manner towards himself,
he could not fix on any thing unkind or
unjust; but from the consciousness of his own
conduct not having been what it ought, his
heart was ill at ease, and he knew not with
what right he felt angry; but yet he did so
feel, and was tempted to inveigh against
the fickleness of woman, while a thought of
Mr. Foley obtruded itself among all the rest,
and shewed him an imaginary rival.
"Can all this," he asked himself, "be only
preparatory to her breaking off her engagement
altogether?"
Such was the mood of mind in which Lord
Albert entered Lady Tilney's drawing-rooms,
and as hardly any of the invited were as yet
come from the Opera, he had leisure unmolested
to walk through them. They were brilliantly
lighted, and filled with all the rifled
sweets of the green-house; sweets, which
seem but ill suited in their fresh purity for the
scene they were brought to adorn.
While the apartments were still empty, he
had an opportunity of examining some of the
works of art with which they were decorated.
He stopped opposite to a Claude, which was
certainly a contrast to the feelings of his own
mind. The glowing sunrise, the dancing
wave, the palace of the Medici, the business
of a sea-port, conveyed him in idea to the
Pitti Palace. "Often as that subject has been
repeated," he said, turning to Mr. Francis
Ombre, "by the same pencil, it is always new,
always redolent of repose and pleasure; the
scintillating sunbeams are still emblematic
of that dancing of the heart, which in the
morning of our days gilds every thing with
beauty: no, there is no after-pleasure which
can equal the sunrise of existence; and if
ever picture conveyed a moral truth, the pictures
of Claude most assuredly have this
power."
"Yes," replied Mr. Ombre, "I love to
sun myself at a Claude, it is the only sun one
does see in this climate." Lord Albert passed
on, sighing as he went, and his attention was
again arrested by an antique bust of Psyche:
"What refinement of tenderness in the eyelid;
what soul in the curvature of the lip!
how the line swells, and then is lost again in
the almost dimpling roundness of the chin!
how child-like, and yet how replete with
meaning, the turn of the head and neck! it is
at once the bud, the flower, the fruit of beauty
amalgamated and embodied in the marble."
It was indeed an emblem of soul. And
of whom did it remind Lord Albert? Of
his own Adeline. His own! there was an
electric touch in the thought--was she _indeed
still his own_, or had he lost her for ever?
Lady Hamlet Vernon had stood unperceived
by him, watching him for some previous
minutes, and by that sense which
never fails to inform a woman in love, she
felt certain from his manner of looking at
the Psyche, that it conveyed more to interest
him than any mere ideas of _virtù_ could possibly
do.
Her agitation was extreme, and she could
scarcely master it so as to wear a semblance
of composure; at length, though the part
she had to play was a difficult one, she determined
on fulfilling her assignation; and
having previously decided how she should
manage what she had to do, she went up to
him, and at the very moment he was asking
himself whether or not he had lost Adeline
for ever, a soft voice awoke him to a sense of
who and where he was: he turned round and
beheld Lady Hamlet Vernon. The recognition
of any one whom we believe has an interest
in us when the heart feels desolate, is a
powerful cordial to the spirits.
Lord Albert greeted her with an animation of
pleasure that he was scarcely himself aware
of, and which elicited from her an answering
sentiment of kindness, that at once cheered
and gave him new life. "I have much to say
to you," he whispered; "let us sit down in
yonder alcove, which is unoccupied, and
where we may have an opportunity of speaking
unheard by others." He offered her his
arm, which she accepted, and they moved to
that part of the apartment. At the same instant
Lady Glenmore entered, leaning on her
husband's arm, and a crowd followed which
filled the room. Among these, Mr. Leslie
Winyard and Lady Tenderden were conspicuous
personages: but Lady Glenmore was
the _nouveauté du jour_. When Georgina Melcomb
was an unmarried girl, nobody looked
at her, or thought about her; but now that
she was to play a part, and in her turn become
a card to play in the game of fashion, all eyes
were fixed upon her. At this moment she
was the very picture of innocent happiness,
and in the countenance of her husband shone
the reflection of her own felicity. There is
something in that sort of happiness which
involuntarily inspires respect, and to all
hearts that are not dead to nature, there is
awakened a simultaneous sensation of pleasure.
But yet there are serpents in the world,
who, envious of such pure bliss, seek only its
destruction. "Really," said Mr. Leslie Winyard
to Lady Tenderden, "that is a fine-looking
creature!" speaking of Lady Glenmore
as she stood talking with animation to
her husband, "and when she has rubbed
off a little of her coarseness, and become
somewhat less conjugally affected, I don't
know but what I may do her the honour to
talk to her sometimes myself." Lady Tenderden
laughed as she replied,
"There is no saying how condescending
you may become--but when do you intend to
begin? don't you see that if she is allowed to
go on in this way, she will never get out Of
it? and as I have undertaken her education
myself, I do beg that you will by some contrivance
unhook her from Lord Glenmore, and
leave me to engage his attention while I make
my pupil over to you for the evening, _vraiment
ça vaut la peine_; only _la jeune Ladi est
tant soit peu maussade et il faut la mettre sur
le bon chemin_."
"With all my heart; if you will only
begin the attack I will follow it up."
"_Allons donc_," she replied, taking his arm
and going towards the Glenmores.
The usual nothings of common-place talk,
the unmeaning greetings, and the self-same
observations on singers and dancers which
have been made a hundred times before, opened
the meditated campaign. "My dear Lord
Glenmore," said Lady Tenderden, "I have
long wished to consult you about a _changement
de décoration_" (and she looked at Mr.
Leslie Winyard) "which I purpose making in
my house in town, and I have some thoughts
of copying in part the Rotunda-room which
is here, only there are some objections to be
made to it, which I wish to avoid if possible,
and I am desirous that you should assist me
with your perfection of taste; have the kindness
for a moment to come with me--but I
could not think of giving Lady Glenmore that
trouble. There, Mr. Winyard, while I run
away with my lord, do you make the _preux
chevalier_, and defend Lady Glenmore from
all dangers."
So saying, she passed her arm through
Lord Glenmore's and led him away. Lady
Glenmore looked for a moment as if she
intended to follow, and even half rose from
her chair for that purpose; but the lessons
Lady Tenderden had given her about
not seeming to pursue her husband recurred
to her, and she sat down again, blushing and
breathless, and evidently discomposed. Mr.
Leslie Winyard enjoyed the scene: "shall I
call Lord Glenmore back again?" he asked,
after fixing his eyes upon her maliciously,
"or will you allow me to conduct you to
him?" and he smiled, evidently in ridicule at
her awkwardness. But she was not a fool,
though ignorant of the ways of the world; and
in a few minutes she recovered herself, and
spoke uncommonly well on common-place
topics, to the astonishment of her hearer:
she even passed upon the set to which he
belonged some very stinging remarks, the
more so from their being uttered as if unconscious
that they were so, or that he was one
of the persons to whom they applied.
"Do you know," said he, gazing at her with
looks of admiration, "do you know you are
a very extraordinary personage? Suffer me to
say that this is all very well in joke, but if
you are _serious_ in your opinions, we must
undergo a great revolution, or we shall not
be at all able to live with you. I do not pretend,"
he said, "to decide who is in the right
or who is in the wrong, but I am very certain
of one thing, a change must take place
somewhere, if your ideas of things in general
are correct." Lady Glenmore replied, "that
she was very certain her ideas would _not_
change;" to which he rejoined, "_nous verrons_."
At that moment a move in the room
announced that every one was going to supper,
and the doors were thrown open into an
adjoining apartment, towards which there
was a general rush. Lady Glenmore again
cast her eye anxiously around, but in vain--her
husband was not to be seen.
"Allow me," said two or three young men,
offering their arm to her, "to hand you to
supper," and in the confusion she took that
of Mr. Leslie Winyard. "But," he observed,
"you seem so uneasy, that if you will allow
me, I will merely see you agreeably placed,
and go in quest of this envied Lord Glenmore."
"You are very good," she replied, "but
I cannot think of giving you that trouble."
"Oh dear, I beg you will not mention it;
and the mission is so new a one, that I am
particularly proud to be employed in executing
it."
"How, new? Is there any thing extraordinary
in wishing to know whether one's husband
chooses one should go home, or whether
he stays supper or not?"
"Yes, Lady Glenmore! most new! most
wonderful! But I do not think it is a fashion
that will generally take. But here is
a table with some seats unoccupied. Will
you allow me to recommend your availing
yourself of it? It seems to be the choice of the
chosen; here is Lady Hamlet Vernon, and
Lord D'Esterre, and the Boileaus, and the
Ellersbys, and Mr. Spencer Newcomb; do
take this seat, and I will go in quest of your
lord and _master_. But see, he has not fallen
into any of the whirlpools or quicksands that
you seem to apprehend for him in these dangerous
regions, for by all that is fortunate
there he is next to Lady Tenderden."
"Where?" cried Lady Glenmore, looking
eagerly around.
"The third table from us, just behind
Lady Baskerville; however, if you are still
_uneasy_, you have only to command me."
"No, it is his intention to remain for supper,
and all is well, for if he had wanted me he
would have sought for me."
"Always depend upon that. And now what
shall I help you to?" Lady Glenmore, in
her own mind, was not at all satisfied as to
the danger of whirlpools and quicksands,
though they were of another sort from those
Mr. Winyard had passed his jokes on; but
again Lady Tenderden's advice recurred to
her, which had acquired consequence from
Lord Glenmore's opinion of that lady, and
she endeavoured to enter into the conversation
of those around her. It was a sort of
dead language as yet to her ears, but she
could perceive that, under disguise, many
allusions were made to herself, and to her untutored
behaviour, which checked her natural
flow of spirits, and she gradually became silent,
and could no longer conceal her anxious
impatience to be once more safe under her
husband's wing. The very first person that
arose afforded her an opportunity of doing so
likewise, and making a sign to Lord Glenmore,
she waited for him in the door-way.
He was not long before he joined her, and
with apparently mutual satisfaction they once
more found themselves together. This difference,
however, existed in their feelings, that
Lord Glenmore, though honourable himself,
and incapable of thinking really ill of others,
however he might consider them trifling, yet
from habit and the manners of the world, had
not an idea of watching his wife's conduct in
public.
Lord Glenmore's character has been already
described; but it has not perhaps been sufficiently
explained how very much his guileless
unsuspecting nature laid him open to become
the prey of others who were the reverse.
Let no man cast a young wife (unprepared for
the dangers she will meet with) upon the
licentious intercourse of the world of _ton_, nor
leave her, unguarded by his presence and
authority, to stem the tide of vice which may
steal in upon her unawares. It is a husband's
duty to be the guide and support of his wife;
and, without tyranny, but with the determined
rectitude of tender solicitude, to watch over
their mutual interests. The maxim so often
quoted, that "the wife whom a man can doubt
is not worthy of his regard," is not always a
true one. Every mortal is liable to err--and
why should woman, the weaker sex, be cast
upon the world, and committed to its dangers,
without stay or support from her natural
guardian and protector?
The fact is, it is a maxim often resorted to
in idleness or indifference, and is more frequently
an apology for bad conduct in those
who make it, than arising from any true nobility
of soul or any moral or religious principle.
Lord Glenmore, from living in the midst of
the world of fashion, and from never having
(a rare instance) been spoiled by such a life,
was less aware than any human being perhaps
of the danger to which he was exposing his
young wife. Had any body told him the terms
upon which she was to be admitted as one of
the _élite_ of _ton_, in plain language, he would
have started with disgust and horror from all
such association; but, like some few, deceived
as he was by specious appearances, he saw
nothing in the set but the airiness of fashion,
and the folly, at worst, of a few months during
the London season; whereas the truth stood
thus.--
The husband of an Exclusive must be exclusively
given to his own devices, without ever
making his wife a party at all concerned in
them; unless, indeed, they arrive at that _acmé_
of exclusive perfection when they boast to
each other of the degrading license of their
lives, and tell of their different favourites,
comparing the relative merits of these with
that of others of the same society. Into the
mysteries of an exclusive _coterie_ no unmarried
woman, that is to say, no girls, are to be admitted--in
order that the conversation may be
unchecked. The more admirers a married
woman has, the higher her reputation amongst
them; and it is never quite complete till some
one _adorateur_ moving in the same circle is the
_ami preféré_. If the cavalier be a man of title,
power, and wealth, then the lady has _the
world--their_ world--at her feet. This arrangement
ensures the latter (whatever her husband's
fortune may be) the advantages of dress
and equipage, from which expense _he_ is then
exonerated; and while he has the credit of
keeping up a tasteful establishment, he is
exempted from all trouble or thought as to the
means by which it is so kept. But as in all
communities there are different degrees of
distinction, so in this,--those who commence
their career have a certain rubicon to pass
through before they arrive at such a height of
perfection.
The first requisite for a newly-initiated
member to know is, how to cut all friends
and relations who are not deemed worthy of
being of a certain _coterie_;--the next, is to dress
after a particular fashion, talk a particular
species of language, not know any thing or
any person that does not carry the mark of the
coterie, and speak in a peculiar tone of voice.
To hold any conversation which deserves that
name is called being prosy;--to understand
any thing beyond the costume of life, pedantic.
Whatever vice or demoralization may exist
in character, providing it exist with what they
call good taste (that idol of their idolatry), is
varnished over. If not approved openly, it is
tacitly assented to, and allowed to pass as a
venial error; whereas whatever takes place
contrary to this _good taste_, though in itself
perfectly innocent, tending it may be to virtue
rather than vice, is insufferable--not to be
named _among them_; and unfits the offending
parties from communication with the Exclusives.
Indignation expressed at crime is voted
vulgar; any natural expression of the feelings,
ill-breeding; and right and wrong, in
short, consists in being, or not being, _one of
the set_. To their choice meetings children
dare not invite parents, or brothers and
sisters of one another, except under their seal
and sign-manual. The husbands and wives,
who are members of the association, are
invariably persons who have separate interests,
separate views, and agree only in this
one point, namely, in being a cloak for each
other's follies or vices.
It is to be hoped, and indeed may be asserted
with truth, that many are ensnared to
tread this Circean circle who are in ignorance
of what it leads to; who see in it only
a brilliant phantasm of pleasure and of pride;
an _ignis fatuus_ that pleases their fancy; but
which terminates too frequently in leading
them on, till some entanglement of fortune,
or virtue, levels them with its worse members;
and from which it is a mercy indeed if
they ever escape.
An open defiance of received laws and customs,
a coarse career of vicious pleasure, a
bold avowal of any illegitimate pursuit, would
startle and astound many a wavering mind;
but the slow-sapping mischief of this love of
exclusiveness, the airy indifference with which
all the safeguards of conduct are broken
down, the cruel heartlessness which lies concealed
under apparently indifferent actions,
the artful weaning of the mind from all fixed
principle of conduct, these are the means
they use; and which, step by step, adulterate
the character, indurate the heart, pollute the
judgment, and are subversive of every thing
that is dignified or amiable in human nature.
It is precisely because the evil works so insidiously,
and under such a variety of masks
(under none more than a placid _insouciance_),
a fortuitous occurrence of accidents--that the
veil should be drawn aside, and that it should
be set forth in its native deformity and danger.
CHAPTER V.
A RURAL EXCURSION.
A brilliant water party had been arranged
among the exclusives, to go to Richmond,
merely to view the scene; it consisted of the
Glenmores, Baskervilles, Lady Tenderden,
Comtesse Leinsengen, Lady Tilney, Lord Boileau,
Sir William Temple, Lord De Chere,
Mr. Winyard, Mr. Spencer Newcomb, Comte
Leinsengen, and a few other young men of
their set.
When the day arrived, Lord Glenmore told
his wife that as he was on a committee of the
House, he should not be able to accompany
her.
"Then I would far rather not go myself."
"Do not be so childish," he said; "for
as we could not, at all events, be together,
you might just as well be at Richmond
as here; and the day is beautiful, so that I
hope you will have a pleasant excursion."
Lady Glenmore sighed, and hung her head,
while a tear came into her eye.
"What is the matter, love?--Has any
thing vexed you?--is it any thing which I
can remedy?--You know you have only to
speak, and your wishes are my laws." He
pressed her fondly to his breast as he said
this, and she replied:
"Nothing; nothing vexes me, except that
we are hardly ever together, as it seems to
me--or never, but when in public; and I long
for the time when we shall be in the country,
and that all our occupations will be mutual;
when you are not with me, I find more pleasure
in music, or in reading, than in going to
parties: for nobody cares for me; and I am
sure I return the compliment."
"Nay, my sweet Georgina, this is really
nonsense. Are you not courted and paid attention
to by every one in the most marked
manner?"
"Do not mistake me," she replied; "I
have not explained what I mean. As to outward
attentions of politeness, oh! yes, I receive
them in abundance; but what I intended
to make you understand is, that the
things I take interest in, and the pleasures I
have in view, seem so entirely different from
those of the generality of the set I live in,
that there is nothing left for me to say; and
I often observe that when I do speak, my conversation
is either laughed at, or they stare at
me as if they did not believe I was serious."
Lord Glenmore smiled, and loved his innocent
little wife a thousand times the more for
her unsophisticated sweetness; nevertheless,
as he was likely always to have a part to play
in the great world, he could not help wishing
that his wife should be able, without putting
any force upon her inclinations, to do so likewise.
He therefore said, and speaking rather
more seriously than he had done: "Retain
always, dearest Georgina, this youth and
purity of character; but, for my sake, learn,
my love, to endure an intercourse with others
who may be of a less pure nature than yourself;
but who are yet, from your situation
and circumstances, likely to be those with
whom you must naturally associate: to please
me, then, my dearest Georgina, begin from
to-day: put on all your smiles, and let me
hear that you are the envy of the women, and
the admiration of the men. Remember, love,
to _please me_."
"Any thing to please you," she replied;
and she decorated herself with more than
usual care. Just as her toilette was about to be
completed, Lord Glenmore entered her room
with a quantity of lilies of the valley. "Here,"
he said, "I have brought you your favourite
flowers; wear them, love, and let their fragrance
remind you of the donor." All this
lover-like attention enchanted the person to
whom it was addressed, and her eyes sparkled
with unwonted brilliancy, and her cheeks were
tinged with the glow of pleasure as she fastened
her _bouquet_ in her breast. Lord Glenmore,
proud of such a wife, as well he might
be, handed her into her carriage, and she
drove to Lady Tilney's, where the party were
to assemble to go to Whitehall stairs.
When she entered the room she found nobody
yet arrived; a servant made Lady Tilney's
apology, saying she should be dressed
shortly. Having played a few airs on the
piano-forte, she took up a novel, and was
busily employed in its pages when Mr. Leslie
Winyard was announced. Lady Glenmore felt
embarrassed in his presence, she knew not
why, but there was something of fear and
flutter that came over her whenever he approached,
which she could not command.
She arose and curtseyed; and then, as though
she had payed him too marked a distinction,
she remained awkwardly standing, as though
she had taken that position by accident--not
in honour of him.
All this was not unobserved by Mr. Winyard.
He was too well practised in the ways
of women's hearts not to read her's at a
glance. At least he occasioned emotion, no
matter what emotion. He was not to be seen
with indifference--that was enough for him;
and he despaired not of turning it to his own
advantage. This advantage, however, was
not, in the present instance, to be obtained by
a _coup de main_; and assuming an air of polite,
but frigid _nonchalance_, he accosted Lady Glenmore
with an expression of surprise at finding
her the first-arrived person; and then examined
one of the miniatures which hung in a
glass cabinet. Lady Glenmore soon recovered
her composure, and entered into conversation
by asking some of those questions which are
merely the opening of conversation. "Yes,
I like music," said Mr. Winyard, in answer
to one of her questions; "it is one of the very
few things which is worth giving one's-self any
trouble about. I once learned to sing; the
only thing I ever learned." Lady Glenmore
laughed; and as her own ingenuous manner
returned, she evinced that propensity to being
amused by the present moment, which is so
natural and so pleasing in youth.
"Will you do me the honour to sing a
duet with me?"
"Oh! certainly," she said; and turning
over some music which lay scattered on the
instrument, she added, "Oh! here is that
delightful little duet, '_Sempre piu_' which,
though not new, is always charming." Mr.
Leslie Winyard had a sort of shuddering at
the idea that, notwithstanding her general
elegance, she might excruciate his ears by an
open English pronunciation, and a drawl by
way of sentiment; but he had embarked in
the danger, and fortunately there was no one
in the way to hear if his own talent should be
marred. He therefore courageously opened
the music leaf; and Lady Glenmore, having
touched a few chords, gave an assurance that
better things were in store. Nor did she disappoint
the promise; her sweet, rich-toned
voice had been tutored by Italian taste, and
swelled or sunk to every intonation, with a
delicacy of feeling which could not be surpassed;
the _sempre piu t'amo_ was uttered in
the purest enunciation of the language; and
Mr. Leslie Winyard thought, if it were only
addressed to him, it would be a triumph, which
the world he had lived in had not yet afforded.
Lady Tilney entered the room while they were
yet singing.
"I am glad to find you have not been
tired," she said, "waiting for me. I beg you
a thousand pardons, Lady Glenmore; but
really I had so many things to do to-day--notes,
those terrible time destroyers; and then
the last number of the Edinburgh Review, together
with Mr. Kirchoffer's last work, have
so entirely occupied me, I totally forgot how
the hours flew past, till Argenbeau told me
that you were arrived. However, I hope you
find the instrument in good order. Mr. Winyard
sings like an angel; and I make no doubt,"
(looking at him, to ask how far she was right
in the assertion) "Lady Glenmore does so
likewise."
Mr. Winyard said, "I assure you, Lady
Tilney, _que voilà ce que l'on appelle chanter_,"
indicating Lady Glenmore with a movement
of his head, "I had no idea any thing not
of the Land of Song could sing in that manner."
"Well, really, you astonish me; why Lady
Glenmore keeps all her perfections to herself!
But she must really be drawn out, and not
suffered to hide her talents in obscurity."
At this moment Lady Tenderden and the
Baskervilles entered, and shortly after the
remainder of the company. "Well, it is time
we should be gone, if we mean to see Richmond,"
observed Mr. Spencer Newcomb,
"though I believe _eating_ Richmond is fully
as interesting, and candle-light at any time
is better worth seeing than the sun-light;
are you not of my opinion, Lady Glenmore?"
He addressed himself in preference to her,
because he thought she was new enough to
be astonished, and astonishment was an homage
paid to his power which he well knew
he could not extract from any of the rest of
the company.
"Both are good," replied Lady Glenmore,
"in their proper season."
"A philosophical answer!" cried Sir William;
"you did not expect that, did you,
Newcomb?"
"No, it is too wise for me," he said, "for
it leaves me nothing to say--it is a truism;
_messieurs et mesdames, je vous avertie_, that as
I do not like the evening fogs of the river I
cannot postpone my departure. Lord Baskerville,
Mr. Winyard, will you come with me?
I have a _voiture a quatre places_, and any
lady may come that likes." Mr. Leslie
Winyard bowed and whispered Lady Glenmore,
"would she go?" Lady Tenderden
whispered her on the other side, "by all
means go, my dear Lady Glenmore, and I will
arrange my party in your carriage."
Lady Tenderden's advice was not to be
slighted, and Lady Glenmore accordingly accepted
Mr. Leslie Winyard's offered arm, and
followed Comtesse Leinsengen, who treating
her as nobody, as she was generally wont to
do every one whom she dared, she entered
her carriage and drove off. At Whitehall-stairs
they found their boat waiting, the best
barge, the most knowing bargemen, and all
things in exquisite order--they take their
places, and, a band of music following, glide
down the stream, and are, or appear to be, in
the most harmonious of humours.
"What is become of Glenmore to-day?"
asked Lord Gascoigne.
"I am sorry to say he was obliged to be
on a committee, and I feel so lonely without
him, half my pleasure is gone," replied Lady
Glenmore. The men looked at one another--the
ladies tittered; there was a pause, and
the speaker felt sadly embarrassed, she knew
not why. Lady Tenderden whispered to her
as they leaned over the boat-side:
"That was a very injudicious speech of
your's, my dear; you must learn not to _affiché_
these tendernesses; for if you really feel them
nobody cares, and people in general only imagine
you affect them by way of being singular."
Poor Lady Glenmore made no answer;
but was again convinced that she should
never like a society in which she was to be
so perfectly unnatural. Mr. Leslie Winyard,
who saw at a single glance the truth and
freshness of Lady Glenmore's character, was
certain that it would not do to attempt to gain
her good graces by any common-place mode
of attack, such as flattery of the person, or
intoxicating representations of power, dissipation,
and pleasure. He therefore took
an opportunity, when the rest of the party
were engaged in their own conversation, to
approach Lady Glenmore, and having found
a seat next to her, he commenced a discourse
which he conceived would be more to her
taste. Music afforded him an opening; it
was a subject on which he spoke elegantly
and well, and she listened with pleased attention.
"After all," he observed, "where science
and taste have done their utmost to produce
perfection, and without these guides certainly
nothing will do; even after they have lent
their assistance, there is a third ingredient
which is _given_ only, and cannot be _acquired_,
without which there will ever remain a flatness,
an _ineffectiveness_, if I may so speak,
which renders the whole vapid and inefficient--I
mean feeling; and there, indeed,
you must know, Lady Glenmore, that you
are not wanting." He fixed his eyes on her
with an expression which made her blush;
but she replied smiling:
"How can _you_ know that, Mr. Winyard?"
"Did I not hear you a short time ago sing
'_Sempre piu t'amo_'?"
"Oh," she replied, "you judge by that?"
"And can I appeal to a more convincing
proof of what I assert? But if I needed any
other proof, surely the words, and the look
which accompanied the words, when you expressed
your regret at Lord Glenmore not
being of the party to-day, would be an
undoubted corroboration of the fact."
"Oh, that was natural," she said; "it
would have been odd could I have done
otherwise. But real feeling is a much deeper
seated quality than can be judged of by singing
a song, or a passing impulse, and I do
not own that you can know any thing about
me or my feelings."
"Perhaps not," replied Mr. Leslie Winyard,
looking grave and humble; "may it be
my good fortune to know more of these, and
to have the honour and advantage of improving
my acquaintance with you."--Here a
louder laugh than was usual among the fastidious
in manners, interrupted this _tête-à-tête_;
"will you not allow us to benefit by the wit?"
asked Mr. Winyard.
"Oh," said Lady Tenderden, "it is only
that Sir William Temple fell asleep, and asked,
when he was awoke, for some more maids
of honour."--"To be sure," he said, "what
does one go to Richmond for, but to eat those
exquisite compositions. If all maids of honour
were like them, I am sure their race would
be more in vogue than it is. I would give a
hundred or two to have the receipt, for notwithstanding
that I have brought my cook
disguised _en valet de chambre_ a thousand
times, he never could find out the secret;
neither has he been able, with all his art, to
produce any precise _fac-simile_."
"Ah!" exclaimed Lord Gascoigne, "that
is the true spirit of philanthropy; a hundred
or two for a receipt to make cheesecakes!
while we have such men in the state we need
not be under any apprehension that the arts
and sciences will fail."
"Yes, arts and sciences, my Lord Gascoigne;
for I affirm that the pleasures of the
table require one to be an adept, both in
order to procure and preserve them in perfection.
Who will deny that the cultivation
and use of the animals, and vegetables, and
elements, that are employed, do not include
all these, not to speak of the _main d'œuvre_."
"I am not disputing the fact," said Lord
Gascoigne; "why did you address yourself to
me? On the contrary, I am so well convinced
of it, that I pay my cook a hundred a year:
but the rascal threatens to leave me if I do
not raise his wages."
"I cannot be surprised at that," said Lord
Baskerville, "for I give mine two, and he is
only a second-rate performer."
"It is vastly extravagant," cried Lady
Tilney; "however, one need not do it if one
does not chuse; and, after all, it is not too
much to pay a man to become a salamander."
"Oh," cried the Comtesse Leinsengen,
"_ils son fait au feu ces gens-là_, they are good
for nothing else, and if you were not to yield
to them, you would have them for half de
money; but you are all _des dupes_ in England.
You think the more you pay, de grander you
are, that is the truth."
"Well, my dear Comtesse," rejoined Lord
Baskerville, "that is all very well to say, but
I am certain that you never would get any
body to serve you if you did not pay him
well; and I must declare that I had rather
give a hundred or two more to my cook, than
to any other servant in my house; for one's
whole domestic comfort depends upon one's
cook, don't you think so, Temple?"
"I was always of opinion that you were a
wise man, and I am now confirmed in that
opinion. Most indubitably one's cook is the
great nucleus upon which one's whole existence,
mental and physical, depends; for if
you eat of a bad greasy ragoût, the _physique_
immediately suffers, and then bilious hypochondria
ensues, and one's friends are the
victims of one's indigestion; and all the economy
of life, in short, goes wrong, if there is a
failure in that department."
"Nobody has ever denied," observed Mr.
Spencer Newcomb, "_que le bonheur est dans
l'estomac_, and that happiness depends very
much on what one eats--and what one
eats depends upon the cook. I hold it to be
an incontrovertible maxim, _que le bonheur des
bonheurs_ is to have a _cordon bleu_ at one's
command--even the ladies will agree with
me."
"Certainly," said Lady Baskerville, "I account
it to be one of the requisites of life."
"Yes," rejoined Mr. Winyard; "for a
lady ought to appreciate the beauty of every
thing, even of a _poulet santé aux truffes_; and
though I cannot endure a woman to have
what is vulgarly called a good appetite--a
sort of beef and cabbage voraciousness--I like
her to know the various flavours and high-wrought
refinements of the palate. Indeed,
I am sure she is always vulgar if she does
not. But here, we are nearly at the landing-place;
and now let us hope to put our theories
in practice, and find in this _rural_ retreat
a change of viands to recreate and stimulate
our somewhat palsied palates."
As the ladies were gathering up their
shawls and reticules, Lady Glenmore stooped
down to arrange a part of her dress, and the
lilies of the valley her husband had given her
fell into the water. She made an exclamation,
and attempted to catch them, but a
breeze bore them beyond her reach. "Oh
my nosegay! I would not lose it for the
world," she cried.
Mr. Leslie Winyard looking in her face, and
seeing that she was eager in her wish to recover
the flowers, hastily darted from another
part of the boat; and in making an effort to
catch them, lost his balance, and fell into the
water. As they were literally on the shore,
there was no sort of danger, besides that
of getting a ducking; but he thought it
might avail him something in Lady Glenmore's
favour: nor was he mistaken. Seeing
him floundering in the water, she cried out,
"for God's sake save his life!" and while
he made the most of the awkwardness of his
situation, he kept brandishing the lilies with
one hand, and would not suffer any body to
touch them till he delivered them safely to
her. She was exceedingly touched by this
effort to oblige her, and for the rest of the
evening, after he had made a fresh toilette,
he reaped the rewards of his gallantry, by
finding that Lady Glenmore listened to him
with a kind of favourable impression, that
he could scarcely have hoped to inspire her
with, had not fortune thus favoured him.
During dinner nothing was talked of but
the merits of a Richmond party:--"there is
surely nothing in the world more beautiful,"
said Mr. Newcomb, "than the view of Richmond
Hill; it is the only _riante_ landscape in
England; a perfect Claude; and for my part,
I never desire to go farther in quest of the
picturesque--it is quite a _gentle_ scene; no
horrors, no rugged rocks or torrents; but a
sweet, soft, sylvan composition."
"Enlivened too," observed Sir William
Temple, "by stage-coaches, and mail-coaches,
and coaches of all sorts, in short; without
which I hold all views to be very wearisome
things _à la longue_."
"Only made for the eyes of the vulgar, depend
upon it," was Lord Baskerville's observation.
"Except during the hunting season,
the country is hateful; but one may bear a
row to Richmond, especially in such company,"--and
he bowed to Comtesse Leinsengen.
"The country is all very well," she rejoined,
"in a _grande chateau bien remplie de tout ce
qu'il y a de mieux en fait de société_; but it
makes me shudder to think of being in one of
your provinces, in a house in the middle of a
shut-up park, with a neighbour or two _pour
tout bien_; no no, I am perished with _ennui_
but to think of it."
"It makes me shudder too," said Lady
Baskerville, smiling at the Comtesse Leinsengen's
broken English; "but, in fact, it
is what nobody does now-a-days; either the
real or the pretended incapacity on the score
of fortune for living at the country-seats, as
they used to be called, gets rid of all that
sort of thing. People live very much now
as they used to do in France, I am told, when
Paris was the only place in that country which
any body lived in."
"Yes," said Mr. Spencer Newcomb, "and
as long as the people don't find out that their
landlords forsake them, and rack them for
their money, which they spend any where
rather than in doing them any good, it is very
agreeable not to be bored with that sort of
useful virtuous life. Long may they continue
to administer to our pleasures--they ought
certainly to be made for nothing else; but,
unfortunately, there came a time in France
when these things were all changed, and the
vulgars took it into their heads that they were
to have their day; and off went heads, and
on went caps of liberty, and all things were
turned upside down, as every body knows.
I wonder now how Lord Baskerville would
like to turn groom, and rub down his own
horses!"
"Ha! ha! ha!" was echoed around.
"So long as you keep a good whip hand,
and de rein in both, you will not be in any danger,"
cried Comtesse Leinsengen; "you have
only to keep down _de canaille_. What sinifie
all these schools of learning? dey are the most
terrible nonsense; good for nothing but to
turn the people's heads, and make them think
themselves wiser than their masters; we do
not do so in my country. When they learn to
sing, they only learn _one note_, so that no
single person is independent of anoder, and
yet they make excellent concerts; these sort
of people should be always kept dat way, so
you see dat keeps all quiet, and the country
goes on from one age to another all de same."
"Capital," said Winyard, "that is worth
putting in print."
"Oh, I am quite of another opinion,"
cried Lady Tilney; "you must pardon me;
but I think that every thing which has not
freedom for its basis, must be wrong; let
every body have a fair chance of becoming
something; above all, let the light of learning
shine every where, in every thing; there will
always be ways and means of keeping people
in their several stations. A country may
have all the blessings of liberty, and yet a
certain set may exist who shall have a superiority
of its own, move in a sphere of its
own, and be kept quite apart from the vulgar
crowd; there is always a way of managing
these things. I uphold liberty and literature;
but that is not to say, that your authors
and your musicians are to mix with certain
societies--quite the contrary. The liberty
of the latter will always keep its ground against
the intrusion of the former, don't you think so,
Sir William?"
"I think, Lady Tilney, that whatever you
say must be right; and when you command,
I feel always inclined to reply, as some body,
I forget who, did to the Queen of France, _si
c'est possible c'est déjà fait, si c'est impossible
ça ce fera_."
"I have always thought," rejoined Mr.
Spencer Newcomb, "that that speech ought
to be the truest that ever was uttered, for
it is exactly the sort of thing a lady would
like to have said, and I am sure it is the
most ingenious that ever was contrived." A
walk was now proposed, previous to which
the ladies withdrew to the drawing-room.
"Well," said Lady Tenderden, "I think
we have had a charming day, do you not Lady
Glenmore?"
"Very much so," she replied, "and if
only----."
"I will finish the phrase for you--if only
Lord Glenmore was here--now my dear, I
thought I had warned you not to indulge in
that infantine habit of saying always what
you think. You cannot conceive what strange
ideas men attach to these sort of declarations;
they are apt to suppose it is a hint to them to
make love to you."
"Impossible!" said Lady Glenmore, colouring.
"Oh, you do not yet know the world, my
dear Lady Glenmore. Be advised at first, and
then afterwards act for yourself."
"I must beg of you, ladies," interrupted
Comtesse Leinsengen, coming up to them, "to
patronize a little _modiste_ who is newly
established, and whom I take under my special
protection. She has all her patterns from
Paris--dey are of the _premier goût_, and have
that particular mark of distinction about
them, which dose who are copied from the
_feuilles des modes_ never so attain. Mademoiselle
Dumesnil has promised me never to
sell certain things but to certain people;
so that one is quite sure of not seeing _le double_
of one's own dress on Mrs. Hoffer, or Lady
Delafont, which is quite sufficient to make
one fall into a syncope, and put one in bad
humour for de whole season."
The Ladies smiled, agreed with her, and
promised compliance with her wishes. "Mademoiselle
Dumesnil's story," continued Comtesse
Leinsengen, "_feroit un roman_; it is
quite touching, and" (she added in a whisper,
as the gentlemen entered the room), "its
hero, _le voilà_," pointing to Mr. Leslie Winyard;
then in a low voice she proceeded to
give the whole particulars to the two Ladies,
Glenmore and Tenderden, who sat next to
her.
The gentlemen now expressed their wish
to know whether the ladies would not profit
by the beauty of the evening to walk out,
and the measure being agreed upon, the party
was so arranged that Lady Glenmore fell to
the lot of Mr. Leslie Winyard, and much as
she now felt averse to accept his arm, after
the particulars she had just heard from Comtesse
Leinsengen, it was impossible for her
to refuse without incurring, as she thought,
Lady Tenderden's animadversions. Lady
Glenmore's silence, however, as they walked
along, attracted her companion's particular
notice. Something, he conceived, must have
occurred, to change her manner so completely
since dinner; but Mr. Leslie Winyard was too
well versed in intrigue to augur from this
circumstance any thing unfavourable to his
wishes, because he knew that to have made
an impression _quelconque_, was the first step
towards attaining his end.
Determined, nevertheless, to ascertain the
reason of this alteration in Lady Glenmore's
manner, he very cautiously, but very adroitly,
contrived to find out that something had been
said which she conceived was to his disadvantage;
and he could be at no loss to guess of
what nature it was, for the affair in which
his name had been mixed up, in Comtesse
Leinsengen's conversation, was of too recent
a date, and too _marquante_, to have escaped the
memories even of that thoughtless circle--it
was, in short, his last.
With this just apprehension of the fact,
therefore, he turned the conversation upon
the subject of scandal, which he deprecated
bitterly; and, as if instancing the effects of it
in regard to a person intimately known to
himself, gave a totally different, but very plausible,
interpretation of the exact story, which
Lady Glenmore had heard detailed half an
hour before by Comtesse Leinsengen.
Lady Glenmore had listened to this artful
language with considerable interest and
surprise. From the generosity of her nature,
she felt much pleasure in thinking that
the evil she had heard, and which made her
uneasy even to be in Mr. Leslie Winyard's
society, was totally without foundation. Her
manner, therefore, gradually relaxed in rigour
towards him; she seemed to have suddenly
recovered her spirits, and her conversation
flowed naturally without any constraint.
The moment the party returned from their
walk she flew up to Lady Tenderden, and
referring to the previous conversation of Comtesse
Leinsengen, repeated that which she
had just heard from Mr. Leslie Winyard, and
which she conceived to be his interpretation
of his own story; commenting, as she related
it, on the injurious effects of speaking evil of
any person without a thorough knowledge of
the fact. Lady Tenderden foresaw, that were
all this carried back to Lord Glenmore, many
impediments would arise in fitting Lady Glenmore
for their exclusive circle, and bringing
her down to a moral level with themselves;
she therefore said, after a minute's pause, "I
make no doubt the Comtesse Leinsengen has
been exceedingly misinformed; but at the
same time the less that is said of these matters
is always best, on every account; and as Mr.
Leslie Winyard is my very particular friend, I
shall esteem it a favour, my dear Lady Glenmore,
that you do not mention this idle
story to Lord Glenmore, who might conceive
some prejudice against him, which
would make me very unhappy. It is, in
fact, of no consequence whatever; but when
things of that nature pass through various
mouths, they accumulate a consequence in
their passage which they have not in themselves;
and therefore promise me, dear
Lady Glenmore, that you will not mention
this matter to any one; besides," she added,
looking very mysterious, "you know Lord
Glenmore's great interests may be much
affected by the Leinsengens; and the knowledge
of her having retailed that sort of
story, and retailed it under a mistaken
point of view, might produce some coolness
between them; for you know Lord Glenmore
is vastly fond of Mr. Leslie Winyard."
Lady Glenmore did _not_ know this, and
hardly comprehended any part of the speech;
in truth, how should she? But she remembered
her husband's having recommended her to
take Lady Tenderden's advice, and therefore
she determined so to do in the present instance.
Shortly after this conversation, it was put to
the vote whether the party should return to
town by land or by water; and with the
exception of Princess Leinsengen and Lord
Baskerville, who preferred a close carriage
for fear of damp, the rest agreed to go as
they had come. It was soon quite night;
but a brilliant moon made the water look very
beautiful; and the soft language of Mr. Winyard,
as he sat by the side of Lady Glenmore
in the boat, fashioned in its phrase to the taste
of his hearer, appeared to her in unison with
the scene, and she thought him the only one
of the party who was at all amusing, or had
given a colouring of any interest to the hours
she had passed with them.
Arrived at Whitehall, Lady Tenderden proposed
their adjourning to her house, where
supper was prepared; but Lady Glenmore,
uneasy at a longer absence from home and
her husband's society, determined for once to
be firm in her refusal; and stepping into
her carriage, which awaited her, drove at
once home. On her arrival there, however,
she was doomed to sustain an unexpected
disappointment, as she found a note from
Lord Glenmore, dated from the House; in
which he told her not to be uneasy if he were
late, for that the business of the morning was
likely to be followed by a protracted debate on
an important question. Lady Glenmore sighed
over this note as she perused it; and, tired
with the day's excursion, yet not sufficiently
composed for rest, she experienced that listlessness
of mind, which admits not of any
active exertion, and yet affords no satisfactory
contemplation whereon to dwell.
Lord Glenmore's attention happened to be
at this moment directed to a high post under
government, which it was more than probable
he would attain. But could he have dreamt
that in this pursuit he was neglecting the
duties of private life, and casting forth an
inexperienced young person, unprotected, amid
all the dangers of a pleasure-loving world,
he would have left all else to guide her
through the perils to which he now so frequently
left her exposed. How often does it
happen, in various instances, that in the blindness
of human wishes, we hurry to the goal of
our desires--even those which we deem innocent
and praiseworthy; but which, when suffered
to lead us on, without a reference to a
higher power, never fail to _mislead_, and prove
fallacious when obtained. Yes, this is that
self-pride of reason, which, confiding too much
in its own merits, and not acting under the
reliance of a superintending Providence, even
when on the point of realizing its fondest
hopes, finds it has grasped at a shadow; and
to an ideal good, sacrificed a permanent happiness.
Had Lord Glenmore paused to reflect, and
had recourse to that unerring light, which
never dazzles to betray--his steps would
have been guided by unfailing wisdom, and
he would have found his chief happiness in
his chief duty; whereas he pursued the phantom
ambition; he did not consider that the
necessary consequence which must follow an
official occupation, was his leaving his young
wife without a natural protector, amid scenes
that were any thing but safe; and he was
desirous that she, too, should play her part,
and by those graces and influences which
have such sway over the destinies of men and
of empires, take an interest and acquire a
power in that vaulting game of ambition in
which he himself delighted to engage. He
considered not how often he must leave her
through the day, and the greater part of the
night, to run this hazardous career, at an
age when caution sleeps and passions are
awake, and in the midst of a set which, though
certainly not wholly devoid of some unblemished
characters, was yet, generally speaking,
in its whole tendency perilous to the
pure and domestic virtues--a woman's only
true glory.
Yet on this precipice was Lady Glenmore
placed, without one real friend to whom
she could look for genuine advice or succour.
Her mother's (Lady Melcomb) absence from
town prevented that natural tie, and had she
been there it would have proved the business of
the exclusives to have prevented that free and
happy intercourse, both on the principle of
not allowing any aged person to mar the
brilliancy of their set, as well as that of excluding
all those who might see through the
drift of the society. On Lady Melcomb's part
it was too early in the day to have any suspicion
of the work of mischief which was carrying
on to separate her from her daughter,
and thus was Lady Glenmore like a lovely
lamb amidst ravening wolves.
Scarcely had she been received amongst
them, when Mr. Leslie Winyard, being at the
moment _desœuvré_, conceived that she was just
put in his way as a fit play-thing for the hour,
and without the least scruple he determined
she should swell the list of his conquests, already
as numerous as those of Don Giovanni
in all lands. He took no pains to conceal this
design from any one save herself, and his intentions
served many of the set as a topic of
conversation, a fit subject for betting on:
"how would Glenmore take the thing;
would he be a wise man or a fool--put on the
cap which fitted him with a good grace, or
make grimaces at it?" Such is the license
with which the most serious delinquencies
were talked over, and though when set down
on paper they may seem exaggerated, yet
certainly the fact is not in the least so; only
people start at things and actions when called
by their right names, which under the title
of venial errors, youthful indiscretions, and the
sanction of custom and habit, are certainly
tolerated, if not commended; _tacitly_ approved,
if not openly avowed. Ought not such a
desperate system to be analyzed? Ought not
language to pourtray in its strongest terms
those deeds and those manners which, under
the semblance of polite terms, and fictitious
representation, and deceptive elegancies,
pass current as being harmless or indifferent.
Let those whose hearts have bled on the
shrine of fashion and of _ton_--who have mourned
the loss of all that was valuable in character,
or beautiful in mental existence, sacrificed
to the insatiable appetite of pleasure,
the degrading occupations of frivolous pursuit,--let
_them_ say if colours can be too deep,
or language too strong, to paint so destructive
an evil as that of the whole false, futile
system of the exclusiveness of _ton_.
Lady Glenmore was evidently one of those
persons marked out to become its victim, and
when the character of Mr. Leslie Winyard is
taken into account, as being the man who
attempted above all others to lead her to her
ruin, it cannot be wondered at, circumstanced
as she was, that the pit of degradation yawned
at her feet. Mr. Winyard was one of those
who to the gentlest manners united the hardest
of hearts: he had not, perhaps, always
merited such a description; but the being who
lives entirely for pleasure, becomes gradually
hardened to every natural sentiment, and selfishness
is the invariable consequence of a life
of idle dissipation. From selfishness springs
every other evil, and as it is the meanest of all
principles of action, when considered in the
baldness of the term, so it is, perhaps, the
most common, and the one which above all
others no person will like to avow--no, not
even Mr. Leslie Winyard.
Yet he was a man who, after having by
every sort of riot and debauchery ruined
himself, proceeded to ruin his own mother
and sister, bringing the grey hairs of the
one to the grave with sorrow, and leaving the
other to work out her existence in a situation
unfitting her rank, but far more honourable
and desirable than the one he filled; yet
this was a man, the beauty of whose personal
appearance, the refinement of whose manners,
the powers of whose understanding
and charm of fascination, were calculated
to destroy every innocent mind; and it was
difficult to arm against such a powerful
enemy--a very Proteus in the power of
becoming all things at pleasure, and suiting
himself precisely to the taste and habits of
the victim whom he was insidiously endeavouring
to undermine.
What could protect an unsuspecting,
youthful mind against such an enemy?
Nothing but religion; nothing but that habitual
looking for wisdom, where alone it
may be found; and perhaps, Lady Glenmore
was in this only security fatally defective;
she was good and pure, in as much as
human nature can be said to be so. And
how totally valueless this goodness is, without
it rests on a firmer basis, may be
seen in her, as in every other person to
whom the same vital want attaches: for her
character was not built on that rock which
when the floods come, and the storm beats,
will remain unmoved by them: she had yet
the greatest of all lessons to learn, not to
depend on _self_.
CHAPTER VI.
RETROSPECTION.
When Lady Hamlet Vernon drew Lord
Albert D'Esterre aside, at Lady Tilney's supper
party, it was, he conceived, with an intention
of explaining to him the words contained
in her note at Restormel alluding to Lady
Adeline Seymour--and he was confirmed in
this idea by the violent agitation which her
manner betrayed, although she strove to retain
that composure which the circumstances of
the time and place particularly demanded.
For several minutes after they had sat down,
she seemed labouring for breath; and Lord
Albert, notwithstanding his own anxiety
and impatience felt exceedingly for her distress.
"My dear Lady Hamlet Vernon," he said,
"I beseech you be not thus agitated; remember,
whatever you have to say, however
painful it may be to me to hear, I am certain
that it must be from friendly motives
alone that you make such communication,
and I must always feel grateful to you for
your intention; but keep me no longer in
suspense I entreat, for I am prepared for
whatever you may have to tell me."
"I have nothing to tell you, Lord Albert."
"What do you mean? what, can you possibly
intend to disappoint me; and, having so
cruelly excited my feelings, cast them back
upon me to prey upon themselves? No, I
never can believe you so inconsequent; so
very--"
"Stay, Lord Albert, and before you condemn,
hear me.--It is true I was on the
point of betraying a trust--of revealing a
secret--of becoming _really dishonourable_--for
what? for the sole purpose of befriending
you--for the sole purpose of snatching _you_
from a danger which it was then time to
prevent your falling into; but since that moment
is past for ever--since it is now in vain
that I should prove useful to you by being
false to another, my lips must for ever be
sealed."
"Strange and unaccountable mystery!
What, you will not tell me--you will not endeavour
to warn me against a danger which
hangs over me--is this friendship? How _can
you_ know that the time is past for pointing
out to me such danger? How can _you_ be so
thoroughly acquainted with the events of my
life--the secrets of my heart, as first to imagine
my fate _was_ in your hands, and then
suddenly be equally well assured it is so no
longer? No, I cannot conceive there is any
friendship in such conduct."
"Ah," said Lady Hamlet Vernon, sighing,
"I see you are like all your sex;
you receive the devotion of a heart as a
thing of course; you take into no consideration
the pain, the remorse I felt, at the
idea of becoming false to a trust for your
sake, when I thought that by so doing I
might save you from misfortune. And now
that I tell you the time is gone by when I
might possibly have been of use, even by the
sacrifice of my own integrity, you still wish
for that sacrifice, although it can avail you
nothing:--is this generous?" Lord Albert
felt confused; he was even moved by the look,
the air, the words of Lady Hamlet Vernon,
but still the disappointment wrung his heart,
and jealousy, with every other feeling, goaded
him on to press for a disclosure of the
secret.
"I am not ungrateful, indeed I am not; I
feel deeply the kind interest you take in me;
but if that interest does not sleep, or rather
if it is not extinguished, I still plead to be
made acquainted with a circumstance so very
nearly affecting my welfare; and when I say
that your disclosing it to me would be like
keeping it in another casket, surely, surely
you will not deny me."
"In this respect, my dear Lord Albert,
I alone can be the judge, and even at the
risk of losing your good opinion, or rather
of losing your friendship for the time being,
I must persist in remaining silent." There
was a long pause, which was at last interrupted
by Lady Hamlet Vernon resuming the conversation.
"Whatever may be your opinion of me, I
must, ere our intercourse altogether ceases,
touch upon one subject, which I believe to be
the prime object of your life, and that to which
all your views tend--I mean the noble career
which lies open to your ambition; may you
pursue it with unbounded success; but remember,
that you are not likely to do so if you have
any secondary interest to clog and drag you
back. If domestic troubles, at least domestic
cares, obtrude themselves upon your higher
aims, what a terrible hindrance to your plans
they must of necessity become. Think well, my
dear Lord Albert, of this--for _le roman de la
vie_ is soon over you know, but life itself goes
on to the end; and whatever women do, men
should look to that alone with a providing
care. We, who are creatures born to suffer
(at least all women who live as most women
do, the slaves of your sex), we indeed may
live upon that illusion, which destroys while
it delights; but it is not in your nature to do
so; public concerns--public applause--public
success--facts, not feelings, must fill up
the measure of a man's existence. Think,
then, what it is to have these great ends
marred, defeated, by some minor power that
corrodes and destroys in detail those thoughts,
those actions, which, if unshackled by petty
duties, would raise you to high consideration
and power; but if tied to a partner wholly
a stranger to your feelings and pursuits, she
must, however amiable in herself, ultimately
poison all your happiness."
Lord Albert had listened to Lady Hamlet
Vernon without a wish to interrupt her, and
with deep and fixed attention, painfully dwelt
upon every word she uttered; he could not
remain in ignorance of the drift of her words,
and they pierced him like swords, yet still he
remained silent.
"If," continued Lady Hamlet Vernon, "a
woman shares her husband's feelings, enters
into his views, goes along with him, not
merely from duty but from habit and inclination,
in all his interests, then indeed it is
possible such a woman might forward, and
not impede his prospects; but where habits,
principles, and prejudices, have all tended to
form a different character, and above all,
where bigotry has fastened chains on the
mind wholly destructive of any active or
useful pursuits, the probability is, that
wretchedness to both ensues." Lord Albert
no longer affected to misunderstand her, and
replied,
"Every thing you have said has been in
allusion to my approaching union with Lady
Adeline Seymour, an engagement you cannot
be ignorant of, as it has been well known to
the world in general for some years past.
Tell me, I adjure you tell me, to what principles,
to what habits do you allude? There
is enough in your words to startle and confound
me; but there lurks yet an unpronounced
sentence in your mind, which I now
implore you to declare. If, indeed, the least
regard for my happiness ever swayed your
breast, be explicit now, for my destiny perhaps
hangs on your open sincerity." Lord
Albert's thoughts were one chaos of uneasiness
and pain; jealousy had fired the train,
which set his whole being in a state of
anarchy, and he lost all command over himself--all
presence of mind, or capability of
sifting truth from falsehood. Poor human
reason, how weak is it even in the strongest
minds! when the passions are roused, who
dares to answer for himself, unless a higher
power assist him in his hour of need?
"Be composed, be calm," said Lady Hamlet
Vernon, "do nothing in haste; suffer me now
to drop this subject, and we may resume it at
a more favourable opportunity, when you
have considered fully the opinions I have now
expressed. All I wish you to remember is,
that when a man chooses a companion for
life, the chief thing to be considered is, not
her amiable qualities, but whether they are
of a kind which will assimilate with his. The
mere obedience which proceeds from duty,
will never satisfy a noble nature: no, it is the
devotion of a glowing heart which beats in
unison--a mind capable of sharing in the
plans and pursuits of an aspiring nature, unwarped
by prejudice, unobscured by fanaticism;
above all, a heart that is wholly and
undividedly its own."
Lord Albert, in listening to these words,
unconsciously compared the happiness of
being united to such a woman as the one he
now heard and beheld, to that of the pure
but infantine mind of Adeline Seymour. "Besides,"
he thought, "is she so pure? has no
preference for another, usurped the allegiance
which she owes wholly to me? Has George
Foley not become more necessary to her than
myself?" And while these imaginations, and
such as these passed rapidly to and fro in his
mind, his eyes were rivetted on Lady Hamlet
Vernon, whose exceeding beauty heightened
by the expression of an interest for himself
which he never before had seen so visibly betrayed,
made him say, in a tone and manner
not devoid of a similar feeling,
"Oh! Lady Hamlet Vernon, you who
can paint happiness so well--you who know
to distinguish, with such enchanting delicacy,
those shades of felicity which my warm imagination
has figured to be the charm of
married life, do not with a pertinacity unlike
yourself, withhold from me the secret on
which my fate depends, and either be my
guardian-angel or--"
"Hold, I beseech you in my turn; I have
already told you that I cannot fully impart all
I know--I may not, must not be explicit.
But this much I will reveal to you, providing
you swear to keep the secret, and never to
probe me further."
"Oh yes, I swear I will never betray so
generous a friend; I will never search further
into what you wish that I should not know."
"Well, then," Lady Hamlet Vernon replied,
after a pause, and trembling with excessive
emotion, "for the sake of the great, the deep
interest I feel for you, and have felt since
I first knew you, receive this pledge and
earnest of my friendship;" saying which, she
placed a ring in his hand, and added at the
same time in a low distinct voice, "you can
never be happy with Lady Adeline Seymour."
There are blows and shocks which strike at
the very vitality of existence--who has not
felt these before he has numbered many
years? and such was the power of these
words on Lord Albert, that he remained for
some minutes motionless; their sound vibrated
in his ear long after the sound itself
had ceased; for strange it is, though true, that
we can sometimes endure to think what we
scarcely can bear to hear uttered. In the one
case the thought seems not to be embodied in
reality; in the latter it has received existence,
and appears actually stamped with the seal of
certainty.
At length, however, he had summoned his
reason to his aid, and was about to speak further
to Lady Hamlet Vernon, when, interrupted
by the quick succeeding questions of many
of the company who were passing the room
in which they sat to go to supper, Lord
Albert offered his arm mechanically to Lady
Hamlet Vernon, and they followed in the
train of others. The noise and gaiety and
brilliancy of the scene could not for a moment
take Lord Albert out of himself; one idea,
one image engrossed him, and all the surrounding
persons and circumstances glanced
before his eye or came to his ear, with the
glitter and the buzz of undistinguishable lights
and sounds. He went through the forms of
the place and scene with the precision of an
automaton, and when the supper ended he
followed Lady Hamlet Vernon about like her
shadow, sometimes absorbed in the deepest
concentration of thought, sometimes endeavouring
to revert to their former conversation,
which had been so abruptly, and to
him so unopportunely broken off; eager to
renew its discussion, as well as to elicit a
disclosure (regardless of his solemn promise)
of that part of the subject on which she refused
all explanation.
In both, however, he wholly failed; and
having been obliged, although reluctantly,
to part from her for that time, he handed
Lady Hamlet Vernon to her carriage
and bent his way home. He felt it a
relief to be alone, in order to take a review
more collectedly of what was passing
in his own breast: but yet, when he commenced
the task, he found a contradiction of
thoughts and feelings which were so involved
that for a time he yielded to them,
and they alternately swayed him in opposite
directions, without his being able to come to
any decision.
On considering the length of time, and the
intimate footing on which Mr. Foley had lived
at Dunmelraise (notwithstanding the peculiar
circumstances in which he was placed, as the
son of Lady Dunmelraise's dearest friend, and
her own _protégé_), on recalling his descriptions
and praises of Lady Adeline when they
met at Restormel, he thought he saw a confirmation
of his worst fears. What, he asked
himself, could induce a young man to seek so
lonely and retired a situation but love? And
Lady Dunmelraise he thought must have approved
his views, or she would not have suffered
such an intimacy to subsist, even though
as her friend's child she received him under
her roof; at least it was evident that she
chose to give her daughter an opportunity of
turning her affections from that quarter to
which they had been originally directed. Adeline's
letters, too, so equable in their expression
of calm content, so lavish in Mr. Foley's
praise, so minute in her detail of his way of
thinking and manner of feeling, showed that
had she not been more than commonly interested
in him, she could not have thus busied
herself with analysing his character.
"It is clear," he said, "Adeline does not
love me; and her mother is no longer anxious
in consequence that our union should take
place!" While this idea prevailed he was desirous
immediately to break off the engagement;
formed a thousand plans for doing this,
in such a way as to appear disinterested and
honourable in their opinion; and worked himself
up to a belief, for the moment, that he
was only acting with that refinement and generosity
due to his own feelings as well as to
Lady Adeline's, by losing no time in putting
this resolve into execution, and then she
would be free. But for himself, would the
same step afford him the same advantage?
Would his heart be really free? were there
no strong ties that bound him to Adeline? no
habit of attachment formed in his breast,
though she had broken through the one, and apparently
could never have cherished the other?
Would he, in short, be free, though she were?
Could he turn the current of his affections
at once towards another object; could he accept
the heart, even were it her's to bestow,
of the person who had shewn such an interest
in his welfare; of one whose beauty was enhanced
by the deep expression which played
over her features--whose manners, talents,
character, were alike formed--could he make
her his wife? Again he paused at that title--it
had never been associated with any save
Adeline, and when coupled now with another,
it made him start from his own thoughts, as
though he were guilty in indulging them.
Struck at this idea, and with the conviction
of what would be the state of his own
mind were he indeed at once to let Lady
Adeline loose from her engagement, his feelings
and his reasonings took another course.
"Should I be justified," he asked himself,
"in the steps I am proposing, without further
proofs of Adeline's inconstancy? My surmises
perhaps have ground sufficient, but something
more than surmise is due to her. It is true,
I am told I shall never be happy with her,"
(and he shuddered as he repeated the words
to himself); "but I very much doubt if ever I
can be happy without her. My own conduct,
too, lately--what has it been? Has it not
carried with it proofs of coldness and neglect?
Why should I expect to receive that constant
and ardent devotion, which I have shewed no
anxiety to retain; and what, on my part, has
occasioned this passive indifference? Has it
not been a growing partiality for the society
of another--and was _this_ Adeline's fault?"
He dwelt on this idea for some moments, and
his self-reproaches were painful. Then again
he thought, allowing that all is as it was
between us, that she loves me in _her_ way,
and I her in mine, is that enough to constitute
lasting happiness? "_No, it is not._ I
should loathe the insipid homage of daily
duties pointedly fulfilled, and weary of a mind
which had not sufficient energy to think for
itself. If I saw that my wife did not enter,
from a similarity of tastes, into my occupations
and pursuits, I should feel no satisfaction
in her doing so to oblige me; and I certainly
have already observed, that Adeline's habits,
and even her principles, have led her to a life
of monotonous tranquillity and insipid cares."
And here again Lady Hamlet Vernon's words
recurred to him with tremendous power.
Would it not then, after all, be more noble to
set her free from an engagement, which
would fail in producing the happiness
that they both had been led to expect? He
mused with painful intensity as his thoughts
rested on this idea; but in the exercise of
analyzing, comparing, and combining these
various views of his situation, his mind was
imperceptibly drawn to the single subject
productive of them--his early attachment to
Adeline; and he fell into a comparatively calm
reverie--that species of calm, which dwelling
upon _one_ feeling generally produces, after the
mind has been tossed about in various contending
conflicts. His youthful and first affections,
together with all the awakening recollections
of early tenderness--the development of their
mutual passion, ere yet they knew they were
destined for each other--the happy prospect
of bliss which had succeeded--all, all recurred
to him, and revived the dying glow of attachment
in his breast. He took out her picture
from his writing-desk--gazed at the well-known
features, yet thought he had never
before been aware of their full and perfect
charm, that union of intelligence with purity
which is supposed to constitute the being
of an angel, that perfect candour, mingled
with quick perception, which this portrait
conveyed, and conveyed but feebly in comparison
with the original,--set the seal to his
conviction, that no one could prove to him
what Adeline had been.
In replacing the portrait, he lifted up some
loose papers, and it chanced that the lock of
Lady Hamlet Vernon's hair, which he had
kept (and never since looked at) on the night
when she had been overturned at his door,
dropped from the paper. He could not but
admire it; its glossy richness--its hue of gold
shining through the depth of its darkness:
it was certainly very beautiful, and he sighed
as he laid it down. "What if, indeed, her
words should be true, and how can they be
_true_ unless in one sense--in that of Adeline's
loving another? It must, it must be so!" and
this fatal conviction broke down once more
all the fabric of happiness which a moment
before he had erected: and in this revived
frenzy of feeling he passed the night. It was
broad daylight ere he could bring himself to
seek repose, nor did he then till worn-out
nature sunk in forgetfulness and sleep.
When he awoke the next day--for morning
was far advanced--it was like one awaking
from the delirium of fever. He felt exhausted,
spent, as though a long illness had shaken
his being--so much will a few hours of mental
agitation unnerve the strongest frame.
The more he tried to collect his thoughts and
bring them to a final result, the less did he
find himself capable of the effort; the energies
of his mind seemed paralyzed; he appeared
to himself to be under the influence of
some spell which impelled all his actions in an
opposite direction to his wishes, as in paralytic
affections, the limb ever moves in a contrary
motion to that which the sufferer would
have it. He was perplexed, amazed, and saw
no clue to guide him through the labyrinth.
The object of all his wishes--she to whom all
his views and plans had had reference from
the moment he could feel at all--now appeared
to have been almost within reach of his attainment,
and yet, by some inimical power,
was placed at a greater and more uncertain
distance than she had ever been. Lord Albert
was not a weak character: but who is not
weak, while they admit passion, and not principle,
to guide their conduct.
At length, after having run over the subjects
of his last night's perturbed reflections,
the decision to which he came was one, that
feeling alone, unaided by moral and religious
principle, was likely to conduct him to; and
he determined to pursue a middle course,
without making known his suspicions. He
resolved to miss no opportunity of observation,
till he should either have his fears dispelled
or confirmed concerning Mr. Foley. He
argued, that to speak openly to Lady Adeline,
would _not_ be to know the truth. Perhaps
she would not break from her engagement,
from a motive of delicacy as a woman, however
much she might wish to do so; and it
was left for him to free her from a chain
which was no longer voluntarily worn.
The more he reflected the more he thought
the intricacy of the case required this delicacy
on his part. She may not, he thought,
be herself aware of the nature of the attachment
she feels for me; compliance with her
parent's wishes, habit, duty, the kindly affection
of a sister's love, may be all that she
has felt towards myself; and now, for the
first time, she may experience the overpowering
nature of love. This must be what Lady
Hamlet Vernon alluded to; and if it is really
so, I should mar her happiness as well as my
own, by leading her to fulfil such a joyless
engagement. Oh, if indeed Lady Hamlet Vernon
has saved me from the wretchedness
which a marriage, under these circumstances,
with Adeline, must have produced, what
do I not owe her--gratitude--friendship--He
hesitated even in thought--he hesitated
to pronounce the word love; but a glow of
feverish rapture passed through his heart as
he recalled Lady Hamlet Vernon's beauty, her
fascination, her evident partiality for himself.
Yes, I must sift this matter to the utmost; I
must have irrefragable proofs of Adeline's
unshaken truth; nay more, of my being the
decided and sole chosen object of her truest
affections: and in the interim I will see her
frequently--see her in the world as well as in
retirement--and not allow myself to be blinded
by the specious veil which hitherto habit,
perhaps, has rendered equally deceptive to
both.
Could Lord Albert have known this to be
the self-same decision that Lady Adeline and
Lady Dunmelraise had come to in regard to
himself, it would have gone far to have settled
his determination at once, and to have
hastened a declaration which must have
confirmed his union with Lady Adeline. The
fatal security however of thinking that, under
all circumstances, Lady Adeline would keep
her engagement with him, whatever he might
ultimately decide upon, made him the more
apprehensive of owing her possession to any
motive save that of pure attachment; and
it may be also (for the heart is deceitful above
all things) that, resting on this very security,
he had allowed his feelings to betray him imperceptibly
into an aberration from their
natural channel, till at length he could not
distinguish truth from falsehood, and would
too certainly deplore his error when the remedy
was past his power.
Under the false but specious reasoning, then,
in which he now indulged, he strengthened
himself in his determination to pursue the plan
he had laid down, namely, of watching the
feelings and conduct of Lady Adeline in silence,
and of endeavouring to elicit from
Lady Hamlet Vernon, in whose friendship
and interest he placed a fatal but implicit
confidence, some of the grounds upon which
her mysterious words rested. With this decision
he prepared to go to South Audley
Street.
CHAPTER VII.
TRUE NOBILITY.
It must not be supposed that Lady Hamlet
Vernon admitted to herself that she was the
mover of _premeditated_ evil. Impelled by violent
impulse, it is true she hesitated not in
adopting means of any kind to attain her
wishes; for she invariably succeeded in reasoning
herself, however falsely, into a belief
that she had at least some apology to gloss
over, if not to justify, the measures she pursued.
Whatever calm she had assumed in her
late interview with Lord D'Esterre, she suffered
in secret the most painful agitation: the
violence she had done her feelings, in concealing
the disappointment she endured on
Lord Albert D'Esterre's leaving Restormel,
and the restraint that those feelings had since
undergone before she found a favourable opportunity
of speaking to him, all contributed
(when at length that opportunity at Lady
Tilney's supper-party did present itself) to
render their indulgence more overwhelming.
When she returned home that night, the
sleepless hours of suffering she passed were
not less painful in degree than those in which
Lord D'Esterre shared; with this difference
only in their nature, that the anguish endured
by him was of a varied and mixed kind;
whereas the whole mass of Lady Hamlet's
wishes were centred in an uncontrolled
passion for him; a passion which, since
she had allowed it to wear its undisguised
character, she found a thousand plausible
reasons for admitting to control her every
thought.
There was no cause, she argued, sufficiently
strong in Lord D'Esterre's engagement
with Lady Adeline to forbid the indulgence
of her love for him; _she_ had no relative
duties to sway her conduct--she was her
own mistress: and in the opinion of the
world--_her_ world at least--she would be
justified, where envy did not bias the
judgment, in endeavouring to form so desirable
a connexion. However Lord Albert
D'Esterre might have been ostensibly considered
by the members of the exclusive
circle as one of themselves, and however
much they affected to deride and despise his
principles and habits, yet as a man whose
talents promised to shine in the senate, and
whose interest was considerable, his actions
were not, in fact, quite so undervalued, or so
indifferent to the leading personages of that
body, as they might on a cursory view appear
to be. He was still, Lady Tilney thought,
too young, in her political way of viewing
every thing, and had not given sufficient
proofs of firmness, as a party man, for any
direct overtures to be made to him on that
score. But in as far as regarded his admission,
in the first instance, to society amongst
her coterie, he owed that distinction to his
youth, his personal appearance, and his high
rank; to his youth especially, as fitting him
to become, under clever tuition, an obedient
satellite; and when his very attractive exterior
and manners, which were at once dignified
and original, were added to the account, it
is not to be wondered that he was reckoned
a person worth courting, and a character
worth forming, which might be incorporated,
in due time, as one of their own.
Still there was a probationary state to pass
through before any one was actually admitted
into the arena of that circle. Lady Hamlet
Vernon, however, who from his first appearance
had marked him with her peculiar approbation,
was very clear-sighted as to the
views which might be formed of others respecting
an appropriation of him to their own
purposes; and she thought she perceived,
almost from the first, in the politic and eager
attentions of Lady Tilney towards him, as
well as in those of her silent but not uninterested
lord, some ulterior object in obtaining
his favour and confidence, which she
imagined might also turn to her own account,
as affording herself means to acquire an influence
over him of another nature.
It is surprising with what quick perception
women will discover the most hidden sentiments
of others, when they have the remotest
reference to the object of their favour and
predilection; and many a man owes his success
in life to the unceasing, and perhaps unknown
endeavours to serve him, of some devoted,
and it may be, unrequited heart. Who
will watch like a woman over those minute
details, which swell the aggregate of greater
means? Who can feel, as a woman can,
those vibrations of circumstances which may
enable her to seize upon favourable moments,
those _mollissima tempora fandi_, when the
current of success may be directed to the
object of her wishes. Lady Hamlet was well
skilled to do all this, and from the first of
Lord Albert's appearance in the circle in
which she moved, her most diligent attention
was ever awake to all that concerned him.
She perceived that whenever he was spoken
of, the Tilneys were particularly cautious
and guarded in giving their opinion; and
she was not mistaken in thence arguing
that they were aware he might become a
man of high consequence, in every sense of
the term, as well as in their own peculiar
acceptation of it.
Lady Hamlet Vernon felt that in this they
had not formed an erroneous view of him,
for she read ambition in his character: and
though the species of that quality of mind
was certainly very different in Lord Albert
and in herself, yet its general nature was no
stranger to her, and she knew it to be too
powerful a lever in human actions to overlook
or disregard it in this instance. On the
contrary, she determined to use it in behalf of
her own views; and from this motive she dwelt
with energy on the subject of Lord Albert's
prospects for the future, while conversing
with him at Lady Tilney's. She then
found she was touching a master-key to open
the secret recesses of his mind and feelings.
In its very first application, she had found it
more than answer her expectations; and the
consciousness that the apparent harmony of
her sentiments with his on this point, had
established an interest in and obtained an
influence over the very main-spring of Lord
Albert D'Esterre's being, inspired her with
the liveliest hope.
No mercenary views, it is true, no mean
love of power for little ends, actuated her,
but a violent and overpowering passion,
which, however, was equally subversive of
rectitude of conduct, since it was neither
guided by principle, nor restrained by moral
or religious control. It was not directly any
selfishness of motive that impelled her to
the course she was pursuing, for she would
have gone blindly forward in any plan the
most contrary to her interests, her habits, or
her feelings, which promised to draw her into
a union of sentiment with the object of her
passion; but those who suffer themselves to
be directed by such impulses, are under complete
delusion respecting the estimate they
form of themselves. Whenever passion obtains
the mastery, the effect is equally certain;
the wholesome freedom of a mind at liberty
is gone; and when once enslaved, it becomes
like a wave of the sea, tossed about in every
direction the sport of winds, and is as liable
to dash into ruin, as to use any power it may
possess to beneficial purposes.
Whilst the fever of agitation swayed Lady
Hamlet Vernon, she gave herself up in
secret to the inebriating delight of dwelling
upon Lord Albert's looks and words, during
their last interview; she recalled the expression
of his eyes, as he gazed at her while she
was speaking; she still seemed to feel the
pressure of his hand thrill through her veins,
as when he received the ring she gave him
in pledge of friendship; but as these intoxicating
sensations subsided, she relapsed again
into fear, lest she should have gone too far at
first; lest any thing she had said or looked
might have appeared too violent, too plainly
have told the tale of her feelings, ere time
had ripened the moment when their disclosure
might be more in unison with his
wishes. Then again she hoped that her agitation
might have been attributed alone to the
caution which she had ventured to give him
respecting Lady Adeline; and that she gave
him such caution, she trusted would have been
ascribed to a friendly feeling for his happiness.
"Yes, his happiness!" she repeated
to herself; "for I could sacrifice my own to
secure that boon for him. It is not from
motives of jealousy that I did so warn him,
for I could bear to see him the husband of
another, providing that other were really
worthy of him, one who would share in his
views, his plans, his feelings; but to unite
himself with a woman wholly unfit for him--a
girl, a weak insipid girl, made up of puritanical
observances and prejudices--no, I could
not see him set the seal to his future misery
by allowing him to remain in ignorance of
a fact which is known to all the world except
himself."
In this sophistical manner did Lady
Hamlet Vernon argue herself into the belief
that no selfish motive impelled her, but
that she was acting a noble part, and as
the end designed was good, the means she
thought were so likewise. In flattering
this belief, she recalled every look and
gesture of Lord Albert D'Esterre, and she
thought she had perceived that he entertained
a feeling of jealousy towards Mr.
Foley. "Perhaps," she said, musing on that
point, to which she had not before given her
full attention, "perhaps his jealousy is not
without foundation. Why is Mr. Foley so
much at Dunmelraise? The circumstance
of Lady Dunmelraise's protection of him
through life, is not sufficient cause. After
all, why should he not marry Lady
Adeline, if she likes him? It would be a
union much more consonant with Mr. Foley's
happiness (inasmuch as he would not care
what were her ways of thinking) than it
would be for the noble-minded, aspiring
D'Esterre."
In this new point of view Lady Hamlet
Vernon found another specious argument in
favour of her own conduct, and her secret
wishes; and if indeed this latter assumption
of a fact were true, she would be doing a
doubly generous action, in forwarding the
wishes of her friend Mr. Foley, while she at
the same time saved Lord D'Esterre from
a step that would inevitably render him
unhappy.
Such were the false reasonings with which
Lady Hamlet Vernon justified her feelings
and her conduct to herself, and under their
sway, she awaited with the utmost anxiety
and impatience for Lord D'Esterre on the
following morning. But it was late before he
came, and he was abstracted and silent when
he did arrive; unlike the animated being
whom she had witnessed speaking to her
with such force and expression of lively
feeling on the previous evening. The fact
is, Lord Albert D'Esterre had been at Lady
Dunmelraise's, where he had found Adeline
alone; and as, in her converse and presence,
there was a soothing calm, a persuasive assurance,
even in her silence, of her perfect
purity and truth, those feelings of jealous
doubt and mistrust that had preyed upon
him before his visit to her, had gradually
subsided while under the influence of her
immediate power. Above all, the interest
she expressed for him, the alarm she declared
she felt on beholding his haggard
look, and suddenly changed appearance,
awoke in his breast all those tender feelings
which it was a second nature for him
to cherish towards her.
He felt indeed that he could have laid his
head on her breast, confessed his folly, and
wept out his fault in having for a moment
suspected her; "but then again," he thought,
"it will be time enough thus to humble
myself when I see proof that my suspicions
are indeed groundless; and I shall not be
acting up to my resolution, if I allow a moment
of tenderness to put it out of my
power to certify the truth of her's."
Mr. Foley's name was not once mentioned
during his visit. Mr. Foley did not appear;
and for the time Lord Albert D'Esterre felt
happy. "We shall see," he said to himself,
"if this fair shew is real; a short time will
serve to prove its truth, and then my happiness
will stand on a secure basis."
He took leave, therefore, of Lady Adeline
with a mind much relieved, and having
impressed her also with the sensation that
he felt towards her, all he had ever felt; but
no sooner did he quit her presence, than,
with that waywardness of spirit, which is too
often apt to embitter our best interests, he
was impelled to call on Lady Hamlet Vernon,
for the sole purpose, as he fancied, of
gathering indirectly from her conversation a
more clear insight into the subject of her
discourse. But in her presence, he in vain
endeavoured to lead her to it; she avoided
all reference, however remote, to the cause
of his inquietude, and when she touched on
the topic of his public career in life, Lord
Albert felt that it was done in so vague
and wary a manner, as to afford him no clue
whatever to what engrossed at that moment
all his thoughts, and he involuntarily became
silent, and manifested an indifference to
all farther converse. When he arose to take
his leave, if he was less happy than when he
had left Lady Adeline, he was not conscious
of any reason why it should be so; but that
of which he could not fail to be conscious, was
the sensation that a spell was spread around
him, whenever he approached Lady Hamlet
Vernon.
To her inquiries if he would join her circle
in the evening, and if he were one of those
invited to the water-party the following day,
he answered with apparent indifference; and,
with a doubtful half-formed promise to attend
her in the evening, he left the house.
He was bewildered and uneasy; dissatisfied
with himself, and consequently with all the
world; and Lady Hamlet Vernon was miserable
on her part at witnessing his change
of manner, and remarking the serious and
preoccupied expression of his countenance,
which seemed totally at variance with her
wishes.
That evening Lord Albert dedicated to a
few hours of quiet in his own apartments;
but the habit, of any kind, which has once
been broken through, is not so easily resumed;
and in particular the power of sober
application to serious pursuits is hardly by
any man to be laid by and recovered at will.
The mind which is suffered to float about,
driven by the winds of chance, becomes unfitted
for fixed attention to any one particular
point; and the effort is painful which must
be made before it can be brought to bear on
reflective subjects, after having been suffered
to follow the vague direction of the feelings,
or the yet more debilitating influence of dissipation.
Lord Albert acknowledged this, as he had
recourse to various books for amusement.
His attention wandered; and now he was at
Lady Dunmelraise's, now at Lady Hamlet
Vernon's--but never was he on the subject
of the leaves which he vainly turned over;
and after an evening spent in vacuity, he felt
as fatigued, and more dispirited than had he
been deeply engaged in some mental effort.
The consciousness of this lowered state of
being was exceedingly uneasy to him. He
was one who, for so young a man, had
learnt thoroughly to know the value of time,
and when it was thus utterly lost or misapplied,
he could not forgive himself for the
irreparable fault.
Lord Albert, too, had an impression fixed
indelibly on his mind, that when we are not
advancing we are retrograding in our mental
or moral course of existence; and fortunately
for him, he was yet keenly sensible to the
reproaches of conscience. His determination
at the moment, therefore, to redeem this
heavy loss was salutary and sincere; and he
felt a renovation in his whole being when he
took his early walk next day to Lady Dunmelraise's,
full of the good resolutions he
had formed the preceding day. To be in the
presence of Lady Adeline Seymour, was like
being in the sunshine of spring. There was
an habitual serenity about her, which seemed
to animate all around her; every thing and
every sentiment of Adeline's was in its right
place--no one took undue precedence of the
other; the harmony of her form and features
was a true reflection of her happily disposed
nature; but that nature owed its very essence
and continuance to the great ruling
feeling of her mind. Every thought, and every
action, were immediately or remotely under
the guidance of pious belief: the nature of
her happiness could not be uprooted by any
earthly power; she might suffer _anguish
here_; but she had a secret and secure joy
that those only know who, like her, fix the
anchor of their trust on an hereafter.
Having spent the greater part of the morning
in such society, Lord Albert tacitly acknowledged
its superiority to that in which
he had lately lived, and the invitation he
received to dine in South Audley Street was
eagerly accepted. The party which he found
assembled at Lady Dunmelraise's consisted
chiefly of her family,--Lord and Lady Delamere,
their two sons and daughter, and a
few other persons who came in the evening.
Lord Delamere was a shy man, and his shyness
had sometimes the effect of pride; but
the estimable points in his character were of
such sterling value, that his friends loved him
with a zeal of attachment which spoke volumes
in his praise; and he was looked up to
by his family, not only as their father, but
their companion: nothing could be more
beautiful than the union which subsisted between
them; nothing more truly worthy of
imitation than the virtuous dignity with which
they filled their high station.
Lady Delamere still possessed great beauty;
and the charm that never dies, the charm
of fascination of manner and of air, defied
the inroads which time makes on mere personal
beauty. She was one of those very
few women, who unite to feminine gentleness
the qualities ascribed to a masculine mind.
At the time she married, her husband's affairs
were so much involved, that nothing but the
utmost self-denial could possibly retrieve
them: and she entered into his plans of retrenchment
with an alacrity and vigour,
which proved her to be a wife indeed; not
the play-thing of an hour, to deck the board,
or gratify the vanity of the possessor, but a
companion, a friend, a helpmate, one who
in retirement possessed resources that could
enliven and cheer the solitary hour: who
knew she was loved, and felt she deserved to
be so, with that security of honest pride,
which the consciousness of desert never fails
to impart in married life, and yet whose
refinement and delicacy of feeling never lost
the elegancies of polished manners, because
there were no novel objects to excite a sickly
appetite for admiration.
To please is certainly the peculiar attribute
and business of woman, in every relation
of life; and those who neglect to foster
and keep alive this power, reject one of the
greatest means which Providence has placed
in their hands to effect mighty operations of
good. But there is a false and spurious kind
of pleasing which must not be confounded
with the true. Every woman will know
how to distinguish these in her own conscience.
When the wish to please is a
mere gratification of vanity, when it lives
always beyond the circle of her own hearth,
and dies as soon as it is called upon for
exercise within domestic walls; then, indeed,
it may be known for what it is: but
when, as in Lady Delamere's case, this virtue
shone most splendidly confined to the sphere
of home, its price was above rubies; in
short it might truly be said of her, "the
heart of her husband doth safely trust in
her."
At the time when Lord Delamere was
in the greatest difficulties, he did not, as
too many do, fly to a foreign country, to continue
the life of self-indulgence which he
could no longer maintain in his own; he
did not make it an excuse for forsaking his
patrimony, and the seat of his ancestors,
that he could not live there in that splendour
which he had formerly done; but with a
spirit of true pride he said: "the land of
my forefathers with bread and water, rather
than banishment and luxuries." He
made no secret of his poverty; and it was a
means of clothing him with honour: for
with patience in his solitude he found content,
and with content all things. His self-denial
enabled him to be generous to others:
and the very act of living on his estates, gave
bread to hundreds. Lady Delamere went
hand in hand with him in all his plans; and
they pursued, for some years, with untiring
step, the path of duty which they had marked
out.
Meanwhile, their family grew up around
them, and every thing prospered--for a
blessing went along with them: they were
adored by their dependents; honoured even
by those who hated them for their superiority;
and with the occasional visit of a
relative or friend their time flowed on, fruitful
in its course, and fraught with real and
substantial happiness.
But in this their retirement they were not
forgotten. It is not those who are fluttering
about their empty shewy existence in the
sunshine of pleasure and splendour whose
memories live longest, even in that very
world they so busily court. All great and
useful works are the fruit of retirement; all
strength of character is formed, not in indulgence
and prosperity, but in retreat, and
under the grave hand of that schoolmaster
Adversity. The corn is not ripened till it receives
the first and the latter rain: neither is
the moral character formed to its great end,
till it has known the storms of adversity. The
Delameres had now reaped the fruits of this
earthly probation, and they shone forth with
lustre, which could not be eclipsed by any
tinsel splendour of mere outward grandeur.
The children of such parents could not be
supposed to be altogether different from
themselves, for though there are anomalies in
nature, it rarely happens that the offspring
are not like either father or mother, still
less that they are not ultimately influenced
by the example of parents.
When Lord Albert D'Esterre found himself
in this happy society, so different, and
yet, as he acknowledged to himself in every
passing moment, so superior to that in which
he had lately lived, he felt as if he also
were of another race of beings; a pleased
sort of self-satisfaction took possession of
him: so much are we affected by outward
things, so much does the mind reflect the
hues by which it is surrounded. Are these,
he thought, the persons whose names I have
been accustomed to hear coupled with ridicule
or condemnation--are these the persons
who are designated vulgar? Strange indeed
is the misnomer! And that there were
many in the same grade, whose characters
shed lustre upon their high stations, many
who constituted the true character of British
nobles, was a truth that Lord Albert had not
sufficiently considered; for where is there
a body in any country more worthy of respect
and admiration than the real nobility
of our land? It is only to be lamented that
the errors of the few, and the assumed superiority
of the _ton_, should have given ground
for a false estimate of those characters of
solid worth, whose virtues and whose ancient
ancestry reflect a mutual value on
each other; and the moral tranquillity of
whose lives is at once a dignified refutation
of the depreciation of high birth, and the
best confirmation of its real consequence.
But the middling classes, those who envy
their superiors, or those who would attain
to a distinction in society to which they have
no immediate claim, are too apt in these days
to form a mistaken judgment, founded upon
newspaper reports or the spurious publications
of the day, in which much false representation
is mingled with some gross truths,
and the delinquency of the few ascribed to the
conduct of the many. Nor is it these alone,
who are thus led into an erroneous opinion.
The public press produces a circulation of
good and evil, of truth or falsehood, universally;
and wherever the latter creeps in, there
ought to be an antidote administered. It
should not be suffered to smoulder and gain
force till it produce some serious mischief.
It should be told that the few individuals,
whose idle and trifling lives, and whose tenour
of conduct lay them open to contumely
and blame, do _not_ constitute the great mass
of English nobility. So far from it, they are
persons whose lives differ as much from the
general existence of their compeers, as does
the life of one individual in any class from
that of another. Vice is not confined to
nobility because a few great names have
sullied its brightness. It is a false conclusion
to consider _them_ as examples of their
caste, any more than the man in inferior
station, whose delinquency is proved, and
who suffers the penalty of the law, is to be
taken as a specimen of the people at large.
In the course of conversation at Lady
Dunmelraise's dinner, the ensuing drawing-room
was spoken of. "I am one of those
old-fashioned persons," said Lady Delamere,
"who feel a real pleasure in the thought of
going to court--for first, I shall have the
gratification of seeing my Sovereign, and of
presenting to him another branch of that
parent stock, who are personally as well as
on principle attached to him and to his
house. And though, doubtless, there are
many who share in these feelings, yet I will
yield the palm of loyalty and zeal to none;
and, in the second place, I do very firmly
believe that, in as far as society goes, a drawing-room
does much moral good. There
are certain lines drawn, which are useful to
remind persons in general, that vice is contemned,
and virtue honoured; and there is
a distinction, too, of time, and place, and situation,
which is not yet laid aside; I heartily
wish there were many more drawing-rooms
than there are."
Lord Delamere fully agreed with his
wife in this opinion--the young people did
not giggle and whisper, "what a bore it
will be," but coincided with their parents.
Lady Mary Delamere too declared, that she
thought there was no occasion better suited
to shew off real beauty to advantage than the
splendour of a mid-day assembly, where
every thing conspired to give people an air of
decorative style which they could not possess
at any other public meeting. "What pleasure,"
she continued, "I shall have in going
with my cousin Adeline, and gathering up all
the stray words of admiration, which I am sure
will abundantly fall in her praise. Do tell
me, love," addressing herself to her in a half
whisper, while the rest of the persons at
table conversed on other matters, "do tell
me of what colour is your dress, and how it
is to be trimmed?"
"Really," replied Lady Adeline, colouring
as though she had committed a crime, "I
have not thought about it. All I begged of
Mamma was, that it might be very simple,
and, I believe, of a rose-colour--for a rose is
my favourite flower."
"Dear child," said the good-natured
Lady Mary, "you must think about it now,
for the day is drawing near, and I shall
be so disappointed if you are not well
dressed."
"You are very kind, sweet cousin, but if
you only knew how very little I care about
the matter;" and she laughed heartily at the
idea of its being a subject of the least importance.
"But, Lord Albert D'Esterre," said Lady
Mary, appealing to him as he sat on the
other side of Lady Adeline, "you will
interfere, will you not? You will not be
pleased, I am sure, lovely as Adeline is, to see
her a _figure_ at a drawing-room."
"What sort of figure do you mean?" he
asked, smiling.
"Oh dear! you know well enough what
I mean--unbecomingly attired."
"I think," he replied, "that although
some figures will always be admired, still
there is no merit in disdaining the usages of
society or the advantages of dress, and that
the neglect of appearance may in a young
person be produced by some causes which
are not desirable." He looked fixedly at
Adeline as he spoke, and she blushed very
deeply; but answered with an unhesitating
voice:
"I shall be always desirous of pleasing
those I love, even in trifles; but I should be
sorry that trifles occupied their thoughts."
Lord Albert was silent; he felt a kind of
chill come over him, for the remembrance
of Lady Hamlet Vernon's instructions recurred
to him; and he thought he saw a
species of puritanical pride in the general
tenour of Lady Adeline's manner of thinking
and speaking, which seemed to justify the observations
she had made upon her character.
Then again he feared, that in other points
he might discover more reason still to be
dissatisfied--points on which his vital happiness
rested. He looked instinctively round
the room; but the person who at that
moment crossed his thoughts was not present,
and he again wrapped himself up in
that mood of suspicion, which is ever on
the alert to seek out the object which
would give it most pain; under this influence
he returned to the subject of Adeline's
presentation dress, and said, addressing
Lady Dunmelraise:
"I am not particularly an advocate for
splendid attire; but I am sure, Lady Dunmelraise,
you will agree with me in thinking,
that there is an affectation in going unadorned
to a court, which is a sort of disrespect
to the place."
"Indeed," said Lady Adeline, in her wild
eager way, "I will not go to much expenditure
on my dress, for I have a plan for doing
some good going on, which will require all
the money I can collect, and I should be very
sorry to see mamma wasting her's on any
thing which I so little prize as my court-dress."
Lady Dunmelraise only smiled, and replied,
"We must all subscribe to Adeline's
toilette, for she is the veriest miser on that
score herself. However, Lord Albert, do not
be uneasy, I think she will not disgrace us,"
and the pleased mother passed on to other
discourse.
This tenacity of Lady Adeline appeared to
be a confirmation of his suspicions; and
when, in the after part of the evening, Mr.
Foley was announced, Lord Albert lost all
command over himself, and under plea of
a bad head-ache, sat silent, that he might
the better watch every look and motion of
Lady Adeline and Mr. Foley. Turning every
indifferent word and gesture into the meaning
with which his jealousy clothed it, he fancied
that they were certainly mutually attached.
Whatever soothing attentions Lady Adeline
shewed to himself, he imagined were put
on for the purpose of deceiving him; and
his manner was so cold and haughty, that
she in her turn began to shrink within herself,
and to wear an abstracted, and somewhat
distressed countenance.
Under this impression, Mr. Foley, with
his _doucereux_ air, whispered Lady Adeline,
"that he was sure she was ill," and asked
her "to cast out the evil spirit by her sweet
power of music."
"Do, my love," said Lady Dunmelraise,
"sing that delightful duet, which is always
charming, '_O Momento fortunato!_' and
then I feel sure we shall be all love and
harmony--shall we not, Lord D'Esterre?"
The chords of the piano-forte relieved
him from the embarrassment of a reply, and
he listened to the impassioned tones of _poi
Doman, poi Doman l'altro_, ascribing to every
intonation and every sentiment of her feeling
voice the dictates of a passion for his
supposed rival.
"That used to be a favourite of yours,
Albert," said Lady Adeline when the duet
was finished; "but I am afraid your head-ache
prevents you from enjoying any thing
to-night."
"I do not feel well," he replied shortly;
"and lest my indisposition should in any
way affect the pleasure of others, I will
hasten away."
"Oh yes, you appear ill, indeed!" said
Lady Adeline, fixing her eyes tenderly on
his; "and, dear Albert, perhaps you had
better go--the noise of company may be
too much for you:" and she held out her
hand to him--"Oh, if you are unwell, by
all means go home," she repeated, with an
anxiety of tender interest, that no one else
could misinterpret to be any thing but
genuine affection, but which to him seemed
to spring from the desire of his absence.
"You shall be obeyed," he said, returning
her look reproachfully; and at the same time
reaching his hat, which happened to lie on
a table beyond Mr. Foley, he almost rudely
snatched it away, and with a celerity of
movement that admitted of no courtesy to
any one present, departed. Lady Dunmelraise
called after him, "Lord Albert, do
you dine here to-morrow?" But he heard
not, or affected not to hear, and with the
gnawing rage of blind jealousy darted into his
carriage, and gave the order, "home."
Soon after the rest of the party broke up;
and when Lady Dunmelraise and her daughter
found themselves once more alone, their
mutual silence proved that they both felt the
strangeness of Lord Albert's manner of departure.
But although the words were on
Lady Dunmelraise's tongue to utter--"_he is
capricious_,"--she restrained, and suffered
them to die away in silence, determined that
her daughter's own unbiassed judgment
should form for herself that opinion of Lord
Albert's character, which would soon now
ultimately decide on her acceptance or rejection
of him as her husband.
CHAPTER VIII.
OFFICIAL LIFE.
It may be recollected, that when Lady
Glenmore returned from the water-party, she
was cruelly disappointed at finding only a
note from her husband. "How little," she
thought, as she sat at her toilette taking off
the dress which in the morning she had not
despised, as having been approved of and
admired by him, but which now she cast aside
with disdain--"how little men know how
to value the affections of a wife! I have been
for many hours in what is called a gay scene,
and during the whole of the time, I cannot
recal one moment when Glenmore was
not present to my fancy; but he, I dare say,
on the contrary, has not given a wish or a
sigh to me." She looked in the glass as she
thought this, and although a tear dimmed
her eye, vanity whispered, "ought this to
be so?"
"I am at least _pretty_; young, no one can
deny; yet I am neglected for a number of
old stupid men, a dull political discussion.
Oh, those vile politics! how I hate them.
And when he comes home, he will look so
grave, so preoccupied! Oh, I wish there was
no such thing in the world as a House of
Lords or Commons. Is life itself long enough
for love?--and must dull, dry business, consume
the hours of youth, pale his cheek,
perhaps blanch his hair, his beautiful hair,
for they say care has whitened the locks even
in one night! how very terrible this is."--And
she arose, and walked to and fro in her
room, and listened to every carriage that
rolled by--then she took up Lalla Rookh--read
some of the most impassioned passages,
and wished herself a Peri.
"I have but one wish," she said, "that
wish is to be loved as I love."--Poor Lady
Glenmore! this beautiful phantom of a young
heart is, nevertheless, in the sense in which
she framed it, a mere deceit. Love such as
her's does _not_ grow by feeding on; there is
a strength of character, a consciousness of
self-dignity, the duties of a rational being,
above all, the duties of a Christian, which
must be cherished and understood, before
any lasting fabric of happiness can be built
on love. This was never more proved than
in the restless impatience, the miserable (for
such hours to such minds are miserable)
anxiety and disappointment, which converted
minutes into hours, and hours into ages, before
Lord Glenmore returned. As she foresaw,
when he did come, though he pressed
her with almost rapturous tenderness to his
heart, and inquired with trusting fondness
at her party, hoping she had been well
amused, he was himself so exhausted and
harassed by business, that he professed himself
unable to talk. "Why did you sit up
for me, dearest?" he asked; "you will fatigue
yourself uselessly; and I must really insist
in future that you do not do so. At least,
if you had been _amusing yourself_, I should,
not be so sorry; but as it is, really Georgina,
love, you must be better behaved in future--but
why did you not go to the supper?"
"I came home to see you," she answered
in a tremulous voice. Lord Glenmore chided
her lovingly, and assured her that he had
not less anxiously desired to return to her;
but he said, smiling,
"You know you have the advantage over
our sex, for _your business_ is love--but our
_business_ is a matter apart from that gentler
care. I long to tell you, my sweet Georgina,
all that has interested me this day, and I
think you will share in my satisfaction; but
I am really unequal to enter into the details
at present: to-morrow, love, you shall know
all." Lady Glenmore only sighed; but with
the sweet docility of her nature, never questioned
his will, and his being with her constituted
in fact all she cared to know. The
truth was, that certain changes in the ministry
had long been talked of, and on that
morning overtures had been made to Lord
Glenmore to take on himself an important
office. The whole of the morning had been
occupied in settling preliminaries, and ascertaining
the sentiments of these public men
with whom he was to act: for Lord Glenmore
was a conscientious man, and would
not mount a ladder, which he intended
afterwards to cast down. It was not place
he sought, but power, for purposes alike
good and great. He felt within himself a
capacity for the honours and distinctions
he aspired to, and knew on principle the
responsibility which attends success in such
measures.
One of the first persons, whom he considered
to be a man of inflexible integrity,
and whom he wished for as a colleague
in office, was Lord Albert D'Esterre;
and since the situation which he
had himself received threw several appointments
into his own hands, Lord Glenmore
lost no time in writing him the following
note:
/#
"MY DEAR D'ESTERRE:--I think that I
shall not be making a proposal unacceptable
to your wishes, or in discrepancy with your
future plans, when I announce to you that
I have accepted the office of ----. The
official appointments immediately connected
with it of course become mine, and it would
afford me the greatest satisfaction in my
arduous undertaking, to have one possessed
of your talents to aid me in the performance
of its duties. Would you accept the
office of under Secretary of State in my
department? I need not express my ardent
hope that you will consent. You know that
our views of public matters coincide thoroughly--let
me therefore hear from or see
you as soon as possible.
/*[5]
"Your's ever most truly,
"GLENMORE."
*/
#/
After despatching this note, Lord Glenmore
sought his wife, and entered into
an account of what passed the previous
day; he spoke of the increased expediency
that would ensue of her living very much
in society, whether he could himself be
present with her or not; and added, that she
must not allow any fears or mistrust, either
of herself or him, to lessen the pleasure
which it was natural, at her age and with her
charm of person, she should derive from the
homage around her.
"It is not mistrust, dearest Glenmore,
that makes me feel joyless in your absence,
for what can I fear?--it is true that I am
uninterested in every thing, when you are
not by to share my pleasure; but indeed
you quite mistake me, love, if you suppose
that I am not all confidence in you. And as to
myself, what is there that can be for a moment
dangerous to my peace, when all my
interest, all my wishes, are centred in your
love?"
"My own best Georgina," he replied,
pressing her to his breast, "be ever thus, and
what can I wish for more. But, love, mark
me--you are now no longer the girl, whose
duties were centred in passive obedience to
her relatives, and whose recreations were the
innocent, but trifling pursuits of girlhood;
you are the wife of a man who is become a
servant of the public--whose high cares must
necessarily debar him frequently from the
enjoyment of those domestic pleasures which
a less busy or responsible life might allow.
It is now become your duty, love, to feel your
own consequence in his--to play _your_ part in
the scale by which his actions must be measured,
and to be aware that many will court
you from an idea of your being wife to a
minister, who would not for your own sake
alone, perhaps, have thought of you; while
others who previously courted you for the
charm of your presence and the beauty of
your outward shew, will now doubly affect your
society, and endeavour, it may be, to use your
influence to undue purposes. All are not pure
and single-hearted like you, my dearest, and
these cautions, believe me, are not given
as to one whose worth I doubt, but, on the
contrary, to one whose very ingenuousness
and worth may prove a snare to her. In all
that concerns mere knowledge of the world
I recommend you to look to Lady Tenderden
and Lady Tilney; they have passed creditably
through the busy throng, and are certainly
in all respects fashionable, and bear
a high consideration in the estimation of
the London world. You cannot do better,
then, than to shape your course by their's
in respect to what the French call _conduite_;
and to the dictates of the heart, and moral duties,
I refer you to your own and your excellent
mother's."
Lady Glenmore scarcely knew why, but
her heart swelled almost to bursting while
her husband spoke thus to her; and it was
with difficulty that she restrained the tears
which seemed at every moment ready to
overflow. The truth was she dwelt upon
his first words, his declaration that his newly
acquired honour would debar him from the
pleasures of home society; and she looked up
timidly as with tender accents she asked,
"whether she was doomed now to be always
absent from him."
"I trust not, dearest; at all events, you
know my best and fondest interests are centred
in you, and you would, I am sure, consider
your husband's advantage and glory to
be of value to you, even though these were
obtained by the sacrifice of his company."
She said "yes," but _felt_ decidedly, that had
she spoken the truth, the "yes" would have
been "no."
Lord Glenmore received several notes,
and with a preoccupied air which prevented
his observing the melancholy depicted
on his wife's countenance, he snatched a
hasty embrace, and was hurrying away,
when looking back he said, "Remember
love, not a word of this to any one, even
to your mother. A few days will release
the restraint I put upon your tongue," he
added, smiling; "but in you I expect to
find the _wonder_, that a woman can keep
a secret;--in all things, I believe in, and
trust you. Adieu, love, adieu." And he was
gone.
That which would have pleased a vain
woman, and gratified an ambitious one, fell
only like lead on the young Georgina's
heart.
"So," she said, sinking down in a chair, "I
am a minister's wife. And am I the happier?
Far, far from it; I am seldom now to see my
husband, and when I do, the concerns of the
public are to form our consideration and
discourse; whereas, hitherto, in the short
sunshine of our marriage, ourselves, our
mutual hopes, our own dear home, have
constituted all our care; and I fondly
trusted, perhaps foolishly hoped, would have
continued to do so. What a desolating
change! But he says I must prepare
for it; and since it is his will that thus
it should be, I will endeavour to hide the
mournful feelings of my heart. My dear
mamma shall not see that I have wept either,
for she will, perhaps, ascribe my tears to my
husband's temper, and that would be worse
still." So saying, she roused herself from the
despondency into which she had fallen, bathed
her face, called up smiles which were _not
genuine_ for the first time in her life; and,
having re-arranged her dress, she said to
herself as she cast a glance at her mirror,
"Am I not now metamorphosed into the
wife of a minister?" Just as she was preparing
to ring her bell for her carriage,
Lady Tenderden arrived.
"How well you are looking, _la belle aux
yeux bleus_," said Lady Tenderden, kissing
her: "there certainly never was any body
who had the azure of the skies so exactly reflected
in her eyes." This might be true;
but it certainly was not true that she was
looking well. To a vague answer given by
Lady Glenmore she made no allusion; but
looking at her very fixedly, so fixedly that
it made her colour deeply, Lady Tenderden
said, "Yet methinks something more than
usual has occurred--is the report true?"
"What report?"
"Nay, now, do not make the _discreet_, for
by to-morrow it will be in the newspapers.
Come, tell me, your friend, am I not to wish
you joy?"
"Of what, I may ask you in return,
Lady Tenderden, for I can sincerely answer,
that no increased cause of joy has befallen
me, that I know of." This was said so
very naturally, that her interrogator was
posed. Judging by herself, Lady Tenderden
conceived it impossible that the report
of Lord Glenmore's having accepted a high
office in government, which would have
been the envy of so many, should be true;
or else she thought the little lady must be
more silly than she ever believed her to
be. She went on, nevertheless, to sound
Lady Glenmore in various ways, expecting
to make out something relative to the subject;
but Lady Glenmore's calm indifference
totally foiled her, as she herself afterwards
confessed; and she set it down in her own
mind that for the present she could not
be of any particular service to her, or derive
any more reflected lustre from her, as being
the friend of a woman whose husband was
in power.
How the simplicity of a genuine character
confounds the pertinacity of a keen worldly
mind! Lady Tenderden was completely at
fault: when another visitor, who came much
on the same errand, afforded an additional
proof of the truth of this observation. Lady
Tilney came up to Lady Glenmore, and after
the first salutation, entered with all her
energy and eloquence upon politics; inveighing
against government measures, and
hoping that now a man of more liberal
principles had come in, some change of
_measures_ at least would be adopted. Lady
Glenmore sat abstracted, and began arranging
her embroidery frame; seeing that there
was no chance of Lady Tilney's speech coming
to a conclusion:
"Well, my dear, and now," the latter said,
"you will really have a part to play: how I envy
you! What interest--what endless business
will devolve on you! Were I you, I would
propose to Lord Glenmore to write all his
private letters for him; by this means you
know you would be _au fait_ of all the state
secrets, and could, in a great measure,
guide things your own way. You write rapidly,
I believe; and your hand is not bad;
it wants a little more character perhaps: but
you know there is the man who advertises to
teach any hand-writing. I do assure you he
is excellent--I tried him myself, and a very
few lessons from him would teach you to give
your writing the firm diplomatic air--and
you would quickly learn that significant
style which means nothing; and by which,
should any thing occur to make you change
your mind (Lord Glenmore's, I mean), you
could twist the phrase into another meaning,
suitable to the occasion. I am sure I am
always for decision and truth; but in certain
cases prudence and caution are necessary;
and therefore these resources are requisite to
be observed in diplomatic writing. If you look
back, you will always see it has been so in
all ministers' letters."
Lady Glenmore, who had sat silent hitherto,
now conceived herself obliged to
speak, and replied, "that she knew nothing
of diplomacy, except the name; that every
thing of the kind always made her yawn, and
she hoped she should never have to copy any
letters of business for any body." Lady Tilney
in her turn stared, and observing that
Lady Glenmore was very young, she said:
"Well, but at all events you will be delighted
to see your name perpetually with
all the people in power; and to hear them
say, that is the minister's beautiful wife! and
the honours of your husband, at least to any
one so domestically inclined, must be a great
delight."
"I do not want Lord Glenmore to have
any more honours than he has, for my own
sake; but whatever pleases him will certainly
please me."
"Oh, oh! so then you do confess it? and
he _is_ minister for ----"
"I am happy to hear it, if it really is to
confer all the honour you seem to think upon
him. But I wish you would tell me what _you_
mean, Lady Tilney, for I do not quite understand
you." There was a sort of real _not
caring_ about Lady Glenmore, which deceived
Lady Tilney, as it had done Lady Tenderden.
It was a thing so totally out of Lady Tilney's
calculation that any one should not be
enchanted at such a situation, that she was
persuaded either that the fact was not so, or
that Lady Glenmore did not know that it was
the case.
Just as this inquisition had ceased, a servant
entered with a few lines written in pencil
on a card, which he gave to Lady Tenderden:
they were from Mr. Leslie Winyard, to say,
that having seen her carriage at the door,
and having something very particular to
communicate to her, he requested ten minutes'
conversation, if he might be allowed
to come up. Lady Tenderden remembered
Lady Glenmore's former scruples about receiving
him, but determined to overcome
them.
"_Chère ladi_," she said, "you must positively,
notwithstanding the fear of Lord
Glenmore, allow me to see Mr. Leslie Winyard;
I will take all the _imminent_ risk of the
danger upon myself; and besides, you know,
visiting _me_ is not visiting _you_." Lady Glenmore
looked exceedingly distressed, and
said, "If you want to speak to Mr. Leslie
Winyard, why can you not speak to him in
your carriage?"
"Oh! that is so uncomfortable. Besides,
Lady Tilney, I appeal to you, was there ever
any thing so strange as Lady Glenmore's
refusing to let Mr. Leslie Winyard come up
stairs to see me, merely because _le tiran de
mari_ does not approve of morning visits
from gentlemen?"
"Pho, pho," said Lady Tilney, "he was
only joking, and that dear little good Georgina
thought he was serious." Then turning
to the servant who was waiting for orders,
"Shew Mr. Leslie Winyard up stairs directly,"
commanding, as she always did, or
tried to do, in every place and every person.
In a few minutes Mr. Leslie Winyard made
his appearance; and having paid his compliments
to Lady Glenmore and Lady Tilney
for some little time, he then stepped aside
with Lady Tenderden, and after conversing
together, apparently engaged on a most interesting
subject, they returned to the other
ladies, and he entered into general conversation
with his usual light and amusing
anecdote. At length, however, Lady Tilney
arose, saying to Mr. Winyard, "well, notwithstanding
your _agrémens_, I must go, for
I have a hundred things to do." Lady Tenderden
echoed this declaration, and they both
went away, leaving Mr. Leslie Winyard, who
seemed determined to sit them out _en tête-à-tête_
with Lady Glenmore.
The consciousness that any thing has been
said on any subject, always creates in an unartificial
mind an awkwardness when the predicament
that has led to the discussion really
occurs;--and Lady Glenmore experienced this
painfully. Every instant the sensation became
stronger, and, of course, was not lost to
the observation of her companion, though he
affected not to perceive it; and by dint of
feigning ignorance, and talking on indifferent
subjects, he arrived at bringing her into
the calm and comfortable frame of mind
he had in view, one in which she would
feel _le diable n'est pas si noir_; and this
he effected with his usual address, till he
evidently saw that she was rather diverted
than otherwise by his conversation.
He then led the discourse to music, and
entreated her once more to sing the _Sempre
più t'amo_ of Caraffa. She readily agreed, and
their voices were in beautiful and thrilling
unison when the door opened, and in came
Lord Glenmore. His wife suddenly stopped,
and rising from the instrument, looked abashed.
Lord Glenmore, with the manners of a man
of the world, addressed Mr. Leslie Winyard,
regretted that he had interrupted the music,
declared that he had some letters to write,
and prayed him to finish the duet. But Lady
Glenmore tried in vain to recommence singing--her
voice faltered, her hand trembled,
as she touched the keys--her eyes wandered to
her husband with an expression of inquiry
and uneasiness; and Mr. Leslie, too much the
man of the world, and too much skilled in
his _métier_ to push matters at an unfavourable
moment, declared that he was exceedingly
sorry, but found himself under the
necessity of going away, having an appointment
on business which he could not put
off. Apologizing, therefore, to Lord Glenmore,
to whom he always took care to pay
particular deference, for not being able to
remain, he hurried out.
Lady Glenmore hastened with considerable
trepidation of manner to explain to her
husband how it had chanced that he found
her singing with Mr. Leslie Winyard; but
Lord Glenmore seemed more deeply engaged
in thinking of the letter he was perusing
than of what she was saying, and only looked
up smilingly in her face, and said, "My dear
love, why are you so agitated about such a
trifle?"--"Is it a trifle?" she said: "well,
then, I need not care, and am quite happy
again." She kissed his forehead; and further
discourse was prevented by a servant's entering,
to inform Lord Glenmore that Lord
D'Esterre requested to see him if he was
disengaged. Lord Glenmore immediately
desired that he might be shewn into his private
apartment; and at the same time gave
orders that no one else might be admitted
except the persons whose names were on the
list; then pressing his wife's hand tenderly,
but evidently much preoccupied in mind, he
left the room.
"Is it possible," thought Lady Glenmore,
looking after him--"can this be _my_ husband,
who so lately appeared to have no thought
save what we mutually shared? and now we
seem suddenly cast asunder: different interests,
different hours, different societies, all
seems to place us, as if by magic, apart, and
to divide us from each other. He too, who
dwelt so particularly on my not receiving
morning visits from young men, now seems
to think it is become a matter of indifference,
or rather not to think about it at all. Has
power then changed him so quickly? What
a horrible thing power is!--how it transforms
every thing into its own heartless self!
Surely, surely, it is the most miserable thing
in the world to be a minister's wife!" To
dissipate the melancholy she felt, she ordered
her carriage, and proceeded to visit
her mother, who she found was ill, having
caught cold in coming out of the Opera.
"Why did you not inform me of your
indisposition before, dearest mamma?--I
would have been here early?"
"I know, love, that you would not have
been remiss in any kindness; but when a
woman is married, her first duty is to her
husband; and I fancy," she added, smiling,
and implying by her manner that she knew
more than she would exactly say; "I fancy
Lord Glenmore will occupy more of your
time than ever, dear Georgina, if what is
reported be true."
"I am sure he will never prevent my
coming to you, under any circumstances; but
really he has so much business, that I see
less and less of him every day."
"Indeed!" said Lady Melcomb, looking
rather blank. Fortunately for both parties,
Lord Melcomb came in from his morning
walk, with a countenance even more bright
and cheerful than was his wont. "So, my
love," he cried, "I fear you must now be
no longer my little Georgy, if the current
news be true, I must look at you in a new
light--eh?" and he examined her countenance.
"I am very sorry to hear that, dearest
papa; I was so happy in the old one, that
nothing can make me wish to change in
your eyes."
"Come come, love, tell us now, has Lord
Glenmore accepted the appointment of ----
or not?"
"Whenever he tells me to say that he
has done so, I shall certainly, my dearest
papa and mamma, make you the first to be
acquainted with the event."
"Well, Georgina, I see how it is: you
need not say more, for you are already quite
diplomatic in your mode of answering. But
you are right, my child: whatever confidence
your husband reposes in you, you ought to
regard it as sacred;" and Lord Melcomb
changed the subject like a good and a sensible
man, who wishes really that his child
should prove a good and faithful wife. "You
have given the best earnest any girl can
give," he said, "my sweet Georgina, of being
an invaluable treasure to your husband, by
having first been such to your parents; and
the obedience you paid us should now be
implicitly transferred to Lord Glenmore. The
woman who has not learnt obedience, is likely
to be very unhappy: for it is surely one of
the first duties in every sphere of a woman's
life. You know the lines, that I have
so often repeated to you, and I am sure you
practise them, my own Georgina, as forming
the great golden rule to be observed by
a married woman: one who
/P
"'Never answers till her husband cools;
And if she rules him, never shows she rules.'
P/
"But when I say obedience, I do not mean
that slavish obedience, which in matters of
conscience must remain a question for conscience
to decide; I mean that system of gentle
acquiescence in all the minor motives of
life, which can alone render the domestic
circle a circle of harmony."
Lady Glenmore assured her father she had
not forgotten, and never should forget his excellent
lessons; and that every thing which he
had recommended her to do, she invariably
called to mind every night and morning. Lord
Melcomb had, during a very busy life, acquitted
himself under all circumstances with
credit both abroad and at home, and if he had
leant to the despotic side of governing in his
own house, he had done it with so much gentleness
as well as firmness, that no one felt
inclined to consider the yoke heavy. His
daughter had never even felt it could be so,
for she was by nature and inclination a docile
gentle being, leaning upon those she loved with
implicit confidence for guidance and support.
It was at this particular moment more than
usually sweet to her to be in the society of her
parents, and she promised that if Lord Glenmore
were engaged in the evening, she would
bring her work, and instead of passing the _then_
dull hours at home, find a sweet solace with
them; they were a happy family, united in
the bond of the strictest union, and even at
a temporary parting felt pain, in proportion as
being together gave them pleasure; but it
was time for Lady Glenmore, she said, to go
home, and they separated.
The interview between Lord Glenmore and
Lord D'Esterre that morning had passed to
their mutual satisfaction; their general opinion
of public affairs, and their views of domestic
happiness were too similar for them not to
draw together; and yet there were points of
difference in their character, which tended to
keep alive an awakening interest, and render
the one more necessary to the other; but in
regard to the great question then agitating the
public mind, Lady Tilney was quite mistaken
in her ideas of his principles, which were at
variance in many respects with what she
called _liberalism_.
The fact was, the minister of the day, having
discovered that those whom he had allowed to
continue in office, on agreed and well-defined
principles as to the line they were to pursue
in their political conduct, were acting out of
the pale of their engagements, and forfeiting
the pledges given to himself; consequently,
with that decision of character, and straightforwardness
of conduct, which formed the leading
feature of his life, he availed himself of the
first favourable opportunity of breaking off a
connexion with men, whose moral complexions
were so very unlike his own. Well
knowing how vast were his resources, he
sought among the rising nobility of England
(who, take them altogether, form perhaps, a
body more talented, and more patriotic than
any other nation in the world) for that support
and coadjutancy which the emergency of
the times demanded, in order to maintain
the constitutional rights of the nation. Lord
Glenmore was one of these, and amongst the
parties whom he, in his turn, named as being
those he wished should co-operate with him
in his individual department, Lord D'Esterre
stood pre-eminent. This happy nomination
met at once with the entire approbation of
the minister, whose discernment was as penetrating,
as it was prompt and decisive.
Lord Albert, it may be, in his acceptance of
office, was not influenced alone by political
views. He felt that, in the uncertain and agitated
state of his mind, some great and commanding
power for exertion was necessary to
him; some influential weight of sufficient magnitude
to poise the fluctuations of a mind,
whose energies he was conscious were wasting
themselves in a diseased state of excitement.
He thought that by engaging in a political
career, where the duties imposed were of an
imperious and absorbing nature, he should
best find that refuge against himself which he
deemed it wise to seek. Men in such cases
have most indubitably great advantages over
women; many a noble career lies open to them.
When they are oppressed by any woe of a private
nature, they may in the exercise of their
powers find arms against a sea of troubles;
but women have only one great lesson to
learn, greater still perhaps if duly entertained--to
suffer resignedly.
Lord Glenmore and Lord Albert prolonged
their discussion to a late hour--so late that
Lord Glenmore pressed him to remain and
dine. "We have no company to-day," he said,
"and Lady Glenmore will excuse your toilette."
The invitation was too acceptable to be
refused, and they passed into the drawing-room,
where they found Lady Glenmore all
smiles and beauty; for the idea of enjoying her
husband's company had again restored her to
her wonted placid happiness.
The conversation took that happy course
which it ever does when similarity of tastes
directs the subjects; and as the minds of these
young men were not only of a superior cast,
but their manners too formed on that refined
model which, when it is accompanied by
intellectual power, gives grace to force, their
social intercourse was truly such, as one likes
to think is the sample of a high-born, high-bred
British nobleman.
Lady Glenmore listened with no insipid
mawkish indifference, even to matters beyond
her ken, and the remark she ventured now
and then to slide in was one that bespoke a
diffident, but not deficient understanding. A
delighted glance of approbation occasionally
escaped from Lord Glenmore, in homage to
his wife, and as Lord Albert beheld this married
happiness, he could not help sighing, as
he thought "such might have been mine;" and
he almost unconsciously drew a parallel between
Lady Glenmore and Adeline, in which
he did not deceive himself in giving the decided
palm to the latter.
When he was preparing to depart, he found
it was so late that he drove home; but when
there, the same incapacity to settle himself
to any occupation which he had before experienced,
returned, and he fancied that he
might yet be in time for an hour of the ballet.
So he ordered his carriage, made a brief toilette,
and drove to the Opera-house. "It is too
late," he thought, "to go to South Audley
Street; I shall disturb Lady Dunmelraise;"
but yet the idea that he had not called upon
her that day haunted him painfully.
Arrived at the Opera, he walked in, and
hearing, as he passed the pit-door, a favourite
air sung by Pasta, he made his way through
the crowd, obtained a tolerable place, and was
listening intently to the music, when he was accosted
by Mr. George Foley. The recollection
of what he had suffered the previous evening
came freshly to his feelings, like a dark dense
cloud, obscuring every other idea. Mr. Foley,
either not seeing, or not choosing to see,
the coldness of his reception, pertinaciously
kept up a conversation with him on various
subjects, precisely in that quiet and self-satisfied
manner, which is so insufferable to a person
under feelings of irritation. Nor did Mr.
Foley cease talking till he suddenly turned
round, and saw some one in the boxes, to
whom he nodded with much apparent familiarity
of interest. Lord Albert mechanically
turned his head also, and beheld Lady Hamlet
Vernon--who kissed her hand to him; and
both of them, as if by mutual consent, proceeded
to join her. She was but just arrived, having
been at a dinner at the Leinsengens, she
said, and her face was lit up with more than ordinary
animation as she greeted them on their
entering; then noticing to Lord Albert to
take the seat next her in front of the box, she
bent towards him, so as to whisper in his
ear, "I heartily congratulate you; I have just
heard of the arrangements at the Leinsengens
where I dined, as I have already told you, and
where I heard all the finest things in the world
said of you, as I have not yet told you; but I
assure you the generality of the persons there
were, I really believe, for once sincere in what
they said. But you do not express any satisfaction
at this event yourself: why are you so
exceedingly indifferent?" and her eyes spoke a
language which was any thing but that of indifference.
"Because," he said, "I do not avow that
the news you have heard is true. We must
wait and see the event publicly announced,
before one can have any feeling about it, one
way or the other." Lady Hamlet Vernon continued
to banter him on his cautious reserve
for some time; but did not press the matter
further, as she saw his dislike to being probed
on the subject.
"Only remember," she whispered, "you
have one friend, who enters into all your joys
and sorrows, and feels every thing that betides
you with a keen perception of interest." After
some vain attempts on her part to unite Mr.
Foley in a conversation with them, which she
resumed aloud, he being perfectly aware
that Lord Albert in fact engrossed her completely,
took an early opportunity of withdrawing.
Lord Albert remained till near the close
of the ballet in earnest conversation with Lady
Hamlet Vernon, interrupted only occasionally
by chance visitors, who seeing the preoccupied
air, and observing the thoughtful expression
of Lord Albert, did not long obtrude themselves.
He would probably have remained
where he was till the entire end of the performance,
had not a sudden movement in the
box opposite, attended with bustle, and some
lady apparently fainting, caught his attention.
He looked eagerly again, and in another minute
recognized Lady Delamere, and thought in the
reclining figure that he could trace a likeness
to Lady Adeline Seymour. Hastily rising, he
rushed out of the box, without making any
apology to Lady Hamlet Vernon, or mentioning
the cause of his very abrupt departure.
When he arrived at the opposite side of the
house, he found his fears and conjectures true;
and his heart smote him in an instant, as he
figured to himself what Lady Adeline's feelings
must have been, in seeing him occupied so long
a time, and his attention so intensely fixed
upon another, as he was conscious his had been
on Lady Hamlet Vernon. Although Lady Adeline
might not know who she was, yet the circumstance
of his not having been near her all
day, the reason of which she could not know,
together with the fact which she saw, namely,
that he preferred the society of another to her's,
were all circumstances that struck him with
self-condemnation, and his look, and manner,
implied the full expression of tender penitence.
But Lady Adeline was still insensible;
she could not see, or observe, _what_ his feelings
then were at beholding her thus; but with
Lady Delamere the case was different; he
thought he read in her cold reception of his
offered services, and the penetrating glance
which she cast upon him, her complete
knowledge of all that had passed in his mind
relative to Lady Hamlet Vernon, and he shrunk
confused from her gaze.
This, however, was neither a time nor place
adapted for explanations; and, indeed, to whom
was he to make them? To no one did he feel
responsible but to Adeline; to no one he felt
would they be satisfactory, save to Adeline. He
knew her mind was truth itself, and so utterly
incapable of deception, that she could not believe
that any one would deceive her; he determined
therefore to unbosom himself to her, and be forgiven.
With these feelings, which were rapid
and almost simultaneous in their effect, though
language is slow in expressing them, he
caught the sinking Adeline in his arms, and
lifted her inanimate form into the corridor,
where a seat being hastily taken from the box,
he supported her, kneeling by her side. At
this moment Mr. Foley appeared, breathless
with haste, bearing some water and a smelling-bottle,
which he proceeded to apply, whilst
Lady Delamere aided him in his efforts to restore
Adeline, and was assisted by several
of their acquaintance who were passing by.
Lord Albert could only partially be of use,
as one arm supported her; but with the other
he tenderly pressed her hand as he bathed it
in the water. Animation, after a few minutes,
returned; she opened her eyes, and gazed vacantly;
but in another moment her senses were
fully restored; and on recognizing Lord Albert,
she quickly closed her eyes again, and a sort
of convulsive throb seemed about to make her
relapse; but struggling to disengage her hand,
which he let drop with an expression of sorrow
and dismay, Lady Adeline made an effort
to recover herself; and half rising, she turned
to Lady Delamere, and said inarticulately,
"I should like, dear aunt, to be taken home."
"Stop, for heaven's sake," cried Lord Albert
D'Esterre, stepping forward, as if to catch her
tottering frame; "wait till you are more recovered."
"No," she said; but speaking still as if to
Lady Delamere, "I shall be better when I
am at home; dear aunt, let me go." Lady
Delamere, judging of Adeline's feelings by
her own observations of the circumstances
which she thought had caused her sudden
indisposition, said coolly, addressing Lord
Albert, "Thank you, Lord Albert, but Adeline
is the best judge of her own feelings."
Then turning to Mr. Foley, she asked him
if he had seen her servants. He answered
in the affirmative; and added, "the carriage
will be up by this time certainly."
"Then," rejoined Lady Delamere, "have
the goodness, Mr. Foley, to give your arm to
my niece;" and she continued, with marked
emphasis, "Adeline dear, I will support you
on the other side." It was impossible for
Lord Albert to mistake what this arrangement
implied; his whole frame was convulsed,
though he betrayed no gesture of suffering,
but stood rooted to the spot, as his eyes gazed
on her, walking away feebly between her two
supporters, without thinking of following her;
and then, by a sudden impulse, he rushed
after her, and arrived at the door just in time
to see Mr. Foley get into the carriage, after
having placed the ladies in safety, and to hear
the word "home" pronounced by the footman
as they drove from the door of the Opera-house.
He mechanically turned round, and with an
agitation of mind that allowed not of reflection,
returned to Lady Hamlet Vernon's box. He
sat down without speaking; and, gazing in
vacancy, remained for some time like one in a
deep reverie. Fortunately there was no one in
the box but themselves; and though Lady Hamlet
Vernon was quite aware of his situation, and
partly guessed the cause, she was too deeply
interested herself in the issue of the event to
press indiscreetly into his feelings at that moment,
but simply asked him "if he were not
well?" "Oh, quite well," he replied; "only
rather astonished.--It was,"--he stopped--seemed
to muse again, and then he added to
himself, "they went away together." Lady
Hamlet Vernon's eyes filled with tears--(tears
will come sometimes to some people when they
are called)--she said, in a low voice, "I must
always grieve for what gives you pain; but
I have thought"--she paused.--Lord Albert
fixed his eyes on her for an instant, as if he
would inquire, "what have you thought?" but
the latter, without appearing to deny that she
_had_ thought, at the same time added, in a
hurried tone,
"Yet, my dear Lord Albert, let not my
thoughts weigh with you; let not a momentary
appearance alone decide on any measure
which may influence your whole life; look
dispassionately on appearances; sound them,
sift them thoroughly, ere you allow yourself
to act upon them." There was a gentle reason
in these words, an expression of heart-felt
interest in the speaker, which at the
present instant was doubly efficacious in turning
the current of his thoughts and feelings in
favour of her who uttered them; and he gave
way to a warmth of expression in his reply which
was joy to her heart. Still she repressed the
triumph she felt at this impassioned answer;
and it was only when he handed her to her
carriage, that the pressure of her hand spoke
a tenderer language, which vibrated through
his frame.
END OF VOL. II.
/*
LONDON:
PRINTED BY J. L. COX, GREAT QUEEN STREET.
*/
THE
COURT JOURNAL.
The whole impression of this new and popular weekly
journal being now stamped, subscribers may receive and
transmit it to their friends, POSTAGE FREE, throughout
all parts of the kingdom.
THE PROPRIETORS of the COURT JOURNAL, with due
acknowledgments for the highly gratifying reception their
work has already met with, beg leave to point out to
readers in general the advantages of their publication in
its present improved form.
The occupations, engagements, and amusements of the
higher classes of society had long required a record; they
found it in the Court Journal. The fête champêtre, the
sumptuous banquet, the concert, the soirée, the ball, the
public and private habits of royal and noble life, those
habits which give the tone to manners throughout the
empire, were depicted with a freshness and accuracy
hitherto unattempted; and, in all instances, with the
most attentive avoidance of injury to personal feelings.
It may be easily imagined that those details could not
have been supplied from ordinary sources,--thus the
connexions of the Proprietors afforded them peculiar
opportunities, and many of the articles of the Court
Journal were contributed by individuals, whose rank and
fashion gave even a pledge at once for the good taste
and the truth of their descriptions.
But something more was still required, to realize the
original idea of the publication. It was hitherto the
Journal of an elevated but exclusive class; the purpose
was to render it available to all classes, retaining its
anecdote, pleasantry, and spirit of high life, to make it
the vehicle of intelligence of every interesting kind; the
companion not only of the boudoir but of the breakfast
table and the study,--a Journal in which not merely the
woman of fashion might find the round of her engagements
for the week brought gracefully before her eye;
but the politician, the student, and the various orders of
intelligent society might find the species of information
suited to their purposes; to make the Court Journal a
WEEKLY NEWSPAPER of the most improved and
valuable nature.
For this object a Stamp was necessary, and the Proprietors
did not hesitate to subject themselves to the
serious additional expense, that they might give the public
their paper in its complete state, feeling confident that
the claims of the work to great popularity and extensive
circulation would be duly estimated by the public at large.
THE COURT JOURNAL is regularly published every
SATURDAY MORNING on a handsome sheet of 16 quarto
pages, containing 48 columns, price 10d. and may consequently
be received on Sunday in all parts of the country.
Published for HENRY COLBURN, by W. Thomas, at the
office, 19, Catherine Street, Strand. Orders are received
by all Booksellers and Newsvenders throughout the kingdom;
and those who desire to become subscribers are
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or Newsman in their own immediate neighbourhood,
as the best mode of receiving it regularly.
N.B.--Advertisements or orders sent from the country
to the office must be accompanied by a reference for
payment in London.
TRANSCRIBER'S NOTE:
Obvious printer errors have been corrected. Otherwise, the author's
original spelling, punctuation and hyphenation have been left intact.
End of the Project Gutenberg EBook of The Exclusives (vol. 2 of 3), by
Charlotte Campbell Bury
*** | {
"redpajama_set_name": "RedPajamaBook"
} | 6,055 |
@implementation VNLPID
@end
| {
"redpajama_set_name": "RedPajamaGithub"
} | 9,363 |
Q: Can you tripleclick() a text to select a paragraph in python selenium? Can you tripleclick() a text to select the paragraph in python selenium? Or doubleclick()? And then copy-paste it with ActionChains? Without having to ctrl+a.
A: from selenium.webdriver.common.action_chains import ActionChains
actionChains = ActionChains(driver)
actionChains.double_click("element variable name").perform()
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 4,845 |
Creative Thinking Skills Result and Explanation: 43, I think I got this result because I can create creative solutions to problems.
Analytical Thinking Skills Result and Explanation: 44, I think I got this result because I am able to analyze problems and ideas and create a logical solution to problems.
Practical Thinking Skills Result and Explanation: 41, I think I got this result because I am able to create a realistic and ethical approach to a problem or conflict.
Self-Management Result and Explanation: 6, I think I got this score because I have self control and I am able to take control of my impulses.
Social-Awareness Result and Explanation: 4, I think I got this score because I am able to have empathy with others, but I still have difficulty being around others.
Self-Awareness Result and Explanation: 7, I think I got this score because I am able to understand my emotions and how they impact my school performance.
Relationship Management Result and Explanation: 3, I think I got this score because even though I can be a good leader in a group, I lack the ability to send clear and concise messages to other individuals.
Explanation- Lately I have been trying to overcome and let go of my anxieties and fears.
Explanation- I want to help people when I am a nurse and I want to be able to change the world for the better.
Explanation- I feel that happiness may be the meaning of existence and I want to pick a career that will keep me happy and challenged.
Logical-Mathematical Result and Explanation: 21, I am highly developed in this area. I think this is because I have always loved science and wondered how things work instead of just relying on facts.
Musical Result and Explanation: 15, I am moderately developed in this area. I think this is because music keeps my mind busy and singing along to it reduces my stress.
Verbal-Linguistic Result and Explanation: 14, I am moderately developed in this area. I think this is because I don't enjoy reading but I do like discussing topics that interest me and writing.
Interpersonal Result and Explanation: 14, I am moderately developed in this area. I think this is because I enjoy helping others and they sometimes come to me for help.
Visual-Spacial Result and Explanation: 19, I am moderately developed in this area. I think this is because I enjoy looking at graphs, diagrams, and models rather than just reading a textbook.
Intrapersonal Result and Explanation: 23, I am highly developed in this area. I think this is because I am always interested in self-improvement and I do not like working with other people on projects.
Bodily-Kinesthetic Result and Explanation: 21, I am highly developed in this area. I think this is because I feel like I get more out of doing hands on activities rather than reading or looking at things.
Naturalistic Result and Explanation: 13, I am underdeveloped in this area. I think this is because I don't often think about how things can fit into categories and I also don't think about how things can relate.
I like to work alone rather than in groups because I feel like I cannot trust anyone or depend on anyone to do their part of the project.
I have always learned and performed better in very cold weather. I often do homework outside in the winter. I also get headaches when it is too bright. It has to be quiet when I am studying or I cannot focus. I like to sit at a desk when I am doing homework and studying because it makes me feel more productive.
Visual Focus- I enjoy looking at diagrams, photographs, and videos so I can further understand the topic and get a more visual view on it.
Hands on Presentation- In classes like Lab and science, I need to physically do experiments and touch and look at models and specimens to fully grasp an idea.
Lecture, Verbal Focus- I prefer to learn with the professor lecturing the entire class. I often take notes before class and this helps me make side notes while the professor is speaking.
Detailed Presentation- I enjoy learning small details about topics. An example of this would be Cellular Respiration in Biology. I would have to know what it is but I also have to know the different types.
Adventurer Result and Explanation: 20, I have a moderate tendency in this dimension. I think this is because I am usually active, fast-paced, and open-minded.
Giver Result and Explanation: 16, I have a moderate tendency in this dimension. I think this is because I am trustworthy, honest, and I am passionate about helping others.
Organizer Result and Explanation: 20, I have a moderate tendency in this dimension. I think this is because I enjoy planning things and being systematic.
Thinker Result and Explanation: 24, I have a moderate tendency in this dimension. I think this is because I enjoy to think logically and realistically. | {
"redpajama_set_name": "RedPajamaC4"
} | 1,732 |
With the 'Extract from SMS message' option, the email address must be present in the message text of the received SMS message. None of the messages in your receive log contain an email address so Diafaan SMS Server forwards the received SMS messages to the default email address.
If you send an SMS message with a valid email address in it (for instance 'Test message to mail@domain.com'), the SMS message will be forwarded to the email address in the SMS message.
i check the config file and no email are present in the reply sms, how could i add an email ?
i try many combinaison by editing the sms template and nothing work for the moment.
According to the configuration file, your Email Connector is now set up to extract the email address from the incoming SMS message and send the message as an email to the email address in the SMS message, or to the default email address if Diafaan SMS Server does not find a valid email address in the SMS message.
If the messages are only sent to the default email address then Diafaan SMS Server cannot find the email addresses in the SMS messages that you receive. It is possible that there is no email message present in the SMS messages but it is also possible that the email address that you use in the SMS message is non-standard and is not recognized by Diafaan SMS Server. If you send me the receive log of Diafaan SMS Server (menu options 'File-Export-Receive log') I will have a look to see which of the two it is.
and a print screen of my setting.
Thanks for sending the configuration file. Your Email Connector is set up to send all incoming SMS messages from the GSM Modem Gateway to the default email address. This is the default setting of the Email Connector. You can change the default routing for received SMS messages in the 'Receive SMS' options of the Email Connector.
If you set the 'Extract from SMS message' option, the email Connector tries to find an email address in the SMS message and send the message to that email address or to the default email address if it can't find an email address in the SMS message.
If you check the 'Reply to email...' option, the Email Connector will try to send the received SMS messages to the 'from' email address that last sent an message to the mobile number where the SMS message came from. Otherwise it will send the message to the default email address (or to the email address in the SMS message).
The Email Connector in Diafaan SMS server can be set up to forward received SMS messages to a default email address, to an email address in the text of the SMS message or as a reply to an email address that previously sent an SMS message to the phone number of the sender of the reply (this last option is not 100% reliable due to the nature of SMS).
If the SMS message text does not contain a valid email address or if it is not recognized as a reply, the message will be forwarded to the default email address.
If you send the configuration file (menu options 'File-Export-Configuration') of Diafaan SMS Server to dms@diafaan.com I will have a look to see if I can find something wrong in the way the Email Connector is set up.
i setup forward received sms message to email , but look like all my sms go to the default adress when i reply.
in my sms , diaaffan is supossed to reply to the from adress. | {
"redpajama_set_name": "RedPajamaC4"
} | 286 |
Written for a Tumblr Prompt: Adam falls asleep on the couch at Monmouth where Ronan had planned on drunkenly crashing.
Pynch fics I've been writing and probably one Glue fic if I decide to put it up here. I previously put these on Tumblr, Fanfic, and Deviantart. Just reposting them here for my own amusement.
Adam gets cold during the night, and Ronan's shirt is there.
Ronan tries to break into Adam's apartment while drunk.
that explains how pretentious it is????
When Ronan Lynch first saw Adam Parrish, really saw him, his fingers tingled with something unknown and his skin felt too tight to bear.
Chapter 36 of The Raven Boys in Ronan's POV, basically the scene where Ronan hits Adam's dad. Kinda Pynch? I'm a big Pynch shipper, so there are hints.
Churches were made for confessions, but Ronan never expected it to be like this—half drunk with Adam on the roof of St Agnes in the middle of the night, his hands shaking at his sides, a memory of a kiss playing at his lips, his body aching more something more. With Adam, he was always wanting something more.
Ronan Lynch has been dreaming more, with schemes on his mind that he can't manifest into reality.
At least he tells himself he can't.
but are in currently different locations and cannot give consent to eachother u feel?
You know how you hear stories all the time about one twin knowing when something bad happens to the other? They can ~feel~ it? Its like that. Ronan somehow know when something is happening to thing's he brought from his dreams.
And alas the world (or perhaps it's Glendower, or maybe the God that Ronan worships) hears his struggle, and laughs. Gansey fears death, dreads it, obsesses over it, craves to know it like nothing else, just to conquer it. And it must be because of this, because Gansey can truly not think of any other reason, because of that interest that the world has decided Richard Gansey III will never meet death.
When Gansey dies, he wakes up.
Gansey could've swore his heart skipped a beat at that expression on her face. The dull ache was back and he automatically brought his hand up to his chest, knowing there was nothing he could do to stop it. He felt this ache when he was alone at night, missing her and waiting to hear her voice on the other end of the phone, and he felt it when she looked at him, her eyes bright and curious.
The gang find out Ronans name means little seal but are unprepared for the effect the nick name has on Ronan.
Adam and Ronan could make the impossible possible, but they couldn't tell each other one simple thing: I need you.
I'm shaking. I'm so angry. I'm so confused. I don't know what's happening right now.
"Don't wait up." I say into the air. I walk outside and get back in the BMW.
I drive to the Barnes.
Or, the one where Adam and Ronan kiss and Adam freaks out and leaves to figure out his feelings. Majority of the fic is Ronans narrative on trying to find Adam. | {
"redpajama_set_name": "RedPajamaC4"
} | 9,168 |
require 'jsduck/logger'
module JsDuck
module Process
# Reports bugs and problems in documentation
class Lint
attr_accessor :relations
def initialize(relations)
@relations = relations
end
# Runs the linter
def process_all!
warn_unnamed
warn_optional_params
warn_duplicate_params
warn_duplicate_members
warn_singleton_statics
warn_empty_enums
end
# print warning for each member or parameter with no name
def warn_unnamed
each_member do |member|
if !member[:name] || member[:name] == ""
warn(:name_missing, "Unnamed #{member[:tagname]}", member)
end
(member[:params] || []).each do |p|
if !p[:name] || p[:name] == ""
warn(:name_missing, "Unnamed parameter", member)
end
end
end
end
# print warning for each non-optional parameter that follows an optional parameter
def warn_optional_params
each_member do |member|
if member[:tagname] == :method
optional_found = false
member[:params].each do |p|
if optional_found && !p[:optional]
warn(:req_after_opt, "Optional param followed by regular param #{p[:name]}", member)
end
optional_found = optional_found || p[:optional]
end
end
end
end
# print warnings for duplicate parameter names
def warn_duplicate_params
each_member do |member|
params = {}
(member[:params] || []).each do |p|
if params[p[:name]]
warn(:dup_param, "Duplicate parameter name #{p[:name]}", member)
end
params[p[:name]] = true
end
end
end
# print warnings for duplicate member names
def warn_duplicate_members
@relations.each do |cls|
members = {:members => {}, :statics => {}}
cls.all_local_members.each do |m|
group = m[:static] ? :statics : :members
type = m[:tagname]
name = m[:name]
hash = members[group][type] || {}
if hash[name]
warn(:dup_member, "Duplicate #{type} name #{name}", hash[name])
warn(:dup_member, "Duplicate #{type} name #{name}", m)
end
hash[name] = m
members[group][type] = hash
end
end
end
# Print warnings for static members in singleton classes
def warn_singleton_statics
@relations.each do |cls|
if cls[:singleton]
cls.find_members({:local => true, :static => true}).each do |m|
warn(:sing_static, "Static members don't make sense in singleton class #{cls[:name]}", m)
end
end
end
end
# print warnings for enums with no values
def warn_empty_enums
@relations.each do |cls|
if cls[:enum] && cls[:members].length == 0
warn(:enum, "Enum #{cls[:name]} defined without values in it", cls)
end
end
end
# Loops through all members of all classes
def each_member(&block)
@relations.each {|cls| cls.all_local_members.each(&block) }
end
# Prints warning + filename and linenumber from doc-context
def warn(type, msg, member)
Logger.warn(type, msg, member[:files][0])
end
end
end
end
| {
"redpajama_set_name": "RedPajamaGithub"
} | 5,130 |
Grammostola gossei is een spinnensoort uit de familie van de vogelspinnen (Theraphosidae).
De wetenschappelijke naam van de soort werd in 1899 als Citharoscelus gossei gepubliceerd door Reginald Innes Pocock.
Vogelspinnen | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 1,923 |
Ukraine war Put Crypto on Putin's Table
Russian x Ukraine conflict Put Crypto on Putin's Table.
Russia attacked Ukraine on February 24, 2022, in a dramatic escalation of the Russo-Ukrainian War, which began in 2014. With more than 7.5 million Ukrainians fleeing the nation since the invasion, Europe's fastest-growing refugee crisis since World War II.
The invasion of Ukraine by Russia constituted an act of aggression that violated the United Nations Charter. In addition, Russia has been accused of committing war crimes and crimes against humanity, as well as conducting war in violation of international law by indiscriminately targeting densely populated areas and causing unnecessary and disproportionate harm to people.
Ukraine filed a case in the International Court of Justice (ICJ) accusing Russia of violating the 1948 Genocide Convention, which both Ukraine and Russia had joined, by inventing bogus genocide charges to justify the invasion. The International Association of Genocide Scholars backed Ukraine's request to the International Court of Justice (ICJ) to order Russia to stop its offensive in Ukraine.
International Sanctions during Russo-Ukrainian War
Following Russia's invasion of Ukraine in late February 2014, a large number of countries, notably the United States, Canada, and the European Union, implemented international sanctions against Russia and Crimea.
Individuals, corporations, and officials from Russia and Ukraine were targeted by sanctions imposed by the US, as well as other countries and international organizations. Russia retaliated by imposing sanctions on a number of countries, including a complete embargo on food imports from Australia, Canada, Norway, Japan, the United States, and the European Union.
Following Russia's invasion of Ukraine in Feb 2022, the US, the EU,and other countries imposed or considerably increased sanctions on Vladimir Putin and other government officials. Selected Russian banks were also shut off from SWIFT. The Russian financial crisis of 2022 was precipitated by the boycott of Russia and Belarus in 2022.
Russia Mulls Allowing Cryptocurrency for International Payments
Due to the threat that digital currencies pose to financial stability, the Bank of Russia has advocated prohibiting cryptocurrency trade and mining. The finance ministry is divided, and President Vladimir Putin has urged officials to reach an agreement.
The Finance Minister Anton Siluanov told state television channel Rossiya-24 that he believed issues could be resolved by the end of the year and a bill governing cryptocurrencies passed. He stated that the final decision would be made by the government.
Allowing crypto to be used as a means of international trade settlement would help Russia combat the effects of Western sanctions, which have hampered Russia's access to traditional cross-border payment systems, according to Chebeskov.
Russia will 'sooner or later' legalize Bitcoin and other cryptocurrencies for payment purposes, according to a ministry. Russia is moving forward with plans to create its own digital rouble. For years, Russian officials have claimed that cryptocurrency may be used to launder money or fund terrorists. Central Bank Governor Elvira Nabiullina had stated in a report released earlier this month that the bank cannot welcome cryptocurrency investments and advocated banning trading and mining. Russian cryptocurrency transactions total roughly $5 billion (€4.76 billion) every year.
What is the USDD coin?
what are the NFTs?
Secret Network, data privacy by default
BabyDoge: A deep dive into this coin
Quant Network: Interoperable Finance! | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 39 |
class AdminController < ApplicationController
http_basic_authenticate_with name: "cemi", password: "cemi85"
end | {
"redpajama_set_name": "RedPajamaGithub"
} | 3,794 |
Léopold Bernhard Bernstamm (20 April 1859 – 22 January 1939), also written as Léopold-Bernhard Bernstam, Léopold Bernard Bernstamm or Leopold Adolfovich Bernstam, was a Baltic German sculptor active in France and Russia. He was one of the official sculptors of the Musée Grévin.
Biography
Bernstamm was born in Riga, now Latvia, where he entered the studio of Prof. David Jensen at age 13, and at 14 entered the Imperial Academy of Fine Arts of Saint Petersburg, where he won several awards.
In the early 1880s he made about thirty busts of celebrated Russians including Fyodor Dostoyevsky (from a death mask, 1881), Denis Fonvizin, Aleksandr Ostrovsky (for the foyer of the Alexandrinsky Theater), and Mikhail Saltykov-Shchedrin (erected at the writer's grave in 1900). These busts established his reputation. He then spent 1884 in Rome and Florence, continuing his studies under a Professor Rivalti.
In 1885 he settled in Paris, often returning to Saint Petersburg. His sculptures of eminent Frenchmen soon made him famous, including portraits of François Coppée, Paul Déroulède, Gustave Flaubert, Ludovic Halévy, Ernest Renan, Victorien Sardou, Émile Zola, and Jean-Léon Gérôme. He also made portraits of Czar Nicholas II of Russia and members of the Imperial family (1896), Anton Rubinstein (1901), and Alexander Pushkin (1911).
His last work for Saint Petersburg was the bust of Czar Alexander III of Russia (erected in the Russian Museum garden, removed in 1918). All told, he sculpted approximately 300 portraits of Russian and European representatives of culture, science and politics, and sculpted some monuments. Bernstamm was made chevalier of the Légion d'honneur in 1891.
References
Further reading
Serge Bernstamm, Leopold Bernstamm. Sa vie – son oeuvre, L. Lapina & Cie, Paris. 1913 (reprinted 2017)
External links
Jewish Encyclopedia article
Saint Petersburg Encyclopedia article
Artnet entry
1859 births
1939 deaths
Artists from Riga
People from Kreis Riga
Latvian Jews
Latvian sculptors
20th-century Russian sculptors
20th-century Russian male artists
19th-century sculptors from the Russian Empire
19th-century male artists from the Russian Empire
Russian male sculptors
20th-century Latvian male artists
19th-century Latvian male artists | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 9,574 |
Q: Javascript for() loop in Drupal not working? I've just spent a long time trying to figure out why my Javascript for() loop won't work in a Drupal block, I feel like I've checked out the syntax - any idea why this isn't working?!
$(document).ready(function() {
var i=0;
while (i<=5)
{
alert(i);
i++;
}
});
That doesn't do anything - and also if I put something like this in- does not work either:
for (var i=0; i<31; i++){
alert(i);
}
Thanks!
A: None of the alerts will happen until the thread is done executing. By that time, i has exceeded your limit. This is a very commonly asked question. You need to learn about closures in JavaScript.
Here's a good overview. There are also many answers to this question in StackOverflow.
http://james.padolsey.com/javascript/closures-in-javascript/
From that article, this code:
for (var i = 0; i < 100; ++i) {
myElements[i].onclick = (function(n) {
return function() {
alert( 'You clicked on: ' + n );
};
})(i);
}
Which is similar to what you want.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 3,032 |
October 1, 2015 — Ariane 5 has been flawlessly launched from Kourou, French Guiana, for the 68th consecutive time, building on the European launcher's track record. Two satellites were launched with a total mass of more than 10 tonnes.
AccuWeather Global Weather Center, July 22, 2015 — AccuWeather, Inc., the global leader in weather information and digital media, today introduced AccUcast™, an exciting interactive crowdsourcing feature available in the AccuWeather universal iOS app where users can now share their local weather updates. | {
"redpajama_set_name": "RedPajamaC4"
} | 2,913 |
* Gestion compte utilisateur : envoi de photos, création d'albums
* Système de follow
* Système de messagerie interne
* Panneau d'administration
* Système de tags et de notation | {
"redpajama_set_name": "RedPajamaGithub"
} | 2,273 |
Well. The repair could be done a lot cheaper than what the dealer is quoting These modules can be had for under $100. Part number 13587704 shows as fitting the 2015 Sierra. I'm sure there is someone on this forum that can program the VIN number for you. eBay or some place like GM Parts Direct is where I'd go. I think even another dealer would probably program it for you. Just my opinion.
I currently have an Extang Trifecta on my 2015 All Terrain. I like the fact that it's easy to install and uninstall. I DO NOT like how it bows in at the front hinge. When I got the cover I sent Extang an email about the hinge pads not being installed properly. They were sitting on top of the weatherstripping creating a bulge in the cover. Waited a few days and never heard anything so I called. The guy goes through his emails and found it. I had attached photos of the pads and of how it bowed at the front hinge. He said that it was normal. He did send new pads for the the hinges. It looks almost like the cover is too tight at the front hinges. Not at all happy with it but I'm outside my 30 days with Autoanything and Extang doesn't act like it's a problem. I had another brand tri-fold on my 2011 Silverado and it wasn't this way and cost half as much. I just didn't like the way the rear latches were They were bad about scratching the bed rail covers up.
Ordered the Extang Trifecta for my 2015 All Terrain from Autoanything last night. I just hope it's as good as the reviews. | {
"redpajama_set_name": "RedPajamaC4"
} | 9,035 |
GlobalNxt University (formerly known as Global e-University) is a pioneering academic institution that delivers online graduate programs through a unique global classroom pedagogy. Bringing together over 75 distinguished faculty members from across 17 countries, GlobalNxt provides an interactive learning approach. Through a state-of-the-art online learning platform, the university offers students access to rigorous, individualised education anytime, anywhere.
The university's globally diverse student population is represented by over 72 different countries. In addition, through close industry partnerships, the university has talent development programs with over 100 multinational companies.
Courses offered by GlobalNxt University include Masters of Business Administration (MBA), Postgraduate Diploma of Business Administration and Master of Science in Information Technology Management. It also develops customised programs for organisations to meet their specific requirements. | {
"redpajama_set_name": "RedPajamaC4"
} | 3,768 |
Aquarium Advice - Aquarium Forum Community - Thermometer reading vs. Heater reading?
Thermometer reading vs. Heater reading?
I have a 55 gallon FW tank. Right now I have 2 kissing gourami, a featherfin catfish & an electric blue jack dempsey. Their temperature range is upper 70's.
I have 2 thermometers, one at each end of the tank. They both read around 82 degrees. My Marineland 300 watt heater is set to 76-77 degrees and the thermometer attached to it actually reads about 75 degrees.
Which reading do I take as fact? What's going on here?
Ya you could go that approach... turn your heater down a few degrees and monitor the temps closely.
The heater shouldn't be broken, it's only about a month old.
Well if the thermometers are correct & the heater's reading is wrong, then I should turn down the heater even more? Right now the heater is set to 76 & the thermometers read 81, so if I lower the heater to 74 or so, my temeprature should be about right.
As long as it's not the sticky thermometers that are on the outside of the tank I would go with the thermometers. I would turn it down and watch temps closely, but first we should see if maybe it's something else that is the problem.
How is your water circulation?
What do you have for lights/other things that could heat the tank?
Is your tank close to a heat vent or a baseboard heater?
What is the temp of the room?
Is the tank near a sunny window?
Is there anything else that might affect the temperature?
They are two regular glass thermometers.
I have undergravel filter trays with two powerheads attached to the up-tubes. I have the powerheads pointed to spray from the corners towards the center of the tank.
I have a few plants in the tank, just some low-mid light things, so I keep the hood lamp on for around 6-8 hours a day. It's just the basic Marineland hood and bulb to fit across the 4 ft length.
Vents and windows are not near it. Room temp is low 70's.
sadly it seems the quility of the Marineland heaters been slipping. i know of a bunch of people with problems with them. go with the glass thermometers.
But for now if you don't have the money (or don't want to spend it) I would go with mfdrookie561's suggestion and just turn the heater down 2-6 degrees till your thermometers read the temp that you would like in your tank.
Or just go to Walmart and pick up a thermometer for very cheap. The glass ones are about 1.99. | {
"redpajama_set_name": "RedPajamaC4"
} | 9,384 |
\section{Introduction}\label{sec: introduction}
Data-driven control has received increasing interest during the past few years. Specifically, this interest has been surging towards optimal control problems\cite{GB-VK-FP:19,ST-BR:19,FC-GB-FP:21,KZ-BH-TB:20}. The Linear Quadratic Gaussian (LQG) is one of the most fundamental optimal control problems, which deals with partially-observed linear dynamical systems in the presence of additive white gaussian noises \cite{KZ-JCD-KG:96}. When the system is known, the LQG problem enjoys an elegant closed-form solution obtained via the separation principle \cite[Theorem 14.7]{KZ-JCD-KG:96}. In the context of data-driven control, however, the LQG problem is less studied in the literature due to some major challenges: (i) the states of the system cannot be directly measured for learning purposes, (ii) the optimal policy is expressed in the dynamic controller form where it is not unique \cite{KZ-JCD-KG:96}, and (iii) the set of stabilizing controllers can be disconnected \cite{YZ-YT-NL:21}. On the other hand, the Linear Quadratic Regulator (LQR) optimal control problem has received more attention in the context of data-driven control \cite{GB-VK-FP:19,ST-BR:19,FC-GB-FP:21}. Some of the reasons that make the LQR problem attractive is that the optimal policy can be expressed as a static feedback gain and it is unique \cite[Theorem 14.2]{KZ-JCD-KG:96}. Moreover, the set of stabilizing feedback gains for the LQR problem is connected and the LQR cost function is gradient dominant \cite{MF-RG-SK-MM:18,HM-AZ-MS-MRJ:19}. These properties are useful for providing convergence guarantees for gradient-based methods for solving the LQR problem \cite{JB-AM-MF-MM:19,IF-BP:21} as well as for model-free policy optimization methods \cite{HM-AZ-MS-MRJ:21}. However, LQR controllers require measuring the full states, and are used in deterministic settings, which limits the use of LQR controllers in practical control applications.\\
In this paper, we make the LQG problem more accessible for data-driven methods. In particular, we show that the optimal LQG controller can be expressed as a static feedback gain by reformulation of the LQG problem in the space of input-output behaviors. Then, we highlight the advantages of having a static LQG gain in the context of data-driven control and gradient-based~algorithms.\\
\textbf{Relared work.} The LQG control problem has been studied extensively in the literature \cite{KZ-JCD-KG:96,DPB:01a}, where fundamental properties have been characterized, such as the existence of optimal solution, how to obtain it using separation principle \cite{KZ-JCD-KG:96}, and its lack of stability margin guarantees in closed-loop \cite{JCD:78}.
However, in the context of data-driven control, the LQR problem has received more attention than the LQG problem.
The landscape properties for the LQR problem with state-feedback control has been studied in \cite{MF-RG-SK-MM:18,HM-AZ-MS-MRJ:19},
which has paved the way for subsequent works investigating convergence properties of gradient methods for solving the LQR problem~\cite{JB-AM-MF-MM:19,IF-BP:21,HM-AZ-MS-MRJ:21}.
Recent studies have revisited the LQG problem in the context of data-driven methods
(e.g. \cite{SL-KA-BH-AA:20,LF-YZ-MK:20,YZ-LF-MK-NL:21}). In \cite{YZ-YT-NL:21}, the authors characterize the optimization landscape for the LQG problem, showing that the set of stabilizing dynamic controllers can be disconnected. In the context of data-driven control, the behavioral approach has
garnered much attention in recent years~\cite{JCW-PR-IM-BLMDM:05,CDP-PT:19,LF-BG-AM-GFT:21,VK-FP:21},
as it circumvents the need for state space representation.
Owing it this fact, it belongs in the same category as the
difference operator representation and ARMAX models~\cite[Sec. 2.3 and Sec. 7.4]{GCG-KSS:14},
and shares several connections with these classes of models. We refer the reader to \cite{IM-FD:21-survey} for a comprehensive overview of the behavioral approach.\\
Despite recent interest in the behavioral approach, a fundamental
understanding of the LQG problem from a behavioral perspective
is still lacking, and our work addresses this gap.
Different from the literature, our work seeks to characterize the optimal behavioral feedback
controllers for the LQG problem and to demonstrate their suitability for
data-driven control and gradient methods for controller design.
More specifically, we show that the optimal LQG controller can be expressed as a
static behavioral-feedback gain, which underlies its advantages
for developing data-driven methods to learn LQG controllers.
\textbf{Contributions.} This paper features three main contributions. First, we introduce equivalent representations for stochastic discrete-time, linear, time-invariant systems and the LQG optimal control problem in the behavioral space (Lemma~\ref{lemma: system in z} and Lemma~\ref{lemma: lqg in z}, respectively). Second, we show that, in the behavioral space, the optimal LQG controller can be expressed as a static behavioral-feedback gain, which can be solved for directly from the LQG problem represented in the behavioral space (Theorem \ref{thrm: solution of lqg in z}). Third, we highlight the advantages of having a static feedback LQG gain over a dynamic LQG controller in the context of data-driven control and gradient-based algorithms (section~\ref{sec: numerical methods}).\\
\textbf{Notation.} A Gaussian random variable $x$ with mean $\mu$ and
covariance $\Sigma$ is denoted as $x\sim\mathcal{N}(\mu,\Sigma)$. The
$n\times n$ identity matrix is denoted by $I_n$. The expectation
operator is denoted by $\mathbb{E}[\cdot]$. The spectral radius and
the trace of a square matrix $A$ are denoted by $\rho(A)$ and
$\Tr{A}$, respectively. A positive definite (semidefinite) matrix
$A$ is denoted as $A\succ 0$ ($A\succeq 0$). The Kronecker product is denoted
by $\otimes$, and vectorization operator is denoted by vec($\cdot$). The left (right) pseudo inverse of a tall (fat) matrix $A$ is denoted by $A^{\dagger}$
\begin{figure*}[!b]
\normalsize
\setcounter{MYtempeqncnt}{0
\setcounter{equation}{4}
\hrulefill
\smallskip
\begin{align} \label{eq: z dynamics}
\begin{aligned}
\underbrace{\left[ \begin{smallarray}{c}
u(t-n+1) \\
\vdots \\
u(t-1)\\
u(t) \\
\hdashline[2pt/2pt]
y(t-n+1)\\
\vdots \\
y(t)\\
y(t+1) \\
\hdashline[2pt/0pt]
w(t-n+1) \\
\vdots \\
w(t-1) \\
w(t) \\
\hdashline[2pt/2pt]
v(t-n+1)\\
\vdots \\
v(t)\\
v(t+1)
\end{smallarray} \right]}_{z(t+1)}=&
\underbrace{\left[ \begin{smallarray}{ccccc;{2pt/2pt}ccccc;{2pt/0pt}ccccc;{2pt/2pt}ccccc}
0 & I & 0 & \cdots & 0 & 0 & 0 & 0& \cdots & 0 & 0 & 0 & 0& \cdots & 0 & 0 & 0 & 0& \cdots & 0\\
\vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \vdots & \ddots & \vdots \\
0 & 0 & 0 & \cdots & I & 0 & 0 & 0& \cdots & 0 & 0 & 0 & 0& \cdots & 0 & 0 & 0 & 0& \cdots & 0\\
0 & 0 & 0 & \cdots & 0 & 0 & 0 & 0& \cdots & 0 & 0 & 0 & 0& \cdots & 0 & 0 & 0 & 0& \cdots & 0\\
\hdashline[2pt/2pt]
0 & 0 & 0 & \cdots & 0 & 0 & I & 0 & \cdots & 0 & 0 & 0 & 0& \cdots & 0 & 0 & 0 & 0& \cdots & 0\\
\vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \vdots & \ddots & \vdots \\
0 & 0 & 0& \cdots & 0 & 0 & 0 & 0 & \cdots & I & 0 & 0 & 0& \cdots & 0 & 0 & 0 & 0& \cdots & 0\\
\multicolumn{5}{c;{2pt/2pt}}{\mathcal{A}_u} & \multicolumn{5}{c;{2pt/0pt}}{\mathcal{A}_y} & \multicolumn{5}{c;{2pt/2pt}}{\mathcal{A}_w} & \multicolumn{5}{c}{\mathcal{A}_v}\\
\hdashline[2pt/0pt]
0 & 0 & 0 & \cdots & 0 & 0 & 0 & 0 & \cdots & 0 & 0 & I & 0 & \cdots & 0 & 0 & 0 & 0& \cdots & 0\\
\vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \vdots & \ddots & \vdots \\
0 & 0 & 0& \cdots & 0 & 0 & 0 & 0 & \cdots & 0 & 0 & 0 & 0 & \cdots & I & 0 & 0 & 0& \cdots & 0 \\
0 & 0 & 0& \cdots & 0 & 0 & 0 & 0 & \cdots & 0 & 0 & 0 & 0 & \cdots & 0 & 0 & 0 & 0& \cdots & 0 \\
\hdashline[2pt/2pt]
0 & 0 & 0 & \cdots & 0 & 0 & 0 & 0 & \cdots & 0 & 0 & 0 & 0 & \cdots & 0 & 0 & I & 0 & \cdots & 0\\
\vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \vdots & \ddots & \vdots\\
0 & 0 & 0 & \cdots & 0 & 0 & 0 & 0& \cdots & 0 & 0 & 0 & 0 & \cdots & 0 & 0 & 0 & 0 & \cdots & I \\
0 & 0 & 0 & \cdots & 0 & 0 & 0 & 0& \cdots & 0 & 0 & 0 & 0 & \cdots & 0 & 0 & 0 & 0 & \cdots & 0
\end{smallarray} \right]}_{\mathcal{A}}
\underbrace{\left[ \begin{smallarray}{c}
u(t-n) \\
\vdots \\
u(t-2) \\
u(t-1) \\
\hdashline[2pt/2pt]
y(t-n)\\
\vdots \\
y(t-1)\\
y(t) \\
\hdashline[2pt/0pt]
w(t-n) \\
\vdots \\
w(t-2) \\
w(t-1) \\
\hdashline[2pt/2pt]
v(t-n)\\
\vdots \\
v(t-1)\\
v(t)
\end{smallarray} \right]}_{z(t)
+\underbrace{\left[ \begin{smallarray}{c;{2pt/0pt}c;{2pt/0pt}c}
0 & 0 & 0\\
\vdots & \vdots & \vdots \\
0 & 0 & 0\\
I & 0 & 0 \\
\hdashline[2pt/2pt]
0 & 0 & 0\\
\vdots & \vdots & \vdots \\
0 & 0 & 0\\
CB & C & I \\
\hdashline[2pt/0pt]
0 & 0 & 0\\
\vdots & \vdots & \vdots \\
0 & 0 & 0\\
0 & I & 0\\
\hdashline[2pt/2pt]
0 & 0 & 0\\
\vdots & \vdots & \vdots \\
0 & 0 & 0\\
0 & 0 & I
\end{smallarray} \right]}_{\left[ \begin{smallarray}{c;{1pt/0pt}c;{1pt/0pt}c}
\mathcal{B}_u & \mathcal{B}_w & \mathcal{B}_v
\end{smallarray}\right]}
\left[ \begin{smallarray}{c}
u(t)\\
w(t)\\
v(t+1)
\end{smallarray} \right],\\
y_z(t)=&\underbrace{\left[\begin{smallarray}{cccc;{2pt/0pt}cccc}
I & 0 & \cdots & 0 & 0 & 0 & \cdots & 0\\
0 & I & \cdots & 0 & 0 & 0 & \cdots & 0\\
\vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \ddots & \vdots \\
0 & 0 & \cdots & I & 0 & 0 & \cdots & 0
\end{smallarray}\right]}_{\mathcal{C}} z(t)
\end{aligned}
\end{align}
\setcounter{equation}{\value{MYtempeqncnt}}
\end{figure*}
\section{Problem setup and main results}\label{sec: setup}
Consider the discrete-time, linear, time-invariant system
\begin{align}
\begin{aligned}\label{eq: system in x}
x(t+1) &= Ax(t) + Bu(t)+w(t), \\
y(t) &= Cx(t) + v(t), \qquad t\geq 0,
\end{aligned}
\end{align}
where $x(t)\in\mathbb{R}^{n}$ denotes the state, $u(t)\in \mathbb{R}^{m}$ the
control input, $y(t)\in\mathbb{R}^{q}$ the measured output, $w(t)$ the
process noise, and $v(t)$ the measurement noise at time $t$. We assume that
$w(t)\sim\mathcal{N}(0,Q_w)$, with $Q_w\succeq 0$, $v(t)\sim\mathcal{N}(0,R_v)$,
with $R_v\succ 0$, and $x(0)\sim\mathcal{N}(0,\Sigma_0)$, with
$\Sigma_0 \succeq 0$, are independent of each other at all
times. For the system \eqref{eq: system in x}, the Linear Quadratic Gaussian
(LQG) problem asks to find a control input that minimizes the cost
\begin{align}\label{eq: LQG cost}
\mathcal{J}\triangleq\lim_{T\rightarrow \infty}\mathbb{E} \left[
\frac{1}{T}\Big(\sum_{t=0}^{T-1}x(t)^{\T}Q_x x(t) +
u(t)^{\T} R_u u(t)\Big) \right] ,
\end{align}
where $Q_x \succeq 0$ and $R_u \succ 0$ are weighing matrices of~appropriate dimension. We assume that $(A,B)$ and $(A,Q_w^{1/2})$ are controllable, and $(A,C)$ and $(A,Q_x^{1/2})$ are observable. As a classic result \cite{KZ-JCD-KG:96}, the optimal control input that solves the LQG problem can be generated by a dynamic controller of the form
\begin{align}\label{eq: compensator}
\begin{aligned}
x_c(t+1) &= Ex_c(t)+Fy(t), \\
u(t) &= Gx_c(t) + Hy(t),
\end{aligned}
\end{align}
where $x_c(t)$ denotes the state at time $t$, and $E\in \mathbb{R}^{n\times n}$,
$F\in \mathbb{R}^{n\times q}$, $G\in \mathbb{R}^{m\times n}$, and
$H\in \mathbb{R}^{m\times p}$ are the dynamic, input, output and
feedthrough matrices of the compensator, respectively.
The optimal LQG controller can be
conveniently obtained using the separation principle by concatenating
the Kalman filter for \eqref{eq: system in x} with the (static)
controller that solves the Linear Quadratic Regulator problem for
\eqref{eq: system in x} with weight matrices $Q_x$ and
$R_u$. Specifically, after some manipulation, the optimal input that
solves the LQG problem reads as \eqref{eq: compensator}, we refer the reader to Appendix \ref{app: optimal LQG controller} for the details.\\
In what follows, we will make use of an equivalent representation of
the system \eqref{eq: system in x}. To this aim, let
\begin{align}\label{eq: z}
z(t)\triangleq[U(t-1)^{\T},Y(t)^{\T},
W(t-1)^{\T}, V(t)^{\T}]^{\T} ,
\end{align}
where
\begin{align}
U(t-1)&\triangleq\left[u(t-n)^{\T}, \cdots, u(t-1)^{\T}\right]^{\T}, \nonumber\\
Y(t)&\triangleq\left[y(t-n)^{\T}, \cdots, y(t)^{\T}\right]^{\T}, \nonumber\\
W(t-1)&\triangleq\left[w(t-n)^{\T}, \cdots, w(t-1)^{\T}\right]^{\T}, \nonumber\\
V(t)&\triangleq\left[v(t-n)^{\T}, \cdots, v(t)^{\T}\right]^{\T}. \nonumber
\end{align}
We can write an equivalent representation of \eqref{eq: system in x} in the behavioral space $z$ as \eqref{eq: z dynamics} (see Appendix \ref{app: behavioral dynamics} for the derivation).
In fact, given a sequence of control inputs and noise values, the state $z$ contains the system output $y$ over time, and can be used to reconstruct the exact value of the system state~$x$. This also implies that a controller for the system \eqref{eq: system in x} can
equivalently be designed using the dynamics \eqref{eq: z dynamics}. In
fact, we show that any \emph{dynamic} controller for
\eqref{eq: system in x} can be equivalently represented as a
\emph{static} controller for \eqref{eq: z dynamics}, see Appendix \ref{app: dynamic to static controller}. Next, we reformulate the LQG problem \eqref{eq: LQG cost} for the behavioral dynamics \eqref{eq: z dynamics} and characterize its optimal solution. The LQG problem \eqref{eq: LQG cost} can be equivalently written in the behavioral space as:
\setcounter{equation}{5}
\begin{align}\label{eq: LQG cost z}
\mathcal{J}_z\triangleq\lim_{T\rightarrow \infty}\mathbb{E} \left[
\frac{1}{T}\Big(\sum_{t=n}^{T-1}z(t)^{\T}Q_z z(t) +
u(t)^{\T} R_u u(t)\Big) \right],
\end{align}
%
subject to \eqref{eq: z dynamics}, where $Q_z$ is presented in Appendix \ref{app: LQG problem in the behavioral space} along with the derivation of \eqref{eq: LQG cost z}, and $R_u$ is as in \eqref{eq: LQG cost}.
The solution to the LQG problem in the behavioral space is given by a static controller, which we characterize next.
\begin{theorem}{\bf \emph{(Behavioral solution to the LQG
problem)}}\label{thrm: solution of lqg in z}
Let $u^*$ be the minimizer of \eqref{eq: LQG cost z} subject to
\eqref{eq: z dynamics}. Then,
\begin{align}\label{eq: behavioral LQG gain}
u^* (t) =& \underbrace{-\left(R_u+\mathcal{B}_u^{\T}M
\mathcal{B}_u\right)^{-1} \mathcal{B}_u^{\T}M \mathcal{A} P
\mathcal{C}^{\T}\left( \mathcal{C} P
\mathcal{C}^{\T}\right)^{\dagger}}_{\mathcal K} y_z (t)
\end{align}
where $M\succeq 0$ and $P \succeq 0$ are the unique solutions of the
following coupled Riccati equations:
\begin{align*}
M &= \mathcal{A}^{\T}M \mathcal{A} -\mathcal{A}^{\T}M\mathcal{B}_u S_M
\mathcal{B}_u^{\T}M \mathcal{A} + Q_z\\
&+\left(I-P\mathcal{C}^{\T}S_P\mathcal{C}\right)^{\T}\mathcal{A}^{\T}M\mathcal{B}_u
S_M\mathcal{B}_u^{\T} M
\mathcal{A}\left(I-P\mathcal{C}^{\T}S_P\mathcal{C} \right)\\
P &=\mathcal{A}P \mathcal{A}^{\T} - \mathcal{A}P\mathcal{C}^{\T}S_P
\mathcal{C}P \mathcal{A}^{\T}+\mathcal{B}_w Q_w\mathcal{B}_w^{\T}
+ \mathcal{B}_v R_v\mathcal{B}_v^{\T}\\
+&\left(I-M\mathcal{B}_u S_M
\mathcal{B}_u^{\T}\right)^{\T}\mathcal{A}P\mathcal{C}^{\T}S_P\mathcal{C}P
\mathcal{A}^{\T} \left(I-M\mathcal{B}_uS_M\mathcal{B}_u^{\T}\right)
\end{align*}
with $S_M\triangleq(R_u+\mathcal{B}_u^{\T}M\mathcal{B}_u)^{-1}$ and
$S_P\triangleq(\mathcal{C}P\mathcal{C}^{\T})^{\dagger}$.\oprocend
\end{theorem}
\smallskip
The proof of Theorem \ref{thrm: solution of lqg in z} is postponed to Appendix \ref{app: proof of theorem}. The gain $\mathcal{K}$ is not unique since $\mathcal{C}P\mathcal{C}^{\T}$ is generally not invertible. In some cases, such as with SISO systems, the gain $\mathcal{K}$ becomes unique, which gives solving for the optimal LQG controller in the behavioral space an advantage over solving for it in the state space. The issue of non-uniqueness of $\mathcal{K}$ stems from the fact that $y_z$ has components that are dependent on each other, which makes the left kernel of $\mathcal{C}P\mathcal{C}^{\T}$ non-empty. We can avoid this issue by carefully choosing the time window of $U$ and $Y$ that form the behavioral space in \eqref{eq: z}, but we leave this aspect for future work. Note that, solving the coupled Riccati equations that characterize the LQG gain in Theorem \ref{thrm: solution of lqg in z} can be challenging. One method to solve for the LQG gain is to solve for the LQR and the Kalman gains, then use \eqref{eq: state space LQG} and Lemma \ref{lemma: static controller in z}.
\begin{example}{\bf \emph{(LQG controller in the behavioral space)}} \label{ex: example_1}
Consider the system \eqref{eq: system in x} with $A=1.1$, $B=1$, $C=1$, $Q_w=0.5$, and $R_v=0.8$. Also, consider the optimal control problem \eqref{eq: LQG cost} with $Q_x=R_u=1$. The Kalman and the LQR gains are $\subscr{K}{kf}=0.5474$ and $\subscr{K}{lqr}=0.7034$, respectively, which can be written as \eqref{eq: compensator} using \eqref{eq: state space LQG} with $E=0.1716$, $F=0.0973$, $G=-0.7034$, and $H=-0.3991$. Using \eqref{eq: z}, we define the behavioral space as $z(t)\triangleq\left[u(t-1), y(t-1), y(t),w(t-1),v(t-1),v(t)\right]^{\T}$ for $t\geq 1$. Using Lemma \ref{lemma: system in z}, we write the equivalent dynamics of \eqref{eq: system in x} in the behavioral space as \eqref{eq: z dynamics} with $\mathcal{A}_u=0.4977$, $\mathcal{A}_y=\begin{bmatrix}0.5475 & 0.6023 \end{bmatrix}$, $\mathcal{A}_w=0.4977$, and $\mathcal{A}_y=\begin{bmatrix}-0.5475 & -0.6023 \end{bmatrix}$. Using Theorem \ref{thrm: solution of lqg in z}, the LQG gain is $\mathcal{K}=[0.1716,0,-0.3991]$. Fig. \ref{fig: example_1}(a) shows the free response of \eqref{eq: system in x} and \eqref{eq: z dynamics} with equal initial conditions. Fig. \ref{fig: example_1}(b) shows the response of \eqref{eq: system in x} and \eqref{eq: z dynamics} to the LQG controllers \eqref{eq: state space LQG} and \eqref{eq: behavioral LQG gain}, respectively.\oprocend
\end{example}
\begin{figure}[!t]
\centering
\includegraphics[width=1\columnwidth,trim={0cm 0cm 0cm
0cm},clip]{example_1}
\caption{This figure shows the free response and the LQG feedback response of \eqref{eq: system in x} and \eqref{eq: z dynamics} for the setting defined in Example \ref{ex: example_1}. In both panels, the solid blue line and the dashed red line represent the output of \eqref{eq: system in x} and the output of \eqref{eq: z dynamics} that corresponds to $y(t)$, respectively. Panel (a) shows the free response of \eqref{eq: system in x} and \eqref{eq: z dynamics}, we observe that the response of both systems are equal, which agrees with Lemma \ref{lemma: system in z}. Panel (b) shows the feedback response of \eqref{eq: system in x} and \eqref{eq: z dynamics} to the LQG controller \eqref{eq: state space LQG} and the behavioral LQG controller in Theorem \ref{thrm: solution of lqg in z}, respectively. We observe that both systems have equal responses, which agrees with Lemma \ref{lemma: lqg in z} and Theorem \ref{thrm: solution of lqg in z}. Notice that the response of \eqref{eq: z dynamics} starts at time $t=n=1$ since we have to wait $n=1$ time steps in order to get the equivalent initial condition for \eqref{eq: z dynamics}.}
\label{fig: example_1}
\end{figure}
\section{Implications of behavioral representation in numerical methods} \label{sec: numerical methods}
In this section, we highlight some implications of our behavioral representation and results. In particular, we provide an analysis of learning the LQG controller from finite expert demonstration, and an analysis of solving for the behavioral LQG gain via a gradient descent method. First, we present the following Lemma regarding the sparsity of the LQG gain in \eqref{eq: behavioral LQG gain}, which we use in our subsequent analysis.
%
\begin{lemma}{\bf \emph{(Sparsity of the optimal LQG gain)}}\label{lemma: sparsity of the LQG gain}
Consider the LQG gain written in the behavioral space as
\begin{align}\label{eq: lqg controller rewritten}
u(t)=\left[ \begin{array}{ccc}
\mathcal{K}_1 & \mathcal{K}_{2} & \mathcal{K}_{3}
\end{array}\right]
\left[\begin{array}{c}
U(t-1)\\
y(t-n)\\
\overline{Y}(t)
\end{array}\right],
\end{align}
where $\overline{Y}(t)\triangleq \left[y(t-n+1)^{\T}, \cdots, y(t)^{\T}\right]^{\T}$. Then $\mathcal{K}_{2}=0$.~\oprocend
\end{lemma}
A proof of Lemma \ref{lemma: sparsity of the LQG gain} is presented in Appendix \ref{app: sparsity of the LQG gain}.
\subsection{Learning LQG controller from expert demonstrations}
Consider the system \eqref{eq: system in x}, assume that the system is stabilized by an expert that uses an LQG controller. We also assume that the system and the noise statistics are unknown. Our objective is to learn the LQG controller from finite expert demonstrations, which are composed of input and output data. In the behavioral representation, this boils down to learning the gain $\mathcal{K}$ of the subspace $u(t)=\mathcal{K}y_z(t)$ for $t\geq n$. Using Lemma \ref{lemma: sparsity of the LQG gain}, we only need to learn $\mathcal{K}_1$ and $\mathcal{K}_3$, which are obtained as $\left[ \mathcal{K}_1 \ \mathcal{K}_3 \right]=U_N Y_N^{\dagger} + \mathcal{K}_{\text{null}}$, where
\begin{align}\label{eq: expert data}
\begin{split}
U_N&\triangleq
\begin{bmatrix}
u(t) & \cdots & u(t+k-1)
\end{bmatrix},\\
Y_N& \triangleq
\begin{bmatrix}
u(t-n) & \cdots & u(t-n+k-1)\\
\vdots & \ddots & \vdots \\
u(t-1) & \cdots & u(t-2+k)\\
y(t-n+1) & \cdots & y(t-n+k) \\
\vdots & \ddots & \vdots \\
y(t) & \cdots & y(t-1+k)
\end{bmatrix},
\end{split}
\end{align}
for $t\geq n$, where $k$ is the number of columns, and $\mathcal{K}_{\text{null}}$ is any matrix with appropriate dimension whose rows belong to the left null space of $Y_N$. Note that $\mathcal{K}_{\text{null}}$ will disappear when multiplied by the feedback $y_z(t)$, i.e., $\mathcal{K}_{\text{null}}y_z(t)=0 $. Therefore, without loss of generality, we set $\mathcal{K}_{\text{null}}=0$.
\begin{lemma}{\bf \emph{(Sufficient number of expert data to compute the optimal LQG gain)}}\label{lemma: number of expert data}
Consider input and output expert samples $U=[u(t),\cdots,u(t+N-1)]$ and $Y=[y(t),\cdots,y(t+N-1)]$
generated by LQG controller to stabilize system \eqref{eq: system in x}, such that $U$ is full-rank.
Then, $N=n+nm+np$ expert samples are sufficient to compute the LQG gain $\mathcal{K}$.\oprocend
\end{lemma}
\smallskip
A proof of Lemma \ref{lemma: number of expert data} is presented in Appendix \ref{app: number of expert data}. We note that the rank condition on the input matrix~$U$ in the statement of Lemma~\ref{lemma: number of expert data}
is a reasonable assumption owing to the fact that system~\eqref{eq: system in x} is driven by i.i.d. process noise~$w$ and that
the measurement noise~$v$ is also i.i.d.
Furthermore, note that we can learn the dynamic controller matrices $E$, $F$, $G$, and $H$ in \eqref{eq: compensator} (up to a similarity transformation) using subspace identification methods for deterministic systems (see \cite{PVO-BDM:96}) with $U$ and $Y$ treated as the output and input signals to \eqref{eq: compensator}, respectively. Using \cite[Theorem 2]{PVO-BDM:96}, we need at least $N= 2(n+1)(m+p+1)-1$ expert samples to learn \eqref{eq: compensator}, which is more than the sufficient number of expert samples to learn $\mathcal{K}$ (Lemma \ref{lemma: number of expert data}).
\begin{example}{\bf \emph{(Learning LQG controller from expert data)}} \label{ex: example_3}
Consider the system in Example \ref{ex: example_1} where the system dynamics and the noise statistics are assumed to be unknown. The system is driven by an expert that uses an LQG policy. According to Lemma \ref{lemma: number of expert data}, we collect $N=n+nm+np=3$ expert input/output samples to form the data matrices
\begin{align*}
U_N=\begin{bmatrix}
-0.2269 & -0.1231
\end{bmatrix},\quad
Y_N=\begin{bmatrix}
1.7878 & -0.2269\\
1.3371 & 0.211
\end{bmatrix}.
\end{align*}
Using the data, we obtain $[\mathcal{K}_1 \ \mathcal{K}_3]=[0.1716 \ -0.3991]$~with $\mathcal{K}_{\text{null}}=0$, which matches the LQG gain in Example \ref{ex: example_1}.\oprocend
\end{example}
\subsection{Gradient descent in the behavioral space}
In this section, we use gradient descent to solve for $\mathcal{K}$:
\begin{align}\label{eq: grad descent}
\mathcal{K}^{(i+1)}=\mathcal{K}^{(i)} - \alpha^{(i)} \nabla\mathcal{J}_z(\mathcal{K}^{(i)})
\quad \text{for } i=0,1,2, \cdots
\end{align}
where the index $i$ refers to the iteration number, $\alpha^{(i)}$ is the step size at iteration $i$, and $\nabla\mathcal{J}_z(\mathcal{K}^{(i)})$ is computed using \eqref{eq: derivative of Jz}. We initialize the gradient descent method with a stabilizing gain $\mathcal{K}^{(0)}$. We determine the step size $\alpha^{(i)}$ by the Armijo rule \cite[Chapter $1.3$]{DPB:95B}: initialize $\alpha^{(0)}=1$, repeat $\alpha^{(i)}=\beta \alpha^{(i)} $ until
\begin{align*}
\mathcal{J}_z(\mathcal{K}^{(i+1)})\leq \mathcal{J}_z(\mathcal{K}^{(i)}) -\sigma\alpha^{(i)} \left\| \nabla\mathcal{J}_z(\mathcal{K}^{(i)})\right\|_{\text{F}}^2
\end{align*}
is satisfied, with $\beta,\sigma \in (0,1)$.
\begin{example}{\bf \emph{(Gradient descent)}} \label{ex: example_4}
We consider the example in \cite{JCD:78} discretized with sampling time $T_s=0.4$,
\begin{align*}
A&=\begin{bmatrix}
1.4918 & 0.5967 \\
0 & 1.4918
\end{bmatrix}, \
B=\begin{bmatrix}
0.1049\\
0.4918
\end{bmatrix},\
C=\begin{bmatrix}
1 & 0
\end{bmatrix},\\
Q_w&=\begin{bmatrix}
4.6477 & 3.7575 \\
3.7575 & 3.0639
\end{bmatrix},\quad
Q_x=\begin{bmatrix}
3.0639 & 3.7575 \\
3.7575 & 4.6477
\end{bmatrix},\quad
\end{align*}
$R_v=2.5$ and $R_u=0.5966$. The LQG gain from Theorem \ref{thrm: solution of lqg in z} is $\mathcal{K}=[-0.0366, -0.103,0,5.8461, -4.7434]$. Using Lemma \ref{lemma: sparsity of the LQG gain}, we only need to do the search over $\mathcal{K}_1$ and $\mathcal{K}_3$ since $\mathcal{K}_2=0$. We use gradient descent in \eqref{eq: grad descent} to solve for the LQG gain. We choose a stabilizing initial gain $\mathcal{K}^{(0)}$ that randomly place the closed-loop eigenvalues within $[0.45,0.92]$. We use the Armijo rule to compute the step size with $\alpha^{(0)}=1$, $\beta=0.8$, and $\sigma=0.7$. We set the stopping criteria to be when the gradient vanishes or when the maximum number of iterations is reached (in this example we set it to $15000$ iterations). For numerical comparison, we use gradient descent to solve for the optimal LQG dynamic controller in the form of \eqref{eq: compensator} as in \cite{YZ-YT-NL:21}, where we optimize the LQG cost \eqref{eq: LQG cost} and apply the gradient search over the control parameters $E$, $F$, $G$, and $H$.{\footnote{In \cite{YZ-YT-NL:21}, $H=0$ since it is assumed that the control input $u(t)$ at time $t$ depends on the history $\{u(0),\cdots,u(t-1),y(0),\cdots,y(t-1)\}$. In this paper, $u(t)$ depends also on $y(t)$, therefore $H$ is nonzero (see Appendix~\ref{app: optimal LQG controller}). We computed the gradient of $\mathcal{J}$ w.r.t. the controller matrices $E$, $F$, $G$ and $H$ as in \cite{YZ-YT-NL:21} adapted to the case where $H$ is nonzero. We have not included the derivations in this paper due to space constraint.}} Fig. \ref{fig: example_4} shows the convergence performance of the gradient descent for different initial conditions. We observe that the gradient descent over $\mathcal{K}$ in Fig. \ref{fig: example_4}(a) converges to $\mathcal{K}^*=[-0.0366, -0.1030,0,5.8460, -4.7434]$ before reaching the maximum number iterations for different initial conditions. Starting from initial conditions equivalent to the ones in Fig. \ref{fig: example_4}(a), the gradient descent over the controller matrices $E$, $F$, $G$ and $H$ in Fig. \ref{fig: example_4}(b) did not converge within~$15000$~iterations.~\oprocend
\end{example}
\begin{figure}[!t]
\centering
\includegraphics[width=1\columnwidth,trim={0cm 0cm 0cm
0cm},clip]{example_4}
\caption{This figure shows the convergence performance (measured by the suboptimality gap) of the gradient descent applied to the system in Example \ref{ex: example_4}. The solid blue line, dashed red line and the dash-dotted green line represent different initial conditions, respectively. Panels (a) and (b) shows the convergence performance of the gradient descent over $\mathcal{K}$ and the gradient descent over over the controller matrices $E$, $F$, $G$ and $H$,~respectively.}
\label{fig: example_4}
\end{figure}
\section{Conclusion and future work}
In this work, we introduced a behavioral space which consists of a window of input and output measurements augmented with a window of the noise history. Then, we derived equivalent representations for discrete-time, linear, time-invariant systems and the LQG problem in the behavioral space. After that, we showed that the optimal LQG controller can be expressed as a static behavioral-feedback gain, which can be solved for directly from the LQG problem in the behavioral space.
Finally, we highlighted the advantages of having a static LQG gain over a dynamic LQG controller in the context of data-driven control and gradient-based algorithms,
which arise from the fact that the behavioral approach circumvents the need for a state space representation and the fact that
the optimal behavioral feedback is a static gain.
There still remain several unexplored questions, including the investigation of the optimization landscape of the LQG problem in the behavioral space, which will pave the way for an improved understanding of the convergence properties of data-driven and gradient algorithms, as well as, for investigating the uniqueness of the optimal LQG gain.
\bibliographystyle{unsrt}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 5,081 |
Jailed Iranian Rapper Reportedly Attempts Suicide
Jailed Iranian dissident rapper Saman Yasin Seydi, who was sentenced to death last month, has attempted suicide in prison, an informed source has told RFE/RL's Radio Farda.
Saman Yasin, who is in the Rajaei-Shahr prison near the Iranian capital of Tehran, tried to take his own life on the evening of December 20 by taking a large number of pills, the source said, adding that the rapper was returned to his prison ward after leaving the prison hospital, where he had his stomach pumped.
A rapper from Kermanshah Province -- a northwestern region with a significant Kurdish population and which has been a focus of the government crackdown -- has been accused of acting against the country's security and "waging war against God." He was sentenced to death on November 7.
The source told Radio Farda that the 24-year-old rapper was denied the right to a lawyer during his interrogation and court sessions, and that the court forced him to accept a public defender.
Iran's judiciary often forces such a scenario, as public defenders rarely have enough time to prepare for a case while some have been known to take the state's side.
The Kurdistan Human Rights Network has said that since his arrest, Saman Yasin has been subjected to "severe" physical and mental torture, including being kept in solitary confinement, being kept in a morgue, being severely abused and thrown from a height, and being forced to make confessions under the pressure of security interrogators.
At least 12 men have been sentenced to death in Iran without due process, according to the Center for Human Rights in Iran. Another 25 face charges that could carry the death penalty.
Carlos Kasper, a member of the German federal parliament and a political sponsor of Saman Yasin, wrote on Twitter that he was "shocked" at the news of the artist's suicide attempt.
"The suicide attempt was prompted by the inhumane prison conditions! This has to stop! I demand his immediate release from prison and access to adequate medical support," Casper added.
Since the death of Mahsa Amini after she was detained for allegedly wearing a head scarf improperly, Iranians have flooded the streets across the country to protest a lack of rights, with women and schoolgirls making unprecedented shows of support in the biggest threat to the Islamic government since the 1979 revolution.
Last month, 227 lawmakers from the 290-seat parliament led by hard-liners urged the judiciary to approve death sentences for some protesters arrested amid the recent wave of demonstrations.
Two public executions have already taken place, according to the authorities, and rights groups say many others have been handed death sentences, while at least two dozen others face charges that could carry the death penalty.
The activist HRANA news agency said that, as of December 13, at least 431 protesters had been killed during the unrest, including 68 minors, as security forces try to stifle widespread dissent.
The Oslo-based Iran Human Rights Organization says the number of executions in Iran has exceeded 500 this year.
Jailed rapper Saman Yasin Seydi attempted suicide in prison after receiving a death sentence a month earlier
https://www.rferl.org/a/iran-rapper-prison-suicide-attempt/32189550.html | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 457 |
El Campeonato Nacional de Clausura "BancoEstado" de Primera División de Fútbol Profesional, año 2004 fue el segundo y último torneo de la temporada 2004 de la primera división chilena de fútbol. Comenzó el 31 de julio y finalizó el 19 de diciembre de 2004.
El trofeo fue ganado por Cobreloa, tras derrotar a la Unión Española por 3-1 en el Estadio Santa Laura y empatar sin goles en el Municipal de Calama, en el partido definitorio. El equipo loíno conseguía de esta manera su octava estrella en el fútbol chileno.
Modalidad
El campeonato se jugó al estilo de los torneos de la Primera división mexicana. Los 18 equipos se enfrentaron en modalidad "todos contra todos", en una Fase Clasificatoria.
Los equipos se agruparon en cuatro grupos (dos de cinco equipos y dos de cuatro), clasificando a segunda ronda (play-off) los tres mejores puntajes de cada grupo. Existió, adicionalmente, la posibilidad de un "repechaje" en caso de que un tercero de grupo obtuviese un mejor puntaje en la clasificación general que el segundo de otro grupo. En este caso se jugaba un partido de repechaje en la cancha del equipo con mayor puntaje durante la fase regular.
Los 12 clasificados se enfrentaron entre sí en partidos de ida y vuelta, dando la oportunidad de clasificación a los seis ganadores y los dos mejores perdedores de esta ronda. Posteriormente, se desarrollaron los cuartos de final, semifinales y final, para dirimir al campeón.
Desarrollo
Dirigidos por el estratega uruguayo Nelson Acosta (que reemplazó en la fase de play-offs al técnico Fernando Díaz, que dirigió la fase regular), los naranjas superaron una irregular primera ronda, tras la cual terminaron séptimos en la Fase Clasificatoria, para eliminar consecutivamente a Audax Italiano (6°), Cobresal (4°) y Colo-Colo (1°) en la fase de play-offs. La final con Unión Española resultó inédita en este tipo de campeonatos, ya que era la primera vez que ninguno de los tres "grandes" del fútbol chileno no llegaba a la instancia desde la instauración de los torneos cortos en 2002.
El elenco minero contó, además, con jugadores clave como Patricio Galaz, goleador del certamen con 19 goles, y el argentino José Luis Díaz. Cobreloa debió superar un muy difícil inicio de campeonato, en lo deportivo, disciplinario e institucional, para poder probarse su octava corona. Considerando fase regular y postemporada, Cobreloa jugó 25 partidos, ganó 10, empató 10 y perdió 5, con un rendimiento de 53.3%.
Equipos por región
<center>
{| border=0
| <div style="position:relative;">
Fase Clasificatoria
Grupo A
Grupo B
Grupo C
<center>
{| class="wikitable sortable" width=65%
|- bgcolor=#006699
! Pos
!width=35%|Equipo
! Pts
! PJ
! G
! E
! P
! GF
! GC
! DIF
|- align=center style="background:#ADD8E6;"
||1||align=left|Colo-Colo
||32||17||9||5||3||28||18||10
|- align=center style="background:#ADD8E6;"
||2||align=left|Universidad de Chile
||30||17||9||6||2||27||14||13
|- align=center style="background:#ADD8E6;"
||3||align=left|Deportes Temuco
||21||17||4||9||4||34||36||2
|- align=center style="background:#ffffff;"
||4||align=left|Palestino
||21||17||6||3||8||23||28||-5
|- align=center style="background:#ffffff;"
||5||align=left|Rangers
||15||17||3||6||8||19||33||-14
|}
</center>
Grupo D
Temporada 2004
Clasificación a Copa Libertadores 2005
Clasificaron a este torneo:
Universidad de Chile: campeón del Apertura 2004, como Chile 1 (fase grupal)
Cobreloa: campeón del Clausura 2004, como Chile 2 (fase grupal)
Colo-Colo: mejor puntaje en la Fase Clasificatoria del Clausura 2004, como Chile 3''' (fase preliminar)
Descensos
En esta temporada no hubo descensos a Primera B, como parte de la política de la directiva del fútbol de expandir el número de equipos en Primera división de 16 a 20 clubes. Ascendieron desde la división de honor el campeón Deportes Melipilla y el subcampeón Deportes Concepción.
Véase también
Primera B de Chile 2004
Referencias
Fuente
RSSSF Chile 2004
2004-C
1ra. Division Clausura | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 7,658 |
\section{Introduction}
Throughout the paper, we work in the usual Zermelo-Fr\ae{}nkel set theory \( {\sf ZF} \), plus the Axiom of Dependent Choices over the reals \( {\sf DC}(\mathbb{R}) \).
Let \( X \) be a Polish space, and let \( \mathcal{F} \) be a \emph{reducibility (on \( X \))}, that is a collection of functions from \( X \) to itself closed under composition and containing the identity \( \operatorname{id} = \operatorname{id}_X \). Given \( A,B \subseteq X \), we say that \( A \) is \emph{reducible} to \( B \) if and only if
\[
A = f^{-1}(B) \text{ for some } f \colon X \to X,
\]
and that \( A \) is \emph{\( \mathcal{F} \)-reducible} to \( B \) (\( A \leq_\mathcal{F} B \) in symbols) if \( A \) is reducible to \( B \) \emph{via a function in \( \mathcal{F} \)}.
Notice that clearly \( A \leq_\mathcal{F} B \iff \neg A \leq_\mathcal{F} \neg B \) (where, to simplify the notation, we set \( \neg A = X \setminus A \) whenever the
underlying space \( X \) is clear from the context).
Since \( \mathcal{F} \) is a reducibility on \( X \), the relation \( \leq_\mathcal{F} \) is a preorder which can be used to measure the ``complexity'' of subsets
of \( X \): in fact, if \( \mathcal{F} \) consists of reasonably simple functions, the assertion ``\( A \leq_\mathcal{F} B \)'' may be understood as ``the set \( A \) is not more
complicated than the set \( B \)'' --- to test whether a given \( x \in X \) belongs to \( A \) or not, it is enough to pick a witness \( f \in \mathcal{F} \) of
\( A \leq_\mathcal{F} B \), and then check whether \( f(x) \in B \) or not. This suggests that the reducibility \( \mathcal{F} \) may be used to form a hierarchy of subsets of \( X \)
in the following way. Say that \( A , B \subseteq X \) are \emph{\( \mathcal{F} \)-equivalent}
(\( A \equiv_\mathcal{F} B \) in symbols) if \( A \leq_\mathcal{F} B \leq_\mathcal{F} A \). Since \( \equiv_\mathcal{F} \) is the equivalence relation canonically induced by \( \leq_\mathcal{F} \), we can
consider the \emph{\( \mathcal{F} \)-degree} \( [A]_\mathcal{F} = \{ B \subseteq X \mid A \equiv_\mathcal{F} B \} \) of a given \( A \subseteq X \), and then order the
collection \( {\sf Deg}(\mathcal{F}) = \{ [A]_\mathcal{F} \mid A \subseteq X \} \) of such \( \mathcal{F} \)-degrees using
the quotient of \( \leq_\mathcal{F} \), namely setting \( [A]_\mathcal{F} \leq [B]_\mathcal{F} \iff A \leq_\mathcal{F} B \) for every \( A, B \subseteq X \). The resulting structure \( {\sf Deg}(\mathcal{F}) = ({\sf Deg}(\mathcal{F}),\leq) \) is then called \emph{\( \mathcal{F} \)-hierarchy on \( X \)}. When considering the restriction
\( {\sf Deg}_{\boldsymbol{\Gamma}}(\mathcal{F}) \) of such structure to the \( \mathcal{F} \)-degrees of sets in a given \( \boldsymbol{\Gamma} \subseteq \mathscr{P}(X) \), we
speak of \emph{\( \mathcal{F} \)-hierarchy on \( \boldsymbol{\Gamma} \)-subsets of \( X \)}.
In his Ph.D. thesis~\cite{Wadge:1983}, Wadge considered the case when \( X \) is the Baire space \( \pre{\omega}{\omega} \) (i.e.\ the space
of all \(\omega\)-sequences of natural numbers endowed with the product of the discrete topology on \(\omega\)) and \( \mathcal{F} \) is either the
set \( \mathsf{W} = \mathsf{W}(X) \) of all continuous functions, or the set \( \L(\bar{d}) \) of all functions which are nonexpansive with respect to the usual metric
\( \bar{d} \) on \( \pre{\omega}{\omega} \) (see Section~\ref{sec:ultrametric} for the definition). Using game-theoretical methods, he was able to show
that in both cases the \( \mathcal{F} \)-hierarchy on Borel subsets of \( X = \pre{\omega}{\omega} \) is \emph{semi-well-ordered}, that is:
\begin{enumerate}[(1)]
\item
it is \emph{semi-linearly ordered}, i.e.\ either \( A \leq_\mathcal{F} B \) or \( \neg B \leq_\mathcal{F} A \) for all Borel \( A,B \subseteq X \);
\item
it is \emph{well-founded}.
\end{enumerate}
Notice that the \emph{Semi-Linear Ordering principle for \( \mathcal{F} \)} (briefly: \( {\sf SLO}^\mathcal{F} \)) defined in (1) implies that antichains have size at
most \( 2 \), and that they are of the form \( \{ [A]_\mathcal{F}, [\neg A]_\mathcal{F} \} \) for some \( A \subseteq X \) such that \( A \nleq_\mathcal{F} \neg A \) (sets with
this last property are called \emph{\( \mathcal{F} \)-nonselfdual}, while the other ones are called \emph{\( \mathcal{F} \)-selfdual}: since \( \mathcal{F} \)-selfduality is
\( \equiv_\mathcal{F} \)-invariant, a similar terminology will be applied to the \( \mathcal{F} \)-degree of \( A \) as well). This in particular means that if we further
identify each \( \mathcal{F} \)-degree \( [A]_\mathcal{F} \) with its \emph{dual} \( [\neg A]_\mathcal{F} \) we get a linear ordering, which is also well-founded when (2) holds.
A semi-well-ordered hierarchy is practically optimal as a measure of complexity for (Borel subsets of) \( X \): by well-foundness, we can associate
to each \( A \subseteq X \) an ordinal rank (the \emph{\( \mathcal{F} \)-rank of \( A \)}), and antichains are of minimal size.%
\footnote{Asking for no antichain at all seems unreasonable by the following considerations: let \( A \) be e.g.\ a proper open subset of a given
Polish space \( X \). On the one hand, checking membership in \( A \) cannot be considered strictly simpler or strictly more difficult than checking
membership in its complement: this means that the degrees of \( A \) and \( \neg A \) cannot be one strictly below the other in the hierarchy. On the other
hand, the fact that open sets and closed sets have in general different (often complementary) combinatorial and topological properties, strongly
suggests that the degrees of \( A \) and \( \neg A \) should be kept distinct. Therefore such degrees must form an antichain of size \( 2 \).}
In fact, in~\cite{MottoRos:2012b, MottoRos:2012c} it is proposed to classify arbitrary \( \mathcal{F} \)-hierarchies on corresponding topological spaces \( X \)
according to whether they provide a good measure of complexity for subsets of \( X \). This led to the following definition.
\begin{definition}
Let \( \mathcal{F} \) be a reducibility on a (topological) space \( X \), and let \( \boldsymbol{\Gamma} \subseteq \mathscr{P}(X) \). The \( \mathcal{F} \)-hierarchy \( {\sf Deg}_{\boldsymbol{\Gamma}}(\mathcal{F}) \) on \( \boldsymbol{\Gamma} \)-subsets of \( X \) is called:
\begin{enumerate}[\( \bullet \)]
\item
\emph{very good} if it is semi-well-ordered;
\item
\emph{good} if it is a well-quasi-order, i.e.\ all its antichains and descending chains are finite;
\item
\emph{bad} if it contains infinite antichains;
\item
\emph{very bad} if it contains both infinite antichains and infinite descending chains.
\end{enumerate}
\end{definition}
Since the pioneering work of Wadge, many other \( \mathcal{F} \)-hierarchies on the Baire space \( \pre{\omega}{\omega} \) (or, more generally, on
\emph{zero-dimensional} Polish space) have been considered in the
literature~\cite{VanWesep:1978,Andretta:2003d,Andretta:2006,MottoRos:2009,MottoRos:2010,MottoRos:2010b}, including Borel functions,
\( \boldsymbol{\Delta}^0_\alpha \)-functions,%
\footnote{Given a countable ordinal \( \alpha \geq 1 \) and a Polish space \( X \), a function \( f \colon X \to X \) is called
\emph{\( \boldsymbol{\Delta}^0_\alpha \)-function} if \( f^{-1}(A) \in \boldsymbol\Sigma^0_\alpha \) for every \( A \in \boldsymbol{\Sigma}^0_\alpha \).}
Lipschitz functions, uniformly continuous functions, functions of Baire class \( < \alpha \) for a given additively closed countable ordinal \(\alpha\),
\( \boldsymbol{\Sigma}^1_n \)-measurable functions, and so on. It turned out that all of them are very good when restricted to Borel sets, or even to larger collections of subsets of \( \pre{\omega}{\omega} \)
if suitable determinacy principles are assumed.
In contrast, it is shown in~\cite{Hertling:1993,Hertling:1996,Ikegami:2012,Schlicht:2012,MottoRos:2012b} that when considering the continuous
reducibility on the real line \( \mathbb{R} \) or, more generally, on arbitrary Polish spaces with \emph{nonzero dimension}, then one usually gets a (very)
bad hierarchy (and the same applies to some other classical kind of reducibilities, depending on the space under consideration)
\footnote{Of course, one can further extend the class of topological spaces under consideration, and analyze e.g.\ the continuous reducibility on them: for example,~\cite{Selivanov:2005} considers the case of \( \omega \)-algebraic domains (a class of spaces relevant in theoretical computer
science), while~\cite{MottoRos:2012b} consider the broader class of the so-called quasi-Polish spaces. Moreover, it is possible to generalize
the notion of reducibility itself by considering e.g.\ reducibilities between finite partitions (see
e.g.~\cite{vanEngelen:1987,Hertling:1993,Selivanov:2005,Selivanov:2007,Selivanov:2010} and the references contained therein).}
Given all these results, one may be tempted to conjecture that all ``natural'' \( \mathcal{F} \)-hierarchies on (Borel subsets of) a zero-dimensional Polish
space \( X \) need to be very good. This conjecture is justified by the fact that every such space is homeomorphic to a closed subset (hence to a
topological retract) of the Baire space, and a well-known transfer argument (see e.g.\ \cite[Proposition 5.4]{MottoRos:2012b}) shows that this
already implies the following folklore result.
\begin{proposition} \label{prop:continuous}
Let \( X \) be a zero-dimensional Polish space, and let \( \mathcal{F} \) be an arbitrary reducibility on \( X \) which contains \( \mathsf{W}(X) \), i.e.\ all continuous functions from
\( X \) to itself. Then the \( \mathcal{F} \)-hierarchy \( {\sf Deg}_{\boldsymbol{\Delta}^1_1}(\mathcal{F}) \) on Borel subsets of \( X \) is very good.
\end{proposition}
\noindent
In fact,~\cite[Theorem 3.1]{MottoRos:2009} (essentially) shows that this result can be further strengthened when \( X \) itself is a closed subset
of \( \pre{\omega}{\omega} \): if \( X \) is equipped with the restriction \( \bar{d}_X \) of the canonical metric \( \bar{d} \) on
\( \pre{\omega}{\omega} \), then \( {\sf Deg}_{\boldsymbol{\Delta}^1_1}(\mathcal{F}) \) is very good as soon as \( \mathcal{F} \) contains the collection \( \L(\bar{d}_X) \) of all
\emph{\( \bar{d}_X \)-nonexpansive} functions.
Despite the above mentioned results, in~\cite[Theorem 5.4, Proposition 5.10, and Theorem 5.11]{MottoRos:2012c} it is shown that there are
various natural reducibilities on \( \pre{\omega}{\omega} \) that actually induce (very) bad hierarchies on its Borel subsets. In particular, it is
shown that
\( \pre{\omega}{\omega} \) can be equipped with a complete ultrametric \( d' \), still compatible with its usual product topology, such that
the \( \mathcal{F} \)-hierarchy on Borel (in fact, even just clopen) subsets of \( \pre{\omega}{\omega} \) is very bad for \( \mathcal{F} \) the collection of all the
\( d' \)-nonexpansive (alternatively: \( d' \)-Lipschitz) functions.
Motivated by these results, in the present paper we continue this investigation by considering various complete ultrametrics on
\( \pre{\omega}{\omega} \) (compatible with its product topology) and, more generally, the collection of all \emph{ultrametric Polish spaces \( X = (X,d) \)}, a very natural and interesting class which includes e.g.\ the space \( \mathbb{Q}_p \) of \( p \)-adic numbers (for every prime \( p \in \mathbb{N} \)).%
\footnote{More generally, the completion of any countable valued field \(K \) with valuation \( | \cdot |_K \colon K \to \mathbb{R} \) and metric \( d(x,y) = | x - y |_K \) (for \( x,y \in K \)) is always an ultrametric Polish space.}
On
such spaces, we then consider the hierarchies of degrees induced by one of the following reducibilities%
\footnote{Notice that since the metric topology on \( X \) is always zero-dimensional, it does not make much sense to consider reducibilities
\( \mathcal{F} \supseteq \mathsf{W}(X) \), because by Proposition~\ref{prop:continuous} they always induce a very good hierarchy on Borel
subsets of \( X \).}
on \( X \):
\begin{enumerate}[\( \bullet \)]
\item
the collection \( \L(d) \) of all nonexpansive functions, where \( f \colon X \to X \) is called \emph{nonexpansive} if \( d(f(x),f(y)) \leq d(x,y) \)
for all \( x,y \in X \);
\item
the collection \( \mathsf{Lip}(d) \) of all Lipschitz functions (with arbitrary constants), where \( f \colon X \to X \) is a \emph{Lipschitz function with constant
\( L \)} (for a nonnegative real \( L \)) if \( d(f(x),f(y)) \leq L \cdot d(x,y) \) for all \( x,y \in X \);
\item
the collection \( \mathsf{UCont}(d) \) of all uniformly continuous functions, where \( f \colon X \to X \) is \emph{uniformly continuous} if for every
\( \varepsilon \in \mathbb{R}^+ \) there is a \( \delta \in \mathbb{R}^+ \) such that \( d(x,y) < \delta \Rightarrow d(f(x),f(y)) < \varepsilon \) for all \( x,y \in X \)
(here \( \mathbb{R}^+ \) denotes the set of strictly positive reals).
\end{enumerate}
The main results of the paper are the following:
\begin{enumerate}[(A)]
\item
The \( \mathsf{UCont}(d) \)-hierarchy on Borel subsets of \( X \) is always very good (Theorem~\ref{th:unifcontandlip}). Since by Proposition~\ref{prop:UCont} it is possible to equip the Baire
space with a compatible complete ultrametric \( d' \) such that \( \L(\bar{d}) \not\subseteq \mathsf{UCont}(d') \) (where $\bar{d}$ is the usual metric on \( \pre{\omega}{\omega} \)), this also
implies that \( \L(\bar{d}) \subseteq \mathcal{F} \) is a sufficient but not necessary condition for the \( \mathcal{F} \)-hierarchy on Borel subsets of
\( \pre{\omega}{\omega} \) being very good (for \( \mathcal{F} \) a reducibility on \( \pre{\omega}{\omega} \)).
\item
If \( X \) is perfect, then the \( \mathsf{Lip}(d) \)-hierarchy on the Borel subsets of \( X \) is either very good (if \( X \) has bounded diameter), or else it is very
bad already when restricted to clopen subsets of \( X \) (if the diameter of \( X \) is unbounded). A technical strengthening of the property of having
(un)bounded diameter (see Definition~\ref{def:nontriviallyunbounded}) works similarly for arbitrary ultrametric Polish spaces
(Theorems~\ref{th:unboundeddiam} and~\ref{th:notunbounded}, Corollary~\ref{cor:unboundeddiam}).
\item
If the range of \( d \) contains an honest increasing sequence (see Definition~\ref{def:honestincreasingsequence}), then the \( \L(d) \)-hierarchy on
clopen subsets of \( X \) is very bad (Theorem~\ref{th:increasingdistances}); in particular, this happens in the special case when \( X \) is perfect and
has unbounded diameter. If instead the range of \( d \) is either finite or a decreasing \(\omega\)-sequence converging to \( 0 \), then the
\( \L(d) \)-hierarchy on Borel subsets of \( X \) is always very good (Theorem~\ref{th:descendingdistances}).
\item
It follows from the second part of (C) that if \( X \) is compact, then both%
\footnote{Since on compact metric spaces continuity and uniform continuity coincide, the \( \mathsf{UCont}(d) \)-hierarchy on Borel subsets of a compact
\( X \) is very good already by Proposition~\ref{prop:continuous}.}
the \( \mathsf{Lip}(d) \)- and the \( \L(d) \)-hierarchy on Borel subsets of \( X \) are very good (Theorem~\ref{th:compact}).
\item
If we assume the Axiom of Choice \( {\sf AC} \), then the \( \mathcal{F} \)-hierarchy on (arbitrary subsets of) an uncountable \( X \) is very bad for every reducibility
\( \mathcal{F} \) such that \( \L(d) \subseteq \mathcal{F} \subseteq \mathsf{Bor} (X) \), where \( \mathsf{Bor}(X) \) is the collection of all Borel functions from \( X \) into itself
(Theorem~\ref{th:illfounded hierarchy}). If we further assume that \( \mathsf{V = L} \), then the \( \mathcal{F} \)-hierarchy on \( X \) is very bad already when
restricted to
\( \boldsymbol{\Pi}^1_1 \), i.e.\ coanalytic,%
\footnote{Equivalently, to \( \boldsymbol{\Sigma}^1_1 \) (i.e.\ analytic) subsets of \( X \).}
subsets of \( X \) (Theorem~\ref{th:illfounded hierarchy in L}).
\end{enumerate}
In particular, the results in (A)--(D) generalize those from~\cite[Section 5]{MottoRos:2012c} and answer most of the questions
in~\cite[Section 6]{MottoRos:2012c}. Moreover, they allow us to construct discrete ultrametric Polish spaces \( X = (X,d) \) whose \( \mathsf{Lip}(d) \)- and
\( \L(d) \)-hierarchies are very bad (Corollaries~\ref{cor:countable} and~\ref{cor:X_1}), a fact which contradicts the conceivable conjecture that
the \( \mathsf{Lip}(d) \)- and the \( \L(d) \)-hierarchy on them need to be (very) good since all subsets of such spaces are extremely simple (i.e.\ clopen).
Notice also that the result mentioned in (E) under the assumption \( \mathsf{V = L} \) (which is best possible for most reducibilities \( \mathcal{F} \) by
Proposition~\ref{prop:continuous} and the comment following it)
can be viewed as an extension of the well-know classical result that if \( \boldsymbol{\Pi}^1_1 \)-determinacy fails then there are proper \( \boldsymbol{\Pi}^1_1 \) sets which are not (Borel-)complete for coanalytic sets.
We end this introduction with two general remarks concerning the results presented in this paper:
\begin{enumerate}[i)]
\item
to simplify the presentation, we will consider only \( \mathcal{F} \)-hierarchies on \emph{Borel subsets} of a given ultrametric Polish space \( X \) (except in Section \ref{sec: choice}): this is
because in this way we can avoid to assume any axiom beyond our basic theory \( {\sf ZF} + {\sf DC}(\mathbb{R}) \). However, as usual in Wadge theory, all our
results can be extended to larger pointclasses \( \boldsymbol{\Gamma} \subseteq \mathscr{P}(X) \) by assuming corresponding determinacy axioms
(more precisely: the determinacy of subsets of \( \pre{\omega}{\omega} \) which are Boolean combinations of sets in \( \boldsymbol{\Gamma} \)).
In particular, under the full Axiom of Determinacy \( {\sf AD} \) (asserting that all games on \(\omega\) are determined), all these results remain true
when considering \emph{unrestricted} \( \mathcal{F} \)-hierarchies \( {\sf Deg}(\mathcal{F}) \) on \( X \);
\item
when showing that a given \( \mathcal{F} \)-hierarchy on \( X \) (possibly restricted to some \( \boldsymbol{\Gamma} \subseteq \mathscr{P}(X) \)) is very bad, we will
actually show that some very complicated partial (quasi-)order on \( \mathscr{P}(\omega) \), like the inclusion relation \( \subseteq \), or even the more
complicated relation \( \subseteq^* \) of inclusion modulo finite sets, embeds into such a hierarchy. This gives much stronger results, as it implies
e.g.\ that the \( \mathcal{F} \)-hierarchy under consideration contains antichains of size the continuum and, in the case of \( \subseteq^* \), that (under
\( {\sf AC} \)) every partial order of size \( \aleph_1 \) embeds into the \( \mathcal{F} \)-hierarchy on (\( \boldsymbol{\Gamma} \)-subsets of) \( X \) (see~\cite{Parovivcenko:1963}).
\end{enumerate}
\section{Basic facts about ultrametric Polish spaces} \label{sec:ultrametric}
Given a metric space \( X = (X,d) \), we denote by \( \tau_d \) the \emph{metric topology (induced by \( d \))}, i.e.\ the topology generated by the
basic open balls \( B_d(x,\varepsilon) = \{ y \in X \mid d(x,y) < \varepsilon \} \) (for some \(x \in X \) and \( \varepsilon \in \mathbb{R}^+ \)). When considered
as a topological space, the space \( X \) is tacitly endowed with such topology, and therefore we will e.g.\ say that the metric space \( X \) is separable
if there is a countable \( \tau_d \)-dense subset of \( X \), and similarly for all other topological notions. The diameter of \( X \) is \emph{bounded} if
there is \( R \in \mathbb{R}^+ \) such that \( \sup \{ d(x,y) \mid x,y \in X \} \leq R \), and \emph{unbounded} otherwise.
A metric \( d \) on a space \( X \) is called \emph{ultrametric} if it satisfies the following strengthening of the triangle inequality, for all \( x,y,z \in X \):
\[
d(x,z) \leq \max \{ d(x,y), d(y,z) \}.
\]
\begin{definition}
An \emph{ultrametric Polish space} is a separable metric space \( X = (X,d) \) such that \( d \) is a complete ultrametric. The collection of all
ultrametric Polish spaces will be denoted by \( \mathscr{X} \).
\end{definition}
Every (\( \tau_d \)-)closed subspace \( C \) of an ultrametric Polish space \( X = (X,d) \) will be tacitly equipped with the metric
\( d_C = d \restriction C \), which is obviously a complete ultrametric compatible with the relative topology on \( C \) induced by \( \tau_d \).
When there is no danger of confusion, with a little abuse of notation the metric \( d_C \) will be sometimes denoted by \( d \) again.
\begin{notation}
Given an ultrametric Polish space \( X = (X,d) \), we set \( R(d) = \{ d(x,y) \mid x,y \in X, x \neq y \} \), the set of all nonzero distances realized in \( X \).
\end{notation}%
A typical example of an ultrametric Polish space is obtained by equipping the Baire space with the usual metric \( \bar{d} \) defined by
\[
\bar{d}(x,y) =
\begin{cases}
0 & \text{if } x = y \\
2^{-n} & \text{if \( n \) is smallest such that } x(n) \neq y(n):
\end{cases}
\]
it is straightforward to check that \( \bar{d} \) is actually an ultrametric generating the product topology on \( \pre{\omega}{\omega} \), and
obviously \( R(\bar{d}) = \{ 2^{-n} \mid n \in \omega \} \). We will keep denoting this ultrametric by \( \bar{d} \) throughout the paper.
We collect here some easy but useful facts about arbitrary ultrametric (Polish) spaces \( X = (X,d) \):
\begin{enumerate}[(1)]
\item
for every \( x,y,z \in X \) two of the distances \( d(x,y) \), \( d(x,z) \), \( d(y,z) \) are equal, and they are greater than or equal to the third
(the ``isosceles triangle'' rule);
\item
for every \( x,y,z \in X \), if \( d(x,z) \neq d(y,z) \) then \( d(x,y) = \max \{ d(x,z),\) \(d(y,z) \} \). In particular, if \( x,y,z,w \in X \) are such that
\( d(x,z),d(y,w) < d(x,y) \) then \( d(z,w) = d(x,y) \);
\item
given a (\( \tau_d \)-)dense set \( Q \subseteq X \), all distances are realized by elements of \( Q \), that is: for every \( x,y \in X \) there are
\( q,p \in Q \) such that \( d(x,y) = d(q,p) \). In particular, if \( X \) is separable then \( R(d) \) is countable;%
\footnote{Vice versa, for every countable \( R \subseteq \mathbb{R}^+ \) there is an ultrametric Polish space \( X = (X,d) \) such that \( R(d) = R \), for example \( X=R\cup\{0\} \) with \( d(x,y)=\max\{x,y\} \) for distinct \( x , y \in X \).}
\item
for every \( x \in X \) and \( r \in \mathbb{R}^+ \) the open ball \( B_d(x,r) \) is actually clopen, and \( B_d(y,r) = B_d(x,r) \) for every \( y \in B_d(x,r) \). In
particular, the topology \( \tau_d \) is always zero-dimensional, and hence if \( X \) is an ultrametric Polish space, then it is homeomorphic to a
closed subset of the Baire space by~\cite[Theorem 7.8]{Kechris:1995} (see also Lemma~\ref{lemma:unifcontandlip});
\item
given \( x,y \in X \) and \( r,s \in \mathbb{R}^+ \), the (cl)open balls \( B_d(x,r) \) and \( B_d(y,s) \) are either disjoint, or else one of them contains the other.
\end{enumerate}
To simplify the terminology, we adapt the definition of family of reducibilities introduced in~\cite[Definition 5.1]{MottoRos:2012b} to the restricted
context of ultrametric Polish spaces.
\begin{definition}
Let \( \mathcal{F} \) be a collection of functions between any ultrametric Polish spaces. For \( X,Y \in \mathscr{X} \), denote by \( \mathcal{F}(X,Y) \) the collection of all
functions from \( \mathcal{F} \) with domain \( X \) and range included in \( Y \). The collection \( \mathcal{F} \) is called \emph{family of reducibilities} (on
\( \mathscr{X} \)) if:
\begin{enumerate}
\item
it contains all the identity functions, i.e.\ \( \operatorname{id}_X \in \mathcal{F}(X,X) \) for every \( X \in \mathscr{X} \);
\item
it is closed under composition, i.e.\ for every \( X,Y,Z \in \mathscr{X} \), \( f \in \mathcal{F}(X,Y) \), and \( g \in \mathcal{F}(Y,Z) \), the function \( g \circ f \) belongs to
\( \mathcal{F}(X,Z) \);
\end{enumerate}
\end{definition}%
Examples of family of reducibilities are the collections of all continuous functions, of all uniformly continuous functions, of all Lipschitz functions,
and of all nonexpansive functions. Notice also that if \( \mathcal{F} \) is a family of reducibilities then \( \mathcal{F}(X) = \mathcal{F}(X,X) \) is a reducibility on the space \( X \)
(for every \( X \in \mathscr{X} \)). The next simple lemma is a minor variation of~\cite[Proposition 5.4]{MottoRos:2012b} and can be proved in a
similar way.
\begin{lemma} \label{lemma:retraction}
Let \( \mathcal{F} \) be a family of reducibilities and \( X,Y \in \mathscr{X} \). Suppose that there is a surjective \( f \in \mathcal{F}(X,Y) \) admitting a right inverse
\( g \in \mathcal{F}(Y,X) \). Then there is an embedding from \( (\mathscr{P}(Y), \leq_{\mathcal{F}(Y)}, \neg) \) into \( (\mathscr{P}(X), \leq_{\mathcal{F}(X)}, \neg ) \).
In particular, if \( \mathcal{F} \) consists of Borel functions
and the \( \mathcal{F}(X) \)-hierarchy on Borel subsets of \( X \) is (very) good, then also the \( \mathcal{F}(Y) \)-hierarchy on Borel subsets of \( Y \) is (very) good.
\end{lemma}
\begin{proof}
The map \( \mathscr{P}(Y) \to \mathscr{P}(X) \colon A \mapsto f^{-1}(A) \) is the desired embedding.
\end{proof}
\section{Uniformly continuous and Lipschitz reducibilities}
In~\cite[Question 6.2]{MottoRos:2012c}, it is asked whether one can equip the Baire space \( \pre{\omega}{\omega} \) with a compatible complete
ultrametric \( d' \) so that \( \L(\bar{d}) \not\subseteq \mathsf{UCont}(d') \), and whether it is possible to strengthen this last condition to: the
\( \mathsf{UCont}(d') \)-hierarchy on \( X \) is (very) bad. We start by answering positively the first part of this question.
\begin{notation}
Given a function \( \phi \colon \omega \to \mathbb{R}^+ \), we denote by \( \operatorname{rg} (\phi) \) the \emph{range of \( \phi \)}, i.e.\ \( \operatorname{rg}(\phi) = \{ r \in \mathbb{R}^+ \mid \exists n \in \omega \, ( \phi(n) = r) \} \).
\end{notation}
\begin{definition} \label{def: d_phi}
Given a function \( \phi \colon \omega \to \mathbb{R}^+ \) with \( \inf \operatorname{rg} (\phi)>0 \), define the metric \( d_\phi \) on \( \pre{\omega}{\omega} \) by setting for every
\( x,y \in \pre{\omega}{\omega} \)
\[
d_\phi(x,y) = \max \{ \phi(x(0)), \phi(y(0)) \} \cdot \bar{d}(x,y).
\]
\end{definition}
It is not hard to check that each \( d_\phi \) is a complete ultrametric compatible with the product topology on \( \pre{\omega}{\omega} \) (and that \( \inf \operatorname{rg}(\phi)>0 \) is necessary for completeness).
\begin{notation}
Given a natural number \( i \in \omega \) and an ordinal \(\alpha\), we denote by \( i^{(\alpha)} \) the constant \(\alpha\)-sequence with value \( i \).
\end{notation}
\begin{proposition} \label{prop:UCont}
Let \( \phi \colon \omega \to \mathbb{R}^+ \colon n \mapsto 2^{n} \). Then \( \L(\bar{d}) \not\subseteq \mathsf{UCont}(d_\phi) \).
\end{proposition}
\begin{proof}
Consider the map \( f \colon \pre{\omega}{\omega} \to \pre{\omega}{\omega} \colon n {}^\smallfrown{} x \mapsto 3n {}^\smallfrown{} x \).
We show that for every \( \varepsilon,\delta \in \mathbb{R}^+ \) there are \(x,y \in \pre{\omega}{\omega} \) such that \( d_\phi(x,y) < \delta \) but
\( d_\phi(f(x),f(y)) > \varepsilon \). Let \( 0 \neq k \in \omega \) be such that \(2^{-k} < \delta \). Then for every \( n \geq k \) we get that setting
\( x = n^{(2n)} {}^\smallfrown{} 0^{(\omega)} \) and \( y = n^{(2n)} {}^\smallfrown{} 1^{(\omega)} \),
\[ d_\phi(x,y) = 2^n \cdot 2^{-2n} = 2^{-n} \leq 2^{-k} < \delta . \]
However,
\[
d_\phi(f(x),f(y)) = 2^{3n} \cdot 2^{-2n} = 2^n,
\]
hence letting \( n \) be large enough we get \( d_\phi(f(x),f(y)) > \varepsilon \), as desired.
\end{proof}
In order to answer the second half of~\cite[Question 6.2]{MottoRos:2012c}, we abstractly analyze the behavior of the \( \mathsf{UCont}(d) \)-hierarchy on
an arbitrary ultrametric Polish space \( X = (X,d) \).
The following lemma uses standard arguments (see e.g.\ the proof of~\cite[Theorem 7.8]{Kechris:1995}), but we fully reprove it here for the
reader's convenience.
\begin{lemma} \label{lemma:unifcontandlip}
Let $X = (X,d)$ be an ultrametric Polish space. Then there is a closed set \( C \subseteq \pre{\omega}{\omega} \) and a bijection
$f \colon (C, \bar{d}) \to (X,d)$ such that $f$ is uniformly continuous and $f^{-1}$ is nonexpansive. Moreover, if $X$ has bounded diameter,
then $f$ is even Lipschitz,
and if \( X \) has diameter \( \leq 1 \) then we can alternatively require \( f \) to be nonexpansive and \( f^{-1} \) to be Lipschitz with constant \( 2 \).
\end{lemma}
\begin{proof}
Let \( Q \) be a countable dense subset of \( X \).
Define the sets \( A_s \subseteq X\) for \( s \in \pre{< \omega}{\omega} \) recursively on \( \operatorname{lh}(s) \) as follows: \( A_\emptyset = X \).
Given \( A_s \subseteq X\),
let \( \{ B_{s,i} \mid i < I \} \) (for some \( I \leq \omega \)) be an enumeration without repetitions of the set of open balls
\( \{ B_d(x,2^{-\operatorname{lh}(s)}) \mid x \in Q \cap A_s \} \), and set \( A_{s {}^\smallfrown{} i} = B_{s,i} \) if \( i < I \) and
\( A_{s {}^\smallfrown{} i} = \emptyset \) otherwise. Since \( d \) is an ultrametric, one can easily check that the family
\( (A_s)_{s \in \pre{< \omega}{\omega}} \) is a Luzin scheme with vanishing diameter consisting of clopen sets, and with the further property that
\( A_s = \bigcup_{n \in \omega} A_{s {}^\smallfrown{} n} \) for every \( s \in \pre{< \omega}{\omega} \). Therefore the set
\( C = \{ x \in \pre{\omega}{\omega} \mid \bigcap_{n \in \omega} A_{x \restriction n} \neq \emptyset \} \) is a closed subset of
\( \pre{\omega}{\omega} \), and the map \( f \colon C \to X \) sending \( x \in C \) to the unique element in
\( \bigcap_{n \in \omega} A_{x \restriction n} \) is a bijection. So it remains only to check that such \( f \) has the desired properties.
Given \( \varepsilon > 0 \), let \( n \in \omega \) be smallest such that \( 2^{-n} \leq \varepsilon \), and set \( \delta = 2^{-n} \). If \( x,y \in C \) are such
that \( \bar{d}(x,y) < \delta \), then \( x \restriction (n+1) = y \restriction (n+1) \), which implies \( f(x),f(y) \in A_{x \restriction (n+1)} \). By definition of
the \( A_s \), this implies that \( d(f(x),f(y)) < 2^{-n} \leq \varepsilon \). This shows that \( f \) is uniformly continuous.
Further assuming that \( X \) be of bounded diameter, we get that \( f \) is Lipschitz with constant \( \max \{ 2,k \} \), where \( k \in \omega \) is an
arbitrary bound to the diameter of \(X \), i.e.\ it is such that \( d(x,y) \leq k \) for every \( x,y \in X \). To see this, fix distinct \( x,y \in C\). If
\(x(0) \neq y(0) \) then \( d(f(x),f(y)) \leq k \leq k \cdot \bar{d}(x,y) \) by our choice of \( k \in \omega \). Let now \( n \neq 0 \) be smallest such that
\( x(n) \neq y(n) \), so that \( \bar{d}(x,y) = 2^{-n} \). Since \( x \restriction n = y \restriction n \) we get that \( f(x),f(y) \in A_{x \restriction n} \),
which implies \( d(f(x),f(y)) < 2^{-(n-1)} \): therefore \( d(f(x),f(y)) < 2 \cdot 2^{-n} = 2 \cdot d(x,y) \).
Now fix \( x,y \in C \), and let \( n \in \omega \) be such that \( \bar{d}(x,y) = 2^{-n} \). Since \( x(n) \neq y(n) \) implies
\( A_{x \restriction (n+1)} \cap A_{y \restriction (n+1)} = \emptyset \), we get that \( d(f(x),f(y)) \geq 2^{-n} \) (because \( d \) is an ultrametric),
and hence \( \bar{d}(x,y) \leq d(f(x),f(y)) \). This shows that \( f^{-1} \) is nonexpansive.
Finally, assume that \( X \) has diameter \( \leq 1 \). In the construction above, redefine the collections \( \{ B_{s,i} \mid i < I \} \) as enumerations
without repetitions of the sets \( \{ B_d(x,2^{-(\operatorname{lh}(s)+1)}) \mid x \in Q \cap A_s \} \), and then use this new sets to define the \( A_s \)'s and the map \( f \).
Arguing as before, one can easily check that \( f \) is now nonexpansive while \( f^{-1} \) is Lipschitz with constant \( 2 \), as required.
\end{proof}
\begin{remark}
The special case of Lemma~\ref{lemma:unifcontandlip} where \( X \) has diameter \( \leq 1 \) already appeared (with the same proof)
in~\cite[Theorem 4.1]{MottoRos:2009b}. However, such a result cannot be literally extended to an arbitrary ultrametric Polish space \( X \), and in fact
the assumptions in Lemma~\ref{lemma:unifcontandlip} are optimal. To see this, note that if \( X \) has unbounded diameter then we cannot
require a map \( f \) as in Lemma~\ref{lemma:unifcontandlip} to be Lipschitz because every Lipschitz image of a space with
bounded diameter (like any set \( C \subseteq \pre{\omega}{\omega} \)) has necessarily bounded diameter too. Similarly, a nonexpansive image of a
set of diameter
\( \leq R \) (for some \( R \in \mathbb{R}^+ \)), has diameter \( \leq R \) too.
\end{remark}
•
\begin{definition}
Let \( X \) be a topological space, \( \mathcal{F} \) be a collection of functions from \( X \) to itself, and \( A \subseteq X \). We call \emph{\( \mathcal{F} \)-retraction} of
\( X \) onto \( A \) any surjection \( f \in \mathcal{F} \) from \( X \) onto \( A \) such that \( f \restriction A = \operatorname{id}_A \); if such a function exists we also say that \( A \)
is an \emph{\( \mathcal{F} \)-retract} of \( X \).
\end{definition}
Recall from \cite[Proposition 2.8]{Kechris:1995} that if \( \emptyset \neq A \subseteq C \) are closed subsets of the Baire space, then there is an
\( \L(\bar{d}_C) \)-retraction (i.e.\ a nonexpansive retraction) of \( C \) onto \( A \) --- a fact that will be repeatedly used throughout the paper.
The next corollary generalizes this result to arbitrary ultrametric Polish spaces, provided that we slightly weaken the requirement that the
retraction be nonexpansive.
\begin{corollary} \label{cor:retraction}
Let \( X = (X,d) \) be an ultrametric Polish space. For every nonempty closed \( A \subseteq X \), there is a uniformly continuous retraction
\( r \colon X \twoheadrightarrow A \). If moreover \( A \) has bounded diameter, then the retraction \( r \) can be taken to be Lipschitz.
\end{corollary}
\begin{proof}
Let \( C \) and \( f \) be as in Lemma~\ref{lemma:unifcontandlip}, with \( f \) uniformly continuous and \( f^{-1} \) nonexpansive. Notice that since
\( f \) is, in particular, a homeomorphism, the set \( A' = f^{-1}(A) \) is a nonempty closed subset of \( C \). Let \( g \colon C \to A' \) be a
nonexpansive retraction: then \( r = f \circ g \circ f^{-1} \colon (X,d) \twoheadrightarrow (A,d) \) is the desired uniformly continuous retraction.
Assume now that \( A \) has bounded diameter, and let \( C \), \( f \), and \( g \) be as in the previous paragraph. Arguing as in the proof of
Lemma~\ref{lemma:unifcontandlip}, one can easily check that \( f \restriction A' \colon (A', \bar{d}) \to (X,d) \) is actually Lipschitz (since \( A \) has
bounded diameter): therefore \( r = (f \restriction A') \circ g \circ f^{-1} \colon (X,d) \twoheadrightarrow (A,d) \) is the desired Lipschitz retraction.
\end{proof}
\begin{remark}
It is not possible in general to strengthen Corollary~\ref{cor:retraction} by requiring the reduction to be nonexpansive, even if we require the entire
\( X \) to have small diameter. To see this, let \( X = \{ 0 \} \cup \left\{ \frac{1}{2} + 2^{-(n+1)} \mid n \in \omega \right\} \), and set
\( d(x,y) = \max \{ x,y \} \) for all distinct \( x,y \in X \). Then \( X = (X,d) \) is a discrete ultrametric Polish space of diameter \( \leq 1 \). Consider the
clopen set \( A = X \setminus \{ 0 \} \), and let \( f \colon X \twoheadrightarrow A \) be a retraction. Let \( n \in \omega \) be such that
\( f(0) = \frac{1}{2} + 2^{-(n+1)} \): then setting \( x = 0 \) and \( y = \frac{1}{2} + 2^{-(n+2)} \) we get that
\[
d(f(x),f(y)) = d(f(x),y) = \frac{1}{2} + 2^{-(n+1)} > \frac{1}{2} + 2^{-(n+2)} = d(x,y),
\]
so \( f \) is expansive.
\end{remark}
\begin{theorem} \label{th:unifcontandlip}
The \( \mathsf{UCont}(d) \)-hierarchy \( {\sf Deg}_{\boldsymbol{\Delta}^1_1}(\mathsf{UCont}(d)) \) on the Borel subsets of an arbitrary ultrametric Polish space \( X = (X,d) \) is always very good. If \( X \) has
bounded diameter, then the \( \mathsf{Lip}(d) \)-hierarchy \( {\sf Deg}_{\boldsymbol{\Delta}^1_1}(\mathsf{Lip}(d)) \) on the Borel subsets of \( X \) is very good as well.
\end{theorem}
\begin{proof}
Let \( C \subseteq \pre{\omega}{\omega} \) and \( f \colon C \to X \) be as in Lemma~\ref{lemma:unifcontandlip}, and let
\( g \colon (\pre{\omega}{\omega}, \bar{d}) \to (C, \bar{d}) \) be a nonexpansive retraction. Then \( f^{-1} \) is a right inverse of \( g \circ f \),
and hence the result follows from Lemma~\ref{lemma:retraction} and the fact that both the \( \mathsf{UCont}(\bar{d}) \)-hierarchy and the
\( \mathsf{Lip}(\bar{d}) \)-hierarchy are very good by~\cite{MottoRos:2010}.
\end{proof}
In particular, this fully answers in the negative the second half of~\cite[Question 6.2]{MottoRos:2012c}. Moreover,
Theorem~\ref{th:unifcontandlip} provides also a negative answer to~\cite[Question 6.1]{MottoRos:2012c}: letting \( \phi \) be as in
Proposition~\ref{prop:UCont}, we get that the set \( \mathsf{UCont}(d_\phi) \) of uniformly continuous functions is a surjective image of \( \pre{\omega}{\omega} \),\footnote{When working in models of \( {\sf AD} \) (as it is often the case when dealing with Wadge-like hierarchies), for technical reasons it is often preferable to express ``cardinality inequality'' using surjections instead of injections. Therefore the stated property should be intended (in any model of \( {\sf ZF} \)) as: the cardinality of \( \mathsf{UCont}(d_\phi) \) is not larger than that of the Baire space. Obviously, further assuming the Axiom of Choice \( \mathsf{AC} \) this just means that \( \mathsf{UCont}(d_\phi) \) has cardinality \( \leq 2^{\aleph_0} \).} it does not contain \( \L(\bar{d}) \),
but it induces a very good hierarchy on the Borel subsets (or, further assuming \( {\sf AD} \), on the collection of all subsets) of
\( \pre{\omega}{\omega} \).
Theorem~\ref{th:unifcontandlip} shows that having a bounded diameter is a sufficient condition for having that the \( \mathsf{Lip}(d) \)-hierarchy on the
Borel subsets of an ultrametric Polish space \( X = (X,d) \) is very good. In fact, we are now going to show that a technical strengthening of this
condition is both necessary and sufficient for that.
\begin{definition} \label{def:nontriviallyunbounded}
Let \( X = (X,d) \) be an (ultra)metric Polish space. We say that the diameter of \( X \) is \emph{nontrivially unbounded} if for every
\( k \in \omega \) and every \( \varepsilon \in \mathbb{R}^+ \) there are \( x,y \in X \) with \( d(x,y) > k \) such that both \( x \) and \( y \) are not
\( \varepsilon \)-isolated.%
\footnote{Recall that a point \( x \) of a metric space is called \emph{\( \varepsilon \)-isolated} (for some \( \varepsilon \in \mathbb{R}^+ \)) if \( B_d(x,\varepsilon) = \{ x \} \).}
\end{definition}
Notice that if \( X \) is \emph{perfect}, then the diameter of \( X \) is nontrivially unbounded if and only if it is unbounded.
\begin{example} \label{xmp:p-adic}
Let \( p \) be a prime natural number, and let \( \mathbb{Q}_p \) be the ultrametric Polish space of \( p \)-adic numbers equipped with the usual \( p \)-adic metric \( d_p \): then \( \mathbb{Q}_p \) has unbounded diameter and is perfect (hence its diameter is nontrivially unbounded). To see the former, given \( k \in \omega \) let \( n \in \omega \) be such that \( n \geq 2 \) and \( k < p^n \): setting \( x = p^{-1} \) and \( y = p^{-n} \) we easily get \( d_p(x,y) = p^n > k \). To see that \( \mathbb{Q}_p \) is also perfect, fix an arbitrary \( q \in \mathbb{Q} \), and given \( \varepsilon \in \mathbb{R}^+ \) let \( l \in \omega \) be such that \( p^{-l} < \varepsilon \): then \( q' = q - p^l \) is distinct from \( q \) and \( d_p(q,q') = p^{-l} < \varepsilon \). This shows that \( q \) is not isolated, and since \( \mathbb{Q} \) is dense in \( \mathbb{Q}_p \) we are done.
\end{example}
\begin{notation}
We let \( \subseteq^* \) denote the relation of inclusion modulo finite sets between subsets of \( \omega \), i.e.\ for every
\( a,b \subseteq \omega \) we set
\[
a \subseteq^* b \iff \exists \bar{k} \in \omega \, \forall k \geq \bar{k} \, (k \in a \Rightarrow k \in b).
\]
\end{notation}
\begin{theorem} \label{th:unboundeddiam}
Let \( X = (X,d) \) be an ultrametric Polish space, and assume that its diameter is nontrivially unbounded. Then there is a map \( \psi \) from
\( \mathscr{P}(\omega) \) into the clopen subsets of \( X \) such that for all \(a,b \subseteq \omega \):
\begin{enumerate}
\item
if \( a \subseteq^* b \) then \( \psi(a) \leq_{\L(d)} \psi(b) \);
\item
if \( \psi(a) \leq_{\mathsf{Lip}(d)} \psi(b) \) then \( a \subseteq^* b \).
\end{enumerate}
In particular, \( (\mathscr{P}(\omega), \subseteq^* ) \) embeds into both \( {\sf Deg}_{\boldsymbol{\Delta}^0_1}(\mathsf{Lip}(d)) \) and \( {\sf Deg}_{\boldsymbol{\Delta}^0_1}(\L(d)) \).
\end{theorem}
\begin{proof}
Let \( (q_n)_{n \in \omega} \) be an enumeration of a countable dense subset \( Q \) of \( X \).
We first recursively construct two sequences \( (r_n)_{n \in \omega} \), \( (s_n)_{n \in \omega} \) of nonnegative reals and two sequences
\( (x_n)_{n \in \omega} \), \( (y_n)_{n \in \omega} \) of points of \( X \) such that for all distinct \( n,m \in \omega \) the following properties hold:
\begin{enumerate}[(a)]
\item
\( d(x_n,x_m) = r_{\max \{ n,m \}} \) and
\( d(x_n,y_n) = s_n \);
\item
\( r_{n+1} > \max \{ n+1, r_n^2 \} \) (in particular, \( (r_n)_{n \in \omega} \) is strictly increasing and unbounded in \( \mathbb{R}^+ \));
\item
\( s_0 < 1 \) and \( s_{n+1} < \frac{s_n}{r_n+1} \) (in particular, \( (s_n)_{n \in \omega} \) is a strictly decreasing sequence).
\end{enumerate}
\begin{claim} \label{claim:nonisolated}
If \( x \in X \) is not \( \varepsilon \)-isolated then there are at least two distinct \( q_i,q_j \in Q \) such that \( q_i,q_j \in B_d(x,\varepsilon) \).
\end{claim}
\begin{proof}[Proof of the Claim]
Since \( x \) is not \( \varepsilon \)-isolated, there is \( y \in B_d(x,\varepsilon) \) such that \( x \neq y \). By density of \( Q \), there are
\( q_i,q_j \in Q \) such that \( q_i \in B_d(x,d(x,y)) \) and \( q_j \in B_d(y,d(x,y)) \). Then \( q_i \neq q_j \) since
\( B_d(x,d(x,y)) \cap B_d(y,d(x,y)) = \emptyset \), while \( q_i,q_j \in B_d(x,\varepsilon) \) because
\( B_d(x,d(x,y)), B_d(y,d(x,y)) \subseteq B_d(x,\varepsilon) \) by \( d(x,y) < \varepsilon \).
\end{proof}
Let \( x\in X \) be not \( 1 \)-isolated (such an \( x \) exists because the diameter of \( X \) is nontrivially unbounded), and let \( q_i,q_j \) be as in
Claim~\ref{claim:nonisolated} for \( \varepsilon=1 \). Then we set \( x_0 = q_i \), \( y_0 = q_j \), \( r_0 = 0 \), and \( s_0 = d(q_i,q_j) \). Now assume that \( x_n \), \( y_n \),
\( r_n \), and \( s_n \) have been defined. Let \( x,y \in X \) be such that \( d(x,y) > \max \{ n+1,r^2_n \} \) and \( x,y \) are not
\( \frac{s_n}{r_n+1} \)-isolated. Then at least one of \( x \) and \( y \) has distance greater than \( \max \{ n+1,r^2_n \} \) from \( x_n \) (and hence
also from all the \( x_m \) for \( m \leq n \)): if not, then we would have \( d(x,y) \leq \max \{ d(x,x_n),d(y,x_n) \} \leq \max \{ n+1,r^2_n \} \),
contradicting our choice of \( x,y \). So we may assume without loss of generality that \( d(x,x_n) > \max \{ n+1,r^2_n \} \) and \( x \) is not
\( \frac{s_n}{r_n+1} \)-isolated. Let \( q_i,q_j \) be as in Claim~\ref{claim:nonisolated} for \( \varepsilon=\frac{s_n}{r_n+1} \), and set \( x_{n+1} = q_i \), \( y_{n+1} = q_j \),
\( r_{n+1} = d(q_i,x_n) \), and \( s_{n+1} = d(q_i,q_j) \). Since \( d(q_i,x) < \frac{s_n}{r_n+1} \leq 1 \leq \max \{ n+1,r^2_n \} \), we have
\( r_{n+1} = d(q_i, x_n) = d(x,x_n) > \max \{ n+1,r^2_n \} \). Moreover, \( s_{n+1} < \frac{s_n}{r_n+1} \) by the fact that
\( q_i,q_j \in B_d(x,\frac{s_n}{r_n+1}) \). Arguing by induction on \( n \in \omega \), it is then easy to check that the sequences constructed in this
way have all the desired properties.
Given \( a \subseteq \omega \), let \( \hat{a} = \{ 2i \mid i \in \omega \} \cup \{ 2i+1 \mid i \in a \} \), so that \( \hat{a} \) is always infinite
and for every \( a,b \subseteq \omega \)
\[
a \subseteq^* b \iff \hat{a} \subseteq^* \hat{b}.
\]
For \( a \subseteq \omega \), set \( \psi(a) = \bigcup_{i \in \hat{a}} B_d(x_i,s_i) \). Clearly, each \( \psi(a) \) is an open subset of \( X \). To see that it is also
closed, observe that \( B_d(x_i,s_i) \subseteq B_d(x_i,1) \) for every \( i \in \omega \) by our choice of the \( s_i \)'s, and that for distinct
\( i,j \in \omega \) the clopen balls \( B_d(x_i,1) \) and \( B_d(x_j,1) \) are disjoint by our choice of the \( x_i \)'s and of the \( r_i \)'s: therefore,
since the open balls in \( X \) are automatically closed we get that
\begin{align*}
X \setminus \psi(a) =& \bigcup \left\{ B_d(z,1) \mid z \notin \bigcup\nolimits_{i \in \hat{a}} B_d(x_i,1) \right\} \\
&\cup \bigcup \{ B_d(x_i,1) \setminus B_d(x_i,s_i) \mid i \in \hat{a} \}
\end{align*}
is open.
Let now \( a ,b \subseteq \omega \) be such that \( a \subseteq^* b \), which in particular implies \( \hat{a} \subseteq^* \hat{b} \), and let
\( 0 \neq \bar{k} \in \omega \) be such that \( \bar{k} \in \hat{a} \) and \( k \in \hat{a} \Rightarrow k \in \hat{b} \) for every \( k \geq \bar{k} \). Define
\( f \colon (X,d) \to (X,d) \) as follows:
\[
f(x) =
\begin{cases}
x_{\bar{k}} & \text{if } x \in B_d(x_i,s_i), i < \bar{k}, i \in \hat{a} \\
y_{\bar{k}} & \text{if } x \in B_d(x_0, r_{\bar{k}}) \setminus \bigcup \{ B_d(x_i,s_i) \mid i < \bar{k}, i \in \hat{a} \} \\
y_i & \text{if } x \in B_d(x_i,s_i), i \geq \bar{k}, i \notin \hat{a} \\
x & \text{otherwise.}
\end{cases}%
\]
It is straightforward to check that \( f \) reduces \( \psi(a) \) to \( \psi(b) \), so we only need to check that \( f \) is nonexpansive, and this amounts to
check that if \( x,y \) are distinct points of \( X \) which fall in different cases in the definition of \( f \), then \( d(f(x),f(y)) \leq d(x,y) \). A careful
inspection shows that the unique nontrivial cases are the following:
\begin{enumerate}[{case} A:]
\item
\( x \in B_d(x_0,r_{\bar{k}}) \), while \( y \notin B_d(x_0,r_{\bar{k}}) \cup \bigcup \{ B_d(x_i,s_i) \mid i \geq \bar{k}, i \notin \hat{a} \} \). Then
\( d(x,y) \geq r_{\bar{k}} \) (by case assumption) and \( d(x,f(x)) = r_{\bar{k}} \) (because
either \( f(x) = x_{\bar{k}} \) or \( f(x) = y_{\bar{k}} \), depending on whether \( x \in B_d(x_i,s_i) \) for some \( i \in \hat{a} \) smaller than
\( \bar{k} \) or not).
Since in the case under consideration \( f(y) = y \), we get that either \( d(f(x),f(y)) \leq r_{\bar{k}} \), or else \( d(f(x),f(y)) = d(f(x),y) = d(x,y) \)
by the isosceles triangle rule: in both cases, \( d(f(x),f(y)) \leq d(x,y) \) as required.
\item
\( x \in B_d(x_0, r_{\bar{k}}) \setminus \bigcup \{ B_d(x_i,s_i) \mid i < \bar{k}, i \in \hat{a} \} \), while \( y \in B_d(x_i,s_i) \) for some
\( i \geq \bar{k} \), \( i \notin \hat{a} \). Then since \( d(x,x_0) < r_{\bar{k}} \) and \( d(x_0,y) = r_i \geq r_{\bar{k}} \), we get \( d(x,y) = r_i \).
Since by case assumption \( f(x) = y_{\bar{k}} \) and \( f(y) = y_i \), either \( f(x) = f(y) \) (in case \( i = \bar{k} \)) or \( d(f(x),f(y)) = r_i \), and
hence we again get \( d(f(x),f(y)) \leq d(x,y) \), as required.
\item
\( x \in B_d(x_i,s_i) \) for some \( i \geq \bar{k} \), \( i \notin \hat{a} \), while also \( y \notin B_d(x_0,r_{\bar{k}}) \cup\) \( \bigcup \{ B_d(x_i,s_i) \mid i \geq \bar{k}, i \notin \hat{a} \} \). Then \( d(x,y) \geq s_i \), \( d(x,f(x)) = s_i \) (because \( f(x) = y_i \)), and \( f(y) = y \): this implies that either \( d(f(x),f(y)) \leq s_i \) or \( d(f(x),f(y)) = d(f(x),y) = d(x,y) \), so that in any case \( d(f(x),\) \(f(y)) \leq d(x,y) \).
\end{enumerate}
This concludes the proof of part (1).
\medskip
We now prove part (2) of the theorem. Given \(a,b \subseteq \omega \), assume that \( f \colon (X,d) \to (X,d) \) is a \( \mathsf{Lip}(d) \)-reduction of \( \psi(a) \) to \( \psi(b) \), and let \( 0 \neq n \in \omega \) be such that \( d(f(x),f(y)) \leq r_n \cdot d(x,y) \) for every \( x,y \in X \) (such an \( n \) exists because \( (r_n)_{n \in \omega} \) is unbounded in \( \mathbb{R}^+ \) by (b) above).
Notice that, necessarily,
\[
f \left(\bigcup \{ B_d(x_i,s_i) \mid i \in \hat{a} \}\right) = f(\psi(a)) \subseteq \psi(b) \subseteq \bigcup_{j \in \omega } B_d(x_j,s_j).
\]
We now argue as in the proof of~\cite[Theorem 5.4]{MottoRos:2012c}.
\begin{claim} \label{claim:Lip1}
Fix an arbitrary \( i \in \hat{a} \). If there are \( x \in B_d(x_i,s_i) \) and \( j \geq n \) such that \( f(x) \in B_d(x_j,s_j) \), then \( f(B_d(x_i,s_i)) \subseteq B_d(x_j,s_j) \).
\end{claim}
\begin{proof}[Proof of the Claim]
Suppose not, and let \( y \in B_d(x_i,s_i) \) and \( j' \neq j \) be such that \( f(y) \in B_d(x_{j'},s_{j'}) \). Then
\[
d(f(x),f(y)) = \max \{ r_j,r_{j'} \} \geq r_j \geq r_n \cdot 1 > r_n \cdot s_i > r_n \cdot d(x,y),
\]
contradicting the choice of \( n \).
\end{proof}
\begin{claim}\label{claim:Lip2}
For every \( i \in \hat{a} \) such that \( i > n \), \( f(B_d(x_i,s_i)) \subseteq B_d(x_j,s_j) \) for some \( j \geq i \).
\end{claim}
\begin{proof}
Suppose towards a contradiction that there are \( x \in B_d(x_i,s_i) \) and \( j < i \) such that \( f(x) \in B_d(x_j,s_j) \), so that, in particular, \( j \in \hat{b} \)
because \( x \in \psi(a) \) and \( f \) reduces \( \psi(a) \) to \( \psi(b) \). Then since \( d(x,y_i) = s_i \), by our choice of the \( s_i \)'s we get
\[
d(f(x),f(y_i)) \leq r_n \cdot s_i \leq r_{i-1} \cdot s_i < s_{i-1} \leq s_j,
\]
and hence \( f(y_i) \in B_d(f(x),s_j) = B_d(x_j,s_j) \subseteq \psi(b) \): but this contradicts the fact that \( f \) is a reduction of \( \psi(a) \) to \( \psi(b) \),
because \( y_i \notin \psi(a) \) while \( B_d(x_j,s_j) \subseteq \psi(b) \) since \( j \in \hat{b} \). Thus, given an arbitrary \( x \in B_d(x_i,s_i) \) there is \( j \geq i > n\) such that \( f(x) \in B_d(x_j,s_j) \): by Claim~\ref{claim:Lip1}, we then get \( f(B_d(x_i,s_i)) \subseteq B_d(x_j,s_j) \), as required.
\end{proof}
Let now \( \bar{\imath} \) be the smallest element of \( \hat{a} \). By Claim~\ref{claim:Lip1}, either \( f(B_d(x_{\bar{\imath}},s_{\bar{\imath}})) \subseteq \bigcup_{j < n } B_d(x_j,s_j) \), or \( f(B_d(x_{\bar{\imath}}s_{\bar{\imath}})) \subseteq B_d(x_j,s_j) \) for some \( j \geq n \). Therefore, in both cases there is \( \bar{k} > \max \{ n, \bar{\imath} \} \) such that \( f(B_d(x_{\bar{\imath}},s_{\bar{\imath}})) \subseteq \bigcup_{j \leq \bar{k}} B_d(x_j,s_j) \): we claim that \( k \in \hat{a} \Rightarrow k \in \hat{b} \) for every \( k \geq \bar{k} \), which also implies \( a \subseteq^* b \).
Fix \( k \geq \bar{k} \) such that \( k \in \hat{a} \). By Claim~\ref{claim:Lip2} and \( \bar{k} > n \), there is \( j \geq k \) such that \( f(B_d(x_k,s_k)) \subseteq B_d(x_j,s_j) \). Assume towards a contradiction that \( j > k \):
then
\[
d(f(x_{\bar{\imath}}),f(x_k)) = r_j > r_k \cdot r_k > r_n \cdot r_k = r_n \cdot d(x_{\bar{\imath}},x_k),
\]
contradicting the choice of \( n \). Therefore \( f(B_d(x_k,s_k)) \subseteq B_d(x_k,s_k) \), which in particular implies that \( \psi(b) \cap B_d(x_k,s_k) \neq \emptyset \) (since \( x_k \in \psi(a) \) and \( f \) reduces \( \psi(a) \) to \( \psi(b) \)): but this means that \( k \in \hat{b} \), and hence we are done.
\end{proof}
Applying Theorem~\ref{th:unboundeddiam} to the space \( \mathbb{Q}_p \) of \( p \)-adic numbers (which is possible by Example~\ref{xmp:p-adic}) we get the following corollary.
\begin{corollary}\label{cor:p-adic}
Let \( p \) be a prime natural number, and let \( d_p \) be the \( p \)-adic metric on the space \( \mathbb{Q}_p \). Then both the \( \mathsf{Lip}(d_p) \)- and the \( \L(d_p) \)-hierarchies are very bad already when restricted to clopen subsets of \( \mathbb{Q}_p \).
\end{corollary}
The condition on the diameter of \( X = (X,d) \) used to prove Theorem~\ref{th:unboundeddiam} is very weak: this allows us to construct extremely simple (in fact: discrete) ultrametric Polish spaces \( X = (X,d) \) with the property that their \( \mathsf{Lip}(d) \)- and \( \L(d) \)-hierarchies are both very bad, despite the fact that all their subsets are topologically simple (i.e.\ clopen).
\begin{corollary} \label{cor:countable}
There exists a discrete (hence countable) ultrametric Polish space \( X_0 = (X_0,d_0) \) such that \( (\mathscr{P}(\omega), \subseteq^*) \) embeds into both the \( \mathsf{Lip}(d_0) \)- and the \( \L(d_0) \)-hierarchy on (the clopen subsets of) \( X_0 \). In particular, \( {\sf Deg}(\mathsf{Lip}(d_0)) = {\sf Deg}_{\boldsymbol{\Delta}^0_1}(\mathsf{Lip}(d_0)) \) and \( {\sf Deg}(\L(d_0)) = {\sf Deg}_{\boldsymbol{\Delta}^0_1}(\L(d_0)) \) are both very bad.
\end{corollary}
\begin{proof}
Let \( X_0 = \{ x^i_n \mid n \in \omega, i = 0,1 \} \) and set
\[
d_0(x^i_n,x^j_m) =
\begin{cases}
0 & \text{if } n = m \text{ and } i = j \\
2^{-n} & \text{if } n = m \text{ and } i \neq j \\
\max \{ n,m \} & \text{if } n \neq m.
\end{cases}%
\]
It is easy to check that \( X_0 = (X_0,d_0) \) is a discrete ultrametric Polish space. Now observe that the diameter of \( X_0 \) is nontrivially unbounded. In fact, given \( n \in \omega \) and \( \varepsilon \in \mathbb{R}^+ \), let \( k \) be minimal such that \( 2^{-k} < \varepsilon \) and \( l = \max \{ n, k \} \): then \( d_0(x^0_l,x^0_{l+1}) = l+1 > n \), and the points \( x^1_l \) and \( x^1_{l+1} \) witness that \( x^0_l \) and \( x^0_{l+1} \) are not \( \varepsilon \)-isolated. Therefore \( X_0 \) is as desired by Theorem~\ref{th:unboundeddiam}.
\end{proof}
The next proposition extends Theorem~\ref{th:unifcontandlip} and shows that the condition on \( X \) in Theorem~\ref{th:unboundeddiam} is optimal.
\begin{theorem} \label{th:notunbounded}
Let \( X = (X,d) \) be an ultrametric Polish space whose diameter is not nontrivially unbounded. Then the \( \mathsf{Lip}(d) \)-hierarchy \( {\sf Deg}_{\boldsymbol{\Delta}^1_1}(\mathsf{Lip}(d)) \) on Borel subsets of \( X \) is very good.
\end{theorem}
\begin{proof}
Let \( n \in \omega \) and \( \varepsilon \in \mathbb{R}^+ \) be such that for every \( x,y \), if \( d(x,y) > n \) then at least one of \( x \) and \( y \) is \( \varepsilon \)-isolated.
Let us first consider the degenerate case in which all points of \( X \) are \(\varepsilon\)-isolated. Since constant functions are always (trivially) Lipschitz, we get that the sets \( X \) and \( \emptyset \) are \( \mathsf{Lip}(d) \)-incomparable, and that they are both (strictly) \( \leq_{\mathsf{Lip}(d)} \)-below any other set \( \emptyset,X \neq A \subseteq X \). Assume now that \( B \subseteq X \) is another set which is different from both \( \emptyset \) and \( X \): we claim that then \( A \equiv_{\mathsf{Lip}(d)} B \). To see this, fix \( \bar{x} \in B \) and \( \bar{y} \in \neg B \), and for every \( x \in X \) set \( f(x) = \bar{x} \) if \( x \in A \) and \( f(x) = \bar{y} \) if \( x \in \neg A \). Then \( f \colon (X,d) \to (X,d) \) reduces \( A \) to \( B \). Moreover, since for all distinct \( x,y \in X \) we have \( d(x,y) \geq \varepsilon \) (because both \( x \) and \( y \) are \(\varepsilon\)-isolated), we get
\[
d(f(x),f(y)) \leq d(\bar{x},\bar{y}) = \frac{d(\bar{x},\bar{y})}{\varepsilon} \cdot \varepsilon \leq \frac{d(\bar{x},\bar{y})}{\varepsilon} \cdot d(x,y),
\]
so that \( f \) is Lipschitz with constant \( \frac{d(\bar{x},\bar{y})}{\varepsilon} \). This shows that \( A \leq_{\mathsf{Lip}(d)} B \). Switching the role of \( A \) and \( B \), we get that also \( B \leq_{\mathsf{Lip}(d)} A \), and hence we are done. Therefore we have shown that the \( \mathsf{Lip}(d) \)-hierarchy on \( X \) is constituted by the two \( \mathsf{Lip}(d) \)-incomparable degrees \( [\emptyset]_{\mathsf{Lip}(d)} = \{ \emptyset \} \) and \( [ X]_{\mathsf{Lip}(d)} = \{ X \} \), plus a unique \( \mathsf{Lip}(d) \)-degree above them containing all other subsets of \( X \), and is thus (trivially) very good.
Assume now that there is a non-\(\varepsilon\)-isolated point \( x_0 \in X \), and set \( X' = B_d(x_0,\) \(n+1) \). By our choice of \( n \) and \( \varepsilon \), we get that \( d(x,y) \geq n+1 \) for every \( x \in X' \) and \( y \in X \setminus X' \), and that each \( y \in X \setminus X' \) is \( \varepsilon \)-isolated (because \( d(x_0,y) > n \) and \( x_0 \) is not \(\varepsilon\)-isolated). We first prove the following useful claim.
\begin{claim} \label{claim:transfer}
Let \( A,B \subseteq X \) be such that \( B \neq \emptyset,X \). If there is a Lipschitz reduction \( f \colon (X',d_{X'}) \to (X',d_{X'}) \) of \( A' = A \cap X' \) to \( B' = B \cap X' \), then \( A \leq_{\mathsf{Lip}(d)} B \).
\end{claim}
\begin{proof}
Let \( f \) be as in the hypothesis of the claim, and let \( 1 \leq k \in \omega \) be such that \( d(f(x),f(y)) \leq k \cdot d(x,y) \) for every \( x,y \in X' \). Fix \( \bar{x} \in B \) and \( \bar{y} \in \neg B \), and extend \( f \) to the map \( \hat{f} \colon (X,d) \to (X,d) \) by letting \( \hat{f}(x) = \bar{x} \) if \( x \in A \setminus X' \) and \( \hat{f}(x) = \bar{y} \) if \( x \in X \setminus (X' \cup A) \). Clearly, \( \hat{f} \) reduces \( A \) to \( B \), and we claim that \( \hat{f} \) is Lipschitz with constant \( c \), where \( c \) is
\[
c = \max \left \{ k , \frac{d(\bar{x},\bar{y})}{\varepsilon}, \frac{d(x_0,\bar{x})}{n+1}, \frac{d(x_0,\bar{y})}{n+1} \right\}.
\]
Fix arbitrary \( x,y \in X \). If \( x,y \in X' \), then
\[
d(\hat{f}(x),\hat{f}(y)) = d(f(x),f(y)) \leq k \cdot d(x,y) \leq c \cdot d(x,y)
\]
by our choice of \( k \in \omega \). If \( x,y \in X \setminus X' \), then \( d(x,y) \geq \varepsilon \) because both \( x \) and \( y \) are \(\varepsilon\)-isolated, and either \( \hat{f}(x) = \hat{f}(y) \) or \( d(\hat{f}(x),\hat{f}(y)) = d(\bar{x},\bar{y}) \). Therefore in both cases
\[
d(\hat{f}(x),\hat{f}(y)) \leq \frac{d(\bar{x},\bar{y})}{\varepsilon} \cdot \varepsilon \leq c \cdot d(x,y).
\]
Let now \( x \in X' \) and \( y \in X \setminus X' \), and assume without loss of generality that \( \hat{f}(y) = \bar{x} \) (the case \( \hat{f}(y) = \bar{y} \) is analogous, just systematically replace \( \bar{x} \) with \( \bar{y} \) in the argument below). Then either \( \bar{x} \in X' \), in which case \( d(\hat{f}(x),\hat{f}(y)) < n+1 \leq d(x,y) \leq c \cdot d(x,y) \) (since \( c \geq k \geq 1 \)), or else
\[
d(\hat{f}(x),\hat{f}(y)) = d(x_0,\bar{x}) = \frac{d(x_0,\bar{x})}{n+1} \cdot n+1 \leq c \cdot d(x,y).
\]
The case \( x \in X \setminus X' \) and \( y \in X' \) can be treated similarly, so in all cases we obtained \( d(\hat{f}(x),\hat{f}(y)) \leq c \cdot d(x,y) \), as required.
\end{proof}
We now want to show that the \( {\sf SLO}^{\mathsf{Lip}(d)} \) principle holds for Borel subsets of \( X \), so let us fix arbitrary Borel \(A,B \subseteq X \). Assume first that \( B = X \). Then either \( A = X \), in which case the identity map on \( X \) witnesses \( A \leq_{\mathsf{Lip}(d)} B \), or else \( \neg A \neq \emptyset \), in which case any constant map with value \( \bar{x} \in \neg A \) witnesses \( B \leq_{\mathsf{Lip}(d)} \neg A \). The symmetric case \( B = \emptyset \) can be dealt with in a similar way, so in what follows we can assume without loss of generality that \( B \neq \emptyset,X \). Moreover, switching the role of \( A \) and \( B \) in the argument above we may further assume that \( A \neq \emptyset , X \). Set \( A' = A \cap X' \) and \( B' = B \cap X' \). Since \( X' \) has bounded diameter, by Theorem~\ref{th:unifcontandlip} there is a Lipschitz function \( f \colon (X',d) \to (X',d) \) such that either \( f^{-1}(B') = A' \) or \( f^{-1}(X' \setminus A') = B' \). Since \( \neg A \cap X' = X' \setminus A' \), applying Claim~\ref{claim:transfer} we get that either \( A \leq_{\mathsf{Lip}(d)} B \) or \( B \leq_{\mathsf{Lip}(d)} \neg A \), as desired.
Finally, let us show that the \( \mathsf{Lip}(d) \)-hierarchy on Borel subsets of \( X \) is also well-founded. Suppose not, and let \( (A_n)_{n \in \omega} \) be a sequence of Borel subsets of \( X \) such that \( A_{n+1} <_{\mathsf{Lip}(d)} A_n \) for every \( n \in \omega \). Notice that this in particular implies that \( A_n \neq \emptyset,X \) for every \( n \in \omega \). By Claim~\ref{claim:transfer} and our choice of the \( A_n \)'s, for all \( i < j \) there is no Lipschitz \( f \colon (X',d_{X'}) \to (X',d_{X'}) \) reducing \( A_i \cap X' \) to \( A_j \cap X' \).
Using Ramsey's theorem, we get that there is an infinite \( I \subseteq \omega \) such that either \( \forall i,j \in I \, (i < j \Rightarrow A_j \cap X' \leq_{\mathsf{Lip}(d_{X'})} A_i \cap X') \), or else \( \forall i,j \in I \, (i < j \Rightarrow A_j \cap X' \nleq_{\mathsf{Lip}(d_{X'})} A_i \cap X') \): in the former case the sequence \( (A_i \cap X' )_{i \in \omega} \) would give an infinite (strictly) descending chain in the \( \mathsf{Lip}(d_{X'}) \)-hierarchy on \( X' \), while in the latter it would give an infinite antichain (in the same hierarchy). Since \( X' \) has bounded diameter and all the sets \( A_i \cap X' \) are clearly Borel in it, both possibilities contradicts Theorem~\ref{th:unifcontandlip}, and hence we are done.
\end{proof}
\begin{corollary}
Let \( X = (X,d) \) be an ultrametric Polish space.Then the following are equivalent:
\begin{enumerate}
\item
the diameter of \( X \) is nontrivially unbounded;
\item
\( (\mathscr{P}(\omega), \subseteq^*) \) embeds into \( {\sf Deg}_{\boldsymbol{\Delta}^0_1}(\mathsf{Lip}(d)) \);
\item
the \( \mathsf{Lip}(d) \)-hierarchy on Borel (equivalently, clopen) subsets of \( X \) is very bad;
\item
the \( \mathsf{Lip}(d) \)-hierarchy on Borel (equivalently, clopen) subsets of \( X \) is not very good.
\end{enumerate}
Hence \( {\sf Deg}_{\boldsymbol{\Delta}^1_1}(\mathsf{Lip}(d)) \) is either very good or very bad.
\end{corollary}
\begin{proof}
By Theorem~\ref{th:unboundeddiam} and Theorem~\ref{th:notunbounded}.
\end{proof}
\begin{corollary} \label{cor:unboundeddiam}
Let \( X = (X,d) \) be a \emph{perfect} ultrametric Polish space. Then
\begin{enumerate}
\item
\( X \) has bounded diameter \( \iff \) the \( \mathsf{Lip}(d) \)-hierarchy on Borel (equivalently, clopen) subsets of \( X \) is very good;
\item
\( X \) has unbounded diameter \( \iff \) the \( \mathsf{Lip}(d) \)-hierarchy on Borel (equivalently, clopen) subsets of \( X \) is very bad, and in fact in this case
the partial order \( ( \mathscr{P}(\omega), \subseteq^*) \) embeds into \( {\sf Deg}_{\boldsymbol{\Delta}^0_1}(\mathsf{Lip}(d)) \).
\end{enumerate}
\end{corollary}
Let us consider again the ultrametrics \( d_{\phi} \) introduced in Definition \ref{def: d_phi}.
\begin{corollary}
Let \( \phi \colon \omega \to \mathbb{R}^+ \) have unbounded range and suppose that \( \inf \operatorname{rg} (\phi) >0 \). Then \( (\mathscr{P}(\omega), \subseteq^*) \) embeds into both the \( \mathsf{Lip}(d_\phi) \)- and \( \L(d_\phi) \)-hierarchy on clopen subsets of \( \pre{\omega}{\omega} \), and therefore both \( {\sf Deg}_{\boldsymbol{\Delta}^0_1}(\mathsf{Lip}(d_\phi)) \) and
\( {\sf Deg}_{\boldsymbol{\Delta}^0_1}(\L(d_\phi)) \)
are very bad. Conversely, if \( \phi \) has bounded range, then the \( \mathsf{Lip}(d_\phi) \)-hierarchy \( {\sf Deg}_{\boldsymbol{\Delta}^1_1}\) \((\mathsf{Lip}(d_\phi)) \) on Borel subsets of \( \pre{\omega}{\omega} \) is very good.
\end{corollary}
\begin{proof}
Observe that \( (\pre{\omega}{\omega},d_\phi) \) is a perfect ultrametric Polish space, and that it has unbounded diameter if and only if the \( \operatorname{rg}( \phi) \) is unbounded in \( \mathbb{R}^+ \); then apply Theorems~\ref{th:unboundeddiam} and~\ref{th:unifcontandlip}.
\end{proof}
\section{Nonexpansive reducibilities}
\begin{definition} \label{def:honestincreasingsequence}
Let \( X = (X,d) \) be an ultrametric Polish space. We say that \( R(d) \) contains an \emph{honest increasing sequence} if it contains a strictly increasing sequence \( (r_n)_{n \in \omega} \) such that for some sequences \( (x_n)_{n \in \omega} \), \( (y_n)_{n \in \omega} \) of points in \( X \)
the following conditions holds:
\begin{enumerate}[(i)]
\item
\( d(x_n,x_m) = r_{\max\{ n,m \}} \) for all distinct \( n,m \in \omega \);
\item
\( d(x_0,y_0) < r_0 \) and \( d(x_{n+1},y_{n+1}) < d(x_n,y_n) \) for all \( n \in \omega \).
\end{enumerate}
\end{definition}
The above condition is somewhat technical, but in case \( X = (X,d) \) is a perfect ultrametric Polish space it is immediate to check that \( R(d) \) contains an honest increasing sequence if and only if one of the following equivalent%
\footnote{To see that these two conditions are indeed equivalent, argue as in the first part of the proof of Theorem~\ref{th:unboundeddiam}.}
conditions are satisfied:
\begin{enumerate}
\item
there is \( X' \subseteq X \) such that \( R(d_{X'}) \) has order type \( \omega \) (with respect to the usual ordering on \( \mathbb{R} \));
\item
there is a sequence \( (x_n)_{n \in \omega} \) of points in \( X \) and a strictly increasing sequence \( (r_n)_{n \in \omega} \) of distances in \( R(d) \) such that \( d(x_n,x_m) = r_{\max \{n,m \}} \) for all distinct \( n,m \in \omega \).
\end{enumerate}
Notice also that if the diameter of an ultrametric Polish space \( X = (X,d) \) is nontrivially unbounded, then \( R(d) \) contains an honest increasing sequence by the first part of the proof of Theorem~\ref{th:unboundeddiam}.
\begin{theorem} \label{th:increasingdistances}
Let \( X = (X,d) \) be a ultrametric Polish space such that \( R(d) \) contains an honest increasing sequence. Then there is a map \( \psi \) from \( \mathscr{P}(\omega) \) into the clopen subsets of \( X \) such that for all \( a,b \subseteq \omega \)
\[
a \subseteq^* b \iff \psi(a) \leq_{\L(d)} \psi(b).
\]
\end{theorem}
\begin{proof}
Argue similarly to Theorem~\ref{th:unboundeddiam}, with the following variations:
\begin{enumerate}[(a)]
\item
let the sequences \( (x_n)_{n \in \omega} \), \( (y_n)_{n \in \omega} \), and \( (r_n)_{n \in \omega} \) constructed at the beginning of the proof of Theorem~\ref{th:unboundeddiam} be witnesses of the fact that \( R(d) \) contains an honest increasing sequence (forgetting about the extra properties required in Theorem~\ref{th:unboundeddiam}), and set \( s_n = d(x_n,y_n) \);%
\footnote{Clearly, the points \( x_n \) and \( y_n \) can again be chosen in any given countable dense set \( Q \subseteq X \).}
\item
given \( a \subseteq \omega \), define \( \psi(a) \) as before, i.e.\ set \( \psi(a) = \bigcup_{i \in \hat{a}} B_d(x_i,s_i) \), where \( \hat{a} = \{ 2i \mid i \in \omega \} \cup \{ 2i+1 \mid i \in a \} \);
\item
to prove the backward direction, use an argument similar to that of Theorem~\ref{th:unboundeddiam}, but dropping any reference to the integer \( n \) (this simplification can be adopted here because we have to deal only with nonexpansive functions). More precisely:
let \( f \) be a nonexpansive reduction of \( \psi(a) \) to \( \psi(b) \). Then for every \( i \in \hat{a} \) there is a unique \( j \in \omega \) such that \( f(B_d(x_i,s_i)) \subseteq B_d(x_j,s_j) \) (because of the choice of the \( x_i \), \( y_i \)'s and the fact that \( f \) is nonexpansive). Arguing as in Claim~\ref{claim:Lip2}, one immediately sees that we cannot have \( j < i \) because in such case \( s_i \leq s_j \). Conclude as in the final part of the proof of Theorem~\ref{th:unboundeddiam}, using the fact that \( r_k < r_j \) for every \( j > k \). \qedhere
\end{enumerate}
\end{proof}
\begin{corollary} \label{cor:X_1}
There is an ultrametric Polish space \( X_1 = (X_1,d_1) \) whose set of nonzero distances \( R(d_1) \) is bounded away from \( 0 \) (hence it is countable and discrete) such that \( (\mathscr{P}(\omega), \subseteq^*) \) embeds into the \( \L(d_1) \)-hierarchy on (clopen subsets of) \( X_1 \). Therefore \( {\sf Deg}(\L(d_1)) = {\sf Deg}_{\boldsymbol{\Delta}^0_1}(\L(d_1)) \) is very bad.
\end{corollary}
\begin{proof}
Let \( X_1 = \{ x^i_n \mid n \in \omega, i = 0,1 \} \) and set
\[
d_1(x^i_n,x^j_m) =
\begin{cases}
0 & \text{if } n = m \text{ and } i = j \\
\frac{1}{2} + 2^{-(n+1)} & \text{if } n = m \text{ and } i \neq j \\
2 - 2^{-\max \{ n,m \}} & \text{if } n \neq m.
\end{cases}%
\]
It is easy to check that \( X_1 = (X_1,d_1) \) is an ultrametric Polish space. Moreover \( r \geq \frac{1}{2} \) for every \( r \in R(d_1) \), hence \( R(d_1) \) is bounded away from \( 0 \). Moreover, the sequences obtained by setting \( r_n = 2 - 2^{-n} \), \( x_n = x_n^0 \), and \( y_n = x_n^1 \) witness that \( R(d_1) \) contains an honest increasing sequence. Hence the result follows from Theorem~\ref{th:increasingdistances}.
\end{proof}
\begin{remark}
Notice that if an ultrametric Polish space \( X = (X,d) \) satisfies the hypothesis of Corollary~\ref{cor:X_1} (i.e.\ it is such that \( R(d) \) is bounded away from \( 0 \)), then its \( \mathsf{Lip}(d) \)-hierarchy is always (trivially) very good by Theorem~\ref{th:notunbounded} and the fact that all its points are \( \varepsilon \)-isolated for \( \varepsilon = \inf R(d) > 0 \).
\end{remark}
\begin{corollary} \label{cor:specialincreasing}
Given \( \phi \colon \omega \to \mathbb{R}^+ \) such that \( \inf \operatorname{rg}( \phi)>0 \), if \( \operatorname{rg}(\phi) \) contains an increasing \( \omega \)-sequence then \( ( \mathscr{P}(\omega), \subseteq^*) \) embeds into the \( \L(d_\phi) \)-hierarchy on clopen subsets of \( \pre{\omega}{\omega} \), and therefore \( {\sf Deg}_{\boldsymbol{\Delta}^0_1}(\L(d_\phi)) \) is very bad.
\end{corollary}
\begin{proof}
Notice that \( (\pre{\omega}{\omega}, d_\phi ) \) is always a perfect Polish space, and that \( R(d_\phi) \) has an honest increasing sequence if and only if \( \operatorname{rg}(\phi) \) contains an increasing \( \omega \)-sequence. Then apply Theorem~\ref{th:increasingdistances}.
\end{proof}
\begin{proposition} \label{prop:descendingdistances}
Suppose that \( X = (X,d) \) is an ultrametric Polish space such that \( R(d) \) is either finite or a descending (\( \omega \)-)sequence converging to \( 0 \), let \( I \leq \omega \) be the cardinality of \( R(d) \), and let \( \rho \) be the unique order-preserving map from \( \{ 2^{-i} \mid i < I \} \) and \( R(d) \). Then there is a closed set \( C \subseteq \pre{\omega}{\omega} \) and a bijection \( f \colon C \to X \) such that for all \( x,y \in X \)
\begin{equation} \tag{$*$} \label{eq:isom}
d(x,y) = \rho(\bar{d}(f^{-1}(x),f^{-1}(y))).
\end{equation}
In particular, the structures \( ( \mathscr{P}(X), \leq_{\L(d)}, \neg ) \) and \( ( \mathscr{P}(C), \leq_{\L(\bar{d})}, \neg ) \) are isomorphic.
\end{proposition}
\begin{proof}
Let us first assume that \( I = \omega \), i.e.\ that \( R(d) \) is a descending (\(\omega \)-)sequence converging to \( 0 \). Inductively define the family
\( (A_s)_{s \in \pre{< \omega}{\omega}} \) of subsets of \( X \) by induction on \( \operatorname{lh}(s) \) as follows. Set
\( A_\emptyset = X \). Then let \( \{ B_{s,j} \mid j < J \} \) (for some \( J \leq \omega \)) be an enumeration without repetitions
of the collection \( \{ B_d(x, \rho(2^{-\operatorname{lh}(s)})) \mid x \in A_s \} \), and set
\( A_{s {}^\smallfrown{} j} = B_{s,j} \) if \( j < J \) and \( A_{s {}^\smallfrown{} j} = \emptyset \) otherwise. It is easy to check that the family \( (A_s)_{s \in \pre{< \omega}{\omega}} \) is a Luzin scheme with vanishing diameter consisting of clopen sets. Hence letting \( C = \{ x \in \pre{\omega}{\omega} \mid \bigcap_{n \in \omega} A_{x \restriction n} \neq \emptyset \} \) and \( f \colon C \to X \) be defined by letting \( f(x) \) be the unique element of \( \bigcap_{n \in \omega} A_{x \restriction n} \), we get that \( C \) and \( f \) are as required.
Assume now that \( I \) is finite, so that, in particular, \( X \) is a discrete space. Inductively define the sets \( A_s \) as above for all \( s \in \pre{< \omega}{\omega} \) of length \( \leq I \). Then if \( \operatorname{lh}(s) = I \) the set \( A_s \) is either empty or a singleton. Letting \( C = \{ s {}^\smallfrown{} 0^{(\omega)} \mid \operatorname{lh}(s) = I, A_s \neq \emptyset \} \) and defining \( f \colon C \to X \) by letting \( f(s {}^\smallfrown{} 0^{(\omega)}) \) be the unique element of \( A_s \) we again have that \( C \) and \( f \) are as required.
For the last part, notice that the map \( \mathscr{P}(X) \to \mathscr{P}(C) \colon A \mapsto f^{-1}(A) \) is the desired isomorphism. To see this, simply notice that~\eqref{eq:isom} implies that \( \L(d) = \{ f \circ h \circ f^{-1} \mid h \in \L(\bar{d}_C) \} \).
\end{proof}
\begin{theorem}\label{th:descendingdistances}
Suppose that \( X = (X,d) \) is an ultrametric Polish space such that \( R(d) \) is either finite or a descending (\( \omega \)-)sequence converging to \( 0 \). Then the \( \L(d) \)-hierarchy \( {\sf Deg}_{\boldsymbol{\Delta}^1_1}(\L(d)) \) on Borel subsets of \( X \) is very good.
\end{theorem}
\begin{proof}
By Proposition~\ref{prop:descendingdistances}, it is clearly enough to show that the \( \L(\bar{d}_C) \)-hierarchy on Borel subsets of \( C \) is very good: but this easily follows from the existence of a nonexpansive retraction of \( (\pre{\omega}{\omega}, \bar{d}) \) onto \( (C, \bar{d}_C) \), Lemma~\ref{lemma:retraction}, and the fact that the \( \L(\bar{d}) \)-hierarchy on the Borel subsets of \( \pre{\omega}{\omega} \) is very good.
\end{proof}
\begin{corollary} \label{cor:specialconvergingto0}
Let \( \phi \colon \omega \to \mathbb{R}^+ \) and suppose that \( \operatorname{rg}(\phi) \) is finite (so that trivially \( \inf \operatorname{rg}(\phi) > 0 \)).
Then the \( \L(d_\phi) \)-hierarchy \( {\sf Deg}_{\boldsymbol{\Delta}^1_1}(\L(d_\phi)) \) on Borel subsets of \( \pre{\omega}{\omega} \) is very good.
\end{corollary}
\begin{proof}
Simply observe that under our assumptions the set \( R(d_\phi) \) is always an \( \omega \)-sequence converging to \( 0 \), and then apply Theorem~\ref{th:descendingdistances}.
\end{proof}
Let us now consider the general problem of determining the character of the \( \L(d_\phi) \)-hierarchy on Borel subsets of \( \pre{\omega}{\omega} \) for an arbitrary \( \phi \colon \omega \to \mathbb{R}^+ \) with \(\inf \operatorname{rg} (\phi) >0\). By Corollary~\ref{cor:specialincreasing}, if \( \operatorname{rg}(\phi) \) contains
an increasing \(\omega\)-sequence, then \( {\sf Deg}_{\boldsymbol{\Delta}^1_1}\) \((\L(d_\phi)) \) is very bad, hence we can assume without loss of generality that \( \operatorname{rg}(\phi) \) has order type%
\footnote{Given a linear order \( L = (L, \leq) \), we denote by \( L^* \) the reverse linear oder induced by \( L \), i.e.\ \( L^* = (L,\leq^{-1}) \). Since \( \alpha = \{ \beta \mid \beta < \alpha \} \) (for every ordinal \(\alpha\)), we tacitly identify \(\alpha\) with the linear order \( \alpha = (\alpha, \leq) \), so that \( \alpha^* = (\alpha, \geq) \).}
\( \alpha^* \) for some countable ordinal \(\alpha\). Corollary~\ref{cor:specialconvergingto0} considered the subcase where \( \alpha \) is finite:
the next proposition considers instead the special (but yet significant) subcase where \( \alpha=\omega \) and \( \phi \) is injective.
\begin{notation}
Given a set \( A \subseteq \pre{\omega}{\omega} \) and a finite sequence \( s \in \pre{< \omega}{\omega} \), let \( s {}^\smallfrown{} A = \{ s {}^\smallfrown{} x \mid x \in A \} \). When \( \operatorname{lh}(s) = 1 \), we simplify the notation by setting \( n {}^\smallfrown{} A = \langle n \rangle {}^\smallfrown{} A \), and
with a little abuse of notation we set \( r {}^\smallfrown{} A = \{ r {}^\smallfrown{} x \mid x \in A \} \subseteq \{ r \} \times \pre{\omega}{\omega} \) also when \( r \) is not a natural number.
Finally, given a family \( (A_n)_{n \in \omega} \) of subsets of \( \pre{\omega}{\omega} \), we set \( \bigoplus_{n \in \omega} A_n = \bigcup_{n \in \omega} n {}^\smallfrown{} A_n \).
\end{notation}
\begin{theorem} \label{prop:specialalpha*}
Let \( \phi \colon \omega \to \mathbb{R}^+ \) be such that \( \inf \operatorname{rg}(\phi) > 0 \), and suppose that \(\phi \) is injective and that \( \operatorname{rg}(\phi) \) has order type \( \omega^* \). Then the \( \L(d_\phi) \)-hierarchy \( {\sf Deg}_{\boldsymbol{\Delta}^1_1}(\L(d_\phi)) \) on the Borel subsets of \( \pre{\omega}{\omega} \) is very good.
\end{theorem}
\begin{proof
Using the usual game-theoretic arguments (see e.g.\ \cite{Andretta:2007}), it is easy to see that if a Borel \( A \subseteq \pre{\omega}{\omega} \) is \( \L(\bar{d}) \)-selfdual, then its \( \L(\bar{d}) \)-degree \( [A]_{\L(\bar{d})} \) is followed by an \( \omega_1 \)-chain of \( \L(\bar{d}) \)-selfdual degrees \( (\mathscr{L}^{(\alpha)} [A]_{\L(\bar{d})})_{\alpha < \omega_1} \), where the \( \mathscr{L}^{(\alpha)} [A]_{\L(\bar{d})} \) are recursively defined as follows:
\begin{enumerate}[(i)]
\item
\( \mathscr{L}^{(0)} [A]_{\L(\bar{d})} = [A]_{\L(\bar{d})} \);
\item
\( \mathscr{L}^{(\alpha+1)} [A]_{\L(\bar{d})} = [0 {}^\smallfrown{} C]_{\L(\bar{d})} \) for some/any \( C \in \mathscr{L}^{(\alpha)} [A]_{\L(\bar{d})} \);
\item
for limit \( \alpha \)'s, \( \mathscr{L}^{(\alpha)} [A]_{\L(\bar{d})} = [ \bigoplus_{n \in \omega} C_n ]_{\L(\bar{d})} \), where \( C_n \in \mathscr{L}^{(\alpha_n)} [A]_{\L(\bar{d})} \) for each \( n \in \omega \), and \( (\alpha_n)_{n \in \omega} \) is some/any increasing sequence cofinal in \(\alpha\).
\end{enumerate}
We will use the following known facts about the Baire space $(\pre{\omega}{\omega},\bar{d})$.
\begin{enumerate}[\( \bullet \)]
\item
A set $A$ is \emph{self-contractible} (i.e.\ reducible to itself via a contraction) if and only if it is $\L(\bar{d})$-nonselfdual; in this case the iterates of the contraction are reductions of $A$ to itself and have a unique common fixed point (see~\cite[Corollary 4.4]{MottoRos:2012c}).
\item
The $\L(\bar{d})$-nonselfdual degrees coincide with the $\mathsf{W}(\bar{d})$-nonselfdual degrees (see e.g.~\cite[Theorem 3.1]{VanWesep:1978}).
\item
Every $\mathsf{Lip}(\bar{d})$-selfdual degree \( [A]_{\mathsf{Lip}(\bar{d})} \) is of the form $\bigcup \{[0^{(n)} {}^\smallfrown{} A']_{\L(\bar{d})} \mid {n<\omega}\}$ for some \( \L(\bar{d}) \)-selfdual set $A'$; if instead \( [A]_{\mathsf{Lip}(\bar{d})} \) is \( \mathsf{Lip}(\bar{d}) \)-nonselfdual, then \( [A]_{\mathsf{Lip}(\bar{d})} = [A]_{\L(\bar{d})} \) (see~\cite{MottoRos:2010}).
\item
If $A<_{\mathsf{Lip}(\bar{d})} B$, then for all $\varepsilon \in \mathbb{R}^+$ there is a Lipschitz reduction of $A$ to $B$ with constant $\varepsilon$ (see the end of Section 4 in~\cite{MottoRos:2012c}).
\item
Let \( \mathsf{W} = \mathsf{W}(\pre{\omega}{\omega}) \) be the set of all continuous functions from \( \pre{\omega}{\omega} \) into itself, which is clearly a reducibility. Then every $\mathsf{W}$-selfdual degree \( [A]_\mathsf{W} \) is of the form $\bigcup \{ \mathscr{L}^{(\alpha)}([A']_{\L(d)}) \mid \alpha < \omega_1 \}$ for some \( \L(\bar{d}) \)-selfdual set $A'$ (see e.g.\ \cite{Andretta:2007}).
\end{enumerate}
Note that $(\pre{\omega}{\omega}, d_{\phi})$ is isometric to the space $Y=\bigcup_{r \in \operatorname{rg}(\phi)} r^\smallfrown \pre{\omega}{\omega} $ equipped with the ultrametric (which with a little abuse of notation will be denoted by \( d_\phi \) again)
\[
d_{\phi} (r^\smallfrown x,s^\smallfrown y) =
\begin{cases}
0 & \text{if } r=s \text{ and }x=y, \\
\max\{r,s\} & \text{if }r \neq s, \\
r \cdot 2^{-(n+1)} & \text{if } r=s \text{ and } n \text{ is least such that } x(n) \neq y(n).
\end{cases}
\]
\begin{claim} \label{claim:normalforms}
Every Borel subset \( \bar{C} \) of $Y = (Y, d_{\phi})$ is $\L(d_\phi)$-equivalent to one of the following \emph{(\(\L(d_\phi) \)-)normal forms \( \bar{A} \)} (where in what follows $A,A_n \subseteq \pre{\omega}{\omega}$ and \( < \) is the usual order on the reals):
\begin{enumerate}
\item
$\bar{A}=\bigcup_{n\in\omega} r_n^\smallfrown A_n$, where the sequence of the \( A_n \)'s is $<_{\mathsf{Lip}(\bar{d})}$-increasing and the sequence $(r_n)_{n \in \omega}$ in $\operatorname{rg}(\phi)$ is strictly \( < \)-decreasing.
\item
$\bar{A}=\bigcup_{n\in\omega} r_n^\smallfrown A_n$, where the sequence of the \( A_n \)'s is $<_{\L(\bar{d})}$-increasing, $A_m\equiv_{\mathsf{Lip}(\bar{d})} A_n$ for all $m,n \in \omega$, and the sequence $(r_n)_{n \in \omega}$ in $\operatorname{rg}(\phi)$ is strictly \( < \)-decreasing.
\item
$A$ is $\L(\bar{d})$-nonselfdual and
\begin{enumerate}
\item
$\bar{A}=r^\smallfrown A $ for some $r \in \operatorname{rg}(\phi)$, or
\item
$\bar{A}=(r_0^\smallfrown A)\cup (r_1^\smallfrown (\neg A))$ for some $r_0, r_1 \in \operatorname{rg}(\phi)$ with $r_0>r_1$, or
\item
\( \bar{A}=\bigcup_{i \in \omega} r_{2i} {}^\smallfrown{} A \cup \bigcup_{i \in \omega} r_{2i+1} {}^\smallfrown{} (\neg A) \) for some strictly \( < \)-decreasing sequence $(r_n)_{n \in \omega}$ in $\operatorname{rg}(\phi)$.
\end{enumerate}
\item
$A$ is $\mathsf{L}(\bar{d})$-selfdual and
\begin{enumerate}
\item $\bar{A}=r {}^\smallfrown{} A $ for some $r\in \operatorname{rg}(\phi)$, or
\item \( \bar{A}=\bigcup_{n\in\omega} r_n^\smallfrown A \) for some strictly \( < \)-decreasing sequence $(r_n)_{n \in \omega}$ in $\operatorname{rg}(\phi)$.
\end{enumerate}
\end{enumerate}
\end{claim}
\begin{proof}[Proof of the Claim]
Let us sketch how to obtain these normal forms. We will often use the following easy fact. Let \( D \subseteq \operatorname{rg}(\phi) \), \( \rho \colon D \to \operatorname{rg}(\phi) \) be a non-\( < \)-increasing map, \( \{ f_r \colon \pre{\omega}{\omega} \to \pre{\omega}{\omega} \mid r \in D \} \subseteq \L(\bar{d}) \), and \( f' \colon \bigcup_{r \in \operatorname{rg}(\phi) \setminus D} (r {}^\smallfrown{} \pre{\omega}{\omega}) \to Y \) be a nonexpansive map (with respect to \( d_\phi \)): then the map \( f \colon Y \to Y \) defined by
\[
f(r {}^\smallfrown{} x) =
\begin{cases}
\rho(r) {}^\smallfrown{} f_r(x) & \text{if } r \in D \\
f'(r {}^\smallfrown{} x) & \text{otherwise}
\end{cases}%
\]
is in \( \L(d_\phi) \).
Now let $\bar{C}=\bigcup_{r\in \operatorname{rg}(\phi)} (r^\smallfrown C_r)$ be an arbitrary Borel subset of $(Y, d_{\phi})$, and set $\mathcal{C}=\{C_r\mid r\in \operatorname{rg}(\phi)\}$, so that each \( C_r \) is a Borel subset of \( \pre{\omega}{\omega} \).
If $\mathcal{C}$ has no $\mathsf{Lip}(\bar{d})$-maximal element, choose a strictly \( < \)-decreasing sequence $(r_n)_{n\in\omega}$ in $\operatorname{rg}(\phi)$ such that $(C_{r_n})_{n\in\omega}$ is strictly $<_{\mathsf{Lip}(d_{\phi})}$-increasing and $<_{\mathsf{Lip}(d_{\phi})}$-cofinal in $\mathcal{C}$. Then $\bar{A}=\bigcup_{n\in\omega} C_{r_n}$ is in the normal form (1), and moreover it is easy to see that $\bar{A}\equiv_{\mathsf{L}(d_{\phi})}\bar{C}$.
Otherwise, if $\mathcal{C}$ has a $\mathsf{Lip}(\bar{d})$-maximal element but no $\mathsf{L}(\bar{d})$-maximal element, then we can similarly find a set $\bar{A}$ in the normal form (2) which is $\mathsf{L}(d_{\phi})$-equivalent to $\bar{C}$.
Now suppose that there is an $\mathsf{L}(\bar{d})$-maximal element \( B \) among the sets in $\mathcal{C}$. Suppose first that \( B \) is $\mathsf{L}(\bar{d})$-nonselfdual.
If there is no $C\in \mathcal{C}$ with $C\equiv_{\mathsf{L}(\bar{d})} \neg B$, then we choose some $r\in \operatorname{rg}(\phi)$ with $C_r\equiv_{\mathsf{L}(\bar{d})} B$. Using the assumption $\inf \operatorname{rg}(\phi)>0$ and the fact mentioned at the beginning of the proof that $\mathsf{L}(\bar{d})$-nonselfdual sets are self-contractible with arbitrarily small Lipschitz constant, it follows that $\bar{A}=r^\smallfrown C_r\equiv_{\mathsf{L}(d_{\phi})} \bar{C}$, and $\bar{A}$ is in the normal form (3a).
(For the nontrivial reduction, for each \( t \in \operatorname{rg}(\phi) \) choose a \( \L(\bar{d}) \)-reduction \( f_t \colon \pre{\omega}{\omega} \to \pre{\omega}{\omega} \) of \( C_t \) to \( C_r \), let \( \varepsilon \in \mathbb{R}^+ \) be such that \( \max \operatorname{rg}(\phi) \cdot \varepsilon \leq \inf \operatorname{rg}(\phi) \), and let \( g \colon \pre{\omega}{\omega} \to \pre{\omega}{\omega} \) be a Lipschitz map with constant \( \varepsilon \) reducing \( C_r \) to itself. Define \( f \colon Y \to Y \) by setting \( f( t {}^\smallfrown{} x) = r {}^\smallfrown{} g(f_t(x)) \) for every \( t \in \operatorname{rg}(\phi) \) and \( x \in \pre{\omega}{\omega} \): it is easy to check that \( f \in \L(d_\phi) \) reduces \( \bar{C} \) to \( \bar{A} \).)
If there is a \( < \)-minimal $s\in\operatorname{rg}(\phi)$ with $C_s\equiv_{\mathsf{L}(\bar{d})} \neg B$, let $r$ be either the \( < \)-minimal element of \( \operatorname{rg}(\phi) \) with $C_r\equiv_{\mathsf{L}(\bar{d})} B$, or the \( < \)-largest element of \( \operatorname{rg}(\phi) \) satisfying both $C_r\equiv_{\mathsf{L}(\bar{d})} B$ and \( r < s \).
Then $\bar{A}=r^\smallfrown C_r\cup s^\smallfrown C_s$ is in the normal form (3b), and arguing as above one can check that $\bar{C}\equiv_{\mathsf{L}_{d(\phi)}} \bar{A}$ using the assumption $\inf \operatorname{rg}(\phi)>0$ and the previously mentioned fact about self-contractions. (For the nontrivial reduction, notice that we can assume without loss of generality that \( r < s \) (otherwise we simply switch the role of \( C_r \) and \( C_s \)). Let \( D = \{ t \in \operatorname{rg}(\phi) \mid C_t \equiv_{\L(\bar{d})} \neg B \} \), so that \( s = \min D \). For \( t \in \operatorname{rg}(\phi) \), set \( \rho(t) = s \) if \( t \in D \) and \( \rho(t) = r \) otherwise. Let \( f_t \) be a \( \L(\bar{d}) \)-reduction of \( C_t \) to \( C_s \) if \( t \in D \) and of \( C_t \) to \( C_r \) otherwise. Let \( \varepsilon \) and \( g \) be as above. Then the map \( f \colon Y \to Y \) defined by \( f(t {}^\smallfrown{} x) = s {}^\smallfrown{} f_t(x) \) if \( t \in D \) and \( f(t {}^\smallfrown{} x) = r {}^\smallfrown{} g(f_t(x)) \) otherwise is an \( \L(d_\phi) \)-reduction of \( \bar{C} \) to \( \bar{A} \).)
If there are unboundedly many $s\in\operatorname{rg}(\phi)$ with $C_s\equiv_{\mathsf{L}(\bar{d})} \neg B$ and an \( < \)-minimal \( r \in \operatorname{rg}(\phi) \) with \( C_r \equiv_{\L(\bar{d})} B \), argue as in the previous paragraph switching the role of \( B \) and \( r \) with, respectively, \( \neg B \) and \( s \).
In the remaining case there are unboundedly many $r\in\operatorname{rg}(\phi)$ with $C_r\equiv_{\mathsf{L}(\bar{d})} B$ and unboundedly many $s\in\operatorname{rg}(\phi)$ with $C_s\equiv_{\mathsf{L}(\bar{d})} \neg B$. In this situation it is easy to see that $\bar{C}$ is $\mathsf{L}(d_{\phi})$-equivalent to a set $\bar{A}$ in the normal form (3c).
Finally, suppose that $\mathcal{C}$ has a $\mathsf{L}(\bar{d})$-maximal element $B$ and that $B$ is \( \L(\bar{d}) \)-selfdual. It follows from the remarks at the beginning of the proof that there is an \( \L(\bar{d}) \)-nonselfdual set $A$ with $B\in\mathscr{L}^{(\lambda+n)}[A\oplus(\neg A)]_{\mathsf{L}(\bar{d})}$ for some $n\in\omega$ and $\lambda=0$ or $\lambda$ a countable limit ordinal. Set \( D = \{ r \in \operatorname{rg}(\phi) \mid C_r \in \bigcup_{j \in \omega} \mathscr{L}^{(\lambda+j)}[A\oplus(\neg A)]_{\mathsf{L}(\bar{d})}\} \), and
define the \emph{index} of any $r\in D$ as $i(r)=r\cdot 2^{-(j+1)}$, where \( j \) is the unique natural number such that $C_r\in \mathscr{L}^{(\lambda+j)}[A\oplus(\neg A)]_{\mathsf{L}(\bar{d})}$. Then for any $r,s\in D$ for which $i(r)\leq i(s)$ there is an \( \L(d_\phi) \)-map \( f \) such that $f(s^\smallfrown {}^\omega \omega) \subseteq r{}^\smallfrown {}^\omega \omega$ and \( f \) reduces $s^\smallfrown C_s$ to $r^\smallfrown C_r$.
Suppose first that there is \( j \leq n \) such that $C_{r_m}\in \mathscr{L}^{(\lambda+j)}[A\oplus(\neg A)]_{\mathsf{L}(\bar{d})}$ for some strictly \( < \)-descending sequence \( (r_m)_{m \in \omega} \) of distances in \( \operatorname{rg}(\phi) \), and let \( k \) be the largest of such \( j \)'s. If \( n = k \), then $\bar{A}=\bigcup_{m\in\omega} r_m^\smallfrown C_{r_m}$ is in the normal form (4b) and $\bar{A}\equiv_{\L(d_{\phi})}\bar{C}$. If \( n>k \), let \( r \) be \( < \)-smallest in \( \operatorname{rg}(\phi) \) such that \( C_r \equiv_{\L(\bar{d})} B \). If
\( \inf \operatorname{rg}(\phi) < r \cdot 2^{n-k} \), then using the fact that \( C_r \) is reducible to each of the \( C_{r_m} \)'s with some Lipschitz function with constant \( 2^{n-k} \) we get that \( \bar{C} \equiv_{\L(d_\phi)} \bar{C}' \), where \( \bar{C}' = \bar{C} \setminus \left ( \bigcup_{t \geq r} t {}^\smallfrown{} \pre{\omega}{\omega} \right ) \). Applying recursively this same procedure, after finitely many steps we will end up with a set \( \bar{C}^* \equiv_{\L(d_\phi)} \bar{C} \) such that either the \( C_{r_m} \) are \( \L(\bar{d}) \)-maximal in \( \mathcal{C}^* \), or else there is an \( < \)-smallest \( r \) such that \( C_r \) is \( \L(\bar{d}) \)-maximal in \( \mathcal{C}^* \), \( C_r \in \mathscr{L}^{(\lambda+n^*)}[A \oplus (\neg A)]_{\L(\bar{d})}\) for some \( k < n^* \leq n \), and \( r \cdot 2^{n^*-k} \leq \inf \operatorname{rg}(\phi) \). In the former case we
again easily get that $\bar{A}=\bigcup_{m\in\omega} r_m^\smallfrown C_{r_m}$ is in the normal form (4b) and $\bar{A}\equiv_{\mathsf{L}(d_{\phi})}\bar{C}^* \equiv_{\L(d_\phi)} \bar{C}$. In the latter case, we get that $\bar{A} = r {}^\smallfrown{} C_r$ is in normal form (4a) and \( \bar{A} \equiv_{\mathsf{L}(d_\phi)} \bar{C}^* \equiv_{\L(d_\phi)} \bar{C} \). (To see that \( \bar{C}^* \leq_{\L(d_\phi)} \bar{A} \), which is the only nontrivial reduction, notice that we
may assume without loss of generality that all the \( C_{r_m} \)'s equal a fixed set \( C \neq \pre{\omega}{\omega} \), that
\( C_r = 0^{(n^*-k)} {}^\smallfrown{} C \), and that for \( t \notin \{ r \} \cup \{ r_m \mid m \in \omega \} \) either \( C_t = \emptyset \) or \( C_t = 0^{(i_t+1)} {}^\smallfrown{} C \) for some \( i_t < n^*-k \).
Fix \( t \in \operatorname{rg}(\phi) \). If \( t \geq r \) then
let \( f_t \colon \pre{\omega}{\omega} \to \pre{\omega}{\omega} \) be a \( \L(\bar{d}) \)-reduction of \( C_t \) to \( C_r \). If \( t = r_m \) for some \( m \in \omega \), define \( f_t \) by setting \( f_t(x) = 0^{(n^*-k)} {}^\smallfrown{} x \) for all \( x \in \pre{\omega}{\omega} \). Finally, if \( t < r \) and \( t \neq r_m \), then let \( f_t \) be a constant
map with value \( 0^{(n^*-k)} {}^\smallfrown{} y \) for some fixed \( y \notin C \) if \( C_t =\emptyset \), and otherwise set \( f_t(x) = 0^{(n^*-k -i_t - 1)} {}^\smallfrown{} x \) for all \( x \in \pre{\omega}{\omega} \). Then the map \( f \colon Y \to Y \) defined by setting
\( f(t {}^\smallfrown{} x) = r {}^\smallfrown{} f_t(x) \) for all \( t \in \operatorname{rg}(\phi) \) and \( x \in \pre{\omega}{\omega} \) is a \( \L(d_\phi) \)-reduction of \( \bar{C} \) to \( \bar{A} \).)
Therefore we may assume without loss of generality that \( D \) is finite. Actually, applying the standard arguments used above it is not difficult to see that we may also assume that there are \( m \in \omega \), a strictly \( < \)-decreasing sequence \( r_0, \dotsc, r_m \in \operatorname{rg}(\phi) \), and a strictly decreasing sequence \( n_0, \dotsc, n_m \in \omega \) such that:
\begin{itemize}
\item
\( C_{r_k} \in \mathscr{L}^{(\lambda+n_k)}[A \oplus(\neg A)]_{\L(\bar{d})} \) for all \( k \leq m \);
\item
\( i(r_k) < i(r_{k+1}) \) for all \( k < m \);
\item
\( C_t = \emptyset \) for all \( t \geq r_m \) which are not of the form \( r_k \) for some \( k \leq m \);
\item
\( C_t <_{\mathsf{Lip}(\bar{d})} C_{r_m} \) for all \( t < r_m \).
\end{itemize}
Assume first that \( \lambda > 0 \). Then without loss of generality we may assume that
\( C_{r_k} = 0^{(n_k)} {}^\smallfrown{} \bigoplus_{l \in \omega} (0^{(l)} {}^\smallfrown{} C'_l) \) for all \( k \leq m \), where the \( C'_l \)'s are strictly \( \L(\bar{d}) \)-increasing
subsets of \( \pre{\omega}{\omega} \) such that their \( \L(\bar{d}) \)-degrees are cofinal below \( \mathscr{L}^{(\lambda)}[A \oplus (\neg A)]_{\L(\bar{d})} \). Notice that in this case \( i(r_k) \) measures the
\( d_\phi \)-distance between each pair of subsets of \( C_{r_k} \) of the form \( 0^{(n_k)} {}^\smallfrown{} l {}^\smallfrown{}0^{(l)} {}^\smallfrown{} C'_l \). Assume first
that there is \( l \in \omega \) such that \( C_t \leq_{\L(\bar{d})} C'_l \) for all \( t < r_m \). Then it is not hard to see that
\( \bar{A} = r_0 {}^\smallfrown{} C_{r_0} \) is in normal form (4a) and \( \bar{A} \equiv_{\L(d_\phi)} \bar{C} \). (An
\( \L(d_\phi) \)-reduction \( f \) of \( \bar{C} \) to \( \bar{A} \) may be defined on sets of the form \( t {}^\smallfrown{} \pre{\omega}{\omega} \) for \( t < r_m \) by
fixing \( l'\geq l \) such that \( 2^{-l'} \leq \inf \operatorname{rg}(\phi) \) and an \( \L(\bar{d}) \)-reduction $f_t$ of \( C_t \) to \( C'_{l'} \), and then setting
\( f(t {}^\smallfrown{} x) = r_0 {}^\smallfrown 0^{(n_0)} {}^\smallfrown{} l' {}^\smallfrown{} 0^{(l')} {}^\smallfrown{} f_t(x) \);
for \( t \geq r_m \), the map \( f \) may be defined on
\( t {}^\smallfrown{} \pre{\omega}{\omega} \) in the obvious way using the property of the \( i(r_k) \)'s mentioned above.) Now assume instead that the
family \( \{ C_t \mid t < r_m \} \) is \( \L(\bar{d}) \)-cofinal below \( \bigoplus_{l \in \omega} C'_l \equiv_{\L(\bar{d})} \bigoplus_{l \in \omega} 0^{(l)} {}^\smallfrown C'_l \in \mathscr{L}^{(\lambda)}[A \oplus (\neg A)]_{\L(\bar{d})} \). Then using arguments similar to the one already
applied, one gets that if \( i(r_0) \leq \inf \operatorname{rg}(\phi) \) then we can again set \( \bar{A} = r_0 {}^\smallfrown{} C_0 \), so that \( \bar{A} \) is in normal
form (4a), and prove that \( \bar{A} \equiv_{\L(d_\phi)} \bar{C} \), while if \( i(r_0) > \inf \operatorname{rg}(\phi) \) then we may choose a strictly decreasing sequence
\( (t_h)_{h \in \omega} \) so that \( t_0 < \min \{r_m, i(r_0) \} \) and the \( C_{t_h} \)'s are \(\leq_{\L(\bar{d})} \)-increasing, all in the same \( \mathsf{Lip}(\bar{d}) \)-degree,
and cofinal below \( \bigoplus_{l \in \omega} 0^{(l)} {}^\smallfrown C'_l \), and then prove that \( \bar{A} = \bigcup_{h \in \omega} t_h {}^\smallfrown{} C_{t_h} \) is in
normal form (2) and \( \L(d_\phi) \)-equivalent to \( \bar{C} \).
Finally, let $\lambda=0$. In this case we may assume without loss of generality that
\( C_{r_k} = 0^{(n_k)} {}^\smallfrown{} (A \oplus \neg A) \) for all \( k \leq m \), and \( i(r_k) \) measures the distance between the copies of $A$ and $\neg A$ in $C_{r_k}$. Let us first suppose that there are arbitrarily small $r,s>\inf \operatorname{rg}(\phi)$ with $C_r\equiv_{\mathsf{L}(d_{\phi})}A$ and $C_s\equiv_{\mathsf{L}(d_{\phi})} \neg A$. If $i(r_0) \leq \inf \operatorname{rg}(\phi)$, we let $\bar{A}=r_0^\smallfrown C_{r_0}$; then $\bar{A}$ is in the normal form
(4a) and arguing as above we get $\bar{A}\equiv_{\mathsf{L}(d_{\phi})}\bar{C}$. If $i(r_0)>\inf \operatorname{rg}(\phi)$, we choose a strictly decreasing sequence $(t_h)_{h\in\omega}$
in $\operatorname{rg}(\phi)$ with \( t_0 < \min \{ r_m, i(r_0) \} \), $C_{t_{2p}}\equiv_{\mathsf{L}(\bar{d})} A$ and $C_{t_{2p+1}}\equiv_{\mathsf{L}(\bar{d})} \neg A$, and let $\bar{A}=\bigcup_{h\in\omega} t_h {}^\smallfrown{} C_{t_h}$. Then $\bar{A}$ is in the normal form (3c) and, arguing as in the case \( \lambda > 0 \), we get $\bar{A}\equiv_{\mathsf{L}(d_{\phi})}\bar{C}$.
Next, let us suppose that there are no $r,s<i(r_0)$ in $\operatorname{rg}(\phi)$ with $C_r \equiv_{\mathsf{L}(\bar{d})} A$ and $C_s \equiv_{\mathsf{L}(\bar{d})} \neg A$. Let $\bar{A}=r_0^\smallfrown C_{r_0}$. Then $\bar{A}$ is in the normal form (4a), and using the self-contractibility of \( A \) and \( \inf \operatorname{rg}(\phi) > 0 \) we again obtain $\bar{A}\equiv_{\mathsf{L}(d_{\phi})}\bar{C}$.
Finally, suppose that there are $r,s<i(r_0)$ in $\operatorname{rg}(\phi)$ with $C_r \equiv_{\mathsf{L}(\bar{d})} A$ and $C_s \equiv_{\mathsf{L}(\bar{d})} \neg A$ and that there is an \( < \)-minimal $r\in\operatorname{rg}(\phi)$ with $C_r \equiv_{\mathsf{L}(\bar{d})} A$ (the analogous situation in which there is a minimal $r\in\operatorname{rg}(\phi)$ with $C_r\equiv_{\mathsf{L}(\bar{d})} \neg A$ can be treated similarly). We consider the \( < \)-smallest $s\in\operatorname{rg}(\phi)$ with $C_s\equiv_{\mathsf{L}(\bar{d})} \neg A$ if this exists, and any $s\in\operatorname{rg}(\phi)$ with $s<r$ and $C_s\equiv_{\mathsf{L}(\bar{d})} \neg A$ otherwise. Then $\bar{A}=r^\smallfrown A\cup s^\smallfrown(\neg A)$ is in the normal form (3b) and $\bar{A}\equiv_{\mathsf{L}(d_{\phi})}\bar{C}$.
\end{proof}
By Claim~\ref{claim:normalforms}, to show that \( {\sf SLO}^{\L(d_\phi)} \) holds for Borel subsets of \( Y \) it is enough to show that for every pair of Borel sets $\bar{A}$ and $\bar{B}$ in $\L(d_\phi)$-normal form, either \( \bar{A} \leq_{\L(d_\phi)} \bar{B} \) or \( \bar{B} \leq_{\L(d_\phi)} \neg \bar{A} \): we are now going to sketch the proof of this fact, by considering all the possible combinations of normal forms.
If $\bar{A}$ is in case (1) of the normal form, then it is \( \L(d_\phi) \)-selfdual, and hence semi-linearity is equivalent to showing that
\( \bar{A} \leq_{\L(d_\phi)} \bar{B} \) or \( \bar{B} \leq_{\L(d_\phi)} \bar{A} \). Let \( A' = \oplus_{n \in \omega} A_n \), so that
\( [ A']_{\L(\bar{d})} = \sup_{n \in \omega} [A_n]_{\L(\bar{d})} \). First assume that \( \bar{B} \) is either in normal form (1) or (2), and let
\( B' = \bigoplus_{n \in \omega} B_n \).
If \( A' <_{\L(\bar{d})} B' \) (equivalently, \( A' <_{\mathsf{Lip}(\bar{d})} B' \)), then
we get \( \bar{A} \leq_{\L(d_\phi)} \bar{B} \), and similarly switching the role of \( A \) and \( B \).
If instead \( A' \equiv_{\L(\bar{d})} B' \), then we get \( \bar{A} \equiv_{\L(d_\phi)} \bar{B} \).
Assume now that \( \bar{B} \) is either in normal form (3) or (4). Then using \( B \) in place of \( B' \) in the argument above (and noticing
that either \( A' \leq_{\mathsf{Lip}(\bar{d})} B \) or else \( B \leq_{\L(\bar{d})} A_n \) for all sufficiently large \( n \in \omega \)) we get again that \( \bar{A} \)
is \( \L(d_\phi) \)-comparable with \( \bar{B} \), as required.
Let now \( \bar{A} \) be in normal form (2). If $\bar{B}$ is in normal form (2) too, arguing as in the previous case we compare $A' = \bigoplus_{n \in \omega} A_n$ and $B' = \bigoplus_{n \in \omega} B_n$ with respect to \( \L(\bar{d}) \).
Similarly, if $\bar{B}$ is in case (3), we compare $A'$ with $B$ with respect to \( \L(\bar{d}) \), and then argue as above again.
Now let us suppose that $\bar{B}$ is in case (4). If $A_n <_{\mathsf{Lip}(\bar{d})} B$ for all $n\in\omega$, then $\bar{A}\leq_{\mathsf{L}(d_{\phi})} \bar{B}$.
Otherwise $B\leq_{\mathsf{L}(\bar{d})} A_n$ for some $n\in\omega$ and thus $\bar{B}\leq_{\mathsf{L}(\bar{d})} \bar{A}$.
We now assume that \( \bar{A} \) is in normal form (3). If $\bar{B}$ is in normal form (3) too, we can prove \( \bar{A} \leq_{\L(d_\phi)} \bar{B} \) or \( \bar{B} \leq_{\L(d_\phi)} \neg \bar{A} \) by first comparing \( A \) and \( B \) with respect to \( \mathsf{Lip}(\bar{d}) \)-reducibility (equivalently, \( \L(\bar{d}) \)-reducibility), using the assumption $\inf \operatorname{rg}(\phi)>0$.
If $\bar{A}\equiv_{\mathsf{Lip}(\bar{d})} \bar{B}$ and both $\bar{A}$ and $\bar{B}$ are in case (3b), then we simply compare the minimum of the values $r_1$ appearing in their normal forms. The comparison is straightforward in all other cases for $\bar{A}$ and $\bar{B}$ in the normal form (3) with $\bar{A}\equiv_{\mathsf{Lip}(\bar{d})} \bar{B}$.
If instead $\bar{B}$ is in case (4), using the assumption $\inf \operatorname{rg}(\phi)>0$,
we simply need to compare the $\mathsf{L}(\bar{d})$-degrees of $A$ and $B$; all possible relationships between these degrees with respect to \( \leq_{\L(\bar{d})} \) can be transferred back to analogous relationships between \( \bar{A} \) and \( \bar{B} \) with respect to \( \leq_{\L(d_\phi)} \).
Let us finally suppose that $\bar{A}$ and $\bar{B}$ are both in case (4); this is the more delicate case. Since $\bar{A}$ and $\bar{B}$ are clearly $\mathsf{L}(d_\phi)$-selfdual, it is again sufficient to show that they are $\mathsf{L}(d_{\phi})$-comparable.
First assume that $\bar{A}$ and $\bar{B}$ are both in case (4a), where $\bar{A}=r^\smallfrown A$ and $\bar{B}=s^{\smallfrown} B$.
If $A$ and $B$ are not in the same ${\mathsf{Lip}(\bar{d})}$-degree, then it is easy to compare \( \bar{A} \) and \( \bar{B} \) with respect to $\leq_{\mathsf{L}(d_{\phi})}$, and if $A\leq_{\mathsf{L}(\bar{d})} B$ then $\bar{A} \leq_{\mathsf{L}(d_{\phi})} \bar{B}$.
Hence we can assume that \( A \equiv_{\mathsf{Lip}(\bar{d})} B \) and $B\leq_{\mathsf{L}(\bar{d})} A$, so that $A\equiv_{\mathsf{L}(\bar{d})} 0^{(n)}{}^\smallfrown B$ for some $n\in\omega$.
Using the assumption $\inf \operatorname{rg}(\phi)>0$, it is now easy to check that $\bar{A}\leq_{\mathsf{L}(d_{\phi})}\bar{B}$ holds if $s\cdot 2^n \leq r$, while $\bar{B}\leq_{\mathsf{L}(d_{\phi})}\bar{A}$ holds if $r\leq s\cdot 2^n$.
Suppose now that $\bar{A}=r^\smallfrown A$ is in case (4a) and $\bar{B}$ is in case (4b).
We have $\bar{A}\leq_{\mathsf{L}(d_{\phi})} \bar{B}$ if $A \leq_{\mathsf{L}(\bar{d})} B$ holds,
and moreover
$B<_{\mathsf{Lip}(\bar{d})} A$ implies that $0^{(n)}{}^\smallfrown B\) \(\leq_{\mathsf{L}(\bar{d})} A$ for all $n\in\omega$ (which in turn implies $\bar{B}\leq_{\mathsf{L}({d_{\phi}})}\bar{A}$).
Thus we can assume that $A\equiv_{\mathsf{Lip}(\bar{d})} B$ and $B\leq_{\mathsf{L}(\bar{d})} A$, so that again $A\equiv_{\mathsf{L}(\bar{d})} 0^{(n)}{}^\smallfrown B$ for some $n\in\omega$, and let $s=\inf \operatorname{rg}(\phi)$.
Arguing similarly to the previous case, it is easy to check that $\bar{A}\leq_{\mathsf{L}(d_{\phi})}\bar{B}$ holds if $s\cdot 2^n < r$, while $\bar{B}\leq_{\mathsf{L}(d_{\phi})}\bar{A}$ holds if $s \cdot 2^n \geq r$.
The last case that needs to be considered is when both $\bar{A}$ and $\bar{B}$ are in case (4b).
We may assume that $A\leq_{\mathsf{L}(\bar{d})} B$ and hence $\bar{A}\leq_{\mathsf{L}(d_{\phi})}\bar{B}$.
This concludes the proof that \( {\sf SLO}^{\L(d_\phi)} \) holds for Borel subsets (in normal form) of \( Y \).
\medskip
It remains to show that the \( \L(d_\phi) \)-hierarchy on Borel subsets of \( Y \) is well-founded, and for this we may again concentrate only on sets in normal form. Assume towards a contradiction that there is a family \( (\bar{A}^{(i)})_{i \in \omega} \) of Borel subsets of \( Y \) in normal form such that \( \bar{A}^{(i+1)} <_{\L(d_\phi)} \bar{A}^{(i)} \) for all \( i \in \omega \). Since there are only finitely many types of normal form, passing to a subsequence if necessary we may further assume that all the \( \bar{A}^{(i)} \)'s share the same type of normal form. We now consider the various possibilities.
First assume that the \( \bar{A}^{(i)} \)'s are all in normal form (1), and set \( (A')^{(i)} = \bigoplus_{n \in \omega} A^{(i)}_n \), where the sets
\( A^{(i)}_n \subseteq \pre{\omega}{\omega} \) are those appearing in the normal form of \( \bar{A}^{(i)} \). Notice that all the \( (A')^{(i)} \) are necessarily \( \L(\bar{d}) \)-selfdual. Then \( (A')^{(i+1)} <_{\L(\bar{d})} (A')^{(i)} \), because otherwise \( (A')^{(i)} \leq_{\L(\bar{d})} (A')^{(i+1)} \), whence one would easily get \( \bar{A}^{(i)} \leq_{\L(d_\phi)} \bar{A}^{(i+1)} \), contradicting the choice of the \( \bar{A}^{(i)} \)'s. Therefore the \( (A')^{(i)} \) are strictly \( \L(\bar{d}) \)-decreasing,
contradicting the fact that the \( \L(\bar{d}) \)-hierarchy on Borel subsets of \( \pre{\omega}{\omega} \) is well-founded.
The case where all the \( \bar{A}^{(i)} \)'s are in normal form (2) can be dealt with in the same way, and a similar argument works also for the other cases with the following minor modifications:
\begin{enumerate}[\( \bullet \)]
\item
When considering normal forms as in (3a), set \( (A')^{(i)} = A^{(i)} \), where \( A^{(i)} \subseteq \pre{\omega}{\omega} \) is the set appearing in the normal form of \( \bar{A}^{(i)} \), and pass to a subsequence if necessary to avoid the situations in which \( A^{(i+1)} \equiv_{\L(\bar{d})} \neg A^{(i)} \);
\item
When considering normal forms as in (3b) or (3c), set \( (A')^{(i)} = (0 {}^\smallfrown{} A^{(i)}) \cup (1 {}^\smallfrown{} (\neg A^{(i))}) \), where \( A^{(i)}, \neg A^{(i)} \subseteq \pre{\omega}{\omega} \) are the sets appearing in the normal form of \( \bar{A}^{(i)} \). In case (3b), we may need to pass to a subsequence \( ((A')^{(i_l)})_{l \in \omega} \) to guarantee that \( A^{(i_{l+1})} <_{\L(\bar{d})} A^{(i_l)} \).
\item
When considering normal forms as in (4), set \( (A')^{(i)} = A^{(i)} \), where \( A^{(i)} \subseteq \pre{\omega}{\omega} \) is the set appearing in the normal form of \( \bar{A}^{(i)} \). In case (4a) it may be necessary to first pass to a subsequence \( ((A')^{(i_l)})_{l \in \omega} \) to guarantee that the sequence of the \( r^{(i_l)} \)'s appearing in the canonical form of \( \bar{A}^{(i_l)} \) is not \( < \)-increasing.
\end{enumerate}
This concludes the proof of the well-foundness of \( \leq_{\L(d_\phi)} \) on Borel subsets of \( Y \), and hence of the entire proposition.
\end{proof}
\iffalse
Theorem \ref{prop:specialalpha*} is optimal in the sense that $\omega$ cannot be replaced with a larger ordinal.
\begin{remark}
There is a map $\phi\colon \omega\rightarrow \mathbb{R}^+$ with $\inf \operatorname{rg}(\phi)>0$ and $\operatorname{rg}(\phi)$ of order type $(\omega\cdot 2)^*$ such hat the $\mathsf{L}(d_{\phi})$-hierarchy on the Borel subsets of $({}^{\omega}\omega,d_{\phi})$ is not semilinear and hence not very good.
To see this, we consider a map $\phi\colon\omega\rightarrow \mathbb{R}^+$ with $\operatorname{rg}(\phi)=\{r_n\mid n\in\omega\}\cup \{s_n\mid n\in\omega\}$ where $(r_n)_{n\in\omega}$ and $(s_n)_{n\in\omega}$ are strictly decreasing and $r_m< s_n$ for all $m,n\in\omega$. Then $({}^{\omega}\omega,d_{\phi})$ is isometric to the space $Y=\bigcup_{r\in \operatorname{rg}(\phi)} r^\smallfrown {}^{\omega}\omega$ equipped with the ultrametric
\[
d_{\phi} (r^\smallfrown x,s^\smallfrown y) =
\begin{cases}
0 & \text{if } r=s \text{ and }x=y, \\
\max(\{r,s\}) & \text{if }r \neq s, \\
r \cdot 2^{-(n+1)} & \text{if } r=s \text{ and } n \text{ is least such that } x(n) \neq y(n).
\end{cases}
\]
In order to define a pair of subsets $\bar{A},\bar{B}$ of $(Y,d_{\phi})$ which contradicts $\mathsf{SLO}^{\mathsf{L}(d_{\phi})}$, let us first choose an $\mathsf{L}(\bar{d})$-nonselfdual and hence \todo{@Luca: please check this} $\mathsf{W}(\bar{d})$-nonselfdual set $C\subseteq {}^{\omega}\omega$.
Let $\bar{A}=r_0^\smallfrown (\neg C)\cup \bigcup_{n\in\omega} s_n^\smallfrown C$ and $\bar{B}=\bigcup_{n\in\omega} r_n^\smallfrown (\neg C) \cup s_0^\smallfrown C$.
Suppose that $f\colon Y\rightarrow Y$ is an $\mathsf{L}(d_{\phi})$-nonexpansive reduction of $\bar{A}$ to $\bar{B}$. Since $C$ is $\mathsf{W}(\bar{d})$-nonselfdual, $f[s_n^\smallfrown {}^{\omega}\omega]\cap s_0^\smallfrown {}^{\omega}\omega\neq\emptyset$ for all $n$. Since $f$ is $\mathsf{L}(d_{\phi})$ is nonexpansive, this implies that $f[r_0^\smallfrown {}^{\omega}\omega]\subseteq s_0^\smallfrown {}^{\omega}\omega$ and hence $\neg C\leq_{\mathsf{W}(\bar{d})} C$, contradicting the assumption on $C$.
Now suppose that $f\colon Y\rightarrow Y$ is an $\mathsf{L}(d_{\phi})$-nonexpansive reduction of $\bar{B}$ to $\neg\bar{A}$. Since $C$ is $\mathsf{W}(\bar{d})$-nonselfdual, $f[s_0^\smallfrown {}^{\omega}\omega]\cap r_0^\smallfrown {}^{\omega}\omega\neq\emptyset$ and $f[r_1^\smallfrown {}^{\omega}\omega]\cap s_n^\smallfrown {}^{\omega}\omega \neq\emptyset$ for some $n$. Let us choose $x,y\in {}^{\omega}\omega$ with $f(s_0^\smallfrown x)\in r_0^\smallfrown {}^{\omega}\omega$ and $f(r_1^\smallfrown y)\in s_n^\smallfrown {}^{\omega}\omega$. Then $d_{\phi}(s_0^\smallfrown x, r_1^\smallfrown y)=r_1<r_0=d_{\phi}(f(s_0^\smallfrown x), f(r_1^\smallfrown y))$, sontradicting the assumption that $f$ is $\mathsf{L}(d_{\phi})$-nonexpansive.
Then the copies of the sets $C_n$ in $r_1^\smallfrown \bigoplus_{n\in\omega} C_n$ have to essentially match up with the sets $r_n^\smallfrown C_n$, in the sense that there are arbitrarily large $n$ such that $f(r_m^\smallfrown x)\in r_n^\smallfrown {}^{\omega}\omega$ for some $m\geq1$ and some $x\in {}^{\omega}\omega$. It follows that $f(y)\in r_0^\smallfrown {}^{\omega}\omega$ for all $y\in r_0^\smallfrown {}^{\omega}\omega$, since otherwise $f$ could not be $\mathsf{L}(\bar{d})$-nonexpansive.
Thus $B\leq_{\mathsf{L}(d_{\phi})} A$, contradicting the assumption.
\end{remark}
\begin{question} Is there a map $\phi\colon \omega\rightarrow \mathbb{R}^+$ with $\inf\operatorname{rg}(\phi)>0$ and $\operatorname{rg}(\phi)$ of order type $(\omega+1)^*$ such hat the $\mathsf{L}(d_{\phi})$-hierarchy on the Borel subsets of $({}^{\omega}\omega,d_{\phi})$ is not semilinear?
\end{question}
\begin{remark} \todo{Check!}
There is a map $\phi\colon \omega\rightarrow \mathbb{R}^+$ with $\operatorname{rg}(\phi)$ of order type $(\omega+1)^*$ such hat the $\mathsf{L}(d_{\phi})$-hierarchy on the Borel subsets of $({}^{\omega}\omega,d_{\phi})$ is not semilinear and hence not very good.
To see this, we consider the map $\phi\colon\omega\rightarrow \mathbb{R}^+$ defined by $\phi(0):=\frac{1}{3}$ and $\phi(n):=\frac{1}{3}+(\frac{2}{3})^{n}$ for $n\geq 1$. Note that $({}^{\omega}\omega,d_{\phi})$ is isometric to the space $Y=\bigcup_{r\in \operatorname{rg}(\phi)} r\smallfrown ({}^{\omega}\omega)$ with the ultrametric
\[
d_{\phi} (r^\smallfrown x,s^\smallfrown y) =
\begin{cases}
0 & \text{if } r=s \text{ and }x=y, \\
\max(\{r,s\}) & \text{if }r \neq s, \\
r \cdot 2^{-n} & \text{if } r=s \text{ and } n \text{ is least such that } x(n) \neq y(n).
\end{cases}
\]
In order to define a pair of subsets $\bar{A},\bar{B}$ of $(Y,d_{\phi})$ which contradicts $\mathsf{SLO}^{\mathsf{L}(d_{\phi})}$, let us first choose a strictly $\mathsf{L}(\bar{d})$-increasing sequence $(C_n)_{n\geq1}$ of $\mathsf{Lip}(\bar{d})$-equivalent $\mathsf{L}(\bar{d})$-selfdual subsets of ${}^{\omega}\omega$ and moreover two $\mathsf{L}(\bar{d})$-selfdual sets $A<_{\mathsf{L}(\bar{d})} B<_{\mathsf{L}(\bar{d})} C_1$.
Let $r_n=\phi(n)$ for $n\in\omega$ and define $\bar{A}=(r_0^\smallfrown A)\cup \bigcup_{n\geq1} (r_n^\smallfrown C_n)$ and $\bar{B}=(r_0^\smallfrown B)\cup (r_1^\smallfrown \bigoplus_{n\in\omega} C_n)$. Then $\bar{A}$ and $\bar{B}$ are $\mathsf{L}(d_{\phi})$-selfdual.
Suppose that $f\colon Y\rightarrow Y$ is an $\mathsf{L}(d_{\phi})$-nonexpansive reduction of $\bar{A}$ to $\bar{B}$. Since the distance between the sets $r_1^\smallfrown n^\smallfrown {}^{\omega}\omega$ for different $n\in\omega$ is $\frac{1}{2} r_1=\frac{1}{2}>\frac{1}{3}$, it follows that there is some $n\in\omega$ such that $r_i^\smallfrown C_i\leq_{\mathsf{L}(d_{\phi})} r_1^\smallfrown n^\smallfrown C_n$ for all but finitely many $i\in\omega$. This contradicts the assumption that $(C_n)_{n\in\omega}$ is strictly $\mathsf{L}(\bar{d})$-increasing.
Now suppose that $f\colon Y\rightarrow Y$ is an $\mathsf{L}(d_{\phi})$-nonexpansive reduction of $\bar{B}$ to $\bar{A}$. Then the copies of the sets $C_n$ in $r_1^\smallfrown \bigoplus_{n\in\omega} C_n$ have to essentially match up with the sets $r_n^\smallfrown C_n$, in the sense that there are arbitrarily large $n$ such that $f(r_m^\smallfrown x)\in r_n^\smallfrown {}^{\omega}\omega$ for some $m\geq1$ and some $x\in {}^{\omega}\omega$. It follows that $f(y)\in r_0^\smallfrown {}^{\omega}\omega$ for all $y\in r_0^\smallfrown {}^{\omega}\omega$, since otherwise $f$ could not be $\mathsf{L}(\bar{d})$-nonexpansive.
Thus $B\leq_{\mathsf{L}(d_{\phi})} A$, contradicting the assumption.
\end{remark}
\fi
Corollaries~\ref{cor:specialincreasing},~\ref{cor:specialconvergingto0}, and Theorem~\ref{prop:specialalpha*} already cover many interesting cases, and using the methods developed in the proof of Theorem~\ref{prop:specialalpha*} it seems plausible to conjecture that if the range of \( \phi \) does not contain increasing \(\omega\)-sequences, then the \( \L(d_\phi) \)-hierarchy on Borel subsets of \( \pre{\omega}{\omega} \) is well-founded.
However, the general problem of determining the character of the \( \L(d) \)-hierarchy on an arbitrary ultrametric Polish space \( X = (X,d) \) remains open:
\begin{question} \label{quesn:notincreasing}
Let $X = (X,d)$ be an ultrametric (perfect) Polish space such that \( R(d) \) does not contain an honest increasing sequence, and assume that \( R(d) \) is neither finite nor a (\( \omega \)-)sequence converging to \( 0 \). Is the $\mathsf{L}(d)$-hierarchy \( {\sf Deg}_{\boldsymbol{\Delta}^1_1}(\L(d)) \) on the Borel subsets of $X$ (very) good?
\end{question}
\begin{remark}
In order to answer Question~\ref{quesn:notincreasing}, it may be useful to note the following.
It is proved in \cite[Theorem 4.1]{Gao:2011} that every ultrametric Polish space $X = (X,d)$ is isometric to a closed subspace of the ultrametric Urysohn space $U_{R(d)}=\{(x_n)_{n \in \omega} \in {}^{\omega}(R(d) \cup \{ 0 \})\mid x_n \geq x_{n+1}$ for all $n$ and $\lim_{n\to \infty} x_n=0\}$ equipped with the complete ultrametric
\[
d_{U_{R(d)}} ((x_n)_{n \in \omega},(y_n)_{n \in \omega}) =
\begin{cases}
0 & \text{if } x_n = y_n \text{ for all }n, \\
\max({x_n, y_n}) & \text{if \( n \) is least such that } x(n) \neq y(n).
\end{cases}
\]
Suppose that $X = (X,d)$ is a perfect ultrametric Polish space and choose a closed subspace $Y$ of $(U_{R(d)},d_{U_{R(d)}})$ such that $Y =(Y, d_{U_{R(d)}})$ is isometric to $X$.
Let $S(Y)=\{y \restriction n\mid y\in Y,\ n\in\omega\}$, and set $D(s)=\{r\in\mathbb{R}^+ \mid \exists x\in {}^\omega(R(d) \cup \{ 0 \}) \, (s {}^\smallfrown{} r {}^\smallfrown{} x\in Y) \}$ for each $s\in S(Y)$. Notice that $S(Y)$ and $D(s)$ are countable since $R(d)$ is countable.
If there is a strictly increasing sequence $(r_n)_{n\in\omega}$ in $D(s)$ for some $s\in S(Y)$, then we obtain an honest increasing sequence in $R(d)$ from the assumption that $(X,d)$ is perfect. If there is no honest increasing sequence in $R(d)$, it follows that the order type of each $D(s)$ is $\alpha_s^*$ for some countable ordinal $\alpha_s$.
\end{remark}
Finally, we want to show that, even if
by Theorem~\ref{th:increasingdistances} it is possible that the \( \L(d) \)-hierarchy \( {\sf Deg}_{\boldsymbol{\Delta}^1_1}(\L(d)) \) on Borel subsets of a given ultrametric Polish space \( X = (X,d) \) with bounded diameter is very bad, a natural (modest) strengthening of the preorder \( \leq_{\L(d)} \) already yields to a semi-linearly ordered hierarchy.
\begin{definition} \label{def:almost}
Suppose $X=(X,d)$ is an ultrametric Polish space, and let $A,B\subseteq X$. Let us write $A\leq_{\mathsf{Lip}(d,L)} B$ if there is a Lipschitz function $f\colon (X,d) \rightarrow (X,d)$ with constant $L \in \mathbb{R}^+$ such that $A=f^{-1}(B)$.
We say that $A$ is \emph{almost nonexpansive reducible} to $B$ ($A\leq_{\mathsf{a}\mathsf{L}(d)}B$ in symbols) if \( A \leq_{\mathsf{Lip}(d,L)} B \) for every $1 < L \in \mathbb{R}^+ $.
\end{definition}
Notice that the relation $\leq_{\mathsf{a}\mathsf{L}(d)}$ is a preorder (for the transitivity use the fact that if \( f,g \colon X \to X \) are Lipschitz functions with constant \( L, L' \), respectively, then \( g \circ f \) is Lipschitz with constant \( L \cdot L' \)). Moreover, \( \leq_{\mathsf{a} \L (d)} \) is strictly between \( \leq_{\L(d)} \) and \( \leq_{\mathsf{Lip}(d)} \). Even if literally \( \leq_{\mathsf{a}\L(d)} \) is not of the form \( \leq_\mathcal{F} \) for some reducibility \( \mathcal{F} \) on \( X \), with a little abuse of notation and terminology we can nevertheless consider the \( \mathsf{a}\L(d) \)-hierarchy on (Borel subsets of) \( X \), the Semi-Linear Ordering principle \( {\sf SLO}^{\mathsf{a}\L(d)} \), and so on (with the obvious definitions).
\begin{proposition}
Let $X=(X,d)$ be an ultrametric Polish space with bounded diameter. Then the $\mathsf{a}\mathsf{L}(d)$-hierarchy on the Borel subsets of $X$ is semi-linearly ordered, and hence not bad.
\end{proposition}
\begin{proof}
Given \( L > 1 \), let $d_L \colon X\times X \to \mathbb{R}^+$ be defined by $d_L(x,y)=\min (\{L^{n}\mid d(x,y)\leq L^n$ and $n\in \mathbb{Z}\})$ if \(x,y \in X \) are distinct, and by \( d_L(x,y) = 0 \) if \( x = y \in X \). Then $d_L$ is a complete ultrametric on $X$ compatible with the metric topology \( \tau_d \), and since we assumed that \( X \) has bounded (\( d \)-)diameter we also have that \( R(d_L) \subseteq \{ L^n \mid n \in \mathbb{Z} \} \) is either finite, or a decreasing sequence converging to \( 0 \). By Theorem~\ref{th:descendingdistances}, this means that the \( \L(d_L) \)-hierarchy on Borel subsets of \( X \) is very good, and hence, in particular, semi-linearly ordered.
Moreover, $\operatorname{id}\colon (X,d)\rightarrow (X,d_L)$ is Lipschitz with constant \( L \), while $\operatorname{id} \colon (X,d_L)\rightarrow (X,d)$
is nonexpansive.
Hence for all subsets $A,B$ of $X$:
\begin{itemize}
\item if $A\leq_{\mathsf{Lip}(d,L')} B$, then $A\leq_{\mathsf{Lip}(d_L, L \cdot L')} B$;
\item if $A\leq_{\mathsf{Lip}(d_L,L')} B$, then $A\leq_{\mathsf{Lip}(d, L\cdot L')} B$.
\end{itemize}
In particular, $A\leq_{\mathsf{L}(d_L)} B$ implies that $A\leq_{\mathsf{Lip}(d,L)} B$.
We claim ${\sf SLO}^{\mathsf{a}\mathsf{L}(d)}$ holds for Borel subsets of $X$. By the observation above and \( {\sf SLO}^{\mathsf{Lip}(d_L)} \), for every fixed \( L > 1 \) we have that either $A\leq_{\mathsf{Lip}(d,L)} B$ or $B\leq_{\mathsf{Lip}(d,L)} \neg A$. If for every \( n \in \omega \) there is \( 1< L \leq 1+2^{-n} \) such that $A\leq_{\mathsf{Lip}(d,L)} B$, then \( A \leq_{\mathsf{a} \L(d)} B \). Similarly, if for every \( n \in \omega \) there is \( 1< L \leq 1+2^{-n} \) such that $B\leq_{\mathsf{Lip}(d,L)} \neg A$, then \( B \leq_{\mathsf{a} \L(d)} \neg A \). Since one of the two possibilities necessarily occurs, we get that either \( A \leq_{\mathsf{a}\L(d)} B \) or \( B \leq_{\mathsf{a} \L(d)} \neg A \), as required.
\end{proof}
\section{Compact ultrametric Polish spaces}
It is well-known that any continuous function between metric spaces is automatically uniformly continuous as soon as its domain is compact (see e.g.\ \cite[Proposition 4.5]{Kechris:1995}). In particular, this means that it does not make much sense to consider the \( \mathsf{UCont}(d) \)-hierarchy on a compact ultrametric Polish space \( X = (X,d) \): since it coincide%
\footnote{In fact in the specific case of the Cantor space \( \mathcal{C} = (\pre{\omega}{2}, \bar{d}_{\mathcal{C}}) \) one can check that, although \( \mathsf{Lip}(\bar{d}_{\mathcal{C}}) \subsetneq \mathsf{UCont}(\bar{d}_{\mathcal{C}})\), the \( \mathsf{Lip}(\bar{d}_{\mathcal{C}}) \)- and the \( \mathsf{UCont}(\bar{d}_{\mathcal{C}}) \)-hierarchies coincide.}
with the \( \mathsf{W}(X) \)-hierarchy on \( X \), its restriction to the Borel sets is always very good by Proposition~\ref{prop:continuous}. However, one may wonder about the character of the \( \mathsf{Lip}(d) \)- and the \( \L(d) \)-hierarchy on (Borel subsets of) such an \( X \): the next results show that they must always be very good as well.
\begin{proposition} \label{prop:R(d)}
Let \( X = (X,d) \) be a compact ultrametric Polish space. Then either \( X \) (and hence also \( R(d) \)) is finite, or else \( R(d) \) is a strictly decreasing (\( \omega \)-)sequence converging to \( 0 \). In particular, \( X \) has bounded diameter.
\end{proposition}
\begin{proof}
It is clearly enough to show that for every \( \bar{r} \in \mathbb{R}^+ \), the set \( R(d)_{\geq \bar{r}} = \{ r \in R(d) \mid r \geq \bar{r} \} \) is finite. To see this, observe that the family \( \mathcal{B} = \{ B_d(x,\bar{r}) \mid x \in X \} \) is a finite covering of \( X \) because \( X \) is compact. Assume towards a contradiction that \( R(d)_{\geq \bar{r}} \) is infinite, let \( (r_n)_{n \in \omega} \) be an enumeration without repetitions of it, and let \( (x_n)_{n \in \omega} \) and \( (y_n)_{n \in \omega } \) be such that \( d(x_n,y_n) = r_n \) for every \( n \in \omega \). Since \( \mathcal{B} \) is finite, there are distinct \( n,m \in \omega \) such that \( d(x_n,x_m), d(y_n,y_m) < \bar{r} \). Since \( r_m \geq \bar{r} \), we get that \( r_n = d(x_n,y_n) = d(x_m,y_m) = r_m \), contradicting the choice of the \( r_n \)'s.
\end{proof}
\begin{theorem} \label{th:compact}
Let \( X = (X,d) \) be a compact ultrametric Polish space. Then both the \( \L(d) \)- and the \( \mathsf{Lip}(d) \)-hierarchy on Borel subsets of \( X \) are very good.
\end{theorem}
\begin{proof}
Use Proposition~\ref{prop:R(d)} together with Theorems~\ref{th:descendingdistances} and~\ref{th:unifcontandlip}.
\end{proof}
In particular, we cannot change the ultrametric on the Cantor space \( \pre{\omega}{2} \) to make its nonexpansive or its Lipschitz hierarchy (very) bad: if \( d' \) is any complete ultrametric compatible with the product topology on \( \pre{\omega}{2} \), then both the \( \mathsf{Lip}(d') \)- and the \( \L(d') \)-hierarchy on Borel subsets of \( \pre{\omega}{2} \) are very good.%
\footnote{However, analogously to~\cite[Section 5]{MottoRos:2012c} it is still possible to define compatible complete ultrametrics \( d_i \), \( i = 0,1 \), on \( \mathcal{C} = \pre{\omega}{2} \) so that \( \L(\bar{d}_{\mathcal{C}}) \not\subseteq \mathsf{Lip}(d_0) \) (hence also \( \L(\bar{d}_{\mathcal{C}}) \not\subseteq \L(d_0) \)), while \( \L(\bar{d}_{\mathcal{C}}) \not\subseteq \L(d_1) \) but \( \mathsf{Lip}(\bar{d}_{\mathcal{C}}) = \mathsf{Lip}(d_1) \).}
\begin{remark}
Albeit Theorem~\ref{th:compact} shows that there is no compact ultrametric Polish space \( X = (X,d) \) with a (very) bad \( \mathsf{Lip}(d) \)- or \( \L(d) \)-hierarchy, Corollaries~\ref{cor:countable} and~\ref{cor:X_1} shows that there are \( \boldsymbol{K}_\sigma \)-spaces%
\footnote{A topological space is \( \boldsymbol{K}_\sigma \) if it is the union of countably many compact subsets.}
\( X_i = (X_i,d_i) \), \( i = 0,1 \), such that:
\begin{enumerate}[\( \bullet \)]
\item
both \( {\sf Deg}_{\boldsymbol{\Delta}^0_1}(\L(d_0)) \) and \( {\sf Deg}_{\boldsymbol{\Delta}^0_1}(\mathsf{Lip}(d_0)) \) are very bad;
\item
\( {\sf Deg}_{\boldsymbol{\Delta}^0_1}(\L(d_1)) \) is very bad, while \( {\sf Deg}_{\boldsymbol{\Delta}^1_1}(\mathsf{Lip}(d_1)) \) is very good.
\end{enumerate}
\end{remark}
Let us now concentrate on the Cantor space \( \mathcal{C} = \pre{\omega}{2} \), and let us briefly consider another kind of reducibility that was analyzed in~\cite{MottoRos:2012c} for the case of the Baire space, namely the collection of all contraction mappings.
\begin{notation}
Let \( \bar{d} = \bar{d}_{\mathcal{C}} \) be the usual metric on the Cantor space. We denote by \( \c(\bar{d}) \) the collection of all \emph{contractions} from \( \mathcal{C} \) into itself, i.e.\ of all Lipschitz functions \( f \colon \mathcal{C} \to \mathcal{C} \) with constant strictly smaller than \( 1 \).
Given two sets \( A,B \subseteq \mathcal{C} \), set
\[
A \leq_{\c(\bar{d})} B \iff A = B \vee \exists f \in \c(\bar{d}) \, (A = f^{-1}(B)).
\]
In fact, \( {\leq_{\c(\bar{d})}} = {\leq_\mathcal{F}} \), where \( \mathcal{F} \) is the reducibility on \( \mathcal{C} \) obtained by adding the identity \( \operatorname{id} = \operatorname{id}_{\mathcal{C}} \) to the set \( \c(\bar{d}) \).
\end{notation}
Using the methods developed in~\cite[Section 4]{MottoRos:2012c}, it is easy to check that the following hold:
\begin{theorem} \label{th:contractions}
Let \( A,B \) be Borel subsets of \( \mathcal{C} \).
\begin{enumerate}
\item
If \( A \not\equiv_{\L(\bar{d})} B \), then \( A \leq_{\c(\bar{d})} B \iff A \leq_{\L(\bar{d})} B \), while if \( A \equiv_{\L(\bar{d})} B \), then \( A \leq_{\c(\bar{d})} B \iff A \nleq_{\L(\bar{d})} \neg A \).
\item
\( A \) is \emph{selfcontractible} (i.e.\ \( A = f^{-1}(A) \) for some \( f \in \c(\bar{d}) \)) if and only if \( A \nleq_{\L(\bar{d})} \neg A \).
\item
If \( A \nleq_{\L(\bar{d})} \neg A \), then \( [A]_{\c(\bar{d})} = [A]_{\L(\bar{d})} \), while if
\( A \leq_{\L(\bar{d})} \neg A \), then \( [A]_{\c(\bar{d})} = \{ A \} \).
\item
\( A <_{\c(\bar{d})} B \iff A <_{\L(\bar{d})} B \).
\end{enumerate}
\end{theorem}
Therefore, to describe the \( \c(\bar{d}) \)-hierarchy on Borel subsets of \( \mathcal{C} \) it is enough to determine how many sets are
contained in each \( \L(\bar{d}) \)-degree of an \( \L(\bar{d}) \)-selfdual Borel subset of \( \mathcal{C} \), and to combine this information
with the well-known description of the \( \L(\bar{d}) \)-hierarchy on Borel subsets of \( \mathcal{C} \) (see~\cite{Andretta:2007}). Let us first briefly describe this last
hierarchy. First of all, the hierarchy is semi-well-ordered. At the bottom we found the \( \L(\bar{d}) \)-nonselfdual pair constituted by
\( [\mathcal{C}]_{\L(\bar{d})} = \{ \mathcal{C} \} \) and \( [\emptyset]_{\L(\bar{d})} = \{ \emptyset \} \). Immediately after each
\( \L(\bar{d}) \)-nonselfdual pair \( \{ [A]_{\L(\bar{d})} , [ \neg A ]_{\L(\bar{d})} \} \) there is the \( \L(\bar{d}) \)-degree
of the \( \L(\bar{d}) \)-selfdual set
\( A \oplus \neg A = (0 {}^\smallfrown{} A) \cup (1 {}^\smallfrown{} (\neg A)) = \{ 0 {}^\smallfrown{} x \mid x \in A \} \cup \{ 1 {}^\smallfrown{} x \mid x \in \neg A \} \).
On the other hand, if \( A \) is \( \L(\bar{d}) \)-selfdual, then immediately
after \( [A]_{\L(\bar{d})} \) there is the \( \L(\bar{d})\)-degree of the selfdual set \( 0 {}^\smallfrown{} A = \{ 0 {}^\smallfrown{} x \mid x \in A \} \).
Finally, at all limit level there is always an \( \L(\bar{d}) \)-nonselfdual pair. Therefore we get the structure represented in Figure~\ref{fig:Lhierarchy}, where bullets
represent \( \L(\bar{d}) \)-degrees and each \( \L(\bar{d}) \)-degree is \( \L(\bar{d}) \)-reducible to another one if and only if it is (strictly) to the left
of it.
\begin{figure}[!htbp]
\centering
\[
\begin{array}{llllllllll}
\bullet & & \bullet & & \bullet & & & \bullet & &
\\
& \smash[b]{\underbrace{\bullet \; \bullet \;
\bullet \; \cdots}_{\omega}} & &
\smash[b]{\underbrace{\bullet \; \bullet \; \bullet \;
\cdots}_{\omega}} & & \smash[b]{\underbrace{\bullet \; \bullet \;
\bullet \; \cdots}_{\omega}} & \cdots\cdots
& & \smash[b]{\underbrace{\bullet \; \bullet \;
\bullet \; \cdots}_{\omega}} & \cdots\cdots
\\
\bullet & & \bullet & & \bullet & & & \bullet & &
\\
& \quad \; \; \; \stackrel{\uparrow}{\makebox[0pt][l]{\text{clopen sets }}} & & & &
& &
\stackrel{\uparrow}{\makebox[0pt][l]{\text{limit levels }}} &
\end{array}
\]
\caption{The \( \L(\bar{d}) \)-hierarchy on Borel subsets of \( \mathcal{C} \).
}
\label{fig:Lhierarchy}
\end{figure}
Notice that the first \(\omega\)-chain of consecutive \( \L(\bar{d}) \)-selfdual degrees contains all nontrivial clopen sets, while the first non-trivial
\( \L(\bar{d}) \)-nonselfdual pair is formed by all proper open and proper closed subsets of \( \mathcal{C} \).
To compute the cardinality of a given \( [A]_{\L(\bar{d})} \) (for \( A \subseteq \mathcal{C} \)), recall first that if \( \emptyset, \mathcal{C} \neq A \)
is clopen, then there is \( 0 \neq n \in \omega \), called the \emph{level of \( A \)} such that \( A \equiv_{\L(\bar{d})} \boldsymbol{\mathrm{N}}_{0^{(n)}}\),
where for an arbitrary \( s \in \pre{< \omega}{2} \) we set \( \boldsymbol{\mathrm{N}}_s = \{ x \in \mathcal{C} \mid s \subset x \} \) --- in fact \( A \) is in the
\( n \)-th \( \L(\bar{d}) \)-selfdual degree of the first \(\omega\)-chain of consecutive \( \L(\bar{d}) \)-selfdual degrees if and only if it is of level \( n \).
\begin{proposition} \label{prop:contractions}
Let \( \emptyset,\mathcal{C} \neq A \subseteq \mathcal{C} \).
\begin{enumerate}
\item
if \( A \) is clopen, then \( [A]_{\L(\bar{d})} \) contains exactly \( 2^{2^{n}} - 2^{2^{n-1}} \)-many sets, where \( n \) is the level of \( A \);
\item
if \( A \) is not clopen, then there is an injection \( j \colon \mathcal{C} \to [A]_{\L(\bar{d})} \).
\end{enumerate}
\end{proposition}
\begin{proof}
For each \( 0 \neq n \in \omega \), the collection of all clopen sets \( \L(\bar{d}) \)-reducible to \( \boldsymbol{\mathrm{N}}_{0^{(n)}} \) consists of all the sets of the
form \( \bigcup_{s \in S} \boldsymbol{\mathrm{N}}_s \) for \( S \) a subset of \( \{ s \in \pre{<\omega}{2} \mid \operatorname{lh}(s) = n \} \): therefore there are \( 2^{2^n} \)-many
such sets. So if \( A \) is a clopen set of level \( n \), then to compute the cardinality of \( [A]_{\L(\bar{d})} \) we have to subtract to \( 2^{2^n} \)
the number of sets which are not \( \L(\bar{d}) \)-equivalent to \( \boldsymbol{\mathrm{N}}_{0^{(n)}} \), i.e.\ \( \emptyset \), \( \mathcal{C} \), and all sets
\( \L(\bar{d}) \)-reducible to \( \boldsymbol{\mathrm{N}}_{0^{(n-1)}} \): since there are \( 2^{2^{n-1}} \)-many such sets, we get that \( [A]_{\L(\bar{d})} \)
contains exactly \( 2^{2^{n}} - 2^{2^{n-1}} \)-many sets.
For the second part, let us first assume that \( A \nleq_{\L(\bar{d})} \neg A \). If \( A \) is a proper open set, then the map
\( j \colon \mathcal{C} \to [A]_{\L(\bar{d})} \colon x \mapsto \mathcal{C} \setminus \{ x \} \) is as required. Therefore we can assume without
loss of generality that \( B \leq_{\L(\bar{d})} A \) for every proper closed set \( B \). By Theorem~\ref{th:contractions}(2), there is
\( f \in \c(\bar{d}) \) such that \( f^{-1}(A) = A \). Let \( i =0,1 \) be such that \( f(\mathcal{C}) \subseteq \boldsymbol{\mathrm{N}}_{\langle i \rangle} \), and consider the map
\[
j \colon \mathcal{C} \to [A]_{\L(\bar{d})} \colon x \mapsto A_x = (A \cap \boldsymbol{\mathrm{N}}_{\langle i \rangle}) \cup \{ (1-i) {}^\smallfrown{} x \}.
\]
Clearly \( j \) is an injection, so it remains only to show that \( A \equiv_{\L(\bar{d})} A_x \) for every \( x \in \mathcal{C} \). For one direction, \( f \)
witnesses \( A \leq_{\L(\bar{d})} A_x \). For the other direction, let \( g_x \in \L(\bar{d}) \) be a reduction of \( \{ (1-i) {}^\smallfrown{} x \} \)
to \( A \): then \( (\operatorname{id}_{\mathcal{C}} \restriction \boldsymbol{\mathrm{N}}_{\langle i \rangle}) \cup (g_x \restriction \boldsymbol{\mathrm{N}}_{\langle (1-i) \rangle}) \) witnesses
\( A_x \leq_{\L(\bar{d})} A \).
Finally, let \( A \) be \( \L(\bar{d}) \)-selfdual. Since by case assumption \( A \) is not clopen, there is an \( \L(\bar{d}) \)-nonselfdual
\( B \neq \emptyset, \mathcal{C} \) and \( n \in \omega \) such that
\( A \equiv_{\L(\bar{d})} 0^{(n)} {}^\smallfrown{} (B \oplus \neg B) \). Let \( j' \colon \mathcal{C} \to [B]_{\L(\bar{d})} \)
be an injective map: then
\[
j \colon \mathcal{C} \to [A]_{\L(\bar{d})} \colon x \mapsto 0^{(n)} {}^\smallfrown{} (j'(x) \oplus \neg j'(x))
\]
is clearly as required.
\end{proof}
Since by Theorem~\ref{th:contractions} the \( \c(\bar{d}) \)-hierarchy on Borel subsets of \( \mathcal{C} \) is the refinement of the
\( \L(\bar{d}) \)-hierarchy obtained by splitting each \( \L(\bar{d}) \)-selfdual degree into the singletons of its elements,
using Proposition~\ref{prop:contractions} we can represent such hierarchy as in Figure~\ref{fig:chierarchy}, where the
bullets represent the \( \c(\bar{d}) \)-degrees and the boxes around them represent the \( \L(\bar{d}) \)-degrees they come from (notice that by Proposition~\ref{prop:contractions}(1) in the second column there are \( 12 \) different \( \c(\bar{d}) \)-degrees, while in the third column we already find \( 240 \) distinct \( \c(\bar{d}) \)-degrees!).
\begin{figure}[!htbp]
\centering
\[
\begin{array}{c}
\phantom{\vdots}\\
\phantom{\bullet} \\
\framebox{$\bullet$} \\
\framebox{$\bullet$} \\
\phantom{\bullet} \\
\phantom{\vdots}
\end{array}%
\smash[b]{
\underbrace{
\framebox{$
\begin{array}{c}
\bullet \\
\bullet
\end{array}
$} \;
\framebox{$
\begin{array}{c}
\bullet\\
\vdots \\
\bullet \\
\bullet\\
\vdots\\
\bullet
\end{array}
$} \;
\framebox{$
\begin{array}{c}
\bullet\\
\vdots \\
\bullet\\
\bullet \\
\bullet\\
\bullet\\
\vdots\\
\bullet
\end{array}
$} \;
\dotsc
}_{\omega}}
\begin{array}{c}
\phantom{\vdots}\\
\phantom{\bullet} \\
\framebox{$\bullet$} \\
\framebox{$\bullet$} \\
\phantom{\bullet} \\
\phantom{\vdots}
\end{array}
\smash[b]{
\underbrace{
\framebox{$
\begin{array}{c}
\vdots \\
\bullet\\
\bullet\\
\bullet \\
\bullet\\
\bullet \\
\bullet \\
\bullet \\
\bullet \\
\vdots
\end{array}
$} \;
\framebox{$
\begin{array}{c}
\vdots \\
\bullet\\
\bullet \\
\bullet\\
\bullet \\
\bullet\\
\bullet \\
\bullet\\
\bullet\\
\vdots
\end{array}
$} \;
\framebox{$
\begin{array}{c}
\vdots \\
\bullet\\
\bullet\\
\bullet \\
\bullet\\
\bullet \\
\bullet \\
\bullet\\
\bullet\\
\vdots
\end{array}
$} \;
\dotsc
}_{\omega}}
\begin{array}{c}
\phantom{\vdots}\\
\phantom{\bullet} \\
\framebox{$\bullet$} \\
\framebox{$\bullet$} \\
\phantom{\bullet} \\
\phantom{\vdots}
\end{array}
\;
\dotsc \dotsc
\]
\vspace{1cm}
\caption{The \( \c(\bar{d}) \)-hierarchy on Borel subsets of \( \mathcal{C} \).}
\label{fig:chierarchy}
\end{figure}
Proposition~\ref{prop:contractions} and Theorem~\ref{th:contractions} also imply the following corollary.
\begin{corollary}\label{cor:contractions}
\begin{enumerate}[(1)]
\item
The \( \c(\bar{d}) \)-hierarchy \( {\sf Deg}_{\boldsymbol{\Delta}^1_1}(\c(\bar{d})) \) on Borel subsets of \( \mathcal{C} \) is bad but not very bad. In fact it contains antichains of size the continuum.
\item
The \( \c(\bar{d}) \)-hierarchy \( {\sf Deg}_{\boldsymbol{\Sigma}^0_1 \cup \boldsymbol{\Pi}^0_1}(\c(\bar{d})) \) on open or closed subsets of \( \mathcal{C} \) is good but not very good.
\end{enumerate}
\end{corollary}
Notice that Corollary~\ref{cor:contractions}(2) gives a partial answer to~\cite[Question 6.3]{MottoRos:2012c}. However, such solution is not completely satisfactory, as we needed to restrict our hierarchy to a very small class of subsets of \( \mathcal{C} \) --- of course it would be more interesting to find a reducibility \( \mathcal{F} \) (on some Polish space \( X \)) inducing a good but not very good hierarchy on the entire collection of Borel subsets of \( X \) (or, under \( {\sf AD} \), even on the entire \( \mathscr{P}(X) \)). This last problem seems to be completely open, but the next example shows that if the requirement that the preorder inducing the hierarchy be of the form \( \leq_\mathcal{F} \) (for some reducibility \( \mathcal{F} \) on \( X \)) is dropped, then one can obtain a ``natural'' hierarchy on the collection of all Borel subsets of \( \pre{\omega}{\omega} \) which is good but not very good .
\begin{example}
Given a set $R\subseteq\mathbb{R}^+$ and $A,B \subseteq \pre{\omega}{\omega}$ such that \( A \leq_{\mathsf{Lip}(\bar{d})} B \), let
\( L_{A,B} = \inf \{ 0 < L \in \mathbb{R}^+ \mid A \leq_{\mathsf{Lip}(\bar{d},L)} B \}\), where \( \leq_{\mathsf{Lip}(\bar{d},L)} \) is as in Definition~\ref{def:almost}. Then set
\[
A \leq_R B \iff A \leq_{\mathsf{Lip}(\bar{d})} B \wedge L_{A,B}\in R\cup\{0,1\}.
\]
Notice that \( \leq_R \) is always reflexive: in fact, either \( A \nleq_{\mathsf{Lip}(\bar{d},L)} A \) for all \( L < 1 \) (in which case the identity function witnesses \( L_{A,A} = 1 \)), or else by considering arbitrarily large powers of any witness of \( A \leq_{\mathsf{Lip}(\bar{d}),L} A \) (for some \( L < 1 \)) we see that \( L_{A,A} = 0\). In contrast, notice that in general $\leq_{R}$ need not to be transitive. However, when \( \leq_R \) actually happens to be a preorder (as in all the relevant cases considered below), then with a little abuse of terminology we can consider the \( \leq_R \)-hierarchy on Borel subsets of \( \pre{\omega}{\omega} \) (with the obvious definition).
Using the methods introduced at the end of~\cite[Section 4]{MottoRos:2012c}, it is easy to see that if \( A,B \subseteq \pre{\omega}{\omega} \) are Borel sets such that \( A <_{\mathsf{Lip}(\bar{d})} B \), then also \( A <_R B \), because in this case \( L_{A,B} = 0 \). Moreover, since by~\cite[Corollary 4.4]{MottoRos:2012c} if \( A \subseteq \pre{\omega}{\omega} \) is \( \mathsf{Lip}(\bar{d}) \)-nonselfdual (equivalently: \( \L(\bar{d}) \)-nonselfdual), then \( A \leq_{\mathsf{Lip}(\bar{d},L)} A \) for every \( L > 0 \), we get that for such an \( A \), \( A \leq_{\L(\bar{d})} B \Rightarrow A \leq_R B \) for every \( B \subseteq \pre{\omega}{\omega} \), and if \( R \subseteq (0,1] \) we in fact have that \( A \leq_{\L(\bar{d})} B \iff A \leq_R B \).
Finally, if \( A \subseteq \pre{\omega}{\omega} \) is \( \L(\bar{d}) \)-selfdual and \( B \in [A]_{\L(\bar{d})} \), then \( A \equiv_R B \) because~\cite[Proposition 4.2]{MottoRos:2012c} implies that all the witnesses of \( A \leq_{\L(\bar{d})} B \) and \( B \leq_{\L(\bar{d})} A \) cannot have Lipschitz constant \( < 1 \).
Summing up, we get that if \( R \subseteq (0,1] \) (and \( \leq_R \) is transitive), then the $\leq_R$-hierarchy refines the $\mathsf{L}(\bar{d})$-hierarchy, and may differ from it only within the $\mathsf{Lip}(\bar{d})$-selfdual degrees.
Let us now concentrate on the canonical examples given by $R_n=(0, 2^{-n}]$ (for \( n \in \omega \)). It is easy to check that if \( n \leq 1 \),
then the \( \leq_R \)-hierarchy coincides with the \( \L(\bar{d}) \)-hierarchy. However, if \( n > 1 \) and $A$ is an $\L(\bar{d})$-selfdual set, then
\begin{equation} \tag{$\dagger$} \label{eq:dagger}
A\leq_{R_n} B \iff A \leq_{\L(\bar{d})} B \wedge (B \equiv_{\L(\bar{d})} A \vee 0^{(n)} {}^\smallfrown{} A \leq_{\mathsf{L}(\bar{d})} B).
\end{equation}
Therefore the restriction of \( \leq_{R_n} \) to the Borel subsets of
\( \pre{\omega}{\omega} \) is always transitive (hence a preorder), and it is also well-founded. Moreover, \eqref{eq:dagger} also implies that the
antichains in $\leq_{R}$ have always size \( \leq n \). Since e.g.\ \( \{ 0^{(i+1)} {}^\smallfrown{} \pre{\omega}{\omega} \mid i < n \} \) is an \( \leq_{R_n} \)-antichain of size precisely \( n \) consisting of clopen sets, we get that for all $n\geq 3$ the $\leq_{R_n}$-hierarchy on Borel subsets of \( \pre{\omega}{\omega} \) is good
but not very good.
\end{example}
\section{Wadge-like reducibilities and the Axiom of Choice} \label{sec: choice}
By (the comment following) Proposition~\ref{prop:continuous}, the \( \L(\bar{d}) \)-hierarchy \( {\sf Deg}_{\boldsymbol{\Delta}^1_1}(\L(\bar{d})) \) on the Borel subsets of
\( \pre{\omega}{\omega} \) is very good, and as already recalled the same is true for larger classes of subsets of
\( \pre{\omega}{\omega} \) if we further assume corresponding determinacy axioms. It is therefore natural to ask what happens if,
instead of assuming such determinacy principles, we assume the Axiom of Choice \( {\sf AC} \) or other strong choice principles.
Similar considerations apply to arbitrary Polish spaces as well. It is shown in~\cite{Schlicht:2012} that for every non-zero-dimensional Polish space
\( X \) the \( \mathsf{W}(X) \)-hierarchy on Borel subsets of \( X \) already contains antichains of size the continuum, and in fact~\cite{Ikegami:2012} shows that
if e.g.\ \( X = \mathbb{R} \) then we can also embed \( (\mathscr{P}(\omega), \subseteq^*) \) into \( {\sf Deg}_{\boldsymbol{\Delta}^1_1}(\mathsf{W}(X)) \) (but this last result
cannot be extended to arbitrary \( X \): as explained in~ \cite[Section 5.1]{MottoRos:2012b}, all continuous functions on the Cook continuum \( X \)
are either constant or the identity, and therefore all chains of subsets of \( X \) with respect to continuous reducibility have length \( \leq 2 \)).
However,~\cite{MottoRos:2012b} shows that for every Polish space \( X \), the \( \mathsf{D}_\alpha(X) \)-hierarchy on Borel subsets of \( X \) (where
\( \mathsf{D}_\alpha(X) \) denotes the collection of all \( \boldsymbol{\Delta}^0_\alpha \)-functions from \( X \) to itself) is always very good for
\( \alpha \geq \omega \), and that the same is true for \( \alpha \geq 3 \) if \( X \) is of dimension \( \neq \infty \). Also these last results extend to larger
classes of subsets of \( X \) under suitable determinacy assumptions, and therefore it is meaningful to ask what happens if instead we assume \( {\sf AC} \).
Not surprisingly, it turns out that under choice all the above mentioned hierarchies of degrees (on arbitrary subsets of \( X \)) become very bad.
Clearly, Borel determinacy forces us to consider non-Borel subsets of \( X \) to get such results: therefore in what follows we will concentrate only on
uncountable (ultrametric) Polish spaces.
\begin{notation}
If $X$ is a set and $A\subseteq X^2$, we denote by \( A_x \) the ``vertical section'' determined by \( x \in X \), i.e.\ we set $A_x=\{y\in X\mid (x,y)\in A\}$. Moreover, for every cardinal \( \mu \) we set $[X]^{\mu}=\{Y\subseteq X \mid |Y|=\mu\}$.
\end{notation}
\begin{lemma} [\({\sf AC} \)] \label{lem:choice}
Let \(\mu\) be an infinite cardinal and $X$ be a set of size $\mu$. Moreover, let $\mathcal{C} \subseteq [X]^{\mu}$, $\mathcal{F}$ be a collection of functions from \( X \) to itself, and suppose that \( | \mathcal{C} | = | \mathcal{F} | = \mu \). Then there is a set $A\subseteq X^2$ such that $A_{x}\cap C \nleq_{\mathcal{F}} A_{y}$ for all distinct $x,y \in X$ and all $C\in\mathcal{C}$.
\end{lemma}
\begin {proof}
We first recursively construct a sequence $ (\{ A_{x,\alpha}, B_{x,\alpha} \mid \ x\in X \})_{\alpha<\mu}$ such that $A_{x,\alpha}\cap B_{x,\alpha}=\emptyset$,
$A_{x,\alpha}\subseteq A_{x,\beta}$,
$B_{x,\alpha}\subseteq B_{x,\beta}$, and
$|A_{x,\alpha}\cup B_{x,\alpha}|\leq |2 \cdot\alpha|$
for all $\alpha \leq \beta<\mu$ and $x\in X$.
Fix a surjection $h\colon \mu\rightarrow \mathcal{C}\times \mathcal{F}\times X^2$, and
set $A_{x,0}=B_{x,0}=\emptyset$ for all $x\in X$. Let now \( 0 < \alpha < \mu \), and assume that all sets of the form \( A_{x,\beta},B_{x,\beta} \) for \( x \in X \) and \( \beta < \alpha \) have already been defined, so that we can set $A_{x,<\alpha}=\bigcup_{\beta<\alpha} A_{x,\beta}$ and $B_{x,<\alpha}=\bigcup_{\beta<\alpha} B_{x,\beta}$. Let \( (C,f,x,y) \in \mathcal{C} \times \mathcal{F} \times X^2 \) be such that $h(\alpha)=(C,f,x,y)$, and let $C_0=C\setminus (A_{x,<\alpha}\cup B_{x,<\alpha})$. Notice that $| C_0| = \mu$ because $|A_{x,<\alpha}\cup B_{x,<\alpha} | < \mu \) and $|C| = \mu$.
We distinguish two cases: if $|f(C_0)|<\mu$, we choose distinct $a,b\in C_0$ such that $f(a)=f(b)$ (this is possible because $| C_0| = \mu > |f(C_0)|$), and then we set $A_{x,\alpha}= A_{x,<\alpha}\cup \{a\}$, $B_{x,\alpha}= B_{x,<\alpha}\cup \{b\}$, and \( A_{z,\alpha} = A_{z, <\alpha} \), \( B_{z, \alpha} = B_{z,<\alpha} \) for all \( z \in X \) distinct from \( x \).
If instead $|f(C_0)|=\mu$, we pick some $a\in C_0$ with $f(a)\notin A_{y,<\alpha}\cup B_{y,<\alpha}$ (which exists because $|A_{y,<\alpha}\cup B_{y,<\alpha}| < \mu$, and hence $f(C_0) \setminus (A_{y,<\alpha}\cup B_{y,<\alpha}) \neq \emptyset$), and then we set $A_{x,\alpha}=A_{x,<\alpha}\cup\{a\}$, $B_{x,\alpha} = B_{x,<\alpha} $, \( A_{y,\alpha} = A_{y,<\alpha} \), $B_{y,\alpha}=B_{y,<\alpha}\cup\{f(a)\}$, and \( A_{z,\alpha} = A_{z,<\alpha} \), \( B_{z,\alpha} = B_{z,<\alpha} \) for all \( z \in X \) distinct from \( x \) and \( y \). This completes the recursive step of our construction, and it is easy to check by induction on \( \alpha < \mu \) that the sets \( A_{x,\alpha} \), \( B_{x,\alpha} \) are as required.
Finally, we set $A_x=\bigcup_{\alpha<\mu} A_{x,\alpha}$, \( B_x = \bigcup_{\alpha < \mu} B_{x,\alpha} \), and \( A = \{ (x,y) \in X^2 \mid y \in A_x \} \), so that, in particular, \( A_x \cap B_x = \emptyset \) for every \( x \in X \). It is straightforward to check that the $\alpha$-th step in the recursive construction above ensures that \( f \) is not a reduction of $A_x\cap C$ to $A_y$, because either there are \( a \in A_x\cap C \) and \( b \in B_x \subseteq X \setminus A_x \) such that \( f(a) = f(b) \), or else there is \( a \in A_x\cap C \) such that \( f(a) \in B_y \subseteq X \setminus A_y \).
\end{proof}
\begin{theorem}[\({\sf AC} \)] \label{th:illfounded hierarchy}
Let $X = (X,d)$ be an uncountable ultrametric Polish space. Then there is a
map \( \psi \colon \mathscr{P}(\omega) \to \mathscr{P}(X) \) such that for all \( a,b \subseteq \omega \)
\begin{enumerate}
\item
if \( a \subseteq b \), then \( \psi(a) \leq_{\L(d)} \psi(b) \);
\item
if \( \psi(a) \leq_{\mathsf{Bor}(X)} \psi(b) \), then \( a \subseteq b \).
\end{enumerate}
In particular, \( (\mathscr{P}(\omega), \subseteq) \) embeds into the \( \mathcal{F} \)-hierarchy on \( X \) for every reducibility \( \L(d) \subseteq \mathcal{F} \subseteq \mathsf{Bor}(X) \), hence \( {\sf Deg}(\mathcal{F}) \) is very bad.
\end{theorem}
\begin{proof}
We apply the Lemma~\ref{lem:choice} letting \( \mu = |X| = 2^{\aleph_0} \), \( \mathcal{C} \) be the set of all \emph{uncountable} Borel subsets of $X$, and \( \mathcal{F} = \mathsf{Bor}(X) \) be the collection of all Borel functions from \( X \) to itself. Thus we obtain a sequence of $\leq_{\mathsf{Bor}(X)}$-incomparable sets $A_n\subseteq X$ (the lemma gives more, but an $\omega$-sequence is sufficient here). Notice that each \( A_n \) is necessarily uncountable and that \( A_n \neq X \), as otherwise in both cases we would easily have \( A_n \leq_{\mathsf{Bor}(X)} A_m \) for every \( m \in \omega \). Now choose a sequence $(X_n)_{n\in\omega}$ of pairwise disjoint uncountable clopen balls in $X$, and fix a Borel isomorphism $h_n\colon X \to X_n$ for every \( n \in \omega \). Given \( a \subseteq \omega \), set $\psi(a)=\bigcup_{n \in a} h_n(A_n)$.
To see that $\psi$ is as required, first suppose that \( a,b \subseteq \omega \) are such that $a \subseteq b$, and for every $n\in b \setminus a$ pick a point \( y_n \in X_n \setminus h_n(A_n) \) (which exists because \( A_n\neq X \)). Then we define \( f \colon X \to X \) by setting
\[
f(x) =
\begin{cases}
y_n & \text{if }x \in X_n \text{ for some } n \in b \setminus a, \\
x & \text{otherwise}.
\end{cases}%
\]
Clearly \( f \) reduces \( \psi(a) \) to \( \psi(b) \), and it is easy to check that since \(d \) is an ultrametric and the \( X_n \) are (cl)open balls, then \( f \in \L(d) \): therefore $\psi(a)\leq_{\L(d)} \psi(b)$, as required.
Now let \(a,b \subseteq \omega \) be such that $\psi(a) \leq_{\mathsf{Bor}(X)} \psi(b)$, let \( f \in \mathsf{Bor}(X) \) be a witness of this, and fix an arbitrary $n\in a$. Notice that \( f( \psi(a)) \subseteq \psi(b) \subseteq \bigcup_{m \in b} X_m \). Since \( A_n \) is uncountable, this means that there is $m \in b$ such that $f^{-1}(X_m)\cap X_n$ is uncountable. Fix \( \bar{y} \in X \setminus A_m \): setting $C=h_n^{-1}( f^{-1}(X_m)\cap X_n)$, we get that \( C \) is an uncountable Borel set, and that the map \( g \colon X \to X \) defined by
\[
g(x) =
\begin{cases}
(h^{-1}_m \circ f \circ h_n) (x) & \text{if } x \in C, \\
\bar{y} & \text{otherwise}
\end{cases}%
\]
witnesses \( A_n \cap C \leq_{\mathsf{Bor}(X)} A_m \). By our choice of the \( A_n \)'s, this implies \( n = m \), whence \( n \in b \). Therefore \( a \subseteq b \), as required.
\end{proof}
\begin{remark}
Notice that to get Lemma~\ref{lem:choice} it is enough to assume that \( X \) is a well-orderable set. Therefore, also in Theorem~\ref{th:illfounded hierarchy} we can weaken the assumption \( {\sf AC} \) by just requiring that \( X \) (equivalently, any uncountable Polish space) is well-orderable.
\end{remark}
Using essentially the same argument, one can also show that a variant of Theorem~\ref{th:illfounded hierarchy} applies to arbitrary uncountable Polish spaces \( X \) (and not only to the ultrametric ones).
\begin{theorem}[\( {\sf AC} \)]\label{corollary: illfounded Borel hierarchy}
Let \( X \) be an uncountable Polish space. Then there is a map \( \psi \colon \mathscr{P}(\omega) \to \mathscr{P}(X) \) such that for every \( a,b \subseteq \omega \)
\begin{enumerate}
\item
if \( a \subseteq b \), then \( \psi(a) \leq_{\mathsf{D}_2(X)} \psi(b) \);
\item
if \( \psi(a) \leq_{\mathsf{Bor}(X)} \psi(b) \), then \( a \subseteq b \).
\end{enumerate}
In particular, \( (\mathscr{P}(\omega), \subseteq) \) embeds into the \( \mathcal{F} \)-hierarchy on \( X \) for every reducibility \( \mathsf{D}_2(X) \subseteq \mathcal{F} \subseteq \mathsf{Bor}(X) \), hence \( {\sf Deg}(\mathcal{F}) \) is very bad.
\end{theorem}
\begin{proof}
In the proof of Theorem \ref{th:illfounded hierarchy}, let $(X_n)_{n\in\omega}$ be a partition of $X$ into uncountable ${\bf \Delta}^0_2$ sets.
\end{proof}
\begin{remark}
In Theorem~\ref{corollary: illfounded Borel hierarchy} we cannot replace \( \leq_{\mathsf{D}_2(X)} \) with continuous reducibility \( \leq_{\mathsf{W}(X)} \): in fact, in the Cook continuum \( X \) (which is uncountable), we cannot hope to embed \( (\mathscr{P}(\omega), \subseteq) \) into \( {\sf Deg}(\mathsf{W}(X)) \) because there are no infinite chains of subsets of \( X \) (with respect to continuous reducibility).
\end{remark}
We now aim to show that if we further assume \( \mathsf{V=L} \), then the map $\psi$ of Theorems \ref{th:illfounded hierarchy} and \ref{corollary: illfounded Borel hierarchy} can be chosen to range in the collection of ${\bf \Pi}^1_1$ (alternatively: \( \boldsymbol{\Sigma}^1_1 \)) subsets of the given (ultrametric) Polish space: this in particular implies that the $\mathsf{L}(\bar{d})$-hierarchy on ${\bf \Pi}^1_1$ (respectively, \(\boldsymbol{\Sigma}^1_1 \)) subsets of $\pre{\omega}{\omega}$ is very bad in $\mathsf{L}$.
To prove this, we will modify the recursion used in the proof of Lemma~\ref{lem:choice} so that membership in each of the sets can be computed in the next admissible set.
\begin{notation}
For \( x,y \in \pre{\omega}{\omega} \), let $\omega_1^{x,y}$ denote the least $(x,y)$-admissible ordinal $\gamma$.%
\footnote{That is, \( \omega_1^{x,y} \) is the least $\gamma>\omega$ such that $L_{\gamma}[x,y]$ is a model of Kripke-Platek set theory.}
To simplify the notation, set also \( \omega_1^x = \omega_1^{x,x} \).
\end{notation}
\begin{theorem}[Spector-Gandy] (see \cite[Theorem 5.5]{Hjorth10}) \label{th:Spector-Gandy}
A set $A\subseteq{}^{\omega}\omega$ is $\Pi^1_1$ in a parameter $p\in {}^{\omega}\omega$ if and only if there is a $\Sigma_1$-formula $\varphi(x)$ such that
$$x\in A\Leftrightarrow L_{\omega_1^{x,p}}[x,p]\vDash \varphi(x,p)$$
for all $x\in {}^{\omega}\omega$.
\end{theorem}
\begin{lemma}\label{lemma:codes}
Let $X$ be a Polish space. Then there is a set $G\subseteq {}^{\omega}\omega\times X^2$ such that:
\begin{enumerate}
\item A set $F\subseteq X^2$ is the graph of a Borel function from $X$ to itself if and only if $F=G_x=\{(y,z)\in X^2\mid (x,y,z)\in G\}$ for some $x\in \operatorname{p}(G)$.
\item The projection $\operatorname{p}(G)$ on the first coordinate is a ${\bf \Pi}^1_1$ set.
\item $G$ is both ${\bf \Pi}^1_1$ and ${\bf \Sigma}^1_1$ on $\operatorname{p}(G)\times X^2$.
\end{enumerate}
\end{lemma}
\begin{proof}[Sketch of proof]
Notice that we can concentrate only on ultrametric Polish spaces \( X = (X,d) \), because the result can then be transferred to an arbitrary Polish space \( Y \) by using a Borel isomorphism between \( X \) and \( Y \). Therefore from now on we fix
an ultrametric
Polish space \( X = (X,d) \). Recall that \( \mathsf{Bor}(X) \) coincides with the collection of all Baire class $\alpha$ functions (for arbitrary $\alpha<\omega_1$), i.e.\ with the closure under pointwise limits of the collection of all Lipschitz functions (see~\cite[Corollary 2.16]{MottoRos:2009b} and~\cite[Theorems 24.3]{Kechris:1995}). Starting with a function $f$ defined on a fixed countable dense set $D\subseteq X$, we form the (pseudo-)limit $\bar{f}$ of \( f \) by setting $\bar{f}(x)=\lim_{n\rightarrow \infty} f(x_n)$ (for an arbitrary sequence $(x_n)_{n\in\omega}$ in $D$ with $\lim_{n\rightarrow \infty} x_n=x$) if
\[
\mathrm{osc}_f(x)=\lim_{n\rightarrow \infty} \sup \{ d(f(y), f(z)) \mid y,z \in X \wedge d(x,y), d(x,z)<2^{-n} \}=0,
\]
and \( \bar{f}(x) = y_0 \) (for \( y_0 \in X \) a fixed value) otherwise. From a countable family of functions \( f \) as above attached to the terminal nodes of a given well-founded tree, we can then build up a Borel function \( g \) by forming (pseudo-)limits (i.e.\ taking the pointwise limit where it exists and some fixed value \( y_0 \in X \) elsewhere) in the obvious way along the tree. The tree is then coded into an element of $x \in {}^{\omega}\omega$, and for all \( x \)'s built in this way we let \( G_x \) be the graph of the corresponding Borel function \( g \).
Notice that the set of codes is ${\bf \Pi}^1_1$ because of the condition that the trees used in the coding are well-founded.
\end{proof}
\begin{theorem}\label{th:illfounded hierarchy in L}
Assume $\mathsf{V=L}$ and let $X = (X,d)$ be an uncountable ultrametric Polish space. Then there is a
map \( \psi \) from \( \mathscr{P}(\omega) \) into the \( {\bf\Pi}^1_1 \) subsets of \( X \) such that for all \( a,b \subseteq \omega \)
\begin{enumerate}
\item
if \( a \subseteq b \), then \( \psi(a) \leq_{\L(d)} \psi(b) \);
\item
if \( \psi(a) \leq_{\mathsf{Bor}(X)} \psi(b) \), then \( a \subseteq b \).
\end{enumerate}
In particular, \( (\mathscr{P}(\omega), \subseteq) \) embeds into the \( \mathcal{F} \)-hierarchy on the ${\bf\Pi}^1_1$ subsets of $X$ for every reducibility \( \L(d) \subseteq \mathcal{F} \subseteq \mathsf{Bor}(X) \), hence \( {\sf Deg}_{\boldsymbol{\Pi}^1_1}(\mathcal{F} ) \) is very bad.
\end{theorem}
\begin{proof} Let $\mathbf{N}_{t}=\{x\in {}^{\omega}\omega\mid t\subseteq x\}$ for $t\in {}^{<\omega}\omega$.
We first assume that $(X,d)=({}^{\omega}\omega,\bar{d})$.
The map \( \psi \colon \mathscr{P}(\omega) \to \mathscr{P}(\pre{\omega}{\omega}) \) is defined by first
constructing a $\Pi^1_1$ set $A\subseteq \mathscr{P}(\omega)\times {}^{\omega}\omega$,\footnote{We freely identify each $a\in\mathcal{P}(\omega)$ with its characteristic function in ${}^{\omega}2$. The notions of $\Pi^1_1$ subsets of $\mathcal{P}(\omega)$ and $\mathcal{P}(\omega)\times{}^{\omega}\omega$ are defined accordingly. Since the identification is computable, the Spector-Gandy Theorem \ref{th:Spector-Gandy} holds for $\Pi^1_1$ subsets of $\mathcal{P}(\omega)\times {}^{\omega}\omega$. } and then letting $\psi (a)= A_a=\{y\in {}^{\omega}\omega \mid (a,y)\in A\}$.
To define the desired \( A \), we will in turn construct by recursion on $\alpha<\omega_1$ a sequence $(A_{n,\alpha}, B_{n,\alpha})_{n<\omega, \alpha<\omega_1}$ such that for all $n<\omega$ and $\alpha \leq \beta <\omega_1$
\begin{enumerate}
\item $A_{n,\alpha}, B_{n,\alpha}\subseteq \mathbf{N}_{\langle n \rangle}$,
\item $A_{n,\alpha}\cap B_{n,\alpha}=\emptyset$,
\item $|A_{n,\alpha}|, |B_{n,\alpha}|<\omega_1$, and
\item $A_{n,\alpha} \subseteq A_{n,\beta}$ and $B_{n,\alpha} \subseteq B_{n,\beta}$.
\end{enumerate}
Given \( a \subseteq \omega \), we then let for \( n < \omega \) and \( \gamma < \omega_1 \)
\begin{align*}
A_{a,\alpha}= & \bigcup\nolimits_{n\in a} A_{n,\alpha}, & B_{a,\alpha}=& \bigcup\nolimits_{n\in a} B_{n,\alpha}, \\
A_{n,<\gamma}=&\bigcup\nolimits_{\alpha<\gamma} A_{n,\alpha}, & B_{n,<\gamma}=&\bigcup\nolimits_{\alpha<\gamma} B_{n,\alpha}, \\
A_{a,<\gamma}=& \bigcup\nolimits_{\alpha<\gamma} A_{a,\alpha}, & B_{a,<\gamma}=&\bigcup\nolimits_{\alpha<\gamma} B_{a,\alpha},
\end{align*}
and finally we set
\begin{align*}
A_n=&\bigcup\nolimits_{\alpha<\omega_1} A_{n,\alpha}, & B_n= &\bigcup\nolimits_{\alpha<\omega_1} B_{n,\alpha}, \\
A_a=&\bigcup\nolimits_{n\in a} A_n, & B_a=&\bigcup\nolimits_{n\in a} B_n.
\end{align*}
Notice that by (1), (2) and (4), we have that for all \( a \subseteq \omega \) and \( \gamma < \omega_1 \), \( A_{a,<\gamma} \cap B_{a,<\gamma} = \emptyset \) and \( A_a \cap B_a = \emptyset \). Finally,
to simplify the notation we will also write $s_{\gamma}=(A_{n,\alpha}, B_{n,\alpha})_{n<\omega, \alpha<\gamma}$ for $\gamma<\omega_1$.
The construction is based on the following claims.
\begin{claim} \label{claim: many reals with large omega1ck}
For every $s\in L_{\omega_1}$ and $l<\omega$, there are uncountably many $x \in \mathbf{N}_{\langle l \rangle}$ with $s\in L_{\omega_1^x}$.
\end{claim}
\begin{proof}
Let \( \alpha < \omega_1 \) and $x\in \mathbf{N}_{\langle l \rangle}$ be such that $s\in L_{\alpha}$ and $\omega_1^x>\alpha$. Then $\omega_1^{x,y}>\alpha$ for all $y\in {}^{\omega}\omega$, so $s\in L_{\omega_1^{x\oplus y}}$ for any $y\in {}^{\omega}\omega$ (where $x \oplus y$ is defined by $(x\oplus y)(2i)=x(i)$ and $(x\oplus y)(2i+1)=y(i)$).
\end{proof}
Let $A_{n,0}= B_{n,0}=\emptyset$.
Let now $\gamma > 0$, and suppose that the $\gamma^{\mathrm{th}}$ element in $<_L$ is of the form $(c,a,b,l)$, where $c\in {}^{\omega}\omega$ is a code for a Borel measurable function $f\colon {}^{\omega}\omega\rightarrow{}^{\omega}\omega$ as in Lemma~\ref{lemma:codes}, $a,b\subseteq\omega$, and $l\in a\setminus b$ (if this is not the case, we simply let $A_{n,\gamma }=A_{n,<\gamma}$ and $B_{n,\gamma}=B_{n,<\gamma}$ for all $n\in\omega$).
\begin{claim}\label{claim: extension in successor step}
There is $x\in \mathbf{N}_{\langle l \rangle}$ such that $s_{\gamma}\in L_{\omega_1^x}[x]$ and $x\notin A_{l,<\gamma}\cup B_{l,<\gamma}$.
\end{claim}
\begin{proof}
Let
$W \subseteq {}^{\omega}\omega$ denote the set of $x\in {}^{\omega}\omega$ with $s_{\gamma}\in L_{\omega_1^x}[x]$. Since $W \cap \mathbf{N}_{\langle l \rangle}$ is uncountable by Claim~\ref{claim: many reals with large omega1ck} and $A_{l,<\gamma}\cup B_{l,<\gamma}$ is countable by (3), there is $x\in (W\cap \mathbf{N}_{\langle l \rangle}) \setminus (A_{l,<\gamma}\cup B_{l,<\gamma})$, as required.
\end{proof}
Let $\bar{x} \in {}^{\omega}\omega$ denote the $<_L$-least element satisfying Claim~\ref{claim: extension in successor step}.
If $f(\bar{x}) \in A_{b,<\gamma}$, let $B_{l,\gamma}=B_{l,<\gamma}\cup\{ \bar{x} \}$.
If $f(\bar{x}) \in B_{b, < \gamma}$, let $A_{l,\gamma}= A_{l,<\gamma}\cup \{ \bar{x} \}$.
Finally, if $f(\bar{x}) \notin A_{b,<\gamma}\cup B_{b,<\gamma}$ and $f(\bar{x})(0) = m$, let $A_{l,\gamma}=A_{l,<\gamma}\cup\{ \bar{x} \}$, and if additionally $m\in b$, then let $B_{m,\gamma}=B_{m,<\gamma}\cup \{f(\bar{x})\}$.
For all remaining $A_{n,\gamma}$'s and $B_{n,\gamma}$'s, let $A_{n,\gamma}=A_{n,<\gamma}$ and $B_{n,\gamma}=B_{n,<\gamma}$.
Note that if $f(\bar{x}) \notin A_{b,<\gamma}\cup B_{b,<\gamma}$ and $m\in b$, then
$f(\bar{x})\notin A_{m,<\gamma}$.
This completes the construction of the \( A_{n,\alpha} \)'s and \( B_{n, \alpha} \)'s, and hence also of the sets \( A_a \) and \( B_a \) for \( a \subseteq \omega \).
\begin{claim}
If $a \subseteq b \subseteq \omega$, then $A_a\leq_{\mathsf{L}(\bar{d})} A_b$.
\end{claim}
\begin{proof}
Let $f(z)=z$ for all $z\in \mathbf{N}_{\langle l \rangle}$ with $l\in a\cup (\omega\setminus b)$, while for $z\in \mathbf{N}_{\langle l \rangle}$ with $l\in b\setminus a$, fix $z_0\notin A_b$ and let $f(z)=z_0$. Clearly \( f \in \L(\bar{d}) \). The result then follows from the fact that \( A_a \cap \mathbf{N}_{\langle l \rangle} = A_l \subseteq \mathbf{N}_{\langle l \rangle} \) for \( l \in a \) and \( A_a \cap \mathbf{N}_{\langle l \rangle} = \emptyset \) otherwise (and similarly for \( a \) replaced by \( b \)).
\end{proof}
\begin{claim}\label{claim: no Borel reduction}
If $a\not\subseteq b$, then $A_a\not\leq_{\mathsf{Bor}({}^{\omega}\omega)} A_b$.
\end{claim}
\begin{proof}
Suppose that $f\colon {}^{\omega}\omega\rightarrow {}^{\omega}\omega$ is a Borel measurable function with $A_a=f^{-1}(A_b)$. Fix $l\in a\setminus b$, and let \( c \) and \( \gamma \) be such that $c$ codes $f$ as in Lemma~\ref{lemma:codes} and $(c,a,b,l)$ is the $\gamma^{\text{th}}$ element in $<_L$.
Suppose that in step $\gamma$ of the construction, $ \bar{x} \in {}^{\omega}\omega$ is the $<_L$-least pair in Claim~\ref{claim: extension in successor step}.
If $f(\bar{x}) \in A_{b , < \gamma}$, then $\bar{x} \in B_l\subseteq B_a$ and $f(\bar{x})\in A_b$. Then $\bar{x}\notin A_a$ because \( A_a \cap B_a = \emptyset \), and hence $A_a\neq f^{-1}(A_b)$.
If $f(\bar{x}) \in B_{b, < \gamma} $, then $\bar{x} \in A_l\subseteq A_a$ and $f(\bar{x})\in B_b$. Then $A_a\neq f^{-1}(A_b)$ because \( A_b \cap B_b = \emptyset \).
Finally, suppose that $f(\bar{x}) \notin A_{b, < \gamma} \cup B_{b, < \gamma}$ and $f(\bar{x})(0) = m$. Then $\bar{x} \in A_l\subseteq A_a$. If $m\notin b$, then $f(\bar{x})\notin A_b$ because \( A_b \cap \mathbf{N}_{\langle m \rangle} = \emptyset \).
If $m\in b$, then $f(\bar{x})\in B_m \subseteq B_b$, so $f(\bar{x})\notin A_b$ by \( A_b \cap B_b = \emptyset \). So in both cases $A_a\neq f^{-1}(A_b)$.
\end{proof}
Note that the condition that $(c,a,b,l)$ is of the required form is $\Pi^1_1$ by Lemma~\ref{lemma:codes}. Since the truth value of $\Pi^1_1$ statements can be calculated in admissible sets by the Spector-Gandy Theorem \ref{th:Spector-Gandy}, this implies that the recursion is absolute between (and definable over) admissible sets.
Let $A=\{ (a,y) \mid y \in A_a \} \subseteq \mathscr{P}(\omega) \times \pre{\omega}{\omega}$.
\begin{claim}
A is $\Pi^1_1$.
\end{claim}
\begin{proof}
We freely identify each \( a \subseteq \omega \) with its characteristic function, so that \( A \) can be viewed as a
subset of \( \pre{\omega}{\omega} \times \pre{\omega}{\omega} \). It is then sufficient to show that there is a $\Sigma_1$ formula $\varphi$ such that $(x,y)\in A$ if and only if $L_{\omega_1^{x,y}}[x,y]\vDash \varphi(x,y)$ for all $x,y\in {}^{\omega}\omega$, by the Spector-Gandy Theorem \ref{th:Spector-Gandy}.
Let $\varphi_l(y)$ state that there is some $\gamma<\omega_1$ and a sequence $s$ of length $\gamma$ constructed according to the recursion
such that $y$ is added to $A_l$ at step $\gamma$.
Let $\varphi(x,y) \iff \exists l \in x\ \varphi_l(y)$.
Suppose that $(x,y)\in A$. Then $y \in A_l\subseteq A_x$ for some $l\in x$. Suppose that $y$ is added to $A_l$ at step $\gamma<\omega_1$ in the construction. Then $s_{\gamma}\in L_{\omega_1^y}[y]$ by the definition in the successor step. Then $L_{\omega_1^{x,y}}[x,y]\vDash \varphi_l(y)$, and hence $L_{\omega_1^{x,y}}[x,y]\vDash \varphi(x,y)$.
Now suppose that $L_{\omega_1^{x,y}}[x,y]\vDash \varphi(x,y)$. Then $L_{\omega_1^{x,y}}[x,y]\vDash \varphi_l(y)$ for some $l\in x$. Since the recursion is absolute between admissible sets, this implies that $y \in A_x$ and hence $(x,y)\in A$.
\end{proof}
Now suppose that $(X,d)$ is an arbitrary uncountable ultrametric Polish space. Fix a sequence $(U_n)_{n\in\omega}$ of disjoint uncountable open balls.
For each $l\in\omega$, fix a \( \boldsymbol{\Pi}^0_2 \) proper subset $C_l$ of $U_l$ homeomorphic to ${}^{\omega}\omega$, together with a homeomorphism $h_l\colon \mathbf{N}_{\langle l \rangle}\rightarrow C_l$ and a point \( x_l \in U_l \setminus C_l \).
Let $h\colon {}^{\omega}\omega\rightarrow X \colon x \mapsto h_{x(0)}(x)$. Then $h\colon {}^{\omega}\omega\rightarrow \bigcup_{l\in\omega} C_l$ is a homeomorphism and $\bigcup_{l\in\omega} C_l$ is a Borel subset of $X$. Finally, let $\psi(a)=h(A_a)$ for $a \subseteq\omega$.
\begin{claim}\label{claim: Lipschitz reduction for ultrametric space}
If $a\subseteq b\subseteq\omega$, then $\psi(a)\leq_{\mathsf{L}(d)}\psi(b)$.
\end{claim}
\begin{proof}
Let $f(z)=z$ for all $z\in C_l$ with $l\in a\cup (\omega\setminus b)$ and all $z\in X\setminus \bigcup_{l\in\omega} U_l$, and let $f(z)=x_l$ for all $z\in U_l$ with $l\in b\setminus a$. Then $f\colon X\rightarrow X$ is Lipschitz and $\psi(a)=f^{-1}(\psi(b))$.
\end{proof}
\begin{claim} \label{claim:nonBorelreduction}
If $a\not\subseteq b$, then $\psi(a)\not\leq_{\mathsf{Bor}(X)}\psi(b)$.
\end{claim}
\begin{proof}
Suppose towards a contradiction that \( f \colon X \to X \) is a Borel map reducing \( \psi(a) \) to \(\psi(b) \). Since \( b \neq \omega \) by \( a \not\subseteq b \), there is \( y_0 \in (\bigcup_{l \in \omega} C_l ) \setminus \psi(b) \). Let \( \bar{f} \colon X \to X \) by letting \( \bar{f}(x) = f(x) \) if \( f(x) \in \bigcup_{l \in \omega} C_l \) and \( \bar{f}(x) = y_0 \) otherwise: then \( \bar{f} \colon X \to \bigcup_{l \in \omega} C_l \) is Borel and still reduces \( \psi(a) \) to \( \psi(b) \). Since the map $h\colon {}^{\omega}\omega\rightarrow \bigcup_{l\in\omega} C_l$ is a homeomorphism, the function \( h^{-1} \circ \bar{f} \circ h \colon \pre{\omega}{\omega} \to \pre{\omega}{\omega} \) is a well-defined Borel function reducing \(A_a \) to \(A_b \), contradicting Claim~\ref{claim: no Borel reduction}.
\end{proof}
This completes the proof of Theorem \ref{th:illfounded hierarchy in L}.
\end{proof}
Similar to Corollary \ref{corollary: illfounded Borel hierarchy}, we now obtain:
\begin{theorem}\label{corollary: illfounded Borel hierarchy in L}
Assume $\mathsf{V=L}$ and let \( X \) be an uncountable Polish space. Then there is a map \( \psi \) from \( \mathscr{P}(\omega) \) into the \( {\bf\Pi}^1_1\) subsets of \( X \) such that for every \( a,b \subseteq \omega \)
\begin{enumerate}
\item
if \( a \subseteq b \), then \( \psi(a) \leq_{\mathsf{D}_2(X)} \psi(b) \);
\item
if \( \psi(a) \leq_{\mathsf{Bor}(X)} \psi(b) \), then \( a \subseteq b \).
\end{enumerate}
In particular, \( (\mathscr{P}(\omega), \subseteq) \) embeds into the \( \mathcal{F} \)-hierarchy on the ${\bf\Pi}^1_1$ subsets of $X$ for every reducibility \( \mathsf{D}_2(X) \subseteq \mathcal{F} \subseteq \mathsf{Bor}(X) \), hence \( {\sf Deg}_{\boldsymbol{\Pi}^1_1}(\mathcal{F}) \) is very bad.
\end{theorem}
\begin{proof}
Define \(\psi\) as at the end of the proof of Theorem~\ref{th:illfounded hierarchy in L} (where the case of an arbitrary ultrametric Polish space is considered). If \( a \subseteq b \subseteq \omega \), we define \( f \colon X \to X \) as in the proof of Claim~\ref{claim: Lipschitz reduction for ultrametric space}: then \( f \) is clearly $\mathsf{D}_2(X)$ and $\psi(a)=f^{-1}(\psi(b))$. The converse direction can be easily proved as in Claim~\ref{claim:nonBorelreduction}, hence we are done.
\end{proof}
The existence of maps \( \widetilde{\psi} \colon \mathscr{P}(\omega) \to {\bf\Sigma}^1_1(X) \) with the properties stated in Theorems~\ref{th:illfounded hierarchy in L} and~\ref{corollary: illfounded Borel hierarchy in L} follows immediately by taking complements, i.e.\ by setting \( \widetilde{\psi}(a) = X \setminus \psi(a) \) for every \( a \subseteq \omega \) (where \( \psi \colon \mathscr{P}(\omega) \to \boldsymbol{\Pi}^1_1(X) \) is as in Theorem~\ref{th:illfounded hierarchy in L} or Theorem~\ref{corollary: illfounded Borel hierarchy in L}).
\begin{remark}
By Borel determinacy, the requirement that \( \psi \) ranges into \( \boldsymbol{\Pi}^1_1 \) (alternatively: \( \boldsymbol{\Sigma}^1_1 \)) subsets of \( X \) in Theorems~\ref{th:illfounded hierarchy in L} and~\ref{corollary: illfounded Borel hierarchy in L} cannot be further improved, and therefore such results are optimal.
\end{remark}
It is well-known that ${\bf \Pi}^1_1$-determinacy implies that e.g.\ the $\mathsf{L}(\bar{d})$-hierarchy on ${\bf \Pi}^1_1$ subsets of \( \pre{\omega}{\omega} \) is very good. In fact, Harrington~\cite{Harrington:1978} (essentially) showed that the following are equivalent:
\begin{itemize}
\item
every \( \boldsymbol{\Pi}^1_1 \) subset of \( \pre{\omega}{\omega} \) is determined;
\item
for all \( x \in \pre{\omega}{\omega} \), \( x^\# \) exists;
\item
\( {\sf SLO}^{\L(\bar{d})} \) holds for \( \boldsymbol{\Pi}^1_1 \) subsets of \( \pre{\omega}{\omega} \).
\end{itemize}
Since sharps do not exist if \( \mathsf{V = L} \), Theorem~\ref{th:illfounded hierarchy in L} can then be regarded as a strengthening of (one direction)
of the above mentioned Harrington's result: under the further assumption \( \mathsf{V=L} \), not only \( {\sf SLO}^{\L(\bar{d})} \) for
\( \boldsymbol{\Pi}^1_1 \) subsets of \( \pre{\omega}{\omega} \) does not hold, but in fact we can embed a reasonably complicated partial order in
\( {\sf Deg}_{\boldsymbol{\Pi}^1_1}(\L(\bar{d})) \). Notice also that since
\( {\sf Deg}_{\boldsymbol{\Delta}^1_1}(\L(\bar{d})) \) needs to be very good by Borel determinacy, Theorem~\ref{th:illfounded hierarchy in L} actually shows that if \( \mathsf{V=L} \), then
\( (\mathscr{P}(\omega), \subseteq) \) embeds into the \( \L(\bar{d}) \)-hierarchy on \emph{proper} \( \boldsymbol{\Pi}^1_1 \) subsets of
\( \pre{\omega}{\omega} \), and Theorem~\ref{corollary: illfounded Borel hierarchy in L} shows that the same partial order embeds also e.g.\ in the \( \mathsf{Bor}(X) \)-hierarchy on \emph{proper} \( \boldsymbol{\Pi}^1_1 \) (alternatively:
\emph{proper} \( \boldsymbol{\Sigma}^1_1 \)) subsets of any uncountable Polish space \( X\). This conclusion considerably strengthen the well-known
fact that if \( \boldsymbol{\Pi}^1_1 \)-determinacy fails then there are proper \( \boldsymbol{\Pi}^1_1 \) subsets which are not (Borel-)complete for that
class.
The next questions essentially asks if it is possible to further strengthen Theorems~\ref{th:illfounded hierarchy} and~\ref{th:illfounded hierarchy in L} by either trying to embed a more complicated quasi-order into the relevant hierarchies, or by weakening the assumption required for those results to \( \boldsymbol{\Pi}^1_1 \)-determinacy.
\begin{question}
Assume $\mathsf{AC}$.
\begin{enumerate}
\item Is there a map \( \psi \colon \mathscr{P}(\omega) \to \mathscr{P} (\pre{\omega}{\omega}) \) such that $a\subseteq^* b \iff \psi(a)\leq_{\mathsf{Bor}(\pre{\omega}{\omega})}\psi(b)$ for all \( a,b \subseteq \omega \)?
\item Does the non-existence of $0^{\#}$
already imply that the $\mathsf{Bor}(\pre{\omega}{\omega})$-hierarchy on ${\bf \Pi}^1_1$ subsets of \( \pre{\omega}{\omega} \) is ill-founded?
\end{enumerate}
\end{question}
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The canton of Montluçon-4 is an administrative division of the Allier department, in central France. It was created at the French canton reorganisation which came into effect in March 2015. Its seat is in Montluçon.
It consists of the following communes:
Lamaids
Lavault-Sainte-Anne
Lignerolles
Montluçon (partly)
Prémilhat
Quinssaines
Teillet-Argenty
References
Cantons of Allier | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 4,094 |
{"url":"https:\/\/ankplanet.com\/physics\/physical-optics\/speed-of-light\/michelsons-method-to-determine-the-speed-of-light\/","text":"# Michelson\u2019s Method to Determine the Speed of Light\n\nAn American physicist Albert A. Michelson determined the speed of light for which he received a Nobel Prize in 1907. Michelson\u2019s method to determine the speed of light uses mirror having various faces rotating in very high speed.\n\nThe simplified version of his experiment is shown in the following figure.\n\nLet $M_1$ be an octagonal steel mirror which can be rotated about a vertical axis through its centre with the help of a motor of variable speed. Light from a source $S$ passes through a slit $S_1$ which falls on the face $1$ of the mirror $M_1$ at an angle of $45\u02da$. The mirror $M_1$ reflects it to a distant concave mirror $M_2$ at a point $A$. The beam is parallel to the axis of the mirror $M_2$. Then, the mirror $M_2$ reflects it to a plane mirror $M_3$ which is at the focus of the mirror $M_2$.\n\nThe mirror $M_3$ again reflects it to a point $B$ of the mirror $M_2$ from where it is reflected parallel to the axis of the mirror $M_2$. If the mirror $M_1$ is at rest, the beam falls to the face $3$ and it will be reflected towards the observer who can see the image of the source through the eyepiece $E$.\n\nThe mirror is set into rotation with a low speed. In this case, the beam of light returning from the mirror $M_2$ to the mirror $M_1$ will not be incident at the angle of $45\u02da$. As a result, the image will not enter the eyepiece.\n\nWhen the speed of the mirror is gradually increased and suitably adjusted, then the image will again appear in the eyepiece. This would happen when the speed of rotation is such that the adjacent face $2$ of the mirror $M_1$ is in position formerly occupied by face $1$, during which light travels from the mirror $M_1$ to mirror $M_2$ and back to mirror $M_1$.\n\n## Mathematical Measurement of Speed of Light by Michelson\u2019s Method\n\nLet the distance between the mirrors $M_1$ and $M_2$ be $d$, then, the time taken by the light to travel from $M_1$ to $M_2$ and back to $M_1$ is,$t=\\frac{2d}{c}\\text{ __(1)}$\n\nDuring this time period, each mirror of octagonal is rotated through an angle $\u03b8$. So, the total angle rotated by all the eight mirrors will be $8\u03b8$ which will be equal to $2\u03c0$ i.e. $8\u03b8=2\u03c0$$\u03b8=\\frac{\u03c0}{4} \\text{ __(2)}$\n\nIf $f$ is the frequency of the rotation of the mirror, then, $\u03c9=\\frac{\u03b8}{t}$ $t=\\frac{\u03b8}{\u03c9}$ $t=\\frac{\u03b8}{2\u03c0f} \\text{ __(3)}$ [Circular Motion]\n\nFrom equations $(2)$ and $(3)$, $t=\\frac{\u03c0\/4}{2\u03c0f}$ $t=\\frac{1}{8f} \\text{ __(4)}$\n\nFrom $(1)$ and $(4)$, $\\frac{2d}{c}=\\frac{1}{8f}$ $c=16fd$\n\nThis can be used to determine the speed of light.\n\nMichelson mounted the revolving mirror on Mount Wilson and placed a reflector on Mount San Antorio, $35$ $\\text{km}$ away. He rotated the mirror with an air jet at $528$ revolutions $\\text{sec}^{-1}$. He measured the frequency by comparing the rotating mirror with an electrically driven tuning fork.\n\nTherefore, putting the value of $f=528$ $\\text{rev\/sec}$ and $d=35\\;\\text{km}=35\u00d710^3\\;\\text{m}$, $c$ can be calculated. Michelson\u2019s result for the speed of light in vacuum was $2.99910\u00d710^8\u00b150$ $\\text{ms}^{-1}$.\n\nModern methods of measuring the speed of light use highly coherent and unidirectional laser beams. Because of this, the speed of light can be measured with very high degree of accuracy. The accepted value of the speed of light in free space is $2.99774\u00d710^8$ $\\text{ms}^{-1}$.\n\n1. The distance $(d)$ between two mirrors is very large (about $35$ $\\text{km}$). So, it gives an accurate measure of speed of light.\n2. The distance travelled by light and the rate of rotation of the mirror can be measured accurately.\n3. Light cannot reach the eyepiece directly from the source.\n4. It is a null deflection method so correction in the measurement of displacement is not required.\n5. The appearance of the image of the slit is abrupt which helps to determine the speed of light accurately.","date":"2022-12-02 06:08:56","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\": 2, \"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\": 0.7558810114860535, \"perplexity\": 242.10095576337787}, \"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-2022-49\/segments\/1669446710898.93\/warc\/CC-MAIN-20221202050510-20221202080510-00529.warc.gz\"}"} | null | null |
Q: When is $\sum_{n \ge 0} g_n(z)$ analytic? Let $D$ be an open subset of $\mathbb{C}$ where $g_n(z)$, $n \in\mathbb{N}$ are analytic. Then $$f(z)=\sum_{n \ge 0}g_n(z)$$
is analytic on $D$ iff $\sum_{n\ge 0}g_n(z)$ is locally uniformly convergent.
Firstly, what is the proof of this, and is there a way of intuiting why pointwise convergence is not strong enough to imply $f$'s analyticity, and why (non-local) uniform convergence is not required?
A: I'm sorry for the discussions in the comments. They are not helpful, so let me try to make my point clear. You asked for intuition on
a) why uniform convergence is needed,
b) why it is sufficient, and
c) why it is only needed locally.
The hardest point here is b), and I'm not sure if I can explain this entirely intuitionally, so let's do the easy stuff first.
Ad a), pointwise convergence only is never good enough to preserve continuity; there are always series of arbitrarily nice (non constant) functions converging pointwise to a discontinuous function. On the other hand, uniform convergence will always results in a continuous function.
Ad c), as I tried to point out in my comment, analyticity is a local property by definition. A function is analytic if it is analytic in any point, i.e. if any point has an open neighbourhood on which the function is a convergent power series.
Therefore, any condition that is good enough to make sure a series converges to an analytic function will work locally too.
For a), finally, we can observe the wonderful miracle of function theory: holomorphic functions are analytic, roughly because their derivatives are again holomorphic, which is pretty strong, but the conditions imposed on their differentiability are not to strong. We only need a continuous function that fulfills a certain condition, where the occurring terms are compatible with uniform convergence.
I'm thinking of Morera's theorem, of course, that is, a continuous function, defined on some open domain in $\mathbb{C}$, is holomorphic on this domain if its contour integral over any closed curve in this domain vanishes.
Now, one is encouraged to gain an intuitive understanding of what it means for a contour integral to vanish. If you know some physics, you could first convince yourself of the relationship between the (converse of the) above statement and Green's theorem and try to combine it with the physical idea behind the latter.
Unfortunately, I don't know this stuff good enough to give good further explanation of this.
(Anyone more familiar with this kind of interpretation may very well add extra information or references where this is explained. Therefore, I made this post community wiki.)
A: Can't help you much on the intuition, but I know one proof that $f$ is analytic using Cauchy's theorem.
Let $f_n = \sum_n g_n$ and pick a point $\alpha \in {\rm Int}(\Gamma)$ where $\Gamma$ is a compact region where $f_n$ converges uniformly. Since $f_n$ is analytic
$$f_n(\alpha) = \frac{1}{2\pi i}\int_\gamma \frac{f_n(z) dz}{z-\alpha}$$
where $\gamma$ is the boundary of $\Gamma$. To show that $f$ is analytic in $\alpha$ we only need to show that $f(\alpha) = I$ where
$$I = \frac{1}{2\pi i}\int_\gamma \frac{f(z) dz}{z-\alpha}$$
But this is easy as
$$|I - f(\alpha)| \leq |I - f_n(\alpha)| + |f_n(\alpha) - f(\alpha)|\\=\frac{1}{2\pi i}\int_\gamma \frac{|f_n(z)-f(\alpha)| dz}{z-\alpha} + |f_n(\alpha) - f(\alpha)| \leq 2|f_n - f|_\infty \to 0 {\rm ~~for~~} n\to \infty$$
by the uniform convergence of $f_n$.
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#include <QtCore/qlist.h>
#ifndef QSTRINGLIST_H
#define QSTRINGLIST_H
#include <QtCore/qalgorithms.h>
#include <QtCore/qregexp.h>
#include <QtCore/qstring.h>
#include <QtCore/qstringmatcher.h>
QT_BEGIN_NAMESPACE
class QRegExp;
class QRegularExpression;
typedef QListIterator<QString> QStringListIterator;
typedef QMutableListIterator<QString> QMutableStringListIterator;
class QStringList;
#ifdef Q_QDOC
class QStringList : public QList<QString>
#else
template <> struct QListSpecialMethods<QString>
#endif
{
#ifndef Q_QDOC
protected:
~QListSpecialMethods() {}
#endif
public:
inline void sort(Qt::CaseSensitivity cs = Qt::CaseSensitive);
inline int removeDuplicates();
inline QString join(const QString &sep) const;
inline QString join(QChar sep) const;
inline QStringList filter(const QString &str, Qt::CaseSensitivity cs = Qt::CaseSensitive) const;
inline QStringList &replaceInStrings(const QString &before, const QString &after, Qt::CaseSensitivity cs = Qt::CaseSensitive);
#ifndef QT_NO_REGEXP
inline QStringList filter(const QRegExp &rx) const;
inline QStringList &replaceInStrings(const QRegExp &rx, const QString &after);
#endif
#ifndef QT_BOOTSTRAPPED
#ifndef QT_NO_REGULAREXPRESSION
inline QStringList filter(const QRegularExpression &re) const;
inline QStringList &replaceInStrings(const QRegularExpression &re, const QString &after);
#endif // QT_NO_REGULAREXPRESSION
#endif // QT_BOOTSTRAPPED
#ifndef Q_QDOC
private:
inline QStringList *self();
inline const QStringList *self() const;
};
// ### Qt6: check if there's a better way
class QStringList : public QList<QString>
{
#endif
public:
inline QStringList() Q_DECL_NOTHROW { }
inline explicit QStringList(const QString &i) { append(i); }
inline QStringList(const QList<QString> &l) : QList<QString>(l) { }
#ifdef Q_COMPILER_RVALUE_REFS
inline QStringList(QList<QString> &&l) Q_DECL_NOTHROW : QList<QString>(std::move(l)) { }
#endif
#ifdef Q_COMPILER_INITIALIZER_LISTS
inline QStringList(std::initializer_list<QString> args) : QList<QString>(args) { }
#endif
QStringList &operator=(const QList<QString> &other)
{ QList<QString>::operator=(other); return *this; }
#ifdef Q_COMPILER_RVALUE_REFS
QStringList &operator=(QList<QString> &&other) Q_DECL_NOTHROW
{ QList<QString>::operator=(std::move(other)); return *this; }
#endif
inline bool contains(const QString &str, Qt::CaseSensitivity cs = Qt::CaseSensitive) const;
inline QStringList operator+(const QStringList &other) const
{ QStringList n = *this; n += other; return n; }
inline QStringList &operator<<(const QString &str)
{ append(str); return *this; }
inline QStringList &operator<<(const QStringList &l)
{ *this += l; return *this; }
inline QStringList &operator<<(const QList<QString> &l)
{ *this += l; return *this; }
#ifndef QT_NO_REGEXP
inline int indexOf(const QRegExp &rx, int from = 0) const;
inline int lastIndexOf(const QRegExp &rx, int from = -1) const;
inline int indexOf(QRegExp &rx, int from = 0) const;
inline int lastIndexOf(QRegExp &rx, int from = -1) const;
#endif
#ifndef QT_BOOTSTRAPPED
#ifndef QT_NO_REGULAREXPRESSION
inline int indexOf(const QRegularExpression &re, int from = 0) const;
inline int lastIndexOf(const QRegularExpression &re, int from = -1) const;
#endif // QT_NO_REGULAREXPRESSION
#endif // QT_BOOTSTRAPPED
using QList<QString>::indexOf;
using QList<QString>::lastIndexOf;
};
Q_DECLARE_TYPEINFO(QStringList, Q_MOVABLE_TYPE);
inline QStringList *QListSpecialMethods<QString>::self()
{ return static_cast<QStringList *>(this); }
inline const QStringList *QListSpecialMethods<QString>::self() const
{ return static_cast<const QStringList *>(this); }
namespace QtPrivate {
void Q_CORE_EXPORT QStringList_sort(QStringList *that, Qt::CaseSensitivity cs);
int Q_CORE_EXPORT QStringList_removeDuplicates(QStringList *that);
QString Q_CORE_EXPORT QStringList_join(const QStringList *that, const QChar *sep, int seplen);
QStringList Q_CORE_EXPORT QStringList_filter(const QStringList *that, const QString &str,
Qt::CaseSensitivity cs);
bool Q_CORE_EXPORT QStringList_contains(const QStringList *that, const QString &str, Qt::CaseSensitivity cs);
void Q_CORE_EXPORT QStringList_replaceInStrings(QStringList *that, const QString &before, const QString &after,
Qt::CaseSensitivity cs);
#ifndef QT_NO_REGEXP
void Q_CORE_EXPORT QStringList_replaceInStrings(QStringList *that, const QRegExp &rx, const QString &after);
QStringList Q_CORE_EXPORT QStringList_filter(const QStringList *that, const QRegExp &re);
int Q_CORE_EXPORT QStringList_indexOf(const QStringList *that, const QRegExp &rx, int from);
int Q_CORE_EXPORT QStringList_lastIndexOf(const QStringList *that, const QRegExp &rx, int from);
int Q_CORE_EXPORT QStringList_indexOf(const QStringList *that, QRegExp &rx, int from);
int Q_CORE_EXPORT QStringList_lastIndexOf(const QStringList *that, QRegExp &rx, int from);
#endif
#ifndef QT_BOOTSTRAPPED
#ifndef QT_NO_REGULAREXPRESSION
void Q_CORE_EXPORT QStringList_replaceInStrings(QStringList *that, const QRegularExpression &rx, const QString &after);
QStringList Q_CORE_EXPORT QStringList_filter(const QStringList *that, const QRegularExpression &re);
int Q_CORE_EXPORT QStringList_indexOf(const QStringList *that, const QRegularExpression &re, int from);
int Q_CORE_EXPORT QStringList_lastIndexOf(const QStringList *that, const QRegularExpression &re, int from);
#endif // QT_NO_REGULAREXPRESSION
#endif // QT_BOOTSTRAPPED
}
inline void QListSpecialMethods<QString>::sort(Qt::CaseSensitivity cs)
{
QtPrivate::QStringList_sort(self(), cs);
}
inline int QListSpecialMethods<QString>::removeDuplicates()
{
return QtPrivate::QStringList_removeDuplicates(self());
}
inline QString QListSpecialMethods<QString>::join(const QString &sep) const
{
return QtPrivate::QStringList_join(self(), sep.constData(), sep.length());
}
inline QString QListSpecialMethods<QString>::join(QChar sep) const
{
return QtPrivate::QStringList_join(self(), &sep, 1);
}
inline QStringList QListSpecialMethods<QString>::filter(const QString &str, Qt::CaseSensitivity cs) const
{
return QtPrivate::QStringList_filter(self(), str, cs);
}
inline bool QStringList::contains(const QString &str, Qt::CaseSensitivity cs) const
{
return QtPrivate::QStringList_contains(this, str, cs);
}
inline QStringList &QListSpecialMethods<QString>::replaceInStrings(const QString &before, const QString &after, Qt::CaseSensitivity cs)
{
QtPrivate::QStringList_replaceInStrings(self(), before, after, cs);
return *self();
}
inline QStringList operator+(const QList<QString> &one, const QStringList &other)
{
QStringList n = one;
n += other;
return n;
}
#ifndef QT_NO_REGEXP
inline QStringList &QListSpecialMethods<QString>::replaceInStrings(const QRegExp &rx, const QString &after)
{
QtPrivate::QStringList_replaceInStrings(self(), rx, after);
return *self();
}
inline QStringList QListSpecialMethods<QString>::filter(const QRegExp &rx) const
{
return QtPrivate::QStringList_filter(self(), rx);
}
inline int QStringList::indexOf(const QRegExp &rx, int from) const
{
return QtPrivate::QStringList_indexOf(this, rx, from);
}
inline int QStringList::lastIndexOf(const QRegExp &rx, int from) const
{
return QtPrivate::QStringList_lastIndexOf(this, rx, from);
}
inline int QStringList::indexOf(QRegExp &rx, int from) const
{
return QtPrivate::QStringList_indexOf(this, rx, from);
}
inline int QStringList::lastIndexOf(QRegExp &rx, int from) const
{
return QtPrivate::QStringList_lastIndexOf(this, rx, from);
}
#endif
#ifndef QT_BOOTSTRAPPED
#ifndef QT_NO_REGULAREXPRESSION
inline QStringList &QListSpecialMethods<QString>::replaceInStrings(const QRegularExpression &rx, const QString &after)
{
QtPrivate::QStringList_replaceInStrings(self(), rx, after);
return *self();
}
inline QStringList QListSpecialMethods<QString>::filter(const QRegularExpression &rx) const
{
return QtPrivate::QStringList_filter(self(), rx);
}
inline int QStringList::indexOf(const QRegularExpression &rx, int from) const
{
return QtPrivate::QStringList_indexOf(this, rx, from);
}
inline int QStringList::lastIndexOf(const QRegularExpression &rx, int from) const
{
return QtPrivate::QStringList_lastIndexOf(this, rx, from);
}
#endif // QT_NO_REGULAREXPRESSION
#endif // QT_BOOTSTRAPPED
QT_END_NAMESPACE
#endif // QSTRINGLIST_H
| {
"redpajama_set_name": "RedPajamaGithub"
} | 4,617 |
Fortune Dogs (Cãezinhos de Sorte no Brasil) é um anime baseado na história em quadrinhos de mesmo nome criada por Shuji Kishihara e Yasuharu Tomono.
É a história de Freddy, um Bulldog Francês que veio ao mundo para cumprir uma missão muito importante - salvar a Árvore da Sorte.
Inicialmente, o pequeno Freddy nada sabe, começando sua jornada apenas por ter se perdido de sua dona, a menina Ai. No caminho de volta para casa, ele ganha alguns companheiros de viagem (Dach, Max e Cocco), além de conhecer outros cães, reencontrar-se com alguns dos filhotes que conviveram com ele no Canil Feliz e vencer muitos obstáculos.
Na segunda parte, após ter se reencontrado com Ai, uma nova aventura começa e a história de Freddy apresenta o ciclo da Jornada do Herói, (Monomito) descrita pelo antropólogo Joseph Campbell em seu livro O Herói de Mil Faces.
Personagens
Cães
Freddy/Alex (フレディー) - Bulldog Francês, cachorrinho que a menina Ai-chan adotou no Canil Feliz. Ela o chama de Alex, mas ele próprio escolheu para si o nome de Freddy, em homenagem ao herói canino do livro As Aventuras de Freddy.
Airi - Setter Irlandês Ruivo, mora com a Princesa Tomiko e a Himechin
Avô do Mook - Old English Sheepdog
Bernado - São bernardo, sábio que prediz que Freddy é um cachorro especial. Vive no povoado Taner junto com Jess e Kinta.
B-chan - Beagle
Bright - Spitz Japonês (?) macho, hóspede na clínica veterinária
Bruto - Buldogue metido a besta, cheio de si, pensa que é o tal.
Cammy (キャミー) - Cavalier King Charles Spaniel fêmea, participa de exposições caninas. Sua dona é uma senhora muito rica, que, apesar de tratá-la com carinho, não a deixa sair de casa, não a leva para passear. Dach se apaixona por ela.
Cão policial - Schnauzer gigante - persegue os cães para prendê-los a mando dos humanos
Cão policial - Dogue alemão, persegue os cães para prendê-los a mando dos humanos
Cão policial - Dogue alemão, persegue os cães para prendê-los a mando dos humanos
Cão policial - Pointer inglês, persegue os cães para prendê-los a mando dos humanos
Cão policial - Pointer inglês, persegue os cães para prendê-los a mando dos humanos
Chibiyama - Chihuahua, filhote no Canil Feliz
Cocco (コッコ) - Cocker Spaniel Americano, companheira de Freddy em parte de sua jornada. Seus donos, um casal jovem, têm um filhinho chamado Sachiko.
Corocorone/Dom Coro (ドン・コロコローネ) - Buldogue. Seu dono (ou seu papai, como Coro gosta de chamá-lo), Panter, é ladrão de jóias; o próprio Coro tem ares de mafioso... Ele é treinado para enganar o faro dos cães da polícia, após algum roubo realizado por Panter, que adora cães e trata muito bem o seu cachorro e qualquer amigo dele que apareça para uma visita - muita comida, e da boa! Conhece Freddy ainda quando está no Canil Feliz e mais tarde se reencontram. Apesar das circunstâncias, Freddy diz que sempre será seu amigo e Dom Coro retribui esta amizade.
Dach (ドッチ) - Dachshund (ou Teckel), companheiro de Freddy em sua jornada,é namorador e especialista em cavar buracos; entre outras coisas, ensina a Freddy como interagir com humanos e a mostrar a barriga para um outro cão quando se sentir ameaçado, dizendo que significa "fora daqui!" (na verdade, é um ato de total submissão...) É um cão que vive nas ruas e não quer saber de dono, após péssimas experiências - primeiro, foi abandonado por sua primeira dona (uma menina que o comprou por impulso em uma feira) numa noite chuvosa, no meio de uma estrada deserta (para Freddy, Dach contou outra história: tinha alguém que cuidava dele quando era pequeno, mas esta pessoa morreu e ele preferiu viver nas ruas); depois, com o menino Jack, meio bruto e desajeitado no trato com os animais. Apaixona-se por Cammy e vive implicando com Cocco (ainda que goste dela).
Daltian (ダルジャ") - Dálmata fêmea
Earnest (アーネスト) - Galgo afegão
Edgar (エドガー) - Fox terrier de Pêlo Duro (?)
Eiji (エージ) - Pastor-alemão, foi cão farejador da polícia; aposentado, foi adotado pelo dono do Canil Feliz. Sua melhor amiga é a gata Grace - que faz gato e sapato dele. Prediz que Alex é um cão especial.
Freezer (フリーザー) - Husky siberiano, rival de Ryoma
Himechin - Spaniel Japonês fêmea, mora com a Princesa Tomiko e o Airi.
Jackie - Jack Russell Terrier, trabalha no Circo Arco-Íris; é um ótimo acrobata, mas muito bobinho e ingênuo. Seu dono (e companheiro de acrobacias) é o Capitão.
Jess - Golden Retriever, mãe de Kinta; quando Maria, sua dona, morreu, foi expulsa de casa pelo viúvo dela, ficando com medo e raiva dos humanos. Vive no povoado Taner.
Kinta - Golden Retriever macho, filho de Jess, adotado pelo menino Bob, vive no povoado Taner.
Kitchen (キッチ) - Rough Collie macho
Kosuke - Norwich Terrier, cãozinho farejador da Polícia, ajuda a caçar o "papai" criminoso de Dom Coro.
Kowalski - Weimaraner, "capanga" de Dom Coro, tem medo e ao mesmo tempo ama o "Chefinho".
Labre - Labrador Retriever, ex-cão de resgate, trabalha como cão de terapia. Sua dona chama-se Loretta.
Leão - Cão de Santo Humberto, o pai de seu dono é arqueólogo.
Lovely
Marcô - Maltês fêmea em tratamento na clínica veterinária.
Martaff (マータフ) - Dobermann que não quer saber de Freddy e Dach em seu território; amigo de Cammy.
Max (マックス) - Boxer, um dos cãezinhos no Canil Feliz, mais tarde companheiro de Freddy em sua jornada. Só pensa em comer; foi expulso de sua casa pela dona, pois era um péssimo cão de guarda - deixou um mesmo ladrão entrar duas vezes na residência, em troca de comida.
Minich
Mook (ムック) - Old English Sheepdog da fazenda de Rogers, tem dificuldade em pastorear ovelhas.
Noppe - Bull Terrier, "capanga" de Dom Coro, tem medo e ao mesmo tempo ama o "Chefinho", como ele o chama.
Pai do Mook - Old English Sheepdog, ferido seriamente em um ataque de coiotes, fica afastado do trabalho de pastorear as ovelhas da fazenda de Rogers.
Pochi - SRD, se sente inferior por não ser de raça. Sempre tristonho, não quer brincar com seu dono e seu narizinho vive escorrendo...
Poppy - Papillon filhote no Canil Feliz.
Poron - Lulu da Pomerânia (ou Spitz alemão), filhote em tratamento na clínica veterinária.
Princesa Tomiko - Shar-Pei; seu dono, um jovem motociclista, abandonou a casa onde morava (por motivo não explicado); Tomiko espera sua volta ansiosa - toda vez que escuta barulho de moto, corre para o portão na esperança de rever seu querido amigo, que costumava dizer que seus tesouros eram sua moto e Tomiko, além de Airi e Himechin.
Pugbou - Pug capturado pelos cães policiais, pede ajuda a Freddy
Rikyu - Akita Inu idoso que fica em frente à estação de trem esperando pacientemente o retorno de seu dono. História baseada no real Akita Hachiko, homenageado com duas estátuas no Japão (Tóquio e Odate).
Ryoma (リョウマ) - Tosa, cão idoso, ensinou Freddy como fazer pipi com a perninha levantada, cumprimentar outros cachorros, marcar territórios.
Sábio - vive numa montanha (cujo topo tem uma bandeira com a cara de um cachorro, e numa das faces, outra cara de cachorro esculpida na pedra); Dom Coro procura sua ajuda quando seu amado papai o abandona.
Sanbe - Welsh Corgi Pembroke, cãozinho sábio, não fala, se comunica apenas pelo pensamento e sabe que Freddy é especial. Sua dona chama-se Loretta.
sem nome - Pointer inglês (?), persegue o pequeno Jackie por este ter tentado roubar alguns ossos enterrados.
Shetlan - Pastor de Shetland da menina Karen.
Shibata - Shiba Inu apaixonado por Shina (que conheceu no Canil Feliz). Vive com sua dona, Alice, uma jovem artista de rua, fazendo malabarismos em feiras e praças; num certo mommento, tem que fazer uma escolha: seguir com sua dona Alice ou ficar com Shina.
Sheena - Shih-Tzu fêmea, apaixonada por Shibata (que conheceu no Canil Feliz), vive com sua dona, uma senhora idosa.
Shunaemon - Schnauzer Standard (?)
Wang - Chow-chow - Cadela que mora na rua
Wealthy - Spitz Japonês ou Samoieda (?)
Welshy - filhote no Canil Feliz
Whity (ホワイティ) - Poodle fêmea da fazenda de Rogers.
Yoko - Yorkshire Terrier fêmea levada na clínica veterinária por sua dona, que fica muito brava com o veterinário por este dizer que ela dá comida demais para a cadelinha.
Yupi - robô
Zenji - apaixonado pela gata Rainha.
Gatos
Grace - grande amiga de Eiji, o Pastor-alemão
Rainha - gatinha por quem Zenji é paixonado
Episódios
Ligações externas
Cãezinhos de Sorte
Joseph Campbell Foundation
Séries de anime | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 9,808 |
{"url":"https:\/\/unofficialaciguide.com\/2020\/03\/01\/understanding-roles-in-aci-mso-and-integrating-with-freeradius\/","text":"### Understanding Roles in ACI & MSO and integrating with FreeRadius\n\nIn this blog we will talk about:\n\n\u2022 Roles in ACI, how to create them and how to use them to achieve your objective.\n\u2022 Roles in MSOs, how to create them and how to use them to achieve your objective.\n\u2022 We will also set up Roles in the IPN\/ISN\u00a0 and a N3K which we use for connecting the ACI Border Leaf to the external world (L3 Out)\n\u2022 Then we will show how to setup a basic Free Radius docker container to test out the roles\n\nLet\u2019s quickly discuss what Roles are:\n\nRoles play a critical part in authorization.\n\nWe\u2019ll do that with a simple lab scenario.\u00a0\u00a0 When a user logs on to the APIC \/ MSO, IPN\/ISN, or the N3K (peering routers) we want to give them appropriate permissions.\u00a0 For instance,\u00a0 we might want a user to be able to work only on their tenant and create objects there but no where else.\u00a0 However mabye we might want them to have read access to the entire fabric or perhaps not even see anything else.\u00a0 Similarly for MSO the same concept applies,\u00a0 we might allow users to create objects in their tenant, but not be able to add sites or create local users, etc, etc.\u00a0\u00a0 Similar concept applies for the NXOS devices or any other network device.\n\nAPIC:\n\nIn ACI the user is associate with\u00a0 one or more Security Domains and each Serucity Domain is associated with one or more Role with either read or write privileges to give appropriate permissions (read or read-write).\n\nIn the example below the user soumukhe is assoicated with Security Domain \u201call\u201d and Role of admin with WritePriv.\n\nIn effect what happened below is that user soumukhe is give full read-write access to the entire fabric\n\nSecurity Domain \u201call\u201d is a built in Security Domain in ACI and every object in ACI (Tenants, Fabric polices, users etc, etc) is automatically a member of the \u201call\u201d Security Domain.\n\nRole \u201cadmin\u201d is a built in Role in ACI that gives object level Privileges (access) to every object in ACI. However, notice that role of \u201cadmin\u201d does not necessarily mean that the role gives you read-write access.\u00a0 All it means is that the user will get access to the objects based on the Security Domain associated with this role.\n\nIf you give the user a security domain of all and role of admin but \u201cread\u201d priv only, then the user can see everything but not be able to create anything.\n\nAlso, keep in mind that you are not allowed to modify the objects that come default with ACI.\u00a0 As an example, the \u201cread-all\u201d role comes with Priv of every object, except for \u201caaa\u201d and \u201cadmin\u201d.\u00a0 Notice below when I try to edit that role and add \u201caaa\u201d to it, it won\u2019t let me.\n\nHow to create Security Domains and Roles in APIC and associate with Local User:\n\nIn APIC, you can create Security Domains and Roles from GUI\/API calls or CLI.\u00a0 We will discuss the GUI method here since it\u2019s easy to demonstrate.\n\nCreating Security Domains and associating with Tenant in ACI:\n\nIn APIC go to\u00a0 \u201cAdmin\/Security\/Security Domains\u201d and create a Role there.\u00a0\u00a0 Give the Role access to the objects as you desire\/need to as shown below. where we are creating a Security Domain called \u201cTestSecurityDomain\u201d\n\nNext, associate the Role with one or more objects (generally Teannt), but you can get more granular and add the security domain to other permitted objects like \u201cphysical-domains\u201d, leafs, or leaf ports, etc, etc.\u00a0 In the example below we are associating the Security Doman that we just created to the Tenant acme.\u00a0 To do this go to Tenant\/acme\/Policy\/SecurityDomain\n\nNext, create the Role:\n\nGo to:\u00a0 Admin\/Security\/Roles\u00a0 and create a new role\n\nIn the example below,\u00a0 I created a CustomTeantRole and gave them access to all objects except admin and aaa\n\nAssociate user (or create new user) with the Security Domain and Role with write priv.\u00a0\u00a0 Below we created a new local user called \u201ctestUser\u201d and associated with the Security Domain and Role we created and gave that user writePriv\n\nThe Result of this is that when \u201ctestUser\u201d logs in, he can only see his Teanat and modify objects in his tenant.\u00a0 Everything else is greyed out (including Fabric Policies, etc, etc)\n\nNow, Let\u2019s see how to create Roles in MSO\n\nThe Concept of Roles is similar in MSO.\u00a0 The difference is that there is no concept of Security Domains.\u00a0\u00a0 The user is directly associated with the Roles.\n\nBelow capture you can see that the user \u201csoumukhe\u201d is associated with the Built In Role of \u201cSite And Tenant Manager\u201d & \u201cSchema Manager\u201d\n\nCreating\u00a0 a new Role for MSO:\n\nIn MSO,\u00a0 new roles cannot be created from the GUI.\u00a0 They have to be created through API push.\u00a0\u00a0\u00a0 Here\u2019s how you do it.\u00a0\u00a0 An easy way to do it is to use the swagger interface as shown below (click on gear and then \u201cView Swagger Docs\u201d and click on \u201cLaunch\u201d for \u201cUser API\u201d\n\nOnce there,\u00a0 click on \u201cGet\u201d for \u201cRole APIs\u201d.\n\nNote:\u00a0 there are 5 methods:\n\n\u2022 GET All \u2013 to get all\n\u2022 POST \u2013 to create new\n\u2022 GET by ID \u2013 to get particular object\n\u2022 PUT by ID-\u00a0 to update particular object\n\u2022 DELETE by ID \u2013 to delete an object\n\nOnce you click on \u201cGET\u201d , click on \u201cTry it Out\u201d and then execute.\u00a0 This will give you the json for all the Roles currently defined in the MSO.\n\nFrom the json output,\u00a0 note the ID for the \u201cPower User\u201d role.\u00a0 In our case it is:\u00a0\u00a0\u00a0 \u201cid\u201d: \u201c0000ffff0000000000000031\u201d\u00a0\u00a0 Copy the ID to buffer, in this case: 0000ffff0000000000000031.\u00a0 Then, click on GET by ID and paste in the \u201cPower-User\u201d Role ID in the id field.\n\nClick Execute and you should get the json output for only the \u201cPower-User\u201d role\n\nNow, copy the json output to buffer and click on POST.\u00a0\u00a0 Click Try it out. copy the ID to the ID field and change the ID ( you need a unique ID for a new object). In my case I\u2019ve changed the last 2 digits to 36.\u00a0 Change the Permission objects to what you need and also change the name, display name and description, etc, etc.\u00a0\u00a0\u00a0 For this demonstration, I\u2019ve just changed the names, display name and Description\n\n\u201cid\u201d: \u201c0000ffff0000000000000036\u201d,\n\u201cname\u201d: \u201cMy-CustomRole\u201d,\n\u201cdisplayName\u201d: \u201cCreated for Demonstration\u201d,\n\u201cdescription\u201d: \u201cElevates this user to \\\u201dMy-CustomRole\\\u201d\u201d,\n\nOnce you are done, you will see the new Custom Role you created show up in GUI and you can apply users to it.\n\nNow Let\u2019s talk quickly on NXOS Roles.\n\nNXOS has built in Roles also, like network-operator (read-only)\u00a0 or network-admin (write-all) and many others.\n\nHowever for our purpose, we will create a new role for both IPN and N3K called \u201ccustomrole\u201d.\u00a0 This can be done by ssh\u2019ing to the router and then do a config t, followed by\u00a0 \u201crole name custom role\u201d\n\nThe customrole defined in IPN\/ISN\u00a0 and N3K is as shown below:\n\nThe Reason for this is that we are getting prepared for Radius Integration.\u00a0\u00a0 We will send the same AV-Pair from Radius Server to both IPN and N3K, namely \u201ccustomrole\u201d, but the authorization for N3K will allow full access and for IPN will give only read access\n\nNow that we\u2019ve spoken about Roles with regard to Authorization, let\u2019s tie all this together by integrating to Free Radius.\u00a0 You can ofcourse use any other commercial Radius Server like Cisco ISE.\u00a0 ISE ofcourse has exteremly rich features and is much more than a simple Radius Server.\u00a0 However, here we are going to show you a very quick way of standing up FreeRADIUS in a docker container.\u00a0 You may want to do this in a lab situation (if you don\u2019t already have ISE) or even for production.\u00a0 After all if all you are looking for is Radius, then\u00a0 FreeRADIUS is very good.\u00a0 It is responsible for authenticating one third of all users on the Internet, so, it\u2019s not just for lab.\n\nHere\u2019s the idea on how all this works:\n\nStep 1)\n\nSetup Free Radius to send the desired Cisco AV Pair, based on the role names you defined in APIC, MSO, IPN and N3Ks\n\nAlso, setup encrypted password ( data at rest encryption) for the usernames in FreeRadius.\u00a0 Now you won\u2019t need to mess with users any more on the local devices.\n\nOptionally integrate FreeRADIUS with your LDAP or ActiveDirectory using proxy or some other method\n\nStep 2)\n\nYou go to each device and configure the device to use Radius and put in the Radius IP on the device.\u00a0 In this case, our devices are:\n\n\u2022 APIC\n\u2022 MSO\n\u2022 N3K (NXOS)\n\u2022 IPN (NXOS)\n\nUnderstanding the format of the AVPairs that RADIUS needs to send to the devices is the first step.\n\nBefore we start setting up the FreeRADIUS server, let\u2019s quickly discuss the format of the AV Pairs that the individual devices expect.\u00a0 You can look up CCO documentation and get fancy,\u00a0 (cisco security configuration guide) but here\u2019s the basic concept:\n\nAPIC:\nshell:domains=SecurityDomainName:x\/y(UID)\nHere x = some Role Name.\u00a0 Putting it in first position (x position) here means \u201cwrite priv\u201d.\ny = some Role Name.\u00a0 Putting it in second position (y position) means \u201cread priv\u201d\nUID = a unique Unix User ID for a user for instance 16001 (used during SSH to device)\n\nHere are some examples.\n\nBelow gives admin access to security domain smTestSD (so, only tenant associated with smTestSD is visible and write)\n\nCisco-AVPair = \u201cshell:domains =smTestSD\/admin\/(16001)\u201d\n\nBelow only gives read access to smTestSD nothing else is viewable, since x is null\nCisco-AVPair = \u201cshell:domains =smTestSD\/\/admin\u201d\n\nsimilarly:\u00a0 Cisco-AVPair = \u201cshell:domains =all\/admin\u201d\u00a0 means full access to everything, whereas : \u201cshell:domains =all\/\/admin\u201d gives read access to everything.\u00a0\u00a0\u00a0 Also,\u00a0 \u201cshell:domains=all\/read-all\u201d\u00a0 means write permissions to all objects that are specified to role \u201cread-all\u201d.\u00a0 (In my opinion, the name read-all for a role is confusing.)\n\nMSO:\n\nSimilar concept holds for MSO:\u00a0 (MSO Config Guide)\n\nThe format for MSO is:\nshell:msc-roles=x\/y\nwhere x is the place holder for write priv and y is the place holder for read priv.\n\nexample:\n\nIn the below example the user associated with the AVPair will get write priv to for the objects defined in SMDefineCustom\u00a0\u00a0 role\n\nCisco-AVPair = \u201cshell:msc-roles=SMDefineCustom\/\u201d\n\nwhereas:\nCisco-AVPair = \u201cshell:msc-roles=\/SMDefineCustom\u201d\nwill get read priv to for the objects defined in SMDefineCustom\u00a0\u00a0 role\n\nStep 1:\n\nSetting Up FreeRADIUS for Docker Container (using docker-compose).\n\nStep 1:\u00a0 Setup CentOS or Ubuntu VM with Docker and Docker-Compose (this is pretty standard and if you are not sure look up docker or kubernetes site\n\nStep2:\u00a0 If you are behind a proxy, make sure to setup proxy properly for docker, apt-get or yum, pip, wget\u00a0 \u2014 all pretty standard stuff\n\nStep3:\u00a0 Showing for Ubuntu (do similar stuff for CentOS)\n\nsudo apt get install docker-compse\u00a0 && sudo apt install freeradius-utils -y\nThis will install docker-compose and the freeradius utilities, so you can encrypt passwords with the radcrypt utility\n\nStep 4\n\nssh and go to your home directory in Ubuntu.\u00a0 In my case my home directory is\u00a0 \/home\/soumukhe.\u00a0 Also for docker-compose service name and container name adjust as follows.\n\nThen follow these steps:\n\n***************Summary of install: ****************\n\nprerequisites:\n\nsoumukhe@worker-1:~\/freeradius\/raddb$cat clients.conf client aci { ipaddr = 0.0.0.0 secret = SomeSecret-notWhatImUsing netmask = 0 nastype = cisco shortname = aci } make your authorize file in freeradius\/raddb\/mods-config\/files\/ soumukhe@worker-1:~\/freeradius\/raddb\/mods-config\/files$ cat authorize\nsoumukhe Crypt-Password := \u201cro2\/uqmzhU8Tk\u201d\nService-Type = NAS-Prompt-User,\nCisco-AVPair = \u201cshell:domains =all\/\/aaa,all\/admin\/\u201d\n\nfor MSC also:\n\nsoumukhe@worker-1:~\/freeradius\/raddb\/mods-config\/files$cat authorize soumukhe Crypt-Password := \u201cro2\/uqmzhU8Tk\u201d Service-Type = NAS-Prompt-User, Cisco-AVPair = \u201cshell:domains =all\/admin\/\u201d, Cisco-AVPair += \u201cshell:msc-roles=powerUser\/\u201d now do the below. In freeradius directory: 1) first do these: soumukhe@worker-1:~\/freeradius$ cat Dockerfile\n\nsoumukhe@worker-1:~\/freeradius$cat docker-compose.yaml version: \u20183\u2019 services: radius: container_name: sm-radius restart: always build: context: . dockerfile: .\/Dockerfile command: [] ports: \u2013 \u201c1812-1813:1812-1813\/udp\u201d volumes: \u2013 \u201c.\/temp_files:\/tmp:rw\u201d 2) docker-compose build docker-compose up -d docker ps \u2014 make sure container is up 3) docker exec -it sm-radius \/bin\/bash cd \/etc\/raddb cp -a * \/tmp \/\/****** remember \/tmp is actually the \/freeradius\/temp_files in your host vm 4) exit container and go to \/freeradius\/temp_files (all is owned by root) sudo cp -a * ..\/raddb 5) cd ..\/raddb do id -a , id -g id -u to see the user:group ids, let\u2019s say it\u2019s 1000:1000 cd mods-config\/files sudo chown 1000:1000 authorize (it was owned by root before) ls -lag to check 6) change the Dockerfile and docker-compose.yaml soumukhe@worker-1:~\/freeradius$ cat Dockerfile\n\nsoumukhe@worker-1:~\/freeradius$cat docker-compose.yaml version: \u20183\u2019 services: radius: container_name: sm-radius restart: always build: context: . dockerfile: .\/Dockerfile command: [] ports: \u2013 \u201c1812-1813:1812-1813\/udp\u201d volumes: \u2013 \u201c.\/temp_files:\/tmp:rw\u201d \u2013 \u201c.\/raddb:\/etc\/raddb:rw\u201d # added 7) go back to freeradius directory docker rm -f sm-radius docker build docker up -d docker ps # verify sm-radius is not crashing 8) everytime you make change to the authorize file, do docker-compose restart radius \/\/ note radius is the service name (look at the docker-compose.yaml file) docker-compose restart will restart all docker-compose containers, so don\u2019t do that done! Here is an example of users that I\u2019m giving full write priv to: Here is an example of users that I\u2019m giving read-only priv to: Note: The passwords are obviously encrypted using freeradius utils \u201cradcrypt\u201d. You have the option of doing cleartext password also, in which case the authorize file would have entries like so: radcrypt usage for encryption:$ radcrypt foobar\nHaX0xn7Qy650Q\n\n\\$ radcrypt -c foobar HaX0xn7Qy650Q\n\nSome Good Ideas on setting up FreeRadius:\n\nYou probably don\u2019t want to vi the authorize file by hand.\u00a0 Mistakes there\u00a0 (like missing a\u00a0 comma) will make your container constantly reboot. You probably want to write a simple python sctipt that will modify the authorize file as you add new entries.\n\nIn my case, I\u2019ve implemented this so lab users can set their own passwords to access the lab ACI Fabric.\n\nHere\u2019s the basic workflow:\n\na)\u00a0 Users subscribe to a mailer (closed mailer group, where we have to approve them before they can become a member)\n\nb)\u00a0 We wrote a flask front-end where users are authorized against their usernames\/passwords for corporate AD (Active Directory), then if they pass, it checks their membership in corporate LDAP using the python ldap3 module\n\nc) if users have CN=lab-users , OU = MAILER,\u00a0 then they get write permission, if not they get read permission and the authorize files are appended to accordingly.\n\nd)\u00a0 The authorize file is written to the flask front-end home directory\n\ne)\u00a0 A simple python script then is run by crontab every 5 minutes, that compares the authorize file in the flask-front end, to the real one running in the raddb\/mods-config\/files\/authorize directory.\u00a0 If it is the same nothing happens and the script exits.\u00a0 If the new authorize file is different,\u00a0 the script copies the new authorize file to another directory for backup and replaces the authorize file in raddb. The script then does a docker-compose restart for the radius server, so the new authorize file is read into the radius server memory.\n\ncrontab entry (from crontab -e)\n\nStep 2:\n\nThe only thing remaining to do is to go to the devices and configure them to use the Radius server we set up:\n\n\u2022 APIC\n\u2022 MSO\n\u2022 N3K (NXOS)\n\u2022 IPN (NXOS)\n\nAPIC:\n\nGo to Admin\/Authentication\/AAA and tie in the Radius Provider\n\nMSO:\n\nGo to\u00a0 Admin\/Login Domain and tie in the Radius Server and make it active\n\nNXOS Devices:\n\nN3K \u2014 read-write\u00a0\u00a0 using customrole\n\nIPN \u2013 write using customrole\n\nradius-server key 7 \u201cvdewwdNjuuoy\u201d\nradius-server host 10.29.198.52 authentication accounting\nip radius source-interface mgmt0\naaa authentication login default group radius local\naaa authentication login console local\n\nResults:\u00a0 Now the user can log into any of the devices and get the appropriate authorization.\n\nEND: I know this is sort of a long post, but I\u2019m hoping it is of help to you.\n\nThis site uses Akismet to reduce spam. Learn how your comment data is processed.","date":"2022-05-22 00:53:50","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\": 0.20965352654457092, \"perplexity\": 5445.79355766712}, \"config\": {\"markdown_headings\": true, \"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-2022-21\/segments\/1652662543264.49\/warc\/CC-MAIN-20220522001016-20220522031016-00214.warc.gz\"}"} | null | null |
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"redpajama_set_name": "RedPajamaC4"
} | 7,262 |
Q: Sort GeoPandas by Name Of Line And By Geographic Location I have a geopandas dataframe created from a shapefile.
I would like to sort my dataframe according to the column: "name" AND the line chunks should also be sorted by geographic location, such that all nearby chunks which have the same name are grouped together.
How can I do this kind of sorting ?
What I have tried:
1. I calculate the mean coordinate fro each linestring:
df['mean_coord'] = df.geometry.apply(lambda g: [np.mean(g.xy[0]),np.mean(g.xy[1])])
*I group the dataframe according to the "name" column and I sort the resulting dataframe according to the mean coordinate:
grouped=df.sort_values(['mean_coord'],ascending=False).groupby('name')
But I am not sure, if this is the best/most elegant or even correct way to do it. Other than that, I don't know how to get back to a pandas dataframe from the grouped element ?
A: First, I'm going show you what I've hard-coded and assumed to be a representative dataset. This is really something you should have provided in the question, but I'm feeling generous this holiday season:
from shapely.geometry import Point, LineString
import geopandas
line1 = LineString([
Point(0, 0),
Point(0, 1),
Point(1, 1),
Point(1, 2),
Point(3, 3),
Point(5, 6),
])
line2 = LineString([
Point(5, 3),
Point(5, 5),
Point(9, 5),
Point(10, 7),
Point(11, 8),
Point(12, 12),
])
line3 = LineString([
Point(9, 10),
Point(10, 14),
Point(11, 12),
Point(12, 15),
])
gdf = geopandas.GeoDataFrame(
data={'name': ['A', 'B', 'A']},
geometry=[line1, line2, line3]
)
So now I'm going to compute the x- and y-coordinates of the centroids of each line, average them, sort by the average and name of the line, the remove the intermediate columns.
output = (
gdf.assign(x=lambda df: df['geometry'].centroid.x)
.assign(y=lambda df: df['geometry'].centroid.y)
.assign(rep_val=lambda df: df[['x', 'y']].mean(axis=1))
.sort_values(by=['name', 'rep_val'])
.loc[:, gdf.columns]
)
print(output)
name geometry
0 A LINESTRING (0 0, 0 1, 1 1, 1 2, 3 3, 5 6)
2 A LINESTRING (9 10, 10 14, 11 12, 12 15)
1 B LINESTRING (5 3, 5 5, 9 5, 10 7, 11 8, 12 12)
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 9,035 |
{"url":"https:\/\/unum.cloud\/post\/2021-01-05-bottlenecks\/","text":"A modern DBMS can be viewed as a composition of the following layers:\n\n1. A key-value binary storage layer such as RocksDB or WiredTiger.\n2. A single-instance logic layer such as Postgres.\n3. A distributed load-balancer and replication manager like GreenPlum.\n\nDepending on your performance goals you may have to optimize one or more layers in this equation. We replaced all of them with custom solutions. Assuming a DBMS implementation size can reach millions of lines of code, there are many design decisions to make. Let\u2019s group them by affected computer components.\n\n## SSD\n\n### Data Layout\n\nCommon Ingredients Our Ingredients\nRow-wise or Column-wise Hybrid and Domain-specific\n\nImagine having a table with 1'000'000 rows and 3 columns. Every row represents your companies customers with columns: first-name, last-name, age. If you often delete or add customers it makes sense to store them in a row-major format. This means, that when you are deleting customer #200'000, you won\u2019t have to shift the following 800'000 entries one row up. The row-major format means that the DB is optimized for row-level operations and can store different rows in different buckets on disk. The benefit is that the updates are fast.\n\nThe downside is that the lookups aren\u2019t very fast and aggregations are painfully slow. If you try computing the mean age of your customers - you would have to jump from one bucket to another only to extract a single number from every entry. This is where columnar databases shine. Some of them are immutable or append-only, but they still have a huge value, as they accelerate the increasingly important analytical queries.\n\nAt Unum, we use a hybrid approach to guarantee good performance across a wide variety of workloads. Our design is perfectly tuned to minimize the number of random jumps on SSD\u00a0(can be as slow as 100 MB\/s) and maximize the number of sequential operations (as fast as 6\u00a0GB\/s). It means you can use UnumDB both as your primary database and analytical database at the same time.\n\nWhy it\u2019s important? If your data scientists have to sample & export the data every time they run experiments, your pipelines become bloated and slow. What\u2019s worse, the experiment results may be irrelevant as the data becomes outdated before the experiments finish.\n\n### Compression\n\nCommon Ingredients Our Ingredients\nZlib, Snappy Custom\n\nIf the SSDs are so slow compared to the rest of your server - the next logical step is to minimize the data that goes to disk. Designing a good compression algorithm is true science. Most of the research in this industry was done in 1980s and boils down to a few key topics: Shannons entropy and Huffman coding. Even today industry-standard libraries such as Snappy and Zlib use the same old ideas.\n\nLuckily for our customers, we actively design and benchmark new specialized compression algorithms for different parts of our DB. Remember, we have a custom Data Layout. This means we have a deeper understanding of how the bytes will be arranged on a disk. \u21d2 We can replace a general-purpose compression engine with a tailored algorithm for our system!\n\n## CPU\n\n### Search Algorithms\n\nCommon Ingredients Our Ingredients\nBNDM, DFA, LSH Custom\n\nThis is where things get repetitive. Let\u2019s take a primitive task of searching for a substring in a bigger string (such as count the inclusions of \u201cthe\u201d in \u201cthe theme\u201d). How hard can it be? Computer Scientists know the answer. If N is the length of the needle and H is the length of the haystack, then:\n\n\u2022 Brute Force algorithm takes up to ~O(N*H) steps,\n\u2022 Rabin\u2013Karp algorithm takes on average ~O(N+H) steps,\n\u2022 Knuth-Morris-Pratt and BNDM algorithms takes up to ~O(H) steps, and so on.\n\nThis is what the textbook says, but if you take a few years off to run a few thousand benchmarks chances are you can find better approaches. Furthermore, search is not just about finding exact string matches. Here is what else we have:\n\n\u2022 fuzzy search for partial matches,\n\u2022 custom RegEx search for complex textual patterns,\n\u2022 custom k-Nearest Neighbors search for high-dimensional vector representations,\n\u2022 on the fly indexing of scalar fields.\n\n### Optimized Implementations\n\nCommon Ingredients Our Ingredients\nHigh-level General-Purpose Language SIMD Assembly, GPGPU\n\nComputer Science is great, but programming is very much an applied field. A new algorithm should not only have a better asymptotic complexity, but also a low constant overhead. This where our systems truly shine. Math and engineering come together in our products.\n\nWhen we identify a hot data-flow path, we start optimizing. We can either process more bytes per-cycle on the CPU or send the data to a specialized accelerator card like GPU. If the data-points are small and low latency is important - we use AVX2 and AVX-512 SIMD instructions on x86 and NEON SIMD instructions ARM.\n\nIf the data batches are in 100 MB - 10 GB range, we often switch to GPUs and implement our kernels with CUDA, OpenCL, Halide, SyCL and all kinds of other heterogeneus computing technologies.\n\n### Parallelism, Concurrency and Serialization\n\nFor the sake of completeness there are a few other technical things we should mention:\n\n1. Multi-processing and multi-threading is not the same thing.\n2. Not every concurrent data-structure is lock-free.\n3. System calls are expensive and synchronous I\/O is avoided at Unum.\n4. We avoid JSON, XML and similar formats for internal use in favor of binary formats.\n5. SQL comes with a hefty parsing overhead and is replaced with higher-level language bindings.\n\n## RAM\n\n### Memory Management\n\nCommon Ingredients Our Ingredients\nMulti-copy, Garbage Collection Bypass OS cache and 0 copies\n\nRAM chips are considered fast, but they are also very expensive. RAM is ~50x more expensive than NAND Flash (per byte). A good database must manage that tiny RAM\u00a0pool very efficiently, not leaving any irrelevant data copies behind. Unfortunately, that\u2019s not always possible. Depending on the I\/O protocols you use - hidden copies of your data can be stored in the OS cache pages, let alone the caches of the database itself.\n\nAnother problem, is that some databases (written in Java or other high-level languages) run in a Garbage-Collecting environment. This means that the developers of such databases rely on a slow and faulty procedure that automatically reclaims memory. It sometimes makes software development easier, but also comes with undeniable drawbacks and inefficiencies.\n\n### Memory Accesses\n\nCommon Ingredients Our Ingredients\nNo Direct Optimization Rare Page-Aligned Accesses\n\nEvery modern computer has separate slots for the CPU and RAM on the motherboard. Every bit must travel through the PCB from RAM to CPU and vice-versa. Those round-trips are orders of magnitude cheaper, than reading from SSD or HDD, but if you have optimized everything else - this is the next place to look.\n\nOperation Energy Costs \u00b9 CPU Cycles \u00b2\nLoad from SRAM 3 pJ 10\nMove 10 mm on-chip 30 pJ\nSend off-chip 500 pJ\nSend to DRAM 1 nJ 200\nRead from NAND Flash SSD \u00b3 2 \u03bcJ\n\nMeans, simply accessing an integer from DRAM can be 1'000x more expensive, than operating on it. HPC software is optimized to reduce the number of memory accesses. A good example of that are the fast matrix multiplication algorithms which achieve sub-cubic complexity through such optimizations. We perform similar micro-optimizations throughout our system.\n\n\u00b9 The numbers vary greatly depending on the litographic process generation, but the ratios generally hold.\n\n\u00b2 Read \u201cApproximate cost to access various caches and main memory\u201d.\n\n\u00b3 SSDs don\u2019t address single bytes. Generally the 2-4 KB pages are grouped into 128 KB blocks.\n\n## WEB\n\n### Communication Protocols\n\nCommon Ingredients Our Ingredients\nTCP\/IP UDP, Infiniband, RDMA\n\nAs soon as you database grows beyond a single server instance, you need those servers to communicate with each other and communication is hard. Not only the synchronization logic must be bullet-proof to guarantee the same ACID transactions in a distributed environment, but also the implementation must be efficient. The growth of the AI sector has accelerated the R&D in high-bandwidth networking and you can often find clusters ready for 100\/200\/400 or even 800 Gb\/s networking.\n\nThe sad reality is that still most of the distributed systems still rely on the TCP\/IP stack for communication. It doesn\u2019t support such speeds, it doesn\u2019t support global memory addressing and introduces a huge bottleneck when encoding\/decoding packets. So if you want to reach our numbers, say goodbye to TCP\/IP and hello to Infiniband RDMA!\n\nThe importance of each optimization above is extremely hard to quantify in a general case, because there is no \u201cgeneral case\u201d. Every part must be meticulously benchmarked in conjunction with underlying hardware to achieve ideal performance! Happy benchmarking!","date":"2022-05-24 18:39:54","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.22141212224960327, \"perplexity\": 2719.1909165916845}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-21\/segments\/1652662573189.78\/warc\/CC-MAIN-20220524173011-20220524203011-00005.warc.gz\"}"} | null | null |
package com.jcabi.github.mock;
import com.jcabi.github.Limit;
import com.jcabi.github.Limits;
import org.hamcrest.MatcherAssert;
import org.hamcrest.Matchers;
import org.junit.Test;
/**
* Test case for {@link MkLimits}.
* @author Yegor Bugayenko (yegor@tpc2.com)
* @version $Id$
*/
public final class MkLimitsTest {
/**
* MkLimits can work.
* @throws Exception If some problem inside
*/
@Test
public void worksWithMockedData() throws Exception {
final Limits limits = new MkGithub().limits();
MatcherAssert.assertThat(
new Limit.Smart(limits.get(Limits.CORE)).limit(),
Matchers.greaterThan(0)
);
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 6,742 |
Nargis Dutt (nascida Fatima Rashid; Prayagraj, 1 de junho de 1929 – Bombaim, 3 de maio de 1981), também conhecida por seu pseudônimo Nargis, foi uma atriz de cinema indiana que atuou em filmes clássicos de Bollywood. Considerada uma das maiores atrizes da história do cinema hindu, estreou na tela em um papel menor aos 5 anos de idade com Talash-E-Haq (1935), mas sua carreira de atriz realmente começou com o filme Tamanna (1942).
Em uma carreira que durou três décadas, Nargis apareceu em inúmeros filmes de sucesso comercial, bem como aclamados pela crítica, muitos dos quais a apresentavam ao lado do ator Raj Kapoor. Ela era a irmã mais nova do famoso ator Anwar Hussain. Seu papel mais conhecido foi o de Radha no filme indicado ao Oscar Mother India (1957), um desempenho que lhe rendeu o Prêmio Filmfare de Melhor Atriz. Ela apareceria com pouca frequência nos filmes durante os anos 60. Alguns de seus filmes desse período incluem o drama Raat Aur Din (1967), pelo qual ela recebeu o prêmio National Film Award de Melhor Atriz.
Nargis se casou com sua co-estrela de Mother India, Sunil Dutt, em 1958. Juntos, eles tiveram três filhos, incluindo o ator Sanjay Dutt. Junto com o marido, Nargis formou a Trupe de Cultura das Artes de Ajanta, que reuniu vários atores e cantores importantes da época e realizou apresentações de palco em áreas de fronteira. No início dos anos 1970, Nargis tornou-se o primeiro patrono da The Spastic Society of India e seu trabalho subsequente com a organização trouxe seu reconhecimento como assistente social e mais tarde uma indicação ao Rajya Sabha em 1980.
Nargis morreu em 1981 de câncer de pâncreas, apenas três dias antes de seu filho, Sanjay Dutt, fazer sua estréia em filmes hindi com o filme Rocky. Em 1982, a Fundação Nargis Dutt Memorial Cancer foi estabelecida em sua memória. O prêmio de Melhor Longa-Metragem sobre Integração Nacional na cerimônia do Annual Film Awards é chamado de Prêmio Nargis Dutt em sua homenagem.
Início da vida e antecedentes
Nargis nasceu como Fatima Rashid em Calcutá, na presidência de Bengala, no Império Britânico da Índia (agora Kolkata, Bengala Ocidental, Índia). Seu pai Abdul Rashid, ex-Mohanchand Uttamchand ("Mohan Babu") Tyagi, era originalmente um rico herdeiro hindu Mohyal Brahmin de Rawalpindi, província de Punjab, que havia se convertido ao islamismo. Sua mãe era Jaddanbai, uma cantora de música clássica hindustani e uma das pioneiras do cinema indiano. A família de Nargis mudou-se para Allahabad de West Punjab. Ela introduziu Nargis na cultura do cinema que se desenrolava na Índia na época. O meio-irmão materno de Nargis, Anwar Hussain (1928-1988), também se tornou um ator de cinema.
Carreira
Fátima fez sua primeira aparição no cinema no filme Talashe Haq de 1935, quando tinha seis anos de idade, creditada como Baby Nargis. Nargis ( [ˈNərɡɪs]) é uma palavra persa que significa narciso, a flor. Ela foi posteriormente creditada como Nargis em todos os seus filmes.
Nargis apareceu em vários filmes depois de sua estréia; ganhou fama duradoura por seus papéis posteriores, adultos, a partir dos 14 anos, em Tahdeer, de Mehboob Khan , em 1943, contracenando com o Motilal. Ela estrelou em muitos filmes populares hindus do final dos anos 1940 e 1950, como Barsaat (1949), Andaz (1949), Jogan (1950), Awaara (1951), Deedar (1951), Anhonee (1952), Shree 420. (1955) e Chori Chori (1956). Ela apareceu no drama épico indicado ao Oscar de Mehboob Khan, Mother India, em 1957, pelo qual ela ganhou o prêmio de Melhor Atriz Filmfare por sua performance. Baburao Patel, da revista cinematográfica Filmindia (dezembro de 1957), descreveu Mother Índia como "o maior filme produzido na Índia" e escreveu que nenhuma outra atriz teria sido capaz de interpretar o papel tão bem quanto Nargis.
Após seu casamento com Sunil Dutt em 1958, Nargis desistiu de sua carreira cinematográfica para se estabelecer com sua família, depois que seus últimos filmes foram lançados. Ela fez sua última aparição no filme de 1967, Raat Aur Din. O filme foi bem recebido e o desempenho de Nargis como uma mulher que tem transtorno dissociativo de identidade foi aclamado pela crítica. Para este papel, ela ganhou um National Film Award de Melhor Atriz e se tornou a primeira atriz a ganhar nesta categoria. Ela também recebeu uma indicação ao Filmfare Best Actress Award para este filme.
Em 2011, a Rediff.com listou-a como a maior atriz de todos os tempos, afirmando: "Uma atriz com estilo, graça e uma presença incrivelmente quente na tela, Nargis é verdadeiramente uma protagonista para se celebrar." ML Dhawan do The Tribune disse: "Em quase todos os seus filmes, Nargis criou uma mulher que poderia ser desejada e deificada. O carisma da imagem da tela de Nargis estava em que oscilava entre o simples e o chique com igual facilidade."
Ela também foi nomeada para o Rajya Sabha (Câmara Alta do Parlamento Indiano) de 1980-81 mas devido ao câncer, ela adoeceu e morreu durante o seu mandato.
Filmografia
Talashe Haq (1935)
Madam Fashion (1936)
Taqdeer (1943)
Humayun (1945)
Bisvi Sadi (1945)
Ramayani (1945)
Nargis (1946)
Mehandi (1947)
Mela (1948)
Anokha Pyar (1948)
Anjuman (1948)
Aag (1948)
Roomal (1949)
Lahore (1949)
Darogaji (1949)
Barsaat (1949)
Andaz (1949)
Pyaar (1950)
Meena Bazaar (1950)
Khel (1950)
Jogan (1950)
Jan Pahchan (1950)
Chhoti Bhabhi (1950)
Babul (1950)
Aadhi Raat (1950)
Saagar (1951)
Pyar Ki Baaten (1951)
Hulchul (1951)
Deedar (1951)
Awaara (1951)
Sheesha (1952)
Bewafaa (1952)
Ashiana (1952)
Anhonee (1952)
Amber (1952)
Shikast (1953)
Paapi (1953)
Dhoon (1953)
Aah (1953)
Angarey (1954)
Shree 420 (1955)
Jagte Raho (1956)
Chori Chori (1956)
Pardesi
Mother India (1957)
Lajwanti (1958)
Ghar Sansar (1958)
Adalat (1958)
Yaadein (1964)
Aditya ₨ (1967)
''Tosa oneira stous dromous' ' (1968)
Atores da Índia
Punjabis
Mortes por câncer de pâncreas
Mortos em 1981
Nascidos em 1929 | {
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} | 4,808 |
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The Faux "Civility" of "Broken Windows"
Posted on December 22, 2014 by Devin Smith | 2 Comments
On December 18, 2014, William Bratton and George Kelling published an op ed in the Wall Street Journal decrying "The Assault on 'Broken Windows' Policing." I'll be writing a broader response to their piece noting their failure to implement "broken windows" enforcement against the elite white-collar criminals who have made Wall Street one of the world's most destructive criminal "hot spots." In this column I point out the implications of their attempt to label criticisms of a NYPD policy they developed and favor as an "assault." They chose the word to be inflammatory and to try to label their critics as inherently illegitimate and pro-crime.
Here, in light of the tragedy of the murder of two NYPD officers and the reactions to that murder I want to point out what Bratton and Kelling asserted explained broken windows policing was effective. "[T]housands of police interventions on the street … restored order and civility across the five boroughs." Except, of course, for Wall Street, where crime has skyrocketed and rudeness and disorder are defining elements of the corrupt culture. But that's my next piece.
Mayor Bill de Blasio, upon his election, recruited Bratton to again take the position of NYPD Commissioner. The Mayor, of course, was well aware of Bratton's intense support for (blue-collar only) broken windows policing. Broken windows policing, therefore, far from being under "assault," is actively being promoted by the NYPD Commissioner and the Mayor. What Bratton and Kelling are so upset about is, first, after the latest police killings of black males, there were continuing protests in many cities. Second, many criminologists and police commissioners criticize broken windows policing (against blue collar criminals only).
The Mayor did not criticize Bratton's blue collar only broken windows policing strategy even after these police killings. The Mayor has, however, made public the advice he and his wife gave to their son in dealings he may have with police.
"The mayor, who is white, appeared on ABC's 'This Week' and talked about what he and wife Chirlane McCray, who is black, have told their 17-year-old biracial son Dante about interacting with the police, which included not reaching for a cellphone because it 'might be misinterpreted if it was a young man of color,' by police. The comments came after a grand jury decided not indict a white cop in the choking death of Eric Garner, who is black, in Staten Island."
The Mayor's comments enraged the NYPD police unions.
"Mayor de Blasio made 'moronic' comments Sunday that prove he 'doesn't belong' in New York, a key police union chief said, further inflaming the war of words between Hizzoner and the NYPD.
The comments from Ed Mullins, head of the Sergeants Benevolent Association…."
If the Mayor "doesn't belong" in NYC because he gave his son that advice, then no black parent with a son "belong[s]" in NYC because some close variant of what the Mayor and his wife advised his son is absolutely normal advice. The fact that parents with minority sons feel that giving such advice is essential does reveal that the black and white experience in America is still distinct in some important ways. While Bratton and Kelling ignored the point, minorities are substantially more likely to be subjected to humiliating "stop and frisk" demands by NYPD police – and the great majority of such demands lead to no criminal case, so there is no reason to assume that the police are successfully targeting criminals in these encounters. I provided the data on these points in a prior article.
The first proverbial bottom line is that the Mayor did not criticize the police, reinforced rather than "assault[ed]" blue collar broken windows policing, and gave advice to his biracial son that is not only normal advice that parents give to minority males but also excellent advice that helps the police and minority males when they interact. The second takeaway is that the police unions don't care – they're feeling abused by the protests. The Mayor did not support the charges of the protestors, all he said is that that peaceful protestors had a constitutionally protected right to do so. In sum, he upheld the rule of law. One of the criticisms of these humiliating stop and frisk encounters that are so disproportionately used against black and Latino males by the NYPD is that they inherently create substantial hostility between the minority community and the police officers. The Mayor was elected in part based on his criticism of the aggressive stop and frisk strategy targeted primarily at black males and Latinos.
This is the context in which the tragic murder of the two police officers by a man with a history of violent crimes and apparent severe psychological problems (including an attempted suicide) occurred. He began by shooting his former girlfriend in Maryland and then bragged on social media that he planned to murder police officers. Unfortunately, he succeeded in killing two NYPD officers who he had never met and who he killed in a fashion that made it impossible for them to put up any defense.
I understand that police officers are dealing with grief and their own fears of being the subject of such murderous assaults and that they cannot be at their best in such circumstances. I understand that many of the protestors are not in the least civil. These factors are both reasons that leaders exist. How did those police leaders demonstrate their commitment to "civility?"
"In a stunning show of disapproval and disrespect, police officers, led by union officials, turned their backs to him on Saturday night when the mayor went to the hospital to talk about the two officers who were killed."
Worse, union leaders have resorted to blood libels against the Mayor.
"Patrick Lynch, president of the Patrolmen's Benevolent Association, laid the blame for the deaths of the officers squarely at the feet of the mayor.
'That blood on the hands starts on the steps of City Hall, in the office of the mayor,' he said."
These statements indicate a crisis in the NYPD. The comments are outrageous, particularly coming from leaders of a group that is quasi-military in its discipline and organization. The Mayor has nothing to do with the murders of the policemen.
It is fine for Bratton to brag in the WSJ about the restoration of civility (again, ignoring Wall Street) by the police, but he is the leader of the NYPD. He needs to restore civility among his officers and then repair the rift that his officers' statements and actions have widened with the community through their vicious comments and "stunning … disrespect." All of us join the NYPD in mourning the murder of the two officers, and the loved ones of the woman who was the criminal's first victim in hoping for her full recovery from her grievous wounds.
This entry was posted in William K. Black and tagged broken windows, NYPD, WSJ. Bookmark the permalink.
2 responses to "The Faux "Civility" of "Broken Windows""
Susan Anderson | December 22, 2014 at 8:24 am |
Excellent post. Regulatory agencies ARE the 'cops' who patrol Wall St., and it was the complete LACK of law enforcement that threw us into the recession.
FWIW, this murderer wanted to get his 15 minutes of fame by inserting himself into a news story. I don't think foR one second that he cared two cents for Garner.
Robert Sadin | December 22, 2014 at 10:36 am |
Outstanding. Yes…broken windows does not apply to Wall Street. In fact, as you have pointed out, nothing in law enforcement applies to Wall Street.
One note:
Use of the term "blood libel" is not appropriate. Blood libel refers to specific attacks on Jews for using Christian blood to make matzoh.
https://www.google.com/webhp?sourceid=chrome-instant&ion=1&espv=2&ie=UTF-8#q=blood%20libel
This blood libel led to mass murder of Jews again and again.
This is not a generic term. To use it as such cheapens the word. | {
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} | 3,540 |
The Astroduck – amerykański film animowany z udziałem Kaczora Daffy'ego i Speedy'ego Gonzalesa.
Wersja polska
Wersja lektorska ITI do kasety Kaczor Daffy z 1990 r.
Wersja polska: ITI Home Video
Tekst:
Tomasz Beksiński
Mariusz Arno-Jaworowski
Wersja dubbingowa z 1992 r.
Wersja polska: Master Film
Występują:
Mieczysław Gajda - Kaczor Daffy
Tomasz Kozłowicz - Speedy Gonzales
oraz
Ryszard Olesiński
Lektor: Roch Siemianowski
Linki zewnętrzne
Amerykańskie filmy komediowe
Animowane filmy krótkometrażowe wytwórni Warner Bros.
Amerykańskie filmy z 1966 roku | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 1,339 |
Q: Питон, разбитие текста на строки Прошу помощи, имею вывод, например
['Отвертка трещоточная с набором бит (636120)', 'Паста-смывка 200мл (973515-4002)', 'Разрушитель ржавчины (973520-4500)', 'Очиститель тормозов (973520-0650)', 'очки защитные Классик (GL-01010)', 'перчатки (PGT-020)', 'GW40 210мл (973520-3210)']
Нужно этот текст превратить в построчный вывод, вот так:
Отвертка трещоточная с набором бит (636120)
Паста-смывка 200мл (973515-4002)
Разрушитель ржавчины (973520-4500)
Очиститель тормозов (973520-0650)
очки защитные Классик (GL-01010)
перчатки (PGT-020)
GW40 210мл (973520-3210)
Элементов может быть разное количество, которое заведомо неизвестно. Прошу помочь кодом на питоне. Спасибо.
A: d = ['Отвертка трещоточная с набором бит (636120)', 'Паста-смывка 200мл (973515-4002)', 'Разрушитель ржавчины (973520-4500)', 'Очиститель тормозов (973520-0650)', 'очки защитные Классик (GL-01010)', 'перчатки (PGT-020)', 'GW40 210мл (973520-3210)']
for a in d:
print(a)
сделано простым перебором списка
A: Вариант без циклов:
d = ['Отвертка трещоточная с набором бит (636120)', 'Паста-смывка 200мл (973515-4002)', 'Разрушитель ржавчины (973520-4500)', 'Очиститель тормозов (973520-0650)', 'очки защитные Классик (GL-01010)', 'перчатки (PGT-020)', 'GW40 210мл (973520-3210)']
print('\n'.join(d))
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 1,981 |
<?php
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default:
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* original function get_the_term_list was in 'wp-include/category-template.php'
*/
function get_my_term_list( $id, $taxonomy, $before = '', $sep = '', $after = '' ) {
$terms = get_the_terms( $id, $taxonomy );
if ( is_wp_error( $terms ) )
return $terms;
if ( empty( $terms ) )
return false;
foreach ( $terms as $term ) {
$link = get_term_link( $term, $taxonomy );
if ( is_wp_error( $link ) )
return $link;
$term_links[] = '<a href="' . esc_url( $link ) . '" rel="tag" class="ui orange tiny label">' . $term->name . '</a>';
}
$term_links = apply_filters( "term_links-$taxonomy", $term_links );
return $before . join( $sep, $term_links ) . $after;
}
/**
* original function get_the_tag_list was in 'wp-include/category-template.php'
*/
function get_my_tag_list( $before = '', $sep = '', $after = '', $id = 0 ) {
return apply_filters( 'the_tags', get_my_term_list( $id, 'post_tag', $before, $sep, $after ), $before, $sep, $after, $id );
}
/**
* original function comment_form was in 'wp-include/comment-template.php'
*/
function my_comment_form( $args = array(), $post_id = null ) {
if ( null === $post_id )
$post_id = get_the_ID();
else
$id = $post_id;
$commenter = wp_get_current_commenter();
$user = wp_get_current_user();
$user_identity = $user->exists() ? $user->display_name : '';
$args = wp_parse_args( $args );
if ( ! isset( $args['format'] ) )
$args['format'] = current_theme_supports( 'html5', 'comment-form' ) ? 'html5' : 'xhtml';
$req = get_option( 'require_name_email' );
$aria_req = ( $req ? " aria-required='true'" : '' );
$html5 = 'html5' === $args['format'];
$fields = array(
'author' => '<p class="comment-form-author">' . '<label for="author">' . __( 'Name' ) . ( $req ? ' <span class="required">*</span>' : '' ) . '</label> ' .
'<input id="author" name="author" type="text" value="' . esc_attr( $commenter['comment_author'] ) . '" size="30"' . $aria_req . ' /></p>',
'email' => '<p class="comment-form-email"><label for="email">' . __( 'Email' ) . ( $req ? ' <span class="required">*</span>' : '' ) . '</label> ' .
'<input id="email" name="email" ' . ( $html5 ? 'type="email"' : 'type="text"' ) . ' value="' . esc_attr( $commenter['comment_author_email'] ) . '" size="30"' . $aria_req . ' /></p>',
'url' => '<p class="comment-form-url"><label for="url">' . __( 'Website' ) . '</label> ' .
'<input id="url" name="url" ' . ( $html5 ? 'type="url"' : 'type="text"' ) . ' value="' . esc_attr( $commenter['comment_author_url'] ) . '" size="30" /></p>',
);
$required_text = sprintf( ' ' . __('Required fields are marked %s'), '<span class="required">*</span>' );
/**
* Filter the default comment form fields.
*
* @since 3.0.0
*
* @param array $fields The default comment fields.
*/
$fields = apply_filters( 'comment_form_default_fields', $fields );
$defaults = array(
'fields' => $fields,
'comment_field' => '<p class="comment-form-comment"><label for="comment">' . _x( 'Comment', 'noun' ) . '</label> <textarea id="comment" name="comment" cols="45" rows="8" aria-required="true"></textarea></p>',
'must_log_in' => '<p class="must-log-in">' . sprintf( __( 'You must be <a href="%s">logged in</a> to post a comment.' ), wp_login_url( apply_filters( 'the_permalink', get_permalink( $post_id ) ) ) ) . '</p>',
'logged_in_as' => '<p class="logged-in-as">' . sprintf( __( 'Logged in as <a href="%1$s">%2$s</a>. <a href="%3$s" title="Log out of this account">Log out?</a>' ), get_edit_user_link(), $user_identity, wp_logout_url( apply_filters( 'the_permalink', get_permalink( $post_id ) ) ) ) . '</p>',
'comment_notes_before' => '<p class="comment-notes">' . __( 'Your email address will not be published.' ) . ( $req ? $required_text : '' ) . '</p>',
#'comment_notes_after' => '<p class="form-allowed-tags">' . sprintf( __( 'You may use these <abbr title="HyperText Markup Language">HTML</abbr> tags and attributes: %s' ), ' <code>' . allowed_tags() . '</code>' ) . '</p>',
'comment_notes_after' => '<p class="form-allowed-tags">' . sprintf( __( 'You may use <a href="http://daringfireball.net/projects/markdown/syntax">Markdown</a> language and realtime preview can see above.' ) ),
'id_form' => 'commentform',
'id_submit' => 'submit',
'title_reply' => __( 'Leave a Reply' ),
'title_reply_to' => __( 'Leave a Reply to %s' ),
'cancel_reply_link' => __( 'Cancel reply' ),
'label_submit' => __( 'Post Comment' ),
'format' => 'xhtml',
);
/**
* Filter the comment form default arguments.
*
* Use 'comment_form_default_fields' to filter the comment fields.
*
* @since 3.0.0
*
* @param array $defaults The default comment form arguments.
*/
$args = wp_parse_args( $args, apply_filters( 'comment_form_defaults', $defaults ) );
?>
<?php if ( comments_open( $post_id ) ) : ?>
<?php
/**
* Fires before the comment form.
*
* @since 3.0.0
*/
do_action( 'comment_form_before' );
?>
<div id="respond" class="comment-respond">
<h3 id="reply-title" class="comment-reply-title"><?php comment_form_title( $args['title_reply'], $args['title_reply_to'] ); ?> <small><?php cancel_comment_reply_link( $args['cancel_reply_link'] ); ?></small></h3>
<?php if ( get_option( 'comment_registration' ) && !is_user_logged_in() ) : ?>
<?php echo $args['must_log_in']; ?>
<?php
/**
* Fires after the HTML-formatted 'must log in after' message in the comment form.
*
* @since 3.0.0
*/
do_action( 'comment_form_must_log_in_after' );
?>
<?php else : ?>
<form action="<?php echo site_url( '/wp-comments-post.php' ); ?>" method="post" id="<?php echo esc_attr( $args['id_form'] ); ?>" class="comment-form ui form"<?php echo $html5 ? ' novalidate' : ''; ?>>
<?php
/**
* Fires at the top of the comment form, inside the <form> tag.
*
* @since 3.0.0
*/
do_action( 'comment_form_top' );
?>
<?php if ( is_user_logged_in() ) : ?>
<?php
/**
* Filter the 'logged in' message for the comment form for display.
*
* @since 3.0.0
*
* @param string $args['logged_in_as'] The logged-in-as HTML-formatted message.
* @param array $commenter An array containing the comment author's username, email, and URL.
* @param string $user_identity If the commenter is a registered user, the display name, blank otherwise.
*/
echo apply_filters( 'comment_form_logged_in', $args['logged_in_as'], $commenter, $user_identity );
?>
<?php
/**
* Fires after the is_user_logged_in() check in the comment form.
*
* @since 3.0.0
*
* @param array $commenter An array containing the comment author's username, email, and URL.
* @param string $user_identity If the commenter is a registered user, the display name, blank otherwise.
*/
do_action( 'comment_form_logged_in_after', $commenter, $user_identity );
?>
<?php else : ?>
<?php echo $args['comment_notes_before']; ?>
<?php
/**
* Fires before the comment fields in the comment form.
*
* @since 3.0.0
*/
do_action( 'comment_form_before_fields' );
foreach ( (array) $args['fields'] as $name => $field ) {
/**
* Filter a comment form field for display.
*
* The dynamic portion of the filter hook, $name, refers to the name
* of the comment form field. Such as 'author', 'email', or 'url'.
*
* @since 3.0.0
*
* @param string $field The HTML-formatted output of the comment form field.
*/
echo apply_filters( "comment_form_field_{$name}", $field ) . "\n";
}
/**
* Fires after the comment fields in the comment form.
*
* @since 3.0.0
*/
do_action( 'comment_form_after_fields' );
?>
<?php endif; ?>
<?php
/**
* Filter the content of the comment textarea field for display.
*
* @since 3.0.0
*
* @param string $args['comment_field'] The content of the comment textarea field.
*/
echo apply_filters( 'comment_form_field_comment', $args['comment_field'] );
?>
<?php echo $args['comment_notes_after']; ?>
<p class="form-submit">
<input name="submit" type="submit" id="<?php echo esc_attr( $args['id_submit'] ); ?>" value="<?php echo esc_attr( $args['label_submit'] ); ?>" class="ui orange submit button" />
<?php comment_id_fields( $post_id ); ?>
</p>
<?php
/**
* Fires at the bottom of the comment form, inside the closing </form> tag.
*
* @since 1.5.2
*
* @param int $post_id The post ID.
*/
do_action( 'comment_form', $post_id );
?>
</form>
<?php endif; ?>
</div><!-- #respond -->
<?php
/**
* Fires after the comment form.
*
* @since 3.0.0
*/
do_action( 'comment_form_after' );
else :
/**
* Fires after the comment form if comments are closed.
*
* @since 3.0.0
*/
do_action( 'comment_form_comments_closed' );
endif;
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 7,204 |
require "slim"
# Assets
set :css_dir, "assets/css"
set :fonts_dir, "assets/fonts"
set :images_dir, "assets/images"
set :js_dir, "assets/js"
set :videos_dir, "assets/videos"
# Partials
set :partials_dir, "layouts/partials"
## Generate pages from data example
#data.products.each do |category|
# category.products.each do |product|
# proxy "/products/#{category.name}/#{product.name}.html", "/products/category_template.html", :locals => { :category => category, :product => product }, :ignore => true
# end
#end
## Blogging
#activate :blog do |blog|
# blog.paginate = true
# blog.per_page = 10
# blog.prefix = "blog"
#end
#page "blog/*", :layout => :article_layout
# Internationalization
#activate :i18n, :mount_at_root => :en
# Reload the browser automatically whenever files change
activate :livereload
# Pretty URLs
activate :directory_indexes
# Relative URLs
set :relative_links, true
activate :relative_assets
# Build-specific configuration
configure :build do
#activate :asset_hash
#activate :gzip
#activate :imageoptim
#activate :minify_css
#activate :minify_html
#activate :minify_javascript
end
| {
"redpajama_set_name": "RedPajamaGithub"
} | 3,125 |
{"url":"https:\/\/math.stackexchange.com\/questions\/2408900\/the-harmonic-logarithm-and-its-relation-to-the-prime-number-theorem","text":"# The Harmonic Logarithm and its relation to the Prime Number Theorem\n\nFor the purposes of this question, the Harmonic Number, $H_n$ is defined by the finite series sum $H_n=\\sum_{k=1}^n\\frac{1}{k}$ (n and k being positive integers) and the Harmonic Logarithm, $hlog(n)$ is here defined as $$hlog(n)=\\left( H_{n^2}-H_{n} \\right)=\\sum_{k=n+1}^{n^2}\\frac{1}{k}$$\n\nThe standard definition for $\\gamma$ the Euler\u2013Mascheroni constant is\n\n$$\\gamma=\\lim_{n \\rightarrow \\infty} \\gamma_n$$ where $\\gamma_n=H_n-\\log(n)$\n\nFollowing Scott [REF2], let $a_n=2\\gamma_n-\\gamma_{n^2}$, giving $\\gamma=\\lim_{n \\rightarrow \\infty} a_n=\\lim_{n \\rightarrow \\infty} \\left(2\\gamma_n-\\gamma_{n^2} \\right)$\n\n\\begin{align}a_n&=2\\gamma_n-\\gamma_{n^2}\\\\ &=2\\left(H_n-\\log(n) \\right)-\\left(H_{n^2}-\\log(n^2) \\right)\\\\ &=2H_n-2\\log(n)-H_{n^2}-2\\log(n)\\\\ &=2H_n-H_{n^2}\\\\ &=H_n-\\left(H_{n^2}-H_n \\right)\\\\ &=H_{n}-hlog(n)\\\\ \\end{align}\n\nTherefore in the $\\lim_{n \\rightarrow \\infty}$ we have $$\\gamma=\\lim_{n \\rightarrow \\infty} \\left( H_n-hlog(n)\\right)$$ or coverging from below using the alternative definition $\\gamma_n=H_n-\\log(n+1)$ $$\\gamma=\\lim_{n \\rightarrow \\infty} \\left( H_n-hlog(n+1)\\right)\\tag1$$\n\nwhere $\\gamma$ is the Euler\u2013Mascheroni constant.\n\n(1) gives a slowly converging rational series for $\\gamma$ $$\\gamma=\\sum_{n=1}^{\\infty} \\frac{1}{n}+\\frac{1}{n+1}-\\sum_{k=n^2+1}^{(n+1)^2}\\frac{1}{k}$$\n\n$$\\gamma=\\frac{5}{12}+\\frac{221}{2520}+\\frac{1517}{2520}+...\\tag2$$\n\nFor the theoretical proofs and background see for example:\n\n[REF1] J. Lambek and L. Moser, (Feb., 1956), Rational Analogues of the Logarithm Function, The Mathematical Gazette, Vol. 40, No. 331, pp. 5-7.\n\n[REF2] J. A. Scott, (Nov., 1996), The Euler Constant \u03b3 without Logarithms, , The Mathematical Gazette, Vol. 80, No. 489, pp. 585-586.\n\n(Aside: The term \"Harmonic Logarithm\" appears to have been first coined by M.F. Egan in the same issue of the Mathematical Gazette as the Lambek and Moser paper, using a slightly different definition to mine above (see pages 8-10). If readers know of any other relevant papers please let me know. The term \"Harmonic Logarithm\" has more recently been applied to a definition involving both rational and transcendental numbers http:\/\/mathworld.wolfram.com\/HarmonicLogarithm.html.)\n\nRelation of the Harmonic Logarithm to the Prime Number Theorem\n\nOne way of stating the prime number theorem is\n\n$$\\pi(N) \\thicksim \\frac{N}{log(N)}$$\n\nwhere $\\pi(N)$ is the prime counting function, being the number of primes less than or equal to N.\n\nA closer approximation to the prime counting function can be found using the $Li(x)=\\int_2^x \\frac{dt}{log t}$ function. However if we calculate $Li((N+1)^2)-Li((N)^2)$ we find that the number of primes between two consecutive squares, $N^2$ and $(N+1)^2$, is also approximately given by $\\frac{N}{log(N)}$\n\nNow one conjecture here involving series (2) is\n\n$$\\gamma_n \\;(n^2)!= \\left( \\frac{1}{n}+\\frac{1}{n+1}-\\sum_{k=n^2+1}^{(n+1)^2}\\frac{1}{k} \\right) (n^2)! = \\frac{C}{p_1 p_2 ... p_n}$$ where C is some arbitrary positive integer and $p_1 p_2 ... p_n$ is a product of all the primes between $n^2$ and $(n+1)^2$, none of which being divisors of C.\n\nHowever perhaps of greater interest is how to tackle the proof of the prime number theorem involving $hlog(N)$ instead of $log(N)$ $$\\pi(N) \\thicksim \\frac{N}{hlog(N)}$$\n\nwithout involving transcendental functions in the proof at all.\n\nI have seen proofs using $H_n$ that generate Chebyshev bounds for the prime counting function, as referenced in the J. Lambek and L. Moser paper above, but surely this can be improved upon using $hlog(N)$ instead.\n\n\u2022 Replacing $\\text{Li}(x)$ by an approximation $f(x) = \\text{Li}(x)+o(\\text{Li}(x))$ won't change the proof of the prime number theorem at all. \u2013\u00a0reuns Aug 29 '17 at 12:35\n\u2022 In hindsight I need to make further changes to make this question clearer. The ultimate aim is to encourage more competent mathematicians than me to try to prove the prime number theorem without any recourse to transcendental functions or numbers. Therefore Li(x) cannot be used in the final proof. Something like $\\sum_{k=2}^n \\frac{1}{hlog(k)}$ seems to track Li(n) quite well, but I have no idea how to relate one to the other at the moment. \u2013\u00a0James Arathoon Aug 29 '17 at 13:13\n\u2022 As I said replacing $\\text{Li}(x)$ by an approximation won't change anything. For the analytic proof of the PNT, $\\text{Li}(x)$ is clearly the better choice because $\\int_2^\\infty (\\pi(x)-\\text{Li}(x)) x^{-s-1}dx$ is analytic at $s=1$ and $\\int_2^\\infty \\text{Li}(x) x^{-s-1}dx$ is easy to understand (it is for good reasons if we write the PNT in term of $\\text{Li}(x)$) \u2013\u00a0reuns Aug 29 '17 at 13:15","date":"2019-10-17 12:43:58","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 1, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8979420065879822, \"perplexity\": 313.6693628436707}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-43\/segments\/1570986675316.51\/warc\/CC-MAIN-20191017122657-20191017150157-00209.warc.gz\"}"} | null | null |
PASIG Prague 2016
tags: archives, digital preservation, libraries, open source, software, technology
National Technical Library, Prague
This is apparently the first time that PASIG has been held in Europe; it's also the first time I've attended. My goal was to come speak about the Ontario Library Research Cloud, both to get the word out about the model we followed as well as to expose that model to a bit of peer critique and commentary. In that regard, it's been a successful journey and I hope that it just started a conversation that will continue on.
It was a very well organized and catered event. The reception even included a chartered vintage tram that took us to Obecní dům, a somewhat touristy yet glorious Jugendstil monument in Prague. Given the modest registration fee, the quality has been remarkable. Many thanks to those who planned and carried out this event.
Digital Preservation Bootcamp
OAIS Model, Theory & Practice
Neil Jefferies, Oxford U
Gave an overview of OAIS. Noted, after the overview, that one of the problems is what he defines as a SIP (submission information package) problem, i.e.- we often don't have good (or any) metadata for certain collections. That's not a good reason not to ingest them into an archival system. Also, DIP (dissemination information package) problems exist. Much of the audience for our data hasn't been born yet, for example, so it's hard to make an economic case for preserving data. DIP also become richer in information over time, so the AIP must be updated to reflect this enrichment. Last, the AIP also have problems. They must be preserved, so as he put it "don't mangle it to fit a standard!" Archived objects are also not static, so the AIP may require updating (incremental curation).
Much of the meaning or significance of a file depends on its context and provenance, not necessarily the intrinsic information in the object. File names are easily changed, as are formats. As he noted, a file "is a meaningless stream of bytes." The metadata can be more meaningful than the data itself, given the primacy of context and provenance.
Pointed out that, contrary to the experience of corporations, our institutions (i.e.- universities) are remarkably durable. They nearly never fail, so we will clearly be migrating systems and entire ways of doing things over time. Long-term preservation is a fact of life, not tied to the rise and fall of a corporate entity.
Made some recommendations for setting up preservation systems. One of his first points was not to "bake decisions into systems at a low level." The more generic/abstract the lower levels are, the more flexible they can be in terms of tolerating multiple formats and techniques for ingest, etc. Best is to use standards and to work modularly and in layers. Break down processes into "simple, small, atomic tasks" so that parts can be deferred, done parallel, etc. Keep originals and metadata, if possible. Use versioning, leave an audit trail.
Trustworthy Digital Preservation Systems
David Minor, U of California San Diego
Trust has to be defined and it's an iterative process. Funding comes and goes, partners change, etc. Questions to ask include institutional commitment, infrastructure demands, technical systems, sustainability, and so forth. If everything else fails, what then? Are there "fail-safe" partners?
Gave an overview of European and (North) American frameworks for certifying repositories. In Europe, it's known as European Framework for Audit and Certification of Digital Repositories, and has three levels: basic, extended, and formal. The first is self-assessment, while the last, of course, involves investigation of claims. Outcomes are Data Seal of Approval (basic), nestor (extended), TRAC/ISO 16363 (formal), and DRAMBORA (mix).
TRAC is divided into three sections: organizational infrastructure, digital object management, and technologies / technical infrastructure / security. In North America, the Center for Research Libraries has been the certifying body and has certified four U.S. and two Canadian (Canadiana.org and Scholar's Portal) entities. Now the push is to make it a standard, hence ISO 16363, which came out in 2012 (TRAC dates to 2007). They are similar, but it has been standardized now. To date, no one has gone through the ISO 16363 process; at present there are no certifying bodies for ISO 16363. There is now a separate standard (from 2014) for becoming an auditing body (ISO 16919) so that's just getting rolling (training for certifiers). Drambora is UK-based and follows similar patterns.
Mentioned the Digital POWRR Tool Evaluation Grid maintained by NIU. POWRR stands for Preserving (Digital) Objects With Restricted Resources. It breaks down tools by the needs they address and provides basic information and evaluations.
Ended by asking if we are moving toward a single tool or framework. He noted that right now one has to ask which certification one needs to pursue since multiples exist.
Applying DP Standards for Assessment and Planning
Bertram Lyons, AVPreserve
Didn't take a lot of notes for his talk, but one key point he emphasized is that the non-technical aspects of preservation–planning, documentation, etc.–are often the areas that require more focus and attention. Technical aspects such as hardware and software decisions and management are areas where we tend to focus our efforts.
Another key point is to break the work into smaller chunks and not to attempt to do everything at once, but rather in a prioritized and planned order.
Long-term Digital Preservation Hardware & Systems
Tiered Storage Architectures
Donna Harland, Oracle
How can you lose data? Cannot find it, cannot read it, cannot validate authenticity, or cannot interpret it. This is not a new problem, she noted. Books have burned, tapes have been lost (e.g. – Apollo 11 moon walk original data feed), etc. We have recreated Alexandria, to use her words: obsolete applications, inaccessible documents, unreadable devices, family photos, etc.
PKX / Practitioners Knowledge Exchange: Case Studies in Preservation & Archiving Architectures and Operations
Qatar National Library
Krishna Chowdhury, QNL
She described the institution and the national digitization program. At one point, she showed a slide detailing their technical stack. Clearly they had a bit of money when putting this together. We could never afford anything so luxurious in our context. Worked closely with Oracle on this, so it's made up of a lot of 'Sun' hardware as well as their storage solutions. One number that caught my eye was that they have 28 physical servers in their second phase deploy. That's a serious deploy, and I wonder what their user population looks like, i.e.- whom does this stack serve/whom will it serve.
Then again, what she had was the opportunity to build this all from scratch. That's a chance to do things right, so to speak, rather than to inherit a broad set of perhaps less than optimal decisions. In that light, it might make sense to build it more robustly than we ever could.
The University of Oklahoma's Galileo's World: Creating New Demands for Digital Archiving & Preservation
Carl Grant, U of Oklahoma
Literally about Galileo. Oklahoma is one of two libraries–the Vatican is the other–to hold first editions of all 12 of Galileo's works, four of which have marginalia in his hand. Wanted to create an exhibition around these works, not least to celebrate OU's 125th anniversary. Another goal was to create an exhibition that would live on into the future, as well as to make it appealing to a wide audience, both scholarly and public.
Evolving the LOCKSS Technology
David S.H. Rosenthal, Stanford U
Started with a brief history of LOCKSS, telling the anecdotes related to how he and Vicky came up with the original idea as an analogue to print preservation, of sorts. Much has been written about LOCKSS, so I won't repeat the details here.
Also gave a tour of the impact of LOCKSS, i.e.- who is using it and how. It has gone well beyond preserving journals. In various contexts, it is being used for dissertations, research data, government information, and so on.
The Ontario Library Research Cloud: Future Considerations and Cost Models
Dale Askey, McMaster University
I gave this talk, so no notes!
Status of Long Term Preservation Service in Finland 2016
Mikko Tiainen, CSC – IT Centre for Science
Noted while speaking about the technical specifications that there is no proprietary software in their entire stack; it's all either open source or CSC-created. It's also highly modularized, so there's no monolithic piece to break down or fail. Individual components can be replaced as needed. Does not use object storage, but rather POSIX. I'll need to consult some colleagues when I return who can explain the implications of this.
In addition to their open source tools, they've got about 15,000 lines of in-house Python code in their production systems, using Python Luigi as their workflow engine. The terminal storage for their data is tape, if I saw correctly, three copies, two tied to the production systems in IBM and Oracle tape vaults, with the third being a dark archive copy.
Project Updates and Digital Preservation Community Developments
EUDAT: a Data Infrastructure for European Research
Rob Baxter, U of Edinburgh
Have built a suite of research data services, all under the name B2xxxxx. They found it challenging to work with a large number of communities, but found it worth doing so rather than just building something and hoping people would come use it. Tying together all of the disparate services (from across Europe) is also a challenge, not least due to varying metadata standards and practices.
Trying to model better research data management. Have now added more B2 services, e.g. B2HANDLE and B2ACCESS, the former creating persistent identifiers and the latter about identity and authorization.
Showed an interesting data pyramid. Base is transient data, which has individual value, the middle is registered data, which has community value, and at the top of the pyramid is citable data, which has societal value. The pyramid model makes sense, as it implies that of the great mass of project-immanent data, only some of it emerges to go to the next stage, and so forth. I often get the sense that researchers here "data sharing" and assume that society is asking them to share everything. Not the case.
The DPC Community: Growth, Progress, and Future Challenges
Paul Wheatley, Digital Preservation Coalition
Gave an overview of the DPC, and also discussed the reaction of the digital preservation community to Vint Cerf's comments last year about no one doing digital preservation and advocacy for print copies.
Also noted the potential impact of trade agreements on copyright and our ability to do digital preservation, noting that the community needs to continue to push back against these.
Tossed in a nice dig at those who mock emulation as a mechanism for preserving software. It's possible to do, and we should be taking on this challenge.
Preservation of Electronic Records
Luis Faria, Keep Solutions
Keep solutions is a spinoff of the University of Minho. Spoke about E-ARK, which is a pan-European effort to create and international methodology for archiving digital records and databases. Talked about the preservation of relational databases, something he noted has not been much discussed as yet at this event. There is a preservation standard known as SIARD 2.0, which they hope will emerge, as he put it, the one standard to rule them all with regard to DB preservation.
To go with the standard, there's a tool to support it, currently in beta and only available as a command line tool: database preservation toolkit.
There is also a push to create pan-European formats for SIP, AIP, and DIP. As he put it, yes, another standard, but necessary. Currently in draft form and open for comment. Tools go with this, too, e.g.- RODA 2.0, currently in alpha. Enables creation of many SIPs easily. Supports both E-ARK and Bagit.
David Rosenthal, Stanford U
Currently, LOCKSS remains financially viable using a "Red Hat" model to sustain itself. A grant a few years back did allow them to increase the amount of programming activity.
Changes in Web architecture, however, necessitate changes in LOCKSS, not least the presence of forms that must be filled to access documents (he suggests that this is done deliberately by some to thwart harvesting) as well as the emergence of AJAX content from some publishers. Other integration desires include Memento as well as Shibboleth for access control by identity rather than by location.
They have also added new polling types to ensure that nodes have identical copies: local, symmetric, proof of possession. Previously, polls looked for disagreement, enabled repairs by agreement. This was remote and asymmetric. Using new metrics, they have shown that the delay between new agreements has dropped from 30 to 10 days.
They are now replacing components of the existing LOCKSS software with open source components, e.g.- Hadoop and OpenWayback. The idea is to make it more scalable, reduce costs, and to contribute to open source technology. To deploy this, it's more difficult than installing a monolithic package, so they are opting to Dockerize the components to make it easier to do.
One Site Among Many: Stanford and Collaborative Technical Development for Web Archiving
Nicholas Taylor, Stanford U
Beyond link rot, there is the issue of content drift, i.e.- content is not stable over time. The link may be fine, but the content has changed. Showed a graph that made visible how serious this issue is.
Most organizations are relying on external providers for Web archiving. There is at least an upward trend in the number of institutions that are doing Web archiving, which is good. Archive-It has contributed to this. As he put it, it's never been easier to Web archive, but the scale of content has made it hard for organizations to do it themselves. Most Web archives are stored internally, i.e.- within the organization.
The Web itself has changed. Less static content, more JS and other dynamic content.
What has Stanford done? They have a Web Archive Portal (SWAP – based on OpenWayback). Pushing those into their general catalogue (SearchWorks, based on Blacklight). They also collect specific sites related to Stanford interests and projects. Their architecture is hybrid. They use Archive-It for capture and push it into their own digital repository.
Wants to push the Web archiving toolkit toward using more APIs and make the stack more actively developed (think I got this right). Better than large systems is to build modular components that can interoperate, not least since none of the large systems cover all facets of the work required. First API they are tackling with their grant is an export API to allow a Web archive to push data out to other repositories. The larger goal is to build a community that roadmaps the creation of further APIs and tools to exploit/deploy them.
Fedora 4
Gave an introduction to Fedora, noting that it is not so much a software application as a system. As a Fedora institution, most of this is familiar to me. Referred to Fedora as middleware (it sits on your storage) and referred to Islandora as a presentation layer. I often tend to think of Fedora as the foundation and Islandora as the middleware, so this is some food for thought.
Core features: linked data platform, Memento versioning, fixity, etc. It is also extensible.
There are pluggable components that use a standardized API, as well as external components. The details of the latter go beyond my technical ken. Scalability tests have shown that one can upload a 1TB file via the REST API.
Random notes
These are some things I heard about that I'll hope to hear/learn more about in the future:
PREFORMA project – EU-funded project to establish standard preservation formats and verification tools for text, graphical, and video objects. Betas available via GitHub.
DMPS – Data Management Planning Systems – Automation of DMP tools, many of which are in development. Currently, they are text documents, but not executable, i.e.- they cannot trigger action in rule-based data management systems. In short, machine readable so that rules and operations can be read from them to ensure/actualize compliance with funding agency requirements. Also referred to it as ADMP (A = Active).
iRODS – Came up in the above talk on DMPS, but have heard about it before, of course. Have never quite wrapped my brain around it or understood if it has potential application in our environment and/or is something we could handle.
AXF – Archive eXchange Format – Oracle uses this, but it's an open standard.
RODA – My question is whether this is a parallel development to Archivematica, or something else entirely. Would it work in our environment, i.e.- with our standards and related tools, or is it "Europeanized"?
Archive-It – Came up numerous times, most clearly in Nicholas Taylor's talk. We are not doing much at all with regard to Web archiving. Should perhaps get started.
← Office holiday gifting
CNI Spring 2016 Notes → | {
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{"url":"https:\/\/bet-kodikos-bonus.com\/amide-formula-jmbmkk\/feit-electric-30-ft-colour-changing-led-outdoor-string-lights-8f854f","text":"Table 1.1 * graph! Calculate and record in lab notebook the [FeSCN2+] in each solution and its absorbance. This condition is expressed in the equilibrium constant Kc. Sto1 Find the initial number of moles of Fe') and SCN) Usc Equation (12) and enter your results into the first two columns of Table 1.2. Thermodynamics and Equilibrium By: Omish Samaroo Introduction The goal of this experiment is to determine the value of an equilibrium constant at different temperatures and use these data to calculate the enthalpy and entropy of reaction. <> Determination of an Equilibrium Constant for the Iron (III) thiocynate Reaction Pre-lab Assignment Before coming to lab: \u2022 Read the lab thoroughly. Determination of an Equilibrium Constant Step 2 Enter the equilibrium value of (FNCS*) from Table 1.1. 17: Chemical Equilibrium and Beer's Law A. Need Help Finishing Lab Report. For a reaction involving aqueous reactants and products, the equilibrium constant is expressed as a ratio between reactant and product concentrations, where each term is raised to the power of its reaction coefficient (Equation \\ref{1}). I (0.310M (0.0200L): : 0.000000 0.6 w 10. The values for [FENCS\") are obtained from the graph of absorbance vs. (FeNCS\" (calibration curve). For the equilibrium in Reaction 5 to exist, there must be some solid PbCl2, present in the system. Write the equilibrium constant expression for the following reaction: Cd (aq) + SCN (aq) = C (SCN)\" (aq) 2. An equilibrium constant can then be determined for each mixture; the average should be the equilibrium constant value for the formation of the FeSCN 2+ ion. Because a large excess of Fe+3 is used, it is reasonable to assume that all of the SCN- is converted to FeSCN2+. Standard Fe(SCN)2+ Concentrations Are: Test Tube 1: 0.00001 Test Tube 2: 0.00002 Test Tube 3: 0.00003 Test Tube 4: 0.00004 Test Tube 5: 0.00005 The expression for the equilibrium constant for a reaction is determined by \u2026 Show Your Work In \u2026 the equilibrium constant expression is as follows: $K_c = \\frac{[C]^c[D]^d}{[A]^a[B]^b}$ To calculate the equilibrium constant (also known as the dissociation constant), the concentrations of each species in the reaction at equilibrium must be measured. Question: Experiment V: Equilibrium Constant Lab Report I. Calibration Curve Record The Absorbance For Each Of The Standard Solutions In The Table Below. 2 0 obj . . <>>> 10 mL of 0.007 M SCN 10 mL of 0.8 M HNO These reagents were mixed and \u2026 University of Illinois at Chicago. Introduction. View desktop site, Determination of an Equilibrium Constant Lab Report Repo 001.10 2020 \u043c \u0442 \u0442 \u043a \u0432 Data Complete Table 1.1. 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L'Eriador est une région de la Terre du Milieu, univers de fiction créé par .
Géographie
L'Eriador est limité par :
à l'est, les Monts Brumeux ;
au nord, la baie glaciale de Forochel ;
à l'ouest, les montagnes d'Ered Luin ;
au sud, la Mitheithel, puis la Gwathló.
Plusieurs fleuves traversent l'Eriador, dans une direction générale Nord-Est / Sud-Ouest : la Lhûn, le Baranduin, la Mitheithel, la Bruinen et la Gwathló. La région est également parsemée de massifs de faible altitude : les collines d'Evendim (), les Hauts du Nord (), les Hauts Lointains (), les Hauts des Galgals (), les Collines du Temps () et les Hauts du Sud ().
La majeure partie de l'Eriador se compose de vastes plaines dégagées. Il subsiste néanmoins quelques bois de petite taille : la Vieille Forêt, les Fourrés des Trolls () ou Eryn Vorn. La région de plaines comprise entre le Baranduin et la Gwathló est appelée Minhiriath, « entre les rivières » en sindarin. Au nord-est de l'Eriador se trouve l'Angmar, une région sinistre, jouxtée au sud par les landes d'Etten (), zone tout aussi peu amicale où vivent notamment des Trolls.
Les établissements peuplés sont rares en Eriador : le principal est la Comté des Hobbits. Parmi les villes humaines, celles fondées par les Dúnedain, Annúminas et Fornost, sont en ruines à la fin du Troisième Âge, et seule Bree et les villages environnants continuent à exister. À l'est, au pied des Monts Brumeux, se trouve Fondcombe, la demeure cachée d'Elrond le Semi-Elfe. Les principales routes traversant la région sont la Grande Route de l'Est () et le Chemin Vert ().
Histoire
L'Eriador est à l'origine couvert de forêts, mais au cours du Second Âge, les Dúnedain déboisent intensivement la région pour alimenter leurs chantiers navals. Après la submersion de Númenor, la majeure partie de l'Eriador est intégrée au royaume d'Arnor fondé par Elendil, partagé par la suite entre les royaumes rivaux de Rhudaur, d'Arthedain et du Cardolan.
Après la Grande Peste de 1636 T.Â. et la ruine du royaume d'Arthedain en 1974 T.Â., l'Eriador devient une région en grande partie désolée et abandonnée, hormis quelques havres de civilisation comme la Comté, Bree ou Fondcombe, que protègent de leur mieux les derniers Dúnedain, surnommés « Rôdeurs ».
Notes et références
Bibliographie
.
Région de la Terre du Milieu
de:Regionen und Orte in Tolkiens Welt#Eriador
lb:Länner a Stied aus Middle-earth#Eriador
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\section{Introduction}
Traditional variational inference (VI) minimizes the ``exclusive'' KL divergence $\mathrm{KL}(q\Vert p)$ between the approximating distribution $q$ and the target $p$. There has been great recent interest in methods to minimize other alpha-divergences, such as the ``inclusive'' KL divergence, $\mathrm{KL}(p\Vert q)$. Some methods employ unbiased gradient estimators \citep{dieng2017chivi, kuleshov2017neural}. These estimators often suffer from a high variance, difficulting optimization \citep{geffner2020difficulty}. Another class of methods estimate a gradient using self-normalized importance sampling \citep{rws, OnIWAE, renyiVI}. While these estimators may control variance, they do so at the cost of some bias. While some positive results have been observed for biased methods (e.g. higher log-likelihoods \citep{renyiVI, dieng2017chivi}), the magnitude of the bias and the effect it has on the distributions they return are not well understood.
In this paper we empirically evaluate biased methods for alpha-divergence minimization.
In particular, we focus on how the bias affects the solutions found, and how this depends on the dimensionality of the problem. Our two main takeaways are (i) solutions returned by these methods appear to be strongly biased towards minimizers of the traditional ``exclusive'' KL-divergence, $\mathrm{KL}(q\Vert p)$. And (ii) in high dimensions, an impractically large amount of computation is needed to mitigate this bias and obtain solutions that actually minimize the alpha-divergence of interest.
Finally, we relate these results to the curse of dimensionality. In high dimensions, it is well known that self-normalized importance sampling often suffers from ``weight degeneracy'' (unless the number of samples used is exponential in the dimensionality of the problem \citep{bugallo2017adaptive, bengtsson2008curse}), resulting in estimates with high bias. We empirically show that weight degeneracy does indeed occur with these estimators in cases where they return highly biased solutions.
\subsection{Estimators considered} \label{sec:estimators}
\textbf{Notation:} $q_\phi$ denotes the variational distribution parameterized by $\phi$. $z_{\phi, k}$ denotes a sample from $q_\phi$ obtained via reparameterization \citep{vaes_welling, doublystochastic_titsias}. $\psi$ denotes the parameters $\phi$ ``protected under differentiation'' (i.e. $\psi = \texttt{stop\_gradient}(\phi)$).
\begin{itemize}
\item For the Renyi alpha-divergence, $\mathrm{R}_\alpha(q||p)$, \citet{renyiVI} proposed the estimator
\[ g_{R_\alpha} = -\sum_{k=1}^K \frac{w_{\alpha, k}}{\sum_{j=1}^K w_{\alpha, j}} \, \nabla_\phi \log \frac{p(x, z_{\phi, k})}{q_\phi(z_{\phi, k})}, \qquad \mbox{where} \qquad w_{\alpha, k} = \left( \frac{p(x, z_{\phi, k})}{q_\phi(z_{\phi, k})} \right)^{1 - \alpha}. \]
This is defined for $\alpha>0$. We use $\alpha=0.5$ in our experiments.
\item For the ``inclusive'' divergence $\mathrm{KL}(p||q)$, the reweighted wake-sleep estimator \citep{rws} (also used in Edward \citep{edward2016}) is given by
\[ g_\mathrm{rws} = \sum_{k=1}^K \frac{w_k}{\sum_{j=1}^K w_j} \, \nabla_\phi \log q_\phi(z_{\psi, k}), \qquad \mbox{where} \qquad w_k = \frac{p(x, z_{\phi, k})}{q_\phi(z_{\phi, k})}.\]
For the same divergence, the ``sticking the landing'' estimator \citep{stickingthelanding} is given by\footnote{This estimator was originally proposed as an estimator for importance weighted variational inference \citep{IWVAE}. \citet{OnIWAE} introduced the view of it being a self-normalized importance sampling estimator for the gradient of $\mathrm{KL}(p||q)$.}
\[ g_\mathrm{stl} = -\sum_{k=1}^K \frac{w_k}{\sum_{j=1}^K w_j} \, \nabla_\phi \log \frac{p(x, z_{\phi, k})}{q_\psi(z_{\phi, k})}, \qquad \mbox{where} \qquad w_k = \frac{p(x, z_{\phi, k})}{q_\phi(z_{\phi, k})}.\]
\item For the chi divergence, $\chi^2(p||q)$, the CHIVI algorithm \citep{dieng2017chivi} uses the estimator
\[ g_\mathrm{chivi} = -\sum_{k=1}^K \left(\frac{w_k}{\max_j w_j}\right)^2 \nabla_\phi \log \frac{p(x, z_{\phi, k})}{q_\phi(z_{\phi, k})}, \qquad \mbox{where} \qquad w_k = \frac{p(x, z_{\phi, k})}{q_\phi(z_{\phi, k})}. \]
(This estimator was used by \citet{dieng2017chivi} in their experiments, but not in their analysis.) For the same divergence, the doubly reparameterized estimator\footnote{It is known that importance weighted VI is equivalent to minimizing the $\chi^2$ divergence in the limit \citep{maddison2017filtering, domke2018importance}. The doubly reparameterized estimator for importance weighted VI was introduced by \citet{doublyrep}, and \citet{OnIWAE} introduced the view of it being a self-normalized importance sampling estimator for the gradient of $\chi^2(p||q)$.} \citep{doublyrep, OnIWAE} is given by
\[ g_\mathrm{drep} = -\sum_{k=1}^K \left(\frac{w_k}{\sum_{j=1}^K w_j}\right)^2 \nabla_\phi \log \frac{p(x, z_{\phi, k})}{q_\psi(z_{\phi, k})}, \qquad \mbox{where} \qquad w_k = \frac{p(x, z_{\phi, k})}{q_\phi(z_{\phi, k})}. \]
\end{itemize}
All of these estimators are asymptotically unbiased in the limit of $K\rightarrow \infty$ except for $g_\mathrm{chivi}$. However, the bias for finite $K$ is not well understood.
\section{Empirical Evaluation}
We now present an empirical evaluation of the estimators described above. We consider two scenarios for the model $p$: a simple Gaussian distribution and logistic regression. In both cases we use Adam \citep{adam} with each of the gradient estimators to minimize the corresponding alpha-divergence, and compare the results obtained against the theoretically optimal ones.
\subsection{Evaluation I: Gaussian Model} \label{sec:gaussian}
\noindent\textbf{Model:} Similarly to \citet{neal2011mcmc}, we set the target $p$ to be a diagonal $d$-dimensional Gaussian with mean zero and variances $\sigma_{p_i}^2 = 0.2 + 9.8 \frac{i}{d}$. So, the variance of the components of $p$ grows linearly from $\sigma_{p_1}^2=0.2$ to $\sigma_{p_d}^2=10$. We ran simulations for dimensionalities $d \in \{10, 100, 1000\}$\vspace{0.1cm}
\noindent\textbf{Variational distribution:} We set $q$ to be a mean-zero isotropic Gaussian with covariance $\sigma_q^2 I$. So, $q$ has a single parameter $\sigma_q$, which we initialize to $\sigma_q^2 = 9$.\vspace{0.1cm}
\noindent\textbf{Optimization details:} We attempt to optimize alpha-divergences by running Adam (step-size $\eta = 0.01$) for $2000$ steps using each of the gradient estimators introduced in Section \ref{sec:estimators}. We repeat this for estimators obtained using $K$ samples, with $K \in \{10, 100, 1000\}$.\vspace{0.1cm}
\noindent\textbf{Baselines:} In this scenario we can compute the optimal $\sigma_q$ to exactly minimize each of $\mathrm{KL}(q\Vert p)$, $\mathrm{KL}(p\Vert q)$, $R_\alpha(q\Vert p)$ and $\chi^2(p\Vert q)$. This gives us a clear way of visualizing the bias induced by each estimator.\vspace{0.1cm}
\noindent\textbf{Results:} Fig.~\ref{fig:target_method_gauss} shows how the parameter $\sigma_q$ evolves as optimization proceeds when using the ``sticking the landing'' estimator $g_{stl}$, which targets the divergence $\mathrm{KL}(p\Vert q)$. For low dimensions ($d = 10$), the optimal value $\sigma_q^*$ is recovered almost exactly as long as $K \geq 100$ samples are used to estimate the gradients. For higher dimensions, the solution is increasingly biased towards the minimizer of $\mathrm{KL}(q||p)$. While this bias can in theory be mitigated by increasing the number of samples $K$ used to estimate the gradients, the number required becomes impractically large in high dimensions.
\begin{figure}[htbp]
\floatconts
{fig:target_method_gauss}
{\caption{\textbf{For high dimensions an impractically large number of samples $K$ is needed to mitigate the estimator's bias.} Optimization results when minimizing $\mathrm{KL}(p||q)$ for the synthetic Gaussian model using the biased gradient estimator $g_{stl}$ obtained using $K$ samples.}}
{\includegraphics[scale=0.3, trim = {0 0 0 0}, clip]{images/opt_klpq_drep_dim10_gauss.pdf}\hspace{0.5cm}
\includegraphics[scale=0.3, trim = {2.1cm 0 0 0}, clip]{images/opt_klpq_drep_dim100_gauss.pdf}\hspace{0.5cm}
\includegraphics[scale=0.3, trim = {2.1cm 0 0 0}, clip]{images/opt_klpq_drep_dim1000_gauss.pdf}
}
\end{figure}
Fig.~\ref{fig:var_dim_gauss} shows that a similar phenomena occurs with all other estimators introduced in Section \ref{sec:estimators}. The plots do not show optimization traces; they show the final $\sigma_q^2$ after 2000 optimization steps as a function of the problem's dimension. (We show raw optimization results for all estimators in Appendix \ref{apd:1}). The same conclusion as the one described above applies for all estimators (except $\g_\mathrm{chivi}$): The methods tend to work well in low dimensions, but return suboptimal solutions that are strongly biased towards minimizers of $\mathrm{KL}(q||p)$ in higher dimensions. Again, while this bias can be mitigated by increasing the number of samples $K$ used to estimate gradients, the value of $K$ required becomes impractically large in high dimensions. ($\g_\mathrm{chivi}$ also yields suboptimal solutions in low dimensions. This is likely because this estimator uses atypical weight normalization and so is not asymptotically unbiased.)
\begin{figure}[htbp]
\floatconts
{fig:var_dim_gauss}
{\caption{\textbf{In high dimensions solutions are strongly biased towards minimizers of $\mathrm{KL}(q||p)$.} Optimization results for all estimators for the synthetic Gaussian model, as a function of the dimensionality of the problem and the number of samples $K$ used to estimate gradients.}}
{\includegraphics[scale=0.3, trim = {0 0 0 0}, clip]{images/finalval_dim_klpq_drep_gauss.pdf}\hspace{0.5cm}
\includegraphics[scale=0.3, trim = {2.7cm 0 0 0}, clip]{images/finalval_dim_klpq_sf_gauss.pdf}\hspace{0.5cm}
\includegraphics[scale=0.3, trim = {2.7cm 0 0 0}, clip]{images/finalval_dim_renyi05_gauss.pdf}\vspace{0.5cm}
\includegraphics[scale=0.3, trim = {0 0 0 0}, clip]{images/finalval_dim_chi2_drep_gauss.pdf}\hspace{0.5cm}
\includegraphics[scale=0.3, trim = {2.2cm 0 0 0}, clip]{images/finalval_dim_chi2_max_gauss.pdf}
}
\end{figure}
We believe that the suboptimality of the solutions returned by biased methods in high dimensions is related to the weight collapse effect (also known as weight degeneracy) suffered by self normalized importance sampling \citep{bengtsson2008curse}. To verify this empirically, we plot the magnitude of the normalized importance weights obtained for different dimensionalities $d$ and number of samples $K$. We observe that the pairs $(d, K)$ for which solutions are highly biased correspond to the cases for which the weight collapse effect is observed (details in Appendix \ref{app:3} and Fig.~\ref{fig:wc} therein).
\subsection{Evaluation II: Logistic Regression}
\noindent\textbf{Model:} Bayesian logistic regression with two datasets: \textit{sonar} ($d= 61$) and \textit{a1a} ($d = 120$).\vspace{0.1cm}
\noindent\textbf{Variational distribution:} We set $q$ to be a diagonal Gaussian, with mean $\mu_q$ and variance $\sigma_q^2$ (vectors of dimension $d$), with components initialized to $\mu_{q_i} = 0$ and $\sigma_{q_i}^2 = 9$. (We parameterize the variance using the log-scale parameters.)\vspace{0.1cm}
\noindent\textbf{Optimization details:} We attempt to optimize alpha-divergences by running Adam (step-size $\eta = 0.01$) for $5000$ steps using each of the gradient estimators introduced in Section \ref{sec:estimators}. We repeat this for estimators obtained using $K$ samples, with $K \in \{10, 100, 1000\}$.\vspace{0.1cm}
\noindent\textbf{Baselines:} We compare against the optimal parameters $(\mu_q^*, \sigma_q^*)$ that minimize $\mathrm{KL}(p||q)$. While these cannot be computed in closed form, we approximate them by minimizing $\mathrm{KL}(p||q)$ using the algorithm proposed by \citet{MarkovianSC}\footnote{The algorithm's main idea involves minimizing $\mathrm{KL}(p||q)$ using samples from $p$ obtained via MCMC. In our case we use Stan \citep{carpenter2017stan} to get reliable samples, making sure to run multiple chains and checking several convergence criteria, such as the value of $\hat R$.}. Again, having these parameters provides a clear way of visualizing the effect of using biased gradient estimates.\vspace{0.1cm}
\noindent\textbf{Results:} Fig.~\ref{fig:opt_log_reg_drep} shows optimization results for the estimator $g_\mathrm{stl}$, which targets $\mathrm{KL}(p\Vert q)$. It can be observed that, for the \textit{sonar} dataset ($d = 61$), distributions that attain near-optimal performance are obtained using gradient estimates computed with $K \geq 100$ samples. In contrast, for the \textit{a1a} dataset ($d = 120$), all values of $K$ tested lead to significantly biased and suboptimal solutions. (Though, as expected, increasing the number of samples $K$ reduces the suboptimality gap.)
\begin{figure}[htbp]
\floatconts
{fig:opt_log_reg_drep}
{\caption{\textbf{For the dataset with higher dimensionality the algorithm returns biased solutions that are suboptimal regardless of the number of samples $K$ used.} The plots show optimization results for minimizing $\mathrm{KL}(p\Vert q)$ for the logistic regression model using the estimator $g_{stl}$ obtained with $K$ samples. The y-axis in the plots show the true $\mathrm{KL}(p\Vert q)$ (up to the additive constant $c = \log p(x)$ -- which can be estimated using samples from $p(z|x)$, obtained using Stan \citep{carpenter2017stan}). The seemingly strange behavior of optimization traces is not due to bad optimization hyperparameters, but to the bias of the gradient estimator.} }
{
\includegraphics[scale=0.35, trim = {0 0 0 0}, clip]{images/logistic-sonar_opt_klpq_klpq_drep_logreg.pdf}\hspace{0.5cm}
\includegraphics[scale=0.35, trim = {1.5cm 0 0 0}, clip]{images/logistic-a1a_opt_klpq_klpq_drep_logreg.pdf}
}
\end{figure}
Fig.~\ref{fig:params_stl_logreg} shows how the optimal parameters $(\mu_q^*, \sigma_q^*)$ compare against the parameters obtained by optimizing using the biased gradient estimator $g_\mathrm{stl}$. We observe two things. First, the optimal mean parameters are well-recovered for both datasets regardless of the number of samples $K$ used to estimate gradients\footnote{This is probably because optimizing $\mathrm{KL}(p||q)$ and $\mathrm{KL}(q||p)$ gives nearly the same mean parameters on these problems.}. Second, the scale parameters recovered are biased towards minimizers of $\mathrm{KL}(q||p)$. For the \textit{sonar} dataset $(d=61)$, this bias can be removed by increasing the number of samples $K$ used to estimate gradients. However, for the \textit{a1a} dataset $(d = 120)$, increasing $K$ to 1000 provides only a tiny improvement, suggesting a huge value for $K$ would be needed.
Results for all other estimators are similar to the ones shown in this section for $\g_\mathrm{stl}$. We show them in Appendix \ref{app:2}.
\begin{figure}[htbp]
\floatconts
{fig:params_stl_logreg}
{\caption{\textbf{In high dimensions optimizing with the biased estimator leads to solutions strongly biased towards minimizers of $\mathrm{KL}(q\Vert p)$, and an impractically large $K$ is needed to mitigate this effect.} Results for the logistic regression model with both datasets, \textit{sonar} $(d = 61)$ and \textit{a1a} $(d = 120)$. The plots show the mean and variance of each component of the variational distribution $q$ obtained by optimizing with the gradient estimator $g_\mathrm{stl}$ with $K$ samples. Components are sorted to facilitate visualization.} }
{\includegraphics[scale=0.363, trim = {0 2cm 0 0}, clip]{images/logistic-sonar_meanparams_klpq_drep_logreg.pdf}\hspace{0.5cm}
\includegraphics[scale=0.35, trim = {1.3cm 2cm 0 0}, clip]{images/logistic-a1a_meanparams_klpq_drep_logreg.pdf}\vspace{0.25cm}
\includegraphics[scale=0.363, trim = {0 0 0 1.15cm}, clip]{images/logistic-sonar_varparams_klpq_drep_logreg.pdf}\hspace{0.5cm}
\includegraphics[scale=0.35, trim = {1.2cm 0 0 1.15cm}, clip]{images/logistic-a1a_varparams_klpq_drep_logreg.pdf}
}
\end{figure}
\section{Conclusions}
All gradient estimators analyzed are asymptotically unbiased (except $g_\mathrm{chivi}$). This means that, if a large enough number of samples $K$ is used to estimate gradients, these methods are guaranteed to return near-optimal solutions. In practice, however, we observe that even for very simple problems, the value of $K$ needed is typically very large.
Interestingly, solutions returned by these methods appear to be biased towards minimizers of $\mathrm{KL}(q||p)$. Upon close examination, it is not obvious why this should be true and to the best of our knowledge no theoretical support for this behavior is known. We find this surprising and consider it to be an appealing property of these methods: Even when they fail to minimize the target alpha-divergence, they do something ``reasonable'', i.e. minimize the traditional divergence $\mathrm{KL}(q||p)$.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 576 |
Mitinskaya () is a rural locality (a village) in Tarnogskoye Rural Settlement, Tarnogsky District, Vologda Oblast, Russia. The population was 5 as of 2002.
Geography
Mitinskaya is located 14 km south of Tarnogsky Gorodok (the district's administrative centre) by road. Podgornaya is the nearest rural locality.
References
Rural localities in Tarnogsky District | {
"redpajama_set_name": "RedPajamaWikipedia"
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\section{Introduction}
It has been known for a while that among the solutions to Einstein's vacuum field equations $\hat{R}_{\mu \nu} = \lambda\,\hat{g}_{\mu \nu}$
with positive cosmological constant $\lambda$ on manifolds with space-sections diffeomorphic
to an orientable, compact 3-manifold
$S$ there is an open (in terms of Sobolev norms on Cauchy data) subset of solutions which are future asymptotically simple in the sense of Penrose \cite{penrose:1965}, i.e. the solutions admit the construction of a conformal boundary ${\cal J}^+$ at their infinite time-like future which is $C^{\infty}$ if the solutions are $C^{\infty}$ and is of correspondingly lower smoothness otherwise (see \cite{friedrich:beyond:2015} for more details and references). This property generalises to the Einstein-$\lambda$ equations coupled to conformally covariant matter field equations with trace free energy momentum tensor. In \cite{friedrich:1991} this has been discussed in detail
for the Maxwell and the Yang-Mills equations, where a procedure has been laid out which applies, possibly which some modifications in specific cases, to other such field equations (see \cite{luebbe:valiente-kroon:2013} for a recent example).
Matter fields with energy momentum tensors which are not trace free were generally expected to lead to difficulties in the construction of reasonably smooth conformal boundaries.
(The emphasis here is on results about the evolution problem, we are not talking about geometric studies near conformal boundaries which postulate properties of energy momentum tensors convenient for their analysis).
It has recently been observed, however, that this need not be the true \cite{friedrich:massive fields}.
In the case of the Einstein-Klein-Gordon equations the conformal field equations with suitably transformed matter field imply evolutions system which are hyperbolic, irrespective of the sign of the conformal factor, if the mass and the cosmological constants are related by the equation $m^2 = \frac{2}{3}\,\lambda$. If this condition is imposed a fairly direct calculation shows that the equation for the rescaled scalar field becomes regular where the conformal factor goes to zero. However, that the conformal equations for the geometric fields become regular in this limit is far from immediate and, as in the case discussed in the following, came as a surprise after various attempts to cast the singular equations into a form which would allow one to draw conclusions about the precise asymptotic behaviour of the solutions
in the presence of singularities.
Leaving aside the questions about the significance of this particular result, the present article is concerned with the analysis of another matter model with non-vanishing trace of the energy momentum tensor. We study in detail the future asymptotic behaviour of solutions to the Einstein-$\lambda$-dust equations.
In a recent article Had$\check{z}$i\'c and Speck have shown that the FLRW solutions
to the Einstein-$\lambda$-dust equations with underlying manifolds of the form $\mathbb{R} \times \mathbb{T}^3$ are future stable, i.e. slightly perturbed FLRW data on $ \mathbb{T}^3$ develop into solutions to the Einstein-$\lambda$-dust equations whose causal geodesics are future complete \cite{hadzic-speck-2015}. The authors use the method proposed in \cite{friedrich:hyperbolic} to control the evolution of a general wave gauge in terms of its gauge source functions. As emphasized in \cite{friedrich:hyperbolic}, it is clear that (under fairly weak smoothness assumptions) any coordinate system can in principle be controlled in terms of its gauge source functions and suitable initial data. But finding gauge source functions which are useful in a specific problem is quite a delicate matter.
The authors manage to identify gauge source functions which allow them to derive estimates that give control on the long time evolution of their solutions (see \cite{ringstroem:2008} for another such case).
It is, however, quite a different question whether the gauge so established lends itself to analyzing the asymptotic behaviour of solutions in detail and to deciding, for instance, whether the differentiable as well as the conformal structure of the solutions admit simultaneously extensions of some smoothness to (future) time-like infinity as required by asymptotic simplicity.
FLRW solutions are known to be future asymptotically simple (see section \ref{FLRW-sols}).
This may be expected to be is just an artifact of the high symmetry
requirements which imply local conformal flatness and hypersurface orthogonality of the flow field.
The present study grew out of attempts to understand what may go wrong under more general assumptions and what kind of obstruction to the asymptotic smoothness of the conformal structure may possibly arise from the presence of a non-vanishing energy density $\hat{\rho}$.
In the article \cite{friedrich98} have been derived hyperbolic evolution equations from the Einstein-dust equation in a geometric gauge based of the flow field. The following analysis may be seen as a conformal version of this discussion. After presenting the Einstein-$\lambda$-dust equations in section \ref{Einstein-lambda-dust}, we derive in section \ref{conf-field-equ} the conformal field equations and suitably transformed matter field equations. It turns out that two equations of the system are singular in the sense that there occur factors of the form $\Omega^{-1}$ on the right hand side, where $\Omega$ is the conformal factor which is positive on the {\it physical} solution space-time and relates the {\it physical} metric $\hat{g}_{\mu\nu}$ there to the {\it conformal} metric $g_{\mu\nu}$ by $g_{\mu\nu} = \Omega^2\,\hat{g}_{\mu\nu}$. Since things are to be arranged such that
$\Omega \rightarrow 0$ at future time-like infinity, where we want to understand the precise nature of the solutions, there arise problems. One of the singularities, namely the one in the transformed (geodesic) flow field equation, was to be expected. Much more serious is a singularity in the equation for the rescaled conformal Weyl tensor
$W^{\mu}\,_{\nu \lambda \rho} = \Omega^{-1}\,C^{\mu}\,_{\nu \lambda \rho}[g]$, which plays a central role in the system. The singularities carry, however, interesting geometric information.
They imply that the (so far formally defined) set $\{\Omega = 0\}$ can only define a smooth conformal boundary of the solution space-time if the flow lines approach this set orthogonally.
Thus, if one wants to approach the problem in terms of estimates, one has to aim for sufficient control to be able to define simultaneously a conformal boundary at time-like infinity, if admitted by the solution at all, and correspondingly control the behaviour of the flow lines.
In the present article we try to exploit the conformal properties of the system in the most direct way. In section \ref{the-reg-rel} it is shown that due to the specific form of the energy momentum tensor for dust the geodesics tangent to the flow field can be identified after a parameter transformation with curves underlying certain conformal geodesics. Since conformal geodesics are invariants of the conformal structure, this opens the possibility to define a gauge which extends regular across the conformal boundary ${\cal J}^+ = \{\Omega = 0\}$ if the latter can indeed be attached in a smooth way to the solution manifold (on which $\Omega > 0$, of course). It turns out that this gauge implies a certain {\it regularising relation} which proves useful in three different contexts. Its first important merit is to render the conformal field equations regular.
In section \ref{hyp-red-equ} it is shown that the conformal field equations imply a
{\it hyperbolic reduced system of evolution equations} which can make sense up to and beyond the conformal boundary at time-like infinity (if it exists). This system is not obtained immediately.
The regularizing relation leads to a system which is hyperbolic where
$\Omega > 0$ but becomes singular where $\Omega \rightarrow 0$.
A further regularization is performed to obtain a system
which is hyperbolic independent of the sign of the conformal factor.
In section \ref{subs-syst} is derived a subsidiary system which implies that solutions to the hyperbolic evolution system for data that satisfy the constraints on a given Cauchy hypersurface (with respect to the metric provided by the evolution system) will satisfy in fact the complete system of conformal field equations. This closes the hyperbolic reduction argument.
To obtain complete information on the class of future asymptotically simple solutions
to the Einstein-$\lambda$-dust solutions we characterize in Lemma \ref{free-data-on-scri}
the possible {\it asymptotic end data} which may be prescribed on the conformal boundary
${\cal J}^+ = \{\Omega = 0\}$ (assumed to be 3-dimensional, orientable, compact) of a solution that admits the construction of such a boundary with sufficient smoothness. As observed already
in \cite{friedrich:1986a} in the vacuum case, the constraints reduce on ${\cal J}^+$
to a linear system of equations. Remarkably, there is a case where the problem of solving the constraints simplifies even further. In the case where the density
$\hat{\rho}$ is positive everywhere certain fields can be prescribed completely freely on ${\cal J}^+$ and
the rest follows by algebra and taking derivatives.
There is no need to solve any differential equation at all
(but see the remarks following Lemma \ref{free-data-on-scri}).
The reduced system of evolution equations is used in section \ref{ex-strong-stab} to derive our main results. Being based on hyperbolic equations, a completely detailed statement of the results should give information about Sobolev norms. Since we only use properties of symmetric hyperbolic systems which can be found
in the literature at various places and because we are mainly interested in solutions of class $C^{\infty}$, we refrain from listing Sobolev indices. We would consider these only be of interest
if the weakest possible smoothness assumptions were needed in the context of some concrete problems.
\begin{theorem}
\label{main-result}
Let $S$ be a smooth, orientable, compact 3-manifold, assume $\lambda > 0$, and denote by
${\cal A}_{\lambda, S}$ the set of standard Cauchy data on $S$
to the Einstein-$\lambda$-dust equations with energy density $\hat{\rho} \ge 0$. Then
\vspace{.1cm}
\noindent
(i) There is an open (with respect to suitable Sobolev norms) subset ${\cal B}_{\lambda, S}$ of data in ${\cal A}_{\lambda, S}$
which develop into solutions that admit the construction of conformal boundaries in their infinite time-like future which are of class $C^{\infty}$ if the data are of class $C^{\infty}$ and of correspondingly lower differentiability if the data are of lower differentiability.
\vspace{.1cm}
\noindent
(ii) The solutions which develop from data in ${\cal B}_{\lambda, S}$ are completely parametrized by the asymptotic end data on $S$ (specified in Lemma \ref{free-data-on-scri}) which correspond to the data induced on the future conformal boundaries ${\cal J}^+$ of the solutions.
\end{theorem}
The case of the Nariai solution, an explicit, geodesically complete solution to the Einstein-$\lambda$-dust equations
with $\hat{\rho} = 0$ that admits not even a patch of a smooth conformal boundary (see \cite{friedrich:beyond:2015}), shows that our reduced evolution system is by itself not sufficient to ensure the existence of a smooth conformal boundary. Some extra information on the Cauchy data is required.
Because the FLRW solutions do admit a smooth conformal future boundary one could consider data close to FLRW data.
Following instead the arguments introduced in \cite{friedrich:1986b} and \cite{friedrich:1991},
a much larger class of suitable reference solutions (which includes the FLRW solutions)
will be constructed in section \ref{ex-strong-stab} by solving a backward Cauchy problem
for the reduced equations with asymptotic end data that are given on a 3-manifold $S$ which in the end will represent the future conformal boundary ${\cal J}^+ = \{ \Omega = 0\}$ of the physical space-time defined on the set $\{\Omega > 0\}$.
In a second step we consider the `physical' standard Cauchy data that are induced by one of these solutions on a `physical' Cauchy hypersurface. It is shown that under sufficiently small perturbations of these data the resulting solutions are {\it strongly stable} in the sense that the smooth extensibility of their conformal structures at future time-like infinity is preserved. This makes use of the fact that a future asymptotically simple solution admits a conformal representation that extends as a smooth solution to the conformal Einstein-$\lambda$-dust equations beyond the conformal boundary into a domain where $\Omega < 0$. The strong stability result follows then as a consequence of the well known Cauchy stability property of hyperbolic equations and the fact that the equation themselves ensure that the set of points where $\Omega = 0$ defines a smooth space-like hypersurface.
Though they lead to the same sets of solutions in the end, it is of interest to distinguish the two different ways of looking at the solutions.
In the construction of the reference solutions some features of asymptotic simplicity are
{\it built in from the start} by using asymptotic end data. In the stability result, however, asymptotic simplicity for the perturbed solution is {\it deduced} as a consequence of the conformal properties of the equations and the reference solution.
In contrast to the approach of \cite{hadzic-speck-2015}, which concentrates on deriving suitable estimates, the emphasis is put in this article on the analysis of the field equations and the explicit
use of their conformal properties.
While the conformal equations may lead to serious difficulties when the conformal structure of the solutions is intrinsically not well behaved at time-like infinity, they give results which are sharp and complete if the conformal structure extends smoothly and only the standard energy estimates for symmetric hyperbolic systems are needed.
Moreover, the information obtained on the equations is in that case of considerable practical interest. The reduced evolution system provides the possibility to calculate numerically - on a finite grid - future complete solutions to Einstein's field equations, including the details of their asymptotic behaviour. In the Einstein-$\lambda$ case this has been successfully demonstrated by the work of Beyer (see \cite{beyer:2008} and the references given there).
Besides the one analysed in \cite{friedrich:massive fields} this is the second example that illustrates that even in cases in which the energy momentum tensor is not trace free the conformal field equations with $\lambda > 0$ and suitably rescaled matter fields can imply hyperbolic evolution equations that are well defined up to and beyond the future time-like infinity of the physical solutions. The two cases are quite different but the
results suggest that the analysis of the asymptotic conformal structure in the presence of matter fields can be more useful than expected.
The possibility to extend solutions to the conformal field equations into a domain in which $\Omega < 0$, where they define another solution to the original equations (see section \ref{ex-strong-stab}), has been used here only as a technical device in the stability argument leading to
Theorem \ref{main-result}. Whether it is of any significance in the context of Penrose's proposal of {\it conformal cyclic cosmologies}
\cite{penrose:2011} is a question not discussed here.
\section{The Einstein-$\lambda$-dust system}
\label{Einstein-lambda-dust}
\vspace{.5cm}
The Einstein-Euler system with cosmological constant $\lambda$ consists of the Einstein equations
\begin{equation}
\label{einst}
\hat{R}_{\mu \nu} - \frac{1}{2}\,\hat{R}\,\hat{g}_{\mu \nu}
+ \lambda\,\hat{g}_{\mu \nu} = \kappa\,\hat{T}_{\mu \nu},
\end{equation}
for a Lorentz metric $\hat{g}_{\mu \nu}$ on a four-dimensional manifold $\hat{M}$ with an energy momentum tensor of a simple ideal fluid
\begin{equation}
\label{spfl}
\hat{T}_{\mu \nu} = (\hat{\rho} + \hat{p})\,\hat{U}_{\mu}\,\hat{U}_{\nu} +
\hat{p}\,\hat{g}_{\mu \nu}.
\end{equation}
Here $\hat{U}^{\mu}$ is the future directed time-like flow vector field, normalized so that
$\hat{U}_{\mu}\,\hat{U}^{\mu} = - 1$, and
$\hat{\rho}$ and $\hat{p}$ denote the total energy density and the pressure as measured
by an observer moving with the fluid. The equations require the relation
$\hat{\nabla}^{\mu}\,\hat{T}_{\mu \nu} = 0$,
which is equivalent to the system consisting of the equations
\begin{equation}
\label{2pfleq}
(\hat{\rho} + \hat{p})\,\hat{U}^{\mu}\,\hat{\nabla}_{\mu}\,\hat{U}_{\nu}
+ \{\hat{U}_{\nu}\,\hat{U}^{\mu}\,\hat{\nabla}_{\mu} + \hat{\nabla}_{\nu}\}\,\hat{p} = 0,
\end{equation}
\begin{equation}
\label{1pfleq}
\hat{U}^{\mu}\,\hat{\nabla}_{\mu}\,\hat{\rho} + (\hat{\rho} + \hat{p})\,\hat{\nabla}_{\mu}\,\hat{U}^{\mu} = 0.
\end{equation}
These equations must be implemented by an equation of state.
In the following we set $\kappa = 1$, assume $\lambda > 0$, and consider solutions on manifolds diffeomorphic to $\hat{M} = \mathbb{R} \times S$ where $S$ is a compact (without boundary), orientable
$3$-manifold which specifies the topology of the time slices. We will be interested in the case where $\hat{p} = 0$ throughout, referred to as
{\it pressure free matter} or, shortly, as {\it dust}. It is supposed that $\hat{\rho}$ does not vanish identically and satisfies
\begin{equation}
\label{pos-en}
\hat{\rho} \ge 0 \quad \mbox{on} \quad \hat{M}.
\end{equation}
Equation (\ref{2pfleq}) reduces then to
$\hat{\rho}\,\,\hat{U}^{\mu}\,\hat{\nabla}_{\mu}\,\hat{U}^{\nu} = 0$.
This will be satisfied without condition on $\hat{U}^{\mu}$ on sets where
$\hat{\rho} = 0$
and implies that the flow is geodesic where $\hat{\rho} \neq 0$. We require
$\hat{U}^{\mu}$ to be geodesic everywhere.
The system to be considered consists then of (\ref{einst}),
\begin{equation}
\label{dust}
\hat{T}_{\mu \nu} = \hat{\rho}\,\,\hat{U}_{\mu}\,\hat{U}_{\nu},
\end{equation}
\begin{equation}
\label{2dust}
\quad \quad \,
\hat{U}^{\mu}\,\hat{\nabla}_{\mu}\,\hat{U}^{\nu} = 0, \quad
\quad \hat{U}_{\mu}\,\hat{U}^{\mu} = - 1,
\end{equation}
\begin{equation}
\label{1dust}
\hat{\nabla}_{\mu}\,(\hat{\rho}\,\,\hat{U}^{\mu}) = 0. \quad \quad \quad \quad
\quad \,\,\,
\end{equation}
\vspace{.3cm}
Let $\hat{S}$ be a hypersurface in $\hat{M}$ which is space-like for $\hat{g}_{\mu\nu}$ and denote by $\hat{n}^{\mu}$ the future directed normal of $\hat{S}$ normalized by $\hat{n}_{\mu}\,\hat{n}^{\mu} = - 1$. Let coordinates $x^{\mu}$ be given near $\hat{S}$ so that
$\hat{S} = \{x^0 = 0\}$ and the $x^{\alpha}$, $\alpha, \beta = 1, 2, 3$, are local coordinates on $\hat{S}$. Denote by
$\hat{h}_{\alpha \beta}$, $\hat{\kappa}_{\alpha \beta}$
the first and the second fundamental form induced on $\hat{S}$ by $\hat{g}_{\mu \nu}$ and by
$\hat{h}_{\mu}\,^{\nu} = \hat{g}_{\mu}\,^{\nu} + \hat{n}_{\mu}\,\hat{n}^{\nu}$ the orthogonal projector onto the tangent spaces of $\hat{S}$. Equations
(\ref{2dust}), (\ref{1dust}) are evolution equations for $\hat{U}^{\mu}$ and $\hat{\rho}$.
Equation (\ref{einst}) induces with (\ref{dust}) on $\hat{S}$ the constraints
\[
0 = R[\hat{h}] - \hat{\kappa}_{\alpha \beta}\,\hat{\kappa}^{\alpha \beta} + (\hat{\kappa}_{\alpha}\,^{\alpha})^2 - 2\,\lambda - 2\,\hat{n}^{\mu}\,\hat{n}^{\nu}\,
\hat{T}_{\mu \nu},
\]
\[
0 = \hat{D}_{\beta}\,\hat{\kappa}_{\alpha}\,^{\beta} - \hat{D}_{\alpha}\,\hat{\kappa}_{\beta}\,^{\beta}
- \hat{n}^{\mu}\,\hat{h}_{\alpha}\,^{\nu}\,\hat{T}_{\mu \nu}.
\]
Setting $a = - \hat{n}^{\mu}\,\hat{U}_{\mu} > 0$,
$\hat{u}_{\mu} = \hat{h}_{\mu}\,^{\nu}\,\hat{U}_{\nu}$, so that
\[
\hat{U}_{\mu} = a\,\hat{n}_{\mu} + \hat{u}_{\mu} \quad \mbox{with} \,\,\,
- 1 = - a^2 + \hat{u}_{\beta}\,\hat{u}^{\beta}
\quad \mbox{where} \quad
\hat{u}_{\beta}\,\hat{u}^{\beta} = \hat{h}^{\beta \gamma}\,\hat{u}_{\beta}\,\hat{u}_{\gamma},
\]
the constraints take the form
\begin{equation}
\label{hat-Ham-constr}
0 = R[\hat{h}] - \hat{\kappa}_{\alpha \beta}\,\hat{\kappa}^{\alpha \beta}
+ (\hat{\kappa}_{\alpha}\,^{\alpha})^2 - 2\,\lambda
- 2\,\hat{\rho}\,(1 + \hat{u}_{\alpha}\,\hat{u}^{\alpha}),
\end{equation}
\begin{equation}
\label{hat-mom-constr}
0 = \hat{D}_{\beta}\,\hat{\kappa}_{\alpha}\,^{\beta} - \hat{D}_{\alpha}\,\hat{\kappa}_{\beta}\,^{\beta}
+ \hat{\rho}\,\sqrt{1 + \hat{u}_{\beta}\,\hat{u}^{\beta}}\,\,\hat{u}_{\alpha}.
\end{equation}
\vspace{.3cm}
It has been shown in \cite{friedrich98}
how to derive from equations (\ref{einst}), (\ref{dust}), (\ref{2dust}), (\ref{1dust})
a symmetric hyperbolic evolution system of equations for all unknowns in a gauge based on the flow vector field $\hat{U}$.
Given $\lambda > 0$ and a sufficiently smooth initial data set
\begin{equation}
\label{phys-in-data-set}
(\hat{S}, \,\hat{h}_{\alpha \beta}, \,\hat{\kappa}_{\alpha \beta}, \,\hat{u}^{\alpha}, \,\hat{\rho}),
\end{equation}
satisfying (\ref{hat-Ham-constr}),
(\ref{hat-mom-constr}) with $\hat{h}_{\alpha \beta}$ a Riemannian metric and
$\hat{\rho} \ge 0$, the evolution system
can be used to construct
a globally hyperbolic solution $(\hat{M},\,\hat{g}_{\mu\nu},\,\hat{U}^{\mu}, \,\hat{\rho})$
to the Einstein-dust equations with cosmological constant $\lambda$ into which the initial data set is isometrically embedded so that $\hat{S}$ represents after an identification a space-like Cauchy hypersurface for
$(\hat{M},\,\hat{g}_{\mu\nu})$. The manifold $\hat{M}$ will then be ruled by the geodesics tangent to $\hat{U}^{\mu}$. The ODE's
\[
\hat{U}^{\mu}\hat{\nabla}_{\mu}\,\hat{\rho} + \hat{\rho}\,\hat{\nabla}_{\mu}\,\hat{U}^{\mu} = 0,
\]
along the geodesics tangent to $\hat{U}^{\mu}$
ensure that $\hat{\rho} > 0$ or $= 0$ along a given geodesic, depending on whether this relation is satisfied at the point where the geodesic intersects $\hat{S}$. Thus
$\hat{\rho} \ge 0$ will hold on $\hat{M}$.
For smooth initial data the evolution system given in \cite{friedrich98} provides a smooth solution in coordinates $x^0 = t$, $x^a$ so that $<dx^a, \hat{U}> \,= 0$, $<dt, \hat{U}> \,= 1$, whence $\hat{U} = \partial_t$. The initial hypersurfac is given by $\hat{S} = \{t = t_*\}$ for some fixed value $t_*$, the metric is of the form
\begin{equation}
\label{phys-coords}
\hat{g} = - (a\,dt)^2 + h_{\alpha \beta}\,(\hat{u}^{\alpha}\,dt +dx^{\alpha})\,
(\hat{u}^{\beta} \,dt+dx^{\beta})
\quad \mbox{on} \quad \hat{M},
\end{equation}
the future directed $\hat{g}$-unit normal to $\hat{S}$ is given by
\begin{equation}
\label{phys-unit-normal}
\hat{n}^{\mu} = \frac{1}{a}(\delta^{\mu} - \hat{u}^{\mu}) \quad \mbox{with shift vector field
$\hat{u}^{\mu}$ so that} \quad \hat{u}^0 = 0,
\end{equation}
and the lapse function $a$ satisfies $- 1 = \hat{g}(\hat{U}, \hat{U})
= - a^2 + h_{\alpha \beta}\,\hat{u}^{\alpha}\,\hat{u}{\beta}$.
If $\hat{U}$ is hypersurface orthogonal we can assume that $a = 1$, $\hat{u}^{\alpha} = 0$ and
the coordinates define a Gauss system. This will not necessarily be assumed in this article.
The questions to be analyzed in the following asks whether there exist a reasonably large set of data for which the solutions can be extended to become future complete, so that $t$ takes values in
$[t_*, \infty[$, and whether these solutions allow us to give a sharp and detailed description of the
asymptotic behaviour of the conformal structure in the expanding direction, where
$t \rightarrow \infty$.
\section{The metric conformal field equations}
\label{conf-field-equ}
Let $\Omega$ denote a positive {\it conformal factor} on $\hat{M}$ and
$g_{\mu \nu} = \Omega^2\,\hat{g}_{\mu\nu}$ the {\it rescaled metric}.
We shall in the following consider the tensor fields
\begin{equation}
\label{1-unknowns}
\Omega, \quad s = \frac{1}{4}\,\nabla_{\mu}\nabla^{\mu}\,\Omega + \frac{1}{24}\,\Omega\,R[g],
\quad
L_{\mu\nu} = \frac{1}{2}\left(R_{\mu\nu}[g] - \frac{1}{6}\,R[g]\,g_{\mu\nu}\right),
\end{equation}
\begin{equation}
\label{2-unknowns}
W^{\mu}\,_{\eta \nu \lambda} = \Omega^{-1}\,C^{\mu}\,_{\eta \nu \lambda}[g],
\end{equation}
where $\nabla_{\mu}$ denotes the Levi-Civita connection of $g$ and
the last two fields denote the Schouten and the rescaled conformal Weyl tensor of $g_{\mu\nu}$ respectively. Moreover, we shall consider the {\it conformal matter fields}
\[
U_{\mu} = \Omega\,\hat{U}_{\mu}, \quad \quad \rho = \Omega^{-3}\,\hat{\rho}.
\]
The vector fields $U^{\mu} = g^{\mu\nu}\,U_{\nu}$ and
$\hat{U}^{\mu} = \hat{g}^{\mu\nu}\,\hat{U}_{\nu}$ are then related by
\[
U^{\mu} = \Omega^{-1}\hat{U}^{\mu} \quad \mbox{so that} \quad
g(U, U) = \hat{g}(\hat{U}, \hat{U}) = - 1.
\]
The tensor fields above satisfy the system of {\it conformal field equations}
(see \cite{friedrich:1991}, \cite{friedrich:massive fields})
\begin{equation}
\label{coord-alg-equ}
6\,\Omega\,s - 3\,\nabla_{\eta}\Omega\,\nabla^{\eta}\Omega -
\lambda = - \frac{1}{4}\,\hat{T},
\end{equation}
\begin{equation}
\label{coord-Omega-equ}
\nabla_{\mu}\,\nabla_{\nu}\Omega + \,\Omega\,L_{\mu\nu} - s\,g_{\mu\nu}
= \frac{1}{2}\,\Omega\,T^*_{\mu\nu},
\end{equation}
\begin{equation}
\label{coord-s-equ}
\nabla_{\mu}\,s + \nabla^{\eta}\Omega\,L_{\eta\mu}
= \frac{1}{2}\,\nabla^{\eta}\Omega\,T^*_{\eta \mu}
- \frac{1}{24\,\Omega}\,\nabla_{\mu}\,\hat{T},
\end{equation}
\begin{equation}
\label{coord-L-equ}
\nabla_{\nu}\,L_{\lambda \eta}
- \nabla_{\lambda}\,L_{\nu \eta} -
\nabla_{\mu}\Omega\,\,W^{\mu}\,_{\eta \nu \lambda}
= 2\,\hat{\nabla}_{[\nu}\,\hat{L}_{\lambda] \eta},
\end{equation}
\begin{equation}
\label{coord-W-equ}
\nabla_{\mu}\,W^{\mu}\,_{\eta \nu \lambda}
= 2\,\Omega^{-1}\,\hat{\nabla}_{[\nu}\,\hat{L}_{\lambda] \eta}.
\end{equation}
The right hand sides are determined by the trace
\begin{equation}
\label{coord-T-trace}
\hat{T} = \hat{g}^{\eta \mu}\,\hat{T}_{\eta \mu} = - \hat{\rho} = - \Omega^3\,\rho,
\end{equation}
and the trace free part
\begin{equation}
\label{coord-T-trace-free-part}
T^*_{\eta \mu} =
\hat{\rho}\left(\hat{U}_{\eta}\,\hat{U}_{\mu} + \frac{1}{4}\,\hat{g}_{\eta \mu}\right) =
\Omega\,\rho \left(U_{\eta}\,U_{\mu} + \frac{1}{4}\,g_{\eta \mu}\right),
\end{equation}
of the energy momentum tensor (\ref{dust}) and the {\it physical Schouten tensor} $\hat{L}_{\mu \nu}$, which
takes with our energy momentum tensor, the field equations, and the rescaled fields the form
\begin{equation}
\label{hat-L-form}
\hat{L}_{\mu \nu} =
\frac{1}{6}\,(\hat{\rho} + \lambda)\,\hat{g}_{\mu \nu} + \frac{1}{2}\,\hat{\rho}\,\hat{U}_{\mu}\,\hat{U}_{\nu}
= \frac{1}{6}\,\lambda\,\hat{g}_{\mu \nu}
+ \Omega\,\rho\left(\frac{1}{2}\,U_{\mu}\,U_{\nu} + \frac{1}{6}\,g_{\mu \nu}\right).
\end{equation}
Taking into account the transformation law of the connection coefficients under conformal rescaling this gives
\[
2\,\hat{\nabla}_{[\nu} \hat{L}_{\lambda] \eta}
= \hat{\nabla}_{[\nu} \hat{\rho} \,\,\hat{U}_{\lambda]} \,\hat{U}_{\eta}
+ \frac{1}{3}\,\hat{\nabla}_{[\nu} \hat{\rho}\,\,\hat{g}_{\lambda] \eta}
+ \hat{\rho}\,(\hat{\nabla}_{[\nu}\hat{U}_{\lambda]} \,\hat{U}_{\eta}
+ \hat{U}_{[\lambda} \,\hat{\nabla}_{\nu|}\hat{U}_{\eta})
\]
\[
= \Omega\left(\rho\,\,(\nabla_{[\nu}\,U_{\lambda]} \,U_{\eta}
+ U_{[\lambda} \,\nabla_{\nu]}\,U_{\eta})
+ \nabla_{[\nu} \rho\,\,U_{\lambda]} \,U_{\eta}
+ \frac{1}{3}\,\nabla_{[\nu} \rho \,\,g_{\lambda] \eta}\right)
\]
\[
+ \rho \left(\nabla_{[\nu} \Omega \,\,g_{\lambda] \eta} + 2\,\nabla_{[\nu} \Omega\,\,U_{\lambda]} \,U_{\eta}
+ U_{[\nu} \,g_{\lambda] \eta}\,g^{\pi \delta}\,\nabla_{\pi}\Omega\,U_{\delta}
\right).
\]
Finally, the geodesic equation (\ref{2dust}) translates into
\begin{equation}
\label{Omega-phys-geod}
\nabla_U U^{\mu} = \frac{1}{\Omega}\,(- g(U, U)\,g^{\mu}\,_{\rho} + U^{\mu}\,U_{\rho})\,\nabla^{\rho}\Omega.
\end{equation}
while equation (\ref{1dust}) for the density $\hat{\rho}$ gives
\begin{equation}
\label{conformal-rho-equ}
\nabla_{U}\,\rho + \rho\,\nabla_{\mu}\,U^{\mu} = 0.
\end{equation}
\vspace{.2cm}
We express the equations in terms of a frame field
$e_k =e^{\mu}\,_k\partial_{x^{\mu}}$, $k = 0, 1, 2, 3$, which has a time-like vector field given by
\[
e_0 = U,
\]
and which is orthonormal, so that $g_{jk} \equiv g(e_j, e_k) = \eta_{jk} = diag(-1, 1, 1, 1)$.
The space-like frame fields are given by the $e_a$, where $a, b, c = 1, 2, 3$ denote spatial indices to which the summation convention applies.
The metric is given by
\[
g = \eta_{jk}\,\sigma^j\,\sigma^k,
\]
where $\sigma^j$ denotes the field of 1-forms dual to $e_k$ so that their coefficients in the coordinates $x^{\mu}$ satisfy
$\sigma^j\,_{\mu}\,e^{\mu}\,_k = \delta^j\,_k$.
The connection coefficients, defined by
$\nabla_je_k \equiv \nabla_{e_j}e_k = \Gamma_j\,^l\,_k\,e_l$,
satisfy $ \Gamma_{jlk} = - \Gamma_{jkl}$ with $ \Gamma_{jlk} = \Gamma_j\,^i\,_k\,g_{li}$,
because $\nabla_i g_{jk} = 0$.
The covariant derivative of a tensor field $X^{\mu}\,_{\nu}$, given in the frame by
$X^i\,_j$, takes the form
\[
\nabla_k\,X^i\,_j = X^i\,_{j\,,\mu}\,e^{\mu}\,_k + \Gamma_k\,^i\,_l \,X^l\,_j -
\Gamma_k\,^i\,_l\,X^i\,_j.
\]
For the covariant version of $U$, i.e.
$U_j = - \,\delta^0\,_j$, equation (\ref{Omega-phys-geod}) implies the form
\begin{equation}
\label{nabla-U}
\nabla_k\,U_l = \Gamma_k\,^0\,_l
= \delta^0\,_k\,\Omega^{-1}\,(\nabla_l\Omega + U_l\,\nabla_0\,\Omega)
+ \delta^a\,_k\,\delta^b\,_l\,\chi_{ab}.
\end{equation}
If $U$ is hypersurface orthogonal and if $\hat{S}$ were chosen to be orthogonal to $U$ so that the vector fields $e_a$ define an orthonormal frame on $\hat{S}$,
the field $\chi_{ab}$ would represent the second fundamental form induced by $g$ on the slice
$\hat{S}$ whence $\chi_{ab} = \chi_{(ab)}$. In general hypersurface orthogonality will not be assumed here.
We shall write $g^{ab}\,\chi_{ab} = \chi_a\,^a$.
\vspace{.1cm}
The metric coefficients and the connection coefficients satisfy the {\it first structural equations}
\begin{equation}
\label{O-torsion-free condition}
e^{\mu}\,_{i,\,\nu}\,e^{\nu}\,_{j}
- e^{\mu}\,_{j,\,\nu}\,e^{\nu}\,_{i}
= (\Gamma_{j}\,^{k}\,_{i} - \Gamma_{i}\,^{k}\,_{j})\,e^{\mu}\,_{k},
\end{equation}
which ensures the connection to be torsion free, and the {\it second structural equations}
\begin{equation}
\label{O-Ricci identity}
\Gamma_l\,^i\,_{j,\,\mu}\,e^{\mu}\,_k - \Gamma_k\,^i\,_{j,\,\mu}\,e^{\mu}\,_l
+ 2\,\Gamma_{[k}\,^{i\,p}\,\Gamma_{l]pj}
- 2\,\Gamma_{[k}\,^p\,_{l]}\,\Gamma_p\,^i\,_j
\end{equation}
\[
= \Omega\,W^i\,_{jkl}
+ 2\,\{g^i\,_{[k}\,L_{l] j} + L^i\,_{ [k}\,g_{l] j}\},
\]
which relates the coefficients (and thus the metric $g_{\mu\nu}$) to the unknowns in the conformal field equations. The conformal field equations read now
\begin{equation}
\label{f-alg-equ}
6\,\Omega\,s - 3\,\nabla_{i}\Omega\,\nabla^{i}\Omega -
\lambda = \frac{1}{4}\,\Omega^3\,\rho,
\end{equation}
\begin{equation}
\label{f-Omega-equ}
\nabla_{j}\,\nabla_{k}\Omega + \,\Omega\,L_{j k} - s\,g_{j k}
= \frac{1}{2}\,
\Omega^2\,\rho\left(U_{j}\,U_{k} + \frac{1}{4}\,g_{j k}\right),
\end{equation}
\begin{equation}
\label{f-s-equ}
\nabla_{k}\,s + \nabla^{i}\Omega\,L_{ik}
= \frac{1}{2}\,\Omega\,\rho\,\nabla^{i}\Omega\left(U_{i}\,U_{k} + \frac{1}{4}\,g_{i k}\right)
+ \frac{1}{8}\,\Omega\,\rho\,\nabla_{k}\,\Omega + \frac{1}{24}\,\Omega^2\,\nabla_{k}\,\rho,
\end{equation}
\begin{equation}
\label{f-L-equ}
\nabla_{k}\,L_{l j}
- \nabla_{l}\,L_{k j} -
\nabla_{i}\Omega\,\,W^{i}\,_{j k l}
\end{equation}
\[
= \Omega\left(\rho\,\,(\nabla_{[k}\,U_{l]} \,U_{j}
+ U_{[l} \,\nabla_{k]}\,U_{j})
+ \nabla_{[k} \rho\,\,U_{l]} \,U_{j}
+ \frac{1}{3}\,\nabla_{[k} \rho \,\,g_{l] j}\right)
\]
\[
+ \rho \left(\nabla_{[k} \Omega \,\,g_{l] j} + 2\,\nabla_{[k} \Omega\,\,U_{l]} \,U_{j}
+ U_{[k} \,g_{l] j}\,g^{p q}\,\nabla_{p}\Omega\,U_{q}
\right),
\]
\begin{equation}
\label{f-W-equ}
\nabla_{i}\,W^{i}\,_{j k l}
=
\end{equation}
\[
\rho\,\,(\nabla_{[k}\,U_{l]} \,U_{j}
+ U_{[l} \,\nabla_{k]}\,U_{j})
+ \nabla_{[k} \rho\,\,U_{l]} \,U_{j}
+ \frac{1}{3}\,\nabla_{[k} \rho \,\,g_{l] j}
+ \frac{1}{\Omega}\,\,\rho\,\,Z_{jkl}
\]
with
\[
Z_{jkl} =
\nabla_{[k} \Omega \,\,g_{l] j} + 2\,\nabla_{[k} \Omega\,\,U_{l]} \,U_{j}
+ U_{[k} \,g_{l] j}\,g^{p q}\,\nabla_{p}\Omega\,U_{q}.
\]
The matter equations are given by
\begin{equation}
\label{f-U-equ}
\nabla_U U^{k} = \frac{1}{\Omega}\,(g^{k}\,_{i} + U^{k}\,U_{i})\,\nabla^{i}\Omega,
\end{equation}
\begin{equation}
\label{f-rho-equ}
\nabla_{U}\,\rho + \rho\,\chi_a\,^a = 0.
\end{equation}
\vspace{.5cm}
Equations (\ref{O-torsion-free condition}) to (\ref{f-rho-equ}) establish a system of differential equations for the unknowns
\begin{equation}
\label{unknnowns}
e^{\mu}\,_k, \quad \Gamma_i\,^j\,_k, \quad \Omega, \quad s, \quad L_{jk}, \quad W^i\,_{jkl}, \quad
U^k, \quad \rho,
\end{equation}
which is (apart from subtleties which may arise in cases of low differentiability) equivalent to the system
(\ref{einst}), (\ref{dust}), (\ref{2dust}), (\ref{1dust}) in domains where $\Omega > 0$.
If the system is to be used to solve Cauchy problems with data given on a space-like hypersurface
$\hat{S}$, one has to restrict the
available gauge freedom.
We shall follow the procedure of \cite{friedrich:1991} and \cite{friedrich:massive fields}, where the conformal freedom is removed be considering the Ricci scalar $R = R[g]$ in a suitable neighborhood of $\hat{S}$ as a prescribed function of the space-time coordinates and by prescribing suitable initial data for $\Omega$ and $\nabla_i\Omega$ on $\hat{S}$. The coordinates $\tau = x^0$ and $x^a$ are chosen near $\hat{S}$ so that
$\tau = \tau_*$ on $\hat{S}$ and $<U, dx^a> \,= 0$, $\,\,<U, d\tau> \,= 1$, whence
\[
U^{\mu} = e^{\mu}\,_0 = \delta^{\mu}\,_0 \quad \mbox{ near $\hat{S}$}.
\]
Apart from a parameter transformation $t = t(\tau)$ these coordinates coincide with the
ones considered in (\ref{phys-coords}).
Precise conditions on the vector fields $e_a$ orthogonal to $U$ will be stated later.
\vspace{.3cm}
Our main interest is the question whether there exist solutions to the system above
on the domain where
$\Omega > 0$ which admit a meaningful (i.e. sufficiently smooth) limit to a boundary where
$\Omega \rightarrow 0$. In that case we write $\{\Omega = 0\} = {\cal J}^+$, and refer to this set as the future {\it conformal boundary} of the solution.
By equation (\ref{f-alg-equ}) the limit of $\nabla^i\,\Omega$ will then define a time-like normal to the set ${\cal J}^+$ so that the latter will define a space-like hypersurface. It represents (future) time-like and null infinity for the `physical'
space-time on which $\Omega > 0$.
There arises an obvious problem with the differential system above.
The right hand sides of equations (\ref{f-W-equ}) and(\ref{f-U-equ}) are formally singular where
$\Omega \rightarrow 0$. This problem will be analyzed in the next section. Here we just point out its geometric nature.
If the fields entering equation (\ref{f-U-equ}) have limits as $\Omega \rightarrow 0$ the term in brackets on the right hand side of (\ref{f-U-equ}) defines a projection operator with kernel generated by the unit vector $U$. The right hand side of (\ref{f-U-equ}) can only admit a limit as $\Omega \rightarrow 0$ if the gradient of $\Omega$ is in the kernel of that operator and thus proportional to $U$, whence
\vspace{.1cm}
\noindent
\hspace*{.9cm}{\it The solutions can only admit a
reasonably smooth {\it conformal boundary} \\
\hspace*{1.5cm}${\cal J}^+$ if the geodesics generated by $\hat{U}$ approach ${\cal J}^+$ orthogonally}.
\vspace{.1cm}
\noindent
Remarkably, the singularity of equation (\ref{f-W-equ}) is of a similar geometric nature. If we want to keep the freedom to have non-vanishing conformal densities $\rho$ on ${\cal J}^+$, the right hand side of (\ref{f-W-equ})
can only assume a limit if $Z_{jkl} \rightarrow 0$ at ${\cal J}^+$. Since this implies that
$U^j\,Z_{jkl} = - \nabla_{[k} \Omega\,\,U_{l]} \rightarrow 0$, which implies in turn that $Z_{jkl} \rightarrow 0$,
the conclusion above follows again.
\section{The regularizing relation}
\label{the-reg-rel}
A conformal geodesic in a given space-time $(\hat{M}, \hat{g})$ is a curve $x^{\mu}(\sigma)$ together with a 1-form field $b_{\nu}(\sigma)$ which satisfy the system of {\it conformal geodesic equations}
\[
\hat{\nabla}_{V} V^{\mu}
+ S(b)_{\lambda}\,^{\mu}\,_{\rho}\,V^{\lambda}\,V^{\rho} = 0,
\]
\[
\hat{\nabla}_{V}b_{\nu} - \frac{1}{2}\, b_{\mu}\,S(b)_{\lambda}\,^{\mu}\,_{\nu}\,V^{\lambda}
- \hat{L}_{\lambda \nu}\,V^{\lambda} = 0,
\]
where $S(b)_{\lambda}\,^{\mu}\,_{\rho} = \delta_{\lambda}\,^{\mu}\,b_{\rho}
+ \delta_{\rho}\,^{\mu}\,b_{\lambda} - \hat{g}_{\lambda \rho}\,\hat{g}^{\mu \nu}\,b_{\nu}$
and $V^{\mu}(\sigma) = \frac{dx^{\mu}}{d\sigma\,}$ denotes the tangent vector of the curve. Sometimes it will be convenient to write these equations in the form
\begin{equation}
\label{a-hat(g)-conf-geod}
\hat{\nabla}_V V + 2<b, V>V - \hat{g}(V, V)\,b = 0,
\end{equation}
\begin{equation}
\label{b-hat(g)-conf-geod}
\hat{\nabla}_Vb\, - <b, V>b + \frac{1}{2}\,\hat{g}(b, b)\,V - \hat{L}(V, \,.\,) = 0,
\end{equation}
where the index position should be clear from the above.
For a conformal geodesic the initial data at a given point consist of its tangent vector and its 1-form at that point.
On a given space-time there exist thus more conformal geodesics than metric geodesics. Moreover, there exists in general no particular relation between conformal and metric geodesics. The problem of interest here is, however, very special in this respect.
\begin{lemma}
\label{geod-conf-geod-lemma}
Let $(\hat{M}, \hat{g})$ be a solution to the Einstein-dust system (\ref{einst}), (\ref{dust}), (\ref{2dust}), (\ref{1dust}). Then the geodesics tangential to the vector field $\,\hat{U}$ coincide after a reparameterization
with the curves underlying certain conformal geodesics.
\end{lemma}
\noindent
{\bf Proof:} Suppose $\bar{x}^{\mu}(t)$ is a $\hat{g}$-geodesic with
$\frac{d\bar{x}^{\mu}}{dt\,} = \hat{U}^{\mu}(\bar{x}(t))$ and $(x^{\mu}(\sigma), b_{\nu}(\sigma))$
a conformal geodesics with $V^{\mu}(\sigma) = \frac{dx^{\mu}}{d\sigma\,}$. Then there exists a parameter transformation $t = t(\sigma)$ so that $\frac{dt}{d\sigma} > 0$ and
$x^{\mu}(\sigma) = \bar{x}^{\mu}(t(\sigma))$ if and only if
\begin{equation}
\label{V-hatU-rel}
V^{\mu}(\sigma) = \omega(\sigma)^{-1}\,\hat{U}^{\mu}(\bar{x}(t(\sigma))) \quad \mbox{with} \quad
\omega^{-1} = \frac{dt}{d\sigma} > 0, \quad \quad
\hat{g}(V, V) = - \omega^{-2}.
\end{equation}
For $x^{\mu}(\sigma)$ to be up to a reparametrization a geodesic we need to have
a relation
\begin{equation}
\label{b-V-rel}
b_{\mu} = \alpha\,V_{\mu},
\end{equation}
with some function $\alpha = \alpha(\sigma)$ so that (\ref{a-hat(g)-conf-geod}) reads
\begin{equation}
\label{V-non-affine-geod}
\hat{\nabla}_{V} V^{\mu} + \alpha\,\hat{g}(V, V)\,V^{\mu} = 0.
\end{equation}
It follows then that
$2\,\omega^{-3}\,\hat{\nabla}_{V}\,\omega = \hat{\nabla}_{V}\,(\hat{g}(V, V)) = - 2\,\alpha\,\omega^{-4}$, whence
\begin{equation}
\label{alpha-omega-rel}
\alpha = - \omega\,\hat{\nabla}_V\,\omega.
\end{equation}
Basic for our result is that relations (\ref{hat-L-form}) and (\ref{V-hatU-rel}) give along $x^{\mu}(\sigma)$
\[
V^{\nu}\,\hat{L}_{\nu \mu} = \frac{1}{6}\,(\lambda - 2\,\hat{\rho})\,V_{\mu}, \quad \mbox{with} \quad
\hat{\rho} = \hat{\rho}(\bar{x}^{\mu}(t(\sigma))).
\]
Inserting this and (\ref{b-V-rel}) into (\ref{b-hat(g)-conf-geod}) and observing (\ref{V-non-affine-geod}),
(\ref{alpha-omega-rel}) gives the equation
\[
\omega\,\,\frac{d^2\omega}{d\,\sigma^2} - \frac{1}{2}\,\left(\frac{d\,\omega}{d \sigma}\right)^2
+ \frac{1}{6}\,(\lambda - 2\,\hat{\rho}(\bar{x}^{\mu}(t(\sigma)))) = 0,
\]
which provides with the relation
\begin{equation}
\label{t-sigma-rel}
\frac{dt}{d\sigma} = \frac{1}{\omega},
\end{equation}
a system of ODE's for $\omega = \omega(\sigma)$ and $t = t(\sigma)$ along
$x^{\mu}(\sigma) = \bar{x}^{\mu}(t(\sigma)))$. Prescribing arbitrary initial data
$t|_{\sigma_*} = t_*$, $\omega|_{\sigma_*}$, and
$\frac{d\omega}{d\,\sigma}|_{\sigma_*}$ with $ \omega_* > 0$ at the point
$x^{\mu}(\sigma_*) = \bar{x}^{\mu}(t_*))$
it can be solved. A straight forward calculation then shows that
\[
V^{\mu}(\sigma) = \frac{1}{\omega}\,\hat{U}^{\mu}(\bar{x}(t(\sigma)),
\quad b_{\nu}(\sigma) = - \frac{d \omega}{d\sigma}\,\hat{U}^{\mu}(\bar{x}(t(\sigma)),
\]
do indeed satisfy equations (\ref{a-hat(g)-conf-geod}) and (\ref{b-hat(g)-conf-geod}). $\Box$
\vspace{.1cm}
\noindent
It will later be important to note that the freedom to prescribe the initial data for $\omega$ gives the freedom to prescribe $\alpha$ arbitrarily at a given point.
\vspace{.1cm}
Conformal geodesics are of interest in the present context because the curves underlying {\it conformal geodesics are conformal invariants of a given conformal structure}:
If $g_{\mu \nu} = \Omega^2\,\hat{g}_{\mu \nu}$, where $\Omega$ is a conformal factor as considered above
and $x^{\mu}(\sigma)$, $b_{\lambda}(\sigma)$ satisfy the conformal geodesic equations with respect to
$\hat{g}_{\mu\nu}$,
then
$x^{\mu}(\sigma)$, $f_{\nu}(\sigma)$ with
\begin{equation}
\label{b-f-transf}
f_{\nu}(\sigma) = b_{\nu}(\sigma) - \Omega^{-1}\nabla_{\nu}\Omega|_{x(\sigma)},
\end{equation}
satisfy the conformal geodesics equations
\begin{equation}
\label{1-g-f-conf-geod}
\nabla_V V + 2<f, V>V - g(V, V)\,f = 0,
\end{equation}
\begin{equation}
\label{2-g-f-conf-geod}
\nabla_Vf\, - <f, V>f + \frac{1}{2}\,g(f, f)\,V - L(V, \,.\,) = 0,
\end{equation}
with respect to $g_{\mu\nu}$, where $\nabla$ and $L$ denote the Levi-Civita connection and the Schouten tensor of $g_{\mu \nu}$
(for this and further properties of conformal geodesics we refer to \cite{friedrich:AdS}, \cite{friedrich:cg on vac}).
If $g(V, V) = - \theta^{-2}$ with
$\theta > 0$ at a given point, equation (\ref{1-g-f-conf-geod}) gives
\[
\nabla_V \theta = \theta<V, f>,
\]
which shows that $\theta$ will stay positive and $x^{\mu}(\sigma)$ will be time-like as long as $V$ and $f$ remain sufficiently smooth. Equations (\ref{1-g-f-conf-geod}), (\ref{2-g-f-conf-geod})
do not see the relation
$g_{\mu \nu} = \Omega^2\,\hat{g}_{\mu \nu}$.
Thus, if $(\hat{M}, \hat{g})$ admits a smooth conformal boundary ${\cal J}^+$, one can arrange time-like conformal geodesics to extend smoothly to ${\cal J}^+$ with finite and non-vanishing tangent vector.
In the following we shall assume $V$ to be a conformal geodesic vector field which is related, as in
(\ref{V-hatU-rel}), to the $\hat{g}$-geodesic vector field $\hat{U}$ by
\begin{equation}
\label{B-V-hatU-rel}
V^{\mu}= \omega^{-1}\,\hat{U}^{\mu}.
\end{equation}
With the notation above we have then
\[
\theta\,V^{\mu} = U^{\mu} = \Omega^{-1}\,\hat{U}^{\mu},
\]
and thus
\begin{equation}
\label{theta-omega-Omega-rel}
\theta = \frac{\omega}{\Omega}, \quad \quad
\nabla_U \theta = \theta<U, f>.
\end{equation}
Since $\theta$ stays smooth and positive if $U$ crosses the conformal boundary this has the remarkable
consequence, used already in \cite{friedrich:AdS},
that $\omega$ goes to zero precisely where $\Omega$ does.
In terms of $U$ equation (\ref{1-g-f-conf-geod}) takes the form
\begin{equation}
\label{first-g-conf-geod-equ}
\nabla_U U + <U,f>U - g(U, U)\,f = 0.
\end{equation}
Replacing in (\ref{2-g-f-conf-geod}) the field $V$ by $U = \theta\,V$ renders that equation
in the form
\begin{equation}
\label{second-g-conf-geod-equ}
\nabla_Uf - <U,f>f +\frac{1}{2}\,g(f,f)U - L(U, \,.\,) = 0.
\end{equation}
This version of the conformal geodesic equations will be assumed from now on. The only effect of the transition is a reparametrization of
$x^{\mu}(\sigma) \rightarrow x^{\mu}(\tau)$, $f_{\nu}(\sigma) \rightarrow f_{\nu}(\tau)$ where
$\sigma$ is replaced by a function $\sigma(\tau)$ so that
\begin{equation}
\label{tau-sigma-rel}
\frac{d \tau}{d \sigma} = \frac{1}{\theta(x(\sigma))}.
\end{equation}
In the following the parameter $\tau$ will be used.
With (\ref{theta-omega-Omega-rel}) and the relations obtained in the proof of
Lemma \ref{geod-conf-geod-lemma} we get
\[
f_{\mu} = b_{\mu} - \Omega^{-1}\,\nabla_{\mu}\Omega =
- \omega\,\nabla_V\omega\,\,\hat{g}_{\mu\nu}\,V^{\nu} - \Omega^{-1}\,\nabla_{\mu}\Omega
\]
\[
=
- (\theta\,\Omega)\,\theta^{-1}\,\nabla_U(\theta\,\Omega)\,\Omega^{-2}\,g_{\mu\nu}\,\theta^{-1}\,U^{\nu}
- \Omega^{-1}\,\nabla_{\mu}\Omega
\]
\[
=
- (\theta^{-1}\,\nabla_U\theta
+ \Omega^{-1}\,\nabla_U\Omega)\,U_{\mu}
- \Omega^{-1}\,\nabla_{\mu}\Omega,
\]
\[
=
- (<U, f>
+ \Omega^{-1}\,\nabla_U\Omega)\,U_{\mu}
- \Omega^{-1}\,\nabla_{\mu}\Omega,
\]
and thus the {\it regularising relation}
\begin{equation}
\label{f-Omega-U-rel}
\nabla_{\mu}\Omega =
- (\nabla_U \Omega + \Omega<U, f>)U_{\mu}
- \Omega\,f_{\mu}.
\end{equation}
This relation will play a critical role. It will be used later to obtain a hyperbolic system of evolution equations which extends in a regular way to the set $\{\Omega = 0\}$ and it will be used to set up a subsidiary system to show that constraints and gauge conditions are preserved by the evolution system.
Here it is used to remove the singularities in equations (\ref{f-W-equ}) and (\ref{f-U-equ}).
In fact, replacing in $Z_{jkl}$ the term $\nabla_k\Omega$ by the right hand side of (\ref{f-Omega-U-rel}), we get (\ref{f-W-equ}) in the form
\begin{equation}
\label{regular-div-W-equ}
\nabla_{i}\,W^{i}\,_{j k l} =
\nabla_{[k} \rho\,\,U_{l]} \,U_{j}
+ \frac{1}{3}\,\nabla_{[k} \rho \,\,g_{l] j}
\end{equation}
\[
+ \rho\,\,(\nabla_{[k}\,U_{l]} \,U_{j}
+ U_{[l} \,\nabla_{k]}\,U_{j}
- f_{[k}\,g_{l]j} - 2\,f_{[k}\,U_{l]}\,U_j - U_{[k}\,g_{l]j}\,U^i\,f_i).
\]
\vspace{.1cm}
\noindent
Using (\ref{f-Omega-U-rel}) to replace $\nabla_k\Omega$ on the right hand side of (\ref{f-U-equ}), the equation takes the form
\begin{equation}
\label{regular-f-U-equ}
\nabla_0 U^{k} + f^k + U^k\,U_i\,f^i = 0,
\end{equation}
which is just (\ref{first-g-conf-geod-equ}) again. Equation (\ref{nabla-U}) is then replaced by the formally regular version
\begin{equation}
\label{reg-nabla-U}
\nabla_k\,U_l = \Gamma_k\,^0\,_l
= (- \delta^0\,_k\,f_b + \delta^a\,_k\,\chi_{ab})\,\delta^b\,_l.
\end{equation}
\vspace{.2cm}
Finally we note that given sufficient asymptotic smoothness
and an arrangement such that $\Omega(x(\tau)) \rightarrow 0$ for some finite value of $\tau$, the relation
\begin{equation}
\label{t-tau-rel}
\frac{dt}{d\tau} = \frac{1}{\Omega(x(\tau))},
\end{equation}
which follows from (\ref{t-sigma-rel}), (\ref{theta-omega-Omega-rel}), (\ref{tau-sigma-rel})
implies with (\ref{coord-alg-equ}) that $t \rightarrow \infty$ as
$\Omega(x(\tau)) \rightarrow 0$.
\section{The hyperbolic reduced equations}
\label{hyp-red-equ}
To extract from our equations a hyperbolic system we need to complete the gauge conditions for the
$g$-orthonormal frame field $e_k$ satisfying $e_0 = U$. The reduction procedure of the Einstein-dust system in \cite{friedrich98} employs a frame that is $\hat{g}$-parallely transported in the direction of
$\hat{U}$. Since the field $U$ is not geodesic with respect to $g$ this cannot be done here.
We use instead a frame whose vector fields $X$ satisfy
the Fermi transport law
\[
0 = \mathbb{F}_UX \equiv \nabla_UX - g(X, \nabla_U U)\,U + g(X, U)\, \nabla_U U,
\]
which has the properties: $\mathbb{F}_UU = 0$ and if $\mathbb{F}_UX = 0$, $\mathbb{F}_UY = 0$ then $\nabla_U(g(X, Y)) = 0$.
On a given space-like hypersurface transverse to the flow line of $U$
we thus choose smooth fields $e_k$ with $e_0 = U$ such that $g_{jk} = g(e_j, e_k) = \eta_{jk}$
and extend the $e_a$ away from the hypersurface by the requirement that $\mathbb{F}_Ue_a = 0$. The smooth orthonormal frame field so obtained is then closely related to the frame considered in \cite{friedrich98}. In fact,
if $\hat{e}_k$ is a $\hat{g}$-orthonormal frame such that $\hat{e}_0 = \hat{U}$ and
$\hat{\nabla}_{\hat{U}}\hat{e}_k = 0$, then $e_k = \Omega^{-1}\hat{e}_k$ is a
$g_{\mu\nu} = \Omega^2\,\hat{g}_{\mu\nu}$-orthonormal frame with $e_0 = U$ and
$\mathbb{F}_Ue_a = 0$.
As a consequence of relation $\mathbb{F}_Ue_k = 0$ the connection coefficients satisfy
\begin{equation}
\label{fermi-transp}
\Gamma_0\,^a\,_b = 0.
\end{equation}
The transport equation for the flow field $U$ is given by
(\ref{first-g-conf-geod-equ}). The coefficients $U^{\mu} = e^{\mu}\,_0 = \delta^{\mu}\,_0$ have been fixed by our choice of coordinates, however, and equation (\ref{regular-f-U-equ}) reduces to
the relation
\begin{equation}
\label{U-acc}
\Gamma_0\,^a\,_0 = - f^a = - g^{ab}\,f_b
\quad \mbox{resp.} \quad \Gamma_0\,^0\,_a = - f_a,
\end{equation}
between the connection coefficients and the acceleration of $U$.
The remaining not necessarily vanishing connection coefficients are then given by
\begin{equation}
\label{U-acc}
\Gamma_a\,^b\,_c
\quad \mbox{and} \quad
\Gamma_a\,^0\,_b = \nabla_a\,U_b = g(\nabla_{e_a}e_0, e_b) \equiv \chi_{ab}
\quad \mbox{resp.} \quad
\Gamma_a\,^b\,_0 = \chi_a\,^b = \chi_{ac}\,g ^{cb}.
\end{equation}
In the case in which $U$ resp. $\hat{U}$ is hypersurface orthogonal, the field $\chi_{ab}$ is symmetric and represents the second fundamental form while the $ \Gamma_a\,^b\,_c$ are the connection coefficients of the intrinsic connection induced on the hypersurfaces orthogonal to $U$ in the frame $e_a$.
We shall now derive the reduced equations for the remaining frame and connection coefficients.
With our gauge conditions and the connection coefficients above the first structural equations (\ref{O-torsion-free condition}) induce the evolution equations
\begin{equation}
\label{frame-evolv}
e^{\mu}\,_{a, \,0} = - f_a\,\delta^{\mu}\,_0 - \chi_a\,^b\,e^{\mu}\,_b,
\end{equation}
for the fields $e^{\mu}\,_a$.
The second structural equations (\ref{O-Ricci identity}) induce the evolution equations
\begin{equation}
\label{space-Gamma-evol}
\Gamma_c\,^a\,_{b,\,0} = f^a\,\chi_{cb}
- \chi_{c}\,^a\,f_b
- \chi_{c}\,^d\,\Gamma_d\,^a\,_b
+ \Omega\,W^a\,_{b0c} - g^a\,_{c}\,L_{0 b} + L^a\,_{ 0}\,g_{c b},
\end{equation}
\begin{equation}
\label{chi-evol}
\chi_{a b,\,0} + D_a f_{b}
= f_{a}\,f_b
- \chi_{a}\,^c\,\chi_{cb}
- \Omega\,W_{0b0a}
+ L_{a b} - L_{ 00}\,g_{a b},
\end{equation}
for $\Gamma_c\,^a\,_b$ and $\chi_{a b}$, where we set
\[
D_a f_b = f_{b\,,\mu}\,e^{\mu}\,_a - \Gamma_{a}\,^c\,_{b}\, f_{c}.
\]
No equation is implied for $\Gamma_0\,^0\,_a = - f_a$ by (\ref{O-Ricci identity}).
Such an equation is provided, however, by
(\ref{second-g-conf-geod-equ}), which takes in our gauge the explicit form
\begin{equation}
\label{f0-evol}
f_{0,\,0} = - \frac{1}{2}\,f_j\,f ^j + L_{00},
\end{equation}
\begin{equation}
\label{fa-evol}
f_{a,\,0} = L_{0a}.
\end{equation}
At this stage arises a problem. We are aiming for a system that is symmetric hyperbolic. The principal part of the coupled
system
\[
\chi_{a b,\,0} + D_a f_{b} = \ldots , \quad \quad \quad f_{a,\,0} = \ldots,
\]
does not satisfy the required symmetry condition. One might think of proceeding as follows.
The structural equations (\ref{O-Ricci identity}) imply after a contraction an
analogue of Codacci's equation, which takes with the convention
$D_c\,\chi_{a b} \equiv \chi_{a b\,,\mu}\,e^{\mu}\,_c - \Gamma_c\,^d\,_a\,\chi_{db}
- \Gamma_c\,^d\,_b\,\chi_{ad}$ the form
\[
D^a\,\chi_{a b} - D_b(\chi_a\,^a) = \ldots \,\,,
\]
(where the index position in the first term has to be respected because $\chi_{ab}$ is not necessarily symmetric). By adding a suitable multiple of this equation
to the second of the equations above one could hope to obtain a symmetric system. A careful analysis shows, however, that this does not work. We skip the details.
Help is again provided by (\ref{f-Omega-U-rel}). By this relation the field
\[
N_k = \nabla_k\Omega + (\nabla_U\Omega +
\Omega <U, f>)\,U_k + \Omega\,f_k,
\]
vanishes in our gauge. While $N_0 = N_k\,U^k = 0$ identically,
the equation $N_a = 0$ with
\[
N_a = \Omega\,f_a + \nabla_a\Omega,
\]
has non-trivial content. The relation
\[
\nabla_j\,N_k = \nabla_j\,\nabla_k\Omega +
\nabla_j\,(\nabla_U\Omega + \Omega <U, f>)\,U_k
\]
\[
+ (\nabla_U\Omega + \Omega <U, f>)\,\nabla_j\,U_k
+ \nabla_j\,\Omega\,f_k
+ \Omega\,\nabla_j\,f_k,
\]
implies in our gauge
\[
\nabla_a\,N_b - N_a\,f_b = \nabla_a\,\nabla_b\,\Omega
+ (\nabla_U\Omega + \Omega <U, f>)\,\chi_{ab}
- \Omega\,f_a\,f_b
+ \Omega\,\nabla_a f_b
\]
\[
= \nabla_a\,\nabla_b\,\Omega
+ (\nabla_U\Omega + \Omega <U, f>)\,\chi_{ab}
- \Omega\,f_a\,f_b
+ \Omega \,(D_a f_b - \chi_{a b}\,f_0) .
\]
\[
= \nabla_a\,\nabla_b\,\Omega
+ \nabla_U\Omega\,\chi_{ab}
- \Omega\,f_a\,f_b + \Omega \,D_a f_b,
\]
which gives with (\ref{f-Omega-equ})
\begin{equation}
\label{N-and-nablaN=0}
\nabla_a\,N_b - N_a\,f_b
= \nabla_U\Omega\,\chi_{ab} + s\,g_{ab} + \Omega \,(D_a\,f_b - f_a\,f_b
- L_{ab} + \frac{1}{8}\,\Omega\,\rho\,g_{ab}).
\end{equation}
Solving the equation $\nabla_a\,f_b - N_a\,f_b = 0$ for $D_a f_b$ and using the resulting expression to replace that term in the evolution equation for $\chi_{ab}$, gives the latter in the form
\begin{equation}
\label{sing-chi-evol}
\chi_{a b,\,0} - \Omega^{-1}\,(\nabla_U\Omega\,\,\chi_{a b} + s\,g_{ab})
=
- \chi_{a}\,^c\,\chi_{cb}
- \Omega\,W_{0a0b} - L_{ 00}\,g_{a b}.
\end{equation}
With the reduced equations obtained so far and the ones that follow below this gives again a symmetric hyperbolic system where
$\Omega \neq 0$.
Let us assume that the solution admits a smooth conformal boundary ${\cal J}^+ = \{\Omega = 0\}$.
To obtain a system which extends in a regular fashion to ${\cal J}^+$ we recall that this would require that $e_0 = U$ approaches ${\cal J}^+$ orthogonally. With (\ref{f-alg-equ}) this would imply
that
\[
\nabla_U\Omega \rightarrow - \nu < 0 \quad \mbox{as}\quad
\Omega \rightarrow 0, \quad \mbox{where}\quad \nu \equiv \sqrt{- \frac{\lambda}{3}},
\]
and thus $\nabla_U\Omega < 0$ also in a neighborhood of ${\cal J}^+$.
In the discussion of the conformal constraints on ${\cal J}^+$ in the next section we shall see that the conformal gauge can be chosen such that $s$ and $\chi_{ab}$ vanish at ${\cal J}^+$. If data on a `physical' initial hypersurface are evolved in the direction of ${\cal J}^+$ it is, however, difficult to decide how the conformal gauge must be chosen such that these fields will vanish at ${\cal J}^+$. This suggests to introduce regularizing unknowns which are derived from fields which go to zero at ${\cal J}^+$ in any conformal gauge. Such unknowns are suggested by the equation $\nabla_a\,f_b - N_a\,f_b = 0$. In fact,
the fields
\begin{equation}
\label{regularizing-unknowns}
\zeta_{ab} \equiv \frac{\chi_{a b} - \frac{1}{3}\,g_{ab}\,\chi_c\,^c}{\Omega}, \quad \quad
\xi \equiv \frac{ \nabla_U\Omega\,\,\chi_c\,^c + 3\,s}{\Omega},
\end{equation}
satisfy for $\Omega \neq 0$ and $\nabla_U\Omega \neq 0$ by (\ref{N-and-nablaN=0})
\begin{equation}
\label{zeta-in-terms-of-f-etc}
\zeta_{ab} =
- (\nabla_U\,\Omega)^{-1} \left(D_a\,f_b - f_a\,f_b - L_{ab}
- \frac{1}{3}\,(D_c\,f^c - f_c\,f^c - L_{c}\,^c)\,g_{ab}\right),
\end{equation}
and
\begin{equation}
\label{xi-in-terms-of-f-etc}
\xi = - D_a\,f^a
+ f_a\,f^a + L_{a}\,^a - \frac{3}{8}\,\Omega\,\rho,
\end{equation}
and can thus be expected to extend smoothly to ${\cal J}^+$.
The original unknown will be recovered from the new ones by
\begin{equation}
\label{chi-in-terms-zeta-xi}
\chi_{a b} = \Omega\,\zeta_{ab} + \frac{1}{3}\, (\nabla_U\Omega)^{-1}\,(\Omega\,\xi - 3\,s)\,g_{ab},
\end{equation}
which will certainly be well defined on neighbourhoods of ${\cal J}^+$ where $\nabla_U\Omega \neq 0$. This will suffice for our purpose because we can use equation
(\ref{sing-chi-evol}) where $\Omega \neq 0$.
The equations we have obtained so far imply equations for the unknowns (\ref{regularizing-unknowns}) that are regular where $\nabla_U\Omega \neq 0$. Indeed, a direct calculation gives with
(\ref{sing-chi-evol}) the equation
\begin{equation}
\label{zeta-evol}
\zeta_{ab\,,0} = - \Omega\,(\zeta_a\,^c\,\zeta_{cb} - \frac{1}{3}\,\zeta^{cd}\,\zeta_{dc}\,g_{ab})
- \frac{2}{3}\,(\nabla_U\Omega)^{-1}\,(\Omega\,\xi - 3\,s)\,\zeta_{ab} - W_{0a0b}.
\end{equation}
From (\ref{f-Omega-equ}) follows
\[
\Omega_{,00} - \Gamma_0\,^a\,_0\nabla_a\Omega = \nabla_0\nabla_0\Omega
= - \Omega\,L_{00} - s + \frac{3}{8}\,\Omega^2\,\rho,
\]
and thus with $0 = N_a = \Omega\,f_a + \nabla_a\Omega$
\[
\Omega_{,00} = \Omega\,f_a\,f^a
- \Omega\,L_{00} - s + \frac{3}{8}\,\Omega^2\,\rho.
\]
Equation (\ref{f-s-equ}) gives with (\ref{f-rho-equ}) and $N_a = 0$
\[
s_{,0} = \nabla_U\Omega\,L_{00} + \Omega\,f^a\,L_{a0} - \frac{1}{4}\,\rho\,\Omega\,\nabla_U\Omega
- \frac{1}{24}\,\rho\,\Omega^2\,\chi.
\]
With these two equations relation (\ref{sing-chi-evol}) implies
\begin{equation}
\label{xi-evol}
\xi_{,0} = (\nabla_U\Omega)^{-1}\,(\Omega\,\xi - 3\,s) \left(- \frac{1}{3}\,\xi + f_a\,f^a - L_{00} + \frac{1}{4}\,\rho\,\Omega\right)
\end{equation}
\[
- \nabla_U\Omega\,\,\Omega\,\,\zeta_{cd}\,\zeta^{dc}
+ 3\,f^a\,L_{a0} - \frac{3}{4}\,\rho\,\nabla_U\Omega.
\]
This completes the evolution system for the metric and the connection coefficients.
To deal with equations of first order we introduce
\[
\Sigma_k = \nabla_k\Omega,
\]
as an unknown and use (\ref{f-Omega-equ}) to get the evolution equations
\begin{equation}
\label{Omega-evol}
\nabla_0\Omega = \Sigma_0,
\end{equation}
\begin{equation}
\label{Sigma-evol}
\nabla_{0}\,\Sigma_{k} = - \,\Omega\,L_{0 k} + s\,g_{0 k}
+ \frac{1}{2}\, \Omega^2\,\rho\left(U_{0}\,U_{k} + \frac{1}{4}\,g_{0 k}\right).
\end{equation}
From (\ref{f-s-equ}) we get
\begin{equation}
\label{s-evol}
\nabla_{0}\,s = - \nabla^{i}\Omega\,L_{i0}
= \frac{1}{2}\,\Omega\,\rho\,\nabla^{i}\Omega\left(U_{i}\,U_{0} + \frac{1}{4}\,g_{i 0}\right)
+ \frac{1}{8}\,\Omega\,\rho\,\nabla_{0}\,\Omega + \frac{1}{24}\,\Omega^2\,\nabla_{0}\,\rho.
\end{equation}
As mentioned above, the Ricci scalar $R = R[g]$ of $g_{\mu\nu}$ will play the role of a conformal gauge source function and thus be prescribed as an explicit function of the coordinates near the initial hypersurface.
Because of the relation
\begin{equation}
\label{L00-def}
- L_{00} + g^{ab}\,L_{ab} = L_j\,^j =\frac{1}{6}\,R,
\end{equation}
it suffices to derive an evolution system for the components
$L_{0a}$, $L_{ab}$, $a, b = 1, 2, 3$, of the Schouten tensor. To simplify the equations we set
\begin{equation}
\label{def-K}
K_{jkl} =
\nabla_{i}\Omega\,\,W^{i}\,_{j k l}
\end{equation}
\[
+ \Omega\left(\rho\,\,(\nabla_{[k}\,U_{l]} \,U_{j}
+ U_{[l} \,\nabla_{k]}\,U_{j})
+ \nabla_{[k} \rho\,\,U_{l]} \,U_{j}
+ \frac{1}{3}\,\nabla_{[k} \rho \,\,g_{l] j}\right)
\]
\[
+ \rho \,(\nabla_{[k} \Omega \,\,g_{l] j} + 2\,\nabla_{[k} \Omega\,\,U_{l]} \,U_{j}
+ U_{[k} \,g_{l] j}\,g^{p q}\,\nabla_{p}\Omega\,U_{q}),
\]
so that (\ref{f-L-equ}) takes the form
\[
\nabla_{k}\,L_{l j} - \nabla_{l}\,L_{k j} = K_{jkl}.
\]
It implies by contraction
\[
\nabla_0\,L_{l 0} - g^{bc}\,\nabla_b\,L_{l c} = \frac{1}{6}\,\nabla_{l}\,R + K^j\,_{jl}.
\]
These equations are used to define the evolution system
\begin{equation}
\label{L0a-evol}
\nabla_0\,L_{0a} - h ^{bc}\,\nabla_b\,L_{a c} = \frac{1}{6}\,\nabla_{a}\,R + K^j\,_{ja}, \quad a = 1, 2, 3,
\end{equation}
\begin{equation}
\label{L11-etc-evol}
\nabla_{0}\,L_{a a} - \nabla_{a}\,L_{0 a} = K_{a0a},\quad a = 1, 2, 3,
\end{equation}
\begin{equation}
\label{L12-etrc-evol}
2\,\nabla_{0}\,L_{a b} - \nabla_{a}\,L_{0 b} - \nabla_{b}\,L_{0 a} = K_{a0b} + K_{b0a},
\quad a, b = 1, 2, 3, a \neq b.
\end{equation}
for the set of unknowns
\[
L_{01}, \quad L_{02}, \quad L_{03}, \quad L_{11}, \quad L_{12}, \quad
L_{13}, \quad L_{22}, \quad L_{23}, \quad L_{33}.
\]
For given right hand sides the system will then be symmetric hyperbolic on a neighborhood of an initial hypersurface on which $e^{\mu}_0 = \delta^{\mu}\,_0$ and on which $e^0\,_a$ is sufficiently small. Moreover, we find with our gauge conditions
\[
K^j\,_{ja} = - \frac{1}{2}\,\rho\,(\Omega\,f_a + \nabla_a\Omega),
\]
\[
K_{a0b} = \nabla_i\Omega\,W^i\,_{a0b}
+ \frac{1}{2}\,\Omega \left(\rho\,\chi_{ba} + \frac{1}{3}\,\nabla_U\rho\,g_{ab} \right),
\]
and thus the important fact that on the right hand sides of the evolution system above
only that derivative of $\rho$ occurs which can be removed by using the equation (\ref{f-rho-equ}), i.e.
\begin{equation}
\label{rho-evol}
\nabla_{U}\,\rho + \rho\,\chi_a\,^a = 0.
\end{equation}
This equation is assumed, of course, to be part of the reduced system.
The following extraction of an evolution system for the rescaled conformal Weyl tensor from equation (\ref{regular-div-W-equ}) is close to the procedure to obtain evolution equations for the conformal Weyl tensor discussed in \cite{friedrich98}, \cite{friedrich:rendall}, to which we refer for more details.
Let
\[
h^j\,_k = g^j\,_k +U^j\,U_k, \quad \quad l^j\,_k = g^j\,_k +2\,U^j\,U_k,
\]
denote the projection operator which maps the tangent spaces onto their subspaces $U^{\perp}$
orthogonal to $U$ and the reflection operator which maps $U$ onto $- \,U$ and induces the identity on
$U^{\perp}$ and consider the totally antisymmetric tensor densities
\[
\epsilon_{ijkl} = \epsilon_{[ijkl]} \quad \mbox{with} \quad \epsilon_{0123} = 1
\quad \mbox{and} \quad \epsilon_{jkl} = U^i\, \epsilon_{ijkl}.
\]
Further, define the $U$-electric part $w_{jl}$ and the $U$-magnetic part $w^*_{jl}$ of $W^{i}\,_{j k l}$ by setting
\[
w_{jl} = W_{i p k q}\,U^i \,h^p\,_j\,U^k\,h^q\,_l, \quad \quad
w^*_{jl} = \frac{1}{2}\,W_{i p m n}\,\epsilon^{mn}\,_{kq}\,U^i \,h^p\,_j\,U^k\,h^q\,_l,
\]
so that these symmetric trace free fields are given in our gauge essentially by their `spatial' components
$w_{ab}$ and $w^*_{ab}$.
It will be convenient to write equation (\ref{regular-div-W-equ}) in the form $F_{jkl} = 0$ with
\begin{equation}
\label{Fjkl-def}
F_{jkl} = \nabla_{i}\,W^{i}\,_{j k l}
- \nabla_{[k} \rho\,\,U_{l]} \,U_{j}
- \frac{1}{3}\,\nabla_{[k} \rho \,\,g_{l] j}
\end{equation}
\[
- \rho\,\,(\nabla_{[k}\,U_{l]} \,U_{j}
+ U_{[l} \,\nabla_{k]}\,U_{j}
- f_{[k}\,g_{l]j} - 2\,f_{[k}\,U_{l]}\,U_j - U_{[k}\,g_{l]j}\,U^i\,f_i).
\]
Inserting the representation
\[
W_{ijkl} = 2\,(l_{i[k}\,w_{l]j} - l_{j[k}\,w_{l]i} - U_{[k}\,w^*_{l] p}\,\epsilon^p\,_{ij}
- U_{[i}\,w^*_{j]p}\,\epsilon^p\,_{kl}),
\]
of the rescaled conformal Weyl tensor into the equations
\begin{equation}
\label{P-ij-def}
0 = P_{ij} \equiv - F_{pkq}\,h^p\,_{(i}\,U^k\,h^q\,_{j)}
+ \frac{1}{3}\,h_{ij}\,h^{kl}\,F_{pmq}\,h^p\,_k\,U^m\,h^q\,_l,
\end{equation}
\begin{equation}
\label{Q-ij-def}
0 = Q_{ij} \equiv - \frac{1}{2}\,F_{mpq}\,h^m\,_{(i}\,\epsilon_{j)}\,^{pq},
\end{equation}
the latter take the explicit form
\begin{equation}
\label{w-evol}
w_{ab,\,0} + D_c\,w^*_{d(b}\,\epsilon_{a)}\,^{cd} = \chi_{(a}\,^c\,w_{b)c} + 2\,\chi^c\,_{(a}\,w_{b)c}
- 2\,\chi_c\,^c\,w_{ab}
\end {equation}
\[
- h_{ab}\,\chi^{cd}\,w_{cd} - 2\,a_c\,w_{d(b}\,\epsilon_{b)}\,^{cd}
- \frac{1}{6}\,\rho\,(3\,\chi_{(ab)} - h_{ab}\,\chi_c\,^c),
\]
\begin{equation}
\label{w*-evol}
w^*_{ab,\,0} - D_c\,w_{d(b}\,\epsilon_{a)}\,^{cd} = \chi^c\,_{(a}\,w^*_{b)c} - \chi_c\,^c\,w^*_{ab}
\end {equation}
\[
+ 2\,a_c\,w_{d(a}\,\epsilon_{b)}\,^{cd} + \chi_{cd}\,w_{ef}\,\epsilon_{(i}\,^{ce}\,\epsilon_{j)}\,^{df},
\]
where we set, as before,
\[
D_a\,w_{bc} = w_{bc, \,\mu}\,e ^{\mu}\,_a - \Gamma_a\,^d\,_b\,w_{dc} - \Gamma_a\,^d\,_c\,w_{bd},
\]
etc. (The slight differences with the analogues equations in \cite{friedrich98}, \cite{friedrich:rendall}
result from the use of the relation ${\cal L}_U\,w_{ij} = w_{ij,\,0} + 2\,\chi_{(i}\,^k \,w_{k)j}$ for $w_{ab}$ and $w^*_{ab}$.) For given right hand side equations (\ref{w-evol}) and (\ref{w*-evol}) represent a symmetric hyperbolic system for $w_{ab}$ and $w^*_{ab}$ if it is ignored that these fields are trace free.
Their trace-freeness will be taken care of by the construction of the initial data and then be preserved by the equations. Again it is important that no derivatives of the field $\rho$ occur on the right hand sides.
\vspace{.3cm}
If on the right hand sides the field $\nabla_k\Omega$ is replaced by $\Sigma_k$,
$\nabla_0\rho$ is removed by using (\ref{rho-evol}),
$\chi_{ab}$ is replaced by (\ref{chi-in-terms-zeta-xi}), and $L_{00}$ is removed where it occurs (also in expressions like
$\nabla_a\,L_{0b} = L_{0b,\,\mu}\,e ^{\mu}\,_a- \Gamma_a\,^k\,_0\,L_{kb} - \Gamma_a\,^k\,_b\,L_{0k}$)
by using (\ref{L00-def}), then equations (\ref{frame-evolv}),
(\ref{space-Gamma-evol}), (\ref{f0-evol}), (\ref{fa-evol}), (\ref{zeta-evol}), (\ref{xi-evol}), (\ref{Omega-evol}),
(\ref{Sigma-evol}), (\ref{s-evol}), (\ref{L0a-evol}), (\ref{L11-etc-evol}), (\ref{L12-etrc-evol}), (\ref{rho-evol}),
(\ref{w-evol}), (\ref{w*-evol}) represent, irrespectively of the sign of $\Omega$, for suitably chosen initial data a quasi-linear symmetric hyperbolic evolution system for the unknowns
\[
e^{\mu}\,_a, \quad \Gamma_c\,^a\,_{b}, \quad f_k, \quad \zeta_{ab}, \quad \xi, \quad
\Omega, \quad \Sigma_k, \quad s, \quad L_{0a}, \quad L_{ab}, \quad \rho, \quad
w_{ab}, \quad w^*_{ab},
\]
where $\nabla_0\Omega \neq 0$. Where $\Omega \neq 0$ such an evolution system can be obtained by
replacing $\zeta_{ab}$ and $\xi$ by $\chi_{ab}$ and using directly equation (\ref{sing-chi-evol}).
The characteristics of the systems so obtained are time-like or null with respect to the solution metric, i.e.
the metric
$g_{\mu\nu}$ that satisfies $g_{\mu\nu}\,e ^{\mu}\,_j\,e ^{\nu}\,_k = \eta_{jk}$.
\section{Asymptotic end data}
\label{as-end-dat}
In section \ref{ex-strong-stab} we shall discuss the natural question how initial data for the reduced field equations are derived from solutions to the constraints (\ref{hat-Ham-constr}, (\ref{hat-mom-constr}) induced by the Einstein-$\lambda$-dust system on `physical' initial hypersurfaces.
The nature of the argument employed in \ref{ex-strong-stab} suggests, however, to consider first asymptotic data.
For solutions to Einstein's field equations with a positive cosmological constant which admit a smooth conformal boundary ${\cal J}^+$
it has been observed in the vacuum case \cite{friedrich:1986a}, in the case of matter models involving conformally covariant matter models with $\hat{g}^{\mu\nu}\,\hat{T}_{\mu\nu} = 0$ \cite{friedrich:1991},
and also in the case of a matter model with $\hat{g}^{\mu\nu}\,\hat{T}_{\mu\nu} \neq 0$
\cite{friedrich:massive fields} that the problem of providing initial data simplifies considerably if solutions to the constraints are constructed on that boundary. There is no need any longer to consider non-linear elliptic equations.
Assuming that the solutions admit a smooth conformal boundary ${\cal J}^+ = \{\Omega = 0\}$,
it will be shown in this section that
the constraints induced on ${\cal J}^+$ by the conformal equations
in the Einstein-dust case with a positive cosmological constant lead to the same simplification. Moreover, in the particular case where $\rho > 0$ on ${\cal J}^+$ they simplify even further. The solutions to the conformal Einstein-dust constraints can then in principle be constructed without solving any differential equation at all.
To construct the {\it asymptotic end data} on a 3-manifold which will later acquire the status of a smooth conformal boundary, let $S$ be a smooth, orientable, compact (though the latter is not really needed in the following discussion)
$3$-manifold. Assume that it represents a smooth conformal boundary ${\cal J}^+$ of an Einstein dust solution with cosmological constant $\lambda > 0$. The conformal constraints induced on it must then be considered with an induced metric which is Riemannian and a conformal factor $\Omega$ which vanishes on $S$.
As seen earlier, the future directed conformal flow field $U$ must be orthogonal to $S$. The conformal field equations will be considered in a frame $e_k$, $k = 0, 1, 2, 3$, on $S$
so that $e_0 = U$ and the $e_a$, $a = 1, 2, 3$, represent a frame on $S$ for the induced metric
\[
h_{ab} = g_{ab} = g(e_a, e_b) = diag(1, 1, 1),
\]
on $S$. The connection coefficients defined by $g$ in the frame $e_k$ are given again by $\nabla_k e _j = \Gamma_k\,^l\,_j\,e_l$.
As before $h^j\,_k = g_j\,^k + U_j\,U^k$ denotes the orthogonal projector onto $S$.
By assumption we have
$\Omega > 0$ in the past and $< 0$ in the future of $S$ and thus $e_0(\Omega) < 0$ on $S$.
Because $e_0$ is orthogonal to $S$ the field
\[
\chi_{ab} = \Gamma_a\,^0\,_b = g(\nabla_{e_a}e_0, e_b),
\]
represents the second fundamental form induced on $S$ and is thus symmetric, while the
$\Gamma_a\,^b\,_c$ define the connection coefficients on $S$ in the frame $e_a$ of the Levi-Civita connection $D$ defined by the intrinsic metric $h_{ab}$.
The electric part $w_{jl} = W_{ipkq}\,U^i\,U^k\,h^p\,_j\,h^q\,_l$ of the rescaled conformal Weyl tensor is then represented by $w_{ab} = W_{0a0b}$ and $w^*_{ab} = \frac{1}{2}\,W_{0acd}\,\epsilon_b\,^{cd}$
represents its magnetic part
$w^*_{jl} = \frac{1}{2}\,W_{ipmn}\,\epsilon^{mn}\,_{kq}\,U^i\,U^k \,h^p\,_j\,h^q\,_l$, where
$\epsilon_{ijkl}$ and $\epsilon_{jkl}$ are defined as before.
With these assumptions equation (\ref{f-alg-equ}) reduces to the condition
\begin{equation}
\label{dOmega-on-S}
\nabla_0\Omega = - \nu,
\quad \nabla^0\Omega = \nu
\quad \mbox{on} \quad S, \quad \mbox{where} \quad \nu = \sqrt{\lambda/3} > 0.
\end{equation}
Equation (\ref{f-Omega-equ}) reduces on $S$ to $\nabla_{i}\,\nabla_{j}\Omega = s\,g_{ij}$.
The only non-trivial condition implied by this relation is
a restriction on the second fundamental form
\begin{equation}
\label{2nd-fund-form-on S}
\nu\,\chi_{a b } = s\,h_{ab}
\quad \mbox{on} \quad S.
\end{equation}
Equation (\ref{f-s-equ}) implies the constraint
\begin{equation}
\label{L0a-on-S}
\nabla_a\,s + \nu\,L_{0 a}
= 0 \quad \mbox{on} \quad S.
\end{equation}
Under the conformal gauge transformation $g \rightarrow \bar{g} = \theta^2\,g$, $\Omega \rightarrow \bar{\Omega} = \theta\,\Omega$ with smooth $\theta > 0$ the function $s$ transforms as
$ s \rightarrow \bar{s} = \theta\,s + g^{\rho \delta}\,\nabla_{\rho}\Omega\,\nabla_{\nu}\theta$.
This shows that for given $\theta > 0$ on $S$ the derivative $\nabla_{\mu}\theta$
can be determined on $S$ such that $\bar{s}$ coincides on $S$ with any prescribed function.
The function $s$ could be carried along as a free function in the following equations but for simplicity the choice
that
\begin{equation}
\label{s=0-etc}
s = 0, \quad \chi_{a b } = 0, \quad \nabla_{i}\,\nabla_{j}\Omega = 0, \quad L_{0a} = L_{a0} = 0 \quad \mbox{on} \quad S,
\end{equation}
will be assumed, which still leaves the freedom to rescale the metric on $S$. It should be observed, however, that the gauge above may not be satisfied if a solution is evolved into $S$ from the domain where $\Omega > 0$.
In that case the more general relations like (\ref{2nd-fund-form-on S}) and (\ref{L0a-on-S}) must be considered.
Because the conformal Weyl tensor $\Omega\,W^i\,_{jkl}$ vanishes on $S$, the curvature tensor of $g$ is determined there by its Schouten tensor $L_{jk}$. Because the second fundamental form vanishes on $S$, the orthogonal projection of the curvature tensor of $g$ onto $S$ coincides by Gauss' theorem with the curvature tensor of $h$, i.e. $R_{abcd}[g] = R_{abcd}[h]$. It follows that the decomposition
of $R_{abcd}[g]$ in terms $g_{ab} = h_{ab}$ and the components $L_{ab}[g]$ of its Schouten tensor
is formally identically with the decomposition of of $R_{abcd}[h]$ in terms $h_{ab}$ and its Schouten tensor $l_{ab}[h] = R_{ab}[h] - \frac{1}{4}\,R[h]\,h_{ab}$. This implies that
\[
L_{ab}[g] = l_{ab}[h],
\]
which can be calculated from $h_{ab}$. The component $L_{00}$ then follows from $\frac{1}{6}\,R[g] = L_j\,^j$ as
\[
L_{00} = - \frac{1}{6}\,R[g] + h^{ab}\,L_{ab},
\]
once the conformal gauge source function $R[g]$ has been prescribed.
Equation (\ref{f-L-equ}) induces the constraint
$\nabla_{a}\,L_{b c} - \nabla_{b}\,L_{a c} = - \nu\,W^{0}\,_{c ab}$ on $S$.
Because the second fundamental form on $S$ vanishes, it can be written in the form
\begin{equation}
\label{Cotton-magn-part}
w^*_{ab} = \frac{1}{\nu}\,\epsilon_a\,^{cd}\,D_c\,l_{db}.
\end{equation}
The equation says that the magnetic part of the rescaled conformal Weyl tensor
is given on $S$ up to a factor by the (dualized) Cotton tensor of $h$.
Equation (\ref{f-L-equ}) induces the further constraint
$\nabla_{a}\,L_{b 0} - \nabla_{b}\,L_{a 0} = 0$ on $S$. This is satisfied as a consequence of
(\ref{s=0-etc}).
\vspace{.2cm}
With $F_{jkl}$ given by (\ref{Fjkl-def}), the constraints induced on $S$ by equation
(\ref{regular-div-W-equ}) are given by (see \cite{friedrich98}, \cite{friedrich:rendall})
\begin{equation}
\label{P-k-and-Q-k-def}
0 = P_k \equiv F_{jpl}\,U^j\,h^p\,_k\,U^l, \quad \quad
0 = Q_k \equiv - \frac{1}{2}\,F_{jpq}\,U^j\,\epsilon_k\,^{pq}.
\end{equation}
They can be written more explicitly in the form
\begin{equation}
\label{asymp-el-part-constr}
D^aw_{ac} = \frac{1}{3}\,D_{c} \rho - \rho\,f_c,
\end{equation}
which is a genuine constraint, and
\begin{equation}
\label{asymp-mag-part-constr}
D^a\,w^*_{ab} = 0,
\end{equation}
which is, consistent with (\ref{Cotton-magn-part}), the differential identity satisfied by the Cotton tensor
and imposes thus no additional restriction.
\vspace{.1cm}
The $1$-form $f_a$ characterizes the deviation of $U$ from hypersurface orthogonality
(see the datum $\hat{u}^{\alpha}$ in (\ref{phys-in-data-set})
and the following discussion of hypersurface orthogonal flows) and can be prescribed freely on $S$. The value of $f_0$ only affects the gauge. It can be prescribed freely and we assume that $f_0 = 0$ on $S$.
\vspace{.1cm}
The initial data for $\zeta_{ab}$ and $\xi$ which follow from
(\ref{zeta-in-terms-of-f-etc}) and (\ref{xi-in-terms-of-f-etc}) are then given on $S$ by
\begin{equation}
\label{c-zeta-in-terms-of-f-etc}
\zeta_{ab} =
\nu^{-1} \left(D_a\,f_b - f_a\,f_b - L_{ab}
- \frac{1}{3}\,(D_c\,f^c - f_c\,f^c - L_{c}\,^c)\,g_{ab}\right),
\end{equation}
and
\begin{equation}
\label{c-xi-in-terms-of-f-etc}
\xi = - D_a\,f^a
+ f_a\,f^a + L_{a}\,^a.
\end{equation}
\vspace{.1cm}
The observations above can be summarized in terms of local coordinates $x^{\alpha}$, $\alpha = 1, 2, 3$, on $S$ as follows.
\begin{lemma}
\label{free-data-on-scri}
Any smooth initial data set for the reduced equations is determined on the set $S = \{\Omega = 0\}$
uniquely by a Riemannian metric $h_{\alpha \beta}$, the density $\rho \ge 0$, the acceleration
$f_{\alpha}$ and a symmetric, $h$-trace free tensor field $w_{\alpha \beta}$, which are arbitrary up to the relation
\begin{equation}
\label{w-constr-on-scri}
D^{\alpha}w_{\alpha \beta} = \frac{1}{3}\,D_{\beta} \rho - \rho\,f_{\beta} \quad \mbox{on} \quad S,
\end{equation}
where $D$ denotes the Levi-Civita operator defined by $h_{\alpha \beta}$.
\end{lemma}
As in the cases mentioned in the beginning there is no need to solve an analogue of the Hamiltonian constraint.
The Riemannian space $(S, h_{\alpha \beta})$ is not subject to any further restriction. The situation even simplifies for
the class of data with $\rho > 0$ on $S$. In that case $h_{\alpha \beta}$, $\rho > 0$, and $w_{\alpha \beta}$ can be prescribed completely freely and $f_{\beta}$ is then determined by reading (\ref{w-constr-on-scri}) as its defining equation. It should be pointed out, however, that if
$f_{\alpha}$ is required to satisfy some extra conditions, as in the hypersurface orthogonal case discussed below, equation (\ref{w-constr-on-scri}) must be read as a differential equation. The situation can then be discussed by the well known splitting techniques
used in the discussion of the standard constraints \cite{bartnik:isenberg}.
\vspace{.1cm}
The gauge requirement $s|_{\{\Omega = 0\}} = 0$ leaves the conformal gauge freedom
\[
\Omega \rightarrow \Omega' = \theta\,\Omega, \quad
g_{\mu\nu} \rightarrow g'_{\mu\nu} = \theta^2\,g_{\mu\nu},
\]
with smooth functions $\theta > 0$ that are arbitrary on $S$.
If $n^{\mu}$ denotes the future directed unit normal to $S$
the conformal gauge transformation above implies associated transformations
\[
\quad
h_{\alpha \beta} \rightarrow h'_{\alpha \beta} = \theta^2\,g_{\alpha \beta}, \quad
n^{\mu} \rightarrow n'^{\mu} = \theta^{-1}\,n^{\mu}, \quad
U^{\mu} \rightarrow U'^{\mu} = \theta^{-1}\,U^{\mu}, \quad
\rho \rightarrow \rho' = \theta^{-3}\,\rho,
\]
and, by the transformation law for the 1-forms associated with conformal geodesics,
\begin{equation}
\label{conf-transf-fa-on-scri}
f_{\alpha} \rightarrow f'_{\alpha} = f_{\alpha} - \theta^{-1}\,D_{\alpha}\theta.
\end{equation}
If $n$ is extended as unit vector field into $\hat{M}$, the relation
$g_{\alpha \mu}\,W^{\mu}\,_{\nu \beta \rho}\,n ^{\nu}\,n^{\rho} = \Omega^{-1}\,g_{\alpha \mu}\,C^{\mu}\,_{\nu \beta \rho}\,n ^{\nu}\,n^{\rho}$ makes sense and suggests on $S$ for $w_{\alpha \beta}$
the transformation law
\[
w_{\alpha \beta} \rightarrow w'_{\alpha \beta} = \theta^{-1}\,w_{\alpha \beta}.
\]
It follows then
\[
h'^{\alpha \beta}\,D'_{\alpha}\,w'_{\rho \gamma}
= \theta^{-3}\,h^{\alpha \beta}\,D_{\alpha}\,w_{\rho \gamma},
\]
whence
\[
D'^{\alpha}\,w'_{\alpha \beta} - \frac{1}{3}\,D'_{\beta}\,\rho' + \rho'\,f'_{\beta}
= \theta^{-3}\,(D_{\alpha}\,w^{\alpha}\,_{\beta} - \frac{1}{3}\,D_{\beta}\,\rho + \rho\,f_{\beta}),
\]
so that
the constraints are preserved.
\subsection{Hypersurface orthogonal flows}
\label{hyp-orthog-flow}
Obviously, the vector field $\hat{U}^{\mu}$ is hypersurface orthogonal where $\Omega \neq 0$ if and only if this is true for $U^{\mu} = \Omega^{-1}\,\hat{U}^{\mu}$. Formally this follows from the relation
$\hat{U}_{[\rho}\,\hat{\nabla}_{\mu}\,\hat{U}_{\nu]} = \Omega^{-2}\,U_{[\rho}\,\nabla_{\mu}\,U_{\nu]}$.
In our gauge the hypersurface orthogonality condition
$\hat{U}_{[\rho}\,\hat{\nabla}_{\mu}\,\hat{U}_{\nu]} = 0$ is equivalent to
\begin{equation}
\label{conf-hyp-ortho-cond}
0 = \nabla_{[a}\,U_{b]} = \chi_{[ab]}.
\end{equation}
From (\ref{chi-evol}) we get with $\sigma_{ab} = \chi_{(ab)}$ along the flow lines of $U^{\mu}$ the ODE
\[
\chi_{[a b],\,0} + D_{[a}\,f_{b]}
= \sigma_{a}\,^c\, \chi_{[cb]} - \sigma_{b}\,^{c}\, \chi_{[ca]}.
\]
It follows that $D_{[a}\,f_{b]} = 0$ if $U^{\mu}$ is hypersurface orthogonal.
If the solution admited a smooth conformal extension, so that $\chi_{[ab]} = 0$ on ${\cal J}^+$,
we could conclude from the equation above that $\chi_{[ab]} = 0$ if we knew that $D_{[a}\,f_{b]} = 0$.
With the gauge condition $\nabla_a\,N_b - N_a\,f_b = 0$ equation (\ref{N-and-nablaN=0}) gives, however, only the relation
\[
0 = \Omega_{,\,0}\,\chi_{[ab]} + \Omega \,D_{[a}\,f_{b]}.
\]
But this combines with the equation above to give
\[
\left(\Omega^{-1}\,\chi_{[a b]}\right)_{,0} =
\sigma_{a}\,^c\,\left(\Omega^{-1}\,\chi_{[c b]}\right)
- \sigma_{b}\,^{c}\, \left(\Omega^{-1}\,\chi_{[ca]}\right).
\]
It follows that $\chi_{[a b]} = 0$ along a given integral curve of $U^{\mu}$ if it vanishes at a point of it where $\Omega \neq 0$.
On the other hand, the relation above shows that $\Omega^{-1}\,\chi_{[a b]}$
assumes the limit $(\nabla_0\Omega)^{-1}\,D_{[a}\,f_{b]}$ on ${\cal J}^+$, which vanishes where the integral curves of $U^{\mu}$ meet ${\cal J}^+$ if and only if
$D_{[a}\,f_{b]} = 0$ there. Observing the discussion of the conformal gauge freedom in the construction of data on the conformal boundary, in particular (\ref{conf-transf-fa-on-scri}), we conclude:
\begin{lemma}
\label{U-hyp-orthog-cond}
Let be given a solution to the Einstein-dust system (\ref{einst}), (\ref{dust}), (\ref{2dust}), (\ref{1dust})
that admits a smooth conformal boundary ${\cal J}^+$.
Then the field $U^{\mu}$ is hypersurface orthogonal if and only if the initial data for the conformal field equations induced on ${\cal J}^+$ in the gauge above are such that
\[
D_{[a}\,f_{b]} = 0 \quad \mbox{on} \quad {\cal J}^+.
\]
If this condition is satisfied and the field $f_a$ can be given on ${\cal J}^+$ as the differential of a function
$f$, then the conformal gauge can be chosen so that $f_a = 0$ on ${\cal J}^+$.
\end{lemma}
\subsection{FLRW-type solutions}
\label{FLRW-sols}
\vspace{.5cm}
In the following we discuss the FLRW solutions along the lines of the previous sections.
The FLRW-type solutions to (\ref{einst}), (\ref{dust}), (\ref{2dust}), (\ref{1dust}) on
$\hat{M} = \mathbb{R} \times S$ with $S = \mathbb{S}^3, \, \mathbb{T}^3$ or $\mathbb{H}^3_*$
(a suitable factor space of hyperbolic $3$-space)
are of the form
\[
\hat{g} = - dt^2 + a^2\,k, \quad \quad \hat{U} = \partial_{t},
\quad \quad \hat{\rho} = \hat{\rho}(t) \ge 0,
\]
with a function $a = a(t) > 0$ and a $3$-metric of constant curvature which is given in local coordinates
$x^{\alpha}$, $\alpha, \beta, \ldots = 1, 2, 3$, on $S$ by
$k = k_{\alpha \beta}\,dx^{\alpha}\,dx^{\beta} $, so that $R_{\alpha \beta \gamma \delta }[k] = 2\,\epsilon\,k_{\alpha [\gamma}\,k_{\beta]\delta} $ where
$\epsilon = 1, 0$ or $-1$. Rescaling the fields with a conformal factor $\Omega = \Omega(t)$
\[
\hat{g} \rightarrow g = \Omega^2\,\hat{g}, \quad \quad
\hat{U} \rightarrow U = \Omega^{-1}\,\hat{U} , \quad \quad
\hat{\rho} \rightarrow \rho = \Omega^{-3}\,\hat{\rho},
\]
and introducing a coordinate $x^0 = \tau(t)$ so that $<U, d\tau> \,\,= 1$, the conformal version of the metric above takes the form
\[
g = - d\tau^2 + l^2\,k, \quad \quad U = \partial_{\tau},
\quad \rho = \rho(\tau),
\]
with some function $l = l(\tau) > 0$.
The non-vanishing Christoffel symbols and the second fundamental form $\chi_{\alpha \beta}$ of the slices $\{\tau = const.\}$ are then given by
\[
\chi_{\alpha \beta} = \Gamma_{\alpha}\,^0\,_{\beta}[g] = l\,l'\,k_{\alpha \beta}, \quad
\Gamma_0\,^{\alpha}\,_{\gamma}[g] = \Gamma_{\gamma}\,^{\alpha}\,_0[g]
= \frac{1}{l}\,l'\,k^{\alpha}\,_{\gamma}, \quad
\Gamma_{\beta}\,^{\alpha}\,_{\gamma}[g] = \Gamma_{\beta}\,^{\alpha}\,_{\gamma}[k],
\]
where $' = \frac{d}{d\tau}$. The Ricci scalar and the Schouten tensor are given by
\[
R[g] = \frac{6}{l^2}\,(\epsilon + l\,l'' + (l')^2),
\]
\[
L_{00}[g] = \frac{1}{2\,l^2}\,(\epsilon - 2\,l\,l'' + (l')^2), \quad
L_{\alpha 0}[g] = L_{0 \alpha}[g] = 0, \quad
L_{\alpha \beta}[g] = \frac{1}{2}\,(\epsilon + (l')^2)\,k_{\alpha \beta}.
\]
Choosing the conformal gauge function as $R[g] = 6\,\epsilon$ on $\hat{M}$, the function $l$ must satisfy $l\,l'' + (l')^2 + \epsilon\,(1 - l^2) = 0$. Using the remaining conformal gauge freedom to achieve $l =1$, $l' =0$ on a slice $\{\tau = const.\}$, it follows that $l = 1$.
The only non-vanishing Christoffel symboly are then given by
$\Gamma_{\beta}\,^{\alpha}\,_{\gamma}[g] = \Gamma_{\beta}\,^{\alpha}\,_{\gamma}[k]$ and
\[
L_{00} = \frac{\epsilon}{2}, \quad
L_{\alpha0} = L_{0\alpha} = 0, \quad
L_{\alpha \beta} = \frac{\epsilon}{2}\,k_{\alpha \beta}.
\]
Where $\Omega > 0$ the physical field is then given by
\begin{equation}
\label{hom-g-hat}
\hat{g} = \Omega^{-2}\,g = - dt^2 + a^2\,d\omega^2,
\end{equation}
\begin{equation}
\label{c-coord-conf-transf}
a(t) = \frac{1}{\Omega(\tau(t))}, \quad \quad
\frac{dt}{d\tau} = \frac{1}{\Omega(\tau)}.
\end{equation}
\vspace{.1cm}
The high symmetry assumptions leads to a simplification of the conformal field equations.
There do not occur singularities any longer in the equations.
In fact, because $U$ is $g$-geodesic and hypersurface orthogonal and $\Omega = \Omega(\tau)$, the singularity in (\ref{Omega-phys-geod}) is gone.
Because the line element $g$ is locally conformally flat it follows that $W^{\mu}\,_{\nu \rho \kappa} = 0$ and thus $\hat{\nabla}_{[\nu}\,\hat{L}_{\lambda] \rho} = 0$
by (\ref{coord-W-equ}).
Moreover, it follows by (\ref{coord-L-equ}) that $\nabla_{[\nu}\,L_{\lambda] \rho} = 0$.
It will be assumed in the following that the conformal time coordinate $\tau$ vanishes on a set
$\{\Omega = 0\}$ and that $\nabla_U \Omega = \Omega' < 0$ there. Equations (\ref{coord-alg-equ}) and
(\ref{coord-T-trace}) then imply
\[
\Omega'(0) = - \nu = - \sqrt{\lambda/3} < 0.
\]
Equation (\ref{conformal-rho-equ})
reduces because of $\nabla_{\mu}\,U^{\mu} = \chi_c\,^c = 0$ to $\rho' = 0$, so that
\[
\rho = \rho_* = const. > 0,
\]
equations (\ref{coord-Omega-equ}) and (\ref{coord-T-trace-free-part}) imply
$s = \frac{\epsilon}{2}\,\Omega - \frac{1}{8}\,\rho_*\,\Omega^2$,
$\Omega'' + \epsilon\,\Omega - \frac{1}{2}\,\rho_*\,\Omega^2 = 0$
and equations (\ref{coord-s-equ}), (\ref{coord-T-trace}), (\ref{coord-T-trace-free-part}) give
$s' = \frac{\epsilon}{2}\,\Omega' - \frac{1}{4}\,\rho_*\,\Omega\,\Omega'$,
which is satisfied by the function $s$ given above. The equations for $s$ are redundant under the given assumptions. So we are left with the initial value problems
\[
\Omega'' + \epsilon\,\Omega - \frac{1}{2}\,\rho_*\,\Omega^2 = 0,
\quad \Omega(0) = 0, \quad \Omega'(0) = - \nu,
\]
which clearly have a smooth solutions near $\{\tau = 0\} = {\cal J}^+$. Where $\Omega' \neq 0$ (thus in particular near ${\cal J}^+$.) the ODE is equivalent to
$(3\,\Omega'^2 + 3\,\epsilon\,\Omega^2 - \rho_*\,\Omega^3)' = 0$,
which implies with the boundary conditions
\begin{equation}
\label{conf-first-order-FRW-equ}
3\,\Omega'^2 + 3\,\epsilon\,\Omega^2 - \rho_*\,\Omega^3 = \lambda.
\end{equation}
The decreasing solutions to this equation cover all the expanding ends of the FRW-type solutions.
With (\ref{c-coord-conf-transf}) the usual (physical) equations (see \cite{hawking:ellis}) for $a(t)$ are implied by (\ref{conf-first-order-FRW-equ}).
\section{The subsidiary system}
\label{subs-syst}
To show that solutions to the reduced equations for data which satisfy the constraints do indeed satisfy the complete set of conformal field equations, it has to be shown that the {\it zero quantities} $N_j$
and
\begin{equation}
\label{zero-quantities}
T_i\,^j\,_k, \quad \Delta^i\,_{jkl}, \quad A, \quad B_j, \quad C_{jl}, \quad D_j, \quad H_{jkl}, \quad F_{jkl},
\end{equation}
vanish as a consequence of the reduced equations and the given initial data. Here
\[
N_j \equiv \Omega\,f_j + U^k \Sigma_k\,U_j
+ \Sigma_j + \Omega\,\,U^k f_k\, U_j,
\]
\[
T_i\,^k\,_j\,e_k \equiv
- [e_i,e_j] + (\Gamma_i\,^l\,_j - \Gamma_j\,^l\,_i)\,e_l,
\]
\begin{equation}
\label{Delta-ijkl-def}
\Delta^i\,_{jkl} \equiv R^i\,_{jkl} - \Omega\,W^i\,_{jkl}
- 2\,\{g^i\,_{[k}\,L_{l] j} + L^i\,_{ [k}\,g_{l] j}\},
\end{equation}
with
\[
R^i\,_{jkl} = \Gamma_l\,^i\,_{j,\,\mu}\,e^{\mu}\,_k - \Gamma_k\,^i\,_{j,\,\mu}\,e^{\mu}\,_l
+ 2\,\Gamma_{[k}\,^{i\,p}\,\Gamma_{l]pj}
- 2\,\Gamma_{[k}\,^p\,_{l]}\,\Gamma_p\,^i\,_j,
\]
\[
A \equiv 6\,\Omega\,s - 3\,\Sigma_i\,\Sigma^i -
\lambda - \frac{1}{4}\,\Omega^3\,\rho,
\]
\[
B_k \equiv \nabla_k\Omega - \Sigma_k
\]
\[
C_{jk} \equiv \nabla_{j}\,\Sigma_{k} + \,\Omega\,L_{j k} - s\,g_{j k}
- \frac{1}{2}\,
\Omega^2\,\rho\left(U_{j}\,U_{k} + \frac{1}{4}\,g_{j k}\right),
\]
\[
D_k \equiv \nabla_{k}\,s + \Sigma^{i}\,L_{ik}
- \frac{1}{2}\,\Omega\,\rho\,\Sigma^{i}\left(U_{i}\,U_{k} + \frac{1}{4}\,g_{i k}\right)
- \frac{1}{8}\,\Omega\,\rho\,\Sigma_{k} - \frac{1}{24}\,\Omega^2\,\nabla_{k}\,\rho,
\]
\[
H_{jkl} \equiv \nabla_{k}\,L_{l j} - \nabla_{l}\,L_{k j} - K_{jkl},
\]
\[
F_{jkl} \equiv \nabla_{i}\,W^{i}\,_{j k l} - M_{jkl},
\]
where
\begin{equation}
\label{def-Mjkl}
M_{j k l}
= \nabla_{[k} \rho\,\,U_{l]} \,U_{j}
+ \frac{1}{3}\,\nabla_{[k} \rho \,\,g_{l] j}
\end{equation}
\[
+ \rho\,\,(\nabla_{[k}\,U_{l]} \,U_{j}
+ U_{[l} \,\nabla_{k]}\,U_{j}
- f_{[k}\,g_{l]j} - 2\,f_{[k}\,U_{l]}\,U_j - U_{[k}\,g_{l]j}\,U^i\,f_i),
\]
\begin{equation}
\label{def-Fjkl}
K_{jkl} = \Sigma_{i}\,W^{i}\,_{j k l} + \Omega\,M_{jkl}.
\end{equation}
\vspace{.1cm}
Some of these quantities vanish trivially because of symmetries, gauge conditions, or the reduced equations. The latter comprise equations (\ref{first-g-conf-geod-equ}), (\ref{second-g-conf-geod-equ}), (\ref{rho-evol})
and
\begin{equation}
\label{a-reduced-equations}
U^i\,T_i\,^k\,_j = 0, \quad
U^k \Delta^i\,_{jkl} = 0, \quad
U^j B_j = 0, \quad U^j C_{jl} = 0, \quad U^j D_j = 0,
\end{equation}
\begin{equation}
\label{b-reduced-equations}
H^j\,_{ja} = 0, \quad H_{a0b} + H_{b0a} = 0,\quad a, b = 1, 2, 3, \quad P_{ij} = 0, \quad
Q_{ij} = 0,
\end{equation}
The zero quantities not in this list correspond to constraints or gauge conditions. Concerning the second of equations (\ref{a-reduced-equations}) we refer to the remarks below.
\vspace{.1cm}
In the following we shall use the covariant derivative operator $\nabla_j$ defined by the connection coefficients $\Gamma_i\,^j\,_k$ that satisfy the gauge conditions and the reduced
equations. This operator is metric in the sense that $\nabla_i\,g_{jk} = 0$ but, as seen from
the first of conditions (\ref{a-reduced-equations}), it is not known a priori whether the connection is torsion free.
In the following arguments will be needed the commutators of covariant derivatives, which are for a function $\phi$ and a vector field $X^i$ in the case of a general metric connection of the form
\[
(\nabla_i\,\nabla_j - \nabla_j\,\nabla_i)\,\phi = - T_i\,^l\,_j\,\nabla_l\,\phi
\]
\[
(\nabla_i\,\nabla_j - \nabla_j\,\nabla_i)\,X^k = R^k\,_{lij}\,X^l - T_i\,^l\,_j\,\nabla_l\,X^j.
\]
To avoid carrying along various non-illuminating terms involving components of the torsion tensor we shall refer to such terms in an equation often in the form $\ldots + P(T)$, where the dots indicate the equation of interest and $P(T)$ is a generic symbol for a polynomial in the components of the torsion tensor that satisfies $P(0) = 0$.
The equation above will then take the form
\[
(\nabla_i\,\nabla_j - \nabla_j\,\nabla_i)\,X^k = R^k\,_{lij}\,X^l + P(T).
\]
The other zero quantities in the list (\ref{zero-quantities}) will be kept explicitly
in an equation if needed to indicate how the calculations goes, otherwise the equations will be written in the form
$\ldots + P(Z)$, where the dots indicate the members of interest and $P(Z)$ is a polynomial in the components of the zero quantities (that may occasionally absorb a $P(T)$) with smooth coefficients that satisfies $P(0) = 0$.
\vspace{.1cm}
The regular system has been obtained from the original version of the conformal field equations by using
the gauge requirements $N_j = 0$ and $\nabla_aN_b = 0$. It needs to be shown that they are preserved by the reduced equations to establish that the original version of the conformal field equations is satisfied. They are needed in particular to show that the equations for $\zeta_{ab}$ and $\xi$ imply the equations $U^i\Delta^0\,_{aib} = 0$, $U^i\Delta^a\,_{0ib} = 0$. The zero quantity $N_j$ plays a particular role because its vanishing follows directly from the reduced equations and the initial conditions.
\vspace{.1cm}
\noindent
{\it If $N_k = 0$ on a hypersurface transverse to the flow lines of $U^{k}$ (which will, for instance, be the case if data are prescribed on $\{ \Omega = 0\}$), this relation
is preserved along the flow lines of $U$ as a consequence of the reduced equations.
}
\vspace{.1cm}
\noindent
In fact, equations (\ref{first-g-conf-geod-equ}) and (\ref{second-g-conf-geod-equ})
imply
\[
\nabla_U N_{i} = U^k\,U^l\,C_{kl}\,U_i + U^k\,C_{ki} + U^kB_k\,(f_i \,+ U^l f_l\,U_i)
- U_{i} \,f^{k} N_{k},
\]
which reduces with (\ref{a-reduced-equations}) to the linear homogeneous ODE
\begin{equation}
\label{N-subs-equ}
\nabla_U N_{i} = - U_{i} \,f^{k} N_{k},
\end{equation}
along the flow lines of $U$. From this the assertion follows. Since the solution to the reduced equations is ruled by the flow lines it follows also that $\nabla_iN_j = 0$ on the solution.
\vspace{.1cm}
\noindent
It can thus be assumed that $N_j = 0$, $\nabla_aN_b = 0$ so that we have indeed
$U^i\Delta^0\,_{aib} = 0$ and the equivalent equation $U^i\Delta^a\,_{0ib} = 0$ as written in (\ref{a-reduced-equations}).
\vspace{.1cm}
The subsequent discussion follows to some extent the derivation of subsidiary systems
in earlier work on the conformal field equations.
It will be convenient to use for the covariant derivative of a given tensor field $X_{ij}\,^k$ the notation
\[
\nabla_l X_{ij}\,^k = e_l(X_{ij}\,^k) + (\Gamma X)_{lij}\,^k,
\]
so that $X_{ij}\,^k \rightarrow (\Gamma X)_{lij}\,^k$ denotes a purely algebraic linear operator
which does not involve derivatives.
\vspace{.1cm}
The connection defined by the $\Gamma_i\,^j\,_k$ and the associated
torsion and curvature
tensor satisfy the first Bianchi identity
\[
\sum_{(jkl)} \nabla_j\,T_k\,^i\,_l =
\sum_{(jkl)} (R^i\,_{jkl} + T_j\,^m\,_k\,T_l\,^i\,_m),
\]
where $\sum_{(jkl)}$ denotes the sum over the cyclic permutation of the
indices $jkl$.
Setting here $j = 0$, observing that the symmetries of $C^i\,_{jkl} = \Omega\,W^i\,_{jkl}$ and
$L_{kl}$ imply
$\sum_{(jkl)} R^i\,_{jkl} = \sum_{(jkl)} \Delta^i\,_{jkl}$ and taking
into account the
reduced equations, we get from this the equation
\begin{equation}
\label{T-subs-equ}
\nabla_0\,T_k\,^i\,_l = - (\Gamma T)_{l0 }\,^i\,_k + (\Gamma T)_{k0}\,^i\,_l
+ 3\,\sum_{(0kl)} (\Delta^i\,_{0kl} + T_0\,^m\,_k\,T_l\,^i\,_m) = P(Z).
\end{equation}
\vspace{.1cm}
To obtain an equation of the desired type for $\Delta^i\,_{jkl}$ we show that the right hand side of the
identity
\[
\nabla_j \Delta^i\,_{mkl} + \nabla_l \Delta^i\,_{mjk} + \nabla_k \Delta^i\,_{mlj} = \frac{1}{2}\,
\epsilon_{njkl}\,\epsilon^{npqr}\,\nabla_p \Delta^i\,_{mqr}.
\]
can be written as a linear expression in the zero quantities. We write (\ref{Delta-ijkl-def}) in the form
\[
R^i\,_{jkl} =
\Delta^i\,_{jkl} + \Omega\,W^i\,_{jkl} + G^i\,_{jkl} + E^i\,_{jkl},
\]
with
\[
G^i\,_{jkl} = L\,g^i\,_{[k}\,g_{l]j}, \quad L = L_i\,^i, \quad
E^i\,_{jkl} = 2\,\{g^i\,_{[k}\,L^*_{l] j} + L^{*i}\,_{ [k}\,g_{l] j}\},
\quad L^*_{l j} = L_{l j} - \frac{1}{4}\,L\,g_{l j},
\]
and use the second Bianchi identity
\begin{equation}
\label{2Bianchi}
\sum_{(jkl)} \nabla_j\,R^i\,_{mkl} =
- \sum_{(jkl)} R^i\,_{mpj}\,T_k\,^p\,_l,
\end{equation}
to obtain
\[
\epsilon^{njkl}\,\nabla_j \Delta_{imkl} = - \epsilon^{njkl}\,\left(\nabla_j\Omega\,W_{imkl}
+ \Omega\,\nabla_j W_{imkl}
\right.
\]
\[
\left.
+ \nabla_jG_{imkl} + \nabla_jE_{imkl} + R_{impj}\,T_k\,^p\,_l\right).
\]
The well known facts that the left and right duals of $W_{ijkl}$ and $G_{ijkl}$ coincide respectively
while the left dual of $E_{ijkl}$ differs from its right dual by a sign then imply with the reduced equations
\[
\epsilon_n\,^{jkl}\,\nabla_j \Delta_{imkl} = \epsilon_{im}\,^{kl}\,\left( \nabla_j\Omega\,W^{j}\,_{nkl}
+ \Omega\,\nabla_j W^{j}\,_{nkl}
\right.
\]
\[
\left.
+ \nabla_jG^{j}\,_{nkl} - \nabla_jE^{j}\,_{nkl} \right) - \epsilon_n\,^{jkl}\,R_{impj}\,T_k\,^p\,_l
\]
\[
= \epsilon_{im}\,^{kl}\,\left( \nabla_j\Omega\,W^{j}\,_{nkl}
+ \Omega\,(F_{nkl} + M_{nkl})
\right.
\]
\[
\left.
2\,\nabla_{[k}L\,g_{l]n} - 2\,\nabla_{[k}\,L_{l]n} - 2\,\nabla_j\,L^j\,_{[k}\,g_{l]n} \right) - \epsilon_n\,^{jkl}\,R_{impj}\,T_k\,^p\,_l
\]
\[
= \epsilon_{im}\,^{kl}\,\left( \nabla_j\Omega\,W^{j}\,_{nkl}
+ \Omega\,(F_{nkl} + M_{nkl})
\right.
\]
\[
\left.
- H_{nkl}
- \Sigma_{i}\,W^{i}\,_{n k l} - \Omega\,M_{nkl}
- 2\,H^j\,_{j[k}\,g_{l]n}\right) - \epsilon_n\,^{jkl}\,R_{impj}\,T_k\,^p\,_l.
\]
In the last step it has been used that $K^j\,_{jl} = 0$. This follows because the tensor $W^i\,_{jkl}$ has vanishing contractions and because equations
(\ref{first-g-conf-geod-equ}) and ({\ref{rho-evol}), which are satisfied as members of the reduced system, imply that $M^j\,_{jl} = 0$.
Using again the reduced equations we finally get
\begin{equation}
\label{Delta-imkl-subs}
\nabla_{0} \Delta^i\,_{mkl} =
- (\Gamma \Delta)_l\,^i\,_{m0k}
+ (\Gamma \Delta)_k\,^i\,_{m0l}
\end{equation}
\[
- \frac{1}{2}\,\epsilon^n\,_{0kl}\left\{
\epsilon^i\,_{m}\,^{kl}\,\left(B_p\,W^{p}\,_{nkl}
+ \Omega\,F_{nkl}
- H_{nkl}
- 2\,H^p\,_{pk}\,g_{l n}\right) - \epsilon_n\,^{jkl}\,R^i\,_{mp0}\,T_k\,^p\,_l
\right\} = P(Z).
\]
\vspace{.1cm}
A direct calculation gives for the quantity
\begin{equation}
\label{alg-zero-q}
A = 6\,\Omega\,s - 3\,\Sigma_i\,\Sigma^i -
\lambda - \frac{1}{4}\,\Omega^3\,\rho,
\end{equation}
the relation
\[
\nabla_j A = 6\,\Omega\,D_j - 6\,\Sigma^i\,C_{ji} + (6\,s - \frac{3}{4}\,\Omega^2\,\rho)\,B_j.
\]
On the initial slice, where the zero quantities on the right hand side vanish by the construction of the initial data, this relation reduces to $\nabla_j A = 0$.
This implies that $A = 0$ on that slice if it holds at one point of it. In the case of `physical' data (i. e.
$\Omega = 1$) the condition $A = 0$ reduces to $0 = 4\,\hat{A} = \hat{R} - 4\,\lambda - \hat{\rho}$, which will be satisfied by the construction of the physical data. Using the freedom to prescribe $\Omega$ and its time derivative on the initial slice the condition $A = 0$ can also be achieved in the transition to conformal data. We recall that the relation $A = 0$ served to determine the value of $\Sigma_j$
in our discussion of the conformal data on $\{\Omega = 0\}$.
With the reduced equations the relation above implies that
\[
\nabla_U A = 0.
\]
{\it We can thus assume that $A = 0$ on the solution manifold}.
\vspace{.1cm}
A straightforward but lengthy calculation shows that
the fields
\[
Z^B_{jk} = \nabla_{[j}\,B_{k]}, \quad Z^C_{jkl} = \nabla_{[j}\,C_{k]l}, \quad Z^D_{jk} = \nabla_{[j}\,D_{k]},
\]
can be expressed as linear (homogeneous) functions of the zero quantities with smooth coefficients. Taking into account the reduced equation $U^j B_j = 0$,
$U^j C_{jl} = 0$, $U^j D_j = 0$ one gets
\[
U^j\,\nabla_{j}B_{k} = 2\,U^j\,Z^B_{jk} + U^j\,\nabla_{k}\,B_{j}
= 2\,U^j\,Z^B_{jk} + \nabla_{k}\,(U^jB_{j}) - (\nabla_{k}U^j)\,B_{j} = P(Z).
\]
Similar calculations give
\begin{equation}
\label{B-C-D-subs}
U^j\,\nabla_{j}B_{k} = P(Z), \quad \quad
U^j\,\nabla_{j}C_{kl} = P(Z), \quad \quad
U^j\,\nabla_{j}D_{k} = P(Z).
\end{equation}
The remaining subsidiary equations are obtained by analyzing the expressions
\[
\nabla_{[l}\,H^i\,_{jk]}
\quad \quad \nabla^j\,F_{jkl},
\]
from two different points of view. As a preparation we observe the algebraic relations
\begin{equation}
\label{M-algebra}
M_{jkl} = - M_{jkl}, \quad \quad M_{[jkl]} = 0, \quad \quad M^j\,_{jl} = 0.
\end{equation}
The first of them follow immediately from the definition while, as pointed out above, the last one follows as a consequence of the reduced equations (\ref{first-g-conf-geod-equ}) and ({\ref{rho-evol}). These relations imply
\begin{equation}
\label{F-algebra}
F_{jkl} = - F_{jkl}, \quad \quad F_{[jkl]} = 0, \quad \quad F^j\,_{jl} = 0,
\end{equation}
and also
\begin{equation}
\label{K-algebra}
K^j\,_{jl} = 0.
\end{equation}
Moreover, a straightforward though fairly lengthy calculation which makes repeatedly use of the reduced equations, shows that
\begin{equation}
\label{nabla-M-rel}
\nabla^jM_{jkl} = P(Z),
\end{equation}
and
\[
\nabla_l K^l\,_{jk} = \nabla_l \Sigma_i\,W^{il}\,_{jk} + \Sigma_i\,\nabla_lW^{il}\,_{jk}
+ \nabla_l\Omega\,M^l\,_{jk} + \Omega\,\nabla_l M^l\,_{jk}
\]
\[
= C_{li}\,W^{il}\,_{jk} + B_l M^l\,_{jk} - \Sigma_l F^l\,_{jk} + \Omega\,\nabla_l M^l\,_{jk} = P(Z).
\]
From this follows the relation
\begin{equation}
\label{nabla-H-contraction}
\nabla_{[l} H^l\,_{jk]} =\Delta^l\,_{p[lj}\,L_{k]}\,^p - \nabla_{[l} K^l\,_{jk]} + P(T) = P(Z).
\end{equation}
Similar calculations, which use that the left and right duals of the conformal Weyl tensor coincide, gives
\[
\epsilon^{qljk}\,\nabla_l H^p\,_{jk} = \epsilon^{qljk}\,(
\nabla_l\nabla_jL_k\,^p - \nabla_l(\Sigma_n\,W^{np}\,_{jk} + \Omega\,M^p\,_{jk}))
\]
\[
= \epsilon^{qljk}\,(\Delta^p\,_{nlj}\,L_k\,^n + W^p\,_{nlj}\,C_k\,^n - B_l\,M^p\,_{jk}) +
\epsilon^{npik}\,\Sigma_n\,F^q\,_{jk}
\]
\[
+ \frac{1}{2}\,\rho\,\Omega^2\,\epsilon^{qljk}\,\,W^p\,_{njk}\,U^n\,U_l
- 2\,\Sigma_l\,M^{(q}\,_{jk}\,\epsilon^{p)ljk} - \Omega\,\nabla_l\,M^p\,_{jk}\,\epsilon^{pljk} .
\]
From equations (\ref{M-algebra}), (\ref{nabla-M-rel}) follows that
\[
\epsilon_{pqmn}\, \nabla_l\,M^p\,_{jk}\,\epsilon^{qljk} = P(Z).
\]
Solving the equation $N_l = 0$ for $\Sigma_l$ and inserting this into the equation above, we thus finally get
\[
\epsilon^{qljk}\,\nabla_l H^p\,_{jk} =
\frac{1}{2}\,\rho\,\Omega^2\,\epsilon^{qljk}\,\,W^p\,_{njk}\,U^n\,U_l
\]
\[
- 2\,\Sigma_l\,M^{(q}\,_{jk}\,\epsilon^{p)ljk} - \Omega\,\nabla_l\,M^{(p}\,_{jk}\,\epsilon^{q)ljk} + P(Z).
\]
\[
\epsilon^{qljk}\,\nabla_l H^p\,_{jk} =
\frac{1}{2}\,\rho\,\Omega^2\,\epsilon^{qljk}\,\,W^p\,_{njk}\,U^n\,U_l
+ 2\,\,\nabla_U\Omega\,\,U_l\,M^{(q}\,_{jk}\,\epsilon^{p)ljk}
\]
\[
+ \Omega\,\{
2\,(f_l \,+ <U,f>U_l)\,M^{(q}\,_{jk}\,\epsilon^{p)ljk}
- \nabla_l\,M^{(p}\,_{jk}\,\epsilon^{q)ljk}\} + P(Z).
\]
A direct calculation shows now that
\begin{equation}
\label{H00-Hab-rel}
\epsilon^{0ljk}\,\nabla_l H^0\,_{jk} = P(Z), \quad \quad
\epsilon^{aljk}\,\nabla_l H^b\,_{jk} = P(Z), \quad a, b = 1, 2, 3.
\end{equation}
\vspace{.1cm}
After solving the $9$ reduced equations for the components $L_{0a}$, $L_{ab}$, they
resume their original form if $1/6\,R$ is replaced again by $L_j\,^j$.
To show that they imply for suitably given initial data the full set $H_{jkl} = 0$, it needs to be shown that
\[
H_{abc} = 0, \quad \quad H_{0ab} = 0, \quad a \neq b.
\]
In fact, the equation $0 = H^j\,_{ja} = - H_{00a} + g^{cd}\,H_{cad}$ implies then that $H_{00a} = 0$ and with the identities
\[
H_{jkl} = - H_{jlk} \,\,\, \mbox{and} \,\,\, \epsilon^{ijkl}\,H_{jkl} = 0, \,\,\, \mbox{i.e.}\,\,\,
\epsilon^{abc}\,H_{abc} = 0 \,\,\, \mbox{and} \,\,\, H_{0ab} + H_{b0a} + H_{ab0} = 0, \,\,\, a \neq b,
\]
and the reduced equation $H_{a0b} + H_{b0a} = 0$ it follows then that
\[
0 = H_{0ab} = - H_{b0a} + H_{a0b} = 2\, H_{a0b} \quad a \neq b,
\]
which exhaust the remaining cases.
\vspace{.1cm}
We derive now the equations for the zero quantities above.
The reduced equation $H^j\,_{ja} = 0$ implies that
$\nabla_k H^l\,_{la} = (\Gamma\,H)_k\,^i\,_{la} = P(Z)$. Observing this in equations
(\ref{nabla-H-contraction}) we an equation of the form
\begin{equation}
\label{1-H-red-equ}
\nabla_0 H_{0ab} - g_{cd}\,\nabla_cH_{dab} = P(Z).
\end{equation}
On the other hand we have by (\ref{H00-Hab-rel})
\[
\nabla_0 H_{dab} + \nabla_b H_{d0a} - \nabla_aH_{d0b} = 3\,\nabla_{[0} H_{|d| ab]} = P(Z)
\]
and
\[
\nabla_d H_{0ab} + \nabla_b H_{0da} + \nabla_aH_{0bd} = 3\,\nabla_{[d} H_{|0|ab]} = P(Z)
\]
(where indices with a modulus sign are exempt from the anti-symmetrization).
Observing the relations $H_{a0b} = - H_{b0a}$ and $2\,H_{c0d} = H_{0cd}$ implied be the reduced equations, one gets from this an equation of the form
\begin{equation}
\label{2-H-red-equ}
2\,\nabla_0 H_{dab} - \nabla_d H_{0ab} = P(Z).
\end{equation}
Equations (\ref{1-H-red-equ}), (\ref{2-H-red-equ}) constitute a system of equations for the unknowns $H_{0ab}$ and $H_{abc}$ which is, for given right hand sides, symmetric hyperbolic.
\vspace{.1cm}
The properties (\ref{F-algebra}) imply in particular the relation $F^a\,_{0a} = F^i\,_{0i} = 0$. The
field $P_{ij}$ and $Q_{kl}$ introduced in
(\ref{P-ij-def}} and (\ref{Q-ij-def}) are thus completely represented by
\[
P_{ab} = - F_{(a |0| b)}, \quad \quad Q_{ab} = - \frac{1}{2}\,F_{(a}\,^{cd}\,\epsilon_{b)cd}.
\]
To discuss the remaining content of the field $F_{jkl}$ we recall the definitions
\[
P_a = F_{0a0}, \quad \quad
Q_b = - \frac{1}{2}\,F_{0cd}\,\epsilon_b\,^{cd},
\]
given in the discussion of the constraints. These fields exhaust the information in $F_{0a0}$ and $F_{0bc}$. Because $F_{a0b}$ is trace free it remains to control its anti-symmetric part. The relation $F_{[jkl]} = 0$ gives
\[
- Q_c\,\epsilon^c\,_{ab} = \frac{1}{2}\,F_{0de}\,\epsilon_c\,^{de}\,\epsilon^c\,_{ab}
= F_{0ab} = F_{a0b} - F_{b 0 a},
\]
whence
\[
F_{a0b} = - P_{ab} - \frac{1}{2}\,\epsilon_{abc}\,Q^c.
\]
Because $F_{abc}\,\epsilon^{abc} = 0$, the field $F_{acd}\,\epsilon_{b}\,^{cd}$ is trace free. Contracting its anti-symmetric part suitably twice with epsilons and using that gives $F^j\,_{jl} = 0$ gives
\[
F_{[a}\,^{cd}\,\epsilon_{b]cd} = - F_d\,^{dc}\,\epsilon_{cab} = - F_{00c}\,\epsilon^c\,_{ab}
= P_c\,\epsilon^c\,_{ab},
\]
and thus
\[
F_{abc} = \frac{1}{2}\,Q_{ad}\,\epsilon_{bc}\,^d - h_{a[b}\,P_{c]}.
\]
Observing now the reduced equations $P_{ab} = 0$ and $Q_{ab} = 0$, the remaining content of
$F_{jkl}$ is then described by the formula
\[
F_{jkl} = 3\,U_j\,P_{[k}\,U_{l]} - g_{j[k}\,P_{l]}
+ Q_i\,(U_j\,\epsilon^i\,_{kl} - \epsilon^i\,_{j[k}\,U_{l]}).
\]
Inserting this into $\nabla^j\,F_{jkl}$ and projecting suitably gives his
\[
(\nabla_U\,P_l)\,h^l\,_i + \frac{1}{2}\,\epsilon_i\,^{kj}\,\nabla_k\,Q_j = \nabla^j\,F_{jkl}\,U^k\,h^l\,_i
+ P(Z),
\]
\[
(\nabla_U\,Q_l)\,h^l\,_i
- \frac{1}{2}\,\epsilon_i\,^{kj}\,\nabla_k\,P_j = \frac{1}{2}\,\nabla^j\,F_{jkl}\,\epsilon_i\,^{kl} + P(Z).
\]
Working then out $\nabla^j\,F_{jkl}$ explicitly and observing (\ref{nabla-M-rel})
one finally gets equations of the form
\begin{equation}
\label{P-subs-equ}
P_{a, \,0} + \frac{1}{2}\,\epsilon_a\,^{bc}\,D_b\,Q_c = P(Z),
\end{equation}
\begin{equation}
\label{Q-subs-equ}
Q_{a,\,0} - \frac{1}{2}\,\epsilon_a\,^{bc}\,D_b\,P_c = P(Z).
\end{equation}
For given right hand sides this is a symmetric hyperbolic system for the fields $P_a$ and $Q_a$.
\vspace{.2cm}
We have seen above that solutions to the reduced equations for suitably arranged initial data
satisfy $N_j = 0$ and $A = 0$.
Equations (\ref{T-subs-equ}), (\ref{Delta-imkl-subs}), (\ref{B-C-D-subs}),
(\ref{1-H-red-equ}), (\ref{2-H-red-equ}), (\ref{P-subs-equ}), (\ref{Q-subs-equ}) constitute a system of differential equations for those of the remaining components of the zero quantities (\ref{zero-quantities})
which do not vanish already because of gauge conditions or the reduced equation. The system is symmetric hyperbolic and has characteristics which are time-like or null with respect to the metric
$g_{\mu\nu}$ that is supplied by the reduced system.
\vspace{.2cm}
\noindent
{\it It follows that a solution to the reduced system for data that satisfy the conformal constraints on the initial slice satisfies on the domain of dependence of the initial slice the gauge conditions and the complete set of conformal Einstein-$\lambda$-dust equations.}
\section{Existence and strong future stability}
\label{ex-strong-stab}
In this section the properties of the conformal field equations derived above and standard results about quasi-linear symmetric hyperbolic systems will be used to draw conclusions on the global structure of solutions to the Einstein-$\lambda$-dust equations.
Since we are mainly interested in $C^{\infty}$ solutions and not in the weakest possible smoothness assumptions on the data we refrain from specifying Sobolev norms.
We refer to \cite{friedrich:1991} for details of the patching arguments in the context of Cauchy stability and for some relevant PDE reference.
\subsection{Existence of asymptotically simple solutions}
To construct solutions to the Einstein-dust equations with positive cosmological constant
$\lambda$ that admit a smooth conformal boundary in their infinite future we consider Cauchy problems for the reduced field equations on $\mathbb{R} \times S$ where data are prescribed on the submanifold $\{0\} \times S$. We identify the latter diffeomorphically
with the manifold $S$ underlying a given {\it asymptotic end data set} as considered in section
\ref{as-end-dat}. The {\it conformal time} variable $\tau$ in the reduced field equations will correspond to the factor $\mathbb{R}$ above and it will be assumed that $\tau = 0$ on $S$.
The conformal gauge source function represented by the Ricci scalar $R[g]$ of the conformal metric $g$ to be constructed will be required to vanish and it is assumed that the condition $R[g] = 0$ is also underlying the construction of the given asymptotic end data.
A fixed gauge source function will in general only work well for some limited time. For our purpose this will suffice, however, because it will be arranged that a finite interval of the conformal time
$\tau$ will cover an interval of {\it physical} time of infinite extent.
\vspace{.1cm}
Since $S$ is compact and may have complicated topology, we use the fact that the hyperbolicity of the reduced equation allows us to obtain a solution on a neighborhood of
$S \sim \{0\} \times S$ in $\mathbb{R} \times S$ by patching together local solutions.
Compactness implies that $S$ can be covered by a finite number of open subsets
$V_A$, $A = 1, 2, \ldots , k$, of $S$ which carry
smooth local coordinates $x ^{\alpha}$, $\alpha = 1, 2, 3$, and a smooth frame field $e_a$, $a = 1, 2, 3$, that satisfies $h_{ab} \equiv h(e_a, e_b) = \delta_{ab}$, where $h$ denotes the 3-metric on $S$ supplied by the asymptotic end data. It can be assumed that their exist shrinkings $V'_A$
with compact closure $\overline{V'}_A$ in $V_A$ so that the $V'_A$ still define an open covering
and the boundary of $V'_A$ in $V_A$ is smooth. Standard results on symmetric hyperbolic systems then imply the existence of smooth solutions to the reduced field equations on open neighbourhoods
${\cal D}_A$ of $V'_A$ in $\mathbb{R} \times S$ which imply on $V'_A$ the data induced on
$V'_A$ by the asymptotic end data on $S$ in the gauge chosen on $V_A$. It can be assumed that the solution
extends smoothly to the closure of ${\cal D}_A$ in $\mathbb{R} \times S$ with
$\det(e^{\mu}\,_k) \neq 0$ so that
${\cal D}_A$ acquires a boundary which consists of (i) smooth hypersurfaces ${\cal H}^{\pm}_A$
in the future/past of ${\cal D}_A$ which are null with respect to the solution metric $g$
and approach
$\overline{V'}_A \setminus V'_A$ in their past/future, (ii) the intersection of $\overline {{\cal D}_A}$
with hypersurfaces $\{\tau = \tau_{\pm}\}$ in $\mathbb{R} \times S$
defined by some constants $\tau_- < 0 < \tau_+$ (which can be chosen to be the same for all
$V'_A$), and (iii) the three 2-dimensional
edges diffeomorphic to $\overline{V'}_A \setminus V'_A$ where these hypersurfaces approach each other. It can be assumed that the solution on ${\cal D}_A$ is globally hyperbolic
with respect to metric $g$.
The subsidiary system then implies that the full set of conformal
Einstein-$\lambda$-dust equations is satisfied on $\overline {{\cal D}_A}$.
If $p \in V'_A \cap V'_B$, there exists an open neighborhood $V_p \subset V'_A \cap V'_B$ of $p$
so that solutions are given in the domain of dependence ${\cal D}_{A, p}$ of $V_p$ in ${\cal D}_A$
as well as in the domain of dependence ${\cal D}_{B, p}$ of $V_p$ in ${\cal D}_B$. On $V_p$ these two solutions can be related to each other because the coordinate and frame transformations which relate the data induced on $V_p$ by the data on $V'_A$
and the data on $V'_B$ respectively are known explicitly. Because the gauge inherent in the reduced equations is evolved by invariant propagation laws along the invariantly defined flow lines of the flow field $U$, the coordinate and frame transformations
extend, independent of $\tau$, and allow us to relate the solution on ${\cal D}_{A, p}$ isometrically to the solution on
${\cal D}_{B, p}$. By extending the argument it follows that the solution induced on the domain of dependence of
$V'_A \cap V'_B$ in ${\cal D}_A$ can be identified isometrically with the solution induced on the domain of dependence of $V'_A \cap V'_B$ in ${\cal D}_B$.
By patching together the local solutions,
we obtain a smooth, globally hyperbolic solution to the
conformal Einstein-$\lambda$-dust equations
on a subset of the form
$M = [\tau_*, \tau_{**}] \times S$ of $\mathbb{R} \times S$
with constants $\tau_* < 0 < \tau_{**}$ so that the conformal factor obtained on $M$ satisfies
$\Omega > 0$ on
$\hat{M} = [\tau_*, 0[ \times S$
while $\Omega < 0$ on $\check{M} = ]0, \tau_{**}] \times S$.
\\
The
hypersurfaces $S_{\sigma} = \{\tau = \sigma = const.\}$ with $\tau_* \le \sigma \le \tau_{**}$ can be required to be space-like. In fact, with the co-normal to $\{\tau = const.\}$ given by
$n_{\mu} = - a\,\tau_{,\mu}$
the future directed normal is given by
\[
n^{\mu} = \frac{ - a\,g^{\mu 0}}{\sqrt{|a^2\,g^{00}|}} = - \frac{\eta^{jk}\,e^{\mu}\,_j\,e^0\,_k}
{\sqrt{|\eta^{jk}\,e^{0}\,_j\,e^0\,_k|}}
= \frac{\delta^{\mu}\,_0 - \eta^{ab}\,e^{\mu}\,a\,e^0\,_b}{\sqrt{1 - \eta^{ab}\,e^{0}\,_a\,e^0\,_b}},
\]
and the condition $n_{\mu}\,n ^{\mu} = - 1$ implies the expression
\begin{equation}
\label{2nd-a-form}
a = \frac{1}{\sqrt{1 - \eta^{ab}\,e^0\,_a\,e^0\,_b}}.
\end{equation}
Moreover,
\begin{equation}
\label{2nd-n-form}
n^{\mu}
= a\,(\delta^{\mu}\,_0 - \eta^{ab}\,e^{\mu}\,_a\,e^0\,_b)
=
\frac{1}{a}\,(U^{\mu}
- u^{\alpha}\,\delta^{\mu}\,_{\alpha})
\quad \mbox{with} \quad u^{\alpha} = a^2\,\eta^{ab}\,e^{\alpha}\,_a\,e^0\,_b.
\end{equation}
We thus require that
\begin{equation}
\label{2nd-e-restr}
e^{0}\,_a\,e^{0}\,_b\,\eta^{ab} \le const. < 1 \quad \mbox{on} \quad M,
\end{equation}
which can be achieved with suitable choices of $\tau_*$ and $\tau_{**}$ because $e^0\,_a = 0$ on $S_0$. The
hypersurfaces $S_{\sigma} $ will then be Cauchy hypersurfaces for $(M, g_{\mu\nu})$.
To simplify things,
so that we only need to consider the regularized reduced equations involving the unknowns
$\zeta_{ab}$ and $\xi$,
it will also be assumed that $\Omega_{,\tau} < 0$ on $M$, which makes sense because $\Omega_{,\tau} = - \nu$
on $S_0$.
\vspace{.1cm}
The metric $g_{\mu\nu}$, the conformal factor $\Omega$, the flow field $U$ and the density function $\rho$ are then such that
the `physical' fields
\begin{equation}
\label{conf-phys-field-rel}
\hat{g}_{\mu\nu} = \Omega^{-2}\,g_{\mu\nu}, \quad
\hat{U}_{\mu} = \Omega^{-1}\,U_{\mu}, \quad
\hat{\rho} = \Omega^{3}\,\rho
\end{equation}
define a solution to the Einstein-$\lambda$-dust equations on the manifold $\hat{M}$ with
$\hat{\rho} \ge 0$ on $\hat{M}$. Extending smoothly to $S_{\tau_*}$, this solution
admits an extension into the past of $S_{\tau_*}$ but we are not interested here in controlling something like a maximal globally hyperbolic solution. What is important for us is that
the set ${\cal J}^+ \equiv S_0 = \{\Omega = 0\}$ defines for the
solution $(\hat{M}, \hat{g}_{\mu\nu})$ a smooth conformal boundary at future time-like infinity.
\vspace{.1cm}
Equations (\ref{O-torsion-free condition}) to (\ref{f-rho-equ}) are invariant under the transformation
which implies the map
\[
\Omega \rightarrow - \Omega, \quad
\nabla_k\Omega \rightarrow - \nabla_k\Omega, \quad s \rightarrow - s, \quad W^i\,_{jkl} \rightarrow - W^i\,_{jkl},
\quad \rho \rightarrow - \rho, \quad \nabla_k\rho \rightarrow - \nabla_k\rho,
\]
but leaves the fields $e^{\mu}\,_k$, $\Gamma_i\,^j\,_k$, $L_{jk}$, and $U^k$ unchanged.
It follows that after performing this transition on $M$ and restricting to $\check{M}$
gives us another solution to the Einstein-$\lambda$-dust equations on the manifold $\check{M}$. It follows, however, that then $\hat{\rho} \le 0$ on $\check{M}$.
For this solution the set $ \{\Omega = 0\}$ defines a smooth conformal boundary in the infinite past. In this article we shall not be interested in this solution any further.
\vspace{.1cm}
Two facts have been used above to obtain solutions whose conformal structures extend smoothly across future time-like infinity so as to define there smooth conformal boundaries: (i)
The Einstein-$\lambda$-dust equations admit conformal representations which imply with suitable gauge conditions systems of evolution equations that are hyperbolic irrespective of the sign of the conformal factor $\Omega$, (ii)
some requirements needed to ensure the existence of smooth conformal extensions
{\it are put in by hand} by starting from asymptotic end data.
The case of the Nariai solution, an explicit, geodesically complete solution to the Einstein-$\lambda$-dust equations
with $\hat{\rho} = 0$, shows that that the property (i) is by itself not sufficient to ensure the existence of a smooth conformal boundary (see \cite{friedrich:beyond:2015}). This raises the question whether the use of asymptotic end data may result in the construction of a very restricted class of solutions.
The following argument, introduced in the vacuum case in \cite{friedrich:1986b} and used in the presence of conformally invariant matter fields in \cite{friedrich:1991},
shows that the existence of smooth asymptotic conformal structures is in fact a fairly general feature of solutions to the Einstein-$\lambda$-dust equations. The smooth extensibility
of the conformal structure across future time-like infinity will
be {\it derived} as a consequence of the property (i) of the Einstein-$\lambda$-dust equations and the existence of a given reference solution that admits a smooth asymptotic structure.
\subsection{Strong future stability of the solutions}
Let
\begin{equation}
\label{B-unknnowns}
\Delta = (e^{\mu}\,_k, \,\, \Gamma_i\,^j\,_k, \,\, \zeta_{ab}, \,\,\xi, \,\,
f_k, \,\, \Omega, \,\,\nabla_i\Omega,
\,\, s, \,\, L_{jk}, \,\, W^i\,_{jkl}, \,\,
U^k, \,\, \rho),
\end{equation}
be one of the solutions constructed above.
The associated physical fields $\hat{g}_{\mu\nu} = \Omega^{-2}\,g_{\mu\nu}$,
$\hat{U}^{\mu} = \Omega\,U^{\mu}$, $\hat{\rho} = \Omega^3\,\rho$
then induce on the Cauchy hypersurface $S' \equiv S_{\tau_*}$ with local coordinates
$x^{\alpha}$, $\alpha = 1, 2, 3$,
standard Cauchy data
$\hat{\delta} = (\hat{h}_{\alpha \beta}, \,\hat{\kappa}_{\alpha \beta}, \,\hat{u}^{\alpha}, \,\hat{\rho})$,
i.e. a solution to the constraints (\ref{hat-Ham-constr}) and (\ref{hat-mom-constr}),
where $\hat{u}^{\alpha}$ denotes the orthogonal projection of $\hat{U}^{\mu}$ onto $S'$.
As a first step towards showing that the asymptotic simplicity of the solution above is preserved under sufficiently small perturbations of the data $\hat{\delta}$, any given standard Cauchy data set on $S'$ needs to be transformed into a suitable Cauchy data set for the conformal field equations. This involves several transformations and a suitable handling of the gauge freedom which will be discussed now by showing how the restriction of $\Delta$ to $S'$ is obtained from $\hat{\delta}$.
Conformal data $\delta = (h_{\alpha \beta}, \,\kappa_{\alpha \beta}, \,u^{\alpha}, \,\rho)$ on $S'$
are obtained from the standard data $\hat{\delta}$
by using the functions $\Omega > 0$ and $\nabla_U\Omega < 0$ on $S'$ to define
\[
h_{\alpha \beta} = \Omega^2\,\hat{h}_{\alpha \beta}, \quad
u^{\alpha} = \Omega^{-1}\,\hat{u}^{\alpha}, \quad
\rho = \Omega^{-3}\,\hat{\rho},
\]
and, using the transformation law of second fundamental forms under conformal rescalings,
\[
\kappa_{\alpha \beta} = \Omega\,(\hat{\kappa}_{\alpha \beta}
+ \hat{h}_{\alpha \beta}\,\nabla_n\Omega).
\]
Here $n$ denotes the future directed unit normal to $S'$ with respect to $g$, which is related to the flow vector field $U$ and its projection $u$ onto $S'$ (that represents the shift vector field on $S'$, see the ADM representation of $g$ below) by the relation
\[
n = \frac{1}{a}\,(U - u) \quad \mbox{with} \quad a = \sqrt{1 + h_{\alpha \beta}\,u^{\alpha}\,u^{\beta}},
\]
where the expression for the positive lapse function $a$ is obtained from
\[
- 1 = g(U, U) = a^2\,g(n, n) + g(u, u)
= - a^2 + h_{\alpha \beta}\,u^{\alpha}\,u^{\beta}.
\]
It follows that
\[
\nabla_n\Omega = \frac{1}{a}\,(\nabla_U\Omega - \Omega_{, \alpha}\,u^{\alpha}),
\]
can be calculated from the data given above.
\vspace{.2cm}
When starting from arbitrarily given standard Cauchy data $\hat{\delta}$ the functions $\Omega > 0$ and $\nabla_U\Omega < 0$ are not given but represent part of the conformal gauge freedom. Suitable choices will be discussed later.
\vspace{.2cm}
As a second step it will be convenient to derive all the unknowns entering the conformal field equations
in a $g$-orthonormal frame $c_k$ on $S'$ which is adapted to $S'$ in the sense that $c_0 = n$.
This frame, which is not needed in the final process, is introduced because it simplifies various discussions.
In a third step all the data will be expressed on $S'$ in terms of the
$g$-orthonormal frame $e_k$ satisfying $e_0 = U$.
\vspace{.2cm}
To remove the gauge freedom in the transition $c_k \rightarrow e_k$,
we prescribe a specific field of Lorentz transformations $K^i\,_j$ on $S'$ which map the $g$-orthonormal frame field $e_k$ with $e_0 = U$ onto a smooth $g$-orthonormal frame $c_j = K^i\,_j\,e_i$ field
with $c_0 = n$ by setting
\begin{equation}
\label{K-def}
K^i\,_j
= \left(
\begin{array}{cc}
K^0\,_0, & K^0\,_b\\
K^a\,_0, & K^a\,_b \\
\end{array}
\right)
=
\left(
\begin{array}{cc}
- g(c_0, e_0)\,\,\,\,, & g(c_0, e_b) \\
\eta^{ad}\,g(c_0, e_d), & \delta^a\,_b + \frac{1}{1 - g(c_0, e_0)} \,\eta^{ad}\,g(c_0, e_d)\,g(c_0, e_b) \\
\end{array}
\right).
\end{equation}
In terms of the frame coefficients $e^{\mu}\,_k$ given by the solution $\Delta$ this reads
\[
K^i\,_j
=
\left(
\begin{array}{cc}
\quad \quad a \quad \quad \,, & - a\,e^0\,_b \\
- a\,\eta^{ac}\,e^0\,_c\,, & \delta^a\,_b + \frac{a^2}{1 + a} \,\eta^{ac}\,e^0\,_c\,e^0\,_b \\
\end{array}
\right).
\]
It follows that indeed
\[
K^i\,_0\,e_i = K^0\,_0\,e_0 + K^a\,_0\,e_a = - g(c_0, e_0)\,e_0 + \eta^{ad}\,g(c_0, e_d)\,e_a
= g(c_0, e_i)\,\eta^{ij}\,e_j = c_0.
\]
In the following considerations (\ref{2nd-a-form}) and
(\ref{2nd-n-form}) will be useful. A direct calculation verifies that $\eta_{ij}\,K^i\,_k\,K^j\,_l = \eta_{kl}$.
The coefficients of the frame
$c_k$ are given in the coordinates $x^{\mu}$ by
\[
c ^{\mu}\,_k = \left(
\begin{array}{cc}
\,\,\,\,\,c^0\,_0\,\,\,\,\,\,, & 0\\
c^{\alpha}\,_0\,, & c^{\alpha}\,_b \\
\end{array}
\right) =
\left(
\begin{array}{cc}
\,\,\,\,\,\frac{1}{a}\,\,\,\,\,\,, & 0\\
- \frac{1}{a}\,u^{\alpha}\,, & e^{\alpha}\,_b + \frac{1}{1 + a}\,u^{\alpha}\,e^0\,_b \\
\end{array}
\right),
\]
and the coefficients of the 1-forms $\mu^k$ that satisfy
$c^{\mu}\,_k \,\mu^k\,_{\nu} = \delta^{\mu}\,_{\nu}$ are so that
\[
\mu^{k}\,_{\nu} = \left(
\begin{array}{cc}
\mu^0\,_0\,, & 0\\
\mu^{a}\,_0\,, & \mu^{a}\,_{\beta} \\
\end{array}
\right),
\]
with
\[
\mu^0\,_0 = a, \quad \quad
u^{\alpha} = (e^{\alpha}\,_b + \frac{1}{1 + a}\,u^{\alpha}\,e^0\,_b)\,\mu^b\,_0,
\]
\[
c^{\alpha}\,_b\,\mu^b\,_{\beta} =
(e^{\alpha}\,_b + \frac{1}{1 + a}\,u^{\alpha}\,e^0\,_b)\,\mu^{b}\,_{\beta} = \delta^{\alpha}\,_{\beta},
\]
whence
\[
\mu^{a}\,_{\alpha}\,(e^{\alpha}\,_b + \frac{1}{1 + a}\,u^{\alpha}\,e^0\,_b)= \delta^{a}\,_{b},
\quad \quad
\mu^{a}\,_{\alpha}\,u^{\alpha} = \mu^{a}\,_0.
\]
The comparison of
\[
g = \eta_{jk}\,\mu^j\,\mu^k
= - a^2\,d\tau^2 + \eta_{ab}\,\mu^a\,_{\alpha}\,\mu^b\,_{\beta}
(u^{\alpha}\,d\tau + dx^{\alpha})\,(u^{\beta}\,d\tau + dx^{\beta}),
\]
with the ADM representation
\[
g = - (a\, d\tau)^2 + h_{\alpha \beta}\,(u^{\alpha}\,d\tau + dx^{\alpha})\,(u^{\beta}\,d\tau + dx^{\beta})
\]
gives then
\[
h_{\alpha \beta} = \eta_{ab}\,\mu^a\,_{\alpha}\,\mu^b\,_{\beta},
\quad \quad
a = \sqrt{1 + u^{\alpha}\,u^{\beta} \,h_{\alpha \beta}}.
\]
With the frame $c_j$ defined above we set
\begin{equation}
\label{K-def}
M^i\,_j =
\left(
\begin{array}{cc}
- g(e_0, c_0)\,\,\,\,, & g(e_0, c_b) \\
\eta^{ad}\,g(e_0, c_d), & \delta^a\,_b + \frac{1}{1 - g(e_0, c_0)} \,\eta^{ad}\,g(e_0, c_d)\,g(e_0, c_b) \\
\end{array}
\right).
\end{equation}
In terms of the frame coefficients $c^{\mu}_k$ this can be written
\[
M^i\,_j =
\left(
\begin{array}{cc}
M^0\,_0, & M^0\,_b\\
M^a\,_0, & M^a\,_b \\
\end{array}
\right)
=
\left(
\begin{array}{cc}
\quad \quad a \quad \quad \,, & u_{\alpha}\,c^{\alpha}\,_b \\
\eta^{ac}\,u_{\alpha}\,c^{\alpha}\,_c\,, & \delta^a\,_b
+ \frac{1}{1 + a} \,\eta^{ac}\,u_{\alpha}\,c^{\alpha}\,_c\,u_{\beta}\,c^{\beta}\,_b \\
\end{array}
\right)
\]
\[
=
\left(
\begin{array}{cc}
\quad \quad a \quad \, ,& a\,e^0\,_b \\
a\,\eta^{ac}\,e^0\,_c\,, & \delta^a\,_b + \frac{a^2}{1 + a} \,\eta^{ac}\,e^0\,_c\,e^0\,_b \\
\end{array}
\right).
\]
A direct calculation shows that $M^i\,_j\,K^j\,_k = \delta^i\,_k$
and $e_k = M^j\,_k\,c_j$.
\vspace{.2cm}
Because the fields $c_a$, $a = 1,2,3$ are tangential to $S'$
we can set
\[
h'_{ab} = h_{\alpha \beta}\,c^{\alpha}\,_a\,c^{\beta}\,_b = \eta_{ab},
\quad \quad
\kappa'_{ab} = \kappa_{\alpha \beta}\,c^{\alpha}\,_a\,c^{\beta}\,_b
\]
where here and in the following a prime is used to indicate when a tensor field is given in terms of the frame $c_k$. Directional derivatives with respect to $c_k$ will also indicated by a prime, so that
$\nabla'_k = \nabla_{c_k}$ etc.
\vspace{.2cm}
When the data for the conformal field equations are to be constructed by starting from standard Cauchy data, the frame $e_k$ is not available. Instead, the frame $c_k$ has to be chosen first and $e_k$ will then be obtained by applying $M^i\,_k$. The
field $c_0$ is uniquely determined as the future directed unit normal to $S'$ but the frame $c_a$ tangent to $S'$ is only determined up to rotations. In the stability argument given below this freedom will have to be removed in a specific way.
\vspace{.2cm}
Connection coefficients with respect to the frame $c_k$ satisfying the relation
$\nabla_{c_i} c_k = \gamma_i\,^j\,_k\,c_j$ with respect to the Levi-Civita connection $\nabla$
given by $g$
can only be defined if the frame is defined near
$S'$. It will be convenient to extend the frame by the requirement $\nabla_{c_0}c_k = 0$ and to define
coordinates $\upsilon = x^{0'}$ and $x^{\alpha'}$
near $S'$ so that $x^{\mu'} = x^{\mu}$ on $S'$ and $<c_0, d\upsilon>\,=1$ and
$<c_0,x^{\alpha'}>\, = 0$.
The coordinates $x^{\mu'}$ are then Gauss coordinates based on $S'$ and the coefficients
$c^{\mu'}\,_k$ satisfy $c^{\mu'}\,_0 = \delta^{\mu'}\,_0$ and $c^{0'}\,_a = 0$ so that
$c_0 = \partial_{\upsilon}$. The coordinates $x^{\mu}$ and $x^{\mu'}$ satisfy
\[
\frac{\partial x^0}{\partial x^{0'}} = \,<n, d\tau> = \frac{1}{a}<U - u, d\tau>\, = \frac{1}{a}, \quad
\frac{\partial x^0}{\partial x^{\alpha'}} = 0,
\]
\[
\,\,\quad \quad \frac{\partial x^{\alpha}}{\partial x^{0'}} = \frac{1}{a}<U - u, d x^{\alpha}>\, = - \frac{1}{a}\,u^{\alpha}
,\quad
\frac{\partial x^{\alpha}}{\partial x^{\alpha'}} = \delta^{\alpha}\,_{\alpha'}
\quad \mbox{on $S'$},
\]
so that the relation $e^{\mu'}\,_k = M^j\,_k\,c^{\mu'}\,_j$ can be used to determine on $S'$
\[
e^{\mu}\,_k = \frac{\partial x^{\mu}}{\partial x^{\mu'}}\,c^{\mu'}\,_l\,M^l\,_k.
\]
The connection coefficients with respect to $c_k$ can now be defined. They satisfy
\[
\gamma_0\,^j\,_k = 0, \quad \gamma_a\,^0\,_b = \kappa'_{ab} = \kappa'_{ba},\quad
\gamma_a\,^c\,_0 = \kappa'_{ab}\,h'^{bc}, \quad
\gamma_a\,^d\,_b\,c_d = D_{c_a} c_b \quad \mbox{on $S'$},
\]
where $D$ denotes the Levi-Civita connection of the metric $h$ on $S'$.
The connection coefficients in the frame $c_k$ are related to the connection coefficients in the frame $e_k$ by
\[
\Gamma_i\,^j\,_k
= K^j\,_n
\left(M^n\,_{k, \,\mu'}\,e^{\mu'}\,_i + \gamma_l\,^n\,_p\,M^l\,_i\,M^p\,_k \right)
\]
\[
= K^j\,_n
\left(M^n\,_{k, \,0'}\,e^{0'}\,_i + M^n\,_{k, \,\alpha'}\,e^{\alpha'}\,_i
+ \gamma_l\,^n\,_p\,M^l\,_i\,M^p\,_k \right).
\]
Apart from $M^n\,_{k, \,0'}$, which can only be determined by taking into account the evolution equations for the frame $e_k$,
all the other terms in the expression above can be calculated from the data available so far.
The relation $e^{\mu'}\,_k = M^j\,_k\,c^{\mu'}\,_j$ implies
\[
e^{\mu'}\,_{k,\,0'} = M^j\,_{k,\,0'}\,c^{\mu'}\,_j + M^j\,_k\,c^{\mu'}\,_{j,\,0'}.
\]
The first structural equation with respect to the frame $c_k$ gives
\[
c^{\mu'}\,_{j,\,0'} = \delta^{\mu'}\,_{\alpha'}\,\delta^a\,_j\,c^{\alpha'}\,_{a,\,0'}
= - \delta^{\mu'}\,_{\alpha'}\,\delta^a\,_j\,\gamma_a\,^b\,_0\,c^{\alpha'}\,_b
= - \delta^{\mu'}\,_{\alpha'}\,\delta^a\,_j\,\kappa'_{ac}\,h'^{bc}\,c^{\alpha'}\,_b
\quad \mbox{on $S'$}.
\]
The field $e_0 = U = U'^k\,c_k$, given on $S'$ by $U = a\,c_0 + u'^a\,c_a$
with $u'^a = \mu^a\,_{\alpha'}\,u^{\alpha'}$, must thus satisfy by (\ref{first-g-conf-geod-equ})
\[
0 = U'^k\,_{, \,\mu'}\,c ^{\mu'}\,_l\,U'^l + U'^l\,U'^j\,\gamma_l\, ^k\,_j
+ <U,f>U'^k + f'^k \quad \quad
\]
\[
\quad = a\,U'^k\,_{, \,0'} + U'^k\,_{, \,\alpha'}\,u ^{\alpha'}
+ U'^l\,U'^j\,\gamma_l\, ^k\,_j + <U,f>U'^k + f'^k \quad \mbox{on $S'$}.
\]
The fields $e_a = e'^k\,_a\,c_k$ must satisfy $\mathbb{F}_U e_a = 0$, which implies with
(\ref{first-g-conf-geod-equ})
\[
0 = a\,e'^k\,_{c,\,0'} + e'^k\,_{c,\,\alpha'}u^{\alpha'} +U'^i\,e'^j\,_c\gamma_i\,^k\,_j
+ f'_l\,e'^l\,_c\,U'^k - U'_l\,e'^l\,_c\,f'^k \quad \mbox{on $S'$}.
\]
These relations determine $c^{\mu'}\,_{j,\,0'}$, $e^{\mu'}\,_{k,\,0'}$ whence
$M^j\,_{k,\,0'}$ and $\Gamma_i\,^j\,_k $ uniquely from the given data on $S'$ once $f'_k$ is
given there.
\vspace{.2cm}
Our gauge requires that the tensorial field
\[
N'_k = \nabla'_k\Omega + (\nabla_U\Omega +
\Omega <U, f>)\,U'_k + \Omega\,f'_k,
\]
vanishes on $S'$. The condition that its orthogonal projection $N'_a$ into $S'$ vanishes gives
\[
f'_a = - \frac{1}{\Omega}\,\{ \nabla'_a\Omega + (\nabla_U\Omega +
\Omega <U, f>)\,u'_a\} \quad \mbox{on $S'$}.
\]
If this is satisfied it follows with $U_k = U'_i\,\,M^i\,_k$, $N_k = N'_i\,\,M^i\,_k$
\[
0 = U^k\,N_k = U'^k\,N'_k = a\,n'^k\,N'_k,
\]
and thus together $N'_k = 0$. The relation
\[
f'_0 = n'^k\,f'_k = \frac{1}{a}\,(<U, f> - u'^a\,f'_a),
\]
shows that $f'_k$ is determined from the data given on $S'$ only up to $f_0 = \,<U, f>$. This is consistent with the fact remarked on earlier
that the quantity $f_0$ is pure gauge and can be chosen arbitrarily. With a suitable choice of $f_0$
(made in a specific way later) we can the set $f_k = f'_j\,\,M^j\,_k$.
\vspace{.2cm}
The Einstein equations and the conformal rescaling of the density imply
$R[\hat{g}] = 4\,\lambda + \Omega^3\rho$. With this the conformal transformation law of the Ricci scalar gives
\[
\nabla_{\mu}\nabla^{\mu}\Omega + \frac{1}{6}\,R[g]\,\Omega
= \frac{2}{\Omega}\,\nabla_{\mu}\Omega\,\nabla^{\mu}\Omega
+ \frac{1}{6\,\Omega}\,R[\hat{g}]
= \frac{2}{\Omega}\,\nabla_{\mu}\Omega\,\nabla^{\mu}\Omega
+ \frac{1}{6\,\Omega}\,(4\,\lambda + \Omega^3\rho).
\]
With the gauge condition $R]g| = 0$ we thus set
\[
4\,s = \nabla'_k\nabla'^k\Omega = \frac{2}{\Omega}\,\nabla'_{i}\Omega\,\nabla'^{i}\Omega
+ \frac{1}{6\,\Omega}\,(4\,\lambda + \Omega^3\rho).
\]
The second equation determines $\partial^2_{\upsilon}\,\Omega = c_0(c_0\,\Omega)$
in terms of known data because
\[
\nabla'_k\nabla'^k\Omega = - \nabla'_0\nabla'_0\Omega + \eta^{ab}\,\nabla'_a\nabla'_b\Omega
= - c_0(c_0\,\Omega)
+ \eta^{ab} (D'_aD'_b\Omega - \kappa'_{ab}\,\nabla_n\Omega)
\quad \mbox{ on $S'$}.
\]
Thus $s$ and $\nabla'_j\nabla'_k\Omega$ are determined on $S'$ from known data and the scalar equation (\ref{f-alg-equ})
is satisfied there. Given $s$ and $\chi_{ab} = \Gamma_a\,^0\,_b$, the fields
$\zeta_{ab}$ and $\xi$ are then defined on $S'$ by (\ref{regularizing-unknowns}).
\vspace{.2cm}
The conformal transformation law of the Schouten tensor, the field equations, and the conformal rescalings of the flow vector field and the density give
\[
L_{\mu\nu} = \hat{L}_{\mu\nu} - \frac{1}{\Omega}\,\nabla_{\mu}\,\nabla_{\nu}\Omega
+ \frac{1}{2\,\Omega^2}\,\nabla_{\rho}\Omega\,\nabla^{\rho}\Omega\,g_{\mu\nu}
\]
\[
= \frac{1}{6}\,\lambda\,\Omega^{-2}\,g_{\mu\nu}
+ \Omega\, \rho\left(\frac{1}{2}\,U_{\mu}\,U_{\nu} + \frac{1}{6}\,g_{\mu\nu}\right)
- \frac{1}{\Omega}\,\nabla_{\mu}\,\nabla_{\nu}\Omega
+ \frac{1}{2\,\Omega^2}\,\nabla_{\rho}\Omega\,\nabla^{\rho}\Omega\,g_{\mu\nu},
\]
and we set
\[
L'_{i j} =
\frac{1}{6}\,\lambda\,\Omega^{-2}\,g'_{i j}
+ \Omega\,\rho\left(\frac{1}{2}\,U'_{i}\,U'_{j} + \frac{1}{6}\,g'_{i j}\right)
- \frac{1}{\Omega}\,\nabla'_{i}\,\nabla'_{j}\Omega
+ \frac{1}{2\,\Omega^2}\,\nabla'_{l}\Omega\,\nabla'^{l}\Omega\,g'_{i j} \quad \mbox{ on $S'$ }.
\]
By the way $\nabla'_0\nabla'_0\Omega$ has been determined above it follows that
$g'^{ik}\,L'_{ik} = \frac{1}{6}\,R[g] = 0$. The appropriate data on $S'$ for the reduced field equations
are then given by $L_{jk} = L'_{il}\,\,M^i\,_j\,M^l\,_k$.
\vspace{.2cm}
To determine the rescaled conformal Wey tensor we observe that the Gauss and the Codazzi equation
with respect to $S'$ read in terms of the frame $c_k$}
\[
R'_{abcd}[g] = R'_{abcd}[h] + \kappa'_{a c}\,\kappa'_{b d} - \kappa'_{a d}\,\kappa'_{b c},
\]
\[
n'^kR'_{kabc}[g] = D'_{c} \kappa'_{b a} - D'_{c}\,\kappa'_{d a},
\]
where the fields on the right hand sides can be determined from the data available so far. With
$L'_{jk}$ as given above, the general relation
\[
R'_{i j k l}[g] =
2\,\{g'_{i[k}\,L'_{l] j} +
\,L'_{i[k}\,g'_{l] j}\}
+ C'_{i j k l},
\]
then allows us to calculate the components
$C'_{abcd}[g]$ and $n'^kC'_{kabc}[g]$ of the conformal Weyl tensor.
The conformal Weyl
tensor admits the decomposition
\[
C'_{ijkl} = 2 \left( k'_{i[k}\,e'_{l]j} - k'_{j[k}\,e'_{l]i}
+ n'_{[k}\,m'_{l]m}\,\epsilon'^{m}\,_{ij}
+ n'_{[i}\,m'_{j]m}\,\epsilon'^{m}\,_{kl} \right).
\]
where $h'_{jk} = g'_{jk} + n'_j\,n'_k$ and $k'_{jk} = g'_{jk} + 2\,n'_j\,n'_k$ and
$e'_{ik} = h'_i\,^m\,h'_k\,^n\,C'_{mjnl}\,n'^j\,n'^l$ and
$m'_{ik} = h'_i\,^m\,h'_k\,^n\,C'^*_{mjnl}\,n^j\,n^l$ with
$C'^*_{ijkl} = \frac{1}{2}\,C'_{ijmn}\,\epsilon'^{mn}\,_{kl}$
denote the electric and magnetic part of the conformal Weyl tensor
{\it with respect to} $n$ in the frame $c_k$ respectively.
It holds $e'_{ij} = e'_{ji}$, $e'_{ij}\,n'^j = 0$, $e'_i\,^i = 0$ and similar
relations hold for $m'_{ij}$.
It follows that
\[
C'_{abcd} = 2\,( h'_{a[c}\,e'_{d]b} + e'_{a[c}\,h'_{d]b})
\quad \mbox{whence} \quad
e'_{bd} = h'^{ac}\,C'_{abcd},
\]
and
\[
n'^k C'_{kbcd} = 2\,(n'_{[i}\,m'_{j]m}\,\epsilon'^{m}\,_{kl})
\quad \mbox{whence} \quad
m'_{ab} = - \frac{1}{2}\,n'^k\,C'_{kbcd}\,\epsilon'_{b}\,^{cd}.
\]
The tensors $C'_{ijkl}$ and $W'_{ijkl} = \Omega^{-1}\,C'_{ijkl}$ whence
$W_{ijkl} = W'_{mnpq} \,M^m\,_i\,M^n\,_j\,M^p\,_k\,M^q\,_l$ can thus be determined from the given data and thus also $U$-electric and -magnetic parts $w_{ij}$ and $w^*_{kl}$ of
$W_{ijkl}$ which enter the reduced conformal field equations.
\vspace{.2cm}
The conformal field equations and their unknowns are derived from the Einstein equations by conformal rescalings, the use of various differential identities, and the use of the frame formalism.
This leaves a coordinate, frame, and conformal gauge freedom which is controlled by suitable initial data and propagation laws for the coordinates, the frame field, and the conformal factor
(controlled here implicitly by the requirement $R[g]= 0$). Following this procedure it follows from the discussion above how to derive from a given smooth solution
$\hat{\delta} = (\hat{h}_{\alpha \beta}, \,\hat{\kappa}_{\alpha \beta}, \,\hat{u}^{\alpha}, \,\hat{\rho})$
to the constraints (\ref{hat-Ham-constr}) and (\ref{hat-mom-constr})
and given smooth gauge dependent fields
\begin{equation}
\label{gauge-rep-fields-on-S'}
\Omega > 0, \,\,\, \nabla_U\Omega < 0, \,\,\, f_0 = \,<U, f>, \,\,\, \mbox{and a smooth
$h$-orthonormal field $c_a$ on $S'$},
\end{equation}
the unknowns $\Delta'_{S'}$ on $S'$ of the conformal field equations in the frame $c_k$ and also the unknowns
\begin{equation}
\label{conf-e-data-onS'}
\Delta_{S'}
= (e^{\mu}\,_k, \,\,\, \Gamma_i\,^j\,_k, \,\,\, \zeta_{ab}, \,\,\, \xi, \,\,\,
f_k, \,\,\, \Omega, \,\,\, \nabla_j\Omega,
\,\,\, s, \,\,\, L_{jk}, \,\,\, W^i\,_{jkl}, \,\,\,
U^k, \,\,\, \rho),
\end{equation}
in the frame $e_k$ on $S'$.
Written in terms of the frame $c_k$ and the frame coefficients $c^{\mu'}\,_k$ as defined above, the conformal field equations allow us to derive from the data $\Delta'_{S'}$ a formal expansion type solution in terms of the coordinate $\upsilon$ so that the complete set of conformal field equations is satisfied at all orders. The constraints are satisfied because of differential identities and the fact that the data $\hat{\delta}$ satisfy the `physical' constraints.
A similar formal expansion is obtained in terms of the coordinate $\tau$ if the equations and the data are expressed in terms of the frame $e_k$.
In this case the expansion coefficients are seen, however, to be the coefficients of a Taylor expansion of an actual smooth solution to the conformal field equations because the equations comprise the hyperbolic system of reduced conformal field equations.
The life time of the solution in the given gauge depends, of course, on the data (\ref{conf-e-data-onS'}) and in particular on the
choice of the free fields in (\ref{gauge-rep-fields-on-S'}). Suppose
\begin{equation}
\label{comparison-solution}
\Delta^{\star}(\tau)
= (e^{\star\,\mu}\,_k, \,\,\, \Gamma^{\star}_i\,^j\,_k, \,\,\, \zeta^{\star}_{ab}, \,\,\, \xi^{\star}, \,\,\,
f^{\star}_k, \,\,\, \Omega^{\star}, \,\,\, \nabla_j\Omega^{\star},
\,\,\, s^{\star}, \,\,\, L^{\star}_{jk}, \,\,\, W^{\star i}\,_{jkl}, \,\,\,
U^{\star k}, \,\,\, \rho^{\star}),
\end{equation}
is one of the solutions to the conformal field equations considered in the previous subsection.
It exists and is smooth for
$\tau_* \le \tau \le \tau_{**}$ with $ \Omega^{\star} \rightarrow 0$ as $\tau \rightarrow 0$
so that $S_0$ represents the conformal boundary at future time-like infinity for the physical solution associated with
$\Delta^{\star}(\tau)$. Denote by
$\Delta^{\star}_{S'} = \Delta^{\star}(\tau_*)$ the data for the reduced equations on $S'$ and by
$\hat{\delta}^{\star} = (\hat{h}^{\star}_{\alpha \beta}, \,\hat{\kappa}^{\star}_{\alpha \beta},
\,\hat{u}^{\star \alpha}, \,\hat{\rho}^{\star})$ the physical data induced by this solution on $S'$.
Let $\hat{\delta} = (S',\,\hat{h}_{\alpha \beta}, \,\hat{\kappa}_{\alpha \beta}, \,\hat{u}^{\alpha}, \,\hat{\rho})$ denote a smooth solution
to the constraints (\ref{hat-Ham-constr}) and (\ref{hat-mom-constr}), $\Delta_{S'}$ the corresponding initial data on $S'$ for the reduced conformal field equations as considered in (\ref{gauge-rep-fields-on-S'}), and $\Delta(\tau)$, where
$\tau \in [\tau_*, \tau_* + \tau^*[$ with some $ \tau^* > 0$, the solution to the conformal field equations determined by these data.
To compare the life times of the solutions $\Delta^{\star}(\tau)$ and $\Delta(\tau)$ the corresponding gauge conditions must be comparable. It will be assumed that the data $\Delta_{S'}$ have been constructed such that
\[
\Omega = \Omega^{\star}, \quad \nabla_U\Omega = \nabla_U\Omega^{\star}, \quad
f_0 = f^{\star}_0 \quad \mbox{on $S'$}.
\]
Let $h^{\star}_{\alpha \beta} = \Omega^{\star 2} \,\hat{h}^{\star}_{\alpha \beta}$,
and $h_{\alpha \beta} = \Omega^{2} \,\hat{h}_{\alpha \beta} = \Omega^{\star 2} \,\hat{h}_{\alpha \beta}$
denote the metric induced on $S'$ by the solution $\Delta^{\star}(\tau)$ and $\Delta(\tau)$ respectively.
As discussed above, the frame $e^{\star}_k$ given by the data $\Delta^{\star}_{S'}$ can be used to define a field of Lorentz transformation $K^{\star j}\,_l$ on $S'$ so that the relation $c^{\star}_k = K^{\star j}\,_k\,e^{\star}_j$ defines a frame field on $S'$
for which $c^{\star}_0$ is normal to $S'$. The fields $c^{\star}_a$, $a = 1, 2, 3$, then define an $h^{\star}$-orthonormal frame
field on $S'$. It will be assumed in the following that the $h$-orthonormal field $c_a$ has been chosen so
that $c_a = c^{\star}_c\,\alpha^c\,_a$ with a $3 \times 3$ matrix $\alpha^c\,_a$ that satisfies
$\alpha^1\,_1 > 0$, $\alpha^2\,_2 > 0$, $\alpha^3\,_3 > 0$, and $\alpha^c\,_a = 0$ if $a < c$.
The frame $c_a$ so defined is smooth and fixed uniquely so that
$\alpha^c\,_a \rightarrow \delta^a\,_c$ precisely if $c_a \rightarrow c^{\star}_a$.
The point of these choices is that the
space-time conditions $R[g^{\star}] = 0$ and $R[g] = 0$ combine with these gauge conditions on $S'$ to ensure that
$||\hat{\delta} - \hat{\delta}^{\star} || \rightarrow 0$ if and only if $|||\Delta_{S'} - \Delta^{\star}_{S'}||| \rightarrow 0$,
where the norms are meant to indicate Sobolev norms on $S'$ which are chosen corresponding to the differentiability order of the fields involved.
\vspace{.2cm}
We can invoke now the Cauchy stability property which holds for hyperbolic equations to conclude that
for data $\hat{\delta}$ sufficiently close to $\hat{\delta}^{\star}$ or, equivalently,
for data $\Delta_{S'}$ sufficiently close to $\Delta^{\star}_{S'}$ the solution $\Delta(\tau)$ of the conformal field equations
that develops from the data $\Delta_{S'}$ also exists in the interval $\tau_* \le \tau \le \tau_{**}$
and the conformal factor $\Omega$ supplied by $\Delta(\tau)$ is negative on $S_{\tau_{**}}$ \cite{kato}.
This conclusion may require repeated patchings (see \cite{friedrich:1991}).
There exists then a map $S \ni q \rightarrow \tau(q) \in ]\tau_*, \tau_{**}[$ so that
$\Omega(\tau(q) , q) = 0$ for $q \in S$
and $\Omega(\tau, q) > 0$ if $\tau_* \le \tau < \tau(q)$.
Equation (\ref{f-alg-equ}) then implies that on the subset
${\cal J}^+ = \{(\tau(q), q), q \in S\}$ of $\mathbb{R} \times S$ the gradient
$\nabla^{i}\Omega$ is time-like
for the metric $g$ supplied by $\Delta(\tau)$. It follows that ${\cal J}^+$ defines a smooth space-like hypersurface
which represents a conformal boundary in the infinite future of the set
$\hat{M} = \{(\tau, q) \in \mathbb{R} \times S \,|\,\tau_* \le \tau < \tau(q)\}$ on which the fields
$\hat{g}_{\mu\nu} = \Omega^{-2}\,g_{\mu \nu}$, $\hat{U}_{\mu} = \Omega^{-1}\,U^{\mu}$, $\hat{\rho}' = \Omega^3\,\rho$ define a
smooth solution to the Einstein-$\lambda$-dust equations. The smooth asymptotic end data induced by its conformal extension
$\Delta(\tau)$ on ${\cal J}^+ \sim S$ belongs then to the class of conformal end data considered in section
\ref{as-end-dat}.
Combining the results of the last two subsection we obtain Theorem \ref{main-result}.
\vspace{.5cm}
\noindent
{\bf Acknowledgements}: I would like to thank the relativity group at Cordoba in Argentina, where this work was begun,
for hospitality and discussions.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 2,492 |
The Land Trust welcomed Tetsuo, Yumiko, Yasuhide, Yuki and Paul to the famous Lady of the North to talk to them about the process of transforming land into biodiverse green space that benefits the local community.
When a Tsunami hit Japan in March 2011 it sparked a disaster at the Fukushima nuclear power plant on the East coast. The region of Yamakiya was evacuated due to high radiation levels soon after and work to decontaminate the area began.
The six-year evacuation period allowed teams to remove around 20 million tonnes of contaminated soil from affected areas and move it into temporary storage. This soil will now remain in storage for over 30 years.
The effects of this disaster have been left on the environment of the area, which once thrived on agricultural trade. They can no longer grow the rice, forage crops, tobacco and Eustoma flowers which were exported from Yamakiya.
In addition to these problems, the social impact on the area has been devastating. Before the disaster the population of Yamakiya was 1,246 but after the evacuation order was cancelled in March 2017 only around 150 people returned.
The scientists visited Northumberlandia as well as Eden Project this week in order to understand the decision making process of the transformation projects and the relationship with neighbouring residents, stakeholders and visitors.
The group were shown how Northumberlandia, which lies adjacent to the active Shotton Surface Mine in Cramlington, was built as part of the restoration of the mine so that local people could enjoy a new landscape while the mine is still operational.
The park has become a gateway to tourism for the area and has become a fantastic resource for the community to enjoy and for nature to flourish. The addition of a visitor centre and café has also boosted the visitor experience of the site, which attracted over 85,000 people in 2017-18.
The Land Trust has a wealth of experience in transforming once contaminated or derelict areas into public open green spaces with community and environmental assets and shared this experience with the visitors.
Alan Carter, Director of Portfolio Management at the Land Trust, said:
"It was fantastic to meet the team and show them the incredible work that has been carried out to transform this active mine for the local community.
"The team can now transfer the information we have given them into actions back in Yamakiya and start making a significant impact on the community there." | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 856 |
Малое Яровое — горько-солёное озеро в Славгородском районе Алтайского края в системе озёр центральной части Кулундинской равнины. Площадь водного зеркала составляет 35,2 км². Площадь водосборного бассейна — 1010 км². Лежит на высоте 98 метров над уровне моря. Средняя глубина озера около 2 метров, максимальная — около 5.
Озеро бессточное, имеет правильную округлую форму, на западном и восточном берегах имеются выходы артезианских вод общим дебитом 16000 л/час. Берега крутые и обрывистые, высотой от 3 до 5 м, у северного, восточного и южного берегов — песчаные отмели. С севера и запада впадают короткие пересыхающие ручьи.
Код водного объекта — 13020000311115200007197.
Примечания
Озёра Алтайского края
Славгород (Россия) | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 9,491 |
\section{Introduction}\label{sec:introduction}
The purpose of this paper is to motivate, analyze and experimentally validate a new preconditioner
for the overlap operator of lattice QCD. Lattice QCD simulations are among today's most demanding supercomputer applications \cite{PRACE:ScAnnRep12,PRACE:ScC12} and substantial resources are spent in these computations.
From a theoretical point of view, the overlap operator is particularly attractive since it respects an
important physical property, chiral symmetry, which is violated by
other lattice discretizations. From a practical point of view, the overlap operator has the disadvantage that its computational cost can be two orders of magnitude larger than when using standard discretizations.
The basic idea of the preconditioner we propose is to use a standard discretization of the Dirac equation to form a preconditioner for the overlap operator. This may be regarded as a variant of the fictitious (or auxiliary) space preconditioning technique \cite{Nepomnyaschikh1991}
that has been used for developing and analyzing multilevel preconditioners for various nonconforming
finite element approximations of PDEs; cf.~\cite{Oswald1996,Xu1996}. In this context, one works with a mapping from the
original space to a fictitious space, yielding an equivalent problem that is easier
to solve. Preconditioning is then done by (approximately) solving this equivalent problem. The convergence
properties of auxiliary space preconditioning depend on the choice of the
fictitious space, and its computational efficiency depends, in addition, on the efficiency of the solver used in that space; cf.\ \cite{Nepomnyaschikh1991}.
For the overlap operator in lattice QCD, choosing its
kernel---the Wilson-Dirac operator---as the auxiliary space preconditioner is facilitated by the fact that both operators
are defined on the same Hilbert space. In this way, the preconditioner for the former can be constructed
using an adaptive algebraic multigrid solver for the latter on the same finite dimensional lattice.
We note that similar approaches are possible for other QCD discretizations.
For example, the direct and strong coupling of the Wilson blocks used in the 5d domain wall operator \cite{Kaplan1992342} suggest that
a similar Wilson auxiliary-space preconditioner (with a more general mapping) may also be effective.
We demonstrate that the technique we develop in this paper is able to reduce the computational cost for solving systems with the overlap operator substantially, reaching speed-ups of a factor of 10 or more in realistic settings. The preconditioning technique thus contributes to making the overlap operator more tractable in lattice QCD calculations.
This paper is organized as follows. We start by explaining some physical background in section~\ref{QCD:sec} where we also introduce the two lattice discretizations of the continuum Dirac equation of interest here, the Wilson-Dirac operator and the overlap operator. In section~\ref{prec:sec} we give a precise mathematical analysis which shows that our preconditioner is effective in an idealized setting where both operators are assumed to be normal. Section~\ref{smearing:sec} shows that current {\em smearing} techniques in lattice QCD can be viewed as methods which drive the discretizations towards normality, thus motivating that we can expect the analysis of the idealized setting to also reflect the influence of the preconditioner in realistic settings. This is then confirmed by large-scale parallel numerical experiments reported in section~\ref{numerics:sec} which are performed for lattice configurations coming from state-of-the-art physical simulations.
\section{The Wilson-Dirac and the Overlap Operator in Lattice QCD}
\label{QCD:sec}
Quantum Chromodynamics (QCD) is a quantum field theory for the strong
interaction of the quarks via gluons and as such part of the standard model of
elementary particle physics. Predictions that can be deduced from this theory
include the masses and resonance spectra of hadrons---composite particles
bound by the strong interaction (e.g., nucleon, pion; cf.~\cite{Durr21112008}).
The Dirac equation
\begin{equation}\label{Dirac_eq}
(\mathcal{D}+m)\psi = \eta
\end{equation}
is at the heart of QCD. It describes the dynamics of the quarks and the
interaction of quarks and gluons.
Here, $\psi
= \psi(x)$ and $\eta = \eta(x)$ represent quark fields. They depend on $x$, the
points in space-time, $x=(x_0,x_1,x_2,x_3)$\footnote{Physical space-time is a
four-dimensional Minkowski space. We present the theory in Euclidean space-time
since this version can be treated numerically. The two versions are equivalent,
cf.~\cite{montvay1994quantum}.}. The gluons are represented in the Dirac
operator $\mathcal{D}$ to be discussed below, and $m$ is a scalar mass parameter.
It is independent of $x$ and sets the mass of the
quarks in the QCD theory.
$\mathcal{D}$ is given as
\begin{equation} \label{Dirac_continuum:eq}
\mathcal{D}=\sum_{\mu=0}^3\gamma_\mu \otimes \left( \partial_\mu + A_\mu \right)\,,
\end{equation}
where $ \partial_\mu = \partial / \partial x_\mu$ and $A$ is the gluon (background) gauge field with
the anti-hermitian traceless matrices $A_\mu(x)$ being elements of $\mathfrak{su}(3)$, the Lie
algebra of the special unitary group $\mathrm{SU}(3)$. The $\gamma$-matrices
$\gamma_0,\gamma_1,\gamma_2,\gamma_3 \in \mathbb{C}^{4 \times 4}$ represent the generators of the
Clifford algebra with
\begin{equation} \label{commutativity_rel:eq}
\gamma_\mu \gamma_\nu + \gamma_\nu \gamma_\mu = \begin{cases} 2 \cdot \mbox{id} &\mu = \nu\\0 & \mu \neq \nu \end{cases} \quad \text{ for } \mu,\nu=0,1,2,3.
\end{equation}
Consequently, at each point $x$ in space-time, the spinor $\psi(x)$, i.e., the quark field $\psi$ at
a given point $x$, is a twelve component column vector, each component corresponding to one of
three colors (acted upon by $A_\mu(x)$) and four spins (acted upon by $\gamma_\mu$).
For future use we remark that $\gamma_5 =
\gamma_0\gamma_1\gamma_2\gamma_3$ satisfies
\begin{equation} \label{gamma_commutativity:eq}
\gamma_5 \gamma_\mu = - \gamma_\mu \gamma_5, \enspace \mu=0,1,2,3.
\end{equation}
The only known way to obtain predictions in QCD from first principles and non-perturbatively, is to
discretize and then simulate on a computer. The discretization is typically formulated on an equispaced lattice.
In a lattice discretization, a periodic $N_t \times N_s^3$ lattice $\mathcal{L}$ with uniform lattice spacing $a$ is used, $N_s$ denoting the number of lattice points for each of the three space dimensions and $N_t$ the number of lattice points in the time dimension. A quark field $\psi$ is now represented by its values at each lattice point, i.e., it is
a spinor valued function $\psi: \mathcal{L} \to \psi(x) \in \mathbb{C}^{12}$.
The {\em Wilson-Dirac} discretization is the most commonly used discretization in lattice QCD simulations. It is obtained from
the continuum equation by replacing the covariant derivatives by centralized covariant finite differences on the lattice together with an additional second order finite difference stabilization term. The Wilson-Dirac discretization yields a local operator in the sense that it represents a nearest neighbor coupling on the lattice. To precisely describe the action of the Wilson-Dirac operator $D_W$ on a (discrete) quark field $\psi$ we introduce the shift vectors $ \hat{\mu} = (\hat{\mu}_0,\hat{\mu}_1, \hat{\mu}_2, \hat{\mu}_3) \in \mathbb{R}^4 $ in dimension $\mu$ on the lattice, i.e.,
$$
\hat{\mu}_{\nu} = \begin{cases} a & \mu=\nu \\ 0 & \text{else.} \end{cases}.
$$
Then
\begin{eqnarray}
\hspace*{-0.7em}(D_W\psi)(x) \, = \, \frac{m_0+4}{a} \psi(x)
&-& \frac{1}{2a}\sum_{\mu=0}^3 \left( (I_4-\gamma_\mu)\otimes U_\mu(x)\right) \psi(x+\hat{\mu}) \nonumber \\
&-& \frac{1}{2a}\sum_{\mu=0}^3 \left( (I_4+\gamma_\mu)\otimes U_\mu^H(x-\hat{\mu})\right) \psi(x-\hat{\mu}), \label{Wilson-Dirac:eq}
\end{eqnarray}
where $U_\mu(x)$ now are matrices from the Lie group SU(3), and the lattice indices $x\pm \hat{\mu}$ are to be understood periodically.
The mass parameter $m_0$ sets the quark mass (for further details, see~\cite{montvay1994quantum}), and we will write $D_W(m_0)$ whenever the dependence on $m_0$ is important. The matrices $U_\mu(x)$ are called
{\em gauge links}, and the collection $\mathcal{U} = \{U_\mu(x): x \in \mathcal{L}, \mu=0,\ldots3\}$ is termed the {\em gauge field}.
From \eqref{Wilson-Dirac:eq} we see that the
couplings in $D_W$ from lattice site $x$ to $x+\hat{\mu}$ and from $x+\hat{\mu}$ to $x$ are given by
\begin{equation} \label{DandDH_entries:eq}
(D_W)_{x,x+\hat{\mu}} = -(I_4-\gamma_\mu) \otimes U_\mu(x) \enspace \mbox{and} \enspace
(D_W)_{x+\hat{\mu},x} = -(I_4+\gamma_\mu) \otimes U^H_\mu(x),
\end{equation}
respectively. Due to the commutativity relations \eqref{gamma_commutativity:eq} we therefore have that
\[
(\gamma_5\otimes I_3) \big(D_W\big)_{x,x+\hat{\mu}} = (\gamma_5 \otimes I_3) \big(D_W\big)_{x+\hat{\mu},x},
\]
implying that with $\Gamma_5 = I_{n_{\mathcal{L}}} \otimes \gamma_5
\otimes I_3 $, $n_\mathcal{L}$ the number of lattice sites, we have
\begin{equation} \label{gamma_5_symmetry:eq}
(\Gamma_5 D_W)^H = \Gamma_5 D_W.
\end{equation}
This $\Gamma_5$-symmetry is a non-trivial, fundamental symmetry
which the discrete Wilson-Dirac operator inherits from a corresponding
symmetry of the continuum Dirac operator
\eqref{Dirac_continuum:eq}. The matrix $\Gamma_5$ is hermitian and
unitary, since $\gamma_5^H = \gamma_5$ and $\gamma_5^2 = id$; see
\cite{FroKaKrLeRo13}, e.g., and \eqref{gamma_commutativity:eq}.
The Wilson-Dirac operator and its clover improved variant (where a diagonal term is added in order
to reduce the local discretization error from $\mathcal{O}(a)$ to $\mathcal{O}(a^2)$) is an adequate
discretization for the numerical computation of many physical observables. It, however, breaks
another fundamental symmetry of the continuum operator, namely {\em
chiral symmetry}, which is of vital importance for some physical
observables like hadron spectra in the presence of magnetic fields,
for example. As was pointed out in \cite{Luscher:1998pqa},
a lattice discretization $D$ of $\mathcal{D}$ which obeys the Ginsparg-Wilson relation~\cite{WilsonGinsparg1982}
\begin{equation} \label{GW:eq}
\Gamma_5D + D\Gamma_5 = aD\Gamma_5 D
\end{equation}
satisfies an appropriate lattice variant of chiral symmetry. It has
long been unknown whether such a discretization exists until Neuberger
constructed it in~\cite{Neuberger1998141}. For convenience, the essentials of the arguments
in \cite{Neuberger1998141} are summarized in the following proposition and its proof.
\begin{proposition} \label{neuberger:prop}
\emph{Neuberger's overlap operator}
\begin{equation*}
D_N = \frac{1}{a} \left( \rho I + D_{W}(m^{\mathit{ker}}_{0})\Big(D_{W}(m^{\mathit{ker}}_{0})^H
(D_{W}(m^{\mathit{ker}}_{0})\Big)^{-\frac12} \right
\end{equation*} fulfills~\eqref{GW:eq} for $\rho=1$, has local discretization
error $\mathcal{O}(a)$, and is a stable discretization
provided $-2 < m_0^\mathit{ker} < 0$.
\end{proposition}
\begin{proof} We write $\mathcal{D}_{\mathcal{L}}$ for the
restriction of the continuum Dirac operator $\mathcal{D}$ from \eqref{Dirac_continuum:eq} to the lattice $\mathcal{L}$, i.e.,
$\mathcal{D}_{\mathcal{L}}$ is the finite dimensional operator which takes the same values as $\mathcal{D}$ at the points from $\mathcal{L}$.
The fact that the Wilson-Dirac operator has first order discretization error can then be expressed as\footnote{For simplicity, we consider here the ``naive'' limit $a \rightarrow 0$. In the full quantum theory
one has $\mathcal{D}_{\mathcal{L}} = D_W(m_0(a)) + \mathcal{O}(a)$ with the mass $m_0(a)$ of order $1/\log(a)$; see~\cite{montvay1994quantum}.}
\begin{equation*}
\mathcal{D}_{\mathcal{L}} = D_W(0) + \mathcal{O}(a),
\end{equation*}
implying
\begin{equation} \label{eq:WilsonDiscError}
\mathcal{D}_{\mathcal{L}} + \frac{m_{0}}{a}I = D_W(m_0) + \mathcal{O}(a)
\end{equation}
for any mass parameter $m_0$.
To construct $D_N$ we first note that any
operator $\widehat{D}$ that is $\Gamma_5$-symmetric and
fulfills~\eqref{GW:eq} can be parametrized by
\begin{equation}\label{eq:NeubergerParametrization}
a\widehat{D} = I+\Gamma_5 S,
\end{equation} with $S^{H} = S$ and $S^2 = I$. Both conditions are fulfilled for
\begin{equation*}
S = \Gamma_{5}D_{W}(m^{\mathit{ker}}_{0})\Big(D_{W}(m^{\mathit{ker}}_{0})^H
(D_{W}(m^{\mathit{ker}}_{0})\Big)^{-\frac12}, \quad -m_{0}^\mathit{ker} \in
\mathbb{R} \setminus \operatorname{spec}(D_{W}(0)).
\end{equation*}
Using~\eqref{eq:WilsonDiscError} we obtain
\begin{eqnarray*}
S &=& \Gamma_{5}\Big(\mathcal{D}_{\mathcal{L}} + \tfrac{m_{0}^\mathit{ker}}{a} I +
\mathcal{O}(a)\Big)\Big(\big(\mathcal{D}_{\mathcal{L}} + \tfrac{m_{0}^\mathit{ker}}{a}I
+ \mathcal{O}(a)\big)^H \big(\mathcal{D}_{\mathcal{L}} + \tfrac{m_{0}^\mathit{ker}}{a}I
+ \mathcal{O}(a)\big)\Big)^{-\frac12} .
\end{eqnarray*}
Since $\mathcal{D}$ is anti-selfadjoint, we have $\mathcal{D}_{\mathcal{L}}^H = -\mathcal{D}_{\mathcal{L}}$ and thus
\begin{eqnarray*} \lefteqn{\Big(\big(\mathcal{D}_{\mathcal{L}} + \tfrac{m_{0}^\mathit{ker}}{a} I + \mathcal{O}(a)\big)^H
\big(\mathcal{D}_{\mathcal{L}} + \tfrac{m_{0}^\mathit{ker}}{a} I + \mathcal{O}(a)\big)\Big)^{-\frac12} }
&& \\
&=& \tfrac{a}{|m_0^\mathit{ker}|} \Big(\big(\tfrac{a}{m_{0}^\mathit{ker}}\mathcal{D}_{\mathcal{L}} + I + \mathcal{O}(a^2)\big)^H
\big(\tfrac{a}{m_{0}^\mathit{ker}} \mathcal{D}_{\mathcal{L}} + I + \mathcal{O}(a^2)\big)\Big)^{-\frac12} \\
&= & \tfrac{a}{|m_{0}^\mathit{ker}|} I + \mathcal{O}(a^2),
\end{eqnarray*}
which in turn yields
\begin{equation}\label{eq:NeubergerOrder}
S = \Gamma_{5}\Big(\frac{a}{|m_0^\mathit{ker}|}\mathcal{D}_{\mathcal{L}} + \operatorname{sign}(m_{0}^\mathit{ker})I + \mathcal{O}(a^{2})\Big).
\end{equation}
Combining~\eqref{eq:NeubergerOrder} with \eqref{eq:NeubergerParametrization} we find
\begin{equation*}
a\widehat{D} = I + \frac{a}{|m_0^\mathit{ker}|}\mathcal{D}_{\mathcal{L}} + \operatorname{sign}(m_{0}^\mathit{ker})I + \mathcal{O}(a^{2})
\end{equation*}
so that for $m_{0}^\mathit{ker} < 0$ we have
\begin{equation*}
\widehat{D} = \frac{1}{|m_0^\mathit{ker}|}\mathcal{D}_{\mathcal{L}} + \mathcal{O}(a)
\end{equation*}
This shows that $\widehat{D}$ is a first order discretization of
$\mathcal{D}$. For it to be stable
one has to choose $-2 < m_{0}^\mathit{ker} < 0$, a result for which we do
not reproduce a proof here, referring to \cite{Neuberger1998141}
instead.
To conclude, note that $D_{N} = \widehat{D} +
{\tfrac{\rho - 1}{a}I}$, so $\rho - 1$ sets the
quark mass (see \eqref{Dirac_eq}) up to a renormalization factor. \qed
\end{proof}
Using the Wilson-Dirac operator as the kernel in the overlap operator is
the most popular choice, even though
other kernel operators have been investigated as well~\cite{deForcrand:2011ak}.
Neuberger's overlap operator has emerged as a popular scheme in lattice QCD over the years.\footnote{The domain wall
discretization satisfies \eqref{GW:eq} approximately and, hence, has
also been the focus of extensive research.} In the literature one
often writes
\begin{equation} \label{overlap_def:eq}
D_N = \rho I + \Gamma_5 {\rm sign}\big(\Gamma_5 D_W(m_0^\mathit{ker})\big)
\end{equation}
with ${\rm sign}$ denoting the matrix extension of the
sign function
\[
{\rm sign}(z) = \left\{ \begin{array}{ll} +1 & \mbox{if } \Re(z) > 0 \\
-1 & \mbox{if } \Re(z) < 0 \end{array} \right. .
\]
We note that ${\rm sign}(z)$ is undefined if $\Re(z) = 0$.
Since $\Gamma_5D_W(m_0)$ is hermitian, see \eqref{gamma_5_symmetry:eq}, the matrix ${\rm sign}(\Gamma_5 D_W(m_0^\mathit{ker}))$ is hermitian, too. Since $\Gamma_5^2 = I$, we also see
that the overlap operator satisfies the same $\Gamma_5$-symmetry as
its kernel $D_W$,
\begin{equation} \label{Gamma_5_symmetry_D_N:eq}
\big(\Gamma_5D_N\big)^H = \Gamma_5 D_N.
\end{equation}
We end this section with a characterization of the spectra of the
Wilson-Dirac and the overlap operator.
\begin{lemma} \label{props_Wilson_Overlap:lem}
\begin{itemize}
\item[(i)] The spectrum of the Wilson-Dirac matrix $D_W(m_0)$ is symmetric to the real axis and to the vertical line $\Re(z) = \frac{m_0+4}{a}$, i.e.,
\[
\lambda \in {\rm spec}\big(D_W(m_0)\big) \Rightarrow \overline{\lambda}, \, {\textstyle 2 \frac{m_0+4}{a}-\lambda} \in {\rm spec}\big(D_W(m_0)\big).
\]
\item[(ii)] The overlap operator $D_N$ is normal. Its spectrum is symmetric to the real axis and part of the circle with midpoint $\rho$ and radius 1, i.e.,
\[
\lambda \in {\rm spec}\big(D_N\big) \Rightarrow \overline{\lambda} \in {\rm spec}\big(D_N\big) \mbox{ and }
|\lambda-\rho| = 1.
\]
\end{itemize}
\end{lemma}
\begin{proof}
Recall that $\Gamma_5^H = \Gamma_5^{-1} = \Gamma_5$. If $D_W(m_0) x = \lambda x$, then by \eqref{gamma_5_symmetry:eq} we have $(\Gamma_5x)^HD_W = x^H(\Gamma_5D_W) = (\Gamma_5D_Wx)^H = \overline{\lambda} (\Gamma_5 x)^H$. This proves the first assertion in (i). For the second assertion,
consider a red-black ordering of the lattice sites.
where all red sites appear before black sites. Then the matrix $D_W(-\frac{4}{a})$ has the block structure
\[
D_W(\textstyle{-\frac{4}{a}}) = \left( \begin{array}{cc} 0 & D_{rb} \\ D_{br} & 0 \end{array} \right).
\]
Thus, if $x = (x_r, x_b)$ is an eigenvector of $D_W(-\frac{4}{a})$ with eigenvalue $\mu$, then $x' = (x_r, -x_b)$ is an eigenvector of $D_W(-\frac{4}{a})$ with eigenvalue $-\mu$. Applying this result to
$D_W(m_0)$ gives the second assertion in (i).
To prove (ii) we first remark that the sign function is its own inverse and that
$\Gamma_5 D_W(m_0)$ is hermitian. This implies that ${\rm sign}(\Gamma_5D_W(m_0))$ is its own inverse and hermitian, thus unitary. Its product with the unitary matrix $\Gamma_5$ is unitary as well,
implying that all its eigenvalues have modulus one. As a unitary matrix, this product is also normal. The term $\rho I$ in \eqref{overlap_def:eq} preserves normality and shifts the eigenvalues by $\rho$.
It remains to show that ${\rm spec}(D_N)$ is symmetric with respect to the real axis, which follows from
the $\Gamma_5$-symmetry \eqref{Gamma_5_symmetry_D_N:eq} of the overlap operator in the same manner as in (i). \qed
\end{proof}
For the purposes of illustration, Figure~\ref{spectra:fig} gives the spectra of the Wilson-Dirac operator and the overlap operator for a $4^4$
lattice. There, as everywhere else from now on, we set $a=1$ which is no restriction since $a^{-1}$ enters $D_W$ simply as a linear scaling. The
matrix size is just $3,\!072$, so all
eigenvalues and the sign function can be computed with standard methods for full matrices. The choice for $m_0$ in the Wilson-Dirac matrix
as a negative number such that the spectrum of $D_W$ lies in the right half plane with some eigenvalues being close to the imaginary axis is
typical. The choice for $m_0$ when $D_W(m_0)$ appears in the kernel of
the sign function is different (namely smaller, see Proposition~\ref{neuberger:prop}).
\begin{figure}
\begin{minipage}[t]{0.65\textwidth}
\centering\scalebox{0.61}{\input{./overlap_spec_wilson.tex}}
\end{minipage}
\hfill
\begin{minipage}[t]{0.34\textwidth}
\centering\scalebox{0.61}{\input{./overlap_spec_overlap.tex}}
\end{minipage}
\caption{Typical spectra of the Wilson-Dirac and the overlap operator
for a $4^4$ lattice.
}
\label{spectra:fig}
\end{figure}
\section{A Preconditioner Based on the Wilson-Dirac Operator} \label{prec:sec}
The spectral gaps to be observed as four discs with relatively few eigenvalues in the
left part of Figure~\ref{spectra:fig} are typical for the spectrum of the Wilson-Dirac operator and become even more pronounced as lattice sizes are increased. In practice, the mass parameter $m_0$ that appears
in the definition of the kernel
$D_W(m_0^\mathit{ker})$ of the overlap operator is chosen such that the origin lies in the middle of the leftmost of these discs. For this choice of $m_0^\mathit{ker}$ we now motivate why the
Wilson-Dirac operator $D_W(m_0^\prec)$ with adequately chosen mass $m_0^\prec$ provides a good preconditioner for the overlap operator.
To do so we investigate the connection of the spectrum of the overlap operator and the Wilson-Dirac operator in the special case that $D_W(0)$ is normal. This means that
$D_W(0)$ is unitarily diagonalizable with possibly complex eigenvalues, i.e.,
\begin{equation} \label{eq_D_normal_dec}
D_W(0) = X \Lambda X^H, \mbox{ with } \Lambda \mbox{ diagonal and $X$ unitary.}
\end{equation}
Trivially, then, $D_W(m_0)$ is normal for all mass parameters $m_0$ and
\begin{equation} \label{eq:D_m_normal}
D_W(m_0) = X(\Lambda + m_0I)X^H.
\end{equation}
To formulate the resulting non-trivial relation between the eigenvalues of
$D_N$ and its kernel $D_W(m_0^\mathit{ker})$ in the theorem below we use the notation ${\rm csign}(z)$ for a complex
number $z$ to denote its ``complex'' sign, i.e.,
\[
{\rm csign}(z) = z/|z| \mbox{ for } z \neq 0.
\]
The theorem works with the singular value decomposition
$A = U \Sigma V^H$ of a matrix $A$ in which $U$ and $V$ are orthonormal, containing the left and right singular vectors as their columns, respectively, and $\Sigma$ is diagonal with non-negative diagonal elements, the
singular values. The singular value decomposition is unique up to choices for the orthonormal basis of singular vectors belonging to the same singular value, i.e., up to transformations $U \to UQ, V \to VQ$
with $Q$ a unitary matrix commuting with $\Sigma$; cf.~\cite{GLMatrix1989}.
\begin{theorem} \label{the:D_D_N_eigs}
Assume that $D_W(0)$ is normal, so that $D_W(m)$ is normal as well for all $m \in \mathbb{C}$, and let $X$ and $\Lambda$ be from \eqref{eq_D_normal_dec}. Then we have
\begin{equation} \label{D_N_normal:eq}
D_N = X\big(\rho I + {\rm csign}(\Lambda + m_0I)\big)X^H.
\end{equation}
\end{theorem}
\begin{proof}
Let
\begin{equation} \label{gamma_D_eigendecomposition:eq}
\Gamma_5 D_W(m) = W_{m} \Delta_{m} W_{m}^H \mbox{ with } \Delta_m \mbox{ diagonal}, W_m \mbox{ unitary},
\end{equation}
be the eigendecomposition of the hermitian matrix $\Gamma_5 D_W(m)$.
We have two different representations for the singular value decomposition of
$\Gamma_5D_W(m)$,
\[
\begin{array}{rcll}
\Gamma_5 D_W(m) &=& \big(\Gamma_5X{\rm csign}(\Lambda+mI)\big) \cdot |\Lambda +mI| \cdot X^H
&\quad \mbox{(from \eqref{eq:D_m_normal})}\; , \\
\Gamma_5 D_W(m) &=& \big( W_m{\rm sign}(\Delta_m)\big) \cdot |\Delta_m| \cdot W_m^H
&\quad \mbox{(from \eqref{gamma_D_eigendecomposition:eq})} \; .
\end{array}
\]
Thus, there exists a unitary matrix $Q$ such that
\begin{equation} \label{themess:eq}
W_m = XQ \mbox{ and } W_m {\rm sign}(\Delta_m) = \Gamma_5 X {\rm csign}(\Lambda + mI) Q.
\end{equation}
Using the definition of $D_N$ in \eqref{overlap_def:eq}, the relations \eqref{themess:eq} give
\begin{eqnarray*}
D_N &=& \rho I + \Gamma_5 {\rm sign}(\Gamma_5 D_m) \\
&=& \rho I + \Gamma_5 W_m {\rm sign}(\Delta_m) W_m^H \\
&=& \rho I + \Gamma_5 \Gamma_5 X {\rm csign} (\Lambda + mI)Q (VQ)^X \\
&=& X(\rho I + {\rm csign}(\Lambda +mI)X^H.
\end{eqnarray*}
\qed
\end{proof}
We remark in passing that as an implicit consequence of the proof above we have that the eigenvectors
of $\Gamma_5 D_W(m) = \Gamma_5 D_W(0) + m\Gamma_5$ do not depend on
$m$. Thus if $D_W$ is normal,
$\Gamma_5$ and $\Gamma_5D_W$ admit a basis of common eigenvectors.
The result in \eqref{D_N_normal:eq} implies that $D_N=\rho I + \Gamma_5{\rm sign}(\Gamma_5D_W(m_0^\mathit{ker}))$ and $D_W(0)$
share the same eigenvectors and that
\[
{\rm spec}(D_N) = \{ \rho + {\rm csign}(\lambda +m^\mathit{ker}_0), \lambda \in {\rm spec}(D_W(0))\}.
\]
Taking $D_W(m_0^\prec)$ as a preconditioner for $D_N$, we would like eigenvalues of $D_N$ which are small in
modulus to be mapped to eigenvalues close to 1 in the preconditioned matrix $D_ND_W(m_0^\prec)^{-1}$.
Since $D_W(m_0^\prec)$ and $D_N$ share the same eigenvectors, the spectrum of the preconditioned matrix
is
\[
{\rm spec}\big(D_N D_W(m_0^\prec)^{-1}\big) = \Big\{ {\frac{\rho + {\rm csign}(\lambda +m_0^\mathit{ker})}{\lambda + m_0^\prec}}, \lambda \in {\rm spec}(D_W(0) \Big\}.
\]
For $\omega > 0$ and $m_0^\prec = \omega\rho+m_0^\mathit{ker}$, the mapping
\[
g: \mathbb{C} \to \mathbb{C}, z \mapsto {\frac{\rho + {\rm csign}(z +m_0^\mathit{ker})}{z + m_0^\prec}}
\]
sends $C(-m_0^\mathit{ker},\omega)$, the circle with center $-m_0^\mathit{ker}$ and radius $\omega$, to one single value $\frac{1}{\omega}$.
We thus expect $D_W(m_0^\prec)$ to be a good preconditioner if we
choose $m_0^\prec$ in such a manner
that the small eigenvalues of $D_W(m_0^\prec)$ lie close to $C(-m_0^\mathit{ker},\omega)$.
Let $\sigma_{\min} > 0$ denote the smallest real part of all eigenvalues of
$D_W(0)$. Assuming for the moment that $\sigma_{\min}$ is actually an eigenvalue, this eigenvalue will lie exactly
on $C(-m_0^\mathit{ker},\omega)$ if we have
\begin{equation} \label{eq:default_m}
\omega = \omega^{\mathit{def}} := -m_0^\mathit{ker}- \sigma_{\min} \mbox{ and thus } m_0^\prec = m_0^{\mathit{def}} := \omega^{\mathit{def}} \rho + m_0^\mathit{ker}.
\end{equation}
For physically relevant parameters, $\omega^\mathit{def}$ is close to 1.
We will take $m_0^{\mathit{def}}$ from \eqref{eq:default_m} as our default choice for the mass parameter when preconditioning
with the Wilson-Dirac operator, although a slightly larger value for $\omega$ might
appear adequate in situations where the eigenvalues with smallest real part come as a complex conjugate
pair with non-zero imaginary part.
Although $D_W(0)$ is non-normal in physically relevant situations, we expect the above reasoning
to also lead to an effective Wilson-Dirac preconditioner in these settings, and particularly so when the deviation of $D_W(0)$
from normality, as measured in some suitable norm of $D_W^HD_W - D_WD_W^H$, becomes small. This is so, e.g.,
when the lattice spacing is decreased while keeping the physical volume constant, i.e., in
the ``continuum limit'', since the Wilson-Dirac operator then approaches the continuous Dirac operator which is normal. Moreover,
as we will show in
section~\ref{smearing:sec}, when {\em smearing} techniques are applied to a given gauge
configuration $U_\mu(x)$, the deviation of $D_W(0)$ from normality is also decreased.
Figure~\ref{fig:spectra_prec} shows the spectrum for the preconditioned matrix with the choice \eqref{eq:default_m}
for $m_0^\prec$ for the same $4^4$ configuration as in Figure~\ref{spectra:fig}. The matrices in
these tests are not normal, nonetheless the spectrum of the preconditioned matrix tends to
concentrate around 0.7.
\begin{figure}
\centering \scalebox{0.7}{\input{./overlap_spec2.tex}}
\caption{Spectra for a configuration of size $4^4$}
\label{fig:spectra_prec}
\end{figure}
In the normal case, the singular values are the absolute values of the eigenvalues, and the singular vectors
are intimately related to the eigenvectors. This relation was crucial to the proof of Theorem~\ref{the:D_D_N_eigs}. In the non-normal case, the relation \eqref{D_N_normal:eq}, which uses the eigenvectors of $D_W(0)$, does not hold. For the sake of completeness we give, for the general, non-normal case, the following result which links the overlap operator to the singular value decomposition of its kernel $D_W(m)$.
\begin{lemma} \label{lem:svd_kernel}
Let $\Gamma_5 D_W(m) = W_m \Delta_m W_m^H$ denote an eigendecomposition of the
hermitian matrix
$\Gamma_5D_W(m)$, where $\Delta_m$ is real and diagonal and $W_m$ is unitary.
Then
\begin{itemize}
\item[(i)] A singular value decomposition of $D_W(m)$ is given as
\[
D_W(m) = U_m \Sigma_m V_m^H \mbox{ with } V_m = W_m, \Sigma_m = |\Delta_m|, U_m = \gamma_5
W_m {\rm sign}(\Delta_m).
\]
\item[(ii)] The overlap operator with kernel $D_W(m)$ is given as
\[
D_N = \rho I + \Gamma_5 {\rm sign}\big(\Gamma_5D_W(m) \big) = \rho I + U_mV_m^H.
\]
\end{itemize}
\end{lemma}
\begin{proof}
Since $\Gamma_5^{-1} = \Gamma_5$, we have the factorization $D_W(m) = \Gamma_5 W_m \Delta_m W_m^H =
\Gamma_5 W_m {\rm sign}(\Delta_m) |\Delta_m| W_m^H$, in which $\Gamma_5 W_m {\rm sign}(\Delta_m)$ and $W_m$
are unitary and $|\Delta_m|$ is diagonal and non-negative. This proves (i). To show (ii), just observe that for the hermitian matrix $\Gamma_5 D_W(m)$ we have ${\rm sign}(\Gamma_5 D_W(m)) = W_m {\rm sign}(\Delta_m)W_m^H$ and use (i).
\end{proof}
\section{Smearing and Normality} \label{smearing:sec}
To measure the deviation from normality of $D_W$ we
now look at the Frobenius norm of $D_W^HD_W-D_WD_W^H$.
We show that this measure can be fully expressed
in terms of the pure gauge action, defined as a sum of path-ordered products of link variables, the {\em plaquettes}, to be defined
in detail below. Based on this connection
we then explain that ``stout'' smearing~\cite{Morningstar:2003gk}, a
modification of the gauge links by averaging with neighboring links, has the effect
of reducing the non-normality of $D_W$, among its other physical
benefits. This result indicates that preconditioning with the Wilson-Dirac operator and using the choice \eqref{eq:default_m} for $m^\mathit{ker}$ is increasingly better motivated as more smearing steps are applied. This observation will be substantiated by numerical experiments in section~\ref{numerics:sec}.
\begin{definition} Given a configuration of gauge links $\{U_\mu(x)\}$, the
{\em plaquette} $Q_x^{\mu,\nu}$ at lattice point $x$ is defined as
\begin{equation} \label{plaquette_def1:eq}
Q_x^{\mu,\nu} = U_\nu(x)U_\mu(x+\hat{\nu})U^H_\nu(x+\hat{\mu})U^H_\mu(x).
\end{equation}
\end{definition}
A plaquette thus is the product of all coupling matrices along a cycle of length 4
on the torus, such cycles being squares in a $(\mu,\nu)$-plane
\[
Q_x^{\mu,\nu} \mathrel{\widehat{=}} \Q{1}{1} \enspace .
\]
Similarly, the plaquettes in the other quadrants are defined as
\begin{equation} \label{plaquette_def2:eq}
Q_x^{\mu,-\nu} \mathrel{\widehat{=}} \Q{1}{-1}, \quad
Q_x^{-\mu,\nu} \mathrel{\widehat{=}} \Q{-1}{1}, \quad
Q_x^{-\mu,-\nu} \mathrel{\widehat{=}} \Q{-1}{-1} \ .
\end{equation}
Note that on each cycle of length four there are four plaquettes which are conjugates of each other.
They are defined as the products of the gauge links along that cycle with different starting sites,
so that we have, e.g., $Q_{x+\hat{\mu}}^{-\mu,\nu} =
U^H_{\mu}(x)Q_x^{\mu,\nu}U_\mu(x)$, etc.
The deviation of the plaquettes from the identity is a measure for
the non-normality of $D$ as determined by the following proposition. Its proof is obtained by simple, though technical, algebra which we summarize in the appendix.
\begin{proposition}\label{Fnorm:prop} The Frobenius norm of $D_W^HD_W-D_WD_W^H$ fulfills
\begin{equation}\label{eq:deviationfromnormality}
\| D^H_WD_W-D_WD_W^H \|_F^2 = 16 \sum_x \sum_{\mu < \nu} \Re(\tr{I-Q_x^{\mu,\nu}})
\end{equation}
where the first sum is to be taken over all lattice sites $x$ and
$\sum_{\mu < \nu}$ is a shorthand for $\sum_{\mu = 0}^3 \sum_{\nu = \mu+1}^3$.
\end{proposition}
As a consequence of Proposition~\ref{Fnorm:prop} we conclude that $D_W$ is normal in the case of the {\em free theory},
i.e., when all links $U_\mu(x)$ are equal to the identity or when $U_\mu(x) = U(x)U^H(x+\hat{\mu})$ for a collection of
$SU(3)$-matrices $U(x)$ on the lattice sites $x$.
For physically relevant configurations, however, $D_W$ is non-normal. The quantity
\[
\sum_x \sum_{\mu < \nu} \mbox{Re}(\tr{I-Q_x^{\mu,\nu}})
\]
is known as the {\em Wilson gauge action}\footnote{To represent a physically meaningful quantity, the Wilson gauge action is usually
scaled with a scalar factor. This is not relevant in the present context.} $S_W(\mathcal{U})$ of the gauge field $\mathcal{U} = \{U_\mu(x)\}$.
Smearing techniques for averaging neighboring gauge links have been studied extensively in
lattice QCD simulations. Their use in physics is motivated by the goal to reduce ``cut-off
effects'' related to localized eigenvectors with near zero eigenvalues.
We now explain why ``stout'' smearing~\cite{Morningstar:2003gk} reduces
the Wilson gauge action and thus drives the Wilson-Dirac operator
towards normality. Other smearing techniques like APE~\cite{Albanese:1987ds},
HYP~\cite{Hasenfratz:2007rf} and HEX~\cite{Capitani:2006ni} have similar effects.
Given a gauge field $ \mathcal{U}$,
stout smearing modifies the gauge links according to
\begin{equation} \label{stout:eq}
U_\mu(x) \to \tilde{U}_\mu(x) = \mathrm{e}^{\epsilon Z^{\mathcal{U}}_\mu(x)} U_\mu(x)\,
\end{equation}
where the parameter $\epsilon$ is a small positive number and
\begin{eqnarray}
Z^{\mathcal{U}}_\mu(x)
& = & -\frac{1}{2}(M_\mu(x)-M^H_\mu(x)) + \frac{1}{6}\tr{M_\mu(x)-M^H_\mu(x)}\,, \label{Zdef:eq}
\end{eqnarray}
where
\begin{eqnarray}
M_\mu(x) &=& \sum_{\nu=0,\nu \neq \mu}^3 Q_x^{\mu,\nu} + Q_x^{\mu,-\nu}. \nonumber
\end{eqnarray}
Note the dependence of $Z^{\mathcal{U}}_\mu(x)$ on local plaquettes associated with $x$.
The following result from \cite{Luscher:2010iy,Luscher:2009eq} relates the {\em Wilson flow} $\mathcal{V}(\tau) = \{V_\mu(x,\tau): x \in \mathcal{L}, \mu=0,\ldots,3\}$
defined as the solution of the initial value problem
\begin{equation}\label{eq:floweq}
\frac{\partial}{\partial \tau}V_\mu(x,\tau) = - \left\{ \partial_{x,\mu} S_\mathrm{W}(\mathcal{V}(\tau))\right\} V_\mu(x,\tau)\,, \quad V_\mu(x,0) = U_\mu(x)\,,
\end{equation}
to stout smearing. Here
$V_\mu(x,\tau) \in \mathrm{SU}(3)$, and $\partial_{x,\mu}$ is the canonical differential operator
with respect to the link variable $V_\mu(x,\tau)$ which takes values in $\mathfrak{su}(3)$, the algebra of $\mathrm{SU}(3)$.
\begin{theorem} Let $\mathcal{V}(\tau)$ be the solution of \eqref{eq:floweq}. Then
\begin{itemize}
\item[(i)] $\mathcal{V}(\tau)$ is unique for all $\mathcal{V}(0)$ and all $\tau\in(-\infty,\infty)$ and differentiable
with respect to $\tau$ and $\mathcal{V}(0)$.
\item[(ii)] $S_W(\mathcal{V}(\tau))$ is monotonically decreasing as a function of $\tau$.
\item[(iii)] One step of Lie-Euler integration with step size $\epsilon$ for \eqref{eq:floweq}, starting at $\tau = 0$, gives the approximation $\widetilde{\mathcal{V}}(\epsilon) = \{\widetilde{V}_\mu(x,\epsilon)\}$ for $\mathcal{V}(\epsilon)$ with
\[
\widetilde{V}_\mu(x,\epsilon) = \mathrm{e}^{\epsilon Z^{\mathcal{U}}_\mu(x)} U_\mu(x),
\]
with $Z^{\mathcal{U}}_\mu(x)$ from \eqref{Zdef:eq}
\end{itemize}
\end{theorem}
We refer to \cite{Luscher:2010iy,Luscher:2009eq} and also \cite{Bonati14} for details of the proof for (i) and (ii).
It is noted in~\cite{Bonati14} that the solution of~\eqref{eq:floweq} moves the gauge configuration
along the steepest descent in configuration space and thus actually minimizes
the action locally. Part (iii) follows directly by applying the Lie-Euler scheme; cf.~\cite{HaLuWa06}.
The theorem implies that one Lie-Euler step is equivalent to a step of
stout smearing, with the exception that in stout smearing links are
updated sequentially instead of in parallel. And since the Wilson
action decreases along the exact solution of \eqref{eq:floweq}, we can
expect it to also decrease for its Lie-Euler approximation, at least
when $\epsilon$ is sufficiently small.
\begin{figure}
\centerline{\scalebox{0.7}{\input{./plaquette}}}
\caption{Illustration of the effect of stout smearing on the average plaquette value~\eqref{eq:avgplaquette}.\label{fig:smearingVSavgplaquette}}
\end{figure}
In Figure~\ref{fig:smearingVSavgplaquette} we illustrate the
relation between iterations of stout smearing and the average
plaquette value $Q_{\mathit{avg}}$ for configuration~\ref{JF_32_32}
(cf.~Table~\ref{table:allconfs}). The average plaquette value is defined by
\begin{equation}\label{eq:avgplaquette}
Q_{\mathit{avg}} = N_Q^{-1}\sum_x \sum_{\mu < \nu}
\mathit{Re}(\tr{Q_x^{\mu,\nu}}),
\end{equation} where $N_Q$ denotes the total number of
plaquettes. In terms of $Q_{\mathit{avg}}$~\eqref{eq:deviationfromnormality}
simplifies to
\[
\norm[F]{D_{W}^HD_W - D_WD_W^H} =
16N_Q(3-Q_{\mathit{avg}}).
\] Figure~\ref{fig:smearingVSavgplaquette} shows that the Wilson action decreases rapidly in the first iterations of
stout smearing.
To conclude this section we note that there are several works
relating the spectral structure and the distribution of plaquette
values. For example, it has been shown in~\cite{Neuberger:1999pz} that the
size of the spectral gap around $0$ of $\Gamma_5 D_W$ is related to
$\Re(\tr{I-Q_x^{\mu,\nu}})$ being larger than a certain threshold
for all plaquettes $Q_x^{\mu,\nu}$. Other studies consider the
connection between fluctuations of the plaquette value and localized zero modes,
see~\cite{Berruto:2000fx,Negele:1998ev,Niedermayer:1998bi}, and the
influence of smearing on these modes~\cite{Hasenfratz:2007iv}.
\section{Numerical Results} \label{numerics:sec}
\begin{table
\centering\scalebox{0.9}{\tabcolsep=0.16cm\begin{tabular}{ccccccc}
\toprule
ID & lattice size & kernel mass & default & smearing & provided by &\\
& $N_t \times N_s^3$ & $m_0^\mathit{ker}$ & overlap mass $\mu$ & $s$ & &\\
\midrule
\ref{JF_32_32}\conflabel{JF_32_32} & $32 \times 32^3$ & $ -1-\frac{3}{4}\sigma_{\min}$ & $0.0150000$ & $\{0,\ldots,6\}$-stout~\cite{Morningstar:2003gk} & generated from~\cite{DelDebbio:2006cn,DelDebbio:2007pz} &\\
\ref{BMW_32_32}\conflabel{BMW_32_32} & $32 \times 32^3$ & $-1.3$ & $0.0135778$ & $3$HEX~\cite{Capitani:2006ni} & BMW-c, based on~\cite{Borsanyi:2012xf,Toth:2014uqa} &\\
\bottomrule
\end{tabular}}
\caption{Configurations used together with their parameters.
See the references for details about their generation.}
\label{table:allconfs}
\end{table}
In physical simulations, gauge fields are generated via a stochastic process and by fixing physical parameters.
The term {\em configuration} designates a gauge field together with the information about its generation and its physical parameters.
In this section we report numerical results obtained on relatively large configurations used in current simulations involving the overlap operator, detailed in Table~\ref{table:allconfs}. The configurations with ID~\ref{JF_32_32} are available with
different numbers $s=0,\ldots,6$ of stout smearing steps applied. Note that $s$ influences $\sigma_{\min}$, the smallest real part of all eigenvalues of $D_W(0)$. The given choice for $m_0^\mathit{ker}$ as a function of $\sigma_{\min}$, used in $D_N = \rho I + \Gamma_5 {\rm sign}(\Gamma_5 D_W(m_0^\mathit{ker}))$ places the middle of the first `hole' in the spectrum of $D_W(m_0^\mathit{ker})$
to be at the origin. The configuration with ID~\ref{BMW_32_32} was
obtained using 3 steps of HEX smearing in a simulation similar in spirit to \cite{Borsanyi:2012xf,Toth:2014uqa} with its physical results not yet published. The value $m_0^\mathit{ker} = -1.3$ is
the one used in the simulation. The middle of the first `hole' in $D_W(m_0^\mathit{ker})$ is thus close to but not exactly at the origin.
To be in line with the conventions from~\cite{Borsanyi:2012xf}, e.g., we express the parameter $\rho\geq 1$ used in the overlap operator
$D_N$ as
\[
\rho = \frac{-\mu/2 + m_0^{\mathit{ker}}}{\mu/2 + m_0^{\mathit{ker}}},
\]
where $\mu >0$ is yet another, ``overlap'' mass parameter.
In our experiments, we will frequently consider a whole range for $\mu$ rather than just the default value from Table~\ref{table:allconfs}.
The default value for $\mu$ is chosen such that it fits to other physically interpretable properties of the respective configurations like,
e.g., the pion mass $m_\pi$. For both sets of configurations used, $m_\pi$ is approximately twice as large than the value observed in nature,
and the ultimate goal is to drive $m_\pi$ to its physical value, which very substantially increases the cost for generating the respective configurations. We would then use smaller values for $\mu$, and the results of our experiments for such smaller $\mu$ hint at how the preconditioning will perform in future simulations at physical parameter values.
Note that smaller values for $\mu$ make $\rho$ become closer to 1, so $D_N$ becomes more ill-conditioned.
All results were obtained on the Juropa machine at J\"ulich
Supercomputing Centre, a cluster with $2,\!208$ compute nodes, each
with two Intel Xeon X5570 (Nehalem-EP) quad-core processors \cite{IntelXeonX5570,wwwJUROPA}.
This machine provides a maximum of $8,\!192$ cores for a single job from which we always use $1,\!024$ in our experiments.
For compilation we used the \texttt{icc}-compiler with the optimization flags
\texttt{-O3}, \texttt{-ipo}, \texttt{-axSSE4.2} and \texttt{-m64}.
In all tests, our code ran with roughly $2$ Gflop/s per core which accounts to $8-9\%$ peak performance.
The multigrid solver used to precondition with $D_W(m_0^\prec)$ (see below) performs at roughly $10\%$ peak.
\subsection{Accuracy of the preconditioner and influence of $m_0^\prec$}
In a first series of experiments, we solve the system
\begin{equation} \label{linsys:eq}
D_N \psi = \eta
\end{equation}
on the one hand without any preconditioning, using GMRES(100), i.e., restarted GMRES with a cycle length of 100.
On the other hand, we solve the same system using $D_W^{-1}$ as a (right) preconditioner. To solve the respective
linear systems with $D_W$ we use the domain decomposition based adaptive algebraic multigrid method
(DD-$\alpha$AMG) presented in \cite{FroKaKrLeRo13}. Any other efficient solver for Wilson-Dirac equations as, e.g.,
the ``AMG'' solver developed in~\cite{MGClark2010_1,MGClark2007,Luescher2007,MGClark2010_2} could
be used as well. In our approach, preconditioning
is done by iterating with DD-$\alpha$AMG until the relative residual is below a prescribed bound $\epsilon^{\prec}$.
Without going into detail, let us mention that DD-$\alpha$AMG uses a red-black multiplicative Schwarz method as
its smoother and that it needs a relatively costly, adaptive setup-phase in which restriction and prolongation
operators---and with them the coarse grid systems---are constructed. We refer to \cite{FroKaKrLeRo13} for further reading.
The setup has to be done only once for a given Wilson-Dirac matrix $D_W$, so its cost becomes negligible when
using DD-$\alpha$AMG as a preconditioner in a significant number of GMRES iterations.\footnote{In all our experiments, the setup
never exceeded 2\% of the total execution time, so we do not report timings for the setup.}
We use GMRES with odd-even preconditioning~\cite{MGClark2010_2} as a solver for the coarsest system.
The whole DD-$\alpha$AMG preconditioning iteration is non-stationary which has to be accounted for by using {\em flexible}
restarted GMRES (FGMRES) \cite{Saad:2003:IMS:829576} to solve \eqref{linsys:eq} instead of GMRES. The restart length for
FGMRES is again $100$.
\begin{figure} [thb]
\hspace*{0.15\textwidth}\scalebox{0.485}{\input{./scan_m0w_32s3_iter}}
\hspace*{0.132\textwidth}\scalebox{0.50}{\input{./scan_m0w_32s3}}
\caption{Preconditioner efficiency as a function of $m_0^\prec$ for two accuracies for the DD-$\alpha$AMG solver (configuration ID~\ref{JF_32_32}, $s=3$). Top: number of iterations, bottom: execution times.}
\label{fig:prec_eff_m0w}
\end{figure}
Figure~\ref{fig:prec_eff_m0w} presents results
for configuration ID~\ref{JF_32_32} with $s=3$ stout smearing steps and the default overlap mass $\mu$ from Table~\ref{table:allconfs}.
We scanned the values for $m_0^\prec$ in
steps of $0.01$ and report the number of iterations necessary to reduce the initial residual by a factor of $10^{-8}$ for each of these values.
We chose two different values $\epsilon^\prec$ for the residual reduction required in the DD-$\alpha$AMG iteration in the preconditioning.
The choice $\epsilon^\prec = 10^{-8}$ asks for a relatively accurate solution of the systems with $D_W(m_0^\prec)$, whereas the choice
$\epsilon^\prec = 10^{-1}$ requires an only quite low accuracy and thus only a few iterations of DD-$\alpha$AMG.
The upper part of Figure~\ref{fig:prec_eff_m0w} shows that there is a dependence of the number of FGMRES iterations on $m_0^\prec$, while at the
same time there is a fairly large interval around the optimal value for $m_0^\prec$ in which the number of iterations required is not more than 20\% larger than the minimum.
These observations hold for both accuracy requirements for the DD-$\alpha$AMG solver, $\epsilon^\prec = 10^{-8}$ and $\epsilon^\prec = 10^{-1}$.
The number of iterations needed without preconditioning was 973.
The lower part of Figure~\ref{fig:prec_eff_m0w} shows that similar observations hold for the execution times.
However, the smaller iteration numbers obtained with $\epsilon^\prec = 10^{-8}$ do not translate into smaller execution times,
since the time for each DD-$\alpha$AMG solve in the preconditioning is substantially higher as for $\epsilon^\prec = 10^{-1}$.
This turned out to hold in all our experiments, so from now on we invariably report results for $\epsilon^\prec = 10^{-1}$.
We also observe that the value of $m_0^\mathit{def}$ from \eqref{eq:default_m} lies within an interval in which iteration numbers and execution times (for both values for $\epsilon^\prec$) are quite close to the optimum. The execution time without preconditioning
was 294s.
\begin{figure}[htb]
\begin{minipage}[t]{0.48\textwidth}
\centering \scalebox{0.68}{\input{./scan_rho_m0w_diff_estimate_32s0_iter}}
\end{minipage} \hfill
\begin{minipage}[t]{0.48\textwidth}
\centering \scalebox{0.68}{\input{./scan_rho_m0w_diff_estimate_32s3_iter}}
\end{minipage}
\centering \scalebox{0.68}{\input{./smearing_scan_m0w_diff_estimate}}
\caption{Quality of $m_0^\mathit{def}$ without smearing (top left), with $s=3$ steps of stout smearing (top right), and for $s=0,\ldots,6$ steps of
stout smearing at fixed $\mu$ (bottom), configuration ID~\ref{JF_32_32}.}
\label{fig:estimate_s0_s3}
\end{figure}
Figure~\ref{fig:estimate_s0_s3} reports results which show that the default value $m_0^\mathit{def}$ is a fairly good choice in general.
For two different configurations (no smearing and 3 steps of stout smearing) and a whole range of overlap masses $\mu$, the plots at the top give
the relative difference $\delta m_0 = (m_0^{\mathit{opt}} - m_0^\mathit{def})/m_0^\mathit{def}$ of the optimal value $m_0^{\mathit{opt}}$ for $m_0^\prec$ and its default value from
\eqref{eq:default_m} as well as the similarly defined relative difference $\delta \mathrm{iter}$ of the corresponding iteration numbers. These
results show that the iteration count for the default value $m_0^\mathit{def}$ is never more than 15\% off the best possible iteration count.
The plot at the bottom backs these findings. We further scanned a whole range of smearing steps $s$ at the default value for $\mu$
from Table~\ref{table:allconfs}, and the number of
iterations with $m_0^\mathit{def}$ is never more than $5\%$ off the optimal value. The large values for $\delta m_0$ in the top right plot for
$\mu = 2^{-3}$ are to be attributed to the fact that the denominator in the definition of $\delta m_0$, i.e.,
$m_0^\mathit{def}$ is almost zero in this case.
These results suggest that \eqref{eq:default_m} is indeed a good choice for $m_0^\prec$.
However, $\sigma_{\min}$ needed to compute $m_0^\mathit{def}$ from
\eqref{eq:default_m} is not necessarily known a priori, and it may be
more efficient to approximate the optimal value for $m_0$ ``on the
fly'' by changing its value from one preconditioned FGMRES iteration
to the next.
In order to minimize the influence of the choice of
$m_0^\prec$ on the aspects discussed in the following sections we will
always use the optimal $m_0^\prec$, computed to a precision of $.01$ by scanning
the range $[-\widetilde{\sigma}_{\min},0]$, where
$\widetilde{\sigma}_{\min}$ is a rough guess at $\sigma_{\min}$ which
fulfills $\widetilde{\sigma}_{\min} > \sigma_{\min}$. This guess can be
easily obtained by a fixed number of power iterations to get an
approximation for the largest real part $\widetilde{\sigma}_{\max}$ of
an eigenvalue of $D$ and then using the symmetry
of the spectrum to obtain $\widetilde{\sigma}_{\min}$ by rounding
$8-\widetilde{\sigma}_{\max}$ to the first digit.
\subsection{Quality and cost of the preconditioner}
We proceed to compare in more detail preconditioned FGMRES($100$) with
unpreconditioned GMRES($100$) in terms of the iteration count. As
before, the iterations were stopped when the initial residual was
reduced by a factor of at least $10^{-8}$.
\begin{figure}[htb]
\begin{minipage}[t]{0.48\textwidth}
\centering \scalebox{0.68}{\input{./scan_s_32}}
\end{minipage} \hfill
\begin{minipage}[t]{0.48\textwidth}
\centering \scalebox{0.68}{\input{./scan_rho_32s3}}
\end{minipage}
\caption{Comparison of preconditioned FGMRES(100) with unpreconditioned GMRES(100) (configuration ID~\ref{JF_32_32}). Left: dependence on the
number of stout smearing steps $s$ for default value for $\mu$, cf.\ Table~\ref{table:allconfs}. Right: dependence on the overlap mass $\mu$ for $s=3$.}
\label{fig:gmres_comparison}
\end{figure}
Figure~\ref{fig:gmres_comparison} gives this comparison, once as a function of the non-normality of the configuration, i.e., the number $s$ of stout smearing steps applied, and once as a function of the overlap mass $\mu$.
We see that for the default value of $\mu$ from Table~\ref{table:allconfs}, the quality of the preconditioner increases with the number $s$ of stout smearing steps,
ranging from a factor of approximately $5$ for $s=0$ over $12$ for $s=3$ up to $25$ for $s=6$. We also see that the quality of the preconditioner increases as $\mu$ becomes smaller, i.e., when $D_N$ becomes more ill-conditioned.
From the practical side, a comparison of the execution times is more important than comparing iteration numbers. Before giving timings, we
have to discuss relevant aspects of the implementation in some detail.
Each iteration in GMRES or preconditioned FGMRES for \eqref{linsys:eq} requires one matrix vector multiplication with $D_N = \rho I + \Gamma_5{\rm sign}(\Gamma_5D_W)$. The matrix $D_N$ is not given explicitly as it would be a full, very large matrix despite $\Gamma_5 D_W$
being sparse. Therefore, a matrix vector multiplication $D_N\chi$ is obtained via an additional ``sign function iteration'' which approximates
${\rm sign}(\Gamma_5 D_W)\chi$ as part of the computation of $D_N\chi$. For this sign function iteration we use the restarted Krylov subspace method proposed
recently in \cite{FrGuSc14b,FrGuSc14} which allows for thick restarts of the Arnoldi process and has proven to be among the most efficient methods to approximate ${\rm sign}(\Gamma_5 D_W)\chi$. The sign function iteration then still represents the by far most expensive part of the overall computation.
A first approach to reduce this cost, see \cite{Cu05a}, is to use relaxation in the sense that one lowers the (relative) accuracy
$\varepsilon_{{\rm sign}}$ of the
approximation as the outer (F)GMRES iteration proceeds. The theoretical analysis of inexact Krylov subspace methods in \cite{SiSz,SlevdE}
shows that the relative accuracy of the approximation to the matrix-vector product at iteration $k$ should be in the order of
$\epsilon/ \| r_k \|$ (with $r_k$ the (F)GMRES residual at iteration $k$) to achieve that at the end of the (F)GMRES iteration the initial residual be decreased by a factor of $\epsilon$. We used this relaxation strategy in our experiments.
A second commonly used approach, see e.g.~\cite{Edwards:1998yw,Giusti:2002sm,vdE02}, to reduce the cost of the sign function iteration is deflation. In this approach the $k$ smallest in modulus eigenvalues $\lambda_1,\ldots,\lambda_k$ and their normalized eigenvectors $\xi_1,\ldots,\xi_k$ are precomputed once. With $\Xi = [\xi_1|\ldots|\xi_k]$
and $\Pi = I-\Xi\Xi^H$ the orthogonal projector on the complement of these eigenvectors, ${\rm sign}(\Gamma_5D_W)\chi$ is given as
\[
{\rm sign}(\Gamma_5D_W)\chi = \sum_{i=1}^k {\rm sign}(\lambda_i) (\xi^H\chi) \xi_i + {\rm sign}(\Gamma_5D_W) \Pi \chi.
\]
The first term on the right side can be computed explicitly and the second term is now easier to approximate with the sign function iteration,
since the $k$ eigenvalues closest to the singularity of ${\rm sign}(\cdot)$ are effectively eliminated via $\Pi$.
\begin{table}[htb]
\centering\scalebox{0.9}{\begin{tabular}{llcc}
\toprule
& parameter & notation & default \\
\midrule
(F)GMRES${}^{dp}$ & required reduction of initial residual & $\varepsilon_{\outer}$ & $10^{-8}$ \\
& relaxation strategy & $\varepsilon_{{\rm sign}}$ & $\frac{\varepsilon_{\outer}}{\| r_k\|}\cdot 10^{-2}$ \\
& restart length for FGMRES & $m_{\mathit{restart}}$ & $100$ \\
\midrule
DD-$\alpha$AMG${}^{sp}$ & required reduction of initial residual & $\varepsilon_{\prec}$ & $10^{-1}$ \\
& number of levels & & $2$ \\
\bottomrule
\end{tabular}}
\caption{Parameters for the overlap solver. Here, {\em dp} denotes double precision and {\em sp} single precision.}
\label{table:allparms1}
\end{table}
Table~\ref{table:allparms1} summarizes the default settings used for the results reported in Figure~\ref{fig:constant_sign_fct_cost}. The superscripts {\em dp} and {\em sp} indicate that
we perform the preconditioning in IEEE single precision arithmetic, while the multiplication with $D_N$ within the (F)GMRES
iteration is done in double precision arithmetic. Such mixed precision approaches are a common further strategy to reduce computing times
in lattice simulations.
\begin{figure}[htb]
\begin{minipage}[t]{0.48\textwidth}
\centering \scalebox{0.68}{\input{./comparison_fixed_sign_cost}}
\end{minipage} \hfill
\begin{minipage}[t]{0.48\textwidth}
\centering \scalebox{0.68}{\input{./fixed_cost_scan_rho_32s3}}
\end{minipage}
\caption{Comparison of execution times for preconditioned FGMRES and GMRES. Left: for $0$ to $4$ steps of stout smearing (configuration ID~\ref{JF_32_32}, default value for $\mu$ from Table~\ref{table:allconfs}), right: different overlap masses $\mu$ for configuration ID~\ref{JF_32_32} and 3-step stout smearing.}
\label{fig:constant_sign_fct_cost}
\end{figure}
For the results reported in Figure~\ref{fig:constant_sign_fct_cost} we tried to keep the cost for a matrix vector multiplication with $D_N$
independent of the number of smoothing steps which were applied to the configuration. To do so, we used the $100$th smallest eigenvalue of $\Gamma_5D_W$ for $s=0$ as
a threshold, and deflated all eigenpairs with eigenvalues below this threshold for the configurations with $s>0$.
The left plot in Figure~\ref{fig:constant_sign_fct_cost} shows that, at fixed default overlap mass $\mu$, we gain a factor of 4 to 10
in execution time using the preconditioner. The quality of the preconditioning improves with the numbers of smearing steps.
The right part of Figure~\ref{fig:constant_sign_fct_cost} shows that for smaller values of $\mu$ we can expect an even larger reduction of
the execution time. For the smallest value considered, $\mu = 2^{-8},$ which is realistic for future lattice simulations, the improvement due to preconditioning is a factor of about $25$.
\subsection{Comparison of optimized solvers}
Physics production codes for simulations with the overlap operator use recursive preconditioning as an additional technique to further
reduce the cost for the matrix vector multiplication (MVM) with $D_N$; cf.~\cite{Cu05a}. This means that the FGMRES iteration is preconditioned by using an additional ``inner'' iteration to approximately invert $D_N$, this inner iteration being itself again FGMRES. The point is that we may require only low accuracy for this inner iteration, implying that all MVMs with ${\rm sign}(\Gamma_5D_W)$ in the inner iteration may be approximated to low accuracy and computed in IEEE single precision, only.
In this framework, we can apply the DD-$\alpha$AMG preconditioner, too, but this time as a preconditioner for the inner FGMRES iteration. In this manner we keep the advantage of needing only a low accuracy approximation to the MVM with ${\rm sign}(\Gamma_5D_W)$, while at the same time reducing the
number of inner iterations and thus the (low accuracy) evaluations of MVMs with ${\rm sign}(\Gamma_5D_W)$.
We denote $\varepsilon_{\mathit{inner}}$ the residual reduction we ask for in the unpreconditioned inner iteration and $\varepsilon_{\mathit{inner}}^{\prec}$ the corresponding accuracy required when using the DD-$\alpha$AMG iteration as a preconditioner. The inner iteration converges much faster when we use preconditioning. More accurate solutions in the inner iteration reduce the number of outer iterations and thus the number of costly high precision MVMs with ${\rm sign}(\Gamma_5D_W)$. When preconditioning is used for the inner iteration, requiring a higher accuracy in the inner iteration comes at relatively low additional cost. It is therefore advantageous to choose $\varepsilon_{\mathit{inner}}^{\prec}$ smaller than $\varepsilon_{\mathit{inner}}$. As an addition to Table~\ref{table:allparms1}, Table~\ref{table:allparms2} lists the default values we used for the inner iteration and which were found to be fairly optimal via numerical testing.
\begin{table}[htb]
\centering\scalebox{0.9}{\begin{tabular}{llcc}
\toprule
& parameter & notation & default \\
\midrule
inner FGMRES${}^{sp}$ & required reduction of initial residual & $\varepsilon_{\mathit{inner}}^{\prec}$ & $10^{-2}$ \\
& (with preconditioning) & & \\
& required reduction of initial residual & $\varepsilon_{\mathit{inner}} $ & $10^{-1}$ \\
& (without preconditioning) & & \\
& relaxation strategy &
& $\frac{\varepsilon_{\mathit{inner}},\varepsilon_{\mathit{inner}}^{\prec}}{\|r_k\|}\cdot 10^{-2}$ \\
& restart length & $m_{\mathit{restart}}^{\mathit{inner}}$ & $100$ \\
\bottomrule
\end{tabular}}
\caption{Parameters for the inner iteration.}
\label{table:allparms2}
\end{table}
Figure~\ref{fig:recursive} shows results for the solvers optimized in
this way. We consider different sizes for the deflation subspace, i.e., the number of
smallest eigenvalues which we deflate explicitly. The computation of these eigenvalues (via PARPACK~\cite{wwwPARPACK}) is costly, so that deflating a
larger number of eigenvalues is efficient only if several system solves with the same overlap operator are to be performed. The figure shows that,
irrespectively from the number of deflated eigenvalues, the preconditioned recursive method outperforms the unpreconditioned method in a
similar way it did in the non-recursive case considered before. When more smearing steps are applied, the improvement grows; improvement factors
reach 10 or more. The figure also shows that in the case that we have to solve only one or two linear systems with the same matrix, it is not advisable to use deflation at all, the cost for the computation of the eigenvalues being too large. We attribute this finding at least partly to
the fact that the thick restart method used to approximate the sign function from \cite{FrGuSc14} is particularly efficient, here.
While all other data in Figure~\ref{fig:recursive} was obtained for configuration ID~\ref{JF_32_32},
the rightmost data on the left plot refers to configuration
ID~\ref{BMW_32_32}. We see a similar high efficiency of our
preconditioner as we did for configuration~\ref{JF_32_32} with 3
smearing steps, an observation consistent with the fact that
configuration ID~\ref{BMW_32_32} was also obtained using 3 steps of
(HEX) smearing, see Table~\ref{table:allconfs}.
\begin{figure}[]
\centering \scalebox{0.58}{\input{./recursive}}
\centering \scalebox{0.58}{\input{./recursive_scaling}}
\caption{Comparison of GMRESR with FGMRESR with different deflation spaces (configuration IDs~\ref{JF_32_32} and \ref{BMW_32_32} with $1,\!024$ processes).}
\label{fig:recursive}
\end{figure}
\section*{Conclusions} The new, fast adaptive algebraic multigrid solvers for the Wilson-Dirac operator $D_W$
now allow to efficiently use this operator as a preconditioner for the overlap operator. We presented a thorough analysis of this auxiliary space preconditioner in the case that $D_W$ is normal. This is not the case in practice, but
the trend in current simulations in lattice QCD is to reduce the non-normality of $D_W$ as one approaches the continuum limit and smearing techniques are applied. For a state-of-the-art parallel implementation
and for physically relevant configurations and parameters we showed that the improvements in time to solution gained through the preconditioning are at least a factor of 4 and, typically, more than 10.
\section*{Acknowledgments}
We thank the Budapest-Marseille-Wuppertal collaboration and Jakob Finkenrath for providing configurations. All numerical results were obtained on Juropa at J\"ulich Supercomputing Centre (JSC) through NIC grant HWU12.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 6,100 |
\c gptest;
select count(*) from sto_altcp1;
-- Alter table rename default partition
Alter table sto_altcp1 rename default partition to new_others;
Insert into sto_altcp1 values(1,10,3,4);
select * from sto_altcp1 order by b,c;
-- Alter table rename partition
Alter table sto_altcp3 RENAME PARTITION FOR ('2008-01-01') TO jan08;
select count(*) from sto_altcp3;
-- Alter table drop default partition
Alter table sto_altcp1 drop default partition;
select * from sto_altcp1 order by b,c;
-- Alter table drop partition
Alter table sto_altcp1 drop partition for (rank(1));
select * from sto_altcp1 order by b,c;
-- Alter table add heap partition
Alter table sto_altcp1 add partition new_p start(6) end(8);
Insert into sto_altcp1 values(1,7,3,4);
select * from sto_altcp1 order by b,c;
-- Alter table add ao partition
Alter table sto_altcp1 add partition p1 start(9) end(13) with(appendonly=true);
Insert into sto_altcp1 values(1,10,3,4);
select * from sto_altcp1 order by b,c;
-- Alter table add co partition
Alter table sto_altcp1 add partition p2 start(14) end(17) with(appendonly=true, orientation=column);
Insert into sto_altcp1 values(1,15,3,4);
select * from sto_altcp1 order by b,c ;
-- Alter table add default partition
Alter table sto_altcp1 add default partition part_others;
Insert into sto_altcp1 values(1,25,3,4);
select * from sto_altcp1 order by b,c;
select count(*) from sto_altcp2;
-- Alter table split partition
Alter table sto_altcp2 split partition p1 at(3) into (partition splitc,partition splitd) ;
select * from sto_altcp2 order by a;
-- Alter table split subpartition
Alter table sto_altcp1 alter partition p1 split partition FOR (RANK(1)) at(2) into (partition splita,partition splitb);
select * from sto_altcp1 order by b,c;
-- Alter table split default partition
Alter table sto_altcp2 split default partition start(15) end(17) into (partition p1, partition others);
select * from sto_altcp2 order by a;
| {
"redpajama_set_name": "RedPajamaGithub"
} | 3,071 |
Q: PrimeNg empty message shown before data is loaded I am using PrimeNg table to show the data and have added the empty message template like the following :
<ng-template pTemplate="emptymessage">
<tr>
<td>
No records found
</td>
</tr>
</ng-template>
and I am using lazy loading as the data is fetched from the server. I have added a loading flag, which is changed when the http call is finished. The code is as below:
this.myService
.myHttpCallFunction(params)
.pipe(
finalize(() => this.loading = false)
)
.subscribe(
(result: JsendResponse) => this.data = result.data,
errors => this.errors = errors
);
I am passing the loading flag to the table and it looks like the following :
<p-table [value]="data?.data" [columns]="settings.columns" [lazy]="true" [loading]="loading">
The table is loaded from a shared compoent and which accepts data as an input parameter. So the declaration of the data in shared component is like
@Input()
set data(data) {
if (data) {
this._data = data;
this.total = data.meta.pagination.total;
}
}
get data(){
return this._data;
}
Now the table will show No Records Found first for a second and then the data is get loaded. I am assuming this is because the table is loaded before the HTTP response is received. But how can I fix this ?
A: You can avoid this problem by setting a [loading] property on the p-table component template.
<p-table [value]="data" [loading]="loading">
<ng-template pTemplate="header">
<tr>
<th>Colmun</th>
</tr>
</ng-template>
<ng-template pTemplate="body" let-c>
<tr>
<td>{{c.value}}</td>
</tr>
</ng-template>
<ng-template pTemplate="emptymessage" let-c>
<tr>
<td [attr.colspan]="c.length">
No records found
</td>
</tr>
</ng-template>
</p-table>
And in your component.ts file:
loading: boolean = true;
And finally, set it to false when the data is fetched:
ngOnInit() {
setTimeout(() => {
// Fetch data here...
this.loading = false;
}, 1000);
}
This way, you won't get the empty message before the data is loaded.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 4,604 |
Top Around the Web
Round Up Of The Apple Event Held On 25 March
by Sahil KapoorMarch 26, 2019
Cupertino based tech titan, Apple held an event on the 25th of March where it announced four brand new subscription services. The list of newly announced services includes Apple TV Plus, Apple Arcade, Apple Card and Apple News Plus.
Apple Arcade is the company's take on a gaming service which spans across the iPhone, iPad, Apple TV and Mac computers. The paid subscription will initially provide access to over 100 gaming titles which will sprout further over time. Noteworthy, it even allows the user to play games in offline mode. This is a significant aspect as its direct competitor, the newly launched Google Stadia lacks support for this notable feature. The service will roll out in autumn this year and its pricing details are yet to be revealed. Exceptionally, family members can also share the subscription across different devices without an additional cost.
Apple card is a virtual credit card service which is developed for iPhone users and is an extension of the Apple Pay and Wallet services. The service is intended towards lowering interest rates, removing unnecessary fees and simplifying the process of applications in obtaining a credit card. According to the company, the service will simplify the use of existing digital payment mediums to a great extent.
Notably, it is just a credit card service and not a banking service. Apple has partnered with Goldman Sachs and MasterCard that handle its financial end. As previously mentioned, the user can directly sign up for a card right from the iPhone which is generated in minutes, unlike conventional service which can take days. Reportedly, the service makes use of machine learning which can precisely reflect the spending patterns of the user and also offer rewards in cash. Distinctly, the service will not charge late fees from the however, its functioning is unclear as of now.
Apple News Plus
As the name suggests, it is a news service from Apple that again requires a paid subscription. It showcases the content in a user-friendly and elegant manner which is enriched with photographs, animations and typography optimised for Apple devices including iPhone, iPad and Mac. The news service incorporates content from over 300 established newspapers, magazines and digital publishers some of which include Vogue, WSJ, GQ, Traveller, Wired and The New Yorker. The service is currently limited to the US and Canada where the users can subscribe it at a monthly fee of $ 9.99 and $ 12.99 respectively. Admirably, six members of a family can share one Apple News Plus subscription.
Also Read: Xiaomi Poco F1 Gets Game Turbo Mode For Better Gaming Performance
Apple TV Plus is the company's attempt at a video streaming service which will take on the likes of Netflix and Amazon Prime. The service is set to launch in over 100 countries which will roll out in Q4 of 2019. According to the company, around 30 original projects are currently in the works which will be exclusive to the platform. Additionally, content from renowned channels including HBO, Showtime, CBS All Access will also be available on the streaming service. Like the Apple Arcade, its price is also currently unavailable.
In addition to the Apple Plus subscription service, the company has also released an all-new Apple TV application which has a completely redesigned interface. The new version of the application is designed to serve on-demand digital content, without advertisement with the option of offline downloads. The new Apple TV app is now available for the Apple TV and iOS devices. While its support for third-party smart TVs from LG, Sony, Samsung and Vizio will be available some.
Apple, Latest, Top Around the Web
Apple Arcade, Apple card, apple news, apple tv, launch
Sahil Kapoor
Boot Camp Ref, Still Doesn't Like Pizza !
AppleLatestTop Around the Web
Apple Launches iPhone 11
Apple Launches iPhone 11 Pro And iPhone 11 Pro Max
Apple Will Sell Its Products Online In India For The First Time Through Its Official Website
Haunted iPhone XS Flashes Every Night Between 3 AM and 4:30 AM
Apple iPhone Pro, New iPads And 16 inch MacBook Pro To Launch Soon | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 7,443 |
Q: Level Shifter Issue with Ground I am using this level shifter spark fun breakout board: https://learn.sparkfun.com/tutorials/using-the-logic-level-converter
My goal is to interface a 5V logic SPI LCD with a 3.3V logic micro controller. To start I have been testing the level shifter board to make sure I can get the correct levels. I apply 3.3 V from the micro controller to the TXI pin and get 5V on the TXO pin no problem. The issue is when I apply 0V to the TXI pin. This results in a 1.4V on the TXO pin. From the spark fun board schematic I see that it is using a MOSFET to apply the conversion and from my understanding when one side of the MOSFET is pulled low it enters into a conducting state and this should result in both sides going low. I don't understand why I'm seeing the 1.4V as opposed to just 0V.
If I'm stuck with the MOSFET going to 1.4 V for low-level logic should I expect my SPI LCD to treat 1.4 V as low-level logic and then operate correctly?
A: Found the issue. I simply wasn't powering the breakout board properly.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 4,720 |
Q: Using loop to insert array type of data in PostgreSQL I'd like to insert array data equip to a table using loop.
var svr = 1;
var equip = [3, 4, 5];
For that I need to insert the data three times.
Looks like this:
INSERT INTO khnp.link_server_equipment_map(svr, equip)
VALUES (1, 3);
INSERT INTO khnp.link_server_equipment_map(svr, equip)
VALUES (1, 4);
INSERT INTO khnp.link_server_equipment_map(svr, equip)
VALUES (1, 5);
Can someone please get me out of this rabbit hole? Thanks
A: You can try unnest:
INSERT INTO khnp.link_server_equipment_map(svr, equip)
VALUES (1, UNNEST(ARRAY[3, 4, 5]));`
A: You can use the INSERT statement to insert several rows.
INSERT INTO table_name (column_list)
VALUES
(value_list_1),
(value_list_2),
...
(value_list_n);
According to your mentioned data example the rows insertion would be done this way
INSERT INTO khnp.link_server_equipment_map(svr, equip) VALUES
(1, 3),
(1, 4),
(1, 5);
Also to avoid adding the array content one by one you can use the UNNEST array function.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 2,288 |
Epidendrum sympetalostele är en orkidéart som beskrevs av Eric Hágsater och Luis M. Sánchez. Epidendrum sympetalostele ingår i släktet Epidendrum och familjen orkidéer.
Artens utbredningsområde är Colombia. Inga underarter finns listade i Catalogue of Life.
Källor
Orkidéer
sympetalostele | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 6,620 |
Sales of the New Nissan Leaf Are Off to a Great Start
It's not so hot in the U.S. yet, but the new Leaf is a hit in Europe and Japan.
By Eric Brandt March 22, 2018
Eric Brandt View Eric Brandt's Articles
brandt004
The Nissan Leaf has been thoroughly refreshed for the 2018 model year and buyers are responding quite positively to the updated model. The range isn't great, but it's an otherwise excellent little EV. The new look, tech, and electric drivetrain of the 2018 Leaf have driven its sales to impressive numbers, especially in Japan and Europe.
According to electric car enthusiast site InsideEVs, only about 900 new Leafs were sold in the U.S. in February, but Nissan sold over 3,700 of them in Japan and even more (3,766) in Europe. That puts it in the top five best-selling Nissans in Europe which is pretty impressive considering EVs made up a minuscule market share worldwide not that long ago. 6.5 percent of all Nissans sold in Europe in February were Leafs (Leaves?).
It's worth noting that Nissan Leaf sales have been consistently declining in the U.S. every year since 2014 according to Car Sales Base. It peaked at 30,200 sales in 2014 and dropped down to 11,230 units sold in 2017. Granted, the sales especially slowed in late 2017 with buyers holding out for the new model after it was announced. Could the new model be popular enough to reverse the trend in Leaf sales in the States?
Regardless, it's fair to say the new Leaf is a hit on a global scale. Japan, Europe, and the U.S. are the top three markets for the Nissan Leaf and considering its strong start in those markets, it could soon be as ubiquitous as the Altima.
The 2018 Nissan Leaf, Unplugged: An Otherwise-Exemplary EV Falls Short in Driving Range
Unless you're cool with a Nissan EV that covers half the miles of Tesla's Model 3, bide your time for the longer-range Leaf coming late this year.
2018 Nissan LEAF Production Begins
The LEAF will also hit U.S. dealerships next month.
Japanese University Project Uses Nissan LEAF Parts on Buses
Kumamoto University is bringing together auto manufacturers, government, and academia to electrify buses.
Nissan Confirms Leaf Nismo for Production
Watch out, Tesla.
This Is the All-New 2018 Nissan Leaf
More power, more range, and more usability—but will it be enough? | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 3,015 |
This project was made towards the end of 2010 for director Chris Cunningham, but was not completed due to budgetary constraints, which is a great shame. The photos below show early tests, and in no way represent a completed artistic vision.
A MIDI drum machine was connected to a box of electronics containing a series of relays, which in turn operate solenoid valves, causing the pneumatic rams to function. The rams were connected to various mechanisms that strike the drums in accordance to the rhythm programmed on the Groovebox. The smoke machine was also controlled via MIDI.
There was also some circuitry added to control the length of the pulse sent to the solenoid valve, so that the time before the air flow in the ram was reversed could be adjusted to create an optimum percussive sound. The rod seals were machined to make the rams act faster.
I learned a lot on this project - please get in touch if you would like to commission a MIDI powered robot of any description. | {
"redpajama_set_name": "RedPajamaC4"
} | 5,688 |
var http = require('http');
var cp = require('child_process');
var fs = require('fs');
var path = require('path');
var os = require('os');
var urlParse = require('url').parse;
var serveStatic = require('serve-static');
var uuid = require('uuid');
var ejs = require('ejs');
var semver = require('semver');
var WebSocket = require('faye-websocket');
var gist = require('./gist');
var DEBUG = false;
var RE_GIST_ID = /^[a-f0-9]+$/;
var config = {
// `child_process.spawn()` options:
encoding: 'utf8',
env: null,
timeout: 5 * 60 * 1000,
address: '0.0.0.0',
jobAbandonTimeout: 30 * 1000,
maxConcurrency: 1,
maxJobSize: 512 * 1024,
maxQueued: 1000,
port: 80,
targetsPath: process.cwd(),
vanilla: true
};
var serve = serveStatic(path.join(__dirname, '..', 'public'), {
index: false
});
var tplPath = path.join(__dirname, '..', 'templates');
var jobTpl = fs.readFileSync(path.join(tplPath, 'job.ejs'), 'utf8');
jobTpl = ejs.compile(jobTpl);
var benchmarkModulePath = JSON.stringify(require.resolve('benchmark'));
var syntaxErrArgs = [ path.join(__dirname, 'syntaxError.js') ];
var queue = [];
var targets = [];
var targetPaths = {};
var noArgs = [];
var mainHTML;
var current;
var server;
var wsOpts;
function noopErrorHandler(err) {}
function exit(code) {
if (arguments.length > 1)
console.error.apply(console, Array.prototype.slice.call(arguments, 1));
process.exit(code);
}
function notifyJobOwners() {
if (DEBUG)
console.log('notifyJobOwners()');
for (var i = 0; i < queue.length; ++i) {
var job = queue[i];
if (!job.stopped && job.ws)
job.ws.send(''+i);
}
}
function isPosNum(val) {
val = Math.floor(val);
return (isFinite(val) && val >= 0);
}
function tryParseJSON(str) {
try {
return JSON.parse(str);
} catch (ex) {}
}
function denyWS(ws, code, err) {
if (typeof code === 'string')
ws.close(code);
else
ws.close(code, err);
}
function finishWork(job) {
// Notify client of new result(s)
if (DEBUG)
console.log('finishWork() for %s', job.id);
if (job.ws)
job.ws.send(JSON.stringify(job.targets));
if (job.stopped ||
(job.pids.length === 0 &&
job.targets.waiting.length === 0)) {
if (job.ws)
job.ws.close(4001);
clearTimeout(job.timeout);
job.stopped = true;
current = null;
processQueue(false);
} else
processQueue(true);
}
function parseSyntaxError(str) {
if (DEBUG)
console.log('parseSyntaxError: %j', str);
var m = /^__benchd_(setup|teardown|bench):(\d+)\r?\n[\s\S]+(SyntaxError[\s\S]+)$/.exec(str);
if (!m)
return false;
var line = +m[2];
if (!isNaN(line))
return { source: m[1], stack: m[3], line: line };
else
return { source: m[1], stack: m[3] };
}
function gatherSyntaxErrors(errors, idx) {
idx || (idx = 0);
if (DEBUG) {
console.log('gatherSyntaxErrors[%d]: current=%s',
idx, errors && require('util').inspect(errors[idx]));
}
var current = errors[idx];
var proc = cp.execFile(current.targetPath, syntaxErrArgs, config, callback);
var jscode = current.job.codes.benchMap[current.benchName];
if (DEBUG) {
console.log('gatherSyntaxErrors[%d]: sending: %j',
idx, jscode);
}
proc.stdin.end(jscode);
function callback(err, stdout, stderr) {
if (DEBUG) {
console.log('gatherSyntaxErrors[%d]: err=%j; stdout=%j; stderr=%j',
idx, err, stdout, stderr);
}
var val = false;
if (!err && stderr.length)
val = parseSyntaxError(stderr.trim() + '\n' + current.stack);
current.result.benchmarks[current.benchName] = val;
if (current.job.ws)
current.job.ws.send(JSON.stringify(current.job.targets));
if (++idx === errors.length)
return finishWork(current.job);
gatherSyntaxErrors(errors, ++idx);
}
}
function processQueue(skipCheck) {
if (DEBUG) {
console.log('processQueue(%j): current? %j; queue.length=%d',
skipCheck, !!current, queue.length);
}
if ((current && !skipCheck) || (!current && !queue.length))
return;
if (!current) {
notifyJobOwners();
current = queue.shift();
}
var job = current;
var nprocs = Math.min(job.concurrency, job.targets.waiting.length);
var versions = job.targets.waiting.splice(0, nprocs);
if (DEBUG) {
console.log('processQueue(): starting %d target(s) for job %s: %j',
nprocs, job.id, versions);
}
versions.forEach(function(v) {
var timer = setTimeout(function() {
if (DEBUG) {
console.log('processQueue(): killing target %s for job %s (timeout)',
v, job.id);
}
child.kill('SIGKILL');
}, config.timeout);
var failed = false;
var result = { fastest: null, benchmarks: {} };
var buffer = '';
var errbuffer = '';
var syntaxErrors = [];
var child = cp.spawn(targetPaths[v], noArgs, config);
var pid = child.pid;
job.pids.push(pid);
job.targets.results[v] = result;
if (job.ws)
job.ws.send(JSON.stringify(job.targets));
if (DEBUG) {
console.log('processQueue(): spawning target %s for job %s; pid=%d',
v, job.id, pid);
}
child.stdout.setEncoding('utf8');
child.stdout.on('data', processOutcome);
child.stderr.on('data', processSyntaxError);
child.on('close', function(code, signal) {
clearTimeout(timer);
if (DEBUG) {
console.log('processQueue(): target %s ended for job %s with code %j;' +
' signal %j; failed %j',
v, job.id, code, signal, failed);
}
var idx = job.pids.indexOf(pid);
if (~idx)
job.pids.splice(idx, 1);
if (code === 25 && !failed) {
// We got a syntax error in setup or teardown code in vanilla mode
// We need to append the stdout buffer because that is where the actual
// syntax error message is
errbuffer = errbuffer.trim() + '\n' + buffer + '\n';
var errinfo = parseSyntaxError(errbuffer);
if (!errinfo)
fail();
else {
var changed = false;
Object.keys(job.codes.benchMap).forEach(function(name) {
if (result.benchmarks[name] === undefined) {
changed = true;
result.benchmarks[name] = errinfo;
}
});
// Only node versions v0.8-v0.10 will have changed results here
if (changed && job.ws)
job.ws.send(JSON.stringify(job.targets));
}
} else if (code !== 0 && !failed)
fail();
if (code === 0 && syntaxErrors.length) {
var first = syntaxErrors.shift();
errbuffer = errbuffer.trim() + '\n' + first.stack;
var errinfo = parseSyntaxError(errbuffer);
result.benchmarks[first.benchName] = errinfo;
if (job.ws)
job.ws.send(JSON.stringify(job.targets));
if (syntaxErrors.length)
return gatherSyntaxErrors(syntaxErrors);
}
finishWork(job);
});
if (DEBUG) {
var stderr = '';
child.stderr.setEncoding('utf8');
child.stderr.on('data', function(data) {
stderr += data;
});
child.on('close', function(code, signal) {
if (stderr.length) {
console.log('target %s for job %s had stderr:', v, job.id);
console.log('================================');
console.log(stderr);
console.log('================================');
}
});
}
child.stdin.end(job.jobCode);
function processOutcome(chunk) {
if (failed)
return;
buffer += chunk;
if (~buffer.indexOf('\n')) {
var lines = buffer.split('\n');
buffer = lines.splice(-1, 1).join('');
for (var i = 0; i < lines.length; ++i) {
if (DEBUG) {
console.log('target %s for job %s had line: %j',
v, job.id, lines[i]);
}
var ret = tryParseJSON(lines[i]);
if (Array.isArray(ret)) {
// Fastest benchmark(s)
result.fastest = ret;
} else if (typeof ret === 'object' && ret !== null) {
// Benchmark result
if (typeof ret.result === 'object' &&
ret.result !== null &&
ret.result.source === undefined &&
ret.result.line === undefined) {
// node versions v0.8-v0.10 suppress multiple syntax errors on
// stderr, so we will have to queue up failed benchmarks for
// independent checking/parsing
if (DEBUG) {
console.log('enqueueing benchmark %j for syntax check',
ret.name);
}
syntaxErrors.push({
targetPath: targetPaths[v],
job: job,
result: result,
stack: ret.result.stack,
benchName: ret.name
});
continue;
}
result.benchmarks[ret.name] = ret.result;
} else {
fail();
child.kill('SIGKILL');
return;
}
}
// Notify client of new result(s)
if (job.ws)
job.ws.send(JSON.stringify(job.targets));
}
}
function processSyntaxError(chunk) {
errbuffer += chunk;
}
function fail() {
if (failed)
return;
failed = true;
result.fastest = false;
for (var j = 0; j < job.benchNames.length; ++j) {
if (result.benchmarks[job.benchNames[j]] === undefined)
result.benchmarks[job.benchNames[j]] = null;
}
child.stdout.removeListener('data', processOutcome);
child.stderr.removeListener('data', processSyntaxError);
}
});
}
server = http.createServer(function(req, res) {
if (req.method === 'GET') {
var parsedUrl = urlParse(req.url, true);
switch (parsedUrl.pathname) {
case '/':
res.writeHead(200, { 'Content-Type': 'text/html' });
res.end(mainHTML);
return;
case '/gist':
var query = parsedUrl.query;
if (query.id) {
if (RE_GIST_ID.test(query.id)) {
// TODO: change url depending on "secret" config value when that is
// supported
var url = 'https://gist.github.com/anonymous/' + query.id;
gist.loadFromGist(url, function(err, obj) {
if (err) {
res.writeHead(500, { 'Content-Type': 'text/plain' });
return res.end(err.message);
}
res.writeHead(200, { 'Content-Type': 'application/json' });
res.end(JSON.stringify(obj));
});
} else {
res.writeHead(400, { 'Content-Type': 'text/plain' });
res.end('Invalid gist id');
}
} else {
res.writeHead(400, { 'Content-Type': 'text/plain' });
res.end('Missing gist id');
}
return;
}
} else if (req.method === 'POST' && req.url === '/gist') {
var buf = '';
req.on('data', function(data) {
buf += data;
}).on('end', function() {
var obj;
try {
obj = JSON.parse(buf);
} catch (ex) {
res.writeHead(400, { 'Content-Type': 'text/plain' });
return res.end('Invalid JSON');
}
try {
gist.saveToGist(obj, function(err, url) {
if (err) {
res.writeHead(500, { 'Content-Type': 'text/plain' });
return res.end(err.message);
}
res.writeHead(200, { 'Content-Type': 'text/plain' });
return res.end(url);
});
} catch (ex) {
res.writeHead(400, { 'Content-Type': 'text/plain' });
return res.end(ex.message);
}
});
return;
}
serve(req, res, function() {
res.statusCode = 404;
res.end();
});
});
server.on('upgrade', function(req, socket, body) {
if (req.url === '/ws' && WebSocket.isWebSocket(req)) {
var remoteAddress = socket.remoteAddress;
var remotePort = socket.remotePort;
if (DEBUG)
console.log('websocket request from %s:%d', remoteAddress, remotePort);
var ws = new WebSocket(req, socket, body, null, wsOpts);
function resetPingTimeout() {
clearTimeout(pingTmr);
if (ws)
pingTmr = setTimeout(closeSocket, 30 * 1000);
}
function closeSocket() {
if (ws)
ws.close(4000);
}
function resetIdTimeout() {
if (ws)
idTmr = setTimeout(closeSocket, 10 * 1000);
}
var pingTmr;
var idTmr;
var job;
ws.on('open', function (event) {
if (DEBUG)
console.log('ws.onopen for %s:%d', remoteAddress, remotePort);
resetIdTimeout();
});
ws.on('error', noopErrorHandler);
ws.on('message', function(event) {
if (DEBUG) {
console.log('ws.onmessage for %s:%d: %j',
remoteAddress, remotePort, event.data);
}
if (typeof event.data !== 'string')
return denyWS(ws, 1003, 'Binary data not supported');
if (event.data.length > config.maxJobSize)
return denyWS(ws, 1009, 'Job size too large');
if (job)
resetPingTimeout();
if (!event.data.length)
return;
if (event.data === 'ping') {
if (job)
ws.send('ping');
return;
} else if (job)
return;
var msg = tryParseJSON(event.data);
if (typeof msg === 'object' && msg !== null) {
// New job
clearTimeout(idTmr);
var concurrency;
var reqTargets;
var jobTargets;
if (queue.length === config.maxQueued) {
denyWS(ws,
4002,
'Cannot accept new benchmark requests at this time.' +
' Please try again later.');
return;
}
if (!Array.isArray(msg.benchmarks) || !msg.benchmarks.length)
return denyWS(ws, 4003, 'Missing JS code');
if (!Array.isArray(msg.targets) || !msg.targets.length)
return denyWS(ws, 4003, 'Missing target(s)');
concurrency = Math.floor(msg.concurrency);
if (!isFinite(concurrency) ||
concurrency < 1 ||
concurrency > config.maxConcurrency)
return denyWS(ws, 4003, 'Missing/Invalid concurrency');
reqTargets = msg.targets;
jobTargets = [];
for (var i = 0; i < reqTargets.length; ++i) {
var reqTarget = reqTargets[i];
for (var j = 0; j < targets.length; ++j) {
var t = targets[j];
if (t === reqTarget && jobTargets.indexOf(t) === -1) {
jobTargets.push(t);
break;
}
}
}
if (!jobTargets.length)
return denyWS(ws, 4003, 'No valid target(s) selected');
var benchmarks = msg.benchmarks;
var benchMap = {};
var benchNames = [];
for (var i = 0; i < benchmarks.length; ++i) {
var bench = benchmarks[i];
if (typeof bench !== 'object')
return denyWS(ws, 4003, 'Malformed benchmark #' + (i+1) + ']');
if (typeof bench.name !== 'string' || !bench.name.length) {
return denyWS(ws,
4003,
'Missing name for benchmark #' + (i+1) + ']');
}
if (~benchNames.indexOf(bench.name)) {
return denyWS(ws,
4003,
'Duplicate name for benchmark #' + (i+1) + ']: ' +
bench.name);
}
benchNames.push(bench.name);
if (config.vanilla)
bench.jscode = '"use strict";' + bench.jscode;
benchMap[bench.name] = bench.jscode;
bench.jscode = JSON.stringify(bench.jscode);
bench.name = JSON.stringify(bench.name);
}
if (msg.setupCode) {
if (typeof msg.setupCode !== 'string')
return denyWS(ws, 4003, 'Wrong type for setup code');
if (config.vanilla)
msg.setupCode = '"use strict";' + msg.setupCode;
msg.setupCode = JSON.stringify(msg.setupCode);
}
if (msg.teardownCode) {
if (typeof msg.teardownCode !== 'string')
return denyWS(ws, 4003, 'Wrong type for teardown code');
if (config.vanilla)
msg.teardownCode = '"use strict";' + msg.teardownCode;
msg.teardownCode = JSON.stringify(msg.teardownCode);
}
job = {
id: uuid.v4(),
codes: {
benchMap: benchMap,
setupCode: msg.setupCode,
teardownCode: msg.teardownCode
},
jobCode: jobTpl({
benchmarks: benchmarks,
benchmarkModulePath: benchmarkModulePath,
vanilla: config.vanilla,
setupCode: msg.setupCode,
teardownCode: msg.teardownCode
}),
benchNames: benchNames,
targets: {
waiting: jobTargets,
results: {}
},
concurrency: concurrency,
pids: [],
ws: ws,
stopped: false,
timeout: null,
remove: function() {
if (DEBUG)
console.log('Job %s abandoned', job.id);
var idx;
if (current !== job && (idx = queue.indexOf(job)) === -1)
return;
// We're still in the queue or currently running
if (DEBUG)
console.log('... killing');
if (idx !== undefined)
queue.splice(idx, 1);
job.stopped = true;
var pids = job.pids;
for (var i = 0; i < pids.length; ++i)
process.kill(pids[i], 'SIGKILL');
job.pids = [];
}
};
queue.push(job);
ws.send('{"id":"' + job.id + '","pos":' + queue.length + '}');
processQueue(false);
return resetPingTimeout();
} else if (msg === undefined) {
// User reconnected
clearTimeout(idTmr);
var id = event.data;
if (current && current.id === id) {
job = current;
clearTimeout(job.timeout);
job.timeout = null;
ws.send('0');
} else {
for (var i = 0; i < queue.length; ++i) {
if (queue[i].id === id) {
job = queue[i];
clearTimeout(job.timeout);
job.timeout = null;
ws.send(''+(i+1));
break;
}
}
}
if (job) {
job.ws = ws;
ws.send(JSON.stringify(job.targets));
resetPingTimeout();
} else
denyWS(ws, 4004, 'Invalid job id');
return;
}
denyWS(ws, 4005, 'Malformed message');
});
ws.on('close', function(event) {
ws = null;
clearTimeout(idTmr);
clearTimeout(pingTmr);
if (DEBUG) {
console.log('ws.onclose for %s:%d; code=%d; reason=%j',
remoteAddress, remotePort, event.code, event.reason);
}
if (job) {
job.ws = null;
if (!job.stopped)
job.timeout = setTimeout(job.remove, config.jobAbandonTimeout);
}
});
}
});
server.on('error', function(err) {
exit(5, 'HTTP server error: %s', err);
});
server.on('listening', function() {
wsOpts = { maxLength: config.maxJobSize };
mainHTML = fs.readFileSync(path.join(tplPath, 'main.ejs'), 'utf8');
mainHTML = ejs.render(mainHTML, {config: config, targets: targets});
var addrport = server.address();
console.log('Targets found:');
targets.forEach(function(t) {
console.log(' * %s @ %s', t, targetPaths[t]);
});
console.log('benchd server listening on %s port %d',
addrport.address, addrport.port);
});
// Initialization ....
function printHelp() {
console.log([
'Usage: benchd [options] [--] [config_file]',
' Valid options:',
' --address <string> Web interface address [default: 0.0.0.0]',
' --jobAbandonTimeout <number> Time to wait for client reconnection before',
' automatically removing queued/running job',
' [default: 30 * 1000ms]',
' --maxConcurrency <number> Max concurrent processes [default: 1]',
' --maxJobSize <number> Max raw JSON size for a job [default: 512KB]',
' --maxQueued <number> Max global queue size [default: 1000]',
' --port <number> Web interface port [default: 80]',
' --targetsPath <string> The path containing node binaries to test with',
' [default: current working directory]',
' --timeout <number> Max execution time allowed for a node binary',
' [default: 5 * 60 * 1000ms]',
' --vanilla [true|false] Executes all code in a more restricted',
' environment with no access to node modules',
' [default: true]',
'',
' Config files are JSON documents that contain the same parameters'
].join('\r\n'));
}
(function() {
// Process command line args
var argv = process.argv;
var configFiles = [];
var cmdLnArgs = {};
var foundFileArg = false;
var cmdLnArgsKeys;
for (var i = 2; i < argv.length; i++) {
if (argv[i] === '--help' || argv[i] === '-h')
return printHelp();
else if (argv[i].slice(0, 2) === '--' && argv[i].length > 2) {
var key = argv[i].slice(2);
if ((i + 1) < argv.length) {
if (argv[i + 1].slice(0, 2) === '--' && argv[i + 1].length > 2)
cmdLnArgs[key] = true;
else {
cmdLnArgs[key] = argv[i + 1];
++i;
}
} else
cmdLnArgs[key] = true;
} else if (!foundFileArg) {
foundFileArg = true;
configFiles.push(argv[i]);
}
}
cmdLnArgsKeys = Object.keys(cmdLnArgs);
if (process.env.BENCHD_CONF)
configFiles.push(process.env.BENCHD_CONF);
var localConfigPath = path.join(process.cwd(), 'benchd.conf');
configFiles.push(localConfigPath);
var config_ = null;
console.log('Checking for config ...');
for (var i = 0; i < configFiles.length; ++i) {
try {
config_ = fs.readFileSync(configFiles[i], 'utf8');
config_ = JSON.parse(config_);
console.log('Using config from %s', configFiles[i]);
break;
} catch (ex) {
config_ = null;
}
}
if (typeof config_ === 'object' && config_ !== null) {
if (cmdLnArgsKeys.length) {
// Merge command line arguments with config file settings, with the former
// taking precedence
for (var i = 0; i < cmdLnArgsKeys.length; ++i)
config_[cmdLnArgsKeys[i]] = cmdLnArgs[cmdLnArgsKeys[i]];
}
} else if (cmdLnArgsKeys.length)
config_ = cmdLnArgs;
if (typeof config_ === 'object' && config_ !== null) {
if (isPosNum(config_.port))
config.port = Math.floor(config_.port);
if (isPosNum(config_.timeout))
config.timeout = Math.floor(config_.timeout);
if (config_.maxConcurrency === '-1')
config.maxConcurrency = os.cpus().length;
else if (isPosNum(config_.maxConcurrency))
config.maxConcurrency = Math.floor(config_.maxConcurrency);
if (isPosNum(config_.maxQueued))
config.maxQueued = Math.floor(config_.maxQueued);
if (isPosNum(config_.maxJobSize))
config.maxJobSize = Math.floor(config_.maxJobSize);
if (typeof config_.targetsPath === 'string')
config.targetsPath = config_.targetsPath;
if (isPosNum(config_.jobAbandonTimeout))
config.jobAbandonTimeout = config_.jobAbandonTimeout;
if (config_.vanilla === false || config_.vanilla === 'false')
config.vanilla = false;
if (typeof config_.address === 'string')
config.address = config_.address;
} else {
console.log('Using default config');
}
var files;
try {
files = fs.readdirSync(config.targetsPath);
} catch (ex) {
exit(1, 'Cannot access path to executables: %s', ex);
}
(function checkExecutable(i) {
if (i === files.length) {
if (targets.length) {
targets.sort(semver.rcompare);
try {
server.listen(config.port, config.address);
} catch (ex) {
exit(5, 'Cannot listen on port for HTTP: %s', ex);
}
} else {
exit(1, 'No usable executables found in %s', config.targetsPath);
}
return;
}
// Skip dot files
if (files[i][0] === '.')
return checkExecutable(i + 1);
var stdout = '';
var filePath = path.join(config.targetsPath, files[i]);
try {
var stats = fs.statSync(filePath);
// Naive executability check (any executable bit set and regular file)
if (stats.mode & 0x49 && stats.mode & 0x8000)
var proc = cp.spawn(filePath, ['-pe', 'process.version']);
else
return checkExecutable(i + 1);
} catch (ex) {
// Skip files that resulted in error during stat()/spawn()
return checkExecutable(i + 1);
}
var timeout = setTimeout(function() {
proc.kill();
}, 10 * 1000);
// Just swallow errors
proc.on('error', noopErrorHandler);
proc.stdout.setEncoding('ascii');
proc.stdout.on('data', function(chunk) {
stdout += chunk;
if (stdout.length > 50)
proc.kill();
});
proc.on('close', function(code) {
clearTimeout(timeout);
if (code === 0 &&
/^v\d+\.\d+\.\d+/.test(stdout) &&
stdout.slice(-1) === '\n') {
var version = stdout.trim();
targets.push(version);
targetPaths[version] = filePath;
}
checkExecutable(i + 1);
});
})(0);
})();
| {
"redpajama_set_name": "RedPajamaGithub"
} | 8,216 |
Q: Deploying angular application with aws-cdk on github actions I am not much of devops person so looking for some help. I have tried a few different things here. So I have a single AWS account which I created access tokens for with correct permissions and am looking to have github actions build and deploy this pretty small angular app into an S3 bucket
name: Deployment
on:
push:
branches:
- main
workflow_dispatch:
env:
AWS_ACCOUNT: ${{ secrets.AWS_ACCOUNT }}
jobs:
build:
name: Build Angular project
runs-on: ubuntu-latest
steps:
- name: Checkout code
uses: actions/checkout@v2
- name: Setup Node.js
uses: actions/setup-node@v2
with:
node-version: "16.x"
registry-url: "https://npm.pkg.github.com"
cache: "npm"
- name: Compile client code
run: |
npm ci
npm run build
rm -f dist/*.map
cd ./cdk
npm ci
cd ..
- name: Upload build artifact
uses: actions/upload-artifact@v3
with:
name: build
path: |
${{ inputs.ng_directory }}/dist
${{ inputs.ng_directory }}/src/configs
retention-days: 1
- name: cdk bootstrap
uses: youyo/aws-cdk-github-actions@v2.1.1
with:
cdk_subcommand: "bootstrap"
cdk_stack: "aws:// ${{ secrets.AWS_ACCOUNT }}/us-east-1"
env:
AWS_ACCESS_KEY_ID: ${{ secrets.AWS_ACCESS_KEY_ID }}
AWS_SECRET_ACCESS_KEY: ${{ secrets.AWS_SECRET_ACCESS_KEY }}
AWS_DEFAULT_REGION: "us-east-1"
- name: cdk synth
uses: youyo/aws-cdk-github-actions@v2.1.1
with:
cdk_subcommand: "synth"
working_dir: "./cdk"
env:
AWS_ACCESS_KEY_ID: ${{ secrets.AWS_ACCESS_KEY_ID }}
AWS_SECRET_ACCESS_KEY: ${{ secrets.AWS_SECRET_ACCESS_KEY }}
AWS_DEFAULT_REGION: "us-east-1"
- name: cdk deploy
uses: youyo/aws-cdk-github-actions@v2.1.1
with:
cdk_subcommand: "deploy"
cdk_stack: "frontend-appName"
cdk_args: "--require-approval never"
working_dir: "./cdk"
actions_comment: false
env:
AWS_ACCESS_KEY_ID: ${{ secrets.AWS_ACCESS_KEY_ID }}
AWS_SECRET_ACCESS_KEY: ${{ secrets.AWS_SECRET_ACCESS_KEY }}
AWS_DEFAULT_REGION: "us-east-1"
build is right now failing at
Install aws-cdk latest
Successful install aws-cdk latest
Run cdk bootstrap "aws:// ***/us-east-1"
⏳ Bootstrapping environment aws:// ***/us-east-1...
❌ Environment aws:// ***/us-east-1 failed bootstrapping: Error: Need to perform AWS calls for account ***, but the current credentials are for ***
at SdkProvider.forEnvironment (/usr/lib/node_modules/aws-cdk/lib/api/aws-auth/sdk-provider.ts:184:60)
at Function.lookup (/usr/lib/node_modules/aws-cdk/lib/api/bootstrap/deploy-bootstrap.ts:31:18)
at Bootstrapper.modernBootstrap (/usr/lib/node_modules/aws-cdk/lib/api/bootstrap/bootstrap-environment.ts:81:21)
at /usr/lib/node_modules/aws-cdk/lib/cdk-toolkit.ts:575:24
at async Promise.all (index 0)
at CdkToolkit.bootstrap (/usr/lib/node_modules/aws-cdk/lib/cdk-toolkit.ts:572:5)
at initCommandLine (/usr/lib/node_modules/aws-cdk/lib/cli.ts:341:12)
I have added the extra step "cdk bootstrap" because previous runs had a failure stating the application was not bootstrapped
Can someone help me troubleshoot this? I am not sure my specific issue but feel like I am very close.
A: Your machine needs to have the access & secret keys of the account which you're trying to bootstrap, but it seems they're incorrectly configured.
I suggest log into the AWS account which you're trying to bootstrap and generate fresh secret & access keys for your user, and then configure AWS CLI your machine to use those keys.
This article explains how to do the same, hope this helps.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 4,279 |
Q: Getting a number of integers which are more than some value using Aggregate I have a task to find a number of integers which are more than some specified value using Aggregate. For example, there is a list of integers
List<int> list = new List<int>() { 100, 200, 300, 400, 500 };
On this list I need to find the number of elements of this list which are more than 250. There should be 3 of them, but I need to use Aggregate there somehow to find this number.
Can anybody help please?
A: You can use Count, which is designed for counting:
using System.Linq;
...
List<int> list = new List<int>() { 100, 200, 300, 400, 500 };
...
int result = list.Count(item => item > 250);
If you insist on Aggregate:
int result = list.Aggregate(0, (s, a) => a > 250 ? s + 1 : s);
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 2,039 |
\section{\label{}}
\section{INTRODUCTION}
\label{intro}
Quantum Chromodynamics is made special among phenomenologically
relevant field theories by the property of confinement. Given that the
lagrangian is written in terms of fields that do not appear in the
construction of the true asymptotic states of the theory, it may seem
surprising that perturbative calculations performed around the trivial
vacuum have any relevance at all. The predictive power of perturbative
QCD, in the presence of a kinematic scale $Q^2$ much larger than the
confinement scale $\Lambda^2$, is rescued by asymptotic freedom,
combined with quantum-mechanical incoherence and gauge invariance.
These are the necessary ingredients entering the proof of
factorization theorems~\cite{Collins:1989gx}, which are the
cornerstones of all PQCD calculations.
Factorization theorems in essence provide a bound for the parametric
size of nonperturbative corrections to high energy inclusive cross
sections. Such corrections are typically suppressed by powers of the
hard scale $Q^2$. It should be emphasized, however, that factorization
theorems are proven perturbatively, by examining the all-order
structure of long-distance singularities in Feynman diagrams. Their
phenomenological relevance must then rely upon the additional (if
plausible) assumption that confinement be a relatively soft process,
happening without a violent rearrengement of momentum configurations,
as colored particles evolve away from the hard scattering event.
Decades of experience with QCD phenomenology have taught us that this
assumption is very well borne out by the data. Indeed, granted this
assumption, the bound on the size of nonperturbative effects provided
by the factorization theorem is the first and simplest case of
nonperturbative information extracted form QCD by purely perturbative
methods.
The general idea that perturbation theory, whenever genuine all-order
information is available, can provide important clues to understand
nonperturbative effects, has subsequently been applied successfully in
a variety of situations. Typically, the perturbative expansion is
found to diverge, and the uncertainty in the physical prediction due
to this divergence is interpreted as a measure of the size of the
expected nonperturbative correction. Specifically, the nonperturbative
contribution must be ambiguous by an amount matching the uncertainty
in the perturbative prediction. The assumption that the actual size of
the nonperturbative corrections should be well represented by this
ambiguity is sometimes referred to as ultraviolet dominance of power
corrections~\cite{Beneke:1997sr}.
The all-order perturbative information required to begin any study of
power corrections has mostly been provided by two complementary
sources: renormalon-type calculations (reviewed in
\cite{Beneke:1998ui}), which roughly speaking target running-coupling
effects by summing up fermion bubble corrections to single gluon
emission, and soft gluon resummations, which make use factorization
and universality to compute leading multigluon contributions in the
soft and collinear limits (for a recent review, see
\cite{Laenen:2004pm}). Recently, it was shown that the two appoaches
can be combined \cite{Gardi:2001di}, yielding a strongly constrained
and rather elegant model of power corrections in the Sudakov region.
Event shape distributions in hard collisions are an especially
interesting class of observables for power correction studies, and
indeed a lot work has been done in the past several years on the
subject, especially in the context of $e^+ e^-$ annihilation and
DIS~\cite{Dasgupta:2003iq,Magnea:2002xt}. Event shape distributions,
in fact, provide a continuous interpolation between processes
featuring mostly hard, perturbative radiation and configurations
dominated by soft and collinear gluon emission. The corresponding
theoretical prediction must then be constructed matching a variety of
tools: NLO perturbative results for hard emissions, soft gluon
resummations when the value of the event shape forces radiation to be
soft, and finally models of power corrections very close to threshold.
In general, models of power corrections involve nonperturbative
parameters or functions, which must be determined from experiment,
much as one does with parton distributions. The predictive power of
these models must then rely on a degree of universality of soft
radiation, which is well understood in perturbation theory, and must
be assumend to hold nonperturbatively as well. By comparing theoretical
predictions for different but related event shapes one can then test
our understanding of QCD at or beyond the strict limits of
applicability of perturbation theory.
The application of these techniques has lead in recent years to
quantitatively testable and quite successful models of power
corrections. Here I will mostly discuss results obtained by Dressed
Gluon Exponentiation (DGE) \cite{Gardi:2001di}, as applied to thrust
\cite{Gardi:2001ny,Gardi:2002bg}, the C-parameter \cite{Gardi:2003iv},
and the class of angularities
\cite{Berger:2003pk,Berger:2004xf}. These examples show that current
tools lead to simple, analytical, quantitative results that can
readily be compared with experimental data. Most strikingly, leading
power corrections to angularity distributions obey a simple scaling
rule as a function of a continuous parameter, which gives a powerful
test of our understanding of soft QCD in electron-positron
annihilation. In Sect.~(\ref{soft}) I will briefly summarize the
formalism of shape functions for event shape distributions, and show
how DGE provides a renormalon model for shape functions incorporating
the constraints of NLL soft gluon resummation. I will use mostly
thrust as a working example, comparing at the end with similar results
obtained for the $C$ parameter. In Sect.~(\ref{angu}) I will discuss
the class of angularities and derive the scaling rule, and finally in
Sect.~(\ref{hadro}) I will briefly comment on possible extensions of
these techniques to hadron-hadron collisions.
\section{SOFT GLUON EFFECTS FOR EVENT SHAPE DISTRIBUTIONS}
\label{soft}
An event shape distribution is a weighted cross section, assigning a
prescribed value to a specific infrared and collinear safe combination
of the momenta of final state particles in a high energy collision. In
the case of $e^+ e^-$ annihilation, let $F_m (p_1, \ldots, p_m)$ be
one such combination, computed for an $m$-parton final state. The
distribution of the associated event shape $f$ is then
\begin{equation}
\frac{d \sigma}{d f} = \frac{1}{2 Q^2} \sum_m \int d {\rm LIPS}_m~
\overline{\left| {\cal M}_m \right|^2} ~\delta \left(f - F_m (p_1,
\ldots, p_m) \right)~,
\label{diste}
\end{equation}
where ${\cal M}_m$ is the appropriate matrix element.
In the following, I will consider event shapes $f$ which vanish in the
limit of a pencil-like two-jet event. A prime and well-known example
is $\tau = 1 - T$, with $T$ the thrust,
\begin{equation}
T = \max_{\hat{n}} \left[\frac{\sum_i \left\vert \vec{p}_i \cdot \hat{n}
\right\vert}{Q} \right]~,
\label{thrust}
\end{equation}
which I will use below to illustrate the general features of the
approach. Other shapes I will consider include the $C$-parameter,
\begin{equation}
C = 3 - \frac{3}{2} \sum_{i, j}
\frac{(p_i \cdot p_j)^2}{(p_i \cdot Q) \, ( p_j \cdot Q)}~,
\label{cpar}
\end{equation}
which does not require a maximization procedure, and the one-parameter
class of angularities,
\begin{equation}
\tau_a = \frac{1}{Q} \sum_i (p_\perp)_i
{\rm e}^{- |\eta_i| (1 - a)}~.
\label{classang}
\end{equation}
where rapidity $\eta_i$ and transverse momentum $p_{\perp,i}$ are
computed with respect to the thrust axis, so that for $a = 0$ one
verifies that $\tau_0 = \tau$.
The common feature of these event shapes, which opens the way to an
all-order perturbative analysis and to studies of power corrections,
is the fact that for small values of $f$ all radiation is constrained
to be soft or collinear. As a consequence, the distributions develop
double logarithmic singularities of Sudakov type, which can (and must)
be resummed thanks to the universal properties of soft radiation and
to the factorizability of the cross section in the Sudakov limit. In
QCD, resummation displays the ambiguity of perturbation theory,
originating from the presence of the running coupling evaluated at
soft scales. This leads naturally to models of power corrections. I
will now illustrate the general features of the method using thrust as
an example.
\subsection{Resummation}
In order to resum singular contributions to the thrust distribution in
the limit $\tau \to 0$, one needs to take a Laplace transform, which
factorizes the $\delta$-function constraint fixing the value of
$\tau$. Logarithmic contributions then exponentiate according to
\begin{equation}
\hspace{-1cm} \int_0^\infty d \, \tau {\rm e}^{- \nu \tau} \frac{1}{\sigma}
\frac{d \sigma}{d \tau} \, = \, \exp \Bigg[ \int_0^1 \frac{d u}{u} \left(
{\rm e}^{- u \nu} - 1 \right) \Bigg( B \left(\alpha_s \left(u Q^2 \right)
\right) \, + \, 2 \, \int_{u^2 Q^2}^{u Q^2} \frac{d q^2}{q^2} \, A
\left(\alpha_s(q^2) \right) \Bigg) \Bigg]~.
\label{genexp}
\end{equation}
The pattern of exponentiation is highly nontrivial, since the double
logarithms of the ordinary perturbative expansion turn into single
logarithms in the exponent. Generically one finds a structure of the
form~\cite{Catani:1992ua}
\begin{equation}
\sum_k \alpha_s^k \sum_p^{2 k} c_{k p} L^p
\rightarrow \exp \Big[ L \, g_1 (\alpha_s L) + g_2 (\alpha_s L) +
\alpha_s g_3 (\alpha_s L) + \ldots \Big]~,
\label{pattern}
\end{equation}
where $L$ is the logarithm of the transformed variable, $L = \log \nu$
in this case. Leading logarithms (LL) are generated to all orders by
the function $g_1$, which is completely determined by the knowledge of
the anomalous dimension $A(\alpha_s)$ to one loop. Next-to-leading
logarithms (NLL), corresponding to the function $g_2$, require the
knowledge of $A(\alpha_s)$ to two loops and $B(\alpha_s)$ to one
loop. NLL accuracy is the common standard for resummation of event
shape distributions. Note however that Sudakov resummation, expressed
here by \eq{genexp}, in general involves one more function, $D
\left(\alpha_s(u^2 Q^2)\right)$. This function is associated with
wide-angle soft gluon emission, and is process-dependent, unlike the
anomalous dimension $A$. To any finite logarithmic accuracy, the
contributions of $D$ can be reproduced by modifying $B$ in a
process-dependent manner, and for the event shapes discussed here $D$
does not give any contribution at NLL level. The fact that $D$ dpends
on the scale $u Q$, however, has important consequences on power
corrections, as discussed below.
\subsection{Power Corrections}
One can deduce from \eq{genexp} (where the integration variable $u$ in
the exponent plays the role of $\tau$) that for small values of $\tau$
there are two relevant momentum scales: $\tau Q^2$ and $\tau^2
Q^2$. This can be understood from the physical picture underlying
Sudakov factorization: at small $\tau$ gluon radiation can be
organized into jets of particles collinear to the primary partons,
with invariant mass proportional to $\sqrt{\tau} Q$, plus the
contribution of wide-angle soft gluons, characterized by their total
energy $\tau Q$. It is natural to expect that power corrections will
be organized by these two scales, and thus be of the form
$(\Lambda^2/(\tau Q^2))^m$ and $(\Lambda^2/(\tau^2 Q^2))^n$
respectively. Clearly, when $\tau \sim \Lambda/Q$ all power
corrections of this second kind become important, and must be
collectively taken into account. Power corrections in the larger
scale, on the other hand, become important only when $\tau \sim
\Lambda^2/Q^2$, a value which is too small to be relevant for LEP
fits. The need for power corrections is apparent in \eq{genexp}, since
the integrals over momentum scales are perturbatively ill-defined
because of the Landau singularity of the running coupling. An elegant
way to summarize the nonperturbative information encoded in
\eq{genexp} was described in \cite{Korchemsky:1999kt}. The basic
assumptions are the applicability of the factorization underlying
\eq{genexp} all the way down to values of $\tau$ such that $\Lambda^2
\sim \tau^2 Q^2 \ll \tau Q^2 \ll Q^2$, and the existence of a
nonperturbative definition of the running coupling rendering the scale
integrals well defined. Consider then, for example, the term in \eq{genexp}
containing the anomalous dimension $A$. In order to disentangle
perturbative and nonperturbative domains, one can simply introduce a
factorization scale $\mu$, switch the order of the $q^2$ and $u$
integrations, and define
\begin{eqnarray}
S (\nu,Q^2) & \equiv & \int_0^1 \frac{d u}{u} \left( {\rm e}^{- u \nu} -
1 \right) \int_{u^2 Q^2}^{u Q^2} \frac{d q^2}{q^2} \,
A \left( \alpha_s (q^2) \right) \, = \,
S_{\rm NP}(\nu/Q,\mu) \, S_{\rm PT}(\nu, Q,\mu)~, \nonumber \\
S_{\rm NP}(\nu/Q,\mu) & \equiv & \int_0^{\mu^2} \frac{d q^2}{q^2}
A \left( \alpha_s (q^2) \right) \int_{q^2/Q^2}^{q/Q}
\frac{d u}{u} \left({\rm e}^{- u \nu} - 1 \right)
\, = \, \sum_{n = 1}^\infty \frac{1}{n!} \left( \frac{\nu}{Q}
\right)^n \lambda_n(\mu^2) \, + \, {\cal O} \left( \frac{\nu}{Q^2}
\right)~.
\label{sudpar}
\end{eqnarray}
The last equality expresses a set of nonperturbative contributions
to the Sudakov exponent in terms of moments of the anomalous dimension
$A$ at low scales. These moments,
\begin{equation}
\lambda_n(\mu^2) = \frac{1}{n} \int_0^{\mu^2} d q^2 \, q^{n - 2}
A \left( \alpha_s (q^2) \right)~,
\label{lamb}
\end{equation}
are not computable in perturbation theory: much like parton
distributions, they should be measured for a given observable at a
given factorization scale, and then used to predict different
observables, based on their universality properties. In general, the
full set of leading nonperturbative corrections will involve also
moments of the function $D$, which also parametrize power corrections
of the form $(\Lambda \nu/Q)^n$. These corrections thus have a
universal component, expressed in terms of the anomalous dimension
$A$, and a process-dependent component given by the function $D$. At
power accuracy, it is natural to disentangle the contributions of $B$
and $D$ by requiring that the function $B$ should be the same
appearing in the resummation formula for DIS structure functions,
where the corresponding $D$ function is known to
vanish~\cite{Gardi:2006jc}. Expressions like \eq{sudpar} provide a
framework to test universality, or to construct specific models of
power corrections. To summarize the effects of the parameters
$\lambda_n (\mu^2)$ one can use them to build up a ``shape function'',
according to
\begin{equation}
\exp \Big[ S_{\rm NP}(\nu/Q,\mu) \Big] \, \equiv \,
\int_0^\infty d \epsilon \, {\rm e}^{- \nu \epsilon/Q} \,
f_{\tau, {\rm NP}}(\epsilon, \mu)~.
\label{shpf}
\end{equation}
Here $\epsilon$ can be interpreted as the total energy carried into the
final state by soft gluons at scales below $\mu$. Confining oneself to
the leading power correction, corresponding to the first moment
$\lambda_1 (\mu^2)$, one recovers the result of the ``tube model''
\cite{Webber:1994zd}: that nonperturbative effects shift the
distribution away from the small $\tau$ region by an amount
proportional to the average energy carried away by soft
radiation. Subleading moments provide additional smearing.
\subsection{Dressed gluon exponentiation}
The shape function idea is very general, and can be used both to test
universality, by connecting power corrections to related event
shapes~\cite{Korchemsky:2000kp}, or to construct models based on
factorization and Lorentz invariance in specific
cases~\cite{Belitsky:2001ij}. One can get more detailed predictions by
making stronger assumptions: for example, one can apply a renormalon
model, and study the corresponding power corrections in the Sudakov
region. This is the basic idea underlying dressed gluon exponentiation
(DGE)~\cite{Gardi:2001di}. I will now summarize the basic steps of
this method, using thrust as an example.
First of all, one computes the single gluon contribution to the event
shape under study, for a gluon of nonvanishing virtuality $\xi =
k^2/Q^2$. This is the characteristic function of the dispersive
approach~\cite{Ball:1995ni,Dokshitzer:1995qm} to power corrections.
Since we are interested in the Sudakov region, we need to retain only
terms that are singular as $\tau \to 0$. Given the characteristic
function ${\cal F}(\xi, \tau)$, one can write a clean representation
of the single gluon cross section by introducing a Borel
representation for the strong coupling. One defines
\begin{equation}
\bar{A}(\xi Q^2) = \int_0^{\infty} d u \, \xi^{- u} \, \left( Q^2/\Lambda^2
\right)^{- u} \, \frac{\sin \pi u}{\pi u} \, {\rm e}^{\kappa u} \,.
\label{axiq}
\end{equation}
This amounts to an analytic continuation of the strong coupling at the
scale $k^2$ from the euclidean to the timelike region, formally valid
in the large-$n_f$ limit. The factor ${\rm e}^{\kappa u}$ is
renormalization-scheme dependent, with $\kappa = 5/3$ in the $\overline{MS}$
scheme. The single dressed gluon cross section is then
\begin{equation}
\frac{1}{\sigma} \frac{d \sigma}{d \tau} (\tau, Q^2) = - \frac{C_F}{2
\beta_0} \, \int_{0}^{1}{d \xi} \, \frac{d {\cal F}(\xi, \tau)}{d \xi}
\, \bar{A} (\xi Q^2) \, \equiv \, \frac{C_F}{2 \beta_0} \, \int_0^{\infty}
d u \, \left( Q^2/\Lambda^2 \right)^{- u} \, B(\tau, u)\,,
\label{sdgf}
\end{equation}
where in the last equality I introduced the Borel function $B(\tau,
u)$, which contains the physical information on the thrust
distribution. The strategy of performing all integrals except the one
over the Borel parameter yields at the end a trasparent representation
for power corrections. In the case of thrust, the terms responsible
for logarithmic enhancements in $\dot{\cal F}(\xi, \tau)$ are given by
\begin{equation}
\left. \dot{\cal F}(\xi, \tau) \right|_{\log} =
2 \, \left(\frac{2}{\tau} - \frac{\xi}{\tau^2} -
\frac{\xi^2}{\tau^3} \right)~,
\label{hjm1}
\end{equation}
which gives a Borel function of the form
\begin{equation}
\left. B(\tau, u) \right\vert_{\rm log} \, = \, 2 \, {\rm e}^{\kappa u} \,
\frac{\sin \pi u}{\pi u} \, \left[ \frac{2}{u} \, \tau^{- 1 - 2 u} -
\tau^{- 1 - u} \left(\frac2u + \frac1{1 - u} +
\frac{1}{2 - u} \right) \right]\,.
\label{Bt}
\end{equation}
Note that already at this stage one can make several useful
observations. Poles in $B(\tau, u)$ at, say, $u = u_0$, would
correspond to renormalon singularities in the distribution, and
expected power corrections of size $(\Lambda/Q)^{2 u_0}$; in fact,
$B(\tau, u)$ has no such poles: the would-be singularities at $u =
1,2$ are cancelled by the factor $\sin \pi u$, and poles at $u = 0$
cancel between the two terms in square brackets, because of the
infrared safety of thrust. Renormalons arise when taking moments of
the distribution, because of the convergence constraints on the Borel
integral at small values of $\tau$. One also notes that the first term
in \eq{Bt} is associated with soft wide-angle radiation, since it
generates in \eq{sdgf} terms proportional to $(\Lambda/(\tau Q))^{2
u}$. Similarly, the second term in \eq{Bt} is associated with the
jet function, contaning collinear as well as soft enhancements.
The key step in DGE is to note that at LL level Sudakov resummation
yields a simple exponentiation of the one-gluon emission cross section
in moment space. One can then retain all large-$n_f$ information, and
the corresponding model of power corrections, in the Sudakov exponent
by simply using the single dressed gluon cross section as kernel of
exponentiation. One defines
\begin{equation}
\left( \frac{1}{\sigma} \frac{d \sigma}{d \tau}
\right)_{\rm DGE} = \int_{ k - {\rm i} \infty}^{k + {\rm i} \infty}
\frac{d \nu}{2 \pi {\rm i}} \, {\rm e}^{\nu \tau} \,
\exp \left[ - E (\nu, Q^2) \right]\,,
\label{tau_DGE}
\end{equation}
where the Sudakov exponent is now given by
\begin{equation}
E (\nu, Q^2) \, = \, \int_0^{\infty} \, d \tau \,
\left( 1 - {\rm e}^{- \nu \tau} \right)
\left( \frac{1}{\sigma} \frac{d \sigma}{d \tau} \right)_{\rm SDG}
\equiv \frac{C_F}{2 \beta_0} \, \int_0^{\infty} d u \,
\left(Q^2/\Lambda^2\right)^{- u} \, B_\tau (\nu, u) \, .
\label{S}
\end{equation}
Here the single dressed gluon cross section is defined by \eq{sdgf},
virtual corrections have been taken into account by subtracting the
value of the Laplace transform at $\nu = 0$, and in the second
equality the Borel function $B_\tau (\nu, u)$ for the Sudakov exponent
has been defined. As usual, all integrals are performed except the one
on the Borel parameter $u$. Although formally similar to \eq{sdgf} for
the single dressed gluon cross section, \eq{S} has a much richer
physical content, displayed by the nontrivial renormalon structure of
the Borel function $B_\tau (\nu, u)$. This is a consequence of the
fact that exponentiation, subject to the constraint of energy
conservation, has promoted the single gluon result to a genuine
approximation for multigluon emission\footnote{In fact, the exponent
$E (\nu, Q^2)$ has a natural interpretation in terms of the Borel
representation of Sudakov anomalous dimensions associated respectively
with soft and collinear radiation~\cite{Gardi:2006jc}. Corrections
subleading in $n_f$ change the the logarithmic behavior of the cross
section as well as the size of the residues of poles in the Borel
plane, but they are not expected to modify the analytic structure of
$B_\tau (\nu, u)$, which determines which power corrections are
actually present.}. In the specific case of the thrust, the result
for the Borel function is
\begin{equation}
B_\tau (\nu, u) \, = \, 2 \, {\rm e}^{\kappa u} \, \frac{\sin \pi u}{\pi u}
\left[\Gamma(- 2 u) \left(\nu^{2 u} - 1 \right) \frac{2}{u}
- \Gamma(- u) \left(\nu^u - 1\right) \left(\frac2u + \frac1{1 - u}
+ \frac{1}{2 - u} \right) \right]\,.
\label{bnut}
\end{equation}
It is useful to compare this result with the one for the
C-parameter~\cite{Gardi:2003iv}
\begin{equation}
B_c (\nu, u) \, = \, 2 \, {\rm e}^{\kappa u} \,
\frac{\sin \pi u}{\pi u} \, \left[\Gamma(- 2 u) \left(\nu^{2 u} - 1
\right) 2^{1 - 2 u} \frac{\sqrt{\pi} \Gamma(u)}{\Gamma(\frac12 + u)}
\, - \, \Gamma(- u) \left({\nu}^{u} - 1 \right)
\left(\frac2u + \frac1{1 - u} + \frac{1}{2 - u} \right) \right]\,.
\label{bnuc}
\end{equation}
Clearly, Eqs.~(\ref{bnut}) and (\ref{bnuc}) are very similar; there
are, however, important differences, which highlight the degree of
universality to be expected in comparing the two distributions, and
which can in principle be tested experimentally. First of all, the
Borel functions contain perturbative information on Sudakov
logarithms: although the exponentiation was performed assuming
independent gluon radiation, it can be shown~\cite{Gardi:2001ny} that
one can upgrade the formalism to NLL accuracy by simply replacing the
running coupling \eq{axiq} with the two-loop expression, and by
changing renormalization scheme, including in the constant $\kappa$
the contribution of terms singular as $x \to 1$ in the NLO
Altarelli-Parisi splitting function (the ``gluon bremsstrahlung''
scheme~\cite{Catani:1990rr}). Beyond NLL, the coefficients of all
subleading logarithms can be computed in the large $n_f$ limit, by
simply expanding the Borel function in powers of $u$, and replacing
$u^n \to n! (b_0 \alpha_s/\pi)^{n + 1}$. Computing subleading
logarithms uncovers the factorial growth of their coefficients, and
can be used to gauge the reliability of perturbative resummation in
different kinematical regimes. Next, one may observe that the infrared
safety of $\tau$ and $C$ is once again reflected in the cancellation
of the poles of both Borel functions at $u = 0$. One also notes that
wide-angle soft radiation and collinear gluons contribute as before
two separate terms, and the jet functions (the terms proportional to
$\Gamma(- u)$) are identical for the two observables\footnote{Note
that $B_c (\nu, u)$ is computed for a rescaled $C$ parameter, $c =
C/6$}. They contribute renormalons at $u = 1,2$, corresponding to
exponentiated power corrections of the form $(\Lambda^2 \nu/
Q^2)^{1,2}$. Soft gluon contributions, on the other hand, have
renormalons at $u = m/2$, for all odd values of $m$, yielding the
leading power corrections $(\Lambda \nu/Q)^m$, and they are
quantitatively different for $\tau$ and $C$, distinguishing the two
observables. Specifically, by taking the ratio of the two ``soft
functions'' (the terms proportional to $\Gamma(- 2 u)$) and expanding
in powers of $u$, one can verify that the two observables begin to
differ perturbatively at NNLL level, as predicted
in~\cite{Catani:1998sf}, but the growth of the coefficients of further
subleading logarithms is weaker for the $C$-parameter than for the
thrust. Similarly, if one boldly takes the large-$n_f$ residues of the
poles of the Borel functions as a reasonable estimate of the size of
the corresponding power corrections, one observes that $(\Lambda
\nu/Q)^m$ corrections are systematically smaller for the $C$
parameter: the two shape functions should therefore differ, and one
expects that the resummed perturbative prediction, as well as the
approximation of the shape function by a constant shift, should work
better phenomenologically for $C$ than for $\tau$.
\section{THE CLASS OF ANGULARITIES}
\label{angu}
The discussion in Sect.~(\ref{soft}) illustrates the predictive power
of DGE. Another interesting application concerns angularities, whose
definition, \eq{classang}, can be rewritten for massless particles as
\begin{equation}
\tau_a \, = \, \frac{1}{Q} \, \sum_i \omega_i \left( \sin
\theta_i \right)^a \left( 1 - \left| \cos \theta_i \right|
\right)^{1 - a}~,
\label{barfdef}
\end{equation}
with $\theta_i$ the angle with respect to the thrust axis.
Angularities (so christened in \cite{Berger:2004xf}) were introduced
in \cite{Berger:2003iw} as auxiliary shape variables used to tame
nonglobal logarithms~\cite{Dasgupta:2001sh} for observables related to
out-of-jet energy flow. They have several remarkable features, which
make them very interesting for our understanding of QCD at the edge of
the perturbative domain . First of all, they are characterized by a
tunable parameter $a$, which can be used to interpolate between
different shapes, or to bring to focus specific momentum
configurations in a continuous way. The parameter $a$ must satisfy $a
< 2$ for infrared safety, and a tighter restriction $a <1$ is required
in order to preserve a relatively simple resummation in the Sudakov
region, in the form of \eq{genexp}: for $1 \leq a < 2$ further
logarithmic singularities associated with jet recoil must be taken
into account~\cite{Dokshitzer:1998kz}. For $a = 1$, one recognizes
that $\tau_1 = B$, the broadening; for $a = 0$, $\tau_0 = 1 - T$; for
negative $a$, events dominated by high rapidity give increasingly
suppressed contributions to $\tau_a$, which in turn suppresses power
corrections of collinear origin; finally, for $a \to - \infty$ the
distribution becomes a $\delta$ function at $\tau_a = 0$, with a
strenght given by the total cross section.
Remarkably, although the relative weights of rapidity and transverse
momentum change with $a$, it is possible to derive a resummation
formula~\cite{Berger:2003iw} of the form of \eq{genexp}, valid for $a
< 1$. Indeed, at NLL accuracy one can write
\begin{equation}
\ln \big[ \tilde{\sigma} \left(\nu, a \right) \big] \, = \,
\int\limits_0^1 \frac{d u}{u} \, \Bigg[ \,
B \left(\alpha_s(u Q^2)\right) \left( {\rm e}^{- u \, \nu^{2/(2 - a)} } -1 \right)
\, + \, 2 \, \int_{u^2 Q^2}^{u Q^2} \frac{d q^2}{q^2} \,
A \left(\alpha_s (q^2) \right) \left( {\rm e}^{- u^{1 - a} \nu
\left( q/Q \right)^a } - 1 \right) \Bigg]~.
\label{thrustcomp}
\end{equation}
The $a$ dependence of Sudakov logarithms is clearly nontrivial: as an
example, the function $g_1 (\alpha_s L)$ responsible for leading
logarithms in \eq{pattern} is given by
\begin{equation}
g_1 ( x, a ) \, = \, - \, \frac{4}{\beta_0} \, \frac{2 - a}{1 - a} \,
\frac{A^{(1)}}{x} \, \Bigg[ \frac{1 - x}{2 - a} \,
\ln \left( 1 - x \right) \, - \, \left(1 - \frac{x}{2 - a}\right)
\ln \left(1 - \frac{x}{2 - a}\right) \Bigg] \, .
\label{g1a}
\end{equation}
Power corrections, however, turn out to have a much simpler $a$
dependence~\cite{Berger:2003pk}. Performing the analysis leading to
\eq{lamb}, one easily finds that in the nonperturbative region all
moments of the anomalous dimension $A$ are multiplied by a simple
common factor
\begin{equation}
\lambda_n^{(a)} (\mu^2) \, = \, \frac{1}{1 - a} \,\frac{1}{n} \,
\int_0^{\mu^2} d q^2 \, q^{n - 2} A \left( \alpha_s (q^2) \right) \,
= \, \frac{1}{1 - a} \, \lambda_n^{(0)} (\mu^2) ~.
\label{lamba}
\end{equation}
As a consequence, the Laplace transform of the shape function defined
by \eq{shpf} obeys a simple and remarkable scaling rule
\begin{equation}
\tilde{f}_{a, {\rm NP}} \left(\frac{\nu}{Q}, \kappa\right) = \left[
\tilde{f}_{0, {\rm NP}} \left(\frac{\nu}{Q}, \kappa\right)
\right]^{1/(1 - a)}~,
\label{rule}
\end{equation}
which should be experimentally testable without great effort using
existing LEP data. Lacking a direct experimental analysis, the scaling
rule was tested against the output of {\tt PYTHIA}, with positive
results~\cite{Berger:2003pk,Berger:2003gr}. The physical picture
underlying the scaling rule is appealing, and once again reminiscent
of the ``tube model''. The relevant feature of the radiation pattern
is boost invariance, which applies to soft gluons emitted in the
two-jet limit, since such emissions are correctly represented by the
eikonal approximation. This means that soft gluons contribute to the
event shape a rapidity-independent amount, and in turn the integration
over rapidity simply measures the size of the region where gluons can
be emitted without strongly affecting the event shape. This region
scales with $(1 - a)^{-1}$. The derivation of the scaling rule
neglects correlations between gluons emitted into the opposite
hemispheres defined by the thrust axis, which however are expected to
become important only for $a \geq 1$, the region excluded by the
present treatment. The main assumption is then that nonperturbative
soft radiation should share the property of boost inveriance with the
relatively harder perturbative component which is treated by
resummation. This is the nonperturbative property that would be
directly tested by an experimental study of \eq{rule}.
One can go further, and study subleading power corrections, mostly
related to radiation collinear to the hard jets. Applying the same
method to the anomalous dimension $B$, and to the subleading terms
generated by $A$, once again one finds a simple pattern: the Laplace
transform of the cross section can be expressed in factorized form,
introducing a subleading shape function, as
\begin{equation}
\tilde{\sigma} \left(\nu, a\right) = \tilde{\sigma}_{\rm PT}
\left(\nu, \kappa, a\right) \, \tilde{f}_{a, {\rm NP}}
\left(\frac{\nu}{Q},\kappa\right) \, \tilde{g}_{a, {\rm NP}}
\left(\frac{\nu}{Q^{2 - a}},\kappa\right)~.
\label{sub}
\end{equation}
Clearly, as $a$ becomes large and negative, collinear power
corrections are expected to become more and more negligible, and the
scaling rule \eq{rule} is expected to hold with increasing accuracy.
In the light of the discussion of Sect.~(\ref{soft}), it is
interesting to verify whether these nice features of angularities are
preserved in specific models for the shape function, such as DGE.
This is particularly relevant in this case, since the key result,
\eq{rule}, is related to boost invariance, which is broken by the
formal introduction of gluon virtuality, which is a necessary tool of
renormalon analysis. The study of angularities with DGE was performed
in~\cite{Berger:2004xf}. It is technically nontrivial, since the
interplay of the parameter $a$ with gluon virtuality and phase space
constraints makes it difficult to extract the $a$ dependence
analytically. The first step is to find an appropriate generalization
of the definition of angularity to the case of single massive gluon
emission. Such definition should have the correct limit as $\xi \to
0$, reduce to known results for thrust as $a \to 0$, and be simple
enough to keep the computational task manageable. At one loop, one
such definition is
\begin{equation}
\tau_a = \frac{(1 - x_i)^{1 - a/2}}{x_i}
\left[(1 - x_j - \xi)^{1 - a/2} (1 - x_k + \xi)^{a/2} +
(j \leftrightarrow k) \right]~,
\label{defo}
\end{equation}
where $x_n = 2 p_n \cdot Q/Q^2$ are the customary energy fraction
variables, and the definition applies to the phase space region where
the gluon is soft; $x_i$ is then the (anti)quark energy fraction.
With this definition, it is possible to construct the Borel function
for the Sudakov exponent for angularities, in analogy with
Eqs.~(\ref{bnut}) and (\ref{bnuc}). Remarkably, the soft component of
the Borel function, responsible for all leading power corrections, is
just the expected rescaling of \eq{bnut},
\begin{equation}
B_a^{\rm soft} (\nu, u) = \frac{1}{1 - a} \left[ 2 \,
{\rm e}^{\kappa u} \, \frac{\sin \pi u}{\pi u} \, \Gamma(- 2 u)
\left(\nu^{2 u} - 1 \right) \frac{2}{u} \right]~,
\label{scale}
\end{equation}
which leads once again to the scaling rule, \eq{rule}. Collinear power
corrections are much more difficult to handle analitycally, and the
collinear counterpart of \eq{scale} can at best be expressed in terms
of a one-dimensional integral representation, which reduces to
combinations of hypergeometric functions for rational values of
$a$. This is however enough to classify the singularities in the Borel
parameter $u$, and thus the pattern of power corrections. One indeed
finds that all these subleading power corrections can be organized in
a single shape function $\tilde{g}$, depending only on the combination
$\nu/Q^{2 - a}$, as in \eq{sub}. DGE thus confirms the general scaling
behavior found from resummation, showing that the introduction of
gluon virtuality does not spoil the effects of boost invariance in the
Sudakov limit. Thrust, jet masses, angularities and the $C$ parameter
are all found to have closely related pattern of power corrections,
highlighted by the scaling rule relating generic angularities to the
thrust. Clearly, this is a highly predictive framework, and our
understanding of soft radiation in the two-jet limit can be put to
stringent tests.
\section{HADRON COLLIDER EVENT SHAPES}
\label{hadro}
As we approach the expected date for the start up of the Large Hadron
Collider at CERN, it is natural and appropriate to ask whether tools
like those described here could be applicable in the environment of
hadron collisions, and, if so, to what extent power corrections might
be relevant to our understanding of the data, at the extreme energies
available at the LHC. Beginning with the second question, one might
naively observe that $\Lambda/Q$ must be a very small number for any
reasonable value of the hard scale $Q$ that one might envisage at the
LHC. It would not be wise, however, to neglect power correction
studies on this basis, for at least three different reasons.
First of all, it has already been shown at the Tevatron that power
corrections have an impact even on observables which are largely
dominated by high $p_\perp$ events, and even at very high
energy~\cite{Mangano:1999sz}. In Ref.~\cite{Mangano:1999sz}, Mangano
considered the single-jet inclusive $E_\perp$ distribution, comparing
data at different CM energies. One sees that ratios of cross sections
at different energies do not have the proper scaling behavior dictated
by NLO QCD, but the correct behavior can be recovered by including a
power correction determined by a single parameter associated with the
normalization of the jet transverse energy. The reason is that even a
small shift in the jet $E_\perp$ is amplified in the distribution by
the fact that the cross section is falling steeply for increasing
$E_\perp$, so that $\delta \sigma/\sigma \sim - n \, \delta E_\perp$
if $\sigma \sim E_\perp^{-n}$. One can expect that in general power
corrections will be important for the determination of jet energy
scales, and in turn accurate knowledge of these scales may well turn
out to be crucial for many high energy studies.
The second reason is that almost any LHC observable will require,
before it can be compared with a theoretical prediction, a subtraction
of all hadronic activity unrelated to the hard scattering, loosely
referred to as ``underlying event''. There is currently very little,
if any, theoretical control on the underlying event, which at the LHC
will contain a mix of multiple parton scattering, beam-beam
interactions and soft radiation associated with the selected hard
process. Even Monte-Carlo methods have difficulties in finding the
proper tuning to describe this kind of physics (see, for
example,~\cite{Alekhin:2005dx}). In this context, the lesson of power
correction studies in the gentler environment of $e^+ e^-$
annihilation or DIS is that we might learn to discriminate between the
different components of soft radiation in hadron collisions. On the
one hand there are soft and collinear gluons associated with the hard
scattering event: their effects are in principle computable in PQCD,
using generalizations of the known techniques, and their distribution
in phase space and in the space of color configurations will be
nontrivial and predictable. On the other hand, there is soft radiation
which is in practice out of reach for the techniques of PQCD, such as
minijets due to multiple parton scattering or soft gluons arising from
beam-beam interactions. This second kind of radiation fills phase
space with a high degree of uniformity, and will have to be modelled
with different techniques, including Monte-Carlo tools. This kind of
statistical modelling is bound to be more successful if we can first
achieve a better understanding of the ``pure'' hard scattering
process, including the energy and color flow that it generates at all
scales.
Finally, it should be emphasized that the physics of event shapes at
hadron colliders is interesting for its own sake, as a probe to
understand hadronization in terms of both momentum and color flow. It
is well known that for hadron collisions, where most processes involve
four partons already at Born level, Sudakov resummation is expressed
in terms of anomalous dimension matrices that tie together color
exchange and momentum flow. This interplay is bound to influence the
pattern of power corrections as well, and studies of this kind may
well lead to deep insights into the mechanics of color neutralization
and hadron formation.
Preliminary studies of resummed event shapes at hadron colliders have
already been performed~\cite{Banfi:2004nk,Zanderighi:2006mx}, and a
generalization of the notion of angularity to a hadronic environment
has been proposed~\cite{Berger:2005ct}. For most of these proposals, a
primary concern is that of suppressing the contributions of particles
close to the beam axis, where beam remnants interfere with all
measurements. It should be noted, however, that soft radiation
associated with the underlying event tends to fill phase space,
including regions separated in rapidity from both the beam and the
high-$p_\perp$ jets. It would therefore be of great interest to find
event shapes designed to focus on this kind of wide-angle radiation,
where it would be most useful to disentangle soft gluons generated by
the hard scattering from the genuine underlying event (a pioneering
study with a similar goal is~\cite{Marchesini:1988hj}). A promising
avenue of investigation might be the use of observables such as those
introduced in~\cite{Berger:2003iw}, joint distributions correlating
energy flow in a chosen angular region $\Omega$ with a standard event
shape such as ordinary angularity. The form of such a correlation is
\begin{equation}
\sigma \left( \epsilon_1, \epsilon_2, a \right) = \frac{1}{2 s} \sum_N
\overline{\left| M(N) \right|^2} \,\,
\delta (\epsilon_1 - f_\Omega (N)) \,\, \delta(\epsilon_2 -
\tau_a^{(1)} (N) - \tau_a^{(2)} (N))~,
\label{esefc}
\end{equation}
where $f_\Omega (N)$ is an observable related to energy flow into the
angular region $\Omega$, away from hard jets, while $\tau_a^{(i)} (N)$
are the contributions to angularity from the two hemispheres defined
by the thrust axis.
In a hadronic environment, one might envisage measuring angularities
with respect to the current jet axis, as suggested
in~\cite{Berger:2005ct}, or even introducing a third parameter
$\epsilon_3$ to constrain radiation near the beam remnants. Tuning the
various parameters and energy threshold in such observables one would
be able to focus on different regions of phase space, forcing the
observable to be more or less inclusive for soft gluons, or for
particles collinear either to the beam or to the current jets. Notice
that boost invariance, underlying the scaling rule for angularities,
is lost in hadron collisions, where correlations between beam jets and
current jets must be taken into account.
Clearly, event shape studies at hadron colliders are in their early
days. I believe that such studies will be both instrumental to
further our understanding of QCD, and very helpful in order to exploit
the full potential of the LHC, a task which will require a solid
understanding of strong interactions as much as good skills in the
building and testing of new physics models.
\begin{acknowledgments}
\noindent I would like to thank Carola Berger and Einan Gardi, who
worked with me on the research described here, and went over the draft
of this contribution, spotting several misprints and one significant
inaccuracy, now fixed. I also thank George Sterman and Mrinal Dasgupta
for sharing their insights, and the organizers and FRIF for the
opportunity granted by the Workshop, which was conducted in a friendly
and constructive atmosphere. This work was supported in part by MIUR
under contract $2004021808\_009$.
\end{acknowledgments}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 7,888 |
\section{Introduction}
\label{sec:introduction}
The problem of combining data from multiple assays is an important
topic in modern biostatistics. For many studies, the researchers have
more data than they know how to handle. For example, a
researcher studying cancer outcomes may have both gene expression and
copy number data for a set of patients. Should that researcher use
both types of predictors in their analysis? Should any care be given
to distinguish the fact that these predictors are coming from
different assays and may have differing meanings? If the researcher
needs to make future predictions based on only gene expression, is
there a way that having copy number data in a training set can help
those future predictions? All of these are important questions that
are still up for debate.
In this paper we propose a method for this problem called ``Collaborative
Regression'', a form of sparse
supervised canonical correlation analysis.
In Section \ref{sec:coll-regr} we define Collaborative
Regression (CollRe) and characterize its solution. This involves explicit
closed form solutions for the unpenalized algorithm, as well as a
discussion of some useful convex penalties that can be applied. Then, in Section
\ref{sec:coaching-variables} we explore the possibility of using
CollRe in a prediction framework. While this may seem like an
intuitive use case, simulations suggest that CollRe is not able to
improve prediction error even over methods that do not take advantage
of the secondary dataset.
We look at using CollRe in a sparse sCCA framework in Section
\ref{sec:mult-canon-corr}, including a simulation study where we
compare CollRe to one of the leading competitors. We
show how the penalized version can be applied to a real
biological dataset in Section \ref{sec:real-data-example}. Finally,
in section \ref{appen} we explore how to efficiently solve the
convex optimization problem given by the penalized form of the algorithm.
\section{Collaborative Regression}
\label{sec:coll-regr}
Collaborative Regression is a tool designed for the scenario where
there are groups of covariates that can
be naturally partitioned and a response variable. Let us assume that we have
observed $n$ instances of $p_x+p_z$ covariates and a response. We can
partition the covariates into two matrices, $X$ and $Z$, that are
$n\times p_x$ and $n\times p_z$ respectively. The response values
are stored in a vector, $\vec{y}$, of length $n$. Then Collaborative
Regression finds the $\hat \vec{\theta_x}$ and $\hat \vec{\theta_z}$ that minimize the
following objective function:
\begin{equation}
\label{crobj}
J(\theta_x, \theta_z)=\frac{b_{xy}}{2} \|\vec{y} - X\vec{\theta_x}\|^2 + \frac{b_{zy}}{2} \|\vec{y} - Z\vec{\theta_z}\|^2 + \frac{b_{xz}}{2}
\|X\vec{\theta_x} - Z\vec{\theta_z}\|^2
\end{equation}
This objective function seems natural for the multiple
dataset situation. Basically, it says that we want to make
predictions of $\vec{y}$ based on $X$ or $Z$, but we will penalize
ourselves based on how different the predictions are. Essentially,
the goal is to uncover a signal that is common to $X$, $Z$, and $\vec{y}$.
Consider trying to maximize the objective function (\ref{crobj}). It is
easy to show using calculus that the optimal solution, $\hat\vec{\theta_x}$ and
$\hat\vec{\theta_z}$ will satisfy the following First Order Conditions:
\begin{equation}\hat\vec{\theta_x} = \frac{1}{b_{xy}+b_{xz}}(X^TX)^{-1}X^T(b_{xy}\vec{y} + b_{xz}
Z\hat\vec{\theta_z})
\end{equation}
\begin{equation}
\hat\vec{\theta_z} = \frac{1}{b_{zy}+b_{xz}}(Z^TZ)^{-1}Z^T(b_{zy}\vec{y} + b_{xz}
X\hat\vec{\theta_x}).
\end{equation}
By substituting for $\hat\vec{\theta_z}$ and solving, we can find a closed form
solution for $\hat\vec{\theta_x}$:
\begin{multline}
\hat\vec{\theta_x} = \left(I - \frac{b_{xz}^2}{(b_{xy} + b_{xz})(b_{zy} +
b_{xz})}(X^TX)^{-1}X^TZ(Z^TZ)^{-1}Z^TX\right)^{-1}\\\left(\frac{b_{xy}}{b_{xy}
+ b_{xz}}(X^TX)^{-1}X^T\vec{y} + \frac{b_{xy}b_{zy}}{(b_{xy}
+ b_{xz})(b_{zy} + b_{xz})}(X^TX)^{-1}X^TZ(Z^TZ)^{-1}Z^T\vec{y}\right)
\end{multline}
In the above we have assumed that $X^TX$
and $Z^TZ$ are non-singular. Assuming they are, and none of
the parameters are zero, then that
guarantees the invertibility of \newline
$\left(I - \frac{b_{xz}^2}{(b_{xy} + b_{xz})(b_{zy} +
b_{xz})}(X^TX)^{-1}X^TZ(Z^TZ)^{-1}Z^TX\right)$. Note that $X^TX$
and $Z^TZ$ will always be nonsingular in the classical case where
$\max(p_x,p_z) < n$.
\subsection{Infinite Series Solution}
\label{sec:inif-seri-solut}
Another way to characterize the optimal solution to the objective
function (\ref{crobj}) is as an infinite series. Instead of solving for
$\hat\vec{\theta_x}$ after substituting, consider instead what would happen if we
just continued substituting for $\hat\vec{\theta_x}$ or $\hat\vec{\theta_z}$ on the RHS. Then, we
get an infinite series representation of $\hat\vec{\theta_x}$. Let $P_X =
X(X^TX)^{-1}X^T$ be the matrix that performs orthogonal projection
onto the column space of $X$ (and let $P_Z$ be defined similarly).
Then we can also write $\hat\vec{\theta_x}$ as:
\begin{multline}
\hat\vec{\theta_x} = \frac{b_{xy}}{b_{xy} + b_{xz}}(X^TX)^{-1}X^T\vec{y} + \frac{b_{xz}}{b_{xy}
+ b_{xz}}\frac{b_{zy}}{b_{zy} + b_{xz}}(X^TX)^{-1}X^TP_Z\vec{y} \\
+ \frac{b_{xz}}{b_{xy} + b_{xz}}\frac{b_{xz}}{b_{zy} + b_{xz}}\frac{b_{xy}}{b_{xy} +
b_{xz}}(X^TX)^{-1}X^TP_ZP_X\vec{y}\\
+ \frac{b_{xz}}{b_{xy} + b_{xz}}\frac{b_{xz}}{b_{zy} + b_{xz}}\frac{b_{xz}}{b_{xy} +
b_{xz}}\frac{b_{zy}}{b_{zy} + b_{xz}}(X^TX)^{-1}X^TP_ZP_XP_Z\vec{y} \dots
\end{multline}
If we let
\begin{align*}
w_i &=&
\begin{cases}
\frac{b_{xy}}{b_{xy} + b_{xz}}\left(\frac{b_{xz}}{b_{xy} +
b_{xz}}\frac{b_{xz}}{b_{zy} + b_{xz}}\right)^i & \text{if }i\text{ is even} \\
\frac{b_{xz}}{b_{xy} +
b_{xz}}\frac{b_{zy}}{b_{zy} + b_{xz}}\left(\frac{b_{xz}}{b_{xy} +
b_{xz}}\frac{b_{xz}}{b_{zy} + b_{xz}}\right)^i & \text{if }i\text{ is odd}
\end{cases}\\
\vec{y}^{(i)} &=&
\begin{cases}
(P_ZP_X)^i\vec{y} & \text{if }i\text{ is even} \\
(P_ZP_X)^iP_Z\vec{y} & \text{if }i\text{ is odd}
\end{cases}
\end{align*}
Then
\begin{equation}
\hat\vec{\theta_x} = (X^TX)^{-1}X^T\displaystyle\sum_{i=0}^\infty w_i\vec{y}^{(i)}
\end{equation}
Looking at the infinite expansion can help build some understanding of
what CollRe actually does. We note that $\sum w_i = 1$, so
essentially CollRe is equivalent to regressing $X$ on the weighted
average of the $\vec{y}^{(i)}$'s. Those $\vec{y}^{(i)}$'s trace out the path of
successive projections onto the column space of $X$ and $Z$. As the
column spaces of $X$
and $Z$ are affine, it is known from Projection onto Convex sets that
the sequence will converge to the projection of $\vec{y}$ onto the
intersection of those two spaces. In the case where the columns of
$X$ and $Z$ are linearly independent, $\vec{y}^{(i)}$ will eventually converge to
0. Thus, CollRe is basically shrinking $\vec{y}$ towards the part that can
be explained by both $X$ and $Z$.
Additionally, we get some picture as to how the parameters $\{b_{xy}, b_{zy}, b_{xz}\}$ affect the solution. $\frac{b_{xy}}{b_{xy} +
b_{xz}}$ acts in large part to control the amount of shrinkage
imposed on $\hat\vec{\theta_x}$, while $\frac{b_{zy}}{b_{zy} + b_{xz}}$ does the same
for $\hat\vec{\theta_z}$.
\subsection{Penalized Collaborative Regression}
\label{sec:penal-coll-regr}
One nice aspect of the objective function (\ref{crobj}) is
that it is convex. This means that the problem can still be easily
solved through convex optimization techniques if we add convex penalty
functions to the objective. Thus, we can define Penalized
Collaborative Regression (pCollRe) as finding the minimizer of the
following objective:
\begin{equation}
\label{pcrobj}
F(\theta_x, \theta_z)=\frac{b_{xy}}{2} \|\vec{y} - X\vec{\theta_x}\|^2 + \frac{b_{zy}}{2} \|\vec{y} - Z\vec{\theta_z}\|^2 + \frac{b_{xz}}{2}
\|X\vec{\theta_x} - Z\vec{\theta_z}\|^2 + P^x(\vec{\theta_x}) + P^z(\vec{\theta_z})
\end{equation}
where $P^x(\vec{\theta_x})$ and $P^z(\vec{\theta_z})$ are convex penalty functions. Note
that some of the convex penalties that may warrant use include:
\begin{itemize}
\item The Lasso: $P^x(\vec{\theta_x})$ is an $\ell_1$ penalty on $\vec{\theta_x}$, namely
$P^x(\vec{\theta_x}) = \lambda_x\|\vec{\theta_x}\|_1$. The lasso penalty is known to
introduce sparsity into $\vec{\theta_x}$ for sufficiently high values of $\lambda_x$.
\item Ridge: $P^x(\vec{\theta_x})$ is a (squared $\ell_2$ penalty on $\vec{\theta_x}$, namely
$P^x(\vec{\theta_x}) = \lambda_x\|\vec{\theta_x}\|^2_2$. Ridge penalties help to smooth
the estimate of $X^TX$ to ensure non-singularity. This can be
especially important in the high dimensional case where $X^TX$ is
known to be singular.
\item The Fused Lasso: $P^x(\vec{\theta_x}) = \displaystyle\sum_{i=2}^{i=p_x}\lambda_x|(\vec{\theta_x})_i
- (\vec{\theta_x})_{i-1}|$. The fused lasso will help to ensure that $\vec{\theta_x}$ is
smooth. This can be helpful if there is reason to believe that the
predictors can be sorted in a meaningful manner (as with copy number
data).
\end{itemize}
In addition to the convex penalties above, situations may also call
for linear combinations of those penalties. For example, the lasso
and ridge penalties are often combined to find sparse coefficients for
predictors that are highly correlated. The lasso and fused lasso are
often combined to find sparse and smooth coefficient vectors. In
Section \ref{appen} we discuss solving pCollRe efficiently in the case where the
penalty terms are asso penalties.
\section{Using CollRe for Prediction}
\label{sec:coaching-variables}
One potentially appealing use of CollRe where we
want to make predictions of $\vec{y}$ for future cases where you will only
have the variables in $X$ available, and $Z$ is only be available for a
training set. Can the information
contained in $Z$ be used to help identify the correct direction in
$X$? There are many practical situations in which this framework might be useful. For example, maybe it is much
more costly to gather data with a lower amount of noise.
Alternatively, it could be that some data is not accessible until
after the fact; autopsy results may be very helpful in identifying
different types of brain tumors, but it is hard to use that
information to help current patients.
CollRe seems like it provides a natural way in which to perform
a regression with additional variables present only in the training
set. Basically, it is saying that we want our future predictions to
agree with what we would have predicted given $Z$. In this framework, CollRe is similar to ``preconditioning'' as
defined by Paul and others (2008) \cite{PBHT2006}. Instead of
preconditioning on $Z$ and then fitting the regression, we are
simultaneously doing the preconditioning and fitting.
Looking at the infinite
series solution in Section \ref{sec:inif-seri-solut} it is clear that
performing CollRe is similar to doing ordinary regression after
shrinking $y$. If that shrinkage on $y$ is done in such a way that it
reduces noise, we may ultimately expect ourselves to do better in
estimating the correct $\hat\vec{\theta_x}$. We investigate this next.
\subsection{Simulated Factor Model Example}
\label{sec:simul-fact-model}
We decided to generate data from a factor model to test CollRe. A
factor model seems natural for this problem,
and is a simply way to create correlations between
$X$, $Z$, and $\vec{y}$. Another reason the factor model was appealing is
because it is relatively easy to analyze, and given $\hat\vec{\theta_x}$ and
$\hat\vec{\theta_z}$ it is easy to compute statistics like the expected
prediction error or the correlations between linear combinations of
the variables. More concretely, given values for parameters $n, p_x,
p_z, p_u, s_u, s_x, s_z,\text{ and } s_y$, we generate data according to the
following method:
\begin{enumerate}
\item $\vec{v_y} \in \mathcal{R}^{p_u}$ distributed MVN(0,$I_{p_u}$)
\item $\vec{v^x_j} \in \mathcal{R}^{p_x}$ distributed iid
MVN(0,$I_{p_x}$) for $j = 1,\dots,p_u$
\item $V_x = [\vec{v^x_1},\dots,\vec{v^x_{p_u}}]$
\item $\vec{v^z_j} \in \mathcal{R}^{p_z}$ distributed iid
MVN(0,$I_{p_z}$) for $j = 1,\dots,p_u$
\item $V_z = [\vec{v^z_1},\dots,\vec{v^z_{p_u}}]$
\item For $i = 1,\dots,n$:
\begin{enumerate}
\item $\vec{u_i} \in \mathcal{R}^{p_u}$ distributed iid
MVN(0,$s^2_uI_{p_u}$)
\item $ y_i = \vec{v_y}^T\vec{u_i} + \epsilon^y_i$ with $\epsilon^y_i$
distributed N(0,$s^2_y$)
\item $\vec{x_i} = V_x\vec{u_i} + \vec{\epsilon_x^i}$ with
$\vec{\epsilon_x^i}$ distributed MVN(0,$s^2_xI_{p_x}$)
\item $\vec{z_i} = V_z\vec{u_i} + \vec{\epsilon_z^i}$ with
$\vec{\epsilon_z^i}$ distributed MVN(0,$s^2_zI_{p_z}$)
\end{enumerate}
\item $X = [\vec{x_1}, \dots,\vec{x_n}]^T, Z = [\vec{z_1},
\dots,\vec{z_n}]^T, \text{ and } \vec{y} = [y_1,\dots,\vec{y}_n]^T$
\end{enumerate}
Thus, steps 1-5 generate the factors ($V = [V_X; V_Z; \vec{v_y}]$) and step
6 generates the loadings ($u_i$) and noise.
In order to test the performance of CollRe in doing prediction, we
generated a set of factors from the above model with $n = 50,
p_x = p_z = 10, p_u = 3, s_u = s_x = s_z = s_y = 1$. Then, for each
of 80 repetitions, we generated loadings and noise before fitting a
range of models. CollRe was fit with $b_{xy} = b_{zy} = 1$ and a
variety of values of $b_{xz}$. Additionally, at each level of $b_{xz}$ we
fit models with a range of ridge penalties. Ridge Regression models were also fit,
which corresponds to $b_{xz} = 0$. To evaluate the success of the fits,
we looked at prediction error based on using just $X$ relative to Ordinary Regression as well
as the sum correlation. Here, by sum correlation, we mean
$\text{cor}(\vec{x_*}^T\vec{\theta_x},y_*) + \text{cor}(\vec{z_*}^T\vec{\theta_z},y_*) +
\text{cor}(\vec{x_*}^T\vec{\theta_x},\vec{z_*}^T\vec{\theta_z})$, where $(\vec{x_*}, \vec{z_*}, y_*)$
is a future observation (corresponding to making another pass through
step 6).
\begin{figure}[]
\label{lowdim}
\centering
\includegraphics[width=\textwidth]{lowdimsim.pdf}
\caption{\em Results of a simulation study to test the effectiveness of
CollRe with $\ell_2$ penalty in a prediction framework. Here, the points all the way to the left correspond to
no $\ell_2$ penalty and the $\ell_2$ penalty increases (simpler models) as we
move right along the x-axis. The first plot shows prediction
error for making future predictions based on $X$ only. The second
plot shows the theoretical sum correlation. Values have been
averaged over 80 repetitions. As we can see, while CollRe
outperforms ordinary regression with no penalty in terms of
prediction error (the far left of
the first plot), Ridge Regression achieves a lower minimum. The
second plot helps illuminate the reason; CollRe does a better job
of maximizing the sum correlation, so it is sacrificing some of
the correlation between $\vec{y}$ and $X\hat\vec{\theta_x}$ in order to get a
larger correlation between $X\hat\vec{\theta_x}$ and $Z\hat\vec{\theta_z}$}
\end{figure}
The results of the simulation, in Figure \ref{lowdim}, sheds some
light on the effectiveness of using CollRe to improve a regression of
$\vec{y}$ on $X$. First, we note that ridge regression outperforms CollRe
at any choice of $b_{xy}$ and $\ell_2$ penalty for this particular problem.
At first, this might seem surprising given the fact that CollRe gets
the advantage of using $Z$ and ridge regression does not. When
looking at the sum correlation though, we see that CollRe
outperforms ridge. This suggests that the reason CollRe is doing worse
on predicting $\vec{y}$ is because it is focusing on the distance between
$X\hat\vec{\theta_x}$ and $Z\hat\vec{\theta_z}$ instead of just the typical RSS.
Essentially, CollRe is giving up a little of the fits involving $\vec{y}$
in order to get a higher correlation between $X\hat\vec{\theta_x}$ and $Z\hat\vec{\theta_z}$.
It seems that CollRe is more naturally suited for
supervised canonical correlation analysis, discussed next.
\section{Supervised Canonical Correlation Analysis (sCCA)}
\label{sec:mult-canon-corr}
Canonical Correlation Analysis (CCA) is a data analysis technique that
dates back to Hotelling (1996) \cite{hotelling1936}.
Given two sets of centered variables, $X$ and
$Z$, the goal of CCA is to find linear combinations of $X$ and $Z$
that are maximally correlated. Mathematically, CCA performs the
following constrained optimization problem:
\[(\hat\vec{\theta_x},\hat\vec{\theta_z}) = \arg\max_{\vec{\theta_x},\vec{\theta_z}} \vec{\theta_x}^TX^TZ\vec{\theta_z} \text{ such that
} \vec{\theta_x}^TX^TX\vec{\theta_x} \le 1, \vec{\theta_z}^TZ^TZ\vec{\theta_z} \le 1\]
In this form, it is possible to derive a closed form solution for
CCA using matrix decomposition techniques. Namely, $\hat\vec{\theta_x}$
will be the eigenvector corresponding to the largest eigenvalue of
$(X^TX)^{-1}X^TZ(Z^TZ)^{-1}Z^TX$. A similar expression can be found
for $\hat\vec{\theta_z}$ by switching the roles of $X$ and $Z$.
CCA might be a useful tool for finding a signal that is common
to both $X$ and $Z$, but there is no guarantee that the discovered signal will
also be associated with $\vec{y}$. To approach this issue, a generalization
of CCA called Multiple Canonical Correlation Analysis (mCCA) was
developed. mCCA allows for more than 2 datasets and seeks to find a
signal that is common to all of the datasets. The case we have, where
the third dataset is a vector, can be thought of as a special case of
mCCA that we will call Supervised Canonical Correlation Analysis
(sCCA).
There are many techniques that approach the mCCA problem. Most of
them focus on optimizing a function of the correlations between the
various datasets. Gifi (1990) \cite{gifi1990nonlinear} provides an overview of many of the
suggestions that have been made for this problem. One example of an
optimization problem that people would call mCCA is based on trying to
maximize the sum of the correlations:
\begin{equation}
\label{mccaobj}
\{\vec{\theta_i}\}_{i = 1,\dots,k} = \arg\max \sum_{i < j} \vec{\theta_i}^TX_i^TX_j\vec{\theta_j} \text{ such that
} \vec{\theta_i}^TX_i^TX_i\vec{\theta_i} \le 1\text{ } \forall i
\end{equation}
Now the optimization problem above is multiconvex as long as each of
the $X_i^TX_i$ are non-singular. This means that a local
optimum can be found by iteratively maximizing over each $\theta_i$
given the current values of the rest of the coefficients.
\subsection{Sparse sCCA}
\label{sec:sparse-scca}
For high dimensional problems (where $p_i >>> n$ for at least one $i$), several issues emerge when doing sCCA.
First, the constraints given in equation (\ref{mccaobj}) are
no longer strictly convex constraints because $X_i^TX_i$ is necessarily
singular for at least one $i$. This means that the problem cannot be
as easily solved by an iterative algorithm.
One approach that some people take to this problem is to add a ridge
penalty on the coefficients. As with ridge regression, adding a ridge
penalty will effectively replace $X_i^TX_i$ with $X_i^TX_i + \lambda_iI$ ($I$ being the identity matrix), which
will then be non-singular. This means that the mCCA problem in
equation (\ref{mccaobj}) can be solved by adding a ridge penalty.
Examples of works where people have pursued this method include
Leurgans and others (1993)
\cite{leurgans1993}. Another approach is pursued by Witten and
Tibshirani (2009) \cite{WittenTibsSAGMB09} where $X_i^TX_i$ is
replaced by $I$ in order to ensure strict convexity of the constraints.
Now, even after adjusting to make sure that the constraints (or
penalties in the Lagrange form) are convex, there is still another
issue that the high dimensional regime adds. For many problems in the
high dimensional regime, the goal of the problem is to do some sort of
variables selection. After all, it is much more useful for a
biologist to uncover 30 genes or pathways that are particularly
important in a process than it is to uncover 30,000 coefficient values
that are all fairly noisy anyway. Another way to state that is that
we want to find coefficients that are sparse (mostly 0). There has
been a lot of work in sparse statistical methods following the
introduction of the lasso by Tibshirani (1996)
\cite{Ti96}.
Witten and Tibshirani (2009) \cite{WittenTibsSAGMB09} offer the following
optimization problem to perform sparse mCCA:
\[\{\vec{\theta_i}\}_{i = 1,\dots,k} = \arg\max \sum_{i < j} \vec{\theta_i}^TX_i^TX_j\vec{\theta_j} \text{ such that
} \vec{\theta_i}^T\vec{\theta_i} \le 1, \|\vec{\theta_i}\| < c_i
\forall i\]
where the $c_i$ can be chosen to impose the desired level of sparsity
on each coefficient vector. Note that further convex constraints (or penalties) can be
added to the above such as the fused lasso or non-negativity
constraints. As with the other methods, this problem is multiconvex
and can be solved through an iterative algorithm.
Note however, that a multiconvex problem may be particularly hard to
solve in a high dimensional space. While we know the algorithm will
converge to a local optimum, we would ideally like to find the global
optimum. For a low dimensional space this can be mostly resolved by
doing multiple starts from random points in the coefficient space.
With enough starts we believe that we can search the space
sufficiently well that our best local optimum is at least close to
globally optimum. This logic breaks down in high dimensional spaces
because it is impossible to sufficiently search the space without
exponentially many starting points. This means that while the above
methods for Sparse mCCA have outputs, we won't know whether those
outputs are even optimizing the criteria in high dimensions.
We generated some data to test the extent to which MultiCCA gets
caught in local optima. These datasets have $n = 50, p_u = 30, s_u =
\sqrt{1/10}, s_x = s_z = s_y = 1$. For $p = 50,500,2000, \text{ and }5000$, we
generated a dataset with $p_x = p_z = p$ and then ran
MultiCCA from 1000 random (uniform on the unit sphere) start locations. Figure \ref{vsdanielaobj}
shows histograms of the resulting objective values. The vertical
lines correspond to the default starting point of MultiCCA, and a
starting point that is based on a penalized CollRe solution. As we
can see, there are many local optima that emerge especially in higher
dimensions. One interesting thing is that the CollRe starts typically
end up in a better solution than the default starts provided by the
MultiCCA function.
\begin{figure}
\label{vsdanielaobj}
\centering
\includegraphics[width=.48\textwidth]{biconissue50.pdf}
\includegraphics[width=.48\textwidth]{biconissue500.pdf}\\
\includegraphics[width=.48\textwidth]{biconissue2000.pdf}
\includegraphics[width=.48\textwidth]{biconissue5000.pdf}
\caption{\em Result of a simulation to see how close the locally
optimal solutions to MultiCCA end up to the global optimum. Histograms of objective values of MultiCCA from $1000$ random
starts. As we can see, the random starts end up at a variety of
local optima, and using the results of CollRe as a starting point
often outperforms the default start which is based on an singular
value decomposition. In each case, $n = 50$.}
\end{figure}
Another option to perform sparse high-dimensional sCCA was suggested
by Witten and Tibshirani (2009) \cite{WittenTibsSAGMB09}. She
suggests that a method of supervision similar to Bair and others
(2006) \cite{BHPT2006}
can be used: before doing a fit, all of the variables are
screened against $\vec{y}$. Only the ones that have correlation above some
threshold will be passed along to a CCA model. This method
can also be used to add a supervised component to any of the methods
that can be used to perform CCA. The main issue with this approach is that it does the
supervision in a way that is completely univariate.
\subsection{Penalized Collaborative Regression as Sparse sCCA}
\label{sec:coll-regr-as}
Consider one of the three terms from our objective function:
\begin{equation}
\label{eq:myterm}
\min \|X\vec{\theta_x} - Z\vec{\theta_z}\|^2 = \vec{\theta_x}^TX^TX\vec{\theta_x} + \vec{\theta_z}^TZ^TZ\vec{\theta_z} - 2\vec{\theta_x}^TX^TZ\vec{\theta_z}
\end{equation}
Now let's compare that to the following version of the CCA objective:
\[\min -\vec{\theta_x}^TX^TZ\vec{\theta_z}\ \text{ such that } \vec{\theta_x}^TX^TX\vec{\theta_x} \le 1,
\vec{\theta_z}^TZ^TZ\vec{\theta_z} \le 1\]
We can convert the CCA problem from its bounded form into the
Lagrange form as follows:
\[\min -\vec{\theta_x}^TX^TZ\vec{\theta_z} + \lambda_x \vec{\theta_x}^TX^TX\vec{\theta_x} + \lambda_z
\vec{\theta_z}^TZ^TZ\vec{\theta_z}\],
where $\lambda_x$ and $\lambda_z$ are chosen appropriately to enforce
the unit variance constraint. In this way CCA can also be characterized as
a penalized optimization problem. The difference between the term from our objective, and the penalized
form of CCA is that instead of using $\lambda_x$ and $\lambda_z$ in
order to enforce unit variance, we choose the values that would result
in the objective being convex instead of merely biconvex.
Now it is worth noting that an unenviable fact about the
penalty used in equation (\ref{eq:myterm}) is that it results in the
minimum being achieved by setting all of the coefficients equal to
zero. Fortunately, CollRe avoids this issue because the two terms
involving $\vec{y}$.
Thus, CollRe with $b_{xy} = b_{zy} = b_{xz} = 1$ is very similar to doing a
sum of correlations mCCA as in the equation (\ref{mccaobj}), with the
exception that we have picked the penalties that allow for convexity
instead of the penalties that correspond to unit variance.
As discussed in Section \ref{sec:penal-coll-regr} one of the
advantages of CollRe is the simplicity with which convex penalties can
be added to the objective function. Thus, it is easy to convert
CollRe into a form that is appropriate for sparse sCCA by adding
penalties just as in Witten and Tibshirani (2009) \cite{WittenTibsSAGMB09}.
To compare CollRe against a competing algorithm for sparse sCCA, we generated data from the above model with $n = 50,
p_x = p_z = 20, p_u = 3, s_u = 1, s_x = s_z = s_y = 1/2$. Then, we
added $40$ variables to both $X$ and $Z$ that were generated from $3$
new factors that have no effect of $\vec{y}$. These $40$ variables act as
confounding variables that reflect an effect we do not want to
uncover. This could correspond to a batch effect in the measurements,
or maybe some other underlying difference among the sampled patients.
Finally, we
added another $440$ columns to $X$ and $Z$ that were just independent
gaussians to act as null predictors. We ran both CollRe with a lasso
penalty and Wittens MultiCCA from the PMA package with an
$\ell_1$
constraint each over a range of parameter values. This process was
repeated 80 times with new loadings and noise each time but the same
factors. Figure \ref{vsdaniela} shows the average (over repetitions) theoretical sum correlation
for future observations, as well as the recovery of true predictors,
against a range of nonzero coefficients that corresponds to a range of
penalty parameters.
\begin{figure}[]
\label{vsdaniela}
\centering
\includegraphics[width=\textwidth]{highdimvsdanielawithconfinal.pdf}
\caption{\em Results of a simulation to compare CollRe and MultiCCA
in performing sparse supervised mCCA. For each repetition, a
dataset with $n = 50, p_x = p_z = 500$ is created. For both $X$
and $Z$, $40$ of the predictors are confounding variables and $20$
of the predictors are true variables (the rest are null). Confounding
variables are the ones that share a signal between $X$ and $Z$, but
not $\vec{y}$. The true variables share a signal between all three
datasets. Values have been averaged over $80$ repetitions.
MultiCCA is much more susceptible to picking up the confounding
variables, and thus has a much harder time achieving high
correlations. Interestingly, while CollRe finds many more true
variables at first, after $70$ or so included variables MultiCCA
starts finding more.}
\end{figure}
From the results, we can see that CollRe does a much better job of
finding coefficients that have high sum correlation. MultiCCA seems
to get caught in the trap set by the confounding variables, which makes it
harder to raise the sum correlation much above 1 (a perfect
correlation between $\vec{x_*}^T\vec{\theta_x}$ and $\vec{z_*}^T\vec{\theta_z}$ with no
relation to $y_*$). Interestingly,
while MultiCCA does worse than CollRe on recovery of true variables
for the first 70 or so variables added, it seems to do a better job of
recovering true variables after that point. It is unclear what
exactly is causing that transition in this problem.
\section{Real Data Example}
\label{sec:real-data-example}
To demonstrate the applicability of penalized CollRe, we also ran it on a high
dimensional biological dataset. We used a neoadjuvant breast cancer
dataset that was provided by our collaborators in the Division of
Oncology at the Stanford University School of Medicine. Details about
the origins of the data can be found at ClinicalTrials.gov using the
identifier NCT00813956. This dataset consists of $n = 74$
patients who underwent a particular breast cancer treatment. Before
treatment, the patients had measurements taken on their gene expression as well
as copy number variation. In all, after some pre-processing, there were $p_x = 54,675$ gene expression
measurements per patient and $p_z = 20349$ copy number variation
measurements. Additionally, each patient was given a RCB score six
months after treatment that corresponds to how effective the treatment
was. The RCB score is essentially a composite of various metrics on
the tumor: primary tumor bed area, overall \% cellularity, diameter of
largest axillary metastasis, etc.
The goal of the analysis is to select a set of gene expression
measurements that are highly correlated with a particular pattern of copy number
variation gains or losses. That said, we are only interested in sets
that also correlate with the RCB value. As such, it is the perfect
opportunity to employ CollRe.
Due to computational limitations and issues with noise in the
underlying measurements, some further pre-processing was done to the data.
First, the gene expression measurements were screened by their
variance across the subjects. Only the top $28835$ gene expression
genes were kept. For the copy number variation measurements we needed
to account for the fact that for each patient the copy number
variation measurements are VERY highly autocorrelated because they had
already been run through a circular binary segmentation algorithm (a
change point algorithm used to smooth copy number variation data). We use a fused lasso penalty to help correct for the
fact that we don't really have gene level measurements. However,
doing fused lasso solves can be very slow for large $p$, so we took
consecutive triples of the copy number variation measurements and
averaged them. This reduced the number of copy number variation
measurements to $6783$. Our new $X$ and $Z$ matrices were scaled and
centered, and then CollRe on the dataset with
$b_{xy} = b_{xz} = b_{zy} = 1$ and the following parameters and penalty terms:
\[P^x(\vec{\theta_x}) = \lambda_x(.9\|\vec{\theta_x}\|_1 + .1\frac{1}{2}\|\vec{\theta_x}\|_2^2)\]
\[P^z(\vec{\theta_z}) = 4\|\vec{\theta_z}\|_1 + 200 \displaystyle\sum_{i=2}^{i=p_z}|(\vec{\theta_z})_i
- (\vec{\theta_z})_{i-1}|\]
We searched a grid of $\lambda_x$ in order to find a
solution with about 50 nonzero coefficients in each set of variables. This
corresponds roughly with the number of genes a collaborator thought
she would be able to reasonably examine for plausible connections.
The penalty terms on $\vec{\theta_z}$ were chosen in a way that the selected
coefficients looked reasonably smooth. The resulting $\hat\vec{\theta_z}$ vector
can be seen in Figure \ref{fig:realex}
\begin{figure}[]
\label{fig:realex}
\centering
\includegraphics[width=\textwidth]{cnv.pdf}
\caption{\em The resulting vector of coefficients for the copy
number variation data from running CollRe on the RCB dataset.
Regions with
positive coefficients (amplification associated with higher RCB)
are darker and appear above the line. Regions with negative
coefficients are lighter and appear below the line. The size of
the bars are proportional to the coefficient values. Missing
chromosomes had no nonzero coefficients. The piece-wise constant
nature of the
coefficient vector is due to the use of a fused lasso.}
\end{figure}
\section{Solving CollRe with Penalties}
\label{appen}
In Section \ref{sec:penal-coll-regr} we mentioned that CollRe is
solvable with a variety of penalty terms added. In fact, due to the
nature of the CollRe objective, it can often be solved for common
penalty terms using out of the box penalized regression solvers. To
make this concrete, let us focus on CollRe with the addition of $\ell_1$ penalties.
Consider then, the objective function with penalty terms:
\begin{multline}
\label{fullobj}
J(\vec{\theta_x},\vec{\theta_z};X,Z,\vec{y}, b_{xy},b_{zy},b_{xz},\lambda_x,\lambda_z) =
\frac{b_{xy}}{2} \|\vec{y} - X\vec{\theta_x}\|^2 + \frac{b_{zy}}{2} \|\vec{y} - Z\vec{\theta_z}\|^2 + \frac{b_{xz}}{2}
\|X\vec{\theta_x} - Z\vec{\theta_z}\|^2 + \\\lambda_x\|\vec{\theta_x}\|_1 + \lambda_z\|\vec{\theta_z}\|_1
\end{multline}
We note that (\ref{fullobj}) is a convex function, so we can optimize
it by iteratively optimizing over $\vec{\theta_x}$ and $\vec{\theta_z}$. For a given value
of $\vec{\theta_z}$, the optimal $\vec{\theta_x}$ is given by:
\begin{equation}
\hat\vec{\theta_x} = \text{LASSO}(X,\vec{y^*},\frac{\lambda_x}{b_{xy}+b_{xz}}) \text{ , where } \vec{y^*}
= \frac{b_{xy}}{b_{xy}+b_{xz}}\vec{y} + \frac{b_{xz}}{b_{xy}+b_{xz}}Z\vec{\theta_z}
\end{equation}
Here, LASSO$(\tilde X,\tilde{\vec{y}},\tilde\lambda)$ is
the solution to the $\ell_1$ penalized regression problem:
\begin{equation}
\hat{\vec{\beta}} = \arg \min_{\vec{\beta}} \|\tilde{\vec{y}}-\tilde X\vec{\beta}\|^2 +
\tilde \lambda \|\vec{\beta}\|_1
\end{equation}
An equivalent solution of $\hat\vec{\theta_z}$ given $\vec{\theta_x}$ can be found
by symmetry. Thus, by iterating back and forth between these two
$\ell_1$ penalized regression problems, we are guaranteed to the optimum of equation
(\ref{fullobj}). This means it is trivial to write a solver for
CollRe using $\ell_1$ penalties as long as you have access to a solver for
regression with $\ell_1$ penalties. Many such functions can be found in R
packages, including the popular \textbf{glmnet} function in the self
titled package.
\subsection{Proof of Correctness of Algorithm (CollRe with $\ell_1$ Penalty)}
\label{sec:proof-algorithm}
Let $\tilde J$ be the LASSO criterion:
\begin{equation}
\tilde J(\tilde{\vec{\beta}}; \tilde X, \tilde{\vec{y}}, \tilde \lambda) = \|\tilde{\vec{y}}-\tilde X\tilde{\vec{\beta}}\|^2 + \tilde\lambda \|\tilde{\vec{\beta}}\|_1
\end{equation}
Then we see that $\tilde J$ has subgradient:
\begin{equation}
\frac{\partial \tilde{J}}{\partial \tilde{\vec{\beta}}} = \tilde X^T\tilde X\tilde{\vec{\beta}} - \tilde X^T\tilde{\vec{y}} + \tilde \lambda s(\tilde{\vec{\beta}})
\end{equation}
Compare this to the subgradient of $J$ with respect to $\vec{\theta_x}$:
\begin{equation}
\label{subgJ}
\frac{\partial J}{\partial \vec{\theta_x}} = (b_{xy} + b_{xz})X^TX\vec{\theta_x} - X^T(b_{xy} \vec{y} +
b_{xz} Z\vec{\theta_z}) + \lambda_x s(\vec{\theta_x})
\end{equation}
Dividing (\ref{subgJ}) by $b_{xy}+b_{xz}$ and substituting $\vec{y^*} =
\frac{b_{xy}}{b_{xy}+b_{xz}}\vec{y} + \frac{b_{xz}}{b_{xy}+b_{xz}}Z\vec{\theta_z}$ completes the
proof.
\subsection{Augmented Data Version}
\label{sec:augm-data-vers}
For some selections of penalties, parameters, and solvers, CollRe
can be fit using an augmented data approach. This means that the
solution can be found in just one call to a solver instead of having to
iterate. In practice, this can increase the rate of convergence and
reduce total computation time.
Let us return to the example of trying to fit CollRe with the addition
of an $\ell_1$ penalty. Consider the following LASSO problem:
\begin{equation}
\tilde X = \left[ \begin{array}{cc}
\sqrt{b_{xy}} X & 0 \\
0 & \sqrt{b_{zy}} \frac{\lambda_x}{\lambda_z} Z \\
\sqrt{b_{xz}} X & - \sqrt{b_{xz}} \frac{\lambda_x}{\lambda_z} Z \end{array}\right],
\tilde \vec{y} = \left[ \begin{array}{c}
\vec{y} \\
\sqrt{b_{zy}}\vec{y} \\
0 \end{array}\right],
\tilde \vec{\beta} = \left[ \begin{array}{c}
\vec{\theta_x} \\
\frac{\lambda_z}{\lambda_z}\vec{\theta_z} \end{array}\right]
\end{equation}
\begin{equation}
\label{lassoaugver}
\hat{\tilde \vec{\beta}} = \arg\min_{\tilde \vec{\beta}} \|\tilde X \tilde{\vec{\beta}} -
\tilde \vec{y}\|^2 + \lambda_x\|\tilde{\vec{\beta}}\|_1
\end{equation}
It can be easily verified that equation (\ref{lassoaugver}) is exactly
the CollRe with $\ell_1$ penalty fit for the parameters given. Essentially, this means that
instead of iterating between LASSO solves with $(\tilde{n} =
n,\tilde{p} = p_x)$ and $(\tilde{n} = n,\tilde{p} = p_z)$ until
convergence, we only do one solve with $(\tilde{n} =
3n,\tilde{p} = p_x + p_z)$. Because we expect $n <<< \max (p_x,p_z)$,
we don't expect tripling $\tilde{n}$ to have much effect on run time.
Further, due to active set rules that are built into packages like the R package \textbf{glmnet}, even if we double $\tilde{p}$ it
should not have too large an effect on run time (Friedman and others
(2010) \cite{FHT2010}).
We ran some simulations that involve generating $X,Z,$ and $\vec{y}$ from
independent standard normal draws. we then fit CollRe with Elastic Net to the data
setting all of the parameters equal to one (except $\lambda^x_2 = \lambda^z_2 = 0$).
For $n = 100$, $p_x = p_z = 2000$ the normal version of CollRe with $\ell_1$
penalty ($\lambda_x = \lambda_z = 1$) takes about 2.2
seconds to run on a 2010 Macbook Pro. The augmented version only takes 0.6
seconds to run. The augmented version also achieves a lower value for
the objective function (8.127974 compared to 8.128410), so the speedup is
not just coming from a premature convergence.
\section{Discussion}
\label{sec:discussion}
In this paper, we introduced a new model called Collaborative
Regression, which can be used in settings where one has two sets of predictors
and a response variable for a set of observations. We explored the
possibility of using CollRe in a prediction framework, but ultimately
decided that it was not particularly well suited for that task.
We then discussed the
problem of sparse supervised Canonical Correlation Analysis, which
seems to be an increasingly interesting problem for biostatistics.
While current approaches to sCCA are biconvex and don't
necessarily lend themselves to a sparse generalization, CollRe does
not suffer from those same issues. We used several simulations and real data to
explore both the issues of biconvexity in high dimensions, as well as
the performance of CollRe.
\section*{Acknowledgments}
The authors thank S. Vinayak, M. Telli, and J. Ford for
comments regarding the use of CollRe as well as providing the dataset used
in section \ref{sec:real-data-example}. We also thank T. Hastie,
J. Taylor, D. Donoho, and D. Sun for comments regarding the development of the
CollRe algorithm.
\input{CollaborativeRegression.bbl}
\end{document}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 8,596 |
\section{Introduction}
Parallel and distributed computing is ubiquitous in science, technology, and beyond.
Key to the performance of a distributed system is the efficient utilization of resources: in order to obtain a substantial speed-up it is of utmost importance that all processors have to handle the same amount of work.
Unfortunately, many practical applications such as finite element simulations are highly ``irregular'', and the amount of load generated on some processors is much larger than the amount of load generated on others.
We therefore investigate \emph{load balancing} to redistribute the load.
Efficient load balancing schemes have a plenitude of applications, including high performance computing \cite{DBLP:journals/ijhpca/ZhengBMK11}, cloud computing \cite{DBLP:journals/access/MohammadianNHD22}, numerical simulations \cite{DBLP:conf/dimacs/Meyerhenke12}, and finite element simulations \cite{DBLP:journals/aes/PatzakR12}.
In this paper we consider \emph{neighborhood} load balancing on arbitrary graphs with $n$ nodes, where the nodes balance their load in each step only with their direct neighbors.
We assume \emph{discrete} load items as opposed to \emph{continuous} (or \emph{idealized}) load items which can be broken into arbitrarily small pieces.
We study \emph{infinite} and \emph{dynamic} processes where new load items are generated in every step.
We consider two different settings.
In the \emph{synchronous} setting $m$ load items are generated on randomly chosen nodes.
Then a matching is chosen and the load of the nodes is balanced (via weighted averaging) over the edges of that matching.
Here we further distinguish between two matching models.
We consider the \emph{random matching model} where linear-size matchings are randomly chosen, and the \emph{balancing circuit model} where the graph is divided deterministically into $d_{\max}$ many matchings.
Here $d_{\max}$ is the maximum degree of any node.
In the \emph{asynchronous model} exactly one load item is generated on a randomly chosen node.
In turn, the node chooses one of its edges at random and balances its load with the corresponding neighbor.
This model can be regarded as a variant of the synchronous model where the randomly chosen matching has size one.
It was introduced by \cite{DBLP:conf/icalp/AlistarhNS20} where the authors show results for cycles assuming continuous load.
Our goal is to bound the so-called \emph{discrepancy}, which is defined as the maximal load of any node minus the minimal load of any node.
\paragraph{Results in a Nutshell}
In this paper we present, for the three models introduced above, bounds on the expected discrepancy and bounds that hold with high probability.
Our bounds for the synchronous model with balancing circuits hold for arbitrary graphs $G$, the bounds for the asynchronous model and the synchronous model with random matchings hold for regular graphs $G$ only.
For the asynchronous model and the model with random matchings our bounds on the discrepancy are expressed in terms of hitting times of a standard random walk on $G$, as well as in terms of the spectral gap of the Laplacian of $G$. For the synchronous model with balancing circuits
we express our bounds in terms of the \emph{global divergence}. This can be thought of as a measure of the convergence speed of the Markov chains modeling a random walk on $G$. However, it does not directly measure the speed of convergence of the chain. It accounts for the time period in which the chain keeps a given distance from the stationary (and uniform) distribution. In physics terminology, it is a measure of total \emph{absement}, which is the time-integral of displacement.
For all three infinite processes our bounds on the discrepancy hold at an arbitrary point of time as long as the system is initially empty.
Otherwise, the bounds hold after an initial time period, its length is a function of the initial discrepancy.
In the following we give some exemplary results assuming that the system is initially empty and $m=n$.
For the synchronous model with random matchings and the asynchronous model
we can bound the discrepancy by ${\operatorname{O}}(\sqrt{n}\log(n))$ for \emph{any} regular graph $G$. Our results show
a polylogarithmic bound on the discrepancy for all regular graphs with a hitting time at most ${\operatorname{O}}(n \poly\log(n))$ (e.g., the two-dimensional torus or the hypercube).
In all models we can bound the discrepancy by ${\operatorname{O}}(\sqrt{n\log(n)})$ for arbitrary constant-degree regular graphs.
For the full results we refer the reader to \cref{thm:main_sync_random}, \cref{thm:main_sync_circuit}, and \cref{thm:main_async}.
We give a detailed overview on the results on specific graph classes in \cref{table:disc:upperbound} in \cref{sec:conclusions}.
All bounds presented in this paper also hold for the corresponding continuous processes without rounding.
The authors of \cite{DBLP:conf/icalp/AlistarhNS20} consider the asynchronous process on cycles in the continuous setting where the load items can be divided into arbitrary small pieces.
They bound the expected discrepancy and show that $\disc(G)=O(\sqrt{n} \log(n))$ for a cycle $G$ with $n$ nodes.
In contrast, we improve that bound for the cycle to $\disc(G)= O(\sqrt{n \log(n)})$.
Note that our result not only bounds the expected discrepancy but it also holds with high probability.
Our main analytical vehicle is a drift theorem that bounds the tail of the sum of a non-increasing sequence of random variables.
Our drift theorem adapts known drift results from the literature, similarly to the Variable Drift Theorem in \cite{DBLP:series/ncs/Lengler20}.
\subsection{Related Work}
There is a vast body of literature on iterative load balancing schemes on graphs where nodes are allowed to balance (or average) their load with neighbors only.
One distinguishes between \emph{diffusion} load balancing where the nodes balance their load with all neighbors at the same time and the \emph{matching model} (or \emph{dimension exchange}) model where the edges which are used for the balancing form a matching.
In the latter model every resource is only involved in one balancing action per step, which greatly facilitates the analysis.
In this overview we only consider theoretical results and, as it is beyond the scope of this work to provide a complete survey, we focus on results for discrete load balancing.
For results about continuous load balancing see, for example,~\cite{DBLP:journals/pc/DiekmannFM99, DBLP:conf/focs/KempeDG03}.
There are also many results in the context of balancing schemes where not the resources try to balance their load but the tokens (acting as selfish players) try to find a resource with minimum load. See~\cite{DBLP:journals/siamcomp/FischerRV10} for a comprehensive survey about selfish load balancing and~\cite{DBLP:journals/dc/AckermannFHS11,DBLP:journals/corr/HoeferS13,DBLP:conf/ipps/BerenbrinkKLM17} for some recent results.
Another related topic is token distribution where nodes do not balance their entire load with neighbors but send only single tokens over to neighboring nodes with a smaller load.
See \cite{DBLP:journals/siamcomp/GhoshLMMPRRTZ99, DBLP:journals/algorithmica/HeideOW96, DBLP:journals/siamcomp/PelegU89} for the static setting and
\cite{DBLP:journals/siamcomp/AnshelevichKK08} for the dynamic setting.
\paragraph{Discrete Models}
The authors of~\cite{DBLP:journals/mst/MuthukrishnanGS98} give the first rigorous result for discrete load balancing in the diffusion model.
They assume that the number of tokens sent along each edge is obtained by rounding down the amount of load that would be sent in the continuous case.
Using this approach they establish that the discrepancy
is at most $O(n^2)$ after $O(\log(Kn))$ steps, where $K$ is the initial discrepancy.
Similar results for the matching model are shown in~\cite{DBLP:journals/jcss/GhoshM96}.
While always rounding down may lead to quick stabilization, the discrepancy tends to be quite large, a function of the diameter of the graph.
Therefore, the authors of~\cite{DBLP:conf/focs/RabaniSW98} suggest to use randomized rounding in order to get a better approximation of the continuous case.
They show results for a wide class of diffusion and matching load balancing protocols and introduce the so-called \emph{local divergence}, which aggregates the sum of load differences over all edges in all rounds.
The authors prove that the local divergence gives an upper bound on the maximum deviation between the continuous and discrete case of a protocol.
In~\cite{DBLP:conf/stoc/FriedrichS09} the authors show several results for a randomized protocol with rounding in the matching model.
For complete graphs their results show a discrepancy of $O(n\sqrt{\log n})$ after $\Theta(\log(Kn))$ steps.
Later, \cite{DBLP:journals/jcss/BerenbrinkCFFS15} extended some of these results to the diffusion model.
In~\cite{DBLP:conf/focs/SauerwaldS12} the authors show that the number of rounds needed to reach constant discrepancy is w.h.p.\ bounded by a function of the spectral gap of the relevant mixing matrix and the initial discrepancy.
In~\cite{DBLP:journals/jpdc/BerenbrinkFH09} the authors propose a very simple potential function technique to analyze discrete diffusion load balancing schemes, both for discrete and continuous settings.
In \cite{DBLP:conf/ipps/BerenbrinkFKK19} the authors investigate a load balancing process on complete graphs.
In each round a pair of nodes is selected uniformly at random and completely balance their loads up to a rounding error of $\pm 1$.
The authors of \cite{DBLP:conf/icalp/CaiS17} study load balancing via matchings assuming random placement of the load items.
The initial load distribution is sampled from exponentially concentrated distributions (including the uniform, binomial, geometric, and Poisson distributions).
The authors show that in this setting the convergence time is smaller than in the worst case setting.
Regardless of the graph's topology, the discrepancy decreases by a factor of $\sqrt[4]{t}$ within $t$ synchronous rounds.
Their approach of using concentration inequalities to bound the discrepancy (in terms of the squared $2$-norm of the columns of the matrices underlying the mixing process) strongly influenced our approach.
\paragraph{Dynamic Models}
There are far less results for the dynamic setting where new load enters the system over time.
In \cite{DBLP:conf/icalp/AlistarhNS20} the authors study a model similar to our asynchronous model.
In each step one load item is allocated to a chosen node.
In the same step the chosen node picks a random neighbor, and the two nodes balance their loads by averaging them (continuous model). The authors show that the expected discrepancy is bounded by $O( n\sqrt{n} \log n)$, as well as a lower bound on the square of the discrepancy of $\Omega(n)$.
The authors of \cite{DBLP:journals/siamcomp/AnagnostopoulosKU05} consider load balancing via matchings in a dynamic model where the load is, in every step, distributed by an adversary.
They show the system is stable for sufficiently limited adversaries.
They also give some upper bounds on the maximum load for the somewhat more restricted adversary.
The authors of \cite{DBLP:journals/algorithmica/BerenbrinkFM08} consider discrete dynamic diffusion load balancing on arbitrary graphs.
In each step up to $n$ load items are generated on arbitrary nodes (the allocation is determined by an adversary).
Then the nodes balance their load with each neighbor and finally one load item is deleted from every non-empty node.
The authors show that the system is stable, which means that the total load remains bounded over time (as a function of $n$ alone and independently of the time $t$).
\section{Balancing Models and Notation} \label{sec:model}
We consider the following class of dynamic load balancing processes on $d$-regular graphs~$G$ with $n$ nodes $V(G) = [n]$.
Each process is modeled by a Markov chain $(\LoadVecT{t})_{t\in\N_0}$, where the \emph{load vector} $\LoadVecT{t} = (\NodeLoadT{i}{t})_{i \in [n]} \in \R^n$ is the \emph{state} of the process at the end of step $t$, and $\NodeLoadT{i}{t}$ is the load of node~$i$ at time~$t$.
We measure a load vector's imbalance by the discrepancy $\discr(\vec{x})$, which is the difference between the maximum load and the minimum load $\discr(\vec{x}) \coloneqq \max_{i \in [n]} x_i - \min_{j \in [n]} x_j$.
We consider two balancing processes, the synchronous process $\textsc{SBal}$ and the asynchronous process $\textsc{ABal}$.
Both processes are parameterized by a \emph{balancing parameter} $\beta$ determining the balancing speed and a matching distribution $\mathcal{D}(G)$.
For $\textsc{SBal}$, $\mathcal{D}(G)$ is a distribution over linear-sized matchings of $G$.
For $\textsc{ABal}$, $\mathcal{D}(G)$ is a distribution over edges of $G$.
$\textsc{SBal}$ is additionally parameterized by the number of load items $m \in \N^+$ allocated in each round.
$\textsc{ABal}$ allocates only one new load item per step.
\paragraph{Synchronous Processes}
The synchronous process $\SyncProc{\mathcal{D}(G)}{\beta}{m}$ works as follows.
The process first allocates $m$ items to randomly chosen nodes.
Then it uses the matching distribution $\mathcal{D}(G)$ to determine the matching which is applied.
Finally it balances the load over the edges of the matching (see Process $\BalProc{\mathbf{m}}{\beta}$ described below).
The parameter $\beta \in (0, 1]$ controls the fraction of the load difference that is sent over an edge in a step.
For the synchronous process $\textsc{SBal}$ we consider two families of matching distributions, random matchings ($\randModel(G)$) and balancing circuits ($\baldModel(G)$).
$\randModel(G)$ is generated according to the following method described in~\cite{DBLP:journals/jcss/GhoshM96}.
First an edge set $S$ is formed by including each edge with probability $1/(4d) - 1/(16d^2) = \Theta(1/d)$, independently from all other edges.
Then a linear-sized matching $\MixMatT{t} \subseteq S$ is computed locally.
We will use capital $\mathbf{M}$ for randomly chosen matchings. The analysis for
the random matching model can be found in \cref{sec:analysis:random:matching}.
In the \emph{balancing circuit model} we assume $G$ is covered by $\zeta$ fixed matchings $\mixMatT{1},\ldots, \mixMatT{\zeta}$.
$\baldModel(G)$ deterministically chooses matchings in periodic manner such that in step $t$ the matching $\mixMatT{t}=\mixMatT{t \bmod \zeta}$ is chosen.
We will use small $\mathbf{m}$ for deterministically chosen matchings.
The analysis for the balancing circuit model can be found in \cref{sec:analysis_balancing_circuit}.
\paragraph{Asynchronous Process}
The asynchronous process $\AsyncProc{\mathcal{D}(G)}{\beta}$ works as follows.
The process first uses $\mathcal{D}(G)$ to generate a matching, this time containing one edge only.
The distribution we consider, $\singdModel(G)$, first chooses a node $i$ uniformly at random and then it chooses one of the nodes' edges $(i,j)$ uniformly at random.
Finally one new token is assigned to either node $i$ or $j$ and then the edge $(i,j)$ is used for balancing (see $\BalProc{\mathbf{m}}{\beta}$).
Note that for $\AsyncProc{\singdModel(G)}{\beta}$ the load allocation heavily depends on the edges which are used for balancing.
This makes the analysis for this model quite challenging.
In contrast, in $\SyncProc{\singdModel(G)}{\beta}{m}$ the load allocation and the balancing are independent.
Note that in the case of $d$-regular graphs $\singdModel(G)$ is equivalent to the uniform distribution over all edges or to choosing a random matching of size one.
We analyze the asynchronous model in \cref{sec:asynchronous}.
\medskip
\noindent\colorbox{black!10}{\begin{minipage}{\textwidth-2\fboxsep}
$\SyncProc{\mathcal{D}(G)}{\beta}{m}$:
In each round $t \in \N^+$:
\begin{enumerate}
\item Allocate $m$ discrete, unit-sized load items to the nodes uniformly and independently at random.
Define $\ell_i(t)$ as the number of tokens assigned to node $i$.
\item Sample a matching $\MixMatT{t}$ according to $\mathcal{D}(G)$.
\item Balance with $\BalProc{\MixMatT{t}}{\beta}$ applied to $X_i(t):=X_i(t)+\NodeAllocT{i}{t}$, $i\in \{1,\ldots n\}$.
\end{enumerate}%
\end{minipage}}
\medskip
\noindent\colorbox{black!10}{\begin{minipage}{\textwidth-2\fboxsep}
$\AsyncProc{\mathcal{D}(G)}{\beta}$:
In each round $t \in \N^+$:
\begin{enumerate}
\item Select an edge $\{i, j\}$ according to $\mathcal{D}(G)$.
\item Allocate a single unit-size load item to either node $i$ or $j$
with a probability of $1/2$.
I.e., with prob. $1/2$ set $\ell_i(t)=1$ and $\ell_k=0$ for all $k\neq i$, otherwise
set $\ell_j(t)=1$ and $\ell_k=0$ for all $k\neq j$.
\item Balance with $\BalProc{\MixMatT{t}}{\beta}$ applied to $X_i(t):=X_i(t)+\ell_i(t)$,
where $\MixMatT{t}$ includes just the edge $\{i,j\}$.
\end{enumerate}
\end{minipage}}
\medskip
\noindent\colorbox{black!10}{\begin{minipage}{\textwidth-2\fboxsep}
$\BalProc{\mathbf{m}}{\beta}$: For each edge $\{i,j\}$ in the matching $\mathbf{m}$ balance loads of $i$ and $j$:
\begin{enumerate}
\item Assume w.l.o.g.\ that $\NodeLoadT{i}{t}\ge \NodeLoadT{j}{t}$.
\item Let $p=\frac{\beta \cdot (\NodeLoadT{i}{t}-\NodeLoadT{j}{t})}{2}-
\left\lfloor\frac{\beta \cdot (\NodeLoadT{i}{t}-\NodeLoadT{j}{t})}{2}\right\rfloor$.
\item Then, node $i$ sends $L_{i,j}$ load items to node $j$ where
\vspace{-1ex}
\[L_{i,j} \coloneqq \begin{cases}
\left\lceil\frac{\beta \cdot(\NodeLoadT{i}{t}-\NodeLoadT{j}{t})}{2}\right\rceil, & \text{with probability } p, \\[5pt]
\left\lfloor\frac{\beta \cdot (\NodeLoadT{i}{t}-\NodeLoadT{j}{t})}{2}\right\rfloor, & \text{with probability } 1-p.
\end{cases}\]
\end{enumerate}
\end{minipage}}
\medskip
\noindent In the idealized setting, where the load is continuously divisible, a load of
${\beta (\NodeLoadT{i}{t}-\NodeLoadT{j}{t})}/{2}$
is sent from node $i$ to node $j$.
\subsection{Notation}
We are given an arbitrary graph $G=(V,E)$ with $n$ nodes.
We mainly assume that $G$ is regular and write $d$ for the node degree.
Recall that the process is modeled by a Markov chain $(\LoadVecT{t})_{t\in\N}$, where $\LoadVecT{t} = (\NodeLoadT{i}{t})_{i \in [n]} \in \R^n$ is the \emph{load vector} at the end of step $t$, and $\NodeLoadT{i}{t}$ is the load of node $i$ at time $t$.
We write $\NodeAllocT{i}{t}$ for the number of load items allocated to node $i$ in step $t$ and define $\AllocVecT{t} = (\NodeAllocT{i}{t})_{i \in [n]}$.
We will use upper case letters such as $\NodeLoadT{i}{t}$ and $\MixMatT{t}$ to denote random variables and random matrices and lower case letters (like $x_i(t)$, $\mixMatT{t}$) for fixed outcomes.
If clear from the context we will omit $t$ from a random variable.
We model the idealized balancing step in round $t$ by multiplication with a matrix $\MixMatBeta{\beta}(t) \in \R^{n \times n}$ given by
\[\MixMatBeta{\beta}_{i,j}(t) \coloneqq \begin{cases}
1,\quad&\textup{if $i=j$ and $i$ is not matched at time $t$,} \\
1-\beta/2,\quad&\textup{if $i=j$ and $i$ is matched at time $t$,} \\
\beta/2,\quad&\textup{if $i$ and $j$ are matched at time $t$,} \\
0,\quad&\textup{otherwise.}
\end{cases}\]
We will omit the parameter $\beta$ if it is clear from context.
With slight abuse of notation we use the same symbol $\MixMatT{t}$ for the matching itself and the associated balancing matrix
and refer to both as just ``matchings''. Furthermore, we write $E(\MixMatT{t})$ for their edges.
For the product of all matching matrices from time $t_1$ to time $t_2$ we write
\[\MixMatTT{t_1}{t_2} \coloneqq \MixMatT{t_2} \cdot \MixMatT{t_2 - 1} \cdot \cdots \cdot \MixMatT{t_1 + 1} \cdot \MixMatT{t_1},\]
where for $t_1 > t_2$ we consider this to be the identity matrix.
We generally refer to these matrices as \emph{mixing matrices}. Moreover, we write $\MixMatSeq{t}$ for the sequence of matching matrices~$(\MixMatT{\tau})_{\tau \in [t]}$ and analogously $\mixMatSeq{t}$ for a fixed sequence of matching matrices ~$(\mixMatT{\tau})_{\tau \in [t]}$.
We will write $\mathbf{M}_{k,\cdot}$ for the vector forming the $k$th row of the matrix $\mathbf{M}$ (which we often treat as a column vector despite it being a row).
In the balancing circuit model we define the \emph{round matrix} $\mathbf{R}\coloneqq \mixMatTT{1}{\zeta}$ as the product of the matching matrices forming a complete period of the balancing circuit.
Note that $\zeta$ has no relation to the minimum or maximum degree,
although we may assume w.l.o.g.\ that each edge is covered by at least one of the matchings.
We write $\SpectralGap(\mathbf{R})$ for the spectral gap of the round matrix $\mathbf{R}$,
i.e., for the difference between the largest two eigenvalues of $\mathbf{R}$.
We write $\RoundingErrVecT{t} \in \R^n$ for the vector of additive rounding errors in round $t$.
Then $\RoundingErrT{k}{t}$ is the difference between the load at node $k$ after step $t$ and the load at node $k$ after step $t$ in an idealized scheme where loads are arbitrarily divisible.
Putting all of this together we can express the load vector at the end of step $t \in \N^+$ as
\begin{equation}\label{eq:load_recurrence}
\LoadVecT{t} = \MixMatT{t} \cdot\left(\LoadVecT{t-1} + \AllocVecT{t}\right) + \RoundingErrVecT{t}.
\end{equation}
We write $\HittingTime$ for the \emph{hitting time} of $G$,
which is the maximum expected time it takes for a standard random walk on $G$ (i.e., the walk moves to a neighbor chosen uniformly at random in each step)
to reach a given node $i$ from a given node $j$,
with the maximum taken over all such pairs of nodes.
We write $\EdgeHittingTime$ for the \emph{edge hitting time} of $G$,
which is defined like the hitting time,
except that the maximum is taken over adjacent nodes only.
We write $\Laplacian(G)$ for the normalized Laplacian matrix of a graph~$G$.
For regular graphs it may be defined as
$\Laplacian(G) \coloneqq \IdentityMat - \AdjacencyMat(G) /d$,
where $\AdjacencyMat(G)$ is the adjacency matrix of $G$.
Writing $\lambda_0 \leq \lambda_1 \leq \ldots \leq \lambda_{n-1}$ for the real eigenvalues of $\Laplacian(G)$,
we let $\SpectralGap(\Laplacian(G)) \coloneqq \lambda_1 - \lambda_0$ be the spectral gap of the Laplacian of $G$.
\section{Random Matching Model}
\label{sec:analysis:random:matching}
In this section we analyze the process $\SyncProc{\randModel(G)}{\beta}{m}$ for $d$-regular graphs $G$, where the matching distribution $\randModel(G)$ is generated by the algorithm given in~\cite{DBLP:journals/jcss/GhoshM96}.
Note that the result (as well as the results for the two other models) holds at any point of time $t$ if the system is initially empty.
Furthermore, we can show the same results in the idealized setting where load items can be divided into arbitrarily small pieces (see \cite{DBLP:conf/icalp/AlistarhNS20}).
For more details we refer the reader to the paragraph directly after \cref{eq:disc_component_sum}.
\begin{theorem}
\label{thm:main_sync_random}
Let $G$ be a $d$-regular graph and define
\(T(G) \coloneqq \min \Big\{\frac{\HittingTime }{n} \cdot \log(n), \sqrt{\frac{d}{\SpectralGap(\Laplacian(G))}},\discretionary{}{}{} \frac{1}{ \SpectralGap(\Laplacian(G))} \Big\}\).
Let $\LoadVecT{t}$ be the state of process $\SyncProc{\randModel(G)}{\beta}{m}$
at time $t$ with $\discr(\LoadVecT{0}) \eqqcolon K \ge 1$.
There exists a constant $c>0$ such that for all
$t \geq c \cdot
\log(K\cdot n)\discretionary{}{}{} /({\SpectralGap(\Laplacian(G)) \cdot \beta})$
it holds w.h.p.\footnote{The expression \emph{with high probability (w.h.p.)} denotes a probability of at least $1-n^{-\Omega(1)}$.} and in expectation
\[ {\discr(\LoadVecT{t}) = {\operatorname{O}}\left(\log(n) \cdot \left(1 + \sqrt{\frac{m}{n} \cdot \frac{\EdgeHittingTime}{n}}\right) + \sqrt{\frac{\log(n)}{\beta} \cdot \frac{m}{n} \cdot T(G)}\right). } \]
\end{theorem}
\begin{proof}
We first expand the recurrence of \cref{eq:load_recurrence} (cf.~\cite{DBLP:conf/focs/RabaniSW98}).
After one step we get
\begin{align*}
\LoadVecT{t} &= \MixMatT{t} \cdot\left(\LoadVecT{t-1} + \AllocVecT{t}\right) + \RoundingErrVecT{t}
\\&= \MixMatT{t} \cdot\Big(\underbrace{\left(\MixMatT{t-1} \cdot\left(\LoadVecT{t-2} + \AllocVecT{t-1}\right) + \RoundingErrVecT{t-1}\right)}_{\LoadVecT{t-1}} + \AllocVecT{t}\Big) + \RoundingErrVecT{t}
\\& =
\MixMatTT{t-1}{t}\cdot \LoadVecT{t-2} + \sum_{\tau=t-1}^{t} \MixMatTT{\tau}{t}\cdot \AllocVecT{\tau} + \sum_{\tau=t-1}^{t} \MixMatTT{\tau+1}{t} \cdot \RoundingErrVecT{\tau}
\end{align*}
We repeatedly expand this form up to the beginning of the process and get
\begin{equation}\label{eq:load:vector:equality}
\LoadVecT{t}
= \underbrace{ \vphantom{\sum_{\tau=1}^{t}} \MixMatTT{1}{t} \cdot \LoadVecT{0}}_{\InitialContribVecT{t}}
+ \underbrace{\sum_{\tau=1}^{t} \MixMatTT{\tau}{t}
\cdot \AllocVecT{\tau}}_{\DynamicContribVecT{t}}
+ \underbrace{\sum_{\tau=1}^{t} \MixMatTT{\tau+1}{t} \cdot \RoundingErrVecT{\tau}}_{\RoundingContribVecT{t}}.
\end{equation}
We write $\InitialContribVecT{t}$, $\DynamicContribVecT{t}$, and $\RoundingContribVecT{t}$ for the three terms as indicated.
Note that in general these terms are vectors of real numbers.
The sum $\InitialContribVecT{t} + \DynamicContribVecT{t}$ can be regarded as the contribution of an idealized process, where $\InitialContribVecT{t}$ is the contribution of the initial load and $\DynamicContribVecT{t}$ is the contribution of the dynamically allocated load.
Thus, $\RoundingContribVecT{t}$ is the deviation between the idealized process without rounding and the discrete process described in \cref{sec:model}.
To bound the discrepancy $\discr(\LoadVecT{t})$ of the load vector $\LoadVecT{t}$ at time $t$ we use the fact that the discrepancy is sub-additive such that $\disc(\vec{x} + \vec{y}) \leq \disc(\vec{x}) + \disc(\vec{y})$ (see \cref{obs:disc_subadditive} in \cref{apx:omitted-proofs-3}).
Hence, to bound $\discr(\LoadVecT{t})$ we individually bound the discrepancies of the three terms in \cref{eq:load:vector:equality}
and get
\begin{equation}\label{eq:disc_component_sum}
\discr(\LoadVecT{t}) \leq \discr(\InitialContribVecT{t}) + \discr(\DynamicContribVecT{t}) + \discr(\RoundingContribVecT{t}) .
\end{equation}
If the system is initially empty, then $\disc(\InitialContribVecT{t}) = 0$.
Moreover, in the idealized setting without rounding $\disc(\RoundingContribVecT{t}) = 0$.
Techniques to bound the first term $\discr(\InitialContribVecT{t})$ and the last term $\discr(\RoundingContribVecT{t})$ are well-established.
We state the corresponding results in \cref{lem:initial:load:vanishes} and \cref{lem:rounding:errors:are:small} directly below the proof of our theorem. The main part of the proof is to bound
$\discr(\DynamicContribVecT{t})$, which will be done in \cref{sec:bound:contribution:dynamcally:allocated:balls}.
Let now $\gamma > 1$.
First, it follows from
\cref{lem:initial:load:vanishes}
that for all $t \geq c \cdot
\log(K\cdot n) /({\SpectralGap(\Laplacian(G)) \cdot \beta})$
we have
$\discr(\InitialContribVecT{t}) \leq 1$
with probability at least $1-n^{-\gamma}$.
Second, it follows from
\cref{lem:disc:dyn}
that
$\discr(\RoundingContribVecT{t}) \leq 2 \sqrt{\gamma\log(n)/\beta}$ with probability at least $1-3\cdot n^{-\gamma +1}$.
Third, it follows from
\cref{lem:rounding:errors:are:small}
that
\[\discr(\DynamicContribVecT{t}) = {\operatorname{O}}\left(
\gamma\log(n) \cdot \left(1 +
\sqrt{\frac{m}{n} \cdot \frac{\EdgeHittingTime}{n}} \right) +
\sqrt{\frac{\gamma \log(n)}{\beta} \cdot\frac{m}{n} \cdot T(G)}\right)\]
with probability at least $1-2\cdot n^{-\gamma+1}$.
The statement of the theorem therefore follows from a union bound over the statements of \cref{lem:initial:load:vanishes}, \cref{lem:rounding:errors:are:small}, and \cref{lem:disc:dyn}.
The bound on expectation follows analogously from the linearity of expectation and the bounds on the expected discrepancies in the aforementioned lemmas.
\end{proof}
Intuitively, \cref{lem:initial:load:vanishes} states that the contribution of the initial load to the discrepancy is insignificant if $t$ is large enough.
We generalize the analysis of Theorem 1~\cite{DBLP:conf/focs/RabaniSW98} (or Theorem 2.9 in~\cite{DBLP:conf/focs/SauerwaldS12}) to establish a bound on the discrepancy of the initial load as a function of $\beta$. For the sake of completeness the proof of \cref{lem:initial:load:vanishes} is given in \cref{proof:lem:initial:load:vanishes}.
\begin{lemma}[name=Memorylessness Property,restate=restateInitialLoadVanishes,label=lem:initial:load:vanishes]
Let $G$ be a $d$-regular graph. Let $K=\discr(\LoadVecT{0})$.
Then there exists a constant $c>0$
such that for all $\gamma > 0$
and $t \in \N$ with $t \geq t_0(\gamma) \coloneqq c \cdot
\max\left\{\gamma \log(n), \log(K\cdot n)\right\} \cdot \smash[b]{\frac{1}{\SpectralGap(\Laplacian(G)) \cdot \beta}}$ we get with probability at least $1 - n^{-\gamma}$ and in expectation
\[ \discr(\InitialContribVecT{t}) \leq 1.\]
\end{lemma}
The next lemma bounds $\discr(\RoundingContribVecT{t})$, the discrepancy contribution of cumulative rounding errors.
Note that this result does not just hold for the random matching model, but for all the three models that we consider in this paper.
In the proof of the lemma we extend then results of Theorem 3.6 in~\cite{DBLP:conf/focs/SauerwaldS12} (which is based on work in~\cite{DBLP:journals/jcss/BerenbrinkCFFS15}) to establish a bound as a function of $\beta$. The proof is given in \cref{p:lem:rounding:errors:are:small}.
\begin{lemma}[name=Insignificance of Rounding Errors,restate=restateRoundingErrorsAreSmall,label=lem:rounding:errors:are:small]
Let $G$ be an arbitrary graph.
Then for all $\gamma >1$, $t \in \N$, and $k \in [n]$ we get with probability at least $1 - 2n^{-\gamma+1}$ and in expectation
\[ \discr(\RoundingContribVecT{t}) \leq 2\cdot\sqrt{{\gamma \log(n)}/{\beta}}. \]
\end{lemma}
To bound $\discr(\DynamicContribVecT{t})$, the discrepancy contribution of dynamically allocated load items we apply the next lemma. It is in fact the core of our work. We prove it in \cref{sec:bound:contribution:dynamcally:allocated:balls}.
\begin{lemma}[Contribution of Dynamically Allocated Load]\label{lem:disc:dyn}
Let $G$ be a $d$-regular graph.
Define $T(G) \coloneqq \min \left\{\HittingTime\cdot \log n/{n}, \sqrt{d/{\SpectralGap(\Laplacian(G))}}, 1/{\SpectralGap(\Laplacian(G))} \right\}$.
Then for all $\gamma > 1$ and $t \in \N$ we get with probability at least $1 - 3n^{-\gamma+1}$ and in expectation
\[
\discr(\DynamicContribVecT{t}) = {\operatorname{O}}\left(
\gamma\log(n) \cdot \left(1 +
\sqrt{\frac{m}{n} \cdot \frac{\EdgeHittingTime}{n}} \right) +
\sqrt{\frac{\gamma \log(n)}{\beta} \cdot\frac{m}{n} \cdot T(G)}\right).
\]
\end{lemma}
\subsection{Bounding the Contribution of Dynamically Allocated Load} \label{sec:bound:contribution:dynamcally:allocated:balls}
In this section we prove \cref{lem:disc:dyn}.
Some of the proofs are omitted and can be found in \cref{sec:Omitted:Proofs:31}.
As a first step, we bound $\discr(\DynamicContribVecT{t})$
using the \emph{global divergence} $\Upsilon(\MixMatSeq{t})$, which is defined over a sequence of matching matrices $\MixMatSeq{t}$
as
\[\Upsilon(\MixMatSeq{t}) \coloneqq \max_{k \in [n]} \Upsilon_k(\MixMatSeq{t}),\quad\textup{where}\quad
\Upsilon_k(\MixMatSeq{t})
\coloneqq \sqrt{\sum_{\tau=1}^t \norm*{\MixMatTT{\tau}{t}_{k,\cdot}- \frac{\vec{1}}{n}}_2^2}.
\]
The global divergence can be regarded as a measure of the convergence speed of a random walk
that uses the matching matrices as transition probabilities.
In~\cite{DBLP:conf/stoc/FriedrichS09,DBLP:conf/focs/SauerwaldS12,DBLP:journals/jcss/BerenbrinkCFFS15} the authors use a related notion which they call the \emph{local $p$-divergence}, also defined on a sequence of matchings $\mixMatSeq{t}$.
The difference lies in the fact that the global divergence, essentially, measures differences between nodes' values and a global average, while the local divergence measures differences between neighboring nodes.
To show \cref{lem:disc:dyn} we first observe the following.
\begin{observation}\label{obs:disc:in_terms_of_one_viation}
It holds that $\discr(\DynamicContribVecT{t}) \leq 2 \cdot \max_{k\in[n]}\abs{\NodeDynamicContribT{k}{t} - t\cdot m/n} $.
\end{observation}
Next we consider a fixed node $k$ and show a concentration inequality on $\NodeDynamicContribT{k}{t}$ in terms of $\Upsilon_k(\mixMatSeq{t})$, where $\mixMatSeq{t}$ is the sequence of matchings applied by our process (\cref{lem:mixing_well_means_balancing_well}).
Note that in the lemma we assume the matchings are fixed and the randomness is due to the random load placement only. Hence, the lemma directly applies to $\baldModel(G)$.
Afterwards, we bound the global divergence of the random sequence of matchings, $\Upsilon_k(\MixMatSeq{t})$ in terms of a notion of ``goodness'' of the used matching distribution $\mathcal{D}$,
for the random sequence of matchings (\cref{lem:glob:div:bound:drift}),
and then bound the ``goodness'' of the distribution $\randModel(G)$ used in the random matching model (\cref{lem:rmdistr_is_good}).
We start with a bound on the deviation of $\NodeDynamicContribT{k}{t}$ from the average load $t \cdot m/n$ in terms of $\Upsilon(\mixMatSeq{t})$.
\begin{lemma}[Load Concentration]\label{lem:mixing_well_means_balancing_well}
Let $\mixMatSeq{t}$ be an arbitrary sequence of matchings.
Then for all $\gamma>0$, $t \in \N$, and $k \in [n]$ we get with probability at most $2 \cdot n^{-\gamma}$
\[
\abs*{\NodeDynamicContribT{k}{t} - t \cdot \frac{m}{n}} \geq \frac{4}{3} \cdot \gamma\log(n) + \sqrt{8\gamma\log(n) \cdot \frac{m}{n}} \cdot \Upsilon_k(\mixMatSeq{t}).
\]
\end{lemma}
\begin{proof}
Our goal is to decompose $\NodeDynamicContribT{k}{t}$ into a sum of independent random variables. Recall that we assume that the matching matrices are fixed and all randomness is due to the random choices of the load items.
This will enable us to apply a concentration inequality to this sum.
For the decomposition observe that
\(\DynamicContribVecT{t} = \sum_{\tau=1}^t \mixMatTT{\tau}{t} \cdot \AllocVecT{\tau},\)
where $\AllocVecT{\tau}$ is the random load vector corresponding to the $m$ load items allocated at time $\tau$.
So the $k$th coordinate of $\DynamicContribVecT{t}$ is
\(
\NodeDynamicContribT{k}{t}
=\sum_{\tau=1}^t\sum_{w\in[n]} \mixMatTT{\tau}{t}_{k,w}\cdot \NodeAllocT{w}{\tau}.
\)
We define the indicator random variable $\BallsRoundNode{\tau}{j}{w}$ for $\tau\in[t], j\in[m]$ and $w\in[n]$ as
\[\BallsRoundNode{\tau}{j}{w} \coloneqq
\begin{cases}
1, & \text{if the $j$-th load item of step $\tau$ is allocated to node $w$, } \\
0, & \mbox{otherwise.}
\end{cases}
\]
Note that for fixed $\tau$ and $j$ we have $\sum_{w\in [n]} \BallsRoundNode{\tau}{j}{w} =1$, $\Pr\left[\BallsRoundNode{\tau}{j}{w}=1\right]=1/n$ and $\E[\BallsRoundNode{\tau}{j}{w}]=1/n$.
Observe that $\NodeAllocT{w}{\tau}$, the load allocated to node $w$ at step $\tau$, can be expressed as $\sum_{j\in [m]} \BallsRoundNode{\tau}{j}{w}$.
Merging this with the value of $\NodeDynamicContribT{k}{t}$ gives
\begin{align*}
\NodeDynamicContribT{k}{t} &=
\sum_{\tau=1}^{t}\sum_{w\in [n]} \mixMatTT{\tau}{t}_{k,w} \cdot \left(\sum_{j\in [m]} \BallsRoundNode{\tau}{j}{w} \right)
=\sum_{\tau=1}^t \sum_{j\in[m] } \underbrace{\left(\sum_{w\in [n]} \left( \mixMatTT{\tau}{t}_{k,w}\cdot \BallsRoundNode{\tau}{j}{w}\right)\right)}_{\eqqcolon\BallRoundContr{k}{\tau}{j}}.
\end{align*}
For a fixed $\tau\in [t]$ and $j\in[m]$ we define
$\BallRoundContr{k}{\tau}{j}\coloneqq\sum_{w\in [n]} \mixMatTT{\tau}{t}_{k,w}\cdot \BallsRoundNode{\tau}{j}{w}$.
This random variable measures the contribution of $j$-th load item of round $\tau$ to $\NodeDynamicContribT{k}{t}$.
Note that the load items are allocated independently from each other.
Since $\mixMatTT{\tau}{t}$ are fixed matrices, then $\BallRoundContr{k}{\tau}{j}$ and $\BallRoundContr{k}{\tau'}{j'}$ are independent for all $\tau$ and $\tau'$ and $j\neq j'$.
To apply the concentration inequality from \cref{thm:upper:bound:on:sum} we need to show that $\BallRoundContr{k}{\tau}{j}\le 1$ and compute an upper bound on $\Var[\BallRoundContr{k}{\tau}{j}]$. Showing the first condition is easy since
exactly one of the indicator random variables $\BallsRoundNode{\tau}{j}{w}$ is one and $\mixMatTT{\tau}{t}_{k,w}$ has a value between zero and one.
It remains to consider the variance of $\BallRoundContr{k}{\tau}{j}$. First note that by linearity of expectation
\begin{align*}
\BigAutoExp{\BallRoundContr{k}{\tau}{j}} =\BigAutoExp{\!\sum_{w\in [n]\!\!\!} \left( \mixMatTT{\tau}{t}_{k,w}\cdot \BallsRoundNode{\tau}{j}{w}\right)} \!\!=\!
\sum_{\!\!\!w\in[n]\!\!\!}\mixMatTT{\tau}{t}_{k,w}\cdot \BigAutoExp{\BallsRoundNode{\tau}{j}{w}}\!=\! \sum_{\!\!\!w\in[n]\!\!\!}\mixMatTT{\tau}{t}_{k,w}\cdot \frac{1}{n} \!=\! \frac{1}{n},
\end{align*}
where the last equality follows form the fact that $\mixMatTT{\tau}{k}$ is doubly stochastic.
Now we get
\begin{align*}
\Var[\BallRoundContr{k}{\tau}{j}]
&= \BigAutoExp{\left(\BallRoundContr{k}{\tau}{j} - \AutoExp{\BallRoundContr{k}{\tau}{j}}\right)^2}
= \BigAutoExp{\Big(\Big(\sum_{w\in [n]} \mixMatTT{\tau}{t}_{k,w}\cdot \BallsRoundNode{\tau}{j}{w}\Big) - \frac{1}{n}\Big)^2}
\\ & =
\sum_{w' \in [n]} \frac{1}{n} \cdot \left(\mixMatTT{\tau}{t}_{k,w'} - \frac{1}{n}\right)^2
= \frac{1}{n} \cdot \norm*{\mixMatTT{\tau}{t}_{k,\cdot} - \frac{\vec{1}}{n}}_2^2,
\end{align*}
where we used that for each $\tau$ and each $j$ exactly one of the $\BallsRoundNode{\tau}{j}{w}$ is one and all others are zero, and each of the $n$ possible cases has uniform probability.
Recall that $\BallRoundContr{k}{\tau}{j}$ and $\BallRoundContr{k}{\tau'}{j'}$ are independent for all $\tau, \tau'$ and $j\neq j'$. Hence we get
\begin{align*}
\BigAutoVar{\sum_{\tau=1}^{t}\sum_{j\in[m]} \BallRoundContr{k}{\tau}{j}}
&= \sum_{\tau=1}^{t}\sum_{j\in[m]} \Var[\BallRoundContr{k}{\tau}{j}]
= \frac{1}{n} \cdot \sum_{\tau=1}^t\sum_{j\in[m]} \norm*{\mixMatTT{\tau}{t}_{k,\cdot} - \frac{\vec{1}}{n}}_2^2
\\&= \frac{m}{n}\cdot \left(\Upsilon_k(\mixMatSeq{t})\right)^2,
\end{align*}
where the final equality uses the definition of the global divergence $\Upsilon_k(\mixMatSeq{t})$.
Applying \cref{thm:upper:bound:on:sum} with $M=1$ and $X=\NodeDynamicContribT{k}{t}=\sum_{\tau=1}^{t}\sum_{j\in[m]} \BallRoundContr{k}{\tau}{j}$ with $\lambda=2\gamma\log(n)/3 + \Upsilon_k(\mixMatSeq{t})\cdot \sqrt{2\gamma m/n}$ results in
\[\Pr\left[{ \NodeDynamicContribT{k}{t}- t\cdot \frac{m}{n}} \geq \frac{2}{3} \cdot \gamma\log(n) + \sqrt{2\gamma\log(n) \cdot \frac{m}{n}} \cdot \Upsilon_k(\mixMatSeq{t})\right]\le n^{-\gamma}.
\]
The lower bound can be established using \cref{thm:lower:on:sum} (with $a_i=0$ and $M=1$) instead of \cref{thm:upper:bound:on:sum}. Via a union bound we get
\[
\BigAutoProb{\abs*{\NodeDynamicContribT{k}{t} - t \cdot \frac{m}{n}} \geq \frac{4}{3} \cdot \gamma\log(n) + \sqrt{8\gamma\log(n) \cdot \frac{m}{n}} \cdot \Upsilon_k(\mixMatSeq{t})}
\leq 2 \cdot n^{-\gamma}. \qedhere \]
\end{proof}
To bound the global divergence of the matching sequence used by the process we use two potential functions.
The \emph{quadratic node potential} $\NodePotential(\vec{x})$ is given by
\[\NodePotential(\vec{x}) \coloneqq \sum_{i \in [n]} \left(x_i - \overline{x}\right)^2,\quad \text{where} \quad \overline{x} \coloneqq \frac{1}{n} \cdot \sum_{j \in [n]} x_j.\]
For a set of edges $S$ on the nodes $[n]$ and a vector $\vec{x} \in \R^n$, the \emph{quadratic edge potential} is
\[\EdgePotential_S(\vec{x}) \coloneqq \sum_{\{i, j\} \in S} (x_i - x_j)^2.\]
We may also write $\EdgePotential_G \coloneqq \EdgePotential_{E(G)}$ whenever $G$ is a graph, and $\EdgePotential_\mathbf{M} \coloneqq \EdgePotential_{E(\mathbf{M})}$ whenever $\mathbf{M}$ is a matching matrix.
The following observation relates the drop of node potential to the edge potential in terms of $\beta$.
\begin{observation}[name=,label=obs:node_potential_change_exact,restate=restateObsPotentialRelation]
Let $\MixMatBeta{\beta}$ be a matching matrix with parameter $\beta \in (0, 1]$.
Then for any $\vec{x} \in \R^n$ we have
$\NodePotential(\vec{x}) - \NodePotential(\MixMatBeta{\beta} \cdot \vec{x})
= \frac{1 - (1-\beta)^2}{2} \cdot \EdgePotential_{E(\MixMatBeta{\beta})}(\vec{x})$.
\end{observation}
We now define a notion of a matching distribution being \emph{good}.
In \cref{lem:glob:div:bound:drift} below we show that the notion is sufficient for showing that matching sequences generated from such distributions have bounded global divergence.
Note that the ``goodness'' of a distribution does not depend on $\beta$ but on graph properties and the random choices with which the matchings are chosen.
Hence, we assume $\beta=1$.
\begin{definition}\label{def:goodness}
Assume $G$ is an arbitrary $d$-regular graph.
Let $g\colon \R_0^+ \to \R^+$ be an increasing function and let $\sigma^2 > 1$.
Then a matching distribution $\mathcal{D}(G)$ is \emph{$(g,\sigma^2)$-good} if the following conditions hold for $\MixMatBeta{1} \sim \mathcal{D}(G)$ and all stochastic vectors $\vec{x} \in \R^n$.
\begin{enumerate}
\item \(\NodePotential(\vec{x}) - \AutoExp{\NodePotential(\MixMatBeta{1} \cdot \vec{x})} \geq g(\NodePotential(\vec{x})).\)
\item \(\AutoVar{\NodePotential(\MixMatBeta{1} \cdot \vec{x})} \leq (\sigma^2 - 1) \cdot \left(\NodePotential(\vec{x}) - \AutoExp{\NodePotential(\MixMatBeta{1} \cdot \vec{x})}\right)^2.\)
\end{enumerate}
\end{definition}
It remains to show two results.
First, assuming a matching distribution is $(g,\sigma^2)$-good, the global divergence of a matching sequence generated by that distribution can be bounded in terms of $g$ and $\sigma$ (\cref{lem:glob:div:bound:drift}). Second, we have to calculate a function $g_G$ and the values of $\sigma_G$ for which the matching distribution $\randModel(G)$ is $(g_G,\sigma_G^2)$-good (see \cref{lem:rmdistr_is_good}).
\begin{lemma}[name=Global Divergence,label=lem:glob:div:bound:drift,restate=restateLemGlobalDivergence]
Assume $G$ is an arbitrary graph. Let $g\colon \R_0^+ \to \R^+$ be an increasing function, $\sigma^2 > 1$, and $\beta \in (0,1]$.
Let $\MixMatSeq{t} = (\MixMatBeta{\beta}(\tau))_{\tau=1}^t$ be an i.i.d.\ sequence of matching matrices generated by $\mathcal{D}(G)$ and
assume $\mathcal{D}(G)$ is a $(g,\sigma^2)$-good matching distribution.
Then for all $\gamma > 0$ and $k \in [n]$ we get with probability at least $1 - n^{-\gamma}$
\[
\left(\Upsilon_k(\MixMatSeq{t})\right)^2 \leq 8 \sigma^2 (\gamma \log(n) + \log(8 \sigma^2)) + \frac{2}{\beta} \cdot \int_0^1 \frac{x}{g(x)}\,{\mathrm{d}x}.
\]
\end{lemma}
\begin{lemma}
\label{lem:rmdistr_is_good}
Assume $G$ is an arbitrary $d$-regular graph. Let
\[g_G(x) \coloneqq \frac{1}{16 d} \cdot \max\left\{
d \cdot \SpectralGap(\Laplacian(G)) \cdot x,
\frac{x^2}{\mathrm{Res}(G)} ,
\frac{4}{27} \cdot x^3\right\} \text{ and }
\sigma_G^2 = 32 \cdot (\EdgeHittingTime / n) + 5.\]
Then $\randModel(G)$ is $(g_G, \sigma_G^2)$-good.
\end{lemma}
\begin{proof}
First, note that the function $g_G(x)$ is increasing in $x$.
Applying the first part of \cref{prop:node_potential_change_statistics} (see below) we get that for any vector $\Vec{x}\in \R^n$ it holds that
\[\NodePotential(\vec{x}) - \BigAutoExp{\NodePotential(\MixMatBeta{1} \cdot \vec{x})}
\ge \frac{1}{16d} \cdot \EdgePotential_G(\vec{x}).\]
From the first two statements of \cref{lem:edge_potential_bounds}
(stated behind \cref{lem:edge_potential_bounds}) we see that for $\MixMatBeta{1} \sim \randModel(G)$ and all stochastic vectors $\vec{x} \in \R^n$
\[
\EdgePotential_G(\vec{x}) \geq \max\left\{d\cdot \SpectralGap(\Laplacian(G))\cdot \NodePotential(\vec{x}), \frac{\NodePotential(\vec{x})^2}{\mathrm{Res}(G)} , \frac{4}{27} \cdot \NodePotential(\vec{x})^3 \right\}.
\]
Hence,
\[\NodePotential(\vec{x}) - \BigAutoExp{\NodePotential(\MixMatBeta{1} \cdot \vec{x})}
\ge \frac{1}{16d} \cdot \max\left\{d \cdot \SpectralGap(\Laplacian(G)) \cdot \NodePotential(\vec{x}),
\frac{\NodePotential(\vec{x})^2}{\mathrm{Res}(G)},
\frac{4}{27} \cdot \NodePotential(\vec{x})^3\right\},\]
and as a consequence, $\NodePotential(\vec{x}) - \AutoExp{\NodePotential(\MixMatBeta{1} \cdot \vec{x})} \geq g_G(\NodePotential(\vec{x}))$ by the definition of $g_G$.
It remains to check the second condition of \cref{def:goodness} with our claimed value~$\sigma_G^2$.
Inserting its value as stated in the lemma, the condition requires that
\[\AutoVar{\NodePotential(\MixMatBeta{1} \cdot \vec{x})} \leq (32 (\EdgeHittingTime / n) + 5 - 1) \cdot \left(\NodePotential(\vec{x}) - \AutoExp{\NodePotential(\MixMatBeta{1} \cdot \vec{x})}\right)^2,\]
which is given in the second part of \cref{prop:node_potential_change_statistics} (see below).
\end{proof}
In \cref{prop:node_potential_change_statistics} we first relate the drop of $\NodePotential$ to the quadratic edge potential $\EdgePotential$. In the second part we bound the variance of the potential drop as a function of the edge hitting time.
\begin{lemma}[label=prop:node_potential_change_statistics,restate=restateLemNodePotentialChangeStatistics]
Let $G$ be a $d$-regular graph, let $\mathbf{M}^1 \sim \randModel(G)$, and let $\vec{x} \in \R^n$,
then
\begin{enumerate}
\item \(\NodePotential(\vec{x}) - \BigAutoExp{\NodePotential(\MixMatBeta{1} \cdot \vec{x})}
\ge \frac{1}{16d} \cdot \EdgePotential_G(\vec{x}).\)
\item \(\BigAutoVar{\NodePotential(\MixMatBeta{1} \cdot \vec{x})}
\leq (32 \cdot (\EdgeHittingTime / n) + 4) \cdot\left(\NodePotential(\vec{x}) - \BigAutoExp{\NodePotential(\MixMatBeta{1} \cdot \vec{x})}\right)^2.\)
\end{enumerate}
\end{lemma}
In \cref{lem:edge_potential_bounds} we relate the size of the quadratic edge potential $\EdgePotential_G$ to
the second-largest eigenvalue of $\Laplacian(G)$, the effective resistance of $G$ and node potential.
To state it,
we need some additional definitions.
For any two nodes $i$ and $j$ of the graph $G$
$\ResistiveDistance{i}{j}$ is the \emph{effective resistance} (or \emph{resistive distance}) between $i$ and $j$ in $G$ (for a detailed definition see \cref{apx:aux}).
Furthermore, we write $\mathrm{Res}(G)$ for the \emph{resistive diameter} of $G$,
i.e., the largest resistive distance between any pair of nodes in $G$,
and write $\mathrm{Res}^*(G)$ for the maximum effective resistance between any pair of nodes adjacent in $G$.
I.e., $\mathrm{Res}(G) \coloneqq \max_{i,j \in [n]} \ResistiveDistance{i}{j}$ and $\mathrm{Res}^*(G) \coloneqq \max_{\{i,j\} \in E(G)} \ResistiveDistance{i}{j}$.
The first part of the following lemma was previously shown in~\cite{DBLP:journals/jcss/GhoshM96,DBLP:conf/focs/SauerwaldS12}.
\begin{lemma}
[label=lem:edge_potential_bounds,restate=restateEdgePotentialBounds]
Let $\vec{x} \in \R^n$, and let $G$ be a connected $d$-regular graph.
\begin{enumerate}
\item \(\EdgePotential_G(\vec{x}) \geq d \cdot \SpectralGap(\Laplacian(G)) \cdot \NodePotential(\vec{x})\).
\item If $\vec{x}$ is stochastic, then
$\EdgePotential_G(\vec{x}) \geq \max\left\{\frac{1}{\mathrm{Res}(G)} \cdot \NodePotential(\vec{x})^2, \frac{4}{27} \cdot \NodePotential(\vec{x})^3\right\}$
\item \(\max_{\{i,j\} \in E(G)} (x_i - x_j)^2 \leq \mathrm{Res}^*(G) \cdot \EdgePotential_G(\vec{x}).\)
\end{enumerate}
\end{lemma}
\subsubsection*{Proof of \cref{lem:disc:dyn}}
\begin{proof}
Define $g_G(x) = \frac{1}{16d} \cdot \max\left\{d \cdot \SpectralGap(\Laplacian(G)) \cdot x, x^2 / \mathrm{Res}(G), 4 x^3 /27\right\}$ and let $\sigma_G^2 \coloneqq 32 \cdot (\EdgeHittingTime/n) + 5$.
Then by \cref{lem:rmdistr_is_good} the matching distribution $\randModel(G)$ is $(g_G, \sigma_G^2)$-good.
By \cref{lem:glob:div:bound:drift} we have for all $t \in \N$, $k \in [n]$
\[\BigAutoProb{\left(\Upsilon_k(\MixMatSeq{t})\right)^2 \leq 8 \sigma_G^2 ((\gamma+1) \log(n) + \log(8 \sigma_G^2)) + \frac{1}{\beta} \cdot \int_0^1 \frac{x}{g_G(x)}\,{\mathrm{d}x}} \geq 1 - n^{-(\gamma+1)}.\]
To bound $\Upsilon_k(\MixMatSeq{t})$ we use the following two claims (see \cref{apx:proof_discr_dyn} for the proof).
\begin{claim}\label{claim:integral_bound}
It holds that $\displaystyle \int_0^1 {x}/{g_G(x)}\,{\mathrm{d}x} = {\operatorname{O}}(T(G))$.
\end{claim}
\begin{claim}\label{claim:edge_hitting_time_lower_bound}
For any $d$-regular graph $G$ it holds that $\EdgeHittingTime / n \geq 1/2$.
\end{claim}
Together we get from \cref{claim:integral_bound} and \cref{claim:edge_hitting_time_lower_bound} that with probability at least $1 - n^{-(\gamma+1)}$
\begin{equation}\label{eqn:concrete_global_div_bound}
\left(\Upsilon_k(\MixMatSeq{t})\right)^2 = {\operatorname{O}}\left(\frac{\EdgeHittingTime}{n} \cdot \left(\gamma\log(n) + \log\left(\frac{\EdgeHittingTime}{n}\right)\right) + \frac{T(G)}{\beta}\right).
\end{equation}
Since $\EdgeHittingTime = {\operatorname{O}}(n^3)$ (Proposition 10.16 in \cite{LevinPeresbook}),
$\log(\EdgeHittingTime / n) = {\operatorname{O}}(\log n)$, and $\gamma > 1$,
\[
\Upsilon_k(\MixMatSeq{t})
= {\operatorname{O}}\left(\sqrt{\gamma\log(n) \cdot \frac{\EdgeHittingTime}{n} + \frac{T(G)}{\beta}}\right) = {\operatorname{O}}\left(\sqrt{\gamma \log(n) \cdot \frac{\EdgeHittingTime}{n}} + \sqrt{\frac{T(G)}{\beta}}\right).
\]
Now \cref{lem:mixing_well_means_balancing_well} states that for any fixed sequence of matching matrices $\mixMatSeq{t}$, with probability at least $1 - 2n^{-(\gamma+1)}$ it holds that
\begin{equation}\label{eqn:concrete_dynamic_concentration_bound}
\abs*{\NodeDynamicContribT{k}{t} - t \cdot \frac{m}{n}} = {\operatorname{O}}\left( \gamma\log(n) + \sqrt{\gamma\log(n) \cdot \frac{m}{n}} \cdot \Upsilon_k(\mixMatSeq{t})\right).
\end{equation}
Applying a union bound over all $k \in [n]$, \cref{eqn:concrete_global_div_bound} and \cref{eqn:concrete_dynamic_concentration_bound} hold for all $k$ with probability at least $1 - 3n^{-\gamma}$. Hence, for all $k \in [n]$
\begin{equation*}\begin{aligned}
\abs*{\NodeDynamicContribT{k}{t} - t \cdot \frac{m}{n}}
&={\operatorname{O}}\left( \gamma\log(n) + \sqrt{\gamma\log(n) \cdot \frac{m}{n}} \cdot \left(\sqrt{\gamma \log(n) \cdot \frac{\EdgeHittingTime}{n}} + \sqrt{\frac{T(G)}{\beta}}\right)\right)
\\
&={\operatorname{O}}\left(\gamma\log(n)\cdot \left(1 + \sqrt{\frac{m}{n} \cdot \frac{\EdgeHittingTime}{n}}\right) + \sqrt{\frac{(\gamma+1)\log(n)}{\beta} \cdot \frac{m}{n} \cdot T(G)}\right).
\end{aligned}\end{equation*}
The high-probability bound now follows from \cref{obs:disc:in_terms_of_one_viation}.
The corresponding bound on $\AutoExp{\discr(\DynamicContribVecT{t}}$ follows readily; see \cref{lem:tail_bound_to_expectation_bound} in \cref{apx:known-results-probability-theory} for the details.
\end{proof}
\section{Balancing Circuit Model}\label{sec:analysis_balancing_circuit}
Here we assume $\beta=1$.
Recall that we assume $G$ is covered by $\zeta$ fixed matchings $\mixMatT{1},\ldots, \mixMatT{\zeta}$.
The matching distribution $\baldModel(G)$ then deterministically chooses the matching $\mixMatT{t}=\mixMatT{t \bmod \zeta}$ in step $t$.
The round matrix is defined as $\mathbf{R} \coloneqq \mixMatTT{1}{\zeta}$ and the mixing matrices are fixed in this model.
Thus, for a sequence of matchings $\mixMatSeq{t}$ the global divergence is
$\Upsilon(\mixMatSeq{t}) \coloneqq \max_{k\in [n]}\sqrt{\sum_{\tau=1}^t \norm*{\mixMatTT{\tau}{t}_{k,\cdot} - 1/n}_2^2}$.
The next theorem provides an upper bound on the discrepancy for this model.
Note that the following theorem holds for arbitrary graphs, while \cref{thm:main_sync_random} only holds for $d$-regular graphs.
\begin{theorem}\label{thm:main_sync_circuit}
Let $G$ be an arbitrary graph and $\LoadVecT{t}$ be the state of process $\SyncProc{\baldModel(G)}{1}{m}$ at time $t$ with $\discr(\LoadVecT{0})\eqqcolon K$.
For all $t \in \N$ with $t \ge \frac{\zeta}{\SpectralGap{(\mathbf{R})}}\cdot\left(\ln(K\cdot n) \right)$
it holds w.h.p.\ and in expectation
\[\discr(\LoadVecT{t})={\operatorname{O}}\left(\log (n)+ \sqrt{ m/n}\cdot\Upsilon(\mixMatSeq{t})\cdot \sqrt{\log (n)} \right).\]
\end{theorem}
\begin{proof}
The proof follows the same line as the proof \cref{thm:main_sync_random}, which is proved via \cref{lem:initial:load:vanishes},
\cref{lem:disc:dyn}, and \cref{lem:rounding:errors:are:small} bounding
$\InitialContribVecT{t}, \DynamicContribVecT{t}$, and $\RoundingContribVecT{t}$, respectively.
\Cref{lem:initial:load:vanishes} is replaced by \cref{lem:erro_bound} below. \Cref{lem:initial:load:vanishes} can also be applied to the balancing circuit model since it only requires that the subgraph used for balancing is a matching.
It remains to replace \cref{lem:rounding:errors:are:small}.
Since the matching matrices are fixed this time the proof is much simpler.
The proof of \cref{lem:mixing_well_means_balancing_well} carries to over to this model giving us a bound on $\abs{\NodeDynamicContribT{k}{t}-tm/n}$ for $k\in[n]$ with probability at least $1-2\cdot n^{-\gamma}$.
Applying the union bound over all nodes $k\in[n]$, together with \cref{obs:disc:in_terms_of_one_viation} (stating that $\discr(\DynamicContribVecT{t}) \leq 2 \cdot \max_{k\in[n]}\abs{\NodeDynamicContribT{k}{t} - t\cdot m/n} $), gives a bound on $\discr(\DynamicContribVecT{t})$ which holds with probability at least $1-2\cdot n^{\gamma+1}$.
\end{proof}
\begin{lemma}[Memorylessness Property] \label{lem:erro_bound}
For all $t\in \N$ with $t \ge {\zeta}/{\SpectralGap{(\mathbf{R})}}\cdot \left(\ln(K\cdot n)\right)$ it holds that $\discr(\InitialContribVecT{t})\le 2$.
\end{lemma}
\begin{proof}
Since $\NodePotential(\vec{x}) \le K^2\cdot n$ it follows from Lemma 2 in \cite{DBLP:conf/spaa/GhoshMS96} that
\begin{equation*}
\NodePotential\left(\mixMatTT{1}{t} \cdot \vec{x}\right) \leq (1-\SpectralGap{(\mathbf{R})})^{2\lfloor t \rfloor/\zeta}\cdot \NodePotential(\vec{x}) \le (1-\SpectralGap{(\mathbf{R})})^{2\lfloor t \rfloor/\zeta}\cdot K^2\cdot n \le e^{-2 \lfloor t \rfloor \cdot \SpectralGap{(\mathbf{R})}/\zeta+ 2\ln(Kn)}.
\end{equation*}
Setting $t\ge (\zeta/\SpectralGap{(\mathbf{R})})\cdot \left(\ln(Kn)\right)$ gives
$\NodePotential\left(\mixMatTT{1}{ t} \cdot \vec{x}\right) \leq 1$ which implies that $\discr(\InitialContribVecT{t}\le 2$.
\end{proof}
Note that a similar statement was shown in~\cite{DBLP:conf/focs/RabaniSW98,DBLP:conf/focs/SauerwaldS12,DBLP:journals/jcss/BerenbrinkCFFS15}.
The next theorem provides a lower bound on the discrepancy for this model. The proof can be found in \cref{apx:analysis_balancing_circuit}.
\begin{theorem}\label{thm:main_sync:lower}
Let $G$ be an arbitrary graph and $\LoadVecT{t}$ be the state of process $\SyncProc{\baldModel(G)}{1}{m}$ at time $t$.
Then for all $t\in \N$ and $m\ge 4n\cdot \log(n)/ \Upsilon(\mixMatSeq{t})$ it holds with constant probability
\[ \discr(\LoadVecT{t})=\Omega\left(\sqrt{m/n}\cdot \Upsilon(\mixMatSeq{t})\right).\]
\end{theorem}
\section{Asynchronous Model}
\label{sec:asynchronous}
The following is our main theorem for the asynchronous model.
The bounds provided by \cref{thm:main_async} for the asynchronous model differ from those in \cref{thm:main_sync_random} for the random matching model in two details.
First, the lower bound on the balancing time is larger by a factor of $n$. This is due to the fact that the asynchronous model balances across just one edge per round in contrast to $\Theta(n)$ edges in the random matching model.
Second, the upper bound on $\discr(\LoadVecT{t})$ is much simpler.
Note, however that setting $m=n$ in \cref{thm:main_sync_random} and further simplifying the result by using $\EdgeHittingTime / n = \Omega(1)$ (see also \cref{claim:edge_hitting_time_lower_bound} in the proof of \cref{lem:disc:dyn}) results in the same asymptotic bound as in \cref{thm:main_async}.
\begin{theorem}
\label{thm:main_async}
Let $G$ be a $d$-regular graph and
define \((T(G) \coloneqq \min \Big\{\frac{\HittingTime}{n} \cdot \log(n), \sqrt{\frac{d}{\SpectralGap(\Laplacian(G))}},\discretionary{}{}{} \frac{1}{\SpectralGap(\Laplacian(G))} \Big\} \).
Let $\LoadVecT{t}$ be the state of process $\AsyncProc{\singdModel(G)}{\beta}$
at time $t$ with $\discr(\LoadVecT{0}) \eqqcolon K \ge 1$.
There exists a constant $c>0$ such that for all
$t \geq c \cdot n \cdot \log(K\cdot n) / (\SpectralGap(\Laplacian(G)) \cdot \beta)$
it holds w.h.p.\ and in expectation
\[ {\discr(\LoadVecT{t}) = {\operatorname{O}}\left(\log(n) \sqrt{\frac{\EdgeHittingTime}{n}} + \sqrt{\frac{\log(n)}{\beta} \cdot T(G)}\right).}\]
\end{theorem}
\begin{proof}[Proof Sketch of \cref{thm:main_async}]
The proof of the theorem follows along the same lines at the proof of \cref{thm:main_sync_random}.
However, there are some major differences.
Most importantly, the proof of \cref{lem:mixing_well_means_balancing_well} (giving a concentration bound on $\NodeDynamicContribT{k}{t}$ in terms of the global divergence of the sequence of matching matrices) can not be applied for $\textsc{ABal}$.
The proof heavily relies on the fact that the load allocation and the matching edges are chosen independently from each other, which is certainly not the case for $\textsc{ABal}$.
Our new lemma (\cref{lem:mixing_well_means_balancing_well_async} in \cref{apx:asynchronous}) carefully analyses the dependency, and it uses a stronger concentration inequality.
In addition, we also have to re-calculate the function $g_G$ and $\sigma_G$ to show that the matching distribution used by $\singdModel$ is $(g_G, \sigma_G^2)$-good (see \cref{lem:asdistr_is_good} in \cref{apx:asynchronous}).
\end{proof}
\iffalse
\section{Application to Balanced Allocations}
We can also adapt our analysis of the asynchronous process
to the following load allocation process on a graph $G$ parameterized by some $\beta \in (0, 1)$,
which we call $\textsc{Adaptive}_\beta(G)$.
Just as before,
we write $\LoadVecT{t} = (X_i(t))_{i \in [n]} \in \R^n$ for the load vector after $t$ steps,
and we let $\LoadVecT{0} = \vec{0}$.
In each step $t \in \N^+$:
\begin{enumerate}
\item Sample an edge of the graph $\{i,j\} \in E(G)$ uniformly at random.
\item Throw a biased coin with probability $p_{\textup{heads}} \coloneqq \min\{1, (1 + \beta \cdot \Delta) / 2\}$,
where $\Delta \coloneqq \abs{X_i(t-1) - X_j(t-1)}$ is the current load difference between $i$ and $j$.
\begin{itemize}
\item If the coin hit heads, allocate a load item to the node in $\{i, j\}$ which has less load (breaking ties uniformly at random).
\item Otherwise, allocate a load item to the node with more load (again breaking ties uniformly at random).
\end{itemize}
\end{enumerate}
Note that $\textsc{Adaptive}_\beta(G)$ does not reallocate any load, and that it is a ``local'' algorithm in the sense that the only information used is the load difference across the edge.
\begin{theorem}\label{thm:main_adaptive}
Consider the process $\textsc{Adaptive}_\beta(G)$ where $G$ is regular,
and let $\LoadVecT{t}$ be the load vector generated by the process after $t \in \N$ load items have been allocated.
Then for all $t \in \N$, there exists a $\beta = \beta(G, t)$
such that, w.h.p.,
\[\discr(\LoadVecT{t}) = {\operatorname{O}}\left(\log(n \cdot t) \cdot \left(\sqrt{\frac{\EdgeHittingTime}{n}} + T(G)\right)\right),\]
where $T(G)$ is defined as in \cref{thm:main_sync_random}.
\end{theorem}
Result given by \cref{thm:main_async} (with $\gamma > 1$):
\[\discr(\LoadVecT{t}) = {\operatorname{O}}\left(\gamma \log(n) \sqrt{\frac{\EdgeHittingTime}{n}} + \sqrt{\frac{\gamma \log(n)}{\beta} \cdot T(G)}\right).\]
Choose $\beta$ such that $1/\beta = c \cdot \gamma \log(n) \cdot \max\{ \sqrt{\EdgeHittingTime / n}, T(G)\}$ for a sufficiently large but constant $c > 0$.
Then, the bound becomes
\[\discr(\LoadVecT{t}) = {\operatorname{O}}\left(\gamma \log(n) \cdot \left(\sqrt{\frac{\EdgeHittingTime}{n}} + \sqrt{\max\left\{\sqrt{\frac{\EdgeHittingTime}{n}}, T(G)\right\} \cdot T(G)}\right)\right).\]
If $\sqrt{\EdgeHittingTime / n} \leq T(G)$,
then the bound is
\[\discr(\LoadVecT{t}) = {\operatorname{O}}\left(\gamma \log(n) \cdot \left(T(G) + \sqrt{T(G) \cdot T(G)}\right)\right)
= {\operatorname{O}}(\gamma \log(n) \cdot T(G)).\]
Otherwise, if $T(G) \leq \sqrt{\EdgeHittingTime / n}$,
the bound is
\begin{align*}
\discr(\LoadVecT{t})
&= {\operatorname{O}}\left(\gamma \log(n) \cdot \left(\sqrt{\frac{\EdgeHittingTime}{n}} + \sqrt{\sqrt{\frac{\EdgeHittingTime}{n}} \cdot \sqrt{\frac{\EdgeHittingTime}{n}}}\right)\right)
\\ &= {\operatorname{O}}(\gamma \log(n) \cdot \sqrt{\EdgeHittingTime / n}).
\end{align*}
Thus,
\[\discr(\LoadVecT{t}) = {\operatorname{O}}\left(\gamma \log(n) \cdot \left(\sqrt{\EdgeHittingTime / n} + T(G)\right)\right).\]
As the load difference between any two nodes is at most the discrepancy,
the load transferred across a given edge in a step is at most
\(\lceil \beta \discr(\vec{\LoadSymb}) / 2\rceil.\)
If $\sqrt{\EdgeHittingTime / n} \geq T(G)$,\todo{finish calculation}
\fi
\section{Drift Result} \label{sec:drift}
In our analysis we use the following tail bound for the sum of a non-increasing sequence of random variables with variable negative drift.
The proof uses established methods from drift analysis.
In particular, it relies one techniques found in the proof of the Variable Drift Theorem in \cite{DBLP:series/ncs/Lengler20}.
The full technical proof can be found in \cref{apx:drift_proof}.
\begin{theorem}[name=,restate=restateLemDrift,label=lem:drift]
Let $(X(t))_{t\ge 0}$ be a non-increasing sequence of discrete random variables with $X(t)\in \R^+_0$ for all $t$ with fixed $X(0) = x_0$.
Assume there exists an increasing function $h\colon \R^+_0 \to \R^+$ and a constant $\sigma > 0$ such that the following holds. For all $t \in \N$ and all $x > 0$ with $\AutoProb{X(t) = x} > 0$
\begin{enumerate}
\item \(\AutoExpCond{X(t+1)}{X(t) = x} \leq x - h(x),\)\label{cond:drift:1}
\item \(\AutoVarCond{X(t+1)}{X(t) = x} \leq \sigma \cdot \left(\AutoExpCond{X(t+1)}{X(t) = x} - x\right)^2.\) \label{cond:drift:2}
\end{enumerate}
Then the following statements hold.
\begin{enumerate}
\item For all $\delta \in (0, 1)$ and any arbitrary but fixed $t$
\[\BigAutoProb{\int_{X(t)}^{x_0} \frac{1}{h(\varphi)}\,{\mathrm{d}\varphi} \leq (1-\delta)t} \leq \exp\left(-\,\frac{\delta^2 t}{2(\sigma + 1)}\right).\]
\item For all $\delta \in (0, 1)$ and $p \in (0,1)$ we define $t_0 \coloneqq \frac{2(\sigma + 1)}{\delta^2} \left(-\log(p) + \log\left(\frac{2(\sigma + 1)}{\delta^2}\right)\right)$. Then \[\BigAutoProb{\sum_{t=t_0+1}^{\infty} X(t) \leq \frac{1}{1-\delta} \cdot \int_0^{x_0} \frac{\varphi}{h(\varphi)} {\mathrm{d}\varphi}} \geq 1 - p.\]
\end{enumerate}
\end{theorem}
\section{Conclusions and Open Problems} \label{sec:conclusions}
In this paper we analyze discrete load balancing processes on graphs.
As our main contribution we bound the discrepancy that arises in dynamic load balancing in three models, the random matching model, the balancing circuit model, and the asynchronous model.
Our results for the random matching model and the asynchronous model hold for $d$-regular graphs, while our analysis for the balancing circuit model applies to arbitrary graphs.
To the best of our knowledge our results constitute the first bounds for discrete, dynamic balancing processes on graphs.
Furthermore, our results improve the work by Alistarh et al.~\cite{DBLP:conf/icalp/AlistarhNS20} who prove that the expected discrepancy is bounded by $\sqrt{n}\log(n)$ in the (arguably simpler) continuous asynchronous process {\def{^{\text{(cont)}}}$\AsyncProc{\singdModel(G)}{1}$}.
We improve their bound to $\sqrt{n \log(n)}$ and additionally show that it holds with high probability.
We conjecture that our results are tight up to polylogarithmic factors.
However, showing tight upper and lower bounds remains an open problem.
\paragraph{Results for Specific Graph Classes}
We show an overview of our bounds on the discrepancy for specific graph classes in \cref{table:disc:upperbound}. The corresponding results are formally derived in \cref{apx:hitting-time_spectral-gap} for the random matching model, \cref{apx:bounds-specific-graphs-C} for the balancing circuit model, and \cref{apx:async-bounds-graph-classes} for the asynchronous model.
\begin{table}[ht]
\caption {Asymptotic upper bounds on the discrepancy in specific graph classes.}
\label{table:disc:upperbound}
\def\hx#1{#1&}
\def1.5{1.5}
\begin{tabularx}{\textwidth}{ Xccc }
\toprule
Graph & $\SyncProc{\randModel(G)}{1}{m}$ & $\SyncProc{\baldModel(G)}{1}{m}$ & $\AsyncProc{\singdModel(G)}{1}$\\[-1ex]
& \cref{apx:hitting-time_spectral-gap} & \cref{apx:bounds-specific-graphs-C} & \cref{apx:async-bounds-graph-classes} \\
\midrule
\hx{$d$-regular graph\newline\small (const. $d$)}
$\log(n) + \sqrt{m\cdot\log(n)}$
&$\log(n) +\sqrt{m\cdot\log(n)}$
& $\sqrt{n\cdot\log(n)}$
\\
\hx{cycle $C_n$}
$\log(n) + \sqrt{m\cdot\log(n)}$
&$\log(n) +\sqrt{m\cdot\log(n)}$
& $\sqrt{n\cdot\log(n)}$
\\
\hx{2-D torus}
$\log(n) + \sqrt{m/n}\cdot \log^{3/2}(n)$
& $(1 + \sqrt{m/n})\cdot\log(n)$
& $\log^{3/2}(n)$\\
\hx{$r$-D torus\newline\small (const. $r \geq 3$)}
$(1 + \sqrt{m/n})\cdot\log(n)$
& $\log(n) + \sqrt{m/n\cdot\log(n)}$
& $\log(n)$ \\
\hx{hypercube}
$(1 + \sqrt{m/n})\cdot\log(n)$
& $(1 + \sqrt{m/n})\cdot\log(n)$
& $\log(n)$ \\
\bottomrule
\end{tabularx}
\end{table}
\paragraph{Open Problems}
We are confident that our results carry over to arbitrary graphs (as opposed to regular graphs), provided that there exists a lower bound on the probability $p_{min}$
with which an edge is used for balancing.
However, to show bounds on the discrepancy one has to overcome fundamental problems such as the bias introduced by high-degree nodes.
Another interesting open question is whether the results carry over to a model where the amount of load that may transmitted over an edge in each step is bounded by a constant.
If only a single load item can be transferred per edge and step the problem is similar to the token distribution problem (see, for example, \cite{DBLP:journals/algorithmica/HeideOW96}).
Finally, we believe that one can also adapt our analysis to variant of a graphical balls-into-bins process.
The process works as follows.
In each step an edge $(i,j)$ is sampled uniformly at random. W.l.o.g.\ assume that the load of $i$ is smaller than the load of $j$ by an additive term~$\Delta$.
Then a biased coin is tossed showing heads with probability $p \coloneqq \min\{1, (1 + \beta \cdot \Delta) / 2\}$ and tails otherwise, where $\beta$ is a suitably chosen and non-constant parameter.
If the coin hits heads one item is allocated to $i$ and otherwise to $j$.
A formal analysis of this allocation process (as well as of other, related balls-into-bins processes) is beyond the scope of our paper and remains an open problem.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 36 |
package tagit2.api.tagcloud;
import java.util.List;
import org.apache.solr.client.solrj.response.FacetField.Count;
import tagit2.util.tagcloud.TagCloudItem;
public interface TagitTagCloudService extends TagitTagCloudServiceLocal, TagitTagCloudServiceRemote {
public List<TagCloudItem> getTagCloud( List<Count> facets );
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 4,704 |
\section{Introduction}
For many decades the phase diagram of the quantum chromodynamics (QCD) was believed to be simple
\cite{Cabibbo:1975ig}. Only two phases were considered: a confined phase, where the relevant degrees
of freedom are hadrons which are build from quarks and gluons, and a deconfined phase, where the
relevant degrees of freedom are quarks and gluons \cite{Rischke:2003mt}.
With increasing computational power, lattice simulations were able to test the above mentioned
picture at nonzero temperature and zero quark chemical potential \cite{Aoki:2006br,Cheng:2006qk}.
Since, at present lattice simulations are not able to access the high quark potential regime,
effective models and QCD-like theories have to be used. These approaches suggest a different
picture, according to which a rich phase diagram with a complicated structure is realized.
At low temperatures and at low densities matter is confined and the relevant degrees of freedom are
hadrons; on the contrary, at very high densities a color superconducting phase should be present
\cite{Rischke:2003mt}, but it is unclear how the transition from nuclear matter to such phases looks
like. Also the critical quark chemical potential for a transition to such a phase is unknown.
Beside these uncertainties, there is also the possibility that further states of matter e.g.
quarkyonic matter \cite{Kojo:2009ha} exist.
All approaches to describe QCD at finite density have to use simplifications. For example the NJL
model \cite{Klevansky:1992qe} takes only quark degrees of freedom into account and therefore is not
the best choice to describe the regime of confined matter. On the other hand, models based on
hadrons have no chances to access the regime where quarks and gluons are the relevant degrees of
freedom.
The general idea of inhomogeneous condensation goes back to Migdal \cite{Migdal:1978az}. He first
introduced inhomogeneous condensation within nuclear matter which was realized via the famous
chiral density wave (CDW). The idea of inhomogeneous condensation never got out of fashion and only
a few years ago the Gross-Neveu model in two dimensions could be solved exactly, analytically
\cite{Schnetz:2004vr} and numerically \cite{Wagner:2007he}. It has be shown that for high densities
and low temperatures inhomogeneous condensation dominates the phase diagram. Based on these results
it was shown that the NJL-model in four dimensions shows the same class of modulations as
the Gross-Neveu model \cite{Nickel:2009wj}.
It is of major interest to also look for the possibility of inhomogeneous condensation within a
fully hadronized model that is able to describe vacuum properties and to describe nuclear matter
saturation. The parity doublet model \cite{Jido:1998av} successfully reproduces vacuum
phenomenology \cite{Gallas:2009qp} and physics at finite density \cite{Zschiesche:2006zj}. The next
step, achieved here, is to test the ground state at nonzero density for the formation of a
inhomogeneous condensation. Furthermore a generalization of the parity doublet model is
straightforward and also exotic matter states like tetraquarks and glueballs can be introduced
\cite{Gallas:2011qp}. In this work however, we use the model studied in Ref.\
\cite{Zschiesche:2006zj}.
\section{The Model}
The mesonic part of the Lagrangian is a $SU(2)$ linear sigma model including vector mesons
\cite{Walecka:1974qa}. It has the following form:
\begin{align}
\mathscr{L}_M &= \frac{1}{2}\partial_\mu \sigma \partial^\mu \sigma + \frac{1}{2} \partial_\mu
\vec{\pi} \partial^\mu \vec{\pi}
-\frac{1}{4} F_{\mu \nu}F^{\mu \nu} \nonumber \\
&+ \frac{1}{2} m^2 (\sigma^2 + \vec{\pi}^2) + \frac{1}{2}m_\omega^2 \omega_\mu \omega^\mu -
\frac{\lambda}{4}(\sigma^2
+ \vec{\pi}^2)^2 + \epsilon \sigma ~,
\end{align}
with the field strength tensor for the vector meson fields $F_{\mu \nu} = \partial_\mu
\omega_\nu-\partial_\nu \omega_\mu$. In the vacuum chiral symmetry is spontaneous broken; this is
achieved in the model via the sign of the parameter $m^2$. The vacuum exception value (v.e.v.) of
the field $\sigma$ is nonzero and its value $\sigma_0 = \varphi$ corresponds at zero temperature
and density to the pion decay constant, $\varphi = f_\pi$.
The baryon part of model considers, besides the nucleon $N$, also its chiral partner $N^*$
\cite{Zschiesche:2006zj}. The Lagrangian is formulated in terms of the bare fields $\psi_1$ and
$\psi_2$: These fields are chiral eigenstates but not mass eigenstates of the model. The physical
fields $N$ and $N^*$ emerge as superpositions of $\psi_1$ and $\psi_2$. The fields $\psi_1$ and
$\psi_2$ transform under $SU_L(2) \times SU(2)_R$ the following way:
\begin{align}
\psi_{1, R} \rightarrow U_R ~\psi_{1, R} ~, ~~~~ \psi_{1, L} \rightarrow U_L ~\psi_{1, L} ~,
~~~~~~~~~
\psi_{2, R} \rightarrow U_L ~\psi_{2, R} ~, ~~~~ \psi_{2, L} \rightarrow U_R ~\psi_{2, L} ~.
\end{align}
Notice that $\psi_2$ transforms in a mirror way with respect to the field $\psi_1$. Besides the well
known kinetic terms and the coupling to mesons, the mirror assignment allows to construct a further
chiral invariant bilinear term:
\begin{align}
\bar{\psi}_{2, L} \psi_{1, R} - \bar{\psi}_{2, R} \psi_{1, L} - \bar{\psi}_{1, L}
\psi_{2, R} +
\bar{\psi}_{1, R} \psi_{2, L}
= \bar{\psi}_2 \gamma_5 \psi_1 - \bar{\psi}_1 \gamma_5 \psi_2 ~.
\end{align}
The full Lagrangian in the parity doublet model is:
\begin{align}
\mathscr{L}_B &= \bar{\psi}_{1} \imath \slashed \partial \psi_{1} -\frac{1}{2} \hat{g}_1
\bar{\psi}_{1}
\left( \sigma + \imath \gamma_5 \vec{\tau} \cdot \vec{\pi} \right) \psi_{1}
+ \bar{\psi}_{2} \imath \slashed \partial \psi_{2} - \frac{1}{2} \hat{g}_2
\bar{\psi}_{2}
\left( \sigma - \imath \gamma_5 \vec{\tau} \cdot \vec{\pi} \right) \psi_{2} \nonumber \\
&- g_\omega^{(1)}\bar{\psi}_{1} \imath \gamma_0 \omega_0 \psi_{1}
- g_\omega^{(2)}\bar{\psi}_{2} \imath \gamma_0 \omega_0 \psi_{2}
+ m_0 \left( \bar{\psi}_{2} \gamma_5 \psi_{1} - \bar{\psi}_{1} \gamma_5 \psi_{2} \right)
+\mathscr{L}_M ~.
\end{align}
The term proportional to $m_0$ generates an additional contribution to the mass of the nucleons as
well as a mixing between $\psi_1$ and $\psi_2$. This leads to the aforementioned difference between
the chiral and mass eigenstates. In fact in order to obtain the physical masses of the nucleons,
$\psi_1$ and $\psi_2$ have to be transformed according to the following transformations:
\begin{align}
\left(\begin{array}{c}
N^* \\
N \\
\end{array} \right) = \frac{1}{\sqrt{2 \text{cosh} \delta } }
\left(\begin{array}{cc}
\exp(\delta/2) & \gamma_5 \exp(-\delta/2) \\
\gamma_5 \exp(-\delta/2) & -\exp(\delta/2) \\
\end{array} \right)
\left(\begin{array}{c}
\psi_1 \\
\psi_2 \\
\end{array} \right) ~.
\end{align}
Resulting from this transformation the now physical masses of the $N^*$ and $N$ resonances read:
\begin{align}
m_N &=
\frac{1}{2}\sqrt{\left( \frac{1}{2}\hat{g}_1 + \frac{1}{2}\hat{g}_2\right)^2 \varphi^2 + 4 m_0^2}
+ \frac{1}{2}\left( \frac{1}{2}\hat{g}_1 - \frac{1}{2}\hat{g}_2\right) \varphi ~, \nonumber\\
m_{N^*} &=
\frac{1}{2}\sqrt{\left( \frac{1}{2}\hat{g}_1 + \frac{1}{2}\hat{g}_2\right)^2 \varphi^2 + 4 m_0^2}
- \frac{1}{2}\left( \frac{1}{2}\hat{g}_1 - \frac{1}{2}\hat{g}_2\right) \varphi ~.
\end{align}
From the equations for the nucleon masses, the relevance of the parameter $m_0$ is clear. It allows
for a mass of the nucleon even if the chiral symmetry is restored, $\varphi = 0$. The chiral
condensate is not solely responsible for the mass of the baryons, but generates the mass splitting
between the nucleon and its chiral partner. Sending the parameter $m_0 \rightarrow 0$ one reobtains
the naive assignment, and the masses are generated only by the chiral condensate.
The parity doublet model combines the linear sigma model with the nucleon $N$ and it's chiral
partner $N^*$. In the mean field approximation \cite{Serot:1984ey} the Lagrangian reduces to
the potential:
\begin{align}
\mathscr{V}_M = - \frac{1}{2} m^2 \varphi^2 - \frac{1}{2}m_\omega^2 \bar{\omega}_0^2 +
\frac{\lambda}{4}\varphi^4 - \epsilon \varphi~,
\end{align}
and the grand canonical potential in the no see approximation reads:
\begin{align}
\Omega/V = \mathscr{V}_M + \sum_j \frac{\gamma_j}{(2 \pi)^3}
\int_{-\infty}^\infty d^3k \left( E_j(k) - \mu_j^* \right) \theta \left(E_j(k) - \mu_j^* \right)~,
\end{align}
where the sum runs over the resonances $N$ and $N^*$ with the corresponding fermionic degeneracy
factor $\gamma_j$. The energy has the known form $E_j = \sqrt{k^2 + m_j^2}$. The $\theta$ function
requires the chemical potential to be $\mu_j^* = \mu_j - g_\omega \bar{\omega}_0 = \sqrt{k^2 +
m_j^2}$. Minimizing $\Omega$ with respect to $\varphi$ and $\bar{\omega}_0$ leads to the nonzero
density behavior. This model has been studied and extended in Refs.\
\cite{Gallas:2009qp,Gallas:2011qp}.
Following the promising results in the above mentioned publications, the model has to be tested for
inhomogeneous condensation. A straightforward approach is the CDW, which can be parameterized in the
following way \cite{Migdal:1978az}:
\begin{align}
\left\langle \sigma \right\rangle = \varphi \cos (2 f x) ~, ~~~~
\left\langle \pi_0 \right\rangle = \varphi \sin (2 f x) ~, ~~~ \text{where $x$ is a spatial
coordinate.}
\end{align}
In the limit $f \rightarrow 0$ the v.e.v. is a constant equal to $\varphi$, and the ordinary parity
doublet model is realized. Inserting this Ansatz the Lagrangian for the baryons as well as for the
mesons changes. For the potential $\mathscr{V}_M$ a further contribution $2 f^2 \varphi^2$ follows.
The contribution arises from the kinetic term for the pions and sigma. As expected, this term
suppresses the formation of a CDW. Due to the fact that the CDW is suppressed in the vacuum the
parameters remain the same as in the case without CDW. Also the explicit symmetry breaking term get
slightly modified at finite baryon chemical potential.
For the baryons the modifications are more demanding. The Lagrangian $\mathscr{L}_B$ has now a
explicit coordinate space dependency. By transforming the fields $\psi_1$ and $\psi_2$
as \cite{Ebert:2011rg}:
\begin{align}
\bar{\psi}_{1} &\rightarrow \bar{\psi}_{1} \exp[-\imath \gamma_5 \tau_3 f x] ~,
\psi_1 \rightarrow \exp[-\imath \gamma_5 \tau_3 f x] \psi_1 ~, \\
\bar{\psi}_{2} &\rightarrow \bar{\psi}_{2} \exp[+\imath \gamma_5 \tau_3 f x] ~,
\psi_2 \rightarrow \exp[+\imath \gamma_5 \tau_3 f x] \psi_2 ~,
\end{align}
the Lagrangian $\mathscr{L}_B$ takes the following form:
\begin{align}
\mathscr{L}_B
=& \bar{\psi}_{1} \imath \slashed \partial \psi_{1}
+ \bar{\psi}_{1} \gamma_1 \gamma_5 \tau_3 f\psi_{1}
+ \bar{\psi}_{2} \imath \slashed \partial \psi_{2}
- \bar{\psi}_{2} \gamma_1 \gamma_5 \tau_3 f\psi_{2}
\nonumber \\
&- \frac{1}{2} \hat{g}_1 \varphi \bar{\psi}_{1} \psi_{1} - \frac{1}{2} \hat{g}_2 \varphi \bar{\psi}_{2} \psi_{2}
+ m_0 \left( \bar{\psi}_{2} \gamma_5 \psi_{1} -\bar{\psi}_{1} \gamma_5 \psi_{2} \right)~.
\end{align}
For illustrative purpose the additional $\bar{\omega}_0$ dependency and baryon chemical potential
$\mu$ is not shown, but it is simple to generalize the expressions. The diagonalization of the
Lagrangian leads to a splitting of the energy eigenstates. In the presence of the CDW four different
energy eigenstates emerge, in contrary to the homogeneous case where only $N$ and $N^*$ are present.
The different states can be written down in the general form $E_k(p) = \sqrt{p^2 +
\bar{m}_k(p_1)^2}$, $k = 1\dots4$, where $\bar{m}_k(p_1)^2$ is a momentum dependent mass which has
to be calculated numerically. The grand canonical potential is given by:
\begin{align}
\Omega/V
&= 2f^2\varphi^2 +\frac{1}{4}\lambda\varphi^4 -\frac{1}{2}m^2\varphi^2- \epsilon\varphi
-\frac{1}{2}m_\omega^2\bar{\omega}_0^2 \nonumber \\
&+\sum_{k=1}^4 \frac{2}{(2 \pi)^2} \int_{-\infty}^\infty dp_1~
\Theta \left[\mu^*- E_k(p_1)\right] \frac{1}{6} \left[3 \mu^* E_k(p_1)^2 - (\mu^*)^3 -2 E_k(p_1)^3
\right] ~,
\end{align}
with $\mu^* = \mu-g_\omega \bar{\omega}_0$. The behavior at nonzero baryon chemical potential can be
found by extremizing the grand canonical potential with respect to the dynamical degrees of
freedom:
\begin{align}
0 \stackrel{!}{=} \frac{\partial (\Omega/V)}{\partial \bar{\omega}_0} ~, ~~
0 \stackrel{!}{=} \frac{\partial (\Omega/V)}{\partial \varphi} ~,~~
0 \stackrel{!}{=} \frac{\partial (\Omega/V)}{\partial f} ~.
\end{align}
The parameters $g_\omega$, $\lambda$, $m$, $\epsilon$ and $m_\omega$ are fixed using the known
vacuum values for the masses of the pions and omega, as well the pion decay constant ($2 \lambda
\varphi^2 = m_\sigma^2 - m_\pi^2$, $2 m^2 = m_\sigma^2 - 3 m_\pi^2$, $\epsilon = m_\pi^2 f_\pi $)
and the conditions for nuclear matter saturation:
\begin{align}
\frac{E}{A} - m_N = -16 ~\text{MeV},~~\text{and} ~~\rho_0 = 0.16 ~\frac{1}{\text{fm}^3} ~.
\end{align}
The first equation ensures a binding energy per nucleon and the second the density at the minimum
(it corresponds to the baryon chemical potential of $\mu = 923 ~\text{MeV}$). The parameter $m_0$
remains as a input parameter. Using in addition the masses for $N$ and $N^*$ the parameters
$\hat{g}_1$ and $\hat{g}_2$ can be fixed.
\section{Results}
The phase diagram of the parity double model shows a rich structure at dense hadronic matter. Since
the model is based on Ref.\ \cite{Zschiesche:2006zj} we use their parameters as a starting
point. For different values of the mass $m_{N^*}$ and different values for the parameter $m_0$ the
chiral condensate $\varphi$ is calculated as a function of the baryon chemical potential $\mu$. A
feature of the model is to achieve nuclear saturation. For all combinations of the values for
$m_{N^*}$ and $m_0$ the parameters $m$ and $g_\omega$ are tuned in a way to fullfill the
conditions for nuclear saturation at a baryon chemical potential of $\mu = 923 ~\text{MeV}$. The
results are shown in Fig.\ 1. On the left hand side $m_{N^*}$ is low, $m_{N^*} = 1200
~\text{MeV}$ and on the right hand side is high, $m_{N^*} = 1500 ~\text{MeV}$. For both cases the
parameter $m_0$ is varied from $600 ~\text{MeV}$ to $700 ~\text{MeV}$ and finally to $800
~\text{MeV}$. A general result for all parameter sets for the behavior of $\varphi$ is the
presence of three different phases.
For small $\mu$ the chiral symmetry is broken and the value for the condensate $\varphi$ remains
constant at the vacuum value $f_\pi$. For an intermediate range for $\mu$ in the order of $\mu = 923
~\text{MeV}$ one obtains a first order phase transition. The value for the condensate $\varphi$
drops from the vacuum value $f_\pi$ to $30 - 60 ~\text{MeV}$, where the ground state is
still homogeneous and the CDW is not favored. In this range nuclear matter is formed. It shows that
for the formation of homogeneous nuclear matter the mass of the pion plays a crucial role. Sending
$m_\pi \rightarrow 0$, the CDW seems to be always the favored ground state. This is unaffected by
the choice of $m_0$. For large $\mu$ a second first order phase transition occurs. The CDW is now
always the favored ground state. Also for asymptotic large values for $\mu$ chiral symmetry will
not be restored and the CDW remains the favored state of matter.
For $m_{N^*} = 1500 ~\text{MeV}$ (left panel of Fig.\ 1) the second phase transition shows a
strong dependence on the parameter $m_0$. For small values of $m_0$ the transition happens at large
$\mu$ and for a larger $m_0$ the intermediate homogeneous ground state shrinks to a small range of
$\mu$. The value for $\varphi$ after the second phase transition also displays a dependence on
$m_0$, still the overall behavior is the same.
For $m_{N^*} = 1200 ~\text{MeV}$ (right panel of Fig.\ 1) the situation is slightly different: the
behavior of $\varphi$ shows no $m_0$ dependency after the second phase transition. Furthermore for
$m_0 = 600 ~\text{MeV}$ and $m_0 = 700 ~\text{MeV}$ the critical $\mu$ is almost the same.
\begin{figure
\includegraphics[scale=0.58] {sigma_vs_mu_m0_Ns_1500_legend.eps}
\includegraphics[scale=0.58] {sigma_vs_mu_m0_Ns_1200_legend.eps}
\caption{The chiral condensate $\varphi$ as a function of the baryon chemical potential $\mu$ for
different values of $m_0$ and $m_{N^*}$. On the left panel with a large mass for the
chiral partner of the nucleon $m_{N^*}=1500 ~\text{MeV}$ and on the right panel a small mass
$m_{N^*}=1200 ~\text{MeV}$. The red color indicates the ground state to be homogeneous, while the
green that the CDW is favored.}%
\label{sigma_of_mu}
\end{figure}
Another interesting point is the dispersion relation in the inhomogeneous phase, see Fig. 2
left panel. It shows the that the spectrum splits from two energy levels in the homogeneous phase
to four in the inhomogeneous one. Moreover, for the lower two the energy decreases with increasing
momentum and the upper two increase with increasing momentum. The parameters are the same as in the
case for $m_0 = 800 ~\text{MeV}$ and $\mu$ is slightly larger than the critical value for the
transition to the CDW. Using the same parameters as in the left plot of Fig.\ 2 the relative
densities are shown. Within the homogeneous region the transition to nuclear matter is found.
\begin{figure
\includegraphics[scale=0.69] {mu1000_m2_p3_v3_sq.eps}
\includegraphics[scale=0.69] {relative_densities_v2.eps}
\caption{The left plot shows the dispersion relation for the CDW. The spectrum shows four
different energy levels in contrast to the homogeneous ones. The right shows the relative density of
the states at finite $\mu$.}%
\label{dis_rel}
\end{figure}
\section{Summary}
Inhomogeneous condensation is a relevant feature of the parity doublet model coupled to the linear
sigma model with vector mesons. In a first step it has been shown that for $m_\pi = 139
~\text{MeV}$ homogeneous nuclear matter is obtained. At a large baryon chemical potential the
CDW is the favored state of matter. Sending the pion mass to zero (chiral limit) the picture
changes. It is now no longer possible to find a homogeneous ground state. Furthermore, in both cases
chiral symmetry does not get restored at high baryon chemical potential and the chiral condensate
even increases for increasing density.
For the future the model has to be further extended. The CDW is just one of many different
realizations of inhomogeneous condensation. A more general ansatz should be used
\cite{Schnetz:2004vr}. The extended linear sigma model (eL$\sigma$m)
\cite{Parganlija:2012fy} has proven to be robust in describing vacuum
phenomenology. Therefore combining both the parity doublet and the eL$\sigma$m would allow to
constrain the existents of CDW. Also the effect to the onset of the CDW regime of exotic matter like
tetraquarks or glueballs \cite{Gallas:2011qp} has to be
tested in a more general framework.
\section{Acknowledgments}
The authors acknowledges support from the Helmholtz Research School on Quark Matter Studies and
thanks D. H. Rischke, M. Wagner and F. Giacosa for valuable discussion.
| {
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{"url":"https:\/\/math.stackexchange.com\/questions\/3068598\/if-a-is-a-symmetric-square-matrix-i-need-to-show-that-it-is-positive-definite-o","text":"If A is a symmetric square matrix. I need to show that it is positive definite only if all eigenvalues are positive.\n\nI understand that a positive definite matrix by the definition is a symmetric matrix where all eigenvalues are positive. I also know that if $$(x,y) = {x^T}{\\cdotp}M{\\cdotp}y$$ then it is positive definite if $${x^T}{\\cdotp}M{\\cdotp}x {\\geq} 0$$ and $${x^T}{\\cdotp}M{\\cdotp}x = 0$$ only if $$x=\\vec{0}$$.\n\nNow I believe I need to go about this by first proving that all eigenvalues of A are positive using $$Ax = {\\lambda}x$$ where x is a real eigenvector and $$\\lambda$$ is a real eigenvalue. After I've done this I believe I need to then answer the question and prove that it is positive definite only if all eigenvalues are positive, but I'm not entirely sure how to do the second part.\n\n\u2022 Sorry i don't know the vector notation in latex, also i'll edit the question a bit sorry \u2013\u00a0L G Jan 10 at 12:56\n\u2022 The definition is not what you say it is! (If that were the definition then there'd be nothing to prove - what you say you want to prove would reduce to $A$ is positive definite if and only if $A$ is positive definite.) \u2013\u00a0David C. Ullrich Jan 10 at 14:32\n\u2022 Obligaroty remark: for beginner questions about positive definiteness, always remember to state the definition of positive definite matrix, because different authors define positive definite matrix differently. E.g. in one old textbook that I've read, a real symmetric matrix $A$ is said to be positive definite if $A=P^TDP$ for some nonsingular matrix $P$ and some positive diagonal matrix $D$. \u2013\u00a0user1551 Jan 11 at 13:25\n\nThat's not quite right. A symmetric real matrix is said to be positive definite if $$x^TMx$$ is positive whenever $$x\\neq\\vec 0.$$ Nothing is said, here, about eigenvalues.\nNow, suppose that $$\\lambda$$ is some eigenvalue of $$M,$$ meaning that there is some vector $$x$$ with $$x\\ne\\vec 0$$ such that $$Mx=\\lambda x.$$ What can you say about $$x^TMx$$? How can you rewrite $$x^TMx$$? What does this let you say about $$\\lambda$$?\n\u2022 Would i be right in introducing a diagonal matrix (let's call it D) and an orthogonal matrix (let's say B) and then saying how $M = BD{B^{-1}}$ \u2013\u00a0L G Jan 10 at 13:06\n\u2022 @LG You certainly could, but there's no need. We can immediately use the definitions to conclude that $x^TMx$ is positive and that $x^TMX=x^T\\lambda x,$ and since $\\lambda$ is a scalar, then $x^TMx=\\lambda x^Tx.$ Can you take it from there? \u2013\u00a0Cameron Buie Jan 10 at 13:32\n\u2022 I think i understand, since ${x^T}Mx$ is positive then ${\\lambda}{x^T}x$ is also positive and since ${x^T}x$ is positive (as ${||x||^2}$) then $\\lambda$ must also be positive \u2013\u00a0L G Jan 10 at 13:32","date":"2019-01-21 20:42:05","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\": 16, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.936882495880127, \"perplexity\": 107.4122184606139}, \"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-2019-04\/segments\/1547583807724.75\/warc\/CC-MAIN-20190121193154-20190121215154-00360.warc.gz\"}"} | null | null |
Home » Italy » Gianna Manzini » Lettera all'editore (Game Plan for a Novel)
Gianna Manzini: Lettera all'editore (Game Plan for a Novel)
Both Manzini and Anna Banti have been called the Italian Virginia Woolf. This book is certainly more Woolfian than, say, Artemisia (Artemisia). Though there is a story – there are in fact two stories – it is her impressions and her feelings, as with Woolf, rather than the plot that matter. For example, her description of a simple bowl of fruit immediately recalls a Cézanne bowl of fruit.
There are two stories. The first is of a couple and their difficult relationship, about which the author is writing. The second story is, as the title implies, about the letters the author is sending to her publisher about her problems in writing the novel (both personal and artistic). She mixes in, not only these two stories and her impressions and feelings but also how she, as the author, sees her characters which, of course, recalls Pirandello, though, of course, his style is very different.
This is not an easy book, just as Woolf's novels are not easy and it needs to be read more than once to get to grips with it. Fortunately, it has finally become available in English translation (only sixty-two years after it was first published in Italian) and, while Woolf's books remain firmly in print in England, this novel tends to go in and out of print in Italy. But it is as important to Italian literature as Woolf is to English literature.
First published in Italian 1946 by Mondadori
First English translation by italica Press in 2008 | {
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The Problem of Good: Is Christianity Necessary?
Nathaniel Schmucker | The Dartmouth Apologia | Fall 2013
Many are familiar with the so-called Problem of Evil. For ages, mankind has observed evil things that happen in this world—hurricanes flooding coastal cities, tornadoes destroying innocent towns, gunmen massacring schoolchildren, bombers turning heroic marathons into national tragedies—and people have asked themselves whether God is really good or if he exists at all. Christian scholars have developed clear responses to the Problem of Evil and have shown that earthly evil is not inconsistent with the concept of a good God.[i]
This essay, however, does not repeat those arguments but instead considers a parallel question, the Problem of Good. Just as people have taken the presence of evil as a reason to question the validity of Christianity, so also people have seen the presence of good as inconsistent with Christianity's message. The skeptic asks: Why do non-Christians do good deeds? Why do they prosper, succeed, and at times act morally superior to their Christian neighbors? Is Christianity really necessary for good behavior? Similarly, many ask: Don't all good people go to heaven? If I can be good and can go to heaven without the nuisance of attending church, praying, and paying tithes, why am I told that I must obey Christian dogma?
For the skeptic to ask these questions is reasonable, for it is undeniable that people can be good without the Christian God. We need look no further than the example given by Mahatma Gandhi, one of the most famous Hindus, who led a large, non-violent campaign for freedom and civil rights in India. Gandhi suffered persecution from the government and spent many months in prison, but he never abandoned his creed of non-violence and truthfulness. His actions brought lasting change to Indian politics and inspired many others to follow his example of peaceful protest.
Similarly, it is undeniable that professing Christians are not always as good as their neighbors and that atrocities have been committed in the name of Christianity. On the broadest scale, Christians have launched wars, crusades, and inquisitions to spread their message. On a smaller scale, churches have sparked disputes in their communities. Individual Christians at times lie, cheat, steal, act selfishly, and grow angry over petty issues.
Thus, one might come to the conclusion that people can be good without Christianity and that Christianity is a failed moral system. If the old adage, "Good people go to heaven; bad people go to hell" holds true, then the logical conclusion is that we can enter heaven without having to take on the label "Christian." The doctrines, catechisms, dogmas, practices, and traditions of Christianity seem superfluous, and life would be much simpler if people abandoned Christianity altogether to seek an easier alternative to moral behavior and eternal life.
Christianity throughout the ages has rejected this line of thinking as a misunderstanding. Unfortunately, this misunderstood version of Christianity has become increasingly common in contemporary America with the rise of what sociologists Christian Smith and Melinda Lundquist Denton call Moralistic Therapeutic Deism. As we shall see, biblical Christianity, as opposed to its Moralistic Therapeutic Deist counterpart, is a religion whose foundation is about identity rather than thought, emotion, or action.
Christianity Affirms the Skeptic's Initial Questions
Christianity readily affirms that non-Christians can do good works and that Christians sometimes sin egregiously. The Christian explanation of why the secular world prospers is part of the doctrine of common grace. God's common grace is an unmerited blessing that he bestows on all people, whether Christian, Muslim, Hindu, atheist, or agnostic. This grace has three functions: it gives blessings, it restrains evil in the world, and it enables all people to do good.[ii]
In its first sense, common grace means that God gives life, joy, success, talents, and other blessings to all people. Christianity holds that God created and actively sustains the entire universe and that thus it is only natural that God bestows relationships, love, capacity for education, prosperity, and other gifts to the Christian and the non-Christian alike. As the book of Matthew says, God "makes his sun rise on the evil and on the good, and sends rain on the just and on the unjust."[iii] It is God who causes the rain to fall on the fields of both the Christian and the non-Christian farmer. Although specific gifts and blessings may differ in magnitude from person to person, by no means are they limited to only Christians.
Secondly, common grace restrains sin in people. In his letter to the Romans, the Apostle Paul says that all people have a "debased mind to do what ought not to be done. They were filled with all manner of unrighteousness, evil, covetousness, malice. They are full of envy, murder, strife, deceit, maliciousness. They are gossips, slanderers, haters of God, insolent, haughty, boastful, inventors of evil, disobedient to parents, foolish, faithless, heartless, ruthless."[iv] Christians believe that God's common grace prevents people from acting with such evil all of the time. Even if the skeptic does not believe the sinful state of human nature described by Paul, to think that people could easily be much worse than they presently are is no stretch of the imagination.
The third aspect of common grace, which is closely paired with the second, is that all people can pursue the good. If the second aspect is 'negative' in that God restrains evil, this third is 'positive' in that God enables all people to do good things. A non-Christian policeman is just as capable as a Christian policeman of giving his life to protect his city, and all children, regardless of religion, could share their lunches with a classmate who forgot hers. Goodness is not reserved for Christians alone. Common grace, then, provides the basis for the Christian understanding of why non-Christians do good deeds and prosper.
Consider next the question of why Christians sin. The Christian explanation for why believers sometimes act in unholy ways lies in the doctrines of human nature and of sanctification. As we have already seen in the book of Romans, and as the church has affirmed throughout history, human nature is broken. As St. Augustine writes in the Enchiridion, mankind before salvation is either ignorant of the Scriptures and "lives according to the flesh [i.e. a sinful life] with no restraint of reason" or is aware of the written law of God, in which state "even if he wishes to live according to the law [of God], he is vanquished—man sins knowingly and is brought under the spell and made the slave of sin."[v] Put another way, in man's natural state, he is non posse non peccare—not able not to sin.
After salvation, man does not instantaneously become perfect and cease sinning. The rest of his life is a long process of sanctification—of growing and maturing in faith. In St. Augustine's words, "although there is still in man a power that fights against him—his infirmity being not yet fully healed—yet the righteous man lives by faith and lives righteously in so far as he does not yield to evil desires."[vi] At salvation, the state of man changes such that he is posse non peccare—able not to sin. Although Christians are no longer slaves of sin living under its inescapable spell and although they are freed to pursue righteousness, they are not above the influence of sin.[vii] The Christian does not always act morally, nor is he expected to always do so. This state of man does not excuse immoral behavior, but suffices to show that Christianity does not bestow moral perfection.
Common grace and the sinful state of human nature thus validate the skeptic's initial questions about the Problem of Good. Doing good deeds without being a Christian is possible, and becoming a Christian does not cause people to cease sinning. The skeptic might here deduce that since good people go to heaven and since we can be good without God, Christianity is not necessary.
The Skeptic's Underlying Assumption
The flaw in the skeptic's thinking is that he has misunderstood the purpose of Christianity. Frequently, the assumption behind the skeptic's questions is that Christianity teaches a moral system that promotes good behavior and that this morality gains people entrance into heaven. In recent decades, this view of Christianity has pervaded American culture.
In 2005, Christian Smith and Melinda Lundquist Denton published a landmark sociological study on the religious state of American adolescents. Their book, Soul Searching: The Religious and Spiritual Lives of American Teenagers, draws upon data from 267 face-to-face interviews with teenagers in 45 states and from a five-year study conducted by the University of North Carolina at Chapel Hill.[viii] Smith and Denton show that teenagers and their parents, although professing adherence to various Protestant denominations, the Catholic Church, Judaism, Islam, Mormonism, or other religions, in practice all share the same religious belief called Moralistic Therapeutic Deism (MTD). This finding revolutionized the understanding of the state of religion of America.[ix]
The five basic tenets of this "de facto dominant religion" are:
A God exists who created and orders the world and watches over human life on earth.
God wants people to be good, nice, and fair to each other, as taught in the Bible and by most world religions.
The central life goal is to be happy and to feel good about oneself.
God does not need to be particularly involved in one's life except when God is needed to resolve a problem.
Good people go to heaven when they die.[x]
MTD focuses on encouraging moral behavior. Most Americans do not have a clear understanding of morality beyond the simple statement, being moral is about doing good deeds. Morality is about fulfilling potential, treating others according to the golden rule, and being generally kind, friendly, and non-disruptive to social norms.[xi] Most think that one of religion's primary functions is helping people make good choices but that religion is not necessary for teaching that good behavior. Religion thus becomes non-essential for achieving its chief purpose.[xii] In the words of one female participant in Smith and Denton's study:
Morals play a large part in religion. Morals are good if they're healthy for society. Like Christianity, which is all I know, the values you get from, like, the Ten Commandments. I think every religion is important in its own respect. You know, if you're Muslim, then Islam is the way for you. If you're Jewish, well, that's great too. If you're Christian, well good for you. It's just whatever makes you feel good about you.[xiii]
The participant's last sentence leads us into the second aspect of MTD, its therapeutic benefits. Religion teaches morality in order to "[make] you feel good about you."[xiv] According to Smith and Denton, religion is "about feeling good, happy, secure, at peace."[xv] Church services are not about the corporate worship of an almighty God, but rather are about socializing with people, singing feel-good songs, and finding encouragement to continue the all-important pursuit of morality. On Sunday afternoons, churchgoers volunteer in a soup kitchen, play on the church softball team, and go home feeling satisfied that they have done their good deeds for the week.
Finally, MTD revolves around a belief in the Christian God or some other divine being. MTD is similar to eighteenth-century Deism, a movement that taught that God was a Divine Watchmaker who made the universe and set it in motion but who leaves it alone without interfering in individual affairs.[xvi] In contrast to Deism, however, MTD teaches that God does interfere in the affairs of mankind, but only to help people out when they are stuck. God is "something like a combination Divine Butler and Cosmic Therapist: he is always on call, takes care of any problems that arise, professionally helps his people to feel better about themselves, and does not become too personally involved in the process."[xvii] God helps with the job interview, the hard exam, and the sick child, but he does little else.
Although Moralistic Therapeutic Deism has become the predominant subconscious religion of many Americans who profess to be Christians, its adherents suffer from the common misconception that Christianity is a religion primarily concerned with right thinking, feeling, or doing.[xviii] Many misunderstandings of Christianity are rooted epistemologically in thinking—we must wrestle with and master a set of philosophical assertions. Others are rooted existentially in feeling—we will find ultimate happiness by conquering our emotions. Still others are rooted pragmatically in doing—we must live and act a certain way. Most misconceptions of Christianity in America are not purely about thinking, feeling, or doing but are some combination of the three.[xix]
MTD reduces Christianity to a combination of feeling and doing; right thoughts are relatively unimportant. Smith and Denton find that most Americans are incredibly inarticulate about their religious beliefs. When asked what they believed and why it mattered, most of the adolescents interviewed could not name any specific beliefs or could at most string together a few fragments of doctrines.[xx] In the 267 interviews, 13 (4.9%) mentioned obeying God or the church, 12 (4.5%) mentioned repentance, and only 6 (2.2%) mentioned salvation.[xxi]
In contrast, experiencing positive emotions was an important aspect of religion, according to those interviewed. 112 teenagers (41.9%) mentioned personally feeling, being, getting, or being made happy, and 99 teenagers (37.1%) mentioned feeling good about oneself or life. Interviews included the phrase "feel happy" well more than 2000 times.[xxii] The average use of the phrase "feel happy" was higher than the total number of times people mentioned salvation. In its therapeutic aspects, MTD is about creating proper feelings.
MTD is also about encouraging proper behavior. As we have already seen, MTD is inherently moralistic, teaching good behavior and adherence to cultural norms. The Bible is nothing more than the Ten Commandments, the church is nothing more than a community service organization, and Christianity is nothing more than a moral system.
MTD is rarely as dramatic or radical as this essay may make it sound; in reality, MTD is a worldview that operates in the background of people's lives. People do not profess to be Moralistic Therapeutic Deists, but they subconsciously adhere to and affirm its tenets. MTD does not always evidence itself as strongly as some of the examples above, for it tends to be subtle and subconscious, but it is always a system that focuses on right feeling and right doing.
Articulating the Biblical Perspective
In contrast to the principles that MTD proclaims as Christian, the principles that orthodox Christianity has passed down for generations proclaim that Christianity is a religion that concerns being, rather than thinking, feeling, or doing. Christianity has never been about right philosophies. The Pharisees, who were the foremost religious scholars of the first-century world and who studied the Scriptures with more diligence than anyone else, received the harshest rebukes from Jesus.[xxiii] Neither is Christianity about creating good feelings. In fact, the Bible promises that Christians will face emotional hardship: "all who desire to live a godly life in Christ Jesus will be persecuted."[xxiv] Neither is Christianity about right actions. When the rich, young ruler approached Christ to ask what he must do to enter the kingdom of heaven, boasting that he had obeyed the Ten Commandments since he was a child, Christ said that actions alone do not merit salvation.[xxv]
Christianity's root is not in thought, emotion, or action, but in identity. When the Bible speaks of salvation and of entrance into heaven, it does so in ontological terms.[xxvi] In the book of Romans, Paul writes that we have an identity either in Adam or in Christ.[xxvii] Through Adam's sin, all people have entered into the state described above by St. Augustine in which we are "brought under the spell and made the slave of sin."[xxviii] In salvation, God restores our sinful natures, changing who we fundamentally are. Our identity transforms us from being in Adam to being in Christ, from having minds that are focused on earthly desires to minds that are focused on God. We are made into "a new creation," for God says, "I will give you a new heart, and a new spirit I will put within you. And I will remove the heart of stone from your flesh and give you a heart of flesh."[xxix] Neither our thoughts, nor our emotions, nor our actions alone can achieve this transformation of our natures, for it requires God's supernatural work.
As the natural consequence of this new being, the Christian pursues right thinking, feeling, and doing. In the words of St. Augustine, the Christian is no longer enslaved to sinful desires but is gradually "conquering them by his love of righteousness."[xxx] The desires and affections of his heart are now pointed heavenwards. The process of sanctification begins, and the Christian is freed to pursue the good and to grow in holiness.
Salvation's ontological root shapes our understanding of the skeptic's question: Since all good people go to heaven, and I can be good without Christianity, why should I bother with it? We should bother with Christianity because it is not simply about encouraging good behavior or making us feel good. While the skeptic is right in thinking that morality is a part of Christianity, he mistakes it for being Christianity's chief purpose. Instead, our being in Christ through God's renewal of our natures allows us to enter heaven, and although we might pursue morality and happiness outside of Christianity, we will not find the necessary change of being through any other worldview or religion.
The adage that good people go to heaven and bad people go to hell still holds true, but the Bible's definition of good is not the same as the skeptic's. In Jesus' words, "No one is good except God alone."[xxxi] If only God is good, then the goodness of MTD is not good enough. Instead, if our identity lies in Christ, his goodness and holiness cover us and present us blameless before God. When we are new beings in Christ, God looks on us and says, "Well done, good and faithful servant…Enter into the joy of your master."[xxxii]
[i] For one such apologetic, see Chris Hauser's article, "Divine Attributes: Why an Imperfect God Just Won't Do," in the Spring 2013 issue of the Apologia.
[ii] Scott Kauffmann, "The Problem of Good," Q, 10 June 2013, < http://www.qideas.org/essays/the-problem-of-good.aspx>.
[iii] Matthew 5:45 (ESV)
[iv] Romans 1:28-31 (ESV)
[v] St. Augustine was a Christian theologian who lived from 354-430 and whose writings remain very influential in Christian scholarship. Augustine of Hippo, Enchiridion, trans. Albert C. Outler, 10 June 2013, < http://www.tertullian.org/fathers/augustine_enchiridion_02_trans.htm>, 31.118. See also Romans 8:1-8.
[vi] Augustine, Enchiridion, 31.118.
[vii] I John 1:8.
[viii] Christian Smith with Melinda Lundquist Denton, Soul Searching: The Religious and Spiritual Lives of American Teenagers (New York: Oxford University Press, 2005), 6.
[ix] Smith and Denton, Soul Searching, 162.
[x] Smith and Denton, Soul Searching, 162.
[xi] Smith and Denton, Soul Searching, 163.
[xii] Smith and Denton, Soul Searching, 155.
[xiii] Smith and Denton, Soul Searching, 163.
[xiv] Smith and Denton, Soul Searching, 163.
[xv] Smith and Denton, Soul Searching, 164.
[xvi] Smith and Denton, Soul Searching, 165.
[xvii] Smith and Denton, Soul Searching, 165.
[xviii] Michael Ramsden, "Understanding the Root of the Gospel," bethinking, 11 June 2013, <http://www.bethinking.org/other-religions/intermediate/understanding-the-root-of-the-gospel.htm>.
[xix] Ramsden, "Root of the Gospel."
[xx] Smith and Denton, Soul Searching, 131-3.
[xxi] Smith and Denton, Soul Searching, 168.
[xxii] Smith and Denton, Soul Searching, 168.
[xxiii] Matthew 23:1-36.
[xxiv] II Timothy 3:12 (ESV). See also Matthew 5:10.
[xxv] Mark 10:17-27.
[xxvi] Ramsden, "Root of the Gospel."
[xxvii] Romans 5:12-21.
[xxviii] Augustine, Enchiridion, 31.118.
[xxix] II Corinthians 5:17; Ezekiel 36:26 (ESV).
[xxx] Augustine, Enchiridion, 31.118.
[xxxi] Mark 10:18 (ESV).
[xxxii] Matthew 25:23 (ESV).
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{"url":"https:\/\/www.zbmath.org\/serials\/?q=se%3A00002121","text":"# zbMATH \u2014 the first resource for mathematics\n\n## Applied Mathematical Finance\n\n Short Title: Appl. Math. Finance Publisher: Taylor & Francis (Routledge), London ISSN: 1350-486X; 1466-4313\/e Online: http:\/\/www.tandfonline.com\/loi\/ramf20\n Documents Indexed: 475 Publications (since 1994) References Indexed: 426 Publications with 10,237 References.\nall top 5\n\n#### Latest Issues\n\n 27, No. 6 (2020) 27, No. 5 (2020) 27, No. 4 (2020) 27, No. 3 (2020) 27, No. 1-2 (2020) 26, No. 6 (2019) 26, No. 5 (2019) 26, No. 4 (2019) 26, No. 3 (2019) 26, No. 2 (2019) 26, No. 1 (2019) 25, No. 5-6 (2018) 25, No. 4 (2018) 25, No. 3 (2018) 25, No. 2 (2018) 25, No. 1 (2018) 24, No. 5-6 (2017) 24, No. 3-4 (2017) 24, No. 1-2 (2017) 23, No. 5-6 (2016) 23, No. 3-4 (2016) 23, No. 1-2 (2016) 22, No. 5-6 (2015) 22, No. 3-4 (2015) 22, No. 1-2 (2015) 21, No. 5-6 (2014) 21, No. 3-4 (2014) 21, No. 1-2 (2014) 20, No. 5-6 (2013) 20, No. 1-2 (2013) 19, No. 5-6 (2012) 19, No. 3-4 (2012) 19, No. 1-2 (2012) 18, No. 5-6 (2011) 18, No. 3-4 (2011) 18, No. 1-2 (2011) 17, No. 5-6 (2010) 17, No. 3-4 (2010) 17, No. 1-2 (2010) 16, No. 5-6 (2009) 16, No. 3-4 (2009) 16, No. 1-2 (2009) 15, No. 5-6 (2008) 15, No. 3-4 (2008) 15, No. 2 (2008) 15, No. 1 (2008) 14, No. 5 (2007) 14, No. 4 (2007) 14, No. 3 (2007) 14, No. 2 (2007) 14, No. 1 (2007) 13, No. 4 (2006) 13, No. 3 (2006) 13, No. 2 (2006) 13, No. 1 (2006) 12, No. 4 (2005) 12, No. 3 (2005) 12, No. 2 (2005) 12, No. 1 (2005) 11, No. 4 (2004) 11, No. 3 (2004) 11, No. 2 (2004) 11, No. 1 (2004) 10, No. 4 (2003) 10, No. 3 (2003) 10, No. 2 (2003) 10, No. 1 (2003) 9, No. 4 (2002) 9, No. 3 (2002) 9, No. 2 (2002) 9, No. 1 (2002) 8, No. 4 (2001) 8, No. 3 (2001) 8, No. 2 (2001) 8, No. 1 (2001) 7, No. 4 (2000) 7, No. 3 (2000) 7, No. 2 (2000) 7, No. 1 (2000) 6, No. 4 (1999) 6, No. 3 (1999) 6, No. 2 (1999) 6, No. 1 (1999) 5, No. 3 (1998) 5, No. 2 (1998) 5, No. 1 (1998) 4, No. 4 (1997) 4, No. 3 (1997) 4, No. 2 (1997) 4, No. 1 (1997) 3, No. 4 (1996) 3, No. 3 (1996) 3, No. 2 (1996) 3, No. 1 (1996) 2, No. 4 (1995) 2, No. 3 (1995) 2, No. 2 (1995) 2, No. 1 (1995) 1, No. 2 (1994) 1, No. 1 (1994)\nall top 5\n\n#### Authors\n\n 8 Benth, Fred Espen 8 Eberlein, Ernst W. 8 Jaimungal, Sebastian 7 Atkinson, Colin 7 Forsyth, Peter A. 7 Siu, Tak Kuen 6 Avellaneda, Marco 6 Cartea, \u00c1lvaro 6 Chiarella, Carl 5 Cherubini, Umberto 5 Elliott, Robert James 5 Hagan, Patrick S. 5 Howison, Sam D. 5 Madan, Dilip B. 5 Rutkowski, Marek 5 Vetzal, Kenneth R. 5 Zagst, Rudi 4 Goard, Joanna M. 4 Kwok, Yue-Kuen 4 Oosterlee, Cornelis Willebrordus 4 Sircar, Ronnie 3 Baldeaux, Jan 3 Bayraktar, Erhan 3 Bermin, Hans-Peter 3 Caginalp, Gunduz 3 Cheang, Gerald H. L. 3 Donnelly, Ryan 3 Escobar, Marcos 3 Fouque, Jean-Pierre 3 Glau, Kathrin 3 G\u00f6tz, Barbara 3 Gu\u00e9ant, Olivier 3 Jamshidian, Farshid 3 Jonsson, Mattias 3 L\u00e9pinette, Emmanuel 3 Lorig, Matthew J. 3 Matsumoto, Koichi 3 Mokkhavesa, Sutee 3 Ninomiya, Syoiti 3 Papanicolaou, George C. 3 Platen, Eckhard 3 Rebonato, Riccardo 3 Vaugirard, Victor E. 3 Woodward, Diana E. 3 Zheng, Wendong 2 Ahn, Hyungsok 2 Albrecher, Hansj\u00f6rg 2 Almgren, Robert F. 2 Bacinello, Anna Rita 2 Baptiste, Julien 2 Bensoussan, Alain 2 Bouchaud, Jean-Philippe 2 Boyle, Phelim P. 2 Buchen, Peter W. 2 Carr, Peter P. 2 Challet, Damien 2 Chesney, Marc 2 Crouhy, Michel G. 2 Dang, Duy Minh 2 Doust, Paul 2 Duck, Peter W. 2 Ekstr\u00f6m, Erik 2 Ericsson, Jan 2 Fabozzi, Frank J. 2 Figueroa, Marcelo G. 2 Forde, Martin 2 Galai, Dan 2 Gamba, Andrea 2 Geman, H\u00e9lyette 2 Grasselli, Matheus R. 2 Gzyl, Henryk 2 Hughston, Lane P. 2 Ishii, Ryosuke 2 Jackson, Kenneth R. 2 Jacquier, Antoine 2 Johnson, Paul V. 2 Joshi, Mark S. 2 Kallsen, Jan 2 Keller, Joseph Bishop 2 Kennedy, Joanne E. 2 Keppo, Jussi 2 Kingdon, J. 2 Konstandatos, Otto 2 Korn, Ralf 2 Ku, Hyejin 2 Labahn, George 2 Leung, Tim 2 Levendorski\u012d, Serge\u012d Zakharovich 2 Lord, Roger 2 Mai, Jan-Frederik 2 Mancino, Maria Elvira 2 Mayer, Philipp A. 2 Nejad, Sina 2 Ortu, Fulvio 2 Pacelli, Graziella 2 Pag\u00e8s, Gilles 2 Papageorgiou, Evan 2 Para\u2019s, Antonio 2 Perez Arribas, Imanol 2 Peskir, Goran ...and 603 more Authors\nall top 5\n\n#### Fields\n\n 473 Game theory, economics, finance, and other social and behavioral sciences\u00a0(91-XX) 116 Probability theory and stochastic processes\u00a0(60-XX) 36 Statistics\u00a0(62-XX) 28 Numerical analysis\u00a0(65-XX) 17 Partial differential equations\u00a0(35-XX) 17 Systems theory; control\u00a0(93-XX) 15 Operations research, mathematical programming\u00a0(90-XX) 8 Calculus of variations and optimal control; optimization\u00a0(49-XX) 4 Computer science\u00a0(68-XX) 2 Integral equations\u00a0(45-XX) 2 Geophysics\u00a0(86-XX) 1 Linear and multilinear algebra; matrix theory\u00a0(15-XX) 1 Real functions\u00a0(26-XX) 1 Measure and integration\u00a0(28-XX) 1 Approximations and expansions\u00a0(41-XX) 1 Integral transforms, operational calculus\u00a0(44-XX) 1 Biology and other natural sciences\u00a0(92-XX) 1 Information and communication theory, circuits\u00a0(94-XX)\n\n#### Citations contained in zbMATH Open\n\n333 Publications have been cited 2,634 times in 2,087 Documents Cited by Year\nOptimal execution with nonlinear impact functions and trading-enchanced risk.\u00a0Zbl\u00a01064.91058\nAlmgren, Robert F.\n2003\nCalibrating volatility surfaces via relative-entropy minimization.\u00a0Zbl\u00a01007.91015\nAvellaneda, Marco; Friedman, Craig; Holmes, Richard; Samperi, Dominick\n1997\nMultigrid for American option pricing with stochastic volatility.\u00a0Zbl\u00a01009.91034\nClarke, Nigel; Parrott, Kevin\n1999\nPricing in electricity markets: a mean reverting jump diffusion model with seasonality.\u00a0Zbl\u00a01134.91526\nCartea, \u00c1lvaro; Figueroa, Marcelo G.\n2005\nA non-Gaussian Ornstein-Uhlenbeck process for electricity spot price modeling and derivatives pricing.\u00a0Zbl\u00a01160.91337\nBenth, Fred Espen; Kallsen, Jan; Meyer-Brandis, Thilo\n2007\nVolatility skews and extensions of the Libor market model.\u00a0Zbl\u00a01013.91041\nAndersen, Leif; Andreasen, Jesper\n2000\nAnalysis of Fourier transform valuation formulas and applications.\u00a0Zbl\u00a01233.91267\nEberlein, Ernst; Glau, Kathrin; Papapantoleon, Antonis\n2010\nPricing volatility swaps under Heston\u2019s stochastic volatility model with regime switching.\u00a0Zbl\u00a01281.91161\nElliott, Robert J.; Siu, Tak Kuen; Chan, Leunglung\n2007\nWeak approximation of stochastic differential equations and application to derivative pricing.\u00a0Zbl\u00a01134.91524\nNinomiya, Syoiti; Victoir, Nicolas\n2008\nGeneral Black-Scholes models accounting for increased market volatility from hedging strategies.\u00a0Zbl\u00a01009.91023\nSircar, K. Ronnie; Papanicolaou, George\n1998\nManaging the volatility risk of portfolios of derivative securities: The Lagrangian uncertain volatility model.\u00a0Zbl\u00a01097.91514\nAvellaneda, Marco; Para\u2019s, Antonio\n1996\nOptimal basket liquidation for CARA investors is deterministic.\u00a0Zbl\u00a01206.91077\nSchied, Alexander; Sch\u00f6neborn, Torsten; Tehranchi, Michael\n2010\nToward real-time pricing of complex financial derivatives.\u00a0Zbl\u00a01097.91530\nNinomiya, S.; Tezuka, S.\n1996\nOn modelling and pricing weather derivatives.\u00a0Zbl\u00a01013.91036\nAlaton, Peter; Djehiche, Boualem; Stillberger, David\n2002\nOn the pricing and hedging of volatility derivatives.\u00a0Zbl\u00a01108.91316\nHowison, Sam; Rafailidis, Avraam; Rasmussen, Henrik\n2004\nOn Markov-modulated exponential-affine bond price formulae.\u00a0Zbl\u00a01169.91342\nElliott, Robert J.; Siu, Tak Kuen\n2009\nOptimal exercise boundary for an American put option.\u00a0Zbl\u00a01009.91025\nKuske, Rachel A.; Keller, Joseph B.\n1998\nEquivalent Black volatilities.\u00a0Zbl\u00a01009.91033\nHagan, Patrick S.; Woodward, Diana E.\n1999\nAn explicit finite difference approach to the pricing of barrier options.\u00a0Zbl\u00a01009.91022\nBoyle, Phelim P.; Tian, Yisong\n1998\nA finite element approach to the pricing of discrete lookbacks with stochastic volatility.\u00a0Zbl\u00a01009.91030\nForsyth, P. A.; Vetzal, K. R.; Zvan, R.\n1999\nStochastic modelling of temperature variations with a view towards weather derivatives.\u00a0Zbl\u00a01093.91021\nBenth, Fred Espen; \u0160altyt\u0117-Benth, J\u016brat\u0117\n2005\nStochastic volatility effects on defaultable bonds.\u00a0Zbl\u00a01142.91523\nFouque, Jean-Pierre; Sircar, Ronnie; S\u00f8lna, Knut\n2006\nMarkowitz\u2019s mean-variance asset-liability management with regime switching: a multi-period model.\u00a0Zbl\u00a01213.91137\nChen, Ping; Yang, Hailiang\n2011\nPricing asset scheduling flexibility using optimal switching.\u00a0Zbl\u00a01156.91361\nCarmona, Ren\u00e8; Ludkovski, Michael\n2008\nOn American options under the variance gamma process.\u00a0Zbl\u00a01160.91346\nAlmendral, Ariel; Oosterlee, Cornelis W.\n2007\nThe dynamic interaction of speculation and diversification.\u00a0Zbl\u00a01113.91019\nChiarella, Carl; Dieci, Roberto; Gardini, Laura\n2005\nSharp upper and lower bounds for basket options.\u00a0Zbl\u00a01138.91457\nLaurence, Peter; Wang, Tai-Ho\n2005\nPrices and asymptotics for discrete variance swaps.\u00a0Zbl\u00a01396.91718\nBernard, Carole; Cui, Zhenyu\n2014\nOptimal financial portfolios.\u00a0Zbl\u00a01151.91542\nStoyanov, S. V.; Rachev, S. T.; Fabozzi, F. J.\n2007\nMean-variance optimal adaptive execution.\u00a0Zbl\u00a01239.91153\nLorenz, Julian; Almgren, Robert\n2011\nValuing volatility and variance swaps for a non-Gaussian Ornstein-Uhlenbeck stochastic volatility model.\u00a0Zbl\u00a01141.91015\nBenth, Fred Espen; Groth, Martin; Kufakunesu, Rodwell\n2007\nGeneral lower bounds for arithmetic Asian option prices.\u00a0Zbl\u00a01134.91394\nAlbrecher, H.; Mayer, P. A.; Schoutens, W.\n2008\nBinomial models for option valuation \u2013 examining and improving.\u00a0Zbl\u00a01097.91513\n1996\nInterpolation methods for curve construction.\u00a0Zbl\u00a01142.91526\nHagan, Patrick S.; West, Graeme\n2006\nStochastic volatility model with time-dependent skew.\u00a0Zbl\u00a01148.91021\n2005\nBond, futures and option evaluation in the quadratic interest rate model.\u00a0Zbl\u00a01097.91525\nJamshidian, Farshid\n1996\nEnergy futures prices: term structure models with Kalman filter estimation.\u00a0Zbl\u00a01016.91033\nManoliu, Mihaela; Tompaidis, Stathis\n2002\nModelling asset prices for algorithmic and high-frequency trading.\u00a0Zbl\u00a01396.91680\nCartea, \u00c1lvaro; Jaimungal, Sebastian\n2013\nValuing the guaranteed minimum death benefit clause with partial withdrawals.\u00a0Zbl\u00a01189.91066\nB\u00e9langer, A. C.; Forsyth, P. A.; Labahn, G.\n2009\nOptimal quantization for the pricing of swing options.\u00a0Zbl\u00a01169.91337\nBardou, Olivier; Bouthemy, Sandrine; Pag\u00e8s, Gilles\n2009\nConvex hedging in incomplete markets.\u00a0Zbl\u00a01151.91537\nRudloff, Birgit\n2007\nOn the distributional characterization of daily log-returns of a world stock index.\u00a0Zbl\u00a01157.91422\nFergusson, Kevin; Platen, Eckhard\n2006\nA matched asymptotic expansions approach to continuity corrections for discretely sampled options. I: Barrier options.\u00a0Zbl\u00a01281.91166\nHowison, Sam; Steinberg, Mario\n2007\nStochastic volatility, smile & asymptotics.\u00a0Zbl\u00a01009.91032\nSircar, K. Ronnie; Papanicolaou, George C.\n1999\nBivariate option pricing with copulas.\u00a0Zbl\u00a01013.91050\nCherubini, U.; Luciano, E.\n2002\nAsymptotic pricing of commodity derivatives using stochastic volatility spot models.\u00a0Zbl\u00a01156.91374\nHikspoors, Samuel; Jaimungal, Sebastian\n2008\nOn cross-currency models with stochastic volatility and correlated interest rates.\u00a0Zbl\u00a01372.91075\nGrzelak, Lech A.; Oosterlee, Cornelis W.\n2012\nStochastic volatility: option pricing using a multinomial recombining tree.\u00a0Zbl\u00a01134.91372\nFlorescu, Ionu\u0163; Viens, Frederi G.\n2008\nMinimizing coherent risk measures of shortfall in discrete-time models with cone constraints.\u00a0Zbl\u00a01090.91054\nNakano, Yumiharu\n2003\nOption pricing and filtering with hidden Markov-modulated pure-jump processes.\u00a0Zbl\u00a01457.91372\nElliott, Robert J.; Siu, Tak Kuen\n2013\nModelling the temperature time-dependent speed of mean reversion in the context of weather derivatives pricing.\u00a0Zbl\u00a01142.91575\nZapranis, A.; Alexandridis, A.\n2008\nConsistent modelling of VIX and equity derivatives using a $$3\/2$$ plus jumps model.\u00a0Zbl\u00a01395.91429\n2014\nFuzzy measures and asset prices: Accounting for information ambiguity.\u00a0Zbl\u00a01009.91006\nCherubini, Umberto\n1997\nAmerican call options under jump-diffusion processes \u2013 A Fourier transform approach.\u00a0Zbl\u00a01169.91340\nChiarella, Carl; Ziogas, Andrew\n2009\nSmall-time asymptotics for an uncorrelated local-stochastic volatility model.\u00a0Zbl\u00a01246.91129\nForde, Martin; Jacquier, Antoine\n2011\nA note on the Flesaker-Hughston model of the term structure of interest rates.\u00a0Zbl\u00a01009.91020\nRutkowski, Marek\n1997\nADI schemes for pricing American options under the Heston model.\u00a0Zbl\u00a01396.91799\nHaentjens, Tinne; in \u2019t Hout, Karel J.\n2015\nNumerical methods for non-linear Black-Scholes equations.\u00a0Zbl\u00a01229.91339\nHeider, Pascal\n2010\nSimulations of transaction costs and optimal rehedging.\u00a0Zbl\u00a00832.90006\nMohamed, Benjamin\n1994\nMulti-asset portfolio optimization with transaction cost.\u00a0Zbl\u00a01106.91319\nAtkinson, C.; Mokkhavesa, S.\n2004\nPhenomenology of the interest rate curve.\u00a0Zbl\u00a01009.91036\nBouchaud, Jean-Philippe; Sagna, Nicolas; Cont, Rama; El-Karoui, Nicole; Potters, Marc\n1999\nThe use and pricing of convertible bonds.\u00a0Zbl\u00a00876.90022\nNyborg, K. G.\n1996\nVariance-optimal hedging for time-changed L\u00e9vy processes.\u00a0Zbl\u00a01232.91668\nKallsen, Jan; Pauwels, Arnd\n2011\nBoundary values and finite difference methods for the single factor term structure equation.\u00a0Zbl\u00a01179.91247\nEkstr\u00f6m, Erik; L\u00f6tstedt, Per; Tysk, Johan\n2009\nPricing of Parisian options for a jump-diffusion model with two-sided jumps.\u00a0Zbl\u00a01372.91100\nAlbrecher, Hansj\u00f6rg; Kortschak, Dominik; Zhou, Xiaowen\n2012\nNumerical methods and volatility models for valuing cliquet options.\u00a0Zbl\u00a01142.91570\nWindcliff, H. A.; Forsyth, P. A.; Vetzal, K. R.\n2006\nA matched asymptotic expansions approach to continuity corrections for discretely sampled options. II: Bermudan options.\u00a0Zbl\u00a01281.91165\nHowison, Sam\n2007\nMean-semivariance efficient frontier: a downside risk model for portfolio selection.\u00a0Zbl\u00a01113.91018\nBallestero, Enrique\n2005\nA numerical PDE approach for pricing callable bonds.\u00a0Zbl\u00a01026.91046\nD\u2019Halluin, Y.; Forsyth, P. A.; Vetzal, K. R.; Labahn, G.\n2001\nOn arbitrage-free pricing of weather derivatives based on fractional Brownian motion.\u00a0Zbl\u00a01087.91020\nBenth, Fred Espen\n2003\nMultiple time scales in volatility and leverage correlations: a stochastic volatility model.\u00a0Zbl\u00a01093.91537\nPerell\u00f3, Josep; Masoliver, Jaume; Bouchaud, Jean-Philippe\n2004\nRobust barrier option pricing by frame projection under exponential L\u00e9vy dynamics.\u00a0Zbl\u00a01398.91672\nKirkby, J. Lars\n2017\nA new approach to pricing double-barrier options with arbitrary payoffs and exponential boundaries.\u00a0Zbl\u00a01188.91210\nBuchen, Peter; Konstandatos, Otto\n2009\nRobust approximations for pricing Asian options and volatility swaps under stochastic volatility.\u00a0Zbl\u00a01233.91272\nForde, Martin; Jacquier, Antoine\n2010\nComputation of Greeks and multidimensional density estimation for asset price models with time-changed Brownian motion.\u00a0Zbl\u00a01233.91315\nKawai, Reiichiro; Kohatsu-Higa, Arturo\n2010\nThe endogenous price dynamics of emission allowances and an application to CO$$_2$$ option pricing.\u00a0Zbl\u00a01372.91079\nChesney, Marc; Taschini, Luca\n2012\nIndifference pricing and hedging for volatility derivatives.\u00a0Zbl\u00a01213.91152\nGrasselli, M. R.; Hurd, T. R.\n2007\nDelta, gamma and bucket hedging of interest rate derivatives.\u00a0Zbl\u00a00831.90012\nJarrow, Robert A.; Turnbull, Stuart M.\n1994\nVarious passport options and their valuation.\u00a0Zbl\u00a01009.91038\nAhn, Hyungsok; Penaud, Antony; Wilmott, Paul\n1999\nA note on arbitrage-free pricing of forward contracts in energy markets.\u00a0Zbl\u00a01101.91323\nBenth, Fred Espen; Ekeland, Lars; Hauge, Ragnar; Nielsen, Bj\u00f8rn Fredrik\n2003\nPricing of swing options in a mean reverting model with jumps.\u00a0Zbl\u00a01156.91377\nKjaer, Mats\n2008\nA note on the suboptimality of path-dependent pay-offs in L\u00e9vy markets.\u00a0Zbl\u00a01179.91085\nVanduffel, Steven; Chernih, Andrew; Maj, Matheusz; Schoutens, Wim\n2009\nExchange options under jump-diffusion dynamics.\u00a0Zbl\u00a01239.91160\nCheang, Gerald H. L.; Chiarella, Carl\n2011\nThe British put option.\u00a0Zbl\u00a01239.91166\nPeskir, Goran; Samee, Farman\n2011\nThe implied market price of weather risk.\u00a0Zbl\u00a01372.91108\nH\u00e4rdle, Wolfgang Karl; Cabrera, Brenda L\u00f3pez\n2012\nLevel-slope-curvature - fact or artefact?\u00a0Zbl\u00a01160.91334\nLord, Roger; Pelsser, Antoon\n2007\nLaplace transforms and American options.\u00a0Zbl\u00a01031.91053\n2000\nPricing stock and bond derivatives with a multi-factor Gaussian model.\u00a0Zbl\u00a01011.91040\nBajeux-Besnainou, Isabelle; Portait, Roland\n1998\nA new approximate swaption formula in the LIBOR market model: an asymptotic expansion approach.\u00a0Zbl\u00a01072.91016\nKawai, Atsushi\n2003\nEnhancing trading strategies with order book signals.\u00a0Zbl\u00a01418.91454\nCartea, \u00c1lvaro; Donnelly, Ryan; Jaimungal, Sebastian\n2018\nMean variance hedging in a general jump model.\u00a0Zbl\u00a01229.91314\nKohlmann, Michael; Xiong, Dewen; Ye, Zhongxing\n2010\nA simple derivation of and Improvements to Jamshidian\u2019s and Rogers\u2019 upper bound methods for Bermudan options.\u00a0Zbl\u00a01186.91194\nJoshi, Mark S.\n2007\nPassport options with stochastic volatility.\u00a0Zbl\u00a01013.91046\nHenderson, Vicky; Hobson, David\n2001\nOn pricing and reserving with-profits life insurance contracts.\u00a0Zbl\u00a01026.91060\nPrieul, David; Putyatin, Vladislav; Nassar, Tarek\n2001\nMulti-asset barrier options and occupation time derivatives.\u00a0Zbl\u00a01089.91031\nWong, Hoi Ying; Kwok, Yue-Kuen\n2003\nDynamic programming and mean-variance hedging in discrete time.\u00a0Zbl\u00a01088.91030\n\u010cern\u00fd, Ale\u0161\n2004\nContingent claim pricing using probability distortion operators: methods from insurance risk pricing and their relationship to financial theory.\u00a0Zbl\u00a01064.91043\n2003\nComparison between the mean-variance optimal and the mean-quadratic-variation optimal trading strategies.\u00a0Zbl\u00a01396.91705\nTse, S. T.; Forsyth, P. A.; Kennedy, J. S.; Windcliff, H.\n2013\nConvergence of a least-squares Monte Carlo algorithm for bounded approximating sets.\u00a0Zbl\u00a01169.91346\nZanger, Daniel Z.\n2009\nEfficient pricing of derivatives on assets with discrete dividends.\u00a0Zbl\u00a01142.91569\nVellekoop, M. H.; Nieuwenhuis, J. W.\n2006\nLimit order books, diffusion approximations and reflected SPDEs: from microscopic to macroscopic models.\u00a0Zbl\u00a01451.91186\nHambly, Ben; Kalsi, Jasdeep; Newbury, James\n2020\nMean-field game strategies for optimal execution.\u00a0Zbl\u00a01410.91498\nHuang, Xuancheng; Jaimungal, Sebastian; Nourian, Mojtaba\n2019\nOptimal asset allocation for retirement saving: deterministic vs. time consistent adaptive strategies.\u00a0Zbl\u00a01410.91413\nForsyth, Peter A.; Vetzal, Kenneth R.\n2019\nGeneralised Lyapunov functions and functionally generated trading strategies.\u00a0Zbl\u00a01430.91089\nRuf, Johannes; Xie, Kangjianan\n2019\nShort maturity forward start Asian options in local volatility models.\u00a0Zbl\u00a01426.91274\nPirjol, Dan; Wang, Jing; Zhu, Lingjiong\n2019\nDual representation of the cost of designing a portfolio satisfying multiple risk constraints.\u00a0Zbl\u00a01426.91262\nBouveret, G\u00e9raldine\n2019\nDeep reinforcement learning for market making in corporate bonds: beating the curse of dimensionality.\u00a0Zbl\u00a01433.91194\nGu\u00e9ant, Olivier; Manziuk, Iuliia\n2019\nPolynomial processes for power prices.\u00a0Zbl\u00a01433.91104\nWare, Tony\n2019\nEnhancing trading strategies with order book signals.\u00a0Zbl\u00a01418.91454\nCartea, \u00c1lvaro; Donnelly, Ryan; Jaimungal, Sebastian\n2018\nOptimal decisions in a time priority queue.\u00a0Zbl\u00a01418.91465\nDonnelly, Ryan; Gan, Luhui\n2018\nExtended Gini-type measures of risk and variability.\u00a0Zbl\u00a01418.91229\nBerkhouch, Mohammed; Lakhnati, Ghizlane; Righi, Marcelo Brutti\n2018\nA non-Gaussian Ornstein-Uhlenbeck model for pricing wind power futures.\u00a0Zbl\u00a01418.91503\nBenth, Fred Espen; Pircalabu, Anca\n2018\nTransition probability of Brownian motion in the octant and its application to default modelling.\u00a0Zbl\u00a01411.91601\nKaushansky, Vadim; Lipton, Alexander; Reisinger, Christoph\n2018\nPortfolio optimization under fast mean-reverting and rough fractional stochastic environment.\u00a0Zbl\u00a01411.91498\nFouque, Jean-Pierre; Hu, Ruimeng\n2018\nOption pricing in illiquid markets with jumps.\u00a0Zbl\u00a01411.91619\nCruz, Jos\u00e9 M. T. S.; \u0160ev\u010dovic, Daniel\n2018\nOptimal expected-shortfall portfolio selection with copula-induced dependence.\u00a0Zbl\u00a01418.91469\nGijbels, Ir\u00e8ne; Herrmann, Klaus\n2018\nDynamic index tracking and risk exposure control using derivatives.\u00a0Zbl\u00a01418.91522\nLeung, Tim; Ward, Brian\n2018\nReal-world scenarios with negative interest rates based on the LIBOR market model.\u00a0Zbl\u00a01411.91591\nLopes, Sara Dutra; V\u00e1zquez, Carlos\n2018\nThe optimal interaction between a hedge fund manager and investor.\u00a0Zbl\u00a01411.91527\nRamirez, Hugo Eduardo; Johnson, Paul V.; Duck, Peter; Howell, Sydney\n2018\nHybrid L\u00e9vy models: design and computational aspects.\u00a0Zbl\u00a01411.91552\nEberlein, Ernst; Rudmann, Marcus\n2018\nRobust barrier option pricing by frame projection under exponential L\u00e9vy dynamics.\u00a0Zbl\u00a01398.91672\nKirkby, J. Lars\n2017\nA dimension and variance reduction Monte-Carlo method for option pricing under jump-diffusion models.\u00a0Zbl\u00a01398.91669\nDang, Duy-Minh; Jackson, Kenneth R.; Sues, Scott\n2017\nPrice manipulation in a market impact model with dark pool.\u00a0Zbl\u00a01398.91529\nKl\u00f6ck, Florian; Schied, Alexander; Sun, Yuemeng\n2017\nRegime-switching stochastic volatility model: estimation and calibration to VIX options.\u00a0Zbl\u00a01398.91593\nGoutte, St\u00e9phane; Ismail, Amine; Pham, Huy\u00ean\n2017\nOptimal market making.\u00a0Zbl\u00a01398.91520\nGu\u00e9ant, Olivier\n2017\nOptimal accelerated share repurchases.\u00a0Zbl\u00a01398.91600\nJaimungal, S.; Kinzebulatov, D.; Rubisov, D. H.\n2017\nMartingale property of exponential semimartingales: a note on explicit conditions and applications to asset price and Libor models.\u00a0Zbl\u00a01398.91574\nCriens, David; Glau, Kathrin; Grbac, Zorana\n2017\nSharper asset ranking from total drawdown durations.\u00a0Zbl\u00a01398.62286\nChallet, Damien\n2017\nThe affine inflation market models.\u00a0Zbl\u00a01398.91619\nWaldenberger, Stefan\n2017\nModelling stochastic skew of FX options using SLV models with stochastic spot\/vol correlation and correlated jumps.\u00a0Zbl\u00a01398.91671\nItkin, Andrey\n2017\nTwo asset-barrier option under stochastic volatility.\u00a0Zbl\u00a01398.91592\nGoetz, Barbara; Escobar, Marcos; Zagst, Rudi\n2017\nSmall-maturity asymptotics for the at-the-money implied volatility slope in L\u00e9vy models.\u00a0Zbl\u00a01396.91731\nGerhold, Stefan; G\u00fcl\u00fcm, I. Cetin; Pinter, Arpad\n2016\nCounterparty credit exposures for interest rate derivatives using the stochastic grid bundling method.\u00a0Zbl\u00a01396.91741\nKarlsson, Patrik; Jain, Shashi; Oosterlee, Cornelis W.\n2016\nMarket calibration under a long memory stochastic volatility model.\u00a0Zbl\u00a01396.91760\nPosp\u00ed\u0161il, Jan; Sobotka, Tom\u00e1\u0161\n2016\nPricing occupation-time options in a mixed-exponential jump-diffusion model.\u00a0Zbl\u00a01396.91713\nAoudia, Djilali Ait; Renaud, Jean-Fran\u00e7ois\n2016\nPitfalls of the Fourier transform method in affine models, and remedies.\u00a0Zbl\u00a01396.91803\nLevendorski\u012d, Sergei\n2016\nIndifference fee rate for variable annuities.\u00a0Zbl\u00a01396.91295\nChevalier, Etienne; Lim, Thomas; Romero, Ricardo Romo\n2016\nAnalysis of VIX markets with a time-spread portfolio.\u00a0Zbl\u00a01396.91758\nPapanicolaou, A.\n2016\nOptimal prediction of resistance and support levels.\u00a0Zbl\u00a01396.60041\nDe Angelis, T.; Peskir, G.\n2016\nOptimal partial proxy method for computing gammas of financial products with discontinuous and angular payoffs.\u00a0Zbl\u00a01396.91801\nJoshi, Mark S.; Zhu, Dan\n2016\nOn the method of optimal portfolio choice by cost-efficiency.\u00a0Zbl\u00a01396.91702\nR\u00fcschendorf, Ludger; Wolf, Viktor\n2016\nPricing timer options and variance derivatives with closed-form partial transform under the 3\/2 model.\u00a0Zbl\u00a01396.91774\nZheng, Wendong; Zeng, Pingping\n2016\nADI schemes for pricing American options under the Heston model.\u00a0Zbl\u00a01396.91799\nHaentjens, Tinne; in ’t Hout, Karel J.\n2015\nPricing path-dependent options with discrete monitoring under time-changed L\u00e9vy processes.\u00a0Zbl\u00a01396.91769\nUmezawa, Yuji; Yamazaki, Akira\n2015\nImplied volatility of leveraged ETF options.\u00a0Zbl\u00a01396.91748\nLeung, Tim; Sircar, Ronnie\n2015\nRecursive marginal quantization of the Euler scheme of a diffusion process.\u00a0Zbl\u00a01396.91805\nPag\u00e8s, Gilles; Sagna, Abass\n2015\nSemi-Markov model for market microstructure.\u00a0Zbl\u00a01396.91218\nFodra, Pietro; Pham, Huy\u00ean\n2015\nOptimal execution and block trade pricing: a general framework.\u00a0Zbl\u00a01396.91687\nGu\u00e9ant, Olivier\n2015\nDimension and variance reduction for Monte Carlo methods for high-dimensional models in finance.\u00a0Zbl\u00a01396.91798\n2015\nSemi-analytical pricing of currency options in the Heston\/CIR jump-diffusion hybrid model.\u00a0Zbl\u00a01396.91710\nAhlip, Rehez; Rutkowski, Marek\n2015\nCorrection to: \u201cExchange option under jump-diffusion dynamics\u201d.\u00a0Zbl\u00a01406.91435\nCaldana, Ruggero; Cheang, Gerald H. L.; Chiarella, Carl; Fusai, Gianluca\n2015\nPricing of spread options on a bivariate jump market and stability to model risk.\u00a0Zbl\u00a01396.91717\nBenth, Fred Espen; Di Nunno, Giulia; Khedher, Asma; Schmeck, Maren Diane\n2015\nA note on dual-curve construction: Mr. Crab\u2019s bootstrap.\u00a0Zbl\u00a01396.91778\nBaviera, Roberto; Cassaro, Alessandro\n2015\nPricing exotic discrete variance swaps under the 3\/2-stochastic volatility models.\u00a0Zbl\u00a01396.91772\nYuen, Chi Hung; Zheng, Wendong; Kwok, Yue Kuen\n2015\nEffect of volatility clustering on indifference pricing of options by convex risk measures.\u00a0Zbl\u00a01396.91744\nKumar, Rohini\n2015\nA new variance reduction technique for estimating value-at-risk.\u00a0Zbl\u00a01396.91802\nKorn, Ralf; Pupashenko, Mykhailo\n2015\nStochastic models for oil prices and the pricing of futures on oil.\u00a0Zbl\u00a01396.91756\nOud, Mohammed A. Aba; Goard, Joanna\n2015\nThe British lookback option with fixed strike.\u00a0Zbl\u00a01396.91742\nKitapbayev, Yerkin\n2015\nPerpetual exchange options under jump-diffusion dynamics.\u00a0Zbl\u00a01396.91722\nCheang, Gerald H. L.; Lian, Guanghua\n2015\nPrices and asymptotics for discrete variance swaps.\u00a0Zbl\u00a01396.91718\nBernard, Carole; Cui, Zhenyu\n2014\nConsistent modelling of VIX and equity derivatives using a $$3\/2$$ plus jumps model.\u00a0Zbl\u00a01395.91429\n2014\nVariational solutions of the pricing PIDEs for European options in L\u00e9vy models.\u00a0Zbl\u00a01395.91497\nEberlein, Ernst; Glau, Kathrin\n2014\nAn extension of the chaos expansion approximation for the pricing of exotic basket options.\u00a0Zbl\u00a01396.91727\nFunahashi, Hideharu; Kijima, Masaaki\n2014\nOptimal trade execution under stochastic volatility and liquidity.\u00a0Zbl\u00a01395.91398\nCheridito, Patrick; Sepin, Tardu\n2014\nOption pricing with transaction costs and stochastic interest rate.\u00a0Zbl\u00a01395.91466\nSengupta, Indranil\n2014\nOptimal execution and price manipulations in time-varying limit order books.\u00a0Zbl\u00a01395.91394\nAlfonsi, Aur\u00e9lien; Acevedo, Jos\u00e9 Infante\n2014\nA radial basis function scheme for option pricing in exponential L\u00e9vy models.\u00a0Zbl\u00a01395.91433\nBrummelhuis, Raymond; Chan, Ron T. L.\n2014\nStochastic correlation and volatility mean-reversion \u2013 empirical motivation and derivatives pricing via perturbation theory.\u00a0Zbl\u00a01395.91439\nEscobar, Marcos; G\u00f6tz, Barbara; Neykova, Daniela; Zagst, Rudi\n2014\nSaddlepoint approximation methods for pricing derivatives on discrete realized variance.\u00a0Zbl\u00a01396.91773\nZheng, Wendong; Kwok, Yue Kuen\n2014\nClosed-form pricing of two-asset barrier options with stochastic covariance.\u00a0Zbl\u00a01395.91443\nG\u00f6tz, Barbara; Escobar, Marcos; Zagst, Rudi\n2014\nApproximate hedging in a local volatility model with proportional transaction costs.\u00a0Zbl\u00a01395.91455\nL\u00e9pinette, Emmanuel; Tran, Tuan\n2014\nTail VaR measures in a multi-period setting.\u00a0Zbl\u00a01395.91512\nKatsuki, Yuta; Matsumoto, Koichi\n2014\nOn the approximation of the SABR with mean reversion model: a probabilistic approach.\u00a0Zbl\u00a01395.91453\nKennedy, Joanne E.; Pham, Duy\n2014\nImplied filtering densities on the hidden state of stochastic volatility.\u00a0Zbl\u00a01395.91441\nFuertes, Carlos; Papanicolaou, Andrew\n2014\nPerpetual options on multiple underlyings.\u00a0Zbl\u00a01396.91726\nDuck, Peter W.; Evatt, Geoffrey W.; Johnson, Paul V.\n2014\nRare shock, two-factor stochastic volatility and currency option pricing.\u00a0Zbl\u00a01396.91770\nWang, Guanying; Wang, Xingchun; Wang, Yongjin\n2014\nA multivariate default model with spread and event risk.\u00a0Zbl\u00a01396.91792\nMai, Jan-Frederik; Olivares, Pablo; Schenk, Steffen; Scherer, Matthias\n2014\nModelling asset prices for algorithmic and high-frequency trading.\u00a0Zbl\u00a01396.91680\nCartea, \u00c1lvaro; Jaimungal, Sebastian\n2013\nOption pricing and filtering with hidden Markov-modulated pure-jump processes.\u00a0Zbl\u00a01457.91372\nElliott, Robert J.; Siu, Tak Kuen\n2013\nComparison between the mean-variance optimal and the mean-quadratic-variation optimal trading strategies.\u00a0Zbl\u00a01396.91705\nTse, S. T.; Forsyth, P. A.; Kennedy, J. S.; Windcliff, H.\n2013\nStock loans in incomplete markets.\u00a0Zbl\u00a01457.91376\nGrasselli, Matheus R.; G\u00f3mez, Cesar\n2013\nAmerican options in the Heston model with stochastic interest rate and its generalizations.\u00a0Zbl\u00a01457.91367\nBoyarchenko, Svetlana; Levendorski\u012d, Sergei\n2013\nOptimal selling of an asset with jumps under incomplete information.\u00a0Zbl\u00a01396.91694\nLu, Bing\n2013\nA simple stochastic rate model for rate equity hybrid products.\u00a0Zbl\u00a01396.91780\nEberlein, Ernst; Madan, Dilip; Pistorius, Martijn; Yor, Marc\n2013\nJoint modelling of gas and electricity spot prices.\u00a0Zbl\u00a01457.91278\nFrikha, Noufel; Lemaire, Vincent\n2013\nDefault times in a continuous time Markov chain economy.\u00a0Zbl\u00a01396.91821\nElliott, Robert J.; van der Hoek, John\n2013\nPricing and hedging of lookback options in hyper-exponential jump diffusion models.\u00a0Zbl\u00a01396.91736\nHofer, Markus; Mayer, Philipp\n2013\nOn cross-currency models with stochastic volatility and correlated interest rates.\u00a0Zbl\u00a01372.91075\nGrzelak, Lech A.; Oosterlee, Cornelis W.\n2012\nPricing of Parisian options for a jump-diffusion model with two-sided jumps.\u00a0Zbl\u00a01372.91100\nAlbrecher, Hansj\u00f6rg; Kortschak, Dominik; Zhou, Xiaowen\n2012\nThe endogenous price dynamics of emission allowances and an application to CO$$_2$$ option pricing.\u00a0Zbl\u00a01372.91079\nChesney, Marc; Taschini, Luca\n2012\nThe implied market price of weather risk.\u00a0Zbl\u00a01372.91108\nH\u00e4rdle, Wolfgang Karl; Cabrera, Brenda L\u00f3pez\n2012\nStochastic expansion for the pricing of call options with discrete dividends.\u00a0Zbl\u00a01372.91107\n\u00c9tor\u00e9, Pierre; Gobet, Emmanuel\n2012\nPricing fixed-income securities in an information-based framework.\u00a0Zbl\u00a01372.91109\nHughston, Lane P.; Macrina, Andrea\n2012\nThe stochastic intrinsic currency volatility model: a consistent framework for multiple FX rates and their volatilities.\u00a0Zbl\u00a01372.91074\nDoust, Paul\n2012\nOn the approximation of the SABR model: a probabilistic approach.\u00a0Zbl\u00a01373.62520\nKennedy, Joanne E.; Mitra, Subhankar; Pham, Duy\n2012\nThe effect of correlation and transaction costs on the pricing of basket options.\u00a0Zbl\u00a01372.91101\nAtkinson, C.; Ingpochai, P.\n2012\nDynamic portfolio optimization in discrete-time with transaction costs.\u00a0Zbl\u00a01372.91094\nAtkinson, Colin; Quek, Gary\n2012\nOptions on realized variance in log-OU models.\u00a0Zbl\u00a01372.91105\nDrimus, Gabriel G.\n2012\nBonds and options in exponentially affine bond models.\u00a0Zbl\u00a01372.91102\nBermin, Hans-Peter\n2012\nBias reduction for pricing American options by least-squares Monte Carlo.\u00a0Zbl\u00a01372.91118\nKan, Kin Hung (Felix); Reesor, R. Mark\n2012\n...and 233 more Documents\nall top 5\n\n#### Cited by 2,678 Authors\n\n 33 Siu, Tak Kuen 29 Benth, Fred Espen 23 Forsyth, Peter A. 23 Zhu, Songping 22 Elliott, Robert James 19 Jaimungal, Sebastian 16 Cartea, \u00c1lvaro 15 Schied, Alexander 14 Cui, Zhenyu 13 Chiarella, Carl 13 Lorig, Matthew J. 12 Ballestra, Luca Vincenzo 12 Sircar, Ronnie 11 Bernard, Carole 11 Company, Rafael 11 Glau, Kathrin 11 Kim, Jeong-Hoon 11 Oosterlee, Cornelis Willebrordus 11 Vetzal, Kenneth R. 11 Westerhoff, Frank H. 10 Eberlein, Ernst W. 10 Escobar, Marcos 10 J\u00f3dar Sanchez, Lucas Antonio 10 Yang, Hailiang 10 Zagst, Rudi 9 Boyle, Phelim P. 9 Dang, Duy Minh 9 Goard, Joanna M. 9 Kawai, Reiichiro 9 Lian, Guanghua 9 Linetsky, Vadim 9 Muhle-Karbe, Johannes 9 Sch\u00f6neborn, Torsten 8 Alfonsi, Aur\u00e9lien 8 Bayer, Christian 8 Bayraktar, Erhan 8 Dieci, Roberto 8 D\u00fcring, Bertram 8 Ekstr\u00f6m, Erik 8 Fouque, Jean-Pierre 8 Gobet, Emmanuel 8 He, Xinjiang 8 Joshi, Mark S. 8 Li, Shenghong 8 Pacelli, Graziella 8 Pham, Huy\u00ean 8 Soner, Halil Mete 8 Tan, Ken Seng 7 Brody, Dorje C. 7 Chan, Leunglung 7 Ching, Wai-Ki 7 Fabozzi, Frank J. 7 Ferrando, Sebasti\u00e1n Esteban 7 Figueroa-L\u00f3pez, Jos\u00e9 E. 7 Grasselli, Martino 7 Gu\u00e9ant, Olivier 7 Kallsen, Jan 7 Kwok, Yue-Kuen 7 Leung, Tim 7 Li, Zhongfei 7 Mamon, Rogemar S. 7 Pagliarani, Stefano 7 Papapantoleon, Antonis 7 Pascucci, Andrea 7 Platen, Eckhard 7 Sgarra, Carlo 7 Vanduffel, Steven 7 Yoon, Ji-Hun 6 Avellaneda, Marco 6 Bouchard, Bruno 6 Carmona, Ren\u00e9 A. 6 Chen, Wenting 6 Dyshaev, Mikha\u012dl Mikha\u012dlovich 6 Fusai, Gianluca 6 Goutte, St\u00e9phane 6 Hobson, David G. 6 Horst, Ulrich 6 Howison, Sam D. 6 Hughston, Lane P. 6 Ivanov, Roman V. 6 Khaliq, Abdul Q. M. 6 Kim, Geonwoo 6 Laurence, Peter 6 Li, Lingfei 6 Ludkovski, Michael 6 Pag\u00e8s, Gilles 6 R\u00fcschendorf, Ludger 6 Rutkowski, Marek 6 \u0160altyt\u0117-Benth, J\u016brat\u0117 6 Schoenmakers, John G. M. 6 Schoutens, Wim 6 \u0160ev\u010dovi\u010d, Daniel 6 Swishchuk, Anatoliy 6 Vanmaele, Mich\u00e8le 6 V\u00e1zquez Cend\u00f3n, Carlos 6 Wong, Hoi Ying 6 Yamazaki, Akira 6 Yao, Haixiang 6 Zanette, Antonino 6 Zhang, Jin E. ...and 2,578 more Authors\nall top 5\n\n#### Cited in 265 Journals\n\n 216 International Journal of Theoretical and Applied Finance 214 Quantitative Finance 142 Applied Mathematical Finance 81 Insurance Mathematics & Economics 71 Mathematical Finance 62 Journal of Computational and Applied Mathematics 62 European Journal of Operational Research 56 Journal of Economic Dynamics & Control 56 SIAM Journal on Financial Mathematics 52 Finance and Stochastics 33 Applied Mathematics and Computation 33 Review of Derivatives Research 31 Computers & Mathematics with Applications 31 Annals of Operations Research 27 The Annals of Applied Probability 23 Stochastic Processes and their Applications 21 Mathematics and Financial Economics 20 Stochastic Analysis and Applications 20 Annals of Finance 19 International Journal of Computer Mathematics 19 Asia-Pacific Financial Markets 19 Stochastics 15 Journal of Econometrics 15 Mathematical Methods of Operations Research 14 Journal of Mathematical Analysis and Applications 14 Journal of Optimization Theory and Applications 13 Mathematics and Computers in Simulation 13 Discrete Dynamics in Nature and Society 13 North American Actuarial Journal 12 SIAM Journal on Control and Optimization 12 Statistics & Probability Letters 12 Decisions in Economics and Finance 11 Advances in Applied Probability 11 Mathematical Problems in Engineering 11 The ANZIAM Journal 10 Chaos, Solitons and Fractals 10 Applied Mathematics and Optimization 10 Journal of Complexity 10 Methodology and Computing in Applied Probability 10 ASTIN Bulletin 10 Journal of Industrial and Management Optimization 9 Mathematics of Operations Research 9 Scandinavian Actuarial Journal 9 Computational Management Science 8 Journal of Applied Probability 8 Numerische Mathematik 8 Operations Research Letters 8 European Journal of Applied Mathematics 8 Monte Carlo Methods and Applications 8 Bernoulli 8 International Journal of Stochastic Analysis 7 Applied Numerical Mathematics 6 Optimization 6 Computers & Operations Research 6 Japan Journal of Industrial and Applied Mathematics 6 Journal of Applied Statistics 6 Journal of Systems Science and Complexity 5 Physica A 5 Communications in Statistics. Theory and Methods 5 Linear Algebra and its Applications 5 Computational Optimization and Applications 5 Abstract and Applied Analysis 5 Communications in Nonlinear Science and Numerical Simulation 5 Journal of Applied Mathematics 5 East Asian Journal on Applied Mathematics 5 4 Theory of Probability and its Applications 4 Nonlinear Analysis. Theory, Methods & Applications. Series A: Theory and Methods 4 Operations Research 4 Systems & Control Letters 4 Communications in Statistics. Simulation and Computation 4 Journal of Statistical Computation and Simulation 4 Computational Statistics and Data Analysis 4 SIAM Journal on Scientific Computing 4 Applied Stochastic Models in Business and Industry 4 Stochastic Models 4 European Actuarial Journal 4 European Series in Applied and Industrial Mathematics (ESAIM): Proceedings and Surveys 3 Lithuanian Mathematical Journal 3 Mathematics of Computation 3 Automatica 3 BIT 3 Fuzzy Sets and Systems 3 Acta Applicandae Mathematicae 3 Statistics 3 Applied Mathematics Letters 3 Mathematical and Computer Modelling 3 M$$^3$$AS. Mathematical Models & Methods in Applied Sciences 3 Automation and Remote Control 3 Theory of Probability and Mathematical Statistics 3 Computational and Applied Mathematics 3 Engineering Analysis with Boundary Elements 3 Chaos 3 Acta Mathematica Sinica. English Series 3 Nonlinear Analysis. Real World Applications 3 Discrete and Continuous Dynamical Systems. 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\section{Introduction}
In the ongoing effort to construct an ultraviolet complete theory of quantum gravity, higher curvature
theories of gravity emerge as corrections to the usual Einstein- Hilbert term. Such theories have now
come to play an important role in cosmology, black hole physics, supergravity, string theory and holography. They provide a framework for understanding which features of gravitational theory
are generic and which are special. Amongst the most important of these theories is Lovelock gravity \cite{Lovelock:1971yv}, which is the natural generalization of General Relativity to $D$ dimensions in two important respects: its field equations are of second-order and it is free of ghosts when expanded on a background of constant curvature \cite{Zwiebach:1985uq}.
One of the most important questions to address in any gravitational theory is the definition of conserved charges. This is a subtle issue in gravitational physics because of the equivalence principle, which makes localization of gravitational energy (and momentum and angular momentum) fraught with ambiguities \cite{MTW}. Approaches for addressing this problem date back six decades
\cite{Arnowitt:1960es,Arnowitt:1962hi} and more, and include a mixture of global \cite{Ashtekar:1984zz,Balasubramanian:1999re,Ashtekar:1999jx,Das:2000cu,Mann:2005yr} and quasilocal \cite{Brown:1992bq,Brown:1994gs,Creighton:1995au,McGrath:2012db,Epp:2013hua} methods. Generalizations
to Lovelock gravity have been carried out \cite{Olea:2006vd}, and a
universal form of the boundary term yielding a background-independent definition of conserved quantities for any Lovelock gravity theory with anti-de Sitter (AdS) asymptotics has been constructed \cite{Kofinas:2008ub}.
The purpose of this paper is to extend our understanding of conserved charges in Lovelock gravity to degenerate solutions of the field equations. The simplest form of such solutions is that of a $k$-fold degenerate AdS vacuum. In general the field equations of $N$-th order Lovelock gravity admit
as many as $N$ distinct vacua of constant curvature, each with their own effective cosmological constant. However these vacua need not be distinct -- depending on the values of the Lovelock coupling
constants, as many as $k$ cosmological constants can be equal to each other, where $k \leq N$; when
this takes place we say that the vacuum is $k$-fold degenerate. This concept extends to solutions, such as black holes, that do not have constant curvature.
Understanding what the conserved charges are for such solutions is rather subtle. The asymptotic falloff conditions differ from those cases in which the solution has no degeneracy, and standard approaches
\cite{Balasubramanian:1999re,Ashtekar:1999jx,Das:2000cu} do not work.
In this paper we address this problem. We obtain an expression for conserved charges in
Lovelock AdS gravity for $k$-fold degenerate solutions. In particular we demonstrate that the mass of
a black hole solution to the field equations on a branch of multiplicity $k$ comes from an expression containing the product of $k$ Weyl tensors. There is thus a link between
the degeneracy of a given vacuum and the nonlinearity of the energy formula. We find that
potentially divergent contributions of the type (Weyl)$^q$, with $1\le q<k$, are in fact suppressed.
Noting the obstruction to linearization of such solutions about a constant curvature background
\cite{Fan:2016zfs,Camanho:2013pda}, our results can provide
useful insight on the holographic properties of degenerate Lovelock gravity. This
may be regarded as a natural generalization of Conformal Mass to these
theories \cite{Ashtekar:1999jx,Das:2000cu}.
The outline of our paper is as follows. We begin in section \ref{sec2} with a review of Lovelock gravity,
highlighting the notion of degenerate solutions. In section \ref{sec3} we review the construction
of conserved charges in the Kounterterm formalism \cite{Olea:2006vd}, and in section \ref{sec4} we
derive a general formula for conserved charges for degenerate solutions (technical details are shown in Appendix A). Although even and odd dimensions need to be treated distinctly, we find that a single expression is valid in any dimension,
and discuss its applications to obtaining the mass of a black hole solution. We summarize our
work in section \ref{sec5}.
\section{Lovelock AdS gravity}
\label{sec2}
\label{Lovelock} Lovelock theory of gravity is the most general theory whose
dynamics is described by a second-order equation of motion. The action on a
D$-dimensional manifold $\mathcal{M}$ endowed by the metric $g_{\mu \nu}$ is
a polynomial in the Riemann curvature,
\begin{equation}
I = \frac { 1 }{ 16\pi G } \int\limits_{\mathcal{M}}{\ d^{ D }x\sqrt { -g }
\sum _{ p=0 }^{ N } \alpha _{ p }\mathcal{L}_p(R)}\,, \label{action}
\end{equation}
where $\alpha_{p}$ are the coupling constants, $N=[(D-1)/2]$ and $\mathcal{L
_{p}$ is the dimensionally continuation of the $p$-th Euler density,
\begin{equation}
\mathcal{L}_p =\frac{1}{2^p}\,\delta _{\nu_1\cdots \nu _{2p}}^{\mu_1\cdots
\mu _{2p}}\,R_{\mu _{1}\mu _{2}}^{\nu _{1}\nu _{2}}\cdots R_{\mu_{2p-1}\mu
_{2p}}^{\nu _{2p-1}\nu _{2p}}\,.
\end{equation}
Here, $\delta _{\nu _{1}\cdots \nu _{m}}^{\mu _{1}\cdots \mu _{m}}=\det
[\delta _{\nu _{1}}^{\mu _{1}}\cdots \delta _{\nu _{m}}^{\mu _{m}}] $
denotes a completely antisymmetric Kronecker delta of rank $m$. We can think
of this series as a modification to the Einstein-Hilbert action with
negative cosmological constant $\Lambda=-(D-1)(D-2)/2\ell^{2}$ with $\ell$
as the bare AdS radius. The Einstein-Hilbert term corresponds to first two
terms in the polynomial (\ref{action}), with $\alpha_{0}=-2\Lambda$ and
\alpha_{1}=1$. Then the variation of the action leads to the following
equations of motion,
\begin{eqnarray}
\mathcal{E}_\mu^\nu=R_{ \mu }^{ \nu }-\frac { 1 }{ 2 } R{\ \delta }_{ \mu
}^{ \nu }+\Lambda \delta _{ \mu }^{ \nu }-H_{ \mu }^{ \nu }=0\,, \label{eom}
\end{eqnarray}
where $H^{\nu}_{\mu}$ is the generalized Lanczos tensor that contains all
the higher-curvature contributions to the Einstein tensor in $D>4$,
\begin{equation}
H_\mu^\nu=\sum _{ p=2 }^{ N } \frac { \alpha _{ p } }{ 2^{ p+1 } }\, \delta
_{ \mu \mu _{ 1 }\cdots \mu _{ 2p } }^{ \nu \nu _{ 1 }\cdots \nu _{ 2p } }\,
R_{ \nu _{ 1 }\nu _{ 2 } }^{ \mu _{ 1 }\mu _{ 2 } }\cdots R_{ \nu _{ 2p-1
}\nu _{ 2p } }^{ \mu _{ 2n-1 }\mu _{ 2p }}\,. \label{LL}
\end{equation}
\subsection{Vacua of the theory}
An alternative way to write the field equations is
\begin{align}
\mathcal{E}_\mu^\nu &= \sum _{ p=0}^{ N } \frac { \alpha _{ p } }{ 2^{ p+1 } } \delta _{ \mu \mu _{ 1 }\cdots \mu _{ 2p } }^{
\nu \nu _{ 1 }\cdots \nu _{ 2p } }\, \left(R_{\nu _{ 1 }\nu _{ 2 } }^{ \mu
_{ 1 }\mu _{ 2 } }+\frac{1}{\ell_{\mathrm{eff}}^{(1) 2 }} \,\delta_{\nu _{ 1
}\nu _{ 2 } }^{ \mu _{ 1 }\mu _{ 2 } } \right)\cdots \left(R_{ \nu _{ 2p-1
}\nu _{ 2p } }^{ \mu _{ 2n-1 }\mu _{ 2p }}+\frac{1}{\ell_{\mathrm{eff}}^{(N)
2}}\,\delta_{ \nu _{ 2p-1 }\nu _{ 2p } }^{ \mu _{ 2n-1 }\mu _{ 2p }}
\right) \nonumber\\
&=0\,, \label{product of AdS}
\end{align}
where
$\ell_{\mathrm{eff}}^{(i)}$, with $i=1,\cdots,N$, are effective AdS
radii. They differ from the bare AdS radius $\ell$ due to the contribution
of higher-order terms through the couplings $\{\alpha_2,\ldots , \alpha_N\}$.
Eq.(\ref{product of AdS}) explicitly shows that maximally-symmetric
spacetimes are particular vacuum solutions of the theory, whose Riemann
curvature is
\begin{equation}
R^{\mu \nu}_{\alpha \beta}=-\frac{1}{\ell_{\mathrm{eff}}^{2}}\,\delta^{\mu
\nu}_{\alpha \beta} \label{maxR}
\end{equation}
which we can also write as $R+\frac{1}{\ell _{\mathrm{eff}}^{2}}
\delta^{[2]} = 0$, suppressing indices using the square-bracket notation.
We can find the form of $\ell_{\mathrm{eff}}$ by inserting the above
expression into $\mathcal{E}_\mu^\nu=0$ and obtain the characteristic
polynomial of order $N$ in the variable $\ell _{\mathrm{eff}}^{-2}$,
\begin{equation}
\Delta (\ell _{\mathrm{eff}}^{-2}) \equiv \sum_{p=0}^{N}{\frac
(D-3)!\,(-1)^{p-1}{\alpha }_{p}}{\left( D-2p-1\right) !}}\left( \frac{1}
\ell _{\mathrm{eff}}^{2}}\right) ^{p}=0\,. \label{Delta}
\end{equation}
The roots of $\Delta$ depend on the parameter set $\{\alpha_p\}$; the simplest
case is $N=1$ with $\Lambda = -(D-1)(D-2)/2/\ell _{\mathrm{eff}}^{2}$, for which
$\ell _{\mathrm{eff}} = \ell$. Writing $\lambda=\ell _{\mathrm{eff}}^{-2}$, if there
are $N$ real roots $\lambda_i$ ($i=1\ldots N$) then the theory has
N$ maximally symmetric vacua, which can be AdS ($\lambda_i >0$), dS ($\lambda_i <0
) or flat ($\lambda_i =0$). If a root $\lambda_i$ is complex then a maximally
symmetric space for this value of $\ell _{\mathrm{eff}}$
is not a particular solution of the theory.
We are interested in a spacetime that is asymptotically AdS, defined as a
branch of the theory by the corresponding AdS radius $\ell_{\mathrm{eff}}$.
The root itself can be simple, or $k$-fold degenerate, describing a vacuum of
multiplicity $k$.
The existence of a vacuum with multiplicity $k>1$ is a defining feature
of degenerate theories. A particular root is simple if its characteristic
polynomial satisfies
\begin{equation}
\Delta^{(1)}(\ell _{\mathrm{eff}}^{-2})=\frac{d\Delta}{d(\ell_{\mathrm{eff}}^{-2})}=\sum_{p=1}^{N}
\frac{(D-3)!\,(-1)^{p-1}p\alpha_{p}}{(D-2p-1)!} \left(\frac{1}{\ell_{\mathrm{eff
}^{2}}\right)^{p-1}\neq 0\,. \label{DeltaPrime}
\end{equation}
The relation $\Delta^{(1)}=0$ defines a \textit{critical} point in the
parameter space, which acts as an obstruction to the linearization of the
theory (see, e.g, ref.\cite{Fan:2016zfs} in the context of
Einstein-Gauss-Bonnet gravity and ref.\cite{Camanho:2013pda} in Lovelock
gravity). This fact prevents the obtention of an energy formula as a linearized charge \cite{Deser:2002jk,Petrov:2019roe}.
The nature of this criticality will depend on how degenerate the
particular branch is.
We will focus on a theory with a vacuum that is $k$-fold degenerate. It is defined
by the fact that all derivatives of $\Delta$ vanish up to order $k$,
\begin{eqnarray}
\Delta^{(q)}(\ell_{\mathrm{eff}}^{-2})&=&{\frac{1}{q!}\frac{d^q\Delta}{d(\ell_{\mathrm{eff}}^{-2})^q}}=0\,,\qquad \mathrm{when} \quad 1 \leq q \leq
k-1\,, \notag \\
\Delta^{(k)}(\ell _{\mathrm{eff}}^{-2})&=& \sum_{p=k}^{N}\binom{p}{k}\frac
(D-3)!\,(-1)^{p-1}\alpha_{p}\,(\ell _{\mathrm{eff}}^{-2})^{p-k}}{(D-2p-1)!
\neq 0\,. \label{DeltaK}
\end{eqnarray}
{The above condition means that the equations of motion (\ref{product of AdS
) are factorizable by $\left( R+\frac{1}{\ell _{\mathrm{eff}}^{2}
\delta^{[2]}\right) ^{k}$.}
Examples of degenerate theories with maximal multiplicity are, in even
dimensions $D=2n$, Born-Infeld AdS gravity,
\begin{equation}
\alpha_p^{\mathrm{BI}}=\binom{n-1}{p}\frac{2^{n-2}(2n-2p-1)!}{\ell^{2(n-p-1)
} \,,\qquad 0\leq p\leq n-1 \,,
\end{equation}
and, in odd dimensions $D=2n+1$, Chern-Simons AdS gravity,
\begin{equation}
\alpha_p^{\mathrm{CS}}=\binom{n}{p}\frac{2^{n-1}(2n-2p)!}{\ell^{2(n-p)}}
\,,\qquad 0\leq p\leq n \, \,.
\end{equation}
A special feature of Chern-Simons AdS gravity is that it has enhanced local symmetry from Lorentz to AdS, so its dynamics and a number of degrees of freedom are different compared to any other Lovelock gravity.
In that sense, Chern-Simons AdS gravity is a truly gauge theory. Similar features hold for Chern-Simons de Sitter and Poincar\'e gravities.
A generalization of the above theories, which considers a unique solution for $\ell_{\mathrm{eff}}$ with an intermediate degeneracy $k$, is the so-called Lovelock Unique Vacuum (LUV) AdS theory in $D$ dimensions. Here, $\Delta ^{(k)}$ is the only nonvanishing coefficient meaning that, apart from
(\ref{DeltaK}), there is also the additional requirement $\Delta ^{(q)}(\ell
_{\mathrm{eff}}^{-2})=0$ when $k<q\leq N$. In this case, the coupling
constants have the form
\begin{equation}
\alpha_p^{\mathrm{LUV}}=\binom{k}{p}\frac{2^{k-1}(D-2p-1)!}{\ell^{2(k-p)}}
\qquad 0\leq p\leq k , \qquad \alpha_{p > k}^{\mathrm{LUV}} = 0 \,. \label{alpha_LUV}
\end{equation}
Einstein-Hilbert, Born-Infeld and Chern-Simons AdS gravities are particular
cases of LUV AdS theories.
Our present goal is to find a mass formula for all those gravity theories
where the degeneracy is higher than one in odd and even spacetime
dimensions. To this end, we have to analyze the asymptotic behavior of the
corresponding solutions. In the next section, we obtain the falloff of the
relevant quantities for static geometries with $1 \leq k \leq N$.
\subsection{Asymptotic behavior of Lovelock AdS solutions}
\label{asymptotic}
A topological, static black hole solution in the local coordinates $x^\mu
=(t,r,y^m)$ ($m=2,\cdots, D-1$) is described by the line element
\begin{equation}
{\ d }s^{ 2 }=-f(r){\ dt }^{ 2 }+\frac { 1 }{ f(r) } {\ dr }^{ 2 }+{\ r
^{2} \gamma_{mn}(y)dy^mdy^n\,, \label{ansatz}
\end{equation}
where $\gamma_{nm}$ is the metric of the transverse section of constant
curvature $\kappa=+1,0,-1$. The metric function $f(r)$ is given by the
algebraic master equation obtained as the first integral of the equations of
motion \cite{Boulware-Deser,CaiEGB,CaiLovelock},
\begin{equation}
\sum _{p=0}^{N}\,\frac{\alpha_p(D-3)!}{(D-2p-1)!}\left(\frac{\kappa-f(r)}{r^
}\right)^p=\frac{\mu}{r^{D-1}}\,, \label{master}
\end{equation}
where $\mu$ is an integration constant related to the mass of the solution, as we shall discuss below. In case of asymptotically AdS spaces,
f(r)$ behaves as
\begin{equation}
f(r)=\kappa+\frac{r^{2}}{\ell_{\mathrm{eff}}^{2}}+\epsilon (r)\,,
\end{equation}
where $f_{\mathrm{AdS}}=\kappa +\frac{r^{2}}{\ell _{\mathrm{eff}}^{2}}$ is
the global AdS space and $\epsilon (r)$ is a function that, in most cases,
decays sufficiently fast. Plugging this into the master equation and
expanding in the asymptotic region ($1/r \rightarrow 0$), we find
\begin{eqnarray}
\frac{\mu }{r^{D-1}} &=&\sum_{p=0}^{N}\frac{\left( -1\right)
^{p}(D-3)!\alpha _{p}}{\left( D-2p-1\right) !}\,\left( \frac{1}{\ell_
\mathrm{eff}}^{2}} +\frac{\epsilon }{r^{2}}\right) ^{p} \notag \\
&=&\sum_{q=0}^{N}\sum_{p=q}^{N}\frac{\left( -1\right) ^{p}(D-3)!\alpha _{p}}
\left( D-2p-1\right) !}\binom{p}{q}\left(\frac{1}{\ell_{\mathrm{eff}}^{2}
\right) ^{p-q}\left( \frac{\epsilon }{r^{2}}\right) ^{q}\,,
\end{eqnarray}
or equivalently
\begin{equation}
\frac{\mu }{r^{D-1}}=\sum_{q=0}^{N}\Delta ^{(q)}\,\left( \frac{\epsilon }
r^{2}}\right) ^{q}\,,
\end{equation
where $\Delta^{(q)}$ is given by eq.(\ref{DeltaK}). We can expand the above
series knowing that, for a $k$-fold degenerate vacuum, the first nonvanishing
term is $\Delta ^{(k)}$ yielding
\begin{equation}
\frac{\mu }{r^{D-1}}=\Delta ^{(k)}\,\left( \frac{\epsilon }{r^{2}}\right)
^{k}+\Delta ^{(k+1)}\,\left( \frac{\epsilon }{r^{2}}\right) ^{k+1}+\cdots \,.
\label{Expansion1}
\end{equation}
Furthermore, if $\Delta^{(k+1)}=0$, it can be shown that $\Delta^{(q)}=0$
for all $q>k$. {Then $\epsilon (r)$ can be solved exactly as $\epsilon
=\left( \frac{\mu }{\Delta ^{(k)}\,r^{D-2k-1}}\right)^{1/k}$.}
For an arbitrary value of $\Delta ^{(k+1)}$, we can assume an asymptotic
behavior for $\epsilon(r)$ of the form
\begin{equation}
\epsilon (r)=\frac{A}{r^{x}}+\frac{B}{r^{y}}+\cdots \,,\qquad y>x \geq
0\,,\qquad A\neq 0\,, \label{e}
\end{equation}
where $A$, $B$, $x$ and $y$ are coefficients to be determined. Inserting eq.
\ref{e}) into (\ref{Expansion1}), we obtain
\begin{equation}
\frac{\mu }{r^{D-1}}=\Delta ^{(k)}\,\left( \frac{A^{k}}{r^{(x+2)k}}+\frac
kA^{k-1}B}{r^{(x+2) (k-1) +y+2}}+\cdots \right) +\Delta ^{(k+1)}\,\left(
\frac{A^{k+1}}{r^{(x+2)(k+1)}}+\cdots \right) .
\end{equation}
At leading order, we find
\begin{equation}
\frac{\mu }{r^{D-1}}=\Delta ^{(k)}\,\frac{A^{k}}{r^{\left( x+2\right) k}}\,,
\end{equation}
and therefore
\begin{equation}
x=\frac{D-2k-1}{k}\,,\qquad A=\left(\frac{\mu }{\Delta ^{(k)}}\right) ^
\frac{1}{k}}\,. \label{x,A}
\end{equation}
Notice that the mass parameter may be negative and that $A$ can have any sign for even $k$. The subleading contributions
become
\begin{equation}
0=\Delta ^{(k)}\,\left( \frac{kA^{k-1}B}{r^{\left( x+2\right) \left(
k-1\right) +y+2}}+\cdots \right) +\Delta ^{(k+1)}\,\left( \frac{A^{k+1}}
r^{\left( x+2\right) \left( k+1\right) }}+\cdots \right) \,.
\label{sub-leading}
\end{equation}
We can distinguish three cases for the coefficients $A$ and $B$:
\begin{itemize}
\item[\textit{a})] When $A=B=0$, the solution becomes the vacuum state of
the theory, $\mu =0$, i.e., global AdS.
\item[\textit{b})] When $A\neq 0$ and $B=0$, the solution exists only if
\Delta ^{(k+1)}=0$. As discussed after eq.(\ref{Expansion1}), $\Delta ^{(k)}$
is the only nonvanishing coefficient and $\epsilon$ can be solved exactly.
This case is that of LUV theories.
\item[\textit{c})] When $A,B\neq 0$, the terms along $\Delta ^{(k)}$ and
\Delta ^{(k+1)}$ are of the same order. That implies
\begin{equation}
y=\frac{2 \left( D-1\right) -2k}{k}\,, \label{y}
\end{equation}
and, in turn, the coefficient $B$ reads
\begin{equation}
B=-\frac{\Delta ^{(k+1)}A^{2}}{k\Delta ^{(k)}}=-\frac{\Delta ^{(k+1)}}
k\Delta ^{(k)}}\left( \frac{\mu }{\Delta ^{(k)}}\right) ^{\frac{2}{k}}.
\label{B}
\end{equation}
\end{itemize}
The cases \emph{a})--\emph{c}) are covered by the formulas (\ref{x,A}), (\re
{y}) and (\ref{B}). In this way, for a degenerate vacuum with multiplicity
k $, we find a general falloff of the metric function as
\begin{equation}
f(r)=\kappa+\frac{r^{2}}{\ell _{\mathrm{eff}}^{2}}+\left( \frac{\mu }{\Delta
^{(k)}r^{D-2k-1}}\right) ^{\frac{1}{k}} -\frac{\Delta ^{(k+1)}}{k {\Delta
^{(k)}}^{2}}\left( \frac{\mu^{2} }{\Delta ^{(k)}r^{2(D-1) -2k}}\right) ^
\frac{1}{k}}+\cdots \,. \label{f(r) K}
\end{equation}
We illustrate the above relation with the exact solutions in LUV gravity
for $k<N$ \cite{Crisostomo:2000bb}
\begin{equation}
f_{\mathrm{LUV}}(r)=\kappa +\frac{r^{2}}{\ell _{\mathrm{eff}}^{2}}-\left(
\frac{2GM}{r^{D-2k-1}}\right) ^{\frac{1}{k}}\,, \label{LUV-BH}
\end{equation
where $G$ is the gravitational constant. In maximally degenerate cases,
these are Chern-Simons and Born-Infeld AdS black holes
\begin{eqnarray}
f_{\mathrm{CS}}(r) &=&\kappa +\frac{r^{2}}{\ell _{\mathrm{eff}}^{2}}-(2GM+1)^{\frac{1}{n}}\,, \notag \\
f_{\mathrm{BI}}(r) &=&\kappa +\frac{r^{2}}{\ell _{\mathrm{eff}}^{2}}-\left(
\frac{2GM}{r}\right) ^{\frac{1}{n-1}}\,. \label{CS,BI-BH}
\end{eqnarray}
The mass parameter in the Chern-Simons case has been redefined
so that the horizon
shrinks to a point when $M \rightarrow 0$. This produces a mass gap, $M=-1/2G $,
between the Chern-Simons black hole, $M\geq 0$, and global AdS space with $M=-1/2G$.
It is worthwhile noting that the identification of the mass parameter $M$ (related to the integration constant $\mu$ in the general formula (\ref{f(r) K})) as the total mass of the black hole was made in ref.\cite{Crisostomo:2000bb} and, for $k=2$, in ref.\cite{Fan:2016zfs}, based on thermodynamic calculations. In \cite{Crisostomo:2000bb}, it has been also obtained as a Hamiltonian mass in the minisuperspace approach. In our method, however, we calculate the mass as the one that comes from Noether theorem, once the action has been supplemented by adequate boundary terms. This notion agrees with the thermodynamic and Hamiltonian mass.
In what follows, we make extensive use of the falloff of the metric (\ref{f(r) K}) to find both the AdS curvature and the corresponding Weyl tensor.
\subsection{AdS curvature and Weyl tensor}
\label{riewf}
The only nonvanishing components of the Riemann tensor for the static
solution (\ref{ansatz}) are
\begin{equation}
\begin{array}{ll}
R_{tr}^{tr}=-\frac{1}{2}\,f^{\prime \prime }\,,\medskip \qquad &
R_{rm}^{rn}=R_{tm}^{tn}=-\frac{f^{\prime }}{2r}\,\delta _{m}^{n}\,, \\
R_{m_{1}m_{2}}^{n_{1}n_{2}}=\frac{\kappa-f}{r^{2}}\,\delta
_{m_{1}m_{2}}^{n_{1}n_{2}}\,. &
\end{array
\end{equation}
Using the falloff (\ref{f(r) K}), we get
\begin{eqnarray}
R_{tr}^{tr} &=&-\frac{1}{\ell _{\mathrm{eff}}^{2}}-\frac{\left(
D-2k-1\right) \left( D-k-1\right) }{2k^{2}}\left( \frac{\mu }{\Delta
^{(k)}r^{D-1}}\right) ^{\frac{1}{k}}+\mathcal{O}\left( r^{-\frac{2(D-1)}{k
}\right) \,, \notag \\
R_{rm}^{rn} &=&R_{tm}^{tn}=\left[ -\frac{1}{\ell _{\mathrm{eff}}^{2}}+\frac
D-2k-1}{2k} \left( \frac{\mu }{\Delta ^{(k)}r^{D-1}}\right) ^{\frac{1}{k}}\right] \delta _{m}^{n}+\mathcal{O}\left( r^{-\frac{2(D-1)}{k
}\right) \,, \notag \\
R_{m_{1}m_{2}}^{n_{1}n_{2}} &=&\left[ -\frac{1}{\ell _{\mathrm{eff}}^{2}
+\left( \frac{\mu }{\Delta ^{(k)}\,r^{D-1}}\right) ^{\frac{1}{k}}\right]
\delta _{m_{1}m_{2}}^{n_{1}n_{2}}+\mathcal{O}\left( r^{-\frac{2(D-1)}{k
}\right) \,. \label{Riem}
\end{eqnarray
} Furthermore, the AdS curvature
\begin{equation}
F_{\alpha \beta }^{\mu \nu }=R_{\alpha \beta }^{\mu \nu } +\frac{1}{{\ell_
\mathrm{eff}}^{2}}}\,{\delta }_{\alpha \beta }^{\mu \nu }\,,
\label{AdScurvature}
\end{equation}
is the only part of the field strength of the local $SO(D-1,2)$ group that differs from zero in a Riemannian geometry. Using eqs.(\ref{Riem}), it
is straightforward to evaluate it as
\begin{eqnarray}
F_{tr}^{tr} &=&-\frac{(D-2k-1)(D-k-1)}{2k^2}\left( \frac{\mu }{\Delta
^{(k)}r^{D-1}}\right)^{\frac{1}{k}} +\mathcal{O}\left( r^{-\frac{2(D-1)}{k
}\right) \,, \notag \\
F_{rm}^{rn} &=&F_{tm}^{tn}=\frac{D-2k-1}{2k}\left( \frac{\mu }{\Delta
^{(k)}r^{D-1}}\right)^{\frac{1}{k}}\, \delta_m^n+\mathcal{O}\left(r^{
\frac{2(D-1)}{k}}\right) \,, \notag \\
F_{m_{1}m_{2}}^{n_{1}n_{2}} &=&\left( \frac{\mu }{\Delta ^{(k)}\,r^{D-1}}
\right) ^{\frac{1}{k}}\delta _{m_{1}m_{2}}^{n_{1}n_{2}} +\mathcal{O
\left(r^{-\frac{2(D-1)}{k}}\right) \,. \label{AdSasymp}
\end{eqnarray
}
We can express the above in a useful way in terms of the Weyl tensor, which is defined in terms of the Riemann tensor and its contractions as
\begin{equation}
W_{\alpha \beta}^{ \mu \nu }=R_{ \alpha \beta }^{ \mu \nu }- \frac { 1 }{
D-2 } \,\delta_{ [\alpha }^{ [\mu }R_{ \beta ] }^{ \nu] } +\frac {R}
(D-1)(D-2) } \,{\delta }_{ \alpha \beta }^{\mu \nu }\,,
\end{equation}
{where the second term is the skew-symmetric product between the Kronecker
delta and the Ricci tensor, $\delta_{[\alpha}^{[\mu } R_{\beta ]}^{\nu ]}
=\delta _{\alpha }^{\mu }R_{\beta }^{\nu}-\delta _{\alpha }^{\nu }R_{\beta
}^{\mu } -\delta _{\beta }^{\mu }R_{\alpha}^{\nu }+\delta _{\beta }^{\nu
}R_{\alpha }^{\mu }$.}
In Einstein AdS gravity, the on-shell Weyl tensor coincides with the AdS
curvature. However in Lovelock AdS gravity the higher-order
contributions modify the above relation such that the on-shell Weyl tensor
can be written as
\begin{equation}
W_{\alpha \beta }^{\mu \nu }=F_{\alpha \beta }^{\mu \nu }+X_{\alpha \beta
}^{\mu \nu }\,,
\end{equation
where the tensor $X_{\alpha \beta }^{\mu \nu }$ is constructed from the
generalized Lanczos tensor (\ref{LL}) and its trace $H=H_{\mu }^{\mu }$,
\begin{equation}
X_{\alpha \beta }^{\mu \nu }=\left( \frac{1}{\ell ^{2}}-\frac{1}{\ell _
\mathrm{eff}}^{2}}+\frac{2H}{(D-1)(D-2)}\right) \delta _{\alpha \beta }^{\mu
\nu }-\frac{1}{D-2}\,{{\delta }_{[\alpha }^{[\mu }H_{\beta ]}^{\nu ]}}\,.
\label{X}
\end{equation
A computation based on the falloff of the Riemann tensor \eqref{Riem} computed above
shows that the components of $X$ are always subleading in $r$ with respect
to the AdS curvature. More precisely, $X_{\alpha \beta }^{\mu \nu }=\mathcal
O}\left( \mu ^{2/k}/r^{2(D-1)/k}\right) $ asymptotically. Therefore, the
leading order of the AdS curvature and the Weyl tensor is
\begin{equation}
W_{\alpha \beta }^{\mu \nu }=F_{\alpha \beta }^{\mu \nu }=\mathcal{O}\left(
\mu ^{1/k}/r^{(D-1)/k}\right) \,. \label{WeylAsymp}
\end{equation}
In the next section we show that, for a given theory with degeneracy $k$,
the conserved charge formula is proportional to the $k$-th power of the Weyl
tensor. This is obtained as a consistent truncation of the charges found in
ref.\cite{kofinas,Kofinas:2008ub} in order to produce a finite energy flux
at the asymptotic region.
\section{Kounterterms and conserved charges}
\label{sec3}
The Lovelock AdS action is infrared (IR) divergent and has to be renormalized by adding
boundary counterterms. Instead of obtaining the local counterterm series
perturbatively, as in standard Holographic Renormalization \cit
{Henningson:1998gx,Balasubramanian:1999re,Mann:1999pc,deHaro:2000vlm}, the idea behind the Kounterterm method
\cite{Olea:2005gb,Olea:2006vd} is that the bulk action is supplemented with
an appropriate boundary term that is linked either to topological
invariants or Chern-Simons forms. In $D=d+1$ dimensions, the renormalized
action defined on the manifold $\mathcal{M}$ reads
\begin{equation}
I_{\mathrm{ren}}={I}_{\mathrm{bulk}}+{c}_{d}\int\limits_{\partial {\mathcal{
}}}{{d}^{d}x\,{B}_{d}}(h,K,\mathcal{R})\,,
\label{Ireg}
\end{equation}
where $B_{d}(h,K,{\mathcal{R}})$ is a scalar density on the boundary
\partial \mathcal{M}$ that depends on the boundary metric, the extrinsic
curvature, and the boundary curvature. The overall factor ${c}_{d}$ is a
given coupling. It has been shown that the method appropriately leads to
finite conserved charges in higher-curvature gravity theories \cite{kofinas,
Giribet:2018hck} and it is also useful for computing holographic quantities
such as entanglement entropy \cite{Anastasiou:2018rla}.
As usual, the charges are expressed as an integral over a co-dimension 2
surface at fixed time and radial infinity. We can proceed taking a radial
foliation of the spacetime $\mathcal{M}$ in Gauss-normal coordinates,
\begin{equation}
ds^{2}=N^{2}(r)\,dr^{2}+h_{ij}(r,x)\,dx^{i}dx^{j}\,,
\end{equation
where $N(r)$ is the lapse function and $h_{ij}$ is the induced metric at a
fixed $r$. In turn, the boundary metric admits a time-like ADM foliation as
\begin{equation}
h_{ij}dx^{i}dx^{j}=-\tilde{N}^{2}(t)dt^{2}+\sigma _{mn}\left( dy^{m}+\tilde{
}^{m}dt\right) \left( dy^{n}+\tilde{N}^{n}dt\right) \,,
\end{equation
where now $\sqrt{-h}=\tilde{N}\sqrt{\sigma }$, with $\sigma _{mn}$ the
co-dimension two metric of the asymptotic boundary $\Sigma _{\infty }$. The
unit normal to the hypersurface is given by $u_{j}=(u_{t},u_{m})=(-\tilde{N
,0)$ and therefore, the conserved charges are given by the surface integral
\begin{equation}
Q[\xi ]=\int\limits_{\Sigma _{\infty }}{{d}^{d-1}y\sqrt{\sigma }\,{u}_{j}
\xi }^{i}}\left( \tau _{i}^{j}+\tau _{(0)i}^{j}\right) \,, \label{Q}
\end{equation
where $\xi ^{i}$ is an asymptotic Killing vector. The charge density tensor
is naturally split in two contributions: $\tau _{i}^{j}$ that, when
integrated, can be identified with the mass and angular momentum of the
black hole, and $\tau _{(0)i}^{j}$ is associated with the vacuum/Casimir
energy in the context of AdS/CFT correspondence \cite{Balasubramanian:1999re}.
In even dimensions $D=2n$, $\tau _{i}^{j}$ has the form \cite{kofinas}
\begin{eqnarray}
\tau _{i}^{j} &=&\frac{1}{2^{n-2}}\,\delta _{{i}_{1}{i}_{2}\dots {i}_{2n-1}}^{j{j}_{2}\dots {j}_{2n-1}}\,K_{i}^{{i}_{1}}\left[ \frac{1}{16\pi
G}\sum_{p=1}^{n-1}{\frac{p{\alpha _{p}}}{(2n-2p)!
\,R_{j_{2}j_{3}}^{i_{2}i_{3}}\cdots
R_{j_{2p-2}j_{2p-1}}^{i_{2p-2}i_{2p-1}}\times }\right. \notag \\
&&\qquad \qquad \times \left. \delta
_{j_{2p}j_{2p+1}}^{i_{2p}i_{2p+1}}\cdots \delta
_{j_{2n-2}j_{2n-1}}^{i_{2n-2}i_{2n-1}}+n{c
_{2n-1}R_{j_{2}j_{3}}^{i_{2}i_{3}}\cdots
R_{j_{2n-2}j_{2n-1}}^{i_{2n-2}i_{2n-1}}\rule{0pt}{17pt}\right] ,
\label{qqeven}
\end{eqnarray
and $\tau _{(0)i}^{j}=0$. The coupling in \eqref{Ireg} is fixed from the action principle,
\begin{equation}
c_{2n-1}=-\frac{1}{16\pi nG}\sum_{p=1}^{n-1}\frac{p\alpha _{p}}{(D-2p)!
\,(-\ell _{\mathrm{eff}}^{2})^{n-p}\,.
\end{equation
In odd dimensions $D=2n+1$, the charge density tensor
reads
\begin{eqnarray}
\tau _{i}^{j} &=&\frac{1}{2^{n-2}}\,\delta _{{i}_{1}{i}_{2}\dots {i
_{2n}}^{j{j}_{2}\dots {j}_{2n}}\,K_{i}^{{i}_{1}}{\delta }_{{j}_{2}}^{{i
_{2}}\left[ \frac{1}{16\pi G}\sum_{p=1}^{n}{\frac{p{\alpha _{p}}}{(2n-2p+1)!
R_{j_{3}j_{4}}^{i_{3}i_{4}}\cdots R_{j_{2p-1}j_{2p}}^{i_{2p-1}i_{2p}}\delta
_{j_{2p+1}j_{2p+2}}^{i_{2p+1}i_{2p+2}}}\cdots \right. \notag \\
&&\hspace{-0.3cm}\left. \cdots \delta _{j_{2n-1}j_{2n}}^{i_{2n-1}i_{2n}}+n{c
_{2n}\int\limits_{0}^{1}{du\left( R_{j_{3}j_{4}}^{i_{3}i_{4}}+\frac{{u}^{2}}
\ell _{\mathrm{eff}}^{2}}\delta _{j_{3}j_{4}}^{i_{3}i_{4}}\right) \cdots
\left( R_{j_{2n-1}j_{2n}}^{i_{2n-1}i_{2n}}+\frac{{u}^{2}}{\ell _{\mathrm{eff
}^{2}}\delta _{j_{2n-1}j_{2n}}^{i_{2n-1}i_{2n}}\right) \right] \label{qqodd}
\end{eqnarray}
and
\begin{equation*}
\tau _{(0)i}^{j}=-\frac{nc_{2n}}{2^{n-2}}\,\int_{0}^{1}du\,u\,\delta
_{{i}_{1}{ i}_{2}...{i}_{2n}}^{j{ j}_{2}...{ j
_{2n}}\left( {\delta }_{{ j}_{2}}^{{ i}_{2}}K_{i}^{{ i}_{1}}+{ \delta
}_{{ i}}^{{ i}_{2}}{ K}_{{ j}_{2}}^{{ i}_{1}}\right){ \mathcal{F}}_{
j}_{3}{ j}_{4}}^{{ i}_{3}{i}_{4}}(u)\cdots {\mathcal{F}}_{{ j
_{2n-1}{ j}_{2n}}^{{ i}_{2n-1}{i}_{2n}}(u)\,,
\end{equation*}
where
\begin{equation}
\mathcal{F}_{lk}^{ij}(u)={\ R}_{lk}^{ij}-\left( u^{2}-1\right) \left(
K_{k}^{i}K_{l}^{j}-K_{l}^{i}K_{k}^{j}\right)+\frac{u^{2}}{\ell _{\mathrm{eff}}^{2}}{\ \delta }_{kl}^{ij}\,.
\end{equation
The coupling constant in this case is
\begin{equation}
c_{2n}=-\frac{1}{16\pi nG}\left[ \int\limits_{0}^{1}du(1-u^{2})^{n-1}\right]
^{-1}\sum_{p=1}^{n}\frac{p\alpha _{p}}{(D-2p)!}\,(-\ell _{\mathrm{eff
}^{2})^{n-p}\,. \label{c_odd}
\end{equation}
The charge in $D$ dimensions is a polynomial of order $N$ in the curvature.
In nondegenerate Lovelock theories, this polynomial can be truncated so that
it becomes linear in the Weyl tensor \cite{Arenas-Henriquez:2017xnr}
\begin{equation}
\tau _{i}^{j}=-\frac { { { \ell _{ \rm{ eff } }\, \Delta ^{ (1) }(\ell_{\rm{eff}}^{-2}) } } }{ 32\pi G\, (D-3) } \, \delta _{ ii_{ 2 }{ i }_{ 3 } }^{ jj_{ 2 }{ j }_{ 3 } }\, { W }_{ { j }_{ 2 }{ j }_{ 3 } }^{ { i }_{ 2 }{ i }_{ 3 } } \,.
\end{equation}
This expression corresponds to Conformal Mass for nondegenerate Lovelock AdS gravity \cit
{Arenas-Henriquez:2017xnr}, as an extension of the concept developed by
Ashtekar, Magnon and Das \cite{Ashtekar:1984zz, Ashtekar:1999jx}. Using the identity
$\delta _{ii_{2}}^{jj_{2}}{W}_{{j}_{2}{j}_{3}}^{{i}_{2}{i}_{3}}=4W_{ri}^{rj}$
the charge becomes proportional to the electric part of the Weyl tensor
\begin{equation}
\tau _{i}^{j}=-\frac{\ell_{\rm{eff}}}{8\pi G} \, \Delta^{(1)}(\ell_{\rm{eff}}^{-2})E^{j}_{i}\,.
\label{conformalmass}
\end{equation}
where in $D$ dimensions
\begin{equation}
E^{j}_{i}=\frac{1}{D-3}\,W^{rj}_{ri}\, .
\end{equation}
Clearly \eqref{conformalmass} fails when $\Delta^{(1)}=0$.
When
the theory has multiplicity $k$, a power-counting argument indicates that the
asymptotic falloff of $\tau_{i}^{j}$ is such that the system has finite global
charges. Namely, the bulk metric behaves as in eq.(\ref{f(r) K}), and so
\sqrt{\sigma}= \mathcal{O}(r^{D-2})$, $u_{j} = \mathcal{O}(r)$ and $\xi =
\mathcal{O}(1)$. Since the charge $Q[\xi]$ is of order $\mathcal{O}(1)$, it
implies that $\tau _{i}^{j}$ should be of order $\mathcal{O}(1/r^{D-1})$. At
the same time, the charge should be zero for global AdS. Hence it must be a nonlinear expression in the Weyl tensor.
\section{Generalized Conformal Mass}
\label{sec4}
In this section we manipulate the general formulas (\ref{qqeven}) and (\re
{qqodd}) for $\tau_{i}^{j}$ to make manifest the dependence on the
degeneracy conditions at different orders.
\subsection{Even dimensions $D=2n$}
The charge density in even dimensions can be conveniently rewritten as
\begin{eqnarray}
\tau _{i}^{j} &=&\frac{{{\ell _{\mathrm{eff}}^{2n-2}}}}{16\pi G\, {2}^{n-2}
\,\delta _{i_{1}\cdots i_{2n-1}}^{jj_{2}\cdots
j_{2n-1}}\,K_{i}^{i_{1}}\sum_{p=1}^{n-1}{\frac{p{\alpha }_{p}}{(2n-2p)!}
\left( \frac{1}{{{\ell _{\mathrm{eff}}^{2}}}}\right)
^{p-1}R_{j_{2}j_{3}}^{i_{2}i_{3}}\cdots
R_{j_{2p-2}j_{2p-1}}^{i_{2p-2}i_{2p-1}}\times \notag \\
&&\hspace{-0.3cm}\left[ {\left( \frac{1}{{{\ell _{\mathrm{eff}}^{2}}}
\right) ^{n-p}\delta _{j_{2p}j_{2p+1}}^{i_{2p}i_{2p+1}}}\cdots \delta
_{j_{2n-2}j_{2n-1}}^{i_{2n-2}i_{2n-1}}-{(-1)
^{n-p}R_{j_{2p}j_{2p+1}}^{i_{2p}i_{2p+1}}\cdots
R_{j_{2n-2}j_{2n-1}}^{i_{2n-2}i_{2n-1}}\right] . \label{echarge}
\end{eqnarray
As it is clear from the above formula, the charge is a polynomial of order
n-1$ in the curvature. For a branch with the degeneracy $k$, this can be
rearranged in order to express the polynomial as a product of $k$ AdS
curvatures times a polynomial $\mathcal{P}(R)$ of order $n-1-k$ in the
curvature
\begin{align}
\tau _{i}^{j}& =\frac{{{\ell _{\mathrm{eff}}^{2(n-1)}}}}{16\pi G\,{2}^{n-2}}\,\delta _{i_{1}i_{2}\cdots {i}_{2k}{i}_{2k+1}\dots {i}_{2n}i_{2n-1}}^{jj_{2}\cdots {j}_{2k}{j}_{2k+1}\dots {j}_{2n}j_{2n-1}}K_{i}^{i_{1}}\left( R_{j_{2}j_{3}}^{i_{2}i_{3}}+\frac{1}{\ell_{\mathrm{eff}}^{2}}\delta _{j_{2}j_{3}}^{i_{2}i_{3}}\right) \times \dots
\notag \\
& \qquad\qquad \dots \times \left( R_{j_{2k}j_{2k+1}}^{i_{2k}i_{2k+1}}+\frac{1}{\ell _{\mathrm{eff}}^{2}}\delta _{j_{2k}j_{2k+1}}^{i_{2k}i_{2k+1}}\right){\mathcal{P}_{j_{2k+2}\cdots j_{2n-1}}^{i_{2k+2}\cdots i_{2n-1}}}(R)\,,
\label{goal}
\end{align}
where $\mathcal{P}(R)$ written as in eq.(\ref{P final}) of Appendix \ref{Fact} with all the indices
\begin{eqnarray}
&&\left. \mathcal{P}{_{j_{2k+2}\cdots j_{2n-1}}^{i_{2k+2}\cdots i_{2n-1}}
(R)=\sum_{s=0}^{n-k-1}\sum_{p=k}^{k+s}\Delta ^{(p)}(C_{k+s,p})
_{j_{2k+2}\cdots j_{2n-2s-1}}^{i_{2k+2}\cdots i_{2n-2s-1}}\times }\right.
\notag \\
&&\hspace{-0.3cm}\left( R_{j_{2n-2s}j_{2n-2s+1}}^{i_{2n-2s}i_{2n-2s+1}}
\frac{1}{\ell _{\mathrm{eff}}^{2}}\delta
_{j_{2n-2s}j_{2n-2s+1}}^{i_{2n-2s}i_{2n-2s+1}}\right) \cdots \left(
R_{j_{2n-2}j_{2n-1}}^{i_{2n-2}i_{2n-1}}+\frac{1}{\ell _{\mathrm{eff}}^{2}
\delta _{j_{2n-2}j_{2n-1}}^{i_{2n-2}i_{2n-1}}\right) \label{P}
\end{eqnarray}
For a detailed construction of this factorization in even dimensions, see Appendix \ref{evenfactorization}.
The charge has to be evaluated in the asymptotic region where we know that
the curvature tensor behaves as in eq.(\ref{Riem}). In turn, the extrinsic
curvature has the asymptotic form
\begin{equation}
K_{j}^{i}=-\frac{1}{\ell _{\mathrm{eff}}}\,\delta _{j}^{i}+\mathcal{O
(1/r^{2})\,. \label{extrinsic}
\end{equation
Furthermore, given the fact that the charge density at large distances is $\tau _{j}^{i}=\mathcal{O}(1/r^{D-1})$, the AdS curvature falloff (\ref{AdSasymp}) constrains the polynomial $\mathcal{P}=\mathcal{O}(1)$ to leading order. This implies that the Riemann curvature
is $R=-\frac{1}{\ell _{\mathrm{eff}}^{2}}\delta^{[2]}$ in the
polynomial so that the only nonzero contribution in eq.(\ref{P}) comes from $s=0,p=k$, corresponding to the coefficient $C_{kk}$ given by
\begin{equation}
(C_{kk}){_{j_{2k+2}\cdots j_{2n-1}}^{i_{2k+2}\cdots i_{2n-1}}}=\frac
(-1)^{k-1}}{2(2n-3)!\ell _{\mathrm{eff}}^{2(n-2)}}\,\delta
_{j_{2k+2}j_{2k+3}}^{i_{2k+2}i_{2k+3}}}\cdots \delta {_{j_{2n-2}\cdots
j_{2n-1}}^{i_{2n-2}i_{2n-1}}}
\end{equation}
from eq.(\ref{cqq}) in Appendix \ref{evenfactorization}.
The above reasoning yields
\begin{equation}
\mathcal{P}_{j_{2k+2}\cdots j_{2n-1}}^{i_{2k+2}\cdots i_{2n-1}}(-\ell_{\mathrm{eff}}^{-2}\delta^{[2]})=-\frac{(-1)^{k}\Delta^{(k)}}{2(2n-3)!\ell_{\rm{eff}}^{2(n-2)}}\,\delta_{{j}_{2k+2}{j
_{2k+3}}^{{i}_{2k+2}{i}_{2k+3}}\cdots {\delta }_{{j}_{2n}{j}_{2n-1}}^{{i
_{2n}{i}_{2n-1}}.
\end{equation
Clearly, this quantity is nonvanishing only in a theory whose vacuum has
multiplicity $k$. Replacing the asymptotic form of the extrinsic curvature
of eq.(\ref{extrinsic}) and the relation between the AdS curvature and the
Weyl tensor of eq.(\ref{WeylAsymp}), we find that the charge density tensor in even dimensions can be consistently truncated up to the order $k$ in the Weyl tensor,
\begin{eqnarray}
\tau _i^j &=&\frac{\ell_{\mathrm{eff}}(-1)^k\,\Delta ^{(k)}}{16\pi G\,{2}^{n-1}(2n-3)!}\,\delta _{ii_{2}\cdots {i}_{2k}{i}_{2k+1}\dots {i
_{2n}i_{2n-1}}^{jj_{2}\cdots {j}_{2k}{j}_{2k+1}\dots {j}_{2n}j_{2n-1}}\,{W}_
{j}_{2}{j}_{3}}^{{i}_{2}{i}_{3}}\cdots {W}_{{j}_{2k}{j}_{2k+1}}^{{i}_{2k}{i
_{2k+1}}\times \notag \\
&&\qquad \qquad \qquad \qquad \qquad \times {\delta }_{{j}_{2k+2}{j
_{2k+3}}^{{i}_{2k+2}{i}_{2k+3}}\cdots {\delta }_{{j}_{2n}{j}_{2n-1}}^{{i
_{2n}{i}_{2n-1}}\,.
\end{eqnarray
Upon a suitable contraction of the Kronecker deltas, the charge becomes
\begin{equation}
\tau _{i}^{j}=\frac{{{\ell_{\mathrm{eff}}{ (-1)}^{k}\,\Delta
^{(k)}(2n-2k-2)!}}}{16\pi G\,{2}^{k}(2n-3)!}\,\delta _{ii_{2}\cdots {i
_{2k}{ i}_{2k+1}}^{jj_{2}\cdots {j}_{2k}{ j}_{2k+1}}\,{W}_{{ j}_{2}{
j}_{3}}^{{ i}_{2}{\ i}_{3}}\cdots {W}_{{ j}_{2k}{j}_{2k+1}}^{{ i}_{2k
{ i}_{2k+1}}\,.
\end{equation
For $k=1$, the last formula reduces to the Conformal Mass \eqref{conformalmass} in nondegenerate Lovelock theories.
In the maximally degenerate case, i.e., Born-Infeld AdS gravity, the charge
is a product of $n-1$ Weyl tensors or AdS curvatures, which matches with the
result in ref.\cite{Miskovic:2007mg}. The degeneracy condition (\ref{DeltaK
) as an overall factor shows the validity of the formula for a particular
gravity theory with multiplicity $k$. Indeed, for theories with the
intermediate degeneracy $1<k<n-1$, we can drop $n-k-1$ curvatures from the
original charge formula (\ref{qqeven}).
\subsection{Odd dimensions $D=2n+1$}
\label{Odd+CS} In odd dimensions, black hole mass is associated with
\tau^{j}_{i}$ given by eq.(\ref{qqodd}), which can be recast as
\begin{equation}
\tau _i^j =\frac{(2n-1)!(-\ell _{\mathrm{eff}}^2)^{n-1}}{2^{3n-4}(n-1)!^2
16\pi G}\, \delta _{ i_{ 1 }\cdots i_{ 2n } }^{ jj_{ 2 }\cdots j_{ 2n } }K_{
i }^{ i_{ 1 } }\delta _{ j_{ 2 } }^{ i_{ 2 } }\sum _{ p=1 }^{ n } \frac {
(-1)^{ p }p\alpha _{ p }\, \ell _{ \mathrm{{eff } }}^{ 2(1-p) } }{
(2n-2p+1)! } \int _{ 0 }^{ 1 } du\, (\mathcal{I}_{p})_{ j_{ 3 }\cdots j_{
2n } }^{ i_{ 3 }\cdots i_{ 2n } }(u)\,, \label{qodd}
\end{equation}
where the tensorial quantity $\mathcal{I}_{p}(u)$ is a polynomial of order
n-1$ in the curvature,
\begin{equation}
\mathcal{I}_{p}(u)=\left( R+\frac{u^{2}}{\ell _{\mathrm{eff}}^{2}}\,\delta
^{\lbrack 2]}\right) ^{n-1}-(u^2-1) ^{n-1}(-R)^{p-1}\left( \frac{1}{\ell _
\mathrm{eff}}^{2}}\,\delta ^{\lbrack 2]}\right) ^{n-p}. \label{I}
\end{equation}
We will restrict our analysis to the theories with $k<n$. We can rearrange
the expression for $\tau_{i}^{j}$ and factorize it in a similar fashion as
in the even dimensional case (see Appendix \ref{oddfactorization}),
\begin{eqnarray}
\tau _i^j&=&\frac{ 2^{2n-2}{(n-1)!}^2\ell _{ \mathrm{eff}}^{ 2(n-1) } }{ 2^{ n-2 }(2n-1)! 16\pi G}\,\delta _{ i_{ 1 }i_{ 2 }\cdots
i }_{ 2k }{ i }_{ 2k+1 }\dots { i }_{ 2n }i_{ 2n-1 } }^{ jj_{ 2 }\cdots
j }_{ 2k }{j }_{ 2k+1 }\dots { j }_{ 2n }j_{ 2n-1 } }K_{ i }^{ i_{ 1 }
}\left( R_{ j_{ 2 }j_{ 3 } }^{ i_{ 2 }i_{ 3 } }+\frac { 1 }{ \ell _{ \mathrm{eff}}^2}\, \delta _{ j_{ 2 }j_{ 3 } }^{ i_{ 2 }i_{ 3 } } \right)
\times \dots \notag \\
&& \qquad \qquad \dots \times \left( R_{j_{ 2k }j_{ 2k+1 } }^{i_{ 2k }i_{
2k+1 } }+\frac { 1 }{ \ell _{\mathrm{{eff } }}^{ 2 } }\, \delta _{j_{ 2k
}j_{2k+1 } }^{i_{ 2k }i_{ 2k+1 } } \right) \mathcal{ P }_{j_{
2k+2 }\cdots j_{ 2n-1 } }^{ i_{2k+2 }\cdots i_{2n-1 } }(R)\,,
\label{qoddfact}
\end{eqnarray}
where $\mathcal{P}(R)$ is a polynomial in the curvature tensor of
order $n-k-1$. It turns out that, again, it has the form of a linear combination of
the derivatives $\Delta ^{(p)}$ of the characteristic polynomial (see eq.
\ref{P final}) in Appendix \ref{Fact})
\begin{equation}
\mathcal{P}_{j_{2k+2}\cdots j_{2n-1}}^{i_{2k+2}\cdots
i_{2n-1}}(R)=\sum_{s=0}^{n-1-k}\sum_{p=k}^{k+s}\Delta
^{(p)}(C_{k+s,p})_{j_{2k+2}\cdots j_{2n-1}}^{i_{2k+2}\cdots
i_{2n-1}}F_{j_{2n-2}j_{2n-1}}^{i_{2n-2}i_{2n-1}}\cdots
F_{j_{2n-2}j_{2n-1}}^{i_{2n-2}i_{2n-1}}\,,
\end{equation}
with some tensorial coefficients $C_{k+s,p}$ that differ from the even dimensional case. Evaluating the polynomial in the
asymptotic region, the only part relevant for the conserved charge is
\begin{equation}
\mathcal{P}_{j_{2k+2}\cdots j_{2n-1}}^{i_{2k+2}\cdots i_{2n-1}}(-\ell _
\mathrm{eff}}^{-2}\delta ^{[2]})=\Delta
^{(k)}(C_{kk})_{j_{2k+2}\cdots j_{2n-1}}^{i_{2k+2}\cdots i_{2n-1}}
\end{equation}
where the
coefficient has the form
\begin{equation}
(C_{kk})_{j_{2k+2}\cdots j_{2n-1}}^{i_{2k+2}\cdots i_{2n-1}}=\frac{\left(
-1\right) ^{n-k-1}2^{2n-3}\left( n-1\right) !^{2}}{(2n-1)!}\,\ell _{\mathrm
eff}}^{2}\,{\delta }_{{j}_{2k+2}{j}_{2k+3}}^{{i}_{2k+2}{i}_{2k+3}}\cdots
\delta }_{{j}_{2n}{j}_{2n-1}}^{{i}_{2n}{i}_{2n-1}}
\end{equation}
as shown in eq.(\ref{cqq-odd}) of Appendix \ref{oddfactorization}.
Knowing the behavior of the extrinsic curvature (\ref{extrinsic}) and the
relation (\ref{WeylAsymp}), we can evaluate the charge asymptotically as
\begin{eqnarray}
\tau _{i}^{j} &=&\frac{{{\ell _{\mathrm{eff}}{(-1)}^{k}\Delta ^{(k)}}}}
16\pi G\,{2}^{n-1}(2n-2)!}\,\delta _{ii_{2}\cdots {i}_{2k}{i}_{2k+1}\dots {i
_{2n}i_{2n-1}}^{jj_{2}\cdots {j}_{2k}{j}_{2k+1}\dots {j}_{2n}j_{2n-1}}\,{W}_
{j}_{2}{j}_{3}}^{{i}_{2}{i}_{3}}\cdots {W}_{{j}_{2k}{j}_{2k+1}}^{{i}_{2k}{i
_{2k+1}}\times \notag \\
&&\qquad \qquad \qquad\qquad \qquad \times {\delta }_{{j}_{2k+2}{j}_{2k+3}}^
{i}_{2k+2}{i}_{2k+3}}\cdots {\delta }_{{j}_{2n}{j}_{2n-1}}^{{i}_{2n}{i
_{2n-1}}\,.
\end{eqnarray}
Finally, contracting the antisymmetric deltas, the charge density becomes
\begin{equation}
\tau ^{ j }_{ i }=\frac { { { \ell _{ \mathrm{{\ eff } }}{\ (-1) }^{ k
}\Delta ^{ (k) }(2n-2k-1)! } } }{ 16\pi G\, {\ 2 }^{ k }(2n-2)! } \, \delta
_{ii_{ 2 }\cdots {i }_{ 2k }{ i }_{ 2k+1 } }^{ jj_{ 2 }\cdots {j }_{
2k }{ j }_{ 2k+1 } }\, {W }_{{ j }_{ 2 }{j }_{ 3 } }^{ { i }_{ 2 }{
i }_{ 3 } }\cdots {W }_{ {j }_{ 2k }{j }_{ 2k+1 } }^{ { i }_{ 2k }{
i }_{ 2k+1 } }\,.
\end{equation}
The last expression is valid for theories whose AdS vacua have a degeneracy
level in the interval $1\leq k \leq n-1$. It is a generalization of the
Ashtekar-Magnon-Das conformal mass formula \cite{Ashtekar:1984zz,Ashtekar:1999jx} to the Lovelock AdS gravity \cit
{Arenas-Henriquez:2017xnr}. Once again, for $k=1$ it reduces to the known
Conformal Mass (\ref{conformalmass}).
In the case of Chern-Simons AdS gravity, the mass term does not fall off as
r\rightarrow \infty$. The fact that the energy for global AdS is not
continuously connected with the spectrum of black holes of the theory
indicates that the energy for the vacuum state cannot be achieved by charge
that is proportional to the Weyl tensor. This presents a qualitative
difference between the theories with degeneracy $k<n$ and Chern-Simons AdS
gravity, which is reflected in their holographic properties \cit
{Banados:2004zt,Banados:2005rz}.
\subsection{Energy of the LUV AdS black hole }
As an example, we calculate the total energy of the LUV AdS black hole
solution described by the metric (\ref{ansatz}) with the metric function
\ref{LUV-BH}). We can identify $\frac{\mu }{\Delta ^{(k)}}=2GM$. Because the
tensor $X_{\alpha \beta }^{\mu \nu }$ given by (\ref{X}) identically
vanishes in this case, the Weyl tensor and the AdS curvature are equal
on-shell, ${W}_{\alpha \beta }^{\mu \nu }={F}_{\alpha \beta }^{\mu \nu }$.
The degeneracy (\ref{DeltaK}) for the LUV gravity becomes
\begin{equation}
\Delta _{\mathrm{LUV}}^{(k)}=2^{k-1}(-1)^{k-1}(D-3)!\,.
\end{equation}
The total energy of the system is the Noether charge (\ref{Q}) for the time
translations $\xi =\partial _{t}$ and the boundary with the unit vector
u_{j}=-\sqrt{f}\,\delta _{j}^{t}$. Furthermore, we have $\sigma
_{nm}=r^{2}\gamma _{nm}(y)$, where the transverse metric $\gamma _{nm}$
depends only on the transverse coordinates $y^{m}$. Then we can find the
Jacobian ${\sqrt{\sigma }=r}^{D-2}\sqrt{\gamma }$ and the volume of the
transverse section $\Omega _{D-2}=\int\limits_{\Sigma _{\infty }}{{d}^{D-2}
\sqrt{\gamma }}$.
With this at hand, we can evaluate the total energy a
\begin{equation}
E=Q[\partial _{t}]=E_{\mathrm{vacum}}-\Omega _{D-2}\left( {r}^{D-2}\sqrt{f
\tau _{t}^{t}\right) _{r\rightarrow \infty }\,,
\end{equation
where the vacuum energy for Lovelock gravity (the energy of the global AdS
space) exists in odd dimensions $D$ only, and was calculated in ref.\cit
{kofinas}.
On the other hand, the charge density tensor $\tau _{i}^{j}$ given by eqs.
\ref{Teven}) and\ (\ref{Todd}) in even and odd dimensions, respectively, has
the form
\begin{equation}
\tau _{t}^{t}=\frac{{{{(-1)}^{k}}}\ell _{\mathrm{eff}}{{{\ }\Delta
^{(k)}(D-2k-2)!}}}{16\pi G\,{2}^{k}(D-3)!}\,\delta _{n_{1}\cdots {n
_{2k}}^{m_{1}\cdots {m}_{2k}}\,{W}_{{m}_{1}{m}_{2}}^{{n}_{1}{n}_{2}}\cdots {
}_{{m}_{2k-1}{m}_{2k}}^{{n}_{2k-1}{n}_{2k}}\,.
\end{equation
The only relevant components of the Weyl tensor read
\begin{equation}
W_{m_{1}m_{2}}^{n_{1}n_{2}}=\left( \frac{2GM}{r^{D-1}}\right) ^{\frac{1}{k
}\delta _{m_{1}m_{2}}^{n_{1}n_{2}}\,. \label{WeylLUV}
\end{equation
Using the expression
\begin{equation}
\delta _{n_{1}\cdots {n}_{2k}}^{m_{1}\cdots {m}_{2k}}\,{W}_{{m}_{1}{m}_{2}}^
{n}_{1}{n}_{2}}\cdots {W}_{{m}_{2k-1}{m}_{2k}}^{{n}_{2k-1}{n}_{2k}}=\frac{2G
}{r^{D-1}}\,\frac{2^{k}\left( D-2\right) !}{\left( D-2k-2\right) !}\,,
\end{equation
it is straightforward to evaluate the energy density as
\begin{equation}
\tau _{t}^{t}=-\frac{\ell _{\mathrm{eff}}{\ }2^{k}\left( D-2\right) !}{32\pi
G\,}\,\frac{2GM}{r^{D-1}}\,,
\end{equation
and the total energy of the syste
\begin{equation}
E=E_{\mathrm{vacuum}}+\frac{\left( D-2\right) !2^{k}\Omega _{D-2}}{16\pi \,
\,M\,. \label{E}
\end{equation}
The total mass, $E-E_{\mathrm{vacuum}}$, is indeed linear in the parameter $M$. Furthermore, in order to have the charge that is directly $M$ as in ref.\cite{Crisostomo:2000bb}, the gravitational action (\ref{action}), (\ref{alpha_LUV}) has to be normalized suitably, by dividing it by $\Omega _{D-2}$, a $D$-dependent factor and introducing the gravitational constant $G_{k}$ which explicitly depends on the multiplicity \cite{Footnote}.
General Relativity ($k=1$) and BI gravity ($k=n-1$) are particular cases of LUV gravity in AdS space.
\section{Conclusions}
\label{sec5}
We have derived the Conformal Mass formula for a branch of Lovelock AdS gravity with
a $k$-fold vacuum degeneracy. This comes as the generalization of the results found in ref.\cite{Arenas-Henriquez:2017xnr}, which indicate that the energy of
black holes in that theory cannot be written as a linear expression in the AdS curvature (\ref{AdScurvature}) or, equivalently, in
terms of the electric part of the Weyl tensor. On the contrary, we find that the conserved quantity must be nonlinear in the Weyl tensor in order
to capture the falloff properties of the mass term in the metric.
To summarize, the charge density tensor has the form
\begin{equation}
\tau _{i}^{j}=\frac{{{{(-1)}^{k}}}\ell _{\mathrm{eff}}{{{\ }\Delta
^{(k)}(D-2k-2)!}}}{16\pi G\,{2}^{k}(D-3)!}\,\delta _{ii_{2}\cdots {i}_{2k}{i
_{2k+1}}^{jj_{2}\cdots {j}_{2k}{j}_{2k+1}}\,{W}_{{j}_{2}{j}_{3}}^{{i}_{2}{i
_{3}}\cdots {W}_{{j}_{2k}{j}_{2k+1}}^{{i}_{2k}{i}_{2k+1}}
\end{equation}
in both even and odd dimensions, respectively, given by eqs.(\ref{Teven}) and (\ref{Todd}).
As an example, we showed that this formula gives the total energy of a static topological black hole in AdS space of multiplicity $k$.
When $k=1$, this tensor becomes proportional to the electric part of the Weyl
tensor (\ref{conformalmass}). In general, when $k>1$, the
appearance of the degeneracy condition in the corresponding surface terms coming from an arbitrary variation of the
renormalized action (\ref{Ireg}) may be useful in understanding holographic properties of degenerate AdS gravity theories. In
particular, it has been claimed \cite{Camanho:2013pda,Bueno:2018yzo} that the $C_{T}$ coefficient in holographic two-point functions is proportional to the first degeneracy
condition (\ref{DeltaPrime}). As the coefficient is linked to the
$a$-charge (type $A$-anomaly), recent results suggest that it is necessary to consider higher degeneracy conditions \cite{Li:2018drw} when one deals with theories with degenerate AdS vacua.
\section*{Acknowledgments}
G.A.H. and R.O. thank to G. Anastasiou, I.J. Araya, C. Arias, P. Bueno, F. Diaz and D. Rivera-Betancour for insightful comments.
This work was supported in parts by Chilean FONDECYT projects N$^{\circ}$1170765 ``Boundary dynamics in anti-de Sitter gravity and gauge/gravity
duality'' and N$^{\circ}$1190533 ``Black holes and asymptotic symmetries'' and in part by
the Natural Sciences and Engineering Research Council of Canada.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 1,746 |
\section{Introduction}
The Planck collaboration~\cite{2018arXiv180706209P, Akrami:2018odb, 2018arXiv180706205P} produced the latest and strongest constraints on the parameters of the standard cosmological model. Their results confirmed previous experiments such as WMAP~\cite{2013ApJS..208...19H, 2013ApJS..208...20B} showing that the 6-parameter $\Lambda$CDM model continues to be the best fit model for the cosmic microwave background (CMB) data at high redshift. In addition, low redshift experiments also point for this concordance model as the best phenomenology to describe the evolution of the universe. Thus, one of the theoretical and observational challenges of present day cosmology is to push our knowledge further back in time into the primordial universe.
Inflation is certainly the most popular model to describe the primordial universe and has become the paradigm dynamics for this era. Inflation alleviates some of the standard model problems such as the flatness problem and provides a consistent origin for the primordial cosmological perturbations~\cite{Guth:1980zm, Linde1982389, Albrecht_inflation, Riotto:2002yw, Lyth:1998xn}. However, there are others competitive models that also solve the same problems of the standard model and have a good fit to the available observational data such as bounce models~\cite{Brandenberger:2012zb, Falciano:2015kya, Falciano:2008gt, Peter:2002cn}, pre-big-bang~\cite{Durrer:2002jn, Enqvist:2001zp}, ekpyrotic~\cite{Khoury:2001zk, Lyth:2001pf, Lehners:2007ac, Martin:2001ue} and string-inspired models~\cite{Brandenberger1989391, String_gas_Battefeld}.
In particular, bounce models figure among the simplest extensions of the standard model. Evidently, in order to produce a bounce one has to violate the null energy condition or to appeal to modified gravity theories. But similarly to inflation, one can also use bounce models as a pure phenomenological scenario. Furthermore, a close analysis shows that the theoretical support for inflation is as good as for bounce models. Therefore, the current status is that there is no reason to privilege one over the other.
Future experiments will allow us to probe deeper into primordial universe physics by measuring the CMB B-mode polarization, primordial non-gaussianities and the spectrum of primordial gravitational waves~\cite{Matsumura:2013aja, Abazajian:2016yjj, Ade:2018sbj, Bouchet:2011ck}. This new data can break the above mentioned degeneracy between different primordial universe scenarios. In this context, an important theoretical challenge is to make concrete predictions that would allow us to to discriminate between these models.
The aim of the present paper is to construct a bounce model that mimics the inflationary model that best fit the observation, namely Starobinsky inflation. Our construction relies on Wands' duality~\cite{Wands:1998yp, Brustein:1998kq, Finelli:2001sr}, which shows that the Mukhanov-Sasaki equation~\cite{Mukhanov:1990me, Brandenberger:2004LNP...646..127B} displays a symmetry transformation by changing appropriately its time-dependent mass term. One of the advantages of this construction is that one can control every contribution to the primordial power spectrum and check how far we can emulate a given primordial model with a different scenario. Thus, the limits of this construction indicate how one can distinguish different primordial universe models. In particular, we show that a quasi-matter bounce can reproduce the same dependence of the scalar spectral index $n_s$ and the tensor-to-scalar ratio $r$ with the slow-roll parameters as happens in Starobinsky inflation but there is a numerical factor that encodes the physical different between these two models.
The paper is organized as follows. Section~\ref{DfCP} briefly review the basic features of linear cosmological perturbation theory both in GR and modified theories of gravity and highlight some important features of Starobinsky's inflation. In section~\ref{MSI} we introduce Wands' duality and use it to construct the appropriate collapsing phase prior to the bounce. Section~\ref{CB} is devoted to the bounce phase, which is realized by quantum effects within the Loop Quantum Cosmology scenario and in section~\ref{Con} we conclude with some final remarks.
\section{Cosmological Perturbations}\label{DfCP}
In this paper, we are interested in linear cosmological perturbations and how they can be connected with cosmological observations. There are different primordial scenarios and for each of them the cosmological perturbations have specific developments. The original formulation of Starobinsky inflation~\cite{Starobinsky:1980te} is a modified theory of gravity~\cite{DeFelice:2010aj, Hwang:1996bc, Hwang:1996xh} and hence has to be treat differently from the conventional single field inflationary models\footnote{The fact that Jordan frame description of Starobinsky inflation is formally equivalent to a single field inflationary model in GR is a nontrivial exception.} in General Relativity (GR). Notwithstanding, the dynamic equations for the first order perturbations are formally very similar. In this section we briefly summarize the theory of cosmological perturbations for a scalar field in GR and for $f(R)$ theories.
\subsection{Cosmological Perturbations in General Relativity}
Cosmological perturbation theory shows that at linear order each type of tensor mode evolves independently and hence we can treat scalar and tensor perturbations separately. Let us consider GR minimally coupled with a scalar field in the comoving gauge. Expanding the action up to second order in the curvature perturbation ${\mathcal{R}}$ \cite{Mukhanov:1990me} gives
\begin{align}
S_{(2)} = \frac{M_{\mathrm{Pl}}^2}{2}\int \dd t \, \dd x^3 \, a^3 \frac{\dot{\varphi}^2}{H^2} \left[\dot{{\mathcal{R}}}^2 - \frac{(\partial_{i} {\mathcal{R}})^2}{a^2} \right]\ , \label{ac2o}
\end{align}
where a dot means derivative with respect to cosmic time and $ M_{\mathrm{Pl}}$ is the reduced Planck mass\footnote{The reduced Planck mass absorbs the $\sqrt{8\pi}$ in its definition, hence $M_{\mathrm{Pl}}=m_{pl}/\sqrt{8\pi}=2.44\times10^{18} {\rm GeV}=4.35\times10^{-6} {\rm g}$ and we use throughout the paper $\hbar=c=1$.}. Defining the Mukhanov-Sasaki variable $v(t,\vec{x})$ and the function $z_{s}(t)$ as
\begin{align}
v \equiv z_{s}{\mathcal{R}} \quad ,\qquad
z_{s} \equiv \frac{a}{H}\sqrt{\rho + p} = a \frac{\dot{\varphi}}{H} \ , \label{eqz}
\end{align}
the action \eqref{ac2o} simplifies to
\begin{equation}
S_{(2)} = \frac{M_{\mathrm{Pl}}^2}{2} \int \mathrm{d}\eta \, \mathrm{d}x^3 \left[v'^2 -(\partial_{i} v)^2 + \frac{z_{s}''}{z_{s}}v^2 \right]\ ,
\end{equation}
where now the prime means time derivative with respect to conformal time given by $\eta=\int \, a^{-1} \dd t $. Variation of the above action with respect to $v(t,\vec{x})$ gives the Mukhanov-Sasaki equation. Using a Fourier decomposition, the mode function ${v}_{\vb{k}}(\eta)$ satisfies the dynamic equation
\begin{align}
{v}''_{\vb{k}} + \left(k^2 - \mu^2_{s}\right)v_{\vb{k}} = 0\ , \ \mbox{with}\quad \mu^2_{s} = \frac{z_{s}''}{z_{s}} \label{mesc} \ .
\end{align}
Eq.~\eqref{mesc} is formally identical to a parametric harmonic oscillator with mass term $\mu_{s}(\eta)$. Its time dependence comes from the background dynamics through the function $z_{s}(\eta)$. Strictly speaking, Wands' duality~\cite{Wands:1998yp} is a variable transformation that leaves this mass term invariant.
The tensor sector of the second order action reads
\begin{equation}
S_{(2)} = \frac{M_{\mathrm{Pl}}^2}{8} \int \dd \eta \, \dd x^3 a^2 \left[({h}'_{ij})^2 - (\partial_{l}{h}_{ij})^2\right]\ ,
\end{equation}
where $h_ {ij}(\eta,\vec{x})$ is the tensor part of the metric perturbation, i.~e. a gauge invariant quantity. Using again a Fourier decomposition for each polarization mode $h^{\lambda}_{\vb{k}}(\eta)$ and defining its associated Mukhanov variable
\begin{equation}\label{mukhtensor}
v^{\lambda}_{\vb{k}} = \frac{a\, M_{\mathrm{Pl}} }{2}h^{\lambda}_{\vb{k}}\ ,
\end{equation}
the resulting Mukhanov-Sasaki equation for each polarization is
\begin{align}\label{mstgr}
{v^{\lambda}_{\vb{k}}}'' + \left(k^2 - \mu^2_{t}\right)v^{\lambda}_{\vb{k}} = 0\ , \ \mbox{with}\quad \mu^2_{t} = \frac{a''}{a} \ .
\end{align}
It is worth recalling that for a quasi-dust domination where $H^2\approx\dot{\varphi}^2\approx$ constant, both mass term are equal $\mu^2_{s}=\mu^2_{t}$. As a result, the scalar and tensor modes have identical power spectrum ${k}$-dependence.
\subsection{Cosmological Perturbations in $f(R)$ theories}
Apart from the degrees of freedom already present in GR, $f(R)$ theories have an extra scalar degree of freedom~\cite{DeFelice:2010aj,Sotiriou:2008rp}. By their formal equivalence with massless scalar-tensor theories, we know that this extra degree of freedom propagates with the speed of light. At the background level, $f(R)$ theories are observationally indistinguishable from the $\Lambda \mathrm{CDM}$ model. It is only in the perturbative level that this two frameworks can be put into test.
Considering a FLRW universe, the background value of the Ricci scalar depends only on time. Contrary to GR where the mechanism to generate inflation resides in the matter field (commonly a scalar field with appropriate potential), in $f(R)$ models of inflation it is the non-linearity of the Ricci scalar that guides the evolution without any scalar field. The extra degree of freedom is encoded in $F = \partial f / \partial R$, which can be decomposed as $F(\eta,\vec{x}) = \bar{F}(\eta) + \delta F(\eta,\vec{x})$, where $\bar{F}(\eta)$ is the background and $\delta F(\eta,\vec{x})$ its perturbation. Expanding the action up to second order in ${\mathcal{R}}$~\cite{DeFelice:2010aj, Hwang:1996bc, Hwang:1996xh} gives
\begin{align}
S_{(2)} &= \frac{1}{2}\int \dd \eta \, \dd^{3}x \, a^2 Q_{s} \left[{{\mathcal{R}}'}^2 - (\partial_{i} {\mathcal{R}})^2 \right], \label{aperfr} \\
Q_{s} &= 3M_{\mathrm{Pl}}^2\frac{{F'}^2/ 2F}{\left[\mathcal{H} + \left(\frac{{F'}}{2F}\right) \right]^2}. \label{defQs}
\end{align}
where $\mathcal{H}\equiv a'/a$ is the Hubble factor in conformal time. The function $Q_{s}$ plays a similar role as $\dot{\varphi}^2/H^2$ in Eq.~\eqref{ac2o}. Therefore, it is straightforward to vary the above action and find
\begin{align}
&{v}''_{\vb{k}} + \left(k^2 - \mu^2_{fs}\right)v_{\vb{k}} = 0 \quad , \quad v_{\vb{k}} = z_{fs} \, {\mathcal{R}}_ {\vb{k}} \quad ,\label{mssfr}\\
& \mu^2_{fs} = \frac{z_{fs}''}{z_{fs}} \quad , \quad z_{fs} = a \sqrt{Q_{s}}\quad .\label{defzfs}
\end{align}
Similar to GR, in this scenario the perturbation has quantum origin. Modified theories of gravity follows the same canonical quantization procedure and impose the same Bunch-Davies initial vacuum state for the variable $v$. The scalar power spectrum is defined as
\begin{align}\label{Pe}
\mathcal{P}_{\mathcal{R}} &= \frac{k^3}{2 \pi^2} |{\mathcal{R}}|^2 = \frac{k^3}{2 \pi^2} \frac{\left | v \right |^2}{a^2Q_s} \quad .
\end{align}
The tensor perturbation expansion is completely analogous to GR since there is no extra tensor degree of freedom. Therefore, the definition of the two polarizations remain identical but there is an additional $F$ term multiplying the second order action that now reads
\begin{align}
S_{(2)} = \frac{M_{\mathrm{Pl}}^2}{8} \int \dd \eta \, \dd x^3 a^2F \left[({h}'_{ij})^2 - (\partial_{l}{h}_{ij})^2\right]\ .
\end{align}
The extra $F$ term is absorbed in the definition of the mass term. Variation of the action gives
\begin{align}
&{v^{\lambda}_{\vb{k}}}'' + \left(k^2 - \mu^2_{ft}\right)v^{\lambda}_{\vb{k}} = 0 \quad , \quad v^{\lambda}_{\vb{k}} = \frac{M_{\mathrm{Pl}}}{2}\, z_{ft} \, h^{\lambda}_{\vb{k}} \quad ,\label{mstfr}\\
&\mu^2_{ft} = \frac{z_{ft}''}{z_{ft}} \quad , \quad z_{ft} = a \sqrt{F}.
\end{align}
Taking into account the polarization states, the spectrum of tensor perturbations is given by
\begin{equation}\label{Pt}
\mathcal{P}_{\mathcal{T}} = 2\times \frac{k^3}{2 \pi^2} \left | h \right |^2= \frac{4k^3}{\pi^2M_{\mathrm{Pl}}^2} \frac{\left | v \right |^2}{a^2F} \quad .
\end{equation}
The dynamic equation for the scalar and tensor linear perturbation in GR and $f(R)$ theories are formally identical. The difference between Eq.'s~\eqref{mesc},\eqref{mstgr},\eqref{mssfr} and \eqref{mstfr} are encoded in the definition of the Mukhanov-Sasaki variables and their mass terms. During reheating and the bounce phase it is expected that the dynamics to be modified by new phenomena characteristic of these periods. Indeed, loop quantum cosmology corrections modify the formal structure of the dynamic equation depending on an energy scale parameter $\rho_{c}$. We recover the Mukhanov-Sasaki dynamics in the limit $\rho_{c}\rightarrow\infty$.
\subsection{Starobinsky inflation}
Assuming the cold inflationary paradigm, the model that best fit the observation data is the Starobinsky inflation. This model can be described as a single field inflation~\cite{Mukhanov:1990me} (Einstein frame) or as a solution of a modified theory of gravity~\cite{Starobinsky:1980te} (Jordan frame). We shall follow its original formulation and described it in terms of a $f(R)$ gravity using the metric formulation~\footnote{Inflationary models work on both frames~\cite{Mukhanov:2005sc} but physical quantities are well defined only in Jordan frame~\cite{DeFelice:2010aj}. There is an extensive discussion on the validity of the two frames in the literature (see~\cite{Kamenshchik:2016gcy} for more details).}. An exact vacuum de Sitter expansion is a solution of the dynamic equations only if $f(R) = c_0 R^2$~\cite{DeFelice:2010aj}. The Starobinsky inflation propose a theory with $f(R) = R +c_0 {R^2}$, and hence, the gravitational sector of the action reads
\begin{align}\label{acstab}
S = \frac{M_{\mathrm{Pl}}^2}{2}\int\dd^4 x \, \sqrt{-g} \left( R + \frac{R^2}{6M^2}\right) \ ,
\end{align}
where $M$ is a mass parameter that gives the energy scale where the dynamics deviates from GR. During the inflationary phase, the scale factor in leading order in the slow-roll parameters can be approximated by~\cite{Starobinsky:1980te,Mukhanov:1990me}
\begin{equation}\label{Stscal1}
a(t) = a_{0}\left(t_{s} - t\right)^{1/2}\exp\left[-\frac{M^2}{12}\left(t_{s} - t\right)^2\right] \ .
\end{equation}
Calculating the next order correction~\cite{Koshelev:2016xqb} gives
\begin{align}
a(t) &= a_{0}\left(t_{s} - t\right)^{-1/6}\exp\left[-\frac{M^2}{12}\left(t_{s} - t\right)^2\right] \ .\label{festab}
\end{align}
Note that the difference between these two orders is just the power of the polynomial. Since the evolution is dominated by the exponential, this modification is very small. The adequate definition of slow-roll parameters in $f(R)$ theories is slightly different than in GR. Following the nomenclature of~\cite{DeFelice:2010aj} we have
\begin{align}
\epsilon_{1} = -\frac{\dot{H}}{H^2} \quad ,
&&
\epsilon_{3} = \frac{\dot{F}}{2HF}\quad ,
&&
\epsilon_{4} =\frac{\ddot{F}}{H\dot{F}}\quad .
\end{align}
Let us calculate these parameters for a scale factor of the form
\begin{equation}\label{starobagener}
a(t) = a_{0}\left(t_{s} - t\right)^{p}\exp\left[-\frac{M^2}{12}\left(t_{s} - t\right)^2\right] \ .
\end{equation}
Straightforward calculation gives
\begin{widetext}
\begin{align}
\epsilon_{1} &= \frac{6}{M^2\left(t_{s} - t\right)^2}\left(1+\frac{6p}{M^2\left(t_{s} - t\right)^2}\right)\left(1-\frac{6p}{M^2\left(t_{s} - t\right)^2}\right)^{-2}=\frac{6}{M^2\left(t_{s} - t\right)^2}+ \mathcal{O}(M^{-4})\quad ,\label{hpsre}\\
\epsilon_{3} &= -\frac{M^2}{6H^2}\left[1+\frac{p}{H\left(t_{s} - t\right)}\left(1+\frac{3(1-2p)}{M^2\left(t_{s} - t\right)^2}\right)\right]\left[1+\frac{M^2}{6H^2}\left(1-\frac{3p}{M^2\left(t_{s} - t\right)^2}\right)\right]^{-1}
=-\frac{6}{M^2\left(t_{s} - t\right)^2}+ \mathcal{O}(M^{-4})\quad ,\label{epsilon3}\\
\epsilon_{4} &=-\frac{M^2}{6H^2}\left[1-\frac{54p(1-2p)}{M^4\left(t_{s} - t\right)^4}\right]\left[1+\frac{p}{H\left(t_{s} - t\right)}\left(1+\frac{3(1-2p)}{M^2\left(t_{s} - t\right)^2}\right)\right]^{-1}
=-\frac{6}{M^2\left(t_{s} - t\right)^2}+ \mathcal{O}(M^{-4})\quad .\label{epsilon4}
\end{align}
\end{widetext}
All three slow-roll parameters are equal in leading order and do not depend on the power $p$ of the polynomial in the scale factor \eqref{starobagener}. Any correction from a different $p$ is at least of order $\mathcal{O}(M^{-4})$.
During a quasi-de Sitter expansion, the general solution of Eq.'s~\eqref{mssfr} and \eqref{mstfr} can be written in terms of Hankel functions of order $\gamma$, which depends on the slow-roll parameters. Assuming a Bunch-Davies initial state and following the standard matching procedure at horizon crossing one can show that
\begin{equation}
v_{\vb{k}}(\eta)=\frac{\sqrt{\pi |\eta|}}{2}e^{i(1+2\gamma )\pi/4}H_{\gamma}^{(1)}(k|\eta|)
\end{equation}
The evolution of the scalar perturbations gives~\cite{DeFelice:2010aj}
\begin{align}
\mathcal{P}_{\mathcal{R}} \approx \frac{1}{Q_{s}}\left(\frac{H}{2\pi}\right)^2\left(\frac{|k\eta_c|}{2}\right)^{n_{{\mathcal{R}}} - 1}&\ ,&\ n_{{\mathcal{R}}} - 1 \approx -4\epsilon_{1} + 2\epsilon_{3} - 2\epsilon_{4}\ ,
\end{align}
where $\eta_c$ is the time when the wave-number $k$ crosses the horizon. The tensor perturbations follows a similar reasoning mutatis mutandis the evolution
\begin{align}
\mathcal{P}_{\mathcal{T}} \approx \frac{2}{\pi^2 F}\left(\frac{H}{M_{\mathrm{Pl}}}\right)^2\left(\frac{|k\eta_c|}{2}\right)^{n_{T}}&\ ,&\ n_{T}\approx -2\epsilon_{1} - 2\epsilon_{3}\ .
\end{align}
Finally, the tensor-to-scalar ratio reads
\begin{equation}
r\equiv \frac{\mathcal{P}_{\mathcal{T}}}{\mathcal{P}_{\mathcal{R}}}\approx\frac{8 Q_s}{M_{\mathrm{Pl}}^2F}\approx 48\epsilon_{3}^2
\end{equation}
It is convenient to express all observables in terms of the number of e-folds. By definition, the total number of e-folds is $N=\log{a_f/a_i}$ where $t_i$ and $t_f$ are respectively, the onset and end of inflation, i.~e.
\begin{align}
N &\approx \frac{M^2}{12}\left(t_{s} - t_i\right)^2\approx\frac{1}{2\epsilon_{1}}\approx-\frac{1}{2\epsilon_3}
\end{align}
Combining these results, the spectral index and the tensor-to-scalar ration read
\begin{align} \label{ns_r_star}
n_{{\mathcal{R}}} - 1 \simeq -4\epsilon_{1} = -\frac{2}{N} \quad ,\quad
r &\simeq 48\epsilon_{3}^2 \simeq \frac{12}{N^2}\quad .
\end{align}
The Planck 2018 release~\cite{Akrami:2018odb} gives a spectral index of $n_{\mathcal{R}}=0.9649\pm0.0042$ at $68\%$ confidence level. This implies that $50< N< 65$. We can recast the spectral index and the tensor-to-scalar ratio as
\begin{align}
&n_{{\mathcal{R}}} - 1 \approx -3,51\times 10^{-2}\left(\frac{N}{57}\right)^{-1} \ ,\label{nsPlanck}\\
& r \approx 3,69\times 10^{-3}\left(\frac{N}{57}\right)^{-2}\ .\label{rPlanck}
\end{align}
The Planck $95\%$ confident level upper limit on the tensor-to-scalar ratio is $r_{0.002}<0.10$. This value is even tightened by a combining analysis with the BICEP2/Keck Array BK14 data that bring the tensor-to-scalar value to $r_{0.002}<0.064$. The predicted value for the Starobinsky inflation Eq.~\eqref{rPlanck} is safely within the observational measurements.
\section{Mimicking Starobinsky Inflation}\label{MSI}
The Starobinsky model describes a universe with a violent quasi-de Sitter expansion. This primordial universe model has several known advantages that we simply summarize here by stating that it is the inflationary model that best fits the data. It can be considered as the archetype of inflationary models. Thus, in order to be considered as competitive, any primordial universe model must fit the data as well as Starobinsky´s model.
Our goal now is to construct a bounce model that encodes the key features of the Starobinsky model in the first perturbative order. The suitable mathematical tools for this is Wands' duality~\cite{Wands:1998yp}. This duality can be understood as a symmetry transformation that leaves the mass term of the Mukhanov-Sasaki equation invariant.
All linear order perturbation equation described in section~\ref{DfCP} has the same structure. They are parametric oscillators with time dependent mass terms. The dynamics of the background enters only on the mass $\mu^2_\alpha\equiv z''_\alpha/z_\alpha$ where the index $\alpha$ designates if we are considering a scalar or a tensor perturbation and if the framework is the vacuum $f(R)$ theory or the scalar field minimally coupled in GR. Any two distinct backgrounds composing the same mass term $\mu_\alpha$ will produce the same evolution for the linear order perturbation. In order to implement this idea, consider a given function $z_\alpha(\eta)$. We can define a new function
\begin{equation}\label{wdual}
\tilde{z}_\alpha(\eta)\equiv c_0 \ z_\alpha(\eta)\, \int_{\eta_{*}}^{\eta}\frac{\dd x}{z_\alpha^2(x)} \ ,
\end{equation}
with $c_0$ and $\eta_{*}$ two arbitrary constants. It can be straightforwardly verified that
\begin{equation}
\tilde{\mu}^2_\alpha(\eta)\equiv \frac{\tilde{z}''_\alpha}{\tilde{z}_\alpha}=\frac{z''_\alpha}{z_\alpha}=\mu^2_\alpha(\eta)\ .
\end{equation}
The arbitrary constant $c_0$ only re-scales the function $z_\alpha$ but has no observational effect, whereas $\eta_{*}$ sets a family of one parameter solutions. Let us consider a specific scenario to exemplify how this duality works. Scalar perturbations with a minimally coupled scalar field in GR are described by Eq.'s~\eqref{eqz} and \eqref{mesc}. An exact de Sitter universe has $\dot{\varphi}/H$ constant, hence, $z_s\propto a\propto -{1}/{\eta}$. Using transformation Eq.~\eqref{wdual} we find that $\tilde{z}_s=\eta^2$, which describes a dust dominated universe. Therefore, an expanding de Sitter universe produce the same mass term for the linear scalar perturbation as a contracting dust dominated universe. As a consequence, both has the same spectrum of solution for the their Mukhanov-Sasaki variable. It is not a coincidence that matter-bounce scenarios produce scale-invariant power spectrum~\cite{Finelli:2001sr,WilsonEwing:2012pu,deHaro:2015wda}.
Generically, a universe dominated by an adiabatic perfect fluid with equation of state given by $p=\omega\, \rho$ (with constant $\omega$) has a scale factor with a power law in cosmic time of the form $a(t) \propto t^{2/3(1+\omega)}$. In terms of conformal time, the scale factor evolves as $a(\eta) \propto \eta^{\frac{1}{2} - \nu}$ with $\nu = \frac{3}{2} -\frac{3(1+\omega)}{1+3\omega}$. In GR, the function $z_s=a\sqrt{\rho+p}/H\propto a$ and apart from a constant factor it coincides with the $z_t$. Thus, both mass term are given by $\mu^2_s=\mu^2_t=a''/a$. A radiation fluid has zero mass term since $a\propto \eta$ and there is no possible duality to be performed. For all other fluids, the mass term and the power spectrum associated with this evolution are given by
\begin{align}
\mu^2 &= \frac{\nu^2 - 1/4}{\eta^2}\quad , \quad \nu = \frac{3}{2} -\frac{3(1+\omega)}{1+3\omega}\quad ,\label{eqnuw}\\
\mathcal{P}_{u} &= \frac{C^2(|\nu|)k^2(-k\eta)^{1 - 2|\nu|}}{4\pi^2}\quad ,
\end{align}
where $C^2(|\nu|)$ is a numeric coefficient. Note that the above power spectrum is invariant under $\nu \rightarrow - \nu$, which can be translated into a transformation of the fluid's equation of state as
\begin{align}\label{dueqsta}
\omega \rightarrow \tilde{\omega} = \frac{1 +\omega}{-1+ 3\omega}\quad .
\end{align}
This transformation has two fixed points at $\omega=-\frac13$ and $1$. For these fixed points, the evolution of the linear perturbations is univocally determined by the background dynamics. For any other value, there are two background dynamics associated with the same perturbed dynamics. Indeed, it is straightforward to verify that two subsequent transformations return to the same equation of state, i.e. $\tilde{\tilde{\omega}}=\omega$. Therefore, in general, there is a pair of adiabatic perfect fluid background dynamics associated with the same evolution for the linear perturbations. Even though de Sitter evolution is not a power law for the scale factor, its duality transformation is still described by Eq.~\eqref{dueqsta}. As already mentioned before, a de Sitter universe, which has $\omega=-1$ is mapped into a dust dominated universe $\omega=0$.
\begin{figure}
\includegraphics[width=0.45\textwidth,height=5.6cm]{Fig_wandsDuality}
\caption{Wands' duality maps an equation of state $\omega$ into $\tilde{\omega}$. There are only two fixed point that mapped into itself given by $\omega=-\frac13$ and $1$. The solid lines represent the map according to Eq.~\eqref{dueqsta}. The dots mark conventional equation of states in cosmology such as $\omega=-1,-\frac13,0,1$.}\label{FigwandsDuality}
\end{figure}
Note, however, that the duality transformation does not specify the theoretical framework. Eq.~\eqref{wdual} map twp distinct $z_\alpha$ functions but does not restrict in which scenario we work with. We can map, for instance, a scalar perturbation in GR into another GR dynamics, $z_{s}\rightarrow \tilde{z}_{s}$, but we can also map a scalar perturbation in GR into a $f(R)$ scenario with the adequate definition of $z_{fs}$, i.e. $z_{s}\rightarrow \tilde{z}_{fs}$.
In the same manner we can map a de Sitter expanding universe into a contracting matter-dominated universe, we shall construct a contracting universe that share the same mass term of the Starobinsky inflation. Though, there is one pitfall. Wands' duality is defined using the conformal time while Starobinsky inflation has an explicitly expression for the scale factor in terms of the cosmic time. The conventional scheme would be to use the definition of conformal time to invert $a(t)$ into $a(\eta)$ but this relation cannot be analytically inverted for Eq.~\eqref{festab}. In order to circumvent this issue, we shall work with a slight modification of Starobinsky's scale factor. As has been argued above, the power of the polynomial in the scale factor is subdominant up to order $\mathcal{O}(M^{-4})$. We shall use this freedom to define a scale factor that allows us to invert the relation and find $a(\eta)$. The appropriate definition of the scale factor which shall be used henceforward is
\begin{align}
a(t) = a_{0} \left(t_{s} - t\right)^{-1} \exp \left[-\frac{M^2}{12}\left(t_{s} - t\right)^2 \right]\quad . \label{fep1}
\end{align}
The associated conformal time is
\begin{align}
\eta &= \int \frac{\mathrm{d}t }{a(t)}=\frac{-6}{a_{0} M^2} \exp \left[\frac{M^2}{12}\left(t_{s} - t\right)^2 \right] \quad .
\end{align}
Thus, the scale factor reads
\begin{equation}
a(\eta) = -\frac{\sqrt{3}}{M\eta}\frac{1}{ \ln^{1/2}\left(\bar{\eta}\right)}\quad .\label{altc}\\
\end{equation}
where we have defined $\bar{\eta} \equiv -{a_{0} M^2\eta}/{6}$, which is a positive quantity. A pure de Sitter universe has $a\propto -1/\eta$, hence, in Starobinsky inflation, the deviation from de Sitter comes from the $\log(\bar{\eta})$ term. Since $M$ is very large, $\log(\bar{\eta})=\log(a_{0} M^2/{6})+\log(-\eta)\approx \log(a_{0} M^2/{6})$, showing that Eq.~\eqref{altc} indeed describes a quasi-de Sitter evolution. Straightforward calculation also gives the Hubble factor and the slow-roll parameters respectively as
\begin{align}
\mathcal{H}(\eta) &= -\frac{1}{\eta}\left(1+\frac{1}{2\ln\left(\bar{\eta}\right)}\right) \quad ,\label{hcaltc}\\
\epsilon_{1}=-\epsilon_{3} &=-\epsilon_{4}= \frac{1}{2\ln\left(\bar{\eta}\right)}\left[1+\mathcal{O}\left(\frac{1}{\ln\left(\bar{\eta}\right)}\right)\right] \quad . \label{e1altc}
\end{align}
Recalling Eq.s~\eqref{defQs} and \eqref{defzfs} we can calculate the $z_s$ function for a scalar perturbation and its associated mass term. Using their definitions we have in leading order
\begin{align}
z_ {s}(\eta) &=-\frac{\sqrt{6}}{\eta} \frac{1}{\ln\left(\bar{\eta}\right)}\left[1+\mathcal{O}\left(\frac{1}{\ln\left(\bar{\eta}\right)}\right)\right]\quad ,\label{zsF}\\
\mu^2_s&= \frac{2}{\eta^2}\left[1 + \frac{3}{2}\frac{1}{\ln\left(\bar{\eta}\right)} +\mathcal{O}\left(\frac{1}{\ln^{2}\left(\bar{\eta}\right)}\right)\right]\quad .\label{musF}
\end{align}
Expression Eq.~\eqref{zsF} can be used to construct a contracting $z^B_s$ function that further will be associated with a bounce model. The duality relation Eq.~\eqref{wdual} gives
\begin{align}
z_{s}^{B}(\eta) &= c_0.z_{s}(\eta) \int_{\eta*}^{\eta} \frac{\mathrm{d}{\eta}'}{z_{s}({\eta}')^2}\nonumber \\
&= \frac{c_0}{3\sqrt{6}}\eta^2 \ln\left(\bar{\eta}\right)\left[1-\frac2{3\ln\left(\bar{\eta}\right)}+\frac{2}{9\ln^{2}\left(\bar{\eta}\right)}\right]+C\left(\eta_{*}\right)\nonumber \\
&= C_1\eta^2 \ln\left(\bar{\eta}\right)\left[1+\mathcal{O}\left(\frac{1}{\ln\left(\bar{\eta}\right)}\right)\right]\label{eqconstc}
\end{align}
where $C_1$ is an arbitrary constant. Is is straightforward to check that $z_{s}^{B}$ and $z_{s}$ produce the same mass term $\mu_s$ up to $\mathcal{O}\left({\ln^{-1}\left(\bar{\eta}\right)}\right)$. Once we have the function $z_{s}^{B}$, we must specify within which scenario the universe is evolving. This extra step is necessary to associate $z_s^{B}$ with a specific background dynamics. For purpose of the present analysis, we choose to immerse this function in a GR contracting solution with the matter content described by a minimally coupled scalar field, hence we have $z_s^{B}=a_B\, \dot{\varphi}/H$.
As argued before, in GR, a quasi-de Sitter inflation is mapped through Wands' duality into a quasi-matter dominated universe. Therefore, we expect that $a^{B}$ should describe a almost matter dominated universe where
\begin{align}\label{approx1}
\dot{\varphi}^2 \simeq 2V&& \Rightarrow&& H^2 \simeq \frac{2V}{3 M_{\mathrm{Pl}}^2}&& \Rightarrow&& \frac{\dot{\varphi}}{M_{\mathrm{Pl}}} \simeq \sqrt{3}H\quad .
\end{align}
As a result, the scale factor $a^B$ should be proportional to the function $z_s^B$. Thus, we have
\begin{align}
a_{B}(\eta) &= a_{B0}\, \eta^2 \ln({\bar{\eta}})\left[1-\frac2{3\ln\left(\bar{\eta}\right)}+\mathcal{O}\left(\frac{1}{\ln^{2}\left(\bar{\eta}\right)}\right)\right] \quad ,\label{fatescesc}\\
\mathcal{H} &= \frac{2}{\eta}\left[1+\frac{1}{2\ln({\bar{\eta}})}+\mathcal{O}\left(\frac{1}{\ln^{2}\left(\bar{\eta}\right)}\right)\right] \quad , \label{hctotal}
\end{align}
In order to find the time dependence of the scalar field and its potential, we can use the exact expression
\begin{align}
\varphi'^2 &= 2\left( {\cal H}^2 - {\cal H}'\right)\quad ,\label{varphiNoapprox}\\
V &= \frac{\left( 2 {\cal H}^2 + {\cal H}'\right)}{a^2}\quad .\label{VNoapprox}
\end{align}
which is valid for a scalar field with arbitrary potential $V$. The approximation Eq.~\eqref{approx1} is sufficient to argue that $\dot{\varphi}/H$ is constant, while Eq.~\eqref{varphiNoapprox} gives the correct numerical factor for $\varphi'$. Using Eq.s~\eqref{varphiNoapprox} and \eqref{VNoapprox}, the time dependence of the potential and of the scalar field read
\begin{align}
V(\eta) &=\frac{6}{a_{B0}^2}\frac{1}{\eta^6 \log ^2(\bar{\eta})}\left[1+\frac{15}{6\ln\left(\bar{\eta}\right)}+\mathcal{O}\left(\frac{1}{\ln^{2}\left(\bar{\eta}\right)}\right)\right]\quad , \label{Veta}\\
\varphi &= -\sqrt{12}\ln \left[\bar{\eta}\ln^{5/12}(\bar{\eta})\right]+\mathcal{O}\left(\frac{1}{\ln\left(\bar{\eta}\right)}\right)\quad .\label{phieta}
\end{align}
As a consistency check we can calculate the effective equation of state given by ratio of pressure and energy density, i.e. $\omega\equiv{p}/{\rho}$ Using the above equations we find
\begin{equation}
\omega=\frac{{\varphi'^2}-{2a^2V}}{{\varphi'^2}+{2a^2V}}=-\frac{1}{6\ln\left(\bar{\eta}\right)}+\mathcal{O}\left(\frac1\ln^{2}\left(\bar{\eta}\right)\right)\quad .\label{weff}
\end{equation}
For $\bar{\eta} \gtrsim 10^4$, the equation of state is close to zero with less than $2\%$. Recall that $\bar{\eta}=-a_0M^2\eta/6$ and the mass parameter is expected to be very large, hence relatively small values of conformal time should already satisfy this condition. It is worth noticing that $\omega \lesssim 0$. This is a crucial property to guarantee a slight redshift in the almost scale invariant power spectrum. A positive equation of state would produce a blueshift that contradicts current observation.
Finally, we can combine the above equations to find the potential in terms of the scalar field $V(\varphi)$. After some simple algebra we find
\begin{align}
V(\varphi) &= V_0 \sqrt{1 - \varphi/\varphi_{\ast}}\, e^{\sqrt{3} \, \varphi}\quad , \label{vphiqmd}
\end{align}
with $V_0$ and $\varphi_{\ast}$ two constant parameters that completely specify the potential. A dust fluid can be described by a scalar field with potential $\exp\left[\sqrt{3} \, \varphi\right]$, hence it is not surprising that $V(\varphi)$ has this kind of exponential dependence. The novelty is the square root correction, which is intrinsically related to the polynomial correction in the scale factor of Starobinsky inflation. We can again check our construction plotting the phase portrait associated with potential Eq.~\eqref{vphiqmd}. Fig.~\ref{fig:phaseplot} shows the trajectories of the scalar field in the $\left(\varphi,\dot{\varphi}\right)$ plane. For relative large values of $\varphi$ the velocity $\dot{\varphi}$ rapidly goes to zero, which is consistent with a dust fluid given the exponential dependence of the potential $V(\varphi)$.
\begin{figure}[h]
\centering
\includegraphics[width=6.5cm]{phasep.png}
\caption{Phase portrait of $\dot{\varphi}$ versus $\varphi$ for the potential Eq.~\eqref{vphiqmd} using the values $V_0=\varphi_{\ast}=1$. One can see that the dynamics generated by this reconstructed potential is very similar to the exact dust $(p=0)$ potential showing that the square root deformation of the exponential potential work as a small correction.}
\label{fig:phaseplot}
\end{figure}
\section{Crossing the Bounce}\label{CB}
Bounce models are a subclass of nonsingular models that commonly has a single contracting phase followed by an expanding phase. By construction, the contracting phase is smoothly connected to the expanding phase, hence the universe is eternal and free of spacetime singularities. However, this does not mean that one should oppose bounce and inflationary models. Even though a pure inflationary mechanism cannot avoid the initial singularity~\cite{Borde:1994PRL72,Borde_IJMP1996}, a nonsingular model can accommodate an inflationary phase~\cite{Falciano:2007yf, Falciano:2008nk}. However, bounce models are frequently understood as alternative to inflation.
There are viable bounce models that are consistent with almost scale-invariant power spectrum and small tensor-to-scalar ratio~\cite{NPN_Peter_Pinho_1,Peter:2008qz,Bojowald_RepProgPhys2015,Cai:2014zga,Craig_Singh_CQG2013,Astekar_PRD74,Singh_PRD.74.2006}. In these models, the dynamic through the bounce influences the observable effects. For instance, the mode mixing of scalar perturbations across the bounce is responsible for producing the almost scale-invariant power spectrum. Therefore, it seems reasonable that in order to consider bounce models as a physically viable scenario for the primordial universe, one should recognize them as alternative to inflation and not just as a complementary phase prior to it.
Bounce and inflation have completely distinct background dynamics. Besides the different concerning the singularity problem, at the background level, inflation and bounce models have different shortcomings and theoretical challenges of their own~\cite{Brandenberger:2000as,Jerome_PhysRevD.63.123501,Battefeld:2014uga,Brandenberger:2009jq}. Notwithstanding, at first order perturbation, bounce and inflation are formally very similar. Indeed, Wands' duality described in section~\ref{MSI} is one manifestation of the mathematical similarity between these two scenarios.
Generically, the dynamics of linear perturbations $\nu_{\vb{k}}$ are described by a parametric oscillator equation like Eq.~\eqref{mesc} where the time-dependent mass term $\mu_\alpha$ encodes the background dynamics. In each case we have a specific definition for $\nu_{\vb{k}}$ and $\mu_\alpha$ but the framework is almost identical. Let us compare some of their features.
In both scenarios, even though for different physical reasons, the initial conditions are set in the most (possible) remote past and have a quantum vacuum fluctuation origin. In inflationary models, we have a quasi-de Sitter expansion, which makes the physical length of interest for present cosmology much smaller than the curvature scale. As a consequence, the perturbations are not influenced by the expansion and the initial state is set as a Minkowski vacuum state. In a bounce model, the initial conditions are given in the far past much before the bounce phase. The universe is immense and with negligible curvature, hence, the initial state is a Bunch-Davies vacuum.
As the universe evolves the relation between the physical length and the Hubble length changes. In both scenarios the ratio between these two lengths increases. In terms of the perturbed dynamic equation, this means that with the background evolution, the mass term increases compared to the wavenumber until they become comparable in magnitude. This moment specifies the crossing from outside to inside the potential for the perturbations. The mass term continues to grow until it reaches a maximum that typically locates the bounce or the reheating period for inflationary models. Then the potential starts to decrease until its value becomes again comparable to the wave number characterizing the crossing outside the potential (inside the Hubble length)\footnote{Note that the description in term of the potential for the perturbation (the time dependent mass term) is the opposite as compared to the relative size of the physical and Hubble lengths. Crossing outside the Hubble length means going inside the potential and vice-verse.}. Thereupon, both scenarios are connected to the FLRW radiation epoch and the dynamics follows the standard model.
It is evident from the above description that the violent quasi-de Sitter expansion phase is related to the long contracting phase of bounce models. Moreover, the reheating phase of inflation should be compare to the physical processes during the bounce phase. Thus, it is not surprising that the reheating and the bounce are the two most speculative periods of the evolution.
Inflationary models often overlook the details of the reheating processes. In a certain sense, this is due to the assumption that whatever physical process taking place in this period should only transfer energy into the matter fields and not significantly modify the other physical quantities such as the almost scale-invariant power spectrum or the tensor-to-scalar ratio\footnote{It is worth mentioning that non-gaussianities encoded in the bispectrum are much more sensitive to reheating.}. This idea has support on Weinberg's theorem~\cite{Weinberg:2003sw} that states that, in the large wavelength limit, the field equations for the cosmological perturbations in the Newtonian gauge always have an adiabatic solution with ${\mathcal{R}}$ constant and nonzero in all eras.
In contrast, bounce models can not avoid examining the bounce phase since one must define the physical mechanism that produces the bounce. In addition, the physics of the bounce remains encoded in the spectrum of primordial perturbations. As we will show in the following, the relation between the scalar spectral index and the tensor-to-scalar ratio depends on the physics of the bounce. The observational data available are not yet sensitive enough to discriminate between different bounce mechanisms but as in the case of non-gaussianities, future experiments might allow us to probe the physics of the bounce.
In order to connect the contracting phase of the model constructed in the last section to the CMB observables, in the following sections we shall describe the bounce as a quantum gravity effect using the Loop Quantum Cosmology (LQC) framework~\cite{WilsonEwing:2012pu,WilsonEwingLCDM}. There are other appealing frameworks such as Wheeler-DeWitt~\cite{Pinto-Neto:2013toa, PintoNeto:2012ug, Vitenti:2012cx, Falciano_PRD_2015, PintoNeto:2005gx, Falciano:2013uaa} or string cosmology~\cite{Brandenberger_2011, Gasperini:2002bn}. However, LQC has analytical bounce solutions for a scalar field mimicking a perfect fluid, hence, from a technical point of view, it is the most direct description to accommodate a previous Starobinsky-like contracting phase.
Loop quantum gravity (LQG) is a non-perturbative, background independent quantum theory of gravity. It is based on a reformulation of GR in terms of the Ashtekar-Barbero variables. The classical variables promoted to operators are the holonomies of the Ashtekar connection and the fluxes of the densitized triads. One important kinematical result of this quantization procedure is the discretization of spacetime, which in turn establishes a minimum of length, area and volume. LQC relies on using loop quantization techniques to quantize the holonomies and the fluxes of homogeneous and isotropic universes. It is not a full quantum gravity theory but an effective approach that hopefully captures the essential features of LQG in a cosmological scenario (for further details see~\cite{AshtekarSingh2011,BojowaldPRD93,Bojowald_RepProgPhys2015,AshtekarPRD73}). The cosmological dynamics can be described by a phenomenological Hamiltonian. Given a flat FLRW metric, the dynamics with respect to cosmic time reads
\begin{align}
H^2&=\frac{M_{\mathrm{Pl}}^{-2}}{3}\rho\left(1-\frac{\rho}{\rho_{c}}\right)\quad ,\label{LQCF1}\\
\dot{H}&=-\frac{M_{\mathrm{Pl}}^{-2}}{6}(\rho+p)\left(1-\frac{2\rho}{\rho_{c}}\right)\quad ,\label{LQCF2}\\
\dot{\rho}&+3H\left(\rho+p\right)=0\label{LQCEC}\quad ,
\end{align}
where $\rho_{c}$ is a critical energy density that establishes the energy scale where quantum corrections are important\footnote{We have used the conservation of energy-momentum as our third dynamic equation but we could instead have used the Klein-Gordon equation for the scalar field. The two system of equations are equivalent.}. This dynamic system has analytical bounce solutions for perfect fluids $p=\omega\rho$ with constant $\omega$~\cite{WilsonEwing:2012pu,MielczarekPLB3,WilsonEwing2}. Furthermore, we can use a scalar field with exponential potentials to model the perfect fluid. Indeed, using the fact that
\[
\rho=\frac12\dot{\varphi}^2+V(\varphi)\quad , \quad p=\frac12\dot{\varphi}^2-V(\varphi)\quad ,
\]
one can show that there is an exact solution
\begin{align}
\rho&=\rho_c \left(\frac{a_B}{a}\right)^{3(1+\omega)}\quad ,\label{LQCsol1}\\
a(t)&=a_B\left(1+
\alpha^2\left(t-t_B\right)^2\right)^{1/3(1+\omega)}\label{LQCsol2}\quad ,\\
\varphi(t)-\varphi_B&=\frac{\sqrt{\rho_c(1+\omega)}}{\alpha}
\arcsinh\bigg(\alpha(t-t_B)\bigg)\quad , \label{LQCsol3}
\end{align}
where $\alpha = \sqrt{3\rho_{c}} (1 + \omega)/2 M_{\mathrm{Pl}}$. The parameters $t_B$ and $a_B$ are respectively the values of the cosmic time and the scale factor at the bounce. Note that the energy density reaches its maximum value at the bounce Eq.~\eqref{LQCsol1}. This is a characteristic feature of symmetric bounces. The scalar field potential for this solution is given by
\begin{equation}\label{VLQC}
V=\frac{\rho_c (1-\omega)}{2}\sech^2\left[
\frac{\alpha\left(\varphi-\varphi_B\right)}{\sqrt{\rho_c(1+\omega)}}
\right]
\quad ,
\end{equation}
where $\varphi_B$ is an arbitrary constant. This solution has two parameters $a_B$ and $\varphi_B$ in addition to the energy density scale $\rho_c$ of LQC. The classical limit is approached in the $\rho_c\rightarrow\infty$. In this limit, Eq.~\eqref{VLQC} tends to $V\sim \exp\left(\sqrt{3(1+\omega)}\varphi/M_{\mathrm{Pl}}\right)$, which corresponds to the scalar field potential that describes a perfect fluid with equation of state $\omega$ in GR.
\subsection{Scalar Perturbations in Bounce Models}\label{ScaPert}
Quantum cosmology is an attempt to include quantum effects in the evolution of the universe. In this manner, we must necessarily consider modifications in the GR equations of motion. However, bounce models generically assume that far from the bounce region we recover the GR dynamics. Therefore, long before and after the bounce the scalar perturbations are described by
\begin{align}
{v}'' + \left(k^2 - \mu^2_{s}\right)v = 0\ , \ \mbox{with}\quad \mu^2_{s} = \frac{z_{s}''}{z_{s}} \label{mesc2} \ ,\\
v \equiv z_{s}{\mathcal{R}} \quad ,\qquad z_{s} \equiv \frac{a}{H}\sqrt{\rho + p} = a \frac{\dot{\varphi}}{H} \ . \label{eqz2}
\end{align}
Using the quasi-matter dynamics of last section Eq.~\eqref{eqconstc}, we find that the classical contracting phase has
\begin{align}
v^{in}(\eta) &= \sqrt{\frac{-\pi \eta}{4}}\, \mathrm{H}^{(1)}_{\gamma}(-k\eta) \label{sclass} \\
\gamma &= \frac{3}{2} + \epsilon_c = \frac{3}{2} + \frac{1}{\ln\left(\bar{\eta}\right)} + \frac{2}{3}\frac{1}{\ln\left(\bar{\eta}\right)^2}\ ,
\end{align}
where $\mathrm{H}^{(1)}_{\gamma}$ is the Hankel function of the first kind and we have defined in the last expression $\epsilon_c\equiv\gamma - \frac{3}{2}$. The $\epsilon_c$ will play a role analogous to a slow-roll parameter, which differs from the matter bounce parameter\cite{Elizalde:2014uba} by being a small quantity $|\epsilon_c|\ll 1$. Indeed, during the period of validity of the above solution, this term is very small compared to unit, hence, we can consider series expansion in its powers. Our task now is to describe the bounce and use matching conditions to connect this contracting phase with the expanding phase of the standard model. The LQC perturbed equations have two modifications with respect to GR. The Mukhanov-Sasaki equation now reads
\begin{align}
v'' + \left[\left( 1 - \frac{2\rho}{\rho_{c}} \right) k^2 - \frac{z''}{z}\right]v = 0 \quad ,\label{ms_lqc}
\end{align}
where the $z$ function is defined as
\begin{align}
z = \frac{a \sqrt{\rho + P}}{ H} = M_{\mathrm{Pl}}\sqrt{\frac{3(1 + \omega)}{1 - \rho/\rho_{c}}}\ a
\quad . \label{zlqc}
\end{align}
Far away from the bounce, the energy density is much less than the critical density, i.e. $\rho/\rho_c\ll 1$ and we recover the classical definitions. Thus, during the contracting phase far away from the bounce, we have~\eqref{sclass}. We need to match this solution with a solution valid during the bounce. Eq.~\eqref{ms_lqc} can be transformed into an integral equation given by
\begin{align}
v(\eta) = &B_{1}z + B_{2}z \int^{\eta} \frac{\dd \bar{\eta}}{z^2} - k^2 \int^{\eta} \frac{\dd \bar{\eta}}{z^2} \int^{\bar{\eta}} \dd \bar{\bar{\eta}} \, z \,v \nonumber \\
&+ \frac{2 k^2}{\rho_{c}} \, z \int^{\eta} \frac{\dd \bar{\eta}}{z^2} \, \int^{\bar{\eta}} \dd \bar{\bar{\eta}} \, z \, v\quad .\label{key}
\end{align}
Close to the bounce, it is the mass term that dominates hence we can series expand the solution in powers of the wave number. The solution Eq.s~\eqref{LQCsol2} and \eqref{LQCsol3} are given in cosmic time. We can interpret the conformal time of the above expression as a function of cosmic time. Using the LQC background solution we find at leading order
\begin{align}
v(t) =&
B_{1}z(t)+
B_{2}z(t)\left( \frac{a_B^{-3}M_{\mathrm{Pl}}^{-2}}{3(1+\omega)}\right)\times \label{slqcq}
\\& \quad \times
\left[\frac{\alpha^2t^3}{3}{}_2F_1\left[\frac32,\frac{2+\omega}{1+\omega},\frac52,-\alpha^2t^2\right]+c_2
\right]
\ ,\quad \nonumber
\end{align}
where ${}_2F_1\left[a,b,c,z\right]$ is the hypergeometric function and $c_2$ is an integration constant that can be chosen conveniently to simplify the matching at the contracting phase. The function $x^3{}_2F_1\left[\frac32,\frac{2+\omega}{1+\omega},\frac52,-x^2\right]$ goes to a constant in the limit $x\rightarrow\pm\infty$, hence we can choose $c_2$ to cancel this constant term in the far past. Consequently, we will have $2c_2$ in the far future after the bounce. Taking the limit $\alpha t\rightarrow -\infty$ we find that $c_2=\frac{\pi}{4\alpha}$. The coefficient $B_{1}$ represents the decreasing mode during Hubble crossing in the contracting phase. We can immediately see from the above expression that due to the behavior of the hypergeometric function the bounce produces a mode mixing transferring the coefficient $B_2$ to the dominant mode after the bounce.
The validity of the contracting solution Eq.~\eqref{sclass} relies on $\epsilon_c$ being almost constant in time and small $\epsilon_c\ll 1$. Thus, we can perform the matching between the contracting phase and the bounce solution well inside the potential for the perturbation but still very far from the bounce. This means that we should take the limit $k\eta\rightarrow0$ in Eq.~\eqref{sclass} and the limit $t\ll -1/\alpha$ in Eq.~\eqref{slqcq}. In addition, our contracting phase has equation of state given by Eq.~\eqref{weff}, hence we must identify $\omega=-\frac16\epsilon_c$. In this limit we can write the scale factor and the cosmic time in terms of the conformal time, i.e.
\begin{align}
a(\eta) &= a_B\left[ \frac{\alpha(1-\epsilon_c/3)}{3} a_B\eta\right]^{2+\epsilon_c}\quad ,\\
\alpha t(\eta) &= \left[ \alpha \left( \frac{1 -\epsilon_c/3}{3}\right) a_B\eta \right]^{3 +\epsilon_c} \ ,\label{tc_lqc}
\end{align}
where we have used $\epsilon_c\ll 1$ and kept only the leading order terms. Using Eq.'s~\eqref{zlqc}-\eqref{slqcq} we find
\begin{align}
z(\eta) =&a_B^{3+\epsilon_c}\left( 1 - \frac{13\epsilon_c}{12}\right)\frac{\sqrt{\rho_c}}{2} \left(\frac{\rho_{c}}{12M_{\mathrm{Pl}}^2}\right)^{(1+\epsilon_c)/2} \eta^{2+\epsilon_c} \quad ,\\
v(\eta) =& B_{1}a_B^{3+\epsilon_c}\left( 1 - \frac{13\epsilon_c}{12}\right)\frac{\sqrt{\rho_c}}{2} \left(\frac{\rho_{c}}{12M_{\mathrm{Pl}}^2}\right)^{(1+\epsilon_c)/2} \eta^{2+\epsilon_c}
\nonumber \\&
\quad -\frac{4B_{2}}{\sqrt{3}}\frac{a_B^{-3-\epsilon_c}}{\sqrt{\rho_c}}
\left( 1 + \frac{5\epsilon_c}{12} \right)
\left(\frac{M_{\mathrm{Pl}}^2}{\rho_c}\right)^{(1+\epsilon_c)/2}
\eta^{-1-\epsilon_c}\quad . \label{vqc_lqc}
\end{align}
This solution has to be matched with the contracting solution Eq.~\eqref{sclass} in the limit $k\eta \ll 1$, namely
\begin{align}
v^{in}(\eta) = \frac{1}{3\sqrt{2}}k^{3/2+\epsilon_c}\eta^{2+\epsilon_c}+\frac{i}{\sqrt{2}}k^{-3/2-\epsilon_c}\eta^{-1-\epsilon_c}\ .
\end{align}
Straightforward comparison shows that
\begin{align}
B_{1} &=
\frac{\sqrt{2}a_B^{-3-\epsilon_c}}{3\sqrt{\rho_c}}\left( 1 + \frac{13\epsilon_c}{12}\right) \left(\frac{\rho_{c}}{12M_{\mathrm{Pl}}^2}\right)^{-(1+\epsilon_c)/2}k^{3/2+\epsilon_c}\ ,
\label{b1lqc}\\
B_{2} &=
-i\frac{\sqrt{3}}{4\sqrt{2}}\frac{\sqrt{\rho_c}}{a_B^{-3-\epsilon_c}}\left( 1 - \frac{5\epsilon_c}{12} \right)\left(\frac{\rho_c}{M_{\mathrm{Pl}}^2}\right)^{(1+\epsilon_c)/2}k^{-3/2-\epsilon_c}
\label{b2lqc}\ .
\end{align}
The solution Eq.\eqref{slqcq} is valid across the bounce. Having defined the coefficients $B_1$ and $B_2$ we can find the solution after the bounce. The expanding phase solution is described by taking the limit $t\gg 1/\alpha$ in Eq.\eqref{slqcq}, i.e.
\begin{align}
v^{out}(\eta)&=
\left[ B_{1}
+B_{2}
\left(
\frac{\pi a_B^{-3}(1+\epsilon_c/3)}{3\sqrt{3\rho_c} M_{\mathrm{Pl}} }
\right)\right] z(\eta)\\
&=\left[ \frac{k^{3/2+\epsilon_c}}{3\sqrt{2}}
-i\frac{\pi \left( 1 - \frac{7\epsilon_c}{6}\right)}{48\sqrt{6}}
\left(\frac{a_B^2\rho_c}{M_{\mathrm{Pl}}^2}\right)^{3/2+\epsilon_c}
k^{-3/2-\epsilon_c}\right] \eta^{2+\epsilon_c}
\quad .\nonumber
\end{align}
In cosmological perturbations we are interested in the small wavenumber limit, hence for very small wavenumber it is the $k^{-3/2}$ that dominates. However, this is true only if the numerical factors are of order one. The parameter $\rho_c$ is expected to be smaller but comparable in at least a few order of magnitude of the Planck energy density, i.e. $\rho_c=10^{-n}\rho_{_{\rm Pl}}$, with $1<n<10$. The value of the scale factor at the bounce must be at least a few order of magnitude higher than the Planck mass, otherwise we could not rely on our quantum cosmology effective scenario, i.e. $a_B=10^m \, l_{_{\rm Pl}}$ with $5>m>2$. The ratio between the two term above is
\begin{align}\label{conds}
&\approx 14.28\times 10^{3(m-n/2)} \, l_{_{\rm Pl}}^{-3} k^{-3}\gg 1\quad .
\end{align}
Therefore, it is indeed the $k^{-3/2}$ the dominant coefficient for all values of interest of wavenumber in cosmology and the scalar perturbation is
\begin{align}
{\mathcal{R}}&=\frac{v}{z}\approx \frac{\pi}{12\sqrt{2}}\sqrt{\frac{\rho_c}{M_{\mathrm{Pl}}^4}}k^{-\frac32-\epsilon_c}\approx 0.185
\sqrt{\frac{\rho_c}{M_{\mathrm{Pl}}^4}}k^{-\frac32-\epsilon_c}\quad , \label{nuout}
\end{align}
with spectral index given by
\begin{align}
n_{s} - 1 = -2\epsilon_c\quad . \label{spec_ind_lqc}
\end{align}
As expected, the power spectrum is almost scale invariant but with a small redshift. Using the Planck 2018 release $n_s=0.9649\pm0.0042$ (see Ref.~\cite{Akrami:2018odb}), we have $0.0196<\epsilon_c<0.0155$.
\subsection{Tensor Perturbations}\label{TensPert}
Similarly to the scalar perturbations, the dynamic equation for tensor perturbations in LQC has quantum corrections proportional to $\rho/\rho_c$. The Mukhanov-Sasaki variable is defined in terms of the tensor perturbations $h=2 v / z_{T} M_{\mathrm{Pl}}$, where function $z_T$ is also modified due to quantum corrections. The Mukhanov-Sasaki equation reads
\begin{align}
v'' + \left[\left( 1 - \frac{2\rho}{\rho_{c}} \right) k^2 - \frac{z_{T}''}{z_{T}}\right] v = 0 \quad ,\label{tms_lqc}
\end{align}
where the function $z_T$ is given by
\begin{align}
z_{T} = \frac{a}{\sqrt{1 - 2\rho/\rho_{c}}}\quad . \label{zt_lqc}
\end{align}
The tensor perturbations in the contracting phase has the same solution as the scalar perturbations, namely
\begin{align}
v^{in}(\eta) = \sqrt{\frac{-\pi \eta}{4}}\, \mathrm{H}^{(1)}_{\gamma}(-k\eta)
\end{align}
where again $\gamma=3/2+\epsilon_c$. Following the same procedure as before, we can transform the differential equation into an integral equation for $\mu$ similar to Eq.~\eqref{key}. The solution across the bounce can be obtained by a series expansion on powers of the wavenumber. At leading order in $k$, the formal solution to its integral form is
\begin{align}\label{mu_lqc_formal}
v(t) = D_1 \, z_{T}(t) + D_2 \, z_{T}(t) \int^{\bar{\eta}} \frac{\dd \eta}{z_{T}(\eta)^2}
\end{align}
where $D_1$ and $D_2$ are two constants of integration. By virtue of Eq.~\eqref{zt_lqc}, the formal solution is
\begin{align}
v(t) &= D_1 \, z_{T}(t) + \frac{D_2}{a_B^3} \, z_{T}(t) \left[ \frac{\alpha^2 t^3}{3} \, _{2}F_{1} \left( \frac{3}{2}, \frac{2+\omega}{1+\omega}, \frac{5}{2}, -\alpha^2 t^2 \right) \right. + \nonumber \\
&\left.- t \times\ _{2}F_{1} \left( \frac{1}{2}, \frac{2+\omega}{1+\omega}, \frac{3}{2}, -\alpha^2 t^2\right) + C \right] \quad .\label{fst_lqc}
\end{align}
As before, we chose the constant $C$ conveniently to cancel the constant term in the far past. As a result we have $C=-\frac{\pi\omega}{2\alpha}$. Recall that $\alpha = \sqrt{3\rho_{c}} (1 + \omega)/2 M_{\mathrm{Pl}}$ and $\omega=-\frac16\epsilon_c$. In order to match this solution with the contracting phase, we must take the limit $t\ll -1/\alpha$ that gives
\begin{align}
v(t) =&\frac{D_1 \left( 1 - \epsilon_c\right)}{a_B^{-3-\epsilon_c}} \left( \frac{ \rho_{c}}{12M_{\mathrm{Pl}}^2} \right)^{1+\epsilon_c/2} \eta^{2 +\epsilon_c} + \nonumber \\
&\qquad + \frac{D_2}{3a_B^3} \left( 1+\frac{2\epsilon_c}{3} \right) \left( \frac{12M_{\mathrm{Pl}}^2}{ \rho_{c}} \right)^{1 +\epsilon_c/2} \eta^{-1-\epsilon_c}\quad .
\label{muqc_lqc}
\end{align}
This expression has to be matched with the limit $k\eta \ll 1$ for the classical solution~\eqref{sclass}, namely
\begin{align}
v^{in}(\eta) =& \frac{1}{3\sqrt{2}}k^{\frac32+\epsilon_c} \eta^{2+\epsilon_c}
- i \frac{1}{\sqrt{2}} \, k^{-\frac32-\epsilon_c} \eta^{-1-\epsilon_c}\ .
\end{align}
Thus, we identify
\begin{align}
D_1 &= \frac{\left( 1 + \epsilon_c\right)}{3\sqrt{2}a_B^{3+\epsilon_c}}\left( \frac{12M_{\mathrm{Pl}}^2}{ \rho_{c}} \right)^{1+\epsilon_c/2}k^{\frac32+\epsilon_c}\quad ,\label{D1}\\
D_2 &=-i\frac{3a_B^3}{\sqrt{2}}
\left( 1-\frac{2\epsilon_c}{3} \right) \left( \frac{\rho_{c}}{12M_{\mathrm{Pl}}^2} \right)^{1 +\epsilon_c/2} k^{-\frac32-\epsilon_c}\ .\label{D2}
\end{align}
The expanding phase is given by taking the limit $t\gg1/\alpha$. Thus, we have
\begin{align}
v^{out} (\eta)&= \left[ D_1 - \frac{D_2}{a_B^3}\frac{\piM_{\mathrm{Pl}}}{3\sqrt{3\rho_{c}}}\epsilon_c \right] \, z_{T}^{c}(\eta) \label{mu_grow}
\end{align}
where $D_1$ and $D_2$ are given by Eq.s~\eqref{D1} and \eqref{D2}. It is worth noting that the term proportional to $D_2$ is linear in $\epsilon_c$, hence the mode mixing in the tensor perturbation depends on how small is the slow-row parameter. To leading order in wavenumber, the tensor perturbation reads
\begin{align}
h&=\frac{2v}{z_TM_{\mathrm{Pl}}} = \frac{2}{M_{\mathrm{Pl}}}\left[ D_1 - \frac{D_2}{a_B^3} \frac{\piM_{\mathrm{Pl}}}{3\sqrt{3\rho_{c}}}\epsilon_c \right] \\
&\approx \frac{i\pi}{6\sqrt{6}} \epsilon_c \sqrt{ \frac{\rho_{c}}{M_{\mathrm{Pl}}^4}} k^{-\frac32-\epsilon_c}
\approx 0.214 i \epsilon_c \sqrt{ \frac{\rho_{c}}{M_{\mathrm{Pl}}^4}} k^{-\frac32-\epsilon_c}
\quad .\label{muout}
\end{align}
Thus, the tensor spectral index is $n_t=-2\epsilon_c=n_s-1$. Finally, using Eq.s~\eqref{nuout} and \eqref{muout}, we find the tensor-to-scalar ratio
\begin{align}
r&=\frac{\mathcal{P}_{\mathcal{T}}}{\mathcal{P}_{\mathcal{R}}}=2\frac{|h|^2}{|{\mathcal{R}}|^2}=\frac83 \epsilon_c^2=\frac23 (n_s-1)^2 \label{r_lqc}
\end{align}
Note that we succeed in obtaining the same relation between $n_s-1$ and $r$ as in the Starobinsky inflation. However, even though with the correct power of the slow-roll parameter $\epsilon_c^2$, there is a numerical factor difference of order unit. Eq.~\eqref{ns_r_star} shows that Starobinsky inflation has a relation between the scalar spectral index and the tensor-to-scalar ratio given by
\begin{equation} \label{rstab_conc}
r = 3\left( n_s - 1 \right)^2 \quad ,
\end{equation}
hence our model is a factor $2/9$ smaller. This difference is a convolution of two contributions coming from the ratio $z_{s}/z_{T}$ but they have completely distinct physical origin.
First, in inflationary models, the ratio $\left(z_{s}/z_{T}\right)^2$ is $1/2\epsilon_{c}^{2}$ larger than its value in bouncing models. Indeed, one can check that in the Starobinsky model we have $\left(z_{s}/z_{T}\right)^2=Q_s/F\approx \frac32 M_{\mathrm{Pl}}^2\epsilon_{c}^2$, while for a matter bounce model we have $\left(z_{s}/z_{T}\right)^2=3M_{\mathrm{Pl}}^2$. The simple fact that the horizon crossing happens in two different background dynamics (quasi-de Sitter for inflation and quasi-matter for bounce) changes the tensor-to-scalar ratio by a factor $1/2\epsilon_{c}^{2}$. The factor ${\epsilon_{c}^{2}}/{9}$ has a completely different physical origin. It comes from the dynamics across the bounce.
Inflationary models with adiabatic perturbations have a decreasing and a constant mode. With the quasi-exponential expansion, it is the constant mode that dominates and gives the almost scale-invariant power spectrum. In contrast, bounce models have a constant and an increasing mode before the bounce. The bounce dynamics makes the latter the dominant mode after the bounce (there is a mode mixing), which has an integral contribution of $z^{-2}$ (see Eq.~\eqref{mu_lqc_formal}). This term carries information across the bounce and depends on the dynamics chosen to describe the bounce. In our case we get a ${\epsilon_{c}^2}/{9}$ contribution from the time integral across the LQC bounce. Another bounce like WDW should give a different numerical factor but the same ${\epsilon_{c}^2}$ contribution.
In summary, there is a crucial difference on how inflation and bounce models obtain a small tensor-to-scalar ratio. Both dynamics start with the same vacuum state but the inflationary dynamics amplifies more\footnote{Note that this amplification difference appears only for the tensor-to-scalar ratio and it is irrelevant for the amplitude of the scalar perturbations since one can always adjust the free parameter $\rho_{c}$ in order to accommodate this difference. This is not the case for $r$ since the scalar and tensor perturbations have the same dependence on $\rho_{c}$ hence it drops out (see Eq.s~\eqref{nuout} and \eqref{muout}).} the scalar perturbations than the quasi-matter contraction by a factor of $1/2\epsilon_{c}^2$. On the other hand, the evolution across the bounce suppresses the tensor perturbations by a factor of ${\epsilon_{c}^2}/{9}$. The net result is the $2/9$ difference factor between the two tensor-to-scalar ratios given by Eq.s \eqref{r_lqc}-\eqref{rstab_conc}.
\section{Conclusions}\label{Con}
In the near future, we expect to have decisive new observational data of the very early universe. The 21 cm redshift surveys together with measurements of the CMB B-mode polarization, non-gaussianities and primordial gravitational waves will enable us to discriminate between different primordial universe scenarios. Therefore, it is pressing to identify signatures of each type of primordial universe scenario that would allow us to make testable predictions.
In the present work, we have used Wands' duality to construct a quasi-matter bounce that mimics the Starobinsky inflation. This map allow us to identify the correct contracting phase dynamics that gives the same time-dependent mass term in the Mukhanov-Sasaki equation. The adequate scalar field potential $V(\varphi)$, given by Eq.~\eqref{vphiqmd}, is a deformation of the exponential potential known to describe a pressureless dust fluid. This result agrees with the fact that a quasi-de Sitter phase should be mapped into a quasi-matter dominated contracting universe. After the linear perturbations cross the horizon, the system must go through a bounce phase. We chose to describe the bounce using LQC inasmuch it is the easiest quantum bounce if the matter field is described by a scalar field.
Our constructive method permit us to discriminate the contribution of each dynamical phase in the primordial power spectrum. In particular, we showed that mapping the Starobinsky inflation into a quasi-matter bounce gives the correct relation between the scalar spectral index $n_s-1$ and the tensor-to-scalar ratio $r$ but it appears a factor $2/9$ of difference. The crucial point is to understand the origin of this numerical factor. It comes from the ratio $z_s/z_T$ and it is a convolution of two distinct contribution. The comparison between this ratio from an inflationary expansion to a quasi-matter contraction gives a factor $(2\epsilon_{c}^{2})^{-1}$, while the dynamics through the LQC bounce results in an additional factor of $\epsilon_{c}^{2}/9$.
An interesting feature of our analysis is to show that the bounce leaves a signature in the primordial power spectrum. The scalar and tensor spectral indexes depend on the background dynamics during the horizon crossing. But the amplitudes of the scalar and tensor power spectrum, hence the tensor-to-scalar ratio, carry information from the dynamics across the bounce.
\begin{acknowledgments}
The authors would like to thank and acknowledge financial support from the National Scientific and Technological Research Council (CNPq, Brazil) and the State Scientific and Innovation Funding Agency of Esp\'\i rito Santo (FAPES, Brazil).
\end{acknowledgments}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 9,258 |
Opeth new record is "all over the place"
As we were telling you a while ago, Opeth are in the studio, heavily working on a 'more sinister' album – frontman Mikael Akerfeldt shared some fresh details regarding their new work, explaining that it is "all over the place".
"I have eight songs written and the direction is going straight into the territory of 'all over the place!'" he kicked off with a laugh.
Influences for this album include Goblin, Mahavishnu Orchestra, and Emerson Lake and Palmer.
Who's the real Prince of Darkness? | {
"redpajama_set_name": "RedPajamaC4"
} | 1,568 |
Department of Medical & Health services, Jaipur (DMHS) released a notification on 9th April 2018 to fill 4514 Rajasthan Nurse Grade II Vacancies. Apply online for DMHS Rajasthan Recruitment 2018 @ www.rajswasthya.nic.in from 12th April 2018 to 12th May 2018. Once, check complete details about DM&HS Jobs 2018 from www.rajswasthya.nic.in Notification.
Latest Sarkari Naukri News!!Out. If you are looking for Rajasthan Govt Jobs. Then you are on the correct page. Because here we have provided the latest employment news which is issued by Department of Medical & Health Services, Jaipur to hire candidates for Nurse Grade II Posts. According to DMHS Rajasthan Latest Notification with Advt No. 327 & 328, the board has 4514 DMHS Vacancies to fill up Nurse posts in scheduled & non-scheduled areas. Therefore, candidates can apply online for DMHS Rajasthan Recruitment to fulfil the dream. In addition to this, we are suggested to submit the filled DM&HS Rajswasthya Job Online Application Forms as soon as possible to avoid last minute server issues.
Here you can get all the information regarding DMHS Rajasthan Jobs in detailed. So, check them once before going to start online registrations in DMHS Portal. The board has 359 DM&HS Nurse Vacancies in scheduled areas & 4155 DMHS Nurse Posts in Non-scheduled areas. For Rajasthan people, this is the golden opportunity to work with DMHS. Hence there are huge vacancies, the competition will be intense. Therefore, hope good & get the www.rajswasthya.nic.in Jobs. Go through the below modules for knowing more requirements like eligibility, selection, pay scale, how to apply online steps, etc.
Aspirants who are interested to apply for Rajasthan DMHS Recruitment 2018 must check the eligibility criteria like age limit & education qualification from the below sections.
Candidates who are willing to apply for Rajswasthya Nurse Grade II Vacancy 2018 should complete 12th & GNM Course or its equivalent qualification from an institute, recognised by the State Government of Rajasthan.
To select the skilled & talented persons for the DMHS Nurse Jobs, the board will calculate the marks obtained in 12th + GNM course & other bonus marks. At last, selected candidates' list will be displayed on the Medical & Health Service Department, Jaipur Portal.
Interested candidates who are dreaming of Rajasthan Govt Jobs can apply online for DMHS Rajasthan Jobs 2018 for the post of Nurse Grade II in Scheduled & Non-Scheduled areas. To fill 4514 Vacancies in Department of Medical & Health Services Jaipur, board invites online applications from the eligible candidates. The commencement for submission of online applications starts from 12th April 2018 and likely to be ended on 12th May 2018. So, follow the below apply online steps for successful registrations & submission of DMHS Online Forms. Otherwise, click on directly below Apply online link to fill the form as early as possible. At last, pay the application fee for Rajswathya Nurse Posts using an online payment gateway.
How to Apply for DMHS Jobs 2018 @ www.rajswasthya.nic.in?
On the home page, click on "DMHS Latest Recruitment for Nurse Gr-II vacancies in Scheduled Areas & Non-Scheduled Areas" links.
Every individual must log in to the site before clicking 'Apply Online' button.
After completion of registration, fill the details in the DMHS Rajasthan Nurse Grade II Online form.
Verify the entered details and upload all the required documents then press the "Submit" button.
In addition to this, make a payment of DMHS Rajasthan application fees by online payment gateway.
Finally, take a print out of the DMHS Jaipur Application form & fee receipt for future purpose.
Rs. 50/- extra charge for the application.
The board of Department of Medical & Health Services Jaipur offers a good amount of remuneration Rs. 26500/- for the selected candidates as DMHS Jaipur Nurse Gr-II Posts.
Hope the information I have given in the above article is beneficial to all the applicants of DMHS Rajasthan Recruitment 2018-19. If any doubts please visit the official notification pdf which is attached above. Meanwhile, bookmark our sarkarijobs.io page for latest Rojghar Samachar, Previous papers, Syllabus & Admit Cards. | {
"redpajama_set_name": "RedPajamaC4"
} | 1,432 |
La ragazza della domenica è un film del 1952 diretto da Robert Z. Leonard.
È un musical statunitense con Gower e Marge Champion, Dennis O'Keefe, Monica Lewis e Elaine Stewart.
Trama
Marito e moglie sono una coppia di danzatori, Chuck e Pamela Hubbard, nella vita felicemente sposati. Il loro sogno è esibirsi nei teatri di Broadway e per questo lavorano a lungo e duramente, ma al momento di raccogliere i frutti del loro impegno, i due scoprono che Pamela è in stato interessante e il loro medico, il Dottor Charles, proibisce tassativamente qualsiasi tipo di esibizione prima del parto. Inizierà una faticosa e rocambolesca ricerca per una sostituta che porterà non pochi problemi nella coppia.
Produzione
Il film, prodotto e sceneggiato da George Wells, ha come soggetto un'idea della sceneggiatrice americana Ruth Brooks Flippen per la Metro-Goldwyn-Mayer e girato nei Metro-Goldwyn-Mayer Studios a Culver City, California.
Distribuzione
Il film fu distribuito negli Stati Uniti dal 31 ottobre 1952 al cinema dalla Metro-Goldwyn-Mayer.
Note
Collegamenti esterni
Film musicali
Film diretti da Robert Z. Leonard | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 8,652 |
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Supporting caregivers in their difficult yet critical role ...
AgingFebruary 18, 2022
Supporting caregivers in their difficult yet critical role
For National Caregivers Day, experts discuss the challenges of caring for people with dementia or Alzheimer's disease
(SACRAMENTO)
Feb. 18 is National Caregivers Day, which honors individuals who provide personal care and physical and emotional support to those who need it most. More than 40 million people provide care nationwide. Their hours are long, the work is stressful and it often takes a toll on the caregiver's health.
Below is a Q&A that addresses a wide range of issues faced by caregivers. Helen Kales is a geriatric psychiatrist and an expert in helping caregivers manage the complex care needed by people with dementia and Alzheimer's disease. Terri Harvath is the director of the Betty Irene Moore School of Nursing Family Caregiving Institute and consults with family caregivers in the new Healthy Aging Clinic.
Caregivers can feel invisible to their family, friends and health care teams. Why is it important to acknowledge their contributions?
HARVATH: Family caregivers are the linchpin in the care for older adults. They are the ones who implement any plan of care that we have for patients when they leave our hospitals and clinics. In addition, they often have a deep understanding of how illness symptoms manifest and can detect subtle changes in condition that are clinically important. A recent study from AARP documents the complexity of care that families often provide with little to no training. To ensure families can provide safe and competent care, we must recognize the vital role they play in the care of frail, older adults and offer them the education and support they so desperately need.
What are some common misconceptions about caregiving?
KALES: Caregiving is often presented as a completely negative thing in our society. But there are a lot of studies that show caretaking has benefits. Some of it depends on what your relationship was like with the person before caretaking, but for a lot of people, it can bring families closer. There can be a lot of meaningful shared experiences, especially if you can laugh. People who can find humor in situations and are able to cope with humor generally look at caretaking more positively.
HARVATH: It is important to note that most families want to provide care to their older relatives and only relinquish that care when the person's needs exceed what the family can provide. We don't adequately prepare family caregivers for the complex role they take on. We need to help them understand how to detect subtle changes in the older person's condition before it becomes a serious threat to their health. We need to teach them how to manage complex care tasks (e.g., catheters, oxygen, injections) so they feel comfortable providing care and know when to ask for more help.
If we support family caregivers and include them as both a member of the older person's care team and a target of our interventions, we can deliver better, family-centered care in the community where most older adults prefer to remain.
Caregivers provide roughly $470 billion in unpaid assistance. Who is at risk when their health needs are not met?
HARVATH: Family caregivers often set their own health needs on the back burner because of the demands of their caregiving situation. This means that they are often sicker when they finally do seek care, adding to the illness burden they experience. This can jeopardize care for their older family member, taxing our long-term care system further with the care families can no longer provide.
Helen Kales has launched a nationwide study of DICE, a tool designed to help caregivers take care of people with dementia.
What are some misconceptions about caring for people with Alzheimer's disease or dementia?
KALES: One misconception is that people think of Alzheimer's disease as a memory problem — which it is. However, it is much more than that. Everyone with dementia has behavioral changes. Those can include agitation, depression, aggression, hallucinations, wandering and others. Behaviors like these are the most difficult, stressful, and costly aspects of care and often cause caregivers the most stress and depression.
What are some of the underlying reasons for these behavioral changes?
KALES: Sometimes, the behavior happens because the person lacks the ability to communicate. Maybe they are hungry, thirsty, or tired, and their behavior is an expression of an unmet need. Or it could be an expression of pathologies, such as being in pain or having an infection or an injury. The change in behavior tells us that something is happening that we need to pay attention to. Rather than memory, behavior is what puts people in nursing homes, and the reason there is an overuse of psychiatric medications for people with dementia. About 70% of people with dementia are put on psychiatric medications; much of that prescribing is for sedation.
What is the DICE method? How can it help caregivers with behavioral changes?
KALES: DICE stands for describe, investigate, create and evaluate (DICE). It is designed to help caregivers take care of people with dementia, but it can be a useful technique for other caregivers. It's very common for people with dementia to be prescribed sedating medication. The goal of DICE is to have their symptoms assessed and treated like any other symptom. For example, if you went to a doctor and said, 'I'm short of breath,' and the doctor immediately said, 'you must have pneumonia,' you would think you were going to a quack. You expect the doctor to investigate your symptoms before arriving at a conclusion. The same should be true for patients with dementia or Alzheimer's disease.
What do you mean by "investigate" the behavior?
KALES: Say, for example, someone suddenly starts getting agitated when it's time for a bath. Maybe it turns out they are very modest and feel uncomfortable with a new caregiver of a different gender. Or maybe the shower or bath is too hot or too cold. By describing and investigating the behavior, it's easier to find a non-pharmacological intervention. When patients are simply sedated, the underlying problem, or pathology, won't be revealed. More information about this method is available at the DICE website. We are also launching a national study using this method, which has been developed into an app that can be used on any smartphone, tablet or desktop device. People can find out more information about the study on the WeCareAdvisor study page.
Terri Harvath provides consultations and resources for caregivers at the Healthy Aging Clinic.
How many of the caregivers that you see at the Healthy Aging Clinic are caring for a family member with cognitive decline and how do you support them?
HARVATH: At the Family Caregiving Institute, we provide 1:1 consultation to family caregivers to better understand the challenges they face. The vast majority of the caregivers we see deal with dementia and usually two to three other chronic conditions (e.g., diabetes, hypertension). We help connect them to services they may be unaware of to ease the burden they are experiencing. In addition, we offer coaching for some of the dementia-related behavior symptoms they struggle with and offer support to try to alleviate some of the stress they experience. We also help them with some of the difficult decisions they face (e.g., can my mom keep driving? Is it time to move dad into a care setting?) to ensure they are making informed choices when safety concerns are in tension with a desire to protect autonomy and quality of life.
The pandemic has added stress for caregivers and people with dementia. Do you have any advice that might help improve mental health?
KALES: One suggestion is to make sure both of you get outside and get some sunlight. It sounds simple, but most older adults get only a fraction of the daily light that they need. Light synchronizes our circadian rhythms and improves mood and sleep, both for caregivers and people who are older. The other piece of advice is to tailor activities to what your loved one enjoys. For example, music can improve someone's mood, but not if you don't like that song or artist.
Any other advice for activities that help manage behavioral symptoms of dementia?
KALES: It's good to tailor activities they like to their current abilities. For example, maybe someone liked to fish but can't anymore. You could still have them look at a fishing magazine, watch a fishing video or organize a tackle box. Also, even if someone is physically or cognitively impaired, everybody wants to feel useful and have a sense of purpose, even it's just tossing a salad or folding laundry. We build helping strategies like those into the DICE approach.
What is the single most important advice you have for caregivers who feel overwhelmed and undervalued?
HARVATH: I like to stress that there is often no single right way to go when caring for an older family member, especially when there is dementia in the mix. It is also important to note that we often make decisions that only provide the illusion of safety and not actual or complete safety. We certainly saw this during the pandemic when some of the nursing homes and assisted living facilities that were supposed to provide safe harbor were not always able to do so. So instead, finding the 'best' path forward (i.e., the path that works the best for this family at this time) often involves trial and error until they land on something that works. In finding that best path, it is important to keep quality of life and the older person's expressed preferences in mind and not only attend to safety.
Learn more about the Family Caregiving Institute, which is dedicated to the well-being of family caregivers through education and research.
Learn more about the DICE approach and the WeCareAdvisor.
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\section{Introduction}
\input{intro}
\section{Background} \label{sec:background}
\subsection{SAT}
The \emph{Boolean satisfiability} or {\ensuremath{\mathsf{SAT}}\xspace} problem is to decide, given a Boolean formula $\varphi(\vec{x})$ over a vector of variables $\vec{x}$, whether or not there is a \emph{satisfying assignment} to $\vec{x}$ that makes the formula true.
In this paper, we make the common assumption that the formula $\varphi$ is in \emph{conjunctive normal form} (CNF): it is a conjunction $\psi_1 \land \dots \land \psi_m$ of \emph{clauses}, where each clause $\psi_i$ is a disjunction $\ell_{i1} \lor \dots \lor \ell_{ik}$.
Here each $\ell_{ij}$ is a \emph{literal}: either a variable from $\vec{x}$ or the negation of such a variable.
We also assume that every clause has the same length $k$.
A formula $\varphi$ satisfying these conditions is called a \emph{$k$-\ensuremath{\mathsf{SAT}}\xspace} formula.
Given a partial assignment $\rho$ to some of the variables $\vec{x}$, the \emph{restriction} $\varphi \lceil_\rho$ of $\varphi(\vec{x})$ to $\rho$ is the formula obtained from $\varphi$ by removing all clauses satisfied by $\rho$ and all literals falsified by $\rho$.
We can apply a restriction to any list of clauses analogously.
The \emph{size} of the restriction is the number of variables assigned by $\rho$.
\subsection{Resolution and CDCL}
\emph{Resolution} \cite{resolution} is a fundamental proof system that underlies modern {\ensuremath{\mathsf{SAT}}\xspace} solving algorithms.
It consists of a single rule stating that from clauses $(v \lor \vec{w})$ and $(\lnot v \lor \vec{u})$ that have occurrences of $v$ with opposite polarities, we may infer the clause $(\vec{w} \lor \vec{u})$.
As we will see in a moment, the importance of resolution for our purposes is that in order to establish that a formula $\varphi$ is unsatisfiable, {\ensuremath{\mathsf{SAT}}\xspace} solvers based on CDCL implicitly construct a \emph{resolution refutation} of $\varphi$: a derivation of a contradiction (the empty clause) from $\varphi$ using the resolution rule.
This effectively means that the runtime of such a solver cannot be shorter than the length of the shortest such refutation.
To make this precise we need to define what we mean by CDCL.
\emph{Conflict-driven clause learning} \cite{sh-cdcl} describes a class of algorithms that extend the Davis--Putnam--Logemann--Loveland (DPLL) algorithm \cite{dpll}.
DPLL is a classical search algorithm that assigns each variable in turn, backtracking if a clause is falsified by the current assignment.
If at any point there is a clause with only a single unassigned variable, then that variable can immediately be given the assignment which satisfies the clause --- a rule called \emph{unit propagation}.
If we eventually assign every variable, then we have found a satisfying assignment; otherwise, the search will backtrack all the way to the top level, every possible assignment will have been tried, and the formula is unsatisfiable.
CDCL-type algorithms augment this procedure by \emph{learning} at every backtrack point a new clause that summarizes the reason why the current partial assignment falsifies the formula \cite{grasp,bayardo-schrag}.
This \emph{conflict clause} $C$ is derived by resolving the falsified clause $F$ with one or more other clauses that were used to assign variables in $F$ by unit propagation.
As a result $C$ is always derivable from the original formula $\varphi$ by resolution, and writing out all clauses learned by CDCL when $\varphi$ is unsatisfiable gives a resolution refutation of $\varphi$ \cite{cdcl-sim}.
So the shortest such refutation gives a lower bound for the runtime of CDCL.
This is true regardless of the heuristics used by the particular CDCL variant to decide which variable to assign and its polarity, how exactly to derive the conflict clause, and when to restart search from the beginning (see \cite{cdcl-sim} for a more precise statement).
We also note that pre- or inprocessing techniques that add no clauses (e.g. blocked clause elimination \cite{bce}) or add only clauses derived via resolution (e.g. variable elimination \cite{var-elim}) will not affect the lower bound.
\subsection{Random SAT Instances}
To study the performance of CDCL on ``typical'' instances, we use the framework of \emph{average-case complexity}, which analyzes the efficiency of algorithms on \emph{random} instances drawn from a particular distribution.
We will be interested in complexity lower bounds that hold for almost all sufficiently large instances:
\begin{definition}
An event $X$ occurs \textbf{with high probability} in terms of $n$ if $\Pr[X] \rightarrow 1$ as $n \rightarrow \infty$.
\end{definition}
For example, if flipping $n$ fair coins, with high probability at least $49\%$ will be heads.
Perhaps the simplest distribution over {\ensuremath{\mathsf{SAT}}\xspace} instances arises from fixing the numbers of variables, clauses, and variables per clause, and then sampling uniformly:
\begin{definition}
$F_k(n, m)$ is the uniform distribution over $k$-CNF formulas with $n$ variables and $m$ clauses.
\end{definition}
This \emph{random $k$-{\ensuremath{\mathsf{SAT}}\xspace} model} has been widely studied, and is known to be difficult on average for CDCL (for clause-variable ratios in a certain range) by the resolution lower bound discussed above: with high probability, a random unsatisfiable instance has only exponentially long resolution refutations \cite{beame}.
As we will discuss shortly, our work extends this result to a more recent random {\ensuremath{\mathsf{SAT}}\xspace} model that favors instances that are ``pseudo-industrial'' in the sense of having good community structure.
\subsection{Community Structure}
The notion of community structure has a long history in many fields \cite{community-history}.
The essential idea is that graphs with ``good community structure'' can be broken into relatively small pieces, \emph{communities}, that are densely connected internally but only sparsely connected to each other.
There are a number of metrics which have been proposed to make this notion formal, of which one of the most popular is \emph{modularity} \cite{modularity}.
We consider unweighted graphs as weighted graphs with all weights $1$.
\begin{definition}
Let $G = (V, E)$ and let $\delta = \{C_1, \dots, C_n\}$ be a vertex partition.
Let $\deg v$ be the degree of $v$, and $w(x,y)$ be the weight of the edge $(x,y)$ or zero if there is no such edge.
The \textbf{modularity} (or $Q$-value) of $G$ is
\[
Q = \max_\delta \sum_{C \in \delta} \left[ \frac{\sum_{x, y \in C} w(x, y)}{\sum_{x, y \in V} w(x, y)} - \left(\frac{\sum_{x \in C} \deg x}{\sum_{x \in V} \deg x} \right)^2 \right].
\]
\end{definition}
While work on community structure in {\ensuremath{\mathsf{SAT}}\xspace} instances has focused on modularity, there are several competing metrics that have been used to measure community structure in other domains.
In Appendix \ref{sec:other-metrics} of this paper, we consider four: silhouette index, conductance, coverage, and performance \cite{best-mod}.
Finally, we introduce notation for two graphs that will be useful in this paper: $K_n$, the complete graph on $n$ vertices, and $K_n^m$, consisting of $m$ disjoint copies of $K_n$.
\subsection{SAT and Community Structure}
Recent work on the community structure of {\ensuremath{\mathsf{SAT}}\xspace} instances begins by associating to each instance its \emph{variable incidence graph} (also known as the \emph{primal graph}).
\begin{definition}
Let $\varphi$ be a CNF formula.
The \textbf{variable incidence graph} (\textbf{VIG}) of $\varphi$ is the graph $G_\varphi = (V, E)$ where $V$ is the set of all variables occurring in $\varphi$ and $E$ is the set $\{(v_1, v_2) : v_1, v_2 \in V$ and they appear together in some clause of $\varphi\}$.
\end{definition}
Some works use a \emph{weighted} version of this graph with $w(v_1,v_2) = \sum_{cl} \left[ 1/\binom{|cl|}{2} \right]$, where the sum is over all clauses in which both $v_1$ and $v_2$ appear \cite{community-sat,giraldez2015modularity}.
This ensures that each clause contributes an equal amount to the total weight of the graph regardless of its length.
Our results apply to both the weighted and unweighted versions.
Obviously, the graph $G_\varphi$ does not preserve all information about the instance $\varphi$.
In particular, the polarities of the literals are ignored.
But the graph does capture significant structural information: for example, if the graph has two connected components on variables $\vec{x}$ and $\vec{y}$ then the formula $\varphi(\vec{x},\vec{y})$ can be split into $\psi(\vec{x}) \land \chi(\vec{y})$ and each subformula solved independently.
In practice a perfect decomposition is rare, but one into \emph{almost} independent parts is more plausible.
This is exactly the idea of community structure, and leads us naturally to consider applying modularity to {\ensuremath{\mathsf{SAT}}\xspace} instances.
\begin{definition}
The \textbf{modularity} of a formula $\varphi$ is the modularity of $G_\varphi$.
\end{definition}
As was mentioned earlier, it has been found empirically that modularity correlates with CDCL performance \cite{newsham}.
This is a claim about the \emph{average} behavior of CDCL over a wide variety of industrial benchmarks, not about its behavior on any specific instance.
Thus it is naturally formalized in the average-case complexity framework discussed above, by giving a distribution that favors instances that are ``industrial'' in character.
One such proposal, based on the idea that the key commonality of industrial instances is their good community structure, is the \emph{community attachment} model of Gir\'aldez-Cru and Levy \cite{giraldez2015modularity}.
In addition to the numbers of variables and clauses, this model has parameters controlling the number of communities and the (expected) fraction of clauses that lie within a single community instead of spanning multiple communities.
\begin{definition}
Let $N$ be a set of $n$ variables. A \textbf{partition of
$N$ into $c$ communities} is a partition $S = \{S_1, \dots, S_c\}$ of $N$ such
that $|S_i| = n / c$. A clause
is \textbf{within a community} if it contains only variables from a single
$S_i$.
A \textbf{bridge clause} is a clause whose variables are all in different communities.
\end{definition}
\begin{definition}[\cite{giraldez2015modularity}] \textbf{(Community Attachment Model)}
Let $n, m, c, k \in \ensuremath{\mathbb{N}}\xspace$ and $p \in [0, 1]$ such that $c$ divides $n$ and $2 \le k \leq c \leq n /
k$. Then $F_k(n, m, c, p)$ is the distribution over $k$-CNF formulas with $n$
variables and $m$ clauses given by the following procedure: first, choose a random
partition of $n$ variables into $c$ communities. With probability $p$, choose a
clause uniformly among clauses within a community, and otherwise choose uniformly among bridge clauses.
Generate $m$ clauses independently in this way.
\end{definition}
\begin{remark}
We define bridge clauses in a way that matches the community attachment model, but as we will discuss below our results also hold for a modified model where a bridge clause is any clause not within a single community.
\end{remark}
Like the random $k$-{\ensuremath{\mathsf{SAT}}\xspace} model $F_k(n,m)$, the model $F_k(n,m,c,p)$ ranges over $k$-CNF formulas with $n$ variables and $m$ clauses, and each clause is chosen independently of the others.
However, in this model the clauses are of two different types: those lying entirely within a community, and those spread across $k$ different communities.
The probability $p$ controls how likely a clause is to be of the first type versus the second.
The idea behind this model is that by picking $c$ and $p$ appropriately, one is likely to obtain instances that decompose into loosely-connected communities, as has been observed in actual industrial instances.
More precisely, the expected modularity of an instance drawn from $F_k(n,m,c,p)$ is lower bounded by $p - (1/c)$, so that for nontrivial $c$ highly modular instances can be generated by setting $p$ large enough \cite{giraldez2015modularity}.
Furthermore, Gir\'aldez-Cru and Levy find experimentally that high-modularity instances generated with this model are solved more quickly by CDCL than by look-ahead solvers, and the reverse is true for low-modularity instances \cite{giraldez2015modularity}.
This parallels the same observation for industrial instances versus random instances.
Thus, they conclude, $F_k(n,m,c,p)$ is a more realistic model of industrial instances than the random $k$-{\ensuremath{\mathsf{SAT}}\xspace} model $F_k(n,m)$.
\section{Worst-Case Hardness} \label{sec:worst-case}
In this section, we propose a simple class of graph metrics that we argue should include most metrics quantifying community structure.
We show that modularity is in fact within the class, as are several other popular graph clustering metrics.
However, we demonstrate that the set of $\ensuremath{\mathsf{SAT}}\xspace$ instances that have ``good community structure'' according to any metric in the class is \ensuremath{\mathsf{NP}}\xspace-hard.
Therefore, no such metric can be a guaranteed indicator of the difficulty of a $\ensuremath{\mathsf{SAT}}\xspace$ instance.
\subsection{A Class of ``Modularity-like'' Graph Metrics} \label{sec:pcm-definition}
We begin by formalizing what we mean by a graph metric.
\begin{definition}
A \textbf{graph metric} is a function $m$ from weighted graphs to $[0,1]$.
Given $m$ and any $\epsilon \in [0,1]$, $\ensuremath{\mathsf{SAT}}\xspace_{m,\epsilon}$ is the class of all \ensuremath{\mathsf{SAT}}\xspace instances $\varphi$ such that $m(G_\varphi) \geq 1 - \epsilon$.
\end{definition}
For example, if $m$ is modularity then $\ensuremath{\mathsf{SAT}}\xspace_{m,\epsilon}$ consists of the ``high modularity'' formulas, where ``high'' means any modularity above $1-\epsilon$.
In general we are interested in graph metrics that represent a notion of community structure, assigning larger values to graphs which have such a structure than those that do not.
For such a metric $m$, consider the following property:
\begin{definition}
A graph metric $m$ is a \textbf{polynomial clique metric} (\textbf{PCM}) if for all $\epsilon > 0$, there is a poly-time computable function $c : \ensuremath{\mathbb{N}}\xspace \rightarrow \ensuremath{\mathbb{N}}\xspace$ with at most polynomial growth and some $n_0 \in \ensuremath{\mathbb{N}}\xspace$ such that for all $n \ge n_0$, if $K$ is $K_n$ with any positive edge weights then $m(K^{c(n)}) \ge 1 - \epsilon$.
\end{definition}
\begin{remark}
If using the unweighted version of the variable incidence graph, our proofs will work using a relaxed definition that applies only to $K = K_n$ with unit weights.
\end{remark}
In essence, the definition states that for any (sufficiently large) size $n$, at most a polynomial number of copies of $K_n$ are needed to produce a graph that $m$ considers to have ``good community structure''.
This is a natural property for modularity-like metrics to have, since copies of $K_n$ are in some sense ideal communities: internally connected as much as possible, with no external edges.
Of course we would not consider a single copy of $K_n$ to have good community structure, so the definition of a PCM only requires that such structure be obtained for \emph{some} number of copies at most polynomial in $n$.
Next we demonstrate that the PCMs are a large class including modularity and several other popular clustering metrics.
While the other metrics have not been experimentally evaluated in the context of SAT, this still supports our claim that the PCM property is a natural one for metrics of community structure to have.
For lack of space, we defer the definitions and analysis of the metrics other than modularity to Appendix \ref{sec:other-metrics}.
\begin{theorem}
Modularity is a PCM.
\end{theorem}
\begin{proof}
Fix any $\epsilon > 0$ and $n \geq 2$.
Let $K$ be $K_n$ with arbitrary positive edge weights, and let $G = K^c$.
Let $\delta$ be the vertex partition that groups two vertices iff they are in the same copy of $K$.
Then since each community is identical, and there are $c$ communities,
$\sum_{x, y \in C} w(x, y) / \sum_{x, y \in V} w(x, y) = 1/c$ and $\sum_{x \in C} \deg x / \sum_{x \in V} \deg x = 1/c$
for any $C \in \delta$.
Therefore, $Q(G) \geq c(1 / c - (1 / c)^2) = 1 - 1 / c$.
Putting $c = 1 / \epsilon$, we have $Q(G) \geq 1 - \epsilon$.
Since $c$ is $O(1)$ with respect to $n$, $Q$ is a PCM.
\qed
\end{proof}
\vspace{-1.7ex}
\begin{restatable}{theorem}{otherMetricsPCM}
Silhouette index, conductance, coverage, and performance are PCMs.
\end{restatable}
\subsection{Hardness of PCM-Modular Instances} \label{sec:pcm-hardness}
Now we show that the $\ensuremath{\mathsf{SAT}}\xspace$ instances which have ``good community structure'' according to a PCM are no easier in the worst case than any other instance.
The PCMs thus form a wide class of metrics which cannot be used as a guaranteed indicator of the difficulty of a $\ensuremath{\mathsf{SAT}}\xspace$ instance.
Our reduction can be viewed as a variation of that suggested by Ganian and Szeider~\cite{ganian-szeider} to show \ensuremath{\mathsf{NP}}\xspace-hardness in the specific case of modularity.
\begin{theorem}
For any PCM $m$, the class $\ensuremath{\mathsf{SAT}}\xspace_{m,\epsilon}$ is \ensuremath{\mathsf{NP}}\xspace-hard for all $\epsilon > 0$.
\end{theorem}
\begin{proof}
Given a $\ensuremath{\mathsf{SAT}}\xspace$ instance $\phi$, we will convert it into an
equisatisfiable instance of $\ensuremath{\mathsf{SAT}}\xspace_{m,\epsilon}$ in polynomial time. Let $V$ be
the set of all variables occurring in $\phi$, along with new variables as necessary so that $|V| \ge n_0$.
Fixing a variable $x$ not in $V$, let $\psi$ be the formula obtained by adding to $\phi$ all clauses of the
form $x \lor y \lor z$ with $y, z \in V$. Clearly, the VIG of $\psi$ is $K_n$ with $n = |V| + 1 \ge n_0$.
Furthermore, $\phi$ and $\psi$ are equisatisfiable, since we can simply assert $x$ to
satisfy all the new clauses. Now letting $\chi$ be the conjunction of $c(n)$
disjoint copies of $\psi$ (i.e. copies with variables renamed so none are
common), the variable incidence graph $G$ of $\chi$ is $K_n^{c(n)}$ (with some positive weights).
By the PCM property, we have $m(G) \ge 1-\epsilon$, so $\chi \in
\ensuremath{\mathsf{SAT}}\xspace_{m,\epsilon}$. Since $\chi$ and $\psi$ are clearly equisatisfiable, so are
$\chi$ and $\phi$, and thus this procedure gives a reduction from $\ensuremath{\mathsf{SAT}}\xspace$ to
$\ensuremath{\mathsf{SAT}}\xspace_{m,\epsilon}$. Finally, the procedure is polynomial-time since $c(n)$ has
at most polynomial growth and can be computed in polynomial time.
\qed
\end{proof}
\section{Average-Case Hardness} \label{sec:average-case}
In contrast to the previous section, we now consider the difficulty of modular instances for a particular class of algorithms, namely those like CDCL which prove unsatisfiability by effectively constructing a resolution refutation.
While these results are therefore more specific, they are also much more powerful: they show that modular instances are difficult not just in the worst case but also on average.
Our argument is largely based on the resolution lower bound of Beame and Pitassi \cite{beame}, which can be used to establish the hardness of instances from the random $k$-{\ensuremath{\mathsf{SAT}}\xspace} model.
In order to use that result, we need to show that most instances from the community attachment model have certain \emph{sparsity} properties used by the proof.
So our main steps, detailed in Sections \ref{sec:defining-distribution}--\ref{sec:cdcl-runtime} below, are as follows:
\begin{enumerate}
\item Define a new distribution $\overline{F}_k(n,m,c,p;m')$ over $k$-CNF formulas that works by taking a \emph{random subformula} of an instance from the random $k$-{\ensuremath{\mathsf{SAT}}\xspace} model $F_k(n,m')$.
\item Show that this new distribution is in fact identical to the community attachment model $F_k(n,m,c,p)$.
\item Observe that the sparsity properties are inherited by subformulas, so the sparsity result in \cite{beame} for the random $k$-{\ensuremath{\mathsf{SAT}}\xspace} model $F_k(n,m')$ transfers to the community attachment model $F_k(n,m,c,p)$.
\item Adapt the Beame--Pitassi argument \cite{beame} to obtain an exponential lower bound on the resolution refutation length.
\item Conclude that CDCL takes exponential time on unsatisfiable formulas from the community attachment model $F_k(n,m,c,p)$ with high probability.
\end{enumerate}
\subsection{Defining the New Distribution} \label{sec:defining-distribution}
We begin by defining our new distribution $\overline{F}_k(n,m,c,p;m')$, which takes an additional parameter $m'$ that we will specify in Section \ref{sec:transfer}.
\begin{definition} \label{def:mod-levy-dist}
Let $n, m, c, k, m' \in \ensuremath{\mathbb{N}}\xspace$ and $p \in [0, 1]$ such
that $2 \le k \leq c \leq n / k$. Then $\overline{F}_k(n, m, c, p;
m')$ is the distribution over $k$-CNF formulas with $n$ variables and $m$
clauses defined by Algorithm \ref{mod_levy} (which is such a distribution by Lemma \ref{lemma:well-defined} below).
\end{definition}
\begin{algorithm}
\caption{defining the distribution $\overline{F}_k(n, m, c, p; m')$}
\label{mod_levy}
\begin{algorithmic}[1]
\State choose $\phi$ from $F_k(n, m')$ \label{line:sample-phi}
\State choose a uniformly random partition of the $n$ variables into $c$ communities
\State $h \gets c\binom{n / c}{k} / \binom{n}{k}$
\State $b \gets (n / c)^k\binom{c}{k} / \binom{n}{k}$
\State $\psi \gets$ the empty formula on $n$ variables
\ForAll {clauses $C$ of $\phi$}
\WithProb {$p$} \label{line:loop-start}
\If {$C$ is within a community}
\State add $C$ to $\psi$
\EndIf
\OtherwiseProb \label{line:otherwise-branch}
\WithProb {$h / b$} \label{line:second-coin}
\If {$C$ is a bridge clause}
\State add $C$ to $\psi$
\EndIf
\EndProb
\EndProb
\If {$|\psi| = m$}
\Return $\psi$ \Comment{the algorithm ``succeeds''} \label{line:success}
\EndIf
\EndFor
\State choose a fresh $\psi$ from $F_k(n, m, c, p)$ \label{line:sample-psi} \\
\Return $\psi$ \Comment{the algorithm ``fails''} \label{line:failure}
\end{algorithmic}
\end{algorithm}
\begin{lemma} \label{lemma:well-defined}
For all parameters satisfying the conditions of Definition \ref{def:mod-levy-dist}, Algorithm \ref{mod_levy} defines a probability distribution over $k$-CNF formulas with $n$ variables and $m$ clauses.
\end{lemma}
\begin{proof}
First we must check that $h/b \le 1$ so that the algorithm is well-defined.
We have
\[
\frac{h}{b} = \frac{c \binom{n/c}{k}}{\left( \frac{n}{c} \right)^k \binom{c}{k}} \le \frac{c \left( \frac{n}{c} \right)^k}{k! \left( \frac{n}{c} \right)^k \left( \frac{c}{k} \right)^k} = \frac{k^k}{k! \; c^{k-1}} \le \frac{1}{(k-1)!} \le 1,
\]
since we assume $c \ge k$.
Algorithm \ref{mod_levy} always terminates, returning a formula $\psi$ from either line \ref{line:success} or \ref{line:failure}.
In the first case, $\psi$ is a subset of $\phi$, which is drawn from $F_k(n,m')$ and so has $k$-CNF clauses over $n$ variables.
Furthermore, the algorithm does not return from line \ref{line:success} unless $\psi$ has $m$ clauses.
In the second case, $\psi$ is drawn from $F_k(n,m,c,p)$, and so again is a $k$-CNF formula with $n$ variables and $m$ clauses.
\qed
\end{proof}
\subsection{Comparing the Distribution to the Community Attachment Model}
Next we prove that our definition via Algorithm \ref{mod_levy} is equivalent to the usual community attachment definition.
Since the algorithm adds each clause independently, in essence this amounts to showing that each clause is within a community with probability $p$.
\begin{lemma} \label{lemma:equiv}
For any $m' \in \ensuremath{\mathbb{N}}\xspace$, the distribution $\overline{F}_k(n, m, c, p ; m')$ is identical to the distribution $F_k(n, m, c, p)$.
\end{lemma}
\begin{proof}
When Algorithm \ref{mod_levy} returns a formula $\psi$ from line \ref{line:failure}, $\psi$ is drawn from $F_k(n,m,c,p)$, and so the two distributions are trivially identical.
So we need only consider the case when the algorithm returns from line \ref{line:success}.
Because the algorithm handles each clause of $\phi$ independently (until $m$ clauses are added), it suffices to show that when a clause is added to $\psi$, it is within a community with probability $p$ and is otherwise a bridge clause.
Starting from line \ref{line:loop-start} of Algorithm \ref{mod_levy}, let $C_\text{comm}$ be the event that the clause $C$ is within a community, $C_\text{bridge}$ the event that $C$ is a bridge clause, and $C_\text{added}$ the event that $C$ is added to $\psi$.
Let $A$ be the event that the algorithm takes the random branch on line \ref{line:loop-start} instead of the branch on line \ref{line:otherwise-branch}.
Then we have
\[
\Pr[ C_\text{comm} | C_\text{added} ] = \Pr[ C_\text{comm} | A, C_\text{added}] \Pr[A | C_\text{added}] + \Pr[ C_\text{comm} | \overline{A}, C_\text{added}] \Pr[ \overline{A} | C_\text{added}] .
\]
The second term is zero because $\overline{A}$ means the algorithm takes the branch on line \ref{line:otherwise-branch} and thus only adds the clause if it is a bridge clause.
Likewise, $\Pr[ C_\text{comm} | A, C_\text{added}] = 1$ because the branch on line \ref{line:loop-start} only adds the clause if it is within a community.
So
\[
\Pr[ C_\text{comm} | C_\text{added} ] = \Pr[A | C_\text{added}] = \Pr[C_\text{added} | A] \Pr[A] \;/\; \Pr[C_\text{added}] .
\]
By straightforward counting arguments, $\Pr[C_\text{comm}] = c\binom{n / c}{k} / \binom{n}{k} = h$ and $\Pr[C_\text{bridge}] = (n / c)^k\binom{c}{k} / \binom{n}{k} = b$.
Since the coin flips on lines \ref{line:loop-start} and \ref{line:second-coin} are independent of $C$, we have $\Pr[C_\text{added} | A] = \Pr[C_\text{comm}] = h$ and $\Pr[C_\text{added} | \overline{A}] = (h/b) \Pr[C_\text{bridge}] = h$.
Also $\Pr[A] = p$, so
\[
\Pr[C_\text{added}] = \Pr[C_\text{added} | A] \Pr[A] + \Pr[C_\text{added} | \overline{A}] \Pr[\overline{A}] = h p + h (1-p) = h .
\]
Plugging these into the expression above we obtain $\Pr[ C_\text{comm} | C_\text{added} ] = p$.
So each clause added to $\psi$ is within a community with probability $p$, and otherwise by construction it must be a bridge clause.
Therefore when Algorithm \ref{mod_levy} returns from line \ref{line:success}, it is equivalent to generating $m$ clauses independently, each of which is a uniformly random clause within a community with probability $p$, and otherwise a uniformly random bridge clause.
So $\overline{F}_k(n, m, c, p ; m')$ is identical to $F_k(n,m,c,p)$.
\qed
\end{proof}
\subsection{Transferring Subformula-Inherited Properties} \label{sec:transfer}
Algorithm \ref{mod_levy} can ``fail'' by adding fewer than the desired number of clauses $m$ to $\psi$, then falling back on the community attachment model as a backup.
Otherwise, the algorithm ``succeeds'', returning on line \ref{line:success} a formula that was built up from clauses of $\phi$ and is therefore a subformula of it.
Since our goal is to have the formulas from this distribution inherit properties from $\phi$, we need to ensure that Algorithm \ref{mod_levy} succeeds with high probability.
We can do this by taking $m'$, the number of clauses in $\phi$, to be large enough: then even if a given clause is only added to $\psi$ with a small probability, overall we are likely to add $m$ of them.
As we will see in the proof, the probability of adding a clause is roughly $1/c^{k-1}$, so taking $m'$ to be slightly larger than $c^{k-1} m$ will suffice.
We use the following standard tail bound.
\begin{lemma} \label{bound} If $B(n,p)$ is the number of successes in $n$ Bernoulli trials each with success probability $p$, then for $k < pn$ we have
\[
\Pr[B(n, p) \leq k] \leq \exp\left(\frac{-(pn - k)^2}{2pn}\right).
\]
\end{lemma}
\begin{lemma} \label{lemma:success}
Suppose that $c$ is $o(n)$, $m \rightarrow \infty$ as $n \rightarrow \infty$, and $m' = (1 + \epsilon)c^{k - 1}m$ for some $\epsilon > 0$.
Then Algorithm \ref{mod_levy} returns from line \ref{line:success} with high probability.
\end{lemma}
\begin{proof}
As shown in Lemma \ref{lemma:equiv}, the probability that starting from line \ref{line:loop-start} the clause $C$ will be added to $\psi$ is $h$.
So the probability that Algorithm \ref{mod_levy} returns from line \ref{line:success} is $\Pr[B(m', h) \geq m] = 1 - \Pr[B(m', h) \leq m - 1]$.
Now observe that
\[
h c^{k-1} = \frac{c^k\binom{n /c}{k}}{\binom{n}{k}} = \frac{n (n-c) \cdots (n-c(k+1))}{n (n-1) \cdots (n-k+1)} \le 1.
\]
Furthermore, we have
\[
\lim_{n \rightarrow \infty} h c^{k-1} = \lim_{n \rightarrow \infty} \frac{c^k\binom{n /c}{k}}{\binom{n}{k}} = \lim_{n \rightarrow \infty} \left[ \frac{\binom{n/c}{k}}{\frac{(n/c)^k}{k!}} \cdot \frac{\frac{n^k}{k!}}{\binom{n}{k}} \right] = \left[ \lim_{n \rightarrow \infty} \frac{\binom{n/c}{k}}{\frac{(n/c)^k}{k!}} \right] \left[ \lim_{n \rightarrow \infty} \frac{\frac{n^k}{k!}}{\binom{n}{k}} \right] = 1,
\]
where in evaluating the second-to-last limit we use the fact that $c$ is $o(n)$ and so $\lim_{n \rightarrow \infty} (n/c) = \infty$.
So for sufficiently large $n$ we have $hc^{k - 1} \ge 1 - \epsilon / 2(1 + \epsilon)$, and therefore
\[
h m' = h(1 + \epsilon)c^{k - 1}m \ge \left(1 - \frac{\epsilon}{2(1 + \epsilon)} \right) (1 + \epsilon) m = (1 + \epsilon/2) m.
\]
Applying Lemma \ref{bound}, we have
\begin{align*}
\Pr[B(m',h) \le m - 1] &\le \exp \left( \frac{-[hm' - (m-1)]^2}{2 h m'} \right) \le \exp \left( \frac{-[(1+\epsilon/2) m - m]^2}{2 h (1+\epsilon) c^{k-1} m} \right) \\
&= \exp \left( \frac{-m (\epsilon/2)^2}{2(1+\epsilon) \cdot h c^{k-1}} \right) \le \exp \left( \frac{-m \epsilon^2}{8 (1+\epsilon)} \right),
\end{align*}
which goes to zero as $m \rightarrow \infty$, and therefore as $n \rightarrow \infty$.
So with high probability, Algorithm \ref{mod_levy} will return from line \ref{line:success}.
\qed
\end{proof}
Now it is simple to show that subformula-inherited properties are indeed passed down from random $k$-{\ensuremath{\mathsf{SAT}}\xspace} instances to instances drawn from our distribution.
Here ``subformula-inherited'' simply means that if $\varphi$ has the property, then any formula made up of a subset of the clauses of $\varphi$ also has the property.
For example, being satisfiable is subformula-inherited, but being unsatisfiable is not.
\begin{lemma} \label{lemma:inherit}
Suppose that $c$ is $o(n)$, $m \rightarrow \infty$ as $n \rightarrow \infty$, $m' = (1 + \epsilon)c^{k - 1}m$ for some $\epsilon > 0$, and $P$ is a subformula-inherited property.
Then if a formula drawn from $F_k(n,m')$ has property $P$ with high probability, a formula drawn from $\overline{F}_k(n, m, c, p ; m')$ has property $P$ with high probability.
\end{lemma}
\begin{proof}
Run Algorithm \ref{mod_levy} to sample from $\overline{F}_k(n, m, c, p; m')$.
Let $P_\psi$ and $P_\phi$ respectively be the events that the returned formula $\psi$ and the formula $\phi$ from line \ref{line:sample-phi} have property $P$.
Also let $R$ be the event that the algorithm returns from line \ref{line:success}.
When the algorithm returns from line \ref{line:success}, $\psi$ is a subformula of $\phi$, and since $P$ is inherited by subformulas we have $\Pr[ P_\psi | R ] \ge \Pr[ P_\phi ]$.
Now as $\phi$ is drawn from $F_k(n,m')$, the event $P_\phi$ occurs with high probability, and so $\Pr[ P_\psi | R ] \rightarrow 1$ as $n \rightarrow \infty$.
By Lemma \ref{lemma:success}, the event $R$ also happens with high probability, so $\Pr[ P_\psi ] \ge \Pr[ P_\psi \land R ] = \Pr[ P_\psi | R ] \cdot \Pr[R] \rightarrow 1$ as $n \rightarrow \infty$.
Therefore $\psi$ has property $P$ with high probability.
\qed
\end{proof}
Together, Lemmas \ref{lemma:equiv} and \ref{lemma:inherit} show that subformula-inherited properties of random $k$-{\ensuremath{\mathsf{SAT}}\xspace} instances are also possessed (with high probability) by instances from the community attachment model.
\subsection{Proving the Resolution Lower Bounds} \label{sec:beame-pitassi}
Now we transition to adapting the argument of Beame and Pitassi \cite{beame}.
The proof uses two types of sparsity conditions.
Both view a clause $C$ as a set of variables, so that another set of variables $X$ ``contains'' $C$ if and only if every variable in $C$ is in $X$.
\begin{definition}
A formula is \textbf{$n'$-sparse} if every set of $s \leq n'$ variables contains at most $s$ clauses. \end{definition}
\begin{definition}
Let $n' < n''$. A formula is \textbf{$(n', n'', y)$-sparse} if every set of $s$ variables with $n' < s \leq n''$ contains at most $ys$ clauses.
\end{definition}
These are both clearly subformula-inherited.
The Beame--Pitassi argument \cite{beame} is broken into three major lemmas, each of which we will use without change.
The last lemma establishes the sparsity properties above for the random $k$-{\ensuremath{\mathsf{SAT}}\xspace} model.
\begin{lemma}[\cite{beame}] \label{big-clause} Let $n' \leq n$ and $F$ be an unsatisfiable
CNF formula in $n$ variables with clauses of size at most $k$ that is both
$n'$-sparse and $(n'(k + \epsilon) / 4, n'(k + \epsilon) / 2, 2 / (k +
\epsilon))$-sparse. Then any resolution proof $P$ of the unsatisfiability of $F$
must include a clause of length at least $\epsilon n' / 2$. \end{lemma}
\begin{lemma}[\cite{beame}] \label{whp-proof} Let $P$ be a resolution refutation of $F$ of
size $S$. Given $\beta > 0$, say the \textbf{large clauses} of $P$ are those clauses mentioning more than $\beta n$
distinct variables. Then with probability at least $1 - 2^{1 - \beta t / 4}S$, a
random restriction of size $t$ sets all large clauses in $P$ to 1. \end{lemma}
\begin{lemma}[\cite{beame}] \label{sparsity} Let $x > 0$, $1 \geq y > 1 / (k - 1)$, and $z
\geq 4$. Fix a restriction $\rho$ on $t \leq \min\{xn / 2, x^{1 - 1 / y(k -
1)}n^{1 - 1/(k - 1)} / z\}$ variables. Drawing $F$ from $F_k(n,m)$ with
\[
m \leq \frac{y}{e^{1 + 1 / y}2^{k + 1 / y}}x^{1/ y - (k - 1)}n ,
\]
then with probability at least $1 - 2^{-t} - (2^k + 1) / z^{k - 1}$, $F\lceil_\rho$ is both $(xn / 2, xn, y)$-sparse and $xn$-sparse.
\end{lemma}
We can now combine these to prove the analog of the main theorem of Beame and Pitassi for modular instances.
Our argument is almost identical to theirs: the only difference is that we apply Lemma \ref{sparsity} to larger instances from the random $k$-{\ensuremath{\mathsf{SAT}}\xspace} model,\footnote{Note that as required by its statement, we are applying Lemma \ref{sparsity} to formulas drawn from $F_k(n,m)$, \emph{not} to formulas drawn from $\overline{F}_k(n,m,c,p;m')$. Lemmas \ref{big-clause} and \ref{whp-proof} work for any formula, so we may use all three lemmas precisely as proved in \cite{beame}.} so that our results above will give us sparsity for modular instances of the correct size embedded in them as subformulas.
\begin{theorem} \label{main}
Let $k \geq 3$, $0 < \epsilon < 1$, and $x$, $t$, $z$, $c$ be functions of $n$ such that $x > 0$, $t$ and $z$ are
$\omega(1)$, $c$ is $o(n)$, and $t$ satisfies the conditions of Lemma \ref{sparsity} for all
sufficiently large $n$. Then with high probability, an unsatisfiable formula drawn from $F_k(n, m, c, p)$ with
\[
m \leq \frac{1}{2^{7k / 2}(1 + \epsilon)c^{k - 1}} x^{-(k - 2 - \epsilon) / 2} n
\]
does not have a resolution refutation of size $\leq 2^{\frac{\epsilon}{4(k + \epsilon)}xt} / 8$.
\end{theorem}
\begin{proof}
Let $S = 2^{\frac{\epsilon}{4(k +
\epsilon)}xt} / 8$ and let $U$ be the set of unsatisfiable $k$-CNF formulas with $n$ variables and $m$ clauses. For each $\varphi \in U$ fix a shortest
resolution refutation $P_\varphi$, and let $W \subseteq U$ be the set of
$\varphi$ such that $|P_\varphi| \leq S$.
Let $R$ be the set of all restrictions of size $t$, and for any formula $\varphi$ and $\rho \in R$ let $L(\varphi, \rho)$ be the indicator function for the event that either $\varphi$ is satisfiable or $P_\varphi \lceil_\rho$ contains a clause of length at least $\epsilon x n / (k + \epsilon)$.
Now for any $\varphi \in W$, by Lemma \ref{whp-proof} with $\beta = \epsilon x / (k + \epsilon)$ we have
\[
\sum_\rho \frac{L(\varphi, \rho)}{|R|} \le 2^{1 - \frac{\epsilon}{4(k + \epsilon)} xt}S = 2^{1 - \frac{\epsilon}{4(k +
\epsilon)} xt}(2^{\frac{\epsilon}{4(k + \epsilon)} xt} / 8) = 1 / 4 .
\]
Let $X$ be a random variable defined over a restriction $\rho$ and equal to $\Pr_\varphi [ L(\varphi, \rho) | \varphi \in W]$, where $\varphi$ is distributed as $F_k(n,m,c,p)$.
Putting a uniform distribution on $\rho$ and writing $q(\psi)$ for the conditional distribution $\Pr_\varphi [ \varphi = \psi | \varphi \in W ]$,
\[
\E_\rho [ X ] = \sum_\rho \frac{1}{|R|} \Pr_\varphi [ L(\varphi, \rho) | \varphi \in W ] = \sum_{\psi \in W} q(\psi) \left[ \sum_\rho \frac{L(\psi, \rho)}{|R|} \right] \le \sum_{\psi \in W} q(\psi) \frac{1}{4} = \frac{1}{4} .
\]
So by Markov's inequality,
\[
\Pr_\rho [X \geq 1 / 2] \leq \frac{\E_\rho[X]}{1 / 2} \leq 1 / 2 ,
\]
and therefore there is some $\rho'$ such that $\Pr_\varphi [ L(\varphi, \rho') | \varphi \in W ] \le 1/2$.
In other words, there is a restriction that eliminates large clauses from a random $\varphi \in W$ with probability at least $1/2$.
Now let $y = 2 / (k + \epsilon)$.
Since $k \ge 3$ and $\epsilon < 1$ we have $y \ge 1 / (k-1)$ and
\begin{align*}
\frac{y}{e^{1 + 1 / y}2^{k + 1 / y}} &= 2 \left[ (k+\epsilon) e^{1 + \frac{k+\epsilon}{2}} 2^{k + \frac{k+\epsilon}{2}} \right]^{-1} \ge 2 (k+\epsilon)^{-1} e^{-\frac{k}{2} - \frac{3}{2}} 2^{-\frac{3k}{2} - \frac{1}{2}} \\
&= 2 (k+\epsilon)^{-1} e^{-3/2} 2^{-1/2} 2^{- k (3 + \log_2 e) / 2} \\
&\ge 2 (k+1)^{-1} e^{-3/2} 2^{-1/2} 2^{- 2.23 k} \ge 2^{-1.23 k} 2^{- 2.23 k} \ge 2^{-7k/2} .
\end{align*}
By our assumption on $m$,
\begin{align*}
(1+\epsilon) c^{k-1} m &\le 2^{-7k/2} x^{-(k-2-\epsilon)/2} n = 2^{-7k/2} x^{1/y - (k-1)} n \\
&\le \frac{y}{e^{1 + 1 / y}2^{k + 1 / y}} x^{1/y - (k-1)} n .
\end{align*}
Finally, since $z$ is $\omega(1)$ we have $z \geq 4$ for sufficiently large $n$, and then all the conditions of Lemma \ref{sparsity} are satisfied by $y$, $z$, $t$,
and $m' = (1 + \epsilon)c^{k - 1}m$.
Therefore for a formula $\varphi$ drawn from $F_k(n, m')$, $\varphi \lceil_{\rho'}$ is simultaneously $(xn / 2, xn, 2 / (k + \epsilon))$-sparse and $xn$-sparse with probability at least $1 - 2^{-t} - (2^k + 1) / z^{k - 1}$.
Since $t$ and $z$ are $\omega(1)$, $\varphi$ has this property with high probability.
Furthermore, the property is inherited by subformulas, so by Lemma \ref{lemma:inherit} it also holds with high probability for formulas drawn from $\overline{F}_k(n, m, c, p; m')$.
Then by Lemma \ref{lemma:equiv} the same is true for formulas drawn from $F_k(n, m, c, p)$.
Now let $n' = 2xn / (k + \epsilon)$.
Since $k + \epsilon \ge 3$, we have $n' \le xn$ and so $xn$-sparsity implies $n'$-sparsity.
Also note that
\[
\frac{xn}{2} = \frac{2xn(k + \epsilon)}{4(k + \epsilon)} = \frac{n'(k + \epsilon)}{4}
\hspace{0.5cm}\text{and}\hspace{0.5cm}
xn = \frac{n'(k + \epsilon)}{2} .
\]
So by Lemma \ref{big-clause}, when drawing an unsatisfiable formula $\varphi$ from $F_k(n,m,c,p)$, with high probability every resolution refutation of $\varphi \lceil_{\rho'}$ has a clause of length at least $\epsilon n' / 2 = \epsilon xn / (k + \epsilon)$.
That is, $\Pr_\varphi [ L(\varphi, \rho') \;|\; \varphi \in U ] \rightarrow 1$ as $n \rightarrow \infty$.
So
\[
\Pr_\varphi [ \varphi \in W | \varphi \in U] = \frac{\Pr_\varphi [\varphi \in W \land \overline{L(\varphi,\rho')} \;|\; \varphi \in U]}{\Pr_\varphi [\overline{L(\varphi,\rho')} \;|\; \varphi \in W]} \le \frac{\Pr_\varphi [\overline{L(\varphi, \rho')} \;|\; \varphi \in U]}{1/2} \rightarrow 0
\]
as $n \rightarrow \infty$.
Therefore with high probability, an unsatisfiable instance drawn from $F_k(n,m,c,p)$ does not have a resolution refutation of size $\le S$.
\qed
\end{proof}
Next we instantiate this general result to obtain exponential lower bounds for the refutation length when the number of communities is not too large.
We use slightly different arguments for $k \ge 4$ and $k = 3$, again following Beame and Pitassi \cite{beame}.
As the computations are uninteresting, we defer the proofs to Appendix \ref{sec:more-proofs}.
The basic idea is to let $x$ go to zero fast enough that the bound on $m$ required by Theorem \ref{main} is satisfied when $m = O(n)$, but slowly enough that the length bound is of the form $2^{O\left( n^\lambda \right)}$.
\begin{restatable}{theorem}{thmBoundKFour} \label{thm:k4}
Suppose that $k \ge 4$, $m = O(n)$, and $c = O(n^\alpha)$ for some $\alpha < \frac{k-2}{4(k-1)}$.
Then there is some $\lambda > 0$ so that with high probability, an unsatisfiable formula drawn from $F_k(n,m,c,p)$ does not have a resolution refutation of size $2^{O\left( n^\lambda \right)}$.
\end{restatable}
\vspace{-1.5ex}
\begin{restatable}{theorem}{thmBoundKThree} \label{thm:k3}
Suppose $m = O(n)$ and $c = O(n^\alpha)$ for some $\alpha < 1/10$.
Then there is some $\lambda > 0$ so that with high probability, an unsatisfiable formula drawn from $F_3(n, m, c, p)$ does not have a resolution refutation of size $2^{O\left(n^\lambda\right)}$.
\end{restatable}
\subsection{Deducing a Lower Bound on CDCL Runtime} \label{sec:cdcl-runtime}
Finally, we can conclude that unsatisfiable random instances from $F_k(n,m,c,p)$ with sufficiently few communities usually take exponential time for CDCL to solve.
\begin{theorem}
If $m = O(n)$ and $c = O(n^\alpha)$ for any $\alpha < 1/10$, the runtime of CDCL on an unsatisfiable formula $\varphi$ from $F_k(n,m,c,p)$ is exponential with high probability.
\end{theorem}
\begin{proof}
If $\varphi$ is unsatisfiable, the runtime of CDCL on $\varphi$ is lower bounded (up to a polynomial factor) by the length of the shortest resolution refutation of $\varphi$ \cite{cdcl-sim}.
If $k = 3$, then the shortest refutation of $\varphi$ is exponentially long with high probability by Theorem \ref{thm:k3}.
If instead $k \ge 4$, the same is true by Theorem \ref{thm:k4}, since $1/10 < \frac{k-2}{4(k-1)}$.
Therefore with high probability, CDCL will take exponential time to prove $\varphi$ unsatisfiable.
\qed
\end{proof}
\begin{remark}
By picking a sufficiently high clause-variable ratio, we can ensure $\varphi$ is unsatisfiable with high probability, so that CDCL takes exponential time on average for formulas drawn from $F_k(n,m,c,p)$ (not just the unsatisfiable ones).
\end{remark}
We also note that our proof technique is not sensitive to the details of how the community attachment model is defined.
For example, changing the definition of a bridge clause so that the variables do not all have to be in different communities requires only minor changes to the proof (detailed in Appendix \ref{sec:more-proofs}).
\begin{restatable}{theorem}{thmModifiedModel}
Let $\widetilde{F}_k(n,m,c,p)$ be the community attachment model modified so that any clause that is not within a single community counts as a bridge clause.
Then if $m = O(n)$ and $c = O(n^\alpha)$ for any $\alpha < 1/10$, the runtime of CDCL on an unsatisfiable formula $\varphi$ from $\widetilde{F}_k(n,m,c,p)$ is exponential with high probability.
\end{restatable}
Thus we have showed that similarly to unsatisfiable random $k$-{\ensuremath{\mathsf{SAT}}\xspace} instances, unsatisfiable random \emph{modular} instances (as formalized by the community attachment model) are hard on average for CDCL as long as they do not have too many communities.
\section{Discussion} \label{sec:discussion}
We have introduced a broad class of ``modularity-like'' graph metrics, the polynomial clique metrics, and showed that no PCM can be a guaranteed indicator of whether a {\ensuremath{\mathsf{SAT}}\xspace} instance is easy (unless $\mathsf{P}=\ensuremath{\mathsf{NP}}\xspace$).
This is perhaps not too surprising in light of the fact that the VIG throws away the Boolean information in the formula.
While the VIG has received the most attention in recent work on community structure, it would be worthwhile to investigate other graph encodings that preserve more information.
Regardless, our result does indicate that it may be difficult to define a tractable class of {\ensuremath{\mathsf{SAT}}\xspace} instances based purely on modularity or its variants.
Furthermore, by setting up a concrete barrier (the PCM property) that must be avoided to obtain a tractable class, our result can help guide future attempts to find a graph metric that does work.
Our result on the community attachment model $F_k(n,m,c,p)$ is more interesting, as it shows that instances from this model are exponentially hard for CDCL even on average (when $c$ is small enough).
An important point is that the result is actually nontrivial when $p < 1$, unlike for $p = 1$.
In the latter case there are no bridge clauses, so the instances consist of $c$ independent problems of size $n/c$, and since we assume $c = O(n^{1/10})$ each problem has size $\Omega(n^{9/10})$.
So by the old results on random $k$-{\ensuremath{\mathsf{SAT}}\xspace} CDCL would take exponential time to solve even the easiest problem, and so likewise for the original instance (with a slightly smaller exponent on $n$).
On the other hand, when $p < 1$ it is conceivable that the bridge clauses could actually make the instances easier, by adding some extra propagation power or easier-to-find contradictions that would make the whole instance easier to solve than any individual community.
Our result effectively says that this happens with vanishing probability as $n \rightarrow \infty$.
The case $p = 1$ also brings out an important caveat when interpreting our result as evidence that community structure doesn't explain CDCL's effectiveness on industrial instances.
Our result shows that such structure isn't enough to bring random formulas down from exponential-time-on-average to polynomial-time-on-average.
However, it could decrease the time from (say) $2^{n^{1/2}}$ to $2^{n^{1/4}}$, which could be the difference between intractability and tractability if $n$ is small enough.
On the other hand, given the enormous size of many industrial instances it isn't clear whether this is really all that is happening.
It would be interesting to do experiments on parametrized families of industrial instances to see whether CDCL actually avoids exponential behavior, or if the point of blow-up is just pushed out far enough that we tend not to encounter it in practice.
Another important aspect of our result is the limit on the number of communities.
It does not apply when communities have logarithmic size, for example, so that $c = \Theta(n / \log n)$.
In fact it is easy to see that the result cannot hold in this case: if one of the communities is unsatisfiable then it will have a polynomial-length resolution refutation, and as $c \rightarrow \infty$ the probability that at least one community is unsatisfiable by itself goes to $1$.
So with high probability the entire instance has a short refutation, and CDCL could in theory solve it in polynomial time.
A clear direction for future work is to see whether improved proof techniques can extend our results to larger numbers of communities, closing the gap between $O(n^{1/10})$ and $\Omega(n / \log n)$.
This is also another way our results can inform future experiments: it would be interesting to explore a variety of growth rates for $c$ above $n^{1/10}$ and see how the performance of CDCL changes.
We have done some preliminary experiments along these lines, sampling instances from $F_3(n,5n,5n^\alpha,0.9)$ for a variety of values of $n$ and solving them with MiniSat 2.2.0 \cite{minisat}.
For each value of $n$ we generated 10 instances using the generator from \cite{giraldez2015modularity} and averaged their runtimes.
Note that every instance had at least 10 communities, so that the expected modularity was at least $p - (1/c) \ge 0.9 - 0.1 = 0.8$.
In Figure \ref{fig:runtime-community-size}, we plot the results as a function of the community size $n / (5n^\alpha) = n^{1-\alpha} / 5$.
\begin{figure}[ht]
\centering
\includegraphics[width=\textwidth]{runtime-community-size}
\caption{Average MiniSat runtimes for instances from $F_3(n,5n,5n^\alpha,0.9)$ for $\alpha = 0.1$ (orange), $0.2$ (blue), $0.3$ (green), and $0.4$ (red), plotted as a function of community size ($n^{1-\alpha}/5$).}
\label{fig:runtime-community-size}
\end{figure}
It is clear from the right half of the graph\footnote{On the left half the growth is much slower and close to linear, but since this occurs only for runtimes on the order of a second or less it may be that parsing the formula and initializing the solver dominate the time needed for the actual search.} that the runtime blows up exponentially in the community size (and thus in $n$) even for $\alpha$ significantly larger than $1/10$.
This suggests that improving our result to larger $\alpha$ is likely possible.
However, it is important to point out that although all values of $\alpha$ are undergoing exponential growth, the lines for the different values of $\alpha$ are quite close together, indicating that community size is a much more important factor in determining runtime than the total formula size.
For example, at a community size of around 270 the $\alpha=0.1$ instances have 3,000 variables total, while the $\alpha=0.4$ instances have 165,000.
So the latter are more than 50 times larger than the former, but their runtimes are only about 3 times longer.
This shows that while larger values of $\alpha$ do not avoid exponential blowup, they can significantly aid performance.
In total, our results indicate that high modularity alone may not be adequate to ensure good performance even on average, but that it could be rewarding to investigate more refined notions of ``good community structure'' that somehow restrict the number of communities.
\noindent
{\bf Acknowledgments.}
The authors thank Vijay Ganesh, Holger Hoos, Zack Newsham, Markus Rabe, Stefan Szeider, and several anonymous reviewers for helpful discussions and comments.
This work is supported in part by the National Science Foundation Graduate Research Fellowship Program under Grant No. DGE-1106400, by the Hellman Family Faculty Fund, by gifts from Microsoft and Toyota, and by TerraSwarm, one of six centers of STARnet, a Semiconductor Research Corporation program sponsored by MARCO and DARPA.
\bibliographystyle{splncs}
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